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International Journal of Thermal Sciences 50 (2011) 532e543

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International Journal of Thermal Sciences

journal homepage: www.elsevier .com/locate/ i j ts

ANN based estimation of heat generation from multiple protruding heatsources on a vertical plate under conjugate mixed convection

Apurv Kumar 1, C. Balaji 2,*

Heat Transfer and Thermal Power Laboratory, Indian Institute of Technology, Madras, India

a r t i c l e i n f o

Article history:Received 21 April 2010Received in revised form3 November 2010Accepted 11 November 2010Available online 31 December 2010

Keywords:Protruding heat sourceEstimationANNCovariance analysisConjugate mixed convection

* Corresponding author. Tel.: þ91 4422574689; faxE-mail address: balaji@iitm.ac.in (C. Balaji).

1 Formerly M.Tech student2 Professor, Department of Mechanical Engineering

1290-0729/$ e see front matter � 2010 Elsevier Masdoi:10.1016/j.ijthermalsci.2010.11.003

a b s t r a c t

Inverse Heat Transfer Problems (IHTP) are characterized by estimation of unknown quantities by utilizingany given information of the system. In this study, the inverse problem of estimation of heat generationin multiple two dimensional protruding heat sources on a vertical plate, a geometry frequentlyencountered in the cooling of electronic equipment, is carried out from the information available on thetemperature distribution on the substrate on which these sources are mounted. A non-iterative methodis applied utilizing Artificial Neural Networks (ANN) and covariance analysis to estimate the heatgeneration in the protruding heat sources on a vertical plate. The forward model involving laminar, twodimensional, steady, incompressible fluid flow and mixed convection heat transfer is numerically solvedwith FLUENT 6.3 for known values of heat generation in the protruding sources and the temperaturedistribution thus obtained on the PCB substrate is utilized to train the ANN for the inverse model.Parametric studies are conducted on the forward model to investigate the effect of Richardson number,Reynolds number, the chip and substrate conductivities on the heat dissipation to the fluid flowing overthe heat sources. The trained networks are finally used to estimate the heat generation from the sourcesfor a given temperature distribution on the substrate wall generated, for known values of the heatgeneration rates, which serve as the “measured” temperature distribution. Use is made of covarianceanalysis in order to identify the important temperature locations sufficient to carry out the inverseanalysis. Finally, a systematic investigation on the effect of noise in the temperature “measurements” onthe estimates also has been carried out.

� 2010 Elsevier Masson SAS. All rights reserved.

1. Introduction

Space programs in the late 50’s and early 60’s have played animportant role in the advancement of theory and application ofInverse Heat Transfer problems (IHTP) where the goal is the esti-mation of heat flux on the surface of the space vehicle during itsreturn to atmosphere. This technique of estimation of unknownquantities from the information available has now become diver-sified. In general, inverse estimation is carried out when directestimation is not feasible. Such techniques are also widely appliedin an estimation of unknown boundary conditions, the geometriccharacteristics, initial conditions or the thermo-physical propertiesof the material or medium, from the available experimental data.Similarly, the thermo-physical properties of anisotropic materialsvary with temperature and space and thus measurement of such

: þ91 4422570509.

, IIT Madras.

son SAS. All rights reserved.

properties under operating conditions becomes complicated and insome cases highly unsatisfactory if conventional methods are used.Inverse techniques greatly simplify such problems and yield satis-factory results.

Convection continues to occupy an important place in contem-porary research in the thermal sciences. Heat treatment, electroniccooling, turbo-machinery and heat exchangers are a few applica-tions where both natural and forced convection are important.Often due to complexity involved in the computations, it becomesincreasingly difficult to analyze convection for complex geometriesespecially when an inverse solution is sought. Inverse convectionproblems generally involve estimation of inlet temperature profilein laminar flows, transient inlet temperature in laminar flows, axialvariation of thewall heat flux and simultaneous estimation of spacewise and time wise variations of the wall heat flux. Unlikeconduction problems, the convection problems involve the highlynon-linear and formidable Navier Stokes equations and theenergy equation. Additionally, in natural convection problems themomentum and energy equations are coupled and invariablythe Boussinesq approximation is invoked to model density.

Nomenclature

ai value predicted by ANNg acceleration due to gravitation, 9.81 m s�2

Gr modified Grashof number (see Section 2.1)k thermal conductivity, W m�1 K�1

L length of the vertical plate, mn number of data pointsP pressure vector, Paq000 heat generation, W m�3

R absolute fraction of variance (see eq. (18))Ri Richardson Number, Gr/Re2

s input signal (see Section 2.3.1)TN fluid temperature, KT temperature, KDTref reference temperature, DTref ¼ q

000L2

ksti value predicted by numerical simulationU dimensionless velocity vector in the domain

u velocity component in x-direction, m s�1

v velocity component in y-direction, m s�1

w weights used in ANNx internal activation (see eq. (14))

Greek Lettersd signal function (see eq. (15))G thermal conductivity ratio, k/kairq internal threshold (see Section 2.3.1)r density of fluid in the domain, kg m�3

Subscriptsch heat sourcef fluidi neuronj neurons substrate

A. Kumar, C. Balaji / International Journal of Thermal Sciences 50 (2011) 532e543 533

In recent years, inverse convectionproblemshave receivedmuchattention. The solution to inverse problems in convection requiresa simultaneous solution to the momentum and energy equations.Moutsoglou [1] reported the very first inverse convection analysiswhere steady state inverse forced convection between parallelplates was investigated. Straight inversion and domain regulariza-tion scheme were used to estimate the heat flux at the boundary.Subsequently, many studies were carried out in inverse convectionproblems both in natural and forced convection for various geom-etries. Different types of inverse heat transfer problems have beenconsidered in literature. Estimation of boundary conditions such asboundary heat flux or temperatures and estimation of geometricalcharacteristics such as boundary design or shape of the domain arethe most widely studied problems. A good number of studies arealso found on inverse analysis in conjugate heat transfer problems.Simultaneous inverse estimation of two boundary heat fluxes inirregularly shaped channels for forced convectionwas carried out byOrlande and Colaco [2]. Three different types of boundary condi-tions were used for different inverse analysis where accurate esti-mates were made for the unknown functions with this approach.The forward model is first numerically solved to obtain the desireddistribution of properties, say the temperature or velocity. Theinversemodel is then developed and solvedwith the input from theforward or directmodel [3]. For convection in particular, the inverseanalysis is difficult as compared to conduction due to the parabolicnature of the solution in contrast with the elliptic nature forconduction. Thus, a proper correlation between the positions ofmeasured and estimated properties of a system is imperative [4]. Forinstance, in convection the estimated and measured propertiesmust be determined along a streamline only, in order to carry outinverse analysis with successful results. Coupling of themomentumand energy equations in natural convection further adds to thecomplexity in inverse analysis. Investigators [5e8] have usedadjoint and conjugate gradient methods for inverse convectionanalysis. Arbitrary heat flux from measured temperatures withouta priori information for natural and forced convection in an enclo-surewas predicted by Payan et al. [8]. In this study, the estimation ofheater strength was carried out as an inverse analysis fromknowledge of the temperature. However, simple geometries werediscussed and iterative methods are required to obtain the solutionto the inverse problem. Being ill-posed in nature, inverse convectionproblems require regularization with heavy damping whichsubsequently dies out when the solution is reached. This makes theinverse problems well posed [3].

Non-iterative methods have also been adopted to solve inverseproblems and they are mainly concerned with problems inconduction [9]. Artificial Neural Network (ANN) is a tool that can beused for non-iterative inverse analysis, given the advantages itcarries over the iterative process in terms of computation cost andaccuracy of the predicted results. Recently, a non-iterative methodwas proposed by Kumar and Balaji [10] where ANN is used in orderto obtain solution to inverse convection problem in a square cavity,a fundamental problem. Estimation of the heat flux on one of theboundary was carried out with the information from the temper-ature distribution from the top and bottom boundaries of thecavity.

The present work focuses on applying the non-iterative methodproposed by the authors [10] for a more practical geometry likeprotruding heat sources on a vertical plate with laminar conjugatemixed convection to estimate the heat generation in the heatsources. Use is made of statistical tools such as covariance analysisin order to make the inverse model more robust. Furthermore, asopposed to [10], with the stacking of protruding heat sources ona vertical wall, the interaction of boundary layers makes both theforward and inverse problems more challenging.

2. Inverse convection problem

2.1. Direct or forward problem

In the presentwork, a practical problem is considered. Protrudingheat sources mounted on a vertical plate is used as the geometry forthe forward model. A PCB substrate of finite conductivity is used asthe vertical plate having finite thickness. Four rectangular heatsources are mounted on the vertical plate over which air flows. Thiscase greatly resembles an electronic circuit board cooling problemwhere electronic chips are mounted on a substrate and need to becooled by flowing air. As the number of components in an electronicboard increases, the exact heat generation is not generally knownwhen frequency or voltage is not at the design condition. Asa thermal designer, itmay become imperative to estimate the correctvalue of heat dissipation when the equipment is under steady stateand has been working for long hours. For thermal management ofelectronic boards, information on the heat generation is critical.A direct method of measurement would interrupt the fluid flow andhence the cooling of the chips. Here, an inverse analysis can estimatethe heat source accuratelywithout interfering in the cooling process.In the present work, the temperature is measured at a remote

A. Kumar, C. Balaji / International Journal of Thermal Sciences 50 (2011) 532e543534

location which does not directly interfere with the fluid flow. Theheat generation is from the electronic chips which are protrudinginto the fluid. For a direct measurement of temperature or any otherproperty, the instruments have to be located on the electronic chipswhich would interfere with the fluid flow. Hence, in this study weconsider a case where the temperatures are measured at the back-side of the adiabatic surfaces, without causing any hindrance to theflow. Even so, it is only by using these temperatures that eventuallythe inverse problem is solved.

Fig. 1 gives the physical model of the present problem. The left,top and bottomwalls of the substrate are adiabatic. Fluid flow takesplace from the bottom and the plate is open to atmosphere.Laminar, two dimensional, incompressible, steady flow withBoussinesq approximation for modeling density variation of themedium is considered.

The fluid flow and heat transfer over the plate are governed bythe continuity equation, Navier Stokes equations and the equationof energy. A separate energy equation for heat transfer throughconduction inside the PCB substrate and the heat sources is alsoconsideredmaking the problem a conjugate one. Thermal buoyancyeffect is taken into consideration. Boussinesq approximation isapplied to model density variation. Viscous dissipation terms,

Heat source 1

Heat source 2

Heat source 3

Heat source 4

L

W

h

s

d

u ,T∞ ∞

X

Y

g

H

Fig. 1. Physical model for heat generation in protruding heat sources on a verticalplate.

compressibility work and radiation effects are not taken intoaccount. No slip and impermeability conditions are considered onall the surfaces. Air is taken as the working fluid. The thermo-physical properties of the heat sources are assumed to be constantand are that of Aluminum. The substrate material is taken as FR-4which is generally used for PCB. FR-4 is National Electrical Manu-facturers Association (NEMA) grade designation for glass reinforcedepoxy printed circuit boards. It is made up of woven fiberglass clothwith an epoxy resin binder that is flame resistant. It is also known toretain its mechanical and electrical insulating qualities in both dryand humid conditions. For the above conditions, the steady stategoverning equations are given by

Continuity:vUvX

þ vVvY

¼ 0 (1)

X-Momentum: UvUvX

þ VvUvY

¼ �vPvX

þ 1ReL

v2UvX2 þ v2U

vY2

!þ Gr

Re2Lf

(2)

Y-Momentum: UvVvX

þ VvVvY

¼ �vPvY

þ 1ReL

v2VvX2 þ v2V

vY2

!(3)

Energy: Uvf

vXþ V

vf

vY¼ 1

ReLPr

v2f

vX2 þv2f

vY2

!(4)

Energy Equation for the solid region:Substrate:

v2f

vX2 þv2f

vY2 ¼ 0 (5)

Heat Sources:

v2f

vX2 þv2f

vY2 þGs

Gch¼ 0 (6)

The above mentioned equations are non-dimensional and areobtained after introducing the following relations into the Navierstokes equations and the energy equation:

X ¼ xL

Y ¼ yL

U ¼ uuN

V ¼ v

uNP ¼ p

r�yfL

�2

f ¼ T � TNDTref

;where DTref ¼ q000L2

ksReL ¼ uNL

yfGr ¼ gbDTrefL

3

n2

Pr ¼ n

af

where the symbols r, g, v, af, and b denote, respectively, the density,gravitational acceleration, kinematic viscosity, thermal diffusivityand the coefficient of volumetric expansion.

The boundary conditions are:

At inletðcX and Y ¼ 0Þ

u ¼ uN (7)

T ¼ TN (8)

For the substrate

Table 2Domain independence study for protruding heat sources mounted on a vertical

A. Kumar, C. Balaji / International Journal of Thermal Sciences 50 (2011) 532e543 535

vTvY

¼ 0 at Y ¼ 0 and for 0 � X � L (9)

plate.

Domain Maximumtemperature, K

% error with respect totop and bottom domain

Extended top and bottom 404 e

Extended only bottom 405 0.48Extended only top 401 0.74No extension 400 0.99

vTvX

¼ 0 at X ¼ 0 and X ¼ L; for 0 � Y � W (10)

�ks

�vTvY

�s¼ �kair

�vTvY

�air

at Y ¼ W and for 0 � X

� H; H þ h � X � H þ hþ s; H þ 2hþ s � X

� H þ 2hþ 2s; H þ 3hþ 2s � X

� H þ 3hþ 3s and H þ 4hþ 3s � X � L (11)

�ks

�vTvY

�s¼�kch

�vTvY

�ch

atY ¼W andfor H�X

�Hþh; Hþhþs�X�Hþ2hþsHþ2hþ2s�X

�Hþ3hþ2s; Hþ3hþ3s�X�Hþ4hþ3s (12)

For the heat sources:

�kch

�vTvY

�ch

¼ �kair

�vTvY

�air

at Y ¼ W þ d and for H � X

� H þ h; H þ hþ s � X

� H þ 2hþ s; H þ 2hþ 2s � X

� H þ 3hþ 2s; H þ 3hþ 3s � X � H þ 4hþ 3s

(13)

A grid independence study is done in order to choose theoptimum grid size. The parameter under consideration is themaximum temperature in the domain. Fine grids were chosen nearthewalls and the heat sources and coarser girds were used at placesaway from these. Table 1 gives the results of the grid independencestudy based on three grids for a typical set of parameters. Thedifference in the maximum temperature between a grid with43 200 cells and that with 67 500 cells is less than 0.5% error. Hence,the former grid is adequate. However, in order to be conservativethe grid with 67 500 cells was used for all subsequentcomputations.

A computational domain independence study was also carriedout in order to obtain a computationally economical domain. Thedomain was extended on the top and bottom together and alsoindividually to see if there is any effect on the maximum temper-ature predicted. Table 2 shows the results of the domain indepen-dence study for a typical set of parameters. As seen from the Table,a computational domain with no extension at the top and bottomgives accurate results with less than 0.99% error as opposed toa “well” extended domain. Fig. 2 shows the computational domainextended only at the top along with appropriate boundaryconditions.

Commercially available FLUENT 6.3 that works on the finitevolume method is used to numerically solve the governing equa-tions. Non-uniformmeshing is employedwith finemeshes near the

Table 1Grid independence study for protruding heat sources mounted on a vertical plate.

Total no. of cellsin the domain

Maximumtemperature, K

% difference of maximumtemperature betweensuccessive grids

24 300 384.6 e

43 200 377.6 1.7967 500 375.8 0.48

walls and coarsermeshes away from them. The SIMPLE algorithm isselected to solve the pressure-velocity coupling. Second orderupwinding is used for the momentum equations and the equationof energy. Convergence criteria for energy, continuity and x and ymomentum equations are 1 � 10�6, 1 � 10�3 and 1 � 10�3

respectively. Upon convergence, energy balance was found to beaccurate to within �0.08%.

2.1.1. ValidationThe results of conjugate heat transfer in a vertical channel with

single protruding heat source for natural convection werecompared with the experimental results of Said and Krane [11] forone protruding heat source, due to paucity of results in the openliterature for the exact problem being considered in this study. Saidand Krane considered a curved protrusion in the vertical channel.The experiments were conducted for uniform temperature alongthe channel walls with the temperature difference between thewall and ambient being 30 K. A Rayleigh number of 2 � 104 wasconsidered with an aspect ratio of 0.27 (defined as the ratio ofchannel width and height of the vertical channel). The presentnumerical model however consists of a rectangular protrusion.Fig. 3 shows a comparison of the local Nusselt number from thepresent numerical model and the values obtained from theexperiments of Said and Krane. As seen from the figure, thatthe numerical model predicts the Nusselt number accurately alongthe channel wall. Due to the difference in shape there is an error inprediction of the local Nusselt number over the protrusion, butthere is generally broad agreement.

2.2. Parametric study

2.2.1. Range of parameters consideredAs the problem under consideration is of high relevance to

electronic cooling industry, a parametric study will throw more

Fig. 2. Extended domain at top along with appropriate boundary conditions.

Fig. 3. Comparison of local Nusselt number along the channel wall for the presentnumerical model and Said and Krane [11].

Fig. 4. Variation of dimensionless temperature in the domain with Reynolds number.

A. Kumar, C. Balaji / International Journal of Thermal Sciences 50 (2011) 532e543536

light on the physics of the conjugate heat transfer taking place inthe geometry. Hence, even though the goal of this work is to solvethe inverse heat transfer problem, first results of a parametric studyare discussed as it is important to know the effect of variousparameters on the heat transfer between various components inthe system such as the electronic chips, substrate wall etc.Accordingly, the effect of Reynolds number, Grashof number andthe thermal conductivity ratios, Gs and Gch on the fluid flow andheat transfer characteristics have been studied. The range ofparameters considered is given in Table 3. The baseline parametersconsidered in the present study: ReL ¼ 7550, Gr ¼ 9.5 � 107 Gch ¼7695 and Gs ¼ 8:75. The Grashof number Gr is actually a modifiedGrashof number as there is no reference DTref available for thisproblem.

2.2.2. Effect of Reynolds numberThe Reynolds number was varied from 500 to 1.8 � 104 for

a constant Grashof number of 9.5 � 107. The effect of increase inReynolds number on the maximum temperature in the domainwhich occurs either on the third or fourth chip from the bottom ofthe domain, was studied. It can be seen in Fig. 4 that with anincrease in the inlet velocity, the maximum temperature decreases.This is due to the enhanced cooling effect caused by the increasedvelocity. Recirculation zones were found on the top of the last chipnear the exit of the domain. At low Reynolds number, the size of therecirculation zones is small and they increase in size as well as innumber with increased velocity. Another recirculation zone origi-nated between the third and the fourth chip at higher Reynoldsnumbers. This circulation of air at the top of the heat source is thereason for the temperature of the top heat source being lower thanthat of the penultimate heat source. Fig. 5 shows the variation ofthe temperature of the heat sources with Richardson numberdefined as GrL/Re2. It can be seen that at high Richardson numbers

Table 3Range of thermal parameters considered in the parametric study for protruding heatsources on a vertical plate.

1.9 � 107 � Gr � 9.5 � 107

500 � ReL � 1.8 � 104

0.8745 � GPCB � 87457.7 � Gch � 77 000

(i) the maximum temperature occurs for the heat source located atthe top and (ii) the mode of heat transfer shifts to naturalconvection. At low Richardson numbers, due to the circulationeffect caused by forced convection at the top of the fourth heatsource, the temperature becomes lower than that of the penulti-mate heat source.

A study was also conducted to evaluate the apportioning of theheat dissipated into the fluid by various surfaces in the domain. Theright substratewall in contact with the air and the four heat sourceswere studied for their contribution towards increasing the enthalpyof the fluid. It is seen that at low Reynolds numbers, the conjugatewall contributed more than the four heat sources. As the inletvelocity is increased, this percentage drops and there is compen-sation by an increase in the heat dissipation by the four heatsources. It can be seen from Fig. 6 that the percentage of heatdissipation into the fluid is low for the heat sources at top andbottom (1st and 4th heat sources) and higher for the sources inbetween (2nd and 3rd heat sources). Despite the low percentage ofheat dissipation into the fluid, the temperature of the first heatsource is found to be lowest. This is due to the conduction of heatinto the substrate from the heat sources. From this study, it is seenthat the percentage of heat transfer into the substrate is highest forthe sources located at top and bottom and lowest for the middleones. The sources at top and bottom are relatively open andtherefore more heat transfer takes place into the substrate wall

Fig. 5. Variation of temperature of the heat sources with Richardson number.

Fig. 6. Plot showing the variation of the percentage of heat dissipated to the air by thesubstrate wall and the four heat sources.

A. Kumar, C. Balaji / International Journal of Thermal Sciences 50 (2011) 532e543 537

from them than from those sources which are in between. Thus, thecombined effect of conduction and convection brings down thetemperature of the first heat source. Fig. 7 shows the temperaturecontours for various Reynolds numbers. As can be seen from thetemperature contours, an increase in the Reynolds number shiftsthe maximum temperature in the domain from the top heat sourceto the penultimate heat source (3rd heat source).

2.2.3. Effect of Grashof numberThe Grashof number was varied by varying the magnitude of

heat generation from 105 to 5 � 105 W/m3. With an increase in theGrashof number, there is a decrease in the maximum non-dimen-sional temperature. However, the dimensional temperatureincreased substantially due to increased heat generation in thesources. The decrease in the maximum non-dimensional temper-ature is due to the definition of the reference temperature DTref ,which is inversely proportional to the heat generation in the heatsources. The percentage of heat dissipated into the air remainsconstant for the range of Grashof number considered in the presentstudy (see Table 4). There is no change observed in the flowpatterns in the fluid for the range of Grashof numbers studied.

2.2.4. Effect of substrate thermal conductivityA thermal conductivity parameter Gs is defined as ks=kair, where

ks is the thermal conductivity of the substrate. Gs is varied in therange of 0.8745e8745. This range is chosen in order to cover theentire range of insulating materials and also study the effects overawide range of thermal conductivities. For FR-4, Gs is of the order of10. The effect of Gs on the heat transfer into the air is also studied.As can be seen from Fig. 8, with an increase in the thermalconductivity of the substrate there is an increase in the percentageof heat dissipated into the air by the substrate wall with a propor-tionate decrease in the percentage of heat dissipated by the heatsources. For high values of thermal conductivity, the substrate wallcontributes to more than 50% of the total heat transfer into the air.The temperature variation in the substrate with variation of Gs canbe seen from Fig. 9. It can be seen that the substrate becomesisothermal for high values of the wall thermal conductivity, asexpected. The temperature variation is measured on the left wall ofthe substrate. For very low values of Gs, there is a significant vari-ation of temperature within the substrate with the peaks occurringin the plot signifying the location of the heat sources. There is alsoan increase in the temperature at the inlet region of the wall. This

causes the heating of the inlet air right from the beginning of thesubstrate wall and thus more amount of heat is transferred whichaccounts to more than 50% for higher values of the parameter Gs, asalready discussed.

2.2.5. Effect of thermal conductivity of heat sourceThe heat source conductivity ratio ðGchÞ defined as kch=kair,

where kch is the thermal conductivity of the heat sources is intro-duced to delineate the effect of the conductivity of the heat sourceon the heat transfer. In the present study, this parameter is variedfrom 7.7 to 77 000 and its effect on the heat transfer into the air isstudied. The present range of thermal conductivity ratio Gch ischosen in order to cover the entire range of metals and also to studyits effect on a wide range of thermal conductivities. For Aluminum,Gch is of the order of 8000. From Fig. 10, it can be seen that with anincrease in the thermal conductivity parameter Gch there is anincrease in the percentage of heat transfer from the heat sources tothe air and a proportionate reduction in the heat transfer from thesubstrate wall. With regard to the temperature distribution in theelectronic chip (or heat sources), it can be observed that there isa significant variation of temperature distribution within chips forlower values of Gch as expected. For higher values of Gch, the chipsbecome isothermal which can be seen from Fig. 11. It can also beobserved that there is a sharp decrease in the temperature at thecorners of the chips. This is due to the sudden contact of the chipswith relatively colder air just above the horizontal portion of theheat sources and also because of the availability of additional areahere for cooling. A typical heat flux plot is shown in Fig. 12.

2.3. Inverse problem

2.3.1. Artificial Neural Network (ANN)Neural Networks are ensembles of interconnected artificial

neurons generally organized into layers of fields. They aremassively parallel adaptive networks of simple non-linearcomputing elements called neurons which are intended to abstractand model some of the functionality of the human nervous systemin an attempt to partially capture some of its computationalstrengths. In literature, parameter retrieval and optimization inconjunction with other techniques like genetic algorithms is usedalong with neural networks [12]. A non-iterative estimation of heattransfer coefficient for inverse heat conduction was carried out byusing ANN [9]. A simple geometry was considered for the inverseanalysis and the mode of heat transfer was limited to conduction.The present work aims at combining covariance analysis with ANNfor the convection problem discussed earlier which has morepractical relevance with conjugate heat transfer involvingconduction and mixed convection.

The information in the form of input signal si is received fromvarious sources (say n in number) by the neuron. These inputsundergo linear weight aggregation and are modified by an internalthreshold, qj. This is done by traversing weighted pathways wij andan internal activation xj is generated, which can be expressed as:

xj ¼Xni¼1

wijsi þ qj (14)

The neuron is subsequently transformed through a signalfunction to generate an output signal sj ¼ dðxjÞ. This signal functioncan be a binary threshold, linear threshold, sigmoidal, Gaussian, orprobabilistic. The sigmoidal function is the most widely used signalfunction in neural networks. It is expressed as

dj�xj� ¼ 1

1þ e�ljxj(15)

Fig. 7. Temperature (in K) distribution in the domain with protruding heat sources on a vertical wall with finite thickness for various Reynolds number.

Table 4Percentage of heat dissipated into the air by different components in the domain.

Grashofnumber

% Heat dissipated in air Maximumtemperature, K

Conjugatewall

1st chip 2nd chip 3rd chip 4th chip

1.90 � 107 20.66 18.86 20.13 19.32 21.24 3233.80 � 107 19.88 18.81 19.99 20.14 21.17 3445.70 � 107 19.51 18.82 20.17 20.32 21.15 3667.59 � 107 19.3 18.83 20.35 20.31 21.19 389

Chip with maximum temperature.Chip with minimum temperature.

A. Kumar, C. Balaji / International Journal of Thermal Sciences 50 (2011) 532e543538

This function has properties like monotony and continuitywhich enable the networks to approximate and generalize onfunctions by learning from data. Hence, this signal function is themost widely used in the hidden layers of the neural network.

For the present inverse model, the estimation of the heatgeneration is to be performed using the information available fromthe temperature distribution on the substrate wall. The neuralnetwork can be trained to predict the heat generation by setting thetemperature distribution as the input and the corresponding heatgeneration parameters as the output. From the present grid size,451 temperatures are obtained at the left substrate wall corre-sponding to the heat generation from the heat sources, usingnumerical simulations. Only three heat sources are considered for

Fig. 8. Percentage variation in heat dissipation by various components in the domainwith thermal conductivity parameter, Gs.

Fig. 10. Variation of percentage of heat transfer by different surfaces in the domainwith the thermal conductivity parameter Gch.

A. Kumar, C. Balaji / International Journal of Thermal Sciences 50 (2011) 532e543 539

the inverse model in order to simplify the application of the inversemethod, as the number of cases to be generated from the forwardmodel is reduced. The procedure can be easily extended for four ormore number of components.

As can be observed from the architecture of the neural networkfor the inverse problem (Fig. 13), the input data consists of 451temperatures and the output is just three parameters. This not onlynecessitates more effort to train the ANN in identifying the patternbut also leads to practical difficulties. In such a case, it would bebeneficial and necessary to reduce the number of input tempera-ture locations to a realistic value. Furthermore, such a high numberof input variables may also lead to overtraining the network. Hence,in order to obtain key temperature locations that would suffice forretrieval of the heat generation values, a covariance analysis is alsocarried out.

2.3.2. Procedure for solving the inverse problemThe forward model together with boundary conditions is

numerically solved using the commercially available FLUENT 6.3 for

Fig. 9. Temperature distribution on the left wall of the substrate with variation of Gs.

various values of heat generation in the three heat sources varyingfrom 105 to 5 � 105 W/m3. In all about 225 data sets were obtainedfor different combinations of the three heat sources, Q1, Q2 and Q3.The temperature distributions on the left substrate wall areobtained from the numerical simulations. A data vector of 451temperatures was selected on the left substrate wall to train anANN. This trained network is then used to predict the outputparameters for any given temperature distribution. The ANNmodelembeds all of the physics associatedwith the problem as it has beendeveloped using a parametric study of the pertinent physicalquantities in the problem that govern the fluid flow and heattransfer.

As already mentioned briefly, the original temperature data setselected may be superfluous. To reveal the correlation the selectedtemperatures have on the parameters Q1, Q2 and Q3, a covarianceanalysis is done. Covariance analysis will bring to light clearly thosetemperatures that have strong correlation on the parameters. In thedesign of experiments, where there is a limitation on the number ofthermocouples that can be mounted on a given geometry, covari-ance analysis will help decide on the optimum positions of the

Fig. 11. Variation of the temperature in the heat sources for different values of Gch.

Fig. 12. A typical variation of heat fluxes on the surface of a heat source.

A. Kumar, C. Balaji / International Journal of Thermal Sciences 50 (2011) 532e543540

thermocouples. Finally, to carry out inverse analysis only thesetemperatures need be used thereby reducing the computationaltime in training the neural network.

All the simulations were carried out on desktop machineequipped with a Core 2 Duo Intel Processor with 2.8 GHz speed and2 GB RAM.

Fig. 13. A typical neural network architecture employed in this study for the inversemodel before covariance analysis.

3. Results and discussion

3.1. Training the inverse neural network and sensitivity analysis

The temperature distribution obtained from the forward modelis set as input into the network and the corresponding parametersQ1, Q2 and Q3 are defined as the output, before and after covarianceanalysis. This inverse model determines the heat generation in theheat sources for a given temperature distribution on the leftsubstrate wall. Once trained, the network will predict the param-eters. Neuron independence studies have been carried out todetermine the optimum network based on the following perfor-mance metrics: Mean Relative Error (MRE), Mean Square error(MSE) and Absolute fraction of variance (R2). These can be mathe-matically expressed as:

MRE ¼ 1n

Xni¼1

jai � tijjtij

(16)

MSE ¼ 1n

Xni¼1

ðai � tiÞ2 (17)

R2 ¼ 1�

26664Pni¼1

ðai � tiÞ2

Pni¼1

ðtiÞ2

37775 (18)

The robustness of the neural network obtained before and aftercovariance analysis is determined by adding a noise to the inputdata and comparing the retrieved parameters with the true valuesof the parameter.

The temperature distribution for this purpose is obtained by thefollowing expression:

Tnoise ¼ T � 3sðrandÞ;where rand is a random number between 0 and 1 ð19Þ

Table 5 shows the results of the neuron independence study beforethe covariance analysis is carried out. From the results it, can beseen that the optimum network given in shaded colour is found toconsist of 1 hidden layer with just 4 neurons. The R2 value for theestimated parameters is of the order of 0.999 which signifies a veryaccurate estimation of the heat generation in the sources. However,when a sensitivity analysis was carried out by adding noise to theinput temperatures meant for testing the network, the estimationwas found to be highly inaccurate for the optimum network.

3.2. Covariance analysis

Covariance analysis was carried out and depending on themagnitude of the covariance between the input temperatures andthe output parameters, the important temperature locations on theleft substrate wall are identified. It is seen that the temperaturelocations near the tip of the wall are relatively less significant thatthose present near the heat sources. This is because most of theinformation regarding the heat source can be obtained from loca-tions near them. The farther the temperature location is from theheat sources higher the inaccuracy in the inverse estimation of theheat sources in the presence of “simulated” noise. This is the reasonwhy the original data vector consisting of 451 temperature loca-tions when used in the network estimates heat generation in thesources inaccurately in the presence of noise. Upon reduction of thesize of the temperature data vector to those temperature locations

Table 5Neuron independence study before covariance analysis.

Number ofhidden layers

Number ofneurons

MRE MSE R2 Time (s)

Q1 Q2 Q3 Q1 � 10�7 Q2 � 10�7 Q3 � 10�7 Q1 Q2 Q3

1 4 0.0224 0.0186 0.0238 3.72 4.59 6.69 0.9996 0.9996 0.9995 601 5 0.0631 0.0491 0.0158 32.3 24.7 2.70 0.9979 0.9981 0.9998 402 3 0.0673 0.0805 0.0560 38.1 53.3 28.0 0.9966 0.9959 0.9979 182 8 0.0279 0.0363 0.0204 6.26 10.1 7.23 0.9994 0.9992 0.9994 7002 10 0.0429 0.0647 0.0231 18.6 42.4 6.16 0.9983 0.9968 0.9995 6503 5 0.0402 0.0158 0.0323 11.0 2.20 15.6 0.9990 0.9998 0.9988 1433 8 0.0360 0.0366 0.0939 9.74 12.8 125.0 0.9991 0.9990 0.9912 2104 5 0.0384 0.0297 0.0426 11.0 8.60 26.2 0.9990 0.9993 0.9981 3544 8 0.0332 0.0304 0.0173 9.37 8.25 4.11 0.9991 0.9993 0.9997 312

Table 6Performance of the optimum networks obtained after reduction of input data points by covariance analysis.

No. of inputdata points

Distance between thedata points (mm)

MRE MSE R2

Q1 Q2 Q3 Q1 � 10�7 Q2 � 10�7 Q3 � 10�7 Q1 Q2 Q3

87 1 0.0173 0.0226 0.0156 2.20 6.2 4.5 0.9998 0.9995 0.999640 2 0.0019 0.0032 0.0067 0.051 0.18 0.66 1.0000 0.9999 0.999923 4 0.0089 0.0085 0.0094 0.60 0.84 1.72 0.9999 0.9999 0.999818 5 0.0048 0.0039 0.0055 0.20 0.37 0.42 0.9999 0.9999 0.999910 10 0.3200 0.1354 0.4780 59.9 21.5 955.0 0.8569 0.8147 0.7546

Fig. 14. Estimated values of heat sources after covariance analysis.

A. Kumar, C. Balaji / International Journal of Thermal Sciences 50 (2011) 532e543 541

which are strongly correlated to the output parameters, a signifi-cant increase in the accuracy of the inverse estimation is found toexist even though a large number of data points are not included.After the initial reduction in the data points to 87 data obtainedfrom covariance analysis, subsequent reductions were again madein order to determine the optimum size of the input data pointswhich yields accurate inverse estimation in presence of simulatednoise. As the number of data points was reduced to 87 from 451, theperformance of the optimum network did not change much. Thedifference, however, was found during the sensitivity analysiswhere the network with reduced set performed much better. TheR2 value for the network tested for sensitivity to noise in input datawas of the order of 0.6 for 3s values ranging from 0.1 to 0.5. Ascompared to this, the networks with reduced data points estimatedthe parameters with R2 value of the order of 0.99e0.94 for 3s valuesranging from 0.1 to 1. Subsequently, the data points were reducedby increasing the distance between them and Table 6 shows theperformance of the network for different data locations consideredbetween X ¼ 0.18 to X ¼ 0.8. The Table also shows the distancebetween the data points considered. It is interesting to see that theaccuracy increases further with a reduction in the number of datapoints. This shows that the initial data points considered wereindeed superfluous. However, as the data points were reducedbeyond 18 the inverse estimation became inaccurate once againdue to the lack of sufficient information of the heat generation fromthe heat sources.

Thus for the present study, the “optimal” network with 18 datapoints was selected for the inverse analysis. The networkwas testedwith 25 simulated temperature data sets which representeda temperature profile on the left wall of the substrate and whichcorresponded to a combination of the there heat generation, Q1, Q2and Q3. These data points were not used in training of the network.Fig.14 shows a parity plot of the estimated heat generationwith thedata points for which the temperature profile was provided.Table 7 shows the effect of “simulated noise” for various s values.Fig. 15 shows the estimation of Q3 for different s values, as Q3 isfound to bemore sensitive to noise during the inverse estimation. Itcan be seen that for s values of 0.1 and 0.2, the inverse estimation isquite accurate. Finally, as an additional validation of the inverse

model, the temperature distribution on the back of the substratewas reconstructed from the estimated value of a particular combi-nation of heat generation values by running full numerical simula-tions (using CFD). Fig. 16 shows a comparison of the temperaturedistribution obtained from inverse estimation and CFD by solvingthe forward model which further corroborates the potency andadequacy of the estimation procedure employed in the study.

In a typical iterative method for inverse analysis, the forwardmodel is solved for various guess values of the boundary conditionsand the estimate is reached after several iterations. At this stage,the error between the solution to the forward model (i.e., thegoverning equation) with the estimated values of the parametersand the measurements (usually temperature in a heat transferproblem) is minimum in a least square sense. The computationalcost for an inverse analysis includes the CPU time consumed tosolve the forward model for various estimated values of theboundary conditions.

Table 7Performance of the optimum network for 18 input data points at various noise levels.

s MRE MSE R2

Q1 Q2 Q3 Q1 � 10�7 Q2 � 10�7 Q3 � 10�7 Q1 Q2 Q3

0.1 0.0329 0.0414 0.0552 14.0 23.1 63.8 0.9987 0.9982 0.99510.2 0.0731 0.0883 0.1166 61.0 110.0 238.0 0.9946 0.9918 0.98070.33 0.0922 0.0623 0.1004 280.0 72.3 173.0 0.9739 0.9943 0.9867

Fig. 15. Effect of noise in inverse estimation of the heat sources.

A. Kumar, C. Balaji / International Journal of Thermal Sciences 50 (2011) 532e543542

The advantage with the use of ANN is the reduction in theinverse analysis time since the forward model is solved prior toactual inverse analysis for various combinations of the boundaryconditions. Once the forward model is solved for, the inverseanalysis for any desired boundary condition can be computed morerapidly and once and for all without going through iterations. Theinputs to the “inverse ANN” are the “experimental temperatures”and the outputs are the parameters to be estimated.

Fig. 16. Comparison of estimated and predicted values of temperature by inversemethod and CFD respectively.

The CPU time on average is 2700 s (45 min) for the forwardsolution to the problem under consideration which involves 3parameters of heat generation (Q1, Q2 and Q3). For the case ofa conventional inverse analysis where iterative solution is sought,the CPU time for, say, 400 cases is 1080000s to estimate a singlecombination of the parameter set (assuming that the inversesolution is obtained with 400 trial solutions). In the case of thepresent method of inverse analysis, the forward model for variouscombinations of parameters is computed prior to inverse analysis.This reduces the time taken for estimating any desired combinationof parameters since the ANN is already trained and gives thedesired values of parameters accurately and directly without iter-ations. The “inverse” ANN can be adequately trained with a muchfewer number of solutions compared to 400. For example 100e150solutions are sufficient for a 3 parameter problem like the oneconsidered here.

The computational gain for the case considered is 400/150which is of the order of 3. This ratio will go up substantially if morenumber of parameters is to be estimated or if the solution to theforward model is exceedingly time consuming.

4. Conclusions

Inverse analysis for a two dimensional laminar conjugate mixedconvection on avertical platewith protruding heat sources has beenconducted to estimate the heat generation in the sources for a giventemperature distribution on the substrate wall. Artificial neuralnetwork is used in conjunctionwith covariance analysis in order toestimate the heat generation in the heat sources inversely given thetemperature distribution on the back of the substrate wall. Thepresentmethod of inversion has proved to be accurate in estimationof heat generation which is an important design consideration fora thermal designer even under realistic values of noise. From theresults obtained, it is evident that the non-iterative method used inthe work is computationally economical and robust. The novelty ofthe present work lies in application of a non-iterative method toa more practical and complex geometry. The ANN used for theinverse model is shown to be robust and highly accurate in theretrieval of heat generation in the heat sources mounted onthe substrate wall. Covariance analysis assists in designing experi-ments and determining the positions where temperatures can berecorded for a successful inverse solution. The inverse model canthus be solved by an optimal choice of the temperatures in the givenproblem using covariance analysis which is practically feasibleduring experiments. Temperature locations were brought downfrom 451 to 87 locations based on covariance analysis which weresubsequently reduced to just 18 for a successful inverse solution.

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