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Numerical Heat Transfer, Part A:Applications: An International Journal ofComputation and MethodologyPublication details, including instructions for authors andsubscription information:http://www.tandfonline.com/loi/unht20

Simultaneous Estimation of BoundaryHeat Flux and Convective Heat TransferCoefficient of a Curved Plate Subjectedto a Slot Liquid Jet Impingement CoolingYu-Ching Yang a , Wen-Lih Chen a , Huann-Ming Chou a & Jose LeonSalazar ba Clean Energy Center, Department of Mechanical Engineering , KunShan University , Taiwan, R.O.C.b Material Engineering School , Costa Rica Institute of Technology ,Cartago , Costa RicaPublished online: 09 May 2014.

To cite this article: Yu-Ching Yang , Wen-Lih Chen , Huann-Ming Chou & Jose Leon Salazar (2014)Simultaneous Estimation of Boundary Heat Flux and Convective Heat Transfer Coefficient of aCurved Plate Subjected to a Slot Liquid Jet Impingement Cooling, Numerical Heat Transfer, PartA: Applications: An International Journal of Computation and Methodology, 66:3, 252-270, DOI:10.1080/10407782.2013.873264

To link to this article: http://dx.doi.org/10.1080/10407782.2013.873264

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SIMULTANEOUS ESTIMATION OF BOUNDARY HEATFLUX AND CONVECTIVE HEAT TRANSFERCOEFFICIENT OF A CURVED PLATE SUBJECTED TOA SLOT LIQUID JET IMPINGEMENT COOLING

Yu-Ching Yang1, Wen-Lih Chen1, Huann-Ming Chou1, andJose Leon Salazar21Clean Energy Center, Department of Mechanical Engineering, Kun ShanUniversity, Taiwan, R.O.C.2Material Engineering School, Costa Rica Institute of Technology, Cartago,Costa Rica

In this study, an inverse algorithm based on the conjugate gradient method and the discrep-

ancy principle is applied to simultaneously estimate the unknown boundary heat flux and

convective heat transfer coefficient in a curved plate cooled by an impinging slot jet from

the knowledge of temperature measurements taken within the plate. Subsequently, the

distributions of temperature in the plate can be determined. It is assumed that no prior

information is available on the functional forms of the heat flux and convective heat transfer

coefficient; hence the procedure is classified as the function estimation in inverse calculation.

The temperature data obtained from the direct problem are used to simulate the temperature

measurements, and the effect of the errors and locations in these measurements upon the

precision of the estimated results is also considered. Results show that an excellent esti-

mation on the heat flux and convective heat transfer coefficient and temperature distributions

can be obtained for the two test cases considered in this study.

1. INTRODUCTION

Jet impingement is one of the most effective heating=cooling methods; especially,its capability for a large amount of heat removal makes it the favorable coolingsolution for devices subjected to extreme heating conditions. Currently, jet impinge-ment has been widely used in many industrial processes; including, cooling of gasturbines, cooling of electrical equipment, deicing of aircraft wings, hardening andquenching of metal parts, tempering of glass, the secondary cooling of continuouscasting of steels, etc. [1]. In these processes, it is essential to understand the heattransfer of jet impingement to prevent equipment failure and further to improvesystem performance. Hence, there has been significant research work published on

Received 27 April 2013; accepted 23 November 2013.

Address correspondence to Wen-Lih Chen, Clean Energy Center, Department of Mechanical

mail.ksu.edu.tw

Color versions of one or more of the figures in the article can be found online at www.tandfonline.

com/unht.

Numerical Heat Transfer, Part A, 66: 252–270, 2014

Copyright # Taylor & Francis Group, LLC

ISSN: 1040-7782 print=1521-0634 online

DOI: 10.1080/10407782.2013.873264

252

Engineering, Kun Shan University, Yung-Kang city, Tainan 710-03, Taiwan, R.O.C. E-mail: wlchen@

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the heat and mass transfer of impinging jets [2]. There are many different ways ofemploying jet impingement, including using single jet, pulsed jets, inclined jets,2-D slot jets, array of discrete jets, and annular jets. These jets can be classified intotwo principal configurations; namely, circular jets and slot jets. Between the twoconfigurations, there have been enormous reports on circular jets [3], but much lesson slot jets in the open literature; yet, slot jets have begun to attract significantattention in recent years. Chen et al. [3] argued that slot jets offer many beneficialfeatures, such as higher cooling effectiveness, greater uniformity, and more controll-ability; therefore, the trend of developing increasingly high heat addition=removaltechnology favors the use of slot jets.

The fluid used for jet impingement can be liquid or gas, and between them,liquid jet impingement produces an overwhelming heat transfer rate that is severalorders of magnitude higher than that of gas jet impingement [4]. To meet the evergrowing demand for rapid heating=cooling techniques, liquid jet impingement standsout as a better choice. According to the difference in ambient fluid, liquid jetimpingement can be categorized into submerged jet impingement and free surfaceimpingement. The former is a liquid jet ejected into the same ambient liquid, andthe latter into ambient gas. Free surface impingement has been a common practiceadopted for quenching hot metal parts, and has been established as a very importanttechnique in material processing industries. The quality of the finished metal productdepends largely on the quenching process, which affects grain size distribution,thermal stresses, and some other properties in metal. Hence, the understanding ofsuch processes is crucial to many industries that involve metal processing. However,the physics of such a quenching process is very complicated. Normally, coolant isdischarged onto a very hot metal surface, with the temperature ranging from 800to 1000�C. Upon contact with such high temperature surfaces, the coolant, mostlywater, could boil via different modes of boiling, including stable film boiling,

NOMENCLATURE

b plate thickness, r2� r1, (m)

D width of the jet slot, (m)

H1 computational domain extension above

the slot opening, (m)

H2 distance between the jet slot opening and

the plate, (m)

h convective heat transfer coefficient on the

outer surface, (Wm�2K�1)

J functional

J0 gradient of functional

k thermal conductivity, (Wm�1K�1)

p direction of descent

q heat flux at the bottom side of the plate,

(Wm�2)

r radial coordinate, (m)

r1 inner radius of the plate, (m)

r2 outer radius of the plate, (m)

s circumference coordinate as shown in

Fig. 1 (m)

T temperature, (K)

T1 cooling fluid temperature, (K)

x spatial coordinate, (m)

rm temperature measurement position, (m)

Y measured temperature, (K)

y spatial coordinate, (m)

D small variation quality

/ angular coordinate, (rad)

b step size

c conjugate coefficient

g very small value

h elevation temperature defined as

h¼T�T1, (K)

k variable used in adjoint problem

r standard deviation

- random variable

Superscripts

K iterative number

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transition boiling, and nucleate boiling [5]. This, together with conjugate heat trans-fer, gives rise to a very complex heat transfer process between coolant and metalparts, and to study such a subject, either experimentally or numerically, presentsa highly challenging task.

Having undergone continuous development in recent decades, inverse analysishas now become a valuable alternative when direct measurement of data is difficultor the measuring process is very expensive; for example, the determination of heattransfer coefficients, detection of contact resistance, estimation of unknown thermal-physical properties of new materials, optimization of geometry, prediction of creviceand pitting in furnace wall, determination of heat flux at the outer surface of avehicle re-entry, etc. Inverse methods have also found their way in optimization ofquantities or geometries, such as determining optimal thickness of insulation layer,optimizing shape of gas channel in fuel cell, and finding optimal control force ina vibration system [6–8]. Due to the complexity of the heat transfer involved inthe quenching of metal parts by impinging jets, the simplicity of estimating impor-tant heat transfer data, such as boundary heat flux and convective heat transfercoefficient, via inverse methods makes them potentially very useful tools to studysuch complicated processes. Therefore, the objective of this study is to develop aninverse method to simultaneously estimate the boundary heat flux and convectiveheat transfer coefficient of a finite-thickness curved plate cooled by an impinging slotjet. To this end, we employ the conjugate gradient method (CGM) [9–14] and thediscrepancy principle [15] to solve the inverse problem in this study. The conjugategradient method with an adjoint equation, also called Alifanov’s iterative regulariza-tion method, belongs to a class of iterative regularization techniques, which meansthe regularization procedure is performed during the iterative processes; thus, thedetermination of optimal regularization conditions is not needed. On the other hand,the discrepancy principle is used to terminate the iteration process in the conjugategradient method.

2. ANALYSIS

2.1. Direct Problem

In a jet impingement heat transfer problem, the heat transfer effect of theimpinging jet on the solid material can be represented by imposing a convective heattransfer coefficient function on the solid surface which is in contact with the jet. Ifthis function is known, the analytical domain can be narrowed down to just the solidmaterial domain without the consideration of the fluid. This practice largely simpli-fies analytical work, because those complicated liquid boiling issues in metal quench-ing processes are no longer relevant. Another important heat transfer quantity is wallheat flux should the solid material be subjected to heating. With the knowledge ofheat flux and convective heat transfer coefficient on its boundaries, the overall tem-perature of the solid material can be computed. Therefore, to study the heat transferof heating=cooling metal plates with an impinging slot jet, heat flux and convectiveheat transfer coefficient are two prominent heat transfer characteristics to be deter-mined. In this case, if a plate with finite thickness is heated from below=inside andcooled by jet impingement on the top=outside, the heat flux on the lower=inner wall

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and convective heat transfer coefficient on the upper=outer wall have to be calculatedor measured. In this study, a curved plate heated from the lower surface and cooledfrom the upper surface by an impinging slot jet is chosen to develop the inversemethod to simultaneously determine heat flux and convective heat transfer coef-ficient on the lower and upper surfaces, respectively. To illustrate the methodologyfor developing expressions for such a problem, a curved plate in the shape of a halfcylinder with inner radius r1 and outer radius r2, as shown in Figure 1, is considered.The length of the plate is assumed infinite; hence the problem can be regarded astwo-dimensional. On the outer surface, the cooling effect of impinging slot jet hasbeen taken into account via a convective heat transfer coefficient function, whereas,on the inner surface, another function of heat flux is imposed to simulate the heatingof the plate. The problem is further assumed steady, and symmetrical to the centralline, thus, only half of the domain needs to be considered. In addition, all physicalproperties of the curved plate are assumed constant. Therefore, the direct problemis a steady two-dimensional heat conduction problem, and the governing equationand boundary conditions can be written as follows [2].

q2hðr;/Þqr2

þ 1

r

qhðr;/Þqr

þ 1

r2q2hðr;/Þ

q/2¼ 0 ð1Þ

qhðr;/Þqn

¼ 0; at / ¼ 0; r1 � r � r2 ð2Þ

qhðr;/Þqn

¼ 0; at / ¼ p=2; r1 � r � r2 ð3Þ

Figure 1. Geometry and coordinate system.

HEAT TRANSFER COEFFICIENT OF A CURVED PLATE WITH JET COOLING 255

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�kqhðr;/Þ

qr¼ qð/Þ; at r ¼ r1; 0 � / � p=2 ð4Þ

�kqhðr;/Þ

qr¼ hð/Þ � hðr;/Þ; at r ¼ r2; 0 � / � p=2 ð5Þ

where the elevation temperature h is defined as h¼T�T1, and k is the thermal conduc-tivity. The direct problem considered here is concerned with the determination of themedium temperature when the heat flux q (/), the convective heat transfer coefficienth (/), thermophysical properties of the plate, and boundary conditions are known.

2.2. Inverse Problem

For the inverse problem, the heat flux function q(/) and the convective heattransfer coefficient h(/) are regarded as being unknown, while everything else inEqs. (1)–(5) is known. In addition, temperature readings taken at (r, /)¼ (rm1, /i)and (r, /)¼ (rm2, /j) are considered available. Here, i¼ 1�M and j¼ 1�N, whileM and N are the total measurement points in the / direction at r¼ rm1 and r¼ rm2,, respectively. The objective of this inverse analysis is to predict the unknownspace-dependent functions q(/) and h(/) simultaneously, merely from the knowledgeof these temperature readings. Let the measured temperatures at the measurementpositions (r, /)¼ (rm1, /i) and (r, /)¼ (rm2, /j) be denoted by Y1i(rm1, /i) andY2j(rm2, /j), respectively. Then, this inverse problem can be stated as follows: by uti-lizing the above-mentioned measured temperature data Y1i(rm1, /i) and Y2j(rm2, /j),the unknowns q(/) and h(/) are to be estimated simultaneously over the specifiedspace domain.

The solutions of the present inverse problem are to be obtained in such a waythat the following functional is minimized.

J½qð/Þ; hð/Þ� ¼XMi¼1

hðrm1;/iÞ � w1iðrm1;/iÞ½ �2þXNj¼1

hðrm2;/jÞ � w2jðrm2;/jÞ� �2 ð6Þ

where w1i (rm1, /i)¼Y1i (rm1, /i)�T1 and w2j (rm2, /j)¼Y2j (rm2, /j)�T1; h (rm1,/i) and h (rm2, /j) are the estimated (or computed) temperatures at the measurementlocations (r, /)¼ (rm1, /i) and (r, /)¼ (rm2, /j), respectively. In this study, h (rm1, /i)and h (rm2, /j) are determined from the solution of the direct problem given pre-viously by using an estimated qK (/) and hK (/) for the exact q(/) and h(/). HereqK (/) and hK (/) denote the estimated quantities at the Kth iteration. In addition,in order to develop expressions for the determination of the unknowns q (/) and h(/), a sensitivity problem and an adjoint problem are constructed, as described below.

2.3. Sensitivity Problem and Search Step Size

Since the problem involves two unknown functions q(/) and h(/), and in orderto derive the sensitivity problem for each unknown, we should perturb the unknownfunctions one at a time.

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First, it is assumed that when q(/) undergoes a variation Dq(/), h(r, /) isperturbed by Dh1(r, /). Then, replacing in the direct problem q(/) by q(/)þDq(/)and h by hþDh1, subtracting from the resulting expressions the direct problem,and neglecting the second-order terms, the following sensitivity problem for thesensitivity function Dh1 can be obtained.

q2Dh1ðr;/Þqr2

þ 1

r

qDh1ðr;/Þqr

þ 1

r2q2Dh1ðr;/Þ

q/2¼ 0 ð7Þ

qDh1qn

¼ 0; at / ¼ 0; r1 � r � r2 ð8Þ

qDh1qn

¼ 0; at / ¼ p=2; r1 � r � r2 ð9Þ

�kqDh1qr

¼ Dqð/Þ; at r ¼ r1; 0 � / � p=2 ð10Þ

�kqDh1qr

¼ hð/Þ � Dh1; at r ¼ r2; 0 � / � p=2 ð11Þ

The sensitivity problem of Eqs. (7)–(11) can be solved by the same method as thedirect problem of Eqs. (1)–(5). Similarly, by perturbing h(/) with Dh(/), the secondsensitivity problem for the sensitivity function Dh2(r, /) can be obtained [10].

2.4. Adjoint Problem and Gradient Equation

To formulate the adjoint problem, Eq. (1) is multiplied by the Lagrangemultiplier (or adjoint function) k1 (r, /), and the resulting expressions are integratedover the correspondent space domains. Then, the results are added to the right handside of Eq. (6) to yield the following expression for the functional J[q(/), h(/)].

J½qð/Þ; hð/Þ� ¼XMi¼1

½hðrm1;/iÞ � w1iðrm1;/iÞ�2þXNj¼1

½hðrm2;/jÞ � w2jðrm2;/jÞ�2

þZ p=2

/¼0

Z ro

r¼ri

r � k1ðr;/Þ � ½q2hqr2

þ 1

r

qhqr

þ 1

r2q2h

q/2�drd/ ð12Þ

The variation DJ1 is derived after q(/) is perturbed by Dq(/) and h(r,/) isperturbed by Dh1(r,/) in Eq. (12). Subtracting from the resulting expression theoriginal Eq. (12) and neglecting the second-order terms, we thus find the following.

DJ1½qð/Þ; hð/Þ� ¼Z p=2

/¼0

Z ro

r¼ri

2XMi¼1

½hðrm1;/iÞ � w1iðrm1;/iÞ�Dh1 � dðr� rm1Þ

� dð/� /iÞdrd/þZ p=2

/¼0

Z ro

r¼ri

2XNj¼1

½hðrm2;/jÞ � w2jðrm2;/jÞ�Dh1 � dðr� rm2Þ

� dð/� /jÞdrd/þZ p=2

/¼0

Z ro

r¼ri

r � k1ðr;/Þ � ½q2Dh1qr2

þ 1

r

qDh1qr

þ 1

r2q2Dh1q/2

�drd/ ð13Þ

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where d(�) is the Dirac function. We can integrate the third double integral term inEq. (13) by parts, utilizing the boundary conditions of the sensitivity problem. Then,DJ1 is allowed to go to zero. The vanishing of the integrands containing Dh1 leads tothe following adjoint problem for the determination of k1(r,/).

q2k1ðr;/Þqr2

þ 1

r

qk1ðr;/Þqr

þ 1

r2q2k1ðr;/Þ

q/2

þ2PMi¼1

½hðrm1;/iÞ � w1iðrm1;/iÞ� � dðr� rm1Þ � dð/� /iÞ

r

þ2PNj¼1

½hðrm2;/jÞ � w2jðrm2;/jÞ� � dðr� rm2Þ � dð/� /jÞ

r¼ 0

ð14Þ

qk1ðr;/Þqn

¼ 0; at / ¼ 0; r1 � r � r2 ð15Þ

qk1ðr;/Þqn

¼ 0; at / ¼ p=2; r1 � r � r2 ð16Þ

qk1ðr;/Þqr

¼ 0; at r ¼ r1; 0 � / � p=2 ð17Þ

�kqk1ðr;/Þ

qr¼ hð/Þ � k1ðr;/Þ; at r ¼ r2; 0 � / � p=2 ð18Þ

Then, the adjoint problem can be solved by the same method as the direct problem.Finally, the following integral term is left.

DJ1 ¼Z

½r � k1ðr;/Þ=k�r¼r1Dqð/Þd/ ð19Þ

Here, ½k1ðr;/Þ�r¼r1denotes k1(r1, /). From the definition used in reference [9], we

have the following.

DJ1 ¼Z

J 01ð/ÞDqð/Þd/ ð20Þ

where J01(/) is the gradient of the functional J1. A comparison of Eqs. (19) and (20)leads to the following form.

J 01ð/Þ ¼ ½r � k1ðr;/Þ=k�r¼r1

ð21Þ

Similarly, Eq. (1) is multiplied by the Lagrange multiplier (or adjoint function)k2(r,/) to derive the adjoint problem for the case when perturbing Dh(/). Followingthe same procedure, eventually, we find that the solutions for adjoint equation ofk2(r,/) are identical to those for k1(r,/). This implies that the adjoint equations needto be solved only once since k2(r,/)¼ k1(r,/). Finally, the gradient equation for h(/)can be obtained as J 0

2ð/Þ ¼ ½�r � k2ðr;/Þ � hð/Þ=k�r¼r2. Again, ½k2ðr;/Þ � hðr;/Þ�r¼r2

denotes k2(r2, /) � h(r2, /).

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2.5. Conjugate Gradient Method for Minimization

The following iteration process based on the conjugate gradient method is nowused for the estimation of q(/) and h(/) by minimizing the above functional J[q(/),h(/)].

qKþ1ð/Þ ¼ qKð/Þ � bK1 pK1 ð/Þ; K ¼ 0; 1; 2; . . . ð22Þ

hKþ1ð/Þ ¼ hKð/Þ � bK2 pK2 ð/Þ; K ¼ 0; 1; 2; . . . ð23Þ

Where bK1 and bK2 are the search step size in going from iteration K to iteration Kþ 1,

and pK1 ð/Þ and pK2 ð/Þ are the direction of descent (i.e., search direction) given by

pK1 ð/Þ ¼ J 0K1 ð/Þ þ cK1 p

K�11 ð/Þ ð24Þ

pK2 ð/Þ ¼ J 0K2 ð/Þ þ cK2 p

K�12 ð/Þ ð25Þ

which is conjugation of the gradient direction J 0K1 ð/Þ and J 0K

2 ð/Þ at iteration K with

the direction of descent pK�11 ð/Þ and pK�1

2 ð/Þ at iteration K� 1. The expressions for

the conjugate coefficient cK1 and cK2 can be found in reference [9]. To perform the

iteration according to Eqs. (22) and (23), we need to compute the step size bK1 and

bK2 and the gradient of functional J 0K1 ð/Þ and J 0K

2 ð/Þ. The bK1 and bK2 are computed

by minimizing J[qKþ 1 (/), hKþ 1 (/)] given by Eq. (6) with respect to bK1 and bK2 ,respectively [10]. On the other hand, the gradient of functional J 0K

1 ð/Þ and J 0K2 ð/Þ

are obtained from the solutions of adjoint problem.

2.6. Stopping Criterion

If the problem contains no measurement errors, the traditional convergencecondition is specified as follows.

J½qKþ1ð/Þ; hKþ1ð/Þ� < g ð26Þ

where g is a small specified number, can be used as the stopping criterion. However,the observed temperature data contains measurement errors; as a result, the inversesolution will tend to approach the perturbed input data, and the solution will exhibitoscillatory behavior as the number of iteration is increased [16]. Computationalexperience has shown that it is advisable to use the discrepancy principle [15] forterminating the iteration process in the conjugate gradient method. Assumingh(rm1, /i)�w1i (rm1, /i)ffi h (rm2, /j)�w2j (rm2, /j)ffir, the stopping criteria g bythe discrepancy principle can be obtained from Eq. (6) as follows.

g ¼ ðM þNÞr2 ð27Þ

where r is the standard deviation of the measurement error. Then, the stoppingcriterion is given by Eq. (26) with g determined from Eq. (27).

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2.7. Computational Procedure

The computational procedure for the solution of this inverse problem may besummarized as follows.

Suppose q(/) and h(/) is available at iteration K.

. Step 1. Solve the direct problem given by Eqs. (1)–(5) for h(r,/).

. Step 2. Examine the stopping criterion given by Eq. (26) with g given by Eq. (27).Continue if not satisfied.

. Step 3. Solve the adjoint problem given by Eqs. (14)–(18) for k1(r, /).

. Step 4. Compute the gradient of the functional J 01ð/Þ and J 0

2ð/Þ.. Step 5. Compute the conjugate coefficient cK1 and cK2 and direction of decent pK1

and pK2 , respectively.

. Step 6. Set Dqð/Þ ¼ pK1 ð/Þ and Dhð/Þ ¼ pK2 ð/Þ, and solve the sensitivity problemgiven by Eqs. (7)–(11) for Dh1ðr;/Þ and Dh2ðr;/Þ.

. Step 7. Compute the search step size bK1 and bK2 .

. Step 8. Compute the new estimation for qKþ1ð/Þ and hKþ1ð/Þ from Eqs. (22) and(23), and return to step 1.

3. RESULTS AND DISCUSSION

The validity of the mathematical model for the direct problem and the accuracyof the numerical solution are two important issues that have to be addressed for thecredibility of an inverse estimation in a real application. In terms of the first issue,Eq. (1) is well-known and has been wildly used in steady-state heat conductionproblems and is capable of returning accurate solutions. The second issue concernsgrid density, and will be verified through the following grid-independency test.The numerical procedure in this article is based on the unstructured-mesh, fullycollocated, finite-volume code, USTREAM developed by the corresponding author.This is the descendent of the structured-mesh, multi-block code of STREAM [17].In the grid-independency test, the material of the curved plate is assumed to be steelwith constant thermal conductivity k¼ 43Wm�1K�1, the geometrical parametersare set as r1¼ 0.04m and r2¼ 0.05m, and the plate is subjected to heating on thelower surface and cooling by a slot impinging jet on the upper surface. The heatingand cooling effects are accounted for by applying a distribution of wall heat flux onthe lower surface and another distribution of convective heat transfer coefficienton the upper surface. The distributions of heat flux and convective heat transfercoefficient are assumed to take the following forms.

qð/Þ ¼ 250; 000 W m�2 ð28Þ

hð/Þ ¼ k

dð100þ 30 sin/Þ W m�2K�1 ð29Þ

where d (¼ 2r2) is the outer diameter of the curved plate=half cylinder. The convectiveheat transfer coefficient function takes a sine functional form in resemblance to the

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Nusselt number distributions reported in reference [2]. Three different grids with gridsizes (in /- and r-directions) of 18� 6, 60� 20, and 90� 30, respectively, areemployed for grid-independency test. Figure 2 shows the grid of 60� 20 in size. Here,denser grid has been allocated near the lower and upper surfaces to facilitate betternumerical resolution. The results, in terms of temperature variations along the lowerand upper surfaces of this test, are given in Figure 3, in which the horizontal coordi-nate s denotes the circumference along the curved plate boundary, as shown inFigure 1. It can be seen that the results by the 60� 20 grid are almost identical to thoseby the 90� 30 grid, indicating that the 60� 20 grid is good enough to returngrid-independent solutions. This grid will be used for all cases in the rest of this study.To this end, the two important issues concerning the credibility of inverse estimationhave been properly addressed.

The objective of this article is to validate the present inverse analysis when usedin simultaneously estimating the unknown heat flux and convective heat transfercoefficient respectively at the lower and upper walls of the curved plate accuratelywith no prior information on the functional forms of the unknown quantities, a pro-cedure called function estimation. In the analysis, we do not have a real experimentalset up to measure the temperatures w1i (rm1, /i) and w2j (rm2, /j) in Eq. (6). Instead,we assume functional forms of q(/) and h(/) and substitute them into the directproblem of Eqs. (1)–(5) to calculate the temperatures at the locations where the tem-perature sensors are placed. The results are taken as the computed temperatures

wexact1i ðrm1;/iÞ and wexact

2j ðrm2;/jÞ. Nevertheless, in reality, the temperature measure-

ments always contain some degree of errors, whose magnitude depends upon theparticular measuring method employed. In order to take measurement errors intoaccount, a random error noise is added to the above computed temperatures to obtain

Figure 2. Computational grid with 60� 20 cells.

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the measured temperatures w1i (rm1, /i) and w2j (rm2, /j). In general, the measuredtemperature w(rm, /) is expressed as follows.

wðrm;/Þ ¼ wexactðrm;/Þ þ -r ð30Þ

Where - is a random variable within �2.576 to 2.576 for a 99% confidence bounds,and r is the standard deviation of the measurement. The measured temperaturegenerated in such way is the so-called simulated measurement temperature.

Figure 3. Temperature distributions on the lower and the upper surfaces with three different grids. (a)

Temperature distributions on the lower surface, and (b) temperature distributions on the upper surface.

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To demonstrate the accuracy of the CGM in simultaneously estimating thewall heat flux and convective heat transfer coefficient of the curved plate inthe present problem, two different sets of functional forms of q(/) and h(/) are con-sidered. In all cases, the following geometrical parameters are adopted: r1¼ 0.04m,r2¼ 0.05m.

3.1. Case 1

The unknown heat flux and convective heat transfer coefficient are assumed theforms given by Eqs. (28) and (29). The inverse temperatures and the estimated valuesof the unknown functions at lower and upper surfaces are illustrated in Figures 4 and5, respectively. These are obtained with the initial guess values q0 (/)¼ 0.0 and h0

(/)¼ 0.0, and under two different measurement errors of deviation r¼ 0.0 and0.01, respectively. For a temperature of unity and 99% confidence, the standard devi-ation r¼ 0.01 corresponds to measurement error of 2.58%. The temperaturesw1i(rm1, /i) and w2j(rm2, /j) are measured at rm1¼ r1 (on the lower surface) andrm2¼ r2 (on the upper surface), respectively, and the locations of sensors are assumedto match the cell coordinates of the cells adjacent to the lower and upper boundaries.Therefore, the numbers of sensors for w1i(rm1, /i) and w2j(rm2, /j) are both 60.Figure 4 suggests that inverse and simulated temperature distributions are almostidentical, indicating that the inverse solutions have been well converged. In the caseof zero measurement error, the results in Figure 5 show that the estimated heat fluxand convective heat transfer coefficient functions are in excellent agreement with theexact functions. The average relative error for the estimated heat flux isERR1¼ 0.07%, and that for the estimated convective heat transfer coefficient isERR2¼ 0.05%, where the relative average errors are defined as follows.

ERR1 ¼

PMi¼1

qiexactð/Þ�qiinvð/Þqiexactð/Þ

������

M� 100% ð31Þ

ERR2 ¼

PNj¼1

hjexactð/Þ�h

jinvð/Þ

hjexactð/Þ

������

N� 100% ð32Þ

WhereM and N are the numbers of sensors at r¼ rm1 and r¼ rm2, respectively. In thecase of r¼ 0.01, it is noticeable from the results that the estimated heat flux byconsidering measurement errors only slightly deviates from the exact value withERR1¼ 2.59%, which is slightly larger than the magnitude of measurement errors.The error of convective heat transfer coefficient, ERR2, on the other hand, remainsvery small, only at 0.1%. This suggests that present algorithm is not sensitive tothe measurement error, and reliable inverse solutions can still be obtained whenmeasurement errors are considered. Since the temperature distributions at themeasurement locations at rm1 and rm2 are affected by heat flux and heat transfercoefficient, the errors in temperature measurements could be rooted in the disagree-ment between estimated and exact heat fluxes, or heat transfer coefficients, or bothheat fluxes and heat transfer coefficients. That is, even though there is excellent

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agreement between estimated and exact values in one quantity, the disagreementin the other quantity alone could still produce some degree of errors on temperaturemeasurements at both rm1 and rm2.

With the heat flux and convective heat transfer coefficient well estimated, thetemperature in the curved plate can be obtained accurately. This is illustrated inFigure 6, where the distributions of temperature along the radial direction at twodifferent angle locations, /¼ 0� and 90�, are shown; the inverse results are obtainedwith r¼ 0.0. Excellent agreement between the simulated and inverse temperaturescan be seen at both angle locations.

Figure 4. Case 1 simulated and inverse temperature distributions on the lower and the upper surfaces

with r¼ 0.0 and r¼ 0.01. (a) Temperature distributions on the lower surface, and (b) temperature

distributions on the upper surface.

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Next, the sensitivity of inverse accuracy on the measurement locations will beexamined. In some inverse problems, inverse accuracy is found to be dependent onmeasurement locations. This would hamper the practical usefulness of an inversemethod because sensors can only be placed at some limited locations to get accurateresults. To test such sensitivity, the measurement locations are set at rm1¼ 0.042mand rm2¼ 0.048m, respectively, that is, all sensors have been imbedded inside thecurved plate. In this case, the relative errors of the returned inverse estimations withr¼ 0.0 are ERR1¼ 0.26% and ERR2¼ 0.04%, which are very small and are insimilar magnitude to the previous case where all sensors are placed on the surface.

Figure 5. Case 1 exact and inverse boundary heat flux and convective heat transfer coefficient distributions,

respectively, on the lower and the upper surfaces with r¼ 0.0 and r¼ 0.01. (a) Heat flux distributions

on the lower surface, and (b) convective heat transfer coefficient distributions on the upper surface.

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The results indicate that the inverse accuracy of the current method is not sensitive tothe measurement locations. The reason is due to the nature of the solution to theelliptic type governing equation of steady-state heat conduction problems. In sucha solution, a disturbance in any point of the domain is broadcasted to the entiredomain. Therefore, the effect of any change at the boundary propagates to all partsof the solid material, that is, sensor can be placed anywhere to pick up variations atthe boundary.

3.2. Case 2

The unknown boundary heat flux and convective heat transfer coefficient takethe following forms.

qð/Þ ¼ 50; 000 sinð/Þ þ 200; 000 W m�2 ð33Þ

hð/Þ ¼ k

dð90 sin4 /þ 45 sin9 2/þ 15Þ W m�2K�1 ð34Þ

As seen in Eqs. (33) and (34), both heat flux and convective heat transfer coefficientfunctions are more complicated than those forms in case 1. The heat flux q(/) nowvaries along the /-direction, and the second term in h(/) creates a secondary peakat /¼ 45�, simulating the cooling effect of a second jet impinging at that location.These are more challenging features to test the inverse method. The inverse tempera-tures and estimated values of the boundary heat flux and convective heat transfercoefficient, with the initial guess values q0 (/)¼ 0.0 and h0 (/)¼ 0.0, and measurementerror of deviation r¼ 0.0 and 0.01, are shown in Figures 7 and 8, respectively.In Figure 7, the excellent agreement between inverse and simulated temperatures

Figure 6. Case 1 simulated and inverse temperature distributions along the radial direction at /¼ 0�

and 90�.

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indicates the inverse solutions have been converged quite well. The distributions ofboundary heat flux and convection heat transfer coefficient in Figure 8 again indicatethat the estimated results are in very good agreement with those of the exact values.The average relative errors are ERR1¼ 0.06%, ERR2¼ 0.05%, and ERR1¼ 2.55%,ERR2¼ 0.06%, respectively for r¼ 0.0 and r¼ 0.01. The estimated heat flux vari-ation without measurement error is almost identical to the exact variation, and themagnitude of relative error of inverse heat flux is similar to that of measurement

Figure 7. Case 2 simulated and inverse temperature distributions on the lower and the upper surfaces with

r¼ 0.0 and r¼ 0.01. (a) Temperature distributions on the lower surface, and (b) temperature distributions

on the upper surface.

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errors when they are considered. To this end, the superb accuracy of the proposedinverse method is further demonstrated through the second test case. This practicealso proves that the current method is capable of accurately estimating boundary heatflux and convective heat transfer coefficient in arbitrary forms.

4. CONCLUSION

An inverse algorithm based on the conjugate gradient method and the discrep-ancy principle was successfully applied for the solution of the inverse problem to

Figure 8. Case 2 exact and inverse boundary heat flux and convective heat transfer coefficient distributions,

respectively, on the lower and the upper surfaces with r¼ 0.0 and r¼ 0.01. (a) Heat flux distributions

on the lower surface, and (b) convective heat transfer coefficient distributions on the upper surface.

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simultaneously determine the unknown heat flux at the lower surface and the con-vective heat transfer coefficient at the upper surface of a finite-thickness curved platein the shape of a half circular cylinder cooled by impinging slot jets, while knowing thetemperature history at some measurement locations. Subsequently, the temperaturedistributions inside the plate material can be calculated. Numerical results confirmthat the method proposed herein can accurately estimate the boundary heatflux, convective heat transfer coefficient, and overall temperature distributions forthe problem even involving the inevitable measurement errors. In addition, theaccuracy of inverse estimation is not sensitive to the sensor location, hence, thereis no strict limitation on where temperature sensors can be placed, a beneficial featurein practical applications. Two sets of boundary heat flux and convective heat transfercoefficient functions have been tested by the current method, and very accurateinverse estimations have been returned for both cases, proving the capability ofthe current method to simultaneously estimate variations of both quantities inarbitrary forms. This method can be developed into a useful tool to study liquidslot jet impingement cooling problems encountered in industrial applications.

FUNDING

This work was supported by the National Science Council, Taiwan, Republicof China under grant numbers NSC-101-2221-E-168-029 and NSC-101-2221-E-168-019.

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