First of all I would like to emphesize that I am not a physicist, our professor gave us this exercise to train our ability of making differential equations.

Since it is related to physics only, I have few questions.

Let $t$ be time, $T(t)$ - temperature inside the building at time $t$, $T_1(t)$ - temperature outside. Speed of heat loss from the building is given by $k_1(T_1(t)-T(t))$ in $[\frac{J}{sec}]$, where $k_1 > 0$ is constant. Speed of heat transfer into building (central heating) is given by $q + k_2(T_0-T(t))$, where $q,k_2>0$ are constants. $T_0$ is fixed temperature (intepretation: $q$ is a speed of heat transfer into the building, such that for fixed outside temperature, the temperature inside remains constant and equal to $T_0$). Let $c$ be heat capacity of the building. Find the differential equation which describes relation between temperature $T$ and time $t$.

Hint: Due to conservation of energy, amount of heat transfered into the building by central heating minus heat loss, between time $t$ and $t+\Delta t$ is given by $c \cdot (T(t+\Delta t) - T(t))$

Since I am not a physicist, I was thinking like this: Given were two rates of change energy in time, firstly the speed of heat loss, secondly the speed of heating up, both in $energy/time$. The total heat transfer in time interval $\Delta t$, for me, will be total speed of heat transfer (given - lost) times time interval $\Delta t$ (Is it correct?).

Then I thought it would be ok to compare $c \cdot (T(t+\Delta t) - T(t))$ with $[k_1(T_1(t)-T(t))+q + k_2(T_0-T(t))]\cdot \Delta t$ . Then, as a mathematician I would like to divide both sides by $\Delta t$ and just let it converge to zero. Is it even close?

Another question I have is about $T_1(t)$ temperature outside. In the heat loss rate it is given by function of time, but in the heat transfer into the building it is said (interpretation given) that we consider fixed outside temperature. How we relate these two?

Moreover, what physics law describes such situation? I bet that given exercise is just one case, so how can we generalize relationship between the speed of temperature change and heat transfer?


I think everything you considered so far is correct except you need to use the mass of the building (that of air and wall and maybe furnitures, an averaged mass is ok but you also need an averaged heat capacity). You don't have to be a physicist to do this. The differential equation is close.

The physics behind the equations is that the building temperature change is due to the net heat energy it receives. The net energy is the energy from central heating system minus the heat loss (plus some heat from Solar radiation etc.).

The outside temperature varies during a day. For the simplicity, you can assuming it is constant. For example, from 11:00am to 2:00pm, it is fair to assume it is constant. To make the differential equation more useful, you can measure the outside temperature by every hour and piece wise fit a curve for you equation.


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