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Rocket Nozzle Plume Heating

This page summarizes an innovative application of C&R's Thermal Desktop for modeling the plume heating of a rocket nozzle. The nozzle is convectively heated by the plume using a heat transfer coefficient that varies along the length and around the circumference of the nozzle. For this example, a portion of the nozzle radiates to a 5°K sink temperature.

Background: Rocket Plume Heat Transfer

C&R Thermal Desktop® and SINDA are commonly used for thermal analyses of spacecraft and propulsion systems. Less frequently, these tools are used for calculating the temperatures in supersonic exhaust nozzles, such as those in rockets or thrusters.

The temperature of the nozzle wall is an important aspect of rocket design. The exhaust-gas temperature typically exceeds the maximum allowable temperature of the nozzle wall material. The ability to estimate the wall temperature allows the design of a cooling system.

Four types of cooling systems can be modeled in Thermal Desktop: heat sink; thermal radiation; even regenerative (using FloCAD®). A difficult part of modeling the cooling system is approximating the heat transfer from the plume to the nozzle wall. The convective film coefficient can be estimated through a number of methods (Bartz equation, TDK boundary layer technique, etc); the coefficient is highly dependent on the axial location within the nozzle.

This example shows one method of applying an axial variation to the heat transfer coefficient within a nozzle by using the Bartz equation to calculate the heat transfer coefficient. The Bartz equation is [1]:

where the subscript 0 represents stagnation conditions and

Pr = the Prandtl number,
, the characteristic velocity,
D* = throat diameter,
rc = throat radius of curvature in a plane which contains the nozzle axis,

and,

where M is local Mach number and w is the exponent of the viscosity-temperature relation maTw. It should be noted that the first term in the Bartz equation (in brackets) is based on stagnation properties and geometry and is, therefore, invariant and requires calculation only once. The heat transfer coefficient, hg, is used in combination with the plume adiabatic wall temperature and the hot wall temperature to obtain the heat transfer rate. Since rocket plumes can have boundary-layer recovery factors greater than 0.9, using the stagnation temperature in place of the adiabatic wall temperature will produce slightly conservative results. An angular variation can be applied to model asymmetrical flow within the nozzle due to injector non-uniformities, also known as “streaking”.

Problem Statement

A parabolic rocket nozzle has the following dimensions:

  • Length: 195.6 cm
  • Maximum diameter: 116.8 cm
  • Expansion ratio: 55:1

The wall is made of niobium alloy C-103 with a silicide coating that gives the nozzle an emissivity of 0.7. The first quarter of the nozzle has a fixed temperature, representing regenerative cooling, and the last three-quarters radiate to a 5°K sink temperature. The oxidizer is NTO (nitrogen tetroxide) and the fuel is MMH (monomethyl-hydrazine) with a mixture ratio of 1.65:1 and a chamber pressure of 8.62 atmospheres.

Thermal Desktop Model

The nozzle, shown in Figure 1, was modeled using a single paraboloid. The top three quarters of the paraboloid, the radiating surface, was modeled with diffusion nodes, which are time-dependent calculation points. The lower quarter, the boundary surface, was modeled with boundary nodes set to 750 K. The emissivity was assumed to be that of silicide, approximately 0.7. For boundary surface, shown in Figure 1, the nodes of that portion of the nozzle were changed to boundary nodes with a fixed temperature of 750 K.  A solid cylinder was used to represent the plume reference temperature for visual clarity; this could have been any surface or solid. The single node for the solid cylinder was over-ridden to be a boundary node set to the stagnation temperature of the plume. For this model the Bartz equation was used, and the stagnation temperature of the exhaust was used as the reference temperature, as discussed above.

Figure 1 - Thermal Desktop® Model

Symbols were used to define the geometry, provide self-documentation and allow for correction or safety factors. The Symbol Manager window is shown in Figure 2.

Figure 2: Symbol Manager Window

The key to modeling the nozzle heat transfer in Thermal Desktop is a feature called “contactors”. Contactors are a means of thermally linking high-order thermal objects, such as surfaces and solids. Contactors can be made between arbitrary objects, so alignment of the node or surfaces is not necessary[2]. The contactor was created from the nozzle to the plume object: the order of a contactor is significant since the geometry of the From object is used for the area of the contactor and the dimensions for scaling. The contactor edit window and some sub-windows are shown in Figure 3.

Scaling can be applied to the contactor for each dimension of the surface, angular and height for the paraboloid. To use scaling in a contactor, the user opens a Tabular Input window by clicking on the appropriate button in the Conduction Coefficient Scaling region of the Contactor Edit form (Figure 3). The scaling factor versus the proportional dimension can be typed or pasted into the Tabular Input window.

The scaling factor table, lower right of Figure 3, was obtained by using CEA[3] to calculate the exhaust properties and local Mach numbers within the nozzle and then using a spreadsheet set up to calculate the heat transfer coefficient as a function of nozzle length using the Bartz equation. The expression used in the Conduction Coefficient field (upper right of Figure 3) allows the chamber pressure, possibly calculated elsewhere in the model, to vary during the solution without significant loss of fidelity. For this purpose the expression must be output to SINDA/FLUINT. The correction factor, Correct, can be used as a safety factor or to correlate the heat transfer coefficient to test data.

If asymmetric heating is present around the circumference of the nozzle, the angular scaling factor can be applied. For this example, a simple sinusoidal function has been applied between 150 degrees and 210 degrees. This procedure is the same as that described above for the height scaling factor.

Figure 3 - Contactor Edit Window with Expression Editor and Tabular
Input for Coefficient Scaling by Height

Results and Discussion

The steady-state solution is presented in Figure 4. Note the increased temperatures on the front right side of the nozzle. The radiating section of the nozzle shows varying wall temperatures as a result of the changing heat transfer coefficient.

When compared to the actual system, the convective heat fluxes for the radiating portion of the nozzle are underestimated by about 20%. These are reasonable results based on simplification of the geometry (no structural reinforcements were included) and assumptions made within CEA. The correction factor mentioned above could be adjusted to remove this error or provide a safety factor: a key benefit of model parameterization.

Figure 4 - Steady-State Results of Plume Convection in a Radiating Nozzle
Using the Bartz Equation with Streaking

Further expansion of this model could be:

  • adding regenerative cooling in place of the fixed-temperature condition
  • adding surfaces representing the throat (a torus, perhaps) and the combustion chamber [4]
  • adding ablative properties to the inner wall of the nozzle or throat
  • adding a second, concentric surface around the nozzle and mapping solid elements between the surfaces to form a heat sink nozzle
  • mapping the results to a NASTRAN or ANSYS structural FE model

Recognition

Special recognition must go to Dave Mohr of D&E Propulsion for his assistance in developing this concept and example and for his review. Also, XiaoYen Wang, of NASA Glenn Research Center, should be recognized for asking the questions that led to developing this example.

1 - This definition and description of the Bartz equation was obtained from Mechanics and Thermodynamics of Propulsion by P. G. Hill and C. R. Peterson, 1970. The original reference is “A Simple Equation for Rapid Estimation of Rocket Nozzle Convective Heat Transfer Coefficients,” D. R. Bartz, Jet Propulsion, 37, 1, January 1957, pages 49-51.

2 - For more information on creating contactors see the Thermal Desktop User’s Manual.

3 - CEA – Chemical Equilibrium with Applications, a computer application written at NASA Glenn Research Center by S. Gordon and B. McBride.

4 - Each surface in series along the length of the nozzle requires its own contactor with its own scaling factor, since the scaling factor is based on the proportional length of a single surface not a group of surfaces.

 





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