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Transport Phenomena of Falling Liquid Film Flow on a Plate with Rectangular Holes Ming Zhu, Chun Jiang Liu,* Wei Wei Zhang, and Xi Gang Yuan State Key Laboratory of Chemical Engineering, Tianjin UniVersity, Tianjin 300072, China
Falling liquid film flow is a sort of ubiquitous transport phenomenon occurring in chemical engineering. Fundamental research on the flow and mass transfer behavior of falling liquid film flow on plates with different structures can help engineers to develop more efficient equipment. In this paper, falling liquid film flow on a plate with rectangular holes is investigated with both numerical and experimental methods. Mass transfer experiments show that opening rectangle-shaped narrow holes on a plate can enhance the mass transfer efficiency. Compared to an ordinary plate, the vapor-liquid mass transfer rate on a holed plate can be increased by 10-20%. To analyze the detailed flow and mass transfer behavior, the computational fluid dynamics model (CFD) based on the volume of fluid (VOF) method is developed. In this model, the Marangoni effect induced by mass transfer is considered. The new model was demonstrated to give more satisfactory results than the previous model. 1. Introduction Falling liquid film flow is a primary phenomenon occurring in nature and industry. Many scientists have delved into this field and many works focused on falling liquid film flow have been published.1-3 Chemical engineer interest is the mass transfer behavior of this kind of flow in process equipment, such as distillation and absorption in packed columns and vapor-liquid reaction in trickle bed reactor. Experimental results show that the flow pattern of this kind of flow is of great importance in determining heat and mass transfer efficiency. By changing the structure of the media on which falling film flow formed, velocity profiles including interfacial velocity distribution can be changed, which will lead to an increase or decrease of mass and heat transfer efficiency. For corrugated structured packing used in distillation column, opening holes on packing sheets can change the falling film flow behavior and increase the vapor-liquid mass transfer efficiency. Fundamental investigation of the flow and mass transfer behavior of the falling film flow on plates with holes is very important for designing a structured packing column and developing novel structured packing. In order to describe the falling liquid film flow behavior in structured packing, the group of Cerro4-8 carried out a number of experimental and theoretical studies. In their works, a film evolution equation with an approximation of the viscous longwave for the Navier-Stokes equations was presented. Using the volume of fluid (VOF)9 method, Szulczewska et al.10 and Gu et al.11,12 investigated the falling film flow on a corrugated plate, corresponding to the surface texture of structured packing. Effects of the plate microstructure, liquid viscosity, surface tension, and gas flow velocity on liquid flow pattern were investigated as well. Also using the VOF method, Raynal et al.13 performed 2D computational fluid dynamics (CFD) simulations to determine the liquid thickness on a wavy plate similar to the packing surface. The results were used to derive the liquid hold-up on the structured packing. Recently, Raynal and RoyonLebeaud14 developed a multiscale approach for CFD calculations of gas-liquid flow within a large size column equipped with structured packing. In their work, simulation results were * To whom correspondence should be addressed. E-mail: cjliu@ tju.edu.cn.
compared with experimental data. Using semianalytical and CFD techniques, Valluri et al.15,16 investigated the dynamic evolution of films over an industrial structured packing surface at moderate Reynolds numbers. In their work, CFD was used as a virtual experiment to verify the semianalytical model. Hoffmann et al.17,18 investigated the two and three phases film flow behavior for packing. For three-phase flow, their qualitative comparison showed good agreement of simulations with experiments. Yuan et al.19 developed a two-fluid CFD model to simulate the two-phase cross/countercurrent flow in the packed column with a novel internal. Their results showed that the pressure drop calculated by the CFD model agrees well with the experimental data at low gas superficial velocity. Yu et al.20 investigated the flow behavior of falling liquid film flow using the laser doppler anemometry method and measured the surface wavelengths and wave velocities. To better understand the enhancement of the mass-transfer process, they also did masstransfer studies. In the paper of Ataki and Bart,21 the flat packing element of Rombopak4M was investigated. CFD simulation results were used as a basis to derive or modify correlations to describe the degree of wetting, the effective area, and the liquid holdup for the Rombopak4M packing. Xu et al.22 investigated the vertical falling film arrangement using the VOF method, in which the wave dynamics and the associated mass transfer phenomena were discussed and compared with experimental empirical relationships. In the work of Alekseenko et al.,23 a method for the measurement of liquid film thickness on corrugated sheets inside a column with a regular packing using a fiber-optic sensor has been developed. It is shown that inside a typical geometrical cell, the maximal thickness of a liquid film is near the contact points of the sheets, where liquid is redistributed over the corrugated surface. Haroun et al.24 investigated the flow and reactive mass transfer behavior of gas-liquid flow on structured packing. Their results showed that mass transfer was increased compared to the transfer for a plane liquid film. Xu et al.25 simulated the falling liquid film flow and its mass transfer behavior using a three-dimensional CFD model. The simulation results showed a good agreement with the experimental data. On the basis of the work of Gu,11 Chen et al.26 simulated the hydrodynamics and mass-transfer behavior in a typical representative unit of the structured
10.1021/ie9020683 2010 American Chemical Society Published on Web 09/29/2010
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Table 1. Physical Properties of Vapor and Liquid Phases applied system
density, F (kg/m3)
viscosity, µ × 105 (kg/m · s)
surface tension, σ × 103 (N/m)
temp (K)
water 2-propanol air
998.3 795.5 1.2050
100.4 231 1.81
72.6 21.3 -
293 293 293
packing. In their work, the c2 - εc model was used for the closure of turbulent mass transfer equation. Although falling liquid film flow behavior has been investigated extensively, very few works have been published concerning falling liquid film flow on plate with holes. But in industry, many types of structured packing with corrugated sheets have holes. So investigation on the flow and mass transfer behavior of this kind of flow is highly desired. On the basis of previous works,11,12,26 the present work aims at studying details of transport phenomena of this kind of flow by CFD and experimental methods. 2. Experimental Facilities and Procedure In order to determine the vapor-liquid mass transfer characteristics of falling film flow on a plate with rectangular holes, a desorption experiment was carried out. In this experiment, air was supplied as vapor phase, and the liquid phase was a diluted aqueous solution of 2-propanol. The 2-propanol was provided by Kewei Chemical Co. Inc. and had a declared purity of 99.7%. The physical properties of the vapor and liquid phases are listed in Table 1, and the schematic diagram of the experimental setup is presented in Figure 1, which includes an air compressor (1), temperature control system (2), vapor-liquid falling film contactor (5), centrifugal pump (7), and piping system. The falling film contactor was made of PMMA (polymethyl methacrylate, Figure 2) in which the desorption process occurred. The vapor phase was supplied into the contactor via the air compressor. In the experiment, two vapor streams with the same flow rate controlled by rotameters were fed into the contactor through vapor inlets at the bottom of the contactor and released to the atmosphere after mass transfer with the liquid phase. The liquid phase was supplied by a centrifugal pump. Two streams of liquid with the same flow rate were fed into the contactor through liquid inlets at the top of the contactor. When liquid flows down in the contactor as falling liquid film on the plate, it would contact counter-currently with the vapor phase, and the desorption process then occurred. After contacting with the vapor phase, the discharged liquid was collected by a tank under the contactor. All experiments were carried out at a pressure of 101.325 kPa and a temperature of 20 ( 0.5 °C, as maintained by a PID controller. Detailed configuration of the falling film contactor is given in Figure 2. It is 520 mm in total height, 80 mm in width, 50
Figure 2. Geometry and boundary condition of the contactor.
mm in length. A stainless steel plate of 80 mm width can be inserted into the middle of the contactor, and it divides the contactor into two symmetrical parts of 25 mm in length. The plate has two margins of 15 mm by which to fix it in the contactor, so the effective mass transfer area in each part is 50 mm in width. When liquid streams are fed into the contactor, it will spread onto the surface of the stainless steel plate through the liquid inlet gap of 0.5 mm in width. Two types of stainless steel plates were used to test their desorption performance: a holed plate with rectangular holes on the plate and an ordinary plate without holes. Both plates are 0.2 mm in thickness, 80 mm in width, and 520 mm in length, but for holed plate, rectangular holes 3 mm in height and 50 mm in width are opened. The distance between two neighbor holes is 12 mm. In our experiment, the surface of the plate was coated with thin paper, a hydrophilic material that can make the liquid film spread evenly. After coating, the reproducibility of the experiment could be guaranteed. A series desorption experiments was carried out with triplicates for each treatment. Sampling of the liquid phase was taken at the inlet and outlet of the contactor, and the concentration was analyzed by gas chromatograph (HP4890) with Porapak-Q packed column. The oven, injector, and detector temperatures of GC were 423, 453, and 453 K, respectively. 3. Theoretical Model and Numerical Methodology
Figure 1. Schematic diagram of the experimental setup for the countercurrent falling film flow: (1) air compressor, (2) temperature control system, (3) valve, (4) rotameter, (5) falling film contactor, (6) liquid receiver, (7) centrifugal pump, and (8) raw material tank.
Using a numerical method, the detailed flow and mass transfer behavior of falling liquid film flow are simulated. In the experiment, the width of the flow channel is magnificently larger than thickness of the liquid film (δ/W , 1), so the falling liquid film flow can be simulated using a two-dimensional model. 3.1. Governing Equations. The basic model equations including continuity equation, volume fraction equation, mo-
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mentum conservation equation, and species conservation equation are continuity equation: volume fraction equation:
∂ (F) + ∇ · (Fu b) ) 0 ∂t ∂ (R ) + b u · ∇(Rq) ) 0 ∂t q
(1) (2)
where q ) L, G. ∂ (Fu b) + ∇ · (Fu bb u) ) ∂t T - ∇P + ∇ · [µ(∇u b + ∇u b )] + Fg b+b F (3)
momentum transport equation:
In published literature,11,25 only the surface tension source term and drag force source term were taken into consideration in describing the falling liquid film flow. The temperature-driven Marangoni effect was considered in Cazabat’s work27 in describing the fingering of liquid film. In Yu’s20 experiment, it was found that the concentration driven Marangoni effect is very important in determining the mass transfer behavior. In this work, mass transfer induced Marangoni effect was taken into account in describing the desorption process of the falling film flow.20 We considered it as a source term in the momentum transport equation. F ) FVOL + FLG + FMa
Fκi∇Ri
)
]
The scalar transport equation in this work is the 2-propanol conservation equation: ∂ (R F w ) + ∇(RqFqwqb u ) ) ∇(Dq∇(RqFqwq)) + SLG ∂t q q q
nˆ ) nˆw cos θw + m ˆ w sin θw
(8)
1 (∂σ/∂C)(∂C/∂xi) 1 ∂σ/∂xi ) 2 δ 2 δ
(12)
SLG ) kGaeMACGT(yAI - yA)
(13)
where ae is effective interfacial area per unit volume. In the simulation, ae is defined as the mass transfer area/the volume of the experimental contactor, which is 40 m2/m3. For equilibrium between vapor and liquid phases, yAI )
βxAI 1 + (β - 1)xAI
m1SLG2 + m2SLG + m3 ) 0
(9)
In Figure 3, the experimental data of surface tension vs concentration is given (using OCA15, Dataphysics Co.). From
(14)
where relative volatility β ) 1.2. Substituting eqs 12 and 13 into eq 14 gives
The Marangoni effect source term FMa is defined as FMa )
SLG ) kLaeMACLT(xA - xAI) or
(7)
where nˆw and m ˆ w are the unit vectors normal to and tangential to the wall, respectively. The contact angle θw is the angle between the wall and the tangent to the interface at the wall. The so-called dynamic boundary condition results in adjustment of the curvature of the surface near the wall. The drag force source term FLG between two phases can be described by the friction pressure drop model. Here, FLG is defined as follows:25 uL - b u G |(u bL - b u G) 1 FfLG | b dp ) dy 2 δ
which describes the mass transfer behavior of falling liquid film flow, where q ) liquid or vapor. In the equation, mass transfer in the vapor-liquid interface can be involved in the source term of the species SLG, where SLG can be defined as
(6)
where nˆ ) n/|n| and n ) ∇Rq The unit surface normal at the live cell next to the wall is replaced by the following equation:
FLG )
(10)
(11)
1 n · ∇ |n| - (∇ · n) |n| |n|
[(
1 (∂C/∂xi) FMa ) A 2 δ
(5)
1 (F + Fj) 2 i
where F is the volume-averaged density, σ is the surface tension coefficient, and κ is the free surface curvature defined in terms of the divergence of the unit normal nˆ as κ ) ∇ · nˆ )
the figure, the experimental data can be fit by σ ) 64.7 0.018C. Thus, it is reasonable to simplify ∂σ/∂C as a constant. Then, eq 9 becomes
(4)
For a two-phase system, the surface tension source term is defined as11 FVOL ) σij
Figure 3. Surface tension σ changes with 2-propanol concentration.
(15)
where m1 )
β-1 kLkGCLTCGT(aeMA)2
(16)
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Figure 5. Simulation data changes with grid size (liquid inlet velocity 0.111 m/s, superficial vapor velocity 0.333 m/s). Figure 4. Two-dimensional hybrid grid around the film flow region and the vapor-liquid interface.
m2 )
[
(β - 1)xA + 1 (β - 1)yA - β kLaeMACLT kLaeMACLT
m3 ) {βxA - [(β - 1)xA + 1]yA}
]
(17) (18)
and the source term SLG can be calculated from SLG )
-m2 + √m22 - 4m1m3 2m1
(19)
The discrepancy between the experimental data and the penetration theory is small, so mass transfer coefficients in eqs 12 and 13 are defined as follows:28 kL ) 2
DL πt
(20)
DG πt
(21)
kG ) 2 and contact time t is t)
l usurf
(22)
3.2. Boundary and Initiative Conditions. Corresponding to the experiment, the simulation model geometry and dimensions are given in Figure 4 and Table 2. Initially, the liquid inlet gap domain (Figure 4) was filled with 2-propanol-water solution, and liquid film with Nusselt film thickness was patched over the surface of the stainless steel plate. For the rest of the calculation domains, it was filled with vapor. 3.3. Grid Strategy. A detailed grid strategy is given in Figure 4. In this work, a hybrid grid strategy is used to decrease the computing cost.29 In the liquid film flow regions and vapor-liquid Table 2. Boundary Conditions velocity liquid inlet liquid outlet vapor inlet vapor outlet wall
volume fraction mass fraction
uL,x ) 0, uL,y ) uL,in RL ) 1, RG ) 0 pressure outlet uG,x ) 0, uG,y ) uG,in RL ) 0, RG ) 1 pressure outlet no slip, contact angle ) 0°
wL ) wL,in
interface regions, quadrilateral grid elements are used. The quadrilateral-shaped grids are either parallel or normal to flow and mass transfer direction; thus, they have the minimum effect on numerical diffusion and can minimize the simulation discrepancy. On the contrary, using other shaped grids such as triangle grids will lead to difficulty in convergence.9 But in regions out of liquid film and vapor-liquid interface regions, unstructured triangle grids are used because the momentum and concentration gradients are very small, and triangle grids are more flexible to suit the computational domain. Furthermore, the accuracy of simulation is not only determined by grid shape but also by grid size. In the liquid film flow region, the momentum and concentration gradients are larger than those in other regions, thus deserving more clustered grid to reduce the discretization errors.29 The size of the grid in the film flow region must be less than the Nusselt film thickness δN (nearly 0.2 mm in this case). A finer grid can lead to more accurate simulation. To balance simulation accuracy and computing cost, four different types of grid strategies are investigated. The simulation results of these grid strategies are compared with experimental results as demonstrated in Figure 5. It shows that as grid size decreases, the simulation results are more consistent with experimental results. The result of grid strategy 3 (as given in Table 3) is closer to experimental data than strategioes 1 and 2, and grid strategy 4 is almost the same as strategy 3, but it needs more computing cost. Therefore, grid strategy 3 is adopted in the rest of this work, which is 0.1 mm along the streamwise direction and 0.025 mm (1/8 of Nusselt film thickness δN) along the mass transfer direction. 3.4. Simulation Scheme. Since the Reynolds number of the liquid phase in the cases considered is lower than 100 and the velocities of the counter-current vapor phase are low, the system can be treated as laminar flow. The FLUENT software package is used to calculate the falling film flow on various plates. Time step sizes are given as 10-4 s. First-order upwind differencing is chosen as the solution of the momentum equation and species conservation equation. The body force weighted method is adopted for pressure discretization, and PISO is used for pressure-velocity coupling. A volume of fluid (VOF) model and a geo-reconstruct method are used to trace the interface of different phases. In addition, the momentum source term is implemented by user defined functions (UDF) into each cell at the interface, which can be determined by a characteristic function of the liquid phase, such as the volume fraction, R. The scalar transport equation (eqs 11) is implemented by the user defined scalar (UDS). The simulation operation procedure is performed in two steps. First, it is carried out for the case
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Table 3. Grid Quality of Computational Domain
liquid film and vapor-liquid interfacial region transient region outer region
grid shape
cell length (mm)
cell width (mm)
grid number
% skewness,
