- 2071 Magh · 4 marks
Thermal resistances: plane wall, cylinder, convection
Short answer
Thermal resistance is R_th = ΔT / Q. For a plane wall: R = L/(kA). For a cylindrical layer: R = ln(r₂/r₁)/(2πkL). For a fluid convection layer: R = 1/(hA), or 1/(h·2πrL) for a cylindrical surface. Series resistances add directly.
Define thermal resistance. Write down expression for thermal resistances for a plane wall, cylindrical layer and convective layer of a fluid.
Thermal resistance is the opposition a layer offers to heat flow. By analogy with Ohm's law (I = V/R), heat flow Q (like current) is driven by a temperature difference ΔT (like voltage) against a thermal resistance R_th:
Q = ΔT/R_th, so R_th = ΔT/Q (unit K/W or °C/W)
| Layer | Resistance | Derived from |
|---|---|---|
| Plane wall (thickness L, area A, conductivity k) | R = L/(kA) | Fourier: Q = kA ΔT/L |
| Cylindrical layer (radii r₁, r₂, length L, conductivity k) | R = ln(r₂/r₁)/(2πkL) | Q = 2πkL ΔT/ln(r₂/r₁) |
| Convective layer of a fluid (film coefficient h, area A) | R = 1/(hA) | Newton: Q = hA ΔT |
Resistances in series add: R_total = R₁ + R₂ + ..., and Q = ΔT_total/R_total.
For a cylinder with a convective surface of radius r: R_conv = 1/(h 2π r L).
✓ Answer checked by Claude Opus 5.5 (high effort)
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