Heat exchanger modeling analysis using artificial neural networks
The model of artificial neural networks (ANN) was utilized to analyze doublepipe heat exchanger model. ANN method was used in many applications of engineering that simulated nonlinear equations. This method did not require knowledge of the input and output such as controlling of dynamic system and prediction of thermal system performance. In last decades, ANN model was used widely in the analysis of heat exchanger models [34]. Neural network stimulates the human intelligence to a high degree which may be trained to find the right outputs and classed the training exercises. Thus, it needs to know the training inputs. After training, the flow parameters can be classed with high accuracy [35].
The two primary types of ANN are as follows:

1.
Feed forward in which the network does not create any loops

2.
The feedback that leads to the formation of one or more loops
Figure 7 showed three layers of the feedforward back propagation. The ANN model was arranged in three layers based on the following: layers for input, hidden, and output. In the hidden layer, many neurons were connected to neurons in the input and output layers. Input layer included four parameters as follows: inlet hot water temperature T_{h, i}, inlet cold air temperature T_{c, i}, hot water mass flow rate\({\dot{\ \textrm{m}}}_{\textrm{h},\textrm{i}}\), and inlet air mass flow rate\({\dot{\ \textrm{m}}}_{\textrm{c},\textrm{i}}\). The output layer included four parameters as follows: average rate of heat transfer \({\dot{\textrm{Q}}}_{\textrm{ave}.}\), Nusselt number Nu, friction factor f, and heat transfer effectiveness ∈. The statistical parametersquared correlation coefficient (R) serves as the foundation for making comparisons between the various algorithms. The orders of ANN’s input values may vary by many magnitude orders, and it may not show the relative importance of the input values, which are within a specific range of (−1, 1) as the maximum and minimum range. The following is a summary of the procedure for using ANN modeling to predict the output parameters [36] as follows:
1) Utilize data sets to identify training patterns.
2) Define the architecture of a neural network.
3) Set the parameters of the network.
4) Execute a program for feedforward back propagation.
5) Analysis and comparison.
There are three equations of experimental correlations to know the number of nodes in the hidden layer N. In this present work and for forward propagation input, the following formula was used [1]:
$$\textrm{N}=\sqrt{0.43\textrm{nm}+0.12{\textrm{m}}^2+2.5\textrm{n}++0.77\textrm{m}+0.86}$$
(21)
where n and m represent the number of nods for the input and output layers respectively.
The hidden layer included seven neurons. In the hidden layer, the function that was selected is the logistic sigmoid (TANSIG), and in the output layer, the function was PURELIN. Summation of the product of each contact weight (W_{jk}) is from a neuron (j) to the neuron (k), input (X_{j}), and the bias b_{i} (additional weight) to obtain the sum of each neuron. Neuron i^{th} has a sum of the weighted input (W_{ij}, X_{j} ) and the net input P_{i} formed by the bais b_{i} as shown in the equation:
$${\textrm{P}}_{\textrm{i}}=\sum\nolimits_{\textrm{j}=1}^{\textrm{n}1}{\textrm{W}}_{\textrm{ij}}{\textrm{X}}_{\textrm{j}}{\textrm{b}}_{\textrm{i}}$$
(22)
where W_{ij} represents the connection strength from input j^{th} to neuroni^{th}, n denotes about the input vectors number, X_{j} represents the input vector, and b_{i} is the bias of the neuron. The sigmoid function is employed to determine the output of the neuron. The training process is important as a first step to form ANN model due to the training process denotes that the weights are corrected to determine the output (target) values. Training means that weights. The training process is a pair of the actual input data (XS, YS), XS represents the vector, and YS represents the corresponding target after the successful training process. When the target values YS are corrected for each vector XS, then the neural network was given a correct prediction for any input X to produce any target value Y depending on the fundamentals of the ANN model. The observed data percentage used in the ANN model is 70% as a training dataset, 15% as test dataset, and 15% as validation dataset. The performance is improved due to the difference between the observed data, and the predicted values filtered back through the system and connecting between the layers. ANN model for the training, test, and validation qualities can be evaluated by applying the correlation of the squared coefficient R as follows:
$$\textrm{R}=1\frac{\sum\nolimits_{\textrm{j}=1}^{\textrm{n}}{\left({\textrm{y}}_{\textrm{i}}{\textrm{y}}_{\textrm{i}}^{\textrm{t}}\right)}^2}{\sum\nolimits_{\textrm{j}=1}^{\textrm{n}}{\left({\textrm{y}}_{\textrm{i}}{\textrm{y}}_{\textrm{o}}\right)}^2}$$
(23)
$${\textrm{y}}_{\textrm{o}}=\frac{1}{\textrm{n}}{\sum\nolimits}_{\textrm{i}=1}^{\textrm{n}}\left({\textrm{y}}_{\textrm{i}}{\textrm{y}}_{\textrm{i}}^{\textrm{t}}\right)$$
(24)
where y_{i} is the output’s value of either i^{th} training, test, or validation and \({\textrm{y}}_{\textrm{i}}^{\textrm{t}}\) represents the value of the corresponding target [37, 38].
MATLAB software programming algorithm
MATLAB programming (version R2015b) was used to predict the experimental data using ANN model. The experimental data of the inputs and outputs values were read as a form of Excel file in workspace window. Function (nn train tool) was used in commend window to manage the data. After the input and output data were imported, the network was created after selecting a set of network properties. After then, the created network was opened to show the network configuration. Start to train the network by pressing on the “train.” It is important to show that the network’s performance was reduced when the over training occurred. Twohundred twentyfour experimental data were considered in the ANN model. The feedforward back propagation neural network was used to put the trained data into action. The rate average of heat transfer, Nusselt number, friction factor, and heat exchanger effectiveness with and without bumpers were predicted together depending on the testing input data, temperature of the air at the inlet, mass flow rate of hot and cold air, and temperature of the hot water at the inlet [39].