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Temperature Gradient

The temperature gradient is a fundamental concept in thermodynamics and heat transfer that quantifies the rate of change of temperature with respect to distance. It describes how the temperature of a substance changes as we move from one point to another within it. In other words, it represents how quickly or slowly temperature varies across a given distance. Before discussing temperature gradient, let us first discuss isothermal surfaces.

Isothermal surface :- Any surface (within a conductor) having its all points at the same temperature, is called an isothermal surface. The direction of flow of heat through a conductor at any point is perpendicular to the isothermal surface passing through that point. Different types of isothermal surfaces are shown in figure below :-

Temperature Gradient :- The rate of change of temperature with distance between two isothermal surfaces is called temperature gradient.

Mathematical Expression of Temperature Gradient

$\displaystyle \frac{d\theta }{dx}=-\frac{\Delta \theta }{\Delta x}$

Note :-

The negative sign shows that temperature decreases as the distance x increases in the direction of heat flow.
For Steady state condition(uniform temperature fall), $\displaystyle \left| \frac{\Delta \theta }{\Delta x} \right|=\frac{{{\theta }_{1}}-{{\theta }_{2}}}{l}$
S.I. Unit : K/m or °C/m
Dimensions :- [L^-1θ]

To find temperature at a distance x from the hotter end

(Temperature Gradient)

Consider a metallic rod in steady state having temperature of end points as T_H and T_L(T_H >T_L). Suppose we have to find out temperature at a distance x from the hotter end.

As the rod is in steady state, the temperature gradient is given by :-

$\displaystyle \frac{dT}{dx}=-\left( \frac{{{T}_{H}}-{{T}_{L}}}{l} \right)$

The negative sign shows that temperature decreases as the distance x increases in the direction of heat flow.

$\displaystyle \Rightarrow dT=-\left( \frac{{{T}_{H}}-{{T}_{L}}}{l} \right)dx$

$\displaystyle \Rightarrow \int\limits_{{{T}_{H}}}^{T}{dT}=-\left( \frac{{{T}_{H}}-{{T}_{L}}}{l} \right)\int\limits_{0}^{x}{dx}$

$\displaystyle \Rightarrow \left[ T \right]_{{{T}_{H}}}^{T}=-\left( \frac{{{T}_{H}}-{{T}_{L}}}{l} \right)\left[ x \right]_{0}^{x}$

$\displaystyle \Rightarrow T-{{T}_{H}}=-\left( \frac{{{T}_{H}}-{{T}_{L}}}{l} \right)\left( x-0 \right)$

$\displaystyle \Rightarrow T={{T}_{H}}-\left( \frac{{{T}_{H}}-{{T}_{L}}}{l} \right)x$

Temperature(T) v/s position(x) graph :-

Example.

The temperature of end A of a rod is maintained at 0°C. The temperature of end B is changing slowly such that the rod may be considered in steady state at all time and is given by T_B = αt ; where α is positive constant and t is time. Temperature of point C, at a distance x from end A, at any time is ____.