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Self-heating in a bioreactor: Coupling of heat and mass transferwith turbulent convection

C.E. Zambra a,⇑, C. Rosales b, N.O. Moraga c, M. Ragazzi d

a Centro de Investigación de Recursos Hídricos (CIDERH), Universidad Arturo Prat, Arturo Prat 2120, Iquique Zip code: 1100000, Chileb Departamento de Ingeniería Mecánica, Universidad Técnica Federico Santa María, Av. España 1680, Valparaíso Zip code: 2340000, Chilec Departamento de Ingeniería Mecánica, Universidad de La Serena, Benavente 980, La Serena Zip code: 1700000, Chiled Department of Civil and Environmental Engineering, University of Trento, Via Mesiano 77, Trento Zip code: I-38050, Italy

a r t i c l e i n f o

Article history:Received 31 December 2010Received in revised form 8 July 2011Accepted 8 July 2011Available online 2 August 2011

Keywords:Self-heating in a bioreactorBio-drying reactorMunicipal waste materialFinite volume simulationj–e turbulence model

a b s t r a c t

A tridimensional mathematical model that considers the turbulent flow inside a bioreactor of municipalwaste material (MWM) is studied. The MWM is self-heating by the biological activity of micro-organisms.The model includes the unsteady solution of the turbulent flow field inside the reactor, the energy trans-port in the flow and the ensuing heat and mass transfer by convection over the MWM load. The MWM istreated as a porous medium and the oxygen depletion and heat generation produced by aerobic bacteriaare also included. Turbulence treatment is done with the j–e model. The numerical simulation showsgood agreement with experimental measurements available. The results also shed light over some designcriteria that can be explored in order to increase the efficiency of the process.

� 2011 Elsevier Ltd. All rights reserved.

1. Introduction

In the last decades, the municipal waste materials (MWM) out-puts in many countries have increased significantly due to thelarge increase in population and industries. These wastes containorganic matter that can be recovered, and through recycling,MWM can be returned to the environment. One option in thatdirection, for example, is to use them as fertilizers. Another inter-esting possibility is to use them as fuels. In this way, in addition tosolving the problem of MWM disposal, the recycling of these mate-rials becomes a useful source of energy.

It is known that self-heating and auto-ignition can occur incompost piles produced in waste water treatment and also inMWM. This phenomenon occurs as a result of biological activity,when the temperatures are low, and by chemical oxidation if thetemperatures are sufficiently high [1,2]. When these waste materi-als are employed as a source of energy, they require usually a pre-treatment to eliminate water, in order to have good combustion.This process is essentially a method for humidity control of thewaste, which has a great impact in combustion efficiency. Severalresearchers have studied different aspects of the drying process.Some have focused on the search for a better drying [3,4] and effi-ciency [5,6] in a bioreactor. Others have studied ideal aeration and

inoculation [7], the release of volatile organic compounds [8] andbehavior of PCDD/F (Polychlorinated Dibenzo-p-Dioxins) duringthe process of municipal solid waste (MSW) bio-drying [9]. Essen-tially, in a drying bioreactor, a flow of air is forced to pass over thesurface of the waste material. The numerical modeling of heat andmass transfer involved in the drying process without self-heatinghas motivated several studies [10–14]. Forced laminar convectionfor drying of rectangular moist objects was studied employing fi-nite differences in two dimensions in Ref. [15]. A two-dimensionalstudy considering various dryer configurations was also reportedin Ref. [16], while a three-dimensional case, but without porousmedia, was analyzed in Ref. [17]. A non-Darcian porous media withfilm evaporation, as a result of convective heat and mass transfer,is presented by Leu et al. [18] using a finite difference numericalmethod. These studies have provided useful insight into the phe-nomenon, but the air flow regime in a typical drying bioreactoris nevertheless turbulent. Therefore there is interest in the analysisand characterization of this process in the fully turbulent flowregime.

It is well known that a 3D turbulent flow poses a challengingproblem in terms of computational resources and model capabili-ties. A large volume of literature is available in connection withthe problem of computational modeling of turbulent flows.Numerical techniques range from simple phenomenologicalapproaches to direct numerical simulation (DNS) of the Navier–Stokes equations. Several novel methods have been studied (e.g.,

0017-9310/$ - see front matter � 2011 Elsevier Ltd. All rights reserved.doi:10.1016/j.ijheatmasstransfer.2011.07.025

⇑ Corresponding author.E-mail address: carlos.zambra.s@gmail.com (C.E. Zambra).

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a recent new variant of the lattice Boltzmann method, such as inRef. [19]). In Ref. [20], a comparison of performance for various tur-bulence models is presented. In this particular application, involv-ing complex three-dimensional flow coupled with heat and masstransfer, a good compromise between a sound description of thephysics of the mean flow and reasonable computer resources isprovided by the so called complete two-equation turbulence mod-els. Among these, the j–e model is the most widely used. It wasinitially introduced by Jones and Launder [21,22], with severaladditional contributions for improvements in the model constantsand boundary conditions treatment (e.g. Refs. [23–28], and refer-ences therein).

In this paper, we study the transient tridimensional turbulentflow inside a MWM bioreactor, including the self-heating of thematerial due to chemical and biological reactions. Mathematicalmodeling and numerical simulation based on a model of coupledheat and mass transfer by forced convection is used to assess thereactor geometry, the self-heating parameters, as well as the airflow and temperature effects on the bioreactor performance. Themathematical modeling includes the convective turbulent flow ofmomentum, energy and oxygen concentration outside the MWM,and the diffusion of energy and oxygen concentration in the satu-rated porous medium within the MWM. The j–e turbulence modelis used for the modeling of air flow within the reactor [29], coupled

with a diffusion model initially proposed by Sidhu et al. [30], andused by Moraga et al. [31], for predicting the auto-ignition in com-post piles and recently extended to consider the humidity insidethe pile [32]. The coupled three-dimensional differential equationscomprised by this mathematical model are solved with a finite vol-ume method (FVM) formulation [33–35] for an experimental bio-reactor. Results obtained with the mathematical model andnumerical simulations show very good agreement with experi-mental data.

2. Physical configuration and mathematical model

2.1. General physical arrangement of the bioreactor

The mathematical model considers the transient coupled three-dimensional transport of momentum and thermal energy in thedrying air, as well as the diffusion of heat and oxygen concentra-tion in the waste material, treated as a porous medium. The phys-ical configuration is shown in Fig. 1, and corresponds essentially toa rectangular chamber which contains the compacted material tobe dried. In order to compare the model results with experimentaldata, we perform the computations for a case with the dimensionsof the experimental bioreactor: 1 m � 1 m � 1 m, in the x, y and

Nomenclature

Ac pre-exponential factor for the oxidation of cellulose(m3/kg s)

A1 pre-exponential factor for the biomass growth (m3/kg s)A2 pre-exponential factor for the inhibition of biomass

growth (J/mol)C constants in the turbulence modelCp specific heat capacity (J/kg K)Cox oxygen concentration (kg/m3)dc distance to the walldþc dimensionless mixing lengthD mass diffusion coefficient (m2/s)E1 activation energy for the biomass growth (J/mol)Ec activation energy for the cellulose (J/mol)E2 activation energy for the inhibition of biomass growth

(J mol/biomass)g gravity (m/s)G production of kinetic turbulent energy (N/m2 s)H height of the compost pile (m)h convective heat transfer coefficient (W/m2 K)k thermal conductivity (W/m K)kT thermal conductivity of the air (W/m K)K von Karman constant (0.4187)L half-length of the compost pile (m)p pressure (Pa)q00 volumetric heat generation (W/m3)Q exothermicity for the oxidation (J/kg)R ideal gas constant (J/K mol)Sc independent source termSp dependent source termT temperature (K)t time (s)u velocity (m/s)Uc tangential velocity to the wall outside of laminar sub

layer (m/s)Uf friction velocity on the wall (m/s)~v velocity vectorxl maximum dimension in x coordinate (m)yl maximum dimension in y coordinate (m)

zl maximum dimension in z coordinate (m)(x, y, z) coordinates (m)

Greek symbolsC diffusion coefficientDt time step (s)e dissipation rate of turbulent kinetic energy (m2/s2)ec dissipation rate of turbulent kinetic energy on the wall

(m2/s2)e porosity of municipal waste materials (MWM)j turbulent kinetic energy (m2/s2)jc turbulent kinetic energy on the wall (m2/s2)q density (kg/m3)l dynamic viscosity (N s/m2)lt turbulent viscosity (N s/m2)d indicator of gravity directionr Prandlt numbersw shearti kinematic laminar viscosity/ dependent variable

Subscript0 initial conditionsair air propertiesBt bottomb biomassc celluloseeff effective propertyin inleti, j vector directionout outletT temperaturewm waste materialse dissipation rate of turbulent kinetic energyj turbulent kinetic energy

Superscript– averaged values

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zcoordinate directions respectively. The walls are assumed adia-batic and impermeable. Air at ambient temperature is forced intothe reactor through an inflow in the plane z = 0. This inflow has asection of 0.05 m � 0.05 m and is located at the center of the lowerwall. The air entering through this opening flows past the block ofMWM inside the reactor fulfilling the drying of this material. Theair outflow is located at the center of the top wall (z = 1 m) andhas the same dimensions of the inflow located below.

2.2. Mathematical model for the bioreactor

The air flow inside the reactor is governed by the Navier–Stokesequation

@

@tðquiÞ þ

@

@xjðqujuiÞ ¼

@p@xiþ @sij

@xij

� qgdi3; ð1Þ

where ui is the velocity vector field, q is air density, p is the pressureand sij is the deviatoric stress tensor. In this section the coordinateswill be denoted with index notation (and implied summation overrepeated indices is used). The last term represents the body forceassociated with the gravitational acceleration g, acting along thex3 (i.e. z) direction. The flow must also comply with mass conserva-tion expressed as

@q@sþ @

@xiðquiÞ ¼ 0: ð2Þ

These equations are solved in their Reynolds-averaged form. Theflow is treated as one of variable density but quasi-incompressible(i.e. low-Mach number approximation). In this way, densitychanges are associated with temperature variations, but they arenot directly coupled with the pressure. This means also that allthe density–velocity correlations in the Reynolds decompositionof the Navier–Stokes equations are negligible, as well as some dila-tational terms. The Reynolds-averaged governing equations thenbecome

@

@tð�q�uiÞ þ

@

@xjð�q�uj�uiÞ ¼

@�p@xiþ @

@xjðlþ ltÞ

@�ui

@xjþ @

�uj

@xi

� �� �� �qgdi3;

ð3Þ

and

@�q@tþ @

@xið�q�uiÞ ¼ 0; ð4Þ

where the bars denote time averages, and the turbulent stresseshave been included through the turbulent viscosity variable lt.The isotropic part of the turbulent stress tensor has been includedinto the pressure term.

The turbulent viscosity is obtained by means of the j–e model.In terms of the turbulent kinetic energy j, and its rate of dissipa-tion e, it can be expressed, by dimensional arguments as

lt ¼ qCuj2

e; ð5Þ

where Cu is a phenomenological constant (see below). The variablesj and e are obtained by solving their corresponding transportequations

@

@tð�qjÞ þ @

@xjð�q�uijÞ ¼

@

@xi

lþ lt

rj

� �@x@xj

� �þ G� �qe; ð6Þ

@

@tð�qeÞ þ @

@xjð�q�uieÞ ¼

@

@xi

lþ lt

re

� �@e@xj

� �þ GC1

ej� C2 �q

e2

j: ð7Þ

The term G in Eq. (6) is the rate of production of turbulent kineticenergy by the action of the mean-flow velocity gradient tensoragainst the turbulent stresses:

G ¼ ðlþ ltÞ@�ui

@�xjþ @

�ui

@�xj

� �@�ui

@xj: ð8Þ

Parameters C1 and C2 are closure constants of the model, as well asrj and re which are equivalent Prandtl numbers for the turbulentdiffusion of kinetic energy and dissipation. The whole set of modelconstants is Refs. [26–28] the following:

C1 ¼ 1:44; C2 ¼ 1:92; Cu ¼ 0:09; rj ¼ 1:0; re ¼ 1:217: ð9Þ

The thermal energy equation for the air can be written as

@

@tð�qTÞ þ @

@xjð�q�uiTÞ ¼

@

@xikT þ

lt

rT

� �@T@xj

� �; ð10Þ

where rT is the turbulent thermal Prandtl number and kT is themolecular thermal diffusion coefficient. The air is considered asan ideal gas mixture and hence the mean density can be obtainedfrom the ideal gas equation of state.

The model to be applied within the volume occupied by theMWM was introduced in Ref. [28], in a two-dimensional applica-tion. Cellulosic oxidation and micro-organism activity inside theMWM are taken into account by the model in the form of a volu-metric heat generation source. As a simplifying hypothesis, localthermal equilibrium is assumed to prevail. This is a commonassumption for porous media and packed particle beds [36]. Theheat transfer equation within the MWM takes thus the form

ðqCpÞeff@T@t¼ keffr2T þ Qcð1� �ÞAcqcCox exp

�Ec

RT

� �

þ Qbð1� �Þqbqc

A1 exp �E1RT

� �1þ A2 exp �E2

RT

� �" #

; ð11Þ

while the oxygen concentration in the MWM is given by

�@Cox

@t¼ Deffr2Cox � ð1� �ÞAcqcCox exp

�E2

RT

� �: ð12Þ

In Eqs. (11) and (12), Ac is the pre-exponential factor for the oxida-tion rate of the cellulose, and Ec, E1, E2, are activation energies forthe cellulose oxidation, biomass growth and inhibition of biomassgrowth, respectively. Heat and mass transfer properties in the por-ous medium are defined in terms of the pile porosity e as

keff ¼ �kair þ ð1� �Þkc; ð13Þ

ðqCpÞeff ¼ �qairCp;air þ ð1� �ÞqcCp;c; ð14Þ

Deff ¼ �Dair;c; ð15Þ

where keff and Deff are respectively the effective thermal conductiv-ity and diffusion coefficient, which are considered independent of

Fig. 1. Physical configuration of the bio-drying reactor.

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the temperature and concentration. In these expressions the sub-script c refers to the cellulose.

The heat produced due to the cellulosic material oxidation isrepresented by the second term on the right hand side of Eq.(11). This term is expressed as a function of the oxygen concentra-tion and it is responsible for the self-heating. The last term in Eq.(11) represents the heat generated by the biological activity withinthe pile, caused by micro-organisms growth. The oxygen content isdetermined by Eq. (12). The initial content of oxygen in the pile ofwaste material corresponds to the same concentration of oxygen inthe external air. The oxygen is depleted by the micro-organismsgrowth process and the organic matter oxidation. This oxygen con-sumption is included through the second term on the right handside of Eq. (12).

Details for the formulation of the term representing the heatgenerated by the biomass have been given by Chen and Mitchell[37]. Most of the parameter values used in the mathematical modelfor the processes that take place inside the porous block of wastematerial were obtained from reference [30], and they are summa-rized in Table 1.

2.3. Initial and boundary conditions

Classical ‘‘wall functions’’ are applied for the flow conditions inthe finite control volumes adjacent to the walls. The velocity com-ponent tangential to the wall, Uc, (outside the laminar sublayer) isassumed to follow the logarithmic law of the wall,

Uc

Uf¼ 1

Kðdþc =dþ0 Þ; ð16Þ

where K is the von Karman constant (K = 0.4187) and dþc is thedimensionless distance from the control volume center to the wall:

dþc ¼ Uf dc=m; ð17Þ

In this expression, m is the kinematic viscosity and Uf is the frictionvelocity defined as Uf ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffisw=rho

pwith sw being the shear stress on

the wall. The parameter dþ0 corresponds to the wall rugosity non-dimensionalized in the form (17) (the value taken is dþ0 ¼ 0:102,which corresponds to relatively smooth walls). It is assumed thatthe turbulence is in local equilibrium in this boundary layer flow,so that the production G equals the local dissipation e. The term Gis dominated by the velocity gradient normal to the wall, whencelt( oUc/odc)2 = qe. Using Eq. (5) to substitute e, and the fact that

ltoUc/odc, is the shear stress in the region near to the wall, the resul-tant expression for the turbulent kinetic energy at the nodes adja-cent to the walls is

jc ¼U2

fffiffiffiffiffiffiCup : ð18Þ

Similarly, the corresponding expression for the dissipation calcu-lated at the wall boundary control volumes is

ec ¼U3

f

Kdc: ð19Þ

Eq. (19) is used directly to prescribe the dissipation at the wall con-trol volumes, while for the turbulent kinetic energy boundary con-dition. Eqs. (18) and (19) combined are used to compute the localvalue of the turbulent viscosity (Eq. (5)), which is then applied tothe source term of the transport equation. The friction velocitycomes from Eq. (16).

The previous relations allow also to calculate the stress on thewall as

sw ¼qKC

14uj

12c

lnðdþc =dþ0 ÞUc; ð20Þ

which substitutes the diffusive flux of momentum across the facetowards the wall, for the boundary control volumes, in the momen-tum conservation equations (3).

The initial velocity is taken as zero: ~vðx; y; z;0Þ ¼ 0. The pre-scribed value for the inlet velocity is 1 m/s in the z coordinate:

uinðx; y;0; tÞ ¼ 0; v inðx; y;0; tÞ ¼ 0; winðx; y;0; tÞ ¼ 1 m=s: ð21Þ

At the outlet, zero-gradient outflow boundary conditions areapplied.

The walls are considered adiabatic and impermeable. The tem-perature of the air at the inlet is constant and an initial linear dis-tribution in the domain was assumed:

Tðx; y;0; tÞ ¼ 283 K; ð22Þ

Tðx; y; z;0Þ ¼ 283� z=0:2: ð23ÞInside the MWM the process is considered essentially as diffusion inporous medium, and therefore the velocities are zero in the regionoccupied by the block of material. Constant oxygen concentration isimposed around the MWM volume,

Coxðx; y;0; tÞ ¼ 0:272 kg=m3 ðfor ðx; y; zÞ outside the MWMÞ:ð24Þ

The initial temperature and oxygen concentration within the MWMare considered homogeneous,

Tðxwm; ywm; zwm;0Þ ¼ 283 K; Coxðxwm; ywm; zwm;0Þ¼ 0:272 kg=m3: ð25Þ

3. Numerical procedure

The system of partial differential equations governing thisproblem was solved numerically using the finite volume method[33–35]. The calculation was carried out with an original finite-vol-ume computer code, written in Fortran. Each governing equationwas treated in the generalized form (26) for a transport equation,with unsteady, convection, diffusion and linearized source terms:

@ðq/Þ@t

þ divðq~u/Þ ¼ divðCgrad/Þ þ Sp/þ Sc ð26Þ

No convection is considered inside the volume occupied by theMWM, since the flow velocities are zero in that region, and diffusionbecomes the only transport mechanism inside this porous medium.

Table 1Constant values used in the mathematical model for theinternal processes in the porous municipal waste material(MWM) [30].

Parameter Value

Ac 1.8 � 104

A1 2 � 106

A2 6.86 � 1032

Cair 1005Cc 3320Do,air 2.4 � 10�7

Ec 1.1 � 105

E1 1.0 � 105

E2 2.0 � 105

Cox,0 0.272Qb 7.66 � 106

Qc 5.5 � 109

kair 0.026kc 0.18e 0.3qair 1.17qb 575qc 1150

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This ‘‘blocked’’ region is implemented by setting a very largenumerical viscosity (1030) for the control volumes enclosed in suchregion, which renders the velocity essentially zero (�10�30) insidethe material. The wall-functions boundary conditions for the flowof air are applied on the surface of this block in the same way asit is done on the reactor walls. The time integration is performedwith an explicit Euler scheme

@u@t¼ /tþDt � /t

Dt: ð27Þ

At each time step, the system of discretized nodal equations foreach main variable (velocity components, temperature, k, e, etc.)is solved iteratively (internal iterations), with a combination ofthe alternating tridiagonal matrix algorithm (TDMA) and Gauss–

Seidel method. The sequential coupling of these main variables(external iterations) is done with the SIMPLE method. Under-relax-ation is applied during these external iterations, with a relaxationcoefficient of 0.5 for all the variables, except for the pressure correc-tion for which a value of 0.8 was used instead.

The bioreactor was discretized with a non-uniform mesh of32 � 62 � 82 finite volumes in x, y and z directions, respectively.Additionally, a buffer zone extending for 1 m beyond the exit ofthe reactor is considered, in order to ensure a zero-gradient condi-tion at the outlet of the computational domain. The waste materialoccupies a volume inside the reactor equivalent to a parallelepipedwith dimensions of 0.5 � 0.5 � 0.6 m3. The discretization of thissub-region of the domain comprises 16 � 40 � 40 finite volumesin the x, y and z directions, respectively. The simulation was carriedout for a total time of 696 h. The time step was 3600 s for most ofthis period. During the first hour a dynamic time step was used(starting with a step of 0.001 s) in order to achieve convergenceat each time step during the initial fast transient period startingfrom the prescribed initial condition.

4. Results and discussion

4.1. Validation

The physical situation presented in this work is the same that isstudied experimentally in Ref. [3], which provides a good sourcefor validation of the general mathematical modeling. In the exper-imental bio-drying reactor analyzed in Ref. [3], three thermocou-ples were installed inside the MWM, along the z direction, atequal intervals of 0.2 m in z direction inside of MWM, to measurethe internal temperature in the material. Fig. 2 presents thenumerical results of temperature evolution obtained with themathematical model and the numerical simulation described inSections 2 and 3 of this article. The numerical results in Fig. 2 wereobtained in the following three (x, y, z) positions: P1(0.49, 0.44,0.19), P2(0.49, 0.44, 0.39) and P3(0.49, 0.44, 0.59). The values forall coordinates are in meters.

The results for the time evolution of the temperature in Fig. 2are consistent with the behavior showed by Rada et al. [3]. Theself-heating, product of the biological and chemical heatgeneration is clearly observable. Initially, a sudden increase of tem-perature occurs in the three sampled positions until 90 h. For alltimes, the temperature at point P2 is greater than the others. This

Time, h

Tem

pera

ture

, K

0 96 192 288 384 480 576 672280

290

300

310

320

330

340

Position1Position2Position3

Fig. 2. Temperature evolution in three positions of the MWM.

Table 2Comparison of temperatures (K) between experimental data [3] and calculatedvalues.

Time,h

Temperature, K

Thermocouple2, Ref. [3]

P1 Thermocouple3, Ref. [3]

P2 Thermocouple4, Ref. [3]

P3

96 317 315 335 337 330 329168 323 325 335 338 335 334360 307 307 331 329 312 317696 290 291 298 299 292 293

Fig. 3. Velocities in z coordinate (xy plane) and three-dimensional streamlines within the bio-drying reactor.

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can be attributed to its more inner position, making more difficultfor the locally generated heat to diffuse to the surface, where is re-moved by the flowing air. Points P1 and P3, being located nearer tothe surface, can release more easily the generated heat to the sur-rounding air. The asymmetry in the temperature at P1 and P3 isdue to the location of P1 on the side subjected directly to theimpinging flow of cold air. After about 96 h, the temperatures atthe three points moderate their rate of increase, and the maximumtemperatures occur at about t 168 h for all these positions. Themaximum temperature reached inside the MWM is close to340 K, at the central position P2. The existence of these maximacan be associated with an intrinsic self-moderation of the rate ofheat 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 popu-lation grows, but after exceeding 318 K approximately this sourceof heat starts to decrease, because of the progressive inhibition inmicro-organism growth as the temperature continues to increase.After 168 h, the temperature decreases in all positions as the inter-nal heat source has become weaker and at the same time a heatdiffusion pattern inside the material has established, allowingand effective removal of the generated heat towards the surround-ing air.

Table 2 shows a comparison between computed temperaturesat the sampled points and experimental data from Ref. [3] at corre-sponding positions. The temperatures calculated in the present

Fig. 4. Streamlines in yz plane, within the bio-drying reactor.

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study approximate very well the experimental values in the threecompared positions. A maximum difference of 5 K can be observedat 360 h in the position P3.

4.2. Numerical results and discussion

Fig. 3 shows the magnitude of the velocity components in the zdirection for three transverse planes xy, and the three-dimensionalstreamlines inside of the bioreactor at t = 696 h. As can be ex-pected, the maximum vertical velocities inside the reactor occurnear the inlet and the outlet convergence. The flow pattern dis-played by the streamlines in Fig. 3 presents an extensive recirculat-ing region around the lateral walls of the MWM block. This flowpattern can be observed with more details in Fig. 4, which showsbidimensional projections of the streamlines at different sectionalyz planes across the reactor. The strong lateral air recirculationover the MWM is due basically to the very compact size of thereactor and the high ratio of occupied volume by the MWM tothe total volume of the chamber. The distance from the inlet tothe MWM lower surface is small, and as a result of this the air

stream coming from the inlet has no time to spread after enteringthe chamber, and impinges directly on the MWM lower surface.This air must flow then in the relatively narrow gap between suchsurface and the bottom wall of the reactor, in an essentially hori-zontal divergent flow structure. The entering air is thus propelledmassively towards the lateral walls of the reactor, and thereforethe free stream of incoming air is flowing away from the MWM lat-eral walls, in an orthogonal direction to them, at the boundarylayer separation points on the lower edges of the MWM block. Be-cause of this, no reattachment of the boundary layer on the MWMwalls is possible, and a strong recirculation bubble is establishedover their whole length. For such reattachment to be possible withthis incoming flow conditions, the length of the block of materialshould be several times its width, which is not the case for the gi-ven geometry of the reactor and its load. This is not a good designcondition for a drying process, since the recirculation of the airwould be detrimental for an adequate removal of moist air awayfrom the surface and its replacement by fresh dryer air.

The distributions of turbulent kinetic energy at two transverseyz planes are shown in Fig. 5. The maximum values occur in the re-

Fig. 5. Spatial distribution of turbulent kinetic energy in a yz plane at t = 696 h.

Fig. 6. Internal distribution of the dissipation of turbulent kinetic energy in a yz plane at t = 696 h.

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gion below the MWM block, particularly around the inlet. This is aresult of the more intense production of turbulent kinetic energy atthose locations, given the strong velocity gradients produced bythe sudden deceleration and deflection of the entering air stream.Similarly, a secondary maximum takes place in the zone surround-ing the outlet, because of the intensification of the velocity gradi-ents, and hence turbulence production, produced by the flowconvergence toward the exit. The intensity of turbulence howeveris quite small in the wake region behind the block of material, as

well as in the lateral zones surrounding it. It can be seen that theturbulent kinetic energy generated below the block is not signifi-cantly advected toward the lateral regions around the block, whichis an indication of a rather quick dissipation. This is confirmed bythe results shown in Fig. 6, which displays the distribution of tur-bulent kinetic energy dissipation at the same x positions shown inFig. 5. Clearly the dissipation is concentrated in the same regionswhere turbulence generation is taking place. Being the dissipationproportional to the square of the strain-rate tensor magnitude, a

Fig. 7. Distribution of temperature in a yz plane at t = 696 h.

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high level of dissipation can be expected in zones of high turbu-lence production by shear stresses. The resultant distributionscan be attributed to the low velocities in most of the field, and cor-respondingly to the generally low level of turbulence intensity, sothat in the regions where the flow is turbulent, such turbulence isdissipated locally, before a significant downstream advection of itcan take place. From the point of view of a drying process it wouldbe desirable a higher level of turbulence intensity, which wouldimprove the mass transfer from the material to the air in suchapplication. For the same geometrical configuration analyzed inthis work an evident way of increasing the turbulence intensityis the increment of the inlet velocity. Nevertheless, achieving high-er levels of turbulence in this reactor only by an increment in theinlet velocity would not be very efficient for the present configura-tion. Given the very obstructed shape of the inlet zone, most of theadditional turbulent kinetic energy will be locally dissipated in thesame region, or it will be initially convected towards the lateralwalls of the chamber, by the horizontal flow generated in thatzone, and dissipated by the near wall region by the strong strainresulting from this second impingement. The analyses of the abovenumerical simulations have shown that the design of the experi-mental bio-drying reactor can be improved.

Fig. 7 shows the temperature distribution in the reactor att = 696 h. The maximum temperatures (295 K) are observed inthe center of the MWM. By this time, the self-heating of the mate-rial has finished and the MWM has cooled. Hence, the air temper-ature is practically unaffected. The closed isotherms for the air onthe side of the MWM are produced by the recirculation of the flowin those areas. The temperature distribution inside the wastematerial is near to the one corresponding to isotropic heatdiffusion.

The final oxygen concentration inside the material at the end ofthe simulated period is shown in Fig. 8, for the two yz planes. Theslight differences in temperature in the material by this time arenot enough to produce significant differences in the oxygen con-centration, by oxidative consumption, and thus this has an essen-tially constant value.

5. Conclusions

A mathematical three-dimensional model considering turbu-lence, chemical and biological reactions inside an experimentalbioreactor and numerical simulations with the finite volume meth-

od have been presented. The numerical simulations have showedaccuracy when are compared to the experimental values. In addi-tion to show the soundness of the model, the results highlight sev-eral key factors that can be improved for an optimal design of areactor of this kind. These factors are related with the design ofthe air inlet zone, which should allow a better spreading of the in-let jet, for a more homogeneous velocity profile in front of the loadmaterial, instead of a concentrated impinging jet, which dissipatesits turbulent kinetic energy very fast. Also, the fraction of volumeoccupied by the load inside the reactor, as well as the aspect ratioof the material (e.g., the ratio of its frontal area to its length) showto be important parameters. A slender shape in the direction of theflow, instead of a compact cubic shape, would be more appropriate.

Acknowledgments

N.M. acknowledges the support provided by CONICY/Chile tothe FONDECYT 1111067 Project. C.R. acknowledges the supportof the Chilean National Commission for Scientific & TechnologicalResearch (CONICYT) through Project FONDECYT 11080025.

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