A STUDY OF THE EFFECT OF CHEMICAL NON-EQUILIBRIUM IN THE FLOW OF A HIGH TEMPERATURE REACTIVE GAS MIXTURE THROUGH A CONVERGING-DIVERGING NOZZLE By Clarence Quentin Ford AN ABSTRACT Submitted to the School for Advanced Graduates Studies. of Michigan State University of Agriculture and Applied Science in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Mechanical Engineering 1959 Approved /’- Z”. jigsaé 1 / . / fl A Clarence Quentin Ford , AN ABSTRACT The object of this thesis was to investigate the recombina- tion reactions of a dissociated high temperature gas mixture, as the mixture eXpands in a converging-diverging nozzle, and what effects these reactions might have on the expansion process. As an approach to the problem, an analytical developnent was carried out, on the basis of collisional properties, to estab- lish relationships which would describe the fluid dynamics and chemical kinetics . These relationships were combined to permit an analysis of the flow of a compressible mixture under- going recombination reactions . The generalized analysis was then reduced to a simplified form, that would show only the effects of reactions occurring at a finite rate. A particular nozzle shape was established, along with a set of initial conditions for a chanically reactive gas mixture, and a set of calculations were carried out to obtain pmrsical properties along the nozzle anis. As a means of comparison, flow processes with recombination reaction rates of zero and infinity were calculated from the same initial conditions and for the same nozzle configuration. A zero reaction rate is equiv- alent to the "frozen" flow, while an infinite reaction rate is comparable to a process always in an equilibrium state. Comparison of the finite reaction rate flow path calcula- tions with those for the frozen and equilibrium paths shows that, Clarence Quentin Ford 2. calculations with those for the frozen and equilibrium paths showed that, at the high temperatures resulting from a combustion process, the finite path remained close to the equilibrium path for the entire nozzle length. As the temperature decreased and velocity increased with expansion, the finite path diverged slowly from the equilibrium path. However, contrary to some published results on similar studies, performed on individual gases rather than a mixture, the flow'path did not assume a.frozen path in the diverging section of the nozzle. Lack of experimental data on chemical reaction kinetics, especially in the high temperature regime, prevented a more thorough analysis of nozzle flow for a chemically reactive fluid. However, it was indicated in the results of this study that an equilibrium flow path, for which tabulated data are available, would approximate actual conditions more closely than the commonly employed frozen flow path. A STUDY OF THE FFFEIJT OF CHEMICAL NON-EQUILIBRIUM IN THE FLOW OF A HIGH TEMPERATURE REACTIVE GAS MIXTURE THROUGH A CONVERGING—DIVFRGING NOZZLE By Clarence Quentin Ford A THESIS Submitted to the School for Advanced Graduate Studies of Michigan State University of Agriculture and Applied Science in partial fulfillment of the requirements for the degree of DOCTOR (F PHILOSOPHY Department of Mechanical Engineering 1959 G 5739 o// 7/1” AC KN OWLEDGM EN TS The author wishes to express his appreciation to Dr. Ralph.Rotty for his help in setting up the original portions of the problem, and to Dr. Joachim Lay for his help in carrying the study to completion. The author extends thanks to Professor L. C. Price, Dr. Rolland Hinkle, and Dr. Charles Wells for their service on the author's committee. Special and most sincere thanks go to Dr. James Dye for his invaluable interest and suggestions throughout the entire course of the study. VITA Clarence Quentin Ford candidate for the degree of Doctor of Philosophy Final examination: May 1h, 1959, 9:00 a.m., Room 308, Olds Hall Dissertation: A Study of the Effect of Chemical Non-Equilibrium in the Flow of a High Temperature Reactive Gas Mixture Through a Converging-Diverging Nozzle Outline of Studies: Major Subject: Mechanical Engineering Minor Subjects: Physical Chemistry, Mathematics Biographical Items : Born: August 6, 1923, Glenwood, New Mexico Undergraduate Studies: United States Merchant Marine Academy, 19112-th, New Mexico State University, Graduate Studies: University of Missouri, 1919-50; Michigan State University, 1956-59. Experience: Engineering Officer, United States Navy, 19142446; Instructor in Mechanical Engineering, University of Missouri, 19149-50, Instructor - Assistant Professor in Mechanical Engineering, Washington State College, 1950-56; Instructor in Mechanical Engineering, Michigan State University, 1956-59. Member of Sigma Xi, Pi Tau Sigma, Pi Mu Epsilon, Gamma Alpha, Registered Professional Engineer. iii TABLE OF CONTENTS NWCLATURE...... INTRODLBTION AND ASSIGNMENT OF ANALYTICAL DEVELOWT . . . Governing Flow Equations Reaction Kinetics. . . Reactive Flow . . . . CALCULATION PROCEDURE . . . Equilibrimn Path . . . Frozen Path. 0 e Temperature-Time Relationships Flow with Finite Reaction Rate PRESENTATION AND DISCUSSION cm REULTS. APPENDIX. 0 o o o e o o BIBLIOGRAH-IY . . . . . . iv 97 Table II III A-I LIST CF TABth MUILIBRIUMFLOW . . . FROZENFLOW. . . . . FINITEFLOW. . . . . COMPOSITIONS, x1, EQUILIBRIUM AND FINITE FLOW. NOZZLE GEOMEI‘I‘K RELATIONS 72 73 88 10 8-1 LIST OF FIGURES Flo." cylimer O O O O O O O O O O O Mollier diagram of equilibrium mixture. Pressure in atmos., temperature in °R . . Residence period of mixture in the converging- diverging nozzle, for equilibrium and frozen flow paths . . . . . . . . . Comparison of flow paths on basis of area ratios I O O O O O O O O O O O 0 Pressure variation for nozzle expansion along equilibrium, finite and.frozen paths. . . Density variations of reactive mixture flowing through converging-diverging nozzle . . . Velocity variation of reactive mixture flowing through converging-diverging nozzle . . . Temperature profile through converging- diverging nozzle for reactive mixture . . Nozzle configuration used in calculating finite flow mths e e o o o o o o 0 Value of A/At as a function of axial length for assumed nozzle configuration . . . . Collisions of molecules of type ”j" with one molecule of type "i” . . . . . . . . vi 12 Sh 61 7h 75 76 77 78 89 90 91 Symbols USed A 2 K W 9Q D' :1: 00 H: '3) tr) :3 NOMENCLATURE frequency factor unit vector net rate of formation of ith species per unit volume per unit time individual velocity of 1th species diffusion velocity of 1th species heat capacity at constant pressure heat capacity at constant volume internal energy activation energy free energy function distribution function extensive variable enthalpy enthalpy per unit mass equilibrium.constant reaction rate constant chemical compound number'density number of moles pressure molar gas constant position vector vii Symbols Deed J5 h< a :a «a at R U k: ‘0 (iv, t: t: 1:) >~"—] ‘TQ EU 2: 'S 2 > \- \ \ NOMENCLATURE (cont .) surface entropy temperature time mass average velocity linear velocity average velocity volmme molecular‘weight mass rate of flow external force mole fraction (ith species) rectangular coordinates nozzle axis reaction velocity pressure tensor extensive variable flux vector extensive variable rate of production transport function mass density stoichiometric coefficient of reactants stoichiometric coefficient of products reaction coordinate viii NOMENCLATURE (cont .) Symbols Used 55 angle of convergence angle of divergence Subscripts Used b backwork specific reaction rate 1‘ forward specific reaction rate i,j species index t throat conditions in nozzle (X initial state for flow conditions T reaction designator Others _~. vector quantity 2'" Eulerian derivative D 2.... + {I L Dt ’ I): . 3 t o a r [M] concentration of chemical compound a» non-equilibrium state S‘! a” ’L’ () L1 INTRODUCTION AND ASSIGNMENT W PROBLEM The purpose of this study is to investigate the effects of chemical reaction as a high-temperature reactive gas mixture flows through a converging-diverging duct device. In practice, such a com- bination would exist in the nozzle arrangement located between a com- bustion chamber and the blading of a gas turbine, or it could be the exhaust nozzle for a jet powered aircraft or missile. The effects of chemical reaction in such cases is to cause the thermodynamics of the gas mixture to deviate from an equilibrium path during the expansion process, and as such, the true path would thus enter into the realm of non-equilibrimn , irreversible, thermodynamics . Classical thermodynamics is based on equilibrium phenomena, conditions which are not necessarily fulfilled in high speed, high temperature flow situations of reactive gas mixtures . Non-equilibrium is brought about primarily by the inability of the molecules in the mixture to react (combine) at a rate comparable to the changes in pressure and temperature, which is referred to as “chemical lag." Another effect comes about in a similar manner, where the molecules are unable to absorb the energy made available to than as fast as this energy is supplied, which is the nrelaxAtion lag ." What is normally sought for in a thermodynamic analysis are the conditions of the system at any point along its flow path, when the initial state is known. Specifically, the quantities of main interest in nozzle flow are the temperature, pressure, composition and 2. enthalpy; from which other quantities such as velocity, thrust, density, etc. can be calculated. Combustion in engines which rely upon a nozzle as a part of their driving mechanism may be assumed as an adiabatic process, and thus results in temperatures in the order of hSOO°R to SODO'R. At temperatures of this magnitude, products resulting from the chemical combustion of a stoichiometric mixture, of a hydrocarbon fuel and air will not be carbon dioxide, water, . and nitrogen; rather, a highly re- active mixture consisting of atomic states, free-radicals, complete molecular forms, free electrons, etc . Thus, the products are actually those of "incomplete" or "partial" combustion, and as such will con- tinue the combustion phenomenon as expansion, with its attendant tem- perature decrease, takes place. The analysis here is to examine the situation of flow with con- tinuing combustion, in conjunction with changing pm'sical conditions. Nozzle flow with a non-reactive system accounts for the increase in kinetic energy by renoving just sufficient sensible energy from the gases through temperature change. With a reactive system, a reduction in temperature brings about a series of chemical recombination re- actions, most of which are exothermic, and thus latent energy is re- leased along with sensible energy. In comparison to non-reactive flow, it is seen that for a given change in kinetic energy, less temperature change is necessary with reactive mixtures by virtue of the additional energy of formation being released. Current practices of thermodynamic analyses of the flow of re- active mixtures either neglects the reactive potential of a high tem- 3. perature, dissociated mixture or assumes that the system is always in a state of equilibrium. Some reactions occur very rapidly, and are thus always in a quasi-equilibrium condition, while other reactions proceed so slowly that for practical purposes, they constitute a non- reactive system, or are so-called "frozen" reactions (more properly a frozen reaction is a "pure mixture”) . A reactive system in equilibrium then must have either an infinitely fast reaction rate or else an in- finitely long period of time available for the reaction to take place. Real reactive systems are not infinitely fast, and only finite times are of interest in applied problens, so it is not correct to neglect the "chemical lag" phenomena, just as it is not correct to neglect the possibility of reaction. Reaction half-times are a function of temper- ature and are very short, in the order of 10"6 seconds for the higher temperatures, but are still long enough to cause chemical lag at high velocities . In fact, at the high velocities and relatively low temper- atures existing at the exit of a nozzle, the reaction might take such a long period of time as to be essentially frozen. Deviation from equilibrium flow results and appears as a lower temperature and velocity; evenso, this temperature and velocity would be higher than for an en- tirely frozen flow. Just where the true intermediate physical properties lie, is the sought for answer. Only a few studies have been reported on re- active flow, and they were made on a highly restricted basis .1’2 The 1Heims, 5. P., Effect 9__f 0::ng Recombination on One-Dimensional Fl___o__w _a__t High Mach Numbers, NA .C.A., TN mm, 1958. 2Bray, K .N C. , De ure from Dissociation Equilibrium in a Hmersonic Nozzle, A .R ..C ,I‘E ch 9 h. major problem is that of being able to adequately describe the recom- bination reactions and postulating the proper reaction rates . There is only partial agreement on the proper theory to describe reaction rates, and even less agreement on what probable sequence of reactions actually occurs in a given recombination process. Experimental data of reaction rates are wholly lacking in the high tenperature domain, reliable values being reported up to only about 2000’R. Extrapolation beyond these limits appears to be quite risky. Mathematically, reactive flow processes can be described com- pletely in a relatively compact notation, but to obtain a solution of the resulting differential equations is extremely difficult, the dif- f iculty arising primarily from a lack of knowledge concerning the attendant physical phenomena. Thus, in order to effect a solution, it is necessary to apply the best available information and synthesize a solution around these conditions. Such a procedure is followed herein, to determine as accurately as possible, the conditions existing in reactive-gas expansion. By this means, it is possible to Obtain a more accurate solution than would be possible with either 'equilibrium' or "frozen" analyses, but much more time consuming. This method also indicates the probable error involved in using either of the simpler methods. In order to study the effects of non-equilibrium during flow, due to chemical reaction, it is necessary to initially describe the flow analytically and then solve the resulting expressions for the situation in question. The approach taken in this study is to first develop the fundamental equations for the fluid dynamics and chenical 5. kinetics in as general terms as possible, and then reduce these equa- tions to a form suitable for nozzle flow. After this system of dif- ferential equations is available, a set of assumed data is used to obtain numerical results. Underlying all of the study is the assumption that the fluid is a high temperature, low pressure, reactive mixture of gases, that has an equation of state like that of an ideal gas. This does not mean that the fluid has all the properties of a thermodynamically ideal gas, but only that p - R p T (l) where p is pressure, p is density, '1‘ is absolute temperature, and R is the gas constant. The use of (l) as such is not important to the general theory, but its implication and dependence on the prin- ciples of kinetic theory does make it important. As chemical kinetics is primarily a collisional phenomenon, using kinetic theory as the basis for the fluid dynamics leads to an easier combination of the two phenomena. Once the dynamics has been established, it is desirable to have a means of measuring the amount of chemical lag. This is done by means of a reaction coordinate, g , which establishes the extent of reaction as the fluid moves through the nozzle. Such a parameter is not a portion of the flow theory as such, but in a sense, gives the effect of finite reaction rates. Combining the fluid, chemical, and reaction coordinate ex- pressions gives the system of simultaneous equations describing re- active flow in general. However, to make this system usable, the final '"3 t? I‘:' i: 01 (1‘ P": (a 12.. equations are expressed in terms of ’the following assumptions: 1 . One-dimensional flow 2. Non-viscous fluid 3 . Adiabatic system 1:. No external forces 5. Steady flow 6 . Thermodynamic equilibrium The first four of these assumptions simply establish the flow as being that of a stream tube internal to the main body of fluid . Steady flow requires that there be no sinks or sources in the flow, that the system is "closed.” Thermodynamic equilibrium requires that there be no grad- ients of pressure or temperature in any plane normal to the direction of flow. The second part of this study is to examine a specific situation to obtain numerical data sufficient to demonstrate the effects of chem- ical reaction during flow. Selection of the fluid, operating conditions, and duct shape is entirely arbitrary: therefore, to make the situation reasonable and simple, the following set of boundary conditions are assumed, which apply in conjunection with the assumptions given above for general developnent. Reactive products of combustion which are flowing are those which would result from the burning of a anZn fuel in air with a stoichiometric ratio. This mixture is presumed to be moving slowly and to be in a chemical and thermodynamic state of equilibrium at the time it enters the nozzle. The nozzle is a uniformly converging- diverging duct, with a configuration as given in Appendix A . Inlet con-i: 50}: p718? conditions are a pressure of ten atmospheres and a temperature of SOOO°R, expanding through the nozzle to an exit pressure of one atnos- phere. Sources of reaction data will be given in a later section where the calculations are described. here specific data are not available, they will be approximated from what data are available or by recomended procedures as appears in the literature. ANALYTICAL DEVELOPER” The first portion of this study is devoted to developing the analytical expressions for reactive flow. Rather than consider the flow and chmical phenomena at the same time, suitable expressions will be developed for each and then these will be brought together into a cormnon analytical tool. The sequence to be followed is: l . Governing Flow Equations 2 . Reaction Kinetics 3 . Reactive Flow Chemical reaction is primarily a collisional phenomenon and for con- sistency in approach, flow analysis is developed on such a basis. Governing Flow Equations A formal solution of arm flow problen with chemical reaction requires not only knowledge of the manner of chanical changes, but also the appropriate forms of the usual conservation equations of equilibrium, non-reactive, flow corrected to account for the reacting gas mixture. Such corrections are required because reactive flow is a non-equilibrium situation, except for infinitely fast or infinitely slow chanical re- action rates, and there will be gradients in density, temperature, and velocity. In amt case, the following fundamental equations are neces- sary to obtain a solution to a flow problan: (1) The continuity equation (2) The equation of motion (3) The conservation of energy equation (1;) The thermal equation of state, P - P (V,T) (S) The caloric equation of state, E - E (V,T) The first three of these equations are known as the "equations of change," and they can be canpactly written so as to completely des- cribe the dynamics of a fluid in many types of phenomena. In this section, these equations of change will be developed in most general terms, from the Boltzmann equation of kinetic theory, and will be ex- pressed in vector notation, even though a one-dimensional application is to be made from them and the assumption of an adiabatic, non-viscous, 1' low will cause several of the terms to drop. It is felt that a devel- opment in general terms is more satisfactory, in that the effects of various assumptions will then be more explicit. 10. Eyen though the equations of change are general and will apply to any fluid, they are, however, only useful under conditions that make it pmrsically meaningful to speak of point properties. Local density, temperature, and velocity can be defined formally, but any such defini- tion would be reasonable only where the fluid behaves as continuum. Where the macroscropic properties have large differences over a distance in the order of a mean free. path, the distribution of velocities of the molecules can no longer be considered as Mmellian, and flux vectors can no longer be expressed in terms of the local density, temperature, and flow velocity and their first derivatives . These equations of change describe the change of the macroscopic properties of the fluid (the local density, stream velocity, and temperature) in terms of the flux of the mass (the diffusion velocity), the flux of the momentum (the pressure tensor), the flux of the energy, and the chemical reaction kinetics. The properties of a gas at low densities can be completely described by a distribution function £( 'r" , {I , t ), defined in such a manner that f( r‘ , 'G , t ) d? d ‘13. is the probable number of molecules which at the time t have position coordinates between '5 and i: ¢ (1 i“ and have a velocity ‘5 between 3 and d t . This distribution function is given as the Boltzmann integro-differential 1,2 equation . Using a form given by Kirkwood, This equation can be expressed as _¥ llfirkwood, J. 6., Journal of Chem. Phys. 15’, 72, 191:7; Journ. of Chane PhySo 18, 817, 19500 2'I-Iirschfelder, J. 0., C. F. Curtiss, R. B. Bird, Molecular Theory 21; Gases 2.3g Li uids, John Wiley and Sons, 195,4. ..8 ( i 'Vfi) . E If (11:5 - 31‘3”” bdde duj (2) An explanation of the terms of (2) is given in Appendix B. When a gas is under non-equilibrium conditions, gradients occur in one or more of the macroscopic pmsical properties of the system: composition, mass average velocity, and temperature. Molecular trans- port of mass (m3), momentum (m:j '53), and kinetic energy (é: m3 6:), through the gas, will result from the effect of these gradients . The velocity ‘3 is the velocity of a molecule of species j with re- spect to an axis moving with the mass average velocity ‘50 . Vel- ocities pertinent to this study are defined as: (a) Linear velocity of a molecule of species j is denoted as {1.3 , with space coordinates of ux, uy, uz . Its magnitude [uj I is the molecular speed “3 . (b) Average velocity i) , for a chemical species 3 present with a number density Ni , is defined as: :2 A 1 a- .) ..s. ..s “J(rrt)'fi3fujfd(rrujrt)duj (3) (c) Mass average velocity, '50 , is the stream or flow velocity and, .8 uo(?,t)-—5—:ijj u3 (u) 12. in which p ( r, t) is the over all density at a particular point. p('z’-‘, t) - 21¢ij (5) (d) Individual velocity, 6:.) , is the velocity of a molecule with respect to an axis moving with the mass average velocity Tia %J(?,Ii‘j,t)-II-TI (6) J o (e) Diffusion velocity, '63 , is the rate of flow of molecules of 3 with respect to the mass average velocity, 63 ( 3:, t) ' =53 ‘30 (7) As the mechanism of transport for each of the above molecular properties can be handed similarly, they will be denoted by A 3’ To investigate these transport phenomena on a microscopic level, visualize a cylinder as in Figure l, which represents a cylinder formed by a surface dS moving through the gas with a velocity do . A unit vector '5 normal to the surface specifies the orientation of the surface . Fig. 1. Flow Cylinder 13. According to the definition of Ej , it is the velocity of molecules of species j with respect to the surface element dS. Thus the cylinder with base dB and generators parallel to '53 and of length [53 | dt ’> must contain all of the molecules of species j which have a velocity in a small range d33 about ‘63 , which pass through dB in a tine interval dt. The cylinder, so specified, would have a volume Volume - 3 d3 ° '53 dt (8) As there are 1‘ 3‘15; molecules per unit volume which have a velocity 63, the number of molecules crossing dS in a time interval dt , is (a‘ . ‘63) r, (13de dt (9) Associated with each of the molecules is some property, denoted above as A J , whose magnitude depends on A3 5 then the amount of this property transported across dS in the time dt, by those molecules in the velocity range d6, about 53 , is (a . Byljrj dadedt (10) The amount which creases per unit area per unit time, the flax, is Ajrjn‘ . Eyed, (11) When the contribution of all molecules in all velocity ranges is summed, the total flux is then A Where, 114. is the "flux vector" associated with the property A 3 . The physical interpretation of this vector is that the component of the vector, in any direction '5. , is the flux of the associated physical properties across a surface normal to ‘3? . Expressing the transport phenomena of mass, momentum, and kinetic energy in terms of flux vectors, there results: (1) Transport of mass If & j - m3 then a. the flux vector related to the transport of mass. It is to be noted that . l r: .- ..-b .5 (ii) Transport of momentum If A: 'mj (u‘J- 'u‘o),then /\J 'mjfcjyfd 0:)ch -NJ mi 03), c3 (16) is the flux vector associated with the transport of the y component of momentum, relative to Tic . This vector has components proportional t° Cir 531’ air any and Jr 3% ' momentum have similar flux vectors, giving a total of three flux The x and 2 components of vectors for momentum transport. This gives a symmetric second order tensor P , whose nine components are 15. Which may be written in closed form as [Pd "a": an (15) and is the tensor related to the partial pressure of the 3th species in the gas. The pressure tensor for the mixture is obtained by sum- ming over all the partial pressure tensors, i.e., P'ZJ‘PJ'XJINJMJCJCJ ‘19) Such a pressure tensor has the pmsical significance that it represents the flux of momentum through a gas . Individual components are interpreted by noting that the diagonal elements pn, pyy’ p22 are normal stresses; or pyy is the force per unit area in the y direction exerted on a plane surface normal to the y direction. The non-diagonal elements are shear stresses: or pm, is the force per unit area in the x direction exerted or a plane surface normal to y. Stresses or pressures represented by such a tensor are those relative to an axis moving with the flow velocity '50 . At equilibrium the shear stresses are zero and the normal stresses are equal and this p11 - pyy - pzz - p (20) (iii) Transport of kinetic energy. 1 .. _. 2 “As-amj(ud-uo) J “ .l 2‘ - .1 .. Then Ad 2%[03cjrjdc:J EmJNchcJ (21) is the flux vector for the transport of kinetic energy for molecules. Smmed over all components of the gas mixture such vectors give the "heat flux vector, " c '6‘ (22) 2 "33 NH" 3 ' Z “‘3 a J Returning now to the Boltzmann equation, (2), it is possible to transform it into a general equation of change known as the Enskog equation. This relationship is for a physical property A i associated with the 1th molecule, in which summation over all i gives the equation of change for the property A . When (2) is multiplied by A 1 and integrated over ii, there results A, 3.243, .Vr)+L(3c’ .Vr.)d“ 3t 1 mi 1 1 ui . Zflffli (1‘ng “f1 f3) 81:) bdb dE dii‘j d iii (23) 1 Taking I“ , iii , and t , as independent variables, and noting that A i may depend on "f and t through uo ( ‘r‘, t), the three left-hand terms may be transformed by use of the relationships: 3‘1 —- of —- f 9A1 ._. fact “i at Aii ‘11 13,0 “1 .M - 1,1431; (21,) 9t 9t [Aiuix axdul'EfAiuixfidui‘jfiul-X gxdui 17. . _N9_:_i (26) i 9 “ix Where only the x components are written for (25) and (26), and the term (A1 f1) is assumed to diminish rapidly for large ml, the transformed equation is then, 9A1 9t 2* - (31; .. VuiAi) =zfmli whether the mixture is reactive or not. In physical terms, an interpretation of (32) shows that the number density or species 1 may change because of the expansion of the fluid, diffusion processes, and due to either the production or elimination of species 1 by chemical reaction. In the case of the mixture, the mass density, p , only changes because the volume of the fluid element is changing. When A i a mi '6. and the ensuing expression is sunmed over 1 i , there results 3(N.€) 3...; a? A Zmi 51171;“ (V '"i “i Ci)‘N1 3 ti-Ni ( Uli ° V61) (I " £(Yi ' vui 6;) '0 (3’4) mi Utilizing the definitions of velocities and of the pressure tensor this reduces to the "equation of motion," D ii If g _. .. (35) where F and “iii are the independent variables of differentiation. This equation shows that the velocity of the fluid element varies because of the gradient of the pressure tensor and be- 20. cause of the external forces acting on the individual species. To obtain the conservation of energy equation, set A1 - 3;- miciz , and in a manner identical to above, there is obtained from the general change equation 1 3(Nici) _.—2 (Sci -. __ O ON C ‘1'“..- E ml[ t (V iuii) Ii at 2 ‘3 i -Ni(fii'vci2)-(;% ' V1,; Ciz) '0 (36) Using the heat flux vector of (22), and the pressure tensor, this last equation, (36) reduces to a A A “fiflWV - Peuo>+t--(V.7)+P (M) This is the "equation of change of g", in differential form. Expressed in terms of the substantial derivative, this last equation becomes, D A PBE--(V°7)*P (hS) The dynamic equations of change can be applied to this equation to give expressions for 7 and F in terms of transport properties and flux vectors. However, as pointed out above, it is extremely difficult, if not impossible, to obtain these expressions in workable form and it is necessary to introduce sufficient simplifications to make the re- lationships cormnensurate with present knowledge of physical data. Reaction Kinetics A one-step chemical reaction, in which reactants are directly forming products, can be written aA+bB+---—->cc+dD+--- (h6) in which there are a moles of component A reacting with b moles of component B, etc ., to form products consisting of c moles of component 21:. C, d moles of component D, etc. To write all reactions in this form would become cumbersome and unwieldy, so a more compact notation will be anployed . The stoichiometric relation describing a one-step chanical reaction of arbitrary complexity may be represented by the reaction equation 2 use —> z: I ’I where the terms U5 and 1/0 represent respectively the stoichiometric coefficients of the reactants and products. For any chemical compound 1 Mi which does not appear as a reactant then 1/,- - 0, and likewise II if a compound is not formed then M - O. In a general sense, ()4?) can adequately describe all possible chemical reactions. In accordance with the _L_§._1_r gt; ga_s_§ AM, the rate of dis- appearance of a chemical species is proportional to the products of the concentration of the reacting chemical species, each concentration being raised to a power equal to the corresponding stoichiometric co- efficient . For the chemical compound represented by the symbol [i , concentration will be noted by [Mi] , with the units of mole/ft3 . Thus reaction rate at: 11" [111]“; (1:8) L is the Law of Mass Action in symbolic form. Introducing a proportion— ality factor k3 , called the specific reaction rate constant or co- efficient, (1:8) , becomes I reaction rate - kj TT [111]“: (’49) i 25. As presented here, (149) is for a homogeneous system and for a given chemical reaction the specific reaction rate k3 is independent of concentration [111], depending only on temperature. In general, k, can be expressed in the form accredited to Arrhenius, k3 - A e (50) where A , the frequency factor, is a quantity that is indeperderrt of or varies relatively little with the temperature and E , the activation energy, is the difference in heat content between the activated and inert molecules . Provided that the temperature range is not large, the quantities A and E can be taken as constant. It may this be assumed that k3 is an empirically determined coefficient which depends only on the temperature for a given chemical reaction. Further dis- cussion of k1 will appear later. All specific reaction rate constants will be symbolized as k j , with j - f for forward reactions and j - b for backward or reverse reactions . A forward reaction is one in which the reactants appear on the left hand side of the chemical equation and products on the right hand side, while backward reactions are the reverse. Net rates of change of the chemical components are the only observable results of a chemical reaction. From (147) the net rate of production of Mi is 33(2)].- dt - U1) (Reaction Rate) ' / - (vi - 1/1) x, 71; [Milu‘ <51) 26. as Ud’lmoles of Hi are formed for every 1/; moles of Mi used up in the reaction. The law of mass action, as expressed by (51), may be applied, in a meaningful way, only to elementary reaction steps which describe the correct reaction mechanism. For a number of simple chemical processes, a plausible reaction mechanism has been deduced by chanical kineticists, but for the most part, only intelligent conjectures can be made by the kineticists regarding probable reaction mechanism. De- 1 tailed studies show that the production of H20 from H2 and 02 involves, among others, the following elementary steps, OH 0 H2 —> H20 4' H H + 02 —-> OH + O O + H2 —-> OH + H to each of which the law of mass action applies, and for each of which there is a specific reaction rate. The stoichiometric coefficients for elementary reaction give information about the number of moles reacting, not about the weights or volumes changing. As a function of time, the number of moles of re- actants will decrease asymptotically toward a limiting value (not necessarily zero) whereas the number of moles of product species will increase asymptotically toward a limiting value. Ikperimental chemical kinetics is a science of considerable complexity because very large observable effects are frequently brought J'I..ewis, B. and G. von Elbe, Combustion, Flames, and Explosion of _C_}_____ases, Academic Press, Inc., New YorE, I931.“ 27. about by uncontrolled factors such as minute concentration of impurities, surface effects, etc. The absolute precision of even the best-brown rate constants is low. Due to the temperature dependence of kj , it is necessary to have extremely precise values of E to yield a reliable prediction for the change of k j with temperature. Most observed data on 1:: are restricted to a relatively narrow temperature range, in which the results can be fitted equally well by a number of different rate laws, and tlms extrapolation of known values to higher temperatures are almost always uncertain by an order of magnitude or more. Chemical reactions can proceed in both the forward direction (reactants forming products, rate constant kf ) and in the reverse direction (reaction products forming the reactants, rate constant kb ), both of which are essentially single-step reactions. At thermodynamic equilibrium, there is no net change in composition, which means that kf and kb must be related through this equilibrium constant, Kc , expressed in terms of the ratio of the concentrations raised to appro- priate powers . A general set of opposing chemical reactions may be expressed by Zus§2fla 7[ P1 7T (a (72) II I (Vi-ngfu-{g 1 Where the quantities pi , Xi , P1 , and Ki denote, respectively, partial pressure, mole fraction, mass per unit volume, and weight fraction of species i if the concentration of species i is [111]. The starred quantities defined above refer to a situation in which equilibrium conditions do not exist at a given point in the system, at least not relative to the existing temperature and pressure at that point. Such parameters are not to be interpreted as indicating a pseudo-equilibrium, but rather as a measure of deviation from equi- l librium. In an article by Penner, on the flow of reactive gases, he lPenner, S. 8., Jour. of Chem. FEE-9.3.: Vol. 19, No. 7, July 1951. 33. points out how such relations may be used for "near-equilibrium" flow conditions, if the deviations from true equilibrium are of small order. Chemical reactions were discussed above in terms of reaction rates and the equilibrium constant, dwelling primarily on "how fast" a reaction might proceed and what the change in concentration might be with respect to time, the implication being that the reaction would, or most certainly could, attain a condition of equilibrium if sufficient time were available . In flow systems there is always the possibility that sufficient time will not be available for the reaction to go to completion, attain an equilibrium state, or the reaction may be intentionally stopped at some arbitrary point, and it is desirable to know just how far the re- action has actually progressed . The amount of energy liberated, the change in concentrations of the components, and many other properties will depend on how far the reaction proceeds. To facilitate keeping track of the amount of reaction progress, it is desirableat this time to introduce a reaction coordinate, g , which will be used to indicate the amount of change. This is similar to the parameter introduced by DeDorxier and referred to as the "extent of reaction." Considering a system in which no transfer of matter takes place between the system and its surroundings, a "closed system," the masses of the various components can vary only as a result of spontaneous chemical reactions occurring within the system. For the single re- action symbolized in general form as I! Z U.M. :2 );U M. (52) 111 ii 3b. where U1 and Vi are the stoichiometric coefficients of the re- actants and products respectively . A sign convention will be applied in which the stoichiometric coefficients of the components formed, 1/: , as the reaction proceeds to the right are positive, while for the com- ponents consumed U; will be taken as negative. This will permit dropping the! superscript and using only the subscript i to denote the 1th component . Increase in mass of component i which is being formed in the reaction is proportional to its molecular weight, '1 , and to its stoichiometric coefficient, 1/1 , in the reaction. Therefore, m1 " m1° " V1'1§ ---- ---- ---- (73) mi " “1° ' U1'1§ in which mio refers to the initial masses of the ith components at time zero, when the extent of reaction, g , is also zero. Initial states are in this way defined by g - 0 5 and a state of g - l 3 corresponds to the conversion of U; . . . . 1/i moles of M1 . . . M1 I reactants to U1 . . . . U1 moles of M1 . . . . Mi products. When a system goes from state § - O to g - 1 it is said to have com- pleted "one equivalent of reaction ." According to the "principle of conservation of mass ," the total mass of a system remains constant in the course of time, and upon sum- ming over all the equations of (73), there results ' . 21: U1 '1 = 0 (7b) This is the stoichiometric equation for the reaction written. 35. In equations (73), for a closed system in which only one re- action is occurring, the variables ml . . . mi can be replaced by the variables g and m1° . . . nLLO . Then the thermodynamic state of a system for which mlo . . . 11110 are given can be determined by two physical parameters, i.e., T and P , and a single chemical variable f. Differentiating equations (73) with respect to time, noting that the initial masses are constant, gives ——— I w. —-—-—- dt 1 1 dt or U 1'1 UZWZ I U 1'1 (75) (76) Rather than defining the chemical state of the system in terms of masses, it is usually better to employ the number of moles of the various components, n1 . . . n1 . Then equation (73) becomes n1 ‘ n1o ' V1§ ni'nio " U1§ Where the original number of moles n10 . . . “1 (77) refer to t - O. 36. Differentiating results in dnl . dn dni d _. - __ - --_ n__ a (78) V1 V2 V1 § Mole fractions of any component can be given by X n1 . “:1.+ Uig n10 +U1§ 1 gm >gni°+§§vi we where 1/ .2111. - 1 Normally the system being studied will undergo a series of _ simultaneous independent reactions, rather than a single reaction as considered above. Here "independent" has the same connotation as in the mathematical sense, in that a series of reactions is described as independent if no one of the stoichiometric equations can be derived from the others by linear combination. For such a set of r reactions (71;) would become g 1/1,]. '1 - 0 ---- ---- mm 21; Visrwi . 0 where 111,1 and U1“. represent respectively the stoichiometric coefficients of the i’"h component in the 18t to rth reaction. Written in more compact form (79) becomes €114,711 -0 ( T -1, 2, —--, r.) (80) 37. and (76) may now be written as —.—...-— - -———-———- I - .. - a ________ - d (81) ”am V2.7 '2 V1.7 '1 £7 Likewise (78) becomes drn], - drnz _ _____ drni .d§T (32) V1.7 V2.7 V1.7 The symbol dTni represents the change in the number of moles of the it'h component in the time dt resulting from the 1' th reaction. The total change in n:L as a result of all the rt’h reactions is wu-d1n1*d2n1*----*drni'zdrni <83) T 01" 391.. . Z” d.__§___'r (an) dt T’ 1’7 dt - In this last expression the relation of rate of change of with respect to time is known as the reaction velocity and will be written as d § N. E 7' (85) dt It is the ratio of the change (positive, zero, or negative) in the ex- tent of reaction during the interval of time dt (always positive) in which this change takes place. 38. The rate of consumption or production of moles of components can be expressed in terms of the reaction velocity, as by (78) and (8h). d 331’ ' U1 /\/ (Single reaction) (86) dni —- . U- Multi 1 reacti dt 2 c, 7’ ”r ( p e on) In terms of the actual masses of the components am: 3": " l: U1,TW1 ”7' (87) By definition, molar concentration is I - “1 88 M1] Vol. ( ) Differentiating this with respect to time gives d[ J . dt V dt v2 dt and from (78) - V1 d “1 dV [1 . I -—- -—-—- — -- -- 0 ‘1] v dt v2 dt (9 ) or - 1/1 “1 av - [ ] . —— /Y - _ _ (91) Ni V v2 dt which gives the change in concentration for each component in a single rea~t‘:=i’.ion . 39. Extending this last equation to a mltiple reaction, the com- bined change in concentration for any 1th component that might appear in more than one reaction, results in : U V v2 dt T which shows that the concentration changes because the number of moles are changing and also, because there may be a change of volune resulting from the reaction. Volume change in this case is attributable directly to the reaction, being a function of the state equation, and not to the spatial volume change connected with container dimension change. ho . Reactive Flow Now that the general theory for the flow of a reactive gas mixture has been developed, the next step is to express these relation- ships in a form applicable to the conditions set forth in the assignment of the problem . This will entail introducing the relevant assumptions to reduce the general equations to ones which can be solved simultaneously, comensurate with available data and boundary conditions . With the flow restricted to a one-dimensional path, all of the space derivatives become functions only of the axial dimension y , there being no changes in the radial directions . Thus, in any plane normal to the y-axis, there are no changes in properties which are functions of position over that plane, 1.6., no gradients are permitted to exist in the plane. Likewise, in this plane normal to the axis, there will be no changes relative to time in those properties which are independent of the effects of chemical reaction, which includes overall density, total internal energy, and total pressure. Even though these are properties which change with a chemical reaction, ard are thus a function of time, the concept of steady-state flow can be applied to the overall flow. It will simply mean that the reaction rates are independent of time, being functions of position, and the properties which are affected by a reaction will be affected in the same manner, relative to time, at any position in the flow device. The reaction rates, which are functions of tanperature for a given re- action, are dependent on position only in that temperature is changing along the flow path and is thus a function of position. ’41. The equations of change give a set of equations which¢iescribe the bulk properties of the flow as it proceeds through a varying area device. In the one-dimensional analysis, in which the boundaries of the channel are surfaces of revolution, described by a relation f(x,z) - y , the following procedure is used to reduce the time inde- pendent equation. Consider a plane normal to the y-axis and let S be that portion of the plane contained within the intersection with the f(x,z) - y surface. If F?) is a vector field which on the surface f(x,z) - y is tangent to the surface, then HW' FT?))dxdz-§— f]? dxdy (93) y s V S where Fy is the y component of i; . In steady state, the three dimensional continuity equation, (33), is x0(V - If.) - o (91;) In the nozzle, the velocity is either zero or tangent to the walls, and (93) may be applied to give, d which, upon integrating with respect to y , gives Jr “by dx d2 - Constant (9S) Noting that the density and velocity are uniform over any cross-section, (9b) can be written 15% “oy‘Ay - ‘w - Constant (96) {5.133. nut-u “an“: 1:2. where w is the mass flow rate, a constant. Since in one dimension it is unnecessary to carry the notation that density, area, and velocity are functions of that dimension, it is permissible to write P 11.0 A - w (97) A one dimensional conservation of energy equation can be ob- tained in a similar manner. Starting with equation (35), and multiply- ing it by P TITO , after which the result is added to equation (39) , there results we. - v Stable molecule E = 0 K cal Atom 0- Molecule -——> Products E - 25 K cal Molecule + Molecule -——-> Products E - 50 K cal No experimental evidence is available to substantiate the assumed activation energies, at the high temperatures involved, but from low temperature data, these seem to be reasonable mean values. Both 3 and A are relatively independent of temperature, and using the above assumed data gives values of k comparable to published values for equal temperatures . lGlasstone, s., x. J. Laidler, and H. hb'ring, _T_h§_ Theory 9; Rate Processes, (New York: McGraw-Hill, 191.1) . 2Amdur, 1., Journal 93 American Chemical Society, 60,23h7,(1935). 67. with this information (72) can be applied to each of the re- actions, giving the A7 nit/db contribution per reaction. It is not necessary to solve Arni/dt for each component in each reaction, since from the reaction coordinate, g , d £7 _ 1 . d‘rni dt Ui’r dt (82) and for the multiple reactions d dt dt T’ In this manner the net rate of change of each component is found by summing over all of the reactions, in this case, twenty-three. Examination of the resulting rates of change showed that several of the reactions were relatively slow and the significant effects could be described by a system of ten reactions. These ten reactions, which are assumed for the rest of the study, are: CO 0 O + M ——) 002 0 M H2 0' O —-) H20 2H 0 ll —-) H2 iv M 2 OH —-) H2 4' 02 OH + H + M ——-) H20 + M 2 N0 —-) N2 + . 02 NO + N -—> N2 4- O 02 + N —-> 0 + NO 2 0 + H ——-> 02 + M o c H + u —-)- 0H + M 68. Contrary to the situations for equilibrium and frozen flow, it is necessary to have a control volume or mass for finite flow in order to know the number of moles of the 1th species. For this study, a control volume was chosen equivalent to the volume in the nozzle bettieen y inlet and y - 0.1: , on the converging side. From the equation of state, with initial conditions, the number of moles of mixture, and each component, may be calculated. In this case there are 6.3601: x 10"5 moles of mixture. with the initial composition thus established, the change in composition due to reaction may be determined up to T - h900’R . This canes from (150) in which the term is taken as the mean value between equilibrium and frozen flow at T *- lt9SO°R , as indicated in Fig. 3. Then n1 - n1 - A n. (151) T T 1 I d and the moles of mixture is obtained from sunning over all i. The new mole fractions at T .- h900 are i '( ) T?-'h900 E: n, L- r -h9oo x (152) After the 11 are found, the mixture molecular weight and heat capacity are determined through Wmixture - Z '1 X1 (153) l 69. and CI) - Z Cpi (15h) i Sufficient information is thus available to obtain a solution of the energ equation, as given by (126). A solution in this case, results in the pressure of the mixture necessary to satisfy the mixture at the temperature in question. As the left hand side of this relation is a statement of the sensible and latent enthalpies, it also gives the total enthalpy change. The enthalpy change, A h , may also be cal- culated from (123). After the A h and the pressure are known, the remaining quantities are calculated in the same manner, as noted above, for equi- librium and frozen flows. Finite flow properties for the remainder of the nozzle are cal- culated in an analogous manner, the major difference being in the application of (72) for d?” rtil/(n . After the first step down the nozzle, the non-equilibrium mole fractions are known and thus K1“. may be calculated directly, without the necessity of using Van't Hoff's equation. In this case, n u * U1 _ U1 19. - 7T (Xi) (155) L for the finite composition. The ’8: term is given by, H 3 U1 .. U1 1:, - IT (Xi’e) (156) L in which Xi e are the equilibrium mole fraction for the same temper- ’ ature . PRESENTATION AND DISCUSSION OF RESULTS Using the reactive gas mixture and method of calculation, as indicated above, an expansion of the mixture is followed through a nozzle with a ten to one pressure differential. As this is a compara- tive study, the equilibrium path and frozen path are used as the "boundary or limiting" conditions 3 because they represent a limit in terms of reaction rates and practically all analyses are made on one or the other bases. Parameters selected for purposes of comparison are: 1. temperature profile 2 . pres sure distribution 3 . density distribution 1:. area ratios, relative to nozzle inlet and throat Values of these parameters, as determined by the indicated calculation procedure, are presented in Tables I, II, and III. Only five points, in addition to the initial conditions, are presented for finite flow. Due to the magnitude of the calculations, it was not practicable to generate more data. Electronic computation is almost mandatory in making a complete analysis, as was done for the data available to equilibrium flow. It is felt that the data computed is representative and while machine calculation would permit considerable refinement, the results would be comparable. Actual comparison of the three flow paths is shown by means of Figures ’4, 5, 6, 7, and 8 . Temperature is used as the basis of com- 71. mHH. 0 Na. N 8H. N am. mm NS. o S. N 030 N2. me. 0 8% 3H. 0 Nmm. N one. N no. 3 SH. H 5. mN 2mm 80 8. H 8mm EH6 0mm. N HmaN 3. me aHH.H 3.3 33 NS SH 89.. 026 93. N 30. m 3. a. NHm.H 3.8 33 New «NH 8? sees as. H an. H as. s as. H when Sam 8e 3H 8% HNN. 0 0%. H as. m 8. ca 8a. H ma. 8 ENm Rm 3. H 8% NmN. o Nam. H 80. a He. we Sa. H Na. N 8cm 4% on. H o8: RN. 0 93H 3... a 3. 8H HcN. N an. N Hmn: an: on. N 83 H26 8m; Hma. s 933 0mm. N 3.8 83 N3 2.. N 8N; acme RH. H EH. m 8. NH Sm. N am. mN .HNQ NR N. m 83 3;. 0 N8. H ems. m 3. RH 34. m mm. 8 N8: mNm om. m 093 8m. 0 omo.H 0%. m mH. 3H mNm. m cm. 3 mmbm EN mm. a 83 Ham. 0 wHo.H coo. c 3. 3H 54.: Na. 3 3mm HNN an. m 83 ones So. H 92. n 3. 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SA 83 N8.H 8.0 3.2: oaN.m 88 9an 8.8 She can 83 NRH aces aN.mHH NNme oacH m.mm 8.8 See me.» 83 .86 8o.H 898 mg; 08 0.0 8H.8 8.0 8.3 08m E can: an}: afiefl coast sages acH m sass 85a a. are} (3. m, \ mNodHK a a < p e do a .H 8a 822 HHH H.549 7h. 75. 76 . 77. 78. 79. parison in constructing the curves of each of the figures; not because temperature has a special significance in the dynamics of nozzle flow, but due to influence on the reaction kinetics . The most significant feature brought out in this study is the nearness of the finite flow path to the equilibrium path throughout the entire length of the nozzle . There is a gradual tendency for the finite flow to deviate from the ideal equilibrium flow, but it appears that this tendency is not as severe as sometimes assumed. Brayl in- vestigated a so—called "Ideal Dissociating Gas" and concluded that the gas would act like a frozen mixture shortly after the throat of a nozzle. Heims2 reported from the study of "air" flow through a nozzle, with a mixture consisting, initially, of undissociated nitrogen and partially dissociated oxygen, that the flow path would be located approximately mid-way between the equilibrium and frozen paths . As the flow proceeds down the nozzle, there is a reduction of temperature, which reduces the reaction rate constant through its ex- ponential term; and at the same time, the velocity is increasing, which decreases the time available for reactions. Both of these effects tend to decrease the amount of net reaction. However, as these two effects are driving the mixture away from equilibrium composition, the ratio of free energy functions, expressed as K*/K, acts as a potential to restore equilibrium. It is the net effect of these three forces which will ultimately predict the finite path location . lBray, K. N. C., 22.: cit. 2Heims, S. P., 92: cit. 80. An examination of Table IV, where compositions are compared at the same temperature and pressure, shows that some of the mixture con— stituents tend to remain very close to equilibrium while others deviate quite appreciably. This comes about from the complex system of re- actions involved in the recombination process, each of which has its own activation energy and collision frequency factor. Thus, the gross mixture properties are made up of the contributions of the individual reactions making up the entire system, some of which may be near equi- librium and some near frozen paths. It is the gross properties which are being compared in this study. If the nozzle used in the above calculations was designed on the basis of frozen flow analysis, as is the current practice in in- dustry, it would have to be of sufficient length to satisfy the area ratios for the converging and diverging section. Finite flow analysis shows that such a nozzle would be about 2% too long in the converging section and 11% too short in the diverging section, for a pressure ratio of ten to one. The designed nozzle would be expected to have a mass flow rate of 3.136 pounds per second, but with finite flow it would only carry 3.358 pounds per second. In addition, there would be a higher temperature and pressure distribution than predicted by the design. The effect of having too long a converging section would be negligible, only wasted material, but the short diverging section would cause an under—expansion of the flow and would result in a shock at the exit. This would represent a loss of nozzle effectiveness, be it measured in terms of velocity, thrust, or impulse. 0n the basis of TABLE COMPCBITIONS, x1, r, ’12 A 00 002 a, H20 112 5000 8.1193 3 3.523 .2 9.225 2 6.5115 3 1.153 1 7.067 1 11900 8.512 3 3.3011 2 9.505 2 5.901 3 1.161 1 7.088 1 8.511 3 3.335 2 9.1158 2 5.820 3 1.160 1 7.087 1 11700 8.553 3 2.6111 2 1.019 1 11.8711 3 1.200 1 7.122 1 8.5113 3 2.818 2 1.002 1 5 .0117 3 1.215 1 7.118 1 1.300 8.622 3 1.562 2 1.130 1 3.015 3 1.238 1 7.205 1 8.6011 3 1.8113 2 1.103 1 2.881 3 1.257 1 7.163 1 3900 8.672 3 7.985 3 1.223 1 1.500 3 1.275 1 7.258 1 8.662 3 9.576 3 1.192 1 1.681 3 1.265 1 7.21.0 1 3500 8.699 3 3.970 3 1.263 1 7.560 11 1.2911 1 7.282 1 8.678 3 6.8911 3 1.203 1 1.031 3 1.290 1 7.2711 1 Note: 8.1193 3 means 0.0081193 IV EQUILIBRIUM AND FINITE FLOW 02 O OH H NO N 1.372 2 1.9611 3 8.761 3 1.931 3 8.959 3 1.121 b 1.275 2 1.620 3 7.750 3 1.620 3 8.105 3 8.785 5 Equil. 1.2914 2 1.726 3 8.023 3 8.353 11 8.558 3 9.930 5 Finite 1.132 2 1.085 3 6.005 3 1.215 3 6.202 3 5.2115 5 Elluil. 1.11111 2 1.107 3 5.379 3 1.107 3 7.2116 3 7.278 5 Finite 6.935 3 11.2110 11 3.210 3 11.615 11 3.290 3 1.515 5 Equil. 7.757 3 8.707 11 11.7119 3 11.116 11 3.688 3 1.890 5 Finite 3.3511 3 9.7110 5 1.285 3 1.263 11 1.1.90 3 2.950 6 Equil. 11.217 3 8.1103 1. 3.139 3 1.961. 11 1.696 3 6.31.2 6 Finite 1.310 3 1.783 5 6.1110 11 2.150 5 7.367 11 3.355 7 Equil. 2.11811 3 11.936 5 2.882 3 2.707 5 1.169 3 5.1177 7 Finite 83. specific thrust, design would indicate a value of 171 pound-force- second per pound mass, while the finite flow would result in a specific impulse of 1.65 pound-force-second per pound mass . This represents a decrease of 3.5 percent in design performance. Had the nozzle design been based on finite flow analysis, the expected thrust would be 170.1 pound-force-second per pound mass, an increase of some 3.7 percent over that developed in a frozen flow designed nozzle . These values are based on the assumptions as set up for the study, and show only the effects of neglecting recombination reactions with their potential energy release. I All of the results of calculation presented here are valid within the frame of reference, as prescribed by the initial assumption, that the nozzle is frictionless, adiabatic, etc. As the purpose of this study is to investigate one specific phenomenon, it is felt that the simplification permitted by these assumptions, is justified. The - eventual goal in nozzle design should be to include all of the possible deviations from ideality, but the analyses of individual effects should prove to be a most useful adjunct to the overall understanding of nozzle problems . With the increase interest in rocketry and Jet pro- pulsion, it is necessary that chemical thermodynamics become a part of fluid dynamic theory. Until a great deal more experimental data and electronic cal- culating equipment are available, it appears that the recommended form of flow analysis, for design purposes, should be based on an equilibrium path. With access to tabulated information, such as that of Powell and 81". Suciu1 and of the Bureau of Standards ,2 an equilibrium analysis is no more difficult than a frozen analysis. Based on results obtained from this study, the following con- clusions may be drawn: 1. When a high temperature mixture of reactive gases is expanded through a converging-diverging nozzle, the thermodynamic path followed approximates an equilibrium path, when the mixture is initially in a state of equilibrium. 2‘. Deviation from the equilibrium path increases with decreasing pressure, but at a decreasing rate. 3. Nozzle performance, based on a finite reaction rate design, was found to be 3 percent better than for a comparable frozen flow design. 14. The effects of chanical reaction should be included before expecting the results to accurately describe -the best performance a properly-designed nozzle might have . 1'Powell, H. N. and S. N. Suciu, 22; cit. 2Rossini, F. D., et al, op. cit. APPENDIX AITENDIX A NOZZLE CONFIGURATION The nozzle shape assumed for purposes of this study is that of two truncated cones, placed small and to small end. In the convergent section, the angle of convergence, fl, is taken as 30°, while in the divergent section, the angle of divergence, 9 , is 7 l/2°. Each cone of the nozzle may be defined by an equation, I dr I ry rt+y—- rt+ytan4 in which y is positive to the left, for the converging section and the z. is ¢ 5 y is positive to the right for the diverging section and the angle is O . A dimensionless ratio of the area at any point y, to that at the throat, A/At , is given by r 2 r +ytanz. 2 A/At- -- - t - (1+Ltant.) rt rt rt 86 87. Table A-l contains a tabulation of some of the properties of A/At for various y , based on a throat radius of unity. Fig. 9 shows the configuration of such a nozzle. Fig. 10 shows the relation- ships of Table 11-1 . 88. 0%. N 8%. 4 80m. 0 m. 4 48. N 0084 008.0 0.4 84. N 0004. 4 0004. o m. m 08. 4 0804 080.0 0. n 80. 4 000m. 4 000m. 0 0. N 30. 0 0040. N 340. 4 0. N N00. 4 84m. 4 84m. 0 0. N 08. 0 440m. N 440m. 4 0. N NE. 4 0044. 4 004m. 0 4. N 400. m 88. N 008. 4 .4. N 304 884 88.0 N. N .34. m 88. N 884 N. N 0&4 884 88.0 0. N 30.4 Sm4. N Sm44 0. N 03. 4 28.4 88.0 04 84.4 N80. N N084 04 3:4 0084 008.0 04 40N. m 0804 88.0 04 00.4. 204. 4 904. 0 4. 4 N8. 0000. 4 M000. 0 4. 4 40m. 4 084. 4 084.0 N. 4 000. N 800. 4 8%. 0 N. 4 48.4 N484 N420 0.4 84. N 5R4 88.0 0. 4 48.4 8444 844.0 0.0 8n. N 8G4 0040.0 0.0 NNN. 4 $04. 4 $04.0 0. 0 84. N 0404. 4 040.4. 0 0. 0 84. 4 N80. 4 88. 0 N. 0 48. 4 304. 4 30.4. 0 N. 0 4044 02.04 008.0 0.0 040. 4 4040.4 408.0 0.0 84. 4 300. 4 800. 0 m. 0 400. 4 $8. 4 4.08. 0 m. 0 004.4 NNm04 800.0 8.0 m4m4 88.4 008.0 4.0 4004 800.4 88.0 no 084 N244 3.4.0 0.0 $0. 4 $84 . $8.0 N.0 .484 mm44.4 mm44.0 N.0 80.4 N304 840.0 4.0 N444 N504 2.8.0 4.0 000.4 0000.4 008.0 0.0 80.4 0804 0000.0 0.0 04.} 0.80.4.4 0:3» 0 04} 08.0.4.4 05...» .4 o «\H h I o N 9443083 com I & $588280 mmngouuaam Hugo HANNoz Alqumg. 90. .APPEMDIX B BOLTZMANN TERMS The terms of equation (2) may best be explained by means of 1 Fig . B-l. Fig. B-l. Collisions of molecules of type "j" with one molecule of type "i" This equation is an expression relating the various molecular forces and momenta involved in the collision of moving molecules of type “3" with a fixed molecule of type "1". To find the probability that an "1" molecule, located at position i" , with velocity Tfi , will experience an impact with a molecule of type "j", in the time interval dt , with the impact parameter in a range db about b . The impact parameter b is the distance of closest approach of the two molecules if they continued to move in straight lines with their initial velocities . The relative velocities of the two types of mole- cules is given by the difference I 33 - fii| and is noted as g” , (s13 - 331). 1 . Rossins, F. D., Hi Speed Aerodx%ics and Jet Propulsion, Vol. I, (Princeton, N. J.7%inceton vers y fie), 1951;. 91 92. During a short interval of time (it , any molecule of type "j", which is located in the cylinder of base bdde and height gijdt , will undergo a collision with molecule 1 . The probable number of molecules of type "3" in this cylinder is fj(?,u,t)g13bdbd€ dt Then the total number of collisions experienced by this molecule i , with molecules of type "3", is obtained by sunning over all the number of collisions characterized by all values of the parameter b and 6 and all the relative velocities g1.j . The result is dt ffjf3(r,u,t)gijbdbd€duj Since the probable number of i molecules in the volume element d 'F about i" with velocities in the range d iii about "111 is 1‘1 ('17 , II , t) d ‘5 d ‘13 , it follows that a collision parameter r135) rm (-) _§ ij 3 J[j]r.fi Is 813 b db dWE d uj By similar reasoning, another collision parameter for con- is + ditions after a collision, I}; ) , is given by r,(") t I A 13 Iffffifjgijbdbdéduj where the primes refer to post-collision properties and the equalities I b-b I 313 a 313 ' d'fiid'fij ad'iii d‘fi J are assumed to hold. 93. It was on the basis of these parameters and relationships that the Boltzmaxm equation, in the form (2), was derived. The term J X1 represents an external force acting on the molecules, which is a function of position and time. APPENDH C . VALIDITY OF EQUATIONS (28) AND (29) The integral I I [ff/15:11} ..rirj) gid bdbd€ dui duj is equivalent to the integral written in terms of inverse encounters, i.e., where different symbols are used to indicate the variables of integration. Thus ’ c I I I t _‘u A: Aiuifd- f1 fj)g13b dbdéduiduj It has already been shown that ' I .3 .3 .a' .A. , b=b , and duidu‘j Idu:l du.j I? 013 - gij which permits rewriting the above equation to I I A _s .. jjjfliuirj -f1fj)gidbdbd€ duiduj As the integrals are equal, they are also each equal to one- half the sum of the two . Such an operation gives I I .. .; [ff/A1611} -f1 f3) gubdbdé duidu:j I I I .¢ «# 'l/foff(A1-A1)(fifJ-fif3)gijbdbd€ duicluJ 9h 95. When A - m the validity is immediately established; for i i I ( A i - A 1) - 0 indicates that the individual masses of the mole- cules are conserved in an encounter. However, the last equation may be smmned over both 1 ad :1 , after which the dummy indices are interchanged to give the identity l/2 ijjdi-Aixri r3- £1163) gijbdbdé d‘fiid'fi‘J 1.1 - lflZ’UUMJ - A3)“; :5 - £1 £3) gij b dde c1111 c1313 3 Ihereupon, ZfMA1(f; r; - r1 £3)g13b db d6 c1711 an: 3 L: . - M: Z fjffdl * A3 ' Xi - am; r; . :1 :3) if A A gijbdbdé duidu‘1 The invariants of an encounter are then defined by the vanishing / I of the term (A1 4» A3 — A1 - A3), or those properties which make (28) and (29) 8Qual to zero. Properties which are conserved in an encounter are thus: mass of the individual molecule ml , manentum _s 2 mi Ci , and kinetic energy 1/2 m1 Ci . BIBLIOGRAPHY 12. 13. BIBLImRAPHY Selected References Amdur, I., Jour. 91.: American Chemical Societ , 60,231.57, 1935. Bray, K. N. 0., De arture from Dissociation Euilibrium in g Mersonic Nozz e, A. F. C. 19,935, iii-ch 9 . Brinkley, 3. R. and B. Lewis, The Therm cs gf_ Combustion Gases: General Consideration, Bureau 0? %es, W52 . Glasstone, S., Thermod amics for Chemists,(New York: D. Van Nostrand 00.5, 1935, p. 538. Glasstone, S., K. J. Laidler, H. Eyring, The Theo of Rate Processes, (New York: McGraw-Hill Boo .5, E915. Helms, S . P., Effect _o_f 939392 Recombination on (hie-Dimensional Flow at High Eh Numbers, N133. TN HIKE, 1958. Hilsenrath, J. , et al, Tables of Thermal Pro ies of: Gases, National Bureau of m m 33E, E553 Hirschfelder, J. 0., C. F. Curtiss, and R. B. Bird, Molecular Theo of; Gases and Liquids, (New York: John Wiley Ea Sans), I935. Kirkwood, J. 9., Jour. of: Chemical sics, 15, 73, 191:? and Jour. 93: Chance-I macs, I8, , SO. Penner, S. 8., Jour. 9!; Chemical msics, Vol. 19, No. 7, July 1951 Powell, H. N. and S. N. Suciu, Pro ies of Combustion Gases, Vol. I and II, (New York: McGrawgfitm BSOE 00.), I953. Rossini, F. D., #3h 3 ed Aergdxné-ngcs and Jet Pro ion, Vol. I, (Princeton, . J., Princeton versTt'i F5383}, $311. Rossini, F. D. et al, Selected Values of Chemical Therm c Properties, National Bureau 3? StaEaHs Cir 00, 9 . 97 98. General References Clarke, J. F.,Eh erg:J Transfer‘l‘hr Thro h a Dissociated Diatomic Gas in Couette echanics, Vol. I, To. 5, 55m.” Evans, J. 3., Method for Calculati Effects of Dissociation _an Flow Varimu" the Ramon Zone BehIEdLomEILhock‘ !_aves, .CA., _TTT'EBEO, Dec . 19%. Freanan, N. C. ., Non-equilibrium Flow of an Ideal Dissocia Jour. of Fluid Nechanics, Vol.75, 1m ‘T‘Wf'fligg Hansen, C. F. and S. P. Heims, A Review of _t___he Therm namic Transart, and Chemical ReactEn Rate Progert es 0 h 21n- Eratur em, _NI MI” 5339, m Helms, S. P. , Prandtl er §£__ansion _o_f_ Chemically React Gases In Local ChanicEI__ _____o_c_1_ynamic -Equilibrium, NA.C 5530: Manes, M., L. J. E. Hofer, and S. Weller, Classical oThermod%_l.cs and Reaction Rates Close to uilibrium, Jour. hemi misfits—WEI, .‘i'BTNo. lo,‘5ot . 9 o. Prigogine, I. and R. Defay, Translation by D. H. Everett, Chemical Thermod s, (New York: Longmans, Green and 00.), l . Prigogine, I. , P. Outer, and Cl. Herbo, Affinit and Reaction Rate Close 19 filuilibrium, Jour. of s. o 1313' W, I953. Ioodley, H. N., Effect 9__f Dissociation on Them c Pro rties _o_f Pur____e_ nami”, 18—3570, § 13% %9 $300.3 USE ONLY m \IIIIIWIIHHIIWH m5 mmmm4 m7 mmmflb mmmfi WWWS m0 m3 Hflflmu 1293