Ind. Eng. Chem. Res. 1992,31, 901-908
is attracted to that nodal point and only the other end can depart from the separatrix line to a significant distance. (This other end not only can but will depart because a residue line joining two nodal points is not a separatrix.) In order to depart from the crossed separatrix significantly in both directions, forming of such a pattern is necessary that the crossed separatrix join two saddle points as in Figure 10. The existence of such a pattern is not theoretically excluded, and some candidate combinations are presented in this article. However, occurrence of such a pattern in nature must be at least rather rare (if anyhow possible), and therefore the working assumption of Doherty and Caldarola (1985) based on the rareness of such cases is well established. Nomenclature A to G and S = singular pointa in the composition triangle t = time or any arbitrarily selected parameter T = temperature of phase equilibrium V = scalar potential V , = liquid volume parameter in the Wilson model x = liquid composition (mole fraction) y = vapor composition (mole fraction) Indices
b = bubble point d = dew point i and j = components n = stage number Abbreviations SDR = simple distillation region
Literature Cited Bushmakin, I. N.; Kish, I. N. 0. razdelyayusshih liniyah destillyatsiyi i rektifikatsiyi troinyh sistem. Zh. W k l . Khim. 1957,30,561-567. Chinikamala, A.; et al. Vapor-Liquid Equilibria of Binary Systems Containing Selected Hydrocarbons with Perfluorobenzene. J . Chem. Eng. Data 1973,18,322-325. Doherty, M. F.; Perkins, J. D. On the Dynamics of Distillation Processes-I. The simple distillation of multicomponent non-reacting homogeneous liquid mixtures. Chem. Eng. Sci. 1978,33, 281-301.
901
Doherty, M. F.; Perkins, J. D. On the Dynamics of Distillation Processes-III. The topological structure of ternary residue curve maps. Chem. Eng. Sci. 1979,34,1401-1414. Doherty, M. F.; Caldarola, G. A. Design and Synthesis of Homogeneous Azeotropic Distillations. 3. The Sequencing of Columns for Azeotropic and Extractive Distillations. Znd. Eng. Chem. Fundam. 1986,24,474-485. Ewell, R. H.; Welch, L. M. Rectification in Ternary Systems Containing Binary Azeotropes. Znd. Eng. Chem. 1945,37,1224-1231. Foucher, E. R.;et al. Automatic Screening of Entrainers for Homogeneous Azeotropic Distillation. AZChE Annual Meeting, Chicago, Nov 11-16; AIChE: New York, 1990, paper 215c. Gmehling, J.; Onken, U. Vapor-Liquid Equilibrium Data Collection; DECHEMA: Frankfort, 1982. Horsley, L. H. Azeotropic Data-ZZfi American Chemical Society: Washington, DC, 1973. Kogan, V. B. Azeotropnaya i ekstraktivnuya rektifikatsiya; Khimiya: Leningrad, 1971. Laroche, L.; et al. Selecting the Right Entrainer for Homogeneous Azeotropic Distillation. AZChE Annual Meeting, Chicago, Nov 11-16 AIChE: New York, 1990;paper 215d. Lloyd, B.; Wyatt, P. A. The Vapour Pressures of Nitric Acid Solutions. Part I. New Azeotropes in the Water-Dinitrogen Pentoxide System. J . Chem. SOC.1955,2248-2252. Matauyama, H.; Nishimura, H. Topological and Thermodynamic Classification of Ternary Vapor-Liquid Equilibria. J. Chem. Eng. Jpn. 1977,10,181-187. Swietoslawski,W. Azeotropy and Polyazeotropy; Pergamon Press: New York, 1963. Tamir, A.; Wisniak, J. Correlation and Prediction of Boiling Temperatures and Azeotropic Conditions in Multicomponent Systems. Chem. Eng. Sci. 1978,33,657-672. Van Dongen, D. B.; Doherty, M. F. On the Dynamics of Distillation Processes V. The Topology of the Boiling Temperature Surface and ita Relation to Azeotropic Distillation. Chem. Eng. Sci. 1984, 39,883-892. Van Dongen, D. B.; Doherty, M. F. Design and Synthesis of Homogeneous Azeotropic Distillations. 1. Problem Formulations for a Single Column. Znd. Eng. Chem. h n d a m . 1985,24,451-463. Wilson, G. M. Vapor-Liquid Equilibria. XI. A New Expression for the excess free energy of mixing. J. Am. Chem. SOC.1964,86, 127-130. Yuan,K. S.; Lu, B. C.-Y.Vapor-Liquid Equilibria. Part I. J. Chem. Eng. Data 1963,8,549-553.
Received for review May 1, 1991 Revised manuscript received September 4, 1991 Accepted November 14,1991
Modeling Zero- Gravity Distillation Eduardo
A.Rarnirez-GonzBlez*
Departamento de Zngenieda Qdmica, Centro de Znuestigacidn e n Quimica Aplicada, Apartado Postal 379, Saltillo, Coahuila 25100, Mexico
Carlos M a r t i n e z a n d J e s h Alvarez Departamento de Zngenierh de Procesps e Hidrbulica, Universidad Autdnoma Metropolitana- Unidad Zztapalapa, Avenida Michoacdn y Purisima, Colonia Vicentina, Apartado Postal 55-534, Mixico, D.F. 09340, Mexico
Zero-gravity distillation (ZGD) is a recently reported operation, where capillarity is the basic driving mechanism. The operation name is due to the fact that gravity forces do not act on the separation process. Previous laboratory-scale experimental work has demonstrated operation feasability. In this work, ZGD is conceptualized in terms of first principles. The resulting model consists of two partial differential equations for liquid and vapor compositions coupled to algebraic equations for gas-liquid interface equilibrium, mass flux,and bubble-point conditions. The model exhibits the interplay of capillarity, mass transfer, and phase equilibrium. With a standard finite-element technique, the partial differential equations are discretized, and the resulting algebraic system is solved with a standard method. Good agreement with reported experimental data is obtained. 1. Introduction Because of ita impact on plant energy requirements, distillation is a current subject of research along three main
directions: (i) energy integration schemes, (ii) improved design of distillation apparatus, and (iii) new ways of carrying out separations. Integration schemes (Andreco-
0888-588519212631-0901$03.00/0 0 1992 American Chemical Society
902 Ind. Eng. Chem. Res., Vol. 31, No. 3, 1992
vich and Westerberg, 1985) offer important improvements but bring more difficult operation and control problems (Fidkowski and Krolikowski, 1986, Morari and Faith, 1980; Waller et al., 1988). It has been reported that improved equipment design (Haselden, 1977; Seader, 1980) reduces energy consumption. Regarding new separation techniques in the past decade, promising alternatives have been brought up. Imperial Chemical Industries (O’Sullivan, 1983) reported use of a spinning squat doughnut-shaped drum with conventional packing material to carry out distillation under high-gravity conditions. Membranes (Darton, 1987),as well as porous and packed systems, have been considered as alternatives that may substitute for conventional distillation towers. Recently, for azeotropic mixtures, pervaporation by means of a nonporous membrane (Redman, 1990) has been posed as a separation operation that offers major advantages over conventional distillation with regard to energy consumption. It has been indicated that, at low concentrations,that operation should not be used. Seok and Hwang (1985) claimed a new unit operation consisting of a horizontal column that operates under the capillarity principles of a heat pipe. Considering that gravity forces do not act on the liquid mixture separations, that was denominated as “zero-gravitydistillation” (ZGD). Because only a fraction of the column length necessitated by conventional distillation is required, ZGD is also referred to as microdistillation. Experimental resulta obtained by Seok and Hwang (1985) in a horizontal ZGD device with ethanol-water and methanol-water mixtures show the feasibility of the operation. Separations were achieved with ‘/&h of the equivalent length required by a vertical packed distillation tower. Modeling was made by posing an analogy with a conventional packed column where a constant equivalent mass transfer coefficient was fitted. By doing so, basic capillarity driving force parameters were lumped into an equivalent mass transfer coefficient. Although good predictions were obtained, an understanding of the operation is not complete. ZGD has a potential for industrial application because it fits two of the major requirements mentioned above. With regard to energy transfer and efficiency, the heat pipe mechanism is recognized as a highly efficient apparatus (Dunn and Reay, 1982). With regard to new separation schemes, distillation driven by interfacial tension forces points toward size reduction in equipment. Current pressures toward more stringent specifications on quality, efficiency, availability, operability, and safety demand more systematic equipment, operation, and control design procedures. On the other hand, it is known that process separation designs which imply high separation degrees and rationalization of energy consumption lead to more difficult operation and control problems. In our opinion, process and control design should be one procedure which should rely as much as possible on understanding and modeling the process in terms of first principles. By doing so, passage from laboratory to plant scale should be more efficient, and one could aim toward more demanding operations with advanced process and control design tools while ensuring adequate safety levels. At this stage, it is necessary to conceptualize the ZGD in terms of first principles where capillarity phenomena must play a central role, and its relationship to known equilibrium and mass transfer effects must be accounted for. In this work, we take a first step in that direction. Our objective is to describe the ZGD operation in terms of basic phenomena: fluid motion driven by porous media capillarity, temperature dependence of interfacial tension, effective medium diffusivity, gas-liquid equilibrium, and mass
FEED
A
. -. -
B
V A P O R FLOW
_-
z -E55 M L A T - E PRODJCT
acc -
*
PLOW
MORE VQATILE pRoovcT
Figure 1. Schematic of the microdistillation apparatus.
transfer through the gas-liquid interface. Without restricting the treatment, and by comparing with reported experimental data, we derive a model for a binary ZGD column. The model consists of partial differential equations for liquid and gas compositions coupled to algebraic expressions for interface equilibrium and mass transfer as well as to bubble-point temperature-composition conditions. With a standard finit element technique, the partial differential equations are discretized, and the resulting nonlinear algebraic system is then solved with standard methods used for distillation simulations. With minor adjustments in the porous medium characteristic factors, the model predicts accurately the reported experimental composition profiles and yields the underlying temperature profile. As a result, ZGD is better explained and understood. 2. Description of Zero-Gravity Distillation Figure 1 shows a schematic of the ZGD column: an external metal pipe heated and cooled at the left and right extremes, respectively, with a concentric porous ring (wick) all along. In the case of Seok and Hwang (1985), the ring was made of fiberglass and was held to the pipe with a metalic mesh. The middle part of the external pipe face was insulated. At some point of the adiabatic zone, the liquid mixture feed was injected. Light and heavy products leave the column at the extremes. Vapor is generated at the hot end and flows toward the cold end, where it condenses. By means of capillary action, liquid flows through the porous medium from the cold to the hot end in flow countercurrent to the vapor. As in a heat pipe, provided there are no liquid and vapor inputs and outputs, liquidvapor countercurrent flow enables heat transport from a heat source (heating zone) to a heat sink (cooling zone). Because at the wet porous medium-gas interface the liquid and vapor phases are away from equilibrium, mass exchange between phases occurs. As the vapor phase moves toward the cold end, it becomes enriched in the more volatile component. Correspondingly, as the liquid phase moves toward the hot end, it becomes enriched in the less volatile component. When the operation is carried out with no input and output, the procedure is equivalent to total reflux operation in a distillation column. 3. Fluid Motion and Interfacial Transfer 3.1. Fluid Motion. The ZGD porous ring enables liquid flow with liquid-gas interface mass exchange. Mostly for pure fluids, such a problem has been addressed recently (Nakayama and Koyama, 1987a,b; Nakayama et al., 1987; Kaviany, 1987; Merker and Mey, 1987; Arastoopour and Semrau, 1989), but results have not been exploited for mixture separation purposes. In this subsection, the aforementioned description for liquid flow in porous media is recalled and posed in a format suited for our ZGD
Ind. Eng. Chem. Res., Vol. 31, No. 3, 1992 903 SOLID
I
PHASE
GAS
FLOW
LIQUID
GAS
PHASE
LIQUID PHASE
(b)
Figure 3. Film mixing promoted by velocity fluctuations according to flow regime.
SOLID
e
-0
where u1 is the liquid velocity, K is the porous medium permeability, and p1is the liquid viscosity. Combination of eqs 1 and 2 yields the liquid velocity in terms of the surface tension gradient:
u1 = (b 1
Figure 2. Contact angle for partial wetting and perfect wetting: (a) definition of contact angle in capillaries; (b) condition for perfect wetting (wall is perfectly wetted).
problem. Because, in comparison to porous ring resistance to axial liquid flow, resistance to axial gas flow is small, the gas pressure drop is negligible, and therefore, the gas pressure Pgis assumed to be constant. Liquid axial convective flow through the porous ring is necessary for functioning of the ZGD column. Therefore, the axial liquid flow is necessarily due to a pressure drop caused by capillarity in the porous medium. Because the ratio of porous ring length to thickness is large, radial pressure gradients can be neglected. uI represents the liquid surface tension, which is a function of temperature and composition, T the temperature, rp the effective pore radius of the ring, and 9 the effective solid-liquid-gas contact angle (see Figure 2a). When B = 0, perfect wetting of the porous medium is attained (see Figure 2b). From Dunn and Reay (1982), at any axial point z, the liquid and gas pressures are related by the following equation: 2u1 COS
e
P1 = Pg- , 0 I 9 I1800 (1) rP On the other hand, the convective liquid flow in porous media is usually approached with the empirical Darcy equation, which Whitaker (1986) derived from first principles by the method of volume averaging:
2K cos 9 dui
w,,
az
(3)
As it is done in heat pipe modeling (Dunn and Reay, 1982),liquid flows are assumed to be small enough 90 there is a constant surface tension gradient, and eq 3 is expressed in terms of concentration-temperature conditions a t the pipes extremes, under perfect wetting conditions: 2~ ulc(Tc) - ule(Te) u1 = (4) Lt Plrp where ulc(TJis the light component surface tension at the condensation zone temperature, ule(Te)is the heavy component surface tension at the evaporator zone temperature, and Lt is the effective pipe length. 3.2. Mass Transfer. In this subsection, the objective is to describe and justify the reported (Seok and Hwang, 1985) high mass-transfer efficiencies attained for ZGD at low Reynolds flow regimes which do not ocm even in short conventional packed columns. To do so, we propose an enhanced mass transfer due to a mixing effect (Back and McCready, 1988) at the gas-porous medium interface induced by the porous ring structure. Back and McCready (1988) studied interfacial mass transport for concurrent gas-liquid flow. Mass-transfer resistance was derived in terms of film thickness and flow regime type. It was shown that mass-transfer coefficienta depend on the magnitude and frequency of the velocity fluctuations normal to the interface. It can be seen (Figure 3) that this concept implies a certain degree of mixing at the film near the interface. As turbulence in the bulk flow increases, interface velocity fluctuation frequency increases, film mixing increases, and therefore, mass-transfer capability grows. Accordingly, a laminar bulk flow provides less velocity fluctuation and, consequently, less film mixing, with a smaller mass-transfer capability.
904 Ind. Eng. Chem. Res., Vol. 31, No. 3, 1992
If it is considered that the maas-transfer rate is promoted by the degree of the filmmixing, independently of how this is achieved, it may be assumed that in the ZGD column there exists a high degree of film mixing. But such a level of mixing is not expected in the actual flow regime. Therefore, we can assume the porous medium is responsible for this mixing, because it consists of many channels that provide random depths for the fluid flow, Le., velocity fluctuations are higher with the porous medium than without it. In this way, effective film resistance is reduced substantially, and the mass-transfer rate is enhanced. This conceptualization is different from the one which attributes a high mass exchange rate to a large interfacial area provided by the packing material. As will be seen below we work only with mass-transfer coefficientsfor each phase. Capillary pressure may also play an important role in this process but will be discussed later. The film mixing concept provides us the expectancy of high mass-transfer coefficients. Now heat transfer and diffusivity are analyzed. 3.3. Heat Transfer. As described, the ZGD column consists basically of three sections: evaporator, adiabatic zone, and condenser. Heat is added to the system a t the evaporator and is extracted at the condenser. At steadystate operation, no heat accumulation exists, so that axial heat flow through the adiabatic zone must be constant. Heat is transported mainly with mass, so that considering that liquid and gas compositions vary along the column, heat transfer between phases must occur, and also a temperature profile must be established. As the gas flows toward the condenser and becomes enriched in the light component, it loses heat, which is gained by the liquid phase, as this flows toward the evaporator. We can see then that mass transfer is accompanied by heat transfer and composition and temperature profiles must be coupled. 3.4. Diffusivity at the Liquid and Gas Phases. An important feature to see before final establishment of the ZGD equations is that composition in the column varies from dilute to concentrated and this affects directly the liquid diffusion coefficient. This can be estimated from the Vignes correlation (Reid et al., 1977, p 585). Dml = (DoABI)~B(DO BAJ"
(5)
Taking this feature into account, the liquid-phase mass-transfer coefficient will not be constant along the pipe. This can be seen from the following correlations for packed systems (Treybal, 1980, p 75). laminar flow kl = 1 . 0 9 ~ ~ / ( E R e ~ / ~ S c ~ / ~ )
(6)
so we may consider it approximately constant along the pipe. An appropriate expression for the gas-phase masstransfer coefficient is
k, = Ug/(SC2/3)
(10)
4. Zero-Gravity Distillation Column Model 4.1. Mathematical Model. In the preceding section, the underlying ZGD phenomena have been established and their associated assumptions and mathematical correlations have been stated. In this section a synthesizing framework is brought up by means of a mathematical model. Then, the numerical solution issue is addressed. In general, the adiabatic, the heating, and the cooling zones must be modeled and solved simultaneously. To be able to compare with Seok and Hwang's (1985) approach and experimental results, as those authors do, here we circumscribe ourselves to model the adiabatic zone and, instead of modeling the condensation and evaporation zones, experimental composition and temperature data of those zones are fed as model boundary conditions. In order to establish the mathematical model, it is important to see that in the liquid phase a component moves in the axial direction because of two factors: (1)diffusion promoted by concentration gradients along the pipe and (2) convective flux due to the natural motion of the fluid. Upon the assumption of perfect mixing in the porous medium there are no concentration gradients in the angular direction. Considering the small radius of the pipe and, even more, the small thickness of the wick, we may assume a flat radial composition profile, letting mass transfer at each point in the column from the liquid to the gas phase (or vice versa), occur through the very thin interfacial film, which separates interfacial composition from bulk composition. In the gas phase, molar fluxes are also promoted in the axial direction by diffusion and convection. In the radial direction we again consider a flat concentration profile due to the small radius of the pipe and that diffusion rates in the gas phase are normally high, enhancing a high degree of mixing. With the above consideration, ZGD conceptualization and assumptions of section 3 translate into the following steady-state model equations: liquid-phase composition
a2xA axA aDeff axA Deff-+ - -- u1--a22
a%
dz
a2
2r1kl 2(xA - XAi) = 0 r22- r1 (11)
vapor-phase composition
turbulent flow
kl = 20.4ul/(~e0.815Sc2/3) considering that SC = f(DmI)
(7)
In the porous medium an effective diffusivity takea place and is defined for isotropic porous media as (Smith, 1986, P 542)
xAi and yAiare interfacial liquid and vapor compositions. Boundary conditions for these equations assume that as total evaporation and condensation occur at the evaporator and condenser, liquid- and gas-phase compositions are the same at these points, so that at the condenser 2 = 0; XA = Y A = x , = yc (13) at the evaporator
where e is the porosity of the medium and 7 the tortuosity factor, which may be approximated as l / t . In the gas phase, the diffusion coefficient is not as sensitive to changes in composition as it is in the liquid phase,
= Ye (14) Equations 11and 12 are coupled through mass transfer at the interface
(8)
2
= L,;
XA
= YA
Ind. Eng. Chem. Res., Vol. 31, No. 3, 1992 906 klCl(XA - XAi) = kgCgbAi - YA)
(15)
and the assumption of interfacial equilibrium (16) YAI = f ( X A i ) Under the previous consideration of a flat radial composition profile and saturated liquid and gas phases, gas and liquid temperatures are very similar to the interfacial temperature at each point in the column. That is, we assume Tl = Tg = Ti (17) which is very ne= true if gas and liquid phases are not far away from equilibrium. This assumption leads to the consideration that heat conduction may be neglected at the interface. Equilibrium considerations at the interface allow us to obtain a temperature profile by means of bubble point calculations: 2
f ( r ) = :C -1 Y j - 1 J-A
Interfacial equilibrium computations were achieved with the Antoine equation for vapor pressures and the Wilson equation for the activity coefficients. For our example, the pure component temperature dependencies surface tension data were interpolated from a heat pipe design handbook (Dunn and Reay, 1982). In general, any suitable &,r) correlation can be used (Reid et al., 1977, Chapter 12). Global mass balance equations were also required. 4.2. Numerical Solution. The ZGD model consists of two composition (liquid and vapor) partial differential equations (PDE) (eqs 11-14) coupled to algebraic interface equilibrium (eqs 15 and 16) and bubble-point equations (eq 18). To discretize (approximate with algebraic equations) the two PDE's, any standard method (finite differences, finite element, collocation, or combinations of them) may be used. Because ZGD is intended for highpurity separations, we need a method which can handle ill-behaved (steep gradients with smooth zones) concentration profiles. To handle poor numerical conditions due to ill-behaved profiles and nonlinearities, as well as to endow the model implementation with simplicity, we have chosen, a local profile approximation based on a linear combination of Hermite spline functions supported on finite domain intervals (Finlayson, 1980). NE XAb)
x(z)
K=l
cXdXK(z)
(19)
NE
Y A ~ )
V z ) = C cyKd~yK(z) K=l
(20)
where (zl, ...,zmj denotes the node mesh abscissa, NT the number of collocation points in each element, and NE the number of elements. dJx and dJy are defined as the hermite cubic polynomials on each element. Substitution of the profile representations (19) and (20) into our mass-transfer equations (11) and (12) yields two residual functions: Rx(z) and Ry(z). Following the collocation method (Finlayson, 1980), it is required that the later residuals vanish at the node mesh Rn(z) = 0, R y ( z ) = 0 (21) By doing so, the partial differential equations have been replaced by NE X NT - 2 algebraic equations for each phase. Fulfillment of boundary conditions provides two additional equations for each phase. Interfacial, equilibrium and bubble-point expressions provide three equations at each point of the mesh. As a result, there is a set of
algebraic nonlinear equations whose solution by means of the Gauss-Seidel iterative method yields the expansion composition coefficients and the interface compositions and temperatures at the node mesh. To obtain the temperature profile, the same composition representation procedure is used. The system was asked for a convergence of 0.0005 tolerance in the difference of calculated values between iterations for each collocation point. In general, convergence is excellent for the adiabatic and condenser zones, where deviations (residuals) are as low as lo-'. Convergence is slow for the gas phase at the evaporator. This is not any surprise because we know that the model is not rigorous for this zone. However, if a tolerance of 0.001 is given, convergence is reached in less than five iterations. At the beginning, all elements were equally sized. Then, considering the small deviations at the condenser element, this first element was increased to about 40% of the total lenath, being the other elements of the same size as a result of &e &t&ution of the rest of the length among the other elements. Runs were done with 6,8,10. and 12 elements. and it was found that 8 elements suffked. 4.3. Model Parameters. In addition to equilibrium, mass-transfer and diffusivity parameters which are used in distillation, ZGD requires additional parameters related to the capillarity effect in the porous wick: porosity (e), pore size (d,), permeability (K), and the wetting angle (0) and temperature dependencies of the surface tensions of both components. For the present model, only the surface tensions at the given condenser and evaporator temperatures are required for the light and heavy components, respectively. Seok and Hwang (1985) do not report the porous wick parameters used, so we had to estimate them from available correlations in literature (Greenkorn, 1981; Morrow, 1970). We know that such estimates are very sensitive with regard to the way in which the bed is constructed. With this in mind, estimated values must be taken with reserve. On the basis of previous discussion and considerations of wetting in porous media (Greenkorn, 1981), 0 is assigned as zero. Examination of the ZGD model shows that a = K / F ,enters as one constant, for which a value of 0.045 cm is suggested from a reported correlation (Greenkorn, 1981). For the actual parameters to be used, we decided to use known data of fluid flows for our case, and, with adjustments of the Darcy velocity and global mass balance equations, a value of 0.355 is estimated for porosity. 4.4. Results and Discussion. Initially, the low (laminar) mass-transfer coefficient correlation (6) was used, and the results were not satisfactory. When the high (turbulent) mass-transfer correlation (7) was used, in general, a good approximation was obtained. Light component profiles are shown in Figures 4-6 for methanolwater and ethanol-water mixtures at different values of G/L,which is the ratio of gas molar flow to liquid molar flow. Some deviations are present near the evaporator. One possibility for this disagreement is that low masstransfer rates occur near this zone, due to the effect of interfacial forces. In fact, the bubble formation process in heat pipes and the passage of this bubble to the gas phase involves additional energy which makes the temperature in the porous medium rise some degrees over the normal boiling point (Dunn and Reay, 1982). This is so because vapor bubbles must move against i n t e r f a d forces with a consequent increase in gas temperature and pressure and a decay in mass-transfer rate. On the basis of this observation, simulations were carried out using the low (6) and high (7) mass-transfer rates correlations for the
906 Ind. Eng. Chem. Res., Vol. 31, No. 3, 1992
J
z
9
\
0 6-
c
:
0 5-
I
Y I
-
0U
-
A/
0 4-
E
-
/
0 7..
0
EXPERIMWAL
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=O 2 METHANO..
L 3
t
PREDICTED
A
c a
W 4.
z
0 3-
W
c
W
$
01 CONDENSER
10
10
20
LENGTH
30
40
50
55
(cm.)
Figure 4. Comparison of predicted profile vs experimental data reported by Seok and Hwang (1985) for the methanol-water system. High mass-transfer rate correlation along the entire column.
-I
0
30
COLUMN LENGTH
9
40
60
5
5
cm 1
Figure 7. Predicted temperature and composition profiles for the methanol-water system with low mass-transfer rate in the evaporator side. Comparison with experimental composition data reported by Seok and Hwang (1985).
t
O9
20
0.91
\
08
z
2I-
07
W
06 2 0
F u
I
0.5
a
04
'
03
z 9 c U
05-
2
04-
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-PREDICTED 0 EXPERlHDlTAL
T)
FEED 0 2 METHANOL G/L
03-
=
4.
121
z 70
0 2-
02 01
10
IO
20
30
40
50
LENGTH ( c m l
I
0.9 0.8
1
dz
9
c W
2
E
-
8
E
0
W
i
PREOIKED
EXPERIMENTAL
FEEU 0.2 EWANOL G / L = 1.1
IO
20
30
40
LENGTH
55
Figure 5. Comparison of predicted profile vs experimental data reported by Seok and Hwang (1985)for the methanol-water system. High mass-transfer rate correlation along the entire column.
20
x)
40
50
55
(cm 1
Figure E. Predicted temperature and composition profiles for the methanol-water system with low mass-transfer rate in the evaporator side. Comparison with experimental composition data reported by Seok and Hwang (1985).
evaporator and condenser half-sides, respectively. As can be seen in Figures 7-9, where composition and temperature profiles are shown, predictions improved substantially. Figures 4 and 7, which correspond to high separation, exhibit some deviation at the condenser side. Inclusion of condensation and evaporation zones should lead to further model prediction improvement, which, for the purposes of the present work, were not further pursued. Regarding mass-transfer rates, turbulent-like conditions seem to occur in a great part of the column because of mixing effects promoted by the porous medium. Yet, experimental verification is necessary in order to obtain information which will allow a conclusive affirmation of the nature of mass-transfer coefficients. 5. Conclusions
50
55
LENGTH ( c m )
Figure 6. Comparison of predicted profile vs experimental data reported by Seok and Hwang (1985) for the ethanol-water system. High mass-transfer rate correlation along the entire column.
For zero-gravity distillation, which is a new promising operation, a conceptualization and its associated mathematical model have been derived and validated. The proposed conceptualization focuses on the novel separation principles which are the porous medium liquid flow caused by capillarity forces due to a surface tension gradient, induced by temperature, and the porous medium prop-
Ind. Eng. Chem. Res., Vol. 31, No. 3, 1992 907
-r
0
0.8 0.9
z
E z
P
0.6
I
3
’. 80
0.5W -I
B
0.4
4c K W
-
0.3.. 0.2 -.
:
PREDICTED
70
0 EXPERIMENTAL
W I-
-
FEED: 0 2 E T H W WL 1.1
0.1
60
10
20
30
40
50
55
LENGTH I c m l
Figure 9. Predicted temperature and composition profiles for the ethanol-water system with low mass-transfer rate in the evaporator side. Comparison with experimental composition data reported by Seok and Hwang (1985).
erties. The conceptualization accounts also for the interplay of the capillarity effect with phase equilibrium, porous diffusivity, and mass transfer. In contrast to earlier modeling by analogy with a conventional packed tower column, the proposed modeling explains the high efficiency attained with ZGD. The model consists of partial differential equations coupled to algebraic ones whose solution is obtained with standard numerical techniques. The resulting model exhibits a high capability of predicting composition profiles. It was evidenced that available correlations for mass transfer in packed beds under low regime flow are not adequate for ZGD modeling. I t was shown that experimental results can only be explained with a combination of low and high mass-transfer correlations. This suggests that the nature of the porous medium may enhance film mixing with a mechanism like the one proposed by Back and McCready (1988). Experimental verification of this conjecture is necessary.
Acknowledgment We are grateful to the National Council of Science and Technology (CONACfI’) of Mexico for its support of this work.
Nomenclature c = coefficient of the trial functions approximation C = concentration, gmol/cm3 DAB = binary diffusion coefficient for system A-B, cmz/s Deff= effective diffusion coefficient, cmz/s G = vapor flow rate, g-mol/s K = wick permeability, cm2 k , = gas-phase mass-transfer coefficient, g-mol/(s cm2) (9mol/cm3) k , = liquid-phase mass-transfer coefficient, g-mol/ (cm2 s) (g-mol/cm3) Lt = total length of the ZGD column, cm L = liquid flow rate, g-mol/s NE = total number of elements NT = total number of collocation points at each element Pg= gas pressure, mmHg P,= liquid pressure, mmHg rl = pipe radius excluding the wick, cm r2 = total internal pipe radius, cm Re = Reynolds number, dimensionless rp = effective pore radius, cm
R = residual function Sc = Schmidt number, dimensionless T = temperature, “C u = fluid superficial velocity, cm/s X = approximation function for liquid-phase composition X A = concentration of component A in liquid, mole fraction x, = concentration of component A in liquid at the condenser end, mole fraction x , = concentration in liquid at the evaporator end, mole fraction Y = approximation function for gas-phase composition yA = concentration of component A in gas, mole fraction yc = concentration in gas at the condenser end, mole fraction y e = concentration in gas at the evaporator end, mole fraction z = axial coordinate Greek Letters a = Darcy equation constant, cm e = porosity 6 = angle of contact p = viscosity, (dyn s)/cm2 u = surface tension, dyn/cm 7 = tortuosity factor 4 = basis functions for polynomial approximation Subscripts
A = component A c = condenser e = evaporator g = gas phase i = interfacial K = element number 1 = liquid phase p = pore X = liquid phase Y = gas phase Superscript * = reference state
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Received for review March 26, 1991 Revised manuscript received November 26, 1991 Accepted December 4, 1991
Simulation of Mass Transfer in a Batch Agitated Liquid-Liquid Dispersion A. H.P.Skelland* and Jeffrey S . Kanelt School of Chemical Engineering, Georgia Institute of Technology, Atlanta, Georgia 30332
A simulation model is developed to compute the transient Sauter-mean drop diameter and the fractional mass transfer in batch agitated liquid-liquid dispersions when the continuous phase, dispersed phase, or both phases offer significant resistance to mass transfer. This model accounts for mass transfer during drop breakup, drop-drop coalescence or rebounding, drop oscillations, and free motion of the drops throughout the vessel. Using 16 different formulations, 8 of which were developed here, the transient fractional mass transfer was predicted with an average deviation of f20% for all four directions of diffusion and dispersion, using no adjustable parameters. However, to enhance the simulation, one constant in the breakage frequency was then varied to improve predictions to f15% for 10 simulations, including 4 with surfactants present. Furthermore, the transient Sauter-mean drop diameter was predicted with an average deviation of &8% for these simulations. The fraction of the total mass transferred due to drop-drop interactions was found to be indeterminate, but small, in this work. However, drop breakup and subsequent damped oscillations accounted for about 5 % of the total transfer in the simulation for dispersed-phasecontrolled systems and insignificantly for continuous-phase-controlled systems. Introduction To design an agitated liquid-liquid contactor, appropriate models for the system must be constructed that accurately describe the phenomena occurring within it. Physical factors that influence the extraction rate are droplet phenomena such as breakage, coalescence, rebounding, oscillations, and free motion throughout the vessel; microscopic interfacial transport processes; and macroscopic hydrodynamics in the vessel. Earlier investigators developed models to predict the drop size distribution for agitated liquid-liquid dispersions in the absence of mass transfer using either drop population balances [Valentas et d. (1966), Valentas and Amundson (1966), Ramkrishna (19741, and Bajpai et al. (1976)] or simulation techniques [Zeitlin and Tavlarides (1972), Coulaloglou and Tavlarides (1977),Molag et al. (1980), and Hsia and Tavlarides (1983)l. When solute transfer was added to drop population balance models, a trivariant drop distribution was required to account for a drop's size, ita Present address: Eastman Chemical Company, Kingsport, TN.
age (time that the droplet has existed in the vessel), and its solute concentration. Both drop population balances and simulation techniques have been used to predict solute-transfer rates in these dispersions; the former are represented by Bayens and Lawrence (1969) and Shah and Ramkrishna (1973) and the latter by Curl (1963),Bapat et al. (19831, Bapat and Tavlarides (1985), and Jeon and Lee (1986). Models for mass transfer in agitated liquid-liquid dispersions are summarized in Table I. These previous models only computed solute transfer during the free movement of droplets through the vessel and assumed that coalescence and breakup occur instantaneously, tacitly neglecting transfer during these processes. However, during drop splitting, Bozorgzadeh (1980) showed a 100-200 7 ' 0 enhancement in the mass-transfer coefficient (accounting for increased surface area) over oscillating drop predictions, and Bately and Thornton (1989) concluded that for drop-drop coalescence "it might be expected that the enhancement over and above stagnant diffusion rates would be appreciable, and the effect of inter-droplet coalescence should not be dismissed without experimental confirmation". Thus, the present model accounted for
0888-5885/92/2631-0908$03.00/00 1992 American Chemical Society
