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CFD simulation and experimental validation for wall effects on heat transfer of finite cylindrical catalyst
International Communications in Heat and Mass Transfer 38 (2011) 1148–1155
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International Communications in Heat and Mass Transfer j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / i c h m t
CFD simulation and experimental validation for wall effects on heat transfer of finite cylindrical catalyst☆ F.S. Mirhashemi, S.H. Hashemabadi ⁎, S. Noroozi Computational Fluid Dynamics Research Laboratory, School of Chemical Engineering, Iran University of Science and Technology, Narmak, Tehran, 16846, Iran
a r t i c l e
i n f o
Available online 19 May 2011 Keywords: Heat transfer coefficient Computational Fluid Dynamics (CFD) Cylindrical particle Wall effect Heat and mass transfer analogy
a b s t r a c t In this paper, heat transfer of single cylindrical particle affected by wall has been investigated numerically and experimentally for Reynolds number range 2000 to 6000. The heat transfer in two different orientations, axial and cross flow over the particle has been considered in simulation with MultiPhysics Software FEMLAB version 2.3. The heat and mass transfer analogy technique has been applied for validation of the simulation results. The coated particle with naphthalene was sublimated to obtain the corresponding Sherwood numbers. The results show that the CFD model can predict the particle-to-fluid heat transfer for two situations due to trivial error (an average error of 6%) compared to experimental values. Influence of wall on heat transfer of particle in seven different bed-to-cylinder diameter ratio (N = 1.66, 2.65, 2.75, 5, 6.66, 12, and 18) have been discussed in different velocities. According to obtaining results, with increasing the bed-to-cylinder diameter ratio over the 12 wall have no significant consequence on Nusselt number. Due to this fact, a CFD based correlation has been proposed to consider the wall effects on particle-to-fluid Nusselt number with an average error of 2.19%. © 2011 Elsevier Ltd. All rights reserved.
1. Introduction Packed-beds are widely used in petrochemical, fine chemical and pharmaceutical industries. Detailed knowledge of flow in the void space of such packed beds is essential for understanding different characteristics of these beds. Heat transfer plays an important role in consideration of packed bed performance, and in view of this fact it had been very essential in previous studies. Consideration of single cylindrical particle heat transfer in cross flow is one of these aspects. There exist hundreds of experimental studies in terms of local and mean heat transfer in this case. The majority of these studies mainly focus on dependency of Nusselt on Reynolds and Prandtl numbers and their effects on heat transfer. Zukauskas and Morgan have completely considered the heat transfer of cylinder [1,2]. Although the effects of free turbulence flow on heat transfer effects have been taken into account in these studies; moreover, wall effect and length to cylinder diameter ratio have been ignored. Churchill et al. [3] have proposed a correlation for calculating mean heat transfer that has been used in many other works in this field as verification case. They claimed that turbulence flow, wall effects, and channel to cylinder diameter ratio have minor effects on the heat transfer. Quarmby and Al-Fakhri [4] have scrutinized the effect of length to cylinder diameter ratio on heat transfer. They experimentally considered a range of length to cylinder diameter ratio ☆ Communicated by W.J. Minkowycz. ⁎ Corresponding author. E-mail address: [email protected] (S.H. Hashemabadi). 0735-1933/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.icheatmasstransfer.2011.05.001
between 1 and 12. In so doing, they found for short cylinders (aspect ratio is smaller than 4), the effect of aspect ratio on mean heat transfer is considerable and when this ratio is bigger than 4 it can be ignored. This fact was considered by Chang and Mills [5] that led to derive a correlation in terms of Nusselt number with respect to length to cylinder diameter ratio. Because of extreme difficulty in measuring fluid flow and heat transfer inside the bed by conventional means without disturbing the packing arrangement, efforts towards improvement in modeling using computational fluid dynamics (CFD) have been recently developed. In general, 2D and 3D CFD models have been used in various fields to simulate flow profiles and heat transfer. The earliest CFD fixed bed simulations used two-dimensional models. Dalman et al. [6] investigated an axisymmetric radial plane. This work gave a first insight in flow structure in fixed beds. McKenna et al. [7] obtained valuable insight into the effect of particle size on particle– fluid heat transfer from 2D CFD study of small spherical particles. They have validated their basic models with Ranz-Marshal (RM) correlation and have shown the weakness of that correlation in predicting heat transfer from small clusters of spherical particles. Lloyd and Boehm [8] determined the sphere spacing effect on the drag coefficient and the particle to fluid heat transfer by studying on linear array of eight spheres in 2D. It was found that the heat transfer from the spheres decreased with reduction of particle spacing. The 3D models have been applied for packed beds more recently. Derx and Dixon [9] used a simple model to obtain the wall heat transfer coefficient. An eight-sphere model, in which eight spheres are located into two layers with four ones without solid-solid contact points, has been studied by Logtenberg and Dixon [10,11]. This model shows that
F.S. Mirhashemi et al. / International Communications in Heat and Mass Transfer 38 (2011) 1148–1155
Nomenclature A Cp D jH jD k L Nu NuD P Pr Re Q V to T y+ x
Area Specific heat capacity Diameter Colbourn j factor (heat transfer coefficient) Colbourn j factor (mass transfer coefficient) Thermal conductivity Length of cylindrical particle Overall Nusselt number, hD/k Circumferentially averaged Nusselt number Pressure Prandtl number Reynolds number Sink or source term in energy equation Velocity vector Thickness of distributor Temperature Dimensionless distance from the wall Cartesian x-coordinate
Greek symbols Cμ, Cε1, Cε2 Empirical constant ρ Density μ Dynamic viscosity ν Kinematic viscosity σk, σ Empirical constant ε Turbulence energy dissipation κ Turbulence kinetic energy Subscripts dis Distributor b Bulk o Orifice p Particle t Turbulent W Wall Superscripts † Transpose
conventional fixed bed models could not describe flow and heat transfer behaviors. Nijemeisland et al. [12,13] modeled the heat transfer of low channel to particle ratio by using CFD. They presented an extensive overview of prior work on modeling and measurement of the characteristics of these packed beds. Furthermore, they showed that CFD method is reliable for modeling flow and heat transfer phenomena inside the packed bed. In similar studies, Guardo et al. [14] drew a comparison between the performance in flow and heat transfer estimation of five different RANS (Reynolds-Averaged Navier–Stokes) turbulence models in a fixed bed composed of 44 homogenous stacked spheres. In one of the few heat transfer studies of cylindrical particles, Nijemeisland et al. [15] simulated heat transfer and fluid flow in a near wall segment of a steam reformer tube-filled with cylindrical particles in which inert packing was heated up, they illustrated that particles with no internal voids appear better than particles with internal voids, in the bed interior. In another investigation, AhmadiMotlagh and Hashemabadi [16] have studied two and three dimensional CFD modeling of heat transfer from discrete cylindrical particles in different situations of flow. They reported a good qualitative as well as reasonable quantitative agreement between CFD results and empirical correlations. Romkes et al. [17] investigated the mass and heat transfer characteristics of a composite structured reactor packing containing spherical particles.
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They showed that the CFD simulations could be used to adequately predict the rate of mass and heat transfer from the catalyst particles to the fluid. In another study, AhmadiMotlagh and Hashemabadi [18] investigated the hydrodynamics and heat transfer characteristics of a randomly packed bed of cylindrical particles with a channel-toparticle diameter ratio of 2. Their results were validated by naphthalene sublimation mass transfer experiments and the particle heat transfer Nusselt number were obtained by the use of analogy between mass and heat transfer. As it mentioned in above studies, the cylindrical catalysts have been the subject of fewer studies rather than spherical catalysts. In spite of the simplicity of modeling the spherical particles, the growing applications of cylindrical catalysts in reaction engineering requires a vital need for further exploration of heat transfer behavior from these particles. Despite the Reynolds and Prandtl number impacts on heat transfer of packed beds, some secondary parameters such as free stream eddies, particle length to diameter ratio, and fraction of channel to cylinder diameter have some influences on heat transfer phenomenon. Correlations that have been presented for heat transfer coefficient in previous works have been mainly ignored the effect of these secondary factors. On the grounds of this fact, in this paper, effects of some parameters such as tube-to-particle diameter ratio and flow direction on cylinderical particles have been illustrated. 2. Theory 2.1. Governing equations In this work compressible turbulence steady state flow heat transfer has been taken into account. The conservation equations of continuity, momentum, and energy can be simplified respectively: ∇⋅ðρV Þ = 0
ð1Þ
h i † ∇⋅ðρVV Þ = −∇P + ∇⋅ ðμ + μt Þ ∇V + ∇V
ð2Þ
∇⋅ −ðk + kt Þ∇T + ρCp TV = Q
ð3Þ
where Q is a heat sink or heat source term which is set to zero in this study, μ t and kt are the turbulent viscosity and thermal conductivity respectively that μ t can be defined as: 2
μ t = ρCμ
κ ε
ð4Þ
and Cμ is a model constant which is equal to 0.09. The standard κ-ε is used in order to take into account of turbulence flow [17]. Where the transport equations for the turbulent kinetic energy κ and the turbulence dissipation rate ε are defined respectively [21]: ∂κ + V⋅∇κ = ∇⋅ ∂t
" ν+
! # 2 Cμ κ2 κ † 2 ∇V + ∇V −ε ∇κ + Cμ σk ε ε
ð5Þ
" ! # 2 Cμ κ2 ∂ε ε † 2 −Cε2 + V⋅∇ε = ∇⋅ ν + ∇ε + Cε1 Cμ κ ∇V + ∇V σε ε κ ∂t ð6Þ where the model constants in the above equations are determined, experimentally [16]: Cε1 = 1:44; Cε2 = 1:92; σκ = 1; σε = 1:3 Turbulent thermal diffusivity (kt/ρCp) is usually related to eddy viscosity via a turbulent Prandtl number (Prt ≈ 0.85–0.9).
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2.2. Boundary conditions and CFD simulation Fig. 1A represents the boundary conditions. For reducing the thermophysical properties changes, the channel walls are adiabatic and the temperature of particle surface is kept at Tp, which was 10 °C warmer than the fluid, during this simulation. Air is used as the cold fluid; furthermore, constant temperature and flow rate is set at inlet (To and Qo). The feed inlet of the column is subjected to inlet velocity boundary condition. Moreover, the outlet of the column is assumed to be fully open to the atmosphere, so the outlet pressure is set at atmospheric pressure. The wall boundaries are set to no slip condition and the standard wall function is applied to near the wall. For turbulent flow the thickness of the cell walls is expressed in the terms of y +. The dimensionless wall distance, y +, is used to verify the
mesh adequacy for this turbulent simulation. A typical value for y + between 30 and 300, or at least between 10 and 1000, is generally recommended for best accuracy [17]. Subsequently, the standard wall function was applied in turbulent flow and mesh independency with respect to y + criteria has been applied and the optimized mesh has been selected. Moreover, the heat transfer inside solid particles is ignored, since particle surfaces are assumed to be isothermal. The particle-to-fluid heat transfer was simulated with surface constant temperature (Ts) for particle which is 10 °C warmer than the fluid (air). As a result of the temperature difference between the particle surface and the fluid, heat is transferred from the particle to the fluid. Above governing equations, Eqs. (1)–(6), with mentioned boundary conditions have been solved with the Thermal-Fluid interaction model of FEMLAB (Multiphysics in MATLAB) software version 2.3
Fig. 1. (A) Schematic of particle in bed, thermal and hydrodynamic boundary conditions, (B) schematic diagram of cross flow (C) schematic diagram of axial flow for cylindrical particle in circular channel.
F.S. Mirhashemi et al. / International Communications in Heat and Mass Transfer 38 (2011) 1148–1155
which implements the finite element methods (FEM). The average particle Nusselt number can be determined by the following equation: dp Ts −Tb
Nu =
∂T ∂n s
ð7Þ
2.3. Heat and mass transfer analogy In this study mass transfer measurements have been done using the naphthalene sublimation technique [18]. On the grounds of the similarity of the governing equations and analogy between the heat and mass transfer, mass transfer results have been applied to heat transfer processes. The constant properties for flow have been assumed which allows one to solve the continuity and momentum equations in an uncoupled set of equations. The assumptions are valid for small differences of temperature and concentration in the flow field. The molar flow rate of the naphthalene from the particle surface to the bulk gas is calculated as follows: Δm = kg Ap ðCs −Cb Þ MΔt
ð8Þ
where Δm is the mass reduction of the naphthalene cylinder during time Δt of an experiment. The conditions of experiments are chosen as the concentration in the bulk flow that can be assumed zero (Cb = 0) within an acceptable error margin. The naphthalene concentration at the surface of the cylinder particles are given by: Cs =
P sat RT
ð9Þ
Eq. (8) can be rewritten as: kg =
Δm Ap Cs MΔt
ð10Þ
where the cylinder mass reduction must be determined, experimentally. Then Sherwood number is obtained by: kg dp D
Sh =
ð11Þ
The Schmidt number is 2.284 in air temperature of 295 K [19]. Therefore, in this work an average diffusion coefficient of naphthalene in air (0.0681 cm 2/s) has been applied. The Chilton–Colburn analogy is applied to relate the heat and mass transfer experiments: jH =
Nu Re Prn
ð12Þ
jD =
Sh Re:Scn
ð13Þ
For similar geometry jH equals jD. Thus, the ratio of Nusselt and Sherwood numbers can be expressed as: Nu = Sh
n Pr Sc
ð14Þ
where in this work, the exponent n is constant that can be determined from the empirical results and has been found in the range of 1/3 up to 0.4.
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naphthalene coatings on test specimens include machining, dipping, spraying, casting or combination of these methods [19]. Equilateral cylinder (D = 0.025) was coated by dipping into the melted naphthalene. The cylinder was placed in a glass-column with a channel-to-particle diameter ratio of N = 2.65 that the air passed through the distributor before the entrance of the channel. Upstream of the column, a turbulent flow of air with temperature of 22 °C (Pr = 0.74) was held to obtain a sufficient high mass transfer rate of naphthalene. Saturated pressure and the naphthalene concentration will be constant in surface of cylinder, providing that temperature of cylinder surface is kept invariable. The mass transfer rate was determined by weighting cylinder right before and after an experimental run at a constant air velocity. The experiments have been run for 4, 5, 6 and 7 m 3/hr air flow rates for two different particles arrangements. Fig. 1B and C present schematic diagram of two arrangements form of cylinder. 4. Results and discussion 4.1. Simulation of distributer For more uniformity of gas flow field through the bed, a distributer has been installed in bottom of the bed adequately far away the cylindrical particle. The entrance velocity profile to bed, after the distributer, has been independently calculated by CFD simulation. The distributer includes 170 orifices with 1 mm diameter and triangular arrangement. Schematic diagram of distributer is shown in Fig. 2A that vividly represents the arrangement of orifices. For simulation of distributer, the κ–ε turbulence model (Eqs. (5) and (6)) is used for prediction of velocity profile and air is considered as an ideal gas. Furthermore, for mesh independency two tetrahedral and hexahedral mesh types have been used with mesh density of 300,000 cells. With respect to the obtaining results, hexahedral mesh type had less divergence compared to the empirical correlation prediction [20]. In the view of this fact, hexahedral mesh is chosen as effective mesh type. Three different hexahedral mesh densities (300,000, 600,000, 700,000 cells) in order to take into account of mesh independency have been applied. According to gaining results the 600,000 mesh density was found as optimized mesh density in terms of accuracy and computational cost. Fig. 2B shows the velocity contour in longitudinal cross section of distributer that vividly shows increasing in velocity in holes because of pressure drop through the distributer. Qureshi and Creasy correlation [20] for pressure drop of distributor is used in this study for verification of CFD simulation results:
ΔP = 1:49
! 1 = 4 ρg V 2 d0 t0 2A2dis
ð15Þ
Fig. 3A shows the pressure drop through the distributer versus the flow rate, obtained by CFD simulation and above mentioned empirical correlation. As it is shown in Fig. 3A, CFD results reveal good agreement with experimental results (mean divergence of 9%). Fig. 3B represents the velocity radial distribution for three different flow rates at distributor's outlet. As shown in this figure, the maximum of velocity is created at the hub of the distributer due to minimum wall drag effect. Now, this velocity profile can be used as entrance velocity profile at bed inlet for study of particle heat transfer. 4.2. Heat transfer of wall affected finite cylinder
3. Experimental work In this work, the particle-to-fluid mass transfer characteristics of wall affected cylindrical particle are investigated using the naphthalene sublimation technique. The techniques available to produce
Fig. 4A and B illustrate the velocity profile for two different arrangements (vertical and parallel form with respect to flow direction). As shown in Fig. 4A, in parallel arrangement the velocity profile after impact with cylindrical particle linearly increases and due
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A
B
Fig. 2. (A) Geometry of distributor in bed, (B) gas velocity contours at distributer upstream and downstream.
to collision with sharp edge, the turbulency increases as well, whereas in other one (Fig. 4B) due to surface curvature, the velocity firstly increases and then decreases gradually. Consequently, the temperature gradient in parallel arrangement is higher compared to cross flow. This difference is likely because of higher velocity in parallel form after collision of flow to cylinder. Thus, it can be expected that the heat transfer or Nusselt number is higher in parallel arrangement in comparison with cross one. Fig. 5 represents a comparison between CFD and experimental work for two arrangements of cylindrical particle in terms of overall Nusselt number. With respect to obtaining experimental data and simulation results, the Sherwood and Nusselt number of cylinder have been derived. As it can be clearly seen, CFD results show an appropriate agreement in comparison with empirical results. Mean deviations from experimental data for axial and cross flow are almost 6 and 4 percent, respectively. On the grounds of this
fact, the CFD model, which has been developed in this study, is proper for simulation of further cases. In order to take into account of wall effect on heat transfer, six different bed to cylinder diameter proportions (N = 1.66, 2.75, 5, 6.66, 12, and 18) have been applied for both arrangements in Reynolds range of 2000 b Re b 6000. Fig. 6A and B illustrate the Nusselt number versus N in different Reynolds number in cross and axial flows, respectively. With regard to these figures, for two arrangements, with enhancement of column to particle diameter ratio (N), the cylindrical particle Nusselt number reduces. With increasing of N, the wall impact on flow around the cylinder decreases, which, in turn, can affect the cylinder heat transfer magnitude. Therefore, the lower N has higher effects on the velocity around the cylinder. On account of this velocity increasing, heat transfer between cylinder and channel increases. On the contrary, the air velocity around the cylinder
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250
200
150
100
50
A 0 3.5
4.5
5.5
6.5
7.5
8 7 6 5 4 3 2 1
B 0 0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
Fig. 3. (A) The effect of air flow rate on distributer pressure drop (comparison of empirical correlation [20] and CFD simulation), (B) velocity radial distribution for three different flow rates at distributer downstream.
decreases with enhancement of bed to particle diameter ratio, which causes the heat transfer coefficient to reduce. According to obtaining results, with increasing the bed to cylinder diameter ratio (N) over the 12, wall has no significant effect on Nusselt number of cylinder; thus, the heat transfer of particle can be analyzed like as a free flow over the cylinder. With respect to the attaining results, trend of data for two arrangements are mainly similar. This means that the difference between two arrangements is only in heat transfer magnitude with fluid and the same tendency can be found in Nusselt number reduction with N enhancement in each arrangement. Fig. 7 shows the contour of temperature in longitudinal crosssection of channel for two different N (N = 1.66, and 12) in cross flow condition. As it can be brightly seen, when N equals 1.66, heat exchanging between fluid and cylinder is very high that makes the fluid temperature almost the same with cylinder temperature. Fig. 7 is also represents that when N equals 12 obtaining Nusselt numbers are very close to free flow and bed's wall has no considerable effects. There exist many of correlations for obtaining a particle Nusselt number in free stream that some of them are listed in Table 1. As shown in Table 1, most of these correlations have not considered the wall effects (bed to cylinder diameter ratio) as a effective factor for calculating the heat transfer coefficient. Fig. 8 represents the comparison of CFD simulation derived Nusselt number and prediction
Fig. 4. Velocity profile around the particle in two arrangements (A) axial flow (B) cross flow.
Fig. 5. Comparison between CFD simulation results and experimental data for axial and cross flows.
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F.S. Mirhashemi et al. / International Communications in Heat and Mass Transfer 38 (2011) 1148–1155 Table 1 Different equations to predict Nusselt number of cylindrical particle in free flow. Correlation
Reference
NuD = 0:148ReD0:633
Morgan and et al. [2]
(16)
Zukauskas and et al. [1] Quarmby and Al-Fakhri [4]
(17)
NuD = 0:23Re0:6 D
5000 bRe b 50000 1000 b Re b20000
0:85 D Re0:792 NuD = 0:123ReD0:651 + 0:00416 D L " 5 = 8 #4 = 5 0:62Re1 = 2 Pr1 = 3 Re NuD = 0:3 + " #1 = 4 1 + 1+
0:4 Pr
2=3
282000
Churchill and et al. [3]
(18) (19)
ation is about 0.5%) up to Reynolds number of 4400. This deviation reaches to 7% in Reynolds range of higher than 4400. Good agreement between CFD results and Quarmby and Al-Fakhri's correlation attributes to this fact that they considered the effect of diameter-tolength ratio for short cylindrical particle in their correlation. On account of this fact that the Quarmby and Al-Fakhri's correlation presented based on finite cylinder in free stream and wall effects have been ignored in this correlation. In this work with respect to simulation results in different bed to cylinder diameter ratios, this correlation has been modified. According to this modification a new correlation for calculation of Nusselt number for a finite cylinder in flow, affected by wall, has been presented as follows:
NuD = 0:0137
0:85 Re0:98 D −0:102 D + 0:992 ReD L N0:231
ð20Þ
Fig. 9 represents comparison between CFD simulation and modified correlation results for N = 1.66 and N = 6.66. According to the results the mean divergence between two gaining sets of data is about 2.19%. 5. Conclusion Fig. 6. The influence of Reynolds number on Nusselt number for different aspect ratio (N) in (A) cross flow, (B) axial flow.
of three other correlations (Table 1) in different Reynolds numbers. According to Fig. 8 the CFD results show an appropriate agreement compared to Quarmby and Al-Fakhri's correlation [4] results (devi-
Fig. 7. The temperature contours in longitudinal cross-section of channel for two different aspect ratios (N).
The steady state heat transfer of wall affected finite cylindrical particle in two arrangements with constant temperature has been studied by CFD simulation. The experimental works have been done for verification of CFD simulation results. In experimental work mass transfer of cylindrical particle coated with naphthalene has been
Fig. 8. Comparison of Nusselt number predictions in cross flow by CFD and different correlations.
F.S. Mirhashemi et al. / International Communications in Heat and Mass Transfer 38 (2011) 1148–1155
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in low aspect ratio (bed to particle diameter ratio). Finally, a modified correlation based on CFD simulations has been presented for prediction of particle Nusselt number that is affected by boundaries (for a wide range of aspect ratio).
References
Fig. 9. Comparison between CFD data and presented correlation (Eq. (20)) for N = 1.66 and N = 6.66.
considered for prediction of cylindrical particle heat transfer (according to heat and mass transfer). Moreover, the particle Nusselt number changes with Reynolds number (between 2000 and 6000) have been considered. Gaining CFD results show appropriate agreement with the experimental work and the mean divergence of 4% has been attained. In addition, the effect of channel to particle diameter ratio has been considered in terms of particle Nusselt number. The results show that this factor plays an important role on particle heat transfer especially
[1] A.A. Zukauskas, Heat transfer from tubes in cross flow, Advances in Heat Transfer 8 (1972) 116–133. [2] V.T. Morgan, The overall convective heat transfer from smooth circular cylinders, Advances in Heat Transfer 11 (1975) 199–264. [3] S.W. Churchill, M. Bernstein, A correlating equation for forced convection from gases and liquids to a circular cylinder in cross flow, Heat Transfer 99 (1977) 300–306. [4] A. Quarmby, A.A.M. Al-Fakhri, Effect of finite length on forced convection heat transfer from cylinders, International Journal of Heat and Mass Transfer 23 (1980) 463–469. [5] B.H. Chang, A.F. Mills, Effect of aspect ratio on forced convection heat transfer from cylinders, International Journal of Heat and Mass Transfer 47 (2004) 1289–1296. [6] M.T. Dalman, J.H. Merkin, C. Mcgeavy, Fluid flow and heat transfer past two spheres in a cylindrical tube, Computers & Fluids 14 (1986) 267–281. [7] T.F. McKenna, R. Spitz, Heat transfer from catalysts with computational fluid dynamics, AIChE Journal 45 (1999) 2392–2410. [8] B. Lloyd, R. Boehm, Flow and heat transfer around a linear array of spheres, Numerical Heat Transfer 26 (1994) 237–252. [9] O.R. Derkx, A.G. Dixon, Determination of the fixed bed wall heat transfer coefficient using computational fluid dynamics, Numerical Heat Transfers 29 (1996) 777–794. [10] S.A. Logtenberg, A.G. Dixon, Computational fluid dynamics studies of fixed bed heat transfer, Chemical Engineering and Processing 37 (1998) 7–21. [11] S.A. Logtenberg, A.G. Dixon, Computational fluid dynamics studies of the effects of temperature-dependent physical properties on fixed bed heat transfer, Chemical Engineering and Processing 37 (1998) 739–747. [12] M. Nijemeisland, A.G. Dixon, Comparison of CFD simulations to experiment for convective heat transfer in a gas-solid fixed bed, Chemical Engineering Journal 82 (2001) 231–239. [13] M. Nijemeisland, A.G. Dixon, CFD study of fluid flow and wall heat transfer in a fixed bed of spheres, AIChE Journal 50 (2004) 906–911. [14] A. Guardo, M. Coussirat, M.A. Larrayoz, F. Recasens, E. Egusquiza, Influence of the turbulence model in CFD modeling of wall-to-fluid heat transfer in packed beds, Chemical Engineering Science 60 (2005) 1733–1742. [15] M. Nijemeisland, A.G. Dixon, Catalyst design by CFD for heat transfer and reaction in steam reforming, Chemical Engineering Science 59 (2004) 5185–5193. [16] A.H. AhmadiMotlagh, S.H. Hashemabadi, CFD based evaluation of heat transfer coefficient from cylindrical particles, International Communications in Heat and Mass Transfer 35 (2008) 674–680. [17] S.J.P. Romkes, F.M. Dautzenberg, C.M. van den Bleek, H.P.A. Calis, CFD modeling and experimental validation of particle-to-fluid mass and heat transfer in a packed bed at very low channel to particle diameter ratio, Chemical Engineering Journal 96 (2003) 3–13. [18] A.H. AhmadiMotlagh, S.H. Hashemabadi, 3D CFD simulation and experimental validation of particle-to-fluid heat transfer in a randomly packed bed of cylindrical particles, International Communications in Heat and Mass Transfer 35 (9) (2008) 1183–1189. [19] R.J. Goldstein, H.H. Cho, A review of mass-transfer measurements using naphthalene sublimation, Experimental Thermal and Fluid Science 10 (1995) 416–434. [20] A.E. Qureshi, D.E. Creasy, Fluidized bed gas distributors, Powder Technology 113 (1979) 22–29. [21] FEMLAB MultiPhysics V2.3, Modeling Guide Manual, 2002.
Contents lists available at ScienceDirect
International Communications in Heat and Mass Transfer j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / i c h m t
CFD simulation and experimental validation for wall effects on heat transfer of finite cylindrical catalyst☆ F.S. Mirhashemi, S.H. Hashemabadi ⁎, S. Noroozi Computational Fluid Dynamics Research Laboratory, School of Chemical Engineering, Iran University of Science and Technology, Narmak, Tehran, 16846, Iran
a r t i c l e
i n f o
Available online 19 May 2011 Keywords: Heat transfer coefficient Computational Fluid Dynamics (CFD) Cylindrical particle Wall effect Heat and mass transfer analogy
a b s t r a c t In this paper, heat transfer of single cylindrical particle affected by wall has been investigated numerically and experimentally for Reynolds number range 2000 to 6000. The heat transfer in two different orientations, axial and cross flow over the particle has been considered in simulation with MultiPhysics Software FEMLAB version 2.3. The heat and mass transfer analogy technique has been applied for validation of the simulation results. The coated particle with naphthalene was sublimated to obtain the corresponding Sherwood numbers. The results show that the CFD model can predict the particle-to-fluid heat transfer for two situations due to trivial error (an average error of 6%) compared to experimental values. Influence of wall on heat transfer of particle in seven different bed-to-cylinder diameter ratio (N = 1.66, 2.65, 2.75, 5, 6.66, 12, and 18) have been discussed in different velocities. According to obtaining results, with increasing the bed-to-cylinder diameter ratio over the 12 wall have no significant consequence on Nusselt number. Due to this fact, a CFD based correlation has been proposed to consider the wall effects on particle-to-fluid Nusselt number with an average error of 2.19%. © 2011 Elsevier Ltd. All rights reserved.
1. Introduction Packed-beds are widely used in petrochemical, fine chemical and pharmaceutical industries. Detailed knowledge of flow in the void space of such packed beds is essential for understanding different characteristics of these beds. Heat transfer plays an important role in consideration of packed bed performance, and in view of this fact it had been very essential in previous studies. Consideration of single cylindrical particle heat transfer in cross flow is one of these aspects. There exist hundreds of experimental studies in terms of local and mean heat transfer in this case. The majority of these studies mainly focus on dependency of Nusselt on Reynolds and Prandtl numbers and their effects on heat transfer. Zukauskas and Morgan have completely considered the heat transfer of cylinder [1,2]. Although the effects of free turbulence flow on heat transfer effects have been taken into account in these studies; moreover, wall effect and length to cylinder diameter ratio have been ignored. Churchill et al. [3] have proposed a correlation for calculating mean heat transfer that has been used in many other works in this field as verification case. They claimed that turbulence flow, wall effects, and channel to cylinder diameter ratio have minor effects on the heat transfer. Quarmby and Al-Fakhri [4] have scrutinized the effect of length to cylinder diameter ratio on heat transfer. They experimentally considered a range of length to cylinder diameter ratio ☆ Communicated by W.J. Minkowycz. ⁎ Corresponding author. E-mail address: [email protected] (S.H. Hashemabadi). 0735-1933/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.icheatmasstransfer.2011.05.001
between 1 and 12. In so doing, they found for short cylinders (aspect ratio is smaller than 4), the effect of aspect ratio on mean heat transfer is considerable and when this ratio is bigger than 4 it can be ignored. This fact was considered by Chang and Mills [5] that led to derive a correlation in terms of Nusselt number with respect to length to cylinder diameter ratio. Because of extreme difficulty in measuring fluid flow and heat transfer inside the bed by conventional means without disturbing the packing arrangement, efforts towards improvement in modeling using computational fluid dynamics (CFD) have been recently developed. In general, 2D and 3D CFD models have been used in various fields to simulate flow profiles and heat transfer. The earliest CFD fixed bed simulations used two-dimensional models. Dalman et al. [6] investigated an axisymmetric radial plane. This work gave a first insight in flow structure in fixed beds. McKenna et al. [7] obtained valuable insight into the effect of particle size on particle– fluid heat transfer from 2D CFD study of small spherical particles. They have validated their basic models with Ranz-Marshal (RM) correlation and have shown the weakness of that correlation in predicting heat transfer from small clusters of spherical particles. Lloyd and Boehm [8] determined the sphere spacing effect on the drag coefficient and the particle to fluid heat transfer by studying on linear array of eight spheres in 2D. It was found that the heat transfer from the spheres decreased with reduction of particle spacing. The 3D models have been applied for packed beds more recently. Derx and Dixon [9] used a simple model to obtain the wall heat transfer coefficient. An eight-sphere model, in which eight spheres are located into two layers with four ones without solid-solid contact points, has been studied by Logtenberg and Dixon [10,11]. This model shows that
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Nomenclature A Cp D jH jD k L Nu NuD P Pr Re Q V to T y+ x
Area Specific heat capacity Diameter Colbourn j factor (heat transfer coefficient) Colbourn j factor (mass transfer coefficient) Thermal conductivity Length of cylindrical particle Overall Nusselt number, hD/k Circumferentially averaged Nusselt number Pressure Prandtl number Reynolds number Sink or source term in energy equation Velocity vector Thickness of distributor Temperature Dimensionless distance from the wall Cartesian x-coordinate
Greek symbols Cμ, Cε1, Cε2 Empirical constant ρ Density μ Dynamic viscosity ν Kinematic viscosity σk, σ Empirical constant ε Turbulence energy dissipation κ Turbulence kinetic energy Subscripts dis Distributor b Bulk o Orifice p Particle t Turbulent W Wall Superscripts † Transpose
conventional fixed bed models could not describe flow and heat transfer behaviors. Nijemeisland et al. [12,13] modeled the heat transfer of low channel to particle ratio by using CFD. They presented an extensive overview of prior work on modeling and measurement of the characteristics of these packed beds. Furthermore, they showed that CFD method is reliable for modeling flow and heat transfer phenomena inside the packed bed. In similar studies, Guardo et al. [14] drew a comparison between the performance in flow and heat transfer estimation of five different RANS (Reynolds-Averaged Navier–Stokes) turbulence models in a fixed bed composed of 44 homogenous stacked spheres. In one of the few heat transfer studies of cylindrical particles, Nijemeisland et al. [15] simulated heat transfer and fluid flow in a near wall segment of a steam reformer tube-filled with cylindrical particles in which inert packing was heated up, they illustrated that particles with no internal voids appear better than particles with internal voids, in the bed interior. In another investigation, AhmadiMotlagh and Hashemabadi [16] have studied two and three dimensional CFD modeling of heat transfer from discrete cylindrical particles in different situations of flow. They reported a good qualitative as well as reasonable quantitative agreement between CFD results and empirical correlations. Romkes et al. [17] investigated the mass and heat transfer characteristics of a composite structured reactor packing containing spherical particles.
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They showed that the CFD simulations could be used to adequately predict the rate of mass and heat transfer from the catalyst particles to the fluid. In another study, AhmadiMotlagh and Hashemabadi [18] investigated the hydrodynamics and heat transfer characteristics of a randomly packed bed of cylindrical particles with a channel-toparticle diameter ratio of 2. Their results were validated by naphthalene sublimation mass transfer experiments and the particle heat transfer Nusselt number were obtained by the use of analogy between mass and heat transfer. As it mentioned in above studies, the cylindrical catalysts have been the subject of fewer studies rather than spherical catalysts. In spite of the simplicity of modeling the spherical particles, the growing applications of cylindrical catalysts in reaction engineering requires a vital need for further exploration of heat transfer behavior from these particles. Despite the Reynolds and Prandtl number impacts on heat transfer of packed beds, some secondary parameters such as free stream eddies, particle length to diameter ratio, and fraction of channel to cylinder diameter have some influences on heat transfer phenomenon. Correlations that have been presented for heat transfer coefficient in previous works have been mainly ignored the effect of these secondary factors. On the grounds of this fact, in this paper, effects of some parameters such as tube-to-particle diameter ratio and flow direction on cylinderical particles have been illustrated. 2. Theory 2.1. Governing equations In this work compressible turbulence steady state flow heat transfer has been taken into account. The conservation equations of continuity, momentum, and energy can be simplified respectively: ∇⋅ðρV Þ = 0
ð1Þ
h i † ∇⋅ðρVV Þ = −∇P + ∇⋅ ðμ + μt Þ ∇V + ∇V
ð2Þ
∇⋅ −ðk + kt Þ∇T + ρCp TV = Q
ð3Þ
where Q is a heat sink or heat source term which is set to zero in this study, μ t and kt are the turbulent viscosity and thermal conductivity respectively that μ t can be defined as: 2
μ t = ρCμ
κ ε
ð4Þ
and Cμ is a model constant which is equal to 0.09. The standard κ-ε is used in order to take into account of turbulence flow [17]. Where the transport equations for the turbulent kinetic energy κ and the turbulence dissipation rate ε are defined respectively [21]: ∂κ + V⋅∇κ = ∇⋅ ∂t
" ν+
! # 2 Cμ κ2 κ † 2 ∇V + ∇V −ε ∇κ + Cμ σk ε ε
ð5Þ
" ! # 2 Cμ κ2 ∂ε ε † 2 −Cε2 + V⋅∇ε = ∇⋅ ν + ∇ε + Cε1 Cμ κ ∇V + ∇V σε ε κ ∂t ð6Þ where the model constants in the above equations are determined, experimentally [16]: Cε1 = 1:44; Cε2 = 1:92; σκ = 1; σε = 1:3 Turbulent thermal diffusivity (kt/ρCp) is usually related to eddy viscosity via a turbulent Prandtl number (Prt ≈ 0.85–0.9).
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2.2. Boundary conditions and CFD simulation Fig. 1A represents the boundary conditions. For reducing the thermophysical properties changes, the channel walls are adiabatic and the temperature of particle surface is kept at Tp, which was 10 °C warmer than the fluid, during this simulation. Air is used as the cold fluid; furthermore, constant temperature and flow rate is set at inlet (To and Qo). The feed inlet of the column is subjected to inlet velocity boundary condition. Moreover, the outlet of the column is assumed to be fully open to the atmosphere, so the outlet pressure is set at atmospheric pressure. The wall boundaries are set to no slip condition and the standard wall function is applied to near the wall. For turbulent flow the thickness of the cell walls is expressed in the terms of y +. The dimensionless wall distance, y +, is used to verify the
mesh adequacy for this turbulent simulation. A typical value for y + between 30 and 300, or at least between 10 and 1000, is generally recommended for best accuracy [17]. Subsequently, the standard wall function was applied in turbulent flow and mesh independency with respect to y + criteria has been applied and the optimized mesh has been selected. Moreover, the heat transfer inside solid particles is ignored, since particle surfaces are assumed to be isothermal. The particle-to-fluid heat transfer was simulated with surface constant temperature (Ts) for particle which is 10 °C warmer than the fluid (air). As a result of the temperature difference between the particle surface and the fluid, heat is transferred from the particle to the fluid. Above governing equations, Eqs. (1)–(6), with mentioned boundary conditions have been solved with the Thermal-Fluid interaction model of FEMLAB (Multiphysics in MATLAB) software version 2.3
Fig. 1. (A) Schematic of particle in bed, thermal and hydrodynamic boundary conditions, (B) schematic diagram of cross flow (C) schematic diagram of axial flow for cylindrical particle in circular channel.
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which implements the finite element methods (FEM). The average particle Nusselt number can be determined by the following equation: dp Ts −Tb
Nu =
∂T ∂n s
ð7Þ
2.3. Heat and mass transfer analogy In this study mass transfer measurements have been done using the naphthalene sublimation technique [18]. On the grounds of the similarity of the governing equations and analogy between the heat and mass transfer, mass transfer results have been applied to heat transfer processes. The constant properties for flow have been assumed which allows one to solve the continuity and momentum equations in an uncoupled set of equations. The assumptions are valid for small differences of temperature and concentration in the flow field. The molar flow rate of the naphthalene from the particle surface to the bulk gas is calculated as follows: Δm = kg Ap ðCs −Cb Þ MΔt
ð8Þ
where Δm is the mass reduction of the naphthalene cylinder during time Δt of an experiment. The conditions of experiments are chosen as the concentration in the bulk flow that can be assumed zero (Cb = 0) within an acceptable error margin. The naphthalene concentration at the surface of the cylinder particles are given by: Cs =
P sat RT
ð9Þ
Eq. (8) can be rewritten as: kg =
Δm Ap Cs MΔt
ð10Þ
where the cylinder mass reduction must be determined, experimentally. Then Sherwood number is obtained by: kg dp D
Sh =
ð11Þ
The Schmidt number is 2.284 in air temperature of 295 K [19]. Therefore, in this work an average diffusion coefficient of naphthalene in air (0.0681 cm 2/s) has been applied. The Chilton–Colburn analogy is applied to relate the heat and mass transfer experiments: jH =
Nu Re Prn
ð12Þ
jD =
Sh Re:Scn
ð13Þ
For similar geometry jH equals jD. Thus, the ratio of Nusselt and Sherwood numbers can be expressed as: Nu = Sh
n Pr Sc
ð14Þ
where in this work, the exponent n is constant that can be determined from the empirical results and has been found in the range of 1/3 up to 0.4.
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naphthalene coatings on test specimens include machining, dipping, spraying, casting or combination of these methods [19]. Equilateral cylinder (D = 0.025) was coated by dipping into the melted naphthalene. The cylinder was placed in a glass-column with a channel-to-particle diameter ratio of N = 2.65 that the air passed through the distributor before the entrance of the channel. Upstream of the column, a turbulent flow of air with temperature of 22 °C (Pr = 0.74) was held to obtain a sufficient high mass transfer rate of naphthalene. Saturated pressure and the naphthalene concentration will be constant in surface of cylinder, providing that temperature of cylinder surface is kept invariable. The mass transfer rate was determined by weighting cylinder right before and after an experimental run at a constant air velocity. The experiments have been run for 4, 5, 6 and 7 m 3/hr air flow rates for two different particles arrangements. Fig. 1B and C present schematic diagram of two arrangements form of cylinder. 4. Results and discussion 4.1. Simulation of distributer For more uniformity of gas flow field through the bed, a distributer has been installed in bottom of the bed adequately far away the cylindrical particle. The entrance velocity profile to bed, after the distributer, has been independently calculated by CFD simulation. The distributer includes 170 orifices with 1 mm diameter and triangular arrangement. Schematic diagram of distributer is shown in Fig. 2A that vividly represents the arrangement of orifices. For simulation of distributer, the κ–ε turbulence model (Eqs. (5) and (6)) is used for prediction of velocity profile and air is considered as an ideal gas. Furthermore, for mesh independency two tetrahedral and hexahedral mesh types have been used with mesh density of 300,000 cells. With respect to the obtaining results, hexahedral mesh type had less divergence compared to the empirical correlation prediction [20]. In the view of this fact, hexahedral mesh is chosen as effective mesh type. Three different hexahedral mesh densities (300,000, 600,000, 700,000 cells) in order to take into account of mesh independency have been applied. According to gaining results the 600,000 mesh density was found as optimized mesh density in terms of accuracy and computational cost. Fig. 2B shows the velocity contour in longitudinal cross section of distributer that vividly shows increasing in velocity in holes because of pressure drop through the distributer. Qureshi and Creasy correlation [20] for pressure drop of distributor is used in this study for verification of CFD simulation results:
ΔP = 1:49
! 1 = 4 ρg V 2 d0 t0 2A2dis
ð15Þ
Fig. 3A shows the pressure drop through the distributer versus the flow rate, obtained by CFD simulation and above mentioned empirical correlation. As it is shown in Fig. 3A, CFD results reveal good agreement with experimental results (mean divergence of 9%). Fig. 3B represents the velocity radial distribution for three different flow rates at distributor's outlet. As shown in this figure, the maximum of velocity is created at the hub of the distributer due to minimum wall drag effect. Now, this velocity profile can be used as entrance velocity profile at bed inlet for study of particle heat transfer. 4.2. Heat transfer of wall affected finite cylinder
3. Experimental work In this work, the particle-to-fluid mass transfer characteristics of wall affected cylindrical particle are investigated using the naphthalene sublimation technique. The techniques available to produce
Fig. 4A and B illustrate the velocity profile for two different arrangements (vertical and parallel form with respect to flow direction). As shown in Fig. 4A, in parallel arrangement the velocity profile after impact with cylindrical particle linearly increases and due
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A
B
Fig. 2. (A) Geometry of distributor in bed, (B) gas velocity contours at distributer upstream and downstream.
to collision with sharp edge, the turbulency increases as well, whereas in other one (Fig. 4B) due to surface curvature, the velocity firstly increases and then decreases gradually. Consequently, the temperature gradient in parallel arrangement is higher compared to cross flow. This difference is likely because of higher velocity in parallel form after collision of flow to cylinder. Thus, it can be expected that the heat transfer or Nusselt number is higher in parallel arrangement in comparison with cross one. Fig. 5 represents a comparison between CFD and experimental work for two arrangements of cylindrical particle in terms of overall Nusselt number. With respect to obtaining experimental data and simulation results, the Sherwood and Nusselt number of cylinder have been derived. As it can be clearly seen, CFD results show an appropriate agreement in comparison with empirical results. Mean deviations from experimental data for axial and cross flow are almost 6 and 4 percent, respectively. On the grounds of this
fact, the CFD model, which has been developed in this study, is proper for simulation of further cases. In order to take into account of wall effect on heat transfer, six different bed to cylinder diameter proportions (N = 1.66, 2.75, 5, 6.66, 12, and 18) have been applied for both arrangements in Reynolds range of 2000 b Re b 6000. Fig. 6A and B illustrate the Nusselt number versus N in different Reynolds number in cross and axial flows, respectively. With regard to these figures, for two arrangements, with enhancement of column to particle diameter ratio (N), the cylindrical particle Nusselt number reduces. With increasing of N, the wall impact on flow around the cylinder decreases, which, in turn, can affect the cylinder heat transfer magnitude. Therefore, the lower N has higher effects on the velocity around the cylinder. On account of this velocity increasing, heat transfer between cylinder and channel increases. On the contrary, the air velocity around the cylinder
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250
200
150
100
50
A 0 3.5
4.5
5.5
6.5
7.5
8 7 6 5 4 3 2 1
B 0 0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
Fig. 3. (A) The effect of air flow rate on distributer pressure drop (comparison of empirical correlation [20] and CFD simulation), (B) velocity radial distribution for three different flow rates at distributer downstream.
decreases with enhancement of bed to particle diameter ratio, which causes the heat transfer coefficient to reduce. According to obtaining results, with increasing the bed to cylinder diameter ratio (N) over the 12, wall has no significant effect on Nusselt number of cylinder; thus, the heat transfer of particle can be analyzed like as a free flow over the cylinder. With respect to the attaining results, trend of data for two arrangements are mainly similar. This means that the difference between two arrangements is only in heat transfer magnitude with fluid and the same tendency can be found in Nusselt number reduction with N enhancement in each arrangement. Fig. 7 shows the contour of temperature in longitudinal crosssection of channel for two different N (N = 1.66, and 12) in cross flow condition. As it can be brightly seen, when N equals 1.66, heat exchanging between fluid and cylinder is very high that makes the fluid temperature almost the same with cylinder temperature. Fig. 7 is also represents that when N equals 12 obtaining Nusselt numbers are very close to free flow and bed's wall has no considerable effects. There exist many of correlations for obtaining a particle Nusselt number in free stream that some of them are listed in Table 1. As shown in Table 1, most of these correlations have not considered the wall effects (bed to cylinder diameter ratio) as a effective factor for calculating the heat transfer coefficient. Fig. 8 represents the comparison of CFD simulation derived Nusselt number and prediction
Fig. 4. Velocity profile around the particle in two arrangements (A) axial flow (B) cross flow.
Fig. 5. Comparison between CFD simulation results and experimental data for axial and cross flows.
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F.S. Mirhashemi et al. / International Communications in Heat and Mass Transfer 38 (2011) 1148–1155 Table 1 Different equations to predict Nusselt number of cylindrical particle in free flow. Correlation
Reference
NuD = 0:148ReD0:633
Morgan and et al. [2]
(16)
Zukauskas and et al. [1] Quarmby and Al-Fakhri [4]
(17)
NuD = 0:23Re0:6 D
5000 bRe b 50000 1000 b Re b20000
0:85 D Re0:792 NuD = 0:123ReD0:651 + 0:00416 D L " 5 = 8 #4 = 5 0:62Re1 = 2 Pr1 = 3 Re NuD = 0:3 + " #1 = 4 1 + 1+
0:4 Pr
2=3
282000
Churchill and et al. [3]
(18) (19)
ation is about 0.5%) up to Reynolds number of 4400. This deviation reaches to 7% in Reynolds range of higher than 4400. Good agreement between CFD results and Quarmby and Al-Fakhri's correlation attributes to this fact that they considered the effect of diameter-tolength ratio for short cylindrical particle in their correlation. On account of this fact that the Quarmby and Al-Fakhri's correlation presented based on finite cylinder in free stream and wall effects have been ignored in this correlation. In this work with respect to simulation results in different bed to cylinder diameter ratios, this correlation has been modified. According to this modification a new correlation for calculation of Nusselt number for a finite cylinder in flow, affected by wall, has been presented as follows:
NuD = 0:0137
0:85 Re0:98 D −0:102 D + 0:992 ReD L N0:231
ð20Þ
Fig. 9 represents comparison between CFD simulation and modified correlation results for N = 1.66 and N = 6.66. According to the results the mean divergence between two gaining sets of data is about 2.19%. 5. Conclusion Fig. 6. The influence of Reynolds number on Nusselt number for different aspect ratio (N) in (A) cross flow, (B) axial flow.
of three other correlations (Table 1) in different Reynolds numbers. According to Fig. 8 the CFD results show an appropriate agreement compared to Quarmby and Al-Fakhri's correlation [4] results (devi-
Fig. 7. The temperature contours in longitudinal cross-section of channel for two different aspect ratios (N).
The steady state heat transfer of wall affected finite cylindrical particle in two arrangements with constant temperature has been studied by CFD simulation. The experimental works have been done for verification of CFD simulation results. In experimental work mass transfer of cylindrical particle coated with naphthalene has been
Fig. 8. Comparison of Nusselt number predictions in cross flow by CFD and different correlations.
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in low aspect ratio (bed to particle diameter ratio). Finally, a modified correlation based on CFD simulations has been presented for prediction of particle Nusselt number that is affected by boundaries (for a wide range of aspect ratio).
References
Fig. 9. Comparison between CFD data and presented correlation (Eq. (20)) for N = 1.66 and N = 6.66.
considered for prediction of cylindrical particle heat transfer (according to heat and mass transfer). Moreover, the particle Nusselt number changes with Reynolds number (between 2000 and 6000) have been considered. Gaining CFD results show appropriate agreement with the experimental work and the mean divergence of 4% has been attained. In addition, the effect of channel to particle diameter ratio has been considered in terms of particle Nusselt number. The results show that this factor plays an important role on particle heat transfer especially
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