Artificial neural network is a calculation tool

which is used to test the data and to create a model by these data. ANFIS

46,47 (Adaptive Neuro Fuzzy Inference System) is an adaptive network which

allows the implementation of neural network topology, together with fuzzy logic

48. When these two systems are combined, they may qualitatively and quantitatively

achieve an appropriate result that will include either fuzzy intellect or

calculative abilities of neural network and uses the advantages of both

methods. The main target of ANFIS is to find a model which can model the inputs

with the outputs accurately. We implemented a Takagi–Sugeno fuzzy inference

system, which contains a five-layered architecture. A schematic diagram for

this architecture and the proposed ANFIS model is shown in Figure.6. For

simplicity, assume that the fuzzy inference system has two inputs namely x

and y and an output f which is associated with the following

rules:

Figure 6. ANFIS architecture based on

Takagi–Sugeno.

Rule 1 If (x

is A1) and (y is B1) then f1=p1x+q1y+r1

Rule 2 If (x

is A2) and (y is B2) then f2=p2x+q2y+r2

Where Ai, Bi

and fi are fuzzy sets and system’s output

respectively. pi, qi and ri

are designing parameters which are obtained during the learning process. The

output of each layer in the ANFIS network consider as Oij (ith

node output in jth layer) so the various layers functions of this

network explain as follows:

Layer 1

In this layer, each node is equal to a fuzzy

set and output of a node in the respective fuzzy set is equal to the input

variable membership grade. The parameters of each node determine the membership

function form in the fuzzy set described by a Gaussian function, so we will

have:

(17)

Where ai, bi

and ci are referred to premise parameters.

Layer 2

In this layer the input signals values into

each node are multiplied by each other and a rule firing strength (wi)

is calculated.

(18)

In which ?Ai is the membership grade of x

in Ai fuzzy set and ?Bi is the membership of y in fuzzy

set of Bi.

Layer 3

Each node in this layer estimates the ratio (wi)

of the ith rule’s firing strength to sum of the firing strength of all

rules. They normalize the firing strength of the previous layer. The output of

each node in this layer is:

(19)

The outputs of this layer are called normalized

firing strengths.

Layer 4

In this layer that name is rule layer, the

output of each node is the product of the previously found relative firing

strength of the ith rule by first order polynomial Sugeno fuzzy rule:

(20)

where pi, qi

and ri are design parameters. The output of this layer is

compressed of a linear combination of the inputs multiplied by the normalized

firing strength W.

Layer 5

Layer 5, this layer is the last layer of the

network and is composed of one node and adds up all inputs of the node. It

computes the overall output as the summation of all incoming signals from layer

4:

(21)

The ANFIS output is computed by using the

consequent parameters in the forward pass. For adaption of premise parameters,

the output error is used by means of a standard back-propagation algorithm. It

has been proven that this hybrid algorithm is highly efficient in training the

ANFIS. Table.3 shows the parameters for ANFIS which are used in this study. More

details about the ANFIS algorithm can be found in Refs. 11,12,21-26. In

general, the performances of the networks are appraised using the statistical

coefficient of correlation coefficients (R2), mean relative

error (MRE) and root-mean square error (RMSE) values, which are

calculated by the following expressions:

(22)

where Qi is the actual value,

Pi is the ANFIS output or predicted value, n is the

number of output data.

The input parameters were tube flattening

ranging from 2 to 10 mm, porous layer thickness ratio from 0 to 1, porosity of

porous layer ranging from 0.1 to 0.9, wall heat flux ranging from 200 to 1000

and entrance flow rate ranging from 0.0024 to 0.0218 were design parameter. The

output parameters were convection heat transfer (h) and wall shear

stress (

). Because

of different hydraulic diameter of flat tubes, in this paper the wall shear

stress used instead of pressure drop. These two parameters are as follows:

(23)

(24)

Where L is length and Dh

is hydraulic diameter of tube. Fifty set of input parameters have been chosen

by the use of DOE technique to construct the models of ANFIS. In present paper, Response Surface Methodology

(RSM) which is a sub method of Design of Experiments (DOE) is used Montgomery,

1991 to design the number of

input–output data in GMDH modeling. There are a total number of 50 input–output

CFD data considering five design variables and two objective functions.

. In order to improve the ANFIS model, about

75% of data are used for training and 25% for testing performance. The ANFIS

models for training and testing data for h and

are shown in Figures 7 and 8. It is observed

that the comparison among the numerical and predicted values of heat transfer

coefficient and pressure drop using ANFIS model are in good agreement with more

than R-square value of 0.999. Furthermore, appropriate difference in error

values between the train and test data set proves the reliability of the model.

It is also observed that the higher relative error of average Nusselt number

was approximately 2.68% (for training) and 2.75% (for testing). The mean relative

error is within the range of 0.1 to 3%. The higher values of relative error of

dimensionless pressure drop were approximately 2.92% (for training) and 3.08%

(for testing) and the mean relative error is within the range of 2.71% to

2.57%.