Optimization
5.2 Multi-objectives functions optimization
Due to the existent trade-off betweenNOxemissions and fuel consumption it was considered manda- tory to perform a multi-objective optimization in order to minimize bothNOx emissions and specific fuel consumptionS, bearing in mind emissions and operating costs optimization. Now, one will have two ob- jective functions: the single-equation that predicts the values forEIN Ox and the specific fuel consump- tion equation. The results for the Rizk and Mongia’sNOxemissions prediction model multi-optimization process will be presented next, while the results for the other two models in chapter 4 may be checked in Appendix A.
5.2.1 Results
For the simultaneous optimization of Rizk and Mongia’sNOxemissions prediction model and specific fuel consumption, the objective functions are represented by equations (4.8) for theNOxemissions and by equation (2.47) forS. EIN Ox equation depends onTt3,pt3andτ, andS equation depends onf,B andΨ. These quantities depend on the design parameters for reference operating conditions. Thus, in this case, the objective function comprises six or seven optimization variables as indicated in equations (5.2) and equations (5.3), with the lower and upper bounds given in Table 5.3 for two- and three-spool turbofan models.
EIN Ox =f[πf R, πcLR, πcHR, BR, Tt4R, τ]2spool
S=f[πf R, πcLR, πcHR, BR, Tt4R]2spool
(5.2)
EIN Ox=f[πf R, πcLR, πcHR, πcSHR, BR, Tt4R, τ]3spool S=f[πf R, πcLR, πcHR, πcSHR, BR, Tt4R]3spool
(5.3)
Optimization variable bounds Optimization variable bounds 2−spool design 3−spool design
1< πf R<2.1 1.15< πf R<2 3< πcLR<5 1.15< πcLR<2 4< πcHR<6 2< πcHR<4 3.7< BR<5.3 3< πcSHR<5 1525K < Tt4R<1800K 4< BR<7
0.1ms < τ <10ms 1525K < Tt4R<1800K 0.1ms < τ <10ms
Table 5.3: Variable bounds for the multi-objective optimization of Rizk and Mongia’s NOx emissions prediction model
The optimization results for thisNOxemissions prediction model are presented in Figure 5.2, using now a logarithmic scale in the abscissas axis. Considering sea-level static conditions and the inputs defined in Table 3.1 for ambient and flight data, and component efficiencies, it is possible to achieve the optimization results presented in Figure 5.2. The value ∆Tpz = 1125K employed was based on the analysis already done in chapter 4. These results were obtained for two- and three-spool turbofan models. The iterative process of the genetic algorithm was interrupted at the fifth generation (see next subsection for an explanation). From the results over the five generations period, it is possible to find a range of optimal solutions forEIN Ox andS. Carefully analysing the results and bearing in mind the simultaneous desiderata of maximum reduction of bothEIN Ox andS, it is possible to present a single solution for these two performance parameters: for a two-spool turbofan engine design the optimal solution would be EIN Ox = 0.119 g/kgf uel and S = 0.489 N/N h; for a three-spool turbofan engine design, the optimal solution would beEIN Ox = 0.095g/kgf uelandS = 0.385N/N h. In order to obtain these values for the emission index ofNOx and specific fuel consumption, the design parameters will have to be set to the values given in Table 5.4.
(a) 2-spool Design (b) 3-spool Design
Figure 5.2: Multi-objective optimization results of Rizk and Mongia’sNOxemissions prediction model
Optimum design parameters Optimum design parameters 2−spool design 3−spool design
πf R= 1.916 πf R= 1.838 πcLR= 3.075 πcLR= 1.383
πcHR= 4.163 πcHR= 2.493
BR= 4.552 πcSHR= 3.879
Tt4R= 1685K BR= 6.218
τ = 0.417ms Tt4R= 1625K
τ= 0.124ms
Table 5.4: Multi-objective optimum design parameters of Rizk and Mongia’s NOx emissions prediction model
5.2.2 Discussion
By performing a multi-objective functions optimization, one may achieve values for optimum design parameters more in agreement with the actual goal of current early-stage turbofan design: maximizing engine performance and achieving significant reductions inNOxemissions. Comparing the results ob- tained for two- and three-spool design, it is possible to conclude that an improvement in the reduction of specific fuel consumption, as well as in the reduction of the value ofEIN Oxis provided by the latter. This can be explained by the fact that the three-spool configuration model enables higher values of bypass and pressure ratios (Table 5.3) and has a slightly lower optimum inlet turbine temperature (Table 5.4).
The optimization variable bounds were set in accordance with typical values used for the design parameters; however, they were also constrained by the possibilities of the modelling. Optimizing six or seven variables together with multi-objectives with the genetic algorithm (each of which, within each iteration, generates a new population with a wide number of individuals) can result in a severe compu- tational effort and may also lead to nonsensical results. In order to compute proper results, the values set for the variable bounds were those defined in Table 5.3. The number of generations iterated is also connected with these problems: in the multi-objective functions optimization the maximum number of generations achieved in the process was only five. The number of generations is significantly lower than the one obtained for single-objective optimization; optimized results were nevertheless achieved.
Via the observation of the optimum values for the design parameters (Table 5.4), it is possible to see that, in order to achieve the optimization goal, two-spool design exhibits higher values of bypass and fan pressure ratios, inlet turbine temperature, and residence time in combustion chamber than three-spool design. On the other hand, three-spool design exhibits higher compressor pressure ratio. This is mainly due to the fact that it includes an extra stage, the super-high-pressure compressor.
By analysing multi-objective optimization results it is clear that one does not obtain an optimum single solution for minimizingEIN OxandS. Instead, there is a range of optimal solutions for both parameters, as can been seen in Figure 5.2. Due to the fact that the results obtained vary among a wide spectrum of values, the abscissas axis is presented in a logarithmic scale. According to the genetic algorithm results, achieving the lowest possible values of specific fuel consumption would lead to extremely high values of NOx emissions; on the other hand, lower values of NOx emissions entail a severe increase in fuel consumption. Bearing in mind the goal of this optimization process, the values presented in subsection
5.2.1 and described as possible single solutions, are those that fulfil best the optimization goal. In the plots of Figure 5.2, these are the results lying closer to the origin: (EIN Ox, S) = (0.119; 0.489)for two- spool design;(EIN Ox, S) = (0.095; 0.385)for three-spool design. By comparing these optimized results with the ones obtain in the chapters 3 and 4 for the same operating conditions, it is possible to see that there is a considerably improvement inNOx emissions reduction and a also a good improvement in the reduction in fuel consumption, specially for the three-spool design.