Abstract

Inverse Convection Problem of Simultaneous Estimation of Two Boundary Heat Fluxes in Parallel Plate Channels

Marcelo José Colaço

Programa de Engenharia Mecânica – COPPE/UFRJ

colaco@newton.com.ufrj.br

Helcio Rangel Barreto Orlande

Programa de Engenharia Mecânica – COPPE/UFRJ

C.P.: 68503 21945-970 Rio de Janeiro, RJ. Brazil

helcio@serv.com.ufrj.br

Introduction

Heat conduction was the first heat transfer mode to be addressed in the inverse problem literature. This is probably because of the interest of researchers involved with the space program in the 50’s and 60’s, aiming at the accurate estimation of thermal properties of heat shields, as well as of the heat flux on the surface of re-entrant vehicles. Some heuristic methods of solution for inverse problems, which were based more on pure intuition than on mathematical formality, were developed in the 50’s. Later in the 60’s and 70’s, most of the methods, which are in common use nowadays, were formalized in terms of their capabilities to treat ill-posed unstable problems (Tikhonov, 1963.a,b, Tikhonov and Arsenin, 1977, Alifanov, 1974, 1977, 1994, Beck, 1962, Beck et al, 1977, 1985). The basis of such formal methods resides on the idea of reformulating the inverse problem in terms of an approximate well-posed problem, by utilizing some kind of regularization (stabilization) technique.

The interest on the solution of inverse heat convection problems is more recent than on the solution of inverse heat conduction problems. To the best of the authors’ knowledge, the first article dealing with an inverse heat convection problem is the one due to Moutsoglou (1989). Afterwards, several other articles involving inverse convection appeared in the literature (Moutsoglou, 1990; Huang and Özisik, 1992; Raghunath, 1993; Bokar and Özisik, 1995; Liu and Özisik, 1996.a,b; Li et al, 1995; Machado and Orlande, 1996, 1997, 1998; Szczygiel, 1997; Moaveni, 1997; Aparecido and Özisik, 1999; Colaço and Orlande, 2000; Huang and Chen, 2000; Zabaras and Nguyen, 1995; Zabaras and Yang, 1997; Yang and Zabaras, 1998; Prud’homme and Nguyen, 1997; Park and Chung, 1999.a,b, 2000.a,b, Li and Yan, 1999, Hsu et al, 2000, Su et al, 2000). In their majority, those papers dealt with forced convection inside tubes or channels, with unknown wall heat flux (Moutsoglou, 1990; Huang and Özisik, 1992; Machado and Orlande, 1996, 1997, 1998; Liu and Özisik, 1996.a; Szczygiel, 1997; Aparecido and Özisik, 1999; Huang and Chen, 2000, Li and Yan, 1999, Su et al, 2000) or unknown inlet condition (Raghunath, 1993; Bokar and Özisik, 1995; Liu and Özisik, 1996.b; Colaço and Orlande, 2000). However, inverse natural convection problems have also been addressed (Moutsoglou, 1989; Zabaras and Nguyen, 1995; Zabaras and Yang, 1997; Yang and Zabaras, 1998; Li et al, 1995; Prud’homme and Nguyen, 1997; Park and Chung, 1999.a,b, 2000.a,b). In all those works dealing with inverse heat convection, a single unknown function was considered in the analysis.

In this paper, we present the solution of the inverse problem of estimating simultaneously the boundary heat fluxes at the two walls of a parallel plate channel. The flow inside the channel is assumed to be laminar and hydrodynamically developed. As the solution technique, we apply the conjugate gradient method of function estimation with adjoint problem (Alifanov, 1974, 1994; Huang and Özisik, 1992; Bokar and Özisik, 1995; Liu and Özisik, 1996.a,b; Machado and Orlande, 1996, 1997, 1998; Colaço and Orlande, 2000; Huang and Chen, 2000; Zabaras and Nguyen, 1995; Zabaras and Yang, 1997; Yang and Zabaras, 1998; Prud’homme and Nguyen, 1997; Park and Chung, 1999.a,b, 2000.a,b; Özisik and Orlande, 2000). Simulated temperature measurements taken inside the channel are used in the inverse analysis, in order to address the accuracy of the present solution technique.

The basic steps of the conjugate gradient method of function estimation include (Özisik and Orlande, 2000): (i) Direct Problem, (ii) Inverse Problem, (iii) Sensitivity Problems, (iv) Adjoint Problem, (v) Gradient Equations, (vi) Iterative Procedure, (vii) Stopping Criterion and (viii) Computational Algorithm. Details of such steps are described next, as applied to inverse problems involving the estimation of: (i) Spatially dependent heat fluxes; (ii) Time-dependent heat fluxes; and (iii) Time and spatially dependent heat fluxes.

Direct Problem

The physical problem considered here involves the laminar hydrodynamically–developed forced convection of an incompressible Newtonian fluid in a parallel plate channel. The fluid physical properties are assumed constant. The energy source term resulting from viscous dissipation and buoyancy effects are supposed negligible. The fluid is initially at the temperature T₀, which is also considered to be the constant fluid inlet temperature. The channel walls, separated by a distance h, are subjected to different heat fluxes, which may vary in time and along the channel. The mathematical formulation of such physical problem is given by:

Nomenclature

c_p = specific heat, J/kg. ºC

d = direction of descent

G = objective functional

h = height of the channel, m

k = thermal conductivity, W/m.K

q₀ = constant with units of heat flux appearing in Eq.(4), W/m2

q₁, q₂ = applied heat fluxes at the boundaries y = 0 and y = h, respectively,W/m2

S = number of sensors

t = time, s

T = estimated temperature, ºC

u = velocity component on longitudinal direction, m/s

w = dummy variable that represents x, t and (x,t) for Problems I, II and III, respectively

x, y = cartesian coordinates

Greek Symbols

b = search-step size

D = variation

DT = sensitivity function

e_RMS = RMS error

g = conjugation coefficient

l = Lagrange multiplier

m = measured temperatures, ºC

r = density, kg/m3

s = standard deviation of the measurement errors

Subscripts

est, ex estimated and exact heat fluxes

s sensor location

1, 2 refer to the boundaries at y = 0 and y = h

Subscripts

k iteration number

The laminar hydrodynamically developed velocity profile is given by:

where u is the mean velocity of the fluid.

The direct problem is concerned with the determination of the temperature distribution inside the channel, from the knowledge of the velocity profile (2), initial and inlet temperature T₀, thermophysical properties and boundary heat fluxes q₁(x,t) and q₂(x,t).

Inverse Problem

The inverse problem under picture in this paper is concerned with the simultaneous estimation of the boundary heat fluxes q₁(x,t) and q₂(x,t), by using temperature measurements taken inside the channel. The inverse problem is reformulated as a minimization problem involving the following objective functional:

where t_f denotes the final time, S is the number of sensors used in the analysis, while m_s(t) and T[x_s , y_s , t ; q₁, q₂] are the measured and estimated temperatures, respectively, at the measurement positions (x_s , y_s), for s = 1, ,S. The estimated temperatures are obtained from the solution of the direct problem, by using estimates for the boundary heat fluxes q₁(x,t) and q₂(x,t).

For the cases examined in this paper, the unknown boundary heat fluxes are supposed to vary in the following form:

where q₀ is a constant with units of heat flux, while f_xj(x) and f_tj(t) are dimensionless functions of the longitudinal position and time, respectively.

Three different types of inverse problems addressed in this paper, depending on the functional dependence of the unknown boundary heat fluxes, include:

Spatially dependent heat fluxes: In this case, the a priori information about the boundary heat fluxes being constant in time is available, so that f_tj(t)=1, for j=1,2. Hence, q_j(x) º q₀f_xj(x), for j=1,2, become the unknown functions and the inverse problem is concerned with the estimation of the spacewise variations of the boundary heat fluxes.

Time-dependent heat fluxes: In this case, the a priori information about the boundary heat fluxes being uniform along the channel is available, so that f_xj(x)=1, for j=1,2. Hence, q_j(t) º q₀f_tj(t), for j=1,2, become the unknown functions and the inverse problem is concerned with the estimation of the timewise variations of the boundary heat fluxes.

Time and spatially dependent heat fluxes: In this case, q_j(x,t) º q₀ f_xj(x) f_tj(t), for j=1,2, become the unknown functions and the inverse problem is concerned with the estimation of both the spacewise and timewise variations of the boundary heat fluxes.

These three types of inverse problems are hereafter designated as Problems I, II and III, respectively. A function estimation approach is used here for the identification of the boundary heat fluxes, for each of these inverse problems. In such an approach, no information regarding the functional form of the unknown function is assumed available for the inverse analysis, except for the functional space that the function belongs to. The unknowns are usually assumed to belong to the Hilbert space of square integrable functions (Alifanov, 1994, Özisik and Orlande, 2000).

A function f(w) in the space of square integrable real valued functions in a domain W , L₂(W ), satisfy the following property:

The inner product of two functions f(w) and g(w) in such functional space is given by:

and the norm of a function f(w) that belongs to L₂(W)is obtained from the inner product as

We note that the domain W appearing in Eqs.(5.a-c) may represent the spatial domain (0,x_f), the time domain (0,t_f), or the combined time and spatial domain (0,t_f)x(0,x_f), depending on which type of inverse problem is under picture, that is, Problems I, II or III, respectively (t_fand x_f denote the final time and total length of the channel, respectively). Similarly, the independent variable w may represent x, t, or (x,t), for Problems I, II or III, respectively.

The minimization of the objective functional (3) is obtained through the conjugate gradient method (Alifanov, 1974, 1994; Huang and Özisik, 1992; Bokar and Özisik, 1995; Liu and Özisik, 1996.a,b; Machado and Orlande, 1996, 1997, 1998; Colaço and Orlande, 2000; Huang and Chen, 2000; Zabaras and Nguyen, 1995; Zabaras and Yang, 1997; Yang and Zabaras, 1998; Prud’homme and Nguyen, 1997; Park and Chung, 1999.a,b, 2000.a,b; Özisik and Orlande, 2000). Auxiliary problems, known as the sensitivity and adjoint problems, are required for the implementation of the iterative procedure of the conjugate gradient method. The derivations of these problems can be found below.

Sensitivity Problems

The sensitivity problem is used to determine the temperature variation due to changes in the unknown quantity. Since the present work deals with the estimation of two unknown functions, two sensitivity problems are required in the analysis. They are derived by considering perturbations in the boundary heat fluxes each at a time, as described next.

Let us consider that the temperature T(x,y,t) undergoes a variation eDT₁(x,y,t), when the boundary heat flux q₁(w) is perturbed by eDq₁(w), where e is a small real number. In order to derive the sensitivity problem for DT₁(x,y,t), we write the direct problem in operator form and apply the following limiting process (Alifanov, 1994; Özisik and Orlande, 2000):

where L_e(q_1e) and L(q₁) are the operator forms of the direct problem written for the perturbed [q₁(w) + eDq₁(w)] and unperturbed q₁(w) heat fluxes at the boundary y = 0, respectively. A similar procedure is used for the derivation of the sensitivity problem for the function DT₂(x,y,t), resultant from the perturbation of the heat flux q₂(w) by eDq₂(w), at the boundary y = h. We then obtain the sensitivity problems for the determination of the functions DT_j(x,y,t), for j = 1, 2, respectively as:

where w may represent x, t, or (x,t), for Problems I, II or III, respectively, and

Adjoint Problem

The adjoint problem is derived by multiplying Eq.(1.a) by the Lagrange Multiplier l (x,y,t) and integrating over the time and space domains. The resulting expression is then added to the functional given by Eq.(3) in order to obtain:

where d (× ) is the Dirac delta function and r_s is the vector with the sensor position (x_s , y_s).

We now perturb q₁(w) by eDq₁(w) and T(x,y,t) by eD T₁(x,y,t) in Eq.(9) and apply the following limiting process to obtain the directional derivative of the functional G[q₁(w), q₂(w)] in the direction of the perturbation Dq₁(w) (Alifanov, 1994, Özisik and Orlande, 2000):

where G_e(q_1e) and G(q₁) denote the functional (9) written for the perturbed [q₁(w) + eDq₁(w)] and unperturbed q₁(w) heat fluxes at the boundary y = 0, respectively. The following expression results:

By employing integration by parts in the second integral term appearing on the right-hand side of Eq.(11), utilizing the initial and boundary conditions of the sensitivity problem for DT₁(x,y,t) and also requiring that the coefficients of DT₁(x,y,t) in the resulting equation vanish, the following adjoint problem is obtained:

We note that the adjoint problem involves outlet and final conditions given by Eqs.(12.b,e), instead of the usual inlet and initial conditions, as well as negative transient and convective terms in the governing Eq.(12.a). However, it can be transformed into a regular forced convection problem by utilizing the following suitable transformations of the independent variables:

A limiting process analogous to Eq.(10) is used in order to obtain the directional derivative of the functional G[q₁(w), q₂(w)] in the direction of the perturbation Dq₂(w) (Alifanov, 1994, Özisik and Orlande, 2000). After performing similar manipulations, we obtain the adjoint problem resulting from the perturbation in q₂(w), which is identical to that given by Eqs.(12.a-e) resulting from the perturbation in q₁(w). Therefore, one single adjoint problem needs to be solved at each iteration of the conjugate gradient method, despite the fact that two unknown functions are to be estimated.

Gradient Equations

In the process of obtaining the adjoint problem resulting from the perturbation in q₁(w), the directional derivative of the functional in the direction Dq₁(w) reduces to

We now invoke the hypothesis that the unknown functions belong to the space of square integrable functions in the domain W of interest, for each of the three types of inverse problems considered here. Let us consider Problem I, where w º x and W º (0,x_f). In this case, the directional derivative of G[q₁(x),q₂(x)] in the direction of the perturbation Dq₁(x) is given by (Alifanov, 1994, Özisik and Orlande, 2000):

Therefore, by comparing Eqs.(14) and (15.a), we obtain the gradient equation for Problem I, resulting from perturbations in q₁(x) as

For Problem II, where w º t and W º (0,t_f), the directional derivative of G[q₁(t),q₂(t)] in the direction of the perturbation Dq₁(t) is given by (Alifanov, 1994, Özisik and Orlande, 2000):

Hence, by comparing Eqs.(14) and (16.a), we obtain the gradient equation for the estimation of q₁(t) for Problem II as:

We now consider Problem III, where both the time and spatial variations of the heat flux are unknown, that is, w º (x,t) and W º (0,x_f)x(0,t_f). For such case, the directional derivative of G[q₁(x,t),q₂(x,t)] in the direction of the perturbation Dq₁(x,t) is given by (Alifanov, 1994, Özisik and Orlande, 2000):

Therefore, from the comparison of Eqs.(14) and (17.a), we obtain the following gradient equation for the estimation of q₁(x,t):

Analogous procedures are used in order to obtain the gradient equations of the functional for the estimation of the function q₂(w). In such cases, we obtain the gradient equations for Problems I, II and III, respectively as:

Iterative Procedure

The iterative procedure of the conjugate gradient method, as applied to the simultaneous estimation of q_j(w), for j=1,2, is given by (Alifanov, 1994, Özisik and Orlande, 2000):

where k is the number of iterations. The directions of descent d_j^k(w) are obtained from

The conjugation coefficients g_j^k can be obtained from the Fletcher-Reeves expression as:

where the L₂ norm in the domain W is defined by Eq.(5.c).

Expressions for the search step sizes b_j^k, for j=1,2, are obtained by minimizing G[q₁^k+1(w), q₂^k+1(w)] with respect to b₁^k and b₂^k, that is,

By linearizing the estimated temperatures T(x_S,y_S,t;q₁^k-b₁^kd₁^k,q₂^k-b₂^k d₂^k) with respect to b₁^k and b₂^k, performing the minimization and solving the resulting linear system of equations, we obtain the following expressions for b₁^k and b₂^k:

where:

In Eqs.(22.a-e), D T₁(r_s , t;d₁^k) and D T₂(r_s , t;d₂^k) are the solutions of the sensitivity problems given by Eqs.(7.a-e) for j = 1, 2, respectively, obtained by setting Dq_j(w)=d_j^k(w).

Stopping Criterion

The iterative procedure of the conjugate gradient method is not capable of providing by itself regularized solutions for inverse problems, as a result of their ill-posed character. In fact, it is generally observed that the random errors present on the measured variables are amplified for the solution of the inverse problem, as estimated temperatures approach the measured ones during the minimization of the functional (3). However, the use of the conjugate gradient method may result on stable solutions if the Discrepancy Principle (Alifanov, 1974, 1977, 1994) is used to specify the tolerance for the stopping criterion of the iterative procedure. In the Discrepancy Principle, the solution is assumed to be sufficiently accurate when the difference between measured and estimated temperatures is of the order of magnitude of the measurement errors, that is,

where s is the standard deviation of the measurement errors, which is assumed constant in the present analysis.

The stopping criterion used here is given by

where G[q₁(w),q₂(w)] is computed with Eq.(3). The tolerance e based on the Discrepancy Principle is then obtained by substituting Eq.(23) into Eq.(3). It results:

For cases involving errorless measurements, stable solutions for the inverse problem can be obtained by specifying the tolerance e as a sufficiently small number, since no perturbation is present in the input (measured) data. However, such is the case only if the sensors are appropriately located in regions where the measurements are sensitive to variations in the sought function.

The iterative procedure presented above, together with the solutions of the direct, sensitivity and adjoint problems, can be suitably arranged in a computational algorithm. Such algorithm is not presented here for the sake of brevity, but it can be found in Özisik and Orlande (2000).

Results and Discussions

For the results presented below, we considered the laminar hydrodynamically developed flow of water with Re = 100, in a channel of height 0.05 m and length 1.4 m. The inlet temperature, which is equal to the initial temperature, is taken as 25ºC. Direct, sensitivity and adjoint problems were numerically solved with finite-volumes, by using a discretization with 100 volumes in each direction and the WUDS interpolation function (Maliska, 1995). The final time was taken as 600 s and a time-step of 1 s was used for the finite-volume solution. Such time-step and number of volumes used for the discretization were chosen by using a grid convergence analysis and by comparing the numerical solution of the direct problem with the benchmark solution presented by Cotta and Özisik (1986). The agreement between our numerical solution for the Nusselt number and that by Cotta and Özisik (1986) was generally better than 2.2 % in the thermally developing region and better than 0.05% in the thermally developed region.

After validating the numerical code developed for the solution of the direct problem, we now proceed to the analysis of the accuracy of the conjugate gradient method of function estimation, as applied to each of the three types of inverse problems addressed in this paper. For the inverse analysis, we used simulated measurements containing random errors, normally distributed, with zero mean and constant standard-deviation (s). Such simulated measurements were obtained by adding a random noise to the solution of the direct problem for a priori established functional forms for the boundary heat fluxes. The solution of the direct problem was computed by considering q₀ = 1000 W/m² and by using the following functional forms containing discontinuities and sharp corners for f_xj(x) and f_tj(t), respectively:

An analysis of Eqs.(12.b,e), (17.b) and (18.c) reveals that the gradient equations for Problem III are null at the final time and at the channel outlet. Therefore, the initial guesses used for the conjugate gradient method remain invariant as the iterative procedure advances, thus generating instabilities on the inverse problem solution in the neighborhood of these points (Özisik and Orlande, 2000). Similar difficulties are encountered in the estimation of the unknown functions at the channel outlet for Problem I and at the final time for Problem II. In order to overcome these difficulties, sensors were not located at the channel outlet and a final time larger than that of interest was used for the inverse analysis. The sensors were uniformly distributed along the channel and near each of the walls, but the first and the last control-volumes were avoided for their positioning. Also, we used measurements taken up to 700 s, but only the results in the time domain 0 < t < 600 s are presented below. Such results were obtained by using a null heat flux as the initial guess for both q₁(w) and q₂(w).

Different test-cases were examined here, depending on the type of inverse problem under picture, number and location of sensors, as well as the level of random errors of the measurements. Such test-cases are summarized in Table 1. Two numbers of sensors, located near each of the channel walls, were tested: 21 and 34. For test-case 2, the sensors were distanced from each of the walls by 5.125 mm; but for all the other test-cases they were distanced by 2.625 mm. Different levels of measurement errors analyzed include: s = 0 (errorless measurements) and s = 0.01 T_max, where T_max is the maximum measured temperature. For the test-cases examined, the maximum measured temperature was never less than 41ºC, which gives a standard-deviation of 0.4ºC. However, we note that the measurement errors are normally distributed, so that the maximum error is at least 1ºC at the 99% confidence-level. For test-cases 1 through 4 we dealt with the solution of Problem I, while the solution of Problem II was analyzed in test-cases 5 and 6, and of Problem III in test-cases 7 and 8.

We also present in Table 1 the RMS errors for the estimated functions q_j(w), for j = 1,2. The RMS errors were computed as

where M and I are the number of control-volumes in the x direction and the number of transient measurements taken per sensor, respectively. The subscript ex and est refer to exact and estimated quantities, respectively.

For the inverse analysis, one measurement was assumed available per sensor at each time step. Also, since the numbers of sensors were smaller than the number of control-volumes used in the discretization, an interpolation procedure was used for the measured temperatures along the x direction in Problems I and III, which involved the estimation of the spatial variation of the boundary heat fluxes. NETLIB’s subroutine GCVSPL, based on the cross-validation smoothing procedure, was used for the interpolation.

An examination of Table1 reveals that the e_RMS errors for the estimation of q₂(w) are generally smaller than those for q₁(w). Such is the case because the functions analyzed for q₁(w) are discontinuous, while those for q₂(w) are continuous. A comparison of the RMS errors for test-cases 1 and 2 shows that the results are very little affected by increasing the distance of the sensors from the channel walls from 2.625 mm to 5.125 mm. This is because the sensors are located inside the thermal boundary layer for both cases. On the other hand, we notice from the comparison of test-cases 1 and 3 that the accuracies of the estimated functions are enhanced by using 34 sensors instead of 21. Hence, we preferred to use for the subsequent test-cases 34 sensors located at a distance of 2.625 mm from each of the boundaries, despite the fact that one single sensor is required for the estimation of time-dependent functions, such as in test-cases 5 and 6 involving Problem II. Table 1 shows that, for Problems I and II, the accuracies of the estimated functions deteriorate as measurements with random errors are used in the analysis, instead of errorless measurements. On the other hand, for Problem III the values of e_RMS are very little affected by the presence of random errors in the measurements, for the levels of error considered (see test-cases 7 and 8). This is because the time and spatially dependent functions of Problem III are much more difficult to estimate than functions of one single independent variable, such as for Problems I and II.

Figures 1-3 show the results obtained for test-cases 4, 6 and 8, respectively. These figures show that the present function estimation approach is capable of recovering spatially dependent, time-dependent, and both time and spatially dependent boundary heat fluxes quite accurately, even for the strict cases under picture here. We note that smoother functions were estimated for the spatial variation than for the time variation of the heat fluxes, as shown by Figs.1 and ² . This is a result of the interpolation procedure utilized for the estimation of spatially dependent functions, which makes use of smoothing techniques, as discussed above.

Conclusions

In this paper we applied the conjugate gradient method for the simultaneous identification of two boundary heat fluxes in a parallel plate channel. A function estimation approach was utilized, where no information was assumed available regarding the functional form of the unknown. Three different types of inverse problems were examined here, involving the estimation of spatially dependent, time-dependent and both time and spatially dependent heat fluxes. Results obtained with simulated temperature measurements reveal that quite accurate estimates can be obtained for the unknown functions with the present inverse problem approach. The sensors need to be located near each of the walls, inside the thermal boundary layer, in order to be sensitive to variations on the wall heat flux.

Acknowledgements

The CPU time for this work was provided by NACAD-UFRJ (Cray-J90) and CESUP-UFRGS (Cray-T94). The support provided by Renault-COPPE Plateau de Recherche for the implantation of the computational facilities of the Heat Engines Laboratory-PEM/COPPE is greatly appreciated. This work was partially funded by the CNPq grant 522.469/94-9 and by the FAPERJ grant E-26/170.344/2000.

Manuscript received: August 2000. Technical Editor: Átila P. S. Freire.

Publication Dates

History