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ENERGY SCIENCE, ENGINEERING AND TECHNOLOGY
Copyright © 2012. Nova Science Publishers, Incorporated. All rights reserved.
HEAT FLUX: PROCESSES, MEASUREMENT TECHNIQUES AND APPLICATIONS
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Heat Flux: Processes, Measurement Techniques and Applications : Processes, Measurement Techniques and Applications, Nova Science
ENERGY SCIENCE, ENGINEERING AND TECHNOLOGY
HEAT FLUX: PROCESSES, MEASUREMENT TECHNIQUES AND APPLICATIONS
GIANLUCA CIRIMELE Copyright © 2012. Nova Science Publishers, Incorporated. All rights reserved.
AND
MARCELLO D'ELIA EDITORS
Nova Science Publishers, Inc. New York
Heat Flux: Processes, Measurement Techniques and Applications : Processes, Measurement Techniques and Applications, Nova Science
Copyright © 2012 by Nova Science Publishers, Inc. All rights reserved. No part of this book may be reproduced, stored in a retrieval system or transmitted in any form or by any means: electronic, electrostatic, magnetic, tape, mechanical photocopying, recording or otherwise without the written permission of the Publisher. For permission to use material from this book please contact us: Telephone 6312317269; Fax 6312318175 Web Site: http://www.novapublishers.com NOTICE TO THE READER The Publisher has taken reasonable care in the preparation of this book, but makes no expressed or implied warranty of any kind and assumes no responsibility for any errors or omissions. No liability is assumed for incidental or consequential damages in connection with or arising out of information contained in this book. The Publisher shall not be liable for any special, consequential, or exemplary damages resulting, in whole or in part, from the readers’ use of, or reliance upon, this material. Any parts of this book based on government reports are so indicated and copyright is claimed for those parts to the extent applicable to compilations of such works.
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Independent verification should be sought for any data, advice or recommendations contained in this book. In addition, no responsibility is assumed by the publisher for any injury and/or damage to persons or property arising from any methods, products, instructions, ideas or otherwise contained in this publication. This publication is designed to provide accurate and authoritative information with regard to the subject matter covered herein. It is sold with the clear understanding that the Publisher is not engaged in rendering legal or any other professional services. If legal or any other expert assistance is required, the services of a competent person should be sought. FROM A DECLARATION OF PARTICIPANTS JOINTLY ADOPTED BY A COMMITTEE OF THE AMERICAN BAR ASSOCIATION AND A COMMITTEE OF PUBLISHERS. Additional color graphics may be available in the ebook version of this book. Library of Congress CataloginginPublication Data Heat flux : processes, measurement techniques, and applications / editors, Gianluca Cirimele and Marcello D'Elia. p. cm. Includes index. ISBN: (eBook) 1. Heat fluxResearch. I. Cirimele, Gianluca. II. D'Elia, Marcello. QC320.36.H43 2011 536'.2072dc23 2011027344
Published by Nova Science Publishers, Inc. New York Heat Flux: Processes, Measurement Techniques and Applications : Processes, Measurement Techniques and Applications, Nova Science
CONTENTS Preface Chapter 1
Measurement of Heat Flux and Heat Transfer Coefficient Jan Taler and Dawid Taler
Chapter 2
Heat Flux Biocalorimetry: A RealTime Analytical Tool for Bioprocess Monitoring M. Surianarayanan, S. Senthilkumar and A. B. Mandal
105
Ability of Soil to Transfer a Large Amount of Heat under Reduced Air Pressure Toshihiko Momose and Tatsuaki Kasubuchi
137
Chapter 3
Chapter 4
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vii
Chapter 5
Contributions to the Net Heat Flux in the Mediterranean Sea: Seasonal and Interannual Variations F. CriadoAldeanueva, J. SotoNavarro, J. GarcíaLafuente, C. Naranjo, C. Calero and E. Bruque Simulation of Heat Flux Transport in the Edge Plasma of Small Size Divertor Tokamak A. H. Bekheit
Chapter 6
Meridional Heat Fluxes in the North Indian Ocean T. Rojsiraphisal and L. Kantha
Chapter 7
Heat Flux and Temperature at the ToolChip Interface in Dry Machining of Aeronautic Aluminium Alloy G. List, D. Géhin, A. Kusiak, J. L. Battaglia and F. Girot
Index
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155
167 187
197
211
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PREFACE Heat flux or thermal flux is the rate of heat energy transfer through a given surface. In this book, the authors present topical research in the study of the processes, measurement techniques and applications of heat flux. Topics discussed include measuring heat flux and heat transfer coefficients; the science and application of heat flux biocalorimetry; measuring the thermal conductivity and the heat flux of soil as a function of air pressure; the net heat flux in the Mediterranean Sea; simulation of heat flux transport in the edge plasma of small size divertor tokamak and meridional heat fluxes in the North Indian Ocean. Chapter 1  Heat flux is defined as the quantity of heat transferred per unit surface area per unit time. Heat flux to or from a surface can be inferred from temperaturetime measurements at the surface or within the body or can be measured directly, using heat flux meters. In the first approach the flux or heat transfer coefficient is determined from the solution of the steadystate or transient inverse heat conduction problem. The operation of the heat flux meters is based on the simple relations between the heat flux and temperature difference between two points located inside the sensor or at its surfaces. Also, the variation of sensor temperature with time is used to determine heat flux to the meter. Uncertainties in the determined heat flux and heat transfer coefficient can be estimated using the variance propagation rule developed by Gauss. Chapter 2  Heatflux calorimeters have extensive applications on monitoring and controlling chemical process systems. Unlike chemical processes, biochemical processes are complex systems involving enzymatic reactions at intra and extracellular levels. A realtime process measurement is indeed required to gain better insight on dynamics of cell growth. As for Biothermodynamics, excessive Gibb’s energy is dissipated in the form of heat during cell growth process. Hence, measurements of metabolic heat flux can provide realtime information on cell physiology and bioprocess behavior. Further, heatflux measurements are robust, nonspecific and noninvasive, irrespective of bioprocess systems. For the past few decades there has been an increasing interest on employing heatflux calorimetry for investigating biochemical reaction systems. Several approaches are available for heat measurement, yet heatflux calorimetry as proven by several research groups, is more suitable for bioprocess monitoring. This review attempts to highlight significant findings of the investigations so far carried out by research groups employing bench scale heatflux biocalorimeter in different biological systems. Technical and design modifications, dealt with heatflux calorimeters, to achieve high resolution of heat flow signal, and major drawbacks observed on their proposed strategies are discussed in a separate section. A brief description
Heat Flux: Processes, Measurement Techniques and Applications : Processes, Measurement Techniques and Applications, Nova Science
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viii
Gianluca Cirimele and Marcello D'Elia
of quantitative biocalorimetric studies on microbial growth process, stoicheometric analysis on cell growth, heat yield coefficient determination and diauxic behavior is given with suitable illustrative calorimetric results. Significance of calorimetric data on estimation of biothermodynamic parameters and their related complexities are summarized in ‘Biothermodynamics’ section. Applications of bench scale heatflux calorimetry for realtime monitoring of different bioprocess systems viz., aerobic, fermentation, anaerobic, photoautotrophic and ecological, are discussed in separate sections. Though extensive work had been done for several years, the real potential of heatflux biocalorimetry in the industrial biotech sector has not yet been exploited. Developing an effective protocol for transforming bioreactors to biocalorimeters can pave the way for the advent of “Large scale calorimetry” and this will certainly reveal the benefits to biotech industrial communities, due to its economic viability. This review will provide the readers an insight to science and applications of heatflux biocalorimetry. Moreover, this contribution may serve as an initiative for academia and industry to employ ‘heatflux biocalorimetry’ as a ‘future generation’ realtime analytical tool for bioprocess monitoring applications. Chapter 3  The thermal conductivity of a twophase soil composed of solid and gas phase decreases as the air pressure is reduced. The small thermal conductivity under reduced air pressure results from the decrease in the heat transfer in soil pore spaces. However, the authors’ previous study has found that the thermal conductivity of a threephase soil, such as an unsaturated soil, increases sharply under reduced air pressure. The maximum thermal conductivity obtained becomes close to the thermal conductivity of some metals such as manganese, mercury and stainless steel. This chapter introduces the authors techniques for measuring the thermal conductivity and the heat flux of soil as a function of air pressure, and describes the mechanism of heat transfer in soils under reduced air pressure. Chapter 4  Several NCEP climatological datasets have been combined to analyse the seasonal and interannual variations of the heat budget in the Mediterranean Sea. The seasonal cycle of the net heat is positive (toward the ocean) between March and September with maximum in June and negative the rest of the year with minimum in December. Although subject to inherent uncertainty, the authors obtain a practically neutral budget of 0.7 Wm2 in a yearly basis. The net heat budget is positive for the western Mediterranean (~12 Wm2) and negative for the eastern Mediterranean (~ 6.4 Wm2) mainly due to the high latent heat losses of this basin. Combining the climatological values with in situ measurements in Espartel sill (Strait of Gibraltar), a heat advection Qa = 3.2±1.5 Wm2 through the Strait of Gibraltar has been obtained that, combined with the longterm averaged surface heat flux, implies that the net heat content of the Mediterranean Sea would have increased in the last decades. Chapter 5  The B2.SOLPES.0.5.2D fluid transport code is applied for modeling SOL (Scrape off Layer) plasma in the small size divertor tokamak. Detailed distributions of the plasma heat flux and other plasma parameters in SOL, especially at the target plate of the divertor are found by modeling. The modeling results show that most of the electron heat flux and small part of ion heat flux arrive at target plate of the divertor, while, a large part of the ion heat flux and part of electron heat flux arrive at the outer wall. Also the simulation results shows the following results (1) large asymmetries in heat flux at targets plates are observed. (2) When strong ITB is formed The reduction of plasma radial heat flux is higher by factor (~ 2.5 ) than neoclassical heat flux (3) the shear of the radial electric is enhanced by increase in temperature heating of plasma due to increase in pressure gradient and large reduction of radial heat flux. This leads to the core confinement is improved which correspond to the ITB
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Preface
ix
formation. (4) The radial heat flux is strongly influenced by toroidal rotation. (5) the amplification of conduction part of radial heat flux imposes nonresilient profile of ion temperature, under which the effect of toroidal rotation on ion temperature profile is strong.(6) the comparison between radial heat flux at different momentum input shows that, the radial ion heat flux with larger ion temperature scale length in the case of coinjection neutral beam is larger than the ion heat flux with smaller ion temperature scale length in the case of counterinjection neutral beam. Chapter 6  The seasonal cycle of the crossequatorial heat transport is of great importance to South Asian monsoons since it is a component of the heat budget and hence determines the SST of the North Indian Ocean. In this chapter, the authors use a dataassimilative model of the North Indian Ocean assimilating altimetry and MCSST data and driven by ECMWF and QuikSCAT wind stresses, to explore the seasonal and interannual variability of the meridional heat fluxes during the years 1993 to 2005. In addition to a strong seasonal variability, the crossequatorial heat transport also exhibits considerable interannual variability. There are also strong, short period intraseasonal fluctuations on time scales of 2050 days. The meridional heat flux values are 2030% higher when the hindcast is driven by observed QuikSCAT wind stresses than when it is driven by ECMWF winds. Chapter 7  The heat flux and temperature rise at the toolchip interface were investigated in the case of dry machining of the aerospace aluminium alloy AA2024 T351. On the one hand, a complete experimental setup allowed us to study in realtime the friction force, the chip geometry (observation by CCD high speed camera) and the heat flux transmitted into the tool (by using an inverse method). On the other hand, numerical simulations of the chip formation were carried out using the finite element method. The obtained values are compared to experimental results to validate the modelling. A good correlation between experiments and numerical simulations was found but the results indicated a strong influence of the contact conditions between the tool and the chip such as seizure or BuiltUp Edge (BUE). From the measured heat flux, the temperature rise was also estimated by using a classic analytical model and was compared to the values found by numerical simulations. The trends are the same, but the analytical calculation tends to overestimate the temperature rise compared to the numerical simulations.
Heat Flux: Processes, Measurement Techniques and Applications : Processes, Measurement Techniques and Applications, Nova Science
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In: Heat Flux Editors: G. Cirimele and M. D'Elia
ISBN 9781614708872 © 2012 Nova Science Publishers, Inc.
Chapter 1
MEASUREMENT OF HEAT FLUX AND HEAT TRANSFER COEFFICIENT Jan Taler1 and Dawid Taler2 1
Department of Thermal Power Engineering, Cracow University of Technology, Al. Jana Pawła II 37, Cracow, Poland 2 Institute of Thermal Engineering and Air Protection, Cracow University of Technology, ul. Warszawska 24, Cracow, Poland
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NOMENCLATURE a inner radius of boiler tube and fluxtube, m, a time at which heat flux is maximal, s a0, a1 constants a, b coefficients A0, Am coefficients of temperaturetime polynomials b outer radius of fluxtube, m, Bi Biot number, Bi ha / k or Bi = hx/k c inner radius of boiler tube, m, c specific heat, J/(kg·K) C1,i, …, C4,i coefficients (derivatives) of the piecewise cubic function D heating duration, s e eccentric, m e maximum temperature error, K E thermocouple depth below heated or cooled surface, m E sensor distance from heated surface, m ET Young modulus, MPa eh standard error mean of the heat transfer coefficient e+ dimensionless error mean f measured temperature at an interior point, oC Fo Fourier number, Fo = αt/L2 Heat Flux: Processes, Measurement Techniques and Applications : Processes, Measurement Techniques and Applications, Nova Science
2
Jan Taler and Dawid Taler fi f Fj h In J Jm k L m m M M1 n n
N N N NC NT q qN qs
ith measured temperature, oC vector of measured wall temperatures, oC, jth equality limitation heat transfer coefficient, W/(m2·K), identity matrix, total number of the data points Jakobian matrix, thermal conductivity, W/(m·K), thickness of the sensor (plate), m number of temperature measurement points, parameter in heat conduction equation (m=0,1 and 2 for a slab, cylinder and sphere, respectively) number of control volumes number of node, in which the first temperature sensor was mounted iteration number number of unknown parameters, number of data points used in digital filtering number of temperature measurement points number of measurement points in moving average filter, N = (2L+1) number of spatial grid points total number of data points heat flux, W/m2 nominal heat flux of the triangular test case, W/m2 surface heat flux, W/m2
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q+ dimensionless heat flux, q
q qN
r radius, m, ri radial coordinate of the ith thermocouple, m, rE rin ro rm s s+ S St
sensor location, m inner radius, m outer radius, m mean radius, m, rm = (rin+ro)/2 dimensionless time coordinate dimensionless mean square error least squares function, K2 time scaling factor
t t T Tf Ti,e Ti T0 Tm
time, s pitch of water wall tubes, m, temperature, C or K fluid temperature, C or K exact temperature, C or K node temperature, C initial temperature, C vector of computed temperatures,
Heat Flux: Processes, Measurement Techniques and Applications : Processes, Measurement Techniques and Applications, Nova Science
Measurement of Heat Flux and Heat Transfer Coefficient u ratio of the outside to the inside radius of the flux tube, u() = ro/a wi weighting factor x xi y zi
cartesian coordinate, m location of the ith temperature sensor smoothed value of measured temperature at an interior point, C ith determined parameter
Greek Symbols thermal diffusivity, = k/(c·), m2/s linear coefficient of thermal expansion, 1/K I ith Lagrange multiplicator x thickness of the control volume, x = L/(M1) r spatial size of control volume, m t time step, s Fo dimensionless time step, Fo = t/rin2 iteration tolerance i random variable of uniform distribution with values in the range
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[1,1] dimensionless parameter, = h2t/(ck) dummy time variable dimensionless temperature smoothing (regularization) parameter Poisson’s ratio
dimensionless parameter,
density, kg/m3 standard deviation root mean square norm axial stress, MPa radial stress, MPa tangential stress, MPa angular coordinate, rad
σ
a r t
x 2 t
i angular coordinate of the ith thermocouple, rad
temperature excess over the fluid temperature, T Tf
view factor
scaled time
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3
4
Jan Taler and Dawid Taler
Subscripts i number of temperature measurement point in inner o outer Heat flux is defined as the quantity of heat transferred per unit surface area per unit time. Heat flux to or from a surface can be inferred from temperaturetime measurements at the surface or within the body or can be measured directly, using heat flux meters [113]. In the first approach the flux or heat transfer coefficient is determined from the solution of the steadystate or transient inverse heat conduction problem. The operation of the heat flux meters is based on the simple relations between the heat flux and temperature difference between two points located inside the sensor or at its surfaces. Also, the variation of sensor temperature with time is used to determine heat flux to the meter. Uncertainties in the determined heat flux and heat transfer coefficient can be estimated using the variance propagation rule developed by Gauss [710].
1. HEAT FLUX METERS Most heat flux sensor measure the total heat flux at the fluidsolid interface due to convection and radiation. Three types of heat flux meters will be discussed:
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slug calorimeters, axial conduction probes, radial conduction gauges.
The use of heat flux meters for measuring heat flux or heat transfer coefficient will be presented in the following.
1.1. Slug Calorimeters In a typical slug calorimeter of high thermal conductivity the front surface is subject to a heat flux while the side and back surfaces are insulated (Figure 1).
1
q
3 2
4 5 Figure 1. A simple slug calorimeter; 1 – slug, 2  casing, 3 and 4 – insulating elements, 5 – thermocouple.
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Measurement of Heat Flux and Heat Transfer Coefficient
The change of slug temperature with time is used to determine the heat flux to or from the front surface. Very often, a single temperature measurement at the insulated rear surface of the slug is made since this surface is easily accessible and the temperature sensor is not exposed to the direct thermal radiation or to the surrounding fluid. The dimensionless temperature θdistribution θ in the plate (slug) (Figure 2) with heated front surface and insulated rear surface is for Fo Fo* 0.35 given by [1113]. 2 T T0 k x 1 x 1 Fo ,
qL
L 2 L
3
Fo Fo*,
(1)
where: T  slug temperature, T0  uniform initial slug temperature, k  thermal conductivity, q heat flux, x  Cartesian coordinate, L  slug thickness. The Fourier number is defined as Fo = αt/L2, where: α = k/(c)  thermal diffusivity, c  specific heat,  density, t  time. T
q T/t=0 =T 0 x
0
L
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Figure 2. A slug heated at front surface with perfect insulation at back surface.
Temperature Tt of the rear surface (x = L) is linear function of time
Tt T0 k Fo qL
1 , 6
Fo Fo 0.35 .
Figure 3. Plate temperature at selected locations,   thin plate or infinite material thermal conductivity. Heat Flux: Processes, Measurement Techniques and Applications : Processes, Measurement Techniques and Applications, Nova Science
(2)
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Jan Taler and Dawid Taler
Figure 4. Temperature distributions over the plate thickness for selected Fourier numbers.
Differentiating Eq. (2) gives
k dTt , qL d t L2 and hence
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q cL
dTt c LvT , dt
(3)
where the symbol vT = dTt /dt = const. denotes constant rate of change of the rear surface temperature. The thickness L of the meter is important for proper measurements of high heat fluxes. The duration of the measurement with the slug calorimeter is limited because the temperature of the exposed surface should not exceed the allowable temperature Tal for the slug material. The duration Δt of the calorimeter use was estimated in [14]. The measurement time period is given by
t tmax tmin .
(4)
The symbol tmin denotes a time period before data logging, after that the quasisteady state temperature distribution is formed inside the slug. After time tmin the rate of temperature change vT and temperature difference (Ts – Tt) over the slug thickness are constant. The time period tmin results from the condition: Fo Fo*
tmin
L2 Fo
.
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(5)
7
Measurement of Heat Flux and Heat Transfer Coefficient
Maximum time tmax of the heat flux measurement is determined from the condition that the temperature Ts of the front surface (x = 0)
Ts
qL t 1 T0 , t tmin k L2 3
(6)
should not exceed the allowable temperature Tal for slug material Ts Tal
(7)
Substituting Eq. (6) into (7) and solving for tmax yields
tmax
Tal T0 k L
q
L2 . 3
(8)
Substituting Eqs (5) and (8) into Eq. (4) gives
t
Tal T0 k q
L2 L2 Fo . 3
(9)
The optimum thickness Lopt that maximizes measurement duration Δt is determined from the necessary condition
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d t 0. dL
(10)
The optimum thickness Lopt obtained from Eq. (10) is given by
Lopt
3k Tal T0
2q 1 3Fo
.
(11)
The thickness Lopt depends on the Fourier number value Fo*. If we assume that Fo* = 0.35, then we have [14]
Lopt
k Tal T0 1.367q
,
(12)
while for Fo*= 0.5 [15] we obtain
Lopt
k Tal T0 1.667q
.
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Jan Taler and Dawid Taler
After substituting Eq. (11) into Eq. (9) we obtain the following expression for the maximum allowable measurement time
3k 2 Tal T0 . 4q 2 1 3Fo 2
tmax
(14)
Slug calorimeters are mainly used for measuring the time invariant heat flux or heat transfer coefficient. However, they can be used for measurements of arbitrary timewise variations in incident heat flux or heat transfer coefficient provided the slag thickness is small. The absorbed heat flux is calculated using the simple formula
q cL
dTt . dt
(15)
To increase measurement accuracy, the formulas based on the solution of the inverse heat conduction problem [12, 13, 16] can be used
dT d 3Tt L2 d 2Tt L4 q(t ) c L t 2 , 120 2 dt 3 dt 6 dt
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T
x 0
Tt
d 3Tt L2 dTt L4 d 2Tt L6 . 2 dt 24 2 dt 2 720 3 dt 3
(16) (17)
The inaccuracies inherent in differentiating the temperature histories Tt(t) can be reduced by using the moving averaging filters, which will be presented in section 2. The heat transfer coefficient is defined as
h
q(t ) T f T (0, t )
(18)
where the surface heat flux q(t) and temperature T(0,t) are given by Eqs (16) and (17), respectively. One of the disadvantages of the slug meter is the need to cool the instrument before reuse.
1.2. Axial Conduction Probes (PlugType Probes) Axial conduction sensors are frequently used to measure high heat fluxes in furnaces and combustion chambers of boilers. The sensors operate by measuring temperatures f1 and f2 at two selected points: x1 and x2 (Figure 5). The effect of temperature on the conductivity of the sensor material is described by the function k = k(T). In the following onedimensional analysis it is assumed that only axial conduction occurs, i.e. in any plane perpendicular to the x axis the temperature within the sensor is uniform. The side surface of the sensor is perfectly
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Measurement of Heat Flux and Heat Transfer Coefficient
insulated. To reduce heat losses such that onedimensional heat flow exists in the axial direction through the cylindrical sensor its side surface can be protected by concentric guard rings, as depicted in Figures 6a and 6b. q f1
= (T)
x1
T(x) f2
x2 L 0
x
Figure 5. Operation principle of conduction heat fluxmeters.
q k (T )
dT dx
(19)
Integration of Eq. (19) gives x2
f2
qdx k (T )dT .
x1
(20)
f1
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Introducing the mean thermal conductivity km, defined as
1 km f1 f 2
f1
k T dT
(21)
f2
the following expression for the heat flux is obtained from Eq. (20)
q km
f1 f 2 . x2 x1
(22)
Equation (22) represents the discrete form of the Fourier law. If the temperature sensors are placed at the exposed and rear surfaces, then
q
km T , L
(23)
where T = f1  f2 is the temperature difference between two thermocouples. For linear variation of thermal conductivity with temperature
k (T ) a b T .
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Jan Taler and Dawid Taler
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Figure 6. Guarded axial conduction sensor in which the radial flow is prevented by annular slots, (a) probe used in furnaces[17], 1 – sensor with protective rings, 2 – furnace wall; (b) probe used in internal combustion engines[18], 3 – cooling water, 4 – sensor; f1 – f6 – measured metal temperatures.
The cooling water flow rate can be adjusted to operate the probe with a prescribed temperature of the front (exposed) surface. The heat flows from the furnace to the cooling water through the sensor. The absorbed heat flux is determined based on measured temperature values f1 and f2 while additional four temperature sensors (f2 – f6) may be used to measure the temperature on the guard ring circumference (Figure 6b). The heat flux is evaluated using Fourier’s law The mean value of the thermal conductivity km between the temperatures f2 and f1 obtained from Eq. (21) is
km a b
f1 f 2 k ( f1 ) k ( f 2 ) a b fm , 2 2
(25)
where
fm
f1 f 2 . 2
(26)
Determining the heat flux q allows us to obtain the temperature distribution in the sensor using Fourier’s law
q (a bT )
dT . dx
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(27)
11
Measurement of Heat Flux and Heat Transfer Coefficient Integrating Eq. (27) from x to x2 f2
x2
qdx (a bT )dT x
(28)
T
gives
b 2 b T aT q( x x 2 ) af 2 f 22 0. 2 2
(29)
The solution to Eq.(29) is
T ( x)
a b
a 2q( x x2 ) , f2 b b 2
(30)
where q is determined with Eqs (22) and (25). When b 0 then the temperature profile T(x) is convex, if b 0 then T(x) is concave. Flat sensors are often applied in portable probes while in cylindrical tubes meters thermocouples are installed on site in the waterwall tubes. In a cylindrical sensor heated at the outer surface, the heat flow rate is
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Q k T
dT dT A r const or Q 2 rLk T const , dr dr
(31)
where L is the length of the sensor. After the separation of variables
Q dr k T dT 2 L r
(32)
and integration Eq.(32), one obtains r
f
Q 1 dr 1 k T dT , 2 L r2 r f2
(33)
r Q ln 1 km f1 f 2 , 2 L r2
(34)
where km is expressed by formula (21). Since on the outer surface r = ro, the heat flux qo is given by
qo
Q , 2 ro L
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(35)
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Jan Taler and Dawid Taler
one obtains from Eq.(34) the following expression
qo
km f1 f 2 ro ln(r1 / r2 )
.
(36)
Equation (36) is used to determine the heat flux qo on the outer surface of the sensor based on the measured temperatures f1 and f2 at the locations r1 and r2, respectively. Axial heat conduction meters can be installed on site (Figure 7 and 8) or are designed as portable probes. A heat flux meter for use in boiler combustion chambers is depicted in Figure 7. 1
2
6 5.5 3 12.7 4
5
6
7 8 9
13 10 11 12
Figure 7. Plug type heat flux gauge welded to a waterwall tube [19]; 1 – protection shield, 2 thermocouple, 3 cylindrical pin, 4 – melted region, 5, 12 – thermocouple protection tube, 6 – protection insert, 7 – insulation gap, 8 – plug sensor, 9 – guard ring, 10  weld, 11, 13 – waterwall tube (dimensions in millimetres).
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1
29 2
5.85
Figure 8. Axial conduction gauge for mounting in the wall [20]; 1 – cooling water, 2 – plug sensor (dimensions in millimetres).
The mode of heat transfer to the gauge is predominantly radiative. Heat flowing into the plug sensor is conducted axially to the waterwall tube through the attachment weld. Installation into the boiler is simple. A small hole is drilled through the boiler casing to pass the thermocouples between adjacent tubes. The meter and thermocouple shield are welded to
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Measurement of Heat Flux and Heat Transfer Coefficient
13
the tube as shown in Figure 7. The axial conduction heat flux probe is inserted into the furnace through a hole in the furnace setting for monitoring the wall heat flux (Figures 9 and 10). The emissivities of the furnace wall and the heat flux meter surface should be the same.To improve the accuracy of the heat flux measurement the temperature of the front and cooled surface are measured (Figs. 10 and 11). If the distance between temperature sensors is larger, then inaccurate placing of the temperature sensors has smaller influence on the error in the determined heat flux [22]. Portable plug heat flux meters can operate continuously at high temperature in the furnace, and in the presence of corrosive combustion products for a long period of time.
q 1
3
2
4
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Figure 9. Portable axial conduction heat flux probe cooled by water [21]; 1 – plug sensor, 2 – hole for thermocouple, 3 – guard ring, 4 – water inlet.
Figure 10. Portable axial conduction heat flux probe cooled by water [22]; 1 – plug sensor, 2 – guard ring, 3 – sealing ring, 4 thermocouple protection tube, 5  casing, 6 – distance pin, 7 – water inlet, 8 – outer tube, 9 – water outlet, 10  bottom, 11 – gland seal, 12 – thermocouple fixing, 13  screw, 14 – thermocouple plug, 15 – thermocouple (dimensions in millimetres).
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Jan Taler and Dawid Taler
Figure 11. Circular plug sensor used in the probe show in Figure 10 [22] (dimensions in millimetres).
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1.3. Radial Conduction Gauges (Gardon Type Gauges) The Gardon heat flux gauge [23, 24] consists of thin disc connected to a heat sink at its periphery. Heat flow absorbed by the front surface is conducted radially to the edge of the disc because its rear surface is thermally insulated. A copper wire is attached to the centre of the constantan disc. A second copper wire is connected to a copper block. A voltage measurement device connected across the two copper ends (Figure 12) is proportional to the temperature difference between the centre and the edge of the constantan disc, which in turn is proportional to the heat flux absorbed by the exposed disc surface. For continuous heat flux measurement the disc edge is cooled by water (Figure 13). Consider a sensor in which temperatures f1 and f2 are measured at points r1 and r2 (Figure 13). The aim is to derive a formula for calculating temperature distribution in the Gardon sensor and heat flux q on the basis of measured temperatures f1 and f2. In order to derive a differential equation, which describes heat conduction in the Gardon sensor, the energy balance equation will be written for a control volume dV = 2πrLΔr shown in Figure 13.
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15
Measurement of Heat Flux and Heat Transfer Coefficient
q 1
2 3
+
4

Figure 12. Gardon foiltype heat flux gauge; 1 – constantan foil, 2 – protective shield, 3 – copper block (heat sink), 4 – copper ends.
q 1
L Q2 r
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2
q
Q3 Q1 r r1 2R
f1
f2
r
r2 s
Figure 13. Gardon heat flux meter cooled by water; 1 – circular sensor, 2 – cooling water, q – heat flux, f1 and f2 – measured temperatures.
Allowing for the variations of thermophysical properties of the disc material with temperature, the energy balance equation is
T Q1 Q2 Q3 , t
(37)
T T Q1 2 rL k T , Q2 2 r r L k T , r r r r r Q3 2 rΔrq.
(38)
c T T 2 rLΔr where
By substituting (38) into (37), one obtains the following for r 0
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Jan Taler and Dawid Taler
c T T
T 1 T q k T r . t r r r L
(39)
In the case of steadystate operation T/t = 0, the temperature distribution is only a function of a single variable r
1 T q k T r . r r r L
(40)
Boundary conditions have the following form:
dT dr
0,
T
r 0
r r1
f1 ,
T
r r2
f2.
(41)
In order to linearize the problem (40)–(41), Kirchhoff’s transformation will be used T
U k T dT .
(42)
0
Since
dU dU dT dT k T , dr dT dr dr
(43)
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equation (40) and boundary conditions (41) can be written as follows
1 d dU r r dr dr
dU dr
0,
q , L
U
r 0
r r1
(44)
U1 ,
U
r r2
U2 ,
(45)
where T1
U1 k T dT , 0
T2
U 2 k T dT .
(46)
0
From equation (44) with the first two boundary conditions in Eq. (45), one obtains 2 2 1 q r r1 U U1 . 4 L
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(47)
17
Measurement of Heat Flux and Heat Transfer Coefficient From the third boundary condition in Eq. (45) it follows that 2 2 1 q r2 r1 U2 U1 , 4 L
(48)
from which we obtain a formula for heat flux q
q
4 U1 U 2 L . r22 r12
(49)
Since T1
T2
T1
U1 U 2 k T dT k T dT k T dT km T1 T2 0
0
(50)
T2
T1
where
km
k T dT
T2
T1 T2
.
(51)
Equation (49) then, after allowing for (51), can be written in the following form
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q
4km T1 T2 L . r22 r12
(52)
If k (T) = a + bT, then the mean thermal conductivity is given by T1
km
a bT dT
T2
T1 T2
1 a T1 T2 b T12 T22 2 T1 T2
(53)
1 a b T1 T2 k Tm , 2 where Tm = (T1 + T2)/2. On the basis of the derived formulae, one can determine the heat flux value. Figure 14 illustrates the section of a Gardon type heat flux meter which is attached to the waterwall tube in the boiler. The design and material requirements of the meter for measurements of the local heat flux to the watersteam tubes of boiler combustion chambers are stated in [25].
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Jan Taler and Dawid Taler
Figure 14. Gardon type heat flux meter for use in boiler furnaces [25]; (a) section of disc heat flux meter, (b) disc flux meter welded to waterwall tube.
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Figure 15 shows a portable Gardon type heat flux meter for measurements of the local heat flux in boiler furnaces.
A
B Figure 15. Continued on next page. Heat Flux: Processes, Measurement Techniques and Applications : Processes, Measurement Techniques and Applications, Nova Science
Measurement of Heat Flux and Heat Transfer Coefficient
19
B
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Figure 15. Portable disc heat flux meter [22]; (a) section of water cooled probe of Gardon type, (b) sensor (disc of constant thickness); 1 – radial sensor (disc), 2 – outer tube, 3 – cover, 4, 5 – inner tubes, 5 – casing , 6 – distance pin, 7, 10, 19 – bottom, 8 – water outlet , 9 – water inlet, 9 – water outlet, 12, 17 – gasket, 13 – screw, 14, 15, 18 – elements of gland seal, 16 – fixing ring (dimensions in millimetres).
The absorbed heat flux is determined on the basis of the measured disc temperature measured at its center and periphery. The temperature of the disc periphery is measured at four points to check that the temperature distribution in the disc is axisymmetric (Figure 15b). The edge of the sensor 1 is cooled by the water flowing through annuli. The uniform temperature of the sensor edge is extremely important for the proper functioning of Gardon meters. The heat flux at the periphery of the circular sensor is very high. Even small differences in thermal resistance of the joint between the sensor and the heat sink cause large errors in determining heat flux with Eq. (52) because the edge temperature is not constant and the temperature distribution in the sensor is asymmetric. Other portable probes for measuring local heat flux in boiler furnaces are presented in the monograph [22], in which the steadystate as well as transient response of various probes was studied.
2. STEADYSTATE INVERSE METHODS FOR DETERMINING HEAT FLUX AND HEAT TRANSFER COEFFICIENT In inverse methods, the heat flux or heat transfer coefficient at the surface of the component is determined based on the measured temperatures at selected points inside the component. The fluid temperature is also measured to calculate the heat transfer coefficient.
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Jan Taler and Dawid Taler
2.1. Methods for Solving OneDimensional Inverse Nonlinear Heat Conduction Problems Encountered in Experimental Determination of Boundary Conditions This section presents the solution to a problem of determining the heat flux density and the heat transfer coefficient, on the basis of temperature measurement at three locations in the flat sensor, with the assumption that the heat conductivity of the sensor material is temperature dependent [26]. Three different methods of determining the heat flux and heat transfer coefficient, with their practical applications, were presented. The uncertainties in the determined values were also estimated.
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2.1.1. Introduction There are numerous measurement techniques used to determine the heat transfer coefficient [2733]. The most popular ones are: the thinfilm naphthalene masstransfer technique [2728], the electrochemical method [2930] and methods using liquid crystals [3133]. In tests conducted in higher temperatures, e.g. in steam boilers, industrial furnaces or in experimental tests of fluid boiling, most often the conduction probes are used [12, 22, 34]. On the basis of temperature measurement conducted at points of various coordinates, the heat flux density and the heat transfer coefficient are determined. In this section, the problem of determining the heat flux q and heat transfer coefficient h, on the basis of temperature measurement at three locations in the flat plate was presented, with the assumption that the thermal conductivity of the material of which the sensor was made is temperature dependent. Three different methods of determining the heat flux q and heat transfer coefficient h, with their practical applications, were presented. The uncertainties in the determined values were also estimated. 2.1.2. The Formulation of the Problem A measurement device presented in Figure 16a was used for the determination of the heat transfer coefficient on the surface of the solid body on which the liquid boiling takes place or over which the fluid flows [34]. The copper block is heated from the bottom with the use of a resistance heater made of chromenickel wire. The side surfaces of the copper block are thermally insulated, thus the heat flow is one dimensional. The heat transfer coefficient h on the surface of the block is determined from the simple formula
h
q , Ts Tf
(54)
where: h – heat transfer coefficient on the surface of the block, Ts – temperature of the surface of the block, Tf – bulk temperature of the fluid or gas, q – heat flux. In order to determine the heat flux, the temperature of the block has to be measured by 2 different thermocouples, located at different positions. For the apparatus presented in Figure 16a, the temperature was measured at 2 points of the coordinates: (L – x3) = 2.54 mm and (L – x2) = 7.44 mm (Figure 16a). This allows to determine the heat flux density q. Considering that
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21
Measurement of Heat Flux and Heat Transfer Coefficient
the heat flux density on the entire height of the block is constant, and assuming that the thermal conductivity k of the copper block is constant, the heat flux density can be easily calculated from the formula
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q
k f 2 f3 , x3 x2
(55)
where: f2 = f(x2) – temperature measured at point x2, f3 = f(x3) – tmperature measured at point x3 (Figure 16a). For the device presented in Figure 16a the distance between the temperature sensors is: x3 – x2 = 7.44 mm – 2.54 mm = 4.9 mm. In addition, to check the determined value of q, the temperature is measured at the third point of the coordinate:(L – x1) = 12.3 mm. This is easy to arrange, as the temperature distribution along the height of the block is linear. Additionally in order to improve the accuracy of the calculations, the third measuring point can be used for the determination of the values of q and h. Since the number of the measuring data points is greater than the number of the unknown parameters, the problem becomes overdetermined and the values of q and h can be determined more precisely. The overdetermined inverse heat conduction problems are also encountered in experimental determination of metal thermal conductivity. Figure 16b depicts a simple device used to measure thermal conductivity [35]. It consists of a hot plate as a heat source. In contact with the hot plate is a 2.5 cm diameter rod made of stainless steel of known thermal conductivity that has thermocouples attached for obtaining temperature readings. Resting on the stainless steel is a 2.5 cm diameter aluminium rod of unknown thermal conductivity that also contains thermocouples [35]. However, it is necessary that the thermal conductivity of stainless steel must be known. The least squares method may be used to determine the heat flow in the axial direction, thermal conductivities of upper and lower rod, surface temperatures of both rods at the contact, and thermal resistance of the contact between two rods. In this section, a more general problem of determining the values of q and h will be analyzed, when the number of the measurement data N is equal or greater than 2 and the thermal conductivity k is temperature dependent. Infinitely long plate or rod, which is thermally insulated on the side surface (Figure 17) is heated with the heat flux q. On the top surface, x = L the heat is absorbed by a liquid of temperature Tf . On the basis of temperature measurement in N 2 locations (in this case N = 3), the sensor heat flux q and the heat transfer coefficient h were determined. The thermal conductivity k of the sensor material, that is the plate or the rod, depends on temperature T. The steadystate heat conduction equation has the following form:
dq 0 dx
(56)
where
q k T
dT dx
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(57)
22
Jan Taler and Dawid Taler
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a
b
Figure 16. Schematics of devices for measuring heat transfer coefficient (a) and for measuring thermal conductivity of metals (b).
Figure 17. Locations of thermocouples in the sensor. Heat Flux: Processes, Measurement Techniques and Applications : Processes, Measurement Techniques and Applications, Nova Science
23
Measurement of Heat Flux and Heat Transfer Coefficient The temperatures at N internal locations are known from measurement
T
x x1
f1
T x x2
f2
T x xN
(58)
fN .
The heat flux q and the heat transfer coefficient h, appearing in Eq. (54) are searched. The boundary conditions are
k
k
dT dx
x 0
dT dx
xL
q,
h T
(59)
x 0
Tf
(60)
The number of unknowns is 2, thus it is lower or equal to the number of the measured data N ≥ 2. The problem is overdetermined and will be solved using the weighed least squares method
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S
N
Ti f i 2
i 1
i2
min ,
(61)
where i is the standard temperature deviation fi measured at the point xi. Because of the 2 weight coefficients wi i in equation (61), thermocouples (xi, fi), which have a great uncertainty are mostly ignored and do not worsen the quality of the approximation. The problem of the least squares (61) can be solved using several methods, depending on the method used for solving equation (56); that is, it is solved differently when the exact analytical solution exists and differently when the temperature distribution T(x) is determined numerically.
2.1.3. Solving the Inverse Problem The heat flux q and the heat transfer coefficient h will be determined using three different methods. In the first one, the T(x) function will be determined analytically and in two next methods, the temperature distribution T(x) will be determined discretely, as a result of the numerical solution. 2.1.3.1. Analytical Determination of Temperature Distribution Taking into consideration that the heat flux is constant, q = const, and assuming the linear dependency of the conductivity k on temperature
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Jan Taler and Dawid Taler
k T a bT
(62)
where a and b are constants and temperature T is expressed in C, equation (56) can be solved analytically. After substituting Eq. (62) to Eq. (56) and after integration we obtain:
1 qx aT bT 2 C 2
(63)
where C is the integration constant. From condition:
T
xL
q Tf h
(64)
we obtain
q C qL a T f h
2
bq Tf . 2 h
(65)
After considering Eq. (65) in Eq. (63) and solving the quadratic equation with regard to T, we obtain 2
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a 2a q a 2q T x x L Tf b b b h b
q Tf h
2
(66)
The temperature distribution T(x) is a nonlinear function of q and h . The values of q and h, for which the sum S as determined from Eq. (61) reaches the minimum, will be determined using the LevenbergMarquardt method [36, 37]. For the thermal conductivity independent of temperature, that is when k = a, then from equation (63) we obtain
qx aT C kT C
(67)
from which it results
T
qx D. k
After substituting (68) to (61) we obtain
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(68)
25
Measurement of Heat Flux and Heat Transfer Coefficient 2
qxi D fi N k min S 2
i
i 1
(69)
The necessary conditions of the existence of the minimum of the sum of squares S are
S 0 q S 0 D
(70)
from which the following system of equations is obtained
q k2
N
xi2
i 1
2 i
D N xi 1 N fi xi , k i 1 i2 k i 1 i2
N N f q N x 1 i2 D 2 i2 . k i 1 i i 1 i i 1 i
(71)
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After solving the system of equations (71) q and D are obtained. The heat transfer coefficient h is calculated from Equation (54), and the surface temperature Ts is calculated from formulation (68), taking x = L.
2.1.3.2. Numerical Determination of Temperature Distribution The control volume method was used to determine the temperature distribution. The division of the wall into control volumes is presented in Figure 18. The problem of determining q and h will be solved using 2 different methods: the Lagrange multipliers method and the LevenbergMarquardt method. 2.1.3.2.1. The Lagrange Multiplipliers Method Assuming that the plate 0 x L is divided into M > N control volumes and N temperature measurement points are situated at nodes M1,..., MN = M1+N1 of the coordinates x1,..., xN, the temperature distribution will be searched firstly in the x1 x xN zone and subsequently, using the extrapolation algorithm, the temperatures in zones 0 x x1 and xN x L will be determined. Next, on the basis of the determined temperature distribution, the heat transfer coefficient h and the heat flux density q will be calculated. Searched temperatures in nodes M1,...,MN should additionally be in compliance with the heat conduction equation (56), what means that for the used control volume method, the heat balance equations for every control volume in the x1 x xN zone should be satified
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Jan Taler and Dawid Taler
k Ti 1 k Ti Ti 1 Ti k Ti k Ti 1 Ti 1 Ti 0, 2 x 2 x i M 1 1, , M 1 N 2. Fi
(72)
Figure 18. Division of the sensor into control volumes.
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Thus, the stated conditions constitute the optimization problem (61) with equality constraints given by Eq. (72). This problem will be solved using the Lagrange multipliers method, according to which, the minimized function assumed the following form
S
M 1 N 1
Ti f i 2
iM 1
i2
M 1 N 2
i Fi min
(73)
i M 11
The minimized function S, given by Eq. (73), is non linear with respect to the searched temperatures TM1,...,TM1+N1. One of the methods, which can be applied to solve such a problem, is the GaussNewton method [12, 36]. Another method is the determination of the set of normal equations
S 0 Ti
i M 1,..., M 1 N 1 ,
(74)
which, together with the equality constraints equations
F j 0,
j M 1 1,..., M 1 N 2
(75)
provide a set of nonlinear algebraic equations, which can be solved using the NewtonRaphson method. As a solution, the set of N temperatures Ti and a set of (N2) Lagrange Heat Flux: Processes, Measurement Techniques and Applications : Processes, Measurement Techniques and Applications, Nova Science
27
Measurement of Heat Flux and Heat Transfer Coefficient
multipliers are obtained. For three measuring points presented in Figure 18 (M1 = 2, N = 3) the sum (73) is given by 4
Ti fi
i 2
i2
S
2
k T2 k T3 T2 T3 k T4 k T3 T4 T3 1 min 2 x 2 x
(76)
The normal equation set (74) assumes the form
2 T2 f 2
2 2
2 T3 f3
2 3
k T2 T2 T3 k T2 k T3 1 0, T 2 x 2 x 2 k T3 T2 2T3 T4 k T2 2k T3 k T4 1 0 2 x 2 x T3 (77)
2 T4 f 4
42
k T4 T4 T3 k T4 k T3 1 0. 2 x T4 2 x
In this case we have only one constraint equation (75), e.g. the heat balance equation for node 3 (Figure 18)
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k T2 k T3 T2 T3 k T4 k T3 T4 T3 0. 2 x 2 x
(78)
By solving the set of equations (7778) using the NewtonRaphson method, the temperatures T2, T3, T4 and the multiplier 1 were obtained. In order to determine the temperature field and subsequently q and h, the extrapolation of the temperature distribution from domain x1 x xN (Figure 18) towards the edges x = 0 and x = L was performed. For this case, from the heat balance equation for node 2 (Figure 18), one obtains
k T1 k T2 T1 T2 k T2 k T3 T3 T2 0. 2 x 2 x
(79)
Thus, using a simple iteration method, the T1 is determined
T1
k 1
T2
k T2 k T3
k T1 k k T2
T2 T3 ,
k 1, 2,3,
(80)
T10 T2 can be assumed as a first approximation in Eq.(80). After a few iterations the solution, satisfying the condition
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Jan Taler and Dawid Taler
T1k 1 T1k T1k 1
(81)
can be obtained, where is the assumed tolerance of the calculations. The T5 can be determined in a similar way. From the heat balance equation for node 4, the following equation is obtained:
k T3 k T4 T3 T4 k T5 k T4 T5 T4 0, 2 x 2 x
(82)
from which, using the simple iteration method, the T5 temperature is determined.
T5
k 1
T4
k T3 k T4
k T5 k k T4
T4 T3 ,
k 1, 2,3,
(83)
T50 T4 can be assumed as a first approximation in (83). Knowing temperatures in all 5 nodes (Figure 18), the heat flux q
q
k T1 k T2 T1 T2 2 x
(84)
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and the heat transfer coefficient h can be determined
h
k (T4 ) k (T5 ) T4 T5 1 . 2 x T5 T f
(85)
2.1.3.2.2. The Implementation of the LevenbergMarquardt Method The practical implementation of the method described in the paragraph 2.1.3.2.1 can be difficult for 2dimensional cases, because the extrapolation outside the inverse zone requires that the temperature should be measured inside the body, along the closed curve. If the temperature measurement points are distributed within the analysed zone, not within the closed curve, then more appropriate would be to apply the least squares method, described below. The parameters q and h, are assumed to be unknowns, as in the first method, and the minimum of the function (61) is searched for using the LevenbergMarquardt method [36, 37]. The temperature distribution at the kth iteration step, for the given values of q(k) and h(k) is determined using the control volume method, from the following set of equations:
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Measurement of Heat Flux and Heat Transfer Coefficient
k T k T q k k T1 k T2 T2 T1 0 k 12 x 2 q T2 T1 0 2 x k Ti 1 k Ti Ti 1 Ti k Ti k Ti 1 Ti 1 Ti 0, i 2, k Ti 1 2 k Ti Ti 1 1 x Ti k Ti 2k Ti 1 Ti x Ti 0, i 2, 2 x k TM 12 k TM TMx1 TM k h T T f M 0 k TM 1 2 k TM TM 1 x TM h k T T 0 f M 2 x
, M 1 , M 1
(86) where M is the number of control volumes. For the division presented in Figure 3 we have: M = 5. For every iteration step, the nonlinear set of algebraic equations is solved using the GaussSeidel method. The values of h and q are determined using the LevenbergMarquardt method [36, 37], in such a way, that the temperatures TM1,...,TM1+N1, determined from the set of equations (86) satisfy the condition:
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S
M 1 N 1
Ti f i 2
iM 1
i2
min
(87)
The advantage of this method of determining q and h on the basis of measured temperatures fM1,...,fM1+N1 is the possibility to consider any dependency of thermal conductivity k on T, not only linear (62), and high precision of q and h determination. In contrary to methods I and II, the temperature distribution from the zone of the direct solution is not extrapolated to the zone of the inverse solution. The described method can be used for the 2 and 3dimensional problems for the piecewise approximation of changes of q and h at the boundary. For this case, the number of determined components of q and h can be significant.
2.1.4. Calculating Uncertainty of Measurement of Heat Flux and Heat Transfer Coefficient Assuming that z1 = q and z2 = h only depend on the precision of the temperature measurement fM1,...,fM1+N1 and assuming that there are no errors in xi coordinates of the thermocouples mounting points, the zi standard deviation can be calculated in accordance to the error propagation rule [710, 3840]
zi where
fj
M 1 N 1
j M 1
z i f j
2
2 f , j
(88)
 standard deviation of fj, which is the measure of the temperature measurement
uncertainty. Partial derivatives zi/fi will be approximated by central differences
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Jan Taler and Dawid Taler
zi zi f M 1 , f M 11 ,..., f M 1 j ,..., f M 1 N 1 f j 2
(89)
zi f M 1 , f M 11 ,..., f M 1 j ,..., f M 1 N 1 2
where is a small positive number.
2.1.5. Example Of Calculations All three, described above methods of measuring q and h will be tested for the probe used for measuring the heat flux density [8, 12] of the thickness 0.016 m. The thermal conductivity of the material of the probe is given by Equation (62); with a = 14.65 W/(m·K) and b = 0.0144 W/(m·K·C). The temperature probes (thermocouples) were installed at 3 locations: x1 = 0.004 m, x2 = 0.008 m and x3 = 0.012 m. The “exact measurement data” will be calculated assuming: q = 274 800 W/m2, h = 2 400 W/(m2 ·K) and Tf = 15C. The temperature distribution over the thickness of the plate and the “nonexact measurement data” were presented in Table 1. The results of the calculations were presented in Tables 1 and 2. The analysis of the obtained results shows that all of them are identical for all methods. The temperature distribution on the thickness of the sensor (plate), determined using the first of the described methods is presented in Figure 19. The agreement of the temperature distribution, determined using the least squares method with the exact distribution presented in Table 1 is very good.
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Table 1. Exact temperature distribution over the thickness of the measurement plate and the results of the calculations x, m 0.000 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.010 0.011 0.012 0.013 0.014 0.015 0.016
Exact Measured temperatures data Ti,e,C fi, C 370.43 356.61 342.65 328.54 314.29 315.30 299.88 285.31 270.58 255.68 252.70 240.60 225.33 209.88 194.23 195.70 178.38 162.31 146.02 129.50 Sum S
wi i2 o
2
C
1
0.25
0.4444
Method I Ti, C 370.99 357.17 343.21 329.10 314.84 300.43 285.86 271.12 256.22 241.13 225.87 210.41 194.76 178.90 162.83 146.54 130.01 3.702
Method II Method III Ti, C Ti, C 370.99
370.99
314.84
314.84
256.22
256.22
194.76
194.76
130.01 3.702
130.01 3.702
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Measurement of Heat Flux and Heat Transfer Coefficient
31
Table 2. Determined values of the heat flux q and the heat transfer coefficient h and other results of the calculations
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q, W/m2 h ,W/(m2 ·K) q ,W/m2 h ,W/(m2 ·K) T1 – T1,e, C T2 – T2,e, C T3 – T3,e, C
Exact data 274 800 2 400
Method I 274 973.09 2 390.76 4 024.669 82.650 0.55 0.54 0.53
Method II 274 973.09 2 390.76 4 022.730 82.815 0.55 0.54 0.53
Method III 274 964.40 2 390.55 4 027.383 82.681 0.55 0.54 0.53
Figure 19. Temperature distribution in the probe: 1experimental data, 2exact temperature distribution (Table 1), 3temperature distribution obtained by the least squares methodtemperature is calculated using analytical expression (66).
2.1.6. Conclusion Three different techniques for determining heat flux and heat transfer coefficient based on temperature measurements inside a solid are presented. Thermal conductivity of the sensor material is temperature dependent. The errors in the determined values of the heat flux and the transfer coefficient are estimated using the variance propagation rule. The third technique, based on the LevenbergMarquardt method, has the advantage that can be easily extended to the overdetermined multidimensional inverse heat conduction problems. From the presented three methods of determination the heat flux q and the heat transfer coefficient h, the third method, in which the temperature distribution in the analysed zone is determined using the GaussSeidel method and the unknown parameters are determined using the LevenbergMarquardt method was proven to be the most versatile. This method of the identification of the boundary conditions can be successfully used for solving multi
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Jan Taler and Dawid Taler
dimensional, steadystate and dynamic problems; for much greater number of searched parameters than two. Additional advantage of this method is its good convergence, even for inaccurate approximation of the initial values of the searched parameters. Standard deviations for the obtained values can be then determined using the Gauss rule of error propagation.
2.2. Inverse Determination of Local Heat Transfer Coefficients
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Two methods for the solution of the nonlinear inverse heat conduction problems encountered in the experimental determination of the local heat transfer coefficient distributions are presented in this subsection. The methods are formulated as linear and nonlinear leastsquares problems. The uncertainties in the estimated heat transfer coefficients are determined for the temperature measurements with known and unknown standard deviations. The main advantage of the presented methods is that they do not require any knowledge, or solution to, the complex fluid flow field.
2.2.1. Introduction There are many different methods for measuring local heat transfer. One common category is mass transfer method [41, 42]. The naphtalene sublimation method is one of the most convenient mass transfer methods, that is particularly useful in complex flows and geometries. Another widely used technique is the transient method [4345]. Liquid crystals have been used extensively for the surface temperature measurements with this method. Both techniques have been widely used with considerable success but they have certain difficulties. The limitations to the naphtalene sublimation method are high temperatures. Liquid crystal thermography has a limited temperature range from –25C to 250C, so it is not feasible for objects at high temperature. An alternative method to obtain the local convective heat transfer coefficient that does not have any disadvantages noted above, is the inverse procedure. Determination of the spacevariable heat transfer coefficient on a complex shape surface requires the solution of the nonlinear inverse heat conduction problem [4648]. The unknown parameters associated with the solution are selected to achieve the closest agreement in a least squares sense between the computed and measured temperatures using the GaussNewton method in conjunction with the singular value decomposition or modified GramSchmidt methods. Hensel and Hills [49] approached the twodimensional steadystate inverse heat conduction problem using the linear leastsquares method. Linearization of the least squares problem is accomplished by assuming an unknown temperatures [49] or temperatures and heat fluxes [48,50] on the boundary. The boundary is divided into large number of elements and temperatures or heat fluxes are assumed to be constant over each element. Having determined the boundary values of temperature and heat flux from the solution of the IHCP, the convective heat transfer coefficients are determined from the Newton`s Law of Cooling. Numerical [4950] and experimental tests demonstrated that spatial distribution of the heat transfer coefficient can be estimated with satisfactory accuracy if the division of the boundary into elements is very fine. If the number of segments on the boundary is too small, than the constant value of temperature or heat flux over an element can not be assumed. In order to solve over
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33
Measurement of Heat Flux and Heat Transfer Coefficient
determined IHCP the number of interior temperature measurement points should be greater than the number of boundary temperature or heat flux components. However, mounting numerous thermocouples inside the solid disturbs the temperature distribution and is impractical. In this section two different problems are considered. In both cases the distribution of heat transfer coefficient is deduced from internal temperature measurements. In the first case the number of temperature measurement points is equal to the number of heat transfer components to be estimated. In the second case of the overspecified IHCP, the number of internal temperature measurements is greater than the number of unknown heat transfer coefficients on the boundary. The thermal conductivity of the solid k(T) may be temperaturedependent. Both linear and nonlinear leastsquares formulations are studied. The leastsquares problems are parametrized by assuming the stepwise changes of heat transfer coefficient on the boundary or expressing the space variations of the heat transfer coefficient in the functional form. The confidence intervals of IHCP solutions are calculated using the error propagation rule of Gauss [51].
2.2.2. Numerical Formulation of the Inverse Heat Conduction Problem The temperature distribution in the body is governed by nonlinear partial differential equation
k T T 0 .
(90)
The unknown boundary conditions may be expressed as: boundary conditions of the first kind
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T f rs on S,
(91)
boundary conditions of the second kind
k T
T q rs on S, n
(92)
boundary conditions of the third kind
k T
T h rs T f Ts , n
(93)
where rs represents points on the boundary, S. In addition to the unknown boundary conditions the internal temperature measurements f i are included in the analysis:
T ri f i , i 1,, m , m n .
(94)
The objective of the present approach is to determine the spatial distribution of the heat transfer coefficient from measured temperatures at m interior locations. The measured Heat Flux: Processes, Measurement Techniques and Applications : Processes, Measurement Techniques and Applications, Nova Science
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Jan Taler and Dawid Taler
temperatures f = (f1, …, fm)T depend on the transfer coefficients and can, therefore, be used for their determination. The problem may be formulated as linear or nonlinear IHCP.
2.2.3. Linear IHCP First, the part of the boundary with unknown conditions is discretized into n intervals. The unknown boundary temperature or heat flux is then approximated by piecewise constant function (stepwise function) (Figure 20). The parameters x are unknown temperatures or heat fluxes. The Kirchhoff transformation can be introduced to transform Eqs. (9092) to linear problem. The new variable is the integral over temperature of the thermal conductivity. For known boundary condition of the third kind, however, the transformed boundary condition remains nonlinear. In this section the method of successive substitution will be used to linearize the problem (9092). The spatial discretization of Eq. (90) subject to boundary conditions (9192) using finite difference, or finite element methods yields the nonlinear system of NE algebraic equations
A T T B T x ,
(95)
where vector x contains n unknown surface temperatures and heat fluxes, A is NE x NE matrix, T is the NE dimensional column vector of node temperatures and B is NE x n matrix. The right side of Eq. (95) Bx contains all of the applied boundary and volumetric source functions. Using the method of successive substitution Eqs. (95) can be written as
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A T k T k 1 B T k x k 1 ,
(96)
where the superscript indicates the iteration number. The algorithm has a reasonably large 0
radius of convergence. After selecting the starting vector T the convergent solution is obtained in about six iterations. If the thermal conductivity is constant iteration is not needed. The linear system (96) has the unique solution
1
T k 1 A T k B T k x k 1 , k = 0,1,.....
(97)
Temperature of the body is measured at m interior locations which are assumed to be coincident with selected nodes in the mesh. The temperature at the jth node, where the temperature sensor is placed can be calculated in the following way
1
T j k 1 eTj A T k B T k x k 1 ,
Heat Flux: Processes, Measurement Techniques and Applications : Processes, Measurement Techniques and Applications, Nova Science
(98)
35
Measurement of Heat Flux and Heat Transfer Coefficient
where ej denotes jth column vector of the identity matrix I N E . The vector of calculated k 1
temperatures Tm
at nodes coincident with sensor locations is then formed as follows
Tm k 1 Cmk x k 1 , k 1 m
where T
(99)
is the mdimensional vector, and C
k m
is the m x n matrix. The object is to
choose x such that computed temperatures agree with certain limits with the experimentally measured temperatures. This may be expressed as
Tm k 1 f 0 , k 1
where Tm
(100)
is given by Eq. (99).
If the number m of measured temperatures is equal to the number of unknown boundary temperatures and heat fluxes n, then the n x n system of linear equations (100) has the unique solution
x k 1 C(mk ) f , 1
k 0,1,
k
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provided the inverse matrix Cm
(101)
1
exists.
If thermal conductivity k does not depend on temperature iteration is not needed. The uncertainty of the calculated parameters x can be determined using error propagation rule [51, 52]. A standard procedure is to take more temperature measurements than the number of unknown parameters xi. The leastsquares method is used to determine x1, …, xn when m > n. To measure how well the calculated temperature agree with data, the chisquare merit function is used
S f Tm k 1
T
G f f Tm k 1 ,
(102)
where Gf is known positive definite matrix. If the measurement errors are known then the matrix Gf is
1 12 0 0 1 22 Gf 0 0
0 0 , 0 1 m2
(103)
where i is the standard deviation of the ith measured temperature fi. The leastsquares solution that minimizes the sum (102) is [52] Heat Flux: Processes, Measurement Techniques and Applications : Processes, Measurement Techniques and Applications, Nova Science
36
Jan Taler and Dawid Taler
x k 1 CTm G f Cmk k
1
C T m
k
Gff .
(104)
Once the boundary temperatures and heat fluxes are determined, the heat transfer coefficient is determined from
h
k T T , T S Tf n S
(105)
where the fluid temperature Tf is known from measurements. The measure of the accuracy of the estimated parameters xˆ is 95% confidence interval. Having finished iteration the 95% confidence intervals are calculated according to
xˆi 2 xi xi xˆi 2 xi , where
(106)
xi Cii .
The notation Cii represents the diagonal element of the covariance matrix
Cxˆ CTmG f Cm . 1
(107)
The limits of 95% confidence interval associated with the estimate Tj can be found from
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Tˆj 2 Tj T j T j 2 T j ,
(108)
where (Tj) are square roots of the diagonal elements of the matrix
CT CmCxˆ CTm .
(109)
In some cases the standard deviations associated with a set of temperature measurements are not known in advance and the sum of squares are used to derive a value . If we assume that Gf = Im the 100(1–)% confidence intervals for xi are calculated using the formulas
xˆi tm 2n
S S Cii xi xˆi tm 2n Cii , mn mn
(110)
2
where the value tm n is the (1–)th quantile of the Student`s tdistribution with (mn) degrees of freedom. The leastsquares sum S is computed from Eq. (102) using the fitted parameters xˆ (boundary temperatures or heat fluxes). The covariance matrix (107) simplifies to
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Measurement of Heat Flux and Heat Transfer Coefficient
Cxˆ CTmCm . 1
(111)
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2.2.4. Nonlinear IHCP If the number of the unknown heat transfer components is equal to the number of the temperature measurement points, the method presented in [46] can be used. The heat transfer coefficients are estimated so that the computed temperatures agree with the experimentally measured temperatures, e.g. Eqs. (94) should be satisfied. The predicted temperatures depend implicitly on the heat transfer coefficients x. In order to solve the system of nonlinear equations (94) the SORsecant method or SORNewton methods can be used [46]. The SOR (successive overrelaxation) algorithm works well in practice and is very efficient in terms of the programming and computing time. When unknown parameters are heat transfer components and m>n then the leastsquares problem becomes nonlinear (Figure 20). The numerical solutions of the nonlinear overdetermined IHCP encountered in determining heat transfer coefficients, are presented in [46,47]. The GaussNewton method in conjunction with the singular value decomposition was used to determine the space components of the heat transfer coefficient. The methods are very efficient provided the starting value x(0) is sufficiently close to the global minimum xˆ . This weakness of the GaussNewton method is not inherent in the LevenbergMarquardt algorithm [36, 52], that is fast and robust. The LevenbergMarquardt method is one of the best generalpurpose GaussNewtonbased algorithms for solving nonlinear least squares problems.
Figure 20. Location of temperature sensors and boundary discretization; Method I – boundary conditions of the first or second kinds – x1,....,xn represent unknown temperatures or heat fluxes, Method II – boundary condition of the third kind  x1,....,xn represent unknown heat transfer coefficients.
The problem of determining spacevariable heat transfer coefficient can be formulated as a parameter estimation problem by selecting the functional form for the heat transfer coefficient or by approximating the spatial changes of the heat transfer coefficient by stepwise function. In both cases, there are n parameters: x = (x1, …, xn)T to be determined
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Jan Taler and Dawid Taler
such that the computed temperatures Ti agree in the leastsquares sense with the experimentally acquired temperatures fi. The LevenbergMarquardt method performs the kth iteration as
x k 1 x k δ k ,
(112)
where
δ k J mk
T
G f J mk k I n
1
J k
T
G f f Tm x ,
(113)
and
Jm
Tm x Ti x xT x j
.
(114)
mxn
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Suppose we have converged to the solution of the nonlinear IHCP. To obtain approximate 100(1–)% confidence intervals for x, Eqs. (106) or (110) may be used as in the case of the method I. However, the matrix Jm should be used in Eqs. (106) and (110) instead of Cm.
2.2.5. Test Cases First numerical experiment with simulated data is presented to demonstrate the method I and II. The local heat transfer coefficient from a hot surface to boiling water is often measured using a heat flux transducer. The transducer body is made of stainless steel that incorporates three thermocouples placed in different positions (Figure 21). Unidirectional steadystate heat transfer is assumed to occur within the transducer between the resistance heater and boiling water. Similar constructions have heat flux meters inserted in inspection ports [12, 22]. Based on temperature measurements at three locations, the heat flux x1 = q on the surface x = 0 and the surface temperature x2 T x L at x = L (method I) or the heat transfer coefficient x2 = h (method II) are determined. Since the heat flux x1 is a constant independent of y, then the heat transfer coefficient in the method I can be calculated from the estimated values of the heat flux x1 sensor surface temperature x2 and from measured fluid temperature Tf:
h
x1 . x2 T f
(115)
The governing heat conduction equation with temperaturedependent thermal conductivity is
d dT k T 0. dy dy Heat Flux: Processes, Measurement Techniques and Applications : Processes, Measurement Techniques and Applications, Nova Science
(116)
39
Measurement of Heat Flux and Heat Transfer Coefficient The objective is to determine the boundary conditions
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T k T y
y 0
x1 ,
(117)
Figure 21. Location of thermocouples in the heat flux sensor (a) and division of the sensor into control volumes (b).
T
yL
x2 , (method I),
or
Heat Flux: Processes, Measurement Techniques and Applications : Processes, Measurement Techniques and Applications, Nova Science
(118)
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Jan Taler and Dawid Taler
T x2 T k T y yL
xL
T f , (method II).
(119)
Using, the interior conditions known from temperature measurements (Figure 21a)
T
y y1
f1 ,
(120)
T
y y2
f2 ,
(121)
T
y y3
f3 ,
(122)
the parameters x1 and x2 will be determined using the methods I and II. The thermal conductivity is temperature dependent and represented as follows
k T 14.65 0.0144 T ,
(123)
where k is expressed in W/(m·K) and T in oC. To simulate “exact” measurement data the direct heat conduction problem was solved using: q 274800 W/m2, h = 2400 W/(m2 ·K), Tf =15oC, L=0.016m, y1=0.004m, y2=0.008m and y3=0.012m. In order to study the effect of
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errors in the measured temperatures, small numbers i , i=1,2,3 were added to “exact” data
f1 T
y y1
1 314.29 1.01 315.3C
f2 T
y y2
2 255.68 2.98 252.7C
f3 T
y y3
3 194.23 1.47 195.7C
(124)
The standard deviations of the measured temperatures are: 1 = 1C, 2 = 2C and 3 = 1.5C. The differential equation (116) with boundary conditions (117)(118) is solved numerically using the finite volume method. The resulting system of nonlinear, algebraic equations is solved using the matrix inversion method in conjunction with successive iteration (method I) or by using the GaussSeidel iteration (method II). The results given in Table 3 are in excellent agreement with the input data: x1 = q = 274800 W/m2 and x2 = h = 2400 W/(m2·K). In the second example the actual experimental data are used. Experiments were performed with an array of vertical finned tubes with two longitudinal fins arranged in staggered pattern. The experimental results reported herein are among the first that show the variation of the local heat transfer coefficients over the circumference of the finned tube. Most data reported previously were acquired for smooth tubes at low temperatures [32, 5359].
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Measurement of Heat Flux and Heat Transfer Coefficient Table 3. Results obtained by the inverse methods
Estimated parameters
Analytical method (exact solution) q = 274800 W/m2, h = 2400 W/(m2·K),
T Standard Deviations
Sum of squares
S xˆ
Method II
xˆ1 qs 275045 W/m2,
xˆ1 q 274964 W/m2,
xˆ2 T
xL
2 129.97 C, xˆ2 h 2390.6 W/(m ·K),
h = 2392.3 W/(m2·K),
T
xL
130.01 C
xˆ1 4082.3 W/m2, xˆ1 4027.4 W/m2, xˆ2 2.28 K
xˆ2 82.7 W/(m2·K)
3.695K2
3.702K2
Temperature distribution T(y), C Analytical method Method I 370.43 371.02 314.29 314.85 255.68 256.21 194.23 194.74 129.50 129.97
Method II 370.99 314.84 256.22 194.76 130.01
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y [m] 0.000 0.004 0.008 0.012 0.016
o x L 129.5 C
Method I
Figure 22. Cross section of the finned tube showing locations of nine thermocouples. Heat Flux: Processes, Measurement Techniques and Applications : Processes, Measurement Techniques and Applications, Nova Science
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Jan Taler and Dawid Taler
With uniform heat flux at the inner surface of the tube placed in a cross flow, heat flows by conduction in the peripheral direction due to the asymmetric nature of the air flow around the perimeter of the tube. The peripheral heat flow affects the wall temperature distribution to such an extent that in some cases significantly different results may be obtained for geometrically identical surfaces [54]. In this study IHCP is solved to account for the peripheral wall heat conduction. Figure 22 shows a cross through the finned tube and dimensions needed in the calculation of the local heat transfer values around the tube periphery [60]. The temperature of the tube was measured at nine locations shown in Figure 22. The finned tube surface was instrumented with the 1 mm in diameter Ktype sheeted thermocouples. The tube with an outside diameter of 24.7 mm was constructed of the K18 low alloy steel and was centrally located at the fourth row in the array. An array of 36 electrically heated tubes was placed in the wind tunnel in the 0.27 m x 0.30 m rectangular test section. The heated length of the tube was 0.30 m. The tubes are mounted in a staggered array with nine successive rows each consisting of five tubes. The pitch of tubes in direction of flow SL and the pitch of tubes in plane perpendicular to flow ST were: SL = 0.031 m and ST = 0.05325 m, respectively. The air temperature at the inlet and outlet of the bundle was measured with Ktype thermocouples of the 1 mm in diameter placed in the two cross sections of the air channel. The Reynolds number was calculated based on the outer diameter and the maximum air velocity in the narrowest cross section of the bundle. The heat flux on the inner surface of the tube q, is given by q = Q/(dinH), where Q denotes the electric power of the resistance heater placed inside the tube, din is the inner diameter and H is the heated length of the tube.
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The local heat transfer coefficient h L is approximated by the following function
5 L h L xi cos i 1 , Lc i 1
(125)
where xi are the coefficients to be estimated, L is the distance from the point 1 at the tip of the fin in direction of flow ( path coordinate along tube ) and Lc is the extended distance between points 1 and 9 in Figure 22. Figures 23 and 24 show spatial distributions of the temperature and heat transfer coefficient as a function of path coordinate along the finned tube for two different values of the Reynolds number. The experiments and calculations were carried out at the following conditions: Q = 103.9 W, Tf = 23.4C for Re = 47962, and Q = 123.5 W, Tf = 29.2C for Re = 14514, where Tf denotes the air temperature. The thermal conductivity of the tube material was k = 53 W/(mK). The unknown coefficients xi, i = 1, …, 5 in Eq. (125) were estimated using the method II. The temperature distribution in the finned tube cross section was calculated using the control (finite) volume method. Because of the symmetry only a half of the tube was considered. It was divided into 75 control volumes. The following 95% confidence intervals for the parameters xi are obtained for Re = 47962: x1 = 154.18.3 W/(m2·K), x2 = 56.44.1 W/(m2·K), x4 = 1.08.5 W/(m2·K), and x5 = 0.976.7 W/(m2·K). The heat transfer conditions within the bundle are dominated by boundary layer separation effects and by wake interactions. The local heat transfer coefficient
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Measurement of Heat Flux and Heat Transfer Coefficient
43
decreases from the fin tip up to angular position, 105 on the tube circumference (Figure 24), where boundary layer separation occurs, then increases due to the turbulence in the wake. Similar results are obtained when spatial variation of the heat transfer coefficient was approximated by the stepwise function.
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Figure 23. Measured temperature at locations 19 and determined temperature distribution versus the path coordinate along tube or the angular position at two different Reynolds numbers.
Figure 24. Spatial distribution of the heat transfer coefficient on the outer surface of the finned tube at two different Reynolds numbers. Heat Flux: Processes, Measurement Techniques and Applications : Processes, Measurement Techniques and Applications, Nova Science
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Jan Taler and Dawid Taler
2.2.6. Conclusion Two inverse heat conduction methods were developed to determine unknown distributions of the local heat transfer coefficient on surfaces of arbitrarily shaped solids. The distribution of the heat transfer coefficient is estimated by utilizing temperature measurements inside the solid. The fluid temperature is also measured. The boundary of the solid is divided into segments, on which temperatures or heat fluxes are assumed to be constant (method I) or the inverse heat conduction problem (IHCP) is parametrized by assuming the stepwise changes of heat transfer coefficient or expressing it as a function of space (method II). The overdetermined IHCP is solved using the linear (method I) and nonlinear (method II) leastsquares methods. The thermal conductivity of the solid may be dependent on temperature. The uncertainties in the estimated components of the heat transfer coefficient or in the estimated parameters are determined for the temperature measurements with known and unknown standard deviations. The determination of the circumferential heat transfer coefficient distribution on the heated tube with two longitudinal fins in cross flow demonstrates the accuracy and the potential use of the second method. The main advantage of the presented methods is that they do not require any knowledge, or solution to, the complex fluid flow field. It should be noted that determining unknown steady distribution of heat transfer coefficients by using the developed methods is inexpensive, since they require only one fluid temperature probe and a few thermocouples for temperature measurements inside the solid.
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2.3. Measurement of Heat Flux to WaterWalls In Boiler Combustion Chambers The tubular type instrument (flux tube) was developed to identify boundary conditions in water wall tubes of steam boilers. The meter is constructed from a short length of eccentric tube containing four thermocouples on the fire side below the inner and outer surfaces of the tube. The fifth thermocouple is located at the rear of the tube on the casing side of the waterwall tube. The boundary conditions on the outer and inner surfaces of the water fluxtube are determined based on temperature measurements at the interior locations. Four Ktype sheathed thermocouples of 1 mm in diameter, are inserted into holes, which are parallel to the tube axis. The nonlinear least squares problem is solved numerically using the Levenberg– Marquardt method. The heat transfer conditions in adjacent boiler tubes have no impact on the temperature distribution in the flux tubes.
2.3.1. Introduction The tubular type instruments (flux tube) [6165] and other measuring devices [66] were developed to identify boundary conditions in water wall tubes of steam boilers. The meter is constructed from a short length of eccentric tube containing four thermocouples on the fire side below the inner and outer surfaces of the tube. The fifth thermocouple is located at the rear of the tube on the casing side of the waterwall tube (Figure 25).
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Measurement of Heat Flux and Heat Transfer Coefficient
45
The boundary conditions on the outer and inner surfaces of the water fluxtube must then be determined from temperature measurements at the interior locations. Four Ktype sheathed thermocouples, 1 mm in diameter, are inserted into holes, which are parallel to the tube axis.
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Figure 25. The heat flux tube placed between two water wall tubes; a – flux tube, b – water wall tube, c – thermal insulation (dimensions in millimeters).
The thermal conduction effect at the hot junction is minimized because the thermocouples pass through isothermal holes. The thermocouples are brought to the rear of the tube in the slot machined in the protecting pad. An austenitic cover plate with thickness of 3 mm welded to the tube is used to protect the thermocouples from the incident flame radiation. A Ktype sheathed thermocouple with a pad is used to measure the temperature at the rear of the fluxtube. This temperature is almost the same as the watersteam temperature. A method for determining fireside heat flux, heat transfer coefficient on the inner surface and temperature of watersteam mixture in waterwall tubes is developed. The unknown parameters are estimated based on the temperature measurements at a few internal locations from the solution of the inverse heat conduction problem. The nonlinear least squares problem is solved numerically using the Levenberg–Marquardt method. The diameter of the measuring tube can be larger than the waterwall tube diameter. The view factor defining the distribution of the heat flux on the measuring tube circumference was determined using exact analytical formulas and compared with the results obtained numerically using ANSYS software. The method developed can also be used for an assessment of scale deposition on the inner surfaces of the waterwall tubes or slagging on the fire side. The presented method is suitable for water walls made of bare tubes as well as for membrane water walls. The heat transfer conditions in adjacent boiler tubes have no impact on the temperature distribution in the flux tubes.
2.3.2. Theory At first, the temperature distribution at the cross section of the measuring tube will be determined, i.e. the direct problem will be solved. Linear direct heat conduction problem can be solved using analytical method. The temperature distribution will be calculated numerically using the finite element method (FEM). In order to show accuracy of a numerical approach, the results obtained from numerical and analytical methods will be compared.
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Jan Taler and Dawid Taler
The following assumptions have been made: thermal conductivity of the flux tube material is constant, heat transfer coefficient on the inner surface of the measuring tube does not vary on the tube circumference, rear side of the waterwall, including the measuring tube, is thermally insulated, diameter of the eccentric flux tube is larger than the diameter of the waterwall tubes, the outside surface of the measuring flux tube is irradiated by the plane flame surface, so the heat absorption on the tube fire side is nonuniform.
The temperature distribution in the eccentric heat flux tube is governed by heat conduction
1 1 k kr 0 r r r r r
(126)
subject to the following boundary conditions
k n r r qm
(127)
o
k
r
h r a
(128)
r a
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The cylindrical coordinate system is shown in Figure 26.
Figure 26. Approximation of the boundary condition on the outer tube surface.
The left side of Eq. (127) can be transformed as follows (Figure 26)
k n r r q r q n o
r ro
T k T k cos 1 sin 1 r r r ro
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(129)
47
Measurement of Heat Flux and Heat Transfer Coefficient
The second term in Eq. (129) can be neglected since it is very small and the boundary condition (127) simplifies to
k
r
r ro
qm cos 1
(130)
Heat flux over the tube circumference can be approximated by the Fourier polynomial qm = q0 qn cos n cos 1 n 1
(131)
where q0
qm d , 0 cos 1 1
qn
2
qm cos n d , 1
cos 0
n 1,...
(132) The boundary value problem (126,128,130) was solved using the separation of variables to give
r , A0 B0 ln r Cn r n Dn r n cos n
(133)
n 1
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where
A0
q0 ro 1 ln a , k Bi
(134)
B0
q0 ro , k
(135)
1 n 1 u Bi n n qn ro n a Cn , 2n k Bi u 1 n u 2 n 1
(136)
1 n u Bi n a n qn ro n Dn . k Bi u 2 n 1 n u 2 n 1
(137)
The ratio of the outer to inner radius of the eccentric flux tube: u = u(φ )= ro(φ) /a depends on the angle φ, since the outer radius of the tube flux surface (Figure 26)
ro e cos b2 esin
2
Heat Flux: Processes, Measurement Techniques and Applications : Processes, Measurement Techniques and Applications, Nova Science
(138)
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Jan Taler and Dawid Taler
is the function of the angle φ. The symbol e and b in Eq.(138) denote the eccentric and the outer radius of the heat flux tube, respectively. Eq. (133) can be used for the temperature calculation when all the boundary conditions are known. In the inverse heat conduction problem three parameters are to be determined:
absorbed heat flux referred to the projected furnace wall surface: x1 = qm, heat transfer coefficient on the inner surface of the boiler tube: x2 = h, fluid bulk temperature: x3 = Tf.
These parameters appear in boundary conditions (127) and (128) and will be determined based on the wall temperature measurements at m internal points (ri,i)
T ri , i fi , i 1,..., m , m 3 .
(139)
In a general case, the unknown parameters: x1, …, xn are determined by minimizing sum of squares
S f Tm f Tm , T
(140)
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where f = (f1, …, fm)T is the vector of measured temperatures, and Tm = (T1, …, Tm)T the vector of computed temperatures Ti = T(ri,i), i = 1, …, m. The parameters x1, …, xn, for which the sum (140) is minimum, are determined using the LevenbergMarquardt method [36, 52]. The parameters, x, are calculated by the following iteration
xk 1 xk δk , k 0,1,...
(141)
where
δ
k
J mk
T
k
Jm
In k
1
J k m
T
(142)
j 1,..., n.
(143)
f Tm x k .
The Jacobian Jm is given by
Jm
Tm x Ti x i 1,..., m xT x j mn
The symbol In denotes the identity matrix of n n dimension, and (k) the weight coefficient, which changes in accordance with the algorithm suggested by Levenberg and Marquardt. The upper index T denotes the transposed matrix. Temperature distribution T(r,,
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Measurement of Heat Flux and Heat Transfer Coefficient
49
x(k)) is computed at each iteration step using Eq. (133). After a few iteration we obtain a convergent solution.
2.3.3. Computational and Boiler Tests Firstly, a computational example will be presented. “Experimental data” are generated artificially using the analytical formula (133). Consider a waterwall tube with the following parameters (Figure 125):
outside radius b = 35 mm, inside radius a = 25 mm, pitch of the waterwall tubes t = 80 mm, thermal conductivity k = 28.5 W/(m·K), absorbed heat flux qm = 200000 W/m2, heat transfer coefficient h = 30000 W/(m2·K), fluid temperature Tf = 318C.
The view factor distribution on the outer surface of waterwall tube was calculated analytically and numerically by means of the finite element method (FEM). The changes of the view factor over the tube circumference are illustrated in Figure 27a. Comparison of analytical and numerical results is presented in Figure 27b. The agreement between the temperatures of the outer and inner tube surfaces which were calculated analytically and numerically is also very good (Figure 28). The small differences between the analytical and FEM solutions are caused by the approximate boundary condition (130). The temperature distribution in the flux tube cross section is shown in Figure 29.
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a)
b)
Figure 27. View factor associated with radiation heat exchange between elemental surface on the flux tube and flame: (a) 1 total view factor accounting for radiation from furnace and boiler setting, 2approximation by the Fourier polynomial of the seventh degree, 3 exact view factor for furnace radiation, 4 view factor from boiler setting, (b) comparison of total view factor calculated by exact analytical and FEM method.
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Jan Taler and Dawid Taler
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Figure 28. Computed temperature distribution in C in the cross section of the heat flux tube; qm = 200000 W/m2, h = 30000 W/(m2·K), Tf =318C.
Figure 29. Temperature distribution on the inner and outer surface of the flux tube calculated by the analytical and finite element method.
The following input data is generated using Eq. (133): f1 = 438.24C, f2 = 434.79C, f3 = 383.52C, f4 = 380.90C, f5 = 321.58.24C. The following values were obtained using the proposed method: qm* = 200000.57 W/m2, h* = 30001.80 W/(m2·K), Tf* = 318.00C. There is only a small difference between the estimated parameters and the input values. The highest temperature occurs at the crown of the fluxtube (Figures 28 and 29).
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Measurement of Heat Flux and Heat Transfer Coefficient
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The temperature of the inner surface of the flux tube is only a few degrees above the saturation temperature of the watersteam mixture. Since the heat flux on the rear side of the tube is small, the circumferential heat flow rate is significant. However, the rear surface thermocouple indicates temperatures of 24C above the saturation temperature. Therefore, the fifth thermocouple can be attached to the unheated side of the tube so as to measure the temperature of the watersteam mixture flowing through the flux tube. In the second example, experimental results will be presented. Measurements were conducted at a 50 MW pulverized coal fired boiler. The temperatures indicated by the flux tube at the height of 19.2 m are shown in Figure 30.
Figure 30. Measured flux tube temperatures; marks denote measured temperatures taken for the inverse analysis.
The heat flux tube is of 20G low carbon steel with temperature dependent thermal conductivity
k T 53.26 0.02376224T 8.67133 106 T 2 ,
(144)
where the temperature T is expressed in C and thermal conductivity in W/(m·K). The unknown parameters were determined for eight time points which are marked in Figure 30. The inverse analysis was performed assuming the constant thermal conductivity k (T ) which was obtained from Eq. (144) for the average temperature: T T1 T2 T3 T4 / 4 . The estimated parameters: heat flux qm, heat transfer coefficient h, and the watersteam mixture Tf are depicted in Figure 31.
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Jan Taler and Dawid Taler
Figure 31. Estimated parameters: absorbed heat flux qm, heat transfer coefficient h, and temperature of watersteam mixture Tf.
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The developed flux tube can work for a long time in the destructive high temperature atmosphere of a coalfired boiler. Flux tubes can also be used as a local slag monitor to detect a build up of a slag. The presence of the scale on the inner surface of the tube wall can also be detected.
2.3.4. Conclusion A new method for determining the heat flux absorbed by a furnace wall was developed.The measuring device is an eccentric tube. The ends of the four thermocouples are located at the fireside part of the tube and the fifth thermocouple is attached to the unheated rear surface of the tube. Using the temperature readings from the thermocouples and knowing the locations at which the termocouples are placed, the heat flux absorbed by the tube, the heat transfer coefficient on the inner surface, and the fluid temperature can be calculated. The meter presented in the section has one particular advantage over the existing flux tubes to date. The temperature distribution in the flux tube is not affected by the water wall tubes, since the flux tube is not connected to adjacent waterwall tubes with metal bars, referred to as membrane or webs. To determine the unknown parameters only the temperature distribution at the cross section of the flux tube must be analysed.
3. TRANSIENT INVERSE METHODS Transient methods for measuring timeconstant heat flux or heat transfer coefficient are based on the solutions of onedimensional heat conduction problems in a semiinfinite body
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53
Measurement of Heat Flux and Heat Transfer Coefficient
with a step increase in surface heat transfer or with a step increase in fluid temperature. From the comparison of the measured and calculated surface temperature at the prescribed time point, the constant heat flux or heat transfer coefficients are determined. If the surface heat flux or fluid temperature are timevarying then the method of superposition (Duhamel’s integral) is used to calculate the surface temperature of a halfspace. If the time changes or surface temperature are known, the surface heat flux is determined using the Duhamel integral. For bodies with finite dimensions, a space marching method for determining surface heat flux based on the measured time history at the interior point of the body will be presented.
3.1. Transient Techniques for Determining Steady Heat Flux or Heat Transfer Coefficient
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The surface heat flux or heat transfer coefficient are inferred from transient measurements of the surface temperature of the semiinfinite medium. One quite accurate method of temperature measurement offer foil resistance temperature detector (RTD). RTD foil sensors may be fabricated by depositing a platinum or metalglas slurry on a ceramic substrate. The time response of platinum film is very quick. Also, liquid crystals are ideal for surface temperature measurements. The cholesteric and nematic liquid crystals are used for thermal measurements. Liquid crystals for surface temperature measurements are in encapsulated form that can be used as a thin film (paint). If temperature of an illuminated liquid crystal increases, the color of the crystal changes in bands [5]. Temperature response at a distance x beneath the substrate surface is given by [11, 13]
q t x2 4 t x T T0 2 e xerfc , 0 x k 2 t
(145)
A plane wall (slab) can be treated as a semiinfinite medium if its back surface temperature changes insignificantly. The time period t in which the heat penetration depth [13] is smaller than the substrate thickness can be determined from the condition
t
L2 . 16
(146)
Surface temperature response is
T
x 0
T0
2q
t k
T0
2q t . kc
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(147)
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Jan Taler and Dawid Taler
The surface temperature (x = 0) of a semiinfinite substrate is measured at time tx to determine a time invariant surface heat flux q. The heat flux q is determined from the condition
T 0, t x f t x ,
(148)
where the symbol f(tx) stands for the measured surface temperature at time tx. Substituting Eq.(147) into Eq. (148) and solving for q gives
k f t x T0 q . 2 tx /
(149)
If the measured temperatures fi are known at N times ti, i = 1,…, N than the overdetermined set of linear equations with respect to q can be solved using the least squares method. Similar method is used for determining the time independent heat transfer coefficient. The temperature distribution in the semiinfinite medium ( in the half space) is given by [11,13]
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h x h 2 t x T T0 T f T0 erfc exp 2 k 2 t k h x erfc t , 0 x . 2 t k
(150)
Function (150) describes temperature distribution in the halfspace with an initial temperature T0, whose surface is subject to (for time t > 0) a liquid with temperature Tf ≠ T0. The halfspace can be heated or cooled by a fluid with a temperature of Tf = const. Halfspace surface temperature x = 0 is defined by the following expression obtained from Eq. (150)
h2 t h T 0, t T0 T f T0 1 erfc t exp 2 k k
(151)
h2 t h T 0, t T0 T f T0 1 exp 2 1 erf t . k k
(152)
or
Expression (152) is also used for the experimental determination of the heat transfer coefficient h on the basis of measured surface temperature at a given time point tx. The heat transfer coefficient h is determined from the condition (148 ), where the surface temperature T(0, tx) is given by (152) and the symbol f(tx) stands for the measured surface temperature at time tx. If the measured temperatures fi are known at N times ti, i = 1,…, N than the overdetermined set of linear equations with respect to h can be solved using the least squares method.
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Measurement of Heat Flux and Heat Transfer Coefficient
In actual practice, the fluid temperature does not change stepwise but is a function of time. Using the Duhamel integral the surface temperature Ts(t) = T(0, t) can be calculated for time dependent fluid temperature Tf (t) [13, 67], which is approximated by a stepwise curve (Figure 32). The coordinates of temporal points i, in which temperature fi = Tf, i = Tf (i) is measured, are indicated in Figure 32.
Figure 32. Approximation of fluid temperature changes Tf (t) by a stepwise line.
Taking into account that (Figure 32)
f1 T f 1,0
(153)
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and
f1 fi 1 Tf i ,i 1 Tf ,i Tf ,i 1 ,
i 2,
M,
(154)
the surface temperature Ts(tM) obtained by using the Duhamel integral is [13, 67]
h 2 tM i Ts tM T0 1 exp 2 k i 1 h tM i T f i ,i 1 . 1 erf k M
(155)
If we introduce the dimensionless variable
h 2 tM M , k2 then formula (155 ) can be written in the form
Heat Flux: Processes, Measurement Techniques and Applications : Processes, Measurement Techniques and Applications, Nova Science
(156)
56
Jan Taler and Dawid Taler M Ts M T0 1 exp M i 1
i 1 erf M 1 tM
i 1 tM
T f i ,i 1 .
(157)
Using formula (157), one can determine heat transfer coefficient h while allowing for the medium’s timevariable temperature Tf (t). By measuring halfspace surface temperature in time tM by means of the liquid crystals and by comparing it to temperature calculated from the expression (157 ), one is able to determine h from the following nonlinear algebraic equation
Ts , c tM Ts , m tM 0 ,
(158)
where Ts,c(tM) is the temperature given by Eq. (157 ), while Ts,m(tM) a measured halfspace surface temperature. For details about the conducted experiment refer to [67]. Good reviews of heat transfer coefficient measurement methods, including manufacturing techniques, are given in [13, 67].
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3.2. Transient Techniques for Determining Time Varying Heat Flux or Heat Transfer Coefficient Unified mathematical procedures of transient methods for measuring surface heat transfer rates will be presented in this subsection. Three heat flux gauges are discussed: thin film, thickwall gauges placed on semiinfinite substrates and thinskin calorimeters. The aim of this subsection is to present a method for a simple and accurate determination of the timevarying heat transfer coefficient (or heat flux) given an accurate temperature history of the body at a selected point beneath the surface. The interior temperature measurements are converted into local instantaneous heat transfer coefficients by solving the inverse heat conduction problem for the gauge. The effect of the inaccuracies in the measurement of the interior temperature was eliminated by cubic spline smoothing or digital filtering of the raw interior temperature data prior to using it in the inverse heat conduction analyses. General case closed form equations for instantaneous surface heat flux, or heat transfer coefficient, are developed.
3.2.1. Introduction Most heat transfer measurements consist of monitoring the temperature of a body at selected points and then relating that temperature history to the one of heat transfer rate. In a general case, the temperatures measured are related in a complex way to the heat transfer coefficients and hence measurements are usually made with geometries such that only one spatial coordinate needs to be considered. The usual quantities of interest are the local heat
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Measurement of Heat Flux and Heat Transfer Coefficient
flux qs(t), and the convective heat transfer h(t). If the radiation can be neglected, the local heat transfer coefficient can be easily related to the heat flux
h
qs , T f Ts
(159)
where Ts represents the material surface temperature and Tf is the fluid temperature far away from the wall. The devices for measuring heat transfer between a flowing fluid and a solid surface can be categorized as gauges: (1) semiinfinite onedimensional gauges, (2) thinskin calorimeter gauges, and (3) thickwall gauges. The first method records instantaneous surface temperature from which instantaneous heat flux rates are deduced using the heat conduction solution for the semiinfinite substrate (Figure 33a). The surface heat flux is obtained by using the onedimensional, semiinfinite medium solution for a step change in surface temperature [1113] and applying Duhamel's superposition integral to give
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qs (t )
ck t 1 df () d , 0 t d
(160)
where f(t) is the measured surface temperature history. Such a onedimensional heat flow can also be achieved by having the conducting material in the form of a rod or a strip that is well insulated from the surrounding model. Onedimensional gauges employing surface thermocouples on such conducting rods fall into this category because they can be considered semiinfinite as the thermal penetration distance during experimental runtimes is small compared to the linear dimension of the gauge [6877]. The most popular measurement method for the surface temperature is the thinfilm metal resistance layers (such as palladium on MACOR or platinum on quartz) [6971, 73] or a surface thermocouples (such as coaxial or eroding thermocouples) [74, 77]. The surface temperature can also be determined from the calibrated liquidcrystal colour [75, 76]. The thinskin method is one of the oldest, simplest and most effective methods of obtaining transient heat flux [7882]. The calorimetric element is very thin (Figure 33b), so the rate of rise of the rear surface temperature, which is usually monitored, is equal to the rate of rise of the mean temperature. The expression used to obtain transient heat flux data is given by
q(t ) cE
df dt
(161)
where f(t) is the back surface temperature history. Equation (161) assumes no heat losses at the back surface and a negligible temperature drop across the calorimeter wall. In the case of thickwall gauges (Figure 33c) the heat received by the gauge is largely stored within the gauge while only a small portion is transferred to the substrate.
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Jan Taler and Dawid Taler
a)
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b)
c) Figure 33. Basic geometries of gauges used in heat transfer measurements; (a) semiinfinite onedimensional gauge, (b) thinwall (calorimeter) gauge, (c) thickwall gauge.
In this case the measured temperature is related in a more complex way to surface heat flux than that for thinskin calorimeters.
3.2.2. Analysis The physical model that may be applied to both the semiinfinite and calorimeter gauge is that of a uniform slab on a semiinfinite substrate composed of different material. The instantaneous surface temperature f(t) of the substrate is measured at x = E . The problem is to calculate the front surface temperature and the heat flux at x = 0 given the measured temperature at point x = E. The problem can be subdivided into two separate problems, one of which is a direct problem as shown in Figure 34. The semiinfinite body from x = E to x can be analyzed
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Measurement of Heat Flux and Heat Transfer Coefficient
as a direct problem because the boundary conditions at both boundaries are known (T2(E, t) = f(t) at x = E, T2/ x = 0 and T2 = T0 at x ).
Figure 34. Subdivision of an inverse heat conduction problem into inverse and direct problems.
When substrate thermal properties are treated as constant, the heat flux qE passing through surface x = E is calculated by converting the measured temperature history to heat transfer rate by using Duhamel’s Theorem (160). The same heat flux qE must leave body 1 (0 x E). Two conditions (temperature and heat flux) are prescribed at x = E in body 1 and none at x = 0. The surface temperature Ts = T1(0, t) and heat flux qs = k1T1/xx
= 0
histories of body 1 must be determined from
conditions at location x = E. Such a problem is referred to as an inverse problem. The general solution of the inverse problem was independently given by Stefan [83], Burggraf [84], and Langford [85]:
1 ( E x) 2 n d n y 1n dt n n 1 (2n)!
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T1 ( x, t ) y (t )
( E x) 1 ( E x ) 2 n d n qE qE (t ) k1 1n dt n n 1 (2n 1)!
qs (t ) k1
2n
T1 x
E 2 n 1 1 d n y n n n 1 (2n 1)! 1 dt
(162)
x 0
qE (t ) k1
n
(163)
E d qE 1 n 1n n 1 (2n)! dt The solution requires that finite order derivatives of the measured temperature y(t) and calculated heat flux qE(t) at interior location x = E must exist. The method is stable for any n
time step, provided d y / dt
n
and d n qE / dt n are bounded. Unwanted oscillations
(oscillatory instability) can be produced in the calculated surface temperature or heat flux if the time derivatives are not calculated with sufficient accuracy. The above series converge quite rapidly that only the first few time derivatives need to be considered. Truncating equations (162163) after the third or second derivative yields the approximate solutions of
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Jan Taler and Dawid Taler
acceptable accuracy. In a heatconduction body, variations in the surface conditions are always damped at interior points. In the inverse heat conduction problems the surface temperature and heat flux histories are obtained from the damped temperature data taken at a subsurface location. Therefore, one cannot hope to find the higher frequency components of the boundary conditions using only interior temperature measurements, especially when the temperature sensor is located far from the surface. To determine rapid variations in the surface conditions, the temperature sensor should be placed as close to the surface as possible.
3.2.3. Heat Flux Transferred to Substrate Materials The temperature measurements at x = E are made at discrete times: t1, t2, .., tM or in general at time ti at which the temperature measurement is denoted fi. If the interface temperature between successive times is assumed to vary linearly with time (Figure 35), Eq. (160) can be integrated analytically to give [13, 68] qE (tM 1 ) k2 2
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2
T2 x
k1 xE
k2 2 c2
k2 2 c2
M
i 1 M
xE
(164)
fi 1 fi tM 1 ti 1 tM 1 ti fi 1 fi i 1 ti
t i 1
T1 x
tM 1 ti 1 tM 1 ti
Figure 35. Piecewise linear interpolation of the timetemperature data.
Polynomial regression data fitting technique can also be applied to smooth the surface temperaturetime response m
y (t ) A0 A1t A2t 2 ,..., Amt m Ati i .
(165)
i 0
Substituting of the polynomial approximation (165) into equation (160) and its integrating yields [86] Heat Flux: Processes, Measurement Techniques and Applications : Processes, Measurement Techniques and Applications, Nova Science
61
Measurement of Heat Flux and Heat Transfer Coefficient qE (t ) 2
m i 1 c2 2 k2 (1)k (2i 1)/2 1 (i 1)! A1 t iAti (2 k 1) k !( i 1 k )! i 2 k 1
(166) If m 7, equation (166) reduces to a simple form
c2 2 k2 1/2 4 3/2 8 5/2 64 7/2 A t A2t A3t A4t 1 3 5 35 128 9/2 512 11/2 1024 13/2 A5t A6t A7t 63 231 429
qE (t ) 2
(167)
The solutions (164) and (166) have several practical limitations. Since the inverse solutions given by Eq. (162163) require continuous first and highorder derivatives of temperature data y(t) and heat flux qE, equation (164) is not appropriate. A major weakness of Eq. (166) is low accuracy of polynomial fitting. Polynomial smoothing allows the calculation of highorder derivatives but may not reproduce the real data points especially when the time spread of the fitted data is large. These restrictions can be avoided by using an alternate procedure based on spline or digital filter smoothing of temperaturetime data. The experimental temperature is represented by a third order spline in the form (Figure 36)
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1 1 yi ( ) C1,i C2,i ( i ) C3,i ( i )2 C4,i ( i )3 , 2 6 i i 1 , i 1, 2,..., M
(168)
where: = St t  scaled time, St  scaling factor. A general cubic polynomial (168) involves four constants
C1,i y ( i ) f ( i ), C2,i y( i )
dyi ( i ) 1 dyi (ti ) , d St dt
C3,i y( i )
d 2 yi ( i ) 1 d 2 yi (ti ) 2 , d 2 St dt 2
C4,i y( i )
d 3 yi ( i ) 1 d 3 yi (ti ) 3 . d 3 St dt 3
(169)
There is a sufficient flexibility in the cubic spline approximation to insure that not only the smoothing spline is continuously differentiable on the interval, but also that it has a continous second derivative on the interval.
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Jan Taler and Dawid Taler
The identical form to Eq.(168) has the orthogonal Gram polynomial of degree 3 used to construct a digital filter. However, the digital filters do not ensure the continuity of the functions yi() and their derivatives at the nodes i. Starting with Eq. (160) and the assumption that the surface temperature response is approximated by cubic splines (168), the surface heat flux qE() can be expressed as
c k M i1 dy () 1 q( M 1 ) 2 2 2 i d St i 1 i d 1 3 5 c k M 1 1 W 3 C 5 2 2 2 2 Vi Pi 2 Ri2 i Pi 2 Ri2 4,i Pi 2 Ri2 i 1 3 10
(170)
1 c2 2 k2 WM 23 C4, M 52 2 PM PM St , VM PM 3 10 M 1, 2,3,..., J 1 ,
2
where
Pi M 1 i , Ri M 1 i 1 ,
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Fi C1,i C2,i Pi
C3,i 2
Pi 2
C4,i 6 C4,i
Pi 3 ,
(171)
Vi
dFi C2,i C3,i Pi Pi 2 , d M 1 2
Wi
d 2 Fi C3,i C4,i M 1 i C3,i C4,i Pi . d M2 1 f y(t) y() 3
y() i fi
yM1()
y() 2 y() 1
1
f3
f2
fM
i 2
1 f1 2 3
yM ()
fi+1 f M+1
i
3
i
M1
i+1
M
M
=St t M+1
Figure 36. Cubic spline interpolation of the timetemperature data. Heat Flux: Processes, Measurement Techniques and Applications : Processes, Measurement Techniques and Applications, Nova Science
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Measurement of Heat Flux and Heat Transfer Coefficient
The time derivatives required in the exact solutions (162163) can be calculated analytically with high accuracy. The first two derivatives are dyM (tM 1 ) dy ( ) St M M 1 StVM , dt d d 2 yM (t M 1 ) St2WM , dt 2 dqE (tM 1 ) dq ( ) St E M 1 dt d 1 1 c k M 1 V 1 W 1 2 2 2 2 i Pi 2 Ri 2 i Pi 2 Ri2 i 1 2 2
3 C4,i 32 c2 2 k2 2 P R i i 2 12
VM 12 WM 1 C4, M 3 3 P PM2 PM2 St2 , 2 M 2 12
M 1, 2,3,... 2 d 2 qE (tM 1 ) 2 d q E ( M 1 ) S t dt 2 d 2 3 1 1 c k M 1 V 3 3W 2 2 2 2 i Pi 2 Ri 2 i Pi 2 Ri 2 4 i 1 4 1 1 ck 3 C4,i Pi 2 Ri2 2 2 2 2 8
1 5 VM 32 3WM 12 3 PM PM C4, M PM2 St2 , 4 4 8
M 1, 2,3,..., ( J 1).
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(172)
3.2.4. Cubic Spline Smoothing of the Measurement Data An observation of the timetemperature curve reveals, that small segments of the curve can be closely approximated by cubic splines (168). A spline is simply a piecewise polynomial, the pieces joined together at points called "knots". In particular, a cubic spline (168) is a piecewise cubic polynomial, constructed in such a way that second derivative continuity is preserved at the knots. The temperature data are known to be in error. Suppose that pairs of temperature data values (ti, fi), i = 1,..., J are observed and we wish to describe the relationship between them with a regression model
fi y(ti ) e i ,
(173)
where i are the uncorrelated errors with zero mean and y(t) is the smoothing polynomial spline of degree 3. The total number of measurements is J >4. The more frequently used method of fitting smoothing splines parallels the least squares curvefitting procedure by minimizing a criterion that depends on a least squares like term plus a term penalizing roughness. A measure of the rapid local variation of a curve can be given by a roughness penalty such as the integrated squared second derivative. The fitted spline is the solution to the optimization problem [8791].
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Jan Taler and Dawid Taler Minimize J
w y( ) f
2
i
i 1
i
i
J
y( ) d 2
(174)
1
where the parameter 0 controls the amount of smoothing. If is too mall, the spline will overfit, in the limiting case as 0, becoming an interpolating spline. As , the smoothing term dominates and removes not only noise but "signal" as well. The correct choice of is of considerable importance. The method of crossvalidation for choosing has also been offered as an option for choosing [9293]. It can be shown [8990] that the curve y has the following properties: 1. 2.
is a cubic polynomial in each interval [i, i+1];
at the measurement point i the curve and its first two derivatives are continuous, but there may be a discontinuity in the third derivative.
Schönberg [88] and also Reinsch [8990] point out that the spline solution y() to (174) has the property that it minimizes J
y( )
d
2
(175)
1
subject to
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J
w y( ) f i 1
2
i
i
i
S,
(176)
where S is a given nonnegative number which controls the extent of smoothing. This of course is a global method, requiring all the data points be available before fitting can begin. The choice of the spline smoothing parameter, S, is somewhat arbitrary. Fortunately, a wide optimum range for the smoothing parameter, S, can be found that gives accurate results. The results obtained for various S differ only in degree of smoothing [46]. If the smoothing parameter, S, is too small, the developed method produces a noise in the estimated surface heat flux or heat transfer coefficient. The noise error can usually be distinguished from true fluctuation in transient heat transfer without a priori knowledge of the noise error with the input signal. Furthermore, the inverse heat conduction problem involves the calculation of surface heat flux and heat transfer coefficient from transient, measured temperature history inside a solid. The higher frequency components of the boundary conditions are damped at interior point at a higher rate than the lower frequency components. In the inverse heat conduction problem the boundary conditions are estimated based on the low frequency components of the input signal.
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The spline smoothing technique or digital lowpass filtering used in this subsection allow to separate the lower frequency components of the true signal from the higher frequency components in the measurement errors. For practical purposes, it is sufficient to choose the smoothing parameter S subjectively, by plotting out a few temperaturetime curves and choosing the one which "eliminates best" high frequency measurement errors from the input data. If the method developed is to be used routinely on a large number of data sets or a part of larger procedure then an automatic choice of the smoothing parameter, S, is essential. In these cases the smoothing of the temperature data can be performed using the spline smoothing methods presented in [9495]. Both these methods attempt to follow trends in the data and are applicable even if the magnitude of error is unknown. The methods do not require explicit specification of control parameters. However, the test calculations show, that the method by Reinsch [89] gives more accurate results. If the time steps ti = ti+1 – ti are too small or too large it often happens that the minimizing procedure for estimating the spline coefficients become illconditioned. One way to overcome this difficulty is to introduce scaled time = St t. The effect of bad scaling in Reinsch method [89, 90] is that values of spline coefficients: J
C1,i, ...,C4,i are affected by roundofferrors and the sum of squares:
w y( ) f i 1
2
i
i
i
is
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not equal to the "a priori" given value S.
3.2.5. Digital Filtering of Measurement Data The smoothing procedure described above is a global method that requires knowledge of all the data points before construction of the approximating splines can begin. However, for online systems, it is frequently necessary to follow and fit the data without knowing its end beforehand [12]. The local method, in which the polynomial pieces are calculated as the data is gathered, is useful for such cases. In the case of a local method, the approximating cubic spline in any interval between data points depends only on a small set of neighboring data points (Figure 37).
Figure 37. Digital filtering of the timetemperature data.
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For example, Npoint moving digital filter is a local approximation. The digital filter approach is important because it is much more computationally efficient than other methods. Heat transfer coefficient measuring devices can incorporate the digital filter and immediate graphical output can be provided. Numerical experiments indicate that for equally spaced data points 7 N 11 is satisfactory. Leastsquares fitting with orthogonal Gram polynomial of degree 3 to eleven (N = 11) equally spaced data points (Figure 37) yields for the centre point i 1 36 fi 5 9 fi 4 44 fi 3 69 fi 2 84 fi 1 89 fi 84 fi 1 429 69 fi 2 44 fi 3 9 fi 4 36 fi 5 , yi y (ti )
dy (ti ) 1 300 fi 5 294 fi 4 532 fi 3 dt 5148 t 503 fi 2 296 fi 1 296 fi 1 503 f i 2 532 f i 3 294 f i 4 300 f i 5 , yi
yi
d 2 y (ti ) 5 2 1 2 3 2 f fi 4 fi 3 fi 2 fi 1 f i 2 i 5 2 dt 5 15 5 5 3 143 t
3 2 1 2 fi 1 fi 2 fi 3 f i 4 f i 5 , 5 5 15 5 yi
d 3 y (ti ) 5 1 11 23 7 7 f i 5 f i 4 f i 3 fi 2 f i 1 f i 1 3 3 dt 5 15 30 15 15 143 t
23 11 1 fi 2 fi 3 fi 4 f i 5 . 30 15 5
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(177) Substituting (177) into (169) gives the coefficients of the spline function (168). In order to treat each data point in the same manner, the scanning (gathering) of the data should start at least (N1)/2 = 5 time steps t before cooling or heating process starts. If this is done, the heat flux qE(t) can be calculated at t = 0.
3.2.6. Application of Procedures In order to examine the accuracy of the proposed procedures three different test cases are solved. In the first example surface temperature measurements are used to estimate instantaneous heat flux to a semiinfinite body. A semiinfinite copper body is exposed to heat flux that varies in time in a triangular fashion: t , D qe (t ) qmax 1
qe (t ) qmax
t a, D t a t D , a 1, 1 a D 0
(178)
where D is heating duration. The maximum heat flux qmax occurs at a. The surface temperature history of a semiinfinite solid initially at the uniform temperature T0 = 0C is given by [96] Heat Flux: Processes, Measurement Techniques and Applications : Processes, Measurement Techniques and Applications, Nova Science
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Measurement of Heat Flux and Heat Transfer Coefficient 3
Tw T
Tw T
x 0
x 0
qmax
D 4 t 2 t , 0 a, kc 3a D D
qmax
3 t 2 3 a D 4 t 2 D , a t 1. kc 3a D 1 a D
(179) The "measured" temperatures at the surface are simulated by adding a random error j to exact temperatures Tj
f j T j e j j = 1, ..., J
(180)
where J is a random variable of a uniform distribution with values in the range [1, 1] and e is the maximum magnitude of the temperature error. This artificial data is then input to the inverse algorithm and its output compared to the original assumed heat flux. Figure 38 shows the artificial data for a particular case: a = 0.5, D = 0.1s, qmax = 100 000 kW/m2, T0 = 0C, kc = 1372.7 (kJ)2/(m4·K2·s), e = 5 K. The simulated measured temperature data, fj containing measurement errors are shown as
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the data points in the Figure 38, while the solid line represents the exact data. The time step t is 1.4286·103 s. There are then 70 time steps (J = 71). The dotted line represents the results of measured data smoothing with cubic splines and the parameter values St = 100 and S = 434 K2. This value of S was subjective adjusted. The resulting splines for temperature data taken at x = 0 shown in Figure 38 give very good compromise between smoothness and goodness of fit. A measure of the errors in the temperature measurements f(t) is the sample root mean square norm. It is given by
f
1 N 1 N 2 2 f T ( t ) e i w i i . N i 1 N i 1
(181)
In this test case: f = 2.783 K. If the discretized computed heat flux component is denoted q(ti) and the true component is qe(ti), in order to measure the error, the root mean square norm is determined as
q
1 N 2 q(ti ) qe (ti ) . N 1 i 2
In this equation the summation is from i = 2 as q(t = 0) is not known.
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(182)
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Figure 38. Surface temperature of semiinfinite body for triangular heat flux; _____  errorless temperature measurements, +  inexact temperature measurements.
Some numerical results for errorless measurements with f = 0 and inexact measurement
with f = 2.783 K are shown in Figures 3940. Figure 39 shows the results for errorless data. Clearly, the agreement between the estimated and the true heat flux is excellent. The spline smoothing approach to interpolating of temperature measurements produces better results than the interpolation by straightline pieces, the difference is not large though. The root mean squared errors in q are q,l = 190.3 kW/m2 and q,s = 70.7 kW/m2 for
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piecewise linear and spline interpolation, respectively.
Figure 39. Calculated surface heat flux with errorless data; _____  known (exact) heat flux and calculated heat flux for spline interpolation of the timetemperature data, +  calculated heat flux  piecewise linear interpolation of the timetemperature data. Heat Flux: Processes, Measurement Techniques and Applications : Processes, Measurement Techniques and Applications, Nova Science
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Figure 40. Calculated surface heat flux for inexact temperature data; ____  known (exact) heat flux, + piecewise linear interpolation of the timetemperature data,   spline interpolation of the timetemperature data.
Figure 41. Calculated surface heat flux for inexact temperature data; ______  known (exact) heat flux, + piecewise linear interpolation of the timetemperature data,   spline approximation of the timetemperature data (S = 434 K2).
The results obtained by using measurements with random errors are also in good agreement with the exact (known) heat flux history, in spite of large errors in the input data (Figure 40). The spline interpolation, in conjuction with Duhamel's integral method, yields results slightly better than more common piecewise linear interpolation (q,s = 3096.5 kW/m2 and q,l = 3577 kW/m2).
However the interpolation of the input data has limited applicability because the estimated heat flux curve is noisy and time derivatives of the surface heat flux qE(t) can not be Heat Flux: Processes, Measurement Techniques and Applications : Processes, Measurement Techniques and Applications, Nova Science
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evaluated with sufficient accuracy. When a spline smoothing technique with S = 434 K2 is applied to the inexact temperaturetime data again excellent results are obtained. Figure 41 shows that the triangular shape of heat flux is quite well reproduced and the results are smooth. The root mean squared error is q,s = 935.1 kW/m2. The results of the application of the three techniques for calculating the surface heat flux clearly indicate that the spline smoothing technique is superior to the piecewise linear and spline interpolation procedures. In the second example a transient technique for measuring heat transfer between a flowing fluid and a solid surface is presented. In the experiments whereas a high temperature surface was spray cooling with water spray, the heat transfer coefficient is evaluated from the transient response of the solid at some interior point x = E. When the solid has a low thermal diffusivity and is very thick or when the transient of interest is very short, the solid temperature response is limited to a thin layer near the surface and the solid may be considered to be a semiinfinite medium. In order to test the accuracy of the method, approximate recovering of heat transfer coefficient h(t) is investigated for semiinfinite solid, initially heated to a uniform temperature T0 and suddenly exposed to convective environment. The heat transfer coefficient h and fluid temperature Ti are assumed constant over the duration of the experiment. The exact data for this problem are generated using the analytical solution given by Eq. (152) which is written in the dimensionless form
T ( x, t ) T T0 T erf exp Bi 1 erf , x 0 , (183) where
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x 2 t
,
Bi
hx , k
h2 t . c k The exact surface heat flux is then calculated as follows
qs q 0, t k
T x
x 0
h T f Ts t ,
(184)
where the surface temperature is obtained from Eq. (183). The result is
Ts t T 0, t T f T0 T f exp 1 erf
.
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(185)
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The transient heat flux at x = E is calculated using the Eq. (170). The temperature field and the surface heat flux are determined using the exact solution (162163). Only two first terms in Eqs.(162) and (163) were evaluated to compute the temperature and heat flux responses with a good accuracy. The "measured" temperatures are obtained from Eq. (180) for e = 2K. Simulated experimental temperatures have been generated for 101 time points with a time step of 1 s. The root mean squared error in the temperature data is f = 1.147 K. A temperature sensor is located at an interior point , E = 0.006 m below the cooled surface. The thermal properties of the test body (a thick steel wall of the PWR reactor) are assumed to be constant and the following values are used: k = 42 W/(m·K), = 11.6·106 m2/s, ck = 158 629 212 J2/(m4·K2·s) Figure 42 shows the exact (solid line) and simulated measured temperatures (crosses) for the parameters values: h = 2000 W/(m2 ·K), T0 = 300oC, Tf = 20oC.
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The effect of inaccuracies in the measurement of the interior temperature is minimized by smoothing the raw interior temperaturetime data prior to the calculation of the heat transfer coefficient. The spline smoothing approach with St = 1 and S = 150 K2 or the elevenpoint digital filter (N = 11) are used to smooth the corrupted data. For this value of smoothing parameter, S, the curve y(t) in Figure 42 looks very good, i.e. is smooth and eliminates random errors. The fluid temperature Tf = 20C is not disturbed by random errors. When a third order spline interpolation (S = 0) is applied to the disturbed temperaturetime data of Figure 42, the results are of little interest because the estimation of the heat transfer coefficient is very poor. Therefore, graphical results for this case are not presented. Notice that though h is actually constant, it is determined as though it were a time varying function h(t). The root mean square norm for h is determined as
h
2 1 J h ti he ti , J 1 i 2
(186)
where h(ti) is the computed heat transfer coefficient and he(ti) is the true value of h(t) at ti. Figure 42 shows the calculated temperature at the surface (x = 0) and at the location x = E/2 = 0.003 m for the errorless data (Sf = 0, S = 0). Cleary, the agreement between the estimated and the exact surface temperature is excelent. The true surface temperature is indicated by empty squares. Figure 43 depicts the estimated heat flux at x = E = 0.006 m and at the surface (x = 0) and the heat transfer coefficient for the same data used in Figure 42. The root mean squared error in h for f = 0 is: h = 37.6 W/(m2·K). Inspection of Figures 42 and 43 indicates that both the true surface heat flux and temperature are in very good agreement, although some oscillations in the heat transfer coefficient are observed near t = 0. The estimated surface heat flux and heat transfer coefficient exhibit greater errors associated with lower accuracy of the spline smoothing for small time. In addition, to achieve higher
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accuracy, the third and higher order derivatives should be retained in the series (5) as t0. The effect of the uncertainty in the input temperature data can be seen in Figures 44 and 45.
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Figure 42. Temperature of semiinfinite body determined from temperature measurements at the location x = E = 0.006 m using spline approximation of the temperaturedata; +  exact data (1), spline interpolation (S = 0) (2),   calculated temperature at x = E/2 = 0.003 m (3),  calculated surface temperature (4),  exact surface temperature (5).
Figure 43. Heat transfer coefficient h and surface heat flux q determined from exact temperature data at x = E = 0.006 m using spline approximation (S = 0) of the temperature data; _____  heat flux at x = E = 0.006 m (1),   surface heat flux (2),  exact surface heat flux (3),  heat transfer coefficient (4). Heat Flux: Processes, Measurement Techniques and Applications : Processes, Measurement Techniques and Applications, Nova Science
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Figure 44. Temperature of semiinfinite body determined from inexact temperature measurements at the location x = E = 0.006 m using spline approximation of the temperature data; +  inexact temperature data (1),  spline approximation (S = 150 K2) (2),   calculated temperature at x = E/2 = 0.003 m (3),  calculated surface temperature (4), □  exact surface temperature (5).
Figure 45. Heat transfer coefficient h and surface heat flux q determined from inexact temperature measurements at the location x = E = 0.006 m using spline approximation of the temperature data; _____ heat flux at x = E = 0.006 m (1),   surface heat flux (2), □  exact surface heat flux (3), ▲  heat transfer coefficient (4).
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Note that the temperature measurement errors affect primarily the estimated heat transfer coefficient (h = 95 W/(m2·K). Compared to the error in the recovered heat flux and heat transfer coefficient, the calculated temperatures are found to be less sensitive to random errors in the data. It should be noted, that prior to applying the inverse heat conduction procedure, the temperaturetime data have been smoothed using the cubic spline with S = 150 K2. When the elevenpoint digital filter is applied to the same disturbed temperaturetime data, the results (Figures 45 and 46) are in fair agreement with the true values (h = 143.1 W/(m2·K)).
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For the case of f = 1.147 K, the results look good but not as smooth as those for the spline smoothing approach. The scatter in the estimated heat flux and heat transfer coefficient is caused primarily by the piecewise approximation used in digital filtering. Unlike the smoothing splines, in digital filtering neither the function nor its first two derivatives are required to be continuous. The advantage of the digital filter approach is that it requires less computer time, particularly as more time steps are considered. In the third example, the data actual measured data from a thermal shock experiment are considered [97].
Figure 46. Temperature of semiinfinite body determined from inexact temperature measurements at the location x = E = 0.006 m using the elevenpoint digital filter for smoothing of the temperature data; + inexact temperature data (1), _____  smoothed temperature using the elevenpoint digital filter (2),  calculated temperature at x = E/2 = 0.003 m (3), _____  calculated surface temperature(4), □  exact surface temperature (5)
The heat transfer to droplets impinging on a heated surface was investigated based on experimentally acquired temperature at a interior location of a semiinfinite body. The rapid cooling of a hot solid surface with an impinging water jet is used in many industrial processes. Typical applications are found in the continuous casting processes of metalurgical industries and the emergency cooling of pressure vessels of PWR reactors [98].
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Figure 47. Heat transfer coefficient h and surface heat flux q determined from inexact temperature measurements at the location x = E = 0.006 m. The temperature data with errors from interval 2 K smoothed using the elevenpoint digital filter; _____  heat flux at x = E = 0.006 m (1),   surface heat flux (2), □  exact surface heat flux (3), ▲  heat transfer coefficient (4).
The experimental study presented in [97] was made in attempting to obtain fundamental information concerning the heat transfer from a heated wall to saturated droplets deposited on it in the postdryout mist flow regime. Measurements of the surface heat flux during the residence time of a droplet on the high temperature surface were made based on the timedependent variation of the wall temperature measured near the surface. The test surface was a 0.01 m dia. and 0.005 m thick stainless steel disc supported by a sheated chromelalumel thermocouple of 0.00065 m O.D. located at the center of the disc (Figure 48). The thermocouple junction is located at distance E = 0.0003 m from the cooled surface.
5 0.65 q
10
E=0.3
1 2 0
x
Figure 48. Stainless steel disc used for measurement of unsteady heat flux; 1 – thermocouple, 2 – disc (dimensions in millimiters).
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Figure 49. Cooling curves of semiinfinite body after the impingement of saturated droplets upon a heated surface; +  temperatura data at x = E = 0.0003 m (1),  spline approximation of the temperature data (S = 2.0 K2) (2), _____  calculated temperature at x = E/2 = 0.00015 m (3), ■ calculated surface temperature (4).
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In the present study the surface temperature and heat flux are calculated from recorded temperature variations at x = E = 0.0003 m assuming that the disc behaves as semiinfinite body with uniform initial temperature. The thermal properties of the disc material are assumed to be constant as follows k = 17.65 W/(m·K), α = 4.26·106 m2/s, ck = 73 127 307 J2/(m4·K2·s). The inverse heat conduction (IHC) calculations are performed on the experimental temperature data shown in Table 4. For these computations, St=10000 is used. Prior to applying the IHC procedure, the time temperature data from the thermocouple are approximated by cubic spline functions using Reinsch method with S = 2.0 K2. In this case the small value of S was chosen as the number of data points is not large (J = 16) and the temperature measurements f(ti) (Figure 49) I = 1, ..., 16 are not noisy. Figure 49 shows the temperature decay in a semiinfinite body (stainless steel disc). The surface temperature decreases sharply on the droplet impingement but its recovery after the droplet rebounding is also rapid. Although the value of E = 0.0003 m is small, the temperature difference between the point x = E and the cooled surface: x = 0 is large. Also the heat flux at location x = E differs very significantly from that at the surface (Figure 50). The heat flux at the disc surface changes in a complicated manner with time on droplet impingement. During the direct contact of impinging droplet with the metal surface the surface heat flux decreases remarkably. Then, the surface temperature increases and the state shifts to film boiling  a spheroidal state with a thin steam film between the droplet and disc surface. The heat transfer rate to the droplet in a spheroidal state is very low [32]. Positive surface heat flux in the time interval 6.5 s t 12 s results probably from errors of the
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77
temperature measurements at x = E = 0.0003 m. Using the method developed, the transient surface heat transfer to droplets impinging on a heated surface can be examined in detail.
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Table 4. Temperature  time variations, measured at a distance of x = E = 0.0003 m on droplet impingement I
t, s
f , oC
1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16.
0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01 0.011 0.012 0.013 0.014 0.015
250.0 249.8 249.3 247.6 245.3 242.3 240.1 238.0 236.6 236.0 235.5 235.3 235.2 235.25 235.3 235.4
i
i
Figure 50. Heat flux results after the impingement of saturated droplets upon a heated surface; ______ heat flux at x = E = 0.0003 m determined by using spline approximation of the temperature data (S = 2.0 K2) (1),  heat flux at x = E = 0.0003 m determined by using piecewise linear interpolation of the temperature data (2), ■  surface heat flux determined by using spline approximation of the temperature data (3).
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3.2.7. Concluding Remarks A technique for determining the transient heat flux and heat transfer coefficient at a solid interface, based on experimentally acquired interior temperaturetime data, is developed. For the analyzed case of a single interior temperature history, the problem is subdivided into two separate problems : a direct problem for the semiinfinite solid and an inverse problem for the flat plate. The heat flux at the location of the temperature sensor is determined from the solution of onedimensional heat conduction using Duhamel’s theorem. Global and local spline approximations are used to smooth measured interior temperaturetime curves. General case closed form equation for the interface heat flux is obtained. Knowing both the temperature and heat flux at a sensor location, the temperature and heat flux at the active surface are determined from the solution of the inverse heat conduction problem using StefanBurggrafLangford method. The total and global cubic spline approximations of the measured temperature data are compared for the same test cases. The methods give generally similar answer but the global spline approximation gives slightly better results, e.g. more accurate and smoother. In terms of computing time, the local spline approximation takes much less time than the global approximation. The local approximation has the advantage that the inverse heat conduction problem can be solved online on personal computer. The developed technique is also very useful for handling very complex geometries. Clearly it is not precise to use equation (160) as formula for calculating transient heat transfer coefficient for nonplanar boundaries, but for short duration experiments, the heat conduction within the solid can be assumed to be locally onedimensional neglecting the heat flow along the boundary surface. The local, timedependent heat transfer coefficients can be determined from measured temperaturetime variations at several subsurface points of the solid. This technique has the following advantages:
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it is economical because a short duration test does not require much time to carry out the experiment, measurements can be obtained with simple, low cost experimental models and equipment, although the data analysis is relatively complicated, the computing time is very small.
3.3. Space Marching Method for Determining Surface Heat Flux A space marching method will be presented for solving the onedimensional nonlinear heat conduction problems. The temperature  dependent thermal properties and boundary condition on an accessible part of the boundary of the body are known. Additional temperature measurements in time are taken with a sensor located in an arbitrary position within the solid, and the objective is to determine the surface temperature and heat flux on the remaining part of the unspecified boundary. The temperature distribution throughout the solid, obtained from the inverse analysis, is then used for the computation of thermal stresses in the entire domain, including the boundary surfaces. The proposed method is appropriate for
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online monitoring of thermal stresses in pressure components. The three presented examples show that the method is stable and accurate.
3.3.1. Introduction The inverse problem considered here is concerned with the estimation of the temperature and the heat flux of a onedimensional body from measured interior temperature histories. The thermal properties of the material are considered to be a function of temperature. The presented method can be easily extended to inverse heat conduction problems (IHCP) in composite media. In the direct heat conduction problem the errors in the surface conditions are damped and delayed when evaluating the interior temperature. In contrast to the direct analysis, in the inverse problem the small random errors in the measured data are amplified as the unknown boundary values are calculated. Thus, large oscillations in the calculated surface conditions can result in the inverse problems from small inaccuracies in the interior temperature data. Consequently, the inverse method, should include special corrective procedure in order to reduce the effect of error growth and propagation. The total analysis interval tt = (2L + 1)t, [12, 99] depends on the time it takes for the surface condition to significantly affect the interior temperature at a given sensor position. The longer the analysis interval tt, the more complete is the information about the surface condition received at the internal thermocouple. Unfortunately, the method can not be able to reproduce accurately the surface heat flux as a function of time, if the tt is too large, because the result of the inverse analysis is averaged over the whole time interval tt. Extensive numerical experiments [12, 100, 101] have shown that for optimal results the distance of the internal sensor from the exposed surface E must be correlated to the total time interval tt through a Fourier number of the form Fo = tt/E2 such that for an optimal result Fo 0.20.5. Beck [100] has stabilized the IHCP using several “future” time steps to reduce the sensitivity to measurement errors. The surface heat flux qs(t) is determined at time t using interior temperatures measured at times greater than t. Beck’s approach permits the use of small time steps t in the heat flux calculations without encountering the instabilities. Many complex methods have been proposed to solve the inverse problems. Space marching methods [102109] have the advantage over other methods since they are simple and can be used to solve the nonlinear one and multidimensional IHCP. In the space marching methods the local interior temperature and heat flux are extrapolated beyond the thermocouple position to the exposed surface. For the IHCP, the whole spatial domain can be divided into a direct region for which boundary conditions are known, and an inverse region with an unknown boundary condition at the surface. The inverse procedures first calculate the temperature distribution in the direct region and then extrapolate it through the inverse region to the exposed surface. The first spacemarching method developed by D’Souza [102] replaces the heat conduction equation by the backwarddifference method. Unlike direct heat conduction problems, the implicit D’Souza method can produced oscillatory or unstable results. The stability problems can partially be overcome by using Weber method [103], which replaces
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the heat conduction equation with a hyperbolic equation, for which the inverse problem becomes wellposed. The equations for surface heat flux and temperature in the space marching methods are numerical analogs of the exact solution [8385] in which time derivatives are replaced by finite difference approximations. The order of the highest time derivative in the equation for the surface heat flux is equal the number of spatial grid points [99, 107]. Refining the grid has the effect of increasing the order of the highest derivatives of measured temperature f(t) and heat flux qE(t) at the location of the temperature sensor. The number of future temperatures involved in the estimation of the surface conditions also increases when the number of spatial grid points is larger. The spacemarching methods are usually stabilized by refining the spatial grid. For example, specifying twenty spatial nodes will result in the usage of 19 future and past measurements in the surface condition estimates. When the number of grid points is too large the numerical solutions becomes smoother and may deviate substantially from the true solution, whilst when it is too small the numerical solution becomes oscillatory. The number of future time steps incorporated in spacemarching techniques is usually to large, since the large number of spatial nodes is taken [103106]. Another source of errors is using noisy data for calculating time derivatives of measured temperatures. The major weakness in the space marching methods results from the numerical evaluation of the time derivatives of the measured temperature history. The accuracy of derivatives computed by finitedifference formulas is affected by roundoff errors when using the exact data. If the data being differentiated are from experimental tests, the errors influence the derivative values calculated by numerical procedures till such extent that they may be meaningless. The numerical differentiation of the noisy data is the main source of errors in the spacemarching methods [102106]. In order to eliminate the difficulty in choosing the number of the control volumes whilst maintaining the stability of the IHCP solution, the semianalytical method is developed[107] It was found that three or four control volumes are sufficient to reduce oscillations and not too small to reproduce the true changes in the surface heat flux. In this study a new spacemarching method for the onedimensional nonlinear heat conduction equation is presented. The final equation for the prediction of temperature incorporates only the first time derivative of the measured temperature. The noisy input data are first smoothed and then used in the inverse procedure. Presented method has not the drawbacks mentioned above. It is very stable and appropriate for temperature dependent thermal properties. It can also be easily extended to two or three dimensional problems.
3.3.2. Analysis of the Inverse Problem The cases to be considered are shown in Figure 51. The onedimensional configurations of interest are the plane wall (m = 0) , the hollow cylinder (m = 1) and the hollow sphere (m = 2). To identify the temperature distribution inside the solid and heat fluxes at the boundary surfaces ro and rin it is necessary to measure the temperatures at least at the two interior locations r1 and r2 (Figure 51a). Boundary conditions given at r1 and r2 result in a direct problem for r1 r r2 and inverse problems in the regions where rin r r1 or r2 r ro. If r1 and r2 are different points then the direct problem is solved first and its solution used to determine the temperature distributions beyond the thermocouple positions, i.e. in intervals
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rin r r1 and r2 r ro. If the distance (r2r1) between two thermocouples is equal to the step size in space r, which is used in the spacemarching method, then the temperature traces at the positions r1 and r2 are directly used in the inverse solutions (Figure 51b). Numerical computations show, however, that in this case the results of the inverse analysis are significantly affected by temperature measurement errors. Consequently, in practical problems the close arrangement of the internal thermocouples is not recommended. A problem that frequently occurs in practical applications consists of estimating the surface temperature and heat flux from one measured temperatures inside of a heat conducting solid (Figure 51c).
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a
b Figure 51. Continued on next page.
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c Figure 51. Onedimensional inverse heat conduction problem; a) and b)  temperature is measured at r1 and r2; c) temperature at r1 and boundary conditions at ro are known.
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The temperature sensor may also be placed at secondary surface of the body. If instead of temperature measurements taken at r = r2 boundary conditions at the outer surface of the body r = ro are known, then the problem can be subdivided into two separate problems, one of which is a direct problem as shown in Figure 51c. The temperature distribution in the direct region is first calculated and then the known solution of the direct problem is extrapolated to points rin r r1 outside the direct region r1 r ro. In this study, the last case shown in Figure 51c will be considered in detail.
3.3.3. Problem Formulation The onedimensional nonlinear heat conduction problem is illustrated in Figure 51c. The surface at r = ro is insulated or the boundary condition at this surface is known. The temperature  dependent properties and initial temperature distribution are considered to be known while the surface conditions are unknown and are to be estimated from interior temperature readings. The temperature sensor is located at the interior position r = rE. The objective is to find the temperature distribution in the inverse region rin r rE based on the temperature readings f(t) at particular times tj , j = 1, 2, ... For a solid with temperature dependent properties , the governing equation is
c(T ) (T )
T 1 T m k (T )r m , r r ro , t 0, t r r r in
where m = 0, 1, 2 for a slab cylinder, or sphere , respectively. The boundary conditions for the direct problem are
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(187)
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Measurement of Heat Flux and Heat Transfer Coefficient
T rE , t f t , t 0,
(188)
T r
(189)
and
0, t 0. r ro
Instead of insulated outer surface other boundary condition can be given. The temperature measurement (188) provides a known boundary condition for the inverse problem. In order to start space marching procedure the temperature at the second spatial node of the direct region is calculated(Figure 52):
T rE r, t f 2 t , t 0.
(190)
The temperature
Ts T rin , t ,t 0
(191)
is to be recovered.
T q s k r
, t 0.
(192)
r rin
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The initial condition is only needed in the direct region
T r ,0 T0 r
Figure 52. Space grid for the solution of the inverse heat conduction problem.
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(193)
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3.3.3.1. Solution of the Direct Problem The direct problem is solved using the method of lines. In this method only the spatial derivative in Equation (187) is discretized. Since temperature  dependent thermal properties make the problem nonlinear, the finite control volume method is chosen. Thus, the partial differential equation is approximated by the following coupled system of ordinary differential equations
TN f t , r ri d 2
(194)
m 1
r ri d 2 m 1
m 1
cTi Ti
dTi dt
r k Ti 1 k Ti Ti 1 Ti ri d 2 2 rd m
(195)
r k Ti 1 k Ti Ti 1 Ti ri d , i N 1, N 2,..., NC. 2 2 rd m
To write a central difference approximation for the boundary condition at the outer surface (189), it is necessary to use a fictious node outside the solid at i = NC+1. If the outer surface is insulated the boundary condition at r = ro can then be written
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TNC 1 TNC 1 0 2rd
(196)
Solving for TNC+1 gives
TNC1 TNC1
(197)
The equations (195) apply for all interior nodes and i = NC. The fictious term TNC+1 is eliminated from Eq. (195) by use of Eq. (197). To start the solution, the initial temperature distribution throughout the solid must be known at each node Ti = T0(ri), i = N, N+1, ..., NC.
(198)
The set of the ordinary differential equations (195) was solved using the RungeKutta method. In order to determine whether the RungeKutta values are sufficiently accurate, the direct problem was additionally solved by fully explicit finite difference method. The thermal property values are lagged one time level. Since slight changes in the values of Ti, i = 2, 3, ..., NC were observed, the RungeKutta results were accepted.
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Measurement of Heat Flux and Heat Transfer Coefficient
3.3.4. Solution of the Inverse Problem A new space marching method for the solution of nonlinear heat conduction will be presented. After calculating the temperature distribution in the direct region the nodal temperatures in the inverse region can be determined. Solving the heat balance equation (195) with r = rd for Ti1 gives r ri 2 k Ti 1 k Ti Ti 1 Ti Ti m r k Ti 1 k Ti ri 2 m
Ti 1
m 1 m 1 r r ri r ri 2 2
m 1 ri r 2
m
2c(Ti ) Ti dTi , i N , N 1,...,2 k Ti 1 k Ti dt
(199)
Because Equation (199) is nonlinear, the fixedpoint iterative technique is used to determine the temperature Ti1.
r ri 2 k Ti 1 k Ti T T Ti ( n11) Ti i 1 i m (n) r k Ti 1 k Ti ri 2 m 1 m 1 r r ri r 2c T T ri i i dTi , 2 2 m k Ti ( n1) k Ti dt r m 1 ri 2 i N , N 1, , 2 n 0,1, 2,
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m
(200) where n is the iteration number; is some small number (tolerance). The iteration process continues until the following condition is met
Ti ( n11) Ti ( n1)
(201)
Starting from the temperature Ti 1 Ti , that is not far from a root Ti1, only three or four (0)
iterations are needed to obtain the tolerance 0.0001 K. If the thermal properties are constant, the iterations are not required. Heat flux qs at the inner surface is determined from the boundary condition at the inner surface
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k T1
T r
k T1 r r1
T2 T0 qs . 2r
(202)
Determining the T0 using equation (202) then yields
T0 T2
2q s r . k T1
(203)
Substitution of equation (203) for T0 into equation (200) for i = 0 yields m r r1 k T1 2 k T2 k T1 qs T2 T1 T1 T2 m 2r r k T0 k T1 r1 2 m 1 m 1 r r r1 r r1 2 2 2cT1 T1 dT1 m k T0 k T1 dt r m 1 r1 2
(204) The iterative scheme takes the form
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T0( 0) T1
(205)
m r r1 k T1 k T2 k T1 2 ( n 1) T2 T1 qs T1 T2 m (n) 2r r k T0 k T1 r1 2 m 1 m 1 r r r1 r r1 2 2 2cT1 T1 dT1 , m k T0( n ) k T1 dt r m 1 r1 2
(206)
T0( n1) T2
2qs( n1) r , n 0,1,... k T1
The iterative process is continued until the following convergence criterion Heat Flux: Processes, Measurement Techniques and Applications : Processes, Measurement Techniques and Applications, Nova Science
(207)
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Measurement of Heat Flux and Heat Transfer Coefficient
q s( n1) q s( n )
(208)
is satisfied, where is some chosen tolerance. If the thermal properties are constant, iterations are not needed. The presented space marching scheme is very straightforward in this case
TN 1 f t
1 r df t , 2 dt
Ti 1 2Ti Ti 1
2
(r ) 2 dTi , i = N1, ... ,2 dt
(209)
(210)
and
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qs
k T1 T2 1 dT c r 1 . r 2 dt
(211)
3.3.5. Smoothing the Temperature Data Random measurement errors of the temperature f(t) have great influence on the estimated temperature, thermal stress distributions and the heat transfer coefficients. If the temperature data are perturbed with random errors, least squares smoothing will be used to reduce the effect of the measurement errors on the calculated time derivatives df/dt or dTi /dt. Least squares approximation is very suitable for the recovery of a smooth function from noisy information. It is possible to choose an appropriate function which is flexible enough to reconstruct the underlying noise free function and its derivative while still orthogonal to the noise, i.e. unable to follow the oscillations in the measured data. The Gram orthogonal polynomials [12] will be used for smoothing the measured timetemperature history f(t) and estimated temperatures Ti(t), i = N1, ..., 1. Assuming that measured fj = f(tj ) and calculated Ti(tj) temperatures are equally spaced in time, we construct piecewise cubic polynomials and then use a least square procedure to estimate appropriate polynomial coefficients. The local piecewise approximation using the Gram polynomials has the advantage that it can be used in online mode. In online monitoring of thermal stresses, it is frequently necessary to follow and fit the data without knowing its end beforehand. Making an assumption that the temperature f(t) is measured with equal time steps t for N = (2L+1) successive time points tj = (j1)t, j = 1, ..., N the local Gram polynomial is constructed.
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Figure 53. Time smoothing of the measured temperatures using Npoint averaging filter.
Introducing a new dimensionless time coordinate (Figure 53)
s
t j t1 t
L,
(212)
the coefficients of the Gram polynomial y(s) are determined using the least squares method sL
ys f s
2
min .
(213)
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s L
The coefficients of the polynomial y(s) are obtained from normal equations. A third order polynomial for the set of N = 9 data points is
1 f 4 f 3 f 2 f 1 f 0 f1 f 2 f3 f 4 9 1 3 1 1 1 1 3 f 4 f 3 f 2 f 1 f1 f 2 f 3 f 4 s 15 4 2 4 4 2 4 1 1 2 17 5 17 2 f 4 f 3 f 2 f 1 f 0 f1 f 2 99 4 7 28 7 28 7 y s
1 7 1 13 9 f3 f 4 3s 2 20 f 2 f 1 f 4 f 3 4 2970 2 14 14 9 13 1 f1 f 2 f 3 f 4 5s 3 20 14 14 2
(214) where s =4,...,0,...,4.
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Measurement of Heat Flux and Heat Transfer Coefficient Differentiating equation (214) with respect to t yields dy s 1 1 3 1 1 1 f 44 f 33 f 22 f 11 ds t 15 4 2 4 4 2 1 2 17 5 17 f 44 f 44 f 33 f 22 f 11 f 00 4 7 28 7 28 33 1 7 1 13 9 9 f33 f 44 s f 44 f 33 f 22 f 11 4 2970 2 14 14 14 1 2 f33 f 44 15s 2 59 2
dy 1 dt t 3 f33 4
1 f11 2 2 f11 7
f 22 f 22
13 f11 f 22 14
(215) where s =4,...,0,...,4. The smoothed temperature and its time derivative are evaluated at the center point t(s = 0). Future and past data appear in equations (214) and (215) , where the points s = 1, ..., 4 are the future data and those s = 1,...,4 are the past data. Having calculated the temperature and its time derivative at the center point t(s = 0) , the whole time interval 8t is moved one time step forward , dropping the last data point s = 4 and adding a new one. When the real process begins, no past data is available. At the first four time points: t = 0, t = t, t = 2t, and t = 3t, y(t) and dy/dt are calculated for s = 4, s = 3, s = 2 and s = 1 respectively.
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At the last four data points t f 3t t t f , y(t) and dy/dt are calculated for s = 1,
s = 2, s = 3, and s = 4, respectively. At the time interval 4t t tf 4t, y(t) and dy/dt are always evaluated only at the center point s = 0 , because the accuracy of the polynomial approximation at the interval center is much more higher than at its ends. The proposed method is stable and gives good accuracy if
Fo
rE
t
rin
2
0.02 0.05
(216)
Lower values of Fo are recommended for the exact data. If the noise to signal ratio is larger, the longer step sizes should be used to obtain stable results.
3.3.6. Thermal Stresses Transient thermal stresses in plates, hollow cylinders and hollow spheres can be calculated if the temperature distribution obtained from the inverse solution is given. The tangential, axial and radial stresses at the inner surface are given by the following expressions:
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t a r 0.
E Tm t T rin , t , 1
(217)
The mean integral temperature over the wall thickness is defined as ro
m 1 m Tm Fo m1 r T r , t dr. ro rinm1 rin
(218)
The calculations of stresses for plates and hollow cylinders are valid for cross sections sufficiently distant from free ends, so that their disturbances can be neglected. Approximating the integral in Equations (218) by using the trapezoidal rule over the inverse and direct regions yields Nc
m 1 Tm t m1 m1 rim1Ti1 ri mTi ri . ro rin i 2
(219)
The wall temperature difference T = Tm(t) T(rin, t) is determined in power stations approximately by measuring the metal temperature in the center of the wall rm and in the location close to the wall inner surface rin + E, where E is the distance of the thermocouple from the inner surface. Thermal stresses are calculated using simple expression:
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t'
ET T rm , t T rin E , t . 1
(220)
Because the distance E from the inner surface to the first measuring point can not be smaller than 3.5 mm due to high pressure inside the construction element, real temperature changes of the inner surface can be significantly different from the measured values. Furthermore, the point of temperature equivalent to the mean temperature Tm(t) does not lie in the wall middle. Therefore, the theoretical estimating of the thermal stresses from the Eq.(218) is much more accurate.
3.3.7. Numerical Examples In order to test the accuracy of the developed method two numerical examples are presented. First one of the standard test cases for comparing results of inverse heat conduction algorithms is performed for a triangular heat flux. The geometry is a flat plate and the sensor is located at x = E = L, which is thermally insulated. This test case is thoroughly discussed by Beck et al. [3]. The heat flux is zero before time zero and is again zero after dimensionless time Fo = 1.2. For Fo between zero and 0.6 it increases linearly with time, and for Fo > 0.6 the heat flux decreases linearly to zero at Fo = 1.2. Both exact temperature data and data with error are considered. Although all experiments have error, analyzing the effects of errorless measurements can give insight into the maximum possible resolution of surface conditions
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Measurement of Heat Flux and Heat Transfer Coefficient
available from the measurements. The only error introduced into the problem is the error due to computer round off. The surface heat flux was calculated for three (Figure 54) and four (Figure 55) control volumes. The input temperature data were prefiltered before using in the developed IHCP algorithm. The heat flux shown in Figure 54a and in Figure 55a is calculated using “exact data”. The exact data is corrupted with additive, normally distributed, and zero mean random errors with a dimensionless standard deviation of T T k / qN L 0.0017 [3] to obtain “noisy” data.
Figure 54b and Figure 55b show the heat flux obtained for the temperature data with measurement errors. The surface heat flux is accurately estimated both with exact and noisy data, including the regions of sudden changes in the heat flux. The high accuracy is obtained even though a small number of control volumes is used. The effect of using more control volumes is seen by comparing the results shown in Figures 54 and 55. The results obtained for four control volumes (Figure 55) are not markedly better than the estimates for three control volumes (Figure 54). In order to compare the results more exactly, the root mean square error of the qs estimates is calculated 1
2 2 1 1 NT e s q q s t j q s t j , q N NT 1 j 1
(221)
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where, qs(tj) is the estimated component and qse(tj) is the true value. The following values of sq+ are obtained:  three control volumes and
sq+ = 0.02194 sq+ = 0.03386
for exact data, for noisy data.
 four control volumes sq+ = 0.01825 for exact data, and sq+ = 0.02579 for noisy data. In the second example the temperature sensor is embedded within a hollow sphere of 0.24145 m outer radius and 0.17955 m inner radius. The objective is to determine the heat transfer coefficient and the thermal stress on the inside of the sphere wall from temperature measurements at the centre of the wall. The outer sphere surface is insulated. The thermal properties are assumed to be temperature dependent and represented as follows:  thermal conductivity  volumetric heat capacity
k T a0 a1T ,
cT T b0 b1 exp T b2 ,
(222) (223)
where: a0 = 23.514 W/(m·C), a1 = 0.00458 W/(m·C2), b0 = 2703690.5 J/(m3·K), b1 = 792008.3 J/(m3·K), b2 = 412.62 C and T is expressed in C.
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a
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b Figure 54. Calculated surface heat flux for triangular heat flux case with exact (a) and noisy (b) input data using three control volumes (NC = 3, r = (rErw)/2); 1  input heat flux, 2  inverse solution.
A method of lines [1213] is used to calculate the simulated temperature data for this test. The direct problem is defined by the nonlinear heat conduction equation (187), the boundary condition (189) at the outer surface and the following boundary conditions at the inner surface
T k T r
h T f t T rin , t .
(224)
r rin
The initial temperature distribution T(r, 0) = T0 = 20C is assumed to be uniform. The hollow sphere is convectively heated at inner surface by a fluid whose temperature changes at first stepwise up to 100C and then varies linearly with time up to 350C. After t = 1250 s the fluid temperature is constant and equal to 350C (Figure 57).
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Measurement of Heat Flux and Heat Transfer Coefficient
Figure 55. Calculated surface heat flux for triangular heat flux case with exact (a) and noisy (b) input data using four control volumes (NC = 4, r = (rErw)/3); 1  input heat flux, 2  inverse solution.
The temperature at the center of the wall obtained from the solution of the direct problem using 10 control volumes simulates the “exact” data for the inverse problem. The “noisy” data f(t) is generated by adding normally distributed, and zero mean random errors j to the exact temperature T(tj) , that is:
f t j T t j j j 1,..., NT ,
(225)
where 0.5 j 0.5C. The standard deviation of errors is f = 1/6C. The solution of the inverse problem begins with the solution of the direct problem for the region rE r ro (Figure 56) to calculate the
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temperature at node 6. Then, the space marching procedure developed for inverse heat conduction can be used. The time step t = 30 s is chosen.
Figure 56. Control volume grid for determining thermal stresses from temperature measurements in the middle of the wall (at node No.4).
The estimated surface temperature heat flux and heat transfer coefficients using exact data are shown in Figure 57. The thermal stresses corresponding to the estimated temperature distribution (NC = 10) are plotted in Figure 58. Since the true value of the heat transfer coefficient is known, the sample standard
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deviation s h and standard error mean e h are calculated using 1
2 1 1 NT 2 s h he ht j he NT 1 j 1
(226)
and h
e
s h
,
NT
(227)
where he is the true value and h(tj) is the estimated value of the heat transfer coefficient. The statistics for the estimated heat transfer are: sh+ = 0.06816 and eh+ = 0.00839. From the inspection of the Figures 57 and 58 can be seen that the accuracy of the calculated temperature, heat transfer coefficient and thermal stress at the inner surface is very good. The obtained results illustrate, that the heat transfer coefficient and the thermal stresses are more difficult to calculate accurately than the surface temperature.
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Measurement of Heat Flux and Heat Transfer Coefficient
95
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Figure 57. Surface temperature and heat transfer coefficient calculated from temperature measurements with normally distributed random errors between 0.5C; 1 –fluid temperature, 2  measured temperature in the middle of the wall , 3  temperature at r = rE + r (node 6 in Figure 56), 4  exact temperature of the inner surface (direct solution), 5  calculated temperature of the inner surface (inverse solution), 6  calculated heat transfer coefficient.
Figure 58. Thermal stresses at the inner surface of the spherical component calculated for data burdened with normally distributed random errors between 0.5C; 1  direct solution, 2  inverse solution.
Finally, let’s consider an example using actual measured data. The thermal stresses at the inner surface of the spherical pressure component of the Benson steam generator are determined using transient temperature measurements in two internal locations: r1 = 0.18305 m and r2 = 0.2105 m . The dimensions of the component are the same as in the previous
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example: rin = 0.17955 m and ro = 0.24145 m. The thermal properties of the component material are assumed to be temperature dependent (Figure 59).
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Figure 59. Thermophysical properties of the X20CrMoV121 steel as a function of temperature; measured value [111],  least squares approximation.
Figure 60. Thermal stresses on the inner surface of the pressure component during startup of the Benson boiler; 1  temperature measured at the distance E = 4 mm from the inner surface, 2 temperature measured in the middle of the wall thickness, 3  thermal stresses calculated from temperature measurements at rE = rin + 0.004 m =0.18355 m, 4  thermal stresses calculated from temperature measurements at rE = rin + 0.5(ro rin) = 0.2105 m, 5  thermal stresses based on the measured temperature difference.
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Measurement of Heat Flux and Heat Transfer Coefficient
97
The temperature data were measured at t = 30 s time steps for a total time period of 24000 s (Figure 60). The calculations are carried out twice, first using the measured temperatures at r = r1 = 0.18305 m and then at r = r2 = 0.2105 m. Figure 60 shows the estimated thermal stresses determined by using the present inverse method and equation (220). It can be seen that the present method produces almost the same results throughout the whole time domain. The effects of the location of internal temperature measurements in relation to the distance from the inner surface where the thermal stresses are determined are only noticeable during fast temperature changes. The large discrepancies between present method and Eq. (220) have already been explained earlier.
3.3.8. Summary A space marching method for the solution of the onedimensional nonlinear, inverse transient heat conduction problem is presented. The method is based on the space discretisation of the heat conduction equation. The method has the advantage that time derivative is not replaced by finite differences and the good accuracy of the method results from an appropriate approximation of the first time derivative using smoothing polynomials. The numerical results obtained using both exact and noisy data show a good stable estimation of the direct solutions. Finally, the extension of the method presented in this study to higher dimensions inverse heat conduction problems is straightforward.
REFERENCES
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[1]
Childs P. R. N., Practical Temperature Measurement. ButterworthHeinemann, Oxford 2001. [2] Han J. Ch., Dutta S., Ekkad S. V., Gas Turbine Heat Transfer and Cooling Technology. TaylorandFrancis, New York 2000. [3] Holman J. P., Experimental Methods for Engineering. Seventh Edition, McGrawHill International Edition, Boston 2001. [4] Venkateshan S. P., Mechanical Measurements. CRC Press, Boca Raton 2008. [5] Lage J. L. Experimental Methods. Chapter 9, pp.913946 in Heat Transfer Handbook, Editors: Bejan A., Kraus A., Wiley, Hoboken 2003. [6] Moffat R. J., Temperature and Heat Transfer Measurements. Chapter 4.6, pp.41824205 in The CRC Handbook of Mechanical Engineering, EditorinChief. Kreith F., CRC Press, Boca Raton 1997. [7] Coleman H. W., Steele W. G., Experimentation, Validation, and Uncertainty Analysis for Engineers. Third Edition, Wiley, Hoboken 2009. [8] Taler D., Measurement of Pressure, Velocity, and Flow Rate of Fluid, Publishing House of University of Science and Technology, Cracow 2006. [9] Wolberg J., Data Analysis Using the Method of Least Squares. Springer, BerlinHeidelberg 2005. [10] Dunn F. P., Measurement and Data Analysis for Engineering and Science. Second Edition, CRC Press, Boca Raton 2010.
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[11] Carslaw H. S., Jaeger J. C., Conduction of Heat in Solids. Second edition, Oxford Science Publications, Oxford 2008. [12] Taler J., Theory and Praxis of Identification of Heat Transfer Processes. Ossolineum, WrocławKrakówWarszawa 1995 (in Polish). [13] Taler J., Duda P., Solving Direct and Inverse Heat Conduction Problems. Springer, BerlinHeidelberg 2006. [14] Kirchhoff, R.H., Calorimetric heatingrate probe for maximumresponsetime interval. AIAA Journal, Vol.2 (1964), No 5, 966967. [15] ASTM. Standard test method for E45796 measuring heat transfer rate using a thermal capacitance (slug) calorimeter. American Society for Testing and Materials, 1997. [16] Burggraf O. R., An exact solution of the inverse problem in heat conduction theory and applications. ASME Journal of Heat Transfer, Vol. 86, August 1964, 373382. [17] Chedailee J., Braud Y., Measurements in Flames, Volume 1, Edward Arnold, London 1972. [18] Hohenberg G. F., Advanced approaches for heat transfer calculations. Society of Automotive Engineers, Paper No. 790825, 27882806. [19] Neal S. B. H. C., Northover E. W., Preece R. J., The measurement of radiant heat flux in large boiler furnaces. II. Development of flux measuring instruments. International Journal of Heat and Mass Transfer, Vol. 23 (1980), No. 7, 10231031. [20] Popiel C. O., van der Meer T. H., Hoogendoorn C. J., Convective heat transfer on a plate in an impinging round hot gas jet of low Reynolds number. International Journal of Heat and Mass Transfer, Vol. 23 (1980), 10551067. [21] Gerashchenko O. A., Basics of Thermal Measurements, Naukova Dumka, Kiev 1971 (in Russian). [22] Seeger M., Taler J.: Konstruktion und Einsatz transportabler Wärmeflußsonden zur Bestimmung der Heizflächenbelastung in Feuerräumen. FortschrittBerichte der VDI Zeitschriften, Reihe 6, nr129, VDI Verlag, Düsseldorf 1983. [23] Gardon R., An instrument for the direct measurement of thermal radiation. Review of Scientific Instruments, Vol.24, May 1953, 366370. [24] Gardon R., An instrument for the measurement of heatflow rate. Transactions of the ASME, Journal of Heat Transfer, November 1960, 396398. [25] Northover E. W., Hitchcock J. A., A heat flux meter for use in boiler furnaces. J. Sci. Instrum., Vol.44 (1967), 371373. [26] Taler D., Grądziel S., Taler J., Measurement of heat flux density and heat transfer coefficient, Archives of Thermodynamics, Vol. 31 (2010), No. 3, 318. [27] Neal S.B.H.: The development of the thinfilm naphthalene masstransfer analogue technique for the direct measurement of heat transfer coefficients, Int. J. Heat Mass Transfer, Vol. 18, pp. 559567, 1975. [28] White F.M.: Heat and Mass Transfer, AddisonWesley, Reading 1991, pp. 651653. [29] Mizushina T.: The Electrochemical Method in Transport Phenomena, Advances in Heat Transfer, Vol. 7, pp. 87161, Academic Press, New York 1971. [30] Lucas D.M., Davis W.A., Gay B.: Evaluation of local and average convective heat transfer coefficients in a furnace using an electrolytic mass transfer model, Journal of the Institute of Fuel, 1975, March, pp. 3137. [31] Matsumoto R., Kikkawa S., Senda M.: Effect of pin fin arrangement on endwall heat transfer, JSME International Journal, Series B, Vol. 40, No. 1, 1997, pp. 142151.
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[32] Baughn J.W., Mayhew J.E., Anderson M.R., Butler R.J.: A periodic transient method using liquid crystals for the measurement of local heat transfer coefficients, Transactions of the ASME, Journal of Heat Transfer, Vol. 120, August 1998, pp. 772777. [33] Wolfersdorf J., Hoecker R., Hirsch C.: A data reduction procedure for transient heat transfer measurements in long internal cooling channels, Transactions of the ASME, Journal of Heat Transfer, Vol. 120, May 1998, pp.314321. [34] Leland J.E., Pais M.R.: Free jet impingement heat transfer of a high Prandtl number fluid under conditions of highly varying properties, Transactions of the ASME, Journal of Heat Transfer, Vol.121, August, 1999, pp.592597. [35] Janna W.S.: Engineering Heat Transfer, Third Edition, CRC Press – Taylor and Francis Group, Boca Raton 2008. [36] Press W.H., Teukolsky S.A., Vetterling W.T., Flannery B.P.: Numerical Recipies in Fortran. The Art of Scientific Computing, Cambridge University Press, Cambridge 2007. [37] FORTRANSubroutines for Mathematical Applications, Vol. 2, IMSL, Visual Numerics, Houston 2008. [38] Szydłowski H.: Physics Laboratory, Polish Scientific Publisher PWN, Warsaw, Poland 1994 (in Polish). [39] Moffat R.J.: Describing the Uncertainties in Experimental Results, Experimental Thermal and Fluid Science 1988, No. 1, pp. 317. [40] Kunze H.J.: Physikalische Messmethoden, Eine Einführung in Prinzipien klassischer und moderner Verfahren, B.G Teubner, Stuttgart 1986. [41] Goldstein R.J.,Cho H.H.: A review of mass transfer measurements using naphthalene sublimation, Experimental Thermal and Fluid Science, 10 (1995), 416434. [42] Kottke V., Geschwind P., Li H. D.: Heat and mass transfer along curved walls in internal flows, ERCOFTAC Bulletin No.32 (1997), 2124. [43] Baughn J. W., Mayhew J. E., Anderson M. R., Butler R. J.: A periodic transient method using liquid crystals for the measurement of local heat transfer coefficients, Transactions of the ASME, Journal of Heat Transfer, 120 (1998), 772777. [44] Du H., Ekkad S., Han J.C.: Effect of unsteady wake with trailing edge coolant ejection on detailed heat transfer coefficient distributions for a gas turbine blade, Transactions of the ASME, Journal of Heat Transfer, 119 (1997) 242248. [45] Taler J.: Theory of transient experimental techniques for surface heat transfer, Int. J. Heat and Mass Transfer, 39 (1992) 37333748. [46] Taler J.: Numerical solutions for general inverse heat conduction problem. Wärmeund Stoffübertragung, 27 (1992), 505513. [47] Taler J.: Nonlinear steadystate inverse heat conduction problem with spacevariable boundary conditions, Transactions of the ASME, Journal of Heat Transfer 114 (1992), 10481051. [48] Martin T. J., Dulikravich G.S.: Inverse determination of steady heat convection coefficient distribution. Transactions of the ASME, Journal of Heat Transfer 120 (1998), 328334. [49] Hensel E., Hills R.: Steady –state twodimensional inverse heat conduction, Numerical Heat Transfer, Part B, 15 (1989), 227240.
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[50] Martin T.J., Dulikravich G.S.: Inverse determination of boundary conditions and sources in steady heat conduction with heat generation. Transactions of the ASME, Journal of Heat Transfer, 118 (1996), 546554. [51] Policy on reporting uncertainties in experimental measurements and results, Transactions of the ASME, Journal of Heat Transfer, 122 (2000), 411413. [52] Seber G. A. F., Wild C. J.: Nonlinear Regression. WileyInterscience, New York 1989. [53] Baughn J. W.: Effect of circumferential wall heat conduction on boundary conditions for heat transfer in a circular tube, Transactions of the ASME, Journal of Heat Transfer, 100 (1978), 537539. [54] Lee Y., Kakade S. G.: Effect of peripheral wall conduction on heat transfer from a cylinder in cross flow. Int. J. Heat and Mass Transfer, 19 (1976), 10311037. [55] Achenbach E.: Total and local heat transfer from a smooth circular cylinder in crossflow at high Reynolds number, Int. J. Heat and Mass Transfer, 18 (1975), 13871396. [56] Aiba S., Yamazaki Y.: An experimental investigation of heat transfer around a tube in a bank, Transactions of the ASME, Journal of Heat Transfer, 108 (1976), 503508. [57] Baughn J. W., Saniei N.: The effect of the thermal boundary condition on heat transfer from a cylinder in crossflow, Transactions of the ASME, Journal of Heat Transfer, 113 (1991), 10201023. [58] Maillet D., Degiovanni A., Pasquetti R.: Inverse heat conduction applied to the measurement of heat transfer coefficient on a cylinder: comparison between an analytical and a boundary element technique. Transactions of the ASME, Journal of Heat Transfer, 113 (1991), 549557. [59] Ervin J., Bizzak D. J., Murawski C., Chyu M. K., MacArtur C.: Surface temperature determination of a cylinder in cross flow with thermographic phosphors, HTDVol. 252, Visualization of Heat Transfer Processes, ASME, New York, 1993. [60] J. Taler, Inverse determination of local heat transfer coefficients, Transactions of the Institute of FluidFlow Machinery, No. 109 (2001), 87100. [61] Taler J., Messung der lokalen Heizflächenbelastung in Feuerräumen von Dampferzeugern, BrennstoffWärmeKraft (BWK) 42 (1990), No. 5, 269277. [62] Taler J., Messung der Wärmebelastung der gasdicht verschweiβten VerdampferRohrwände in Dampferzeugern, VGB Kraftwerkstechnik 70 (1990), No. 8, 644650. [63] Taler J., Taler D.,Tubular Type Heat Flux Meter for Monitoring Internal Scale Deposits in Large Steam Boilers, Heat Transfer Engineering 28 (2007), No. 3, 230239. [64] Taler J., Taler D., Kowal A., Measurements of absorbed heat flux and waterside heat transfer coefficient in water wall tubes. [65] Dzierwa P.: Measurement of local heat flux in boiler combustion chambers, Energetyka, (682) 2011, No. 5, 622624 (in Polish). [66] Albrecht M. J., Hawk S. E., Attachable heat flux measuring device, Patent US 6,485,174 B1, Date of Patent: Nov. 26. 2002. [67] Metzger D. E., Larson D. E. Use of melting point surface coatings for local convection heat transfer measurements in rectangular channel flows with 90deg turns. Transactions of the ASME, Journal of Heat Transfer, Vol. 108 (1986), 4854. [68] W.J. Cook and E.J. Felderman, Reduction of data from thinfilm heattransfer gauges ; a concise numerical technique, AIAA J. 4(3), 561562 (1966). [69] H. Consigny and B.E. Richards, Short duration measurements of heattransfer rate to a gas turbine rotor blade, ASME Journal of Engineering for Power 104, 542551 (1982).
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[70] D.M. Kercher, R.E. Sheer, Jr. and R.M.C. So, Short duration heat transfer studies at high freestream temperatures, ASME Journal of Engineering for Power 105, 156166 (1983). [71] Ch.G. Miller, Refinement of an "alternate" method for measuring heating rates in hypersonic wind tunnels, AIAA J. 23(5), 810812 (1985). [72] J.E. Doorly and M.L.G. Oldffield, The theory of advanced multilayer thin film heat transfer gauges, Int. J. Heat Mass Transfer 30(6), 11591168 (1987). [73] P.R.A. Lyons and S.L. Gai, A method for the accurate determination of the thermal product (ck)1/2 for thin film heat transfer or surface thermocouples gauges, J. Phys. E : Sci. Instrum. 21, 445448 (1988). [74] A.H. George and J.L. Smalley, An instrumented cylinder for the measurement of instantaneous local heat flux in high temperature fluidized beds, Int. J. Heat Mass Transfer 34(12), 30253036 (1991). [75] J.W. Baughn, P.T. Ireland, T.V. Jones and N. Saniei, A comparison of the transient and heated  coating methods for the measurement of local heat transfer coefficients on a pin fin, ASME Journal of Heat Transfer 111(4), 877881 (1989). [76] S.A. Hippensteele and P.E. Poinsatte, Transient liquid  crystal technique used to produce highresolution convective heat  transfer  coefficient maps, HTD  Vol. 252, Visualization of Heat Transfer Processes. ASME, New York 1993. [77] T.E. Diller and D.P. Telionis, Time  resolved heat transfer and skin friction measurements in unsteady flow. In Advances in Fluid Mechanics Measurement (Edited by M. GadelHak), pp. 323355. Springer, New York 1989. [78] W.D. Harvey, Instrumenting models for aerodynamic heat transfer studies involving transient heating rates. ISA Transactions 6(1), 4246 (1967). [79] T.V. Jones, Heat transfer, skin friction, total temperature and concentration measurements. In Measurements in Heat Transfer (Edited by E.R.G. Eckert and R.J. Goldstein), pp. 63102. HemisphereMcGrawHill, New York 1977. [80] C.T. Kidd, Thinskin technique heattransfer measurement errors due to heat conduction into thermocouple wires. ISA Transactions 24(2), 19 (1985). [81] C.T. Kidd, Lateral heat conduction effects on heattransfer measurements with the thinskin technique. ISA Transactions 26(3), 718 (1987). [82] R.C. Mehta, T. Jayachandran and V.M.K. Sastri, Finite element analysis of conductive and radiative heating of a thin skin calorimeter. Wärme und Stoffübertragung 22, 227230 (1988). [83] J. Stefan, Über die Theorie der Eisbildung insbesondere über Eisbildung im Polarmeere. Sitzungsberichte der Kaiserlichen Akademie Wiss. Wien, Math.Natur. 98(2a), 965983 (1889). [84] O.R. Burggraf, An exact solution of the inverse problem in heat conduction theory and applications. ASME Journal of Heat Transfer 86C(3), 373382 (1964). [85] D. Langford, New analytic solutions of the onedimensional heat equation for temperature and heat flow rate both prescribed at the same fixed boundary (with applications to the phase change problem). Quart. Appl. Math. 24(4),315322 (1967). [86] J. Taler and W. Zima, Measurement of timevarying heat transfer coefficient. Inżynieria Chemiczna i Procesowa (Chemical and Process Engineering) 4, 535554 (1993) (in Polish).
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[87] E.J. Wegman and I.W. Wright, Spline in statistics. Journal of the American Statistical Association 78(382), 351365 (1983). [88] I.J. Schoenberg, Spline functions and the problem of graduation, Proceedings of the National Academy of Sciences of the U.S.A., 52, 947950 (1964). [89] C.H. Reinsch, Smoothing by spline functions I, Numerische Mathematik 10, 177183 (1967). [90] C.H. Reinsch, Smoothing by spline functions II, Numerische Mathematik 16, 451454 (1971). [91] C. de Boor, A Practical Guide to Splines, Springer, New York 1978. [92] P. Craven, G. Wahba, Smoothing noisy data with spline functions: estimating the correct degree of smoothing by the method of generalized crossvalidation, Numerische Matematik 31, 377403 (1979). [93] G.Wahba, How to smooth curves and surfaces with splines and crossvalidation, Proceedings of the 24th Conference on the Design of Experiments, U.S. Army Research Office, Report 792 (1979). [94] M.J.D. Powell, Subroutine VC03, Harwell Subroutine Library. Computer Science and Systems Division, Harwell Subroutine Library. Computer Science and Systems Division, Harwell Laboratory, Oxfordshire, England 1985. [95] K.Ichida, T.Kiyono, Curve fitting by a onepass method with a piecewise cubic polynomial. ACM Transactions of Matematical Software, 3(2) 164174 (1977). [96] P.J. Schneider, Conduction. In Handbook of Heat Transfer (Edited by W.M. Rohsenow and J.P. Hartnett), Section 3. McGrawHill, New York 1973. [97] T. Ueda, T. Enomoto and M. Kanetsuki, Heat transfer characteristics and dynamic behavior of saturated droplets impinging on heated vertical surface, Bulletin of the JSME 22 (167), 724732 (1979). [98] J. Taler, Notkühlsimulation der Reaktordruckbehälter von Druckwasser reaktoren. Forschungsheft der Technischen Universität Kraków, Nr. 151, pp. 1123, Kraków 1993. [99] J.Taler (1996) XVI Polish Numerical method for the solution of nonlinear transient inverse heat conduction problems. Conference on Thermodynamics, KoszalinKołobrzeg, Conference Proceedings, Vol.2, pp.425433, (in Polish). [100] J.V. Beck, B.Blackwell, C.R.St.Jr (1985) Inverse Heat Conduction.. Wiley, New York. [101] O.M.Alifanov (1995) Inverse Heat Transfer Problems. .SpringerVerlag, Berlin. [102] N.D’Souza (1974) Numerical solution of onedimensional inverse transient heat conduction by finite difference method. ASME Paper No. 75WA/HT81. [103] C.F.Weber (1981) Analysis and solution of the illposed inverse heat conduction problem. Int. J.Heat Mass Transfer 24, 17831792. [104] E.Hensel , R.G.Hills (1986) An initial value approach to the inverse heat conduction problem. ASME J.Heat Transfer 108, 248256. [105] M.Raynaud ,J.V. Beck (1988) Methodology for comparison of inverse heat conduction methods. ASME Journal of Heat Transfer 110,3037. [106] J.Vogel, L.Sara, L.Krejci (1993)A simple inverse heat conduction method with optimization. Int. J. Heat Mass Transfer 36,42154220. [107] J.Taler (1996) A seminumerical method for solving inverse heat conduction problems. Heat and Mass Transfer 31, 105111. [108] J.Taler, W.Zima (1999) Solution of inverse heat conduction problems using control volume approach. Int. J. Heat Mass Transfer, 42 (1999), 11231140.
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[109] J. Taler, A new space marching method for solving inverse heat conduction problems. Forschung im Ingenieurwesen (Engineering Research) 64(1999),296306. [110] G.A.Korn, T.M.Korn (1968) Mathematical Handbook. .McGraw, New York. [111] Warmfeste und hochwarmfeste Stähle. MannesmannRöhrenWerke (1985) Düsseldorf.
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In: Heat Flux Editors: G. Cirimele and M. D'Elia
ISBN 9781614708872 © 2012 Nova Science Publishers, Inc.
Chapter 2
HEAT FLUX BIOCALORIMETRY: A REALTIME ANALYTICAL TOOL FOR BIOPROCESS MONITORING M. Surianarayanan , S. Senthilkumar and A. B. Mandal Chemical Engineering Department Central Leather Research Institute Chennai, India
ABSTRACT
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Heatflux calorimeters have extensive applications on monitoring and controlling chemical process systems. Unlike chemical processes, biochemical processes are complex systems involving enzymatic reactions at intra and extracellular levels. A realtime process measurement is indeed required to gain better insight on dynamics of cell growth. As for Biothermodynamics, excessive Gibb’s energy is dissipated in the form of heat during cell growth process. Hence, measurements of metabolic heat flux can provide realtime information on cell physiology and bioprocess behavior. Further, heatflux measurements are robust, nonspecific and noninvasive, irrespective of bioprocess systems. For the past few decades there has been an increasing interest on employing heatflux calorimetry for investigating biochemical reaction systems. Several approaches are available for heat measurement, yet heatflux calorimetry as proven by several research groups, is more suitable for bioprocess monitoring. This review attempts to highlight significant findings of the investigations so far carried out by research groups employing bench scale heatflux biocalorimeter in different biological systems. Technical and design modifications, dealt with heatflux calorimeters, to achieve high resolution of heat flow signal, and major drawbacks observed on their proposed strategies are discussed in a separate section. A brief description of quantitative biocalorimetric studies on microbial growth process, stoicheometric analysis on cell growth, heat yield coefficient determination and diauxic behavior is given with suitable illustrative calorimetric results. Significance of calorimetric data on estimation of biothermodynamic parameters and their related complexities are summarized in ‘Biothermodynamics’
Email: [email protected]
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M. Surianarayanan, S. Senthilkumar and A. B. Mandal section. Applications of bench scale heatflux calorimetry for realtime monitoring of different bioprocess systems viz., aerobic, fermentation, anaerobic, photoautotrophic and ecological, are discussed in separate sections. Though extensive work had been done for several years, the real potential of heatflux biocalorimetry in the industrial biotech sector has not yet been exploited. Developing an effective protocol for transforming bioreactors to biocalorimeters can pave the way for the advent of “Large scale calorimetry” and this will certainly reveal the benefits to biotech industrial communities, due to its economic viability. This review will provide the readers an insight to science and applications of heatflux biocalorimetry. Moreover, this contribution may serve as an initiative for academia and industry to employ ‘heatflux biocalorimetry’ as a ‘future generation’ realtime analytical tool for bioprocess monitoring applications.
Keywords: heatflux, biothermodynamics
biocalorimetry,
bioprocess
monitoring,
ecobiocalorimetry,
NOMENCLATURE Symbols
s , x , P Reluctance degree of substrate, biomass and product Specific growth rate (h1) max Maximum specific growth rate (h1) s
Substrate concentration (g l1)
H Enthalpy efficiency of growth Copyright © 2012. Nova Science Publishers, Incorporated. All rights reserved.
Tbl
Baseline temperature gradient (°C)
∆Hi′ Molar or Cmolar heat of combustion of compound i, kJ C mol1 or kJ mol1 ∆G  Gibbs free energy change (kJ mol1) ∆G0X  Gibbs energy dissipated per Cmole of new biomass grown (kJ C mol1) ∆H  Enthalpy change (kJ mol1) ∆Ha, ∆Hc  Anabolic and catabolic enthalpy changes (kJ mol1) ∆S  Entropy change (kJ mol1K1)
Notations A Heat transfer area (m2) C Carbon C x Biomass concentration (g l1)
C p f Specific heat capacity of feed (J g1 K1) D  Constant H Hydrogen
Heat Flux: Processes, Measurement Techniques and Applications : Processes, Measurement Techniques and Applications, Nova Science
Heat Flux Biocalorimetry Hmet Metabolic heat generated (kJ)
K e Lumped heat transfer coefficient (W m2 K1) m f Mass flow rate of feed (g h1) .
mgi Mass flow rate of inlet gas (g h1)
Mt Torque (Nm) N Nitrogen O Oxygen q Heat evolution rate (W)
qj
Heat flow through the reactor wall to the jacket oil (W)
qa Heat flow of the acid or base addition and
q ac Heat accumulation in the bulk (W) qbl
Baseline heat (W)
qCO2 Heat flow of the CO2 vaporization (W). qe Heat flow to the environment through the nonjacketed part of the reactor (W)
qf
Heat flow of the feed (W)
qr
Heat flow of the running process
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qs Heat generated through agitation (W) Q Heat liberated (kJ) ri′ Conversion rate of compound i (Cmol s1 L1) rj′ Conversion rate of compound j (Cmol s1 L1)
RH i Relative humidity of inlet gas (g/g) RH o Relative humidity of exit gas (gg1) S 0 Initial substrate concentration (g l1) S  Substrate concentration (g l1) t Time
T f Temperature of feed (K)
Te Temperature of environment (K) Tgi Temperature of inlet gas (K) Tgo Temperature of exit gas (K) Tj TR U V X
Temperature of jacket oil (K) Temperature of reactor contents (K) Global heat transfer coefficient (Wm2K1) Volume (L) Biomass concentration (g l1)
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YQ i Heat yield coefficient with respect to compound i (kJ g1) YQ X Heat yield coefficient with respect to biomass (kJ g1) YC S CO2 yield coefficient (mol Cmol1) YN S Nitrogen yield coefficient (mol Cmol1) YO S Oxygen yield coefficient (mol Cmol1) YQ O Heat yield coefficient with respect to O2 (kJ Cmol1) YQ S Heat yield coefficient with respect to substrate (kJ Cmol1) YX S Biomass yield coefficient (mol Cmol1)
Y
i j R
Yield coefficient of component I with respect to j for pure respiratory growth (mol Cmol1)
WWater
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ABBREVIATIONS BioRC1 BOD COD GCMS I LCMS MSM NB OUR P PTFE RC1 RQ WWT

Biological Reaction Calorimeter Biochemical Oxygen Demand Chemical Oxygen Demand Gas Chromatography Mass Spectrometry Integral Controller Liquid Chromatography Mass Spectrometry Mineral Salt Media Nutrient Broth Oxygen uptake rate Proportional controller Poly Tetra Fluoro Ethylene Reaction Calorimeter Respiratory Quotient Wastewater Treatment
1. INTRODUCTION Thermodynamics of life states that all biochemical reactions of living systems involve alteration in heat and it is proved that their stored internal energy is dissipated in the form of heat with surroundings to sustain their metabolism [1]. Therefore, heat measurement gives an overall estimation of biological activity of any living system. Of all living species microbes are considered as powerful sources for heat generation in contrast with other higher level organisms and hence calorimetry finds significant application on fingerprinting their metabolic activities [2]. This principle finds application in areas usually referred to as
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“Biochemical or Biological” calorimetry. In general, calorimetry, being nonspecific, noninvasive and insensitive to the electrochemical and optical properties of the investigated system, can serve as an analytical tool to measure exothermic heat in the bioprocess [3]. Control of biotechnological processes requires reliable and robust sensors capable of providing realtime information on the main variables of the processes. Nowadays, sensors to monitor pH, oxygen or carbon dioxide and ammonia are plentiful in use for bioprocess applications [4]. Inline measurements of biomass can be done using sophisticated biosensors [5]. But, the evaluation of metabolic activity of organisms is now dealt offline after analyzing the kinetic data obtained from the experimental results. Biocalorimetry has proved to be an efficient tool [6] for the inline monitoring of bioprocesses, i.e. studying the growth and metabolic activity of cell cultures (or) for the detection of biological key components [7]. Redl and Tiefenbrunner [8] reported that it was possible to employ the measured heat flow rate signal to characterize a bioprocess system based on cellular activity response with a specific substrate, where a quick rise in heat production indicated a welladapted system, and a slow rise indicated a poorly adapted one. Moreover, many metabolic events occurring in a biological system, such as shifts from one substrate to another (diauxic growth) or change in the cell metabolism, e.g. from oxidative to fermentative [9], can be identified by measuring the heat evolution patterns. The advantage of calorimetry is that it measures the total energy flow; under certain conditions this can also be measured by respirometry. In contrast to calorimetry, however respirometry is restricted to aerobic conditions. On combining calorimetry with other inline bioprobes, it is feasible to achieve significant goals in bioprocess engineering [10]. Duboc et al [11] provided an extensive introduction to quantitative calorimetry. The application of calorimetric techniques for bioprocesses monitoring is related to the sensitivity of the instrument, to the net enthalpy change of the bioreaction under study and to its rate. Two main types of calorimeters have been intensively developed during the past few decades: Microcalorimeters [12, 13] and bench scale calorimeters [14]. Microcalorimeters were developed first and attained sufficient sensitivity for monitoring biological processes. In the late 1960’s, following the pioneering work of Calvet in France and Benzinger in the USA, the development of modern isothermal microcalorimeters began to accelerate [15]. Lena Gustaffson [16] employed microcalorimetry to study the aerobic growth of the yeast S.cervisiae with glucose as the only carbon and energy source and reported that a continuously changing proportion of the respiratory catabolism in relation to fermentative catabolism and explained the mixed respiratoryfermentative metabolism. The use of microcalorimeters including flowthru measurements, which achieve the necessary sensitivity to measure a low heat flow rate signal, was hindered by the technical difficulties in fulfilling the biological environmental needs (oxygen and substrates supply, a controlled pH, mixing, etc.) in the small volume of the micro calorimeter measuring cell (typically around 1 ml) and sample transfer time [1719]. These technical limitations led to development of benchscale biocalorimeters. Benchscale biocalorimeters were broadly classified in to three major categories viz., dynamic calorimetry, continuous calorimetry and heat flux calorimetry based on heat measurement principle. A more detailed explanation of principle of operation of these three modes of benchscale calorimetry is available in a review by von Stockar and Marison [2]. Heat flux calorimetry has a significant role in bioprocess application since its inception into market by 1980. The design of a benchscale heat flux calorimeter was first developed by CibaGeigy AG, Basel, Switzerland and later commercialized as a ‘Reaction Calorimeter
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RC1’ by Mettler Instruments AG, Greifensse, Switzerland [20].Reaction calorimeters have proved suitable for cultivation of different cell lines mimics the real time process conditions of a bioreactor. They have become powerful tools for quantitative thermodynamic studies and reliable monitoring and control of many bioprocess systems [21]. Bench scale heat flux biocalorimeters are high performing (bio) reactors, whose potential as tools for quantitative monitoring and control of (bio) processes is now well established [22]. The operating principle of a bench scale heatflux biocalorimeter could be briefly explained as follows: In the isothermal mode, the reactant temperature (Tr) was maintained constant by controlling the jacket temperature (Tj) by circulating lowviscosity silicone oil through the reactor jacket at a higher rate (2 l/s). The jacket temperature was carefully controlled by blending oils from a ‘hot’ and a ‘cold’ circuit, via an electronically controlled metering valve. Thus, an exothermic or endothermic process would decrease or increase Tj leading to a temperature gradient across the reactor wall which was directly proportional to the thermal flux liberated by the process (q) according to the following equation
q UA(Tr T j )
(1)
The above equation facilitates realtime measurement of heat flow rate resultant due to metabolic process in a biocalorimeter. However, the low sensitivity of the measured heat flow rate signal has hindered their application to weakly exothermic bioprocess systems.
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2. BIOCALORIMETRIC SENSITIVITY IMPROVEMENT The advent of benchscale heat flux calorimeters over the last 20 years has brought major improvements in its sensitivity. Currently, heat flux calorimetric systems can be operated in isothermal, isoperibolic or adiabatic modes, of which the isothermal mode is most suited for biological studies [23]. Reaction calorimeter known as ‘RC1’ developed by MettlerToledo AG; Switzerland for chemical processes was modified (sensitivity enhanced) version ‘BioRC1’ for bioprocess monitoring (Figure 1) applications. The heat flow rate signal measured from BioRC1 confirmed the role of biocalorimetry in biotechnology and proved its importance for a thermodynamic analysis of biological systems. However, the resultant heat flow rate signal due to biochemical reaction needed to be deduced from an online energy balance accounting for heat gain/loss terms viz., stirring, dosing, accumulation environment and exit gas. Menoud et al. [24] calculated the power generated due to stirring action by measuring torque, online, using a torque meter. Baseline heat flow rate signal was accounted for power added by stirrer and corrected heat flow rate signal was used for effective bioprocessmonitoring. Marison et al. [25] made a detailed review on methods of improving the sensitivity of commercial standard version RC1 to monitor weakly exothermic biochemical reactions. In this review a detailed online energy balance was deduced for heat flows involved in a biochemical reaction. Resolution of heat signal was improved by elimination of shortterm noise, achieved by high resolution temperature controllers and pre saturating, thermostating inlet gas to reactor with a bubble column. The longterm noise issues on measured heat signal due to ambient temperature fluctuations were minimized by applying suitable smoothening techniques to filter out the noise from the
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signal. GarciaPayo et al. [26] improved the resolution of benchscale calorimeter to the range 412 mW/l (compared to former 50 mW/l reported on Thermochimica acta 309 (1998) by von Stockar et al.), which has been regarded as a significant milestone in biocalorimetric literature. This low resolution is comparable to micro calorimetric resolution and paved the way for employing benchscale heat flux calorimeter to monitor weakly exothermic reactions viz., anaerobic and animal cell growth processes. In their work, the sensitivity of calorimeter was improved by obtaining optimal values for parameters P and I of PI temperature algorithm. Though tuning of P and I values to optimum level resulted in improved heat flow signal sensitivity, electronic noise generation and decrease in response time emerged as major setbacks to implementing it in practical operation.
Figure 1. Setup of BioRC1: 1, Motor for stirring; 2, Reactor; 3, Oil circulation tank; 4, Calibration heater; 5, Tr sensor; 6, Tj sensor; 7, Float valves; 8, Coolant connection; 9, Pump motor; 10, T c sensor; 11, Connection for dielectric heating; 12, Drain valve; 13, Cooling oil container; 14, Expansion tank; [73].
Environmental heat losses (from head plate and reactor wall surface to surrounding) further influenced the electronic noise formation. Thermostating head plate with operating reaction temperature (Tr) and provision of thermostatic reactor shields contributed to considerable reduction on electronic noise signal generation along with measured heat flow signal. Although the sensitivity of heat flow signal was increased by thermostating head plate, these above mentioned provisions, and change in bioprocess conditions (agitation and gas flow rates) caused significant fluctuations on heat flow signal sensitivities. This may pose difficulties in employing calorimeter on monitoring obligate aerobic cultures. Extensive research on improving electronic hardware settings (temperature controller) of RC1 paved the
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M. Surianarayanan, S. Senthilkumar and A. B. Mandal
way for evolution of highly sensitive BioRC1, suitable for monitoring all type of biological systems. However design modifications of RC1 and provision of additional accessories (head plate thermostating, bubble column and thermostated reactor shield) further complicated the evaluation of heat flow signal.
Baseline Heat Flow Rate Estimation Method Estimation of baseline heat flow (accounting all heat gain and loss effects) at defined operating conditions of a bioprocess system seems to be less complicated and has been widely adopted in almost all heatflux based biocalorimetric measurements. A basic description of calorimetry is presented here to facilitate the understanding of the measurement principle and modifications made to suit our measurement. Heat flows are defined here as positive if heat is released inside the bioreactor. For an aerobic process, the heat balance around the reactor can be written as:
qac qr q j qs qg qe q f qa qCO 2 In the above equation 2, running process, power,
q ac is the heat accumulation in the bulk, q r the heat flow of the
q j the heat flow through the reactor wall to the jacket oil, qs the stirring
q g the heat flow induced by aeration, qe the heat flow to the environment through the
nonjacketed part of the reactor, Copyright © 2012. Nova Science Publishers, Incorporated. All rights reserved.
(2)
base addition and
qCO
2
q f the heat flow of the feed, qa the heat flow of the acid or
the heat flow of the CO2 vaporization.
For bioprocesses with an efficientbioreactor temperature controller, the heat accumulation term can usually be neglected. The jacket heat flow is the most important to monitor since it contains all other heat flows, especially the heat flow generated by the (bio) processes. In the isothermal mode, it will be varied by the temperature controller to keep the bioreactor temperature constant. In the case of RC1 it can be written:
q j UA.Tr T j UA.T
(3)
When all other heat flows are constant, the heat transfer coefficient can be determined with a known or measured calibration heat flow ( qe ). In case of the BioRC1, calibration was carried out by means of an electrical heater which releases a measured quantity of heat in the reaction (20 W). The heat flow due to stirring is important compared with the process heat flow and is usually not directly measured. In our experiment a torque meter was used to measure it and the heat flow can be calculated as:
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113
Heat Flux Biocalorimetry
q s 2 .
S .Mt 60
(4)
The heat flow induced by aeration is a complex function depending on parameters described as . q g f Tgi ; Tgo ; RH i ; RH o ; mgi
(5)
The heat losses to the environment can be calculated with a single relation, considering one lumped heat transfer coefficient for the upper part of the bioreactor. It has to be used only for high ambient temperature variations.
qe K e .Tr Te
(6)
Any matter added to the bioreactor will also induce a heat flow. The heat flow for substrate feed is given below:
q f m f .C p f .Tr T f
(7)
During fermentations, carbon dioxide is produced and the associated heat term is given as follows:
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qCO2 nCO2 .H r
(8)
The heat flow of an ongoing (bio) process, qr, is the result of the energy dissipated by all biochemical reactions. It cannot be directly measured, but can be determined from the following relation.
qr q j qs qc q g qe q f qa qCO
(9)
When there is no bio chemical activity in the bio reactor, the sum of all the heat flows on the right side of equation 9, should always be zero, provided they are properly measured and calibrated. In practice, equation can be reduced by lumping all the constant heat flows in a “baseline” term, qbl , which has to be determined before and after any experiment. If culture conditions are not varied, qS , q g , qC and q f are potentially constant. Finally, for monitoring purposes,
qa
and qCO can be lumped in qr , since they relate to metabolic activity of cells. 2
They have to be separated only for the thermodynamic evaluation of qr . Equation 9 can be reduced to Equation 10, which is used with the basic (BioRC1) setup:
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M. Surianarayanan, S. Senthilkumar and A. B. Mandal
qr q j qbl
(10)
When using the basic BioRC1 setup, the process heat flow was obtained from equation 10 and the following procedure is applied. The bioreactor was maintained under culture conditions and stabilized to determine the initial baseline ( Tbl ). A UA calibration was made immediately before inoculation (Figure 2). UA calibration was performed during experiment at regular time intervals by means of in situ calibration heater (20 W) and
q
accordingly the measured heat signal ( r ) was corrected. When the culture conditions were modified (aeration rate, temperature, stirring speed etc.), the baseline stabilization and UA calibrations were repeated for each set of culture conditions. Heat effects due to inlet aeration, stirring speed and other heat losses were eliminated by prethermostatting and insulating the respective streams. The exhaust gas stream was allowed to another membrane filter (0.2 m) to ensure aseptic conditions inside the lab.
2.5
Power, W
2.0
1.5
1.0
Innoculum addition
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0.5
Baseline 0.0
qb=0.05 W 0
1
2
3
4
5
6
7
8
9
Time, h
Figure 2. Baseline heat evaluation for growth of P.aeruginosa in a complex growth media on BioRC1 (Sivaprakasam et al. Unpublished).
After the experiment, cells had to be inactivated and the RC1 stabilized to determine the final baseline. A second UA calibration was then made for the offline evaluation of the process heat flow. Several situations could occur, 1. UA did not vary and the final baseline heat flow ( q bl ) was equal to the initial one. In this case,
qr was not corrected offline.
2. UA varied but the final
q bl
was equal to the initial one. In this case, UA was re
calculated linearly with time or volume (if available) to correct
qr offline.
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Heat Flux Biocalorimetry
115
3. UA did not vary but the final q bl was different from the initial one. In this case, q bl was recalculated linearly with time or ambient temperature (if available) to correct
qr offline. 4. UA varied and the final q bl was different from the initial one. In this case, both UA and
q bl were recalculated as described under points 2 and 3.
However, the heat flow evaluation was carried out using a userfriendly evaluation software (supplied with the equipment) known as WINRC. Heat loss terms and calibration for obtaining heat flux data were done in an automatic mode of the instrument.
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2.1. NonInvasive UA Estimation In heatflux type biocalorimeters, especially with BioRC1 the overall heat transfer coefficient ‘UA’ is generally measured using insitu calibration heaters (5 – 20 W). This is a direct and precise method for measuring ‘UA’ value which is vital for heat rate estimation. Continuos measurement of UA values is indeed required in a bioprocess system as there will be change in rheology of reaction broth due to cell growth. However, continuos switching on/off of the calibration heater questions isothermal process conditions and attributes disturbance to the measured heat rate signal. So, noninvasive estimation of UA is a need of the hour for heatflux based biocalorimeters. BouDiab et al. [27] described a novel Oscillation Reaction Calorimeter (ORC) method for precise calculation of UA value and illustrated it with a fed batch bioprocess. ORC method allows continuous evaluation of UA by inducing sinusoidal oscillation on jacket temperature based on a mathematical model suggested by Tietze et al. [28]. ORC method did shown positive results when employed to study fed batch process using Saccharomyces cerevisiae. This method proved to be effective in comparison to conventional oneanchor (heat capacity constant throughout reaction) and two anchor methods (heat capacity calculation at start and end of reaction). But ORC method involves complex mathematical models, which may cast doubts on its practical application. On further development with UA estimation methods, Voisard et al. [29] made an attempt to increase the sensitivity of reaction calorimeter with a slight design modification to reaction vessel. Frequent changes in overall heat transfer coefficient (UA) values during the course of bioreaction were eliminated by forcing heat transfer to occur through a constant area by insertion of a PTFE sleeve on reactor wall. This modified reaction vessel design showed significant improvement on accuracy and operating flexibility of system. However the presence of PTFE sleeve on reactor vessel can decrease the working volume, increase complexity in calorimeter operation, cause change in mixing pattern and reduction of head space, which may add to the problem in release of exit gas/vapor from reaction under study. Thanks to the recent developments in thermal sensors field, now real time UA estimation is possible with the latest heatflux type calorimeter ‘RTCal’ designed by Mettler Toledo. This improved design of reaction calorimeter is equipped with vertical and horizontal sensor
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M. Surianarayanan, S. Senthilkumar and A. B. Mandal
bands positioned on the reaction vessel. These sensors bands are thermocouples impregnated in a polymer matrix, and facilitate continuous UA measurement. The principle of operation and design of ‘RTCal’ calorimeter is well reported [30]. This real time UA evaluation methodology will definitely put forward the heat flux calorimetry on to next generation bioprocess applications. More research groups have undertaken extensive studies on different bioprocess systems employing benchscale heat flux calorimetry and they can be broadly classified as, a) Quantitative studies on kinetics of microbial growth b) Biothermodynamic studies c) Monitoring and control of bioprocesses
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3. QUANTITATIVE STUDIES ON KINETICS OF MICROBIAL GROWTH A good number of groups have undertaken the investigation of microbial growth using heatflux calorimeters with focus on different aspects, for instance: influence of environmental conditions (19972002), degradation of harmful substances (19931995), biopharmaceuticals and biocides production (1990 and 1998), and fermentation processes (1991 and 1998). The first review in the use of calorimetry in biotechnology dates back to 1989 by von Stockar and Marison [31]. The authors have dealt in detail with microbial heat generation pointing out how it depends or specific ways of growth, biomass yield, maintenance metabolism, nature of substrate and energetic efficiency of growth, oxygen uptake and product formation. It is possible to resort to heat measurements to gain information on any of the above factors. In the review it has been demonstrated that heat flow rate measurements could be used to determine the monitoring of general microbial activity and finger printing metabolic events such as substrate limitations, inhibition and diauxics. Substrate limitation effect on biomass growth could be better exemplified by monitoring the metabolic heat production rate during growth of P. aeruginosa. Cultivation of a salt tolerant P.aeruginosa in heatflux biocalorimeter (BioRC1) using glucose limited minimal media and resulted in distinct shaped heat flow signal [32] as shown in Figure 3. From Figure 3, it is evident that the biomass concentration increased exponentially until glucose concentration was depleted completely. The heat flow rate, q, and the biomass growth profiles were observed to exhibit similar pattern during all the phases of biomass growth. Such behavior reflected the significance of heat flow rate measurements in predicting the biomass concentration without the need for offline analysis. The heat flow rate signal fell down rapidly as soon as the culture reached the endogenous phase. Heat continued to release at a constant rate until glucose concentration was exhausted. A marked increase in heat signal was observed during the shift from lag phase to exponential phase. In a similar way, after glucose got exhausted, there was a sudden drop in heat value, which corresponded to the biomass decay. Both heat and OUR profiles seemed to follow the similar pattern. The sudden drop in glucose concentration followed by a similar drop in heat value showed that this strain possessed greater affinity to glucose. This case study of P. aeruginosa growth illustrates the significance of heat rate measurement on detecting substrate limited biomass growth.
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Heat Flux Biocalorimetry 3.0
Power OUR Biomass
70
0.7
0.6
Glucose
1
60 50
0.4 40
1
0.3 30
0.2
20
0.1
10 0 0
2
4
6
8
10
0.0 14
12
2.0
1.5
Power, W
0.5
1
2
Glucose X 10 gl
2.5
Biomass, gl and OUR mgl .s
80
1.0
0.5
Time, h
Figure 3. Growth of P.aeruginosa in a glucoselimited (0.1 %) mineral salts medium at optimized conditions in BioRC1 [44].
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3.1. Elemental and Enthalpy Balance for Analyzing Microbial Growth Processes If heat flux calorimetry is to play a more important role in quantitative engineering related studies and in bioprocess monitoring, the ‘black box’ model for cell growth process need be revealed. The quantitative relationship of the heat evolution rate with other relevant process variables, such as biomass concentration, growth rate, CO2 evolution rate, and so on must be elucidated. Heat flow has often been considered as non specific information, which may account for some prejudices in the field of biocalorimetry. However, it has been shown in [3335] that specific, quantitative information on the above mentioned parameters may readily be deduced from heat evolution measurements on analysing them in terms of combined enthalpic and elemental balances of microbial growth. The stoicheometry of a general growth process under pure aerobic conditions (no side product) may be described in terms of a “chemical equation” as follows: CH O N Y O Y NH Y CH O N S1
S2
S3
O/S
2
N/S
3
X /S
X1
X2
X3
Y CO Y H O C/S
2
W /S
2
(11) This equation has been formulated in terms of Cmoles, which means that each chemical formula has been reduced to the basis of one carbon atom. In this notation, the letters appearing in the subscript stand for substrate (S), biomass (X), O2 (O), NH3 (N), CO2 (C) or H2O (W), whereas the numbers designate the elements H (1), O (2), and N (3). The stoicheometric coefficients, or Cmolar yields are defined as the ratio of the heat flow rate to conversion rates ri (Cmol s1 L1) of either substrates ( ri 0) or products ( ri 0):
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M. Surianarayanan, S. Senthilkumar and A. B. Mandal
r Yi/ j i rj
(12)
Six yield coefficients found in growth stoicheometric Eq. (11) satisfy C, H, O and N balances at all times. Two of them can be determined independently with experimentation and the rest are represented in the form of experimentally determined yield values. The relationship of the heat flow rate q (WL1) to conversion rate of any component ‘i’ (involved in reaction), could be expressed by a similar ratio:
q YQ / i ri
(13)
The proportionality coefficients defined by Eq. (13) can be regarded as “heat yields” or “energetic yields” and determined from calorimetric experimental results; this further reduces the need for determining unknown yield value, since by measuring heat rate using calorimetry, one can obtain the entire growth stoicheometry with estimation of heat yields. In order to use the combined elemental and enthalpy balances discussed above, one needs the chemical formula for all compounds listed in Eq. (11) as well as data on their heats of combustion. It appears difficult to estimate the enthalpic content of dried microbial biomass from the literature because of a considerable scatter of data [3642] and complexity on estimation by combustion calorimetry. An attractive alternative to combustion calorimetry is
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therefore the estimation H X on the basis of elemental composition. Cordier et al. [43] critically reviewed several mathematical models relating the heat of combustion of dried biomass to its elemental composition. For a given type of reaction, standard correlations were developed based on degree of reduction (
i
) to estimate heat yields theoretically. In case of pure respiration process with
no fermentation product, the heat yield correlations were simplified and represented using degree of reduction as follows: Rate of heat flow per unit mass of substrate consumed, catabolic heat yield,
Y
Q/S
q Q r 0
S
X
Y X /S
(14)
R
S
Rate of heat flow per unit mass of biomass generation, anabolic heat yield,
Y Q/ X
q Q r Y
S
0
X
X /S
X
R
Rate of heat flow per mole of oxygen consumed, Oxycalorific coefficient,
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(15)
119
Heat Flux Biocalorimetry
YQ / O 4Q0 460 kJ mol1
(16)
Significance of calorimetry on determination of growth stoicheometry and prediction of yield coefficients can be explained in detail by considering a case study on aerobic growth of P. aeruginosa [44]. In the growth stoicheometric equation, compositions of all the components were known except the elemental composition of grown biomass. CHN analyzer was employed for estimation of mass fraction of elements and finally elemental compositions were arrived from standard correlations. In a similar way, the molecular formula for the bacterial strain P. aeruginosa deduced as CH 1.747 N 0.21O0.55 and estimated values were found in accordance with the values available in literature [45]. Stoicheometric equation for pure aerobic growth of P. aeruginosa was obtained as follows:
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C6 H12O6 5.93O2 0.01428NH 3 0.068CH1.747 N 0.21O0.55 5.932CO2 5.962H 2 O
(17)
Table 1 shows the comparison of theoretical and experimental heat yields (BioRC1) for growth of P.aeruginosa in a glucoselimited mineral salt medium. A good agreement was observed between their values. This proves the efficiency of heatflux calorimetry (BioRC1) on monitoring metabolic activity of P.aeruginosa and suggests its further application on several other bioprocess systems. Coupling biokinetics and bioenergetics may help understand more of microbial growth process and pinpoint the unknown anomalies occurring during the biodegradation process [46]. From this brief discussion one can understand that most of the research groups have so far made an approximation of growth stoicheometry to a simple equation (with one or two byproducts apart from new biomass). Since metabolic pathway of any organism comprises a large number of parallel reactions, it is impossible to provide a well defined stoicheometry for all reaction steps. Heat evolved from living systems is considered as overall output of all metabolic actions i.e. all parallel reactions; it is justifiable to approximate growth stoicheometry to a single overall reaction since most other reactions are either endothermic or non contributors to all over heat. Table 1. Comparison of predicted and experimental yield coefficients of aerobic strain P.aeruginosa cultivated at glucose limited growth media (Senthilkumar et al. 2007) Glucose (g/l) 1
YX
S
(th) 0.53
YX
S
(exp) 0.420
YQ
X
(th) 329.6
YQ
X
(exp) 407.8
YQ S
YQ S
YQ O
YQ O
(th) 243.53
(exp) 267.6
(th) 460
(exp) 413.54
Recent developments in metabolic flux analysis modelling could help in obtaining a promising pathway and elucidating a stoicheometric growth equation for complex bioprocess systems. Moreover, of late, a number of naturally derived/complex substrates are in use in many and their molecular formulae have not yet been determined. Future improvement on instrumentation techniques may provide opportunities for researchers to determine molecular formulae for substrates and develop a database there of. One can expect biocalorimetrists to do analysis of the complete metabolic pathway of organisms based on heat flow profiles.
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3.2. Realtime Estimation of Specific Growth Rate from Heat Rate Data Specific growth rate is a key parameter in most of the bioprocess systems for determining the product yield and process limitations. Moreover, the value of specific growth rate can provide information to a biochemical engineer about start and termination of fedbatch operation while operating a bioreactor. So far offline analysis and inline bioanalytical tools have been in use for measuring cell concentration and estimation of specific growth rate. They are time consuming and invasive restricting the operator from taking immediate action during the course of a reaction. As seen in the previous section, the metabolic heat production can be related to cell growth process by means of heat yield coefficient and this allows rapid estimation of specific growth rate. Surianarayanan et al. [47] attempted to predict specific growth rate from metabolic heat flow rate values measured during aerobic cultivation of P. aeruginosa. The mathematical model suggested in their work can be useful for real time estimation of biokinetic parameters from heat flow data. Heat rate based model for specific growth rate estimation has been briefly discussed here uses cultivation of P. aeruginosa in a substrate limited growth medium as an illustrative example. Metabolic heat generated due to biomass growth can be well explained by stating a heat balance at any instant of time say ‘t’. Let H S , and H X be heat liberated on burning a unit mass of substrate and cell concentration respectively. By Hess's law (for pure aerobic process),
H
S
( S S ) H
t
X
0
(X X ) q 0
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On differentiating Eq. (18), solving for
q
met
H X
S
Y Y
X S
q
met
0
met
t dt
(18)
,
H
X S
X
(19)
Biomass growth rate can be represented in terms of heat rate as follows,
r X
dX H dt
q Y Y
met
S
H
X S
X
X S
(20) Let denote the denominator in Eq. (19) as D. The value of all the parameters in the denominator ‘D’ are known theoretically from literature Finally Eq. (20) can be simplified after substituting for X (t ) as follows,
q (t ) ( DX H met
0
met
(t ))
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(21)
Heat Flux Biocalorimetry
121
Figure 4 showed a linear profile when plotting values of
q t against DX H (t ) and 0
met
met
the slope of this profile yielded the value of specific growth rate. This finding agrees well with thermodynamic analogies predicted by von Stockar et al. [48] for estimation of biomass yield and its growth rate. 2.4
Specific growth rate=0.04041
2.2 2.0
qmet
1.8 1.6 1.4 1.2 1.0 0.8 4255
4260
4265
4270
4275
4280
4285
4290
4295
DX0+Hmet
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Figure 4. Heat yield due to growth of P. aeruginosa at 0.1 % glucose concentration under optimized conditions [47].
Specific growth rate for P.aeruginosa cultivated in 0.1 % (w/v) glucose limited MSM media under optimized conditions was observed to be 0.04041. The linearity of the profile (R2= 0.989) shown in Figure 4 further proved the feasibility of applying calorimetric results for estimation of biokinetic parameters. Recently, some research groups employed heat rate based models for real time estimation of specific growth rate in aerobic and fermentative bioprocess systems. Their significant findings and applications are discussed briefly in the ‘Section 5’ (Monitoring and Control of Bioprocess Systems) of this chapter.
3.3. Diauxic Behavior Diauxic mode of growth is a common feature in cell growth process when using growth media comprising mixed substrates. Specific affinity of a particular substrate in contrast with another one by organism under study can be influenced by factors such as substrate limitation, nature of substrate etc. Fingerprinting the metabolic shift in an organism to switch from one substrate to another and detecting the phase at which it occurs is challenging and enable progress on of the bioprocess without growth inhibition/substrate limitation. Though several online tools (Biomass Monitor, Respirometry and Exhaust gas Analyzer) are presently in use, calorimetry was proven to be more efficient for instantaneous detection of bioprocess anomalies especially of diauxic phenomenon. Calorimetric investigation on growth of P. aeruginosa in a complex growth medium can better illustrate the diauxic behavior and its rapid detection based on heat rate signal.
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0.6
7
2.6 2.4
0.5
2.2
0.5
6
2.0 0.4
3
2
1.8
1
0.4
0.3
0.3
Glucose
0.2
0.2
Peptone CDW Heat Flux
1
CDW, gl 1 Glucose, gl
4
Heat Flux, W
1
5
Peptone, gl
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Biocalorimetric experiments were performed in BioRC1 to investigate the aerobic growth of P. aeruginosa, isolated from tannery saline wastewater. Calorimetric batch trials were performed on growth of P.aeruginosa in a mixture of complex growth media (Nutrient broth) and glucose [27]. A comparative plot of heat flow, cell dry weight, peptone uptake and glucose uptake of P. aeruginosa in biocalorimeter on glucose (0.3 % by wt) limitedNB media is shown in Figure 5. Heat rate (Power) curve indicated the current activity of the cell culture, and changes in slopes of the curve showed the effects of limiting factors on metabolic activity of P. aeruginosa. On the basis of the powertime curve, the behaviour of the cell culture was deduced. Phase 1 in Figure 5(0–10 h) comprises both lag and earlier exponential growth phases, where the culture adapted to the NB media and effectively utilised the glucose. In this phase, 70 % of glucose was metabolized, and a low uptake of peptone (20 %) was observed. This indicates that P.aeruginosa initially adapts in NB media slowly and rapidly metabolizes glucose present in the media. Glucose being a wellknown reduced substrate, the bacteria under study utilized glucose effectively and cometabolized peptone present in NB media at slow rate. The peptone depletion profile in Figure 5 further confirmed the slow adaptation of P. aeruginosa in NB media in phase 1. From Figure 5, it can be seen that there is no marked increase in growth of P. aeruginosa in phase 1. Maximum heat flux (0.5 W) observed in phase 1 further suggests that glucose dissimilation by P.aeruginosa contributes the major part of heat generation. A sudden rise in growth profile in phase 2 (Figure 5) Corresponds to initiation of exponential growth of P. aeruginosa. In phase 2 (10– 14 h), P.aeruginosa utilized peptone as the sole carbon source for growth, even though considerable amount of glucose (1.08 gl1) was present in the growth medium. Presence of excess amounts of glucose (0.3 %) in growth medium did not cause marked increase in growth of P.aeruginosa and thus 0.2 % was considered the optimum glucose dosage for better growth. This kind of diauxic trend was observed in growth curve (phases 1 and 2) and powertime curve. Figure 6 depict comparative heat flux profiles due to the growth of P. aeruginosa at varying glucose concentrations, 0.1 %, 0.2 % and 0.3 %.
1.6 1.4 1.2 1.0 0.8 0.6
0.1
0.1
0.4 0.2
0.0
0 0
2
4
6
8
10
12
14
0.0
16
Time, h
Figure 5. A Comparative plot showing heat release profiles, cell density, glucose uptake and peptone uptake of P.aeruginosa grown in 0.3 % glucose – NB media at optimized growth conditions [32].
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Heat Flux Biocalorimetry 0.6
NB + 0.1 % Glucose
1
NB + 0.2 % Glucose NB + 0.3 % Glucose
0.5
1
Heat flux(qr), W
0.4
2
1 0.3
2
2 0.2
0.1
0.0 0
2
4
6
8
10
12
14
16
Time, h
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Figure 6. Comparative heat flux profiles generated by the growth of P.aeruginosa grown in glucoselimited NB media [32].
All these heat flux profiles exhibit two different phases namely 1 and 2. Phase 1 corresponds to rapid glucose uptake by P.aeruginosa and slower adaptation of culture to the growth media. Phase 2 corresponds to diauxic growth nature of P.aeruginosa, where the bacterium utilizes peptone as its carbon source after complete exhaustion of glucose in phase 1 and also a rapid increase in growth of P. aeruginosa. After complete consumption of glucose, heat flux dropped to a lower value (at end of phase 1). Switching from limiting (glucose) to excess nutrient concentration (peptone) resulted in specific intermittent profiles of heat flux. The subsequent increase in heat flux was due to a change in metabolic pathway. P.aeruginosa excretes alkaline proteases to hydrolyze the peptone present in the cultivation medium. Therefore heatflux calorimetry allows for instantaneous detection of diauxic nature of organism noninvasively and provides information on limitation of substrate of interest.
4. BIOTHERMODYNAMIC STUDIES Just as thermodynamic concept on predicting feasibility of newly formulated chemical reactions, biothermodynamics may be applied to bench scale biochemical reactions and this will enable us to estimate technical feasibility as well as economic viability for up scaling it. The first review on the thermodynamic considerations for constructing the energy balances in cellular growth was done by von Stockar et al. [49] in 1993. They considered the biological species as a chemical compound in a given thermodynamic state as defined by the state of aggregation. They assumed that physical transition from one species to another and transformations between different species, catalyzed by living cells were described by one (or) several overall processes with fixed stoicheometry. The resulting enthalpy balance was generally applicable to closed adiabatic and closed isothermal systems, open systems at steady state and also to open system in transient stages. It’s possible to determine the enthalpy
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M. Surianarayanan, S. Senthilkumar and A. B. Mandal
growth efficiency in a very simple way by calorimetry. The free energy efficiency can be computed based on the knowledge of growth stoicheometry, which remains constant at 60 % for K.fragilis under aerobic, facultative and anaerobic conditions. Since 1997 a rigorous research programme on the thermodynamic analysis of microbial growth was undertaken by von Stockar et al. They concluded that the amount of Gibbs energy dissipated per Cmole of new biomass grown, ∆G0X, was the key parameter for understanding the thermodynamics of microbial growth. Furthermore, they stated that on one hand it was linked to the rate of metabolism and therefore regarded as a driving force for growth and on the other the Gibbs energy balance (∆RG0X) determined the biomass yield (Yx/s). i.e. higher the value of ∆RG0X , lower would be the biomass yield. In a similar way the aerobic growth of P.aeruginosa in glucoselimited mineral salt medium on BioRC1 was analyzed thermodynamically. Growth stoicheometric equation (Eq.11) can be split in two forms representing catabolic and anabolic processes. Gibb’s free energy change of growth reaction can be calculated as:
G H TS
(22)
The enthalpy change of the overall reaction can be found as the sum of the enthalpy changes of the catabolic and the anabolic reactions.
H r
1 H C H A 40299.177 KJmol 1 YX
(23)
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D
Change in entropy for anabolic and catabolic process of growth was calculated from respective stoicheometric equations using standard values of entropy of formation of reactants and products involved in reaction. Substituting values for enthalpy change (∆H) and entropy change (∆S) in Eq. (22), free energy change was calculated as 41519.766 kJ mol1. Overall, the growth process of P.aeruginosa is purely exothermic (Figure 7) as free energy change in metabolic process is dissipated mostly in the form of heat (enthalpy change of 97.06 %). This suggests that calorimetry is most suitable for inline monitoring and control of pure aerobic growth process (or) metabolic activity of crab treenegative organism. An attempt to analyze thermodynamic feasibility of biochemical reaction was made for the first time by von Stockar et al. Gibbs energy change (∆rG) for each reaction step in a metabolic pathway has to be negative for the reaction to be thermodynamically feasible. Based on this principle, glycolysis pathway was analyzed and found to be thermodynamically not feasible. It has been concluded that uncertainty of values of ∆rG, and concentrations of the metabolites, negligence of pH and activity coefficients might render the proposed approach as not feasible [50]. Hence it could be understood that prediction of feasibility was rather cumbersome for biochemical reactions compared with chemical reactions.
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Figure 7. Enthalpy change in overall free energy change on aerobic growth of P.aeruginosa (Surianarayanan et al. Unpublished).
von Stockar et al reported the calorimetric investigation of an extreme case of entropydriven (endothermic) microbial growth [51]. In a recent review [52] of the thermodynamics of microbial growth and metabolism, it was stated that wide variations of biomass yield reported for different microbial growth system could be explained on thermodynamic reasoning. These variations appeared to be the result of an evolutionary adaptation of the amount of Gibb’s energy dissipation towards a reasonable compromise between growth efficiency and growth rate. Most importantly, the database concerning the Gibb’s energy change of the chemicals of life and the biochemical reactions, and knowledge on intracellular chemicals affecting the forces during these reactions must prove it. This includes more accurate and more detailed data on the metabolites, whose concentrations have a decisive impact on the thermodynamic calculations. Also, von Stockar et al [53] assumed a blackbox model for cell growth involving only two metabolic processes, catabolism and anabolism. They evolved a simple relation for prediction of biomass yield as given below:
Y
X /S
G G G
(24)
0
a
0
r
X
0
b
However, this relationship suffers from inability to predict other yield values for e.g., product yield [54]. In overall von Stockar et al. made an extensive research on biothermodynamics for overall growth reaction. Correlations and thermodynamic models suggested by them allow the assumption on exclusion of intermediate metabolic pathways involved in overall growth metabolism (discussed in detail in Section 3.1). The developments in current bioprocess analytical instrumentation and metabolic flux analysis may provide researchers an opportunity to establish a complete metabolic pathway for a growth process and a detailed biothermodynamic model i.e. White (or) structured model can be developed with help of calorimetric results.
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5. MONITORING AND CONTROL OF BIOPROCESSES Application of calorimetry to bioprocess monitoring began at the time it was applied to chemical reaction systems during the mid 1980’s. A number of authors reported the possibility of employing isothermal and heat flux calorimetry for investigating aerobic [55], fermentative processes [56, 57] and anaerobic systems [46]. Since heat is a global parameter for all living species to far analysis of their metabolic activity, employment of calorimetry for inline monitoring and control of bioprocesses is gaining importance since the last decade. Recently Jungo et al. [58] employed bench scale calorimetric investigations on fedbatch fermentation of Pichia pastoris for recombinant protein production. Through the heat flow profile responses, it was observed that in a mixed substrate feed P.pastoris fedbatch fermentation, sorbitol was found to be a suitable cosubstrate compared with glycerol. The interesting finding of their research work is prediction of the start of induction phase at low specific growth rate based on heat rate measurements. High specific growth rate rendered high heat generation, caused limitation on oxygen and cooling water supply. On extension of this work, some more calorimetric trials were performed by Jungo et al. to study mixed feed strategy of methanol and glycerol on fed batch fermentations of P.pastoris for avidin production [59] and alcoholic oxidase expression [60]. Change in composition of methanol in feed and shift in substrate uptake was well depicted by corresponding change in heat flow profile. These findings proved the application of calorimetry for effectively monitoring fed batch fermentation systems. Most of their calorimetric results were focused on discussing the pattern of heat flow fluctuations for change in biomass growth, substrate limitation, diauxic growth and type of substrate (e.g., readily metabolized, complex long chain). Sivaprakasam et al [61] employed calorimetry to correlate enzymatic activity with heat flow profile as it was an inherent parameter of metabolism. Calorimetric trials were performed on BioRC1 to investigate proteolytic activity of P.aeruginosa cultivated in a peptoneenriched medium (Figure 8). Phase 1 corresponded to exponential growth phase, phase 2 was the stationary phase and endogenous phase was designated as phase 3. In phase 1, both biomass growth and heat rate profiles exhibited a steep rise in values with subsequent utilization of peptone and glycerol respectively. The comparative analysis of heat profiles, biomass growth, OUR, protease activity and substrate uptake results defined that P. aeruginosa utilized peptone effectively and glycerol acted as a cosubstrate. Respirogram closely followed power–time curve in all the phases of growth and a perfect linear correlation between respirometric and calorimetric data was achieved. Low oxycalorific coefficient (355 kJ mol−1) value and low peptoneheat yield (6 kJ g−1) showed the existence of fermentation coupled metabolism of P. aeruginosa. A biomass yield of 13.4g gmol−1 of oxygen consumption showed that dissolved oxygen as an inevitable substrate for optimum biomass growth and protease secretion. Biochemical reactions involving protease production account for higher heat generation compared to cell culture growth and break down of substrates. This study revealed that both growth and nongrowth related reactions involved in this cell culture metabolism could be monitored efficiently by calorimeter and the heat yield values can be used for better design of fermentors and their scale up. Liu et al. proved the existence of endothermic microbial growth by cultivation of acetotrophic methanogen, Methanosarcina barkeri in BioRC1. This was the first reported study in ‘anaerobic process monitoring’ area employing heat flux calorimeter
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[62]. Heat evolution curve was observed to be endothermic and indicated the different phases of growth during cultivation of M.barkeri. Moreover, the heat profile provided realtime information on growth limitation effect well in advance compared to other process parameters. This study proved the capability of heat flux calorimetry to monitor even a slow growth process with low heat evolution rate. The success witnessed above ensures the application of heatflux calorimetry as a potential tool for monitoring commercially significant anaerobic bioprocesses (e.g., biofuel production) in near future. Control of biotechnological process is rather important in industrial perspective in order to achieve high product yield. Fedbatch mode is the commonly adopted bioprocess strategy, in which the limiting substrate is fed in to process system based on a measured key process parameter. 2.6
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1
12
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2.4
0.6 0.4 0.2
0.0 36
0.0
Time, h
Figure 8. A comparison plot showing heat flow per unit volume with concentration of cell mass (as dry weight), Peptone uptake, glycerol uptake and Protease secretion of a fedbatch culture of P.aeruginosa grown in a peptoneenriched media at optimized conditions [61].
Hence, there is a need of reliable measurement to directly measure (or) estimate the key process parameter. As seen from the previous sections, heat evolution rate is observed to be the robust signal providing realtime information on cell growth, cell physiology, process limitations and product yield. Control based on calorimetric signal started a decade before, where highsensitive BioRC1 is employed for monitoring fedbatch growth of S. cerevisae [63]. Here, heat rate is used as an indicative parameter to track the metabolic state of S. cerevisae and the feeding rate of substrate is controlled by means of Respiratory Quotient (RQ), derived from heat rate values. Recently, Sivaprakasam et al. employed directly the heat rate measurements for estimating cell concentration and specific growth rate [64]. A feedback control strategy was designed based solely on heat rate measurements and employed for realtime feed control. The results suggested that heat rate signal was more
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robust than other measured signals and have wide scope for application in different industrially relevant bioprocess systems.
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5.1. Photoautotrophic Bioprocess Systems Photoautotrophic cultures (e.g., Microalgae) are now gaining more attention among research groups due to their wide application in bioprocess engineering and environmental biotechnology. Technical and design limitations remained a challenge on employing BioRC1 (or) developing photobiocalorimeters for monitoring photosynthetic reactions. Calorimetry research group under the supervision of Prof.von Stockar improved the sensitivity of existing BioRC1 as a photobiocalorimeter. Photosynthetic efficiency of algal cultures namely Chlorella vulgaris, Chlorella sorokiniana was investigated in the highly sensitive photobiocalorimetric RC1 setup, an improved design of BioRC1 [65]. The principle of the photobiocaorimeter is measurement of change in heat flow attributed due to change in stored chemical energy (converted from incident light) of algal biomass. Calorimetric results showed that heat flow data could be employed for monitoring phototrophic biochemical reactions but with limitations. Drifts in baseline heat flow signal caused by temperature oscillations in jacket temperature were to be corrected. Algal biomass yield was observed to be low on contrast with incident high amount of photons from LED. This confirmed possibility that a considerable amount of photons might be not participated in photosynthetic process and could contribute in measured heat flow signal. In their recent study, they verified this phenomenon with batch cultivations of Chlorella sorokiniana on a photobiocalorimeter [64]. In their research work, they observed that the photosynthetic efficiency decreased with increase in algal biomass density due to dark zone formation inside the reactor. At this instant, major fraction of incident light energy was dissipated in the form of heat (nonphotosynthetic quenching), which hindered the feasibility of continuous monitoring of photoautotrophic cultures. Stringent procedures to obtain a temperature controlled environment were required for acquiring stability on heat flow signals. Hence design complications on BioRC1 setup and suitable light source remained a bottleneck for development of photobiocalorimeters for practical use.
5.2. Biological Wastewater Treatment Systems For past few decades, several research groups were actively involved in monitoring aerobic, anaerobic, fermentation and photoautotrphic growth processes using calorimetry as elaborated in previous chapters. But only a few groups were engaged on employing heat flux calorimetry for wastewater treatment monitoring applications. Environment related issues viz., land, water pollution and global warming presently pose great threat to health of different living communities existing all over the world. Precise monitoring and control methodologies are indeed needed for analyzing biochemical reactions involved in wastewater treatment systems. In wastewater treatment, efficiency and status at any time in the bioreactor are now monitored by measuring the Chemical Oxygen Demand (COD) and Biochemical Oxygen Demand (BOD). These two tests are not standardized, time consuming and so it becomes difficult to control the treatment process on a routine basis. Although several
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biosensors (N2 probes, DO probes, Infra red analyzer for CO2 and CH4 detection) are presently under use, they do not provide real status of system under study and require careful calibration. There is need adequate modelling for considering the effect of masstransfer limitations and liquidphase chemical equilibrium. Therefore, a more sophisticated inline biosensor is required to monitor the bioreactor (for wastewater treatment) performance and study the metabolic efficiency of bacterial consortia. Dermoun et al [67], reviewing the flow calorimeter developed by Monk and Wadso [68], pointed out that measurements could be unreliable, due to the growth on the wall of the measuring cell, the change in metabolism during transfer from the culture flask to the calorimetric vessel and the oxygen deficit due to increasing cell density. Beaubien and Jolicoeur [69] observed that microcalorimetry allowed the evaluation of the biodegradable organic content of an influent. Very few studies are found in literature [70] about the applicability of bench scale calorimetry to biological wastewater treatment processes without using flowcalorimeters. Aulentaa et al used a high resolution batch calorimeter to assess the biological activity of sludge from a fullscale WWTP and estimated biokinetic parameters [71]. Daverio et al. [72] analyzed acidogenic and methanognic phases of anaerobic granular sludge with the help of flow signal. For repeated glucose spikes, feeble heat flow signal was observed and this decrease in heat flow was attributed to nutrient limitation. Sivaprakasam et al. [73] employed BioRC1 for investigating biological treatment of tannery saline wastewater by salt tolerant bacterial consortia. A wellacclimatized bacterial consortia (at endogenous phase) on tannery saline wastewater was taken in BioRC1 and spiked with known volume of wastewater sample of same origin (Figure 9). Wastewater degradation was observed by an instantaneous increase in heat production rate and oxygen uptake rate, probably due to the presence of a small amount of readily biodegradable COD. After reaching the maximal, both heat and OUR profiles fell gradually. 1.90
0.6
OUR
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OUR, mgl s
1 1
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Heat Flux
0.3 1.75 0.2 1.70 0.1 1.65 0.0
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Figure 9. Heat flux and OUR profiles acquired during tannery saline wastewater degradation (added after endogenous phase) [73]. Heat Flux: Processes, Measurement Techniques and Applications : Processes, Measurement Techniques and Applications, Nova Science
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Wastewater additions were repeated and a satisfactory reproducibility was observed for oxygen and heat measurements. During the experiments, endogenous respiration dropped from 0.5 to 0.1mg l1min1 as a result of repetitive wastewater pulses. The shortterm biological oxygen demand, evaluated as the area under the respirogram for each wastewater pulse, was found to be constant. For 100 ml wastewater addition and a total reaction volume of 1 l the average value was 64 mg l1. The oxycalorific coefficient (YQ/O ) for tannery saline wastewater was found to be 414.2 kJ mol O21. This value was nearer to the theoretical value for heterotrophic aerobic metabolism (460 kJ mol O21). A linear correlation observed between metabolic heat and COD for acetate spike on salt tolerant consortia further proved the success of employing BioRC1 for instantaneous monitoring of wastewater treatment systems (Figure 10). In this research work, the authors also indicated that heat data could be used to determine indirectly BOD value of wastewater samples instantaneously and a novel correlation was yet to be developed. Further research is needed focuss on real time estimation of biomass sludge age and its activity using heat flow rate measurements. 22000 20000 18000
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Figure 10. Linear plot between heat production (Jl1) and acetate consumption (mg COD l1) [73].
Successful completion of the above mentioned research tasks will be useful for commercial wastewater treatment facilities in using heat flow rate signal as a monitoring tool for planning immediate action and preserve working biomass sludge from sudden wastewater loads, toxic inhibition etc.
6. LARGESCALE HEAT FLUX BIOCALORIMETRY The special aspect of benchscale calorimetry and its huge potential for exploitation in large scale industrial bioreactors is termed ‘Macro (o) Largescale biocalorimetry’. The rate
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of heat production in heat flux calorimetry is proportional to surface available for heat transfer. As we move from benchscale to industrial scale bioreactors, the ratio of surface area to volume decreases. Hence the rate of heat production will be high in large scale bioreactors and quality of heat signal will be also good [74]. So, several biocalorimetrists are now interested in modifying the existing bioreactors to biocalorimeters. Turker et al.[74] and Voisard et al. [75] made significant efforts on exploring the development of a large scale biocalorimeter to monitor and control bioprocess and concluded that it was indeed possible to apply realtime quantitative calorimetry at pilot to production scale for easy online monitoring and control. In order to achieve this it was necessary to precisely measure nonbiological heatflows viz. accumulation inside reactor, stirring power, exit gas and loss to the environment. A cascade control platform was suggested by Voisard et al. for a 300 L capacity pilot scale bioreactor to maintain the reactor temperature (Tr) at set point by regulating jacket side coolant temperature (Tj). The sensitivity of the measured heat signal was observed comparable to benchscale heat flux calorimeter. Schubert et al. in their recent contribution proposed an approach for conversion of bioreactor irrespective of its size to a biocalorimeter by integrating suitable calorimetric measurement principles [76]. With this approach they achieved a high sensitivity of heat signal 50 mW L1, stability of 0.2 mW L1 and response time of 12 min for a biocalorimeter modified from a standard bioreactor. The above characteristics of measured heat signal are comparable with heat signal measured from currently available expensive benchscale calorimeters. Further progress in development of highsensitive temperature sensors and data acquisition tools will eventually render wide open the doors for heat flux calorimetry to flourish at industrial scale as an inevitable analytical tool.
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7. FUTURE PROSPECTS OF BENCH SCALE CALORIMETRY Real time monitoring and control of bioprocess is currently gaining more importance in the biochemical engineering field. Due to its nonspecific, noninvasive and robust characteristics, bench scale heat flux calorimetry is regarded advantageous compared to existing biosensors in use. However, sensitivity and resolution of measured heat flow signal restrict its application on weakly exothermic bioprocess systems. Presence of electronic noise in measured heat flow signal is a major hurdle in employing heat flux calorimeters for weakly exothermic reactions. Although several research groups have made significant contributions on filtering noise from output heat flow signal (for e.g., changing electronic hardware interface of temperature controllers of the caloirmeter, design modifications in reactor setup and mathematical treatment of heat flow signal), complete elimination of noise interference is cumbersome. Stability and response time of heat flow signal are also observed to be prominent problems witnessed from heat flux calorimetric experimentations. Both manufacturers and research groups involved with benchscale heat flux calorimeter need to consider these technical issues and actively involve groups on design and development of an ultra sensitive BioRC1 meeting the requirements for future needs. This will enable the calorimetrists to extend the heat flux calorimetry application for monitoring animal cell bioprocess systems too. For the past few decades, extensive bench scale calorimetric investigations had been done on pure aerobic, fermentation and anaerobic bioprocess systems.
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But some other interesting bioprocess systems still need to be investigated viz., solid state fermentation, immobilized bioprocess systems and enzymatic biochemical reactions. This can exploit the real potential of heat flux calorimetry exhibiting its versatility of application to all process systems. Conversions of existing bioreactors to biocalorimeters may be an attractive option in contrast with employing custommade heatflux calorimeters (e.g., BioRC1) for investigating bioreaction systems it is also regarded as an economical option. Only a few groups (von Stockar et al, Turker et al. and Schubert et al.) made significant contributions to this kind of large scale biocalorimetry. Further advancementof research attempts on large scale biocalorimetry in the near future can substantiate the value of heat based sensor in bioprocess industry. Thanks to calorimetric research group’s contributions to biological wastewater treatment (WWT) systems, ecobiocalorimetry is becoming an emerging research area with a lot of scope for heat flux calorimetry. Green house gases discharge (especially from anaerobic WWT systems) is a serious threat for environment because of its contribution to global warming. Calorimetric investigation on monitoring exothermicity of anaerobic reactions may provide useful results to control and prevent green house gases discharge from treatment plants. Biofuels are regarded as future generation eco friendly fuels and commercialization of their bioprocesses is getting more importance. Inline monitoring of biological activity on biofuel conversion employing heat flux calorimetry can provide an enhanced yield. Simultaneous development of an ultra sensitive heat flux calorimetry at benchscale level and a robust large scale heatflux biocalorimeter design can extend applications to different bioprocess systems and certainly reveal its significance to biotechnological communities in both academia and industry.
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[62] Liu, J.S., Marison, I.W. and von Stockar, U. (), Microbial Growth by a Net Heat Uptake:A Calorimetric and Thermodynamic Study on Acetotrophic Methanogenesis by Methanosarcina barkeri Biotechnology and Bioengineering, 75. [63] von Stockar, U., Duboc, P., Menoud, L. and Marison, I.W. (1997), Online calorimetry as a technique for process monitoring and control in biotechnology. Thermochimica acta, 300, pp. 225236. [64] Sivaprakasam, S., Schuler, M., Hama, A., Hughes, K.M. and Marison, I.W. (2011), Biocalorimetry as a process analytical technology process analyser; robust inline monitoring and control of aerobic fedbatch cultures of crabtreenegative yeast cells, Journal of Thermal Analysis and Calorimetry, 104, pp. 7585. [65] Janssen, M., Pati˜no, R. and von Stockar, U. (2005) Application of benchscale biocalorimetry to photoautotrophic cultures. Thermochimica acta, 435, pp.18–27. [66] Janssen,M., Wijffels, R. and von Stockar, U. (2007) Biocalorimetric monitoring of photoautotrophic batch cultures. Thermochimica acta. 458, pp.54–64. [67] Dermoun, Z., Boussand, R., Cotton, D. and Belaich J.P. (1985), A new batch calorimeter for aerobic growth studies. Biotechnology and Bioengineering, 27, pp. 9961004. [68] Monk, P. and Wadso, I. (1968), A flow micro reaction calorimeter. Acta Chem. Scand, 22, pp. 18421852. [69] Beaubien, A. and Jolicoeur, C. (1985), Application of flow microcalorimetry to process control in biological treatment of industrial wastewater. Journal of Water Pollution Control Federation, 57, pp. 95100. [70] Ligthart, J. and Daverio, E. (2003), Application of calorimetric measurements for biokinetic characterization of nitrifying population in activated sludge. Water Research, 37, pp. 27232731. [71] Aulentaa, F., Bassanib, C., Ligthart, J., Majonea, M. and Tilche, A. (2002), Calorimetry: a tool for assessing microbial activity under aerobic and anoxic conditions. Water Research, 36, pp. 12971305. [72] Daverio, E., Spanjers, H., Bassani, C., Ligthart, J. and Nieman, H. (2003), Calorimetric Investigation of Anaerobic Digestion Biomass Adaptation and Temperature Effect, Biotechnology and Bioengineering, 82, pp.5. [73] Sivaprakasam S., Mahadevan S. and, Swaminathan G. (2008), Biocalorimetric and respirometric studies on biological treatment of tannery saline wastewater, Appl. Microbiology Biotechnology, 78, pp.249–255. [74] Turker, M. (2004), Development of biocalorimetry as a technique for process monitoring and control in technical scale fermentations. Thermochimica acta, 419, pp. 7381. [75] Voisard, D., Pugeaud, P., Kumar, A.R., Jenny, K., Jayaraman, K., Marison I.W. and von Stockar U. (2002), Development of a largescale biocalorimeter to monitor and control bioprocesses. Biotechnology and Bioengineering, 80, pp. 125137. [76] Schubert, T., Breuer, U., Harms, H. and Maskow, T. (2007), Calorimetric bioprocess monitoring by small modifications to a standard benchscale bioreactor, Journal of Biotechnology, 130, pp.24–31.
Heat Flux: Processes, Measurement Techniques and Applications : Processes, Measurement Techniques and Applications, Nova Science
In: Heat Flux Editors: G. Cirimele and M. D'Elia
ISBN 9781614708872 © 2012 Nova Science Publishers, Inc.
Chapter 3
ABILITY OF SOIL TO TRANSFER A LARGE AMOUNT OF HEAT UNDER REDUCED AIR PRESSURE Toshihiko Momose1 and Tatsuaki Kasubuchi2 1
Department of Hydrogeology, Bavarian Environment Agency, Hof, Germany 2 Faculty of Agriculture, Yamagata University, Tsuruoka, Japan
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ABSTRACT The thermal conductivity of a twophase soil composed of solid and gas phase decreases as the air pressure is reduced. The small thermal conductivity under reduced air pressure results from the decrease in the heat transfer in soil pore spaces. However, our previous study has found that the thermal conductivity of a threephase soil, such as an unsaturated soil, increases sharply under reduced air pressure. The maximum thermal conductivity obtained becomes close to the thermal conductivity of some metals such as manganese, mercury and stainless steel. This chapter introduces our techniques for measuring the thermal conductivity and the heat flux of soil as a function of air pressure, and describes the mechanism of heat transfer in soils under reduced air pressure.
1. INTRODUCTION Thermal conductivity of a twophase soil composed of solid and gas phase decreases as the gas pressure decreases. This phenomenon results from the decrease in heat transfer in soil pore spaces, which is caused by the mean free path of the gas molecules exceeding the distance between soil particles with air pressure reduction. Measurements of the thermal conductivity of twophase soils under reduced air pressure have been used for clarifying the mechanisms of heat transfer in twophase soil (e.g. Woodside and Messmer, 1961; Momose
Correspondence: Toshihiko Momose. Email: [email protected]
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and Kasubuchi, 2004). Over the last 50 years, experimental and theoretical studies have been made on the heat transfer in threephase soils, such as an unsaturated soil (Gurr et al., 1952; Philip and de Vries, 1957; Cary, 1965; Jackson et al., 1965; Jury and Letey, 1979; Cass et al., 1984; Campbell et al., 1984; Hiraiwa and Kasubuchi, 2000; Sakaguchi et al., 2007). These investigations have been conducted mainly at atmospheric pressure. The heat transfer phenomenon in a moist soil under reduced air pressure is little known. The relative humidity in the pore spaces of most soil is nearly equal to 1.0, even under fairly dry conditions at a water potential of 103 K kg1. In soil pores, the water vapour migrates from hot to cold side. When the water molecules evaporate at the hot side, they absorb a significant amount of latent heat; then, they condense releasing the latent heat at cold side. The vapour transfer from hot to cold side accompanies the latent heat, enhancing the heat transfer in soil and increasing the thermal conductivity. Under atmospheric pressure, the water vapour transfer must be suppressed by the air molecules such as nitrogen and oxygen, since nitrogen and oxygen molecules are heavier than the water vapour and occupy most of the pore spaces. As the air pressure decreases, the density of nitrogen and oxygen molecules decrease; the water vapour density is independent of the air pressure at a given water content and temperature. Therefore, the thermal conductivity of moist soils is supposed to increase under reduced air pressure. A difficulty of measurements of thermal conductivity of moist soils under reduced air pressure is to keep the soil water content constant. Our previous studies have overcome the difficulty, and have measured the thermal conductivity and the heat flux of soils over a wide range of water content under reduced air pressure. This chapter introduces our techniques for measuring the thermal conductivity and the heat flux of moist soils as a function of air pressure, and describes the mechanism of heat transfer in soils under reduced air pressure.
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2. MATERIALS AND METHODS 2.1. Soil Samples We used three typical Japanese soils: Ando soil, Red Yellow soil and Toyoura sand. The soil samples were air dried and sieved through 2mm mesh. Table 1 shows the physical properties of soils. Following procedures reported by Hiraiwa and Kasubuchi (2000), we prepared soil samples over a wide range of water content for the thermal conductivity measurements (Table 2). Table 1. Physical properties of the samples Table 1 Physical properties of the samples Volumetric solid content
Particle density
(m3 m3)
(g cm3)
Clay loam
0.35
2.44
Light clay
0.40
2.70
Sand
0.60
2.63
Soil name
Soil texture
Ando soil Red Yellow soil Toyoura sand
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Table 2 Volumetric water content prepared for the measurements of thermal conductivity and heat flux Volumetric water content prepared
Soil name
(m3 m3)
Red Yellow soil Toyoura sand
Light clay
0.40
2.70
Sand
0.60
2.63
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139
Table 2. Volumetric water content prepared for the measurements of thermal conductivity and heat flux. Table 2 Volumetric water content prepared for the measurements of thermal conductivity and heat flux Volumetric water content prepared
(m3 m3)
Soil name for transient heat probe method
for steadystate apparatus
Ando soil
0.10, 0.15, 0.20, 0.25, 0.30, 0.34, 0.40
0.15, 0.20, 0.30, 0.40
Red Yellow soil
0.11, 0.15, 0.20, 0.25, 0.30, 0.34, 0.40
Toyoura sand
0.00, 0.02, 0.05, 0.07, 0.10, 0.19, 0.28
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2.2. Measurements of Thermal Conductivity of Soil 2.2.1. Transient Heat Probe Method Figure 1 shows a soil container and a heat probe (a thermal conductivity sensor). The soil samples were packed into the container (6.0 cm in diameter and 5.5 cm tall) capped with hard plastic plate, which was bound with a silicon sealant. The centre of the top cap has a hole (9.0 mm in diameter) for insert of the heat probe. Installed in the top cap, a spiral stainlesssteel tube (1.5 mm outer diameter, 0.8 mm inner diameter and 1m long) had functions to equilibrate the air pressure of the container with that of outside and to suppress water loss by diffusion. We confirmed that the soil water content did not change throughout a series of measurements. The heat probe was made of a stainlesssteel tube with the following dimensions: 0.10 cm outer diameter, 0.05 cm inner diameter, and 5.0 cm long. The heating wire (enamelled constantan wire, 0.01 cm in diameter) and a thermocouple (enamelled copper and constantan wires, 0.01 cm in diameter) were placed into the tube, and the remaining space was filled with epoxy resin. The cold junction of the thermocouple was placed into a lead block (1.2 kg in mass) to keep a constant temperature under reduced air pressure at a given temperature. Figure 2 is a schematic diagram of the apparatus for measuring the thermal conductivity under reduced air pressure, developed in our previous paper (Momose and Kasubuchi, 2002). The soil container was put into a decompression box, which was exposed to the range of air pressure from atmospheric pressure to saturated vapour pressure at a given temperature. The air pressure was regulated by a vacuum pump through a pressure regulator (Model series 44, Moore Products Co.) with sensitivity of ±3 Pa. The air pressure was measured by a digital monometer (MT110 Yokogawa Co. Ltd). The decompression box was placed in a constant temperature chamber, controlled to ±0.1 ºC. The temperature in the chamber was set at 10, 25, 45, 65, 75 ºC. The thermal conductivity was measured by the improved twin heat probe method (Kasubuchi, 1992; Kasubuchi and Hasegawa, 1994). The voltage applied to the heater wire was regulated to raise the temperature of the heat probe by less than 1.0 K. The temperature changes of the heat probe were recorded every 1.5 seconds with a data acquisition system (GK88, ESD Co. Ltd) with a preamplifier (DCA902, Tokyo Riko Co. Ltd). This system has a resolution of the temperature measurement of 0.005 K. Fifty time – temperature data pairs were collected for each heating and cooling process.
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Figure 1 A soil container and a heat probe (thermal conductivity sensor)
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Toshihiko Momose and Tatsuaki Kasubuchi
Thermocouple
Power supply Spiral stainlesssteel tube
Hard plastic plate
Thermocouple junction
Lead block
Probe wall
Heater Figure 2 Schematic diagram of an apparatus for measuring the thermal conductivity of soils
Figure 1.under A soil container and a heat probe (thermal conductivity sensor). reduced air pressure Digital manometer
Vacuum pump Pressure regulator Preamplifier
Data acquisition system
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DC power supply Decompression box
Constant temperature chamber
Computer
Figure 2. Schematic diagram of an apparatus for measuring the thermal conductivity of soils under reduced air pressure.
The improved twin heat probe method requires the measurement of time – temperature data in a standard material with known thermal conductivity. As a standard material, we used water dissolving Nacarboxymethyl cellulose (3% by weight). Then, comparing the time –temperature data of the standard material to those of soils, we obtained the thermal conductivity of soils. According to the line heat source theory, the thermal conductivities for heating (heating) and cooling (cooling) processes are respectively expressed as
λ heating
q ln t , and 4π ΔTheating
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(1)
Ability of Soil to Transfer a Large Amount of Heat …
λ cooling
141
q lnt t t 1 , 4π ΔTcooling
(2)
where q is the heat strength unit the length of heat probe (W m1), t is elapsed time after the heating process starts (s), t1 is the period of the heating process (s), and T is the temperature change of the heat probe (K). For a standard material and soils, we apply the same voltage for heater wire; the q values for both measurements can be the same. For heating process, the ratio of the thermal conductivity of soil (heatingsoil) to that of standard material (standard) is expressed as
λ heatingsoil λ standardl
q ln t 4π ΔT heatingsoil
q ln t 4π ΔT heatingstandard
Therefore, the thermal conductivities of soil in heating process is expressed as
λ heatingsoil λ standard
ΔTheatingstandard ΔTheatingsoil
(4)
Similarly, the thermal conductivity of soil in cooling process is expressed as
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λ coolingsoil λ standard
ΔTcoolingstandard ΔTcoolingsoil
In computing the thermal conductivity, we deleted the first five points from the time – temperature data for each heating and cooling process; then we determined a gradient of the regression line between Tstandard and Tsoil for each process. If its correlation coefficient is less than 0.99, the data were disregarded. The thermal conductivity of soil is the average of those of heating and cooling processes.
2.2.2. SteadyState Method Figures 3 and 4 show a schematic diagram of an apparatus and a sample container for measuring the onedimensional steadystate heat flux under reduced air pressure, developed in our previous paper (Momose et al., 2008). The soil sample was packed into the acrylic container (50 * 50 * 100 mm) that was placed between the heat sink and the copper plate (5 mm thick) in which the thermistor was embedded. A thermomodule (6301/071/030, Ferrotec Corp.) was placed between the heat source and the copper plate, and was used as a Peltier device to control the electrical heat supplied to the sample and generate a fixed temperature difference between its ends over a wide range of thermal conductivity. The heat source and sink were hollow copper boxes (60 * 60 * 15 mm) of which temperature were respectively kept at 45 and 35 ºC by the water circulating through the bathes. The lateral surfaces of the acrylic container were covered by the stainless steel plates (2 mm thick), the ends of which were attached to the heat source and sink. The stainless steel plates were also covered with 30
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Toshihiko Momose and Tatsuaki Kasubuchi
mm thick Styrofoam. The sample container was placed in a decompression box, which was put into a constant temperature box set at 40 ºC. The air pressure was controlled by the vacuum pump and the pressure regulator, the system of which was the same to that used for the transient heat probe method. The spiral stainless steel tube was installed to the sample container to equalize the air pressure of the sample container with that of the decompression box and to reduce water loss by diffusion. To keep both temperatures of the heat source and the copper plate, electrical heat is supplied from the Peltier controller (DPC100, Ferrotec Corp.). The temperature at hot end of the sample can be assumed to be equal to that of the heat source. As the cold end of the sample touches the heat sink, the temperature difference between the ends of the sample is fixed regardless of the change in the thermal conductivity of the sample. Under steadystate, a liner temperature gradient is generated in the stainless steel plates, covering the sample container, independent of external temperature, and is similar to that in the sample. The stainless steel plates have the effect of minimizing the heat exchange through the lateral surfaces of the sample. The electrical heat generated from the thermomodule can be equivalent to the heat flux in sample under steadystate. Since the electrical resistance of the thermomodule is constant under a given temperature, the electrical heat generated from the thermomodule is a function of the voltage supplied to the thermomodule. The voltage supplied to the thermomodule was measured every 5 seconds and its mean voltage was recorded every 10 minutes with the data acquisition system (CR10X, Campbell Scientific Inc.). Using four standard samples with a wide range of thermal conductivity from 0.6 to 8.0 W m1 K1 (1% agar gel, Toyoura sand saturated with water, lead beads saturated with water, and bismuth), we obtained the relationship between the voltage supplied to the thermomodule and the heat flux in the samples, which was used to determine the heat flux in soil under reduced air pressure. Figure 3 Schematic diagram of an apparatus for measuring onedimensional Copyright © 2012. Nova Science Publishers, Incorporated. All rights reserved.
steadystate heat flux of soils under reduced air pressure Digital manometer
Vacuum pump Pressure regulator Peltier controller
Data acquisition system Spiral stainless steel tube
Constant temperature chamber （40℃） Decompression box Pump
Water bath (45℃)
Water bath (35℃)
Figure 3. Schematic diagram of an apparatus for measuring onedimensional steadystate heat flux of soils under reduced air pressure. Heat Flux: Processes, Measurement Techniques and Applications : Processes, Measurement Techniques and Applications, Nova Science
Figure 4 Sample container for measuring the onedimensional steadystate heat flux of soils
Ability of Soil to Transfer a Large Amount of Heat …
143
Thermomodule Thermistor
Heat source
Heat sink
Acrylic container Stainlesssteel plate Thermal insulation Copper plate
Figure 4. Sample container for measuring the onedimensional steadystate heat flux of soils.
3. RESULTS AND DISCUSSIONS
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3.1. Thermal Conductivity of ThreePhase Soil under Reduced Air Pressure 3.1.1. Effect of Reduced Air Pressure on Soil Thermal Conductivity Complete data for the thermal conductivity over a wide range of temperature can be obtained from our previous paper (Momose and Kasubuchi, 2002). In this chapter, we focus on the thermal conductivity data at 65 °C for Ando soil and Red Yellow soil and 45 °C for Toyoura sand, at the temperature conditions of which we obtained the maximum thermal conductivity. Figure 5 shows the relationship between the air pressure and the soil thermal conductivity. The thermal conductivity can be separated into two categories at a particular volumetric water content for each soil, which is 0.23, 0.23 and 0.06 m3 m3 for Ando soil, Red Yellow soil and Toyoura sand, respectively. Above theparticular water content, the thermal conductivity increases sharply with the air pressure reduction; below the value, it remains almost constant. The maximum thermal conductivity obtained are 4.8 W m1 K1 (at 65°C) for Ando soil, 13.0 W m1 K1 (at 65°C) for Red Yellow soil, and 8.0 W m1 K1 (at 45°C) for Toyoura sand, which are similar to the thermal conductivity of some metals, such as manganese (8 W m1 K1), mercury (7.8 W m1 K1) and stainless steel (14 W m1 K1) (National Astronomical Observatory, 2000). The large thermal conductivity results from the increase in heat transfer in soil pores. Heat transfer by air molecules in the pores of moist soil involves both conductive and latent heat transfer.
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144
Fig. 5 Thermal conductivity of threephase soil under reduced air pressure
Toshihiko Momose and Tatsuaki Kasubuchi
(a) Ando soil at 65 ºC
(b) Red Yellow soil at 65 ºC 14
0.10 4
0.15 0.20
3
0.25 0.30
2
0.11
12
Thermal conductivity / W m1 K1
Thermal conductivity / W m1 K1
5
0.34 0.40
1
0.15 0.20
10
0.25 8
0.30 0.35
6
0.40
4 2 0
0 0
20
40
60
80
100
0
120
20
40
60
80
100
120
Air pressure / kPa
Air pressure / kPa
(c) Toyoura sand at 45 ºC 9
Thermal conductivity / W m1 K1
8
0.00 0.02
7
0.05 6
0.07 0.10
5
0.19
4
0.28
3 2 1 0 0
20
40
60
80
100
120
Air pressure / kPa
Copyright © 2012. Nova Science Publishers, Incorporated. All rights reserved.
Figure 5. Thermal conductivity of threephase soil under reduced air pressure.
Nitrogen and oxygen molecules participate in the conductive heat transfer, and water molecules participate in the latent heat transfer. Nitrogen and oxygen molecules are heavier than water molecules and occupy most of the soil pores under atmospheric pressure; thus, nitrogen and oxygen molecules suppress the latent heat transfer. As the air pressure decreases the density of nitrogen and oxygen decreases; on the other hand, the density of water vapour is independent of air pressure at constant temperature and soil water content. The latent heat transfer increases with increase in the abundance ratio of the water molecule in soil pores. However, the latent heat effect is minimal below the particular water content.
3.1.2. Theory of Coupled Vapour and Heat Transfer in Soils The theory of couple of vapour and heat transfer in soils under temperature gradient has been described in literatures. (e.g. Cass et al., 1984; Miyazaki, 1993; Hiraiwa and Kasubuchi, 2000). Vapour flux according to Fick’s low is expressed by the equation: Jv =  D ∇
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(6)
Ability of Soil to Transfer a Large Amount of Heat …
145
where Jv is the vapour flux (g m2 s1), D is the diffusion coefficient of water vapour in air (m2 s1), and ∇is the gradient of water vapour density (g m4). For the vapour flux in soils, Equation (6) is written as: Jv =  h D ∇ where is the volumetric airfilled porosity (m3 m3), is a dimensionless tortuosity factor, and h is the relative humidity. In our experimental conditions, the relative humidity can be assumed to be 1. Over a constant soil water potential, Equation (7) is transformed as Jv =  h D (d/dT)∇
where T is temperature (ºC). Heat flux in soils (Js) is given by Js =  c∇ + H Jv,
(9)
where c is the thermal conductivity due to heat conduction (W m1 K1), H is the latent heat (J g1), and H Jv is the latent heat flux (W m2). Substitution of Equation (8) for Jv yields Js =  (c + v) ∇ where v is the thermal conductivity due to latent heat transfer (W m1 K1), and it is expressed as
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v = h D H (d/dT).
(11)
Although both values of and are less than 1, it has been pointed out that the vapour flux in soil is larger than the vapour flux estimated from Equation (8) (Gurr et al., 1952). To eliminate the discrepancy, Philip and de Vries (1957) introduce a mechanistic enhancement factor, : v = h D H (d/dT).
(12)
Alternatively, canbe replaced by a phenomenological enhancement factor, , introduced by Cary (1965): v = h D H (d/dT).
(13)
The D value depends on the air pressure. According to the kinetic theory of gases, the D value at a given pressure is expressed as D = 1/3 vL,
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(14)
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Toshihiko Momose and Tatsuaki Kasubuchi
where v is the mean molecular velocity (m s1)and L is the mean free path (m). The mean molecular velocity does not depend on the air pressure; the mean free path is inversely proportional to the air pressure. Therefore, the following relationship can be obtained: D = D0 P0/P,
(15)
where D0 is the diffusion coefficient of water vapour in air at atmospheric pressure, P0, and P is a given pressure. Substituting Equation (15) into Equation (13), we obtain the equation describing the dependence of air pressure on v: v = h D0 H (P0/P) (d/dT).
(16)
Substitution of Equation (16) into Equation (10) yields Js =  (c + h D0 H (P0/P) (d/dT)) ∇
Finally, we obtain the equation explaining the dependence of air pressure on the soil thermal conductivity, : = c + h D0 H (P0/P) (d/dT).
(18)
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The soil thermal conductivity should be linearly correlated to the reciprocal of air pressure. Required for the analysis, the parameters (D0, H, and d/dT) are determined by the following equations: D0 = 5 * 1010 * T2 + 1.9 * 107 * T + 2.34 * 105
(19)
H = 2.44*T + 2501.9
(20)
d/dT = 20.67 * 107 * T2 – 4.42 * 105 * T + 9.97 * 104 (kg m3 °C1) (21) Equation (19) is based on the vapour diffusion coefficient data from Miyazaki (1993). Equation (20) and (21) are obtained from National Astronomical Observatory (2000).
3.1.3. Analysis of the Thermal Conductivity Data Figure 6 shows the relationship between the reciprocal of air pressure and the thermal conductivity. Below the particular water content, the thermal conductivity data are expressed as liner functions of the reciprocal of the air pressure. Above the water content, the thermal conductivity data are linearly correlated to the reciprocal of air pressure in the range of the air pressure more than 40 kPa (corresponding to the reciprocal of air pressure of 0.025 kPa1) for Ando soil and Red Yellow soil and 16 kPa (0.064 kPa1) for Toyoura sand; however, the linear relation no longer holds below the air pressure. Focusing on the pressure range more than 40 kPa for Ando soil and Red Yellow soil and 16 kPa for Toyoura sand, we make the regression line for each water content, as shown in Figure 6. The intercept of the line is the
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Ability of Soil to Transfer a Large Amount of Heat …
147
value of c, and the slope of the line is the value of HD0P0hd/dT, according to Equation (18). Therefore, the value can be the ratio of the slope to HDhd/dT. Figure 7 shows the relationship between the volumetric water content and the value of . The values exponentially increase with the water content, and the maximum values are obtained at the volumetric water content of 0.30, 0.25 and 0.10 m3 m3 for Ando soil, Red Yellow soil and Toyoura sand, respectively. Then, the values decrease with the increase in the water content. The maximum values are 0.50, 0.85and and for Ando soil, Red Yellow Fig. 6 Relationship between the reciprocal of air pressure the0.70 thermal conductivity soil and Toyoura sand, respectively. (a) Ando soil at 65 ºC
(b) Red Yellow soil at 65 ºC 14
5
0.11
4
Thermal conductivity / W m1 K1
Thermal conductivity / W m1 K1
0.10 0.15 0.20 0.25 3
0.30 0.34
2
0.40
1
0 0.00
0.01
0.02
0.03
0.04
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12
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10 8
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0.01
0.02
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0.04
0.05
Reciprocal of air pressure / kPa1
Reciprocal of air pressure / kPa1
(c) Toyoura sand at 45 ºC
9 0.00 Thermal conductivity / W m1 K1
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8
0.02
7
0.05
6
0.07
5 4
0.10 0.19 0.28
3 2 1 0 0.00
0.02
0.04
0.06
0.08
0.10
0.12
Reciprocal of air pressure / kPa1
Figure 6. Relationship between the reciprocal of air pressure and the thermal conductivity.
The values are found to be less than 1, meaning that the vapour flux in soil is smaller than that in bulk air. Soil components, such as soil solids and liquid water, possibly prevent the vapour transfer. Absorbing the latent heat the water vapour migrates from hot to cold side, where the water vapour condenses releasing the latent heat. The heat released transfers to colder side by heat conduction through soil solids and liquid water, and induces the vapour transfer in another soil pore. The thermally induced water vapour transfer in soil repeats a series process: evaporation, latent heat transfer, condensation, heat conduction and evaporation. The
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processes of the phase change and the heat conduction prevent the vapour transfer, causing Fig. 7 Relationship between and volumetric water content the delay in heat transfer, which represents the values. 1.0
Ando soil Red Yellow soil Toyoura sand
0.8
0.6
0.4
0.2
0.0 0.0
0.1
0.2
0.3
0.4 3
0.5
3
Volumetric w ater content /m m
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Figure 7. Relationship between and volumetric water content.
The thermal conductivity data, shown in Figure 6, exceed the regression line in the range of air pressure less than 40 kPa for Ando soil and Red Yellow soil and 16 kPa for Toyoura sand. Taking into account for the temperature of the thermal conductivity measurements (65 °C for Ando soil and Red Yellow soil and 45 °C for Toyoura sand), we calculate the abundance ratio of water vapour in soil pores at the pressure, the value of which is equivalent to a ratio of the saturated vapour pressure to the air pressure, Pv / P. The values of Pv at 65 °C and 45 °C are 25 kPa and 9.58 kPa, respectively. We realize that the Pv / P values correspond to approximately 0.6. Above the Pv/P value of 0.6 and the particular water content, the vapour transfer is larger than that estimated from Equation (18), suggesting that another mode of vapour transfer, as well as vapour diffusion, takes place. Focusing on the Pv/P range more than 0.6, we calculate values, the difference between the measured thermal conductivity and the values estimated from Equation (18): c Dhd/dT
(22)
As shown in Figure 8, we find out that the values are linearly correlated to vapour mass flow in bulk air, Dhd/dT. The value is the mass flow factor, P / (PPv), introduced by Miyazaki (1993). Therefore, when the Pv / P is more than 0.6, the can be expressed as Dhd/dT,
(23)
where is a gradient of versus Dhd/dT. Note that the value is zero below the particular water content and in the range of Pv / P less than 0.6. Substituting Equation (23)
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into Equation (22), we obtain the following equation indicating that both of the vapour mass flow and the vapour diffusion participate in the latent heat transfer: Fig. 8 Relationship between HhD0(P0/P) (d/dT) and
= c + ( + )Dhd/dT. (a) Ando soil at 65 ºC
(b) Red Yellow soil at 65 ºC
1.0
5.0 0.25
0.25
0.30
0.30
0.8
4.0
0.34
0.35 0.40
0.40 0.6
/ W m1 K1
/ W m1 K1
(24)
0.4
0.2
3.0
2.0
1.0
0.0
0.0
0
5
10
15
20
25
30
0
HDh(d/dT) / W m1 K1
5
10
15
20
25
30
HDh(d/dT) / W m1 K1
(c) Toyoura sand at 45 ºC 1.0 0.07 0.10 0.8
0.19
/ W m1 K1
0.28 0.6
0.4
0.2
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0.0 0
5
10
15
20
25
30
HDh(d/dT) / W m1 K1
Figure 8. Relationship between HhD0(P0/P) (d/dT) and .
Figure 9 shows the relationship between the volumetric water content and the value. For Ando soil and Red Yellow soil, the maximum values are observed at the volumetric water content of 0.30 and 0.25 m3 m3, respectively. These values correspond to the water content giving the maximum value. For Toyoura sand, we cannot see a clear peak of the value. The values for three soils are less than 1, meaning that the vapour mass flow in soil is smaller than that in bulk air. Also, the values are even smaller than the values. Resulting from the vapour mass flow as well as the vapour diffusion, the latent heat transfer in soil is considered to repeat the following series process: evaporation, latent heat transfer, condensation, heat conduction and evaporation. The processes of the phase change and the heat conduction cause the delay in the vapour mass flow, representing that the values are smaller than 1. Since the vapour mass flow transfers more latent heat than the vapour diffusion does, the processes of the phase change and the heat conduction can be bigger thermal resistance for the vapour mass flow than the vapour diffusion. As a result, the values can be smaller than the values. Although the vapour mass flow in soil is smaller
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Toshihiko Momose and Tatsuaki Kasubuchi
than that in bulk air, it causes the large heat transfer in soil under reduced air pressure. However, when theFig.soil water content than particular water content, the vapour 9 Relationship between is andlower volumetric waterthe content mass flow does not take place. 0.25 Ando soil Red Yellow soil Toyoura sand
0.20
0.15
0.10
0.05
0.00 0.0
0.1
0.2
0.3
0.4
0.5
Volumetric w ater content / m3 m3
3.1.4. A Factor Determining the Particular Water Content Figure 10 shows the relationship between the volumetric water content and the hydraulic diffusivity. The particular water content corresponds to a hydraulic diffusivity of the order of 108 m2 s1, suggesting that the liquid water mobility is a primary factor causing the vapour Fig. 10 Relationship betweenheat hydraulic volumetric water content mass flow. Absorbing the latent thediffusivity water and vapour migrates from hot to cold side, where the water vapour condenses releasing the latent heat. The condensed water might return to hot side above the particular water content. If the above thought is correct, the large latent heat transfer is able to maintain. The mechanism of the large latent heat transfer can be clarified by measuring the soil heat flux under steadystate and reduced air pressure. 105
Hydraulic diffusivity / m2 s1
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Figure 9. Relationship between and volumetric water content.
Ando soil Red yellow soil Toyoura sand
6
10
107
108
109
1010 0
0.05
0.1
0.15
0.2
0.25
0.3
Volumetric water content / m3 m3
Figure 10. Relationship between hydraulic diffusivity and volumetric water content. Heat Flux: Processes, Measurement Techniques and Applications : Processes, Measurement Techniques and Applications, Nova Science
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3.2. SteadyState Heat Flux of Soil under Reduced Air Pressure 3.2.1. Heat Transfer Phenomena in Soil Figure 11 shows the relationship between the air pressure and the steadystate heat flux of 11 Relationship between air pressure and soil fluxflux data shown in Figure 3 are Ando soil overFig. different volumetric water content. Allheat heat confirmed to keep constant for 11 hours, the period of the experiment (Sakaguchi et al., 2009). 200
160
0.15
Heat flux / W m2
0.20 0.30
120
0.40 80
40
0 0
20
40
60
80
100
120
Air pressure / kPa
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Figure 11. Relationship between air pressure and soil heat flux.
The results measured by the steadystate apparatus are consistent with those obtained by the transient heatprobe method: the heat transfer phenomena fall into two categories at the particular water content. Below the volumetric water content of 0.20 m3 m3, the heat flux is almost constant; above the volumetric water content of 0.30 m3 m3, it increases sharply as the air pressure decreases.
3.2.2. Analysis of the Heat Flux Data According to Equation (17), the soil heat flux should be expressed as a linear function of the reciprocal of air pressure. Figure 12 shows the relationship between the reciprocal of air pressure and the soil heat flux. Below the volumetric water content of 0.20 m3 m3, the heat flux data are linearly correlated to the reciprocal of air pressure. Above the volumetric water content of 0.30 m3m3, the linear relation is observed in the range of air pressure more than 15 kPa (corresponding to the reciprocal of air pressure of 0.07 kPa1), and the regression lines are made in the graph; however, at 10 kPa the measured heat flux data exceed the regression line. The difference between the measured heat flux data and the regression line can be the latent heat flux due to the vapour mass flow. The large vapour transfer is observed by the steadystate apparatus, proving the continuity of the large vapour transfer from hot to cold side. In soil pores where the large amount of heat transfers, the water is circulating. The mechanism of large vapour transfer in soil is similar to the principle of heat pipe operation.
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Fig. 12
152
Relationship between qv and soil heat flux
Toshihiko Momose and Tatsuaki Kasubuchi 200
0.15 0.20
Heat flux / W m2
160
0.30 0.40
120
80
40
0 0.00
0.02
0.04
0.06
0.08
0.10
0.12
Reciprocal of air pressure / kPa1
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Figure 12. Relationship between qv and soil heat flux.
3.2.3. Heat Pipe Phenomenon in Soil The principle of heat pipe operation is comprehensively described in literatures in engineering field. A common heat pipe is a vacuum tight chamber of cylindrical shape with an inner wall of wick structure and a working fluid. Heat pipes are divided axially into three sections: evaporator, transport and condenser. At the evaporator section, the working fluid absorbs the heat from the outside, and changes its phase to vapour, forming the difference in vapour pressure between evaporator and condenser sections. Due the pressure gradient, the vapour migrates through the transport section to the condenser section, where it condenses releasing the latent heat to outside of the heat pipe. The condensed water returns back to hot side through the inner wall of wick structure. By the circulation of the working fluid, the heat pipe can continuously transfer a large amount of heat. The working principle of the heat pipe can be applied to the mechanism of the large vapour transfer in soil. (1) Absorbing the latent heat the water vapour migrates from hot to cold side of soil pores. (2) At the cold side, the water vapour condenses releasing the latent heat. (3) The condensed water returns to the hot side, and maintain the amount of water there. The above cycle is starting all over again. Each soil pore functions as a microheat pipe. Released by the condensation at the cold side, the heat transfers to colder side by heat conduction through soil solids and liquid water, and triggers the vapour transfer in another pore. As a result, the microheat pipes link each other, and a series heat pipe phenomenon takes place throughout soil.
CONCLUSION We have measured the thermal conductivity of soils over a wide range of water content under reduced air pressure using a transient heat probe method. The heat transfer phenomena are separated into two categories at a particular water content for each soil corresponding to the hydraulic diffusivity of the order of 108 m2 s1. Below the particular water content, the thermal conductivity is almost constant over the air pressure range; above the water content, the thermal conductivity increases sharply as the air pressure decreases. The maximum
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thermal conductivity values obtained are similar to the thermal conductivity of some metals, such as manganese, mercury and stainless steel. The large thermal conductivity results from the latent heat transfer due to the vapour mass flow. Under atmospheric pressure, nitrogen and oxygen molecules, heavier than water vapour molecule, occupy most of soil pores and suppress the latent heat transfer. As the air pressure decreases, the density of nitrogen and oxygen molecules decrease; the density of water vapour is independent of air pressure at constant temperature and soil water content. The abundance ratio of water vapour in soil pores increases with air pressure reduction. When the abundance ratio of water vapour reaches 0.6, the vapour mass flow takes place resulting in the sharp increase in the latent heat transfer. This large latent heat transfer is an intrinsic phenomenon in soil, which was confirmed by the results from the steadystate apparatus. Namely, the large latent heat transfer maintains. Our experimental results have revealed a new mechanism of the heat transfer in soil: Soil functions as a heat pipe above the particular water content. Absorbing the latent heat the water vapour migrates from hot to cold side of soil pores, where the water vapour condenses and releases the latent heat. The condensed water returns to the hot side, maintaining the amount of water there. Each soil pore functions as a microheat pipe. Released by the condensation, the heat transfers to colder side by heat conduction through soil solids and liquid water, and triggers the vapour transfer in another pore. As a result, the microheat pipes connect each other, and a series heat pipe phenomenon takes place throughout soil. The heat pipe effect contributes to the ability of soil to transfer the large amount of heat under reduced air pressure.
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REFERENCES Campbell, G.S., Jungbauer, J.D. Jr., Bidlake, W.R. and Hungerford, R.D. (1994) Predicting the effect of temperature on soil thermal conductivity. Soil Science, 158, 307313. Cass, A., Campbell, G.S. and Jones, T.L. (1984) Enhancement of thermal water vapor diffusion in soil. Soil Science Society of America Journal, 48, 2532. Cary, J.W. (1965) Soil heat transducers and water vapor flow. Soil Science, 100, 168175. Gurr, C.G., Marshall, T.J. and Hutton, J.T. (1952) Movement of water in soil due to a temperature gradient. Soil Science, 74, 335345. Hiraiwa, Y. and Kasubuchi, T. (2000) Temperature dependence of thermal conductivity of soil over a wide range of temperature (575 ˚C). European Journal of Soil Science, 51, 211218. Jackson, R.D., Rose, D.A. and Penman, H.L. (1965) Circulation of water in soil under a temperature gradient. Nature, 205, 314316. Jury, W.A. and Letey, J. (1979) Water vapor movement in soil: reconciliation of theory and experiment. Soil Science Society of America Journal, 43, 823827. Kasubuchi, T. (1992) Development of insitu soil water measurement by heatprobe method. Japan Agricultural Research Quarterly, 26, 178181. Kasubuchi, T. and Hasegawa, S. (1994) Measurement of spatial average of field soil water content by the long heat probe method. Soil Science and Plant Nutrition, 40, 565571.
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Miyazaki, T. In: Water Flow in Soils; Miyazaki, T., Ed.; Marcel Dekker: New York, 1993, pp 169196. National Astronomical Observatory 2000. In: Rikanenpyo (Chronological Scientific Tables); Maruzen Co., Tokyo, 2000, pp. 488489. Philip, J.R. and de Vries, D.A. (1957) Moisture movement in porous materials under temperature. Transactions of the American Geophysical Union, 38, 222232. Sakaguchi, I., Momose, T., Mochizuki, H. and Kasubuchi, T. (2009) Heat pipe phenomenon in soil under reduced air pressure. European Journal of Soil Science, 60, 110115. Sakaguchi, I., Momose, T. and Kasubuchi, T. (2007) Decreases in thermal conductivity with increasing temperature in nearly dry sandy soil. European Journal of Soil Science, 58, 9297. Momose, T. and Kasubuchi, T. (2002) Effect of reduced air pressure on soil thermal conductivity over a wide range of water content and temperature. European Journal of Soil Science, 53, 599606. Momose, T. and Kasubuchi, T (2004) Estimation of the thermal separation of soil particles from the thermal conductivity under reduced air pressure. European Journal of Soil Science, 55, 193199. Momose, T., Sakaguchi, I. and Kasubuchi, T. (2008) Development of an apparatus for measuring onedimensional steadystate heat flux of soil under reduced air pressure. European Journal of Soil Science, 59, 982989. Woodside, W. and Messmer, J.H. (1961) Thermal conductivity of porous media I. Unconsolidated sands. Journal of Applied Physics, 32, 16881699.
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In: Heat Flux Editors: G. Cirimele and M. D'Elia
ISBN 9781614708872 © 2012 Nova Science Publishers, Inc.
Chapter 4
CONTRIBUTIONS TO THE NET HEAT FLUX IN THE MEDITERRANEAN SEA: SEASONAL AND INTERANNUAL VARIATIONS F. CriadoAldeanueva, J. SotoNavarro, J. GarcíaLafuente, C. Naranjo, C. Calero and E. Bruque Physical Oceanography Group, Department of Applied Physics, University of Málaga, Spain
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ABSTRACT Several NCEP climatological datasets have been combined to analyse the seasonal and interannual variations of the heat budget in the Mediterranean Sea. The seasonal cycle of the net heat is positive (toward the ocean) between March and September with maximum in June and negative the rest of the year with minimum in December. Although subject to inherent uncertainty, we obtain a practically neutral budget of 0.7 Wm2 in a yearly basis. The net heat budget is positive for the western Mediterranean (~12 Wm2) and negative for the eastern Mediterranean (~ 6.4 Wm2) mainly due to the high latent heat losses of this basin. Combining the climatological values with in situ measurements in Espartel sill (Strait of Gibraltar), a heat advection Qa = 3.2±1.5 Wm2 through the Strait of Gibraltar has been obtained that, combined with the longterm averaged surface heat flux, implies that the net heat content of the Mediterranean Sea would have increased in the last decades.
1. INTRODUCTION The Mediterranean Sea (Figure 1), a semienclosed basin that extends over 3000 km in longitude and over 1500 km in latitude with an area of 2.5∙1012 m2, communicates with the Atlantic Ocean through the Strait of Gibraltar and with the Black Sea through the Turkish Bosphorus and Dardanelles Straits. The Sicily Channel separates the western and eastern
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Mediterranean basins. Evaporative losses (E) are not balanced by precipitation (P) and river runoff (R) and an Atlantic inflow through the Strait of Gibraltar is necessary to balance the freshwater and salt budgets. The net heat budget consists of two radiation components (solar shortwave radiation absorbed by the sea and longwave radiation emitted by the sea) and two turbulent contributions (latent and sensible heat fluxes). In the longterm, vertical heat fluxes integrated over the basin must be balanced by heat transport through the Strait of Gibraltar. Macdonald et al. (1994), using in situ current and temperature observations estimated an annual average heat transport from the Atlantic to the Mediterranean of 5.2±1.3 Wm2. Other authors have also obtained the longterm heat flux through Gibraltar from estimates of the volume transport and the temperatures of the inflow and outflow. Results range from 8.5 Wm2 (Béthoux, 1979) to 5 Wm2 (Bunker et al., 1982). Since the uncertainty of these results is rather low, they can be used as a reference for the evaluation of the surface heat flux budget. Several studies (Bunker et al., 1982; Garrett et al., 1993; Schiano et al., 1993; Gilman and Garrett, 1994) have compared long term averages of vertical heat fluxes with the heat transport through the Strait of Gibraltar obtaining discrepancies of up to 30 Wm2. The reasons given for the disagreement are the different periods covered and the different bulk formula parameterisations or the wind forcing fields (Ruti et al., 2008). More recently, Ruiz et al. (2008) have examined 44 years (19582001) of HIPOCAS model data to report a value of 1 Wm2 for the vertical heat flux (heat loss from the ocean). They attribute the difference with respect to the heat gain through the Strait to an increase in the net heat content of the Mediterranean Sea during the last decades.
Figure 1. A) Map of Mediterranean Sea. The main basins and subbasins are indicated. B) Zoom of the Strait of Gibraltar region. The rectangle indicates the area selected as representative for Espartel (ES).
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Semienclosed basins such as the Mediterranean are suitable for the characterisation of heat fluxes since they make a budget closure feasible. In this work we combine several datasets to analyse the seasonal and interannual variations of the components of heat budgets and compare the long term means with direct measurements in the Strait of Gibraltar. The work is organised as follows: section 2 describes the data and methodology; in section 3 the main results are presented and discussed for the heat fluxes. Finally, section 4 summarises the conclusions.
2. DATA
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Monthly means from January 1948 to February 2009 of surface heat fluxes have been retrieved from the National Center for Enviromental PredictionNational Center of Atmospheric Research (NCEPNCAR) reanalysis project (NCEP hereinafter, Kalnay et al., 1996), which is run at T62 spectral resolution (approximately a grid size of 1.9ºx1.9º) with 28 sigma levels. Seasonal means have been computed by averaging JFM (winter), AMJ (spring), JAS (summer) and OND (autumn) monthly data. In situ measurements of the outflow through the Strait of Gibraltar have been collected in the frame of the INGRES 12 projects. Data from a CT probe placed over Espartel sill, at 35º 51.70N, 5º 58.60W and 5 m above the seafloor between September 2004 and December 2009 have been used in this work to characterise its temperature. MEDATLAS database provided historical ConductivityTemperatureDepth (CTD) profiles over Espartel in order to determine the inflow properties. The region within 35º 48.6’N – 35º 53.9’N / 05º 56.7’W – 06º 00.8’W (see Figure 1B) has been considered to be representative for the Espartel area. 48 CTD profiles spanning all seasons have been identified, most of them from the field work carried out during the Gibraltar Experiment (1986).
3. RESULTS AND DISCUSSION 3.1. Surface Heat Fluxes in the Mediterranean Sea 3.1.1. Spatial Climatologies and Seasonal Cycle Figure 2 displays the seasonal climatology of the different components of the net heat budget. Sensible heat flux, Qh (panel A) concentrate higher losses during autumn and winter with maxima above 60 Wm2 in the Aegean and Adriatic and slightly lower values in the Gulf of Lions and the Levantine subbasin (~50 Wm2). Elsewhere, the spatial distribution is rather uniform in these seasons with losses of some 20 Wm2. Heat gains up to 20 Wm2 occur in spring and summer in the Aegean, the Levantine subbasin and some areas of the northAfrican coasts. Latent heat flux, Qe (panel B) is larger in the eastern basin with losses up to 160 Wm2 in the Levantine area in autumn and winter. In the western Mediterranean, the highest losses are located in the Gulf of Lions and the Balearic subbasin (~130 Wm2). Minimum fluxes take place in spring with a more uniform spatial distribution and lower values in the Adriatic and the westernmost area (~20 Wm2). The solar shortwave net radiation, Qs (panel C) depicts a northsouth, westeast gradient in all seasons, with maxima
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in spring in the Levantine subbasin (more than 250 Wm2) and minima in autumn in the western European coasts (~50 Wm2). The longwave net radiation Qb (panel D) is rather independent of seasonal variations. Higher losses of some 90 Wm2 concentrate in summer in the Aegean Sea and Levantine subbasin whereas lower values correspond to the Balearic and Tyrrhenian subbasins in spring (~60 Wm2). The combination of these four contributions produces a marked seasonaldependent net heat flux Qn (not shown), with losses in autumn and winter and gains in spring and summer.
Figure 2. Continued on next page.
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Contributions to the Net Heat Flux in the Mediterranean Sea
159
Figure 2. Seasonal climatology of the four components of the net heat budget in the Mediterranean for 19482009 (Wm2, positive toward ocean): sensible (panel A), latent (panel B), solar shortwave (panel C) and terrestrial longwave (panel D). In all panels: winter (topleft), spring (topright), summer (bottomleft) and autumn (bottomright).
Higher losses are observed in the Levantine subbasin, the Aegean, the northern Adriatic and the Gulf of Lions (>150 Wm2) in autumn, that favours the formation of intermediate and deep waters in these areas (Tziperman and Speer 1994; Candela 2001; Schroeder 2009). Mean values for each contribution in the eastern and western basins are presented for all seasons in Table 1. The Mediterraneanaveraged climatological seasonal cycle for each component is presented in Figure 3.
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F. CriadoAldeanueva, J. SotoNavarro, J. GarcíaLafuente et al.
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Figure 3. Mediterraneanaveraged climatological seasonal cycle for sensible heat (Qh, grey dashed line), latent heat (Qe, grey solid line), shortwave (Qs, black dasheddotted line) and longwave (Qb, black dashed line) contributions for the period 19482009. Net heat seasonal cycle is also presented (Qn, black solid line). Bars are the standard deviation.
For the sensible heat flux Qh, the values are negative all year round, with a range of variation of 34 Wm2, a maximum of 2 Wm2 in June and a minimum of 36 Wm2 in December. The latent heat flux Qe, is minimum (125 Wm2) in November and maximum (50 Wm2) in May. The seasonal cycle of the shortwave radiation Qs, positive all the year, has a range of variation of 196 Wm2, a maximum of 281 Wm2 in June and a minimum of 85 Wm2 in December. Finally, the net longwave radiation Qb does not exhibit a clear seasonal cycle but a rather uniform value between 75 and 80 Wm2. These results are in reasonable good agreement with those obtained by Matsoukas et al (2005), who derive the radiative components by a radiation transfer model instead of bulk formulae. The seasonal cycle of the net heat shows positive values (heat gain by the ocean) between March and September with maximum in June (143 Wm2) and negative values during the rest of the year. It shows a minimum in December (152 Wm2) and a range of variation of 295 Wm2, which is slightly less than the 330 Wm2 obtained by Ruiz et al. (2008) and close to the lower limit of the interval reported by Garrett et al. (1993), 280 Wm2360 Wm2. The obtained phase is in agreement with both works and slightly different from that obtained by Matsoukas et al (2005), who situate the maximum in May. Solar radiation and latent heat are the major contributions to the net heat flux. The heat flux Qn is the time derivative of the heat content H, Qn = dH/dt, responsible of the thermosteric anomaly. If we assume a harmonic function for the annual cycle of Qn, then H will also have a harmonic shape but delayed π/2 (3 months) and therefore the thermosteric sea level cycle is expected to peak in September, in agreement with previous works (FenoglioMarc et al., 2006; García et al., 2006; CriadoAldeanueva et al., 2008).
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Table 1. Mediterranean (Med) long term mean heat fluxes contributions (Wm2). Values for the western (Wm) and eastern (Em) basins are shown for each season Med
Mean Wm Em
Med
Winter Wm Em
Med
Spring Wm
Em
Med
Summer Wm Em
Med
Autumn Em Wm
Qh
15.1
13.3
16.2
27.6
22.3
30.8
1.5
2.0
1.2
3.8
3.7
3.8
27.6
25.0
29.1
Qe
93.5
78.4
103.4
99.4
82.3
110.5
57.9
48.5
64.0
93.9
Qs
186.3
176.7
192.3
133.9
125.3
1390.2
254.6
246.2
259.8
245.0
75.9
105.6
123.1
107.2
133.5
233.7
252.0
112.3
101.9
Qb
76.9
73.1
79.3
76.9
74.1
78.6
75.9
71.5
78.5
118.8
78.8
73.9
81.8
76.2
72.9
Qn
0.73
11.7
6.4
70.0
53.5
80.7
120.0
120.0
120.0
78.2
69.6
80.1
61.1
110.0
100.0
120.0
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Table 2. Mediterranean long term mean heat budget (Wm2) estimated by different authors. The periods to which the estimates refer are also indicated Authors Bethoux (1979) Bunker et al. (1982) (1) Bunker et al. (1982) (2) May (1986) Garrett et al. (1993) Gilman and Garrett (1994) Castellari et al. (1998) Matsoukas et al. (2005) Ruiz et al. (2008) This work
Qh 13 13 11 11 7 7 13 11 8 15
Qe 120 101 130 112 99 99 122 90 88 93
Qs 195 202 202 193 202 183 202 186 168 186
Qb 68 68 68 68 67 77 78 63 73 77
Qn 6 20 7 2 29 0 11 22 1 1
Period Not specified 19411972 19411972 19451984 19461988 19461988 19801988 19842000 19582001 19482009
st.com/lib/multco/detail.action?docID=3020499.
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3.1.2. BasinAveraged Annual Means and LongTerm Fluctuations Figure 4A displays the yearly, Mediterraneanaveraged, time series of the different contributions and the net heat flux. Solar shortwave radiation is the only positive contribution with a mean value of ~186±4 Wm2. The other contributions are negative with mean values about 93±6 Wm2, 77±2 Wm2 and 15±3 Wm2 for latent, longwave and sensible heat, respectively. As a result, we obtain a nearly neutral budget of 0.7 Wm2. The mean values have also been computed for each basin (Table 1): the net heat budget is positive (~12 Wm2) for the western Mediterranean and negative for the eastern Mediterranean (~ 6.4 Wm2) due to the high latent heat losses (up to 100 Wm2). The longterm averages of each component are compared with previous estimates in Table 2. The value for shortwave radiation is the same as the one obtained by Matsoukas et al (2005) from a radiation transfer model, a value lower than most previous estimations except for those of Gilman and Garrett (1994) and Ruiz et al. (2008) who computed a contribution 10% lower from 19582001 HIPOCAS reanalysis data, probably due to a different parameterisation scheme. The latent heat flux is also lower than previous estimations and similar to that of Matsoukas et al. (2005) and Ruiz et al. (2008), which is thought to be rather accurate due to the higher spatial resolution of HIPOCAS dataset. The value for longwave radiation is the same as the one obtained by Gilman and Garrett (1994) and Castellari et al. (1998) and is close to that of Ruiz et al. (2008). The computed sensible heat flux is greater than all previous estimations, although it is not far from values reported by Bethoux (1979), Bunker et al. (1982) and Castellari et al. (1998). The net heat flux is in the range of previous studies, especially close to those of May (1986), Gilman and Garrett (1994) and Ruiz et al. (2008). Although there is no significant trend in the series, Figure 4B reveals three different periods in the heat flux anomalies: from early 50s to mid 60s, a negative trend of 1.6±0.6 Wm2y1 is observed. Trend changes to positive (1.1±0.3 Wm2y1) until late 80s when it changes sign again (0.9±0.6 Wm2y1). Maximum heat gain of about 20 Wm2 is observed in 1989 and maximum losses of the same order in 1963 and 2005. Since fluctuations in the net budget do not appear to be random, discrepancies with previous estimations could be related to the different periods analysed. It is interesting to remark that fluctuations in the net heat flux closely follow those of the latent heat (trends of 1.1±0.5 Wm2y1, 0.7±0.2 Wm2y1 and 0.7±0.4 Wm2y1 are observed for the same periods referred above), suggesting that this contribution is the main source of interannual variability. The visual inspection of Figure 4B also suggests a 40year period multidecadal oscillation of 11±2 Wm2 and 7.5±1.4 Wm2 amplitude for net and latent heat fluxes, respectively, probably related to longterm atmospheric forcing. However, longterm variability is the less reliable aspect of reanalyses datasets so some caution is necessary here. Although general good agreement is found with the results of Mariotti (2010) based on different datasets, this author reports an increase in the recent period compared to the 1960s (i.e. a trend superposed to the decadal variability) that sheds doubts on the 40year oscillation. Longer time series will be of great help to clarify this issue.
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Figure 4. A) Yearly Mediterraneanaveraged time series of sensible heat (Qh, grey dashed line), latent heat (Qe, grey solid line), shortwave (Q s, black dasheddotted line), longwave (Qb, black dashed line) and net heat flux (Qn, black solid line) for the period 19482009. B) Latent (grey) and net (black) heat anomalies (yearly means, dotted; 5year running means, solid). A multidecadal oscillation is clearly suggested.
3.2. Budgets and Exchange through the Strait of Gibraltar We now compute heat advection through the Strait of Gibraltar from in situ measurements and historical MEDATLAS CTD profiles (see section 2 for details). A mean temperature of To=13.25 ± 0.07 ºC has been obtained for the outflow from the CT probe. A spatially (within 35º 48.6’N – 35º 53.9’N / 05º 56.7’W – 06º 00.8’W, see Figure 1B) and depthaveraged temperature above the mean depth of the interface (186 m, SanchezRoman et al., 2009) of Ti=15.6 ± 1.1 ºC has been obtained for the inflow which implies a temperature difference of 2.4ºC. With these values and our mean estimation of 0.82 Sv for the inflow (CriadoAldeanueva et al., 2011), a result of Qa = 3.2 ± 1.5 Wm2 is obtained for the heat advection. Although the value of 186 m for the mean depth of the interface is a welldocumented choice (SánchezRomán et al., 2009), the result for the heat advection is fairly robust and only small variations (less than 10%) have been observed for a wide range (150200 m) of the mean interface.
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This value is lower than historical reports that range from 8.5 Wm2 (Béthoux, 1979) to 5 Wm2 (Bunker et al., 1982) but is thought to be realistic since it comes from reliable datasets. The discrepancies with other results are probably due to a previous overestimation of the inflow (usually set to values above 1 Sv) since the temperature difference is rather similar. When combined with the longterm averaged surface net heat flux, this implies that the net heat content of the Mediterranean Sea would have increased in the last decades. This is compatible with the increment of deep water temperature reported by different authors (Rohling and Bryden, 1992; Bethoux and Gentili, 1999; LópezJurado et al., 2005; Font et al., 2007) and also with a positive thermosteric sea level trend (CriadoAldeanueva et al., 2008). In any case, considering the uncertainty inherent to the estimation of surface heat fluxes, this result must be considered with caution.
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4. SUMMARY AND CONCLUDING REMARKS We have used climatological datasets to analyse the seasonal and interannual variations of the components of heat budget and compare the long term means with direct measurements in the Strait of Gibraltar. The seasonal cycle of the net heat shows positive values (toward the ocean) between March and September with a maximum in June and negative values the rest of the year with a minimum in December. On a yearly basis, we obtain a nearly neutral budget of 0.7 Wm2. The net heat budget is positive (~12 Wm2) for the western Mediterranean and negative for the eastern Mediterranean (~ 6.4 Wm2) mainly due to the high latent heat losses of this basin (up to 100 Wm2). Reanalyses are useful for a comprehensive description of climate and related water/energy cycles, especially for describing climatological characteristics. However there is no constrain on the closure of the water and energy budgets at the level of the Mediterranen Sea, so there are uncertainties associated to results based on these products. Longterm variability is the less reliable aspect of reanalyses datasets as variability on these timescales may be affected by artifices (e.g. deriving from nonstationary data inputs). Despite of these caveats, the good agreement with other previous results in the literature makes them reliable for the estimation of the heat exchange through the Strait of Gibraltar. A heat advection of Qa = 3.2 ± 1.5 Wm2 through the Strait of Gibraltar is obtained. This value, although lower than historical, is thought to be realistic, the discrepancies with other estimates being attributable to a previous overestimation of the inflow. This heat advection, along with the longterm averaged surface net heat flux, implies that the net heat content of the Mediterranean Sea would have increased in the last decades. This result, although subject to the uncertainty of the surface heat fluxes estimation, is compatible with the findings of Rohling and Bryden (1992), Bethoux and Gentili (1999), LópezJurado et al. (2005) and Font et al. (2007) who report an increment of deep water temperature and also with the positive thermosteric sea level trend observed by CriadoAldeanueva et al. (2008).
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ACKNOWLEDGMENTS This work has been carried out in the frame of the P07RNM02938 Junta de Andalucia Spanishfunded project. JSN acknowledges a postgraduate fellowship from Conserjería de Innovación Ciencia y Empresa, Junta de Andalucía, Spain. NCEP data have been provided by the NOAA/OAR/ESRL PSD, Boulder, Colorado, USA, from their website at http://www.esrl.noaa.gov/psd/.
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REFERENCES Bethoux, J.P, 1979. Budgets of the Mediterranean Sea: their dependence on the local climate and on the characteristics of Atlantic waters. Ocean. Acta 2, 157163. Bethoux, J.P. and Gentili B., 1999. Functioning of the Mediterranean Sea: Past and Present Changes realed to freshwater input and climatic changes. J. Mar. Syst. (20), 3347. Bunker, A.F., Charnock, H., Goldsmith, R.A., 1982. A note on the heat balance of the Mediterranean and Red Seas. J. Mar. Res. 40, 7384, Suppl. Candela, J., 2001. Mediterranean water and global circulation, in Ocean Circulation and Climate, edited by G. Siedler, J. Church, and J. Gould, pp. 419–429, Academic Press, San Diego, Ca. Castellari, S., Pinardi, N., Leaman K., 1998. A model study of airsea interactions in the Mediterranean Sea. J. Mar. Sys. 18, 89114. CriadoAldeanueva F., Del Río Vera J., GarcíaLafuente J., 2008. Steric and mass induced sea level trends from 14 years altimetry data. Global and Planetary Change, (60), 563575. CriadoAldeanueva, F., SotoNavarro, J. and GarcíaLafuente, J., 2011. Seasonal and interannual variability of surface heat and freshwater fluxes in the Mediterranean Sea: budgets and exchange through the Strait of Gibraltar. International Journal of Climatology, doi: 10.1002/joc.2268 (in press). FenoglioMarc, L., Kusche J., Becker M., 2006. Mass variations in the Mediterranean Sea from GRACE and its validation by altimetry, steric and hydrologic fields. Geophys. Res. Lett., 33, L19606, doi:10.1029/2006GL026851. Font, J., P. Puig, J. Salat, A. Palanques, M. Emelianov, 2007. Sequence of hydrographic changes in NW Mediterranean deep water due to the exceptional winter of 2005. Scientia Marina 71(2), 339346. García, D., Chao, B.F., Del Río, J., Vigo, I. 2006. On the steric and massinduced contributions to the annual sea level variations in the Mediterranean Sea, J. Geophys. Res., 101, C09030, doi:10.1029/2006JC002956. Garrett, C., Outerbridge, R., Thompson, K., 1993. Interannual variability in Mediterranean heat and buoyancy fluxes. J. Climate 6, 900910. Gilman, C., Garrett, C., 1994. Heat flux parameterization for the Mediterranean Sea: the role of atmospheric aerosols and constraints from the water budget. J. Geophys. Res. 99 (C3), 51195134. Kalnay, E and coauthors (1996). The NCEP/NCAR 40year reanalysis project. Bull. Amer. Meteor. Soc., 77:437471.
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LópezJurado, J.L., GonzálezPola, C., VélezBelchí, P. 2005. Observation of an abrupt disruption of the longterm warming trend at the Balearic Sea, Western Mediterranean Sea, in summer 2005, Geophy. Res. Lett., 32, L24606, doi:10.1029/2005GL024430. MacDonald, A.M., Candela, J., Bryden H.L, 1994. An estimate of the net heat transport through the Strait of Gibraltar, in Seasonal and Interannual variability of the Western Mediterranean Sea, Coastal Estuarine Stud., vol. 46, edited by P.E. LaViolette, pp. 1232, AGU, Washington, D.C. Mariotti A, 2010. Recent Changes in the Mediterranean Water Cycle: A Pathway toward LongTerm Regional Hydroclimatic Change? Journal of Climate 23 (6), 15131525. Matsoukas, C., Banks, A. C., Hatzianastassiou, N., Pavlakis, K. G., Hatzidimitriou, D., Drakakis, E., Stackhouse, P. W., Vardavas, I., 2005. Seasonal heat budget of the Mediterranean Sea. J. Geophys. Res., 110, C12008, doi:10.1029/2004JC002566. May, P.W., 1986. A brief explanation of Mediterranean heat and momentum flux calculations. NORDA code 322, NSTL, MS 39529. Rohling, E.J., Bryden, H.L., 1992. Maninduced salinity and temperature increases in western Mediterranean deep water. Journal of Geophysical Research 97 (C7), 11191–11198. Ruiz S., Gomis D., Sotillo M. G., Josey S. A., 2008. Characterization of surface heat fluxes in the Mediterranean Sea from a 44year highresolution atmospheric data set. Global and Planetary change, (63), 256274. Ruti, P. M., Marullo, S., D'Ortenzio, F., Tremant, M., 2008 Comparison of analyzed and measured wind speeds in the perspective of oceanic simulations over the Mediterranean basin: Analyses, QuikSCAT and buoy data. Journal of Marine Systems 70, 3348. SanchezRomán A., Sannino G., GarcíaLafuente J., Carillo A. and CriadoAldeanueva F., 2009. Transport estimates at the western section of the Strait of Gibraltar: A combined experimental an numerical modeling study. Journal of Geophysical Research. (114), C06002, doi:10.1029/2008JC005023. Schiano M.E., L. Santoleri, F. Bignami, R.M. Leonardo, Marullo and Bohm, 1993. Airsea interactions measurements in the west Mediterranean Sea during the Thyrrhenian Eddy MultiPlatform Observations Experiments, J. Geophys. Res., 98(C2), 24612474. Schroeder K., A. Ribotti, M. Borghini, R. Sorgente, A. Perilli, G.P. Gasparini, 2009. An extensive western Mediterranean deep water renewal between 2004 and 2006. Geophysical Research Letters 35(18), L18605. Tziperman E. and Speer K., 1994. A study of mass transformation in the Mediterranean Sea: analysis of climatological data and simple threebox model. Dynamics of Atmospheres and Oceans. (21), 5382.
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In: Heat Flux Editors: G. Cirimele and M. D'Elia
ISBN 9781614708872 © 2012 Nova Science Publishers, Inc.
Chapter 5
SIMULATION OF HEAT FLUX TRANSPORT IN THE EDGE PLASMA OF SMALL SIZE DIVERTOR TOKAMAK A. H. Bekheit Plasma and Nuclear Fusion department, Nuclear Research Centre, Atomic Energy Authority, Cairo, Egypt
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ABSTRACT The B2.SOLPES.0.5.2D fluid transport code is applied for modeling SOL (Scrape off Layer) plasma in the small size divertor tokamak. Detailed distributions of the plasma heat flux and other plasma parameters in SOL, especially at the target plate of the divertor are found by modeling. The modeling results show that most of the electron heat flux and small part of ion heat flux arrive at target plate of the divertor, while, a large part of the ion heat flux and part of electron heat flux arrive at the outer wall. Also the simulation results shows the following results (1) large asymmetries in heat flux at targets plates are observed. (2) When strong ITB is formed The reduction of plasma radial heat flux is higher by factor (~ 2.5 ) than neoclassical heat flux (3) the shear of the radial electric is enhanced by increase in temperature heating of plasma due to increase in pressure gradient and large reduction of radial heat flux. This leads to the core confinement is improved which correspond to the ITB formation. (4) The radial heat flux is strongly influenced by toroidal rotation. (5) the amplification of conduction part of radial heat flux imposes nonresilient profile of ion temperature, under which the effect of toroidal rotation on ion temperature profile is strong.(6) the comparison between radial heat flux at different momentum input shows that, the radial ion heat flux with larger ion temperature scale length in the case of coinjection neutral beam is larger than the ion heat flux with smaller ion temperature scale length in the case of counterinjection neutral beam.
INTRODUCTION Two dimensional edge plasma modeling has become a standard tool for computational studies of tokamak edge plasma physics and divertors. Many modeling codes had been
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A. H. Bekheit
developed [1, 2] for two dimensional edge plasma. The physics models and numerical implementation of most of the computational models which are used for boundary plasma transport studies were described by Vold et al. [3]. The B2 code [4] is one of the three main modeling codes (B2, EDGE2D, and UEDGE) [5] for edge plasma modeling which solved the full two dimensional problem of SOL multifluid transport equations. Many authors had carried out a series of edge plasma research by using the B2code or combination of the B2 code and the EIRENE neutral gas Monte Carlo code [6, 7], B2 EIRENE code was used for operating or planned tokamaks [8–15]. The code B2.SOLPES.0.5.2D [16, 17] which is a version of B2 which includes drifts effects. The B2.SOLPES.0.5.2D has become an important tool for edge plasma modeling and divertor designs for KSTAR [18], ASDEXUPGRAD [17, 19], SST [20], ITER [21] and the small size divertor tokamaks. We present here the results of simulations of the edge plasma of the small size divertor tokamak (has dimension R = 0.3 m, a = 0.1 m, the torodial magnetic field BT = 1.7 T, Toroidal current I = 50 kA and ion temperature Ti = 75 eV) by means of the B2.SOLPES.0.5.2D code based on a reduced form of the transport equations. The equations and results of simulation correspond to the single ion species. It is demonstrated that, most of the electron heat flux and small part of ion heat flux arrive at target plate of the divertor, while, a large part of ion heat flux and small part of electron heat flux arrive at the outer wall. Also the comparison between radial heat flux at different momentum input shows that, the radial ion heat flux with larger ion temperature scale length in the case of coinjection neutral beam is larger than the ion heat flux with smaller ion temperature scale length in the case of counterinjection neutral beam.
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BASIC TRANSPORT EQUATIONS The 2D fluid equations [22] (Braginskii transport equations) are written on a curvilinear orthogonal coordinate system. The x coordinate varies along flux surfaces, y coordinate varies perpendicular to flux surface and z is the toroidal direction coordinate (see figure 1).
Bz y x
Xpoint
Figure 1. coordinate system and simulation mesh: x is the poloidal coordinate; y is the radial coordinate orthogonal to the flux surfaces. The directions of magnetic field and plasma current correspond to normal operation conditions of SSD tokamak (B drift of ions directed towards the xpoint). Heat Flux: Processes, Measurement Techniques and Applications : Processes, Measurement Techniques and Applications, Nova Science
Simulation of Heat Flux Transport in the Edge Plasma… The metric coefficients are hx = 1/x , hy = 1/y , hz = 1/z and
169 g hx h y hz .one
can replace hz by 2R, where R is local major radius of tokamak. The physical components of vector are used. The ions of single species are considered with Z = 1 so that ne = ni = n. The subscript '' denotes the direction perpendicular to both the magnetic field B and y axis bx = Bx / B and bz = Bz / B.
(1) Particle Balance Equation In the particle balance equation [22] we take into account the fact that the diamagnetic velocity is almost divergence free. Therefore, instead of diamagnetic velocity it is possible to
~ ( dia)
introduce the effective velocity V the particle flux .we have thus
, which gives the same contribution to the divergence of
1 g na 1 g n ax ay S a t g x hx g y hy
(1)
where a = e.i, ax, ay are poloidal and radial particle fluxes determined by
(2)
1 na 1 naTi ~ n p ay V y(aE ) V y(adia) na D AN D AN a a h y y h y y
(3)
1 na 1 naTi ~ ) n p ax bxV a bzV( E ) bzV( dia na D AN D AN a a a hx x hx x
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(E)
The velocities V
V( E )
~ ( dia)
and V
are given by:
B 1 1 1 (E) z2 , Vy B hy y B hx x
Ti Bz 1 TB 1 ~ ( dia) ~ ) V( dia i z 2 ,Vy a a Z a e hx x B 2 Z a eb z h y y B
(4)
(5)
physically, velocity equation(5) represents vertical guiding centre drift of ions caused by B. n, p D AN , S(n) are anomalous diffusion coefficients and volume particle source.
(2)Parallel Momentum Balance for Ions Combining the inertia and gyroviscosity terms in the Braginskii equations, we find [22] Heat Flux: Processes, Measurement Techniques and Applications : Processes, Measurement Techniques and Applications, Nova Science
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A. H. Bekheit
mi
niVi 1 hz g m 1 hz g m ix iy t hz g x hx hz g y h y
bx ni Ti 1 ln hz bx eni E x e i ne me (Vi Ve ) ReTi i  bx ni miVi 2 S im hx x hx x
(6) where i x , i y are poloidal and radial momentum fluxes given by: m
m
V i ~ ) imx miV i i x bz ni miV iV( dia 2 i hx x
(7)
V i ~ imy miV i i y ni miV iV y(idia) 2 hy y
(8)
n 2 nmi DAN is anomalous viscosity coefficient. The electric and pressure gradient
forces are given by 1st and 2nd terms in RHS of equation (6). The 3rd and 4th terms in the RHS of equation (6) represent the ionelectron friction and thermal friction forces. The 5th term in RHS of this equation represents the parallel projection of parallel ion viscosity divergence
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 i 
12 B V 3 0 i bx i  4 bx B 2 3 hx x B 2 12 ( 0 ) B qi  3 bx B 2 bx hx x i i B 2 hx
B ( dia) B ( E ) Vi x V x Bx Bx hx x B ( dia) qi x Bx x
(9)
nd 0i 0.96ni Ti / i i is Braginskii parallel viscosity coefficient and the 2 term of equation (9)
corresponds to the parallel viscosity driven by ion heat flux where qi 
( 0)
i  bx
1 Ti , ( dia) 5 ni Ti Bz Ti qi x hx x 2 eB 2 h y y m
Finally the 6th term in the RHS of equation (6) and S i ll corresponds to the centrifugal force and volume momentum source.
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Simulation of Heat Flux Transport in the Edge Plasma…
171
(3) Parallel Momentum Balance for Electrons Poloidal electric field is calculated from the parallel momentum balance for electrons equation. This equation has the standard form given by (22)
1 neTe 1 T j  Re i  bx E x ene hx x ene
(10)
2 with  e ne , j is parallel (toroidal) current (3.4)
me e i
(4) Current Continuity Equation Equation for charge conservation is [22]
1 g 1 g jx j 0. g x hx g y hy y
(11)
j x bz j bx j . Expressions for j and j y components are obtained from the radial and poloidal
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projections of the total momentum balance equations (the sum of electron and ion momentum balance equations). The resulting current is a sum of contributions from pressure gradient
( dia )
(diamagnetic current) j
( s) friction j :
( in )
, inertia and gyroviscosity j
( vis)
, viscosity j
~ ( dia) (in) (vis ) ( s ) j j j j j j
and ionneutral
(12)
Effective diamagnetic current corresponding to B drift can be chosen in a form (18)
~ ( dia ) 1 n e Te ni Ti B z 1 j , bz hy y B2
n T niTi Bz 1 ~ ( dia ) jy e e hx x B2 Current driven by parallel viscosity can be reduced to the form:
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(13)
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A. H. Bekheit
B ~ (vis ) j x 10i
1
B 2 Vi 
B ( dia ) B ( E ) Vi x V x Bx Bx hx x
3B 2
B x 0 i ~ ( vis ) jy bz 1
1
B 2 Vi 
B ( dia) B ( E ) Vi x V x Bx Bx hx x
3B 2
1 hy y B 2
1 hx x B 2
(14)
(15)
The current produced by components of viscosity tensor which are connected with heat fluxes can be written in the same form [23]:
~ ( visq ) j
( 0) B ( dia ) 1 qi  qi x B 2 B x 1 0.24 B 1 x hx x hy y B 2 2 B ii
0.24 B ~ ( visq) jy bz 1 x
q i(0)
B 2 i i
1 B ( dia) 2 q i x B Bx 1 h x x h x x B 2
(16)
(17)
Expression for inertial current contains only the contribution from the c entrifugal force
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is:
j (in )
1 ln hz mi niVi 2 B hy y
j y(in )
1 ln hz mi niVi2 B hx x
(18)
(19)
The contribution from perpendicular viscosity is taken into account for the main ions component only and in the radial direction: j y( vis)
g 1i Vi g y h y 2 B y
1
where perpendicular velocity is reduced to the form:
Va V( E ) Va(dia) .
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(20)
Simulation of Heat Flux Transport in the Edge Plasma…
173
This current corresponds to pure cylindrical approximation. It is important in the separatrix vicinity. In the core region the currents driven by parallel viscosity are more important. The total current caused by viscosity is:
( vis ) ( vis) ( vis ) ( visq ) j j j j
(21)
The contribution from ion  neutral friction results in the comparatively small current and m
m
B
B
for impurity ions is negligible. For the main component j ( s ) S i y , j ( s ) S i , where y
S im ,
S imy
are the momentum sinks due to ionneutral friction. After the standard evaluations
for these currents we obtain [23]
1 ni Ti ~ (s) ~ (s) j x j bz i N bz2 i N bz 2 i N B zV yN hx x eni x
(22)
1 ni Ti ~ (s) j y i N iN i N B V N h y y eni h y y
(23)
where i N is the ionneutral perpendicular conductivity i N
ni mi Vi N ex n N 2B 2
.
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(5) Energy Balance Equation for Electrons In order to obtain the energy balance equations we combine the convective heat flux with the diamagnetic heat flux. Energy balance for the electrons can be written in the form [22]:
3 n e Te 1 g ~ 1 g ~ n e Te g b x q ex q ey Ve 2 t g x hx g y hy g x hx 1 1 1 Qe n e Te B h x h y y x B 2 x y B 2 (24) where 3 5 B 1 5 n 1 ni 1 niTi p Te q~ex ne bzV( E ) bxVe neTe z DAN 2 bz DAN i i 2 2 e h y B 2 h x h y x x x 2 2 1 Te q bx qbz ebx e bz hx x
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A. H. Bekheit 3 5 B 1 5 n 1 ni 1 niTi p q~ey neVy( E ) neTe z DAN Te 2 DAN i i 2 e hx x B 2 hy y hy y 2 1 Te qy e hy y
Here e is the Braginskii heat conductivity. The perpendicular heat conductivity e is taken anomalous. The last terms in the energy fluxes correspond to effective diamagnetic velocity of the electrons.
Z aTe ~ ( dia) ) V~e(dia Va, y ,y Ti In the last term of the R.H.S. of equation (24) we took into account the fact that E B drift velocity is almost divergence free. The electron heat source consists of energyexchange term ( Q ), radiation and ionization losses ( Q
rad
, Q ion ), and Ohmic heating source ( Qu ):
Qe Qerad Qeion Qu Q
Q a
3me ne ea Te Ti ma
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(6) Energy Balance Equation for Ions The ion energy balance equation is [22]
3 Ti ni 1 g ~ 1 g ~ ni Ti g qix qiy Vi bx 2 t g x hx g y h y g x hx 1 1 1 Qi Ti B ni , hx h y y x B 2 x y B 2 (25) where heat fluxes are: 3 5 n 1 n i 1 n i Ti p q~ix n i b z V ( E ) b xVi b z D AN D AN i i 2 h x x h x x 2 1 Ti ( i  b x2 i b z2 ) h x x
5 ~ b z n i Vi (dia) Ti 2
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Simulation of Heat Flux Transport in the Edge Plasma… 3 5 n 1 n i 1 n i Ti p q~iy V y( E ) n i D AN D AN i i 2 hy y hy y 2 1 Ti i . hy y
175
5 n i V~i (ydia) Ti 2 a
As for electrons, parallel heat conductivity is classical, while perpendicular heat conductivity coefficient is taken anomalous.
(7) Boundary Conditions Whether one deals with diamagnetic or EB perpendicular drift velocity, special core must be taken to avoid unphysical flows at the plates. The standard boundary condition is to require that the flow be at least sonic at the entrance [24], which translate into:
Vll C s bz / bx V
(26)
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where CS is the local sound speed and (bz / bx) is the inverse of the field line pitch. However, for very small angles as in stellrators the V term may over power the sound velocity [24] and the boundary condition would require a plasma flow exiting the plate. Therefore, the pitch angle at the plates is limited to be no less than 1 degree, motivated by the engineering limits met when attempting to align the divertor tiles with the magnetic field. The solution procedure of the code was also modified so as ensure that no unphysical flows were being created, and solves the parallel momentum equation using the updated electric potential. Boundary conditions for the current equation at the plate correspond to sheath current – voltage characteristics j x en bx cs bx
1 2
e Te exp 1 me Te
(27)
where is the secondary electron emission coefficient. At the inner (core) flux surface the currents are set either to the divergent part of the diamagnetic current or zero. At walls, the same conditions on the normal current components were imposing as for the inner core. The electron and ion heat fluxes to the target plates are: q~ex bx
n 2
e 1 Te exp 1 Te e me 1 Te
3 q~ix nTi cs bx 2
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(28) (29)
176
A. H. Bekheit
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THE MAIN RESULTS OF SIMULATION The B2.SOLPES.0.5.2D code [25, 26] is used for present modeling. The code employed to solve the full two dimensional problem of the SOL multifluid transport equations (1,6,11,24,25) uses an edge geometry and input assumptions for plasma transport based on a comparison with experimental data from other tokamaks. Multifluid plasmas consist of neutral particles, ions and electrons with various physical processes (e.g. ionization, recombination and charge exchange) (23). For small size divertor tokamak (a = 0.1m, R = 0.3 m, BT = 1.7 T, I = 50 k A) operation, the simple anomalous crossfield transport is characterized by a particle diffusivity D (AN) = 0.5 m2/s and heat diffusivities e,i = 0.7 m2/s. The parallel heat and momentum transport are classical but flux limited. The computational region for modeling is based on SN (Single null) and covers the outer SOL and the divertor below the midplane plus a small segment of the region of closed flux surfaces and the private flux region(see figure1). The boundary conditions for modeling are given in section of basic equations. Target profiles of ion density ni, ion temperature Ti, poloidal current density J and total heat flux are represent in Figure (25). Figure (2) Presents a plot of ni in the divertor regions for the case with normal toroidal magnetic field. It shows the ability of B2SOLPS0.5.2D code to derive the main tendency for higher density plasma in the inner divertor plate noticed in normal BT discharges (this result agree with the result of [27] for normal direction of toroidal magnetic field). Density asymmetries derived by the drift are accompanied by Ti asymmetries, with the plasma being much cooler at outer plate in normal direction of toroidal magnetic field case, consistent with [28]. Larger Ti asymmetries lead to larger thermoelectric current density [26] in normal direction of toroidal magnetic field BT case. The asymmetry in profile of poloidal current density is observed in Figure (3).The poloidal current density change it sign and hence the poloidal current should flow from the plasma to plates, the positive direction in Figure (3), and from plates to the plasma , the negative direction in Figure (3). However, in separatrix the poloidal current is nonmonotonic. Here, mainly the current driven by perpendicular viscosity [26] Jy  ( 2 E / y 2) balance the poloidal current. The non – monotonic poloidal current is responsible for the positive and negative spike at separatrix Figure (3). Also we notice large asymmetry of total heat flux at targets plates. The total heat flux flow, integrated over each target, is 1.51 kw/m2 and 2.36 kw/m2 to the inner and outer target, respectively, giving a ratio 0.64 for normal direction of toroidal magnetic field [29]. The second result of simulation shows the changes in Ti and Te profiles were simultaneously observed as shown in Figure(6), which shows local drop in radial heat flux, and then ITB were formed as show in Figure(6). However, radial (ion and electron) heat flux at ITB is close to neoclassical level as shown in Figures (7, 8). The method to calculating the neoclassical heat flux q (NEO) has described in [25, 30]. This result shows that, plasma has strong ITB and the reduction of transport level is higher by a factor of 2.5 (for ion heat flux) than neoclassical level as show Figures. (7, 8)[31].
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Simulation of Heat Flux Transport in the Edge Plasma… 120
19
2.6x10
(2)
19
1.8x10
3
100 90 80 70
19
1.6x10
19
1.4x10
outer plate
19
2.0x10
inner plate
19
2.2x10
(3)
110
Ti ( e V )
2.4x10
outer plate
inner plate
19
ni ( m )
177
60 50 40 30 20
TOP
1.2x10
19
1.0x10
TOP
19
10 0
18
8.0x10
10 0
20
40
60
80
0
100
20
40
60
80
100
X (cm)
X (cm) 11000 40
20
5000 4000 3000 2000
40
1000 0
50
1000
60 0
20
40
60
80
100
0
20
40
60
80
100
X ( cm )
X ( cm )
Figure 25. Distribution of ion density, ion temperature, current density and total heat flux in separatrix as function of poloidal coordinate. 220
(6)
200
Ti
180 160 140
Ti ( eV )
Te
120 100 80 60
Separatrix
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outer plate
6000
Top
30
7000
2
inner plate
0 10
inner plate
8000
qt x ( W / m )
10
J ( KA/m )
9000
outer plate
20
2
(5)
10000
(4)
30
40 20 0 0
5
10
15
20
25
y ( cm )
Figure 6. Radial distribution of electron and ion and ion temperatures versus radial coordinate y.
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A. H. Bekheit 0.025
25 20 15 10
0.020
0.015
0.010
0.005
Separatrix
6
qiy ( K W / m
2
)
30
( d Er / hydr ) * 10 ( s )
Neoclassical ion heat flux Radial ion heat flux Electric field shear
(7)
35
1
5 0.000 0 0
5
10
15
20
25
y ( cm )
Figure 7. Distribution radial electric fiel shear, ion heat flux, neoclassical ion heat flux as function of radial coordinate y. 0.025
(8)
2
10
qe
20
0.010
0.005
1
30
0.015
6
Electron heat flux Neoclassical electron heat flux Electric field shear
0.020 ( d Er / hy dr )*10 (s )
(KW/m )
0
Separatrix 0.000 0
5
10
15
20
25
y ( cm )
Figure 8. Distribution radial electric field shear, electron heat flux, neoclassical electron heat flux as function of radial coordinate y. 0.06
(9)
0.05 0.04
LT ( m )
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40
0.03 0.02 0.01 0.00 0
100 200 300 400 500 600 700 800 6
1
Electric field shear * 10 ( S )
Figure 9. The relation between radial electric field shear and characteristic temperature length. Heat Flux: Processes, Measurement Techniques and Applications : Processes, Measurement Techniques and Applications, Nova Science
Simulation of Heat Flux Transport in the Edge Plasma…
0.016
179
(10)
0.014 0.012
LV (m)
0.010 0.008 0.006 0.004 0.002 0.000 100
0
100 200 300 400 500 600 700 800
Electric field shear * 10
6
1
(s )
Figure 10. The relation between radial electric field shear and characteristic velocity length.
(11)
70 60 50
0.06 0.05 0.04 0.03 0.02
40
0.01
30 20
0.07
i
Ti ( e V )
80
0.08
L T ( cm )
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90
Counterinjection NB
110 100
ion temperature temperature scale length
120
Separatrix 0
5
10
0.00 15
20
25
y ( cm ) Figure 11. The radial distribution of ion temperature and ion temperature scale length as function of radial coordinate for discharge with counterinjection NB.
Fourth result of simulation: The radial electric field shear near separatrix is calculated as follow:
d Ey h y dy
d E NEO h y dy
d Ti h y dy e
ln n ln Ti 2.7 y y
bx V B
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d E NEO 180 h y dy h y dy d Ey
A. H. Bekheit
d Ti h y dy e
ln n ln Ti 2.7 y y
bx V B (30)
Neglect relatively small correction [32] (dBx/dy, dB/dy) assuming linear profile close to separatrix with
LTi L n (where LTi is characteristic temperature scale length
LTi ln T / h y y
1
and L n ln n / h y y
1
is characteristic density scale length)
we get:
d Ey hy d y
2.7 Ti Bx V LV e L2Ti
(31)
The dependence of the radial electric field shear on the characteristic temperature scale length and characteristic toroidal velocity scale length L V
d ln V
/ h y dy
1
is show
in Figure (910). Figure (910) shows the drop in LTi, L V was observed, which indicated to strong ITB is
(12)
110 100
Ti ( eV )
80 70 60 50 40
Coinjection NB
90
0.030 0.025 0.020 0.015 0.010 0.005
30
Separatrix
20 10
0.035
i
ion temperature temperature scale length
120
L T ( cm )
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formed. The possible physical processes contribute in the formation of ITB are explains as follow: the shear of the radial electric is enhanced by increase in temperature heating of plasma due to increase in pressure gradient and large reduction of radial heat flux. This leads to the core confinement is improved which correspond to the ITB formation [31].
0
5
10
0.000 15
20
25
y ( cm ) Figure 12. The radial distribution of ion temperature and ion temperature scale length as function of radial coordinate for discharge with coinjection NB.
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Simulation of Heat Flux Transport in the Edge Plasma… 12000
0.08 radial heat flux charateristic temperature scale length
10000
6000
4000
0.04
0.02
Separatrix 0
5
10
i
Counter injection NB
2
8000
0.06
LT ( cm )
qiy ( W / m )
181
0.00 15
20
25
y ( cm ) Figure 13. The radial distribution of ion heat flux and temperature scale length in the edge plasma of small size divertor tokamak in case of Counter NBI. 16000 0.035
radial heat flux charateristic temperature scal length
14000
0.030
6000
i
8000
0.020
Coinjection NB
2
10000
0.025
LT ( cm )
qiy ( W / m )
12000
0.015 0.010 0.005
4000
Separatrix 0.000 0
5
10
15
20
25
y ( cm ) Figure 13. The radial distribution of ion heat flux and temperature scale length in the edge plasma of small size divertor tokamak in case of Co NBI. 12000
0.08 radial heat flux charateristic temperature scale length
6000
4000
Counter injection NB
2
8000
0.04
0.02
Separatrix 0
0.06
5
10
i
qiy ( W / m )
10000
LT ( cm )
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2000
0.00 15
20
25
y ( cm ) Figure 14. The radial distribution of ion heat flux and temperature scale length in the edge plasma of small size divertor tokamak in case of Counter NBI. Heat Flux: Processes, Measurement Techniques and Applications : Processes, Measurement Techniques and Applications, Nova Science
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Fifth result of simulation: The toroidal torque inputs in the two direction (in the directions of Co and Counter NB injection) leads to wide variation in ion temperature Ti and ion temperature characteristic scale length LTi ln T / h y y in the edge plasma of small size divertor tokamak (see Figure (1112)). Also this result indicates that, the amplification of conduction part of radial heat flux imposes nonresilient profile of ion temperature, under which the effect of toroidal rotation [32] on ion temperature profile is strong as shown. Sixth result of simulation: the radial heat flux at different momentum input shows that, the radial ion heat flux with larger ion temperature scale length in the case of coinjection neutral beam is larger than the ion heat flux with smaller ion temperature scale length in the case of counterinjection neutral beam as shown as in Figure (1314). Seventh result of simulation: Typical profile of radial and poloidal heat flux are shown in Figure (1518). The steep radial heat fluxes profile correspond to barrier formation and drop of diffusion and thermal conductivity coefficients inside this barrier. The poloidal heat flux varies explicitly according to the direction of toroidal momentum as shown in Figures (1518), and the reduction in poloidal and radial heat flux transport are obtained as the toroidal torque are injected in Co and Counter neutral beam[32]. 12000
Counterinjection NB
12000
2
9000
7500
6000
10500 9000 7500
3000 0
5
10
Separatrix
6000 4500
Separatrix
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(16)
13500
(15) qiy( W / m )
2
qiy ( W / m )
10500
Coinjection NB
15000
4500 3000 15
y ( cm )
20
25
0
5
10
15
20
25
y ( cm )
Figure 1516. The radial distribution of heat flux for Co and Counterinjection NB in edge plasma of small size divertor tokamak.
Figure (1518) shows the radial distribution of poloidal heat flux profile . In those Fig's we can see a viscous layer exist, where the poloidal heat flux deviates significantly from neoclassical heat flux(10) as shown in Figures (1920). In this layer the parallel flux flows in the edge plasma of small size divertor tokamak.
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Simulation of Heat Flux Transport in the Edge Plasma…
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60000
Separatrix 5
10
15
20
Separatrix
0
0
30,000
2
qix ( W / m )
Coinjection NB
2
qix ( W / m )
30000
30000
(18)
0
Counterinjection NB
(17)
60,000
90,000 0
25
5
10
15
20
25
y ( cm )
y ( cm )
Figure 1718. The radial distribution of poloidal heat flux for Coand Counter injection neutral beam in the edge of small size divertor tokamak. 12
4
150
100
Separatix
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0
5
10
15
20
25
12 10 8 6
50
4
Separatrix
0
120
Coinjection NB
2
( kW/m )
6
i ( NEO)
8
2
q
80
(20)
q
60
14
200
qix ( KW / m )
40
10
2
Counterinjection NB
2
(19)
100
16 Neoclassical Code
qix ( K W / m )
(KW/m )
20
i (NEO)
250
Neoclassical Code
0
2 0
y ( cm )
5
10
15
20
25
2
y ( cm )
Figure 1920.The radial distribution of ion heat flux and comparison with neoclassical in the edge plasma of small size divertor tokamak.
CONCLUSION The simulation results by transport code B2SOLPS0.5.2D provide the following results: 1. The electron heat flux and small part of ion heat flux arrive at target plate of the divertor; while, a large part of the ion heat flux and part of electron heat flux arrive at the outer wall. 2. Large asymmetries in heat flux at targets plates are observed. 3. When strong ITB is formed the reduction of plasma radial heat flux is higher by factor (~ 2.5) than neoclassical heat flux. 4. The shear of the radial electric is enhanced by increase in temperature heating of plasma due to increase in pressure gradient and large reduction of radial heat flux.
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A. H. Bekheit This leads to the core confinement is improved which correspond to the ITB formation. 5. The radial heat flux is strongly influenced by toroidal rotation. 6. The amplification of conduction part of radial heat flux imposes nonresilient profile of ion temperature, under which the effect of toroidal rotation on ion temperature profile is strong. 7. The comparison between radial heat flux at different momentum input shows that, the radial ion heat flux with larger ion temperature scale length in the case of coinjection neutral beam is larger than the ion heat flux with smaller ion temperature scale length in the case of counterinjection neutral beam.
REFERENCES [1] [2] [3] [4] [5] [6]
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[7] [8] [9]
[10]
[11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21]
D. Reiter, J. Nucl. Mater. 80, 196–198 (1992) M. Baelmans, Core improvement and applications of a 2D edge plasma model for toroidal devices. Report KFAJUL2891 Kernforschungaszentrum Julich (1994). E.L. Vold, F. Najmabadi, R. Conn, W. Phys. Fluids B3, 3132 (1991). B.J. Braams, A multifluid code for simulation of the edge plasma in tokamaks. Report EUR_FU/XII80/87/68 commission of the European Communities, Brussels (1987). M. Fichtmuller, Coreedge coupling and the effect of the edge on overall plasma performance report jetP (98) 69 jet joint undertaking. Abingdon, Oxfordshire (1998) D. Reiter, The EIRENE code, version Jan. 92, user manual report JUL2599. Kernforschungaszentrum Julich (1992) M. Baelmans, Report JUL1947 Kernforschungaszentrum Julich (1984) M.A. Ulrickson et al., TRX tokamak physics experiment design report 93930325PPPL/GNeillson01, Princeton Plasma Physics Laboratory, NJ, chap 8 (1993). D.N. Hill, K.H. Im, B.J. Lee, H. Neilson, H.K. Park, Power and particle control requirements for KSTART. KSTART design point definition workshop, Princeton, presented paper (1997) Detail of the ITER outline design report (the ITER machine) presented by the ITER director for the 4th meeting of the technical advisory committee, vol 2, San Diego Joint work Site, CA(1994) M. Warrier, S. Jaishankar, S. Deshpande, SST divertor modeling report. Institute for plasma research, Bhat Gandhinager (1996–97) D.N. Hill et al., Fusion Technol. 21, 1267 (1992) R.V. Budny et al., J. Nucl. Mater. 196–198, 462 (1992) A. Kulushkin et al., J. Nucl. Mater 241–243, 462 (1992) D.P. Coster et al., J. Nucl. Mater 241–243, 690 (1997) B.J. Braams, Contrib. Plasma Phys. 36, 276 (1996) V.A. Rozhansky, M. Tendler, Rev. Plasma Phys. 19, 147 (1996) R. Zagorski, H. Gerhauser, H.A. Claasen, Contrib. Plasma Phys. 38, 61 (1996). V.A. Rozhansky, A.A. Ushakov, S.V. Voskoboynikov, Nucl. Fusion 39, 613 (1999) M. Tendler, V. Rozhansky, Comments Plasma Phys. Control Fusion 13, 191 (1990) F.L. Hinton, Y.B. Kim, Nucl. Fusion 34, 899 (1994)
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[22] E. G. Kaveeva "Simulation of electric field and current in tokamak plasma edge", M.sc. thesis, university of Saint Petersburg State Politechnical, (2001). [23] Y. Feng, F.Sardei, and J. Kisslinger, "J. Nucl. Mater." 226 269, 812, (1999). [24] A. H. Bekheit; J. Fusion Energy ;27,(4), 338345, (2008). [25] R. Schneider, V. Rozhansky, V. Voskoboynikov; Nucl. Fusion; 41, 387, (2001). [26] N. Asakura al., "30th EPS Conference on Contr. Fusion and Plasma Phys., St. Petersburg, 711 July " ECW Vol. 27A, p2.153, (2003). [27] A. H. Bekheit ; J. Fusion Energy ;27,(4), 321326, (2008). [28] A. H. Bekheit ; J. Fusion Energy ;29,(4), 261266, (2010). [29] S. Ide et.al ; Nucl. Fusion; 44, 876882, (2004). [30] A. H. Bekheit ; J. Fusion Energy ;29,(285289), (2010). [31] A. H. Bekheit ; J. Fusion Energy ;29,(360364), (2010). [32] A. H. Bekheit ; J. Fusion Energy ;26,(4), 331335, (2007).
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Copyright © 2012. Nova Science Publishers, Incorporated. All rights reserved. Heat Flux: Processes, Measurement Techniques and Applications : Processes, Measurement Techniques and Applications, Nova Science
In: Heat Flux Editors: G. Cirimele and M. D'Elia
ISBN 9781614708872 © 2012 Nova Science Publishers, Inc.
Chapter 6
MERIDIONAL HEAT FLUXES IN THE NORTH INDIAN OCEAN T. Rojsiraphisal1 and L. Kantha2 1
Department of Mathematics, Burapha University, Thailand 2 Department of Aerospace Engineering Sciences, University of Colorado, Boulder, Colorado, US
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ABSTRACT The seasonal cycle of the crossequatorial heat transport is of great importance to South Asian monsoons since it is a component of the heat budget and hence determines the SST of the North Indian Ocean. In this chapter, we use a dataassimilative model of the North Indian Ocean assimilating altimetry and MCSST data and driven by ECMWF and QuikSCAT wind stresses, to explore the seasonal and interannual variability of the meridional heat fluxes during the years 1993 to 2005. In addition to a strong seasonal variability, the crossequatorial heat transport also exhibits considerable interannual variability. There are also strong, short period intraseasonal fluctuations on time scales of 2050 days. The meridional heat flux values are 2030% higher when the hindcast is driven by observed QuikSCAT wind stresses than when it is driven by ECMWF winds.
1. INTRODUCTION The overwhelming determinant of the circulation in the North Indian Ocean are the seasonally reversing monsoon winds. During the boreal summer, the winds blow from the southwest. The Southeast Trade Winds south of the equator cross the equator and continue into the Arabian Sea in the form of a strong, southwesterly lowlevel jet along the Somali and Oman coasts. The currents in the surface layers across the equator are northward. During the boreal winter, the winds are exactly in the opposite direction to summer monsoon winds, northeasterly. This gives raise to southward crossequatorial currents in the upper layers. Thus, there is a strong seasonal component to the meridional heat, mass and salt fluxes across
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T. Rojsiraphisal and L. Kantha
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the equator in the upper layers. However, the net crossequatorial transport of heat depends on the currents in the entire water column. A numerical model can shed light on the seasonal and interannual variability of the meridional fluxes. We have used a dataassimilative model of the North Indian Ocean assimilating altimetry and MCSST data and driven by ECMWF (EC henceforth) and QuikSCAT (QS henceforth) wind stresses, to explore this variability during the years 1993 to 2005. The numerical model used for the hindcast is the University of Colorado version of the Princeton Ocean Model (CUPOM). It is a primitive equation model using topographically conformal coordinate in the vertical and orthogonal curvilinear coordinates in the horizontal. The sea surface height is calculated explicitly using the splitmode technique. CUPOM includes a second moment closurebased model (Kantha and Clayson, 1994) of turbulent mixing in the upper and bottom layers. More details about the basic features of CUPOM can be found in Kantha and Clayson (2000b). The model has 38 sigma levels in the vertical, with the levels closely spaced in the upper 300 m. The sigma levels are 0, 1, 2, 5, 10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 110, 120, 130, 140, 150, 160, 170, 180, 190, 200, 210, 220, 230, 240, 250, 270, 300, 400, 700, 1000, 2000, 3000, 4000, 5000 m in a water column of 5000 m depth. The high vertical resolution in the upper 300 m is deliberately chosen to better simulate the nearsurface circulation. Figure 1 shows the model domain, along with bottom topography. The model is forced by 6hourly EC winds. It assimilates altimetric sea surface height anomalies and weekly composite MCSST using a simple optimal interpolationbased assimilation technique. The model was run at two different values of horizontal resolution: 1/2o and 1/4o to explore the sensitivity of the results to model resolution.
Figure 1. The North Indian Ocean showing the bottom topography and the model domain.
The model, and the assimilation methodology based on conversion of SSH anomalies into pseudoBT anomalies for adjusting the model temperature field via optimal interpolation, have been described in detail by Lopez (1998), Lopez and Kantha (2000a and b) and Kantha
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et al. (2008), and will not be repeated here. For details of the model and the hindcast, including comparisons of model results with observational data, see Kantha et al. (2008).
2. MERIDIONAL HEAT FLUXES
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The surface transport is generally southward on both sides of the equator during summer, but northward in winter. The net annual heat flux across the equator is southward and occurs predominantly via the crossequatorial cell (Schott and McCreary, 2001). Figure 2 shows the meridional heat fluxes in PW (1 PW = 1015 W) across various zonal transects (5 oS, 0 oN, 5 o N, 10 oN, 15 oN and 20 oN) averaged over the entire hindcast period, compared with results of other studies in the past. Both 1/2o and 1/4o results are shown. The hindcast results are in good agreement with the values of Hastenrath and Greischar (1993) derived from airsea flux calculations, and Lee and Marotzke (1998) derived from a numerical model, except that the zero heat flux occurs at about 1213 oN instead of 18oN. The observed QS wind stresses produce a higher value of meridional heat fluxes than the EC winds obtained from the coarse resolution Numerical Weather Prediction (NWP) model in general. The net southward meridional heat flux across the equator varies between 0.18 and 0.55 PW.
Figure 2. Meridional heat flux across zonal transects from various studies redrawn from Scott and McCreary, 2001 (H – Hsiung, 1985; HG Hastenrath and Greischar, 1993; LM – Lee and Marotzke, 1998; GS – Gasternicht and Schott, 1997; WP – Wacongne and Pacanowski, 1996; EC – Present study with ECMWF winds; QS Present study with QuikSCAT wind stresses).
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Figure 3. Meridional heat flux (in PW) across the equator from 1/2o (top panel) and 1/4o (bottom panel) models compared to that from ECCO (blue dotted line) and SODA (green) global models. Both the results from the model driven by ECMWF winds (black line) and QuikSCAT wind stresses (red line) show considerable seasonal and interannual variability, with the latter being significantly larger than the former. Note the anomalously weak fluxes between 1998 and 2000.
st.com/lib/multco/detail.action?docID=3020499.
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Figure 3 shows the 7day running mean of the crossequatorial heat flux for the years 1993 to 2005. While the seasonal signal is dominant, considerable intraseasonal and interannual variability are present. The values from the University of Maryland SODA model, which uses the GFDL MOM2.b driven by monthly winds derived from COADS (19501991) and NCEP (19922001), are also shown (green line). SODA assimilates hydrographic data, SST and altimetric SSHA (Carton et al. 2000). The dotted blue line shows values from the ECCO project that uses the MIT nonhydrostatic GCM driven by NCEP products (Koehl 2002). Both ECCO and SODA have coarse 1°x1° horizontal resolutions, although SODA’s resolution increases to 0.5ox1o in the tropics. There is general agreement between all three model results, although the overall mean values differ (1/2o EC: 0.42 PW, 1/2o QS: 0.52 PW, 1/4o EC: 0.26, 1/4o QS: 0.31, SODA: 0.41 PW, ECCO: 0.19 PW). Note the anomalously weak fluxes between 1998 and 2000, with a reduction of about 40% in peaktopeak values compared to the normal years.
Figure 4a. Monthly meridional heat fluxes (in PW) from three different studies (adapted from Schott and McCreary 2001). Heat Flux: Processes, Measurement Techniques and Applications : Processes, Measurement Techniques and Applications, Nova Science
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Figure 4b. Monthly meridional heat fluxes (in PW) from the hindcast.
Figures 4a and b show the mean monthly meridional heat fluxes (averaged over the simulation years, 12 years for QC and 5 years for QS) compared with those from Hastenrath Heat Flux: Processes, Measurement Techniques and Applications : Processes, Measurement Techniques and Applications, Nova Science
Meridional Heat Fluxes in the North Indian Ocean
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and Greischar (1993), Garternicht and Schott (1997) and Hsiung, Newell and Houghtby (1989). The pattern of variability is in good agreement. However, the heat fluxes are considerably larger, especially for the QSdriven hindcasts. This is not surprising because the QS wind stresses are observed values and EC winds are coarse resolution NWP modelgenerated values. Figure 5 shows meridional heat fluxes for the year 1996 (a normal year) and 1998 (an anomalous year), in comparison with the 12year monthly means from the 1/4o model. The anomalous wind patterns during the anomalous year of 1998 give rise to stronger fluxes overall, compared to the normal year.
Figure 5. Monthly mean Meridional heat fluxes for the normal year 1996 and the anomalous year 1998 compared with the monthly mean over the 12year hindcast period for the 1/4o model driven by ECMWF winds.Finally, Figure 6 shows annual and seasonal means of the crossequatorial heat flux from various models for years 1993 to 2005. The annual mean shows considerable yeartoyear variability, with values ranging from 0.1 PW to slightly above 0.6 PW southward. The 3monthly means also exhibit considerable variability but with larger values of as large as 2.3 PW. Overall, all models indicate a northward transport during JanuaryFebruaryMarch and OctoberNovemberDecember, but southward during the rest of the year.
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Figure 6. Annual (top panel) and seasonal (middle and bottom panels) means of the crossequatorial heat flux (in PW) from 1/2o and 1/4o models compared to those from ECCO and SODA global models. The results from both the hindcasts driven by ECMWF winds and QuikSCAT wind stresses are shown.
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CONCLUSION Using a simple 2½ layer model of McCreary et al. [1993], Loschingg and Webster [2000] found that the SST in the North Indian Ocean is regulated by strong advection across the equator, with a crossequatorial heat flux of about 1.0 PW northward during the boreal winter and 0.51.0 PW southward during the boreal springsummer, and a net transport southward of 0.2 PW. The strong seasonal variability (along with a small net southward transport) they infer is consistent with our findings here, reinforcing the notion that the crossequatorial oceanic heat transport is essential to the seasonal cycle of the heat balance of the North Indian Ocean. As can be seen from the results presented above, in addition to a strong seasonal variability, the crossequatorial heat transport also exhibits considerable interannual variability. The anomalously weak fluxes are evident during 19982000, following the anomalous 199798 warming event. There are also strong, short period intraseasonal fluctuations on time scales of 2050 days. The meridional heat flux values are 2030% higher when the model is driven by observed QuikSCAT wind stresses than when it is driven by ECMWF winds. Since most modeling studies use similar coarse resolution NWP winds, this point is of considerable interest in issues related to the oceanic heat transport in the global oceans.
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ACKNOWLEDGMENTS LK acknowledges partial support by ONR through grant N000140610287. Our sincere thanks to Dr. Rena Schoenefeldt for providing us with the ECCO and SODA crossequatorial heat flux data.
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REFERENCES Carton, J. A., Chepurin, G., Cao X., and Giese, B. (2000). A Simple Ocean Data Assimilation analysis of the global upper ocean 195095. Part I: Methodology. J. Phys. Oceanogr., 30, 294309. Garternicht, U., and Schott, F. (1997). Heat fluxes of the Indian Ocean from a global eddyresolving model. J. Geophys. Res., 102, 2114721159. Hastenrath, S. and Greischar, L. (1993). The monsoonal heat budget of the hydrosphereatmosphere system in the Indian Ocean sector. J. Geophys. Res., 98, 68696881. Hsuing, J. (1985). Estimates of global oceanic meridional heat transport. J. Phys. Oceanogr., 15, 14051413. Hsuing, J., Newell, R. E., and Houghtby, T. (1989).The annual cycle of oceanic heat storage and oceanic meridional heat transport. Quart. J. Roy. Meteorol. Soc., 115, 128. Kantha,L.H.,and C.A.Clayson (1994), An improved mixed layer model for geophysical applications, J. Geophys. Res., 99, 25,235–25,266. Kantha, L. H., and C. A. Clayson (2000), Numerical Models of Oceans and Oceanic Processes, 940 pp., Academic, SanDiego, Calif. Kantha, L., T. Rojsiraphisal, and J. Lopez, 2008. The North Indian Ocean circulation and its variability as seen in a numerical hindcast of the years 1993 to 2004. Prog. Oceanogr., 76, 111147. Koehl, A., Lu, Y., P. Heimbach, P., Cornuelle, B., Stammer, D., and Wunsch, C. (2002). The ECCO 1 degree global WOCE Synthesis, ECCO Report No.20, See also http://www.ecco–group.org/ reports.html. Lee, T., and Marotzke, J. (1998). Seasonal cycles of meridional overturning and heat transport of the Indian Ocean. J. Phys. Oceanogr., 28, 923943. Lopez, J. W. (1998). A study of physical processes of the northern Indian Ocean using a comprehensive primitive equation numerical model. Ph. D. dissertation, Aerospace Engineering Sciences, University of Colorado, Boulder, CO. pp.144. Lopez, J. W., and Kantha, L. H. (2000a). Results from a numerical model of the northern Indian Ocean: Circulation in the South Arabian Sea. J. Mar. Syst., 24, 97117. Lopez, J. W., and Kantha, L. H. (2000b). A dataassimilative model of the North Indian Ocean. J. Atmos. and Oceanic Tech., 17, 15251540. Loschnigg, J., and P. J. Webster, (2000). A coupled oceanatmosphere system of SST modulation for the Indian Ocean, J. Climate, 13, 33423360. McCreary, J. P. Jr., Kundu, P. K., and Molinari, R. L. (1993). A numerical investigation of dynamics, thermodynamics and mixed layer processes in the Indian Ocean. Prog. Oceanogr., 31, 181244.
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Schott, F., and McCreary, J. P. (2001). The monsoon circulation of the Indian Ocean. Prog. Oceanogr., 51, 1123. Wacongne, S., and Pacanowski, R. (1996). Seasonal heat transport in a primitive equations model of the tropical Indian Ocean. J. Phys. Oceanogr., 26, 26662699. Submitted March 6, 2008
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In: Heat Flux Editors: G. Cirimele and M. D'Elia
ISBN 9781614708872 © 2012 Nova Science Publishers, Inc.
Chapter 7
HEAT FLUX AND TEMPERATURE AT THE TOOLCHIP INTERFACE IN DRY MACHINING OF AERONAUTIC ALUMINIUM ALLOY G. List1, D. Géhin2, A. Kusiak3, J. L. Battaglia3 and F. Girot2 1
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Laboratoire de Physique et Mécanique des Matériaux, UMR CNRS N° 7554, ISGMP, Université de Metz, Ile du Saulcy, Metz, France 2 Laboratoire Matériaux Endommagement Fiabilité et Ingénierie des Procédés. LAMEFIPENSAM de Bordeaux. Esplanade des Arts et Métiers, Talence Cedex, France 3 Laboratoire inter établissement ‘TRansferts Ecoulements FLuides Energétique’, UMR 8508, Ecole Nationale Supérieure d'Arts et Métiers, Esplanade des Arts et Métiers, Talence cedex, France
ABSTRACT The heat flux and temperature rise at the toolchip interface were investigated in the case of dry machining of the aerospace aluminium alloy AA2024 T351. On the one hand, a complete experimental setup allowed us to study in realtime the friction force, the chip geometry (observation by CCD high speed camera) and the heat flux transmitted into the tool (by using an inverse method). On the other hand, numerical simulations of the chip formation were carried out using the finite element method. The obtained values are compared to experimental results to validate the modelling. A good correlation between experiments and numerical simulations was found but the results indicated a strong influence of the contact conditions between the tool and the chip such as seizure or BuiltUp Edge (BUE). From the measured heat flux, the temperature rise was also estimated by using a classic analytical model and was compared to the values found by
Corresponding author: Tél. : (+33) 387315363
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G. List, D. Géhin, A. Kusiak et al. numerical simulations. The trends are the same, but the analytical calculation tends to overestimate the temperature rise compared to the numerical simulations.
Keywords: dry machining, heat flux, temperature, aluminium alloy, ccd high speed camera
1. INTRODUCTION In order to control wear phenomena and optimize the cutting process, the knowledge of the temperature at the toolchip interface is essential. During machining of aeronautic aluminium alloys such as AA2024, temperature plays a significant role in adhesive and diffusion wear, (List, 2005). During a cutting process, heat is generated by (1) plastic work done in the first and secondary shear zones, (2) friction at the tool and the chip at the toolchip interface and between the tool and the workpiece et the flank face. The heat flows into the chip, the workpiece and the tool by an unequal way depending on thermal properties and cutting conditions. The dissipation of heat in the different parts can be investigated by considering the total energy uT consuming by the chip formation:
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uT u S u F1 u F 2
(1a)
where u S , u F 1 and u F 2 are respectively the energies consumed in the primary shear zone, the friction zone between the tool and the chip and in the friction zone between the tool and the workpiece. Introducing the partitioning factors R1, R2 and R3, the total energy can be written after (Loewen, 1954) in the following form: uT R1u S R2 u F 1 (1 R1 )u S (1 R3 )u F 2 (1 R1 )u F 1 ( R3 )u F 2 u u u c w t (1b) where u C , uW and u t are the energies under the form of heat going into the chip the workpiece and the tool as shown in Figure 1. Numerous experimental methods were used in order to measure the temperature rise at the tool rake face (Komanduri, 2001). The major techniques are thermocouples, thermovision or metallographic analyses. The main difficulties of measurement are related to the very thin zone where the temperature gradient is located and the accessibility to the deformation zone during cutting. In most of studies, the temperature at the rake face is estimated by using analytical models. The mean temperature at the interface T int can be expressed by:
T int T S T f
(2)
where T S is the mean temperature in the primary shear zone and T f the mean temperature rise due to the friction phenomena at the tool chip interface. One method used to
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Heat Flux and Temperature at the ToolChip Interface…
calculate the temperature rise T f consists in considering the chip as a band sliding on the tool with a length 2l and a width m. A classical equation given by (Loewen, 1954) using the solution from (Jaeger, 1942) is:
T f
qt .l .A kt
(3)
where qt is the surfacic heat flux due to the friction going to the tool, kt the thermal conductivity of the tool and A a geometrical factor depending on
A
m : l
2 2m 1 l 1 ln l 3 m 2
(4)
The total heat generated by friction is deduced from the rate energy PF1 , (Loewen 1954, Chao 1955):
PF1 FT Vchip
(5)
where Vchip is the chip velocity assumed equal to the relative sliding velocity between the tool and the chip, and FT the friction force:
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FT F c sin 0 F f cos 0 where Fc and Ff are the cutting forces, see Figure 2b.
Figure 1. Heat sources in orthogonal cutting. Heat Flux: Processes, Measurement Techniques and Applications : Processes, Measurement Techniques and Applications, Nova Science
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With computer progress, numerical methods such as finite element method (FEM) become more and more common to simulate the chip formation and calculate the temperature rise. The needs for FEM modelling are the knowledge of the local friction law at the toolchip interface and the mechanical behaviour at high strain, high strain rate and high temperature. In the present paper, the heat flux in the tool is measured by using an inverse method in order to compare and validate FEM or theoretical calculations.
2
0
Chip
3
Fc 1
Ff Tool t1
a
b
Figure 2. Experimental setup for orthogonal cutting. (a) with (1) Kistler dynamometer, (2) CCD high speed camera and (3) long distance microscope. (b) Example of video recording 0 =30°, t1=0.3 mm.
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2. EXPERIMENTAL SETUP Cutting tests were performed on a planer machine GSP 2108 R20 providing a perfect orthogonal cutting configuration, see Figure 2. The chip formation is observed in real time by using a CCD highspeed video camera Phantom v4 coupled to a long distance microscope Questar QM100 (Figure 2a) with a recording speed of 1000 pictures per second. Thanks to this device, it is possible to measure directly the chip thickness t2 and the tool–chip contact length lc. During chip formation, cutting forces Fc and Ff are also measured with the frequency of 1 kHz using a 9257 B Kistler dynamometer fixed on the tool holder. The workpiece is a AA2024T351 aluminium alloy bar with a Tsection. The inserts consist of WC tungsten carbide with cobalt as binder. (94% WC, 6% Co). Three different tools offering a different rake angles (0°, 15° and 30°) were used. The clearance angle is kept constant to 11° for all tools and all cutting conditions. The cutting speed Vc is equal to 60 m/min and the width of cut w is equal to 4 mm. The selected cutting conditions and the corresponding results are presented in Table 1.
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Heat Flux and Temperature at the ToolChip Interface…
201
Figure 3. Detail of a tool (0 =15°) with the thermistor.
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Table 1. Cutting forces and contact length measured for different conditions Test 1 2 3 4 5 6 7 8 9
Vc (m/min) 60 60 60 60 60 60 60 60 60
0 (°) 0 0 0 15 15 15 30 30 30
t1 (mm) 0.05 0.1 0.3 0.05 0.1 0.3 0.05 0.1 0.3
Fc (N) 250 500 1200 250 450 1200 250 400 1100
Ff (N) 150 300 600 125 200 400 100 100 125
lc (mm) 0.2 0.3 0,6 0.15 0,25 0.6 0.12 0.25 0.46
The mean heat flux in the tool Q (W) is estimated by an inverse method. It requires temperature measurement at one location in the tool and a model describing heat transfer in the tool (Battaglia, 2001, Kusiak, 2006). In the present study, the temperature measurement was carried out by using thermistors located just under the insert, see Figure 3. The thermistor sensitivity is 1000 more than a thermocouple and its temperature measurement range, from −50 to 150 °C, is sufficient for the present configuration. The dimensions of the sensors type micro series, length 4 mm and diameter 0.46 mm confer a sufficient fast response time of 250 ms order. The holes where the sensors are located were made by electro erosion. The thermistors are maintained by an epoxy adhesive containing silver to provide an excellent thermal conductivity.
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3. HEAT FLUX ESTIMATION IN THE TOOL 3.1. Inverse Method 3.1.1. Principle According to previous results obtained in the field of system identification concerned with the diffusion process (Battaglia, 2000, 2001, Rech, 2004) it has been demonstrated that the transfer function F s , that relies the heat flux in the tool to the temperature at the sensor is of the fractional form: L
F s
k L0 M
k
k M0
k
s k s
k
,
, M0 1
1 2
(7)
with
Tc s F s Qs
(8)
The inverse Laplace transforms applied on the previous relations lead to express the model in the continuous time domain as: M
k
D k Tc t
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k M 0
D f t
L
k
D k Qt
(9)
k L0
d f t
, ( IR), denotes the fractional derivative of function f t with d t respect to variable t . This operator can be view as the generalization of the classical
derivative of integer order. The summation bounds M 0 , M , L0 , L , in Equation (9), essentially depend on the location of the sensor from the heated surface.
3.1.2. Identification Method The parameters k et k in relation (9) must be identified for each tool in laboratory by applying a controlled heat flux Q(t) at the tool rake face, see Figure 4. In order to simulate the same thermal conditions than those occurring on the tool at the chiptool interface during machining tests, a specific apparatus is used (Rech 2004, Kusiak, 2005). The flux is provided from a micro resistor formed from a platinum wire deposited on an alumina plate 250 µm thickness and (32) mm dimensions. These small dimensions allow to local heating of the zone loaded during the cutting process and confer a weak response time of the heating system close to 100 ms. The micro resistor is supplied with a current generator.
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Heat Flux and Temperature at the ToolChip Interface…
Assuming there is no loss of heat with the ambient, the measured electric power in the heating resistor corresponds to the heat flux dissipated in the tool. The micro resistor is held in contact with the tool using a silver based stick. The stickcutting edge contact area corresponds approximately to the chip friction surface against the insert during machining, according to the chosen cutting parameters (depth of cut and feed). Nevertheless, it has been demonstrated that the sensitivity of the temperature at the sensor according to a 50% variation of the heated surface is weak (less than 5%). This means that the heat flux in the tool is well estimated even if the heated surface is approximately known. Using the identification method, the following relations between the temperature of the sensor Tc(t) and the heat flux in the tool Q(t) are found for respectively the tools with the rake angle : 0 = 0°, 15° and 30°. 1 0,79 D 12 0,65 D 1 0,024 D 3 2 T t 0,258 1,32 D 12 1,79 D 1 Qt c
(10)
1 0,64 D 12 0,83 D 1 0,033 D 3 2 T t 0,31 1,64 D 12 2,36 D 3 2 Qt c
(11)
1 0,36 D 12 0,4 D 1 0,018 D 3 2 T t 0,23 0,93 D 12 1,04 D 3 2 Qt c
(12)
6
temperature/heat flux (°C/W)
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5
4
3
2
1
0
0
10
20
30
40 time (s)
50
60
70
80
Figure 4. Imposed heat Q(t) for the determination of the transfer function F(s) and the thermal response Tc(t).
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3.2. Estimation of Heat Flux by FEM 2D numerical modelling by finite element method was carried out by using the software Thirdwave AdvantedgeTM which is dedicated to the simulation of Metal Cutting. The code uses a Lagrangian formulation for the chip formation described in (Marusich, 1995). The workpiece is discretized by sixnode quadratic triangular elements. Adadaptative remeshing is integrated to resolve the elements distortion problem. The model in planestrain considers a deformable workpiece and a rigid tool. Contact between the tool and the chip is based on a Coulomb law:
f min( p, crit )
(13)
where f is the friction stress, p the pressure at the toolchip interface and µ the friction coefficient. The formulation for material modelling is a thermoviscoplastic behaviour including a threshold strain rate changing the strain rate sensibility:
p p , p , T 0 1 0 if
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m1 p .1 .(T ) O
n
p t .1 1 O O
(14a)
where
n
p t
p , p , T 0 1 if
t which separates the behaviour in two regimes by
p 0
m1
m1 m 2
.(T )
p t
(14b)
is the effective von Mises stress, σ0 is the initial yield stress at the reference 0p p
temperature T0, is the accumulated plastic strain, is a reference plastic strain rate, n is the hardening exponent, m1 and m2 are respectively low and high strainrate sensitivity exponents. is a thermal softening function which has the typical form :
(T ) 1 a(T T0 )
(15)
where a is a real and T0 the reference temperature. The rate of heat supply due to the plastic deformation is estimated as:
s .W p
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Heat Flux and Temperature at the ToolChip Interface…
p
where W is the plastic power per unit deformed volume and the TaylorQuinney coefficient ( = 0.91). The heat generated by the friction is given by Equation (5). The heat is given to each of the two contacting bodies (chip and tool) by equal proportions. Moreover, the workpiece loses heat to the environment due to convection according to the following relation:
qh h(TW T0 )
(17)
where TW is the workpiece surface temperature, T0 the reference temperature (room temperature) and h is the convection heat transfer coefficient of the workpiece. The mean surfacic heat flux (W/mm2) can also be determined from the temperature field given by the simulation, see (Figure 5) by using the Fourier’s law in the direction perpendicular to the rake face:
qt
FEM
k t .
T n
(18)
k
where t is the thermal conductivity of the tool and T the temperature in the tool. For all simulations, material properties and mechanical behaviour were those provided by default by AdvantedgeTM except for the friction coefficient µ. According to (List, 2005), µ was chosen equal to 0.8. The physical properties of the tool and the workpiece are given in Table 2.
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Table 2. Physical properties of the tool and workpiece
Tool Workpiece
(g/cm3) 15 2.78
kt (W/m.°K) 100 120
Cp (J/kg.°K) 240 856
a Figure 5. Continued on next page. Heat Flux: Processes, Measurement Techniques and Applications : Processes, Measurement Techniques and Applications, Nova Science
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b Figure 5. Temperature distribution estimated by finite element method, 0 =15°, t1=0.1 mm, Vc=60m/min.
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4. RESULTS AND DISCUSSION Figures 6, 7 and 8 show the heat flux transmitted to the tool determined from the measured temperature Tc(t) at the thermistor. The cutting time duration is about 1.5 s for all tests during which the transient temperature Tc(t) at the thermistor increases while the mean flux Q(t) is constant. The flux Q(t) were found between 18 W and 78 W according to the cutting conditions. The values are consistent with the flux used in (Stephenson, 1992) in similar cutting conditions. The heat flux increases with the feed t1 and decreases with the rake angle 0 according to the variation of the friction force FT and the contact length lc. Table 3 shows the surfacic heat flux qt simulations and qt
qt
theo
theo
(1 R2 ).
mes
obtained by experiments, qt
FEM
obtained by numerical
calculated by using Equation (19):
FT .Vchip l c .w
(19)
The chip velocity Vchip is calculated by the conservation Equation (20):
Vc t1 Vchip t 2
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(20)
Heat Flux and Temperature at the ToolChip Interface…
207
Figure 6. Temperature and flux for 0 =0° and Vc=60m/min (a) Temperature measured Tc(t) at the thermistor. (b) Mean heat flux Q(t) dissipated in the tool.
The results of experiments and numerical simulations are similar showing a good correlation between the two methods. For the theoretical solution, as shown in Equation (19), the result strongly depends on the value of R2. A value close to 0.85 needs to be chosen in order to obtain values similar to qt
FEM
or qt
mes
. Owing to the presence of the BuiltUp
Edge (BUL), the most important differences between qt
mes
and qt
FEM
are observed for the
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combination of a weak feed (t1 = 0.05 mm – 0.1 mm) and the lower rake angle (0 = 0°), see Figure 9. BUL modifies strongly the effective rake angle and the tribological conditions which are not reproduced in FEM.
Figure 7. Temperature and flux for 0 =15° and Vc=60m/min (a) Temperature measured Tc(t) at the thermistor. (b) Mean heat flux Q(t) dissipated in the tool.
Figure 8. Temperature and flux for 0 =30° and Vc=60m/min (a) Temperature measured Tc(t) at the thermistor. (b) Mean heat flux Q(t) dissipated in the tool.
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G. List, D. Géhin, A. Kusiak et al.
a
b
Figure 9. Observation of chip formation in real time by high speed video for 0=0° and Vc=60m/min (a) t1=0.05 mm. (b) t1=0.1 mm.
The temperature rise can be estimated from the measured or theoretical heat flux by using Equation (3) with m = w =4 mm and l =lc/2. Results are presented in Table 3 where T the temperature rise calculated from qt
qt
theo
and T
FEM f
mes
, T
theo f
est f
is
the temperature rise calculated from
temperature rise deduced from FEM modelling. The trends are similar
for the three quantities showing the increase with the feed and the decrease with the rake angle, but the values are different.
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Table 3. Heat flux into the tool and temperature rise at the toolchip interface 0
mes
T
est
theo
T f
theo
T f
FEM
FEM
(W/mm ) (°C)
lcFEM (mm)
TmaxFEM (°C)
10.5
35
0.19
170
45
60
0.32
200
163
38.5
105
0.56
255
36
71
11
40
0.16
158
111
23,8
95,8
38
65
0.23
190
23,5
171
27,2
198
34.5
110
0.55
240
0,05
37,5
77,5
36
75,8
44.4
45
0.09
155
30
0,1
26
96.5
24,9
88,7
29
55
0.18
168
30
0,3
27
164
38,3
231
32
110
0.42
235
qt
(°)
t1 (mm)
1
0
0,05
(W/mm ) (°C) 28 70
2
0
0,1
67
3
0
0,3
4
15
5
test
f
qt
(W/mm ) 22
(°C) 54.7
154
32
73
32
230
22,5
0,05
37.5
74
15
0,1
30
6
15
0,3
7
30
8 9
2
2
qt
2
The results using Equation (3) seem to overestimate the temperature rise at the toolchip interface. Indeed, the analytical model considers the chip as a rigid body sliding on the rake face with a velocity at the interface equal to Vchip. However, experimental results for dry machining of aluminium alloys AA2024 have shown that the seizure condition at the tool chip interface has an important effect, (List, 2005). Thus, the velocity at the toolchip interface is probably not equal to the chip velocity and the secondary shear zone cannot be neglected. Seizure is characterized by a weak sliding velocity in a large part of the rake face.
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Heat Flux and Temperature at the ToolChip Interface…
FEM modelling tends to the same results because a high value of the friction coefficient was chosen. The friction stress f reaches rapidly the critical stress and a large zone of sticking is found. To take into account the overestimation in the analytical model, a corrective factor K can be introduce in Equation (3) such as: mes
T
f
K.
qt .l c .A 2.k t
(21)
Based on the results obtained in the finite element methods, a factor K=0.6 allows to get a good estimation of the temperature rise caused by the friction phenomena at the toolchip interface.
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CONCLUSION The heat flux going into the tool was measured by an experimental method during machining of aeronautic aluminium alloy AA2024 T351. In this approach, it is not necessary to know the sliding velocity at the tool chip interface or the part (1R2) of the generated heat which goes into the tool. These two last parameters are indeed not easy to determine by experiments and are a great source of error in modelling. For different cutting conditions, the measured heat flux was compared to FEM results and a good correlation between experiments and numerical simulations was found. Results analysis also indicated that the heat transmitted to the tool strongly depends on the cutting conditions (feed and rake angle) which change the tribological conditions at the toolchip interface. The present approach shows that the measured heat flux can be used in the aim of validate numerical or analytical calculations except for the cases where an important BuiltUp Edge (BUE) appears. The phenomenon of BUE is important when weak feeds are combined with low rake angles. Moreover, the study has shown that the calculation of the temperature increase by using an analytical method trends to overestimate the values compared to the finite element methods. Based on the FEM results, a factor K=0.6 can be used to correct the overestimation.
REFERENCES Battaglia J. L, Cois O., L. Puigsegur, A. Oustaloup, (2001), “Solving an inverse heat conduction problem using a noninteger identified model », International Journal of Heat and Mass Transfer, 44, , 26712680. Battaglia J.L., Le Lay L., Batsale J.C., Oustaloup A., Cois O., (2000), “Heat flux estimation through inverted non integer identification models,” International. Journal of Thermal Science, 39, .374389. Chao BT, Trigger KJ.(1955), “Temperature distribution at the toolchip interface in metal cutting”. Transactions of the ASME; 77, 11071121. Jaeger JC.(1942), “Moving sources of heat and the temperature at sliding contacts”,. Proceedings of the Royal Society of Royal Society of New South Wales, 76, 203224.
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Kusiak A,. Battaglia J.L, Marchal R. (2006), “Influence of CrN coating in wood machining from heat flux estimation in the tool”, International Journal of Thermal Sciences, 45, 10351044. Komanduri R., Hou Z.B, (2001), “A review of the experimental techniques for measurement of heat and temperatures generated in some manufacturing processes and tribology”, Tribology International, 34, 653682. List G,. Nouari M., Géhin D., Gomez S., Manaud J. P; Le Petitcorps Y., Girot F., (2005) “Wear Behaviour of Cemented Carbide Tools in Dry Machining of Aluminium Alloy”, Wear, 259, 11771189. Loewen EG, Shaw MC., (1954), “On the analysis of cutting tool temperatures”, Transactions of the ASME, 76, 217231. Marusich T.D., Ortiz M., (1995), “Modelling and Simulation of High Speed Machining”, International Journal for Numerical Methods in Engineering, 38, 36753694. Rech J., Kusiak A and Battaglia J. L, (2004), “Tribological and thermal functions of cutting tool coatings”, Surface and Coatings Technology, 186, 364371. Stephenson D.A., Ali A., (1992), “Tool Temperature in Interrupted Metal Cutting, Journal of Engineering for Industry”, Transaction of the ASME , 114, 127136.
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INDEX
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A accessibility, 198 accounting, 49, 110, 112 acid, 107, 112 adaptation, 122, 123, 125 Aegean Sea, 158 aerosols, 165 aerospace, ix, 197 agar, 142 age, 130 aggregation, 123 air temperature, 42 alcohol oxidase, 135 algorithm, 25, 34, 37, 48, 67, 91, 111 aluminium, ix, 21, 197, 198, 200, 208, 209 ammonia, 109 amplification of conduction, ix, 167, 182, 184 amplitude, 162 anabolism, 125 anomalous diffusion, 169 aseptic, 114 assessment, 45 assimilation, 188 asymmetry, 176 atmosphere, 52, 195 atmospheric pressure, 138, 139, 144, 146, 153 attachment, 12
B bacteria, 122, 134 bacterium, 123 base, 107, 112, 113, 114, 133, 188 bench scale heatflux biocalorimeter, vii, 105, 110 benefits, viii, 106, 135
biochemical processes, vii, 105 biodegradation, 119 biofuel, 127, 132 biokinetics, 119 biological activity, 108, 129, 132 biological processes, 109 biological systems, vii, 105, 110, 112 biomass, 106, 108, 109, 116, 117, 118, 119, 120, 121, 124, 125, 126, 128, 130, 132, 134 biomass growth, 116, 120, 126 biosensors, 109, 129, 131 biotechnology, 110, 116, 128, 132, 133, 134, 136 biothermodynamics, vii, 105 bismuth, 142 Black Sea, 155 boilers, 8, 20, 44 boundary surface, 78, 80 boundary value problem, 47 bounds, 202 BuiltUp Edge (BUE), ix, 197, 209 byproducts, 119
C Cairo, 167 calibration, 112, 114, 115, 129, 133 calorimetric measurements, 136 calorimetry, vii, 105, 108, 109, 112, 116, 117, 118, 119, 121, 123, 124, 126, 127, 128, 130, 131, 132, 133, 134, 136 carbon, 51, 109, 113, 117, 122, 123 carbon dioxide, 109, 113 carboxymethyl cellulose, 140 case study, 116, 119 casting, 74 catabolism, 109, 125 category b, 57
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Index
cell culture, 109, 122, 126 cell line, 110 cell metabolism, 109 cellulose, 140 ceramic, 53 chemical, vii, 105, 110, 113, 117, 118, 123, 124, 126, 128, 129, 133 chemical reactions, 123, 124, 133 chemicals, 125 circulation, 111, 152, 165, 187, 188, 195, 196 climate, 164, 165 closure, 157, 164, 188 coal, 51, 52 coatings, 100, 210 cobalt, 200 color, 53 combustion, 8, 10, 12, 13, 17, 100, 106, 118, 134 commercial, 110, 130 communities, viii, 106, 128, 132 comparative analysis, 126 complexity, 115, 118 compliance, 25 complications, 128 composition, 118, 119, 126, 134 compounds, 118 computation, 78 computer, 74, 78, 91, 200 computing, 37, 78, 141 condensation, 147, 149, 152, 153 conduction, vii, ix, 2, 4, 8, 9, 10, 12, 13, 14, 20, 21, 25, 31, 32, 38, 40, 42, 44, 45, 46, 48, 52, 56, 57, 59, 60, 64, 74, 76, 78, 79, 80, 82, 83, 85, 90, 92, 94, 97, 98, 99, 100, 101, 102, 103, 145, 147, 149, 152, 153, 167, 182, 184, 209 conductivity, vii, viii, 2, 4, 5, 8, 9, 10, 17, 20, 21, 22, 23, 24, 29, 30, 31, 33, 34, 35, 38, 40, 42, 44, 46, 49, 51, 91, 137, 138, 139, 140, 141, 142, 143, 144, 145, 146, 147, 148, 152, 153, 154, 173, 174, 175, 182, 199, 201, 205 configuration, 200, 201 confinement, viii, 167, 180, 184 conservation, 171, 206 constant rate, 6, 116 construction, 65, 90 consumption, 123, 126, 130 convergence, 32, 34, 86 conversion rate, 117, 118 cooling, 10, 12, 15, 66, 70, 74, 99, 126, 139, 140, 141 cooling process, 139, 141 copper, 14, 15, 20, 21, 66, 139, 141, 142 correlation, ix, 126, 130, 141, 197, 207, 209 correlation coefficient, 141
correlations, 118, 119, 135 cost, 78 covering, 142 crossvalidation, 64, 102 crown, 50 crystals, 20, 32, 53, 56, 99 cultivation, 110, 120, 123, 126, 134 culture, 113, 114, 116, 122, 123, 126, 127, 129, 134 culture conditions, 113, 114 cutting force, 199, 200 cycles, 164, 195
D data analysis, 78 data set, 65, 166 database, 119, 125, 157 DCA, 139 decay, 76, 116 decomposition, 32, 37 deficit, 129 deformation, 198, 204 degradation, 116, 129 deposition, 45 depth, 1, 53, 163, 188, 203 derivatives, 1, 29, 59, 61, 62, 63, 64, 69, 72, 74, 80, 87 detection, 109, 121, 123, 129, 133 deviation, 3, 23, 29, 35, 91, 93, 94, 160 differential equations, 84 diffusion, 139, 142, 145, 146, 148, 149, 153, 169, 182, 198, 202 diffusion process, 202 diffusivity, 3, 5, 70, 150, 152, 176 direct measure, 98, 157, 164 discharges, 176 discontinuity, 64 discretization, 34, 37 dissolved oxygen, 126 distribution, 3, 6, 10, 14, 16, 19, 21, 23, 24, 25, 27, 28, 29, 30, 31, 32, 33, 36, 41, 42, 43, 44, 45, 46, 48, 49, 50, 52, 54, 67, 78, 79, 80, 82, 84, 85, 89, 92, 94, 99, 157, 177, 179, 180, 181, 182, 183, 206, 209 divergence, 169, 170, 174 divertor, vii, viii, 167, 168, 175, 176, 181, 182, 183, 184 DOI, 134 dosage, 122 dosing, 110
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Index
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E ecology, 133 Egypt, 167 electric field, 171, 178, 179, 180, 185 electrical resistance, 142 electron, viii, 167, 168, 170, 171, 174, 175, 176, 177, 178, 183 electrons, 171, 173, 174, 175, 176 emergency, 74 emission, 175 employment, 126 endothermic, 110, 119, 125, 126 energy, vii, 14, 15, 105, 106, 108, 109, 110, 113, 123, 124, 125, 128, 134, 135, 164, 173, 174, 198, 199 energy efficiency, 124 engineering, 109, 117, 128, 131, 133, 152, 175 England, 102 entropy, 124, 125, 135 environment, 70, 107, 110, 112, 113, 128, 131, 132, 205 environmental conditions, 116 enzymatic activity, 126 EPS, 185 equality, 2, 26 equilibrium, 129 equipment, 78, 115 erosion, 201 evaporation, 147, 149 evolution, 107, 109, 112, 117, 127 exclusion, 125 experimental condition, 145 exploitation, 130
F fermentation, viii, 106, 116, 118, 126, 128, 131, 132, 133 filters, 8, 62 finite element method (FEM), ix, 34, 45, 49, 50, 197, 200, 204, 206, 207, 208, 209 flame, 45, 46, 49 flank, 198 flexibility, 61, 115 flow field, 32, 44 fluctuations, ix, 110, 111, 126, 162, 187, 194 fluid, viii, 2, 3, 4, 5, 19, 20, 32, 36, 38, 44, 48, 49, 52, 53, 54, 55, 57, 70, 71, 92, 95, 99, 152, 167, 168, 184 fluidized bed, 101 force, ix, 124, 170, 172, 197, 199, 206
formation, ix, 111, 116, 124, 128, 159, 167, 180, 182, 184, 197, 198, 200, 204, 208 formula, 8, 11, 14, 17, 20, 21, 49, 55, 56, 78, 117, 118, 119, 156 France, 109, 197 free energy, 106, 124, 125 freedom, 36 freshwater, 156, 165 friction, ix, 101, 170, 171, 173, 197, 198, 199, 200, 203, 204, 205, 206, 209 fungi, 134
G gel, 142 geometry, ix, 90, 176, 197 Germany, 137 Gibbs energy, 106, 124, 135 Gibraltar, viii, 155, 156, 157, 163, 164, 165, 166 gland, 13, 19 global warming, 128, 132 glucose, 109, 116, 117, 119, 121, 122, 123, 124, 129, 134 glycerol, 126, 127, 135 glycolysis, 124 graph, 151 growth, vii, 79, 105, 106, 108, 109, 111, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128, 132, 133, 134, 135, 136 growth rate, 106, 117, 120, 121, 125, 126, 127
H health, 128 heat capacity, 91, 106, 115 heat conductivity, 20, 174, 175 heat loss, viii, 9, 57, 111, 113, 114, 155, 156, 162, 164 heat release, 122, 147 heat transfer, vii, viii, 1, 2, 4, 8, 12, 19, 20, 21, 22, 23, 25, 28, 31, 32, 33, 36, 37, 38, 40, 42, 43, 44, 45, 46, 48, 49, 51, 52, 53, 54, 56, 57, 58, 59, 64, 70, 71, 72, 73, 74, 75, 76, 78, 87, 91, 94, 95, 98, 99, 100, 101, 107, 112, 113, 115, 131, 137, 138, 143, 144, 145, 147, 149, 150, 151, 152, 153, 201, 205 heatflux calorimetry, vii, 105, 119, 123, 127 heating rate, 101 height, 21, 51, 188 history, 53, 56, 57, 59, 64, 66, 69, 78, 80, 87 humidity, 107, 138, 145 hydrosphere, 195
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Index
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I ideal, 53 identification, 31, 202, 203, 209 identity, 2, 35, 48 imitation, 2, 129 improvements, 110 in vitro, 134 India, 105 induction, 126 industries, 74 industry, viii, 106, 132 inertia, 169, 171 inhibition, 116, 121, 130 initiation, 122 inoculation, 114 insertion, 115 insulation, 5, 12, 45 integration, 11, 24 interface, ix, 4, 60, 78, 131, 163, 197, 198, 200, 202, 204, 208, 209 interference, 131 inversion, 40 ion heat flux, viii, 167, 168, 170, 175, 176, 178, 181, 182, 183, 184 ion temperature, ix, 167, 168, 176, 177, 179, 180, 182, 184 ionization, 174, 176 ions, 168, 169, 172, 173, 176 Ireland, 101 issues, 110, 128, 131, 194 iteration, 2, 3, 27, 28, 29, 34, 35, 36, 38, 40, 48, 49, 85
J Japan, 137, 153
K kinetics, 116 knots, 63
L Lagrange multipliers, 25, 26, 27 Lagrangian formulation, 204 lead, 139, 142, 176, 202 least squares, 87 LED, 128 light, 128, 188
linear function, 5, 24, 151 liquid crystals, 20, 53, 56, 99 logging, 6 low temperatures, 40
M magnetic field, 168, 169, 175, 176 magnitude, 65, 67 manganese, viii, 137, 143, 153 manufacturing, 56, 210 Maryland, 191 mass, 20, 32, 98, 99, 118, 119, 120, 127, 129, 139, 148, 149, 150, 151, 153, 165, 166, 187 material surface, 57 materials, 154 matrix, 2, 34, 35, 36, 38, 40, 48, 116 matter, 113 measurement, vii, 2, 4, 5, 6, 7, 8, 13, 14, 20, 21, 23, 25, 28, 29, 30, 33, 35, 37, 40, 53, 56, 57, 60, 64, 65, 67, 68, 71, 74, 75, 79, 81, 83, 87, 91, 98, 99, 100, 101, 105, 108, 109, 110, 112, 115, 116, 127, 128, 131, 139, 140, 153, 198, 201, 210 measurements, vii, viii, 4, 6, 8, 17, 18, 31, 32, 33, 35, 36, 38, 40, 44, 45, 48, 53, 56, 58, 60, 63, 66, 67, 68, 69, 72, 73, 74, 75, 76, 77, 78, 80, 82, 90, 91, 94, 95, 96, 97, 99, 100, 101, 105, 109, 112, 116, 117, 126, 127, 129, 130, 132, 133, 136, 138, 139, 141, 148, 155, 157, 163, 164, 166 media, 79, 114, 116, 119, 121, 122, 123, 127, 135, 154 Mediterranean, v, vii, viii, 155, 156, 157, 159, 160, 161, 162, 163, 164, 165, 166 melting, 100 mercury, viii, 137, 143, 153 metabolic, 107, 120, 122, 126, 134 metabolic pathways, 125 metabolism, 108, 109, 116, 124, 125, 126, 129, 130, 132, 133, 135 metals, viii, 22, 137, 143, 153 meter, vii, 4, 6, 8, 12, 15, 17, 18, 19, 44, 52, 98, 110, 112 methanol, 126, 135 methodology, 116, 157, 188 microcalorimetry, 109, 129, 132, 133, 136 microorganisms, 134 microscope, 200 mixing, 109, 115, 188 models, 78, 101, 115, 118, 119, 121, 125, 129, 168, 190, 193, 194, 197, 198, 200, 204, 208, 209 modifications, vii, 105, 112, 131, 136 modulus, 1 mole, 106, 118, 124
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Index molecules, 137, 138, 143, 144, 153 momentum, ix, 166, 167, 168, 170, 171, 173, 175, 176, 182, 184 multidimensional, 31, 79 multiplier, 27
N naphthalene, 20, 98, 99 NCEP climatological datasets, viii, 155 nematic liquid crystals, 53 neutral, viii, ix, 155, 162, 164, 167, 168, 171, 173, 176, 182, 183, 184 New South Wales, 209 next generation, 116 nickel, 20 nitrogen, 138, 144, 153 nodes, 25, 28, 34, 35, 62, 80, 84 North Indian Ocean, v, vii, ix, 187, 188, 194, 195 null, 176 nutrient, 123, 129
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O oceans, 194 oil, 107, 110, 111, 112 one dimension, 20 opportunities, 119 optical properties, 109 optimization, 26, 63, 102 ordinary differential equations, 84 organism, 119, 121, 123, 124 oscillation, 115, 133, 162, 163 oxygen, 109, 116, 118, 126, 129, 130, 138, 144, 153 oxygen consumption, 126
P palladium, 57 parallel, 44, 45, 119, 170, 171, 173, 175, 176, 182 parameter estimation, 37 pathways, 125, 135 photons, 128 physical properties, 15, 96, 138, 205 physics, 167, 184 physiology, vii, 105, 127 pitch, 2, 42, 49, 175 plants, 132 plasma current, 168 plastic deformation, 204 platform, 131 platinum, 53, 57, 202
Poland, 1, 99 pollution, 128 polymer, 116 polymer matrix, 116 population, 136 porosity, 145 porous materials, 154 porous media, 154 precipitation, 156 pressure gradient, viii, 152, 167, 170, 171, 180, 183 principles, 131 probe, 10, 13, 14, 19, 30, 31, 44, 98, 139, 140, 141, 142, 151, 152, 153, 157, 163 process control, 136 programming, 37 project, 157, 165, 191 propagation, vii, 4, 29, 31, 32, 33, 35, 79 proportionality, 118 protection, 12, 13 Pseudomonas aeruginosa, 134, 135 PTFE, 108, 115
Q quantitative biocalorimetric, viii, 105 quartz, 57
R radial distribution, 179, 180, 181, 182, 183 radial heat flux, viii, 167, 168, 176, 180, 182, 183, 184 radiation, 4, 5, 45, 49, 57, 98, 156, 157, 160, 162, 174 radius, 1, 2, 3, 34, 47, 48, 49, 91, 169 random errors, 69, 71, 74, 79, 87, 91, 93, 95 reactant, 110 reactants, 124 reaction temperature, 111 reactions, vii, 105, 108, 110, 113, 119, 123, 124, 125, 126, 128, 131, 133 real time, 110, 115, 120, 121, 130, 200, 208 reasoning, 125 recombination, 176 reconciliation, 153 recovery, 76, 87 Red Sea, 165 regression, 60, 63, 141, 146, 148, 151 regression line, 141, 146, 148, 151 regression model, 63 relaxation, 37 requirements, 17, 131, 184
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researchers, 119, 125 resistance, 19, 20, 21, 38, 42, 53, 57, 142, 149 resolution, vii, 90, 101, 105, 110, 129, 131, 133, 139, 157, 162, 166, 188, 189, 191, 193, 194 respiration, 118, 130, 133 response, 19, 53, 60, 62, 70, 98, 109, 111, 131, 201, 202, 203 response time, 111, 131, 201, 202 restrictions, 61 rheology, 115 rings, 9, 10 rods, 21, 57 room temperature, 205 root, 3, 67, 68, 70, 71, 85, 91 roots, 36 roughness, 63 Royal Society, 209 runoff, 156
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S salinity, 166 salts, 117 saturation, 51 scaling, 2, 61, 65, 123 scatter, 74, 118 science, vii, viii, 106 scope, 128, 132 sea level, 160, 164, 165 seasonal component, 187 seasonal flu, ix, 187, 194 secretion, 126, 127 seizure, ix, 197, 208 sensitivity, 79, 109, 110, 111, 115, 128, 131, 133, 139, 188, 201, 203, 204 sensor temperature, vii, 4 sensors, 8, 9, 10, 11, 13, 21, 37, 53, 109, 115, 131, 201 shape, 32, 70, 152, 160 shear, viii, 167, 178, 179, 180, 183, 198, 208 shock, 74 showing, 41, 122, 127, 188, 207, 208 signals, 128 silicon, 139 silver, 201, 203 simulation, vii, viii, 167, 168, 176, 179, 182, 183, 184, 192, 204, 205 simulations, ix, 166, 168, 197, 205, 206, 207, 209 skin, 56, 57, 58, 101 slag, 8, 52 sludge, 129, 130, 136 smoothing, 3, 56, 61, 63, 64, 65, 67, 68, 70, 71, 74, 87, 88, 97, 102
software, 45, 115, 204 soil particles, 137, 154 solid state, 132 solution, vii, 4, 8, 11, 20, 23, 26, 27, 29, 32, 34, 35, 38, 41, 44, 45, 49, 57, 59, 63, 64, 70, 71, 78, 80, 82, 83, 84, 85, 89, 92, 93, 95, 97, 98, 101, 102, 175, 199, 207 sound speed, 175 South Asia, ix, 187 Spain, 155, 165 species, 108, 123, 126, 168, 169 specific heat, 1, 5 Spring, 161 St. Petersburg, 185 stability, 79, 80, 114, 128, 131 standard deviation, 3, 29, 32, 35, 36, 40, 44, 91, 93, 94, 160 standard error, 1, 94 state, vii, 4, 6, 16, 19, 21, 32, 38, 76, 99, 123, 127, 132, 141, 142, 143, 150, 151, 153, 154 states, 108 statistics, 94, 102 steel, viii, 21, 38, 42, 51, 71, 75, 76, 96, 137, 139, 141, 142, 143, 153 storage, 195 stress, 3, 87, 91, 94, 204, 209 structure, 152 substitution, 34 substrate, 53, 54, 57, 58, 59, 106, 107, 108, 109, 113, 116, 117, 118, 120, 121, 122, 123, 126, 127, 135 substrates, 56, 109, 117, 119, 121, 126, 134 supervision, 128 surface area, vii, 4, 131 surface layer, 187 Switzerland, 109, 110 symmetry, 42
T target, viii, 167, 168, 175, 176, 183 techniques, vii, viii, 20, 31, 32, 56, 70, 80, 99, 109, 110, 119, 137, 138, 198, 210 technology, 136 temperaturetime measurements, vii, 4 Thailand, 187 thermal expansion, 3 thermal properties, 59, 71, 76, 78, 79, 80, 84, 85, 87, 91, 96, 198 thermal resistance, 19, 21, 149 thermodynamic calculations, 125 thermodynamics, 124, 125, 133, 135, 195 third boundary condition, 17 time series, 162, 163
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Index tokamak, vii, viii, 167, 168, 169, 176, 181, 182, 183, 184, 185 total energy, 109, 198 transducer, 38 transformation, 16, 34, 166 transformations, 123 transport, vii, viii, ix, 152, 156, 166, 167, 168, 176, 182, 183, 187, 188, 189, 193, 194, 195, 196 treatment, 128, 130, 131, 136 tribology, 210 triggers, 152, 153 tungsten, 200 tungsten carbide, 200 turbulence, 43 turbulent mixing, 188
U uniform, 3, 5, 8, 19, 42, 46, 58, 66, 67, 70, 76, 92, 157, 160 USA, 109, 165, 187
V
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vacuum, 139, 142, 152 validation, 64, 102, 165 valve, 110, 111 vapor, 115, 153 variables, 11, 47, 109, 117 variance propagation rule, vii, 4, 31
variations, viii, 8, 15, 33, 60, 76, 77, 78, 113, 125, 133, 155, 157, 158, 163, 164, 165 vector, 2, 34, 35, 48, 169 velocity, 42, 146, 169, 172, 174, 175, 179, 180, 199, 206, 208, 209 versatility, 132 vessels, 74 viscosity, 110, 170, 171, 172, 173, 176
W Wales, 209 wall temperature, 2, 42, 48, 75, 90 Washington, 166 wastewater, 122, 128, 129, 130, 132, 136 water, 2, 10, 11, 12, 13, 14, 15, 17, 18, 19, 38, 44, 45, 46, 49, 51, 52, 70, 74, 100, 126, 128, 138, 139, 140, 141, 142, 143, 144, 145, 146, 147, 148, 149, 150, 151, 152, 153, 154, 164, 165, 166, 188 water vapor, 153 weakness, 37, 61, 80 wear, 198 wind speeds, 166 wires, 101, 139 wood, 210
Y yeast, 109, 133, 136 yield, viii, 105, 108, 116, 118, 119, 120, 121, 124, 125, 126, 127, 128, 132, 135, 204
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