# Why heat flux is not related with specific heat capacity?

Rate of heat transfer across a solid material kept between two environments of different temperature (Assume a wall of a house) is mentioned to be dependent on the temperature gradient, length, and thermal conductivity.

But we know that providing energy at same rate to two different substances having different specific heat capacities can result in the one having lower specifc heat capacity to feel hotter faster.

So, doesn't specific heat capacity of a material affect the rate of heat transfer/flux ?

• The key word is “faster.” During the initial transient period, the heat capacity does matter. But once the object reaches steady state, the temperature profile is no longer changing, and the heat capacity no longer matters. Sep 16, 2020 at 15:52
• Sep 16, 2020 at 17:46

So, doesn't specific heat capacity of a material affect the rate of heat transfer/flux ?

It does: just look at Fourier's equation of heat conduction:

$$\frac{\partial T}{\partial t}=\alpha \Big(\frac{\partial^2 T}{\partial x^2}+\frac{\partial^2 T}{\partial y^2}+\frac{\partial^2 T}{\partial z^2}\Big)$$ where $$T$$ is the temperature and: $$\alpha=\frac{k}{\rho c_p}$$

is the thermal diffusivity of the conducting material. $$k$$ is the heat conductivity and $$c_p$$ is the heat capacity of that material.

I'll illustrate how this equation works with a simple $$1D$$ example. A uniform rod of length $$L$$ is clamped with one end at $$T_1$$ and one end at $$T_2$$. The rod is insulated along its length. What's the temperature evolution $$T(x,t)$$ of the rod?

The solution is$$^{\dagger}$$ given by:

\begin{align*}T\left( {x,t} \right) & = {T_1} + \frac{{{T_2} - {T_1}}}{L}x + \sum\limits_{n = 1}^\infty {{B_n}\sin \left( {\frac{{n\pi x}}{L}} \right){{\bf{e}}^{ - \alpha{{\left( {\frac{{n\pi }}{L}} \right)}^2}\,t}}} \end{align*}

The thermal diffusivity $$\alpha$$ appears in the time-based exponential decay factors:

$$e^{-\alpha\Big(\frac{n\pi}{L}\Big)^2t}$$

So the higher the value of $$\alpha$$, the faster that function decays to $$\text{zero}$$, which is for lower values of $$c_p$$.

$$^{\dagger}$$ note that the author confusingly used $$k$$ where it should be $$\alpha$$.

• Then why is this equation (images.app.goo.gl/69xLfGt5Yuj2UGRS9) widely shown as heat transfer equation?
– Zam
May 31, 2021 at 7:50
• @Zam It is not 'widely shown as heat transfer equation'. It's a specific example of steady state heat conduction through a wall (plane of uniform thickness and composition)
– Gert
May 31, 2021 at 12:48