Heat Transfer Mathematical Modelling Numerical Methods and Information Technology Part 15 docx

Thermal Characterization of Solid Structures during Forced Convection Heating 549 (, ,) (, ,) (, ,) (, ,) (, ,) (, ,) (, ,) (, ,) (, ,) (, ,) (, ,) (, ,) (, ,) x hxyz xyz x hxyz xyz y hxyz xyz hx y z y hxyz xyz z hxyz xyz z hxyz xyz Tnnn Tnnn T nnn Tnnn Tnnn Tnnn Tnn n TnnnTnnn Tnnn Tnnn T nnn Tnnn − − − − − − = − − − ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ (32) where (, ,) x hxyz Tnnn is the heater gas temperature at (,,) x xyz Annn, etc. To achieve a compact description, the temperature differences between a given cell and its adjacent cells are also collected into assistant vectors: ( 1,,) (,,) ( 1,,) (,,) (, 1,) (, ,) (, ,) (, 1,) (, ,) (, , 1) (, ,) (, , 1) (, ,) xyz xyz xyz xyz xy z xyz Kx y z xy z xyz xyz xyz xyz xyz Tn nn Tnnn Tn nn Tnnn Tn n n Tn n n Tnn n Tn n n Tn n n Tnnn Tnnn Tnnn Tnnn − − +− −− = +− −− +− ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ (33) We need only those values of (, ,) Kx y z Tnn n where adjacent cells exist. Therefore (, ,) Kx y z Tnn n vectors will be filtered by the (, ,) xyz Gn n n vectors. In those positions where the (, ,) xyz Gn n n vector is zero, the value of the (, ,) Kx y z Tnn n will be disregarded. The solid-gas interfaces are chosen by (, ,) xyz hn n n . By the application of the assistant vectors and matrixes (29–33) this gives us the general expression of the differential equation system which best describes the model: (, ,) (, ,) (, ,) (, ,) (, ,) d( , , ) d(,,) TT xyz xyz xyz xyz Kxyz h xyz xyz hnnn Annn Tnnn Gnnn Tnnn Tn n n tCnnn ⋅⋅ + ⋅ = (34) If we do not maximize the number of the cell neighbours in the model then the assistant vectors (29), (30), (31) and (33) would be matrices with 6xN dimensions (N is the cell number), therefore Eq. (34) and its implementation would be more complex. 5.3 Application example In the last section we present an application of our model which is the convectional heating of a surface mounted component (e.g. during the reflow soldering) in Fig 15. We investigate how change the temperature of the soldering surfaces of the component if the heat transfer coefficients are different around the component (see more details in (Illés & Harsányi, 2008)). The model was implemented using MATLAB 7.0 software. We have defined a nonuniform grid with 792 thermal cells (the applied resolution is the same as in Fig 15.b), the x-y projection in the contact surfaces can be seen in (Fig. 16.a). In this grid, the 13 contact surfaces are described by 31 thermal cells. But the cells which represent the same contact surface can be dealt with as one. The examined cell groups are shown as squares in Fig. 16.a. Heat Transfer - Mathematical Modelling, Numerical Methods and Information Technology 550 We use the following theoretical parameters: T(0)=175ºC; T h =225ºC; h x =40W/m.K; h -x =75W/m.K; h y = h -y =100W/m.K; h z = h -z =80W/m.K. We investigate an unbalanced heating case when there is considerable heat transfer coefficient deviation between right and left faces of the component (Fig. 16.a). The applied time step was dt=10ms. In Fig. 16.b the temperature of the different contact surfaces can be seen at different times. After 3 seconds from the starts of the heating there are visible temperature difference between the investigated thermal cells occurred by their positions (directly or non-directly heated cells) and different heat conduction abilities. The temperature of the cell groups which are heated directly (1a, 1b, 2a, 2b and 3a–3c) by the convection rises faster than the temperature of the cell groups which are located under the component (4a–4f) and the heat penetrates into them only by conduction way. This temperature differences increase during the heating until the saturation point where the temperature begins to equalize. The effect of the unbalanced heating along the x direction ( xx hh − ≠ ) can also be studied in Fig. 16.b. The cell groups under the left side of the component (1a, 2a, 3a, 4a, 4c and 4e) are heated faster compared to their cell group pair (1b, 2b, 3c, 4b, 4d and 4f) from the left side. This kind of investigations are important in the case of soldering technologies because the heating deviation results in a time difference between the starting of the melting process on different parts of the soldering surfaces. This breaks the balance of the wetting force which results in that the component will displace during the soldering. According to the industrial results, if the time difference between the starting of the melting on the different contact surfaces is larger than 0.2s the displacement of the component can occur (Warwick, 2002; Kang et al., 2005). Fig. 16. a) x − y projection of the applied nonuniform grid; b) temperature distribution of the contact surfaces Thermal Characterization of Solid Structures during Forced Convection Heating 551 Comparing the abilities of our model with a general purpose FEM system gave us the following results. The data entry and the generation of the model took nearly the same time in both systems, but the calculation in our model was much faster than in the general purpose FEM analyzer. Tested on the same hardware configuration, the calculation time was less than 3s using our model, while in the FEM analyzer it took more than 52s. 6. Summary and conclusions In this chapter we presented the mathematical and physical basics of fluid flow and convection heating. We examined some models of gas flows trough typical examples in aspect of the heat transfer. The models and the examples illustrated how the velocity, pressure and density space in a fluid flow effect on the heat transfer coefficient. New types of measuring instrumentations and methods were presented to characterize the temperature distribution in a fluid flow in order to determine the heat transfer coefficients from the dynamic change of the temperature distribution. The ability of the measurements and calculations were illustrated with examples such as measuring the heat transfer coefficient distribution and direction characteristics in the case of free streams and radial flow layers. We presented how the measured and calculated heat transfer coefficients can be applied during the thermal characterization of solid structures. We showed that a relatively simple method as the thermal node theory can be a useful tool for investigating complex heating problems. Using adaptive interpolation and decimation our model can improve the accuracy of the interested areas without increasing their complexity. However, the time taken for calculation by our model is very short (only some seconds) when compared with the general FEM analyzers. Although we showed results only from one investigation, the modelling approach suggested in this chapter, is also applicable for simulation and optimization in other thermal processes. For example, where the inhomogeneous convection heating or conduction properties can cause problems. 7. Acknowledgement This work is connected to the scientific program of the " Development of quality-oriented and harmonized R+D+I strategy and functional model at BME" project. This project is supported by the New Hungary Development Plan (Project ID: TÁMOP-4.2.1/B- 09/1/KMR-2010-0002). The authors would like to acknowledge to the employees of department TEF2 of Robert Bosch Elektronika Kft. (Hungary/Hat- van) for all inspiration and assistance. 8. Reference Barbin, D.F., Neves Filho, L.C., Silveira Júnior, V., (2010) Convective heat transfer coefficients evaluation for a portable forced air tunnel, Applied Thermal Engineering 30 (2010) 229–233. Bilen, K., Cetin, M., Gul, H., Balta, T., (2009) The investigation of groove geometry effect on heat transfer for internally grooved tubes, Applied Thermal Engineering 29 (2009) 753–761. Blocken, B., Defraeye, T., Derome, D., Carmeliet, J., (2009) High-resolution CFD simulations for forced convective heat transfer coefficients at the facade of a low-rise building, Building and Environment 44 (2009) 2396–2412. Heat Transfer - Mathematical Modelling, Numerical Methods and Information Technology 552 Castell, A., Solé, C., Medrano, M., Roca, J., Cabeza, L.F., García, D. (2008) Natural convection heat transfer coefficients in phase change material (PCM) modules with external vertical fins, Applied Thermal Engineering 28 (2008) 1676–1686. Cheng, Y.P., Lee, T.S., Low, H.T., (2008) Numerical simulation of conjugate heat transfer in electronic cooling and analysis based on field synergy principle, Applied Thermal Engineering 28 (2008) 1826–1833. Dalkilic, A.S., Yildiz, S., Wongwises, S., (2009) Experimental investigation of convective heat transfer coefficient during downward laminar flow condensation of R134a in a vertical smooth tube, International Journal of Heat and Mass Transfer 52 (2009) 142–150. Gao, Y., Tse, S., Mak, H., (2003) An active coolant cooling system for applications in surface grinding, Applied Thermal Engineering 23 (2003) 523–537. Guptaa, P.K., Kusha, P.K., Tiwarib, A., (2009) Experimental research on heat transfer coefficients for cryogenic cross-counter-flow coiled finned-tube heat exchangers, International Journal of Refrigeration 32 (2009) 960–972. Illés, B., Harsányi, G., (2008) 3D Thermal Model to Investigate Component Displacement Phenomenon during Reflow Soldering, Microelectronics Reliability 48 (2008) 1062– 1068. Illés, B., Harsányi, G., (2009) Investigating direction characteristics of the heat transfer coefficient in forced convection reflow oven, Experimental Thermal and Fluid Science 33 (2009) 642–650. Illés, B., (2010) Measuring heat transfer coefficient in convection reflow ovens, Measurement 43 (2010) 1134–1141. Incropera, F.P., De Witt, D.P. (1990) Fundamentals of Heat and Mass Transfer (3rd ed.). John Wiley & Sons. Inoue, M., Koyanagawa, T., (2005) Thermal Simulation for Predicting Substrate Temperature during Reflow Soldering Process, IEEE Proceedings of 55 th Electronic Components and Technology Conference, Lake Buena Vista, Florida, 2005, pp.1021-1026. Kang, S.C., Kim C., Muncy J., Baldwin D.F., (2005) Experimental Wetting Dynamics Study of Eutectic and Lead-Free Solders With Various Fluxes, Isothermal Conditions, and Bond Pad Metallization. IEEE Transactions on Advanced Packaging 2005; 28 (3):465–74. Kays, W., Crawford, M., Weigand, B., (2004) Convective Heat and Mass Transfer, (4th Ed.), McGraw-Hill Professional. Tamás, L., (2004) Basics of fluid dynamics, (1 st ed.), Műegyetemi Kiadó, Budapest. Wang, J.R., Min, J.C., Song, Y.Z., (2006) Forced convective cooling of a high-power solid- state laser slab, Applied Thermal Engineering 26 (2006) 549–558. Warwick, M., (2002) Tombstoning Reduction VIA Advantages of Phased-reflow Solder. SMT Journal 2002; (10):24–6. Yin, Y., Zhang, X., (2008) A new method for determining coupled heat and mass transfer coefficients between air and liquid desiccant, International Journal of Heat and Mass Transfer 51 (2008) 3287–3297. 22 Analysis of the Conjugate Heat Transfer in a Multi-Layer Wall Including an Air Layer Armando Gallegos M., Christian Violante C., José A. Balderas B., Víctor H. Rangel H. and José M. Belman F. Department of Mechanical Engineering, University of Guanajuato, Guanajuato, México 1. Introduction At present the design of efficient furnace is fundamental to reducing the fuel consumption and the heat losses, as well as to diminish the environment impact due to the use of the hydrocarbons. To reduce the heat losses in small industrial furnaces, a multi-layer wall that includes an air layer, which acts like a thermal insulator, can be applied. This concept is applied in the insulating of enclosures or spaces constructed with perforated bricks to maintain the comfort, without using additional thermal insulator in the walls (Lacarrière et al., 2003; Lacarrière et al., 2006). Besides it has been used to increase the insulating effect in the windows of the enclosures (Aydin, 2000; Aydin, 2006). Nevertheless, the thickness of the air layer must be such that it does not allow the movement of the air. Free movement of the air is favorable to the formation of cellular flow patterns that increase the heat transfer coefficient, provoking natural convection heat transfer through the air layer, reducing the insulating capacity of the multi-layer wall due to the transition from conduction to convection regime. A way to diminish the effect of the natural convection is to apply vertical partitions to provide a larger total thickness in the air layer (Samboua et al., 2008). In Mexico there are industrial furnaces used to bake ceramics, in which the heat losses through the walls are significant, representing an important cost of production. In order to understand this problem, a previous study of the conjugate heat transfer was made of a multi-layer wall (Balderas et al., 2007), where it was observed that a critical thickness exists which identified the beginning of the natural convection process in the air layer. This same result was obtained by Aydin (Aydin, 2006) in the analysis of the conjugate heat transfer through a double pane window, where the effect of the climatic conditions was studied. For the analysis of the multi-layer wall a model applying the volume finite method (Patankar, 1980) was developed. This numerical model, using computational fluid dynamics (Fluent 6.2.16, 2007), allowed to study the natural convection in the air layer with different thicknesses, identifying that one which provides major insulating effect in the wall. 2. Model of the multi-layer wall Industrial furnaces have diverse forms according to the application. The furnace consists of a space limited by refractory walls which are thermally isolated. In the furnace used to bake ceramics the forms are diverse, depending on the operating conditions of the furnace, also, Heat Transfer - Mathematical Modelling, Numerical Methods and Information Technology 554 the thermal isolation is inadequate requiring redesign of the furnace to adapt it to the conditions of production and operation (U.S.A. Department of Energy, 2004) and to obtain the maximum efficiency of the process. To solve this problem a multi-layer wall including an air layer to reduce the heat losses and improving the furnace efficiency is proposed. The Figure 1(a) shows the configuration of the multi-layer wall, where the thickness of the air layer, L, varies while the height of the wall is constant. The materials of the multi-layer wall and H are defined according to the requirements of the furnace. When the thickness of the air layer is increased, some partitions are applied to obtain the insulating effect in the wall. The Figure 1(b) shows the air layer with partitions. The thickness of the air layer, L, varies from 0 to 10 cm. In each case, the heat flow towards the outside of the furnace wall is calculated to identify the thickness that allows the minimal heat losses, defined as the optimal air layer thickness. Common Brick Ceramic Fiber Air Firebrick g Y X HEAT Common Brick Firebrick Ceramic Fiber Air L/n L L/n L/n (a) (b) Fig. 1. Composition of the multi-layer wall. When the thickness of the air layer is increased, the natural convection leads to the formation of cellular flow patterns that increase the heat transfer coefficient and reduce the isolation capacity (Ganguli et al., 2009). Therefore an air layer with vertical partitions in order to maintain the maximum insulation is proposed. In this air layer, each partition has a thickness near the optimal thickness, which is obtained from the analysis of the conduction and convection heat transfer. According to this analysis, a multi-layer wall with 8 and 10 cm of thickness and two, three or four partitions is applied. The configuration of the air layer with partitions is identified according to the following nomenclature. Analysis of the Conjugate Heat Transfer in a Multi-Layer Wall Including an Air Layer 555 N [ ] N thickness of p artitions o f the air layer the air la y er Ln (1) where n is the partitions number and L is the thickness of the air layer. 3. Mathematical formulation The analysis of the multi-layer wall considers the solution of the conjugate heat transfer in steady state between the solid and the air layer in a vertical cavity, which was obtained by CFD (FLUENT®), where a model in two dimensions with constant properties except density is applied, the Boussinesq and non- Boussinesq approximation (Darbandi & Hosseinizadeh, 2007) for the buoyancy effects were used. The work of compressibility and the terms of viscous dissipation in the energy equation were neglected. The thermal radiation within the air layer was neglected. In the non-Boussinesq approximation, the fluid is considered as an ideal gas. The governing equations for the model are: Continuity equation 0 y x u u xy ∂ ∂ + = ∂∂ (2) Momentum equations component x: 22 22 xx xx xy uu uu uu xy xy ν ⎛⎞ ∂∂∂∂ += + ⎜⎟ ⎜⎟ ∂∂ ∂∂ ⎝⎠ (3) component y: 22 22 yy yy x yy uu uu p uu g xyy xy ν ρ ⎛⎞ ∂∂ ∂∂ ∂ ⎜⎟ +=−+ ++ ⎜⎟ ∂∂∂ ∂∂ ⎝⎠ (4) momentum equation with Boussinesq approximation () 22 0 22 yy yy xy y uu uu uu gTT xy xy βν ⎛⎞ ∂∂ ∂∂ ⎜⎟ +=−+ + ⎜⎟ ∂∂ ∂∂ ⎝⎠ (5) Energy equation 22 22 xy TT TT uu xy xy α ⎛⎞ ∂∂ ∂∂ += + ⎜⎟ ⎜⎟ ∂∂ ∂∂ ⎝⎠ (6) the pressure gradient in the momentum equation in component x is not considered due to the small space between the vertical walls. The velocity in x direction is important only in the top and bottom of the cavity due to the cellular pattern forms by the natural convection (Violante, 2009). Heat Transfer - Mathematical Modelling, Numerical Methods and Information Technology 556 3.1 Density model According to the conditions of the problem, is necessary to apply a model of density. The first model applied was the Boussinesq approximation, where the difference of density is expressed in terms of the volumetric thermal expansion coefficient and the temperature difference. For this approximation, the momentum equation in y is expressed by equation 5, where the buoyancy effect depends only on the temperature. Nevertheless, this approximation is only valid for small temperature differences (Darbandi & Hosseinizadeh, 2007). According to the temperature difference in the vertical cavity, it could be possible to use the ideal gas model to calculate the density in the air layer. Therefore, a model where the density is a function of the temperature applied to the problems of natural convection in vertical cavities subject to different side-wall temperatures (Darbandi & Hosseinizadeh, 2007) is: o p w P R T M ρ = (7) 3.2 Boundary onditions and heat flux The temperature used in baking of ceramics is between 1123 K (850 °C) and 1173 K (900 ºC), and the outside temperature of the furnace may be 300 K (27 ºC). Then, the boundary conditions are: bottom and top adiabatic boundary, left and right isothermal boundary and no-slip condition in the walls (including the partitions). In the solid-gas interface, the temperature and the heat flux must be continuous. The solution can be obtained for laminar flow; this consideration is contained in the Rayleigh number and the aspect ratio, whose ranges are 1000 < Ra L < 10 7 and 10< AR (H/L) < 110 are applied (Ganguli et al., 2009). The Rayleigh number for thin vertical cavities is defined as (Ganguli et al., 2009): () 3 0 37 10 10 i L gTTL Ra β να − <= < (8) To determine the heat losses, the heat flux through the multi-layer wall is calculated by applying the following equation: 3 12 eff " io brick iso firebrick TT q l ll L kkk k − = ++ + (9) where l´s is the thickness of each material used in the wall and e ff k is the effective conductivity obtained from the combination of the conduction and natural convection effects present in the air layer, which is define as: eff air y kkNu= (10) where the average Nusselt number is related to the average of heat transfer coefficient in the vertical cavity, which is obtained numerically, y air hH Nu k = (11) Analysis of the Conjugate Heat Transfer in a Multi-Layer Wall Including an Air Layer 557 4. Results and discussion The results were obtained for a multi-layer wall where the inside temperature, T i , of the furnace is 1173 K (900º C) and the outside temperature, T o , is 300 K (27º C). The multi-layer wall is formed by four different materials; see Figure 1(b), whose properties are showed in the Table 1 (Incropera & DeWitt, 1996). Common Brick Firebrick cb ρ 1920 kg/m 3 f b ρ 2050 kg/m 3 _ p cb C 835 J/kg K _ pf b C 960 J/kg K cb k 0.72 W/m K f b k 1.1 W/m K Ceramic Fiber Air (750 K) c f ρ 32 kg/m 3 air ρ 0.4880 kg/m 3 _ p c f C 835 J/kg K _ p air C 1081 J/kg K c f k 0.22 W/m K air k 0.05298 W/m K air μ 3.415x10 -5 kg/m s air α 100.4x10 -6 m 2 /s Table 1. Properties of the materials in the multi-layer wall. According to the properties of the fluid and assuming the film temperature of 750 K, the maximum Rayleigh number is 6 1.37 10 L Ra x= , this value is related to the range of the equation (8), where the laminar flow governs the movement of the fluid. In order to quantify the natural convection, an analysis with both models of the density was done. For the model using the approach of Boussinesq, the equation of momentum in the y direction is equation (5). In the second model with no-Boussinesq approximation, the equations (4) and (7) are applied, where the pressure changes inside the vertical cavity are neglected. The results obtained with both density models are showed in the Table 2, where the ideal gas model is used to analyze the conjugate heat transfer through the multi-layer wall because the Boussinesq approximation fails to predict the correct behavior of the natural convection when the temperature gradient is large (Darbandi & Hosseinizadeh, 2007). Configuration Heat flux (W/m2) Boussinesq approximation Ideal gas model 8[1] 621.31 715.37 8[2] 463.94 565.86 8[3] 423.10 465.49 8[4] 463.29 429.57 10[1] 629.31 715.37 10[2] 468.28 576.42 10[3] 389.91 470.07 10[4] 458.80 400.84 Table 2. Heat flux to Boussinesq and ideal gas model. Heat Transfer - Mathematical Modelling, Numerical Methods and Information Technology 558 The Figure 2 shows the heat flux by conduction and convection through the air layer for different thickness, obtained from equation (9). The conduction heat flux curve shows the insulating effect of the air layer without movement, with the Nusselt number equal to unity, where the heat losses through the multilayer wall continuously decrease for any thickness. The natural convection heat flux curve shows an asymptotic behavior with a constant minimal heat flux, the natural convection heat is produced by cellular flow patterns inside the air layer where the Nusselt number increases. The minimal heat flux is present in thicknesses greater than 3 cm, identifying this value as the optimal thickness to maintain the insulating capacity of the air layer inside the multi-layer wall. Q convection Q conduction 600 650 700 750 800 850 900 950 1000 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 Thickness m q" W/m2 Fig. 2. Heat flow through the air layer. 0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00 4.50 5.00 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 The avera ge Nusselt number Thickness m Fig. 3. Average Nusselt number in the air layer. [...]... Experimental unsteady characterization of heat transfer in a multi-layer wall including air layers—application to vertically perforated bricks Energy and Buildings, 38, 232-237, ISSN: 0378-7788 Patankar S.V (1980) Numerical Heat Transfer and Fluid Flow, Hemisphere, ISBN: 0-07-0487405, New York 566 Heat Transfer - Mathematical Modelling, Numerical Methods and Information Technology Samboua V.; Lartiguea B.;... b z Tb∞(t), ub∞(t) Fig 1 Physical model and coordinate system For the one-dimensional case shown in Fig.1, heat and moisture move along the z axis only When the effect of the moisture content (or potential) gradient in the energy equation is 570 Heat Transfer - Mathematical Modelling, Numerical Methods and Information Technology neglected, the transient heat and moisture diffusion equations for the... configuration we can see the flow 562 Heat Transfer - Mathematical Modelling, Numerical Methods and Information Technology patterns, where it is verified that the greater velocity is in the center of the cavity and this one falls when the partitions are added in the air layer Nevertheless, the best parameter to identify the insulating effect of the air layer is the heat transfer through the multi-layer... double-layer plate subjected 568 Heat Transfer - Mathematical Modelling, Numerical Methods and Information Technology to time-varying hygrothermal loadings at the external surfaces, and analytical solutions for the temperature and moisture fields are presented The solutions are explicitly derived without complicated mathematical procedures such as Laplace transform and its inversion by applying an... air layer and the heat transfer by conduction is dominant in that region of the thin cavity; however, convection becomes important at the top and bottom corners of the cavity (Ganguli et al., 2009) (a) (b) Fig 7 (a) and (b) Temperature contours in the multi-layer wall 564 Heat Transfer - Mathematical Modelling, Numerical Methods and Information Technology (c) (d) Fig 7 (c) and (d) Temperature contours... (19b) (19c) 574 Heat Transfer - Mathematical Modelling, Numerical Methods and Information Technology In order to obtain Eqs (18) and (19), the following orthogonal relationship with discontinuous weight functions in terms of the eigenfunction Rim(z) was used: 2 Λi ∑L ∫ i =1 i Zi Z i −1 ⎧const (m = k ) Rim ( z ) Rik ( z )dz = ⎨ (m ≠ k ) ⎩ 0 (20) 3 Numerical results and discussion Numerical calculations... 19-42 578 Heat Transfer - Mathematical Modelling, Numerical Methods and Information Technology Sugano, Y & Chuuman, Y (1993 a) An analytical solution of hygrothemoelastic problem in multiply connected region Transactions of the Japan Society of Mechanical Engineers A 59, 151 9 -152 5 Sugano, Y & Chuuman, Y (1993 b) Analytical solution of transient hygrothermoelastic problem due to coupled heat and moisture... annular cylinder by linear theory of coupled heat and moisture Applied Mathematical Modelling 21, 721-734 An Analytical Solution for Transient Heat and Moisture Diffusion in a Double-Layer Plate 577 Chang, W J & Weng, C I (2000 a) An analytical solution to coupled heat and moisture diffusion transfer in porous materials International Journal of Heat and Mass Transfer 43, 3621-3632 Chang, W J & Weng,... 4(a) shows the velocity profiles in the vertical 560 Heat Transfer - Mathematical Modelling, Numerical Methods and Information Technology direction in the air layer for the configurations 8 [1] and 10 [1] According to equation (8) these configurations correspond to the air layer without partitions, where the velocity is greater near the vertical walls and practically zero in the center of the air layer,... analytical solution of a boundary-value problem of heat conduction for tribosystem, consisting of the homogeneous semi-space, sliding uniformly on a surface of the strip deposited on a semi-infinite substrate, was obtained in paper (Yevtushenko and Kuciej, 2009a) 580 Heat Transfer - Mathematical Modelling, Numerical Methods and Information Technology Heat conditions on the friction surface of two elastic . 2. Heat flux to Boussinesq and ideal gas model. Heat Transfer - Mathematical Modelling, Numerical Methods and Information Technology 558 The Figure 2 shows the heat flux by conduction and. convective heat transfer coefficients at the facade of a low-rise building, Building and Environment 44 (2009) 2396–2412. Heat Transfer - Mathematical Modelling, Numerical Methods and Information Technology. Fig. 7. (a) and (b) Temperature contours in the multi-layer wall. Heat Transfer - Mathematical Modelling, Numerical Methods and Information Technology 564 (c) (d) Fig. 7. (c) and (d)