# How to model temperature rise of a room with walls of thermal mass and thermal conductivity?

Is this a reasonable approach?

I want to model change of room temperature based on a change in outside temperature.

Room walls have thermal conductivity $$k$$ and thermal mass $$C_p M$$

Where $$C_p$$ is wall heat capacity and $$M$$ is wall mass.

Heat or energy flow is found by $$W = {k A \over d} (T_o-T_i)$$

Where $$T_o$$ is outside temperature, $$T_i$$ is inside temperature, $$k$$ is thermal conductivity of walls $$W \over m K$$, $$A$$ is wall area, and $$d$$ is wall thickness.

Ignoring ceiling and floor, total energy added to room in $$s$$ seconds is $$E=W s$$ joules

Room walls also have a thermal mass $$C_{th}$$ - meaning they absorb energy and $$C_{th} = C_p M$$

Therefore temperature rise of room (ignoring air) is $$\delta K = {W s/1000 \over C_{th}}$$

• Do you want to model the transient (time-dependent) changes in room temperature, or do you just want to model the change after the room has again re-equilibrated with the outside temperature? – Chet Miller Mar 6 at 12:35
• @Chester Miller, Transient: because I'd also like to place energy emitters in room as well as either ventilation or heat pumps to cool room. I'd like to 'see' impact over time with varying external temperatures to evaluate failure of heat pumps (for example). – philcolbourn Mar 6 at 20:21
• To handle the external walls, you are going to have to solve the transient heat conduction equation(a partial differential equation), or have an approximation to the internal heat transfer coefficient. There are also external heat transfer coefficients for heat exchange with the room air and the outside air. What I'm saying is that it takes some experience to do this kind of thing. – Chet Miller Mar 6 at 23:07