A nozzle flow is driven by water in a cubical pressurized tank,3m in height. The

Question

A nozzle flow is driven by water in a cubical pressurized tank,3m in height. The top 0.5 m is air at three times atmospheric pressure( 101.3 kPa), while the rest is filled with water (density = 1000 kg/m ,mu = 0.001kg/m -s).To reach the nozzle, the flow passes through a 0.5 m long tube with a 5 cm diameter .The nozzle opening has an area of 10 cm^2 .

a) Assuming that head losses are negligible in the system,determine the volume flow rate through the nozzle.

b) The jet hits a vane which reverses the flow by 180 .Assuming there is no loss in velocity in the jet during this process, what force does the flow exert on the vane? Express your answer as a force vector (including units).

c) Now, assume there are head losses from the entrance to the tube (not rounded) from the tank and from the tube itself, given a roughness of ks = 0.1mm. Assuming the flow is of sufficiently high Re such that the flow is considered fully turbulent, determine the outflow velocity.

d) Based on the answer from (c), what is theRe for the flow through the tube and was the assumption of fully turbulent flow a valid one?

Please be specific on each step and label each letter a, b, c, and d.

10. A nozzle flow is driven by water in a cubical pressurized tank, 3min height. The top 0.5 m is air at three times atmo spheric pressure(p 101.3kPa), while the rest is filled with water (p 1000 kg m s, u 0.001 kgm -s). Toreach the nozzle, the flow passesthrough a 0.5mlong tube with a 5 cm diameter .The nozzleopening has an area of 10 cm3. a Assuming that head losses are negligible in thesystem,determine the volume flow rate through the nozzle. b The jet hits a vane which reverses the flow by 180 Assuming 1 there is no loss in velocity in the jet during this process,what force 3pa 0.5 m does the flow extert on the vane Express youranswer as a force vector (including units). c Now, assume there are head losses from the entrance to the tube 3 m (not rounded) from the tank and from the tubeitself, given a roughness of k, 0.1 mm. Assuming the flow is of sufficiently high Re such that the flow is considered fully turbulent, determine the outflow velocity d) Based on the answer from (c), what is the Re for the flow through the tube and was the assumptionof fully turbulent flow a valid one?

Explanation / Answer

a)

Absolute pressure at bottom of tank = 3*Pa + rho_w*g*h

= 3*101.3*103 + 1000*9.81*(3 - 0.5)

= 328425 Pa

Applying Bernoulli theorem between pipe entry and exit

P1 + 1/2*rho*V12 + z1 = P2 + 1/2*rho*V22 + z2

We have z1 = z2..since pipe is horizontal.

328425 + 1/2*1000*V12 = 101.3*103 + 1/2*1000*V22

V22 - V12 = 454.25.............eqn1

Applying mass conservation, A1*V1 = A2*V2

3.14/4 * 52 * V1 = 10*V2

V2 = 1.963*V1................eqn2

Solving eqns 1 and 2, we get V1 = 12.613 m/s, V2 = 24.766 m/s

Hence flow rate Q = A2*V2

= 10*10-4 * 24.766

= 0.024766 m3/s

b)

Mass flow rate, m = rho*Q

= 1000 * 0.024766

= 24.766 kg/s

Force exerted F = Rate of change of momentum

= m * (V2 i - (-V2 i))..........where i denotes the unit vector towards right.

= 2 * 24.766 * 24.766 i

= 1226.7 i N

c)

For sharp entrance, minor loss coeff K = 0.5

Relative roughness, k/D = 0.1 mm / 50 mm = 0.002

From Moody diagram, for relative roughness = 0.002, for sufficiently high Reynolds numbers we get

Friction factor f = 0.025

Pressure drop = (K + fL/D)*1/2*rho*V12

= (0.5 + 0.025*0.5 / 0.05)*1/2*1000*V12

= 375*V12

Modifying Bernoulli eqn for pressure drop we get,

328425 - 375*V12 + 1/2*1000*V12 = 101.3*103 + 1/2*1000*V22

4*V22 - V12 = 1817.............eqn3

Solving eqn 2 and 3, we get V1 = 11.23 m/s, V2 = 22.04 m/s

d)

Re = rho*V1*D / u

= 1000 * 11.23 * 0.05 / 0.001

= 561500

Since Re > 2300, flow is fully turbulent.

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