See the PDF file for example 8.4 for the MathCad details

Example 8.4 Determine the hollow fiber membrane surface area for a blood oxygenator

operating under the following conditions. Assume the fibers are made from microporous

polypropylene. The gas membrane permeabilities are then based on diffusion through a

stagnant layer of gas trapped within the pores of the membrane. The length of each fiber is

50 cm with a wall thickness of 50 microns. The inside diameter of a fiber is 400 microns.

The blood flowrate through the lumen of the fibers in the device is 5,000 ml min-1 and the

gas flowrate on the outside of the fibers is 10,000 ml min-1, both at 37 C and 1 atm. The

pO2 of the entering blood is 40 mmHg and the amount of oxygen

transported into the blood as it passes through the oxygenator must equal 250 ml min-

1(BTP). The pCO2 of the entering blood is 45 mmHg and the amount of carbon dioxide

removed from the blood must equal 200 ml min-1 (BTP). The entering gas is saturated with

water (pH2O = 47 mmHg at BTP) and contains a small amount of carbon dioxide in order to

decrease the driving force for carbon dioxide transport so that a proper respiratory

exchange ratio (R) can be achieved and for the calculated surface areas for oxygen and

carbon dioxide to be the same. The respiratory exchange ratio is defined as the ratio of

carbon dioxide output to oxygen uptake and should be equal to 0.8. Estimate the surface

area required to deliver 250 ml min-1 of oxygen to the blood and remove 200 ml min-1 of

carbon dioxide under these conditions. The gas side mass transfer resistance can be

considered negligible because of the low solubility of oxygen and carbon dioxide in blood.

As discussed earlier the permeability of these polypropylene hollow fiber membranes is

also very high resulting in negligible mass transfer resistance. Hence the bulk of the mass

transfer resistance is a result of the boundary layer formed within the blood.

SOLUTION. The solution algorithm is based on first identifying all of the relevant

dimensions and physical properties. There are also three unknowns that serve as

iteration variables. The first iteration variable is the membrane area required for oxygen

and carbon dioxide transport. The second iteration variable is the average value of the

blood pO2 as it flows through the hollow fiber. This value is used to calculate, by Equation

6.37, the value of m which is the slope of the oxygen hemoglobin dissociation curve. The

average pO2 is chosen to make the oxygen mass transfer on the gas and blood sides balance.

Hence the amount of oxygen transferred to the blood must equal the amount of oxygen lost

from the flowing gas. The third iteration variable is the pCO2 in the incoming gas. This

value is adjusted to make the oxygen and carbon dioxide membrane surface areas,

calculated by Equations 8.41 and 8.50, come out to be the same value. In addition, an

overall mass balance on oxygen and carbon dioxide is performed such that their respective

transport rates are 250 ml min-1 and 200 ml min-1 at body temperature and pressure (BTP).

This calculation gives a pO2 of the blood exiting the hollow fibers of 90 mmHg and and a

pCO2 of 41.55 mmHg.

Once the three iteration variables are assumed, the calculation approach is then to calculate the value of m by Equation 6.37. From this value of m the effective diffusivity of oxygen in blood is found from Equation 8.46. Next, the number of hollow fibers based on the assumed membrane area

is determined, i.e. Ld

AN

outside

membranefibers

, where Amembrane is the assumed membrane area, doutside is

the external fiber diameter, and L is the fiber length. Knowing the number of hollow fibers, the

average velocity of blood in a given hollow fiber can be found, i.e. 2

4

dN

QV

fibers

bloodblood

, where

Qblood is the total volumetric flowrate of the blood and d is the internal diameter of a hollow fiber. In this case the velocity was found in the converged solution to be 23.2 cm sec-1. Next the value of Re and Sc are found and from these the blood side mass transfer coefficient can be found from Equation 5.53. The pO2 of oxygen in the exiting gas can then be found from Equation 8.39, which in this case is found to be 664.8 mmHg.

Next, the surface area of the membrane for oxygen transport is found from Equations 8.41

and 8.42. This process is then repeated for carbon dioxide and ends with the calculation of

the membrane area required for carbon dioxide transport using Equations 8.50 and 8.51. For

carbon dioxide it is found that the exiting pCO2 in the gas is equal to 44.26 mmHg.

Following this, the amount of oxygen and carbon dioxide transport is calculated from an

overall mass balance for the gas and for the blood. The surface areas for oxygen and carbon

dioxide transport must also come out to be the same for each respective gas. In addition, the

oxygen and carbon dioxide mass balances must be equal between the gas and blood phases.

If the solution has not converged in this manner then adjustments need to be made to the

iteration variables until convergence is achieved. In this case the membrane area required

for oxygen and carbon dioxide transport was found to equal 2.25 m2. The partial pressure of

carbon dioxide in the feed gas was found to be 29.1 mmHg and the average pO2 used for

calculating the value of m was found to be 59.5 mmHg. The respiratory exchange ratio also

equals 0.80.

Use of a mathematical model to predict oxygen transfer rates inHollow fiber membrane oxygenators, Vaslef et al. ASAIO J 1994(read the paper as well, the following summarizes the key points)

Now go to PDF forthe derivation

Other oxygenatorshad similar results