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Expert Systems with Applications 36 (2009) 1142–1154

Expert Systemswith Applications

Adaptive fuzzy model identification to predict the heat transfercoefficient in pool boiling of distilled water

Mihir K. Das a, Nand Kishor b,*

a Department of Mechanical Engineering, Motilal Nehru National Institute of Technology, Allahabad, Indiab Department of Electrical Engineering, Motilal Nehru National Institute of Technology, Allahabad, India

Abstract

In this paper, a modeling technique based on fuzzy system is used to predict the heat transfer coefficient in pool boiling of distilledwater. To achieve this, an experimental investigation has been carried out for saturated boiling of distilled water from plain copper heat-ing tube surface at atmospheric and sub-atmospheric pressures. An empirical correlation has been established to predict the boiling heattransfer coefficient. Further, the experimental data has been compared with those determined from the zero-order adaptive fuzzy modelwith heat flux as input variable. The representation accuracies of the pool boiling heat transfer coefficient for fuzzy model is very high asindicated from the performance index. The prediction performance of zero-order adaptive fuzzy model has also been compared withMATLAB based ANFIS function. Further, the statistical attributes are determined and compared with the experimental data set of poolboiling of distilled water available in open literatures.� 2007 Elsevier Ltd. All rights reserved.

Keywords: Pool boiling; Heat transfer coefficient; Model identification; Fuzzy systems; Prediction

1. Introduction

The heat transfer coefficient is one of the most impor-tant parameter for a number of applications. The heattransfer coefficient is often calculated with empirical corre-lations between dimensionless numbers. In fact a prelimin-ary tentative model is assumed and it undergoes successivemodifications to correct discrepancies in the value of con-stants following ‘trial-and-error’ approach. The analyticalterms of these correlations are purely empirical and aredeveloped through a long series of modifications in them.This approach is obviously restricted to the range of condi-tion for which the correlation was determined. The calcula-tion of the heat transfer coefficient involves fitting someprocess equations to the experimental data, with h beingthe only factor to determine by the regression. The directdetermination of this coefficient is limited by thermal

0957-4174/$ - see front matter � 2007 Elsevier Ltd. All rights reserved.

doi:10.1016/j.eswa.2007.10.044

* Corresponding author.E-mail address: nand_research@yahoo.co.in (N. Kishor).

metrology problems because it is necessary to measure atthe same location a temperature and a heat flux. A precisemeasure of the heat convection flux is often difficult, nota-bly because of the presence of thermal discontinuitiesbetween the flux meter and the surface, generally of differ-ent materials, and of the superposition of a heat conduc-tion flux, tangential of the surface, due to the presence ofspatial gradients of temperature. To overcome these con-straints in the determination of heat transfer coefficient, ithas been proposed to use methods like inverse techniques,including the sequential function specification method, theextended Kalman filter and the conjugate gradient method.

Recent publications (Scalabrin, Condosta, & Marchi,2006a, 2006b; Wang, Zhao, & Zhang, 2006) have suggestedthe use of neural network approach in prediction of heattransfer coefficient for various applications. Their studiesindicate comparatively better results in determining theheat transfer coefficient than the traditional approach ofobtaining the same from the empirical correlations.

In fact, NNs constitute a powerful interpolation tooland it is always possible to develop a structure that

Nomenclature

do outer diameter of heating tube, mh heat transfer coefficient, W/m2 �Ckw wall-thermal conductivity, W/m �CL effective length of heating tube, mP pressure, kN/m2

Q heat input, Wq heat flux, W/m2

T temperature, KDTw wall superheat, ð¼ ðT w � T lÞÞKT w average wall temperatureT l average liquid temperaturedTw temperature drop over thickness of the cylinder,

KAj;ijðxjÞ ij = 1, 2, . . . ,Mjth antecedent fuzzy set referring

to the jth inputaj;ij cores of fuzzy set Aj;ij

bi1;...;in consequent crisp constant (singleton) on thei1, . . . , inth rule

ai consequent parameter vectorbi scalar offsetk time instantMj number of fuzzy sets on the jth input domainn number of inputs of the fuzzy modelRi1;...;in i1,. . .,inth fuzzy implication (rule)yadap adaptive zero-order TS fuzzy model outputyref model reference outputy predicted model outputbi1;...;in truth (weight) of the i1,. . .,inth ruleNr number of rulesuss steady-state input

M.K. Das, N. Kishor / Expert Systems with Applications 36 (2009) 1142–1154 1143

approximates a function with a given precision. However,there is still distrust about the NNs identification capabilityin some applications (Castro & Miranda, 2002).

Fuzzy set theory plays an important role in dealing withuncertainty in modeling applications. The present researchwork focuses on the fuzzy model identification, which canpredict the pool boiling heat transfer coefficient on a plaintube as close to the experimental values at differentpressures.

2. Experimentation

Fig. 1 shows an experimental set-up used to obtain datafor boiling of distilled water (Das, 2006). It consists of acylindrical vessel, a heating tube, a heater, a condenser, aliquid level indicator, a vacuum pump, wall and liquid ther-mocouples and an instrumentation panel board containinga digital multimeter, digital watt meter, digital voltmeter,and digital ampere meter. Two view ports are provided atdiametrically opposite positions of the vessel to observebubble dynamics over heating tube. The vessel is ade-quately insulated by wrapping asbestos rope and thick lay-ers of paste of plaster of Paris and magnesia powder overits outer surface. A threaded socket is welded at one sideof the vessel for holding the heating tube in horizontal posi-tion inside the vessel.

Present investigation requires proper measurement ofliquid and heating tube surface temperature. Therefore,laboratory made calibrated PTFE coated copper constan-tan thermocouple wires of 30 gauges are used. Wall tem-perature of the heating tube is measured at its top,bottom and two side positions by drilling holes circumfer-entially equispaced at 90� in the wall thickness. For mea-surement of liquid temperature, thermocouple probes areinserted through fittings welded in the body of vessel corre-sponding to wall thermocouple positions.

The heater inside the vessel is energized with a singlephase 230 V, 50 Hz, AC supply. An autotransformer regu-lates input voltage to heater. The heating tube is thermallystabilized and subsequently liquid is degasified to get accu-rate and reproducible experimental data. Standard proce-dure is followed to obtain data for the boiling of distilledwater at atmospheric and sub-atmospheric pressures. Heatflux is varied from 16,228.70 W/m2 to 41,324.90 W/m2 fora given pressure and pressure varies from 44.81 kN/m2 to98.12 kN/m2. Since wall thermocouples have been placedin the wall thickness of heating tube, they do not measurethe temperature of outer surface of the tube directly. So, atemperature drop, dTw across the thickness betweenthermocouple location and outer tube surface has beencalculated by the use of following equation for heat con-duction in a thin cylinder:

dT w ¼qdo

2 kw

lndo

dh

� �ð1Þ

The temperature drop, dTw so obtained, has been sub-tracted from the recorded wall temperature of heating tubemeasured by thermocouples to obtain the outer surfacetemperature.

Heat flux q is obtained from the following equation:

q ¼ Q=pdoL ð2Þ

Heat transfer coefficient, is obtained from the followingequation:

h ¼ q=ðT w � T lÞ ð3Þ

where, T w and T l represent average value of wall tempera-ture and liquid temperature, respectively. The averagevalue of wall temperature is calculated by taking arithmeticmean of temperature at top, two sides and bottomposition of heating tube and similarly liquid temperature

Fig. 1. Experimental set-up to determine the pool boiling heat transfer coefficient.

1144 M.K. Das, N. Kishor / Expert Systems with Applications 36 (2009) 1142–1154

as measured by liquid thermocouple probe are processed toobtain average value of liquid temperature.

The data obtained in each experimental run are pro-cessed. An uncertainty analysis of each experimental runhas been carried out and the maximum uncertainty associ-ated with heat transfer coefficient has been found to be ofthe order of ±1.05%. It is worthwhile to mention here thattemperature drop across thickness of copper coated layerhas not been included in the determination of heat transfercoefficient. In other words, calculation of heat transfercoefficient on coated tube is based on substrate surfacetemperature only.

3. Fuzzy based model identification

System identification deals with the problem of develop-ing mathematical models of dynamical systems based onobserved data from the system and tries to determine thesimplest model capable of exhibiting the observed charac-teristics. In most cases the structural estimation is not gen-erally made. The choice of an appropriate model structureis most crucial for a successful identification application.

System identification, whether on-line or off-line, is anessential part of any control system design. In general, iden-tification of linear systems is a well-established area andthroughout the years, a plethora of approaches have beendeveloped. However, in many practical situations, theassumption of linearity may not hold due to the existenceof nonlinear and/or time-varying elements. In such cases,it is difficult to apply the conventional quantitative methodsto describe adequately the nonlinear and time-varying char-acteristic of the plant. In recent years, rapid development ofintelligent control methodologies such as NNs, fuzzy logictheory, and rule-based expert system, have provided alter-native tools to tackle the problem of system identification.

In particular, since the introduction of fuzzy sets, manyresearchers have shown interest to apply this theory to sys-tem identification. Fuzzy identification has become a veryimportant area in fuzzy system theory.

Many approaches have been proposed to obtain fuzzymodels from input–output measurement data. Few toname them, local linear tree method, tree constructionalgorithms, or variants of neuro-fuzzy approaches (Jang,Sun, & Mizutani, 1997) are included. A mathematical tool,which in some way uses fuzzy sets, is called a fuzzy model.In fact, this model consists of If-then rules with fuzzy ante-cedents and mathematical functions in the consequent part.The relationships between variables are represented bymeans of If-then rules with imprecise predictions, such as:

If heating is high then temperature increase is fast.

This rule defines in a rather qualitative way the relation-ship between the heating and the temperature in a room.To make such a model operational, the meaning of theterms ‘high’ and ‘fast’ is defined by using fuzzy sets, i.e. setswhere the membership is changing gradually rather than inan abrupt way. Fuzzy sets are defined through their mem-bership functions, which map the elements of the consid-ered universe to the interval [0,1]. The extreme values 0and 1 denote complete membership and non-membership,respectively, while a degree between 0 and 1 means partialmembership in the fuzzy set.

The fuzzy models are classified according to the struc-ture of the If-then rules as: the Mamdani (linguistic) andthe Takagi–Sugeno (TS) model. The Mamdani model istypically used in knowledge-based (expert) systems. A gen-eral class of fuzzy systems that includes the functionalfuzzy systems is referred as Takagi–Sugeno fuzzy systems.The fuzzy rules of these two methods are illustrated inFig. 2. In data-driven identification, the TS fuzzy model

x y x y

μ μμμ

iA iAiB ( )if x

i iIf x is A then y is B ( )i i

If x is A then y f x=a b

Fig. 2. Illustration of fuzzy rules: (a) Mamdani model and (b) Takagi–Sugeno model.

M.K. Das, N. Kishor / Expert Systems with Applications 36 (2009) 1142–1154 1145

is quite popular (Babuska & Verbruggen, 2003). Thismodel consists of If-then rules with fuzzy antecedents andmathematical functions in the consequent part. A TS fuzzymodel consists of a set of fuzzy rules, each describing alocal linear input–output relationship defined as

Ri: If x1 is Ai1 and . . . and xn is Ain then

yi ¼ aixþ bi i ¼ 1; 2; . . . ;N r ð4Þ

where ai is the consequent parameter vector and bi is a sca-lar offset.

Here Ri is the ith rule, x ¼ ½x1; . . . ; xn�T 2 X is the inputvector (antecedent) variable, Ai1, . . . ,Ain are fuzzy setsdefined in the antecedent space and yi is the model output.Nr denotes the number of rules. The model combines alinguistic description with standard functional regression:the antecedents describe fuzzy regions in the inputspace in which the consequent functions are valid. Theaggregated output of the model y 2 Y is computed by tak-ing the weighted average of the individual rule’scontributions:

y ¼PN r

i¼1biðxÞyiPN r

i¼1biðxÞ¼PN r

i¼1biðxÞðaTi xþ biÞPNr

i¼1biðxÞð5Þ

where bi(x) is the degree of fulfillment of the ith rule givenby

biðxÞ ¼Yn

j¼1

lAijðxjÞ; i ¼ 1; 2; . . . ;N r

and lAijðxjÞ : R! ½0; 1� is the membership function of the

fuzzy set Aij in the antecedent Ri. In TS systems, the conse-quent part is a crisp function, which gives the approxima-tor the ability to incorporate knowledge about the model.The TS fuzzy model is regarded as a smooth piece-wise lin-ear approximation of a nonlinear function or a parameter-scheduling model. As the consequent parameters are first-order polynomials in the input variables, model Eq. (4) issometimes also called first-order TS fuzzy model. While azero-order TS fuzzy model is expressed as:

Ri: If x is Ai then yi ¼ bi i ¼ 1; 2; . . . ;N r

In case of zero-order TS fuzzy model, the input–outputEq. (5) reduces to form:

y ¼PN r

i¼1biðxÞbiPN r

i¼1biðxÞð6Þ

where, the consequents are constant (zero-orderpolynomial).

3.1. Zero-order adaptive TS fuzzy model

The zero-order fuzzy model has been developed as analternative to higher-order TS models. Though higher-order fuzzy systems have better functional mapping ability,however, at the same time, more prone to result into ‘‘localmodels” that cannot be interpreted as local description ofthe underlying system. From the model application pointof view, it should be noted that first- and higher-orderTS fuzzy models could not be in general inverted analyti-cally, while for zero-order TS fuzzy models the inversionis straightforward. The system identification performancedepends on the inference method used, the parameter esti-mation criterion, the overlap of the antecedent membershipfunctions, etc. While zero-order TS models are less sensi-tive to these issues.

Often the rules of the fuzzy system are designed a prioriand the parameters of the membership functions areadapted in the learning process from input–output datasets. In the conventional adaptive fuzzy inference systems,the rules of the model may lose their meanings that wereinitially assigned to them. In the work as presented here,fuzzy model adaptation to maintain the interpretation ofthe adaptive fuzzy inference system during learning hasbeen considered.

The steady state behavior of the plant is represented by asingleton or zero-order TS fuzzy model represented by fol-lowing set of rules (Abonyi, Nagy, & Szeifert, 1999):

Ri1;...;in : If z1 is A1;i1 and . . . and zn is An;in then yadap

¼ bi1;...;in ð7Þ

where,Ri1;...;in denotes the fuzzy rule,Aj,ij(zj) the ij = 1,2,. . .,Mjth antecedent fuzzy set referring

to the jth input, whose membership functions aredenoted by the same symbols as the fuzzy values

Mj number of the fuzzy sets on the jth input domain.

2z

2z

with Applications 36 (2009) 1142–1154

T

0 1 1z

1

0

1z

21A

11A

25A

24A

23A

22A

15A14A13A12A

Fig. 4. Partition of the input domain for number of input, n = 2, numberof fuzzy sets, M1 = 5, M2 = 5.

For a given input, z = [z1, . . . ,zn] , i.e. z ¼ ½uss; x�, theoutput of the fuzzy model, yadap is inferred by computingthe weighted average of the rule consequents:

yadap ¼PM1;Mn

i1¼1;in¼1bi1;...;in bi1;...;inPM1;Mni¼1;in¼1bi1;...;in

ð8Þ

where uss is the steady state input,PM1;Mn

i1¼1;in¼1 �PM1

i1¼1 . . .PMnin

and the weight, bi1 ; . . . ; in > 0, represents the overalltruth value (degree of fulfillment) of the i1, . . . , inth rule cal-culated based on the degrees of membership values

bi1;...;in ¼Yn

j¼1

Aj;ijðzjÞ; j ¼ 1; 2; . . . ; n ij ¼ 1; 2 . . . ;Mj

ð9ÞThe triangular membership functions for each fuzzy lin-guistic value as shown in Fig. 3 is used in the work, how-ever, the results are independent of the type of themembership functions (Abonyi et al., 1999). The triangularmembership functions are defined as

Aj;ijðzjÞ ¼zj � aj;ij�1

aj;ij � aj;ij�1

; aj;ij�1 6 zj < aj;ij ð10Þ

Aj;ijðzjÞ ¼aj;ijþ1 � zj

aj;ijþ1 � aj;ij

; aj;ij 6 zj < aj;ijþ1 ð11Þ

Aj;ijðzjÞ ¼ 0; otherwise ð12Þ

aþj;ij denotes the cores of fuzzy set Aj;ij :

aj;ij ¼ coresðAj;ijÞ ¼ fzjjAj;ijðzjÞ ¼ 1g ð13Þ

The cores of the adjacent fuzzy sets determine the supportof a set thus ensuring that the fuzzy sets on a universe ofdiscourse always form a fuzzy partition, while keepingthe sum of the membership degrees equal to 1. This con-straint helps to obtain an interpretable and grid-type parti-tioning rule-base. Fig. 4 illustrates a typical grid partitionin a two-dimensional input space. The intersections of thegrids divide the input domain mean of the membership

1146 M.K. Das, N. Kishor / Expert Systems

0

1

, 1jj iA − , jj i

A, 1jj i

A +

, 1jj ia − , jj i

a, 1jj i

a + jz

Fig. 3. Membership functions used for adaptive zero-order TS fuzzymodel.

functions on the If-part of fuzzy rules. Since the productoperator is used for and- connective, the sum operator isused for aggregation and the fuzzy c-means method fordefuzzification. The complete rule base is given by

XM1;Mn

i1¼1;in¼1

bi1;...;in ¼ 1 ð14Þ

Thus at a given observed point, zj 2 ½aj;mj ; aj;mjþ1� Eq. (8)can be simplified as

yadap ¼XM1;Mn

i1¼1;in¼1

Yn

j¼1

Aj;ijðzjÞ !

� bi1;...;in

" #ð15Þ

3.2. Gradient-descent adaptation algorithm

The consequent parameters are identified by the LSEmethod, while the antecedent partition obtained as a carte-sian product of univariate triangular membership functionsis optimized by gradient-descent techniques.

With a given set of input–output data, it is possible togenerate fuzzy model on measured data. The aim of theadaptive algorithm is to tune the parameters of the fuzzymodel in order to minimize the prediction error at eachsample instant between the model and the measured out-put. The tuning of the parameters is used in fuzzy inferenceas numerical optimized technique. Among the conven-tional method, gradient-descent (GD) adaptation method(Babuska, 1998) permits accurate learning of all parame-ters of the fuzzy model. It is simple, requires low memoryand computational cost and thus allows building real-timelearning algorithms.

M.K. Das, N. Kishor / Expert Systems with Applications 36 (2009) 1142–1154 1147

The fuzzy system is parameterized by the followingparameters at the k-th time instant:

HðkÞ ¼ faj;ij ; bi1;...;in ; j ¼ 1; 2; . . . ; n; ij ¼ 1; 2; . . . ;Mjgð16Þ

where, Mj is the number of the antecedent fuzzy sets at thejth input domain.

The GD method seeks to decrease the value of the qua-dratic objective function based on the instantaneous modeloutput error:

EðkÞ ¼ 1

2½yadapðkÞ � yrefðkÞ�

2 ð17Þ

where, yref(k) is the reference for the model output,yadap(k).

The parameter set h(k) of the fuzzy model is changed viathe following iterative rule:

Hðk þ 1Þ ¼ HðkÞ þ DHðkÞ ¼ HðkÞ � aloEðkÞoHðkÞ ð18Þ

where al is learning-rate parameter, which controls howmuch the parameters are altered at each iteration.

The detail of gradient descent adaptation algorithm isgiven (Abonyi et al., 1999).

3.3. ANFIS based fuzzy model

A specific approach in neuro-fuzzy development is theadaptive neuro-fuzzy inference system (ANFIS), whichhas been proved to have significant results in modelingnonlinear functions. In ANFIS, the membership functionsare extracted from a data set that describes the systembehavior. The ANFIS learns features in the data set andadjusts the systems parameters according to a given errorcriterion, i.e. ANFIS are fuzzy models put in the frame-work of adaptive systems to facilitate learning and adapta-tion. The MATLAB toolbox (Fuzzy logic toolbox for usewith MATLAB�, 2000) implements anfis function todevelop fuzzy model. The MATLAB toolbox function gen-

Fig. 5. Experimental data analysis as a function of heat flux for boiling(b) Percentage error variation.

fis1 generates an initial single output Sugeno-type FIS forANFIS training using a grid partition on data.

4. Result and discussion

Present study involves computation of heat transfercoefficient for saturated boiling of distilled water fromplain copper heating tube surface by conducting experi-ment at atmospheric and sub-atmospheric pressures. Theatmospheric pressure is denoted as Pr-1 (98.12 kN/m2),and sub-atmospheric pressures are represented by Pr-2(71.47 kN/m2), Pr-3 (57.96 kN/m2) and Pr-4 (44.81 kN/m2). Fig. 5(a) shows the variation of heat transfer coeffi-cient with heat flux for saturated boiling of distilled water.Pressure is a parameter in this plot. From this plot, follow-ing salient features are obtained:

(i) Heat transfer coefficient increases with an increase inheat flux irrespective of pressure. The variationbetween the two can be described by power law,h / q0.7.

(ii) At a given value of heat flux, an increase in pressuretends to increase the value of heat transfer coefficient.

The above two observations are consistent and are inaccordance with the physics of boiling phenomena. It isunderstood that the heat transfer coefficient for boiling ofa liquid on a plain heating tube depends upon heat fluxand pressure. Therefore, using experimental data for theboiling of distilled water, following relationship:

h ¼ C1q0:7p0:32 ð19Þ

is determined by least squares method within a maximumerror of ±7.5%, where, C1 is a constant whose value is com-puted as 0.5673 for distilled water.Fig. 5(b) shows the var-iation of percentage error with heat flux, for saturatedboiling of distilled water at different values of pressures.The percentage error is defined as:

of water on plain heating tube. (a) Heat transfer coefficient variation.

Fig. 6. Modeling performance of zero-order fuzzy model as a function of number of membership functions. (a) Experimental data. (b) Empirical data.

1148 M.K. Das, N. Kishor / Expert Systems with Applications 36 (2009) 1142–1154

½Experimental data� Empirical data�Experimental data

� 100% ð20Þ

In the development of prediction model for heat transfercoefficient, boiling heat transfer coefficient obtained fromexperiment is referred as experimental data, while the valueof same parameter determined from Eq. (19) is referred asempirical data.

To identify the fuzzy model for prediction of heat trans-fer coefficient, the input–output variable (heat flux-heattransfer coefficient) obtained/computed at a given pressureis divided into two halves as training and validation dataset, each consisting of 40 samples. The samples are com-puted through extrapolation from intermediate data val-ues. The performance of fuzzy model is validated/assessed by means of performance indexes (PI). The vari-ance accounted for (VAF) and the root mean square(RMS) error is used to indicate the identified modelaccuracies.

VAF ¼ 1� varðy � yÞvarðyÞ

� �� 100% ð21Þ

RMSE ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPNv

i¼1½yi � yi�2

N v

sð22Þ

where var denotes the variance, yi is the experimental/empirical output, yi is the predicted output of model andNv is the number of data sample for validation. Nv canbe replaced with N for computation at training data set.A higher VAF indicates the better closeness between theexperimental/empirical and predicted outputs. And lowerthe value of RMSE, the better the performance of model.

4.1. Zero-order adaptive fuzzy model identification

The triangular membership function is used in the modelidentification study, since the results are independent of thetype of the membership functions (Abonyi et al., 1999).The one-step ahead prediction accuracy of heat transfer

coefficient is measured in terms of above defined PI. Thevariation of PI with the number of fuzzy sets, i.e. numberof membership functions at each pressure is shown inFig. 6. As indicated the prediction accuracy improves withincrease in the number of membership functions, i.e. num-ber of fuzzy rules. It is observed that there exists a marginaldifference in terms of prediction accuracy at different pres-sures with same number of membership function. How-ever, the values of PI as computed from empirical dataare comparatively superior to experimental data with samefuzzy rules. The VAF computed at each pressure canhardly be distinguished for model obtained from empiricaldata. It is desired to determine a fuzzy model with sufficientnumber of membership functions, which can predict theheat transfer coefficient at different pressure within consid-erable limits of error. Thus with the above discussion it issaid that the zero-order adaptive fuzzy model identifiedfrom the data at a given pressure is capable to predictthe heat transfer coefficient at other pressure too. As anillustration, the experimental data and predicted heattransfer coefficient with the corresponding heat flux asinput for zero-order adaptive fuzzy model with two num-bers of membership functions at pressure Pr-4 is shownin Fig. 7(a). The corresponding triangular membershipfunctions are shown in Fig. 7(b).

4.2. ANFIS mode identification

MATLAB based anfis function is further utilized todetermine the model for heat transfer coefficient prediction.The ANFIS fuzzy model can be identified with differenttypes and numbers of membership function. The trainingdata and validation data consisting of 40 samples corre-sponding to pressure Pr-1 and Pr-2 respectively is used.The data sample at pressure Pr-3 and Pr-4 is also used astraining data set. The PI for various types of membershipfunctions against the number of membership functionsfrom experimental data is presented in Fig. 8. As observed,

Fig. 8. Performance with variation in number of membership functions for various types of membership functions obtained from experimental data: (a)RMS error value and (b) VAF value.

Fig. 7. Zero-order adaptive fuzzy model response with two number of membership functions at pressure Pr-4. (a) Prediction response (b) Membershipfunctions.

M.K. Das, N. Kishor / Expert Systems with Applications 36 (2009) 1142–1154 1149

the PI improves with increase in number of membershiprules for pi as membership function at all pressure condi-tions considered here. With use of different types of mem-bership function and their numbers, it is suggested that thepredicted value would vary significantly with respect toexperimental data. The same qualitative analysis is alsocarried out from empirical data. The prediction perfor-mance of this study is given in Fig. 9. A similar behaviortrend is observed in this case too as discussed above. Theuse of pi type membership function for fuzzy model identi-fication gives the best PI values among all the types, how-ever these are still poorer than those as given in Fig. 6.

To decide the appropriate number of epoch for ANFISfuzzy model training, the model performance in terms of PIas a function of number of epoch with four number ofmembership functions is shown in Figs. 10 and 11. Thefuzzy model with 50 or more number of epochs in training

do not lead to significant effect on performance. The com-puted PI values are found to be quantitatively inferior tothose from empirical data. Further, it is observed that theprediction of a given fuzzy model with pi and triangulartype membership function gives a comparatively bettervalue of PI at all pressure conditions. However, an excep-tion is observed with triangular membership function fromempirical data at pressure Pr-1. It is understood from theabove discussion that ANFIS model trained at a givenpressure condition does not predict satisfactorily at otherpressure conditions.

4.3. Comparison of zero-order adaptive fuzzy model and

ANFIS model with experimental data

The comparative percentage error at different pressuresbetween: (i) Experimental–Empirical data, (ii) Experimen-

Fig. 9. Performance with variation in number of membership functions for various types of membership functions obtained from empirical data. (a) RMSerror value and (b) VAF value.

Fig. 10. Performance with variation in number of epoch for various types of membership functions obtained from experimental data. (a) RMS error valueand (b) VAF value.

1150 M.K. Das, N. Kishor / Expert Systems with Applications 36 (2009) 1142–1154

tal-Zero-order adaptive fuzzy model predicted data and(iii) Experimental-ANFIS fuzzy model predicted data isshown in Fig. 12. The percentage error for case (i) isdefined by Eq. (20), whereas for case (ii) and case (iii) itis defined by equation given below:

½Experimental data-Fuzzy modelðZero-order adaptive or ANFISÞpredicted data�Experimental data

� 100% ð23Þ

Fig. 12 clearly shows that the percentage error in case (i)lies in the range of (±7.5%) at different pressure conditionsconsidered in the study. For case (ii) and case (iii), the pre-

dicted data for zero-order adaptive and ANFIS fuzzymodel is determined with two and six number of triangulartype membership functions. It is observed that for zero-order adaptive with two and six membership functions,the variation in percentage error at pressure, Pr-1, lies in

the range of (±0.2%). At pressure, Pr-2, error increasesto (±0.4%) for the same model comprising two member-ship functions, while with six membership functions, it

Fig. 11. Performance with variation in number of epoch for various types of membership functions obtained from empirical data. (a) RMS error valueand (b) VAF value.

Fig. 12. Prediction performance from correlation and fuzzy model; with ANFIS fuzzy model using triangular type membership.

M.K. Das, N. Kishor / Expert Systems with Applications 36 (2009) 1142–1154 1151

remained unaltered. It is further noted that the percentageerror does not increase beyond this range even at pressures,Pr-3 and Pr-4. The ANFIS model (with two and six mem-bership function) is found to have prediction inaccuracy inthe range of (±2.5%) at pressure, Pr-1. The percentageerror is not shown at pressure Pr-2 since the ANFIS modelis being validated with data obtained at this condition. Atpressure, Pr-3, the prediction inaccuracy varies in the range(±4%) and (±2%) with two and six membership functions

respectively. However, the variation at pressure, Pr-4, is inthe range of (±3%) for both two and six membershipfunctions.

From the above analysis, it is found that the zero-orderadaptive fuzzy model successfully predicts the heat transfercoefficient at different conditions of pressure considered inthe present study.

The qualitative comparison of zero-order adaptive fuzzymodel (triangular type membership function) with ANFIS

Fig. 13. Prediction performance from correlation and fuzzy model; with ANFIS fuzzy model using pi-type triangular type membership.

1152 M.K. Das, N. Kishor / Expert Systems with Applications 36 (2009) 1142–1154

model (pi type membership function) is shown in Fig. 13. Itis found that the variation in percentage error reduces to(±2%) at pressure, Pr-1. Further, at pressure, Pr-3, the pre-diction performance of ANFIS model with two member-ship functions improves and is the range of (±2%) whileit remains unchanged with six membership function. How-ever, at pressure Pr-4, the variation is similar to thoseobtained using triangular type membership function.

4.4. Prediction accuracy of zero-order fuzzy model on

experimental data available in open literatures

Numerous authors have conducted experimental studyof pool boiling of distilled water on plain tube at atmo-spheric pressure. In the light of determining a generalizedfuzzy model, it is felt that the analytical prediction analysisto be addressed on the experimental data of different

Table 1Performance index computation with data set of different authors

Authors Two membership functions

RMS VAF STD

GDB 21.0509 99.9182 0.4299KUM 939.0659 98.5453 8.5006MIH 5.8428 99.9895 0.1854SSP 96.5592 99.4956 3.9487VIT 14.5716 99.9588 0.4199WAD 501.7282 99.5570 4.2227YOH 78.0438 99.9755 2.1915

authors. In this section, prediction performance of zero-order fuzzy model on the experimental data of VIT (Vitt-ala, Gupta, & Agarwal, 2000), SSP (Hsieh & Hsu, 1994),KUM (Kurihara & Myers, 1960), YOH (Young & Hum-mel, 1964) GDB (Bansal, 1985), WAD (Wang & Dhir,1993) and MIR (Das, 2006) is discussed. Table 1 reportsthe performance index; RMS, VAF and standard deviation(STD) of the fuzzy model with two and six number ofmembership functions. It looks reasonable to consider thatthe obtained prediction performances with two member-ship functions are quite close to the experimental qualityof the data, except in case of Kurihara and Meyers. Fur-ther it is worth to note that a higher (six number) member-ship function enhances the approximation. In fact thesestatistical figures demonstrate an accurate representation.The qualitative analysis of prediction performance withtwo and six membership functions in terms of percentage

Six membership functions

RMS VAF STD

1.4852 � 10�12 100 3.0163 � 10�14

8.4883 � 10�12 100 1.6809 � 10�13

2.0337 � 10�13 100 5.4635 � 10�15

2.8761 � 10�13 100 1.5164 � 10�14

5.3806 � 10�13 100 1.7043 � 10�14

40.1587 99.9972 1.30312.4977 � 10�12 100 5.9623 � 10�14

Fig. 14. Zero-order fuzzy model performance with two membershipfunctions on data set of different authors: (a) Error variation. (b) Zoomed:Error variation

Fig. 15. Zero-order fuzzy model performance with six membershipfunctions on data set of different authors. (a) Error variation and (b)zoomed: Error variation.

M.K. Das, N. Kishor / Expert Systems with Applications 36 (2009) 1142–1154 1153

error is illustrated in Figs. 14 and 15, respectively.Fig. 14(a) shows the prediction error deviation as high as±18% on experimental data of Kurihara and Meyers.The said deviation for other authors lies the range of±0.2% to about ±6%, which suggests fair agreement ofpredicted data with experimental one. Zero-order fuzzymodel with six membership functions has an effect toimprove the accuracy as suggested from the error variationshown in Fig. 15. Amongst the experimental data tested,performance on WAD data is found to be inadequate togive satisfactory agreement with maximum error deviationup to ±3%, however, on remaining authors, the predictionaccuracy is reasonably good with deviation in the range of±10�13%.

From the above analysis, it can be understood that thezero-order adaptive fuzzy model not only has a good agree-ment with experimental data of MIR, but also demon-strates correct representation on other authors.

5. Conclusion

A modeling approach based on fuzzy systems had beenconsidered to predict the boiling heat transfer coefficient.The data obtained through the pool boiling experimentwas used to identify the fuzzy model. The empirical corre-lation developed was found to correlate experimental datawithin a maximum error of (±7%).

The zero-order adaptive fuzzy model suggested thatwith increase in the number of membership functions, theprediction accuracy had improved. The prediction errorof zero-order adaptive fuzzy model at all the pressure con-ditions remained up to (±0.5%).

But, for ANFIS fuzzy model with two numbers of trian-gular type membership function, it increased from (±2.5%)at pressure, Pr-1 to (±4%) and (±3%) at pressures, Pr-3and Pr-4 respectively. However, the use of pi type

1154 M.K. Das, N. Kishor / Expert Systems with Applications 36 (2009) 1142–1154

membership function, had reduced the error variation to(±2%) at pressure conditions Pr-1 and Pr-3, but the sameremained unchanged at pressure Pr-4, i.e. (±3%).

The study has shown that the adopted modelingapproach is effective in prediction of heat transfer coeffi-cient on experimental data of other authors too.

Thus the above conclusion suggested the feasibility ofzero-order adaptive fuzzy model in the application for pre-diction of boiling heat transfer coefficient.

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