Problem 4.6.2.

Air is flowing through a 3-in diameter pipe at constant temperature of 110 degrees Fahrenheit. At the entrance of the pipe, velocity is 320 feet per second and pressure is 30 psi. What is the maximum pipe length and pressure drop for this flow? How much heat is being removed to maintain isothermal flow in the pipe?

Solution:

For compressible fluids, velocity is conveniently represented by Mach number, which is a ratio of the velocity of fluid to velocity of sound at same temperature. Velocity of sound given by sqrt of k R g_c T where k is the ratio of specific heat at constant pressure and specific heat at constant volume. R is the gas constant expressed as per unit mass basis. For air this value 53.3 ft lb-force per pound-mass per degrees Fahrenheit.

Velocity of sound is calculated as 1170.3 feet per second. Mach number at entering condition is equal to 320 over 1170.3 or 0.273. Maximum pipe length is a function of N, friction factor f, and diameter d of the pipe. N is a parameter that is a function of k, and Mach number M. With k as 1.4, and M 0.273, N is 6.324. Now maximum length, L_max is N d over f or 87.8 feet.

Maximum Mach number which could be achieved under isothermal conditions is equal to one over sqrt k or 0.845. So velocity at the end of the pipe is 0.845 times 1170.3 or 989 feet per second. Under isothermal conditions, p₂ over p₁ is equal to M₁ over M₂. So pressure at the end of the pipe is 0.273 over 0.845 times 30 or 9.69 psi. Pressure drop is equal to 30 minus 9.969 or 20.31 psi.

Heat that needs to be provided is (V₂² – V₁²)/2g_c or 12,598 foot pound-force per pound-mass or 17.47 Btu per pound.