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Thermal Resistance Calculator

Find R_th = L/(kA) and heat rate Q = ΔT/R_th for steady conduction

Parameters

W/(m·K)ⓘ
mⓘ
m²ⓘ
Kⓘ
Show Trail

Controls

xⓘ

Calculated Values

Thermal Resistance R_th:
0.25;K/W0.25;K/W
Heat Flow Rate Q:
80.00;W80.00;W

Examples

Wall insulation

k=0.04, L=0.1 m, A=10 m², ΔT=20 K.

    Metal rod

    k=400, L=0.5 m, A=0.001 m², ΔT=50 K.

      Visualization

      Thermal Resistance — Steady-State Heat Conduction

      Fourier's law of heat conduction: heat flux q = −k(dT/dx), where k is thermal conductivity (W/(m·K)) and dT/dx is the temperature gradient. For steady one-dimensional conduction through a plane wall of thickness L and area A with uniform properties: Q = kAΔT/L (W), where ΔT is the temperature difference across the wall.

      Thermal resistance is defined analogously to electrical resistance: R_th = L/(kA) in K/W, giving Q = ΔT/R_th. Higher R_th means less heat flow for the same ΔT — good for insulation. Composite walls add resistances in series: R_total = R₁ + R₂ + … (like resistors in series).

      Thermal conductivity k varies widely: copper ~400 W/(m·K), aluminum ~237, steel ~50, glass ~1, fiberglass ~0.04, still air ~0.026, polyurethane foam ~0.02. Metals conduct well; porous insulators conduct poorly because trapped air has low k.

      Doubling thickness L doubles R_th and halves Q (steady state). Doubling area A halves R_th and doubles Q. The electrical analogy extends to thermal capacitance (transient problems) and thermal circuits with convection resistances 1/(hA).

      US building R-value (ft²·°F·h/BTU) is related to SI R_th = L/k per unit area. Double-pane windows use an air gap (high R_th layer) between glass panes. Contact resistance at interfaces can add significant R_th in heat sinks and electronics.

      This steady-state formula does not apply when temperature changes with time — use the heat equation ρc∂T/∂t = k∇²T for transients. Convection and radiation require separate models or combined thermal circuit analysis.

      Key Concepts

      • Fourier: Q = kAΔT/L (steady 1D conduction)
      • R_th = L/(kA) in K/W; Q = ΔT/R_th
      • Series composite: R_total = Σ R_th,i
      • Low k → high R_th → good insulator
      • Q ∝ A, Q ∝ ΔT, Q ∝ 1/L
      • Electrical analogy: ΔT ↔ V, Q ↔ I, R_th ↔ R

      Real-World Applications

      • Building wall and attic insulation design
      • Heat sink and CPU thermal management
      • Cookware handles and thermal barriers
      • Cryogenic vessel multi-layer insulation
      • Class 11 heat conduction and R-value problems
      • Composite materials and layered pipe insulation

      Explore Further

      More thermodynamics tools

      Physics Equations

      Thermal Resistance:
      Rth=LkAR_{th} = \frac{L}{kA}
      Heat Rate:
      Q=ΔTRth=kAΔTLQ = \frac{\Delta T}{R_{th}} = \frac{kA\Delta T}{L}

      Step-by-Step Solution

      See how the main results are calculated.

      1

      Step 1: Thermal Resistance Definition

      Equation:

      Rth=LkAR_{th} = \frac{L}{kA}

      Explanation:

      Analogous to electrical resistance; L = thickness, k = conductivity.

      2

      Step 2: Compute R_th

      Calculation:

      Rth=0.10.04×10=0.250000 K/WR_{th} = \frac{0.1}{0.04 \times 10} = 0.250000 \text{ K/W}

      Result:

      Rth=0.250000K/WR_th = 0.250000 K/W
      3

      Step 3: Fourier's Law Form

      Equation:

      Q=ΔTRth=kAΔTLQ = \frac{\Delta T}{R_{th}} = \frac{kA\Delta T}{L}

      Explanation:

      Heat flows from hot to cold side.

      4

      Step 4: Heat Rate

      Calculation:

      Q=200.250000=80.0000 WQ = \frac{20}{0.250000} = 80.0000 \text{ W}

      Result:

      Q=80.0000WQ = 80.0000 W
      5

      Step 5: Insulation

      Lower k or larger L reduces Q.

      Explanation:

      Fiberglass has low k; thick walls lower heat loss.

      6

      Step 6: R-Value (building)

      R-value ≈ L/k (US units differ).

      Explanation:

      Higher R-value means better insulation.

      Frequently Asked Questions (FAQ)

      Difference from convection?

      Conduction through solid; convection is fluid motion carrying heat.

      Contact resistance?

      Interfaces add extra R_th — important in heat sinks.

      Convert R-value?

      US R-value ≈ imperial units; SI uses R_th = L/k in K/W per area.

      Transient conduction?

      Needs heat equation ρc∂T/∂t = k∇²T — not steady formula.

      Windows double pane?

      Air gap between panes adds high R_th layer.

      Practice MCQs

      1. Doubling wall thickness L:
      2. Better insulator has:
      3. R_th equals:
      4. Two layers in series:
      5. Larger area A at same ΔT:
      6. Steady state means:

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