Critical Radius Insulation⚠ unverified
Physics / Heat Transfer · Compute the critical radius of insulation for a cylinder
Parameters
| Input | Symbol | Unit | Default | Description |
|---|---|---|---|---|
| k | k | — | 1.0 | Thermal conductivity of the insulation, in watts per metre-kelvin (W/(m*K)) |
| h | h | — | 1.0 | Convective heat transfer coefficient at the outer surface, in watts per square metre-kelvin (W/(m**2*K)) |
| Output | Symbol | Unit | Description |
|---|---|---|---|
| result | rcrit | m | Critical radius, in metres (m). Returns 0.0 when h is not positive |
The science & history
Understanding the Parameters
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$k$, $h$ — larger $k/h$ → larger critical radius. For many pipes $r_{\mathrm{crit}}$ is millimetres; for small wires it can exceed the bare radius.
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Registry may label $k$, $h$ dimensionless — they are W/(m·K) and W/(m²·K).
Derivation (Approaching a Proof)
Total resistance from inner surface to ambient: $R = \ln(r_o/r_i)/(2\pi k L) + 1/(h 2\pi r_o L)$. For fixed $r_i$, minimise/maximise heat loss vs $r_o$ by $dR/dr_o = 0$ ⇒ $r_o = k/h$. That stationary point is a maximum of $Q$ (minimum of $R$).
History
Critical-radius insulation is a classic heat-transfer teaching example and a practical check for small-diameter lines.
Related Concepts: Conduction Cylindrical, Convection Heat Transfer, Thermal Resistance Conduction
Notes: Registry calculator critical-radius-insulation (unverified). Outer convection on a cylinder.