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Chapter 4

Q. 4.P.11

Air is flowing at the rate of 30 kg/m²s through a smooth pipe of 50 mm diameter and 300 m long. If the upstream pressure is 800 kN/m², what will the downstream pressure be if the flow is isothermal at 273 K? Take the viscosity of air as 0.015 mN s/m² and assume that volume occupies 22.4 m³. What is the significance of the change in kinetic energy of the fluid?

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(G / A)^2 \ln \left(P_1 / P_2\right)+\left(P_2^2-P_1^2\right) / 2 P_1 v_1+4\left(R / \rho u^2\right)(l / d)(G / A)^2=0       (equation 4.55)

The specific volume at the upstream condition is:

v_1=(22.4 / 29)(273 / 273)(101.3 / 800)=0.098  m ^3 / kg

G / A=30  kg / m ^2 s

∴                  \operatorname{Re}=(0.05 \times 30) /\left(0.015 \times 10^{-3}\right)=1.0 \times 10^5

For a smooth pipe, R / \rho u^2=0.0032 from Fig. 3.7.
Substituting gives:

(30)^2 \ln \left(800 / P_2\right)+\left(P_2^2-800^2\right) \times 10^6 /\left(2 \times 800 \times 10^3 \times 0.098\right)

+4(0.0032)(300 / 0.05)(30)^2=0

and the downstream pressure, P_2=\underline{\underline{793  kN / m ^2}},

The kinetic energy term =(G / A)^2 \ln (800 / 793)=7.91  kg ^2 / m ^4  s ^2
This is insignificant in comparison with 69,120  kg ^2 / m ^4 s ^2 which is the value of the other terms in equation 4.55.