A rod of length l (laterally thermally insulated) of uniform...

A rod of length $l$ (laterally thermally insulated) of uniform cross-sectional area $A$ consists of a material whose thermal conductivity varies with temperature as $K=\frac{K_{o}}{a+b T}$, where $K_{0}, a$ and $b$ are constants. $T_{1}$ and $T_{2}\left(<T_{1}\right)$ are the temperature of two ends of rod. Then rate of flow of heat across the rod is (A) $\frac{A K_{0}}{b l}\left(\frac{a+b T_{1}}{a+b T_{2}}\right)$ (B) $\frac{A K_{0}}{b l}\left(\frac{a+b T_{2}}{a+b T_{1}}\right)$ (C) $\frac{A K_{0}}{b l} \ln \left[\frac{a+b T_{1}}{a+b T_{2}}\right]$ (D) $\frac{A K_{0}}{a l} \ln \left[\frac{a+b T_{2}}{a+b T_{1}}\right]$

A rod of length $l$ (laterally thermally insulated) of uniform cross-sectional area $A$ consists of a material whose thermal conductivity varies with temperature as $K=\frac{K_{o}}{a+b T}$, where $K_{0}, a$ and $b$ are constants. $T_{1}$ and $T_{2}\left(<T_{1}\right)$ are the temperature of two ends of rod. Then rate of flow of heat across the rod is
(A) $\frac{A K_{0}}{b l}\left(\frac{a+b T_{1}}{a+b T_{2}}\right)$
(B) $\frac{A K_{0}}{b l}\left(\frac{a+b T_{2}}{a+b T_{1}}\right)$
(C) $\frac{A K_{0}}{b l} \ln \left[\frac{a+b T_{1}}{a+b T_{2}}\right]$
(D) $\frac{A K_{0}}{a l} \ln \left[\frac{a+b T_{2}}{a+b T_{1}}\right]$

Key Concepts

Fourier's Law of Heat Conduction

This is the fundamental principle that describes the conduction of heat through a material. It states that the heat flux is proportional to the negative temperature gradient and the material's thermal conductivity. In mathematical terms, the steady state one-dimensional heat flow is given by q = -K * A * (dT/dx), which forms the basis for analyzing how heat flows in materials under a temperature difference.

Temperature-Dependent Thermal Conductivity

In many realistic scenarios, the thermal conductivity of a material varies with temperature. Instead of being a constant, it can be a function K(T). This adds complexity to the analysis because the rate of heat flow now changes along the material as the temperature changes, which requires modifying the standard Fourier’s law to integrate over the temperature gradient appropriately.

Steady-State Heat Conduction

This concept involves the assumption that the temperature distribution along the rod remains constant in time, meaning that the net heat flow into any segment of the rod equals the net flow out. Under steady-state conditions, the temperature profile does not change with time, allowing for the integration of variable properties such as temperature-dependent thermal conductivity to obtain a relationship between the heat flow and the boundary temperatures.

Integration of Differential Equations in Conduction Analysis

When properties like thermal conductivity depend on temperature, differential equations governing heat conduction become non-linear and typically require integration over the spatial or temperature domain. By separating variables and integrating, one can derive expressions that relate the heat flow rate to the boundary temperatures and other parameters, which is crucial for handling problems where material properties vary along the temperature gradient.

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