Relevant diagram loosely based on my real problem:
Description of problem: A fan creates a pressure sink that drives fluid flow through a gently diverging pipe (please note that the diagram is not drawn anywhere near to scale!). The continuity equation allows for calculation of velocities $V_1$, $V_2$ and $V_3$ as $Q_3$ is equal to the max airflow of the fan:
$Q = V A$
So: $V_1 > V_2 > V_3$
I'd like to find the pressures $P_1, P_2, P_3$. I have tried applying Bernoulli's principle to find these as follows:
$P_\text{fan}+ P_3+ 1/2 ρV_3^2+ρgz_3= P_2+ 1/2 ρV_2^2+ρgz_2$
Where $P_\text{fan}$ is the max depression generated by the fan, $ρ$ is the density of air (incompressible fluid) and the height z is the same on both sides so can be ignored. This doesn't find me $P_3$ or $P_2$ but gives the pressure difference. However if I then repeat the process between section 1 and 3, I get a smaller pressure difference due to $V_1 > V_2$.
This'd imply that $P_3 < P_2$. If this was the case I'd expect a reverse flow. I'd greatly appreciate someone pointing out my error of thinking and how I can truly find the section pressures.
Assumptions: 1-D flow, Incompressible fluid, Steady flow.
EDIT
Having thought about it a bit: perhaps the pressure sink generated by the fan should be regarded as negative (relative to atmospheric) as well as the kinetic and pressure terms in the Bernoulli equation. Taking a control volume beyond the fan and with any other cross section would yield:
$-P_\text{fan} = -P_i -1/2 ρV_i^2 - (\text{losses})$
Thus as the velocity of the fluid flow in the system decreases, so the pressure term must become more negative and continue the propagation of flow towards the fan. However negative kinetic pressure doesn't seem right. I'd greatly appreciate any thoughts - even if it's to say I'm barking up the wrong tree!
