On the Prediction of Limiting Flux in Laminar Ultrafiltration of Macromolecular Solutions Joseph J. S. Shen and Ronald F. Probstein’ Lkpartment of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02 139
The transport phenomenon of limiting flux in the steady, parallel plate laminar ultrafiltration of macromolecular solutions with concentration dependent transport properties is investigated. A semiempirical formula for limiting flux is developed that embodies the effect of variable diffusion coefficient in a correction, which modifies the widely used formula developed by Michaels and which is a function of the diffusion coefficient at the gel concentration. An exact numerical calculation is also carried out in which the variation in both diffusion coefficient and viscosity with concentration is taken into account. The semiempirical formula and numerical calculation for limiting flux for a bovine serum albumin solution of known properties are shown to be in excellent agreement between themselves. In addition, the semiempirical formula requires minimum rheological knowledge of the solution, is easy to apply, and is demonstrated to provide a simple method for determining diffusion coefficient at the gel concentration. Predictions using the semiempirical formula are shown to be in good agreement with experimental ultrafiltration data in the literature for bovine serum albumin solutions.
Introduction Ultrafiltration of macromolecular solutions has become an increasingly important separation process. A large number of studies have reported on the application of the process to the commercial concentration or purification of solutions and to the extraction of solvents. At the same time, a number of studies have been published on the nature of the transport phenomena in ultrafiltration, and it is this problem category to which the present paper is addressed. Since ultrafiltration is a pressure-driven membrane separation process, the pressure applied to the working fluid provides the driving potential to force the solvent to flow through the membrane. Typical driving pressures for ultrafiltration systems are in the range of 10 to 100 psi. For small applied pressures, the solvent flux through the membrane is observed to be proportional to the applied pressure. However, as the pressure is increased further, the flux begins to drop below that which would result from a linear flux-pressure behavior. Eventually a limiting flux is reached where any further pressure increase no longer results in any increase in flux. This limiting flux phenomenon is a common feature in ultrafiltration performance and is illustrated in Figure 1with data taken from Blatt et al. (1970). Clearly, the limiting flux behavior imposes a performance limitation on any system, so that it is of importance to be able to predict this limiting flux for a given macromolecular solution to be ultrafiltered, in order to be able to optimize the design and operating characteristics of any system. The commonly accepted explanation for the flux-pressure behavior, and the one which we adopt here, is that the nonlinear behavior is a consequence of concentration polarization. I t is a feature of other membrane separation processes and results from the buildup of rejected solute a t the membrane surface which increases the local osmotic pressure and leads to a lower effective driving pressure and hence, lower flux. In the ultrafiltration of macromolecular solutions, however, the rheological behavior of the solution generally affects the level of the limiting flux, that is, the value of thellux for which increased pressure no longer increases the flux. This is associated with the accumulation of the macromolecular solute a t the membrane surface which can reach a concentration where gelation begins to form a gel layer which offers the limiting hydrodynamical resistance. Thereafter, the limiting flux
ceases to be pressure dependent but becomes flow, or masstransfer, dependent. The model we have described was put forward by Michaels (1968). On the basis of this model he derived a simple expression for the value of the limiting flux as a function of solute gelling concentration and indirectly as a function of the flow characteristics and the bulk solution diffusivity. Although the qualitative behavior of the result appears essentially correct, when compared with experiment the quantitative agreement is less satisfactory. We propose here that the physical model of Michaels is essentially correct and that the disagreement is principally a consequence of (i) not considering a variable solution diffusivity in the model and (ii) the use of diffusion coefficient values which may not have been sufficiently accurate. The present paper concentrates on the first point. In connection with the second point we suggest that the relatively simple expression we derive for the limiting flux may provide an accurate means of measuring diffusion coefficients a t the solute gelling concentration. Approximate Solution with Constant Diffusivity In Michaels’ (1968) analysis he determined the limiting flux by assuming a steady, one-dimensional thin film mass transfer model in which streamwise convection parallel to the membrane surface is neglected. Assuming the solute to be completely rejected we may write a relation for the solute flux in the direction normal to the membrane surface which expresses the fact that the solute is completely held back by the membrane and only the solvent passes through. Balancing the limiting convective solute flux against the back diffusion from the solute concentration gradient normal to the surface dc uwlirnc= D dY where uWim is the limiting flux with the subscript w denoting conditions a t the “wall” or membrane surface. Here, c is the solute concentration and D in this paper is taken to be the molecular diffusion coefficient of the solute in the solvent. Assuming a constant diffusion coefficient, eq 1 can be integrated over the thin film or concentration boundary layer of thickness 6 to give Michaels’ result
vWim = k In % C,
Ind. Eng. Chern., Fundarn., Vol. 16, No. 4, 1977
458
6ol'/ I
0
0
0 t
P
50
A x
I
I '/a OSA 5%0SA 5%0S4 10% BSA 'C% BS4 io% n s 4
V
I IO
I
20
I 30
I
SOIUtlD",
"Y
0
I
200cc/m,n 200cc/mln Solulian, 500cc I m i n S a l u l , o n , 2 0 0 c e / rn8" S o l u t i o n , 500cc I min SoI",,on,IOOOCC/ rn," SolU,,an,
._-_/----
I 40
I 50
I
60
I 70
'I
1
__----
I I
SO
Applied Pressure ( p s i )
Figure 1. Experimental flux-pressure data from laminar ultrafiltration of bovine serum albumin solutions from Blatt et al. (1970). c, ( gm BSA/ IOOcc 1
where k = D/6. The so1ut.econcentration is a maximum at the wall and decreases to the bulk feed concentration at the edge of the film. The wall concentration is therefore taken to be the solute gelling concentration and is denoted by cg, while the bulk feed concentration is denoted by c., The gel concentration is a quantity which can be determined by extrapolating data for the limiting flux as a function of bulk concentration to zero values of the flux with all other parameters held fixed. The basis of the extrapolation is that complete gellation of the bulk fluid by definition reduces the flux to zero. An extrapolation of the type described is shown in the semilog plot of Figure 2, which represents a cross-plot of the data of Blatt et al. (1970) shown in Figure 1. For the same solution the results do give a consistent value for the gel concentration and, moreover, the logarithmic dependence on the bulk concentration, as predicted by eq 2, is seen to be borne out. The coefficient k appearing in eq 2 is a mass transfer coefficient. In the case of laminar flow with a parallel plate geometry Blatt et al. (1970) evaluated this coefficient from the corresponding Leveque heat transfer solution. The resulting expression for the limiting flux is
Figure 2. Experimental determination of gelling concentration by extrapolating limiting flux data. Limiting flux data taken from Figure 1.
tration diffusion equation with a concentration-dependent diffusion coefficient D ( c ) for this case is written (4)
with x measured longitudinally along the membrane surface from the channel inlet and y measured transverse or normal to the membrane surface. The respective velocity components are u and u . Corresponding to the thin film approximation we assume a high Schmidt number system sufficiently close to the channel inlet that the concentration polarization layer is thin compared with the channel width. Within the growing diffusion layer the streamwise velocity is then essentially linear in distance from the wall, while the transverse component may be taken approximately constant a t the limiting flux value (see, e.g., Gill et al., 1971). It follows that we may write approximately
(3)
Here, U is the mean bulk flow velocity, D , is the diffusion coefficient a t the bulk solute concentration, h is the channel half-width, and L is the active membrane length in the main flow direction. The dependence on the one-third power of the fluid shear per unit length ( U / h L )has also been verified by experiment (Blatt et al., 1970; Porter, 1972). Despite the qualitative agreement of eq 2 and 3 with experiment the quantitative predictions have been less satisfactory when similarly compared (see, e.g., Porter, 1972; Grieves et al., 1973). That eq 2 and 3 have the proper parametric dependences would seem to indicate the essential correctness of Michaels' model. We are therefore led to conclude that the quantitative differences between theory and experiment must be accounted for principally by the variable transport properties of the macromolecular solution normal to the membrane surface in the concentration polarization layer and, in particular, the variability of the diffusion coefficient. Approximate Solution with Variable Diffusivity In order to first assess the effects of a variable diffusion coefficient we consider the corresponding problem treated by Blatt et al. (1970), that is, the steady, Newtonian, fully developed laminar flow in a parallel plate system. The concen460
Ind. Eng. Chem., Fundam., Vol. 16, No. 4, 1977
u=
(E)
y and u = vWlim
(5)
with ( d u l d y ) , the velocity gradient at the edge of the concentration diffusion layer. The relevant boundary conditions a t the wall are that the solute concentration is the gelling concentration and the condition of complete solute rejection, i.e.
At the edge of the diffusion layer the concentration takes on the bulk value, i.e. c = c,
(fory
-+
m)
Introducing the dimensionless variables = y (3xD-/
and
($),)-'"
(7)
du u = (3xD..(dy)m
2
)113a
the diffusion equation reduces to the ordinary differential form
._....
d
-dF dF D( a 2 + V,)-=O dq) da The boundary conditions become
G(
+
F(7) = F, = c,/c..
and F(7) = 1
(for a
(7 = 0)
-
m)
(114
(12)
Following the procedure of Kozinski and Lightfoot (1972), eq 10 may be rearranged and integrated twice using the boundary conditions eq 11and 12 to yield 1
1 - - = V,
F,
S,-
exp
v, ( - S, D da) da “2+
as the gelling limit is approached. I t is therefore necessary in any complete solution aimed a t determining the effect of variable transport properties that the effect of viscosity variations also be considered. However, if the basic model of Michaels (1968) is accepted, we would not expect any appreciable effect of a variable viscosity, since the problem is essentially one of mass transfer and is essentially uncoupled from the momentum transfer. This statement is valid so long as the diffusion layer grows in a fully developed velocity profile, that is, so long as we may approximate the velocity profile by a constant shear in the thin concentration boundary layer (Acrivos, 1962). The constant shear approximation is equivalent in the constant viscosity case to the linear velocity profile of eq 5 and is expressed
(13)
The above equation represents an implicit quadrature for V,, the solution of which cannot be obtained without specifying the variation of with 7 through its dependence on c . Since the critical region for mass transfer is adjacent to the membrane surface we follow Kozinski and Lightfoot (1972) and expand 1lD as a function of concentration in a Taylor series from the wall and retain only first-order terms. Ih terms of the stretched coordinate 7 we may then write 1 1 1 (14) 5 ---DJO, + (%),a +... Substituting this expansion into eq 13, integrating, and expanding the exponential terms to consistent order, weobtain after some manipulation that V, is proportional to DW2l3or in dimensional quantities
The result obtained shows that the dominant correction for a variable diffusion coefficient manifests itself through the replacement of the bulk diffusion coefficient by the diffusion coefficient evaluated at the gel concentration. This result and the form of eq 13 suggests that the effect of a variable diffusion coefficient can be accounted for by replacing D.. by D, in the constant property result of eq 3. This then leads to the following expression for the limiting flux in a parallel channel laminar ultrafiltration system
Here, n is a constant which could be somewhat different from the value of 1.18 obtained from a constant property heat transfer correlation. We would emphasize here that we do not represent eq 16 as a rigorous solution. Rather it serves to provide a convenient and physically sound semiempirical modification of the present widely used constant property relation for predicting limiting flux in ultrafiltration. The question of how accurate eq 16 is will be resolved when we compare the results of both eq 3 and 16 with complete numerical solutions, which are presented below, and with experiment.
This relation expresses the momentum transfer and to it must be added the continuity equation
Equations 17 and 18 together with eq 4 represent the coupled partial differential system governing the concentration layer growth with variable transport properties. To these equations must be added the boundary conditions of eq 6 and 7 along with the no-slip condition u =0
(19)
With the momentum transfer considered it is necessary to somewhat modify the dimensionless velocity variables of eq 9 to the form
Using eq 20 and the change of variables previously defined by eq 8 the system of equations reduces to the ordinary differential form
iL($+1)=1
da
(21c)
where we have defined
L ( F ) = F(c)lPm(cm) (22) The boundary conditions are given by eq 11 along with the no-slip condition G(7) =
0
(at 17 = 0)
(23)
Note that eq 21c reduces to the previously examined eq 10 for G = 0. The above system of equations is similar to one presented in Brown et al. (1971). We are interested principally in evaluating the magnitude of the limiting flux. This can be done by averaging over the membrane length L the transverse velocity of eq 20b a t the membrane wall. This gives UWlim
Formulation f o r Variable Diffusivity a n d Viscosity Not only the diffusion coefficient but also the viscosity can have strong variations in the concentration boundary layer
(at y = O )
= - J L[ D , 2 ( d ~ / d y ) ~ / 3 x ] ~ / ~ V , d r
L
o = 1 . U 5 D( , ~ 113 ) V,
Ind. Eng. Chern., Fundam., Vol. 16, No. 4, 1977
461
'01 9
I
1
I
I
I
I
I
t
0 0 I M b u f f e r , p H 4 7 K e l l e r et 0 1 0 0 15M NaCl, pH 7 4 Thowork
11/11
*-
O l 5 M NaCl. pH 7 1 - 7 7 Ibid
5 4 .Q *-
I
I
V
Philliesetol
0 I M NoCI
Doherty
(
Phlllles
(19761
8 Benedrk
3 1
L
1 Albumin Concentration (grn/IOOcc)
Figure 3. Diffusion coefficients of bovine serum albumin solu-
tions. which may be compared in functional form with eq 3 and 16. Of course, V, is not known in advance and is determined from a solution of the differential equations in a manner as to satisfy the boundary conditions. These solutions can only be obtained numerically and then only after the solution properties have been specified. Bovine S e r u m Albumin Solution To solve numerically our system of equations and boundary conditions for the case of both varying viscosity and diffusion coefficient, it is first necessary to define the transport and rheological properties of the macromolecular solution, Le., eq 8c, 22, and c, in eq l l a . Bovine serum albumin (mol wt -69 000) solution is chosen to serve as the working fluid because experimental ultrafiltration data and rather complete physical properties of this Newtonian fluid are available in the literature. Viscosity measurements of BSA solutions over a broad range of concentration have been presented by Kozinski and Lightfoot (1972). Their data show negligible dependence of viscosity on the pH value of the solution and the buffer used. One of the correlations they gave for the viscosity is p
= 0.01 exp(O.00244~~)
(25)
where c is expressed in g of BSA/100 cm3 of solution (g%) and is in dyn/cm2 s. Diffusion coefficient measurements present a more complicated picture. Many measurements of BSA diffusivity at dilute concentrations (usually below 1g%) have been reported in the literature. Diffusivity of dilute BSA solutions a t 25 "C ranges from about 6.6 X to 7.1 X cm2/s from a number of investigators (e.g., the early measurements of Creeth, 1952; Charlwood, 1953). The data show little effect of buffer type, pH value (electrical charge on the macro-ions), and ionic strength of the solution. However, as the concentration increases, these factors cannot be ignored as the interaction among the charged macro-ions becomes more pronounced. Data by Doherty and Benedek (1974) show a strong dependence of BSA diffusivity on the solution ionic strength and the average protein surface charge. Keller et al. (1971) carried out measurements in a diaphragm diffusion cell for solution concentrations up to 31 g %. The BSA solution was 462
Ind. Eng. Chem., Fundam., Vol. 16, No. 4,1977
prepared in 0.1 M acetate buffer at pH 4.7, the isoelectric point of bovine serum albumin. The correlation they gave for D in cmz/s is 7.1 x 10-7 D= tanh (0.159~) (26) 0.159~ Recently, Phillies et al. (1976) presented data for BSA diffusion coefficients in 0.3 M phosphate buffer, 0.25 M acetate buffer, and 0.15 M saline water over the pH range 4.3-7.6 using a spectroscopic technique. Their data showed a significant dependence of the diffusion coefficient on BSA concentration, pH value, and buffer used. However, their data scattered appreciably and disagreed in two important instances with previous measurements. First, the diffusivity data by Phillies et al. and Keller et al. in acetate buffer a t similar pH value and ionic strength do not agree. Secondly, Phillies et al. reported a value for the dilute concentration diffusivity for BSA in 0.15 M saline water at about pH 7.2 of 4.95 X cm2/s. This is significantly below the literature value. For these reasons, the data of Phillies et al. were not used in the numerical solution discussed below. However, the data are valuable for providing an insight into the behavior of the diffusivity in concentrated BSA solutions. All of the available diffusivity data for BSA in saline water are plotted in Figure 3, including one point obtained by us to be discussed in what follows. It should be noted that reported physical properties for BSA solutions are known only up to about 40-45 g %, so that it was necessary to extrapolate these properties to somewhat higher concentrations for our purposes. The gelling concentration of a BSA solution was taken to be 58 g/100 cm3, which is consistent with the Kozinski and Lightfoot (1972) measurement of 58.5 g/100 cm3 and the value of 57.5 g/100 cm3 obtained from the extrapolated experimental data of Figure 2. Numerical Solution f o r Variable Diffusivity a n d Viscosity With the physical properties of the BSA solution defined as described above, eq 21 can be solved numerically by a trial-and-error method. To begin the procedure, a value of V, in eq I l b is assumed and we know from the boundary conditions, eq l l a and 23, that F ( 0 ) = F , and G ( 0 ) = 0. Values of G ( 7 ) and V ( 7 ) are numerically computed by marching out from 7 = 0. ,Ef the assumed value of V, is too small, F ( a)will converge to a value greater than unity, which means physically that the assumed convective transport at the membrane is not large enough to balance the back-diffusion and results in an equilibrium concentration greater than the bulk concentration. On the other hand, if the assumed value of V, is too large, F ( a ) will converge to a value smaller than unity. Physically, this means that the back-diffusion is not large enough to balance ,the assumed convective transport and results in an equilibrium concentration smaller than the bulk concentration. For each bulk concentration under consideration, there is only one value for V , such that the assumed convection and back-diffusion are just in balance and F( a)converges exactly to unity, thereby satisfying the last boundary condition of eq 12. In Figure 4 are presented the calculated values of V, over a wide range of practical bulk concentrations, from 0.5 g of BSA/100 cm3 to 12 g of BSA/100 cm3 in 0.1 M acetate buffer at pH 4.7. Additional calculations have shown that V, is not very sensitive to the gelling concentrations assumed. Using cg = 53 and 63 g/100 cm3 produces at most only about a 5% deviation from the V, calculations assuming cg = 58 g/lW cm3. The sensitivity of the dependence of V , on viscosity has also been investigated by assuming the viscosity of the solution to be independent of the concentration, i.e., ji = 1. This leads to
I .2
"II\
l
I .o
0.7
l
l
l
l
l
l
l
I
I
i
\
0"ld C
0.8
-+
.
rn N
-2 -
>* 0.6
n o C
- BSA
acetate buffer a t p H 4 7, C g = 5 8 g m BSA/lOOcc
0 321
in 0.1 M
"t L 0
I
2
3
4
,C
5
6
7
8
9
1
I1
0.4
,I
a:
- p H 4.7 S o l u t i o n , n:
I . 18
1
I
0.2
I
1
l
l
i
l
1
1
-1
i
cm ( g m B S A / 1OOcc)
L Figure 5. Ratio. of approximate to exact solutions for limiting flux 0 1 1 1 2
(gmBSA/lOOcc)
Figure 4. V , of bovine serum albumin solution at pH 4.7 as a function of bulk concentration.
results for V , which are a t most 50% higher than the calculations when is taken to vary from 1to infinity in the concentration boundary layer as F increases from 1 to F,. Therefore, as expected, it is seen that the limiting flux is much more dependent on the variation of diffusion coefficient than viscosity coefficient. This finding further confirms the assumption of constant viscosity which was adopted in deriving the approximate solution of eq 16. Using Figure 4 for V,, it is a simple matter to predict from eq 24 the limiting flux for the laminar ultrafiltration of bovine serum albumin once the other operating parameters are specified. Results and Conclusions To evaluate the accuracy of the approximate result of eq 16 we have plotted in Figure 5 the ratio of the approximate result to the exact numerical solution with variable physical properties as expressed by eq 24. This ratio is given by
(27) To achieve better agreement, that is, a value of R close to unity for all bulk concentrations, n is taken equal to 1.18 for the pH 4.7 data, which is the same value used in eq 3. It can be seen that the simple approximate expression of eq 16 agrees remarkably well with the exact numerical solution. This bears out the point that it is not the variation in the value of the diffusion coefficient through the diffusion layer which is of importance but rather only its value corresponding to the gel concentration at the membrane surface. This is not surprising since within a simple linear model it is the driving gradient at the wall which defines the back-diffusion, counteracting transverse flux. The Blatt et al. (1970) experimental laminar flux-pressure data of Figure 1for bovine serum albumin solutions have already been briefly discussed. Their solution was reported to be in 0.09 saline water (ionic strength about 0.15 M); however, the pH value was not specified. Our experience with a similar BSA solution gave a pH value of about 7.0. Therefore, the diffusivity data of Keller et al. (1971), which are the most complete available, are not applicable due to the large variation in BSA diffusivity between pH 4.7 and pH 7.0 (see Figure
of bovine serum albumin a t pH 4.7 as a function of bulk concentration.
3). The data of Phillies et al. (1976) were taken at solution ionic strength and pH value similar to the solution of Blatt et al., but because of discrepancies in the data that were mentioned earlier, we are unable to utilize the data fully. However, since the only unknown in eq 16 is the value of D,, in order to predict the limiting flux we need only obtain the value of the diffusivity for BSA in saline solution at about pH 7.0 a t the gel concentration. We have assumed here that n is taken to be 1.18 as evaluated from Figure 5. To obtain D, we carried out ultrafiltration tests of bovine serum albumin in 0.15 M saline water in a thin parallel channel system at 25 O C . The solution pH was adjusted to 7.4 by adding 0.5 N sodium hydroxide solution. The channel was 3 in. wide and 0.38 cm in height. The active membrane area was 365.8 cm2based on a width of 3 in. and a length of 48 cm. The channel had an entry length equal to about 40 channel heights to assure fully developed laminar flow on reaching the membrane region for system Reynolds number less than about 650. The ultrafiltration membrane used in this experiment is the polysuphone type, supplied to us by Abcor of Wilmington, Mass. The solution was prepared from bovine albumin, fraction V powder purchased from Miles Research Products of Elkhart, Tnd. BSA concentrations of the feed solution and the collected permeate were determined by ultraviolet light absorption with a spectrophotometer. Better than 99.3% rejection of macromolecules was achieved by the polysuphone membrane used; complete rejection was therefore assumed for our purpose. Our flux-pressure data are presented in Figure 6. Since the shear rate in our apparatus is much lower than in the study of Blatt et al., the concentration polarization is more pronounced and the limiting fluxes at the two cases of 1.0 and 2.0 L/min volume flow rate can be ascertained readily. The data at higher volume flow rates were not used due to the need to extrapolate them, which could compromise the accuracy of the observed limiting fluxes. Limiting permeate fluxes a t Q = 1 and 2 L/min age about 0.0154 and 0.0182 cm3of permeate/cm2 of membrane/min, respectively. Using eq 16 with n = 1.18 and the appropriate bulk concentration in each case, we find D , to be about 6.7 X 10-7 cm2/s in both cases. It is this value which is shown plotted in Figure 3. Because Blatt et al. used a very thin channel system, the fluid shear a t the membrane surface was very high and quite effective in reducing concentration polarization a t the wall. Ind. Eng. Chem., Fundam., Vol. 16, No. 4, 1977 463
I
Table I. Limiting Flux in Ultrafiltration of Bovine Serum Albumin Solutions
I
___-----
0.02
II
0217 gm%BSA, bliter/min a239 g m % B S A , 4liter/min 0 2 2 1 om%BSA. 2liter/min o 2 0 8 gm BSA; I Iiter/min
///
200
52
200 500
33 43
58 35 47.5
200
22
25
I
I
I
I
I
10
20
30
40
50
c, = 10.0 60
A p p l i e d Pressure ( p s i )
Figure 6. Experimental flux-pressure data for laminar ultrafiltration of bovine serum albumin in 0.15 M saline water at pH 7.4.
I t can be seen in Figure 1that the applied pressures were not high enough to exhibit the complete flattening-out of the flux-pressure curves. However, limiting fluxes can be determined by extrapolation, a t least for the lower flow rate and higher bulk concentration cases. It is recognized that a small error might be introduced by the data extrapolation. Table I presents a comparison of the limiting fluxes so determined with the predictions using eq 16. It may be seen that the predictions of eq 16 are in good agreement with the experimental results, considering the possible experimental error involved and the uncertainty of data extrapolation. We note that the D , value used was deduced from experimental data in a pH 7.4 solution. This may be a slightly higher pH than the pH in the Blatt experiments, so that the diffusivity used may be a bit high (cf. Figure 3). The predictions are seen to be consistently higher than the experimental data by a small amount, and the higher value of the diffusivity used may be the reason. The success of our simple analytic expression eq 16 in predicting the limiting fluxes with reasonable accuracy makes it appear that bulk concentration and variable diffusivity effects on the limiting flux can indeed be separated out in the manner indicated. In effect, we have also demonstrated that eq 16 would seem to provide a simple means of measuring the gel concentration diffusion coefficient from flux-pressure measurements of the kind discussed. The value of 6.7 X cm2/s for the D , of BSA in 0.15 saline water a t pH 7.4 seems quite reasonable when compared with the other data in Figure 3. We would also note that Phillies (1974) derived a theoretical expression for the diffusion coefficient of hard sphere macromolecules as
where DT is the macromolecular tracer diffusion coefficient, A is the osmotic pressure of the solution, k is Boltzmann's constant, T is the absolute temperature, and is the solute volume fraction. Using the tracer diffusion coefficient data of Keller et al. (1971) and the osmotic pressure data of Scatchard et al. (1946), Phillies et al. (1976) determined from eq 28 the diffusion coefficient of bovine serum albumin in 0.15 M saline water near pH 7.2 as a function of albumin concentration. It appears that from their theoretical prediction the diffusion coefficient at very high concentration is about 6.7 X 10-7 cm2/s. Although their D , prediction agrees almost exactly with our determination, we must note that there exists some question concerning eq 28 (for example, Anderson and Reed, 1976). Moreover, the diffusivity at infinite dilution was assumed by them to be 4.95 X cm2/s from their measurements, rather than the literature consensus of about 7.0 X 10-7 cm2/s. Equation 28, therefore, was not incorporated into our numerical solution scheme. Nevertheless, it furnishes 464
1.0
c m = 5.0
c, =
0.01
0
Limiting flux, gal/ft2 day Approx. variable Bulk concn, Flow rate, Exptl," prop. solution eq 16b g of BSA/100 cm3 cm3/min Blatt et al. (1970)
Ind. Eng. Chem., Fundam., Vol. 16, No. 4, 1977
30 34 39 43 a Experimental results extrapolated from Figure 1.Data taken in a thin channel system 16 in. long, 1/4 in. wide, and 10 mils high. Solvent was 0.9% saline water and solution is estimated to have cm2/s,as evaluated from been at pH = 7.0. b D(c,) = 6.7 X data at pH 7.4, and n = 1.18. 500 1000
some insight into the behavior of the diffusivity at high concentration and indicates that our determination of the value of D , equal to 6.7 X lov7cm2/s is quite reasonable. Of course, the accuracy of our prediction of D,, the choice of n,and the success of eq 16 compared to complete numerical solutions can and should be further tested as new experimental measurements of BSA diffusivity under appropriate solution conditions become available. At this time, predictions by eq 2 and eq 24 cannot be compared precisely because of the unavailability of reliable BSA diffusivity data in high ionic strength saline water near p H 7.0. Note should be made of the work of Kozinski and Lightfoot (1972), who analyzed ultrafiltration in a rotating disk system using similarity transformations analogous to those used in the approach of this paper. They also extended their results to parallel plate systems and obtained analytical expressions for the limiting fluxes which contain rather complicated physical correction factors that render the results somewhat difficult to apply. Not shown in Table I is their numerical prediction of 11.6 gal/ft2 day corresponding to the case of c, = 10 g of BSA/100 cm3 and a flow rate of 200 cm3/min. The reason we have not specifically cited their value is that the underestimate of their prediction is probably attributable in large part to the low diffusion coefficient employed (the results of Keller et al., 1971) rather than to any deficiency in the theory. We summarize by observing that only one dominant physical property correction factor is required to modify the widely used formula first advanced by Michaels (1968) in order to accurately describe the limiting flux phenomenon. This modification requires minimum rheological knowledge of the macromolecular solutions, and appears to provide a simple method for determining diffusion coefficients at high concentrations of macromolecular solutions. Nomenclature c = concentration of the solution (g of solute/100 cm3 of solution) O_ = diffusion coefficient, cm2/s D = dimensionless diffusion coefficient, eq 8c F = dimensionless solution concentration, eq 8b G = dimensionless longitudinal velocity variable, eq 20a h = channel half width, cm k = mass transfer coefficient, D/6, cm/s L = longitudinal length of the membrane, cm n = aconstant Q = volume flow rate, L/min R = dimensionless ratio, eq 27 Re = system Reynolds number, dimensionless U = mean longitudinal bulk velocity, cm/s
u = longitudinal velocity, cm/s u = transverse velocity, cm/s or solvent flux, gal/ft2 day V = dimensionless transverse velocity, eq 20b x = longitudinal coordinate, cm
Technology," J. E. Flinn, Ed., p 47, Plenum Press, New York, N.Y., 1970. Brown, C. E.,Tulin, M. P., Van Dyke, P., Chem. Eng. frog. Symp. Ser., 67, No. 144, 174 (1971). Charlwood, P. A., J. Phys. Chem., 57, 125 (1953). Creeth, J. M., J. Biochem., 51, 10 (1952). Doherty, P.. Benedek, G. B., J. Chem. Phys., 5426 (1974). Gill, W. N., Derzansky, L. J.. Doshi, M. R., "Surface and Colloid Science," Vol. IV, E. Matijevic, Ed., p 262, Wiley, New York, N.Y., 1971. Grieves, R. B., Bhattacharyya, D., Schomp, W. G., Bewley, J. L., AlChEJ., 19, 766 (1973). Keller, K. H., Canales, E. R., Yum, S. I., J. Phys. Chem., 75, 379 (1971). Kozinski, A. A., Lightfoot, E. N., AlChEJ., 18, 1030 (1972). Michaels, A. S., Chem. Eng. frog., 64 (12), 31 (1968). Phillies, G. D. J., J. Chem. Phys., 60, 976 (1974). Phillies, G. D. J., Benedek, G. B., Mazer, N. A,, J. Chem. Phys., 65, 1883 (1976). Porter, M. C., Ind. Eng. Chem., Prod. Res. Dev., 11, 234 (1972). Scatchard, G., Batchelder, A. C., Brown, A., J. Am. Chem. SOC., 68, 2320 (1946).
y = transverse coordinate, cm
Greek Letters thickness of concentration boundary layer, cm dimensionless transverse coordinate, eq 8a absolute viscosity of the solution, dyn/cm2 s dimensionless viscosity, eq 22 T = shear stress, dyn/cm2
6 = q = p = p =
Subscripts g = a t gelling condition lim = at limiting flux condition w = atmembranewall 00 = a t bulk or feed condition
Received for review April 12, 1976 Accepted June 29,1977
Literature Cited Acrivos, A., Chem. Eng. Sci., 17, 457 (1962). Anderson, J. L.. Reed, C. C., J. Chem. Phys., 64, 3240 (1976). Blatt. W. F., Dravid, A,, Michaels, A. S., Nelson, L., "Membrane Science and
T h i s research was supported by t h e Office o f Water Research a n d Technology under G r a n t No. 14-31-0001-7506. One o f us (J.S.) was also supported t h r o u g h a N a t i o n a l Science F o u n d a t i o n Energy Traineeship.
Fluidized-Bed Coal Combustion with Lime Additives. The Phenomenon of Peaking of Sulfur Retention at a Certain Temperature Ralph T. Yang,' C. R. Krlshna, and M. Stelnberg Department of Applied Science, Brookhaven National Laboratory, Upton, New York 11973
Preliminary kinetic results and the mechanistic implications of the reactions between calcium sulfate and coal ash are presented. Calcium silicates were identified in the reaction products. These reactions also produce SO2 and have been shown to have a high temperature dependency. Taken with the published kinetic data on the sulfation reaction of limestone, this leads to a possible contributory mechanism toward the phenomenon of peaking of sulfur retention at a certain temperature in fluidized-bed coal combustion with lime additives.
Introduction In this communication, we present some of the results obtained in the experimental study of the following reaction
Cas04
+ coal ash
-
silicates
+ SO2 + V 2 0 2
(1)
This reaction can be of some importance in the combustion of coal in a fluidized bed of limestone. The consequences of such a reaction occurring there along with the following reaction of the sulfating of lime CaO
+ SO2 + l / 2 0 2
-
Cas04
will be discussed after presentation of the experimental results. Experimental Section The rates of reaction 1 were measured gravimetrically. Detailed procedures and the apparatus used for the measurements have been described elsewhere (Yang and Steinberg, 1975, 1976). An alumina sample holder was used in all the experiments. The total amount of the sample mixture was about 0.5-1 g. The reactant sample mixture con-
tained approximately 1 CaS04:1.5 Si02 (molar) in all cases. A gas mixture was passed over the packed sample surface a t approximately 1cm/s. The composition of the gas mixture was 2.9% H20,3% S02,5% 0 2 and the balance N2. The water vapor content was so chosen that direct comparisons can be made with the published results on reaction 2 (Yang et al., 1975). Rates are expressed as (l/W)(dM/dt) where W is the instantaneous mass of the remaining Cas04 in grams, M is the number of gram-moles of SO2 evolved, and t is the time in seconds. The initial rates were obtained by the method described previously (Yang et al., 1975). Drierite (manufactured by Hammond Co.) was used as the Cas04 without purification. The coal ash was obtained by oxidizing Illinois No. 6 bituminous (Hvbb) coal with air at 1000 "C for 12 h. A representative analysis of the ash is given in Table I. The materials were ground, sized, and mixed evenly at the desired proportions for rate measurements. Two series of experiments were performed with two different ash sizes with the same size of Cas04 of the size range of 710-1000 p. The sizes of the ash particles were 250-425 p in the first series and less than 75 p in the second. Contact between the ash and Cas04 particles was obviously more intimate with the Ind. Eng. Chem., Fundam., Vol. 16, No. 4, 1977
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