Ind. Eng. Chem. Res. 2000, 39, 1369-1380
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CFD Modeling of Mass-Transfer Processes in Randomly Packed Distillation Columns F. H. Yin, C. G. Sun, A. Afacan, K. Nandakumar,* and K. T. Chuang Department of Chemical and Materials Engineering, University of Alberta, Edmonton, Alberta, Canada T6G 2G6
The volume-averaged equations for velocity and concentration fields have been used to simulate the hydrodynamics and mass-transfer processes in randomly packed distillation columns. This approach is regarded as a second-generation computational fluid dynamics (CFD) based model, and a significant departure from the traditional one-dimensional, first-generation models. The model has ability to capture radial and axial variations in flow and mass-transfer conditions. The spatial variation of void fraction has been included to take into account the effect of bed structures. The simulation results have been compared with experimental data reported by Fractionation Research, Inc. (FRI) which performed their tests in a 1.22-m-diameter column with a packed bed height of 3.66 m. For validation, we have used data obtained with 15.9-, 25.4-, and 50.8-mm metal Pall rings at various operating conditions. Good agreement between CFD predictions and published experimental data has been obtained. This is regarded as an encouraging sign that CFD models can play a useful role in studying separation processes. Introduction Packed columns have been widely used in many industrially important separation processes such as distillation, absorption, and stripping where intensive contacting of gas and liquid is required. Compared with the tray columns, packed columns have shown some distinct advantages such as low pressure drop and low liquid hold-up. With more and more high efficiency, modern packings available commercially, it is often justifiable to replace an existing tray column with a packed column since the cost of revamping can be recovered quickly through lower operating expense. Design, scale-up, and performance analysis procedures for packed columns have been traditionally based on macroscopic mass balances, assuming homogeneous conditions along the radial direction. Such simple models lead naturally to design procedures based on the unidirectional variation of concentration in the axial direction and hence to the concepts of HTU (Height of a Transfer Unit) and NTU (Number of Transfer Units). It is important to realize that these concepts rely on observations made at the macroscopic (or equipment) scale, i.e., only inlet and outlet flow rates and concentrations are measured. The standard design equation may be written as
Z)
G KGaepMA
∫yy
z
0
dy ) HTU × NTU y - y*
(1)
By using macroscopic data and assuming the transfer processes (both flow conditions and interfacial mass transfer) to be uniform throughout the packed column, and by using models for the interfacial mass transfer (kL and kG) and the characteristics of the packing medium (area, ae), the performance of the column can be investigated. As a consequence the overall mass* To whom correspondence should be addressed. Phone: 780-492-5810.Fax: 780-492-2881.E-mail: kumar.nandakumar@ ualberta.ca.
transfer coefficient KGae is assumed to be uniform not only over a given cross-sectional area, but also with height of the column. The influence of the flow conditions on mass transfer is captured only indirectly through KGae. If the flow conditions do not scale-up properly with increasing equipment size, then a corresponding error in KGae will occur. Successful design and scale-up of packed columns require a model that captures the basic transport phenomena on the correct length scale. Concepts based on HTU and NTU obtained from inlet and outlet concentrations alone are not adequate for scale-up purposes as both HTU and NTU depend on the flow conditions and the interface transfer taking place on the scale of the packing size and not on the scale of the equipment. Current design and scale-up procedures for packed columns are less reliable because they do not take into account such variations on the scale of the packing. Height Equivalent to a Theoretical Plate (HETP), which is normally used to characterize the mass-transfer efficiency of a packed column, varies very strongly with changes in the type and size of packings, the type of the distributor, the liquid and gas flow conditions, and even concentrations. For example, a 2-3-fold variation in HETP was reported for 25.4-mm packings.1,2 The complex mechanisms that influence HETP have not been modeled in a rigorous way in the first generation models that assume conditions within a column to be homogeneous in the radial direction and neglect dispersion in all directions. Therefore, we propose to use CFD models to simulate and analyze the mass-transfer processes in randomly packed distillation columns. Having the ability to capture radial and axial variations in flow and masstransfer conditions, the models can predict the overall separation efficiency based on the detailed local conditions. The CFD models, however, require the specification of a number of closure models to capture the information lost during the averaging process. These closure models should provide the interaction of trans-
10.1021/ie990539+ CCC: $19.00 © 2000 American Chemical Society Published on Web 03/31/2000
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port processes between the phases on a scale smaller than the volume averaging scale. In this study we assemble a preliminary set of such models from literature that describe (a) the void fraction variation within the bed, (b) the pressure drop of two-phase flows, (c) the volume fraction dispersion in a packed bed, (d) the interface mass-transfer parameters, and (e) the mass dispersion coefficients in a packed bed. Clearly there is room for improving these models in future work. The ultimate objective of experimental and modeling efforts in this area must be to develop a set of closure models that remains truly scale invariant. The simulation results using the preliminary set of closure models have been compared with the data from FRI. Since the experiments have been performed under carefully controlled conditions of initial liquid distribution, it can be assumed that the inlet distribution is uniform, thus permitting axisymmetric (or two-dimensional) simulations. This assumption has been subsequently tested by a limited number of runs of three-dimensional simulation. A comparison between two- and three-dimensional simulations shows that our assumption is reasonable for this set of experiments. The radial variations in flow and concentration fields can be created by void fraction variations in the bed, the tendency of trickling liquid to gather near the wall and varying magnitudes of radial and axial dispersion coefficients. In general, the simulation results for HETP, concentration profiles, and pressure drop across a packed bed agree well with the experimental data. Mathematical Models Fluid Dynamic Equations. The separation efficiency of packed columns depends strongly on the flow conditions in the packings. The detailed flow distributions for two-phase flow in a packed column containing random packings can be determined by solving the volume-averaged fluid dynamic equations as given by
Continuity Equations ∂ (γ F ) + ∇‚(γRFRUR - ΓR∇γR) ) 0 ∂t R R
R ) L, G
(2)
Momentum Equations ∂ (γ F U ) + ∇‚{γR[FRUR X UR - µeR(∇UR + ∂t R R R (∇UR)T)]} ) γR(BR - ∇p) + FR R ) L, G (3) where γ represents the volume fraction of liquid phase or gas phase (for solid phase, γ ) 0), F is the fluid density, µeR the effective viscosity, U the interstitial velocity vector, B the body force (including the gravity and the flow resistance offered by the packing elements), p the pressure, F the interface drag force, Γ the volume fraction dispersion coefficient, and R the phase index. The volume fraction γ obeys
γ L + γG ) 1
(4)
implying that these volume fractions are based on the pore volume. Mass-Transfer Equations. Transport equations for mass fractions YiR in the general form can be written as
N
∂
(γRFRYiR) + ∇‚[γR(FRURYiR - ΓiR∇YiR)] )
∂t
R ) L, G i ) 1, ..., NC
∑ m˘ Rβi β)1,β*R
(5)
where Yi is the mass fraction of the ith component in a particular phase, NC is the number of components in each phase, ΓiR is the effective mass dispersion coefi is the interphase massficient for species i, and m ˘ Rβ transfer rate for the ith component from phase R to phase β. Note that ΓR in eq 2 is for modeling the dispersion of a phase in the porous medium, while ΓiR in eq 5 is for modeling the dispersion of individual species within a phase. Equations 2, 3, and 5 represent, respectively, the mass, momentum, and species conservation equations on a volume-averaged basis. This implies that the variables such as velocity, volume fraction, and concentrations are averaged on a scale larger than packing size, but smaller than equipment size. Hence these equations require closure models to capture the information that is lost in the averaging process on scales smaller than packing size. But they can capture largescale maldistribution in packed beds arising out of porosity variations or poor liquid and gas distributors. In this set of equations, the first term on the left-hand side represents the accumulation term, which is set to zero for steady-state flow simulations. The second term on the left-hand side represents the convection and diffusion phenomena. Closure Models. Equations 2-5 do not form a closed system. The closure models are required for quantities ˘ Rβ, ΓR, ΓiR, (for drag forces, body forces, such as FR, BR, m interface mass transfer, phase dispersion, and species dispersion). In the present study, these models are adopted and developed from existing empirical correlations. Many of these empirical correlations have been developed from experiments conducted on a macroscopic (or equipment) scale, i.e., based only on observations of input/output quantities. Use of these correlations in a local sense is akin to the use of Darcy’s law on a microscale, which is based on macroscopic observations of flow rates vs pressure drop data for packed beds. There is clearly room for further refinement of the required closure models, as more refined experimental data on flow and concentration distributions become available. The clear advantage of the CFD model is in its natural ability to track inhomogenieties within the column, provided the closure models remain valid on the pore scale. Interphase Drag Force F. The interphase drag force between the gas and the liquid phase is modeled by
FG ) CGL(UL - UG)
(6)
FL ) -FG
(7)
where CGL is the interphase drag coefficient. In the model equations, it is prescribed in such a way that it incorporates experimentally measured correlations for pressure drops in counter current flow through packed columns. There are several correlations available in the literature that can be used to predict the pressure drop for two-phase flows through a random packed column. These include the Leva correlation3,4 and the Robbins
Ind. Eng. Chem. Res., Vol. 39, No. 5, 2000 1371
correlation.5 The Leva correlation states that the total pressure drop is proportional to the square of the gas velocity. This is reasonable in the low range of operating conditions. However, when the liquid load is high, the interaction between the two phases will be larger. Robbins modified the Leva equation and proposed the following correlation to predict the pressure drop for a packed column5
(
Lf ∆p ) C1Gf2 × 10C2Lf + 0.774 z 20000
)
0.1
(C1Gf2 ×
|UL - UG| )
x(UL - UG)2 + (VL - VG)2 + (WL - WG)2
for three-dimensional flow. Here U, V, and W are axial, radial, and angular components of the velocity, respectively. For two-dimensional flow, W ) 0. Body Force B. In addition to the gravitational body force, the increased resistance to flow due to the presence of the packing elements is also treated as a body force and can be modeled as
10C2Lf)4 (8) where C1 ) 4.002 × 10-2 , and C2 ) 1.99 × 10-2 . Gf and Lf are the gas and liquid loading factors and can be calculated as
{
[ ][ ] [ ][ ]
1.2 FG Gf ) 1.2 G FG G
{
0.5
0.5
Fpd 65.62 Fpd 65.62
[ ][ ] [ ][ ]
1000 Fpd FL 65.62 Lf ) 1000 65.62 L FL Fpd
× 100.0187FG
UR ) -RR-1‚∇p
for p > 1 atm (9)
0.5
L
0.5
µL0.1 for Fpd g 15 (10)
µL0.1 for Fpd < 15
where RR-1 is the inverse tensor of RR, and ∇p is the gradient of pressure. Hence, it is possible to estimate RR from the measured pressure drop. For the gas phase, the flow resistance offered by the solid packing elements can be modeled by utilizing the dry pressure drop part of the Robbins equation (that is, eq 12)
{
RG )
[ ][ ] [ ][ ]
C12FG2γG2
C12FG2γG2
1.2 Fpd |UG|I for p e 1 atm FG 65.12 1.2 Fpd [100.0374FG]|UG|I for p > 1 atm FG 65.12
∇p + (f1 + f2|U|)U ) 0
( )
) C1Gf2
(12)
and
( )
wet
(
)
0.1
2
C2Lf 4
(C1Gf × 10
)
(13)
Here (∆p/z)wet allows us to determine the interphase drag coefficient, that is,
CGL )
f2 )
150(1 - )2µ deq22 1.75(1 - )F deq
(20)
(21)
where deq is the equivalent diameter of the packing element, defined as
) C1(10C2Lf - 1)Gf2 + Lf 0.774 20000
(19)
(11)
with
dry
(18)
where I is the second-order unit tensor. |U| is the absolute magnitude of the interstitial velocity. For the liquid phase, the well-known Ergun equation is suitable to model the flow resistance term.7 For the maldistributed flow, the Ergun equation must be written in the vector form
f1 )
(∆pz)
(17)
where
∆p ∆p ∆p ) + z z dry z wet
( )
(16)
where R is the resistance tensor, representing the flow resistance offered by the porous medium to the fluids. From Darcy’s theory,
for p e 1 atm
Equation 8 can be considered to consist of two parts. The first part, C1Gf210C2Lf , allows for the estimation of pressure drop through the packings in the preloading region, while the second part, 0.774(Lf/20000)0.1(C1Gf210C2Lf)4, takes into account the increase in the pressure drop due to the stronger interaction between gas and liquid phases in the loading regime. The presence of liquid in the packings will reduce the flow space of the gas phase, thus leading to a higher pressure drop. The parameter Fpd, also called the packing factor, represents the effect of packing size and shape on the pressure drop and has been documented for almost all the commonly used random packings.2 The total pressure drop can be expressed as5,6
∆p z
BR ) FRg + RR‚UR
0.5
0.5
(15)
(∆pz)
wet
|UL - UG|
where |UL - UG| is the slip velocity, defined as
(14)
deq )
6(1 - ) ap
(22)
The flow resistance RL can thus be calculated as
RL ) (f1 + f2|UL|)I
(23)
Dispersion Coefficient Γ for Volume Fraction. Liquid spreading in packed columns is due in part to spatial variation in flow resistance. This implies that if a certain flow channel, formed within a packed bed, offers less resistance to flow than other channels of
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equal cross sectional area, liquid will tend to move toward this channel of lower resistance, thus causing a higher liquid hold up (or volume fraction) in this channel. Spatial variation of flow resistance is generated mainly from two sources: the spatial variation of void fraction, and the consequential nonuniform liquid distribution. It is assumed that the dispersion coefficient for volume fraction is linearly proportional to the adverse gradient of the axial flow resistance
Γ ) -Kc∇Rz
2
(25)
The first term on the right side of this equation represents the effect of the bed structure (i.e., the spatial void fraction variation) on liquid spreading, and the second term states the fact that even for homogeneous packed beds the liquid spreading can occur if the initial distribution of liquid is nonuniform. Another important cause for the liquid spreading is the unstable turbulent flow. Evidently, the turbulent motion of fluids will enhance the rate of liquid spreading. To account for this effect, an additional term, ΓT, is introduced to the right side of eq 25. ΓT is the turbulent dispersion coefficient and can be calculated on the basis of the eddy diffusivity hypothesis as
ΓT )
µT σt
(27)
m ˘ ALG ) kGaeMA(yIA - yA)
(28)
The interfacial concentrations xIA and yIA are in equilibrium
RxIA
yIA )
(26)
where µT is the turbulent viscosity of the liquid phase and σt is the turbulent Prandtl number, with a value of 0.01. The spreading of gas is a much faster process than that of liquid, as is observed experimentally by Kouri and Sohlo,8,9 Stikkelman and Wesselingh,10 Stikkelman et al.,11Stoter et al.12 Below the loading point, the gas flow pattern and its spreading depend mainly upon the liquid flow behavior. Compared with the liquid phase, the turbulence intensity is much higher in the gas phase due to its lower viscosity and density. Thus, it can be assumed that the spreading for the gas phase is dominated by the turbulent dispersion. The turbulent dispersion coefficient of gas phase is defined in the same manner as eq 26, but µT represents the gas turbulent viscosity. Interphase Mass Transfer. For binary distillation, both the liquid and vapor phases contain two components, A and B, with A being the more volatile component. If phase equilibrium has not been reached, then mass transfer between liquid and vapor phases will occur. According to the two-film theory, the masstransfer rate can be calculated by one of the following two equations
(29)
1 + (R - 1)xIA
(24)
where Kc is a proportionality constant and can be determined by fitting experimental data. The resistance to liquid flow offered by the packing along the direction of main flow is given by eq 19. Normally the inertial component provides the major resistance to flow in packed columns under normal operating conditions. Thus, taking the inertial term of the Ergun equation as Rz and differentiating it result in
(1 - )FU FU ∇ - 3.5Kc ∇U Γ ) 1.75Kc 2 deq deq
m ˘ ALG ) kLaeMA(xA - xIA)
Combining eqs 27, 28, and 29 results in
[
(R - 1)yA - R R-1 (m ˘ ALG)2 + 2 kLaeMA kLkG(aeMA) (R - 1)xA + 1 A m ˘ LG + {RxA - [(R - 1)xA + 1]yA} ) 0 kLaeMA (30)
]
that is, the local mass-transfer rate m ˘ ALG is related to the local concentrations xA, yA, individual mass-transfer coefficients kL, kG, and the interfacial mass-transfer area ae. The Onda’s formulas13,14 were used to calculate kL and kG
kL ) 0.0051
( )( )( ) ( ) ( ) µLg FL
1/3
kG ) kp(apDG)
L awµL
G apµG
2/3
7/10
µL FLDL
µG F G DG
-1/2
(apdp)2/5 (31)
1/3
(apdp)-2.0 (32)
where kp is the model parameter. According to Onda et al.,13,14 the value of kp depends on the size of packing. For the ring size greater than 15 mm, kp ) 5.23, otherwise, kp ) 2.0. The wetted surface area aw is calculated as
[
()( )
σc aw ) 1 - exp -1.45 ap σ
3/4
L apµL
1/10
( ) L2ap
-1/20
×
FL2g
( ) L2 FLσap
1/5
]
(33)
where σc is the critical surface tension of the packing material. For steel σc ) 75.0 dyn/cm2. The effective vapor-liquid interfacial mass-transfer area is assumed to be equal to the wetted area in the Onda correlations, that is, ae ) aw. Dispersion Coefficient Γir for Mass Transfer. Dispersion tends to reduce the mass-transfer driving force, thus having an adverse effect on the separation efficiency. Normally the dispersion in a packed bed is measured by the tracer method and correlated in terms of Peclet number. For the ring type of packings, the axial dispersion in the liquid and gas phases can be modeled as15,16
PeL ) 5.337 × 10-4
( ) ( ) () d eL µL
PeG )
0.472
deG µG
uG 22uL + uG
0.293
de D
-0.867
(34)
(35)
Ind. Eng. Chem. Res., Vol. 39, No. 5, 2000 1373
where de is the equivalent diameter of the packed bed and defined as
de )
4 ap
(36)
The dispersion coefficient for liquid and gas phases can then be calculated as
ΓL )
FLuLde PeL
(37)
ΓG )
FGuGdp PeG
(38)
Void Fraction Distribution. One of the important characteristics of a randomly packed bed is that the average void fraction approximates to 1.0 near the column wall. This is thought to contribute to the collection of liquid at the wall. The literature abounds with studies on the void fraction distribution for packings of regular shapes, such as equal or unequal sized spheres17-20 or solid cylindrical particles.21 However, there is almost no theoretical and experimental studies for packed beds containing commercially important packings such as Pall rings available in the literature. In view of this, our research group has measured the void fraction variation for metal Pall rings using a γ-ray technique and found that the following model best represents the radial void fraction profile:
[ ( ) ]}
{
) 1 - (1 - b) 1 - exp -2
R-r dp
2
(39)
Determination of Mass-Transfer Efficiency. There are two conventional models for packed column masstransfer rate analysis. The HETP model, which treats a packed column as a series of theoretical stages, is based on the equilibrium-stage assumption. Although the HETP concept is widely used for column design, it does not model in detail the fundamental mechanism involved in the mass transfer occurring in a packed column. The HTU model, on the other hand, is more physically sound because it does model the masstransfer rate between the liquid and vapor phases. On the basis of the HTU model, the packing height can be calculated as, Z ) HTU × NTU. From the simulation, detailed information about the flow distribution and mass concentration distribution within a packed column is available; therefore, the local HTU and NTU can be calculated as follows
NOGl )
∫yy
dy yA* - yA
AZ+∆Z
AZ
(40)
and
HOGl )
∆Z NOGl
(41)
where HOG and NOG are the height of an overall gasphase transfer unit and the number of overall gas-phase transfer units, respectively.
Figure 1. A sketch of the packed column and the simulation domain.
The local HETP at a point can be determined from local HOG
( )
ln ml
Gl Ll
HETPl ) HOGl Gl ml - 1 Ll
(42)
where ml is local slope of the equilibrium line, defined as
ml )
R [1 + (R - 1)xA]2
(43)
In the above discussion, the global flow rates are related to the local quantities by
L ) FLγL|UL|
(44)
G ) FGγG|UG|
(45)
Boundary Conditions The flow boundaries of the packed column are at the top and bottom as indicated in Figure 1. The liquid is introduced at the top of the column via a liquid distributor while the vapor is fed into the column at the bottom. Hence, the inlet conditions for both the liquid and gas are known from the process operating conditions. In the simulation, the following boundary conditions are specified. (1) At the top of the column, the “inlet” boundary is specified. At this boundary the appropriate values for velocity components, volume fractions, and component concentrations must be specified for both phases. (2) At the column wall, a “nonslip” boundary condition is specified for the liquid and gas velocities. For other quantities, no flux boundary is specified. (3) At the bottom of the column, the “mass flow boundary” is used to ensure the global mass conservation. (4) At the column axis all variables are mathematically symmetric and no diffusion occurs across this
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Table 1. Physical Properties of System Studied system C6/n-C7
P (kPa) 33.3 165.5
FL (kg/m3) 713.4 636.7
FG (kg/m3) 1.162 4.907
µL (Pas) 10-3
0.44 × 0.23 × 10-3
µG (Pas) 10-6
7.2 × 8.5 × 10-6
DL (m2/s)
DG (m2/s)
10-9
10-6
2.7 × 6.2 × 10-9
7.6 × 2.1 × 10-6
σ (N/m) 10-3
20 × 12 × 10-3
R 1.9 1.6
Table 2. Characteristics of Metal Pall Rings packing metal Pall ring
nominal specific area void packing size (mm) (m2/m3) fraction factor (1/m) 15.9 25.4 50.8
341 207 102
0.933 0.940 0.951
262 174 79
Figure 2. A comparison of predicted and measured HETP at an operating pressure of 165.5 kPa.
Figure 4. A comparison of predicted and measured composition of C6 along the packed bed height at an operating pressure of 165.5 kPa for the 50.8-mm Pall rings: (b) experiment; (s) prediction.
Figure 3. A comparison of HETPs between experimental data and predictions from CFD models and traditional models at an operating pressure of 165.5 kPa, 25.4 mm Pall rings.
boundary. Therefore, axisymmetry boundary condition is used at r ) 0. The specific form of the equations are described in full in the CFX manual22 and hence they are not repeated here.
of a packed column. In these regions, sufficiently fine grids should be utilized to give a more accurate prediction. In this study, the geometric progression (GP) is used to generate grids in the radial direction with the smallest cell size of about one-eighth packing diameter adjacent to the column wall. In the axial direction the symmetric geometric progression (SYM GP) is used to generate grids with the smallest cell size of about onehalf packing diameter in the top and the bottom of the packed column.
Numerical Methodology
Simulation Results and Discussion
The governing equations are solved numerically by means of a finite volume method, using the CFD package-CFX 4.2 from AEA technology.22 The choice of a CFD package should not be critical. The use of a CFD package makes it easier to treat the complicated geometry and boundary conditions. The well-known SIMPLEC algorithm23 is employed to solve the pressurevelocity coupling in the momentum equations. The most significant variations in the flow field are expected to appear in the top, bottom, and wall regions
The CFD simulation results are compared with the data gathered by Fractionation Research, Inc. (FRI).24 (Additional data were obtained through private communication with the authors.) FRI performed the distillation test using cyclohexane/n-heptane (C6/C7) system under total reflux conditions and at two operating pressures (33.3 and 165.5 kPa). The physical properties of the system at these conditions are listed in Table 1. The test column had a diameter of 1.22 m and a packed bed height of 3.66 m. The packings used were 15.9-,
Ind. Eng. Chem. Res., Vol. 39, No. 5, 2000 1375
Figure 5. The predicted liquid flow distribution in a randomly packed column at various heights. Operating pressure is 165.5 kPa; packing, 25.4-mm Pall rings; F-factor, 1.18 (m/s)(kg/m3)0.5.
25.4-, and 50.8-mm metal Pall rings. Their characteristics can be found in Table 2. HETP is one of the most often used measures to characterize the separation efficiency of a packed column in industry. The success of any hydrodynamics and mass-transfer model for packed columns relies on its ability to predict HETP accurately. Figure 2 shows comparisons between the measured HETPs and the average HETPs determined from the simulation at an operating pressure of 165.5 kPa and a packed bed height of 3.66 m for the 15.9-, 25.4-, and 50.8-mm Pall rings, respectively. To calculate the vapor phase mass-transfer coefficient, the value of kp in eq 32 was taken as 5.23 for 25.4-mm Pall rings and 50.8-mm Pall rings, and 2.0 for 15.9-mm Pall rings. In developing their masstransfer models, Onda et al.13,14 employed experimental data for packings with sizes greater than 25.4 mm and less than 15 mm. There is no data for packings with sizes in the range of 15-25.4 mm. Since 15.9 mm is closer to 15 mm than to 25.4 mm, it is more reasonable to use 2.0 for the value of kp instead of 5.23. Figure 2 is a plot of HETP as a function of the F-factor, which is defined as uGxFG, where uG is the superficial vapor velocity over the column cross section. From this figure, it can be clearly seen that the simulation can predict the separation efficiency quite well over the range of operating conditions for all three packing sizes. It is obvious that our simulation can track the HETP variation with the loading, which is one of the most important concerns for any mass-transfer
modeling effort. The more detailed comparison between experimental data and simulation results is given in Figure 4 in terms of composition profile for three different operating conditions: F-factor ) 0.758, 1.02, 1.52 (m/s)(kg/m3)0.5. The composition on the y axis is plotted as ln(x/(1 - x)) to show the composition ratio variation with the packed bed height. This figure shows the data for the 50.8-mm Pall rings. The composition profiles for the 15.4-mm Pall rings and 25.4-mm Pall rings, although not shown here, are very similar to the profile of 50.8-mm Pall rings. In Figure 4, each predicted composition x is an average value based on the detailed flow distribution over the column cross section at a fixed axial position. The good agreement between simulation and experimental results indicates that the closure models that were used can capture the flow hydrodynamics and mass-transfer processes accurately. While, HETP is a single scalar measure at an operating condition, the composition profiles show the most significant internal variations in the column. It is encouraging that both the quantities agree well with the FRI data. Figure 3 shows a comparison between the predicted HETPs from CFD simulations and traditional models for an operating pressure of 165.5 kPa. In the traditional model, the uniform radial distributions are assumed for both the liquid and vapor phases. It can be seen that the predicted HETPs from traditional models are lower than the measured HETPs for all the F-factors studied. Therefore, the flow maldistribution must be considered
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Figure 6. The profiles predicted by CFD at a column pressure of 165.5 kPa for a packed bed using metal Pall rings of size 25.4 mm at a F-factor of 0.795 (m/s)(kg/m3)0.5: (a) volume fraction in liquid phase, (b) concentration of C6 in liquid phase, (c) liquid velocity vector, and (d) gas velocity vector.
to obtain a better prediction of mass-transfer efficiency. A similar comparison for an operating pressure of 33.3 kPa is shown later in Figure 8. The other significant variation in the column, for which experimental data is often lacking is the liquid velocity along the radial and axial directions. Figure 5 shows the predicted flow distribution of liquid phase in a packed column. As can be seen, the liquid flow rate in the wall region is relatively high compared to that in the bulk region of the packed bed. The formation of wall flow is one of the important characteristics associated with randomly packed columns. As mentioned previously, the void fraction reaches its highest value of unity at the column wall due to the effect of the column wall. The higher void fraction in the near wall region offers lower flow resistance, thereby causing the liquid to flow preferentially along the column wall. This
kind of flow distribution makes the usual assumption of one-dimensional, plug flow in a packed bed to be invalid and one must resort to the CFD based models to predict the flow distribution. After the liquid is uniformly introduced at the top of the packed column, the flow will redistribute itself along the packed bed height due to the liquid spreading ability of the packing elements. The detailed flow pattern development along the packed bed height is also shown in Figure 5. It can been seen that the liquid continues to accumulate on the column wall as it flows downward. At the top part of the packed bed, the build-up of liquid on the wall is rapid, after about 2.5 m, the wall flow still increases with the height, but at a relative slow speed. The CFD simulation is capable of predicting detailed spatial variation of volume fraction, concentration, and velocities of each phases. Experimental data on such a
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Figure 7. A comparison of pressure drop between experimental data and simulation results at an operating pressure of 165.5 kPa.
detailed scale are not yet available, although modern experimental techniques (based on magnetic resonance imaging techniques, X-ray tomography, etc.) might produce such data soon. In Figure 6 we present a sample plot of spatial variations in flow quantities for a typical simulation, at an F-factor of 0.795, a column pressure of 165.6 kPa, and a packed bed height of 3.66 m using 25.4-mm metal Pall rings as packing. Figure 6a shows the shaded (and line) contours of liquid volume fraction, while Figure 6b shows the concentration of C6 in the liquid phase. Although the inlet condition is uniform in both cases, note that a strong variation near the wall zone develops as the liquid flows down the column. Figure 6c shows the liquid-phase velocity, which is uniform at the top, but develops a strong wall flow as it flows down the column. This is a qualitative view of the quantitative information presented in Figure 5. Figure 6d shows the gas velocity profile, which is rather uniform all over the packed bed. The pressure drop across the packed bed is another important design parameter. Figure 7 shows a comparison between the measured pressure drop and calculated pressure drop for three different sized Pall rings. Over the whole range of operating conditions, the simulation can capture the pressure drop variation with the Ffactor pretty well, that is, the pressure drops increase with the increase in the F-factor for all three sized Pall rings. The increase in the F-factor means the increase of gas flow rate and liquid flow rate simultaneously. In addition to the flow rates, the packing size also plays an important role in determining the pressure drop across the bed. The smaller the packing size, the lower the void fraction, and the higher the pressure drop. This trend is correctly captured in our simulation as shown in Figure 7. Depending on the separation requirements and system characteristics, packed distillation columns may be operated under pressure or vacuum conditions. To study the effect of operating pressure on the separation process, FRI ran the column at two different pressures, that is, 165.5 and 33.5 kPa for the C6/C7 system. In both of these cases, the packed bed height was kept the same (3.66 m). Figure 8 shows comparisons between predicted and measured HETPs at an operating pressure of 33.3 kPa for the 25.4-mm Pall rings. For this case the agreement between CFD simulation results and experimental data is again reasonably good. Comparing the results presented in Figures 2 and 8, one can see that the improvement of separation efficiency with an in-
Figure 8. A comparison of HETPs between experimental data and predictions from CFD models and traditional models at an operating pressure of 33.3 kPa, 25.4-mm Pall rings.
Figure 9. Predicted HETP variation along the packed bed height at an operating pressure of 165.5 kPa, F-factor ) 1.02 (m/s)(kg/ m3)0.5, 50.8-mm Pall rings.
crease in pressure for the C6/C7 system has been correctly captured by CFD simulations. This favorable effect of operating pressure on separation efficiency for the studied system has also been reported by Zuiderweg and Nutter,16 Gualito et al.25 and Wagner et al.26 and can be explained by the improvement of the relevant mass-transfer parameters with the pressure. As shown in Table 1, the decrease of liquid surface tension with pressure means better wetting of the packing, thus producing more mass-transfer area and then better separation efficiency. On the other hand, both the increase in liquid diffusion coefficient and decrease in liquid viscosity tend to increase the mass-transfer coefficient in liquid phase. Figure 9 shows a variation of predicted HETP along the packed bed height. A gradual increase in HETP occurs within a packed bed height of about 5 m due to the build up of liquid wall flow. Beyond that bed height, the HETP continues to increase, but at a much slower rate. The build up of liquid wall flow has an adverse effect on separation efficiency and should be controlled. In practice, the amount of liquid accumulated on the column wall can be reduced by using liquid redistributors after a certain packed bed height. For the case studied here, 6 m may be an appropriate height to install a liquid re-redistributor. The above results were generated from the twodimensional, axisymmetric simulations. This was based
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Figure 10. A comparison of liquid velocity profiles from two- and three-dimensional simulations. Operating pressure is 165.5 kPa; packing, 25.4-mm Pall rings; F-factor, 1.18 (m/s)(kg/m3)0.5; (s) 2D simulation; (‚‚‚) 3D simulation.
on the assumption that the liquid distributor used in the FRI experiments could distribute the liquid uniformly over the top of the packings. To test the validity of this assumption, three-dimensional simulation was also carried out with the 25.4-mm Pall rings. In FRI tests, a tubed drip pan (TDP) type distributor was used. This distributor had 121 drip points. In the threedimensional simulation, the liquid initial inlet condition was specified in such a way that it exactly simulated the number of drip points, that is, there was liquid flow under the drip points, and there was no liquid flow between the drip points. In the two-dimensional simulation, however, the liquid was assumed to be distributed uniformly across the column cross section. A comparison between three-dimensional and two-dimension simulation of the liquid distribution is presented in Figure 10. As can be expected, for the three-dimensional simulation, the liquid velocity profile is highly irregular near the top of the column. The local high values in the velocity profile correspond to the drip points of the liquid distributor. As the liquid flows down the column, these local irregularities smooth out rapidly. At the bed depth of 0.4 m from the top of the packing, the two velocity profiles generated from two-dimensional simulation and three-dimensional simulation are very similar. The difference between these two simulations is even smaller
after that development length. For a packed bed of 3.66 m high, that small developing length represents only about 10%. However, the time consumed to run a threedimensional simulation is unduly long compared with that needed to run a two-dimensional simulation. The two-dimensional simulation is thus a good comprise when considering the computational time needed and reasonably good predictions obtained as presented in the above graphs. We find that CFD is a relatively new and powerful tool to assess the performance of packed columns. Although the use of simple closure models based on macroscopic empirical equations for pressure drop, mass-transfer coefficients, effective areas, etc. has allowed us to make inroads in evaluating the CFD based models, there is clearly room for further improvement in the closure models. This study also points out to the experimentalists, the type of detailed predictions that are possible with the CFD models, thus suggesting the need to refine experiments to measure detailed spatial distributions of flow and concentration fields. Conclusions The volume-averaged momentum and mass-transfer equations have been solved to obtain the detailed
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velocity and concentration field within a randomly packed column. The effect of bed structure (void fraction variation) on the flow distribution has been included in the model. Also, the model has taken account of the dispersion effect in both the liquid and vapor phases. It has been shown through comparison with the experimental data over a wide range of operating conditions that this model can predict the mass-transfer efficiency quite well. Through simulation, the effects of packing size, and the operating pressure have been studied. The increase in HETP with the packing size has been correctly captured in simulation. For the cyclohexane/n-heptane system, the experimentally observed favorable effect of the operating pressure on the separation efficiency has also been confirmed in this study. Furthermore, this approach can also give reasonably good prediction of pressure drop across the packed bed for different packing sizes. Three-dimensional simulation results show that it is reasonable to assume the uniform liquid inlet distribution when the distributor drip point density is greater than 100. Acknowledgment The authors are grateful to the Environmental Sciences and Technology Alliance Canada (ESTAC) and the Natural Science and Engineering Research Council of Canada (NSERC) for funding the project. Authors are also grateful to Dr. Ahmad Shariat of FRI, for making available the public domain data on large-scale comprehensive studies of pressure drop and mass transfer. Nomenclature A ) component A ae ) effective interfacial area per unit volume, m-1 ap ) total surface area of packings per unit volume, m-1 aw ) wetted surface area per unit volume, m-1 B ) body force vector, N m-3 B ) component B C ) interphase drag coefficient, kg s-1 m-3 C1, C2 ) constants in eq 8 D ) diffusivity, m s-2 deq ) equivalent diameter of the packing, m de ) equivalent diameter of the packed column, m dp ) size of packing, m F ) inter-phase drag force vector, N m-3 Fpd ) packing factor, m-1 F-factor ) defined as uGxFG, (m/s)(kg/m3)0.5 G ) flow rate of gas per unit cross-sectional area, kg m-2 s-1 Gf ) gas loading factor, kg m-2 s-1 g ) acceleration due to gravity, m s-2 HETP ) height equivalent to a theoretical plate, m HOG ) height of an overall gas-phase transfer unit, m HG ) height of a gas-phase transfer unit, m HTU ) height of an overall gas-phase transfer unit, m I ) second-order unit tensor Kc ) parameter in eq 24 KG ) overall mass-transfer coefficient, kmol s-1 m-2 Pa-1 kG ) gas-phase mass-transfer coefficient, kmol s-1 m-2 kL ) liquid-phase mass-transfer coefficient, kmol s-1 m-2 kp ) model parameter in eq 32 L ) flow rate of liquid per unit cross-sectional area, kg m-2 s-1 Lf ) liquid loading factor, kg m-2 s-1 M ) molecular weight, kg kmol-1
m ) slope of equilibrium line m ˘ ) mass-transfer rate, kg s-1 m-3 NC ) number of components in each phase NG ) number of gas-phase transfer units NOG ) number of overall gas-phase transfer units NTU ) number of transfer units p ) pressure, Pa ∆p ) pressure drop, Pa Pe ) Peclet number R ) resistance tensor R ) radius of the column, m r ) radial coordinate, m t ) time, s U ) interstitial velocity vector, m s-1 U ) interstitial axial velocity, m s-1 u ) superficial velocity, m s-1 V ) interstitial radial velocity, m s-1 W ) interstitial angular velocity, m s-1 x ) mole fraction of a component in liquid phase xI ) interfacial mole fraction of a component in liquid phase Y ) mass fraction of a component in vapor phase or liquid phase y ) mole fraction of a component in vapor phase y* ) equilibrium mole fraction of a component in vapor phase yI ) interfacial mole fraction of a component in vapor phase Z ) packed bed height, m z ) axial coordinate, m Greek Symbols R ) relative volatility in eq 29, phase index in eqs 2 to 5 β ) phase index in eq 5 ) void fraction Γ ) dispersion coefficient for volume fraction, kg m-1 s-1 Γi ) dispersion coefficient for mass fraction, kg m-1 s-1 γ ) volume fraction σ ) surface tension, N m-1 σc ) critical surface tension of packing materials, N m-1 σt ) Prandtl number µ ) viscosity, kg m-1 s-1 F ) density, kg m-3 Subscripts 0 ) inlet A ) component A av ) average B ) component B b ) bulk region of packed bed G ) gas or vapor phase L ) liquid phase l ) local T ) turbulent flow z ) outlet
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Received for review July 22, 1999 Revised manuscript received December 21, 1999 Accepted January 11, 2000 IE990539+
