Air is cooled as it flows through a 30-cm-diameter duct. The inlet conditions are Ma1 = 1.2, T01 = 350 K, and P01 = 240 kPa and the exit Mach number is Ma2 = 2.0. Disregarding frictional effects, determine the rate of cooling of air.
The rate of cooling of air.
Answer to Problem 147RP
The required rate of cooling of air is
Explanation of Solution
Write the formula of ratio of stagnation temperature to the static temperature at inlet of the duct.
Here, the inlet static temperature is
Write the formula of ratio of stagnation pressure to the static pressure at inlet of the duct.
Here, the actual (static) pressure at the inlet of duct is
Write the formula for inlet density of air.
Here, the pressure of air at the inlet is
Write the formula for velocity of sound at the inlet conditions.
Here, speed of sound at the inlet condition is
Write formula for the velocity of air at inlet.
Write the formula for mass flow rate of air with inlet conditions of air.
Here, the mass flow rate of air at the inlet is
Write the formula for stagnation temperature ratio of exit to inlet.
Write the formula for heat transfer rate
Here, the specific heat at constant pressure is
Refer Table A-34, “Rayleigh flow functions for an ideal gas with
The Rayleigh flow function of inlet stagnation temperature to the critical stagnation temperature corresponding to the inlet Mach number of
The Rayleigh flow function of exit stagnation temperature to the critical stagnation temperature corresponding to the exit Mach number of
Refer Table A-, “Molar mass, gas constant, and critical2point properties”.
The gas constant
Refer Table A-2, “Ideal-gas specific heats of various common gases”.
The specific heat ratio
Conclusion:
Substitute
Substitute
Substitute
Equation (III).
Substitute
The cross sectional area
Substitute
Equation (V).
Thus, the rate of cooling air is
Substitute
Substitute
Here, the negative sign indicates that the air requires cooling in order to be accelerated.
Thus, the required rate of cooling of air is
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Chapter 17 Solutions
Thermodynamics: An Engineering Approach
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