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A banquet hall ($150 m^3$) is to be used for a formal dinner for 20 persons. Each person occupies $0.075 m^3$ of space and has an average heat transfer rate of $500 \frac{kJ}{hr}$. If the AC system fails, determine the increase in the internal energy and temperature of the air in the hall during the first 15 minutes of failure. Take the hall to be well insulated from outside. If the hall and all its contents are considered as system how much will be the increase in the internal energy of the system.

I am trying to use $Q= U + W$, don't have an idea about the data related to each person, why it is given. Thanks, please help.

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  • $\begingroup$ What is your assessment so far? $\endgroup$ – Chet Miller Sep 27 at 14:21
  • $\begingroup$ If the room is initially at 20 C, and the pressure is 1 bar, how many moles of air are there in the room? $\endgroup$ – Chet Miller Sep 27 at 14:25
  • $\begingroup$ @ chet miller sir, trying to use q=u +w, taking 500kj/hr X 15 as heat supplied, don't have idea about person data, I think work done should be zero here $\endgroup$ – Millie Sep 27 at 14:27
  • $\begingroup$ Correct and correct. So how much heat do the people add to the air during the 15 minutes. And how many moles of air are there in the room? $\endgroup$ – Chet Miller Sep 27 at 14:38
  • $\begingroup$ Mole of air was not given in the question sir $\endgroup$ – Millie Sep 27 at 14:39
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If the walls are rigid then the system is isolated the total change in internal energy of the system (humans +room air) would be zero.

But when we consider the air in the room as the system the humans are then the surroundings. Any work done by or on the air in the room is due to the expansion and contraction of the human chests during breathing and would cancel out for a net work of zero on the room air. Therefore for the air in the room $\Delta U=Q$ where $Q$ is the heat transfer from the humans to the air in the room.

If we can assume the humans inhale and exhale the same volume of gas then the number of moles of gas is in the room is constant and can be calculated based on the initial condition information given (T,P,V) and ideal gas law. Finally, $\Delta U=nc_{v}\Delta T$ for an ideal gas, which we can assume for the air even though its composition is changing slightly (increase in carbon dioxide and decrease in oxygen).

Hope this helps

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