Computational Framework
The previous sections of this website provide mathematical details on how to model compressible flow and heat transfer in the SSME. For the most part, these topics were treated individually in their respective sections. The goal of this section is to examine the problem from a broader viewpoint and show how these models couple to form a comprehensive solution. The flow chart below describes the process.
Figure 6.3.1: Computational Flow Diagram
The combustion model developed in Section 3 is used to define the fluid properties at the injector face. The model established in Section 5.3 is then used to predict combustion gas properties as it flows down the LTMCC axis. For the initial iteration, the effects of heat transfer and friction are neglected. After the combustion gas properties through the LTMCC are known, the LTMCC heat transfer model (Section 6.1) is used to determine wall temperatures and heat flux. A second iteration of the flow model is then initiated, using the velocity and heat flux profiles developed in the first iteration as inputs. This allows the flow model to account for stagnation pressure drop due to viscosity and stagnation temperature drop due to heat transfer through the chamber walls. This process was looped until convergence was obtained. There was little appreciable change beyond the second iteration of this procedure, as shown below on Figure 6.3.2. This behavior is similar to that reported in Reference [69].
Figure 6.3.2: Global convergence of LTMCC heat flux. First iteration in blue, second iteration in orange.
Local convergence at each wall segment $dx$ in the heat transfer models took quite a bit more computational effort. Both the LTMCC and the nozzle were discretized into 200 segments, with the heat transfer calculations iterated over each segment until the residuals of Equations 6.1.22 - 6.1.23, and Equations 6.2.6 -6.2.7 were less than 1 $W$. The minimum amount of heat transferred into any segment $dx$ in both the LTMCC and nozzle proper was 297 $W$. This corresponds to a maximum local error of 0.3%. The figure below plots the the number of local iterations required for convergence as a function of station number, $dx$. LTMCC stations run from 1-201, and nozzle stations run from 202-402.
Figure 6.3.3: Iterations for Local Convergence
The local iteration count appears to be sensitive to changes in geometry. The spikes in the LTMCC section align with changes in coolant passage dimensions (see Figure 6.1.2), and the large discontinuity in the nozzle proper corresponds to the re-designed coolant tubes (see Figure 6.2.13).
Global Results
The following plots display global results from the both the compressible flow and heat transfer models with the effects of entropy included.
Figure 6.3.4: Global Wall Temperature Profile
Figure 6.3.5: Global Heat Flux Profile
Figure 6.3.6: Global Static Pressure Profile
Figure 6.3.7: Global Mach Number Profile
Figure 6.3.8: Global Stagnation Temperature Profile
Figure 6.3.9: Global Static Temperature Profile
Properties at the nozzle exit plane are tabulated below for both the isentropic and non-isentropic solutions. As expected, nozzle performance degrades when accounting for the effects of friction and heat transfer. Both of these models tend to under predict thrust and specific impulse. This occurs because both models must be provided an injector face Mach number to initiate the calculation. In turn, this Mach number sets the mass flow rate through the system. Referring back to this discussion in Section 5.1, the mass flow rate through the engine is very sensitive to chamber stagnation pressure. If the reported stagnation pressure of 2871 psi is used, the Mach number required to initiate the compressible flow model corresponds to a mass flow rate of 520 kg/sec. This is approximately 6 percent too large, and would cause a corresponding 6 percent over-prediction in the amount of thrust generated. To impose the correct mass flow rate through the nozzle, chamber stagnation pressure was reduced to from 2871 psi to 2725 psi (isentropic case) and 2740 psi (with entropy). This allows the model to predict the correct mass flow rate, but causes the model to under-predict thrust and specific impulse. This adjustment accounts for the fact that the thermodynamic properties of the combustion gas flowing through the nozzle are functions of temperature and pressure. As shown above, these values vary widely between the injector face and nozzle exit plane. As these values change, so does the rate at which the chemical species in the exhaust gas dissociate and re-combine with each other. This means that due to these ongoing chemical reactions, the molecular weight and specific heat ratio of the combustion gas changes as a function of axial position within the nozzle. The model used on this website does not account for equilibrium effects like this. A “frozen flow” model is used instead that assumes exhaust composition is fixed at the injector face. Accounting for equilibrium effects would likely allow the model to run with both a chamber stagnation pressure 2871 psi and a mass flow rate of 492 kg/sec. Nevertheless, the results presented here are acceptable for a conservative preliminary design.
| Isentropic Model | Non-Isentropic Model | Actual Value | |
|---|---|---|---|
| Mass Flow Rate $kg/sec$ | 493 | 493 | 492[2] |
| Chamber Pressure, $psi$ | 2725 | 2740 | 2871[2] |
| Exit Mach Number | 4.577 | 4.534 | 4.540[66] |
| Exit Pressure, $Pa$ | 20,250 | 20,044 | - |
| Exit Temperature, $K$ | 1176 | 1172 | - |
| Thrust, $lbf$ | 490,200 | 484,444 | 491,900[2] |
| Specific Impulse $[sec]$ | 450.3 | 445.5 | 452[2] |
Table 6.3.1