Kuipers, J.A.M.

Peters, E.A.J.F.

Tavassoli, H.

English

Engineering Conferences International

Engineering Conferences InternationalECI Digital ArchivesThe 14th International Conference on Fluidization– From Fundamentals to Products Refereed Proceedings2013Direct Numerical Simulation of Fluid-Particle HeatTransfer in Dense Arrays of Non-Spherical ParticlesH. TavassoliEindhoven University of Technology, NetherlandsE.A.J.F. PetersEindhoven University of Technology, NetherlandsJ.A.M. KuipersEindhoven University of Technology, NetherlandsFollow this and additional works at: http://dc.engconfintl.org/fluidization_xivPart of the Chemical Engineering CommonsThis Article is brought to you for free and open access by the Refereed Proceedings at ECI Digital Archives. It has been accepted for inclusion in The14th International Conference on Fluidization – From Fundamentals to Products by an authorized administrator of ECI Digital Archives. For moreinformation, please contact franco@bepress.com.Recommended CitationH. Tavassoli, E.A.J.F. Peters, and J.A.M. Kuipers, "Direct Numerical Simulation of Fluid-Particle Heat Transfer in Dense Arrays ofNon-Spherical Particles" in "The 14th International Conference on Fluidization – From Fundamentals to Products", J.A.M. Kuipers,Eindhoven University of Technology R.F. Mudde, Delft University of Technology J.R. van Ommen, Delft University of TechnologyN.G. Deen, Eindhoven University of Technology Eds, ECI Symposium Series, (2013). http://dc.engconfintl.org/fluidization_xiv/96Direct Numerical Simulation of fluid-particle heat transfer indense arrays of non-spherical particlesH. Tavassoli E.A.J.F. Peters J.A.M. KuipersMultiphase Reactors Group, Dep. of Chemical Engineering & Chemistry, EindhovenUniversity of Technology, P.O. Box 513, 5600 MB Eindhoven, the Netherlands.AbstractDirect numerical simulations (DNS) are conducted to characterize the fluid-particleheat transfer in dense arrays of non-spherical particles with a sphero-cylindrical shape.The objective of this study is to examine the suitability of the DNS for the predictionof the average heat transfer coefficient in such systems. The results are compared tothe Gunn correlation for heat-transfer of spherical particles by introducing an effectivediameter for the non-spherical particles.1 IntroductionNon-isothermal packed and fluidized bed reactors are widely used in a variety of processindustries. Detailed knowledge of the velocity and temperature fields inside the equipmentis necessary to optimally design and accurately predict the performance of such systems.Many investigations on heat and momentum transfers in packed and fluidized bedreactors have been performed to develop Nusselt number and friction-factor correlationsfor such systems. Although these correlations are expected to predict the flow and heattransfer characteristics in beds with spherical particles reasonably well, their validity fornon-spherical particles is questionable. It is well-known that the pressure drop in a porousbed packed with spherical particles can be estimated satisfactory with the use of empiricalErgun correlation. However, the extension of the Ergun equation to a bed packed by non-spherical particles is not straightforward. An accepted approach which has its foundation inthe Carman-Kozeny approximation, is to use an effective diameter for non-spherical particlesin the Ergun equation.Our current knowledge of the heat transfer characteristics for these systems followsmainly from experiments (e.g. Gunn [1] ,Wakao and Kaguei [2] and Kunii and Levenspiel[3]). Although these well-known correlations are widely used to predict the heat performanceof the packed and fluidized beds, their applicability to non-spherical particles has not beenassessed yet. In addition, the heat transfer between fluid and particles is strongly affected bythe local fluid-solid flow structure and varies spatially for non uniform structures. Therefore,the effect of the packing material shape needs to be considered for accurate prediction ofheat transfer characteristics.In order to gain detailed information at the small scale, DNS of flow and heat transferprocesses through a random array of non-spherical particle were conducted. Although theparticles do not move the results are relevant for fluidized beds because particles do nottouch as in a packed bed, but are separated as in a fluidized bed. In this study, we1(a)LD(b)Figure 1: Illustration of the IB-method (a) A two-dimensional staggered Cartesian grid withIB. Locations of ux and uy are represented by horizontal and vertical arrows, respectively.Scalar variables are positioned at the center of each cell (filled squares). Lagrangian pointson IB are shown with filled circles. (b) Representation of a sphero-cylinder by Lagrangianpoints.employed the Immersed Boundary (IB) method proposed by Uhlmann [4] to simulate non-isothermal flow through a complex geometry (i.e. a random array of sphero-cylinders).Non-overlapping sphero-cylinders (100) are distributed in a cubic domain by a standardMonte Carlo procedure for hard cylinders. Based on the predicted flow and temperaturefields, the average heat transfer coefficient can be estimated numerically as function ofparticle shape, Reynolds number and porosity.2 Immersed Boundary methodSince the complete description of the IB method can be found in literature (e.i. Uhlmann[4], Wang et al. [5], Feng et al. [6]), we only provide a brief description of the method inthis section. In IB methods, a fixed Eulerian grid and moving Lagrangian boundary pointsare employed (Fig. 1(a)). The fluid is assumed to occupy the full domain, even inside theparticles. The particles are represented by Lagrangian points which are distributed over theouter boundary of the particles (Fig. 1(b)). In this method, IB forces and energy sourceterms are introduced into the momentum and thermal energy equations, respectively, todescribe the mutual hydrodynamic and thermal interactions between the particle and thefluid. The source terms are evaluated iteratively such that the desired boundary conditionsare fulfilled on the IB.The following equations can be used in the entire domain for unsteady incompressiblefluid flow with constant properties and negligible viscous heating effect:∂u∂t+ u · ∇u = −ρ−1∇p+ ν∇2u+ f , ∇ · u = 0, (1)ρcp∂T∂t+ ρcpu · ∇T = k∇2T + q. (2)2In the above equations, ρ, ν, cp and k are the density, kinematic viscosity, specific heatand thermal conductivity of the fluid, respectively; u is the vector of velocity field, T is thefluid temperature and p is the fluid pressure.In the momentum equation (Eq.1), the additional volume forcing term f , compared tothe Navier-Stokes equations, is determined such that the no-slip velocity boundary conditionis enforced on the immersed boundary. Similarly to the momentum interaction, an additionalvolume heat source term, q, is added to satisfy the temperature boundary condition at theIB. In fact, f and q are the mutual force and energy interactions, respectively, between fluidand IB and are non-zero only over the IB interface.The momentum and energy equations were discretized by a finite difference schemebased on a staggered Eulerian grid. The equations were integrated in time using thefractional-step method. The nonlinear convection terms were approximated by the explicitsecond-order Adams-Bashforth method. The viscous and conduction terms in the mo-mentum and energy equations, respectively, were treated in time using the Crank-Nicolsonscheme.The sources defined at the Lagrangian points (i.e., forces and heat-sources) are coupledto the Eulerian grids by means of regularized delta-functions. Values for the velocitiesand temperatures defined on the Eulerian grid are interpolated to give the values at theLagrangian points using the same delta-functions. An iterative procedure was used todetermine the source-terms such that boundary conditions at the Lagrangian points areobeyed.3 Problem definitionIn this section, we introduce the non-spherical particle used in this study. Following this,Reynolds number and particle size definitions are presented. The non-spherical particleused in present study is the sphero-cylinder with the diameter D and length L (Fig. 1(b)).Three different aspect ratios (L/D) of 2,3 and 4 are considered to investigate the effect ofthe aspect ratio on the fluid-particle heat transfer.3.1 Simulation conditionsA typical particle configuration used in this study is shown in Fig. 2. The computationaldomain consists of inlet, packed and outlet sections. N = 100 non-overlapping sphero-cylinders are distributed in the packed section via a standard Monte Carlo procedure forhard cylinders. The center of each particle is inside the packing section and particles donot touch the inlet and outlet boundaries. The desired voidage ε is calculated as the ratioof empty space between particles to the volume of packed section.The fluid flows through the duct in direction z. Periodic boundary conditions areimposed in other directions (x and y) in order to avoid wall effects. The particles aremaintained at a constant temperature Ts = 320 K whereas the temperature of the fluidat the inflow is set at T∞ = 275 K. The Pr number was fixed to 1 for all simulations. Sothe fluid is in fact a gas.The boundary conditions for this simulation are set as follows:• At the inlet, uniform axial velocity Us and temperature T∞ of the fluid are imposed:ux = uy = 0, uz = Us and T = T∞. (3)3Figure 2: Geometrical configuration constructed by a random array of sphero-cylinder par-ticles (L/D=2 and ε=0.9).• At the outlet, the boundary conditions are:∂u/∂z = 0, ∂T/∂z = 0 (4)• On the periodic boundaries:u(0, y, z) = u(X, y, z), T (0, y, z) = T (X, y, z)u(x, 0, z) = u(x, Y, z), T (x, 0, z) = T (x, Y, z)(5)where X and Y are the sizes of domain into the x and y directions, respectively.• On the particle surface, no-slip velocity and isothermal temperature are assumed:u = 0, T = Ts (6)In Fig. 3 an example of the distributions of non-dimensional temperature ((T (x, y, z) −T∞)/(Ts − T∞)) and non-dimensional velocity magnitude (|u|/Us) of the fluid inside arandom array of sphero-cylinders when Re=13.572, L/D=2 and ε = 0.9 is shown.4 Heat transfer coefficient for a fluidized bedIn our simulations, it is assumed that the system has constant physical properties wherethe individual particles maintain a fixed temperature, Ts. According to these assumptions,the following equation can be used to predict the fluid thermal behavior in the bed:UsρcpdTbdz= hf ·As · (Ts − Tb) (7)where hf is local heat transfer coefficient, As is specific fluid-particle heat transfer surface(total surface area of the particles divided by the volume of the bed) and Tb is the cup-mixing temperature of the fluid. An average heat coefficient hf can found by solving Eq.7 for a constant heat-transfer coefficient,hf = −UsρcpAsZln[Ts − Tb2Ts − Tb1], (8)400.250.50.751(a)00.751.52.252.56(b)Figure 3: Computed fluid [a] non-dimensional temperature ((T (x, y, z)− T∞)/(Ts − T∞))and [b] non-dimensional velocity magnitude (|u|/Us) distributions in a random array ofsphero-cylinderical particles. (Re=13.572, L/D=2 and ε=0.9)where Tb1=275 K and Tb2 are enter and exit cup mixing temperatures of the fluid, respec-tively, and Z the length of the bed. This relation is used to determine hf from simulationdata.In order to enable a reliable comparison between CFD generated results and empiricalcorrelations, the mean Nusselt number was computed for different values of the Reynoldsnumber, ε and aspect ratio’s. The value of the Nusselt number and the Reynold numberdepends on the definition used for the effective diameter, De,Nu = hfDe/k, and Re = ρUsDe/µ. (9)where µ is the fluid viscosity, Us is the superficial velocity and De is the effective particlediameter. The definition of the particle diameter for non-spherical particles, however, isambiguous. Two often used effective diameter definitions, that can be computed fromgeometric properties of a particle, are the Sauter mean diameter Ds and the equivalentvolume sphere diameter Deq,Ds =6VpApand Deq =(6Vpπ)1/3, (10)respectively, where Vp and Ap are the volume and surface area of the particle. Bothdefinitions give the diameter for a spherical particle, and both definition scale linearly ofthe size of a particle is scaled equally in all special directions. For non-spherical particlesboth parameters, scale differently when changing the aspect ratio. In this study, these twodefinitions are employed to define dimensionless groups such as Re and Nu.For spherical particles in packed and fluidized beds the fluid-particle heat-transfer coef-ficient is predicted reasonably well by empirical correlations of Gunn:NuGunn = (7− 10ε+ 5ε2)(1 + 0.7Re0.2Pr1/3) + (1.33− 2.4ε+ 1.2ε2)Re0.7Pr1/3for 0 < Re < 105, 0.35 < ε < 1. (11)5For sphero-cylinders there is an extra dimensionless group, namely, the aspect ratio. Inorder to extend the Gunn correlation applicability to non-spherical particle, an acceptedapproach can be introducing a shape factor in Eq.11 and replacing the constants of theequation with new constants based on experimental data. However, the estimation of theseconstants for different particles is a difficult task and probably a variation in constants willbe observed due to the diversity in shape and size distributions of particles in the bed.In this study, we investigate if the constants of Eq. 11 can still be used when asuitable effective diameter De is used. The hope is that the aspect ratio dependenceincorporated in the definition of Ds and Deq in Eq. (10) accounts for the shape dependence.Now, the intriguing question is what effective particle diameter should be used in thecalculation. To answer this question, DNS have been performed to investigate the heattransfer characteristic in a bed packed with the sphero-cylinder particles.5 Results and discussionFigs. 4, 5 and 6 report Nusselt numbers generated by the DNS and according to the Gunncorrelation (Eq. 11). The different figures correspond to the different aspect ratios of theparticles considered. An overall agreement is observed in the trends of the numerical andthe experimental results. However, it seems that the Gunn correlation must be modifiedfor high Reynolds number (Re > 100) and dense systems.The effect of the choice De on the estimation of the correlation of the Nusselt numberis shown. The Nusselt numbers that are reported in figures by circle and square symbolsindicate De = Deq and De = Ds, respectively. Although the correspondence betweenthe Gunn correlation and the simulation results are not that good one can make anotherobservation.Suppose that there is a correlation that only depends on Nu(Re, ε,Pr) for a suitablychosen definition of De. In that case the results for the different aspect ratio’s L/D at thesame ε should lie on the same master curves. So, when comparing the three graphs thebest definition of De is the one for which the points, of a specific color, remain on the samecurve. When comparing the graphs the circles De = Deq seem to yield a better field.5.1 ConclusionsMotivated by uncertainty in the heat transfer correlations for a fluidized bed with non-spherical particles, DNS have been conducted to characterize heat transfer in a fluidizedbed with sphero-cylinder particles. The results show that the immersed boundary methodapproach is indeed well suited to compute heat-transfer properties, also for non-sphericalparticles.The simulation results are shown to correspond with the Gunn correlation as far astrends is concerned. The quantitative correspondence with the Gunn correlation is verywell at low void fraction and higher Reynolds numbers.Here we presented our first results on the heat transfer in beds of non-spherical particles.From these results it is hard to conclude if good Nusselt correlations can be formulated fornon-spherical particles by the introduction of a suited effective diameter. Many more simu-lations are needed such that the parameter space can be better sampled. The developmentof improved correlations for fluid-particle heat transfer is the final goal.610 30 50 70 90 110 13001020304050ReNusseltNo.Gunn (ε=0.9)Gunn (ε=0.592)Gunn (ε=0.299)Figure 4: The mean Nusselt number in random arrays of sphero-cylinder obtained fromGunn correlation and numerical simulations when L/D = 2. The DNS results are shownby circles and squares when De = Deq and De = Ds, respectively.10 30 50 70 90 110 130 15001020304050ReNusseltNo.Gunn (ε=0.9)Gunn (ε=0.592)Gunn (ε=0.329)Figure 5: The mean Nusselt number in random arrays of sphero-cylinder obtained fromGunn correlation and numerical simulations when L/D = 3. The DNS results are shownby circles and squares when De = Deq and De = Ds, respectively.710 30 50 70 90 110 130 150 17001020304050ReNusseltNo.Gunn (ε=0.9)Gunn (ε=0.592)Figure 6: The mean Nusselt number in random arrays of sphero-cylinder obtained fromGunn correlation and numerical simulations when L/D = 4. The DNS results are shownby circles and squares when De = Deq and De = Ds, respectively.NotationAp Surface area of the particle (m2) p Fluid pressure (N/m2)cp Specific heat (W/kg.K) T Fluid temperature (K)De Effective particle diameter (m) u Vector of fluid velocity (m/s)Deq Equivalent volume sphere diameter (m) Us Superficial velocity (m/s)Ds Sauter mean diameter (m) Vp Volume of the particle (m3)hf Average heat coefficient (W/m2.K) ρ Density (kg/m3)k Thermal conductivity (W/m.K) ν Kinematic viscosity (m2/s)AcknowledgmentsThe authors would like to thank the European Research Council for its financial support,under its Advanced Investigator Grant scheme, contract number 247298 (Multiscale Flows).References[1] D.J. Gunn and J.F.C De Souza. Heat transfer and axial dispersion in packed beds,chemical engineering science. Chemical Engineering Science, 29:1363–137, 1974.[2] N. Wakao, S. Kaguei, and T. Funazkri. Effect of fluid dispersion coefficients on particle-to-fluid heat transfer coefficients in packed beds. Chemical Engineering Science, 34:325–336, 1979.[3] Kunii D. and Levenspiel O. Fluidization engineering. Butterworth-Heinemann, 1991.[4] M. Uhlmann. An immersed boundary method with direct forcing for the simulation ofparticulate flows. J. Comput. Phys, 209:448–476, 2005.[5] Z. Wang, K. Fan, J.and Luo, and K Cen. Immersed boundary method for the simulationof flows with heat transfer. International Journal of Heat and Mass Transfer, 52:4510–4518, 2009.8[6] Z.-G. Feng and E.E. Michaelides. Heat transfer in particulate flows with direct numericalsimulation. Int. J. Heat Mass Transfer, 52:777–786, 2009.9

Direct Numerical Simulation of Fluid-Particle Heat Transfer in Dense Arrays of Non-Spherical Particles

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Direct Numerical Simulation of Fluid-Particle Heat Transfer in Dense Arrays of Non-Spherical Particles

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