Concept explainers
A 4-m × 5-m × 6-m room is to be heated by an electric resistance heater placed in a short duct in the room. Initially, the room is at 15°C, and the local atmospheric pressure is 98 kPa. The room is losing heat steadily to the outside at a rate of 150 kJ/min. A 200-W fan circulates the air steadily through the duct and the electric heater at an average mass flow rate of 40 kg/min. The duct can be assumed to be adiabatic, and there is no air leaking in or out of the room. If it takes 25 min for the room air to reach an average temperature of 25°C, find (a) the power rating of the electric heater and (b) the temperature rise that the air experiences each time it passes through the heater.
(a)
The power rating of the electric heater.
Answer to Problem 100P
The power rating of the electric heater is
Explanation of Solution
Consider the entire room as system and the air circulates the in the room 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 room to the surrounding. The mass of air circulates in the room 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 power rating of the electric heater is
(b)
The temperature rise that the air experiences each time it passes through the heater.
Answer to Problem 100P
The temperature rise that the air experiences each time it passes through the heater 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:
Substitute
Thus, the temperature rise that the air experiences each time it passes through the heater is
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Chapter 5 Solutions
Thermodynamics: An Engineering Approach
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