Compressed natural gas is fed into a small underground pipeline having an inside diameter of 5.9 in. The gas enters at an initial temperature of 120°C and constant pressure of 132.3 psig. The metered volumetric flow rate at standard pressure of 14.7 psia and standard temperature of 25°C is 195.3 ft³/min. The ground temperature is constant at 15°C, as it serves as an ‘infinite sink” for heat transfer. 

a. Show that the mass flow rate is 0.0604 kg/sec. volumetric flow rate is 0.01 m³/sec at 50°C and 10 atm. the total system pressure is 10 atm, and the gas density, assuming ideal gas behavior, is 6.04 kg/m³ at 50°C. Other properties (not dependent on pressure) for methane at 50°C are k = 0.035 W/m · K, C_p = 220 J/kg · K. and µ = 1.2 ×10^-5 kg/m-sec. 

b. Develop an energy balance model to predict the steady-state temperature profile of the natural gas in the pipe.

c. Is the gas flow laminar, or turbulent?

d. What are the Prandtl, Nusselt, and Stanton numbers for methane in the pipe? What is a reasonable average temperature for estimation the thermophysical properties? 

e. What is the heat-transfer coefficient for methane in the pipe?

f. Finally, how far from the pipe entrance will the methane gas reach a temperature of 50°C? How much heat is transferred at this point?