full question is in description

Question

Suppose you are transporting a viscous fluid through a set of 15.0 circular pipes, each with diameter 1.0 cm. If you now replace this set of pipes with one large circular pipe which has a cross sectional area equal to the sum of the 15.0 small pipes, by what factor does the flow rate increase, assuming ΔP and L are unchanged?

Answer 1

Poiseuille's Law relates the volumetric flow rate Q of a fluid in a circular pipe to several other physical parameters:

 

Q = [Πr⁴/8η]*[Δp/L] 

 

r: radius of pipe

η: fluid viscosity

Δp/L: pressure drop across the length of the pipe.

 

You initially have an array of 15 small pipes with 1-cm diameters, and from Poiseuille's law you can calculate their combined Q in terms of pressure drop and viscosity.

 

The second part of the question tells you to replace the 15 small pipes with one large pipe that has the same cross-sectional area as all the small pipes combined. From that information, you can back-solve for the radius of the equivalent larger pipe. Using Poiseuille's Law again, this time with the new radius, a different value for Q will be obtained (still in terms of pressure drop and viscosity since they haven't changed). 

 

All this question is asking for is the ratio of the larger pipe's value of Q to the Q corresponding to the array of small pipes.