Article pubs.acs.org/IECR
Modeling Flow-Guided Sieve Tray Hydraulics Using Computational Fluid Dynamics Qunsheng Li, Lun Li, Manxia Zhang, and Zhigang Lei* State Key Laboratory of Chemical Resource Engineering, Beijing University of Chemical Technology, Box 266, Beijing, 100029, China ABSTRACT: A computational fluid dynamics (CFD) model was used to predict the hydraulics and flow patterns of flow-guided sieve tray, which combined common sieve holes with flow-guided holes. The model considered the three-dimensional two-phase flow of gas and liquid, and each phase was treated as interpenetrating continua that had separate transport equations. Interaction between the two phases was considered via an interphase momentum transfer term, which was incorporated into the CFD model. On the basis of experimental measurement, the new correlations of clear liquid height and gas holdup were developed. The velocity distribution, clear liquid height, and foam height were predicted and found to be in good agreement with the experimental data. Distribution of clear liquid height and liquid holdup proved that flow-guided holes can promote the liquid flow and decrease the clear liquid height. The CFD method is a useful tool for optimizing the novel flow-guided sieve tray.
1. INTRODUCTION Tray column is an important gas−liquid contact equipment in separation processes. The accurate description of a tray’s hydrodynamics is of great significance to industrial practice. On the basis of the hydrodynamics, the overall performance and separation efficiency of the tray can be predicted for a given set of tray geometry, system properties, and operating conditions. Flow-guided sieve tray, as a modification of Linder-sieve tray, has shown its superior performance especially in dealing with the high viscous mixtures.1−3 For example, the distillation for removing vinyl acetate from a polyvinyl acetate polymer solution with a dynamic viscosity of more than 50 Pa s caused the previous column with sieve trays to be blocked or flooded 1−2 times every month in practice. In the modification, the problem was solved only by replacing the sieve tray with flowguided sieve tray.1 This innovative tray has already obtained the Chinese patents (nos. 2001134859.3 and 2001220319X) due to its excellent performance in separation industry. Although studies on the hydrodynamics of flow-guided sieve tray have been reported,1−4 only empirical correlations were derived based on experimental data, and thus were only suitable for certain separation systems. To optimize the design in distillation, an in-depth understanding of the hydrodynamics and complex behaviors of the gas and liquid flows on the flowguided sieve tray for the given tray geometry and operating conditions is urgently needed. Computational fluid dynamics (CFD) is becoming a powerful research and design tool to model two-phase flow. Many researchers attempted to simulate the sieve tray’s hydraulics by this tool.5−18 Liu et al.8 and Yu et al.9 simulated the two-phase flow behavior on the traditional sieve tray. However, they only described the hydrodynamics of the liquid phase and ignored the variations in the direction of gas flow along the dispersion height. Fischer and Quarini10 tried to model the three-dimensional transient gas−liquid hydrodynamics, assuming a constant momentum exchange or drag coefficient of 0.44. This drag force is appropriate for the uniform bubbly flow, but not for describing the hydrodynamics © 2014 American Chemical Society
of the trays operating either in the froth or in the spray regime. Krishna’s research group11−13 simulated the hydrodynamics of sieve trays by establishing a new momentum exchange coefficient on the basis of the correlations of Bennett et al.14 Li et al.15 and Jiang et al.16 obtained a new correlation based on the clear liquid height suitable for a specific tray. Zarei et al.17 simulated MVG tray and compared the flow result with that of Solari and Bell’s sieve tray.18 Alizadehdakhel et al.19 combined the CFD and experimental approaches to investigate the relationship between the valve weight and gas−liquid interface using VOF (volume of fluid) method. Nevertheless, so far there has been no study about the hydraulics of flow-guided sieve tray using CFD to describe such a type of flow. In this work, a three-dimensional transient CFD model was developed for describing the hydraulics of flow-guided sieve tray, within the two-phase Eulerian−Eulerian framework. The simulations have been carried out with varying superficial gas velocity, liquid load, and weir height, and then were compared to experimental data. The object of this work is to extend the application of CFD model to interpret the macroscopic phenomenon taking place on the flow-guided sieve tray.
2. TRAY GEOMETRY AND EXPERIMENTAL SETUP The geometry of flow-guided sieve tray adopted in this work is shown in Figure 1. The tray is composed of 939 sieve holes and 28 flow-guided holes. The gas, containing certain kinetic energy, jets from the flow-guided holes, which can promote the liquid flow across the tray in the horizontal direction. Additionally, a bubble promoter was set nearby the liquid inlet to enhance the liquid bubbling. The hydrodynamics of flow-guided sieve tray were investigated in a plexiglas column with an inner diameter of Received: Revised: Accepted: Published: 4480
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3. MODEL DEVELOPMENT 3.1. Model Equations. The model considered both gas and liquid phases as interpenetrating continua with separate transport equations in the Eulerian−Eulerian framework. In the simulation, the gas and the liquid were treated as disperse phase and continuous phase, respectively. For either the gas (subscript G) or the liquid (subscript L) phase on the tray, the volume-average mass and momentum conservation equations were given by ∂(αGρG ) ∂t ∂(αLρL ) ∂t
+ ∇·(αGρG uG) = 0
(1)
+ ∇·(αLρL u L) = 0
(2)
∂(αGρG uG)
+ ∇·(αGρG uGuG)
∂t
= ∇·(αGμG (∇uG + (∇uG)T )) − αG∇pG + αGρG g
Figure 1. The flow-guided sieve tray tested in the experiment (the length unit: 10−3 m).
+ MGL ∂(αLρL u L)
0.57 m. Air and water at ambient pressure and temperature were used as the gas and liquid, respectively. The experimental setup is shown in Figure 2. Various weir heights of 0.03, 0.04,
∂t
(3)
+ ∇·(αLρL u Lu L)
= ∇·(αLμL (∇u L + (∇u L)T )) − αL∇pL + αLρL g − MGL
αG + αL = 1
(4) (5)
where αk, ρk, uk, μk, and pk represent the volume fraction, macroscopic density, velocity vector, viscosity, and pressure of the kth phase, respectively; MGL represents the momentum exchange between the gas and liquid phases; and the two phases share the same pressure, pL = pG.20 3.2. Closure Relationships. For gas−liquid bubbly flows on the tray, the interphase momentum transfer term includes drag force, virtual mass, and lift forces. Krishna and Baten11 pointed out that the virtual mass and lift forces have limited effect on the bubbly flows and can be ignored in comparison with drag force. Thus, the equation for MGL is written as
Figure 2. Schematic of experimental setup to measure hydrodynamics of flow-guided sieve tray: (1) air blower, (2) orifice meter, (3) column, (4) tested trays, (5) gas distribution board, (6) tube with scale, (7) static pressure meter, (8) valves, (9) liquid storage, (10) rotameter, (11) pump.
MGL =
3 CD αGρL |uG − u L|(uG − u L) 4 dG
(6)
where CD is the interphase momentum transfer coefficient or drag coefficient, and dG is the diameter of gas bubble. As suggested by Krishna et al.,13 the drag coefficient for the special case of rising bubble swarms in the churn turbulent regime is given as 4 ρL − ρG 1 CD = gdG 2 3 ρL Vslip (7)
and 0.05 m with the same downcomer clearance of 0.024 m and tray spacing of 0.35 m were used in the experiments. The clear liquid height was measured by installing a static pressure tube with scale. One limb of the tube was connected with five pressure tappings, which were installed at different positions of the tray, including the center, both sides, and the liquid inlet and outlet zones of the tray, while the other limb was connected to the air space above the liquid on the tray. During the experiment, the superficial gas velocity, measured by an orifice meter, ranged from 0.53 to 2.56 m/s. The liquid flow rate was controlled by a rotameter, and the liquid flow rate per unit length of weir, QL/Lw, ranged from 1.50 × 10−3 to 6.76 × 10−3 m3/(m·s).
where Vslip is the slip velocity of the bubble swarm with respect to the liquid phase, and can be estimated from superficial gas velocity UG and the gas holdup αGaverage: Vslip =
UG average αG
(8)
Substituting eqs 7 and 8 into eq 6, and taking the influence of liquid holdup αL into consideration, the interphase momentum transfer equation becomes: 4481
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Article
(αGaverage)2 |u G − u L | (1 − αGaverage)UG2
Table 1. Sensitivity of Clear Liquid Height in Simulation to Grid Size grid size
(9)
This relation can be directly used in the simulation because it is independent of bubble diameter. The only input is the value of average gas holdup αGaverage, which, however, has not been reported on modeling flow-guided sieve tray so far. In this work, αGaverage was obtained from the new correlations based on experiment measurement, as mentioned below (see section 4.2). For both the continuous and the dispersed phases, the turbulent contribution to the stress tensor was evaluated by using the standard k−ε model, with the default two-phase parameters (Cμ = 0.09, C1ε = 1.44, C2ε = 1.92, C3ε = 1.3, σk = 1, σε = 1.3, and σpq = 0.75).21,22 3.3. Flow Geometries and Mesh Generation. To reflect the real experimental device, the whole tray and tray spacing, as well as downcomer, were considered. The flow geometry, boundary conditions, and mesh generation used in the CFD simulation are shown in Figure 3. The fractional free-area used
around flow-guided holes
below the weir (m)
0.005 m × 0.005 m × 0.005 m 0.005 m × 0.005 m × 0.005 m 0.004 m × 0.004 m × 0.003 m 0.004 m × 0.004 m × 0.003 m 0.004 m × 0.004 m × 0.003 m
0.006
487 367
0.0260
0.005
597 201
0.0245
0.005
734 438
0.0234
0.004
1 020 814
0.0228
0.003
1 283 247
0.0232
no. of grid cells
clear liquid height in simulation (m)
entered through the downcomer clearance, that is, αL = 1. Thus, the following equations can be obtained: u L, x =
QL hgapLw
(10)
where hgap and Lw represent the height and length of downcomer clearance, respectively, and QL is the liquid flow rate across the tray. 3.4.2. Gas Inlet. The gas inlet velocity should be modified in terms of different coefficients of resistance of sieve holes and flow-guided holes, the values being, respectively, ξsieve = 1.54 and ξfg = 2.02, which have been measured and used in a previous dry plate pressure drop model.4 On the basis of the same pressure drop and the general rule that the coefficient of resistance is in inverse proportion to the square of velocity, the velocity ratio of sieve hole to flow-guided hole was derived. usieve : u fg = Figure 3. Flow geometry, boundary conditions, and mesh generation (the point (x, y, z) = (0, 0, 0) is at the center of the bottom tray).
1 ξsieve
:
1 = 1:0.87 ξfg
(11)
A uniform gas velocity inlet was supposed in the two kinds of holes, while the formulations of gas velocity were derived respectively. Also, only gas entered through the holes on the tray, that is, αG = 1. When gas gets through sieve holes in the vertical direction, the gas inlet velocity is given by uG, x = uG, y = 0 (12)
in the computations was the same as in the experiments, except that square holes were used in the simulations rather than circular holes because a rectangular Cartesian coordinate system was employed in this work. Krishna et al.13 also pointed out that the use of square holes instead of circular holes does not impact the simulation results in the case that Eulerian framework is used for describing either fluid phase. The Gambit software (version 2.3.16) was used to mesh the computational domain. In the x-, y-, and z-directions, 0.005 m grid cells were chosen below the weir. Yet around flow-guided holes, the grids with automatic size of 0.004 m × 0.004 m × 0.003 m were used. Above the weir, to save the computation time, various densities of meshes were employed because the liquid holdup decreased from the bottom to the top of tray spacing. Several simulations were carried out to eliminate the effect of grid number on the model results. The sensitivity of clear liquid height in simulation to grid size was tested, as given in Table 1, where UG = 1.19 m/s, QL/Lw = 0.006 m3/(m·s), and weir height hw = 0.03 m. Finally, a total of 734 438 unstructured grid cells within the computational space were generated. 3.4. Boundary Conditions. 3.4.1. Liquid Inlet. For the experimental flow-guided sieve tray, a uniform liquid inlet velocity boundary condition was recommended. Only the liquid
uG, z =
QG A sieve + 0.87A fg
(13)
When gas gets through flow-guided holes in the horizontal direction, the gas inlet velocity is given by uG, y = uG, z = 0 (14) uG, x =
0.87Q G A sieve + 0.87A fg
(15)
where QG is the gas flow rate across the tray, and Asieve, Afg are the total areas of sieve holes and flow-guided holes, respectively. 3.4.3. Outlet Boundary. Both liquid and gas outlet boundaries were specified as the pressure boundaries. The specification assumed that only gas left the gas outlet and only liquid left the liquid outlet. 3.4.4. Wall. All walls for gas and liquid phases were specified as no-slip wall boundaries. 4482
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C = α1 + α2 exp| − α3hw |
3.4.5. Flow-Field Initialization. At a standard atmospheric pressure, water and air were selected as the primary and secondary phases, respectively. At the beginning of simulations, the tray configuration shown in Figure 3 was filled with a uniform gas−liquid dispersion with 50% liquid holdup. 3.4.6. Solution Algorithms. The mesh file was input into the commercial FLUENT software (version 6.3.26) to solve the two-fluid mixture’s equations of continuity and momentum. FLUENT is a finite volume solver, and all variables are calculated at the cell centers.20 The pressure−velocity coupling was obtained by the phase coupled SIMPLE (semi-implicit method for pressure-linked equations) algorithm. The firstorder upwind spatial discretization scheme was used for all differential equations. During the simulations, the time increment was 0.002 s. The clear liquid height in the system was monitored, and quasi-steady state was assumed to prevail when the value of clear liquid height remained constant for a sufficiently long time. The output parameters were obtained by averaging the values during this period. Typically, a steady state was achieved in about 7 s after the start of the simulations, as shown in Figure 4. The clear liquid height was calculated as the
(17)
⎡ ⎛ ρG αLB = exp⎢ − a4⎜⎜UG ⎢ ρL − ρG ⎝ ⎣
a5 ⎤
⎞ ⎟ ⎥ ⎟ ⎥ ⎠ ⎦
(18)
On the basis of the experimental data, the unknown parameters a1−a5 were obtained as follows: a1 = 0.00439, a2 = 0.0277, a3 = 100.0, a4 = 12.53, and a5 = 0.898. The average error was 0.98%, and the mean absolute error was 6.43%. The experimental clear liquid height versus the calculated values using the new correlations was plotted in Figure 5. It can be seen that most
Figure 5. Calculated results of clear liquid height obtained by the new correlations versus experimental data.
errors of the calculated data are less than 10% in comparison with the experimental data. Thus, the new correlations for clear liquid height for the flow-guided sieve tray were rewritten as: ⎛ (Q /Lw ) ⎞2/3⎤ + C⎜ L B ⎟ ⎥ ⎠ ⎥⎦ ⎝ αL
(19)
C = 0.00439 + 0.0277 exp| − 100.0hw |
(20)
⎡
hCL = Figure 4. Transient holdup monitored as a function of time after injection of gas.
αLaverage⎢hw ⎢ ⎣
while the average gas holdup αG
tray spacing multiplied by the volume-averaged liquid fraction. The simulations were performed on a HP Z800 workstation running with 8 processing cores at 2.8 GHz. It took about 20 h to simulate 18 s of tray hydrodynamics.
average
becomes:
αGaverage = 1 − αLaverage ⎡ ⎛ ρG = 1 − exp⎢⎢ − 12.53⎜⎜UG ρL − ρG ⎝ ⎣
4. RESULTS AND DISCUSSION 4.1. Clear Liquid Height and Gas Holdup. There were several correlations for the clear liquid height and liquid holdup for sieve tray,14,23 but they were not suitable for flow-guided sieve tray, which, as a new tray, has been widely used in the chemical and petrochemical industry. In this work, an average clear liquid height correlation for flow-guided sieve tray was developed. For flow-guided sieve tray, the clear liquid height is mainly affected by gas velocity UG, liquid load QL/Lw, and weir height hw. The correlation for clear liquid height is of the following form:14 ⎡ ⎛ (Q /Lw ) ⎞2/3⎤ hCL = αLB⎢hw + C ⎜ L B ⎟ ⎥ ⎢ ⎠ ⎥⎦ ⎝ αL ⎣ (16)
⎞0.898⎤ ⎥ ⎟ ⎟ ⎥ ⎠ ⎦
(21)
4.2. Comparison between Simulated Results and Experimental Data. In this work, eq 21 obtained from experimental measurement was incorporated into the interphase momentum transfer term MGL , and thus CFD simulations about two-phase hydraulics of flow-guided sieve tray can be carried out under varying operating conditions. The interphase momentum transfer term was coded by UDF (UserDefined Function) of FLUENT software. To validate the simulated results against the experimental data, the clear liquid height was modeled and compared between them. Figure 6 shows the comparison of clear liquid height between simulated results and experimental data with varying superficial gas velocity, liquid load, and weir height, respectively. The simulated values of clear liquid height were obtained after averaging over a sufficiently long time interval, once quasisteady-state conditions were established to determine the cumulative liquid holdup within the x-, y-, and z-directions of
where the assumed functionality for C and the liquid holdup αLB are calculated by 4483
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impurities tend to prevent gas coalescence leading to a higher gas holdup as well as a lower clear liquid height. 4.3. Hydraulics and Flow Patterns Inside the Tray. Figures 7−9 show the snapshots and the liquid velocity profiles
Figure 7. Snapshots of the weir views of liquid flow field at different elevations. The structure and operating conditions are: UG = 1.19 m/s; QL/Lw = 0.006 m3/(m·s); and weir height hw = 0.04 m.
of liquid flow patterns viewed from the weir, front, and top of the tray simulated at the superficial gas velocity UG = 1.19 m/s, liquid weir load QL/Lw = 0.006 m3/(m·s), and weir height hw = 0.04 m. The snapshots at different directions allow for understanding the flow pattern on the tray. It is seen that near the bottom of the tray, the liquid is carried upward and then dropped down. The liquid disengages itself from the dispersed gas and travels down the sides, resulting in circulation cells, which are evident in both the weir view (see Figure 7) and the front view (see Figure 8). However, the liquid recirculation patterns are less prominent from the top view (see Figure 9), indicating that the flow-guided sieve tray is effective for relieving the backflow to some degree. 4.3.1. Velocity Distribution of Liquid Flow Field in xDirection. Figure 10 shows the x-component of liquid velocity below, at and above the weir, operating at UG = 1.19 m/s, QL/ Lw = 0.006 m3/(m·s), and hw = 0.04 m. The values of liquid velocity were also derived after averaging over a sufficiently long time interval once quasi-steady-state conditions were established. It can be seen that the x-component of liquid velocity nearby the liquid inlet is negative above the weir, displaying the backflow phenomenon, which is enhanced with the increase of dispersion height. Additionally, there also exists
Figure 6. Clear liquid height as a function of (a) superficial gas velocity, (b) liquid load, and (c) weir height.
computational domain. It can be seen that the clear liquid heights simulated by CFD have the same trends as experimental data. That is to say, the clear liquid height decreases with the increasing of superficial gas velocity at the given liquid load and weir height, as shown in Figure 6a. Meanwhile, it increases with the increasing of liquid load at the given weir height and superficial gas velocity, as shown in Figure 6b. It also increases with the increasing of weir height at the given superficial gas velocity and liquid load, as shown in Figure 6c. As a whole, the CFD simulations tend to overestimate the clear liquid height. The reason may be that both the new correlations and the CFD simulations were established for water containing no impurity. However, Baten and Krishna12 got the same conclusions, and thought that small 4484
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Figure 8. Snapshots of the front views of liquid flow field at different elevations. The structure and operating conditions are: UG = 1.19 m/s; QL/Lw = 0.006 m3/(m·s); and weir height hw = 0.04 m. Figure 9. Snapshots of the top views of liquid flow field at different elevations. The structure and operating conditions are: UG = 1.19 m/s; QL/Lw = 0.006 m3/(m·s); and weir height hw = 0.04 m.
backflow phenomenon especially at the edge of the tray when the liquid flows from the center to the outlet weir (from x = 0 m to x = 0.17 m). As shown in Figure 10, the plots for y = 0 and y = 0.1 almost overlap at x < 0 because in this case plug flow predominates. However, backflow becomes serious at x > 0, and thus clear deviation is observed. According to Lewis Case I, plug flow, which we want to achieve, may bring the highest mass transfer efficiency in theory.24 As a result, the configuration of flow-guided holes can be optimized in terms of velocity distribution of liquid flow on the tray. That is to say, the hydrodynamic performance of the tested tray can be improved by setting more flow-guided holes near the edge and outlet weir of the tray. 4.3.2. Clear Liquid Height. The clear liquid height profiles determined by the average of liquid volume fraction on the vertical slice above the tray floor are shown in Figure 11. It can be seen from Figure 11a that the clear liquid heights are larger nearby the liquid entrance and outlet weir, supporting the conclusions made earlier concerning the effect of calming zones.7 Moreover, in the bubble area, the clear liquid height decreases due to the increase of gas holdup. On the other hand, the larger clear liquid heights are also found on the tray where no flow-guided hole is set (see Figure 11b). That is to say, the greater is the number of flow-guided holes, the lower are the clear liquid heights on the tray. Therefore, the chemical engineers can set the approximate number of flow-guided holes in the place where the high liquid height occurs.
4.3.3. Liquid Holdup and Foam Height. Figure 12 shows the typical simulated results for the variation of liquid holdup along the dispersion height in z-direction at different weir heights. The liquid holdup was obtained after averaging along the x- and y-directions over a sufficiently long time interval once the quasi-steady-state conditions were established. It is seen that at a given weir height, the liquid holdup first increases and then decreases as the dispersion height increases. Yet as the weir height increases, the liquid holdup also increases. The height of foam layer is usually defined as the height at which the average liquid holdup is below 10%. By this definition, the simulated results indicate that the height of the foam layer increases with the increasing of weir height at the given superficial gas velocity and liquid load. Moreover, it can be seen from Figure 12 that the foam heights are 0.086, 0.098, and 0.110 m for hw = 0.03, 0.04, and 0.05 m, respectively, at the superficial gas velocity UG = 1.19 m/s and liquid load QL/Lw = 0.006 m3/(m·s). For flow-guided sieve tray, the average liquid holdup exhibits a sharp down below z < 0.025 m. However, for common sieve tray, it always decreases with the increase of dispersion height, which is consistent with the previous finding.7,12 This means that the flow-guided holes with gas kinetic energy can promote the liquid flow and decrease the average liquid holdup near the 4485
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Figure 11. The clear liquid height profiles along the x- and ydirections. (a) Area-averaged clear liquid height over y- and zdirections; and (b) area-averaged clear liquid height over x- and zdirections. The bars represent the number of flow-guided holes. The operating conditions are: UG = 1.19 m/s; and QL/Lw = 0.006 m3/(m· s).
Figure 10. The x-direction-component velocity profiles of liquid flow field at different elevations. (a) z = 0.03 m; (b) z = 0.04 m; and (c) z = 0.05 m. The structure and operating conditions are: UG = 1.19 m/s; QL/Lw = 0.006 m3/(m·s); and hw = 0.04 m.
Figure 12. Distribution of liquid holdup along the dispersion height in the z-direction at different weir heights.
tray surface. This result coincides with the industrial practice that flow-guided sieve tray can deal with the high viscous polymer by reducing the possibility of self-polymerization, which often happens on the tray surface. Yet the quantitative validation of simulated results should be addressed in future work.
method. A three-dimensional two-phase flow of gas and liquid in the Eulerian−Eulerian framework was used to model this new type of tray. The liquid- and gas-phase equations were coupled through an interphase momentum transfer term, which was estimated by using the drag coefficient correlation of, and the new correlations for gas holdup for flow-guided sieve tray were obtained on the basis of the experimental measurement. The predictions of clear liquid height and liquid holdup from the simulations showed a trend similar to that of the
5. CONCLUSION The hydraulics and flow patterns of flow-guided sieve tray were predicted by means of computational fluid dynamics (CFD) 4486
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Superscripts
experimental data with varying superficial gas velocity, liquid load, and weir height. In addition, the CFD model has been used to predict the velocity distribution, clear liquid height, and foam height, because the experimental work on trays is proved to be expensive and time-consuming, and few experimental data have been reported to determine the fluid flow patterns inside the tray. It was obtained from the CFD model that the arrangement of flow-guided holes is necessary for promoting the liquid flow and decreasing the clear liquid height at a specific position. The simulation by the CFD model overcomes the limitations associated with the experimental work. This studies show that CFD can be a powerful tool for the design and optimization of this new type of tray.
■
■
average = average B = from Bennett correlations
REFERENCES
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AUTHOR INFORMATION
Corresponding Author
*Tel.: +86 10 64433695. E-mail:
[email protected]. Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS We are thankful for financial support from the Central College Research Fund (JL1101) and the National Nature Science Foundation of China under grant nos. 21121064 and 21076008.
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NOMENCLATURE a1−a5 = parameters in eqs 18−20 Aholes = area of all holes including sieve holes and flow-guided holes (m2) C = constant in eq 19 CD = drag coefficient dG = diameter of gas bubble (m) g = acceleration due to gravity (9.81 m/s2) hgap = height of downcomer clearance (m) hCL = clear liquid height (m) hw = weir height (m) Lw = weir length (m) M = interphase momentum exchange term (N/m3) p = pressure (Pa) QG = gas flow rate across tray (m3/s) QL = liquid flow rate across tray (m3/s) t = time (s) u = velocity vector (m/s) UG = superficial gas velocity (m/s) Vslip = slip velocity between gas and liquid (m/s) x = coordinate (m) y = coordinate (m) z = coordinate (m)
Greek Symbols
α = volume fraction of phase γ = angle of bubble promoter (5.7°) μ = viscosity of phase (Pa s) ρ = density of phase (kg/m3) ξ = coefficients of resistance
Subscripts
fg = short for flow-guided G = referring to gas phase GL = between gas and liquid k = index referring to one of the two phases (liquid or gas) L = referring to liquid phase 4487
dx.doi.org/10.1021/ie402008c | Ind. Eng. Chem. Res. 2014, 53, 4480−4488
Industrial & Engineering Chemistry Research
Article
Part I: Detailed Numerical Simulations. Chem. Eng. Sci. 1999, 54, 2273−2284. (22) Borchers, O.; Busch, C.; Sokolichin, A.; Eigenberger, G. Applicability of the Standard k-ε Turbulence Model to the Dynamic Simulation of Bubble Columns. Part Π: Comparison of Detailed Experiments and Flow Simulations. Chem. Eng. Sci. 1999, 54, 5927− 5935. (23) Colwell, C. J. Clear Liquid Height and Froth Density On Sieve Trays. Ind. Eng. Process Des. Dev. 1981, 20, 298−307. (24) Lewis, W. K. Rectification of Binary Mixtures. Ind. Eng. Chem. 1936, 28, 399−402.
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