International Journal of Multiphase Flow 57 (2013) 29–37

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International Journal of Multiphase Flow

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Direct numerical simulation of particulate flow with heat transfer

0301-9322/$ - see front matter � 2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.ijmultiphaseflow.2013.06.009

⇑ Corresponding author. Tel.: +31 402473122.E-mail address: e.a.j.f.peters@tue.nl (E.A.J.F. Peters).

H. Tavassoli, S.H.L. Kriebitzsch, M.A. van der Hoef, E.A.J.F. Peters ⇑, J.A.M. KuipersMultiphase Reactors Group, Department of Chemical Engineering & Chemistry, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands

a r t i c l e i n f o a b s t r a c t

Article history:Received 28 November 2012Received in revised form 21 June 2013Accepted 29 June 2013Available online 17 July 2013

Keywords:Particulate flowHeat transportImmersed boundary method

The Immersed Boundary (IB) method proposed by Uhlmann for Direct Numerical Simulation (DNS) offluid flow through dense fluid-particle systems is extended to systems with interphase heat transport.A fixed Eulerian grid is employed to solve the momentum and energy equations by traditional computa-tional fluid dynamics methods. Our numerical method treats the particulate phase by introducingmomentum and heat source terms at the boundary of the solid particle, which represent the momentumand thermal interactions between fluid and particle. Forced convection heat transfer was simulated for asingle sphere and an in-line array of 3 spheres to assess the accuracy of the present method. Non-isothermalflows past stationary random arrays of spheres are investigated to assess the capability of our simulationmethod for dense particulate systems. All results are in satisfactory agreement with reported experimen-tal and numerical results.

� 2013 Elsevier Ltd. All rights reserved.

1. Introduction

Non-isothermal gas–solid flows are widely used in a variety ofindustrial applications such as packed and fluidization bed reac-tors. In order to arrive at an optimal design and control of such sys-tems, a precise prediction of the temperature distribution, as wellas the flow field inside the equipment is necessary. Over the pastdecades, a lot of studies have focused on the heat transfer ingas–solid flows and many experiments have been conducted toestablish empirical heat transfer coefficients (HTC). Usually theseexperiments were conducted under a specific range of operatingconditions and the results are interpreted with the help of simpli-fied models. Although these correlations have been used success-fully for design purposes, these are not generally applicable fordifferent systems and a wider range of operating conditions. Besidethese issues, the correlations have limitations in providing insightinto the complex thermal dynamics. For example, the contributionof each of the three basic heat transfer mechanisms (convectionfrom fluid, conduction from particles and radiation) on the totalHTC is difficult to determine. Moreover, the HTC depends stronglyon the gas–solid flow pattern. Also the effect of structural informa-tion on the HTC is not straightforward and cannot be quantifiedeasily with an empirical heat correlation. In other words, theempirical correlations only provide a description of the averagethermal dynamic behavior of a system.

With the increase of computational power, computational fluiddynamics (CFD) simulation becomes an attractive and popularmethod for gaining in depth knowledge on fluid behavior. In CFD

the actual packing geometry is taken as input and the detailed flowand heat transfer patterns are given as output. The well-knownCFD techniques for simulation of particulate flows proposed inthe literature can be classified in three groups: Two fluid method(the motion of each phase is governed by separate Navier–Stokesequation; the interaction between the phases is approximated byempirical correlations), Discrete element method (the fluid motionis described by the Navier–Stokes equation and each particle is de-scribed in terms of Lagrangian equations of motion; the interactionbetween the phases is represented with empirical correlations)and Direct Numerical Simulations (the fluid and particulate phasesare treated by considering the Navier–Stokes equation and theLagrangian equations of motion, respectively; the interaction be-tween the phases is enforced through the no-slip boundary condi-tion at the surface of the particle, and hence there is no need forempirical closures).

At present the Two Fluid and Discrete Particle methods aremost widely used to study the local gas–solid flow structure andthermal behavior of the particulate phase. Although they are com-putationally less demanding than DNS, they suffer from uncertain-ties of the boundary conditions for the particulate phase (e.g. thethermal conductivity of the solid phase in the bed core differs fromthat in the near wall region) and, as indicated above, from the lim-itations that are inherent to the use of the empirical correlations.

With the rapid improvement of computational power, DNS hasattracted considerable interest for the simulation of particulateflow. DNS methods basically fall into two classes: boundary-fittedand non-boundary-fitted approaches. In the boundary-fittedapproach, e.g., the arbitrary Lagrangian–Eulerian methodFeng et al. (1994), the flow is solved on a boundary-fitted mesh,and thus re-meshing is required when the particles move.

30 H. Tavassoli et al. / International Journal of Multiphase Flow 57 (2013) 29–37

Non-boundary-fitted methods, such as the Immersed Boundarymethod (IB) Peskin (1977), employ a fixed Cartesian mesh for thefluid and moving Lagrangian points for the particles (see Fig. 1).In general the non-boundary-fitted methods are more efficientthan the boundary-fitted methods since no remeshing is required.The IB method was first proposed by Peskin (1977) for simulationsof systems with a moving complex boundary. In this method an IBforcing term is introduced into the momentum equation to de-scribe the mutual interaction between the IB and the fluid. Thenthis force is distributed to the Eulerian grid to enforce the no-slipboundary condition on the IB. Various formulations of the IB forc-ing have been derived so far. Goldstein et al. (1993) and Saiki andBiringen (1996) proposed a scheme in which the IB forcing is afeedback on the difference between the calculated velocity andthe desired velocity. The main drawback of this scheme is that itcontains parameters that must be tuned. Mohd-Yusof (1997) intro-duced a direct forcing scheme to calculate the interaction force be-tween IB and fluid, which requires no parameters. In this methodthe IB forcing term is set by the difference between the interpo-lated velocity on the Lagrangian point and the desired boundaryvelocity. Feng and Michaelides (2005), and later Uhlmann (2005)combined the advantages of a direct forcing scheme with the IBmethod to study the particulate flows in multiphase system.

Although the extension of the IB method to heat transfer prob-lems is relatively straightforward, only few computational resultshave been published yet Kim and Choi (2000) proposed an IBfinite-volume method for heat transfer in complex geometries forfixed particles. Feng and Michaelides (2009, 2008) developed anIB fully explicit finite-difference method for heat transfer in parti-cle laden flows. Wang et al. (2009) developed a direct forcing IBprocedure called the ‘‘multi-direct heat source scheme’’. In thismethod the IB forcing terms are calculated with the help of an iter-ative procedure to enforce the Dirichlet boundary condition at theimmersed boundary. Recently Deen et al. (2012) applied the IBmethod to study the HTC of dense non-isothermal fluid-particlesystems and compared these with experimental correlations for arandom array of particles with porosity = 0.7 and Reynolds num-ber = 36, 72, 108 and 144. That study is the first numerical valida-tion of one of the most well known heat correlations (the Gunncorrelation). However, the numerical results are obtained for onlyone porosity value (0.7) and one Prandtl number (0.8).

The objective of this paper is to investigate non-isothermal par-ticulate flow using a direct forcing IB method, for a wider range of

(a)Fig. 1. Illustration of IB (a) A two-dimensional staggered Cartesian grid with an IB. LocatioPressure and temperature are positioned at the center of each cell ( ). Lagrangian poinLagrangian point. (b) Representation of a sphere by Lagrangian points.

Reynolds numbers and porosities. In this method the momentumand energy equations are solved in the whole domain, includingthe regions that are occupied by the particles. The momentumand heat exchange between the phases is accounted for by intro-ducing momentum and energy source terms in the governingequations. The source terms are evaluated iteratively such thatthe velocity and temperature boundary conditions on the IB aresatisfied.

The method is validated by (i) comparing the temperature pro-files for well-defined heat conduction problems with available ana-lytical solutions and/or literature results in Sections 3.1, 3.2, 3.3,and (ii) comparing the results for the convective heat transfer coef-ficients for flow past an array of spheres with well-establishedempirical correlations in Section 3.4. The paper is organized as fol-lows. First, the governing equations of momentum and heat trans-fers in the particulate flow are introduced. Then the numericalsolution method is discussed. Some numerical experiments areconducted to validate the accuracy of the present method. Thesummary and conclusion are given in the final section.

2. Numerical method

2.1. Governing equations for fluid flow

In the IB method, the governing equations for unsteady incom-pressible fluid flow with constant properties and negligible viscousheating effects are:

qf@u@tþ qf u � ru ¼ �rpþ lfr2uþ f; with r � u ¼ 0; ð1Þ

@T@tþ u � rT ¼ afr2T þ q: ð2Þ

In the above equations, qf, lf and af are the density, viscosity andthermal diffusivity of the fluid. u, p and T are the vector of velocityfield, pressure and temperature of the fluid, respectively.

In the momentum equation (Eq. (1)), the additional volumeforcing term f, compared to the Navier–Stokes equation, is deter-mined such that the velocity boundary condition is enforced atthe fluid–IB interface. Similarly a heat source term q is added to en-ergy equation (Eq. (2)) to satisfy the temperature boundary condi-tion at the fluid-IB interface. f and q are non-zero only at the IBinterface. In fact, f and q are the mutual momentum and heat ex-change, respectively, between the fluid and the IB.

(b)ns of ux and uy are represented by horizontal and vertical arrows ( , ), respectively.ts on IB are shown with filled circles ( ). DVk is a volume that is assigned to each

H. Tavassoli et al. / International Journal of Multiphase Flow 57 (2013) 29–37 31

2.2. Determination of IB Heat source term

The implementation of the IB source terms for the momentumand energy balance equations are quite similar. We follow thescheme proposed by Uhlmann (2005) to determine the forcingterm that is required to impose a desired velocity udat the bound-ary. In this section we will focus on the implementation of the IBsource term for the heat transport equation (Eq. (2)), becauseheat-transport is the main topic of this steady.

For the temperature we impose a fixed temperature Td on theparticles. As in the Uhlmann approach for the forcing term, theheat source term is determined by this boundary condition, whichis imposed at the Lagrangian points which are located on theboundaries of the particles (Fig. 1). To transfer information be-tween the Lagrangian points and the Eulerian grid use is made ofso called regularized delta function D(x,xk),

TkðxkÞ ¼Xx�X

TðxÞ Dðx;xkÞ; qðxÞ ¼XN

k¼1

Q kðxkÞ Dðx;xkÞDVk

h3 : ð3Þ

Here the subscript k is used to indicate Lagrangian points and non-subscripted coordinates are quantities defined on the Eulerian grid.Furthermore, N is the number of Lagrangian points, and DVk is thevolume that belongs to Lagrangian point k. Since D(x,xk) is nonzeroonly in the supporting domain, f(x) and heat q(x) are obtained fromLagrangian points that are located inside the supporting domain ofD(x,xk).

The choice of D(x,xk) must fulfill certain criteria, as discussed byPeskin (1977). A variety of functions has been proposed in the lit-erature. We have analyzed the accuracy of regularized delta func-tions by selecting three types of them (Peskin (1977),Deen et al.(2007) and Tornberg and Engquist (2004)). No significant differ-ences were found in the results using these regularized delta func-tions. Therefore, we use the regularized delta function introducedby Deen et al. (2007), since it is computationally cheaper thanthe others.

Using the iterative scheme as depicted in Fig. 2 the heat sourceterms Qk at the Lagrangian points are determined. By means of the

Fig. 2. Solution scheme of the IB method to calculate the temperature field.

regularized delta function this source terms are transferred to theEularian grid. Using these source terms the temperatures at theEulerian grid points are computed. By means of the regularizeddelta function the temperatures on the Lagrangian points can becomputed. The iteration ends as TðaÞk � Td, i.e., when the boundarycondition is obeyed at the Lagrangian points, as in Wang et al.(2009). For imposing the no-slip boundary condition a similar iter-ative method is used.

2.3. Numerical details

The governing equations were solved using a finite differencescheme based on a staggered Eulerian grid. The set of equationsfor the momentum and heat advection is integrated in time usingthe fractional-step method. The nonlinear convection terms in themomentum and energy equations, respectively, are treated by theexplicit second-order Adams–Bashforth method. For the convec-tive term in the heat equation, CT = u � rT, we have

Cnþ1

2T � 3

2Cn

T �12

Cn�1T : ð4Þ

The viscous and conduction terms in the momentum and energyequations, respectively, are discretised in time using the Crank–Nicolson scheme. For the conduction term, ST = �afr2T, this gives,

Snþ1

2T � 1

2Snþ1

T þ SnT

� �¼ �1

2af

h2 ðL Tnþ1 þ L TnÞ: ð5Þ

where L represents the discretised Laplace operator and h the gridsize.

It must be noted that since the physical properties of the fluidare considered to be constant in this work, the momentum and en-ergy equations are decoupled. Therefore, first the momentumequations are solved and subsequently the temperature field is ob-tained. However, extension of this algorithm to coupled momen-tum and energy equations can be readily achieved by solving themomentum and energy equations simultaneously. Also, the simu-lation presented here are for particles with fixed positions. Theimplementation is, however, more general. Forces and torques onparticles are computed and the equation of motion for the particlescan be solved.

3. Verification

The proposed numerical method is validated first by comparingthe numerical results for the heat conduction around a stationaryspherical particle immersed in an infinite stationary fluid withthe analytical solution. In addition the computed Nusselt numberfor forced convection around a hot spherical particle is comparedto well-known empirical correlations. The effect of neighboringparticles is examined for a linear array of three spherical particlesand computed heat transfer coefficients are compared with the re-sults reported by Maheshwari et al. (2006). Finally, in order todemonstrate the reliability of the developed technique for complexgeometry problems, forced convection heat transfer for a randomarray of spherical particles is investigated. In this case the numer-ical results are compared with a well-known empirical correlationpublished in literature.

3.1. Cooling of a sphere in contact with an unbounded fluid

A solid sphere of radius R at a constant temperature Ts is sud-denly immersed at time t = 0 in unbounded fluid of temperatureT1. The radial distribution of the fluid temperature T(r, t), r > R, fol-lows from the heat diffusion equation in spherical coordinates:

Fig. 4. Radial non-dimensional temperature profile of the unbounded fluid,comparison between numerical and analytical solutions (h = Dp/25).

32 H. Tavassoli et al. / International Journal of Multiphase Flow 57 (2013) 29–37

@Tðr; tÞ@t

¼ ar2

@

@rr2 @Tðr; tÞ

@r

� �; ð6Þ

where a is thermal diffusivity. The analytical solution of Eq. (6) is:

Tðr; tÞ � T1Ts � T1

¼ Rr

1� erfr � Rffiffiffiffiffiffiffiffi

4atp� �� �

: ð7Þ

The Nusselt number, defined as Nu = hfDp/kf, where hf, Dp and kf

represent the heat transfer coefficient, particle diameter and ther-mal conductivity of the fluid, respectively, for the cooling of asphere is then:

NuAnalyticalðtÞ ¼ 2þ 2ffiffiffiffipp Rffiffiffiffiffi

atp : ð8Þ

We consider a geometry in which the sphere is positioned at thecenter of a cubic computational domain with edges equal to8Dp. The sphere is 1 mm in diameter. In this simulation a timestep of 10�4 s is used whereas the thermal diffusivity is setto 10�6 m2/s. The initial non-dimensional temperatureT ¼ ðTðr; tÞ � T1Þ=ðTs � T1Þ of sphere and fluid are 1 and 0, respec-tively. A far field boundary condition (temperature is assumed tohave zero-normal derivative) is imposed on all domain boundaries.Four grid sizes h = Dp/15, h = Dp/20, h = Dp/25 and h = Dp/30 areused to show grid convergence.

As time advances, the heat diffuses into the fluid and the tem-perature continuously increases in the vicinity of the sphere.Fig. 3 shows the instantaneous non-dimensional temperature dis-tribution around the sphere at t = 0.6 s. A comparison between theanalytical solution (Eq. (7)) and the numerical results in terms ofthe radial non-dimensional temperature distribution is presentedin Fig. 4 at different times. Table 1 reports the analytical (Eq. (8))and computed Nusselt numbers at four different instantaneoustimes. Table 1 shows that computed Nusselt numbers do notchange noticeably if were obtained from gird sizes of h = Dp/25and h = Dp/30. The simulation results are in good agreement withthe analytical values.

3.2. Forced convection around a stationary sphere

Numerical simulations of an isothermal hot sphere placed in aflowing cold gas were performed to validate the proposed IB meth-od for forced convection heat transfer. The calculated mean Nusseltnumber for a sphere can be compared to the available empiricalcorrelations. Six different Reynolds numbers (Re = 20, 30, 40, 50,60 and 100) based on the free-stream gas velocity U1 and spherediameter Dp = 1 mm are considered. The fluid density, viscosityand the Prandtl number Pr are 1 kg/m3, 10�5 kg/(ms) and 1, respec-

Fig. 3. Non-dimensional temperature distribution of the unbounded fluid aroundthe hot sphere at t = 0.6 s.

tively. Since the Re number is high enough, natural convection canbe neglected.

We would like to obtain the heat-transfer characteristics for theflow past a sphere in an infinitely extended fluid. This means thatthe computational domain needs to be large enough for boundaryeffects are negligible. Nijemeisland (2000) proposed that a domainwith 8Dp � 8Dp � 16Dp in size is proper for removing the wall ef-fect on the velocity and temperature profiles. Guardo et al.(2006) showed that a domain with size of 4Dp � 4Dp � 16Dp canstill be considered as a infinite domain for this problem. We havetaken a size of 8Dp � 8Dp � 15Dp. To study the influence of themesh size were presented with using 3 different mesh sizesh = Dp/10, h = Dp/15 and h = Dp/20.

The boundary conditions for this flow are outlined below:

� At the surface of the sphere, no slip (u = 0) and prescribed tem-perature boundary conditions (T = Ts) are imposed.� At the inlet, uniform axial velocity (U1 = uz, ux = uy = 0) and

temperature (T = T1) of the fluid are imposed.� At the outlet, the boundary conditions are:

Table 1Nusselt

Time

0.40.60.81.0

@u@z¼ 0;

@T@z¼ 0: ð9Þ

� Free slip boundary condition is used for other boundaries:

@u@y¼ @u@x¼ 0;

@T@x¼ @T@y¼ 0: ð10Þ

The inlet and outlet boundaries are located at r = 2Dp and 13Dp,respectively, where r is the distance from the center of the sphere.Fig. 5 shows the distribution of the non-dimensional gas tempera-ture, T ¼ ðTðx; y; zÞ � T1Þ=ðTs � T1Þ, accompanied with the velocityvector around the sphere at the Z–Y plane. The Nusselt number isestimated by evaluating the convective heat transfer coefficienthf around the surface of sphere,

number versus time for cooling of the hot sphere in the unbounded fluid.

(s) Estimated Nusselt No. Nusselt No.(analytical value(Eq. (8)))

h = Dp/15 h = Dp/20 h = Dp/25 h = Dp/30

3.01 2.97 2.95 2.93 2.892.83 2.80 2.78 2.77 2.732.73 2.70 2.68 2.67 2.632.66 2.63 2.61 2.60 2.56

Fig. 5. Distribution of non-dimensional gas temperature around a heated spheretogether with velocity field at Re = 40 and Pr = 1.

Fig. 6. Nusselt number versus Reynolds number for forced convection over asphere.

Fig. 7. Schematic of flow over a row of three spheres.

Table 2Parameters used in simulations of heat transfer from an in-line array of three spheres.

s/Dp

Nx � Ny � Nz Dp(m) Dt/Dp

r1/Dp

r2/Dp

Grid size (m) Pr

2 150 � 150 � 195 10�4 10 4.5 4.5 6.666 � 10�6 0.744 150 � 150 � 255 10�4 10 4.5 4.5 6.666 � 10�6 0.74

Fig. 8. Distribution of non-dimensional gas temperature around heated spherestogether with the velocity field at Re = 10 and s/Dp = 2.

H. Tavassoli et al. / International Journal of Multiphase Flow 57 (2013) 29–37 33

hf ¼Q T

ðT1 � TsÞ Ap; ð11Þ

where QT and Ap are total heat flux and surface area of sphere.Fig. 6 reports the calculated Nusselt number with the empirical

results of Ranz and Marshall (1952), Whitaker (1972) and Feng andMichaelides (2000) for a spherical particle, respectively:

Nu ¼ 2:0þ 0:6 Re0:5Pr0:33 for 10 < Re < 104; Pr > 0:7: ð12Þ

Nu ¼ 2:0þ ð0:4 Re0:5 þ 0:06 Re2=3Þ Pr0:4

for 3:5 < Re < 7:6� 104; 0:7 < Pr < 380: ð13Þ

Nu ¼ 0:992þ Pe1=3 þ 0:1 Re1=3Pe1=3 Pe ¼ Re� Pr;for 0:1 < Re < 4000; 0:2 < Pe < 2000: ð14Þ

The sensitivity of the numerical solutions to grid sizes can beexamined from Fig. 6. There are almost no differences betweenthe results from grid sizes h = Dp/15 and h = Dp/20 when Reynoldsnumber is lower than 50 (around 1%). This difference is around 3%when Reynolds number is 100. The calculated Nusselt numbers arein reasonable agreement with the results obtained from empiricalcorrelations.

3.3. Effect of blockage on flow and heat transfer from an in-line arrayof three spheres

In this section, we perform computations for flow over an arrayof three spheres as shown schematically in Fig. 7 together with the

details of the computational domain. In order to investigate thecombined effects of blockage and of sphere–sphere interactions,the flows past an in-line array of three spheres have been studiedfor three different Reynolds numbers (Re = 1, 10, 50) and fortwo values of the center-to-center spacing between thespheres, namely, s = 2Dp and 4Dp. The same values were used byRamachandran et al. (1989) and Maheshwari et al. (2006) in simu-lations of heat transfer for unconfined in-line arrays of threespheres.

The parameters used in our simulations study are provided inTable 2. The density and viscosity of the gas are 1 kg/m3 and10�5 kg/(ms), respectively. According to the results of mesh inde-pendency tests for forced convection around a sphere (Fig. 6), thenumerical results are mesh independent (when Re 6 50) if the gridsize h is set to Dp/15. The boundary conditions for these simula-tions are the same as the boundary conditions used in the simula-tion of force convection around a stationary sphere in the previoussection. As in all presented simulations, the thermo-physical prop-erties are taken to be temperature independent.

The distribution of the non-dimensional gas temperature aswell as the velocity field around the spheres in the z–y plane areshown in Fig. 8. The present results are compared with results ofRamachandran et al. (1989) and Maheshwari et al. (2006) for air(Pr = 0.74) in Table 3 for the two values of the sphere-to-sphereseparation. According to the numerical results, higher Reynoldsnumbers will increase the heat exchange rate between gas andparticles. Moreover, a larger sphere-to-sphere separation ratioleads to the higher heat transfer rates.

The overall agreement is acceptable, but around 5–7% deviationis observed for the first and third spheres. A possible reason is the

Table 3Comparison between the present results of Nusselt number for an in-line array of three spheres and that of Ramachandran et al. (1989) and Maheshwari et al. (2006) for air(Pr = 0.74).

Re S/Dp Present simulation Ramachandran et al. Maheshwari et al.

1st 2nd 3rd 1st 2nd 3rd 1st 2nd 3rd

1 2 2.09 1.58 1.62 2.12 1.81 1.63 2.03 1.83 1.631 4 2.31 1.96 1.82 2.17 2.03 1.63 2.20 1.94 1.64

10 2 3.45 2.40 2.21 3.37 2.32 2.03 3.32 2.34 2.0510 4 3.51 2.83 2.62 3.28 2.79 2.49 3.33 2.72 2.5350 2 5.72 3.55 3.19 5.50 3.39 2.98 5.42 3.44 3.0850 4 5.80 4.21 3.81 5.40 4.18 3.60 5.40 4.11 3.52

34 H. Tavassoli et al. / International Journal of Multiphase Flow 57 (2013) 29–37

size of the entrance and exit domains (e.g. r1 and r2) on solutions.These sizes are not given by Maheshwari et al. (2006) and wefound that changing them influences the results for the first andthird spheres significantly (up to 15% difference). An additionalreason for the deviation might be the presence of thin boundarylayers, especially for the first sphere, and insufficient grid resolu-tion to resolve these. The accurate simulation of this problem willresult in high computational costs since a uniform grid is used andupon refinement a refined grid must be used for the whole domain.Hence, in this study we do not consider using a larger domain or amore refined grid for this problem.

3.4. Non-isothermal flows through random static arrays of spheres

As a final example non-isothermal flows through random staticarrays of spheres with a diameter Dp are studied to verify the abil-ity of the proposed method to handle complicated geometries andenhance our understanding of convective heat transfer in particu-late flow. The gas flows through a 3-dimensional duct in direction z(streamwise). Periodic boundary conditions are imposed in span-wise (x and y) directions in order to avoid wall effects. The spheres,which are considered as heat sources, are maintained at a constanttemperature Ts = 320 K, whereas the temperature of the gas at theinflow is set at T1 = 275 K. As in previous examples, the gas phys-ical properties are assumed to be independent of temperature.

A typical particle configuration used in this study is shown inFig. 9. The computational domain consists of inlet, packed and out-let sections. N = 54 non-overlapping spherical particles are distrib-uted in the packed section via a standard Monte Carlo procedurefor hard spheres. In fact only the packed section is active in heattransfer.

The desired packing fraction / is calculated as the ratio of totalvolume of particles to the volume of packed section. The sizes ofinlet and outlet sections are equal and set to Nx � Ny � 10 for allsimulations.

In order to enable a comparison between CFD generated resultsand empirical correlations, the mean Nusselt number is computed

Fig. 9. A typical particle configuration used in the simulations for / = 0.1.

for 3 values of the Reynolds number (10, 50, 100) and packing frac-tion / (0.1, 0.3, 0.5). The final estimation of the mean Nusselt num-ber and its standard deviation at a desired Reynolds number and /,are obtained from 5 independent configurations. Tables 4 and 5provide parameters and data that were used for our simulations.

The boundary conditions for this simulation were set asfollows:

� At the inlet, uniform axial velocity U1 and temperature T1 ofthe fluid are imposed:

ux ¼ uy ¼ 0; uz ¼ U1; and T ¼ T1: ð15Þ

� At the outlet, the boundary conditions are:

@u@z¼ 0;

@T@z¼ 0: ð16Þ

� On the periodic boundaries:

uð0; y; zÞ ¼ uðX; y; zÞ; Tð0; y; zÞ ¼ TðX; y; zÞuðx;0; zÞ ¼ uðx;Y ; zÞ; Tðx;0; zÞ ¼ Tðx;Y; zÞ;

ð17Þ

where X and Y are the size of domain in x and y directions,respectively.� At the particle surface, Dirichlet conditions were used for the

velocity (no-slip) and temperature,

u ¼ 0; T ¼ Ts: ð18Þ

As an example we show in Fig. 10 the distributions of the com-puted temperature and gas velocity in a random array of spheres atRe = 50 and / = 0.5.

The local heat transfer rate Qlocal from the solid phase to the gasphase is expressed as:

Q localðzÞ ¼ Uqf CfdTbðzÞ

dz¼ hf-localApðTs � TbðzÞÞ; ð19Þ

where Cf is the heat capacity of the fluid and Ap is specific fluid-particle heat transfer surface (Ap = 6 //Dp), Tb is the cup-mixingtemperature of the fluid and defined as:

Tb ¼R

AcquCf T ds

_mCf; ð20Þ

where Ac is the cross sectional area of the domain.The mean convective heat transfer coefficient hf is expected to

be constant for a thermally fully developed flow in a random arrayof particles and defined by:

lnTs � Tb2

Ts � Tb1

� �¼ � hf ApZ

U1qf Cf; ð21Þ

where Tb1 = 275 K and Tb2 are inlet and exit cup-mixing tempera-tures of the gas, respectively, and Z is the domain size in z direction.

The grid independency of solutions has been tested by runningsimulations using 3 different grid sizes h = Dp/10, h = Dp/15 andh = Dp/20. The total heat fluxes through the bed are reported in

Table 4Parameters used in the simulations of random static arrays of spheres.

Parameter Value Unit

Particle diameter 0.001 mFluid density 1 kg/m3

Fluid viscosity 10�5 kg/(m s)Pr number 1 –

Table 5Data used for the simulations of random static array of spheres (Dp = 20h).

Re / Nx � Ny � Nz Dt (s)

10 0.1 139 � 139 � 159 10�5

10 0.3 96 � 96 � 116 10�5

10 0.5 81 � 81 � 101 10�5

50 0.1 139 � 139 � 159 10�5

50 0.3 96 � 96 � 116 10�5

50 0.5 81 � 81 � 101 10�5

100 0.1 139 � 139 � 159 10�5

100 0.3 96 � 96 � 116 5 � 10�6

100 0.5 81 � 81 � 101 10�6

(a)

(b)Fig. 10. Computed (a) temperature and (b) gas velocity distributions in a randomarray of spherical particles. (Re = 50, / = 0.5).

Table 6Total heat flux through the static random arrays with different grids.

Re / QTotal � 104 (W/m3)

h = Dp/10 h = Dp/15 h = Dp/20

10 0.1 16.05 16.34 16.4510 0.5 7.13 7.49 7.7450 0.1 44.69 43.72 43.2350 0.5 36.21 35.12 34.57

100 0.1 69.16 65.69 64.04100 0.5 69.55 65.60 63.86

Fig. 11. The mean Nusselt numbers in random arrays of spheres obtained from theGunn correlation and numerical simulation (extrapolated solutions obtained byusing Eq. (22)). The standard deviations are obtained from five independentconfiguration at finest grid (h = DP/20).

H. Tavassoli et al. / International Journal of Multiphase Flow 57 (2013) 29–37 35

Table 6 in terms of grid size, Reynolds number and packing frac-tion. In the case of low Reynolds number and packing fraction,the solutions seem to show a good convergence for the finest gridsize. In the case of high Reynolds number and packing fraction, adependence on the grid spacing exists. Since the mean Nusseltnumber is dependent on the grid size, an extrapolation methodcan be used to obtain more accurate results.

When solutions on different grids are available, a more accuratesolution XE in comparison to the solution on the finest grid Xf can

be obtained with the help of Richardson extrapolation (Roache(1994)),

XE ¼ Xf þ Chp; ð22Þ

where p is the order of the scheme and C is assumed to be indepen-dent of h. p is calculated by the solution of the following equation:

X1 � X2

X2 � X3¼ hp

1 � hp2

hp2 � hp

3

ð23Þ

where X1, X2 and X3 are the three results with grid spacing h1, h2 andh3, respectively.

According to Eq. (23) at least three grids are required to esti-mate the discretization error. Three results X1, X2 and X3 (the meanNusselt number in this case) are determined on three grids h = Dp/10,Dp/15 and Dp/20. Then by using Eqs. (23) and (22), the extrapolatedmean Nusselt number is determined.

Historically, extensive experiments have been conducted tostudy heat transfer process in the particulate flows. In this study,the well-known equation of convective heat transfer coefficient gi-ven by Gunn is employed for making comparisons with numericalresults. Gunn (1972) proposed the following correlation of meanNusselt number for packed and fluidized beds as function of voi-dage eg = 1 � / and Reynolds number:

NuGunn ¼ 7� 10eg þ 5e2g

� �ð1þ 0:7Re0:2Pr1=3Þ

þ 1:33� 2:4eg þ 1:2e2g

� �Re0:7Pr1=3;

for 0 < Re < 105; 0:35 < eg < 1: ð24Þ

In Fig. 11, we show both the mean Nusselt numbers obtained fromthe Gunn correlation and our numerical simulations (extrapolatedsolutions obtained by using Eq. (22)). Fig. 11 shows that the devia-tion between numerical and empirical Nusselt numbers increaseswith increasing / and is positive for all cases. The deviation from

Fig. 12. Distribution of the particle Nusselt number in the random arrays of spheres(Re = 10, / = 0.1). The mean Nusselt number of the bed is equal to 3.44 (calculatedby using Eq. (21)).

36 H. Tavassoli et al. / International Journal of Multiphase Flow 57 (2013) 29–37

the Gunn correlation lies well within the accuracy of this correla-tion (see Fig. 5 in Gunn (1972)). Our numbers are also consistentwith results by Tenneti et al. (2013) that appeared in press duringthe reviewing stage of the current paper.

One of the large advantages in using DNS for heat-transfer prob-lems is the ability to calculate HTC’s at a microscopic level. The dis-tribution of HTC’s of individual particles in a bed is helpful toassess the accuracy of the average HTC, since the average HTC isused to characterize the heat transfer in a bed. The HTC of an indi-vidual particle can be estimated experimentally by measuring thedynamic temperature of a hot sphere, using an attached thermo-couple, immersed in the bed. However, the contribution of eachheat transfer mechanisms to the overall heat transfer is difficultto characterize. According to the proposed DNS method in thisstudy, we are able to estimate the convective HTC of an individualparticle in a bed. The HTC of an individual particle hp can be de-fined by the following equation:

hp ¼Q p

pD2p � ðTs � Tf Þ

ð25Þ

where Qp and Tf are the heat exchange rate between the particle andsurrounding fluid and the local average temperature of the fluid,respectively. Obtaining Tf is not so straightforward but as explainedby Deen et al. (2012) using the following equation leads to accept-able accuracy in the estimation of Tf,

Tf ðrpÞ ¼RRR

Vfgðjry � rpjÞTðryÞdVyRRRVf

gðjry � rpjÞdVy; where

gðrÞ ¼ exp � rDp

� �; 0 < r < 2Dp:

ð26Þ

Here g(r) covers the fluid volume in a cubic box of size 4Dp with itscenter coinciding with the center of the sphere. As an example, thedistribution (probability density function) of the particle Nusseltnumber (Nup = hpDp/kf) for Re = 10 and / = 0.1 is shown in Fig. 12.For this case the mean Nusselt number is 3.44. Fig. 12 shows a con-siderable variation of Nup in the bed.

4. Conclusions

In the present paper, the IB method with a finite differencescheme proposed by Uhlmann is extended to heat transfer applica-tions. In this method, a direct heat source term is introduced to en-force the temperature Dirichlet boundary condition at an IB and

estimated in an iterative manner. Thus it inherits the advantagesof IB method as well in heat transfer applications and can be usedefficiently in non-isothermal complex geometry problems. For ourcomputational scheme, the implicit second order fractional stepmethod and the discrete element particle approach are employedto simulate the behavior of fluid and solid phases, respectively.

The accuracy of proposed numerical method was validated bycomparing results obtained for several heat transfer problemsinvolving single and multi-particle systems with well-known corre-lations. First, the heat conduction problem around an isothermal hotsphere immersed in an infinite stationary fluid was simulated. Thecomputed temperature profile and associated Nusselt numberagreed well with analytical solutions. Then the forced convectionaround a single sphere and an in-line array of three spheres for sev-eral Reynolds numbers were simulated. The estimated Nusselt num-bers were compared with the data published in literature. Finally,non-isothermal flows around stationary arrays of spheres with con-stant temperature were considered. According to calculated velocityand temperature distributions, the average Nusselt numbers wereestimated for several Reynolds numbers and packing fractions. Inaddition, the distribution of particle HTC in a random array was ob-tained and compared with the mean HTC of the bed.

These numerical results, never reported in the literature up tonow, describing convective heat transfer in the packed and fluid-ized beds and were compared with the empirical correlation ofGunn. These simulations prove the efficiency of the proposednumerical method in non-isothermal problems with complexgeometry.

Acknowledgments

The authors thank the European Research Council for its finan-cial support, under its Advanced Investigator Grant Scheme, Con-tract Number 247298 (Multiscale Flows).

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