Homework # 4

125:303; Fall 2004 (P. Moghe)

Biomedical Transport Phenomena

Assigned Friday, October 1, 2004

Solutions Due in Class, Friday, November 5, 2004

A major problem in the successful design of implantable tissues is the availability of oxygen for respiring tissues, which is determined by the spatial access of tissues to blood capillaries that bring oxygen carrying red blood cells. A classic model in this field is the Krogh Cylinder (Krogh, 1919). Imagine a tissue space with cells surrounding a cylindrical capillary. Oxygen and other metabolites arriving into the capillary axially due to fresh flow of oxygenated blood will diffuse from the capillary radially toward the tissue, where they will be consumed by the cells. The solution of the Krogh cylinder problem yields an expression for the critical distance into the tissue beyond which no more solute is available, denoted by .

where ;

Parameters given are (Fournier, 1999):

DT, the metabolite tissue diffusivity=2x10-6 cm2/s

V, the blood plasma velocity=0.005 cm/s

rc=capillary radius=0.0005 cm

tm=capillary wall thickness=5x10-5 cm

Ko=overall metabolite mass transfer rate=5.75 x 10-5 cm/s

Co=5 mmole/cm3

Using the Newton-Raphson method in the MATLAB environment, solve the above equation for as a function of z. Vary z from 0.001 to 0.1 cm in increments of 0.01 cm. Plot versus z.

Provide your systematic code for the Newton-Raphson method and show your work and the results and graphs clearly and methodically.