Heat transfer Rate

[chocolatecat]

The equation for heat transfer rate, q=-kA (delta T/ delta x). Given that plastic with the same dimension as a metal has a low conductivity than the metal and the heat transfer rate in the materials would be different. Why was the coefficient 'k' thermal conductivity of a material added to the equation to make them equal?


We would discover that this proportionality is true when switching the material (for example, from metal to plastic).
We would also discover, though, that for equal values of A, x,

and T, the plastic's qx value would be lower than the metal's. This implies that by adding a coefficient that represents a measure of the material behavior, the proportionality may be transformed into equality. Therefore, we write qx=-k(delta T/delta x)

 

[jack action]

In mathematics, proportionality is presented as:
$$q \propto -A\frac{\Delta T}{\Delta x}$$
In physics, this is usually found by observation. It just means that if you begin with one value for the right side and one value for the left side, doubling one means that the other will double as well.

It also means that dividing one by the other will lead to a constant:
$$\frac{q}{-A\frac{\Delta T}{\Delta x}} = k$$
Because whenever I multiply one by $X_1$ or $X_2$, I also multiply the other by $X_1$ or $X_2$ leading to:
$$\frac{(X_n)q}{-(X_n)A\frac{\Delta T}{\Delta x}} = \frac{(X_n)}{(X_n)}\frac{q}{-A\frac{\Delta T}{\Delta x}} = (1)\frac{q}{-A\frac{\Delta T}{\Delta x}} = \frac{q}{-A\frac{\Delta T}{\Delta x}} = k$$
So now by introducing this constant $k$, we can set the equality between both sides, not just proportionality:
$$q = -kA\frac{\Delta T}{\Delta x}$$
This constant $k$ could be universal but, as in this case, it can englobe a myriad of factors that can be set constant based on simple criteria, like what type of material is used.

 

[Lnewqban]

Welcome, @chocolatecat !

Please, see:
https://www.tec-science.com/thermodynamics/heat/thermal-conductivity-fouriers-law/

https://www.thermal-engineering.org/what-is-fouriers-law-of-thermal-conduction-definition/