On FVM Transport Phenomena Prediction in Porous Media with Chemical/Biological Reactions or Solid-Liquid Phase Change

The aim of the chapter is to show the FVM capabilities in the accurate and efficient prediction of transport phenomena in porous media including either biological, chemical reactions or liquid-solid phase transformations. Four applied technological problems are solved with the FVM. Problem 1 is related to heat and mass diffusion in a saturated porous media where chemical and biological reactions occur. Such a situation can be found in compost piles resulting from contaminated water treatments. Field experimental data are used to assess the quality of FVM calculations for temperature and oxygen concentration distribution time variations along the prediction of thermal explosions (Moraga, 2009) and effect of the moisture in compost piles self-heating (Zambra, 2011). Problem 2 deals with the improvement of thermal energy efficiency and pollution reduction in hydrocarbons combustion. Methane combustion with air in a cylindrical porous burner is investigated by solving 2D unsteady continuity, linear momentum, energy, and chemical species governing equations with the FVM. Sensibility studies performed via numerical tests allowed to obtain numerical results for unsteady velocity and temperature distributions, along to the displacement of the combustion zone. The effects of inlet reactants velocity (methane and air) in the range 0.3-0.6 m/s; excess air ratios between 3 and 6 and porosities of 0.3 up to 0.6 in the fluid dynamics, forced convection heat transfer and combustion process are described (Moraga, 2008). Problem 3 is devoted to characterize 3D natural convection and heat conduction with solidification inside a cavity filled with a porous media, in which a Darcy-Brinkman-Forchheimer flow model is used. This infiltration technique can be applied to produce new materials with enhanced physical properties. A fixed grid method is used along the FVM to solve, with a temperature dependent liquid fraction, the moving boundary problem by using a power law model for binary non Newtonian alloys (Moraga, 2010; Moraga, 2010). Problem 4 describes 3D turbulent convective heat transfer in a bioreactor, including the self-heating of the porous material due to chemical and biological reactions. FVM simulation based on a coupled heat mass transfer external forced convection model is used to assess the effects of reactor geometry, selfheating parameters, air flow and temperature in the bioreactor performance. The mathematical model includes the convective turbulent flow of momentum, energy and oxygen concentration, with

the κ-ε turbulence model, and the diffusion of energy and oxygen concentration in the saturated porous medium. Numerical results for the dependent variables are successfully validated with experimental data. Issues such as the use of dynamic time steps, under-relation of dependent variables and local refined meshes are discussed in each one of the problems solved. The pressure-velocity-temperature-chemical species coupling is discussed and a novel PSIMPLER method for the FVM is presented . The stability, rate of convergence and efficiency of the PSIMPLER method is determined by solving natural, forced and mixed h e a t c o n v e c t i o n i n s i d e c a v i t i e s b y c o m p a r i s on with the solution obtained by using the standard SIMPLE algorithm. Improvements achieved in convergence rates by modifying the predictor-corrector schemes used to solve the discretized fluid mechanics, heat and mass transfer equations are discussed in some of the numerical experiments presented. (Patankar, 1980) Convective fluid dynamics/heat and mass transfer, for either laminar or turbulent flows of Newtonian or non-Newtonian fluids, with phase change in porous media is described by partial differential equations. Systems of nonlinear second order partial differential equations can be efficiently solved numerically using the finite volume method. Each governing equation is treated in the generalized form for a transport equation, with unsteady, convection, diffusion and linearized source terms:

General scheme for the classic Finite Volume Method
where  is the dependent variable,  density, t time, u  velocity vector,  diffusion coefficient, S c independent source term and S p dependent source term. No convection is considered inside the volume occupied by the MWM, since the flow velocities are zero in that region, and diffusion becomes the only transport mechanism inside this porous medium. This "blocked" region is implemented by setting a very large numerical viscosity (10 30 ) for the control volumes enclosed in such region, which renders the velocity essentially zero (~10 -30 ) inside the material.
The time integration is performed with an explicit Euler scheme: At each time step, the system of discretized nodal equations for each main dependent variable (velocity components, temperature and mass fraction) is solved iteratively by internal iterations, with a combination of the alternating tri-diagonal matrix algorithm (TDMA) and Gauss-Seidel method. The sequential coupling of these main variables is accomplished by the SIMPLE method in the external iterations. Under-relaxation is always applied to the dependent variables during these external iterations.
gravitational and the source term effects, and is introduced a new term  D  , the follow equation can be used: The volumetric water content  is the quotient between water volume and the total sample volume, so it is has not unit and its values are between 0 and 1.
The effects of the porosity and type of soil should be introduced by the  D  parameter. A non-linear equation for this parameter is reported for Serrano (Serrano, 2004).
The constants 12 ,,   and  , may be obtained by experimental field test. The Eqs. (3) and (4) are used when the specific hydraulic properties of the compost pile are not available. Oxidation and microorganism activity inside the pile are incorporated in the model by volumetric heat generation. For simplicity, local thermal equilibrium is assumed, which is a common assumption for porous medium and packed particle beds (Nield & Bejan, 1992). The equations for the temperature and the oxygen concentration are : In Eqs. (5) and (6)  A and c A are the pre-exponential factor for the oxidation of the cellulose and bio-mass growth, respectively. The coefficients 1  and c  are parameters that allows relations between the variables T and ox C which are function of the moisture and oxygen concentration at time t. The constants c E , 1 E , 2 E , are the activation energy for the cellulose, bio-mass growth and inhibition of biomass growth, respectively. The effects of the vaporization of water in the internal energyare calculated with the third term of the right hand side of Eq. (5), where v L is the vaporization enthalpy, va  is the water vapor density, () q  is the mass water flux and v X is the vapor quality. The total porosity ( fl  ) is calculated in terms of the apparent density and real density, w  and air  are the fraction of water and air into the pore respectively.
;; 1 real app fl flwa i r fls app Thermodynamics equilibrium and ideal mixture between oxygen and water are assumed in the porous medium, while the oxygen concentration within the pile is described by In equations (3) and (4), c A is the pre-exponential factor for the oxidation rate of the cellulose, and. Heat and mass transfer properties in the porous medium are defined in terms of the pile porosity  as where eff K and eff D are respectively the effective thermal conductivity and diffusion coefficient, which are considered independent of the temperature and concentration. In these expressions the subscript c refers to the cellulose. Details for the formulation of the term representing the heat generated by the biomass have been given by Chen and Mitchell (Chen & Mitchell, 1996). The parameter values used in the mathematical model for the processes that take place inside the porous block of waste material were obtained from reference (Sidhu et al., 2007) and are presented in Table 1. Table 1.Constant values used in the mathematical model for the internal processes in the compost pile.

Numerical results of temperature time evolution and grid study
A 2D case of self-heating in a rectangular porous pile, with 2.5 m height (H) and 5 m length (L), was investigated using three grids, with 100x100, 200x200 and 300x300 nodes, and three time steps: 300s, 600s and 3600s.The temperature time evolution was calculated in the three positions: f) H/4, L/4; g) H/2, L/2 and h) 3H/4, 3L/4.Results of the time needed to cause auto-ignition, in days, are shown in  Table 2.Days before the self-ignition in positions f), g) and h) within the compost pile. Fig. 1 shows the time evolution results of temperature in the f) position: H/4, L/4, calculated with a grid of 300x300 nodes using two time steps: 300s and 3600s. A typical heating curve for the temperature is observed in the Figure 1a. Auto-ignition at H/4, L/4 occurs after 247 days. The temperature increased suddenly in one day from 370 to 515K. A complex system of solid, liquid and gaseous fuels as a final result of the cellulosic oxidation originated a volumetric heat generation causing the self-ignition process. Temperature decreasing in time, characterized the last stage, in which the fuel reserves in the location are exhausting. Figure 1b, a zoom view of Fig. 3a for the time interval between days 240 and 255, shows that a time step reduction from 3600s to 300s allows to determine a more accurate prediction (within 1 day) for the time needed to insatiate the self-ignition in a 2.5m height compost pile. Due to the previous analysis, a mesh with 300x300 nodes and a dynamic time step with 300s during the auto-ignition and 3600s in other states may be used in the calculation of the processes of self-heating with thermal explosion. .

Comparison between experimental and numerical results.
The experimental data and numerical results in 2D obtained with the FVM for a 2.5 m high pile and trapezoidal form, are compared in

Inhibition of the self-ignition in the sewage sludge waste water treatment
The coupled heat and mass diffusion equations system of partial differential equation (11) (Moraga and Zambra, 2008) proposed a novel method for the inhibition of the self-ignition in large piles. A uniform mesh with 94x32x62 nodes in x, y and z directions was found to be adequate by comparison of results between plane 2D and the central plane 3D (x and y directions) in a trapezoidal pile. Figure 3 compares the numerical results 2D and 3D in three positions inside the pile. The small differences of the values are produced for the third dimension incorporated in the simulations. The main differences in the temperature values occur when the self-ignition reached the tested position.

Numerical simulation of a cylindrical porous medium burner
A porous media combustor, built on base of alumina spheres placed inside of an axisymmetric cylindrical quartz tube of 0.52 m in length and 0.076 m in diameter, is shown in Fig. 7. Methane and air mixture enter to the combustor at ambient temperature with uniform velocity. To start the combustion a temperature profile of one step type, with a maximum temperature of 1150 K and a thickness of 4 cm is assumed to simulate the ignition by means of an external energy source. In the combustion zone, the products: CO 2 , H 2 O, O 2 and N 2 are generated. Air, gas and products are assumed to behave as ideal gases and hence www.intechopen.com density is calculated in terms of temperature from the ideal gas state equation. Burners based on this technology has been investigated and tested for many industrial applications (Foutko et al., 1996, Zhdanok et al., 1995. The assumptions used to build the mathematical model include: single-step chemical reaction, laminar 2D flow of Newtonian fluid of ideal gases. The mathematical model includes the porosity terms in both the energy equations for the solid as well as for the gas; similarly, also the continuity, linear momentum and fuel mass fraction equations are included. All physical properties are variable with temperature, and it is postulated that density varies according to the ideal gases state equation. The chemical reaction for methane is considered to be in a single step, with excess air included.
Continuity and ideal gas equations Linear momentum in radial r and axial z directions: Fuel mass conservation equation Two energy equations were used for gas and solid in the porous medium where the chemical reaction speed and the effective conductivity of the solid including radiation are: The coefficient of convective heat transfer between solid and gas is calculated as follows Conjugate boundary conditions between the porous burner section and the annular one are used in the internal and external areas of the inner tube: The parameters used in the simulation included the Stephan-Boltzmann constant (  ), combustion enthalpy ( COMB h  ), frequency factor (K), activation energy (E a ), and the universal gas constant (Ro), whose respective values are:

Solution procedure
The coupled, strongly non-linear system of partial differential equations was solved numerically using the FVM, with the SIMPLE algorithm (Patankar, 1980). A fifth power law was used to calculate the convective terms while the diffusion terms were determinated by linear interpolation functions for the dependent variables between the nodes. Each one of the governing equations was written in the general form of the transport equation, with unsteady, convective, diffusion and linearized source terms: A non-uniform grid with 622 x 15 nodes, in axial and radial directions, respectively, was found by a trial and error procedure to be efficient to solve the discretized model with accuracy and a reasonable computation time . The iterative solution procedure was based on the use of the line by line method,that combines the TDMA algorithm with the under-relaxed Gauss-Seidel algorithm. The under-relaxation coefficients used were equal to 0.1, for velocity components, and 0.5, for gas and solid temperatures and for the fuel ratio. A strategy based on a dynamic time step was implemented to calculate the unsteady terms. The initial time step used in the calculation procedure was equal to 0.00001s until t = 0.001s, and then was increased by and order of magnitude as time increased one order of magnitude, to end up with a time step equal to 0.1s when time was over 0.1s.

Results and discussion
Numerical experiments were performed to assess the effect of porosity, inlet gas velocity and excess air coefficient on the fluid dynamics and heat transfer in the porous media burner. The effect of increasing porosity from 0.3 to 0.6 on the axial velocity at four axial locations is shown in figure 8a, while the influence of the inlet velocity is depicted in figure  8b. The increments of porosity causes a reduction in axial velocity and in the velocity gradients near the walls. Axial velocity increases with time and a maximum value is reached for ε = 0.3, when t = 600s, at 0.15m from the inlet. Secondary flows are observed near the wall (z = 0.21m) when ε = 0.3. The axial velocity profile increases with time, when the initial velocity Uo = 0.3m/s, reaching a maximum at 900s, at z= 0.15m and up to 1500s for z = 0.21m.
Superadiabatic combustion in the porous media combustor causes temperature increaments from 300 K, near the inlet and close to the oulet, to 1600 K in the flame region, as depicted in figure 9. A displacement of the combustion front toward the middle of the combuster is achieved by increasing the air excess ratio ψ from 3.0 to 6.0. Similar effects in the temperature distribution can be noticed when porosity increased from ε = 0.3 to ε = 0.6 and when the inlet velocity increased from Uo = 0.3 m/s to Uo = 0.3 m/s. The displacement of the combustion zone, typical for porous combustor with uniform porosity and cross section, in the direction of the burner exit has been shown to be strongly influenced by increments in the values of: excess air ratio, porosity and inlet reactants velocity. The porous combustor design requires a reduced combustion front displacement. The results obtained (table 2) show that the displacement velocity of the combustion front decreases when the porosity and inlet reactants are reduced, and when the excess air ratio increases. Table 3 shows that a change in the mathematical model, from 1D to 2D, caused increments of about 5% in the gas temperature, 7% in the solid temperature and up to 3% in the combustion front velocity.   Table 3. Comparison between results of 1D and 2D models for porous media burner.

Physical situation and mathematical model
The design of a porous post-combustor for a wood stove is investigated with FVM in order to reduce emissions. Figure 11 shows the post-combustor location in the gas exhaust tube along with some of the boundary conditions used in the analyis.Primary air inlet is located in the lower section of the stove front, secondary air enters in the upper section of the lateral walls through a three ways valve that allows high, medium or low secondary air injection and a terciary air inlet located in the upper section on the back, that allows for pre-heating of the auxiliary air. The alumina porous combustor with a length of 300mm and a diameter of 150mm, was designed with two sections with different porous diameter of 1.52mm (length equal to 100mm) and 5.6 mm (with a length of 200mmm). An ignition temperature of 1150K was assumed in the simulations.

Solution procedure
A two stages procedure was used in the numerical study. In the first stage, a 3D turbulent ke model was used to describe, with ANSYS/Fluent, the fluid mechanics and the convective heat transfer in the stove. Temperature and velocity distributions at the entrance of the exhaust stove pipe were found from the 3D model. Then, in the second stage, a secondary combustion process was described from a 2D laminar forced convection model in the porous post-combustor.
a. 3D k-e model for the turbulent gas flow inside the stove: Continuity, ideal gas, linear momentum, energy, species transport, turbulent kinetic energy and rate of dissipation equations: ;P r 0 . 8 5 Pr The values for the five constants used in the k-e turbulence model are those suggested by Launder and Spalding, 1974. 123 The air flow for the three operational modes inside the stove combustion chamber is shown in figure 12.

Inyector primary air
Reductor secondary air Tempering secondary air Combustion gas The 3D model for the combustion zone of the stove was discretised with 736,767 tethahedral, 18,032 wedge and 280 pyramidal elements. Under-relaxation coefficients for pressure were: 0.2 for pressure, 0.3 for the three velocity components, 0.7 for the kinetic energy and for the dissipation rate of the kinetic energy and 0.9 for temperature. Axial discretization for the 2D porous postcombustor included 40 nodes in the pre-heating section, from z = 0 to z = 2cm, 450 nodes in the flame region, up to z= 24cm, 100 nodes in the post-secondary combustion zone (from z=24 to z=30cm) and 30 nodes in the last zone (from z= 30cm to z= 60cm in z direction). The convergence was assumed when the maximum deviation for each dependent variable, at each control volume and for all time steps, Φ k i,j -Φ k-1 i,j was smaller than 0.0001 for the velocity components and smaller than 0.001 for solid and gas temperature. Under-relaxation factors were equal to 0.1 for the velocity components, 0.5 for gas and solid temperature and equal to 0.3 for pressure.

Results and discussion
Velocity and temperature distributions in the central plane of the wood stove are described in Figure 13, for three air operation modes. Higher velocities, in the order of 2m/s are found in the left lower section and in the central part of the stove. A 30% velocity increment is obtained in the low mode. Temperature in the primary combustor is the range between 800K and 1365K, with higher temperature reached in the low mode.  www.intechopen.com Figure 14, in the left side, depicts the air and gas trajectories in the primary combustor, where higher velocities in the range of 2.3m/s are found for the three operation modes. Time evolution for the temperature distribution in the porous post-combustor, is shown in the right hand side of Figure 13. A hot region, with temperatures in the range between 800K and 1150K can be observed to last for 5 minutes and a zone with temperatures higher than 850K can be noticed during the first 50 minutes of operation.

Alloy solidification and natural non-Newtonian convection predicted with a porous model
Liquid to solid phase change is a relevant process in many industrial applications, such as: polymer casting moulding, pure metals and alloys solidification, solar energy storage and food freezing and thawing. The pourpose of this section is to describe numerical solutions obtained with a porous media model and the FVM that have been applied to solidification of pure metals and alloys. The sequential solution of the discretized system of fluid mechanics and convective heat transfer is accomplished by the PSIMPLER algorithm .

Physical situation and mathematical model
The physical situation related to each one of the three cases studied is shown schematically in Figure 15. The first case corresponds to the solidification of pure aluminum, and the second is the solidification of aluminum alloys with 1.7% Si. In all cases the liquid to solid phase change occurs inside a square cavity with the right vertical wall and the horizontal ones being adiabatic. The left-side wall is subjected to a convective condition, of the Robin type, for case 1, while in case 2 an imposed temperature condition is assumed, of the Dirichlet type. The fluid mechanics in the mushy zone and in the liquid phase is based on laminar flow, with a power law non-Newtonian model (n=0.5). Density is assumed to vary linearly with temperature, according to the approximation of Boussinesq. A porous medium model, with a Darcy numberDa=1.37×10 -5 [26], for an average pore diameter dm = 1.2x10 -5 is proposed to describe the fluid motion in the mushy zone, along with the general Darcy-Brinkman-Forchheimer porous flow model. Fig. 15. Solidification of a pure metal, a binary and a terniary alloy in a square cavity.
The mathematical model includes continuity, linear momentum and energy equations in porous media 0 uv xy where the liquid phase change fraction, being equal to 0 when T<T S and equal to 1 when T > T L is calculated from The dynamic apparent viscosity is η = η l /f pc , for Newtonian and non-Newtonian fluids, with η l and the deformation rate for the power law model defined as follows

Computational implementation
The mathematical models presented in the previous equations were solved with the finite volume method and the PSIMPLER algorithm developed and programmed in FORTRAN. The PSIMPLER algorithm is a mixture of two algorithms, SIMPLER and PISO . In all the cases, the grid used was of the overlapping type, and 40x40, 60x60 and 80x80 grid sizes were evaluated. The results are presented for a 60x60 grid, which was efficient in time and accurate enough. The values for the under-relaxation factors used for the two velocity components, temperature and pressure were different for pure metals and for binary and ternary alloys α u = α v = 0.2 ; α T =0.1 ; α P = 0.7 (pure metal) and α u = α v = 0.5 ; α T = 0.3 ; α P = 0.9 (alloys) The iterative procedure was finished when the difference between Φ k i,j -Φ k i,j at two successive iterations was smaller or equal to , for all control volumes and at each time step, with=10 -6 for velocity and =10 -3 for temperature.

Results and discussion
The first case describes the solidification of pure aluminum metal, with Ra = 10 5 , by using the proposed Darcy-Brinkman-Forchheimer porous model for the mushy zone and three alternative temperature dependent liquid phase fractions, defined by changing the exponent www.intechopen.com m in Eq. (44). Figure 16 shows that the time evolution for the temperature distributions obtained with the FVM, for m = 0.5, 1.0 and 2.0, in dashed lines is in agreement with the values calculated by the finite element method and the classical mathematical model, shown in continuos lines (Cruchaga et al., 2000). The time evolution of the solidification front for the aluminum, calculated with the general porous media model is shown in figure 17 to be in agreement with the results obtained with the classical model. In case 2, the melt characterization by a non-Newtonian power law model, with a power index n = 0.5, is investigated along the use of the DBF porous model for the mushy zone, to describe the solidification of the binary Al-1.7wt%Si, when Ra=2.5x10 5 . Figure 18 describes the evolution in time of the isotherms (at the left side) and of the steamlines in the liquid phase and in the mushy zone (at the rigth side), calculated with FVM by assuming either a Newtonian or a non-Newtonian power law fluid models. The pseudoplastic fluid assumption (n=0.5) originates a slighty faster convection, requiring lower time to complete the solid to liquid phase transformation.

Mixed turbulent convection and diffusion in a bioreactor
In the last decades, the municipal waste materials (MWM) outputs in many countries have increased significantly due to the large increase in population and industries. These wastes contain organic matter that can be recovered, and through recycling, MWM can be returned to the environment. One option in that direction, for example, is to use them as fertilizers.
Another interesting possibility is to use them as fuels. In this way, in addition to solving the problem of MWM disposal, the recycling of these materials becomes a useful source of energy. For this, the MWM require usually a pre-treatment to eliminate water, in order to have good combustion. This process is essentially a method for humidity control of the waste, which has a great impact in combustion efficiency.
The numerical modeling of heat and mass transfer involved in the drying process without self-heating has motivated several studies using finite difference methods (Kaya et al., 2006;Kaya et al., 2008, Chandra & Talukdar, 2010Sheng et al., 2009). These studies have provided useful insight into the phenomenon, but the air flow regime in a typical drying bioreactor is nevertheless turbulent. Therefore there is interest in the analysis and characterization of this process in the fully turbulent flow regime. In this case also the selfheating of the material due to chemical and biological reactions is described.

Physical situation of an experimental bioreactor
In order to compare the model results with experimental data, we perform the computations for a case with the dimensions of the experimental bioreactor: 1 m × 1 m × 1 m, in the x, y and z coordinate directions respectively (see Fig. 19). The walls are assumed adiabatic and impermeable. Air at ambient temperature is forced into the reactor through an inflow in the plane z = 0. This inflow has a section of 0.05 m × 0.05 m and is located at the center of the lower wall. The air entering through this opening flows past the block of MWM inside the reactor fulfilling the drying of this material. The air outflow is located at the center of the top wall (z = 1 m) and has the same dimensions of the inflow located below. The physical situation presented in this work is the same that is studied experimentally in reference (Rada et al., 2007), which provides a good source for validation of the general mathematical modeling.

Mathematical model for the bioreactor
The κ-ε turbulence model, equations (28) to (35), is used for the modeling of air flow within the reactor (Launder & Spalding, 1974), coupled with a diffusion model proposed by Sidhu et al. (Sidhu et al., 2007), and presented in equations (11) to (15).
The air flow inside the reactor is governed by continuity and by the Navier-Stokes equations.
The initial velocity is taken as zero:  

Numerical simulation with finite volume method
The sequential coupling of these main variables (external iterations) is done with the SIMPLE method. Under-relaxation is applied during these external iterations, with a relaxation coefficient of 0.5 for all the variables, except for the pressure correction for which a value of 0.8 was used instead.
The bioreactor was discretized with a non-uniform mesh of 32×62×82 finite volumes in x, y and z directions, respectively. Additionally, a buffer zone extending for 1 m beyond the exit of the reactor is considered, in order to ensure a zero-gradient condition at the outlet of the computational domain. The waste material occupies a volume inside the reactor equivalent to a parallelepiped with dimensions of 0.5×0.5×0.6 m 3 . The discretization of this sub-region of the domain comprises 16×40×40 finite volumes in the x, y and z directions, respectively. The simulation was carried out for a total time of 696 hr. The time step was 3600 s for most of this period. During the first hour a dynamic time step was used (starting with a step of 0.001 s) in order to achieve convergence at each time step during the initial fast transient period starting from the prescribed initial condition.

Comparison with experimental data
In the experimental bio-drying reactor analyzed in reference (Rada et al, 2007), three thermocouples were installed inside the MWM, along the z direction, at equal intervals of 0.2 m in z direction inside of MWM, to measure the internal temperature in the material. Figure 20 presents the numerical results of temperature evolution obtained with the mathematical model and the numerical simulation described in previous sections. The numerical results in Fig. 20 Fig. 20 are consistent with the behavior showed by Rada et al. (2007). The self-heating, product of the biological and chemical heat generation is clearly observable. Initially, a sudden increase of temperature occurs in the three sampled positions until 90 h. For all times, the temperature at point P2 is greater than the others. This can be attributed to its more inner position, making more difficult for the locally generated heat to diffuse to the surface, where is removed by the flowing air. Points P1 and P3, being located nearer to the surface, can release more easily the generated heat to the surrounding air. The asymmetry in the temperature at P1 and P3 is due to the location of P1 on the side subjected directly to the impinging flow of cold air.
After about 96 h, the temperatures at the three points moderate their rate of increase, and the maximum temperatures occur at about t = 168 h for all these positions. The maximum temperature reached inside the MWM is close to 340 K, at the central position P2. The existence of these maxima can be associated with an intrinsic self-moderation of the rate of heat generation, given by the last term in Eq. (11). That term, describing the heat released by the biological activity of micro-organisms initially increases with the temperature, as their population grows, but after exceeding 318 K approximately this source of heat starts to decrease, because of the progressive inhibition in micro-organism growth as the temperature continues to increase. After 168 h, the temperature decreases in all positions as the internal heat source has become weaker and at the same time a heat diffusion pattern inside the material has established, allowing and effective removal of the generated heat towards the surrounding air.
The table 4 shows a comparison between computed temperatures at the sampled points and experimental data from Ref. (Rada et al., 2007)

Conclusions
The FVM capabilities to produce efficient and accurate prediction of fluid mechanics, heat and mass transfer in porous media, including biological and chemical reactions or liquid to solid phase transformations, have been shown by solving four practical examples. In the first case, auto-ignition of compost piles was studied and field experimental data were used to assess the quality of FVM results for the evolution of temperature and oxygen concentration distributions. The conclusion found for this case was that a pile heigth equal to 1.7m is a critical value to produce self-ignition. Combustion in porous media, predicted by FVM, allows to conclude in the second case, that improvements on thermal energy efficiency and pollution reduction in a methane porous combustor and in a wood stove, can be achieved. The use of the FVM and a generalized DBF flow porous media model for the mushy zone was found to describe convective cooling during solidification of non-Newtonian melted binary alloys in the third case. A conclusion found for this type of phase change processes was that the CPU time requiered for the numerical simulation can be reduced one order of magnitude by using a new improved predictor-corrector sequential algorithm, PSIMPLER, for the pressure-velocitytemperature-concentration calculation procedure. In the fourth case, the FVM along the k-ε turbulence model, were used to describe 3D turbulent convective heat transfer, with self heating of the porous media due to chemical and biological reactions, in a bioreactor. The conclusions found in this case were that the FVM simulations for the dependent variables were successfully validated with experimental values and that the effects of reactor geometry, selfheating parameters, air flow and temperature in the bioreactor perfomance can be evaluated. Finally, it is concluded that the use of FVM and adequate mathematical models along to experimental physical results can be used to investigate physical, biological and chemical coupled problems in order to achieve improved thermal efficiency, adequate use of energy resources and pollution reduction in these processes.