Write a mass balance equation for urea in the system

1. (33pts) The following setup is a simplified version of hemodialysis:

Q= flow rate of fluids (mL/min)

B = Concentration of urea in the blood

D = Concentration of Urea in dialysate

Subscripts I = in, O = out

a. Identify the system of the dialyzer and draw boundaries around it

b. Write a mass balance equation for urea in the system

c. Knowing that this is an actual system with a patient connected to the dialysis machine, which two variable would you have control over

d. Because urea is a waste product toxic to the body if accumulated, what would you set Di, the concentration of urea in the dialysate, to be?

e. At a particular moment, suppose the patient’s blood flow rate out to the dialysis machine is 300 mL/min, with a urea concentration of 300 mG/L. The dialysate flow rate has been set to 1400 mL/min, and you measured the urea concentration of the existing dialysate to be 54 mG/L. What concentration of urea does the blood going back to the patient contain?

f. We have analyzed the system of the dialyzer, but we are more interested in knowing the concentration of urea within the patient over time. Set up a mas balance equation for the system of the patient. Consider the patient as a control volume remembering:


2. (33pts)Design dialyzer and take into account maximum shear stress on blood.

Our dialysis system uses a cartridge containing hollow fibers (dialysis tubing) with an inner diameter (ID) of 200um and the length of 0.5m



· u = velocity

· L = Diameter (for cylindrical tube)

· Laminar flow when Re <2000 ,

· Transient 2,000<Re<3,000

· Turbulent for RE>3,000

· The density of blood: ρ = 1.06g/cm3

· The viscosity of blood: µ = 3.5 cP (.035 g/cm-s)

a. Under what average fluid velocity can we assume laminar flow through the fibers?

b. If this was a Baxter CT190G, what would the Overall blood flow rate be? (Start by refer to table b4 and use than information to determine the number of fibers)


c. Is it safe to assume laminar flow in our analysis/design of a hemodialysis cartridge then?


3. (33pts) Let us consider the shear stress on blood cells that may result from hemodialysis.

In his review titled “Flow Induced Trauma to Blood Cells”, Sutera says “the Designer of components for extracorporeal circuits should have no difficulty staying well below the lower stress threshold of 1,500 dyn/cm2.” (Sutera, 1977).

In class and homework we’ve been using the Hagan Poiselle Flow equation . It is often written with isolated.

a. Rewrite the equation to isolate

b. Given that the shear stress at a particular r is calculated by and shear stress is greatest at the wall of the tube (r=R). Write an equation for the maximum shear stress by plugging your equation for from into the equation for at the wall, and simplify.

c. What is the maximum flow rate of the blood that will not induce trauma to the blood cells during hemodialysis for your chosen dialysis tubing (ID still 200m)?

d. And the for the ¼” ID connecting tubing going from the patent to the dialyzer?

e. Which portion of dialysis system is more of a threat when considering shear stress on the blood cells?


4. (33pts) Now you are tasked with designing a dialyzer using the same Baxter CT190G dialysis membrane used in the previous problem. You are provided the chart below to work with and asked to maximize the extraction ratio under known performance conditions.

Mathematical model for the efficiency of a dialysis unit. The extraction ratio, E, is shown for different flow rates (QB/CD) and values of relative membrane permeability (KS/QB)

Extraction ratio

= relative rate of permeation through the membrane to blood flow (KS/QB)

K = permeability

Z = relative rate of bloodflow to dialysate flow (Qb/Qd)


a) What relative membrane permeability () will you target?

b) What will be your blood flow (QB) through the dialyzer?

c) Your goal is to create the lowest cost, high performance (Extraction ratio of at least 0.9) dialyzer on the market. One way to cut costs is to use the lowest flow pump that meets your specs. What is your minimum flow spec for your diasalyte pump?

d) In a hypothetical world, several journal articles have been published by as many independent research laboratories in the past year which strongly indicate micro-damage to white blood cells activates a slow, but chronic autoimmune response that leads to the organ damage. This micro-damage results from shear stresses as low as 1.5 dyne/cm2. Your boss ask you to redesign the device so blood shear stress’ fall under this level. What would be your new QB? Is this reasonable? Why or why not?

















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