# Proof that pressure rises in a fixed control volume

I am looking for a proof using Bernoulli's equation that that for $$Q=(Area)(Velocity)$$, $$Q_{in}>Q_{out} \Rightarrow P_{inside}>P_{outside}$$. I read that for a computer, if the case fans blow a higher volumetric flow rate in than out, then the PC case is positive pressure, however I seem unable to prove this. My attempt is shown below, with the assumption that $$z_1=z_2$$, $$P_{1,g}=0$$, and the subscript g indicates gage pressure. My control volume is the inside of a computer case, and subscript 1 represents outside of the control volume (flowing in), and subscript 2 represents inside the control volume (flowing out):

\begin{align} \frac{P_1}{\rho}+\frac{V_1^2}{2}+z_{1}&=\frac{P_2}{\rho}+\frac{V_2^2}{2}+z_{2} \\ \frac{V_1^2}{2}&=\frac{P_{2,g}}{\rho_{air}}+\frac{V_2^2}{2} \Rightarrow \\ \frac{P_{2,g}}{\rho_{air}}&=\frac{V_1^2-V_2^2}{2} \Rightarrow \\ P_{inside,g}&=\rho_{air} \frac{V_{entering}^2-V_{leaving}^2}{2} \end{align}

Since $$V=\frac{Q}{A}$$, then the pressure inside is also dependent on the area of a fan, which contradicts what I read saying $$Q_{in}>Q_{out} \Rightarrow P_{inside}>P_{outside}$$. This is a bit confusing to me and I'd appreciate any answers that I can get.

• Can you define more of your notation? What does the $g$ subscript represent, and are $1$ and $2$ the same as "in" and "out" respectively? – Aaron Stevens Dec 17 '18 at 2:16
• Are you assuming that the air is incompressible? You must not be assuming this, because, otherwise Qin=Qout. Also. Is the fan inside the control volume or outside? – Chet Miller Dec 17 '18 at 2:34
• Is heat being released within the control volume? – Chet Miller Dec 17 '18 at 2:55
• In the particular case you mentioned Bernoulli equation is not applicable because viscous effects are dominant. – Deep Dec 17 '18 at 5:12
• @Josh Can you give the source of where you read it plz ? – J.A Dec 17 '18 at 9:38