Article
Pressure Drop Model on Rotating Zigzag Bed as a New High-Gravity Technology Yumin Li, Jianbing Ji,* Zhichao Xu, Guangquan Wang, Xiaohua Li, and Xuejun Liu Zhejiang Province Key Laboratory of Biofuel, College of Chemical Engineering & Materials Science, Zhejiang University of Technology, Hangzhou, Zhejiang, 310014, China S Supporting Information *
ABSTRACT: The pressure drop across “Higee”, a synonym for high-gravity technology, is one of the most important characteristics, and the liquid flow gives rise to intriguing and unusual effects on the pressure drop. The pressure drop of the wet Higee can be substantially lower than that of the dry one. This trend is quite different from the conventional packed and trayed tower. And the unusual effects also occur in the rotating zigzag bed (RZB), a novel type of Higee. The pressure drop of RZB was studied by two methods of the pressure drop model, based on gas tangential velocity and empirical correlation. By use of an air− water system, experiments were carried out in a RZB with a rotor that had an inner diameter of 214 mm, outer diameter of 486 mm, and axial height of 104 mm. The radial distribution of gas tangential velocity in the RZB rotor was measured by a five-hole pitot probe. Lliquid flow reduced the gas tangential velocity due to the drag exerted by liquid droplets on the gas. The gas tangential velocities were correlated by an empirical equation on which the centrifugal pressure drop was predicted accurately. The correlation of the frictional pressure drop was subsequently obtained. As a result, the pressure drop model based on gas tangential velocity was established, from which the phenomenon that the wet rotor pressure drop is lower than the dry rotor pressure drop was reasonably explained. The rotor and overall pressure drops of RZB were correlated by empirical equations with good agreement. Deviation of the rotor pressure drops based on experiments and on gas tangential velocity was less than that based on experiments and on empirical equations. Variation of the overall pressure drop of RZB was similar to that of the rotor pressure drop. The pressure drop model based on gas tangential velocity can also be applied to the conventional rotating packed bed, which is beneficial to the theory of the pressure drop of Higee.
1. INTRODUCTION By rotating a rigid bed, “Higee”, a synonym for high-gravity technology, adopts higher centrifugal acceleration instead of gravitational acceleration in mass and heat transfer operations to achieve process intensification. The rotating packed bed (RPB), invented by Ramshaw and Mallison,1,2 is one of the earliest and classical Higee. At present, Higee is taking the place of traditional packed and trayed columns in absorption,3,4 stripping,5 and distillation,6−9 due to its attractive and striking advantage of dramatic reduction in equipment volume stemming from greatly intensified mass transfer of gas−liquid, reduced tendency to flooding, very short liquid residence time, and resistance to moderate disturbance, etc.10,11 The theoretical and experimental studies of Higee are more difficult than that of packed and trayed tower, because the hydrodynamics and mass transfer are more complex due to the rotation of the rigid bed, and intrusion of probes or sensors in Higee for measuring experimental parameters is difficult due to the obstruction of rotating internals. So the fundamentals of Higee are still poorly understood. The pressure drop across Higee, as one of the most important characteristics of the Higee performance, has been reported by numerous papers.5−7,11−21 In general, the pressure drop across Higee was predicted by the mathematical model15,17,18 and empirical correlation.7,14 The mathematical model was established by motion differential equations. A Higee setup is divided into three sections: a casing section, a rotor section, and a liquid distributor section. So the pressure © 2013 American Chemical Society
drop across Higee is the sum of the three zonal pressure drops. The zonal pressure drop of the rotor section is made up of momentum gain, centrifugal, and frictional pressure drops as the solutions to the motion equations. The empirical correlation of pressure drop across Higee was established by resistance coefficient. The resistance coefficient, as a function of gas, rotational, and liquid Reynolds numbers, was correlated by experimental data. In addition, the liquid flow gives rise to intriguing and unusual effects on the pressure drop across Higee.11,12,14,15,19 As the liquid enters the Higee, the pressure drop reduces suddenly and then reduces slightly with increasing liquid rates, which is impossible to occur in conventional packed and trayed column. Some researchers14,19 suggested that the vacuum formed by liquid drops thrown by centrifugal force facilitates the gas flow inward, and as a result, the pressure drop reduces. However, the explanation lacks experimental verification. Zheng et al.15 thought that the liquid jets in the eye of the rotor reduce the gas tangential velocity in the eye, leading to the reduction in pressure drop. This paper proposes that the liquid flow reduces the gas tangential velocity in the rotor and then the pressure drop, so the detailed studies are carried out. Received: Revised: Accepted: Published: 4638
May 9, 2012 January 28, 2013 February 19, 2013 February 19, 2013 dx.doi.org/10.1021/ie301207e | Ind. Eng. Chem. Res. 2013, 52, 4638−4649
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Figure 1. Single-rotor RZB: (a) photograph of stationary disc and baffles; (b) photograph of rotating disc and rotational baffles; (c) sketch of singlerotor RZB. (1) Rotating disc; (2) rotational baffles; (3) gas inlet; (4) stationary disc; (5) stationary baffles; (6) perforations on rotational baffles; (7) gas outlet; (8) liquid inlet; (9) casing; (10) liquid outlet; (11) hatlike liquid distributor; (12) mechanical seal; (13) bearing.
2. ROTATING ZIGZAG BED The rotating zigzag bed (RZB), patented by Ji et al.,22−24 is first to really achieve noncolumn distillation with only one RZB unit as well as a condenser, reboiler, etc.,9 because of the unique and excellent structure of RZB rotor.8,19 The RZB rotor coaxially combines a rotating disc and a stationary disc, to which two sets of alternative concentric circular metal sheets are attached. One set is called rotational baffles with upper parts perforated, and the other set is called stationary baffles. The rotating disc with its rotational baffles is driven by a motor through a vertical shaft. The stationary disc with its stationary baffles is fixed to the casing. The upper clearance between rotational baffles and stationary disc, as well as the lower clearance between stationary baffles and rotating disc, offers the zigzag flow channel for gas and liquid phase. A hatlike liquid distributor is located at the center of the rotor, offering a flow channel for gas and liquid phase at the eye of the rotor, as shown in Figure.1. The complex flow of gas and liquid occurs in the RZB rotor as shown in Figure 2. Entering into the casing tangentially to the rotor rotating direction, the gas flows radially inward in a staggered way alternately through the perforations and the upper clearance of the rotational baffles and the lower clearance of the stationary baffles with the upward and downward flow in the axial direction, flowing circumferentially in the annular space between the rotational and stationary baffles. So the gas flow is a three-dimensional flow with radial, axial, and tangential components of velocity. Leaving the rotor, the gas flows over the hatlike liquid distributor and is discharged through the gas outlet with the swirl flow. On entering the liquid inlet at the eye of the casing and the liquid distributor at the eye of the rotor, the liquid climbs upward along rotational baffles as a thin film and is dispersed and tangentially thrown into fine liquid droplets with high rotational speed because of the perforations on rotational
Figure 2. Gas and liquid flow in RZB: (1) liquid flowing outward inside the liquid distributor; (2) liquid climbing upward along rotational baffles as a thin film; (3) liquid fine droplets thrown by rotational baffles; (4) liquid film traveling circumferentially and falling down along stationary baffles; (5) liquid leaving stationary baffles downward and tangentially; (6) gas flowing through the lower clearance of stationary baffles; (7) gas flowing upward in the axial direction; (8) gas flowing through the perforations of rotational baffles; (9) gas flowing through the upper clearance of rotational baffles; (10) gas flowing downward in the axial direction.
baffles. Impacting on the stationary baffles, the droplets agglomerate to form liquid film traveling circumferentially and falling downward along stationary baffles by the action of gravity. The liquid leaves the stationary baffle downward and tangentially and reaches the next rotational baffles, climbing upward and being dispersed again. So the liquid experiences repeated dispersion and agglomeration. In addition, installing multiple rotors coaxially in series in one casing can achieve a multirotor RZB for a significant increase in the theoretical plate number.8,9,19 However, the flow of gas and liquid in multirotor RZB becomes more complex. 4639
dx.doi.org/10.1021/ie301207e | Ind. Eng. Chem. Res. 2013, 52, 4638−4649
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where Lp is the perimeter of the rotational baffle channel in contact with fluid and n0 the is total number of small holes in the perforated zone of the rotational baffle, which is calculated by
Because of the unique structure of RZB rotor and complex flow of gas and liquid, it is necessary to make a detailed investigation of the hydrodynamic behavior of RZB. We will attempt to investigate the pressure drop of RZB. Exploration of the pressure drop of RZB will enrich the hydrodynamic theory of Higee.
n0 =
3. EXPERIMENT AND MODEL Experiments were carried out in a single-rotor RZB with a transparent plexiglass casing for an air−water system. The rotor of the RZB, which had an inner diameter of 214 mm, outer diameter of 486 mm. and height of 104 mm, was composed of four coupled rotational−stationary baffles. Small holes with diameters of 1 mm and with 2 mm distance between holes were found at the upper parts of the rotational baffles. The gas inlet and outlet had diameters of 240 and 152 mm, respectively. The dimensional diagram of the RZB is shown in Figure 3. The gas
2πrR h1′φ0 (π /4)d0 2
(3)
Substituting eq 3 into eq 2 gives dh =
2(h1c + h1′φ0) 1 + (2h1′φ0 /d0)
(4)
Because the cross-sectional area of 2πrH changes with radius r, the average of superficial gas velocity uGr,avg is employed and defined as14 ro G 1 dr ro − ri ri 2πrH r G = ln o 2πH(ro − ri) ri G = 2πHrm
∫
uGr,avg =
(5)
where G is the gas rate and rm is the logarithmic average radius of the RZB rotor, written as r − ri rm = o ln(ro/ri) (6) Similarly, the average of superficial liquid velocity uLr,avg is defined as
Figure 3. Schematic dimensional diagram of RZB: (1) perforated zone of rotational baffle; (2) nonperforated zone of rotational baffle; (3) gas inlet located below the rotating disc of the rotor.
uLr,avg =
2πrR (h1c + h1′φ0) h1c + h1′φ0 S = = 2πrR H 2πrR H H
(1)
where S is the area of the rotational baffle channel through which fluid passes in a radial direction, rR is the radius of the rotational baffle, H is the axial height of the RZB rotor, and φ0 is the fractional hole area of the perforated zone of the rotational baffle. Hydraulic diameter of rotational baffle dh based on the crosssectional area 2πrRH is defined as dh = 4
2πrR (h1c + h1′φ0) S =4 Lp 2·2πrR + n0πd0
∫r
i
ro
L L dr = 2πrH 2πHrm
(7)
where L is the liquid rate. 3.1. Measurement of Pressure Drop, Local Pressure, and Gas Tangential Velocity. Experiments were carried out as follows: (i) The rotor pressure drop ΔPR across the rotor, as a main component of the overall pressure drop ΔPT, was measured by a U-tube water manometer connected to the pressure taps at location 1, close to the inner edge of the rotor, and location 6, close to the outer edge of the rotor, as shown in Figure 4a. The pressure tap at location 6 was a 12-mm diameter static pressure tube to measure the static pressure at location 6. Six small holes with diameters of 1.5 mm, which were distributed uniformly along the circumference of the tube, were found at a distance of 40 mm from the hemisphere head of the static tube, in order to reduce the effect of deflection between the directions of tube axis and gas flow on the measurement of static pressure. According to the specifications of the static tube, the deviation of static pressure was less than 1% with deflection of 10°. The pressure tap at location 1 was a circular tube with an inner diameter of 8 mm. The tube axis was perpendicular to the upper surface of liquid distributor with the downward tube mouth. When the gas flowed through the lower clearance of the stationary baffle, the flowing direction of the gas was parallel to the upper surface of liquid distributor at location 1. As a result, the static pressure at location 1 was measured by the circular tube.
inlet was located below the rotating disk of the rotor to improve circumferential distribution of gas tangential flow in RZB rotor. The rotational baffle was divided into two zones. One zone was the upper part of the rotational baffle as a perforated zone; and the other was the lower part of the rotational baffle as a nonperforated zone. The centrifugal blower blew air tangentially into the RZB casing. The centrifugal pump was used to pump water into the RZB rotor from the liquid inlet located at the center of RZB. The liquid flow rates were measured by a rotameter, and gas flow rates were measured by an orifice meter. The rotor was driven by a motor using a pulley. The cross-sectional area through which gas flows radially is 2πrH. Fractional opening area of rotational baffle φ based on the cross-sectional area 2πrRH is defined as φ=
1 ro − ri
(2) 4640
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Figure 4. Schematic diagram of experimental apparatus: (a) measurement of rotor pressure drop and gas tangential velocity; (b) measurement of overall pressure drop.
five-hole pitot probe, the flying liquid droplets might block the holes, which obviously reduced the pressure of the pressure line tubes connected to the holes and then led to the deviation of the measuring values of gas tangential velocities. If the unexpected situation occurred, a syringe was used to solve the problem. With the piston, the air in the syringe was forced into the pressure line tube through a slim hose, which then discharged the liquid in the holes, as shown in Figure 4a, eliminating the measuring deviation. The syringe was always used to eliminate the effect of the liquid droplets before pressure data were taken from the micromanometers. The gas rate G ranged from 0.057 to 0.234 m3/s and the liquid rate L ranged from 0 to 0.000 444 m3/s with three rotational speeds of 670, 850, and 1033 rpm (iii) The overall pressure drop ΔPT across the RZB unit was measured at different liquid rates, gas rates, and
(ii) The gas tangential velocities at locations 1−5 were measured by a 5-mm diameter spherical five-hole pitot probe. The five holes with diameters of 0.5 mm as pressure taps (denoted a−e) were drilled on the surface of the spherical probe, with pressure tap a placed centrally and other taps b−e placed circumferentially. Pressure tap a faced the direction of tangential component of gas flow. The probe was fitted inside a stainless steel tube to provide the necessary protecting and structural integrity. The pressure line tubes connected to the five holes on the probe surface were fed through the stainless steel tube to five micromanometers with an accuracy of 0.01 mm of water column. The five-hole pitot was close to the inner wall of stationary baffles. Introduction of the liquid to the RZB might influence the measurement results of the five-hole pitot probe. Reaching and staying at the surface of the 4641
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Figure 5. Radial distribution of gas tangential velocity in the absence of liquid: (a) at various rotational speeds and (b) at various gas rates.
The radial friction factor f r in eq 9c is given by
rotational speeds by a U-tube water manometer connected to pressure taps at the gas inlet and gas outlet, as shown in Figure 4b. The gas rate G ranged from 0 to 0.234 m3/s and the liquid rate L ranged from 0 to 0.000 333 m3/s with three rotational speeds of 670, 850, and 1033 rpm. 3.2. Model of Rotor Pressure Drop. From the work of Chandra et al,18 we obtain ρ vr 2 dp v2 v2 − = −ρG r − ρG θ − fr G dr r r 2dh
fr =
ro
∫r
i
vr 2 dr + ρG r
∫r
i
ro
vθ 2 dr + r
Re f =
i
ro
fr
ρG vr
ΔPf =
∫r
i
ΔPc = ρG
∫r
ro
i
ΔPf =
∫r
ro
dr
=
vr dr r
(9a)
vθ 2 dr r
(9b)
2dh
i
dr
ΔPm = ρG
∫r
i
2 ρG ⎛ G ⎞ ⎛ 1 vr 2 1 ⎞ dr = ⎜ ⎟ ⎜ 2 − 2⎟ r 2 ⎝ 2πHφ ⎠ ⎝ ri ro ⎠
ro
fr
ρG vr 2 2dh
dr
⎛1 ρG ⎛ G ⎞2 ⎡ 2πHμG ro 1 ⎞⎤ ⎜ ⎟ ⎢α β ln + − ⎜ ⎟⎥ ri ro ⎠⎥⎦ 2φ 2dh ⎝ 2πH ⎠ ⎢⎣ Gdh ρG ⎝ ri (13)
The gas tangential velocity vθ is a function of radial distance r, rotor angular velocity ω, gas rate G, and liquid rate L. If vθ is measured and its correlation is obtained, the centrifugal pressure drop ΔPc, as a function of ω, G, and L, will be predicted accurately by eq 9b. So the frictional pressure drop ΔPf is obtained from the measured ΔPR by experiment minus the predicted ΔPc by eq 9b and the calculated ΔPm by eq 10, and then the coefficients α and β in eq 13 are obtained. Consequently, ΔPm, ΔPc, and ΔPf can all be predicted by their equations, and their sum ΔPR can be obtained. This just is the model of rotor pressure drop based on the gas tangential velocity. And the pressure drop model is also applied to the conventional rotating packed bed (RPB), if the gas tangential velocity vθ in PRB can be obtained. 3.3. Empirical Correlation of Pressure Drop. To empirically correlate the pressure drop of RZB, the empirical correlation of pressure drop from Liu et al.14 is modified as follows:
(9c)
where ΔPR is the rotor pressure drop, ΔPm is the pressure drop due to momentum gain by the gas that flows toward the center of the rotor, ΔPc is the centrifugal pressure drop due to centrifugal force, and ΔPf is the frictional pressure drop due to friction generated by the rotational and stationary baffles. Substituting vr = G/(2πrHϕ) into eq 9a and integrating from ri to ro gives ro
(12)
(9)
2
ρG vr 2
fr
∫r
i
that is, ΔPm = ρG
μG
Substituting eqs 11 and 12 into eq 9c and integrating from ri to ro gives
2
2dh
= ΔPm + ΔPc + ΔPf ro
vrdh ρG
(8)
∫r
(11)
where α and β are coefficients depending on liquid rate L and rotational speed n, and Ref is the Reynolds number of the friction factor, which is defined as
where dp/dr is the radial pressure gradient in RZB rotor, vr = G/(2πrHϕ) is the gas radial velocity, vθ is the gas tangential velocity, and f r is the radial friction factor. Integration of eq 8 gives ΔPR = ρG
α +β Re f
2 1 ⎛ uGr,avg ⎞ fd (ro − ri) ΔPd = ρG ⎜ ⎟ 2 ⎝ φ ⎠ dh
(10) 4642
(14)
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Figure 6. Radial distribution of gas angular momentum in the absence of liquid: (a) at various rotational speeds and (b) at various gas rates.
where ΔPd is the pressure drop of dry RZB (in the presence of gas flow only) and fd is the resistance coefficient as follows: fd = a11ReGb11 + a12 RebG12Rebω13
When torque T pushes on the gas, the angular momentum mGvθr increases. Conversely, when torque T retards the gas, the angular momentum mGvθr decreases. From the angular momentum mGvθr obtained at locations 1− 5, the radial distribution of mGvθr at various rotational speeds and gas rates is shown in Figure 6. Gas enters the RZB casing tangentially, with its vθ lower than the rotor outer edge velocity, so the rotor exerts torque T to push on the gas, giving rise to the fact that the gas angular momentum mGvθr increases and vθ increases rapidly with the reduction in radius r. When the gas enters the rotor and flows radially inward, the gas tangential velocity vθ increases until it equals or exceeds the rotor velocity ωr. At that time, the rotor exerts no torque T on the gas, so the gas angular momentum mGvθr is invariant, but vθ increases continually due to the reduction in radius r. As soon as the gas tangential velocity vθ exceeds the rotor velocity ωr rather greatly, the rotor exerts torque T to retard the gas, causing the gas angular momentum mGvθr to decrease. As a result, the tangential velocity vθ begins to decrease from the maximum value with decreasing radius r. It is noticed that the deviation of vθ and rotor velocity ωr, namely, the relative tangential slip velocity vθ − ωr, is nearly constant in the inner portion of the rotor. The relative tangential slip velocity vθ − ωr increases with increasing gas rate, independent of rotational speed, as shown in Figure 5. The gas tangential velocity vθ increases with increasing rotational speed, as shown in Figure 5a. With increasing rotational speed, the rotor exerts greater torque T to push on the gas, leading to the increase in gas tangential velocity vθ. However, the gas tangential velocity vθ at location 5 in RZB casing, equaling the gas velocity in gas inlet, is independent of rotational speed, because the rotor does not have enough time to exert torque T on the gas entering the casing from the gas inlet. The velocity vθ at location 5 depends only on the gas rate. The gas tangential velocity vθ increases first and then increases slightly with increasing gas rate, as shown in Figure 5b. From eq 18, we obtain
(15)
where a11, a12, b11, b12, and b13 are the coefficients of eq 15; ReG denotes the gas Reynolds number [= (dhuGr,avgρG)/μG]; and Reω denotes the rotational Reynolds number [= (ωrm2ρG)/μG]. For wet RZB, a small part of the perforations of rotational baffles through which the gas passes is occupied by the liquid, leading to the slight reduction in fractional opening area φ in eq 14. For simplification of the correlation of pressure drop, φ at wet RZB is assumed to be equal to that at dry RZB. Thus, the pressure drop is 2 1 ⎛ uGr,avg ⎞ fw ΔPw = ρG ⎜ (ro − ri) ⎟ 2 ⎝ φ ⎠ dh
(16)
where ΔPw is the pressure drop of wet RZB and f w is the wetted resistance coefficient as follows: fw = a 21RebG21RebL22Rebω23 + a 22 RebL24
(17)
where a21, a22, b21, b22, b23, and b24 are the coefficients of eq 17 and ReL denotes the liquid Reynolds number [= (dhuLr,avgρL)/ μL]. The rotor pressure drop ΔPR and the overall pressure drop ΔPT are correlated empirically. So ΔPd, ΔPw, fd, and f w in eqs 14−17 are changed into ΔPRd, ΔPRw, f Rd, and f Rw for the rotor pressure drop ΔPR and are changed into ΔPTd, ΔPTw, f Td, and f Tw for the overall pressure drop ΔPT.
4. RESULTS AND DISCUSSION 4.1. Gas Tangential Velocity. Based on the measurement of gas tangential velocity vθ at locations 1−5, the radial distribution of vθ and rotor velocity ωr at various rotational speeds and gas rates is shown in Figure 5. The gas tangential velocity vθ increases, exceeding the rotor velocity ωr, to maximum value and then decreases as the radius r reduces, which can be explained by the angular-momentum equation. A gas stream with mass rate mG, on which torque T is exerted, flows circumferentially from radial position r1 with its tangential velocity vθ1 to radial position r2 with its tangential velocity vθ2, so the angular-momentum equation is written as mGvθ 2r2 − mGvθ1r1 = T (18)
vθ 2 =
v r T + θ1 1 mGr2 r2
(19)
According to eq 19, increasing vθ1 leads to the increase in vθ2, when the gas stream flows circumferentially from radial position r1 with its tangential velocity vθ1 to radial position r2 with its tangential velocity vθ2. Similarly, in RZB, the increasing gas tangential velocity vθ at location 5 with increasing gas rate leads to the increase in gas tangential velocity vθ at locations 1−4.
where mGvθ1r1 is gas angular momentum at radial position r1 and mGvθ2r2 is gas angular momentum at radial position r2. 4643
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Figure 7. Effect of liquid rate on gas tangential velocity: (a−c) at various gas rates and (d−f) at various rotational speeds.
FD = C D A pρG (vθ − vθ L)2 /2
However, as the gas rate goes on increasing, the torque T to push on the gas decreases and the torque T to retard the gas increases, leading to the slight increase in vθ. When the liquid is introduced to RZB, the gas tangential velocity vθ drops suddenly and reduces slightly with increasing liquid rate, as shown in Figure 7. The reason is that the gas tangential velocity vθ is greater than the rotor velocity ωr, that is, greater than rotational baffle velocity vθR (= ωrR). The liquid droplets are thrown from the rotational baffles with their tangential velocity vθL (= vθR = ωrR < vθ). When the gas flows past the liquid droplets, the liquid droplets exert drag on the gas, as shown in Figure 8, giving rise to the reduction in gas tangential velocity vθ. The drag FD is written as
(20)
where Ap is the projected area of liquid droplets in the direction of gas flow. When the liquid rate continually increases, the amount and size of liquid droplets increase. But Ap increases slightly because the gas flow has the same direction as the movement of liquid droplets. As a result, the gas tangential velocity vθ reduces slightly. The radial distribution of vθ with liquid flow is similar to that with no liquid flow. The gas tangential velocity vθ is greater than rotor velocity ωr in the inner portion of the rotor with the fact that the relative tangential slip velocity vθ − ωr still approximately stays constant. The relative velocity vθ − ωr is nearly constant in the inner portion of the rotor, that is, when r < rs, in the absence of 4644
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vθ − aG ωr[b − c1(r /ri)c2 (ReG)c3 (Reω)c4 ] = 1 − d1(ReG)d2 (ReL)d3 (Reω)d4
(22c)
Based on the experimental data, the correlation for gas tangential velocity vθ is written as vθ − 37.11G ωr[1.300 − 0.04512(r / ri)4.7415 Re−G0.1471Re−ω0.01200] = 1 − (2.266 × 10−5)Re−G0.3255Re0.2829 Re0.8321 ω L
Figure 8. Gas flow in the space between rotational and stationary baffles: (1) stationary baffle; (2) liquid droplets thrown by rotational baffles; (3) drag exerted by liquid droplets on the gas stream; (4) gas stream traveling around the rotational baffle; (5) gas flowing obliquely through the perforations of rotational baffle at an angle γ; (6) rotational baffle.
(G > 0) vθ = ωr
(r < rs)
(G = 0 and L = 0)
(21a)
ΔPc = ρG
that is,
∫r
vθ 2 dr = 1129.585ρG G2 + ρG ω 2 r
ro
i
vθ − aG −1=0 ωr
(23b)
The average deviation of experimental and calculated vθ is about 5.39%, with the greatest deviation of 17.21%. When G = 0 and L = 0, eq 23a becomes vθ = 1.3ωr, not being simplified to eq 23b because of the combination of eqs 21b and 21c. The correlation of vθ, that is, eq 23a,23b, is excessively complex, so it is expected that a reduced correlation of vθ will be developed. 4.2. Rotor Pressure Drop. Substituting eqs 23a and 23b into eq9b and integrating from ri to ro gives
liquid, where rs is the radius of the shift in relative velocity vθ − ωr in the RZB rotor, depending on the gas rate G and independent of the rotational speed n, as shown in Figures 5 and 7. So it is suggested that vθ − ωr is directly proportional to the gas rate G, which can be expressed as follows: vθ − ωr = aG
(23a)
A2 (0.04022 − 0.05001Re−G0.1471Re−ω0.01200
(r < rs)
+ 0.02500Re−G0.2942Re−ω0.02400) + 74.22ρG Gω
(21b)
When r > rs, the variation of vθ with radius r depends on the gas rate G and the angular velocity ω, so we obtain
A(0.1768 − 0.09248Re−G0.1471Re−ω0.012)
(24a)
⎛ r − rs ⎞c2 ′ vθ − aG − 1 = −c1′⎜ ⎟ (ReG)c3′(Reω)c4 ′ ωr ⎝ ri ⎠ (r > rs)
where A = 1 − (2.266 × 10−5)Re−0.3255 Re0.2829 Re0.8321 . G L ω ΔPc = ρG
(21c)
∫r
vθ 2 dr = ρG r
ro
i
2
2
2
ω (ro − ri )
Combining eqs 21b and 21c gives vθ − aG − 1 = b′ − c1(r /ri)c2 (ReG)c3 (Reω)c4 ωr
∫r
ro
i
(ωr )2 1 dr = ρG r 2
(G = 0 and L = 0)
(24b)
From eqs1−7 and the RZB dimensions, we obtain that ϕ = 39.47% and dh = 0.00489m. Therefore, eqs 10 and 13 become
(22a)
that is,
ΔPm =
vθ − aG = b − c1(r /ri)c2 (ReG)c3 (Reω)c4 ωr
(G > 0)
(22b)
where b = b′ + 1. The introduction of liquid leads to downward translation of the radial distribution curve of vθ. The translational distance depends on G, L, and ω. So
ΔPf =
2 ρG ⎛ G ⎞ ⎛ 1 1 ⎞ 2 ⎜ ⎟ ⎜ 2 − 2 ⎟ = 635.708G 2 ⎝ 2πHφ ⎠ ⎝ ri ro ⎠
(25)
⎛1 ρG ⎛ G ⎞2 ⎡ 2πHμG ro 1 ⎞⎤ ⎜ ⎟ ⎢α β + − ln ⎜ ⎟⎥ ri ro ⎠⎥⎦ 2φ 2dh ⎝ 2πH ⎠ ⎢⎣ Gdh ρG ⎝ ri
= 3.051αG + 9657.640βG2
(26)
Table 1. Values of α and β at Various Liquid Rates L and Rotational Speeds n n = 670 rpm
n = 850 rpm
n = 1033 rpm
L (m3/s)
α
β
α
β
α
β
0 0.000 111 1 0.000 166 7 0.000 222 2 0.000 277 8 0.000 333 3 0.000 444 4
1000.7 368.65 606.55 656.52 658.76 721.4 846.06
2.006 3.21 2.747 2.658 2.686 2.584 2.341
1384.5 956.84 936.83 1006.9 1102 1153.9 1221.3
1.5067 1.9513 2.116 1.9924 1.8266 1.7831 1.7855
831.27 673.57 691.98 867.26 927.45 939.83 1080
2.641 2.568 2.569 2.204 2.101 2.121 1.934
4645
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As the liquid rate L increases, ΔPc first drops and then reduces slightly because the gas tangential velocity vθ has a similar trend. The main contribution to ΔPR stems from ΔPf. The slope of the curves of ΔPf is sensitive to gas rate G and turns steeper at higher G in Figure 9b, which shows the contribution of ΔPf exceeding the contribution of ΔPc by far. ΔPf is nearly invariable between rotational speeds of 670 and 1033 rpm in Figure 9c. And it is surprising that ΔPf reduced slightly and then increases slightly with increasing liquid rate L in Figure 9a. The reason may arise from the gas tangential velocity vθ. Traveling circumferentially around rotational baffles, the gas obliquely flows through the perforations on the wall of rotational baffles at an angle γ with the tangent to the rotational baffles because the gas tangential velocity vθ is greater than rotational baffle velocity vθR, as shown in Figure 8. As the liquid enters the rotor, the angle γ increases with increasing liquid rate L due to the reduction in vθ, thereby lowering dissipation and ΔPf. However, the larger the liquid rate L is, the greater the friction between the gas flow and liquid droplets, thus raising ΔPf. The interaction of the two effects determines the trend of ΔPf. As a result, the rotor pressure drop ΔPR, as the sum of ΔPm, ΔPc, and ΔPf, reduces suddenly up to a certain value when the liquid rate increases from 0 and then increases slightly with increasing liquid rate, as shown in Figure 9a. The fact clearly shows that wet ΔPR (in the presence of liquid) is lower than dry ΔPR (in the absence of liquid). This trend is quite different from the conventional packed and trayed tower, where the wet pressure drop is certainly greater than the dry pressure drop. The unusual phenomenon of wet pressure drop of RZB is explained reasonably by the variation of the gas tangential velocity vθ in RZB rotor. 4.3. Radial Pressure Gradient in RZB Rotor. In the absence of liquid (L = 0 m3/s), substitution of eqs 23a, 11, and 12 into eq 8 gives
Frictional pressure drop ΔPf is obtained as experimental rotor pressure drop ΔPR minus centrifugal pressure drop ΔPc from eq 24a and momentum-gain pressure drop ΔPm from eq 25. By regression analysis of eq 26, coefficients α and β were obtained as shown in Table 1. The calculated ΔPR is the sum of ΔPm from eq 25, ΔPc from eq 24a, and ΔPf from eq 26. The average deviation of experimental and calculated ΔPR is about 2.42%, with the greatest deviation of 7.85%. The rotor pressure drop ΔPR, with its three compoments ΔPm, ΔPc, and ΔPf, is shown in Figure 9 as a function of liquid rate L, gas rate G, and rotational speed n. ΔPm depends only on the gas rate G, and its contribution to ΔPR is so small that it is ignored. The secondary contribution to ΔPR results from ΔPc. ΔPc increases with increasing gas rate G and rotational speed n.
that is,
The radial pressure gradient dp/dr increases with increasing G and ω, according to eq 28. Variation of the radial distribution of dp/dr with gas rate G is shown in Figure 10a. The fact that dp/dr monotonically decreases with increasing radius r indicates the pressure in RZB rotor increases rapidly and then slightly with increasing radius r from ri to ro, as shown in Figure 10b, which shows the rotor pressure drop is concentrated in the inner portion of the rotor. 4.4. Empirical Correlation of Rotor Pressure Drop. From eqs 1−7 and the RZB dimensions, we obtain ϕ = 39.47%, dh = 0.00489m, rm = 0.1658m, uGr,avg = 9.235G, and uLr,avg = 9.235L. With the experimental results and regression analysis, the correlations of dry and wet rotor pressure drop were obtained as follows:
Figure 9. Rotor pressure drop ΔPR and the contributions of its components, ΔPm, ΔPc, and ΔPf, and their variations with (a) liquid rate, (b) gas rate, and (c) rotational speed. 4646
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Figure 11. Comparison of rotor pressure drop based on experiment, gas tangential velocity, and empirical correlation.
(rotor pressure drop), and in the casingand the rotor pressure drop is a major contributor to the overall pressure drop. Variation of the overall pressure drop with gas rate, liquid rate, and rotational speed is similar to that of the rotor pressure drop, as shown in Figure 12. The overall pressure drop increases with increasing gas rate and rotational speed. The wet overall pressure drop is lower than the dry overall pressure drop. The reason for this phenomenon finally is ascribed to the fact that gas tangential velocity with no liquid flow is greater than that with liquid flow. With the experimental results and regression analysis, the correlations of dry and wet overall pressure drop were obtained as follows:
Figure 10. Radial pressure gradient and pressure in RZB rotor: (a) distribution of radial pressure gradient in RZB rotor and (b) radial distribution of pressure in RZB rotor.
Dry rotor pressure drop of RZB ΔPRd =
2 1 ⎛ uGr,avg ⎞ fRd ρG ⎜ (ro − ri) ⎟ 2 ⎝ φ ⎠ dh
Dry overall pressure drop of RZB
(29)
where
ΔPTd =
fRd = (1.023 × 109)Re−G3.711 + 0.007155Re−G0.801Re 0.975 ω
fTd = (1.013 × 107)Re−G4.004 + 0.001942Re−G0.903Re1.151 ω
Wet rotor pressure drop of RZB ΔPRw
(31)
where
(29a) 2 1 ⎛ uGr,avg ⎞ fRw = ρG ⎜ (ro − ri) ⎟ 2 ⎝ φ ⎠ dh
2 1 ⎛ uGr,avg ⎞ fTd (ro − ri) ρG ⎜ ⎟ 2 ⎝ φ ⎠ dh
(31a)
Wet overall pressure drop of RZB
(30)
where
ΔPTw =
fRw = 0.1180Re−G1.700Re 0.0426 Re1.101 + 3.937Re−L 0.0305 L ω
2 1 ⎛ uGr,avg ⎞ fTw ρG ⎜ (ro − ri) ⎟ 2 ⎝ φ ⎠ dh
(32)
where
(30a)
The average deviation of experimental and calculated ΔPRd and ΔPRw is about 4.61% and 4.35%, with the greatest deviations of 12.87% and 11.03%, respectively. Comparison of rotor pressure drops based on experiment, gas tangential velocity, and empirical correlation is shown in Figure 11. The deviation of the rotor pressure drops based on experiment and on gas tangential velocity is obviously less than that based on experiment and on empirical equation, which indicates that prediction of rotor pressure drops based on gas tangential velocity is successful. 4.5. Empirical Correlation of Overall Pressure Drop. The overall pressure drop of RZB is partitioned into three partspressure drop at the gas outlet, across the RZB rotor
fTw = 0.1159Re−G1.308Re 0.0384 Re 0.936 + 3.819Re−L 0.0331 L ω (32a)
The average deviation of experimental and calculated ΔPTd and ΔPTw is about 4.02% and 3.20%, with the greatest deviations of 13.38% and 11.32%, respectively. From eqs 29−32, the values of ΔPRd/ΔPTd are above 82.6%, and the values of ΔPRw/ΔPTw are above 86.1%.
5. CONCLUSION (1) The gas tangential velocity in RZB rotor increases up to maximum value and then decreases with decreasing radial distance, and it is greater than RZB rotor velocity 4647
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(3) The rotor pressure drop is correlated by an empirical equation with good agreement. Deviation of the rotor pressure drops based on experiments and on gas tangential velocity is obviously less than the deviation based on experiments and on empirical equation. (4) The overall pressure drop of RZB is composed of rotor, casing, and gas outlet pressure drops. The unusual phenomenon that its dry pressure drop is greater than its wet rotor pressure drop is attributed to the fact that the gas tangential velocity with no liquid flow is greater than that with liquid flow. The overall pressure drop of RZB is correlated by an empirical equation with good agreement. (5) The radial pressure gradient of RZB rotor reduces with increasing radius, which shows the rotor pressure drop is concentrated in the inner portion of the rotor.
■
ASSOCIATED CONTENT
* Supporting Information S
Two tables, listing RZB dimensions and locations 1−5 where gas tangential velocities were measured. This material is available free of charge via the Internet at http://pubs.acs.org.
■
AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. Notes
The authors declare no competing financial interest.
■ Figure 12. Variation of overall pressure drop of RZB with (a) gas rate and (b) liquid rate.
in the main portion of the rotor, because the gas flow has to observe the angular-momentum equation. The gas tangential velocity increases with increasing gas rate and rotational speed and reduces suddenly as the liquid enters RZB and then reduces slightly with increasing liquid rate, because the liquid droplets exert drag on the gas because the tangential velocity of gas is greater than that of liquid droplets. An empirical equation was proposed to correlate the gas tangential velocity with good agreement. (2) Based on the empirical equation of gas tangential velocity, the centrifugal pressure drop ΔPc is predicted accurately. Correlation of the frictional pressure drop ΔPf is subsequently obtained. The momentum-gain pressure drop ΔPm is very small and is ignored. So the pressure drop model is established, and it is found that introduction of the liquid leads to sudden reduction in ΔPc and slight reduction in ΔPf. Thus the rotor pressure drop ΔPR, as the sum of ΔPc, ΔPf, and ΔPm, has the intriguing and unusual phenomenon that the dry rotor pressure drop is greater than the wet rotor pressure drop. The pressure drop model based on gas tangential velocity is also applied to the conventional rotating packed bed. 4648
NOMENCLATURE a, b, b′, c1, c2, c3, c4, d1, d2, d3, d4 = coefficients of eqs 22a, 22b, and 22c a11, a12, b11, b12, and b13 = coefficients of eq 15 a21, a22, b21, b22, b23, and b24 = coefficients of eq 17 A = constant of eq 24a Ap = projected area of liquid droplets, m2 CD = drag coefficient c1′, c2′, c3′, c4′ = coefficients of eq 21c d0 = hole diameter of perforated zone of rotational baffle, m dh = hydraulic diameter of rotational baffle based on crosssectional area 2πrRH, m fd = dry resistance coefficient f Rd, f Td = dry resistance coefficients for rotor and overall pressure drop f r = radial friction factor f w = wet resistance coefficient f Rw, f Tw = wet resistance coefficients for rotor and overall pressure drop FD = drag exerted by liquid droplets on gas, N G = gas rate, m3/s h0 = height of innermost stationary baffle, m h0c = clearance between innermost stationary baffle and hatlike liquid distributor, m h1 = height of rotational baffle, m h1′ = height of perforated zone of rotational baffle, m h1′′ = height of nonperforated zone of rotational baffle, m h1c = clearance between rotational baffle and stationary disc, m h2 = height of stationary baffle, m h2c = clearance between stationary baffle and rotational disc, m H = axial height of RZB rotor, m dx.doi.org/10.1021/ie301207e | Ind. Eng. Chem. Res. 2013, 52, 4638−4649
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Lp = perimeter of rotational baffle channel in contact with fluid, m L = liquid rate, m3/s mG = mass rate, kg/s n0 = total number of small holes in the perforated zone of rotational baffle P = pressure in RZB rotor, Pa ΔPc = centrifugal pressure drop due to the centrifugal force, Pa ΔPd = pressure drop of dry RZB, Pa ΔPRd, ΔPTd = rotor and overall pressure drops of dry RZB, Pa ΔPf = frictional pressure drop due to friction generated by rotational and stationary baffles, Pa ΔPm = pressure drop due to momentum gain by gas flowing toward the eye of the rotor, Pa ΔPR = rotor pressure drop of RZB, Pa ΔPT = overall pressure drop of RZB, Pa ΔPw = pressure drop of wet RZB, Pa ΔPRw, ΔPTw = rotor and overall pressure drops of wet RZB, Pa r = radius, m ri = inner radius of RZB rotor, m rm = logarithmic average of radius of RZB rotor, m ro = outer radius of RZB rotor, m rR = radius of rotational baffle, m rs = radius of shift in relative tangential slip velocity in RZB rotor, m R0 = radius of RZB casing, m S = area of rotational baffle channel through which fluid passes in a radial direction, m2 T = torque exerted on the gas, N·m uLr,avg = average of superficial liquid velocity, m/s uGr,avg = average of superficial gas velocity, m/s vr = gas radial velocity, G/(2πrHϕ), m/s vθ = gas tangential velocity, m/s vθR = rotational baffle velocity, ωrR, m/s vθL = tangential velocity of liquid droplets thrown from rotational baffles, m/s Greek Symbols
α, β = coefficients γ = angle at which gas obliquely flows through perforations of rotational baffles ρG = gas density, kg/m3 ρL = liquid density, kg/m3 φ0 = fractional hole area of perforated zone of rotational baffle φ = fractional opening area of rotational baffle based on cross-sectional area 2πrRH μG = gas viscosity, Pa·s μL = liquid viscosity, Pa·s ω = angular velocity, rad/s
Dimensionless Groupings
Ref = Reynolds number of friction factor, (vrdhρG)/μG ReG = gas Reynolds number, (dhuGr,avgρG)/μG ReL = liquid Reynolds number, (dhuLr,avgρL)/μL Reω = rotational Reynolds number, (ωrm2ρG)/μG
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REFERENCES
(1) Ramshaw, C.; Mallinson, R. H. Mass Transfer Apparatus and its Use. Eur. Patent 0,002,568, 1979. (2) Ramshaw, C.; Mallinson, R. H. Mass Transfer Process. U.S. Patent 4,383,255, 1981. 4649
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