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
(a) What are the units of 10
‐4
?
(b) What is the absolute pressure at the bottom of the tank in psia when the tank is 50% full?
(c) What is the gauge pressure at the bottom of the tank in psig when the tank is 50% full?
(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.
(e) Determine the inlet flow rate needed to keep the water level at 50% full.
(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?
Think carefully…is the flowrate going out constant?
(g) How long does it take for the initially empty tank from part (f) to reach 50% full?
a. If dm/dt is kg/second, then
kg/second= ? * P= ?*nt/m^2
or ? has units of kg*m^2/Ns
but N is kg*m/s^2
or ?= kg*m^2/(kg*m/s^2 * s)
?= second/meter
(a) The units of 10^-4 are determined by the units of dm/dt and P_bot in the given equation.
dm/dt represents the rate of change of mass over time and has units of kg/s.
P_bot represents pressure and has units of Pascal (Pa).
Since 10^-4 is being multiplied by P_bot, the units of 10^-4 must cancel out the units of P_bot to give a final result in kg/s. Therefore, the units of 10^-4 must be 1/Pa.
(b) To find the absolute pressure at the bottom of the tank in psia (pounds per square inch absolute) when the tank is 50% full, we can use the following equation:
P_abs = P_atm + (ρ * g * h)
Where:
P_abs is the absolute pressure in psia at the bottom of the tank,
P_atm is the atmospheric pressure in psia,
ρ is the density of water in kg/m^3,
g is the acceleration due to gravity in m/s^2,
h is the height of the water column in meters.
Given that the tank is 10 m tall and is 50% full, the water column height is 0.5 * 10 m = 5 m.
Substituting the given values:
P_abs = P_atm + (1000 kg/m^3 * 9.81 m/s^2 * 5 m)
To convert the pressure to psia, you'll need to multiply by the conversion factor of 0.000145038.
(c) To find the gauge pressure at the bottom of the tank in psig (pounds per square inch gauge) when the tank is 50% full, we can use the following equation:
P_gauge = P_abs - P_atm
Substitute the values of P_abs and P_atm obtained in part (b) to calculate the gauge pressure.
(d) To derive an equation that gives the inlet flow rate (in kg/s) required to maintain the water height (h) at a given level, we need to consider the balance between the flow rate in and out of the tank.
At steady state, the flow rate in must be equal to the flow rate out to maintain a constant water level. The flow rate out is given by dm/dt = (10^-4) * P_bot, as given in the question.
The flow rate in can be calculated using the equation:
dm_in/dt = ρ * A * v
Where:
dm_in/dt is the mass flow rate into the tank,
ρ is the density of water in kg/m^3,
A is the cross-sectional area of the tank (π * r^2),
v is the velocity of the water entering the tank.
Since the tank has a constant cross-sectional area, we can determine the inlet flow rate by setting the flow rate in equal to the flow rate out:
dm_in/dt = dm/dt = (10^-4) * P_bot
Substituting the values and rearranging the equation will give the desired equation for the inlet flow rate required to maintain the water height.
(e) To determine the inlet flow rate needed to keep the water level at 50% full, substitute the given values into the equation derived in part (d).
(f) Assuming the tank is initially empty and the inlet flow rate calculated in part (e) is turned on at time t=0, the tank will start filling up. Since the flow rate out depends on the pressure at the bottom of the tank, the flow rate out is not constant.
To determine how long it takes for the tank to reach 30% full, you need to analyze the rates of change of the water level. In other words, you need to solve the differential equation that describes the change in water level with respect to time, using the given flow rate equation and the fact that the tank is initially empty.
(g) To determine how long it takes for the initially empty tank from part (f) to reach 50% full, you need to use the same approach as in part (f). However, this time you need to solve the differential equation for the change in water level with respect to time up until the tank reaches 50% full. The initial condition will be that the tank is initially empty.