# Different definitions for Thermal Resistance

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.

• Thermal resistance, a concept relying on a daft analogy, really only finds good use in the case of composite walls and such like. Then the thermal resistances are in series. engineeringenotes.com/thermal-engineering/heat-conduction/…
– Gert
Dec 1, 2021 at 19:31
• In the above definitions, you failed to consider that $\dot{Q}=\dot{q}A$. Dec 1, 2021 at 20:09

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.