i DEPARTMENT OF ENGRING RESEARC UNIVERSITY OF MIHGAN UNIVERSITY OF MICHIGAN Ann Arbor EXTERNAL MEMORANDUM NO.9 Project MX-794 (AAF Contract W33-038 ac-14222) Project "Wizard" The Calculation of Enthalpy-Entropy Diagrams for and the Specific Impulse of Rocket Fuel Systems Prepared by Approved by R.' H. Bol A LY A. S. Foust Associate Professor of Chemical Engineering August 20, 1947

Lithoprinted from copy supplied by author by Edwards Brothers, Inc. Ann Arbor, Michigan, U.S.A. 1947

DEPARTMENT OF ENGIEERING RESEARCH Report No. UMM 9 UNIVERITY OF MICHIGAN Page i CONTENTS INTRODUCTION AND SUMMARY.................. LIST OF SYMBOLS.................. PART I. CALCULATION OF ENTHALPY-ENTROPY DIAGRAMS FOR PRODUCTS OF COMBUSTION. Generalized Equilibrium Calculation........... Calculation of Enthalpy and Entropy of Gaseous Mixtures.. Preliminary Cross Plots for Diagrams............ Materials....................... * Enthalpy of Unburned Propellant............. Theory and Use of Enthalpy-Entropy Diagrams....... * 0 * 0 1 3.. 5. 12. 28.. 28 *. 29. 46 PART II. CALCULATION OF THEORETICAL SPECIFIC IMPULSE Thermodynamic Approach........... Methods for Calculating Theoretical Specific Impulse, Discussion.................... EEREENCES.................... APPENDICES I. Derivation of Generalized Equilibrium Calculation II. Derivation of Entropy Calculation Equation... III. Enthalpy-Entropy Diagrams from German Data... * *0 * 0 * * * * * 0 ** * * * * 0 Scheme.* * *0 * 0 * 0 0 0 * 56 61 67 69 70 75 77

L.I.DEPARTMENT OF ENGNEERING RESEARCH Report No. UMM 91 UNIVERSITY OF MICHIGAN Pae ii LIST OF FIGURES Figure Number 1-8 Title Entropy Change on Mixing - F(X) = 4.5753Xlog X.... Page 18 - 25 9 Cross Plot for.80 lb RFNA -.20 lb Aniline, H-T... * * 10 Cross Plot for.80 lb RFNA -.20 11 Cross Plot for.80 lb RFNA -.20 12 Cross Plot for.80 lb RFNA -.20 13 Enthalpy-Entropy Diagram,.70 lb 14 Enthalpy-Entropy Diagram,.75 lb 15 Enthalpy-Entropy Diagram,.80 lb 16 Volume Constant, PV - KT,.80 lb 17 Enthalpy-Entropy Diagram,.85 lb lb Aniline, H-P..... lb Aniline, S-T..... lb Aniline, S-P.... RFNA -.30 lb Aniline.. 34 35 36 37 38 RFNA RFNA RFNA RFNA -.25 -.20 -.20 -.15 lb Aniline.. lb Aniline.. lb Aniline.. lb Aniline.. 39 41 43 45 49 50 18 Theoretical Specific Impulse from H-S Diagram. 19 Flow with Friction and Heat Transfer..... 20 Variation of Theoretical Specific Impulse with Ratio, RFNA - Aniline............ 21 Variation of Theoretical Specific Impulse with Chamber Pressure, RFNA - Aniline.... 22 Variation of Theoretical Specific Impulse with Chamber Pressure, Oil - Oxygen..... 23 Variation of Theoretical Specific Impulse with Chamber Pressure, Hexane - Oxygen.... 24 Variation of Theoretical Specific Impulse with Chamber Pressure, Ethanol - Oxygen... 25 Variation of Theoretical Specific Impulse with Chamber Pressure and Method of Calculation.. 26 Variation of Theoretical Specific Impulse with ~ ~ ~ * 0 * Mixture * ~ 0 * * Combustion * * 0 * * Combustion ~* ~ * ~ Combustion * * * * * * Combustion * * 0 * * * Combustion Mixture 51 52 53 54 55 65 66 78 Ratio and Method of Calculation 0 0 0 0 0 0 0 * 0 6 * * 27 Enthalpy-Entropy Diagram for the Products of Combustion of 0.69 KG Oxygen and 0.31 KG Hexane.......... 28 Enthalpy-Entropy Diagram for the Products of Combustion of 0.559 KG Oxygen and 0.441 KG 76% Ethanol........ 79

DEPARTMENT OF ENGINEERING RESEARCH Report No UWM 9 UNIVERSTY OF MICHIGAN Page iii LIST OF TABLES Ni amber Title Pa 1 Parameters for Equilibrium Calculations........... 9 2 LoglO Kf.............. 0... 3 Enthalpy and Entropy of Gases................ 14 4 Molal Specific Heats, Cp~, of Gaseous Molecules....... 16 5 Form for H-S Calculation................... 26 6 Composition Summary -0.80 lb 6.5% RFNA with 0.20 lb Pure Aniline........................... 30 7 Enthalpy-Entropy Summary - 0.80 lb 6.5% RFNA with 0.20 lb Pure Aniline........................ 33 8 Theoretical Performance of the RFNA - Aniline System by Three Methods of Calculation for Four Conditions of Chamber Pressure and Mixture Ratio................. 64

' - "1 ' ~ 'D E P A RT te A Tllc h fi l _ R t att' t^ R EAESR C H Report No. TUJvI 9 UDART RSITY OF MICHIGAN Page 1 INTRODUCTION AI$D SUMIARY It is the purpose of this report to present the methods and fundamental data necessary for calculating the theoretical performance of rocket motors in such a way as to be immediately useful to the research engineer. The ultimate aim of such calculations is to provide an enthalpyentropy diagram for the products of combustion of a fuel-woxidlnt system. These diagrams facilitate the investigation of the effect of operating variables of combustion chamber pressure, fuel temperature, expansion ratio, etc., upon the performance of rocket engines. The calculations need not, however, be carried to completion to be useful. If, for instance, the adiabatic flame temperature and average molecular weight of a propellant system are desired, they may be obtained i'n what is believed to be a minimum of time through the data and methods presented here. The methods may be appropriately applied to any combustion process involving carbon, hydrogen, oxygen, and nitrogen, vrith particular reference to the many combinations commonly found in present-day rocket motors. If, however, the calculations are carried to completion, they will yield enthalpy-entropy diagrams fundamentally similar to those of Hottel (Reference 2),which are based on the assumption of thermochemical equilibrium throughout the combustion cycle. Since this assumption is not certain, the specific impulse (pounds of thrust per pound of propellant burned per second) indicated by such diagrams is compared with that calculated by other assumptions and nmthods in Part II of this report. The comparison shows surprisingly little difference in theoretical specific impulse, some difference in predicted optimum rixture ratio, and a considerable difference in the theoretical exhaust temperature. The preparation of an enthalpy-entropy diagrire starts logically I I mmvwmwww~~~~~~~~~~~~~~~~~~~~ - -, - -~ ~ ~ ~ ~ ~ ~ ~~~~~~~~~~~~

........... lie-I.. DEPARkIm91tR ) -LNAZEKiNG RESEARCH! Page 2 UNIVERSITYOF MICHIGAN tReport No. U1 9 with the calculation of the chemical composition of the reaction products at appropriate temperatures and pressures. The calculated compositions allow calculations of the enthalpy and entropy of the equilibrium mixture at corresponding temperatures and pressures. These calculated data may then be expanded through cross-plotting and graphical interpolation to give the complete enthalpy-entropy diagram. A detailed explanation of these steps follows in Part I. I I

L - I. _d it I-. An DEPARTMENT ( IUOUUER nZ55&CRCH Report No. UTJlI 9 UNiVERIrI Of MICHIGAN Page 3 LIST OF SYMBOLS 0I ~, ~, (, gram atoms of carbon, oxygen, hydrogen, and nitrogen, respectively, per pound of propellant. C Specific heat at constant pressure, cal per gm mol OK F Function G Weight rate of flow, lb per sec H iEnthalpy, energy per lb I Specific impulse, sec J Conversion factor, 778 ft lb per Btu K Gas constant, psia ft cubed per OR or atmos ft cubed per OR K1 'Equilibrium constant based on partial pressure for reaction (1).M Average molecular weight N Total moles in system, gram moles per unit mass P Total pressure, atmos or psia, as noted R Universal gas constant S Entropy, energy per degree per unit mass T Temperature, OR or OK as noted V Volume, ou ft X Gram moles of component in system f Area (g) Gaseous state g Acceleration due to gravity, ft per sec per sec h Molal enthalpy p Partial pressure q Heat transferred to the system from the surroundings, per pound of fluid floving s Molal entropy

t DPDEPARTMMNT F I NENGINEIG RESEARCH Page 4 UNWt oPTctttf-'AMA Report No. UIM 9 u Axial component of velocity, ft per sec v Velocity ws Work done by system on surrounding other than PV work Wf Useful work converted into heat ^ ~Increase, final condition minus initial condition e ~Ratio of specific heats, cp/oC < ~Ratio of oxidant to total of oxidant plus fuel E SSummation Superscripts o Standard pressure state, one atmosphere Subscripo a i m n c e 1 2 f bs Surrounding atmosphere i th component Throat condition n th component Combustion chamber condition Exhaust condition Initial condition Final condition Friction.!, LogaritIms to the base e are denoted by In; logarithms to the base 10 are denoted by log.

DEPARTMET M X"InI t BUSEARCH Report No. UMJ V 9 III'UN IGN____ ag e 5 PART I. - THE CALCULATION OF ENTHALPY-ENTROPY DIAGRAMS FOR PRODUCTS OF COMBUSTION Generalized Equilibrium Calculation. The euilibrium composition of a gaseous mixture is governed by the simultaneous equations of conservation of atomic species and of mass action. With complex systems such as are encountered in the combustion chambers of rocket motors, these equations present a baffling problem in their solution% One of the objects of this portion of the work has been the development of a straightforward series consisting of the minimum number of steps of calculation, which could be executed by one not versed in physical chemistry, homogeneous equilibrium, and the other aspects of thermochemistry involved. The procedure resulting from this work is presented herein. This method has been developed to permit an accurate evaluation of the equilibrium composition at a predetermined temperature and pressure. The procedure has been generalized to apply to any system comrposed entirely of atoms of carbon, oxygen, hydrogen, and nitrogen. For a complete derivation of these equations, the reader is referred to Appendix I. It is convenient, although not necessary, to select 453.6 grams (one pound) of propellant mixture as the system for study. This is expedient because the design engineer prefers to work in English units and because composition calculations on this basis are made with easily written numbers (10 to 0.0001). This results in compositions which are, in fact, gram moles per pound, which, when used for summation of the enthalpy and entropy of the mixture, yield those energy functions per pound. Conversion from gram calories to BTU is the only conversion involved in starting with metric units, which are widely available, and ending with the desired English units for the enthalpy-entropy diagram. This rather unusual unit, I

RESTRICTED DEPARTMENT OF ENGINEERING RESEARCH | Page 6 UNIVERSITY OF MICHIGAN Report No. U13i 9 the gram molecule or energy quantity per 453.6 grams, is not necessary to the method but will give much more easily handled figures leading to the desired result with the minimum of arithmetic manipulation. It should be noted that the system referred to includes both fuel and oxidant. Thus a 3:1 ratio of oxidant to fuel means 0.75 pounds (340.2 grams) of oxidant plus 0.25 pounds (113.4 grams) of fuel. The ratio having been chosen, the atomic composition of the system is calculated in gram atoms according to the principles of elementary stoichiometry. Let these quantities be; ~ - gram atoms carbon in the system, gram atoms per pound (0 - gram atoms oxygen in the system, gram atoms per pound ) - gram atoms hydrogen in the system, gram atoms per pound - gram atoms nitrogen in the system, gram atoms per pound. With the above quantities at hand, it remains only to choose the temperature and pressure at which the equilibrium composition is desired, in order to fix the composition (temperature, pressure, and atomic concentration being determined) of the system,-that is, there can be only one equilibrium composition of the system. This must be calculated by trial and error according to the simultaneous equations of mass action and of conservation of atomic species. These equations have been reduced through algebraic manipulation to a systematic scheme. The concentration values depend only upon the atomic concentrations and certained constants determined by the temperature and pressure. As such, the solution becomes a problem in algebra, requiring no knowledge of chemistry outside of elementary stoichiometry. Nonetheless, some explan ation of the meaning of the symbols will be helpful in arriving at a speedy solution. The molecular symbols written in brackets, [H20], [I2], etc., -i I.

RESTRICTED..t........9 DEPARTMENT OF ENGINEERING RESEARCH Report MNo. m 9 EUNI RSrrY OF MICHIGAN Page 7 represent the concentration in gram formula weight per unit mass of the various species. When the algebraic trial and error scherae has been balanced, the obtained values of these symbols will be their respective concentrations. [Hi and N must be assumed at the outset of the calculation. [H2] obviously cannot exceed J — nor can it reach zero. ( [I21] may become negligible but for the purposes of this scheme it cannot be zero). N represents the total number of moles in the system at equilibrium and as a first approximation may be taken as [ ~ + + JP)] for mixtures lean in fuel and [i~+ ~1 + ~]o for rich mixtures. Within these ranges, accuracy of the first assumption will be improved only with considerable experience. The second assumption of [H2] should be adjusted up or down from the first assumption as the computed value is greater or less than the assumption. Plotting facilitates convergence. Occasionally, for very lean mixtures, extreme accuracy of assumption is necessary for convergence. The necessity of maldking the double assumption of N as well as [H2i] does not introduce any added difficulties since N is large, reasonably constant, and relatively independent of [H2I]. N will normally be determined with final aqouracy on the second or third trial, whereas [1H23 will usually not be determined until the fourth or fifth trial. This requires about three hours time, on the average. With these facts in mind, it is now fitting to proceed to the schematized equations used for composition calculations. The values 2-/jP, 2 /K g, K8 /2/p 5 I* l5,and K10 are parameters calculated from the equilibrium constants for appropriate reactions, with subscripts following Lewis and Von Elbe's tabulation of the values at different temperatures. (See Table 2). Their values may be taken from Table 1, "Parameters for Equilibrium Calculations".

RESTRICTED DEPARTMENT OF ENGINEERING RESEARCH Page 8 UNIVERSITY OF MICHIGAN Report No. TIE 9 The scheme is as followst Assume: [H2] and N. Calculate: 2[H] = 2KE1 /P -N [H2] [OH] + 2EH201 () - 2 [H2] - [HI [OHI [oH] + z2 zo[ [OH1 =. 1 + (2f'P/ K9) [H2]/ [E201 = ([OH] + 2[H201 - [OH]) [O021 (K82/P) N ([Ho20] / [2])2 CO1 f K2 / P N I ]CO2 [NO] - K15 2 J2j ~ - iNO] [CO2] -- @ - ~ - - 2[O] - [OH] 2 - 0 J NO Cco] = ~ - [c02] Cheok: [co] [o]20 [1][2 1 K[10 Cheek: N + = ~ + ~ * ~ + [[H] + [OH] + [No] + o] + [02] Revise assumptions and repeat if necessary. When the above scheme has been balanced, the composition values from the last trial constitute the desired composition of the system.

Table 1: Parameters for Equilibrium Calculations Values for P= 1 atmosphere (Based on Log Kp values reported by Lewis and Von Elbe, see Reference 11) Temp. ~K 2-VK/P -2 Kg|/K9 K82/P | IK2 /P K15 K10 3200.59640 10.139.0085114.22648 15597.13883 3000.33656 19.953.0023988.11777.12389.14454 2800.17521 43.752 5.4954 x 104.056562 095726.15170 2600.08252 109.90 1 x 10-4.024324.07080.16069 2400.03440 309.74 1.5136 x 10-5.0091312.050122.17298 2200.01228 1099.0 1.5136 x 10-6 1.002848.033192.19187 2000 3.5732 x 10- 5023.4 9.1201 x 10-8 7.2 x x 10-4.020186.21979 1800 7.9352 x 10-4 30,974 28.84 x l10 5.2004 x 104.010991.26303 1600 1.2206 x 10-4 315,960 3.9811 x 10-11 1,5669 x 10-5.0051884.32734 1400 1.1119 x 10-5 5,902,050 2.0893 x 10-13 1.0352 x 10-6 1.968 x 103.45604 0 _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ (D 0 a. 0 CII I eis five are included Note: Three figures are significant; to smooth calculation. Cd CD to

I RESTRICTED DEPARTMENT OF ENGIN ERNG RESEARCH Page 10' UNIVERSTY OF MCImGAN Report No. UM 9 I Table lb Correction Factors for Pressures other than 1 Atmos. Total Pressure _-f- fP, Atm. P 40.15811 6.3246.0250 30.18258 5.4772.0333 20.22361 4.4721.0500 10.31623 3.1623.1000 5.44721 2.2361.2000 2.5.63247 1.5811.4000 1 1.00000 1.0000 1.0000 0.5 1.4142.70711 2.0000 0.25 2.0000.50000 4*0000 0.10 3.1623.31623 10.0000 _~~ 1 _ J,~,- _ _, i, I 1

RESLRICTED DEIPARTMENT OF ENGINEERING RESEARCH i Report No. U1M 9, UNIVERSITY OF MICHIGAN Page 11 Table 2: Log10 Kf (Reference 11) aA + bB = cC + dD c d fc fD f f a f b () H2 = 2H ~ H20 0 OH + H2 Note: _f a P and ~02 - 20 ~ CO + Ho = C002 + H2 Kf a Kp, if ___ H2 2 + ix-T NO perfect gases ) H20 = H2 + 02 ( 102 + 2 ' NO are assumed. Temp _ __ _________)_) () _."..".:p K 300 400 600 800 1000 1200 1400 1600 1800 2000 2200 2400 2600 2800 3000 3200 3500 4000 -70.23 -51.35 -32.41 -22.88 -17.13 -13.28 -10.51 -8 429 -6.803 -5.496 -4.424 -3.529 -2.769 -2.115 -1.548 -1.051 -0.409 +0.449 -80.2 -58.6 -36.9 -26.1 -19.48 -15.10 -11.97 -9.61 -7.772 -6.298 -5.091 -4.078 -3.228 -2.495 -1.858 -1.290 -0.577 +0.379 -10.05 -7.90 -6.34 -5.20 -4.27 -3.52 -2.91 -2.41 -2.00 -1.63 -1.31 -10.53 -8.17 -6.47 -5.20 -4.19 -3.40 -2.74 -2.19 -1.74 -1.34 -0.99 +4.947 +3. 167 +1.433 +0.610 +0.147 -0.145 -0.341 -0.485 -0.580 -0.658 -0.717 -0.762 -0.794 -0.819 -0.840 -4.052 -3.267 -2.706 -2.285 -1.959 -1.695 -1.479 -1.300 -1.150 -1.019 -0.907 -0.807 -0.680 -0.513

[ RESTRICTED ~Page 12 ~DEPARTMENT OF ENGINEERING RESEARCH.? ~Page~ 12 U l_ __ UNIVERSITY OF MICHIGAN Report o. 9 I Calulation of Enthalpy and Entropy of Gaseous Mixtures Assumptions: (1) No enthalpy change on mixing. (2) Zero Joule-Thomson effect. (3) PV = NRT (equation of state) Reference States: Processes involving changes in chemical composition as well as changes in temperature are best referred to chemical individuals in definite states of temperature and purity. These are arbitrary, but the chemicals must be such that balanced chemical equations may be written which will form any and all possible constituents of the mixture in prospect. Accordingly, a reference state has been chosen as the pure elements; C (graphite), H2 (g), N2 (g), and 02 (g), at 2980K and one atmosphere. Thus the enthalpy of any constituent is equal to its heat of formation as a gas at 298~K plus its sensible heat above 298~K. Enthalpy: In accordance with the above assumptions, the enthalpy of a gaseous mixture is given by: i- n H:= i hi i;1 I I - 16 where H xr Values of Entropy: - enthalpy of mixture, calories per unit mass, = moles of component i in mixture, gram moles per unit mass, = molal enthalpy of component i at temperature of mixture, calories per gram mole. hi may be found in Table 3, Enthalpy and Entropy of Gases. The entropy of a gaseous mixture may be expressed* by: i =n i n p S =j E isi - X. (4.5753) log X - N(4.5753) log i 1 i 1 - * For derivation, see Appendix 2.!........ i, 1 1 1, ' ' 11 ~ ' 11 ~ ~ ~ ~ ~~~~~

L RESTRICTED DEPARTMENT OF ENGINEERING RESEARCH Report No. U.I 9 UNIVERSITY OF MICHIGAN Pae 13 where S = entropy of mixture at temperature and pressure, calories per OK unit mass Xi = moles of component i in mixture, gram moles per unit mass Si= molal entropy of component i at one atmosphere partial pressure, calories per OK mole P = total pressure, atmospheres N = total moles in mixture, gram moles per init mass i - n -= > Xi, gram moles per unit mass. l 1 Values of Si~ may be found in Table 3, Enthalpy and Entropy of Gases. Values of the function F(X) = 4.5753 X log X may be read from Figures 1 - 8. For ease of computation, forms may be made up similar to Table 5. (Values of H and S are taken from Table 3). For the sake of completeness, Cp values calculated by the original investigators are included in Table 4. These values are the basis for the preparation of enthalpy and entropy tables. I i I

-- I — Table 3: Enthalpy and Entropy of Gases Reference States - Pure 02, H, N2, and C (graphite) at 298~K and 1 atm. pressure. Units: - H~ cal. per gm. mole; S~ cal. per gm. mole per OK. Temp H2 02 N2 H20 CO2 O~KH H S~ H S~ H~ S~ f Ho So Ho~ S I......... _....... _..,...I 1 I 298 400 600 1000 1200 1400 1600 1800 2000 2200 2400 2600 2800 3000 3200 3500 0 + 709 + 2,106 + 4,943 + 6,407 + 7,911 + 9,450 +11,033 +12,652 +14,304 +15,981 +17,690 +19,423 +21,170 +22,930 +25.583 0. 2.040 4.874 8.494 9.826 11.003 12.005 12.945 13.799 14.580 15.320 15.997 16.630 17.244 17.810 18.614 0 + 725 + 2,212 + 5,430 + 7,116 + 8,837 +10,587 +12,361 +14,154 +15,971 +17,811 +19,669 +21,545 +23,437 +25,352 +28.253 0 2*103 5.099 9.184 10.715 12.040 13.208 14.258 15.204 16.063 16.867 17.602 18.293 18.961 19.57 20*42 0 + 711 + 2,127 + 5,132 + 6,724 + 8,357 +10,020 +11,713 +13,431 +15,158 +16,898 +18,646 +20,407 +22,177 +23,956 +26,627 0 2,052 4.920 8.746 10.345 11.465 12.553 13.565 14.468 15.285 16.045 16.740 17.388 18.011 18.575 19.382 -57,796 -57,102 -55,288 -51,603 -49,582 -47,465 -45,248 -42,935 -40,543 -38,095 -35,598 -33,055 -30,478 -27,876 i I I I I -10.641 - 8.270 - 4.878 - 0.200 + 1.638 + 3.280 + 4.779 + 6.12 + 7.39 + 8.56 + 9.64 +10.66 +11.61 +12.51 I I -94,031 -93,072 -90,943 -86,036 -83,382 -80,644 -77,845 -74,995 -72,109 -69,195 -66,253 -63,284 -60,300 -57,297 i I I I I i i I i II i I 4 1 1 i II I II II I I + 0.696 + 3.452 + 7.751 +13.980 +16.397 +18.507 +20.389 +22.036 +23.57. +24.95 i. +26.24 { +27.45 +28.61 +29.71 +30.79 +31.93 I i I F" biW o X O Mi Eg xti v 1 r is o II II I a I-........ - -.. - -. - -.- II I s I I _. l ~ ~~~~I _, _ I3 II I

I______ _ A I I i - Temp CO 01 NO OH HI OK HO S~ Ho So H~ Ho So H~ So 298 400 600 1000 1200 1400 1600 1800 2000 2200 2400 2600 2800 3000 3200 3500 -26,380 -25,668 -24,243 -21,195 -19,582 -17,929 -16,248 -14,541 -12,813 -11,069 - 9,318 - 7,560 - 5,792 - 4,015 - 2,233 + 450 +21.435 +23.491 +26.379 +30.258 +31.655 +33.020 +34.115 +35,130 +36.040 +36.861 +37.627 +38.327 +38.977 +39.605 +40.163 +40.981.59,110 +59,617 +60,611 +62,599 +63,593 +64,587 +65,581 +66,575 +67,569 +68,563 +69,557 +70,551 +71,545 +72,539 +73,533 +75,024 +13 965 +15.429 +17.443 +19.982 +20.889 +21.654 +22.318 +22,904 +23.426 +23.901 +24.333 +24.732 +25.099 +25.442 +25.763 +26.209 +21,530 +22,259 +23,720 +26,853 +28,500 +30,188 +31,900 +33,633 +35,381 +37,142 +38,915 +40,696 +42,484 +44,278 +46,078 +48,787 + 2.944 + 5.043 + 8.002 +11.986 +13.484 +14.784 +15.921 +16.951 +17.869 +18.698 +19.476 +20.184 +20.837 +21.471 +22.034 +22.834 - 5,929 - 5,205 - 3,794 929 + 551 + 2,088 + 3,670 + 5,283 + 6,925 + 8,593 +10,300 +12,023 +13,765 +15,534 +17,325 +20,026 + 3.776 + 5.866 + 8.725 +12.378 +13.728 +14.914 +15.961 +16.916 +17.786 +18.577 +19.322 +20.007 +20.649 +21.267 +21.826 +22.616 +51,911 +52,418 +53,412 +55,400 +56,394 +57,388 +58,382 +59,376 +60,370 +61,364 +62,358 +63,352 +64,346 +65,340 +66,334 +67,825 +11.786 +13.250 +15.264 +17.803 +18.710.19.475 +20.139 +20.725 +21.247 +21.722 +22.154 +22.553 +22.920 +23.263 +23.584 +24.030 SI' 0 co o 9 0 51 s| l| g iS 0100 a0 C9 Original data are from reference oited by Lewis and Von Elbe (Reference 11), graphically interpolated where necessary. Slight corrections for temperature base were made by using the Cp equations of K. K. Kelley, (Reference 9). AH of formation was taken from Perry (Reference 12), AS of formation from original works.

v RESTRICTED DEPARTMENT OF ENGINEERING RESEARCH Page 16 UNIVERSITY OF MICHIGAN Report No. UMM9 Table 4: Molal Specific Heats, Cp, of Gaseous Molecules Temp (4) (7) (4) (6) (3) (5) (8) (1) oK CO 02 N2 OH NO H2 C2 H20 I L 1 10 20 30 40 50 100 150 200 250 298.1 300 350 400 450 500 550 600 650 700 750 800 850 900 i 6.950 5.120 6.945 6.973 7.115 7.363 7.590 7.733 7.468 7.292 7.129 6.954 6.954 6.962 6.962 6.955 6.955 6.955 6.961 6.956 7.195 6.772 I 7.018 7.139 7.144 8.000 6.964 6.960 6.896 8.908 8.002 7.098 7.140 6.950 7.066 7.013 7.197 6.991 7.072 7.168 6.974 9.885 8.155 8,260 7.122 7.434 7.071 7.047 7.294 6.992 10.676 8.379 8.504 7.279 7.675 7.200 7.053 7.476 7.008 11.324 8.635 8.771 7.455 7.890 7.355 7.087 7.663 7.035 11.862 8.910 9.053 7.629 8.069 7.516 7.150 7.840 7.079 12.312 9.199 9.347 7.792 8.216 7.676 7.236 7.996 7.141 12.689 9.497 I I

I.. PRESTRICTED DEPARTMENT OF ENGINEERING RESEARCH Report No. TIv 9 UNIVERSITY OF MICHIGAN Page 17 Table 4: Cont. Temp (4) (7) (4) (6) (3) (5) (8) (1) OK CO 2 N2 OH NO H2 CO2 H20 I I 1000 1050 1100 1125 1150 1200 1250 1300 1500 1750 2000 2250 2400 2500 3000 3500 4000 4500 5000 7.936 8.339 7.821 7.336 8.132 7.220 13.005 9.799 9.948 13.27 10.095 7.476 8.273 8.558 7.613 8.389 8.269 8.422 8.564 8.667 8.169 8.702 8.334 8.489 8.604 8.761 8.863 8.935 8.990 9.037 9.077 7.488 13.50 13.60 13.69 7.718 14.00 7.963 14.3 8.181 14.5 7.881 8.114 8.311 8.561 8.686 8.774 10.240 10.382 10.522 10.656 11.153 11.67 12.09 12.4 12.7 13.1 8.807 8.900 8.964 9.016 9.060 9.099 8.611 8.896 8.531 8.844 8.982 8.796 8.997 9.165 9.108 9.155 9.286 9.509 9.209 9.392 14.8 15.0 15.2 I I -- -~- -- - Note: Numbers in parentheses at heads of columns indicate References from which quoted values were obtained. I ' -

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Report No. UMM 9 RESTRICTED Page 21 CL 11 - (0 U (Y) (X)J

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Report No. UMM 9 RESTRICTED Page 23 (0 0 _0 () M (X)J

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'RESTRICTED DEPARTMENT OF ENGINEERING RESEARCH Page -26 ____ UNIVERSITY OF MICHIGAN Report No. tg 9 Table 5s Form for H-S Calculation Temp 3200 ~K P Bo SO X!O X XH fx 4.57_,3_X log X H20 -25,224 13.37 H2 22,930 17.810 02 25,352 19.57 0 73,533 25.763 H 66,334 23.584 OH 17,325 21.826 CO2 -54,289 30.79 CO - 2,233 39.174 NO 46,078 22.034 N2 23,956 18.575 P /N ~ f log P /N = -4.5753 N log P/s A H _______ Btu / Ib S = _ oal/453.6 gmnK S Btu / lb OR Notes Values in the X column are composition coerrfficients from an equilibrium calculation, with units of gram moles per 453.6 grams; their summation is equal to N. The summation of the XH~ product column is the enthalpy in calories per 453.6 gm; it is converted into Btu per pound by the factor 0.0039683 Btu / lb per cal / 453.6 gm. Values in the fx column are read from Figures 1. through 8. The summation of fx is then written (with a minus sign) directly below the summation of the product XSo. The sum of the values (XSO) plus (-Zfx) plus (-4.5753 N log P/N) is the entropy in oal/453.6 gm ~K; it is converted to entropy in Btu per lb OR by dividing by 453.6. See the sample calculation on Page 27.

RESTRICTED DEPARTMENT OF NGINEEING RSEARCH Report No. UMM 9UNIVERSITY OF MICHIGAN Page 27 Sample Caloulation Form for H-S Calculation Temp 2800 ~K P 40.X80 no x B xSo fx H0 1 8~~ O L X jX13f~ JS~ 7~ 4.57'53X log X H20 -30,478 11.61 6.0113 -183,212.4 69.791 21.420 H12 19,423 16.630.2337 4,539.2 3,886 - 0.697 02 21,545 18.293.1454 3,132.6 2.660 - 0.559 0 71,545 25.099.0136 973.0.341 - 0.118 H 64,346 22.920.0268 1,724.5.614 - 0.192 OH 13,765 20.649.3595 4,948.5 7.423 - 0.727 C2 -60,300 28.61 4.6531 -280,581.9 133.125 14.420 CO - 5,792 38.977 1.1924 - 6,906.4 46.476 0.420 NO 42,484 20.837.0671 2,850.7 1.398 - 0.360 N2 20,407 17.388 5.3444 68,249.2 58.152 8.020 S_ >~: - 1 _16.0473 -384,283.0 323.866 41.627 PA 2.4926 -sx a -41.627 log P/N.39665 -4.5753 N log P/N a -29.123 H - -1,525.0 Btt / lb S * 253.116 oal/453.6 gmn~ S a.55802 Btu / lb ~R

RESTPICTED DEPARTMENT OF ENGINEERING -RESEARCH Page 28 UNIVERSITY OF MICHIGAN, Report No. &I. 9 Preliminary Cross Plots for Diagram An enthalpy-entropy diagram for the products of combustion of a given fuel system is begun by calculation of equilibrium compositions for several points of temperature and pressure. These compositions are used to calculate enthalpy and entropy points. If a complete diagram is desired, about twenty such points must be calculated. These are cross plotted; enthalpyy versus temperature with lines of constant pressure, enthalpy versus pressure writh lines of constant temperature, and so on, until the final H-S plot is completed. Less complete diagrams, which are sufficient for some purposes, may be made without cross plotting from about six compositions at carefully chosen temperatures and pressures. Table 6 illustrates the results of equilibrium calculations and Table 7 shows the enthalpy and entropy for these compositions. Figures 9 through 12 illustrate the cross plots and Figures 13 through 17 are final results. Volume constants are determined from the total moles per unit mass and the perfect gas equation (pv - nRT). Ilaterials Red Fuming Nitric Acid (RFNA) specifications were obtained from E. I. du Pont de Nemours and Co., Inc. as follows: NO3 - 98 20 - 2 % NO2 - 6.5% The above is herein referred to as 6.5 % RFNA, meaning 6.5%o NO2' or simply as RFNA. Aniline, unless otherwise noted, is pure aniline.

RESTRICTED I DEPARTMENT OF ENGINEERING RESEARCH Report No. U 9 UNIVERSIY OF MICHIGAN Page 29 Enthalpy of Unburned Propellant The enthalpy of the unburned propellant must be referred to the same datum as the products of combustion. The datun chosen for the diagrams presented herein is the pure gaseous elements at 298~K and one atmosphere pressure. This makes the enthalpy of the fuel (or oxidant) equal to its heat of formation at 2980K, plus its sensible heat above 2980K, plus its pressure-volume energy above one atmosphere. The heats of formation for aniline, nitric acid, nitrogen dioxide, and water were calculated from the heats of combustion given by Lange (Reference 17). These are 143 Btu per pound for aniline and -1200 Btu per pound for 6.5 % RFNA (taking no account of the heat of solution of NO2 and HgO) at 25~C. Specific heats were obtained from Iougan and Watson (Reference 18). From these data the enthalpy of the unburned propellant for any given fuel-oxidant ratio may be calculated as a function of temperature. This has been done for each of the mixture ratios presented and the function appears on the upper, left-hand, corner of the enthalpy-entropy diagrams. No consideration of P-V energy is included, since it is very small, but it may be added by the user if desired.

Table 6: Composicion unmmary.80.20 lb 6.5% RFNA lb pure aniline (D <D 0 Units: gram moles per one pound total ]3200~K 30000K 28000K 2600~K 2200~K 1800~K I400~K 1000~K 600~K 40 5.4033 5.7578 6.0113 30 5*3038 5.6936 5.9731 20 5.5909 5.9126 6.1197 H20 10 4.8300 5.3801 5.7869 6.0553 5 5.6278 5.9716 6.2354 1 5.6699 6.1911 6.2217 6.1301 5.9042 5.9042.5 6.1582 6.2210.25 6.1121 6.2196 40.4550.3255.2337 30.4927.3498.2482 20.3891.2713.1989 2 10.6698.4700.3201.2231 5.3816.2555.1765 1.3744.1904.2143.3080.5339.5339 *5.2020.2142.25.2199.2143 40.5088.3083.1454 30.5623.3495.1707 20.4129.2113.0753 02 10.7864.5344.2947.1177 5.3968.1748.0060 1.3784.0253.5.0445.25.0741.0002.1052 I: 40 30 20 10 5.1052.1282.2677.0420.0516.0691.1122.0136.0172.0234.0394.0649 0.0060.0106.0182 o co,.0004

0 c ont 1.5.25.0608.0018.0034.0062 PD 0 ti o ct-.0001 H 40 30 20 10 5 1.5.25.1310.1580.3253.0616.0738.0958.1504.0268.0320.0410.0634.0986.0164.0246.0374.1026.0046.0106.0156.0230. 0007.0010.0015 40 1.0286.6480.3595 30 1.1254.7156.4016 20.8203.4674.2227 OH 10 1.5514 1.0203.5988.2948 5.7588.3846.0478 1.6850.1026.0034.5.1402.0048.25.1892.0069 40 3.6393 4.2047 4.6531 30 3.5020 4.1025 4.5894 20 3.9447 4.4887 4.8610 C02 10 2.9241 3.6444 4.2854 4.7549 5 4.0403 4.6171 5.0952 1 4.1429 5.0398 5.1685 5.2634 5.4913 5.4913.5 4.9919 5.1693.25 4.9231 5.1684 S I i0 t..: I CO 40 30 20 10 5 1.5.25 2.2062 1.6408 2.3435 1.7430 1.9008 2.9214 2.2011 1.1924 1.2556 1.3568 1.5601 1.8052.9845 1.0906 1.2284 1.7026 (D H-.7502.8057.8536.9224.6770.6762.6771.5801. 3542.3542

Table 6, continued P 0 atm 3200~K 3000~K 2800~K 2600~K 2200~K 1800~K 1400~K 1000~K 600~K 40.2014.1264.0671 30.2116.1331.0728 20.1447.0808.0357 NO 10.2495.1645.0956.0446 5.1100.0544.0046 1.0800.0096.0002,5.0128.0002.25.0166.0002 40 3.2772 3.3147 3.3444 30 3.2721 3.3114 3.3415 20 3.3055 3.3375 3.3600 N2 10 3.2531 3.2957 3.3301 3.3556 5 3.3229 3.3507 353756 1 3.3379 3.3731 33,777 3.3779 3.3779 3.3779.5 3.3715 3,3778.25 3.3696 3.3778............................... *..... t 0 I is 0 W Pt (1tg IwI ) c) 0 m 3 X 10 N l 40 30 20 10 5 1.5.25 i 16.9560 16.4298 17.0995 16.5239 16.6738 17.7787 16.9783 16.0473 16.1026 16.1908 16.3745 16.6070 15.8801 15.9718 16.0927 16.5345 15.6963 15.7500 15.7937 15.8562 15.6635 15.6646 15.6661 15.6615 15.6615 15.6615 <D 0 c. O o i

RESTRICTED DEPARTMENT OF ENGINEERIG RESEARCH Report No. UMI 9 UN RSY OF MICHGAN Page 33 Table 7: Enthalpy-Entropy Summary.80 lb 6.5% RFNA P - atmospheres pressure N - gram moles per lb.20 lb pure aniline H- Btu per lb K - psia cu ft per OR S - Btu per lb per OR Temp OK P = 40 P 30 P= 20 P _ 10 P a 5 P - 1 P.5 P.25 H -767.6 -698.6 -374.0 3200 S.7005.7340.8764 N 16.9560 17.0995 17.7786 K.40116.40456.42062 H -1181.8 -1135.6 -1063.0 -916.2 3000 S.6275.6575.7010.7815 N 16.4298 16.5239 16.6739 16*9783 K.38871.39094.39449.40169 H -1525.0 -1497.9 -1454.9 -1365.5 -1252.9 2800 S.5580.5840.6224.6901.7604 N 16.0473 16.1026 16.1908 16.3745 16.6070 K.37966.38097.38306.38740.39291 H -1762.4 -1717.6 -1.658.6 -1444.3 2600 S.5559.6147.6769.8396 -N 15.8801 15.9717 16.0927 16.5345 K.37571.37787.38074.39119 H -2157.7 -2131.9 -2110.7 -2080.4 2200 S.5588.8765.7301.7865 N 15.6963 15.7500 15.7937 15.8562 K.37136.37263,37366.37515 H -2470.6 -2470.5 -2469.9 1800 S.5812.6288.6763 N 15.6645 15.6646 15.6661 K.37061,37061.37064 H -2759.3 -2759.3 -2759.3 1400 S.4809.5285.5760 N 15.6615 15.6615 15.6615 K.37054.37054.37054 H -3034.3 -3034.3 -3034.3 1000 S.3500.5976.4452 N 15.6515 15.6615 15.6615 K.57054.37054.37054 H -3290.5 -3290.5 -3290.5 600 S.2920.3396.3872 N 15.5615 15.6615 15.5615 K.37054.37054.57054 L -

Page 34 'DV frp T~ Report No,* UlM 9. Lti I I I L................. - - - - - -............... Is......... I I I................ 1A miimmmi I A I 1. Wr AIA I it t I I Li - - - - - - - - - - - - - - - - - -.......... I I A.......... If ImI I L I I I I................ I I I I I I I. I - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -............................................................................ Li Li III if I E l If 11 I I....... M M Al Is..........................L t w.............Fi.g..9 0

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Report No. UMI 9 RE.STRICTED Page 45 Fig.17

L RE;STRICTED DEPARTMENT OF ENGINEERING RESEARCH H Page 46 UNVERSITY OF MICHIGAN Report No. U)m9 Theory and Use of Enthalpy-Entropy Diagrams Basic Theory - For any flow process, the laws of thermodynamics give: AH + 2+ A =q- ws (1) 2gJ J where A H = increase in enthalpy, per lb fluid flowing v2 IA - = increase in directed kinetic energy per lb fluid flowing 2gJ Z -J increase in position energy, per lb fluid flowing q = heat added to system by surrounding, per lb fluid flowing Ws = work done by system on surroundings other than opv, per lb fluid flowing In the case of an adiabatic rocket motor, this reduces to AH + A t =- 0 (2) 2gJ moreover, for the combustion process itself, AH = (3) Since the combustion process and possibly also the expansion process entails a change in chemical composition, it is necessary that enthalpy be referred to definite chemical compounds at an arbitrary but definite temperature. The reference state used in the above diagram is such that the enthalpy of any constituent is its heat of formation at 2980 K plus its sensible heat above 298~ K. For the expansion process, Equation 2 may be rearranged and solved letting vc 0 to give (since A v2 2- vc) 2gJ 2gJ - 6.95 -AH = I g

TRESSTRIC TED DEPARTMENT OF ENGINEERING RESEARCH Report No, UMM9 UNIVERSITY OF MICHIGAN Page 47 where Ve = exhaust velocity, ft per sec g = acceleration due to gravity, 32.2 ft per sec2 z H = increase in enthalpy, BTU per lb I = specific impulse, sec (Note: See Part II for discussion of specific impulse) Utilization - Figure 18 illustrates the use of the H-S diagrams in calculating theoretical specific impulse. Flow with friction and heat transfer may be approximated if these are known as follows (See Figure 19). By definition: fTdS - q + wf TavAS (5) Rearranging Equation 5, we obtain q12 + wf12 n t~~~~~~~~~~sl2-,~~ r(6) S12 T12 (6v av where wf = (1 - energy efficiency) AH isentropic. Thus if efficiency and heat transfer are known over a pressure interval, the expansion path may be plotted. The velocity path may be calculated by rearranging Equation 1: v22 - vi2 - 2g (q2 - ANH12) (7) The following theoretical specific impulse curves have been obtained using isentropic expansion to one atmosphere with fuel temperature of 5200 R (Figures 20-24). L I

Report No. ULM 9 RESTRICTED Page 49 THEORETICAL SPECIFIC IMPULSE FROM H-S DIAGRAM _ rol FUEL TEMPERATURE Enter herel I I -j w aD -J Iz I I 7.1 C I,/ -I I 4a riust Condition Specific Impulse 6.95 JHFi ENTROPY, BTU PER LB 'R Fig. 18

Page 50 RESTRICTED Report No. US9 9 FLOW WITH FRICTION AND HEAT TRANSFER w m O. D.4 I z LU,ondition 3 I / I I I Is b c AS - 62 9 (ql,-, L It2 5 6 9-.ILt- - 15-56.2 g (qua-WHO ) M ENTROPY, BT Fig. 19 'U PER LB 'R

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I RESTRICTED DEPARTMENT OF ENGINEERING RESEARCH Page 56 UNIVERSITY OF MICHIGAN Report No. UM9 PART II CALCULATION OF THEORETICAL SPECIFIC IMPULSE Thermodynamic Approach In general, specific impulse is given by: I = U -f (Pe - Pa) (see reference 10) g G where I = specific impulse, sec ue = axial component of exit velocity g = units conversion factor, 32.2 ft per sec2 fe= exit area, ft2 G weight rate of flow, lbs per sec P = exhaust pressure, lbs per ft2 Pa = surrounding atmospheric pressure, lbs per ft2 For the present analysis, however, it will be assumed that the nozzle is so designed that Pe = Pa making specific impulse equal to: I =- ' g Furthermore, the difference between ue and ve will be neglected since this is a function of the particular nozzle used, not of the fuel system. Thus, for the purposes of fuel system analysis, specific impulse will be taken as: g The prediction of rocket motor thrust from theoretical considerations requires some assumptions in regard to the state of chemical reactions and physical conditions through the motor, and this situation will obtain until considerably more is known of reaction mechanisms and rates for the complex products of reaction. It is universal practice to assume that equilibrium is reached in the combustion chamber, and to build the combustion chamber large enough to accomplish this for the fuel system contemplated. This i I

1 RESTRICTED DEPARTMENT OF ENGINEERING RESEARCH Report No. UW9 UNIVERSITY OF MICHIGAN Page 57 assumption permits the evaluation of composition and enthalpy content of materials entering the nozzle. The rate and extent of conversion of this enthalpy to velocity energy depends in part on the kinetics of chemical reactions for which a shift in equilibrium composition may occur during expansion. The most optimistic assumption is that equilibrium is maintained through the nozzle, with the resultant release of a considerable amount of dissociation energy to velocity energy. If this is true, the general thermodynamic energy balance is the basis for calculating theoretical specific impulse; AH+ Av2 =q-ws 2g and in an adiabatic and frictionless nozzle, this becomes ZAv2 = _ 2g AH = Ve2 - v2 - e2 (essentially) and I =V AH g g where units are consistent. The assumption of thermochemical equilibrium of the gases entering the nozzle is not subject to too serious criticism. After the expansion process, however, the state of the system is not so easily assumed, since the gases are at a very high temperature and have undergone a very rapid decrease in temperature and pressure. Knowledge of the degree of attainment of equilibrium will be necessary for accurate evaluation of aH. Inspection of the meager kinetics data available indicates that the water-gas reaction, for instance, will not maintain equilibrium concentrations under such rapidly changing conditions of temperature and pressure. Association of atoms of hydrogen and oxygen will probably!

RESTRICTED DEPARTMENT OF ENGINEERING RESEARCH | Page 58 UNIVERSITY OF MICHIGAN! Report No. UWK9 approach equilibrium more closely, and these represent a considerably greater energy release than the sluggish water-gas reaction. Considerably more confidence can be placed in computations based on analysis of individual reaction rates rather than the assumption that equilibrium will be maintained down to exit temperature, or some arbitrarily chosen intermediate temperature, or that composition of the combustion chamber gases will be "frozen." Attainment of this will require a comprehensive study of kinetics of reactions, and the half-lives of various atomic, molecular, and radical species involved in the complex combustion process. A Mollier-type diagram may be constructed for the products of combustion, assuming complete equilibrium or frozen composition at combustion conditions or at any intermediate extent of reaction believed to obtain. This diagram gives immediately the LH between any two conditions included therein. The complete thermodynamic enthalpy function may be applied: L A H = /TdS + /VdP + etc where T = Temperature S = Entropy V = Volume P = Pressure, = Chemical Potential M Mass etc = all other energy effects such as electrical, surface, nuclear, etc. If isentropic expansion at "frozen" composition is assumed, with the miscellaneous energy effects zero, this reduces to A H /VdP This integral can be readily evaluated between limits if a relation --

RESTRICTED DEPARTMENT OF ENGINEERING RESEARCH Report No. UMM 9 UNIVERSITY OF MICHIGAN Page.59 between P and V is known. For the case of perfect gases with constant specific heats and zero Joule-Thomson effect PV = PcVc where 5 = ratio specific heat at constant pressure to that at constant volume, i.e., cp/cv. Appropriate substitution and integration permits evaluation of A H, hence specific impulse according to the procedure designated as Method (1). It should be noted that this integration is possible only if 6 is assumed constant. Since both specific heat at constant pressure and specific heat at constant volume change with temperature, an average value of their changing ratio is necessary. Various methods of averaging are explored in the variations of Method (1). Another procedure is possible if frozen composition is assumed, which obviates the uncertain assumption of constant,. dH - ( dT + (H)T dP For zero Joule-Thomson effect, this becomes: dH = (.H)p dT - cp dT or AH =/Cp dT Since accurate specific heats (cp) for gases formed in combustion processes are available, it is possible to prepare tables or graphs of the enthalpy (and entropy) as a function of temperature for these gases. (e.g. Table 3) These tables may be used in evaluating AH through calculation of two isobars of an H-S diagram (P Pc and P = P) in the proper entropy region so that L\S = O. This procedure is used in Method (3). The most favorable assumption of maintenance of thermochemical and thermophysical equilibrium is used in Method (2), using AH values read

R:STRICTED... 6DEPARTENT OF ENGINEERING RESEARCH F Page 60 __ l_ UNIVERSITY OF MICHIGAN Report No. UMM9 from an equilibrium enthalpy-entropy diagram. Much work is being done to clarify the points of uncertainty mentioned. In the meantime, the researcher must evaluate new fuel combinations, and the designer must plan new motors without accurate knowledge of the theoretical maximum energy which can be extracted in the expansion process. Moreover, unless each knows the theoretical maximum, there remains the danger of overlooking an important possibility such as a new fuel, or of wasted effort on the part of the designer to improve a motor already performing at a maximum. Accurate comparison of experimental results with theoretical prediction is impossible without accurate knowledge of friction losses and proper evaluation of the lateral components of velocity of the jet gases. There follow more detailed elucidations of these various methods, with comparative results tabulated and graphed. It will be seen by inspection of these results that: 1. Specific impulses predicted by the various methods show surprisingly little spread. 2. Predicted fuel-oxidant ratio for optimum thrust may be affected considerably by method of calculation. 3. Exit gas temperature as predicted by various methods differ considerably. The thermal, chemical, and physical processes involved in rocket motors require further study for determination of: 1. Theoretical maximum specific impulse with a greater degree of certainty. 2. Optimum mixture ratio 3. Temperature of gases for analysis of the required cooling system. i

RESTRI CTED DEPARTMENT OF ENGINEERING RESEARCH Report No. UNIVERSITY OF MICHIGAN P Page 61 Methods for Calculating Theoretical Specific Impulse Calculations of theoretical specific impulse differ primarily in assuming either frozen or shifting equilibrium. Three methods will be outlined here covering both assumptions. Method (1) - Adiabatic Expansion Formula (Reference 10) Assumptions - 1. Frozen equilibrium at the combustion chamber composition. 2. Perfect gases. 3. Constant ratio of specific heats. Procedure - The combustion chamber composition and temperature is calculated. Average molecular weight is calculated and average ratio of specific heats is estimated. Exhaust temperature is calculated and this assumption verified by: T P C-1 Tc _ c1 Te.Pe Specific impulse is given by: I2r P I 6.95 M{. 1 - Pe) 1i Where Tc = combustion chamber temperature, OR Te = exhaust temperature, OR Pc = combustion chamber pressure P -= exhaust pressure y = ratio of specific heats I = specific impulse, sec M = average molecular weight The average r required may be obtained by several different methods:

I RESTRICTED DEPARTMENT OF ENGINEERING RESEARCH R Page 62 tJNIVERSITY OF MICHIGAN Report No. U9 a. Arithmetic average S between Tc and Te b. Obtain ' from arithmetic average Cp c. Obtain 6 from arithmetic average temperature between Tc and Te d. Obtain a from integrated average cp Method (2) Enthalpy Entropy Diagram Assumptions - 1. Shifting equilibrium throughout nozzle i.e., thermochemical equilibrium is maintained. 2. PV = NRT equation of state Procedure - Compositions, enthalpies, and entropies are calculated for two or more temperatures at the combustion chamber pressure, When properly chosen, these points "straddle" the flame temperatures. The same is done for the exhaust pressure except that the points are chosen to straddle the combustion chamber entropy. A partial H-S diagram is constructed (see Figures 12, 17, 18) and the proper interpolation gives the enthalpy change for the expansion process. Theoretical specific impulse is given by I = 6.95 - H where I = specific impulse, sec A H = enthalpy change on expansion, Btu per lb The enthalpy-entropy diagram may be as complete as desired by calculation of more points and by cross plotting (see Figures 14-16). This has the advantage of allowing ready investigation of temperature and volume conditions with changing pressure and fuel conditions. Efficiency and heat transfer may also be taken into account (see Figure 19). This method has been used in Germany (References 13 and 16) but apparently very little in this country. Method (3) H-S diagram - "frozen" equilibrium This method has not, to the best of the writer's knowledge, been used at all, but is proposed as an improvement on Method (1). - I

RESTRICTED. IDEPARTENT OF ENGNEERIN G 'RESEARCH Report No. UDM9 UNIVERSITY OF MICHIGAN Page 63 Assumptions: 1. "Frozen" equilibrium 2. PV = NRT Procedure - The flarne temperature and composition are calculated as in Method (1). The enthalpy and entropy of this state are calculated as in Method (2). Using this composition, two or three points (H,S) are calculated at the exhaust pressure so as to straddle the combustion chamber entropy. In other words the assumptions of Method (1) are used with the procedure of Method (2) with the exception of the averaging of specific heats. By use of properly constructed enthalpy-entropy tables the uncertainty introduced by averaging the ratio of specific heats is eliminated. A partial H-S diagram is constructed, and I is calculated as in Method (2). Tabular and graphical comparisons of these methods follow. m * - -----

RESTRICTED DEPARTMENT OF ENGINEERING RESEARCH Page 64 UNIVERSITY OF MICHIGAN _ Report No. T13M 9 Table 8: Theoretical Performance of the RFSN-Aniline System by Three Methods of Calculation for Four Conditions of Chamber Pressure and Mlixture Ratio. Method la lb lc ld 2 3 (av ) (av C) (av temp) (integral (shifting (frozen of av Cp) r-3) iiS) 7h.75.75.75.75.75.75 Pc (atm) 40 40 40 40 40 40 T 5364~R 5364~R 5364~R 5364~ 5364~ 534R 5364~R Te 2671~R 2678~R 2713~R 2693~R 2981~R 2700~R M 25.176 25.176 25.176 25.176 -- 25.176 y'av 1.2331 1.2320 1.2269 1.2294 -_ I 233.9 233.9 234.6 234.2 237 233.7 Method lb lc 2 3 X -..80 *80 --.80.80 P (atm) -- 40 40 -- 40 40 T -o 5616~R 56 561 616~R 56160R T _ - 2923OR 2952~R - 3917~R 2943~R Me - 27.129 27.129 -- -- 27.129 av - 1.2151 1.2112 - I -- 232.7 233.3 - 243 232.2 Method la lb lc ld 2 3 x --.76.75 --.75.75 Pc (atm) -- 20 20 -- 20 20 T - 5292~R 52920R 5292R 5292 9R Tc - 3028~R 3047~R - 3360~R 3044~R MI -- 25.133 25.133 -- - 25.133 -av - 1.2291 1.2259.. am I -- 216.1 216.4 - 220 215.5 Method la lb lc ld 2 3 A --.80.80 --.80.80 P, (atm) - 20 20 -- 20 20 T - 5494~R 5494~R ' 5494~R 5494~R Te - 3236~R 3247~R - 4200~R 3241~R T -_ 26.9664 26.964 - - 26.964 -av 1.2144 1.2129. I -- 214.0 214.2 - 222 213.0 ITote: Pe = 1 atm. in all cases Values from Table 8 are plotted on Figures 25 and 26 for easy comparison.

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RESTRICTED Ro DEPARTMENT OF ENGINEERING RESEARCH Report No. UNIVERSITY OP MICHIGAN Page 67 Discussion Every method of calculation presented in this report involves at least one questionable assumption. The assumption that chemical equilibrium is attained in the combustion chamber is basic to any method of computation now in use. The maintenance of an equilibrium mixture during the expansion is one of the uncertain points in explanation of which much study of combustion kinetics is required. The H-S diagram (Method 2) assumes maintenance of equilibrium, while computation from isentropic perfect gas formulae assumes "frozen" composition. The latter, however, required the further assumption of constant ratio of specific heats over a temperature and pressure range. This last assumptionis obviously dangerous, whether composition changes or not. A true average ratio for any interval is extremely elusive. Inspection of Table 8 shows that Methods (la), (lb), (lc), (ld), and (3) do not differ considerably in results. These methods are based on the same assumption (namely frozen equilibriumr) the difference being only in the use of constant ratio of specific heats. Method (3) obviates this assumption and is therefore theoretically sounder. The difference in results obtained using frozen or shifting equilibrium assumptions is illustrated in Table 8 and Figure 26. Exhaust temperature is the largest point of variance, being considerably higher with shifting equilibrium (Method 2). Figure 26 indicates some difference in optimum mixture ratio as predicted from the two assumptions but specific impulses that are substantially the same. For the fuel systems herein reported, there are certain ratios of fuel to oxidant at which theoretical specific impulses from the two methods described check well. At leaner mixtures, the L-S diagram evaluation indicates

tRESTRICTED DEPARTMENT OF ENGINEERING RESEARCH Page 68 hgemaif UNIVERSITY OF MICHIGAN. Report No. UW gs a higher maximum specific impulse at a leaner fuel ratio than perfect-gas equations. There is considerable difference in the temperatures and sane difference in volumes at exit as predicted by the tvo methods. The labor of evaluating one set of conditions is considerably greater for the H-S diagram, but for a comprehensive study of a system, the labor involved is not greatly different. The difference in theoretical results introduces some question into theoretical analysis of perforrrance, with separation of friction losses, chemical inefficiencies, and nozzle disturbances as the objective. It appears that much study is required on each of these factors before complete understanding is attained,

i RESTRICTED DEPARTMENT OF ENGINEERING RESEARCH Report No. UM2 9 ' UNIV OF MICHIGAN Page 69 REFERENCES 1. Gordon, J. G., J Chem. Phys. 2, 65, 549 (1934) 2. Hottel, H. C., Hershey, R. L., and Eberhardt, J. E., Soo. Auto. Engrts. J. 39, 409-424 (Oct. 1936) 3. Johnston and Chapman, A.T., J.A.C.S., 55 153 (1935) 4. Johnstonand Davis, J.A.C.S., 5 271 (1934) 5. Johnston and Davis, J.A.C.S., 56, 1045 (1934) 6. Johnston and Dawson, J.A.C.S., 55, 2744 (1933) 7. Johnston and Walker, J.A.C.S., 57, 682 (1935) 8. Kassel, J.A.C.S., 56, 1838 (1934) 9. Kelley, K.K., U.S. Bur. Mines Bul. 371 (1934) 10. Lemmon, A.W. Jr., "Fuel Systems for Jet Propulsion," JPIvP of OSRD No. 267 (May 1945) 11. Lewis and Von Elbe, "Combustion Flames and Explosion of Gases, London (1S38) 12. Perry, J.H., "Chemical Engineers' Handbook," 2nd Ed., New York (1941) 13. Sanger and Brebt, "A Rocket Drive for Long Range Bombers," Ainring (Obererbaren) (Aug 1944) 14. WVeo er, R.R., "Thermochemical Calculations," New York (1941) 15. Wilson, B.E. Jr., "The Present State of the Statistical Method of Calculating Thermodynamic Functions," Chem. Rev., 27, 17-37 (1940) (179 References on specific compounds). 16. German Microfilm, PGi.8, Arch 20/1 17. Lange, "Handbcok of Chemistry", 6th Ed. Handbook Publishers Inc., Sandusky, Ohio (1946) 13. liougen and Watson, "Industrial Chemical Calculations", Boston (1936) I --

L RE1STRICTED DEPARTMENT OF ENGINEERING RESEARCH Pagte f ____70 l_ _ UNIVERSITY OF MICHIGAN Report No. UMNI 9 APPE3TDIX I Derivation of Generalized Equilibrium Calculation Scheme The system to be considered is one of unit mass composed entirely of atoms of carbon, oxygen, hydrogen, and nitrogen. By preliminary inspection of equilibrium constant data, it was estimated that ten molecular species should be included. Subsequent check showed this estimate to be substantially correct up to 32000K. These species are: H20, H2, 02, O, I, OH, C02, CO, NO, and N2. The equilibrium composition of such a system is governed by the laws of conservation of atomic species, conservation of mass, and mass action. Since there are four atomic species there arefour equations of their conservation: (30) ~. [CO2] + [CO] (31) @ [H2 H] 2 [021 + [0] + [OH] + 2[C021 + [CO] + [1O] (32) () H2o] + 2[1] + [H] + [ OH] (33) ~) = [iNO] + 2[N2] The equation of conservation of mass is: (34) 1: [ oOl + rIl + ro1 + rol + [09 + roE[ + rcoo + rcot where [12c [ an L 6 J- L 4.- L - r JI- L - L L ' L l J L J + [NO]J + [N2] ). gan atoms of carbon in the system )) * gm atoms of oxygen in the system D " gm atoms of hydrogen in the system = gm atoms of nitrogen in the system I total number of gm moles in the system )] gm moles of water in the system at equilibrium,] - gm moles of hydrogen in the system at equilibrium, etc. *equations are arbitrarily numbered here to avoid confusion with mass action eauation-, whzich are nuim-bered to correspond to those of Lewis and-Von Elbe. (Reference 11)

RESTRI CTED R.eo N.. I DEPARTMENT OF ENGINEERING RESEARCH Report lNo.' UX3i 9 _ UNIVERSITY OF MICHIGAN Page 71 The introduction of N, total number of moles in the system at equilibrium, introduces an eleventh unknown. Thus an algebraic scheme of eleven simultaneous equations and eleven unknowns is being built up. The remaining six equations must be six independent mass action equations. The following six hypothetical reversible chemical reactions will yield mass action equations satisfying this requirement: The numbering is from Lew-is and Von Elbe (Reference 11). (1) H1 2 2H (9) H20o OH + H2 (2) 02 20 (10) C + H20 C02 + H2 (8) H O-H2 + 2 02 (15) o2 + 2 N2z NO Thye mass action equations are: (rp)) 0o11 2 ( N P) (2) K2f T N )( [02] ) (8) L8 S f[W( 1021 p)(p1 ) (.lo K10o:,c]...22... LN Ar1

RESTRICTED ( are he aropriate libri cOi nstants I a pressuDre in atmospheres P In foraulatin the above equaions, ideal gas behavior has beoen assumed allowing total prssure times mole fraction to be used in place of the mors bxaco fugacity. T1is is not an unraasonable asslumption in the range of interest since t+ie templerature is high wlhere the pressure is high, nid the pressure is low where the temperature is low. An algebraic Sy1st-n of eleven equations and eleven unkmowns has now been formulated. it remains to manipulate themir into solvable form: Solving (1) for [;o]: (1,1) [H] - VIA 1 IT Roarranging equation (32): (32.1) [L20] + 1 [oI = @~ - [21] - [iT] iearralnging (9): (5?i lowv[G - [c]3 @[iy iWritin; en identi t: (234)%,, [L 0] + [- [20 + And substituting the Lv;lur of [2O] from (9,l1) into thle left-hand side of (34): ().[c [CJ] [n] [o1 [01;] 4 Soc ling (3I ) for [!C:I

" RESTRI CTED DEPARTMENT OF ENGINEERING RESEARCH Report No. UWA 9 ___ _ UNnIRSITY OF MIClIGAN Page 73 [H2O] + J [OH] (35.1) 1 [H] 2(I [-2 -i1+ Kg - Solving (8) (8.1) for [02] [O2]: (K82/) N ( [HO0]/[H2] )2 Solving (2) for [O]j (2.1) [o] _ {KFA fi [N [2] Solving (15) for [No]: (15.1) [NO]- K15 STO2 N1J Substituting from (33) into (15.1) (15.2) -[o0]= _K5 1 2 -2 [N ] Subs tituting (30) into (31) (31.1) @ - ~: [H20] + 202] + [0 Solving (31.1) for [co2] s (31.2) [COl: @ - [20o] - [~ Rearranging (30): (30.1) [CO] - ~ - [co2 Solving (10) for [121s (10.1) [H] [co ] 10 0CO I- -1 K ] + [OH] + [C 2] + [NO] I HI - 2[02] - [] - [NO] Substituting (30), (31), and (33) into (34) (34.1) N = @~ + I ~ + } [1] + NO[H] + 40[02] The equations are now ready for stepwise trial and error calculation: Assumes [2] and N Calculate 2 [H] (1.1) 2 [H]: 2 K N [] Caloulate [OH] + 2 [20]: (32.1) [OH] + 2[BO]. - 2[H2] - [I]

RESTRICTED DEPARTMENT OF ENGINEERING RESEARCH Page 741 | UNIVERSITY OF MICHIGAN _ Report No JM 9 Caloulate [OH]: (35.1) [Oi] [o1L] +. 2 [ 20] 1 + (2 f/G)- v[H2] f' Calculate [H12O] [HO2] 1 ([OH] + 2 [H20] - [OH]) Calculate [02] (8.1) [02] = (K2/^) N ([I20]/ [2] )2 Calculate [0]: (2.1) [0]: ~K2/P i -[g02] Calculate [NO]: (5.2) [NO] = K15 1 / 1j ~ - 1 [NO] Calculate [Co2]: (31.1) [C02] = ) - ~ - [H2O] - [OH] - 22]- [o] -NO] Calculate[CO]: (30.1) [CO]: - [Co21 Check [H2] (10.1) [E2 ] [10- ] Check N: (34.1) N = + 1(~+ t [H] + 1 [OH] +2 [No ] + () [o R2 2 i aui2 2 i c Revise assumptions and repeat if necessary. Thus a systematic method is established for solving the simultaneous equations governing the equilibrium composition of a gaseous mixture*

RESTRICTIED DEPARTMENT OF ENGINEERING RESEARCH Report N*o. II2 9, UNIVERSITY OF MICHIGAN Page 75 AP.P TDX II Derivation of Entropy Calculation Equation The entropy of a gaseous mixture is usually given (Reference 14) by: i=in S - S xi Si (1) i=sl where S - entropy of mixture at temperature and total pressure Xi m number of moles of component i present in mixture Si - molal entropy of component i at temperature and partial pressure of compoi.Lnent i. For a pure constituent or a mixture of constant composition, the change in entropy with pressure at constant temperature is given by: S, -ITRR In P2/P1 (2) where: AST - increase in entropy at constant temperature N number of moles in system R- universal gas constant P2 = final pressure P1 ' initial pressure If N' represents the total number of moles in the system, the partial pressure of component i is: Pi a -p (3) N wlhere Pi - partial pressure of corapo.ient i Xi number of moles of component i in system N = total nolos in system P _ total pro ssure The m.olal entropy of coinponent i at the temperature of the mixture and 1

L RESTRICTED DEPARTMENT OF ENGINEERING RESEARCH Page 76 UNIVERSITY OF MICHIGAN Report No UMM 9 partial pressure of component i is given by: Si s S~ - R iln i (4) N P~ where Si~ - molal entropy of component i at temperature and reference pressure P~ PO _ reference pressure, taken here as 1 atm Therefore; Si a S~ - R ln P (4.1) 1 N Si from Equation 4.1 may now be substituted into Equation 11 S. ' Z Xi (Si. R ln r ) (5) Suppose, for the moment that the total pressure is numerically equal to N. (Composition to remain constant) Equation 5 becomes: i-n S. (s1oxi - X. R In X.) (5.1) il 1 1 1 isl Now correcting the total pressure of the mixture as given by (5.1) to the actual pressure P by Equation 2 gives: i=n isn S3 xiSi - _ Xi R In Xi - N R In P/4 (6) i-l i-l Substituting numerical value for R and changing logarithm base gives the eqaation used in computation: i=n isn S =XiSi~O -2 4. 5753 Xi log Xi - N 4.5753 log P/N (7) i-l ial

i RESTRICTED DEPARTMNT OF ENGINEERING RESEACH Report No UMM 9 NVERSITY OF MICHIGAN Page 77 APPENDIX III Enthalpy-Entropy Diagrams from German Data Captured German documents have provided us with several enthalpyentropy diagrams very similar in principle and construction to those described in Part I of this report. The differences are as follows: Thermochemical equilibrium is assumed above 20000K; below this temperature, the composition is assumed invarient at the 2000~K value. This assumption should cause only slight differences between what they have calculated and what we would calculate since the only reaction occuring- to any extent below 20000K is the water-gas reaction, and this introduces only small changes in enthalpy and entropy. The German diagrams are based on the arbitrary reference state of H20 (g), C02 (g), and 02 (g) at 298~K for enthalpy and H20 (g), C02 (g), and 02 (g) at 2000~K for entropy. Thus the enthalpy of the unburned propellant is equal to its heat of combustion at 298~K plus its sensible heat above 298~K. The tables of enthalpy and entropy used in constructing the diagrams have not yet been located, but presumably they have been prepared from the same original works as the enthalpy-entropy tables presented in Part I of this report. Thus the following German diagrams, when entered at the unburned propellant enthalpy consistant with the reference state for the diagram, should give theoretical performance characteristics almost exactly as do our own. m - -- --

PAGE 78 FIG, 27

I R RESTRICTED DEPARTMENT OF ENGINEERING RESEARCH Report No. UDMI 9 i U R OF MICHIGAN Page 81 DISTRIBUTION This report is distributed in accordance with AN-GM Mailing List No. 3, dated May, 1947, as corrected, including Part A, Part C, and Part DP.

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