Industrial Applications of Genetic Algorithms:Optimization of a Porous Liner for Transpiration Cooling Using a Genetic Algorithm

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Equation (5.2) contains the parameters that were described above as being important variables for the GA to consider. These parameters are the diameter of the capillaries, D, and the thickness of the capillaries, L. Thus, the GA will need to manipulate the thickness of the material as well as the diameter of the capillaries to develop an optimum solution.

The pressure drop across the porous liner is to be minimized. The pressure drop must satisfy two requirements. The first condition is based on a general engine design assumption of a maximum pressure drop of 20% of the chamber pressure. The second condition is a minimum pressure drop of 5% of the chamber pressure to allow for pressure fluctuations in the engine nozzle.

The injection velocity of the coolant is extremely important with respect to the hot-gas flow. If the coolant is injected at too high a velocity, it will create disturbances in the hot-gas flow field, causing decreased performance and increased structural stress. Thus, the injection velocity, calculated by Equation (5.3), must be low enough to avoid such a circumstance.

Equation (5.3) shows the inverse relationship between the injection velocity and the open area. Thus, the porosity and capillary diameter variables will be optimization factors with respect to the injection velocity.

In summary, the objective of the GA is to select values of channel height and width, porosity, and capillary diameter such that the coolant system effectively cools the thrust-chamber with a minimum pressure drop and low velocity through the porous wall.

PRELIMINARY CALCULATIONS

ODE Computer Code

A beneficial tool in the calculation of properties used in this project is a Lewis Research Center heat transfer program provided by MSFC entitled ODE [7]. This code is used to generate the hot-gas properties along the thrust-chamber nozzle assembly at desired locations. Using the engine’s mixture ratio, inlet pressure, and the desired locations along the assembly, expressed as area ratios, the program provides the properties necessary for calculation of the required coolant flowrates. However, care must be taken to run the ODE program with subsonic area ratios from the chamber to the throat and supersonic area ratios from the throat to the exit.

Mass Flow Calculations

In order to obtain the coolant flowrates for the coolant system design, the hot-gas flow must first be examined. Here, formulae are given to determine the coolant flowrate required to cool the test engine to a wall-gas temperature of 1000 °F.

The given information for this procedure is the shape coordinates of the liner. Also included are the required operating conditions for chamber pressure and the desired wall gas temperature. The equation to determine the mass flow rate is as follows:

Using the shape coordinates given by the liner design, the surface area to be cooled can be estimated using a method of conical areas. Knowing the equation for the surface area of a cone, an approximate value can be determined for the total surface area of the liner design.

The values for hgx are given by Equation (5.5). The value for Cpc can be located from a table of hydrogen properties. Care must be taken to get a Cpc at a pressure above chamber pressure so that the hydrogen flows in one direction.

where

The temperature requirements for the liner are both calculated and determined from desired information. The wall gas temperature desired is entered into the program. The coolant temperature for the hydrogen can be entered from design values for the coolant system. The next thing to calculate is the adiabatic wall temperature.

In order to calculate Taw for the liner, the engine requirements are used in the ODE code. This code will take the mixture ratio, required fuels, and necessary area ratios, and use these values to determine various flow information at each specified point in the liner’s shape. The object of this is to determine the recovery factor at each increment in the liner so that Taw can be found using the following equation:

In this equation, rc is the recovery factor calculated as follows:

This will provide the needed values to calculate rc. Once the rc is determined, Taw can be calculated at each increment. These calculations complete the necessary values needed to determine the mass flow rate at each increment. These mass flow rates can be added at each point, assuming no losses, and a total mass flow rate can be determined.

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