There are two rooms completly isolated from the outer world (a closed system, no heat losses). They are connected with each other through a ventilation tube. Temperature in room A is 32 degrees Celsious (for example), temperature in room B is 15 degrees Celsious. Ventilator in the tube starts to transfer hot air from room A to room B. I need to make an equation which would describe the process of heating room B in time. My later plans are to model this in MATLAB.
As far as I understand the equation should include volumes of the rooms, speed of the air flow, etc.
I myself have little knowledge of these things, any help, links, clarifications will be highly appreciated!
Thanks!
Update 1: I imagine MATLAB model as a combination of room A, B, and the ventialtor between them. Room A has a volume, and temp, room B as well, during each point in time after ventilator starts working with a certain speed I would see drop of temperature in room A and an increase in room B.
Update 2: There should also be a second tube with a ventilator that would take cold air from room B to room A.
Best Answer
If you have a volume of air $V$ at temperature $T_B$, then you replace a part of that air with air of volume $\Delta V$ and temperature $T_A$, then the new average temperature is a weighted average of the temperatures of the room's air and the new air.
$$T_B(t+\Delta t) = \frac{(V-\Delta V) T_B(t) + \Delta V T_A(t)}{V}$$
We get a symmetrical expression for the air in the other room:
$$T_A(t+\Delta t) = \frac{(V-\Delta V) T_A(t) + \Delta V T_B(t)}{V}$$
Simplifying...
$$T_B(t+\Delta t) - T_B(t) = \frac{\Delta V}{V} (T_A(t) - T_B(t))$$ $$T_A(t+\Delta t) - T_A(t) = \frac{\Delta V}{V} (T_B(t) - T_A(t))$$
If we divide both sides by the time interval $\Delta t$ it takes for this volume $\Delta V$ to transfer,
$$\frac{T_B(t+\Delta t) - T_B(t)}{\Delta t} = \frac{\Delta V}{\Delta t} \frac 1 V (T_A(t) - T_B(t))$$ $$\frac{T_A(t+\Delta t) - T_A(t)}{\Delta t} = \frac{\Delta V}{\Delta t} \frac 1 V (T_B(t) - T_A(t))$$
That is an equation you can model in MATLAB. Of if you prefer, you can take the limit as $\Delta t$ approaches zero of both equations, and deal with it as a differential equation:
$$\frac{dT_B}{dt} = \frac Q V (T_A(t) - T_B(t))$$
$$\frac{dT_A}{dt} = \frac Q V (T_B(t) - T_A(t))$$
Where $Q$ is the volumetric flow rate $\frac{dV}{dt}$.