Question

Liquid water enters a nozzle at 20ºC with a speed of 1 m/s and exits at 20ºC, 100 kPa. The exit diameter of the nozzle is one-half of the inlet diameter. Determine the pressure at the inlet, in kPa. (You will need to be careful with units.)

Answer 1

Let us consider

The inlet is point 1 and the outlet is point 2

Then from the given data we can take

Inlet velocity V1 = 1 m/s

Inlet temperature T1 = 20 ºC

Exit pressure P2 = 100 kPa

Exit temperature T2 = 20 ºC

Density of water = 1000 kg/m3

Exit dia of nozzle is one half of inlet dia which gives D2 = 1/2 D1 or D2 = 0.5 D1

The Continuty Equation says that in a steady flow the mass flow rate will be constant through out the fluid flow

Therefore 1 = 2

1 * (volume flow rate at 1) = 2 * (volume flow rate at 2)

1 * A1 * V1 = 2 * A2 * V2

Since the inlet and exit temperatures are equal there will be no change in the density of fluid over the flow,

then 1 = 2

Therefore the continuty equation becomes A1 * V1 = A2 * V2

   () * D12 * V1 = () * D22 * V2

() * D12 * V1 = () * (0.5*D1)2 * V2

up on simplification we get V1 = 0.25 * V2

   V2 = 4 * V1

V2 = 4 * 1 = 4 m/s

The Bernoulli Equation gives

P1 + 1/2 * 1 * V12 +  1 * g * H1 = P2 + 1/2 * 2 * V22 +  2 * g * H2

As the density and height at both inlet and exit of nozzle are same we can reduce the equation to

P1 + 1/2 * * V12 = P2 + 1/2 * * V22

P1 - P2 = 1/2 * * (V22 - V12)

P1 - 100 = 1/2 * 1000 * (42 - 12)

Therefore P1 = 7600 kPa

So, from the calculations we can conclude that the inlet pressure P1 is 7600 kPa