9 reveals the obtained optimum designpoints as a Pareto front of those two objective functions. Four optimal designpoints, designated by A, B, C and D, can be observed in this figure, whosecorresponding design variables and objective functions have been presented inTable 6. It is obvious that all the optimum points in the Pareto front have nodominancy to each other, meaning that no objective function can be betterwithout worsening at least another objective function and there is not two pointswhere one of their objective functions is the same and the other one isdifferent.
In Figure11, the design points A and D exhibitthe best pressure drop and the best heat transfer, respectively. In point D theheat transfer is maximum, however, it should be noted that the pressure loss isalso maximum under these conditions. Conversely, in point A both of thepressure loss and heat transfer is minimum. Moreover, the other points, B and Cfrom Figure11 known as the break points.The design point, B shows important optimumdesign concepts. This point compare withpoint A indicates about 51% increases in and h. Similarly, point C compare with point D, hincreases a little bit(about 24.
1%), while improves by a higher value (about 102.3%). Ingeneral, it is desire to find out an optimum design points compromising bothobjective functions. To detect that point, the mapping method was utilized 38.
For this purpose, first the value of function is reversed (because the minimumvalue of this objective is desired). Then the reversed and Nu functions of all non-dominated pointsare mapped into interval 0 and 1, and calculate the norm of these functions.Using the sum of the mapped values, the trade-off point (point C) is theminimum sum of those values that satisfies both objective functions of heattransfer and pressure loss. For a useful comparison, the optimal data receivedfrom the Pareto front are compared and figured along with the existing numericaldata. Figure12 shows the overlap of the Pareto front and the related numericaldata.
This figure shows that the Pareto front has recognized very accuratelythe best boundary of the CFD data with respect to the lowest pressure drop andhighest heat transfer coefficient which confirms the validity of the multi objectiveoptimization approach presented in present work. Figure 11. Pareto optimal points for h and for optimal design points.
Table.7. Pareto corresponding design variablesand objective functions Point H(mm) Hp ? Q(m3/h) q(W/m2) h A 4 0.25 0.
7 2000 0.0073 858.722 0.06665 B 8 0.75 0.7 2000 0.
0073 1300 0.10047 C 4 0.75 0.3 2000 0.
0073 2330.54 0.33576 D 2 0.
5 0.5 1500 0.01215 2900 0.67936 Figure12. Overlap graph of the obtained optimal Pareto front with the CFD simulation data.1.
ConclusionsIn this paper, modeling and multi-objectiveoptimization of the parameters of flow in horizontal flat tubes equipped withporous insert has been successfully performed using the combination of CFD,ANFIS, GMDH and NSGAII algorithm. The design variables were H, HP,?, Q and q” and the essential target was tosimultaneously enhancement the heat transfer coefficient and reduce the pressuredrop in flat tubes. First, CFD techniques were used to solve the flow in severalflat tubes. After validating the results, the CFD data of this step were usedfor the modeling of objective functions h and by ANFIS and GMDH type ANN. In ANFIS model, allthe training and testing data have been selected randomly and about 75% of dataare used for training and 25% for testing performance. By utilization of variousstatistical parameters, the high accuracy of GMDH polynomials was represented.
Finally,these polynomials were used for the multi-objective optimization of the parametersin flat tubes partially with porous media and the derivation of the Pareto frontby the use of NSGAII algorithm. The Pareto front contained important designinformation regarding the flat tubes and porous layer, which could not be obtainedexcept by combining CFD, GMDH and the multi-objective optimization method. The following conclusionsare derived from this study:1. Theexcellent adaption between the predicted heat transfer coefficient and the numericalresults shows that ANFIS is a reliable technique for modeling and theprediction of results due to its high accuracy.
2. The GMDHmodel presents simple mathematical equations without any need to thecomplicated numerical model. In other words, this model helped us to convertnumerical results to algebraic equations precisely.3. TheResults showed that the ANFIS Model is more accurate than the GMDH Network. The MRE% for training data of ANFIS and GMDHmodels are 2.68 and 3.23 and for testing data are 2.
75 and 3.68 for heattransfer coefficient and for training data of ANFIS and GMDH for wall shearstress are 2.92 and 6.
19 and for testing data are 3.08 and 9.60.4. In the Pareto front, four numbers ofdistinguished points were specified include the point with the highest heattransfer and the lowest pressure loss. The trade-off point that satisfies bothobjective functions of heat transfer and pressure loss obtained by the mappingmethod.