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enter image description hereIf I give heat with a certain source will the change of temperature difference change the heat flow rate?

Suppose I have a aluminium rod which has a weight of 25 gram. I heated it up to 40 degree Celsius and kept it under 40 gram of room temperature water.Then I heated up another aluminium rod of same weight to 90 degree Celsius and kept it under the same amount of room temperature water. Will the heat flow rate from aluminium rod to water be same in those two cases.(Room temperature is 30 degree Celsius)

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Yes, heat flow depends on the temperature gradient.

Though the equation is not precise, many metals follow it quite accurately.

Take one face of a slab at $T_1$ & another at $T_2$ such that $T_2\lt T_1.$ The heat flow is proportional to the area $A$ of the faces & inversely proportional to $d$ the distance between the faces. This can be written as $$\begin{equation}J \propto \Delta T \frac{A}{d}\\ \implies J= \kappa(T_2 - T_1)\frac{A}{d}\end{equation} $$ where $$J=\; \text{thermal energy per unit time}\;\\ \& \\ \kappa= \;\text{proportionality constant}.$$

From this, we can easily deduce the relation for infinitesimal area cases viz:

$$\begin{align}\frac{d J}{d A}= \kappa \frac{dT}{ds} \\ \implies \bf{h}= -\kappa\vec{\nabla} \it{T}.\end{align}$$

The '$-$' sign indicates the flow of thermal energy opposite to the gradient.

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    $\begingroup$ for practical materials $ \kappa = \kappa (T) $ $\endgroup$ – hyportnex Oct 26 '15 at 11:13

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