The objective of this experiment is to measure the overall mass transfer coefficient for a hollow fiber membrane dialyzer and determine the individual contributions of the lumen, membrane, and shell mass transfer resistances.
1. Introduction
This experiment involves the study of dialysis, which is the transport of molecules across a membrane through diffusion. Diffusion is a naturally occurring process where a solute passes from an area of high concentration to an area of lower concentration; the concentration differences are the driving force since this is a mass transfer situation. The diffusion will continue until both phases have reached an equilibrium concentration. In this experiment, a particular application of dialysis will be explored, namely artificial kidneys for the purification of blood.
Hemodialyzers are the name of the artificial kidneys, and they are a tube packed with thousands of hollow fiber hollow fiber membranes. At each end of the tube, there are cases known as tubesheets which seal the ends of the fibers. There is then a port from the tubesheets that allows a fluid, blood during normal operation, to flow through the interior of the hollow fibers, also known as the fiber lumens. There are also ports that allow a fluid to flow on the shell side (external) of the hollow fiber membranes. This fluid is the dialysate in operation, and it always flows counter-currently to the blood. The counter-current flow is analogous to a heat exchanger and minimizes size due to greater concentration differences at each end, which cannot be obtained with co-current flow. The hollow fiber membranes are porous with pores large enough to allow cell metabolic waste to pass through but not large enough to allow the passage of proteins, blood cells, and plasma.
2. Theory
The behavior of mass transfer across a membrane has been well studied and is well characterized. As a result, it is known that the moles of solute passing across the membrane under steady state conditions are represented by equation 1, as given in the laboratory guidelines:
= rate of solute passing across the membrane (mol/s)
= overall mass transfer coefficient (m/s)
= the mass transfer area (m2)
= logarithmic mean concentration difference (mole/m3)
The logarithmic mean of concentration difference between the inlet and outlet streams of the membrane is specified by the laboratory guidelines and given by equation 2:
= concentration of lumen flowing entering (mol/m3 = mM)
= concentration of lumen flowing leaving (mol/m3)
= concentration of dialysate flowing entering (mol/m3)
= concentration of dialysate flowing leaving (mol/m3)
When inlet concentrations, flow rates, and target solute removal rate are specified, the membrane can be sized using equation [1]. As a result, it becomes apparent the size of the membrane is directly related to the overall mass transfer coefficient; the larger the value of ko, the smaller membrane area required. The smaller area has health implications because it will minimize the amount of blood removed, which can be life threatening if too much blood is removed at initiation of the procedure. The overall mass transfer coefficient is dependent on three resistance values: lumen side or feed resistance, Rl, membrane resistance, Rm, and shell side or dialysate resistance, Rs. From the laboratory guidelines, it can be seen the sum of these resistances gives the overall resistance, Ro, which relates to ko by equation 3:
= mass transfer resistance of the lumen phase (s/m)
= mass transfer resistance of the membrane phase (s/m)
= mass transfer resistance of the dialysate phase (s/m)
Additionally, the laboratory guidelines show the lumen phase resistance and dialysate phase resistance are both related to the inverse of their corresponding mass transfer coefficient by equations 4 and 5:
OD = outer diameter of one fiber (m)
ID = inner diameter of one fiber (m)
According to Cussler (2009), at high flow rates of dialysate, the dialysate resistance can be neglected, which gives equation 6:
Additionally, the same logic from Cussler (2009) that led to equation 6 can prove that high flow rates of lumen means that the resistance due to the lumen can be neglected, yielding equation 7:
As the dialysate and lumen flow rates are maximized, the boundary layers at both interfaces begin to diminish as the fluid flow becomes more turbulent. With maximum flow, any resistances from the boundary layers are negligible and the overall mass transfer coefficient can be related to the membrane resistance as shown in equation 8:
However, not all runs will have conditions as specified by equation 8. In those instances, a solver program must be used to analyze the contribution of all three resistances. The experimental values can then be compared to the calculated values to see how closely the system is behaving compared to the theoretical operation.