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

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RESULTS

Table (5.1) below shows a simple tabulated set of general results for the GA. These results include the weighting factors for each fitness parameter and the number of bits used for each parameter to determine discretization. Note that the weighting factor for the injection velocity is much higher than that of the pressure drop. This is due to the fact that, although the pressure drop is the more important of the two parameters, the injection velocity must maintain some adequate ratio in relation to the pressure drop so that the velocity can make some contribution to the overall GA results.

First, an analysis of the fitness values of the GA leads to the development of Figures 5.2 and 5.3 below. These charts graph a typical fitness variation as a function of the two independent fitness parameters in the system. Figure 5.2 indicates the variation of average fitness values with increasing generations. Figure 5.3 indicates the change in the summation of fitness values from generation to generation. Both of these charts show an initial sharp increase in fitness values followed by a gradual leveling out of the graph as the number of generations increases.

Table 5.1 Genetic algorithm parameter characteristic
Weighting Factors
| Pressure Drop | W=0.1
| Injection Velocity | W=3.0
Parameter Bit Lengths
| Capillary Diameter | 10 bits
| Porosity | 7 bits
| Thickness | 9 bits
| Channel Height | 8 bits
| Channel Width | 8 bits
Other GA Parameters
| Generations | 50
| Population Size | 50

Figure 5.2 Average Fitness Variation.

With respect to the liner thickness, one may notice a general decrease in thickness from the throat to the exit (Figure 5.4). This is primarily due to the decrease in temperature and pressure requirements toward the exit of the nozzle. Some slight errors occur near the throat possibly due to large hot-gas temperature fluctuations in this region.

Figure 5.3 Sum of Fitness Variation.

Figure 5.4 Two-Dimensional Representation of Liner Thickness.

Analyzing Figure 5.5 below, the porosity is generally higher at the throat due to increased hot-gas temperatures in this region. At a point, shown in Figure 5.5 at about X=35 inches, the porosity drops to a value of approximately one. This severe drop is determined to be due to the large decrease in temperature of the hot-gas flow near the nozzle exit. The fluctuations in the region of the graph (Figure 5.5) near the throat are most likely due to the close relationships between the porosity and the capillary diameter.

Figure 5.5 Porosity Variation Along the Nozzle X-Axis.

Figure 5.6 Capillary Diameter Variation Along the Nozzle X-Axis.

The next graph, Figure 5.6, of the capillary diameter versus X-coordinate along the nozzle axis shows a similar fluctuation of values near the throat. This further solidifies the hypothesis that the porosity and capillary diameter are closely related. Looking further at Figure 5.6, as the X-coordinate increases, the capillary diameters steady out at about 31 micrometers. Once again, this can be attributed to the reduction in temperature and thus the reduction in coolant requirements towards the nozzle exit. The capillary diameter and porosity level out, leaving the optimization to occur with variations in the liner thickness.

CONCLUSIONS

Figures 5.2 and 5.3 show that the GA has achieved the goals set forth. The GA met all of the requirements and maintained a low injection velocity across the entire nozzle. Thus, success has been achieved in developing an effective liner.

Figures 5.4, 5.5, and 5.6 reveal information about the nozzle and its coolant requirements with respect to position along the axis. Generally, near the throat, the porosity and capillary diameter are closely related and most important in the optimization of that region. However, as the liner moves towards the nozzle exit, the porosity and capillary diameter level out, leaving optimization as a function of the liner thickness.

The greatest question to be considered in this effort is, “How good a solution is it?” Analytically, the solution presented here meets all of the requirements of the design team for this NASA program. The true test of the design’s efficiency will lie in manufacturing and testing of a nozzle. This test liner would serve to dispel or verify the analytical solution presented here and/or the theory of transpiration cooling as developed in the 1950s and 1960s. Therefore, the next step is development of a test article to perform verification tests.

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