Mass conservation in a conical tank

[sandmike_83]

Good Morning to all

I saw this problem in one of the courses that I am taking this semester. It is very simple, it consists of an open conical tank being filled in the upper part with an stream (which is assumed to be cylindrical) of water (flow Qi through an area Ai). At the bottom of the tank there is an opening (Ae) through which the water leaves the tank, following the torricelli equation: Qe=Ae*SQRT(2*g*h)

h being the water level in the tank measured from the lower part.

I was solving the problem using mass conservation law for control volume using fixed and deformable boundaries (to exercise a bit the control volume approach), and for both cases I got the same solution:
(Derivative of the volume of the cone with time)=(flow in)-(flow out)
d( (1/3)*Pi*(r^2)*h)/dt=Qe-Qout

However, if you see the pdf attached the professor's solution is a bit different, and there is an extra term (dh/dt)*Ai.

What is your opinion about it? What did I do wrong?

Thanks in advance

PS. I didn't post this topic under homework because I didn't got this problem as homework, I just saw it in the course notes and I wanted to do the derivation on my own to see If I got the same result.

 

[Chestermiller]

If the hole is small, the difference is insignificant. He just subtracted the volume of the cone projected below the hole opening.