Volumetric flow rate - Help needed

ryancalif

I posted this in the DE forum as well since it is related.
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Hi all,

I've got a fairly straight forward problem to solve here regarding flow rate out of a tank with uniform cross-sectional area. I am treating this is a volumetric flow problem since there is assumed to be volume flow out of the tank, and volume flow into the tank.

I have two Qout terms (one out of a hole in the bottom of the tank, and one from fluid consumed by an engine), as well as one Qin term (via a pump feeding fluid into the tank at a constant rate).

Qouthole (Qh) = a*C*sqrt(2*g*z)
Qoutengine (Qe) = constant
Qin (Qi) = constant

NOTE: a = exit hole area
C = energy loss coefficient
g = gravity
z = head height in tank

My DE looks like:

-A*(dh/dt) = Qh - Qi + Qe

Separating the terms and solving for time by integrating h from Hi to Hf and t from 0 to T, I get this...

((a*C)/A)*sqrt(2*g)*(2/3)*((Hf^(3/2))-(Hi^(3/2))) = (1/A)q*t

I combined the Qi and Qe terms early on since they are constants (easier to integrate), therefore, q in the above equation equals Qi-Qf.

The problem is, solving for t eliminates the tank cross-sectional area (A) since it is divided out.

I feel confident with my general equation, but may have made a mistake somewhere in solving for t. Any help would be GREATLY appreciated! Thank you