Problem 4.6.3.

Nitrogen is flowing through a 1.8-in diameter pipe at constant temperature of 80 degrees Fahrenheit. Inlet and exit pressures are 200 psi and 160 psi, respectively. What is the pipe length if mass flow rate is 3 pounds per second? k is 1.4.

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 nitrogen this value 55.2 ft lb-force per pound-mass per degrees Fahrenheit.

Velocity of sound is calculated as 1159 feet per second. Cross-sectional area of the pipe is 0.01777 square foot. Mass velocity is w over A or 169.77 pounds per square foot per second.

Density of nitrogen at inlet is p₁ over RT. Plugging in values of pressure (200 time 144 psf), gas constant (55.2 ft lb-force per pound-mass per degrees Fahrenheit) and temperature (540 degrees Rankine), density is found to be 0.966 pounds per cubic foot.

Velocity at pipe inlet is 169.77 over 0.966 or 175.5 foot per second. Mach number at entering condition is equal to 175.7 over 1159.2 or 0.15. 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 28.36. Now maximum length, L_max is N d over f or 265 feet.

Density of nitrogen at exit is p₂ over RT or 0.7728 pounds per cubic foot. So velocity at the end of the pipe is 169.7 over 0.7728 or 219.6 feet per second. Mach number at exit is 0.15 time 200 over 160 or 0.1875. Plugging in the value of M₂ and k, N is found from the complicated function as 17. This corresponds to a maximum length is N₂ d over f or 159.4 feet. So length of the pipe is 265 minus 159.4 or 106 feet.