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MULTICOMPONENT MASS TRANSFER
Chapter
9
MULTICOMPONENT MASS T R A N S F E R
The chapter shows how multicomponent mass transfer can be calculated from the multicomponent diffusion coefficients, the velocity field, and the b o u n d a r y conditions. The chief mathematical result is that multicomponent mass-transfer coefficients can be predicted from binary mass-transfer correlations and from the multicomponent diffusion coefficients. If the binary calculation successfully pre dicts mass transfer in the binary case, the multicomponent calculation will do so, t o o ; but if the binary calculation is inadequate, the ternary will be a disaster. This is, of course, the generalization of the case of the concentration profiles for specific experimental geometries discussed in Section 3.E, where a binary concentration profile and multicomponent diffusion coefficients were sufficient to allow these multicomponent profiles to be found by experiment. T h e calcula tion in this chapter is a close parallel but is in matrix notation, so that the earlier algebraic tedium is avoided. One may show that a binary correlation is sufficient to calculate multicomponent behavior for more macroscopic concepts as well. F o r example, one can predict multicomponent distillation eflSiciencies from a binary expression for efficiencies and from multicomponent mass-transfer co efficients (Toor, 1964). The multicomponent mass-transfer coefficients k used here are those suggested by Stewart and Prober (1964): nj = ^ . A £ o +
£iVi
(9.1)
where « i are the interfacial fluxes and ACQ are the concentration differences across the interface. The convective terms civi are not necessary at low mass-transfer rates, when the k are not a function of Vj. However, at moderate rates, k is definitely a function of Vj. The theory making this relation explicit is developed below. The development of this multicomponent theory proceeds in the same way as the more common binary developments (Geankopolis, 1972; Skelland, 1974). The concentration profiles involved are calculated in Section 9.A and used to find expressions for the mass-transfer coefficients in Section 9.B. As expected, multicomponent mass transfer is important when multicomponent diffusion is signi ficant. The largest effects will occur in concentrated solutions of strongly inter acting components of widely differing mobility, just as suggested by Table 5.1. F o u r examples of these effects are given in Section 9.C. The most important is the study of Standart et al (1975), which clearly shows the experimental im-
142 portance of multicomponent mass transfer. Other examples included are of the variation of multicomponent mass transfer with solution concentration, and of free convection caused by multicomponent diffusion. Finally, the combination of multicomponent mass transfer and chemical reaction is briefly discussed in Section 9.D. All four of these Sections provide the framework for further studies of multicomponent mass transfer. 9.A Concentration profiles The calculation of multicomponent mass-transfer coefflcients begins with con tinuity and flux equations, determines concentration profiles, and then a d a p t s these profiles to mass-transfer situations. It has been developed largely t h r o u g h the independent efforts of Toor (1964a, 1964b) and of Stewart and Prober (1964). F o r the development here, we again choose the generalized F i c k ' s law form of the flux equations, although the solution of the generalized Stefan-Maxwell equations is b o t h available and completely equivalent (Stewart and Prober, 1964). Our choice of flux equations is, in matrix notation (Toor, 1964) - ; = D.yc
(9.2)
The continuity equations for this case are - | - +
( V
» £ ) = - / ? ·
These are subject to the initial and boundary conditions: Ac (x, y,z,t
= 0) = Aco
(9.4)
Ac ( 5 , 0 = 0
(9.5)
(¿,0 = 0
(9.6)
ds
where Β and b represent two boundaries of the system. These boundary conditions are written as differences so that the resulting mathematical forms are somewhat simpler (Toor, 1964); they could just as easily be written as concentrations. How ever, the calculation given here requires that the boundary conditions on all con centrations must be written in the same functional form. This is a serious restric tion only in the case of simultaneous chemical reaction discussed in Section 9.D. The reformulation
of the basic
equation
We now assume that there exists a non-singular matrix / which can diagonalize D:
143 σι
0
0
0
^2
0
0
0
^3
(9.7)
where is the inverse of tj and σ is the diagonal matrix of the eigenvalues of the diffusion coefficient matrix ^ . The assumption that D can be diagonahzed is not necessary for a general mathematical solution, but since this assumption is valid for all cases encountered in practice, it is used here. Defining a new com bined concentration Ψ a s : =
(9.8)
/.Ψ
we can combine eqns.(9.2), (9.3), and (9.8) and premultiply the equation by t-^ to obtain : (9.9)
dt which represents a set of scalar equations
(9.10)
et
In this operation, we have assumed that and hence b o t h ί and ^ , are not func tions of composition. This is a good assumption and has been used t h r o u g h o u t the rest of the book. However, it means that these mathematical results based on this assumption are inferior to exact solutions of special cases (e.g. Toor, 1957; Hsu and Bird, 1960; Billingsley, 1974). The initial and b o u n d a r y conditions can also be written in terms of the new com bined concentration Ψ Δ Ψ (χ, y, ζ,Ο) = Δ Ψ ο
=^-'·Δ5ο
(9.11)
Δ Ψ (Β, ί) = Ο
(9.12)
(b, 0 = 0
(9.13)
ds
Thus, a set of coupled differential equations has been separated into uncoupled equations written in terms of the new concentration Ψ.
144 The solution
of the
equations
Equations (9.10)-(9.13) have exactly the same form as t h e associated diffusion p r o b l e m :
binary
de. dt which has the same initial and b o u n d a r y conditions for each species given in eqns. (9.4)-(9.6). If this binary problem has t h e solution: Ac I = FiD,x,t)Aci^
(9.15)
eqns,(9.10)-(9.13) must have the solution: ΔΨί = ^ ( σ ί , χ , Ο Δ Ψ ^ ο
(9.16)
where the eigenvalue σ-^ is substituted everywhere that the binary diffusion co efficient occurs in the binary solution. If we rewrite our solution in terms of the actual concentrations by combining eqns.(9.8), (9.11)-(9.13), and (9.16) Ac = tjF(a_, X, O-^-'-Aco
(9.17)
Thus, we know the concentration profiles in a multicomponent system in terms of the binary solution to the problem, where the binary diffusion coefficient is replaced in t u r n by each eigenvalue of the diffusion coefficient matrix. This derivation is presented briefly because it is the complete analogue of the ternary case developed algebraically in Section 3.E. In t h a t case, we were inte rested in specific experimental geometries capable of measuring ternary diffusion coefficients. Since some experimentaHsts may be unfamiliar with matrices, those problems are solved in completely algebraic terms. Because, in this Section, we are interested in mass-transfer coefficients u n d e r a wide variety of conditions, we have used the b r o a d e r mathematical formulation given by matrices. 9.B Multicomponent mass-transfer coefficients A r m e d with these concentration profiles, we now want t o calculate t h e rates of mass transfer at a boundary. T h e binary calculation of these rates takes three steps. First, the flux relative to fixed coordinates m is calculated from the con centration profile. Second, this flux is evaluated at the interface. Third, the result ing equation is forced into the form: nil = kiAcio
+
ViCii
where the mass-transfer coefficient sion coefficient, the velocity field, similar. Initially, the fluxes n, are binary-like flux equation. Then the
(9.18) is now a k n o w n function of the binary diffu etc. T h e multicomponent calculation is very translated into another form which follows a three steps above can be paralleled to give ex-
145 pressions for a binary-like mass-transfer coefBcient. Finally, the fluxes tin are recovered to give eqn.(9.1), where the various mass-transfer coefficients are now known functions of the multicomponent diff'usion coefficients. The reformulation
of the
fluxes
Quantitatively, the fluxes m are translated into a binary-like form by defining a new set of fluxes η ^t^^-n
(9.19)
= _^-U; + v«c) F r o m eqns.(9.2) and (9.8) we see that these fluxes have a binary-hke form : η = σ.γΨ+ν'Ψ
(9.20)
which represents a set of scalar equations m = σινΨί + ν ^ Ψ ί
(9.21)
In the preceding Section, we found that the concentration profile of Ψ has a binary form; hence the mass-transfer rate equation in terms of Ψ must also have a binary form parallel to eqn.(9.18): η,, = κ^ΔΨίο + νχΨίι
(9.22)
where the are mass-transfer coefficients which are the same functions of eigen value σι, velocity field, etc., as the binary value k is of D, velocity field, etc., for the particular geometry in question. In other words, if: k=f{D,c,,,v\B,
...)
(9.23)
then: = / ( σ i , Ψ i I , V ^ 5 , ...)
(9.24)
where / is exactly the same function in b o t h cases. In matrix form, eqn.(2.22) is: ^ , = κ . Δ Ψ ο + νρΨι
(9.25)
where κ is a diagonal matrix. At low mass-transfer rates, κ will not be a function of velocity Vj. F r o m the multicomponent diff'usion coefficients and a binary correlation, we have thus calculated the mass-transfer coefficients for the combined concentrations.
146 Individual mass-transfer
coefficients
T h e multicomponent mass-transfer coefficients in terms of the actual concen tration differences in the system can be found by combining eqns.(9.8), (9.11), (9.19), and (9.25) to rewrite the mass-transfer rates in terms of the fluxes and con centration gradients of the species physically present in the solution: i h = £ ^ κ · r ' ] · Δ c o + Vi£i
(9.26)
where the elements of ^ a r e the multicomponent mass-transfer coefficients equal t o the quantity in the square bracket. M o r e explicitly, for the ternary case, this result is:
\ ^2
^22
- σι /
=
w h e r e / i s again the analogue to the binary correlation suggested by eqns.(9.23)(9.24). If the cross-term diffusion coefficients are small, J will approximate a unit matrix and fc_will a p p r o a c h a diagonal matrix. As in diffusion, the m o r e impor tant cases, with large off-diagonal terms, will occur when the solutions contain strongly interacting solutes of widely different mobility, or when one concentra tion difference is much larger t h a n any others. Overall mass-transfer
coefficients
In many practical situations, we may wish to calculate the overall mass-transfer r a t e from one fluid phase into another. W e then must develop a scheme for com bining the mass-transfer coefficients on one side of the interface with those on t h e other side, i.e. we want a way t o add the resistances to mass transfer on b o t h sides of the interface. The physical significance of this calculation is simplified if we rewrite the flux equations in terms of mole fractions in the two phases rather t h a n concentrations «i =
V(7-2;,)
«I =
K
-'χ)
(9.31) (9-32)
147 The values of jt, are equal in these two equations, and the experimentally in accessible interfacial concentrations are interrelated: yi = M^xi
(9.33)
The equihbrium constant matrix M , a type of Henry's law coefficient, will com monly be a function of concentration, since currently it implies only linear variations between the two mole fractions. T h e mole fractions y* in equilibrium with X will also be determined by J * = Μ·~χ
(9.34)
If we ignore the concentration dependence of Μ and assume an average value, we can combine eqns.(9.31)-(9.34) to eliminate j j , .Vj, and x : «i=£,.(7~^*)
(9.35)
where K^-i
= k^-i
+ M^.ky^
(9.36)
Thus the single phase resistances are additive in matrix terms, just as they were additive in algebraic terms in the binary case. As in this binary case, one resis tance controls if the resistance in the other phase is negligible; however, the con ditions for this control are m o r e stringent in multicomponent systems. However, this theoretical calculation has not been adequately checked experimentally a n d t h u s , should be viewed with caution (von Behren et al., 1972; Stewart, 1973). 9.C Examples of multicomponent mass transfer Ternary
liquid-liquid
extraction
One dramatic example of multicomponent mass transfer occurs in the system glycerol-water-acetone (Standart et al., 1975). Since this system is not miscible in all proportions, large multicomponent diffusion effects should occur in con centrated solution, particularly near the critical point (cf. Table 5.1 and Section 6.B). These effects are illustrated in Fig.9.1. In this Figure, two solutions whose initial compositions are shown by stars are allowed to interdiffuse. If the multicomponent mass-transfer matrix k_ contains only diagonal terms and if equili brium exists across the interface, then t h e solution concentrations will change with time as shown by the straight dotted lines. However, the solution concentrations actually follow the curved solid lines shown. Moreover, t h e curvature of t h e lines is in the opposite direction t o t h a t expected for interfacial concentrations which are not at equilibrium (Standart
148
Fig. 9.1. An example of multicomponent mass transfer. Two solutions whose compositions are given by the stars are allowed to diffuse together. According to binary theory, their compositions should follow the short dashed lines; in fact, they follow the solid lines. The long dashed line is a tie line. (Redrawn, with permission, from Standart et al., 1975)
et al., 1974). This behavior can only be explained by large m u h i c o m p o n e n t effects. M o r e explicitly, it requires large off-diagonal terms in the m u h i c o m p o n e n t diffusion matrix, i.e. ky^^ ¡^j must be non-zero. Typical values of these coefficients are shown in Table 9.1 for two different concentrations of this system. This Table includes four coefficients in the acetonerich region and four more in the glycerol-rich region. However, these should not be regarded as eight arbitrarily adjustable parameters because they were TABLE 9.1 Ternary mass-transfer coefficients for gycerol(l)-water(2)-acetone(3)« Acetone-rich region avg. mol. frac. " 1 " avg. mol. frac. *'2"
Arn
""The A^jj are in 10-^ cm/sec.
0.04 0.10 4.7 10.3 8.0 27.0
Glycerol-rich region 0.45 0.37 15.7 - 0.7 - 5.3 4.7
149
evaluated assuming that the Onsager reciprocal relations are valid. Just like the ternary diffusion coefficients, these ternary mass-transfer coefficients are strong functions of concentration. However, this concentration dependence does not cause serious problems, as shown in the following. Ternary
concentration
profiles
In the previous Section, as well as m u c h of the rest of the book, we have often assumed t h a t the matrix D is independent of composition, even t h o u g h we know this is an approximation. Because this assumption is so important, it has been thoroughly tested and found accurate, especially in liquids (Fujita and Gosting, 1960; Edwards et al, 1966). This suggests that the k matrix can also be assumed to be independent of composition. However, because diffusion coefficients in liquids do not frequently vary sharply with concentration, a m u c h m o r e stringent test can be m a d e in the gaseous system m e t h a n e - a r g o n - h y d r o g e n (Arnold and Toor, 1967). Because hydrogen is much m o r e mobile t h a n the other two gases, this system shows very large cross-term diffusion coeflScients which are also strong functions of concentration. A typical experiment with this system uses two tubes. One tube initially con tains an equimolar mixture of hydrogen and methane, while the other contains an equimolar mixture of argon and methane. To start an experiment, the ends of the tube are connected; to conclude it, the tubes are separated. While no concen tration difference of methane exists initially, one develops with time. The size of this methane concentration difference can be calculated in two ways. First, it can be exactly predicted by numerically integrating the Stefan-Maxwell equations (eqn.(3.1)). Second, it can be approximately predicted by first finding D from eqns.(5.7)-(5.10), assuming this matrix is independent of composition, and finally calculating the concentration difference from the theory in Section 3.E or 9.A. Both approaches give the same resuhs, as shown in Fig.9.2. Indeed, the two calculations are almost indistinguishable, and b o t h agree closely with experimental results. A similar experiment shown in Fig. 1.2 gives equivalent results. Accord ingly, the variation of D with composition is not a major problem. Average values give satisfactory results. Convection
caused by
diffusion
Multicomponent diffusion can cause free convection even when the heaviei solution is initially below the fighter solution. In principle, this effect could be used to expedite extractions hke that mentioned earlier in this Section. However, in spite of its large size, it has been treated as an experimental annoyance rather t h a n a potential way of accelerating mass transfer. T o illustrate how this effect can occur, we consider the ternary free diffusion experiment shown in Fig.9.3. In this experiment, solute " Γ ' initially has a large
150
050
0.48
(J
0.46
0.44 10
20 30 40 Time, mi ñutes
50
60
50
60
0.42
0.38 h 034
030
026 10
20
30
40
Time, minutes
Fig. 9.2. Concentration changes for methane(I)-argon-(2)-hydrogen (3). Tubes containing equimolar mixtures of argon-methane and of hydrogen-methane were connected and then separated after a given time. The data for this situation are compared with the numerical solution of the Stefan-Maxwell equations (eqn.(3.1) and solid lines) and the solution of the generalized Fick's law equations with constant (eqn.(9.2) and dotted lines). Both predictions in this non-reacting system are excellent. Initial profiles
- Density
Profiles after α given time
-Solute'T - Solute "2" - Gravity
Density
Gravity Fig. 9.3. Free convection caused by ternary diffusion. An unstable density gradient can result from ternary diffusion, so that mass transfer can be greatly expedited.
151 concentration gradient b u t solute " 2 " has no initial gradient. As the experiment proceeds, the concentration gradient of solute " 1 " becomes smaller in m o r e or less the expected fashion. However, this gradient of solute " 1 " in turn generates a considerable flux of solute " 2 " t o produce the concentration profile shown. This ternary eff'ect produces a density profile with a m a x i m u m and a minimum. If this density profile now involves a lighter solution lying below a heavier one, free convection is possible. Free convection has, in fact, been observed in the systems sodium sulfate-sulfuric acid-water (Wendt, 1965) a n d polystyrene-cyclohexanetoluene (Cussler, 1965). Two equal
eigenvalues
The final example of the theory above concerns a mathematical anomaly rather t h a n a physically significant event. W h e n there are repeated eigenvalues in a ter nary system, it may be impossible to diagonalize the matrix D. However, this diagonalization turns out to be unnecessary to solve the problem. Details of the general solution are given elsewhere (Toor, 1964); in expanded form, the ternary result is Ac, =
F(a)
+ {Dn
-
σ)
dF bo
Acio
+
(9.37)
+
dF 2
da
Ac 20
The calculation of mass-transfer rates from this profile is straightforward. W e must re-emphasize that this situation is very unlikely to occur in practice. 9.D Multicomponent mass transfer with chemical reaction Although multicomponent mass transfer can couple strongly with chemical reaction, these interesting and potentially large effects have not been fully studied experimentally. The effects can be large because reactions occur with such a wide variety of rates and mechanisms. Diffusion coefiicients are depressingly similar, rarely varying by more t h a n an order of magnitude in any given system, and crossterm diffusion coefficients are often small. Even a simple second-order reaction offers strong coupling and a widely varying rate constant. However, in contrast with the clean and complete derivation for mass transfer without reaction, the theory in this case is approximate and limited. This lack of generality results from two serious restrictions. First, the only reactions which can be included have isothermal first-order mechanisms, since these imply linear equations which can be handled by general matrix techniques. This means t h a t two of the most interesting cases, second-order reversible reactions and first-order catalytic C'Michaelis-Menton") mechanisms, have not been treated. Second,
152 description of this situation requires b o t h a matrix of diffusion coefficients and a matrix of reaction-rate coefficients; these two matrices cannot be simultaneously diagonahzed, except in special cases. Heterogeneous
reactions
Cases with either multicomponent diffusion or coupled heterogeneous first-order reaction are simpler than the case where b o t h are coupled. Problems in which chemical reactions are coupled, but diffusion follows the binary form of Fick's law, are very completely covered in the chemical engineering literature for a wide variety of reaction mechanisms {e.g. Astarita, 1967). Problems in which crossterm diffusion coefficients are present and the chemical reactions have the form of eqns.(9.4)-(9.6), can be handled with the theory in Section 9.A. When b o t h heterogeneous reaction and diffusion are coupled, the continuity e q u a t i o n s : dc it + V
V«f = f
d^c ^
(9-38)
are subject to the initial and boundary conditions: Ac = Acp a t / ( ; c , r) = 0
(9.39)
^
(9.40)
= ^K-fat;c = 0
where is the matrix of reaction-rate coefficients, limited in his case to firstorder reactions. The general solution of these equations requires diagonalizing b o t h D a n d ^ R , which is possible only if b o t h have the same eigenvectors and, hence, commute. This is very unlikely physically. A complete solution for coupled multicomponent diffusion with heterogeneous reaction is available for the special case of one-dimensional unsteady semi-in finite diffusion without convection (Hudson, 1967). The method of solution is first to diagonalize eqn.(9.38) with ν = 0, and then to take the Laplace trans form o f t h e result. The value of c at the boundary can then be evaluated without diagonalizing ¿R. While the method does not work for more general problems, its solution for this special case is superior to the approximate calculation discussed next. An approximate general solution in this case (Parkin, 196^) can be found by writing the overall flux equations as shown in eqn.(9.1), and then assuming that the mass-transfer coefficients are independent of the interfacial concentrations, but are given by the same empirical expressions as in the case of n o reaction, i.e. by eqn.(9.26). This approximation is only qualitatively successful as shown in Fig.9.4. The system studied was the reaction in the presence of helium of gaseous iodine with solid germanium fixed to the surface of a rotating disc, p r o ducing gaseous germanium iodine (Olander, 1967). The agreement between the approximate solution and the experimental results differs by as much as 3 0 % .
153
o
0.2
04
06
0.8
10
Concentration of Iodine in Bulk
Fig. 9.4. Coupled ternary diffusion and chemical reaction. The agreement between calculated and observed fluxes is sharply poorer when chemical reactions are present than when they are not {cf. Fig. 9.2).
The decomposition of iron pentacarbonyl in the presence of argon gave similar errors (Carlton and Oxley, 1965). Homogeneous
reactions
Problems involving coupled diffusion and homogeneous reaction have also not been solved in general. Again, the mathematical difficulty comes from having both a matrix of diffusion coefficients and one of reaction-rate constants. The ease where homogeneous reactions are coupled but where the off-diagonal diffusion coefficients are negligible is probably the most frequently encountered. If t h e reactions are isothermal and first-order, the system is described by the equations: ^
+ V V V
= ^'V^
+ k^'C
(9.41)
where is now the matrix of the homogeneous first-order reaction rate constants. The boundary conditions for this problem are again those in eqns.(9.4)-(9.6). These equations can be solved analytically only for two special cases. The first case occurs at steady state with no convection (deLancey and Chiang, 1970; Toor, 1965) 0 = v'c+
Z)-i.^R.^
(9.42)
This equation can be diagonalized and solved by paralleling the development in Section 9.A.. Problems of this type when D is a diagonal matrix have been carefully analyzed by Wei (1962).
154 The second special case of eqn. (9.41) is that of diffusion without convection
Ir
=fVf+ÍK-c
(9-43)
Mass transfer in or out of the system can also be included in (Gmitro a n d Scriven, 1966). Solutions can be found from harmonic analysis ( G m i t r o a n d Scriven, 1966; Othmer and Scriven, 1969), or, for the one-dimensional case, by a separation-of-variables matrix technique (Toor, 1965). The chief feature of these solutions is a dependence on a new set of eigenvalues which, in turn, depend on combinations of elements of D and k^. These new eigenvalues apparently allow a wide variety of behavior. F o r thermodynamically ideal systems near equilibrium, the second law and the principle of microscopic reversibility imply that these eigenvalues are real and positive, and hence that oscillatory behavior is impossible (Toor, 1965). However, derivations based on microscopic reversibihty are not generally successful for treating chemical reactions. F o r systems far from equilibrium, sustained oscillations are possible (Solomon and H u d s o n , 1971). Such a mechanism has been suggested to explain the oscillatory movements of surfaces such as the periodic movements observed in living systems (Gmitro and Scriven, 1966; Othmer and Scriven, 1969; Desimone et al., 1973). 9.£ Conclusions Most of the conclusions to this chapter are less definite than others in this book. The most definite is that the multicomponent diff'usion coefficients may be assumed constant without serious approximation. Beyond this, the situation is cloudier. M u h i c o m p o n e n t mass-transfer coefficients can apparently be predicted from binary mass-transfer correlations and multicomponent diff'usion coefficients. These coefficients can produce major eff'ects, at least in the few chemical systems studied to date. When b o t h mass transfer a n d chemical systems occur simuUaneously, other events are possible, including oscillating concentrations. However, the entire subject depends on so much elaborate mathematics and so few definitive experiments that these conclusions must be somewhat tentative.
9
MULTICOMPONENT MASS T R A N S F E R
The chapter shows how multicomponent mass transfer can be calculated from the multicomponent diffusion coefficients, the velocity field, and the b o u n d a r y conditions. The chief mathematical result is that multicomponent mass-transfer coefficients can be predicted from binary mass-transfer correlations and from the multicomponent diffusion coefficients. If the binary calculation successfully pre dicts mass transfer in the binary case, the multicomponent calculation will do so, t o o ; but if the binary calculation is inadequate, the ternary will be a disaster. This is, of course, the generalization of the case of the concentration profiles for specific experimental geometries discussed in Section 3.E, where a binary concentration profile and multicomponent diffusion coefficients were sufficient to allow these multicomponent profiles to be found by experiment. T h e calcula tion in this chapter is a close parallel but is in matrix notation, so that the earlier algebraic tedium is avoided. One may show that a binary correlation is sufficient to calculate multicomponent behavior for more macroscopic concepts as well. F o r example, one can predict multicomponent distillation eflSiciencies from a binary expression for efficiencies and from multicomponent mass-transfer co efficients (Toor, 1964). The multicomponent mass-transfer coefficients k used here are those suggested by Stewart and Prober (1964): nj = ^ . A £ o +
£iVi
(9.1)
where « i are the interfacial fluxes and ACQ are the concentration differences across the interface. The convective terms civi are not necessary at low mass-transfer rates, when the k are not a function of Vj. However, at moderate rates, k is definitely a function of Vj. The theory making this relation explicit is developed below. The development of this multicomponent theory proceeds in the same way as the more common binary developments (Geankopolis, 1972; Skelland, 1974). The concentration profiles involved are calculated in Section 9.A and used to find expressions for the mass-transfer coefficients in Section 9.B. As expected, multicomponent mass transfer is important when multicomponent diffusion is signi ficant. The largest effects will occur in concentrated solutions of strongly inter acting components of widely differing mobility, just as suggested by Table 5.1. F o u r examples of these effects are given in Section 9.C. The most important is the study of Standart et al (1975), which clearly shows the experimental im-
142 portance of multicomponent mass transfer. Other examples included are of the variation of multicomponent mass transfer with solution concentration, and of free convection caused by multicomponent diffusion. Finally, the combination of multicomponent mass transfer and chemical reaction is briefly discussed in Section 9.D. All four of these Sections provide the framework for further studies of multicomponent mass transfer. 9.A Concentration profiles The calculation of multicomponent mass-transfer coefflcients begins with con tinuity and flux equations, determines concentration profiles, and then a d a p t s these profiles to mass-transfer situations. It has been developed largely t h r o u g h the independent efforts of Toor (1964a, 1964b) and of Stewart and Prober (1964). F o r the development here, we again choose the generalized F i c k ' s law form of the flux equations, although the solution of the generalized Stefan-Maxwell equations is b o t h available and completely equivalent (Stewart and Prober, 1964). Our choice of flux equations is, in matrix notation (Toor, 1964) - ; = D.yc
(9.2)
The continuity equations for this case are - | - +
( V
» £ ) = - / ? ·
These are subject to the initial and boundary conditions: Ac (x, y,z,t
= 0) = Aco
(9.4)
Ac ( 5 , 0 = 0
(9.5)
(¿,0 = 0
(9.6)
ds
where Β and b represent two boundaries of the system. These boundary conditions are written as differences so that the resulting mathematical forms are somewhat simpler (Toor, 1964); they could just as easily be written as concentrations. How ever, the calculation given here requires that the boundary conditions on all con centrations must be written in the same functional form. This is a serious restric tion only in the case of simultaneous chemical reaction discussed in Section 9.D. The reformulation
of the basic
equation
We now assume that there exists a non-singular matrix / which can diagonalize D:
143 σι
0
0
0
^2
0
0
0
^3
(9.7)
where is the inverse of tj and σ is the diagonal matrix of the eigenvalues of the diffusion coefficient matrix ^ . The assumption that D can be diagonahzed is not necessary for a general mathematical solution, but since this assumption is valid for all cases encountered in practice, it is used here. Defining a new com bined concentration Ψ a s : =
(9.8)
/.Ψ
we can combine eqns.(9.2), (9.3), and (9.8) and premultiply the equation by t-^ to obtain : (9.9)
dt which represents a set of scalar equations
(9.10)
et
In this operation, we have assumed that and hence b o t h ί and ^ , are not func tions of composition. This is a good assumption and has been used t h r o u g h o u t the rest of the book. However, it means that these mathematical results based on this assumption are inferior to exact solutions of special cases (e.g. Toor, 1957; Hsu and Bird, 1960; Billingsley, 1974). The initial and b o u n d a r y conditions can also be written in terms of the new com bined concentration Ψ Δ Ψ (χ, y, ζ,Ο) = Δ Ψ ο
=^-'·Δ5ο
(9.11)
Δ Ψ (Β, ί) = Ο
(9.12)
(b, 0 = 0
(9.13)
ds
Thus, a set of coupled differential equations has been separated into uncoupled equations written in terms of the new concentration Ψ.
144 The solution
of the
equations
Equations (9.10)-(9.13) have exactly the same form as t h e associated diffusion p r o b l e m :
binary
de. dt which has the same initial and b o u n d a r y conditions for each species given in eqns. (9.4)-(9.6). If this binary problem has t h e solution: Ac I = FiD,x,t)Aci^
(9.15)
eqns,(9.10)-(9.13) must have the solution: ΔΨί = ^ ( σ ί , χ , Ο Δ Ψ ^ ο
(9.16)
where the eigenvalue σ-^ is substituted everywhere that the binary diffusion co efficient occurs in the binary solution. If we rewrite our solution in terms of the actual concentrations by combining eqns.(9.8), (9.11)-(9.13), and (9.16) Ac = tjF(a_, X, O-^-'-Aco
(9.17)
Thus, we know the concentration profiles in a multicomponent system in terms of the binary solution to the problem, where the binary diffusion coefficient is replaced in t u r n by each eigenvalue of the diffusion coefficient matrix. This derivation is presented briefly because it is the complete analogue of the ternary case developed algebraically in Section 3.E. In t h a t case, we were inte rested in specific experimental geometries capable of measuring ternary diffusion coefficients. Since some experimentaHsts may be unfamiliar with matrices, those problems are solved in completely algebraic terms. Because, in this Section, we are interested in mass-transfer coefficients u n d e r a wide variety of conditions, we have used the b r o a d e r mathematical formulation given by matrices. 9.B Multicomponent mass-transfer coefficients A r m e d with these concentration profiles, we now want t o calculate t h e rates of mass transfer at a boundary. T h e binary calculation of these rates takes three steps. First, the flux relative to fixed coordinates m is calculated from the con centration profile. Second, this flux is evaluated at the interface. Third, the result ing equation is forced into the form: nil = kiAcio
+
ViCii
where the mass-transfer coefficient sion coefficient, the velocity field, similar. Initially, the fluxes n, are binary-like flux equation. Then the
(9.18) is now a k n o w n function of the binary diffu etc. T h e multicomponent calculation is very translated into another form which follows a three steps above can be paralleled to give ex-
145 pressions for a binary-like mass-transfer coefBcient. Finally, the fluxes tin are recovered to give eqn.(9.1), where the various mass-transfer coefficients are now known functions of the multicomponent diff'usion coefficients. The reformulation
of the
fluxes
Quantitatively, the fluxes m are translated into a binary-like form by defining a new set of fluxes η ^t^^-n
(9.19)
= _^-U; + v«c) F r o m eqns.(9.2) and (9.8) we see that these fluxes have a binary-hke form : η = σ.γΨ+ν'Ψ
(9.20)
which represents a set of scalar equations m = σινΨί + ν ^ Ψ ί
(9.21)
In the preceding Section, we found that the concentration profile of Ψ has a binary form; hence the mass-transfer rate equation in terms of Ψ must also have a binary form parallel to eqn.(9.18): η,, = κ^ΔΨίο + νχΨίι
(9.22)
where the are mass-transfer coefficients which are the same functions of eigen value σι, velocity field, etc., as the binary value k is of D, velocity field, etc., for the particular geometry in question. In other words, if: k=f{D,c,,,v\B,
...)
(9.23)
then: = / ( σ i , Ψ i I , V ^ 5 , ...)
(9.24)
where / is exactly the same function in b o t h cases. In matrix form, eqn.(2.22) is: ^ , = κ . Δ Ψ ο + νρΨι
(9.25)
where κ is a diagonal matrix. At low mass-transfer rates, κ will not be a function of velocity Vj. F r o m the multicomponent diff'usion coefficients and a binary correlation, we have thus calculated the mass-transfer coefficients for the combined concentrations.
146 Individual mass-transfer
coefficients
T h e multicomponent mass-transfer coefficients in terms of the actual concen tration differences in the system can be found by combining eqns.(9.8), (9.11), (9.19), and (9.25) to rewrite the mass-transfer rates in terms of the fluxes and con centration gradients of the species physically present in the solution: i h = £ ^ κ · r ' ] · Δ c o + Vi£i
(9.26)
where the elements of ^ a r e the multicomponent mass-transfer coefficients equal t o the quantity in the square bracket. M o r e explicitly, for the ternary case, this result is:
\ ^2
^22
- σι /
=
w h e r e / i s again the analogue to the binary correlation suggested by eqns.(9.23)(9.24). If the cross-term diffusion coefficients are small, J will approximate a unit matrix and fc_will a p p r o a c h a diagonal matrix. As in diffusion, the m o r e impor tant cases, with large off-diagonal terms, will occur when the solutions contain strongly interacting solutes of widely different mobility, or when one concentra tion difference is much larger t h a n any others. Overall mass-transfer
coefficients
In many practical situations, we may wish to calculate the overall mass-transfer r a t e from one fluid phase into another. W e then must develop a scheme for com bining the mass-transfer coefficients on one side of the interface with those on t h e other side, i.e. we want a way t o add the resistances to mass transfer on b o t h sides of the interface. The physical significance of this calculation is simplified if we rewrite the flux equations in terms of mole fractions in the two phases rather t h a n concentrations «i =
V(7-2;,)
«I =
K
-'χ)
(9.31) (9-32)
147 The values of jt, are equal in these two equations, and the experimentally in accessible interfacial concentrations are interrelated: yi = M^xi
(9.33)
The equihbrium constant matrix M , a type of Henry's law coefficient, will com monly be a function of concentration, since currently it implies only linear variations between the two mole fractions. T h e mole fractions y* in equilibrium with X will also be determined by J * = Μ·~χ
(9.34)
If we ignore the concentration dependence of Μ and assume an average value, we can combine eqns.(9.31)-(9.34) to eliminate j j , .Vj, and x : «i=£,.(7~^*)
(9.35)
where K^-i
= k^-i
+ M^.ky^
(9.36)
Thus the single phase resistances are additive in matrix terms, just as they were additive in algebraic terms in the binary case. As in this binary case, one resis tance controls if the resistance in the other phase is negligible; however, the con ditions for this control are m o r e stringent in multicomponent systems. However, this theoretical calculation has not been adequately checked experimentally a n d t h u s , should be viewed with caution (von Behren et al., 1972; Stewart, 1973). 9.C Examples of multicomponent mass transfer Ternary
liquid-liquid
extraction
One dramatic example of multicomponent mass transfer occurs in the system glycerol-water-acetone (Standart et al., 1975). Since this system is not miscible in all proportions, large multicomponent diffusion effects should occur in con centrated solution, particularly near the critical point (cf. Table 5.1 and Section 6.B). These effects are illustrated in Fig.9.1. In this Figure, two solutions whose initial compositions are shown by stars are allowed to interdiffuse. If the multicomponent mass-transfer matrix k_ contains only diagonal terms and if equili brium exists across the interface, then t h e solution concentrations will change with time as shown by the straight dotted lines. However, the solution concentrations actually follow the curved solid lines shown. Moreover, t h e curvature of t h e lines is in the opposite direction t o t h a t expected for interfacial concentrations which are not at equilibrium (Standart
148
Fig. 9.1. An example of multicomponent mass transfer. Two solutions whose compositions are given by the stars are allowed to diffuse together. According to binary theory, their compositions should follow the short dashed lines; in fact, they follow the solid lines. The long dashed line is a tie line. (Redrawn, with permission, from Standart et al., 1975)
et al., 1974). This behavior can only be explained by large m u h i c o m p o n e n t effects. M o r e explicitly, it requires large off-diagonal terms in the m u h i c o m p o n e n t diffusion matrix, i.e. ky^^ ¡^j must be non-zero. Typical values of these coefficients are shown in Table 9.1 for two different concentrations of this system. This Table includes four coefficients in the acetonerich region and four more in the glycerol-rich region. However, these should not be regarded as eight arbitrarily adjustable parameters because they were TABLE 9.1 Ternary mass-transfer coefficients for gycerol(l)-water(2)-acetone(3)« Acetone-rich region avg. mol. frac. " 1 " avg. mol. frac. *'2"
Arn
""The A^jj are in 10-^ cm/sec.
0.04 0.10 4.7 10.3 8.0 27.0
Glycerol-rich region 0.45 0.37 15.7 - 0.7 - 5.3 4.7
149
evaluated assuming that the Onsager reciprocal relations are valid. Just like the ternary diffusion coefficients, these ternary mass-transfer coefficients are strong functions of concentration. However, this concentration dependence does not cause serious problems, as shown in the following. Ternary
concentration
profiles
In the previous Section, as well as m u c h of the rest of the book, we have often assumed t h a t the matrix D is independent of composition, even t h o u g h we know this is an approximation. Because this assumption is so important, it has been thoroughly tested and found accurate, especially in liquids (Fujita and Gosting, 1960; Edwards et al, 1966). This suggests that the k matrix can also be assumed to be independent of composition. However, because diffusion coefficients in liquids do not frequently vary sharply with concentration, a m u c h m o r e stringent test can be m a d e in the gaseous system m e t h a n e - a r g o n - h y d r o g e n (Arnold and Toor, 1967). Because hydrogen is much m o r e mobile t h a n the other two gases, this system shows very large cross-term diffusion coeflScients which are also strong functions of concentration. A typical experiment with this system uses two tubes. One tube initially con tains an equimolar mixture of hydrogen and methane, while the other contains an equimolar mixture of argon and methane. To start an experiment, the ends of the tube are connected; to conclude it, the tubes are separated. While no concen tration difference of methane exists initially, one develops with time. The size of this methane concentration difference can be calculated in two ways. First, it can be exactly predicted by numerically integrating the Stefan-Maxwell equations (eqn.(3.1)). Second, it can be approximately predicted by first finding D from eqns.(5.7)-(5.10), assuming this matrix is independent of composition, and finally calculating the concentration difference from the theory in Section 3.E or 9.A. Both approaches give the same resuhs, as shown in Fig.9.2. Indeed, the two calculations are almost indistinguishable, and b o t h agree closely with experimental results. A similar experiment shown in Fig. 1.2 gives equivalent results. Accord ingly, the variation of D with composition is not a major problem. Average values give satisfactory results. Convection
caused by
diffusion
Multicomponent diffusion can cause free convection even when the heaviei solution is initially below the fighter solution. In principle, this effect could be used to expedite extractions hke that mentioned earlier in this Section. However, in spite of its large size, it has been treated as an experimental annoyance rather t h a n a potential way of accelerating mass transfer. T o illustrate how this effect can occur, we consider the ternary free diffusion experiment shown in Fig.9.3. In this experiment, solute " Γ ' initially has a large
150
050
0.48
(J
0.46
0.44 10
20 30 40 Time, mi ñutes
50
60
50
60
0.42
0.38 h 034
030
026 10
20
30
40
Time, minutes
Fig. 9.2. Concentration changes for methane(I)-argon-(2)-hydrogen (3). Tubes containing equimolar mixtures of argon-methane and of hydrogen-methane were connected and then separated after a given time. The data for this situation are compared with the numerical solution of the Stefan-Maxwell equations (eqn.(3.1) and solid lines) and the solution of the generalized Fick's law equations with constant (eqn.(9.2) and dotted lines). Both predictions in this non-reacting system are excellent. Initial profiles
- Density
Profiles after α given time
-Solute'T - Solute "2" - Gravity
Density
Gravity Fig. 9.3. Free convection caused by ternary diffusion. An unstable density gradient can result from ternary diffusion, so that mass transfer can be greatly expedited.
151 concentration gradient b u t solute " 2 " has no initial gradient. As the experiment proceeds, the concentration gradient of solute " 1 " becomes smaller in m o r e or less the expected fashion. However, this gradient of solute " 1 " in turn generates a considerable flux of solute " 2 " t o produce the concentration profile shown. This ternary eff'ect produces a density profile with a m a x i m u m and a minimum. If this density profile now involves a lighter solution lying below a heavier one, free convection is possible. Free convection has, in fact, been observed in the systems sodium sulfate-sulfuric acid-water (Wendt, 1965) a n d polystyrene-cyclohexanetoluene (Cussler, 1965). Two equal
eigenvalues
The final example of the theory above concerns a mathematical anomaly rather t h a n a physically significant event. W h e n there are repeated eigenvalues in a ter nary system, it may be impossible to diagonalize the matrix D. However, this diagonalization turns out to be unnecessary to solve the problem. Details of the general solution are given elsewhere (Toor, 1964); in expanded form, the ternary result is Ac, =
F(a)
+ {Dn
-
σ)
dF bo
Acio
+
(9.37)
+
dF 2
da
Ac 20
The calculation of mass-transfer rates from this profile is straightforward. W e must re-emphasize that this situation is very unlikely to occur in practice. 9.D Multicomponent mass transfer with chemical reaction Although multicomponent mass transfer can couple strongly with chemical reaction, these interesting and potentially large effects have not been fully studied experimentally. The effects can be large because reactions occur with such a wide variety of rates and mechanisms. Diffusion coefiicients are depressingly similar, rarely varying by more t h a n an order of magnitude in any given system, and crossterm diffusion coefficients are often small. Even a simple second-order reaction offers strong coupling and a widely varying rate constant. However, in contrast with the clean and complete derivation for mass transfer without reaction, the theory in this case is approximate and limited. This lack of generality results from two serious restrictions. First, the only reactions which can be included have isothermal first-order mechanisms, since these imply linear equations which can be handled by general matrix techniques. This means t h a t two of the most interesting cases, second-order reversible reactions and first-order catalytic C'Michaelis-Menton") mechanisms, have not been treated. Second,
152 description of this situation requires b o t h a matrix of diffusion coefficients and a matrix of reaction-rate coefficients; these two matrices cannot be simultaneously diagonahzed, except in special cases. Heterogeneous
reactions
Cases with either multicomponent diffusion or coupled heterogeneous first-order reaction are simpler than the case where b o t h are coupled. Problems in which chemical reactions are coupled, but diffusion follows the binary form of Fick's law, are very completely covered in the chemical engineering literature for a wide variety of reaction mechanisms {e.g. Astarita, 1967). Problems in which crossterm diffusion coefficients are present and the chemical reactions have the form of eqns.(9.4)-(9.6), can be handled with the theory in Section 9.A. When b o t h heterogeneous reaction and diffusion are coupled, the continuity e q u a t i o n s : dc it + V
V«f = f
d^c ^
(9-38)
are subject to the initial and boundary conditions: Ac = Acp a t / ( ; c , r) = 0
(9.39)
^
(9.40)
= ^K-fat;c = 0
where is the matrix of reaction-rate coefficients, limited in his case to firstorder reactions. The general solution of these equations requires diagonalizing b o t h D a n d ^ R , which is possible only if b o t h have the same eigenvectors and, hence, commute. This is very unlikely physically. A complete solution for coupled multicomponent diffusion with heterogeneous reaction is available for the special case of one-dimensional unsteady semi-in finite diffusion without convection (Hudson, 1967). The method of solution is first to diagonalize eqn.(9.38) with ν = 0, and then to take the Laplace trans form o f t h e result. The value of c at the boundary can then be evaluated without diagonalizing ¿R. While the method does not work for more general problems, its solution for this special case is superior to the approximate calculation discussed next. An approximate general solution in this case (Parkin, 196^) can be found by writing the overall flux equations as shown in eqn.(9.1), and then assuming that the mass-transfer coefficients are independent of the interfacial concentrations, but are given by the same empirical expressions as in the case of n o reaction, i.e. by eqn.(9.26). This approximation is only qualitatively successful as shown in Fig.9.4. The system studied was the reaction in the presence of helium of gaseous iodine with solid germanium fixed to the surface of a rotating disc, p r o ducing gaseous germanium iodine (Olander, 1967). The agreement between the approximate solution and the experimental results differs by as much as 3 0 % .
153
o
0.2
04
06
0.8
10
Concentration of Iodine in Bulk
Fig. 9.4. Coupled ternary diffusion and chemical reaction. The agreement between calculated and observed fluxes is sharply poorer when chemical reactions are present than when they are not {cf. Fig. 9.2).
The decomposition of iron pentacarbonyl in the presence of argon gave similar errors (Carlton and Oxley, 1965). Homogeneous
reactions
Problems involving coupled diffusion and homogeneous reaction have also not been solved in general. Again, the mathematical difficulty comes from having both a matrix of diffusion coefficients and one of reaction-rate constants. The ease where homogeneous reactions are coupled but where the off-diagonal diffusion coefficients are negligible is probably the most frequently encountered. If t h e reactions are isothermal and first-order, the system is described by the equations: ^
+ V V V
= ^'V^
+ k^'C
(9.41)
where is now the matrix of the homogeneous first-order reaction rate constants. The boundary conditions for this problem are again those in eqns.(9.4)-(9.6). These equations can be solved analytically only for two special cases. The first case occurs at steady state with no convection (deLancey and Chiang, 1970; Toor, 1965) 0 = v'c+
Z)-i.^R.^
(9.42)
This equation can be diagonalized and solved by paralleling the development in Section 9.A.. Problems of this type when D is a diagonal matrix have been carefully analyzed by Wei (1962).
154 The second special case of eqn. (9.41) is that of diffusion without convection
Ir
=fVf+ÍK-c
(9-43)
Mass transfer in or out of the system can also be included in (Gmitro a n d Scriven, 1966). Solutions can be found from harmonic analysis ( G m i t r o a n d Scriven, 1966; Othmer and Scriven, 1969), or, for the one-dimensional case, by a separation-of-variables matrix technique (Toor, 1965). The chief feature of these solutions is a dependence on a new set of eigenvalues which, in turn, depend on combinations of elements of D and k^. These new eigenvalues apparently allow a wide variety of behavior. F o r thermodynamically ideal systems near equilibrium, the second law and the principle of microscopic reversibility imply that these eigenvalues are real and positive, and hence that oscillatory behavior is impossible (Toor, 1965). However, derivations based on microscopic reversibihty are not generally successful for treating chemical reactions. F o r systems far from equilibrium, sustained oscillations are possible (Solomon and H u d s o n , 1971). Such a mechanism has been suggested to explain the oscillatory movements of surfaces such as the periodic movements observed in living systems (Gmitro and Scriven, 1966; Othmer and Scriven, 1969; Desimone et al., 1973). 9.£ Conclusions Most of the conclusions to this chapter are less definite than others in this book. The most definite is that the multicomponent diff'usion coefficients may be assumed constant without serious approximation. Beyond this, the situation is cloudier. M u h i c o m p o n e n t mass-transfer coefficients can apparently be predicted from binary mass-transfer correlations and multicomponent diff'usion coefficients. These coefficients can produce major eff'ects, at least in the few chemical systems studied to date. When b o t h mass transfer a n d chemical systems occur simuUaneously, other events are possible, including oscillating concentrations. However, the entire subject depends on so much elaborate mathematics and so few definitive experiments that these conclusions must be somewhat tentative.