Statistical Approach to Constructing Predictive Models for Thermal

Article pubs.acs.org/IECR

Statistical Approach to Constructing Predictive Models for Thermal Resistance Based on Operating Conditions Hiromasa Kaneko,† Susumu Inasawa,† Nagisa Morimoto,‡ Mitsutaka Nakamura,‡ Hirofumi Inokuchi,‡ Yukio Yamaguchi,† and Kimito Funatsu*,† †

Department of Chemical System Engineering, The University of Tokyo, Hongo 7-3-1, Bunkyo-ku, Tokyo 113-8656, Japan Mitsubishi Chemical Corporation, 3-10 Ushiodori, Kurashiki, Okayama 712-8054, Japan

‡

ABSTRACT: We have constructed statistical models that predict thermal resistance after fouling layer formation in a heat exchanger, in which a slurry of stearic acid in toluene was cooled. Chemoinformatics was used, and the initial rate of increase in thermal resistance (dU−1/dt) was calculated from experimental conditions such as coolant flow rate and the degree of supersaturation. We then constructed models for thermal resistance at a steady state using calculated values of dU−1/dt and experimental conditions. Our model gives a good correlation with the experimental results. The contribution of operating conditions to fouling layer formation was discussed semiquantitatively on the basis of linear regression coefficients that were obtained from our model. Because only operating conditions and set values were used as input, our approach is very practical for prediction of thermal resistance given certain operating conditions.

1. INTRODUCTION The formation of a fouling layer on a heat exchanger can result in an undesired increase in thermal resistance.1 Once a fouling layer has formed, it must be removed by physical cleaning. Because regular cleaning and maintenance are expensive and time-consuming, operating conditions that inhibit fouling-layer formation are preferred. Therefore, an understanding of ways to prevent fouling formation is desirable and has been extensively studied in various systems such as crude oil fouling2−6 and fouling formation by inorganic/organic materials.7−18 The formation mechanisms of the fouling layer are widely discussed in these studies. While there is no doubt about the importance in understanding the fouling mechanism, a computational model that gives accurate predicted values of thermal resistance under certain operating conditions is practically useful, even if we do not understand the full mechanism of fouling. Some groups have previously used a neural network approach to predict heat transfer coefficients in heat exchangers.19,20 However, because it is reported that neural networks give good performance in fitting but may lead to poor performance in prediction,21 another method should be used to obtain better predictive performance. In addition, because of the nonlinearity and the complexity in neural network methods, we are not able to analyze how each parameter contributes to fouling formation. In this study, we use chemoinformatics to construct statistical models to predict thermal resistance from certain operating conditions.22,23 Chemoinformatics uses informatics methods to solve chemistry problems and is used to study a variety of topics, such as quantitative structure−property relationships,24,25 quantitative structure−activity relationships,25−27 reaction design,28−30 drug design,25,31 and process systems.32−34 In our study, two methods are used to construct regression models. One is a partial least-squares (PLS)35 method, and the other is a support vector regression (SVR)36 method. PLS models are based on a linear regression method, and we can © 2012 American Chemical Society

obtain semiquantitative regression coefficients (or contribution ratios) of each considered parameter to predict thermal resistance. On the other hand, SVR models are based on a nonlinear regression method and are reported to have more predictive accuracy than that of neural networks in some cases.21 Our final target is to construct a statistical model that accurately predicts heat transfer coefficients at various operating conditions using operating conditions and set values. We have demonstrated that the combination of two models is useful in constructing the fouling model. Crystallization and deposition fouling on a cooling tube by stearic acid in toluene solvent was chosen as a model fouling system.

2. METHOD In this section, we briefly introduce two general analysis methods: the PLS method and the SVR method. 2.1. PLS Method.35 PLS is a method for relating explanatory variables, X, and an objective variable, y. Using a linear multivariate model, it goes beyond traditional regression methods in that it also models the structures of X and y. In PLS modeling, the covariance between y and the score vector ti is maximized. A PLS model has higher predictive power than ordinary least-squares models.37 A PLS model consists of the following two equations: X = TP′ + E

(1)

y = Tq + f

(2)

Received: Revised: Accepted: Published: 9906

February 6, 2012 June 17, 2012 June 25, 2012 June 25, 2012 dx.doi.org/10.1021/ie300315t | Ind. Eng. Chem. Res. 2012, 51, 9906−9912

Industrial & Engineering Chemistry Research

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where T is a score matrix, P is an X-loading matrix, q is a yloading vector, E is a matrix of X residuals, and f is the vector of y residuals. The PLS regression model is as follows: y = Xb + const

(3)

b = W(P′W)−1q

(4)

The lower the RMSE value, the more accurate is the prediction obtained with the constructed model.

3. EXPERIMENTAL SECTION Fouling observations were conducted using a semibatch system, a schematic illustration of which is shown in Figure 1. A stearic

where W is an X-weight matrix and b is a vector of regression coefficients. 2.2. SVR Method.36 The SVR method applies a support vector machine (SVM) to a regression analysis and can be used to construct nonlinear models by applying a kernel trick as well as a SVM. The primal form of SVR can be shown to be the following optimization problem: Minimize 1 || w ||2 + C ∑ |yi − f (x i)|e 2 i

(5)

|yi − f (x i)|e = max(0, |yi − f (x i)| − e)

(6)

where yi and xi are training data, w is a weight vector, e is a threshold, and C is a penalizing factor that controls the trade-off between a training error and a margin. By minimization of eq 5, we can construct a regression model that has a good balance between generalization capabilities and the ability to adapt to the training data. The kernel function in our application is a radial basis function: 2

K (x , x′) = e−γ ∗|x − x ′|

(7)

where γ is a tuning parameter controlling the width of the kernel function. In this study, LIBSVM38 is used as the machine learning software. 2.3. Statistics. To construct a highly predictive model, the number of components in the PLS models and tuning parameters in the SVR models must be chosen appropriately. The r2 and q2 values are used as measured and defined as follows: r2 = 1 −

2

q =1−

Figure 1. Schematic illustration of our handmade semibatch crystallizer system.

acid/toluene slurry was sealed in a glass reservoir, and the temperature in the reservoir was controlled by a surrounding water jacket. A draft tube was used around the stirrer to obtain a steady flow during stirring. The slurry flowed upward along the reservoir wall and flowed down at the center as shown in Figure 1a, and the flow velocity was controlled by the stirring rate. A U-bend stainless tube was inserted in the reservoir with an inner diameter of 6 mm and wall thickness of 1 mm. Coolant water flowed through the tube, and the coolant temperature at two different points (shown in Figure 1b) was measured by directly inserting two platinum resistance temperature detectors into the tube. The distance between each measurement point was about 8 cm, and the maximum error in the temperature measurements was ±0.3 °C at most. The stearic acid/toluene slurry solution was prepared as follows. Stearic acid and toluene were introduced into the reservoir and stirred for several hours at 50 °C until the stearic acid dissolution was complete. The initial weight ratio of stearic acid/toluene was 37/100 for all cases. The clear solution was then gradually cooled to the desired temperature of 25, 30, or 35 °C. As the solubility of stearic acid in toluene is lower than the initial weight ratio at temperatures below 40 °C, precipitation of stearic acid occurred in the course of cooling and the solution became turbid. The slurry was kept at the same temperature for at least 30 min prior to examination. We started the fouling measurements by changing the coolant water flow from the bypass route so that it passed through the U-bend. The time t = 0 for the measurement was

∑ (yobs − ycalc )2 ∑ (yobs − y ̅ )2

(8)

∑ (yobs − ypred )2 ∑ (yobs − y ̅ )2

(9)

where yobs is the measured y value, ycalc is the calculated y value, and ypred is the predicted y value in the cross-validation procedure. In this study, the leave-one-out method is used in the calculation of ypred. In the above equations, r2 represents the fitting accuracy of the constructed models and q2 represents the predictive accuracy of the constructed models. Values close to unity for both r2 and q2 are favorable. The r2 and q2 values must both be compared using models constructed with the same objective variables data. The root-mean-square error (RMSE) of ycalc and ypred is defined as follows: RMSE =

∑ (yobs − ycalc , pred )2 n

(10) 9907

dx.doi.org/10.1021/ie300315t | Ind. Eng. Chem. Res. 2012, 51, 9906−9912


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determined by considering the residence time of the coolant water between two temperature measurement points. We obtained the overall heat transfer coefficient U as:

U=

Q AΔTav

(11)

where Q is the thermal energy transferred to the coolant water per unit time, A is the surface area of the stainless tube between the two measurement points, and ΔTav is the logarithmic mean temperature difference. In this calculation, we assumed that the slurry temperature was constant and that the slurry and coolant water had countercurrent flows. The experimental ranges of the other parameters were summarized in Table 1. We obtained 23 data points by changing these parameters. Table 1. Process Parametersa no.

symbol

process parameters

1 2 3 4 5

Y1 Y2 X1 X2 X3

Uf−1 dU−1/dt linear velocity coolant temperature flow rate of coolant water solubility of stearic acid at coolant temperature initial concentration of precipitated solute degree of supersaturation

6

X4

7

X5

8

X6

range

20−40 8−32 9−18 −3

5 × 10 −0.14 0.19−0.35

unit m2 K W−1 m2 K W−1 s−1 cm s−1 °C mL min−1 kg solvent-kg

Figure 2. (a) An example for an overall heat transfer coefficient during heat exchange. (b) The time evolution of U−1. We defined two characteristic values, Uf−1 and dU−1/dt, as shown in this figure.

In fact, these two values show a linear relationship as shown in Figure 3, indicating that if we measure the initial increasing rate

−1

kg solvent-kg−1

0.5−45

Note that we used a unit of “kg solvent-kg−1” for the concentration of solute because “wt %” is sometimes confusing. Here, 0.1 kg solventkg−1 means that 0.1 kg of solute is dissolved (X4) or precipitated (X5) in 1 kg of solvent. a

4. RESULTS AND DISCUSSION 4.1. Fouling Behavior during Heat Exchange. A typical example of an overall heat transfer coefficient U is shown in Figure 2. The value of U rapidly decreased in the initial 1000 s after introduction of the coolant water into the U-bend tube due to the formation of the stearic acid fouling layer. The rate of decrease in U gradually decreased after 1000 s and, eventually, after 6000 s; no significant change in U was seen (Figure 2a). The inverse of the overall heat transfer coefficient U−1 is frequently used to evaluate the thermal resistance caused by fouling. In Figure 2b, the time evolution of U−1 is shown. The thermal resistance curve shows a kind of “Kern-Seaton behavior” in which thermal resistance is described as:39 U −1 = (Uf −1 − U0−1)(1 − exp( −t /τ )) + U0−1 −1

Figure 3. Dependence of Uf−1 on dU−1/dt.

in Uf−1, we are able to predict late values of Uf−1. However, as already mentioned, our goal is to predict Uf−1 using operating conditions and set values; therefore, both values of Uf−1 and dU−1/dt should be predicted from operational conditions. To achieve this, we used chemoinformatics to construct relevant prediction models. 4.2. Statistical Analysis and Predictive Model for Thermal Resistance. As shown in Figure 3, dU−1/dt is one of the key factors that should be considered in predicting Uf−1. We first constructed statistical models for dU−1/dt from experimental conditions; then, we constructed a predictive model for Uf−1 using calculated values of dU−1/dt and experimental parameters. Before model construction, we checked the variance of each parameter measured in the experiment and the correlation coefficient for each pair of parameters. We then eliminated two types of parameter: parameters that had zero variance and parameters with correlation coefficients larger than 0.95. This procedure is necessary because zero variance of a parameter means that the parameter has no effect on the objective variables. Furthermore, consideration of parameters with high correlation coefficients decreases the predictive accuracy of a model.25 After this procedure, six parameters were selected as explanatory variables: the linear velocity of the slurry (X1), the

(12)

U0−1

where Uf is thermal resistance at steady state, is the initial thermal resistance, and τ is the characteristic time for fouling layer formation. From eq 12, the initial increasing rate in U−1 is given by: dU −1 dt

= t=0

Uf −1 − U0−1 τ

(13)

In this fouling behavior, the initial increasing rate is an important factor in predicting Uf−1. Therefore, we chose two characteristic values, Uf−1 and dU−1/dt at t = 0 (see Figure 2b). 9908



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coolant temperature (X2), the coolant flow rate (X3), the solubility of stearic acid at the coolant temperature (X4), the initial concentration of precipitated solute (X5), and the degree of supersaturation (X6). We used the degree of supersaturation to quantitatively evaluate the driving force for precipitation and fouling, as it is used in crystallization studies.40−42 The degree of supersaturation ΔS is defined as: ΔS =

C(Tsl) − Cs(Tcool) Cs(Tcool)

Although the SVR method gives a better predictive model than the PLS method, the value of q2 is still smaller than 0.5, which indicates poor predictive accuracy in the model. One of the possible reasons is that the explanatory variables that we used are not enough to describe dU−1/dt, and certain important factors, such as surface roughness of the tube or crystal sizes of stearic acid, should be considered to obtain a more predictive model. We constructed predictive models for Uf−1 using explanatory variables plus measured or calculated values of dU−1/dt. Table 3 shows the modeling results. As a reference, the model

(14)

where C and Cs are the concentration of dissolved solute and the saturated concentration of the solute, and Tsl and Tcool are the temperatures of the slurry and the coolant water, respectively. In our definition, ΔS = 0 corresponds to a thermodynamically saturated state. We used these six parameters as input variables, which we call explanatory variables, in our analysis. All explanatory variables we considered are summarized in Table 1. Two other experimental parameters, the temperature of the slurry and the solubility of solute at the slurry temperature, were precluded by the above procedure. As all the variables in Table 1 had different numerical ranges, we used autoscaling, a combination of mean-centering and scaling, so that each variable has an equal chance of contribution to the overall analysis.22 In mean-centering, each variable is subtracted by the averaged variable; in scaling, each variable is divided by its own standard deviation. The SVR method gives a better predictive model for dU−1/dt as shown in Table 2. This indicates that the experimental

Table 3. Model Construction Results for Uf−1 Using Six Experimental Parameters with/without dU−1/dt Uf−1 method

explanatory variables

r2

RMSE (×10−4)

q2

RMSE (×10−4)

PLS

including measured dU−1/dt including calculated dU−1/dt without dU−1/dt

0.875

2.51

0.781

3.33

0.851

2.74

0.736

3.66

0.789

3.27

0.710

3.83

construction results without considering dU−1/dt are also shown in the same table. Compared with the result without dU−1/dt, both r2 and q2 values in the other two PLS models increased and the corresponding RMSEs decreased. This means that the predictive accuracy increases when dU−1/dt is included as an explanatory variable as expected. Figure 5 shows a plot of

Table 2. Model Construction Results for dU−1/dt Using Six Experimental Parameters dU−1/dt method

2

−7

r

RMSE (×10 )

q2

RMSE (×10−7)

PLS SVR

0.398 0.638

7.19 5.57

0.104 0.341

8.77 7.52

conditions have nonlinear effects on formation of the fouling layer in the initial stages, and the nonlinear regression method is more suitable in describing the initial phenomena. Interaction between the naked surface of the cooling tube and precipitated solute crystals might have a nonlinear effect on the initial fouling formation. Figure 4 shows a plot of measured and calculated dU−1/dt values in the SVR model. The plot is scattered but shows an almost linear trend along the diagonal.

Figure 5. Relationship between measured and calculated Uf−1 in PLS modeling, which includes calculated dU−1/dt.

the measured and calculated Uf−1 values in the PLS model, which includes calculated dU−1/dt as input. The plot shows a tight cluster of calculated values along the diagonal, meaning that the model accuracy is good. Compared with the result with measured dU−1/dt, the model with calculated dU−1/dt has slightly lower values of r2 and q2; however, they are still higher than the model without dU−1/dt. This means that we successfully constructed a good predictive model for Uf−1 from operating conditions and set variables. The value of q2 is much larger than 0.5 in both cases, which means that the models are generally acceptable with enough predictive accuracy.43 In fact, the averaged difference between measured and calculated values is about 3% in the model with calculated dU−1/dt. This result also supports that our model has enough accuracy for practical application. Predictive models for Uf−1 using the SVR method gave us almost the same result.

Figure 4. Model construction result for dU−1/dt using the SVR method. 9909



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imentally obtained values of τ. We used calculated values of dU−1/dt and Uf−1 from our models to obtain τcalc while measured values of dU−1/dt and Uf−1 are used for τobs. The values of U0−1 were taken from the initial conditions of experiments, which are in the range of 1.8 × 10−3 to 5.7 × 10−3 [m2 s K J−1]. The plot shows almost linear trend along the diagonal, except lower predicted values of τcalc at high τobs. The correlation coefficient between τcalc and τobs is 0.861, which shows that the correlation is high. In addition to it, the averaged difference between τcalc and τobs is 13.7%. Therefore, we can say that the characteristic time for the increase in fouling is also predictable within ca. 14% errors, using calculated values of dU−1/dt and Uf−1 from our statistical models. However, we have to point out that prediction of τ is still limited because we used the Kern-Seaton relation in eq 13 to obtain τcalc. Actually, we have constructed a statistical model to predict τcalc from operating conditions; however, the accuracy of the model was not sufficient. For this reason, prediction of τ is possible if the observed fouling shows a well-known behavior and can be described by mathematical form, at present. While our statistical models are successful to predict dU−1/dt and Uf−1, further investigation is still necessary to predict τ from operating conditions. As fouling is a widely observed phenomenon, applicability of our models or methods to other systems should be discussed. We have shown that classical statistical approaches are useful to construct practical models for fouling prediction. Since explanatory variables which should be considered vary depending upon systems, we have to say that usage of constructed models shown here is limited in other systems. However, we used general statistical methods in model construction, and therefore, we believe that these classical statistical approaches are applicable in other systems to construct a model for fouling prediction. Furthermore, consideration of time-dependent variation in operating conditions in a model is also an important issue because most process conditions are time-varying in real chemical plants. However, because operating conditions were kept constant in our experiments, we do not consider dynamic properties of explanatory variables X in this paper. Since a statistical approach to consider dynamic properties of X in chemical plants has been reported in ref 44, it would be possible to perform online prediction of fouling formation by inputting measured values of X into a statistical fouling model. Finally, we note the initial rate of fouling. In our experiment, we directly measured the initial rate of fouling. However, in some practical processes, measurement and definition of the onset of fouling formation may be difficult due to fluctuation in measured thermal resistance. This would also restrict direct application of our proposed model in other systems. Another point is the effect of surface condition for the initial rate of fouling. In our study, we used dU−1/dt as an important explanatory variable. However, physical meaning of the initial rate of fouling may be different when fouling occurs on the fresh surface of a cooling tube or it occurs on an already deposited layer on a cooling tube. For more precise statistical analysis of fouling formation, we consider that it would be necessary to distinguish these two situations. In this sense, fundamental understanding of the initial stage of fouling is still quite important in both practical and scientific meanings.

One of the advantages in the PLS method is that contribution of each parameter to Uf−1 can be studied semiquantitatively on the basis of standard regression coefficients. Figure 6 shows the standard regression coefficients

Figure 6. Standard regression coefficients in the Uf−1 models obtained in PLS modeling, which includes calculated dU−1/dt. Each error bar indicates three standard deviations of the corresponding regression coefficient. The symbols of X1 to X6 are summarized in Table 1.

of the Uf−1 model in PLS modeling. The symbols in Figure 6 are summarized in Table 1. Standard regression coefficients of four explanatory variables, X1, X2, X3, and X4 are negative, indicating that Uf−1 becomes small when these four values are large. A high slurry line velocity (X1) and a large flow rate of coolant water (X3) result in insufficient temperature decrease of slurry because of the short contact time between the two fluids. In addition, a high coolant water temperature (X2) and a high solute solubility at the coolant water temperature (X4) reduce the difference in solubility when the slurry is cooled. Thereby, contribution of these four variables to Uf−1 is negative. On the other hand, the coefficients of the remaining three variables, X5, X6, and Y2, are positive. Because adherence of precipitated solute on the cooling tube causes fouling layer formation, Uf−1 becomes large when X5 is large. Furthermore, the positive value of ΔS means that an excess solute, compared with the saturated solubility, is dissolved and the system is thermodynamically unstable. In that situation, precipitation of dissolved excess solute is preferred. Therefore, X6 also contributes positively to Uf−1. Thus, a statistical approach using the PLS model is quite practical to analyze complicated fouling phenomena since we are able to extract the contribution ratio of each parameter to an objective variable in a simple way. Another practically important problem is the time to reach the steady state value of Uf−1. As already described, because thermal resistance in our experiment shows “Kern-Seaton” behavior, we can obtain the characteristic time, τ, from eq 13. Figure 7 shows comparison between calculated and exper-

Figure 7. Comparison between τcalc and τobs. 9910



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(16) Zettler, H. U.; Weib, M.; Zhao, Q.; Muller-Steinhagen, H. Influence of Surface Properties and Characteristics on Fouling in Plate Heat Exchangers. Heat Transfer Eng. 2005, 26, 3. (17) Saikhwan, P.; Geddert, T.; Augustin, W.; Scholl, S.; Paterson, W. R.; Wilson, D. I. Effect of Surface Treatment on Cleaning of a Model Food Soil. Surf. Coat. Technol. 2006, 201, 943. (18) Lakhdar, M. B.; Cerecero, R.; Alvarez, G.; Guilpart, J.; Flick, D.; Lallemand, A. Heat Transfer with Freezing in a Scraped Surface Heat Exchanger. Appl. Therm. Eng. 2005, 25, 45. (19) Malayeri, M. R.; Müller-Steinhagen, H. Initiation of CaSO4 Scale Formation on Heat Transfer Surfaces under Pool Boiling Conditions. Heat Transfer Eng. 2007, 28, 240. (20) Aminian, J.; Shahhosseini, S. Neuro-based Formulation to Predict Fouling Threshold in Crude Preheaters. Int. J. Heat Mass Transfer 2009, 36, 525. (21) Yang, S. S.; Lu, W. C.; Chen, N. Y.; Hu, Q. N. Support Vector Regression Based QSPR for the Prediction of Some Physicochemical Properties of Alkyl Benzenes. J. Mol. Struct. 2005, 719, 119. (22) Gasteiger, J.; Engel, T. Chemoinformatics − A Textbook; WileyVCH: Weinheim, 2003. (23) Gasteiger, J.; Funatsu, K. Chemoinformatics − An Important Scientific Discipline. J. Comput. Chem. Jpn. 2006, 5, 53. (24) Kaneko, H.; Arakawa, M.; Funatsu, K. Development of a New Regression Analysis Method Using Independent Component Analysis. J. Chem. Inf. Model. 2008, 48, 534. (25) Miyao, T.; Arakawa, M.; Funatsu, K. Exhaustive Structure Generation for Inverse-QSPR/QSAR. Mol. Inf. 2010, 29, 111. (26) Arakawa, M.; Hasegawa, K.; Funatsu, K. Tailored Scoring Function of Trypsin−Benzamidine Complex Using COMBINE Descriptors and Support Vector Regression. Chemom. Intell. Lab. Syst. 2008, 92, 145. (27) Hasegawa, K.; Koyama, M.; Arakawa, M.; Funatsu, K. Application of Data Mining to Quantitative Structure−Activity Relationship Using Rough Set Theory. Chemom. Intell. Lab. Syst. 2009, 99, 66. (28) Corey, E. J.; Cheng, X. M. The Logic of Chemical Synthesis; John Wiley & Sons: New York, 1989. (29) Corey, E. J. The Logic of Chemical Synthesis: Multistep Synthesis of Complex Carbogenic Molecules (Nobel Lecture). Angew. Chem., Int. Ed. Engl. 1991, 30, 455. (30) Satoh, H.; Itono, S.; Funatsu, K.; Takano, K.; Nakata, T. A Novel Method for Characterization of Three-dimensional Reaction Fields Based on Electrostatic and Steric Interactions toward the Goal of Quantitative Analysis and Understanding of Organic Reactions. J. Chem. Inf. Comput. Sci. 1999, 39, 671. (31) Arakawa, M.; Miyao, T.; Funatsu, K. Systematic Generation of Chemical Structures for Rational Drug Design based on QSAR models. Curr. Comput.-Aided Drug Des. 2011, 7, 1. (32) Kaneko, H.; Arakawa, M.; Funatsu, K. Development of a New Soft Sensor Method Using Independent Component Analysis and Partial Least Squares. AIChE J. 2009, 55, 87. (33) Kaneko, H.; Arakawa, M.; Funatsu, K. Applicability Domains and Accuracy of Prediction of Soft Sensor Models. AIChE J. 2011, 57, 1506. (34) Kaneko, H.; Arakawa, M.; Funatsu, K. Novel Soft Sensor Method for Detecting Completion of Transition in Industrial Polymer Processes. Comput. Chem. Eng. 2011, 35, 1135. (35) Wold, S.; Sjöström, M.; Eriksson, L. PLS-regression: a Basic Tool of Chemometrics. Chemom. Intell. Lab. Syst. 2001, 58, 109. (36) Vapnik, V. N. The Nature of Statistical Learning Theory; Springer: New York, 1999. (37) Faber, K.; Kowalski, B. R. Propagation of Measurement Errors for the Validation of Predictions Obtained by Principal Component Regression and Partial Least Squares. J. Chemom. 1997, 11, 181. (38) Chang, C. C.; Lin, C. J. LIBSVM: a Library for Support Vector Machines, 2001. Software available at http://www.csie.ntu.edu.tw/ ∼cjlin/libsvm. (39) Melo, L. F.; Bott, T. R.; Bernardo, C. A. Fouling Science and Technology; Kluwer Academic Publishers: Dordrecht, 1988.

5. CONCLUSION We constructed a predictive model for Uf−1 using operating conditions and set values. The initial layer formation would contain nonlinear phenomena, and the SVR method is more suitable for constructing a model for dU−1/dt. The effects of the experimental parameters on Uf−1 were analyzed using standard regression coefficients obtained by the PLS method. Our results suggest that the statistical approach used in this study would be a practical tool for the analysis and prediction of thermal resistance induced by fouling-layer formation.

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AUTHOR INFORMATION

Corresponding Author

*Tel: +81-3-5841-7751. Fax: +81-3-5841-7771. E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS H.K. is grateful for a Research Fellowship of the Japan Society of Promotion of Science (JSPS) for financial support (No. 21·1337).

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REFERENCES

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