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Ind. Eng. Chem. Res. 1997, 36, 5392-5398
Distillation Lines for Multicomponent Separation in Packed Columns: Theory and Comparison with Experiment† Sami Pelkonen, Ruth Kaesemann, and Andrzej Go´ rak* Universita¨ t Dortmund, D-44221 Dortmund, Germany
Distillation lines that are commonly used to predict feasible product regions are examined, focusing on the theoretical investigation and on the experimental verification of different models. Distillation lines based on the equilibrium stage model, residue curves, fluctuation transport theory, and multicomponent diffusion theory are derived and compared with experiments. The comparison with experiments shows that the consideration of the diffusional interactions between the components may be of essential importance when the column is operated close to a distillation boundary. Introduction Many research works have been focused on the investigation of feasible separation regions in distillation columns. For this purpose distillation lines, distillation diagrams, and residue curve maps are being used. The extensive literature treating the applications of distillation diagrams has been reviewed by Wahnschaft et al. (1992) and recently by Widagdo and Seider (1996). They report that the new developments have been concentrated on the separation of nonideal mixtures, with emphasis on azeotropic distillation. Stichlmair and Herguijuela (1992) developed a method for the design and analysis of processes for complete separation of binary azeotropic mixtures by using an entrainer (azeotropic and extractive distillation). According to them, the knowledge of the separation regions in the distillation diagrams allows for the development of the generalized process and the formulation of criteria for entrainer selection. What seldom has been considered is that different models predict different distillation lines and boundaries. Therefore, a short discussion relating to studies on the mass-transfer mechanism and distillation lines is given below. Commonly, the distillation lines acquired by the equilibrium stage concept or the residue curves are used for the prediction of the feasible distillation regions (Stichlmair and Herguijuela, 1992). Laroche et al. (1991) derived from differential balances that the set of differential equations describing the simple distillation process, i.e., the residue curves, is identical with the one for the composition profiles in packed columns. Van Dongen and Doherty (1985) also demonstrated that the results yielded by a differential column model and by a stage-by-stage equilibrium calculation are very similar. However, Laroche et al. (1991), Stichlmair and Herguijuela (1992), and van Dongen and Doherty (1985) ignored the influence of the diffusional interactions between the single species. Another approach is that of Hampe et al. (1991), who asserted that diffusion should not be expected to be the governing masstransfer mechanism for processes which involve fluids at saturation, such as distillation. Instead, mass trans* To whom correspondence should be addressed. Current address: Universita¨t-Gesamthochschule Essen, D-45141 Essen, Germany. Telephone: + 49 201 183 2663. Fax: + 49 201 183 3298. E-mail:
[email protected]. † Dedicated to Professor Dr.-Ing. Alfons Vogelpohl on the occasion of his 65th birthday. S0888-5885(97)00333-3 CCC: $14.00
fer is assumed to occur in the phase boundary by fluctuant transport of molecular clusters. This is in contrast to the opinion of Taylor and Krishna (1993) and that of Go´rak (1995), who based their mass-transfer models on Maxwell-Stefan relations for multicomponent diffusion. The latter approach was supported by the experimental investigations carried out independently by Ronge (1995) and Pelkonen et al. (1996). On the basis of distillation lines, they showed that the diffusional interactions are present in distillation. This paper is concentrated on two aims. First, a profound theoretical study is performed, focusing on the distillation lines obtained with different simulation models. Second, the model-predicted distillation lines are compared with those obtained from experiments to get information of the mass-transfer mechanism present in distillation. The effect of maldistribution on the distillation lines is omitted. Residue Curves and Distillation Lines at Total Reflux Total reflux experiments with multicomponent mixtures enable investigations of the mass-transfer mechanism, of the mass-transfer resistance, and of the accuracy of the vapor-liquid equilibrium models (Ronge, 1995). For further understanding it is essential to distinguish between residue curves and rectification lines that can be designated by the collective term distillation lines (Vogelpohl, 1993; VDI-Richtlinie 2761). The residue curves are trajectories of the composition of a liquid L remaining in a still as vapor V is continuously boiled off in a single-stage distillation process (Figure 1). The profile of residue lines is determined solely by the thermodynamic equilibrium. For an n-component mixture the residue curve is determined by the (n - 1)-dimensional vector of the form (Vogelpohl, 1993; Widagdo and Seider, 1996) dx/dt = y * – x
(1)
At any time the change of the liquid composition with a differential amount of vapor being removed occurs along the direction of the vapor-liquid equilibrium vector. Equation 1 can be expressed as an (n - 2)-dimensional equation system (Vogelpohl, 1993)
dxi y* i (xi) - xi ) dxn-1 y*n-1(xn-1) - xn-1 © 1997 American Chemical Society
i ) 1, ..., n - 2 (2)
Ind. Eng. Chem. Res., Vol. 36, No. 12, 1997 5393
are calculated from the rate equations that are presented below. For the sake of simplicity the derivation is presented only for the vapor phase, whereas the derivation for the liquid phase is analogous. The molar fluxes over the phase interface for an n-component mixture are given as N = J + y Nt
Figure 1. Simple distillation still.
(4)
The total molar flux Nt over the phase interface is obtained from the energy balance around the interface and must be equal to the sum of the component molar fluxes.
Nt ) Figure 2. Equilibrium rectification line and its tie lines.
qL - qV
n
λy
∑ i)1
n
(λi - λn)J Vi )
Ni ∑ i)1
(5)
The heat of vaporization is calculated as
λi ) H Vi - H Li
(6)
n
λy )
λiyi ∑ i)1
(7)
and the heat transfer fluxes as
Figure 3. Residue curve and equilibrium rectification line through point P.
The rectification lines should give the real composition path in a distillation column. It is obvious that their prediction depends on the model that has been chosen for simulation. In this paper a distinction is made between the equilibrium, fluctuation, and diffusion rectification lines. If the equilibrium stage concept is chosen, the rectification line is provided by discrete composition points that represent a sequence of equilibrium states. The points can be joined by piecewise linear segments (tieline curve) or by spline interpolation through the points (Figure 2). Stichlmair et al. (1989) call the spline interpolations distillation lines; here they are called the equilibrium rectification lines. The equation for the tie lines is given as (Stichlmair et al., 1989)
qV ) hV(T V - T I)
(9)
J = cVt [kV](y – y I)
(10)
where the mass-transfer coefficient matrix is given by
[k] ) [R]-1[Γ]
(11)
with
(3)
It is important to notice that neither the equilibrium rectification lines nor the tie-line curves coincide with the residue curves. Following the presentation of Stichlmair and Herguijuela (1992), the difference and correspondence between the discrete tie lines and the differential residue curves are illustrated in Figure 3. In the following it is studied how the distillation lines behave when two different mass-transfer models are employed. The diffusion rectification lines following Taylor and Krishna (1993) and Ronge (1995) are derived from the multicomponent diffusion theory. The fluctuation rectification lines are obtained on the basis of the fluctuation transport theory of Hampe et al. (1991). Diffusion Rectification Lines. The diffusion rectification lines are derived from the multicomponent mass-transfer theory according to Taylor and Krishna (1993) that is based on the Maxwell-Stefan equations and on the two-film theory. The mass and energy rates
(8)
The contribution of the heat fluxes to the total molar flux is likely to be small in comparison to the average latent heat term λy. The diffusional molar fluxes are obtained from the Maxwell-Stefan equations for multicomponent mass transfer
Rii ) xm +1 – ym* = 0
qL ) hL(T I - T L)
yi
n
∑ k)1
+ κVin
k*i
yk V κik
i ) 1, ..., n - 1
(12)
and
Rij ) -yi
(
1 1 - V V κ ij κ in
)
i, j ) 1, ..., n - 1; i * j (13)
The binary Maxwell-Stefan mass-transfer coefficients κijV result from semiempirical correlations. The thermodynamic correction matrix is obtained from
Γij ) δij + yi
|
∂ ln φi ∂yi
i, j ) 1, ..., n - 1 (14)
T,p,y(i*j)
Based on eqs 4-14 and on the material balance around a differential element, the diffusional rectification lines may be calculated. The material balance is given as (see Figure 4)
5394 Ind. Eng. Chem. Res., Vol. 36, No. 12, 1997
dx1 ) dxn-1 V V V k11 ∆y1 + k12 ∆y2 + ... + k1(n-1) ∆y(n-1) V V V k(n-1)1 ∆y1 + k(n-1)2 ∆y2 + ... + k(n-1)(n-1) ∆y(n-1)
dx2 ) dxn-1 V V V k21 ∆y1 + k22 ∆y2 + ... + k2(n-1) ∆y(n-1)
Figure 4. Two-phase differential element.
V + (L + dL) ) (V + dV) + L
(15)
dV ) dL ) -Nt dA
(16)
V V V k(n-1)1 ∆y1 + k(n-1)2 ∆y2 + ... + k(n-1)(n-1) ∆y(n-1)
(25)
where
l l
The component material balance for the vapor phase gives Vy – N dA = (V + dV)(y + dy)
V dy = –N dA + y(Nt dA)
dxn-2 ) dxn-1 V V V k(n-2)1 ∆y1 + k(n-2)2 ∆y2 + ... + k(n-2)(n-1) ∆y(n-1) V V V k(n-1)1 ∆y1 + k(n-1)2 ∆y2 + ... + k(n-1)(n-1) ∆y(n-1)
(17)
where
Combining eqs 4 and 17 gives V dy = –J dA – y(Nt dA) + y(Nt dA) = –J dA
(18)
∆yi ) (y*(xi) - yi) ) (y*(xi) - xi) i, j ) 1, ..., n - 1
and yields to
(26) V dy = ctV[kV](y I – y) dA
(19)
In the same way an expression for the liquid phase may be written as L dx = ctL [kL](x – x I) dA
(20)
The rectification lines can be derived from eqs 19 and 20. Assuming that the resistance to mass transfer is concentrated only in the vapor phase, eq 19 is given as V dy = ctV[kV](y *(x) – y) dA
(21)
If the resistance to mass transfer is considered only in the liquid phase, eq 20 is expressed as L dx = ctL [kL](x – x *(y)) dA
(22)
For the special case of total reflux and total reboil operation of the column, the following equations are obtained from the material balances
The elements kij, where i, j ) 1, ..., n - 1, are obtained from eqs 11-14. Fluctuation Rectification Lines. If all the components are assumed to have the same facility to mass transfer, like in the case if the fluctuation transport theory of Hampe et al. (1991) is applied, the matrix [k] reduces to a scalar value, i.e.
[k] ) ke[I]
There is no information in the literature on how the fluctuation transport coefficient ke can be quantified from the fluctuation transport theory for distillation. In this work ke is calculated as the average values of the mass-transfer coefficients for all binary subsystems. In addition, it is not clear what the driving force is. Following the results obtained by Ronge (1995), the sssf driving force is described here by (y*(x) - b x) at total reflux column operation. For the fluctuation transport theory, eq 24 reduces to V dy
L = V, x = y,
dx = dy
= L dx
= ctVke (y *(x) – x) dA
(28)
(23)
In this case the fluctuation rectification line is obtained from eq 2, meaning that it is identical with the residue curve. Equation 2 can be written out to yield
Equations 21 and 23 can be combined to yield V dy = L dx = ctV[kV](y *(x) – x) dA
(27)
(24)
if the resistance to mass transfer is considered only in the liquid phase, the expression is obtained from eqs 22 and 23. If mass-transfer resistance in both phases is considered, the composition profile can be expressed by means of eqs 19, 20, and 23. Derived from eq 24, the basic equations for the diffusion rectification lines for an n-component system can be obtained from the following system of equations (total reflux and reboil operation, complete resistance to mass transfer in the vapor phase)
dx1 (y*(x1) - x1) ) dxn-1 (y*(x(n-1)) - x(n-1)) dx2 (y*(x2) - x2) ) dxn-1 (y*(x(n-1)) - x(n-1)) l l dxn-2 (y*(x(n-2)) - x(n-2)) ) dxn-1 (y*(x(n-1)) - x(n-1))
(29)
Ind. Eng. Chem. Res., Vol. 36, No. 12, 1997 5395
Case Study: Rectification Lines for a Ternary Mixture at Total Reflux Operation of the Column. A case study with a nonideal ternary mixture is performed to investigate which parameters affect the profile of the rectification line. A total resistance to mass transfer in the vapor phase at total reflux and reboil operation of the column is assumed. Equation 24 can be rewritten to yield
V V V + y2κ13 + y3κ12 SV ) y1κ23
Combining eqs 30, 32, and 36, the following equation for the composition profile along the column height is obtained dx dz
dx
= MV[kV](y*(x – x)
(30)
(41)
V = MV A [TV]VV V S
(42)
The direction of the rectification line is now defined by s sf
the product of the matrix [TV] and the vector VV that
where
sssf
MV )
gives a new vector MSV
cVt
dA V
(31) MSV =
The vector describing the driving force for mass transfer s sf
VV may be written as – y1 K1x1–x1 P = (y*(y) – y) = (32) K2x2–x2 γ2p02y2 – y2 P
Due to the ideality of the vapor phase, the thermodynamic correction factor matrix equals the unity matrix (see Taylor and Krishna, 1993) and the matrix [kV] is given as
[kV] ) [RV]-1
[(
(43)
V V V V T21 V1 + T22 V2
When the diffusive interactions are taken into account, γ1p01y1
VV = (y*(x) – x) =
VVV + TVVV T11 1 12 2
(33)
]
V 2/3 V 2/3 - 13 - 23 MSV1 ) [(-D ) {y1(-D ) +
(
(
(
)
)
(34)
(36)
where the elements of the matrix [TV] are defined by V V 2/3 V 2/3 V 2/3 -- 13 T 11 ) (D ) {y1(--D23 ) + (1 - y1)(--D12 ) } (37) V V 2/3 V 2/3 V 2/3 T 12 ) y1(--D23 ) {(--D13 ) - (--D12 ) }
(38)
V V 2/3 V 2/3 V 2/3 ) y2(--D13 ) {(--D23 ) - (--D12 ) } T 21
(39)
V V 2/3 V 2/3 V 2/3 -- 23 T 22 ) (D ) {y2(--D13 ) + (1 - y2)(--D12 ) } (40)
and the constant
(
V 2/3 V 2/3 V 2/3 - 23 - 13 - 12 [(-D ) {y2(-D ) + (1 - y2)(-D ) }]
)
γ2p02y2 - y2 p (45)
the vector MSV, which depends on the saturation vapor pressures of the pure components, activity coefficients, and binary Maxwell-Stefan diffusion coefficients. If the diffusion coefficients are equal or the diffusional interactions are neglected, the direction of the rectification s sf
line is determined by the vector VV depending on the pure-component saturation vapor pressures and activity sssf
(35)
where is a function of the physical properties of the mixture and the packing specific constants. If masstransfer coefficients are derived from film model κijV ∝ -- ij or from the penetration theory κijV ∝ D -- ij1/2. Here we D follow eq 35 because it is used in practice. Carrying out the inversion of the matrix [RV], eq 34 reduces to
AV V [T ] SV
)
γ2p02y2 - y2 + p
sssf
AV
[kV] )
)
the direction of the rectification line is determined by
-1
Nearly all mass-transfer correlations for κijV quoted in the literature follow the relationship
κVij ) AV--Dij2/3
(
V 2/3 V 2/3 V 2/3 -- 13 -- 23 -- 12 ) {(D ) - (D ) }] MSV2 ) [y2(D
For a ternary system eq 33 may be written as
y y 1 y1 1 + V2 + V3 - y1 V - V V κ κ κ κ12 κ13 [kV] ) 13 1 12 1 13 y y y 1 -y2 V - V V + V2 + V3 κ12 κ23 κ12 κ23 κ23
)
γ1p01y1 - y1 + p γ1p01y1 V 2/3 V 2/3 V 2/3 -- 23 -- 13 -- 12 - y1 (44) [y1(D ) {(D ) - (D ) }] p V 2/3 -- 12 (1 - y1)(D ) }]
coefficients. The angle φ between the vectors MSV and s sf
VV (Figure 5) is given as cos ϕ =
MSV • VV MSV VV
=
(MSV1 )(VV1 ) + (MSV2 )(VV2 ) (MSV1 )2 + (MSV2 )2
(VV1 )2 + (VV2)2
(46)
Experimental Verification The simulation results of experiments that are shown in this paper represent the conclusions that have been drawn based on the whole experimental data base available at the University of Dortmund. More information is available in Pelkonen (1997) and Pelkonen et al. (1997). The simulation of the experimental data was performed with both the equilibrium stage model and the nonequilibrium stage model with the simulator ChemSep (Taylor et al., 1994). The two different physical descriptions for the mass-transfer models were investigated in the nonequilibrium stage model simulations: the multicomponent diffusion theory according to Maxwell-Stefan relations (Taylor and Krishna, 1993; Go´rak, 1995) and the fluctuation transport theory according to Hampe et al. (1991). The consideration of the liquid-
5396 Ind. Eng. Chem. Res., Vol. 36, No. 12, 1997
ssf
Figure 5. Angle between the vector MV and the vector V B for a ternary system A-B-C.
phase resistance had practically no influence on the diffusion rectification lines when applying the masstransfer relationships provided by Bravo et al. (1985). Hence, the liquid-phase mass-transfer resistance was omitted as was suggested based on experimental data by Ronge (1995) and Pelkonen et al. (1996). The vaporphase diffusion coefficients were calculated according to Fueller et al. (Reid et al., 1988) and vapor-liquid equilibrium using the parameters from Gmehling and Onken (1997). Go´rak (1991) and Pelkonen (1997) found out that the composition profiles are very sensitive to the vapor-liquid equilibrium models and to the parameter data sets applied. Therefore, not only the influence of the investigated models but also that of the vaporliquid equilibrium model selection on the rectification lines is investigated. Ternary Experiments with Methanol-Isopropyl Alcohol-Water For a ternary mixture containing an azeotrope there may be a distillation boundary that cannot be crossed by normal distillation at total reflux operation (Stichlmair et al., 1989). Such a system is, for instance, methanol-isopropyl alcohol-water. On the other hand, the system ethanol-water-ethylene glycol has also an azeotrope but exhibits no distillation boundary. Here we investigate only the first type of system containing a distillation boundary. Comparison of Different Models with Experiments. First, a model comparison is performed by calculating the vapor-liquid equilibrium using the UNIQUAC model with the parameter data set determined by Verhoye and Schepper (Gmehling and Onken, 1977). Also other UNIQUAC data sets such as from Ochi et al. (Gmehling and Onken, 1977) give similar results. These parameter data sets are based on VLE measurements with the whole ternary system. Figure 6 shows the experimental and simulated rectification lines for experiments with methanolisopropyl alcohol-water at total reflux (Go´rak, 1991; Pelkonen, 1997). Presented are the diffusional, fluctuation, and equilibrium rectification lines. Six different experimental compositions have been fixed in the simulations, so that each distillation line goes through the same point. It should be noted that the fixed compositions lie close to the distillation boundary connecting the azeotrope between isopropyl alcohol-water and the low boiling component methanol. For each investigated case the Maxwell-Stefan theory predicts that the reboiler is filled with water as is the case in the experiments. Contrarily, both the fluctuation transport theory and the equilibrium stage concept anticipate a completely different direction: the reboiler is filled with isopropyl alcohol. In other words, when the Maxwell-Stefan approach is applied the experimental compositions lie “below” the distillation boundary. Using the equilibrium stage concept or the fluc-
Figure 6. Experimental (circles) and simulated rectification lines for experiments with methanol-isopropyl alcohol-water at total reflux. Presented are the diffusional (thick dashed line), the fluctuation (thin dashed line), and the equilibrium (full line) rectification lines. Six different experimental compositions have been fixed. In the simulation the UNIQUAC model is used to calculate vapor-liquid equilibrium. The black square presents the azeotrope between water and isopropyl alcohol.
tuation transport theory, the experimental compositions lie “above” the distillation boundary. Obviously, as derived here using the theory of multicomponent diffusion, the direction of the rectification line and the location of the distillation boundary may be influenced not only by the thermodynamics but also by the diffusional interactions between the components. Based on these results, it can be concluded that the feed composition region that leads to a water product is obviously larger when the diffusion rectification lines are used than when the “equilibrium” or “fluctuation” rectification lines are applied. In other words, any feed composition lying in the gap between the boundaries predicted by the different models will lead to different product compositions. Since the distillation boundary is also influenced by the description of the phase equilibrium, it is investigated in the following. Influence of Different VLE Models. The influence of the VLE model and the parameters on the rectification line prediction is illustrated in Figure 7. The simulated rectification lines are obtained using Wilson and NRTL models with the parameters derived by Verhoye and Schepper (Gmehling and Onken, 1977). The experiment is the same as in Figure 6 where the UNIQUAC model was used. The use of the NRTL model results in nearly identical rectification lines as was obtained with the UNIQUAC model: the diffusive rectification line shows the best agreement with the experiment. On the other hand, the experimental trajectory is predicted in a similar way by each model when the Wilson model is applied. This stresses the sensitivity of the profiles and of the distillation boundary to the VLE calculation. Therefore, the
Ind. Eng. Chem. Res., Vol. 36, No. 12, 1997 5397
Figure 7. Experimental (circles) and simulated rectification lines for an experiment with methanol-isopropyl alcohol-water at total reflux. Presented are the diffusional (3, thick dashed line), the fluctuation (2, thin dashed line), and the equilibrium (1, full line) rectification lines. In the simulation the Wilson and NRTL models are used to calculate vapor-liquid equilibrium. The black square presents the azeotrope between water and isopropyl alcohol. Figure 9. Simulated mass-transfer rates along the column height for an experiment with acetone-methanol-isopropyl alcoholwater.
Figure 8. Experimental (circles) and simulated rectification lines for an experiment with acetone-methanol-isopropyl alcoholwater at total reflux. Presented are the diffusional (dashed lines) and fluctuation (full lines) rectification lines. Each ternary diagram presents a side of an unfolded tetrahedron.
investigation with a different system with one more component is carried out with different VLE parameters and is presented in the following. Quaternary Experiments with Acetone-Methanol-Isopropyl Alcohol-Water To study how well the results based on the experiments with methanol-isopropyl alcohol-water apply for another system, experiments with a quaternary system acetone-methanol-isopropyl alcohol-water were performed. The vapor-liquid equilibrium was calculated with the UNIQUAC model. No parameter data sets based on quaternary VLE measurements were found in the literature. Therefore, those obtained on the basis of binary experiments were chosen (Gmehling and Onken, 1977). Figure 8 shows the experimental and simulated distillation lines for this mixture. Each ternary composition diagram presents a side of the unfolded tetrahedron formed by the four components. The simulation is carried out following the Maxwell-Stefan theory for multicomponent diffusion and the fluctuation transport theory. The experimental composition in the liquid distributor at the column height of 2 m is fixed in the simulation. This composition is chosen because the liquid phase is fully mixed (no maldistribution) and because it is close to the distillation boundary. The fluctuation transport theory predicts that the reboiler contains only isopropyl alcohol, whereas the theory of multicomponent diffusion gives the direction
of the experiment: the reboiler is filled with water. To illustrate the difference between the mass-transfer theories, the mass-transfer rates over the phase interface below the fixed composition in the column are shown in Figure 9. For acetone and methanol the models show the same direction for the mass-transfer rates, from the liquid into the vapor phase. Also for isopropyl alcohol and water both models predict the mass-transfer rates from the vapor into the liquid phase between 2 and about 1.5 m packing height. For these latter two components below 1.5 m packing height the mass-transfer rates undergo major changes. The Maxwell-Stefan relations predict that water is transferred from the vapor into the liquid phase. This is opposite for the fluctuation transport theory: isopropyl alcohol is transferred from the vapor into the liquid phase. Conclusions A theoretical derivation of distillation lines based on different methods for the estimation of mass transfer in packed distillation columns is performed. The traditional equilibrium stage concept assumes that the path of the distillation line depends only on the thermodynamic equilibrium. The fluctuation transport theory assumes that each component has an equal facility to mass transfer. Therefore, the path of the distillation line is only based on the thermodynamic equilibrium and how the driving force for mass transfer is described. If the concentration difference in the vapor phase is taken as the driving force, the fluctuation rectification lines obtained at total reflux operation of the column coincide with the residue curves. According to Maxwell-Stefan relations and two-film theory, the trajectory of the distillation lines depends on the thermodynamic equilibrium between the phases, on the distribution of the mass-transfer resistance between the phases, and on the diffusional interactions between the single species. The region close to the distillation boundary was found to be very sensitive to the activity coefficient models and their parameter data sets, to the experimental accuracy, and to the mathematical models applied. Therefore, simulations started close to the distillation boundary show that different products may be obtained when different simulation models or VLE models are used. The comparison with experiments was performed by simulating with the investigated mathematical models, different VLE models, and parameter data sets. For each investigated case the location of the
5398 Ind. Eng. Chem. Res., Vol. 36, No. 12, 1997
distillation boundary was correctly predicted with the Maxwell-Stefan theory for multicomponent diffusion. In most cases a controversial result was obtained with the equilibrium stage model and the fluctuation transport theory, however, depending very much on the VLE model selected. It remains to be seen which influence the diffusional interactions might have on the column design, for example, in azeotropic distillation. Moreover, how large the gap is between the distillation boundaries obtained with or without consideration of the diffusional interactions is still an unknown territory. These two topics are being further investigated by us. Symbols A ) interfacial area for mass transfer, m2 AV ) quantity defined by eq 35 ct ) mixture molar density, kmol/m3 D ) Maxwell-Stefan diffusivity for a binary pair, m2/s E ) energy transfer flux in a stationary coordinate frame of reference, W/m2 h ) heat-transfer coefficient, W/(m2 K) H ) enthalpy, J/kmol J ) molar diffusion flux of a component relative to the molar average velocity, kmol/(m2 s) K ) equilibrium ratio (K value) L ) liquid flow rate, kmol/s MV ) quantity defined by eq 31 N ) component molar transfer flux referred to a stationary coordinate reference frame, kmol/s Nt ) mixture molar transfer flux referred to a stationary coordinate reference frame, kmol/s q ) conductive heat flux, W/m2 SV ) quantity defined by eq 41 t ) time, s T ) temperature, K V ) vapor flow rate, kmol/s x ) liquid mole fraction in bulk phase, kmol/kmol xI ) liquid mole fraction at the phase interface, kmol/kmol y ) vapor mole fraction in bulk phase, kmol/kmol yI ) vapor mole fraction at the phase interface, kmol/kmol z ) packing height, m ∆z ) height of a packing segment, m Greek Letters R ) angle according to eq 46 δij ) Kronecker delta 1, if i ) k; 0 if i * k γ ) activity coefficient in solution Γ ) thermodynamic factor for a binary pair κ ) Maxwell-Stefan mass-transfer coefficient, m/s λ ) latent heat of vaporization, J/kmol φ ) fugacity coefficient in solution (n - 1)-Dimensional Vectors sssf
MSV ) vector defined by eq 43-45
s sf
VV ) vector defined by eq 32 (n - 1) × (n - 1) Dimensional Matrices [Γ] ) matrix of thermodynamic correction factors [I] ) identity matrix [k] ) matrix of multicomponent mass-transfer coefficients, m/s [R] ) matrix function of inverted binary mass-transfer coefficients, s/m [TV] ) matrix defined by eqs 37-40, m4/3/s2/3
L ) liquid m ) stage V ) vapor * ) equilibrium Mathematical Symbols d ) differential operator ∇ ) gradient ) arithmetic mean value ) vector B [ ] ) matrix
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Received for review May 9, 1997 Revised manuscript received August 19, 1997 Accepted August 21, 1997X IE970333D
Indices I ) interface i, j, k ) component
X Abstract published in Advance ACS Abstracts, November 15, 1997.
