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(Again, repeating the previous equation) $$\Delta x\Delta y\frac{\partial T}{\partial t} = \Delta y\alpha\left(\frac{\partial T}{\partial x}\right)_e - \Delta y\alpha\left(\frac{\partial T}{\partial x}\right)_w + \Delta x\alpha\left(\frac{\partial T}{\partial y}\right)_t - \Delta x\alpha\left(\frac{\partial T}{\partial y}\right)_t + S\Delta x\Delta y.$$
In term (1) we apply the conservation law, which is the first law of thermodynamics: $$\frac{dE}{dt} = \dot{Q} + \dot{W} + S\Delta x\Delta y.$$
$$\dot{Q} = -\int_{CS}\vec{q}\cdot\vec{n}d\mathcal{A} = -\left[\int_b() + \int_t() + \int_e() + \int_w()\right] = -[q_e\Delta y - q_w\Delta y + q_t\Delta x - q_b\Delta x].$$
Now, substitute the terms back in (and flipping the equation order):
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