By network are used as inputs for NSGA-II

By applying the regression
technique, this method generates quadratic polynomial functions in a
feed-forward network. This approach has been used by many researchers 10,38-41.
The outputs of this neural network are used as inputs for NSGA-II
multi-objective optimization. NSGA II algorithm is one of the best and the most
complete multi-objective optimization algorithms, which will be used in this
paper as well. This algorithm was first introduced by Deb et al. 42, and it
has been utilized in various engineering-related applications in recent years 38,39,43.
Multi objective optimization is utilized in order to achieve a set of optimal
solutions, called Pareto solutions.

In the present work, the
simultaneous effects of porous layer and flattening the tube on flow heat
transfer and pressure drop is studied using CFD technique. The tube partially
filled with a porous medium and the wall is subjected to the constant wall heat
flux. Flow field in different flattened tube with The
Darcy–Brinkman–Forchheimer model is used. ANFIS model was used to accurately
predict the heat transfer and pressure drop in tube. For training the ANFIS
model, we used CFD data from previous section. ANFIS model developed with five
input parameters and two outputs to predict the Nusselt number and pressure
drop. Next, genetically optimized GMDH-type neural networks are used to obtained
polynomial models. The obtained simple polynomial models are then used in a
Pareto based multi-objective optimization approach to find the best possible
combinations of p and, known as the Pareto front. The results obtained from
GMDH were compared to those of the ANFIS. In optimization process, the maximum
heat transfer and the minimum friction factor were treated as the
multi-objective optimization problem due to the presence of two conflicting
objectives. The tube
flattening, porous
layer thickness ratio, porosity
of porous layer, wall heat fluxes and entrance flow rate were design
parameter variables. The corresponding variations of design variables, known as
the Pareto set, establish some important design principles.

We Will Write a Custom Essay Specifically
For You For Only $13.90/page!


order now

1.     
 Problem statement

 In this paper, the thermal flow in a
horizontal flat tube with a constant heat flux and partially filled with porous
medium is considered. The geometry of present simulations contains five
horizontal flat tubes with different flattening and the same perimeter and the
porous layers are shown in Figure 1. Porous material is placed along the centerline
of the tube. The same perimeter is a constraint which was observed by the other
researchers who have studied flat tubes 7,9. Some important geometrical
parameters of flattened tubes are compared in Table 1. The fluid flow enters
the tube with constant and uniform velocity and temperature.

It should be noted that because the hydraulic
diameter of tubes are different so the results should not be presented in the
non-dimensional form. Therefore, instead of using non-dimensional parameters
such as Nu, Re, Cf, H/Dh
and Hp/Dh, their dimensional parameters
such as h, Qin,

, H
and Hp are used.

 

Figure 1. Schematic of Flat tube with Porous
layer

 

Table 1. Geometrical parameters of flattened tubes and porous
layers

Flat tube
No.

 
H (mm)

 
W (mm)

 
Dh

1

10

0

10

2

8

3.14

9.6

3

6

6.28

8.4

4

4

9.42

6.4

5

2

12.56

3.6

Porous Layer ratios
No.

 
Hp

 

 

1

0

 

 

2

0.25H

 

 

3

0.5H

 

 

4

0.75H

 

 

5

H

 

 

 

 

1.1.
Governing equations and boundary conditions

The thermophysical
properties of solid and fluid phases are assumed to be constant. Steady, incompressible, laminar and fully developed velocity and temperature
profiles are considered and natural
convection, radiative heat transfer and gravitational effects ignores.
Darcy-Brinkman model is utilized to model the momentum
equation in porous material, homogeneous and isotropic characteristics are assumed
for the porous structure, LTNE
model between the solid and fluid phases in the porous medium is taken into
account.

Under these conditions the governing equations
are expressed 44. These yields, continuity

(1)

momentum in the void region

(2)

momentum
in the porous region based of Brinkman-Forchheimer-extended Darcy equation

(3)

fluid phase of energy equation in the clear
region:

(4)

fluid phase of energy equation in the porous
region:

(5)

solid phase of energy equation in the porous
region:

(6)

The subscripts ‘f’ and ‘s’ denote the fluid and
solid phases, respectively. T is temperature, V is the fluid velocity and P is
the pressure.

 

and

 are 
respectively density, viscosity and specific heat capacity of the fluid.
F is the inertial coefficient and

 is the porosity. The permeability of the
porous media “K” can be written as 45:

(7)

Where dp is particle diameter. The
inertial coefficient is expressed as follows:

(8)

The effective conductivities of the porous
media and the fluid are respectively kse and kfe.  These two geometrical functions of the porous
media are expressed as follows 45:

(9)

Specific surface area in the energy equations declared
as 45:

(10)

The fluid-to-solid heat transfer coefficient is
expressed as:

(11)

Pr is the Prandtl number and Rep
is Reynolds number of particle:

(12)

At the entrance, Z = 0, V = 0, T
= Ti and V = Vi. The gradients of V
and T in Z direction are zero. In summary, the boundary
conditions are:

At Z=0:

(13)

At wall:

(14)

At the interface between the fluid and porous
media:

(15)

The conditions for heat transfer at the
boundary between the fluid and porous medium expressed as 45:

(16)

Where qinterface is the heat
flux at the interface of solid and fluid that separately receives equal heat
flux from the external fluid.