MASS TRANSFER COEFFICIENTS FOR USE IN
REVERSE OSMOSIS PROCESS DESIGN S H O J l
K I M U R A ' A N D S. S O U R I R A J A N
Division of Applied Chemistry, National Research Council, Ottawa, Canada
Analytical expressions are given for predicting concentration polarization effects in reverse osmosis using feed solutions such as sodium chloride-water. The cases considered are laminar and turbulent flow between flat parallel or in tubular porous cellulose acetate membranes for experimental conditions defined by the dimensionless parameters y (ratio of osmotic pressure o f feed solution to the operating pressure), 0 (characteristic of the particular membrane-solution system), and X (ratio of the mass transfer coefficient to the solute transport parameter). The partial differential equations describing the concentration distribution on the brine side of the membrane have been solved for the laminar flow case for different values of a, (a dimensionless diffusivity parameter), y, and 0, and the results are transformed such that the mass transfer coefficient i s expressed in the form of Sherwood number ( N s ~as ) a function of longitudinal position ( X ) . The correlation of cu,Nsh/4 vs. .$p ( = X / 3 a , 2 ) for the flat membrane case, which i s identical with that of a,Nsh/2 vs. f r ( =X/4cr,2) for the tubular membrane case, i s in good agreement with that of Sherwood et a/.for the limiting case y = 0 and 0 = 0. Analysis of the characteristics of two typical films shows that the variations of average solute separation, fractional water withdrawal, and boundary concentration with X obtained by the solution of the partial differential equations are in good agreement with those given by the trial and error solution of the basic transport equations using local or average mass transfer coefficients obtained from the above correlations. Using the experimental results of Shaw et a/. and the Leveque solution, the turbulent flow case has been analyzed. On the basis of these analyses, explicit expressions for calculating local and average mass transfer coefficients are given for both laminar and turbulent flow between flat parallel and in tubular membranes.
paper extends the work reported earlier (Kimura and Sourirajan, 1968a) on the concentration polarization effects in reverse osmosis using the Loeb-Sourirajan type of porous cellulose acetate membranes, specified in terms of the pure water permeability constant, A , and the solute transport parameter, DAM/K6, a t a given operating pressure, for solution systems such as sodium chloride-water. T h e procedure for predicting the effect of concentration polarization on membrane performance involves either the computer solution of a partial differential equation (PDE) for laminar flow conditions, or the direct or indirect experimental determination of the mass transfer coefficient applicable for the particular reverse osmosis operating unit, which may not always be easy. I t is the object of this paper to obtain correlations and explicit expressions to calculate mass transfer coefficients which can be used in the basic transport equations (Kimura and Sourirajan, 1967) for predicting membrane performance for both the laminar and turbulent flow cases, using flat or tubular membranes with particular reference to systems such as sodium chloride-water. THIS
Analysis
Basic Equations. T h e following equations, which utilize the mass transfer coefficient, k , and predict the salt concentration in the product, membrane throughput rate, and the concentration polarization, have been derived (Kimura and Sourirajan, 1968a). 1 Present address, Department of Chemical Engineering, University of Tokyo, Tokyo, Japan.
[l vw Vw=L'100=
-y
(CZ
- Call
(3)
where (4)
(5)
and
Equations 1, 2, and 3 assume that the molar density of the solution, c, is essentially constant, the osmotic pressure of the solution is proportional to the mole fraction of the solute, and the solute flux through the membrane is small compared to the 1. These assumptions are reasonsolvent flux-Le., X A Z
