Sections 5.1 and 5.2 define the geometry of the SSME combustion chamber and nozzle. The analysis in Section 4.5 provides the temperature, pressure, and composition of the gas inside the chamber. Now we will use this information to predict what happens to the combustion products as they depart the LTMCC and accelerate toward the nozzle exit plane. At the conclusion of this section we will have a model that predicts flow properties as a function axial position through the length of the nozzle, taking in to account the effects of heat transfer and friction. This information will be used to verify the estimate of specific impulse from Section 3.4 and set the conditions for an analysis of the regenerative cooling circuit in Section 6.
We could begin by using the isentropic relations from any standard compressible flow textbook and develop expressions for how flow properties vary as a function of cross-sectional area. In fact, that is exactly how we determined the dimensions of the nozzle throat and exit in Section 5.1. Unfortunately this approach neglects to account for the effects of friction or heat transfer. We could obtain a solution that accounts for friction by using a Fanno flow analysis. A basic assumption of Fanno flow is that the cross-sectional area of the duct in question remains constant. That is clearly not true in the case in the SSME nozzle. A Rayleigh flow analysis could help account for heat transfer, but this model is also based on the assumption of constant cross-sectional area. To complicate matters further, both the Rayleigh and Fanno solutions are independent of each other. Fanno flow assumes there is no heat transfer, and Rayleigh flow assumes no friction. [28]
To simultaneously account for each of these effects we can establish a “quasi one-dimensional” model of steady flow through a variable area duct. Flow properties are assumed to be constant across each cross-section of the duct. Only velocities normal to the duct cross section are considered (i.e. there are no velocity gradients perpendicular to the flow axis). Heat transfer is assumed to occur instantaneously and uniformly through each cross-section. Making these assumptions allows us to model all the flow variables as a function of distance travelled through the duct, $x$. Combining these assumptions with the ideal gas equation of state and the conservation equations for mass, momentum, and energy enables us to reduce the problem to a system of differential equations: [62]
Consistent with previous nomenclature, $P$ and $T$ represent pressure and temperature. $\gamma$ is the specific heat ratio and $R_s$ is the specific gas constant. These equations introduce several new variables as well. $A$ represents the cross-sectional area of the nozzle. $N$ is defined as the square of the local Mach number:
Figure 5.3.1
Fluid enters this cross section with velocity $V$ and mass flow rate $\dot m$. As it moves through the cross section it transfers heat into the nozzle walls. This heat flux, $q$", is shown in red. Heat flux has units of watts per square meter. Therefore if we multiply the heat flux by the surface area it is transmitted through we will obtain the number of watts transferred. We also know the nozzle is cylindrical with radius $r(x)$. Therefore we can calculate the relevant surface area by multiplying the cross-section’s circumference by the width $\Delta x$. This leads to the following expression for $Q(x)$
Expressions for $r(x)$ are listed in Table 5.1.2 and Equation 5.2.2. The only remaining unknown is an expression for how heat flux, $q$", varies as a function of $x$. We will leave this question unsolved the moment, but will return to it shortly.
Shifting our focus back to the governing set of differential equations, we still have one additional variable to introduce: $F$. This variable represents the amount of power a theoretical pump would be required to provide in order to overcome the pressure losses associated with friction (again on a per mass basis). The amount of pressure lost as fluid travels through a pipe is also known as head loss, $h_L$. It is described by the following equation:[3]
$D$ is the cross-section diameter, $L$ is the length of the duct, and g is the acceleration of gravity. The amount of power required to overcome this pressure drop can be determined by multiplying $h_L$ by the mass flow rate through nozzle and the acceleration of gravity:
To arrive at the variable $F$ required for use in Equations 5.3.1 -5.3.3 this power requirement must be divided by the mass flow rate through the nozzle. The $\dot m$ and $g$ terms cancel out and we reach the following conclusion:
The only unfamiliar variable in this equation is the Darcy Friction factor, $f$. It is a dimensionless quantity that can be thought of as the ratio of viscous forces to inertial forces acting on the interior surface of the pipe. It is a function of the surface roughness of the pipe, and Reynolds number, $R_e$, of the flow. It can be determined by reading the Moody chart (Figure 5.3.2) or calculated by solving the Colebrook equation with an iterative root finding scheme:[63]
Figure 5.3.2[64]
Now we find ourselves in an interesting situation. One goal of of this section was to solve for the velocity, density, pressure, and temperature profiles through the nozzle. But in order to do this we need to solve for the friction factor, which is a function of Reynolds number. As shown above in Figure 5.3.2, Reynolds number, $R_e$, is itself a function of these flow variables. We are in a situation where we can’t solve for friction factor without knowing velocity, and we can’t solve for velocity without knowing the friction factor. To make matters worse, we still don’t have an expression for heat flux through the combustion chamber walls!
This brings us to the crux of the issue. Modeling the effects of entropy in a rocket combustion chamber requires a coupled analysis between a flow model and a thermal model. The flow model model predicts the properties of the combustion gas, while the thermal model analyzes the amount of heat transferred from the combustion gas to the nozzle walls and into the regenerative cooling system. Both the flow model and the thermal model must be solved simultaneously.[65]
In order to move forward we will assume that $Q(x) = F(x) = 0$. This eliminates many terms in Equations 5.3.1 - 5.3.3. By doing this we are left with only terms dependent on $A(x)$, which is known from Sections 5.1 and 5.2. But this is the isentropic solution described at the start of this section! This isentropic solution will be used to initiate the first first iteration of the coupled solution scheme. By setting $Q(x)$ and $F(x)$ equal to zero we obtain enough data to solve Equations 5.3.1-5.3.3 with a fourth-order Runge-Kutta algorithm.[15] We then take the resulting velocity, temperature, and pressure profiles and plug them in to the thermal analysis model described in Section 6. This analysis returns the heat flux, $q"(x)$, along the length of the combustion chamber. We can then return to Equations 5.3.1-5.3.3 with known velocity and flux profiles. This allows us to run the Runge-Kutta algorithm again and re-solve Equations 5.3.1-5.3.3, but this time including the $Q(x)$ and $F(x)$ terms. The resulting flow profile can then be plugged back in to the thermal model. This process is iterated until the system converges.
Results from the initial isentropic solution are presented below. The data call-outs on the chart show properties at the injector face, throat, and nozzle exit plane.
| Calculated Value | Actual Value | Relative Error | |
|---|---|---|---|
| Exit Mach Number | 4.577 | 4.540[66] | 0.8% |
| Exit Pressure $[Pa]$ | 20,250 | - | - |
| Exit Temperature$[K]$ | 1176 | - | - |
| Thrust $[lbf]$ | 490,200 | 491,900[2] | 0.34% |
| Specific Impulse $[sec]$ | 450.3 | 452[2] | 0.37% |
Table 5.3.1
Figure 5.3.3
Figure 5.3.4
Figure 5.3.5
It is worth mentioning that the injector face temperatures and pressures shown in Figures 5.3.4 and 5.3.5 differ slightly from those listed in Table 4.5.4. The properties listed in Table 4.5.4 are stagnation values. The values used to initiate the flow model are static values which are obtained from the stagnation values by using the isentropic flow relations and Mach number at the injector face. The injector face Mach number is adjusted until the throat Mach number becomes extremely close to one. If the injector face Mach number is too small, supersonic exit velocities are never achieved. If the injector face much number is too large, shock waves will develop in the nozzle which produce discontinuities in the resulting flow field. The throat Mach number cannot be exactly one, or else the solution routine will fail completely due to the $1/(N-1)$ terms in Equations 5.3.1 - 5.3.3. The figures below show how even a small deviation from the required injector face Mach number of 0.22602 will cause the algorithm to fail.
Figure 5.3.6: Injector Face Mach Number =0.225
Figure 5.3.7: Injector Face Mach Number =0.22602
Figure 5.3.8: Injector Face Mach Number =0.227