Hydrodynamics in a Rotating Packed Bed. II. A Mathematical Model

Article pubs.acs.org/IECR

Hydrodynamics in a Rotating Packed Bed. II. A Mathematical Model Zuo-yi Yan, Qi Ruan,* and Cheng Lin Department of Chemical Engineering, Fuzhou University, Fuzhou 350108, Fujian, P. R. China ABSTRACT: A mathematical model for air−liquid two-phase flow in rotating packed beds (RPB) is established to describe the real condition of the fluid flow in the RPB. Results show that the model can better and reasonably predict the distribution of the air−liquid two-phase flow field, the pressure drop, the liquid film thickness, the proportion of the turbulent flow in the liquid film and some other complex hydrodynamic characteristics in the RPB. The results provide a theory basis for the gas−liquid twophase heat transfer and mass transfer.

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INTRODUCTION Rotating packed beds (RPB) have been widely applied in industrial processes (such as distillation, absorption, and extraction operation, etc.) because it greatly enhances the mass-transfer coefficient. Therefore, many researchers have conducted research on the mass-transfer process and estimated the mass transfer coefficient by using different models or formulas. Researchers such as Kelleher et al.1 and Lin et al.2 estimated the mass transfer coefficient by using some correlations. Dipendu et al.3 estimated the mass transfer coefficient by using artificial neural networks, and so on. In fact, the mass-transfer coefficient is greatly influenced by the hydrodynamics of gas and liquid phase in the RPB, which is very different from that in the conventional column. Therefore, in order to describe the real condition of the fluid flow in the RPB, many researchers have done a lot of work in the hydrodynamic modeling. The representative models include the following: rotating disk model and rotating blade model, which were established by the Munjal et al.4,5 and Thomas et al.; 6 the flying droplets model, which was established by Zhang et al.,7 Guo et al.,8 and Guo et al.;9 and the model for the liquid flow around a circular cylinder, which was established by Guo et al.10 However, the real condition of air−liquid two-phase flow in RPB packed with random packings is far different from these models. According to the experimental results in Part I of our work, the fluid in the RPB has distinctive flow characteristics. So far, there is no appropriate mathematical model to veraciously describe the actual wetting, the false wetting, the liquid trajectories, the condition of the frequent hittings between liquid and packings, the turbulent flow for the liquid in the RPB, etc. This paper aims to solve these problems.

Figure 1. Area of the wetting region varies with the total input of the liquid: (a) liquid trajectories for the actual wetting; (b) liquid trajectories include the actual wetting and false wetting.

the false wetting region, and it is not thrown away by centrifugal force. Therefore, there are no liquid droplets and liquid film flow on the false wetting region. The false wetting region is added into the “dry region” in this paper, and the actual wetting region equals to the “wet region”. The gas−liquid effective interfacial area has close relationship to the size of the “wet region”. Thus, hydrodynamic characteristics of both regions will be studied, and their corresponding mathematical models will be established respectively. 1. Motion Characteristics of the Gas in the Unirrigated Bed. The motion characteristics of gas in the “dry region” are as the same to the motion characteristics of the gas when it passes through the unirrigated bed because the gas directly contacts the packings. Since the density of gas is very low, the gravitational effect is ignored. Thus, the equation of the N−S in the noninertial system of coordinates for gas in the unirrigated bed can be written as

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MODEL DEVELOPMENT According to the experimental results in Part I of our work, RPB is clearly divided into two different parts: entry region and development region. The development region accounts for quite a large proportion of the bed and will be discussed first. The liquid trajectories in Figure 1a reflect the actual wetting because the operating time is short. However, there is some false wetting on Figure 1b (the liquid trajectories start to change because the operating time is long enough). The liquid adheres to the surface of the packings by the molecular force on © 2012 American Chemical Society

Received: Revised: Accepted: Published: 10482

October 2, 2011 June 23, 2012 July 13, 2012 July 13, 2012 dx.doi.org/10.1021/ie202258f | Ind. Eng. Chem. Res. 2012, 51, 10482−10491

Industrial & Engineering Chemistry Research D(ρG V ) Dt

= ρG (fp − ae − 2ω × V ) + ∇·P

⎛ ⎞ 1 P = −pI + τ = −pI + 2μ⎜E − ∇·VI ⎟ ⎝ ⎠ 3

Article

(1)

(2)

where P is the stress tensor, τ is the deviator stress tensor, and E is the strain rate tensor. As there exist disorderly and unsystematic packings in the beds, the strain rate tensor E is difficult to be calculated. Therefore, a three-dimensional cylinder coordinate system is introduced to describe the gas motion, and the hypotheses are listed as follows. Zheng et al.,11 Sandilya et al.,12 and Rao et al.13 studied the angular velocity of gas in the RPB and found that the gas goes round and round in the RPB and is keeping pace with the packings all the time. This assertion is supported by the work of Liu et al.14 They found that the frictional pressure drop is independent of angular velocity. These observations indicate that there is no significant slip in the θ direction between the packings and the gas, that is, Vθ = 0

Figure 3. Velocity distribution of the gas in the channel approximated by using the 1/n law.

τrr = λ

(8)

V̅ r is the average velocity of the gas at the radial position of r, can be obtained by

Vr̅ =

(3)

VG 2πrhε

(9)

Under the gravitational field, Blasius equation is λ = 0.3164Rem (m = −0.25). However, the equation can not be used in the RPB because there exist the centrifugal force and Coriolis force in rotational systems. The relationship between two forces (centrifugal force and Coriolis force) and λ can be described by the exponent sign function

Ignoring the axial moving of the gas, then Vz = 0

ρVr̅ 2 8

(4)

When the gas (single phase) passes through the packings, the packings constitute a group of complex disorderly channels (The wall of the channel is structured by the random packings, as shown in Figure 2). In order to quantify the dimensions of

⎛ V ̅ d ⎞m λ = k ⎜ r e ⎟ (ω 2r )q (ωVr̅ )s ⎝ ν ⎠

(10)

The virtual tube mentioned is straight; however, the real gas channel is winding. An additional body force f p is introduced to describe the acting force impacted by the winding channel, which is formed by the random packings. In order to calculate conveniently, f p is considered as a constant (regarded as an average value). The acceleration of moving space ae can be written as ae =

Figure 2. Wall of the channel structured by the random packings.

4 × 2πrhε dr 4ε = 2πrha dr a

dV0 =0 dt

(5)

⎛ 1⎞ ide ≤ z ≤ ⎜i + ⎟de ⎝ 2⎠

i = 0, 1, 2, 3...

∂ρG ∂t

⎛ 1⎞ i + ⎟de ≤ z ≤ (i + 1)de ⎝ 2⎠

≈0

p=−

i = 0, 1, 2, 3...

(12)

∂ρG ∂r

≈0

(13)

By solving the simultaneous equations 1−13, the pressure distribution along the radial direction can be obtained

(6)

1/ n ⎛ 2(z − ide) ⎞ Vr , z = Vr ,max ⎜1 − ⎟ de ⎝ ⎠ ⎜

dω =0 dt

The process can be considered as a steady flow and the gas density can be considered not varying because the pressure drop of the PRB is less than 1000 Pa in the experiments in Part I of our work, that is,

The velocity distribution of the gas in the virtual tube can be used to approximate the real gas velocity distribution in the packings space by using the 1/n law (shown in Figure 3) ⎛ 2(z − ide) ⎞1/ n Vr , z = Vr ,max ⎜ ⎟ de ⎝ ⎠

(11)

where V0 is the speed of the origin of noninertial system, and ω is the rotational velocity of noninertial system of the coordinates.

these disorderly channels, an element is taken along the radial direction. As dr approaches zero, the channel can be described as virtual tube. de, the hydraulic diameter of the virtual tube, is de =

dV0 dω + × r + ω × (ω × r ) dt dt

+

(7)

and

ρG ⎛ VG ⎞2 ⎛ de ⎞m ρG ⎛ VG ⎞m + s + 2 2q + s q ⎜ ⎟ + k⎜ ⎜ ⎟ ⎟ ω r ⎝ ν ⎠ 8 ⎝ 2πhrε ⎠ 2 ⎝ 2πrhε ⎠

1 ρ ω 2r 2 + ρG fpr r + c1 2 G

(14)

and 10483

dx.doi.org/10.1021/ie202258f | Ind. Eng. Chem. Res. 2012, 51, 10482−10491

Industrial & Engineering Chemistry Research Δp =

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involves complex air−liquid two-phase flow, and the motion characteristics of liquid is particularly complicated. According to the research of Burns et al.,15 the liquid exists in the form of film flow (on the surface of the packings), and liquid particles (droplets and filaments in the packings space) when the rotor speed is larger than 600 rpm Currently, the experimental data for the size of the liquid droplets in the RPB were obtained by using the high-speed stroboscopic photography in the work of Zhang et al.7 Some experimental formulas were also proposed by Zhang et al.7 and Guo et al.8,10 However, there are still few studies on the liquid flow pattern and the turbulent degree of liquid in the RPB. Combined with the results of the Part I of our work, the fluid flow can be described as follows. At time 0: One part of the liquid travels as a thin film flow on the surface of packings, while the other part travels as flying particles (droplets and filaments) in the packings space. As the size of the liquid film, droplets, and filaments is far smaller than that of the packings, the ratio of the liquid film flow and flying particles depends on the porosity of the packings. The chance for the liquid to enter the space among the packings is equal to porosity (i.e., β = ε). During 0∼Δt: The liquid particles (droplets and filaments) pass through the packings space and the liquid film moves along the solid surface. The liquid film and liquid particles travel the distance of Δl without any interference along the radial direction duringΔt. At timeΔt: The liquid particles (droplets and filaments) hit the solid surface of the packings. The hitting also happens between liquid particle and liquid film flow; then, one part of the liquid reclothes the film on the packing surface, and the other part of the liquid reenters the packings space. Repeat these steps until the liquid is thrown out of the RPB. The liquid flows along the liquid trajectories, and the width of the trajectories does not change along the radial direction, while the gas fills the space of the rotor. Therefore, the orthogonal Cartesian axis is suitable for the liquid, and the column coordinate is suitable for the gas. As the coordinate system for gas is different from that for liquid, equations for the gas−liquid two-phase flow during Δl can be written as vector expressions based on the momentum balance. Liquid:

ρG ⎛ VG ⎞2 −2 ⎛ de ⎞m ρG ⎛ VG ⎞m + s + 2 −2 ⎜ ⎟ (r ⎜ ⎟ ⎟ 1 − r2 ) + k ⎜ ⎝ ν ⎠ 8 ⎝ 2πhε ⎠ 2 ⎝ 2πhε ⎠ 1 ω 2q + s(r2 q − m − s − 2 − r1q − m − s − 2) + ρG ω 2(r2 2 − r12) 2 + ρG fpr (r2 − r1)

(15)

Where, Δp is the pressure drop of the whole bed. According to the experiment results in Part I of our work, the total pressure drop generally ranges from tens to several hundred Pascal. The first term of eq 15 is often less than 1 Pascal and takes a small proportion of the total pressure Δp. Therefore, eq 15 can be simplified by ⎛ d ⎞ m ρ ⎛ V ⎞ m + s + 2 2q + s q − m − s − 2 ω (r2 Δp = k ⎜ e ⎟ G ⎜ G ⎟ ⎝ ν ⎠ 8 ⎝ 2πhε ⎠ 1 − r1q − m − s − 2) + ρG ω 2(r2 2 − r12) + ρG fpr (r2 − r1) 2 (16)

The gas flow rate, VG, can be obtained from eq 16 VG = 2πhε ⎡ ⎤1/ m + s + 2 1 2 2 2 Δ p − ρ ω ( r − r ) − ρ f ( r − r ) 2 1 1 ⎥ ⎢ G pr 2 2 G ⎢ ρG de m 2q + s q − m − s − 2 ⎥ q−m−s−2 − r1 ) ⎥⎦ ⎢⎣ k 8 ν ω (r2

( )

(17)

To describe the velocity and pressure distribution in the RPB, we need the solution of the q, s, k, and f pr, which can be determined by the experiment data. In the experiment in Part I of our work, Δp, VG, and ω can be obtained through measurement. According to eq 16, lg[Δp − (1/2)ρGω2(r22 − r12) − ρG f pr(r2 − r1)] is linearly proportional to lg VG, the slope is m + s + 2, and the ordinate of the origin is ⎤ ⎡ ⎛ d ⎞m ρ ⎛ 1 ⎞m + s + 2 ⎟ ω 2q + s(r2 q − m − s − 2 − r1q − m − s − 2)⎥ lg⎢k ⎜ e ⎟ G ⎜ ⎦ ⎣ ⎝ ν ⎠ 8 ⎝ 2πhε ⎠ i.e., m + s + 2 = const

(18)

⎡ ⎛ d ⎞m ρ ⎛ 1 ⎞m + s + 2 ⎤ ⎟ ω 2q + s(r2 q − m − s − 2 − r1q − m − s − 2)⎥ lg⎢k ⎜ e ⎟ G ⎜ ⎝ ⎠ ⎣ ⎝ ν ⎠ 8 2πhε ⎦ = const

ρL

(19)

Similarly, lg[Δp − (1/2)ρGω2(r22 − r12) − ρG f pr(r2 − r1)] is linearly proportional to lg ω. The slope is 2q + s, and the ordinate of the origin is

ρG

⎡ ⎛ d ⎞ ρ ⎛ V ⎞m + s + 2 ⎤ lg⎢k ⎜ e ⎟ G ⎜ G ⎟ (r2 q − m − s − 2 − r1q − m − s − 2)⎥ ⎣ ⎝ ν ⎠ 8 ⎝ 2πhε ⎠ ⎦

D(V ′) = ρG (fp + fL − ae − 2ω × V ′) + ∇·P Dt

(23)

The continuity equation of boundary layer flow for liquid is (24)

∇·U = 0 (20)

A three-dimensional Cartesian coordinate system (x, y, z) is introduced to describe the liquid trajectories and a column coordinate system (r, θ, z) for the gas (shown in Figure 4). When x = r, the liquid film flow is divided into two parts, as shown in Figure 5: boundary layer (z ≤ δL1) and undisturbed flow (δL1 < z ≤ δL1 + δL2). The width of the liquid trajectories is not changed along the radial direction:

⎡ ⎛ d ⎞m ρ ⎛ V ⎞m + s + 2 ⎤ lg⎢k ⎜ e ⎟ G ⎜ G ⎟ (r2 q − m − s − 2 − r1q − m − s − 2)⎥ ⎣ ⎝ ν ⎠ 8 ⎝ 2πhε ⎠ ⎦ = const

(22)

Gas:

m

i.e., 2q + s = const

D(U ) = ρL (f p′ + fG − ae − 2ω × U ) + ∇·P Dt

(21)

The four unknown parameters (q, s, k, and f pr) can be obtained by solving eqs 18−21. 2. Air−Liquid Two-Phase Flow in the Irrigated Bed. The irrigated bed includes “dry region” and “wet region”, which

Uy = 0 (boundary layer) 10484

(25)


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during Δl. Since Δl is very short, the value of φ of eq30 can be considered as a constant for calculating easily. As shown in Figure 22 in Part I of our work, the equivalent exposed roughness is H1 (H1 = H − Le) when the gas passes an irrigated bed. The studies on the relationship between the friction factor and exposed roughness are relatively mature and abundant. The relationship between the friction factor and exposed roughness approximately is √λ ∝ H1. However, the situation in the irrigated RPB is very special because the bed is not completely wetted. Therefore, the friction factor is modified by λ′/λ ∝ kλH1/H

(31)

where kλ is the modification factor. If the bed is completely wetted,(λ′λ)1/2 ∝ H1/H, then, kλ = 1. If the bed is completely dry, (λ′/λ)1/2 = 1, then, kλ = H/H1. Therefore, it is feasible to consider that kλ is linearly proportional to the rate of the wetted area in the total area of the RPB. As the rate of the wetted area in the total area of RPB at the radial position r is ((nWL2e )/ (2πrhε)), kλ can be written as

Figure 4. Coordinate system that the gas and the liquid adopt.

2 ⎛ H ⎞ nW Le H kλ = ⎜1 − + ⎟ H1 ⎠ 2πrhε H1 ⎝

where kλ is considered as a constant between r and (r + Δl). The volume occupied by liquid in the rotor is equal to εnWL2e (r2 − r1), approximately. The volume occupied by gas is [επh(r22 − r12) − εnWL2e (r2 − r1)]. Therefore, εnWL2e (r2 − r1)f G and [επh(r22 − r12) − εnWL2e (r2 − r1)f L] are action and reaction forces between liquid and gas. As the gas and liquid are under countercurrent along the radial direction, the coordinate component of the f L and f G can be written as

Figure 5. Schematic of the liquid film flow.

Uy = 0

Uz = 0 (undisturbed flow)

(26)

The velocity distribution in the boundary flow is supposed as the parabola: Ux , z =

2Ur ∞ ⎛ z2 ⎞ ⎜z − ⎟ δ L1 ⎝ 2δ L1 ⎠

(z ≤ δ L1);

Ux , z = Ur ∞

(δ L1 < z ≤ δ L1 + δ L2)

(27)

where Ur∞ is the velocity of the undisturbed liquid flow at the radial position of r (x = r) and δL1 is the thickness of the boundary layer of the liquid flow. The continuum equation in AB distance for the whole liquid film in RPB is as follows: n w LeUr ∞δ L2 +

∫0

δ L1

2πhεr − Q /Ur̅ 2πhεr

fGz = 0

(33)

fp |fp |

=

f p′ |f p′ |

(34)

The gas is keeping pace with the packings Vθ = 0

(35)

Ignoring the axial moving of the gas, then Vz = 0

(29)

(36)

The process can be considered as a steady flow, and gas density can be considered not varying because the pressure drop of the total bed is less than 1000 Pa in the experiment in Part I of our work, that is,

According to the analysis in Part I of our work, the passing section of gas decreases with the liquid flow rates in the irrigated bed. Therefore, parameter φ is defined to estimate the ratio of the passing section of the gas in the irrigated bed to that in the unirrigated bed. φ can be given by ϕ≈

fGθ = 0;

It is considered that f p has the same direction as f ′p, that is,

where δL2 is thickness of the undisturbed liquid flow. nw is number of the liquid trajectories. The continuum equation in BC distance for the whole liquid film in the RPB is

∫0

fLz = 0

=0

2Ur ∞ ⎛ z2 ⎞ ⎜z − ⎟n w Ledz = (1 − β)Q δ L1 ⎝ 2δ L1 ⎠

2Ur ∞ ⎛ z2 ⎞ ⎜z − ⎟n w Ledz = (1 − β)Q δ L1 ⎝ 2δ L1 ⎠

fLy = 0;

[επh(r22 − r12) − εnW Le2(r2 − r1)]fLr + εnW Le2(r2 − r1)fGx

(28)

δ L1

(32)

∂ρG ∂t ∂ρL

(30)

∂t

where U̅ r is the average velocity of the liquid at the radial position of r. According to eq 30, φ varies with the radius. Equations for the gas−liquid two-phase flow are established

≈ 0, ≈0

∂ρG ∂r ∂ρL ∂r

≈0

≈0

(37)

(38)

According to the results in Part I of our work, the pressure drop in irrigated bed is almost the same as that in unirrigated 10485



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bed and it does not vary with the liquid flow rate under the condition that the rotor speed and the gas flow rate keep constant, then ΔpQ′ = Δp

m 1 − (kλH1/H )2 (1/ϕ)m + s + 2 ⎛ de ⎞ ⎛ VG ⎞m + s + 2 ⎟ k⎜ ⎟ ⎜ ⎝ ν ⎠ ⎝ 2πhε ⎠ 8Δl

fLr =

ω 2q + s[(r + Δl)q − m − s − 2 − r q − m − s − 2]

(39)

(47)

3. Characteristics of the Entry Region. As the liquid first enters the entry region, there is a great relative movement between the liquid and the packings along the circumferential direction. The packings are almost wetted by the liquid totally. This phenomenon can be proved by Figure 6, which is obtained

where Δp′Q is the pressure drop of the total bed at the liquid flow rate of Q. By calculating the simultaneous equations 22−39, both ΔpQ′ and the thickness of the boundary layer of the liquid flow δL1 can be obtained m+s+2 ⎛ d ⎞m ρ ⎛ V ⎞ ΔpQ′ = (kλH1/H )2 k ⎜ e ⎟ G ⎜ G ⎟ ⎝ ν ⎠ 8 ⎝ 2πhεϕ ⎠

ω 2q + s[(r + Δl)q − m − s − 2 − r q − m − s − 2] 1 + ρG ω 2[(r + Δl)2 − r 2] + ρG Δl(fpr + fLr ) 2 (40)

⎛ 1 δ L1 = ⎜⎜ 19 ⎝ Ur ∞

∫

⎞0.5 30μ 18 Ur ∞ dr + c 2⎟⎟ ρL ⎠

(41)

and

Figure 6. Characteristics of the entry region.

⎡ ρ ⎛ V ⎞2 ⎛ d ⎞ m ρ ⎛ V ⎞ m + s + 2 2q + s q ω r Ur ∞ = ⎢ G ⎜ G ⎟ − k ⎜ e ⎟ G ⎜ G ⎟ ⎝ ν ⎠ 4ρL ⎝ 2πhrε ⎠ ⎢⎣ ρL ⎝ 2πrhε ⎠ ⎤ ⎛ ρ ⎞ 2ρ + ⎜⎜1 − G ⎟⎟ω2r 2 − G fpr r + 2f p′x r + 2fGx r + c3⎥ ρL ⎠ ρL ⎝ ⎦⎥

by the novel experimental method in Part I of our work. Under this condition, it brings a lot of conveniences to calculate when the packings are considered completely wetted in the entry region. Therefore, the thickness of the liquid film in entry region, δE, can be approximately calculated by

0.5

(42)

The value of c2 can be determined by the boundary condition δ L1(r ) = 0

δE =

(43)

The value of c3 also can be determined by the boundary condition. For example, when t = 0, r is on the border of the entry region and development region, and the value of c3 can be determined by U̅ r1+l, which is the initial velocity of the liquid at the entrance of the development region (U̅ r1+l can be calculated by eq 50 approximately). Substituting eq 41 into eq 28 yields δL =

(1 − β)Q 2⎛ 1 + ⎜⎜ 19 3 ⎝ Ur ∞ n w LeUr ∞

∫

Sr = 2πrha

3 (1 − β)Q 2 n w LeUr ∞

(44)

(in BC distance)

(45)

where δL is the whole thickness of liquid film flow along the radial direction between r ∼ (r + Δl). As the value of the δL in eq 44 and eq 45 is the same, the position of B is determined, which means the length of the AB and BC can be determined by eq 46: 2⎛ 1 ⎜ 3 ⎝ Ur19∞

∫A

B

⎞0.5 30μ 18 1 (1 − β)Q Ur ∞ dr + c 2⎟ = 2 n w LeUr ∞ ρ ⎠

(48)

(49)

To obtain the value of δE, the liquid velocity in the entry region should be known. The circumferential relative velocity between the liquid and the packings is ωr1 at the radial position of r1. The radial relative velocity between the liquid and the packings is considered as a small value for the liquid−solid collisions causing the most momentum loss. When the liquid leaves the entry region, the circumferential relative velocity decreases to zero, and the radial relative velocity increases to maximum U̅ r1+l. As l is very short, the centrifugal acceleration changes slightly and the variation laws of liquid velocity in the entry region can be considered as accelerating from zero to U̅ r1+l in a constant-acceleration rate. U̅ r1+l can be approximately determined by eq 50

Substituting eq 41 into eq 29yields δL =

(r1 < r < r1 + l)

where l is the length of the entry region along the radial direction, and Sr is the length of the wetted perimeter at the radial position of r in the entry region. Sr can be obtained by

⎞0.5 30μ 18 Ur ∞ dr + c 2⎟⎟ ρL ⎠

(in AB distance)

(1 − β)Q SrUr̅

Ur̅ 1+ l ≈

■

Q nW Le2

(50)

RESULTS AND DISCUSSION Figure 7 shows the relationship between lg[Δp − (1/2)ρGω2(r22 − r12) − ρG f pr(r2 − r1)] and lg VG in the Cartesian coordinate system. According to eq 15, lg[Δp − (1/2)ρGω2(r22 − r12) − ρG f pr(r2 − r1)] is linearly proportional to lg VG under a certain rotor speed and the slope of the line does not change with the rotor speed. As shown in Figure 7, There is a better linear

(46)

The proportion for the turbulent flow in the whole liquid film flow can be obtained from ((|AB|)/(|BC|)). Integrating eqs 14, 33, and 39, f Lr can be obtained by 10486



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Figure 7. Relationship between lg[Δp − (1/2)ρGω2(r22 − r12) − ρG f pr(r2 − r1)] and lg VG.

relationship between lg[Δp − (1/2)ρGω2(r22 − r12) − ρG f pr(r2 − r1)] and lg VG. The lines under different rotor speeds make a family of parallel lines approximately. The slope of the parallel lines is m + s + 2. The value of m is given so that the value of s can be determined. The values of q, k, and f pr can be obtained by solving eq 18−21 using the trial and error method. Figure 8 shows the relationship between lg ω and lg[Δp − (1/2)ρGω2(r22 − r12) − f pr(r2 − r1)] in the Cartesian coordinate

Figure 9. Comparison of experimental values of with the results calculated using eq15.

obtained by using the high-speed stroboscopic photography in the work of Guo et al.9 and also similar to the measure and simulate results obtained by Munjal et al.,4,5 who studied the liquid mechanics in the high gravity field by using the rotating disk and rotating blade. As shown in Table1, the average velocity of liquid in RPB is about 1−6 m/s. That means the residence time for the liquid in RPB is about 0.01−0.1 s, which is consonant with the experiment data obtained by Guo et al.9 The velocity of the undisturbed flow hardly changes with the liquid flow rate. However, the thickness of the liquid film and the residence time of the liquid decrease with the liquid flow rate. In addition, the velocity of liquid is greatly varying with the rotor speed. This result suggests that the effect of the rotor speed on the residence time is greater than that of the liquid flow rate. This characteristic is also consonant with the experiment data obtained by Guo et al.9 Figure 10 displays the proportion of the turbulent flow in the liquid film as a function of the liquid flow rate ranging from 100 L/h3 to 300 L/h3. As shown in Figure 10, the proportion of the turbulent flow increases with the liquid flow rate under different rotor speeds. The reason is that the thickness of the whole liquid film increases with the liquid flow rate, and the thickness of the undisturbed flow increases because the thickness of the boundary layer is almost unchanged. Thus, the proportion of the turbulent flow layer increases with the liquid flow rate. Figure 11 displays the proportion of the turbulent flow in the liquid film as a function of the rotor speed ranging from 600 to 1200 rpm. As shown in Figure 11, the proportion for the turbulent flow increases with the rotor speed under different liquid flow rates. The flow velocity of the liquid increases with the rotor speed and the turbulent degree of the liquid is increased. It is found an interest phenomenon both in Figure 10 and in Figure 11 that curve ① is steeper than other curves. The proportion of the turbulent flow in the liquid film reflects the liquid turbulent degree in RPB. The mass transfer effect in RPB has close relationship to the liquid flow pattern and liquid turbulent degree. This means the mass transfer effect may be greatly improved by increasing the rotor speed under large

Figure 8. Relationship between lg ω and lg[Δp − (1/2)ρGω2(r22 − r12) − f pr(r2 − r1)].

system. As shown in Figure 8, lg[Δp − (1/2)ρGω2(r22 − r12) − f pr(r2 − r1)] is linearly proportional to lg ω. The lines under different gas flow rates also make a family of parallel lines. Figures 7 and 8 prove that the principle and mathematical model of the unirrigated bed are correct. Figure 9 shows a comparison of the experimental values of Δpwith the results calculated by using eq 15. As observed in Figure 9, most of the experiment results lie within ±10% of the values estimated by eq 15. It is verified that the assumption in this work is reasonable and acceptable. As the pressure drop in the irrigated bed is almost the same as that in the unirrigated bed and it does not vary with the liquid flow rate under the condition that the rotor speed and the gas flow rate are certain, eq 15 also can be used to predict the pressure drop in irrigated bed. Table 1 shows that the liquid film thickness, the velocity distribution of the liquid, and the proportion for the turbulent flow in the whole film flow in different positions and different operating conditions of the RPB in microscale. As shown in Table1, the magnitude of the thickness of the liquid film is about 10−6−10−5 m, which is similar to the measurements 10487



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Table 1. Simulated Results of Some Hydromechanical Characteristic in the RPB

Figure 10. Effect of the liquid flow rate on the proportion of the turbulent flow in the liquid film.

Figure 11. Effect of the rotor speed on the proportion of the turbulent flow in the liquid film.

liquid flow rate. This conclusion may be supported by the mass transfer experiment in the work of Lin et al.16 (O2 stripping process) and in the work of Guo et al.10 (ammonia absorption). By comparing Figure 10, Figure 11, Figure 12, and Figure 13, it is found that curve ① is steeper than other curves in all these figures. This indicates that the proportion of the turbulent flow in the liquid film, which is calculated by the model, can indirectly reflect the mass transfer effect. Figure 14 displays the proportion of the turbulent flow in the liquid film varying with the radius of the RPB. As the centrifugal force and the liquid velocity increase with radius, the liquid film

gets thinner, which shortens the time for the boundary layer spreading from bottom to top in the liquid film. Therefore, the thickness of the liquid film decreases with the radius of the RPB. Thus, the length of AB decreases with the radius and the proportion for the turbulent flow in RPB is getting smaller.

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CONCLUSION A hydrodynamic model of the RPB has been proposed, and the results of its analytical solution are in accordance with the experiment data, which proves that the model can well and 10488



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Figure 13. Mass transfer experiment in the work of Guo et al.:10 (a) effect of liquid flow rate on Kxa; (b) effect of rotor speed on Kxa. Figure 12. Mass transfer experiment in the work of Lin et al.:16 (a) effect of rotor speed on KLa for O2 stripping; (b) effect of liquid flow rate on KLa for O2 stripping.

reasonable describe the air−liquid two-phase flow in the RPB. The flow field distribution of the air−liquid two-phase flow, the pressure drop, the liquid film thickness, the proportion of the turbulent flow in the liquid film, and some other complex hydrodynamic characteristics of the RPB can be calculated by the model, which provides the theory basis for the gas−liquid two-phase heat transfer and mass transfer for RPB. The liquid flow pattern and the turbulent degree of the liquid in the RPB have a large influence on the mass and heat transfer process. Through simulation, it is found that the mass transfer effect may be greatly improved by increasing the rotor speed under high liquid flow rate. This conclusion can be verified by the mass transfer experiment data in the work of Lin et al.16 (O2 stripping) and in the work of Guo et al.10 (ammonia absorption). The results indicate that the proportion of the turbulent flow in the liquid film, which is calculated by the model, can indirectly reflect the mass transfer effect rate.

Figure 14. The proportion of the turbulent flow in the liquid film varying with the radius of the RPB.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest. 10489



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ACKNOWLEDGMENTS

This work was supported by the fund projects of Fujian Province Education Office (Nos. JB08001 and JB06047)

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NOMENCLATURE a = specific surface area of the packing (m2/m3) ad = surface area of one liquid particle (m2) ae = acceleration of moving space aE = gas−liquid interfacial area in the entry region (m2) AG = the area of the gas passing section in the “wet region” (m2) c1, c2, c3 = constants de = hydraulic diameter (m) E = strain rate tensor f G = body force applied by the gas (N/m3) f Gx = component of f G in x direction, when x = r, f Gx = f Gr (N/m3) f Gy = component of f G in y direction (N/m3) f Gz = component of f G in z direction (N/m3) f L = body force applied by the liquid (N/m3) f Lr = component of f L in r direction (N/m3) f Ly = component of f L in y direction (N/m3) f Lz = component of f L in z direction (N/m3) f p = additional body force applied by the random packing when the gas pass through the RPB (N/m3) f ′p = additional body force applied by the random packing when the liquid pass through the RPB (N/m3) h = height of the bed (m) H = equivalent exposed roughness in the unirrigated bed (m) H1 = equivalent exposed roughness in the irrigated bed (m) I = unit tensor in the form of second order k = constant kλ = modification factor l = length of the entry region along the radial direction (m) Le = width of the liquid trajectory (m) Δl = moving distance of the liquid during Δt along the radial direction (m) m = constant nW = number of the liquid trajectories p = static pressure (Pa) P = stress tensor Δp = pressure drop in unirrigated bed (Pa) Δp′ = pressure drop in irrigated bed (Pa) q = constant Q = liquid flow rate (m3/s) r = radius (m) r1 = inner radius of the rotor (m) r2= outer radius of the rotor (m) s = constant Sr = length of the wetted perimeter at the radial position of r in the entry region (m) t = time (s) Δt = time for the liquid particle passing through the space of the packing (s) U = velocity of the liquid in the irrigated bed (m/s) Ur∞ = velocity of the undisturbed liquid flow at the radial position of r (m/s) U̅ r = average velocity of the liquid at the radial position of r in the RPB (m/s) U̅ r1+l = initial average velocity of the liquid at the entrance of the development region (m/s)

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Ux = component of U in the x direction, when x = r, Ux = Ur (m/s) Ux,z = velocity of the liquid whose coordinate position is (x, z) (m/s) Uy = component of U in the y direction (m/s) Uz = component of U in the z direction (m/s) v = kinematic viscosity (m2/s) V = velocity of the gas in the unirrigated bed (m/s) V′ = velocity of the gas in the irrigated bed (m/s) V0 = speed of the origin of noninertial system (m/s) VG = gas flow rate (m3/s) Vr,z = velocity of the gas whose coordinate position is (r, z) (m/s) V̅ r = average velocity of the gas at the radial position of r (m/ s) Vr,max = maximal velocity of the gas at the radial position of r (m/s) Vz = component of V in the axial direction (m/s) Vθ = component of V in the tangential direction (m/s) β = chance for the liquid enters the space among the packing δA = thickness of the liquid film at the position A (m) δC = thickness of the liquid film at the position C (m) δE = approximate thickness of the liquid film in entry region (m) δL = whole thickness of liquid film flow (m) δL1 = thickness of the boundary flow (m) δL2 = thickness of the undisturbed liquid flow (m) ε = porosity φ = ratio of the passing section of the gas in the irrigated bed to it in the unirrigated bed λ = drag coefficient in unirrigated bed λ′ = drag coefficient in irrigated bed μ = fluid viscosity (kg/m·s) ρG = gas density (kg/m3) ρL = liquid density (kg/m3) τ = deviator stress tensor ω = rotational speed (rad/s)

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