Poiseuille's Law Examkrackers 2015

[An7icipa7e]

Hi all,

There's a question in the new 2015 mcat EK books (Reasoning Skills) and I'm having a bit of trouble understanding it. Here's the question and answer.

Water flows through a pipe with a 14 meter radius at 2 L per second. An engineer wishes to increase the length of the pipe from 10 m to 40 m without changing the flow rate or the pressure difference. What radius must the pipe have?
(They give you Poiseuille's Law, followed by these answer choices)

A. 12.1
B. 14.0
C. 19.8
D. 28.0

Answer & Explanation:
"The only way to answer this question is with proportions. Most of the information is given to distract you. Noticed that the difference in pressure between the ends of the pipe is not even given and the flow rate would have to be converted to meters^3 per second. To answer this question using proportions, multiply L by 4 and r by x. Now pull out the 4 and the x. We know by definition, (Poiseuille's law); thus, (x^4)/4 must equal 1. (<- FYI, this is the sentence I don't get. How does the P's law equal 1?) Solve for x, and this is the change in the radius. The radius must be increased by a factor of about 1.4. 14×1.4 = 19.6. The new radius is approximately 19.6 cm. The closest answer is C."

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[chemist16]

Here's what Poiseuille's Law says: volume flow rate = pi*(pressure difference)*radius^4/(8*viscosity*length). In other words, if you want to keep the volume flow rate constant and still change the length, you have to change the radius correspondingly. Which is what the book means by setting up ratios. In other words, initial radius^4 divided by initial length must equal final radius^4 divided by final length. Using that proportion, you can solve for final radius.

In terms of the answer you're giving, you can say that volume flow rate is equal to r^4/L times some constant since all other factors are kept constant. r and L here are not variables but rather the initial radius and length. Keep in mind that volume flow rate isn't changing in this problem. So what they did was they quadrupled the length and changed r by some unknown factor. Let's call this factor x. So, volume flow rate is equal to (rx)^4/(4L). This is the same thing as (x^4/4)*(r^4/L). Now I just said that the volume flow rate doesn't change. So the initial and final expressions are equivalent, or (x^4/4)*(r^4/L) = r^4/L. The only way for this to be true is if x^4/4 = 1.

 

[An7icipa7e]

@chemist16 Thank you man. I think I understand now. For this question, the other factors in this law really are distractions! Cheers.