Thermal Resistance Calculator
Calculate thermal resistance for any material shape or heat-transfer network. Choose conduction through a flat plate, hollow cylinder, or hollow sphere, or switch to convection resistance, series networks, or parallel networks. Pick from 14 built-in materials or enter a custom thermal conductivity, and the calculator instantly shows thermal resistance, heat flow rate, and temperature drop.
Formula
Worked example
A copper pipe has inner radius 50 mm, outer radius 60 mm, and length 1 m. Thermal conductivity of copper is 401 W/(m·K). R = ln(0.06/0.05) / (2 * pi * 1 * 401) = ln(1.2) / 2519 = 0.1823 / 2519 = 0.0000724 K/W. At a 20 K temperature difference, heat flow Q = 20 / 0.0000724 = 276,200 W - confirming copper pipes offer negligible resistance.
What is thermal resistance?
Thermal resistance (R, measured in K/W) quantifies how strongly an object or layer opposes the flow of heat. It is the thermal analogue of electrical resistance in Ohm's law: just as voltage drives current through a resistor, temperature difference (deltaT) drives heat flow through a thermal resistance. The higher the resistance, the harder it is for heat to pass - which is desirable for insulation and undesirable for heat sinks. Thermal resistance depends on material properties (thermal conductivity k), geometry (thickness, area, or radius), and in convection cases on the flow conditions (heat transfer coefficient h). A low thermal resistance means the material conducts heat easily; a high resistance means it insulates effectively.
Formulas for each geometry
For a flat plate or slab, thermal resistance is R = L / (k * A), where L is the thickness in metres, k is thermal conductivity in W/(m·K), and A is the cross-sectional area in m². For a hollow cylinder (common in pipes and insulated tubing), R = ln(r2/r1) / (2 * pi * L * k), where r1 and r2 are inner and outer radii. For a hollow sphere, R = (r2 - r1) / (4 * pi * k * r1 * r2). Convection resistance at a surface is R = 1 / (h * A), where h is the heat transfer coefficient and A is the surface area. Series resistance (like several wall layers in sequence) is simply R_total = R1 + R2 + R3. Parallel resistance (like multiple heat paths side by side) follows 1/R_total = 1/R1 + 1/R2 + 1/R3.
Series and parallel thermal networks
Real thermal problems almost always involve multiple resistances. A building wall is a series network: outer convection boundary, brick layer, insulation layer, inner convection boundary, all in sequence. Total resistance is the sum, so adding insulation in series directly adds to R_total. Fins and heat sinks create parallel paths: heat has multiple routes to the fluid, and the total resistance is lower than any single path. When designing a thermal circuit, identify whether layers are in series (same heat flux passes through all of them) or in parallel (heat flow splits between them). Series networks are dominated by the largest single resistance, making that layer the best target for improvement.
Thermal resistance in electronics cooling
In power electronics, thermal resistance is critical for keeping semiconductors within their rated junction temperatures. The thermal path from a chip junction to ambient air is a series of resistances: junction to case (R_jc, given in the datasheet), case to heat sink (R_cs, affected by thermal interface materials), and heat sink to air (R_sa, determined by heat sink design and fan airflow). Total R_ja = R_jc + R_cs + R_sa. If an IC dissipates P watts, junction temperature T_j = T_ambient + P * R_ja. Reducing any resistance in the chain - using a better thermal paste, a larger heat sink, or a fan - lowers T_j proportionally.
Typical thermal resistances and heat transfer coefficients
| Scenario | Typical R or h | Notes |
|---|---|---|
| Still air gap (25 mm) | R = 0.18 K/W per m² | Very effective insulator for thin gaps |
| Still air, natural convection | h = 5-25 W/(m²·K) | Free convection at vertical surfaces |
| Forced air convection | h = 25-250 W/(m²·K) | Fan-cooled electronics, HVAC ducts |
| Water natural convection | h = 200-1000 W/(m²·K) | Immersion cooling, passive water |
| Water forced convection | h = 1000-15000 W/(m²·K) | Cold plates, liquid cooling loops |
| Glass fibre insulation (100 mm) | R ~2.5 K/W per m² | Building insulation batt, k=0.04 |
| Brick wall (230 mm) | R ~0.32 K/W per m² | Common brick, k=0.72 |
| Copper plate (1 mm) | R ~0.0000025 K/W per m² | Essentially no resistance, k=401 |
Representative values for engineering estimation. Actual values depend on temperature, surface finish, and flow conditions.
Frequently asked questions
What units does thermal resistance use?
Thermal resistance is expressed in kelvins per watt (K/W), sometimes written degC/W since a 1 kelvin difference equals a 1 degree Celsius difference. A resistance of 2 K/W means that for every watt of heat flowing through the object, there is a 2 K temperature rise across it.
What is the difference between thermal resistance and thermal conductivity?
Thermal conductivity (k, in W/(m·K)) is a material property that describes how readily a substance conducts heat, independent of size or shape. Thermal resistance (R, in K/W) is a system property that depends on both the material's conductivity and its dimensions: R = L / (k * A) for a slab. A material with high k gives low R for a given size; a very thin layer of a poor conductor can still have low resistance if its area is large.
Why does a hollow cylinder use a logarithm in its formula?
In a hollow cylinder, the area through which heat flows is not constant. Heat passes through concentric cylindrical shells, each with a larger area than the one inside it. This variable area leads to a logarithmic relationship between the inner and outer radii. The formula R = ln(r2/r1) / (2 * pi * L * k) captures this geometry exactly. For a thin-walled tube where r2 is only slightly larger than r1, ln(r2/r1) is approximately (r2-r1)/r1, so the resistance is small. For thick insulation, the ratio r2/r1 is large and resistance increases significantly.
What is the critical radius of insulation?
For cylindrical pipes with convection on the outer surface, adding insulation does not always reduce heat loss. Insulation increases conduction resistance but also increases the outer surface area available for convection, which reduces convection resistance. The critical radius r_cr = k_insulation / h is the outer radius at which heat loss is actually maximised. Only when the outer radius exceeds r_cr does more insulation start to reduce heat loss. For most practical insulation materials (low k) and typical convection coefficients, r_cr is a few centimetres - so for large pipes this is rarely an issue, but it matters for small electrical cables or thin pipes.
How do I combine conduction and convection resistances?
Place them in series. For a wall with convection on both sides, the total resistance is R_total = R_conv,inner + R_wall + R_conv,outer = 1/(h_i*A) + L/(k*A) + 1/(h_o*A). Once you have R_total, heat flow Q = deltaT_overall / R_total, where deltaT_overall is the temperature difference between the indoor air and outdoor air, not between the wall surfaces.
What is thermal conductance and how is it related to resistance?
Thermal conductance (U, in W/K) is simply the reciprocal of thermal resistance: U = 1/R. Where resistance describes opposition to heat flow, conductance describes the ease of heat flow. A resistance of 0.5 K/W corresponds to a conductance of 2 W/K. In building science, the U-value of a wall (in W/(m²·K)) is a normalised form of conductance per unit area, commonly called the overall heat transfer coefficient.