Thermal resistance is one of the most useful concepts in thermal engineering. Understanding both the theory behind it and the practical application of thermal resistance allows engineers to simplify complex heat transfer problems, identify limiting factors in a design, and make better decisions faster.
What Is Thermal Resistance
Thermal resistance is a convenient way of analyzing heat transfer problems using an electrical analogy, making complicated systems easier to visualize. It is based on Ohm’s law:
In Ohm’s law, “V” is the voltage that drives a current of magnitude “I”. The amount of current that flows for a given voltage is proportional to the resistance (Relec). For an electrical conductor, resistance depends on material properties (copper has lower resistance than wood, for example) and physical configuration (thick, short wires have less resistance than long, thin ones).
For one-dimensional, steady-state heat transfer problems with no internal heat generation, heat flow is proportional to a temperature difference:
Where Q is the heat flow, k is the thermal conductivity of the material, A is the area normal to heat flow, Δx is the distance heat travels, and ΔT is the driving temperature difference.
By treating electrical current as analogous to heat flow and voltage as analogous to the temperature difference driving that flow, the heat flow equation takes a form similar to Ohm’s law:
Here, Rth is the thermal resistance, defined by the thermal resistance equation:
Application of Thermal Resistance
The application of thermal resistance becomes clear when you consider a practical example. Suppose we want to calculate heat flow through a wall made of three different materials, knowing the surface temperatures TA and TB , and the material properties and geometries.
We could write the conduction equation for each material:
That gives us three equations and three unknowns: T1, T2, and Q. Using the thermal resistance analogy, however, we can solve for Q in a single step:
where:
Combining Thermal Resistances
The example above shows how to combine multiple thermal resistances in series, which follows the same structure as the electrical analog:
Thermal resistances can also be combined in parallel, or in series and parallel together:
Beyond Conduction
So far, thermal resistance has been applied to conduction through a plane wall. For steady-state, one-dimensional problems, other modes of heat transfer can be expressed in the same format. Newton’s Law of Cooling for convection, for example:
where Q is the heat flow, h is the convective heat transfer coefficient, A is the area over which heat transfer occurs, Ts is the surface temperature, and Tinf is the free-stream fluid temperature. The thermal resistance for convection is:
For radiative heat transfer from a gray body:
where ε is the surface emissivity, σ is the Stefan-Boltzmann constant, Ts is the surface temperature, and Tsurr is the temperature of the surroundings. Factoring the temperature expression, the thermal resistance for radiation becomes:
Advantage: Easy Problem Setup
Thermal resistance formulations simplify the setup of complex problems. Consider calculating heat flow from a liquid stream through a composite wall to an air stream, with both convection and radiation on the air side. If material properties, heat transfer coefficients, and geometry are known, the equation setup is direct:
Solving this particular problem may require iteration, since the radiative thermal resistance contains the surface temperature. The setup itself, however, is straightforward.
Advantage: Problem Insight
One of the most valuable aspects of the application of thermal resistance is the visibility it gives into which components of a system actually control heat transfer and which are negligible. Take the composite wall example with the following known values:
Liquid side: 20 K/W. Plastic layer (1 mm): 40 K/W. Steel layer (2 mm): 0.5 K/W. Convection to air: 200 K/W. Radiation to surroundings (emissivity = 0.5): 2,500 K/W.
From these values, several conclusions become immediately clear. Since radiation resistance is in parallel with a much smaller convection resistance, its effect on the overall value is small. Increasing emissivity to 1.0 would improve total thermal resistance by only 5%. Ignoring radiation entirely introduces an error of just 6%.
The steel layer, in series but tiny relative to the other resistances, has almost no influence regardless of material. Replacing steel with pure copper, for instance, would improve overall thermal resistance by only 0.2%. The same logic applies to aluminum thermal resistance in similar conduction-dominated layers: when the convective resistance dominates by two orders of magnitude, base material choice has little effect on system-level performance.
The dominant resistance in this example is convection on the air side. Doubling the convection coefficient, by increasing air velocity, for instance, would reduce overall thermal resistance by 36% on its own. That kind of insight is exactly what the thermal resistance framework is built to provide.
Beyond Plane Wall Conduction
Thermal resistance applies to other conduction geometries as well, as long as they can be treated as one-dimensional. For a cylindrical geometry:
where L is the axial length of the cylinder, and r1 and r2 are the inner and outer radii.
For a spherical geometry:
with r1 and r2 as inner and outer radii.
Let’s Build Your Solution
Thermal resistance is a powerful tool for analyzing problems that can be approximated as one-dimensional, steady-state, and free of internal heat generation. It simplifies complex setups, exposes the components that actually drive system performance, and supports faster, more informed design decisions.
Contact Celsia with your next thermal design challenge. We specialize in the design and manufacture of heat sinks using two-phase devices: heat pipes and vapor chambers.
