International Journal of Heat and Mass Transfer 54 (2011) 5001–5009

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International Journal of Heat and Mass Transfer

journal homepage: www.elsevier .com/locate / i jhmt

A simplex search method for a conductive–convective fin with variable conductivity

Ranjan Das ⇑Department of Mechanical Engineering, Tezpur University, Tezpur 784028, Assam, India

a r t i c l e i n f o

Article history:Received 1 February 2011Received in revised form 4 July 2011Accepted 5 July 2011Available online 28 July 2011

Keywords:Conduction–convectionConvective–conductive parameterVariable conductivity parameterSimplex search method

0017-9310/$ - see front matter � 2011 Elsevier Ltd. Adoi:10.1016/j.ijheatmasstransfer.2011.07.014

⇑ Tel.: +91 3712 267007x5860; fax: +91 3712 2670E-mail address: ranjandas81@gmail.com

a b s t r a c t

An inverse problem is solved for simultaneously estimating the convection–conduction parameter andthe variable thermal conductivity parameter in a conductive–convective fin with temperature dependentthermal conductivity. Initially, the temperature field is obtained from a direct method using an analyticalapproach based on decomposition scheme and then using a simplex search minimization algorithm aninverse problem is solved for estimating the unknowns. The objective function to be minimized is repre-sented by the sum of square of the error between the measured temperature field and an initially guessedvalue which is updated in an iterative manner. The estimation accuracy is studied for the effect of mea-surement errors, initial guess and number of measurement points. It is observed that although very goodestimation accuracy is possible with more number of measurement points, reasonably well estimation isobtained even with fewer number of measurement points without measurement error. Subject to selec-tion of a proper initial guess, it is seen that the number of iterations could be significantly reduced. Therelative sensitiveness of the estimated parameters is studied and is observed from the present work thatthe estimated convection–conduction parameter contributes more to the temperature distribution thanthe variable conductivity parameter.

� 2011 Elsevier Ltd. All rights reserved.

1. Introduction a given differential equation. Such problems are known as the

A fin is an extended surface for minimizing the heat loss andfind wide range of applications for enhancement of heat transferbetween multiple surfaces [1–4]. To study the physics, perfor-mance and behavior of fins, a good amount of experimental andnumerical investigations have been done and in the process of con-tinual research. In one hand, the experimental analysis has got sev-eral advantages and is necessary to know the exact behavior of asystem. However, on other hand they require large manpower,considerable financial investment and consume time. Therefore,numerical investigations have received a considerable attentionin almost all fields of science and engineering. In similar connec-tion, a good amount of work has been done in developing efficientalgorithms for studying the heat transfer characteristics of ex-tended surfaces/fins. Yeh [5] has used an analytical approach basedon Harper Brown approximation to determine the optimal dimen-sions of different types of fins. Heggs and Ooi [6] presented designcharts for maximizing the effectiveness of rectangular fins. A widevariety of fins were numerically analyzed by Aziz and Fang [7]. Anoptimization analysis based on analytical method for a pin fin withvariable base thickness was done by Kang [8].

For known values of thermophysical parameters and boundaryconditions, the numerical/analytical solution is obtained by solving

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05.

direct/forward problems and are mathematically well-posed [9].In problems dealing heat transfer from extended surfaces, the finalresult is generally in the form of either temperature or heat flux.However, there are instances when the result is available a priori,but either some values of the thermophysical parameters orboundary condition(s) are unknown. The estimation of correct val-ues of the unknowns invariably requires the solution of a problemreferred to as the inverse problem and is mathematically ill-posed[10]. In past, the inverse problems dealing with parameter retri-evals in fins have been studied extensively. Chen et al. [11] haveestimated the heat flux at the base of a pin fin from the knowledgeof temperature distributions. In their investigation, the conjugategradient method (CGM) was used as an optimization tool. Usingsuperposition theorem, the boundary heat flux in an inverse con-duction problem was estimated by Fang et al. [12]. Maillet et al.[13] compared the analytical and the boundary element methodin estimating the heat transfer coefficient for a plate fin on a cylin-der. From the knowledge of dimensionless temperature data bysolving a direct problem, using the least-squares-error method(LSM) the boundary conditions were determined by Chen et al.[14]. The finite difference method (FDM) was used as a directmethod in their analysis. Using quasi steady-state theory and theLSM the heat transfer coefficient in a conductive–convective finwith insulated tip was estimated by Cole at al. [15]. Chang et al.[16] have estimated the local heat transfer coefficient in a fin thebase of which was subjected to oscillated temperature. The FDMin conjunction with the LSM was used in their inverse problem.

Nomenclature

A non-linear term for second order derivative in thedecomposition method

B non-linear term for first order derivative in the decom-position method

C contraction coefficientc arbitrary constant in the decomposition method (inte-

gral constant)D highest order derivative in the decomposition methodE expansion coefficiente measurement error ð¼

Pnp¼1epÞ

F objective functionh heat transfer coefficientI iteration of the simplex search algorithmJ magnitude of error in the decomposition method

(decomposition error)j sensitivity coefficientk thermal conductivityN2 convection–conduction parameter, ð2hX2Þ=ðk1tÞ

h i¼

Bi ðBiot numberÞn number of measurement pointsp number of variables in the simplexQ heat fluxq point/vertex of the simplexR reflection coefficientS shrinkage coefficientT temperature

t thickness of finX length of the fin geometryx any location along the fin length

Greek symbolsc variable thermal conductivity parameter c0ðTW � T1Þc0 variable thermal conductivity coefficienth non-dimensional temperature

Superscriptscond. conductionconv. convection⁄ index for non-dimensional distance� index for exact temperaturej index for iteration of the simplex search algorithm

Subscripts� index for guessed temperaturee estimatedi index for number of variables in the simplexk index for a particular term in the decomposition meth-

odm measuredp index for measurement pointi index for any point in the simplex1 atmospheric condition

5002 R. Das / International Journal of Heat and Mass Transfer 54 (2011) 5001–5009

It is observed that most of the available studies dealing with theheat conduction assume the thermal conductivity to be constantand its variation with temperature is neglected. This assumptionis not universally true and in such cases the assumption of constantthermal conductivity yields inaccurate results [17]. Few studieshave reported the assumption of variable thermal conductivity inthe numerical solution of a heat transfer through a fin. Aziz andEnamul Hug [18] and Aziz [19] have used the perturbation basedanalytical method for solving the conductive–convective heattransfer equation involving variable thermal conductivity. Muzzio[20] has applied the Galerkin method for obtaining approximatetemperature distributions in a conductive–convective fin. Zubairet al. [21] have obtained the optimal dimensions in circular finshaving temperature dependent thermal conductivity. In connec-tion with inverse problems dealing with the estimation of param-eters in extended surfaces/fins, no work dealing with variable/temperature dependent conductivity has been found. Therefore,in the present work, an inverse method is applied for estimatingthe convective–conductive parameter and the variable conductiv-ity parameter from the knowledge of the temperature distributionsin a conductive–convective rectangular fin. The temperature distri-butions have been obtained from the solution of a forward/directproblem from a decomposition method based on Adomian scheme[22–26]. Subject to incorporation of measurement errors, an in-verse problem is then solved for estimating the desired unknowns.For minimization of the desired objective function, the simplex-search method [27,28] based upon Nelder–Mead optimizationalgorithm is used in the present work. The following section de-scribes the formulation and the solution procedure for obtainingthe temperature distributions using the direct method.

Fig. 1. Physical geometry and boundary conditions of the problem considered forparameter estimation.

2. Formulation

Consider a rectangular fin geometry with the details as shownin Fig. 1. The base of the fin is kept at a uniform temperature, TW

while its tip is well insulated. Assuming an azimuthally symmetriccase, the steady-state energy balance between locations x andx + Dx of Fig. 1 gives the following equation

Q cond:;x � Q cond:;xþDx � Q conv: ¼ 0 ð1Þ

where the subscripts ‘‘cond.’’ and ‘‘conv.’’ represents conductionand convection heat transfer and obey Fourier law and Newton’slaw of cooling, respectively. In the above equation (Eq. (1)), thethermal conductivity, k, is assumed to be temperature dependentaccording to the following equation:

k ¼ k1 1þ c0ðT � T1Þf g ð2Þ

In order to carry out the analysis easily, the non-dimensionalizationis done in the following manner,

h ¼ ðT � T1Þ=ðTW � T1Þ; c ¼ c0ðTW � T1Þ

N ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið2hX2Þ=ðk1tÞ

q; x� ¼ x=X

ð3Þ

R. Das / International Journal of Heat and Mass Transfer 54 (2011) 5001–5009 5003

It can be observed from Eq. (2) that a positive value of variable con-ductivity parameter (c) indicates an increase in the value of thethermal conductivity (k). Similarly, zero and negative values of var-iable conductivity parameter (c) represent, respectively, a constantand a decrease in the values of the thermal conductivity (k). Besidesthis, as mentioned earlier (in the nomenclature) that the parameter(N) is nothing but square root of Biot number (Bi), thus, a higher va-lue of the parameter (N) represents more involvement of the con-vective heat transfer than the conductive heat transfer. Differentcombinations of heat transfer coefficient, thermal conductivity,length and thickness of fin can satisfy a given value of the parame-ter (N). Therefore, the governing energy equation (Eq. (1)) can bewritten in non-dimensional form as follows:

ð1þ chÞ d2h

dðx�Þ2þ c

dh

dx�

� �2

� N2h ¼ 0 ð4Þ

Using the problem description as mentioned above, the non-dimen-sional form of the boundary conditions can be represented asfollows:

West boundary : x� ¼ 0;dh

dx�¼ 0 ðaÞ

East boundary : x� ¼ 1; h ¼ 1 ðbÞð5Þ

Fig. 2. Flowchart of simp

Using the decomposition scheme as given in [22–26], the non-dimensional energy equation (Eq. (4)) is expressed as given below:

Dh ¼ N2h� chd2h

dðx�Þ2� c

dh

dx�

� �2

ð6Þ

where, D represents the derivative of the highest order. Taking aninverse operator D�1 on both sides, the above equation can be re-written as follows [22–26]:

h ¼ h0 þ N2D�1h� cD�1A� cD�1B ð7Þ

where, A and B are given by the following equation [22–26]:

A ¼ hd2h

dðx�Þ2

B ¼ dh

dx�

� �2ð8Þ

Using the boundary condition (Eq. (5(a))) the value of h0 in (Eq. (7))is given by the following [22–26],

h0 ¼ c þ x�dh

dx�

����x�¼0

ð9Þ

lex search algorithm.

5004 R. Das / International Journal of Heat and Mass Transfer 54 (2011) 5001–5009

where, c is an arbitrary constant (integral constant) in the range[0, 1]. Physically, the integral constant c represents corrected-tem-perature which would be required mathematically to satisfy theboundary condition(s). It can be easily computed by taking a suffi-ciently good number of values between [0, 1] and then evaluatingthe respective temperatures using the insulated boundary condition(Eq. (5(a))). Subsequently, the relative magnitude of errors (decom-position errors) J is obtained between the calculated temperaturesand the other boundary condition which is at a fixed temperature(Eq. (5(b))). The value of c which contribute to the least decomposi-tion error (J � 0) is then taken as the accepted value. In order to pro-cess Eq. (7), the information about the non-linear terms (A, B) asgiven in Eq. (8) is required and the same can be written as follows[22–26]:

A ¼X1k¼0

Ak; B ¼X1k¼0

Bk ð10Þ

where, k represents the index for a particular term. Once the valueof c is known, the temperature distribution for a particular term canbe easily computed by solving Eq. (7) and are given by the followingequations [22–26],

hk¼1 ¼12

c2N2x� ðaÞ

hk¼2 ¼1

24cN4ðx�Þ4 � 1

2cc2N2ðx�Þ2andsoon ðbÞ

ð11Þ

Finally, the temperature distribution can be computed by adding allk terms and for a particular location, x⁄ the same is given as below:

Integral constant, c

Dec

ompo

sitio

ner

ror,

J

00

0.2 0.4 0.6 0.8 1

0.2

0.4

0.6

0.8

1

γ = 0.6γ = 0.0γ = -0.6

N = 0.316 (Bi = 0.1)

(a)

(b)

Integral constant, c

Dec

ompo

sitio

ner

ror,

J

0 0.2 0.4 0.6 0.8 10

1

2

3

4

5

6

7

γ = 0.6γ = 0.0γ = -0.6

N =2.0 (Bi = 4)

Fig. 3. Determination of the unknown constant, c

hðx�Þ ¼X1k¼0

hk ð12Þ

From the knowledge of thermophysical properties and boundaryconditions, the temperature, h obtained from the above forward/di-rect methodology and is taken as the exact temperature field, ~h. Theabove method is a well-posed direct problem and the solution is un-ique. Next, through an inverse analysis, simultaneous estimation oftwo parameters such as the convection–conduction parameter, Nand the variable thermal conductivity parameter, c is accomplishedwhich are considered as unknowns. The problem then becomesmathematically ill-posed and solution of the same requires somekind of regularization/optimization. Initially a guessed temperaturefield, h

�corresponding to some arbitrary values of the unknowns is

considered and then the square of the error between the exactand guessed field is minimized. This is generally referred to as theobjective function and is mathematically represented in the follow-ing manner:

F ¼Xn

p¼1

h�p � h

�p

� �2ð13Þ

where, n is number of points at which temperature distributions areobtained. If the exact field, h

�involves any error, e then the temper-

ature field is known as the measured temperature field. The objec-tive function (Eq. (12)) then gets transformed as below:

F ¼Xn

p¼1

h�p � h

�p þ ep

� �n o2ð14Þ

Integral constant, c

Dec

ompo

sitio

ner

ror,

J

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

γ = 0.6γ = 0.0γ = -0.6

N = 1.0 (Bi = 1)

Integral constant, c

Dec

ompo

sitio

ner

ror,

J

0 0.2 0.4 0.6 0.8 110-2

10-1

100

101γ = 0.6γ = 0.0γ = -0.6

N = 3.16 (Bi = 10)

, using minimum error approach; (0 6 c 6 1).

R. Das / International Journal of Heat and Mass Transfer 54 (2011) 5001–5009 5005

where, eP represents the error for a given measurement point. In or-der to minimize the objective function (Eq. (14)), the Nelder–Meadbased simplex search method is used in the present work. Followingsection describes the working procedure of the algorithm.

3. Nelder–Mead algorithm

The working procedure of the simplex search method is basedon the Nelder–Mead algorithm [28] and is widely used for mini-mizing several types of objective functions [27,28]. The method be-longs to the category of direct search algorithms and carries someinherent advantages such as easy implementation, requirement ofless computational time and compatibility with problems wheregradients are either unavailable or difficult to evaluate [29,30].However, it is worth to mention here that the direct search meth-ods have slow convergence [29,30]. The simplex-search algorithmis defined by four coefficients such as the reflection coefficient, R,the expansion coefficient, E, the contraction coefficient, C, andthe shrinkage coefficient, S satisfying the following conditions[27,28],

Distance,x*

Tem

per

atur

e,θ

0 0.2 0.4 0.6 0.8 10.4

0.5

0.6

0.7

0.8

0.9

1

Direct method (present work)Aziz and Hug [18]; Chiu and Chen [26]Muzzio [20]

γ = -0.6

γ = 0.0

γ = -0.6

(a)

(b)

Distance,x*

Tem

per

atur

e,θ

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Direct method (present work)Aziz and Hug [18]; Chiu and Chen [26]Muzzio [20]

γ = 0.6

γ = 0.0

γ = -0.6

Fig. 4. Comparison of the temperature distributions in the medium for validation ofthe forward problem, (a) N = 1.0, (b) N = 2.0.

R > 0; E > 1; E > R and 0 6 ðC; SÞ 6 1 ð15Þ

Initially (j = 0), depending upon the number of variables, p an initialsimplex, Ij is constructed with p + 1 vertices. Therefore, in the pres-ent problem (p = 2), the simplex becomes a triangle. Subsequently,for each iteration the function values Fj

q are evaluated at each vertexand arranged as per the following order,

Fjq¼1 6 � � � F

jq¼pþ1 ð16Þ

In the above equation (Eq. (16)), the point corresponding to Fjq¼1

represents the best vertex. For a objective function consisting oftwo variables, usage of the above equation (Eq. (16)) yields the bestfunction value as Fq ¼ 1j. Thus, the corresponding best point/vertexis represented by qj

q¼ðpð¼1ÞÞ ¼ q1. Similarly, the worst point/vertex isgiven by qj

q¼ðpð¼2þ1ÞÞ ¼ q3. If the objective function value is not foundto satisfy the convergence criteria, the algorithm proceeds to thenext iteration and a new simplex, Ijþ1–Ij is thus created. The flow-chart of an iteration of the algorithm is shown in Fig. 2. Dependingupon the value of the objective function, it is observed from the fig-ure that the algorithm requires evaluation of different points suchas the reflection point, the expansion point, the contraction pointand the shrinkage point and is expressed as follows [27,28],

Reflection point : ðqRÞ ¼Xp

i¼1

qi=pþ RXp

i¼1

qi=p� qpþ1

!ð17Þ

Expansion point : ðqEÞ ¼Þ ¼Xp

i¼1

qi=pþ E qR �Xp

i¼1

qi=p

!ð18Þ

Outside contraction point : ðqCÞ¼Þ¼Xp

i¼1

qi=pþC qR�Xp

i¼1

qi=p

!ðaÞ

Inside contraction point : ðqC0 Þ¼Xp

i¼1

qi=p�CXp

i¼1

qi=p�qpþ1

!ðbÞ

ð19Þ

Shrinking points : q1; qjþ12 ; . . . qjþ1

pþ1 ð20Þ

where,Ppþ1

i¼2 qjþ1i represents the new vertices at next iteration level

and is represented in the following manner [27,28],

Xpþ1

i¼2

qjþ1i ¼ q1 þ S

Xpþ1

i¼2

qi � q1

!ð21Þ

The following section presents the results and discussions forobtaining the temperature distributions from the forward problemand simultaneous estimation of unknown parameters using an in-verse method.

4. Results and discussion

For the problem described as described earlier in Section 2, forknown values of thermophysical properties and prescribed bound-ary conditions, initially a forward problem is solved for obtainingnon-dimensional temperature, h distributions using an analyticallybased decomposition scheme. As described earlier that the evalua-tion of temperature at successive locations is dependent upon thecorrect determination of integral constant, cð0 6 c 6 1Þ and evalu-ation of the same is done using a minimum error approach.Depending upon the number of locations (measurement points,n), the entire range of integral constant, c is discretized. For thepresent study, number of measurement points, n = 100 have beenfound sufficient to yield independent solutions. For different dis-cretized values of integral constant, c, corresponding temperaturesare computed and the respective error difference (decomposition

Table 1Details of the cases considered for different initial guess;(0 6 N 6 5.0; �1.0 6 c 6 1.0), x⁄(X) = 1.0.

Run Exact value, (N, c) Initial guess/point, (Ni, ci)

1 (2.0, �0.6) (1.5, �1.0)2 (0.5, �1.0)3 (2.5, �1.0)4 (2.5,1.0)5 (3.5, 1.0)6 (5.0, 1.0)

0 10 20 30 40 50 6010

-6

10-5

10-4

10-3

10-2

10-1

100

101

102

Iterations

Fu

nct

ion

val

ue

(b)

(a)

0 10 20 30 40 50 60

10-2

10-1

100

101

Iterations

Fu

nct

ion

val

ue

Fig. 5. Convergence history of the objective function, F for estimating the unknownparameters in minimum number of iterations, (a) e = 0, run 3; (b) e = 1.0; run 4.

5006 R. Das / International Journal of Heat and Mass Transfer 54 (2011) 5001–5009

error), J with respect to the boundary condition is obtained in themanner as discussed in Section 2. Fig. 3 presents the variation ofthe decomposition error, J for different values of the integral con-stant, c The comparison is done for four values of the convec-tion–conduction parameter (N = 0.316, 1, 2 and 3.16) and threevalues of the variable conductivity parameter (c = �0.06, 0, 0.06).It is observed from Fig. 3 that in each case only a particular valueof integral constant, c yields the least decomposition error (J � 0)It is worth to mention here that the value of the integral constant,c directly contributes to the local temperature distribution. It isalso observed from Fig. 3 that for the low value of the convec-tion–conduction parameter (N = 0.316) i.e. Biot (Bi = 0.1), the exactvalues of the integral constant, c are relatively higher than that ofhigh value of the convection–conduction parameter (N = 3.16) i.e.Biot (Bi = 10). This is due to increase in heat loss by convection be-tween the fin and the outside atmosphere at higher values of eitherthe convection–conduction parameter, N or Biot, Bi.

Once the correct values of the integral constant, c is determined,a forward problem is solved and temperature, h distributions areobtained using known values of the properties and boundary con-ditions. In order to study the accuracy of the forward method, inFig. 4, a comparison of the steady-state temperature, h field ismade along the fin length, x⁄. For convection–conduction parame-ter, N = (1, 2) i.e. Biot (Bi = 1, 4) and three different values of thevariable conductivity parameter (c = �0.6, 0, 0.6), the results havebeen benchmarked with those available in literatures [18,20,26].It is observed that the results of the forward method based onthe analytical decomposition scheme agree well with those avail-able in Refs. [18,20,26]. In addition to the above, it is observed thatfor all range of the variable conductivity parameter, c, a higher va-lue of the convection–conduction parameter, N i.e. Biot number, Bi(Fig. 4(b)) provides lower local temperature, h distributions. Thisresult is in line with that obtained earlier in Fig. 3 and thus, a sim-ilar explanation holds good in this case too. In addition to this, it isobserved in both cases (Figs. 4a–b), for a fixed value of the convec-tion–conduction parameter, N, the local temperature, h distribu-tions increase with increase in the variable conductivity

Table 2Comparison of the estimated values of the unknown parameters for different runs andmeasurement errors; (0 6 N 6 5.0; �1.0 6 c 6 1.0), x⁄(L) = 1.0.

Run Measurement error,e ¼

Pnp¼1ep

Exact value,(N, c)

Estimated value,(Ne, ce)

No. ofiterations

1 0.0 (2.0, �0.6) (1.87, �0.98) 572 (1.87, �0.98) 763 (1.87, �0.98) 544 (2.0, �0.6) 745 (2.0, �0.59) 896 (2.0, �0.0.58) 871 1.0 (1.50, �2.91) 652 (1.50, �2.91) 743 (1.50, �2.91) 794 (2.29, 0.40) 495 (2.29, 0.40) 596 (2.29, 0.40) 63

parameter, c, which is due to rise in magnitude of the conductiveheat transfer. After obtaining the temperature, h distributions fromthe forward problem, the following pages describes the results forthe simultaneous estimation of unknown parameters using the in-verse method.

In the inverse method, two parameters such as the convection–conduction parameter, N and the variable conductivity parameter,c have been simultaneously estimated. In order to achieve the task,an objective function as given in Eq. (12) /Eq. (14) is minimizedusing Nelder–Mead based simplex search method. The conver-gence limit is taken either when the percentage change in thevalue of the objective function, F between successive iterations be-comes negligible O( 6 10�5) or when the objective function attainszero value. Table 1 presents runs of the inverse algorithm per-formed using different initially guessed values for the unknowns,(Ni, ci). For each run considered in Table 1, the comparison of theexact values of the parameters and corresponding estimated valuesis made in Table 2. The investigation is done for number ofmeasurement points, n = 100 and simulated temperature fieldswith/without measurement errors. The number of iterations re-quired for the convergence is also shown. It is observed from Table2 that for simulated temperature field without having any mea-surement error, run 4 yields the best estimation and requires 74iterations. For other runs, a better estimation accuracy is obtainedfor the convection–conduction parameter, N compared to that for

R. Das / International Journal of Heat and Mass Transfer 54 (2011) 5001–5009 5007

the variable conductivity parameter, c However, for simulatedtemperature field with measurement error e = 1.0; i.e. ep = 0.01,run 4 requires minimum iterations as compared to other cases.In all cases, the estimated values are observed to differ from the ex-act ones. This indicates the sensitiveness of the estimated parame-ters (N, c) to introduce a particular amount of error in thetemperature field. In this case too, the accuracy of the convec-tion–conduction parameter, N is found to be better than the vari-able conductivity parameter, c Therefore, it suggests that the

Measured temperature, θm

Est

imat

edte

mp

erat

ure

,θe

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

45o lineEstimated values

Runs: 1-3e = 0.0N = 2.0γ = -0.6Ne = 1.87γe = -0.98

((a)

Measured temperature, θm

Est

imat

edte

mpe

ratu

re,θ

e

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

45o lineEstimated values

Run: 5e = 0.0N = 2.0γ = -0.6Ne = 2.0γe = -0.59

(

(e)

(c)

Measured temperature, θm

Est

imat

edte

mpe

ratu

re,θ

e

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

45o lineEstimated values

Runs: 1-3e = 1.0N = 2.0γ = -0.6Ne = 1.5γe = -2.91

Fig. 6. Comparison of the estimated and the measured tem

convection–conduction parameter, N influence the temperaturefield more than the variable conductivity parameter, c.

In order to study the variation in objective function, F with iter-ations a comparison is done in Fig. 5. For both cases of measure-ment error, e =0, i.e. ep = 0.0 and e=1, i.e. ep = 0.01, thecomparisons are made for the case requiring minimum numberof iterations. It is observed that in both the cases, minimum re-quired number of iterations for the convergence lie between 50and 60. However, for the case involving measurement error,

b)

Measured temperature, θm

Est

imat

edte

mpe

ratu

re, θ

e

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

45o lineEstimated values

Run: 6e = 0.0N = 2.0γ = -0.6Ne = 2.0γe = -0.58

d)

(f)

Measured temperature, θm

Est

imat

edte

mpe

ratu

re,θ

e

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

45o lineEstimated values

Runs: 4-6e = 1.0N = 2.0γ = -0.6Ne = 2.29γe = 0.4

Measured temperature, θm

Estim

ated

tem

pera

ture

,θe

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

45o lineEstimated values

Run: 4Σe = 0.0N = 2.0γ = -0.6Ne = 2.0γe = -0.6

peratures for different runs with measurement errors.

Distance, x*

Tem

per

atu

re,

θ

0 0.2 0.4 0.6 0.8 10.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

MeasuredEstimated

Distance, x*

Tem

per

atu

re,

θ

0 0.2 0.4 0.6 0.8 10.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

MeasuredEstimated

(b)

(c) (d)

(a)

Distance, x*

Tem

per

atu

re,

θ

0 0.2 0.4 0.6 0.8 10.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

MeasuredEstimated

Distance, x*

Tem

per

atur

e,θ

0 0.2 0.4 0.6 0.8 10.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

MeasuredEstimated

Fig. 7. Effect of number of measurement points with measurement errors on the temperature distribution; (a) n = 100, e = 0, (b) n = 100, e = 1.0, (c) n = 10, e = 0 (d) n = 10,e = 1.0; N = 2.0, c = �0.6.

5008 R. Das / International Journal of Heat and Mass Transfer 54 (2011) 5001–5009

e = 1.0 (Fig. 5b), the convergence is relatively slower and thusrequire more iterations of the optimization algorithm than the casecorresponding to the exact temperature measurement, e = 0(Fig. 5a). This is due to increase in the difference between theinitially guessed and the measured temperatures with the mea-surement error. It is also noticed from Fig. 5 that in both cases,beyond approximately 35 iterations there is no significant changein the value of the objective function and thus could yield the re-sults close to that obtained at convergence.

Fig. 6 presents a comparison of the measured temperature field,hm and the field computed using the estimated values of theparameters which can be termed as the estimated temperature,he. For each measurement error, the study has been done for allcases as considered in Table 2. For the simulated temperature fieldwithout any error, it is observed from Figs. 6(a)–(d) that the esti-mated temperature field agrees very well with the measured field.However, for the simulated temperature field involving measure-ment error, it is observed from Figs. 6(e) and (f) that the estima-tions agree very well although a little deviation is observed athigh temperature range. Thus, it is noticed from Fig. 6 that evenwith different estimated values of the parameters, a very goodagreement occurs between the measured temperature field, hm

and the estimated temperature field, he. Thus, the ill-posed natureof the present inverse problem can be easily observed.

To study the effect of number of temperature measurementpoints, n on the measured and the estimated temperature field, acomparison is made in Fig. 7. For each case of measurement error

and using the case requiring least iterations as obtained earlier inTable 2, the study has been done for two sets of measurementpoints, (n = 100, 10). It is noticed that with accurate temperaturemeasurement (Fig. 7a, c), the estimated temperature field com-pares very well. However, for the case involving measurement er-ror, i.e. with e = 1 (Fig. 7b, for ep = 0.01; Fig. 7d for ep = 0.1), theagreements are relatively poor due to considerable propagationof the effect of measurement error with the number of measure-ment points and is negligible for exact temperature field. There-fore, for simultaneously estimating the parameters such as theconvection–conduction parameter, N and the variable conductivityparameter, c, subject to accurate measurement, the number ofmeasurement points can be significantly reduced.

At last, for studying the effect of the sensitivity of the measuredtemperature (h) to the change in variable conductivity parameter(c) at different locations (x⁄) of the fin, a comparison is made inFig. 8. The sensitivity is represented by sensitivity coefficients½jc ¼ ð@h=@cÞp�; where p = 1, 2, 3 . . . n [31,32]. The comparisons aremade for convective–conductive coefficient (N = 1), i.e. for Biot(Bi = 1.0) and for 100 measurement locations. For computing thesensitivity coefficients (jc), in the present work, the forward differ-ence scheme [32] is adopted. It is observed from the figure that fora non-zero value of the variable conductivity parameter (c), theabsolute magnitudes of the sensitivity coefficients (jc) are higherwhen the locations are nearer to the hot boundary, i.e. (x⁄? 1.0).This is due to the reason of increase in heat loss at higher temper-ature which needs an accurate measurement of temperatures at

Distance,x*

[Sen

sitiv

ityco

effic

ien

t,j γ

]X10

4

0 0.2 0.4 0.6 0.8 1-3

-2

-1

0

1

γ = -0.6γ = 0γ = 0.6

Fig. 8. Comparison of the sensitivity coefficients at various measurement locationsfor change in variable conductivity parameter, c; N = 1.0.

R. Das / International Journal of Heat and Mass Transfer 54 (2011) 5001–5009 5009

these locations [31]. This finding is also in line with the results pre-sented in Fig. 7. In addition to this, it is well-known that for a con-stant thermal conductivity (c = 0) the temperature field remainsunaffected with variable conductivity parameter (c), the sensitivitycoefficients (jc)are zero, which can also be observed from Fig. 8.

5. Conclusions

In this study, simultaneous estimation of two parameters suchas the convective–conductive parameter and the variable thermalconductivity parameter has been done in a conductive–convectiverectangular fin. The exact/measured temperature field was calcu-lated from a forward/direct method using an analytical approachbased upon decomposition scheme. Subsequently, an inverse prob-lem is solved using Nelder–Mead based simplex search methodol-ogy. A very good agreement is observed between the measured andthe estimated temperature fields. Multiple combinations of theestimated parameters have been found to satisfy a given fieldand hence the ill-posed nature is observed. The investigation isdone for studying the effect of the measurement error, initialguess, number of measurement points and the required numberof iterations. The study of sensitivity coefficients has also beendone. The relative sensitiveness of the estimated parameters tothe temperature field has been also reported. Based upon thestudy, the following conclusions are made:

(a) A higher value of the convection–conduction parameter pro-vides lower local temperature distributions.

(b) A higher value of the variable conductivity parameter pro-vides higher local temperature distributions.

(c) Convection–conduction parameter influences the tempera-ture field more than the variable conductivity parameter.

(d) Subject of accurate measurements, number of measurementpoints could be considerably reduced.

(e) With reduced number of measured data and due to consid-erable propagation of the effect of measurement error withthe number of measurement points, the estimations are rel-atively poor.

References

[1] Q.D. Kern, D.A. Kraus, Extended Surface Heat Transfer, McGraw-Hill, New York,1972.

[2] A.D. Kraus, A. Aziz, J.R. Welty, Extended Surface Heat Transfer, John Wiley andSons, New York, 2001.

[3] F. Incropera, D.P. DeWitt, T.L. Bergman, A.S. Lavine, Fundamentals of Heat andMass Transfer, John Wiley and Sons, New York, 2007.

[4] A.D. Kraus, A. Aziz, J.R. Welty, Extended surface heat transfer, Appl. Mech. Rev.54 (5) (2001) B92.

[5] R.H. Yeh, An analytical study of the optimum dimensions of rectangular finsand cylindrical pin fins, Int. J. Heat Mass Transfer 40 (15) (1997) 3607–3615.

[6] P.J. Heggs, T.H. Ooi, Design charts for radial rectangular fins in terms ofperformance ratio and maximum effectiveness, Appl. Therm. Eng. 24 (8-9)(2004) 341–1351.

[7] A. Aziz, T. Fang, Alternative solutions for longitudinal fins of rectangular,trapezoidal and concave parabolic profiles, Energy Convers. Manage. 51 (11)(2010) 2188–2194.

[8] H.S. Kang, Optimization of a pin fin with variable base thickness, J. HeatTransfer 132 (3) (2010) 034501-1–034501-4.

[9] E.T. Jaynes, The well-posed problem, Foundation Phys. 3 (4) (1973) 477–492.[10] M.N. Ozisik, H.R.B. Orlande, Inverse Heat Transfer, Fundamentals and

Applications, Taylor and Francis, New York, 2000.[11] U.C. Chen, W.J. Chang, J.C. Hsu, Two-dimensional inverse problem in

estimating heat flux of pin fins, Int. Commun. Heat Mass Transfer 28 (6)(2001) 793–801.

[12] Z. Fang, D. Xie, N. Diao, J.R. Grace, C.J. Lim, A new method for solving theinverse conduction problem in steady heat flux measurement, Int. J. Heat MassTransfer 40 (16) (1997) 3947–3953.

[13] D. Maillet, A. Degiovanni, R. Pasquetti, Inverse heat conduction applied to themeasurement of heat transfer coefficient on a cylinder: comparison betweenan analytical and a boundary element technique, J. Heat Transfer 113 (3)(1991) 549–557.

[14] C.K. Chen, L.W. Wu, Y.T. Yang, Estimation of time-varying inlet temperatureand heat flux in turbulent circular pipe flow, J. Heat Transfer 128 (1) (2006)44–52.

[15] K.D. Cole, C. Tarawneh, B. Wilson, Analysis of flux-base fins for estimation ofheat transfer coefficient, Int. J. Heat Mass Transfer 52 (1–2) (2009) 92–99.

[16] M.H. Chang, J.H. Lin, C.K. Chen, The inverse estimation of local heat transfercoefficient in a vertical plate fin with its base subjected to periodicallyoscillated temperature, Numer. Heat Transfer A 40 (3) (2001) 253–271.

[17] T.M. Chen, Numerical solution of hyperbolic heat conduction in thin surfacelayers, Int. J. Heat Mass Transfer 50 (21-22) (2007) 4424–4429.

[18] A. Aziz, S.M. Enamul Hug, Perturbation solution for convecting fin withvariable thermal conductivity, J. Heat Transfer 97 (2) (1975) 300–301.

[19] A. Aziz, Application of perturbation techniques to heat transfer problems withvariable thermal properties, Int. J. Heat Mass Transfer 19 (3) (1976) 271–276.

[20] A. Muzzio, Approximate solution for convective fins with variable thermalconductivity, J. Heat Transfer 98 (4) (1976) 680–682.

[21] S.M. Zubair, A.Z. Al-Garni, J.S. Nizami, The optimal dimensions of circular finswith variable profile and temperature-dependent thermal conductivity, Int. J.Heat Mass Transfer 39 (16) (1996) 3431–3439.

[22] G. Adomian, A review of the decomposition method in applied mathematics, J.Math. Anal. Appl. 135 (2) (1988) 501–544.

[23] G. Adomian, Nonlinear Stochastic System Theory and Application to Physics,Kluwer Academic Publisher, Dordrecht, Netherlands, 1988.

[24] G. Adomian, Solving frontier problems modelled by nonlinear partialdifferential equations, Comput. Math. Appl. 22 (8) (1991) 91–94.

[25] G. Adomian, Solving Frontier Problems in Physics: The decomposition method,Kluwer, Academic Publishers, Dordrecht, Netherlands, 1994.

[26] C.H. Chiu, C.K. Chen, A decomposition method for solving the convectivelongitudinal fins with variable thermal conductivity, Int. J. Heat Mass Transfer45 (10) (2002) 2067–2075.

[27] J.C. Lagarias, J.A. Reeds, M.H. Wright, P.E. Wright, Convergence properties ofthe Nelder–Mead simplex method in low dimensions, SIAM J. Optimiz. 9 (1)(1996) 112–147.

[28] J.A. Nelder, R. Mead, A simplex method for function minimization, Comput. J. 7(4) (1965) 308–313.

[29] O. Nelles, Nonlinear system identification, Springer-Verlag, Berlin, 2001.[30] Convergence of the Nelder–Mead simplex method to a nonstationary point,

SIAM J. Optimiz. 9 (1) (1996) 148–158.[31] J.V. Beck, R.B. Dinwiddie. Parameter estimation method for flash thermal

diffusivity with two different heat transfer coefficients, in: Proceedings of 23rdInternational Thermal Conductivity Conference, Springer, Berlin, 1996.

[32] M.N. Ozisik, H.R.B. Ozisik, Inverse Heat Transfer, Taylor and Francis, New York,2000.