ARTICLE IN PRESS. Chemical Engineering Science

Chemical Engineering Science 65 (2010) 726 738 Contents lists available at ScienceDirect Chemical Engineering Science journal homepage: www.elsevier.com/locate/ces Computational study o orced convective heat transer in structured packed beds with spherical or ellipsoidal particles Jian Yang a, Qiuwang Wang a,, Min Zeng a, Akira Nakayama b a State Key Laboratory o Multiphase Flow in Power Engineering, Xi an Jiaotong University, Xi an, Shaanxi 710049, PR China b Department o Mechanical Engineering, Shizuoka University, 3-5-1 Johoku, Hamamatsu 432, Japan article ino Article history: ceived 2 December 2008 ceived in revised orm 30 July 2009 Accepted 11 September 2009 Available online 25 September 2009 Keywords: Computational luid dynamics Packed bed Ellipsoidal particle Laminar low Turbulence Heat transer abstract Randomly packed bed reactors are widely used in chemical process industries, because o their low cost and ease o use compared to other packing methods. However, the pressure drops in such packed beds are usually much higher than those in other packed beds, and the overall heat transer perormances may be greatly lowered. In order to reduce the pressure drops and improve the overall heat transer perormances o packed beds, structured packed beds are considered to be promising choices. In this paper, the low and heat transer inside small pores o some novel structured packed beds are numerically studied, where the packed beds with ellipsoidal or non-uniorm spherical particles are investigated or the irst time and some new transport phenomena are obtained. Three-dimensional Navier Stokes equations and RNG k e turbulence model with scalable wall unction are adopted or present computations. The eects o packing orm and particle shape are studied in detail and the low and heat transer perormances in uniorm and non-uniorm packed beds are also compared with each other. Firstly, it is ound that, with proper selection o packing orm and particle shape, the pressure drops in structured packed beds can be greatly reduced and the overall heat transer perormances will be improved. The traditional correlations o low and heat transer extracted rom randomly packings are ound to overpredict the pressure drops and Nusselt number or all these structured packings, and new correlations o low and heat transer are obtained. Secondly, it is also revealed that, both the eects o packing orm and particle shape are signiicant on the low and heat transer in structured packed beds. With the same particle shape (sphere), the overall heat transer eiciency o simple cubic (SC) packing is the highest. With the same packing orm, such as ace center cubic (FCC) packing, the overall heat transer perormance o long ellipsoidal particle model is the best. Furthermore, with the same particle shape and packing orm, such as body center cubic (BCC) packing with spheres, the overall heat transer perormance o uniorm packing model is higher than that o non-uniorm packing model. The models and results presented in this paper would be useul or the optimum design o packed bed reactors. & 2009 Elsevier Ltd. All rights reserved. 1. Introduction Packed beds, due to their high surace area-to-volume ratio, are widely used in chemical process industries, such as catalytic reactors, chromatographic reactors, ion exchange columns and absorption towers (Caulkin et al., 2007). In industrial applications, randomly packed beds are prevailing or decades because o their low cost and ease o use compared to other packing methods. Nijemeisland and Dixon (2004) studied the low and wall heat transer perormance in a randomly packed bed o spheres. CFD method was used or obtaining the detailed Abbreviations: SC, simple cubic; FCC, ace center cubic; BCC, body center cubic; CFD, computational luid dynamics Corresponding author. Tel./ax: +86 29 82663502. E-mail address: wangqw@mail.xjtu.edu.cn (Q. W. Wang). velocity and temperature ields or gas lowing through a periodic wall segment test cell. They ound that, local heat transer rates did not correlate statistically with the local low ield but related to larger-scale low structures in the bed. Single phase reacting lows in a randomly packed bed reactor were investigated by Freund et al. (2003) with LBM method. They ound that, not only the local behavior but also the integral quantities, such as the pressure drop, depended remarkably on the local structural properties o the packings. Guardo et al. (2005) presented a comparison between the perormance in low and heat transer estimation o ive dierent RANS turbulence models in a randomly packed bed composed o 44 homogeneous stacked spheres. They ound that, the Spalart Allmaras turbulence model was better than the two-equation RANS models or pressure drop and heat transer parameter estimation. The computational results o pressure drop and heat transer coeicient agreed well 0009-2509/$ - see ront matter & 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.ces.2009.09.026

J. Yang et al. / Chemical Engineering Science 65 (2010) 726 738 727 with those calculated by traditional empirical correlations (Ergun, 1952; Wakao and Kaguei, 1982). Guardo et al. (2006, 2007) perormed similar studies or randomly packed beds later. CFD method was proved to be a reliable tool when modeling convective heat transer phenomena in packed beds. However, the low and heat transer perormances o randomly packed beds may not be optimal, the pressure drops in such packed beds are usually much higher than those in other packed beds (Calis et al., 2001), and the overall heat transer perormances may be greatly lowered. In order to reduce the pressure drops and improve the overall heat transer perormances o packed beds, structured packed beds may be used instead o traditional randomly packed beds. Susskind and Becker (1967) measured the pressure drop o water when it lowed through a bed o stainless steel ball bearings packed in ordered rhombohedra geometry. It revealed that, an increase in the relative horizontal spacing o balls in the same geometric array had a very signiicant eect in decreasing the pressure and the eect o ball spacing on pressure drop was more important than that o voidage. Nakayama et al. (1995) studied the low within a threedimensional spatially periodic array o cubic blocks with dierent low directions. They ound that, the macroscopic correlation orm o pressure drop obtained by their model was similar to that o Ergun equation, but the value o inertial coeicient in the correlation was much lower. Calis et al. (2001) and Romkes et al. (2003) investigated the low and heat transer characteristics in composite structured packed beds o spheres with CFD and experimental methods. Five dierent packing orms were studied. The computational results agreed well with the experiment results. The results showed that, the eects o packing orms on macroscopic low and heat transer characteristics were remarkable. By using composite structured packings, the pressure drop could be greatly reduced. The traditional correlations or low and heat transer obtained rom random packings were unavailable or composite structured packings. Considering our dierent periodically repeating arrangements o particles, Gunjal et al. (2005) studied the low and heat transer in an array o spheres with unit-cell approach. When in turbulent low region, the Ergun equation (Ergun, 1952) overpredicts the drag coeicient and Wakao equation (Wakao and Kaguei, 1982) overpredicts the Nusselt number o particle to luid. Furthermore, Larachi et al. (2003) and Petre et al. (2003) developed a combined mesoscale microscale predictive approach to apprehend the aerodynamic macroscale phenomena in structured packings. They advocated CFD as a successul, rapid and economic tool or design and optimization, in terms o energy dissipation, o new structured packing shapes. Similar studies or structured packed beds were also presented by Suekane et al. (2003), Nijemeisland and Dixon (2001) and Lee et al. (2007). All these studies demonstrate that, CFD method is reliable or modeling low and heat transer phenomena inside packed beds o particles. Not only local behavior but also macroscopic characteristics o low and heat transer are signiicantly aected by the internal structural properties o packed beds. The pressure drops can be greatly reduced by using structured packings and the traditional correlations are questionable or ormulating low and heat transer perormances in structured packed beds. According to these studies, we can believe that, with proper selection o packing orm and particle shape, the overall heat transer perormances o packed beds would be improved. In order to give a comprehensive understanding o the transport processes in structured packed beds and optimize the overall heat transer perormances, the low and heat transer inside small pores o novel packings, such as non-uniorm packing and packings o ellipsoid particles, are studied in this paper. According to authors knowledge, almost no attentions have been paid to such packings yet and some new transport phenomena would be obtained. Three-dimensional Navier Stokes and energy equations are solved by using commercial code CFX10. The eects o ynolds number, packing orm and particle shape are studied in detail. 2. Physical model and computational method 2.1. Physical model and dimensions As shown in Fig. 1a, the packed bed consisting o large number o particles (more than 1000) is orderly stacked in a square channel, where the ratio o channel height to particle diameter (N) is larger than 10. The channel walls are adiabatic and the temperature o particle suraces is kept at T p. Air is used as the cold luid and the inlet temperature and velocity are kept at T 0 and u 0, respectively. For current computational condition, direct simulation o low and heat transer or the entire packed bed with so large number o particles is still ineasible. Thereore, representative packed channel with appropriate boundary conditions is selected or present study (see Fig. 1b). The computational domain consists o inlet block, packed channel and outlet block. The packed channel is composed o 8 packed cells to guarantee periodically developed low and heat transer inside. The symmetry boundary conditions are adopted or top, bottom and side walls and the outlet low and heat transer are considered to be ully developed. As shown in Fig. 2, six dierent kinds o packed cells, including three kinds o packing orms (simple cubic, body center cubic and ace center cubic packing) and three kinds o particle shapes (spherical, lat ellipsoidal and long ellipsoidal particle), are selected or investigating the eects o coniguration. The body center packing is divided into uniorm and non-uniorm packings. The non-uniorm body center packing is composed o eight big particles at eight corners and one small particle at body center, where eight big particles contact with each other and the small particle contacts with all big particles. Based on these packed cells, six dierent packed channels are also constructed as presented in Fig. 3, where each packed channel is composed o 8 packed cells. The physical dimensions o computational domain and dierent packed cells are presented in Table 1. Fig. 1. Physical model: (a) structured packed bed and (b) representative computational domain.

728 J. Yang et al. / Chemical Engineering Science 65 (2010) 726 738 Fig. 2. Dierent packed cells:(a) SC (Sphere);(b) BCC (Uniorm sphere);(c) BCC-1 (Non-uniorm sphere);(d) ;(e) and () FCC-2 (Long ellipsoid). 2.2. Governing equations and computational methods The low in the computational domain is considered to be incompressible and steady. Three-dimensional Navier Stokes and energy equations are employed or the computation. The RNG k e turbulence model and scalable wall unction are adopted or internal turbulent low when 4300. The RNG k e turbulence model is applicable to the small-scale eddies, which are independent o the larger-scale phenomena that create them, and it is more suitable or modeling turbulent low in complex geometries (Yakhot and Orszag, 1986; Nijemeisland and Dixon, 2004). Because the computational domain is symmetrical according to y and z axis, only 1/4 part o the computational domain is inally used or the simulation. Furthermore, since particle suraces are assumed to be isothermal, the heat transer inside solid particles is not considered. The conservation equations or mass, momentum, and energy are as ollows: Continuity: r ~ V ¼ 0 ð1þ Momentum: r ð ~ V r ~ V Þ¼ rpþr ½ðm þm t Þðr ~ V þðr ~ V Þ T ÞŠ Energy: r ð V ~ rtþ¼r k c p þ m t s T ðrtþ ð2þ ð3þ Fig. 3. Dierent packed channels: (a) SC (Sphere);(b) BCC (Uniorm sphere);(c) BCC-1 (Non-uniorm sphere);(d) ;(e) FCC-1(Flat ellipsoid) and () FCC- 2 (Long ellipsoid). The transport equations or RNG k e model are as ollows (CFX 10, 2005): 8 k : r ð V ~ rkþ¼r m þ m t >< rk þp s k r e k e : r ð V ~ reþ¼r m þ m t re þ c e1e s e k P e k c e2 r 2 ð4þ >: k

J. Yang et al. / Chemical Engineering Science 65 (2010) 726 738 729 Table 1 Geometry parameters or physical models. Packing model a (mm) b(w) (mm) c(h) (mm) L 1 (mm) L 2 (mm) L 3 (mm) d p (mm) d h (mm) SC (sphere) 12.12 12.12 12.12 30 96.96 80 0.492 12.00 7.75 BCC (uniorm sphere) 14.00 14.00 14.00 30 123.96 80 0.340 12.00 4.12 BCC-1 (non-uniorm sphere) 12.12 12.12 12.12 30 108.96 80 0.293 10.64 3.00 FCC (sphere) 17.14 17.14 17.14 30 149.12 80 0.282 12.00 3.14 FCC-1 (lat ellipsoid) 21.58 21.58 10.79 30 187.73 80 0.281 12.00 2.86 FCC-2 (long ellipsoid) 27.21 13.61 13.61 30 236.72 80 0.282 12.00 2.92 where P k is the shear production o turbulence; m t is the turbulent viscosity; c e1 and c e2 are the turbulence model constants in e equation; s T,s k and s e are the Prandtl numbers in energy, k and e equations. Their deinitions and values are as ollows: P k ¼ m t ðr V ~ þðr V ~ Þ T Þ : r V ~ ; k m t ¼ r 2 c m e ðc m ¼ 0:085Þ; Zð1 Z=4:38Þ c e1 ¼ 1:42 Z Z ¼ 1þ0:012Z ; Z ¼ P k ; 3 r c m e c e2 ¼ 1:68; s T ¼ 1:0; s k ¼ s e ¼ 0:718: Boundary conditions: 8 x ¼ 0 T ¼ T 0 ; u ¼ u 0 ; v ¼ w ¼ 0; k ¼ k 0 ; e ¼ e 0 @T x ¼ L @x ¼ @u @x ¼ @v @x ¼ @w @x ¼ @k @x ¼ @e @x ¼ 0 >< @T y ¼ 0; H=2 @y ¼ @u @y ¼ @w @y ¼ @k @y ¼ @e @y ¼ 0; v ¼ 0 @T z ¼ 0; W=2 @z ¼ @u @z ¼ @v @z ¼ @k @z ¼ @e @z ¼ 0; w ¼ 0 >: Particle walls T ¼ T p ; u ¼ v ¼ w ¼ k ¼ e ¼ 0 The governing equations are solved by using commercial code CFX10 and the convective term in momentum equations is discretized with high resolution scheme. The continuity and momentum equations are solved together with coupled algorithm based on inite control volume method. For convergence criteria, the relative variations o the temperature and velocity between two successive iterations are demanded to be smaller than the previously speciied accuracy levels o 1.0 10 6. 3. Grid independence test and model validation Beore proceeding urther, the grid used or present study is checked at irst. The SC packing with uniorm spheres is selected or the test, as shown in Fig. 2a. Both laminar (=10) and turbulent (=1000) lows are studied. As shown in Fig. 4, the unstructured grids with tetrahedral elements are used and the grids are intensiied on particle walls. In order to avoid generating poor quality meshes at contact points between particles, the particles are stacked with very small gaps (1% o d p ) instead o contact points between each other. According to the studies o Calis et al. (2001) and Nijemeisland and Dixon (2001), these gaps have very small inluence on the low and heat transer (o0.5%). The total numbers o grid elements vary rom 389 247 to 819 694, and the computational results are shown in Table 2, where the Richardson extrapolation approach (Celik and Zhang, 1995) is also adopted to get higher order accuracy results. It shows that, the grid with total element number o 625 795 is good enough or both laminar and turbulent lows, where the maximum length o the grid element is less than 1 mm or central low and 0.2 mm or near wall low regions. For turbulent low, with scalable wall unction method, the value o dimensionless distance o the wall grid elements is o y + 411.06. Thereore, similar grids to that o the test case are inally employed or the ollowing studies. The total numbers o grid elements or dierent packings are listed in Table 3. Furthermore, in order to validate the reliability and accuracy o present computational models and methods, similar problem reported by Calis et al. (2001) and Romkes et al. (2003) are restudied in this section. For the selected problem, as shown in Fig. 5, eight uniorm spheres are orderly stacked within a square channel, where the ratio o channel height to particle diameter (N) equals 1. The channel walls are adiabatic and the temperature o particle suraces is kept at T p. Air is used as the cold luid and the inlet temperature and velocity are kept constant. The computational model and method used or this problem are similar to those presented in Section 2. The predicted riction actor () and Nusselt number o particles to luid (Nu s ) are compared with those reported in the above reerences (Calis et al., 2001; Romkes et al., 2003) infig. 6. The average deviation o is less than 5% and the average deviation o Nu s is less than 10%. This indicates that, the computational models and methods presented in this study are reliable and capable o modeling low and heat transer phenomena inside packed beds o particles. 4. sults and discussion 4.1. Basic hydrodynamic and heat transer correlations The macroscopic hydrodynamic and heat transer perormances in packed beds are usually ormulated by traditional correlations as ollows (Ergun, 1952; Wakao and Kaguei, 1982): Dp Dx ¼ 1 2 r Fig. 4. Part o computational grid or SC (Sphere) packing model. Nu s ¼ h s d P k j ~ V D j! 2 1 d h ¼ c 1 þc 2 ð5þ n ¼ a 1 þa 2 Pr 1=3 n d P ð6þ d h where is the riction actor. c 1, c 2 are the model constants in riction actor correlation, with c 1 =133 and c 2 =2.33 in Ergun equation (Ergun, 1952). h s is the heat transer coeicient o

730 J. Yang et al. / Chemical Engineering Science 65 (2010) 726 738 Table 2 Total pressure drop and heat quantity or SC packing (sphere) with dierent grids (Pr=0.7, =10, 1000). Grid (total elements) Richardson extrapolation 389 247 513 393 625795 819 694 Dp total; ¼ 10 =Pa 0.0489 0.0527 0.0539 0.0546 0.0556 Q total; ¼ 10 =W 0.0167 0.0175 0.0179 0.0182 0.0186 Dp total; ¼ 1000 =Pa 33.125 35.110 36.261 36.625 37.132 Q total; ¼ 1000 =W 0.864 0.897 0.915 0.930 0.951 Table 3 Computational grids or dierent packing models. Packing model SC (sphere) BCC (uniorm sphere) BCC-1 (non-uniorm sphere) FCC (sphere) FCC-1 (lat ellipsoid) FCC-2 (long ellipsoid) Grid (total elements) 625795 1263 527 885603 2312 012 2444 331 2373 971 Experiment (Calis et al., 2001) Computation (Calis et al., 2001) (N=1) Fig. 5. Physical model used by Calis et al. (2001) and Romkes et al. (2003). particle to luid. a 1, a 2 and n are the model constants in heat transer equation, with a 1 =2.0, a 2 =1.1 and n=0.6 in Wakao equation (Wakao and Kaguei, 1982). The ynolds number (), pore scale hydraulic diameter (d h ) and equivalent particle diameter (d p ) are deined as ollows: ¼ r ðj~ V D j=þd h m ; d h ¼ 4 1 V p A p ; d p ¼ 2 3V p cell 4p 1=3 ð7þ 10 3 where ~ V D is the Darcy velocity vector; is the porosity; V p is the particle volume; A p is the area o particle surace. The subscript cell means the value is obtained within packed cell. Furthermore, the low in packed beds can be considered as low in porous media, the macroscopic hydrodynamics can be modeled by Forchheimer extended Darcy model (Nield and Bejan, 2006) as ollows: Nu s Experiment (Romkes et al., 2003) Computation (Romkes et al., 2003) (N=1) dp dx ¼ 1 m V r K ~ D þ p cf iii j V ~ D j V ~ D ð8þ K Compare Eq. (8) with Eq. (5), the permeability (K) and the Forchheimer coeicient (c F ) can be deined as ollows: K ¼ d2 h c 1 =2 ; c F ¼ p c 2=2 iiiiiiiiii 3=2 : ð9þ c 1 =2 In the present study, the microscopic results obtained in the 5th to 7th packed cells are integrated and averaged to extract the macroscopic results, where the periodic low and heat transer are ormed inside. The macroscopic pressure drop (Dp=Dx), heat transer coeicient (h s ) and overall heat transer eiciency (g) are 10 3 Fig. 6. Variations o riction actor and Nusselt number o particle to luid: (a) and (b) Nu s.

J. Yang et al. / Chemical Engineering Science 65 (2010) 726 738 731 Fig. 7. Temperature ields in central diagonal cross-sections o dierent packings with spherical particles (T 0 =293 K, T p =333K, Pr=0.7, =1000). deined as ollows: Dp Dx ¼ 1 Z Z Xcell7 1 ðp 3 ðx out x in ÞA in p out ÞdA; in A in h s ¼ 1 3 cell5 Xcell7 cell5 r c p RRA in uðt ;out T ;in ÞdA ; g ¼ h s =Dp ð10þ A p ðt p T Þ where Dp and T are the average pressure drop and temperature o luid in packed cell, respectively, and their deinitions are as ollows: Dp ¼ 1 Z Z Xcell7 1 ðp 3 A in p out ÞdA; cell5 in A in T ¼ 1 RR! A in uðt ;out þt ;in ÞdA RR ð11þ 2 A in uda cell where the subscript in and out means the inlet and outlet sections o packed cell, respectively. 4.2. The eect o packing orm Firstly, the eect o packing orm is examined and three dierent kinds o packing orms are studied in this section (SC, BCC and FCC packings with uniorm spheres, see Fig. 2a, b and d). The ynolds number () is varying rom 1 to 5000 with Pr=0.7, T 0 =293 K and T p =333K. In order to produce periodically developed low and heat transer in packed channels, each packed channel is composed o 8 packed cells. The temperature distributions in central diagonal cross-sections o dierent packings are shown in Fig. 7. It shows that, the luid temperature in the FCC packed channel increases most quickly along main low direction. The average temperatures o luid in the 6th packed cells o SC, BCC and FCC packings are 308, 317 and 327 K, respectively, which indicates that, the heat transer o the FCC packing would be the highest and it would be the lowest or the SC packing. The coniguration o the FCC packing is the most compact and its surace area-to-volume ratio is the highest, which leads to the best heat transer perormance. The local streamline distributions in the 6th and 7th packed cells o dierent packings are shown in Fig. 8, where the periodically developed lows are ormed inside. Under the same ynolds number, the luid velocity in the FCC packed cells is the highest and it is the lowest or the SC packing. Large vortices are ormed in the SC packed cells, and such vortices become smaller in the BCC Fig. 8. Local streamline distributions in dierent packings with spherical particles (Pr=0.7, =1000): (a) SC (Sphere) (b) and (c).

732 J. Yang et al. / Chemical Engineering Science 65 (2010) 726 738 and FCC packed cells as porosity decreases. Thereore, the pressure drop o the FCC packing would be the highest because o the lowest porosity, but the local tortuosity o the SC packing would be the highest because o large vortices inside. The variations o pressure drop ðdp=dxþ and riction actor () with ynolds numbers are shown in Fig. 9. InFig. 9a, it shows that, as increases, the pressure drops o dierent packings increase. The pressure drop o the FCC packing is the highest and it is the lowest or the SC packing, which agrees well with the analysis o local low variations. When is small (o10), the viscosity eects dominate and the predicted pressure drops o structured packings agree well with those obtained by the Ergun equation. As increases (410), the inertial eects are increasing and become dominant, and the Ergun equation overpredict the pressure drops or structured packings. These results show that, the hydrodynamics o structured packings and random packings are quite dierent. The tortuosities o structured packings are much lower and the pressure drops would be greatly reduced. In Fig. 9b, it shows that, as increases, the riction actors decrease at irst and then keep constant when 43000, which agrees well with traditional studies. When o10, the riction actors o dierent packings are almost same. When 410, the deviations are becoming large. The riction actor o the SC packing is the highest and it is the lowest or the FCC packing. The model constants c 1, c 2 in riction actor correlation b a Δ p.δx -1 / Pa m -1 10 6 10 5 10 4 10 3 10-2 Ergun Equation (Random) SC (Sphere) 10 3 SC (Sphere) Ergun equation (Random) SC (Sphere) 10 3 Fig. 9. Variations o pressure drops and riction actors or dierent packing orms (sphere): (a) Dp.Dx -1 /Pa m -1 and (b). (in Eq. (5)) are obtained with non-linear itting method and the average itting deviation is less than 5%. The predicted values o c 1, c 2 are compared in Table 4. For dierent structured packings, the values o c 1 are similar, while the values o c 2 are quite dierent, which is the highest or the SC packing and the lowest or the FCC packing. This conirms that, the tortuosity o the SC packing is the highest and it is the lowest or the FCC packing. The variations o Nusselt number (Nu s ) and overall heat transer eiciency (g) o particle to luid with ynolds numbers are shown in Fig. 10. InFig. 10a, it shows that, as increases, the Nusselt numbers o dierent packings increase. The value o Nu s Table 4 Predicted values o c 1, c 2 (sphere). Packing model d h (mm) c 1 c 2 SC (sphere) 0.492 7.75 143.88 0.882 BCC (sphere) 0.340 4.12 129.81 0.374 FCC (sphere) 0.282 3.14 164.12 0.297 Random (Ergun, 1952) 133 2.33 Nu s γ/ W m -2 K -1 Pa -1 SC (Sphere) SC (Sphere) Wakao equation (Random) SC (Sphere) 10 3 10 3 10 3 Ergun equation & Wakao equation (Random) SC (Sphere) Fig. 10. Variations o Nusselt numbers and overall heat transer eiciencies o particle to luid or dierent packing orms (sphere): (a) Nu s and (b) g/wm -2 K -1 Pa -1.

J. Yang et al. / Chemical Engineering Science 65 (2010) 726 738 733 or the FCC packing is the highest and it is the lowest or the SC packing, which agrees well with the analysis o local temperature variations. The Wakao equation is ound to overpredict Nu s or structured packings. The model constants a 1, a 2 and n in Nu s correlation (in Eq. (6)) are obtained with non-linear itting method and the average itting deviation is less than 10%. The predicted values o a 1, a 2 and n are compared in Table 5. For dierent structured packings, the values o a 1 and n are similar; while the values o a 2 are quite dierent, which is the highest or the FCC packing and the lowest or the SC packing. In Fig. 10b, it shows that, as increases, the overall heat transer eiciencies o dierent packings decreases. The value o g or the SC packing is the highest and it is the lowest or the FCC packing. These results indicate that, high heat transer perormance o the FCC packing model is obtained at cost o high pressure drop, and the overall heat transer perormance is low. With the same physical parameters, the overall heat transer eiciencies o the BCC and FCC packings are much higher than that o random packings when 4100, however, the overall heat transer eiciency o the SC packing is lower than that o random packing. Thereore, or high porosity case, the random packing is recommended, while or low porosity case, the structured packing orms, such as BCC and FCC packings are recommended or applications. 4.3. The eect o particle shape In this section, the eect o particle shape is investigated. Three dierent kinds o particles are compared, including spherical, lat ellipsoidal and long ellipsoidal particles. Since the heat transer capacity o the FCC packing is the highest, this packing orm is selected or present study (FCC (sphere), FCC-1 (lat ellipsoid) and FCC-2 (long ellipsoid) packings, see Fig. 2d ). The ynolds number () is varying rom 1 to 5000 with Pr=0.7, T 0 =293 K and T p =333 K. The temperature distributions in the central diagonal crosssections o dierent particle models are shown in Fig. 11. It shows that the temperature distributions o dierent particle models are similar under the same packing orm. Thereore, the heat transer Table 5 Predicted values o a 1, a 2 and n (sphere). Packing model d h (mm) d p (mm) a 1 a 2 n SC (sphere) 0.492 7.75 12.00 1.73 0.16 0.70 BCC (sphere) 0.340 4.12 12.00 1.80 0.40 0.63 FCC (sphere) 0.282 3.14 12.00 1.60 0.41 0.67 Random (Wakao and Kaguei, 1982) 2.00 1..60 Fig. 11. Temperature ields in central diagonal cross-sections o FCC packings with dierent particles (T 0 =293 K, T p =333K, Pr=0.7, =1000). Fig. 12. Local streamline distributions in FCC packings with dierent particles (Pr=0.7, =1000): (a) ; (b) and (c) FCC-2 (Long ellipsoid). perormances o dierent particle models may be close under the same packing orm. The local streamline distributions in the 6th packed cells o dierent particle models are shown in Fig. 12. It shows that, under the same ynolds number, the luid velocity in the FCC packed cell is the highest and it is the lowest or the FCC-2 model. For the intrinsic characteristics o particle shape, the tortuosity o the FCC model is the highest and it is the lowest or the FCC-2 model. These results indicate that, under the same packing orm, the tortuosity and the pressure drop would be reduced with proper selection o particle shape, and the overall heat transer eiciency may be improved. The variations o pressure drop ðdp=dxþ and riction actor () with ynolds numbers are shown in Fig. 13. In Fig. 13a, it shows that, the pressure drops o the FCC and FCC-1 models are similar, which are higher than those o the FCC-2 model. The Ergun equation is ound again to overpredict the pressure drops o dierent particle models. With proper selection o particle shapes, the pressure drop can be reduced with the same packing orm. In Fig. 13b, it shows that, as increases, the riction actors o dierent particle models decrease at irst and then keep constant. The riction actor o the FCC model is the highest and it is the lowest or the FCC-2 model. The predicted values o c 1, c 2 or riction actor correlations are compared in Table 6, which are the

734 J. Yang et al. / Chemical Engineering Science 65 (2010) 726 738 Δ p Δ x -1 / Pa m -1 10 7 10 6 10 5 10 4 10 3 Ergun equation (Random) Nu s Wakao equation (Random) 10 3 10 3 10 3 Ergun equation (Random) 10 3 Fig. 13. Variations o pressure drops and riction actors or FCC packings with dierent particles: (a) Dp.Dx -1 /Pa m -1 and (b). γ/ W m -2 K -1 Pa -1 10-2 Ergun equation & Wakao equation (Random) 10 3 Fig. 14. Variations o Nusselt numbers and overall heat transer eiciencies o particle to luid or FCC packings with dierent particles: (a) Nu s and (b) g/wm -2 K -1 Pa -1. Table 6 Predicted values o c 1, c 2 (FCC). Packing model d h (mm) c 1 c 2 FCC (sphere) 0.282 3.136 164.12 0.297 FCC-1 (lat ellipsoid) 0.281 2.855 110.43 0.198 FCC-2 (long ellipsoid) 0.282 2.915 80.35 0.069 Random (Ergun, 1952) 133 2.33 highest or the FCC model and the lowest or the FCC-2 model. This conirms that, the tortuosity or the FCC model is the highest and it is the lowest or the FCC-2 model. The variations o Nusselt number (Nu s ) and overall heat transer eiciency (g) o particle to luid with ynolds numbers are shown in Fig. 14. In Fig. 14a, it shows that, the Nusselt numbers o the FCC and FCC-1 models are similar, which are higher than those o the FCC-2 model. The Wakao equation is ound again to overpredict Nu s or dierent particle models. The predicted values o a 1, a 2 and n or Nu s correlations are compared in Table 7. For dierent particle models, the values o a 1 and n are close to those o Wakao equation, while the values o a 2 are much lower. In Fig. 14b, it shows that, as increases, the overall heat transer eiciencies o dierent particle models decrease. With the same Table 7 Predicted values o a 1, a 2 and n (FCC). Packing model d h (mm) d p (mm) a 1 a 2 n FCC (sphere) 0.282 3.136 12.00 1.60 0.40 0.67 FCC-1 (lat ellipsoid) 0.281 2.855 12.00 1.70 0.34 0.67 FCC-2 (long ellipsoid) 0.282 2.915 12.00 1.60 0.32 0.65 Random (Wakao and Kaguei, 1982) 2.00 1..60 physical parameters, the values o g or dierent particle models are much higher than those o random packings when 4100. The overall heat transer eiciencies o the FCC and FCC-1 models are close, which are lower than that o the FCC-2 model. This conirms that, with the same packing orm, the overall heat transer perormance can be improved with proper selection o particle shape. The overall heat transer perormance o the long ellipsoidal particle model (FCC-2) is the highest and this particle shape is recommended or applications.

J. Yang et al. / Chemical Engineering Science 65 (2010) 726 738 735 10 7 10 6 10 5 Ergun equation (Random) Fig. 15. Temperature ields in central diagonal cross-sections or uniorm and nonuniorm packings with spherical particles (T 0 =293 K, T p =333K, Pr=0.7, =1000). Δ p Δx -1 / Pa m -1 10 4 10 3 BCC-1 (Non-uniorm sphere) BCC (Uniorm sphere) 10 3 BCC (Uniorm sphere) BCC-1 (Non-uniorm sphere) Ergun equation (Random) BCC-1 (Non-uniorm sphere) BCC (Uniorm sphere) 10 3 Fig. 17. Variations o pressure drops and riction actors or uniorm and nonuniorm packings with spherical particles: (a) Dp.Dx -1 /Pa m -1 and (b). Fig. 16. Local streamline distributions in uniorm and non-uniorm packings with spherical particles (Pr=0.7, =1000): (a) BCC (Uniorm sphere) and (b) BCC-1 (Non-uniorm sphere). 4.4. Perormance comparison or uniorm and non-uniorm packing models Finally, the perormances o uniorm and non-uniorm packings are compared. Since the non-uniorm BCC packing orm is stable and easy to be constructed, the BCC packing orm is adopted or present study (BCC (uniorm sphere) and BCC-1(nonuniorm sphere) packings, see Fig. 2b and c). The ynolds number () is varying rom 1 to 5000 with Pr=0.7, T 0 =293 K and T p =333 K. The temperature distributions in central diagonal crosssections o the BCC and BCC-1 packed channels are shown in Fig. 15. It shows that, the luid temperature in the BCC-1 packed channel increases more rapidly than that in the BCC packed channel along main low direction. The average luid temperatures in the 6th packed cells o the BCC and BCC-1 packings are 317 and 323 K, respectively, which indicates that, the heat transer o the BCC-1 model would be higher than that o the BCC model. The coniguration o the BCC-1 packing is more compact and its surace area-to-volume ratio is higher, which leads to the better heat transer perormance. The local streamline distributions in the 6th and 7th packed cells o the BCC and BCC-1 packings are shown in Fig. 16. With the same ynolds number, the luid velocity in the BCC-1 packed cells is much higher than that in the BCC packed cells. There exist pairs o small vortices in the BCC packed cells, but larger vortices are ormed in the BCC-1 packed cells. Thereore, both the tortuosity and pressure drop o the BCC- 1 packing would be higher. The variations o pressure drops ðdp=dxþ and riction actors () o the BCC and BCC-1 packings are shown in Fig. 17. InFig. 17a, it shows that, the pressure drop o the BCC-1 packing is higher than that o the BCC packing. The Ergun equation is ound to overpredict the pressure drops or both the BCC and BCC-1 packings. Although the pressure drop is higher in the BCC-1 packing, it is still much lower than that in random packings. The variations o riction actors are shown in Fig. 17b and the predicted values o correlation constants c 1, c 2 are compared in

736 J. Yang et al. / Chemical Engineering Science 65 (2010) 726 738 Table 8 Predicted values o c 1, c 2 (BCC). Packing model d h (mm) c 1 c 2 BCC (uniorm sphere) 0.340 4.12 129.81 0.374 BCC-1 (non-uniorm sphere) 0.293 3.00 172.53 0.540 Random (Ergun, 1952) 133 2.33 Nu s γ/ W m K -1 Pa -1 BCC (Uniorm sphere) BCC-1 (Non-uniorm sphere) BCC (Uniorm sphere) BCC-1 (Non-uniorm sphere) Wakao equation (Random) BCC-1 (Non-uniorm sphere) BCC (Uniorm sphere) 10 3 BCC-1 (Non-uniorm sphere) Ergun equation & Wakao equation (Random) BCC (Uniorm sphere) 10 3 Fig. 18. Variations o particle wall Nusselt numbers and overall heat transer eiciencies or uniorm and non-uniorm packings with spherical particles: (a) Nu s and (b) g/wm -2 K -1 Pa -1. Table 9 Predicted values o a 1, a 2 and n (BCC). Packing model d h (mm) d p (mm) a 1 a 2 n BCC (uniorm sphere) 0.340 4.12 12.00 1.80 0.40 0.63 BCC-1 (non-uniorm sphere) 0.293 3.00 10.64 1.80 0.49 0.63 Random (Wakao and Kaguei, 1982) 2.00 1..60 Table 8. The riction actor and the values o c 1, c 2 or the BCC-1 packing are higher due to the higher tortuosity inside, which agrees well with the analysis o local low variations. The variations o Nusselt numbers (Nu s ) and overall heat transer eiciencies (g) o particle to luid or the BCC and BCC-1 packings are shown in Fig. 18.InFig. 18a, it shows that, the Nusselt number o the BCC-1 packing is higher than that o the BCC packing, which agrees well with the analysis o local temperature variations. The predicted values o a 1, a 2 and n or Nu s correlations are compared in Table 9. The values o a 1 and n or the BCC-1 packing are the same as those o the BCC packing, while the value o a 2 is higher. In Fig. 18b, it shows that, as increases, the overall heat transer eiciency or both packings decreases. With the same physical parameters, the values o g or both packings are much higher than those o random packings when 4100. The overall heat transer eiciency o the BCC packing is higher than that o the BCC-1 packing. This indicates that, with the same packing orm and particle shape, the heat transer perormance can be improved by using non-uniorm packing while the overall heat transer perormance o such packing is lower due to its higher pressure drop. Thereore, with the same packing orm and same particle shape, the uniorm packing orm is recommended or applications. 5. Conclusions A direct simulation o low and heat transer process inside small pores o some novel structured packed beds has been carried out in this paper, where the packed beds with ellipsoidal or non-uniorm spherical particles are investigated or the irst time and some new transport phenomena are obtained. The eects o packing orm and particle shape are investigated in detail. The low and heat transer perormances in uniorm and non-uniorm packed beds are also compared with each other. The major indings are as ollows: Firstly, it is ound that, with proper selection o packing orm and particle shape, the pressure drop in structured packed beds can be greatly reduced and the overall heat transer perormance will be improved. The traditional correlations are ound to overpredict the pressure drops and Nu s or all the structured packings, and new correlations o low and heat transer are obtained. The orms o these new correlations can it well with those o traditional correlations, but some model constants, such as c 2 and a 2, are much lower. Secondly, it is revealed that, both the eects o packing orm and particle shape are signiicant on the low and heat transer in structured packed beds (a) with the same particle shape, the overall heat transer eiciency o the SC packing is the highest, which is the lowest or the FCC packing. With the same physical parameters, the overall heat transer eiciencies o the BCC and FCC packings are much higher than those o random packings, while the overall heat transer eiciency o the SC packing is lower than that o random packing. Thereore, or high porosity case, the random packing is recommended, while or low porosity case, the structured packing orms, such as the BCC and FCC packings are recommended or applications. (b) with the same packing orm, such as FCC packing, the variations o pressure drops and heat transer perormances o the spherical (FCC) and lat ellipsoidal particle (FCC-1) models are similar, which are higher than those o the long ellipsoidal particle model (FCC-2), while the overall heat transer perormance o the FCC-2 model is higher than those o the FCC and FCC-1 models. Furthermore, with the same packing orm and particle shape, such as BCC packing with

J. Yang et al. / Chemical Engineering Science 65 (2010) 726 738 737 spheres, the pressure drop and heat transer o the nonuniorm packing (BCC-1) are higher than that o the uniorm packing (BCC), while the overall heat transer perormance o the BCC-1 packing is lower. Thereore, with the same packing orm, the long ellipsoidal particle and uniorm packing models are recommended or applications. r s k s T s e density, kg m 3 Prandtl number in turbulent kinetic energy equation Prandtl number in energy equation Prandtl number in turbulence dissipation rate equation porosity These results would be helpul or urther understanding the low and heat transer characteristics in structured packed beds and they would also be useul or the optimum design o packed bed reactors. Notation a length o packed cell, m a 1, a 2 model constants in Nusselt number correlation (in Eq. (6)) A Area, m 2 b width o packed cell, m c height o packed cell, m c 1, c 2 model constants in riction actor correlation (in Eq. (5)) c F Forchheimer coeicient c p speciic heat at constant pressure, J kg 1 K 1 c e1, c e2 model constants in turbulent kinetic energy equation (in Eq. (4)) c m model constant in turbulent viscosity correlation d h pore scale hydraulic diameter, m d p equivalent particle diameter, m riction actor Z RNG k e turbulence model coeicient h s heat transer coeicient o particle to luid, Wm 2 K 1 H height o computational domain, m K permeability, m 2 k turbulent kinetic energy, m 2 s 2 k thermal conductivity o luid, W m 1 K 1 L total length o computational domain, m L 1 length o inlet block, m L 2 length o packed channel, m L 3 length o outlet block, m n model constant in Nusselt number correlation (in Eq. (6)) N ratio o channel height to particle diameter Nu s Nusselt number o particle to luid p pressure, Pa P k shear production o turbulence, kg m 1 s 3 Pr Prandtl number ynolds number T temperature, K u velocity in x direction, m s 1 V volume, m 3 ~V velocity vector, m s 1 ~V D Darcy velocity vector, m s 1 W width o computational domain, m x, y, z coordinate directions, m y + dimensionless wall distance Greek letters g e Z m m t overall heat transer eiciency, W m 2 K 1 Pa 1 turbulence dissipation rate, m 2 s 3 RNG k e turbulence model coeicient dynamic viscosity, kg m 1 s 1 turbulent viscosity, kg m 1 s 1 Subscripts 0 inlet section o packed channel cell value obtained in packed cell Fluid in inlet section o packed cell out outlet section o packed cell p particle Acknowledgments We would like to acknowledge inancial support or this work provided by the National Natural Science Foundation o China (No. 50806056) and National Deense Science and Technology Key Laboratory Foundation o China (No. 9140C7102010804). erences Calis, H.P.A., Nijenhuis, J., Paikert, B.C., Dautzenberg, F.M., van den Bleek, C.M., 2001. 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