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So how is it that the brown plate increases in temperature if the white plate is always colder and there is never any heat flux from the gray to the brown plate?

So how is it that the brown plate increases in temperature if the gray plate is always colder and there is never any heat flux from the gray to the brown plate?

This set of two ordinary differential equations for $\bar{T_b}$ and $\bar{T_g}$ have a single parameter $C_g/C_b$ that must be specified along with initial temperatures for each plate. Let’s consider the case where the gray plate is gone and the brown plate has reached its steady state temperature of $\bar{T_b}=1$, and at time $\bar{t}=0$ the gray plate is introduced at a temperature of $\bar{T_g}=0$. Let’s also take $C_g/C_b = 1$. For this case the time evolution of the temperatures of the brown and gray plates is shown below. Note that the steady state temperatures for the plates are $\bar{T_b}=2^{1/4} T_0$ and $\bar{T_g} = T_0$.

This set of two ordinary differential equations for $\bar{T_b}$ and $\bar{T_g}$ have a single parameter $C_g/C_b$ that must be specified along with initial temperatures for each plate. Let’s consider the case where the gray plate is gone and the brown plate has reached its steady state temperature of $\bar{T_b}=1$, and at time $\bar{t}=0$ the gray plate is introduced at a temperature of $\bar{T_g}=0$. Let’s also take $C_g/C_b = 1$. For this case the time evolution of the temperatures of the brown and gray plates is shown below. Note that the steady state temperatures for the plates are $T_b=2^{1/4} T_0$ and $T_g = T_0$.