Question

An artificial kidney uses porous hollow fibers to purify blood. Blood containing urea flows through the fibers and urea diffuses out, through the porous fiber wall. Tens of thousands of such fibers were used in an artificial kidney. We would like to determing the fraction of the urea removed as blood flows through one such fiber. The fiber has a radius of R = 100 µm and length L = 10 cm. The flow rate of blood (viscosity 0.03 g/cm/s and density 1.06 g/ml) at the enterance of the fiber is Q₀=10 µl/min. Because saline also leaks through the porous wall of the fiber, the blood flow rate decreases along the length of the fiber. The leakage flow rate is V_w= 0.25 ml/hour/cm². The diffusion coefficient of urea in blood is D = 1 x 10^-5 cm²/s. You can ignore the urea that is carried through the tube wall by the leakage flow. Assume steady state condition.

a) Given the blood flow rate, should axial diffusion be ignored compared to convection? (answer with calculation of Peclet number)

b) Use the approach of mass conservation, derive the equation and find the solution of the volumetric flow rate of blood in the tube considering the leakage flow.

c) Urea is transported from the flow to the wall due to mass transfer. For this case, the Sherwood number for the transfer of urea from the flow in the tube (with mean concentration C) to the wall (concentration C_w) is: Sh = (k_fR)/D = 1.58 + 2.5(V_wR)/D, where k_f is the mass transfer coefficient that relates the average flux to the wall as N = k_f(C-C_w). Then urea is transported across the porous membrane via diffusion. Assume that the flux of urea out through the fiber wall is N_w(x) = PC_w(x) at a location along the fiber length with a permeability P = 1.3 x 10-³ cm/s for urea. C_w(x) is the urea concentration at the wall, 0<x<L. Find the expression for the total flux of urea from the blood out through the wall. (hint: the processes act in a prallel fashion similar to the two layered diffusion problem.)

d) Based on mass conservation and the result from part c), develop the transport equation of urea in the tube, and find the bulk concentration of urea along the tube length.

e) Calculate the fraction of urea removed as blood flows through one such fiber.

Answer 1

Venous blood 15 m/min Dialysace 1600 MAM (ml/min a) 'tdatex tenoval gate ω。uld be 2.00-195 . 5 ㎖/min Rate of- mea em ova (200 1'75 (195 38.8 mg uemin sg.gm3 trea(win C- --6-o2 58 mg u%.X 1505 mR /min 266 mi'n demoved