# Is a larger, heated space easier or harder to ventilate if the heat source and ventilation are the same?

Imagine a well insulated room with a heat source inside. The room has two vents, which each have a fan. Cool air is drawn into the room through one vent, and warm air is pushed out the other vent. If all the specifics of the heat source and ventilation system remain the same, would the room be easier to keep cool if it were smaller or larger in volume?

• Define "easier". The fans will do the same work in both cases. – yohBS Jun 22 '15 at 21:15
• Would a small room be cooler on the whole? – Carl Smith Jun 22 '15 at 21:20
• You know, the more I looked at this, the more I realized it was more complex than I first thought. It requires setting several parameters and making some assumptions. I'd be curious to see how folks analyze it. – Terry Bollinger Jun 23 '15 at 4:00

The total heat flux out (or in) to the room is $$P + k(T_{in}-T_{out})$$ Where $T_{in/out}$ are the temperatures of the incoming and outgoing air respectively, $P$ is the heat flux of the heat source, and $k$ is a coefficient that has to do with the heat capacity of air, the flow speed, etc. The room will reach a steady (=time independent) temperature distribution when the total flux vanishes (actually that's only a necessary condition, but forget about this for now). If you assume that the incoming air is at a fixed temperature (because the environment outside is too big to be affected by the room) then this happens when
$$T_{out}=\frac{P}{k}-T_{in}$$