(18) Estimated from endothermicity of the reaction and the activation energy of the reverse reaction. (19) Estimated by analogy to reaction 17. (20) Estimated by analogy to reaction 2. (21) Estimated by analogy to reaction 3. (22) Six-center, secondary hydrogen isomerization step (see ref 5). (23) Estimates from values of the reverse reaction 27 and thermochemical data. (24) Approximated from the analogous reaction 23, and estimated thermochemical data. (25) R. K . Brinton, J . Chem. Phys., 29,781 (1958). (26) Frequency factor taken from estimated value from Benson (ref 1) and the activation energy is estimated from endothermicity and activation energy of reverse, react,ion 30; e.g., E = 25.5 8.5 = 34 kcal/mol, where AH0 = 25.5 kcal/mol and E r e v e r s e = 8.5 kcal/mol. (27) Estimated by analogy to reaction 17.
+
Literature Cited Daubert, T E., Jones, J. H., Fenske, M. R.. J. Chem. Eng. Data, 8, 261 (1963). Jones, J. H., Fenske, M. R., lnd. Eng. Chem., 51, 262 (1959).
Jones, J. H., Allendorf, H D , Hutton, D. G., Fenske, M . R., J . Chem. Eng. Data, 6, 620 (1961a). Jones. J. I - . , Fenske, M . R.. Hutton, D. G., Allendorf, H. D., J. Chem. Eng Data, 6, 623 (1961b) Jones, J. H . , Daubert, T. E., Fenske, M . R., lnd. Eng. Chem., Process Des. Develop., 8. 17 (1969a). Jones, J. H.. Daubert, T. E.. Fenske, M. R . . lnd. Eng. Chem., Process Des. Develop., 8, 196 (1969b). Jones, J. W.. Daubert, T. E.. Fenske, M . R., Sandy, C . W.. Lau. P. J,, lnd. Eng. Chem.. Process Des. Develop., 9, 127 (1970) Jones, J. H.. Fenske, M . R., Rusk. R. A . , lnd. Eng. Chem. Prod. Res. Deveiop., 10, 57 (1971a). Jones, J. H., Fenske. M. R., Belfit, R. W., l n d . Eng. Chem., Prod. Res. Develop.. 10,410 (1971b). Svoboda, K . G., Daubert. T. E , lnd. Eng. Chem.. Prod. Res. Deveiop., 11, 337 (1972).
Receiued for reuiea June 26, 1974 Accepted December 9, 1974 Supplementary Material Available. A detailed derivation of the carbon-path mechanism will appear following these pages in the microfilm edition of this volume of the journal. Photocopies of the supplementary material from this paper only or microfiche (105 X 148 mm, 24X reduction, negatives) containing all of the supplementary material for the papers in this issue may be obtained from the Journals Department, American Chemical Society, 1155 16th St., N . W., Washington, D. C . 20036. Remit check or money order for $4.50 for photocopy or $2.50 for microfiche, referring to code number PROC-75-159.
Effects of Forced and Natural Convection during Ultrafiltration of Protein-Saline Solutions and Whole Blood in Thin Channels William J. Huffman,*’ Robert M. Ward,’ and Richard C. Harshman Department of Chemical Engineering, Ciemson Univers/ty, Clemson. South Carolina
The ultrafiltration of 5 % bovine albumin-saline solution and whole blood were studied in a parallel-plate flow cell with one porous wall. Hydrostatic pressure gradients from 55 to 593 mm Hg were applied to obtain 2-85% volumetric separation in channels ranging from 0.020 to 0.165 c m in height. For forced convection conditions, protein-saline solution data were correlated using a modified Leveque-Graetz equation. Increased ultrafiltration rates up to 200% were obtained when natural convection was superimposed on bulk flow. For whole blood, the Leveque-Graetz correlation only applied at low separations. Increased ultrafiltration rates were also observed for whole blood when natural convection flow was permitted.
The extracorporeal removal of toxic substances in blood using a dialysis apparatus or hemodialyzer has been one of the successes of medical science and engineering. Many devices for this treatment have been developed over the years, but widespread use has been restricted because many of the designs have yielded bulky units requiring constant supervision and expensive membrane replacement. More recent improvements using solid adsorbants minimize some of the disadvantages, but the problem with expensive membrane replacement still exists (Hunt, 1973). Further improvements have been proposed through the use of alternate membrane separation processes, and ultrafiltration has been investigated (Bixler, et al., 1968). As is well known, the driving force for the ultrafiltration or reverse osmosis process is the difference between the applied hydrostatic pressure gradient and the osmotic pres-
’
Address correspondence to this author at the Department of Chemical Engineering, Texas Tech University, Lubbock, Texas 79409 201efin-Plastics Dept.. Dow Chemical, Freeport, Texas 77541
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Ind. Eng. Chem., Process Des. Dev., Vol. 14, No. 2, 1975
sure exerted at the membrane surface by nonpermeating components. Thus, for a given hydrostatic pressure, the rate of ultrafiltration is governed by the removal of stagnant components from the membrane surface (Sourirajan, 1970). In normal engineering practice, this surface concentration polarization may be reduced by mechanical agitation or turbulent flow (Michaels, 1968; Shenvood, 1965; Reilly, 1969), but with fragile materials such as blood these two methods of intense mixing cannot be employed. In addition, the build-up of a gelatinous layer or sedimentation during whole blood ultrafiltration has been reported which further reduces the rate of transfer (Bixler, e t al., 1968) and could limit application due to physiological effects. On this basis, we decided to investigate the contribution of mixing by natural convection to reduce the concentration polarization and minimize deposits or gel layers during ultrafiltration. Experimental Section Chemicals. Bovine albumin was powdered, Cohn Frac-
tion V, control lot 1249 from Nutritional Biochemicals, Corp. Sodium chloride was ACS reagent grade from J. T. Baker Chemical Co. Pooled human serum was used to standardize UV-absorption measurements and obtained from Warner Chilcott Lab., “Versatol” Lot No. 0864127. Water was obtained from a distilling apparatus which provided water a t 0.3 ppm solid with a resistivity of 1.7 megohms/cm. Blood was outdated human blood containing approximately 8% citrate-phosphate dextrose anticoagulant solution. Membranes were UM-1, neutral charge, ultrafiltration membranes from Amicon Corp. with a 10,000 molecular weight cut-off. Apparatus a n d Procedure. For the investigation, a parallel-plate flow cell with one porous wall was constructed from Plexiglas as shown in Figure 1. Supporting equipment included conventional manometers, valves, filters, and feed tanks. The flow cell was designed to simulate a parallel plate unit with one permeable wall and was mounted on a 360” swivel in a constant temperature bath (27°C). This spatial arrangement permitted natural convection effects to be clearly defined by simply rotating the cell. Ultrafiltration experiments were run with the permeable wall above or below the axial flow (flow cell in a horizontal position) or with the permeable wall parallel to gravitation forces (vertical position). A 5-6 wt % protein-saline solution was used for initial studies to simulate osmotic-mass transfer characteristics of diluted whole blood to provide a pseudo-binary system to minimize experimental problems normally encountered when working with whole blood. Flow maldistribution effects in the cell were minimized by specifying entrance and exit volumes four times greater than the flow channel volume. One-inch entrance sections were also provided both upstream and downstream of the filtration zone to ensure fully developed flow patterns over the range of velocities investigated (Knudsen and Katz, 1958). Channel height was always less than 2% of the channel width to simulate parallel plate flow per the theoretical calculations of Hubbard (1965). The combined pressure drop across filtrate tubing, membrane support, and inactive membrane backing was less than ylooo of the resistance to permeation of saline solution through the membrane. Protein concentrations in both the saline solution and whole blood were determined by ultraviolet absorption using the difference method developed by Murphy and Kies (1960). The spectrophotometer was calibrated using the method developed by Haupt (1952). The flow cell was assembled by positioning a wet membrane on the porous wall and aspirating to hold it in place. The nonporous wall was then added along with appropriate Mylar spacers, and cell bolts were tightened (normally less than 100 in.-lb) to eliminate leaks. Air bubbles were removed by flushing with distilled water for 6-8 hr. Membrane permeability to saline solution was then determined at axial-flow and nonflow conditions to ensure that all equipment was operational and a material balance could be closed within experimental error. Volume and time measurements were employed to determine both axial and filtrate flow rates. Required pressures were measured using U-tube manometers containing either mercury or 1.75 sp. gr. oil. Values of all measurements were recorded after steady-state conditions were achieved; initial runs demonstrated that steady state could be achieved within 10-20 times the normal residence time of axial flow over the active membrane area. Dynamic permeability measurements incorporating all experimental variables were determined within *4%. An error of *170 was desired for calculation of mass transfer
FC- I Lower Plexiglas Plate
FC- 2 Porous Membrane Support ~ c - 3Ultrafiltration Membrane FC- 4 Bottom Mylar Spacer FC-5 Mylar Spacers
FC- 6 Upper Plexiglas Plate FC-7 Lover Aluminum Support
FC-8 Upper Aluminum Support FC-9 Feed Port Fc-IO Ex11 Port
FC-I 1 Ultrafillrate Port
Figure 1. Assembly diagram of ultrafiltration cell with one POrous wall. Not to scale.
coefficients; however, this accuracy was not warranted for this particular study and would have required individual measurements four to ten times more accurate than those used. Channel height measurements were determined with YO using a micrometer caliper. At channel heights less than 0.04 cm, membrane compressibility a t the edges caused a significant decrease in actual channel height compared to spacer thickness. Small channel heights were correlated within YO using pressure drop-flow data for saline solution and the porous-wall, parallel-plate channel equations developed by Donoughe (1956). Results An example of the ultrafiltration data obtained with protein-saline solutions is shown in Figure 2 (Huffman, 1970). These data illustrated the fractional removal of saline solution (f) as a function of the dimensionless distance downstream (X) and the flow cell channel height for two membrane orientations perpendicular to the gravitational force: membrane below the axial cell flow (designated “down”) and membrane above the axial flow (designated “up”). A comparison of the two data sets with the membrane in the down position demonstrate the normal effect of diffusion path length. With the membrane in the up position, however, the data showed that channel height had a second effect which could reduce the diffusion path resistance and increase the rate of ultrafiltration 10-200% compared to molecular diffusion alone, depending upon initial conditions. With the membrane parallel to gravitational lorce and the axial flow either co- or countercurrent to gravity (vertical flow), the ultrafiltration rate was increased approximately 10% compared t o the down position (horizontal flow). These vertical orientations were not investigated any further as they were relatively unimportant compared to the horizontal arrangement. We note, however, that a 20% increase would be obtained for a cell with two porous walls and the resulting increase in separation could be significant when processing large volumes of liquids. All orientation effects on the ultrafiltration rate agreed with observations for heat transfer (McAdams, 1954) and suggested that mixing due to natural convection flow could be a contributing factor in the protein-saline solution separation. Correlation Equations for D a t a The data in Figure 2 are plotted in terms o f f and X to coincide with the definition of the point mass transfer Ind. Eng. Chem., P r o c e s s D e s . D e v . , Vol. 14, No. 2, 1975
167
z 0
r
.
3 0 W
z
lo 4 U. 0
z 0
P
..-U.
X , D I M E N S I O N L E S S A X I A L DISTANCE
Figure 2. Fractional removal of a saline solution from proteinsaline solution at 286 mm Hg hydrostatic pressure. Initial concentration of protein was 5 g/100 g of saline solution. Channel height, 0.16 cm: 0 , membrane up; 0 , membrane down. Channel height, 0.02 cm: 0 , membrane up; m, membrane down.
Figure 3. Leveque-Graetz correlation of protein-saline ultrafiltration data a t forced convection conditions (membrane in down position). The numbers after the symbols indicate the channel height, cm, and hydrostatic pressure, mm Hg, respectively: 0 , 0.097, 94; A , 0.097, 593; 8 , 0.165, 92; 4 , 0.165, 286; 7 , 0.020, 286; X , 0.020, 55; +, 0.040, 612; 0, 0.040, 320; A, 0.040, 175. The x data have been shifted three decades horizontally and one decade vertically to compress the graph.
coefficient or Sherwood number used for correlation purposes is shown in Figure 3. The data were correlated using a modified Leveque-Graetz equation. after incorporating the dimensionless treatment suggested by Dresner (1964). Equation 1 was obtained by combining a material balance for the saline solvent
(2)
N, = -(l/W)(dn/dx)
with the chemical potential definition for steady-state, unidirectional, one-component-stagnant diffusion N m ( l - x,) = k ' a ( p / R T ) / a h
(3)
along with the definition of osmotic pressure -RT In a, = TU,
(4)
The osmotic effect of protein in a saline solution was approximated by 71 = crc(l0 + c ) ( 5) using the data of Scatchard (1948). In the analysis, fluid density was assumed constant for a dilute binary mixture (protein-saline) and the hydrostatic gradient across the channel were neglected. The mass transfer coefficient used in the Sherwood number was defined as
k , = 2aD/BRTh
(6)
where D is the diffusion coefficient for protein given by k 'x,Jp. The osmotic pressure of protein a t the membrane surface was calculated using N, = -k,(AP - rm)
(7)
Nmo = -k,(AP -
(8)
and 710)
to yield T,
= (AI'
- no)(df/dx)
(9)
Equations 1, 5, and 9 were used to calculate the Sherwood number, Sh, reported below for both the forced and forced-plus-natural correction data obtained in the investigation. Discussion of Forced Convection Data
A summary of all data obtained in the study of proteinsaline solutions with the membrane in the down position 168
Ind. Eng. Chem., Process Des. Dev., Vol. 14, No. 2, 1975
Sh/Pe = sGz'/3Pe'1/3
(10)
with s = 0.2 f 35%. The introduction of the P e - l term was interpreted as a Schmidt number effect to account for the change in diffusivity and viscosity over the range of surface concentrations obtained (5 to 47 g of protein/100 g of saline solution). Using the definition of the mass transfer coefficient in eq 6 and the theoretical values of Kimura and Sourirajan (1968), the calculated value of the proportionality constant was 0.21. More recent calculations (Gooding, et a / , 1972) using Brian's reverse osmosis model (1965) show that the same data can be correlated within =k16%. Thus, the forced convection, ultrafiltration data reported here are thought to be reasonably well behaved and follow the basic principles of one-componentstagnant diffusion. Our results are not in agreement with the work of Bixler, et al. (1968), who concluded that reverse osmosis theory (Brian, 1965) was inadequate for the protein system and interpreted their data using conventional macroscopic filtration theory. We have not been able to develop a full explanation of the conflicting data but do note that neglecting the nonlinear osmotic pressure-concentration term in eq 5 could lead t o spurious results. For example, a t a surface osmotic pressure of 400 mm Hg, the linear or van't Hoff correlation predicts a protein concentration of 160 g/lOO g of saline solution whereas the nonlinear estimate is only 35 g/100 g. In addition, any failure to approach a true steady state is thought to be a key factor (Gooding, et a1 , 1972). The same flow cell also was used to investigate the ultrafiltration of whole blood (Ward, 1971). As shown in Figure 4, the Sherwood number for the whole blood experiments exhibited a significant deviation from the proteinsaline solution data. At constant Pe, the data demonstrated a decreasing Sherwood number with increasing X The trend was attributed to build-up of a gel-layer or platelet sedimentation and was in agreement with the observations on whole blood ultrafiltration reported by Bixler, et a! (1968). In general, the results illustrated in Figure 5 demonstrated that protein solutions cannot be used to approximate the ultrafiltration of whole blood except at very low rates of removal.
IO'
Io3
102
-1-0 1 Io4
x (Pe2) / 3
Io4
lo5
Ra / Pe
Figure 4. Forced convection data for ultrafiltration of whole blood. The numbers after the symbols indicate the channel height, cm, and hydrostatic pressure, mm Hg, respectively: 0 , 0.106, 192;A, 0.106, 468;-, correlation from Figure 3. T = 37°C.
Ra/Pe
Figure 5. Correlation of natural convection data for protein-saline solutions. Data symbols are for operating conditions cited in Figure 2 .
Discussion of Natural Convection Data The correlation of mass transfer coefficients calculated from eq 1 for protein-saline solutions with the flow cell in the up position is shown in Figure 5. The plot is based upon the vector addition model proposed by Martinelli and Boelter (1965) which was verified by Kubair and Pei's (1965) more general analysis of non-Newtonian Fluids. Other investigators have used vector addition powers ranging from 3 to 6 (Acrivos, 1966; Martinelli and Boelter, 1965), but we could find no significant improvement in our data beyond the square addition indicated in Figure 5. The Rayleigh number used in the correlation was defined using a fictitious density gradient which would exist if the secondary or natural convective flow was instantaneously stopped. Specifically, the definition was
R a = ygh4G/pDq
(114
G = VfCb/D
(1W
where Vf is the Ultrafiltration velocity that would develop from forced convection alone a t the existing CI,. As shown in Figure 5, the data were adequately represented by
S k c = 0.038(Ra/Pe)1/3
lo3
(12)
with these data sets scattered about the correlation in a manner consistent with the forced convection results shown in Figure 3. Because the 1/3-power law correlation with the Rayleigh number was within the 0.25-0.50 range of exponents reported for heat transfer (Catton, 1966; Sil-
Figure 6. Natural convection data for ultrafiltration of whole blood. Data symbols are for operating conditions cited in Figure 4 . Solid lines are from Figure 5 . T = 37°C.
verston 1959; Oliver, 1962; Globe and Dropkin, 1959), Figure 5 was considered to be a reasonable representation of the data. Catton's theoretical calculations (1966) suggest a lh-power correlation be used below Ra = lo4 and this correlation line has also been shown in Figure 5 for comparison purposes. The application of heat transfer correlations to this ultrafiltration or one-component-stagnant-diffusion process was shown to be valid by using the linear perturbation theory discussed by Chandrasekhar (1961). In particular, the principle of exchange of stabilities was demonstrated: the assumed density gradients would be steady and nonoscillatory with a critical point equivalent to that for heat transfer, as long as an average concentration gradient was used (Huffman, 1970). No critical point for the onset of natural convective motion could be determined from the data of Figure 5, but the results did suggest a reduction in convective motion for a Rayleigh number less than lo4 which is consistent with the accepted critical value of 1708 between two rigid boundaries (Chandrasekhar, 1961). The channel heights used throughout this study were always less than 0.16 cm and the correlation of data in terms of natural correction agreed with theoretical and experimental heat transfer studies noted above. Chandra (1938) has reported, however, that the critical point for natural convection no longer exists for channel heights less than 1 cm. This disagreement may be explained by the unsteady-state heating procedure employed by Chandra (see also Sutton, 1950). We concluded, therefore, that the experimental data reported in Figure 5 could be interpreted in terms of natural convective flow due to density gradients. Example data for whole blood processing (Ward, 1971) are shown in Figure 6. As with the protein-saline solutions, the increase in ultrafiltration rate could be interpreted using a natural convection, mass transfer coefficient; however, a correlation was not developed because the range of data was limited. Even with the natural convection, the over-rate or combined mass transfer rate for whole blood was always less than the forced convection ultrafiltration of the protein-saline solution which indicated that the gel-layer or platelet sedimentation was not completely destroyed.
Conclusions Over the range of data reported, we concluded that the ultrafiltration of protein-saline solutions could be described by a one-component-stagnant diffusion analysis. This same conclusion has been shown to be applicable to Ind. Eng. Chem., Process Des. Dev., Vol. 14, No. 2, 1975
169
reverse osmosis (Sourirajan, 1970); thus, both processes may be analyzed on the same basis. In addition, higher rates of ultrafiltration were observed when the membrane was oriented to favor secondary flow by natural convection. The higher rates of ultrafiltration were correlated using the principles of heat transfer in the presence of natural convection. During the ultrafiltration of whole blood, a reduced rate of mass transfer was observed. The reduction was attributed to a gel-layer build-up or platelet sedimentation which was partially destroyed by natural convection flow superimposed on bulk axial flow.
Nomenclature a = thermodynamic activity B = (mol wt of solvent)/(100 x mol wt of protein) c = concentration, g of protein/100 g of solvent D = diffusion coefficient f = fraction of saline solution remaining = n/no g = gravitational constant Gz = Graetz number = l/(Pez)(X) h = ultrafiltration cell channel height, cm k' = proportionality constant, mol/cm sec k = mass transfer coefficient, mol/cm2 sec n = total moles of solvent or saline, mol N = mass transfer flux, mol/cm2 sec T = osmotic pressure, m m Hg 1P = hydrostatic pressure, m m Hg Pe = Peclet number = Voh/D R = gas-lawconstant Ra = Rayleighnumber s = experimental proportionality constant S h = Sherwood number = k h / D / i T = absolute temperature. OK u = axial velocity, cm/sec u = partial molal volume, cm3/mol; ultrafiltration velocity, cm/sec LL: = ultrafiltration cell channel width, cm x = axial distance, cm; mole fraction X = dimensionless axial distance = uox/uoh ck = experimental constant, (mm Hg) (g of solute)/(100 g of solvent) = 0.25 = density, mol/cm3 7 = coefficient of expansion 4 = kinematic viscosity, cm2/sec p = chemical potential
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Subscripts 0 = inlet condition m = membrane or membrane surface b = bulk solution s = salinesolution f = forced convection nc = natural convection Literature Cited Acrivos, A., Chem. Eng. Sci., 21, 343-352 (1966). Bixler, H. J., Nelsen, L. M., Besarab, A , , Chem. Eng. Prog. Symp. Ser., 64, No. 84. 90-103 (1968). Brian, P. L. T., Ind. Eng. Chem., Fundam., 4, 439-445 (1965). Catton, I., Phys. Fluids, 9, 2521-2522 (1966). Chandra, K., Proc. Roy. SOC.Ser. A, 164, 231-242 (1938). Chandraseklar, S., "Hydrodynamic and Hydromagnetic Stabiiity," Oxford University Press, New York, N.Y., 1961. Donoughe, P. L., N.A.C.A. Tech. Note No. 3759, (1956). Dresner, L., Oak Ridge National Laboratory, ORNL-3621, OSW, Department of the Interior. May 1964. Globe. S..Dropkin, D . , J. Heat Transfer, 81, 24-28 (1959) Gooding, C. H., Melsheimer, S. S., Harshman, R. C.. Paper 4d, 73rd National A.1.Ch.E. Meeting, Aug 27-30, 1972. Haupt, G. W . , J. Res. Nat. Bur. Std., 48A, 414-423 (1952). Hubbard. D. W..A.I.Ch.E. J . , 14, 354-355 (1965). Huffman, W. J . , Ph.D. Dissertation, Clemson University, Clemson, S.C.. 1970. Hunt. R. E., Industry Comment Letter, L730602, Arthur D . Little, Inc., June 13, 1973. Kimura. S.. Sourirajan, S., Ind. Ens. Chem., Process Des. Dev., 7 , 539 (1968). Knudsen, J. G., Katz, D . L.. "Fluid Dynamics and Heat Transfer," McGraw-Hill, New York, N.Y., 1958. Kubair, V . G.. Pei, D. C., Int. J. Heat Mass Transfer. 1 1 , 855-869 (1965). McAdams, W. H . , "Heat Transmission," pp 242-245, McGrawHill, New York, N.Y.. 1954. Martinelli, R. C.. Boeiter. L. M. K . , "Heat Transfer Notes," pp 412-417, L. M. K . Boelter. Ed., McGraw-Hill, New York, N.Y., 1965. Michaels, A. S., "Progress In Separation and Purification," E. S. Perry, Ed., Vol. 1, pp 297-298, Interscience, New York. N.Y., 1968. Murphy, J. B.. Kies, M. W., Biochem. Biophys. Acta.. 45, 382384 (1960) Oliver, D. R.. Chem. Eng. Sci., 17, 335-348 (1962). Reilly. J. E., M.S. Thesis, Clemson University, Clernson, S.C.. 1969. Scatchard, G., Batcheider, A . C., Brown A,, Zosa, M., J. Am. Chem. Soc., 68,2160-2612 (1948). Sherwood. T. K . , Brian, P. L. T., Fisher, R . E., Dresner, L., Ind. Eng. Chem.. Fundam., 4, 113-118 (1965). Siiverston, P. L., Forsch. Eng. Wes., 24, 29-32, 59-69 (1959). Sourirajan, S., "Reverse Osmosis," Chapters 1, 4, Academic Press, NewYork, N.Y., 1970. Sutton, 0. G., Proc. Roy. Soc., Ser. A, 204, 297-309 (1950). Ward, R . M.. M.S. Thesis, Clemson University, Clemson, S.C., 1971
Received f o r reuieu July 26, 1974 Accepted December 12, 1974
