# Propulsion and Thermodynamic Parameters of van der Waals Gases in Rocket Nozzles

Authors: Fernando S. Costa; Gustavo A. A. Fischer
Journal: International Journal of Aerospace Engineering
 (2019)

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## Abstract

Propellants or combustion products can reach high pressures and temperatures in advanced or conventional propulsion systems. Variations in flow properties and the effects of real gases along a nozzle can become significant and influence the calculation of propulsion and thermodynamic parameters used in performance analysis and design of rockets. This work derives new analytical solutions for propulsion parameters, considering gases obeying the van der Waals equation of state with specific heats varying with pressure and temperature. Steady isentropic one-dimensional flows through a nozzle are assumed for the determination of specific impulse, characteristic velocity, thrust coefficient, critical flow constant, and exit and throat flow properties of He, H2, N2, H2O, and CO2 gases. Errors of ideal gas solutions for calorically perfect and thermally perfect gases are determined with respect to van der Waals gases, for chamber temperatures varying from 1000 to 4000 K and chamber pressures from 5 to 35 MPa. The effects of covolumes and intermolecular attraction forces on flow and propulsion parameters are analyzed.


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## Body

## 1. Introduction

The classical equations for the calculation of rocket propulsion parameters are based on one-dimensional flows of ideal gases with constant properties [1]. The NASA CEA (Chemical Equilibrium and Applications) code is largely used for the preliminary design of rockets and calculates nozzle properties and propulsion parameters considering frozen or equilibrium one-dimensional flows of thermally perfect ideal gases [2]. Nevertheless, propellants and combustion products in advanced or conventional propulsion systems, such as electrothermal, microwave, laser, nuclear, and chemical rockets, can attain relatively high temperatures and pressures [3–6]. In general, at elevated temperatures, the effects of molecular attraction are negligible and compressibility factors of fluids are close to unity. However, pressures and temperatures decrease rapidly and at different rates along the nozzle, and consequently, real gas effects and variations of properties can become significant. Chamber conditions also can be affected by real gas behavior, such as equilibrium composition and temperature, reaction rates, and other combustion parameters, especially at high pressures [7, 8]. Other processes can affect the performance of nozzles and propulsion parameters, such as viscous losses, chemical kinetics, heat transfer, boundary layer, flow separation, shock waves, two-phase flow, and three-dimensional effects [9–13]. Correction factors are often used to estimate the performance of real nozzles; however, there is limited information about the effects of real gases and variable properties.The main propulsion parameters used to evaluate performance characteristics of rockets are specific impulse, characteristic velocity, and thrust coefficient. Specific impulse relates the total impulse to the total weight of a burned or ejected propellant. The characteristic velocity is a measure of propellant performance and motor design quality, whereas the thrust coefficient indicates nozzle design efficiency. Another important parameter, mainly used for a nozzle design, is the critical flow constant which determines the mass flow rate from chamber stagnation conditions [1].Several studies have considered one-dimensional flows of perfect gases or imperfect gases with variable properties in nozzles, in general without focusing on the determination of propulsion parameters. Johnson [14] investigated the effects of real gases on the critical flow constant, using virial equations based on curve fits of thermodynamic tables. Critical flow constants were determined for air, N2, O2, normal-H2, para-H2, and steam; for temperatures 389-833 K; and for pressures 0-300 atm. Witte and Tatum [15] developed a computer code to determine thermally ideal gas properties, using specific heats approximated as NASA fourth-order polynomials of the temperature. The AGARD-AR-321 report [16] presented real gas discharge coefficients, based on experimental and numerical data from Masure and Johnson, in order to correct the mass flow rate and thrust in air nozzles for stagnation temperatures up to 344 K and stagnation pressures up to 100 bar, and the results were then compared to calorically perfect ideal gas solutions. Kim et al. [17] investigated the flow of high-pressure hydrogen gas through a critical nozzle, using the Redlich-Kwong equation of state with the axisymmetric, compressible Navier-Stokes equations to account for the intermolecular forces and molecular volume of hydrogen. They verified that the critical pressure ratio and the discharge coefficient for the ideal gas assumption are significantly different from those of the real gas, as the Reynolds number happens to exceed a certain value. Yoder et al. [18] have made numerical simulations of air flow through a nozzle assuming calorically perfect air, a calorically perfect gas mixture, and a frozen gas mixture. Thus, they have managed to determine performance parameters such as mass flow rate, gross thrust, and thrust coefficient. Górski and Rabczak [19] have compared experimental data with theoretical results of the critical flow constant for dense gases. The 1D nonlinear model used parameters of the isentropic flow of real gases, analogous to an ideal gas flow. Nagao et al. [20] have investigated the real gas effects on discharge coefficient and thermodynamics properties through a critical nozzle by using H2, N2, CH4, and CO2, with the help of a CFD method. Ding et al. [21] have adopted equations of state based on the Helmholtz energy to describe the flow of hydrogen through a critical nozzle and compared theoretical results with experimental data and CFD simulations with different equations of state. Costa [22] has investigated propulsion parameters of Noble-Abel gases and verified that covolume and variation of specific heats with temperature can influence significantly the propulsion parameters, depending on the pressures and temperatures considered.The present work extends previous studies and derives new analytical solutions for rocket propulsion parameters and nozzle flow thermodynamic parameters of real gases obeying the van der Waals equation of state and provides data for a broad range of stagnation pressures and temperatures. The effects of covolumes and intermolecular attraction forces are analyzed, and the percent errors of propulsion parameters and flow thermodynamic properties of calorically perfect and thermally perfect ideal gases are calculated with respect to van der Waals gases. Steady isentropic one-dimensional frozen flows of He, H2, N2, H2O, and CO2 are considered for vacuum expansion with chamber temperatures 1000-4000 K, chamber pressures 5-35 MPa, and nozzle expansion ratios 50-200. At high pressures, the specific heats are assumed to depend on pressure and temperature, whereas at low pressures they are calculated based on fourth-order polynomials of temperature [23], as adopted by NASA CEA code [2]. These low pressure-specific heats are adjusted from experimental and theoretical data from different sources, for temperatures from 0 to 20000 K. Lower temperatures (1000-2500 K) are usually reached in electrothermal and catalytic augmented thrusters while higher temperatures (2500-4000 K) can be attained in chemical and nuclear rockets.


## 2. Theoretical Analysis

The van der Waals equation of state (VDW EOS) has the following form:(1)P+av2v−b=RT,where P is the pressure, T is the temperature, v is the specific volume, R is the specific gas constant, a is the coefficient for intermolecular attraction, and b is the covolume, an effective molecular volume. Constants a and b are, in general, determined in terms of critical constants, as presented in Table 1. Alternatively, these constants can be determined by curve fits of experimental data or from more accurate equations of state for specific ranges of pressure and temperature. Nevertheless, experimental data for fluids at high pressures are, in general, limited to temperatures lower than 2000 K [24]. In the case of mixtures of propellants or combustion products, the VDW constants a and b can be estimated from a=∑Xiai and b=∑Xibi, where Xi is the molar fractions of propellant/product i or can be estimated from pseudocritical constants, as described by Walas [25]. Maximum densities considered in this work are far from the critical density values, where the van der Waals equation is less accurate. Table 2 shows the maximum and critical densities.Table 1
van der Waals constants.
He
H2
N2
H2O
CO2
Tcr (K)
5.19
33.18
126.10
647.13
304.19
Pcr (MPa)
0.23
1.31
3.39
22.06
7.38
a=27RTcr2/64Pcr (Pa·m6·kg−2)
215.9
6016.8
174.1
1706.3
188.7
b=RTcr/8Pcr (m3/kg)
0.0059
0.0130
0.0014
0.0017
0.0010Table 2
Maximum densities (1000 K and 35 MPa) and critical densities (Data from [24]).
He
H2
N2
H2O
CO2
ρmax (kg/m3)
16.9
7.95
104.92
83.25
170.60
ρcr (kg/m3)
69.64
31.26
313.30
322.00
467.60It is worth mentioning that the adoption of more accurate and detailed equations of state, using attraction parameters dependent on temperatures, would require full numerical solutions in order to determine the propulsion and thermodynamic parameters.The evaluation of thermodynamic properties is required for the derivation of analytical solutions for the flow properties and propulsion parameters. Initially, the specific heats of a real gas at constant pressure and volume,cp and cv, respectively, are obtained from the thermodynamic relations:
(2)cv−cv0=T∫0v∂2P∂T2vdv,(3)cp−cv=T∂v∂TP∂P∂Tv,where the “0” superscript denotes ideal gas property. Replacing equation (1) into equations (2) and (3) provides cv=cv0 and cp−cv=R/1−∆ with
(4)∆=2ε1+ε1−b∗<1,where b∗=b/v and ε=a/Pv2 are, respectively, the nondimensional covolume and nondimensional attraction parameter. Consequently, cp>cv and the ratio of specific heats, γ=cp/cv, of a VDW gas are
(5)γ=γ0−∆1−∆>γ0=cp0cv0.A differential entropy variationds can be calculated from
(6)ds=cvdTT+∂P∂Tvdv=cv0dTT+Rdv−bv−b.Considering an isentropic process,ds=0, and using cp0−cv0=R, the integration of equation (6) yields the variation of a specific volume:
(7)v−bvc−b=TTcexp∫TTccp0RdTT,where subscript “c” denotes chamber conditions. Combining equations (1) and (7), the pressure variation along the nozzle is determined:
(8)P+a/v2Pc+a/vc2=exp−∫TTccp0RdTT.A differential enthalpy changedh can be obtained from
(9)dh=cvdT+T∂P∂Tv−Pdv+dPv.Then, integrating equation (9) from h1T1,v1 to h2T2,v2 yields
(10)h2−h1=∫T1T2cp0−RdT+av1−av2+P2v2−P1v1,which is similar to the enthalpy equation presented by Cramer and Sen [26].Flow velocity is calculated from the energy equationu2=2hc−h for steady one-dimensional adiabatic flow with no work performed and no potential energy variation. Consequently, substituting equation (10), it results the following:
(11)u=2∫TTccp0dT+bRTcvc−b−bRTv−b−2avc+2av1/2.Assuming small pressure variations, the speed of sound is given byc=∂P/∂ρs1/2 where subscript “s” denotes the constant entropy. Applying Maxwell thermodynamic relations, it follows that
(12)us=−γv2∂P∂vT1/2,and for the VDW EOS, it yields
(13)us,VDW=γRT1−b/v2−2aγv1/2.Equation (13) shows that larger covolumes and smaller intermolecular forces increase the speed of sound. Combining the differential forms of the mass conservation equation and Euler equation, with the speed of sound definition, implies that flow velocity and speed of sound are equal at the nozzle throat:
(14)ut=us,t,where subscript “t” indicates the throat conditions. Replacing equations (11) and (13) into equation (14) yields the nozzle throat temperature:
(15)Tt=1−b/vt2R2γt∫TtTccp0dT+bRTcvc−b−bRTtvt−b−2avc+2avt+2avt,which can be solved iteratively using equation (7) for a specific volume calculated at the nozzle throat. Equation (15) is significantly simplified for a Noble-Abel (NA) gas, i.e., with a=0 or ε=0:
(16)Tt=1−bt∗2R2γt∫TtTccp0dT+bPc−Pt,and in the case of thermally perfect ideal gases, a=b=0; therefore, Tt=2/γt∫TtTccp0/RdT.After throat conditions are determined, the critical mass flow rate constant can be calculated:Γ=ṁRTc/AtPc, where ṁ is the mass flow rate of propellants. Since, at the nozzle throat, ṁ=ρtutAt=us,tAt/vt, then,
(17)ΓVDW=γtRTtvt−b2−2avt31/2RTcPc.Defining the characteristic velocity of propellants byc∗=PcAt/ṁ and replacing equation (17), for a VDW gas, it yields
(18)cVDW∗=RTcΓVDWThe thrust coefficient is defined bycF=F/PcAt, where F is the rocket thrust. Assuming no divergence losses, F=ṁue+Pe−PaAe, where subscript “e” denotes the nozzle exit conditions and subscript “a” denotes the ambient conditions. Once the mass flow rate is calculated by
(19)ṁ=ΓAtPcRTcand the exit velocity of propellants, or combustion products, is obtained from equation (11), then
(20)cF=ΓRTcue+PePc−PaPcAeAt.The exit pressurePe depends on the nozzle area ratio Ae/At, since ṁ=ΓAtPc/RTc=ueAe/ve which yields
(21)AeAt=ΓVDWPcRTcveue.Therefore, the thrust coefficient of a VDW gas is(22)cF,VDW=ΓVDWRTcue+Pe−Paveue,where
(23)ve=b+vc−bTeTcexp∫TeTccp0RdT,(24)Pe=−ave2+Pc+avc2exp−∫TeTccp0RdT,(25)ue=2∫TeTccp0dT+bRTcvc−b−bRTeve−b−2avc+2ave1/2.Exit specific volume, exit pressure, and exit velocity can be calculated for a given exit temperature. Assuming perfect expansion in vacuum, the exit-specific volume will approach infinity, and exit pressure and exit temperature will approach zero; then the optimum thrust coefficient in vacuum can be calculated by(26)cF,VDW,opt=ΓVDWRTc2∫0Tccp0dT+bRTcvc−b−2avc1/2.The specific impulse of a rocket is defined byIsp=F/ṁgo, where go is the standard gravity acceleration at a sea level. Since Isp=c∗cF/go, the optimum specific impulse in vacuum is
(27)IspVDW,opt=cVDW∗cF,VDW,optgo=1go2∫0Tccp0dT+bRTcvc−b−2avc1/2.Equation (27) indicates that larger covolumes and smaller intermolecular attraction forces increase the optimum specific impulse, if their influences on specific heat and combustion temperature can be neglected.


## 3. Simplified Solutions

The previous analytical solutions can be simplified in special cases. The critical flow constant and characteristic velocity of a thermally perfect NA gas [22] are calculated, respectively, by
(28)ΓNA=γt,NATcTt,NAexp−∫Tt,NATccp0RdT,cNA∗=RTc/ΓNA,where γNA=cp0/cv0 and γt,NA=cp0Tt,NA/cv0Tt,NA. Simplifying equation (20)–(25) yields the thrust coefficient and nozzle area expansion ratio of a rocket propelled by a NA gas:
(29)cF,NA=ΓNA2Tc∫TeTccp0RdT+bRPc−Pe+PePc−PaPcAeAt,AeAt=ΓNAbPc/RTc+Te/TcPc/Pe2/Tc∫TeTccp0/RdT+2bPc/RTc1−Pe/Pc.SinceIspNA=cNA∗cF,NA/go, the optimum thrust coefficient and the optimum specific impulse, for a NA gas, are given, respectively, by
(30)cF,NA,opt=ΓNA2Tc∫0Tccp0RdT+bPcR,IspNA,opt=cNA∗cF,NA,optgo=1go2∫0Tccp0dT+bPc.If intermolecular attraction forces and covolumes are neglected,a=0 and b=0, and the VDW EOS becomes the thermally perfect ideal gas equation of state (IG-TP EOS) and the previous results for NA gases are further simplified. Assuming constant specific heats, calorically perfect solutions for NA gases and ideal gases can be derived.


## 4. Results and Discussion

Rocket propulsion parameters and flow thermodynamic properties for different rocket chamber conditions and nozzle expansion ratios were obtained for vacuum expansion. Steady isentropic one-dimensional frozen flows of He, H2, N2, CO2, H2O, and gases following the VDW EOS were considered. Frozen flows are assumed when flow residence time in a nozzle is shorter than reaction times, whereas equilibrium flows require a longer residence time. In the case of hydrogen recombination kinetics, the losses of specific impulses decrease with increasing pressures [9].At low pressures, the specific heats were assumed to obey fourth-order polynomials of temperature [23] with coefficients ai, i=1,⋯,7:
(31)cp0R=a1T−2+a2T−1+a3+a4T+a5T2+a6T3+a7T4.The percent errors of propulsion parameters∅=Isp, Γ, etc. of calorically perfect (CP) and thermally perfect (TP) ideal gases (IG) in relation to VDW gases were calculated from
(32)ε∅=100∅IG−∅VDW∅VDW.Figure1 depicts percent errors of propulsion parameters of calorically perfect ideal gases (CP-IG), assuming Pc=10 MPa, chamber temperatures Tc=1000 K–4000 K, and frozen flow expansion in vacuum for Ae/At=100. Table 3 shows the maximum and minimum percent error values of propulsion and some flow thermodynamic parameters for CP-IG.Figure 1
Errors of propulsion parameters of calorically perfect ideal gases with respect to VDW gases at different chamber temperatures, for vacuum expansion,Ae/At=100 and Pc=10 MPa.Table 3
Maximum and minimum error values of propulsion and flow parameters of calorically perfect ideal gases forPc=10MPa, Tc=1000 K–4000 K and frozen flow expansion in vacuum with Ae/At=100.
ϕ
εmax (%)
εmin (%)
CO2
H2O
N2
H2
He
CO2
H2O
N2
H2
He
Γ
−0.227
−0.590
0.032
0.033
0.590
−6.496
−11.528
−2.152
0.008
0.165
c∗
6.947
13.023
2.200
−0.008
−0.165
0.227
0.593
−0.032
−0.033
−0.587
cF
0.049
0.870
1.872
3.142
0.034
−3.682
−7.074
0.163
0.874
0.020
Isp
3.010
5.029
2.607
3.111
−0.144
0.277
1.204
0.973
0.847
−0.560
Pe
43.649
51.564
42.491
48.689
2.067
4.672
16.104
21.983
26.396
0.456
Me
−1.157
−3.806
−5.676
−8.822
−0.236
−8.669
−12.403
−12.172
−13.369
−0.966
Pt
0.536
0.361
1.454
1.117
0.844
0.273
0.047
0.357
0.506
0.213
Tt
0.418
0.744
0.722
0.560
0.339
0.067
0.093
0.127
0.263
0.085As seen in Figure1, critical flow constant errors of N2, H2O, and CO2 (CP-IG) have negative values and their absolute values decrease monotonically from 1000 K to 4000 K, while Γ errors of H2 (CP-IG) are approximately zero and Γ errors of He are slightly positive and approach zero for larger temperatures. Characteristic velocity errors have similar values, but with opposite sign of the critical flow constant errors, since c∗ varies inversely with Γ.Thrust coefficient errors of H2 and N2 (CP-IG) are positive, and present maximum values of +3.14% at 2710 K and +1.87% at 1790 K, respectively, whereas the cF error of He (CP-IG) is approximately zero along the temperature range considered. The cF error of H2O (CP-IG) is −7.07% at 1000 K and reaches a maximum of +0.87% at 2570 K, whereas the cF error of CO2 (CP-IG) increases monotonically from −3.68% at 1000 K to +0.05% at 4000 K.Isp errors of all CP ideal gases, except He, are positive within the temperature range considered. Isp errors of H2 and N2 (CP-IG) show maximum values of +3.11% at 2680 K and 2.61% at 1310 K, respectively, whereas Isp errors of H2O and CO2 (CP-IG) decrease monotonically from 5.03% to 1.2% and 3.01% to 0.28% from 1000 K to 4000 K. Isp errors of He (CP-IG) vary monotonically from −0.56% to −0.14% between 1000 K and 4000 K.Exit pressure errors of CP ideal gases are positive and reach maximum values above 40%, except He that presentsPe errors less than 2.1%. Exit Mach number errors for all CP ideal gases are negative, and their absolute values are larger than 8%, except He that shows absolute values of Me errors less than 1%. Errors of throat pressures and throat temperatures of all CP ideal gases are positive and reach maximum values lower than 1.5% and 0.8%, respectively.Figure2 depicts errors of propulsion parameters of thermally perfect ideal gases (TP-IG), assuming Pc=10 MPa, Tc=1000 K–4000 K, and frozen flow expansion in vacuum through a nozzle with Ae/At=100. Table 4 presents the maximum and minimum error values of propulsion parameters and some flow parameters of TP ideal gases.Figure 2
Errors of propulsion parameters of thermally perfect ideal gases with respect to VDW gases at different chamber temperatures, for vacuum expansion,Ae/At=100 and Pc=10 MPa.Table 4
Maximum and minimum error values of propulsion and flow parameters of thermally perfect ideal gases forPc=10 MPa, Tc=1000 K–4000 K and frozen flow expansion in vacuum with Ae/At=100.
ϕ
εmax (%)
εmin (%)
CO2
H2O
N2
H2
He
CO2
H2O
N2
H2
He
Γ
−0.212
−0.556
0.065
0.228
0.590
−6.428
−11.358
−1.966
0.142
0.165
c∗
6.870
12.813
2.006
−0.142
−0.165
0.213
0.559
−0.065
−0.227
−0.587
cF
−0.284
−0.546
−0.062
0.038
0.034
−5.857
−9.782
−2.097
−0.230
0.020
Isp
0.610
1.778
−0.127
−0.104
−0.144
−0.071
0.010
−0.206
−0.388
−0.560
Pe
8.413
6.401
3.606
1.850
2.067
1.343
1.472
0.740
0.411
0.456
Me
−0.248
−0.187
−0.252
−0.170
−0.236
−1.121
−0.529
−0.946
−0.800
−0.966
Pt
0.413
0.134
0.967
0.794
0.844
0.242
−0.394
0.274
0.196
0.213
Tt
0.305
0.430
0.370
0.243
0.339
0.046
0.041
0.068
0.044
0.085Critical flow constant errors of TP ideal gases are similar to critical flow constant errors of CP ideal gases.Γ errors of N2, H2, and CO2 (TP-IG) have negative values, and their absolute values decrease monotonically from 1000 K to 4000 K, while Γ errors of H2 (TP-IG) are approximately zero and Γ errors of He (TP-IG) are slightly positive and approach zero for larger temperatures. Characteristic velocity errors have opposite sign and are similar to the critical flow constant errors. H2O (TP-IG) presents the largest c∗ errors which vary from about +12.81% to 0.56% between 1000 K and 4000 K.Properties of He (TP-IG) and He (CP-IG) are equal since specific heats of helium do not vary with temperature in the range considered.Thrust coefficient errors of TP ideal gases show similar behavior to critical flow constants of TP ideal gases. Absolute values ofcF errors of H2O, CO2, and N2 (TP-IG) decrease monotonically from 1000 K to 4000 K, whereas the values of cF errors of H2 and He (TP-IG) are approximately zero.Specific impulse errors of H2O and CO2 (TP-IG) decrease monotonically from +1.78% to 0.01% and from +0.61% to −0.07%, respectively, between 1000 K and 4000 K. Meanwhile, the absolute values of the Isp errors of H2 and He (TP-IG) decrease monotonically from 0.39% to 0.10% and 0.56% to 0.14%, respectively, in the range considered. N2 (TP-IG) presents a minimum Isp error of −0.21% at 1560 K.Exit pressure errors of TP ideal gases are positive, decrease monotonically, and are significantly lower than exit pressure errors of CP ideal gases in the temperature range considered. The exception would be, again, helium, which presents equal values for TP and CP ideal gases.Exit Mach number errors of the TP ideal gases considered are negative, and their absolute values are lower than the absolute values of the exit Mach number errors of CP ideal gases, except for helium which maintains the same values for TP and CP gases.Throat pressure errors of TP ideal gases analyzed are positive, except H2O (TP-IG) that shows negative errors up to 1500 K. Throat temperature errors of TP ideal gases analyzed are positive and decrease monotonically for increasing chamber temperatures.Figure3 presents the effects of pressures (Pc=5 to 35 MPa) on percent errors of propulsion parameters of thermally perfect ideal gases in relation to van der Waals gases, for frozen vacuum expansion through a nozzle with area ratio 100, assuming Tc=2500 K.Figure 3
Errors of propulsion parameters of thermally perfect ideal gases with respect to VDW gases versus chamber pressures, for vacuum expansion,Ae/At=100 and Tc=2500 K.Despite being negative in many cases, the absolute values of the critical flow constant errors, characteristic velocity errors, and thrust coefficient errors of TP ideal gases increase approximately linearly with increasing chamber pressures for the temperature considered, since nonideal effects become more significant at higher pressures. Specific impulse errors of TP ideal gases are negative, except for H2O, and show a slightly parabolic variation with increasing pressures. He (TP-IG) presents the largest absolute values of Isp errors, whereas H2 and N2 (TP-IG) present similar Isp behavior for the chamber temperature and chamber pressure range considered.Figures4 and 5 show errors of propulsion and flow parameters of thermally perfect ideal gases in relation to van der Waals gases, respectively, for flow expansion in vacuum through nozzles with area ratios 50 and 200, assuming a chamber pressure of 25 MPa and chamber temperatures 1000 K – 4000 K.Figure 4
Errors of propulsion parameters and flow parameters of thermally perfect ideal gases with respect to VDW gases at different chamber temperatures, for vacuum expansion,Ae/At=50 and Pc=25 MPa.Figure 5
Errors of propulsion parameters and flow parameters of thermally perfect ideal gases with respect to VDW gases at different chamber temperatures, for vacuum expansion,Ae/At=200 and Pc=25 MPa.Thrust coefficient errors and specific impulse errors of TP ideal gases are not significantly affected by the nozzle area ratio variation, but their absolute values decrease significantly for increasing chamber temperatures. On the other hand, exit Mach number errors and exit pressure errors of TP ideal gases are strongly affected by the variation of a nozzle area ratio and their absolute values decrease monotonically with increasing temperatures, except for H2O, which presents a minimum exit Mach number around 1200 K.AGARD-AR-321 [16] presented experimental and theoretical data for the critical flow constant of air for pressures up to 40 atm, showing that errors of the calorically perfect solutions vary linearly with chamber stagnation pressures and inversely with chamber stagnation temperatures in the temperature and pressure ranges considered. Johnson [27] has presented numerical data, based on virial equation solutions, of the critical flow constants of N2 and He, considering stagnation temperatures 100-400 K and stagnation pressures 0-300 atm. The N2 critical flow constant error presented quite slight variations with increasing pressures, for temperatures approaching 400 K. In the case of He, the critical flow constants varied linearly with pressure in all temperatures considered. Similar tendencies have been observed in the present results for the critical flow constants, for both calorically and thermally perfect ideal gases compared to VDW gases.


## 5. Conclusions

Real gas effects and the variation of fluid properties can significantly affect propulsion parameters of rockets and thermodynamic parameters of nozzles. In general, the influence of real gas effects is more meaningful for lower chamber temperatures and higher chamber pressures. However, the flow through a nozzle presents large pressure and temperature variations which yield significant effects on exhaustion properties. New analytical solutions for propulsion parameters were derived, considering gases obeying the van der Waals equation of state and assuming specific heats varying accordingly to pressure and temperature. Equations for specific impulses, thrust coefficients, characteristic velocities, critical flow constants, and throat and exit properties were determined, considering one-dimensional isentropic frozen flows through a nozzle. Errors were calculated for the vacuum expansion of calorically perfect and thermally perfect ideal gases, in comparison to solutions for the expansion of van der Waals gases. Data were presented for He, H2, N2, H2O, and CO2, for chamber temperatures 1000-4000 K and chamber pressures 5-35 MPa, with different nozzle expansion ratios. Correction factors for the effects of real gases and variable properties, in general, are small; however, they can be larger than other correction factors usually adopted for design and performance analysis of real nozzles, depending on chamber conditions and propellant choice. Equations for the optimum thrust coefficients and optimum specific impulses were derived indicating the influences of covolumes and attraction parameters. Larger covolumes and smaller intermolecular forces increase the optimum specific impulse, disregarding the influences on specific heats and combustion temperature. Further analysis may consider mixtures of gases or combustion products, more accurate equations of state, equilibrium, and nonequilibrium flows, heat transfer, viscous losses, boundary layer formation, flow separation, presence of shock waves, and three-dimensional losses.



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