# 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. Jun 22, 2015 at 21:15
• Would a small room be cooler on the whole? Jun 22, 2015 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. Jun 23, 2015 at 4:00

## 2 Answers

As I said in the comments, you need to define what you mean by "easier".

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

$$T_{out}=\frac{P}{k}-T_{in}$$

Since all the stuff on the right hand side are fixed numbers, we conclude that the temperature just near the outgoing fan is also fixed.

In a bigger room, the steady temperature distribution that has this temperature on the wall will have a larger temperature in the middle. In this sense, the larger room will be "hotter". Whether this means "easier" I cannot tell.

Define "well insulated", do you mean no heat at all escapes, or comes in from outside? Which is physically impossible, so the smaller the room volume, the less heat will escape. I am assuming you are pulling in exactly as much heat as you are pushing out per unit time. Smaller is better for keeping warm, larger is better for keeping cool.
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