Question

Air flows through a constant area duct. The pressure and temperature of the air at the inlet to the duct are P1 = 100 kPa absolute, and T1 = 298 K, respectively. Inlet Mach number is M1 = 0.1. Heat is transferred to the air as it flows through the duct and as a result the Mach number at the exit increases.

a) Find the pressure and temperature at the exit, while the exit Mach number changes between M=0.2 to 0.99 with

increments of ?M=0.01.

b) Find the amount of heat that is transferred to the air per unit mass of air.

Answer 1

Given, P_1 = 100 K p_a = 100,000 P_a T_1 = 298 K M_1 = 0.1 = 33.13 M/S (V_1) M_2 = 0.2 = 66.26 M/S Now using Gas Equation P_1 V_1 = M R T or M/V_1 = P_1/R T-1 or rho _1 = 100,00/287 times 298 = 1.169 kg/m^3 Now using continuity Equation, Inlet mass flow Rate = outlet mass flow rate or rho_1 A V_1 = rho _2 A V_2 or rho_2 = rho_1 V_1/V_2 = 0.5845 kg/m^3 So p_2 = rho_2 times R times T_2 or, P_2 = 0.5845 times 287 times T_2 = 167.7515 T_2 (1)\r\n Using Bernoulli's Equation Energy balance, P_1/rho_1 g + V_1^2/2g + Q/g = P_2/rho_2 + V_2^2/2g = [Z_1 = Z_2] or 8719.999 + 55.427 + Q/g = P_2/0.5845g + 2195.193/g or 86091.988 + Q = 1.710 P_2 + 2195.193 OR, Q = 1.710 P_2 - 83,896.795 (11) Now using Thermodynamic relation for flow of fluid\r\n