Single-Stage Dialysis

Membrane separation of one or more dissolved species from a dilute aqueous stream is known as dialysis. The “artificial kidney,” which substitutes periodically (typically three times per week) for the kidney function in patients with end-stage renal dis- ease, is a dialysis process in which 250–400 cm3_{/min of blood is passed through the}

membrane system for the removal of urea and other toxins. Dialysis is also used in emergency situations to remove drugs or poisons from the system. Most dialysis processes consist of long flat or tubular membranes in channels with large aspect ratios, but it is convenient for our purposes to start with an idealized situation in which two well-mixed volumes are separated by a membrane, and a dissolved species is transported across the membrane because of a concentration difference, as shown schematically inFigure 5.1.Water may also migrate across the membrane because of the different states on the two sides, but we will ignore this phenomenon because the amount of water transfer will generally be small in the absence of a significant pressure difference between the two sides of the membrane.

The stream from which we wish to remove the dissolved species is known as the raffinate (from the French raffinere, to refine); we denote the volumetric flow rate of the raffinate by R and the concentration of the dissolved species by cR. The

stream to which the species is transferred is known as the permeate (a noun form

* _{This description is a bit simplistic, but it suffices for our introductory discussion here. In reality,}

the resistance consists of the membrane permeability per se, and resistance to mass transfer within each fluid phase in the neighborhood of the surface; the latter resistance depends on the flow rates and certain physical properties of the particular species. The actual membrane permeability may contribute less than half of the total resistance in liquid systems. Nonetheless, the functional form will remain the same; it is only the calculation of that is affected.

P R cPf VP P cP R c Rf cR VR

Figure 5.1. Schematic of a single-stage dialy- sis process, with two well-stirred sections sep- arated by a semipermeable membrane.

not found in standard dictionaries of English usage, from the Latin verb permeare, to go through); we denote the volumetric flow rate of the permeate by P and the concentration by cP. We assume that the amount of material transferred across the

membrane is too small to affect the volumetric flow rates, so the outflow values of P and R, respectively, are the same as the inflow values. (This assumption is not necessary, but it is realistic and it simplifies the algebra greatly, making it easier to understand the key features of the process.) The volumes are denoted VRand VP,

respectively, and the area of the membrane is denoted A. The rate at which the dissolved species leaves the raffinate and enters the permeate is thenA(cR− cP).

Each volume, VRand VP, is a distinct control volume, and we write the balance

equations separately for each control volume. The overall mass balances are replaced by the assumptions that the raffinate and permeate flow rates into and out of the stages are the same. The balance equations for the dissolved species in the two volumes are then as follows:

VR dcR dt = R(cRf − cR)− A(cR− cP), (5.1a) VP dcP dt = P(cP f − cP)+ A(cR− cP). (5.1b) The first term on the right side of each equation represents the difference between the inflow and outflow rates of the dissolved species, whereas the second term represents the rate of transfer across the membrane. The transfer term is negative in the raffinate equation and positive in the permeate equation, since solute transfer is from the raffinate (outflow) to the permeate (inflow). We are interested in the design equations, for which it suffices to consider only the steady state, so we set dcR/dt = dcP/dt = 0 and get

R(cRf− cR)− A(cR− cP)= 0, (5.2a)

P(cP f− cP)+ A(cR− cP)= 0. (5.2b)

Equations5.2a and5.2b can be added together to obtain a simple relation between the concentrations, which can be rearranged to the form

cP= cP f+

R

We can then solve Equation 5.2a to obtain the net separation in the raffinate stream, cRf− cR=  1 1+ R_P+_AR  (cRf− cP f)≡ M(cRf − cP f). (5.4)

Equation5.4is the basic design equation for the single-stage dialyzer. Notice that some useful estimates follow immediately. First, consider the case A→ ∞; that is, there is no limit on the available membrane surface area for transport. Equation 5.4can then be solved to obtain the minimum value of P/R that can ever be used (i.e., the lower bound, which is never attainable in practice) in order to effect a given separation:  P R  min = cRf − cR cR− cP f . (5.5)

From Equation5.3,we see that this limit corresponds to cP = cR; that is, to equal

concentrations on both sides of the membrane, which is clearly the best that we can do. If the flow rate R of the raffinate stream is specified, as will often be the case, then Equation5.5leads to an explicit expression for the lower bound on the minimum permeate rate:

Pmin= R

cRf − cR

cR− cP f.

(5.6)

Similarly, the minimum surface area will correspond to the case in which the per- meate flow is so rapid that the transferred material is removed instantaneously (P/R → ∞), causing the driving force to be maximized; we therefore obtain

Amin= R   cRf− cR cR− cP f  = Pmin  . (5.7)

We can usually assume that cPf= 0, so these limits can be expressed solely in terms

of the separation factor, sR= cRf/cR: Pmin= R(sR− 1) and Amin= R(sR− 1)/.