Ind. Eng. Chem. Process Des. Dev. 1982, 21, 277-282
277
Predictability of Membrane Performance for Mixed Solute Reverse Osmosis Systems. System Cellulose Acetate Membrane-D-Glucose-D,L-Malic Acid-Water Palanlappan Malalyandl, Takeshl Matsuura, and S. SourIraJan’ Division of Chemistry, National Research Council of Canada, Ottawa, Canada K1A OR9
This paper presents an analytical technique by which the reverse osmosis performance of a cellulose acetate membrane can be predicted for an aqueous feed solution containing D-glucose and partially ionized o,L-malic acid, from only a single set of experimental reverse osmosis data for a NaCI-H,O reference feed solution. The valiii of the analytical technique is confirmed by good agreement between experimental and calculated reverse osmosis data on solute separations and product rates for such feed solution systems and cellulose acetate membranes of different surface porosities. This work is of significance both to the engineering science of reverse osmosis transport and to its practical utilization in process design and development particularly in the field of food technology.
Introduction The applicability of the basic transport equations (Sourirajan, 1970a) for predicting membrane performance in reverse osmosis for a variety of mixed-solute aqueous feed solutions has been illustrated (Agrawal and Sourirajan, 1970; Matsuura and Sourirajan, 1971; Matsuura et al., 1974b; Rangarajan et al., 1978,1979; Sourirajan, 1970b). Extending the above work, this paper is concerned with the problem of predicting membrane performance for aqueous solutions involving food sugar-food acid mixed solutes. This problem is particularly relevant to the application of reverse osmosis for the concentration of fruit juices and/or fruit juice aroma compounds, several aspects of which have already been discussed (Matsuura et al., l973,1974a, 1975a;Matsuura and Sourirajan, 1978; Pereira et al., 1976; Baxter et al., 1980). For purposes of illustration, D-glUCose and D,L-maliC acid were chosen to represent the food sugar and the food acid, respectively, which are commonly present in natural fruit juices such as apple juice. This paper presents the analytical technique by which the reverse osmosis performance of a cellulose acetate membrane can be predicted for an aqueous feed solution containing D-glucose and D,L-malic acid from only a single set of experimental data for a reference feed solution such as sodium chloride-water. The validity of the analytical technique formulated in this work is confirmed by the good agreement obtained between experimental reverse osmosis data and those calculated on the basis of the analytical technique presented in this work. Experimental Section Reagent grade glucose and D,L-malic acid were used without any further purification. Laboratory-made 316 (10/30)-type cellulose acetate membranes (Pageau and Sourirajan, 1972) were used throughout this work. The apparatus used and the general experimental details were the same as those reported earlier (Matsuura and Sourirajan, 1973). In series I of reverse osmosis experiments the molal ratio of glucose to D,L-malic acid was maintained a t 4 to 1, while in series I1 the ratio was 7 to 1. The concentration of glucose ranged from 0.03 to 0.83 m in series I and from 0.28 to 1.67 m in series 11, respectively. The operating pressure for the reverse osmosis experiments was 6895 Wag (= lo00 psig) in all cases. All membranes were subjected to a pressure of 8274 kPag (= 1200 psig) for 2 h with pure water as feed, prior to reverse osmosis experiments. The specifications (Sourirajan, 1970a) of Ol96-4305/82/l12l-0277$01.25/0
membranes used are given in Table I in terms of pure water permeability constant A (in kg-mol of H20/m2s P a ) and solute transport parameter (Dm/K6)NacI(treated as a single quantity in m/s) for sodium chloride. All reverse osmosis experiments were of the short run type, and they were carried out a t the laboratory temperature (23-25 “C) using a feed flow rate of 400 cm3/min. Mass transfer coefficients corresponding to the flow rate used are listed in Table I for each membrane used. In each experiment, the fraction solute separation, f , defined as (molality in feed) - (molality in product) (1) f= molality in feed with respect to glucose and D,Lmalic acid and product rate (PR)and pure water permeation rate (PWP),in kg/h per given area of film surface, 13.2 cm2 in this work, were determined under the specified experimental conditions. In all experiments, the terms “product” and “product rate” refer to membrane permeated solutions, and the reported permeation rates are those corrected to 25 “C using the relative viscosity and density of pure water. The concentration of glucose and malic acid in the feed and the product solutions were determined as follows: the analysis of glucose was carried out colorimetricallyby using 3,5-dinitrosalycylate as color developing reagent and measuring absorbance a t 540 pm (Clark, 1964). The analysis of D,L-malic acid was conducted by a titration method with sodium hydroxide solution as the neutralizing reagent .
Theoretical Section Transport Equations. The basic transport equations describing the system sugar-acid mixture is in essence the combinations of those developed for separations of partially dissociated organic acids (Matsuura et d., 1976b) and for separations of undissociated organic solutes in concentrated sugar solutions (Matsuura and Sourirajan, 1978). They are briefly outlined below. All the symbols used are defined a t the end of the paper. The equation for solvent water flux through the membrane can be written as NB = A(P - ~2 + “3) (2) Assuming that each solute species is transported through the membrane without interaction with each other, 0 1982 American Chemical Society
Ind. Eng. Chem. Process Des. Dev., Vol. 21, No. 2, 1982
278
Table I. Film Specification film no. 1
operating pressure, 68.9 X lo2 kPa (1000 psig) pure water permeability constant, A, (kg-mol H,O/mZ s kPa) X l o 7 solute transwrt Darameter. (DAM/KG jN& m/s x l o 7 mass transfer coefficient,a hNaC1, m/s x l o 6 solute sewration. % b product ;ate, kg/h X l o 3 Feed flow rate, 400 cm3/min.
2
1.058
1.245
1.458
4.572
18.22 97.9 58.29
15.40 92.5 66.38
3
4
5
6
2.079
2.487
3.140
3.762
12.68
22.84
75.85
119.7
25.94 88.2 111.55
27.40 81.5 132.83
33.88 62.0 167.83
32.08 47.4 208.15
Area of film surface, 13.2 em2;feed concentration, 0.06 m.
Table 11. Solute Transport Parameters of Glucose and Dissociated and Undissociated D,L-Mdic Acid film no.
In C*NaCl In ( c * N a d A ) In A * ( 6 *%)/(S *CES)h, (DAM/K~ )g, m/s X 10' (DAM/K~ k, m/s X l o 7 (DAM/K6 h,, m/s X l o 7
1
2
3
4
5
6
-17.11 -1.05 0.0 1.0 0.233 0.540 6.378
-15.97 -0.07 -0.40 0.95 0.621 1.687 17.00
-14.95 0.44 -0.70 0.90 1.557 4.680 40.97
-14.36 0.85 -0.90 0.76 4.319 8.440 89.38
-13.16 1.81 -0.92 0.66 26.13 28.02 400.6
-12.70 2.09 -0.92 0.65 50.05 43.94 715.5
equations for solute fluxes of glucose, ionized malic acid, and nonionized malic acid can be written as (3) Ng = (DAM/K~),(C&~,~ - C3xg,3)
Ni = ( D A M / K ~ ) ~ ( -CCJi,3) ~X~,~
(4)
scribed later in detail together with data characterizing membranes and a set of data representing operating conditions, unknown quantities such as Xg,3,Xm3,NB, Ng,and N, can be obtained. Using above quantities and equations
(5) Nu = (DAM/K8)u(C2Xu,2- C,Xu,J where Nk,Ni and Nu denote the flux of glucose, ionized, and nonionized malic acid, respectively. Equations expressing boundary concentrations can be written for each solute species as xg,2
=
Xg,3 + (Xg,1- Xg,J exp[(NB+ Ng + Ni + Nu)/kgcll (6) Xi,2 = Xi,3 + xu,2
(xi.1-
Xi,3) exp[(NB+ Ng + Ni
=
+ Nu)/kmc11
Xu,3 + ( X ~-J Xu,3) exP[(NB + Ng + Ni
(7)
+ Nu)/kmclI
(8) Equations for calculating product mole fractions for each solute species involved can be written as (9) Xg,3 = Ng/(NB + Ng + Ni + Nu)
+ Ng + Ni + Nu)
(10)
Xu,3 = N u / ( N + ~ Ng + Ni + Nu)
(11)
Xi,3 = Ni/(NB
In addition to the above equations, dissociation equilibria of malic acid in feed and product solutions have to be taken into consideration by Xi,l = [(f/4K2+ loooc~x,,lKa)- f/~Ka]/loooc~ (12) and xi,3 =
+
[(f/4K,2 1000C3Xm,3Ka) - l/zKa]/1oo0C3
and
(PR) = [(NBX MB) + (NgX M g ) + (N,
S X 3600 (19) it is possible to calculate the solute separations of glucose, fg, and of malic acid, f,, as well as the product permeation rate, (PR). Solute Transport Parameters. As shown by eq 3,4, and 5, the flux of each solute species through the membrane is given by the product of solute transport parameter and the difference in solute concentrations at high and low pressure sides of the membrane. In order to make the calculation possible, numerical values have to be provided with respect to the solute transport parameter of each species under consideration. The procedure of calculating the solute transport parameter for different solutes for a given membrane on the basis of that for sodium chloride was illustrated in the earlier work (Sourirajan and Matsuura, 1979) and is outlined briefly as follows. With reference to sodium chloride solute, the solute transport parameter, DAM/Kb, can be written as X
M,)]
X
(13)
Further, since the quantity of total malic acid is the sum of ionized and nonionized species Xm,1 = Xi,l + Xu,1 (14) (15) Xm,3 = Xi,3 + Xu,3 N, = Ni + Nu (16) By solving eq 2 to 16 simultaneously by the method de-
where [-AAG/RT], called free energy parameter, represents the contribution from each ionic species to the solute transport parameter, and the numerical value of In C*NscI is the corresponding expression for the porous structure of the membrane. The numerical values of the free energy
Ind. Eng. Chem. Process Des. Dev., Vol. 21, No. 2, 1982 279
parameter (-AAGIRT) for Na+ and C1- ions are known to be 5.79 and -4.42, respectively, for cellulose acetate Eastman E-398-3 material used in this work (Matsuura et al., 1975b). Using the above constants and the experimental (Dm/K6)NaCI values for the given membranes as listed in Table I, the values of In C*,," for each membrane can be calculated. The results obtained are shown in Table 11. The value of (DAMIK6) for any polar organic solute corresponding to the above (DAM/ K 6 ) N a c i value obtained from the sodium chloride experiment is then given by the following expressions. For ionized solutes In ( D m / K 6 ) = In C*NaC1+ C(-AAG/RT)i (21) for nonionized solutes In (D-/K6) = In C*NaC1 In A* (-AAG/RT)
+
+
+ 6*CE, t w*Cs* (22)
Referring to eq 21, the numerical value of C(-AAG/RT)i for ionized D,L-malic acid is 0.37 from the literature (Matsuura et al., 1976b). Referring to eq 22, the quantity In A* (which represents a scale factor) is a function of In (C*NaC1/A) as indicated in Figure 5 of Sourirajan and Matsuura (1979). The applicable value of (-AAGIRT) for use in eq 22 can be calculated on the basis of the molecular structure of solute using the equation AAG = AGI - AGB (23) and the following additivity equations AGB = CyB(structuralgroup)
AGr = CyI(structural group)
+ YB,O
(24)
+ yI,o
(25)
Data on yB(structural group), Y B , ~ ,yI(structural group), and applicable for glucose and nonionized malic acid are available in the literature with respect to cellulose acetate Eastman E-398-3 material (Matsuura et al., 1976a; Sourirajan and Matsuura, 1979). The numerical values of (-AAGIRT) thus obtained for glucose and nonionized malic acid are 4.95 and 6.41, respectively. The quantity 6*CE, in eq 22 may be treated as a single quantity for solute molecules involving polyfunctional groups such as glucose and malic acid. This quantity for a particular membrane-solute system can be calculated by multiplying the ratio 6*CE0/(6*CE,), by the quantity (C?*CE~ The ) ~correlation . of In ( C * N a c l / A ) vs. the above ratio has been experimentally established as given in Figure 2 of Sourirajan and Matsuura (1979) for cellulose acetate Eastman E-398-3 material. The quantity (6*CE,)h to be used in conjunction with this correlation can be obtained from (6*CEJr, = C$(structural group) + $, (26) The applicable data on $(structural group) and $o are listed in Table IV of Sourirajan and Matsuura (1979). Further, $(structural group) values for -COOH functional group of 1.23, when the solute molecule is branched, and of 1.50, when it is straight, were developed by the analysis of data on some carboxylic acids. The values of ( 6 * x E s ) b so obtained for glucose and nonionized malic acid were -5.42 and -3.57, respectively. The nonpolar contribution to In ( D A M / K 6 ) involved in eq 22 can be omitted since both solute molecules do not contain more than three straight-chain carbon atoms not associated with a polar functional group. Numerical values of solute transport parameter calculated by the procedure described above are listed in Table I1 with respect to glucose and ionized and nonionized malic
acid for all membranes used in this work. Physicochemical Data Applicable for the MixedSolute System Glucose-Malic Acid-Water. Data on osmotic pressure, kinematic viscosity, and molar density for the above mixed solute system, data on mass transfer coefficient for the solutes involved, and data on dissociation constant for malic acid in glucose-water solutions are needed for use in predicting calculations on membrane performance on the basis of the transport equations formulated above. Such physicochemical data are not available in the literature explicitly. Hence appropriate practical techniques, adequate for the purpose of this work, were established for obtaining the required data from those available in the literature for the corresponding single solute systems. These techniques, along with the assumptions involved, may be outlined briefly as follows. Data on Osmotic Pressure. In the range of solute concentrations involved in this work, the osmotic pressure of the solution was assumed to be directly proportional to the mole fraction of solute (XJ, so that T(XA) = B X A (27) where E is a constant. From experimental osmotic pressure data for aqueous glucose solutions available in the literature (Weast, 1977), the value of B for glucose (= Bg) was found to be 1.40 X lo5kPa. Similar experimental data for aqueous malic acid solutions are not available in the literature. However, the available osmotic pressure data (Timmermans, 1960) show that the values of B are 1.46 X lo5 kPa for aqueous tartaric acid solutions and 1.48 X lo5 kPa for aqueous citric acid solutions. On the basis of chemical similarity of the above solutions with those of malic acid, the value for B for aqueous malic acid solutions (= Em)was assumed to be 1.47 X lo5 kPa. The practical validity of this assumption is justified later in this discussion. For the mixed-solute system glucose-malic acid-water, the osmotic pressure was assumed to follow the simple additivity principle so that T(Xg,Xm) = B g X g + B m X m (28)
Data on Kinematic Viscosity. Viscosity data as a function of solute concentration with respect to glucose and malic acid a t 25 "C are available in the literature (Timmermans, 1960). The data fit the polynomial expressions tlg/fw tlm/tlw
=1+ =1
+ PgX,2
(29)
+ a m x m + PmXm2
(30)
agxg
with numerical values of 2.27 X 10,4.14 X lo2, 1.91 X 10, and 3.49 X lo2 for the coefficients cyg, Pg, am,and Pm, respectively. On the basis of the above expressions, assuming the additivity principle, the viscosity of the mixed-solute system glucose-malic acid-water (Tt) was expressed as
( 1 + agXg
PgXg
+ Pmxm
xg+xm
j(xg+x..i.=
+ a m x m + (Pgxg + PmXm)(Xg + X m )
(31)
Density data as a function of solute concentration with respect to glucose and malic acid are also available in the literature (Timmermans, 1960). Assuming again the additivity principle, the density of the mixed solute system glucosemalic acid-water (pJ was expressed by the relation (32) Pt = ~w + ( P g ( X g ) - ~ w + ) ( P m ( X m ) - PW)
280
Ind. Eng. Chem. Process Des. Dev., Vol. 21, No. 2, 1982
Using the qt and pt values obtained from eq 31 and 32, the kinematic viscosity of the mixed-solute system (USwas obtained from the relation Vt
= ?t/Pt
(33)
Data on Molar Density. Assuming again additivity principle, the molar density (c, in kg-mol/m3) of the mixed solute system was calculated from the relation c = Pt/[MgXg + MmXm + M B ( -~ Xg - Xm)l (34) Data on Mass Transfer Coefficients. Using the Wilke-Chang equation (Wilke and Chang, 1955), diffusivities of glucose and malic acid in dilute aqueous solutions (D,,%and D,,w, respectively) were calculated to be 0.755 X 10 and 0.815 X lo4 m2/s, respectively. Following Reid and Sherwood (1958), the diffusivity of glucose (D,) and that of malic acid (D,) at higher concentrations were obtained from the relations
D, = D,,,.-
4w
(35)
4t ?W
D, = Dm,w-4t
Using the numerical values of D , D,, and ut obtained above, data on mass transfer coefficient for glucose (k,) and those for malic acid (k,) in the mixed solute system were obtained from the following relations generated on the basis of correlations similar to those reported earlier by Hsieh et al. (1979a,b)
k, = D , 2 / 3 [ ( k ~ a ~ 1 / ( D ~ ) ~ -a ~1.393 1 2 / 3x) 10' x ( U t 8.963 x io-')] (37) k, = D,2/3[(kNa~1/(DAB)NaC12/3) - 1.393 X IO' x (vt 8.963
X
(38)
where (DM)NaC1 is taken to be the diffusivity of sodium m2/s) in 0.06 m aqueous solution. chloride (= 1.53 X Data on kN&l applicable for the membranes used in this study are included in Table I. Data on Dissociation Constant for Malic Acid in Glucose-Water Solutions. Dissociation constants of malic acid are independent of acid concentration in aqueous solutions; the corresponding data in glucose-water solutions are lower because of the lower dielectric constant of the latter solutions (Timmermans, 1960) (Table 111). Data on dissociation constant for malic acid in different glucose-water solutions can be calculated from the relations (Monk, 1961) -In K, = 13.5474 + 3 log Izizjl/tT + log Q(b) (39) and b = Izizjle2/atkT
(40)
The ionic valencies (ziand zj) are both unity for malic acid. The quantity Q(b) as a function of b, defined by eq 40, is given in Table 14.4a of Monk (1961). Using eq 39 and 40 and K, in water at 25 OC, the values of a and b were calculated to be 0.039 nm and 18.12, respectively. Using the value of a so obtained, the values of K, were calculated for different glucose-water solutions according to their values. Numerical Solution of Transport Equations and Comparisons with Experimental Results. By using basic transport equations developed above, solute separations of glucose and malic acid and product rates can be calculated for a given membrane at a given set of operating
Table 111. Physicochemical Properties of Glucose-Water and D,L-Malic Acid-Water Solutions mole molal- fraction" p g x itya X lo3 kg/m3
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.2 1.4 1.6 1.8 2.0
0 1.798 3.590 5.375 7.154 8.927 10.693 12.453 14.207 15.955 17.696 21.160 24.600 28.016 31.408 34.777
0.9971 1.0050 1.0130 1.0185 1.0241 1.0298 1.0355 1.0413 1.0470 1.0529 1.0582 1.0693 1.0799 1.0899 1.0989 1.1065
pm
x
&/m3
g
0.9971 0.9994 1.0000 1.0005 1.0010 1.0014 1.0021 1.0025 1.0031 1.0035 1.0041 1.0052 1.0061 1.0070 1.0079 1.0089
78.54 78.14 77.73 77.35 76.95 76.56 76.20 75.80 75.42 75.08 74.73 74.05 73.39 72.72 72.12 71.51
f
" Data refer to single solute systems. conditions such as operating pressure, solute concentration in feed, and feed flow rate when a single set of reverse osmosis data with an aqueous sodium chloride solution is given, as described below. By applying the basic transport equations (Sourirajan, 1970a) to sodium chloride experimental data, data on A , (Dm/K6)NaC1, and kNaclare obtained as listed in Table I for the membranes used in this study. Application of eq 20 to 22 for (Dm/k6)NaC1thus obtained enables one to calculate (DM/K6) values for glucose, ionized malic acid, and nonionized malic acid, as shown in Table I1 for the particular membranes listed in Table I. The operating pressure, P, is given as one of the operating conditions. The mole fraction of glucose and malic acid in the feed solution, designated here as Xg,land Xm,l,respectively, are calculated from molality data of both solute components in the feed solution. The mole fraction data of ionized and nonionized malic acid are then calculated from eq 12 and 14. Since mass transfer coefficient for reference solute (kNaCJ and solute mole fractions Xgtland Xm,lin feed are now known, mass transfer coefficients with respect to glucose and malic acid can be calculated by using eq 29 to 33 and 35 to 38. All numerical values obtained above characterize membranes and operating conditions used for the reverse osmosis treatment of the aqueous solution of glucose and malic acid and enable us to solve eq 2 to 19 numerically by following steps. Step 1: Assume numerical values of NB, Xg,3,and X,,,. Step 2: Calculate Xi,3and Xu,3by eq 13 and 15. Step 3: Calculate N,, Ni, Nu, and N, by using eq 9 to 11 and 16. Step 4: Calculate pt,l and c1 by using data for X,,l and Xm,lfrom eq 32 and 34. Step 5: Calculate Xg,2,Xi,2,and Xu,2from eq 6 to 8. Step 6: Calculate 7r2 and 7r3 using eq 28. Step 7: Calculate NB, N,, N, from eq 2 to 5 and 16. Quantities c2 and c3 used in eq 3 to 5 can be obtained from eq 32 and 34 along with quantities Xg,2,Xm,2,Xg,3,and Xm,3*
Step 8 The values of NB, N,, and N , calculated by step 7 are compared with those assumed by step 1and obtained by step 3. When agreement is reached go to the next step; otherwise go back to step 1. Step 9: The solute separations with respect to glucose and malic acid and the product rate are calculated by eq 17 to 19.
Ind. Eng. Chem. Process Des. Dev., Vol. 21, No. 2, 1982 281 GLUCOSE MOLALITY : MALIC ACID M O L A L I T Y 4 : I
perimental results in Figure 1 and Figure 2 for two series of experiments where glucose to malic acid molality ratios were maintained a t 4 to 1 and 7 to 1, respectively. The agreement of calculated and experimental data is good, which testifies to the validity of the analytical technique used for predicting membrane performance. The excellent agreement obtained between the experimental and calculated product rate data justifies the numerical value of B, assigned earlier to malic acid solutions for the calculation of osmotic pressures.
FILM No
Conclusion I
I
'
2
I
I
I
I
60L
2
I
The feed solution involved in this study contained Dglucose and D,L-mdiC acid; while the former remained as a nonionized solute in all cases, the latter remained in different degrees of ionization depending on experimental conditions. Thus the mixed solute system involved in this work is essentially one of three solutes in aqueous solutions. The analytical technique presented in this study for predicting data on reverse osmosis performance for such mixed solute systems makes a useful contribution both to the engineering science of reverse osmosis transport and to its practical utilization in process design and development, particularly in the field of food technology.
I
I
I
V
40-x2x,x-x
++
0
++
L
-
+
I
I
I
I
I
0.5
10 .
I
I:.
I
f 1.5
GLUCOSE MOLALITY
Figure 2. Glucose molality in feed vs. solute separations of glucose and D,L-malic acid and product rate. Membrane, cellulose acetate 316 (10/30); glucose mo1ality:malic acid molality, 7:l; operating pressure, 6895 kPa gauge (= lo00 psig); feed flow rate, 400 cm3/min.
Data on solute separations for glucose and malic acid and product rate thus calculated were compared with ex-
Nomenclature a = the distance of closest approach of two ions, nm A = pure water permeability constant, kg-mol/m2 s kPa B = proportionality constant defined by eq 27, kPa b = quantity defined by eq 40 c = molar density of solution, kg-mol/m3 C*NaC1 = quantity defined b eq 20 D = diffusivity of solute, mH/ e ( D ~ ) N , =c ~diffusivity of reference sodium chloride in water, m2/s ( D A ~ / K 6=) solute transport parameter (treated as single quantity), m/s e = electron charge, C E,, X E S = Taft's steric parameter f = fraction solute separation AGB = free energy of hydration of solute in bulk phase, J/ kg-mol AGI = free energy of hydration of solute in interfacial phase, J/ kg-mol A A G I R T = polar free energy parameter k = Boltzmann constant, JK-' k = mass transfer coefficient for the solute on the high pressure side of the membrane, m/s kNacl= k for reference solute sodium chloride, m/s K , = dissociation constant, kg-mol/m3 M = molecular weight N = solute flux through membrane, kg-mol/m2 s N B = solvent flux through membrane, kg-mol/m2 s P = operating pressure, kPa (PR) = product rate through given area of membrane surface, kg/h (PWP)= pure water permeation rate through given area of membrane surface, kg/h Q(b) = function of b associated with eq 39 S = membrane area, m2 s*, Cs* = modified Small's number (nonpolar parameter) T = absolute temperature, K X = mole fraction of solute z,, z, = ion valency Greek Letters a = constant defined by eq 29 and 30
p = constant defined by eq 29 and 30
yB(structural group) = structural contribution to AGB, J/ kg-mol yI(structuralgroup) = structural contribution to AG1, J/kg-mol Y ~ = ,constant ~ in eq 24 yI,o= constant in eq 25 6* = coefficient associated with steric parameter E,, or CE,
Ind. Eng. Chem. Process Des. Dev. 1982, 27, 282-289
282
(6*CE,), = limiting value of 6*CE, for low porosity membrane surface In A* = quantity defined by eq 22 when polar, steric, and nonpolar effects are each set equal to zero e = dielectric constant O(structura1group) = structural contribution to (6*CE,)li, O,, = constant in eq 26 vt = kinematic viscosity of mixed solution, mz/s 7r = osmotic pressure, kPa p = density, kg/m3 pt = density of mixed solution, kg/m3 pw = density of water, kg/m3 w* = coefficient associated with nonpolar parameter, s* and
cs*
q = viscosity, kPa s qt = viscosity of mixed qw
solution, kPa s
= viscosity of water, kPa s
Subscripts 1 = bulk feed solution 2 = concentrated boundary solution on the high pressure side of membrane 3 = membrane permeated product solution on the low pressure side of membrane g = glucose m = D,L-malic acid i = ionized (dissociated) D,L-malic acid u = nonionized (undissociated) D,L-malic acid Literature Cited Agrawai, J. P.; Sourirajan, S. I d . Eng. Chem. Process Des. Dev. 1870, 9 , 12. Baxter, A. G.; Bednas, M. E.; Matsuura, T.; Sourirajan, S. Chem. Eng. Commun. 1980, 4 , 471. Clark, J. M., Jr., Ed. "Experimental Biochemistry", W. H. Freeman: San Francisco, 1964. Hsieh, F.; Matsuura, T.; Sourirajan, S.Ind. Eng. Chem. Process Des. Dev. 18788, 18, 414.
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Received for reuiew December 1, 1980 Accepted October 10, 1981
Issued as NRC No. 19974.
Correlation for Viscosity Data of Liquid Mixtures Manuchehr Dlzechl Teledyne Inef, Torrance, California 90509
Ekkehard Marschall' Department of Mechanical and Environmental Engineering, University of California, Santa Barbara, California 93 106
McAllister's theory of viscosity for liquid mixtures was modified in order to account for temperature dependencies different than that predicted by the Eyring model. The resulting equation was tested with the experimentally obtained viscosity data of 41 binary and 16 ternary mixtures. The improved version of the McAllister equation correlates viscosities of liquid mixtures very well, whether they contain polar compounds or not.
Introduction A reliable and generally valid theory for the quantitative prediction of viscosities of liquid mixtures from the properties of the pure components has not been established. For this reason, information on viscosity data of liquid mixtures, especially of mixtures containing polar components, continues to be based on experimental investigations. Numerous semitheoretical equations have 0196-4305/82/112 1-0282$01.25/0
been proposed for the correlation of experimental viscosity data. There are several desirable properties which such equations should have. The equations should correlate experimental data as closely as possible; that is, they should allow for only a very small standard deviation. The number of necessary experimental data should be kept to a minimum in order to reduce.costly experiments. Finally, the equation should be generally applicable and should 0 1982 American Chemical Society
