# New Contour Design Method for Rocket Nozzle of Large Area Ratio

Authors: Dechuan Sun; Tianyou Luo; Qiang Feng
Journal: International Journal of Aerospace Engineering
 (2019)

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## Abstract

A rocket engine for space propulsion usually has a nozzle of a large exit area ratio. The nozzle efficiency is greatly affected by the nozzle contour. This paper analysed the effect of the constant capacity ratio in Rao’s method through the design process of an apogee engine. The calculation results show that increasing the heat capacity ratio can produce an expansion contour of smaller expansion angle and exit area ratio. A simple modification of Rao’s method based on thermally perfect gas assumption was made and verified to be more effective. The expansion contour designed by this method has much thinner expansion section and higher performance. For the space engine, a new extension contour type for the end section of the nozzle is proposed. The extension curve bent outward with increasing expansion angle increases the vacuum specific impulse obviously.


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## Body

## 1. Introduction

A de Laval nozzle of large area ratio is generally used in the rocket engines for space propulsion, such as apogee rocket engines that use hypergolic bipropellant combinations. The vacuum specific impulse (Isv) of apogee engines is usually above 300 seconds. For example, S400-15 has the specific impulse of 321 seconds at the nominal point [1], R-4D-14 (HPLAE) has demonstrated a specific impulse of 322.2±2 seconds [2], and the Isv of China’s second-generation 490 N apogee engine is 315 seconds [3, 4].For a space engine, to improve the specific impulse is still the goal of engine design because the higher specific impulse means a longer life cycle. To achieve this goal, we can increase the combustion efficiency or nozzle efficiency. However, it is hard to increase the combustion efficiency because higher combustion efficiency means higher combustion temperature, which increases the risk of failure of material or coating system [5–7]. In contrast, increasing the nozzle efficiency is more effective.The efficiency of a rocket nozzle is affected by many parameters. Manski and Hagemann studied the influence of rocket design parameters on nozzle efficiencies using the nozzle shapes of SSME and Vulcan engines [8]. Their research reveals that increasing chamber pressure leads to an increase of nozzle efficiency and that the kinetic losses raise up to 1% at lower chamber pressure. Compared with first-stage rocket engines, space engines have lower chamber pressure and smaller size, so their nozzle efficiencies are lower. Arrington et al. tested two small nozzles [9]. Their study shows that the bell nozzle has higher performance than the conical nozzle for the throat Reynolds number range from 2×104 to 4×104.For the design of the real rocket nozzle, TIC (Truncated Ideal Contoured), TOC (Thrust Optimized Contoured), TOP (Thrust Optimized Parabolic), and CTIC (Compressed TIC) methods are most commonly used. As summarized by Östlund and Muhammad-Klingmann, the TOC nozzle has the maximum thrust performance except some low altitude application [10]. Therefore, the most widely used design method of the bell nozzle is still the TOC method developed by Rao [11, 12], called Rao’s method. Most of the nozzle contours of the bipropellant liquid rocket engines are designed by this method. Östlund and Bigert [13] and Semenov et al. [14] proposed some configurations based on slight modifications of classical bell-shaped nozzles, using Rao’s method indirectly. Although Rao’s method is quite simple, the assumption of inviscid isentropic flow makes it deviate from the real flow. Thus, some direct optimization methods based on computation fluid dynamics (CFD) were studied [15–18]. These methods take into account the viscosity and chemical kinetics in the real flow and show better performance in applications. However, these methods not only are much more complicated than Rao’s method but also have little increase on performance of the nozzle of large area ratio. For this reason, we restudied the nozzle design process, hoping that the engine performance can be significantly improved only by the nozzle design.In this study, taking the 750 N engine as an example, we made a simple and effective modification on Rao’s method. The expansion nozzle contour designed by this method has a slightly higher specific impulse and a smaller exit area ratio. We also proposed a new method to design the end section of the nozzle for the space rocket engine, which can effectively increase the vacuum specific impulse.


## 2. Approaches

### 2.1. Design Procedure

The design procedure is illustrated in Figure1. First, an initial nozzle contour is designed by using Rao’s method (where γ, the heat capacity ratio, is calculated by one-dimensional chemical equilibrium code, ODE). Then, its coordinates are inputted into a one-dimensional kinetic code (ODK) to calculate the nonequilibrium flow and the properties of the mixture gas. The specific heat capacity of the real gas is calculated and then substituted into the code of modified Rao’s method to redesign a new contour. Finally, a modified nozzle extension section is added to further improve the specific impulse. In the design procedure, the key steps are Rao’s method and its modification.Figure 1
Design procedure.


### 2.2. Equations of Method of Characteristic

Rao’s method is based on the method of characteristic (MOC), which is a typical method for calculating the supersonic flow and is discussed in detail by Arrington et al. [9]. MOC uses compatibility equations for the rotational axisymmetric flow:
(1)ρVdV+dp=0,(2)dp+c2dρ=0,(3)M2−1dpρV2±dθ+sinθyMcosθ±α=0.Compatibility Equations (1) and (2) are held along the streamline, with the characteristic equation:
(4)dydx=tanθ,while Equation (3) holds along the left- and right-running Mach lines given by
(5)dydx=tanθ±α.


### 2.3. Procedure of Rao’s Method

The nozzle design procedure based on Rao’s method is described in Reference [9]. As shown in Figure 2, the initial-value line is established by Kliegel’s method because it is in reasonable agreement with the measurements of Back et al.; then, the kernel region for the flow field is determined by solving the characteristic line originating from the initial-value line and the wall of throat downstream defined by downstream wall radius (Rd) and expansion half angle (β). After that, the last characteristic line, BD, is obtained. To determine the points on the expansion contour, the conservation of mass flow is used; that is, the mass flow through BD1 is equal to the mass flow through D1E1. When the discrete points D1, D2, D3, etc. are sequentially selected on BD, the points E1, E2, E3, etc. on the contour can be determined. Finally, according to the criteria of the exit point, the whole expansion contour can be obtained.Figure 2
Design procedure of Rao’s method: (a) determining the kernel region; (b) determining contour points.
(a)
(b)


### 2.4. Modification of Rao’s Method

In Rao’s method, the heat capacity ratio (γ) is constant, and the thermodynamic parameters (temperature, pressure, density, and sonic velocity) are all corresponding to the magnitude of velocity. Equations (6a), (6b), (6c), and (6d) give the order of solving these parameters:
(6a)T=Ts−V22cp,(6b)c=γRT,(6c)M=Vc,p=psTTsγ/γ−1,(6d)ρ=pRT.However, for the thermally perfect gas assumption, we solve the temperature through a state equation, and the solving order of parameters is modified as follows:(7)T=pρR,c=γRT,M=Vc,where the heat capacity ratio is calculated by
(8)γ=cpcp−R.In Equation (8), cp is a fitting function of temperature through calculations of ODK. When calling the above subroutine, pressure and density should be predetermined.


### 2.5. Numerical Simulation Method

We used Fluent, a commercial CFD software, to solve the two-dimensional axisymmetric Navier-Stokes equations to simulate the flow field of the nozzle. An implicit second-order Roe-FDS scheme was used to solve the steady flow equations. Turbulent flow was modelled by using the nonlinear renormalization group (RNG) model, and the enhanced wall function was introduced to treat the near-wall region flow.


## 2.1. Design Procedure

The design procedure is illustrated in Figure1. First, an initial nozzle contour is designed by using Rao’s method (where γ, the heat capacity ratio, is calculated by one-dimensional chemical equilibrium code, ODE). Then, its coordinates are inputted into a one-dimensional kinetic code (ODK) to calculate the nonequilibrium flow and the properties of the mixture gas. The specific heat capacity of the real gas is calculated and then substituted into the code of modified Rao’s method to redesign a new contour. Finally, a modified nozzle extension section is added to further improve the specific impulse. In the design procedure, the key steps are Rao’s method and its modification.Figure 1
Design procedure.


## 2.2. Equations of Method of Characteristic

Rao’s method is based on the method of characteristic (MOC), which is a typical method for calculating the supersonic flow and is discussed in detail by Arrington et al. [9]. MOC uses compatibility equations for the rotational axisymmetric flow:
(1)ρVdV+dp=0,(2)dp+c2dρ=0,(3)M2−1dpρV2±dθ+sinθyMcosθ±α=0.Compatibility Equations (1) and (2) are held along the streamline, with the characteristic equation:
(4)dydx=tanθ,while Equation (3) holds along the left- and right-running Mach lines given by
(5)dydx=tanθ±α.


## 2.3. Procedure of Rao’s Method

The nozzle design procedure based on Rao’s method is described in Reference [9]. As shown in Figure 2, the initial-value line is established by Kliegel’s method because it is in reasonable agreement with the measurements of Back et al.; then, the kernel region for the flow field is determined by solving the characteristic line originating from the initial-value line and the wall of throat downstream defined by downstream wall radius (Rd) and expansion half angle (β). After that, the last characteristic line, BD, is obtained. To determine the points on the expansion contour, the conservation of mass flow is used; that is, the mass flow through BD1 is equal to the mass flow through D1E1. When the discrete points D1, D2, D3, etc. are sequentially selected on BD, the points E1, E2, E3, etc. on the contour can be determined. Finally, according to the criteria of the exit point, the whole expansion contour can be obtained.Figure 2
Design procedure of Rao’s method: (a) determining the kernel region; (b) determining contour points.
(a)
(b)


## 2.4. Modification of Rao’s Method

In Rao’s method, the heat capacity ratio (γ) is constant, and the thermodynamic parameters (temperature, pressure, density, and sonic velocity) are all corresponding to the magnitude of velocity. Equations (6a), (6b), (6c), and (6d) give the order of solving these parameters:
(6a)T=Ts−V22cp,(6b)c=γRT,(6c)M=Vc,p=psTTsγ/γ−1,(6d)ρ=pRT.However, for the thermally perfect gas assumption, we solve the temperature through a state equation, and the solving order of parameters is modified as follows:(7)T=pρR,c=γRT,M=Vc,where the heat capacity ratio is calculated by
(8)γ=cpcp−R.In Equation (8), cp is a fitting function of temperature through calculations of ODK. When calling the above subroutine, pressure and density should be predetermined.


## 2.5. Numerical Simulation Method

We used Fluent, a commercial CFD software, to solve the two-dimensional axisymmetric Navier-Stokes equations to simulate the flow field of the nozzle. An implicit second-order Roe-FDS scheme was used to solve the steady flow equations. Turbulent flow was modelled by using the nonlinear renormalization group (RNG) model, and the enhanced wall function was introduced to treat the near-wall region flow.


## 3. Preparation for Design

In this paper, we used an apogee engine of 750 N thrust to describe the design process. The engine uses NTO/MMH combination, and the total mixture ratio is 1.65, which are identical with those of China’s 490 N engines and S400 series of Airbus Space systems. Figure3 shows the initial contour of the nozzle, which was designed by Rao’s method. The chamber pressure pc is 0.85 MPa, and the test vacuum specific impulse Isv is 320.4 s. The nondimensional upstream and downstream wall radii of the nozzle (Ru/Rt and Rd/Rt) are 1.633 and 0.816. The expansion half angle β is 36.5°.Figure 3
Structure of the nozzle with the area ratio of 210 : 1. It is an 80% bell-shape nozzle (compared with standard conical nozzle):Rt=12.25 mm, Ru/Rt=1.633, Rd/Rt=0.816, and β=36.5°.


### 3.1. Chemical Equilibrium and Kinetic Calculation

For nozzle design and its performance evaluation, chemical equilibrium and kinetic calculation are the key procedures. These calculations can provide all of the data needed in the design and evaluation process. We used an in-house code to calculate the nozzle’s theoretical performance and the properties of gas [19]. A C-H-N-O reaction mechanism was used in the calculation because of the use of NTO/MMH combination. The element reactions and the corresponding coefficients of the reaction rate can be found in References [20, 21].Figure4 shows the theoretical Isv at different exit area ratios under the chamber pressure of 0.85 MPa. For the nozzle with the exit area ratio of 210 : 1, its corresponding theoretical Isv is about 347.6 s, while the test value is 320.4 s; thus, the total efficiency is about 320.4/347.6=92.2%. We can see that in order to increase the specific impulse, the exit area ratio must be increased.Figure 4
Vacuum specific impulses at different area ratios. To increase theIsv (arearatio=210:1) by 3 s, the area ratio should be increased to 325 : 1 at least.Figure5 shows the variation of heat capacity ratio γ at different nozzle area ratios. γ obtained by using the chemical equilibrium assumption is quite different from that obtained by using the frozen or kinetic flow assumption and presents a smaller value and tends to be constant. The reason is that the composition of the gas changes instantaneously with the temperature under the equilibrium assumption. However, it is impossible in the real nozzle flow. The smaller the nozzle size is, the slower the change of the gas composition is than the flow, and the closer the gas property is to the frozen flow. Thus, the result of kinetics is closest to the real state. Because the calculation of chemical kinetics is related to the size of the nozzle, we used the initial nozzle contour to calculate the chemical reaction flow in the nozzle under the assumption that the nozzle contour has little change and fitted the relationship between the specific heat at constant pressure (cp) and temperature:
(9)cp=a0+a1T+a2T2+a3T3+a4T4.Figure 5
Comparison of the heat capacity ratio obtained with different assumptions.It is a fitting polynomial. ForT≥1000 K, a0=1188.27296, a1=0.74502, a2=‐2.15067E‐4, a3=2.61593E‐8, and a4=‐7.06741E‐13, and for T<1000 K, a0=1476.10931, a1=‐0.14838, a2=7.94836E‐4, a3=‐4.60444E‐7, and a4=8.11403E‐11. It should be noted that although the expression of this cp is identical with that of JANNAF database [21], the composition of the gas varies. In modification of Rao’s method, this expression is substituted into Equation (8) to solve γ.Viscosity of the gas was also treated as a function of temperature:(10)μ=5.65752×10−6+3.71243×10−8T−5.00581×10−12T2+4.3062×10−16T3.


### 3.2. Verification of Numerical Simulation

To verify the accuracy of the numerical simulation method and grid independence, we calculated the flow field of the initial nozzle with different numbers of the grid and then compared them with the test. Table1 lists the calculation results. We can see that the vacuum thrusts Fv calculated by different grids are almost equal. This means that these grids can meet the requirement of calculation. Thus, we selected the grid of 288×80 to simulate all the cases.Table 1
Calculation results with different grid numbers.
Grid number
Fv (N)
Fm (N)
Fp (N)
288×60
735.897
715.096
20.801
288×80
735.895
715.092
20.803
288×100
735.901
715.108
20.793
288×120
735.926
715.133
20.793By integrating the parameters at the nozzle exit, we can obtain axial momentum thrustFm=715.092 N and pressure thrust Fp=20.803 N. Thus, the total vacuum thrust Fv=Fm+Fv=735.895 N, and the vacuum specific impulse Isv=Fv/ṁ=735.895/0.235=3131.468N·s/kg=319.3s. This is very close to the average value of hot fire tests, 320.4 s, and the deviation is -0.34%. Therefore, the numerical simulation results can be used to evaluate the nozzle design.


## 3.1. Chemical Equilibrium and Kinetic Calculation

For nozzle design and its performance evaluation, chemical equilibrium and kinetic calculation are the key procedures. These calculations can provide all of the data needed in the design and evaluation process. We used an in-house code to calculate the nozzle’s theoretical performance and the properties of gas [19]. A C-H-N-O reaction mechanism was used in the calculation because of the use of NTO/MMH combination. The element reactions and the corresponding coefficients of the reaction rate can be found in References [20, 21].Figure4 shows the theoretical Isv at different exit area ratios under the chamber pressure of 0.85 MPa. For the nozzle with the exit area ratio of 210 : 1, its corresponding theoretical Isv is about 347.6 s, while the test value is 320.4 s; thus, the total efficiency is about 320.4/347.6=92.2%. We can see that in order to increase the specific impulse, the exit area ratio must be increased.Figure 4
Vacuum specific impulses at different area ratios. To increase theIsv (arearatio=210:1) by 3 s, the area ratio should be increased to 325 : 1 at least.Figure5 shows the variation of heat capacity ratio γ at different nozzle area ratios. γ obtained by using the chemical equilibrium assumption is quite different from that obtained by using the frozen or kinetic flow assumption and presents a smaller value and tends to be constant. The reason is that the composition of the gas changes instantaneously with the temperature under the equilibrium assumption. However, it is impossible in the real nozzle flow. The smaller the nozzle size is, the slower the change of the gas composition is than the flow, and the closer the gas property is to the frozen flow. Thus, the result of kinetics is closest to the real state. Because the calculation of chemical kinetics is related to the size of the nozzle, we used the initial nozzle contour to calculate the chemical reaction flow in the nozzle under the assumption that the nozzle contour has little change and fitted the relationship between the specific heat at constant pressure (cp) and temperature:
(9)cp=a0+a1T+a2T2+a3T3+a4T4.Figure 5
Comparison of the heat capacity ratio obtained with different assumptions.It is a fitting polynomial. ForT≥1000 K, a0=1188.27296, a1=0.74502, a2=‐2.15067E‐4, a3=2.61593E‐8, and a4=‐7.06741E‐13, and for T<1000 K, a0=1476.10931, a1=‐0.14838, a2=7.94836E‐4, a3=‐4.60444E‐7, and a4=8.11403E‐11. It should be noted that although the expression of this cp is identical with that of JANNAF database [21], the composition of the gas varies. In modification of Rao’s method, this expression is substituted into Equation (8) to solve γ.Viscosity of the gas was also treated as a function of temperature:(10)μ=5.65752×10−6+3.71243×10−8T−5.00581×10−12T2+4.3062×10−16T3.


## 3.2. Verification of Numerical Simulation

To verify the accuracy of the numerical simulation method and grid independence, we calculated the flow field of the initial nozzle with different numbers of the grid and then compared them with the test. Table1 lists the calculation results. We can see that the vacuum thrusts Fv calculated by different grids are almost equal. This means that these grids can meet the requirement of calculation. Thus, we selected the grid of 288×80 to simulate all the cases.Table 1
Calculation results with different grid numbers.
Grid number
Fv (N)
Fm (N)
Fp (N)
288×60
735.897
715.096
20.801
288×80
735.895
715.092
20.803
288×100
735.901
715.108
20.793
288×120
735.926
715.133
20.793By integrating the parameters at the nozzle exit, we can obtain axial momentum thrustFm=715.092 N and pressure thrust Fp=20.803 N. Thus, the total vacuum thrust Fv=Fm+Fv=735.895 N, and the vacuum specific impulse Isv=Fv/ṁ=735.895/0.235=3131.468N·s/kg=319.3s. This is very close to the average value of hot fire tests, 320.4 s, and the deviation is -0.34%. Therefore, the numerical simulation results can be used to evaluate the nozzle design.


## 4. Results and Discussion

As descripted in Figure4, higher performance needs a higher area ratio; the aim area ratio is 330 : 1. In our study, the length of the expansion section was limited to be 80% length of a 15-degree conical nozzle with the area ratio of 330 : 1, xe/Rt≈51.5.


### 4.1. Influence of Constantγ

First, we used various constantγ values in Rao’s method to design the expansion contours, which are shown in Figure 6. With the increase of γ value, the expansion half angle β and the exit area ratio of the contour become smaller.Figure 6
Expansion contours designed by various constantγ values.Table2 lists the calculation results of the nozzle flow filed with the contours in Figure 6. The contour designed with γ=1.2 shows the smallest Isv and Fv. For larger γ from 1.238 to 1.3, different contours show nearly the same Isv and Fv, and the one designed with γ=1.275 shows the maximum performance. With the increase of γ, the contour becomes thinner. The calculation results also show that Rao’s method with constant γ can only increase Isv by 1.78 s (321.08-319.3); this value is less than the theoretical prediction.Table 2
Performance of the nozzle with various contours.
γ
Isv (s)
Fv (N)
Fm (N)
Fp (N)
1.2
320.19
737.90
718.29
19.61
1.238
321.03
739.83
720.49
19.39
1.25
321.06
739.83
720.43
19.41
1.275
321.08
739.89
720.34
19.55
1.3
321.05
739.80
720.11
19.69
1.32
320.93
739.59
719.76
19.84


### 4.2. Effect ofγ

As described in Section3.1, heat capacity of the gas changes rapidly due to the change of temperature near the nozzle throat. Thus, using γ calculated by Equation (8) is more reasonable than using constant γ.Figure7 shows the comparison of the contours designed by Rao’s method with constant γ and modified Rao’s method with variable γ. The new contour designed with variable γ has a smaller expansion angle (β), larger exit angle (θe), and smaller exit area ratio as shown in Figure 7.Figure 7
Comparison of contours designed with constant and variableγ.Table3 lists the CFD results of these contours, where “Var.” represents variable γ and “Var.+dis.” represents the contour designed with variable γ and boundary layer displacement. We can see that Isv can increase one second more than the nozzle contour with the area ratio of 280 : 1 and 2.7 s more than the original contour with the area ratio of 210 : 1, by using modified Rao’s method.Table 3
Performance of the nozzle designed by different methods.
γ
Isv (s)
Fv (N)
Fm (N)
Fp (N)
1.275
321.08
739.89
720.34
19.55
Var.
322.00
742.07
723.57
18.50
Var.+dis.
322.09
742.28
724.08
18.20Figure8 compares the distribution of the Mach number in different nozzles designed by variable and constant γ. We can see that the Mach number in the nozzle designed by variable γ is more uniform although it is smaller than that of constant γ near the exit zone. Figure 9 compares the pressure and axial velocity at the nozzle exit. We can observe that the axial velocity at the exit is relatively uniform and almost the same, but the pressure at the exit of the nozzle by variable γ is more uniform, thus generating higher pressure thrust.Figure 8
Comparison of Mach numbers in different nozzles.Figure 9
Comparison of Mach numbers in different nozzles.
(a)
Pressure
(b)
Axial velocity


### 4.3. The Off-Design Performance

For space propulsion, rocket engines generally work on nominal condition. However, they work on off-design conditions sometimes, and the mass flow rates are greater than or less than the nominal value. Thus, we compared the performance of the nozzles designed by Rao’s method and modified Rao’s method. Table4 shows the calculation results. We can observe that Isv increases slightly with the increase of the mass flow rate. For all off-design conditions, the nozzle designed by modified Rao’s method presents higher performance, and the highest increase of Isv is on the design point.Table 4
Performance of the nozzle designed by different methods.
Mass flow
γ=1.275
Var.γ
Fv (N)/Isv (s)
Fv (N)/Isv (s)
-20%
590.98/320.55
592.22/321.22
-10%
665.73/320.97
667.14/321.65
Nominal
739.89/321.08
742.07/322.00
+10%
813.99/321.10
815.85/321.83
+20%
888.78/321.38
890.58/322.04


### 4.4. Extension of the Nozzle

In this section, we studied how to achieve an obvious increase ofIsv because the above effort can only increase Isv by 2.7 s. A longer nozzle is the only selection. However, since the expansion contour has a smaller expansion angle at the exit of the nozzle, the convex or straight extension curve can bring a smaller area ratio increment and lead to additional viscous losses. Therefore, the increase of Isv caused by the convex extension curve is not large. We proposed a new idea that a smooth curve with an increasing expansion angle will have a larger benefit. Figure 10 shows four different extension curves studied in this paper, marked as L1, L2, L3, and L4. L1 is straight, and the others curve outward. All these extension contours were connected smoothly with the 80% length nozzle designed by modified Rao’s method.Figure 10
Comparison of various extension curves.Table5 lists the CFD results of these contours. Compared with the maximum thrust contour with the area ratio of 256 : 1, the straight extension line (L1) shows an increase of Isv by only 0.42 s. Moreover, the other extension curves have an evident increase of Isv. Among all the curves, L3 has the highest Isv. By comparing each column data in Table 5, we can observe that the increase of thrust mainly depends on the increase of momentum thrust Fm and the variation of pressure thrust Fp, and viscous drag of the wall Dw changes little.Table 5
Performance of the nozzle with various extension contours.
Isv (s)
Fv (N)
Fm (N)
Fp (N)
Dw (N)
Var.+dis.
322.09
742.28
724.08
18.20
36.97
L1
322.42
743.03
724.69
18.34
37.65
L2
323.32
745.10
726.79
18.31
37.68
L3
323.71
746.02
727.85
18.18
37.70
L4
323.64
745.84
727.67
18.16
37.71This phenomenon can be explained by the Prandtl-Meyer flow downstream of pointE. The expansion wave (in the case of L2, L3, and L4) originating from the curve accelerates the main flow to increase Fm and Fr. If the expansion angle is too large, the axial thrust will decrease (L4). Figure 11 shows the distribution of axial momentum force (ρVx) for L1 and the differences between other extension curves and L1, marked as L2-L1, L3-L1, and L4-L1. It can be observed that the axial momentum force at the exit increases with the extension curve moving outward (L2-L1 and L3-L1) and then decreases in the core region (L4).Figure 11
Axial momentum force along the radius.


## 4.1. Influence of Constantγ

First, we used various constantγ values in Rao’s method to design the expansion contours, which are shown in Figure 6. With the increase of γ value, the expansion half angle β and the exit area ratio of the contour become smaller.Figure 6
Expansion contours designed by various constantγ values.Table2 lists the calculation results of the nozzle flow filed with the contours in Figure 6. The contour designed with γ=1.2 shows the smallest Isv and Fv. For larger γ from 1.238 to 1.3, different contours show nearly the same Isv and Fv, and the one designed with γ=1.275 shows the maximum performance. With the increase of γ, the contour becomes thinner. The calculation results also show that Rao’s method with constant γ can only increase Isv by 1.78 s (321.08-319.3); this value is less than the theoretical prediction.Table 2
Performance of the nozzle with various contours.
γ
Isv (s)
Fv (N)
Fm (N)
Fp (N)
1.2
320.19
737.90
718.29
19.61
1.238
321.03
739.83
720.49
19.39
1.25
321.06
739.83
720.43
19.41
1.275
321.08
739.89
720.34
19.55
1.3
321.05
739.80
720.11
19.69
1.32
320.93
739.59
719.76
19.84


## 4.2. Effect ofγ

As described in Section3.1, heat capacity of the gas changes rapidly due to the change of temperature near the nozzle throat. Thus, using γ calculated by Equation (8) is more reasonable than using constant γ.Figure7 shows the comparison of the contours designed by Rao’s method with constant γ and modified Rao’s method with variable γ. The new contour designed with variable γ has a smaller expansion angle (β), larger exit angle (θe), and smaller exit area ratio as shown in Figure 7.Figure 7
Comparison of contours designed with constant and variableγ.Table3 lists the CFD results of these contours, where “Var.” represents variable γ and “Var.+dis.” represents the contour designed with variable γ and boundary layer displacement. We can see that Isv can increase one second more than the nozzle contour with the area ratio of 280 : 1 and 2.7 s more than the original contour with the area ratio of 210 : 1, by using modified Rao’s method.Table 3
Performance of the nozzle designed by different methods.
γ
Isv (s)
Fv (N)
Fm (N)
Fp (N)
1.275
321.08
739.89
720.34
19.55
Var.
322.00
742.07
723.57
18.50
Var.+dis.
322.09
742.28
724.08
18.20Figure8 compares the distribution of the Mach number in different nozzles designed by variable and constant γ. We can see that the Mach number in the nozzle designed by variable γ is more uniform although it is smaller than that of constant γ near the exit zone. Figure 9 compares the pressure and axial velocity at the nozzle exit. We can observe that the axial velocity at the exit is relatively uniform and almost the same, but the pressure at the exit of the nozzle by variable γ is more uniform, thus generating higher pressure thrust.Figure 8
Comparison of Mach numbers in different nozzles.Figure 9
Comparison of Mach numbers in different nozzles.
(a)
Pressure
(b)
Axial velocity


## 4.3. The Off-Design Performance

For space propulsion, rocket engines generally work on nominal condition. However, they work on off-design conditions sometimes, and the mass flow rates are greater than or less than the nominal value. Thus, we compared the performance of the nozzles designed by Rao’s method and modified Rao’s method. Table4 shows the calculation results. We can observe that Isv increases slightly with the increase of the mass flow rate. For all off-design conditions, the nozzle designed by modified Rao’s method presents higher performance, and the highest increase of Isv is on the design point.Table 4
Performance of the nozzle designed by different methods.
Mass flow
γ=1.275
Var.γ
Fv (N)/Isv (s)
Fv (N)/Isv (s)
-20%
590.98/320.55
592.22/321.22
-10%
665.73/320.97
667.14/321.65
Nominal
739.89/321.08
742.07/322.00
+10%
813.99/321.10
815.85/321.83
+20%
888.78/321.38
890.58/322.04


## 4.4. Extension of the Nozzle

In this section, we studied how to achieve an obvious increase ofIsv because the above effort can only increase Isv by 2.7 s. A longer nozzle is the only selection. However, since the expansion contour has a smaller expansion angle at the exit of the nozzle, the convex or straight extension curve can bring a smaller area ratio increment and lead to additional viscous losses. Therefore, the increase of Isv caused by the convex extension curve is not large. We proposed a new idea that a smooth curve with an increasing expansion angle will have a larger benefit. Figure 10 shows four different extension curves studied in this paper, marked as L1, L2, L3, and L4. L1 is straight, and the others curve outward. All these extension contours were connected smoothly with the 80% length nozzle designed by modified Rao’s method.Figure 10
Comparison of various extension curves.Table5 lists the CFD results of these contours. Compared with the maximum thrust contour with the area ratio of 256 : 1, the straight extension line (L1) shows an increase of Isv by only 0.42 s. Moreover, the other extension curves have an evident increase of Isv. Among all the curves, L3 has the highest Isv. By comparing each column data in Table 5, we can observe that the increase of thrust mainly depends on the increase of momentum thrust Fm and the variation of pressure thrust Fp, and viscous drag of the wall Dw changes little.Table 5
Performance of the nozzle with various extension contours.
Isv (s)
Fv (N)
Fm (N)
Fp (N)
Dw (N)
Var.+dis.
322.09
742.28
724.08
18.20
36.97
L1
322.42
743.03
724.69
18.34
37.65
L2
323.32
745.10
726.79
18.31
37.68
L3
323.71
746.02
727.85
18.18
37.70
L4
323.64
745.84
727.67
18.16
37.71This phenomenon can be explained by the Prandtl-Meyer flow downstream of pointE. The expansion wave (in the case of L2, L3, and L4) originating from the curve accelerates the main flow to increase Fm and Fr. If the expansion angle is too large, the axial thrust will decrease (L4). Figure 11 shows the distribution of axial momentum force (ρVx) for L1 and the differences between other extension curves and L1, marked as L2-L1, L3-L1, and L4-L1. It can be observed that the axial momentum force at the exit increases with the extension curve moving outward (L2-L1 and L3-L1) and then decreases in the core region (L4).Figure 11
Axial momentum force along the radius.


## 5. Conclusions

This paper made a modification of Rao’s method and proposed a new idea on design of the nozzle extension. Through a concrete design process of the nozzle of an apogee engine, the methods were verified to be more effective than the traditional method. Based on the calculation results, we draw some conclusions.The heat capacity ratio used in Rao’s method plays a great role in design. For a given length of nozzle, the nozzle contour designed with a larger specific heat ratio has a significantly smaller area ratio, and the nozzle performance is almost identical, which is very beneficial to reduce the nozzle weight.The expansion contour designed by using the variable heat capacity ratio or thermally perfect gas assumption has a much thinner expansion section than that designed by using the constant capacity ratio or caloric perfect gas assumption. Compared with the original design, the exit area ratio of the nozzle designed by this method not only reduces significantly from 330 : 1 to 256 : 1 but also has an increase of the vacuum specific impulse by about 1 second at the design point. The calculation results of off-design performance also show the effectiveness of the method. Furthermore, this method is simple and easy to implement.For the space rocket engine, we proposed an innovative nozzle extension section, which is an extension curve bent outward with the increasing expansion angle. Mounting such a section at the end of the nozzle will significantly increase the vacuum specific impulse, resulting in an additional gain of 1.5 s or even higher.



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*Source: 4926413-2019-12-20.xml*