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