Liquid Propellant Rockets 9781400879953

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PRINCETON AERONAUTICAL PAPERBACKS 1. LIQUID PROPELLANT ROCKETS David Altman, James M. Carter, S. S. Penner, Martin Summerfleld. High Temperature Equilibrium, Expansion Processes, Combustion of Liquid Propellants, The Liquid Propellant Rocket Engine. 196 pages. $2.95 2. SOLID PROPELLANT ROCKETS Clayton Huggett, C. E. Bartley and Mark M. Mills. Combustion of Solid Propellants, Solid Propellant Rockets. 176 pages. $2.45 3. GASDYNAMIC DISCONTINUITIES Wallace D. Hayes. 76 pages. $1.45 4. SMALL PERTURBATION THEORY W. R. Sears. 72 pages. $1.45 5. HIGHER APPROXIMATIONS IN AERODYNAMIC THEORY. M. J. Lighthill. 156 pages. $1.95 6. HIGH SPEED WING THEORY Robert T. Jones and Doris Cohen. 248 pages. $2.95 PRINCETON UNIVERSITY PRESS · PRINCETON, N. J.

NUMBER 1 PRINCETON AERONAUTICAL PAPERBACKS COLEMAN duP. DONALDSON, GENERAL EDITOR

LIQUID PROPELL AMT ROCKETS BY DAVID ALTMAN JAMES M. CARTER, S. S. PENNER, AND MARTIN SUMMERFIELD

PRINCETON, NEW JERSEY PRINCETON UNIVERSITY PRESS

1960

© COPYRIGHT, 1956, 1959, I960, BY PRINCETON UNIVERSITY PRESS

L. c. CARD 60-12048

Reproduction, translation, publication, use, and dis­ posal by and for the United States Government and its officers, agents, and employees acting within the scope of their official duties, for Government use only, is per­ mitted. At the expiration of ten years from the date of publication, all rights in material contained herein first produced under contract Nonr-03201 shall be in the public domain.

PRINTED IN THE UNITED STATES OF AMERICA

HIGH SPEED AERODYNAMICS AND JET PROPULSION

BOARD OF EDITORS THEODORE VON KARMAN, Chairman HUGH L. DHYDEN HUGH S. TAYLOR COLEMAN DUP. DONALDSON, General Editor, 1956Associate Editor, 1955-1956 JOSEPH V. CHARYK, General Editor, 1952-

Associate Editor, 1949-1952 MARTIN SUMMERFIELD, General Editor, 1949-1952 RICHARD S. SNEDEKER, Associate Editor, 1955-

I. II. III. IV. V. VI. VII. VIII. IX. X. XI. XII.

Thermodynamics and Physics of Matter. Editor: F. D, Rossini Combustion Processes. Editors: B. Lewis, R. N. Pease, H. S. Taylor Fundamentals of Gas Dynamics. Editor: H. W. Emmons Theory of Laminar Flows. Editor: F. K. Moore Turbulent Flows and Heat Transfer. Editor: C. C. Lin General Theory of High Speed Aerodynamics. Editor: W. R. Sears Aerodynamic Components of Aircraft at High Speeds. Editors: A. F. Donovan, H. R. Lawrence High Speed Problems of Aircraft and Experimental Methods. Editors: A. F· Donovan, H. R. Lawrence, F. Goddard, R. R. Gilruth Physical Measurements in Gas Dynamics and Combustion. Editors: R. W. Ladenburg, B. Lewis, R. N. Pease, H. S. Taylor Aerodynamics of Turbines and Compressors. Editor: W. R. Hawthorne Design and Performance of Gas Turbine Power Plants. Editors: W. R. Hawthorne, W. T. Olson Jet Propulsion Engines. Editor: Ο. E. Lancaster PRINCETON, NEW JERSEY PRINCETON UNIVERSITY PRESS

PREFACE The favorable response of many engineers and scientists throughout the world to those volumes of the Princeton Series on High Speed Aerody­ namics and Jet Propulsion that have already been published has been most gratifying to those of us who have labored to accomplish its completion. As must happen in gathering together a large number of separate contributions from many authors, the general editor's task is brightened occasionally by the receipt of a particularly outstanding manuscript. The receipt of such a manuscript for inclusion in the Prince­ ton Series was always an event which, while extremely gratifying to the editors in one respect, was nevertheless, in certain particular cases, a cause of some concern. In the case of some outstanding manuscripts, namely those which seemed to form a complete and self-sufficient entity within themselves, it seemed a shame to restrict their distribution by their inclusion in one of the large and hence expensive volumes of the Princeton Series. In the last year or so, both Princeton University Press, as publishers of the Princeton Series, and I, as General Editor, have received many enquiries from persons engaged in research and from professors at some of our leading universities concerning the possibility of making available at paperback prices certain portions of the original series. Among those who actively campaigned for a wider distribution of certain portions of the Princeton Series, special mention should be made of Professor Irving Glassman of Princeton University, who made a number of helpful sug­ gestions concerning those portions of the Series which might be of use to students were the material available at a lower price. In answer to this demand for a wider distribution of certain portions of the Princeton Series, and because it was felt desirable to introduce the Series to a wider audience, the present Princeton Aeronautical Paperbacks series has been launched. This series will make available in small paper­ backed volumes those portions of the larger Princeton Series which it is felt will be most useful to both students and research engineers. It should be pointed out that these paperbacks constitute but a very small part of the original series, the first seven published volumes of which have averaged more than 750 pages per volume. For the sake of economy, these small books have been prepared by direct reproduction of the text from the original Princeton Series, and no attempt has been made to provide introductory material or to eliminate cross references to other portions of the original volumes. It is hoped that these editorial omissions will be more than offset by the utility and quality of the individual contributions themselves. Coleman duP. Donaldson, General Editor PUBLISHER'S NOTE: Other articles from later volumes of the clothbound series, High Speed Aerodynamics and Jet Propulsion, may be issued in similar paperback form upon completion of the original series in 1961.

CONTENTS A. High Temperature Equilibrium

1. 2. 3. 4. 5. 6.

James M. Carter, Aerojet Engineering Corporation, Azusa, California David Altman, Jet Propulsion Laboratory, California Insti­ tute of Technology, Pasadena, California Problems in Combustion Determination of Equilibrium Composition and Thermo­ dynamic Properties Determination of Heat Release and Flame Temperature Gas Imperfection Failure to Maintain Equilibrium in Combustion Cited References and Bibliography

B. Expansion Processes

1. 2. 3. 4. 5. 6.

David Altman, Jet Propulsion Laboratory, California Insti­ tute of Technology, Pasadena, California James M. Carter, Aerojet Engineering Corporation, Azusa, California Classification of Flow Processes Thermodynamic Relations for Flow Processes Determination of Performance Parameters for Isentropic Flow Nonequilibrium Effects Two-Phase Flow Cited References

L. Combustion of Liquid Propellants

3

3 4 15 17 22 24 26

26 28 40 45 52 62 64

David Altman and S. S. Penner, Jet Propulsion Laboratory, Cahfornia Institute of Technology, Pasadena, Cahfornia Chapter 1.

Ignition Phenomena in Bipropellant and Monopropellant Systems

1. Introduction 2. Experimental Methods Used for Measuring Ignition Delay in Bipropellant Systems 3. Representative Ignition Delay Measurements on Spon­ taneous Bipropellant Systems 4. Ignition of Representative Nonspontaneous Bipropellants 5. Ignition of Monopropellants

64 68 75 81 82

CONTENTS

Chapter 2.

Motor Performance of Selected Monopropellants

6. General Characteristics of Monopropellants 7. Classification of Monopropellants 8. Performance Characteristics of Monopropellants Chapter S.

83 84 86

Combustion of Selected Bipropellant Systems

9. Classification of Oxidizers and Fuels 97 10. Combustion of Bipropellants in Rocket Engines 98 11. Performance Characteristics of Several BipropeIlant Com­ binations 101 12. Modern Trends in Combustion Research on Liquid-Fuel Rocket Engines 105 13. Cited References 106 G. The Liquid Propellant Rocket Engine

1. 2. 3. 4. 5. 6. 7. 8.

108

Martin Summerfield, Department of Aeronautical Engineer­ ing, Princeton University, Princeton, New Jersey Introduction 108 Performance Analysis of the Ideal Rocket Motor 109 Departures from Ideal Performance 122 Theoretical Specific Impulse Calculations 133 Combustor Design Principles 144 Cooling of Rocket Motors 159 Liquid Rocket Systems 179 Bibliography 186

SECTION A 111·'

HIGH TEMPERATURE EQUILIBRIUM JAMES M. CARTER DAVID ALTMAN A,l. Problems in Combustion. There are two principal problems in determining the nature of high temperature combustion. The first is the determination of the nature and amount of the combustion products which may exist and also the thermodynamic properties of the mixture at one or more sets of combustion conditions. These combustion conditions are always idealized for purposes of calculation. Thus in a rocket or jet motor, steady state conditions are assumed; combustion during starting or stopping is considered separately by approximate methods from a knowledge of steady state conditions. In order for thermochemistry and thermodynamics to apply to com­ bustion, either certain other simplified conditions must be assumed, or much more detailed information is required than is usually available. Thus, in effect, calculations on combustion in rocket and jet motors are made on the assumption that the combustion process occurs in a small mass element which is injected into a motor near the center of an infinite stream of identical elements and which then travels through the motor with negligible velocity and at constant pressure. The combustion chamber of the motor is assumed long enough to permit equilibrium to be attained, and the walls are assumed far enough removed to exert no effect since the element in question is surrounded by identical elements. Similarly in explosions such as may occur in guns, or in intermittent devices, calculations are actually made for a mass element of given density, surrounded by identical elements of the same density, all in a container large enough to have no effect on the element being considered, except to keep its density constant. It is obvious that in most cases these conditions are far from fulfilled, and that effects due to combustion speed, mixing, loss of heat to walls, and other phenomena must be considered. However, the simplified condi­ tions assumed make calculation much simpler and do provide an upper limit for the efficiency of combustion processes. When these conditions are met, the reaction equilibria among the substances present, as dis­ cussed in I,A and C, do determine the combustion products and their thermodynamic properties.

A · HIGH TEMPERATURE EQUILIBRIUM

Further simplification is obtained by assuming that combustion products are perfect gases, which is nearly true for most processes treated in rockets or jet motors. Deviations which may occur at high pressures, or in ultrafast processes, are considered in I,A and C and in other texts on thermodynamics [1,2,8]. The second problem is the determination of the heat which may be released by the combustion, the maximum combustion temperature which can be reached, or the work which may be extracted, subject to the possible compositions as determined above and to the conditions under which the combustion occurs. The usual conditions assumed are those for adiabatic combustion at constant pressure or at constant volume. Other conditions (those involving heat transfer or external work) are considerably more complicated. Discussion of these conditions has been given by Lewis and von Elbe [4] and by Hirschfelder [5]. The combustion process may be represented by combustibles (px> Vi1 Γι) = combustion products (pi, Vi, Ti) + Q kilocalories (1-1) with Q determined by the expression E ν,,Ti - Ev uti = -Q -

VdV

In the above, the composition of the combustibles and the initial state are known; the composition of the products and their state, together with Q, are to be determined. This problem may conveniently be divided into two parts: the deter­ mination of equilibrium composition and thermodynamic properties; and the determination of flame temperatures and heat released or work done for various final states of the combustion products. A,2. Determination of Equilibrium Composition and Thermo­ dynamic Properties. The composition and thermodynamic properties of the equilibrium products of combustion are uniquely determined by the atomic composition, the temperature, and the pressure (or volume of the system), as is shown in I,A and C. In particular, at a specified tempera­ ture and pressure they do not depend on the heats of formation of the combusting materials or on their heat of reaction. (However, these quan­ tities will determine the temperature range over which the composition and thermodynamic properties are of interest, since they determine the initial energy of the system.) For example, the equilibrium composition and thermodynamic properties of the combustion products of the combustible mixtures repre­ sented.; by 6C + 3H2 + 402, 3C2H2 + 402, and C6H6 + 402 are iden­ tical at any chosen temperature and pressure. Flame temperatures, heat

A,2 · EQUILIBRIUM COMPOSITIONS

released, or work available, differ widely because of different initial energies of the three systems. These energies are determined by the basic conservation-of-energy equations. EQUILIBRIUM COMPOSITION. Equilibrium conditions in the combus­ tion products can be obtained from thermodynamic data and from the total amounts of each atomic species involved. Formany reactions, values of the equilibrium constants are available either from experimental measurements or as calculated values from spectroscopic data. Other equilibrium constants may be calculated from free energy data. The theory of equilibrium conditions is treated in I,A and sources of data are also included in I,A. There is always an element of arbitrariness involved in choosing the equilibria regarded as significant. For instance, a choice frequently made is to neglect all equilibria involving species present in less than 0.01 or 0.1 per cent of the total. But this is not all-conclusive; obviously an equilibrium involving 0.01 per cent of the total moles present for a reac­ tion with an energy change of 150 kilocalories per mole is more significant, in determining the state of the system, than an equilibrium involving 0.1 per cent of the moles for a reaction with an energy change of 10 kilo­ calories per mole. A rough calculation which involves the heat given off in a combustion process resulting in products stable at ambient temperature, and the specific heats of these products, serves to set an upper limit on the com­ bustion temperature. At this temperature, the equilibria which may exist among products involving all the atomic species present are examined. For example, the combustion of gasoline (taken as octane C8H18) with oxygen may be used. A typical reaction for rocket motors may be ap­ proximated by C8H18 + 6Ο2 -» 8CO + 4H20 + 5Hj (2-1) (In combustion reactions involving carbon, hydrogen, and oxygen, a useful rule in estimating reaction products is, (1) to oxidize carbon to CO, (2) to use remaining oxygen to oxidize hydrogen to water, and (3) if any oxygen remains, to oxidize CO to CO2.) If this reaction occurred at 300°K, with the products remaining at 300°K, the heat released would be about 425 kilocalories. Cooling the products to O0K would release an additional 35 kilocalories. Thus the total heat released can be expressed as C8H18 (1) + 6Ο2 (g, 300°K) = 8CO (g) + 4H20 (g) + 5H2 (g, O0K) + 460 kcal Enthalpy tables for the products establish a temperature near 3000°K, if the entire heat release is employed in raising the temperature.

A · HIGH TEMPERATURE EQUILIBRIUM

Examination of the equilibrium constants for the various reactions which might be involved shows that the water-gas reaction is such that CO2 is present, that steam will dissociate to OH, H2, and O2; H2 and Os will dissociate to H and O respectively, but that because of the large excess of Hj, Oi and especially O will be present in very small amounts. There­ fore an initial estimate of the species present would include CO, CO2, H2, H2O, OH, H. In the calculation of equilibrium compositions, the starting point is a series of equations; these equations will equal in number all of the un­ knowns to be determined. These equations are of two kinds: the material balance equations, η = ^ni

(2-2)

(M) = Yj (Uia) • w,· (2-3) (N) = Yj («,*) · Hi etc., and the equilibrium equations, K

< f = f i ' 4 1. ι

(2- 4 )

etc. The first equation states that the total number of moles is equal to the sum of the number of moles of the molecular components. The second set of equations expresses the fact that the total number of gram atoms of each atomic constituent, denoted by (M), (N), etc., is distributed among the molecular components «,·, with atoms of M in the com­ ponent i. The third set of equations are the usual equilibrium expressions for the gas reactions aA + 6B + · · · *=? cC + d D + • · · In the equilibrium equations, / is the fugacity. Conversion to the usual forms for pressure, mole fraction, etc. is made by use of appropriate multiplying factors Qp, Qx, etc. involving fugacity as a function of pressure or density. These relations are given in I,A and C. The following equi­ librium expressions are those most frequently involved in combustion involving carbonaceous fuels. The numbering used follows that given by Hirschfelder [5], and all subsequent references to equilibrium constants or reactions follow this numbering. It should be emphasized that while the expressions are given in terms of pressure, this is accurate only insofar as the perfect gas laws apply. When appreciable deviations occur, pressures must be replaced by fugacities.

A , 2 • EQUILIBRIUM

COMPOSITIONS

In the simplest cases, when there are only one or two atomic species and two or three molecular species, these equations may be solved simply by algebraic means. I n general, however, the number and complexity of the equations make this impossible. For the reaction given in Eq. 2-1, the equilibrium equations would be set up as follows: (2-2a)

(2-3 a) and if the perfect gas laws hold, so that

(2-4a)

The above set of seven equations involving seven unknowns does not lend itself to easy algebraic solution. I n this case, which is relatively simple, the algebraic solution can be obtained, but ordinarily, with more molecular species and more equations involved, such a method is virtually impossible. Recourse must then be had to one of a number of approximate methods. These may be classified as (1) trial and error methods, (2) iterative methods, (3) graphical methods and use of published tables, and (4) punched-card or machine methods. < 7 >

A • HIGH TEMPERATURE EQUILIBRIUM

1. Trial and error methods. The straightforward method of solving for an equilibrium composition may be accomplished by the insertion into Eq. 2-4 of trial values consistent with the material balance Eq. 2-3 until all equations are satisfied. Simple rules, such as the one given previously in the example employing octane and oxygen, may be employed to assign initial trial values. This direct method is the least efficacious and is recom­ mended only in those cases where the number of components is small and a reasonably good guess can be made of the composition. The trial and error method, however, can effectively be applied to more complicated systems if the number of working equations are first reduced by algebraic substitutions. A general method for handling C, H, 0, and N systems of up to ten components is demonstrated in the follow­ ing treatment. Let the number of moles of the components be represented by the following symbols: a = na,

f = wNl

b — WiijO

Q — Wno

c = nco

h = Woh

d — Wco1

ί = WH

e = Wo1

j = Wo

These ten unknowns are related by means of Eq. 2-3 and 2-4 as follows: (2-5) (2-6)

(2-7) (2-8)

(2-9) (2-10)

c + d = (C)

(2-11)

(O)

(2-12)

(H) 2 / + 0 = (N)

(2-13) (2-14)

b+ c + 2 d + 2e+ g + h + j =

2a -|-

2b + h + i —

A,2 · EQUILIBRIUM COMPOSITIONS

where (C), (0), (H), and (N) represent the total number of gram atoms of the elements, η is the total number of moles in the equilibrium mixture, and ρ is the total pressure. The subscripts on the K's denote the specific equilibria as given in Eq. 2-4. Although η is accurately given by the equation n = a + b + c + d + e + f + g + h + i + j

the concentrations of the components are not sensitive to small variations in n. This fact permits the choice of an approximate η value in Eq. 2-6 to 2-10 with only very little loss in accuracy. The above ten equations may now be simplified to yield the following three working equations: h =

(H) - 2a - g,(n/p)ta» ffia(ra/p)* +

» - + Y ^>(E% - E ° ) t + η

dp

(2-18)

dV

(2-19)

A , 2 • EQUILIBRIUM

COMPOSITIONS

Here, w< is the number of moles of component i present at temperature T and A//,0 or AEf is the heat or energy of formation of the component i at 0°K, from the elements at 0 ° K E% - E\ or E% - E% is the integral of the specific heat of component i from 0°K to T. If the elements at ambient temperature (298°K) are taken as a base point, Eq. 2-18 becomes (2-18a) and similarly for Eq. 2-19. If a base of stable composition at ambient temperature is used, the first term on the right of Eq. 2-18a is replaced by £An,Aff?(298), where An,- is the change in component i in going from the composition stable at 298°K to the equilibrium composition at T. (It should be noted as an advantage of the base at 0°K that A = Aand HI = jE{J.) The last terms in the equations correct for the changes in enthalpy or internal energy with pressure or volume. For perfect gases these terms are zero. In the general case, as listed by Bridgman [14] for these and other thermodynamic formulas,

(2-20) (2-21) The numerical values of the expressions above are determined from the equation of state for the gas mixture. Entropy values for combustible mixtures may be determined in a number of ways. For many substances entropy tables are available for the standard state at a pressure of one atmosphere [iS]. The entropy of the mixture is then given by the expression (2-22) For other materials, tables ofthe free energy function are available [IS]. By use of the relation the entropy may be determined as (2-23) In both the above equations, the term is the entropy of mixing all the components, each originally present at one atmosphere, to give the mixture at one atmosphere. To obtain the entropy at the desired pressure or volume, there must be added to the above values the change of entropy with pressure or < 13 >

A • HIGH

TEMPERATURE

EQUILIBRIUM

volume. This is given by the equation (2-24) For most of the applications met with in jet and rocket propulsion, gases may be assumed to obey the perfect gas laws, so that the above equations reduce to (2-25) Specific heats for the combustion products are obtained from the expressions (2-26) In the above, it is necessary to keep in mind that the changes in enthalpy or internal energy include not only the amounts due to change in temperature for each species, but also those due to the change in equilibrium composition with temperature, according to the equations

(2-27)

(2-28) The first two terms in the above give the heat involved in changes in composition, the third terms give the perfect gas value of the specific heat at fixed composition, and the last terms give the effect due to gas imperfections. In view of the complex nature of these equations, a convenient method of obtaining average values for the specific heats over a range of temperatures is by differences from tables of enthalpy or internal energy, if these values are determined at fairly close intervals, both in temperature and in pressure (or volume). For some purposes, is required. This can be obtained from enthalpy and energy tables as above. For more accurate determinations, use is made of the thermodynamic relationship (2-29) * 14 >

A,3 · HEAT RELEASE AND FLAME TEMPERATURE

This can be cast into the form

r (dp\ I = 9l - ι -1 y

Cp

1

(av\

CΓpΡ \ Θ Τ / ν \ Θ Τ / Ρ

(2-30)

where C p is the heat capacity at constant pressure, T is the absolute tem­ perature, F is the volume, and ρ is the pressure. All quantities must be expressed in consistent units. The partial derivatives are obtained from the compositions and the equation of state for the particular mixture under consideration. In particular for nonreacting mixtures obeying the perfect gas law the above expression reduces to

Another thermodynamic property which is useful in dealing with propellants is the velocity of sound. This is given by the expression (2-32) It is shown in thermodynamics texts [i] that this reduces to (2-33) Here again all of the required quantities may be obtained from those previously calculated and from the equation of state of the gas mixture. The extent of the calculations made for composition, equation of state, and thermodynamic properties will vary with the information desired. This· may range from a fairly rough estimate of performance characteristics for one particular mixture at one pressure or density, to detailed calculations for a range of mixtures, at a number of pressures or densities, expanded over a series of expansion conditions. The latter in effect requires the preparation of a number of Mollier or equivalent charts, one for each mixture investigated. If anything more than a single performance value is required, it is surprising to one who has not worked with the calculations before to see how little additional effort is required to cover a wide range of combustion and expansion conditions. A,3.

Determination of- Heat Release and Flame Temperature.

As has been stressed above, the equilibrium molecular composition and thermodynamic properties of any particular atomic composition do not depend on the initial state of the mixture. In other words, the particular nature, heats of formation, etc., of a combustible mixture have no effect on the ultimate composition or thermodynamic properties of the mixture

A · HIGH T E M P E R A T U R E EQUILIBRIUM

after combustion, when these properties are taken as functions of the temperature and pressure or density. However, it is just these factors which determine the heat released in a combustion and the flame temperature which may be reached. To evaluate these effects it is necessary to return to Eq. 1-1 and to separate this into two parts. Eq. 1-1 reads: combustibles (pi, T h Fi) = combustion products (p2, T 2 , F2) + Q

kilocalories

This may be separated into two steps: combustibles (pi, 2\, F1) —> products, standard (p0, Ta, Fo) + Q0 kilocalories

(3-1)

and products, standard (p B ,

T0,

F0) —> products ( p 2 ,

T2,

F2)

+ Qt kilocalories

(3-2)

In the preceding articles, the thermodynamic quantities involved in the second of these processes have been discussed. Q2 is simply the negative of the enthalpy or internal energy at p2, T72, F2. All that is required to determine performance, flame temperatures, and other required data is to evaluate the thermodynamic quantities involved in the first process. The heat or energy values involved in the first process above will of course depend on the thermochemical properties of the combustibles, on their initial state, and on the base taken for p0, To, Fo- It is here that the advantage of the base taken at ambient conditions is most apparent. In such a case Q0 is merely the heat or internal energy involved in the com­ bustion at constant temperature. If the base at 0°K is taken it is neces­ sary to add to the usual heat or energy of combustion the heat involved in cooling the combustion products from ambient temperature to 0°K. Several specific processes may be mentioned in relation to the above. Maximum flame temperature is determined by the relation Qo + Q 2 = 0 . At either constant pressure or volume, if all of the heat or energy released in going from the combustibles to the base state is absorbed in going from the base state to the combustion conditions, no heat or energy is lost in the process, and the maximum temperature is attained. Several examples are given in Sec. L and M. Furthermore, the heat or energy converted to kinetic energy or work is directly obtainable from the above equations. If the final state of the combustion products has been determined after expansion from the com­ bustion conditions as shown in Sec. B, the extent of energy conversion is determined by W = Q0 + Q2

A,4 · GAS IMPERFECTION

where Qi denotes the heat or energy involved in going from standard con­ ditions to final condition. It should be noticed that this equation applies not only to expansion in which equilibrium composition of the combustion products is maintained during expansion, but also when the composition is assumed frozen, either at expansion conditions or at some intermediate temperature, since Q2 is determined by composition as well as by tem­ perature, according to Eq. 2-18 and 2-19. A,4. Gas Imperfection. The effect of gas imperfection in modifying equilibrium compositions and thermodynamic properties has been indi­ cated in some of the equations above. For actual calculations involving gas imperfection, it is necessary to employ an equation of state for the gases involved. By the use of such an equation, the relations between pressure, volume, and temperature of a gas are determined over the range of interest, and corrections can be applied to the equilibrium constants and to the thermodynamic properties. Equations of state. There are various equations of state which have been proposed to correct for deviations from the perfect gas laws. Among these are the van der Waals equation [la], Berthelot's equation [16], the Beattie-Bridgeman equation [17], the Hirschfelder equation [5], particu­ larly adapted to high temperature and pressure, and numerous more or less empirical equations. Graphical representations have also been employed. These are treated in I,C. A general representation of an equation of state for a pure gas may be given by pV - RT ^l + y + ψ-2 + ~ + · ·

(4-1)

where b, c, d, etc. may be functions of temperature. The values of b, c, d, etc., are determined by the nature of the gas. They are usually determined from experimental measurements, although in some cases they may be calculated from theoretical considerations. For a single pure gas, Eq. 4-1 shows a definite relationship involving temperature, volume, and pressure. Furthermore, the fugacity, on which corrections to the equilibrium constants depend, can be calculated for a pure gas. The exact definition of fugacity and consideration of its application is given in I,A. It is sufficient to note here that fugacity is determined by the expression (T = const)

(4-2)

It can be seen that for a perfect gas /2//1 = p*/pi, but that in general for

A · HIGH TEMPERATURE EQUILIBRIUM

imperfect gases this is not true. Iim pi = fi pi—»0 Real difficulty arises with gas mixtures, both in determining ρ, V, T relationships and to a greater extent in determining fugacity. Experi­ mental data for the ρ, V, T behavior of gas mixtures is not abundant, and in any case the entire range of compositions cannot be covered. It is not immediately apparent how the constants in Eq. 4-1 for pure gases should be combined to give an equation for a gas mixture. Further, since fugacities are usually required for individual components of a mixture, there is difficulty in choosing the quantities and limits in the integral in Eq. 4-2. Early attempts to compute the ρ, V, T relationships of gas mixtures were expressed in Dalton's and in Amagat's laws. Dalton's law states that the pressure exerted by a gas mixture is the sum of the individual pressures each gas would exert if it occupied the total volume alone ρ = ni- pi(T, F) + n2 • p 2 (T, V) + · · ·

(4-3)

Amagat's law states that the volume occupied by a gas mixture is the sum of the individual volumes which would be occupied by each gas alone at the same total pressure: F = H1 • Fi(T, p)+n 2 • V 2 (T, p) + · · ·

(4-4)

For perfect gases, both laws are true. Amagat's law is in general somewhat more accurate, but neither is a good approximation at even moderate pressures. Closer approximations require methods for combining the con­ stants in the equation of state Eq. 4-1 for pure gases to represent the behavior for gas mixtures. Consideration of these questions is given in I1C. Evaluation of constants. Constants in the equations of state may be evaluated in various ways. A series of ρ, V, T measurements over a wide range furnishes data for evaluation of the constants; they can also be evaluated from critical data and from data on intermolecular forces. A fairly extensive treatment of the calculation of the constants of the equation of state for combustion gases is given by Hirschfelder, et al [5], The equation of state for gases at high temperatures adopted for this pur­ pose is a somewhat simplified van der Waals equation: VV - ι _l BT

b

_L 0.62562 , 0.286963 , 0.192864 72 "τ" γ* + yi

,. .. (4-5)

which at high densities merges into the limiting form ^ = ^ 1-0 . 6 9 6 2 ^ ]

1

(4-6)

A,4 · GAS IMPERFECTION

In these equations attractive forces between molecules are disregarded (terms involving a in the van der Waals equation). The numerical coeffi­ cients are derived theoretically for the overlapping of rigid spheres. In the range of combustion temperatures (1500 to 5000°K) values of b are nearly independent of temperature and are assumed constant. Values of b for individual gases are evaluated from theoretical considerations on intermolecular forces. For gas mixtures, values of b may be calculated approxi­ mately at moderate pressures according to the relation b = J Xib i

where X, is the mole fraction of component i. Fowler [-/5] gives consider­ ations for calculation of constants involving interaction between different molecular species. With such simple equations as those above, the ρ, V, T relations for gas mixtures are obtained rather simply. Also, the changes in enthalpy, internal energy, and entropy with pressure or density at con­ stant temperature are readily obtained by the usual thermodynamic formulas:

\dp

JT

( M ) \dVj T

(~)

\dp/T

(\dV)r -)

= F =

-'(Sf).

(2-20)

T

(2-21)

-m,

(4-7)

(¾

(4-8)

Changes in equilibrium constants with pressure or density are, however, more complicated since the fugacities of the individual components are involved. To obtain reasonably simple results, it is convenient to assume that a gas mixture is a perfect solution [iS] or that wt- 1




(2-21)

B • EXPANSION

PROCESSES

in which can be identified with dSi, the entropy gained through frictional heating. Integration from Tx and pi to T and p yields (2-22) with for all frictional processes. Fig. B,2b and B,2c show the p, T and H, S diagrams for isentropic flow and flow with friction. The

Fig. B,2b.

pT diagram for isentropic flow and flow with friction.

Fig. B,2c.

HS diagram for isentropic flow and flow with friction.

irreversible entropy gained, ASi, can be evaluated from Eq. 2-23 in terms of T and T0 to yield (2-23) ( 32 )

B,2 • THERMODYNAMIC

RELATIONS

The adiabatic efficiency ij, defined as the ratio of kinetic energy with friction to that without friction, is given by the expression (2-24) The expansion law may be obtained from Eq. 2-24 by multiplying both sides of the equation by and i d e n t i f y i n g a s The result is3 (2-25) This law is sometimes represented with a polytropic expansion constant [6,6] which is an approximate form of Eq. 2-25. Since Eq. 2-25 may be expanded to yield

(2-26) Again, since it will be observed that Eq. 2-26 gives the first two terms of the expansion for and so there finally results (2-27) which may also be expressed in the equivalent form (by use of the perfect gas law): (2-28) where is the polytropic constant. These polytropic expansion laws have also been derived [5,6] by means of the equation and Eq. 2-21. However, it should be noted that this relation for dqi is only approximate since the difference in the expansion work terms jVdp for the viscous and inviscid gas between the fixed pressure levels has been neglected. 2. Combustion and nonequilibrium reaction during flow. The case of combustion during adiabatic flow, unlike that of equilibrium chemical reassociation, is nonisentropic because the chemical composition is not a unique function of the equation of state of the gas. The energy equation governing such flow is given by Eq. 1-4 with dq = dWf = 0 for adiabatic frictionless flow.4 The heat content H is given by Eq. 2-13 presented * Strictly speaking ij may not be constant during flow, but for approximate calculations the assumption of constancy is fairly good [5,6]. * Some authors prefer to treat combustion during flow as nonadiabatic, identifying the chemical heat with q [7], This treatment will yield equivalent results if the reference state for H is defined as is done for a nonreactive gas.

< 33 )

B • EXPANSION

PROCESSES

earlier: (2-13) where [A.fft(298)].' is the heat of formation of the ith species at 298°K, the reference state being the elements at 298°K. For perfect gases, H is a function only of T and n ( , and taking the differential of II gives

(2-29) where is the heat of formation at the temperature of the gas. The energy equation may now be written as (2-30) where Dividing through by nT and denoting the average specific heat of gas results in

as

(2-31) This equation is not integrable since «,• cannot be expressed as a unique function of the state of the gas because of its dependence on time. For a fixed geometry, however, the parameters T and p can be related to the velocity and hence to the time. If the reaction rate constants of the various combustion reactions are known, then expressing drw/dT as (drii/dt) (dt/dT) will permit solution of Eq. 2-31 because these derivatives can be expressed analytically. This method of treatment has been applied to the dissociation of N O during flow through a nozzle [5]. Approximate solutions to Eq. 2-31 can be obtained if one knows the total chemical heat released andassumes the heat to be generated uniformly. The quantity is replaced by 7"j) where AQ is the total chemical heat of reaction, n is the average number of moles of gas considered, and T't is a tentative value of the final temperature. I t will be observed that the quantity has the dimensions of heat capacity and can be denoted by C'v. Eq. 2-31 now integrates to (2-32) with an apparent

The low value of the apparent y will lead to higher stream temperatures as shown in Fig. B,2d. ( 34 )

Β,2 • THERMODYNAMIC RELATIONS

3. Throttled flow: isenthalpic. Steady state adiabatic flow from a high pressure region pi through a valve or porous plug to a low pressure region p2 occurs at constant enthalpy. This result was first obtained in the classical experiments of Joule and Thomson in their famous porous plug experiments.

Q.

Temperature Fig. B,2d.

pT diagram for isentropic flow and flow with combustion.

Employing the integrated form of Eq. 1-1 in which friction is neg­ lected, the energy change per mole of gas in going from pi to p2 is given by the equation E 2 - E 1 = (P1F1 - p2F2) (2-33) where the net work produced by the gas is the work done by the gas at pi, i.e. piVi, minus that done on the gas at p2) i.e. p2F2. Noting that the enthalpy H is given by the equation, H = E + pV, it is seen from Eq. 2-33 that the quantity if2

=

Ei H- p2F2 = Ei + P1Fi — Hi

(2-34)

remains constant. If the gas is perfect so that H is a function only of T, the process is also isothermal. If the gas is imperfect, or if chemical equilibria exist which are pressure dependent, the process is still isen­ thalpic but not isothermal. This principle has found application in gas generation devices where the gas is formed at a high pressure and does work at a lower pressure. 4. Free expansion: isoenergetic. Free expansion of a gas occurs in the transfer of the gas from a higher to a lower pressure level without the appearance of external work. If the system is adiabatic and frictionless, it is obvious from Eq. 1-1 that the over-all process occurs at constant internal energy for the system, i.e. AE = 0. For imperfect gases, E is a

B · EXPANSION PROCESSES

function of both T and ρ and so the energy change associated with the change in ρ must be exactly compensated by a change in T. At low tem­ peratures, this effect results in a cooling of the gas. This phenomenon 1¾ made use of in one method for the liquefaction of low boiling-point gases. NONADIABATIC FLOW. In any real system, the assumption of adiabatic behavior can only be employed when the heat transferred through the walls is small with respect to heat effects produced in the gas. Nonadiabatic flow, on the other hand, is favored when (1) wall surface per unit mass of gas is large, (2) thermal conductivity of the gas and wall material is high, (3) wall temperatures are widely different from that of the gas, and (4) the flow velocity is low. The degree of irreversibility of the heat transfer process may be measured by the difference in tempera­ ture between the wall and the gas. In a thermodynamic sense, this irreversibility resulting from the transfer of heat is given by the entropy increase of the entire system: AS1 + AS, = ^

= q^r > O

(2-35)

where Ti and Ti are the lower and higher temperatures, respectively, and Δ5ι and ASt the corresponding entropy changes. As AT —> O in Eq. 2-35, the process becomes isothermal and reversible. The conditions for this behavior are discussed under Isothermal expansion on page 39. Heat transfer. It is assumed that the heat is transferred from the walls into the gas stream normal to the flow velocity and that the heat flow along the flow path is negligible. It is obvious that in any real ease, a temperature gradient will necessarily become established normal to the flow because of the limitation of thermal conductivity. The treatment given here will be Umited to the thermodynamic be­ havior of a small element of fluid in which average properties can be chosen. The relations so obtained can be applied quite successfully to low velocity turbulent streams with large I/D ratios. Application may also be made to systems where the heat transferred, q, is much less than the enthalpy potential CpT. It can be shown that subject to this limitation, specific consideration of the cross-sectional temperature contour will give rise only to second order corrections to the flow parameters. This fact is of practical significance since it permits the direct application of the following treatment to heat transfer in supersonic nozzles where q