2-49Chapter 2 Solute Transport In Biological Systems 2.14 Design of An Artificial Kidney Utilizing Urease in Polymeric Beads For the treatment of uremia discussed in section 2.11, fresh dialysate is used in kidney dialysis to maintain a concentration gradient across the membrane that separates the blood from the dialysate. As a result, metabolic waste products diffuse from the blood into the dialysis fluid that might be processed to eliminate the waste products if it is going to be reused. We will discuss a possible design of a system to reduce the use of dialysate. In this design the artificial kidney acts as a reactor where the dialysis fluid is mixed with the enzyme encapsulated in solid polymeric beads. The toxic species diffuse through the membrane into the dialysate becoming a part of an agitated slurry of beads as shown in Figure 2.14-1. The enzyme/polymer bead slurry is used to keep the concentration of toxic materials low since any toxic species diffused into the beads will be metabolized into harmless products. Flow inQ,Cb UBFlow out Q, CbUMCURC(z)UStirrerHollow fiberDialysate Figure 2.14-1 Schematic of solute diffusion and reaction in a blood dialysis system. We will now focus our attention on the removal of urea that diffuses from the blood to the dialyate. At steady state the concentration of urea in the dialysate (CUR) is a constant since the amount of urea diffuses from the blood to the dialysis fluid will be equal to the amount diffuses into and then metabolized by the enzyme urease encapsulated in solid polymeric beads. In order to design this type of dialyzer we need to specify the volume of the dialysis fluid, the bead volume fraction, the amount of enzyme in the reactor, and the bead size. We also need to obtain experimental data for the kinetics and mass transfer for this system. The polymeric beads will be spherical and will contain a uniform distribution of the enzyme urease. The beads will be contained in a well-mixed container and surrounded by the dialysis fluid with a uniform concentration of urea (CUR). The urea must dissolve in the bead and2-50diffuse into the interior, where reaction occurs so that the concentration of urea in the dialysate (CUR) can be maintained at a constant value. Making a mole balance on a spherical bead at steady state we have Molar rate of urea reacted within the bead = Molar rate of urea enter the bead at the surface Molar rate of urea reacted within the bead = 4πR2(−DURrUdrdC=) RrUrea reactedwithin the bead Urea entersthe bead Figure 2.14-2 Conversion of urea by urease within the bead. Hence we need to find the concentration profile CU(r) of urea within the bead. If we assume that the kinetic of the conversion of urea is first order, we can easily solve the differential species balance equation to obtain CU(r). In the next section we will use a shell balance to obtain a second order differential equation describing the diffusion and reaction of urea within a spherical bead. 2.14-1 Diffusion with Homogeneous First Order Reaction Rrr+dr Figure 2.14-3 Illustration of a spherical shell 4πr2dr We will consider the diffusion of A into a spherical polymeric bead where homogeneous chemical reaction occurs. The one-dimensional molar flux of A is given by the equation2-51 "AN = − DAdrdCA (2.14-1) Applying a mole balance on the spherical shell shown in Figure 2.14-1 yields for steady state 4πr2rAN" − 4πr2drrAN+" + RA4πr2dr = 0 Dividing the equation by the control volume (4πr2dr) and taking the limit as dr → 0, we obtain − 21rdrd(r2"AN ) + RA = 0 (2.14-2) For a first order reaction, RA = − kCA and substituting the molar flux from equation (2.14-1) into the above equation, we have − 21rdrd−drdCrAAD2 − kCA = 0 DA21r drddrdCrA2 − kCA = 0 (2.14-3) In this equation, DA and k are constants independent of r. We want to transform this equation into the form 22dryd − α2y = 0 (2.14-4) where the solution to the homogenous ODE has two forms 1) y = C1e-αr + C2eαr 2) y = B1sinh(αr) + B2cosh(αr) The first exponential form (1) is more convenient if the domain of r is infinite: 0 ≤ r ≤ ∞ while the second form using hyperbolic functions (2) is more convenient if the domain of r is finite: 0 ≤ r ≤ R. The constants of integration C1, C2, B1, and B2 are to be determined from the two boundary conditions. Let α2 = AkD, we can transform equation (2.14-3) into the form of equation (2.14-4) by the following algebraic manipulations r1drddrdCrA2 − α2 rCA = 02-52 r1+2222drCdrdrdCrAA − α2 rCA = 0 2drdCA + 22drCdrA − α2 rCA = 0 Since drd)(ArCdrd = drd+drdCrCAA = drdCA + drdCA + 22drCdrA, the above equation becomes drd)(ArCdrd− α2 rCA = 0 Let y = rA, the equation has the same form as equation (2.14-4) with the solution y = B1sinh(αr) + B2cosh(αr) or rCA = B1sinh(αr) + B2cosh(αr), where α2 = AkD The two constants of integration B1 and B2 can be obtained from the boundary conditions At r = 0, CA = finite or drdCA = 0 At r = R, CA = CR (a known value) Applying the boundary at r = 0 yields 0 = B2 Applying the boundary at r = R yields RCR = B1sinh(αR) ⇒ B1 = )sinh( RRCRα Therefore the concentration profile for species A within the spherical bead is CA = CRrR)sinh()sinh(Rrαα (2.14-5) At the center of the bead, the concentration is given by2-53 CA(r = 0) = CR)sinh( RRαα 2.14-2 Parameters Specification for an Artificial Kidney We want to design a hemodialyzer that will maximize the utilization of the enzyme and permit the conversion of urea to proceed at a rate that is not limited by mass transfer. We encapsulate the enzyme in spherical beads for which the diameter must be specified. If the total amount of enzyme is fixed and if the concentration of enzyme in each bead in independent of the bead radius then the total bead volume is a constant M = N34πR3 = constant (2.14-6) In this equation N is the number of beads and R is the bead radius. When R varies, N will change. However the total bead volume M remains a constant. The total molar rate of urea metabolized within the beads is given by Total rate of urea reacted within the beads = 4π NR2(−DURrUdrdC=) The urea concentration within the bead is given by equation (2.14-5) rewritten with the subscript U denoting urea CU =