C K Tan
email: ctan@glam.ac.uk.

J Ward

S J Wilcox

R Payne

Artificial Neural Network Modelling of the Thermal Performance of a Compact Heat Exchanger

Keywords: Neural Networks, Non-linear Regression, Heat Exchangers, Fault Detection, Modelling

This paper reports the use of artificial neural network models to simulate the thermal performance of a compact, fin-tube heat exchanger with air and water/ethylene glycol anti-freeze mixtures as the working fluids. The model predictions were compared with experimental data over a range of flow rates and inlet temperatures and with various ethylene glycol concentrations. In addition, the inlet air flow was distorted by obstructing part of the inlet ducting near the front face of the exchanger. The artificial neural networks were able to predict the overall rate of heat transfer in the exchanger with a high degree of accuracy and in this respect were found to be superior over conventional nonlinear regression models in capturing the underlying non-linearity in the data. Moreover the detailed spatial variations in outlet air temperature were also adequately predicted. The results indicate that appropriately trained neural networks can simulate both the overall and "local" characteristics of the compact heat exchanger. In addition the paper demonstrates how an alternative type of neural network, the so-called Self-Organising-Map (SOM), can be employed for heat exchanger condition monitoring by identifying and classifying the deterioration in exchanger performance which, in this case, was associated with different levels of inlet obstruction.

Introduction

Heat exchanger systems are essentially nonlinear in behaviour so that they can be difficult to control effectively. Consequently accurate prediction of the steady state and dynamic performance of heat exchangers is vitally important for optimum system design and heat recovery. The performance is often predicted using empirical convective heat transfer correlations based on best fit "least squares" analysis of experimental data (which may only be applicable to the particular exchanger and flow conditions), and corrected logarithmic mean temperature differences. This method, however, does not provide reliable predictions if the thermal and transport properties of the fluids cannot be specified accurately or if there is substantial mal-distribution of the inlet flows. This paper presents an alternative approach based on artificial neural network (ANN) models to represent the behaviour of the exchanger. These networks can be "trained" to represent complex, non-linear physical processes and data with a high degree of accuracy without requiring an explicit knowledge of the relationships between the inputs and outputs of the system. A neural network has a computational structure, which is analogous to a biological neural system in that it consists of a series of interconnected nodes or neurons. These are arranged in input, output and one or more so-called hidden layers linked by non-linear transfer functions. During "training" the network is presented with data for the process which can be obtained experimentally or, if this is not feasible, by using predictions from an appropriate, rigorous mathematical model of the process. The weightings between nodes and the detailed configuration of the ANN are varied until any differences or errors between the predicted and actual values of the output variables are reduced to a specified level. The networks have only modest computational requirements so that they have been employed as "black box" models in the analysis of a wide range of thermal applications including air-water spray systems [START_REF] Oliveira | Neural Network Analysis of Experimental Data for Air/Water Spray Cooling[END_REF], heat transfer data analysis [START_REF] Thibault | A Neural Network Methodology for Heat Transfer Data Analysis[END_REF][START_REF] Jambunathan | Evaluating Convective Heat Transfer Coefficients Using Neural Networks[END_REF][START_REF] Mazzola | Integrating Artificial Neural Networks and Empirical Correlations for the Prediction of Water Cooled Critical Heat Flux[END_REF], heat pumps [START_REF] Bechtler | Neural Networks -a New Approach to Model Vapour Compression Heat Pumps[END_REF], and solar domestic water heating systems [START_REF] Kalogirou | Long Term Performance Prediction of Forced Circulation Solar Domestic Water Heating Systems Using Artificial Neural Networks[END_REF]. A comprehensive review of the use of these networks in thermal systems can be found in Sen and Yang [START_REF] Sen | Applications of Artificial Neural Networks and Genetic Algorithms in Thermal Engineering[END_REF].

More specifically neural networks have also been applied to heat exchangers. Diaz et al [START_REF] Diaz | Simulation of Heat Exchanger Performance by Artificial Neural Networks[END_REF] studied single phase compact fin-tube systems and concluded that, for a wide range of exchanger geometries, a neural network was able to predict the overall heat transfer rate between the two fluids with much less scatter than the conventional technique based on available correlations. In addition Pacheco-Vega et al [START_REF] Pacheco-Vega | Analysis of Fin Tube Evaporator Performance with Limited Experimental Data Using Artificial Neural Networks[END_REF] successfully employed manufacturer's data to train and test an ANN to model the thermal performance of a range of compact evaporators used for refrigeration. Also Pacheco-Vega et al [START_REF] Pacheco-Vega | Heat Rate Predictions in Humid Air-Water Heat Exchangers Using Correlations and Neural Networks[END_REF] analysed heat transfer in compact fin-tube condensers using data from McQuiston [START_REF] Mcquiston | Heat, Mass and Momentum Transfer Data for Five Plate Fin Tube Heat Transfer Surfaces[END_REF] and found that the use a neural network yielded more accurate predictions than those derived by conventional regression analysis of the data. More recently neural networks have again been used for performance prediction in a further refrigeration application, Islamoglu et al [START_REF] Islamoglu | Performance Prediction for Non-Adiabatic Capillary Tube Suction Line Heat exchanger: an Artificial Neural Network Approach[END_REF]. This ability to model the behaviour of heat exchangers indicates that neural networks can be used for control purposes particularly since the network model can be updated on-line to take into account fouling or other changes in the characteristics of the system. Consequently a number of authors have investigated neural control of heat exchangers including Ayoubi [START_REF] Ayoubi | Dynamic Multi-Layer Perceptron Networks: Application to the Non-Linear Identification and Predictive Control of a Heat Exchanger[END_REF], and Diaz et al [START_REF] Diaz | Artificial Neural Network Control of an Experimental Heat Exchanger Facility[END_REF][START_REF] Diaz | Adaptive Neurocontrol of Heat Exchangers[END_REF]. The present paper is concerned initially with the neural network prediction of the overall and detailed heat transfer characteristics of a compact fin tube exchanger under distorted flow conditions. The experiments were conducted with air and water/ethylene glycol anti-freeze mixtures as the exchanger fluids over a wide range of flow rates and inlet temperatures. The inlet air flow was distorted by partially obstructing the cross section of the inlet duct. In practice non-uniform air flows can arise from the presence of sharp bends before the exchanger inlet, partially open dampers or other flow control valves in the inlet flow ducting, system leakage or accidental ducting collapse or distortion, see Chiou [START_REF] Chiou | Thermal Performance Deterioration in a Cross Flow Heat Exchanger due to Flow Non-Uniformity[END_REF] and Sparrow and Ruiz [START_REF] Sparrow | Effect of Blockage Induced Flow Maldistribution on the Heat Transfer and Pressure Drop in a Tube Bank[END_REF], and generally result in a reduction in thermal performance of the exchanger. The paper also examines the use of an alternative type of neural network, the so-called Self-Organising-Map (SOM), to identify and classify the deterioration in exchanger performance associated with different degrees of inlet obstruction. This ability to classify exchanger characteristics provides a potential method of detecting changes in the performance of the heat exchanger, due for example to fouling or sudden changes in fluid composition, using data only from the normal operating conditions. Consequently, artificial neural networks can offer an attractive method for identifying both sudden and gradual degradation in the performance of complex multiple heat exchanger systems.

The Experiments

Experimental Set-up and Test Procedure

A schematic representation of the overall arrangement of the experimental facility used in the tests is presented in Figure 1. The compact fin-tube heat exchanger was constructed with 8 tube passes and the tubes were fitted with 945 fins/m. The cross sectional dimensions of the exchanger were 274 mm wide by 136 mm high and it had a depth of 55 mm. The test liquids (water or various mixtures of water and ethylene glycol) were supplied to the exchanger from a heated reservoir and were contained in a closed flow circuit. The liquid in the reservoir was heated by two 3 kW electrical heating elements one of which was operated continuously whilst the power input to the other was controlled to maintain a specified constant inlet temperature to the exchanger (in the range 80 to 90°C). The mass flow rate of the liquid was varied during the tests from 0.032 to 0.185 kg/s and the proportion of ethylene glycol (anti-freeze) in the mixture ranged from 0 to 40%. The flow rates were measured by a rotameter which had been previously calibrated by direct collection and weighing of the test liquids. The inlet and outlet liquid temperatures were measured in the pipework connections to the heat exchanger using previously calibrated thermistor temperature sensors.

The air flow was supplied by a centrifugal fan and controlled by a damper at the fan inlet. A honeycomb flow straightener was mounted in the inlet ducting to the exchanger to provide a relatively uniform air flow in the undistorted flow tests. The air flow rate was measured by a standard orifice plate arrangement positioned in a circular duct well downstream of the test section and the flows were varied from 0.050 to 0.091 kg/s during the tests. The outlet ducting from the exchanger incorporated a gradual transition from the rectangular test section to the circular outlet and sufficient duct lengths were employed before and after the orifice plate in accordance with British Standards for flow measurement. A second honeycomb device was mounted downstream of the heat exchanger to "straighten" the flow before entry to the air flow measurement section. The whole length of the inlet and outlet ducting was insulated to minimise heat losses to the surroundings.

The inlet flows were distorted by inserting a steel plate or baffle through the side of the duct downstream of the initial honeycomb section near the inlet face of the exchanger. The percentage of the cross sectional area of the inlet, which was obstructed in this fashion, varied from 0% (the undistorted case) to 50%. The positions of the thermistor sensors, which were used to measure the air temperatures, are indicated in Figure 1. The inlet temperature was measured 10 mm downstream of the initial flow straightener and the outlet temperature was measured at several positions downstream of the exchanger. The most reliable exit temperature data were obtained from the sensor positioned after the second honeycomb where mixing resulted in an essentially uniform value. An additional sensor was placed after the orifice plate to provide further information for temperature correction of the flow rate. The detailed air exit temperature distribution was determined using a vertical array of eight equally spaced thermistors, which was traversed in ten 27 mm steps across the duct at a distance of 5 mm downstream from the outlet face of the heat exchanger. All the thermistors were individually calibrated in an environmental chamber by comparison with the temperatures measured by a platinum resistance thermometer, which had an accuracy of 0.1°C over the temperature range studied in the tests. Data were collected when the system achieved steady-state conditions and overall 359 test conditions were examined.

Selected Experimental Results and Discussion

This section is included to illustrate the main effects of obstructing the inlet air flows into the heat exchanger by briefly presenting some of the data gathered in the tests. However, to assess the consistency and reliability of the results the rates of heat transfer to the air were compared with the corresponding values from the liquid, see Figure 2, which presents the comparisons for the "unobstructed" cases. In these calculations the specific heats of the antifreeze mixtures were determined using manufacturer's data. It is evident that there is generally satisfactory agreement between the heat transfer rates with significantly less than 5% difference in virtually all cases. Similar good agreement was obtained with all the obstructed flows even at an obstruction of 50% when the resultant non-uniformity of flow downstream of the exchanger would have been more likely to lead to potential errors in the measurement of the mean exit air temperature. Consequently it appears that the experimental technique is capable of yielding consistent, reliable data. In the remainder of the paper the mean of the air and liquid side rates are employed as the rate of heat transfer in the exchanger.

Obstructing the inlet air flow by a baffle results in a substantial reduction in the performance of the system, see Figure 3 which presents heat transfer data for various air flow rates at constant fluid inlet temperatures and a water flow rate of 0.085 kg/s. Similar deteriorations in performance were observed at other liquid flow rates and for all water/anti-freeze mixtures. The inlet blockage "channels" the air flow so that velocities, mass flow rate and heat transfer coefficients are all increased in the unobstructed section of the exchanger. However a relatively stagnant separated flow ensues behind the baffle and in this region the mass flows, velocities and heat transfer coefficients are reduced. This effect may be exacerbated at very high levels of obstruction when the momentum of the relatively high velocity air leaving the unblocked side of the exchanger can result in a recirculation or reverse flow of "hot" exiting air back through part of the obstructed region. Overall it is clear that the performance of the heat exchanger deteriorates at an increasing rate as the degree of obstruction increases. The presence of the inlet baffle also markedly affects the outlet air temperature distribution, see Figure 4, which presents data for an unobstructed flow and a 50% blockage. The normal unobstructed flow results in a relatively flat distribution, except at locations close to the side wall, whilst there are sharp discontinuities with the obstructed flow. The exit air temperatures associated with the section "covered" by the baffle (positions 5-10) are significantly higher because of the relatively low mass flow rates in this region, whereas the temperatures over the remainder of the outlet face are reduced as a result of the enhanced mass flow rates through the unobstructed exchanger passages. The addition of ethylene glycol anti-freeze in the liquid mixture also reduces the specific heat capacity which in turn reduces the overall thermal performance of the compact heat exchanger.

Modelling of the Performance of the Heat Exchanger

Artificial Neural Network Models of the Overall Performance

A Multi-Layer Feed-Forward ANN [START_REF]User's Guide: 'Neural Network Toolbox For Use with MATLAB [END_REF] was utilised in this study. This type of network contains a layer of input nodes or neurons which are interconnected to the nodes in one or more hidden layers which are linked in turn to the nodes in the output layer see Figure 5. This type of neural network is "trained" by using input and the corresponding output data derived from the experimental measurements. The input values are fed forward through the network and during transmission are modified by the weighting and transfer function (a tan-sigmoid1 transfer function was used in the present study) associated with each link. Each neuron therefore receives multiple inputs from other neurons and generates a single output, which is then fed on to the neurons in the next layer. The eventual predicted values, which are fed into the output nodes, are compared with the corresponding measured results and any differences, i.e the "errors", are propagated back through the network and used to adjust the weightings of the connections. This process is known a single training cycle or iteration. The cycle is repeated sequentially using this back propagation algorithm so that training proceeds iteratively until the mean square error between the predicted outputs and corresponding measured values is reduced to an acceptable level. The trained network can be subsequently "tested" and independently validated by using previously unseen input and output data to assess the ability of the ANN to generalise the process or system it has been trained to represent.

In the current study, a single hidden layer was found to be appropriate and the number of neurons in this layer was varied to optimise the performance of the neural network. The performance of an ANN is heavily influenced by the number of neurons in the hidden layer. The use of too few neurons can result in poor representation of the process because the network failing to "learn" the underlying patterns between the input and output data. Conversely the use of a large number of nodes in this layer can result in "over fitting" whereby the individual training data points are accurately fitted, but substantial errors occur when the network is asked to generalise the patterns associated with new unseen input data. An ANN was developed initially to represent the overall behaviour of the heat exchanger over the whole range of flow rates, inlet temperatures, liquid compositions and blockage ratios studied in the experiments. The inlet temperatures of the liquid (80 to 90°C), the air inlet temperatures (20 to 24°C), liquid mass flow rates (0.032 to 0.185 kg/s), air mass flow rates (0.050 to 0.091kg/s), inlet obstructions (0, 16.5, 33, 41.5, 50%) and ethylene glycol anti-freeze mass concentrations (0, 10, 20, 30, 40%) were used as input vectors to the ANN whilst the single output neuron represented the overall rate of heat transfer between the two test fluids. In this case at total of 6 hidden neurons were found to be most suitable by a "trial and error" process.

For an un-biased assessment of the network performance, the 359 sets of experimental measurements were randomly sampled into 4 equal-sized subsets of data. The first two subsets (training data) were used to train the network whilst the third subset of data (test data) was used to monitor the likelihood of over-fitting the model during training. Unlike the training data, these test data do not influence the adjustment of the network weights and biases but solely provide an indication of the suitability of the network to generalise unseen results. The resultant network was then independently validated using the remaining fourth subset of data (validation data). To provide a more severe test of performance of the ANN model, the results obtained at specific test conditions were subsequently eliminated from the data used for training. In this case all data corresponding to ethylene glycol anti-freeze mass concentrations of 20% and 40% at inlet obstructions of 33% and 50% were used as independent validation data. Similarly, the test data corresponded to zero ethylene glycol anti-freeze at inlet obstructions of 33% and 50%. In this manner, the ANN was trained to extrapolate to conditions which fell outside the region of the training data.

Nonlinear Regression (NLR) Models of the Overall Performance

This section describes the application of a model based on a nonlinear "least squares" regression technique to predict the overall thermal performance of the heat exchanger. An overall analytical model of the exchanger is complex so that the NLR model assumes that the rate of heat transfer between the two fluids can be represented as the product of a series of independent variables so that

f e d c b a ) OBS 100 ( ) AFC 100 ( M M ) T T ( Q A W Ai Wi          (1)
where, Q = rate of heat transfer in the exchanger (kW) T wi = hot stream inlet temperature (C) T Ai = cold stream inlet temperature (C) M w = hot stream mass flow rate (kg/s) M A = cold stream mass flow rate (kg/s) AFC = antifreeze mass concentration (%) OBS = inlet obstruction (%)

The unknown parameters a, b, c, d, e and f in this nonlinear equation were estimated by minimising the sum of the squared differences between the measured heat transfer rates and the corresponding model predictions. This involved a gradient based iterative procedure using a Gauss-Newton algorithm with Levenberg-Marquardt modifications [START_REF]User's Guide: 'Statistics Toolbox For Use with MATLAB [END_REF] to achieve convergence between the measured and predicted heat transfer rates. However, a multivariate nonlinear model of this type can be difficult to fit to the available data, unless the iteration process is started with "good" initial estimates of the values of the unknown parameters. Moreover gradient-based optimisation was made even more difficult in the present study as there were discontinuities in the data gathered in the tests (as is the case in many other experimental studies). To overcome these difficulties, a genetic algorithm (GA) was first of all used to yield preliminary values of the unknown parameters. Genetic algorithms are loosely based on the principles of biological evolution such as "survival of the fittest".

A number of sets of initial individual values of the parameters, a to f in equation ( 1), (the so-called initial population) were selected in a random fashion. These sets or cases are then assessed in terms of their ability to predict the rate of heat transfer in the exchanger. A proportion of the individual sets which provided the "best fit" was then retained and together with an appropriate number of new sets of parameter values constituted the population for the second generation. The process was repeated until an appropriate agreement between predicted and measured heat transfer rates was achieved.

The genetic algorithm therefore essentially searched a wide range of possible solutions in a computationally efficient manner. In this present study it was not necessary for the GA to fully converge since it was only employed to provide preliminary estimates of the unknown parameters in the regression equation which were reasonably close to the optimum values. These GA-estimated parameters can then be further refined by means of the gradient-based technique to yield the optimal solution. A comprehensive review of the use of genetic algorithms in thermal systems can be found in Sen and Yang [START_REF] Sen | Applications of Artificial Neural Networks and Genetic Algorithms in Thermal Engineering[END_REF]. The NLR model was trained, tested and independently validated in the same manner as the neural network model. The overall performance of the ANN and NLR models is discussed in section 3.3 below.

Comparison of ANN and NLR Model's Predictions

Figure 6 presents the performance of the ANN model when all the data are used in the selection of the training, test and validation data (Simulation-1). There is very good agreement between the predicted and measured overall heat transfer rates. The differences can be more clearly seen from the corresponding error histograms which indicate that the ANN model was able to accurately generalise the experimental data with only a very small number of predictions (2 out of a total of 359 training, test and validation results) exceeding a 5% error margin. Overall, the Mean Absolute Error (MAE) of the ANN's predictions was 0.6%, 0.9% and 0.9% respectively for the training, test and validation data sets. In comparison, there was considerably greater scatter in the NLR model predictions with 17% (61 out of 359 results) in error by more than 5%, see Figure 7. This regression model is clearly less able to represent the experimental overall performance of the heat exchanger since the MAE for the training, test and validation data sets were 2.9%, 2.8% and 3.2% respectively.

When subjected to the more severe test described earlier in section 3.1 (Simulation-2) the performance of the ANN model was slightly degraded, although, even in this situation still only a relatively small proportion (4 out of 359) of the predictions exceeded the 5% error margin. In this more severe test the mean absolute prediction errors were 0.7%, 2.3% and 1.8% respectively for the training, test and validation data sets. In comparison, the performance of the NLR model was significantly poorer than that of the ANN model with a total of 72% of the predictions exceeding the 5% margin of error. In this case, the mean absolute prediction errors were 8.2%, 5.2% and 5.1% respectively for the training, test and validation data sets. Consequently the neural network model with its inherent ability to represent complex non-linear data provides superior predictions of the overall performance of this compact fin-tube heat exchanger. Various other performance indicators such as the Root Mean Square Error (RMSE) and correlation-coefficient (corr-coef) are also compared for the two models in Table 1 and it is again clear that the neural network model provides better representation of the behaviour of the exchanger.

Neural Network Representation of the Detailed Behaviour of the Exchanger

Whilst it is clear that an ANN can successfully predict the overall heat transfer rate, the ability to represent the local distorted exit air temperature distribution provides a more severe test of a neural model of the exchanger. Consequently a neural network was developed to predict the ten temperatures measured by the thermistor sensor as it was traversed in steps across the exit of the exchanger at the mid-height of the outlet duct. The input parameters used to train the network were the same as those specified in section 3.1 whilst the ten output nodes corresponded to the individual temperatures measured across the exit face of the exchanger. The optimised network in this case contained 7 neurons in the hidden layer so that it had a 6-7-10 arrangement of neurons. The ability of this trained ANN to predict this detailed behaviour of the plate-fin system was tested at inlet obstructions of 16.5% and 41.5% and ethylene glycol anti-freeze concentrations of 10% and 30% all of which conditions were not included in the training process, see Figure 8, which presents typical results. The overall error in predicting the output exit air temperatures (in C) was 2.5%. The distribution at the lower blockage of 16.5% was predicted more accurately since this temperature distribution is relatively undistorted and so is likely to be reasonably similar to that for the unobstructed case, which formed part of the training set. Nevertheless there is reasonably good agreement between the measurements and predictions even with the highly distorted flow, which occurs with an obstruction of 41.5%. Consequently an ANN appears to be capable of adequately representing the detailed thermal characteristics of the plate fin exchanger as well as its overall performance.

Diagnosis of the "Condition" of the Heat Exchanger.

As described previously, the performance of the heat exchanger can be markedly degraded as a result of flow maldistribution. In addition, over a longer term, fouling (i.e the gradual build-up of deposits on the heat exchanger surfaces) can also result in a similar significant deterioration in performance. This deterioration can be difficult to detect if the inlet conditions (temperatures, flow rates and fluid compositions) are subject to frequent variations. Neural networks with their inherent ability to represent complex non-linear systems have been used in fault diagnosis and condition monitoring applications so would appear to provide a potential means of identifying and categorising the deterioration in heat exchanger performance in the present study. The deteriorations in overall exchanger performance resulting from the insertion of different inlet obstructions are analogous those due to more common causes such as fouling. Consequently the ability of a trained ANN to detect and classify the level of obstruction (i.e. the degree of deterioration in heat exchanger performance) in the present tests was investigated.

A so-called Self-Organising-Map (SOM) neural network [START_REF] Vesanto | SOM Toolbox for Matlab 5[END_REF] was used in this study since this type of network can readily recognise and classify patterns in input data. Unlike the FFMLP network used previously the SOM network only has input and output layers. This latter layer also known as the competitive layer, see Figure 9, is organised as a two-dimensional hexagonal or rectangular grid and each input neuron is connected to each and every output neuron. Each competitive neuron is represented by a n-dimensional weight vector (a so-called codebook vector), where n is equal to the dimension of the input data vector. In a similar manner to that for the FFMLP network, input data (vectors) are presented via the input layer. However, the neurons on the competitive layer behave in a completely different fashion from those in error back-propagation networks. When an input vector is presented to the SOM, the Euclidean distances between each and every competitive neuron and the input vector are computed. The neuron whose weight vector is closest to the input vector (the socalled Best-Matching Neuron (BMN), is updated so that it is moved closer to the input pattern, a process known as vector quantisation). The topological neighbours of the BMN are updated similarly so that they are also moved closer to the input pattern, see Figure 10. Successive presentation of the input data results in a group of neurons becoming associated with the pattern of the data. Training of the network is relatively simple and fast so that, the SOM is more robust than a FFMLP network and hence is more suited for online applications.

The SOM in this study consisted of a 15x15 rectangular grid of neurons and half of the experimental data were randomly selected and used for training this network. The four input variables to the SOM were the inlet air temperature, the air mass flow rate, the ethylene glycol concentration and the product of the liquid mass flow rate and the temperature difference between the inlet and outlet liquid temperatures. During the training phase each input vector was also labelled to identify its class i.e. the particular obstruction with which it is associated. The data were presented to the SOM in a batch form and the competitive neurons "organised" themselves in response to the patterns of input vectors so that, upon convergence, specific groups of neurons were associated with each class, see Figure 11. Once the SOM has been trained and all the neurons were labelled with classes (corresponding to inlet obstructions of 0, 16.5, 33, 41.5, and 50%) of the training data with which they were associated, the map was then used to determine the degree of membership of any new 'unseen' input vector to each of the categories. This was achieved by first of all determining the five closest neurons and their corresponding neighbours (i.e. the best matching neurons) in the SOM to each new unseen input vector pattern. The Euclidean distances between this input vector and these best matching neurons were then calculated and used as input to a further Radial-Basis-Function (RBF) neural network. This final network then used these distances to calculate the probability of the input vector "belonging" to the particular groups of neurons associated with each of the obstructions. Consequently the RBF network determined the "degree of membership", as a fuzzy vector, of each new input pattern to each of the obstruction classes. The fuzzy output of the RBF network was subsequently "defuzzified" by allocating the input vector to the class with the highest probability of membership.

The ability of the SOM network to classify the degree of inlet air obstruction was simulated by training it with 50% of the test measurements (selected randomly) and so leaving the remaining 50% of the data to test the network performance. Results suggested that the SOM network was able to classify correctly the unseen test data with a reasonably high degree of success rate of 79.3% in comparison to a conventional Multiple Linear Regression model which only able to deliver a success rate of 20.1%. Therefore the SOM network was better able to represent the non-linearity in the heat exchanger data patterns. Moreover a level of success of approximately 79% is likely to be satisfactory for a monitoring system since remedial action to correct the deterioration in exchanger performance would not be instigated by an individual prediction of poor performance but would require a significantly large number of predictions falling in severe categories over a period of time. Therefore an artificial neural network may be capable of monitoring the condition of an exchanger even for situations in which there are substantial variations in the composition, temperatures and flow rates of the individual fluid streams.

In a separate exercise, the effect of increasing the amount of training data was also investigated. In these tests the percentage of total data used for training the SOM network was increased from 50% to 70% and then 90%, thus leaving the remaining 30% and 10% of data respectively for testing the performance of the network. The results (see Table 2) showed that the percentage of successful classifications of the test data was considerably improved from 79.3% to 88.6%. Therefore it can be concluded that the performance of the neural network monitoring system is sensitive to the amount of representative training data.

Conclusions

This study has demonstrated that, once trained, an artificial neural network was able to predict the overall heat transfer rate between the liquid and air steams in a compact fin-tube heat exchanger with a high degree of accuracy. The neural network predictions were in much closer agreement to the experimental data than corresponding predictions derived by the use of a conventional non-linear regression model. A SOM neural network was also able to classify the degree of inlet air obstruction with a success rate of approximately 80 to 90%. This level of success is likely to be satisfactory for a system to monitor the deterioration in heat exchanger performance since any consequent remedial action would not be instigated by an individual prediction of poor performance but would require a significantly large number of predictions indicating severe underperformance over a period of time. Therefore the application of artificial neural networks appears to have potential for monitoring the condition of an exchanger even for situations in which there are substantial variations in the composition, temperatures and flow rates of the individual fluid streams.

Figure1:Figure 2 :

 2 Figure1: A schematic representation of the heat exchanger system

Figure 3 :Figure 4 :Figure 5 :Figure 6 :Figure 7 :Figure 8 :

 345678 Figure 3: Effect of Inlet Obstructions on the Heat Exchanger Performance at Different Air Flow Rates

Figure 9 .

 9 Figure 9. Typical SOM with Rectangular Grid of 5  5 Competitive Neurons

Figure 10 :

 10 Figure 10: Updating the BMN and Its Neighbours towards an Input Pattern x. The Solid and Dashed Lines Correspond to the Neurons' Position Before and After Updating respectively.

Figure 11 :

 11 Figure 11: Architecture of the ANN for Classification of the Inlet Obstructions

Table 1 :

 1 Comparison of ANN and NLR Model Predictions

	Performance indicators	Training data ANN NLR	Simulation 1 Test data ANN NLR	Validation data ANN NLR
	MAE (%)	0.6	2.9	0.9	2.8	0.9	3.2
	RMSE	0.017	0.086	0.030	0.077	0.032	0.092
	Corr-coef	0.999	0.980	0.998	0.984	0.997	0.975
	Performance indicators	Training data ANN NLR	Simulation 2 Test data ANN NLR	Validation data ANN NLR
	MAE (%)	0.7	8.2	2.3	5.2	1.8	5.1
	RMSE	0.022	0.227	0.062	0.163	0.053	0.143
	Corr-coef	0.999	0.980	0.991	0.935	0.995	0.946

Table 2 :

 2 The Effect of the Size of the Training Data on the Performance of the SOM network.

	% of data used for training	Number of mis-classifications	% success rate
	50	36 out of 174	79.3
	70	17 out of 104	83.6
	90	4 out of 35	88.6

tan-sigmoid transfer function is an exponential function that maps the input to the interval (-1,1)


V Bianco

F Chiacchio

O Manca

S Nardini

NUMERICAL INVESTIGATION OF NANOFLUIDS FORCED CONVECTION IN CIRCULAR TUBES

Keywords: Nanofluids, Nanoparticles, Discrete phase model, Laminar convection

In this paper, developing laminar forced convection flow of a water-Al 2 O 3 nanofluid in a circular tube, submitted to a constant and uniform heat flux at the wall, is numerically investigated.

A single and two-phase model (discrete particles model) is employed with either constant or temperaturedependent properties. The investigation is accomplished for size particles equal to 100 nm. The maximum difference in the average heat transfer coefficient between single and two phase models results is about 11%.

Convective heat transfer coefficient for nanofluids is greater than that of the base liquid. Heat transfer enhancement increases with the particle volume concentration, but it is accompanied by increasing wall shear stress values. Higher heat transfer coefficients and lower shear stresses are detected in the case of temperature dependents models. The heat transfer always improves, as Reynolds number increases, but it is accompanied by an increase of shear stress too.

Moreover a comparison with data present in the literature is carried out.

Convective heat transfer is very important for many industrial heating or cooling equipments. The heat convection can passively be enhanced by changing flow geometry, boundary conditions or by enhancing fluid thermophysical properties. An innovative way of improving the thermal conductivities of fluids is to suspend small solid particles in the fluid. Maxwell [START_REF] Maxwell | Electricity and Magnetism[END_REF][START_REF] Maxwell | A Treatise on Electricity and Magnetism[END_REF] showed the possibility of increasing thermal conductivity of a mixture by more volume fraction of solid particles. These fluids containing colloidal suspended nanoparticles have been called nanofluids. Several investigations revealed that nanofluid heat transfer coefficient could be increased by more than 20% also in the case of very low nanoparticles concentrations [START_REF] Li | Heat transfer enhancement of nanofluids[END_REF][START_REF] Kang | Estimation of thermal conductivity of nanofluid using experimental effective particle volume[END_REF].

Nowadays there is a fast growth of research activities in this heat transfer area [START_REF] Khanafer | Buoyancy-driven heat transfer enhancement in a two-dimensional enclosure utilizing nanofluids[END_REF][START_REF] Khaled | Heat transfer enhancement through control of thermal dispersion effects[END_REF][START_REF] Das | Heat Transfer in Nanofluids-A Review[END_REF][START_REF] Buongiorno | Convective Transport in Nanofluids[END_REF][START_REF] Wang | Heat transfer characteristics of nanofluids: a review[END_REF], because the impact of nanofluid technology is expected to be relevant considering that heat transfer performance of heat exchangers or cooling devices is fundamental in many industries [START_REF] Murshed | Thermophysical and electrokinetic properties of nanofluids -A critical review[END_REF]. Recently an industrial application was presented by Kulkarni et al. [START_REF] Kulkarni | Application of aluminum oxide nanofluids in diesel electric generator as jacket water coolant[END_REF], that proposed aluminum oxide nanofluids as jacket water coolant in a diesel engine for electric generation, showing that the efficiency of waste heat recovery heat exchanger increased due to nanofluid, because of its superior convective heat transfer coefficient. Moreover Nnanna et al. [START_REF] Nnanna | Assessment of thermoelectric module with nanofluid heat exchanger[END_REF] developed a nanofluid heat exchanger for electronic cooling devices. They showed how the system performance improved with respect to conventional equipments.

It was demonstrated that solid nanoparticle colloids are extremely stable and exhibit no significant settling under static conditions, even after weeks or months [START_REF] Wang | Heat transfer characteristics of nanofluids: a review[END_REF][START_REF] Eastman | Novel thermal properties of nanostructured materials[END_REF]. However, the development of nanofluids is still hindered by several factors such as the lack of agreement among experimental results from different research groups, poor characterization of suspensions and the lack of theoretical understanding of the heat transfer mechanisms [START_REF] Wang | Heat transfer characteristics of nanofluids: a review[END_REF]. Different concepts and models have been proposed to explain the enhancement of heat transfer [START_REF] Buongiorno | Convective Transport in Nanofluids[END_REF][START_REF] Xuan | Heat transfer enhancement of nanofluids[END_REF][START_REF] Xuan | Conceptions for heat transfer correlations of nanofluids[END_REF][START_REF] Keblinski | Mechanisms of heat flow in suspensions of nanosized particles (nanofluid)[END_REF][START_REF] Wang | Thermal conductivity of nanoparticle-fluid mixture[END_REF].

Theoretical and experimental investigations have been accomplished to estimate the effective thermal conductivity of nanofluids. Some experimental studies [START_REF] Choi | Anomalous thermal conductivity enhancement in nanotube suspension[END_REF][START_REF] Yoo | Study of thermal conductivity of nanofluids for the application of heat transfer fluids[END_REF] show that the measured thermal conductivity of nanofluids is much larger than the classical theoretical predictions [START_REF] Hamilton | Thermal conductivity of heterogeneous two-component systems[END_REF]. Other experimental investigations [START_REF] Putman | Thermal conductivity of nanoparticle suspensions[END_REF][START_REF] Zhang | Effective thermal conductivity and thermal diffusivity of nanofluids containing spherical and cylindrical nanoparticles[END_REF] revealed that the thermal conductivity has not shown any anomalous enhancement and for lower volume fractions, the results agree well with the classical equations [START_REF] Hamilton | Thermal conductivity of heterogeneous two-component systems[END_REF][START_REF] Yamada | Effective thermal conductivity of dispersed materials[END_REF]. Many attempts have been made to formulate efficient theoretical models for the prediction of the effective thermal conductivity, but this topic is still seriously incomplete [START_REF] Xue | Model for effective thermal conductivity of nanofluids[END_REF][START_REF] Xuan | Aggregation structure and thermal conducting of nanofluids[END_REF][START_REF] Chon | Empirical correlation finding the role of temperature and particle size for nanofluid (Al2O3) thermal conductivity enhancement[END_REF].

Relatively, few theoretical and experimental investigations have been reported on convective heat transfer in confined flows, as also reviewed in [START_REF] Das | Heat Transfer in Nanofluids-A Review[END_REF][START_REF] Buongiorno | Convective Transport in Nanofluids[END_REF][START_REF] Wang | Heat transfer characteristics of nanofluids: a review[END_REF][START_REF] Daungthongsuk | A critical review of convective heat transfer of nanofluids[END_REF].

Experimental results were obtained on convective heat transfer for laminar and turbulent flow of a nanofluid inside a tube in [START_REF] Xuan | Heat transfer enhancement of nanofluids[END_REF][START_REF] Pak | Hydrodynamic and heat transfer study of dispersed fluids with submicron metallic oxide particles[END_REF][START_REF] Xuan | Investigation on convective heat transfer and flow features of nanofluids[END_REF]. Correlations for the Nusselt number, using nanofluids composed of water and Cu, 56 nm)/water based nanofluids flowing through a copper tube in laminar regime was reported in [START_REF] Wen | Experimental investigation into convective heat transfer of nanofluids at the entrance region under laminar flow conditions[END_REF]. It was observed that the improvement of the heat transfer coefficient is particularly large at the entrance region, and it decreases with the axial distance. The heat transfer performance of CNT nanofluids in a tube was investigated in [START_REF] Ding | Heat transfer of aqueous suspensions of carbon nanotubes (CNT nanofluids)[END_REF]. Results showed that the enhancement of heat transfer coefficient is significantly higher than the increase in the effective thermal conductivity.

Numerical investigations on nanofluids are carried out using two approaches. The first approach assumes that the continuum assumption is still valid for fluids with suspended nanosize particles, while the other approach considers a two-phase model for better description of both the fluid and the solid phases. Another approach is to adopt the Boltzmann theory.

The single phase model with physical and thermal properties, all assumed to be constant with temperature, was employed in [START_REF] Maiga | Heat transfer behaviours of nanofluids in a uniformly heated tube[END_REF][START_REF] Maiga | Heat transfer enhancement by using nanofluids in forced convection flows[END_REF][START_REF] Maiga | Heat transfer enhancement in turbulent tube flow using Al2O3 nanoparticle suspension[END_REF][START_REF] Roy | Numerical investigation of laminar flow and heat transfer in a radial flow cooling system with the use of nanofluids[END_REF]. The hydrodynamic and thermal characteristics of nanofluids, flowing through a uniformly heated tube, in both laminar and turbulent regimes with adjusted properties, was investigated in [START_REF] Maiga | Heat transfer behaviours of nanofluids in a uniformly heated tube[END_REF]. The advantages of nanofluids with respect to heat transfer were discussed in [START_REF] Maiga | Heat transfer enhancement by using nanofluids in forced convection flows[END_REF], but it was also found that the inclusion of nanoparticles introduced drastic effects on the wall shear stress. A new correlation was proposed in [START_REF] Maiga | Heat transfer enhancement in turbulent tube flow using Al2O3 nanoparticle suspension[END_REF] to describe the thermal performance of Al 2 O 3 -water nanofluids under turbulent regime and a numerical study of heat transfer for water-Al 2 O 3 nanofluids in a radial cooling system was accomplished in Roy et al. [START_REF] Roy | Numerical investigation of laminar flow and heat transfer in a radial flow cooling system with the use of nanofluids[END_REF]. They found that addition of nanoparticles in the base fluid increases the heat transfer rates considerably.

Laminar forced convection flow of nanofluids between two coaxial and parallel disks with central axial injection was considered using temperature dependent nanofluid properties in [START_REF] Palm | Heat transfer enhancement with the use of nanofluids in radial flow cooling systems considering temperature dependent properties[END_REF]. The single phase flow model was solved numerically. Results have shown that considerable differences are obtained when using constant property nanofluids with respect to nanofluids with temperature dependent properties. It has been found that to an increase in wall heat flux corresponds an increase in the average heat transfer coefficient whilst the wall shear stress decreases when temperature-dependent nanofluid properties are used. A numerical study on fully developed laminar mixed convection of a nanofluid consisting of water and Al 2 O 3 in a horizontal curved tube was carried out in [START_REF] Akbarinia | Numerical study of laminar mixed convection of a nanofluid in horizontal curved tubes[END_REF]. Three-dimensional elliptic governing equations were used and the single phase model was employed. It was observed that the fraction of nanoparticles volume does not have a direct effect on the secondary flow, axial velocity and the skin friction coefficient. For a given Reynolds number, a negative effect of buoyancy force on the Nusselt number is obtained while the concentration of nanoparticles has a positive effect on the heat transfer enhancement and also on the reduction of skin friction.

The conjugate heat transfer problem for microheat sinks, considering two types of nanofluids, was numerically investigated in [START_REF] Koo | Laminar nanofluid flow in micro-heat sinks[END_REF]. The effect of Brownian motion on the effective fluid viscosity was considered and found to be less significant than that on the effective thermal conductivity. A numerical investigation on the cooling performance of a microchannel heat sink with nanofluids was carried out in [START_REF] Jang | Cooling performance of a microchannel heat sink with nanofluids[END_REF]. A theoretical model of thermal conductivity of nanofluids that accounts for the fundamental role of Brownian motion was used.

Results showed that nanofluids reduced both the thermal resistance and the temperature difference between the heated microchannel wall and the coolant.

The two phase approach seems a better model to describe the nanofluid flow. In fact, the slip velocity between the fluid and particles might not be zero [START_REF] Xuan | Heat transfer enhancement of nanofluids[END_REF] due to several factors such as gravity, friction between the fluid and solid particles, Brownian forces, Brownian diffusion, sedimentation and dispersion. The two-phase approach provides a field description of the dynamics of each phase or, alternatively, the Lagrangian trajectories of individual particles coupled with the Eulerian description of the fluid flow field [START_REF] Gidaspow | Multiphase Flow and Fluidization[END_REF][START_REF] Fan | Principle of Gas-Solid Flows[END_REF]. A two phase mixture model was applied to study the turbulent forced convection flow of a nanofluid in a uniformly heated tube [START_REF] Behzadmehr | Prediction of turbulent forced convection of a nanofluid in a tube with uniform heat flux using a two phase approach[END_REF] and more recently Namburu et al. [START_REF] Namburu | Numerical study of turbulent flow and heat transfer characteristics of nanofluids considering variable properties[END_REF] investigated the turbulent convection flow of nanofluids inside a tube considering variable properties.

In this paper, developing laminar forced convection flow of a nanofluid in a circular tube is numerically investigated. Steady state of a two dimensional axial symmetric flow is considered and the circular tube is heated at uniform heat flux. The study is carried out for water with alumina particles with a spherical size of 100 nm diameter. The CFD commercial code, Fluent [START_REF]Fluent 6.2 User Manual[END_REF], is employed to solve the problem by means of finite volume method. Single phase and discrete phase approaches are employed to evaluate the developing laminar forced convection flow, taking the constant and temperature variable thermophysical properties into account. A comparison between results obtained by two different models is accomplished in terms of temperature and velocity distributions and Nusselt number profiles. The numerical simulation results are also compared with experimental data of Wen and Ding [START_REF] Wen | Experimental investigation into convective heat transfer of nanofluids at the entrance region under laminar flow conditions[END_REF] and Heris et al. [START_REF] Heris | Experimental investigation of convective heat transfer of Al 2 O 3 /water nanofluid in circular tube[END_REF].

The aim of this paper is just to test the response of this model, evaluating its performance with smaller particles, because it has the great advantage to require just the thermophysical properties of the base fluid and particles as input. Nothing is required about the whole mixture, so it could be used as a first step analysis to check the performance of a new nanofluid. axis so that half of the tube is considered.

MATHEMATICAL MODELLING

The single phase model, which has been used frequently for nanofluids, is also implemented to compare its predictions with the mixture model. The following equations represent the mathematical formulation of the single phase model [32-36, 43, 46] and of the continuous phase of the two-phase model [START_REF] Das | Nanofluids science and technology[END_REF][START_REF] Minkowycz | Handbook of numerical heat transfer[END_REF]:

Conservation of mass: 0 ) ( = → V div ρ (1)
Momentum equation:

m S V gradP V V div + ∇ ⋅ ∇ + - = → → → µ ρ ) ( (2)
Energy equation:

e S kgradT div CpT V div + = → ) ( ) (ρ (3)
The compression work and the viscous dissipation are assumed negligible in the energy equation; the source/sink terms S m and S e represent the integrated effects of momentum and energy exchange with base fluid, as shown in the following, and they are equal to zero in the case of single phase model.

Discrete phase is made of spherical particles following the model given by Ounis et al. [START_REF] Ounis | Brownian diffusion of submicrometer particles in the viscous sublayer[END_REF]. Accordingly, motion equation is expressed in a Lagrangian form, to obtain the following expression [START_REF] Das | Nanofluids science and technology[END_REF][START_REF] Minkowycz | Handbook of numerical heat transfer[END_REF][START_REF] Ounis | Brownian diffusion of submicrometer particles in the viscous sublayer[END_REF]:

F g V V F dt V d p p p D p + - + - = ρ ρ ρ ) ( ) ( (4)
where F is an additional term that can eventually include important additional forces under determined circumstances (i.e. forces that arise due to rotation of reference frame, thermophoretic force, Brownian force), F D (V -V p ) is resistance force per particle mass unit. Eq. ( 4) has a general validity, because it is simply the expression of a force balance on a particle immersed in a fluid. To solve Eq. (4a), it needs to specify the drag coefficient F D and it can be done using the Stokes' law. At this point a first limitation is imposed to the model, because the Stokes' law is valid for Re d 0.1 [START_REF] Asano | Mass Transfer -From fundamentals to modern industrial applications[END_REF], where Re d is defined as: where λ is the particle mean free path. Cunningham correction is necessary to apply the Stokes' resistance law to submicrometer particles [START_REF] Das | Nanofluids science and technology[END_REF][START_REF] Ounis | Brownian diffusion of submicrometer particles in the viscous sublayer[END_REF].

Once solved Eq. ( 4), it is possible to evaluate the momentum transfer between particles and base fluid, computed by examining the change in momentum of a particle as it passes through each control volume in the model. This momentum change is calculated as [START_REF]Fluent 6.2 User Manual[END_REF][START_REF] Minkowycz | Handbook of numerical heat transfer[END_REF]:

= np p p m dt V d V m S δ (2a)
where V is the cell volume and np is the number of particles within a cell volume and those cells with np=0 are assigned a zero value for the source terms.

The same approach used for momentum equation can be employed for energy equation and for spherical particles the following equation is obtained [START_REF] Das | Nanofluids science and technology[END_REF][START_REF] Minkowycz | Handbook of numerical heat transfer[END_REF]:

( ) p p p p T T d h dt dT Cp - = 6 ρ ( 6
)
where h is calculated from the Ranz and Marshall correlation [START_REF] Ranz | Evaporation from Drops, Part I[END_REF][START_REF] Ranz | Evaporation from Drops, Part II[END_REF]:

3 / 1 2 / 1 Pr Re 6 . 0 0 . 2 ⋅ ⋅ + = ⋅ = d bf k d h Nu (6a)
valid for 1<Re d •Pr 2/3 <5•10 4 , where Re d is defined in Eq. 5a.

Following the same approach used for the momentum equation, it is now possible to calculate the source term, S e , for the energy equation [START_REF]Fluent 6.2 User Manual[END_REF][START_REF] Minkowycz | Handbook of numerical heat transfer[END_REF]:

⋅ = np p p e dt dT Cp V m S δ (3a)
The main approximation of the DPM model applied to the nanoparticles is represented by the Ranz and Marshall correlation, which was developed for submicrometer particles and, moreover, in the present case Re d is slightly outside from the lower limit, being around 0.5.

Boundary conditions

At the tube inlet, profiles of uniform axial velocity V 0 , temperature T 0 (=293 K) prevail. Moreover, in the case of temperature dependent properties, the reference viscosity value for Re calculation is taken at T 0 . At the tube exit section, the fully developed conditions prevail, that is to say that all axial derivatives are zero. No-slip conditions and uniform heat flux are imposed on the tube wall. As noted earlier, both the flow and thermal fields are assumed symmetrical with respect to the vertical plane passing through the tube main axis.

Physical properties of the nanofluids for single phase model approach

The determination of nanofluid properties is, as previously mentioned, at the center of current nanofluid research. Base nanofluid properties have been published over the past few years. However, only recently have some data on temperature-dependent properties been provided, even though they are only for nanofluid effective thermal conductivity and effective absolute viscosity and for a few particle loadings.

For single phase model, the following formulas were used to compute the thermal and physical properties of the considered nanofluid:

Density

In the absence of experimental data for nanofluid densities, constant-value temperature independent values, based on nanoparticle volume fraction, are used:

p bf nf ϕρ ρ ϕ ρ + - = ) 1 ( (7)

Specific heat

Similarly, in the absence of experimental data relative to nanofluids, it has been suggested that the effective specific heat can be calculated using the following equation as reported in [START_REF] Pak | Hydrodynamic and heat transfer study of dispersed fluids with submicron metallic oxide particles[END_REF][START_REF] Maiga | Heat transfer behaviours of nanofluids in a uniformly heated tube[END_REF][START_REF] Maiga | Heat transfer enhancement by using nanofluids in forced convection flows[END_REF][START_REF] Maiga | Heat transfer enhancement in turbulent tube flow using Al2O3 nanoparticle suspension[END_REF][START_REF] Roy | Numerical investigation of laminar flow and heat transfer in a radial flow cooling system with the use of nanofluids[END_REF][START_REF] Palm | Heat transfer enhancement with the use of nanofluids in radial flow cooling systems considering temperature dependent properties[END_REF][START_REF] Das | Nanofluids science and technology[END_REF]:

p bf nf Cp Cp Cp ϕ ϕ + - = ) 1 ( (8a)
Other authors suggest an alternative approach based on heat capacity concept [START_REF] Eastman | Novel thermal properties of nanostructured materials[END_REF][START_REF] Xuan | Conceptions for heat transfer correlations of nanofluids[END_REF]:

( ) ( ) ( ) p bf nf Cp Cp Cp ρ ϕ ρ ϕ ρ + - = ) 1 ( (8b)
These two formulations may of course lead to different results for specific heat. Due to the lack of experimental data, both formulations can be considered equivalent in estimating nanofluid specific heat capacity [START_REF] Palm | Heat transfer enhancement with the use of nanofluids in radial flow cooling systems considering temperature dependent properties[END_REF]. Moreover the difference in the Cp calculation using the two approach is less than 10% for the cases considered in this present work. In this paper, Eq. (8a) is considered for the calculations.

Dynamic viscosity

In this work, dynamic viscosity dependence only on is considered in one case, then the variability with the temperature is taken into account in the other case.

In the first case, in order to evaluate nanofluid dynamic viscosity, a least-square curve fitting, based on some scarce experimental data available in [START_REF] Wang | Thermal conductivity of nanoparticle-fluid mixture[END_REF][START_REF] Masuda | Alteration of thermal conductivity and viscosity of liquid by dispersing ultra-fine particles (dispersion of γ-Al 2 O 3 , SiO 2 and TiO 2 ultra-fine particles)[END_REF][START_REF] Lee | Measuring thermal conductivity of fluids containing oxide nanoparticles[END_REF] was performed by Maiga et al. [START_REF] Maiga | Heat transfer behaviours of nanofluids in a uniformly heated tube[END_REF][START_REF] Maiga | Heat transfer enhancement by using nanofluids in forced convection flows[END_REF], leading to the following equation:

1 3 . 7 123 2 + + = = ϕ ϕ µ µ µ bf nf r (9a)
In the second case it is assumed that temperature dependant nanofluid properties will yield even better predictions with respect to constant properties. In the present work, the dynamic viscosity is evaluated by means of the polynomial curve fitting proposed in [START_REF] Palm | Heat transfer enhancement with the use of nanofluids in radial flow cooling systems considering temperature dependent properties[END_REF], based on the data reported in [START_REF] Putra | Natural convection of nanofluids[END_REF]. The resulting equations were used to compute the nanofluid effective viscosity, as a function of the local temperature, expressed in K as given in [START_REF] Palm | Heat transfer enhancement with the use of nanofluids in radial flow cooling systems considering temperature dependent properties[END_REF]

⋅ + ⋅ - ⋅ = → = µ ϕ (9b)

Thermal conductivity

For the thermal conductivity determination, the same criteria used for the dynamic viscosity is considered, thereby introducing the following equations, as given in [START_REF] Maiga | Heat transfer behaviours of nanofluids in a uniformly heated tube[END_REF][START_REF] Maiga | Heat transfer enhancement by using nanofluids in forced convection flows[END_REF] [START_REF] Murshed | Thermophysical and electrokinetic properties of nanofluids -A critical review[END_REF] obtained using the model proposed by [START_REF] Hamilton | Thermal conductivity of heterogeneous two-component systems[END_REF], assuming spherical particles. Such model, which was first developed based on data from several mixtures containing relatively large particles (i.e. millimeter and micrometer size particles) is believed to be acceptable for use with nanofluids, although it may give underestimated values of thermal conductivity [START_REF] Maiga | Heat transfer behaviours of nanofluids in a uniformly heated tube[END_REF][START_REF] Maiga | Heat transfer enhancement by using nanofluids in forced convection flows[END_REF].

In the second case, temperature dependent thermal conductivity is determined by the equation suggested in [START_REF] Palm | Heat transfer enhancement with the use of nanofluids in radial flow cooling systems considering temperature dependent properties[END_REF], derived from the experimental data given in [START_REF] Putra | Natural convection of nanofluids[END_REF]. The following equations, with T in K, are considered For the first case, the thermophysical properties of base fluid are: bf =998.2 kg/m 3 ; Cp bf =4182 J/(kg K); k bf =0.597 W/(m K); µ bf =9.93x10 -4 kg/(m s) For the second case, the thermophysical properties of base fluid, with T expressed in K, are: where and Cp are independent of temperature and equal to the previous value.

Numerical method

The computational fluid dynamic code FLUENT [START_REF]Fluent 6.2 User Manual[END_REF] is employed to solve the present problem. The governing equations (1-3) are solved by control volume approach. This method is based on the spatial integration of the conservation equations over finite control volumes, converting the governing equations to a set of algebraic equations. The algebraic "discretized equations", resulting from this spatial integration process, are sequentially solved throughout the physical domain considered. FLUENT solves the systems resulting from discretization schemes using a numerical method. The residuals resulting from the integration of the governing equations (1-3) are considered as convergence indicators.

In order to ensure the accuracy as well as the consistency of numerical results, several non-uniform grids were subjected to an extensive testing procedure for each of the cases considered.

Results obtained for a particular test case showed that, for the tube flow problem under consideration, the 10 x 12 x 400 non-uniform grid appears to be satisfactory to ensure the precision of numerical results as well as their independency with respect to the number of nodes used. Such grid has 10, 12 and 400 nodes along the radial, tangential and axial directions respectively, with highly packed grid points in the vicinity of the tube wall, especially at the entrance region.

The computer model has been successfully validated with correlation reported in [START_REF] Churchill | Correlations for forced convection with uniform heating in flow over a plate and in developing and fully developed flow in a tube[END_REF] for thermally and hydraulically developing flow, showing an average error less than 2%, as reported in Fig. 1b, where local Nusselt number is calculated according to the following definition:

) ( ) ( k D z h z Nu ⋅ = (14)
and h(z) is defined as:

b w z T z T q z h ) ( ) ( ) ( - = (15)
From the previous equation the h is calculated as follow:

= L av dz z h L h 0 )
( 1 [START_REF] Keblinski | Mechanisms of heat flow in suspensions of nanosized particles (nanofluid)[END_REF] and the average Nusselt number is defined as:

0 k D h Nu av av ⋅ = (17)

RESULTS

Results were carried out employing the single phase and discrete phase models for ϕ=1% and 4%, Re=250, 500, 750 and 1050 and q=5000, 7500 and 10000 W/m 2 for both constant and temperature-dependent properties. In all cases, the size of the spherical particles is considered equal to 100 nm.

Figure 2a shows the profiles of axial velocity along tube radius up to z/L=0.2 for Re=250, q=5000 W/m 2 and ϕ=4%. It can be observed that nanofluid motion is fully developed at exit section which is located 1.00 m after the inlet section. For single phase model, when properties are not dependent on temperature, velocity profile and hydrodynamic entrance length are nearly independent on concentration value. Velocity axial profiles are nearly identical at z/L=0.2 for ϕ=0%, 1% and 4%, as shown in Fig. 2b.

Thermal entrance length depends on Prandtl number too, so when concentration increases Pr number also increases and consequently, thermal entrance length becomes greater. Dimensionless temperature of the fluid at several axial locations along the radius is reported in Fig. 3 for ϕ=0% and 4%. For both the base fluid and the nanofluid, the motion is not thermally developed. As the concentration increases the thermal entrance length rises, as it is also noticed by a higher slope of relative local heat transfer coefficient, h r , for ϕ=4% shown in Fig. 4a. This figure also clearly shows the enhancement of convective heat transfer due to the presence of nanoparticles. At the exit section, the increment of the heat transfer coefficient is 14% for constant properties and 17% for temperature dependent properties, in the case of ϕ=4%.The advantage is particularly great at the entrance section.

Relative local coefficient is always decreasing with axis location when fluid properties are constant, whereas temperature-dependent properties make h r to increase with z. This happens because, in the case of temperature dependent properties, there is a linear increase of thermal conductivity with temperature, and therefore, a better heat transfer between wall and fluid exists. Consequently, there is a decrease in the temperature difference between the wall and bulk temperature and with the heat flux on the wall being constant, there is an increase in the heat transfer coefficient, as shown in Figs. 4b and4c.

From Fig. 4a, in the case of temperature dependent properties, it is possible to observe an increase in the curves slopes, which is due to the fact that the increase in h nf is greater than h bf . If the two coefficients (h nf and h bf ) had the same increase, the curves of Fig. 4a for constant and temperature dependent properties would have overlapped.

The case with variable properties presents, with respect to the case with constant properties, the ratio h nf /h bf with a larger increase because the thermal conductivity of the nanofluid has a higher increment than the simple water.

In Fig. 5a andb radial temperature profiles at z/L=1.0 for Re=250 and q=5000 W/m 2 are reported for the investigated models for constant and non-constant properties respectively. For constant properties, Fig. 5a, temperature is strongly dependent on concentration ϕ. When concentration increases fluid temperature decreases quickly, particularly near the tube wall; furthermore, the difference between temperature values, for base fluid and for nanofluid, increases as r increases, indicating that higher heat transfer rate, with nanoparticles, is achieved. For ϕ=1%, two phase model gives lower temperatures than single phase model for r/r 0 <0.15. Moreover, the difference between temperature at the axis (r/r 0 =0) and surface (r/r 0 =1) is lower for two-phase model. When the volume concentration of particles is higher (4%), temperature profile for twophase model is very similar to the one for single phase model and, on the surface, there is a temperature increase with respect to the reference value of 29 K in both models. When properties are temperaturedependent, Fig. 5b, for ϕ=1%, temperatures difference between the two models decreases with respect to the case of constant properties and at tube wall, the temperature for two phase model is about 1 K lower than the one for single phase model. Moreover, in the core region (r/r 0 <0.2) there is a clear existence of a uniform temperature fluid zone that becomes more visible for higher value of ϕ.

Figures 6a andb show wall and bulk temperature profiles along tube axis for Re=250 and q=5000 W/m 2 for the single and two phase models and for constant and variable properties respectively. It can be noticed that the decrease of wall and bulk temperatures for a nanofluid, with respect to the base fluid, increases with the z coordinate. For a concentration of 1%, wall and bulk temperatures for the single phase model are higher than the ones for the two-phase model, whereas for ϕ=4%, temperature profiles for the two models are very similar particularly for constant properties.

These results have indicated the beneficial effects due to nanoparticles effects that may be mainly explained by the fact that, with the presence of such particles, the thermal properties of the resulting mixture have considerably improved, moreover additional effects such as gravity, drag on the particles, diffusion, Brownian forces play an important role [START_REF] Das | Nanofluids science and technology[END_REF]. In fact considering temperature constant properties, in the single phase model such effects are not considered and the increase in the average heat transfer coefficient is very similar to that of nanofluid vs. base fluid thermal conductivity. While, in the case of discrete phase model, where gravity and drag are taken into account, there is a higher increase of the average heat transfer coefficient, especially for ϕ=1%, as shown in Fig. 7a. Therefore, the nanofluid offers, as expected, higher thermal capability than the base fluid. It is also noted that with higher thermal conductivity of the mixture, the convective heat transfer between wall and fluid should consequently be more efficient.

Average heat transfer coefficient and Nusselt number are reported in Table 1 together with the relative increase of the total heat transfer rates and Nusselt number as a function of the nanoparticle volume fraction . As can be noticed, significant increases of the total heat transfer rates can be found with the use of suspended nanoparticles. For example, for = 4%, heat transfer rate and Nusselt number increase of about 20% and 16% respectively. Slight discrepancies are noticed between the models. Table 2 presents the average wall shear stress value, τ av , and the nanofluid-to-base fluid wall shear stress ratio (defined as τ r = τ nf /τ bf ) for the case Re =250 and q=5000W/m 2 and various particle concentrations. For pure water, τ av value is lower when properties are temperature-dependent because viscosity decreases with temperature. When ϕ=1%, τ av values for constant properties are higher than those for variable properties and in both cases τ av values, for two-phase model are higher than those for single phase model. When ϕ=4%, τ av values for constant properties are again higher than those for variable properties, whereas τ av values for twophase model are lower than those for single phase model. However, the difference between the values for single and two-phase models is small. The values of τ r, for ϕ=1%, are similar for the single and two-phase models, whereas for ϕ=4%, all the four models give very similar values.

Relative heat transfer coefficients are reported in table 3 for the investigated models, for the same Reynolds number (Re=250) and different heat fluxes (q=5000, 7500 and 10000W/m 2 ). The coefficients h r in the case of ϕ=1% and 4% and for non-dependent temperature properties are practically constant with heat flux and, although the coefficients are very similar, in the single phase model, they tend to increase slightly when properties are temperature dependent, while for the two-phase model, the coefficients do not change with the heat flux, but they result slightly higher when properties are constant. An analogous behaviour is observed for the relative wall shear stress τ r , as shown in Table 4. Wall shear stress decreases with heat flux in all the investigated cases because when heat transfer increases, bulk temperature rises too hence fluid viscosity decreases. As previously noticed the presence of nanoparticles increases the wall shear stress. For ϕ=1%, τ r is almost constant with heat flux, whereas for ϕ=4%, τ r slightly increases with q.

In general, τ r increases considerably with the particle volume fraction, ϕ, and this adverse effect is somehow expected, as it results from a strong increase of the mixture viscosity due to inclusion of nanoparticles.

The effect of Reynolds number on average heat transfer coefficient is shown in Figures 7a andb for the single and two phase model, at q=10000 W/m 2 . When the properties are constant, Fig. 7a, for ϕ=1%, some differences, in the range of about 8%-11%, are detected between the single and two phase model, particularly, In this paper the hydrodynamic and thermal behaviours of water-Al 2 O 3 nanofluids flowing inside a uniformly heated tube were numerically investigated in stationary conditions and for laminar flow. Four models were employed: single and two-phase models with either constant or temperature-dependent properties.

Results clearly showed that the inclusion of nanoparticles produced a considerable increase of the heat transfer with respect to that of the base liquid. Heat transfer enhancement was increasing with the concentration of particle volume. However it was accompanied by increasing wall shear stress values.

It can be said that temperature dependent models present higher values of heat transfer coefficients and Nusselt numbers because the difference between wall and bulk temperature is minimized, thereby maximizing the heat transfer. On the other hand, another positive effect is due to the decrease of shear stress, because viscosity decreases with temperature.

For each investigated concentration value, the heat transfer enhancement is higher for the greatest Reynolds number. The effect of Reynolds number is greater for temperature dependent properties. A good relationship was shown with the data from Wen and Ding [START_REF] Daungthongsuk | A critical review of convective heat transfer of nanofluids[END_REF], Maiga et al. [START_REF] Maiga | Heat transfer enhancement by using nanofluids in forced convection flows[END_REF] and Heris et al. [START_REF] Heris | Experimental investigation of convective heat transfer of Al 2 O 3 /water nanofluid in circular tube[END_REF].

As for the comparison between single and two phase models, their results are quite similar (the maximum difference detected is 11% for the average heat transfer coefficient), especially for ϕ=4% and in the case of variable properties. In our opinion, this is a good result, as this model could be used to test new nanofluids, since it requires information about just particles and base fluid, with no reference to the mixture. Therefore,it may result as a "winning first glance" approach in the study of new mixtures, considering also the low cost connected to it.

Figure 1a

TiO 2 and

 2 Al 2 O 3 nanoparticles were proposed. Enhancement of heat transfer performance over the base fluid for the assigned Reynolds number is observed. Experimental results for the convective heat transfer of Al 2 O 3(27-

Figure 1a shows the

  Figure 1a shows the geometrical configuration under consideration. It consists of a tube with a length (L) of 1.0 m and a diameter (D) of 0.01 m. The nanofluid considered is composed of water and Al 2 O 3 particles. The fluid enters with uniform temperature and axial velocity profiles at the inlet section. The tube has appropriate length in order to obtain fully developed profiles (velocity and thermal) at the outlet section (L/D=100). The condition of the axially and circumferentially uniform wall heat flux is considered in this study. Also, the flow and the thermal field are assumed to be symmetrical with respect to the vertical plane passing through the tube main

  the cases considered in the present work Re p 0.01, so the following form of the Stokes resistance law is considered[START_REF] Das | Nanofluids science and technology[END_REF][START_REF] Ounis | Brownian diffusion of submicrometer particles in the viscous sublayer[END_REF][START_REF] Asano | Mass Transfer -From fundamentals to modern industrial applications[END_REF]: factor C c is the Cunningham correction[START_REF] Das | Nanofluids science and technology[END_REF][START_REF] Ounis | Brownian diffusion of submicrometer particles in the viscous sublayer[END_REF]:

  :

  two phase model, the thermophysical properties of Al 2 O 3 are: p =3880 kg/m 3 ; Cp p =773 J/(kg K); k p =36 W/(m K)

  0

Figure 1 .

 1 Figure 1. (a) geometrical configuration under study and (b) grid validation by means of Churchill and Ozoe correlation [55].

Figure 2 .

 2 Figure 2. (a) profiles of axial velocity at several locations for Re=250, q=5000W/m 2 and ϕ=4% and (b) profiles of axial velocity at z=0.2m, for Re=250, q=5000W/m 2 and several concentration values.

Figure 3 .

 3 Figure 3. Dimensionless temperature for Re=250 and q=5000 W/m 2 at several locations and for: a) ϕ=0%; b) ϕ =4%.

Figure 4 .

 4 Figure 4. (a) Increase in nanofluid heat transfer coefficient along tube axis for Re=250 and q=5000 W/m 2 for constant and variables properties, (b) heat transfer coefficient for constant properties, (c) heat transfer coefficient for temperature dependent properties.

Figure 5 .

 5 Figure 5. Profiles of temperature along tube radius at z/L=0.99 for several concentrations and for Re=250 and q=5000 W/m 2 :a) constant properties; b) variable properties.

Figure 6 .

 6 Figure 6. Profiles of wall and bulk temperature along tube axis for several concentrations and for Re=250 and q=5000 W/m 2 : a) constant properties; b) variable properties.

Figure 7 .

 7 Figure 7. Average heat transfer coefficient as a function of Re for the single and two phase model for q=10000 W/m 2 : a) constant properties; b) variable properties.

Figure 8 .

 8 Figure 8. Nusselt number as function of Re for a) ϕ=1% and b) ϕ =4%.

Table 1 .

 1 Heat transfer coefficients for Re=250, q=5000W/m 2 and the investigated models.

	Table(s)
	Model	φ	[W/m 2 K] h av	h nf /h bf Nu av Nu nf /Nu bf
		Cst. 0%	350.8	1	5.88	1
	H 2 O
		Var. 0%	375.5	1	6.29	1
		Cst. 1%	363.9	1.037 5.93	1.009
	1-phase
		Var. 1%	397.7	1.059 6.48	1.030
		Cst. 1%	395.9	1.129 6.45	1.098
	2-phase
		Var. 1%	420.7	1.120 6.86	1.090
		Cst. 4%	413.8	1.180 6.75	1.148
	1-phase
		Var. 4%	443.6	1.181 7.23	1.150
		Cst. 4%	422.0	1.203 6.88	1.171
	2-phase
		Var. 4%	445.8	1.187 7.27	1.155

Table 2 .

 2 Average and relative wall shear stress for Re=250, q=5000W/m 2 and the investigated models.

	Model	φ	τ av [Pa] τ nf /τ bf
	Cst.	0% 0.0203 1
	H 2 O
	Var.	0% 0.0145 1
	Cst.	1% 0.0232 1.145
	1-phase
	Var.	1% 0.0171 1.180
	Cst.	1% 0.0280 1.381
	2-phase
	Var.	1% 1.419
	Cst.	4% 0.0403 1.987
	1-phase
	Var.	4% 0.0290 2.000
	Cst.	4% 0.0363 1.790
	2-phase
	Var.	4% 0.0278 1.920

Table 3 .

 3 Values of h nf /h bf for Re=250, several heat fluxes and the investigated models.

		φ=1%			φ=4%
		1-phase	2-phase	1-phase	2-phase
	q
		cst.	var. cst. var. cst.	var. cst.	var.
	[W/m 2 ]
	5000	1.037 1.059 1.129 1.120 1.180 1.181 1.203 1.187
	7500	1.037 1.069 1.129 1.120 1.180 1.195 1.205 1.184
	10000 1.037 1.079 1.130 1.120 1.180 1.209 1.205 1.183

Table 4 .

 4 Values of  nf / bf for Re=250, several heat fluxes and the investigated models.

		φ=1%		φ=4%
		1-phase	2-phase	1-phase	2-phase
	q
		cst. var.	cst. var.	cst. var.	cst. var.
	[W/m 2 ]
	5000	1.145 1.180 1.380 1.419	1.987 2.000	1.790 1.920
	7500	1.145 1.203 1.383 1.431	1.987 2.054	1.790 1.966
	10000 1.145 1.234 1.380 1.447	1.987 2.135	1.791 2.012

Table 5 .

 5 Comparison between numerical data from this work and experimental data from Wen and Ding [27]. Nu Nu nf /Nu bf Nu Nu nf /Nu bf Nu Nu nf /Nu bf Nu Nu nf /Nu bf Nu Nu nf /Nu bf

	Wen & Ding	1-phase		2-phase
	[22]		Cst.	Var	Cst.	Var
	z/D 26 12.3 1.15	8.89 1.01	9.65 1.03	10.30 1.18	10.87 1.16
	63 8	1.13	6.60 1.01	7.44 1.04	7.75 1.19	8.35 1.17

ACKNOWLEDGMENTS

This work was supported by MIUR with Articolo 12 D. M. 593/2000 Grandi Laboratori "EliosLab".

the two phase model which leads to overestimated values. When the concentration increases, these differences tend to reduce. In fact, for ϕ=4% they are in the range of about 2%-7%, with the highest deviation for Re=1050. When properties are temperature-dependent, Fig. 7b, the differences between the single and two phase models reduce for both examined concentrations. In fact when ϕ=1%, the differences between the two models are contained between 4% and 8%, while for ϕ=4%, they are between 2% and 4%, with the highest deviation always for Re=1050.

In Fig. 8a andb, Nusselt number as function of Re for the considered concentrations is reported. In the figure a comparison with the correlation given by Maiga et al. [START_REF] Maiga | Heat transfer enhancement by using nanofluids in forced convection flows[END_REF] and the experimental data of Heris et al. [START_REF] Heris | Experimental investigation of convective heat transfer of Al 2 O 3 /water nanofluid in circular tube[END_REF] is also carried out. In Fig. 8a it is possible to observe the Nu av behaviour for ϕ=1% and, except for Re=250, a good agreement is found with the correlation given in [START_REF] Maiga | Heat transfer enhancement by using nanofluids in forced convection flows[END_REF]. Moreover, a maximum deviation of about 17%, in the case of Re=1050, for the single phase model, is estimated. The experimental data presented in [START_REF] Heris | Experimental investigation of convective heat transfer of Al 2 O 3 /water nanofluid in circular tube[END_REF] are obtained for a tube with constant wall temperature, they are therefore corrected by means of a 20% increase in Nussult number. This correction derives from the fact that Nu number in a developing laminar flow in a circular tube is averagely 20% higher for constant heat flux boundary condition with respect to constant temperature [START_REF] Bejan | Convection Heat Transfer[END_REF].

The corrected data given in [START_REF] Heris | Experimental investigation of convective heat transfer of Al 2 O 3 /water nanofluid in circular tube[END_REF] are very close to the ones obtained with the single phase model and constant properties with an error of 2.3% and 0.1%, respectively, for Re=500 and 750. In Fig. 8b, average Nusselt number for ϕ=4% is shown and also in this case, except for Re=250, a good accord with the correlation given in [START_REF] Maiga | Heat transfer enhancement by using nanofluids in forced convection flows[END_REF] is confirmed. In fact, the maximum deviation is equal to 12% for Re=750 and single phase model with constant properties.

Finally, a comparison with the experimental data from Wen and Ding [START_REF] Wen | Experimental investigation into convective heat transfer of nanofluids at the entrance region under laminar flow conditions[END_REF] is carried out for Re=1050 for ϕ=1%. In Table 5, local Nusselt number and its ratio with local Nusselt number for base fluid are reported at z/D = 26 and 63 and for all the investigated models. The lower differences between experimental and numerical data are for two-phase model with temperature-dependent properties, the maximum gap being equal to 3.5% at z/D=63.