Steady Conduction and Thermal Resistance

Heat moves by conduction, convection and radiation. Conduction is heat flow through a solid (or a still fluid) from hot to cold, and in steady state it behaves exactly like current in a circuit of resistors.

Fourier's law

q = −k A dT/dx

k is the thermal conductivity (W/m·K): about 400 for copper, 50 for steel, 1 for brick, 0.04 for insulation. Metals conduct well; gases poorly.

Thermal resistances

Heat flow Q = ΔT/R, just like I = V/R.

Resistances in series add; in parallel, their reciprocals add.

Overall heat transfer coefficient U: Q = U A ΔT_overall, with 1/(UA) = ΣR.

Temperature profiles

• Plane wall (constant k, no heat generation): linear.
• Cylinder: logarithmic.
• With uniform heat generation q̇ in a plane wall of half-thickness L, both faces at T_s: T_max = T_s + q̇L²/(2k). In a solid cylinder of radius R: T_max = T_s + q̇R²/(4k).

Critical radius of insulation

Adding insulation to a small pipe or wire can increase heat loss, because the outer surface area grows faster than the conduction resistance. Heat loss is maximum at the critical radius:

• Cylinder: r_c = k/h
• Sphere: r_c = 2k/h

If the pipe's radius is below r_c, a thin layer of insulation increases heat loss (useful for cooling electric wires).

Worked example

A furnace wall has 0.2 m of firebrick (k = 1.0) and 0.1 m of insulation (k = 0.1), per m². Inside 900 °C, outside 50 °C. Find the heat loss.

• R = 0.2/1.0 + 0.1/0.1 = 0.2 + 1.0 = 1.2 K/W per m².
• q = 850/1.2 = 708.3 W/m².
• Interface temperature = 900 − 708.3 × 0.2 = 758.3 °C.

Practice