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In general, the idea looks like this: you keep track of a "stuff" (in this case both air and energy) as it flows into and out-of some "perfectly mixed reactors" (in this case "reactor" means a tank where things which react are often stored). The "stuff" can be anything which obeys a conservation law in that context, so momentum and energy and volume (at ...

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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 ...

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Okay, the "cylinder" thing means that we can think of this as a 2D system; we can assume that the temperature of the cylinder has reached a steady state. Then there are three constraints at play here: Circular symmetry. The temperature of the disk is uniform as the disk is rotated about its center. Equilibrium. When you draw a circle around the center, the ...

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The units for $u$ should be $K$, $m$ and $s$ aren't involved here. The partial derivatives introduce the factor of $T$ and $L$, so it should read, in terms of the units, $$\frac Ks+\frac{m^2}{s}\cdot\frac{1}{m}\cdot\left(\frac{1}{m} \cdot K\right)=\frac Ks\tag{1}$$ which all the terms in the middle reduce to $K/s$. Moving the $\alpha$ to inside the ...

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