2 An airtight cylindrical tank has its axis vertical as show

2. An airtight cylindrical tank has it\'s axis vertical as shown in the figurel. The tank has water filled in to to a height of 10.0 m. The gage pressure in the tank above the water surface is 2 atmospheres. One end of a hose is connected to a smooth circular opening at the bottom surface of the tank such that the other end of the hose discharges water vertically upward into the atmosphere. Neglecting mechanical energy losses, find the maximum height to which the water jet could possibly rise. Take acceleration due to gravity 10 m/s2, density of water p 103 kg/m 1 atm 105 N/m

Solution

The height to which the level of water rises from the pipe is found using the hydrostatic eqn,

P1 - P2 = rho*g*(h2- h1), where P1 - 1 bar and P2 - 2 bar, h1 - 10m

Solve for h2,

h2 = h1 + [(P1 - P2) / rho*g ] = 10 + [100000 / 1000*10] = 20 m

The water rises to a max height of 20m.

a) The volumetric flow rate of water from the pipe is given by,

nu = A2*V2 where A2 - area of pipe, V2 - velocity of flow.

The V2 of the pipe is found using Bernaulli\'seqn for pipe flow neglecting the losses,

P1 + 1/2*rho*V1^2 = P2 + 1/2*rho*V2^2

Rearrange and solve for V2

(P1 - P2) = 1/2*rho*(V1^2 - V2^2)

V2^2 = 2(P1-P2) / rho [ 1- (V2^2 / V1^2)] ---- eqn 1

The conservation of mass defines the mass flow rate through the pipe as,

A1*V1 = A2*V2

V1/V2 = A2/A1

Now, replace the above relation in the eqn 1,

V2^2 = 2(P1-P2) / rho [ 1- (A1^2 / A2^2)]

V2 = sqrt {2(P1-P2) / rho [ 1- (A1^2 / A2^2)]}

The final volumetric flow rate is given as,

nu = A2 * V2 = A2 * sqrt {2(P1-P2) / rho [ 1- (A1^2 / A2^2)]}

b) The mass flow rate of air through the pipe is found using the relation,

dot m_air = rho_air * A2 * V2

dot m_air = rho_air * A2 * sqrt {2(P1-P2) / rho [ 1- (A1^2 / A2^2)]}

Replace the pressure and density values as P1 - 2*10^5 N/m2 and P2 - 10^5 N/m2 and rho - 1.2 kg/m3

dot m_air = 1.2 * A2 *sqrt {2(10^5) / 1.2 [ 1- (A1^2 / A2^2)]}

dot m_air = 489.89 * sqrt {1/ (A2^2 - A1^2)}

The volume flow rate is given as,

nu = dot m_air / rho_air

nu = { 489.89 * sqrt [1/ (A2^2 - A1^2)] } / 1.2

nu = 408.248 * sqrt [1/ (A2^2 - A1^2)]

The volume flow rate, nu is 408.248 * sqrt [1/ (A2^2 - A1^2)]