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.

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.