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This paper presents the applicability of the radial basis function network for prediction of natural convection heat transfer from a confined horizontal elliptic tube. The RBF structure is developed and trained with the help of data obtained by a Mach-Zehnder interferometer. It is expensive and time consuming to do experimental work with changing all variables. The radial basis function network is developed with tube axis ratio, distance from the center of tube and rayliegh number as inputs and average nusselt number as desired output. We used the radial basis function network to simulate the steady condition of heat transfer rate distribution in described geometry. The results of network have an excellent agreement with experimental data. Therefore, the network can be used to predict the unseen data points within the range of experimental results.
Artificial neural networks (ANNs) have seen an explosion of interest over the last few years, and are being successfully applied across an extraordinary range of problem domains, in areas such as diverse as finance medicine, engineering, geology and physics. Indeed, anywhere that there are problems of prediction, classification or control, neural networks are being introduced. The study of heat transfer is one of the most important problems in engineering applications. For example, in electrical engineering field, in places like transmission lines and wires carrying the current and are located compactly, if their distance are not chosen correctly, the heat generated by them will have adverse effect on their actions. It causes melting of their isolation and it is in turn may lead to occur undesirable effects. Moreover, a wide range of practical applications involve the analysis of heat transfer, for example in heat exchangers, reactors etc. . For a better thermal design of such heat exchangers, it is essential to study heat transfer around heated elliptic tube of different cross section confined between two walls. Heat transfer around circular cylinder as a special case of elliptic tube has been previously studied widely [2-7]. To enhance the heat transfer rate around elliptic tube more research has to be done. Fieg and Roetzel in  showed that the elliptical deformation increases heat transfer coefficient during their analytical investigation on laminar film condensation on inclined elliptical tube. A.O.Elsayed et. al., studied free convection from a constant heat flux elliptic tube experimentally . To improve heat transfer from the tube surface, a technique was employed to confine the tube between two adiabatic walls . In the work done so far , an individual ANN network for each access ratio was developed, so an ANN is needed for each axis ratio, thereby this method is time consuming and its computational speed is being reduced. In order to overcome such problems, i.e. to have only one ANN for all axis ratio, a model was developed in  using multilayer perceptron neural network in such a way that axis ratio will be considered as one of the input to the network. The aim of this paper is to create an ANN model to predict the average nusselt number for heat transfer from elliptic tube cross sections confined between two adiabatic walls using radial basis function network.
a major axis(m)
AAD Average Absolute Deviation
b minor axis(m)
output of each neuron
bias vector of each neuron
f transfer function
H wall length
H1 Distance from top to center of tube
H2 Distance from bottom to center of tube
n transfer function input
number of point
Nu average Nusselt number
p input vector
Ra Rayleigh number
t wall spacing (m)
âˆ†NU relative error (Nusselt difference)
%âˆ†NU error deviation (Nusselt difference in percentage)
W weight vector
Y target activation of the output layer
R number of elements in input
ARTIFICIAL NEURAL NETWORK
Artificial neural networks are mathematical or computational models based on biological neural networks. They are being used greatly to model complex relationships between inputs and outputs or to find patterns in data. By using this capability, an ANN model for our purpose has been constructed. Fig. 1 shows a schematic of the proposed ANN model.In this model, the average nusselt number is adopted as a function of three variables namely:
t/b, wall spacing to tube minor axis ration
Ra, Rayleigh number
b/a, Axis ratio
Therefore an ANN model as shown in Fig. 1 is developed with tube axis ratio, distance from the center of tube and rayliegh number as inputs and average nusselt number as desired output.
Fig. 1. Input-output schematic of system.
Radial basis function
Radial basis functions (RBFs) can fit erratic data. RBF networks have been widely applied in many science and engineering fields due to their good approximation capabilities, faster learning algorithms and simpler network structures.
The RBF has a feed forward structure and in its most basic form consists of three separate layers called input layer, hidden layer and output layer as shown in Fig. 2. The transformation from input to hidden layer is nonlinear and from hidden to output layer is linear. All training data points are represented to the network and the interpolating surface has to pass through all of them.
The output from jth neuron of the hidden layer is given by:
K is a strictly positive radially symmetric function (kernel). It has a unique maximum at its center, i.e., and decreases to zero away from the center. is the width of the receptive field in the input space from unit j and k is the number of neurons in the hidden layer. This indirectly indicate that has an desired value only when the distance is smaller than the .
Fig. 2. Radial basis function neural network.
The output layer consists of a set of summation units and provides the response of the network. For an input vector, the output of the mth neuron in the output is defined by:
Fig. 3 shows a schematic of elliptic tube confined between two adiabatic walls . In this figure, H1 and H2 are fixed and equal to 32 mm and 24 mm respectively. The tested tubes are made up of aluminum with the length of 160 mm and major. Their minor axes are chosen to obtain the same periphery. The dimensions of walls are 56Ã-20Ã-160mm and the tube is placed symmetrically between the two walls. Heating wire is placed inside the tube and it is heated using a variable power supply. The Mach-Zehnder Interferometer used in our experiment is the same as the one used by Ashjaee et. al. . The details of the Mach-Zehnder Interferometer setup and the data reduction procedure are fully explained in .
We acquire the experimental data points from  and a good agreement between the experimental results and results of the other researches have been observed, therefore comparisons are not reexpressed here.
Fig. 3 Schematic of the problem.
simulation with ann
In this study, a RBF neural network model was implemented to predict average nusselt number from an isothermal horizontal cylinder of elliptical cross section confined between two adiabatic walls. As mentioned before the experiment needs a considerable amount of time to get accurate results. The aim is to construct a RBF model which is capable of qualified prediction of average nusselt number.
To evaluate the prediction accuracy of the proposed model, error difference ( âˆ†NU) and error deviation (âˆ†NU%) for the average nusselt number are calculated as:
exp and pred represent experimental and predicted values, respectively. Also, the Average Absolute Deviation (AAD %) is defined as:
where is the number of points. The AAD % for the RBF model implemented in the present work and for the MLP model which we created previously in  are shown in Table 1.
The Average Absolute Deviation (AAD %)
As it can be observed from Table 1, there is a good agreement between experimental and predicted data. The Comparison of experimental average nusselt number (Nu exp.) with predicted average nusselt number (Nu pre.), error difference, error deviation for test data (unseen data) of the RBF and MLP models are shown in Table 2. The result show that the predicted values are in good agreement with experimental data. The comparison between experimental and predicted values for training and testing sets in RBF model are shown in Fig. 4 and 5 respectively. These figures also shows the predicted values are very close to experimental values with least error.
Fig. 4. Comparison between experimental and predicted values for training set.
Fig. 5. Comparison between experimental and predicted values for testing set.
Also, the comparison between average nusselt numbers obtained from experiment and those predicted with neural network for tested data, as a function of t/b ratio for some selected Rayleigh numbers for each b/a=0.53, 0.67, 0.8, are shown in Fig. 6. This figure also compares the accuracy of RBF and MLP  models.
Fig. 6a. Comparison between experimental and predicted values for b/a=0.53.
Comparison of experimental average nusselt ( Nu exp.) with predicted average nusselt number(Nu pre.), error difference, error deviation.
Fig. 6b. Comparison between experimental and predicted values for b/a=0.67.
Fig. 6c. Comparison between experimental and predicted values for b/a= 0.8.
It is observed from Fig. 6, that the average nusselt number increases with the increase of the Rayleigh number for each wall spacing. There is an optimum wall spacing for a constant Rayleigh number, where the heat transfer from the elliptic cylinder is maximum. When the wall spacing increases from its optimum value, the average nusselt number decreases and approaches to the value of average nusselt number for a tube in infinite medium. Also, for each Rayleigh number, decrease of the wall spacing from its optimum value makes a sharp decrease in the average nusselt number. On the other hand, Fig. 6 shows that results obtained from neural network have fitted well with the results of experiment ,.
The results showed that the model could be used in this problem for prediction of average nusselt number, which is important in free convection application.
In this work, an accurate RBF model was constructed to predict the average nusselt number for heat transfer from elliptic tube cross sections confined between two adiabatic walls. The network was trained using experimental data. The maximum absolute error for trained and tested values are 0.265% and 2.101% respectively and Average Absolute Deviation (AAD %) are 0.0361% and 0.2580% for train and tested data respectively. A comparative study of soft computing models for load forecasting shows that RBF is more accurate and effective as compared to MLP. The results shows predicted values are very close to experimental values.
The results obtained clearly demonstrate that RBF is more accurate and reliable for the prediction of free coefficient of convection heat transfer.
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