f g gt h
hi ho k
L 1 m N no np
p pc p p
p, Q RF Tbl
T,,
Tbm T, U
W wi xi zi
Fanning friction factor specific gravity, ft./(hr.)2 polynomial number of design variables inside film coefficient, B.t.u./(sq. ft.)(hr.)(O F.) outside (condensing) film coefficient, B.t.u./(sq. ft.) (hr.) (” F.) = thermal conductivity, B.t.u.j(ft.) (hr.) (” F.) = tube length, ft. = tube wall thickness, ft. = number of variables in problem = number of tubes in condenser = number of terms in objective function = total number of terms in objective function and constraints = number of primal constraints = rate of depreciation, l/year = plant factor-on-stream time, hr./year = probability = condenser heat load, B.t.u./year = fouling resistance, (sq. ft.)(hr.) (” F.)/B.t.u. = inlet bulk temperature, O F. = outlet bulk temperature, O F. = mean bulk temperature, ” F. = steam temperature, F. = dual objective function = flow rate inside tubes, lb./hr. = design variables = primal variables = operating variables
= = = = = =
GREEKLETTERS coefficient in steam cost equation, $/B.t.u.
(YO
=
ci
= coefficient in steam cost equation, $/B.t.u. = positive coefficient
Pj
=
cy1
e3 r yk
AP
(” F.)
upper bound on coefficients
AT* = temperature rise of fluid in condenser, F. ATrn, = mean temperature drop through inside tube film,
” F.
A T m p = mean temperature drop through inside tube fouling, O
F.
A T,o = mean temperature drop through condensing film, O F. 6j = dual variable q = over-all p u m p efficiency X = latent heat of evaporation, B.t.u./lb. p = viscosity, lb./(ft.)(hr.) p = density, lb./cu. ft.
SUBSCRIPTS = bulk = physical property evaluated a t outside film temperaf ture b
SUPERSCRIPTS
*
= optimal = variable cost
Literature Cited
Avriel, M., IVilde, J. D., IND.ENG. CHEM.PROCESS DESIGN DEVELOP. 6, 256 (1967a). Avriel, M., IVilde, D. J., “Stochastic Geometric Programming,” Proceedings of International Symposium on Mathematical Programming, Princeton, N. J., 1967b. Clasen, R. J., “Numerical Solution of the Chemical Equilibrium Problem,” Rand Gorp., Research Memorandum RM-4345-PR (1965). Duffin, R. J., Peterson, E. L., Zener, C., “Geometric Programming,” IViley, New York, 1966. Rudd, D., \Vatson, C. C., “Strategy of Process Engineering,” Jt‘iley, New York, 1968. IVilde, D. J., Beightler, C. S., “Foundations of Optimization,” Chap. 4,Prentice Hall, Englewood Cliffs, N . J., 1967. RECEIVED for review May 21, 1968 ACCEPTEDOctober 15, 1968
= geometric mean of coefficients
suboptimum = sum of dual variables in kth constraint = total pressure drop, lb./sq. ft. =
Research supported in part by the Office of Saline Water Grant 14-01-0001-699.
REVERSE OSMOSIS SEPARATION OF UREA IN AQUEOUS SOLUTIONS USING POROUS CELLULOSE ACETATE MEMBRANES H A R U H I K O O H Y A
A N D S. S O U R I R A J A N
Division of Applied Chemistry, .Vational Research Council of Canada, Ottawa, Canada
The transport characteristics of the Loeb-Sourirajan type porous cellulose acetate membranes in reverse osmosis process for the system urea-water are similar to those reported for the systems sodium chloride) for different solutes offer a useful means of expressing water and glycerol-water. The ( D A M / K ~ values selectivity of the membranes in reverse osmosis. A system of general equations for stagewise reverse osmosis process design is presented, and their application is illustrated with a set of numerical data for water renovation from aqueous urea solutions. HE characteristics of the Loeb-Sourirajan type porous Tcellulose acetate membranes (Loeb and Sourirajan, 1963, 1964; Sourirajan and Govindan, 1965) for the reverse osmosis separation of several inorganic and organic solutes in aqueous solution have been reported (Kimura and Sourirajan, 1967, 1968b; Sourirajan and Kimura, 1967). This paper presents similar data for the system urea-water. The reverse osmosis
separation of urea from aqueous solutions is of interest from the point of view of water renovation in space ships. Using the general equations for reverse osmosis process design given by Ohya and Sourirajan (1968), a method for stagewise reverse osmosis process design is presented for the above application along with a detailed set of illustrative calculations. VOL. 8
NO. 1
JANUARY 1969
131
~
~
Table 1. Osmotic Pressure, Density, Kinematic Viscosity, and Diffusivity Data for System Urea-Water at 25’ C. Concentration of Urea Molar Kinematic Osmotic Density of Mole Dt@ivity, fraction Weight, Pressure, Solution, i?O%C. Sq%l%C. Sq. Cm./Sec. P.S.I. G./Cc. x 102 x 102 x 106 Molality x 708 % n 0 0.9971 5.535 0.8963 1.3817 0.9986 5.520 0.8983 36 1.3739 71 1 ,0002 5.506 0,8998 1 ,3663 1.0017 0,9005 106 5.492 1.3591 141 1 ,0033 5.478 0.9025 1.3520 176 0.5 8,927 2.92 1 ,0048 5.464 0,9037 1.3453 210 10.693 1,0063 5.450 0.9058 0.6 1.3380 3.48 244 12.453 1 ,0078 5.435 0.9074 4.03 0.7 1.3311 278 14,207 1,0092 5.422 0.9096 4.58 0.8 1.3243 312 1 ,0107 5.409 0.9117 5.13 0.9 15.955 1.3180 346 1.0121 5.395 0.9114 5.67 17.696 1. o 1.3110 412 21.160 1,0150 5.369 6.72 1.2 0.9187 1.2987 47 8 24.600 1.0178 5.343 7.76 1.4 0.9236 1.2874 28.016 542 1 ,0206 5.318 0.9284 8.77 1.6 1 ,2763 5,292 607 1.0232 0.9338 9.76 31.408 1.2703 1.8 2 0 34 777 10 72 671 1.0259 5 267 0 9402 1 2544. 1.0322 13.05 826 5.206 2.5 43.096 0,9533 1 ,2301 979 1.0381 5.147 3.0 51,273 15.27 0.9686 1 ,2085 17.37 1128 1.0439 5.090 3.5 59.312 0.9848 1.1893 1276 1 ,0495 4.0 67.216 19.37 5.036 1.0000 1.1709 21.28 1421 1.0548 4.5 74.987 4.983 1.0154 1,1555 1564 1 ,0599 5.0 82.631 23.09 4.932 1.0312 1,1413 1.0647 24.83 1704 4.883 1.0468 5.5 90.149 I , 1289 1847 1.0694 4.835 26.49 6.0 97.545 1.0656 1.1174 ~~
Experimental Details
Reagent grade urea and porous cellulose acetate membranes (designated here as CA-NRC-18 type films) made in the laboratory were used. These films were cast at -10’ C. in accordance with the general method described earlier (Loeb and Sourirajan, 1963, 1964; Sourirajan and Govindan, 1965) using the following composition (weight per cent) for the film casting solution: acetone 68.0, cellulose acetate (acetyl content = 39.8%) 17.0, water 13.5, and magnesium perchlorate 1.5. The film details, the apparatus, and the experimental procedure have been reported (Sourirajan, 1964; Sourirajan and Govindan, 1965). Membranes shrunk a t different temperatures were used to give different levels of solute separation a t a given set of operating conditions. The aqueous urea solution (feed) was pumped under pressure past the surface of the membrane held in a stainless steel pressure chamber provided with two separate outlet openings, one for the flow of the membrane-permeated solution, and the other for that of the concentrated effluent. A porous stainless steel plate, specified to have pores of average size equal to 5 microns, was mounted between the pump and the cell to act as a filter for dust particles which might otherwise clog the pores of the membrane surface. Unless otherwise stated, the experiments were of the short-run type, each lasting for about 2 hours; they were carried out a t the laboratory temperature, and the feed rates were maintained a t 380 cc. per minute. The reported product rates are those corrected to 25’ C. using the relative viscosity and density data for pure water. In each experiment, the solute separation, f, defined as
f=
0.1 to 6.OM a t 25’ C. The osmotic coefficient data of Scatchard et al. (1938), the density and viscosity data given in the literature (Gucker et al., 1938; International Critical Tables, 1929; Timmermans, 1960), and the diffusivity data of Gosting and Akeley (1952) were used in obtaining the data given in Table I. Basic Equations and Correlations. The Kimura-Sourirajan analysis gives rise to the following basic equations relating to the pure water permeability constant, A , the transport of solvent water through the membrane, Ne, the solute transport parameter ( D A M / K 8 ) ,and the mass transfer coefficient, k , applicable to the high pressure side of the membrane :
molality of feed (ml)- molality of product (m3) molality of feed (ml)
-
-
FILM TYPE : CA NRC I 8 PURE WATER PERMEABILITY DATA
the product rate, [PR], and the pure water permeability, [PWP], in grams per hour per 7.6 sq. cm. of effective film area were determined a t the preset operating conditions. In all cases the terms “product” and “product rate” refer to the membrane-permeated solutions. The concentrations of the solute in the feed and product solutions were determined by refractive index measurements using a precision Bausch and Lomb refractometer. The accuracy of the separation data is within 1%, and that of the [PR] and [PWP] data is within 3% in all cases. Results and Discussion
Osmotic Pressures, Molar Densities, Kinematic Viscosities, and Solute Diffumvities for the System Urea-Water. These data are given in Table I for the concentration range 132
l & E C PROCESS D E S I G N A N D DEVELOPMENT
0
I
I
I
I
20
40
60
80
OPERATING
PRESSURE
1
IO0
- om.
Figure 1. Effect of operating pressure on pure water permeability constant
F I L M T Y P E : CA-NRC-18 SYSTEM UREA-WATER FEED MOLALITY FEED RATE
380
2 0 M CC /minut0
F I L M TYPE SYSTEM
1
CA-NRC-I8
UREA-WATER
350
FEED RATE
-
410 c c /mlnul.
1500 P
OPERATING PRESSURE
I ,
0
F I L M NO
06
SYSTEM
U R E A - WATER
0 5 , lo-,
P
0 4 L0 2 -
o-o~ooo-~-~opo
O
0
lo-?
CA- NRC-18
FILM TYPE
-
I Q-o-o-$
103
- v-8-0-V' -
o-$-o-'h%$
99 v-v-v-v-v-
/ 0 °
P OO '
0 I
I
1
I
40 60 8 0 100 OPERATING PRESSURE-otm.
cm2sec
Figure 2. Effect of operating pressure on solute transport parameter for system urea-water
0.25
otm.
Figure 3. Variation of pressure effect on solute transport parameter with initial porous structure of membrane for system urea-water
101 rJ-&o-o-oo
0-0
[""= I
20
I
0.7
0.5
FEED
2
3
4
MDLhLITY
Figure 4. Effect of feed concentration 3n solute transport parameter for system urea-water at 1500 p.s.i.g.
FILM T Y P E C A - N R C - ~ S
SYSTEM UREL-WLTER FEED RATE' 3 5 0 - 4 0 0 c c / m i n u f e OPERATING PRESSURE 100 p I
I
q
FILM TYPE CAI-NRC-IE S Y S T E M UREA-WATER FEED MOLALITY
O.5M 0 0 0 V 2.OM O A m V
OPERATING PRESSURE I 5 0 0 P S
10-2
vv-
v-;-
-
v 99 '0- v-v
I O
r--
-8-v-v
FILM NO
--r-v-99
-.-0-0
5x106'
5x
-
0.2
0.5
2
3
4
FEED M O L A L I T Y
Figure 5. Effect of feed concentration on solute transport parameter for system urea-water at 500 p.s.i.g.
o
NQ33 sc
-,A-AA-A
,
100 FEED
RbTE
200
300 400
SYSTEM UREA-WATER F E E 0 MOLALITY 0.5 1 0
where A , is the extrapolated value of A when P = 0, and a is a constant, and
4 "U
1
1
1
70
1
100
I
I 200
I
I
I
I I I I
300 400
E
NR.
cc.lmInute
Figure 6. Effect of feed flow rate on solute transport parameter for system urea-water
(5)
'50
600
-
/
-50
D
*
NSh -
Ib-A-IQp
I
*
o
The above equations have been derived (Kimura and Sourirajan, 1967) and they follow from a n application of the simple film theory and pore diffusion model for reverse osmosis transport. The correlations of A , (DAM/Kb),and k as functions of the operating conditions for the system urea-water obtained with a set of the Loeb-Sourirajan type porous cellulose acetate membranes are illustrated in Figures 1 to 8. Figures 1 and 2 illustrate that the plots of log A us. P and log ( D A M / K bus. ) log P are straight lines for all the films tested. They confirm the general validity of the relations
A = Aoe-aP
DIFFUSION CURRENT METHOD ILM 500 l 1 0 0 p i 0 . & A O B V T
Figure 7. Mass transfer coefficient correlation for system urea-water
(%)
0:
P-8
where p is a constant, a t all levels of solute separations. Figure 3 illustrates the relationship between A , and p which depends on the initial porous structure of the film; p increases with increase in A , u p to a point, and then remains essentially constant. Figures 4, 5, and 6 illustrate that ( D A M / K 6 )is independent of feed concentration and feed flow rate a t a given operating pressure. Figure 7 illustrates that the mass transfer coefficient correlation obtained from the experimental reverse osmosis separation data for the system urea-water is in fair agreement with that obtained by the diffusion current method (Sourirajan and Kimura, 1967). Figure 8 illustrates that the relationship between X A Z and X A is~ uniquely fixed VOL. 8
NO. 1
JANUARY 1969
133
Table II.
Membrane Specifications’
(Film type, CA-NRC-18. Operating prrssure, 1500 p.s.i.g.) A x 106 G. d2rlole H20 Sq. Cm. Sec. A t m .
Film ivo .
1
a
99 100 101 102 103 104 From Figure 10.
t
IO
( D A M / K ~x) 106, Cm./Sec. For .VaCI For urea
1.142 4.277 6.288 1.258 2.013 4.481 2.602
0.900 175.4 684.2 3,433 14.73 341.3 53.5
(41 )a
569.0 1249 78.0 165.1 756.3 314.7
1
I
!
l
l
l
1
l
20
30
40
50
60
70
80
90
I
and Kimura (1967), using the generalized correlation to obtain the required mass transfer coefficient. The data for the system urea-water for films 99 to 104 are obtained experimentally. The results show that under otherwise identical experimental conditions, the level of solute separation obtained for the system urea-water is much lower than that obtained for the system sodium chloride-water. Relative Scale of Membrane Selectivity. Between the quantities A and (DAM/K6)for a solute, which specify a membrane a t a given operating pressure, while the quantity A may change because of membrane compaction under conditions of continuous operation, (DA.M/K6)remains constant, provided the surface pore-structure of the membrane remains the same. This has been illustrated and discussed (Kimura and Sourirajan, 1968a). Thus the variation in the values of ( D A M / K 6 ) for different solutes for a given membrane offers a method of expressing membrane selectivity for different solutes. Figure 10 gives a log-log plot of (DAM/K8)for NaCl us. (DA,v/’ K6) for urea for different membranes a t different operating pressures. Figure 10 represents a useful relative scale of membrane selectivity. If A and (DAM/K8)for NaCl are given for a particular membrane? the performance of that membrane for the system urea-water can be predicted using Figure 10 and the generalized mass transfer coefficient correlation. This is illustrated in Figure 9 for film 1. General Equations for Stagewise Reverse Osmosis Process Design. The possibility of separating urea from aqueous solutions in a stagewise reverse osmosis operation is of practical interest from the point of view of water renovation in space ships. A system of general equations is presented below for stagewise reverse osmosis process design for the above and similar applications. Let Figure 11 represent a unit stage. For the purpose of this analysis, the molar density of the solution is assumed constant. A material balance for the unit stage may be written as follows. Considering an initial unit volume of solution,
xAZxio3 CAi’
= &a3
Figure 8. Concentration of solute in boundary solution vs. that in product for system urea-water
for each film a t a given operating pressure, whatever be the feed concentration-feed flow rate combination used in the experiment. Thus the correlations of the experimental data for the system urea-water given in Figures 1 to 8 show that the general behavior of the above system is identical to that of the systems sodium chloride-water (Kimura and Sourirajan, 1967) and glycerol-water (Sourirajan and Kimura, 1967). Membrane Specifications. O n the basis of the correlations illustrated above, it was pointed out earlier (Sourirajan and Kimura, 1967) that the specification of A and ( D A . M / K ~for ) a membrane a t a given operating pressure is sufficient to predict membrane performance a t that pressure for the entire applicable range of feed concentrations and feed flow rates, provided the appropriate mass transfer coefficient data are available. Table I1 gives the specifications of the films used in this lvork, in terms of A and (DA,/K6) for the solutes sodium chloride and urea a t the operating pressure of 1500 p.s.i.g. Performance of Porous Cellulose Acetate Membranes for Systems Sodium Chloride-Water and Urea-Water. The effect of feed concentration on solute separation and product rate for the systems sodium chloride-water and urea-water is shown in Figure 9 for comparison. The data for the system sodium chloride-water are those predicted from the membrane specifications given in Table I1 by the method of Sourirajan 134
l & E C PROCESS D E S I G N A N D DEVELOPMENT
+ (1 - A ) c A ~
(7)
where 63 and C1 are the respective concentrations expressed in dimensionless units defined as follows:
and
c1 =
CA1 ~
CAl’
For a given reverse osmosis system, of the four quantities C A ~ O , CAI in Equation 7, only any two of them are independent variables; or referring to Equation 8 ) of the three quantities A, 63, and C1, only any one of them is an independent variable. This has been demonstrated (Ohya and Sourirajan, 1968). A diagramatic representation of a multistage reverse osmosis unit is shown in Figure 12. I t consists of a feed stage, and a number of unit stages in the concentration and purification sections. Subscript f outside the brackets refers to the feed stage; superscripts 1,2,. . . j ,, , , w outside the brackets refer to the consecutive unit stages in the concentration section; and subscripts 1 , 2 , , , .i, , , . p outside the brackets refer to the consecutive unit stages in the purification section. C A I O , C A I , and Cas refer to the concentrations indicated in Figure 11 for each ] ~product rate and average unit stage. Let P and [ c ~be~ the
A, EA^, and
FILM TYPE : C A - N R C - I8 OPERATING PRESSURE 1500 p.s.1.9. F I L M AREA: 7.6 sq.cm.
I .o
-I
Figure 9. Effect of feed concentration on solute separation and product for systems urea-water and sodium chloride-water
product concentration, respectively, in the final stage p in the purification section, and let W and [ C A I ] % be the flow rate and concentration, respectively, of the concentrated solution on the high pressure side of the membrane, leaving the final stage w in the concentration section. Let F (= P W) be the rate of the make-up feed solution. The material balance equations for the multistage unit may be written as follows:
+
f (1 - A W ) [ ~ ~ ~ I " (1 - A ~ ) [ G A I ] ~ ' A / [ F A , I , f (1 - AI) [CAlI/ Ai[C~a]if (I - Ai) [CAlIi
[ G A I ~ ]=~ A W [ C ~ 3 1 W
[cA~O]~
= Aj[Ca3]f f
[CAI']~
= =
[CAl'Ip
= AP[CAS]P
[CAIo]/
+ (1 -
(11)
A~)[cAl]p
The following flow rate equations also apply for the multistage unit. VOL. 8
NO. 1
JANUARY 1969
135
30
Figure 10. Relative scale of membrane selectivity for systems ureawater and sodium chloride-water
UNIT STAGE
Figure 1 1 .
Unit stage
I
f
Assuming that the solution streams entering each unit stage have the same composition for economic operation, the following additional equations can be written:
P
+ +
Equation 11 has 2 (w p 1) degrees of freedom; when the constraints of Equation 13 are applied to Equation 11, the latter has only 2 degrees of freedom. Referring to Figure 12 again,
Figure 12.
Multistage reverse osmosis unit
Further, since
Equations similar to Equation 19 can be written for each unit stage in the multistage unit. I t then follows that
136
I&EC PROCESS DESIGN A N D DEVELOPMENT
OF y CASE2. ANYVALUES
The application of one or more of Equations 7 to 26 offers a convenient method for stagewise reverse osmosis process design for a specified reverse osmosis system. Reverse Osmosis System Specification and Performance. T h e performance of a reverse osmosis unit can be predicted by specifying the solution-membrane-operating system in terms of the dimensionless parameters y,0, and X defined as follows:
+ + c3e
(rc3 e)*
T
=
Sc: [
and z = y(C3 - C30)
e, A N D X
AND
1'
exp
= w
(-2)
dC3
+ (I + e )
In-c3
- 1n[
c 3
c 3 0
c 3 0
Y
=
BXAIO ~P
(41)
c 3
\]
+ +-
(27)
CASE3. y = 0, ANYVALUE OF
c1 =
c 3
e, AND X
[4
= w
i + e
(43)
For given values of y, e, and A, the relationships among the following quantities have been derived as follows (Ohya and Sourirajan, 1968). Defining e3
CAI ci= 6.4 1'
T = -
Sv, * t
VI0 and
u,*
=
AP
C
(32)
the following cases are of particular interest. OF y, e, AND A. CASE1. ALL VALUES
(34) (35) A = 1 - exp
(-2)
(36)
=
1
-
(1 - A ) ' + ' A
(46)
Since feed concentration C A ~ O is usually given, it is convenient to fix C A I and hence CI for design purposes. Thus for given values of y,e, A, and CI, the values of 4 and T can be calculated using the above relations. The corresponding value of 63 can then be calculated using Equation 8. A set of such C1 us. A, C3, and T correlations is illustrated in Figures 13 to 15 for reverse osmosis systems specified as y = 0, 0.1, and 0.3, 8 = 0.20 and 0.32, and X = 50 and m . T is a dimensionless time parameter for the hatchwise reverse osmosis process where S is the surface area of membrane and t is time. The value of T calculated from the above relations is identical to that of X defined by the relation
where x is the longitudinal distance along the length of the membrane, Go is the average velocity of feed solution a t membrane entrance, and 1l h is the membrane area per unit volume of fluid space in the reverse osmosis unit in a flow process. Thus Figures 13, 14, and 15 give the basic performance data for the specified reverse osmosis systems. Illustrative Calculations. A hypothetical problem of water renovation from aqueous urea solution is considered here for some illustrative calculations. The make-up feed solution is assumed to contain 277.77 moles, the mole fraction of urea being 7 x these data correspond to 2.35 \veight % urea and about 5070 cc. of solution a t 25' C. The object is to VOL. 8
NO. 1
JANUARY 1 9 6 9
137
Figure 13. A vs. T (or X ) , C1,and ?3 for = 0, 0.1, and 0.3, 0 = 0.32, and X = 50 and 01
:::i A
A
Figure 14. A vs. 7 (or X ) , CI,and E 3 for y = 0 0.1, and 0.3, 0 = 0.20, and X = 03
l
0
0.2
1
t
0.4
l
0.6
I
I
0.8
I
I
A recover water containing less than 200 p.p.m. of urea from the above solution. The mole fraction of urea in the final concentrated solution is assumed to be 6 x 10+ (= 17.55 weight %) so that the mole fraction of urea in the final product (E 200 p.p.m.). Only one of the water can be 6 x above two concentrations can be fixed independently in the reverse osmosis process; for the sake of convenience, let the former be fixed. The number of stages needed to accomplish the above separation by the reverse osmosis process can be found by a set of stepwise calculations using a trial and error procedure. For this purpose, the reverse osmosis operating system must first be specified in terms of y, e, and X. In the concentration range of interest, the plot of osmotic pressure us. mole fraction of solute for the system urea-water is essentially a straight line, and the osmotic pressure of a 2.35 weight 7,urea solution is about 140 p.s.i. (Table I). Hence, a t an operating pressure of 1500 p.s.i.g. the value of y for the feed stage is (140/1500) = 0.093. The average molar density, c, of the solution in the concentration range of interest may be 138
l & E C PROCESS D E S I G N A N D D E V E L O P M E N T
A
taken as 5.53 x mole per cc. From the data on membrane specifications given in Table 11, the values of e for the films 1 and 101 are 0.19 and 0.34, respectively, a t the operating pressure of 1500 p.s.i.g. For purposes of illustration, the performance of two films whose 0 values are 0.32 and 0.20 are considered and y is assumed equal to 0.1. These values are close to those obtained for films 101 and 1, respectively. If a high mass transfer coefficient can be assumed on the high pressure side of the membrane, X may be approximated as m . A detailed set of calculations is given below for the reverse osmosis system specified as y = 0.1 (feed stage), 0 = 0.32, and h = m. The first step is to make a very rough estimate of the number of stages required. This may be done as follows. For the case y = 0, X = m , c 3
c1 when
e
-
e i + e
= 0.32, C3/C1 = 0.24 for each stage or (0.24)n for n
1.6'
-
0.9
19
0.8
17
0.7
15
0.6
13
0.5
II
CI
c3 0.4
9
0.3
7
0.E
5
0.1
3
1
0.2
0
0.4
0.6
A Figure 15.
A vs.
7
0.8
1.0
0
0.2
0.6
0.4
1.0
0.8
A
-
(or X), C1, and C3 for y = 0,0.1, and
0.3,0
= 0.20, and
X
= 50
y = 0.1
U N I T STAGE
A, = 0.957
.:
[C,], =
1 T
[CI],
=
1 ~
0.655
- 1.527
y = 0.0655 A1 = 0 . 4 3
P
P
(a)
(b)
[e311
Figure 16. Multistage reverse osmosis units for water renovation from-aqueousurea solutions
[EA311
= 0.311 =
[63]1'
[cAlOl/
= 0.2037 T
stages. Approximating C1 to initial feed concentration, the C3/'C1 ratio needed for the purification section is 1/100, and approximating C3 to the initial concentration, the c3/c1 ratio needed for the concentration section is about 1/10, indicating 4 stages for the purification section and less than 2 stages for the concentration section. Therefore, for the first trial, a feed stage and a number of purification stages are assumed. Figure 16a represents such a multistage system. The performance of each unit stage in the system is then determined step by step as follows. Figure 13 is used to obtain A, e3, and 7 values corresponding to any given value of C1.
IC,], =
[CAII, [CAlOlf
6 X 7 x
X loT3
STAGE2 [c1]2. [e311
:.
[C,]2
= 1 =
1 T
[Cali
*
- 3.215 = __ 0.311
y = 0,0204 A2
= 0.73
[C3I2= =
-
= 1.426
= 0,455
0.4075
[ E A A ~= ] ~ [ 6 3 ] 2 *[6311.
FEEDSTAGE
[cA~O]/
8.6 T
[ c 3 1 , * [CAI']/
0 . 0 8 3 [ c ~ l o ] , = 0.5811 X l o p a
= 0.820 VOL. 8
NO. 1
JANUARY 1 9 6 9
139
[C1]3
=
1
[CI]P
1 [Calr
[ C1 ]4
= -- 2.454
0.4075
Any six of the above eight equations can be used to calculate the six values of VIO. Since the A values were read from the graphical correlations, some differences should be expected in the VI' values, depending on the choice of the six equations used to determine them. I n this illustration, Equations 12h, 12g, l2f, 12e, 12d, and 12b are used to calculate [VIO]~,[ V 1 ° ] 4 , respectively, and Equations [ V l o ] 3 , [Vlo]z,[V1O]1, and [ V I o ] , , 12a and 12c are used to check the values of A, and [ Vl'],, respectively. The value of VI0 obtained for each stage following the above procedure is given in Table 111. The values of A,, and [VlO], obtained from Equations 12a and 12c are 0.9568 and 746.05, which check well with those given in Table I, Example 1. The membrane area required in each stage can be calculated from the 7 values (defined by Equation 31). For this purpose the values o f t and ow* must be fixed. Assuming an operational time, t , of 1 hour, and uw* = 2.32 x cm. per second (which corresponds to data of film IOl), and a n average molar density of solution = 5.53 x mole per cc., the membrane area required in each stage has been calculated and the data are given in Table 111, Example 1. The results of similar calculations for three other examples are also given in Table 111. In examples 2 and 3, the performance of a film whose B cm. per second, is value is 0.20 and uw* value is 2.11 X given for conditions X = m and X = 50, respectively. In both cases five stages are needed, and the membrane area required in the latter case is only about 4% more than that required for the limiting case X = m . Example 4 is a recalculation of Example 1 where the multistage unit is made u p of a concentration stage, w,in addition to the feed stage, f,and a number of purification stages (Figure 166). For this case, the calculation procedure for the concentration stage is as follows.
1
-- 2.740 0.365
----I.
y = 0.0030 = 0.738
A4 [(?3]4
= 0.375
[EA314
=
[ 6 3 I 4 . [ 6 3 ] 3 . [(?3]2*
= 0.0114 [CA~'],
[6311'[63Ir.
= 7.95
[CAl0]f
x
y = 0.740
STAGE5 [Ci],. [ e 3 1 4 [Cl]s =
1
1 1
= - = 2.67
[C3]4
0.375
y = 0.0011
A , = 0.726 [e315
= 0.360
[Faals =
[ 6 3 ] 5 * [ 6 3 ] 4 . [ 6 3 ] 3 * [(?312.
[6s]i[63]/.
[C~iOlt
=
0.00408 [cA~']/ = 2.856 X lo-' ( ~ 9 p.p.m.) 5 7
= 0.726
Since urea concentration in the product leaving stage 5 is only 95 p.p.m., no further stages are necessary. From a material balance, the following data can be obtained :
F
= 277.77 moles, mole fraction of urea = 7
W
=
P
= 245.50 moles, mole fraction of urea = 2.856
A trial and error procedure is used to find AWinitially, assuming an appropriate value of y for the concentration stage, which is checked later. Let y = 0.17. From Figure 13, by trial and error A- = 0.91 corresponding to (C,/63) = 8 . 6 . Using the relation 1 =
X lo-*
32.27 moles, mole fraction of urea = 6 X low2
AW[63lW
f (1
- A"')[CI]"'
[Q3p= 0.594 and
[Cl]W = 8.6
[63Iw
= 5.108
X lop6
T o find the VI0 value for each unit stage, Equation 12 can be rewritten as follows for the particular multistage unit under consideration.
Now check for the y value.
= 1 .68[cal0]/
Since y for feed stage is 0.1, y for the concentration stage = 0.168, which is close to the assumed value of 0.17. For y = 0.17 and Aw = 0.91, 7 = 1.24 from Figure 13. The calculations for the feed and the successive purification stages are then similar to those illustrated for Example 1. 140
I&EC PROCESS DESIGN A N D DEVELOPMENT
Table 111.
Results of Numerical Calculations for Stagewire Reverse Osmosis Process Design for Water Renovation from Aqueous Urea Solutions
Basis. One hour operation Make-up feed 277.77 moles Mole fraction of urea in feed 7 X 10-3 Mole fraction of urea in concentrate 6 X 10-2 Operating pressure 1500 p.s.i.g. System
Spcczj5cation
e
x
C1
0.32 0.32 0.32 0.32 0.32 0.32
m m m m m
8.6 1.527 3.215 2.454 2.740 2,670
0,001
0.20 0.20 0.20 0.20 0.20
m m m m m
8.6 2.020 4.082 3.030 3.478
3
0.1 0,052 0.013 0,005 0,001
0.20 0.20 0.20 0.20 0.02
50 50 50 50 50
8.6 1.923 3.922 2.857 3 I344
4
0.17 0.10 0,034 0.013 0.005 0.002
0.32 0.32 0.32 0.32 0.32 0.32
m m m m m m
5.108 1.684 2.963 2,469 2.734 2,649
Example
Stage
Y
1
f
0.1 0.066 0.020 0,008 0.003
2 3 4 5
0,001
W
458.50 142.60 58.11 21.21 7.95 2.86
s,
A
VlO, Moles
7
Sq. Cm.
0.957 0.430 0.790 0.695 0.738 0,726
746.2 821.9 513.8 526.0 458.2 338.2
1.165 0,455 0.820 0.700 0.740 0.726
1882 810 912 797 734 532
0.935 0.585 0.855 0.775 0.815
492,O 516.1 389.4 388,6 301.2
1.170 0.615 0.870 0.775 0.815
1370 756 807 717 584
0.940 0.550 0.850 0.754 0.815
528.3 556.6 404.4 399.5 301.2
1.178 0.585 0,867 0.755 0.815
1482 775 835 718 584
0.910 0.510 0,765 0.695 0.740 0.725
355.6 725.8 530.1 524.6 457.7 338.7
1.240 0.560 0.820 0,705 0.745 0.727
955 880 941 801 738 533
( = 95 p.p.m.)
2
0.1 0.05 0.012 0.004
346.50 84.89 28.01 8.05 2.42 ( = 81 p.p.m.)
364.00 92.82 32.50 9.71 3.15 ( = 105 p.p.m.)
700.00 236.25 95.68 34.92 13.18 4.90 ( = 163 p.p.m.)
The results of Example 4 show that even though six stages are still needed, the total membrane area requirement is considerably less in the new configuration than that given in Example 1.
Nomenclature
Conclusions
While the transport Equations 1 to 4 are general and applicable for any reverse osmosis system involving aqueous solutions, the correlations of A , ( D A M I K B ) , and k have to be experimentally determined for each membrane-solution system. T h a t the transport characteristics of the Loeb-Sourirajan type porous cellulose acetate membranes for the system urea-water are similar to systems such as sodium chloride-water and glycerol-water reported earlier is useful information from the point of view of water renovation from aqueous urea solutions. The plots of the type given in Figure 10 suggest a useful method of expressing membrane selectivity for different solutes. The equations for reverse osmosis process design derived and illustrated in this paper, along with those reported earlier (Ohya and Sourirajan, 1968)) offer an effective means of treating reverse osmosis as a general separation process in chemical engineering. Acknowledgment
The authors are grateful to A. G. Baxter and Lucien Pageau for their valuable assistance in the progress of these investigations. One of the authors (H.O.) thanks the National Research Council of Canada for the award of a postdoctoral fellowship.
Fa 3
pure water permeability constant, g. mole HzO/sq. cm. sec. atm. value of A a t P = 0 molar density of solution, g. mole per cc. molar density of feed solution, concentrated boundary solution, and product solution, respectively, g. mole per cc. local solute concentration of feed solution, concentrated boundary solution, and product solution, respectively, a t any time, mole fraction or g. mole per cc. values of CAI, c A ~ , c A 3 , respectively, at time 0 in a batch process or a t membrane entrance in a flow process average solute concentration in product corresponding to a given A, in mole fraction or in g. mole per cc. cA~/cA~Q and c A 3 / C A l o , respectively CAPO/CA~O and cA30/cA10, respectively FA3/cAlo
diameter of membrane, cm. diffusivity of solute in water, sq. cm. per sec. solute transport parameter, cm. per sec. solute separation = (ml- m3)/m1 quantity of make-up feed solution entering a unit stage, cc. or moles membrane area per unit volume of fluid space, cm.-' mass transfer coefficient, cm. per sec. molality of feed solution and product solution, respectively molecular weight of water solvent water flux through membrane, g. mole/ sq. cm. sec. Reynolds number = d i i O / y VOL. 8
NO. 1
JANUARY 1969
141
P
= Schmidt number = v / D = Sherwood number = k d / D = total number of purification stages excluding
P
= operating pressure, atm.
SUBSCRIPTS
P [PRI
= quantity of final product, cc. or moles = product rate, grams per hour per 7.6 sq. cm.
f, 1,2,. . .i,. . . p
[PWP1
= pure water permeability, grams per hour per
NSO
Nsh
= osmotic pressure corresponding to solute con-
T
= quantity defined by Equation 31
centration X A , atm.
feed stage
of film area
S t
a0 VuC v1
v10
u:
W X
X XA1,XAz,XAs
XAl
d X A )
7.6 sq. cm. of film area = membrane surface area, sq. cm. = time, sec. = average fluid velocity at membrane entrance, cm. per sec. = quantity defined by Equation 32, cm. per sec. = quantity of solution on high pressure side of membrane at any time, cc. or moles = value of Vl at time 0 = total number of concentration stages excluding feed stage = quantity of final concentrated effluent from reverse osmosis unit, cc. or moles = longitudinal distance along length of membrane from channel entrance, cm. = quantity defined by Equation 47 = mole fraction of solute in feed solution, concentrated boundary solution, and product solution, respectively, at any time = value of X A , at time 0
2
GREEKLETTERS = constant = constant P = quantity defined by Equation 27 Y A = fraction product recovery defined by Equation 30 e = quantity defined by Equation 28 x = quantity defined by Equation 29 = kinematic viscosity, sq. cm. per sec. V cy
= feed stage, and purification stages 1,2,. . .i,
. . .p, respectively
SUPERSCRIPTS
1,2,. . .j,. . .w
= concentration stages 1,2,. . .j,.
. .w,respec-
tively Literature Cited
Gosting, L. J., Akeley, D. F., J . A m . Chem. SOC. 74, 2058 (1952). Gucker, F. T., Gage, F. W., Moser, C. E., J . A m . Chem. SOC.60, 2582 (1938). International Critical Tables, Vol. V, p. 22, McGraw-Hill, New York, 1929. Kimura, S., Sourirajan, S.,A.1.Ch.E. J . 13, 497 (1967). Kimura, S.,Sourirajan, S., IND.ENG.CHEM.PROCESS DESION DEVELOP. 7 , 197 (1968a). Kimura, S., Sourirajan, S.,IND. ENG.CHEM.PROCESS DESIGN DEVELOP. 7 , 548 (1968b). Loeb, S.,Sourirajan, S., Advan. Chem. Ser., No. 38, 117 (1963). Loeb, S., Sourirajan, S., U.S. Patent 3,133,132 (May 12, 1964). Ohya, H., Sourirajan, S., “Some General Equations for Reverse Osmosis Process Design,” A.1.Ch.E. J., in press, 1968. Scatchard, G., Hamer, W. J., Wood, S.E., J . A m . Chem. Sod. 60, 3061 (1938). Sourirajan, S.,2nd. Eng. Chem. Fundamentals 3, 206 (1964). Sourirajan, S., Govindan, T. S.,Proceedings of First International Symposium on Water Desalination, Washington, D. C., October 1965 (U.S. Dept. Interior, Office of Saline Water, Vol. I, pp. 251-74, 1967). Sourirajan, S., Kimura, S., IND.ENG.CHEM.PROCESS DESIGN DEVELOP. 6, 504 (1967). Timmermans, J., “Physicochemical Constants of Binary Systems in Concentrated Solutions,” Vol. 4, pp. 115-17, Interscience, New York, 1960. RECEIVED for review June 10, 1968 ACCEPTED September 25, 1968 Issued as N.R.C. No. 10494
CORRESPONDENCE ALGORITHM FOR OPTIMIZATION OF ADIABATIC REACTOR SEQUENCE WITH COLD-SHOTS COOLING SIR: In a recent note Malengt and Villermaux (1967) quesrioned the algorithm of Lee and Aris (1963) for the optimization of a multibed adiabatic reactor sequence with cold-shot cooling, on the basis of the incorrect use of the bypass parameter, AB. They have evolved a new algorithm wherein X1 has been explicitly included as a decision variable in the optimization function. I t appears that the inadvertent omission of an important operational feature of the last stage in a reactor sequence of this type has resulted in the rather controversial formulation of algorithms for optimal operation of a reactor sequence by application of the principles of dynamic programming. Once this fact is recognized, an analytical design scheme for the case of cold-shot cooling emerges which is as elegant as that evolved for the heat-exchanger cooling, if not as gt neral. Prior to entry to reactor 1, the mixing of the main stream from reactor 2 and the bypass stream provides the final complete feed to the first reactor and therefore the algorithm for the 142
I & E C P R O C E S S D E S I G N A N D DEVELOPMENT
optimization of the first reactor cannot distinguish between the type of cooling effected between the first and second stages. In other words, the optimal design for the first reactor is a common design for either type of interstage cooling and can be evolved by following the procedure for one-stage optimization with heat exchanger cooling, after neglecting the cooling costs. For any given feed state (XZ, T z ) ,therefore, the optimum Xz is not an independent parameter and hence does not appear in the algorithm. In Figure 1, 71 and 71 are drawn by the usual maximizing procedure for p l , the profit function for the first reactor. If the feed state (XZ, T?)is designated by D or any point D’ on the ray joining origin 0 (fresh feed) to D, the optimal inlet state will be given by B Tl),where the ray intersects the locus i l . In either case A 2 will be calculated as the ratio of segments, OB/PD or OBIPD’. When D’ coincides with B , X Z is unity, which means that there is no bypass of feed and B is a common point specifying inlet feed state to reactor 1, irrespective of whether it is reached for exchanger
(XI,
