World Academy of Science, Engineering and Technology International Journal of Mechanical and Mechatronics Engineering Vol:5, No:1, 2011

Simulation of Heat Transfer in the Multi-Layer Door of the Furnace U. Prasopchingchana

workers [7] examined the combined conduction and mixed Abstract—The temperature distribution and the heat transfer convection flow field and temperature distribution in an rates through a multi-layer door of a furnace were investigated. The enclosure in the presence of vertical partition with finite inside of the door was in contact with hot air and the other side of the thickness and thermal conductivity. Nouanegue et al. [8] door was in contact with room air. Radiation heat transfer from the investigated conjugate heat transfer by natural convection, walls of the furnace to the door and the door to the surrounding area conduction and radiation in open cavities in which a uniform was included in the problem. This work is a two dimensional steady heat flux is applied to the inside surface of the solid wall state problem. The Churchill and Chu correlation was used to find local convection heat transfer coefficients at the surfaces of the facing the opening. Chiu and Yan [9] carried out a numerical furnace door. The thermophysical properties of air were the functions study in the radiation effect on the characteristics of the mixed of the temperatures. Polynomial curve fitting for the fluid properties convection fluid flow and heat transfer in inclined ducts. The were carried out. Finite difference method was used to discretize for effect of conduction of horizontal walls on natural convection conduction heat transfer within the furnace door. The Gauss-Seidel heat transfer in a square cavity is numerically investigated by Iteration was employed to compute the temperature distribution in Mobedi [10]. Molla and Hossain [11] examined the effect of the door. thermal radiation on a study two-dimensional mixed The temperature distribution in the horizontal mid plane of the convection laminar flow of viscous incompressible optically furnace door in a two dimensional problem agrees with the one thick fluid along a vertical wavy surface. dimensional problem. The local convection heat transfer coefficients This research aims to study the coupling between multi- at the inside and outside surfaces of the furnace door are exhibited. layer conduction, natural convection and surface radiation on Keywords—Conduction, heat transfer, multi-layer door, natural the multi-layer door of the furnace. The work used the convection Churchill and Chu [12] empirical correlation in the simulation to compute the average convection heat transfer coefficients. I. INTRODUCTION Later, the local convection heat transfer coefficients were EAT loss from a multi-layer door of a furnace is a determined. Then, radiation heat transfer coefficients were Hcombination problem of multi-layer conduction, natural computed. Finally, the temperature distribution and total heat convection and radiation. At present, several research works loss from the inside of the furnace to room air through the [1-11] have been devoted to investigate the conduction, multi-layer door of the furnace were investigated. natural convection and radiation phenomena. Prasopchingchana and Laipradit [1] performed a simulation of II. MODEL GOMETRY heat transfer in the glass panel of an oven using an empirical This research has been treated as a two dimensional steady- correlation for natural convection heat transfer on the surface state problem. There was hot air on the inside surface of the of the glass panel. Bilgen [2] carried out a numerical and multi-layer door and room air on the outside surface of the experiment study of conjugate heat transfer by conduction and multi-layer door. natural convection on a heated vertical wall. Simulation of natural convection and radiation in the gas cavity and heat Furnace conduction in the walls of an enclosure was performed by Kuznetsov and Sheremet [3]. Nouanegue and Bilgen [4] Tsur carried out a numerical study on vertical channel solar chimneys with a practical geometry, which consist of two Room air Hot air International Science Index, Mechanical and Mechatronics Engineering Vol:5, No:1, 2011 waset.org/Publication/1781 dimensional channel boarded by two solid walls. Xaman et al. T∞,2 [5] study laminar and turbulent natural convection combined T∞,1 with surface thermal radiation in a square cavity with a glass y wall. Zrikem et al. [6] numerically investigated coupled heat Tw Multi-layer door transfers by conduction, natural convection and radiation in x hollow structures heated from below or above. Oztop and co-

Fig. 1 Model geometry The top and bottom sides of the multi-layer door were U. Prasopchingchana is with the Mechanical Engineering Department, assumed to be insulated. The multi-layer door of the furnace is Burapha University, Chonburi 20131 THAILAND (phone: 66-3810-2222 Ext. 50 cm. wide, 50 cm. high and 20.3 cm. thick. The multi-layer 3385 Ext. 104; fax: 66-3874-5806; e-mail: [email protected]).

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door of the furnace consists of 3 layers: fire clay brick, glass Radiation heat transfer from the furnace wall surfaces to the fiber insulation and steel sheet. The thermal conductivity inner door surface and from the outer door surface to the values [13] and the thickness values of each layer are shown surrounding area was included in the problem. The nonlinear in Table 1. equations of radiation heat transfer were linearized. Thus, the radiation heat transfer equations are TABLE I ⎛ ()k ⎞ qr,w = hr,1A⎜Tw − T ⎟ (6) PROPERTIES OF THE MULTI-LAYER DOOR ⎝ s,1 ⎠ Layer Material Thickness Thermal and (m) conductivity ⎛ ()k ⎞ (W/m⋅K) qr,sur = hr,2 A⎜T − Tsur ⎟ (7) ⎝ s,2 ⎠ 1 Fire clay brick 0.1 1.09 2 Glass fiber insulation 0.1 0.038 where hr,1 is the radiation heat transfer coefficient between the 3 Steel sheet 0.003 28 furnace wall surfaces and the furnace door surface, and hr,2 is the radiation heat transfer coefficient between the furnace

door surface and the surrounding area. The radiation heat The emissivity values [13] at the surfaces of the multi-layer transfer coefficients are door inside and outside of the furnace are 0.75 and 0.066 ⎛ ()k −1 ⎞ 2 ()k −1 2 respectively. The temperatures at the furnace wall surfaces hr,1 = ε1σ ⎜Tw + T ⎟(Tw + []T ) (8) ⎝ s,1 ⎠ s,1 equaled the hot air temperatures in the cavity. Also, the and temperatures of the surrounding area equaled the room air 2 temperatures. ⎛ ()k −1 ⎞⎛ 2 ()k −1 ⎞ hr,2 = ε 2σ ⎜Tsur + T ⎟⎜Tsur + []Ts,2 ⎟ (9) ⎝ s,2 ⎠⎝ ⎠ III. MATHEMATICAL MODELS where ε is emissivity and σ is the Stefan-Boltzmann constant. Newton’s law of cooling was employed to determine the All properties of hot air and room air around the inside and the outside of the furnace door surfaces were obtained from heat transfer rates at the surfaces of the furnace door. To NIST (National Institute of Standards and Technology) determine the average Nusselt number, the work used the Standard Reference Database 23, Version 7.0 by polynomial Churchill and Chu empirical correlation [12]: curve fitting: 2 ⎧ 1/ 6 ⎫ 2 n−1 ⎪ 0.387Ra y ⎪ x = a1 + a2T f + a3T f + K + anT f (10) Nu y = ⎨0.825 + ⎬ (1) 9 /16 8/ 27 where x are the air properties, a are the polynomial curve ⎩⎪ []1+ ()0.492 / Pr ⎭⎪ n fitting coefficients and T is the film temperature of air. The where Ra is the Rayleigh number and Pr is the Prandtl f y errors of the properties derived from polynomial curve fitting number at the film temperatures. The film temperatures are are less than 0.1 percent. For all properties, n is 10. T f = (()Ts ave + T∞ )/ 2 (2) The volumetric thermal expansion coefficient is where ()Ts is the average temperature at the door surfaces 1 ⎛ ∂ ρ ⎞ ave β = - ⎜ ⎟ and T is the ambient temperature. The average convection ⎜ ⎟ ∞ ρ ⎝ ∂ T ⎠P heat transfer coefficients were obtained from 1 2 n−2 β = − (a2 + 2a3T f + 3a4T f + K + (n −1)anT ) (11) k f ρ f P h = Nu y (3) y y Within the thickness of the furnace door, the heat diffusion equation was used to calculate the temperature where kf is thermal conductivity of air. The local convection heat transfer coefficients could be obtained from distribution: ∂ 2T ∂ 2T 1 ⎛ y y+δy ⎞ + = 0 (12) h = ⎜ h δy + h δy⎟ 2 2 y+δy ⎜ ∫ y ∫ δy ⎟ ∂x ∂y y + δy ⎝ 0 y ⎠ The finite difference method was used to discretize Eq. (12) ⎛ 1 y ⎞ ⎛ 1 y+δy ⎞ and it yielded ()y + δy h = y⎜ ∫ h δy⎟ + δy⎜ ∫ h δy⎟ y+δy ⎜ y ⎟ ⎜ δy ⎟ ⎡ ⎤ ⎝ y 0 ⎠ ⎝ δy y ⎠ T ()k = Δy 2 ⎛T (k−1 )+ T ()k ⎞+ Δx 2 ⎛T (k−1 )+ T ()k ⎞ / International Science Index, Mechanical and Mechatronics Engineering Vol:5, No:1, 2011 waset.org/Publication/1781 ⎢ ⎜ ⎟ ⎜ ⎟⎥ i, j ⎣ ⎝ i+1, j i−1, j ⎠ ⎝ i, j+1 i, j−1 ⎠⎦ ()y + δy hy+δy = yhy + δyhδy 2()Δy 2 + Δx 2 (13) ()y + δy hy+δy − yhy The finite difference equation for the nodes close to the hδy = (4) δy interfaces as shown in Fig. 2 is Newton’s law of cooling was used to determine the convection heat transfer rates at the furnace door surfaces. The convection heat transfer rates at the furnace door surfaces are

qδy = hδywδy(T∞ − Ts ) (5) where w is the furnace door width.

International Scholarly and Scientific Research & Innovation 5(1) 2011 230 scholar.waset.org/1307-6892/1781 World Academy of Science, Engineering and Technology International Journal of Mechanical and Mechatronics Engineering Vol:5, No:1, 2011

⎛ T (k) T (k−1) T (k) T (k−1) ⎞ ()k ⎡ kΔx ()k kΔx ()k−1 kΔy ()k−1 (k) ⎜ i−1, j i+1, j i, j−1 i, j+1 ⎟ T = ⎢ T + T + T Ti, j = ⎜ + + + ⎟ / 1, j ⎣2Δy 1,J −1 2Δy 1,J +1 Δx 2,J ⎜ R1 R2 R3 R3 ⎟ ⎝ ⎠ ⎤ + ()h1 j ΔyT∞,1 + ()hr,1 ΔyTw / ⎛ 1 1 1 1 ⎞ j ⎦⎥ ⎜ + + + ⎟ (14) ⎝ R1 R2 R3 R3 ⎠ ⎡kΔx kΔy ⎤ ⎢ + + ()h1 j Δy + ()hr,1 Δy⎥ (17) where R1, R2 and R3 are the thermal resistances. The thermal ⎣ Δy Δx j ⎦ resistances are and Δx Δx R1 = A + B ()k ⎡ kΔx ()k kΔx ()k−1 kΔy ()k k Δyw k Δyw T = ⎢ T + T + T A B im, j ⎣2Δy im,J −1 2Δy im,J +1 Δx im−1,J Δx R2 = B ⎤ + ()h2 j ΔyT∞,2 + ()hr,2 ΔyTsur / k B Δyw j ⎦⎥ If Δx A is less than ΔxB , R3 is ⎡kΔx kΔy ⎤ ⎢ + + ()h2 Δy + ()hr,2 Δy⎥ (18) Δy Δy Δx j j R3= ⎣ ⎦ k B Δxw The finite difference equations for the four corners of the furnace door are If Δx A is more than ΔxB , R3 is −1 ()k ⎡ kΔx ()k−1 kΔy ()k−1 ⎛ k A ()()Δx / 2 − ΔxB w k B ()ΔxB + ()Δx / 2 w ⎞ T = ⎢ T + T R3 = ⎜ + ⎟ 1,1 ⎣2Δy 1,2 Δx 2,1 ⎝ Δy Δy ⎠ + ()h ΔyT + ()h ΔyT ⎤ / 1 j ∞,1 r,1 j w ⎦⎥ Material A Material B ⎡kΔx kΔy ⎤ ⎢ + + ()h1 j Δy + ()hr,1 Δy⎥ (19) ⎣ Δy Δx j ⎦ i,j+1 ()k ⎡ kΔx ()k kΔy ()k −1 T = ⎢ T + T Interface 1, jm ⎣2Δy 1, jm−1 Δx 2, jm + h ΔyT + h ΔyT ⎤ / ()1 j ∞,1 ()r,1 j w ⎥ ΔxA ΔxB Δx ⎦ ⎡kΔx kΔy ⎤ + + h Δy + h Δy ⎢ ()1 j ()r,1 j ⎥ (20) i-1,j i,j i+1,j ⎣ Δy Δx ⎦ ()k ⎡kΔx ()k−1 kΔy ()k Δy T = ⎢ T + T im,1 ⎣ Δy im,2 Δx im−1,1 + ()h ΔyT + ()h ΔyT ⎤ / 2 j ∞,2 r,2 j sur ⎦⎥ ⎡kΔx kΔy ⎤ i,j-1 ⎢ + + ()h2 Δy + ()hr,2 Δy⎥ (21) Δy Δx j j Fig. 2 Nodes close to the interface ⎣ ⎦ and The top and bottom sides of the furnace door were assumed ()k ⎡kΔx ()k kΔy ()k T = ⎢ T + T to be insulated. The finite difference equations for the top and im, jm ⎣ Δy im, jm−1 Δx im−1, jm bottom sides are ⎤ 2 ()k 2 (k−1 )2 (k−1 ) + ()h ΔyT + ()h ΔyT / Δy T +Δy T +Δx T 2 j ∞,2 r,2 j sur ⎦⎥ T ()k = i−1,1 i+1,1 i,2 (15) i,1 2 2 ⎡kΔx kΔy ⎤ 2(Δy + Δx ) + + ()h2 Δy + ()hr,2 Δy (22) International Science Index, Mechanical and Mechatronics Engineering Vol:5, No:1, 2011 waset.org/Publication/1781 ⎢ j j ⎥ and ⎣ Δy Δx ⎦ The Gauss Seidel Iteration was used to determine the Δy 2T ()k +Δy 2T (k−1 )+Δx 2T ()k i−1, jm i+1, jm i, jm−1 temperature distribution on the furnace door. The iteration T ()k = (16) i, jm 2()Δy 2 + Δx 2 was terminated when the errors in the temperatures were less than 0.00001. The errors in the temperatures were obtained At the furnace door surfaces, there are three modes of heat from: transfer: conduction, convection and radiation. The finite (k ) (k−1) difference equations are E = Ti, j − Ti, j (23)

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The grid independence tests were performed at 69×168, The distribution of the local convection heat transfer 103×251 and 204×501. The uniform grid size of 69×168 was coefficients on the surfaces of the furnace door for internal o used. The heat transfer balance at the inside and outside temperature and ambient temperature at 1000 and 25 C, surfaces of the furnace door was carried out as shown in Table respectively, is shown in Fig. 5. The coefficient values near 2. the top of the furnace door surface inside of the furnace are higher than those in the other regions of the furnace door TABLE II surface, while the coefficient values of the furnace door GRID INDEPENDENCE TESTS. surface outside of the furnace are inverse to the furnace door Grid size Heat transfer rates at the surfaces of surface inside of the furnace. the furnace door (W) Inner surface Outer surface 69×168 84 84 103×251 84 84 204×501 84 84

The assumptions of this research were the following: 1. The problem was a steady-state condition. 2. Heat conduction in the furnace door was a two dimensional problem. 3. The top and bottom surfaces of the furnace door were insulated. 4. The properties of the furnace door were constant.

IV. RESULTS AND SISCUSSIONS The temperature distribution on the horizontal mid plane of the furnace door at 1000oC internal temperature and 25oC ambient temperature is shown in Fig. 3. The result in the two dimensional problem agrees with the one dimensional

problem. Fig. 4 Heat transfer rates at 20, 25 and 30oC ambient temperature.

International Science Index, Mechanical and Mechatronics Engineering Vol:5, No:1, 2011 waset.org/Publication/1781 Fig. 3 Temperature distribution on the horizontal mid plane of the furnace door at 1000oC internal temperature and 25oC ambient Fig. 5 Local convection heat transfer coefficient distribution on the temperature in one and two dimensional problems. furnace door surfaces.

The heat transfer rates at 20, 25 and 30oC ambient The temperature contour in the furnace door at 1000oC temperature are shown in Fig. 4. The temperatures of the internal temperature and 25oC ambient temperature is shown furnace wall surfaces in the x axis vary from 1000oC to in Fig. 6. The temperature gradients vary with the layers of the 1300oC. The heat transfer rates increase directly with the furnace door. temperatures of the furnace wall surfaces.

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j Number of nodes in the vertical direction jm Maximum number of nodes in the vertical direction r Radiation s Surface sur Surrounding area w Wall surface of the furnace y In y direction ∞ Ambient 1 Inside the furnace 2 Outside the furnace Superscript k Number of iteration

REFERENCES [1] U. Prasopchingchana and P. Laipradit, “Simulation of Heat Transfer in the Glass Panel of an Oven Using an Empirical Correlation for Natural Convection Heat Transfer on the Surfaces of the Glass Panel”, in Proc. the 23rd Conference of Mechanical Engineering Network of Thailand, Chiang Mai, Thailand. 2009. Fig. 6 Temperature contour in the furnace door at 1000oC internal o [2] E. Bilgen, “Conjugate heat transfer by conduction and natural temperature and 25 C ambient temperature. convection”, Applied Thermal Engineering, vol. 29, pp. 334-339. 2009. [3] G. V. Kuznetsov and M. A. Sheremet, “Conjugate natural convection V. CONCLUSION with radiation in an enclosure”, International Journal of Heat and Mass Transfer, vol. 52, pp. 2215-2223. 2009. The temperature gradients in the different materials have [4] H. F. Nouanegue and E. Bilgen, “Heat transfer by convection, different values. The temperature distribution in the horizontal conduction and radiation in solar chimney systems for ventilation of mid plane of the two dimensional problem agrees with the dwellings”, International Journal of Heat and Fluid Flow, vol. 30, pp. 150-157. 2009. result from the one dimensional problem. The highest values [5] J. Xaman, J. Arce, G. Alvarez and Y. Chavez, “Laminar and turbulent of the local heat transfer coefficients occur at the top of the natural convection combined with surface thermal radiation in a square inside surface of the furnace door, and the bottom of the cavity with a glass wall”, International Journal of Thermal Sciences, vol 47, pp. 1630-1638. 2008. outside surface of the furnace door. [6] T. Ait-taleb, A. Abdelbaki and Z. Zrikem, Z., “Numerical simulation of coupled heat transfers by conduction, natural convection and radiation in NOMENCLATURE hollow structures heated from below or above”, International Journal of Thermal Sciences, vol. 47, pp. 378-387. 2008. Symbols [7] H. F. Oztop, Z. Zhao and B. Yu, B., “Conduction-combined forced and h Local convection heat transfer coefficient (W/m2⋅K) natural convection in lid-driven enclosures divided by a vertical solid partition”, International Communications in Heat and Mass Transfer, Average convection heat transfer coefficient h 2 vol. 36, pp. 661-668. 2009. (W/m ⋅K) [8] H. Nouanegue, A. Muftuoglu and E. Bilgen, “Conjugate heat transfer by k Thermal conductivity (W/m⋅K) natural convection, conduction and radiation in open cavities”, Average Nusselt number International Communications in Heat and Mass Transfer, vol 51, pp. Nu 6054-6062. 2008. Pr Prandtl number [9] H. C. Chiu and W. M. Yan, “Mixed convection heat transfer in inclined q Heat transfer rate (W) rectangular ducts with radiation effects”, International Journal of Heat and Mass Transfer, vol. 51, pp. 1085-1094. 2008. Ra Rayleigh number [10] M. Mobedi, “Conjugate natural convection in a square cavity with finite T Temperature (K) thickness horizontal walls”, International Communications in Heat and x Distance in horizontal direction (m) Mass Transfer, vol. 35, pp. 503-513. 2008. y Distance in vertical direction (m) [11] M. M. Molla and M. A. Hossian, “Radiation effect on mixed convection laminar flow along a vertical wavy surface”, International Journal of β Volumetric thermal expansion coefficient Thermal Sciences, vol. 46, pp. 926-935. 2007 ρ Density [12] S. W. Churchill, and H. H. S. Chu, “Correlating equations for laminar Subscripts and turbulent free convection from a vertical plate”, International ave Average Journal of Heat and Mass Transfer, vol. 18, pp. 1323-1329. 1975. [13] J. P. Holman, Heat Transfer, 10 ed., McGraw-Hill, New York. 2010. International Science Index, Mechanical and Mechatronics Engineering Vol:5, No:1, 2011 waset.org/Publication/1781 f Fluid

i Number of nodes in the horizontal direction im Maximum number of nodes in the horizontal direction

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