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Permeation of water at low pressure in cellulose acetate membranes
Permeation of Water at Low Pressure in Cellulose Acetate Membranes I. Experimental Results and Their Variation with Temperature C. F E R N A N D E Z - P I N E D A 1 AND J. I. M E N G U A L Departamento de Termolog~a, Facullad de Ciencias F~sicas, Universidad Complutense, Madrid, Spain Received April 21, 1976; accepted October 25, 1976 An experimental device is described for the measurement at low pressure of different transport phenomena in membranes. The water permeability was measured in 15 cellulose acetate membranes, using cellulose acetate with a degree of acetilation of 2.7. The manufacture of membranes was based on solutions of cellulose acetate in acetone, whose composition varied in the range of 35 to 110 nag of cellulose acetate in 25 cm3 of acetone. The maximal pressure difference was 50 cm of water and the temperatures were made to vary from 30 to 50°C. A significant correlation between temperature and permeability was found. This dependence can be described either by exponential or linear equations with practically the same degree of accuracy. INTROD UCTION Recently (1), the s t u d y of t r a n s p o r t phenomena in membranes a t low pressure has acquired a certain interest, f u n d a m e n t a l l y due to the fact t h a t the results obtained refer to unaltered structures. Nevertheless, for a few years we have been working on t r a n s p o r t p h e n o m e n a through cellulose acetate membranes. I n this respect, we have constructed an experimental device which permits the s t u d y of different t r a n s p o r t phenomena of liquids through membranes, among t h e m the effect of t e m p e r a t u r e on water p e r m e a t i o n a n d thermoosmosis. I n essence, the technique used permits the control of the pressure difference as well as the t e m p e r a t u r e s between b o t h sides of the membrane. T h e m a x i m a l pressure difference used has been 50 cm of water. This work presents the results obtained for the p e r m e a b i l i t y of water (isothermic conditions) as a function of t e m p e r a t u r e for 15 different cellulose acetate membranes. These experimental results represent an i m p r o v e m e n t on similar investigations, since the published 1 Present address : Departamento de Fisica. Facultad de Ciencias. Universidad de Ni~Iaga, Spain.
d a t a (1-4) do not give the relationship between p e r m e a b i l i t y and temperature. I n a second p a p e r we will see which one of the following models is b e t t e r suited for the experimental results : (a) viscous flow; (b) diffusive flow; (c) combined viscous-diffusive flow.
MATERIALS An adequate and critical study of the literature (5-11), as well as the experience acquired with other glass experimental devices, has led to the design of a new cell, as seen in Fig. 1. I n essence it consists of two cylindrical chambers of the same dimensions m a d e of stainless steel. The length is 16 cm and the diameter 1.8 cm. I n order to keep a constant temperature, each chamber is placed inside another cylinder and the water coming from the t h e r m o s t a t is allowed to flow between them. The membrane is fixed between the chambers with the help of two disks of metacrilate (Fig. 2). Each chamber is provided with a pair of holes; one of t h e m is to tlold a t h e r m o m e t e r and the other one is to hold a glass tube in a 95
Copyright ~ 1977 by Academic Press, Inc. All rights of reproduction in any form reserved.
Journal of Colloid and Interface Science, Vol. 61, No. i, August 1977 ISSN 0021-9797
96
FERN~_NDEZ-PINEDA AND MENGUAL
I C
I C
L
L C c
T
CT --~
c
T
[ '
~/r
'
FIG. 1. Scheme of test apparatus: AC, ambient temperature control; WR, water reservoir; C, conduction tubes; V, glass tubes; L, three-way shutoff; CT, control thermostat; T, thermometers; M, membrane; TG, temperature gauge. vertical position. The connection between the tube and the chamber is by means of a system of O-rings similar to the glass-metal joints used in high-vacuum techniques. The sections of the membranes exposed to the water were 2.54 or 7.07 cm 2, depending on the experimental conditions. Three pairs of glass tubes were used whose radii were determined by the mercury drop technique. The measured radii and their corresponding errors (standard deviations) are: (128 4- 2) X 10-a, (131 -4- 1) X 10-a, (99
4 - 2 ) X 10-3 , ( 9 7 4 - 2 ) X 10-3 , (634 4- 2) X 10-4, and (661 -4- 4) X 10-4 cm. In the first part of the work, the temperature was measured by copper-constantan thermocouples. In the second part platinum resistance thermometers were used since they provide a faster and simpler method of measurement. The thermocouples as well as the resistance thermometers were calibrated by comparison with a control platinum resistance thermometer, calibrated in the N.P.L.(London). In all cases the error in the temperature was 4-0.1°C. Once the membrane was mounted, the cell was filled with water, first deionized and later bidistilled in the presence of potassium permanganate, and finally degassed. The initial height of water in the glass tubes was controlled by means of three-way shutoffs which connected the chambers with a water tank located at a higher level. The complete unit (cell, tubes, etc.) was contained in a large ambient thermostat (volume, 150 X 100 X 50 cm 3) in order to maintain the selected temperature. The membranes were obtained by dissolving different amounts of cellulose acetate (acetilation degree, 2.7) in acetone (pro-analysis, Merck) and then by evaporating the solvent by means of a flow of dry air at 30°C in a specially designed chamber. The membranes were formed on the flat bottom of four cylindrical recipients, with diameters of 57.5 mrn each. The amount of cellulose acetate varied O
0
t
u
u
--1"
T-
I
!
Fio. 2. Membrane cell assembly: M, membrane in its holders; I, input of thermostabilized water; O, output of thermostabilized water; U, union of the cell with the glass tubes; T, holes for thermometer probes. Journal of Colloid and Interface Science, VoL 61, No, 1, August 1977
PERMEABILITY VERSUS TEMPERATURE
97
between 35 and 110 mg a n d the volume of acetone was kept constant at 25 cm 3. The reason for working in the range of 35 to 110 mg of cellulose acetate was that below 35 mg the m e m b r a n e cannot be properly cast and above 110 mg the flow of water was too small to be detected in the range of pressures used. The evaporation process lasted a b o u t 8 hr. Despite the fact that the membranes were always cast under the same conditions, a n uncontrolled a m o u n t of the cellulose acetate crept up the walls of the container. Thus, it had not been incorporated into the membrane tested. This point will be discussed further. As the observed permeability measurements were n o t reproducible, different procedures were tried to overcome this inconvenience. The method finally used has been described pre-
the lowest to the highest again. The lower limit was fixed by the room temperature, and the upper limit was 50°C because the membranes broke when they were m a i n t a i n e d for more t h a n 24 hr at temperatures higher t h a n 50°C. The thickness of the m e m b r a n e 8 was determined b y means of a micrometer which was accurate within 4-1 gin. The fractional void volume e was obtained b y the difference between the weights per unit area of the wet and dry samples obtained from the central portion of the membrane which had been exposed to the water. These measurements were performed at least eight times in each case using circular samples. The sample diameters were measured with a microscope whose precision was =t=1 gin. Each of four samples was
viously (12).
placed in six different positions. With these
The
measurements of permeability were
results the calculated value of the area was
carried out in the temperature range from 30 to
2.109 4- 0.001 cm 2. Table I shows the results
50°C, with steps of 5°C. I n each case, two or
for the 15 membranes studied; they are num-
three runs were performed. The first one was performed b y increasing the temperature from
bered in increasing order together with the
the lowest to the highest, the second from
were cast. Also, the corresponding values of 5,
the highest to the lowest, a n d the third from
M,~/S (mass per unit area), and e are listed.
concentration of the solution from which they
TABLE I Geometric Parameters of the Membranes Membrane No,
Solution composition ( m g / 2 5 cm*)
1 2 3 4
35 40 45 50
5
55
6 7 8 9 10 11 12 13 14 15
60 65 70 75 80 85 90 95 100 110
~a X 104 (cm)
(M~/S) X lO S
7 4- 2 6 -4- 2 10 4- 2 9 4- 3 10 4- 2 12 4- 2 12 4- 2 13 4- 2 15 4- 1 17 4- 2 17 4- 2 20 4- 2 18 4- 2 18 4- 2 23 4- 2
0.94 0.75 1.12 0.89 1.27 1.40 1.55 1.63 1.82 2.14 2.22 2.36 2.30 2.44 2.74
(g/era 2)
4.75 X 10-2 1.03 X 10 -1 1.28 X 10-1 4.22 X 10 -2 5.23 X 10_2 5.94 X 10-3 9.51 X 10-e 1.17 X 10-1 1.17 X 10-1 1.03 X 10-1 1.17 X 10-1 1.09 X 10-~ 7.92 X 10-~ 3.14 X 10-1 1.05 X 10-1
a The errors appearing in the values of ~ correspond to the maximal deviation with respect to the mean value of the measurements and are always greater than the precision of the micrometer used. Journal of Colloid and Interface Science, Vol. 61, No. 1, A u g u s t 1977
98
FERNfi~NDEZ-PINEDA AND MENGUAL METHOD
The phenomenological equation which describes the permeation process is JM
=
q AP,
A
-
El"]
where JM is the flow of material in moles of fluid which pass per unit of time through a cross section q of the membrane, AP is the difference in hydrostatic pressure on the two sides of the membrane, and A is a phenomenological coefficient, permeability, which depends on the following parameters: (a) the nature of the membrane; (b) the nature of the permeant fluid; (c) the temperature of the system; and (d) the average pressure of the system. Assuming that permeability is constant and writiI~g AP = p g ( h ' - - h ' ) , Eq. [-13 becomes h" -
h' = (h"o -- h'o)e -'/~
[2"]
where p is the water density, g is the acceleration of gravity, h" and h' are functions describing the variation in time of the water level in the tubes, h"0 and h'0 are the initial heights, t is the time, and r is a constant related to the permeability described by the equation, ~q0 1 A --- - -, [-31 2gqM r
where q0 is the section of glass tube used and M is the molar mass of the water. Taking into account that h " + h ' - - h " 0 + h'0, Eq. E23 can be written h'o H"
-
h'o
-
e- t / ~
E4:~
2 and an equivalent equation for the other tube,/1", is the corrected water level, h'o + h"o H " = h" 2
From Eq. E4-] it follows that the semilogarithmic plot of H " versus t is a straight
line whose slope is equal to - 1 / r . The values of - 1 / r have been obtained by adjusting the experimental data to an exponential function by the least-squares method. In the most unfavorable cases the value of the coefficients of correlation obtained was 0.99 in runs of at least 20 points. This confirms that the assumptions that led to Eq. [-4] were correct in the range of pressures used. RESULTS For most of the membranes three runs were performed in order to obtain the permeability at different temperatures. In two cases (membranes 1 and 8), only two runs were performed, as has been mentioned earlier. In Figs. 3 and 4 the semilogarithmic plot of A (mole g-1 sec) against T (°C) shows that all membranes behave similarly. A careful examination of the figures permits us to draw some conclusions: First, the results obtained can be reproduced after an adequate treatment of the membranes. These results practically coincide for all the runs performed. Second, since the points are on quite straight lines, it seems that the variation of permeability with temperature follows an exponential law. Later on this assumption will be discussed in more detail. In Figs. 3 and 4 some abnormalities in the expected behavior of several membranes are observed. For example, the permeability does not always decrease with the number of the membrane, i.e., with the concentrations of the solutions from which they have been obtained. This is the case for the first five membranes (Fig. 3) and membranes 12 and 13 (Fig. 4). Since the membranes were always obtained under exactly identical experimental conditions, it may be thought that after the solvent is evaporated, uncontrolled amounts of cellulose acetate are deposited on the rims. Thus, the concentration of the original solution should only be taken as a rough indication and the membranes should be characterized by other parameters, such as thickness, weight
Journal of Colloid and Interface Science, Vol. 61, No. 1, August 1977
PERMEABILITY VERSUS TEMPERATURE per area unit, and the fractional void volume, rather than by the initial concentration of the solution. In fact, in Table I it can be seen that membrane 12 has a value of Mm/S greater than membrane 13, and therefore, it may be expected that the first one is less permeable, precisely what we found experimentally. This interpretation allows us to order the membranes from 6 to 15 with respect to the relation between permeability and mass per unit area. Nevertheless, discontinuities are still observed in the first five membranes. It may be due to the fact that their concentration range is very close to the formation limit and errors may have considerable influence.
99
IO'lZ'! 'i P 3
-mo~oi~-_a:D 10_14i[:~
~
3i
7
2~
-~
oa--
--ceo--~o------
@ (~
"
~
~
~
@
-
I1 [
10-" l
10-16~
-c~-
-man-- @ jO-I'r ~ _ 30
f 35
, 40
_
T 45
r 50 T(°C)
3 -clo~o~--
czla-
10-12 __~.______ ~ ~ t x ~ - - - - - - - - o ~
~ 0 ~ -
FIG. 4. Permeability of water through membranes versus temperature in semilogarithmic plot.
@
8
30oc_~
o 30oC--* A 50°C. 50oC--*
DISCUSSION
4j;
[ I 10451 , 30
~
T
,
35
40
45
50
T(°C )
FIG. 3. Permeability of water through membranes versus temperature in semilogarithmic plot. [] o 0 A o 30°C --~ 50 C --* 30°C --~ 50 C.
The study of the behavior of permeability with temperature has been performed by a relatively small number of workers and always at nearly ambient temperatures with small temperature intervals. In work on reverse osmosis it has been found that the product of permeability and viscosity of the permeant is independent of the temperature for each membrane (13). It must also be remarked that the average pressures considered by these workers were in the range of many atmospheres, too far from the range of pressures used in this work. Here it has been observed that in a majority of the membranes, there is a slight
Journal of Colloid and Inierface Science, Vol. 61, No. 1. August 1977
100
FERNfitNDEZ-PINEDA AND MENGUAL
decrease of the product of permeability and viscosity as a function of temperature. This shows that, in our range of pressures, the above-mentioned results are approximately true. The exponential dependence observed in Figs. 3 and 4 has been verified by adjusting the experimental data to an exponential function by the least-squares method. The results appear ill Table II, where the coefficient of correlation for each membrane together with the values of the parameters with their standard deviations are shown. The same experimental data have been used to adjust a straight line by the least-squares method (see Table III). Also, experimental data found by various authors (1--4) have been treated in the same way. The results are presented at the bottoms of Tables II and III. The values of the coefficients of correlation show that the adjustments for both an exponential function and a straight line f i t equally well. It suggests that, in this temperature range, they should coincide. This coincidence is due to the fact that the values of the
m coefficients are small (between 10-2 and 10.-3 °C-1). If the exponential was expanded in a power series, the second-order term should be very small compared with the first. In order to check the above assumption, the series expansion corresponding to the exponential fit shown in Table II was calculated. By considering only the first-order term, this resulted in a linear relationship of permeability to temperature. The slope and intercept of this relationship can be compared with the data in Table III. Since all membranes behave in a similar manner, it will be discussed only for one of them, No. 2, for instance. In Table III, the corresponding values for this membrane are: slope = 3.3 X 10-14 and intercept = 2.8 >( 10-12 mole g-1 sec. For the exponential, in first order, they are 3.1 )< 10-~4 and 2.8 ;< 10-~2 mole g-1 sec, respectively. The discrepancy between the values of the slopes is within the standard error of the tables. Since the temperature interval is small, it may be thought that the fits found considering various functions (e.g., A = Aoe mr, A = A'o + aT, and A = A"oe m''/r) could give similar
TABLE II Exponential A d j u s t m e n t , A = A oemr Membrane No.
Correlation coefficient
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Ref. (1) Ref. (2) Ref. (3) Ref. (4)
0.96 0.98 0.98 0.95 0.98 0.97 0.96 0.96 0.97 0.97 0.99 0.98 0.99 0.97 0.99 0.97 0.99 0.98 0,99
m (°C-1) (1.32 (8.0 (6.4 (1.46 (9.0 (1.58 (9.0 (1.47 (4.61 (5.25 (5.2 (6.02 (1.32 (2.34 (3.09 (1.9 (2.6 (4.65 (1.83
4. 0.03) ~: 0.1 ) 4. 0.3 ) 4. 0.01) 4. 0.3 ) 4. 0.05) 4. 0.1 ) 4. 0.03) q- 0.07) 4- 0.09) 4. 0.1 ) 4. 0,05) -4- 0.01) -4- 0.03) -4- 0.02) 4. 0.1 ) 4. 0.2 ) -4- 0.09) 4. 0.05)
Journal of Collold and Interface Science, Vol. 61, No. 1. August 1977
A0 (mole g-1 sec) X )< )< X )< X X X )< )< )< X X X X X X X X
104 10-3 10.3 10- 2 10-3 10-2 10-3 10-2 10-3 10-3 10-3 10-3 10-~ 10-2 10-2 10-2 10-2 10-2 10-2
4.62 3.01 1.50 5.70 1.95 2.86 2.80 2.09 2.54 1.32 6.46 6.08 1.57 1.91 1.07 1.49 2.50 1.69 3.56
X 10-13 X 10- TM X 10-12 )< 10-13 X 10-12 X 10-13 )< 10-13 X 10-14 X 10-14 X 10- u X 10-15 X 10-16 )< 10-~5 X 10-15 X "10-17 X 10-13 X 10-15 )< 10-15 X 10-15
PERMEABILITY
101
VERSUS T E M P E R A T U R E T A B L E 111
Straight Adjustment, A -= A 'o + aT Membrane No.
Correlation coefficient
a (mole g-a sec °C -~)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Ref. (1) Ref. (2) Ref. (3) Ref. (4)
0.96 0.98 0.98 0.96 0.98 0.96 0.95 0.97 0.97 0.97 0.98 0.98 0.99 0.95 0.98 0.97 0.98 0.96 0.99
(1.0 4-0.1 ) X 1 0 -14 (3.3 4- 0.2 ) )< 10-14 (1.24 4- 0.07) )< 10-14 (1.5 4- 0.1 ) X 10-14 (2.5 4- 0.1 ) 5( 10-14 (8.9 ± 0 . 8 ) X 10-is (3.7 :k 0.4 ) )< 10-I5 (5.5 4- 0 . 5 ) X 10-1G (1.4 4- 0.1 ) X 10-~6 (8.6 4- 0.2 ) X 10-~7 (4.2 4- 0.2 ) X 10-~7 (4.7 4- 0.3 ) X I O -is (3.6 4- 0.2 ) X 10-iv (1.2 4-0.1 ) X 10-~v (1.19 4- 0.08) X 10-18 (5.3 4- 0.4 ) X 10-~5 (1.5 4-0.1 ) X 10-~s (4.1 4- 0.8 ) X 10-1~ (1.2 4- 0.1 ) >( 10-~6
satisfactory
results.
Nevertheless,
the
func-
t i o n A = A"oe "~''/~ h a s b e e n c h e c k e d . T h i s l e a d s t o c o e f f i c i e n t s of c o r r e l a t i o n t h a t a r e , in general, worse than
those listed in Tables
II and III. Finally, a conclusion can be drawn: In the range of temperatures and pressures used, the relationship between the permeability A and t h e t e m p e r a t u r e T is p r a c t i c a l l y l i n e a r . REFERENCES 1. BLUMBERG,A. A., HADDADIN, E. S., CHMIELEWSKI, M. E., AND SrROZ, D. G., ]. Colloid Interface Sci. 49, 24 (1974). 2. MADRAS S., MCINToSH, R. L., AND MASON, S. G., Canad. Y. Res. 27B, 764 (1949). 3. HAASE, R., AND STEIi"~ERT, C., Z. Phys. Chem. N. F. 21, 270 (1959).
A ~0 (mole g-1 sec)
( 3.7 4- 0 . 5 ) ) < 1 0 -13 ( 2.82 4- 0.08) X 10-le ( 1.44 4- 0.03) X 10-~2 ( 4.3 =t= 0.5 ) )< 10-18 ( 1.80 i 0.06) X 10-12 ( 1.8 4- 0.3 ) X 10-13 ( 2.5 =t= 0.1 ) )< 10-13 ( 1.6 4- 0.2 ) X 10-14 ( 2.49 :k 0.04) X 10-~4 ( 1.29 :k 0.08) X 10 -~4 ( 6.30 4- 0.09) X 10-15 ( 5.9 4- 0.1 ) X 10-Is ( 1.24 4- 0.07) X 10-1~ ( 1.3 4- 0.5 ) X 10-~6 (--1.0 4- 0.3 ) X 10-17 ( 1.0 4- 0.2 ) X 10-l~ ( 1.1 :t: 0.4 ) X 10 -~5 (--5 4- 3 ) )< 10-~5 ( 2.7 5 : 0 . 4 ) X 10-~6
4. HAASE, R., AND DE GREIFF, H. J., Z. Naturforsch A 26a, 1773 (1971). 5. RENKIN', E. M., J. Gen. Physiol. 38, 225 (1955). 6. GARY-BoBo, C. M . DIPOLO, R., AND SOLOMON, A. K., J. Gen. Physiol. 54, 369 (1969). 7. ALEXANDER, K. F., AND WIRTZ, K., Z. Phys. Chem. N. F. 195, i65 (1950). 8. CARE, C. Wl, AND SOLLNEF, K., J. Electrochem. Soc. 109, 616 (1962). 9. HAASE, R., A~D DE GREI~'~', H. J., z. Phys. Chem. N. F. 44, 301 (1965). 10. RAS~OGI, R. P., AND .[HA, K. M., Trans. Faraday Soc. 62, 585 (1966). 11. SHERRILL, B. C., ALBRIGTI-I, ft. G., AND DIErSCH¥, J. M., Biochim. Biophys. Acta 311, 261 (1973). 12. FERNANDEz-PINEDA, C., AND MENGUAL, ~. I., An. Fis. 72, 79 (1976). 13. SOURI~AJA~, S., "Reverse Osmosis," p. 228. Academic Press, New York, 1970.
Journal of Colloid and Inlerface Science, Vol. 61, No. 1, August 1977
d a t a (1-4) do not give the relationship between p e r m e a b i l i t y and temperature. I n a second p a p e r we will see which one of the following models is b e t t e r suited for the experimental results : (a) viscous flow; (b) diffusive flow; (c) combined viscous-diffusive flow.
MATERIALS An adequate and critical study of the literature (5-11), as well as the experience acquired with other glass experimental devices, has led to the design of a new cell, as seen in Fig. 1. I n essence it consists of two cylindrical chambers of the same dimensions m a d e of stainless steel. The length is 16 cm and the diameter 1.8 cm. I n order to keep a constant temperature, each chamber is placed inside another cylinder and the water coming from the t h e r m o s t a t is allowed to flow between them. The membrane is fixed between the chambers with the help of two disks of metacrilate (Fig. 2). Each chamber is provided with a pair of holes; one of t h e m is to tlold a t h e r m o m e t e r and the other one is to hold a glass tube in a 95
Copyright ~ 1977 by Academic Press, Inc. All rights of reproduction in any form reserved.
Journal of Colloid and Interface Science, Vol. 61, No. i, August 1977 ISSN 0021-9797
96
FERN~_NDEZ-PINEDA AND MENGUAL
I C
I C
L
L C c
T
CT --~
c
T
[ '
~/r
'
FIG. 1. Scheme of test apparatus: AC, ambient temperature control; WR, water reservoir; C, conduction tubes; V, glass tubes; L, three-way shutoff; CT, control thermostat; T, thermometers; M, membrane; TG, temperature gauge. vertical position. The connection between the tube and the chamber is by means of a system of O-rings similar to the glass-metal joints used in high-vacuum techniques. The sections of the membranes exposed to the water were 2.54 or 7.07 cm 2, depending on the experimental conditions. Three pairs of glass tubes were used whose radii were determined by the mercury drop technique. The measured radii and their corresponding errors (standard deviations) are: (128 4- 2) X 10-a, (131 -4- 1) X 10-a, (99
4 - 2 ) X 10-3 , ( 9 7 4 - 2 ) X 10-3 , (634 4- 2) X 10-4, and (661 -4- 4) X 10-4 cm. In the first part of the work, the temperature was measured by copper-constantan thermocouples. In the second part platinum resistance thermometers were used since they provide a faster and simpler method of measurement. The thermocouples as well as the resistance thermometers were calibrated by comparison with a control platinum resistance thermometer, calibrated in the N.P.L.(London). In all cases the error in the temperature was 4-0.1°C. Once the membrane was mounted, the cell was filled with water, first deionized and later bidistilled in the presence of potassium permanganate, and finally degassed. The initial height of water in the glass tubes was controlled by means of three-way shutoffs which connected the chambers with a water tank located at a higher level. The complete unit (cell, tubes, etc.) was contained in a large ambient thermostat (volume, 150 X 100 X 50 cm 3) in order to maintain the selected temperature. The membranes were obtained by dissolving different amounts of cellulose acetate (acetilation degree, 2.7) in acetone (pro-analysis, Merck) and then by evaporating the solvent by means of a flow of dry air at 30°C in a specially designed chamber. The membranes were formed on the flat bottom of four cylindrical recipients, with diameters of 57.5 mrn each. The amount of cellulose acetate varied O
0
t
u
u
--1"
T-
I
!
Fio. 2. Membrane cell assembly: M, membrane in its holders; I, input of thermostabilized water; O, output of thermostabilized water; U, union of the cell with the glass tubes; T, holes for thermometer probes. Journal of Colloid and Interface Science, VoL 61, No, 1, August 1977
PERMEABILITY VERSUS TEMPERATURE
97
between 35 and 110 mg a n d the volume of acetone was kept constant at 25 cm 3. The reason for working in the range of 35 to 110 mg of cellulose acetate was that below 35 mg the m e m b r a n e cannot be properly cast and above 110 mg the flow of water was too small to be detected in the range of pressures used. The evaporation process lasted a b o u t 8 hr. Despite the fact that the membranes were always cast under the same conditions, a n uncontrolled a m o u n t of the cellulose acetate crept up the walls of the container. Thus, it had not been incorporated into the membrane tested. This point will be discussed further. As the observed permeability measurements were n o t reproducible, different procedures were tried to overcome this inconvenience. The method finally used has been described pre-
the lowest to the highest again. The lower limit was fixed by the room temperature, and the upper limit was 50°C because the membranes broke when they were m a i n t a i n e d for more t h a n 24 hr at temperatures higher t h a n 50°C. The thickness of the m e m b r a n e 8 was determined b y means of a micrometer which was accurate within 4-1 gin. The fractional void volume e was obtained b y the difference between the weights per unit area of the wet and dry samples obtained from the central portion of the membrane which had been exposed to the water. These measurements were performed at least eight times in each case using circular samples. The sample diameters were measured with a microscope whose precision was =t=1 gin. Each of four samples was
viously (12).
placed in six different positions. With these
The
measurements of permeability were
results the calculated value of the area was
carried out in the temperature range from 30 to
2.109 4- 0.001 cm 2. Table I shows the results
50°C, with steps of 5°C. I n each case, two or
for the 15 membranes studied; they are num-
three runs were performed. The first one was performed b y increasing the temperature from
bered in increasing order together with the
the lowest to the highest, the second from
were cast. Also, the corresponding values of 5,
the highest to the lowest, a n d the third from
M,~/S (mass per unit area), and e are listed.
concentration of the solution from which they
TABLE I Geometric Parameters of the Membranes Membrane No,
Solution composition ( m g / 2 5 cm*)
1 2 3 4
35 40 45 50
5
55
6 7 8 9 10 11 12 13 14 15
60 65 70 75 80 85 90 95 100 110
~a X 104 (cm)
(M~/S) X lO S
7 4- 2 6 -4- 2 10 4- 2 9 4- 3 10 4- 2 12 4- 2 12 4- 2 13 4- 2 15 4- 1 17 4- 2 17 4- 2 20 4- 2 18 4- 2 18 4- 2 23 4- 2
0.94 0.75 1.12 0.89 1.27 1.40 1.55 1.63 1.82 2.14 2.22 2.36 2.30 2.44 2.74
(g/era 2)
4.75 X 10-2 1.03 X 10 -1 1.28 X 10-1 4.22 X 10 -2 5.23 X 10_2 5.94 X 10-3 9.51 X 10-e 1.17 X 10-1 1.17 X 10-1 1.03 X 10-1 1.17 X 10-1 1.09 X 10-~ 7.92 X 10-~ 3.14 X 10-1 1.05 X 10-1
a The errors appearing in the values of ~ correspond to the maximal deviation with respect to the mean value of the measurements and are always greater than the precision of the micrometer used. Journal of Colloid and Interface Science, Vol. 61, No. 1, A u g u s t 1977
98
FERNfi~NDEZ-PINEDA AND MENGUAL METHOD
The phenomenological equation which describes the permeation process is JM
=
q AP,
A
-
El"]
where JM is the flow of material in moles of fluid which pass per unit of time through a cross section q of the membrane, AP is the difference in hydrostatic pressure on the two sides of the membrane, and A is a phenomenological coefficient, permeability, which depends on the following parameters: (a) the nature of the membrane; (b) the nature of the permeant fluid; (c) the temperature of the system; and (d) the average pressure of the system. Assuming that permeability is constant and writiI~g AP = p g ( h ' - - h ' ) , Eq. [-13 becomes h" -
h' = (h"o -- h'o)e -'/~
[2"]
where p is the water density, g is the acceleration of gravity, h" and h' are functions describing the variation in time of the water level in the tubes, h"0 and h'0 are the initial heights, t is the time, and r is a constant related to the permeability described by the equation, ~q0 1 A --- - -, [-31 2gqM r
where q0 is the section of glass tube used and M is the molar mass of the water. Taking into account that h " + h ' - - h " 0 + h'0, Eq. E23 can be written h'o H"
-
h'o
-
e- t / ~
E4:~
2 and an equivalent equation for the other tube,/1", is the corrected water level, h'o + h"o H " = h" 2
From Eq. E4-] it follows that the semilogarithmic plot of H " versus t is a straight
line whose slope is equal to - 1 / r . The values of - 1 / r have been obtained by adjusting the experimental data to an exponential function by the least-squares method. In the most unfavorable cases the value of the coefficients of correlation obtained was 0.99 in runs of at least 20 points. This confirms that the assumptions that led to Eq. [-4] were correct in the range of pressures used. RESULTS For most of the membranes three runs were performed in order to obtain the permeability at different temperatures. In two cases (membranes 1 and 8), only two runs were performed, as has been mentioned earlier. In Figs. 3 and 4 the semilogarithmic plot of A (mole g-1 sec) against T (°C) shows that all membranes behave similarly. A careful examination of the figures permits us to draw some conclusions: First, the results obtained can be reproduced after an adequate treatment of the membranes. These results practically coincide for all the runs performed. Second, since the points are on quite straight lines, it seems that the variation of permeability with temperature follows an exponential law. Later on this assumption will be discussed in more detail. In Figs. 3 and 4 some abnormalities in the expected behavior of several membranes are observed. For example, the permeability does not always decrease with the number of the membrane, i.e., with the concentrations of the solutions from which they have been obtained. This is the case for the first five membranes (Fig. 3) and membranes 12 and 13 (Fig. 4). Since the membranes were always obtained under exactly identical experimental conditions, it may be thought that after the solvent is evaporated, uncontrolled amounts of cellulose acetate are deposited on the rims. Thus, the concentration of the original solution should only be taken as a rough indication and the membranes should be characterized by other parameters, such as thickness, weight
Journal of Colloid and Interface Science, Vol. 61, No. 1, August 1977
PERMEABILITY VERSUS TEMPERATURE per area unit, and the fractional void volume, rather than by the initial concentration of the solution. In fact, in Table I it can be seen that membrane 12 has a value of Mm/S greater than membrane 13, and therefore, it may be expected that the first one is less permeable, precisely what we found experimentally. This interpretation allows us to order the membranes from 6 to 15 with respect to the relation between permeability and mass per unit area. Nevertheless, discontinuities are still observed in the first five membranes. It may be due to the fact that their concentration range is very close to the formation limit and errors may have considerable influence.
99
IO'lZ'! 'i P 3
-mo~oi~-_a:D 10_14i[:~
~
3i
7
2~
-~
oa--
--ceo--~o------
@ (~
"
~
~
~
@
-
I1 [
10-" l
10-16~
-c~-
-man-- @ jO-I'r ~ _ 30
f 35
, 40
_
T 45
r 50 T(°C)
3 -clo~o~--
czla-
10-12 __~.______ ~ ~ t x ~ - - - - - - - - o ~
~ 0 ~ -
FIG. 4. Permeability of water through membranes versus temperature in semilogarithmic plot.
@
8
30oc_~
o 30oC--* A 50°C. 50oC--*
DISCUSSION
4j;
[ I 10451 , 30
~
T
,
35
40
45
50
T(°C )
FIG. 3. Permeability of water through membranes versus temperature in semilogarithmic plot. [] o 0 A o 30°C --~ 50 C --* 30°C --~ 50 C.
The study of the behavior of permeability with temperature has been performed by a relatively small number of workers and always at nearly ambient temperatures with small temperature intervals. In work on reverse osmosis it has been found that the product of permeability and viscosity of the permeant is independent of the temperature for each membrane (13). It must also be remarked that the average pressures considered by these workers were in the range of many atmospheres, too far from the range of pressures used in this work. Here it has been observed that in a majority of the membranes, there is a slight
Journal of Colloid and Inierface Science, Vol. 61, No. 1. August 1977
100
FERNfitNDEZ-PINEDA AND MENGUAL
decrease of the product of permeability and viscosity as a function of temperature. This shows that, in our range of pressures, the above-mentioned results are approximately true. The exponential dependence observed in Figs. 3 and 4 has been verified by adjusting the experimental data to an exponential function by the least-squares method. The results appear ill Table II, where the coefficient of correlation for each membrane together with the values of the parameters with their standard deviations are shown. The same experimental data have been used to adjust a straight line by the least-squares method (see Table III). Also, experimental data found by various authors (1--4) have been treated in the same way. The results are presented at the bottoms of Tables II and III. The values of the coefficients of correlation show that the adjustments for both an exponential function and a straight line f i t equally well. It suggests that, in this temperature range, they should coincide. This coincidence is due to the fact that the values of the
m coefficients are small (between 10-2 and 10.-3 °C-1). If the exponential was expanded in a power series, the second-order term should be very small compared with the first. In order to check the above assumption, the series expansion corresponding to the exponential fit shown in Table II was calculated. By considering only the first-order term, this resulted in a linear relationship of permeability to temperature. The slope and intercept of this relationship can be compared with the data in Table III. Since all membranes behave in a similar manner, it will be discussed only for one of them, No. 2, for instance. In Table III, the corresponding values for this membrane are: slope = 3.3 X 10-14 and intercept = 2.8 >( 10-12 mole g-1 sec. For the exponential, in first order, they are 3.1 )< 10-~4 and 2.8 ;< 10-~2 mole g-1 sec, respectively. The discrepancy between the values of the slopes is within the standard error of the tables. Since the temperature interval is small, it may be thought that the fits found considering various functions (e.g., A = Aoe mr, A = A'o + aT, and A = A"oe m''/r) could give similar
TABLE II Exponential A d j u s t m e n t , A = A oemr Membrane No.
Correlation coefficient
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Ref. (1) Ref. (2) Ref. (3) Ref. (4)
0.96 0.98 0.98 0.95 0.98 0.97 0.96 0.96 0.97 0.97 0.99 0.98 0.99 0.97 0.99 0.97 0.99 0.98 0,99
m (°C-1) (1.32 (8.0 (6.4 (1.46 (9.0 (1.58 (9.0 (1.47 (4.61 (5.25 (5.2 (6.02 (1.32 (2.34 (3.09 (1.9 (2.6 (4.65 (1.83
4. 0.03) ~: 0.1 ) 4. 0.3 ) 4. 0.01) 4. 0.3 ) 4. 0.05) 4. 0.1 ) 4. 0.03) q- 0.07) 4- 0.09) 4. 0.1 ) 4. 0,05) -4- 0.01) -4- 0.03) -4- 0.02) 4. 0.1 ) 4. 0.2 ) -4- 0.09) 4. 0.05)
Journal of Collold and Interface Science, Vol. 61, No. 1. August 1977
A0 (mole g-1 sec) X )< )< X )< X X X )< )< )< X X X X X X X X
104 10-3 10.3 10- 2 10-3 10-2 10-3 10-2 10-3 10-3 10-3 10-3 10-~ 10-2 10-2 10-2 10-2 10-2 10-2
4.62 3.01 1.50 5.70 1.95 2.86 2.80 2.09 2.54 1.32 6.46 6.08 1.57 1.91 1.07 1.49 2.50 1.69 3.56
X 10-13 X 10- TM X 10-12 )< 10-13 X 10-12 X 10-13 )< 10-13 X 10-14 X 10-14 X 10- u X 10-15 X 10-16 )< 10-~5 X 10-15 X "10-17 X 10-13 X 10-15 )< 10-15 X 10-15
PERMEABILITY
101
VERSUS T E M P E R A T U R E T A B L E 111
Straight Adjustment, A -= A 'o + aT Membrane No.
Correlation coefficient
a (mole g-a sec °C -~)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Ref. (1) Ref. (2) Ref. (3) Ref. (4)
0.96 0.98 0.98 0.96 0.98 0.96 0.95 0.97 0.97 0.97 0.98 0.98 0.99 0.95 0.98 0.97 0.98 0.96 0.99
(1.0 4-0.1 ) X 1 0 -14 (3.3 4- 0.2 ) )< 10-14 (1.24 4- 0.07) )< 10-14 (1.5 4- 0.1 ) X 10-14 (2.5 4- 0.1 ) 5( 10-14 (8.9 ± 0 . 8 ) X 10-is (3.7 :k 0.4 ) )< 10-I5 (5.5 4- 0 . 5 ) X 10-1G (1.4 4- 0.1 ) X 10-~6 (8.6 4- 0.2 ) X 10-~7 (4.2 4- 0.2 ) X 10-~7 (4.7 4- 0.3 ) X I O -is (3.6 4- 0.2 ) X 10-iv (1.2 4-0.1 ) X 10-~v (1.19 4- 0.08) X 10-18 (5.3 4- 0.4 ) X 10-~5 (1.5 4-0.1 ) X 10-~s (4.1 4- 0.8 ) X 10-1~ (1.2 4- 0.1 ) >( 10-~6
satisfactory
results.
Nevertheless,
the
func-
t i o n A = A"oe "~''/~ h a s b e e n c h e c k e d . T h i s l e a d s t o c o e f f i c i e n t s of c o r r e l a t i o n t h a t a r e , in general, worse than
those listed in Tables
II and III. Finally, a conclusion can be drawn: In the range of temperatures and pressures used, the relationship between the permeability A and t h e t e m p e r a t u r e T is p r a c t i c a l l y l i n e a r . REFERENCES 1. BLUMBERG,A. A., HADDADIN, E. S., CHMIELEWSKI, M. E., AND SrROZ, D. G., ]. Colloid Interface Sci. 49, 24 (1974). 2. MADRAS S., MCINToSH, R. L., AND MASON, S. G., Canad. Y. Res. 27B, 764 (1949). 3. HAASE, R., AND STEIi"~ERT, C., Z. Phys. Chem. N. F. 21, 270 (1959).
A ~0 (mole g-1 sec)
( 3.7 4- 0 . 5 ) ) < 1 0 -13 ( 2.82 4- 0.08) X 10-le ( 1.44 4- 0.03) X 10-~2 ( 4.3 =t= 0.5 ) )< 10-18 ( 1.80 i 0.06) X 10-12 ( 1.8 4- 0.3 ) X 10-13 ( 2.5 =t= 0.1 ) )< 10-13 ( 1.6 4- 0.2 ) X 10-14 ( 2.49 :k 0.04) X 10-~4 ( 1.29 :k 0.08) X 10 -~4 ( 6.30 4- 0.09) X 10-15 ( 5.9 4- 0.1 ) X 10-Is ( 1.24 4- 0.07) X 10-1~ ( 1.3 4- 0.5 ) X 10-~6 (--1.0 4- 0.3 ) X 10-17 ( 1.0 4- 0.2 ) X 10-l~ ( 1.1 :t: 0.4 ) X 10 -~5 (--5 4- 3 ) )< 10-~5 ( 2.7 5 : 0 . 4 ) X 10-~6
4. HAASE, R., AND DE GREIFF, H. J., Z. Naturforsch A 26a, 1773 (1971). 5. RENKIN', E. M., J. Gen. Physiol. 38, 225 (1955). 6. GARY-BoBo, C. M . DIPOLO, R., AND SOLOMON, A. K., J. Gen. Physiol. 54, 369 (1969). 7. ALEXANDER, K. F., AND WIRTZ, K., Z. Phys. Chem. N. F. 195, i65 (1950). 8. CARE, C. Wl, AND SOLLNEF, K., J. Electrochem. Soc. 109, 616 (1962). 9. HAASE, R., A~D DE GREI~'~', H. J., z. Phys. Chem. N. F. 44, 301 (1965). 10. RAS~OGI, R. P., AND .[HA, K. M., Trans. Faraday Soc. 62, 585 (1966). 11. SHERRILL, B. C., ALBRIGTI-I, ft. G., AND DIErSCH¥, J. M., Biochim. Biophys. Acta 311, 261 (1973). 12. FERNANDEz-PINEDA, C., AND MENGUAL, ~. I., An. Fis. 72, 79 (1976). 13. SOURI~AJA~, S., "Reverse Osmosis," p. 228. Academic Press, New York, 1970.
Journal of Colloid and Inlerface Science, Vol. 61, No. 1, August 1977