Artificial simplicity, assume that the fuzzy inference

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

Figure 6. ANFIS architecture based on

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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


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.


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:


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:


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


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:


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:



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