Figure. 9 reveals the obtained optimum design

points as a Pareto front of those two objective functions. Four optimal design

points, designated by A, B, C and D, can be observed in this figure, whose

corresponding design variables and objective functions have been presented in

Table 6. It is obvious that all the optimum points in the Pareto front have no

dominancy to each other, meaning that no objective function can be better

without worsening at least another objective function and there is not two points

where one of their objective functions is the same and the other one is

different.

In Figure11, the design points A and D exhibit

the best pressure drop and the best heat transfer, respectively. In point D the

heat transfer is maximum, however, it should be noted that the pressure loss is

also maximum under these conditions. Conversely, in point A both of the

pressure loss and heat transfer is minimum. Moreover, the other points, B and C

from Figure11 known as the break points.

The design point, B shows important optimum

design concepts. This point compare with

point A indicates about 51% increases in

and h. Similarly, point C compare with point D, h

increases a little bit(about 24.1%), while

improves by a higher value (about 102.3%). In

general, it is desire to find out an optimum design points compromising both

objective functions. To detect that point, the mapping method was utilized 38.

For this purpose, first the value of

function is reversed (because the minimum

value of this objective is desired). Then the reversed

and Nu functions of all non-dominated points

are 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 the

minimum sum of those values that satisfies both objective functions of heat

transfer and pressure loss.

For a useful comparison, the optimal data received

from the Pareto front are compared and figured along with the existing numerical

data. Figure12 shows the overlap of the Pareto front and the related numerical

data. This figure shows that the Pareto front has recognized very accurately

the best boundary of the CFD data with respect to the lowest pressure drop and

highest heat transfer coefficient which confirms the validity of the multi objective

optimization approach presented in present work.

Figure 11. Pareto optimal points for h and

for optimal design points.

Table.7. Pareto corresponding design variables

and 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

Figure

12. Overlap graph of the obtained optimal Pareto front with the CFD simulation data.

1.

Conclusions

In this paper, modeling and multi-objective

optimization of the parameters of flow in horizontal flat tubes equipped with

porous 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 to

simultaneously enhancement the heat transfer coefficient and reduce the pressure

drop in flat tubes. First, CFD techniques were used to solve the flow in several

flat tubes. After validating the results, the CFD data of this step were used

for the modeling of objective functions h and

by ANFIS and GMDH type ANN. In ANFIS model, all

the training and testing data have been selected randomly and about 75% of data

are used for training and 25% for testing performance. By utilization of various

statistical parameters, the high accuracy of GMDH polynomials was represented. Finally,

these polynomials were used for the multi-objective optimization of the parameters

in flat tubes partially with porous media and the derivation of the Pareto front

by the use of NSGAII algorithm. The Pareto front contained important design

information regarding the flat tubes and porous layer, which could not be obtained

except by combining CFD, GMDH and the multi-objective optimization method. The following conclusions

are derived from this study:

1. The

excellent adaption between the predicted heat transfer coefficient and the numerical

results shows that ANFIS is a reliable technique for modeling and the

prediction of results due to its high accuracy.

2. The GMDH

model presents simple mathematical equations without any need to the

complicated numerical model. In other words, this model helped us to convert

numerical results to algebraic equations precisely.

3. The

Results showed that the ANFIS Model is more accurate than the GMDH Network. The MRE% for training data of ANFIS and GMDH

models are 2.68 and 3.23 and for testing data are 2.75 and 3.68 for heat

transfer coefficient and for training data of ANFIS and GMDH for wall shear

stress are 2.92 and 6.19 and for testing data are 3.08 and 9.60.

4. In the Pareto front, four numbers of

distinguished points were specified include the point with the highest heat

transfer and the lowest pressure loss. The trade-off point that satisfies both

objective functions of heat transfer and pressure loss obtained by the mapping

method.