传热传质

o o T l pr to lts tot ty a eter in and output targets of the training set, respectively, and together. The results show that the presented and accordingly it may help the manufacturer design an effective and energy-saving defrosting control strategy. rfaces ts in a d ulti keep t ing dev ral par characteristics [2–6]. Based on frost property correlations, the mathematical models are developed by solving the mathematic equations that describe the frost growth [6–12]. The experimental correlations and mathematical models have been successfully ap- plied in predicting frost growth. However, they usually present poor accuracy and generalization due to the frost growth typically SVM in the refrigeration for the frost growth prediction has been reported in open literature. In view of the mentioned above, the SVM technique is introduced to predict the frost growth in this study. The remainder of this study is organized as follows. In Section 2, a brief introduction of SVM for regression is described. Section 3 elaborates the application of SVM for the frost growth prediction, including data source, performance criteria, prediction results, sen- sitivity analysis of the parameters used in SVM model, and exper- imental results for Gaussian noise. Section 4 provides the concluding remarks. * Corresponding author. Tel.: +86 34 20 6260; fax: +86 21 34 20 6814. E-mail addresses: cao_zhikun@sjtu.edu.cn, cao_zhikun@163.com (Z. Cao), Applied Thermal Engineering 29 (2009) 2320–2326 Contents lists availab Applied Therma ev gubo@sjtu.edu.cn (B. Gu). accurately predicting the frost growth characteristics will help the manufacturer effective and energy-saving control of defrosting processes and better design of the air-to-refrigerant heat transfer equipment. Generally speaking, the methods of frost growth prediction can be divided into two general groups: experimental correlations and mathematical models. The experimental correlations are devel- oped by using experimental data fitting technique. Different corre- lations have been developed to calculate the frost growth machine (SVM) which implements the SRM principle (structured risk minimization), which demands fewer samples to obtain a good performance, i.e., less time and cost to obtain the experimental samples. In literature, one can find considerable amount of research about the application of SVM in various fields, such as pattern recognition, bioinformatics, text categorization, etc. [15]. And in all these studies, SVM has shown great capabilities in handling the complex real-world problem. However, no application of 1. Introduction Frost formation on refrigerated su tions occurs inevitably, which resul and an increase in pressure drop, an the equipment performance [1]. To operating condition, various defrost strategies must be built as an integ 1359-4311/$ - see front matter � 2008 Elsevier Ltd. A doi:10.1016/j.applthermaleng.2008.11.015 � 2008 Elsevier Ltd. All rights reserved. under operating condi- reduction in heat flux mately deterioration of he system in a desired ices with their control t of the system. Hence, characterized by nonlinearity and rather difficult to be described mathematically. Considering the limitation of the method mentioned above, artificial neural algorithms have been lately preferred for thermal application [13]. Recently, Liu and Tang [14] successfully intro- duced the artificial neural network model to simulate the frost growth. Unlike the traditional artificial neural algorithms that implement the ERM (empirical risk minimization), support vector Support vector machine (SVM) Prediction model model is very suited to the frost growth prediction with high accuracy and good robust against noise, A novel prediction model of frost growth vector machine Zhikun Cao *, Hua Han, Bo Gu, Neng Ren Institute of Refrigeration and Cryogenics, School of Mechanical Engineering, Shanghai Jia a r t i c l e i n f o Article history: Received 14 August 2008 Accepted 18 November 2008 Available online 3 December 2008 Keywords: Refrigeration Frost growth a b s t r a c t This paper presents a nove machine. The dataset used erature. The predicted resu relative error 1.82% for the thickness. Then, a sensitivi operating condition param is selected as an example to journal homepage: www.els ll rights reserved. n cold surface based on support ong University, 800 Dong Chuan Road, Min Hang, Shanghai 200240, PR China ediction model of frost growth on cold surface based on the support vector develop and validate the presented model is obtained from the public lit- are found to be in good agreement with the experimental data, with mean al heat flux, 2.65% for the frost mass concentration, and 5.15% for the frost nalysis of the frost growth model is used to investigate the effects of the s that influence frost growth. Finally, the total heat flux prediction model vestigate the models’ roughness by adding white noise in the input vectors le at ScienceDirect l Engineering ier .com/locate /apthermeng Engi 2. A brief introduction of SVM for regression The basic SVM deals with two-class problems, in which the data are separated by a hyper-plane defined by a number of support vectors. However, with the introduction of Vapnik’s e-insensitive loss function and kernel function, SVM has been extended to solve nonlinear regression estimation and noise characterized problems. For regression problems, SVM nonlinearly maps the input data x into a higher-dimensional feature space F (Hilbert space) to yield and solve a linear regression problem in the feature space. The regression approximation addresses the problem of estimating a function based on a given training set G ¼ fxi; aigni , where xi de- notes the input vector, ai denotes the actual value, and n denotes the total number. In SVM, the regression function is approximated by the following function: y ¼ w/ðxÞ þ b ð1Þ where b is the scalar threshold, w is the weight coefficient, and /(x) is called the feature nonlinearly mapped from the input space x. Nomenclature a the actual value b the scalar threshold C penalty coefficient d humidity ratio ERM empirical risk minimization l distance from the leading edge (m) m the frost mass concentration (kg m�2) MAPE mean absolute percentage error N the number of the dataset p the predicted value q0 0 heat flux (Wm�3) r correlation coefficient R a certain constant used only in the sigmoid kernel RBF radial basis function RMSE root mean square error (Wm�2) R-RMSE relative root mean square error SRM structured risk minimization SVM support vector machine Z. Cao et al. / Applied Thermal The coefficients w and b are estimated by minimizing: minRðCÞ ¼ Remp þ 12 jjwjj 2 ¼ C 1 N XN i¼1 Leðai; yiÞ þ 1 2 jjwjj2 ð2Þ Leða; yÞ ¼ ja� yj � e ja� yjP e 0 others � ð3Þ where both C and e are prescribed parameters, and Le(a,y) is called the e-insensitive loss function. Parameter C calculates the penalty which determines the trade-off between the empirical risk and the regularization term of the model when an error occurs. Param- eter e controls the width of the e-intensive zone used to fit the train- ing data. After the positive slack variables f and f*, representing the dis- tance from the actual values to the corresponding boundary values of e-insensitive, are introduced, Eq. (2) is transformed to the fol- lowing function: minRðw; f; f�Þ ¼ 12wwT þ C� PN i¼1 fþ f�ð Þ � � subject to w/ðxiÞ þ bi � di 6 eþ fi �w/ðxiÞ � bi þ di 6 eþ f�i i ¼ 1;2; . . . ;N fi; f � i P 0 8>< >: ð4Þ Finally, by introducing Lagrange multipliers and kernel function, and maximizing the dual function of Eq. (4), the regression function given by Eq. (1) has the following explicit form: f ðx;ai;a�i Þ ¼ XN i¼1 ðai � a�i ÞKðxi; xjÞ þ b ð5Þ where, K(xi,xj) is called the kernel function. The value of the kernel equals the inner product of two vectors xi and xj in the feature space /(xi) and /(xj), meaning that K(xi,xj) = /(xi)T/(xj). The kernel function is intended to handle any dimension feature space with- out the need to calculate /(x) accurately. And any function that can satisfy Mercer’s condition [16] can be used as the kernel func- tion. Currently, the typical kernel functions in the machine learning theories are as follows [17]: Polynomial kernel: Kðxi; xjÞ ¼ xTi xj þ 1 � �d ð6Þ Radial basis function ðRBFÞ kernel: Kðxi; xjÞ ¼ exp �cjjx � x jj2 � � ð7Þ Ts cold plate temperature (�C) Ta air temperature (�C) v air velocity (m s�1) w the weight coefficient x the input vector c the bandwidth of RBF kernel Greek symbols c free parameter (the bandwidth of RBF kernel) d the frost thickness (mm) e loss coefficient ny Gaussian noise f, f* slack variables r standard deviation s time (s) neering 29 (2009) 2320–2326 2321 i j Sigmoid kernel : Kðxi; xjÞ ¼ tanhðxTi xj þ RÞ ð8Þ where d represents the degree of the polynomial kernel, c repre- sents the bandwidth of RBF kernel, and R is a certain constant used only in the sigmoid kernel. Generally, using RBF kernel function will yield better prediction performance [18], and it was used as the SVM model’s kernel in this study accordingly. 3. Application of SVM for frost growth prediction 3.1. Data collection and preprocessing The dataset used in this study was obtained from a public lit- erature authored by Mao et al. [2], which is an experimental investigation of frost growth characteristic on a cold flat surface under typical freezer operating conditions. It consists of 480 samples under the steady operating conditions with different range of environmental parameters, such as distance from lead- ing edge, test surface temperature, supply air temperature, sup- ply air humidity ratio, supply air velocity, inlet Reynolds number et al. Scaling the features is very important before applying SVM. The main advantage of scaling is to avoid attributes in greater numeric ranges dominating those in smaller numeric ranges. Large attribute values might cause numerical problem because kernel values usu- ally depend on the inner products of the feature vectors, so another advantage of it is to avoid numerical difficulties during the calcu- lation. It is recommended to linearly scale each attribute to the range [�1,1] or [0,1]. Likewise, before testing, the same way is ap- plied to scale the testing data. In order to develop the SVM presented, the available dataset is split into training and validation groups. The SVM model was trained using randomly selected 400 samples while the remaining 80 samples were used to test the SVM model. The input vectors of the SVM model are the experimental measured attributes such as cold plate temperature Ts, humidity ratio d, air velocity v and tem- R-RMSE ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPN i¼1½ðai � piÞ=ai�2 N s � 100% ð9Þ MAPE ¼ PN i¼1jai � pij=pi N � 100% ð10Þ RMSE ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPN i¼1ðai � piÞ2 N � 1 s ð11Þ where N is the number of the dataset, ai and pi are the actual values and predicted values, respectively. 3.4. Prediction results A summary of the performance of the three SVM models is pre- sented in Table 2. From Table 2, one can see that the performance of the three SVMmodels is satisfying: the prediction error is small; the correlation coefficients are very close to 1.0, which indicates good agreement between the experimental and the predicted re- sults. Due to the roughness of frost surface [2], the measured data of frost thickness usually contain much more noise, which makes the performance of the frost thickness prediction model inferior to that of the other two. Fig. 3 shows the predicted results of both the three SVMmodels and the NLR models (proposed by Mao et al. [2]) for the total heat flux, the frost mass concentration and the frost thickness. It is obvi- ously that the prediction accuracy of the SVMmodels ismuch better than that of the NLR models. Generally, with exception of few data, the discrepancy between the predicted and the experimental results is less than5% for the totalheatfluxand the frostmass concentration, and less than 10% for the frost thickness. By comparison, the error q 128 11 0.001 m 512 1.3 0.001 d 512 1.5 0.001 2322 Z. Cao et al. / Applied Thermal Engi perature Ta, distance from the leading edge l and time s. The out- put vector is one of the three measured frost growth characteristics (the heat flux q0 0, the frost mass concentration m, and the frost thickness d). The three SVM models have similar structure, as shown in Fig. 1. And how to get the parameters of the SVM model is described in the following section. 3.2. Parameters determination Fig. 2 illustrates the process of parameters optimization for the SVM model. For SVM model, there exists no standard procedure to determine the free parameters C and c. Here, the technique of cross-validation and grid-search [19] was applied to obtain SVM optimal parameters of C and c. In v-fold cross-validation (10-fold cross-validation was used in this study), the training set is first divided into v subsets of equal size. Then one of the subsets is tested using the classifier trained on the remaining (v � 1) subsets. Thus, each subset of the whole training set is predicted once to obtain SVM optimal parameters when the MAPEcross-validation of v-fold cross-validation is at its minimum. Once the optimal parameters set (summa- rized in the Table 1) have been obtained, the SVM models are developed. 3.3. Performance criteria The statistical metrics such as R-RMSE (relative root mean square error), MAPE (mean absolute percentage error), RMSE (root mean square error) and r (correlation coefficient) were used to evaluate the performance of the SVM models. R-RMSE, RMSE and MAPE were used to measure the deviation between the actual and the predicted values: the smaller the values of R-RMSE, RMSE and MAPE, the closer were the predicted values to the actual values. And the correlation coefficients r was adopted to measure the consistency of the experimental data and the predicted results. As it approaches to 1.0, the prediction accuracy improves. The indices are shown as follows: Fig. 1. The architecture of support vector machine. Fig. 2. Flow chart of parameters optimization for support vector regression models. Table 1 Parameters of the SVM models. Model C c e 0 0 neering 29 (2009) 2320–2326 band of the NLR model is ±30% for the total heat flux, ±40% for the frost mass concentration and frost thickness prediction. SE Engi Table 2 Performance evaluation of the SVM models. SVM model Training performance MAPE (%) RMSE r R-RM q0 0 0.22 1.03 (W/m2) 0.999 0.27 m 0.72 1.83 (g/m2) 0.998 1.86 d 0.53 0.7 (mm) 0.989 0.83 Z. Cao et al. / Applied Thermal 3.5. Sensitivity of parameters used in the SVM model A sensitivity analysis of the model was performed to examine the effects of separately changing the five parameters (plate tem- perature Ts, humidity ratio d, air velocity v, air temperature Ta and time s), shown in Table 3. Only one parameter is varied in each study, and comparisons are made with the experimental results for the base case (Test No. 3). Fig. 4 shows the simulation results for the effect of the changing parameters, and they are summarized in Table 3. As shown in Fig. 4 and Table 3: (1) more moisture in the supply airflow results in more frost accumulation on the cold plate, and less heat flux Fig. 3. Experimental vs. predicted values for: (a) the total heat fl Table 3 Parameters used in the sensitivity analysis and their sensitivity in the model. Parameters Base case (Test No. 3) Sensitivity analysis Summary of sensitivity analysis (%) q0 0 m d d (kg/kg) 7.30 � 10�4 1.22 � 10�3 8.0 22.8 22.1 v (m/s) 1 4 80.6 162.6 75.2 Ta (�C) �10.1 �24 �37.3 �76.7 �37.0 Ts (�C) �39.22 �25 �44.7 �36.0 �13.2 s (min) 60 240 0.1 241.1 434.4 Testing performance (%) MAPE (%) RMSE r R-RMSE (%) 1.82 16.71 (W/m2) 0.998 3.05 2.65 5.58 (g/m2) 0.999 3.82 5.15 1.4 (mm) 0.987 8.58 neering 29 (2009) 2320–2326 2323 through the cold plate; (2) increasing the air flow velocity acceler- ates the heat flux and frost accumulation; (3) reducing the temper- ature difference between the cold plate and the supply air temperature (by the means of either decreasing the air tempera- ture or increasing the cold plate temperature) makes a decrease in the total heat flux through the frost and the frost accumulation on the cold plate; (4) it is found that time changing has little effect on the total heat flux, but it does have much effect on the frost thickness and frost mass concentration – 2.4 times higher and 4.3 times higher than that for the base case, respectively; (5) com- pared to the base case and for all unspecified properties the same as the base case, which shows that changing the parameters usu- ally has more effect on the frost thickness and frost mass concen- tration than that of the total heat flux as a whole. 3.6. Experimental results for Gaussian noise In this section, the robustness of SVM model was investigated by adding white noise in the training set. Since the SVM approach is not sensitive to a particular noise distribution [20], Gaussian noise was selected here. For the effect of the white noise on the prediction of the frost growth characteristics shows similar ten- dency, the following analysis is based on the total heat flux predic- tion as an example. ux; (b) the frost mass concentration; (c) the frost thickness. 0 .0 0 .1 0 .2 0 .3 0 .4 0 .5 15 0 30 0 45 0 60 0 Base case d v Ta Ts τ To ta l h ea t f lu x , q' ' W /m 2 Position, l (m) 0 .0 0 .1 0 .2 0 .3 0 .4 0 .5 0 .0 0 0 .0 5 0 .1 0 0 .1 5 0 .2 0 Base case d v Ta Ts τ M as s co n ce n tr at io n , m kg /m 2 Position, l (m) 0 .0 0 .1 0 .2 0 .3 0 .4 0 .5 0 1 2 3 4 Base case d v Ta Ts τ Th ic kn es s, m m Position, l (m) a b c Fig. 4. Sensitivity analysis of the SVM models for the operation condition parameters: (a) the total heat flux, (b) the frost mass concentration; (c) the frost thickness. 006005004003002001 100 200 300 400 500 600 006005004003002001 100 200 300 400 500 600 006005004003002001 100 200 300 400 500 600 006005004003002001 100 200 300 400 500 600 ξy~N 0, 0.02 2 MAPE=2.82% RMSE=27.56 W/m2 R-RMSE=4.73% r=0.993 Pr ed ic te d q' ' W /m 2 Experimental q'' W/m2 +5% -5% ξy~N 0, 0.10 2 MAPE=7.95% RMSE=67.86 W/m2 R-RMSE=10.76% r=0.889 Pr ed ic te d q' ' W /m 2 Experimental q'' W/m2 +25% -25% ξy~N 0, 0.08 2 MAPE=7.55% RMSE=54.30 W/m2 R-RMSE=10.67% r=0.977 Pr ed ic te d q' ' W /m 2 Experimental q'' W/m2 +20% -20% (c) ξy~N 0, 0.04 2 MAPE=4.61% RMSE=34.81 W/m2 R-RMSE=6.34% r=0.991 Pr ed ic te d q' ' W /m 2 Experimental q'' W/m2 +10% -10% a b c d Fig. 5. Effect of Gaussian noise with different level in the output targets of the training set on the prediction performance: (a) ny � N(0,0.022); (b) ny � N(0,0.042); (c) ny � N(0,0.082); (d) ny � N(0,0.102). 2324 Z. Cao et al. / Applied Thermal Engineering 29 (2009) 2320–2326 In the following implementation, the previous training set and testing set were used, and the values of free parameters (C, c and e) in Table 1 were used and fixed when developing the prediction model using noisy training set. Firstly, each data point of the output targets of training set was added by a Gaussian noise (ny) with standard deviation of r (r = 0.02, 0.04, 0.08, 0.10). The effect of Gaussian noise on the pre- diction results is shown in Fig. 5. From Fig. 5, it is found that the presented model is well robust against Gaussian noise: for the noise level r 6 0.02, the prediction results show the same error band and a little decrease in prediction accuracy compared to the previous model (Fig. 5a); when the noise level r increases to 0.04, the predicted values are mostly within ±10% error band, and the other statistical metrics such as r, MAPE, RMSE and R- RMSE also indicate good agreement between the experiment