- Email: [email protected]
CFD modeling of pressure drop and drag coefficient in fixed beds: Wall effects
Particuology 8 (2010) 37–43
Contents lists available at ScienceDirect
Particuology journal homepage: www.elsevier.com/locate/partic
CFD modeling of pressure drop and drag coefficient in fixed beds: Wall effects Rupesh K. Reddy, Jyeshtharaj B. Joshi ∗ Department of Chemical Engineering, Institute of Chemical Technology, Matunga, Mumabi 400 019, India
a r t i c l e
i n f o
Article history: Received 16 January 2009 Accepted 5 April 2009 Keywords: Computational fluid dynamics Fixed bed Wall effects Pressure drop Drag coefficient
a b s t r a c t Simulations of fixed beds having column to particle diameter ratio (D/dp ) of 3, 5 and 10 were performed in the creeping, transition and turbulent flow regimes, where Reynolds number (dp VL L /L ) was varied from 0.1 to 10,000. The deviations from Ergun’s equation due to the wall effects, which are important in D/dp < 15 beds were well explained by the CFD simulations. Thus, an increase in the pressure drop was observed due to the wall friction in the creeping flow, whereas, in turbulent regime a decrease in the pressure drop was observed due to the channeling near the wall. It was observed that, with an increase in the D/dp ratio, the effect of wall on drag coefficient decreases and drag coefficient nearly approaches to Ergun’s equation. The predicted drag coefficient values were in agreement with the experimental results reported in the literature, in creeping flow regime, whereas in turbulent flow the difference was within 10–15%. © 2009 Chinese Society of Particuology and Institute of Process Engineering, Chinese Academy of Sciences. Published by Elsevier B.V. All rights reserved.
1. Introduction Fixed beds having low D/dp ratios (<10) are often used in the applications of exothermic and endothermic reactions in chemical industries. The flow complexities in these beds have so far prevented the detailed understanding of the flow structure in the interstices between the particles. This important subject has become amenable due to an increase in the computational power and the parallel developments in the numerical techniques. In the present work the effect of the column wall on the pressure drop and drag coefficient in fixed beds having different D/dp ratios have been simulated using FLUENT 6.2 commercial CFD software. 1.1. Previous work For the case of a fully developed flow in a fixed bed, Ergun (1952) has proposed the following semi-empirical correlation by linking the Kozeny–Carman equation for the creeping flow regime and the Burke–Plummer equation for the turbulent regime: 1.75L VL2 ∈ S P 150L VL ∈ 2S + . = 2 3 2 L dp ϕ dp ϕ ∈ 3L ∈L
(1)
The above equation holds for the case of large D/dp ratio (>15) where the condition of near uniformity prevails in the void fraction
∗ Corresponding author. Tel.: +91 22 414 0865; fax: +91 22 414 5614. E-mail address: [email protected] (J.B. Joshi).
throughout the bed. For the case of D/dp < 15, the voidage tends to be greater near the wall than in the bulk region. Under these conditions a significant deviation in pressure drop occurs as compared to Eq. (1) (Di Felice & Gibilaro, 2004; Eisfeld & Schnitzlein, 2001; Foumeny, Benyahia, Castro, Moallemi, & Roshani, 1993; Mehta & Hawley, 1969; Nield, 1983; Reichelt, 1972). Various researchers have addressed the effect of wall on the pressure drop in the low D/dp ratio beds. Earlier studies have been performed by Carman (1937) and Coulson (1949) under creeping flow conditions, and describe the wall effects by including the surface area of the column in the definition of the drag coefficient. Mehta and Hawley (1969) have studied the wall effects on pressure drop in packed beds having 7 < D/dp < 91 and Reynolds number less than 10. Similar type of experiments have been carried out by Chu and Ng (1989) in fixed beds having D/dp ratio 2.9 to 24 and Re < 5. These experiments have shown that the pressure drop (P) behaves according to Eq. (1) only when D/dp > 15. Below this value, at every Re, P was found to increase (compared with Eq. (1)) with a decrease in the D/dp ratio. In the turbulent flow regime the pressure drop in low D/dp ratio fixed beds having spheres, cylinders and rings has been measured by Leva (1947). Foumeny et al. (1993) have made measurements in the turbulent region with D/dp ratio in the range of 3–24. In these cases also, the pressure drop was found to follow Eq. (1) when D/dp > 15. Below this value, at any Re, the pressure drop was found to decrease with a decrease in the D/dp ratio. These wall effects have been comprehensively reviewed by Eisfeld and Schnitzlein (2001) by analyzing all the experimental results in the published literature. The authors have confirmed
1674-2001/$ – see front matter © 2009 Chinese Society of Particuology and Institute of Process Engineering, Chinese Academy of Sciences. Published by Elsevier B.V. All rights reserved.
doi:10.1016/j.partic.2009.04.010
38
R.K. Reddy, J.B. Joshi / Particuology 8 (2010) 37–43
Nomenclature surface area of the particle (m2 ) drag coefficient including both hydrodynamic drag force on particles and the column wall model parameter in equation model parameter in equation model parameter for k–ε model column diameter (m) particle diameter (m) drag force on single particle, (kg m/s2 ) turbulent kinetic energy (m2 /s2 ) length of the bed (m) pressure (N/m2 ) time averaged pressure (N/m2 ) pressure drop (N/m2 ) Reynolds number (dp VL L /L ) surface area of particles (m2 ) surface area of wall (m2 ) instantaneous velocity of component i, where i = 1, 2, 3 corresponds to radial, axial and tangential component of velocity (m/s) time average of velocity (m/s) superficial velocity (m/s) volume of the particle (m3 )
AP CD C1 C2 C D dp FD k L Pi Pi P Re Sp Sw ui
ui VL Vp
Greek symbols ε energy dissipation rate (m2 /s3 ) ∈L voidage of bed ∈S fractional solid hold-up molecular viscosity of fluid (kg/(m s)) L L density of fluid (kg/m3 ) kinematic viscosity (m2 /s) t eddy viscosity (m2 /s) ϕ shape factor model parameter k model parameter ε Subscript and superscript i, j, k co-ordinates in generalized form with value 1, 2 and 3 corresponding to radial, axial and tangential direction L liquid phase S solid phase
the foregoing discussion it may be emphasized that the work of Eisfeld and Schnitzlein (2001) is empirical in nature. For predicting the wall effects on the pressure drop in low D/dp ratio fixed beds a two zone (wall and bulk zones) model have been proposed by Di Felice and Gibilaro (2004). Their model overpredicts the experimental results of Leva (1947) and Foumeny et al. (1993) in the turbulent flow regime. As regards to mathematical modeling, CFD simulations give very detailed flow information in the complex geometry like fixed beds. Dalman, Merkin, and McGreavy (1986) began the 2D CFD simulations by considering only two particles and resolved the flow pattern around the particles for Reynolds number up to 200. Lloyd and Boehm (1994) extended the CFD simulation for the case of eight spheres in a row. In this study the influence of the sphere spacing on the drag coefficient was investigated. However, as expected, 2D simulations were not sufficient to resolve the flow complexities. Nijemeisland and Dixon (2004) have reported 3D simulation (using FLUENT) of fixed bed of D/dp = 4 and 72 particles. Guardo, Coussirat, Larrayoz, Recasens, and Egusquiza (2005) have investigated the wall-to-fluid heat transfer and pressure drop for the case of D/dp = 3.92 (44 particles) and over a Reynolds number (Re) range 100–1000. Calis, Nijenhuis, Paikert, Dautzenberg, and van den Bleek (2001) have simulated the pressure drop and drag coefficient in square channels with particles, having D/dp ratio in the range of 1–2 (8–40 particles) by using CFX 5.3 commercial CFD software. The values obtained from CFD simulation were shown to agree with the experimental measurements on laser Doppler anemometer (LDA). Though this study has not explained the wall effect in low D/dp fixed beds, it serves as a good beginning. Recently Reddy and Joshi (2008) have predicted the wall effects in fixed and expanded beds having only one D/dp ratio of 5. From the foregoing discussion, there is a clear need to understand the wall effects in fixed bed having low D/dp ratios and over a wide range of Reynolds number (0.1–10,000) covering creeping, transition and turbulent regimes by resolving the flow around each individual particle. The present “shorter communication” is in the continuation of our earlier work (Reddy & Joshi, 2008) where, the effect of wall at one D/dp ratio of 5 has been studied. In this communication, the effect of wall on the pressure drop and drag coefficient in fixed beds having different D/dp ratios of 3, 5 and 10 has been presented. However, the available computational resources restricted the simulations for D/dp ratios up to 10.
2. CFD modeling the observations of Mehta and Hawley (1969), Reichelt (1972) and Foumeny et al. (1993) in the creeping and turbulent regimes respectively. Further they conclude that the Reichelt (1972) correlation, which corrects the Ergun’s equation for wall effects, is the most promising one. The correlation is given below: K1 A2W L VL ∈ 2S BW L VL2 ∈ S P = + , L dp ϕ ϕ2 dp2 ∈ 3L ∈ 3L
(2)
where AW = 1 + BW =
2 , 3(D/dp )( ∈ S ) 1 2
[k1 (dp /D) + k2 ]
2
(3) .
(4)
The coefficients K1 , k1 and k2 have been obtained by fitting the experimental data. For spheres, they proposed, K1 = 154, k1 = 1.5 and k2 = 0.88 and for cylinders K1 = 190, k1 = 2 and k2 = 0.77. From
2.1. Computational geometry and grid generation All computational geometries were generated by using bottomup technique (volumes were generated from surfaces and edges) by using commercial software GAMBIT 2.0.4 (Reddy & Joshi, 2008). In all the fixed bed geometrical models having the particle size of 25.4 mm, the total number of particles and the height of the bed were varied for covering three different D/dp ratios (3, 5 and 10). In order to study the effect of D/dp ratio on the pressure drop and drag coefficient all the fixed beds (D/dp = 3, 5 and 10) were constructed in such a way that the void fraction remains constant (0.439). It is important that only one variable is considered at one time. In all geometries, surface of the all particles are well refined (up to 1200 surface nodes) for getting accurate predictions. In the present study the simulations were restricted to particle bed, as a result, the distributor at the inlet and the bed limiter at the outlet have not been simulated. For simplicity, the inlet boundary condition was considered to be a flat velocity profile, whereas the
R.K. Reddy, J.B. Joshi / Particuology 8 (2010) 37–43
39
Table 1 Details of geometrical models of fixed beds. Parameters
D/dp = 3
D/dp = 5
D/dp = 10
Average voidage Column diameter (D), mm Particle diameter (dp ) after 1% reduction, mm Inlet and outlet diameter, mm Number of particles Fluid Bed height, mm Distance between inlet and particle bed starting point, mm Distance between outlet and particle bed ending point, mm Total height of the geometry, mm Wall to particles surface area ratio (Sw /Sp ) Mesh Mesh size, mm Number of controlled volumes
0.439 76.3 25.15 76.3 55 particles in 8 layers Water 179.2 35 35 249.2 0.392 Unstructured tetrahedral 1 0.75 × 106
0.440 127 25.15 127 151 particles in 8 layers Water 177.3 35 35 247.3 0.235 Unstructured tetrahedral 1 2.17 × 106
0.440 254 25.15 254 1120 particles in 15 layers Water 329 35 35 399 0.118 Unstructured tetrahedral 1 4.33 × 106
outlet boundary condition was taken to be a constant pressure. Further, the inlet and the outlet planes were located 35 mm each below and above the constructed beds, respectively. The geometric details of all the fixed beds are given in Table 1. Once the geometry is created the next important step is mesh generation. In the present study we have been restricted to use tetrahedral mesh due to complex geometry. Unstructured tetrahedral mesh generation in complex geometries like fixed beds is a complex task because of triangulation, in which elements are generated, to follow or resolve certain regions (particle-particle and particle-wall contact points) of the complex geometry. Generally triangle faces of tetrahedra degenerate at these contact points, when the Voronoi points (which are the centers of the circumcircles of the triangle vertices) lie outside of the triangles. For avoiding this problem in present study the 1% gap has been created. This is a well practiced method (Calis et al., 2001; Nijemeisland and Dixon, 2004; Reddy & Joshi, 2008). It should be noted that this is not an interstitial gap. This is a modification of the real contact point, for reliable simulations. By conducting two separate simulations with 1% and 2% reduction spheres, it has been confirmed that the fluid velocity is zero at these points (Reddy & Joshi, 2008). Therefore, all the spheres were reduced by 1% of their original size after a complete fixed bed was built. A grid independence study was carried out in the fixed bed of D/dp = 3 with five different mesh sizes (0.2, 0.5, 1, 2 and 3 mm). It was observed that the pressure drop varies by 11% when the grid size is changed from 3 to 2 mm. Further, when the grid resolution was increased from 2 to 1 mm, only 3% increase in the pressure drop was observed. No change in the pressure drop was observed when the grid resolution was increased from 1 to 0.5 mm and further to 0.2 mm. Therefore, it was confirmed that, below the grid size of 1 mm the pressure drop is independent of mesh size. Hence, for further studies grid size of 1 mm was selected. 2.2. Model formulation The equation of continuity and motion for a three dimensional system can be represented in the following form: ∂ui = 0, ∂xi ∂ui uj ∂ui 1 ∂Pi ∂ + =− + ∂xi ∂t ∂xj ∂xj
∂u i ∂xj
(5) .
For the case of creeping flow these equations are discretized by control volume formulation (Patankar, 1980). For the turbulent regime, Eq. (5) on Reynolds averaging reduces to the
following form: ∂ui = 0, ∂xi ∂ui ∂ui 1 ∂Pi ∂ + uj =− + ∂xi ∂t ∂xj ∂xj
( + t )
∂ui ∂xj
(6) .
The modeling of t can be done by using standard k–ε model. The equations of k and ε are given below: modeled k equation ∂k ∂k + uk = t ∂t ∂xk
∂ui ∂xk
2 −ε+
∂ ∂xk
modeled ε equation ∂ε ∂ε ε ε2 ∂ = C1 P − C2 + uj + k k ∂t ∂xj ∂xj
+
t k
t + ε
∂k ∂xk
∂ε ∂xj
,
,
(7)
(8)
where t = C
k2 , ε
(9)
C = 0.09, C1 = 1.44, C2 = 1.92, k = 1.0, ε = 1.3.
These standard values of five empirical constants have been shown to perform better in resolving the flow field in packed beds by Calis et al. (2001) and Guardo et al. (2005) and therefore have also been used in the present work. For turbulent flow the thickness of the cell walls is expressed in terms of y+ . In the case of standard wall functions, which were employed in the present study, y+ should preferably be between 20 and 400. As pointed out by Calis et al. (2001), due to large deviations of the local velocity near the solid surfaces, y+ values varies over the sphere surface, when a homogeneous surface mesh is used. These values can be improved by using a non-uniform surface mesh on the spheres. However, this itself is a separate study. In our simulations average y+ , value varies from 33 (at Re = 1000) to 330 (at Re = 10,000). Flow was assumed to be steady and with a flat velocity profile at the inlet, constant pressure at the outlet (101,325 Pa), and no-slip boundary condition was employed at the wall as well as on the particle surfaces. All terms of the governing equations for steady state were discretized using the second-order upwind differencing scheme. The SIMPLE algorithm has been employed for the pressure-velocity coupling. The convergence criterion (sum of
40
R.K. Reddy, J.B. Joshi / Particuology 8 (2010) 37–43
normalized residuals) was set at 5 × 10−4 for all the equations. The simulations have been performed using a 64-bit machine with 16 dual processors each having a clock speed of 2.4 GHz. We have used commercial CFD software FLUENT 6.2. The simulations have been performed for the fixed beds having D/dp of 3, 5 and 10 in creeping, transition and turbulent regimes, with the Reynolds number range of 0.1–10,000. The CFD simulations predict three components of mean velocity, pressure and the turbulence characteristics. 3. Results and discussion 3.1. Pressure drop The pressure difference between the bottom and top of the bed is the “pressure drop (P)”. The profiles of pressure have been simulated over the Reynolds number range of 0.1–10,000 in D/dp of 3, 5 and 10. One such profile in a fixed bed, having D/dp of 10, at Re = 5000 along the length of the column (399 mm height, which includes the inlet section (35 mm), fixed bed (329 mm) and outlet section (35 mm)), is shown in Fig. 1. The constancy of pressure at the both ends and constant pressure gradient in the bed indicate the nonexistence of entrance and exit effects. Therefore, it may be concluded that the geometrical model of 15 layers with 1120 particles can sufficiently represent a fully developed fixed bed having the same D/dp ratio. Further, the observation of no entrance effect was also confirmed in all fixed beds over the entire range of Reynolds number covered in this work. The variation of pressure drop (P) with respect to Reynolds number in fixed beds having D/dp ratios 3, 5 and 10 are shown in Fig. 2. It can be observed that, in comparison with Ergun’s equation, the simulated pressure drop values, for all D/dp ratio beds, are overestimated in the creeping flow regime (Fig. 2A) and underestimated in the turbulent flow regime (Fig. 2B). Further, it can be observed that magnitude of variation of the pressure drop depends on the D/dp ratio of the bed. The predicted values of pressure drop for fixed beds of D/dp = 3 and 10 have also been compared with Reichelt (1972) correlation in Fig. 3. It has been observed that, in creeping flow regime
Fig. 2. Comparison of simulated pressure drop with Ergun’s equation: (A) Re from 0.1 to 10; (B) Re from 100 to 10,000. 1, Ergun’s equation; 2, CFD (D/dp = 3); 3, CFD (D/dp = 5); 4, CFD (D/dp = 10).
(Re = 0.1–1) the simulated pressure drop values are in agreement with the Reichelt (1972) correlation (Fig. 3A). Whereas, in turbulent flow regime (Re > 1000) a deviation of 10–27% has been observed. The deviations in the turbulent flow regime can be attributed to the unstructured tetrahedral mesh in narrow interstitial gaps between the particles, where very high flow velocities (0.5–1.5 m/s) occur. 3.2. Drag coefficient Drag coefficient (CD ) is defined by the following equation (Joshi, 1983; Pandit & Joshi, 1998): CD =
FD /AP (1/2)L VL2
,
(10)
where Fig. 1. Axial pressure profile at Re = 5000 in fixed bed of D/dp = 10.
FD =
P VP . L ∈S
(11)
R.K. Reddy, J.B. Joshi / Particuology 8 (2010) 37–43
41
Fig. 4. Comparison of predicted CD values with experimental results in the range of Re from 0.1 to 100: 1, Ergun’s equation; 2, CFD (D/dp = 3); 3, CFD (D/dp = 5); 4, CFD (D/dp = 10). () Chu and Ng (1989), D/dp = 2.9; (䊉) Coulson (1949), D/dp = 6.2; () Mehta and Hawley (1969), D/dp = 7.7; () Coulson (1949), D/dp = 8; () Coulson (1949), D/dp = 12.8.
imental results of Coulson (1949), Mehta and Hawley (1969) and Chu and Ng (1989). For all D/dp ratios studied (3, 5 and 10) it can be observed that the predicted CD values are in agreement with the experimental values. In the range of Re from 100 to 10,000 the graph of CD vs. Re is shown in Fig. 5. In turbulent flow (Re > 1000), the predicted CD values in fixed beds having D/dp ratios 3, 5, and 10 are compared with the experimental results of Leva (1947) and Foumeny et al. (1993). A deviation of 10–15% have been observed when the predicted drag coefficient values of D/dp = 3 were compared with the experimental results (without reducing sphere sizes but they have identical voidage (0.439) to that selected in the present work with Fig. 3. Comparison of simulated pressure drop with Reichelt (1972) correlation: (A) Re from 0.1 to 10; (B) Re from 100 to 10,000. 1, Reichelt (1972) correlation (D/dp = 3); 2, CFD (D/dp = 3); 3, Reichelt (1972) correlation (D/dp = 10); 4, CFD (D/dp = 10).
From Eqs. (10) and (11) CD can be written as: CD =
(P/L)(dP /6) (1/2)L VL2 ∈ S
.
(12)
Note that the drag coefficient (CD ) includes both hydrodynamic drag forces on particles and the column wall. In all fixed beds the drag coefficient was calculated from Eq. (12) by using the simulated values of pressure drop at Re = 0.1–10,000. The drag coefficient from Ergun’s equation can be obtained by substituting the value of P/L from Eq. (1) in Eq. (12). 3.3. Comparison with experimental data In order to test the accuracy of drag coefficient (CD ) predictions, comparison was made with the experimental results reported in the literature (without reducing sphere sizes but keeping an identical voidage (0.439) to that selected in the present work with D/dp of 3, 5 and 10). In the creeping and transition regimes the variation of drag coefficient (CD ) in fixed beds with respective to Reynolds number (Re) is shown in Fig. 4. In creeping flow regime (Re = 0.1–1) the predicted CD values in fixed bed were compared with the exper-
Fig. 5. Comparison of predicted CD values with experimental results in the range of Re 100 to 10,000: 1, Ergun’s equation; 2, CFD (D/dp = 3); 3, CFD (D/dp = 5); 4, CFD (D/dp = 10). () Leva (1947), D/dp = 2.67; () Foumeny et al. (1993), D/dp = 3.12; () Leva (1947), D/dp = 5.32; (䊉) Foumeny et al. (1993), D/dp = 4.62; () Leva (1947), D/dp = 9.91; () Foumeny et al. (1993), D/dp = 12.6.
42
R.K. Reddy, J.B. Joshi / Particuology 8 (2010) 37–43
D/dp of 3, 5 and 10) of Leva (1947) at D/dp = 2.67 and Foumeny et al. (1993) for D/dp = 3.12. These experimental results were in the bed having D/dp almost equal to 3. Similarly predicted drag coefficient values of D/dp = 5 and 10 were compared with Leva (1947) for D/dp = 5.32; and Leva (1947) for D/dp = 9.91 respectively and found 10–15% deviation (Fig. 5). The deviations in the turbulent flow regime can be attributed to the unstructured tetrahedral mesh in narrow interstitial gaps between the particles, where very high flow velocities (0.5–1.5 m/s) occur. Further, the present study employs k–ε model, which is an isotropic turbulence model. It is well known that this model cannot capture curvature effects and the boundary layer separation especially at high Reynolds number. Therefore, all these limitations led to differences between the experimental values and the CFD predictions in the turbulent flow regime. For resolving this problem, further work is in progress. 3.4. Comparison with Ergun’s equation The predicted drag coefficient values have been compared with the CD obtained from Ergun’s equation in Fig. 6. For all D/dp ratios simulated in creeping flow regime, especially at Re = 0.1 and 1, it can
be observed that the predicted drag coefficient (CD ) was more comparable with Ergun’s equation (Fig. 6A). It was due to the wall effects where the wall friction and the channeling in the high void regions near the wall may affect the overall pressure drop. In our recent study (Reddy & Joshi, 2008) it has been confirmed that increasing drag coefficient in creeping flow is mainly due to wall friction and the effect of channeling is negligible in creeping flow. It can be observed that at Re = 0.1 and D/dp = 3, drag coefficient increase by 51% whereas at same Re and D/dp of 10, the percentage increase in CD is only 13.2. Further, from Fig. 6A it can be observed that the effect of wall friction (increase in CD ) can be decreases when D/dp increases from 3 to 10. The reason for this is that the ratio of frictional surface area of wall to the surface area of particles (Sw /Sp ) decreases from 0.392 to 0.118 (Table 1) when the D/dp ratio of the bed increases from 3 to 10. Thus the contribution of wall area decreased, when compared to that of particles, with an increase in the D/dp ratio. Therefore, drag coefficient approaches the Ergun’s equation at high D/dp ratios (at D/dp = 10 deviation is 13.2%). In turbulent flow regime (Re > 1000), the predicted CD values were observed to be lower than that predicted by the Ergun’s equation (Fig. 6B). This is due to flow channeling which can occurs in the high voidage (least resistance) regions near the wall (it has been confirmed that in turbulent flow the effect of wall friction can be negligible when compared with the channeling (Reddy & Joshi, 2008)). It can be observed that the effects of channeling (reduction in CD ) decrease as D/dp increases from 3 to 10. This can be attributed to a corresponding decrease in the channeling flow area from 42% (D/dp = 3) to 11% (D/dp = 10). In transition regime (Re = 1–500) the values of predicted CD were found to be within 10% of those estimated by the Ergun’s equation. This is probably due to the countering effects of wall friction and channeling near the wall which perhaps compensate the effects of each other. 4. Conclusions (1) The CFD simulations of pressure drop and drag coefficient were performed in the fixed beds having D/dp ratios 3, 5 and 10 in the entire range of Re (0.1 < Re < 10,000), that is in creeping, transition and turbulent flow regimes. The predicted drag coefficient values were in agreement with the experimental results in creeping flow regime, whereas in turbulent flow the difference was within 10–15%. (2) When the CFD simulation was compared with Ergun’s equation, the predicted CD values in the creeping flow region were found to be higher due to wall friction. As D/dp ratio increase from 3 to 10 the effect of the wall friction decreases and the CFD prediction nearly approaches to the Ergun’s equation (at D/dp = 10 deviation is 13.2%). (3) In turbulent flow region the predicted CD values were found to be lower due to channeling. The effect of channeling decreases when the D/dp ratio increases from 3 to 10. In the transition region (10 < Re < 500) the agreement was found to be within 10% of those estimated by the Ergun’s equation. Acknowledgement One of us (Rupesh Kumar Reddy Guntaka) acknowledges the fellowship support given by the university Grant Commission (UGC), Government of India. References
Fig. 6. Comparison of predicted CD values with Ergun’s equation: (A) Re from 0.1 to 30; (B) Re from 30 to 10,000. 1, Ergun’s equation; 2, CFD (D/dp = 3); 3, CFD (D/dp = 5); 4, CFD (D/dp = 10).
Calis, H. P. A., Nijenhuis, J., Paikert, B. C., Dautzenberg, F. M., & van den Bleek, C. M. (2001). CFD modeling and experimental validation of pressure drop and
R.K. Reddy, J.B. Joshi / Particuology 8 (2010) 37–43 flow profile in a novel structured catalytic reactor packing. Chemical Engineering Science, 56, 1713–1720. Carman, P. C. (1937). Fluid flow through granular beds. Transactions of the Institution of Chemical Engineers, 15, 150–166. Chu, C. F., & Ng, K. M. (1989). Flow in packed bed tubes with small tube to particle diameter ratio. AIChE Journal, 35, 148–158. Coulson, J. M. (1949). The flow of fluids through granular beds: Effect of particle shape and voids in streamline flow. Transactions of the Institution of Chemical Engineers, 27, 237–257. Dalman, M. T., Merkin, J. H., & McGreavy, C. (1986). Fluid flow and heat transfer past two spheres in a cylindrical tube. Computers & Fluids, 14, 267–281. Di Felice, R., & Gibilaro, L. G. (2004). Wall effects for the pressure drop in fixed beds. Chemical Engineering Science, 59, 3037–3040. Eisfeld, B., & Schnitzlein, K. (2001). The influence of confining walls on the pressure drop in packed beds. Chemical Engineering Science, 56, 4321–4329. Ergun, S. (1952). Fluid flow through packed columns. Chemical Engineering Progress, 48, 89–94. Foumeny, E. A., Benyahia, F., Castro, J. A. A., Moallemi, H. A., & Roshani, S. (1993). Correlations of pressure drop in packed beds taking into account the effect of confining wall. International Journal of Heat and Mass Transfer, 36, 536–540. Guardo, A., Coussirat, M., Larrayoz, M. A., Recasens, F., & Egusquiza, E. (2005). Influence of the turbulence model in CFD modeling of wall-to-fluid heat transfer in packed beds. Chemical Engineering Science, 60, 1733–1742.
43
Joshi, J. B. (1983). Solid-liquid fluidised beds: Some design aspects. Chemical Engineering Research and Design, 61, 143–161. Leva, M. (1947). Pressure drop through packed tubes. Part I. A general correlation. Chemical Engineering Progress, 43, 549–554. Lloyd, B., & Boehm, R. (1994). Flow and heat transfer around a linear array of spheres. Numerical Heat Transfer, Part A: Applications, 26, 237–252. Mehta, D., & Hawley, M. C. (1969). Wall effect in packed columns. Industrial and Engineering Chemistry, Process Design and Development, 8, 280–282. Nield, D. A. (1983). Alternative model for wall effect in laminar flow of a fluid through a packed column. AIChE Journal, 29, 688–689. Nijemeisland, M., & Dixon, A. G. (2004). CFD study of fluid flow and wall heat transfer in a fixed bed of spheres. AIChE Journal, 50, 906–921. Pandit, A. B., & Joshi, J. B. (1998). Pressure drop in fixed, expanded and fluidized beds, packed columns and static mixers—a unified approach. Reviews in Chemical Engineering, 14, 321–371. Patankar, S. V. (1980). Numerical heat transfer and fluid flow. New York: Mcgraw-Hill Book Company. Reichelt, W. (1972). Zur Berechnung des Druckverlustes einphasig durchstroK mter Kugel- und ZylinderschuK ttungen. Chemie-Ingenieur-Technik, 44, 1068– 1071. Reddy, R. K., & Joshi, J. B. (2008). CFD modeling of pressure drop and drag coefficient in fixed and expanded beds. Chemical Engineering Research and Design, 86, 444–453.
Contents lists available at ScienceDirect
Particuology journal homepage: www.elsevier.com/locate/partic
CFD modeling of pressure drop and drag coefficient in fixed beds: Wall effects Rupesh K. Reddy, Jyeshtharaj B. Joshi ∗ Department of Chemical Engineering, Institute of Chemical Technology, Matunga, Mumabi 400 019, India
a r t i c l e
i n f o
Article history: Received 16 January 2009 Accepted 5 April 2009 Keywords: Computational fluid dynamics Fixed bed Wall effects Pressure drop Drag coefficient
a b s t r a c t Simulations of fixed beds having column to particle diameter ratio (D/dp ) of 3, 5 and 10 were performed in the creeping, transition and turbulent flow regimes, where Reynolds number (dp VL L /L ) was varied from 0.1 to 10,000. The deviations from Ergun’s equation due to the wall effects, which are important in D/dp < 15 beds were well explained by the CFD simulations. Thus, an increase in the pressure drop was observed due to the wall friction in the creeping flow, whereas, in turbulent regime a decrease in the pressure drop was observed due to the channeling near the wall. It was observed that, with an increase in the D/dp ratio, the effect of wall on drag coefficient decreases and drag coefficient nearly approaches to Ergun’s equation. The predicted drag coefficient values were in agreement with the experimental results reported in the literature, in creeping flow regime, whereas in turbulent flow the difference was within 10–15%. © 2009 Chinese Society of Particuology and Institute of Process Engineering, Chinese Academy of Sciences. Published by Elsevier B.V. All rights reserved.
1. Introduction Fixed beds having low D/dp ratios (<10) are often used in the applications of exothermic and endothermic reactions in chemical industries. The flow complexities in these beds have so far prevented the detailed understanding of the flow structure in the interstices between the particles. This important subject has become amenable due to an increase in the computational power and the parallel developments in the numerical techniques. In the present work the effect of the column wall on the pressure drop and drag coefficient in fixed beds having different D/dp ratios have been simulated using FLUENT 6.2 commercial CFD software. 1.1. Previous work For the case of a fully developed flow in a fixed bed, Ergun (1952) has proposed the following semi-empirical correlation by linking the Kozeny–Carman equation for the creeping flow regime and the Burke–Plummer equation for the turbulent regime: 1.75L VL2 ∈ S P 150L VL ∈ 2S + . = 2 3 2 L dp ϕ dp ϕ ∈ 3L ∈L
(1)
The above equation holds for the case of large D/dp ratio (>15) where the condition of near uniformity prevails in the void fraction
∗ Corresponding author. Tel.: +91 22 414 0865; fax: +91 22 414 5614. E-mail address: [email protected] (J.B. Joshi).
throughout the bed. For the case of D/dp < 15, the voidage tends to be greater near the wall than in the bulk region. Under these conditions a significant deviation in pressure drop occurs as compared to Eq. (1) (Di Felice & Gibilaro, 2004; Eisfeld & Schnitzlein, 2001; Foumeny, Benyahia, Castro, Moallemi, & Roshani, 1993; Mehta & Hawley, 1969; Nield, 1983; Reichelt, 1972). Various researchers have addressed the effect of wall on the pressure drop in the low D/dp ratio beds. Earlier studies have been performed by Carman (1937) and Coulson (1949) under creeping flow conditions, and describe the wall effects by including the surface area of the column in the definition of the drag coefficient. Mehta and Hawley (1969) have studied the wall effects on pressure drop in packed beds having 7 < D/dp < 91 and Reynolds number less than 10. Similar type of experiments have been carried out by Chu and Ng (1989) in fixed beds having D/dp ratio 2.9 to 24 and Re < 5. These experiments have shown that the pressure drop (P) behaves according to Eq. (1) only when D/dp > 15. Below this value, at every Re, P was found to increase (compared with Eq. (1)) with a decrease in the D/dp ratio. In the turbulent flow regime the pressure drop in low D/dp ratio fixed beds having spheres, cylinders and rings has been measured by Leva (1947). Foumeny et al. (1993) have made measurements in the turbulent region with D/dp ratio in the range of 3–24. In these cases also, the pressure drop was found to follow Eq. (1) when D/dp > 15. Below this value, at any Re, the pressure drop was found to decrease with a decrease in the D/dp ratio. These wall effects have been comprehensively reviewed by Eisfeld and Schnitzlein (2001) by analyzing all the experimental results in the published literature. The authors have confirmed
1674-2001/$ – see front matter © 2009 Chinese Society of Particuology and Institute of Process Engineering, Chinese Academy of Sciences. Published by Elsevier B.V. All rights reserved.
doi:10.1016/j.partic.2009.04.010
38
R.K. Reddy, J.B. Joshi / Particuology 8 (2010) 37–43
Nomenclature surface area of the particle (m2 ) drag coefficient including both hydrodynamic drag force on particles and the column wall model parameter in equation model parameter in equation model parameter for k–ε model column diameter (m) particle diameter (m) drag force on single particle, (kg m/s2 ) turbulent kinetic energy (m2 /s2 ) length of the bed (m) pressure (N/m2 ) time averaged pressure (N/m2 ) pressure drop (N/m2 ) Reynolds number (dp VL L /L ) surface area of particles (m2 ) surface area of wall (m2 ) instantaneous velocity of component i, where i = 1, 2, 3 corresponds to radial, axial and tangential component of velocity (m/s) time average of velocity (m/s) superficial velocity (m/s) volume of the particle (m3 )
AP CD C1 C2 C D dp FD k L Pi Pi P Re Sp Sw ui
ui VL Vp
Greek symbols ε energy dissipation rate (m2 /s3 ) ∈L voidage of bed ∈S fractional solid hold-up molecular viscosity of fluid (kg/(m s)) L L density of fluid (kg/m3 ) kinematic viscosity (m2 /s) t eddy viscosity (m2 /s) ϕ shape factor model parameter k model parameter ε Subscript and superscript i, j, k co-ordinates in generalized form with value 1, 2 and 3 corresponding to radial, axial and tangential direction L liquid phase S solid phase
the foregoing discussion it may be emphasized that the work of Eisfeld and Schnitzlein (2001) is empirical in nature. For predicting the wall effects on the pressure drop in low D/dp ratio fixed beds a two zone (wall and bulk zones) model have been proposed by Di Felice and Gibilaro (2004). Their model overpredicts the experimental results of Leva (1947) and Foumeny et al. (1993) in the turbulent flow regime. As regards to mathematical modeling, CFD simulations give very detailed flow information in the complex geometry like fixed beds. Dalman, Merkin, and McGreavy (1986) began the 2D CFD simulations by considering only two particles and resolved the flow pattern around the particles for Reynolds number up to 200. Lloyd and Boehm (1994) extended the CFD simulation for the case of eight spheres in a row. In this study the influence of the sphere spacing on the drag coefficient was investigated. However, as expected, 2D simulations were not sufficient to resolve the flow complexities. Nijemeisland and Dixon (2004) have reported 3D simulation (using FLUENT) of fixed bed of D/dp = 4 and 72 particles. Guardo, Coussirat, Larrayoz, Recasens, and Egusquiza (2005) have investigated the wall-to-fluid heat transfer and pressure drop for the case of D/dp = 3.92 (44 particles) and over a Reynolds number (Re) range 100–1000. Calis, Nijenhuis, Paikert, Dautzenberg, and van den Bleek (2001) have simulated the pressure drop and drag coefficient in square channels with particles, having D/dp ratio in the range of 1–2 (8–40 particles) by using CFX 5.3 commercial CFD software. The values obtained from CFD simulation were shown to agree with the experimental measurements on laser Doppler anemometer (LDA). Though this study has not explained the wall effect in low D/dp fixed beds, it serves as a good beginning. Recently Reddy and Joshi (2008) have predicted the wall effects in fixed and expanded beds having only one D/dp ratio of 5. From the foregoing discussion, there is a clear need to understand the wall effects in fixed bed having low D/dp ratios and over a wide range of Reynolds number (0.1–10,000) covering creeping, transition and turbulent regimes by resolving the flow around each individual particle. The present “shorter communication” is in the continuation of our earlier work (Reddy & Joshi, 2008) where, the effect of wall at one D/dp ratio of 5 has been studied. In this communication, the effect of wall on the pressure drop and drag coefficient in fixed beds having different D/dp ratios of 3, 5 and 10 has been presented. However, the available computational resources restricted the simulations for D/dp ratios up to 10.
2. CFD modeling the observations of Mehta and Hawley (1969), Reichelt (1972) and Foumeny et al. (1993) in the creeping and turbulent regimes respectively. Further they conclude that the Reichelt (1972) correlation, which corrects the Ergun’s equation for wall effects, is the most promising one. The correlation is given below: K1 A2W L VL ∈ 2S BW L VL2 ∈ S P = + , L dp ϕ ϕ2 dp2 ∈ 3L ∈ 3L
(2)
where AW = 1 + BW =
2 , 3(D/dp )( ∈ S ) 1 2
[k1 (dp /D) + k2 ]
2
(3) .
(4)
The coefficients K1 , k1 and k2 have been obtained by fitting the experimental data. For spheres, they proposed, K1 = 154, k1 = 1.5 and k2 = 0.88 and for cylinders K1 = 190, k1 = 2 and k2 = 0.77. From
2.1. Computational geometry and grid generation All computational geometries were generated by using bottomup technique (volumes were generated from surfaces and edges) by using commercial software GAMBIT 2.0.4 (Reddy & Joshi, 2008). In all the fixed bed geometrical models having the particle size of 25.4 mm, the total number of particles and the height of the bed were varied for covering three different D/dp ratios (3, 5 and 10). In order to study the effect of D/dp ratio on the pressure drop and drag coefficient all the fixed beds (D/dp = 3, 5 and 10) were constructed in such a way that the void fraction remains constant (0.439). It is important that only one variable is considered at one time. In all geometries, surface of the all particles are well refined (up to 1200 surface nodes) for getting accurate predictions. In the present study the simulations were restricted to particle bed, as a result, the distributor at the inlet and the bed limiter at the outlet have not been simulated. For simplicity, the inlet boundary condition was considered to be a flat velocity profile, whereas the
R.K. Reddy, J.B. Joshi / Particuology 8 (2010) 37–43
39
Table 1 Details of geometrical models of fixed beds. Parameters
D/dp = 3
D/dp = 5
D/dp = 10
Average voidage Column diameter (D), mm Particle diameter (dp ) after 1% reduction, mm Inlet and outlet diameter, mm Number of particles Fluid Bed height, mm Distance between inlet and particle bed starting point, mm Distance between outlet and particle bed ending point, mm Total height of the geometry, mm Wall to particles surface area ratio (Sw /Sp ) Mesh Mesh size, mm Number of controlled volumes
0.439 76.3 25.15 76.3 55 particles in 8 layers Water 179.2 35 35 249.2 0.392 Unstructured tetrahedral 1 0.75 × 106
0.440 127 25.15 127 151 particles in 8 layers Water 177.3 35 35 247.3 0.235 Unstructured tetrahedral 1 2.17 × 106
0.440 254 25.15 254 1120 particles in 15 layers Water 329 35 35 399 0.118 Unstructured tetrahedral 1 4.33 × 106
outlet boundary condition was taken to be a constant pressure. Further, the inlet and the outlet planes were located 35 mm each below and above the constructed beds, respectively. The geometric details of all the fixed beds are given in Table 1. Once the geometry is created the next important step is mesh generation. In the present study we have been restricted to use tetrahedral mesh due to complex geometry. Unstructured tetrahedral mesh generation in complex geometries like fixed beds is a complex task because of triangulation, in which elements are generated, to follow or resolve certain regions (particle-particle and particle-wall contact points) of the complex geometry. Generally triangle faces of tetrahedra degenerate at these contact points, when the Voronoi points (which are the centers of the circumcircles of the triangle vertices) lie outside of the triangles. For avoiding this problem in present study the 1% gap has been created. This is a well practiced method (Calis et al., 2001; Nijemeisland and Dixon, 2004; Reddy & Joshi, 2008). It should be noted that this is not an interstitial gap. This is a modification of the real contact point, for reliable simulations. By conducting two separate simulations with 1% and 2% reduction spheres, it has been confirmed that the fluid velocity is zero at these points (Reddy & Joshi, 2008). Therefore, all the spheres were reduced by 1% of their original size after a complete fixed bed was built. A grid independence study was carried out in the fixed bed of D/dp = 3 with five different mesh sizes (0.2, 0.5, 1, 2 and 3 mm). It was observed that the pressure drop varies by 11% when the grid size is changed from 3 to 2 mm. Further, when the grid resolution was increased from 2 to 1 mm, only 3% increase in the pressure drop was observed. No change in the pressure drop was observed when the grid resolution was increased from 1 to 0.5 mm and further to 0.2 mm. Therefore, it was confirmed that, below the grid size of 1 mm the pressure drop is independent of mesh size. Hence, for further studies grid size of 1 mm was selected. 2.2. Model formulation The equation of continuity and motion for a three dimensional system can be represented in the following form: ∂ui = 0, ∂xi ∂ui uj ∂ui 1 ∂Pi ∂ + =− + ∂xi ∂t ∂xj ∂xj
∂u i ∂xj
(5) .
For the case of creeping flow these equations are discretized by control volume formulation (Patankar, 1980). For the turbulent regime, Eq. (5) on Reynolds averaging reduces to the
following form: ∂ui = 0, ∂xi ∂ui ∂ui 1 ∂Pi ∂ + uj =− + ∂xi ∂t ∂xj ∂xj
( + t )
∂ui ∂xj
(6) .
The modeling of t can be done by using standard k–ε model. The equations of k and ε are given below: modeled k equation ∂k ∂k + uk = t ∂t ∂xk
∂ui ∂xk
2 −ε+
∂ ∂xk
modeled ε equation ∂ε ∂ε ε ε2 ∂ = C1 P − C2 + uj + k k ∂t ∂xj ∂xj
+
t k
t + ε
∂k ∂xk
∂ε ∂xj
,
,
(7)
(8)
where t = C
k2 , ε
(9)
C = 0.09, C1 = 1.44, C2 = 1.92, k = 1.0, ε = 1.3.
These standard values of five empirical constants have been shown to perform better in resolving the flow field in packed beds by Calis et al. (2001) and Guardo et al. (2005) and therefore have also been used in the present work. For turbulent flow the thickness of the cell walls is expressed in terms of y+ . In the case of standard wall functions, which were employed in the present study, y+ should preferably be between 20 and 400. As pointed out by Calis et al. (2001), due to large deviations of the local velocity near the solid surfaces, y+ values varies over the sphere surface, when a homogeneous surface mesh is used. These values can be improved by using a non-uniform surface mesh on the spheres. However, this itself is a separate study. In our simulations average y+ , value varies from 33 (at Re = 1000) to 330 (at Re = 10,000). Flow was assumed to be steady and with a flat velocity profile at the inlet, constant pressure at the outlet (101,325 Pa), and no-slip boundary condition was employed at the wall as well as on the particle surfaces. All terms of the governing equations for steady state were discretized using the second-order upwind differencing scheme. The SIMPLE algorithm has been employed for the pressure-velocity coupling. The convergence criterion (sum of
40
R.K. Reddy, J.B. Joshi / Particuology 8 (2010) 37–43
normalized residuals) was set at 5 × 10−4 for all the equations. The simulations have been performed using a 64-bit machine with 16 dual processors each having a clock speed of 2.4 GHz. We have used commercial CFD software FLUENT 6.2. The simulations have been performed for the fixed beds having D/dp of 3, 5 and 10 in creeping, transition and turbulent regimes, with the Reynolds number range of 0.1–10,000. The CFD simulations predict three components of mean velocity, pressure and the turbulence characteristics. 3. Results and discussion 3.1. Pressure drop The pressure difference between the bottom and top of the bed is the “pressure drop (P)”. The profiles of pressure have been simulated over the Reynolds number range of 0.1–10,000 in D/dp of 3, 5 and 10. One such profile in a fixed bed, having D/dp of 10, at Re = 5000 along the length of the column (399 mm height, which includes the inlet section (35 mm), fixed bed (329 mm) and outlet section (35 mm)), is shown in Fig. 1. The constancy of pressure at the both ends and constant pressure gradient in the bed indicate the nonexistence of entrance and exit effects. Therefore, it may be concluded that the geometrical model of 15 layers with 1120 particles can sufficiently represent a fully developed fixed bed having the same D/dp ratio. Further, the observation of no entrance effect was also confirmed in all fixed beds over the entire range of Reynolds number covered in this work. The variation of pressure drop (P) with respect to Reynolds number in fixed beds having D/dp ratios 3, 5 and 10 are shown in Fig. 2. It can be observed that, in comparison with Ergun’s equation, the simulated pressure drop values, for all D/dp ratio beds, are overestimated in the creeping flow regime (Fig. 2A) and underestimated in the turbulent flow regime (Fig. 2B). Further, it can be observed that magnitude of variation of the pressure drop depends on the D/dp ratio of the bed. The predicted values of pressure drop for fixed beds of D/dp = 3 and 10 have also been compared with Reichelt (1972) correlation in Fig. 3. It has been observed that, in creeping flow regime
Fig. 2. Comparison of simulated pressure drop with Ergun’s equation: (A) Re from 0.1 to 10; (B) Re from 100 to 10,000. 1, Ergun’s equation; 2, CFD (D/dp = 3); 3, CFD (D/dp = 5); 4, CFD (D/dp = 10).
(Re = 0.1–1) the simulated pressure drop values are in agreement with the Reichelt (1972) correlation (Fig. 3A). Whereas, in turbulent flow regime (Re > 1000) a deviation of 10–27% has been observed. The deviations in the turbulent flow regime can be attributed to the unstructured tetrahedral mesh in narrow interstitial gaps between the particles, where very high flow velocities (0.5–1.5 m/s) occur. 3.2. Drag coefficient Drag coefficient (CD ) is defined by the following equation (Joshi, 1983; Pandit & Joshi, 1998): CD =
FD /AP (1/2)L VL2
,
(10)
where Fig. 1. Axial pressure profile at Re = 5000 in fixed bed of D/dp = 10.
FD =
P VP . L ∈S
(11)
R.K. Reddy, J.B. Joshi / Particuology 8 (2010) 37–43
41
Fig. 4. Comparison of predicted CD values with experimental results in the range of Re from 0.1 to 100: 1, Ergun’s equation; 2, CFD (D/dp = 3); 3, CFD (D/dp = 5); 4, CFD (D/dp = 10). () Chu and Ng (1989), D/dp = 2.9; (䊉) Coulson (1949), D/dp = 6.2; () Mehta and Hawley (1969), D/dp = 7.7; () Coulson (1949), D/dp = 8; () Coulson (1949), D/dp = 12.8.
imental results of Coulson (1949), Mehta and Hawley (1969) and Chu and Ng (1989). For all D/dp ratios studied (3, 5 and 10) it can be observed that the predicted CD values are in agreement with the experimental values. In the range of Re from 100 to 10,000 the graph of CD vs. Re is shown in Fig. 5. In turbulent flow (Re > 1000), the predicted CD values in fixed beds having D/dp ratios 3, 5, and 10 are compared with the experimental results of Leva (1947) and Foumeny et al. (1993). A deviation of 10–15% have been observed when the predicted drag coefficient values of D/dp = 3 were compared with the experimental results (without reducing sphere sizes but they have identical voidage (0.439) to that selected in the present work with Fig. 3. Comparison of simulated pressure drop with Reichelt (1972) correlation: (A) Re from 0.1 to 10; (B) Re from 100 to 10,000. 1, Reichelt (1972) correlation (D/dp = 3); 2, CFD (D/dp = 3); 3, Reichelt (1972) correlation (D/dp = 10); 4, CFD (D/dp = 10).
From Eqs. (10) and (11) CD can be written as: CD =
(P/L)(dP /6) (1/2)L VL2 ∈ S
.
(12)
Note that the drag coefficient (CD ) includes both hydrodynamic drag forces on particles and the column wall. In all fixed beds the drag coefficient was calculated from Eq. (12) by using the simulated values of pressure drop at Re = 0.1–10,000. The drag coefficient from Ergun’s equation can be obtained by substituting the value of P/L from Eq. (1) in Eq. (12). 3.3. Comparison with experimental data In order to test the accuracy of drag coefficient (CD ) predictions, comparison was made with the experimental results reported in the literature (without reducing sphere sizes but keeping an identical voidage (0.439) to that selected in the present work with D/dp of 3, 5 and 10). In the creeping and transition regimes the variation of drag coefficient (CD ) in fixed beds with respective to Reynolds number (Re) is shown in Fig. 4. In creeping flow regime (Re = 0.1–1) the predicted CD values in fixed bed were compared with the exper-
Fig. 5. Comparison of predicted CD values with experimental results in the range of Re 100 to 10,000: 1, Ergun’s equation; 2, CFD (D/dp = 3); 3, CFD (D/dp = 5); 4, CFD (D/dp = 10). () Leva (1947), D/dp = 2.67; () Foumeny et al. (1993), D/dp = 3.12; () Leva (1947), D/dp = 5.32; (䊉) Foumeny et al. (1993), D/dp = 4.62; () Leva (1947), D/dp = 9.91; () Foumeny et al. (1993), D/dp = 12.6.
42
R.K. Reddy, J.B. Joshi / Particuology 8 (2010) 37–43
D/dp of 3, 5 and 10) of Leva (1947) at D/dp = 2.67 and Foumeny et al. (1993) for D/dp = 3.12. These experimental results were in the bed having D/dp almost equal to 3. Similarly predicted drag coefficient values of D/dp = 5 and 10 were compared with Leva (1947) for D/dp = 5.32; and Leva (1947) for D/dp = 9.91 respectively and found 10–15% deviation (Fig. 5). The deviations in the turbulent flow regime can be attributed to the unstructured tetrahedral mesh in narrow interstitial gaps between the particles, where very high flow velocities (0.5–1.5 m/s) occur. Further, the present study employs k–ε model, which is an isotropic turbulence model. It is well known that this model cannot capture curvature effects and the boundary layer separation especially at high Reynolds number. Therefore, all these limitations led to differences between the experimental values and the CFD predictions in the turbulent flow regime. For resolving this problem, further work is in progress. 3.4. Comparison with Ergun’s equation The predicted drag coefficient values have been compared with the CD obtained from Ergun’s equation in Fig. 6. For all D/dp ratios simulated in creeping flow regime, especially at Re = 0.1 and 1, it can
be observed that the predicted drag coefficient (CD ) was more comparable with Ergun’s equation (Fig. 6A). It was due to the wall effects where the wall friction and the channeling in the high void regions near the wall may affect the overall pressure drop. In our recent study (Reddy & Joshi, 2008) it has been confirmed that increasing drag coefficient in creeping flow is mainly due to wall friction and the effect of channeling is negligible in creeping flow. It can be observed that at Re = 0.1 and D/dp = 3, drag coefficient increase by 51% whereas at same Re and D/dp of 10, the percentage increase in CD is only 13.2. Further, from Fig. 6A it can be observed that the effect of wall friction (increase in CD ) can be decreases when D/dp increases from 3 to 10. The reason for this is that the ratio of frictional surface area of wall to the surface area of particles (Sw /Sp ) decreases from 0.392 to 0.118 (Table 1) when the D/dp ratio of the bed increases from 3 to 10. Thus the contribution of wall area decreased, when compared to that of particles, with an increase in the D/dp ratio. Therefore, drag coefficient approaches the Ergun’s equation at high D/dp ratios (at D/dp = 10 deviation is 13.2%). In turbulent flow regime (Re > 1000), the predicted CD values were observed to be lower than that predicted by the Ergun’s equation (Fig. 6B). This is due to flow channeling which can occurs in the high voidage (least resistance) regions near the wall (it has been confirmed that in turbulent flow the effect of wall friction can be negligible when compared with the channeling (Reddy & Joshi, 2008)). It can be observed that the effects of channeling (reduction in CD ) decrease as D/dp increases from 3 to 10. This can be attributed to a corresponding decrease in the channeling flow area from 42% (D/dp = 3) to 11% (D/dp = 10). In transition regime (Re = 1–500) the values of predicted CD were found to be within 10% of those estimated by the Ergun’s equation. This is probably due to the countering effects of wall friction and channeling near the wall which perhaps compensate the effects of each other. 4. Conclusions (1) The CFD simulations of pressure drop and drag coefficient were performed in the fixed beds having D/dp ratios 3, 5 and 10 in the entire range of Re (0.1 < Re < 10,000), that is in creeping, transition and turbulent flow regimes. The predicted drag coefficient values were in agreement with the experimental results in creeping flow regime, whereas in turbulent flow the difference was within 10–15%. (2) When the CFD simulation was compared with Ergun’s equation, the predicted CD values in the creeping flow region were found to be higher due to wall friction. As D/dp ratio increase from 3 to 10 the effect of the wall friction decreases and the CFD prediction nearly approaches to the Ergun’s equation (at D/dp = 10 deviation is 13.2%). (3) In turbulent flow region the predicted CD values were found to be lower due to channeling. The effect of channeling decreases when the D/dp ratio increases from 3 to 10. In the transition region (10 < Re < 500) the agreement was found to be within 10% of those estimated by the Ergun’s equation. Acknowledgement One of us (Rupesh Kumar Reddy Guntaka) acknowledges the fellowship support given by the university Grant Commission (UGC), Government of India. References
Fig. 6. Comparison of predicted CD values with Ergun’s equation: (A) Re from 0.1 to 30; (B) Re from 30 to 10,000. 1, Ergun’s equation; 2, CFD (D/dp = 3); 3, CFD (D/dp = 5); 4, CFD (D/dp = 10).
Calis, H. P. A., Nijenhuis, J., Paikert, B. C., Dautzenberg, F. M., & van den Bleek, C. M. (2001). CFD modeling and experimental validation of pressure drop and
R.K. Reddy, J.B. Joshi / Particuology 8 (2010) 37–43 flow profile in a novel structured catalytic reactor packing. Chemical Engineering Science, 56, 1713–1720. Carman, P. C. (1937). Fluid flow through granular beds. Transactions of the Institution of Chemical Engineers, 15, 150–166. Chu, C. F., & Ng, K. M. (1989). Flow in packed bed tubes with small tube to particle diameter ratio. AIChE Journal, 35, 148–158. Coulson, J. M. (1949). The flow of fluids through granular beds: Effect of particle shape and voids in streamline flow. Transactions of the Institution of Chemical Engineers, 27, 237–257. Dalman, M. T., Merkin, J. H., & McGreavy, C. (1986). Fluid flow and heat transfer past two spheres in a cylindrical tube. Computers & Fluids, 14, 267–281. Di Felice, R., & Gibilaro, L. G. (2004). Wall effects for the pressure drop in fixed beds. Chemical Engineering Science, 59, 3037–3040. Eisfeld, B., & Schnitzlein, K. (2001). The influence of confining walls on the pressure drop in packed beds. Chemical Engineering Science, 56, 4321–4329. Ergun, S. (1952). Fluid flow through packed columns. Chemical Engineering Progress, 48, 89–94. Foumeny, E. A., Benyahia, F., Castro, J. A. A., Moallemi, H. A., & Roshani, S. (1993). Correlations of pressure drop in packed beds taking into account the effect of confining wall. International Journal of Heat and Mass Transfer, 36, 536–540. Guardo, A., Coussirat, M., Larrayoz, M. A., Recasens, F., & Egusquiza, E. (2005). Influence of the turbulence model in CFD modeling of wall-to-fluid heat transfer in packed beds. Chemical Engineering Science, 60, 1733–1742.
43
Joshi, J. B. (1983). Solid-liquid fluidised beds: Some design aspects. Chemical Engineering Research and Design, 61, 143–161. Leva, M. (1947). Pressure drop through packed tubes. Part I. A general correlation. Chemical Engineering Progress, 43, 549–554. Lloyd, B., & Boehm, R. (1994). Flow and heat transfer around a linear array of spheres. Numerical Heat Transfer, Part A: Applications, 26, 237–252. Mehta, D., & Hawley, M. C. (1969). Wall effect in packed columns. Industrial and Engineering Chemistry, Process Design and Development, 8, 280–282. Nield, D. A. (1983). Alternative model for wall effect in laminar flow of a fluid through a packed column. AIChE Journal, 29, 688–689. Nijemeisland, M., & Dixon, A. G. (2004). CFD study of fluid flow and wall heat transfer in a fixed bed of spheres. AIChE Journal, 50, 906–921. Pandit, A. B., & Joshi, J. B. (1998). Pressure drop in fixed, expanded and fluidized beds, packed columns and static mixers—a unified approach. Reviews in Chemical Engineering, 14, 321–371. Patankar, S. V. (1980). Numerical heat transfer and fluid flow. New York: Mcgraw-Hill Book Company. Reichelt, W. (1972). Zur Berechnung des Druckverlustes einphasig durchstroK mter Kugel- und ZylinderschuK ttungen. Chemie-Ingenieur-Technik, 44, 1068– 1071. Reddy, R. K., & Joshi, J. B. (2008). CFD modeling of pressure drop and drag coefficient in fixed and expanded beds. Chemical Engineering Research and Design, 86, 444–453.