Concept explainers
A building with an internal volume of 400 m3 is to be heated by a 30-kW electric resistance heater placed in the duct inside the building. Initially, the air in the building is at 14°C, and the local atmospheric pressure is 95 kPa. The building is losing heat to the surroundings at a steady rate of 450 kJ/min. Air is forced to flow through the duct and the heater steadily by a 250-W fan, and it experiences a temperature rise of 5°C each time it passes through the duct, which may be assumed to be adiabatic.
- (a) How long will it take for the air inside the building to reach an average temperature of 24°C?
- (b) Determine the average mass flow rate of air through the duct.
FIGURE P5–173
(a)
The time taken to attain the building’s average temperature of
Answer to Problem 173RP
The time taken to attain the building’s average temperature of
Explanation of Solution
Consider the entire building as system and the air circulates the in the building itself. There is no leakage to the surrounding.
The air flows at steady state through one inlet and one exit system (pipe and duct flow). Hence, the inlet and exit mass flow rates are equal.
Write the energy balance equation.
Here, the heat transfer is
In this system two work inputs are involved namely, the work input to the electric heater
The Equations (I) reduced as follows.
Here, there is no mass leakage from the building to the surrounding. The mass of air circulates in the building itself. Hence, inlet and exit enthalpies are neglected.
The change in internal energy is expresses as follow.
Here, the specific heat at constant volume is
Neglect the inlet and exit enthalpies and substitute
Equation (II).
Express the Equation (III) with respect to change of time and rearrange it to obtain
Write the formula for mass of air
The mass flow rate
Here, the change in time or time interval is
Refer Table A-1, “Molar mass, gas constant, and critical-point properties”.
The gas constant of air
Refer Table A-2, “Ideal-gas specific heats of various common gases”.
The specific heat at constant volume
Conclusion:
Substitute
Substitute
Substitute
Thus, the time taken to attain the building’s average temperature of
(b)
The average mass flow rate of air through the duct.
Answer to Problem 173RP
The average mass flow rate of air through the duct is
Explanation of Solution
Consider the heating duct with fan and heater only as the system. The air passes through in it steadily.
The system is at steady state. Hence, the rate of change in net energy of the system becomes zero.
The heating duct is an adiabatic duct. Hence, there is no heat loss.
The Equations (II) reduced as follows.
Express the Equation (VII) with respect to change of time as follows.
The change in enthalpy is expresses as follow.
Here, the specific heat at constant pressure is
Substitute
Refer Table A-2, “Ideal-gas specific heats of various common gases”.
The specific heat at constant pressure
Conclusion:
It is given that the temperature rise is
Substitute
Thus, The average mass flow rate of air through the duct is
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Chapter 5 Solutions
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