Solution of radiative inverse boundary design problem in a combined radiating-free convecting furnace

International Communications in Heat and Mass Transfer 45 (2013) 130–136

Contents lists available at SciVerse ScienceDirect

International Communications in Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ichmt

Solution of radiative inverse boundary design problem in a combined radiating-free convecting furnace☆ B. Mosavati a,⁎, M. Mosavati a, F. Kowsary b a b

Mechanical Engineering Department, University College of Engineering, Islamic Azad University Science and Research Branch, Tehran, Iran Mechanical Engineering Department, University College of Engineering, University of Tehran, Tehran, Iran

a r t i c l e

i n f o

Available online 26 April 2013 Keywords: Inverse problem Conjugate gradients method Natural convection–radiation Monte Carlo method

a b s t r a c t In this paper an inverse boundary design problem of combined natural convection–radiation is solved. The aim of this paper is to find the strength of heaters in a step-like enclosure to produce desired temperature and heat flux distribution on the design surface. The finite volume method for transition flow (which causes a faster convergence) is used as the direct solver of the energy and momentum equations. The SIMPLE algorithm is utilized to satisfy pressure–velocity coupling in order to solve the free convection heat transfer. Also, the backward Monte Carlo method is employed in order to be able to compute the distribution factors and carry out the radiant exchange calculations. Finally, the goal function which is defined on the basis of square root error is minimized by means of conjugate gradients method. The effects of variation of range of parameters such as the Rayleigh number, temperature ratio, radiation conduction parameter and the emissivity coefficient of insulated surfaces on the relative root mean square and heat flux are investigated and results are compared. The results demonstrate the efficiency and the accuracy of the proposed method. © 2013 Elsevier Ltd. All rights reserved.

1. Introduction The inverse boundary design problem has found its way to the literature since mid 1990s. It has many industrial applications in heat treatments, drying, backing food, and rapid processing chambers. In this type of problem the heaters are set at optimal setting so that they provide a uniform heat and temperature over the “design surface” throughout the heating process. In inverse methods, an objective function, expressed as the sum of square residuals between estimated and desired heat fluxes over the design surface, must be minimized in order to obtain the unknown parameter [1,2]. Since the inverse radiation boundary design problem is mathematically ill-posed, which means that the solution is not unique and is highly sensitive to input fluctuations, a stable solution requires some kind of regularization techniques. These techniques have been well established and addressed comprehensively by many investigators such as; Erturk et al. [3], Howell et al. [4], Emery [5], kowsary [6], and Pourshaghaghy et al. [7]. More recently, the “inverse design problems” has been extended to cases in which radiative heat and natural convection exist simultaneously, though very little work has been carried out on the subject.

☆ Communicated by W.J. Minkowycz. ⁎ Corresponding author at: Sciences & Research Branch, Hesarak, Tehran, Iran, P.O. Box: 1477893855. E-mail addresses: [email protected] (B. Mosavati), [email protected] (M. Mosavati), [email protected] (F. Kowsary). 0735-1933/$ – see front matter © 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.icheatmasstransfer.2013.04.011

Harutunian et al. [8] developed an inverse design methodology for enclosures composed of diffuse-gray walls containing a non-participating medium, where the ill-conditioned set of equations is formed using discrete configuration factors and were solved using modified truncated singular value decomposition. This technique was later extended to treat radiant enclosures containing participating media (Morales et al., França and Goldstein [9,10]) and problems involving multimode heat transfer (França et al. [11]). Daun et al. [12] proposed an optimization methodology for designing radiant enclosures containing specularly reflecting surfaces. They successfully implemented an efficient methodology to design the geometry of two radiant enclosures containing specularly-reflecting surfaces. The variable metric method (VMM) was utilized by Kowsary et al. [13] to investigate radiative boundary design problem in a twodimensional furnace filled with absorbing, emitting and scattering gas. In the first test case, design surface is a step-like geometry over which a uniform dimensionless heat flux of −1 and a uniform dimensionless temperature of unity is employed whereas other enclosure surfaces are assumed to be re-radiating. In the second test case an eccentric cylindrical surface is the design surface over which certain uniform heat flux and temperature distributions are desired. The lower surfaces are considered to be adiabatic and the remaining enclosure walls are heater surfaces. They concluded that VMM, when using a “regularized” estimator, is more accurate as compared to CGM. Mossi et al. [14] have studied boundary radiation in a two-dimensional cavity with turbulent flow and working fluid taking part in radiation. Hong-Liang et al. [15] proposed a hybrid ray-tracing method to solve the radiative transfer inside a plane-parallel absorbing–emitting–scattering medium with one specular surface and another diffuse surface which both of surfaces are considered to be

B. Mosavati et al. / International Communications in Heat and Mass Transfer 45 (2013) 130–136

131

2. Problem description Nomenclature A Dij Erms f g J Ks L ND NH Nr q Q Qrad Qconv Ra → s T

area distribution factor relative root mean-square error objective function   acceleration due to gravity, m =s2 sensitivity matrix thermal conductivity of fluid, W/m k length of the cavity (m) total no. of the elements on design surface total no. of elements on heater surface 4 d L radiation conduction interaction parameter, Nr ¼ σTKsΔT dimensionless heat flux total heat flux (Qconv + Qrad), W/m 2 radiative heat flux, W/m 2 free convection heat flux, W/m 2 the Rayleigh number direction vector temperature at any location in the computational domain, K d T⁎ temperature ratio, TΔT ΔT = (Th − Td) modified temperature difference, (Qd L/Ks), K

Greek symbols ε emissivity σ Stefan Boltzmann constant (5.67 × 10 −8Wm −2 K −4) λ geometric path length θ dimensionless temperature (θ = ((T − Td))/(ΔT))

Subscripts D design e estimated h heater surface k iteration number

semitransparent or opaque. Salinas [16] used an inverse analysis for estimation of the temperature distribution for a gray emitting, scattering, two-dimensional rectangular medium and the inverse problem is solved by using the conjugate gradients method (CGM). Mosavati et al. [17] solved the inverse radiation boundary problem with using a “backward Monte Carlo method (MCM)” for cases where radiation is the dominant mode of heat transfer (i.e., radiative equilibrium). Payan et al. [18] proposed an inverse methodology which is employed to estimate the unknown strengths of heaters on the heater surface of a square cavity with free convection from the knowledge of the desired temperature and heat flux distributions over a given design surface. The approach in this paper is to employ combination of the backward MCM and the finite volume method (FVM) for unsteady free convection flow. To the authors' best knowledge, in none of the previous works on inverse radiation–convection problems these two methods (Backward Monte Carlo and FVM) have been used simultaneously in inverse boundary design calculations. In this work, we have attempted to explore use of these methods in inverse solution. The CGM is used to solve the inverse boundary design in a step-like enclosure. In this paper the problem is solved for different parameters such as the Rayleigh number and the optimum case is obtained. Finally, evaluated temperatures of heaters lead to a uniform heat fluxes on the design surfaces which are also very close to the desired heat flux.

The design problem under consideration in this paper is depicted in Fig. 1. In this two-dimensional furnace, the length and width have been chosen as a unit length. In this test case, height of the step-like design surface is 0.5 m wide which is located at distance of 0.25 m from enclosure side walls. The top wall and side walls are considered as heater surfaces and all the other walls except the step surfaces are insulated. All walls are considered to be diffuse-grey and medium is transparent to thermal radiation. One type of boundary condition (temperature or heat flux) is specified over each boundary surface, except for the heater surface which no boundary condition is specified. The aim of the inverse problem is to find the temperature (or heat flux) distribution over the heater surfaces in such a way that the desired uniform heat flux profile is recovered over the uniform temperature-specified design surface. 3. Formulation of the direct problem In this section design surfaces and heater surfaces are discretized into elements and heater surfaces discrete into elements that points to the calculation of heat flux transferred from heaters to the design surfaces. The FVM is used to solve the momentum equation in X and Y directions. The SIMPLE algorithm is considered to satisfy continuity and coupled pressure and velocity equations. In order to discretize the convective and diffusive terms a hybrid method is used. Meshing is fully rectangular and a staggered grid is used. Moreover, equations are considered to be time dependent and due to the fact that the transition flow is considered, less time is needed as compared to the unsteady flow. According to Mahan [16] the net heat transfer to a given surface in an N-surface enclosure filled with a nonparticipating media, using the distribution factors can be formulated as: Q i−j ¼ ∑ε σ Di−j


 4 4 Ti –Tj

ð1Þ

The distribution factor Dij is defined as the fraction of diffuse radiation leaving the ith surface, which arrives at the jth surface both directly and by all possible specular reflections. In derivation of Eq. (1), the reciprocity rule for the distribution factors i.e., ε i Ai Dij ¼ εj Aj Dji

ð2Þ

has been used. The distribution factors are most easily determined using the MCM as detailed by Mahan [19]. Briefly, to obtain these factors, the ensuing steps are followed:

Fig. 1. Geometry and boundary condition used in inverse problem.

132

B. Mosavati et al. / International Communications in Heat and Mass Transfer 45 (2013) 130–136

1- The arbitrary points, from which discrete rays are launch from any given surface, are selected by two random numbers, for uniform distribution of points across the surface. 2- The directions of diffuse emissions (i.e., the zenith and the azimuthal angles) are determined randomly using the “tangent sphere method” as described by Kowsary 1999. [20] 3- The intersection point of the rays with the enclosure interior surface is determined using the “micro voxel” algorithm as described by Naeimi (2012). [21] 4- Whether the ray is reflected or absorbed by the intersecting surface is determined by drawing a random number and comparing it to the absorptivity of the surface. If the drawn random number is less than the absorptivity value, the ray is absorbed and its history is terminated. In which case, the counter keeping track of the rays absorbed by the intersecting surface is incremented by one. 5- If the ray is to be reflected (random number is greater than the absorptivity of the surface), the manner of the reflection (i.e., diffuse or specular) is determined by drawing a second random number and comparing it with the “specularity ratio” of the surface. If the reflection is to be a diffuse type (as is the case in this paper) the diffuse direction is determined by going back to step 2 and repeating the procedure until the ray is absorbed by one of the surfaces of the enclosure. Thus, the ratio of the number of rays absorbed by a given ith surface to those from the launch surface is a good measure of Dij if sufficiently large numbers of rays are launched from the ith surface. As can be seen the main computation load is devoted to calculation of the distribution factors. For an N-surface enclosure, N 2 distribution factors must be determined in order to perform radiative exchange calculations. However, similar to diffuse view factors, by employing the summation and the reciprocity rules, this number can be reduced to N (N−1)/2 (Mahan [19]). In solving problems where free convection and radiation are simultaneously present, boundary conditions on insulated walls are not explicitly known. The major point is computation of the temperature of the insulated surfaces by means of a trial and error method. The solution process involves guessing a radiative heat flux on insulated surfaces and solving the free convection problem to calculate insulated wall temperature and then using these wall temperatures to solve the radiative problem in order to update radiative heat flux on insulated surfaces. These new heat flux values are compared with initial guess. This procedure is repeated till convergence is achieved. In this study ΔT = (Qd L/Ks) and temperature is nondimensionalized as (θ = ((T − Td))/(ΔT)). Dimensionless form of Eq. (1) is formulated as: 
.
 4 θj qrad ¼ Q i−j = −σTd ¼ ε Dij 

Td

.

T

4
. θ þ1 − i

T

4  þ1

ð3Þ

Where; T ¼ ðΔTÞ In order to solve the combined convection and radiation problem two different dimensionless parameters (T⁎ and Nr) are used. The boundary condition on insulated surfaces is given as: Y ¼ 0; for 0bxb0:25 and 0:75bxb1:0

∂

. θ y ¼ ðN r qrad Þ ∂

ð4Þ

4. The inverse boundary design formulation The overall trend of solving the inverse problem is as follows: the temperature (Td) and the heat flux (qd) on the design surfaces are assumed to be known (i.e. desired). Distribution of heaters temperature (Th) or heat flux on the heater surface (qh) is to be calculated to reach the desired condition on the design surface. In order to estimate the unknown heat flux vector (qh) the temperature of the design surface

is set to be the actual boundary condition of the direct radiation– convection exchange problem. Then by assuming an initial guess for the heat flux at the heater surface direct solution method is invoked to calculate heat flux distribution on the design surface. These calculated values (qe,i) are then compared with the desired value of the heat flux on the design surface, and if necessary, the heat flux distribution on the heater surface is varied and the procedure is repeated until sufficient proximity between the “calculated” and the “design” value of the heat flux is reached. This procedure is facilitated by defining an objective function which is given as: h i2 ND f ðqh Þ ¼ ∑i¼1 qd;i −qe;i

ð5Þ

In order to minimize this function, the derivative of f(qh) with respect to each of unknown qh,1, qh,2,…,qh,nh is set equal zero. ∂f ðqhÞ ∂f ðqhÞ ∂f ðqhÞ ¼ ¼…¼ ¼0 ∂q1 ∂q2 ∂qNH

ð6Þ

In this equation NH is the number of heater surfaces elements. In a matrix form this equation can be written as: "

# i ∂qeT h qd;i −qe;i ¼ 0 ∇ f ðqh Þ ¼ 2 − ∂qh →

ð7Þ

Where ∇f is the gradient vector of the objective functions. In the above equation the term containing the first-order derivate is given as: 2

3 ∂ 6 7 6 ∂qh; 1 7 6 7 ∂ T 6 7 ∂qe 6 7 ¼ 6 ∂qh; 2 7½ qe; 1 6 7 ∂qh 6 7 ⋮ 6 7 4 5 ∂ ∂qh; NH

qe; 2

…

qe; ND 

ð8Þ

The sensitivity matrix is given as the transpose of the above matrix. Therefore, " Jðqh Þ ¼

T

∂qe ∂qh

# ð9Þ

The elements of the sensitivity matrix are sensitivity coefficients. The sensitivity coefficient Jij is defined as the first derivative of i'th heat flux on design surface with respect to j'th heat flux on heaters surfaces. J ij ¼

∂qe; i ∂qh; i

ð10Þ

In fact these coefficients provide a sense for the effect of heater values on design values. 4.1. Calculation of sensitivity coefficients In linear problems, the differential equations for obtaining sensitivity coefficients are very similar to the governing heat transfer equation; but they differ only in boundary conditions. However, the solution for nonlinear problem requires an iterative procedure, which is obtained by linearizing the vector of estimated heat flux using a Taylor series expansion. There is an easier approach to find sensitivity coefficients which is by using numerical differentiation. Using this approach for evaluation of sensitivity coefficients, the direct problem is solved once, and then the parameters are varied

B. Mosavati et al. / International Communications in Heat and Mass Transfer 45 (2013) 130–136

slightly, and the direct problem must be solved once again. The sensitivity coefficients are thus obtained using: h
i  Jij ¼ qe;i qh1 ; qh2 ; …; qh;j þ εqh;j þ … þ qhNH −qei ðqh1 ; qh2 ; …; qhNH Þ =εqh;j ð11Þ

4.2. The CGM In order to minimize the goal function in each iteration, a reduction direction and a proposed optimal step is needed. In CGM this direction is a linear combination of goal function gradients directions in current iteration with reduction direction of last iteration. If q is assumed as variable parameters vector in k'th iteration, then its value for (k + 1)'th iteration will be obtained as follows: → kþ1 qh

k →

→k

¼ qh þ λ S

ð12Þ →

In Eq. (12) λ k is the optimal step size, S is the descent direction and k is iteration index. Vector S is computed as a linear combination of gradients of the objective function in step k and k − 1. Eq. (7) is used to evaluate gradients as:   →k →k →k1 →k1  S S ¼ −∇ f þ  −∇ f =−∇ f 

→k

ð13Þ

→

The estimated heat fluxes ( q k1 h ), can be linearized with a Taylor expansion method and then the minimization with respect to step size, λ k is performed to yield the following expression for the optimal step size. Thus in order to evaluate λ k, the following relation is used:   i →k →k T →k →k  →k →k T h→ → k q d −q e = J S J S λ ¼ J S

ð14Þ

The complete procedure for computing λ k is explained in [1]. Finally the following steps are followed: →k

1. Find the sensitivity coefficients matrix J as described previously. → 2. Start with an initial guess for q kh (as initial point). Set iteration number as k = 1. → 3. Compute the gradient of the objective function (∇ f ) at the point →k q h from T h i → →k ∇ f ðqh Þ ¼ −2 J qd;i −qe;i 4. 5. 6. 7.

ð15Þ

→k

Obtain S vector from Eq. (13). →k Find the optimal step length (λ k) in the direction S from Eq. (14). → Evaluate the new estimated heat fluxes (q h kþ1 ) from Eq. (12). A convergence criterion which is defined as the norm of the gradients of the objective function must be checked. If it is in the convergence domain the procedure is completed, otherwise Set the new iteration number k = k + 1, and return to step (3).

The difference between the estimated heat flux distribution on the design surface and desired heat flux is well described by the root mean square of relative error (Erms) as;

Erms ¼

1

. ND

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi " #2 u u ND ðqd −qei Þ t∑  100 i¼1 q}d

133

temperatures is imposed (non dimensional) on the design surface. In this paper dimensionless heat flux is assumed to be 1 and the design surface's temperature is considered to be zero. As mentioned earlier, the Erms values shows the difference between the calculated and the desired heat flux where continuous descending pattern until reaching a constant position proves the accuracy of inverse solution. The sample problem discussed in this paper is a step cavity on which the effect different parameters such as variation of Rayleigh number are studied. The cavity is two-dimensional for which step surfaces are assumed as design surfaces and heaters configuration and insulated surfaces are illustrated in Fig. 1. In this problem both radiation and free convection are present together. Results are discussed by considering 12 heater surfaces (NH = 12) and the 6 design surfaces (ND = 6). It is clear that these results can be expanded for much more number of heater and design surfaces. Calculations have been performed for the typical ranges of the various parameters that may occur in a practical situation, as shown in Table 1. Variations of these parameters are discussed and their effects are mentioned in sections. Specific case of (Nr = 0.5) has been chosen for the detailed study of the effects of other parameters on the heat transfer characteristics of the enclosure. As whole, results in each section are categorized into two sections. In the first part, the constant value for Rayleigh number (Ra = 10 4) is considered to survey the effect of variation of the temperature ratio (T⁎). In the second part the constant value for T⁎(T⁎ = 1) is considered, in order to discuss the effect of variation of the Rayleigh number.

5.1. Error analysis The effects of various temperature ratios (T⁎) on root mean square of relative error are plotted in Fig. 2a. This figure is drawn for specific radiation–conduction interaction parameter and definite Rayleigh number (Nr = 0.5,Ra = 10 4) for instance. (The plots for other ranges of Nr and Rayleigh number which are specified in Table 1 have similar trend with Fig. 2a which are ignored to be drawn). This figure shows that the root mean square error in the evaluation of heat flux decreases. Also, Erms descends continuously and approaches the final desired answer. Different values of the T⁎ are considered in Fig. 2a, and by comparing root mean square error values, we can consider decreasing the values of T⁎ will increase the speed of convergence. This is evident due to enhancement effects of radiation heat transfer which causes an increase in the difference between hot and cold temperatures. The effect of different Rayleigh number on error function is shown in Fig. 2b. This figure is considers a specific Nr and T⁎ in order to compare effects of different Rayleigh number values. It shows that increasing the free convection heat transfer due increase the Rayleigh number will cause the error function to grow. Error percent values with various (Nr) and (T⁎) are tabulated in Table 2. As can be seen, the error percentage is increased as the Rayleigh number is increased. This is due to enhancement of the free convection as compared radiation. It is clear in Table 2 by considering specific T⁎ (T⁎ = 1) and increases of the Rayleigh number (until Ra = 10 6) Erms grows from 5.558 × 10 1 to 1.219 because of increasing the free convection effects.

ð16Þ

The above relation represents that small values of Erms that correspond to a more accurate result in comparison with desired value. 5. Results and discussion As mentioned previously, in inverse problems the purpose is to obtain a uniform (non dimensional) heat flux when uniform

Table 1 Typical ranges of parameters for the combined convection and radiation inverse problem. Parameter

Range

Rayleigh number Radiation conduction interaction parameter Temperature ratio Emissivity(insulated surfaces) Emissivity(design and heater surfaces)

104 ≤ Ra ≤ 106 0.5 ≤ Nr ≤ 1.5 0.3 ≤ T⁎ ≤ 3 0.2 ≤ εinsulated surfaces ≤ 1.0 εd = εh = 0.5

134

B. Mosavati et al. / International Communications in Heat and Mass Transfer 45 (2013) 130–136

a

Erms (%)

20

simultaneously decrease the radiation heat flux on design surfaces. Also, the effects of increasing the Rayleigh number on free convection and radiative heat fluxes are plotted in Fig. 3b. As can be seen in Fig. 3b when the Rayleigh number grows, free convection heat fluxes become greater in comparison to lower Rayleigh number. However, by comparing Fig. 3a and b, it is clear that when free convection heat flux decrease, this reduction tends to be compensated for by radiative transfer between the hot and cold walls. As obtaining convergence requires a larger number of meshes, increasing the Rayleigh number will increase the runtime. Clearly, in constant value (T⁎) addition of radiation and free convection heat flux is very near to desired heat flux (which its value considered 1 dimensionless)

T*=3 T*=1 T*=0.5 T*=0.3

15

10

5

0 1

2

3

4

5

6

7

8

5.3. The effect of Rayleigh number on heater temperature

Number of iteration

b

Ra=1045 Ra=106 Ra=10

16

Erms (%)

Fig. 4 shows a plot of estimated temperature of heaters by considering the different values for temperature ratio (T⁎) in specific Rayleigh number (Ra = 10 4). It is evident from Fig. 4 that for this case, the temperature of heaters increases when the temperature ratio grows. This is due to strengthening of the free convection as compared radiation. Also it is clear from Fig. 4 that with decreasing the temperature ratio, the temperature of heaters is smoother as compared to larger values of (T⁎). It should be noted that both of this plot has been made for previously mentioned optimum root mean square error percent parameter.

20

12

8

4

0

1

2

3

4

5

6

7

8

Number of iteration

T*=3 (qconv) T*=1 (qconv) T=0.3 (qconv) T*=3 (qrad) T*=1 (qrad) T*=0.3 (qrad) qdesign

a

Fig. 2. Reduction trend of Erms versus number of iteration for a) Nr = 0.5, Ra = 104, b) Nr = 0.5, T⁎ = 1.

5.2. Variation of heat flux on design surfaces and effects of Rayleigh number on heat flux

q"(w/m2)

1

In any case the values of Erms are acceptable for engineering applications.

The heat flux on design surfaces is calculated by assuming diffuse and grey walls with considering free convection and radiation heat transfer. Radiative and free convection heat flux for a constant values (Nr = 0.5, Ra = 10 4) are demonstrated in Fig. 3a. As seen in this figure increasing the (T⁎) will enhance the free convection and Table 2 Relevant Erms for the test case with considering various (Nr), (T⁎).

10

4

T⁎

Erms (%)

0.5

1 0.3 1 0.3 1 0.3 1 0.3 1 0.3 1 0.3 1 0.3 1 0.3 1 0.3

5.558 2.360 4.613 2.018 3.318 1.380 8.127 4.432 7.183 3.881 5.322 2.214 1.219 8.702 1.151 5.402 9.435 4.355

1.5 105

0.5 1.0 1.5

106

0.5 1.0 1.5

0.5

1

× × × × × × × × × × × ×

qconv(Ra=104) qconv(Ra=1056) ) qconv(Ra=10 qrad(Ra=1045) qrad(Ra=106) qrad(Ra=10 ) qdesign

1.2 10−1 10−1 10−1 10−1 10−1 10−1 10−1 10−1 10−1 10−1 10−1 10−1

× 10−1 −1

× 10 × 10−1 × 10−1

1.5

Xd(m)

b

Nr

1.0

0

q"(w/m2)

Ra

0.5

0.8

0.4

0 0.5

1

1.5

Xd(m) Fig. 3. The calculated free convection and radiative heat flux on step-like design surface for a) Nr = 0.5, Ra = 104, b) Nr = 0.5, T⁎ = 1.

B. Mosavati et al. / International Communications in Heat and Mass Transfer 45 (2013) 130–136

135

5.4. Effects of insulated surface emissivities

Ra=1045 Ra=10 Ra=106

0.07

Fig. 5 depicts the variations of Erms with respect to the emissivity of insulated surfaces for different Rayleigh number values. As seen from this figure, the value of Erms increases by increasing the emissivity of insulated surfaces (from 0.2 to 1.0) which results in less uniform temperature distribution and heat flux. It is also seen that for greater Rayleigh number, the variation of the emissivity has no considerable influence on the Erms, and this result will be governed for greater Rayleigh number. However, by increasing the Rayleigh number, the effect of emissivity becomes greater.

0.06

Erms (%)

0.05 0.04 0.03 0.02

5.5. Comparison of isotherm plots for combined natural convection and surface radiation with different Rayleigh number

0.12

46 0.0

0.02

1

0.14

a

0.1

4

0.02

0.08

0.16

0.04

0.8

0.2 0.18 0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0

0.06

0.08

0.04

0.1

4

0.0 0.02 4

0.1

2

0.1

0.06

0.6

0.0 8

0.02

0.14 6 0.0

0.0

0.1

4

0.08

0.16

0.0

0.2

0.4

0.1

0.1

6

8

0.12

0.16

0.2 0.08

0.14

0.1

0 0

0.2

0.4

0.6

0.8

1

X 1

0.02

0.03

0.1

0.01 0.03 0.04

0.0

9

0.01

0.07

0.1

0.6

0.16 0.15 0.14 0.13 0.12 0.11 0.1 0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01 0

0.08

0.06 0.05

0.08 0.09

0.1

0.3

0.05

0.02

0.04

0.06

0.8

0.07

T*=3 T*=1 T*=0.5 T*=0.3

0.03

b

1 0.0

0.0

6 0.01

4

0.02

6

Xh (m) Fig. 4. The estimated temperature of heaters versus heaters number Nr = 0.5, Ra = 104.

4

0.1

0.14 0.16

0.2

0.4

0.15 0.09

0.1

1

0.08

0.1

0.14

0.0 5

0.07 0.11 0.12 0 13

0 0

0.13

3

0.12

2

6 0.0

0.09

0.07

0.13

1

0.09 0.08

6

0.0 0.08 0.1

0

0.0

0.0

0.2

8

9

3

0.1

0.1

5

0.07

0.0

5 0.0 0.07

1

0.0

0.1

0.0

0.4

0.0

Y

2

2

0.1 0.13

0.0

0.2

0.11 0.12

Temperatures of heaters (K)

1

3) Studying the effects of the Rayleigh number on the value Erms shows that decreasing this dimensionless number will cause the error function to grow. This is due to enhancement of the free convection as compared to radiation. Also, by increasing the Rayleigh number, it is clear that the convection heat transfer flux enhances, and heat flux on design surfaces become less uniform. 4) Increasing the emissivity of insulated surfaces degrades the heat flux uniformity on the design surfaces. So, the optimum case is obtained with the lowest value of emissivity of insulated surfaces.

0.1

1) Using the backward MCM with finite volume yields accurate results in considered sample problem cases; the small value of Erms computational can clearly indicate this fact. On the other hand, using the MCM decreases the time. 2) Decreasing the temperature ratio results in smoother temperature distribution on the heater surfaces. Moreover, decreasing the temperature ratio causes a greater radiative heat transfer flux as compared to convection heat transfer flux.

0.8

Fig. 5. The influence of insulated surface's emissivity on the Erms(Nr = 0.5, T⁎ = 1).

0.12

In this paper boundary inverse design in a step-like enclosure with gray wall and transparent media has been studied. The temperatures of heaters have been defined in order to reach constant heat flux accompanied by uniform temperature on design surfaces. The method to solve the direct free convection problem is the FVM using the SIMPLE algorithm. Finally by means of backward MCM radiative calculations were performed. The minimization of the goal function was carried out using the CGM. From the present study, the following conclusions are made:

0.6

Emissivity coefficient

0.12

6. Summary and conclusion

0.4

Y

The isotherms for this cases which was described are shown in Fig. 6a and b. As seen, with increasing the Rayleigh number, free convection will become more effective and can be seen that the fluid is bulk temperature will increase and isothermal lines will become more distorted as a result of high velocities and a more powerful free convection. The isotherms near the design surface tend to be flat; therefore, the heat flux distributions over the design surface tend to be uniform. Also with increasing the Rayleigh number the temperatures of heaters will be smaller. These isothermal lines are drawn for specific (Nr = 0.5, T⁎ = 1) and they can expand for another values (The results are similar, so it is ignored to be drawn).

0.2

01

0.6

0.8

2

1

X Fig. 6. The isotherms for inverse problem, for (a) Ra = 105 and (b) Ra = 106.

136

B. Mosavati et al. / International Communications in Heat and Mass Transfer 45 (2013) 130–136

References [1] M.N. Ozisik, H.R.B. Orlande, Inverse Heat Transfer: Fundamentals and Applications, first ed. Taylor & Francis, New York, 2000. [2] S.S. Rao, Engineering Optimization: Theory and Applications, 2nd ed. Wiley Eastern Limited, New Delhi, London, 1995. , (9th reprint). [3] H. Ertürk, O.A. Ezekoye, J.R. Howell, Comparison of three regularized solution techniques in a three-dimensional inverse radiation problem, Journal of Quantitative Spectroscopy and Radiation Transfer 73 (2) (2002) 307–316. [4] J.R. Howell, k. Daun, H. Ertürk, M. Gamba, M. Hosseini Sarvari, The use of inverse methods for the design and control of radiant sources, JSME International Journal 46 (4) (2003) 470–478. [5] A.F. Emery, Some thoughts on solving the radiative transfer equation in stochastic media using polynomial chaos and wick products as applied to radiative equilibrium, Journal of Quantitative Spectroscopy and Radiation Transfer 93 (1) (2005) 61–77. [6] F. Kowsary, A. Behbahaninia, A. Pourshaghaghy, Transient heat flux function estimate utilizing the variable metric method, International Communications in Heat and Mass Transfer 33 (2006) 800–810. [7] A. Pourshaghaghy, K. Pooladvand, F. Kowsary, K. Karimi-Zand, An inverse radiation boundary design problem for an enclosure filled with an emitting-absorbing, and scattering media, International Communications in Heat and Mass Transfer 33 (3) (2006) 381–390. [8] V. Harutunian, J.C. Morales, J.R. Howell, Radiation exchange within an enclosure of diffuse-gray surfaces: the inverse problem, Proc. ASME/AIChE 35th National Heat Transfer Conference, Portland, OR, 1995. [9] J.C. Morales, V. Harutunian, M. Oguma, J.R. Howell, Inverse design of radiating enclosures with an isothermal participating medium, in: M. Pinar Menguç (Ed.), Radiative Transfer 1: Proc. of the First International Symposium on Radiation Transfer, Bengell House, New York, NY, 1996, pp. 579–593. [10] F.H.R. França, L. Goldstein, Application of zoning method in radiative inverse problems, Proc. Brazilian Congress of Engineering and Thermal Sciences, ENCIT 96, Florianopolis, Brazil, 1996.

[11] F.H.R. França, O.A. Ezekoye, J.R. Howell, Inverse boundary design combining radiation and convection heat transfer, Journal of Heat Transfer 123 (5) (2001) 884–891. [12] K.J. Daun, D.P. Morton, J.R. Howell, Geometric optimization of radiant enclosures containing specular surfaces, Journal of Heat Transfer 125 (2003) 845–851. [13] F. Kowsary, K. Pooladvand, A. Pourshaghaghy, Regularized variable metric method versus the conjugate gradients method in solution of radiative boundary design problem, Journal of Quantitative Spectroscopy and Radiative Transfer 108 (2007) 277–294. [14] A.C. Mossi, H.A. Vielmo, F.H.R. França, J.R. Howell, Inverse design involving combined radiative and turbulent convective heat transfer, International Journal of Heat and Mass Transfer 51 (2008) 3217–3226. [15] Hong-Liang Yi, Bing Zhen, He-Ping Tan, Timothy W. Tong, Radiative heat transfer in a participating medium with specular–diffuse surfaces, International Journal of Heat and Mass Transfer 52 (2009) 4229–4235. [16] T. Carlos Salinas, Inverse Radiation Analysis in two-dimensional gray media using the discrete ordinates method with a multidimensional scheme, International Journal of Thermal Sciences 49 (2) (2010) 302–310. [17] M. Mosavati, F. Kowsary, B. Mosavati, A Novel, non iterative inverse boundary design regularized solution technique, using the Backward Monte Carlo Method, ASME HT-1331, 2013. [18] S. Payan, S.M.H. Sarvari, H. Ajam, Inverse boundary design of square enclosures with natural convection, International Journal of Thermal Sciences 48 (2009) 682–690. [19] J.R. Mahan, Radiation Heat Transfer: Statistical Approach, Wiley, New York, 2002. [20] F. Kowsary, A computationally efficient method for monte carlo simulation of diffuses radiant emission or reflection, International Journal of Heat and Mass Transfer 42 (1999) 193–195. [21] H. Naeimi, An efficient Monte Carlo method to simulate radiation heat transfer in vacuum furnaces. , PhD thesis Department of Mechanical Engineering, University of Tehran, Tehran, 2012.