Desalination of Water by Reverse Osmosis
Charles E. Hechl
Brooklyn College, The City University of New York, Brooklyn
A n exercise in thermodynamic calculation
Desalinat~on . . of water by reverse osmosis ,
is a fairly new1, application of one of the classical thermodynamic colligative properties of solutions and a brief treatment of it could be of interest in physical chemistry and thermodynamics courses. The fugacity of the solvent in a solution is always lowered below that of the pure solvent under the same pressure as the solution. If there is a membrane present between the solution and pure solvent which will pass solvent molecules exclusively (or at least preferentially) there will be a net flow of pure solvent int,o the solution. The osmotic pressure of the solution is by definition the net pressure that must be applied over and above the pressure on the solvent to bring about a state of equilibrium in the situation described above, such that the fugacities of the solvent in the solution and in the pure state become equal resulting in no net solvent flow into the solution. If the pressure on the solution is raised above the equilibrium osmotic pressure then pure solvent will flow out of the solution and this process can be used in the desalination of water. I t is obvious that this method will have an inherent advantage as t,o energy costs over distillation and freezing processes of desalination since it does not require a phase change. Progress in reverse osmosis work is dependent upon the availability of suitable semipermeable membranes which can stand up well over time to the required pressures and st,ill give an appreciable flow of potable water. Results2 have been very encouraging and a pilot plant has been built by the Aerojet-General Corporation of Sacramento, Calif. and design studies are proceeding wit,h respect to the const,ruction of a large production plant. We may qualitatively investigate t,he pressures necessary. A standard derivation3gives:
in which: ovmutic pressure partial molal volume of solvent (cvmponeut "1") assumed independent of spplied pressure, P R = gas constant T = absolute temperature = iugneity of pure solvent under pressure P ,f,'(P) f,(P, S T )= ftlgaeity vf solvent in the solution under. total pressme P and in which its mole frnctim is X1 7
=
I 7
=
The ratio of fugacities is equal to the ratio of artivitiex:
f,'/f,
=
a,'/a,
and using the usual convention that a,',
(2) =
1 we have:
Instead of putting the activity of solvent, al. in the form of an activity coefficient times the solvent mole fraction it is more convenient to use the practical osmotic coefficient,C$ of the solventkdefined as:
in which: chemical potential of p w e solvent chemical potential of solvent in standard state of nnit activity = chemical potential of solvent in the solution = moleculsr weight of the solvent
pi' = pio = = pio
%,
and the factor
Ern, is a sum over all the molalities of
solute species (distinct from solvent) present and for a single electrolyte giving v ions:
Hence:
The origin of the term "osmotic coefficient'' is made clear if we recall that in very dilute solution:
and:for ideal solutions:
Normal sea water for our purposes can be taken as 0.7 molal in NaCl since such a solution is easy to treat and has an ionic strength, s, typical of sea water as exemplified in Table I. The q5 for XaC1 solutions is known very acrurat,ely from freezing point measurement,~. Use of eqn. (;,) for 0.7 molal NaC1 for which6 V, ~ D G EB., F., American Scialisl, 48, 476 (1060). Saline Water Conversion Report, 1964 (United States Department of the Interior, Office of Saline Water, available from Superintendent of Documents, TJ. S. Government Printing Office, Washington, D. C.) E. A., Thermodynamics (North Holland PnhGUGGENHEIM, lishing Co., Amsterdam, 3rd Edition (1957) p. 236. HARNED, H. S., and OWEN,B. B., The Physical Chemistry of Electrolyte Solutions (Reinhold Publishing Corp., New York, 3rd Edition (1958)) p. 13. ?
Volume 44, Number I,
January
1967 / 53
Table 1.
Makeup of Typical Sea Water Sample of Ionic Strength s
Component
wt. %
Molality (m)
NaC1 MgCln MgS0, CaSO, H?0
2.60 0.31 0.22 0.12 96.75
0.460 0.034 0.019 n . no01
s =
xm.zi2
=
where m' is the molality of NaCl present. recovery r2' becomes r2 where:
After y%
100
y1
=
(mq)
T*,
(9)
and
0.67
(10)
i
is 18.9 cma and @ = 0.93 gives an osmotic pressure of 30 atrn (440 psi) at 2 5 T . In general for y% recovery of pure water from the sea water by increasing the pressure on the sea water at 25'C above 30 atm it is convenient to consider the mole ratio r2 (using primes to denote the original condition) : m'M,
"'
=
1000
+ rn' (1 - M,)
54 / Jaurnol o f Chemical Education
(8
Thus for 50% = 0.96 . - recovery r l r = 1.4 and using" and assuming Vl constant, a = 62 atm (910 psi) and for 75% recovery ni = 2.7 arid usings q5 = 1.03, a = 128 atm (1.9 X loa psi). These figures are, of course, ideal equilibrium results and for equal y values actual pressures would need bo be considerably higher due to irreversible effects. Nevert,heless these numbers are qualitatively significant. Tootnote (4) pp. 402 and 416.
