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Let's say there is a main pipe containing $100\: \mathrm{m^3/hr}$ a fluid of density $750\: \mathrm{kg/m^3}$ and it's gonna be branched into 2 pipes (Pipe A and Pipe B) of the same diameter. If Pipe A is much shorter than Pipe B, then the pressure drop across Pipe B will definitely be higher right?

Will this condition affect the volumetric flow rate of the fluid in Pipes A and B?

If it doesn't, will the volumetric flow rate of both pipes be the same, that is $50\: \mathrm{m^3/hr}$?

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What happens exactly, depends on what happens downstream of pipes A and B.

Suppose both pipes flow into the open air, then the pressure at both outlets has to be the same, e.g, at ambient pressure $p_0$ (ignore altitude etc effects). The pressure at the branch is some pressure $p_1$, which is the same for both pipes. In other words $$\Delta p_A = \Delta p_B $$.

This pressure drop, follows from, e.g. the Hagen-Poiseuille equation,

$$\Delta p \propto L Q, $$

with $L$ the length of the pipe, and $Q$ the flow rate. The scaling can be exactly derived for laminar flow, but for turbulent flow (which you probably have), it relation is similar, with a different proportionality. In other words, the lengths and flowrates in both pipes, are relates as follows.

$$\frac{L_A}{L_B}=\frac{Q_B}{Q_A}$$

Example: If pipe A is three times longer than pipe B, than three times more liquid will flow through pipe B. Thus $75 m^3/h$ through B, and only $25 m^3/h$ through pipe A.

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  • $\begingroup$ What if friction is involved and both pipes are NOT flowing into open air? $\endgroup$ – user36045 Dec 19 '13 at 7:23
  • $\begingroup$ Friction: if both pipe are the same, then it doesn't matter. If the pressure at the outlets is not the same you have to be more specific, i.e. you have to add contributions from other components that influence the total pressure drop. (Consider it as a electrical circuit) $\endgroup$ – Bernhard Dec 19 '13 at 7:41
  • $\begingroup$ But if one pipe is longer than the other, wouldn't the pipe losses for the longer pipe be higher, thus the pressure drop is larger too? $\endgroup$ – user36045 Dec 20 '13 at 3:56
  • $\begingroup$ Yes, but that is due to it being longer, not due to the friction itself: the pressure drop scales the same way. $\endgroup$ – Bernhard Dec 20 '13 at 6:40
  • $\begingroup$ So with the higher pressure drop for the longer pipe, the flow rate would be lower right? $\endgroup$ – user36045 Dec 21 '13 at 15:07
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In short if pipe A is much shorter than B, and both A and B have the same size and outlet pressure (aka back pressure), then pipe A and B will have identical pressure drop and pipe A will have a higher flow rate.

Given a fixed inlet and outlet pressure, the pressure drop is also fixed ($\Delta P = P_{inlet} - P_{outlet}$) and therefore the flow in the two pipes would have to adjust to balance this constraint. It is assumed that the inlet pressure is the same as both pipes branch of the same line at the same point in a Tee configuration. It is assumed that the back pressure on pipes A and B is the same in absence of better information.

Pressure drop along a pipe is dependent on several factors and may be calculated using the equation below:

$$ \Delta P = f \frac{L}{D} \frac{\rho v^2}{2} $$

Where:

$P$ is the pressure drop.

$f$ is the friction factor.

$L$ is the pipe length.

$D$ is the pipe internal diameter.

$\rho$ is the fluid density.

$v$ is the fluid velocity.

The fluid properties will be identical in each branch as they are fed from the same main line and therefore we do not need to consider the effect of $\rho$.

If pipes A and B are of identical construction (same size and schedule) they will have the same internal diameter and therefore will do not need to consider the effect of $D$.

The pipe friction factor is a complex function of the Reynolds number and the pipe wall roughness. Assuming the pipes are constructed from the same material and have similar histories (i.e. manufactured and installed at the same time) the wall roughness of the two pipes should be similar. Therefore we would only need to consider the effect of Reynolds number (more precisely the velocity) on the friction factor. However due to its lesser effect on the pressure drop compared to fluid velocity and pipe length and the complexity of its relationship we will not ignore the friction factor moving forward.

At this point we can see pressure drop is primarily a function of the fluid velocity and the pipe length.

$$ \Delta P \propto L v^2 $$

Given the assumption that that $\Delta P$ is fixed, if we reduce $L$, $v$ will be increased to maintain the 'fixed' pressure drop. Similarly if we increase $L$, $v$ will be reduced.

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