ChE 201 Introduction to Chemical Engineering
Report due last class day.
The kidneys perform many functions in our bodies. The primary functions are elimination
of cell metabolism products (nitrogenous compounds produced by protein metabolism) and
maintenance of the proper balance of blood plasma constituents (water and electrolytes).
When the kidneys fail, these functions must be replaced by either a transplant or some
form of artificial treatment. Unfortunately, a lack of sufficient donors has created a great demand
for artificial treatment. The principal form of this is hemodialysis, or one of its variants, with an
artificial kidney.
For this design project, you will specify a dialysis process for the removal of one
component of cell metabolism: urea. The book describes this process in Problem 4.22 and a
schematic diagram is shown below:
Dialysate out
Dialysate in
Artificial Kidney
Blood in
Blood out
The artificial kidney consists of a number of parallel hollow fibers. Its design permits blood to
flow through the center or lumen of the fibers while a fluid called the dialysate flows around the
outside of the fibers or the shell. As the blood flows through the fibers, urea diffuses from the
blood within the fibers to the dialysate outside the fibers. When the blood exits, it contains much
less urea than when it enters while the dialysate contains much more.
To determine the blood urea concentration change during dialysis, a mass balance around
the patient is required. As a simple model of the body is one is which all of the body’s fluid is
contained in a tank; one can withdraw fluid from the tank by withdrawing blood. Consequently,
the dialysis process is represented schematically as shown below:
Dialysate out
Dialysate in
Artificial Kidney
Blood in
Blood out
Body fluid
To determine changes in the body fluid urea concentration during dialysis, one can write a
material balance for urea in the body:
(1)
d  cb V 
dt
 Qb  cbo  cb 
where cb is the blood urea concentration leaving the tank or body (which equals the inlet blood
concentration to the artificial kidney), V is the urea distribution volume of the body (body fluid
volume), and Qb is the rate at which blood is withdrawn from the body and cbo is the blood urea
concentration entering the tank or body (which equals the outlet blood concentration from the
artificial kidney). This neglects urea generation during dialysis treatment and assumes the body
fluid is well mixed as indicated in the figure above. A good estimate of V in liters is 0.58*M
where M is the body mass in kg.
To determine the rate at which urea is removed, one may perform a set of experiments in
which the flow rates of dialysate and blood are varied as well as the concentration of urea the
dialysate and blood. If the inlet dialysate urea concentration is zero, the results of these
experiments may be correlated using:
(2)
 1  exp  N 1  Z  
Qb  cb  cbo   Qb cb 

 1  Z exp  N 1  Z  
where N=kA/Qb , Z=Qb/Qd , k is a mass transfer coefficient, A is the membrane area in the
module and Qd is the dialysate flow rate. By combining Equations (1) and (2), one can calculate
how the body fluid urea concentration changes with time during dialysis.
In between dialysis treatments, urea is generated by the body at a rate of r=(0.25*M+10.4)
g/day. Thus in-between dialysis treatments, the material balance for urea in the body becomes:
(3)
d  cb V 
dt
r
For the design problem you are to answer the following questions:
1. How long should each dialysis treatment last?
2. What are the flow rates of blood and dialysate?
3. What is the body fluid urea concentration at the beginning of treatment? at the end of
treatment?
4. How much dialysate is used during each treatment?
5. How long can a person wait in-between treatments?
6. If you could change the mass transfer coefficient and membrane area of the artificial kidney
how would you change them?
7. Is the assumption of no urea generation during dialysis a good one?
When choosing dialysis conditions, you should strive to minimize the time required for each
treatment (to reduce treatment costs and improve patient well being), maximize the time inbetween treatments (also to reduce treatment costs and improve patient well being), and
minimize the amount of dialysate used (to reduce dialysate costs). Note that you may not be able
to accomplish all of these goals simultaneously.
In your analysis, you may assume:
1.
2.
3.
4.
5.
6.
7.
8.
The average patient has a body mass of 135 lbm.
The body fluid urea concentration cannot exceed 3 g/L (normal levels are 0.5 g/L).
The body fluid volume does not change in time.
During dialysis, blood and dialysate flow rates are held constant.
Blood cannot be withdrawn from the body at a rate greater than 400 ml/min.
Dialysate cannot be introduced into the artificial kidney at a rate greater than 800 ml/min.
The dialysate fed to the artificial kidney is urea free.
The available artificial kidneys possess 1 m2 of membrane area and a mass transfer
coefficient of 0.010 cm/min.
9. Blood and dialysate flow rates do not change significantly from inlet to outlet since only urea
is transported across the membrane and urea is present as a dilute component (the mass
fraction of urea is much smaller than that of the other components of blood and the dialysate).
As an aid to your analysis, I suggest you create a spreadsheet that allows you to readily
change the problem variables and evaluate their impact. In the analysis, you only need to consider
one cycle starting with a dialysis treatment and ending with the beginning of the next treatment;
the urea concentration decreases from its highest value at the beginning of dialysis to its lowest
value at the end of dialysis and then increases slowly back to the same initial value before the
next dialysis treatment.
I would recommend that you analyze the two parts of the cycle (the dialysis treatment
itself and the time in-between dialysis treatments) separately using Equations (1) and (3). The
two analyses are coupled by the fact that the concentration at the end of the dialysis treatment
time is the initial concentration for the time in-between dialysis treatments. Likewise, the
concentration at the end of the time in-between treatments is the initial concentration at the start
of the next dialysis treatment.