What Is Thermal Resistance?
Thermal resistance is a measure of how strongly a material or boundary layer opposes the flow of heat. The concept is directly analogous to electrical resistance — and this analogy is extremely useful for solving multi-layer heat transfer problems by building a thermal resistance network, similar to a circuit diagram.
The general definition of thermal resistance is:
R = ΔT / Q
Where R is thermal resistance (K/W), ΔT is the temperature difference (K), and Q is the heat flow rate (W).
Thermal Resistance Formulae for Each Mode
Conductive Resistance (Flat Wall)
R_cond = L / (k · A)
Where L is wall thickness (m), k is thermal conductivity (W/m·K), and A is cross-sectional area (m²).
Convective Resistance
R_conv = 1 / (h · A)
Where h is the convective heat transfer coefficient (W/m²·K).
Radiative Resistance
R_rad = 1 / (h_r · A), where the linearised radiation coefficient h_r = ε · σ · (T_s + T_surr)(T_s² + T_surr²) allows radiation to be included in the network for moderate temperature differences.
Cylindrical Conduction (Pipe Wall)
R_cyl = ln(r_o / r_i) / (2π · k · L)
Where r_o and r_i are outer and inner radii, and L is pipe length. This form is essential for insulated pipe calculations.
Series and Parallel Resistance Networks
Resistances in Series
When heat must pass through each layer sequentially (e.g. a wall with plasterboard, insulation, and brick), resistances add directly:
R_total = R_1 + R_2 + R_3 + ... + R_n
The same heat flow rate passes through each layer. This is the most common configuration for flat walls and pipe insulation calculations.
Resistances in Parallel
When heat can travel along multiple simultaneous paths (e.g. a wall with both insulated panels and structural steel studs), resistances combine as:
1/R_total = 1/R_1 + 1/R_2 + ... + 1/R_n
Steel studs, being far more conductive than insulation, can dominate the parallel network and dramatically reduce the effective insulation performance — a phenomenon known as thermal bridging.
Worked Example: Insulated Pipe
Consider a steam pipe with the following specifications:
- Inner radius: 50 mm, outer radius: 55 mm, steel (k = 50 W/m·K)
- Insulation: 40 mm thick mineral wool (k = 0.04 W/m·K)
- Inner convection coefficient h_i = 3000 W/m²·K
- Outer convection coefficient h_o = 10 W/m²·K (natural convection in air)
- Steam temperature: 200°C, ambient: 20°C, pipe length: 1 m
The four resistances in series per metre of pipe are:
- R_inner_conv = 1/(h_i · 2π · r_i · L) = 1/(3000 × 2π × 0.05 × 1) ≈ 0.00106 K/W
- R_pipe_wall = ln(0.055/0.05)/(2π × 50 × 1) ≈ 0.0000303 K/W
- R_insulation = ln(0.095/0.055)/(2π × 0.04 × 1) ≈ 2.19 K/W
- R_outer_conv = 1/(10 × 2π × 0.095 × 1) ≈ 0.168 K/W
R_total ≈ 2.36 K/W. Heat loss Q = ΔT/R = 180/2.36 ≈ 76 W/m. The insulation layer dominates — as intended. Without insulation, the outer convection resistance would drop and Q would be several kilowatts per metre.
Contact Resistance
At the interface between two solid surfaces, microscopic surface roughness means actual contact occurs only at discrete asperities. The air gaps between them act as additional conductive resistance — called thermal contact resistance. In precision electronics and power devices, this is a critical design parameter often addressed with thermal interface materials (TIMs) such as thermal paste, pads, or phase-change materials.
Key Takeaways
- Thermal resistance networks turn complex multi-mode heat transfer problems into simple circuit algebra.
- Series resistances add; parallel resistances combine as reciprocals.
- The dominant (largest) resistance controls the heat flow rate — identifying it guides design decisions.
- Cylindrical geometry requires the logarithmic form of conductive resistance.
- Contact resistance at interfaces can be significant in electronic cooling applications.