Suppose the chunks have temperatures T1, T2 at time t and...

Suppose the chunks have temperatures $T_{1}, T_{2}$ at time $t$ and $T_{1}-d T_{1}, T_{2}+d T_{2}$ at time $d t+t .$ Then $\quad C_{1} d T_{1}=C_{2} d T_{2}=\frac{\kappa S}{l}\left(T_{1}-T_{2}\right) d t$ Thus $\quad d \Delta T=-\frac{\kappa S}{l}\left(\frac{1}{C_{1}}+\frac{1}{C_{2}}\right) \Delta T d t$ where $\Delta T=T_{1}-T_{2}$ Hence $\quad \Delta T=(\Delta T)_{0} e^{-t / \tau}$ where $\frac{1}{\tau}=\frac{\kappa s}{l}\left(\frac{1}{C_{1}}+\frac{1}{C_{2}}\right)$

Suppose the chunks have temperatures $T_{1}, T_{2}$ at time $t$ and $T_{1}-d T_{1}, T_{2}+d T_{2}$ at time $d t+t .$
Then $\quad C_{1} d T_{1}=C_{2} d T_{2}=\frac{\kappa S}{l}\left(T_{1}-T_{2}\right) d t$
Thus $\quad d \Delta T=-\frac{\kappa S}{l}\left(\frac{1}{C_{1}}+\frac{1}{C_{2}}\right) \Delta T d t$ where $\Delta T=T_{1}-T_{2}$
Hence $\quad \Delta T=(\Delta T)_{0} e^{-t / \tau}$ where $\frac{1}{\tau}=\frac{\kappa s}{l}\left(\frac{1}{C_{1}}+\frac{1}{C_{2}}\right)$

Key Concepts

Thermal Conductivity

Thermal conductivity is a material property that measures how efficiently heat is transferred through a substance. It plays a crucial role in the analysis of heat flow between bodies, as it quantifies the rate at which thermal energy is conducted per unit area and per unit temperature difference, ultimately affecting how quickly temperature differences are equalized.

Heat Capacity

Heat capacity is the measure of the amount of heat required to change the temperature of a system by a given amount. In thermal processes, it governs how much the temperature of a body changes when energy is transferred to or from it, and it appears in equations that describe the dynamic response of a system to heat exchange.

Exponential Decay

Exponential decay refers to a process in which a quantity decreases at a rate proportional to its current value. In the context of thermal processes, it describes how the temperature difference between two interacting bodies decreases over time, leading to a rapid initial change that slows as equilibrium is approached.

Time Constant in Thermal Processes

The time constant is a parameter that characterizes the speed at which a thermal system responds to heat exchange, indicating the time required for the system to significantly reduce the temperature difference by a set fraction. It combines material properties, heat capacities, and geometric factors, providing a concise representation of the system's thermal response.

Differential Equations in Thermal Processes

Differential equations are used to mathematically model the time evolution of temperature distributions in thermal systems. They express the relationship between the rate of change of temperature and the driving forces, such as heat conduction, thereby providing a framework to predict the system's dynamic behavior over time.

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