Simultaneous inverse estimation for boundary heat and moisture fluxes of a double-layer annular...

Applied Mathematics and Computation 176 (2006) 594–608

www.elsevier.com/locate/amc

Simultaneous inverse estimation for boundary heatand moisture fluxes of a double-layer annular

cylinder with interface resistance

Yu-Ching Yang *, Win-Jin Chang

Department of Mechanical Engineering, Kun Shan University, No. 949,

Da-Wan Road, Yung-Kong City, Tainan 71003, Taiwan, ROC

Abstract

Based on the conjugate gradient method, this study presents a means of solving the inverse boundary value problem ofcoupled heat and moisture transport in a double-layer annular cylinder. While knowing the temperature and moisture his-tory at the measuring positions, the unknown time-dependent boundary heat and moisture fluxes can be simultaneouslydetermined. It is assumed that no prior information is available on the functional form of the unknown heat and moisturefluxes. The accuracy of this inverse heat and moisture transport problem is examined by using the simulated exact andinexact temperature and moisture measurements in the numerical experiments. Results show that excellent estimationon the time-dependent boundary fluxes can be obtained with any arbitrary initial guesses. In addition, the contact resis-tance of heat and moisture transfer at the interface of the two layers is considered in the analysis.� 2005 Elsevier Inc. All rights reserved.

1. Introduction

Physically there exists a coupling effect between temperature and moisture transport in most materials.This coupling effect is especially significant for some porous and composite materials. For a direct heatand moisture transport problem, it is concerned with the determination of temperature and moisture atinterior points of a region when the initial and boundary conditions and thermophysical properties arespecified. In contrast, the inverse heat and moisture transport problem considered in this study involvesthe determination of the unknown time-dependent fluxes of heat and moisture transfer in a double-layerannular cylinder from the knowledge of the temperature and moisture measurements taken within thecylinder.

For the conventional inverse heat transfer problems, the estimation of unknown thermal boundary condi-tions is always the main concerns. On the other hand, the technique of conjugate gradient method (CGM) [1]

0096-3003/$ - see front matter � 2005 Elsevier Inc. All rights reserved.

doi:10.1016/j.amc.2005.10.006

* Corresponding author.E-mail address: [email protected] (Y.-C. Yang).

mailto:[email protected]

Nomenclature

C dimensionless concentration of moistureD equilibrium diffusion coefficient of moisture contentDm material parameterht contact heat transfer coefficienthm contact mass transfer coefficientJ functionalk material parameterL equilibrium diffusion coefficient of temperatureM total number of measuring positions in r directionp direction of decentq1 heat fluxq2 moisture fluxr dimensionless radiusr1 dimensionless inner radius of cylinder (r1 = 1)r2 dimensionless intermediate radius of cylinder (r2 = 1.6)r3 dimensionless outer radius of cylinder (r3 = 2)t dimensionless timeT dimensionless temperature

Greek symbols

b step sizeD small variation qualitye very small valuek coupling coefficient due to heat conductionm coupling coefficient due to moisture migrationr standard deviation- random variable

Subscripts

f final state0 reference state

Superscripts

K iterative number0 initial state

Y.-C. Yang, W.-J. Chang / Applied Mathematics and Computation 176 (2006) 594–608 595

has been shown its potential for these kinds of problems and has been applied to many applications [2–5]. TheCGM derives basis from the perturbational principles [6] and transforms the inverse problem to the solutionof three problems; namely, the direct problem, the sensitivity problem, and the adjoint problem, which will bediscussed in detail in the text.

In this paper, based on the measurements of temperature and moisture, two unknown time-dependentfluxes of heat and moisture transfer for the heat and moisture transport of a two-layer annular cylinder areto be estimated simultaneously. For this reason, two sensitivity problems and two search step sizes are neededin the study. In addition, the measuring positions may be either interior points or points at the surfaces. More-over, the contact resistance of heat and moisture transfer at the interface of the two layers is considered in theanalysis.

596 Y.-C. Yang, W.-J. Chang / Applied Mathematics and Computation 176 (2006) 594–608

2. Direct problem

To illustrate the methodology for developing expressions for use in simultaneously estimating the unknowntime-dependent fluxes of heat and moisture transfer, we consider the following inverse problem. An infinitelylong two-layer annular cylinder, as shown in Fig. 1, having inner, intermediate, and outer radii r�1; r�2, and r�3,respectively, is subjected to heat flux q�1ðt�Þ and moisture flux q�2ðt�Þ at the inner and outer boundary surfaces,respectively. The governing equations can be expressed as follows [7]:

Lir2T �i ðr�; t�Þ ¼o

ot�½T �i ðr�; t�Þ � miC

�i ðr�; t�Þ�; i ¼ 1; 2; ð1aÞ

Dir2C�i ðr�; t�Þ ¼o

ot�½C�i ðr�; t�Þ � kiT �i ðr�; t�Þ�; i ¼ 1; 2; ð1bÞ

where i = 1,2 refers to the inner and outer cylinders. mi and ki are the coupling coefficients, Li and Di are theequivalent diffusion coefficients of temperature and moisture, respectively, and T �i and C�i are temperature andmoisture concentration, respectively.

Considering the contact resistance of heat and moisture transfer at the interface of the two layers, the asso-ciated hygrothermal boundary and initial conditions may be the following:

� k1

oT �1ðr�1; t�Þor�

¼ q�1ðt�Þ; t� > 0; ð2aÞ

oC�1ðr�1; t�Þor�

¼ 0; t� > 0; ð2bÞ

k1oT �1ðr�2; t�Þ

or�¼ ht½T �2ðr�2; t�Þ � T �1ðr�2; t�Þ�; t� > 0; ð2cÞ

*1r

*2r

θ

z

*r*3r

Fig. 1. Geometry and coordinates of the two-layer annular cylinder.


Dm1


¼ hm½C�2ðr�2; t�Þ � C�1ðr�2; t�Þ�; t� > 0; ð2dÞ

k2


¼ ht½T �2ðr�2; t�Þ � T �1ðr�2; t�Þ�; t� > 0; ð2eÞ

Dm2


¼ hm½C�2ðr�2; t�Þ � C�1ðr�2; t�Þ�; t� > 0; ð2fÞ


¼ 0; t� > 0; ð2gÞ

Dm2


¼ q�2ðt�Þ; t� > 0; ð2hÞ

T �i ðr�; 0Þ ¼ T �0; i ¼ 1; 2; ð3aÞC�i ðr�; 0Þ ¼ C�0; i ¼ 1; 2. ð3bÞ

If the following dimensionless quantities are defined:

r ¼ r�

r�1; r1 ¼

r�1r�1; r2 ¼

r�2r�1; r3 ¼

r�3r�1; t ¼ L1t�

r�21

; T i ¼T �i � T �0

miðC�f � C�0Þ;

Ci ¼C�i � C�0C�f � C�0

; q1 ¼r�1q�1

k1m1ðC�f � C�0Þ; q2 ¼

r�1q�2Dm2ðC�f � C�0Þ

.

Here T �0 and C�0 are the reference temperature and moisture concentration, respectively; C�f is the final equi-librium moisture concentration. Then, the dimensionless formulation of this heat and moisture transportproblem can be expressed as

Li

L1

r2T iðr; tÞ ¼o

ot½T iðr; tÞ � Ciðr; tÞ�; i ¼ 1; 2; ð4aÞ

Di

L1

r2Ciðr; tÞ ¼o

ot½Ciðr; tÞ � kimiT iðr; tÞ�; i ¼ 1; 2; ð4bÞ

� oT 1ðr1; tÞor

¼ q1ðtÞ; t > 0; ð5aÞ

oC1ðr1; tÞor

¼ 0; t > 0; ð5bÞ

oT 1ðr2; tÞor

¼ H t1½T 2ðr2; tÞ � T 1ðr2; tÞ�; t > 0; ð5cÞ

oC1ðr2; tÞor

¼ H m1½C2ðr2; tÞ � C1ðr2; tÞ�; t > 0; ð5dÞ

oT 2ðr2; tÞor

¼ H t2½T 2ðr2; tÞ � T 1ðr2; tÞ�; t > 0; ð5eÞ

oC2ðr2; tÞor

¼ H m2½C2ðr2; tÞ � C1ðr2; tÞ�; t > 0; ð5fÞ

oT 2ðr3; tÞor

¼ 0; t > 0; ð5gÞ

oC2ðr3; tÞor

¼ q2ðtÞ; t > 0; ð5hÞ

T iðr; 0Þ ¼ 0; r1 < r < r3; i ¼ 1; 2; ð6aÞCiðr; 0Þ ¼ 0; r1 < r < r3; i ¼ 1; 2; ð6bÞ

where

H t1 ¼r�1 � ht

k1

; H t2 ¼r�1 � ht

k2

; H m1 ¼r�1 � hm

Dm1

; H m2 ¼r�1 � hm

Dm2

.


It is obvious that the present problem is coupled in the governing differential equations. The direct problemconsidered here is concerned with calculating the medium temperature and moisture when the heat flux q1(t)and moisture flux q2(t), thermal properties, and boundary and initial conditions are given. A hybrid numericalmethod of Laplace transformation and finite difference used in our previous work [8] can be applied to solvethe direct problem.

3. Inverse problem

For the inverse problem considered here, everything except the heat flux q1(t) and moisture flux q2(t) in Eqs.(4)–(6) is known. In addition, the measured temperature and moisture distributions within the cylinder at anytime are considered available.

Let the measured temperature at position rm1 and measured moisture at position rm2 at time t be denoted byY1(rm1, t) and Y2(rm2, t), respectively. Then this inverse problem can be stated as follows: by utilizing the abovementioned measured temperature and moisture data Y1(rm1, t) and Y2(rm2, t), estimate the unknown heat fluxq1(t) and moisture flux q2(t) over the specified time domain.

The solution of the present inverse problem is to be obtained in such a way that the following functional isminimized:

J ½q1ðtÞ; q2ðtÞ� ¼Z tf

0

½T ðrm1; tÞ � Y 1ðrm1; tÞ�2 dt þZ tf

0

½Cðrm2; tÞ � Y 2ðrm2; tÞ�2 dt

¼Z r3

r1

Z tf

0

½T ðrm1; tÞ � Y 1ðrm1; tÞ�2dðr � rm1Þdt dr

þZ r3

r1

Z tf

0

½Cðrm2; tÞ � Y 2ðrm2; tÞ�2dðr � rm2Þdt dr. ð7Þ

Here T(rm1, t) and C(rm2, t) are the estimated (or computed) temperature and moisture at positions rm1 and rm2

at time t. These quantities are determined from the solution of the direct problem given previously by using theestimated heat flux q1(t) and moisture flux q2(t). d(•) is the Dirac delta function.

4. Conjugate gradient method for minimization

The following iteration process based on the conjugate gradient method [1] is now used for the estimationof q1(t) and q2(t) by minimizing the above functional J[q1(t),q2(t)]:

qKþ11 ðtÞ ¼ qK

1 ðtÞ � bK1 pK

1 ðtÞ; K ¼ 0; 1; 2; . . . ; ð8aÞqKþ1

2 ðtÞ ¼ qK2 ðtÞ � bK

2 pK2 ðtÞ; K ¼ 0; 1; 2; . . . ; ð8bÞ

where bK1 and bK

2 are the search step size in going from iteration K to iteration K + 1, and pK1 ðtÞ and pK

2 ðtÞ arethe direction of descent (i.e., search direction) given by

pK1 ðtÞ ¼ J 0K1 ðtÞ þ cK

1 pK�11 ðtÞ; ð9aÞ

pK2 ðtÞ ¼ J 0K2 ðtÞ þ cK

2 pK�12 ðtÞ; ð9bÞ

which is a conjugation of the gradient direction J 0K1 ðtÞ and J 0K2 ðtÞ at iteration K and the direction of descentpK�1

1 ðtÞ and pK�12 ðtÞ at iteration K � 1. The conjugate coefficient is determined from

cK1 ¼

R tf0½J 0K1 ðtÞ�

2 dtR tf0½J 0K�1

1 ðtÞ�2 dt; with c0

1 ¼ 0; ð10aÞ

cK2 ¼

R tf0½J 0K2 ðtÞ�

2 dtR tf0½J 0K�1

2 ðtÞ�2 dt; with c0

2 ¼ 0. ð10bÞ


To perform the iteration according to Eqs. (8) and (9), we need to compute the step size bK1 and bK

2 and thegradient of the functional J 0K1 ðtÞ and J 0K2 ðtÞ. In order to develop expressions for the determination of these twoquantities, two sensitivity problems and an adjoint problem are constructed as described below.

5. Sensitivity problems and search step sizes

Since the problem involves two unknown boundary heat flux q1(t) and moisture flux q2(t), and in order toderive the sensitivity problem for each unknown function, we should perturb the unknown function one at atime.

Firstly, it is assumed that when q1(t) undergoes a variation Dq1(t), T(r, t) and C(r, t) are perturbed by DT 0

and DC 0. Then replacing in the direct problem q1 by q1 + Dq1, T by T + DT 0, and C by C + DC 0, subtractingfrom the resulting expressions the direct problem, and neglecting the second-order terms, the following sen-sitivity problem for the sensitivity functions DT 0 and DC 0 can be obtained.

Li

L1

r2DT 0iðr; tÞ ¼o

ot½DT 0iðr; tÞ � DC0iðr; tÞ�; i ¼ 1; 2; ð11aÞ

Di

L1

r2DC0iðr; tÞ ¼o

ot½DC0iðr; tÞ � kimiDT 0iðr; tÞ�; i ¼ 1; 2; ð11bÞ

� oDT 01ðr1; tÞor

¼ Dq1ðtÞ; t > 0; ð12aÞ

oDC01ðr1; tÞor

¼ 0; t > 0; ð12bÞ

oDT 01ðr2; tÞor

¼ H t1½DT 02ðr2; tÞ � DT 01ðr2; tÞ�; t > 0; ð12cÞ

oDC01ðr2; tÞor

¼ H m1½DC02ðr2; tÞ � DC01ðr2; tÞ�; t > 0; ð12dÞ

oDT 02ðr2; tÞor

¼ H t2½DT 02ðr2; tÞ � DT 01ðr2; tÞ�; t > 0; ð12eÞ

oDC02ðr2; tÞor

¼ H m2½DC02ðr2; tÞ � DC01ðr2; tÞ�; t > 0; ð12fÞ

oDT 02ðr3; tÞor

¼ 0; t > 0; ð12gÞ

oDC02ðr3; tÞor

¼ 0; t > 0; ð12hÞ

DT 0iðr; 0Þ ¼ 0; i ¼ 1; 2; ð13aÞDC0iðr; 0Þ ¼ 0; i ¼ 1; 2. ð13bÞ

Similarly, by perturbing q2(t) with Dq2(t), the second sensitivity problem can be obtained as

Li

L1

r2DT 00i ðr; tÞ ¼o

ot½DT 00i ðr; tÞ � DC00i ðr; tÞ�; i ¼ 1; 2; ð14aÞ

Di

L1

r2DC00i ðr; tÞ ¼o

ot½DC00i ðr; tÞ � kimiDT 00i ðr; tÞ�; i ¼ 1; 2; ð14bÞ

oDT 001ðr1; tÞor

¼ 0; t > 0; ð15aÞ

oDC001ðr1; tÞor

¼ 0; t > 0; ð15bÞ

oDT 001ðr2; tÞor

¼ H t1½DT 002ðr2; tÞ � DT 001ðr2; tÞ�; t > 0; ð15cÞ

oDC001ðr2; tÞor

¼ H m1½DC002ðr2; tÞ � DC001ðr2; tÞ�; t > 0; ð15dÞ


oDT 002ðr2; tÞor

¼ H t2½DT 002ðr2; tÞ � DT 001ðr2; tÞ�; t > 0; ð15eÞ

oDC002ðr2; tÞor

¼ Hm2½DC002ðr2; tÞ � DC001ðr2; tÞ�; t > 0; ð15fÞ

oDT 002ðr3; tÞor

¼ 0; t > 0; ð15gÞ

oDC002ðr3; tÞor

¼ Dq2ðtÞ; t > 0; ð15hÞ

DT 00i ðr; 0Þ ¼ 0; i ¼ 1; 2; ð16aÞDC00i ðr; 0Þ ¼ 0; i ¼ 1; 2. ð16bÞ

It is noted that the above sensitivity problems can also be solved by the same method as the direct problem ofEqs. (4)–(6).

The functional J ½qKþ11 ðtÞ; qKþ1

2 ðtÞ� for iteration K + 1 is obtained by rewriting Eq. (7) as

J ½qKþ11 ðtÞ; qKþ1

2 ðtÞ� ¼Z tf

0

T rm1; t; qK1 � bK

1 pK1 ; q

K2 � bK

2 pK2

� �� Y 1ðrm1; tÞ

� �2dt

þZ tf

0

C rm2; t; qK1 � bK

1 pK1 ; q

K2 � bK

2 pK2

� �� Y 2ðrm2; tÞ

� �2dt; ð17Þ

where we replace qKþ11 ðtÞ and qKþ1

2 ðtÞ by the expression given by Eqs. (8a) and (8b). If the estimated temper-ature T rm1; t; qK

1 � bK1 pK

1 ; qK2 � bK

2 pK2

� �and moisture C rm2; t; qK

1 � bK1 pK

1 ; qK2 � bK

2 pK2

� �are linearized by a Taylor

expansion, Eq. (17) takes the form

J qKþ11 ðtÞ; qKþ1

2 ðtÞ� �

¼Z tf

0

T rm1; t; qK1 ; q

K2

� �� bK

1 DT 0ðrm1; tÞ � bK2 DT 00ðrm1; tÞ � Y 1ðrm1; tÞ

� �2dt

þZ tf

0

C rm2; t; qK1 ; q

K2

� �� bK

1 DC0ðrm2; tÞ � bK2 DC00ðrm2; tÞ � Y 2ðrm2; tÞ

� �2dt; ð18Þ

where T ðrm1; t; qK1 ; q

K2 Þ and Cðrm2; t; qK

1 ; qK2 Þ are the solutions of the direct problem by using estimated

qK1 ðtÞ and qK

2 ðtÞ at time t. The sensitivity functions DT 0(rm1, t), DC 0(rm2, t) and DT00(rm1, t), DC00(rm2, t) are takenas the solutions of problems Eqs. (11)–(16) at time t by letting Dq1ðtÞ ¼ pK

1 ðtÞ in Eq. (12b) and Dq2ðtÞ ¼ pK2 ðtÞ in

Eq. (15d), respectively.Eq. (18) is differentiated with respect to bK

1 and bK2 , respectively, and equating them equal to zero to obtain

two independent equations. After solving these two equations, the search step sizes can be determined as

bK1 ¼ ðA3A5 � A2A4Þ=ðA3A3 � A1A2Þ; ð19aÞ

bK2 ¼ ðA3A4 � A1A5Þ=ðA3A3 � A1A2Þ; ð19bÞ

where

A1 ¼Z tf

0

½DT 0ðrm1; tÞ�2 þ ½DC0ðrm2; tÞ�2h i

dt; ð20aÞ

A2 ¼Z tf

0

½½DT 00ðrm1; tÞ�2 þ ½DC00ðrm2; tÞ�2�dt; ð20bÞ

A3 ¼Z tf

0

½DT 0ðrm1; tÞ � DT 00ðrm1; tÞ þ DC0ðrm2; tÞ � DC00ðrm2; tÞ�dt; ð20cÞ

A4 ¼Z tf

0

T rm1; t; qK1 ; q

K2

� �� Y 1ðrm1; tÞ

� �� DT 0ðrm1; tÞ

�þ C rm2; t; qK

1 ; qK2

� �� Y 2ðrm2; tÞ

� �� DC0ðrm2; tÞ

�dt; ð20dÞ

A5 ¼Z tf

0

T rm1; t; qK1 ; q

K2

� �� Y 1ðrm1; tÞ

� �� DT 00ðrm1; tÞ

�þ C rm2; t; qK

1 ; qK2

� �� Y 2ðrm2; tÞ

� �� DC00ðrm2; tÞ

�dt. ð20eÞ


6. Adjoint problem and gradient equation

To obtain the adjoint problem, Eqs. (4a) and (4b) are multiplied by the Lagrange multipliers (or adjointfunctions) /1i(r, t) and /2i(r, t), respectively, and the resulting expression is integrated over the time and cor-respondent space domains. Then the result is added to the right hand side of Eq. (7) to yield the followingexpression for the functional J[q1(t),q2(t)]:

J ½q1ðtÞ; q2ðtÞ� ¼Z r3

r1

Z tf

0

½T ðrm1; tÞ � Y 1ðrm1; tÞ�2dðr � rm1Þdt dr

þZ r3

r1

Z tf

0

½Cðrm2; tÞ � Y 2ðrm2; tÞ�2dðr � rm2Þdt dr

þX2

i¼1

Z riþ1

ri

Z tf

0

r � /1iðr; tÞ �Li

L1

r2T iðr; tÞ �o

ot½T iðr; tÞ � Ciðr; tÞ�

� �dt dr

þX2

i¼1

Z riþ1

ri

Z tf

0

r � /2iðr; tÞ �Di

L1

r2Ciðr; tÞ �o

ot½Ciðr; tÞ � kimiT iðr; tÞ�

� �dt dr. ð21Þ

The variation DJ1 is obtained by perturbing q1(t) by q1(t) + Dq1(t), T(r, t) by T + DT 0, and C(r, t) by C + DC 0

in Eq. (21). Subtracting from the resulting expression the original Eq. (21) and neglecting the second-orderterms, we thus find

DJ 1½q1ðtÞ; q2ðtÞ� ¼Z r3

r1

Z tf

0

2½T ðrm1; tÞ � Y 1ðrm1; tÞ�DT 0ðrm1; tÞdðr � rm1Þdt dr

þZ r3

r1

Z tf

0

2½Cðrm2; tÞ � Y 2ðrm2; tÞ�DC0ðrm2; tÞdðr � rm2Þdt dr

þX2

i¼1

Z riþ1

ri

Z tf

0

r � /1iðr; tÞ �Li

L1

r2DT 0iðr; tÞ �o

ot½DT 0iðr; tÞ � DC0iðr; tÞ�

� �dt dr

þX2

i¼1

Z riþ1

ri

Z tf

0

r � /2iðr; tÞ �Di

L1

r2DC0iðr; tÞ �o

ot½DC0iðr; tÞ � kimiDT 0iðr; tÞ�

� �dt dr. ð22Þ

We can integrate the third and fourth double integral terms in Eq. (22) by parts, utilizing the boundary andinitial conditions of the sensitivity problem. Then DJ1 is allowed to go to zero. The vanishing of the integrandsleads to the following adjoint problem for the determination of /1i(r, t) and /2i(r, t):

Li

L1

r2/1iðr; tÞ þo

ot½/1iðr; tÞ � kimi/2iðr; tÞ� þ 2½T ðrm1; tÞ � Y 1ðrm1; tÞ�

dðr � rm1Þrm1

¼ 0; i ¼ 1; 2; ð23aÞ

Di

L1

r2/2iðr; tÞ þo

ot½/2iðr; tÞ � /1iðr; tÞ� þ 2½Cðrm2; tÞ � Y 2ðrm2; tÞ�

dðr � rm2Þrm2

¼ 0; i ¼ 1; 2; ð23bÞ

o/11ðr1; tÞor

¼ 0; t > 0; ð24aÞ

o/21ðr1; tÞor

¼ 0; t > 0; ð24bÞ

L1

L1

o/11ðr2; tÞor

¼ L2

L1

H t2/12ðr2; tÞ �L1

L1

H t1/11ðr2; tÞ; t > 0; ð24cÞ

D1

L1

o/21ðr2; tÞor

¼ D2

L1

H m2/22ðr2; tÞ �D1

L1

H m1/21ðr2; tÞ; t > 0; ð24dÞ

L2

L1

o/12ðr2; tÞor

¼ L2

L1

H t2/12ðr2; tÞ �L1

L1

H t1/11ðr2; tÞ; t > 0; ð24eÞ

D2

L1

o/22ðr2; tÞor

¼ D2

L1

H m2/22ðr2; tÞ �D1

L1

H m1/21ðr2; tÞ; t > 0; ð24fÞ


o/12ðr3; tÞor

¼ 0; t > 0; ð24gÞ

o/22ðr3; tÞor

¼ 0; t > 0; ð24hÞ

/1iðr; tfÞ ¼ 0; i ¼ 1; 2; ð25aÞ/2iðr; tfÞ ¼ 0; i ¼ 1; 2. ð25bÞ

The adjoint problem is different from the standard initial problems in that the final time conditions at timet = tf are specified instead of the customary initial conditions. However this problem can be transformed toan initial value problem by the transformation of the time variables as s = tf � t. Then the adjoint problemcan be solved by the same method as the direct problem.

Finally the following integral term is left:

DJ 1 ¼Z tf

0

/11ðr1; tÞDq1ðtÞdt. ð26Þ

From the definition used in Ref. [1], we have

DJ 1 ¼Z tf

0

J 01Dq1ðtÞdt; ð27Þ

where J 01 is the gradient of the functional J1. A comparison of Eqs. (26) and (27) leads to the following form:

J 01½q1ðtÞ� ¼ /11ðr1; tÞ. ð28Þ

Similarly Eqs. (4a) and (4b) are multiplied by the Lagrange multiplier (or adjoint function) /3i(r, t) and /4i(r, t)to derive the adjoint problem for the case when perturbing Dq2(t). Following the same procedure as describedpreviously, eventually, we find that the solutions for adjoint equations of /3i(r, t) and /4i(r, t) are identical tothose for /1i(r, t) and /2i(r, t). This implies that the adjoint equations need to be solved only once since/1i(r, t) = /3i(r, t) and /2i(r, t) = /4i(r, t). Finally the gradient equation for q2(t) can be obtained as

J 02½q2ðtÞ� ¼D2

L1

� r3 � /22ðr3; tÞ. ð29Þ

7. Stopping criterion

If the problem contains no measurement errors, the convergence condition for the minimization of the cri-terion is

J ½q1ðtÞ; q2ðtÞ� < g; ð30Þ

where g is related to the accuracy of the direct problem solution. However the measured moisture data maycontain measurement errors. Therefore we do expect the functional Eq. (7) to be equal to zero at the final iter-ation step. Following the experience of the authors [1–5], we use the discrepancy principle as the stopping cri-terion, i.e., we assume that the residuals for moisture may be approximated by

T ðrm1; tÞ � Y 1ðrm1; tÞ ¼ Cðrm2; tÞ � Y 2ðrm2; tÞ � r; ð31Þ

where r is the stand deviation of the measurements, which is assumed to be a constant.Substituting Eq. (31) into Eq. (7), the following expression is obtained for g:

g ¼ r2tf . ð32Þ

Then, the stopping criterion is given by Eq. (30) with g determined from Eq. (32).


8. Computational procedure

The computational procedure for the solution of this inverse problem may be summarizes as follows:Suppose qK

1 ðtÞ and qK2 ðtÞ is available at iteration K.

Step 1. Solve the direct problem given by Eqs. (4a)–(6b) for T(rm1, t) and C(rm2, t), respectively.Step 2. Examine the stopping criterion given by Eq. (30) with g given by Eq. (32). Continue if not satisfied.Step 3. Solve the adjoint problem given by Eqs. (23)–(25) for /1i(r, t) and /2i(r, t), respectively.Step 4. Compute the gradient of the functional J 01½q1ðtÞ� and J 02½q2ðtÞ� from Eqs. (28) and (29), respectively.Step 5. Compute the conjugate coefficients cK

1 and cK1 and the direction of decent pK

1 and pK2 from Eqs. (10)

and (9), respectively.Step 6. Set Dq1ðtÞ ¼ pK

1 and Dq2ðtÞ ¼ pK2 , and solve the sensitivity problem given by Eqs. (11)–(16) for

DT 0(rm1, t), DC 0(rm2, t) and DT00(rm1, t), DC00(rm2, t).Step 7. Compute the search step size bK

1 and bK2 from Eq. (19).

Step 8. Compute the new estimation for qKþ11 ðtÞ and qKþ1

2 ðtÞ from Eq. (8) and return to Step 1.

9. Results and discussion

The objective of this article is to show the validity of the CGM in simultaneous estimating the time-depen-dent boundary heat and moisture fluxes q1(t) and q2(t) for heat and moisture transport problem with no priorinformation on the functional form of the unknown functions, which is the so-called function estimation. Toillustrate the accuracy of the CGM in simultaneous predicting the q1(t) and q2(t) with inverse heat and mois-ture transport analysis from the knowledge of measured transient temperature and moisture distributions, weconsider the exact value of the time-dependent heat and moisture fluxes q1(t) and q2(t) as

q1ðtÞ ¼ sin pt þ sin 3pt=3þ sin 5pt=5þ sin 7pt=7; ð33Þq2ðtÞ ¼ 0:3� sinð2ptÞ þ 0:25� sinð4ptÞ þ 3t � ð1:1� tÞ. ð34Þ

The material properties and radius of the two-layer cylinder structure are listed as follows [7]:

L1 = 5L2 = 7.78 · 10�6/3600 cm2 s�1, D1 = D2 = 7.78 · 10�7/3600 cm2 s�1,m1 = m2 = 0.5 cm3 g�1 K, k1 = k2 = 0.5 g cm�3 K�1,ht = 0.8000 W cm�2 K�1, hm = 800 cm s�1,k2 = 0.2k1, Dm2 = Dm1,r1 = 1.0, r2 = 1.6, r3 = 2.0.

The total measurement time is chosen as tf = 1.0. Besides, the space and time increments used in the numer-ical calculations are taken as Dr = 0.05 and Dt = 0.01 (i.e., 100 discreted time), respectively.

In order to compare the results with situations involving random measurement errors, a random noise isadded to the simulated exact moisture values to generate the measured moisture Y, that is

Y ¼ Y exact þ -r; ð35Þ

where Yexact is the solution of the direct problem with an exact q1(t) and q2(t), r is the standard deviation of themeasurement, and - is a random variable within �2.576 to 2.576 for a 99% confidence bounds.

The inverse analysis is first performed by assuming exact measurement (i.e. r = 0.0) and by using measure-ment data at measuring positions rm1 = 1.1 and rm2 = 1.9. By setting initial guess q0

1ðtÞ ¼ 0:3 and q02ðtÞ ¼ 0:3,

the exact and estimated q1(t) and q2(t) obtained at the 30th iteration are shown in Fig. 2; while the measuredand estimated temperature at rm1 = 1.1, and measured and estimated moisture at rm2 = 1.9 are shown inFig. 3. It can be found in Fig. 2 that the estimations for q1(t) and q2(t) are very accurate except near final time.Moreover, in Fig. 3 we learn that there is a good agreement between the measured and estimated temperatureand moisture.

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Time, t

Hea

t & M

oist

ure

Flux

es

0 0.2 0.4 0.6 0.8 10

0.3

0.6

0.9

1.2

q1 (exact)

q1 (estimated)

q2 (exact)

q2 (estimated)

+

Fig. 2. Estimated heat and moisture fluxes at 30th iteration with initial guesses q01ðtÞ ¼ 0:3 and q0

2ðtÞ ¼ 0:3, r = 0.0, and measured atrm1 = 1.1 and rm2 = 1.9.

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++++++++

++

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Time, t

Tem

pera

ture

& M

oist

ure

0 0.2 0.4 0.6 0.8 10

0.3

0.6

0.9

1.2

1.5

1.8

T at r = 1.1 (measured)T at r = 1.1 (estimated)C at r = 1.9 (measured)C at r = 1.9 (estimated)

+

Fig. 3. Measured and estimated temperature and moisture at 30th iteration with initial guesses q01ðtÞ ¼ 0:3 and q0

2ðtÞ ¼ 0:3, r = 0.0, andmeasured at rm1 = 1.1 and rm2 = 1.9.


The average errors for q1(t) and q2(t) are calculated as ERR1 = 3.29% and ERR2 = 2.41%, respectively,where the average errors for the estimated q1(t) and q2(t) are defined as

ERR1% ¼X100

j¼1

q1ðjÞ � ~q1ðjÞq1ðjÞ

��

" #� 100� 100%; ð36Þ

ERR2% ¼X100

j¼1

q2ðjÞ � ~q2ðjÞq2ðjÞ

��

" #� 100� 100%. ð37Þ


Here j represents the index of discreted time, q1(j) and q2(j) denote the exact values, and ~q1ðjÞ and ~q2ðjÞ denotethe estimated values of temperature and moisture fluxes.

Then the inverse analysis is performed again by using the same parameters as those in Figs. 2 and 3 exceptthe measure positions. The results of Figs. 4 and 5 are obtained using the measurement data measured atrm1 = 1.0 and rm2 = 2.0. The estimations for q1(t) and q2(t) are still very accurate and are shown in Fig. 4.Besides, the measured and estimated temperature and moisture are also in a good agreement and are shownin Fig. 5. In this case, the average errors for q1(t) and q2(t) are calculated as ERR1 = 1.22% and

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Time, t

Hea

t & M

ositu

re F

luxe

s

0 0.2 0.4 0.6 0.8 10

0.3

0.6

0.9

1.2

q1 (exact)

q1 (estimated)

q2 (exact)

q2 (estimated)

+



++

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++

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++++

++

++

++++++++

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+

+

+

Time, t

Tem

pera

ture

& M

oist

ure

0 0.2 0.4 0.6 0.8 10

0.3

0.6

0.9

1.2

1.5

1.8


+


2ðtÞ ¼ 0:3, r = 0.0, andmeasured at rm1 = 1.0 and rm2 = 2.0.


ERR2 = 1.56%, respectively. Next, in order to find the effect of the initial guess values q01ðtÞ and q0

2ðtÞ on theaccuracy of the estimations, we then take q0

1ðtÞ ¼ 0:001 and q02ðtÞ ¼ 0:001 with other parameters the same as

those in Figs. 2 and 3 and the results are plotted in Figs. 6 and 7, which are measured at rm1 = 1.1 andrm2 = 1.9. It indicates that the variation of the initial guess values has a small effect on the estimation, andthe average errors for q1(t) and q2(t) are calculated as ERR1 = 1.90% and ERR2 = 2.36%, respectively.

Finally we will discuss the influence of the measurement errors on the inverse solutions. First, the measure-ment error for the data measured at rm1 = 1.1 and rm2 = 1.9 is taken as r = 0.005; then, it is increased to

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Time, t

Hea

t & M

ositu

re F

luxe

s

0 0.2 0.4 0.6 0.8 10

0.3

0.6

0.9

1.2

q1 (exact)

q1 (estimated)

q2 (exact)

q2 (estimated)

+



+++

++

++

++

+++++

++

++

++

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++

++

++

++

++++

++

++

++++++++

+

+

+

+

Time, t

Tem

pera

ture

& M

oist

ure

0 0.2 0.4 0.6 0.8 10

0.3

0.6

0.9

1.2

1.5

1.8


+


2ðtÞ ¼ 0:001, r = 0.0,and measured at rm1 = 1.1 and rm2 = 1.9.


r = 0.01. The estimated q1(t) and q2(t) are shown in Figs. 8 and 9, respectively. The initial guess values areq0

1ðtÞ ¼ 0:3 and q02ðtÞ ¼ 0:3, and the iteration number is also 30 for both cases. For r = 0.005, the average

errors for q1(t) and q2(t) are calculated as ERR1 = 2.40% and ERR2 = 2.32%, respectively. On the otherhand, for r = 0.01, the average errors for q1(t) and q2(t) are calculated as ERR1 = 2.79% andERR2 = 3.53%, respectively. The results in Figs. 8 and 9 imply that reliable inverse solutions can still beobtained when measurement errors are considered.

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Time, t

Hea

t & M

ositu

re F

luxe

s

0 0.2 0.4 0.6 0.8 10

0.3

0.6

0.9

1.2

q1 (exact)

q1 (estimated)

q2 (exact)

q2 (estimated)

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Time, t

Hea

t & M

ositu

re F

luxe

s

0 0.2 0.4 0.6 0.8 10

0.3

0.6

0.9

1.2

q1 (exact)

q1 (estimated)

q2 (exact)

q2 (estimated)

+




10. Conclusions

The conjugate gradient method was successfully applied for the solution of the heat and moisture transportproblem to estimate the unknown time-dependent boundary heat and moisture fluxes q1(t) and q2(t) for a dou-ble-layer annular cylinder by utilizing simulated temperature and moisture readings obtained from differentmeasuring positions and different initial guess values. The contact resistance of heat and moisture transferat the interface of the two layers is considered in the analysis. Numerical results confirm that the method pro-posed herein can accurately estimate the heat and moisture fluxes q1(t) and q2(t) even involving the inevitablemeasurement errors. In addition, the CGM does not require prior information for the functional form of theunknown quantities to perform the inverse calculation and excellent estimated values can be obtained for theconsidered problem.

References

[1] O.M. Alifanov, Solution of an inverse problem of heat conduction by iteration methods, J. Eng. Phys. 26 (1974) 471–476.[2] Y.C. Yang, S.S. Chu, W.J. Chang, Thermally-induced optical effects in optical fibers by inverse methodology, J. Appl. Phys. 95 (2004)

5159–5165.[3] H.L. Lee, H.M. Chou, Y.C. Yang, The function estimation in predicting heat flux of pin fins with variable heat transfer coefficients,

Energ. Convers. Manage. 45 (2004) 1749–1758.[4] W.J. Chang, T.H. Fang, C.I. Weng, Inverse determination of the cutting force on nanoscale processing using atomic force microscopy,

Nanotechnology 15 (2004) 427–430.[5] W.J. Chang, J.C. Hsu, T.H. Lai, Inverse calculation of the tip–sample interaction force in atomic force microscopy by the conjugate

gradient method, J. Phys. D: Appl. Phys. 37 (2004) 1123–1126.[6] O.M. Alifanov, Inverse Heat Transfer Problem, Springer-Verlag, Berlin, 1994.[7] W.J. Chang, C.I. Weng, An analytical solution to coupled heat and moisture transfer in porous materials, Int. J. Heat Mass Transfer

43 (2000) 2621–2632.[8] Y.C. Yang, H.L. Lee, E.J. Wei, J.F. Lee, T.S. Wu, Numerical analysis of two-dimensional pin fins with non-constant base heat flux,

Energ. Convers. Manage. 46 (2005) 881–892.

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