The artificial kidney uses the principle of to purify the blood of patients whos

Question

The artificial kidney uses the principle of to purify the blood of patients whose own kidneys have failed. The diagram below shows patient on a kidney dialysis machine, Alvin's kidney contains 50 ml of solution, in which there is 10 grams of dissolved protein. A solution of 1/2 gram per ml is piped into Alvin's kidney at 2 ml per minute, and mixes with Alvin's kidney content and piped into dialysis machine at e ml per minute. The mixed solution is then expelled out of the machine into the sewer at 2 ml per minute, but 1 ml/min is piped back into Alvin's kidney. Assume dialysis container holds initially 100 ml of water. Let x=grams of dissolved protein in alvin's kidney y= grams of dissolved protein in Dialysis machine Write a system of differential equations that describe the above situation

Explanation / Answer

3 equation will be there for these situation

2 equation mass balance in kidney and dialysis machine

1 equations for component balance of x in kidney

1 equations for component balance of y in dialysis machine

so

now component balance equation of x in kidney

amount of protein in alvin kidney at time t + amount of protein already in kidney = amount of protein entering kidney - amount of protein leaving kidney + amount of protein entering through recycling

that is

d ( x/ 50 ) / dt + 10/50 = 1/2 /2ml/min - x/3ml /min + y/ 1 ml/min

where x is protein content in kidney

y in dialysis machine

so this is differential equation at any time for x (protein content in kidney ) this is a component balance

now for componet balance of y in dialysis machinne

amount of protein in dialysis mahcine at time t = amount of protein entering machine - amount of protein leaving machine - amount of protein leaving machine through recycling

d(y/100) /dt = x/3 ml/min - y / 2 ml/min -y/ 1 ml/min

both the equation needed to be simultaneously with initial and final condition given

since this is time variant which need time based condition

and since differential equation are coupled need to solved together

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