Flow rate depending on pressure with constant inlet rate.

[blehxpo]

A 10 m high, 5 diameter cylindrical water tank has an inlet at the top and a drain at the bottom. The flow rate out of the tank depends on the pressure at the bottom of the tank via the following:
dm/dt = (10^-4)*(P_bot)
where P_bot is the absolute pressure in Pascal at the bottom of the tank. The tank is open to atmosphere at the top (P_atm) The density of water is 1000kg/m^3

My Attempt:
a. What are the units of 10^-4
- If P_bot is in Pascals then it should just be [meter*second]

b.What is the absolute pressure at the bottom of the tank is psia when the tank is 50% full?

- P = P_atm + (density*g*h)
= 14.7 psi + (49000*145.04*10^-6) psi = 21.8 psi

c.What is the gauge pressure at the bottom of the tank is psig when the tank is 50% full?

- P_gauge = P_abs -P_atm
= 21.8-14.7 = 7.1 psi

d. At steady state, there is a specific inlet flow rate required to maintain a given water level in the tank.
Derive an equation that gives the inlet flow rate (in [kg/s]) required to maintain the water height (h) at a given level, where h [m] is measured from the bottom of the tank upwards.

inlet flow rate[kg/s] = outlet flow rate = F(h) = (10^-4)(P_atm + density*g*h)
I'm not sure about this one.

e. Determine the inlet flow rate needed to keep the water level at 50% full.

10^-4(101325 Pa + 1000(9.8)(5m) = 15.03 kg/s

f. Assume that the tank is initially empty. At time t=0, the inlet is turned on at the flow rate you found in part (e). How long does it take the tank to reach 30% full.

This is where I got stuck.
I would assume inlet rate is constant at 15.03 kg/s. But I have no idea how to calculate the out flow rate.

g. How long does it take for the initially empty tank from part (f) to reach 50% full?

And I guess this is exactly like f.

 

[nvn]

blehxpo: Your answers are correct on items (a) through (e). On item (d), you might want to fill in the known numeric constants, but your choice.

(f) Hint 1: q_out(h) = (1/rho)*F_out(h), where F_out(h) is the function you derived in item (d).
dV/dt = q_in - q_out(h).