Ind. Eng. Chem. Res. 2000, 39, 829-834
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Pressure Drop of Centripetal Gas Flow through Rotating Beds† Chong Zheng,* Kai Guo, Yuanding Feng, and Cun Yang HIGRAVITEC Center, Beijing University of Chemical Technology, Beijing, 100029 China
Nelson C. Gardner* Case Western Reserve University, Cleveland, Ohio 44106
The pressure drop across packed beds that are rotating differs in significant and unexpected ways, compared to the pressure drop behavior across conventional packed beds. For a wide range of operating conditions, (1) the pressure drop across an irrigated bed can be significantly lower than that across a dry bed and (2) for a fixed outer diameter, increasing the depth (i.e., more transfer units) of a rotating bed can result in a reduction in the pressure drop across the bed. Furthermore, the power required to spin the rotor diminishes as the vapor flow rate increases. Previously published correlations on the pressure drop across rotating beds do not explain or predict these phenomena. Here, a semiempirical sectional pressure drop model is presented, where all of these phenomena are related to the gas phase conserving its angular momentum as it flows from the shell through the rotating bed toward the centrally located exit. Introduction Operating a packed bed in a high gravitational field can significantly intensify mass transfer processes. This can be accomplished by spinning a torus shaped (but rectangular in cross section), rigid foam or wire mesh bed and spraying the liquid on the inner surface of the spinning bed.2-4,10-13 The high centrifugal fields generated by the rotation pull the liquid through the bed. By a suitable arrangement of seals, the gas is forced to flow through the bed countercurrent to the liquid flow, entering the rotating bed at the outer periphery and exiting the bed at the inner periphery. Average heights of transfer units for rotating beds are typically a few centimeters, even for viscous materials. The development of modern rotating beds (RB) “Higee” was carried out at ICI by Ramshaw and Maillinson.15 Papers by Ramshaw,14 Lockett,8 Kelleher and Fair,5 and Zheng et al,17 represent a good overview of rotating bed technology. Advantages of this technology include intensified mass transfer with high process rates, short residence times, fast response speed, and reduced capital and operating costs. Commercial application of this technology has been slow in developing in Western countries. However in China, water deareation rotating bed machines varying in capacity from 10 to 300 tons per hour have been successfully operated, and more are planned.18 Applications under investigation include the use of rotating beds for absorption, stripping, particulate removal, reactive precipitation, biooxidation, and polymer devolatilization. Background Although rotating beds show promise for intensified separations and multiphase reactions, the fundamentals * To whom correspondence should be addressed. Fax: 8610-64436609. E-mail:
[email protected] (Zheng). Fax: 1-216368-3016. E-mail:
[email protected] (Gardner). † Project supported by China State Commission of Science and Technology and by China National Natural Science Foundation.
of hydrodynamics and mass transfer in a rotating bed are still poorly understood. Concerning the hydrodynamics of rotating beds, Keyvani and Gardner,6 working with foam metal beds with specific surface areas from 600 to 3000 m-1 and porosities of 92%, reported unexpected phenomena in that (1) for significant operating ranges the pressure drop across an irrigated bed can be substantially less than the pressure drop across a dry rotating bed and (2) the power required to spin the rotor diminishes with increasing gas velocity. Martin and Martelli9 presented pressure drop data for a rotating bed used for distillation studies. Kumar and Rao7 reported pressure drop and mass transfer data on wire mesh sheets with a bed porosity of 0.95 and a surface area of 4000 m-1. Basic and Dudukovic1 reported hydrodynamics and mass transfer data in rotating beds of glass beads. Singh et al.16 published pressure drop data on rotating metal sponge bed of voidage of 0.95 and specific area of 2500 m-1 and on wire gauze of voidage of 0.934 and specific area of 2067 m-1. Lockett8 published the effect of rotating speed on pressure drop and the typical variation of pressure drop with superficial gas velocity at a fixed rotating speed and liquid velocity on RBs packed with corrugated aluminum sheets of specific area 1770 m-1. Liu et al.19 reported data of pressure drops measured across beds packed with plastic of cubic and elliptic cylindrical grains. Most of the previous investigations correlated data based on the assumptions adopted by Kumar and Rao,7 that the total pressure drop across the rotor arises mainly due to the centrifugal and frictional forces and that liquid flow through the bed is in the form of thin films. They also assumed that the gas flows in the rotating bed with tangential velocity Vθ less than the tangential velocity rω of the bed. Kelleher and Fair5 reviewed previously published correlations on RB pressure drops and noted that the pressure drop increased with the square of the rotational speed and linearly with the gas kinetic energy term, while liquid flow rate effects are minimal. Their model was developed based solely upon fundamentals with no empirical parameters.
10.1021/ie980703d CCC: $19.00 © 2000 American Chemical Society Published on Web 02/10/2000
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Figure 1. Table 1. Dimensions of the Foam Metal Beds no.
inner diameter (mm)
1 2 3
45a 150 280
Table 2. Experimental Ranges gas flow rate G liquid flow rate L rotation speed N
0-160 m3/h 0-0.6 m3/h 0-1090 rpm
a Includes a 0.75 cm thick packing at the inner diameter with a specific surface area of 2000 m2/m3.
The difficulty with these prior studies are the following: (1) All terms in these models are positive in the sense that they add to the pressure drop and thus fail to account for a decrease in pressure drop with increasing liquid flow rate. This phenomenon has been confirmed by this work and Liu et al.19 (2) The power measurements of Keyvani6 suggest that the angular momentum of the gas is conserved as the gas is pushed from the outer diameter of the rotor to the exit pipe, and this may have effects on the pressure drop that are not negligible. (3) The pressure drop measurement extends across the entire machine, not just the rotating bed. Machine configurations were not taken into account, and the measured pressure drop data was presumed to result solely from the packed bed. The purpose of this paper is to address these issues. Experiments and Results Experiments were carried out on a prototype machine that is shown schematically in Figure 1 and was loaned by the Glistch Company. The foam metal packing was constructed in the form of a torus and was rectangular in cross section. The packing has a specific surface area of 1000 m2/m3, a void fraction of 0.94, and a pore size on the order of 1-2 mm. Three beds were tested. Their dimensional specifications are given in Table 1. Two plates on each side of the bed prevent transverse flows of gas and liquid through the packing. A perforated cylinder, sandwiched between the two side plates at the outer diameter, contains the bed. The axial thickness is 48 mm. The inner diameter is open. Gas inters into the machine case tangentially to the bed rotating direction. The liquid feed enters at the center and is diverted to two feed pipes with spray holes facing the bed as shown in Figure 1. A variable speed motor drove the rotating packed bed. Gas and liquid flow rates were measured by orifice meters and rotameters. The pressure drop (∆P) across the machine was measured by a water manometer with pressure taps at the gas inlet and outlet. Air and tap
Figure 2. Pressure drop vs the square of the gas flow rate at various rotational speed.
water were used as gas and liquid, respectively. The experimental conditions were varied in the ranges shown in Table 2.The experimental data on packed bed number 2 are typical and will be discussed in the following to illustrate the trends observed in rotating bed. Figures 2 and 3 show the overall pressure drop across the dry machine versus gas flow rate at different rotating speeds. As can be seen from Figure 2: (1) ∆P versus G2 at N ) 0 for a dry bed is an inclined straight line through the origin. It implies that the gas flow can be regarded as totally turbulent with an effective overall drag coefficient of approximately equal to 1.9 × 10-3. (2) The ∆P versus G2 plots for N * 0 are a family of curves with the intercepts on the ordinate increasing with an increase in rotation speed. The curves show a large slope at low gas rates, and then the slope diminishes, forming a curve in transition, then finally going nearly straight and nearly parallel to that of the line for N ) 0. Figure 3 shows the pressure drop versus gas rate at different liquid feed rates with the bed rotating at N ) 1090 rpm. The ∆P for a wet bed can be smaller than that for a dry bed, a phenomenon reported before,6 that for a large range of operating conditions the larger the
Ind. Eng. Chem. Res., Vol. 39, No. 3, 2000 831
Figure 3. Pressure drop vs gas flow rate at various liquid flow rate.
the exit section IV from boundary 4 to the gas outlet pressure tap, boundary 5. The pressure drop in section IV should also include the local pressure drop resulting from entrance enlargement and all other local losses that have not been encountered in the models of sections I-III. The details of gas flow in a RB machine are complicated. To understand the trend of changes of gas velocity and pressure drop though the machine, a sectional model based on mass and momentum conservation is developed in cylindrical coordinates with the following assumptions of simplification: (i) the N-S equation is applicable; (ii) the flow is cylindrically asymmetric in sections I-III; (iii) boundary IV is arbitrarily determined, for the sake of calculation, to be at 0.0225 m in the radial direction from the central axis, with the assumption that before this boundary there is no axial flow (sections I-III) and after it there is only axial flow; (iv) terrestrial gravitational force is neglected; (v) wall effects of the packing’s top and bottom cover plates are neglected; and (vi) forces exerted on the gas by liquid droplets in the annular between bed and case and in the bed are neglected. For section I, the following equations apply: the mass balance equation
∂ (ru ) ) 0 ∂r r
(1)
and the momentum balance equations, first, in the θ direction
(
F ur
) (
)
∂uθ uruθ ∂2uθ 1 uθ )µ + - 2 + ∂r r r r ∂r2
(2)
and then in the r direction,
(
Figure 4. Pressure drop vs gas flow rate. Rotor outer diameter ) 0.3 m, N ) 816 rpm, and L ) 0.
liquid flow rate, the smaller the pressure drop across the bed. This is never observed in conventional packed towers. Figure 4 shows the pressure drop for three dry rotating beds of the same outer diameter but different inner diameters as shown in Table 1. Even more surprising is the observation that the thinner the packed bed, the greater the pressure drop! Finally, it has been previously reported that the shaft horsepower required to rotate the bed goes down as the gas flow is turned on, and the greater the flow, the less the power required.6 It is worthwhile to put these salient features together for analysis: (1) the shaft power goes down when gas is turned on, (2) the pressure drop across the bed may diminish when liquid is turned on, and (3) the pressure drop may diminish with increasing bed thickness, if the outer diameter is fixed. Are these interesting phenomena interrelated? What are their causes? Answers to these questions are given in the modeling section that follows. Modeling and Simulation Referring to Figure 1, the pressure drop across the rotating bed is the sum of the drop across four sections, the shell annular section I from boundary 1 to boundary 2, the rotating bed section II between boundaries 2 and 3, the eye section III between boundaries 3 and 4, and
F ur
)
∂ur uθ2 dp )∂r r dr
(3)
In addition, the boundary conditions are
uθ ) uθ1 ur ) -
G 2πR1H1
uθ ) uθ2
at r ) R1
(4)
at r ) R1
(5)
at r ) R2
(6)
where uθ1 and uθ2 are the gas velocities in the θ direction at the entrance of shell and at the outer periphery of the rotating bed, respectively. H1 is the gap width between upper and lower shell walls of the machine case. Solving eqs 2, 4, 5, and 6 for uθ, we obtain
uθ1 )
( )[
1 R2 1R1
β
( )
where
β) When
( )
uθ2R2 - uθ1R1 R2 β r r uθ2R2 R2 β uθ1R1 + r r r
FG -2 2πH1µ
]
(7)
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Ind. Eng. Chem. Res., Vol. 39, No. 3, 2000
β.1
uθI )
uθ1R1 r
(8)
∆PII ) -
(
G ∫RR ξ2RF h 2πrH 2 2
3
)
2
FG2 4π2H22
dr -
-∆PI )
∫
(
)
1 FG + 8π2H12 R22 - R12
uθ12
∫RR F 1
2
r
dr
(9)
For this section II, the mass balance equation is
∂ (rur) ) 0 ∂r
(10)
where is the free voidage of packing. The momentum balance equations, considering that the flow is fully turbulent, are as follows: in the θ direction,
(
F ur
)
∂uθ uruθ + ) -fθII ∂r r
(11)
and in the r direction,
(
F ur
)
∂ur uθ2 dp ) - - frII ∂r r dr
(12)
In addition, the boundary conditions are
G ur ) 2πR2H2 uθ ) uθ2
at r ) R2at r ) R2+
(14)
(15)
The forces in the θ and r directions exerted on a unit volume of gas by the packing, fθII and frII, are given by
uθ - rω 1 F (uθ - rω)2 -fθII ) ξ Rh 2 |uθ - rω|
(
)( (
G F -∆PII ) ξ 2Rh 2πrH2
) )
1 1 + R3 R2
2
1 FG2 + 2 2 2 3 8π H2 R2 - R22
uθII2 dr (19) r
∫RR F 2
3
For section III, the mass balance equation is
∂ (ru ) ) 0 ∂r r
(20)
The momentum balance equation are developed as follows: Because the projection area of liquid spray leaving the nozzle slot in the r direction is small compared with the inner peripheral free area of rotating packed bed where gas flows out, the retarding force on gas in the r direction exerted by liquid spray, frIII, is considered negligible for simplicity. However, in the θ direction, gas flow will be retarded by liquid spray curtains; fθIII should therefore be considered. In the θ direction,
(
F ur
)
∂uθ uruθ ) -fθIII + ∂r r
(21)
In the r direction,
(
F ur
)
∂ur uθ2 dp )∂r r dr
(22)
The boundary conditions are
G ur ) 2πR3H2
at r ) R3at r ) R3+
uθ ) uθ3
(23) (24)
fθIII can be calculated from, first, for the dry bed,
( )
(16)
2
1 Fur -frII ) ξ Rh 2
R2
When bed is dry or liquid holdup is small, eq 18 can be integrated as
(13)
where H2 is the axial thickness of the packed bed and uθ2 is the peripheral velocity of gas at R2 (assumed to be R2ω). From the mass and momentum balance equations, we can obtain
G ur ) 2πrH2
2
3
uθII2 dr (18) F R3 r
and 2
∫RR 21r3 dr +
fθIII ) -ξ1
∂uθ ∂r
2
and then for the wet bed,
(17)
When uθ < rω, fθII is positive, which means that the gas is forced to flow in the θ direction by the rotating bed. If fθII is negative, i.e., when uθ > rω, the bed is pushed by gas flow, or in other words, the bed retards the gas flow in the θ direction. ξ, Rh, and are the drag coefficient, hydraulic radius, and free voidage of wetted packings, respectively. Because the liquid holdup is small,1,20 the values of ξ, Rh, and can be approximated by that of the dry packing and can be determined by measurement. Equations 11 and 16 can be solved for uθ2; then, ∆PII is given by
( )
fθIII ) -ξ1
∂uθ ∂r
2
- ξ2Luθ2
where ξ1 and ξ2 are parameters that are regressed for each rotor. uθ can be solved numerically from eqs 21 and 24, and thus,
-∆PIII )
(
1 FG2 1 + 8π2H22 R42 R23
∫R
R4
uθIII2 dr (25) r
F
3
For section IV, for the sake of simplicity of solving the model with minimal regressed parameters, ∆PIV is
Ind. Eng. Chem. Res., Vol. 39, No. 3, 2000 833
Figure 6. Simulated pressure drop with various liquid flow rate.
Figure 5. uθ vs radius r.
combined into ∆PIII, and the equation for ∆PIV is ignored. The total pressure drop ∆P is then the sum of ∆PI to ∆PIII.
-∆PIII ) ∆PI + ∆PII + ∆PIII
(26)
ξ1 and ξ2 in ∆PIII are the only regressed parameters of the model. They can be optimized to give good agreement, within 5% of relative error, of total pressure drop across rotating bed machine with experimental results. Curves 1-4 of Figure 5show the calculated uθ distribution along the r direction for bed #2 at various liquid flow rates. Gas enters the machine case (section I) with an initial tangential velocity around 5 m/s that increases gradually due to rotor drag, approaching the rotor’s outer peripheral velocity. As soon as the gas reaches and enters the rotor (section II), uθ jumps suddenly to the rotor’s outer peripheral velocity. As the gas moves radially inward, it maintains a small excess velocity, larger than rω (see line 6 in Figure 5), as it drops down through the rotating packed bed. The packing actually retards the gas as it rotated toward the rotor eye. Thus, the gas is pushing on the bed in the θ direction as it moves radially inward, explaining the reduction in the required shaft horsepower with increasing flow rate. Then, uθ increases immediately after the gas leaves the packed bed and enters the eye (section III). The velocity gradient here is very steep, as uθ goes up to the point where the fictitious boundary point of no axial flow is set. The lines 2-4 show the “curtain effect” on uθ of water injection in the rotor eye space, which actually plays the role of a guide vane, changing the gas flow from the θ to the r direction. The uθ then decreases with the increases of liquid rate, and ∆P in section III also drops accordingly (see lines 2-4 in Figures 5 and 6). This explains the second observation mentioned in the previous section. Also shown in Figure 5, curve 5, is the calculated uθ distribution for bed #3, the thin bed. Now it is also possible to explain the third observation that the pressure drop diminishes with increasing bed thickness. When the packed bed has a larger inner diameter, uθ at the entrance of eye space will be higher than that for a bed with smaller inner diameter (see line 5 in Figure 5), causing a higher pressure drop in eye space and thus higher overall pressure drop across the machine (line 5 in Figures 6 and 7). Figure 7 shows how the pressure drop is distributed in sections I-III at different gas flow rates without any liquid feed. It is seen that the larger the gas rate, the
Figure 7. Simulated pressure drop with various gas flow rate.
larger the pressure drop in different sections. As is shown in Figure 5, uθ is larger than rω in the rotor, which means that the rotor is being pushed as discussed above. Conclusions and Discussion A semiempirical one-dimensional sectional model has been established based on conservation of mass and momentum. The predicted velocity profile and pressure distribution across a rotating bed machine explains the essential features of the intriguing pressure drop behavior of rotating beds. All of these behaviors are related to the gas phase conserving its angular momentum as it flows from the shell through the rotating bed toward the centrally located exit. The explanations of these unusual phenomena may be not complete. This is not only due to the simplicity of modeling but mainly because the pressure drop is not a function of state but of path. Conservation of mass and momentum are necessary but not sufficient requirements for determination of the pressure drop behavior. Unfortunately, the real flow path of the gas in a rotating bed is extremely complicated, and therefore, the pressure drop model is empirically dependent. Liquid injection into the packing will certainly effect the pressure drop of gas flow through packing. One can imagine that the pressure drop will increase drastically during liquid rate approaches to flooding. Liquid flow effects on ∆PII and all local pressure drops have been immersed in the regressed parameters ξ1 and ξ2 in modeling. Whether or not the second and third phenomena are observed
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depends on the configuration of the packing, the rotating speed, and the liquid and gas flow rates. The standard correlation used to predict the pressure drop across packed beds may not be used for rotating beds. The fact that the bed, liquid, and gas phases are rotating causes major deviations from the usual correlation not only in magnitude but also direction. Notation Nomenclature a: specific area of packing (m2/m3) fθII, fθIII: force in the θ direction exerted on unit volume of gas at II and III, respectively (N/m3). frII: force in the θ direction exerted on unit volume of gas at II (N/m3). G: volumetric gas flow rate (m3/h). H2: axial thickness of packing (m). H1: axial width of machine case (m). L: volumetric liquid flow rate (m3/h). r: radius (m). R1: inner radius of machine case (m). R2: outer radius of machine rotor (m). R3: inner radius of packed bed (m). R4: distance from the center of rotor to the fictitious boundary of gas axial flow (m). Rh: hydraulic radius of wetted packing (m). uz: gas velocity in the z direction (m/s). ur: gas velocity in the r direction (m/s). uθ: gas velocity in the θ direction (m/s). uθ1: gas periphery velocity at the entrance of machine case (m/s). uθ2: gas peripheral velocity at the outer periphery of rotor before entering into packed bed (m/s). uθ3: gas peripheral velocity at the inner periphery of packed bed before leaving into eye (m/s). uθI: gas velocity in the θ direction in annular space of rotor and machine case (m/s). uθII: gas velocity in the θ direction in packed bed (m/s). uθIII: gas velocity in the θ direction in eye (m/s). F: gas density. µ: gas viscosity. -∆PI: pressure drop in section I. -∆PII: pressure drop in section II. -∆PIII: pressure drop in section III. : voidage of wetted packing. ξ: drag coefficient of wetted packing. ω: angular speed of rotation. (s-1) ξ1, ξ2: parameters.
(2) Bucklin, R. W.; Won K. W. Higee Contactors For Selective H2S Removal and Superhydration. Laurance Reid Gas Conditioning Conference. University of Oklahoma, March 24, 1987. (3) Fowler, R. A.; Gerdes, K. F.; Nygaard, H. F. CommercialScale Demonstration of Higee for Bulk CO2 Removal and Gas Dehydration. 21th Annual OTC; Houston, Texas, May 1-4, 1989. (4) Fowler, R.; Khan, A. S. VOC Removal with a Rotary Air Stripper. AIChE Annual Meeting; New York, 15-17 Nov, 1987. (5) Kelleher, T.; Fair, J. R. Distillation Studies in a HighGravity Contactor. Ind. Eng. Chem. Res. 1996, 35, (12), 46464655. (6) Keyvani, M.; Gardner, N. Operating Characteristics of Rotating Beds. Chem. Eng. Progr. 1989, 48-52. (7) Kumar, M. P.; Rao, D. P.; Studies on a High-Gravity GasLiquid Contactor. Ind. Eng. Chem. Res. 1990, 29, 917-920. (8) Lockett, M. Flooding of Rotating Structured Packing and Its Application to Conventional Packed Columns. Trans. Inst. Chem. Eng. 1995, 73 (A) 379-384. (9) Martin, C. L.; Martelli, M. Preliminary Distillation Mass Transfer and Pressure Drop Results Using a Pilot Plant Scale Higee Gravity Contacting Unit. AIChE Spring National Meeting; New Orleans, LA., 1992. (10) Mohr, R. J.; Khan, A. S. Higee-A New Approach in Groundwater Clean-up AQTE Conference; Montreal, Canada, 1987. (11) Ramshaw, C. “HIGEE” Distillation - An Example of Process Intensification. Chem. Engr. 1983, 13-14. (12) Ramshaw, C. Process Intensification by Miniature Mass Transfer. Proc. Eng. 1983, 29-31. (13) Ramshaw, C. Opportunities for Exploiting Centrifugal Fields. Chem. Eng. 1987, 17-21. (14) Ramshaw, C. Opportunities for Exploiting Centrifugal Fields. Heat Recovery Sys. CHP 1993, 13 (6), 493-513. (15) Ramshaw, C.; Mallinson, R. H. Mass Transfer Process. U.S. Patent 4283255, Aug 11, 1981 (same as EP0002568, June 27, 1979). (16) Singh, S. P.; Wilson, J. H., et al. Removal of Volatile Organic Compounds from Groundwater Using a Rotary Air Stripper. Ind. Eng. Chem. Res. 1992, 31, 574-580. (17) Zheng, C.; Ai, D.; Guo, K.; Chen, J.; Feng, Y. Recent Progress of High Gravity Technology The First International Workshop of High Gravity Engineering and Technology; Beijing, 1996. (18) Zheng, C.; Guo, K.; Song, Y.; Ai, D.; Xing, W.; Gardner, N. C. Industrial Practice of HIGRAVITEC in Water Deareation. The Second International Conference of Process Intensification; Antwerpen, Belgium, 1997. (19) Liu, H. S.; Lin, C. C.; Wu, S. C.; Shu, H. W. Characteristics of a Rotating Packed Bed. Ind. Eng. Chem. Res. 1996, 35, 35903596. Gou, K. A Study on Liquid Flowing inside the Higee Rotor. Doctoral dissertation, Beijing University of Chemical Technology, 1996.
Literature Cited (1) Basic, A.; Dudukovic, M. P. Hydrodynamics and Mass Transfer in Rotating Packed Bed. Heat and Mass Transfer in Porous Media; Elsevier Science Publishers: 1992.
Received for review November 10, 1998 Accepted December 15, 1999 IE980703D
