Prediction method of thermal conductivity of nanofluids based on radial basis function


Accurately predicting the thermal conductivity of nanofluids under various thermodynamic conditions is of great importance to promote the industrial application of nanofluids. Unfortunately, the accuracy and applicability of the current theoretical or empirical models cannot meet the demand, due to the inherent complex of nanofluids. In this study, an intelligent model, named radial basis function artificial neural network (RBF–ANNs), is developed to predict the thermal conductivity of nanofluids under various conditions. Five parameters including nanoparticle volume concentration, temperature, nanoparticle diameter, thermal conductivity of nanoparticle and thermal conductivity of base fluid are selected as the input variables. A total of 1444 experimental data samples are collected to optimize the structure of model. The RBF model is compared with six theoretical models and three intelligent models through statistical and graphical analyses. Also, trend analysis and sensitivity analysis are conducted to evaluate the influencing mechanism of nanoparticle concentration, temperature, nanoparticle size, thermal conductivities of base fluids and nanoparticle on the thermal conductivity of nanofluids. Meanwhile, the quality of the experimental data is evaluated by means of leverage algorithm. Results indicate the superiority of the RBF, especially when the data size is large. The overall correlation coefficient (R2), average absolute relative deviation (AARD%) and root-mean-squared error of the developed model are 0.9931, 2.715 and 0.0316, respectively. Among the five input parameters, the volume fraction of nanoparticles has the greatest impact on the thermal conductivity of nanofluid. The results of outlier detection demonstrate that the proposed RBF model and data samples are statistically valid.

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This work was supported by Yunnan Provincial Department of Education Science Research Fund Project (Grant No. 2018JS551, 2019J0025) and Scientific Research Foundation of Kunming Metallurgy College (Grant No. Xxrcxm201802).

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Correspondence to Hui Huang or Xiaobo Long.

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Appendix 1

See Tables 6 and 7.

Table 6 Details of the experimental data used in this study
Table 7 The weights and biases of the optimized RBF model

Appendix 2

Multilayer perceptron artificial neural networks (MLP–ANNs)

MLP has been widely applied for solving complex nonlinear problem with high accuracy. An MLP contains one input layer, one output layer and at least one hidden layer. Each layer consisted of numerous units (neuros) connected to the neurons in the following layers. The difference between RBF and MLP is the way that the neurons process the information [10]. The inputs to each neuron are multiplied by the weights (w), and the results are added with each other and with the bias (b) [21]. Then, the resultant value is obtained through a suitable activation function (f):

$$y_{\text{j}} = f\left( {\sum\limits_{i = 1}^{n} {w_{\text{ji}} x_{\text{i}} + b_{\text{j}} } } \right)$$

where xi is the ith input vector, wji is the connection weight between the jth neuron in hidden and the ith neuron in input layer, n is the number of neurons in hidden layer, yj is the output of the jth neuron, bj is the bias of jth neuron, and f represents the transfer function. Many researchers suggested that linear (purelin) function was suitable for the output layer, while the tangent sigmoid (tansig) function was suggested for the hidden layer due to their higher ability [10, 21, 64].

Optimization algorithm used for training the model plays a key role in the performance of MLP. Through training algorithm, the values of weight and bias are adjusted until the minimum deviation between predicted and experimental values is obtained. Backpropagation (BP) is the most common algorithm for training the MLP network. The inputs are feed-forward propagating in neural network. The weights and biases between neurons are optimized according to the errors between the predicted and experimental data, during the backward propagation process. Among the BP training algorithm, Levenberg–Marquardt backpropagation (LM) algorithm is selected to train the MLP model [21]. For now, there is no general method to determine the number of hidden layers in MLP and the number of neurons in hidden layer, and it is usually determined by trial-and-error method. For a complex system with larger amount of data, a MLP with two hidden layer was found to be more appropriate [10, 64]. Through the comprehensive calculation, the best structure of LM–MLP with the lowest error is found to be 5–12–8–1. The values of weights and biases are listed in Tables 8 and 9.

Table 8 The weights and biases for LM-MLP (part 1)
Table 9 The weights and biases for LM-MLP (part 2)

Least square support vector machine (LSSVM)

With the development of artificial intelligence (AI), support vector machine (SVM) has been developed to solve complex problems (classification, nonlinear regression, pattern recognition, etc..) with high accuracy and efficiency [10, 72, 73]. In an SVM, the inputs are mapped to a high-dimensional feature space using nonlinear mapping function. However, the efficiency of conventional SVM is less when a wide range of experimental data is being used, due to the inequality constraints. To overcome the difficulties of SVM, a modified version of SVM, named least square support vector machine (LSSVM), has been proposed. In LSSVM, regression error is added to the optimization constraints [10], and by this way, the convergence speed of LSSVM increases. LSSVM is suitable for solving the nonlinear problems, especially when the number of data is large.

In LSSVM, the penalized function is defined as

$$Q_{\text{LSSVM}} = \frac{1}{2}w^{\text{T}} w + \frac{1}{2}\gamma \sum\limits_{i = 1}^{n} {e_{\text{i}}^{2} }$$

subjected to the following equality constraints:

$$y_{\text{i}} = w^{\text{T}} g(x_{\text{i}} ) + b + e_{\text{i}} ,\quad i = 1,2,3, \ldots ,n$$

where ei is the error variable, γ is the regularization constant which shows the summation of regression errors, g (x) is the mapping function, w is the regression weight, b is the bias, n is the number of data, and superscript T is the transpose matrix.

The weight coefficient (w) is expressed as

$$w = \sum\limits_{i = 1}^{n} {\alpha_{\text{i}} x_{\text{i}} }$$

The Lagrangian multiplier (αi) is expressed as

$$\alpha_{\text{i}} = 2\gamma \cdot e_{\text{i}}$$

Considering the dependent and independent parameters in LSSVM is linearly separable, the equation will be modified as

$$y = \sum\limits_{i = 1}^{n} {\alpha_{\text{i}} } x_{\text{i}}^{\text{T}} x + b$$

Then, the Lagrange multiplier (αi) can be obtained as:

$$\alpha_{\text{i}} = \frac{{y_{\text{i}} - b}}{{x_{\text{i}}^{\text{T}} x + (2\gamma )^{ - 1} }}$$

For nonlinear regression problems, the Kernel function is introduced to combine with Eq. (B6)

$$y = \sum\limits_{i = 1}^{n} {\alpha_{\text{i}} } K(x_{\text{i}} ,x) + b$$

where the Kernel function K (xi, x) is defined as:

$$K(x_{\text{i}} ,x) = g(x_{\text{i}} ) \cdot g(x)^{\text{T}}$$

One of the most common used Kernel functions in LSSVM is Gaussian radial basis function because it has lower tuning parameters and less numerical adversity [73]. In this work, Gaussian Kernel is used for developing the LSSVM, as shown below

$$K(x_{\text{i}} ,x) = \exp \left( { - \frac{{\left\| {x_{\text{i}} - x} \right\|^{2} }}{{\sigma^{2} }}} \right),\quad i = 1,2,3, \ldots ,n$$

where σ2 is a tuning parameter that should be optimized by optimization algorithm during LSSVM training.

To now, there are two tuning parameters, namely σ2 and γ, which should be optimized to find the global optimum of the problem. In this study, cross-validation is employed for optimizing the parameters of LSSVM. The optimum values of σ2 and γ are 0.4886 and 46.821, respectively.

Adaptive neuro-fuzzy inference system (ANFIS)

Adaptive neuro-fuzzy inference system (ANFIS) is the integration of ANNs and fuzzy logic algorithms, and it has been known as a useful tool for solving intricate problem in many fields [22]. The advantages of ANNs and fuzzy logic algorithms are mutually complemented, and the deficiencies in either of the two methods are minimized by the other method [74]. The combination process is carried out through introducing fuzzy if–then rules and specific functions called membership functions. These MFs are subjected to tuning processes by utilizing the ability of ANNs. The ANFIS structure is composed of introductory part and concluding part, which are linked by a set of rules. For a simple Takagi–Sugeno ANFIS with two input variables (x1 and x2), the model can be constructed by the following rules:

$$\begin{aligned} {\text{Rule}}\;1\;{\text{If}}\left( {x_{1} \;{\text{is}}\;A_{1} } \right)\;{\text{and}}\;\left( {x_{2} \;{\text{is}}\;B_{1} } \right)\;{\text{then}}\;y_{1} = p_{1} x_{1} + q_{1} x_{2} + r_{1} \hfill \\ {\text{Rule}}\;2\;{\text{If}}\left( {x_{1} \;{\text{is}}\;A_{2} } \right)\;{\text{and}}\;\left( {x_{2} \;{\text{is}}\;B_{2} } \right)\;{\text{then}}\;y_{2} = p_{2} x_{1} + q_{2} x_{2} + r_{2} \hfill \\ \end{aligned}$$

where Ai and Bi are the fuzzy sets of corresponding input parameters. yi is the output of the system. pi, qi and ri are designing parameters which are obtained from learning.

Briefly speaking, the ANFIS structure consists of five distinct layers with each layer comprised of neurons, which forms a multilayer network [75]. The first layer is an adaptive layer in which fuzzy formation takes place. In the second layer, the input signal values in each neuron are multiplied by each other and fuzzy rules are performed. In the third layer, the firing strength of rules is normalized. The fourth layer is the conclusive part of fuzzy rules which is obtained from multiplication of normalized firing strength. The fifth layer contains single node, which is responsible for calculating the overall output of network. A more detailed information about ANFIS can be found in Mehrabi et al. [74].

The structure identification of ANFIS involves (1) selecting the input parameters, (2) input space partitioning, (3) choosing the appropriate number and the kind of membership functions for inputs, (4) creation of fuzzy rules, (5) premise and conclusion parts of fuzzy rules and (6) selection of the initial parameters for membership functions [76]. The ANFIS employs training methods developed from an ANNs for finding the suitable fuzzy rules and fuzzy membership functions. Considering the situation of data used for modeling as well as the capability of the network, subtractive clustering method is used to construct the ANFIS due to its independency to dimension of problem and faster training process [77]. The optimum value of influence radius (RD) is 0.4542, and Gaussian membership function is selected as it is associated with the minimum value of RMSE. The adaptive and consequent parameters set in ANFIS are obtained via the integration of backpropagation algorithm with least squares method. Details of the proposed ANFIS for predicting keff are listed in Table 10.

Table 10 Details of the proposed ANFIS for predicting keff

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Zhang, S., Ge, Z., Fan, X. et al. Prediction method of thermal conductivity of nanofluids based on radial basis function. J Therm Anal Calorim 141, 859–880 (2020).

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  • Effective thermal conductivity
  • Nanofluid
  • Radial basis function
  • Neural network
  • Sensitivity analysis
  • Outlier detection