Flow through parallel pipes with different diameters

[berkanozturk]

Homework Statement

Consider an apparatus design in which two tubes of different diameters are connected in parallel to each other.
The flow is steady and fully developd in both tubes. Do your analysis regarding
a)the governing differential equations for the case that in both tubes laminar flows occur.

Homework Equations

Asked

The Attempt at a Solution

Hello everyone,
For one pipe, we can simply use Reynolds transport theorem for a differential volume, and then get the related conservation of mass and momentum equations for 1 PIPE. However, when there are two pipes, how should I relate it? I could not solve it, hope you can help me. thanks for reading so far...

best regards
berkan

 

[Astronuc]

Assuming the pipes are connected to the same manifolds (headers), then the pressure drops across the pipes (lengthwise) would be the same. The flow rates would be different because the flow resistance would be different (including the cross-sectional areas).

The electrical analog is two resistors in parallel. Both resistors have the same potential drop across their terminals, but the current (charge flow rate) is different.

 

[orkun]

using bernoullli equation;

P/p + 1/2(V)^2=constant, (P:pressure, p:density, V:velocity)

lets say pipe1 is carrying the main stream and pipe2 and pipe3 are the parallel pipes with different diameters,

then it must be;

P1/p + 1/2(V1)^2 = P2/p + 1/2(V2)^2 = P3/p + 1/2(V3)^2

Remember;

Q=A*V

for different diameters crossectional area difference also changes V (as i know, correct me if i m wrong), so;

P2=P3 can not be achieved.

when it comes to may be

P1=P2 or P1=P3 ?

it depends on V1, V2 and V3 again. the same V will result the same P if material properties (heat, density etc.) and level of pipes not changed.