International Transaction Journal of Engineering, Management, & Applied Sciences & TechnologiesInternational Transaction J...
1. Introduction     Microwave heating of a porous medium are widely implemented in industries such asheating/drying food, ...
operating at a frequency of 2.45 GHz and 300 W of power. From figure 1(b), magnetron (no.1)generates microwave and transmi...
Table 1: The electromagnetic and thermo physical properties used in the computations                                      ...
∂E y       ∂H x ∂H z                                    ε           =       −     − σE y                                  ...
∂E y           ∂E y                                       = ±υ                                               (9)          ...
3.2.1 Flow field equation      The porous medium is assumed to be homogeneous and thermally isotropic. The saturatedfluid ...
the water layer, respectively. The permeability κ and geometric F function are (Abdul-Rahim et.al. 2001; Chamkha et. al. 2...
defined as                 Q = 2πfε 0 ε r tan δ (E y )                                             2                      ...
4.1 Electromagnetic equations and FDTD discretization     The electromagnetic equations are solved by using FDTD method. W...
Figure 3. The temperature distributions taken at 30 seconds are shown as to compare thenumerical solutions with the experi...
wavy behavior corresponding to the resonance of electric field. For the non-uniform porosity, theheating rate is noticeabl...
Fig. 6. Velocity vectors from the two cases of porous medium (a) Non-uniform (b) Uniform.     In terms of flow characteris...
properties of the porous packed bed that affect the heating process markedly.7. Acknowledgement     The authors are gratef...
Nithiarasu P, Seetharamu KN, Sundararajan T. Natural convective heat transfer in a Fluid       Saturated variable porosity...
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Numerical Model of Microwave Heating in a Saturated Non-Uniform Porosity Medium Using a Rectangular Waveguide

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The microwave heating of a porous medium with a non-uniform porosity is numerically investigated, based on the proposed numerical model. A variation of porosity of the medium is considered. The generalized non-Darcian model developed takes into account of the presence of a solid drag and the inertial effect. The transient Maxwell’s equations are solved by using the finite difference time domain (FDTD) method to describe the electromagnetic field in the wave guide and medium. The temperature profile and velocity field within a medium are determined by the solution of the momentum, energy and Maxwell’s equations. The coupled non-linear set of these equations are solved using the SIMPLE algorithm. In this work, a detailed parametric study is conducted for a heat transport inside a rectangular enclosure filled with saturated porous medium of constant or variable porosity. The numerical results agree well with the experimental data. Variations in porosity significantly affect the microwave heating process as well as convective flow pattern.

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Numerical Model of Microwave Heating in a Saturated Non-Uniform Porosity Medium Using a Rectangular Waveguide

  1. 1. International Transaction Journal of Engineering, Management, & Applied Sciences & TechnologiesInternational Transaction Journal of Engineering, Management, & Applied Sciences & Technologies http://www.TuEngr.com, http://go.to/ResearchNumerical Model of Microwave Heating in a Saturated Non-Uniform PorosityMedium Using a Rectangular Waveguide a* aWatit Pakdee , and Phadungsak Rattanadechoa Department of Mechanical Engineering, Faculty of Engineering, Thammasat University, THAILANDARTICLEINFO A B S T RA C TArticle history: The microwave heating of a porous medium with a non-Received 23 August 2010Received in revised form uniform porosity is numerically investigated, based on the proposed23 September 2010 numerical model. A variation of porosity of the medium isAccepted 26 September 2010 considered. The generalized non-Darcian model developed takesAvailable online26 September 2010 into account of the presence of a solid drag and the inertial effect.Keywords: The transient Maxwell’s equations are solved by using the finiteMicrowave heating difference time domain (FDTD) method to describe theNon-uniform porositySaturated porous media electromagnetic field in the wave guide and medium. TheRectangular wave guide temperature profile and velocity field within a medium areMaxwell’s equation determined by the solution of the momentum, energy and Maxwell’s equations. The coupled non-linear set of these equations are solved using the SIMPLE algorithm. In this work, a detailed parametric study is conducted for a heat transport inside a rectangular enclosure filled with saturated porous medium of constant or variable porosity. The numerical results agree well with the experimental data. Variations in porosity significantly affect the microwave heating process as well as convective flow pattern. 2010 International Transaction Journal of Engineering, Management, & Applied Sciences & Technologies. Some Rights Reserved.*Corresponding author (W.Pakdee). Tel/Fax: +66-2-5643001-5 Ext.3143. E-mail addresses:wpele95@yahoo.com. 2010. International Transaction Journal of Engineering, Management, & AppliedSciences & Technologies. Volume 1 No. 1. eISSN: 1906-9642 19Online Available at http://tuengr.com/V01-01/01-01-019-033{Itjemast}_WPakdee.pdf
  2. 2. 1. Introduction  Microwave heating of a porous medium are widely implemented in industries such asheating/drying food, ceramic, biomaterial, concrete etc. Since microwave energy has manyadvantages such as short time process, high thermal efficiency, friendly with environment andhigh product quality. Microwave radiation penetrates into a material and heats the material by adipolar polarization that occurs millions times of each second. The examples of previous studiesabout microwave heating of porous medium are as follow. Chen et al. (1985) developed themodel for predicting local moisture content, density and pressure. Ni et al. (1999) developed amultiphase porous media model to predict moisture transport during an intensive microwaveheating of biomaterials. Zhang et al. (2001) studied microwave sterilization of solid foods usingnumerical modeling and experimental. Heating patterns changed qualitatively with geometry(shape and size) and properties (composition) of the food material, but optimal heating waspossible by choosing suitable combinations of these factors. Ratanadecho et al. (2001, 2002)studied the influence of moisture content on each mechanism (vapor diffusion and capillary flow)during microwave drying process of unsaturated porous material including a multi-layer porousmedium. The packed bed of glass beads was used for their experiment. It was found that thesmall bead size led to much higher capillary forces resulting in a faster drying time. Additionally,location of the medium relative to location of the microwave source had a considerable impact onthe heating process (Cha-um et al., 2009). In the present study, we propose the numerical model for the microwave heating of a porouspacked bed. The variation of the bed porosity is considered since it was proved that the porositydecays away from the wall (Benenati, Brosilow, 1962; Vafai, 1984). Researchers found that theporosity variation could affect characteristics of flow and heat transfer (Abdul-Rahim AK,Chamkha AJ, 2001; Al-Amiri AM, Vafai, K, 1994). To the best knowledge of the authors, noattention has been paid to the porosity variation within the packed bed that is heated bymicrowave energy.2. Experimental Setup  Figure 1 shows the experiment apparatus of microwave heating of saturated porous mediumusing a rectangular wave guide. The microwave system is a monochromatic wave of TE10 mode 20 Watit Pakdee, and Phadungsak Rattanadecho
  3. 3. operating at a frequency of 2.45 GHz and 300 W of power. From figure 1(b), magnetron (no.1)generates microwave and transmits along the z-direction of the rectangular wave guide (no.5)with inside cross-sectional dimensions of 109.2 × 54.61 mm2 that refers to a testing area (circled)and a water load (no. 8) that is situated at the end of the wave guide. On the upstream side of thesample, an isolator is used to trap any microwave reflected from the sample to prevent damagingto the magnetron. Fiberoptic (no. 7) (LUXTRON Fluroptic Thermometer., model 790, accurate to± 0.5oC) is employed for temperature measurement. Figure 1: The microwave heating system with a rectangular wave guide. The fiberoptic probes are inserted into the sample, and situated on the x-z plane at y = 25mm. Due to the symmetrical condition, temperatures are measured for only one side of plane.The samples are saturated porous packed beds that compose of glass beads and water. Acontainer with a thickness of 0.75 mm is made of polypropylene which does not absorbmicrowave energy. In our present experiment, a glass bead 0.15 mm in diameter is examined. The averaged (freestream) porosity of the packed bed corresponds to 0.385. The dielectric and thermal properties ofwater, air and glass bead are listed in Table 1.*Corresponding author (W.Pakdee). Tel/Fax: +66-2-5643001-5 Ext.3143. E-mail addresses:wpele95@yahoo.com. 2010. International Transaction Journal of Engineering, Management, & AppliedSciences & Technologies. Volume 1 No. 1. eISSN: 1906-9642 21Online Available at http://tuengr.com/V01-01/01-01-019-033{Itjemast}_WPakdee.pdf
  4. 4. Table 1: The electromagnetic and thermo physical properties used in the computations (Ratanadecho et al., 2001). Properties Air Water Glassbead Heat capacity, 1007 4190 800 ( C p Jkg −1 K −1 ) Thermal conductivity, 0.0262 0.609 1.0 λ (Wm −1 K −1 ) (Density, ρ kgm −3 ) 1.205 1000 2500Dielectric constant, ε r 1.0 88.15-0.414T+(0.131×10-2)T2-(0.046×10-4)T3 5.1Loss tangent, tan δ 0.0 0.323-(9.499×10-3)T+(1.27×10-4)T2-(6.13×10-7)T3 0.013. Mathematical Formulation 3.1 Analysis of electromagnetic field  Since the electromagnetic field that is investigated is the microwave field in the TE10mode, there is no variation of field in the direction between the broad faces of the rectangularwave guide and is uniform in the y-direction. Consequently, it is assumed that two dimensionheat transfer model in x and z directions would be sufficient to identify the microwave heatingphenomena in a rectangular wave guide (Rattanadecho et al., 2002). The other assumptions are asfollows: 1) The absorption of microwave by air in a rectangular wave guide is negligible. 2) The walls of rectangular wave guide are perfect conductors. 3) The effect of sample container on the electromagnetic and temperature field can be neglected The proposed model is considered in TE10 mode so the Maxwell’s equations can be writtenin term of the electric and magnetic intensities 22 Watit Pakdee, and Phadungsak Rattanadecho
  5. 5. ∂E y ∂H x ∂H z ε = − − σE y (1) ∂t ∂z ∂x ∂H z ∂E y μ =− (2) ∂t ∂x ∂H x ∂E y μ = (3) ∂t ∂z where E and H denote electric field intensity and magnetic field intensity, respectively.Subscripts x, y and z represent x, y and z components of vectors, respectively. Further, ε is theelectrical permittivity, σ is the electrical conductivity and μ is the magnetic permeability. Thesesymbols are ε = ε 0ε r (4) μ = μ0 μr (5) σ = 2πfε tan δ (6) When the material is heated unilaterally, it is found that as the dielectric constant and losstangent coefficient vary, the penetration depth and the electric field within the dielectric materialvaries as well. The boundary conditions for TE10 mode can be formulated as follows: 1) Perfectly conducting boundary. Boundary conditions on the inner wall surface of wave guide are given by Faraday’s law and Gauss’s theorem: E = 0, H ⊥ = 0 (7) where subscripts and ⊥ denote the components of tangential and normal directions, respectively. 2) Continuity boundary condition. Boundary conditions along the interface between sample and air are given by Ampere’s law and Gauss’s theorem: E = E , H = H (8) 3) Absorbing boundary condition. At both ends of rectangular wave guide, the first order absorbing conditions are applied:*Corresponding author (W.Pakdee). Tel/Fax: +66-2-5643001-5 Ext.3143. E-mail addresses:wpele95@yahoo.com. 2010. International Transaction Journal of Engineering, Management, & AppliedSciences & Technologies. Volume 1 No. 1. eISSN: 1906-9642 23Online Available at http://tuengr.com/V01-01/01-01-019-033{Itjemast}_WPakdee.pdf
  6. 6. ∂E y ∂E y = ±υ (9) ∂t ∂z where ± is represented forward and backward direction and υ is velocity of wave. 4) Oscillation of the electric and magnetic intensities by magnetron. For incident wave due to magnetron is given by Ratanadecho et al., (2002) ⎛ πx ⎞ E y = E yin sin ⎜ ⎟ sin (2πft ) ⎜L ⎟ (10) ⎝ x⎠ E yin ⎛ πx ⎞ Hx = sin ⎜ ⎟ sin (2πft ) ⎜L ⎟ (11) ZH ⎝ x⎠ where E yin is the input value of electric field intensity, L x is the length of the rectangularwave guide in the x-direction, Z H is the wave impedance.3.2 Analysis of temperature profile and flow field  The physical problem and coordinate system are depicted in Figure 2. The microwave iscoming to the x-y plane while the transport phenomena on the x-z plane are currentlyinvestigated. Figure 2. Schematic of the physical problem. To reduce complexity of the problem, several assumptions have been offered into the flowand energy equations. 1) Corresponding to electromagnetic field, considering flow and temperature field can be assumed to be two-dimensional plane. 2) The effect of the phase change for liquid layer can be neglected. 3) Boussinesq approximation is used to account for the effect of density variation on the buoyancy force. 4) The surroundings of the porous packed bed are insulated except at the upper surface where energy exchanges with the ambient air. 24 Watit Pakdee, and Phadungsak Rattanadecho
  7. 7. 3.2.1 Flow field equation  The porous medium is assumed to be homogeneous and thermally isotropic. The saturatedfluid within the medium is in a local thermodynamic equilibrium (LTE) with the solid matrix (El-Refaee et al., 1998; Nield, Bejan, 1999; Al-Amiri, 2002). The validity regime of local thermalequilibrium assumption has been established (Marafie, Vafai, 2001). The fluid flow is unsteady,laminar and incompressible. The pressure work and viscous dissipation are all assumednegligible. The thermophysical properties of the porous medium are taken to be constant.However, the Boussinesq approximation takes into account of the effect of density variation onthe buoyancy force. The Darcy-Forchheimer- Brinkman model was used to represent the fluidtransport within the porous medium (Nithiarasu et al., 1997; Marafie, Vafai, 2001). TheBrinkmann’s and the Forchheimer’s extensions treats the viscous stresses at the bounding wallsand the non-linear drag effect due to the solid matrix respectively (Nithiarasu et al., 1997).Furthermore, the solid matrix is made of spherical particles, while the porosity and permeabilityof the medium are varied depending on the distant from a wall. Using standard symbols, thegoverning equations describing the heat transfer phenomenon are given by Continuity equation: ∂u ∂w + =0 (12) ∂x ∂z Momentum equations: 1 ∂u u ∂u w ∂u 1 ∂p υ ⎛ ∂ 2u ∂ 2u ⎞ μu + 2 + 2 =− + ⎜ + ⎟− ε ∂t ε ∂x ε ∂z ρ f ∂x ε ⎝ ∂x 2 ∂z 2 ⎠ ρ f κ (13) − F (u2 + w ) 2 1/ 2 1 ∂w u ∂w w ∂w 1 ∂p υ ⎛ ∂ 2 w ∂ 2 w ⎞ wμ + 2 + 2 =− + ⎜ + ⎟− ε ∂t ε ∂x ε ∂z ρ f ∂z ε ⎝ ∂x 2 ∂z 2 ⎠ ρ f κ (14) − F (u 2 + w2 )1/ 2 + g β (T − T0 ) where ε, ν and β are porosity, kinematics viscosity and coefficient of thermal expansion of*Corresponding author (W.Pakdee). Tel/Fax: +66-2-5643001-5 Ext.3143. E-mail addresses:wpele95@yahoo.com. 2010. International Transaction Journal of Engineering, Management, & AppliedSciences & Technologies. Volume 1 No. 1. eISSN: 1906-9642 25Online Available at http://tuengr.com/V01-01/01-01-019-033{Itjemast}_WPakdee.pdf
  8. 8. the water layer, respectively. The permeability κ and geometric F function are (Abdul-Rahim et.al. 2001; Chamkha et. al. 2002) d pε 3 2 κ= (15) 175(1 − ε ) 2 1.75(1 − ε ) F= (16) d pε 3 The porosity is assumed to vary exponentially with a distance of wall (Benenati, Brosilow1962; Amiri, Vafai 1994; Poulikakos, Renken 1987). Based on these previous studies, weproposed the variation of porosity within three confined walls of the bed; a bottom wall and twolateral walls. The expression that considers the variation of porosity in two directions in the x-zplane is given by ⎡ ⎧ ⎛ bx ⎞ ⎛ b(W − x) ⎞ ⎛ bz ⎞ ⎫⎤ ε = ε s ⎢1 + b ⎨exp ⎜ − ⎟ + exp ⎜ − ⎟ + exp ⎜ − ⎟ ⎬⎥ (17) ⎢ ⎣ ⎩ ⎝ dp ⎠ ⎝ dp ⎠ ⎝ dp ⎠ ⎭⎥ ⎦ where dp is the diameter of glass bead, εs known as a free-stream porosity is the porosity faraway from walls, W is the width of packed bed and b and c are empirical constants. Thedependencies of b and c to the ratio of the bed to bead diameter is small (Vafai, 1984). Vafai(1984) suggested 0.98 and 1.0 for the values of b and c.3.2.2 Heat transfer equation  The temperature of liquid layer exposed to incident wave is obtained by solving theconventional heat transport equation with the microwave power absorbed included as a localelectromagnetic heat generation term: ∂T ∂T ∂T ⎛ ∂ 2T ∂ 2T ⎞ σ +u +w =α ⎜ 2 + 2 ⎟+Q (18) ∂t ∂x ∂z ⎝ ∂x ∂z ⎠ [ε ( ρ c p ) f + (1 − ε )( ρ c p ) s ] where specific heat ratio σ = , α = ke/(ρcp)f is the thermal diffusivity. (ρcp ) f The local electromagnetic heat generation term that is a function of the electric field and 26 Watit Pakdee, and Phadungsak Rattanadecho
  9. 9. defined as Q = 2πfε 0 ε r tan δ (E y ) 2 (19) Boundary and initial conditions for these equations: Since the walls of container are rigid, the velocities are zero. At the interface between liquidlayer and the walls of container, no-slip boundary conditions are used for the momentumequations. 1) At the upper surface, the velocity in normal direction (w) and shear stress in the horizontal direction are assumed to be zero, where the influence of Marangoni flow (Ratanadecho et al., 2002) can be applied: ∂u dξ ∂T η =− (20) ∂z dT ∂x 2) The walls, except top wall, is insulated so no heat and mass exchanges: ∂T ∂T = =0 (21) ∂x ∂z 3) Heat is lost from the surface via natural convection and radiation: ∂T −λ = hc (T − T∞ ) + σ rad ε rad (T 4 − T∞ ) 4 (22) ∂z 4) The initial condition of a medium is defined as: T = T0 at t = 0 (23)4. Numerical Procedure  The description of heat transport and flow pattern within a medium (equations (13)-(18))requires specification of temperature (T), velocity component (u, w) and pressure (p). Theseequations are coupled to the Maxwell’s equations (equation (1)-(3)) by equation (19). Itrepresents the heating effect of the microwaves in the liquid-container domain.*Corresponding author (W.Pakdee). Tel/Fax: +66-2-5643001-5 Ext.3143. E-mail addresses:wpele95@yahoo.com. 2010. International Transaction Journal of Engineering, Management, & AppliedSciences & Technologies. Volume 1 No. 1. eISSN: 1906-9642 27Online Available at http://tuengr.com/V01-01/01-01-019-033{Itjemast}_WPakdee.pdf
  10. 10. 4.1 Electromagnetic equations and FDTD discretization  The electromagnetic equations are solved by using FDTD method. With this method theelectric field components (E) are stored halfway between the basic nodes while magnetic fieldcomponents (H) are stored at the center. So they are calculated at alternating half-time steps. Eand H field components are discretized by a central difference method (second-order accurate) inboth spatial and time domain.4.2 Fluid flow and heat transport equations and finite control volume field  Equations (13)-(18) are solved numerically by using the finite control volume along with theSIMPLE algorithm developed by Patankar. The reason to use this method is advantages of fluxconservation that avoids generation of parasitic source. The basic strategy of the finite controlvolume discretization method is to divide the calculated domain into a number of controlvolumes and then integrate the conservation equations over this control volume over an intervalof time [t , t + Δt ] . At the boundaries of the calculated domain, the conservation equations arediscretized by integrating over half the control volume and taking into account the boundaryconditions. The fully Euler implicit time discretization finite difference scheme is used to arriveat the solution in time. The calculation conditions are as follows:1) Grid size: Δx = 1.0922 mm and Δz = 1.0000 mm2) Time steps: Δt = 2 × 10−12 s and Δt = 0.01 s are used corresponding to electromagnetic field and temperature field calculations, respectively.3) Relative error in the iteration procedures of 10−6 was chosen.5. Results and Discussion  To examine the validity of the mathematical model, the numerical results were comparedwith the experimental data. The computed data, for both uniform and non-uniform, cases wereextracted at 30 and 50 seconds. 28 Watit Pakdee, and Phadungsak Rattanadecho
  11. 11. Figure 3. The temperature distributions taken at 30 seconds are shown as to compare thenumerical solutions with the experimental result. Figure 4. The temperature distributions taken at 50 seconds are shown as to compare thenumerical solutions with the experimental result. The comparisons of temperature distributions on the x-z plane at the horizontal line z = 21mm are shown in Figure 3 and Figure 4 at 30 and 50 seconds respectively. The results show anappreciably improved agreement when variation of porosity within the packed bed is considered.In both of the figures, it is clear that temperature is highest at the middle location since thedensity of the electric field in the TE10 mode is high around the center region in the wave guide. Figure 5 displays temperature contours as a function of time of the two cases which exhibit a*Corresponding author (W.Pakdee). Tel/Fax: +66-2-5643001-5 Ext.3143. E-mail addresses:wpele95@yahoo.com. 2010. International Transaction Journal of Engineering, Management, & AppliedSciences & Technologies. Volume 1 No. 1. eISSN: 1906-9642 29Online Available at http://tuengr.com/V01-01/01-01-019-033{Itjemast}_WPakdee.pdf
  12. 12. wavy behavior corresponding to the resonance of electric field. For the non-uniform porosity, theheating rate is noticeably slower than that for the uniform porosity. The reason behind this is thatmore water content exists near the bottom wall in the case of non-uniform porosity because thereis a higher water-filled pore density closer to the wall. Since water is very lossy, large amount ofenergy can be absorbed as both the incoming waves and the reflected waves attenuate especiallyat the bottom area. This occurrence results in a resonance of weaker standing wave with smalleramplitude throughout the packed bed. The weaker standing wave dissipates less energy which isin turn converted into less thermal energy, giving relatively slow heating rate. Fig. 5. Time evolutions of temperature contour within the porous bed at 20, 40 and 60seconds for uniform case (a-c) and non-uniform case (d-f). 30 Watit Pakdee, and Phadungsak Rattanadecho
  13. 13. Fig. 6. Velocity vectors from the two cases of porous medium (a) Non-uniform (b) Uniform. In terms of flow characteristic, in is observed in Figure 6 that flow velocities in the variable-porosity medium are lower than those in the uniform-porosity medium. This result is attributed tohigher porosities near walls in the non-uniform case. Higher porosity medium corresponds tohigher permeability which allows greater flow velocity due to smaller boundary and inertialeffects. The difference is clear in the vicinity of walls where high velocities carry energy from thewall towards the inner area.6. Conclusion  The microwave heating of a porous medium with a non-uniform porosity is carried out,based on the proposed numerical model. A variation of porosity of the medium is considered tobe a function of the distant from the bed walls. The transient Maxwell’s equations are employedto solve for the description of the electromagnetic field in the wave guide and medium. Thegeneralized non-Darcian model that takes into account of the presence of a solid drag and theinertial effect is included. The numerical results are in a good agreement with the experimentaldata. In addition to the effect on the fluid flow velocity that is larger in the non-uniform case, it isfound that the variation of porosity near the wall has an important influence on the dielectric*Corresponding author (W.Pakdee). Tel/Fax: +66-2-5643001-5 Ext.3143. E-mail addresses:wpele95@yahoo.com. 2010. International Transaction Journal of Engineering, Management, & AppliedSciences & Technologies. Volume 1 No. 1. eISSN: 1906-9642 31Online Available at http://tuengr.com/V01-01/01-01-019-033{Itjemast}_WPakdee.pdf
  14. 14. properties of the porous packed bed that affect the heating process markedly.7. Acknowledgement  The authors are grateful for the financial support provided by the Thailand Research Fundunder Contract No. MGR5180238.8. References Abdul-Rahim AK, Chamkha AJ. Variable porosity and thermal dispersion effects on coupled heat and mass transfer by natural convection from a surface embedded in a non-metallic porous medium. Int. J. Numer. Methods for Heat & Fluid Flow 2001; 11:413-429.Al-Amiri, A.A., Nature convection in porous enclosures: The application of the two-energy equation model, Numerical Heat Transfer Part A, 41, 817-834.Al-Amiri AM, and K.Vafai, Analysis of dispersion effects and non-thermal equilibrium, non- Darcian, variable porosity incompressible flow through porous media. Int. J. Heat Mass Transfer 1994; 37(6):939-954.Benenati RF, Brosilow CB. Void fraction distribution in pack beds. AIChE J. 1962; 8:359-361.Chamkha AJ, Issa C, Khanafer K. Natural convection from an inclined plate embedded in a variable porosity porous medium due to solar radiation. Int. J. Therm. Sci. 2002 ; 41:73- 81.Cha-um W, Pakdee W, Rattanadecho P. Experimental Analysis of Microwave Heating of Dielectric Materials Using a Rectangular Wave Guide (MODE: TE10) (Case study: water layer and saturated porous medium). Experimental Thermal and Fluid Science 2009; 33(3):472-481.Chen Kou W, Davis HT, Davis EA, Joan G. Heat and Mass Transfer in Water-Laden Sandstone: Microwave Heating. AIChE Journal 1985; 31(5):842-848.El-Refaee MM, Elsayed MM, Al-Najem NM, Noor, AA. Natural convection in partially cooled tilted cavities. Int. J. Numerical Methods 1998; 28:477-499.Marafie A, Vafai K. Analysis of non-Darcian effects on temperature differentials in porous media. Int. J. Heat Mass Transfer 2001; 44:4401-4411.Ni, H, Datta AK, Torrance KE. Moisture transport in intensive microwave heating of biomaterials: porous media model. Int. J. Heat and Mass Transfer 1999; 42:1501-1512.Nield DA. Bejan A. Convection in Porous Media. Springer, New York, USA. 1999. 32 Watit Pakdee, and Phadungsak Rattanadecho
  15. 15. Nithiarasu P, Seetharamu KN, Sundararajan T. Natural convective heat transfer in a Fluid Saturated variable porosity medium. Int. J. of Heat Mass Transfer 1997; 40:3955-3967.Poulikakos D, Renken K. Forced convection in a channel filled with porous medium, including the effects of flow inertia, variable porosity and Brinkman frction. ASME J. Heat Transfer 1987; 109:880-888.Ratanadecho, P, Aoki K, Akahori M. Experimental and Numerical study of microwave drying in unsaturated porous material. Int. Comm. Heat Mass Transfer 2001; 28(5):605-616.Ratanadecho, P, Aoki K, Akahori M. Influence of Irradiation Time, Particle Sizes, and Initial Moisture Content During Microwave Drying of Multi-Layered Capillary Porous Materials. J. Heat Transfer 2002; 124(1):151-161.Vafai K. Convective flow and heat transfer in variable-porosity media. J. Fluid Mech. 1984; 147:233-259.Vafai K, Tien CL. Boundary and inertia effects on flow and heat transfer in porous media. Int. J. Heat Mass Transfer 1981; 24:195-203.Zhang H, Datta AK, Taub IA, Doona C. Electromagnetics, heat transfer, and thermokinetics in microwave sterilization. AIChE Journal 2001; 47(9):1957-1968. W. Pakdee is an Assistant Professor of Department of Mechanical Engineering at Thammasat University. He received his Ph.D. (Mechanical Engineering) from the University of Colorado at Boulder, USA. in 2003. Dr. Pakdee has published 8 articles in notable international journals. He has been working in the area of numerical thermal sciences focusing on heat transfer and fluid transport in porous media, turbulent combustion and microwave heating. P. Rattanadecho earned his Ph.D. in Mechanical Engineering from the Nagaoka University of Technology, Japan in 2001. He is currently a Professor at Department of Mechanical Engineering, Thammasat University. He has published more than 40 articles in professional journals and over 70 papers in conference proceedings. Professor Rattanadecho’s research area includes Modern Computational Techniques, Computational Heat and Mass Transfer in Unsaturated Porous Media, Computational Heat and Mass Transport in Phase Change Materials (Moving Boundary Problems and etc.), Computational Heat and Multi-Phases Flow in Porous Media under Electromagnetic Energy, Analysis of Transport Phenomena in Highly Complex System by Statistical Modeling(LBM: Lattice Boltzmann Modeling, Random Walk and Monte-Carlo Technique), Computational Transport Phenomena in HumanBody and Tissue Membrane.*Corresponding author (W.Pakdee). Tel/Fax: +66-2-5643001-5 Ext.3143. E-mail addresses:wpele95@yahoo.com. 2010. International Transaction Journal of Engineering, Management, & AppliedSciences & Technologies. Volume 1 No. 1. eISSN: 1906-9642 33Online Available at http://tuengr.com/V01-01/01-01-019-033{Itjemast}_WPakdee.pdf

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