Question

Water fills a cylindrical tank to a depth h. The tank has a diameter D. The water flows out at an average velocity V, through a hole in the bottom with area A,. Apply conservation of mass for a control volume to develop an expression for the rate of change of the water height in the tank

Answer 1

The expression for the rate of change of water height in a cylindrical tank, derived using conservation of mass, is dh/dt = -(A/A) v, where A is the cross-sectional area of the tank, A is the area of the hole, and v is the average velocity of water flowing out.

Step-by-step explanation:

To derive an expression for the rate of change of water height in a cylindrical tank using conservation of mass, we first acknowledge that the mass of water entering the system must equal the mass leaving the system.

Considering that the water is incompressible and the tank has a constant cross-sectional area given by A = (πD2)/4 where D is the diameter of the tank; the volume V of water in the tank at any time is V = Ah where h is the height of the water level.

The flow rate, Q, which is the volume of water flowing out per unit time, is given by Q = Av, where A is the area of the hole and v is the flow velocity through the hole.

Conservation of mass dictates that the rate of change of the water volume in the tank is equal to the negative of the outflow rate:

dV/dt = -Q
d(Ah)/dt = -Av
A dh/dt = -Av

Isolating dh/dt, we get:

dh/dt = -(A/A) v

This represents the rate of change of the water height in the tank.