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Ind. Eng. Chem. Process Des. Dev. 1984, 2 3 , 320-329
Analytlcal Design Equations for Multisolute Reverse Osmosis Processes by Tubular Modules Ravl Prasad and Kamalesh K. Slrkar’ Department of Chemistry and Chemical Engineering, Stevens Institute of Technology, Hoboken, New Jersey 07030
For “straight-through” tubular modules with turbulent flow inside tubes, simple analytical equations have been developed to determine the number of membrane tubes required and the permeate concentrations of solutes if the dilute feed has two hlghly rejected solutes and a given fractional solvent recovery is desired. Extension to three or more solutes is straightforward. These equations have high predictive accuracy for seawater type feeds with low water recovery. The analytical equations of the design methods based on breaking up the tubular RO plant into two or three sections for calculational purposes achieve high predictive accuracies for brackish water type feeds with high water recoveries. The predictive accuracies of these methods have been checked by comparing the calculated results from these analytical equations with those from a numerical solution of the exact set of equations based on a solution-diffusion type of model of multicomponent reverse osmosis.
Introduction Reverse osmosis (RO) has emerged in recent years as a highly economical and efficient separation process for large-scale purification of water containing microsolutes as well as for concentration of microsolutes from aqueous solutions. A number of recent process design studies (Rogers et al., 1981; Fosberg et al., 1981; Renshaw et al., 1982) attest to the expanding circle of potential process applications of RO dominated still by its beginnings in desalination. Such widening activity suggests development of simple and accurate design equations for making rapid initial estimates or preliminary process design calculations. Availability of simple analytical design equations for any unit is especially useful in parametric studies of processes having a large number of interconnected units due to the extraordinary savings in computer time. Sirkar and Rao (1981) have recently developed simple analytical design equations for tubular reverse osmosis desalination with one highly rejected solute only. Sirkar et al. (1982) have also developed analytical design equations for single-solute RO desalination with spiral wound modules under both turbulent as well as laminar flow conditions. Most process streams to be treated by RO, however, have more than one microsolute. We have, therefore, focussed our efforts here toward obtaining analytical design equations for multisolute reverse osmosis with a straight-through tubular reverse osmosis module. For fouling feeds, tubular modules with turbulent flow are preferred. In a later paper, we will illustrate the design equations for multisolute RO with a spiral-wound module which work best with nonfouling clean feeds. The reverse osmosis membrane transport with mixed solutes in aqueous feed solutions have been studied by Erickson et al. (19601, Agarwal and Sourirajan (1970), Sourirjan (1970), Hodgson (1970), Eliash and Bennion (1976), Rangarajan et al. (1978), Rangarajan et al. (1979), and Malalyandi et al. (1982). An exact irreversible thermodynamic analysis of multicomponent membrane transport in RO with mixed electrolytes is highly complicated (Eliash and Bennion, 1976). However, two observations from the latter work are relevant for our objective of developing simple analytical design equations with highly rejecting membranes and dilute feed solutions. First, the water flux for mixed electrolyte feed solutions can be predicted reasonably satisfactorily from data on single salt feed solutions. Secondly, ion-ion interactions 0196-4305/84/7 123-0320$01.50/0
increase the rejection of, say, MgC12 in the presence of NaCl by no more than 1%over that observed in the absence of NaC1. A similar behavior is observed in the extensive tabulated data in Sourirajan (1970) as long as the total salt molality is less than or around 1and the membrane is highly rejecting. If one chooses to use a solution-diffusion type of model (or the preferential sorption and capillary transport model of Sourirajan (1970), it being mathematically equivalent to a solution-diffusion model, Soltanieh and Gill,1981) one could utilize the above results in an uncoupled engineering approximation framework and propose to use only three fundamental parameters for a two solute system: the pure water permeability constant A and the salt transport parameters (DZM/K26) and (D3M/K36)of microsolutes 2 or 3, respectively. Srinivasan and Tien (1970) in their analysis of concentration polarization in multicomponent RO had in fact used a pure water permeability constant A. Since they had chosen perfect membranes, the salt transport parameters were not required. Malalyandi et al. (1982) have recently demonstrated the validity of this model for a three noninteracting solute RO system using parameters A , (Dz,/K26), ( D 3 ~ / K 3 8and ) , (DdM/Kdb) with cellulose acetate membranes. Srinivasan and Tien (1970) had also suggested that the brine channel mass transfer characterization of an individual solute in. terms of a local Sherwood number was essentially a function of the Schmidt number of the given solute and that the cross-diffusion terms were negligible. When one combines these assumptions with the explicit flux expression approach developed by Rao and Sirkar (1978) and valid for turbulent flow in tubular membranes, it becomes possible to develop a simple analytical expression for the number of tubular membranes required for a given fractional water recovery with high rejection RO membranes, turbulent flow, and multiple solutes. We develop such an analytical design equation for a two-solute system here, the extension to more than two solutes being straightforward. We have next developed analytical design equations to predict the permeate concentrations of each of the two solute species. Again, the development of such equations for a three or four solute system is quite simple. The predictions from these design equations have been compared with those from a numerical solution of the complete set of governing equations using a solution-diffusion type model. The numerical design procedure closely 0 1984 American Chemical Society
Ind. Eng. Chem. Process Des. Dev., Vol. 23, No. 2, 1984
follows that developed by McCutchan and Goel (1974) for turbulent tubular RO with a single solute. The boundary layer analysis of multicomponent RO in conduits by Srinivasan and Tien (1968) was not considered relevant for comparison. It is known for Sirkar and Rao (1981,1983) that higher predictive accuracies for single-solute systems are achieved by using a split-module design procedure employing the design equations in a two- or three-step calculation process. These multistep yet simple analytical design procedures have also been developed here for multisolute systems. The examples include both seawater type and brackish water type feeds encountered in desalination. These equations and design procedures are useful for turbulent tubular RO process design with any kind of noninteracting microsolutes as long as the solutions are dilute and the membrane has high solute rejections. Simplified Flux Expressions and Polarization Relations for Multisolute RO with High Rejection Membranes The complexities of an exact irreversible thermodynamic formulation of multisolute RO are well documented by Eliash and Bennion (1976) as well as Bennion and Pintauro (1980). We adopt here the solution-diffusion type of model for the fluxes of the solvent and the solutes. We restrict our treatment to two solutes, species 2 and 3 only-its extension to more than two solutes being straightforward. The corresponding flux expressions under conditions of concentration polarization are N1r = A [ U - =wall + apemeatel (1) N2r
= (D2M/K28)[C22 - C2d
(la)
Ob) Assume now that the osmotic pressure of any dilute solution of microsolutes is linearly proportional to the total molar concentration of the microsolutes due to ideal solution behavior *wall = bCwd = b[C22 + C32I; =permeate - bcpermeate b[C23r + C33rI (2) Further for dilute solutions, the molar density C of the solution may be assumed constant throughout the module and C21f = cx21J; c21L = cx21L; c22= cx22; c23 = CX23; CV1 P 1; C31f = CX31f; C31L = CX31L; C32 = CX32; C33 = CX33; C31 = CX31; Czl = CX21 (3) These routine assumptions lead to the following flux expressions for a 2-solute case N1, = A[AP - bCX22 - bCX32 + bCX23, + bCX33J (4) N2r = (D2M/K26)C[X22 - x 2 3 r l (44 N3r
= (D3M/K36)[C32 - c3311
= (D3M/K36)C[X32- X33rl (4b) In tubular RO with turbulent flow, film theory expressions are commonly used for estimating concentration polarization for single-solute cases (Brian, 1966; Kimura and Sourirajan, 1967; McCutchan and Goel, 1974; Harris et al., 1976). We propose to use Brian’s (1966) film theory expression separately for concentration polarization for each solute neglecting cross-diffusion terms although the solvent flux couples the expressions for each solute exp [Nlr 91/kl,I 2- - x 2 2 -c2= (5) c 2 1 X2l r2 + (1 - r2) exP[Nlr~l/kl,l N3r
exP[NlrWkl,1 c32 _ - -x32- c31 x31 r3 -k (l - r3) exp[N1rvl/k131
(54
321
Here kl, and k13are the turbulent mass transfer coefficients for solute species 2 and 3, respectively, with the membrane solute rejections defined by r2=1--
231 x 2 2
r3=1--
X331
x32
Due to pure crossflow in tubular RO (Rao and Sirkar, 1978), the permeate concentrations are related to the species fluxes by
and
For given operating conditions at any location, i.e. CZ1,C31, k k AP with known values of A , (DZM/k28),and (&M/g38), simultaneous numerical solution of eq 4,4a, 4b, 5, Sa, 7, and 7a will yield the local values of Nlr,N2,, N3r, C22, C32, C Z ~and ~ , C331. These are the exact solutions in the framework of a solution diffusion (or preferential sorption-capillary transport) model, film theory of concentration polarization and no cross-diffusion effects in brine channel. Note that the film theory expressions 5 and 5a used here incorporate only Nlrin the exponential terms instead of (Nlr+ N,, + N3r). For dilute solutions this is entirely reasonable. Approximate analytical design equations can be developed only on the basis of some assumptions and approximations. We summarize these now for the flux expressions and the concentration polarization relations. We first assume that the membranes under considerations have high salt rejection so that
