A Contribution to the Modelling of Fouling Resistance in Heat Exchanger-Condenser by Direct and Inverse Artificial Neural Network

The aim of this study was to predict the fouling resistance (FR) using the artificial neural networks (ANN) approach. An experimental database collected from the literature regarding the fouling of condenser tubes cooling seawater of a nuclear power plant was used to build the ANN model. All models contained 7 inputs: dimensionless condenser cooling seawater temperature, dimensionless inside overall heat transfer coefficient, dimensionless outside overall heat transfer coefficient, dimensionless condenser temperature, dimensionless condenser pressure, dimensionless output power, and dimensionless overall thermal efficiency. Dimensionless fouling resistance was the output. The accuracy of the model was confirmed by comparing the predicted and experimental data. The results showed that ANN with a configuration of 7 input neurons, 7 hidden neurons, and 1 output neuron presented an excellent agreement, with the root mean squared error RMSE = 3.6588 ∙ 10−7, average absolute percentage error MAPE = 0.1295 %, and high determination coefficient of R2 = 0.99996. After conducting the sensitivity analysis (all input variables had strong effect on the estimation of the fouling resistance), in order to control the fouling, an inverse artificial neural network (ANNi) model was established, and showed good agreement in the case of different values of dimensionless condenser cooling seawater temperature.


Introduction
In the process industry, many heat exchangers-condensers are used for refrigeration. The accumulation of unwanted deposits on the heat exchanger surfaces is generally called fouling. Fouling is a major problem in condensers and heat exchangers. The existence of these deposits resists the transfer of heat, and thus decreases its effectiveness. All industrial circuits cooled with water are affected by the phenomenon of biological fouling consisting of the growth of biofilms and the settlement of several types of living organisms.
There are few articles published in the literature to identify the fouling phenomenon and its impacts, and few have tried to quantify them.
Eguía et al. 1 attempted to show the growth of biological fouling inside the cooling tubes of an exchanger-condenser, keeping the temperature of the wall constant and using seawater. Zubair et al. 2 presented a discussion based on the economic aspects of the fouling of heat exchangers, followed by the thermoeconomic analysis based on the risks of the heat exchanger. Casanueva et al. 3 studied the development, growth, and control of marine biological fouling in condenser heat exchangers in order to assess the impact of new regulations on the particular plant and site. Qureshi et al. 4 studied the effect of fouling on the thermo-hydrau-lic characteristics of the heat exchanger. Durmayaz et al. 5 proposed a theoretical model to determine the functional relationship between the temperature of the cooling water and the pressure of the condenser on the efficiency of a pressurized water reactor nuclear power plant. Nebot et al. 6 presented a model for predicting the maximum value of the thermal resistance of the fouling layer on the steam condensers of power plants cooled with seawater under different conditions: effect of the water velocity and the material of the tube. Walker et al. 7 presented a study with three specific objectives: (1) to quantify the economic impact of the fouling condensers for a 550 MW power plant with moderate temperature and demand profiles, (2) to determine the relative impact on the same power plant knowing the high temperature and demand profiles, and (3) assessing the relative impact on the condensers with varied design parameters and cleaning schedules. Ibrahim et al. 8 presented a parametric study that illustrates the impact of the fouling factor of the seawater-cooled condenser in a range of 1.5-3.5 cm 2 K W −1 , and the temperature in a range of 15-30 °C. Zhang et al. 9 proposed a novel method to calculate the effective thermal conductivity of particulate fouling based on the Image-Pro-Plus image processing and calculated its thermal conductivity using the finite element method.
The artificial neural network (ANN) technique has been used in many scientific domains, such as in solar energy 10,11 and in solubility of solid drugs. 12 The ANN was used by Melzi et al. to develop the predictive models to estimate the molecular diffusion coefficients of various gases at multiple pressures over a large field of temperatures. 13 They experimentally illustrated two optimisation objectives through developing two neural network models. The calculation of maximum flux as well as minimum resistance to fouling throughout ultra-filtration of industrial wastewater was anticipated by means of the genetic algorithm and assembled ANNs by Soleimani et al. 14 They provided a review consisting of three objectives on the thermal analysis of heat exchangers: (1) summarising the studies carried out by ANN, (2) comparing the network architectures, and (3) identifying the supplementary research boundaries and needs of ANN. In the work of Mohanraj et al. 15 ANN gave an exceptional alternative methodology for the thermal analysis of heat exchangers. From the experimental data, Jradi et al. 16 developed three accurate and reliable models of fouling phenomenon in three types of heat exchangers using three methods: Kern and Seaton model, Partial Least Squares model, and Artificial Neural Network model, in order to make a comparative study based on some statistical indices among these different models in regard to the modelling, and the losses prediction of heat exchangers performances due to the fouling phenomenon. The inputs to the ANN model used for the three types of exchangers were inlet and outlet temperatures, steam temperature, volume flow rate, density, and time. Davoudi and Vaferi 17 developed an ANN model for the estimation of the degree of fouling. The inputs for the network were density, surface temperature, fluid temperature, diameter of fluid passage, velocity of fluid, concentration of dissolved oxygen in fluid, and time. Kashani et al. 18 developed an ANN model to estimate the online monitoring and prediction of crude oil fouling behaviour for industrial shell and tube heat exchangers. The input data to the network were tube-side crude oil flow rate, tube-side inlet temperature, and shellside inlet temperature. Aminian and Shahhosseini 19 employed an ANN model for developing a set of mathematical formulations in order to identify regions where there was less/no fouling for pre-heat exchangers of a crude oil by using input variables such as Reynolds number (Re), Prandtl number (Pr), and surface temperature.
Hence, the scientific contribution of this research is to predict the fouling resistance in heat exchanger-condenser (FR * ) using the ANN approach, and based on dimensionless inputs and output. To facilitate the use of the developed ANN model, MATLAB graphical interface was developed to facilitate the calculation for users. Furthermore, a strategy will be developed to estimate the optimal operating conditions (inputs) from fouling resistance (FR * ) using inverse artificial neural network (ANNi).

Artificial neural networks approach
A neural network is an extremely parallel distributed processor that has a natural propensity for storing experiential knowledge and availing it for use. 20 ANN is an adaptive non-linear statistical data modelling technique consisting of interconnected artificial neurons processing data in parallel. In this study, the multi-layer perceptron (MLP) network type was employed (Fig. 1), and the supervised learning process was used for the network parameters adjustment.
The input and output data were normalised within the range of [−1, 1] using a Mapminmax algorithm given by Eq. (1) that performs a normalisation of the maximum and minimum value of each row: 21 After examination of a considerable number of differently structured neural networks, the adequate ANN selected in this paper has a single hidden layer with seven neurons, and an output layer with one neuron. The hidden layer has a tansig transfer function. The output layer has a purelin transfer function. Typical structure of the ANN is shown in Fig. 1.
input layer hidden layer output layer The experimental database regarding the fouling of condenser tubes cooling seawater of a pressurized water reactor nuclear power plant was collected from the literature. 8 Based on the effect of the environmental conditions and the change in fouling on the thermal performance of the condenser and the thermal efficiency of a nuclear power plant, the dimensional experimental inputs were transformed into dimensionless form. This base was transformed to dimensionless numbers including the different values of condenser cooling seawater temperature, inside overall heat transfer coefficient, outside overall heat transfer coefficient, condenser temperature, condenser pressure, output power, overall thermal efficiency, and the fouling resistance.
In this paper, a methodology was given for selecting ANN model to predict the fouling resistance of condenser. The objective of this paper was to select the best model. The methodology started with an extensive search in order to select a model with minimal complexity and optimal performance. The number of observed data used in the ANN was 600, which were divided into three sections: the training set (480 data), the test set (60 data), and the validation set (60 data). Training, test, and validation subsets of the ANN were obtained by selecting 80 % of the dataset as training, 10 % of the dataset as test, and 10 % of the dataset as validation subsets. A summary of the range of different variables is shown in Table 1. The difference between the observed and predicted values was filtered back through the system and used to adjust the connections between the layers, thus the performance improved. The root mean square error (RMSE) was the main criterion to evaluate the performance of ANN, which is defined as follows: 22,23 ( ) The statistical quality of the ANN for the three training, test, and validation sets was then evaluated using the coefficient of determination (R 2 ), relative absolute error (RAE), mean absolute percentage error (MAPE), mean square error (MSE), precision factor (A f ), bias factor (B f ), and acceptability criteria K and K'. The accuracy factor value, A f (Eq. (9)), will always be equal to or greater than one as all variances are positive.
In addition, to ensure greater validity of the developed models, some statistical criteria were used as follows: where 0.85 ≤ K ≤ 1.15; 0.85 ≤ K'≤ 1.15 was acceptability criteria.
y i represents the i th trained test or validation output value, and y i t represents the corresponding target value, with n being the number of input vectors.

Modelling with neural networks
The objective of this work was to build a model to predict the resistance of fouling in heat exchanger-condenser. ANN was chosen as the main modelling tool to perform the task.
Different values of dimensionless condenser cooling seawater temperature, dimensionless inside overall heat transfer coefficient, dimensionless outside overall heat transfer coefficient, dimensionless condenser temperature, dimensionless condenser pressure, dimensionless output power, and dimensionless overall thermal efficiency were input arguments for network (independent variables), while dimensionless fouling resistance was a target argument (dependent variable). The structure of the database used for the model is shown in Table 2 and Fig. 2.
In this part, listed are all the important stages that led to the solution of the optimisation of neural networks. The following parameters were examined and optimised during the development of the best network for prediction of fouling resistance: selection of input and output data, possible transfer functions, learning mode, stopping criteria, learning algorithm, normalisation technique, number of hidden layers, number of hidden nodes, and performance evaluation measures. The results are summarised in Table 3.
Observed and predicted (dimensionless fouling resistance) breakthrough curves shown in Fig. 3 indicate that the ANN describes the experimental data well.

Neural networks
A learning algorithm is defined as a procedure of adjusting the coefficients (weight and bias) of a network to minimise an error function (usually quadratic) between the outputs of the network for a given set of inputs and the correct outputs (already known). If nonlinearities are used, the gradient of the error function can be calculated by the conventional backpropagation procedure. 25 To establish the best algorithm of backpropagation learning, a set of fourteen backpropagation algorithms were studied. In addition, thirty neurons were used in the hidden layer for all backpropagation algorithms. Table 4 presents a comparison of different retro-propagation training algorithms. In the present study, the Levenberg-Marquardt algorithm was considered as the learning algorithm because it might have a smaller maximum relative absolute error (RAE max ) compared to the other learning algorithms.    Fig. 3 -Plot between experimental and ANN predicted dimensionless fouling resistance for training data set, for testing data set, for validation data set, and for all data set To obtain the optimal number of neurons in the hidden layer, a series of topologies were used, using the scaled conjugate gradient algorithm such as the number of neurons ranging from 1 to 30 (Fig. 4) in order to maximise RAE max by looking for a set of weights and connection bias that cause the ANN to produce outputs equal or close to the target values.
To approximate the actual data points, a statistical measure that gives the quality and efficiency of the fit of the regression line of a model was verified by R 2 . R 2 of 0.99996 means that the regression line matches the data.
The forecasts of the results by ANN with respect to the experimental results for training, test, and validation data sets are plotted in Fig. 5 for the fouling resistance. The performance of the selected networks {7-7-1} is described in Fig. 5. The ANN was well formed because R 2 was greater than 0.9999 for the forecast network.

Sensitivity analysis
In order to study the effects of the selected input parameters on the planned outputs, a sensitivity study was performed, where the model that was chosen to study had 7 inputs, 1 output, and 7 neurons in its hidden layer. The Min-Max method was used as a normalisation technique and the Levenberg-Marquardt as the learning algorithm. Once the network had been trained and optimised, weights matrix was generated (Table 5).
In order to evaluate the relative importance of the input variables, the weights matrix was exploited in Eq.  (12) where I j represents the relative importance of the j th input variable on the output variable, N i and N h are the set of input and hidden neurons, respectively, W is the connection weights, the exponents 'i', 'h' and 'o' refer to the input, hidden and output layers, respectively, and 'k', 'm' and 'n' refer to the input, hidden, and output neurons, respectively. Note that the numerator in Eq. (12) describes the sum of the products of the absolute weights for each entry. However, the denominator in Eq. (12) represents the sum of all the weights feeding the hidden unit, taking the absolute values. 26 A summary of the results obtained is presented in

Interface for dimensionless fouling resistance
Our model based on the mathematical non-linear formula of the optimised neural network (ANN) is presented by Eq. (13) linking the inputs to the output ( Table 2).  (13) After having optimised the neural network model, a computer program was developed in MATLAB (Fig. 7). This allows the user to have all the inputs needed to run the model in order to calculate the fouling resistance in the condenser. The interface was designed to provide more flexibility in the use of ANN model for quick and easy calculation of fouling resistance.

Optimal performance by means of ANNi
According to the ANN model, developed was a method of estimating input parameters that influence the resistance of fouling when the input parameters are well known. 12,27,28 Table 5 gives the obtained parameters (w i , w ij , b i , and b j ) of the best fit for seven neurons in the hidden layer. These parameters are used in the proposed model to simulate the FR * value.  A general network is constituted by transfer function (tansig and purelin). Then, the output is given as function of inputs Eq. (14), where n is the number of neurons in the hidden layer (n = 7), m is the number of the input (m = 7), w is the weight, and b is the bias.
Eq. (13) can be expressed as: At this step, the function was obtained that had to be minimised at zero to obtain optimal input parameters (n) In (m=x) : . .
where x are the different values of dimensionless condenser cooling seawater temperature value to be computed. The proposed ANN model had seven neurons in the hidden layer. It was therefore essential to apply an optimisation method. 29,30 Optimised input parameter was established using Nelder-Mead simplex algorithms under the different conditions to prove the feasibility of using ANNi. A test was done with different data to optimise one of the inputsfor instance, different values of dimensionless condenser cooling seawater temperature (T * seawater ) on the one hand, and on the other hand, the simulated parameters were compared to the experimental data in order to check the accuracy of ANNi at way of the error given in the Eq. (16). − = ⋅ exp sim error 100 exp (16) Case1. A group of parameters was provided for fouling resistance predicted process with k = 1, n = 7, and m = 7.
The input value (for example T * seawater ) can be calculated through ANNi. The experimental conditions for this test, for a required output value FR * = 0.00015, with input values: U * ic = 0.855440, U * oc = 0.860584, T * c = 1.07401318, P * c = 1.161460, PW * c = 0.991597, η * c = 0.991542, by minimising the nonlinear function given in Eq. (17). (20) ( ) = − + + + + + + With weights and biases, the optimum T * seawater of the process can be calculated for the required output, using MATLAB software with the optimisation Toolbox. 31 The outcome value (T * seawater ) simulated by ANNi was 0.6. Furthermore, using Eq. (17), the calculated value had an error of 0 % regarding the experimental result (Table 6).

Comparison with other models
In order to evaluate the importance of the obtained results, they were compared with similar studies cited in the Introduction, [16][17][18][19] especially with the models that had the same or mostly the same inputs as we used. It was found that, in most models, the goal was to predict the fouling in heat exchangers. The results obtained from these mentioned models and results in this work are collected in Table 7.

Conclusion
The ANN modelling method has many advantages, such as speed, adaptability, generalisation and simplicity, making it an interesting choice for the modelling of nonlinear complex systems (fouling of heat exchanger-condenser).
During the training process, several configurations of neural networks were used. It started with the use of 14 training algorithms by observing the effect of each of them on the network performance. Therefore, the top three of them (trainlm, trainbr, trainbfg) were chosen, and improvement of their performance was attempted by using one normalisation technique. Among the three obtained networks, the best predicting network was selected, which had also been tested with one hidden layer and several hidden nodes. It was found that one hidden layer with 7 neurons can provide a better prediction.
Our model was developed by MLP back-propagation architecture with direct action with the Levenberg-Marquardt training algorithm. The results confirm that the MLP neural networks are competent enough to predict fouling resistance when using the heat exchanger-condenser operating conditions as inputs.
It was observed that the network with a hidden layer of {7-7-1} gave a correlation coefficient greater than 0.9999 with a small error (RMSE = 3.6588 • 10 −7 ) for the whole database, meaning that the neural network was well formed. The ANN is an adequate interpolation tool for the non-linear behaviour of FR * for all the operating conditions mentioned above, in which an excellent prediction was obtained. Based on the sensitivity analysis, it was found that all the input variables had a significant effect on the estimation of fouling resistance. In addition, based on the model, a methodology was developed to estimate the optimal operating conditions from ANNi when necessary to obtain an optimal result for the known inputs for the required fouling resistance FR * by applying the Nelder-Mead simplex optimisation method. Nevertheless, the mathematical calculation of ANNi was carried out in Case 1.
All ANN inputs were estimated using ANNi with an average relative deviation not exceeding 0.095 %. By comparing these results to other studies in the same context, it may be concluded that our model gave better performance compared to many other models.  the financial support provided. The authors also express their gratitude for useful and interesting discussions with researchers at the Birine Nuclear Research Center (CRNB).