The conjugate gradient method is used to minimize an objective function which is defined by the sum of square errors between estimated and desired heat fluxes over the design surface

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The 6^th. International Chemical Engineering Congress & Exhibition IChEC 2009 16 – 20 November 2009, Kish Island, Iran

An Inverse Boundary Design Problem in a Radiant Enclosure A. Karimipour^*1,S.M.H. Sarvari², Ali Karimipour³, E. Abedini⁴

1Ph.D. candidate, Islamic Azad Uni., Najaf Abad Branch, Email: [email protected]

2Associate professor, University of Sistan & Baluchestan, Email: [email protected]

3B.S., Sima Esfahan Company, Email: [email protected]

4Ph.D student, University of Sistan & Baluchestan, Email: [email protected] Abstract

This work presents an inverse radiative design problem in which the aim is to determine the distribution of heater strengths over the heater surface to produces a desired temperature and heat flux distributions over the design surface of a radiant oven. The walls are assumed to be diffuse-gray and the medium is considered to be transparent. The conjugate gradient method is used to minimize an objective function which is defined by the sum of square errors between estimated and desired heat fluxes over the design surface. The inverse estimation consists of three problems; the direct problem, the inverse problem and the sensitivity problem. The direct problem and the sensitivity problem are solved by the net radiation method, and the configuration factors are calculated by the Hottel’s crossed-string method. Our investigation shows that the presented algorithm is able to estimate heater strengths accurately.

Keywords: Radiation, Inverse boundary design, Conjugate Gradient Method (CGM).

Introduction

Radiation is the important mode of heat transfer in high temperature devices such as combustion chambers and furnaces. Inverse radiation design problems are concerned with the determination of unknown heater settings over the heater surface or unknown heat source distribution in the medium to produce both desired heat flux and temperature distributions over some part of the radiant enclosure, namely the design surface. The desired conditions over the design surface are dependent on the process for which the thermal system is built. The design conditions can be reached by controlling the thermal conditions in other parts of the system, namely the heater surface. Therefore, the ultimate goal in thermal design problems is to find the required powers of heaters over the heater surface to satisfy both specified boundary conditions over the design surface. The forward design procedure is based on the knowledge of one and only one thermal condition on each element of the system. Therefore, the forward design process involves a trial-and-error procedure, which requires a great deal of effort and time. Using the inverse design methodology is in fact an approach to avoid the trial and error procedure to meet the design goal. A comprehensive review of radiative heat transfer in combustion systems has been given by Viskanta and Menguc [1]. Inverse analysis of radiative transfer is concerned with the determination of the radiative properties, boundary conditions and the temperature field or source term distribution from different kinds of radiation measurements. A thorough review of inverse radiation problems has been given by McCormick [2]. A lot of work has been reported on the estimation of radiative properties. Many researchers have also dealt with the inverse problems for determining the temperature profile or source term in media. Li and Ozisik [3], and Liu [4] have reconstructed the temperature profiles or source terms in plane-parallel, spherical, and cylindrical media by the inverse analysis from the data of the radiation intensities exiting the boundaries. Sarvari et al. [5,6,7]

reported a methodology for designing radiant enclosures containing absorbing-emitting media to find the appropriate heater settings. Sarvari and Mansouri [8] used the optimization procedure to determine the heat source distribution in participating media. This procedure was extended by Sarvari [9] to determine the heat source distribution in conductive- radiative media.

In this paper an inverse radiation boundary design problem for an enclosure filled with transparent medium is investigated numerically. The configuration factors are calculated by the Hottel’s crossed-string method and the CGM is used to minimize an objective function which is defined by the sum of square errors between estimated and desired heat fluxes over the design surface. The direct problem and the sensitivity problem are solved by the net radiation method.

Problem Description

The geometry and the coordinate system of the radiant oven with diffuse gray walls and two insulated side walls, is illustrated in Fig. 1. The product surface (design surface) is placed on the bottom wall, whereas the heaters are placed

*corresponding author

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on the heater surface at the top of the oven. The medium is transparent and all the boundary conditions over the oven’s walls are specified except for the heater surface. In order to meet the design goal, both the temperature and the heat flux require having uniform distributions over the product. The goal of the design problem is to identify the heater flux,q_h, that produce the desired uniform heat flux, q_d, over the temperature specified design surface with uniform temperature of T_d (or with uniform emissive power, E_d =εσT_d⁴). Inverse radiative design method is applied to determine the distribution of temperature and heat flux sources on the heater surface, according to the distribution of desired temperature and heat flux on the design surface.

Fig. 1. The Schematic shape of the radiant oven with gray-diffuse walls and transparent medium

In first part of this paper the values of E_d =1.0 ( /w m²) and q_d has the profile is shown in Fig. 2 , so they are known parameters and the unknowns areE_h =?, q_h =?, and in the second part, the values of unknownsE_h =?, q_h =? are calculated while E_d =1.0 ( /w m²) and q_d = −2.7 ( /w m²) are known and constant.

Numerical Procedure

The inverse estimation consists of three problems:

1-Direct problem: Suppose an enclosure with K discrete internal surfaces involving two types of boundary conditions, the surfaces with specified temperature and the surfaces with specified heat flux, where the objectives are to analyze the radiation exchange between the surfaces. The net radiation method is used to solve the radiation exchange in a radiant enclosure. In this method, the boundary is subdivided into surface elements. The equation of radiation exchange for surface elements with specified temperature (emissive power) and for other surface elements with specified heat flux can be described by the following equations:

( )

1

1 , 1

K

kj k k j j k k

j

F J E k K

δ ε ₋ ε

=

 − −  = ≤ ≤

 

∑

^and

⁽ ⁾

¹

1

, 1

K

kj k j j k

j

F J q K k K

δ ₋

=

− = + ≤ ≤

∑

(1a,b) where δ_kj is the Kronicker delta. The set of Eq. (1) is solved to calculate the outgoing heat fluxes,J , j_j =1,,K, then

the unknown boundary condition (emissive power or heat flux) is determined by the following equation:

k

k k k

k

1 ε q J E , 1 k K ε

− + = ≤ ≤ (2)

The configuration factors are calculated by the Hottel’s crossed-string method.

2-Inverse problem: For the inverse problem considered here, the heat flux distribution over the heater surface,q z_h( ), is regarded as unknown, and the desired heat flux distribution over the design surface, q_d( )x , is available for the analysis. The heat flux distributions over the design and the heater surfaces may be expressed as vectors of discrete elemental values, such as

{ }

qh= q_h,n n=1,, N , ^q^d⁼

{

^q^{d ,m} ^m⁼^1,^^,M

}

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where N and M are the number of heaters and the number of surface elements on the design surface, respectively. The estimated heat fluxes by the direct method over the design surface is expressed as ^q^e^{( x )}^d ⁼

{

^q^e,m ^m⁼^1,^^,M

}

^{. The}

solution of the inverse problem is based on the minimization of an objective function given by:

( )

^q^h ^q^d ^q^e

( )

^q^h ^T ^q^d ^q^e

( )

^q^h

G = −    −  (3) The minimization procedure is performed using the conjugate gradient method. Iterations are built in the following manner:

k 1 k k k

h h

q ⁺ =q +β C and ^C^k ^{= ∇}^G

( )

^q^h^κ ⁺^α^{κ κ}^C ⁻¹^and ^{S C} ^q^d ^q^e

( )

^q^h

S C S C

T

κ κ κ

κ

κ κ

β

 

  −

   

=     

(4a,b,c)

where C^k is the direction of descent vector and β^kis the step size.

Here, S , ^∇^G

( )

^q^h^κ ^and^α^k are the sensitivity matrix, gradient direction vector and conjugate coefficient, respectively, which are defined as follows:

( )

^h

S _mn _e,m q _h,n

m n

S q ^κ q

×

 

= = ∂ ∂  and ^∇^G

( )

^q^h^k ^{= −}²^S^T^_^q^d^k⁻^q^e^k

( )

^q^h^k ^_ ^(5a,b)

( ) ( )

^h^h ^h^h

q q

T

0

1 T 1

G G

with 0

G G

κ κ

κ

κ κ

α ∇ ₋ ∇ ₋ α

= =

∇ ∇ (5c)

3-Sensitivity problem: To minimize the objective function given by Eq. (3), we need to calculate the components of the sensitivity matrix, S_mn, defined by Eq. (5a). The sensitivity problem is obtained by differentiating the direct problem given by Eqs. (1) with respect to the elemental heat fluxes over the heater surface,q_h,n. Hence, for temperature specified surface elements we have Eq. (6a) and for surface elements with specified heat flux we have Eq. (6b)

( )

K

kj k k j j h,n 1

j 1

1 F J q 0 ,1 k K

δ ε ₋

=

 − −  ∂ ∂ = ≤ ≤

   

∑

^and ^K ^kj ^{k j} ^j ^h,n ^kn ¹

j 1

(δ F ₋ ) J q δ ,K 1 k K

=

 

− ∂ ∂ = + ≤ ≤

∑

^(6a,b)

The magnitudes of ∂J_j ∂q_h,n are calculated by solving Eqs. (6), then the elements of sensitivity matrix can be obtained by differentiation of Eq. (2) with respect to the elemental heat fluxes over the heater surface, q_h,n.

m

mn e,m h,n m h,n

m

S q q J q , 1 m M

1 ε

ε  

= ∂ ∂ = − − ∂ ∂  ≤ ≤ (7)

The n-th column of the sensitivity matrix is obtained by solving Eq. (7). The set of equations (6) and (7) must be solved N times in order to complete the sensitivity matrix. Because the right hand sides of equations (6a) and (6b) are constants, then the magnitudes of ∂J_m ∂q_h,n is not a function of time; hence, the sensitivity matrix,S, can be obtained before starting the iterative procedure of the inverse problem.

Results and Discussion

In this paper an inverse radiation boundary design problem is investigated numerically for an enclosure with unknowns

h ?

E = , q_h =?, on the heater surface while E_d =1.0 ( /w m²)and q_d (desired) are known (Fig. 2). Our investigation is done by writing a computer code with a useful computational algorithm to apply the conjugate gradient method. In Fig.

2 the comparison between the estimated heat flux, q_e, and desired heat flux , q_d( /w m²), over the design surface is shown.

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This figure shows a good agreement between the values of q_eandq_d, so that we can find the distribution of the emissive power and the heat flux on the heater surface which lead to this profile ofq_e. In Fig. 3 and Fig. 4, the distribution of the heat flux, q_h, and the emissive power on the heater surface are shown.

In the next part of this paper the profile value of unknownsE_h =?, q_h =? are calculated while E_d =1.0 ( /w m²) and 2.7 ( / 2)

qd = − w m are known. In Fig. 5 the comparison between the estimated heat flux, q_e, and desired heat flux , qd, over the design surface is shown. This figure shows when theE_d and q_d have the constant value which any relationship to the X, the inverse radiation method only can estimate the q_e profile which is close to value of q_d and so its accuracy is acceptable.

Conclusion

In this paper an inverse radiation boundary design problem in enclosure with diffuse gray walls was investigated numerically. The conjugate gradient method was used to minimize an objective function which is defined by the sum of square errors between estimated and desired heat fluxes over the design surface. It was shown that the method used in this paper for inverse problem would have been a good accuracy in order to apply in experimental and industrial problems.

1 2 3 4 5 6

-3.2 -3.0 -2.8 -2.6 -2.4 -2.2 -2.0

X q

qd qe

0

Fig. 2.Comparison between the estimated heat flux q w m_e( / ²) and desired heat fluxq_d( /w m²) on the design surface

2 3 4 5

4.10 4.12 4.14 4.16 4.18

Z qh

1

Fig. 3. Distribution of q w m_h( / ²)on the heater surface

2 3 4 5

9.999 10.000 10.001

Z Eh

1

-2.72 -2.71 -2.70 -2.69 -2.68

q

X

q_e q_d

0 1 2 3 4 5 6

Fig.

5.Comparison between the estimated heat flux q w m_e( / ²) and

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Fig. 4. Distribution of E w m_h( / ²)on the heater surface desired heat fluxq_d( /w m²) on the design surface

References

[1]- Viskanta, R., Menguc, M.P., “Radiation heat transfer in combustion Systems”, Prog. Energy Combust. Sci. 13, pp.

97–160, 1987.

[2]- McCormick, N. J., “Inverse radiative transfer problems: a review”, Nucl.Sci. Eng. 112, pp. 185– 98, 1992.

[3]- Li, H. Y., Ozisik, M. N., “Identification of the temperature profile in an absorbing, emitting, and isotropically scattering medium by inverse analysis”, J. Heat Transfer 114, pp. 1060–1063, 1992.

[4]- Liu, L. H., Tan, H. P., Yu, Q.Z., “Simultaneous identification of temperature profile and wall emissivities in one- dimensional semitransparent medium by inverse radiation analysis”, Numer. Heat Transfer, Part A 36, pp. 511–525, 1999.

[5]-Sarvari, S. M. H., Mansouri, S. H., Howell, J. R., “Inverse Boundary Design Radiation Problem in Absorbing- Emitting Media with Irregular Geometry”, Num. Heat Trans., Part A, Vol. 43, pp. 565-584, 2003.

[6]-Sarvari, S. M. H., Howell, J. R., Mansouri, S. H., “Inverse Boundary Design Conduction-Radiation Problem in Irregular Two-Dimensional Domains”, Num. Heat Trans., Part B, Vol. 44, pp. 209-224, 2003.

[7]- Mehraban, S., Sarvari, S. M. H., Farahat, S., “A QUASI-STEADY METHOD FOR INVERSE DESIGN AND CONTROL OF A TWO-DIMENSIONAL RADIANT OVEN IN TRANSIENT STATE”, Proceedings of CHT-08 ICHMT International Symposium on Advances in Computational Heat Transfer, Marrakech, Morocco, May 11-16, 2008.

[8]-Sarvari, S. M. H., Mansouri, S. H., “Inverse Design for Radiative Heat Source in an Irregular 2-D Participating Media”, Num.l Heat Trans., Part B, Vol. 46, pp. 283-300, 2004.

[9]-Sarvari, S. M. H., “Inverse Determination of Heat Source Distribution in Conductive-Radiative Media with Irregular Geometry”, J. Quant. Spect. and Rad. Trans., Vol. 93, pp. 383-395, 2005.