Building Physics: Heat, Air and Moisture, Fundamentals and Engineering Methods with Examples and Exercises, Second Edition 9783433030271, 9783433602379

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Hugo Hens

Building Physics Heat, Air and Moisture Fundamentals and Engineering Methods with Examples and Exercises 2nd Edition

Hugo Hens Building Physics Heat, Air and Moisture Fundamentals and Engineering Methods with Examples and Exercises 2nd Edition

Professor Hugo S. L. C. Hens University of Leuven (KU Leuven) Department of Civil Engineering Building Physics Kasteelpark Arenberg 40 3001 Leuven Belgium

Coverphoto: Courtesy of InfraTec GmbH (www.infratec.de). This high resolution thermography image of an apartment building was taken with a VarioCAM® high resolution megapixel-thermography camera from Infratec GmbH, Dresden (Germany). Library of Congress Card No.: applied for British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available on the Internet at http://dnb.d-nb.de. © 2012 Wilhelm Ernst & Sohn, Verlag für Architektur und technische Wissenschaften GmbH & Co. KG, Rotherstr. 21, 10245 Berlin, Germany All rights reserved (including those of translation into other languages). No part of this book may be reproduced in any form – by photoprinting, microfilm, or any other means – nor transmitted or translated into a machine language without written permission from the publishers. Registered names, trademarks, etc. used in this book, even when not specifically marked as such, are not to be considered unprotected by law. Coverdesign: Sophie Bleifuß, Berlin, Germany Typesetting: Manuela Treindl, Fürth, Germany Printing and Binding: betz-druck GmbH, Darmstadt, Germany Printed in the Federal Republic of Germany. Printed on acid-free paper. Print ISBN: ePDF ISBN: ePub ISBN: mobi ISBN: oBook ISBN:

978-3-433-03027-1 978-3-433-60234-8 978-3-433-60235-5 978-3-433-60236-2 978-3-433-60237-9

To my wife, children and grandchildren

In remembrance of Professor A. De Grave who introduced Building Physics as a new discipline at the University of Leuven (KU Leuven), Belgium in 1952

This second edition represents a complete revision of the first edition, published in 2007. Where appropriate, the text was corrected, reworked, and extended. The exercises have been reviewed and solutions for all 33 problems added.

Preface Overview Until the first energy crisis of 1973, building physics was a rather dormant field in building engineering, with seemingly limited applicability. While soil mechanics, structural mechanics, building materials, building construction and HVAC were seen as essential, designers only demanded advice on room acoustics, moisture tolerance, summer comfort or lighting when really needed or problems arose. Energy was not even a concern, while thermal comfort and indoor environmental quality were presumably guaranteed thanks to infiltration, window operation and the HVAC system. The crises of the 1970s, persisting moisture problems, complaints about sick buildings, thermal, visual and olfactory discomfort, and the move towards more sustainability changed all that. The societal pressure to diminish energy consumptions in buildings without degrading usability opened the door for the notion of performance based design and construction. As a result, building physics and its potentiality to quantify performances suddenly moved into the frontline of building innovation. As with all engineering sciences, building physics is oriented towards application. This demands a sound knowledge of the basics in each of the branches encompassed: heat and mass transfer, acoustics, lighting, energy and indoor environmental quality. Integrating the basics on heat and mass transfer is the main objective of this book, with mass limited to air, water vapour and moisture. It is the result of thirty years of teaching architectural, building and civil engineers, and forty-four years of experience, research and consultancy. Input and literature from over the world has been used, documented after each chapter by an extended literature list. An introductory chapter presents building physics as a discipline. The first part concentrates on heat transport, with conduction, convection and radiation as main topics, followed by concepts and applications which are typical for building physics. The second part treats mass transport, with air, water vapour and moisture as the most important components. Again, much attention is devoted to the concepts and applications which relate to buildings. The last part discusses combines heat, air, moisture transport, who act as a trio. The three parts are followed by exemplary exercises. The book is written in SI-units. It should be usable for undergraduate and graduate studies in architectural and building engineering, although also mechanical engineers studying HVAC, and practising building engineers who want to refresh their knowledge, may benefit. The level of presentation presumes the reader has a sound knowledge of calculus and differential equations along with a background in physics, thermodynamics, hydraulics, building materials and building construction. Acknowledgements A book of this magnitude reflects the work of many persons in addition to the author. Therefore, we would like to thank the thousands of students we had during the thirty years of teaching building physics. They provided the opportunity to test the content. It is a book which would not been written the way it is, without standing on the shoulders of those in the field who preceded. Although I started my career as a structural engineer, my predecessor, Professor Antoine de Grave, planted the seeds that fed my interest in building physics. The late Bob Vos of TNO, the Netherlands, and Helmut Künzel of the Fraunhofer Institut für Bauphysik, Germany, showed me the importance of experimental work and field testing for understand-

VIII

Preface

ing building performance, while Lars Erik Nevander of Lund University, Sweden, taught that complex modelling does not always help in solving problems in building physics, mainly because reality in building construction is much more complex than any model may be. During four decades at the Laboratory of Building Physics, many researchers and Ph. D.students got involved in the project. I am very grateful to Gerrit Vermeir, Staf Roels Dirk Saelens and Hans Janssen who became colleagues at the university; to Jan Carmeliet, now professor at the ETH-Zürich; Piet Standaert, a principal at Physibel Engineering; Jan Lecompte, at Bekaert NV; Filip Descamps, a principal at Daidalos Engineering and part-time professor at the Free University Brussels (VUB); Arnold Janssens, associate professor at the University of Ghent (UG); Rongjin Zheng, associate professor at Zhejiang University, China, and Bert Blocken, professor at the Technical University Eindhoven (TU/e), who all contributed by their work. The experiences gained by working as a structural engineer and building site supervisor at the start of my career, as building assessor over the years, as researcher and operating agent of four Annexes of the IEA, and Executive Committee on Energy Conservation in Buildings and Community Systems forced me to rethink the engineering based performance approach time and time again. The idea exchange we got in Canada and the USA from Kumar Kumaran, Paul Fazio, Bill Brown, William B. Rose, Joe Lstiburek and Anton Ten Wolde was also of great help. A number of reviewers took time to examine the first edition of this book. We would like to thank them, too. Finally, I thank my wife Lieve who managed living with a busy engineering professor, and my three children who also had to live with that busy father, not to mention my many grandchildren who do not know their grandfather is still busy. Leuven, March 2012

Hugo S. L. C. Hens

Contents Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VII 0

Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

0.1

Subject of the book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

0.2 0.2.1 0.2.2 0.2.2.1 0.2.2.2 0.2.2.3 0.2.2.4 0.2.2.5

Building Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Comfort . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Health . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Architecture and materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Economy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sustainability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

0.3

Importance of Building Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

0.4 0.4.1 0.4.2 0.4.3 0.4.4 0.4.5 0.4.6 0.4.7

History of Building Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Heat, air and moisture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Building acoustics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Lighting. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Thermal comfort and indoor air quality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Building physics and building services . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Building physics and construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . What about the Low Countries? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

0.5

Units and symbols. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

0.6

Literature. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1

Heat Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.1

Overview. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.2 1.2.1 1.2.2 1.2.2.1 1.2.2.2 1.2.3 1.2.3.1 1.2.3.2 1.2.3.3 1.2.3.4 1.2.4 1.2.4.1 1.2.4.2 1.2.4.3 1.2.4.4

Conduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conservation of energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fourier’s laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . First law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Second law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Steady state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . What is it? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . One dimension: flat assemblies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Two dimensions: cylinder symmetric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Two and three dimensions: thermal bridges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Transient regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . What? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Flat assemblies, periodic boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . Flat assemblies, random boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . Two and three dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 2 2 3 3 3 3 5 5 5 6 6 7 7 8

15 15 16 16 17 18 18 18 26 27 32 32 33 45 48

X

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1.3 1.3.1 1.3.2 1.3.3 1.3.3.1 1.3.3.2 1.3.4 1.3.4.1 1.3.4.2 1.3.4.3 1.3.5 1.3.5.1 1.3.5.2 1.3.5.3

Convection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Heat exchange at a surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Convective heat transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Convection typology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Driving forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Flow type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Calculating the convective surface film coefficient . . . . . . . . . . . . . . . . . . . . . . . Analytically. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Numerically . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dimensional analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Values for the convective surface film coefficient . . . . . . . . . . . . . . . . . . . . . . . . Flat assemblies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cavities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pipes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

49 49 50 52 52 52 53 53 53 54 56 56 59 61

1.4 1.4.1 1.4.2 1.4.3 1.4.4 1.4.5 1.4.5.1 1.4.5.2 1.4.5.3 1.4.5.4 1.4.6 1.4.6.1 1.4.6.2 1.4.7 1.4.8

Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . What is thermal radiation? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Reflection, absorption and transmission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Radiant surfaces or bodies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Black bodies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Characteristics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Radiant exchange between two black bodies: the view factor . . . . . . . . . . . . . . . Properties of view factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Calculating view factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Grey bodies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Characteristics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Radiant exchange between grey bodies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Coloured bodies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Practical formulae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

61 61 62 62 64 65 65 67 69 69 72 72 73 75 75

1.5 1.5.1 1.5.1.1 1.5.1.2 1.5.1.3 1.5.2 1.5.2.1 1.5.2.2 1.5.2.3 1.5.3 1.5.4 1.5.4.1 1.5.4.2 1.5.5 1.5.6 1.5.6.1 1.5.6.2 1.5.7 1.5.8

Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 Surface film coefficients and reference temperatures. . . . . . . . . . . . . . . . . . . . . . 77 Overview. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 Indoor environment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 Outdoor environment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 Steady state, one dimension: flat assemblies . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 Thermal transmittance and interface temperatures . . . . . . . . . . . . . . . . . . . . . . . . 84 Thermal resistance of a non ventilated, infinite cavity . . . . . . . . . . . . . . . . . . . . . 88 Solar transmittance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 Steady state, cylindrical coordinates: pipes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 Steady state, two and three dimensions: thermal bridges . . . . . . . . . . . . . . . . . . . 94 Calculation by the control volume method (CVM) . . . . . . . . . . . . . . . . . . . . . . . 94 Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 Steady state: windows. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 Steady state: building envelopes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 Overview. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 Average thermal transmittance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 Transient, periodic: flat assemblies. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 Heat balances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

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Contents

1.5.9 1.5.9.1 1.5.9.2 1.5.9.3

Transient, periodic: spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Steady state heat balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Harmonic heat balances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

102 102 102 103

1.6

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

1.7

Literature. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

2

Mass Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

2.1 2.1.1 2.1.2 2.1.3 2.1.4 2.1.5 2.1.6 2.1.7

Generalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Quantities and definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Saturation degrees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Air and moisture transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Moisture sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Air, moisture and durability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Link between mass and energy transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conservation of mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

123 123 125 126 128 129 130 131

2.2 2.2.1 2.2.2 2.2.2.1 2.2.2.2 2.2.2.3 2.2.3 2.2.4 2.2.4.1 2.2.4.2 2.2.4.3 2.2.4.4 2.2.4.5 2.2.5 2.2.5.1 2.2.5.2 2.2.6 2.2.6.1 2.2.6.2 2.2.6.3 2.2.6.4 2.2.6.5 2.2.7 2.2.7.1 2.2.7.2

Air transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Overview. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Air pressure differences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Wind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Stack effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fans . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Air permeances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Air transfer in open-porous materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conservation of mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Flow equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Air pressures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . One dimension: flat assemblies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Two and three dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Air flow across permeable layers, apertures, joints, leaks and cavities . . . . . . . Flow equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conservation of mass: equivalent hydraulic circuit . . . . . . . . . . . . . . . . . . . . . . Air transfer at building level . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Thermal stack . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Large openings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conservation of mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Combined heat and air transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Open-porous materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Air permeable layers, joints, leaks and cavities . . . . . . . . . . . . . . . . . . . . . . . . .

132 132 133 133 134 135 135 139 139 139 139 140 142 143 143 143 144 144 145 145 146 148 151 151 157

2.3 2.3.1 2.3.1.1 2.3.1.2 2.3.1.3

Vapour transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Water vapour in the air . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Overview. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Maximum vapour pressure and relative humidity . . . . . . . . . . . . . . . . . . . . . . .

160 160 160 161 161

XII

Contents

2.3.1.4 2.3.1.5 2.3.1.6 2.3.1.7 2.3.2 2.3.2.1 2.3.2.2 2.3.2.3 2.3.2.4 2.3.2.5 2.3.3 2.3.4 2.3.4.1 2.3.4.2 2.3.4.3 2.3.4.4 2.3.5 2.3.6 2.3.6.1 2.3.6.2 2.3.6.3

Changes of state in humid air . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Enthalpy of humid air . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Measuring air humidity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Water vapour in open-porous materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Overview. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sorption isotherm and specific moisture ratio . . . . . . . . . . . . . . . . . . . . . . . . . . Physics involved . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Impact of salts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Consequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Vapour transfer in the air . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Vapour transfer in materials and assemblies . . . . . . . . . . . . . . . . . . . . . . . . . . . . Flow equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conservation of mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Vapour transfer by ‘equivalent’ diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Vapour transfer by ‘equivalent’ diffusion and convection . . . . . . . . . . . . . . . . . Surface film coefficients for diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Diffusion resistance of a cavity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cavity ventilation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Water vapour balance in a space: surface condensation and drying . . . . . . . . . .

166 166 167 167 172 172 173 174 177 177 177 179 179 182 182 197 204 207 207 207 210

2.4 2.4.1 2.4.2 2.4.2.1 2.4.2.2 2.4.2.3 2.4.2.4 2.4.3 2.4.3.1 2.4.3.2 2.4.3.3 2.4.3.4 2.4.4 2.4.4.1 2.4.4.2

Moisture transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Overview. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Moisture transfer in a pore . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Capillarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Water transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Vapour transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Moisture transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Moisture transfer in materials and assemblies . . . . . . . . . . . . . . . . . . . . . . . . . . Transport equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conservation of mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Starting, boundary and contact conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Remark . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Simplifying moisture transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

211 211 211 211 213 222 224 224 224 227 227 228 228 228 230

2.5

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245

2.6

Literature. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263

3

Combined heat-air-moisture transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267

3.1

Overview. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267

3.2 3.2.1 3.2.2 3.2.3 3.2.3.1

Material and assembly level . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conservation laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

267 267 267 268 268

XIII

Contents

3.2.3.2 3.2.4 3.2.4.1 3.2.4.2 3.2.4.3 3.2.5 3.2.5.1 3.2.5.2 3.2.5.3 3.2.6 3.2.6.1 3.2.6.2 3.2.6.3 3.2.7 3.2.7.1 3.2.7.2

Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Flow equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Heat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mass, air . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mass, moisture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Equations of state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Enthalpy/temperature, vapour saturation pressure/temperature . . . . . . . . . . . . . Relative humidity/moisture content . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Suction/moisture content . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Starting, boundary and contact conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Starting conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Contact conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Two examples of simplified models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Non hygroscopic, non capillary materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hygroscopic materials at low moisture content . . . . . . . . . . . . . . . . . . . . . . . . .

269 272 272 272 273 273 273 273 273 274 274 274 274 275 275 276

3.3 3.3.1 3.3.2 3.3.2.1 3.3.2.2 3.3.2.3 3.3.2.4 3.3.3 3.3.3.1 3.3.3.2 3.3.3.3 3.3.3.4 3.3.3.5 3.3.4 3.3.4.1 3.3.4.2

Building level . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Overview. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Balance equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Vapour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Air . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Heat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Closing the loop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hygric inertia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Generalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sorption-active thickness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Zone with one sorption-active surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Zone with several sorption-active surfaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . Harmonic analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Consequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Steady state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Transient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

277 277 277 277 279 279 282 283 283 283 286 287 288 289 289 289

3.4

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292

3.5

Literature. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303 Postscript . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305 Problems and Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307

0

Introduction

0.1

Subject of the book

This is the first of a series of four books: y Building Physics: Heat, Air and Moisture y Applied Building Physics: Boundary Conditions, Building Performance and Material Properties y Performance Based Building Design 1 y Performance Based Building Design 2 The notion of ‘Building Physics’ is somewhat unusual in the English speaking world. ‘Building Science’ is the more commonly used. However, the two differ somewhat. Building science is broader in its approach as it encompasses all subjects related to buildings that claim to be ‘scientific’. This is especially clear when looking at journals that publish on ‘Building Science’. The range of subjects treated is remarkably wide, ranging from control issues in HVAC to city planning and organizational issues. In ‘Building Physics: Heat, Air and Moisture’, the subject is the physics behind heat, air and moisture transfer in materials, building assemblies and whole buildings. The second book, ‘Applied Building Physics’, deals with the inside and outside climate as boundary condition, followed by the performance rationale for the heat, air, moisture related performances at building part level. Extended tables with material properties are also included. Performance Based Building Design 1 and 2 then apply the performance rationale as an instrument in the design and execution of buildings. As such, they integrate the fields of building construction, building materials, building physics and structural mechanics.

0.2

Building Physics

0.2.1

Definition

Building Physics is an applied science that studies the hygrothermal, acoustical and light-related properties, and the performance of materials, building assemblies (roofs, façades, windows, partition walls, etc.), spaces, whole buildings, and the built environment. At the whole building level, the three sub-fields generate subjects such as indoor environmental quality and energy efficiency, while at the built environment level building physics is renamed ‘Urban Physics’. Basic considerations are user requirements related to thermal, acoustic and visual comfort, health prerequisites and the more-or-less compelling demands and limitations imposed by architectural, material, economics and sustainability-related decisions. The term ‘applied’ indicates that Building Physics is directed towards problem solving: the theory as a tool, not as a purpose. As stated, the discipline contains in essence three sub-fields. The first, hygrothermal, deals with heat, air, and moisture transport in materials, assemblies, and whole buildings and, the heat, air and moisture interaction between buildings and the outdoor environment. The specific topics are: thermal insulation and thermal inertia; moisture Building Physics: Heat, Air and Moisture. Fundamentals and Engineering Methods with Examples and Exercises. Second edition. Hugo Hens. © 2012 Ernst & Sohn GmbH & Co. KG. Published 2012 by Ernst & Sohn GmbH & Co. KG.

2

0 Introduction

and temperature induced movements, strains and stresses; moisture tolerance (rain, building moisture, rising damp, sorption/desorption, surface condensation, interstitial condensation); salt transport; air-tightness and wind resistance; net energy demand and end energy consumption; ventilation; indoor environmental quality; wind comfort, etc. The second sub-sector, building acoustics, studies noise problems in and between buildings, and between buildings and their environment. The main topics are airborne and impact noise transmission by walls, floors, outer walls, party walls, glazing and roofs, room acoustics and the abatement of installation and environmental noises. Finally, the third sub-sector, lighting, addresses issues with respect to day-lighting, as well as artificial lighting and the impact of both on human wellbeing and primary energy consumption.

0.2.2

Criteria

Building Physics deals with a variety of criteria: on the one hand, requirements related to human comfort, health, and well-being, on the other hand restrictions because of architecture, material use, economics, and sustainability demands. 0.2.2.1 Comfort Comfort is defined as a state of mind that expresses satisfaction with the environment. Attaining such a condition depends on a number of environmental and human factors. By thermal, acoustic and visual comfort we understand the qualities human beings unconsciously request from their environment in order to feel thermally, acoustically and visually at ease when performing a given activity (not too cold, not too warm, not too noisy, no large contrasts in luminance, etc.). Thermal comfort connects to the global human physiology and psychology. As an exothermal creature maintaining a constant core temperature of about 37 °C (310 K), humans must be able, under any circumstance, to release heat to the environment, be it by conduction, convection, radiation, perspiration, transpiration, and breathing. Air temperature, air temperature gradients, radiant temperature, radiant asymmetry, contact temperatures, relative air velocity, air turbulence, and relative humidity in the direct environment determine the heat exchange by the six mechanisms mentioned. For a certain activity and clothing, humans experience some combinations of these environmental parameters as being comfortable, others as not, though the possibility to adapt the environment to one’s own wishes influences satisfaction. Acoustic comfort strongly connects to our mental awareness. Physically, young adults perceive sound frequencies between 20 and 16,000 Hz. We experience sound intensity logarithmically, with better hearing for higher than for lower frequencies. Consequently, acoustics works with logarithmical scales and units: the decibel (dB), 0 dB standing for the audibility threshold, 140 dB corresponding to the pain threshold. We are easily disturbed by undesired noises, like those made by the neighbours, traffic, industry, and aircraft. Visual comfort combines mental with physical facts. Physically, the eye is sensitive to electromagnetic waves with wavelengths between 0.38 and 0.78 —m. The maximum sensitivity lies near a wavelength of § 0.58 —m, the yellow-green light. Besides, eye sensitivity adapts to the average luminance. For example, in the dark, sensitivity increases 10,000 times compared to daytime. Like the ear, the eye reacts logarithmically. Too large differences in brightness are disturbing. Psychologically, lighting helps to create atmosphere.

0.3 Importance of Building Physics

3

0.2.2.2 Health Health does not only mean the absence of illness but also of neuro-vegetative complaints, psychological stress, and physical unease. Human wellbeing may be compromised by dust, fibres, (S)VOC’s, radon, CO, viruses and bacteria in the air, moulds and mites on surfaces, too much noise in the immediate environment, local thermal discomfort, etc. 0.2.2.3 Architecture and materials Building physics has to operate within an architectural framework. Floor, façade and roof form, aesthetics and the choice of materials are all elements which shape the building, and whose design is based on among others the performance requirements building physics imposes. Conflicting structural and physical requirements complicate solutions. Necessary thermal cuts, for example, could interfere with strength and the need for stiffness-based interconnections. Waterproof and vapour permeable are not always compatible. Acoustical absorption opposes vapour tightness. Certain materials cannot remain wet for a long time, etc. 0.2.2.4 Economy Not only must the investment in the building remain within budget limits, total present value based life cycle costs should also be as low as possible. In that respect, energy consumption, maintenance, necessary upgrades, and building life expectancy play a role. A building which has been designed and constructed according to requirements that reflect a correct understanding of building physics, could generate a much lower life cycle cost than buildings built without much consideration about fitness for purpose. 0.2.2.5 Sustainability Societal concern about local, countrywide, and global environmental impact has increased substantially over the past decades. Locally, building use produces solid, liquid, and gaseous waste. Nationally, building construction and occupancy accounts for 35–40% of the primary energy consumed annually. A major part of that is fossil fuel related, which means also the CO2-release by buildings closely matches that percentage. In terms of volume, CO2 is the most important of the gases emitted that are responsible for global warming. That striving for more sustainability is reflected in the increasing importance of life cycle analysis and of certification tools such as LEED, BREEAM and others. In life cycle analysis, buildings are evaluated in terms of environmental impact from ‘cradle to grave’, i.e. from material production through the construction and occupancy stage until demolition and reuse. Per stage, all material, energy and water inflows as well as all polluting solid, liquid, and gaseous outflows are quantified and their impact on human wellbeing and the environment assessed. Certification programmes in turn focus on the overall fitness for purpose of buildings and urban environments.

0.3

Importance of Building Physics

The need to build a comfortable indoor environment that protects humans against the vagaries of the outside climate, defines the role of building physics. Consequently, the separation between the inside and the outside, i.e. the building envelope or enclosure (floors, outer walls, roofs) is

4

0 Introduction

submitted to various climatologic loads and climate differences (sun, rain, wind and outside noise; differences in temperatures, in partial water vapour pressure and in air pressure). An appropriate envelope design along with correct detailing must consider these loads, attenuate them where possible or use them if possible in order to guarantee the desired comfort and wellbeing with a minimum of technical means and the least possible energy consumption. In earlier days, experience was the guide. Former generations of builders relied on a limited range of materials (wood, straw, loam, natural stone, lead, copper and cast iron, blown glass), the knowledge of how to use those materials increasing over the centuries. They applied standard details for roofs, roof edges, and outer walls. From the size and orientation of the windows to the overall layout, buildings were constructed as to limit heating in winter and overheating in summer. Because noise sources were scarce, sound annoyance outside urban centres was unknown, and our ancestors saved on energy (wood) by a lifestyle adapted to the seasons. A new era began with the industrial revolution of the 19th century. New materials inundated the marketplace: steel, reinforced and pre-stressed concrete, nonferrous metals, synthetics, bitumen, insulation materials, etc. More advanced technologies created innovative possibilities for existing materials: cast and float glass, rolled metal products, pressed bricks, etc. Better knowledge of structural mechanics allowed the construction of any form and span. Energy became cheap, first coal, then petroleum and finally natural gas. Construction exploded and turned into a supply-demand market. The consequence was mass building with a minimum of quality and, in the early 20th century, a ‘modern school’ of architects, who experimented with alternative structural solutions and new materials. These experiments had nothing to do with former knowledge. Architects designed buildings without any concern for either energy consumption or comfort, nor any understanding of the physical quality of the new outer wall and roof assemblies they proposed. Typical was the profuse application of steel, concrete, and glass, which are all difficult materials from a hygrothermal point of view. The results were and are, severe damage as well as premature restoration, which could have been avoided by a better knowledge of building physics. Figure 0.1 is an example of that: a house designed

Figure 0.1. House Guiette, designed by Le Corbusier, after restoration.

0.4 History of Building Physics

5

by Le Corbusier, built in 1926 and rebuilt at the end of the 1980s, included an upgrade of the overall thermal insulation. Before, end energy consumption for heating could have reached 20,000 litres of fuel a year if all rooms were kept at a comfortable temperature. Fortunately inhabitants adapted by heating rooms only used in the daytime. Building physics is essential if we want to achieve high quality buildings that are fit our purposes. The field should replace time-consuming learning by experience, which cannot keep pace with the rapid evolution in technology and the changes in architectural fashion.

0.4

History of Building Physics

Building physics originated at the crossroads of three application-oriented disciplines: applied physics, building services and building construction.

0.4.1

Heat, air and moisture

In the first half of the 20th century, attention focused on thermal conductivity. In the nineteen thirties, measuring the diffusion resistance gained importance after Teesdale of the US Forest Products Laboratory published a study in 1937 on ‘Condensation in Walls and Attics’. In 1952, an article by J. S. Cammerer on ‘Die Berechnung der Wasserdampfdiffusion in der Wänden’ (Calculation of Water Vapour Diffusion in Walls) appeared in ‘Der Gesundheitsingenieur’. At the end of the 1950s, H. Glaser described a new calculation method for interstitial condensation by vapour diffusion in cold storage walls in the same journal. Others, among them K. Seiffert, applied that method to building assemblies, albeit he used highly unrealistic climatic boundary conditions. His book ‘Wasserdampfdiffusion im Bauwesen’ (Water vapour diffusions in buildings) led to what today is called the vapour barrier phobia, even more that the text overlooked the most important cause of interstitial condensation: air displacement in and across building assemblies. From the sixties on, more researchers studied combined heat and moisture transport, among them O. Krischer, J. S. Cammerer and H. Künzel in Germany, A. De Vries and B. H. Vos in the Netherlands, L. E. Nevander in Sweden and A. Tveit in Norway. That air transport figured as main cause of interstitial condensation, was stated first in Canada, a timber-framed construction country. In a publication in 1961, A. G. Wilson of NRC wrote ‘One of the most important aspects of air leakage in relation to the performance of Canadian buildings is the extent to which it is responsible for serious condensation problems. Unfortunately this is largely unrecognized in the design and construction of many buildings, and even when failures develop the source of moisture is often incorrectly identified’.

0.4.2

Building acoustics

In the early 20th century, physicists started showing interest in applying noise control to building construction. In 1912, Berger submitted a Ph. D. thesis at the Technische Hochschule München entitled ‘Über die Schalldurchlässigkeit’ (Concerning sound transmission). Sabine published his well-known reverberation time in indoor spaces formula in 1920. In the years after, room acoustics became a favourite subject with studies about speech intelligibility, optimal reverberation times, reverberation time in anechoic rooms, etc. One decade later, L. Cremer was responsible for a break-through in sound transmission. In his paper ‘Theorie

6

0 Introduction

der Schalldämmung dünner Wände bei schrägem Einfall’ (Theory of sound insulation of thin walls at oblique incidence), he recognized that the coincidence effect between sound waves in the air and bending waves on a wall played a major role. Later he studied impact noise in detail with floating floors as a solution. Other German engineers, such as K. Gösele and M. Heckl, established the link between building acoustics and building practice. In the USA, Beranek published his book ‘Noise and Vibration Control’ in 1970, a remake of ‘Noise reduction’ of 1960. It became the standard work for engineers, who had to solve noise problems.

0.4.3

Lighting

The application of lighting to buildings and civil engineering constructions came later. In 1931, a study was completed at the Universität Stuttgart, dealing with ‘Der Einfluss der Besonnung auf Lage und Breite von Wohnstraßen’ (The influence of solar irradiation on the location and width of residential streets). Later, physicists used the radiation theory to calculate illumination on surfaces and luminance contrasts in the environment. In the late 1960s, the daylight factor was introduced as a quantity to evaluate natural illumination indoors. More recently, after the energy crises of the 1970s, the relation between artificial lighting and primary energy consumption surged in importance as a topic.

0.4.4

Thermal comfort and indoor air quality

In the 19th century, engineers were especially concerned with housing and urban hygiene. A predecessor was Max von Pettenkofer (1818–1901, Figure 0.2) who was the first to perform research on the relation between ventilation, CO2-concentration and indoor air quality. The 1500 ppm threshold, which still figures as limit between acceptable and unacceptable, is attributed to him. He is also attributed with the notion of ‘breathing materials’, the result of an erroneous explanation of the link he assumed between the air permeability of bricks and stone and the many health complaints in stony dwellings. In the twentieth century, thermal comfort and indoor air quality became important topics. In the service of comfort, research by Yaglou, sponsored by ASHVE (American Society of

Figure 0.2. Max von Pettenkofer.

0.4 History of Building Physics

7

Heating and Ventilation Engineers), a predecessor of ASHRAE (American Society of Heating, Refrigeration and Air Conditioning Engineers) lead to the notion of ‘operative temperature’. Originally, his definition overlooked radiation. This changed after A. Missenard, a French engineer, critically reviewed the research and saw that radiant temperature had a true impact. The late P. O. Fanger took a large step forward with the publication in 1970 of his book ‘Thermal Comfort’. Based on physiology, on the heat exchange between the clothed body and the environment and the random differences in terms of comfort perception between individuals, he developed a steady state thermal model of the active, clothed human. Since then, his ‘Predicted Mean Vote (PMV) versus Predicted Percentage of Dissatisfied (PPD)’ curve has become the basis of all comfort standards worldwide. After 1985, the adaptive model gained support worldwide as an addition to Fanger’s work. In terms of indoor air quality, a growing multitude of pollutants were catalogued and evaluated in terms of health risks. Over the years, the increase in fully air-conditioned buildings simultaneously increased the number of sick building complaints. This reinforced the call for an even better knowledge of the indoor environment, although the claim ‘better’ did not always result from a sound interpretation of data. One overlooked for example too often discontent with the job as a reason to simulate symptoms. Also here, P. O. Fanger had a true impact with his work on perceived indoor air quality, based on bad smell and air enthalpy.

0.4.5

Building physics and building services

In the 19th century, building service technicians were searching for methods to calculate the heating and cooling load. They took advantage of the knowledge developed in physics, which provided concepts such as the ‘thermal transmittance of a flat assembly’. Already quite early, organizations such as ASHVE and VDI had technical committees, which dealt with the topics of heat loss and heat gain. An active member of ASHVE was W. H. Carrier (1876–1950), recognized in the USA as the ‘father’ of air conditioning. He published the first usable psychometric chart. In Germany, H. Rietschel, professor at the Technische Universtät Berlin and the author of a comprehensive book on ‘Heizungs und Lüftungstechnik’ (Heating and Ventilation Techniques), also pioneered in that field. Heat loss and heat gain through ventilation was one of his concerns. He and others learned by experience that well-designed ventilation systems did not function properly when the building envelope lacked air-tightness. This sharpened the interest in air transport. Moisture became a concern around the period air-conditioning (HVAC) gained popularity. The subject had already been a focus of interest, mainly because moisture appeared to be very damaging to the insulation quality of some materials and could cause health problems. Sound attenuation entered the HVAC-field due to the noisiness of the early installations. Lighting was focussed on as more HVAC engineers received contracts for lighting design. From 1973 on, energy efficiency emerged as a challenge. That HVAC and building physics are twins is still observed in the USA where ‘building science’ often is classified with mechanical engineering.

0.4.6

Building physics and construction

The link with construction grew due as designers of the new building tradition wrestled with complaints about noise and moisture. Building physics then became an application oriented field that could help avoid the failures which new solutions designed and built according to the existing ‘state-of-the art’ rules could provide.

8

0 Introduction

It all started in the early 1930s with peeling and blistering of paints on insulated timber framed walls (insulation materials were rather new at that time). This motivated the already mentioned Teesdale to his study on condensation. Some years later, ventilated attics with insulation on the ceiling became the subject of experimental work by F. Rowley, Professor in Mechanical Engineering at the University of Minnesota, USA. His results contained the first instructions on vapour retarders and attic ventilation. In Germany, the Freiland Versuchsstelle Holzkirchen that was created in 1951 did pioneering work by using building physics as a tool to upgrade building quality. In 1973, when energy efficiency became a hot topic and insulation a necessity, the knowledge developed at Holzkirchen, proved extremely useful for the development of high quality, well-insulated buildings, and the manufacturing of glazing systems with better insulating properties, lower solar transmittance, and better visual transmittance. In the nineties of last century, the need for a better quality resulted in a generalized performance approach, as elaborated by the IEA-ECBCS Annex 32 on ‘Integral Building Envelope Performance Assessment’ and promoted thanks to the certification tools introduced worldwide. Also from a building acoustics point of view, the theory got translated in easy to use methods rationales for constructing floors, walls, and roofs with high sound transmission loss for airborne and contact noise. Examples are tie-less cavity party walls, composite light-weight walls, floating screeds and double glass with panes of different thickness, the air space filled with a heavy gas and one of the panes assembled as a glass/synthetic foil/glass composite.

0.4.7

What about the Low Countries?

At the University of Leuven (KU Leuven), Belgium, lecturing in building physics started in 1952. This made that university the pioneer in the Benelux. At the TU-Delft, already before World War II Professor Zwikker gave lectures on the physics of buildings, but a course named ‘Building Physics’ only started in 1955 with Professor Kosten of the applied physics faculty. In 1963, Professor Verhoeven succeeded him. At the TU/e in Eindhoven Professor P. De Lange occupied the chair on Building Physics in 1969. Ghent University waited until 1999 to nominate an assistant professor in building physics. During the early years at the University of Leuven, building physics was compulsory for architectural engineers and optional for civil engineers. Half-way through the 1970s the course also became compulsory for civil engineers. Since 1990, after building engineering started, building physics is one of the basic disciplines for those entering that programme. From that date on, a strong link was also established with the courses on building services and performance based building design. The first building physics professor was A. de Grave, a civil engineer and head of the building department at the Ministry of Public Works. He taught from 1952 until 1975, the year that he passed away. In 1957 he published his book ‘Bouwfysica’ (Building Physics), followed by ‘Oliestook in de woning’ (Oil heating at home). He was a practitioner, not a researcher. Former students still remember his enthusiastic way of teaching. In 1975, the author of this book took over. In 1977, we founded the laboratory of building physics. From the start, research and consultancy focussed on the physical properties of building and insulation materials, on upgrading the performances of well-insulated assemblies, on net energy demand, end energy consumption, and primary energy consumption in buildings, on indoor environmental quality, on airborne and contact noise attenuation and on room acoustics. Later-on, urban physics were added with wind, rain, and pollution as main topics.

9

0.5 Units and symbols

0.5

Units and symbols

y The book uses the SI-system (internationally mandatory since 1977). Base units: the meter (m); the kilogram (kg); the second (s); the Kelvin (K); the ampere (A); the candela. Derived units, which are important when studying building physics, are: Unit of force: Unit of pressure: Unit of energy: Unit of power:

Newton Pascal Joule Watt

(N); (Pa); (J); (W);

1N 1 Pa 1J 1W

= 1 kg · m · s–2 = 1 N/m2 = 1 kg · m–1 · s–2 = 1 N · m = 1 kg · m2 · s–2 = 1 J · s–1 = 1 kg · m2 · s–3

y For the symbols, the ISO-standards (International Standardization Organization) are followed. If a quantity is not included in these standards, the CIB-W40 recommendations (International Council for Building Research, Studies, and Documentation, Working Group ‘Heat and Moisture Transfer in Buildings’) and the list edited by Annex 24 of the IEA, ECBCS (International Energy Agency, Executive Committee on Energy Conservation in Buildings and Community Systems) are applied. Table 0.1. List with symbols and quantities. Symbol

Meaning

Units

a

Acceleration

m/s2

a

Thermal diffusivity

m2/s

b

Thermal effusivity

W/(m2 · K · s0.5)

c

Specific heat capacity

J/(kg · K)

c

Concentration

kg/m3, g/m3

e

Emissivity

–

f

Specific free energy

J/kg

Temperature ratio

–

g

Specific free enthalpy

J/kg

g

Acceleration by gravity

m/s2

g

Mass flow rate, mass flux

kg/(m2 · s)

h

Height

m

h

Specific enthalpy

J/kg

h

Surface film coefficient for heat transfer

W/(m2 · K)

k

Mass related permeability (mass may be moisture, air, salt …)

s

l

Length

m

l

Specific enthalpy of evaporation or melting

J/kg

m

Mass

kg

n

Ventilation rate

s–1, h–1

p

Partial pressure

Pa

q

Heat flow rate, heat flux

W/m2

r

Radius

m

s

Specific entropy

J/(kg · K)

10

0 Introduction

Table 0.1. (continued) Symbol

Meaning

Units

t

Time

s

u

Specific latent energy

J/kg

v

Velocity

m/s

w

Moisture content

kg/m3

x, y, z

Cartesian co-ordinates

m

A

Water sorption coefficient

kg/(m2 · s0.5)

A

Area

m2

B

Water penetration coefficient

m/s0.5

D

Diffusion coefficient

m2/s

D

Moisture diffusivity

m2/s

E

Irradiation

W/m2

F

Free energy

J

G

Free enthalpy

J

G

Mass flow (mass = vapour, water, air, salt)

kg/s

H

Enthalpy

J

I

Radiation intensity

J/rad

K

Thermal moisture diffusion coefficient

kg/(m · s · K)

K

Mass permeance

s/m

K

Force

N

L

Luminosity

W/m2

M

Emittance

W/m2

P

Power

W

P

Thermal permeance

W/(m2 · K)

P

Total pressure

Pa

Q

Heat

J

R

Thermal resistance

m2 · K/W

R

Gas constant

J/(kg · K)

S

Entropy, saturation degree

J/K, –

T

Absolute temperature

K

T

Period (of a vibration or a wave)

s, days, etc.

U

Latent energy

J

U

Thermal transmittance

W/(m2 · K)

V

Volume

m3

W

Air resistance

m/s

X

Moisture ratio

kg/kg

Z

Diffusion resistance

m/s

11

0.5 Units and symbols Table 0.1. (continued) Symbol

Meaning

Units

D

Thermal expansion coefficient

K–1

D

Absorptivity

–

E

Surface film coefficient for diffusion

s/m

E

Volumetric thermal expansion coefficient

K–1

K

Dynamic viscosity

N · s/m2

T

Temperature

°C

O

Thermal conductivity

W/(m · K)

P

Vapour resistance factor

–

Q

Kinematic viscosity

m2/s

U

Density

kg/m3

U

Reflectivity

–

V

Surface tension

N/m

W

Transmissivity

–

I

Relative humidity

–

D, I, š

Angle

rad

[

Specific moisture capacity

kg/kg per unit of moisture potential

©

Porosity

–

©

Volumetric moisture ratio

m3/m3

§

Heat flow

W

Table 0.2. List with currently used suffixes. Symbol

Meaning

Indices A

Air

c

Capillary, convection

e

Outside, outdoors

h

Hygroscopic

i

Inside, indoors

cr

Critical

CO2, SO2

Chemical symbol for gases

m

Moisture, maximal

r

Radiant, radiation

sat

Saturation

s

Surface, area, suction

12

0 Introduction

Table 0.2. (continued) Symbol

Meaning

rs

Resulting

v

Water vapour

w

Water

I

Relative humidity

Notation [ ], bold

Matrix, array, value of a complex number

Dash

Vector (ex.: a )

0.6

Literature

[0.1]

Beranek, L. (Ed.) (1971). Noise and Vibration Control. McGraw-Hill Book Company.

[0.2]

CIB-W40 (1975). Quantities, Symbols and Units For the Description of Heat and Moisture Transfer In Buildings: Conversion factors. IBBC-TNP, Report No. BI-75-59/03.8.12, Rijswijk.

[0.3]

Winkler Prins Technische Encyclopedie, deel 2 (1976). Article on Building Physics. Uitgeverij Elsevier, Amsterdam, pp. 157–159 (in Dutch).

[0.4]

Northwood, T. (Ed.) (1977). Architectural Acoustics. Dowden, Hutchinson & Ross, Inc.

[0.5]

ISO-BIN (1985). Standards series X02-101 – X023-113.

[0.6]

Donaldson, B., Nagengast, B. (1994). Heat and Cold: Mastering the Great Indoors. ASHRAE Publication, Atlanta, 339 p.

[0.7]

Kumaran, K. (1996). Task 3: Material Properties. Final Report IEA EXCO ECBCS Annex 24, ACCO, Louvain, pp. 135.

[0.8]

Hendriks, L., Hens, H. (2000). Building Envelopes in a Holistic Perspective. IEA-ECBCS Annex 32, ACCO, Leuven.

[0.9]

Künzel, H. (2001). Bauphysik. Geschichte und Geschichten. Fraunhofer IRB-Verlag, 143 p. (in German).

[0.10]

Rose, W. (2003). The rise of the diffusion paradigm in the US, Research in Building Physics (J. Carmeliet, H. Hens, G. Vermeir, Eds.). A. A. Balkema Publishers, p. 327–334.

[0.11]

Hens, H. (2008). Building Physics: from a dormant beauty to a key field in building engineering. Proceedings of the Building Physics Symposium, Leuven.

[0.12]

USGBC (2008). LEED 2009 for New Construction and Major Renovations.

[0.13]

BRE (2010). BREEAM bespoke 2010.

1

Heat Transfer

1.1

Overview

Thermodynamics gives a first definition of heat. The discipline divides the world into systems and environments. A system can be anything: a material volume, a building assembly, a building, a part of the heating system, etc. We can even look at a city as a system. ‘Heat’ indicates how energy is transferred between a system and its environment. Whereas ‘work’ is purposeful and organized, heat is diffuse and unorganized. Particle physics offers a second definition, by which heat denotes the statistically distributed kinetic energy of atoms and free electrons. In both cases, heat is the least noble, or most diffuse form of energy, to which each nobler form degrades according to the second law of thermodynamics. Temperature as ‘potential’ determines the quality of heat. Higher temperatures mean higher quality, which refers to a higher mechanical energy of atoms and free electrons and the possibility to convert more heat into power via a cyclic process. It in turn is synonymous with higher exergy. Lower temperatures give heat a lower quality, which means less mechanical energy on the atomic scale and less exergy. Higher temperatures demand warming a system, i.e. adding heat. Lower temperatures require cooling a system, i.e. removing heat. Like any other potential, temperature is a scalar. Heat and temperature cannot be measured directly. Yet, temperature is sensed and indirectly measurable because a great deal of material properties depends on it: y Thermal expansion: for a mercury thermometer, we use the volumetric expansion of mercury as a measure for temperature y Change of electrical resistance: for a Pt100 resistance thermometer, the electrical resistance of a platinum wire functions as a measure for temperature y Change of contact potentials: this is the basis for the measurement of temperature with a thermocouple The SI-system uses two temperature scales: y Empiric Degree Celsius (indicated as °C, symbol T ). 0 °C is the triple point of water, 100 °C the boiling point of water at 1 Atmosphere y Thermodynamic Degree Kelvin (indicated as K, symbol: T ). 0 K is the absolute zero, 273,15 K the triple point of water The relation between both is T = T + 273.15. Temperatures are given in °C or K, temperature differences in K. The USA uses degree Fahrenheit (°F) as temperature scale. The relation with °C is: °F = 32 + 9 / 5 °C. Heat exists in sensitive form, which means temperature-related, or in latent form, which means as transformation heat. Sensitive heat is transferred by: Conduction. The energy exchanged when vibrating atoms collide and free electrons move collectively. Heat moves by conduction between solids at different temperatures in contact Building Physics: Heat, Air and Moisture. Fundamentals and Engineering Methods with Examples and Exercises. Second edition. Hugo Hens. © 2012 Ernst & Sohn GmbH & Co. KG. Published 2012 by Ernst & Sohn GmbH & Co. KG.

14

1 Heat Transfer

with each other and between points at different temperatures within the same solid. The mode also intervenes when heat is exchanged in gases and liquids and in the contacts between gases and liquids at one side and solids at the other. Conduction always occurs from points at higher to points at lower temperature (2nd law of thermodynamics). The mode needs a medium and induces no observable macroscopic movement. Convection. The displacement of molecule groups at a different temperature. Convection is by nature a consequence of macroscopic movement (transfer of enthalpy) and occurs in a pronounced way close to the contact between liquids and gases at one side and solids at the other. We distinguish forced, natural, and mixed convection depending on whether or not an external force, a difference in fluid density, or both cause the movement. In forced convection, work exerted by an exterior source may compel heat to flow from low to high temperatures. Convection needs a medium. Actually, in liquids and gases, convection always includes conduction. Also in convection mode, heat transfer between molecules occurs by conduction. Radiation. Heat transfer, caused by the emission and absorption of electromagnetic waves. At temperatures above 0 K, each surface emits electromagnetic energy. Between surfaces at different temperatures, that emission results in heat exchanges. Heat transfer through radiation does not need a medium. On the contrary, it is least hindered in vacuum and follows physical laws, which diverge strongly from conduction and convection. Latent heat moves along with a carrier, independent of temperature. Each time that carrier undergoes a change of state, related latent heat converts into sensitive heat or vice versa. For example, when water evaporates, it absorbs sensitive heat in a quantity equal to its latent heat of evaporation. The water vapour then diffuses to a cooler spot (= transfer) where it may condense with reemission of the latent heat of evaporation as sensible heat. These conversions influence the temperature course as well as the flow of sensitive heat in solids and building assemblies. Why are heat and temperature so important in building physics? Heat flow means end energy consumption. Thermal comfort requires keeping the operative temperature in buildings at the right level. That requires heating and cooling. Both require the combustion of fossil fuels, converting work into heat, or transforming thermal electricity, hydraulic electricity or electricity generated by wind turbines and PV-cells into heat. In every developed country, heating and cooling of buildings has a substantial share in the primary energy consumption. Fossil fuels will eventually become scarce, and their combustion is responsible for environmental problems such global warming. Thermal electricity in turn is energy-intensive to generate. For all these reasons, the design and construction of buildings with low net heating and cooling demand has become a necessity. That presumes restricted heat flows traversing the envelope. Temperatures influence comfort and durability. Sufficiently high surface temperatures contribute to a feeling of thermal well-being. However, summer indoor temperatures, which are too high, affect building usability. Further, the higher the temperature differences between layers in a building assembly, the larger the differential movements, the larger the thermal stresses and the higher crack risk. Temperatures below freezing in turn may damage moist porous materials, while high temperatures accelerate the chemical breakdown of synthetic materials. Differences in temperatures also induce moisture and dissolved salt displacement in porous materials, etc. Whether or not we can control these temperature effects, depends on how building assemblies are designed and constructed. This section deals with heat transfer by conduction, convection, and radiation. It ends with typical building physics related applications on the building assembly and whole building level. But first, some definitions:

15

1.2 Conduction

y Amount of heat Symbol: Q Units: Joule (J) Quantifies the energy exchange in the form of heat. Because energy is a scalar, heat is also a scalar. y Heat flow Symbol: § Units: Joule per second (J/s) = Watt (W) The heat per unit of time. Heat flow is a measure for ‘power’. Similar to heat, heat flow is a scalar. y Heat flow rate Symbol: q Units: Watt per m2 (W/m2) The heat per unit of time that passes a unit area perpendicular to the flow direction. Heat flow rate is a vector with the same direction as the surface vector. Components: qx, qy, qz in Cartesian co-ordinates, qR, qI, q4 in polar co-ordinates. Solving a heat transfer problem means the determination of two fields: that of temperatures and that of heat flow rate vectors. Consequently, we have two unknown quantities: T (x, y, z, t) and q (x, y, z, t). One is scalar (T ), the other vectorial (q). Computing them demands a scalar and a vector equation.

1.2

Conduction

1.2.1

Conservation of energy

For a first relation between heat flow rate (q) and temperature (T ) we go back to one of the axioms of classic physics: conservation of energy. In the case of pure conduction, no mass motion occurs between the system and the environment. An infinitely small part dV of a material now figures as system, with the remaining volume forming the environment. Hence, for the energy equilibrium in that system, one has: d§  d ©

dU  dW

(1.1)

where d§ is the resulting heat flow by conduction between the system and the environment and d© the dissipation energy released or absorbed per unit of time in the system. Dissipation can take many forms: heat produced by an exothermic chemical reaction, heat absorption by an endothermic chemical reaction, the Joule effect coupled to an electric current, the release, or absorption of latent heat, etc. dU represents the change per unit of time of the internal energy in the system. Finally, dW is the labour exchanged per unit of time between the system and the environment (dW = P d (dV) = P d2V with P pressure). The equation states that the heat flowing between the environment and the system (= d§) plus the heat released or absorbed in the system is responsible for a change in internal energy in the system and an exchange of labour between the system and the environment. If the process is isobaric (P constant), than (1.1) can be written as d (U + P dV ) = dQ + dE. In this equation, U + P dV represents the enthalpy H in the system. If we replace d§ by –div (q) dV, dH by [w (U cp T) / wt] dV with cp specific heat capacity at constant pressure and U the density

16

1 Heat Transfer

of the material (in kg/m3) and, d© by §c dV, §c being the dissipated heat per unit of time and unit of volume, positive for a source, negative for a sink, than energy conservation becomes:





§ w U cp T · ¨ div q  § c  ¸ dV ¨© ¸¹ wt

0

(1.2)

In the case of solids and liquids, the specific heat capacity at constant pressure (cp) hardly differs from the one for other changes of state. This is why we call cp simply ‘specific heat capacity’. Symbol c, units J/(kg · K). The product U c is the volumetric specific heat capacity, units J/(m3 · K). On the other hand for gases, the specific heat capacity depends on the change of state, reason why the following relation couples specific heat capacity at constant pressure to the one at constant volume (cv): cp = cv + R with R the specific gas constant (in Pa · m3/(kg · K)). As (1.2) holds for each infinitely small volume of material (here ‘the system’), the equation becomes: div q



wU c T  §c wt

(1.3)

i.e., the scalar relationship between q and T we need.

1.2.2

Fourier’s laws

1.2.2.1 First law As vector relation between q and T, we have the empirical conduction law, which the French physicist Fourier formulated in the 19th century: q

 O grad T

 O grad T

(1.4)

The equation states that the conductive heat flow rate at a point in a solid, a liquid or a gas is proportional to the temperature gradient at that point. The proportionality factor (O) is called the thermal conductivity or, in short, the O-value of the solid, liquid or gas (units W/(m · K)). The minus symbol in the equation indicates heat flow rate and temperature gradient oppose each other. Thermodynamics in fact teaches that without external work, heat flows from high to low temperatures. In the opposite case, entropy would decrease without energy supply from the environment, which is physically impossible. Instead, a gradient is positive from low to high temperatures. Equation (1.4) is known as Fourier’s first law. The following experimental observation supports it. If surfaces linking all points at the same temperature in a material are constructed, then the heat flow develops perpendicular to them and increases as they lie closer to each other (Figure 1.1). This increase happens proportionally to a property, which is material-specific: the thermal conductivity. For reasons of simplicity, that property is supposed constant and scalar, even though for building and insulating materials this is untrue. For porous ones, thermal conductivity is a function of temperature, moisture content, thickness, and sometimes age. In anisotropic materials, the property is even a tensor instead of a scalar. Fortunately, for first order calculations, the assumption ‘scalar and constant’ suffices. In right-angled Cartesian coordinates [x, y, z], equation (1.4) gives as heat flow rates along each of the axes:

17

1.2 Conduction

Figure 1.1. Lines of equal temperature (called isotherms) and equal heat flow rate (called isoflux lines).

qx

q x ux

qy

q y uy

qz

q z uz

wT wx wT O wy wT O wz O

wT wx wT O wy wT O wz O

The elementary heat flow across an area dA with direction n becomes: d§n

q dAn

O

wT dA u 2 wn n n

O

wT dA wn n

Along the three axes, that equation turns into: d§x d§ y d§z

wT dA wx x wT O dA wy y wT O dA wz z O

1.2.2.2 Second law The system, formed by the energy balance (1.3) and the conduction law (1.4) allows calculating the temperature field. Combining both gives: div  O grad T

wU c T  §c wt

(1.5)

18

1 Heat Transfer

If the thermal conductivity (O) and volumetric specific heat capacity (U c) is constant, then the equation simplifies to: ’ 2T

§ U c · wT § c  ¨© ¸ O ¹ wt O

(1.6)

with ’2 the Laplace operator. In Cartesian coordinates: w2 w2 w2   wx 2 wy 2 wz 2

’2

(1.7)

Equation (1.6) is known as Fourier’s second law. Further study of heat conduction now focuses on solving the system formed by Fourier’s two laws for a series of cases that are relevant for building physics.

1.2.3

Steady state

1.2.3.1 What is it? Steady state means that temperatures and heat flow rates are independent of time. In this respect, one needs constant boundary conditions, time independent material properties and constant energy dissipation. All these, but especially the constant boundary conditions, distort reality. In case thermal conductivity (O), volumetric specific heat capacity ( U c) and energy dissipation (§c ) are set constant by definition, a necessary and sufficient condition for steady state is: wT / wt = 0. Fourier’s second law then becomes (T instead of T ): ’2T



§c O

(1.8)

1.2.3.2 One dimension: flat assemblies In one dimension, (1.8) converts into (temperature changes parallel to the x-axis): §c d2T  O dx 2 which without dissipation becomes d2T / dx2 = 0. Solution: T

C1 x  C2

(1.9)

(1.10)

where C1 and C2 are the integration constants, defined by the boundary conditions. The linear equation (1.10) is the basis of the study of conduction through flat assemblies with the two surfaces at constant, although different temperature. Buildings contain numerous flat assemblies: low-sloped roofs, the sloped roof pitches, outdoor walls, floors, partition walls, glass surfaces, etc. In cross-section, flat assemblies can be single-layered or composite. Single-layered assemblies Assume thermal conductivity of the material (O) and thickness of the assembly (d) are known. Boundary conditions: x = 0: T = Ts1; x = d: T = Ts2 with Ts1 < Ts2, with Ts1 and Ts2, the temperatures on both surfaces, are also given. As a result, we obtain: C2 = Ts1, C1 = (Ts2 – Ts1) / d. (1.10) then becomes:

19

1.2 Conduction

T

Ts2  Ts1 x  Ts1 d

(1.11)

According to Fourier’s first law the heat flow rate is: q

 O grad T

O

dT dx

O

Ts1  Ts2 d

(1.12)

which means that the heat per square meter flows in the negative x-direction, from high to low temperature. In absolute terms: q

O

Ts2  Ts1 d

(1.13)

According to the equations (1.11) – (1.13), the temperature in a flat, single-layered assembly, which is in steady state and contains no heat source or sink, varies linearly between the temperatures at both surfaces (Figure 1.2). The heat flow rate is a constant, proportional to the thermal conductivity of the material and the temperature difference between the two surfaces, and inversely proportional to the thickness of the material. For a given thickness and given temperature difference, a lower thermal conductivity decreases the heat flow rate, which means less heat lost or gained. This is the reason why materials with very low thermal conductivity are called thermal insulation materials.

Figure 1.2. Single-layered assembly.

Equation (1.13) now rewrites as: q

'T d O

(1.14)

where d / O is the specific thermal resistance of the single-layered assembly, in short called the thermal resistance, symbol R, units m2 · K/W. The higher the thermal resistance, the smaller the heat flow rate for a given temperature difference, or, which is the same, the better the assembly insulates. A higher thermal resistance is obtained by increasing the assembly’s thickness or by constructing it using an insulating material, the latter being the most economical option.

20

1 Heat Transfer

The inverse of the thermal resistance is called thermal conductance, symbol P, units W/(m2 · K). The quantity tells how much heat per unit of time and unit of area passes across a flat singlelayered assembly for a temperature difference of 1 K between both surfaces. Composite assembly Each assembly made up of two or more plane-parallel material layers is called ‘composite’, see Figure 1.3. Most building assemblies are composite, the simplest example being a plastered partition wall: three layers, two of plaster, and one of masonry.

Figure 1.3. Filled cavity wall as example of a composite assembly.

Heat flow rate In steady state, without heat sources or sinks, the heat flow rate must be identical in each layer. If not, conservation of energy will allow for thermal storage or discharge, so the regime would become time-dependant, i.e. transient. Suppose the temperature on assembly surface 1 is Ts1 and on assembly surface 2, Ts2. If thermal conductivity Oi of the materials and thickness di of all layers is known and if possible contact resistances between layers may be disregarded, which is true for non-metallic materials, then per layer we can write (Ts1 < Ts2): Layer 1

q

O1

T1  Ts1 d1

Layer 2

q

O2

T 2  T1 d2

Layer n – 1

q

O n 1

Layer n

q

On

… T n 1  T n  2 d n 1

Ts2  T n 1 dn

21

1.2 Conduction

where T1, T2, …, Tn – 1 are the unknown interfaces temperatures and q the constant heat flow rate across the assembly. Rearrangement and summation gives: q

d1 O1

T1  Ts1

q

d2 O2

T 2  T1

! q

dn On

Ts2  T n 1

n §d · Sum: q ¦ ¨ i ¸ i 1 © Oi ¹

or:

q

Ts2  Ts1

Ts2  Ts1 n

¦  di

Oi

(1.15)

i 1

We call the sum 6 (di / Oi ) total thermal resistance of the composite assembly, symbol RT, units m2 K/W. The ratio di / Oi represents the thermal resistance of layer i, symbol Ri, units m2 K/W. The higher the total thermal resistance, the lower the steady state heat flow rate, and the better the assembly insulates. A high thermal resistance is obtained by incorporating a sufficiently thick insulation layer. This procedure is known as ‘insulating’. Since the energy crisis of 1973–1979 and the fear of global warming, a high performing thermal insulation of all assemblies in the building envelope is seen as best way to lower net energy demand for heating and the coupled CO2-release, if one uses fossil fuels for heating. Assembly design fixes thermal resistance. Because total thermal resistance is the sum of the thermal resistances of the separate layers, the commutation property applies: the sequence of the terms, which means the layers, does not change the value of the sum. As a result, inside insulation should be equivalent to outside insulation or to insulation within the assembly. From a building physics point of view, this mathematical conclusion is not correct. Indeed, for the same insulation thickness, inside insulation, outside insulation and insulation within will differ in overall performance. However, a one dimensional steady state model does not allow highlighting these differences. At this point, the analogy between the equations (1.14) and (1.15) and the equation for the current (i) in an electrical circuit with one or more electric resistances (Rei ) in series, subjected to a voltage difference 'V (i = 'V / 6 Rei ), is quite instructive. In fact, temperature replaces voltage, conductive heat flow rate electric current and thermal resistance electrical resistance. This allows converting thermal problems into an electric analogy (Figure 1.4). The conduction equation is indeed mathematically identical to the equation for the electric field.

Figure 1.4. The electrical analogy.

22

1 Heat Transfer

Temperatures When both surface temperatures are known, then temperature in all interfaces can be calculated. In fact, as for a single-layered assembly, temperatures in each layer form a straight line between the interface temperatures with the neighbour layers. These in turn follow from a rearrangement of the equations (1.15): T1

Ts1  q

d1 O1

Ts1   Ts2  Ts1

T2

T1  q

d2 O2

T1  q R2

R1 RT

T s1   Ts2  Ts1

 R1  R2 RT

… i

Ti

T i 1  q

di Oi

¦ Ri

Ts1   Ts2  Ts1 i

1

RT n 1

T n 1

Tn  2

d  q n 1 O n 1

Ts1   Ts2  Ts1

¦ Ri

i 1

RT

So, if we know the thermal conductivity (Oi ) and thickness (di ) of all layers and the temperature at both assembly surfaces, all interface temperatures can be calculated. Reshuffling Ti = Ti – 1 + q di / Oi into (Ti – Ti – 1) / di = q / Oi , shows the temperature gradient in each layer is inversely proportional to thermal conductivity. Hence, gradients are large in insulating materials (low thermal conductivity) and small in conductive ones (high thermal conductivity). i

¦ Ri

The equation Ti Tx Rs1x

Ts1   Ts2  Ts1 i

Ts1  q Rs1x

1

RT

can be rewritten as: (1.16)

with the thermal resistance between surface s1 and the interface x in the assembly. If the calculation starts at surface s2, then (1.16) becomes: T x Ts2  q Rs2x . In a [R, T] coordinate system (abscissa R, ordinate T, further called the [R, T]-plane or [R, T]-graph), both relations represent a straight line through the points (0, Ts1) and (R, Ts2) with the heat flow rate q as slope. Or, in a [R, T]-plane, a composite assembly behaves as a single-layered one. The graphical construction of the temperature line then becomes an easy job. Draw the assembly in the plane with the layers as thick as their thermal resistance, mark on surface s1 temperature Ts1, on surface s2 temperature Ts2, and trace a straight line between them. The correct geometric temperature curve is then obtained by transposing the intersections between that straight line and the successive interfaces to the thickness graph and linking the successive points with line sections. Of course, the layers must keep their correct sequence (Figure 1.5).

23

1.2 Conduction

Figure 1.5. Graphical construction of the temperature line in a composite assembly.

Heat flow For single-layered as well as composite assemblies, the product of the heat flow rate across with the assembly’s surface (A) gives the heat flow: §

qA

(1.17)

Single-layered assembly with variable thermal conductivity Each time we have a temperature difference over, or a moisture content profile across an assembly, thermal conductivity (O) becomes a function of temperature (T , thus of the ordinate (x) (Figure 1.6).

Figure 1.6. Moisture profile turning thermal conductivity into a function of x.

24

1 Heat Transfer

In case thermal conductivity in a single-layered assembly varies linearly with temperature (O = O0 + a T ), the heat flow rate follows from (steady state, x = 0, T = Ts1; x = d, T = Ts2): Ts2

d

Ts1

0

³  O0  a T dT ³ q dx

giving as a solution: O0  Ts2  Ts1 



2 2 a Ts2  Ts1



2

qd

or

q

O T m

Ts2  Ts1 d

(1.18)

with O(Tm ) thermal conductivity at the average temperature in the assembly. Thermal resistance becomes d / O(Tm ). The temperature curve is a parabola of the form: a T2  O0 T 2

q xC

If the assembly contains a moisture content of w(x), so that O = F [w(x)] = f (x), then the heat flow rate becomes (x = 0: T = Ts1; x = d: T = Ts2): d

q

dx

³ O  x 0

Ts2

³

dT

Ts1

If moisture is distributed in such a way that thermal conductivity increases in proportion to x (O = O0 + a x), then that equation gives as a solution: q

Ts2  T s1 1 § O0  a d · ln a ¨© O0 ¸¹

(1.19)

Also here, the denominator represents thermal resistance. The temperature curve is: Tx

ª 1 § O  a x ·º Ts1  q « ln ¨ 0 ¸» ¬ a © O0 ¹ ¼

(1.20)

Single-layer assembly with heat source or sink spread over the thickness In building assemblies, processes may take place that dissipate heat. Hydration of concrete, carbonation of lime plaster, condensation of water vapour, drying, etc., are all examples of that. Suppose a single-layered assembly releases or absorbs §c joule per unit of time and unit of volume in. If §c is a constant (which is not the case for most processes), the equation for the steady state heat transfer becomes: d2T dx 2



§c O

with as boundary conditions: x = 0: T = Ts1; x = d: T = Ts2 (Ts1 < Ts2). The solution is: T



§c 2 x  C1 x  C2 2O

(1.21)

25

1.2 Conduction

Figure 1.7. Uniform steady state heat source or sink in a single-layered assembly: temperature lines.

i.e. a parabolic temperature curve. For a heat source, this parabola is convex, for a sink, it is concave (Figure 1.7). The heat flow rate looks like: q

O

dT dx

§ c x  C1 O

(1.22)

Opposite to the case with no heat dissipation, now the heat flow rate changes from point to point in the assembly. The boundary conditions give as value for the integration constants: C1

Ts2  Ts1 § d c ,  d 2O

C2

Ts1

Composite assembly with local heat source or sink If condensate is deposited in the interface between two layers in an assembly, heat of evaporation is released there. Drying afterwards causes the heat of evaporation to be absorbed again. As a result, a local heat source and sink is activated. Assume that qc W/m2 heat is released and that the surface temperature Ts1 is higher than surface temperature Ts2 (Figure 1.8). Writing a steady state heat balance at the interface x where the source or sink is located then gives the temperatures in the assembly. We presume heat flows from both assembly surfaces to x. Hence, according to conservation of energy, the sum of all heat flow rates in x should be zero. As heat flow rates we have: Ts1  T x x y Conduction between surface s1 and x: qs1 Rs1x Ts2  T x x y Conduction between surface s2 and x: qs2 Rs2x y Dissipated heat qc in x. Drying makes qc negative, condensation positive.

26

1 Heat Transfer

Figure 1.8. Composite assembly, heat source in the interface between two layers.

In the two equations, Tx is the unknown temperature in interface x. Sum zero gives: Rs2x T s1  Rs1x Ts2  q c Rs1x Rs2x

Tx

(1.23)

Rs1x  Rs2x

We obtain the conductive heat flow rates between surfaces s1, s2 and x by introducing (1.23) x x into the equations for the heat flow rates qs1 and qs2 : § Ts1  Ts2 · q c Rs2x ¨© R ¸¹  R T T

x qs1

x qs2

ª § T  Ts2 · q c Rs1x º  « ¨ s1 ¸ R » T ¼ ¬ © RT ¹

(1.24)

Here RT ( Rs1x  Rs2x ) is the total thermal resistance of the assembly. If the dissipated heat qc is positive, the ingoing heat flow rate decreases and the outgoing heat flow rate increases compared to no dissipation. If the dissipated heat qc is negative, the inverse is true. 1.2.3.3 Two dimensions: cylinder symmetric In cylinder co-ordinates, cylinder symmetric problems behave as if they were one-dimensional. Hung heating pipes are an example of that. Some questions to be answered are: what is the heat loss per meter run, pipe temperature and how efficient is pipe insulation? Consider a pipe with inner radius r1 and outer radius r2. The temperature at the inner surface is Ts1, the temperature at the outer surface Ts2. Heat loss per meter run follows from conservation of energy (Figure 1.9): in steady state, in the absence of dissipation, the same heat flow must pass each cylinder concentric to the pipe’s centre. With that centre as origin, the balance per meter run gives: § = –O (2 S r) dT / dr = C te. Integration turns that expression into: §

r2

³

r1

dr r

2 S O

Ts2

³

Ts1

dT

or

§

Ts1  Ts2 §r · ln ¨ 2 ¸ © r1 ¹ 2S O

(1.25)

27

1.2 Conduction

Figure 1.9. The pipe problem.

ln (r2 / r1) / 2 S O is the equivalent of the thermal resistance R of a flat assembly. We call it the thermal resistance per meter run of pipe, units m · K/W. For a composite pipe, analogous reasoning as for a composite flat assembly allows writing the heat flow per meter run as: §

Ts1  Ts2 ª § ri 1 · º ln ¨ » n « © r ¸¹ « ¦ « 2 S Oi »» i 1 ¬ i ¼

(1.26)

Temperatures follow from:

T i 1

i

Ts1  § ¦

i 1

ª § ri 1 · º « ln ¨ ¸» « © ri ¹ » «¬ 2 S Oi »¼

(1.27)

1.2.3.4 Two and three dimensions: thermal bridges Overview When looking in detail at the heat transfer across outer walls, roofs, floors, partition walls, etc., it is clear that the hypothesis ‘flat’ does not apply everywhere. What about lintels above windows? What about window reveals? What about junctions between two outer walls? What about corners between two outer walls and a low-sloped roof (Figure 1.10)? Also, the surfaces of flat assemblies are not necessarily isothermal (Figure 1.11). To study steady state heat transfer in these cases, we have to return to: w2 T w2 T w2 T   wx 2 wy 2 wz 2

r

§c O

or, without dissipation: w2 T w2 T w2 T   wx 2 wy 2 wz 2

0

For some very simple cases (single material, easy geometry, simple boundary conditions), this partial differential equation can be solved analytically. For the majority of building details, however, a numerical solution is the only option left.

28

1 Heat Transfer

Figure 1.10. Construction details where heat transfer develops in two and three dimensions.

Figure 1.11. Flat assembly with non-isothermal inside surface.

Control Volume Method (CVM) Many building details are composed of rectangular material volumes. To apply CVM in such cases we mesh the detail in elementary cube or beam-like control volumes and write energy conservation per control volume. In steady state: the sum of heat flows from all neighbouring volumes to each central control volume is zero (Figure 1.12). When constructing the mesh, either of two options with respect to the interfaces between materials can be chosen: y Meshing coincides with the interfaces. In that case, all control volumes are materialhomogenous. However the calculations do not directly give the interface temperatures y Meshing is done so control volume centres lie on the interfaces. Now the volumes are not material-homogenous but the calculations directly give the interface temperatures.

29

1.2 Conduction

Figure 1.11. Flat assembly with non-isothermal inside surface.

In practice, the second method is preferred. Along the x-, y- and z-axis, the side of the control volume around a mesh point equals the sum of half the distances in negative and positive direction along the axes to the neighbouring mesh points (in three dimensions, six in total: 2 in x-, 2 in y- and 2 in z-direction/in two dimensions four in total: 2 in x- and 2 in y-direction). To give some examples: Central and neighbouring control volumes situated in the same material For a mesh width a along the three axes, the equations are: Heat flow of (l – 1, m, n) to (l, m, n) (= central control volume) §ll, m1,,mn , n

O

Tl 1, m, n  Tl , m, n a 2 a



a O Tl 1, m, n  Tl , m, n



Heat flow from the five neighbouring control volumes: §ll, m1,,mn , n

O

§ll,,mm, n1, n

O

§ll,,mm, n1, n

O

§ll,,mm,,nn1

O

§ll,,mm,,nn1

O

Tl 1, m, n  Tl , m, n a 2 a

Tl , m 1, n  Tl , m, n a 2 a

Tl , m 1, n  Tl , m, n a 2 a

 T l , m , n 1  T l , m , n a 2 a

 T l , m , n 1  T l , m , n a 2 a





















a O Tl 1, m, n  Tl , m, n a O Tl , m 1, n  Tl , m, n a O Tl , m 1, n  Tl , m, n a O T l , m , n 1  T l , m , n a O T l , m , n 1  T l , m , n

30

1 Heat Transfer

Adding and sum zero gives: Tl 1, m, n  Tl 1, m, n  Tl , m 1, n  Tl , m 1, n  Tl , m, n 1  Tl , m, n 1  6 Tl , m, n

0

i.e. a linear equation with seven unknowns: temperature in the central and temperatures in the six neighbouring control volumes. In two dimensions, five unknowns are left: Tl 1, m  Tl 1, m  Tl , m 1  Tl , m 1  4 Tl , m

0

Central control volume spanning the interface between two materials If the interface is parallel with the [x, y] plane, and thermal conductivity of both materials is O1 and O2 then for a mesh width a along the three axes, the following equations are obtained: Heat flow from (l – 1, m, n) to (l, m, n) (= central control volume) §ll, m1,,mn , n or: §ll, m1,,mn , n

O1

Tl 1, m, n  Tl , m, n a 2  O Tl 1, m, n  Tl , m, n a 2 2

2a



a  O1  O2 Tl 1, m, n  Tl , m, n

2a



2

Heat flow from the five neighbouring control volumes: §ll, m1,,mn , n §ll,,mm, n1, n §ll,,mm, n1, n §ll,,mm,,nn1 §ll,,mm,,nn1



a  O1  O2 Tl 1, m, n  Tl , m, n



2



a  O1  O2 Tl , m 1, n  Tl , m, n



2



a  O1  O2 Tl , m 1, n  Tl , m, n



2

 a O 2  T l , m , n 1  T l , m , n a O1 Tl , m, n 1  Tl , m, n

Adding and sum zero gives:

 O1  O2 Tl 1, m, n  Tl 1, m, n  Tl , m 1, n  Tl , m 1, n 2  O2 Tl , m, n 1  O1 Tl , m, n 1  3  O1  O2 Tl , m, n

0

Once again a linear equation with seven unknowns: temperature in the central and temperatures in the six neighbouring control volumes. Two dimensions give a linear equation with five unknowns:

 O1  O2 Tl 1, m  Tl 1, m 2

 O2 Tl , m 1  O1 Tl , m 1  2  O1  O2 Tl , m, n

0

31

1.2 Conduction

Central control volume lying on the intersection between three materials Presume the interfaces are parallel to the [x, y] and [y, z]-plane. Thermal conductivity of the three materials is O1, O2 and O3. The sum found is:

 O2  O3

Tl 1, m, n 2

  O1  O2  O3

  O2  O3

Tl 1, m, n

2 Tl , m, n 1  Tl , m, n 1 4

 O3 Tl , m 1, n   O1  O2

 3 O1  3 O2  6 O3

Tl , m 1, n

Tl , m, n 2

2 0

i.e. a linear equation with seven unknowns. In two dimensions, the result is a linear equation with five unknowns: O3 Tl 1, m   O1  O2   O1  O2  O3 Tl , m

Tl 1, m 2 0

  O2  O3

Tl , m 1 2

  O1  O3

Tl , m 1 2

A solution for all other cases is found the same way. In three dimensions, a control volume may include a maximum of eight materials. In two dimensions, that maximum is four. Per control volume, we obtain a linear equation with seven (three dimensions) or five unknowns (two dimensions). For p control volumes, the result is a system of p equations with p unknowns: the p temperatures in the p control volumes. Solving the system gives the temperature distribution. The above equations then allow calculating the heat flow rate along the x and y or the x, y, and z-axis. That gives the components of the heat flow rate vectors. CVM leads to the following general algorithm. Suppose Ps is the surface-linked thermal conductance between two neighbouring control volumes (units W/K). If the two lie in the same material, then we have: Ps = (O / d) A. If they lie in different materials, then the surface-linked thermal conductance consists of a serial and/or parallel circuit of conductances (Figure 1.13). Thus, for the heat flow from adjacent to central control volume, the following applies: §ll, m1,,mn , n



Ps l, l, m1,, nm, n Tl 1, m, n  Tl , m, n



Figure 1.13. Thermal conductances around the central point in a meshing volume.

(1.28)

32

1 Heat Transfer

Summing for all adjacent control volumes gives:

¦

i l , m, n j r1

ª¬ Ps , i  j Ti  j º¼  Tl , m, n

¦

Ps , i  j

0

(1.29)

i l , m, n j r1

Consequently, the equations only demands calculating all surface-linked thermal conductances Ps. Control volumes with known temperature act as known terms and are transferred to the right. In such a manner, each building detail is converted into a system of equations having the form [Ps]p,p [T]p = [Ps,i,j,k Ti,j,k]p with [Ps]p,p the p rows, p columns the conductance matrix, [T]p the p unknown temperatures the column matrix and [Ps,i,j,k Ti,j,k]p the p known temperatures column matrix. The accuracy of a CVM-calculation depends on mesh width. The smaller these widths, the closer the numeric solution approaches the exact one. However, finer meshing gives a larger system of equations. An infinitely fine mesh produces the exact solution. The price, however, is an infinite number of equations. Therefore, a compromise between accuracy and calculation time has to be sought. An adequate mesh choice (large widths where small temperature gradients are expected and small widths where large temperature gradients are expected) minimizes the difference between the numeric and exact solution. How to apply CVM is the subject of EN ISO 10211-02. During the past decades, powerful software packages allowing thermal evaluation of complex building details have become available (Trysco®, Heat®, etc.).

1.2.4

Transient regime

1.2.4.1 What? Transient means that temperatures and heat flow rates change with time. Varying material properties, time-dependent heat dissipation, and time-depending boundary conditions may be responsible for that. In the latter case, Fournier’s second law becomes: ’2T

U c wT O wt

Figure 1.14. Periodical changes in temperature.

33

1.2 Conduction

Figure 1.15. Non-periodical change in temperature.

Boundary conditions (surface temperature and heat flow rate) may vary periodically or nonperiodically with time, periodically when they fluctuate regularly, and non-periodically when a onetime only change occurs. For example, the outside temperature changes over a period of 1 year, n days, 1 day, and 1 / x days (Figure 1.14). Heating-up instead is responsible for a sudden increase of the inside temperature and surface heat fluxes (Figure 1.15). Such a ‘jump’ is a non-periodical classic, another being the Dirac impulse. 1.2.4.2 Flat assemblies, periodic boundary conditions For a flat assembly, Fourier’s second law simplifies to: w2 T wx 2 or, with a a

U c wT O wt O : Uc

w2 T wx 2

wT wt

a is called the thermal diffusivity of a material. Units m2/s. The quantity indicates how easily a local temperature change spreads all over a material. The higher its value, the faster it spreads. A large thermal diffusivity requires a light material with high thermal conductivity or a heavy material with low specific heat capacity. None of these exists. Light materials have a low thermal conductivity, while specific heat capacity does not differ substantially between light and heavy. The majority of materials therefore have similar thermal diffusivities. Metals are the only exception. If we substitute the ordinate x by the thermal resistance R (= x / O), which means multiplying both terms with O2, then Fourier’s second law becomes: w2 T wR 2

Uc O

wT wt

(1.30)

34

1 Heat Transfer

with a heat flow rate of: q



wT wR

Boundary conditions Periodic means the temperature and heat flow rate at both assembly surfaces fluctuate more or less regularly with time. Each periodic signal with base period T now is transformable into a Fourier series. For the surface temperature: Bs0 f ¦ 2 n 1

Ts

ª §2 n St· § 2 n S t ·º « Asn sin ¨© T ¸¹  Bsn cos ¨© T ¸¹ » ¬ ¼

with: 2T §2 n St· ³ T t sin ¨© T ¸¹ dt T 0 s

Asn

Bsn

2T §2 n St· ³ T t cos ¨© T ¸¹ dt T 0 s

Bs0 / 2 being the average value over the base period T, while As1, As2 … Asn, Bs1, Bs2 … Bsn are the harmonics of 1st, 2nd, …, nth order. Euler’s formulas allow rewriting the harmonics in complex form: sin  x

exp i x  exp  i x 2i

cos  x

exp i x  exp  i x 2

or, for x = 2 n S t / T: Ts t

1 2

f

¦

n f

ª § 2 i n S t ·º ¸¹ » « Dsn exp ¨© T ¬ ¼

(1.31)

where Dsn = Bsn – i Asn, Dsn being the nth complex surface temperature with as amplitude and phase shift: Amplitude

Bs2n  As2n

Phase shift

§ A · atan ¨  sn ¸ © Bsn ¹

The amplitude reflects the size of the complex temperature, whereas the phase shift represents the time delay with respect to a cosine function with period T / (2 n S (in radians). Take as an example: the thirty-year mean air temperature at Uccle, Belgium. Figure 1.16 approximates the monthly values as: Te

ª 2St º 9, 45  7,18 cos «   2,828 » 365, 25 ¬ ¼

with t time in days (365.25 for a period of 1 year, corrected for the leap years), 9.45 average annual temperature Tem, 7.18 annual amplitude Te1 and –2.829 related phase shift Ie1 (in radians). Written as a Fourier series, the equation becomes:

35

1.2 Conduction

Figure 1.16. Thirty year monthly average air temperature in Uccle, Belgium. The black circles show the measured values for the period 1901–1930. The thick line gives the result of a Fourier analysis with one harmonic. The thinner lines represent the annual mean and the sine and cosine terms.

Te

§ 2St · § 2St · 9, 45  2, 21 sin ¨  6,83 cos ¨ © 365, 25 ¸¹ © 365, 25 ¸¹

where B0 = 18.9; A1 = –2.21 and B1 = –6.83. Amplitude and phase shift are deduced as follows: § 2St · § 2St · 2.21 sin ¨  6.83 cos ¨ © 365.25 ¸¹ © 365.25 ¸¹ ª § 2St · § 2 S t ·º T e1 «sin  Me1 sin ¨ ¸¹  cos  Me1 cos ¨© ¸ © 365.25 365.25 ¹ »¼ ¬ for Te1 sin (Ie1) = –2.21 and Te1 cos (Ie1) = –6.83. Solution of this system gives: tg  Me1 Tˆ e1

0.335 or Me1

2.828 rad

2.21 2  6.83 2

7.18 °C

Graphically, 7.18 cos [2 S t / 365.25 – (–2.828)] is a rotating vector with value 7.18 (= Te1), which starts at an angle –2.828 (= Ie1) with the real axis. As a complex number, this vector looks like: De1

T e1 cos  Me1  i T e1 sin  Me1

B1  i A1

We call De1 the annual complex outside air temperature for Uccle. Such complex temperature contains all information on the annual fluctuation: amplitude 7.18 °C and phase shift –2.828 rad.

36

1 Heat Transfer

Solution Suppose a periodic temperature or periodic heat flow rate activates one or both surfaces of a flat assembly from time zero on. The assembly’s response is twofold: a transient, which ebbs slowly, and a periodic. The latter includes the same harmonics as the surface signal (an assembly cannot compress (smaller period) nor extend (larger period) thermal signals). However, the amplitudes of both the temperature and the heat flow rate dampen as the signal traverses the assembly. At the same time, while moving, both signals gradually run behind compared to the signal at the surface. A phase shift develops. Or, the two signals behave as complex numbers. So, a Fourier series, in which the complex temperature and complex heat flow rate are functions of thermal resistance, can be represented by the solution: 1 2

T  R, t

f

¦

n f

dT  R, t  dR

q t

ª § 2 i n S t ·º ¸¹ » « Dn  R exp ¨© T ¬ ¼ 1 2

f

¦

n f

(1.32)

ª § 2 i n S t ·º ¸¹ » « Dscn  R exp ¨© T ¬ ¼

The accent in Dcsn is a reminder that the complex heat flow rate equals the first derivative of the complex temperature to thermal resistance. Its amplitude and phase shift are: Amplitude

Bscn2  Ascn2

Phase shift

§ Ac · atan ¨  sn ¸ © Bscn ¹

Single-layer assembly Inserting the temperature equation (1.32) into equation (1.30) gives: f

¦

n f

2 º °­ ª d Dn  R 2 U c O i n S § 2 i n S t · °½ Dn  R » exp ¨  ®« ¸¹ ¾ 2 © T T »¼ ¯° «¬ dR ¿°

0

(1.33)

The zero sum holds if all coefficients of the exponential functions of time are zero: d 2 Dn  R dR

2



2 Uc Oi nS Dn  R T

0

(1.34)

Or, the partial differential equation of second order breaks up into 2 ’ + 1 differential equations of second order, with the complex temperature Dn(R) as the dependent and the thermal resistance R as the independent variable. These equations hold for all values of n between –’ and +’. Because the solutions between 0 and –’ and 0 and +’ mirror each other, we are left with ’ + 1 equations. Because n intervenes as a parameter and the equations are identical, the solution for one value of n suffices to get them all: Dn  R C1 exp  Zn R  C2 exp   Zn R with Z2n

2 Uc Oi nS and 0 ” n ” ’. T

(1.35)

37

1.2 Conduction

The quantity Z is called the ‘thermal pulsation’. Converting (1.35) using the formulas sinh  x

1 ªexp  x  exp   x ¼º 2¬

cosh  x

1 ªexp  x  exp   x ¼º 2¬

gives as complex temperature and complex heat flow rate: Dn  R

C1  C2 sinh  Zn R  C1  C2 cosh  Zn R

Dcn  R

dD dR

Zn ª¬C1  C2 cosh  Zn R  C1  C2 sinh  Zn R º¼

The integration constants (C1 – C2) and (C1 + C2) follow from the boundary conditions: y At the assembly surface R = 0, the complex temperature is Dsn(0), or: Dsn 0

C1  C2 0  C1  C2

C1  C2

y At the assembly surface R = 0, the complex heat flow rate is Dcsn (0) , or: Dscn 0

Zn ª¬C1  C2  C1  C2 0º¼

Zn C1  C2

That converts the complex temperatures and complex heat flow rates into: sinh  Zn R Zn

Dn  R

Dsn 0 cosh  Zn R  Dscn 0

Dnc  R

Dsn 0 Zn sinh  Zn R  Dscn 0 cosh  Zn R

(1.36)

(1.36) forms a system of two equations, which allows solving two unknowns. Inclusion into the Fourier series (1.32) gives the time-related equations for the temperature and heat flow rate in the single-layered assembly. In general, most of the attention goes to the relationship between the complex quantities at both assembly surfaces. Suppose the surface at R = 0 is inside. Hence, R = RT is then the outside surface for envelope assemblies, and the surface at the other side for partition assemblies (Figure 1.17). In general, for R = RT the system equation becomes: ª¬ Asn  RT º¼ Wn ¬ª Asn 0 ¼º

(1.37)

Figure 1.17. Single-layered wall: the surfaces R = 0 and R = RT.

38

1 Heat Transfer

where [Asn (0)] is the column matrix of the unknown complex quantities at the one surface, and [Asn (RT)] the column matrix of the known complex quantities at the other surface. Wn, the coefficient matrix, is called the system matrix for the nth harmonic of a single-layered assembly. It depends solely on the assembly properties (thickness and material data), the base period, and the harmonic considered. In periodic regime, that matrix replaces the steady state thermal resistance, though it contains much more information. In order to interpret that information, we examine some specific cases. Harmonic order zero (n = 0) For n = 0, only the terms Ds0 and Dcs0 are left, with Ds0 / 2 and Dcs0 / 2 the average temperature and average heat flow rate at the inside surface over the period T. Thermal pulsation for the average is Z02  n Ds0  R

Ds0 0  Ds0 c 0

0 0

0

2i Uc O0S T

Ds0 c R

0 , so that (1.36) becomes:

Ds0 0 0  Ds0 c 0

Ds0 c 0

The second equation shows that the average incoming and outgoing heat flow rates are identical. For a value of R smaller than the maximum RT, i.e. at a spot somewhere in the assembly, we arrive at the same conclusion. Consequently the heat flow rate Dcs0 is a constant, as it is in steady state. The undefined ratio 0 / 0 in the first equation is solved using the de l’Hopitâl’s rule: ª Z R º lim «sinh 0 » Z0 o 0 Z0 ¼ ¬

lim ª¬ R cosh  Z0 R º¼

Z0 o 0

R

giving Ds0 ( R) Ds0 (0)  Ds0 c (0) R . For the thermal resistance R at a spot somewhere in the assembly, one gets the same result. Consequently, in the [R, T] plane, the average temperatures in the assembly all lie on a straight line with the heat flow rate as slope. This also concurs with steady state, or both results extend the steady state concept from ‘invariable over time’ to ‘concerns the average over a sufficiently long period’. Harmonics order different from zero (n  0) (1) Suppose the inside surface temperature is constant. Then all complex inside surface temperatures are zero and the first equation (1.36) becomes: Dsn  R

Dscn 0

sinh  Zn R Dsn  R or Zn Dscn 0

sinh  Zn R Zn

(1.38)

The function sinh (Zn R) / Zn then represents the ratio between the complex temperature at the outside surface or the surface at the other side and the complex heat flow rate at the inside surface. In steady state, we call that ratio, thermal resistance. Analogically, the complex number sinh (Zn R) / Zn is named the dynamic thermal resistance for the nth harmonic, symbol Dqn , and units m2 · K/W. The amplitude gives the size, the argument Iqn arg [sinh ( Zn R) / Zn ] the time shift between the complex temperature at the one and the complex heat flow rate at the other surface. The hypothesis behind the definition of dynamic thermal resistance may seem theoretical. However reality indicates the case is applicable to a building kept at constant temperature. We can easily determine the limit values for a zero and infinitely long period:

39

1.2 Conduction

Infinite period: means thermal pulsation zero. Thus: ª sinh 0 º lim « no0 ¬ 0 »¼

ª Dqn º ¬ ¼ Iqn

ª sinh 0 º lim «arg no0 ¬ 0 »¼

lim ¬ª R cosh 0 ¼º

no0

R

f

Or, the dynamic thermal resistance equals the thermal resistance in that case. An infinite period stands for steady state. Zero period: means thermal pulsation infinite. Thus: ª sinh  f º lim « no f ¬ f »¼

ª Dqn º ¬ ¼ Iqn

sinh  f º ª lim «arg no f ¬ f »¼

lim ¬ª RT cosh  f ¼º

no f

f

0

Or, for ever-shorter fluctuations, the dynamic thermal resistance turns infinite, i.e. the assembly dampens the signal completely. The two limit values show the dynamic thermal resistance is always larger than the thermal resistance. Thus, it suffices to impose a high thermal resistance in order to obtain a high dynamic one. (2) Suppose the heat flow rate at the inside surface is constant. Then all complex heat flow rates at that surface go to zero and the first equation (1.36) becomes: Dsn  R

Dsn 0 cosh  Zn R or

Dsn  R Dsn 0

cosh  Zn R

(1.39)

The function cosh (Zn RT) then represents the ratio between the complex temperature at the outside surface or the surface at the other side and the complex temperature at the inside surface. We call that function temperature damping of an assembly for the nth harmonic, symbol DTn , no units. The amplitude again gives the size, the argument §Tn arg [cosh ( Zn R )] the time shift between the complex temperatures at both surfaces. The hypothesis behind the definition of temperature damping seems fictitious. Indeed, the property has no equivalent in steady state, although it quantifies the spontaneous climate control capacity of an assembly. In dry, hot regions, it is an important characteristic. If solar gains are low and ventilation is restricted, then an enclosure with high temperature damping will create a stable inside climate. Hence, in such regions, specifications should be quite demanding with respect to that property. In this sense, it is a performance characteristic. Limit values for a zero and infinite period are: Infinite period: means thermal pulsation zero. Thus: ª DTn º ¬ ¼

lim ¬ª cosh 0 ¼º

no0

1

ITn

lim ª¬arg ¬ªcosh 0 ¼º º¼

no0

f

Or, for long lasting fluctuations temperature damping approaches one. In fact, in a steady state, no temperature difference can exist between the inside and other surface of an assembly when no heat moves across.

40

1 Heat Transfer

Zero period: means thermal pulsation infinite. Thus: lim ¬ªcosh  f ¼º

ª DTn º ¬ ¼

no f

f

lim ªarg ¬ªcosh  f ¼º º¼

ITn

no f ¬

0

Or, for ever-shorter fluctuations, temperature damping goes to infinite. (3) Suppose the temperature at the outside surface or at the surface at the other side remains constant. Then all complex temperatures at that surface go to zero, converting the first equation of the system (1.36) into (after inclusion of Dqn and DTn ): 0

DTn Dsn 0  Dqn Dscn 0 or

Dscn 0 Dsn 0



DTn Dqn

 Zn cotgh  Zn R

(1.40)

The ratio DTn / Dqn then relates the complex heat flow rate at the inside surface to the complex temperature there. We call it the admittance of an assembly for the nth harmonic, symbol Adn, units W/(m2 · K). The amplitude gives the size, the argument InAd ITn  Iqn the time shift between the complex heat flow rate and complex temperature at the inside surface. Also the admittance looks like a fictitious quantity. At any rate the property shows to what extent assemblies absorb heat. The higher its amplitude, the larger that absorption under fluctuating surface temperature conditions. Thus, the expression ‘capacitive building’ points to an enclosure or to inside partitions and floors with high admittance. Since the thermal pulsation Zn also writes as Zn i U c O 2 n S  T , large admittances obviously require large values of U c O . This square root of the product of volumetric specific heat capacity with thermal conductivity is called the contact coefficient or effusivity of a material, symbol b and units J/(m2 · s–1/2 · K). The larger the contact coefficient, the more active a material becomes as a heat storage medium. The usage of heavy materials without insulating finish results in a high effusivity as well as in a large admittance. The limit values for a zero and infinite period are: Infinite period: means thermal pulsation zero. Thus: ª Ad n º ¬ ¼

§ Dn · lim ¨ Tn ¸ no0 © D ¹ q

1 R

InAs

ª § Dn ·º lim «arg ¨  Tn ¸ » no0 « ¬ © Dq ¹ »¼

f

Or, for long lasting fluctuations, the admittance approaches the thermal conductance (P). In a steady state, the heat flow rate entering or leaving is indeed equal to P 'Ts. Zero period: means thermal pulsation is infinite. Thus: ª nº ¬ Ad ¼

§ Dn · lim ¨ Tn ¸ no f © D ¹ q

f

InAs

ª § Dn ·º lim «arg ¨ Tn ¸ » no f « ¬ © Dq ¹ »¼

0

Or, for ever-faster fluctuations, the admittance becomes infinite. The properties defined allow rewriting the complex system matrix as: Wn

ª DTn « « Z2n Dqn ¬

Dqn º » DTn »¼

41

1.2 Conduction

From complex to real system matrix Thermal pulsation can be written as:

Zn

ib

2nS T

2nS T

1  i b

1  i b

2

nS T

Conversion of i in the formula comes from (1 + i)2 = 2 i or right, the product Zn R becomes:

Zn R

1  i b x

nS T

O

1  i x

i

(1  i ) / 2 . With Zn set

nS aT

where x is the ordinate value (going from zero to the thickness of the assembly) and a (=O2 / b2) thermal diffusivity. If we name Xn the product x n S / a T , then thermal pulsation Zn simplifies to (1 + i) Xn / R and the term DTn in the system matrix writes as: DTn

cosh  Zn R

or, with cosh i X n cosh  Zn R

cosh ¬ª1  i X n ¼º

cos  X n and sinh i X n

cosh  X n cosh i X n  sinh  X n sinh i X n i sin  X n :

cosh  X n cos  X n  i sinh  X n sin  X n

Analogously: sinh  Zn R Zn

^

R ªsinh  X n cos  X n  cosh  X n sin  X n º¼ 2 Xn ¬

`

 i ª¬cosh  X n sin  X n  sinh  X n cos  X n º¼ Zn sinh  Zn R

^

Xn ªsinh  X n cos  X n  cosh  X n sin  X n º¼ R ¬  i ¬ªcosh  X n sin  X n  sinh  X n cos  X n ¼º

One now defines as functions Gni: Gn1

cosh  X n cos  X n

Gn 2

sinh  X n sin  X n

Gn3

1 ªsinh  X n cos  X n  cosh  X n sin  X n º¼ 2 Xn ¬

Gn 4

1 ªcosh  X n sin  X n  sinh  X n cos  X n º¼ 2 Xn ¬

Gn5

X n ª¬sinh  X n cos  X n  cosh  X n sin  X n º¼

Gn 6

X n ª¬cosh  X n sin  X n  sinh  X n cos  X n º¼

`

42

1 Heat Transfer

This allows rewriting dynamic thermal resistance, temperature damping, and the other term as: cosh  Zn R

Gn1  i Gn 2

sinh  Zn R Zn

R Gn3  i Gn 4

Zn sinh  Zn R

Gn5  i Gn 6 R

When amplitude and time shift of the dynamic thermal resistance is Dqn , the temperature damping DTn , the admittance Adn of a single-layer assembly become (R = RT): Amplitude ª Dqn º ¬ ¼

RT

Phase shift Gn23  Gn24

Mqn

atan

Gn 4 Gn3

atan

Gn 2 Gn1

ª DTn º ¬ ¼

Gn21  Gn22

Mqn

ª Ad n º ¬ ¼

ª DTn º ¬ ¼ ª Dqn º ¬ ¼

MnAd

MTn  Mqn

The nth 2 × 2 complex system matrix is now converted into a 4 × 4 real matrix:

Wn

ª Gn1 « G « n2 « Gn5 « R « T « Gn 6 « ¬ RT

Gn 2 Gn1 Gn 6 RT

RT Gn3  RT Gn 4

Gn5 RT

Gn 2

Gn1

RT Gn 4 º RT Gn3 » » » Gn 2 » » » Gn1 » ¼

For the phase shift, the rules given in Figure 1.18 for the sign of the G-functions apply:

Figure 1.18. Periodic thermal regime, phase shift: sign of the G-functions.

1.2 Conduction

43

Composite assemblies A composite assembly consists of series-connected single-layer assemblies, in the case with different layers, each with a system matrix Wn,i , now called the layer matrix, whereas the assembly as a whole has a system matrix Wn,T, for which the following applies (Figure 1.19): ª¬ As, n  RT º¼ Wn,T ª¬ As, n 0 º¼

(1.41)

with [As,n (RT)] the column matrix of the nth complex temperature and heat flow rate at the one bounding surface, and [As,n (0)] the column matrix of the nth complex temperature and heat flow rate at the other bounding surface. As relationship between the complex temperature and the complex heat flow rate at the interfaces j and j + 1 we have [An, j + 1] = Wn, j [An, j]. The latter can be written for each layer, starting at surface R = 0 which, as agreed upon, coincides with inside: ª¬ An,1 º¼ Wn,1 ª¬ As, n 0 º¼ ª¬ An,2 º¼ Wn,2 ª¬ An,1 º¼ ! ª¬ An, m 1 º¼ Wn, m 1 ª¬ An, m  2 º¼ ª¬ As, n  RT º¼ Wn, m ª¬ An, m 1 º¼

Figure 1.19. Composite assembly, system matrix Wn,T.

(1.42)

44

1 Heat Transfer

Importing each preceding equation in the next gives: ª¬ As, n  RT º¼ Wn, m Wn, m 1 "Wn,2 Wn,1 ª¬ As,n 0 º¼ Combination with (1.41) results in: m

Wn,T

– Wn, j

(1.43)

j 1

or, the system matrix of a composite assembly equals the product of the layer matrixes. Multiplication starts at the inner layer and ends with the outer one. Each successive layer matrix must be multiplied with the product of the preceding ones. Because the commutation property does not apply for a product of matrixes, one has to respect this sequence. Translated to design and construction practice, contrary to the thermal resistance, the transient response of a composite assembly changes with layer sequence! At any rate, each layer matrix in the system matrix keeps the meaning it has for a single-layer assembly. Temperatures If two of the four complex boundary conditions (two complex temperatures and two complex heat flow rates) at the surfaces are known, the two others follow from the matrix equation: [Asn (RT)] = WnT [Asn (0)]. The complex temperatures and heat flow rates in the interfaces are then found by ascending or descending the equations (1.42) in the correct sequence. For single-layered assemblies, equations (1.32) gave the temperatures and heat flow rate evolution with time for the thermal resistance varying between 0 and RT. Of course, for ease of numerical calculation, we best use a composite assembly approach. For that purpose, the single layer is divided into m layers with thickness 'x, layer matrix Wn 'x, and system matrix (Figure 1.20): Wn

Wn 'x

m

For all interfaces created that way, the system (1.42) applies, allowing the calculation of the complex temperatures Dn x and the complex heat flow rates Dcn x in each of them.

Figure 1.20. Single-layered assembly, calculating the periodic temperatures and heat flow rates.

45

1.2 Conduction

1.2.4.3 Flat assemblies, random boundary conditions Dirac impulse With a Dirac impulse, temperature or heat flow rate at a surface is zero between time t = –’ and time t = +’ except for an infinitesimally short interval dt, when it is one (Figure 1.21).

Figure 1.21. Dirac impulse.

Response factors Fourier’s second law can be solved for a Dirac impulse. Suppose temperature Ts1 at the bounding surface s1 undergoes a pulse. As a result, a variation qs1(t) of the heat flow rate at that surface and a variation Ts2(t) of the temperature and qs2(t) of the heat flow rate at the other bounding surface s2 will ensue. When all material properties are constants, the functions qs1(t), Ts2(t) and qs2(t) define the response factors of the heat flow rate qs1, the temperature Ts2 and the heat flow rate qs2 for a temperature impulse Ts1 at bounding surface s1. Symbols: I T1 q1

I T1 T2

I T1 q 2

Analogously we have: I q1 T1

I q1 T2

I q1 q 2

I T2 q1

I T2 T1

I T1 q 2

I q 2 q1

I q 2 T1

I q 2 T2

Response factors are again a function of layer sequence. Hence, I T1 T2 differs from I T2 T1 , etc. Convolution integrals Consider now a temperature pulse at bounding surface s1 with value T0. If all response factors are known, then temperature and heat flow rate response at bounding surface s2 and the heat flow rate response at bounding surface s1 become:

46

1 Heat Transfer

Impulse ¹0 at surface s1

Impulse q0 at surface s1

qs1

T 0 I T1 q1

Ts1

q0 I q1 T1

Ts2

T 0 I T1 T2

Ts2

q0 I q1 T2

qs2

T 0 I T1 q 2

qs2

q0 I q1 q 2

An impulse at bounding surface s2 results in analogue relationships. Any random signal Ts1(t) can now be split into a continuous series of successive pulses T0 (t) 't. For the response of Ts2 at bounding surface s2 on a signal Ts1 at bounding surface s1, the following applies (Figure 1.22): t t

0 't

Ts2 0 0 Ts2  't Ts1 t

t

2 't

Ts2  2 't

Ts1 t

t

3 't

Ts2 3 't

Ts1 t

0 I T1 T2 t

 Ts1 t  Ts1 t ! t

n 't

Ts2  n 't

Ts1 t  Ts1 t  Ts1 t

or:

Ts2  n 't

n 1

0 I T1 T2 t

0 I T1 T2 t

't 2 't  Ts1 t 3 't

't I T1 T2 t

2 't I T2 T2 t

0 I T1 T2 t

2 't 't

n 't

't I T1 T2 »t

n  1 't ¼º 2 't I T1 T2 ¬ªt  n  2 't ¼º  !

¦ Ts1  j 't I T1 T2 ¬ªn  j 't ¼º

j 0

Figure 1.22. Convolution, the principle.

't I T1 T2 t

't

47

1.2 Conduction

In integral form: Ts2 t

t

³ Ts1  W I T1 T2 t  W dW 0

This integral of the product of the signal scanned clockwise with the response factor scanned counter clockwise, is called the convolution integral of temperature Ts2 for the signal Ts1(t). In general, convoluting the other three variables with the appropriate response factors gives the answer to a signal the fourth variable gives. Response factors and convolution-integrals are calculated numerically. Step function at the surface of a semi-infinite medium Suppose a semi-infinite medium at uniform temperature Ts0. At time zero, the surface temperature jumps from that temperature Ts0 to a value Ts0 + 'Ts0 (Figure 1.23). The solution of Fourier’s second law with t = 0, 0 ” x ” ’: T = Ts0 as initial condition and t • 0, x = 0: Ts = Ts0 + 'Ts0 as boundary condition results from a separation of variables.

Figure 1.23. Temperature step at the surface of a semi-infinite medium.

The outcome is: T  x, t

Ts0  'Ts0

§ ¨ 2 ¨ ¨ S ¨©

f

³

q



exp  q 2 x

2 at

· ¸ dq ¸ ¸ ¸¹

(1.44)

The term between brackets is the inverse error function while a is the thermal diffusivity of the medium. As heat flow rate at the surface (x = 0) we get: q

O

dT dx

ª 2 'Ts0 § x2 ·º exp ¨  O « ¸» © 4 a t ¹ ¼» x ¬« 2 S a t

 0

'Ts0 b St

(1.45)

where b is the contact coefficient of the medium. Applying the definition of response factors on equation (1.45) gives:

48

1 Heat Transfer



I T1 q1

b St

Semi-infinite mediums do not exist, but the soil or a very thick material layer is a close approximation. In any case, equation (1.45) illustrates the meaning of the contact coefficient. A high value gives large heat absorption and heat release when the surface temperature suddenly rises or drops, while a low value means small heat quantities in such case. Hence, materials with high contact coefficient are active as storage media. For so-called passive solar buildings, for which temporary storage of solar gains in the fabric is a design concern, partitions and floors of sufficiently thick material layers with high contact coefficient are a good choice. The heat, flowing in or out a semi- infinite medium per m2 of surface between time zero and time t equals: t

Q

³ q dt

2 b 'Ts0 S

0

t

Aq

t

(1.46)

The quantity Aq is called the heat absorption coefficient of the material, units J/(m2 · K · s1/2). If two materials, one at temperature T1 and the other at temperature T2 contact each other, the heat flow rate in the interface totals (T1 > T2): Material 1

qs1

T1  Tc

b1 St

Material 2

qs2

Tc  T 2

b2 St

with Tc the contact temperature. Since both rates must be equal in absolute terms (qs1 = qs2), its value becomes: Tc

b1 T1  b2 T 2 b1  b2

(1.47)

The instantaneous contact temperature between materials apparently depends on both their temperature and contact coefficient. This reality affects comfort when touching materials. A material with high contact coefficient (a capacitive material) feels cold or hot while a material with low contact coefficient feels comfortably warm. Indeed, in the first case the skin temperature suddenly moves from 32–33 °C to the material temperature, while in the second case the contact surface adapts to the skin temperature. We will for example experience concrete as unpleasant and wood as pleasant to touch. To indicate that feeling, the terms cold and warm are used. The contact coefficient is an important criterion when choosing floor coverings or chair finishes. 1.2.4.4 Two and three dimensions On spots where heat transfer evolves in two or three dimensions, one has to return to equation (1.6). Analytical solutions for building details do not exist, which is why CVM or FEM is used. The basic principles of CVM were explained above. In a transient regime, a heat capacity U c 'V is assigned to each control volume 'V. According to energy conserva-

49

1.3 Convection

tion without dissipation, resulting heat flow per control volume must now equal the change in heat content, or:

¦ §m 't

U c 'V 'T

where §m is the average heat flow from each neighbouring to each central control volume during time step 't. If we assume §m = p §t + 't + (1 – p) §t with 1 > p > 0, i.e. any heat flow being a weighted average between the heat flows at time t (= 1) and time t + 't (= 2), the balance becomes (i = l, m, n and j = ±1):





p ¬ª ¦ Ps, i  j Ti  j  Tl , m, n  2

U c 'V

1

¦ Ps,i  j ¼º

 2





 1  p ¬ª ¦ Ps,i  j Ti  j  Tl , m, n

¦ Ps,i  j ¼º

1

Tl , m, n  Tl , m, n 't

For p = 0, the system simplifies to a series of successive forward difference equations, while p = 1 gives a system of backward difference equations. For p = 0.5, i.e. when using the arithmetic average according to the Cranck-Nicholson scheme, rearranging the terms gives: ª 2 U c 'V § « ¦ Ps, i  j Ti  j  Tl , m, n ¨© ¦ Ps, i  j  't ¬





·º ¸¹ » ¼

 2

2 U c 'V · º ª § ¸¹ » «  ¦ Ps, i  j Ti  j  Tl , m, n ¨© ¦ Ps, i  j  't ¬ ¼





1

In this equation, the actual temperatures (2) are the unknown and those of the preceding time step (1) the known quantities. That way, for each time step, we obtain a system of m equations with m unknowns (m equal to the number of control volumes). If the starting and boundary conditions are given, then one can solve these systems. Questions to decide upon beforehand are mesh, finer in materials with a high thermal diffusivity, and time step, to be chosen in accordance to mesh density, time step between the successive boundary condition values and desired information density. Bad choices may result in unstable solutions.

1.3

Convection

1.3.1

Heat exchange at a surface

Until now, we assumed the surface temperatures were known. However, in most cases, not these but the air temperatures and sometimes the heat flow rate at a surface are the known quantities. Every weather station registers the outside air temperature while measuring indoor air temperatures is not very complicated. Logging surface temperatures proves to be more difficult. From now on, the focus is on heat transfer from the environment at one side across a building assembly to the environment at the other side. That means we have to know how heat reaches a surface. The answer is twofold: by convection between surface and air and by radiant exchange with all surfaces facing the one considered. Thus, the inside surface of wall A in Figure 1.24 faces the five other surfaces in the room – three partition walls, the floor and the ceiling – the radiator, most furniture, etc. It exchanges radiation with each of these, and at the same time, the air in the room transfers convective heat to it.

50

1 Heat Transfer

Figure 1.24. heat exchange by radiation and convection between surface A and all other surfaces in a room.

1.3.2

Convective heat transfer

In the paragraphs that follow, the term ‘convective’ is used for heat transferred between a fluid and a surface. In such a case, the convective heat flow rate and heat flow write as: qc

hc  T fl  Ts

§c

hc  T fl  Ts A

(1.48)

with Tfl the temperature in the ‘undisturbed’ fluid, Ts surface temperature and hc the convective surface film coefficient, units: W/(m2 · K). Equation (1.48) is known as Newton’s law. At first sight, convection looks simple: a linear relationship between the heat flow rate and the driving temperature difference (Tfl – Ts ). The equation, however, must be read as a definition of the convective surface film coefficient, which accommodates the whole complexity of convection. In convective mode, heat and mass transfer go hand in hand. Combining the scalar law of mass conservation with the vector law of conservation of momentum describes mass flow. Mass (scalar) div  U v

0

Momentum (Navier-Stokes, vector) d  U v dt

U g  grad P  P ’ 2 v

where U is density and P dynamic viscosity of the fluid, P total pressure, and U g the gravity gradient. Unknown in both equations are the velocity components vx, vy, vz and total pressure P. In turbulent flow conditions, turbulence equations add. One often uses the (k, H) model, where k is turbulent kinetic energy and H turbulent energy dissipation. Both require an extra scalar equation. Along with the three scalar equations and the one vector equation, whose split into the x-, y- and z-directions produces three scalar equations, energy conservation intervenes.

51

1.3 Convection

In case of constant properties, negligible kinetic energy and hardly any friction heat produced, and the equation becomes: dT dt

a ’2T

(1.49)

with a thermal diffusivity of the fluid. In both the Navier-Stokes and the energy equation, d / dt accounts for the total derivative: d dt

w wx w wy w wz w    wx wt wy wt wz wt wt

w w w w vx  vy  vz  w x w y w z w t N N N (2)

(1)

(3)

The terms (1), (2) and (3) describe the enthalpy transport, i.e. the fact that a fluid moving at a temperature T with a velocity v (components vx, vy, vz) carries U c T units of heat. Energy conservation adds temperature T as the seventh unknown quantity. All information about convection follows from the solution of that system of seven scalar Partial Differential Equations (PDE’s). Once the temperature of the fluid known, the heat flow rate at a surface follows from Fourier’s first law (when the surface is airtight!). Indeed, at the surface, convection becomes conduction across a laminar boundary layer. Consequently, the following applies for the convective surface film coefficient: hc

 Ofl

grad T s

(1.50)

T fl  Ts

where (grad T)s is the temperature gradient in the boundary layer and Tfl the temperature of the undisturbed flow. Where that is, has to be agreed upon on a case by case basis. Solving the seven PDE’s is analytically possible for simple problems, such as natural convection by laminar flow along a semi-infinite vertical surface or forced convection by laminar flow along a semi-infinite horizontal surface. In the last case, the formulas simplify to: wv x wv y  wx wy

vx

0

wv y wv x  vy wx wy

Qfl

w2 vx wy

2

vx

wT wT  vy wx wy

afl

w2 T wy 2

giving similar velocity and temperature equations. Moreover if thermal diffusivity afl equals kinematic viscosity Qfl, then the two are identical. Therefore, we may presume that convective heat transfer is a copy of the fluid flow pattern. Solving the system gives as convective surface film coefficient: 0, 664 Ofl hc, x

vf x Qfl

x

where x is the distance from the surface free edge and v’ is fluid velocity outside the boundary layer. This equation shows that the convective surface film coefficient is not a constant but a function of the position along the horizontal surface. Any attempt to calculate convective heat transfer at the walls in a room, at the outside surfaces of a building or any other construction analytically, is doomed to fail. Yet, even in extremely complex situations, convection depends on all parameters and properties, which determine flow and heat transfer:

52

1 Heat Transfer

Fluid properties

Flow parameters

Thermal conductivity (Ofl ) Density (Ufl ) Specific heat capacity (cfl ) Kinematic viscosity (Pfl / Ufl ) Volumetric expansion coefficient

Geometry Surface roughness Temperature difference (Tfl – Ts ) Nature and direction of the flow Velocity components vx, vy, vz

1.3.3

Convection typology

The driving forces and the kind of flow define which type of convection will develop. 1.3.3.1 Driving forces When differences in fluid density caused by gradients in temperature and/or concentration are the driving force, then natural convection develops. In natural convection, the flow pattern follows the field of temperatures and/or concentrations. In buildings, this form of convection governs the situation indoors. If on the contrary the driving force is an imposed pressure difference, forced convection follows. In such a case, the flow pattern is independent of temperature and concentration. This form of convection is seen in HVAC systems. Wind also causes it. Forced and natural convection, however, never occur in a clearly separate way. An important natural component adds to low velocity forced convection. An example is convection by wind around a building. Generally, wind speed is too low to eliminate buoyancy flow. On the other hand, pure natural convection is more an exception than a rule. Even indoors, there is always some forced air momentum (think of opening and closing doors). Together, they generate mixed convection. 1.3.3.2 Flow type The flow can be laminar, turbulent or in transition. A laminar flow consists of diverging and converging streamlines which never cross. Particle and flow velocity are equal. In turbulent flow, momentum becomes chaotic and creates whirling eddies with particle velocities in every direction. Particle and flow velocity differ. Turbulent kinetic energy (k) first builds up in the eddies, after which the whirling motion dies away by turbulent dissipation (H). A description of the exact flow pattern is impossible, even though a fractional eddy approach could be used. Transient flow occurs between laminar and turbulent. Small disturbances along the flow path suffice to switch from the one to the other, after which the turbulences die away and the flow turns laminar again. Intense mixing during turbulent flow accounts for a larger heat transfer than laminar flow gives, where heat transfer perpendicular to the flow direction proceeds by conduction only. Transient flow lies in-between. To summarize: Driving force Density differences Density and pressures Pressures

Flow Laminar

Transient

Turbulent

× × ×

× × ×

× × ×

53

1.3 Convection

1.3.4

Calculating the convective surface film coefficient

1.3.4.1 Analytically See above. The few cases which can be solved analytically show that the convective surface film coefficient differs from spot to spot. If heat transfer is the point of interest, using a surface average circumvents that complication: hc

1 ³ h dA A cA

From now on the ‘convective surface film coefficient hcc will typically stand for that average. To calculate surface temperatures, however, local values should be used. 1.3.4.2 Numerically The use of CFD (computerized fluid dynamics) has helped better understand convection. Yet, a problem remains. In CFD, surface functions describe the velocity profiles near a surface. These have to be assumed, which means that the boundary layer is not calculated but modelled beforehand. Table 1.1 and Figure 1.25 gives an example of CFD-results. Table 1.1. Convective heat transfer coefficient outdoors along a rectangular, detached building: CFD generated correlations for wind velocities between 1 and 15 m/s and a surface to air temperature difference of 10 °C (Emmel et al., 2007). Surface to wind angle ° Outer walls

Roofs

Convective surface film coefficient W/(m2 · K)

0

5.15 vw0.31

45

3.34 vw0.34

90

4.78 vw0.71

135

4.05 vw0.77

180

3.54 vw0.76

0

5.11 vw0.78

45

4.6 vw0.79

90

3.76 vw0.85

54

1 Heat Transfer

Figure 1.25. Convective heat transfer coefficient outdoors for a rectangular building: correlations generated using CFD.

1.3.4.3 Dimensional analysis Most information on convection has been and is still obtained from experiment and dimensional analysis. That technique determines which dimensionless ratios between fluid properties at one side and geometric plus kinematic parameters defining the movement at the other side must have the same value in the experiment as in reality to allow extrapolation of measured data to real world situations. The dimensionless ratios are obtained directly from the differential equations (ex.: Q/a) or follow from Buckingham’s S-theorem. That theorem assumes that if a problem depends on n single-valued physical properties, which involve p basic dimensions, then n – p dimensionless numbers are required to find the solution. The description of forced convection, for example, needs seven single physical properties (L, Ofl, vfl, Ufl, Pfl, cfl, hc (see above)). These involve four basic dimensions: [L (length)] [t (time)] [M (mass)] [T (temperature)]. Consequently, three dimensionless numbers (S1, S2, S3) are required, expressed as S1 = f (S2, S3). The three follow from rewriting the three S-functions, which look in general as S La Obfl vflc Ufld Pefl cflf hcg , as an equation that only contains the intervening basic dimensions: S

> L@

a

b

ª M L º ª L ºc « 3 » « » ¬t T¼ ¬t ¼

f

e

d 2 ªM º ª M º ª L º ª M º « «¬ L3 »¼ « L t » t 2 T » « t 3 T » ¬ ¼ ¬ ¼ ¼ ¬

g

Because S is dimensionless, the sum of the exponents for each dimension should be zero, or: b

for M +

b

+

c

for t

–

3b

–

c

for ¹

–

b

for L

a

+

d

+

e

–

3d

–

e

+

2f

–

e

–

2f

–

–

f

–

+

g

=

0

=

0

3g

=

0

g

=

0

55

1.3 Convection

The three dimensionless numbers are then found by recalculating that general S-function three times, first for g = 1, c = 0, d = 0, than for g = 0, a = 1, f = 0 and finally for g = 0, e = 1, c = 0: Solution 1

a = 1, b = –1, e = 0,

f=0

Solution 2

b = 0, c = 1,

Solution 3

a = 0, b = –1, d = 0, f = 1

d = 1, e = –1

or

S1

hc L Ofl

or

S2

Ufl vfl L Pfl

or

S3

cfl Pfl Ofl

For natural convection four dimensionless numbers are needed, two of which can be combined into one. The three (forced convection) or four (natural convection) thus are: The Reynolds number Re

vL ( S2 ) Q

(1.51)

with v the velocity, Q the kinematical viscosity of the fluid (equals viscosity Pfl divided by density Ufl) and L the characteristic length representing the geometry of the problem. For pipes, the characteristic length is the hydraulic diameter. For walls, it is the dimension in the flow direction. For geometries that are more complex it is a calculated length. The Reynolds number reflects the ratio between the inertia force and viscous friction. If Re is small, then viscous friction gains and the result is laminar flow. If Re is large, inertia wins and turbulent flow occurs. Hence, Re determines the nature of the flow: laminar for Re ” 2000, turbulent for Re • 20,000, transient for 2000 < Re < 20,000. The Nusselt number Nu

hc L ( S1 ) Ofl

(1.52)

with Ofl the thermal conductivity of the fluid. Multiplying the left and right hand side in equation (1.50) with the characteristic length gives: hc L / Ofl = (grad T)s / [(Tfl – Ts) / L], which shows that Nusselt represents the ratio between the temperature gradient in the fluid at the surface and the average temperature gradient along the characteristic length. A large Nusselt number means an important gradient at the surface and a small gradient along the characteristic length, which is the case for high fluid velocities. Physically, Nu underlines that conduction governs the heat transfer close to the surface. Even in turbulent flow a laminar boundary layer remains, whose thickness becomes smaller with increasing fluid velocity but never goes to zero. Practically, the Nusselt number underlines the importance of heat transfer by convection compared to conduction.

56

1 Heat Transfer

The Prandl number Pr

Q ( S3 ) a

(1.53)

The Prandl number combines heat and mass transfer by rating two analogous quantities, thermal diffusivity a, which determines the ease with which a local temperature change spreads in the fluid, and kinematic viscosity Q, which determines the ease with which a local velocity spreads in the fluid. The Grasshof number Gr

E g L3 'T ( S2 in case of natural convection) Q2

(1.54)

with E the volumetric expansion coefficient, g acceleration by gravity and 'T the representative temperature difference. The Grasshof number replaces Reynolds in the case of natural convection. Velocity (v) indeed is then the result of density differences (E g), mainly caused by temperature differences ('T ). As velocity in turn influences the temperature difference ('T ), L3 and Q2 replace L and Q. The Rayleigh number Ra

Gr Pr

(1.55)

The Rayleigh number has no physical meaning. It was introduced because in many formulae for natural convection Gr and Pr appear as product (Gr Pr). All experimental, numerical, and analytical expressions for the convective surface film coefficient can now be written as: y Natural convection y Mixed convection y Forced convection

Nu = c (Ra)n Nu = F (Re / Gr1/2, Pr) Nu = F (Re, Pr)

where the coefficient c, the exponent n and the function F( ) differ from geometry to geometry, while depending on the nature of the flow and in case of natural connection, on the flow direction. The relations allow mutating between model and reality. As already said, model and reality couples where Nu1 = Nu2, Ra1 = Ra2 or Nu1 = Nu2, Re1 = Re2, Pr1 = Pr2 are identical.

1.3.5

Values for the convective surface film coefficient

1.3.5.1 Flat assemblies Natural convection (subscript L is the characteristic length) The first question is what characteristic length to use? For vertical surfaces, it is the height, for square horizontal surfaces the side and for rectangular horizontal surfaces the average of length and width. The second question is what temperature to take? For all properties involved, it is the value at a temperature equal to the average between wall surface and undisturbed fluid temperature: (Tfl + Ts) / 2. Examples of relations are:

57

1.3 Convection

Conditions

Functions

RaL ” 10

NuL = 0.56 Ra1/4 L

RaL > 109

NuL = 0.025RaL2/5

105 < RaL ” 2 · 107

NuL = 0.56 Ra1/4 L

2 · 107 < RaL < 3 · 1010

NuL = 0.138 Ra1/3 L

3 · 105 < RaL < 1010

NuL = 0.27 Ra1/4 L

9

Vertical surfaces Horizontal surfaces Heat upwards Heat downwards

Forced convection (subscript L is the characteristic length) Examples of relations are: Conditions Laminar flow

Functions

Pr > 0.1 ReL < 5 · 105

Turbulent flow

1/3 NuL = 0.644 Re1/2 L Pr

Pr > 0.5 ReL > 5 · 105

NuL = 0.036 Pr1/3 (ReL4/5 – 23,200)

In buildings with air-based HVAC-systems, the convective surface film coefficients for mixed convection indoors are often related to the air change rate (n), which describes how many times an hour the air in a room is exchanged for air delivered by the system (Fischer, 1995):

Walls Floor Ceiling Walls

Configuration

Convective surface film coefficient in a rectangular room W/(m2 · K)

Forced convection, ceiling air diffusers, isothermal room

–0.199 + 0.18 n0.8

Forced convection, wall air diffusers, isothermal room

Floor Ceiling

0.159 + 0.116 n0.8 –0.166 + 0.484 n0.8 –0.110 + 0.132 n0.8 0.704 + 0.168 n0.8 0.064 + 0.00444 n0.8

Simplified expressions In and around buildings, the fluid involved is air at atmospheric pressure. This allows simplifying the equations at ambient temperature to: y Natural convection: hc

§ 'T · a¨ ¸ © L ¹

b

58

1 Heat Transfer

where 'T is the surface to undisturbed air temperature difference. For a, b and L following values apply:

Vertical surfaces

Conditions

a

b

L

10 < L 'tT ” 7

1.4

1/4

Height

7 < L3 'T ” 103

1.3

1/3

1

–4

3

Horizontal surfaces heat upwards heat downwards

10–4 < L3 'T ” 0.14

1.3

1/4

eq. side

0.14 < L3 'T ”200

1.5

1/3

1

2 · 10–4 < L3 'T ” 200

0.6

1/4

1

y Forced convection: v d 5 m/s

hc

5.6  3.9 v

v ! 5 m/s

hc

7.2 v 0.78

For v d 5 m/s natural convection plays a role. From there the constant is 5.6.

In both formulae, v is the wind speed is measured at the closest weather station. Clearly, the surface to air temperature difference largely determine the surface film coefficient for natural convection, while for forced convection, air velocity is the main parameter. The increase with air velocity is a direct consequence of the decreasing boundary layer thickness (see the Nusselt number). In any case, the equations given only apply to air flowing along freestanding surfaces. Angles between two surfaces and corners between three surfaces have a disrupting effect. Moreover, if surfaces and angles form a room, the overall flow must satisfy the continuity equation. All this makes convection so complex that, for the sake of simplicity, most standards impose a constant convective surface film coefficient. Heat loss

Surface temperatures

Natural convection (= inside surfaces, European standard) Vertical surfaces

3.5

2.5

Heat upwards (Ĺ)

5.5

2.5

Heat downwards (Ļ)

1.2

1.2

19.0

19.0

Horizontal surfaces:

Forced convection (= outside surfaces)

In a room, the reference temperature is the air temperature, 1.7 m above floor centre. Outdoors, it is the air temperature measured at the closest weather station. When calculating surface temperatures using local convective surface film coefficients, the reference is the air temperature just outside the boundary layer. In case very correct calculations are needed (large temperature differences, complex geometry or protected surfaces), convection should be quantified using the formulae given in the tables or in literature.

59

1.3 Convection

1.3.5.2 Cavities The word cavity refers to an air or gas layer with a thickness that is small compared to the other dimensions. In cavities, heat transfer is a combination of conduction, convection, and radiation (Figure 1.26).

Figure 1.26. At the left, heat exchange between the surfaces in a cavity, on the right possible convective loops.

Convection occurs between both cavity surfaces and the gas in the cavity. At the warmer surface, heat flow rate totals qc1 = hcl (Ts1 – Tc ), at the colder surface qc2 = hc2 (Tc – Ts2 ) with Tc gas temperature in the middle of the cavity. Without cavity ventilation, both rates are on average equal and the mean convective heat flow rate becomes: qc

hc1 hc2 Ts1  Ts2 hc1  hc2

(1.56)

Assuming hc1 = hc2 = hc, (1.56) simplifies to qc = (hc / 2) (Ts1 – Ts2 ). Yet, as conduction and convection in cavities cannot be separated, one mostly uses the conduction equation, multiplying the thermal conductivity (Ofl ) of the gas by the Nusselt number: q  qc

 Ofl Nu

'Ts d

hcc 'Ts

(1.57)

In that equation d is cavity thickness in meter and 'Ts the warmer to colder surface temperature difference in °C. Infinite cavity In horizontal cavities, convection sees Bénard cells develop. Those are circular, symmetric local eddies. In vertical cavities, air rotation occurs. In both cases, the Nusselt number is: Nu d with:

ª m Ra dr º max «1, 1  » Ra d  n ¼ ¬

(102 d Ra d d 108 )

(1.58)

60

1 Heat Transfer

m

n

r

Horizontal cavity heat transfer downwards

0

heat transfer upwards

0.07

3,200

1.33

Vertical cavity

0.024

10,100

1.39

heat transfer upwards

0.043

4,100

1.36

heat transfer downwards

0.025

13,000

1.36

Tilted cavity, slope below 45°

Temperature difference 'T in the Raleigh number is the one between both cavity surfaces, while cavity thickness d figures as characteristic length. Ra below 100 means conduction, Nud = 1. Finite cavity In finite cavities, convection diverges strongly from the infinite ones. The Nusselt numbers are (with d the cavity thickness in m, H the cavity height in m, L the cavity length in m, lam a superscript for laminar, turb a superscript for turbulent, transient and a superscript for transient flow): Nud Vertical cavity Boundaries for the applicability of Nud H/d 5 20 40 80





turb transient , with max Nu lam d , Nu d , Nu d

100

Nu lam d

§ Ra d · 0.242 ¨ d ¸ © H ¹

Nu dturb

0.0605 Ra 0.33 d

Nu dtransient

Ramax

108

2 · 106

2 · 105

3 · 104

ª « « «1  « « «¬

0.273

§ · ¨ 0.293 ¸ ¨ 0.104 Ra d ¸ 1.36 ¸ ¨ ¨ § 6310 · ¸ ¨© 1  ¨© Ra ¸¹ ¸¹ d

3 º 0.33

» » » » » »¼

1.2 · 104

Horizontal cavity Heat transfer upwards

Heat transfer downwards Tilted cavity See literature

Rad ” 1708

1

Rad > 1708

ª max «1.1537 d 2 « «¬ 1

1º § 'T · 4 » ¨© ¸¹ » L »¼

61

1.4 Radiation

1.3.5.3 Pipes From experimental and semi-experimental work on convection between a pipe and the ambient fluid, the following formulae have been derived: Natural convection (Characteristic length is the outside diameter of the pipe) Vertical pipe: Ra d d 109 Ra d ! 109

Nu L Nu L

0.555 Ra1/d 4 0.021 Ra d2/ 5

Horizontal pipe: Ra d d 109 Nu L

0.530 Ra1/d 4

Forced convection Red  500 Red ! 500

Nu d Nu L

0.43  0.48 Re1/d 2 0.46  0.00128 Red

In these equations, all properties assume a temperature equal to the average between the value in the undisturbed fluid around the pipe and the pipe surface: (Tfl + Ts) / 2.

1.4

Radiation

1.4.1

What is thermal radiation?

Thermal radiation differs fundamentally from conduction and convection. In fact, electromagnetic waves cause radiant heat exchange. Every surface warmer than zero Kelvin emits them, while absorption by other surfaces induces thermal agitation of atoms and electrons there. The term ‘thermal’ includes all electromagnetic waves with a wavelength between 107 and 10–3 m: ultraviolet (UV), light (L) and infrared (IR), wavelength (O) following from the ratio between propagation speed in m/s (c) and frequency in Hz (f ). Electromagnetic radiation encompasses a very large interval of wavelengths: O ” 10–6 —m 10–6 < O ” 10–4 —m 10–4 < O ” 10–2 —m

Cosmic radiation Gamma rays x-rays

10–2 < ¼ ” 0.38 —m 0.38 ” ¼ ” 0.76 —m 0.76 < ¼ ” 103 —m

UV-radiation Visible light (L) IR-radiation

103 < O

Radio waves

Because of its electromagnetic nature, thermal radiation does not require a medium. Instead, it develops unhindered in a vacuum where the electromagnetic photons move at a velocity of 299,792.5 km/s. Each surface acts as emitter, with the emission depending upon its nature and its temperature. Resulting heat transfer is only possible between surfaces at different temperature.

62

1.4.2

1 Heat Transfer

Quantities

Table 1.2 outlines the way thermal radiation is quantified. Per variable, the spectral derivative stands for ‘derivative to wavelength’. A single wavelength characterizes monochromatic radiation, several wavelengths characterize coloured radiation. Table 1.2. Radiant heat transfer, variables. Variable

Definition + units

Radiant heat QR

The amount of heat emitted or received as electromagnetic waves. QR is a scalar, units: J.

Radiant heat flow §R

The amount of radiant heat per unit of time. §R is a scalar, units: W.

dQR dt

Radiant heat flow rate qR

The radiant heat flow per unit of surface. qR is a scalar (because a surface emits radiation in and receives radiation from all directions) with units W/m2. The term irradiation, symbol E, is used to characterize the incoming, the term emittance, symbol M, to characterize the emitted radiant heat flow rate.

d 2 QR dA dt

Radiation intensity I

The radiant energy emitted in a specific direction. I is a vector with units W/(m2 · rad). dZ is the elementary angle in the direction considered.

dqR d 2 §R or dZ dA dZ

Luminosity L

The ratio between the radiant heat flow rate in a direction I and the apparent surface, seen from that direction. L is a vector with units W/(m2 · rad). The luminosity describes how a receiving surface sees an emitting surface.

d 2 §R cos  I dA dZ

1.4.3

Equations

Reflection, absorption and transmission

If a radiant heat flow rate (qRi), emitted by a surface at temperature T, touches another surface then (Figure 1.27):

Figure 1.27. Radiant heat exchange, reflection, absorption and transmission at a surface.

63

1.4 Radiation

Figure 1.28. Thermal radiation, specular reflection.

Part is absorbed:

D

qRa qRi

(1.59)

Part is reflected:

U

qRr qRi

(1.60)

Part is transmitted:

W

qRt qRi

(by transparent layers)

(1.61)

with D, U and W the average absorptivity, reflectivity and transmissivity at temperature T. Conservation of energy dictates their sum must be one, or: D + U + W = 1. If the three belong to different temperatures, this does not hold. Put another way, never add absorbed, reflected, and transmitted radiant heat flow rates, which are emitted at different temperatures. Diffuse reflection differs from specular reflection. The latter obeys the laws of optics: incident and reflected beam in the same plane, reflection angle Ir equal to the incident angle Ii (Figure 1.28). Most surfaces reflect diffusely, which means that the reflected radiation is scattered in all directions. Reflectivity can also be defined in relation to the radiation intensity incident on a surface under an angle I. For diffuse reflection, the result is UI = IRr D / IRi I where IRrD is the reflected intensity in direction D. For specular reflection, reflectivity is UI = (IRr / IRi )I, with UI a function of the angle of incidence. Most building and insulation materials are non-transparent for thermal radiation (W = 0). The incoming radiation is absorbed in a very thin surface layer, for metals 10–6 m, for other materials 10–4 m thick. For that reason, the term ‘absorbing surface or body’ and not ‘absorbing material’ is used. At the other hand, most gases, most fluids, and some solids (glass, synthetics) are selectively transparent (W ” 1). They also show selective mass absorption described by their extinction coefficient a (Figure 1.29):

Figure 1.29. Thermal radiation, mass absorption in transparent materials.

64

1 Heat Transfer

dqR qR

 a dx

(1.62)

For a transparent layer with thickness d, the transmitted radiant heat flow rate then becomes qRt = qRi exp (–a d), where qRi is the incoming radiant heat flow rate. The absorbed heat flow rate is qRa = qRi – qRt = qRi [1 – exp (–a d)]. Absorptivity and transmissivity become: D = 1 – exp (–a d) W = exp (–a d). Specular reflectivity is given by: U

Ir Ii

ª n1 cos  Ii  n2 cos  It º « » ¬ n1 cos  Ii  n2 cos  It ¼

2

where n1 and n2 are the refractive indexes of the media at both sides of the incident surface (n = 1 for air), Ii is the angle of incidence in the first medium and It the angle of transmittance into the second medium. As already mentioned, for all materials, absorptivity, reflectivity, and transmissivity vary with wavelength and, consequently, temperature. If the three are defined as radiation intensity ratios, then also the angle of incidence intervenes. The wavelength can have an important impact. An example is glass, where for visible light, the transmissivity is large and the absorptivity small, while for UV and IR transmissivity approaches 0 and absorptivity 0.9, even more. These remarkable differences in properties cause the greenhouse effect. Solar radiation (shortwave radiation, high radiant temperature), which the glass transmits, is absorbed by all inside surfaces and re-emitted as low temperature radiation (IR radiation, low radiant temperature). IR now is absorbed by the glass at its inside surface, leaving conduction across as the only mean to return part of the short wave gain to the outside. Most solar heat is thus absorbed by the inside environment and released slowly. As a consequence, the inside temperature may reach uncomfortably high values. Transparent synthetic materials have analogous characteristics although some are also transparent for IR.

1.4.4

Radiant surfaces or bodies

The black body, a surface which absorbs all incident radiation (D = 1, U = 0, W = 0, D  f (O, I)), figures as the reference. Although just as non-existing as ideal gases or pure elastic materials, the black body helps to understand real surfaces. These divide in: y Grey bodies. Per wavelength absorptivity is constant, independent of direction (D = C t, < 1). y Blank bodies. Are gray but, with absorptivity zero. y Coloured bodies. Absorptivity depends upon wavelength (temperature) and direction. Real bodies are by definition coloured. Therefore the blank and grey ones are just models to keep absorption, reflection, and transmission independent of wavelength and direction. In all further discussions, we assume all bodies are blank or grey. In order not to put too much strain on reality, a difference is made between solar shortwave and ambient long wave radiation. For both, bodies are considered grey, although with different absorptivity and reflectivity, which is expressed by using an index L for long wave and S for shortwave.

65

1.4 Radiation

1.4.5

Black bodies

1.4.5.1 Characteristics Of all surfaces, black bodies emit the highest radiant energy flow rate, independent of temperature. According to the second law of thermodynamics their emissivity e (for the definition, see ‘grey bodies’) is thus 1. In fact, in closed systems, black bodies at different temperatures must evolve irrevocably towards temperature equilibrium. Because each emits the same amount of radiation as it absorbs at equilibrium, the emissivity of a black body must equal its absorptivity, i.e. 1. With respect to direction, a black body obeys Lambert’s law: luminosity (Lb, suffix b for ‘black’) constant. Hence, radiation intensity becomes (Figure 1.30): Ib I

Lb cos  I

(1.63)

Figure 1.30. Cosine law for radiation intensity.

This equation is known as the cosine law. That law offers a simple relation between emittance and luminosity. From the definitions, it follows: Mb

Lb ³ cos  I dZ Z

where

³

is the integral over the hemisphere.

Z

The elementary angle dZ can be calculated if one assumes a hemisphere with radius r0 above the surface dA with emittance Mb. Then the following applies: dZ ro2 sin  I dI d- while on the hemisphere, the intensity decreases from IbI to I b I / ro2 , or: S

2S 2

Mb

Lb

³ ³ 0

0

cos  I ro2

ro2 sin  I dI d-

S

ª  S Lb cos 2  I º 2 ¬ ¼0

This reduces to (Figure 1.31): Mb

S Lb

(1.64)

Spectral density of the emittance is given by Planck’s law: MbO

2 S c 2 h O5 § ch · 1 exp ¨ © k O T ¸¹

(1.65)

66

Figure 1.31. Cosine law proof.

Figure 1.32. Emittance: spectral density, left at rather environmental temperatures, right for the sun as a black body.

1 Heat Transfer

67

1.4 Radiation

where c is the speed of light in m/s, h is Planck’s constant (6.624 · 10–34 J · s) and k is Boltzmann’s constant (1.38047 · 10–23 J/K). The products 2 S c2 h and c h / k are called the radiation constants for a black body, symbols C1 (3.7415 · 10–16 W · m2) and C2 (1.4388 · 10–2 m · K). Figure 1.32 gives the spectral density for different values of the absolute temperature. The emittance increases quickly with temperature (= surface under the curve), while the spectral maxima occur at ever smaller wavelengths. These maxima have as geometric locus in the [O, MBO]-coordinate system a hyperbole of fifth order. Their wavelengths satisfy Wien’s law: 2898 ( OM in Pm)

OM T

(1.66)

For example, at 20 °C the maximum lies in the infrared part of the spectrum with OM = 2898 / 293.15 = 9.9 —m. For the sun, with a radiant temperature of 5800 K, the maximum climbs to the middle of the visible light part, OM = 2898 / 5800 = 0.5 —m. The emittance Mb follows from the integration of (1.65) to the wavelength, with O = 0 and O = ’ as boundaries: f

Mb

2 S5 k 4 4 T 15 c 2 h3

³ M b O dO 0

V T4

(1.67)

V stands for Stefan’s constant, 5.67 · 10–8 W/(m2 · K4). Equation (1.67) is known as the StefanBoltzmann law. This and Wien’s law preceded Planck’s law. For this law, one had to wait until quantum mechanics surfaced as a theory. Most of the time, the Stefan-Boltzmann law is expressed as: Mb

§ T · Cb ¨ © 100 ¸¹

4

(1.68)

where Cb is the black body constant, 5.67 W/(m2 · K4), and T / 100 is the reduced radiant temperature. Luminosity and radiation intensity then become: Lb

Mb S

Cb § T · ¨ ¸ S © 100 ¹

4

Ib I

d 2 §R b d A dZ

Lb cos  I

4

Cb § T · ¨ ¸ cos  I S © 100 ¹

1.4.5.2 Radiant exchange between two black bodies: the view factor If two black bodies with surface dA1 and dA2 are located as in Figure 1.33 without a medium in-between, then the elementary radiant heat flow between both is:

Figure 1.33. View factor, calculation.

68

1 Heat Transfer

d 2 §R,1o 2

M b1

I b1 dA1 dZ1

S

cos  I1 dA1 dZ1

where dZ1 is the angle under which dA1 sees dA2. dZ1 equals dA2 cos (I2) / r2, or: d 2 §R,1o 2

M b1 S

cos  I1 cos  I2 dA1

dA2 r2

cos  I1 cos  I2 dA1

dA2 r2

Analogously: d 2 §R, 2 o1

Mb2 S

The resulting radiant heat flow between the two bodies then becomes: From body 1 to body 2 d 2 §R,12

d 2 §R,1o 2  d 2 §R, 2 o1

M b1  M b 2 cos  I1 cos  I2 dA1 dA2 S r2

From body 2 to body 1 d 2 §R, 21

d 2 §R, 2 o1  d 2 §R,1o 2

M b 2  M b1 cos  I1 cos  I2 dA1 dA2 S r2

If both are finite in shape: From body 1 to body 2 §R,12

ª

M b1  M b 2 A1 «« S1A ¬

³ ³

1 A1 A2

cos  I1 cos  I2 dA2 dA1 º » r2 »¼

(1.69)

cos  I1 cos  I2 dA1 dA2 º » r2 ¼»

(1.70)

From body 2 to body 1 §R, 21

ª

M b 2  M b1 A2 «« S 1A ¬

³ ³

2 A2 A1

The term between brackets in both formulas is called the view factor, symbol F. Other names are angle factor, shape factor or configuration factor. If A1 is presumed to be facing A2, the view factor writes as F12. For the opposite case, F21 is used. View factors are pure geometrical quantities indicating which fraction of the radiant heat flow emitted by a body reaches another. They depend upon the size of each body, their form, the distance, and the angle at which both face each other. Its maximum value is one. In that case, all emitted radiation falls on the other.

69

1.4 Radiation

1.4.5.3 Properties of view factors y Reciprocity exists in a sense that A1 F12 = A2 F21 (this follows directly from the definition) y If surface A2 surrounds A1, then the view factor from A1 is 1 (Figure 1.34). That extends to each surface, surrounded by n – 1 others, when together they form a closed volume: n

¦ F1 j

1

j 2

Figure 1.34. View factor between surface 1, surrounded by surface 2 (on the left) or the surfaces 2 to 6 (on the right). In that case, surface 1 radiates to itself.

1.4.5.4 Calculating view factors An analytical calculation is possible for some simple configurations: y Two infinitely sized, parallel surfaces (e.g. the two surfaces of a cavity): F12

F21

1

y A point (equivalent to an infinitesimal sphere) at a distance D from a rectangle with sides L1 and L2. The view factor can be calculated as the ratio between the angle, at which the point faces the rectangle, and the total angle (4 S) (point = A1, rectangle = A2): F12

1 4S

³

cos  I r2

A2

dA2

For a point situated above a corner of the rectangle, the integral becomes (cos (I) = D / r and r2 = D2 + x2 + y2): F12

1 4S

L1 L2

³ ³ 0

0

D

D

2

x  2

3 y2 2



with solution of: F12

D L12  L22 1 1  atan 8 4S L1 L2

dy d x

70

1 Heat Transfer

Figure 1.35. View factor, point at distance D of surface A2.

Each other position of the point converts to the corner case by the construction of Figure 1.35. The resulting view factor is: F12 = F1a + F1b + F1c + F1d. Radiation between the head and a ceiling is an example of that. y An infinitesimal surface dA1 at an orthogonal distance D from a parallel rectangle with sides L1 and L2 and surface A2. The formula for the view factor in that case is: F12

1 dA1

³ ³

cos  I1 cos  I2 dA2 dA1

dA1 A2

S r2

The way dA1 sees each infinitesimal surface dA2 on A2 is independent its position, or: F12

1 S

³

A2 seen by dA1

cos  I1 cos  I2 dA2 r2

Suppose surface dA1 lies under corner (0, 0, D) of rectangle A2 (Figure 1.36).

Figure 1.36. View factor, infinitesimal small surface dA1 parallel to a surface A2 at distance D.

71

1.4 Radiation

The angle factor then becomes: F12

1 S

L2 L1

³ ³ 0

cos  I1 cos  I2 r2

0

dx d y

As cos I1 = cos I2 = D / r and r2 = D2 + x2 + y2, that equation simplifies to: F12

1 S

L1 L2

³ ³ 0

0

D

D 2 dy dx 2

 x2  y 2



2

with a solution of: F12

ª § L1 L2 1 « atan ¨ 2 2 ¨© D 2  L2 2S « D L 1 1 ¬

· ¸ ¸¹

§ L1 atan ¨ 2 2 2 ¨© D  L2 D  L2 2 L2

·º ¸» ¸¹ » ¼

For any other position of dA1 the same method as for the [point to surface]-configuration applies. All other configurations demand a numerical solution. That is the case for an infinitesimal surface dA1, perpendicular to a rectangle surface A2 with sides L1 and L2. Under no circumstances does dA1 face that part of surface A2 which lies behind the intersection of the plane crossing dA1 with A2, leaving a quarter of a globe as reference angle. The numerical formula for the angle factor then becomes:

F12

ª D 'x 'y « 2S « «x ¬

¦

'x 'x to L1  step 'x 2 2

¦

y

'y 'y to L2  step 'y 2 2



º » 2» x2  y 2  D2 » ¼ x2  y 2



where D is the orthogonal distance between the surfaces dA1 and A2. Other configurations, which are common in rooms and whose view factor can only be calculated numerically, are two identical surfaces in parallel and two surfaces with a common side, perpendicular to each other. The numerics are easily programmed, using spreadsheet software. The view factor allows writing the radiant heat flow and heat flow rate as: IR,12 IR, 21

 A2 F21  M b 2  M b1

A1 F12 M b1  M b 2

qR,12 qR, 21

 F21  M b 2  M b1

F12 M b1  M b2

(1.71)

For n black bodies radiating to another, both heat flow and heat flow rate per body become: IR 1 n

n

A1

¦

j 2





ª F1 j M b1  M b j º qR 1 n ¬ ¼

n

¦

j 2





ª F1 j M b1  M b j º ¬ ¼

where 1 is the body considered and 2 to n the n – 1 others.

(1.72)

72

1 Heat Transfer

1.4.6

Grey bodies

1.4.6.1 Characteristics The laws for grey bodies are similar to the ones for black bodies. Only radiation exchange changes: y For each wavelength and each direction, grey bodies emit the same constant fraction of radiation as a black body. The ratio M / Mb is called emissivity e. Conservation of energy tells that absorptivity D must equal emissivity e. Reflectivity U is given by U = 1 – D = 1 – e. Grey bodies with reflectivity 1 are blank. y A grey body follows Lambert’s law (L = C t ). Consequently, radiant heat flow rate obeys the cosine law, while the emittance is given by: M = S L. y Spectral emittance obeys Planck’s law, now multiplied by the emissivity e. Total emittance is: M

§ T · e Cb ¨ © 100 ¸¹

4

(1.73)

y Each grey body emits and reflects radiation. Hence, the radiant heat flow rates between numbers of grey bodies can be described as follows: let M be the emittance of one of them and E the irradiation by all others. Radiosity of the one grey body, i.e. the radiant impression it gives, is then: Mc

e Mb  U E

(1.74)

The emitted radiant heat flow rate now equals the difference between radiosity and irradiation, or: qR

Mc E

(1.75)

Eliminating the unknown irradiation between both equations gives: qR

Mc

M c  e Mb U

e M b  M c U

(1.76)

The received radiant heat flow rate then equals: qR

e M c  M b U

(1.77)

Equations (1.76) and (1.77) can be read as follows: treat a grey body as an equivalent black body with a grey filter in front and in between a radiant resistance equal to the ratio between reflectivity and emissivity (U / e). The black body has an emittance Mb, the grey filter a radiosity M c.

73

1.4 Radiation

1.4.6.2 Radiant exchange between grey bodies Two bodies For two grey bodies, separated by a transparent medium, the resulting radiant flow from body 1 to body 2 is:





Equivalent black body 1 ĺ grey filter 1:

§R,11

e1 M b1  M1c A1 U1

ĺ grey filter 2:

§R,12

F12  M1c  M 2c A1

Grey filter 1

The second equation is identical to the one for the radiant heat flow between two black bodies. Indeed, both the emittance of a black body and the radiosity of a grey body represent diffuse radiation that obeys Lambert’s law. Grey filter 2 ĺ equivalent black body 2:

§R, 22





e2 M 2c  M b 2 A2 U2

In the three equations, the radiant heat flows §R,11, §R,12 and §R,22 are identical (energy conservation). After elimination of the unknown radiosities M1c and M 2c , the common value is:

§R,12

ª º « » 1 « » M b1  M b 2 A1 « U1  1  U2 A1 » «¬ e1 F12 e2 A2 »¼





(1.78)

For the radiant flow from body 2 to body 1 we get:

§R, 21

ª º « » 1 « » M b 2  M b1 A2 « U2  1  U1 A2 » «¬ e2 F21 e1 A1 »¼





(1.79)

The term between brackets is called the radiation factor, symbol FR. If A1 is seen as the emitting body, then it writes as FR,12. If this is A2, we use FR,21. Dividing both equations by the surface seen as emitting gives the radiant heat flow rate. In (1.78) this is A1, in (1.79) A2. Frequent configurations of two grey bodies are: y Two infinitely large, parallel isothermal surfaces. In that case: F12 = F21 = 1, A1 = A2 and (1.78) becomes (U = 1 – e):

qR,12

§R,12 A1

ª º « » 1 « » M b1  M b 2 « 1  1  1» «¬ e1 e2 »¼





(1.80)

In building construction, the term between brackets reflects the radiation factor in an infinite cavity. If one of the bounding surfaces in that cavity is blank (say surface 1), then FR,12 = 1 / (1 / 0 + 1 / e2 – 1) = 0. If one is black (i.e. surface 2), FR,12 = 1 / (1 / e1 + 1 / 1 – 1) = e1.

74

1 Heat Transfer

y An isothermal surface 1 surrounded by an isothermal surface 2. Now F12 is 1 and (1.78) turns into: §R,12

e1 e2 e U A e2 U1  e1 e2  1 2 1 A2

M b1  M b2 A1

If both bodies are almost black (e > 0.9), then the denominator equals § 1, and we get: §R,12





e1 e2 M b1  M b 2 A1

When moreover the surrounded surface A1 is very small compared to the surrounding surface A2 (A1 / A2 § 0), that equation further simplifies to: §R,12





e1 M b1  M b 2 A1

(1.81)

In such case, the resulting radiant heat flow only depends on the emissivity of body 1. Multiple surfaces In case of multiple isothermal grey bodies facing each other, all at different temperatures, the radiant heat flow between body 1 and the n – 1 others can be expressed as:





Equivalent black body 1 ĺ grey filter 1:

§R,11

e1 M b1  M1c A1 U1

Grey filter 1 ĺ n – 1 other grey filters:

§R,1 j

¦

n

j 2





F1 j M1c  M cj A1

Because of energy conservation, both flows are identical (§R,11 = §R,1j), or: M b1

§ U1 ¨1  e © 1

n

¦

j 2

· U F1 j ¸ M1c  1 e1 ¹

n

¦

j 2

 F1 j M cj

(1.82)

For the 2 to n surface surrounding surface 1, (1.82) converts to: M b1

M1c U1  e1 e1

n

¦

j 2

 F1 j M cj

(1.83)

For eachgrey body, we obtain an equation (1.82) or (1.83), whereby the radiosities M cj are the unknown and the (black body) emittances Mb1 the known quantities. The result is a system of n equations with n unknown: ª M b j º > F @ n.n ª M cj º ¬ ¼n ¬ ¼n

(1.84)

[F]n.n is called the radiation matrix, in the above case of n isothermal grey bodies. The solution of the system of n equations gives the radiosities M cj as a function of the n emittances Mbj. The radiant heat flows then follow from inserting M cj in the equations for the heat exchange between the different equivalent black bodies and their grey filters.

75

1.4 Radiation

1.4.7

Coloured bodies

For coloured bodies, the emissivity, absorptivity, and reflectivity depend on wavelength, which is a function of temperature, and direction. For each, Kirchhoff’s law (e = D) applies. Lambert’s law, however, is no longer applicable since it requires an emission independent from direction. Per wavelength, the spectral emittance differs from a black body. The average emissivity at a temperature T follows from the ratio between the emittances of the coloured body and the emittance of a black body at the same temperature. In order to simplify things, we consider coloured bodies as grey, but with temperature dependent emissivity. For the emittance and irradiance at strongly different temperatures, Kirchoff’s law no longer applies. This is the case for ambient and solar radiation. There, DS  eL with DS short wave absorptivity for sunlight and eL long wave emissivity for the ambient radiation (take polished aluminium: DS = 0.2 to 0.4 and eL = 0.05).

1.4.8

Practical formulae

As shown, thermal radiation can be modelled quite accurately. However, calculating all angle factors is cumbersome and the number of equations in system (1.84) may become very large. A simpler approach is therefore welcomed. In a first step, a (fictive) system of two radiant surfaces replaces each configuration of n surfaces: the one considered and the remaining n – 1, which form the environment. In a second step, we suppose the environment black at temperature Tr, the radiant temperature facing the surface under consideration. This is the temperature the environment as a black body should have, if the surface considered, hereafter called surface 1, is to receive or lose the same radiant flow as in reality. Solving the system of equations (1.84) for radiation between all surfaces in the environment gives that radiant temperature. This way, we get the radiosity M1c of surface 1 as a linear combination of the (black body) emittances of all n surfaces engaged, thus including surface 1: M1c

n

¦ ar i

Mbi

i 1

Inserting the result in (1.77) and equating with (1.81) quantifies the radiant temperature: Tr

4

· 1 §n ¦ ar T 4  e1 T14 ¸¹  273.15 U1 ¨© i 1 i i

In analogy to convection, the radiant heat flow and heat flow rate now are expressed as: qr



hr Ts1  T r



§r





hr Ts1  T r A

(1.85)

In both formulas, hr represents the surface film coefficient for radiation with units W/(m2 · K). Ts1 is the temperature of surface 1 while Tr is the radiant temperature of the environment facing surface 1. The surface film coefficient for radiation varies with the two radiant surfaces configuration advanced as the reference. If the environment surrounds surface 1, a situation which is the reference indoors, then the surface film coefficient follows from equalling (1.85) and (1.81) with Mb = Cb (Ts1 / 100)4:

76

1 Heat Transfer

hr

ª § T ·4 § T ·4 º « ¨ s1 ¸  ¨ r ¸ » « © 100 ¹ © 100 ¹ » e Cb « » Ts1  T r « » « » ¬ ¼

(1.86)

The term between brackets is called the temperature ratio for radiation, symbol FT: FT

Tm 5000

3 ª§ T ·2 § T ·2 º 4 § Tm · « ¨ s1 ¸  ¨ r ¸ » | ¨ ¸ «¬ © 100 ¹ © 100 ¹ »¼ 100 © 100 ¹

with Tm = (Ts1 + Tr) / 2. Calculations now show that the temperature ratio for radiation varies little between –10 °C and 50 °C. Hence, the simplified expression at the right hand side usually suffices. That allows considering equation (1.85) as linear. Inserting the temperature ratio for radiation in (1.86) gives as the surface film coefficient for radiation: hr

e1 Cb FT

(1.87)

Other reference combinations than a surface, surrounded by the environment, are possible: y Both bodies act in parallel. If the environment, i.e. the other surface, is isothermal, temperature Ts2, then the detour via the radiation temperature is superfluous and the radiant surface film coefficient can be written directly as: qr



hr Ts1  Ts 2



with hr

5, 67 FT 1 1  1 e1 e2

y Only part of the environment participates in the radiant exchange, which happens when some surfaces are at the same temperature as surface 1. The radiant temperature might then only include all surfaces at a different temperature ( T rc ) . In such case, the surface film coefficient for radiation becomes: qr



hr Ts1  T cr



with hr

e1 Cb F12 FT e1  U1 F12

F12 is thereby the view factor between surface 1 and the surfaces at different temperature. If surface 1 is almost black, the denominator in hr turns 1, and: hr

e1 Cb F12 FT

5.67 e1 F12 FT

(1.88)

Take for example a surface in an exterior corner formed by two identical walls. Both walls are at the same temperature. That results in a radiant exchange with only half the space, i.e. the part not shielded by the other wall, view factor: 1/2. The surface film coefficient for radiation then equals: FT 2,84 e1 FT 2 Of course, surfaces at the same temperature may also be included in the radiant temperature. The view factor then remains the same but the radiant temperature changes. hr

e1 Cb

77

1.5 Applications

1.5

Applications

1.5.1

Surface film coefficients and reference temperatures

1.5.1.1 Overview The three modes of sensible heat transfer, we treated separately until now, occur together in reality. Consider a massive outer wall. Heat crosses that wall by conduction. However between the inside environment and the inside surface and between the outside environment and the outside surface, it is convection and radiation, which allow for the heat exchange. Both are expressed as the product of a surface film coefficient (hc, hr ) and a temperature difference. But, at any surface they are so intertwined that, where possible, their separate values are combined into one coefficient for heat transfer inside (hi ) and one coefficient for heat transfer outside (he ), both defined with respect to a reference temperature (Tref,i and Tref,e ). Of course, we can also separate convection and radiation. In such a case, with an air and a radiant temperature, the last considered faceing the surface, will characterize the environment. 1.5.1.2 Indoor environment Average inside surface film coefficient (hi) Calculation Suppose surface j is isothermal with temperature Tsi. The average convective heat flow rate between the air and that surface is:

> qci @ j



hci, j Ti,j  Tsi, j



with hci, j the area-averaged convective surface film coefficient and Ti, j the average air temperature outside the boundary layer. If not that temperature, but the air temperature in the centre of the room at a height of 1.7 m (Ti ) figures as the reference (hereafter called the central air temperature), then the average convective heat flow rate changes into (suffix j omitted): qci

hci  Ti  Tsi

where hci is the average surface film coefficient for convective heat transfer with the central air temperature as reference: hci

§ T i, j  Tsi · hci, j ¨ © Ti  Tsi ¸¹

(1.89)

For the radiant heat flow rate between the inside environment and surface j, one has: qri

hri  T ri  Tsi

where Tri is the radiant temperature facing surface j. Total heat flow rate between the inside environment and surface j then becomes: or:

hci  Ti  Tsi  hri  T ri  Tsi

qi

qci  qri

qi

hci  hri ¨ «

§ ª hci Ti  hri T ri º · »  Tsi ¸ h h  ©¬ ¹ ¼ ci ri

(1.90)

78

1 Heat Transfer

The sum hci + hri is called the inside surface film coefficient (for heat transfer). Symbol hi , units W/(m2 · K). The term between brackets is the reference temperature facing surface j, symbol Tref,i . That reference temperature equals the weighted average between central air temperature and radiant temperature j, with the respective surface film coefficients for convection and radiation as weighting factors. Values Standard values for the convective surface film coefficient hci are (W/(m2 · K), EN-standard): § 3.5

Vertical surfaces Horizontal surfaces Heat flow upwards (q Ĺ) Heat flow downwards (q Ļ)

5.5 1.2

The surface film coefficient for radiation equals (surface surrounded by ‘the environment’): hri = 5.67 eL FT, where eL is the long wave emissivity of the surface and FT the temperature ratio for radiation in the interval Tsi – Tri. Inside, that ratio scatters around 0.95, or hri § 5.4 eL. Since most finishes have a long wave emissivity 0.8 to 0.9, one gets: 4.3 ” hri ” 4.95 W/(m2 · K). According to the EN-standards the inside surface film coefficient are then (W/(m2 · K)): 7.7

Vertical surfaces Horizontal surfaces Heat flow upwards (q Ĺ) Heat flow downwards (q Ļ)

10 6

The ASHRAE Handbook of Fundamentals gives a more complete set of values: Position

Direction of the heat flow

hi (W/(m2 · K) Emissivity of the surface considered 0.90

0.20

0.05

Horizontal

Upward

9.26

5.17

4.32

Sloping 45°

Upward

9.09

5.00

4.15

Vertical

Horizontal

8.29

4.20

3.35

Sloping 45°

Downward

7.50

3.41

2.56

Horizontal

Downward

6.13

2.10

1.25

But not all standard values are accurate for cases, which deviate substantially from normal: for the EN-data the use of a reflective finish, large temperature differences, etc. In such cases, one should return to the theory, define case-relevant inside surface film coefficients, and, if necessary, decouple radiation and convection.

79

1.5 Applications

Interpreting the reference temperature Calculating the radiant temperature Tri is quite complex. That is why, provided the room is beam shaped, and all inside surfaces are grey with long wave emissivity § 0.9, the area-weighted average surface temperature usually replaces the real value:

 Ak Ts k

n

¦

k 1

T ri

(1.91)

n

¦

Ak

k 1

y Vertical, sloped, and horizontal envelope parts, the last with upward heat flow If the surface of the part considered behaves as a grey body, the reference temperature becomes: T ref,i

0.44 Ti  0.56 T ri

That value is close to a temperature which governs thermal comfort in residential buildings, office buildings, and schools: the average between the central air and the radiant temperature, a value called the effective temperature, symbol To: Ti  T ri (1.92) 2 For reflective surfaces, the effective temperature equals more or less the central air temperature: To § Ti. In such a case, convection prevails. T ref,i

To

y Horizontal inside partitions and horizontal envelope parts, downward heat flow For partitions and parts with are grey bodies, the reference temperature expresses as: T ref,i

0.2 Ti  0.8 T ri

(1.93)

y Vertical, sloped and horizontal inside partitions, the last with upward heat flow The larger the impact the ‘colder’ outer walls assemblies have on the radiant temperature the inside partitions face, the less evident the above simplifications are for the inside reference temperature. Local inside surface film coefficients The average inside surface film coefficient as defined pertains to area-averaged phenomena, such as heat loss and heat gain. When surface temperatures at specific spots or surface parts are the quantities of interest, then we need local inside surface film coefficients. In general, the following holds (left and right hand side describe the same heat transfer):



hix T ref,i  Tsix



hcix  Tix  Tsix  hrix  T rix  Tsix

(1.94)

with hix the local surface film coefficient linked to a reference temperature Tref,i , Tsix the local surface temperature, hcix the local convective surface film coefficient, Tix the local air temperature just outside the boundary layer, hrix the local surface film coefficient for radiation, and Trix the radiant temperature of the environment facing the spot considered. If an equivalent thermal resistance (Rc) from the inside to the environment at the other side can be linked to that spot, then the following holds:

80

1 Heat Transfer



hix T ref , i  Tsix



Tsix  Tref , j

(1.95)

Rc

Equation (1.95) is an approximation. In fact, the equivalent thermal resistance depends on the distribution of local inside surface film coefficients (hix ) over the whole surface. Tref, j, the reference temperature at the other side, is the outside temperature ( j = e) for the envelope, and the reference temperature in the adjacent room for inside partitions. Eliminating Tsix from (1.94) and (1.95) and solving for the local surface film coefficient (hix ) gives: hix

hcix  hrix  pT 1  R c pT

(1.96)

where: pT







hcix T ref , i  Tix  hrix T ref , i  T rix T ref , i  T ref , j



(1.97)

If the reference temperature Tref,i , the relationship between Tref,i and Tix, the relationship between Tref,i and Trix and, the local inside surface film coefficients hcix and hrix are known, then the combination of (1.96) with (1.97) gives the combined local inside surface film coefficient (hix) at the spot considered, on condition however that the equivalent thermal resistance Rc is calculated in advance. To give an example: 1. Reference temperature inside (Tref,i ) The central air temperature in the room, 1.7 m above floor level. 2. Relationship between Tref,i and Tix Suppose the central air temperature in the room increases linearly with height. The gradient then becomes smaller as the room is better insulated and the heating produces less convection. In a formula: Tix  T ref , j T ref , i  T ref , j

1  0, 2 pc U m  y  1, 7

(1.98)

with y the height ordinate, Tref,i the reference temperature inside, Tref, j the reference temperature in the neighbouring room or outside, pc the heating convection factor (1 for air heating, 0.9 for convectors, 0.4 to 0.8 for radiators, 0.4 for floor heating) and Um the average thermal transmittance (see next paragraph) of the room, given by: Um

¦ ai U i ¦ Ai

Ai

(1.99)

where ai is a weighing factor, for the inside partitions equal to (Tref,i – Tref, j) / (Tref,i – Tref,e), Tref, j being the reference temperature in the neighbouring room. Equation (1.98) reflects the results of a series of measurements in a test room, whose heating system and insulation of the outer walls were varied.

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1.5 Applications

3. Relationship between Tref,i and Trix Now, a uniform radiant temperature Tri, which is proportional to the reference temperature with a gradient depending on the local convective surface film coefficient, the convection factor of the heating, and the average thermal transmittance of the room, replaces the radiant temperature facing the spot considered: T ri  T ref , i T ref , i  T ref , j

hcix

hcix  p  0, 4 U m  c 0, 6

(1.100)

For the interpretation of all parameters in the formula, see above. Equation (1.100) resulted from computer simulations of the radiant heat exchange in rooms of different shape. 4. Local convective surface film coefficient Set hcix equal to 2.5 W/(m2 · K). 5. Local surface film coefficient for radiation hrix Corner between three envelope assemblies or two envelope assemblies and an inside partition 5,5 eL

Surfaces at more than 0,5 m from the corner line and corner point

3,4 eL

Surfaces within the 0,5 m strip around the corner line but at more than 0,5 m from the corner point

2,2 eL

Surfaces within the 0,5 m strip around the corner point

Edge between two envelope assemblies or an envelope assembly and an inside partition 5,5 eL

Surfaces at more than 0,5 m from the corner line

3,4 eL

Surfaces within the 0,5 m strip around the corner line

Envelope assembly or inside partition 5,5 eL

–

6. Remark A surface film coefficient of 2 W/(m2 · K) marks furniture against an outer wall. 1.5.1.3 Outdoor environment Calculation Four heat flow rates intervene now: 1. Convection between the outside environment and the surface The average convective heat flow rate is: qce



hce, j Te, j  Tse



82

1 Heat Transfer

with hce, j the average convective surface film coefficient and Te, j the average temperature of the air outside the boundary layer. If one uses the outside temperature measured at the closest weather station instead, which is the air temperature logged in open field under a thermometer hut 1.7 m above grade, then the average heat flow rate becomes: qce

hce  T e  Tse

where hce is the average convective surface film coefficient for that reference temperature: hce

hce, j

Te, j  Tse T e  Tse

2. + 3. Long wave radiation between surface, terrestrial environment and the sky The surface, the terrestrial environment and the sky form a system of three radiant bodies, the sky black and the others grey. Radiant balances: From the surface (s) to the terrestrial environment (e) and the sky (sk) M bs

ª ULs º U  Fse  Fssk » M sc  Ls  Fse M ec  Fssk M bsk «1  eLs eLs ¬ ¼

From the terrestrial environment (e) to the surface (s) and the sky (sk) M be

ª ULe º U  Fes  Fesk » M ec  Le  Fes M sc  Fesk M bsk «1  eLe eLe ¬ ¼

In both equations, eLs and ULs are the long wave emissivity and reflectivity of the surface and eLe and ULe the average long wave emissivity and reflectivity of the terrestrial environment. Fse is the view factor between the surface and the terrestrial environment, Fssk the view factor between the surface and the sky, Fes the view factor between the terrestrial environment and the surface and Fesk the view factor between the terrestrial environment and the sky. Mbs, Mbe and Mbsk are the black body emittances of respectively the surface, the terrestrial environment, and the sky, while M sc and M ec are the radiosities of the surface and the terrestrial environment. The sum of the view factors Fse and Fssk now equals 1 (the terrestrial environment and the sky surround the surface), while the view factor Fes is close to 0 as the surface is infinitely small compared to the terrestrial environment, and the view factor Fesk is close to 1 as nearly all radiation from the terrestrial environment reaches the sky. The two balances thus simplify to: M bs

U 1 M c  Ls  F M c  Fssk M bsk eLs s eLs se e

M be

U 1 M c  Le M bsk eLe e eLe

Solving the equations for M sc and inserting the result in qrse  qrssk gives (Fse + Fssk = 1, or eLs (Fse + Fssk) Mbs = eLs Mbs): qrse  qrssk

eLs ( M bs  M sc ) / ULs

ª º § · F eLs Fse  M bs  eLe M be  eLs Fssk « M bs  ¨ ULe se  1¸ M bsk » Fssk © ¹ ¬ ¼

An approximation replacing this ‘exact’ result presumes the terrestrial environment is a black body at outside temperature. Using the experimental fact that during clear nights, the sky

83

1.5 Applications

temperature is some 21 °C below the air temperature in the atmospheric boundary layer, the radiant heat flow rate between the surface, the terrestrial environment and the sky simplifies to: qrse  qrssk

eLs Cb ª¬ Fse FTse  Fssk FTssk  T e  Tse  21 Fssk FTssk 1  0.87 c º¼

with FTse the temperature ratio for radiation between the surface and the terrestrial environment, FTws the temperature ratio for radiation between the surface and the sky and c the cloud factor, 0 for a clear and 1 for an overcast sky. 4. Solar radiation Solar radiation (EST) combines three components: beam, diffuse and reflected. A square meter of surface absorbs the product DS EST, with DS the short waved absorptivity. Even though solar radiation is a daytime reality, its impact is large. The resulting ‘simplified’ heat balance so becomes: qe

hce  T e  Tse  5, 67 eLs  Fse FTse  Fssk FTssk  T e  Tse  120 eLs Fssk FTssk 1  f c  DK EST

With the term 5.67 eLs (Fse FTse + Fssk FTssk) called the outside surface film coefficient for radiation (hre), the equation rewrites as:

qe

­ ½ °° ª °° DK EST  eLs 120 Fssk FTssk 1  f c º hce  hre ® « Te  »  Tse ¾ hce  hre ° ¬ °  

¼ °¯ °¿

(1.101)

The term above the brace, which has degree centigrade as dimension, figures as the reference sol-air temperature, symbol T*e . The sum hce + hre is called the outside surface film coefficient he with units of W/(m2 · K). Thus, (1.101) is simply expressed as: qe



he T*e  Tse



(1.102)

Values The number 19 W/(m2 · K) is the standard value for the convective surface film coefficient. If the temperature ratios for radiation FTse and FTssk are set equal to FT, then, knowing Fse + Fssk = 1, we get as surface film coefficient for radiation (hre): hre

5.67 eL FT

(1.103)

Since the temperature factor for radiation FT is between 0.8 and 0.9, the most probable value interval is: 4.4 eL ” hre ” 5.1 eL W/(m2 · K). Provided outside surfaces are grey with long wave emissivity (eL) 0.9, then a value 4 W/(m2 · K) is very likely. Thus, the standard outside surface film coefficient for heat transfer is 19 plus 4, or: he

23 W/(m 2 ˜ K)

(1.104)

The European Standards (EN) advance 25 W/(m2 · K), while the ASHRAE Handbook of Fundamentals makes a difference between winter and summer:

84

1 Heat Transfer

Winter (wind speed 6.7 m/s) Summer (wind speed 3.4 m/s)

Direction of heat flow

he (W/(m2 · K)

Any Any

34.0 22.7

The calculation of the outside surface film coefficient just given is based on a series of assumptions. If more accurate numerics are needed, one should return to the complete heat balance. Measurements on site at the leeward side of an existing building for example showed the coefficient had much lower values there than the ones listed. Interpreting the reference temperature The sol-air temperature (T*e ) does not exist. In fact, it is a fictive air temperature, which ensures the heat exchanged with the outside surface remains the same as the one by solar radiation, long wave radiation and convection, provided the outside convective surface film coefficient equals he. The sol-air temperature depends on the radiant properties of the surface, its inclination and orientation, the average climate data, the specific value of the outside surface film coefficient he, etc. Its value differs according to the application.

1.5.2

Steady state, one dimension: flat assemblies

1.5.2.1 Thermal transmittance and interface temperatures The use of the surface film coefficients hi and he simplifies the calculation of the steady state heat flow rate, environment to environment, across a flat assembly. Outer walls, roofs and floors, the last separating the inside from outside Consider the outside wall of Figure 1.37. Inside, the reference temperature is To, outside, it is T*e . Assume the inside environment is heated while it is cold outside as is the case during the heating season in cold and moderate climates. From the inside environment to the inside surface the heat flow rate is q1 = hi (To – Tsi ) with Tsi the inside surface temperature. The heat flow rate across the assembly is q2 = (Tsi – Tse ) / RT with Tse the outside surface temperature and RT total thermal resistance of the assembly. From the outside surface to the outside environment, the heat flow rate is q3 he (Tse  T*e ) . In steady state, the three flow rates have to be equal. If their common value is q, rearrangement and addition gives:

Figure 1.37. Outer wall, thermal transmittance.

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1.5 Applications

q hi

T o  Tsi

q RT q he

Tsi  Tse Tse  T*e

§1 1· q ¨  RT  ¸ he ¹ © hi

T o  T*e

This sum is now written as q U (T o  T*e ), where: U

1 1 1 R he hi

(1.105)

U is called the thermal transmittance of the flat assembly, units W/(m2 · K). The smaller its value, the less heat flows on average across the assembly. Hence, the thermal transmittance is a measure for the insulation quality of a wall, a roof, or a floor separating the inside from the outside. The inverse (1 / U) is called the thermal resistance environment-to-environment, symbol Ra, units m2 · K/W. The thermal transmittance shows that calculation from environment to environment, i.e. taking radiation and convection at the inside and outside surface into account, means the addition of two extra thermal resistances: a surface resistance 1 / hi at the inside, marked Ri, standard values according to the EN-standard 0.13 m2 · K/W for vertical surfaces, 0.1 m2 · K/W for sloped and horizontal surfaces with the heat flowing upwards and 0.17 m2 · K/W for horizontal surfaces with the heat flowing downwards. At the outside, the surface resistance is 1 / he, marked Re, standard value 0.04 m2 · K/W independent of slope and heat flow direction. The thermal transmittance as defined often gets the prefix ‘clear’ followed by the type of part involved (wall, floor, roof), because possible two- and three-dimensional effects, which complicate things, are not accounted for. This, of course, may be quite far from reality. Even heat transfer across a masonry wall develops three-dimensionally. Masonry in fact is a composite of mortar joints and bricks, the first not homogeneously filled but showing voids, the latter fast ones with vertical perforations, see Figure 1.38.

Figure 1.38. Masonry, far from a one-dimensional assembly.

86

1 Heat Transfer

Inside partitions Here, hi acts as surface film coefficient at both sides. Thermal transmittance then becomes: 1

U

R

(1.106)

2 hi

Temperatures The surface resistances Ri and Re figure as thermal resistances of a fictive surface layer, 1 m thick, with thermal conductivity hi or he. That way the reference temperatures become ‘surface temperatures’ and heat transfer becomes steady-state conduction from fictive surface to fictive surface. Hence, for an envelope assembly, temperature in the [R, T]-plane is a straight line through [0, T*e ] and [Ra, To]. For inside partitions, that straight line goes through [0, To,1] and [Ra, To,2]. Yet, when constructing the temperature line, one must respect the correct layer sequence, included the surface resistances! The slope of the line represents the heat flow rate across the assembly (Figure 1.39). As temperature at the (real) inside surface we have: T o  T*e Ra

Envelope assembly

Tsi

T o  Ri

Inside partition

Tsi

T o.1  Ri

T rs 

T o,1  T o,2 Ra

U hi hi

T o,1.

T

U hi

Figure 1.39. Composite assembly (filled cavity wall): temperature line.

hi

o

 T*e



To,1  To,2

(1.107)

87

1.5 Applications

The suffix hi added to the (clear wall) thermal transmittance underlines the necessity to calculate it with the same surface film coefficient as the one in the denominator. The temperatures at the interfaces are T x T o  q ( Ri  Rsix ). Average thermal transmittance of assembly parts in parallel Consider a flat assembly with surface AT, composed of n parts in parallel with surfaces Ai . Each part is different, think of a window in a wall. If lateral conduction between the parts is negligible, then the partial heat flow through each of them is: §i = Ui Ai 'T (Figure 1.40). Total heat flow (§T) across the assembly equals then the sum of the partial flows: §T

n

¦ §i

'T

i 1

n

¦ U i

Ai

(1.108)

i 1

Rewriting gives: §T

'T U m

n

¦ Ai

(1.109)

i 1

where Um is the average clear wall thermal transmittance of the n parts in parallel: n

n

¦  Ai U i ¦  Ai U i

Um

i 1

i 1

n

¦ Ai

(1.110)

AT

i 1

That average for the assembly thus equals the surface weighted average of the (clear wall) thermal transmittances of the composing parts.

Figure 1.40. Thermal transmittance of an assembly, composed of n parallel parts.

Electrical analogy Equation (1.110) resembles the electrical conductivity of a circuit of n parallel electrical conductivities. Conversion to resistances gives: AT Ram

n

¦

i 1

Ai Rai

or Ram

AT A ¦ Ri i 1 ai n

(1.111)

88

1 Heat Transfer

Figure 1.41. Cavity wall, electrical analogy accounting for the ties that perforate the fill.

As long as lateral conduction in the contacts between the different parts in parallel is negligible, the electrical analogy allows solving quite complex problems. Take a cavity wall with the fill perforated by the ties. Let At be the tie section, Rt their thermal resistance, Ais total surface of the wall and Ris thermal resistance of the insulation (Figure 1.41). For the resulting thermal resistance (R) of insulation and ties, the following applies: R

Ais Ais  At At  Ris Rt

If R1 + Ri is the thermal resistance between the insulation and the inside and R2 + Re the thermal resistance between the insulation and outside, then total thermal resistance of the cavity wall equals the sum of these three series coupled resistances (Ri + R1, R and R2 + Re): RT

 Ri  R1  R   R2  Re  Ri  R1 

Ai   R2  Re Ai  At At  Ris Rt

1.5.2.2 Thermal resistance of a non ventilated, infinite cavity Heat transfer in a non-ventilated, infinite cavity combines conduction, convection, and radiation:

qT

§ · ¨ Og Nu ¸ Cb FT  ¨ ¸  T c1  T c2 1 1 ¨ d   1 ¸¸ ¨© eL1 eL2 ¹

In the formula, Og is the thermal conductivity of the cavity gas and eL1, eL2 the long wave emissivities of and Tc1, Tc2 the temperatures at the bounding surfaces. As for a flat assembly, the thermal resistance represents the ratio between temperature difference over and heat flow rate across the cavity or:

89

1.5 Applications

Rc

§ · ¨ Og Nu ¸ Cb FT  ¨ ¸ 1 1 ¨ d   1 ¸¸ ¨© eL1 eL2 ¹

1

(1.112)

Figure 1.42 gives the thermal resistance (Rc) at an average temperature of 10 °C for an infinite, air-filled vertical cavity as a function of cavity width and temperature difference. The one bounding surface is grey, the other grey, or reflecting. Note the prevalence of radiation in the heat transferred (thermal resistance increases quickly with decreasing long wave emissivity of one of the boundary surfaces), and the absence of any resistance gain for widths above 20 mm (radiation is independent of cavity width while more convection at larger cavity widths compensates for decreasing conduction). At low emissivity, even temperature difference over the cavity gains importance. For first order calculations on non-ventilated cavities with limited dimensions, the thermal resistances of Table 1.3 apply.

Figure 1.42. Thermal resistance of an infinite cavity.

90

1 Heat Transfer

Table 1.3. Cavity, thermal resistance. Thickness (mm)

Vertical cavity 2

Rc (m K/W) Both surfaces grey

Horizontal cavity 2

Rc (m K/W) One surface reflecting

Rc (m2 K/W) Heat flow upwards

Rc (m2 K/W) Heat flow downwards

0 0.001

45 f0

1

1.1

1

139

2.2 Air transfer

2.2.4

Air transfer in open-porous materials

2.2.4.1 Conservation of mass For air, we have (neither source nor sink): wwa (2.20) wt Air content wa depends on total open porosity ©o, saturation degree Sa, air pressure Pa and temperature T: Wa = ©0 Sa Ua = ©0 Sa Pa / (Ra T ). Whereas the gas constant Ra and the open porosity ©o are invariable, saturation degree Sa, air pressure Pa and temperature T may vary. Therefore, the differential of air content with respect to time turns into: div g a

wwa wt



© 0 § wPa wS S P wT ·  Pa a  a a ¨S ¸ wt Ra T © a wt T wt ¹

If the saturation degree is constant, then the equation simplifies to: wwa wt

© 0 Sa § wPa Pa wT ·  ¨ ¸ Ra T © wt T wt ¹ wT 0 (that of course excludes thermal stack): wt § © S · wP ¨ 0 a ¸ a © Ra T ¹ wt

In isothermal conditions, div ga

(2.21)

The ratio ©0 Sa / (Ra T) is called the isothermal volumetric specific air content, symbol ca, units kg/(m3 · Pa). Both the open porosity and saturation degree for air now cannot pass 1. Replacing Ra by its value, 287 J/(kg · K) so gives: ©0 Sa / (Ra T ) < 0.00348 / T, a very small number. 2.2.4.2 Flow equation See equation (2.16). 2.2.4.3 Air pressures Inserting equation (2.16) in the isothermal mass balance (2.21) gives: wPa wt In case the air permeance ka is constant, we can write: div  ka grad Pa

’ 2 Pa

ca wPa k a wt

ca

(2.22)

In (2.22) the ratio ka / ca stands for the isothermal air diffusivity of the material, symbol Da, units m2/s. That value is generally so high each adjustment in air pressure happens within a few seconds. Only for pressures that fluctuate over extremely short periods, such as wind gusts, will the air pressure response show damping and time shift. With respect to periods of some minutes or longer, damping turns to one and the time shifts to zero. Take for example mineral wool with a density of 40 kg/m3. Properties:

140

2 Mass Transfer

Thermal

Air

O = 0.036 W/(m · K)

ka = 8 · 10–5 s

U c = 33,600 J/(m3 · K)

ca = 1,17 · 10–5 kg/(m3 · Pa)

This gives a thermal diffusivity 1,1 · 10–6 m2/s, and an isothermal air diffusivity 6,8 m2/s. That is the latter is 6,355,000 times larger! Thus, when combining air transfer with much slower heat and moisture transfers, the isothermal equation (2.22) reduces to steady-state: ’ 2 Pa

0

With the flow equation (ga = –ka grad Pa)), it forms a copy of steady state heat conduction. In non-isothermal condition, i.e. included stack, the same conclusion holds: steady state suffices. Yet, the isobaric volumetric specific air capacity is larger than the isothermal one (§ 3525 / T 2), resulting in more inertia but for periods beyond minutes, air pressure damping remains one. As thermal stack interferes now, the equations are best explicitly written: ’ 2  Pao  Ua g z

0

ga

 ka grad  Pao  Ua g z

showing that a solution is only possible in combination with the thermal balance. The same holds for non-isothermal flow through air permeable layers, joints, cavities, openings, etc. 2.2.4.4 One dimension: flat assemblies The results are analogical as for steady state heat conduction. Thus, in a single-layered assembly, the air pressures form a straight line (Figure 2.6) while the air flow rate is given by: ga

ka

'Pa d

'Pa d ka

Figure 2.6. Airflow across a single-layered flat assembly.

(2.23)

141

2.2 Air transfer

The ratio d / ka is called the (specific) air resistance of the assembly; symbol W, units m/s. The inverse gives the air conductance or permeance; symbol Ka, units s/m. For the air flow rate across a composite assembly, we have: ga

'Pa d ¦ ki i 1 a,i

(2.24)

n

where 6 di / ka,i is total air resistance of the composite assembly, symbol WT, units m/s. The inverse 1 / WT is called the total air permeance KaT. di / ka,i represents the air resistance of layer i, symbol Wi. The inverse 1 / Wi = ka,i / di is called the air permeance Ka,i of layer i. Air pressure on a random interface in the wall is given by: j

¦ Wi

Pa, j

Pa1  i

1

WT

 Pa2  Pa1

Pa1  g a W1 j

(2.25)

This result is also found graphically, after transposing the composite assembly into the [W, Pa] plane, where it behaves as if single-layered, and transferring the intersections with the interfaces there to the assembly in the [d, Pa] plane (Figure 2.7). As air pressure difference 'Pa, j over any layer j we have: 'Pa, j

Wj WT

 Pa2  Pa1

showing that the largest air pressure difference in a composite assembly is found over the most air-tight layer (Figure 2.8). Including such a layer diminishes air permeance substantially, which is why it is called an ‘air barrier’ or ‘air retarder’. Its purpose is to limit the air flow across the assembly. Of course, an air barrier must be able to withstand wind load.

Figure 2.7. Airflow across a composite assembly. The graph on the left gives the air pressure in a [Wa, Pa]-plane, the graph on the right in the [da, Pa]-plane.

142

2 Mass Transfer

Figure 2.8. Effect of an airtight layer on the air pressure distribution in an assembly. Attracts total pressure difference.

Is such flat assembly approach of practical relevance? The answer is no. The prerequisite to one-dimensional flow excludes stack, which by definition gives a pressure profile along the height. Small air layers in the interfaces may additionally activate stack flow along the height. Also wind pressure is not uniform along the height. And, unintentional cracks can disrupt one-dimensionality. More than a theoretical value should not be attributed to the terms ‘flat’ and ‘one dimension’. 2.2.4.5 Two and three dimensions For isotropic open porous materials the calculation is a copy of two and three dimensional heat conduction. Of course, mass conservation substitutes energy conservation: the sum of air flows from the neighbouring to each central control volume is zero: 6 Ga, i +j = 0. The algorithm then becomes:

¦

i l , m, n j r1

 Ka,c i  j Pa,i  j  Pa,l , m, n i

¦

K a,c i  j

0

l , m, n j r1

with K a,i c  j the air permeance between each of the neighbouring and the central control volumes. Value within a material: ka A / a, with A the contact surface of any control volume with each of the neighbouring control volumes and a the distance along the mesh between their central points (Figure 2.9). For p control volumes, the result is a system of p equations with p unknown:

Figure 2.9. Conservation of mass: sum of airflows from adjacent to central control volume zero.

143

2.2 Air transfer

> Kac @ p˜ p > Pa @ p

¬ª K a,c i , j , k Pa, i , j , k ¼º p

(2.26)

where [ K ac ] p ˜ p is a p rows, p columns permeance matrix, [Pa]p a column matrix of the p unknown air pressures and [ K a,c i , j , k Pa, i , j , k ] p a column matrix containing the p known air pressures and/or air flows. Once the air pressures are calculated, then the air flows follow from: Ga, i , i  j



K a,c i , i  j Pa, i  j  Pa, i



(2.27)

For anisotropic materials the same algorithm applies, though per mesh line the direction-related K ac -value has to be taken. Besides, it is implicitly understood the mesh lines coincide with the main directions of the air permeability tensor. Under non-isothermal conditions, thermal stack intervenes and the thermal and air balances have to be solved simultaneously. Most of the time this requires iteration.

2.2.5

Air flow across permeable layers, apertures, joints, leaks and cavities

2.2.5.1 Flow equations See above. For operable windows and doors, the air permeance exponent b is typically set at 2/3: Ga

a L 'Pa2 3

with L the casement length in m and a the air permeance per meter run of casement, units kg/(m · s · Pa2/3). 2.2.5.2 Conservation of mass: equivalent hydraulic circuit Most constructions combine air-open layers with open porous materials, joints and cavities. In such a case, expressing the conservation law as a partial differential equation is impossible. An approach consists of transforming the construction into an equivalent hydraulic circuit, i.e., a combination of well chosen points, connected by air permeances (Figure 2.10). In each point the sum of the airflows from the neighbouring points must be zero, or: 6 Ga = 0. As each of the air flows is given by Ga K ax 'Pa , insertion in the conservation law results in:

¦

i l , m, n j r1

K

x a, i  j



Pa, i  j  Pa, l , m, n

¦

K a,x i  j

0

(2.28)

i l , m, n j r1

Figure 2.10. Equivalent hydraulic circuit.

144

2 Mass Transfer

With p points, a system of p non-linear equations ensues. The p unknown air pressures are found by solving for the known boundary conditions. Insertion of the air pressures found into the flow equations then gives the airflows between points. However, non-linearity makes iteration necessary, starting by assuming values for the p air pressures, calculating related air permeances and solving the system. The p new air pressures allow recalculating the air permances and solving the system again. Iteration that way continues until standard deviation between preceding and present air pressures reaches a suitable value (for example < 1/1000). As soon as flat assemblies contain air permeable layers, leaks, joints and cavities, that hydraulic circuit method applies. In the exceptional case we nevertheless have one-dimensional flow, and the assembly dissolves in a series circuit of linear and non-linear air permeances. As the same airflow rate ga migrates across each, the following holds: 1

Layer 1

ga

K a1 'Pa1

a1 'Pab1 1

§ g · b1 or ¨ a ¸ ©a ¹

K a 2 'Pa 2

a2 'Pab2 2

§ g · b2 or ¨ a ¸ ©a ¹

K a n 'Pa n

b an 'Pa n n

§ g · bn or ¨ a ¸ ©a ¹

1

'Pa1

1

Layer 2 g a

2

'Pa 2

! 1

Layer n

Sum:

ga

1 § 1 · bi ¨ ga ¸ ga ¨ ¦ 1 ¸ ¨ b ¸ ai i ¹ ©

'Pa

n

'Pa n

(2.29)

where 'Pa is the air pressure difference over the flat assembly. Equation (2.29) requires iteration. Once the air flow rate ga determined, the air pressure distribution in the flat assembly then follows from the layer equations.

2.2.6

Air transfer at building level

2.2.6.1 Definitions In buildings, a distinction is made between inter-zone and intra-zone airflow. Inter-zone considers the air transfer between spaces and between spaces and the outside. To model, we replace the building and its environment by a grid of zone points, linked by flow paths, and write the mass equilibrium per point. Intra-zone considers the air movements within a space. Is air looping from the ceiling along the windows to the floor or vice versa? How does the zone and ventilation air mix? Are there corners, which lack air washing? All this demands application of conservation of energy, mass and momentum to the air filling the space, which demands applying CFD. Here, we limit the discussion to inter-zone airflow.

145

2.2 Air transfer

2.2.6.2 Thermal stack As mentioned, wind, stack and fans induce pressure differences in buildings and between buildings and the environment. Of these, thermal stack is only vertically active: Pa  g Ua z

Pa  z

§1 1· g Pa § 1 1 ·  ¸ | Pa  3460 z ¨  ¸ ¨ Ra © Te Ti ¹ © Te Ti ¹

(2.30)

with Ti the temperature in the space considered, Te outside temperature, both in K and with z the height coordinate. Between spaces at different height but at the same temperature Ti, whose zone points are linked by a flow path, pressure difference is: '12  Pa  U g z

§1 1· Pa1  Pa2  3460  z1  z2 ¨  ¸ © Te Ti ¹

where z1 and z2 are the vertical distances to the horizontal reference plane chosen. If at different temperature, that equation changes into: '12  Pa  U g z

ª §1 §1 1 · 1 ·º   z 2  z0 ¨  Pa1  Pa2  3460 « z0  z1 ¨  ¸ ¸» © Te Ti,1 ¹ © Te Ti,2 ¹ ¼» ¬«

with z0 the vertical distance to the horizontal reference plane (0) at which temperature changes from Ti,1 to Ti,2. 2.2.6.3 Large openings Open doors, open windows and staircases form large openings whose air permeances (Ka ) change with the driving force. While wind and fans give a uniform flow, thermal stack along vertical and sloped large openings gives a double flow: from warm to cold above and from cold to warm below, with a zero sum (Figure 2.11). In other words, the same amount of air moves in both direction. The question is how to calculate the air permeance in both cases?

Figure 2.11. Large opening, on the left left with uniform flow caused by wind and fans, on the right a stack induced flow.

146

2 Mass Transfer

For wind and fans, the airflow is (kg/s): Ga

2 Ua 'Pa

Cf B H

(2.31)

giving as air permeance: Ka

C

f



2 Ua 'Pa0.5

BH

(2.32)

In both equations, Cf is a flow factor with a value between 0.33 and 0.7. B is width and H height of the opening. With thermal stack, the airflow in both direction (kg/s) looks like: Ga1

Ga2

Cfc B 3

Ua g Pa H 3 § 1 1 · ¨© T  T ¸¹ Ra 1 2

(2.33)

The air permeance then is ( 'pT,max g Pa H / Ra (Te1  T21  Te1  T11 ) § 3450 H (T11  T21 ) ): K a1

 K a2

Cfc B H 3

Ua ª § 1 1 ·º «3450 H ¨  ¸ » © T1 T2 ¹ ¼ ¬

0.5

(2.34)

with Ra the gas constant of air and Ccf the flow factor for two-way displacement. (2.34) reflects a parabolic air velocity distribution along the opening’s heigth, a direct consequence of Bernouilli’s law. Indeed, along a flow path at height z above the floor we have: Ua1 g z

Ua2 va,2 z 2

 Ua2 g z

which gives: § U  Ua2 · 2 g z ¨ a1 . © Ua2 ¸¹

va, z

Equation (2.34) then follows from integrating the air velocity up and down over half the height, followed by converting density into temperatures (see the ideal gas law) and including the flow factor Cc. f 2.2.6.4 Conservation of mass If in each zone or node we have n inflows and m outflows, their algebraic sum must be zero. If not, large air pressure fluctuations should happen, which is never observed. As an equation: mn

¦

Ga, i

0

(2.35)

i 1

Take the living room (zone 2) with the two windows in the ground floor view of Figure 2.12. Doors separate it from the kitchen (= zone 1) and the hall (= zone 3). All partitions and opaque outer walls are assumed airtight (which may not be the case in real life). In a first step the building has no fan-driven ventilation system and temperature inside is the same as outside.

147

2.2 Air transfer

Figure 2.12. Ground floor.

In such case only wind induces airflow. To and from the living room we have inflows and outflows across the operable windows (Ga, 2 – e1, Ga, 2 – e2) and in or outflows across the door bays with the kitchen (Ga, 2 – 1) and hall (Ga, 2 – 3). Because the direction of the four flows is unknown, all are assumed entering the living room. Per flow, we have: Ga,2  e1 Ga,2  e2 Ga,2 1 Ga,2  3

 K a,2  e2  Pa,e2  Pa,2 K a,2 1  Pa,1  Pa,2 K a,2  3  Pa,3  Pa,2

K a,2  e1 Pa,e1  Pa,2

Summing and reshuffling gives:





 K a,2  e1  K a,2  e2  K a,2 1  K a,2  3 Pa,2  K a,2  e1 Pa,e1  K a,2  e2 Pa,e2  K a,2 1 Pa,2  K a,2  3 Pa,3 or:  Pa,2

¦  K a,i  j / e  ¦  K a,i  j

Pa, j





 ¦ K a, i  e Pa, e

0



i.e. an equation with three unknowns: air pressure in the living room (Pa2), air pressure in the kitchen (Pa1) and air pressure in the hall (Pa3). Besides the living room, kitchen and hall, we have a garage and toilet on the ground floor. The hall in turn links the ground floor to the first floor with three sleeping rooms and a bathroom. For each, we can repeat the above: all flows assumed to point to that space, the sum zero. The result is a system with as many equations as separate spaces in the building. In matrix notation:

> Ka @n, n > Pa @n

ª¬ K a,e Pa,e º¼ n

(2.36)

148

2 Mass Transfer

Solving this system of non-linear equations again demands iteration. If the building is warmer or colder than outdoors, all leaks have to be located first. Then, nodes are positioned in both neighbouring spaces at each leak’s height. After, related thermal stack with outside against a common reference height complements the pressure difference between neighbouring nodes at different height, with application of very large air permeances in vertical direction between nodes in the same space. Do we know the temperatures in each space, then thermal stack figures as known terms in the balance equations. Temperatures in non-heated spaces, however, depend most of the time on the temperature of the air entering. That requires combining the air balances with conservation of energy. Correct temperatures then follow from alternatively solving the heat and air balances using the temperatures and airflows found in the preceding iteration as known quantities, and to proceed that way until the differences between two successive iterations drop below a preset value. In case ventilation is fan-driven, the fans figure as sources with known pressure/flow characteristic. 2.2.6.5 Applications One opening Airflow across one horizontal opening is impossible. Vertical or sloped openings instead activate air exchange. Not only may thermal stack cause twinned flow, also turbulency by fluctuations in wind pressure activate airflow, not from a small trickle vent above a window, but certainly when operable windows are ajar. Openings in series Take two openings, one with air permeance Ka,1 and the other with air permeance Ka,2. Pa,1 is the air pressure in front of the one, Pa,2 the air pressure beyond the other. In isothermal conditions we have (Figure 2.13):





 K a,1  K a,2 Pa, x  K a,1 Pa,1  K a,2 Pa,2

0

with Pa,x the unknown air pressure in the space in between. Airflow is only possible if Pa,1 differs from Pa,2. This is probably the case if both openings sit are in outer walls with different orientations. Yet, due to wind pressure increasing with height, also openings in the same outer wall but at different heights cause airflow. How large that flow is, depends on both air permeances. If independent of the air pressure difference across them, mass balance gives: Ga,1

Ga,2

Ga



K a,1 Pa,1  Pa, x

Figure 2.13. Two openings in series.





1 Pa,1  Pa,2 1 1  K a1 K a2



149

2.2 Air transfer

If the function of the air pressure difference is across though with same permeance exponent, the resulting airflow becomes: Ga

1

 Pa,1  Pa,2

b

§ 1 1 · ¨ 1b  1b ¸ a2 ¹ © a1

b

If the permeance exponent also differs, quantifying the airflow requires iteration starting from:



1 Pa,1  Pa,2 1 b1 1 b2 º ª b b §G · 2 » « § Ga · 1 ¨ a¸ « ¨© a ¸¹ » © a2 ¹ «¬ 1 »¼

Ga,next



First we guess a value for the airflow (Ga ) on the right hand side, for example, by assuming the permeance exponents to be equal to one. Then with these values we calculate the airflow (Ga,next ) on the left hand side, use that value on the right hand side to recalculate the airflow on the left hand side, and proceed until the previous and actual values obey a pre-determined accuracy criterion. The three equations anyhow show that, if one of the air permeances is very small, the airflow will be minimal. If we have several spaces coupled in series with an air inlet in the outer wall of the first, an air outlet in the outer wall of the last and flow through openings in all partitions in between, then the resulting air flow across all spaces equals: All air permeances constant Ga

1 n

¦

j 1

1 K a,j

 Pa,1  Pa,n 1

All air permeance exponents b identical Ga

1

 Pa,1  Pa, n 1

b

§ n 1 · ¨¦ 1b ¸ © j 1 aj ¹

b

Also the air permeance exponents different 1

Ga n

¦

j 1

1 b j ª « § Ga · b j «¨ ¸ «© a j ¹ «¬

º » » » »¼

 Pa,1  Pa, n 1

with Pa,1 air pressure in front of the inlet and Pa, n + 1 air pressure past the outlet. It must be clear that the smallest air permeance again defines the magnitude of the air flow.

150

2 Mass Transfer

Figure 2.14. Openings at different height in a same outer wall, thermal stack intervening.

If we have openings at different height, and the temperatures at both sides are different, then thermal stack intervenes. Take a space and assume windless weather outdoors, so that only thermal stack is left. One of the outer walls has two openings, the one a height 'H above the other. Both have constant air permeance. In the nodes at the horizontals across both openings the air pressure is Pa,1, respectively Pa,2. A very large but fictitious permeance Ka,z links the two (Figure 2.14). The mass balances are:





ª § 1 1 ·º  K a,1 Pa,e  K a,z «3460 'H ¨  ¸ » © Te Ti ¹ ¼ ¬





ª § 1 1 ·º  K a,2 Pa,e  K a,z «3460 'H ¨  ¸ » © Te Ti ¹ ¼ ¬

Node 1:  K a,1  K a,z Pa,1  K a,z Pa,2

Node 2:  K a,2  K a,z Pa,2  K a,z Pa,1

With the reference air pressure outdoors (Pa,e) zero and the air permeance between both nodes infinite (Ka,z = ’), the airflow becomes: Ga

ª § 1 1 ·º 1 ª0.043 'H  Ti  T e ¼º 3460 'H ¨  ¸ » | « 1 1 ¬ 1 1 ¬ © Te Ti ¹ ¼   K a1 K a2 K a1 K a2 1

Or, the two behave as if they were series coupled. Thus, the smallest will fix the airflow. Openings in parallel Several openings in a partition between two spaces at same temperature act in parallel. If the air permeance per opening is constant but the air pressures at both sides differ, then the resulting airflow is:

¦ K a, j  Pa,1  Pa,2 n

Ga

j 1

In the case the air permeances depend on the air pressure difference over the openings, but the permeance exponents are identical, the resulting airflow can be expressed as: n

Ga

¦ aj j 1

 Pa,1  Pa,2

b

151

2.2 Air transfer

If the permeance exponents are also different, then we get: n

Ga

¦

j 1

a

b j 1

j

'Pa

P

a,1

 Pa,2



That result proves the airflow between both spaces increases the more openings or leaks are present in the partition wall. Put another way, a large opening short circuits a small installed trickle vent.

2.2.7

Combined heat and air transfer

2.2.7.1 Open-porous materials Air displacement across an open-porous material adds enthalpy flow to conduction, enthalpy flow being given by q = H = ca ga T, with ca specific heat capacity, for air 1008 J/(kg · K), and ga airflow rate. If we assume air progresses so slowly in the pores that material matrix and air keep the same temperature anywhere, then energy conservation for an elementary volume dV becomes: div q  ca ga

T 

w  U c  Ua ca Sa © o T r §c wt

with Ua ca the volumetric specific heat capacity of air, Sa the saturation degree for air and ©o the open porosity of the material. In most cases the product Ua ca Sa ©o is much smaller than the volumetric specific heat capacity U c of the material, which allows simplifying the balance to: div q  ca ga

T 

w UcT r §c wt

Introducing Fourier’s heat conduction law and expanding the operator term (div ga = 0) gives (thermal conductivity O and volumetric heat capacity U c constant):  O ’ 2 T  ca ga ’T

U c

wT r §c wt

(2.37)

Steady state As mentioned earlier, boundary conditions in ‘steady state’ do not depend on time, heat sources and sinks never change and the material properties remain constant. In such cases (2.37) becomes: O ’ 2 T  ca ga ’T

r§ c

(2.38)

One dimension: flat assemblies If the air and heat flow rate develop perpendicular to the assembly’s surface, then (2.38) becomes (neither heat source nor sink): d 2 T ca ga dT  O dx dx 2

0

i.e., a second order differential equation with a solution of:

(2.39)

152

2 Mass Transfer

T

§c g x· C1  C2 exp ¨ a a ¸ © O ¹

(2.40)

where C1 and C2 are the integration constants. They are determined by the boundary conditions. The heat flow rate by conduction, the enthalpy flow rate (also called convective heat flow rate) and the total heat flow rate are then given by: dT dx

Conduction

q

O

Enthalpy (convection)

H

qc

Total

qT

qH

§c g x· ca g a C2 exp ¨ a a ¸ © O ¹

ª § c g x ·º ca g a «C1  C2 exp ¨ a a ¸ » © O ¹¼ ¬ q  qc

ca g a C1

While the conduction and enthalpy flow rate change exponentially over the assembly, total heat flow rate remains constant. In other words, the assembly functions as a heat exchanger between conduction and enthalpy flow, thereby heating or cooling the migrating air. Single-layered flat assemblies Let d be the thickness of the assembly. We rewrite (2.40) as: T

§c g d x · C1  C2 exp ¨ a a ¸ © O d¹

(2.41)

The dimensionless ratio ca ga d /O is called the Péclet number, symbol Pe. The Péclet number links the enthalpy to the conduction heat flow rate. The more air flows across 1 m2 and the higher the thermal resistance of the assembly, the higher the number. As boundary conditions we can have: (1) Type 1: the air flow across the assembly and the temperatures at both assembly surfaces are known: x = 0: T = Ts1; x = d: T = Ts2 (2) Type 2: the air flow across the assembly and the heat flow rates by convection and radiation at both assembly surfaces are known: x = 0, q = h1 (T1 – Ts1) + ca ga TV; x = d, q = h2 (T2 – Ts2) + ca ga TV. Type 1 In this case the integration constants follow from: Ts1 = C1 + C2, Ts2 = C1 + C2 exp (Pe), or: C1

Ts2  Ts1 exp  Pe 1  exp  Pe

C2

Ts1  Ts2 1  exp  Pe

Temperatures in the assembly and the conductive, enthalpy and total heat flow rate become: T

Ts1  F1  x  Ts2  Ts1

q

F2  x O

Ts2  Ts1 d

qc

Pe

OT d

qT

O

Ts2 F2 0  Ts1 F2  d d

(2.42)

153

2.2 Air transfer

Figure 2.15. Single-layered assembly, combined conduction and enthalpy flow. Upper temperature curve for warm air mitigating to the cold side, lower temperature curve for cold air mitigating to the warm side.

with: F1  x

x· § 1  exp ¨ Pe ¸ © d¹ 1  exp  Pe

F2  x

x· § Pe exp ¨ Pe ¸ © d¹ 1  exp  Pe

Both F1(x) and F2(x) depend on material characteristics and air flow rate. If the air moves from thickness d to thickness 0, then the flow rate gets a negative sign. In the opposite case, it has a positive sign. The air is responsible for an exponential change in temperature, convex when the air goes from warm to cold, concave when flow develops from cold to warm (Figure 2.15). The higher the Péclet number, i.e., the more air passes across, the more concave or convex the exponential is. Compared to pure conduction (q0), the change in total heat flow rate as a percentage is (Ts1 = 0 °C (x = 0), Ts2 = 1 °C (x = d)): 100 (qT – q0) / q0 = 100 [abs (F2(0)) – 1]. For air flowing from warm to cold abs (F2(0)) is larger than 1. In such case, enthalpy transfer forces total heat transfer to increase. For air flowing from cold to warm, abs (F2(0)) becomes smaller than 1 and enthalpy transfer diminishes total heat transfer. The conductive heat flow rate however changes in the opposite way: smaller when air flows from warm to cold, larger when air flows from cold to warm. Type 2 Boundary conditions of type 2 are converted into type 1 one by adding a fictitious surface film layer to each side of the assembly, one with a thermal resistance 1 / h1 in x = 0 and one with a thermal resistance 1 / h2 in x = d. The two fictitious layers are 1 m thick and have a thermal conductivity h1 or h2. As a consequence, the reference temperatures T1 and T2 turn into fictitious surface temperatures while the assembly changes from single-layered to composite. Such a composite behaves as single-layered in the [T, R]-plane (Tfor temperature, R for thermal resistance), a fact which not only applies for pure conduction but also for the combination of enthalpy flow and conduction. Writing equation (2.39) with T and R as co-ordinates, gives:

154

2 Mass Transfer

d2T dT  ca g a dR dR 2

0

(2.43)

with a solution of: T = C1 + C2 exp (ca ga R). Boundary conditions are: R

0: T

T1

R

1 1 R : T h1 h2

RT

T2

And integration constants: C1

T 2  T1 exp ca g a RT 1  exp ca g a RT

C2

T1  T 2 1  exp ca g a RT

Temperatures in the assembly, heat flow rate by conduction, enthalpy flow rate and total heat flow rate become: T

T1  T 2  T1 F1  R

q

F2  R T 2  T1

qc

ca g a T

T 2 F2 0  T1 F2  RT

qT

(2.44)

with: F1  R

1  exp ca g a R F2  R 1  exp ca g a RT

ca g a exp ca g a R 1  exp ca g a RT

(2.45)

In these equations, R is the thermal resistance between the fictitious assembly surface at R = 0 and the chosen interface in the assembly. Again, the functions F1 and F2 depend on the material characteristics and the air flow rate. Composite flat assemblies The solution for a single-layered assembly with boundary conditions of type 2 also applies for a composite assembly. The boundary conditions (outside part, R = 0 is outside, R = RT inside) are: R

0: T

Te

R

RT

1 1 ¦R : T h1 h2

Ti

For the temperatures in the interfaces one gets (surface 1 sees the outside environment, interface 1 is closest to surface 1, etc.):

etc.

§1· T e   Ti  T e F1 ¨ ¸ © h1 ¹

Surface 1

Ts1

Interface 1

T1

§1 · T e   Ti  T e F1 ¨  R1 ¸ h © 1 ¹

Interface 2

T2

§1 · T e   Ti  T e F1 ¨  R1  R2 ¸ © h1 ¹

155

2.2 Air transfer

Figure 2.16. Composite flat assembly, temperatures for warm inside air flowing to the cold side.

The heat flow rates are (Te (R = 0) < Ti (R = RT)): q

F2  R  Ti  T e

qc

ca g a T

qT

Ti F2 0  T e F2  RT

with: F1  R

1  exp ca g a R 1  exp ca g a RT

F2  R

ca g a exp ca g a R 1  exp ca g a RT

In the [T, R]-plane, the temperature curve is similar to the one in a single-layered assembly (Figure 2.16). In outflow mode (air migrates from the inside, where it is warmer, to the outside, where it is colder), the curve becomes convex, while in the inflow mode, it is concave. With inside insulation for example, outflow increases the wall temperature at the cold side of the insulating layer. What are the consequences for buildings with an air leaky envelope? There, infiltration occurs (i.e. air flow from outside to inside) at the windward side while outflow develops (i.e. air flow from inside to outside) at the leeward side and at the sides more or less parallel to the wind direction. Yet, comparison of the sum of the total heat flow at the windward and the other sides (= §T1 + §T2) with the total heat flow one should have, if conduction (§G = |(TR1 – TR2) (A1 / R1 + A2 / R2)|) and infiltration (§V = |ca Ga (TR1 – TR2)| (with Ga the infiltration flow) occurred separately, shows that §T1 + §T2 is always smaller than §G + §V. In other words, the passage of air through the envelope results in energy conservation. Indeed, part of the higher conduction losses at the windward side are used to warm the inflowing air, while the warm, outflowing air diminishes the conduction losses at the leeward side. However this advantage is no compensation because air passage creates too many other unwanted problems. Two and three dimensions In two and three dimensions, analytical solutions for (2.31) hardly exist. When using CVM or FEM, first the air flows are calculated and then the temperatures and heat flow rates solved. If thermal stack intervenes, iteration is needed between the air and heat balances. The heat flows between adjacent points and a central point may be expressed in two ways:

156

2 Mass Transfer

y By using an integral numerical technique. In such a case, the heat flow between two adjacent points becomes: Air flow from 2 to 1 §21 Air flow from 1 to 2 §21

P21c T 2  T1  ca Ga T 2

P21c T 2  T1  ca Ga T 2

T 2  P21c  ca Ga  T1 P21c

(2.46)

T 2 P21c  T1  P21c  ca Ga

where Pc21 is the surface related thermal permeance. y By using the analytical solution for flat assemblies. For a series connection of surface related thermal resistances: §21

T1 F2 0  T 2 F2  R21 c

where Rc21 is the surface related thermal resistance between both neighbouring points. In case of a parallel connection of series paths between the two points, the equation is used to calculate the flow per series path. The total heat flow then is: §21

2

i ¦ §21

i 1

Transient regime Equation (2.37) is the starting point. For a constant air flow rate and periodically varying temperatures in a flat assembly, the complex calculation method, dealt with in heat conduction, is applicable. For each harmonic: T R

D  R exp

2i n Át T

where D(R) is the complex temperature. In the absence of a heat source or sink, the complex temperature is derived from: d2 D dD 2 i n S U c O  ca g a  D 2 dR T dR

0

In this equation, i is the imaginary unit and T the period. The solution is: D  R C1 exp r1 R  C2 exp r2 R

(2.47)

where r1 and r2 are the roots of the quadratic equation r 2  ca g a r  r1

1ª «c g  2¬ a a

ca

ga 

8ni S Uc Oº » T ¼

r2

1 2

ª «ca g a  ¬

ca

ga 

8niS Uc T

2

2

2niS Uc O T

0:

Oº » ¼

(2.46) can thus be written as: D R

§1 ·ª exp ¨ ca g a R ¸ «C1  C2 sinh ©2 ¹¬

§1 · ¨© a R ¸¹  C1  C2 cosh 2

§1 ·º ¨© a R ¸¹ » 2 ¼

157

2.2 Air transfer

ca

with a

ga  2

8niS Uc O . T

For an air flow rate (ga ) zero, the solution for conduction is obtained. The integration constants C1 and C2 follow from the boundary conditions (example: on surface R = 0 the complex temperature Ds and the complex heat flow rate Dcs are known). It is up to the readers to make these and other calculations. The most important result they will find is that inflow decreases damping and time shift of an assembly, while outflow increases it. For two and three dimensional details, CVM or FEM help in finding the solution. For CVM, the energy balance per control volume 'V becomes:

¦ §m |

U c 'V

'T r § c 'V 't

where §c is a heat source or sink and §m the heat flow by conduction and enthalpy transfer, during time interval 't from each of the neighbouring points to the central point. 2.2.7.2 Air permeable layers, joints, leaks and cavities In general In the case of air permeable layers, leaks, joints and cavities, steady state and transient air and heat transfer problems can be solved by combining the equivalent hydraulic network approach with CVM for the heat balances, using for the heat flows equation (2.46). Ventilated cavities Consider the cavity of Figure 2.17 in a flat assembly. Cavity width is dcav m, its length or height L m (ordinate z) and, air velocity in it v m/s. As the convective surface film coefficient at both bounding surfaces, we have hc (W/(m2 · K)). The thermal resistance between environment 1 and bounding surface 1 is R1 m2 · K/W, between environment 2 and bounding surface 2 R2 m2 · K/W. The heat balance for each surface is:

Figure 2.17. Ventilated cavity in a flat assembly, a combination of heat conduction through both leafs with enthalpy flow in the cavity.

158

2 Mass Transfer

Bounding surface 1:

T1  Ts1  hc  T cav  Ts1  hr  Ts2  Ts1 R1

0

Bounding surface 2:

T 2  Ts2  hc  T cav  Ts2  hr  Ts1  Ts2 R2

0

with hr the surface film coefficient for radiation. For the cavity, the following heat balance applies (the cavity stretches out along the z-axis): ª¬ hc  Ts1  T cav  hc  Ts2  T cav º¼ dz

Ua ca dsp v dT cav

The temperatures Ts1, Ts2 and Tcav are unknown in the three equations, i.e., the temperature in the cavity and at both bounding surfaces. The solution is found by first calculating the temperatures Ts1 and Ts2 from the surface balances as a function of cavity temperature Tcav , and then introducing both in the cavity heat balance. To do so, the constants D, A1, A2, B1, B2, C1, C2 are introduced: D

§ 1 ·§ 1 · 2 ¨© hc  hr  R ¸¹ ¨© hc  hr  R ¸¹  hr 1 2 hc  hr 

A1

C1

1 R2

D R1

A2

hr D R1

§ 1 · hc ¨ hc  hr  h h R2 ¸¹ r c © D

B1

C2

hr D R2

hc  hr  B2

1 R1

D R2

§ 1· hc ¨ hc  hr  ¸  hr hc R1 ¹ © D

The surface temperatures then become: Ts1

A1 T1  B1 T 2  C1 T cav

Ts2

A2 T1  B2 T 2  C2 T cav

(2.48)

For the cavity balance, we get: ª  A1  A2 T1   B1  B2 T 2 º  T cav » dz « 2  C1  C2 ¬  ¼ 

a1

Ua ca d cav v dT cav hc 2  C1  C2  

b1

with a solution of: z b1

§ a  T cav ·  ln ¨ 1 ¸¹ © C

The integration constant C follows from the boundary conditions. For z infinite, the ratio z / b1 becomes infinite, turning the solution into:

159

2.2 Air transfer

§ a1  Tcav, f · ln ¨ ¸¹ C ©

f

which equals a1 – Tcav,’ = 0 or a1 = Tcav,’. Consequently, a1 represents the temperature asymptote in an infinite cavity. That value does not differ from the temperature in a nonventilated cavity. In fact, z / b1 becomes also infinite for an air velocity zero (v = 0)! § Tcav, f  T cav,0 · For z = 0, Tcav equals Tcav,0, giving ln ¨ ¸¹ 0 or C Tcav, f  T cav,0 . C © The cavity temperature thus equals: T cav

§ z· Tcav, f  Tcav, f  T cav,0 exp ¨  ¸ © b1 ¹





Or, the temperature in a ventilated cavity changes exponentially, from the inflow value to a value approaching the one without ventilation. The term b1 has the dimensions of a length. It is named the ‘ventilation length’. The higher the air velocity in the cavity, the larger that length. The larger however the heat transfer to the cavity is, the lower it is. The temperatures at both bounding surfaces (Ts1, Ts2) are found by introducing Tcav into the equations (2.48): Ts1

ª § z ·º A1 T1  B1 T 2  C1 « Tcav, f  Tcav, f  T cav,0 exp ¨  ¸ » © b1 ¹ ¼ ¬

Ts2

ª § z ·º A2 T1  B2 T 2  C2 « Tcav, f  Tcav, f  T cav,0 exp ¨  ¸ » © b1 ¹ ¼ ¬









What is the influence of cavity ventilation on the heat loss through an outer assembly? Let ‘2’ be the inside leaf. For the heat flow across, the following applies: §

1 R2

L

1 R2

L

³ T 2  Ts2 dz 0

³

0

­° ª § z · º ½° ® A2 T1  1  B2 T 2  C2 « Tcav, f  Tcav, f  T cav,0 exp ¨  ¸ » ¾ dz © b1 ¹ ¼ ¿° ¬ ¯°





T 2  Ts2f L  C2 b1 Tspf  Tsp0 ªexp §  L ·  1º R2

R2

« ¬

¨© b ¸¹ 1

» ¼

The average U-value thus is: U

§ L T1  T 2

Uo 





C2 b1 Tspf  Tspo ª § L ·º «1  exp ¨  ¸ » L R2 T1  T 2 ¬ © b1 ¹ ¼

Since the temperature ratio (Tsp’ – Tspo) / (T1 – T2) can be approximated by: R1 / Ra = R1 Uo, this formula simplifies to: U

­° C b R U o ®1  2 1 1 L R2 ¯°

ª § L · º ½° «1  exp ¨  ¸ » ¾ © b1 ¹ ¼ ¿° ¬

(2.49)

160

2 Mass Transfer

Hence, the effect of ventilation depends on the thermal resistances R2 respectively R1 of the inside and outside leaf. If R2 is large and R1 small, i.e., if a partial fill lines up with the inside leaf, then ventilation hardly has an impact. If in turn R2 is small and R1 large, i.e. if the fill lines up with the outside leaf, then the effect is significant. Improper workmanship when installing the insulation may be responsible for that. At the air intake, the heat flow rate equals (T2 – Ts2,0) / R2.

2.3

Vapour transfer

2.3.1

Water vapour in the air

2.3.1.1 Overview Air contains water vapour. Proof of that is the dampness on single glazed windows during winter. Those who wear glasses also experience it when entering a heated, crowded room coming from the cold outdoors. For convenience, moist air is considered a mixture of two ideal gases: dry air and water vapour, even though the term ‘ideal’ is less accurate for water vapour than for dry air. Previous chapter presented the gas equations of dry air. For water vapour, we have: p V = mv R T, where p is the partial water vapour pressure in Pa, V the volume considered in m3, T the ‘water vapour’ temperature in K, mv the water vapour mass filling the volume V in kg and R the gas constant for water vapour, equal to 461.52 J/(kg · K). More accurate is the equation: pV nv R0 T

1

Bvv C  vvv 2 §V · §V · ¨© n ¸¹ ¨ ¸ v © nv ¹

where nv are the moles of water vapour filling volume V, R0 is the general gas constant (8314.41 J/(mol · K)) and Bvv, Cvvv are the viral coefficients for water vapour. Vapour concentration as derived from the ideal gas law is: Uv

mv V

p RT

(2.50)

Normally1 the gases in dry air do not react with water vapour. As a result, the Dalton law applies for moist air: total air pressure in volume V equal to the sum of the partial dry air and the partial water vapour pressure: Pa

pl  p

(2.51)

In the built environment, Pa is the atmospheric pressure, at sea level § 101.3 kPa (1 Atm). Thanks to its limited solubility in water, the presence of dry air does not influence the balance between water, water vapour, and ice (Raoult and Henry’s law). Hence, in air, the diagram of state for water is still applicable: the triple point at 0 °C, etc. However kinetics changes. In air, evaporation and condensation evolves slower than in a vacuum. 1

When SO2, NOx, Cl contaminate air, there is an interaction. SO2 reacts with water vapour to form sulphur acid (H2SO3). However, the concentration of SO2 is so much lower than the concentration of water vapour that the reaction does not have a significant impact.

161

2.3 Vapour transfer

2.3.1.2 Quantities The following quantities describe the presence of water vapour in the air: y Partial water vapour pressure Symbol: p Units: Pa Together with temperature and total air pressure a fundamental variable of state. y Water vapour concentration Symbol: Uv Units: kg/m3 Mass of water vapour per unit volume of air. As a derived state variable, vapour concentration is a function of partial water vapour pressure and temperature (see equation (2.50)). y Water vapour ratio Symbol: x Units: kg/kg Mass of water vapour per mass-unit of dry air. Also the vapour ratio is a derived variable of state. Following relations exist between water vapour ratio x and partial water vapour pressure p: x

Uv Ul

Ra p Rv  Pa  p

0, 62 p Pa  p

(2.52)

The indications ‘partial’ and ‘water’ are omitted in the text that follows. 2.3.1.3 Maximum vapour pressure and relative humidity As stated, the presence of air hardly changes the equilibrium between water, vapour, and ice. Therefore, for each air temperature, the equilibrium vapour pressure between liquid, solid and vapour figures as the maximum attainable, also called the vapour saturation pressure.

Figure 2.18. Vapour saturation pressure psat(T) as function of temperature, line of 50% relative humidity.

162

2 Mass Transfer

Related vapour concentration and vapour ratio are also saturation values. All increase with temperature. At 100 °C, the saturation pressure equals the standard atmospheric pressure at sea level. Saturation values get the suffix ‘sat’: psat, Uv,sat, xsat. Figure 2.18 and Table 2.4 give the saturation pressure as a function of temperature. For temperatures between –30 °C and 41 °C, the table lists the values in steps of 0.1 °C. For temperatures between 45 °C and 95 °C, 5 °C is used as a step. Vapour saturation pressure follows a more or less exponential curve. Table 2.4. Vapour saturation pressure in Pa. Temperatures between 0 °C and –30 °C in steps of 0.1 °C ¹ (°C) –0 –1 –2 –3 –4 –5 –6 –7 –8 –9 –10 –11 –12 –13 –14 –15 –16 –17 –18 –19 –20 –21 –22 –23 –24 –25 –26 –27 –28 –29 –30

–0.0

–0.1

–0.2

–0.3

–0.4

–05

–0.6

–0.7

–0.8

–0.9

611 562 517 475 437 401 368 338 310 284 260 237 217 198 181 165 150 137 125 113 103 96 84 76 69 62 56 51 46 41 37

606 558 513 471 433 398 365 335 307 281 257 235 215 196 179 164 149 136 123 112 102 92 84 76 68 62 56 50 45 41 36

601 553 509 467 430 394 362 332 304 279 255 233 213 195 178 162 148 134 122 111 101 91 83 75 68 61 55 50 45 40 36

596 548 504 464 426 391 359 329 302 276 253 231 211 193 176 160 146 133 121 110 100 90 82 74 67 60 55 49 44 40 36

591 544 500 460 422 388 356 326 299 274 251 229 209 191 174 159 145 132 120 109 99 90 81 73 66 60 54 49 44 39 35

586 539 496 456 419 384 353 323 296 271 248 227 207 189 173 158 143 131 119 108 98 89 80 73 66 59 53 48 43 39 35

581 535 492 452 415 381 350 321 294 269 246 225 206 188 171 156 142 129 118 107 97 88 80 72 65 59 53 48 43 38 35

576 530 488 448 412 378 347 318 291 267 244 223 204 186 170 155 141 128 116 106 96 87 79 71 64 58 52 47 42 38 34

572 526 484 444 408 375 344 315 289 264 242 221 202 184 168 153 139 127 115 105 95 86 78 70 64 57 52 47 42 38 34

567 522 479 441 405 371 341 312 286 262 240 219 200 183 167 152 138 126 114 104 94 85 77 70 63 57 51 46 41 37 33

163

2.3 Vapour transfer Temperatures between 0 °C and 40 °C in steps of 0.1 °C ¹ (°C) 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40

0.0 611 657 706 758 813 872 935 1001 1072 1147 1227 1312 1401 1497 1597 1704 1817 1936 2063 2196 2337 2486 2643 2808 2983 3166 3360 3564 3779 4004 4242 4491 4753 5029 5318 5621 5939 6273 6622 6989 7372

0.1 615 662 711 763 819 878 941 1008 1079 1155 1235 1320 1411 1506 1608 1715 1829 1949 2076 2210 2351 2501 2659 2825 3001 3185 3380 3585 3801 4028 4266 4517 4780 5057 5347 5652 5972 6307 6658 7026 7412

0.2 620 666 716 768 824 884 948 1015 1087 1163 1243 1329 1420 1516 1618 1726 1840 1961 2089 2224 2366 2516 2675 2842 3019 3204 3400 3606 3823 4051 4291 4543 4807 5085 5377 5683 6004 6341 6694 7064 7451

0.3 624 671 721 774 830 890 954 1022 1094 1171 1252 1338 1429 1526 1629 1737 1852 1973 2102 2237 2381 2532 2691 2859 3037 3224 3420 3627 3845 4074 4315 4569 4835 5114 5407 5715 6037 6376 6730 7102 7491

0.4 629 676 726 779 836 896 961 1029 1101 1178 1260 1347 1439 1536 1639 1748 1864 1986 2115 2251 2395 2547 2708 2877 3055 3243 3440 3649 3868 4098 4340 4595 4862 5143 5437 5746 6071 6410 6767 7140 7531

0.5 634 681 731 785 842 903 967 1036 1109 1186 1269 1356 1448 1546 1650 1760 1876 1999 2128 2265 2410 2563 2724 2894 3073 3262 3461 3670 3890 4122 4365 4621 4889 5171 5467 5778 6104 6445 6803 7178 7571

0.6 638 686 736 790 848 909 974 1043 1117 1194 1277 1365 1458 1556 1661 1771 1888 2011 2142 2280 2425 2579 2741 2912 3092 3281 3481 3692 3913 4145 4390 4647 4917 5200 5498 5810 6137 6480 6840 7217 7611

0.7 643 691 742 796 854 915 981 1050 1124 1203 1286 1374 1467 1566 1671 1782 1900 2024 2155 2294 2440 2595 2758 2929 3110 3301 3502 3713 3935 4169 4415 4673 4945 5229 5528 5842 6171 6516 6877 7255 7652

0.8 647 696 747 802 860 922 987 1057 1132 1211 1294 1383 1477 1577 1682 1794 1912 2037 2169 2308 2455 2611 2774 2947 3129 3321 3522 3735 3958 4193 4440 4700 4973 5259 5559 5874 6205 6551 6914 7294 7692

0.9 652 701 752 807 866 928 994 1065 1139 1219 1303 1392 1487 1587 1693 1805 1924 2050 2182 2322 2470 2627 2791 2965 3148 3340 3543 3757 3981 4218 4466 4727 5001 5288 5590 5907 6239 6587 6951 7333 7733

164

2 Mass Transfer

Temperatures between 45 °C and 95 °C in steps of 5 °C ¹ (°C) 45 50 55

pc (Pa) 9,582 12,335 15,741

¹ (°C) 60 65 70

pc (Pa) 19,917 25,007 31,156

¹ (°C) 75 80 85

pc (Pa) 38,550 47,356 57,800

¹ (°C) 90 95

pc (Pa) 70,108 84,524

The most accurate approximation for temperatures between 0 and 50 °C is: psat

ª § T ·º pc,sat exp « 2, 3026 N ¨1  c ¸ » © T ¹¼ ¬

(2. 53)

with Tc the critical temperature of water, i.e. the temperature above which water only exists as vapour, and pc,sat the related saturation pressure. (Tc = 647.4 K, pc,sat = 217.5 · 105 Pa). The parameter N depends on temperature in K: 2

N

§ T · § T · § T · 4.39553  6  2442 ¨  9.953 ¨  5  151 ¨ © 1000 ¸¹ © 1000 ¸¹ © 1000 ¸¹

3

Less accurate approximations are: 10 d T d 50 °C

psat

30 d T d 0 °C psat 0 d T d 40 °C psat

7066.27 ª º exp «65.8094   5.976 ln T » T ¬ ¼

 611 exp 72.5 ˜ 10

611 exp 82.9 ˜ 103 T  288.1 ˜ 106 T 2  4.403 ˜ 106 T3 3

T  288.1 ˜ 106 T 2  0.79 ˜ 106 T3

T d 0 °C psat

§ 22.44 T · 611 exp ¨ © 272.44  T ¸¹

T ! 0 °C psat

§ 17.08 T · 611 exp ¨ © 234.18  T ¸¹

0 d T d 80 °C

psat





exp ¬ª 23.5771  4042.9 T  37.58 ¼º

The fourth and fifth formula are of interest because they allow calculating of the related temperature for the vapour saturation pressure known. Normally, moist air contains less water vapour then saturation. The relationship between the actual ( Uv) and saturation concentration ( Uv,sat ) at the same temperature, is called relative humidity (RV, I, in %.): I 100

Uv Uv,sat

(2.54)

Implementing (2.50) gives: I 100

Uv Uv,sat

100

p psat

(2.55)

165

2.3 Vapour transfer

or, relative humidity also equals the ratio between vapour pressure and vapour saturation pressure at the same temperature. For example, in Figure 2.18, relative humidity in Q [T = 20 °C, p = 1169 Pa] is 50%. All points with relative humidity 50% lie on a curve passing Q, similar in shape to the saturation curve, but at a 0.5 ratio. Equation (2.54) and (2.55) show relative humidity cannot exceed 100%. Indeed, for each temperature, related saturation pressure and saturation concentration ( Uv,sat ) are the highest attainable. If an attempt is made to go beyond, the pixel representing the moist air will stop at saturation and, condensation will start, meaning vapour will precipitate as water while releasing the heat of evaporation (lb). Evaporation of water, in turn, requires that heat be supplied. For water at 0 °C, the heat of evaporation (lbo) equals 2.5 · 106 J/kg. Among relative humidity, air pressure and air temperature, a simple relation exists. Suppose an air mass at relative humidity I1, air pressure Pa1 and temperature T1. Assume the air pressure changes from Pa1 to Pa2 and temperature from T1 to T2, while the quantity of water vapour in the air volume remains constant. Relative humidity I2 in the new situation then becomes: I2

I1

psat1 Pa2 psat2 Pa1

The proof is as follows. Because the quantity of dry air and water vapour does not change, the ratio mv / (m1 + mv) also does not change. In the situation [Pa1, T1] we have:

mv ml  mv

p1 V1 R T1 Pa1 V1 Ra T1

p1 Ra Pa1 R

I1 psat1 Ra P R a1 

(4)

In the situation [Pa2, T2] that ratio becomes:

mv ml  mv

p2 V2 R T2 Pa2 V2 Ra T2

p2 Ra Pa2 R

I2 psat2 Ra Pa2 R  

(4)

In both formulae, the moist air gas constant Ra is (mv R + ml Rl ) / ma. Equating (4) in both proves the relation. Under atmospheric conditions, all changes of state are isobaric (Pa1 = Pa2), converting the relation into: I2

I1

psat1 psat2

(2.56)

That simple equation shows what happens with relative humidity when temperature changes. Because the vapour saturation pressure increases with temperature, cooling will achieve an increase, warming a decrease in relative humidity. For example, raising the temperature of the cold ventilation air in winter to § 20 °C means a lower relative humidity, while cooling that air in summer gives a higher relative humidity indoors than outdoors.

166

2 Mass Transfer

For a correct quantification of any mixture of two gases, three variables of state must be known. For moist air, the obvious choices are total air pressure, vapour ratio, and temperature. In most building physics applications, total air pressure deviates hardly from atmospheric pressure. Thus, two variables remain. Put another way, the relative humidity does not suffice to describe the state of moist air. The only thing known at that moment is the point representing the air in a diagram of state lies on a curve of equal relative humidity. In order to locate that point and to describe the dry air/vapour mixture, we also need temperature, vapour pressure, vapour saturation pressure or any other variable of state. 2.3.1.4 Changes of state in humid air Let us go back to Figure 2.18. From point Q, there are an infinite number of possibilities to reach saturation. Two are specific: y If, without change in air temperature, vapour is added, Q moves parallel to the vapour pressure axis to Qc on the saturation curve. Continuing to inject vapour will block Q in Qc, meaning that the vapour pressure will remain equal to the vapour saturation pressure psat, Q c and that the extra will condense. This change of state is called isothermal. y Instead of adding vapour, the air temperature is lowered. Now, Q moves parallel to the temperature axis to reach Qs on the saturation curve. At this point, the temperature touches a value Td. Any further decrease gives condensation with Qs descending along the saturation curve. The related difference in saturation pressure ( psat, Q  psat, Q c ) drives the change from vapour to liquid, without telling anything about the kinetics. That change of state is called isobaric with the temperature Td representing the dew point of the air. Figure 2.18 shows all points on the same parallel as Q have identical dew point. Thus, just as for relative humidity, the dew point alone does not suffice to describe moist air. We need a second parameter. Mathematically, the dew point is found as follows: the relative humidity for each point on Q Qs is IQ psat,Q / 100. When the temperature equals the dew point, relative humidity reaches 100%, meaning that psat, Q cc IQ psat, Q / 100 psat, Q . In other words, the dew point is the temperature for which the actual vapour pressure becomes saturation pressure (psat (Td ) = p). Each time temperatures fall below the dew point, condensation occurs. An isobaric change of state explains in part phenomena such as soiling, mould growth, and surface condensation (see below). Real changes of state consist of a combination of isothermal and isobaric, i.e., both humidification or vapour removal and temperature change. 2.3.1.5 Enthalpy of humid air The enthalpy of air, containing x kg of water vapour per kg of (dry) air, is given by (reference temperature 0 °C): h

ca T  x cv T  lbo

(2.57)

where ca and cv are the isobaric specific heat of dry air (§ 1008 J/(kg · K)) and water vapour (§ 1860 J/(kg · K)), and lbo is the heat of evaporation at 0 °C.

2.3 Vapour transfer

167

2.3.1.6 Measuring air humidity Measuring the presence of water vapour in the air can be done directly or indirectly. Directly y Mirror dew point meter. Measures the temperature and the dew point y Psychrometer. Measures the dry bulb (T) and wet bulb air temperature (Tw). Air at dry bulb temperature T and relative humidity I reaches the wet bulb value when cooled by adiabatic saturation. Knowing the dry and the wet bulb temperature allows deducing all other variables of state. For example vapour pressure: p

psat  T n  66, 71  T  T n

(2.58)

Indirectly y Hair hygrometer. Determines the relative humidity from the hygric expansion or contraction of a hair bundle. y Capacitive meter. Determines the relative humidity from the change in dielectric constant of a hygroscopic polymer An instrument widely used to analyse the moist air changes of state is Mollier’s diagram, of which several versions exist (ASHRAE, VDI). 2.3.1.7 Applications Vapour balance indoors In spaces without full air conditioning, the heat, air, vapour balance determines vapour pressure and relative humidity. Here, we limit the analysis to the simple case of a space at constant temperature, without sorption-active surfaces (with, is discussed later), where the ventilation air comes from outside or from neighbouring spaces and mixes ideally with the air present. In such a case, air temperature Ti, vapour pressure pi and relative humidity Ii in the middle of the space, 1.8 m above floor level, describe the overall air condition. First, we quantify the airflows between that space, neighbouring spaces and the outside (see the air transfer at building level). Then, conservation of mass for vapour applies: vapour entering the space with the inflowing air, in addition to the vapour released or removed in that space (GvP) including vapour added by surface drying or withdrawn by surface condensation (Gvc/d ) equals the water vapour removed by the out-flowing air (xvi 6 Gai, j) and by diffusion across the fabric, in addition to the water vapour stored in the room air (dxvi Ml / dt). In case the inflow comes from outside and withdrawal by diffusion is negligible, the balance is (Figure 2.19):

Figure 2.19. Vapour balance in a space. Storage in the air is not shown.

168

2 Mass Transfer

xve Ga  GvP  Gvc/d

xvi Ga 

d  xvi M l dt

(2.59)

The air mass Ml in the room equals Ua V with V room volume in m3. The production term GvP includes vapour from people, animals and plants, from activities such as cooking, washing, and cleaning, from evaporation of exposed water surfaces (an indoor swimming pool) etc. Sometimes it also contains the water vapour released by construction parts wetted by rising damp, rain, or building moisture. When neither surface condensation nor surface drying occurs (a case we discuss in the section on water vapour surface film coefficients), then Gvc/d = 0. The ideal gas law now becomes: xve Ga

Uve Uae  n V Uae

pe n V R Te

xvi Ga

Uvi Uai  n V Uai

pi n V R Ti

while the storage term becomes: d  xvi Uai V dt

§U · d ¨ vi Uai V ¸ © Uai ¹ dt

V dpi R Ti dt

In these equations, n is the ventilation rate, i.e. the air supplied per time unit, expressed as number of space volumes exchanged, pe the vapour pressure outside, pi vapour pressure inside, Ti the inside temperature in K and R the gas constant of water vapour. The supply undergoes two changes of state: warming from Te to Ti and humidification or dehumidification from xve to xvi. Warming is then: xve Ga

pe n V R Ti

These transformations convert equation (2.59) into a differential equation of first order with the inside vapour pressure as a dependent variable: pe 

R Ti GvP nV

pi 

1 dpi n dt

(2.60)

Solutions In steady state (GvP = C te, n = C te, pe = C te, dpi / dt = 0) – also called average regime, the inside vapour pressure and relative humidity become: pi

pe 

R Ti GvP nV

Ii

100

pi psat,i

pe R Ti  GvP psat,i psat,i n V

(2.61)

With vapour released inside, the average vapour pressure never drops below the average value outside. At the same time, the excess increases proportionally to the average release and inversely in proportion to the average ventilation rate. This means the additional benefit of ventilation slows down quickly when the rate increases (Figure 2.20).

169

2.3 Vapour transfer

Figure 2.20. Effect of better ventilation on vapour pressure excess inside (space of 400 m3, daily vapour release (GvP) 14 kg, 0 °C outside, 580 Pa outside, 20 °C inside).

In a transient regime, various cases demand consideration: 1. Vapour release inside increases suddenly while vapour pressure outside and ventilation rate remain constant (pe = C te, n = C te). Initial conditions then are: t < 0: GvP = GvP1, t • 0: GvP = GvP2. The solution is: pi





pi f  pi 0  pi f exp   n t

(2.62)

where pi 0 is vapour pressure inside for t = 0 and pi ’ steady state vapour pressure inside for a vapour release GvP2. After a sudden increase or decrease in vapour release, vapour pressure inside lags behind, especially when the ventilation is high (the time constant of the exponential is 1/n). Or, less ventilation seems suddenly positive. The higher end value reached of course completely annuls this advantage (Figure 2.21).

Figure 2.21. Inside vapour pressure, sudden increase in vapour release. Inertia effect because of air buffering (space of 400 m3, daily vapour release (GvP) 14 kg, 0 °C outside, 580 Pa outside, 20 °C inside).

170

2 Mass Transfer

2. Vapour pressure outside changes suddenly without any variation in ventilation rate and vapour release inside (n = C t, GvP = C t). Initial conditions now are: t < 0 pe = pe1, t • 0: pe = pe2. The solution does not differ from (2.62). Now, pi 0 is the vapour pressure inside at the moment t = 0, while pi ’ is the steady state vapour pressure inside for a value pe2 outside. Vapour release time lag remains. 3. The ventilation rate remains constant (n = C t) but vapour release inside and vapour pressure outside change periodically. The solution in such a case looks like: pi

pi 

R Ti GvP · 1 § ª § 2 i S t ·º pe  Re «exp ¨ ¸ ¨ ¸ © Z © nV ¹ T ¹ »¼ ¬

(2.63)

with pi the vapour pressure inside at average vapour release indoors and average vapour pressure outdoors, pe complex vapour pressure outdoors, GvP the complex vapour release inside and Z the complex damping (the term complex is used to pinpoint the analogy with complex temperatures). The three are given by: 1

Z

ª § 2 S ·2 º 2 «1  ¨ ¸ » «¬ © n T ¹ »¼

pe

pˆ e exp i M pe



arg  Z



GvP

§2S· atan ¨ © n T ¸¹



Gˆ vP exp i MGvPe



where Z, pˆ e and Gˆ vP are the amplitudes of the damping factor, the vapour pressure outdoors and the vapour release indoors. M pe is the time lag vapour pressure outdoors experiences, MGvPe the time lag vapour release indoors experiences and arg (Z) the time shift. Compared to the average response (Z = 1, arg (Z ) = 0), periodical changes in vapour release indoors and vapour pressure outdoors give a dampened and time shifted vapour pressure indoors. Both damping and time shift increase with decreasing ventilation rate. 4. If all parameters are variable, a finite difference calculus gives the solution. Dust mites, mould, and surface condensation The three are connected to the relative humidity at surfaces. The number of dust mites colonizing a surface increases explosively when relative humidity passes 52 + 1.2 (Ts – 15) for quite a long time (for Dermatophagoides farinae). Mould risk nears one when relative humidity passes 80% at a four-week mean basis, while surface condensation occurs each time relative humidity at a surface touches 100%. When all three happen indoors, things turn nasty. So, in case of ideal air mixing, mould will develop when the four-week mean vapour pressure indoors ( pi ) passes 0.8 times the monthly mean saturation pressure ( psat,s) at a surface. For surface condensation to happen, vapour pressure inside has to equal that saturation pressure or, put another way, surface temperature must pass the dew point indoors (Figure 2.22). Vapour saturation pressure at an inside surface now depends on surface temperature, which for a spot anywhere on the envelope is given by: Ts

T e  f h i  Ti  T e

(2.64)

with f h i local temperature factor. Clearly, surface temperature and vapour saturation pressure decrease and mite, mould and surface condensation risk increase when:

2.3 Vapour transfer

171

Figure 2.22. Surface condensation on the ceiling of a water closet.

Figure 2.23. Edge between concrete low-sloped roof and two non-insulated outer walls, acting as a thermal bridge. The consequence is mould in the inside corner.

1. It is colder outdoors (Te lower) 2. One heats less (Ti lower) 3. The temperature factor at the spot considered is lower ( f h i lower) While the temperature outdoors is weather-related and beyond human control, the temperature indoors in moderate and cold climates depends on heating habits. Thermal quality of the enclosure at the spot considered in turn defines the temperature factor (How well insulated is a potential thermal bridge? See Figure 2.23), as do the ratio between the outer and total assembly and partition surface in the space (which defines the radiant part in the surface film coefficient hi ) and cupboards or other furniture against outside walls (lowers the surface film coefficient hi ). According to equation (2.61) vapour pressure indoors (pi ) also changes with vapour pressure outdoors, vapour release indoors, and ventilation. The higher the vapour pressure indoors, the greater the likelihood of mite overpopulation, mould problems and long lasting surface condensation. Negative boundary conditions are: 4. Higher vapour pressure outdoors. This is the case during summer in moderate and cold climates. Of course, temperature outdoors is also higher then 5. More water vapour released indoors (GvP / V higher) 6. On average, less ventilation (n lower)

172

2 Mass Transfer

Again, no human has an impact on vapour pressure outdoors. The number of users and their living habits however mainly determine water vapour release, as does the function the building has (dwellings, offices, swimming pool). The ventilation rate finally depends too often on infiltration and airing habits, although purpose designed ventilation is a better choice. Classification of the six factors in terms of their links with building design gives: y Factors 1 and 4? Outdoor climate. Not much can be done about this y Factor 5? Defined by living habits and function. Not much can be done about this. y Factor 2 and 6? Partly defined by living habits, partly by design and construction. An energy wasting design almost automatically leads to lower inside temperatures, simply because comfortable heating is so expensive, direct rebound behaviour goes for less. If ventilation is not purpose designed, ventilating adequately becomes a random variable y Factor 3? Mainly design and workmanship defined. Think about insufficient thermal insulation that is carelessly mounted without any attention given to thermal bridging and air tightness.

2.3.2

Water vapour in open-porous materials

2.3.2.1 Overview Under atmospheric circumstances, air fills all open pores in a dry porous material. Thus, it seems logical to apply the theory for humid air to the air volume ©o V, with ©o open porosity and V the material volume in m3. The amount of water vapour in the open pores and the moisture content in equilibrium with the relative humidity in the air would then be: Gm

Uv © o V

p ©o V 462 T

wH

Gm V

p ©o 462 T

The equilibrium moisture content (wH ) should thus vary linearly with relative humidity, whereas temperature dependence should reflect the vapour saturation curve and rise with increasing temperature. To give an example, concrete has an open porosity of approximately 15%. At 20 °C and 65% relative humidity, 1 m3 should have a moisture content of: wH

Gm V

p ©o 462 T

0.15 ˜ 2340 462 ˜ 293.16

0.0017

kg m3

At 50 °C and 65% of relative humidity, the moisture content should be: wH

Gm V

p ©o 462 T

0.15 ˜ 12, 335 462 ˜ 293.16

0, 0081

kg m3

The reality, however, is completely different. At 20 °C and 65% relative humidity, moisture equilibrium in 1 m3 of concrete reaches 40 to 50 kg! Furthermore, it does not vary linearly but S-shaped with relative humidity, while a higher temperature results in a very limited drop instead of an increase in moisture content. It looks as if the behaviour of the air/vapour mixture changes drastically in the pores of an open-porous material.

173

2.3 Vapour transfer

2.3.2.2 Sorption isotherm and specific moisture ratio As a function of relative humidity a series of equilibrium moisture contents which lie on an S-curve, called the sorption isotherm, with a hysteresis between humidification and dehumidification, characterize each material. See Figure 2.24.

Figure 2.24. Sorption isotherm. The two mechanisms that bind water molecules are (1) adsorption of water molecules at the pore walls at low relative humidity, (2) capillary condensation on the water menisci formed by the adsorbed layers at higher relative humidity.

Each point on and between the two curves represents a possible equilibrium moisture content at the relative humidity abscissa value. What the value will be, depends on the moisture history of the material. At any rate, the equilibrium at 98% is called the maximum hygroscopic moisture content. Each material has its own sorption isotherm. Nevertheless, those showing hardly any sorption, except at a very high relative humidity, are called non-hygroscopic. Examples are bricks, synthetics, most insulation materials, etc. Instead, the ones showing considerable sorption at low relative humidity are called hygroscopic. Examples are cement bonded and timber-based materials. The literature gives mathematical expressions for the sorption isotherm. Although none is universal, it seems that, provided the roots of the denominator lie outside the interval [0, 1], the following one applies for a relative humidity below 95%: wH

I aH I  bH I  cH 2

(2.65)

with aH, bH and cH material-specific constants that differ between sorption and desorption. Another expression, applicable above 20% relative humidity, is:

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2 Mass Transfer

ª ln  I º wc «1  b »¼ ¬

wH



1 c

(2.66)

where wc is the capillary moisture content and b, c are material-specific constants. When the sorption isotherm is known, specific moisture ratio can be calculated, a property representing the amount of moisture a material absorbs or desorbs per unit change in relative humidity (on a scale from 0 to 1). As a formula: dX H dI

[I

(2.67)

with XH moisture ratio. Units kg/(kg · RH). The specific moisture ratio is comparable to the specific heat capacity. It fixes the moisture storage per kg of dry material at any relative humidity. Multiplication by density gives the equivalent of the volumetric specific heat capacity: U [I

dwH dI

(2.68)

where wH is hygroscopic moisture content. U [I is called the specific moisture content, units kg/(m3 · RH). Both properties vary with relative humidity as differentiating the equations (2.65) and (2.66) shows: U [I

dwH dI

§c · wH2 ¨ H2  aH ¸ ©I ¹

U [I

dwH dI

wH cbI

ª ln  I º «1  b » ¬ ¼

1

2.3.2.3 Physics involved Water molecules that diffuse into the pores and stick to the pore walls build up hygroscopic moisture. At low relative humidity (0 to 40%), molecular adsorption fixes the molecules. That explains why the hygroscopic curve is convex there. At high relative humidity (40 to 100%): capillary condensation at the water menisci formed in the pores by the adsorbed water layers makes the curve gradually concave (see Figure 2.24). Molecular adsorption occurs in two steps. First, we have monolayer adsorption of water molecules. The Langmuir equation describes that process: wH

M w Ap

CI Aw N 1  C I

2, 62 ˜ 107 Ap

CI 1 C I

(2.69)

where Mw is the mass of 1 mol of water (0.018016 kg), Aw the surface taken by 1 water molecule (11.4 · 10–20 m2), N the Avogadro number (6.023 · 1023 molecules/mol), Ap the specific pore surface in the material (m2/m3), I relative humidity on a scale from 0 to 1 and C the heat exchange between a water molecule and the pore wall (J/kg), given by: C

§l  l · k exp ¨ a b ¸ © RT ¹

with k the adsorption constant, la the heat of adsorption (J/kg), lb the heat of evaporation (J/kg) and R the gas constant of water vapour.

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2.3 Vapour transfer

Once the monolayer layer is formed at a relative humidity of § 20%, multi-layer adsorption, described by the BET equation, starts: wH

2, 62 ˜ 107 Ap

CI 1 I

ª1   n  1 In  n In 1 º « n 1 » «¬ 1  C  1 I  C I »¼

(2.70)

with n the number of water molecule layers. Both the Langmuir and BET equation show hygroscopic moisture content at low relative humidity increases with heat exchange and specific pore surface, although a higher temperature tempers the increase somewhat (heat exchange C lower). For a given open porosity ©o, a higher specific pore surface demands more very small pores. Total open porosity of sand-lime stone for example is about the same as for medium-weight bricks (§ 33%). However, the average pore diameter is only 0.1 —m, while for bricks it is 8 —m. Sand-lime stone therefore has a § 6000 times larger specific pore surface than brick and is, as measurements confirm, far more hygroscopic at low relative humidity. Capillary condensation starts in the smallest pores, when the adsorbed water layers touch each other. At that moment, surface tension rearranges the molecules into the more stable form of a water-filled capillary with a meniscus at each end. Vapour saturation pressure above a water surface now not only depends on temperature but also on the kind of meniscus formed. Indeed, for a molecule, escaping from a concave meniscus is more difficult than from a flat one. The reverse is true for a convex meniscus. Therefore, the vapour phase above a concave meniscus contains less, above a convex meniscus more molecules than above a flat meniscus. Conversely, condensation on a concave meniscus happens for fewer molecules in the vapour phase than condensation on a flat meniscus. Water deposition will thus start at a relative humidity below 100%, while a convex meniscus needs a relative humidity above 100%. In equilibrium, Thompson’s law describes the phenomenon: psat c

ª V cos - § 1 1 · º psat exp «  w ¨  ¸» ¬ Uw R T © r1 r2 ¹ ¼

(2.71)

with pcsat saturation pressure above the curved meniscus, psat saturation pressure above a flat meniscus, Uw density of water, Vw surface tension of water, - the contact angle between the meniscus and the pore wall and r1 and r2 the curvature radii of the meniscus (see capillarity). If we consider each pore circular with diameter deq, then (2.71) simplifies to psat c

§ 4 V w cos - · psat exp ¨  ¸ © Uw R T d eq ¹

Equating the ratio psat c / psat to relative humidity (I) gives (see Figure 2.25): ln  I



4 V w cos Uw R T d eq

(2.72)

The wider the pores, the higher the relative humidity needed for capillary condensation. Under 20%, the equivalent diameter deq drops below 10–9 m, i.e., below the sphere of influence of a water molecule. In such small pores, molecules no longer behave statistically and the condensation theory loses sense. At 100% relative humidity, the equivalent pore diameter turns infinite. All open pores in a material should then be filled with water, which means saturation.

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2 Mass Transfer

Figure 2.25. Thompson’s law at a temperature of 20 °C.

Hence, air in the pores hinders this. The moisture content will stay at a value, which prohibits air from escaping the porous system. We named that value ‘capillary moisture content’. However, when a 100% relative humidity situation continues for decennia, dissolution of entrapped air will slowly increase moisture content, up to saturation. Temperature of course also plays. The lower, the lower the relative humidity at which capillary condensation starts. The term –4 V w cos (-) / deq in Thompson’s law equals capillary suction pc (or s) in a circular pore with diameter deq. Thus, (2.72) also writes as: ln  M

pc URT

(2.73)

which shows that an unequivocal relationship exists between two apparently different quantities: capillary suction and relative humidity. In a pore with equivalent diameter deq, capillary condensation starts when the relative humidity in the environment equals 100 psat c / psat . That process should be understood as follows: in all pores with an equivalent diameter smaller than or equal to deq (I), condensation occurs, in all pores with an equivalent diameter larger than 2 deq (I), multi-layer adsorption continues whereas in pores with an equivalent diameter in between, condensation on the adsorbed layers develops. The large share of wide pores in the total pore volume causes the very strong increase in hygroscopic moisture content beyond a relative humidity of 90%. Hysteresis between sorption and desorption finally has many causes. Among others, water uptake and release are so slow testing may stop too early, resulting in pseudo equilibriums with too low moisture content in sorption and a too high value in desorption. Also, the interaction between the liquid and the air in the pores differs during sorption and desorption. Indeed, testing in a vacuum gives a much smaller hysteresis. Further, sorption gives other menisci then desorption, mainly because the contact angle - at the pore walls changes a little between both modes.

177

2.3 Vapour transfer

2.3.2.4 Impact of salts With salts in the pores, sorption augments. This is a direct consequence of the salt-specific vapour saturation depression which a saturated salt solution causes: Saturated salt solution

Equilibrium-RV (%)

MgCl2 NaCl KaCl

33 75 86

Thus, with NaCl, a sudden increase in hygroscopic moisture content will show-up at 75%. As a result, masonry, wetted by seawater, hardly dries in a climate where the average relative humidity stays above 75%. A mixture of different salts lifts the whole hygroscopic curve. Conversely, increased sorption is used to demonstrate the presence of salts in a porous material. 2.3.2.5 Consequences Sorption/desorption is an important factor looking to the hygric response of materials and assemblies. In open-porous materials, relative humidity is the factual moisture potential with the sorption isotherm as a link to the moisture content. Sorption further induces hygric inertia with the specific moisture content playing the same role in hygric processes as the volumetric specific heat capacity in thermal processes. A favourable effect of hygric inertia is its inhibitory impact on so-called interstitial condensation. What a steady-state calculation assumes for condensation often is only an increase in hygroscopic moisture content. Hygroscopic moisture has also undesired effects. Material dimensions depend on it and, thus, on relative humidity. Decreasing relative humidity causes shrinkage, increasing relative humidity expansion. Wood-based materials are very sensitive to that, but also cement bonded and even burnt materials react that way. For thin layers, hygric expansion and shrinkage causes most of the cracking. Above a relative humidity of 75 to 80%, the amount of water adsorbed at surfaces suffices to stimulate biological activity.

2.3.3

Vapour transfer in the air

Suppose the dry air and vapour concentrations in moist air are Uda (= ml / V ), respectively Uv (= mv / V ). Moist air concentration is then: mda  mv Uda  Uv V Convection now is the main driver behind moist air movement. Let va be the moist air velocity. When moving, the dry air and vapour component may have their own velocities, different from the moist air. With vda the dry air and vv the vapour velocity, the relationship between the three becomes: Ua

va

Uda vda  Uv v v Uda  Uv

Uda vda  Uv v v Ua

The dry airflow rate is gda = Uda vda, the vapour flow rate gv = Uv vv and the moist airflow rate ga = Ua va, the sum of the dry air and vapour flow rates. A combination of two displacement modes causes the difference in velocities:

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2 Mass Transfer

y Bulk flow, provoked by convection Vapour: gv1

U v va

Dry air: gda1

Uda va

y A diffusive flow coupled to concentration equalization by Brown’s movement in mixtures of liquids or gases. Diffusion in moist air starts whenever differences in vapour to dry air ratio exist. Diffusive flows steer many processes, such as carbonatation of lime plaster, mortar and concrete (CO2 /moist air), the lime to gypsum transformation (SO2 /moist air) and chlorine corrosion in concrete (Cl2 /moist air). In a right-angled coordinate system, which moves along at the same velocity va as the moist air, Fick’s empirical law for diffusion holds: Vapour: gv2

§U ·  Ua Dv,da grad ¨ v ¸ © Ua ¹

Dry air: gda2

§U ·  Ua Dv,da grad ¨ da ¸ © Ua ¹

In both formulas, Dv,da is the binary diffusion coefficient vapour/dry air, units m2/s, given by Schirmer’s equation: 1,81

Dv,da

2.26173 § T · ¨© ¸ Pa 273.15 ¹

(2.74)

For a moist air pressure of 1 Atmosphere (1 Bar, 101,300 Pa) that relation converts into Dv,da = 8.69 · 10–10 T 1.81. Total vapour flow rate gv thus becomes: gv

gv1  gv2

Uv va  Ua Dv,da grad

Uv Ua

In that sum, convective usually dominates. A diffusive vapour flow in one direction now generates an equally large opposite diffusive dry airflow. So, if diffusion is the only mode, neither resulting moist air transfer nor a change in moist air concentration is noted. At atmospheric conditions and temperatures below 50 °C (Pa / p, the sum p /R + pda / Rda is close to constant. That simplifies the equation for the total vapour flow rate:

gv

p Dv,da § p pda · p R  ga  grad p pda Ua R T T ¨© R Rda ¸¹  R Rda

gv |

Dv,da ga p grad p Ua R T RT

or: (2.75)

Equation (2.75) describes vapour transfer in moist air under atmospheric conditions, with vapour pressure p as driving force. Vapour flow rate thus depends on the driving force and its gradient, as in combined heat conduction and enthalpy flow. Two specific cases are worth considering:

179

2.3 Vapour transfer

1. Convection and diffusion push vapour in the same direction, while the dry airflow rate remains zero. In such case the moist airflow rate ga consists of vapour only, converting (2.75) into (§ if Ra § Rda): gv



Dv,da pa 1 Dv,da grad p |  grad p Uv R T R T pa  p 1 Ua

Evaporation through the boundary layer above water is an example of that (Figure 2.26).

Figure 2.26. Evaporation from a water surface (ga = 0, gv > 0).

2. Pure diffusion, which demands a moist airflow rate ga zero. Vapour flow and dry airflow rate then become: Vapour: gv | 

Dv,da RT

Dry air: gv | 

grad p

Dv,da Rda T

grad pda

(2.76)

To keep the analogy with heat and air transport, the ratio Dv,da / (R T) is called the vapour permeability of air, symbol Ga , units s. The vapour flow rate in general and for the two specific cases then becomes: gv

ga p  Ga grad p Ua R T

gv

 Ga

Pa grad p Pa  p

gv

 Ga grad p

The last expression resembles Fourier’s law for heat conduction. We just have to replace vapour permeability Ga by thermal conductivity O and vapour pressure p by temperature T. However, while thermal conductivity could be taken as a constant, vapour permeability is an explicit function of total air pressure and temperature.

2.3.4

Vapour transfer in materials and assemblies

Moisture transport only develops in open-porous materials. At low moisture content, displacement happens mostly, if not only, in the vapour phase. For non-capillary materials, vapour transport is even the only mode until saturation. In other words, talking about vapour flow in materials makes sense. 2.3.4.1 Flow equation At first sight, vapour flow in an open-porous material should not differ from air. However, displacement is only possible now in the pores accessible for water vapour molecules. At the same time, convection plays a much smaller role, while hygric buffering is of great importance. As in air, the much smaller convective vapour flow rate becomes:

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2 Mass Transfer

gv1

ga p Ua R T

(2.77)

while the vapour diffusion (T < 50 °C, Pa / p) looks like: gv2

 G grad p

(2.78)

with G the vapour permeability of (the porous system in) the material (units: s). The part per unit surface of material occupied by open pores varies from smaller to much smaller than the unit surface. Thus, vapour flow rate across a unit surface of material must be less than across a unit surface in air, or, vapour permeability is surely lower than for air. Instead of vapour permeability, Krischer introduced the vapour resistance factor P as material characteristic. It indicates how many times the vapour permeability of a material is lower than in stagnant air, at the same conditions of temperature and total pressure. In a formula: P

Ga [] G

(2.79)

The lowest value of the vapour resistance factor is one, the highest infinite (for a material with only closed pores or no pores at all). The actual value depends on the pore system. The following factors come into play: (1) Open pore area Ao per unit surface of material. A larger open pore area lowers while a smaller one heightens the vapour resistance factor (Figure 2.27). Thus: P y 1 / Ao. For a zero open pore area, vapour resistance factor turns infinite. In such material, neither diffusion nor convection plays.

Figure 2.27. The P-value increases when the open-pore area per unit-surface of material decreases.

(2) Path length lo. The average distance the diffusing molecules have to travel in the pores compared to the thickness of the layer (do). The higher the ratio lo / do, the higher the diffusion resistance factor. The closer the ratio is to one, the smaller the diffusion resistance factor (Figure 2.28). Hence: P y lo / do.

Figure 2.28. P-value increases with path length lo.

181

2.3 Vapour transfer

(3) Tortuosity or ‘deviousness’ of the pore system (Figure 2.29).

Figure 2.29. The P-value increases with tortuosity.

In general, the vapour resistance factor equals the ratio between tortuosity ©l and total open porosity ©o: P

©l ©o

For hygroscopic materials, relative humidity in the pores also plays. A hygroscopic material in fact does not experience pure Fickian diffusion. In pores with a diameter hardly larger than the free path length of a water molecule, friction diffusion, called Knudsen diffusion, occurs. Moreover, at high relative humidity, water transfer in the adsorbed water layers and in the pores filled by capillary condensation adds to diffusion. With increasing hygroscopic moisture content (i.e. with increasing relative humidity), diffusion in the pores also turns into diffusion between the water islet (Figure 2.30), shortening the diffusion path length that way. The three together give a decrease of the vapour resistance factor at higher relative humidity (in fact: moisture content). The first two plus changes in pore structure link the value to temperature.

Figure 2.30. Diffusion length shortened by capillary condensation.

Implementing the vapour resistance factor in the diffusion equation gives: gv2



Ga grad p P

Yet, water vapour permeability of air (Ga) is a very small number (§ 1.87 · 10–10 at 20 °C). Since small quantities give the impression the phenomena they reflect are unimportant, the inverse number, called the diffusion constant, is used (symbol N, units s–1, see Figure 2.31): gv2



1 grad p PN

(2.80)

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2 Mass Transfer

Figure 2.31. Diffusion constant N as a function of temperature and air pressure (upper curve: Pa = 110,000 Pa, middle curve: Pa = 101,300 Pa, lower curve: Pa = 90,000 Pa, lowest curve: Pa = 80,000 Pa).

2.3.4.2 Conservation of mass Mass conservation for water vapour gives (with vapour content converted into vapour pressures): div gv r Gcc



w § ©o p · wt ¨© R T ¸¹

(2.81)

Whenever vapour condensates, evaporates, or sublimates, the source term Gcc differs from zero. On the right, open porosity ©R decreases when water vapour condenses. If water present evaporates, it increases. For a relative humidity permanently below 100%, the source term becomes zero. In hygroscopic materials, changes in hygroscopic moisture encompass sources, sinks and vapour content. If we assume isothermal conditions and a relative humidity below a value IM (< 100%), which marks the end of combined vapour transfer and surface flow, the mass equilibrium may be written as: div gv



wwH wt

 U [I

wM wt

(2.82)

At a relative humidity beyond IM (exact value is unknown, 95 to 98% is often postulated, even though measurements may suggest a lower threshold), water flow develops, which means (2.82) is no longer applicable. 2.3.4.3 Vapour transfer by ‘equivalent’ diffusion The term ‘equivalent’ emphasizes that so-called diffusion in porous materials combines molecular and friction diffusion, surface flow and water transfer in the small pores, filled by capillary condensation. Because the vapour phase dominates, we describe that complex reality by the Fickian diffusion law. Whenever convection is negligible, that kind of diffusion

183

2.3 Vapour transfer

provides a model to evaluate ‘vapour’ transfer across assemblies. Of course, no convection presumes the assembly does not contain air layers, shows no cracks, etc., a restriction commonly overlooked but limiting the applicability of the diffusion approach to compact, barely air permeable assemblies. For that, at least one layer in the assembly should have a very low air permeance, i.e. consist of a material without open pores or containing micro-pores only. Unless they otherwise demonstrate high open porosity, such materials are also vapour tight. Additionally, air pressure differences across assemblies are best absent, an assumption that is hypothetical, because different temperatures at both sides of a vertical or inclined assembly automatically activate air pressure gradients. Hence, so-called diffusion is in no way the rule. Therefore the following paragraphs have less a practical than a theoretical value. This is even more true because the approach assumes each assembly to be dry at the start, while in reality chemical reactions, wind driven rain, melting snow and mixing water may push the initial relative humidity in the layers beyond the value IM. Steady state In steady state, the derivative ˜wH / ˜t becomes zero. That converts (2.82) into div gv = 0. Introducing the diffusion equation then gives: § 1 · div ¨ grad p ¸ ©PN ¹

0

One dimension: flat assemblies In flat assemblies, the equation further simplifies: d § 1 dp · dx ¨© P N dx ¸¹

0

Let us assume the vapour resistance factor P is a constant. The diffusion constant N, however, remains an explicit function of temperature and total air pressure. Under atmospheric conditions, temperature still might vary. This obliges us to differentiate between isothermal and non-isothermal situations. When isothermal, the diffusion constant N also takes a fixed value. Single-layered assembly, isothermal conditions A constant N allows simplifying the equation to d2 p / dx2 = 0 with a solution of: p

C1 x  C2

If the vapour pressures at both surfaces ( ps1, ps2 ), the thickness of the assembly (d) and the vapour resistance factor of the material ( P) are known, then from the boundary conditions x = 0: p = ps1, x = d: p = ps2, ps1 < ps2 (Figure 2.32) we get: C2 = ps1, C1 = ( ps2 – ps1) / d. The vapour pressure in the assembly thus becomes: ps2  ps1 x  ps1 d As vapour flow rate (gv), we get: p

gv



1 dp P N dx



ps1  ps2 , PN d

(2.83)

absolute value: g v

ps2  ps1 PN d

(2.84)

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2 Mass Transfer

Figure 2.32. Diffusion under isothermal conditions in a single-layered assembly.

Thus, the steady state vapour pressure in a single-layered assembly in isothermal regime changes linearly, while the vapour flow rate is a constant, proportional to the vapour pressure difference and inversely proportional to thickness and vapour resistance factor. We call the product P N d the diffusion resistance of the assembly, symbol Z, units m/s. The larger the diffusion resistance, the smaller the vapour flow rate. A large diffusion resistance means a large assembly thickness or the use of a material with high vapour resistance factor i.e., one that acts as a vapour retarder! The diffusion constant N in (2.84) equals § 5.4 × 109 s–1. That large number allows concluding diffusion only generates very small vapour flow rates. Displacement of ‘significant’ quantities of water vapour thus requires a lot of time. In other words, diffusion is a very slow process. Composite assembly, isothermal conditions We assume the vapour pressure at the two assembly surfaces ( ps1, ps2 ), the thickness of all layers (di ) and their vapour resistance factors ( Pi ) known. Contrary to heat conduction, the likelihood to have a notable contact resistance Zci in each interface is high now. Its size depends on how layers contact each other (glued, cast, etc.) As a result, each interface i generates two unknown vapour pressures: pi1 and pi2. For each layer and for the contact resistances, the following holds: Layer 1: g v

p11  ps1 P1 N d1

Contact resistance 1:

gv

p12  p11 Z c1

Layer 2: g v

p21  p12 P2 N d 2

Contact resistance 2:

gv

p22  p21 Z c2

Contact resistance n  1:

gv

! Layer n : g v

ps2  pn 1,2 Pn N d n

pn 1,2  pn 1,1 Z c, n 1

185

2.3 Vapour transfer

In these equations, p11, p12, …, pn–1,1, pn–1,2 are the unknown interface vapour pressures and gv is the unknown but constant vapour flow rate. Rearranging and adding gives: gv

ps2  ps1

N

(2.85)

¦  Pi di  ¦ Z ci

The denominator in (2.85) is called the diffusion resistance of the composite assembly, symbol ZT, units m/s. Pi N di is the diffusion resistance and Pi di the diffusion thickness of layer i, the last with units m. The higher the diffusion resistance of an assembly, the smaller the vapour flow rate. Inserting a layer with high diffusion thickness may achieve a large diffusion resistance ZT. If this is a thin foil with high vapour resistance factor (P), the terms vapour retarder or vapour barrier are used (the latter when its diffusion resistance nears infinity). Gluing layers over their whole interface provides large contact resistances. Vapour pressures follow from the layer and contact resistance equations for the vapour flow rate. The algorithm found is: px

ps1  g v Z s1 x

px

ps2  g v Z s2 x

(2.86)

In the [Z, p]-plane, this are straight lines through the points (0, ps1) and (ZT, ps2) with the vapour flow rate gv as slope. The vapour pressure curve in the [x, p]-plane is obtained by transposing the intersections with the interfaces of layers and contact resistances from the [Z, p]- to the [x, p]-plane and connecting the successive intersections with line segments (Figure 2.33). The highest difference in vapour pressure occurs over the most vapour-tight layer or the highest contact resistance, which in such a case acts as a vapour retarder. In the [Z, p]-plane, a composite assembly apparently resembles a single-layered one. Contact resistances Zci are usually unknown and, in practice, ignored. This, however, is only correct for free contacts with an extremely thin air gap between layers.

Figure 2.33. Diffusion under isothermal conditions in a composite assembly. Remark the contact resistances between the different layers.

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2 Mass Transfer

Single-layered assembly, non-isothermal conditions In non-isothermal steady-state conditions, temperature is a linear function of the ordinate x. Therefore, the diffusion constant N, which is a function of temperature, will also depend on x. Vapour flow rate Dividing the assembly into infinitesimally thin layers, each with a thickness dx and a temperature T (x) = T1 + (T2 – T1) x / d gives the vapour flow rate. This being done, we get (Figure 2.34): gv



1 dp P N ¬ª T  x ¼º dx

In steady state, the rate is constant, or: d xº ª g v P ³ N « T1  T 2  T1 » dx d ¬ ¼ 0



ps2

³

dp

(2.87)

ps1

The diffusion constant N is still given by: N

5.25 ˜10 P T

RT Dvl

6

0.81

a

Implementing that function into (2.87) and solving the integrals results in: gv



ps2  ps1

P d 5.25 ˜ 106 Pa F T

(2.88)

with: F T

T

0.19 s2

 Ts10.19



0,19 Ts2  Ts1

Figure 2.34. Diffusion under non-isothermal steady state conditions in a single-layered assembly.

187

2.3 Vapour transfer

Figure 2.35. F (T) as a function of the mean temperature in °C. A straight line (T2 = 0 °C).

Numerically, that expression hardly differs from the integral of a straight line between Ts1 and Ts2, divided by the integration-interval (Ts2 – Ts1) (Figure 2.35). As temperature changes linearly with x (T = aT x + bT), the outcome equals the mean (Tm) of the surface temperatures. The vapour flow rate then becomes: gv

ps2  ps1 P d N T m

(2.89)

That equation resembles the isothermal one, however with the diffusion constant calculated for the average temperature in the single-layered assembly. Vapour pressures Vapour pressures ensues from: x

p

ps1  P g v ³ N  x dx 0

with the integral equal to: 0.19 ­° ª ½° xº 5.25 ˜ 106 Pa d ® «Ts1  Ts2  Ts1 »  Ts10,.9 ¾ d¼ ¯° ¬ ¿°





0.9 Ts2  Ts1

an expression which closely matches the product N (Tm) x, or: p

ps2  ps1 x  ps1 d

(2.90)

As in isothermal regime, vapour pressure changes almost linearly in the assembly (almost, because the product N (Tm) x is only a close approximation of the correct function).

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2 Mass Transfer

Composite assembly, non-isothermal conditions Again, we assume vapour pressure at the assembly surfaces ( ps1, ps2 ), the boundary temperatures (T*e , Ti ) , the thickness (di ), the vapour resistance factor ( Pi ) and thermal conductivity (Oi ) of all layers known. Vapour flow rate For each layer in the assembly, the diffusion constant to be used is the one for the average temperature in that layer. For the contact resistances, it is the contact temperature. The vapour flow rate so becomes: gv

ps2  ps1

¦



ª¬ Pi di N T mi º¼  ¦ Z ci Ti, i 1



(2.91)

As for isothermal conditions, 6 [ Pi di N (Tmi)] + 6 Zci (Ti, i+1) represents the diffusion resistance ZT of the composite assembly, while Pi di N ( Pmi ) is the diffusion resistance Zi of layer i. Vapour pressures Vapour pressure course hardly differs from the isothermal one, provided the diffusion constant per layer equals the one for the layer’s average temperature and the diffusion constant per contact resistance is the one for the contact temperature. As an algorithm: px

ps1  g v Z s1 x

px

ps2  g v Z s2 x

(2.92)

In the [Z, p]-plane, vapour pressures again lie on a straight line through the points (0, ps1) and (Z, ps2) with the vapour flow rate gv as slope. The curve in the [x, p]-plane is found by transposing the successive intersections at all from the [Z, p]- to the [x, p]-plane and connecting them with line segments. The unknown contact resistances Zci are currently set at zero. The vapour pressure line calculated that way, however, is physically right only if it does not pass saturation pressure in the assembly. Intersection in fact changes things (Figure 2.36 (1)). A phenomenon called ‘interstitial condensation’ then occurs, a situation demanding further analysis. Interstitial condensation That analysis starts by redrawing the assembly in the [Z, p]-plane (as said, contact resistances are set at zero). The temperature course is calculated, related saturation pressures traced and the vapour pressures at both surfaces connected with a straight line. If intersecting saturation pressure, a fictitious situation appears. In fact, between the crossings, vapour pressures pass saturation, which is impossible! The correct solution follows from conservation of mass. If the construction part is initially dry, then the mass balance in the condensation zone is: d § 1 dpsat · dx ¨© P N dx ¸¹

Gcc

(2.93)

with Gcc the condensation rate per m3 and psat local saturation pressure. Outside the condensation zones, the balance returns to: d § 1 dp · dx ¨© P N dx ¸¹

0

(2.94)

2.3 Vapour transfer

189

Figure 2.36. Diffusion under non-isothermal steady state conditions across a composite assembly, tangent method. (1) Vapour pressure passes saturation pressure, meaning interstitial condensation. (2) Transpose the assembly into the [Z, p]-plane. (3/4) Trace vapour pressure ( p) and saturation pressure ( psat) in the assembly. They intersect, so interstitial condensation is a fact. (5) Correct solution in case of intersection: vapour pressure lines replaced by the tangents.

190

2 Mass Transfer

In the [Z, p]-plane, both equations are transformed to: y In a condensation zone d 2 psat dZ

r

2

Gcc PN

y Outside the condensation zones d2 p dZ 2

0

Outside the condensation zone, vapour pressure remains a straight line. Inside the liquid deposited (in kg/(m3 · s)) looks proportional to the second derivative of the saturation line. Suppose now the correct vapour pressure combines the straight lines outside the condensation zones with the saturation curves within. Where both meet, vapour flow rate, which equals the slope of both lines, must be the same whether approached along the vapour pressure or the saturation line. If the straight vapour pressure line does not change slope, this is only possible when the contacts accommodate a vapour source, which conflicts with the assumption of an initially dry assembly. Thus, the straight vapour pressure segments must change direction. As the derivatives of both that line and the saturation curve represent vapour flow rates in the [Z, p]-plane, the following holds: § dpsat · ¨© ¸ dZ ¹ c o

§ dp · ¨© ¸¹ dZ c m

(2.95)

meaning that in each contact, the gradient of the saturation curve must equal the slope of the vapour pressure lines, which is only possible if these become tangent to the saturation curve (Figure 2.36 (2), (3/4), (5)). Mathematically, these points of tangency are found when (equations are written for the inward tangent, starting at the surface ZT, called 2): 1. Vapour and saturation pressure are equal, or: psat, c2

ps2  g v2  Z T  Z c2

with gv2 the unknown inward vapour flow rate 2. Vapour pressure line in the non-condensation zone and saturation pressure curve have the same gradient in the contact: g v2

§ dpsat · ¨© ¸ dZ ¹ c2

§ dpsat q · ¨© dT O P N ¸¹ c2

with q heat flow rate across the assembly, O thermal conductivity and P the vapour resistance factor of the layer containing the point of tangency. For the ingoing as well as the outgoing tangent, these points coincide most of the time with an interface between two layers, which means they are singular points on the vapour saturation curve. Moreover, in many cases the two coincide, restricting the number of condensation interfaces to one.

191

2.3 Vapour transfer

The tangent line solution is named after its developer, the ‘Glaser’ method. It provides an answer to the following questions: y What are vapour pressures in the case of interstitial condensation? Outside the condensation zones vapour pressures coincide with the tangent lines. Inside, they coincide with the vapour saturation curve y Condensation zones? Condensate is deposited in all zones where the vapour and saturation pressure coincide, i.e., between the points of tangency. When the in- and outgoing vapour pressure tangent touch the saturation curve at the same interface, the condensate deposits there. y What is the condensation rate? Determined by the difference in slope between the in- and outgoing tangent. If the diffusion resistance between assembly surface 1 (Z = 0) and the point of tangency for the outgoing line is Zc1, between assembly surface 2 (Z = ZT) and the point of tangency for the ingoing line Zc2 and saturation pressure in the points of tangency psat,c1 and psat,c2, the condensation rate (gc ) is: gc

p2  psat,c2 Z T  Z c2



psat,c1  p1

(2.96)

Z c1

In the [x, p]-plane, the distribution of water deposited in a condensation zone is given by: dg c dx

Gcc

PN

d 2 psat dZ 2

1 d 2 psat P N dx 2

1 d 2 p c § dT · ¨ ¸ P N dT 2 © d x ¹

2

(2.97)

The amount of condensate and the moisture tolerance of materials in the condensation zones determine acceptability. y How to avoid the problem. Analyzing the [Z, p]-graph suggests three possible measures: 1. Drop the vapour pressure at the warm side to the extent vapour and saturation pressure in the assembly no longer intersect. A better ventilation of the building and/or a lower vapour release indoors are the actions to be taken (Figure 2.37 (a)). 2. Increase the diffusion resistance at the warm side to the extent vapour and saturation pressure in the assembly no longer intersect. This option demands the addition of a vapour resistance 'Z (= Zvapour retarder) to the existing resistance ZT – Zc2 the between condensation interface closest to the outside and indoors. The value to be added appears on the Z-axis between the diffusion resistance of the wall and the intersection of the outgoing vapour pressure tangent with the horizontal through the vapour pressure at the warm side ( p2, Figure 2.37 (b)). 'Z

Z vapor retarder t

p2  p1 Z c1  Z T psat,c1  p1

(2.98)

The measure implies the inclusion of a vapour retarding foil at the warm side of the thermal insulation.

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2 Mass Transfer

Figure 2.37. Tangent method (Glaser). (a) Highest vapour pressure inside, (b) diffusion resistance added at the warm side of the thermal insulation, (c) minimum diffusion resistance of an outside finish, all directed to avoid interstitial condensation.

3. Lower the diffusion resistance at the cold side of the condensation zone (Zc1) to the extent i sits on vapour and saturation pressure in the assembly no longer intersect. That value Z c1 the Z-axis between the point of contact of the outgoing vapour pressure tangent and the intersection of the ingoing vapour pressure tangent with the horizontal through the vapour pressure at the cold side ( p1) (Figure 2.37 (c)). Analytically: Z c1 c

Z outside cladding d

psat,c1  p1 p2  psat,c1

 ZT  Zc1

The measure implies the usage of a vapour permeable outside finish.

(2.99)

2.3 Vapour transfer

193

Figure 2.38. Interstitial condensation, simple risk evaluation. (a) Temperature convex and vapour pressure concave, risk zero. (b) Temperature concave and vapour pressure convex, high risk. (c) Temperature and vapour pressure convex or concave, risk unclear.

Evaluating diffusion as a cause of interstitial condensation can be done beforehand with the wall drawn on the thickness axis (Figure 2.38): 1. No interstitial condensation ‘means no intersection between vapour and saturation pressure’ in the assembly. A convex saturation and a concave vapour pressure line from warm to cold are needed. Because saturation pressures and temperatures are identical in shape, this implies a convex temperature and concave vapour pressure line. The rule of the thumb is thus simple. Be sure thermal conductivity and vapour resistance factor decrease from warm to cold, i.e. locate the best insulating but more vapour permeable layers at the cold and the non-insulating vapour retarding layers at the warm side of the assembly. 2. ‘Interstitial condensation hardly avoidable’ translates into ‘most probably intersection between vapour and saturation pressure in the assembly’. Needed are a concave saturation and a convex vapour pressure line from warm to cold. Because saturation pressures and temperatures are identical in shape, this implies a concave temperature and convex vapour pressure line. The rule of the thumb again is simple. Condensation may be a problem if thermal conductivity and the vapour resistance factor increase from warm to cold, i.e. if the best insulating but more vapour permeable layer is located at the warm and the least insulating but vapour retarding layer at the cold side of the assembly. Of course, whether interstitial condensation will occur or not, still depends on the boundary conditions. 3. ‘Interstitial condensation perhaps likely’ is obtained when vapour and saturation pressure line are either concave or convex from warm to cold. Whether interstitial condensation will occur in such case, depends on the boundary conditions and the thermal conductivity and vapour resistance factor sequence in the assembly.

194

2 Mass Transfer

The simple measures analysed, let alone the rules of the thumb, however, do not guarantee moisture tolerance of an assembly. Many vapour permeable layers in fact are unsuitable as outside finish (not rain-tight, too capillary, not strong or rigid enough), while the so-called warm and cold side switches with the seasons. In a moderate climate, the outside surface forms the cold side in winter, but, in summer, the inside may be ‘colder’. In warm and humid climates, the outside surface is the warm one in air-conditioned buildings. Quite some assemblies are also air permeable, meaning that air infiltration, air exfiltration, wind washing, indoor air washing and air looping may overthrow all measures based on a diffusion. Or, the diffusion theory generates rules (applying a vapour retarder, using a vapour permeable outside lining, etc.), which may be far from adequate. Single-layered assembly with variable vapour resistance factor Such assembly behaves as an n layers composite, each with infinitesimal thickness dx. Two and three dimensions Steady state, isothermal regime The CVM-approach does not differ from heat conduction. Conservation of mass – the sum of the vapour flows from the neighbouring to the central control volumes zero – leads to equations of the form:

¦

Gv, i , j

0

i l , m, n j l r1, m r1, n r1

The vapour flows Gv, i,j now equals the vapour pressure difference between any neighbouring and a central control volume, multiplied by the connecting vapour permeances ( Pcd ) . See also heat conduction. Steady state, non-isothermal regime CVM is more complicated. First, we have to calculate the temperature field and convert it into saturation pressures. Then, using the same mesh, vapour pressures are solved. If they do not pass saturation in any of the control volumes, interstitial condensation is excluded. If they pass, it will occur. To find the condensation zone, the amounts condensing and the correct vapour pressures, we assume equality between vapour and saturation pressure in all volumes where saturation is exceeded, and introduce the condensation rate as new variable (Gcc 'V ) . That changes mass equilibrium there into:

¦

Gv, i , j

Gcc 'V

(2.100)

i l , m, n j l r1, m r1, n r1

giving a new system of n equations with n unknown (p or Gcc ). Solving offers a first distribution of vapour pressures and condensation rates. If in some volumes, the condensation rate Gcc turns negative because the already accumulated deposit drops below zero, Gcc is set at zero again and vapour pressure reintroduced as unknown. Solving that system results in a new distribution of vapour pressures and condensation rates. One has to repeat the cycle until part of the control volumes show a positive condensation rate or accumulated deposit and the others show a vapour pressure below saturation. Only then, the condensation zone, the amounts condensing, and the correct vapour pressures are known.

195

2.3 Vapour transfer

Transient In transient regime, the equations (2.81) and (2.82) are to be used: y Non-hygroscopic materials div gv r Gcc

w § ©0 p · wt ¨© R T ¸¹

(2.81)

y Hygroscopic materials div gv



wwH wt

 U [I

wI wt

(2.82)

with: 

gv

1 grad p PN

Hygroscopic materials Below a relative humidity IM the diffusion equation reshuffles to: 

gv

1 grad  psat I PN



dpsat 1 § · grad T ¸ ¨ p grad I  I ¹ P N © sat dT

(2.101)

with temperature and relative humidity as driving forces. As for temperature, provided there are no contact resistances, the relative humidity in all interfaces is continuous and unequivocal, making it a real potential. The step from relative humidity and temperature to moisture content goes via the sorption isotherm. If psat / (P N) is called DI, and I / [P N] dpsat / dT DT, then equation (2.81) changes to:



div DI grad IM  DT grad T





w U [I I wt



(2.102)

The transport coefficients DI and DT and the specific moisture content U [I, are functions of relative humidity and temperature. Therefore (2.102) has to be solved together with the heat balance, using FEM or CVM. Under isothermal conditions, the balance reduces to:



div DI grad I





w U [I I wt



In theory, isothermal moisture transfer is only possible in a material with infinite thermal conductivity. Otherwise, the conversion of latent to sensitive heat irrevocably results in temperature differences. If we assume a constant transport coefficient DI and constant specific moisture content U [I, then the balance further simplifies to: ’2 I

U [ I wI DI wt

(2.103)

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2 Mass Transfer

Figure 2.39. Simplified hygroscopic curve. At relative humidity 0%, moisture content jumps from 0 to b, then increases linearly up to IM%. Beyond IM (set at 95%), it follows a second line, up to capillary.

Constant specific moisture content U [I turns the sorption isotherm into a straight line between relative humidity 0 and IM. The constant value ensues from a least square analysis on the isotherm in the interval 30 to 86%, see Figure 2.39. However, only non-hygroscopic materials have a constant transport coefficient! Indeed, the vapour resistance factor of hygroscopic materials decreases and their transfer coefficient DI increases with relative humidity. That limits the applicability of (2.103) to small differences in relative humidity. Even then, the transport coefficient DI still depends on temperature via the saturation pressure ( psat ) and the diffusion constant (N). Equation (2.103) resembles Fourier’s second law of heat conduction. Solving is done the same way. For a periodic change in relative humidity, the assembly characteristics per harmonic and per temperature become: y System matrix for diffusion Wnd ; y Moisture damping DIn with phase shift IIn , for single-layered assemblies given by: DIn

§Z d · cosh ¨ vn ¸ © DI ¹

IIn

§ Z d· arg ¨ cosh vn ¸ DI ¹ ©

y Dynamic moisture resistance Dgn with phase shift Ign , for single-layered assemblies given by: §Z d · Z d· § sinh ¨ vn ¸ sinh vn ¨ DI ¸ © DI ¹ ¸ Dgn Ign arg ¨ Zn Zn ¨ ¸ ¨© ¸¹ y Hygric admittance Ad nv with phase shift InAdv , for single-layered assemblies given by: Ad nv

DIn Dgn

InAdv

IIn  Ign

197

2.3 Vapour transfer

In these equations, Zvn is the complex hygroscopic pulsation: 2 DI U [I i n S

Zvn

T

For composite assemblies, the same product rules as for heat conduction govern layer and system matrixes. The change in relative humidity inside a semi-infinite solid caused by a sudden increase in relative humidity at its surface ('Is) in turn is expressed as: I

ª Is0  'Is «1  erf ¬«

x º » 4 aH t ¼»

where aH is moisture diffusivity in m2/s, equal to DI / U [I. For the vapour flow rate at the surface, we have: gv

 DI grad I x

0

'I

U [ I DI St

with U [ I DI the water vapour sorption coefficient or vapour effusivity, units kg/(m2 · s1/2). For two- and three-dimensional problems, the analysis demands the use of FEM or CVM. Non-hygroscopic materials Introducing the diffusion equation into (2.81) gives: § 1 · div ¨ grad p ¸ r Gcc ©PN ¹

w § © 0 psat · wt ¨© R T ¸¹

That differential equation can only be solved in conjunction with the heat balance and the equation of state psat (T). If the vapour pressure equals saturation pressure, then the condensation/evaporation term Gcc differs from zero: rGcc

§ 1 · w §© p ·  div ¨ grad psat ¸  ¨ 0 sat ¸ ©PN ¹ wt © R T ¹

In an isothermal, dry construction, the partial differential equation simplifies to: ’2 p

§ © P N · wp ¨ 0 © R T ¸¹ wt

(2.104)

(2.103) and (2.104) are, as equations, identical. They generate analogous characteristics, now called vapour diffusivity R T / (©0 P N ) and vapour effusivity © 0 / (R T P N . In a periodic regime, the properties that define the response of a flat assembly are vapour pressure damping, dynamic diffusion resistance and vapour admittance. 2.3.4.4 Vapour transfer by ‘equivalent’ diffusion and convection As stated, pure diffusion is the exception rather than the rule. Combined convection and diffusion, called advection, delivers a better picture of reality. In fact, many assemblies are

198

2 Mass Transfer

more or less air permeable. Wind, stack, and fans in operation then suffice to activate air inflow and outflow, wind washing, inside air washing and air looping. The mass balance is: div gv r Gcc



w § ©0 p · wt ¨© R T ¸¹

with gv

gv1  gv2

(2.105)

where gv1 is the convective and gv2 the diffusive vapour flow rate. Introducing the flow equations in (2.105) gives: § 1 · ga grad p  div ¨ p ¸ r Gcc Ua R T ¹ ©PN

w § ©0 p · wt ¨© R T ¸¹

(2.106)

Solving (2.106) is not straightforward. Indeed, a solution with CVM or FEM is only possible in combination with the air and heat balances. Further discussion is therefore limited to flat assemblies in steady state. Also, assemblies with a ventilated cavity will be addressed. One dimension: flat assemblies In steady state, the derivative with time in (2.106) disappears. We also set all contact resistances to zero. To get a one-dimensional situation, air and vapour flows must develop perpendicular to the assembly. In such case, (2.106) simplifies to: · ga d § 1 dp  p¸ ¨ dx © P N dx Ua R T ¹

0

(2.107)

Let the vapour resistance factor ( P) be a constant, i.e., no layer in the wall is hygroscopic. Normally, the diffusion constant N is a function of temperature. However, as for diffusion, N may be set at a constant (§ 5.4 · 109 s–1). Isothermal regime Equation (2.107) is transposed to the [Z, p]-plane: ga d2 p dp  2 U R T Z d dZ a

d2 p dp  aP 2 Z d dZ

0

with aP

ga 0.62 g a | Ua R T Pa

The result is a differential equation of second order in Z, which holds for single-layered as well as composite assemblies. The solution is: p

C1  C2 exp aP Z

(2.108)

The integration constants C1 and C2 ensue from the boundary conditions. For the diffusive vapour flow rate, the convective vapour flow rate and total vapour flow rate, the result is: dp dZ

 aP C2 exp aP Z

Diffusion

gv



Convection

g vc

aP p

Total

g vT

g v  g vc

aP ª¬C1  C2 exp aP Z º¼ aP C1

199

2.3 Vapour transfer

While the diffusion and convection flow rates vary along the wall, total vapour flow rate remains constant. With known vapour pressures on the assembly surfaces (Z = 0: p = ps1; Z = ZT = 6 (P N d): p = ps2), solving the system: ps1 = C1 + C2, ps2 = C1 + C2 exp (aP ZT) gives as integration constants: C1

ps2  ps1 exp aP Z T 1  exp aP Z T

C2

ps2  ps1 1  exp aP Z T

Vapour pressure and vapour flow rate in the assembly then become: p gv

ps1   ps2  ps1 Fv1  Z Fv2  Z  ps2  ps1

with: Fv1  Z

g vc

aP p

g vT

ps2 Fv2 0  ps1 Fv2  Z T

1  exp aP Z aP exp aP Z Fv2  Z 1  exp aP Z T 1  exp aP Z T

(2.109)

Advection changes vapour pressure in the [Z, p]-plane from a straight line into an exponential between [0, ps1] and [ZT, ps2]. When the air flows from high to low vapour pressure, then the curve is convex. In the opposite case, it is concave (Figure 2.40). The higher the coefficient aP, i.e., the larger the airflow, the more concave or convex the curves. Vapour pressure in the [x, p]-plane is obtained by transferring the successive intersections between the exponential and the interfaces from the [Z, p]-plane and linking them with exponential segments.

Figure 2.40. Advection under isothermal steady state conditions. At the left vapour pressure in a single layered assembly in the [x, p]-plane. At the right vapour pressure in a composite assembly in the [Z, p]-plane.

200

2 Mass Transfer

At first sight, the results are similar to those for combined heat and airflow. However, there is an important difference. Take a single-layered assembly made of mineral fibre, 10 cm thick, thermal conductivity 0.036 W/(m · K), air permeability 8 · 10–5 s, and a vapour resistance factor of 1.2. For an air pressure difference of 1 Pa, the airflow rate is 0.0008 kg/s (= 2.4 m3/h). The exponent for x = d then becomes: thermal 2.2, hygric 3.2, or, even for a well-insulated, extremely vapour-permeable assembly, vapour pressure is more influenced by airflow than temperature. Thus, the vapour pressure line will be more convex or concave than the temperature line. For less insulating, air permeable assembly, that difference gets even more pronounced. The increase in total vapour flow rate compared to diffusion only (gvo) becomes: 100 (gvT – gv0) / gv0 = 100 {ZT abs [Fv2 (ZT)] – 1}. If the air migrates from high to low vapour pressure, then abs [Fv2 (ZT)] passes 1 and total vapour flow rate increases. When the air takes the opposite direction, then even a limited inflow may invert the resulting vapour flow, from low to high vapour pressure. Non-isothermal conditions Both saturation ( psat ) and vapour pressure, calculated assuming the diffusion constant is a real constant, have to be considered now, the first as a transposition of the exponential temperature curve. If the two do not intersect, condensate does not deposit and the isothermally calculated vapour pressures remains. If, however, both intersect, condensation is a fact. In case the rate is small enough to neglect the heat of evaporation (heat source), only vapour pressure in the assembly changes, not temperature and saturation pressure. Indeed, at the spot where interstitial condensation begins, vapour pressure must equal saturation pressure with identical in and outgoing vapour flow. The same holds at the spot where condensation stops. Therefore, at the borders of a condensation zone, we have: p

psat (1)

dpsat  aP psat dZ

dp  aP p (2) dZ

(2.110)

Saturation pressure and diffusion resistance Z are unknown in both equations. Combining the two simplifies the second to: ª dp º «¬ dZ »¼

ª dpsat º «¬ dZ »¼

(2.111)

Hence, where condensation starts and ends, vapour and saturation pressure must not only be equal but also have a common tangent. In practice, the solution goes as follows: 1. First, the saturation and the vapour pressure curves are calculated. 2. Then, all interfaces where vapour pressure exceeds saturation pressure are tested for condensation, starting with the interface where the excess is highest. Let this be interface j. The vapour pressure line from surface s1 (= inside surface) to interface j then becomes: p





ps1  psat, j  ps1 Fv1  Z s1

with Fv1  Z s1

1  exp ª¬ aP Z s1z º¼ 1  exp ª aP Z s1j º ¬ ¼



with Z s1j the diffusion resistance between surface s1 and interface j and, Z s1z the diffusion resistance from s1 to an interface Z between s1 and j. If that new vapour pressure does not intersect the saturation pressure between s1 and j, then it figures as the ingoing exponential.

201

2.3 Vapour transfer

For the outgoing vapour pressure, an analogous reasoning applies, except that the boundary conditions are now the saturation pressure psat,j in interface j and the vapour pressure at the outside surface s2: p





psat, j  ps2  psat, j Fv1  Z

with Fv1  Z

 1  ex p  a

1  exp aP Z zj



s2 P Zj

z with Z s2 j the diffusion resistance between interface j and surface s2 and, Z j the diffusion resistance from j to an interface Z, now between j and s2. Without intersection with the saturation pressure, this new vapour pressure line figures as the outgoing exponential.

3. In case the ingoing vapour pressure exponential between s1 and j intersects saturation pressure, a systematic procedure, testing each interface in between, allows finding the correct solution. If the intersection zone contains no interfaces, then a point l on the saturation curve between s1 and j is looked for, where:

 psat,l  ps1

l º aP exp ª¬ aP Z s1 ¼ l º 1  exp ª¬ aP Z s1 ¼

ª dpsat º «¬ dZ »¼ l

If present, condensation starts there. The same reasoning is followed between j and s2, giving additional condensation interfaces or the endpoint k of the condensation zone:

 a

aP exp aP Z ks2

 ps2  psat, k 1  exp

s2 P Zk



ª dpsat º «¬ dZ »¼ k

If k and l coincide or delimit a continuous zone, the assembly has only one condensation interface/zone. Additional vapour pressure tangents create extra condensation interfaces/ zones. With air flowing from low to high temperature and low to high vapour pressure, the more concave vapour pressure exponential never intersects the less concave saturation curve, making interstitial condensation impossible. When instead the air flows from high to low temperature and high to low vapour pressure, then a high enough vapour pressure at the warm side unavoidably ends in interstitial condensation, see Figure 2.41. The airflow also changes reaction time, from very slow for diffusion to almost instantaneously. The advection method gives an answer to the following questions: y Vapour pressure line in case of interstitial condensation? See above. y Condensation zones? Air ingress may increase the number of condensation zones in a wall compared to diffusion. y Condensation flow rate? In case of a single condensation interface, the rate is given by (where s2 is outside and s1 inside):

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2 Mass Transfer

Figure 2.41. Composite assembly, advection under non-isothermal steady state conditions. Vapour pressures (thin line) intersect saturation. The assembly thus suffers from interstitial condensation. Correct vapour pressure gives condensate at the backside of the veneer (on the left in a [Z, p]-plane, on the right as [d / p]-picture.





sat, x º ª psat, x Fv2 0  ps2 Fv2 Z s2 ¬ ¼

gc









sat, x sat,x sat,x º  ª ps1 Fv2 Z s2  Z s2  psat, x Fv2 Z sls2  Z s2 ¬ ¼

with Fv2 0

aP



sat,x 1  exp aP Z s2



sat, x Fv2 Z s2



 1  exp a

sat, x aP exp ap Z s2 sat, x P Z s2



sat,x sat,x Fv2 Z s2  Z s2



Fv2 Z sls2





sat, x Z s2







aP sat, x º ª 1  exp aP Z sls2  Z s2 ¬ ¼



 

sat, x aP exp ª aP Z sls2  Z s2 ¬ sat, x 1  exp ª aP Z sls2  Z s2 ¬



º¼ º¼

2.3 Vapour transfer

203

where psat.x is saturation pressure in the condensation interface. Compared to diffusion only, the amounts of condensate increase significantly with air outflow. At the same time, temperature in the condensation interface increases, gradually slowing down the increase in deposit, until a maximum is reached, after which the amounts drop quickly to reach zero once the temperature in the condensation interface equals the dew point at the warm side (Figure 2.42).

Figure 2.42. Interstitial condensation by advection in a composite assembly, deposit in kg/week as function of the outflow rate.

Figure 2.42. Interstitial condensation by advection in a composite assembly, deposit in kg/week as function of the outflow rate.

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2 Mass Transfer

y How to avoid this. Advection provides several options. A radical one is rigorously assuring that the airtight envelope assemblies with the composing layers correctly are ordered in terms of diffusion resistance. With respect to energy efficiency, this is an excellent choice. An alternative seems making the envelope assemblies so air permeable the air egress heats potential condensation interfaces beyond the dew point indoors. That option of course does not serve energy efficiency. In moderate climates, combining good ventilation with low vapour release indoors may of course keep vapour pressure in the building below the saturation value noted at the most likely condensation interfaces. Under-pressurizing the building in cold and moderate climates finally turns air outflow to inflow, which excludes interstitial condensation in winter. In warm, humid climates instead, the building must be over-pressurized. In the diffusion mode, the Glaser diagram allowed fixing the vapour pressure at the warm side that excluded interstitial condensation. Also, the vapour retarder needed at the warm side or the vapour permeance of the outside finish could be deduced. Hence, in advection mode the only measure remaining is limiting vapour pressure at the warm side to the maximum allowable. Vapour retarders and vapour permeable outside finishes completely lose their effect if air-tightness fails. The best and most important measure thus is: assuring the envelope is as airtight as possible.

2.3.5

Surface film coefficients for diffusion

Until now, we assumed the vapour pressures at the bounding surfaces of an assembly are known. Yet, measuring vapour pressure at a surface is an impossible task. Logging vapour pressures in the air through temperature and relative humidity instead is not a problem. The preference therefore goes to using vapour pressures in the air as a boundary condition and describing vapour transfer across an assembly environment to environment. The question then becomes: does water vapour, and in general, each gas, experience resistance when migrating from the air to a surface and vice versa? The answer is yes. A laminar boundary layer, in which diffusion is the only mode possible, sticks to each surface, whereas up to this layer, vapour transfer occurs mainly by convection. Vapour migration across this boundary layer is now described by: gv

E  p  ps

(2.112)

where p is vapour pressure outside the boundary layer (the reference in practice is the vapour pressure in the centre of the floor at a height of 1.7 m), and ps vapour pressure at the surface. E is called the surface film coefficient for diffusion, units s/m. The reverse 1 / E is the surface resistance for diffusion, symbol Z, units m/s. The suffix i will be used to indicate inside (Ei), the suffix e to indicate outside (Ee). The surface film coefficient for diffusion is calculated the same way as the surface film coefficient for heat transfer by convection. The basic equations are the mass balance for air, momentum equilibrium as expressed by the Navier-Stokes equations, the turbulence equations, and conservation of water vapour. Solving this system of seven partial differential equations can be done analytically, numerically or by means of similarity. The last option generates a set of dimensionless numbers, which quantify the surface film coefficient for diffusion:

205

2.3 Vapour transfer

Forced convection

Mixed convection

Free convection

Reynolds number: vL Re Q

Re Gr1 2

Grashoff’s number: 'Ua g L3 Gr Q2

EL , replaces the Nusselt number Ga

The Sherwood number, Sh The Schmidt number, Sc

Q , replaces the Prandl number R T Ga

Like the Nusselt number, the Sherwood number stands for the relationship between total vapour transfer and vapour transfer by diffusion. Schmidt’s number shows the field of flow velocities to be similar in shape to that of vapour pressures. Identical relations as those found for convective heat transfer exist between the numbers. Thus, forced laminar flow along a horizontal flat wall results in a surface film coefficient for diffusion, given by: Sh L

0.664

1 ReL2

1 Sc 3

That similarity with heat convection leads to the following relationship between both surface film coefficients (Lewis equation): E

§ Ra T cp G · hc ¸¹ Ua Ra T cp ¨© Oa

0.67

(2.113)

where Ua is density, Ra the gas constant, cp specific heat at constant pressure and Oa thermal conductivity of air. Under atmospheric circumstances, i.e., for most building applications, (2.113) simplifies to: E

hc

1 Oa N

(2.114)

(2.114) ensues from the boundary layer theory, assuming that layer is equally thick for heat and vapour transfer. This condition is met when the Schmidt and Prandl numbers have the same value. Let d be the common thickness (Figure 2.43).

Figure 2.43. Surface film coefficient for diffusion, boundary layer.

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2 Mass Transfer

For heat, we have: Oa

q

T  Ts d

hc  T  Ts or d

Oa hc

Vapour gives: gv

p  ps Nd

E  p  ps or d

1 N E

Equating results in (2.114). For air, with diffusion constant N § 5.4 · 109 s–1 and a thermal conductivity of Oa § 0.025 W/(m · K), (2.114) reduces to: E | 7, 7 ˜ 109 hc

(2.115)

Also see Table 2.5. Table 2.5. Inside and outside surface film coefficient for diffusion. Surface film coefficient for diffusion ³i (pa = 1 Atm, 0 ” Ti ” 20 °C)

Ei

Surface film coefficient for diffusion ³e (pa = 1 Atm, –20 ” Ti ” 30 °C)

Ti – Tsi K

E · 10–9 s/m

va m/s

Ee · 10–9 s/m

2

28.6

0 + dt, the kinetics simplify to:

vm

dl dt

§ · ¸ ª 2 V w cos º Uw r 2 ¨ 1  g cos D» ¨ ¸ «  L  l ¸ ¬ Uw r l 8 ¨ ¼ K  Ka © w l ¹

(2.129)

The term with the dynamic viscosities of water (Kw) and air (Ka ) now acts as a variable viscosity, infinite large at the start and approaching the value for water as suction proceeds. The equation proves the existence of a maximum suction height hmax. Indeed, velocity turns zero for lmax = 2 Vw cos - / ( Uw g r cos D), giving hmax = lmax cos (D) (Figure 2.51).

Figure 2.51. Maximum suction height.

2.4 Moisture transfer

217

Figure 2.52. Wet height versus time for a 15 m vertical pore (both on a logarithmic scale). r = 1 —m, hmax = 14.8 m / r = 10 —m, hmax = 1.48 m / r = 100 —m, hmax = 0.148 m.

The general solution of (2.129) is: t

ª  Kw  Ka lmax  Ka L º § 8  Kw  Ka °­ l · °½ ® l  « » ln ¨1  ¸¾ 2 Kw  Ka Uw g r cos D °¯ ¬ ¼ © lmax ¹ °¿

(2.130)

giving time as a function of the wet length. Figure 2.52 shows that relationship for a vertical pore. Horizontal pore Water suction by a horizontal pore is seen as representative for rain absorption by a capillary outer finish such as a veneer wall or stucco (Figure 2.53). For a pore with length L, radius r and the cosine of the slope zero, (2.129) becomes:

Figure 2.53. Rain absorption.

218

2 Mass Transfer

dl dt

dx dt

r V w cos 4 ª¬ Kw  Ka x  Ka L º¼

with the quadratic equation 2 (Kw – Ka ) x2 + 4 Ka L x – r Vw cos - t = 0 as a solution. The positive root is: 2

x



§ K L · Ka L r V w cos  ¨ a  t Kw  Ka 2  Kw  Ka © Kw  Ka ¸¹

So, wet thickness x not only depends on time, but also on the radius and the length of the pore. The longer and narrower it is, the slower it sucks water. In other words, the velocity at which a porous façade material picks up rain decreases with thickness and finer pore structure. If we neglect friction in the air column ahead, the solution becomes: x

r V w cos 2 Kw

(2.131) t

For shallow but wide pores, air friction is of little influence. For finer pores, the error when neglected increases substantially, especially at the start of the suction process. The underlined term in equation (2.131) is named the capillary water penetration coefficient, symbol B, units m/s1/2. Multiplying both sides of the equation by the water density gives as absorbed water in kg/m2: mw Uw B t A t when A = Uw B, and A is the capillary water sorption coefficient, units kg/(m2 · s1/2). The smaller the capillary water sorption coefficient, the slower a horizontal pore sucks and the longer it takes before water reaches the other side. Both the correct and simplified t -relationship shows horizontal suction never halts. Whatever the depth of a pore is, the other end will be reached. Once there, the water meniscus straightens, turning suction to zero, which equilibrates the system. An outflow of water by capillary suction is therefore impossible. Thus, rain absorbed by porous materials may cause wet stains inside but never water flow along the inner surface. Only outside pressures or gravitational forces cause seeping. If this happens with rain, gravity is the likely culprit (Figure 2.54). The model of a horizontal pore with invariable section is easily expanded to pores with variable section. As an illustration, take plaster on a substrate. This two-layer-system can be modelled as two serial connected pores, which contact each other. The first pore, the plaster, has a radius r1, a contact angle -1 and a length l1. The second pore, the substrate, has a radius r2, a length l2 and a contact angle -2.

Figure 2.54. Seeping rain.

219

2.4 Moisture transfer

Figure 2.55. Serial two pores model in case of water contact (for example: stucco on whatever substrate).

First, we neglect air friction. Assume the first pore is the widest (Figure 2.55). As long as the meniscus remains in pore 1, m A1 t applies. A smaller capillary water sorption coefficient (A1) thus slows water absorption, i.e. for the pore becoming finer and/or the contact angle larger. When the meniscus reaches the second pore, this takes over suction while the first now acts as a hydraulic resistance. If instead the second pore is widest, then at contact, the meniscus in the first will straighten out until its curve fits with the second. The latter then will take over, while the first pore again will add hydraulic resistance. With the water in the second, total friction is: § 8 S Kw l1 r22  8 S Kw ¨ r12 ©

Ffw

· dx x¸ ¹ dt

(2.132)

x being the wet length in the second. Solving the force equilibrium Ffw = Fc for x then gives: § 2 l r2 · § r V cos - · x 2  ¨ 12 2 ¸ x  ¨ 2 w ¸¹ t © 2 Kw © r1 ¹

0

with a positive root of: 2

x



§ l1 r22 · l1 r22 2  ¨ 2 ¸  B2 t r12 © r1 ¹ 

Clearly, water absorption by the second pore (x) slows down when the ratio above the brace increases, i.e. when the first pore is longer and finer then the second. Hence, a coarse plaster cannot increase the rain resistance of a wall made of fine porous material. Only a water repellent plaster (- § 90°, B very small) may move wetting time beyond the maximum duration of wind driven rain event. With air friction, the solution of the quadratic equation for the wet length in the first pore is:

x



Ka d1 Ka d 2 r1  Kw  Ka  Kw  Ka r22   

  

(1) (2)

2

Ka d 2 r1 º r V w cos ª Ka d1 « K  K  K  K r 2 »  2 K  K t  w  w  w a a 2 » 

a  «

«¬ (1) »¼ (2)

220

2 Mass Transfer

For the radius r2 below the radius r1 and the layer thickness d2 much smaller than the layer thickness d1, term (2) is many times larger than term (1), showing the meniscus now moves a lot slower in the first, wide pore than it did without air friction. A coarse plaster on a fine porous substrate will thus absorb water slower than the same plaster without substrate. However, be careful with this conclusion. While the pore model used is one-dimensional, the pore system in any porous material is three dimensional, whereas surface wetting never occurs uniformly. Even the outflow of air is not one-dimensional. Vertical pore Water suction by a vertical pore is considered representative for rising damp in a capillary material (Figure 2.56). For a vertical pore, the cosine of the slope D is 1 and equation (2.129) converts into: vm

dz dt

Uw r 2 8

§ 2 V w cos ·  g¸ L  z ¨© Uw r z ¹ Kw  Ka z 1

Velocity goes to zero for 2 Vw cos - / (Uw r z) = g, i.e. at the maximum suction height: z

hmax

2 V w cos Uw g r

(2.133)

This maximum increases with smaller pore diameter and/or smaller contact angle. Thus, the finer the pores and the more hydrophilic a material, the higher sucked water rises. Instead, for a material with a contact angle of 90°, that height is zero. The same holds for very wide pores and materials without pores. Rising damp in capillary walls, in other words, can be prevented in three ways: (1) by including a coarse porous layer (r large) above grade, (2) by injecting a water repellent substance above grade which moves the contact angle in the material above 90° (cos - between 0 and –1), (3) by inserting a waterproof layer with zero porosity above grade. Suction exerted by the coarse porous layer or the water repellent zone should preferentially remain lower than Uw g h, with h the distance between the topside of that layer and grade (Figure 2.56).

Figure 2.56. Rising damp, some prevention possibilities.

221

2.4 Moisture transfer

Isothermal water transfer in a pore after interrupting the water contact Constant contact angle, constant section Consider a pore with circular section in contact with water. After a time, the contact is interrupted. The result is a water isle in the pore with a concave meniscus at both sides (Figure 2.57). In a horizontal pore, both meniscuses pull identically, which excludes movement and water from being transferred. By inclining the pore, gravity makes the lower meniscus less and the upper more concave. Flowing will start when the water isle is heavy enough to straighten out the lower meniscus, i.e. when its length equals maximum rise (= hmax / cos D). Hence, the wider a pore, the easier gravitational flow and external pressure flow develop.

Figure 2.57. Water transport in a pore after water contact is interrupted. Isothermal, constant contact angle. On the left constant, on the right variable section.

Constant contact angle, variable section Assume the pore now has a variable section (Figure 2.57). In that case, the meniscus with the smaller diameter d1 (the other is d2) pulls the most. As a result, the water isle moves in the direction of the largest suction (4 Vw cos - / d1 > 4 Vw cos - / d2). This movement lasts until arriving at a location where both meniscuses mirror each other. There, the difference in suction turns zero and the water stops flowing. The flow rate increases when the water isle shortens and the difference in suction enlarges, i.e. when the suction gradient increases. The flow rate can thus be written as: gw

§ U d2 · ¨ w ¸ grad pc © 32 Kw ¹

 k w grad s

(2.134)

with kw the water permeability of the pore, units s. A water isle located partly in a narrow and partly in a wider pore reacts the same way, with the narrow pore draining the wider one. Gravity and external pressures can hinder as well as favour this water motion. Variable contact angle, constant section In this case, the water isle moves towards the smaller contact angle, i.e. in the direction of the largest suction, as shown by equation (2.134). Variable contact angle, variable section The isle again follows the largest suction. This convenes with the largest ratio cos - / d.

222

2 Mass Transfer

Non-isothermal water transfer in a pore after interrupting the water contact Constant contact angle, constant section Suction increases with surface tension. As surface tension of colder water enlarges, temperature differences will force a water isle to the colder side. Suction becomes: s

so

Vw V wo

(2.135)

with s0 the suction at a reference temperature, f.e. 20 °C, explains this (Vw = Vwo – 0.17 · 10–3 T). Suction gradient then becomes: grad s

Vw s grad so  0.17 ˜ 103 o grad T V wo V wo

(2.136)

Since the so-gradient becomes zero at reference temperature in a pore with constant section, the flow equation becomes: gw

ª s § U d 2 ·º  «0.17 ˜ 103 o ¨ w » grad T V wo © 32 Kw ¸¹ ¼» ¬«

 K Tw grad T

(2.137)

We call KT,w the thermal water permeability; units kg/(m · s · K). Variable contact angle, variable section Now, both driving forces act. The equation becomes: gw

 k w grad s  K T,w grad T

(2.138)

Remark As already discussed, apart from temperature and suction gradients two other forces drive water in a pore: gravity and pressure. In narrow pores, the two can be neglected. Not so in wider pores. There, suction only becomes: grad s + grad ( Uw g z) + grad p. Also remember suction stands for the difference in water and air pressure over a meniscus. When air is expelled, air pressure there rises somewhat, whereas moving water isles causing air ingress lower it a little. 2.4.2.3 Vapour transfer Moving water isles cause water transfer. Vapour transfer develops in the air inclusions between isles. The two therefore exist in series. A succession of isles and air inclusions occurs when the contact with water is interrupted repeatedly. Thompson’s law hides the two causes of vapour transfer: gradients in pore diameter and gradients in temperature. Isothermal vapour transfer Constant contact angle, constant section As vapour saturation pressure is the same all over the pore ( pcsat = psat (T) exp [s / ( Uw R T )]), no vapour transfer develops (Figure 2.58). However, vapour exchange with the environment goes on if the relative humidity there diverges from 100 ( psat c / psat ) .

223

2.4 Moisture transfer

Constant contact angle, variable section The variable section induces suction differences between the meniscuses of the sequential water isles (Figure 2.58). This results in a difference in saturation pressure with vapour displacement in between. The equation is: gv

 Ga grad psat c

 Ga

wpsat c grad s ws

 Ga

Uv grad s Uw

(2.139)

Thompson’s law in fact states that, for a meniscus in equilibrium, vapour pressure and suction are interchangeable. In principle, transfer disturbs equilibrium. However, for small water flow rates, the disturbance is too small to exclude exchange.

Figure 2.58. Vapour transport in the air inclusions between water isles, isothermal conditions, constant contact angle, constant and variable section.

Variable contact angle, constant section/variable contact angle, variable section Again, we have a difference in suction between the meniscuses of the sequential water isles with vapour transfer in the air inclusions in between as a result. See equation (2.139). Non isothermal vapour transfer Constant contact angle, constant section Differences in temperature cause differences in saturation pressure pcsat between the meniscuses of sequential water isles, giving as a vapour flow rate: gv

 Ga grad psat c

ª dp psat § ws s · º  Ga I « sat  ¨  ¸ » grad T U w R T © wT T ¹ ¼ ¬ dT   

(1)

Only for very small pore diameters, is term (1) in the formula of the same order as I dpsat / dT. The equation usually simplifies to: gv

 Ga I

dpsat grad T dT

(2.140)

Variable contact angle, variable section Now, both causes join forces, turning the general equation for vapour flow rates in the air inclusions, between the first isle and the environment at one side, and between the last isle and the environment at the other side into:

224

2 Mass Transfer

gv

ªU º dp  Ga « v grad s  I sat grad T » dT ¬ Uw ¼

Of course, that equation also can be expressed as: gv

 Ga grad p

dp § ·  Ga ¨ psat grad I  I sat grad T ¸ © ¹ dT

(2.141)

2.4.2.4 Moisture transfer Moisture transfer in a pore consists of a serial transfer of liquid and water vapour: liquid in the water isles and vapour in between. As a consequence, the isles with the smallest meniscuses at lowest temperature grow, while those with the larger meniscuses at higher temperature narrow. Transfer stops once the isles rearrange in such way equilibrium is reached.

2.4.3

Moisture transfer in materials and assemblies

2.4.3.1 Transport equations Capillary porous materials, moisture content between zero and capillary For capillary porous materials, we distinguish three moisture intervals from dry to saturated: (1) dry to hygroscopic at a relative humidity IM, (2) IM to capillary, (3) capillary to saturation. This section deals with intervals 1 and 2. Each capillary porous material forms a matrix containing a labyrinth of pores. These are neither circular nor straight. Their section changes continuously. They deviate, come together, split, etc. Yet, each so-called ‘representative material volume’ contains the whole labyrinth. Moisture transfer has the same driving forces as in a pore. However, it is neither possible to differentiate between vapour and water nor between series and parallel. If we were to make a cut across a representative volume, then for some pores it would pass a water isle, for others an air inclusion. Hence, each cut sets liquid and vapour flow in parallel, both activated by suction and temperature gradients. By replacing water permeability and thermal water permeability of a pore in equation (2.134) by water permeability and thermal water permeability of the material and vapour permeability of air in the vapour transfer equation (2.141) by vapour permeability of the material, moisture transfer becomes: gw  gv gm

 k w grad s  K T,w grad T U dp  G v grad s  GI sat grad T dT Uw

(2.142)

  km grad s  K T grad T

With: km

§ Uv · ¨© k w  G U ¸¹ w

KT

K T,w  G I

dpsat dT

km is called the moisture permeability, units s, and KT the thermal moisture diffusion coefficient, units kg/(m · s · K) of the material.

225

2.4 Moisture transfer

Unsaturated water transfer plays a leading role in moisture permeability, whereas vapour transfer dominates thermal flow to the extent that the thermal moisture diffusion coefficient is usually set at zero. For most capillary-porous materials, both properties are tensors. As temperature, suction s is a real potential, unequivocally defined in each interface, at least in case of ideal contact between materials. All materials Besides hygroscopic capillary materials (type 1) and non-hygroscopic capillary materials (type 2), we have a group of non-hygroscopic, non-capillary materials (type 3) and a group of hygroscopic, non-capillary materials (type 4). Type 3 includes materials with pores that are so wide that capillary transfer hardly intervenes, while gravity and pressure first come into play when they are soaked. Moisture flow is thus limited to vapour advection. Type 4 consists of materials with very fine pores. For such materials, flow resistance is too high for capillarity, gravity and external pressures to move water, while vapour transfer by convection is also improbable. Only vapour diffusion remains. In other words, in type 3 and type 4, capillary suction as inappropriate as potential. Relative humidity, however, remains usable. Hence, with relative humidity, we get expressions which are more universal than equation (2.142). No advection

gm

Advection

gm

   kI,m grad I  K T grad T  6.21 ˜ 106 ga I psat  kI,m grad I  K T grad T

(2.143) (2.144)

For the transfer coefficients, the following applies: Material

kI,m

Type 1, 2: capillary porous

km

Type 3: non-hygroscopic, non-capillary Type 4: hygroscopic, non-capillary

KT ds dI

GI

dpsat dT

G psat

GI

dpsat dT

G psat

GI

dpsat dT

In type (1) and type (2) materials, moisture transfer below relative humidity IM includes (equivalent) diffusion only. Type (4) materials also do not permit advection. Moisture content above capillary Above capillary, equation (2.142) applies for drying. In all other cases except for external water heads, water transfer is activated by vapour diffusion between water islands and the dissolution of air inclusions in the pore water. External water heads of course directly activate saturated water flow, which the following equation describes: gm

 k w grad Pw

where Pw is water head and kw water permeability at saturation (sometimes called the Darcy coefficient). Saturated water flow lacks inertia, making the solutions steady state. Thus, saturated water flow rate gw in kg/(m2 · s) across a single-layered assembly with thickness d) is:

226

2 Mass Transfer

gw

kw

'Pw d

'Pw d kw

'Pw Ww

with Ww the water resistance of the assembly, units m/s. For a composite assembly, we have: Flow rate (kg/(m2 · s))

'Pw

gw

n

¦ Ww,i

'Pw WwT

i 1

Water pressures (Pa)

x Pw1  g w Ww,1  Pw1  Pw2

Pw, x

i.e. analogous equations as for steady state heat conduction, diffusion and air flow. To make an assembly waterproof, a layer of infinite water resistance must be inserted. As an alternative, equalling water movement to an interface somewhere in an assembly to the drying flow rate from that interface to the side requiring dryness prevents the water front from reaching this side. Moisture permeability (km) For capillary porous materials, moisture permeability depends on suction/relative humidity. The higher the suction (i.e. the finer the pores and the smaller the contact angles) or the lower the relative humidity, the more moisture permeability drops. Above, this dependence was highlighted for a single pore, where for the water part it looked like: kw

Uw d 2 32 Kw

Uw V 2w ª cos - º 2 Kw «¬ s »¼

2

Figure 2.59. Suction characteristic of calcium-silicate, as measured at 6 laboratories.

227

2.4 Moisture transfer

In principle, the property can be determined experimentally as the product of moisture diffusivity Dw (see hereinafter) and specific moisture content, the derivative of the relation between moisture content and suction/relative humidity, also called suction characteristic w(s), see Figure 2.59. An alternative is using a pore model of the material. Assuming, which we did, a thermal moisture diffusion coefficient of zero, makes suction (s) and thus moisture permeability of a capillary porous material a function of temperature. 2.4.3.2 Conservation of mass In general Conservation of mass for a representative material volume (Vrep) gives:

div gm r G c dVrep

§ ww · ¨©  ¸ dV wt ¹ rep

with w the average moisture content in the volume, Gc the moisture source or moisture sink in it and div the average divergence to or from the representative volume. In detail No advection. Applies for type (1), type (2) and type (4) materials. If the representative volume is assumed infinitely small, then combining conservation of mass with the general transfer equation gives:





div kI,m grad I  K T grad T r G c

ww wt

Uw [ I

wI wt

(2.145)

With advection. Applies to type (3) materials. The result is:





div ª¬ kI,m grad I  K T grad T  6.21 ˜ 106 g a I psat º¼ r G c

Uw [I

wI wt

(2.146)

In both cases, Uw [I is the specific moisture content. This property is found by taking the derivative of the suction characteristic and multiplying by the derivative of s for I: [I = [s ˜s / ˜I. The result should equal the derivative of the full hygroscopic curve between 0 and 100%. Type (3) materials have as specific moisture content: § I psat · w¨ © T ¸¹ ©o Uw R wI In type (1) and type (4) materials, hygroscopic moisture also embodies the source as the sink term (Gc). That way, local interstitial condensation and local evaporation disappear as separate phenomenon. 2.4.3.3 Starting, boundary and contact conditions Starting conditions In most cases, the starting condition is an assembly with moisture content far beyond hygroscopic for IM (95%?) (so-called building moisture).

228

2 Mass Transfer

Boundary conditions y Contact with moist air. As vapour flow rate at a surface, we have: gv = E ( p – ps ) where p is vapour pressure in the air, ps vapour pressure at the surface and E the surface film coefficient for vapour diffusion. Vapour pressure expressed as psat,s I with psat,s saturation pressure at the surface, introduces ‘relative humidity’ as potential. ‘Contact with moist air’ acts as a boundary condition when drying, sorption, desorption and interstitial condensation are considered. y Known flow rate. Typical for precipitation without water runoff. y Contact with water. Relates to surface condensation and precipitation, the last once a water film forms. Contact conditions y Suction contact. Continuity of water and vapour flow characterizes ideal suction contact. This seldom happens. Most of the time suction contacts are non-ideal, i.e. introduce a contact resistance, a typical example being the mortar/brick interface. y Diffusion contact. Continuity of vapour flow. Applies when small air gaps separates layers. Also non-capillary materials experience diffusion contacts. y Contact with water. As a result of interstitial condensation and rain penetration, water can fill gaps between layers. Gravity and water pressure may force run off, while capillary forces may immobilize this water layer. 2.4.3.4 Remark Equations (2.145) and (2.146) are correct as long as a capillary-porous material does not contact liquid water. If so, then water moves from the wider to the narrower pores. If it does, wider pores can suck water from narrower ones (see isothermal water transfer in a serial two-pore system that contacts water). That difference in moisture movement is accounted for by valuing moisture permeability and specific moisture content between dry and capillary and not dry and saturated. For non-capillary materials, water contact means a 100% relative humidity in the contact plane, while for capillary materials, it means zero suction, 100% relative humidity and moisture content equal to capillary.

2.4.4

Simplifying moisture transfer

2.4.4.1 The model All properties in equations (2.145) – (2.146) are suction, relative humidity and temperature dependent. This excludes analytical solutions. Applying CVM or FEM of course is possible, which requires a software tool, such as Wufi, Match, Delphin, HygIRC and others. A disadvantage of these tools is their lack of black boxes, hardly of help to understand the physics. This opens the door for simplified models, which are easier to grasp. For that purpose, we introduce moisture content as an ‘improper’ driving force, meaning its value changes in the interfaces between materials. Water transfer then becomes: gw

 Dw grad w

with Dw water diffusivity of a material, units m2/s.

(2.147)

229

2.4 Moisture transfer

Figure 2.60. Moisture diffusivity as function of moisture content, simplified.

The link with water permeability is: Dw

kw

kw U w [s

ds dw

(2.148)

For water diffusivity as function of moisture content, a preset relation is often advanced: 2

Dw

§ A· § w· a ¨ ¸ exp ¨ b © wc ¹ © wc ¸¹

with A the capillary water sorption coefficient, wc capillary moisture content, and a, b the material constants. The simplified model proposed reduces this relation to a step function, zero for moisture contents below critical (wcr) (the moisture content at a relative humidity IM ) and constant above (Figure 2.60). Below critical, moisture transfer is vapour only (gw = –G grad p). Above, unsaturated water flow dominates, with vapour transfer only intervening when non-isothermal: w < wcr

w • wcr

Isothermal

gv

 G grad p

gw

 Dw grad w

Non-isothermal

gv

 G grad p

gw

 Dw grad w  G

dpsat grad T dT

Typical for the model is the sharp moisture front appearing between wet and air-dry in partially above-critical moist materials (Figure 2.61). In fact, if we assume a one-dimensional, hygrically steady-state, isothermal regime, then each elementary material layer must see the same moisture flow rate passing. Thus, between sub(1) and above-critical (2) the rates are equal. When gm1 = –G (grad p)1 = –G dp / dw (grad w)1

230

2 Mass Transfer

Figure 2.61. Moisture front between above-critical wet and air-dry.

= –Dw1 ( grad w)1 and gm1 = gm2 = –Dw2 ( grad w)2, with Dw1 (§ 0) extremely small compared to Dw2, (grad w)1 must be much larger than (grad w)2. Therefore, moisture content at the sub-critical side of the contact must increase suddenly at the contact, while staying very flat above-critical. This is the named moisture front between ‘dry’ and ‘wet’, as simple observations confirm. An ink stain on blotting paper spreads out for a while and then stops. At that moment, we have critical ink content in and very little ink outside the stain. In the wave outflow zone at beaches, the contact between dry and wet sand is clearly drawn. We see a boundary between dry and wet on a brick wall, etc. Even though the transfer equations seem similar, the moisture front creates a clear difference between solving for unsaturated water flow and solving for saturated water flow, vapour diffusion, air transfer and heat conduction. Handling the simplified model analytically for example is only possible when assuming a moisture profile shape. 2.4.4.2 Applications Capillary suction Capillary suction is one of the most important mechanisms wetting materials and assemblies. Think of rising damp and wind driven rain. Hydrophilic, homogeneous material If a sample of a hydrophilic material is weighed regularly while sucking water, then, just as for a pore, a linear relation between the quantities sucked per m2 of contact surface and the square root of time ( t ) emerges (Figure 2.62): mc

A t

(2.149)

where A is the water sorption coefficient, units kg/(m2 · s1/2). When the moving moisture front is monitored, then the following relationship applies between location and time: h

B t

(2.150) 1/2

with B the water penetration coefficient, units m/s .

231

2.4 Moisture transfer

Figure 2.62. Capillary suction, mw

A t.

As soon as the moisture front reaches the other side of the sample, suction switches to a second [ t , m] line with a much smaller slope. Now, suction is the result of diffusion and solution of air inclusions in the pore water. That line is called the line of secondary water sorption. The related slope gives the coefficient of secondary water sorption Ac. The intersection with the capillary line fixes the capillary moisture content (wc). Between the trio of water sorption coefficient, water penetration coefficient and capillary moisture content, the following relationship is assumed: A | B wc

(2.151)

For that relation to be correct, the capillary moisture content should extend over the whole height of the sample. The water penetration coefficient and capillary moisture content are pure material characteristics. The larger the penetration coefficient, the more capillary a material. The larger the capillary moisture content, the more moisture it buffers. The capillary water sorption coefficient instead is a composite property, comparable with the contact coefficient in the case of heat transfer. The number shows how easily a material absorbs water, just as the contact coefficient shows how easily a material absorbs heat. The linear relationship [ t , mc ] does not account for air transfer. As has been shown for a pore, air egress induces some anomaly. This plays when high samples of fine-porous materials or samples sealed at the top are tested. In the first case the curve bends towards the time axis to form a kind of concave parabola. In the second case, water sorption retards as time progresses while, at the water side, air bubbles out of the sample. Hydrophilic, non- homogeneous material In non-homogeneous materials, the suction curve deviates from linear in

t.

Hydrophobic materials Capillary moisture uptake by hydrophobic materials advances very slowly, if at all, without linear linkage to t . Calculating a water sorption coefficient is impossible, except as the slope of tangents to the suction curve.

232

2 Mass Transfer

Simulation with the simplified model Does the simplified model predict the linear relationship between water uptake and square root of time? The answer is as follows. We assume one-dimensional capillary flow. Application of mass conservation then gives: Dw

w2 w wx 2

ww wt

(2.152)

i.e., an equation, analogous to heat conduction in plane layers without source or sink. Capillary water sorption can be compared to a sudden temperature increase at the surface of a semi infinite medium with the temperature step replaced by a moisture content step from hygroscopic to capillary. However, the existence of a critical moisture content complicates the solution. Besides the starting and boundary condition t ” 0, 0 ” x ” ’: w = wH; t • 0, x = 0: w = wc, the moisture front (xfr) imposes two additional conditions: (1) during suction, moisture content there is always critical, (2) related water flow rate supplies the moisture needed to move the front over a distance dxfr per unit of time, or: t ! 0, x

xfr :

(1) w

wcr ,

§ ww · (2)  Dw ¨ ¸ © wx ¹ x

xfr

§ dx · wkr ¨ fr ¸ © dt ¹ x

xfr

Mathematically, x = xfr is a singular point. This excludes analytical solutions. Anyhow, two extreme cases allow an approximate one. Critical moisture content close to zero If hygroscopic (wH ) and critical moisture content (wcr ) are practically zero, then, for the step to capillary moisture content (wc ) at the surface, moisture content at a distance x from that surface in an infinitely long sample becomes:

w

§ · f ¨ 2 ¸ wc ¨ exp  q 2 dq ¸ ³ x ¨ Sq ¸ ¨© ¸¹ 2 Dw t  



(2.153)

The term above the brace is the inverse error function, see Figure 2.63. In the wet zone, critical moisture content is found at the moisture front. At the water contact, moisture content is capillary. For the quantity of water absorbed between t = 0 and t = t, the following holds: mc

2 wc

Dw S

t

Or, between capillary water uptake and square root of time a linear relationship emerges. A 2 wc Dw / S gives the water sorption coefficient, which convenes with a moisture diffusivity equal to: Dw

S§ A· 4 ¨© wc ¸¹

2

(2.154)

233

2.4 Moisture transfer

Figure 2.63. Capillary suction, moisture content in the material when critical moisture content is close to zero (wcr . wc).

For moisture content equal to critical, equation (2.153) gives the position of the moisture front. The water penetration coefficient becomes: B

§w · 2 erf c 1 ¨ cr ¸ © wc ¹

Dw

Materials which follow the assumption, however, are rare. Critical and capillary moisture content are hardly different Let us assume the moisture content profile is linear between sucking surface and moisture front. The mass balance is: wcr  wc dx 2 For x (Figure 2.64): x

2

Dw

Dw

wc  wcr dt x

wc  wcr wc  wcr

t

i.e. again the experimentally found linear relationship between moisture front and the square root of time. The water penetration coefficient is: B

2

Dw

wc  wcr wc  wcr

(2.155)

234

2 Mass Transfer

Figure 2.64. Capillary suction, moisture content in the material when critical and capillary hardly differ (wcr § wc)

Absorbed water between t = 0 and t = t equals (mc, kg/m2): mc

x

wc  wcr 2

Dw  wc  wcr  wc  wcr

t

The water sorption coefficient thus becomes:



Dw wc2  wcr2

A



giving as moisture diffusivity above critical: Dw

w

2 c

A2  wcr2



(2.156)

Moisture diffusivity function of moisture content In reality moisture diffusivity is a function of moisture content. Rewriting (2.152) gives: w § ww · ¨© Dw ¸ wx wx ¹

ww wt

(2.157)

If moisture diffusivity is unequivocally moisture content related, if the material volume is close to semi-infinite, if moisture content is w0 at time zero and if w(x = 0) remains constant for t • 0, then a Boltzmann transformation O x / t allows converting the partial differential equation into a differential equation of second order: 

O dw 2 dO

d dO

dw º ª «¬ Dw  w dO »¼

(2.158)

235

2.4 Moisture transfer

with moisture content depending on the Boltzmann variable O. In a [O, w] Cartesian coordinate system, all time related curves so melt together into one curve, with as surface to the O-axis: Om

³

w  O dO

C te

(2.159)

0

Moisture absorbed between t = 0 and t = t is: mw t

ª xm º « ³ w  x dx » «¬ 0 »¼ t

with xm Om t . As the integration for O is independent of time, combination with (2.159) gives as water sorption coefficient: A

C te

Om

³

w  O dO

(2.160)

0

This proves the relation mc A t is general provided that air flow has no influence and the function Dw (w) is unequivocal. When assuming a profile w(O), (2.160) allows calculating the moisture diffusivity as a function of water sorption coefficient and moisture content. Wind driven rain In general Wind driven rain absorption creates two successive boundary conditions. As long as the moisture content at the wetted surface stays below capillary, the sucked water flow rate (gws) remains equal to the wind driven rain intensity. Once the surface capillary is wet, a water film forms and capillary sorption occurs (Figure 2.65).

Figure 2.65. Rain absorption by a surface before a water film forms (wcr . wc, ws < wc until ws = wc).

236

2 Mass Transfer

Simulation with the simplified model Critical moisture content much lower than capillary As long as the moisture content at the surface (ws) lies below capillary, the boundary conditions are: t ” 0, 0 ” x ” ’: w = 0; t • 0, x = 0: –Dw (dw / dx)x = 0 = gws where gws is the wind driven rain intensity in kg/(m2 · s), assumed to be constant. The solution of the balance equation (2.152) then is: w

§ § 2 g ws ª Dw t x2 · x exp ¨   erf c ¨ « ¸ Dw ¬« S © 4 Dw t ¹ 2 ©2

x Dw

·º ¸» t ¹ ¼»

At the wetted surface (x = 0), moisture content equals: ws

2 g ws Dw

Dw t S

(2.161)

Once capillary, a water film forms and water uptake changes into horizontal suction from a water surface. The moment of film formation is given by: tf

S Dw wc2 2 4 g ws

S 2 A2 2 16 g ws

(2.162)

In other words, the smaller the water sorption coefficient of the material and the higher the wind driven rain intensity, the quicker rain runs off, resulting in an extra water load on the lower parts. Brick veneers and stuccoed walls reflect this difference. Most bricks have a high water sorption coefficient (0.2 ” A ” 0.8 kg/(m2 · s1/2 )), making run off rather exceptional. Water repellent stucco instead has a low water sorption coefficient (A < 0.005 kg/(m2 · s1/2 )), giving an almost instant run off, with water spreading over the lower parts. Until run off starts, the quantity of water absorbed is mw = gws tf . Just before film formation starts, the water flow rate absorbed (and wind driven rain intensity) thus is: g ws

wc

S Dw 4

1 tf

0.89 wc

Dw tf

0.79

A tf

If the solution for capillary sorption should apply for run off, then, after film formation, water sorption becomes: g ws2

wc

Dw S

1 tf

0.56 wc

Dw tf

A 2 t

g ws

2 S

or, when the film forms, the material suddenly absorbs less water. Physically, such discontinuity is impossible. Therefore, after film formation, a transition regime will occur between ‘constant water flow rate’ and ‘suction from a water surface by a uniformly wet material’. Mathematically, this transition results from the condition at the moment of film formation: no dry or uniformly wet material but one with a moisture profile. Critical and capillary moisture content hardly different We assume moisture content in the material changes linearly from the wetted surface down to the critical value at a depth x. As long as the sprayed surface (ws) is below capillary (wc), the water flow rate in the material is:

237

2.4 Moisture transfer

g ws

Dw

ws  wcr x

From the mass balance gws dt = ws dx, the position of the moisture front in the material becomes: x

Dw wcr g ws

2 § · 2 g ws t  1¸ ¨ 1 2 ¨© ¸¹ Dw wcr

while moisture content at the surface (ws) equals wcr + gws x / Dw. Once capillary wet (ws = wc), a water film forms. At this moment, the moisture front penetrated the material to a depth xfr = [Dw (wc – wcr)] / gws or, with Dw A2 / ( wc2  wcr2 ) : xfr

A2  wc  wcr g ws

The moment of film formation is given by: tf

A2 2 2 g ws

Drying Experiment Drying occurs when the relative humidity in the environment drops below the relative humidity at a surface. Experimentally, drying is studied by sealing five of the six sides of a water saturated sample, hanging it in a climate chamber at constant temperature and relative humidity and weighing regularly. When depicting the drying flow rate (= weight decrease per time unit) as a function of the average moisture content left in the sample, one obtains the curve of Figure 2.66. Two drying stages appear with a transition zone in between: a first at higher average moisture content with quasi constant drying flow rate and a second at lower average moisture content with rapidly decreasing drying flow rate.

Figure 2.66. Isothermal drying, the two stages.

238

2 Mass Transfer

Simulation with the simplified transfer model Does the simplified transfer model explain the experimental curve? To find out, take an above critical moist sample. The lateral sides and the rear side are vapour-sealed, resulting in one-dimensional drying at the front side. The sample hangs in a climate chamber at constant temperature (T) and relative humidity (I). In the saturated sample, relative humidity is 100%. As vapour pressure at the drying surface we have the saturation pressure for the surface temperature (psat,s Ts)). Drying flow rate to the environment is: g vd



E psat,s  I psat



(2.163)

with E the surface film coefficient for diffusion, I relative humidity in the chamber and psat saturation pressure at the chamber temperature. Since drying means evaporation, the flow rate carries the heat of evaporation. As a result, temperature and saturation pressure at the drying surface drop below those in the chamber. For the purpose of simplicity, we now assume the sample has an infinite thermal conductivity, while no heat transfer by radiation occurs. In such a case, saturation pressure at the drying surface equals the one in the chamber and (2.163) can be expressed as: g vd

E psat 1  I

(2.164)

Apparently, whatever the material is, drying only depends on the conditions in the chamber: air temperature, relative humidity and air velocity, this via the surface film coefficient. The higher the temperature, the lower relative humidity and the higher the air velocity, the faster the sample dries. People apply this principle unreflectively. Laundry is hung outside when the weather is sunny (high temperature) and dry (= low relative humidity). When windy, ( E higher), drying goes even faster. Drying decreases moisture content at the front surface. As a consequence, a moisture content gradient initiating moisture flow develops in the sample towards that surface, where at any time, the following applies: x = 0, –Dw (dw / dx)x = 0 = E psat (1 – I). The situation of capillary moisture flow to the drying surface, evaporation there and decreasing moisture content in the sample – is called the first drying stage. That stage continues until moisture content at the drying surface turns critical. Then, the moisture diffusivity there ((Dw)x = 0) drops to zero and moisture transfer at that surface becomes vapour only. The moisture front then detaches and shifts into the sample (Figure 2.66), changing the drying flow rate into: g vd

psat 1  I 1 PN x E

(2.165)

with P the vapour resistance factor of the sample material and x the distance between drying surface and moisture front in the sample. The second drying stage starts. As the distance increases, drying rate decreases. This lasts until the sample regains hygroscopic equilibrium with the environment. Once in the second stage, the sample quickly appears superficially dry. Too often, this is taken as proof the material is dry, while in fact it may still contain quite some moisture. The first drying stage is described by equation (2.152) with starting and boundary conditions of (x = 0: dry surface, x = d: vapour tight surface): t ” 0, 0 ” x ” ’: w = w0; t • 0, x = 0: dw / dx = –gvd / Dw, x = d: dw / dx = 0. The solution is:

239

2.4 Moisture transfer

w

w0 

g vd t g vd d  ˜ d Dw

­ 3 x2  6 d x  2 d 2 2 ª  1 n § § n S d  x · º ½ n 2 S2 t · °   D exp cos « ¦ ¨ w ¸ ¨© ¸¹ » °¾ ® d S2 «¬ n 2 d2 ¹ 6 d2 © »¼ ° °   

¯ ¿ The term above the brace describes how the moisture gradient moves from the drying surface to the vapour-tight rear surface. Once there, it becomes negligible and the sample is left with a parabolic moisture profile: w

w0 

g vd t g vd d § 3 x 2  6 d x  2 d 2 ·  ¸ d Dw ¨© 6 d2 ¹

(2.166)

The moisture content is lowest at the drying surface (x = 0): w

w0 

g vd t g vd d  d 3 Dw

(2.167)

As average moisture content at any moment, we have: w

§ g t· w0 ¨1  vd ¸ © w0 d ¹

(2.168)

The second drying stage starts when the drying surface turns critically moist: tsecond

§ w  wcr d ·  d¨ 0 3 Dw ¸¹ © g vd

(2.169)

Moisture then left in the sample establishes the transition moisture content (wtr): wtr

wcr 

g vd d 3 Dw

(2.170)

Equation (2.170) shows the difference between critical and transition moisture content increases at higher drying rates. Hence, faster drying during the first stage does not guarantee earlier dryness. The second stage simply starts at a higher transition moisture content. For highly capillary materials (Dw large), the difference between transition and critical is smaller than for less capillary materials (Dw small) at equal drying flow rate. Low diffusion resistance factor, large moisture diffusivity and low critical moisture content constitute favourable conditions for easy drying. Bricks are an example of this. Instead, less capillary materials with critical moisture content near capillary, such as concrete, show a short first and a very long second drying stage. They dry slowly. The second drying stage is approached as follows. Assume transition and critical moisture content (wtr – wcr) are close, which simulates reality the best for a low drying flow rate during the first stage. Then, drying causes a withdrawing moisture front at critical moisture content. Mass equilibrium consequently advances (x is the moisture front, x = 0 the drying surface):

240

2 Mass Transfer

psat 1  I dt 1 PN x E

 wcr  wH dx

(2.171)

with wH hygroscopic moisture content at the prevailing climate conditions. In steady-state and with the vapour resistance factor ( P) constant, the position of the moisture front (x) becomes: x

2 psat 1  I P N E 2 1 § ¨ 1 P N E ¨© wcr  wH

· t  1¸ ¸¹

(2.172)

Once the vapour resistance P N x is much larger than the surface resistance 1 / E, the equation simplifies to: x

2 psat 1  I P N  wcr  wH

t

(2.173)

Or the drying results in a square root relation between the position of the moisture front and time. In a [ x, t ] axis system, a straight line represents the second drying stage. Multiplying equation (2.173) on the left and right by wcr – wH gives the moisture quantity drying: md

2 psat 1  I  wcr  wH PN

t

Ac t

(2.174)

When both critical and hygroscopic moisture content are known, this equation allows calculating the vapour diffusion factor (P) from a drying test. A higher value clearly slows drying. Despite all simplifications, the model is quite usable at low drying rates and easy heat supply. For high drying rates, the isothermal character gets lost. The heat of evaporation lowers temperature and flow rate at the drying front, while this temperature varies. Indeed, as the front shifts deeper into the material and drying decreases, less heat of evaporation is absorbed and a move back to md Ac t occurs. Hence, the [md , t ] curve will first bend towards the t -axis to change afterwards into a straight line. Drying of a wet layer in a composite assembly The drying model is easily extended to composite assemblies, which have a wet interface between two non-capillary layers (interface j) or which contain one or more moist layers (moisture content beyond critical, boundaries k, l ). In case of ‘equivalent’ diffusion only, the vapour flow rate between interface j or boundaries k and l and the environment at both sides (steady-state) looks like (Figure 2.67): Wet interface g md

§ psat, j  p1 psat, j  p2 · ¨  ¸¹ Z1 Z2 ©

p2  psat, j

g md

§ psat, k  p1 psat, l  p2 · ¨  ¸¹ Z1 Z2 ©

p2  psat, l

Z2



psat, j  p1 Z1

(2.175)

Wet zone Z2



psat, k  p1 Z1

(2.176)

2.4 Moisture transfer

241

Figure 2.67. On the left drying of a moist interface, on the right drying of a moist layer in an assembly with drying condensation as a consequence.

with psat, j vapour saturation pressure in interface j and psat,k and psat, l vapour saturation pressures at the boundaries k and l. In the wet zone, relative humidity is 100% and vapour pressure equals saturation pressure. Z1 and Z2 are the diffusion resistances between j and the two environments at both sides of the assembly, or, between boundary k and environment 1 and between boundary 1 and environment 2. p1 is vapour pressure in environment 1, p2 vapour pressure in environment 2. In the [Z, p] plane, the two terms in both equations represent the slope of the line interconnecting the saturation pressure in the wet interface j or the boundaries k and l with the points [0, p1] and [ZT, p2]. At the cold side (assume this is the environment 1) the line may intersect saturation pressure. Part of the vapour that diffuses from j or k then condenses elsewhere in the assembly. We call this drying condensation. Where, how much and how to correct vapour pressure is given by the tangents to the saturation curve with saturation pressure psat, j or

242

2 Mass Transfer

psat,l and vapour pressure p1 as starting points (Figure 2.67). Drying condensation could be a problem in any assembly that contains building moisture, where one or more layers absorbed rain or which suffered in two or more interfaces or zones from interstitial condensation of vapour, produced inside. If advection supplements diffusion, drying occurs faster but drying condensation amplifies. Interstitial condensation by diffusion of vapour, released inside As explained, vapour flow by ‘equivalent’ diffusion or worse, advection across composite assemblies, can cause interstitial condensation. The question then becomes, does condensation go on forever? Take an assembly made of a capillary (Dw > 0), vapour permeable material with low thermal conductivity and diffusion resistance Z2 and a vapour retarding outer finish, diffusion resistance Z1. Advection is excluded. Consider yearly average climate conditions. Temperature and vapour pressure at the warm side (T2, p2, Z = Z1 + Z2) causes interstitial condensation behind the finish (Z = Z1), condensation flow rate (gwc): g wc

g v2  g v1

p2  psat, Z1 Z2



psat, Z1  p1 Z1

With p1 vapour pressure at the cold side and psat,Z1 vapour saturation pressure at Z1. The capillary material absorbs this condensate, initiating a water flow rate, given by: Dw

w  wcr x

p2  psat, Z1 Z2



psat, Z1  p1 Z1

As a result, a wet zone develops in the capillary layer, extending slowly from the the finish back to the warm side. In this zone, moisture content is slightly above critical, while at the moisture front it is critical (wcr). In the wet zone, vapour pressure equals saturation. The moving front forces the ingoing vapour pressure line in the [Z, p]-plane to shift and rotate from the ‘vapour pressure inside/saturation pressure in the interface Z1’ contact to the ‘vapour pressure inside/ saturation pressure at the moisture front ( psat, f )’ contact (Figure 2.68). As a result, the ingoing

Figure 2.68. Yearly accumulating condensate. Limit state in the case when a capillary, vapour permeable material has a non-capillary, vapour retarding finish at the cold side.

243

2.4 Moisture transfer

vapour flow rate gv2 drops. Once the moisture front arrives at the point where the ingoing rate gv2 equals the outgoing one (gv1), condensation stops. A limit state is reached. In the wet zone of course, a capillary water flow from cold to warm persists in equilibrium with an identical vapour flow rate from warm to cold. The ingoing vapour flow rate left equals: g v2 f

g v1

p2  psat, f

psat,f  psat,Z1

Z2  P N dw

Pw N d w

 Dw

wZ1  wcr dw

(2.177)

with wZ1 moisture content in interface Z1, psat, f vapour saturation pressure at the moisture front, P vapour resistance factor of the dry capillary material, Pw vapour resistance factor of the critically wet material and dw thickness of the wet zone. Solving for wZ1 gives: wcr 

wZ1

· 1 § psat, f  psat, Z1  g v1 d w ¸ ¨ Dw © Pw N ¹  

(2.178)

The term above the brace is now less than a fraction of critical moisture content or, wZ1 | wcr . Limit state hence is a critically moist zone in the capillary material of thickness dw, extending from the interface with the finish inward. The distance between the intersection with saturation and the backside of the finish (Z = Z1) gives its thickness dw, which increases with higher vapour pressure at the warm side. Drawing a parallel with the outgoing tangent through the vapour pressure indoors in a Glaser diagram, approximates correct thickness fairly well. The time to reach limit state calculates from: wcr dxw

psat, Z1  p1 · § p2  psat, f ¨© Z  P N x  ¸¹ dt Z1 2

(2.179)

where x is thickness of the wet zone. To solve the equation, the relationship between saturation pressure (psat, f) and temperature at the moisture front and thus, between psat, f and x, has to be known. An approximate solution assumes: psat, f

a Z  psat, Z1 , psat, Z1

C te , Z

PN x

with as result: PN t wcr

a1 Z  a2 ln 1  a3 Z

(2.174)

where: a1

g v1

1 a  g v1 psat, Z1  p1 Z1

a2

a  g v1 Z 2  g v2  g v1 g v2

a3

Z2 a  g v1

p2  psat, Z1 Z2

Diffusion resistance (Zw) and thus thickness of the wet zone (xw ) are then known as an implicit function of time. For the condensation deposit in kg/m2 after t days, we have: mc = 86,400 wcr Z(t) / (P N). Table 2.6 Lists limit states for other layer combinations.

244

2 Mass Transfer

Table 2.6. Limit state in case of annual accumulation of condensate. Layer Cold side

Limit state

Warm side

Case (2) Vapour retarding, noncapillary

Vapour permeable, non-capillary

The layer at the warm side saturates slowly over a thickness dw from the interface between both inwards with dw approximately equalling the distance between the initial condensation interface and the intersection with the saturation curve of a tangent parallel to the outgoing one across the vapour pressure inside.

Vapour permeable, non-capillary

The layer at the cold side gets critically wet over a thickness dw from the interface between both outwards with dw approximately equalling the distance between the initial condensation interface and the intersection with the saturation curve of a tangent parallel to the ingoing one across the vapour pressure outside.

Vapour permeable, non-capillary

Interstitial condensation starts in the interface between vapour retarding finish and capillary layer, the interface between capillary and non-capillary layer at its warm side, (3) the capillary layer itself. At a limit state, the capillary layer is wet above capillary, while in the interface with the non-capillary layer at the warm side water drips or runs off.

Vapour permeable, capillary

Both layers turn critically wet by back-sucking condensate from the interface in between. At a limit state, the thickness of the critically moist layer in both is such that ingoing and outgoing vapour pressures in the [Z, p]-plane have the same slope. Limit state cannot be determined graphically. We assume:

Case (3) Vapour permeable, capillary

Case (4) Vapour permeable, capillary layer, finished at its cold side with a vapour retarding lining Case (5) Vapour permeable, capillary

d1

1 § A12 · , d2 wkr,1 ¨© A12  A22 ¸¹

1 § A22 · wkr,2 ¨© A12  A22 ¸¹

The assessment whether or not interstitial condensation is acceptable therefore not only depends on the nature of the material in or against which condensation occurs and the quantity deposited, but also on the limit state. Case (4) is most negative. We have it each time a capillary layer sits between a non-capillary thermal insulation at the warm side and a vapour retarding finish at the cold side. An example in cold climates is the deck of a ventilated low-sloped roof. A mineral wool insulated timber framed wall with vinyl paper finish on the inside gypsum board lining and a vapour permeable outside finish causes problems in warm, humid climate.

245

2.5 Problems

2.5

Problems

(17) Outside, the monthly average temperature is 0 °C for a dew point of –2 °C. Inside, we have 22 °C. The lowest temperature factor along the envelope is 0.4, while the daily average vapour release indoors reaches 20 kg. What outside air flow in m3/h is needed to keep mould risk close to zero (which means a surface relative humidity below 80%)? Answer Vapour pressure outside: pe

§ 22.44 T d,e · 611 exp ¨ ¸ © 272.44  T d,e ¹

518 Pa (T d,e is the dew point outside)

Lowest inside surface temperature: Ts

0  22 ˜ 0.4

8.8 °C

Allowable vapour pressure inside: pi

§ 17.08 Ts · 0.8 ˜ 611 exp ¨ © 234.18  Ts ¸¹

907 Pa

Ventilation flow needed: Va

Rv Ti Gv,P pi  pe

462 ˜ 295.15 ˜ 907  518

20 24

292 m3 /h

If risk of surface condensation was to be avoided, then 184 m3/h should suffice, i.e., a ventilation flow which is only 63% of the one needed to exclude mould from developing. (18) Assume the ventilation flow calculated in problem (17) is delivered by an extraction fan. The designer forgot to provide the necessary air inlets. As a result, air leakage through the envelope has to do the job. Air permeance equals 0.083 'Pa0.37 kg/(s ˜ Pa) . What pressure excess indoors is needed to generate the necessary ventilation flow? Answer The relation between air flow and air pressure difference is given by: Ga

K a 'Pa

0.083 'Pa0.63

A volumetric flow of 292 m3/h means a weight related flow of 345.3 kg/h or 0.095922 kg/s. Pressure excess indoors: 'Pa = (0.095922 / 0.083)1 / 0.67 = 1.24 Pa. That result allows concluding that the envelope must be highly air permeable. In fact, at a pressure difference of 50 Pa, we must get a volumetric air flow of 3468.5 m3/h, which is far too high. (19) Repeat problem (18) in case the air tightness of the envelope is 10 times higher, but with the same air pressure difference exponent. What is your conclusion?

246

2 Mass Transfer

(20) Outside, the temperature equals 34 °C for a relative humidity 80%. Indoors, the air conditioning provides 24 °C at a relative humidity of 60%. The building has a volume of 600 m3. The ventilation rate is 0.6 h–1. The daily mean vapour release equals 15 kg. How much vapour must be removed by the system (in g/h) to keep to 60%? Could we influence the quantity by selecting appropriate glazing? Answer Let us start from the formula describing the inside vapour pressure excess: R Ti X nV

'pi,e

with X the vapour flow to be removed from the outside air. In the case at hand, vapour pressure excess reaches ª § 17.08 ˜ 24 · § 17.08 ˜ 34 · º 611 «0.6 ˜ exp ¨ ¸¹  0.8 ˜ exp ¨© ¸ © 234.18  24 234.18  34 ¹ »¼ ¬

2468 Pa

which gives us a total removal of 6471 g/h vapour from the outside air, plus the 625 g/h of vapour released inside! The glazing system chosen has no impact on that number. In fact, as the outside temperature surpasses the inside temperature, the inside surface temperature will be higher than the temperature inside. Condensation inside on the glass is prevented that way. (21) Take a dwelling where the bathroom is directly accessible from the master bedroom. The bathroom has a volume of 15 m3, while the daily vapour release there reaches 2 kg. In the bedroom, with a volume of 50 m3, 1 kg/day additionally is produced. What average vapour pressure will be measured in the bedroom when the ventilation air which enters the bathroom at a rate of 1.7 ach (related to the air volume in the bathroom) moves to the bedroom and mixes there with an additional outside air flow of 30 m3/h. Outside, we have a temperature –10 °C and a relative humidity 90%. The temperature in the bathroom is 24 °C, in the bedroom 18 °C. Answer First we calculate the vapour flow which moves to the bedroom from the bathroom. Vapour pressure in the bathroom is: pi,bath

§ Gv,P · pe  R Ti ¨ © n V ¸¹ bathroom

with: pe

§ 22.44 ˜ 10 T d,e · 0.9 ˜ 611 exp ¨ © 272.44  10 ¸¹

234 Pa

So, pi,bath becomes: pi,bath

§ 2 · ¨ ¸ 234  462 ˜ 297.15 ¨ 24 ¸ 1.7 ˜ 15 ¨© ¸¹

683 Pa

247

2.5 Problems

with a vapour flow from the bathroom to the bedroom equalling 0.127 kg/h. The vapour balance in the bedroom then becomes: 0.127 

30 ˜ 234 1 pi,bed 30  1.7 ˜ 15   462 ˜ 291.15 24 462 ˜ 291.15

0

giving as vapour pressure: 1 · § 0.127  30 ¨ ¸ § · 24 234 ¨  462 ˜ 291.15 ¨ © 30  1.7 ˜ 15 ¹¸ 30  1.7 ˜ 15 ¸ ¨© ¸¹

pi,bed

535 Pa

Thus, between bathroom and bedroom vapour pressure drops. Main reason is the additional ventilation flow in the bedroom. (22) Repeat problem (21) for the case when the only ventilation in the bedroom comes from the airflow leaving the bathroom. Conclusions? (23) Given a cavity wall with section: Layer

Thickness m

¼-value W/(m · K)

Air permeance m3/(m2 · s · Pa)

Inside leaf (no fines blocks)

0.14

1.00

33 · 10–4 'Pa–0.42

Cavity fill (mineral fibre)

0.10

0.04

0.00081

Air layer

0.01

0.067

’

Brick veneer

0.09

1.00

0.32 · 10–4 'Pa–0.19

Calculate the air flow rate across the wall, knowing that wind induces a 6 Pa average pressure difference over it (higher outdoors than indoors) and assuming the flow develops perpendicular to the wall’s surface. Answer The air flow rate is given by: 6

ga

1 1 0.58 ga 1 33 ˜ 104 0.58







1  0.00081

1 1 0.81 ga



0.32 ˜ 104



1 0.81

That formula has to be solved by iteration, starting from an assumed air flow rate in the right hand term. As a starting value we use the rate in case the wall consisted of a fill only: 6 · 0.00081 = 0.00486 m3/s. Successive iterations give as final result: ga = 0.000133 m3/s. The figure below shows that already an excellent approximation is found after a few iteration steps. Thus, 0.48 m3 of air passes through each square meter of cavity wall per hour!

248

2 Mass Transfer

Air pressures differences across the successive layers are found by introducing air flow rate into the three flow equations (ga = Ka 'Pa ). The interface air pressures are then given by: 5

Pa

Pa,o  ¦ 'Pa i 1

See the figure below on the left. Apparently the veneer wall bears the largest pressure difference, which means excessive wind load may threaten its stability.

Rain will also easily seep from the head joints and run off at the veneer’s backside. In other words, such situation should be avoided! A much better solution is to have the largest pressure difference over the inside leaf. This can be realized by plastering. Adapted section:

249

2.5 Problems

Layer

Thickness m

¼-value W/(m · K)

Air permeance m3/(m2 · s · Pa)

Render

0.01

0.3

Inside leaf (no fines blocks)

0.14

1.00

}

Cavity fill (mineral fibre)

0.10

0.04

0.00081

Air layer

0.01

0.067

’

Brick veneer

0.09

1.00

0.32 · 10–4 'Pa–0.19

0.1 · 10–4 'Pa–0.23

That measure decreases the infiltration by 75%, down to 0.12 m3/(m2 · h). Also the interface air pressures change substantially as the figure up on the right shows. In fact, the plaster faces the largest difference now, while the veneer wall is left with some 20% of the total. This creates a much better situation: less wind load and less rain seeping from the head joints. (24) Let us return to the cavity wall, problem (23) started with. The temperature outdoors is –10 °C. Indoors it is 21 °C. Calculate the temperatures in the wall and evaluate the heat loss by conduction at the inside surface in case the wall faces a 4 m/s wind, resulting in a 6 Pa pressure difference over the wall (higher outside than inside). The surface film coefficient indoors is 8 W/(m2 · K), outdoors 22.2 W/(m2 · K) Answer As a temperature difference over a wall in windy weather invokes combined heat and air flow, temperatures across the assembly become: T

T e   Ti  T e F1  R with F1  R

1  exp ca g a R 1  exp ca g a RT

The original cavity wall shows an infiltration rate of 0.000133 m3/s or 0.00016 kg/(m2 · s), giving unit enthalpy flow of 0.1602 W/(m2 · K) (ca = 1008 J/(kg · K)). Temperatures in the interfaces then become:

Layer

d m

¼-value W/(m · K)

R F1(R) cR m2 · K/W m2 · K/W –

Temp. °C 21.0

Surface film resistance (Ri )

0.125

0.125

0.0514

19.4

Inside leaf (no fines blocks)

0.14

1.00

0.14

0.265

0.1077

17.7

Cavity fill (mineral fibre)

0.10

0.04

2.5

2.765

0.9260

–7.7

Air layer

0.01

0.067

0.15

2.915

0.9654

–8.9

Brick veneer

0.09

1.00

0.09

3.005

0.9885

–9.6

0.045

3.05

1

Surface film resistance (Re )

–10

250

2 Mass Transfer

As a figure:

The full line gives the results in case infiltration intervenes. The dashed line accounts for conduction only. Clearly, infiltration cools down the wall a little. Heat flow rate by conduction at the inside surface is: q

F2  Ti  T e

with F2

ca g a exp ca g a RT 1  exp ca g a RT

RT = 3.05 m2 · K/W and ca ga = 0.1602 W/(m2 · K) give: 12.9 W/m2. Without infiltration, the result should have been 10.2 W/m2. Infiltration increases conduction. Simultaneously, an enthalpy flow of 3.36 W/m2 passes the wall. (25) Turn now to the cavity wall rendered at the inside, see problem (23). Calculate the temperatures across the wall and heat flow rate by conduction at the inside surface for the same conditions indoors and outdoors as in problem (24). What are the conclusions? (26) The cavity wall of problem (23) is 2.7 m high. Temperature outdoors is –10 °C while indoors it is 21 °C. Given the wall is air permeable, thermal stack flow develops. What happens if a window-less one storey high wall is built that way? Answer In the case considered, the neutral plane for stack sits at mid-height, resulting in a triangular pressure profile. Just above the floor an underpressure of 1.8 Pa forces outside air to infiltrate. Above mid-height inside air exfiltrates, with a maximum just below the ceiling at an overpressure of 1.8 Pa.

251

2.5 Problems

The air flow rate at different heights is calculated the same way as done in (23). For the result: see the table and figure below. Height m

Air flow rate m3/(m2 · h)

2.7

0.180

2.4

0.147

2.1

0.112

1.8

0.074

1.5

0.030

1.2

–0.030

0.9

–0.074

0.6

–0.112

0.3

–0.147

0.0

–0.180

252

2 Mass Transfer

The air flow profile, stack generates, is close to linear. Total infiltrating and exfiltrating air flow equals 0.13 m3/h per meter run, which is not much. As a consequence, the heat flow by conduction on the inner surface will vary along the height: highest just above the floor, pure conduction at mid-height and lowest under the ceiling. See the figure above. Calculating those heat flow rates does not differ from the way done in problem (24). (27) Given a timber framed wall assembly: Density kg/m3

¼-value W/(m · K)

‡-value –

Layer

d m

Gypsum board lining

0.012

–

0.1

Insulation (mineral fibre)

0.15

20

0.04

OSB-sheathing

0.01

400

0.13

15

3.5 · 10–4 'Pa–0.31

Building paper

0.0005

–

0.2

200

4.2 · 10–4 'Pa–0.4

Cavity

0.03

–

0.18

0

’

Brick veneer

0.09

1600

0.9

5

0.32 · 10–4 'Pa–0.1

Air permeability m3/(m2 · s · Pa) 3.1 · 10–5 'Pa–0.19

12 1.2

0.00081

Will the design suffer from unacceptable interstitial condensation? If so, what vapour retarder should be added where? Boundary conditions to be considered (north-east orientation): J

F

M

A

2.3

3.3

5.9

8.8

M

J

J

A

S

O

N

D

8.8

5.5

2.9

Outdoors Air temp. (°C)

13.5 16.1 16.8 15.7 13.5

Eq. temp. (°C)

3.1

4.2

7.2

10.6 16.1 19.1 19.9 18.6 16.1 10.6

6.7

3.8

Vapour press. (Pa)

619

675

825

997 1277 1427 1470 1406 1277 997

804

653

Air temp. (°C)

20.0 20.3

21

21.8 23.1 23.8

Vapour press. (Pa)

1224 1253 1330 1418 1561 1638 1660 1627 1561 1418 1319 1242

Indoors 24

23.7 23.1 21.8 20.9 20.2

Surface resistances for heat transfer: inside 0.13 m2 · K/W, outside 0.04 m2 · K/W. Calculations, if any, are done with the equivalent temperature. That value accounts for solar gains, undercooling and the exponential relationship between temperature and vapour saturation pressure Answer Suppose ‘equivalent’ diffusion is the only mechanism. To determine if interstitial condensation occurs, we apply a Glaser analysis using the climate data for the coldest month, January.

253

2.5 Problems

Step 1: Temperatures in all interfaces Thickness m

R m2 · K/W

¤R m2 · K/W

Temp. °C

0.0000 0.0000 0.0120 0.1620 0.1720 0.1720 0.2025 0.2925 0.2925

0.00 0.13 0.12 3.75 0.08 0.00 0.17 0.10 0.04

0.00 0.13 0.25 4.00 4.08 4.08 4.25 4.35 4.39

20.0 19.5 19.0 3.9 3.6 3.6 2.9 2.5 2.3

Step 2: Vapour saturation pressure in all interfaces Temp. °C

psat Pa

20.0

2348

19.5

2273

19.0

2206

3.9

808

3.6

791

3.6

791

2.9

754

2.5

732

2.3

724

Step 3: Vapour pressure in all interfaces Thickness

½d

¤½d

m

m

m

Vapour pressure Pa

0.0000 0.0000 0.0120 0.1620 0.1720 0.1720 0.2025 0.2925 0.2925

0.00 0.13 0.12 3.75 0.08 0.00 0.17 0.10 0.04

0.00 0.13 0.25 4.00 4.08 4.08 4.25 4.35 4.39

20.0 19.5 19.0 3.9 3.6 3.6 2.9 2.5 2.3

254

2 Mass Transfer

Step 4: Water vapour saturation and water vapour pressure intersect, or, interstitial condensation is a fact. To get the condensation interface, the correct water vapour pressures and the deposit, we redraw the wall in a [vapour diffusion resistance/vapour pressure]-plane and construct the tangents to the saturation line. For the result, see the figure below. Condensation is deposited at the backside of the OSB-sheathing and wets it stepwise.

Amounts for January: gc

842  619 § 1224  842 · 2  ¨© ¸ ˜ 3600 ˜ 24 ˜ 31 0.41 kg/m 9 1.72 ˜ 10 5.62 ˜ 109  1.72 ˜ 109 ¹

As the sheathing is 1 cm thick and has a density of 400 kg/m3, that result reflects an increase in moisture ratio by weight of 10%! The condensation/drying cycle over a year looks like: J

F

M

A

M

J

J

A

S

O

N

D

Deposit

0.41 0.33 0.16 –0.16 –0.95 –1.48 –1.72 –1.43 –0.92 –0.17 0.18 0.38

Acc.

0.98 1.31 1.46 1.30 0.35

As a figure:

0

0

0

0

0

0.18 0.56

255

2.5 Problems

The deposit accumulates during winter up to a moisture ratio by weight of 36.5% which is unacceptable for a timber-based material. Drying next springtime goes fast. In summer, nothing seems to happen, at least, as long as hygroscopicity and wind-driven rain are not accounted for. As the deposit is far too high, the question then comes what vapour retarder is needed to return to acceptable levels. A rule of the thumb is deposit should not give a moisture ratio increase beyond 3% kg/kg in timber based materials, which allows an end of the winter maximum of 0.12 kg/m2. An iterative calculation shows a retarder with vapour permeance below 3.3 · 10–10 s/m at the inside of the thermal insulation suffices. This is far from extreme. Expressed in terms of equivalent diffusion thickness, we need 0.56 m, a value a vapour-retarding paint may give. Does this result reflect reality? Three important elements are overlooked: advection, hygric inertia and wind driven rain sucked by the capillary veneer. Correction 1 Assume the difference in monthly mean inside and outside air temperature to be the main cause for air egress across the upper part of the 2.7 m high wall. For reasons of simplicity, the neutral plane is fixed at mid height. The maximum air pressure difference below the ceiling then becomes: J 'Pa (Pa)

F

M

A

M

J

J

A

S

O

N

D

0.41 0.33 0.16 –0.16 –0.95 –1.48 –1.72 –1.43 –0.92 –0.17 0.18 0.38

Applying steady state advection across a flat composite assembly gives as condensation deposit per month and accumulated deposit: J

F

M

A

M

J

J

A

S

O

N

D

Deposit (kg/m2)

0.54 0.43 0.25 –0.10 –0.92 –1.47 –1.71 –1.42 –0.89 –0.10 0.28 0.50

Acc. (kg/m2)

1.32 1.75 2.00 1.90 0.98

As a figure:

0

0

0

0

0

0.28 0.78

256

2 Mass Transfer

Although the continuous gypsum lining ensures good air tightness, exfiltration remains strong enough to increase the accumulated condensate behind the OSB-sheathing on top of the wall with some 21%, resulting in a change in moisture ratio by a weight of 44%, i.e. far too high. Improving air tightness is thus the first thing to do, in combination with a better vapour resistance at the inside of the thermal insulation, as calculated above. At mid-height diffusion remains the only driving force intervening, while just above floor level, infiltration will lower the amounts. Correction 2 Calculating the impact of sorption/desorption as correctly as possible demands an appropriate software tool such as Wufi, Match or others. A first approximation is possible, assuming gypsum board and OSB have constant material properties, included a constant specific moisture content, and by modelling the timber framed wall as a serial network of two capacitances, coupled by resistances, whereby the cavity between building paper and brick veneer is supposed to be so well ventilated that the air passing through is at the same vapour pressure as outside. Equations (forward differences): Capacitance 1: gypsum board t t t I1t It2  psat,1 I1t pit  psat,1 psat,2  N d gyps 1 N d gyps N d OSB Pgyps Pgyps   Pinsul N dinsul  POSB 2 2 2 Ei

Ugyps [gyps d gyps

I1t  't  I1t 't

Capacitance 2: OSB t t psat,1 I1t  psat,2 It2

Pgyps

N d gyps 2

 Pinsul N dinsul  POSB

N d OSB 2

 POSB

t t psat,e Ite  psat,2 It2 N dOSB 1  Pbp N d bp  Ecavity 2

It2 't  It2 't Solving that system of two equations demands knowledge of the boundary conditions. We refer to the climate data above. Each month is modelled as a step function with the temperatures and saturation pressures invariable. If, in such a case, we reach the steady state vapour pressure solution before the month ends, then Glaser reflects vapour addition in winter quite well. The time interval used is one day. UOSB [OSB d OSB

Additional material properties: Layer

U¿d kg/m2

Gypsum board lining Insulation (mineral fibre) OSB-sheathing Building paper

0.46 0 0.93, 2.83 beyond 90% RH 0

257

2.5 Problems

Starting conditions: 80% relative humidity in the OSB, 50% relative humidity in the gypsum board. The solution shows that, if calculation starts on January the 1st, hygroscopic moisture uptake progresses until the 29th of that month, absorbing 0.43 kg/m2. See the figure.

Afterwards, condensation at the backside of the OSB starts, depositing 0.048 kg/m2 during the two days left. That deposit causes droplets to be formed, ending in runoff to the sill plate. In February and March the total of all moisture deposits as condensate that runs off is 0.485 kg/m2. In April, drying starts. Correction 3 Inserting an air and vapour retarder at the inside of the thermal insulation is valid as a solution as long as wind driven rain cannot wet the brick veneer. Take a south-west oriented wall, the rain direction in Western Europe. In cool, humid climates, drying of a brick veneer, wetted by successive rain events, proceeds so slowly that it stays at 100% relative humidity, surely from November till March but many times year around. Assume the building paper with an equivalent diffusion thickness of 0.1 m has been replaced by a high permeance wrap with an equivalent diffusion thickness of 0.01 m. Boundary conditions are: J

F

M

A

M

J

J

A

S

O

N

D

Air temp. (°C)

2.3

3.3

5.9

8.8

Eq. temp. (°C)

5.5

7.0

9.2

8.8

5.5

2.9

12.3 17.3 20.0 20.8 19.7 17.3 12.3

8.8

6.1

Vapour press. (Pa)

619

675

825

997 1277 1427 1470 1406 1277 997

804

653

Air temp. (°C)

20.0 20.3

21

21.8 23.1 23.8

Vapour press. (Pa)

1224 1253 1330 1418 1561 1638 1660 1627 1561 1418 1319 1242

Outdoors 13.5 16.1 16.8 15.7 13.5

Indoors 24

23.7 23.1 21.8 20.9 20.2

In winter, things hardly change. Glaser gives some condensation at the backside of the OSB, with an accumulated maximum of 0.47 kg/m2. In reality, that amount includes hygroscopic wetting of the OSB, followed by a little condensation deposit. When applying the method, the brick veneer should be kept at water vapour saturation pressure from November until March.

258

2 Mass Transfer

In summer, however, when the veneer has first been wetted by rain to heat up then by the sun, solar-driven inward vapour flow develops. As a result, condensation deposits on the wrap. On a sunny day, the amount reaches 0.16 kg/m2, which is close to the vapour flow adsorbed by the OSB during the whole month of December! Without wrap, that condensate wets the OSB! The figures below show that the wet veneer situation leads to very high vapour saturation pressures during sunny summer weather.

(28) Given a pitched roof assembly: Layer

d m

Density kg/m3

Lathed ceiling Insulation (mineral fibre) Underlay (fibre cement) Cavity

0.01 0.15 0.0032 0.06

450 20 – –

Tiles

0.012

1800

¼-value W/(m · K)

‡-value –

Air permeability m3/(m2 · s · Pa)

0.14 0.04 0.2

85 1.2 45 0

2 · 10–4 'Pa–0.32 0.00081 4.210 · 10–4 'Pa–0.4 ’

20

1.610 · 10–2 'Pa–0.5

Oeq = 0.33 1

259

2.5 Problems

Will that design suffer from unacceptable interstitial condensation? If so, what vapour retarder should be added where? Boundary conditions to be considered (slope: 30°, looking SW): J

F

M

A

Air temp. (°C)

2.3

3.3

5.9

8.8

Eq. temp. (°C)

3.9

5.2

Vapour press. (Pa)

619

M

J

J

A

S

O

N

D

8.8

5.5

2.9

8.7

12.6 19.1 22.5 23.5 22.1 19.1 12.6

8.2

4.7

675

825

997 1277 1427 1470 1406 1277 997

804

653

Air temp. (°C)

20.0 20.3

21

21.8 23.1 23.8

Vapour press. (Pa)

1224 1253 1330 1418 1561 1638 1660 1627 1561 1418 1319 1242

Outdoors 13.5 16.1 16.8 15.7 13.5

Indoors 24

23.7 23.1 21.8 20.9 20.2

Surface resistance for heat transport: inside 0.13 m2 · K/W, outside 0.04 m2 · K/W. Stack height above the neutral plane 3 m. Calculations, if any, have to be done with the equivalent temperature. This monthly mean accounts for solar gains, under-cooling and the exponential relationship between temperature and vapour saturation pressure. The underlay is capillary. A tiled roof cover is permanently wind washed. Do not account for this in the diffusion only mode. Take it into account when considering advection. Then, the cavity below, the tiled cover and the norm outside surface film resistance merge into one thermal resistance of 0.1 m2 · K/W, and a vapour resistance of zero. Answer It is up to the reader to solve the problem and to formulate comments. (29) Assume the timber framed wall of problem (26) was designed for a cold climate with a gypsum board lining, finished with vinyl wallpaper as air and vapour retarder. The assembly now is moved to a hot and humid climate, where cooling and air drying is needed to keep the building liveable. The indoors, however, is cooled to an air temperature not exceeding 20 °C. What problems could be expected in case the veneer stays dry or is wetted daily by wind driven rain. Layer properties: ¼-value W/(m · K)

Layer

d m

Density kg/m3

Vinyl paper

–

–

–

Gypsum board lining

0.012

–

0.1

‡-value –

1 12

Insulation (mineral fibre)

0.15

20

0.04

OSB-sheathing

0.01

400

0.13

15

Building paper

0.0005

–

0.2

200

Cavity

0.03

–

0.18

Brick veneer

0.09

1600

0.9

‡ d-value m

1.2

0 5

260

2 Mass Transfer

The maximum amount of moisture that can stick to the building paper without running-off is 100 g/m2, while the capillary moisture content of gypsum board equals 150 kg/m3. Surface resistances for heat transport is 0.13 m2 · K/W indoors and 0.04 m2 · K/W outdoors. Boundary conditions (North orientation): J

F

M

A

M

J

J

A

S

O

N

D

Outdoors Air temp. (°C)

22.5 23.2 25.3 28.1 30.9 32.9 33.6 32.9 30.9 28.1 25.3 23.2

Vapour press. (Pa)

2185 2279 2585 3047 3579 4008 4168 4008 3579 3047 2585 2279

Indoors Air temp. (°C) Vapour press. (Pa)

20

20

20

20

20

20

20

20

20

20

20

20

1910 1913 1895 1808 1656 1512 1455 1512 1656 1808 1895 1913

Answer It is up to the reader to solve the problem and to make comments. (30) A masonry cavity wall. Layer properties are: Layer

d m

Gypsum plaster

0.01

Inside leaf (fast bricks)

0.14

¼-value W/(m · K)

‡-value –

980

0.3

7

1400

0.5

5 1.2

Density kg/m3

Insulation (mineral fibre)

0.12

30

0.04

Cavity

0.03

400

0.18

Brick veneer

0.09

1600

0.9

‡ d-value m

0 5

The cavity is ventilated. For that purpose, two head joints per meter run are left open at the bottom and top of the 3 m high wall. The head joints section is: 1 × 6.5 cm, depth: 9 cm. In some, the insulation is correctly mounted. In others, it sits centrally in the cavity with an air layer of 1.5 cm left at both sides. In a few, the insulation touches the brick veneer. The tray at the bottom guarantees an open contact between the cavities or remaining air layers and the open head joints, while at the top, the insulation stops just below the open head joints. This of course results in some thermal bridging which is not considered. To what extent is the U-value in the three cases affected by the ventilation flow when on average a 4 Pa air pressure difference exists between the head joints at the top (highest air pressure) and bottom (lowest air pressure)? Surface film coefficients: he = 25 W/(m2 · K), hi = 8 W/(m2 · K). Indoor temperature: 20 °C, outdoor temperature: –10 °C.

261

2.5 Problems

Answer We solve the problem for the case when the insulation is centrally located in the cavity. For the other two cases it is up to the reader to find the answer. Airflow The airflow is fixed by the two open head joints at the top and bottom per meter run. Their hydraulic resistance in fact outweighs the hydraulic resistance of the two 1.5 cm thick air layers at both sides of the insulation. The hydraulic equation is: 'Pa

2 2 ª · § · º 96 Q Ua 2 Ahead joint L U § Ga Ga  U 2 «1.5 a ¨ 0.42 ¸ ¨ ¸ » « Ga d H d H a © Ua 2 Ahead joint ¹ » 2 © Ua 2 Ahead joint ¹ ¬ ¼ 2

§ Ga · § Ga · | 0.32 ¨  0.08 ¨ ¸ © 0.00065 ¹ © 0.00065 ¸¹

757, 400 Ga2  12.3 Ga

For 'Pa = 4 Pa, the airflow from top to bottom is 0.0023 kg/s. As the air layers at both sides of the insulation are of equal thickness, that flow splits in two identical ones, the one washing the air layer between the insulation and the brick veneer, the other washing the air layer between the insulation and the inside leaf. Temperatures and equivalent U-value As the insulation layer has a thermal resistance 3 m2 · K/W, for reasons of simplicity we set the temperature in the air layer between insulation and brick veneer equal to the temperature outdoors. That way the problem does not differ from calculating the average U-value of a wall with a ventilated cavity between two leafs. Thermal resistances R1 = 3.07 m2 · K/W (the insulation and half of the air layer) R2 = 0.438 m2 · K/W (the plastered inside leaf, included the inside surface resistance) Constants See ‘Cavity ventilation’ D = (3.33 + 4.45 + 1 / 3.07) · (3.33 + 4.45 + 1 / 0.438) – 4.452 = 61.9 In D, 3.33 W/(m2 · K) is the convective and 4.45 W/(m2 · K) the radiant surface film coefficient in the cavity. As for a 1.5 cm wide air layer, the Nusselt number may be taken as 1, and the convective surface film coefficient becomes: 0.025 / 0.015 · 2 = 3.33 W/(m2 · K). A1 = 0.0531, A2 = 0.0235, B1 = 0.1643, B2 = 0.2993, C1 = 0.7826, C2 = 0.6773 Cavity surface temperatures Ts1 = 2.8 + 0.7826 Tcav Ts1 = 5.7 + 0.6773 Tcav Temperatures along the cavity See figure.

262

2 Mass Transfer

It is interesting to see how radiation impacts the bounding surface temperatures. In fact, just below the outside air inlet, the bounding warm surface as well as the cold one are both warmer than the air in the cavity. Equivalent thermal transmittance (U) Is given by: U

§ L · º °½ °­ C b R ª U o ®1  2 1 1 «1  exp ¨  ¸ » ¾ L R2 ¬ © b1 ¹ ¼ °¿ °¯

b1

1008 Ga hc 2  C1  C2

with:

Result: U = 0.55 W/(m2 · K). The clear wall value without that 1.5 cm thick air layer behind the thermal insulation should have been 0.27 W/(m2 · K)! Hence, an increase over 100%, which of course is unacceptable.

2.6 Literature

263

2.6

Literature

[2.1]

Langmuir, I. (1918). The adsorption of gases on plane surfaces of glass, mica and platinum. Journal of the American Chemical Society, n° 40. Brunauer, Emmett, Teller (1938). Adsorption of Gases in Multimolecular Layers. Journal of the American Chemical Society Vol. 60. Cranck, J. (1956). The mathematics of Diffusion. Clarendon Press, Oxford. De Grave, A. (1957). Bouwfysica 1. Uitgeverij SIC, Brussel (in Dutch). Glaser, H. (1958). Wärmeleitung und Feuchtigkeitsdurchgang durch Kühlraumisolierungen. Kältetechnik, n° 3 (in German). Glaser, H. (1958). Temperatur- und Dampfdruckverlauf in einer homogene Wand bei Feuchtigkeitsausscheidung. Kältetechnik, n° 6 (in German). Glaser, H. (1958). Vereinfachte Berechnung de Dampfdiffusion durch geschichtete Wände bei Ausscheiden von Wasser und Eis. Kältetechnik, n° 11 & n° 12 (in German). Glaser, H. (1959). Grafisches Verfahren zur Untersuchung von Diffusionsvorgängen. Kältetechnik, n° 10 (in German). Devries, D. A. (1962). Een mathematisch-fysische behandeling van het transport van warmte en vocht in poreuze media. De Ingenieur, No. 28 (in Dutch). Cammerer, J. S. (1962). Wärme- und Kälteschutz in der Industrie. Springer-Verlag, Berlin, Heidelberg, New York (in German). Krischer, O. (1963). Die wissenschaftlichen Grundlagen der Trocknungstechnik. 2. Auflage, Springer-Verlag, Berlin (in German). Luikov, A. (1966). Heat and Mass Transfer in Capillar Porous Bodies. Pergamon Press, Oxford. Harmathy, T. Z. (1967). Moisture sorption of building materials. Technical Paper No. 242, NRC, Division of Building Research, Ottawa. Vos, B. H., Coelman, E. J. W. (1967). Condensation in structures. Report Nr. BI-67-33/23 TNO-IBBC, Rijswijk. Vos, B. H., Tammes, E. (1969). Moisture and Moisture Transfer in Porous Materials. Report Nr. B1-69-96/03.1.001, TNO-IBBC, Rijswijk. Welty, Wicks, Wilson (1969). Fundamentals of Momentum, Heat and Mass Transfer. John Wiley & Sons, New York. Bomberg, H. (1971). Waterflow through porous materials, Part 1: Methods of water transport measurements. Report 19, Lund Institute of Technology, Section of Building Technology. Bomberg, H. (1971). Waterflow through porous materials, Part 2: Relative suction model. Report 20, Lund Institute of Technology, Section of Building Technology. Van der Kooy, J. (1971). Moisture transport in cellular concrete roofs. Uitgeverij Waltman, Delft. Gösele, Schüle (1973). Schall, Wärme, Feuchtigkeit. 3. Auflage, Bauverlag GMBH, Wiesbaden, Berlin (in German). Klopfer, H. (1974). Wassertransport durch Diffusion in Feststoffen. 1. Auflage, Bauverlag GMBH, Wiesbaden-Berlin (in German). Hens, H. (1975). Theoretische en experimentele studie van het hygrothermisch gedrag van bouw- en isolatiematerialen bij inwendige condensation en droging, met toepassing op de platte daken. Doctoraal proefschrift, KU Leuven, 1975. Kreith, F. (1976). Principles of Heat Transfer. Harper & Row Publishers, New York.

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264 [2.24] [2.25] [2.26] [2.27] [2.28] [2.29] [2.30] [2.31] [2.32] [2.33] [2.34] [2.35] [2.36] [2.37] [2.38] [2.39]

[2.40]

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2 Mass Transfer Feynman, R., Leighton, R., Sands, M. (1977). Lectures on Physics, Vol. 1. Addison-Wesley Publishing Company, Reading, Massachusetts. TI-KVIV (1976, 1979, 1980, 1981, 1983, 1985). Kursus Thermische Isolatie en Vochtproblemen in Gebouwen (in Dutch). TU-Delft, Faculteit Civiele Techniek, Vakgroep Utiliteitsbouw-Bouwfysica (1975–1985). Bouwfysica, naar de colleges van Prof A. C. Verhoeven (in Dutch). Hens, H. (1978, 1981). Bouwfysica, Warmte en Vocht, Theoretische grondslagen, 1e en 2e uitgave. ACCO, Leuven (in Dutch). Nevander, E. L., Elmarsson, B. (1981). Fukt-handbok. Svensk Byggtjänst, Stockholm (in Swedish). Kiessl, K. (1983). Kapillarer und dampfförmiger Feuchtetransport in mehrschichtigen Bauteilen. Dissertation, Essen (in German). Kohonen, R., Ojanen, S. (1985). Coupled convection and conduction in two dimensional building structures. 4th conference on numerical methods in thermal problems, Swansea. Pato, Sectie Bouwkunde (1986). Syllabus van de leergang: Leidt energiebesparing tot vochtproblemen. Delft 30 sept – 1 okt. (in Dutch). Kohonen, R., Ojanen, T. (1987). Coupled diffusion and convection heat and mass transfer in building structures. Building Physics Symposium, Lund. Lutz, Jenisch, Klopfer, Freymuth, Krampf (1989). Lehrbuch der Bauphysik. B. G. Teubner Verlag, Stuttgart (in German). Taveirne, W. (1990). Eenhedenstelsels en groothedenvergelijkingen: overgang naar het SI. Pudoc, Wageningen (in Dutch). IEA-Annex 14 (1990). Condensation and Energy: Guidelines and Practice. ACCO, Leuven. Pedersen, C. R. (1990). Combined heat and moisture transfer in building constructions. Ph. D. Thesis, Technical University of Denmark. Chaddock, J. B., Todorovic, B. (1991). Heat and Mass Transfer in Buiding Materials and Structures. Hemisphere Publishing Corporation, New York. Hens, H. (1992, 1997, 2000). Bouwfysica 1, Warmte en Massatransport, 3e, 4e en 5e uitgave. ACCO, Leuven, (in Dutch). Garrecht, H. (1992). Porenstrukturmodelle für den Feuchtehaushalt von Baustoffen mit und ohne Salzbefrachtung und rechnerische Anwendung auf Mauerwerk. Dissertation, Universität Karlsruhe, 267 p. (in German). Künzel, H. M. (1994). Verfahren zur ein- und zweidimensionalen Berechnung des gekoppelten Wärme- und Feuchtetransports in Bauteilen mit einfachen Kennwerten. Dissertation, Universität Stuttgart, 104 p. + Tab. + Fig. (in German). Trechsel, H. R. (Ed.) (1994). Moisture Control in Buildings. ASTM Manual Series, MNL 18. Krus, M. (1995). Feuchtetransport- und Speicherkoeffizienten poröser mineralischer Baustoffe. Theoretische Grundlagen und Neue Meßtechniken. Dissertation, Universität Stuttgart, 106 p. + Tab. + Fig. (in German). de Wit, M. (1995). Warmte en vocht in constructies. Diktaat TUE, Eindhoven (in Dutch). Hens, H. (1996). Modelling, Vol. 1 of the Final Report Task 1, IEA-Annex 24. ACCO, Leuven, 90 p. Time, B. (1998). Hygroscopic Moisture Transport in Wood. Doctors thesis, NUST Trondheim. Arfvidsson, J. (1998). Moisture Transport in Porous Media, Modelling Based on Kirchhoffs Potentials. Doctoral Dissertation, Lund University.

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Time, B. (1998). Hygroscopic Moisture Transport in Wood. Doctors thesis, NUST Trondheim.

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Arfvidsson, J. (1998). Moisture Transport in Porous Media, Modelling Based on Kirchhoffs Potentials. Doctoral Dissertation, Lund University.

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Brocken, H. (1998). Moisture Transport in Brick masonry: the Grey Area between Bricks. Doctoraal proefschrift, TUE, Eindhoven.

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Janssens, A. (1998). Reliable Control of Interstitial Condensation in Lightweight Roof Systems. Doctoraal proefschrift, KU Leuven.

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Häupl, P., Roloff, J. (Eds.) (1999). Proceedings of the 10. Bauklimatisches Symposium, Band 1 & 2. T. U. Dresden, Institut für Bauklimatik.

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Roels, S. (2000). Modelling Unsaturated Moisture Transport in Heterogeneous Limestone. Doctoraal proefschrift, KU Leuven.

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Sedlbauer, K. (2001). Vorhersage von Schimmelpilzbildung auf und in Bauteilen. Dissertation, Universität Stuttgart, 158 p. + Tab. + Fig. (in German).

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Hagentoft, C. E. (2001). Introduction to Building Physics. Studentlitteratur, Lund.

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Holm, A. (2001). Ermittlung der Genauigkeit von instationären hygrothermischen Bauteilberechnungen mittels eines stochastischen Konzeptes. Dissertation, Universität Stuttgart, 196 p. + Tab. + Fig. (in German).

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Janssen, H. (2002). The influence of soil moisture transfer on building heat loss via the ground. Doctoraal proefschrift, KU Leuven.

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Häupl, P., Roloff, J. (Eds.) (2002). Proceedings of the 11. Bauklimatisches Symposium, Band 1 & 2. T. U. Dresden, Institut für Bauklimatik.

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Blocken, B., Hens, H., Carmeliet, J. (2002). Methods for the Quantification of Driving Rain on Buildings. ASHRAE Transactions, Vol. 108, Part 2, pp. 338–350.

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Hall, C., D’Hoff, W. (2002). Water transport in brick, stone and concrete. Spon Press, London and New York, 318 pp.

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Van Mook, J. R. (2003). Driving Rain on Building Envelopes. TU/e Bouwstenen 69, 198 p.

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Carmeliet, J., Hens, H., Vermeir, G. (Eds.) (2003). Research in Building Physics. Balkema publishers, Lisse, Abingdon, Exton, Tokyo, 1020 p.

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Blocken, B. (2004). Wind-driven rain on buildings. Ph. D. Thesis, KU Leuven.

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Kalagasidis, A. (2004). HAM-tools, An Integrated Simulation Tool for Heat, Air and Moisture Transfer Analysis in Building Physics. Ph. D. Thesis, Chalmers University of Technology.

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Rose, W. B. (2005). Water in Buildings. John Wiley and Sons, Inc., 270 pp.

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Woloszyn, M., Rode, C. (2008). Modelling Principles and Common Exercises. Final Report IEA ECBCS Annex 41, ACCO, Leuven.

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Abuku, M. (2009). Moisture stress of wind-driven rain on building enclosures. Ph. D. Thesis, KU Leuven.

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3

Combined heat-air-moisture transfer

3.1

Overview

With the exception of the paragraphs on combined air and heat transfer, internal condensation and drying, we have kept heat-air-moisture separate. In reality, they are intertwined: y Mass transfer means enthalpy and, thus, heat transfer. This was explained in Chapter 2. y Temperature differences induce mass displacement, i.e. natural convection, water vapour diffusion and pore water movement by changes in surface tension. y Thermal characteristics (volumetric specific heat capacity, thermal conductivity) which are a function of moisture content. y Moisture transfer characteristics asa function of temperature. These interrelationships are the reason why a separate analysis oversimplifies reality, resulting in a loss of information and understanding. Therefore, in this last part, we look to combined heat-air-moisture transfer (HAM).

3.2

Material and assembly level

3.2.1

Assumptions

An exact mathematical treatment of combined heat-air-moisture transfer is not possible. Precipitation of soluble substances, crystallization/hydration of salts and diffusion of solid substances alter the material matrix, modify pores in form and equivalent diameter, and change the specific pore surface. These effects are overlooked. Thus, a first assumption is that any material matrix is invariable. Consequently, material characteristics, such as dry density, pore distribution and specific pore surface do not change. Furthermore, each porous material is assumed to be composed of an infinite number of infinitely small representative volumes (Vrep ), each possessing all characteristics the material has. This of course is untrue. Representative volumes are finite in nature, which is why in the best case a continuum approach mirrors a calculation using and giving average values of all quantities at the representative volume level.

3.2.2

Solution

Any combined HAM-problem is solved when the temperature, air pressure and moisture potentials and the heat, air and moisture flow rates are known in each point of the ‘solution space’, here the material or the assembly. While potentials are scalars, flow rates are vectors. Consequently, we need a system of three scalar and three vector equations, completed with the necessary equations of state. These describe the thermodynamic equilibriums that exist between separate potentials and between potentials and certain material characteristics. In addition, solving any system demands perfect knowledge of the geometry and the initial, boundary and contact conditions. Building Physics: Heat, Air and Moisture. Fundamentals and Engineering Methods with Examples and Exercises. Second edition. Hugo Hens. © 2012 Ernst & Sohn GmbH & Co. KG. Published 2012 by Ernst & Sohn GmbH & Co. KG.

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3.2.3

3 Combined heat-air-moisture transfer

Conservation laws

The conservation laws of classic physics govern combined heat-air-moisture: conservation of mass, energy, and momentum. For mass flow in porous materials, empirical equations substitute conservation of momentum. Convection in cavities and air volumes and mass movement in highly permeable materials may necessitate reference to the Navier-Stokes equations, combined with a turbulence model. 3.2.3.1 Mass The pores of a humid material contain at least three and often four mass components: for temperatures up to 0 °C air, liquid water, water vapour, ice and many times dissolved salts and other substances; for temperatures above 0 °C air, liquid water, water vapour and many times dissolved salts and other substances. To simplify things, we assume salts and other substances absent. For liquid water, water vapour, ice, and air, the quantities per unit volume of material are determined by using water content, vapour content, ice content, and air content: ¹•

¹