1차원 열전도의 역문제

김민찬*, 김 신**, 이윤준**

*제주대학교 화학공학과, **제주대학교 에너지공학과

Inverse Heat Conduction Problem in One-Dimensional Medium

Min Chan Kim*, Sin Kim** & Yoon-Joon Lee**

*Department of Chemical Engineering, **Department of Nuclear and Energy Engineering, Cheju National University

Introduction

The estimation of the thermal conductivity is an interesting subject in the engineering application as well as in the inverse problem. For inhomogeneous medium, the thermal conductivity varies with position and an accurate estimation of the spatially varying thermal conductivity is necessary in many thermal management system. Also, this kind of problem may be encountered in geological waste disposal and also have applications in petroleum engineering.

Many engineering and mathematics researchers have considered the inverse heat conduction problem (IHCP). Some researches focused on the temperature-dependent thermal properties [1,2] and others dealt with the space-dependent ones [3,4]. They used the data measured at interior medium in their inverse algorithms. In this study, the space-dependent thermal conductivity in one-dimensional time-dependent heat conduction medium is obtained based on sensitivity equation approach. Especially, the algorithm in this work does not require any interior measurements. In practical application such as process tomographic technique, the boundary measurements are more common and practical than the interior measurements.

In present study, modified Newton-Raphson method is introduced to obtain the inverse solution. Many numerical experiments are performed for various spatially varying thermal conductivities to compare the estimated ones with the exact ones. In the comparison, the proposed inverse algorithm shows reasonably good predictability.

Mathematical Model

In order to estimate the spatially varying thermal conductivity in one-dimensional domain [eqn: k(x)], the following linear form of thermal conductivity is assumed:

[eqn: k(x) ~=~ sum from n=1 to N k_n ` phi_n (x) ~,]

(1)

where [eqn: phi_n (x)] can be any first order derivative continuous function in the problem domain and [eqn: k_n] are the unknown coefficients. [eqn: N] is an integer. Thus, the present inverse problem is converted into the problem to identify the coefficient [eqn: k_n~].

Consider an infinite slab with thickness [eqn: bar L] and suppose that this slab is a time-dependent heat conduction medium. Initially, the temperature distribution in the slab is uniform and [eqn: bar T_0 ``]. At a specific time, [eqn: bar t ``=`` 0], a heat flux [eqn: bar q_1] is applied to the front surface at [eqn: bar x =0] and another heat flux [eqn: bar q_2] is applied to the back surface at [eqn: bar x = bar L ``]. The heat conduction equation can be non-dimensionalized with [eqn: bar L ``], [eqn: bar T_0 ``], reference thermal conductivity [eqn: bar k_r ``], and heat capacity per unit volume [eqn: bar rho ` bar c_p ``]. The dimensionless governing equation becomes

[eqn: partial over partialx left( k partialT over partialx right) ~=~ partialT over partialt ~,~~~ x ~in~ (0,1) ~,~ t``>``0]

(2)

with an initial condition

[eqn: T(x `, 0 ) ~=~ 1 ~,~~~ x ~in~ [0,1] ~,~ t ``=``0]

(3)

and boundary conditions

[eqn: -k partialT over partialx &=~ q_1 ~,~~~ x=0 ~,~ t``>``0 ~, ## k partialT over partialx &=~ q_2 ~,~~~ x=1 ~,~ t``>``0 ~.]

(4)

In order to decide the search step in each iteration, the sensitivity equations are introduced6). The sensitivity equations are defined as the derivative [eqn: partial / partial k_n] of Eqs. (2)∼(4) and it can be expressed as

[eqn: partial over partialx left( k partialX_n over partialx right) + partial over partialx left[ phi_n (x) partialT over partialx right] = partialX_n over partialt ~,~~~ x ~in~ (0,1) ~,~ t``>``0]

(6)

with an initial condition

[eqn: X_n (x, 0 ) ~=~ 0 ~,~~~ x ~in~ [0,1] ~,~ t ``=``0]

(7)

and boundary conditions

[eqn: - k partialX_n over partialx &=~ - { phi_n (0) } over k q_1 ~,~~~ x=0 ~,~t``>``0 ~,## k partialX_n over partialx &=~ - { phi_n (1) } over k q_2 ~,~~~ x=1 ~,~t``>``0]

(8)

where the sensitivity [eqn: X_n ``=`` partialT / partialk_n ``]. These sensitivities can be obtained if [eqn: k(x)] and [eqn: T] are given.

The forward and sensitivity equations are spatially discretized using the finite element method and the temporal discretization is achieved by Crank-Nicholson representation.

The search step is determined by the following procedure. At first, the discrepancy function is defined as

[eqn: T_D (l,m) ~=~ T_cal (l,m) ``-`` T_meas (l,m)]

(9)

where [eqn: T_cal] and [eqn: T_meas] are calculated and measured temperature at spatial coordinate [eqn: l] ([eqn: l = 1 ldots L]) and temporal coordinate [eqn: m] ([eqn: m = 1 ldots M `]). The discrepancy vector is defined as a column vector of dimension [eqn: LM``] whose components are the discrepancies between the calculated and the measured temperatures at both end points (that is [eqn: L = 2]),

[eqn: \{ T_D \} ~=~ \{ T_cal \} ``-`` \{ T_meas \}]

and the column vector [eqn: \{ k \} = \{ k_1 , k_2 , cdots , k_N \}^T] should satisfy

[eqn: \{ T_D \} ~=~ \{ T_cal \} ``-`` \{ T_meas \} ~=~ 0 ~.]

(10)

This will be expressed as a least square problem to find the coefficient vector [eqn: \{ k \}] minimizing the objective function

[eqn: f( k ) ~=~ 1 over 2 \{ T_D \}^T \{ T_D \} ~.]

(11)

Many researches constructed the object function by using the data measured at interior medium in their inverse algorithms7-8. However, in practical application, it is better to use the data which measured at the boundaries.

The modified Newton-Raphson method is implemented to solve the above nonlinear least square problem and the subroutine DUMIDH of the IMSL is used. The Jacobian of the objective function is calculated from the sensitivity solution:

[eqn: partialf over partialk_n ~=~ \{ T_D \}^T LEFT { { partialT_cal` (l,m) } over partialk_n RIGHT } ~=~ \{ T_D \}^T LEFT { X_n (l,m) right }]

(12)

In this, [eqn: \{ X_n (l,m) \}] is a column vector whose components are the sensitivity value at spatial coordinate [eqn: l] and temporal coordinate [eqn: m], and these are obtained from the sensitivity equations, Eqs. (6)-(8).

Examples

For the evaluation of the proposed algorithm, three examples are introduced. The model function to approximate the thermal conductivity is assumed to be a fourth-order polynomial ([eqn: N=5]).

The boundary heat fluxes are taken as [eqn: q_1 = 10] and [eqn: q_2 = 10], that is, at both sides the heat is added to the medium. The initial values of thermal conductivity coefficients are set to unity, [eqn: k_n = 1].

The followings are the description of the illustrative examples:

Example 1. [eqn: k(x) ~=~ 1+2x+3x^2]

Example 2. [eqn: k(x) ~=~ 1 + sqrt x]

Example 3. [eqn: k(x) ~=~ 1 + 1 over 2x+1]

For all examples, the spatial domain is equally discretized into 10 quadratic elements and the calculation is terminated at [eqn: t=1]. The measurement is assumed to be made 20 times (i.e. [eqn: M = 20]) only at the end points (i.e. [eqn: L=2]).

The predicted and real thermal conductivities for Examples 1 and 2 are compared  in Figs. 1 and 2, respectively. In all examples, the fourth-order polynomial can reproduce the exact thermal conductivity distribution satisfactorily. As shown in these figures, the positions at which maximum local errors occur are dependent on the real conductivity profiles.

Conclusion and Further Studies

A numerical algorithm is presented to estimate the space-dependent thermal conductivity in one-dimensional time-dependent heat conduction medium without interior measurements. The algorithm is composed of two phases, namely the forward problem phase and the inverse problem phase, and they are iteratively applied. In each phase, the heat conduction equation and the sensitivity equations are solved, respectively. The forward and inverse solutions are used to find the search step by modified Newton-Raphson method. Several examples with thermal conductivities are introduced to demonstrate the usefulness of the proposed algorithm and the estimated thermal conductivities are successfully compared with the exact ones. The proposed method may be applicable to the estimation of the spatial composition in heat conduction medium.

Acknowledgement

This work was supported by the Korea Institute of Science and Technology Evaluation and Planning (KISTEP).

References

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2. Yang, C.-Y.: Appl. Math. Modelling, (23), 469 (1999).

3. Flach, G. P. and □zisik, M. N.: Num. Heat Transfer, Part A, 16, 249 (1989).

4. Huang, C.-H. and □zisik, M. N.: Int. J. Hat Fluid Flow, 11, 262 (1992).