Different definitions for Thermal Resistance

Question

I have been reading on the topic of thermal resistance and I came across these two different definitions multiple times.

The first one: $R = \Delta T/\dot{Q} = L/kA $

and the second one $R = \Delta T/\dot{q} = L/k$

I am just guessing the definition will change depending on what we use to solve the problem, either q or Q. But it strikes me as odd, since in electric circuits the units are always Ohms.

Answer 1

As already noted by @Chet Miller, $\dot Q=\dot q A$. Therefore the first equation can be written as

$$R=\frac{\Delta T}{\dot Q}=\frac{\Delta T}{\dot q A}=\frac{L}{kA}$$

making it identical to the second equation when multiplying each side by $A$.

But it strikes me as odd, since in electric circuits the units are always Ohms.

$R$ in these equations is thermal resistance with units of Kelvins per watt ($K/W$). The electrical analog for steady heat transfer through a wall is Ohms law which gives the relationship between current ($I$) voltage ($V$) and resistance ($R$), where $R$ is electrical resistance with units of Ohms ($\Omega$).

With regard to the electrical analogs for the other parameters:

1. $k$ is thermal conductivity with units watts per meter Kelvin ($W/(m\cdot K$)). The electrical analog is electrical conductivity with units ohm meters ($\Omega\cdot m$).

2. $\Delta T$ is the temperature difference between the two walls in degrees Kelvin ($K$). The electrical analogy is potential difference if volts ($V$), or Joules per Coulomb ($J/C$)

3. $\dot Q$ is heat transfer rate in watts, or Joules per second ($J/s$). The electrical analogy is current in amperes ($A$) or Coulombs per second ($C/s$)

4. $\dot q$ is heat flux in watts per square meter ($W/m^2$). The electrical analog is current density in amperes per square meter ($A/m^2$).

Hope this helps.