Then I had an idea for a simulation game, in which the temperature of the building would effect agent happiness. So I did a prototype in order to find a good way of representing heat flow, that would look "realistic" enough for a game, but still be fast enough to do in real time. What I came up with is not really physically accurate, but it looks nice and behaves in a way people probably expect.
It essentially works by splitting the world into discreet tiles, each with a specific heat capacity and temperature. Every step the average of the temperatures around a cell weighted by the specific heat capacity is taken to be the new heat of the cell. In the case of a heating or cooling element (which can have any temperature) the new heat will always be the heat that the element is set to, but the average heat is used to determine how much energy the heating element requires. the following are annotated screenshots of this prototype. 
