Thermal resistance

Heat flows from warm to cold; every layer it has to pass slows it down (a resistance). Same structure as $I = V/R$ → Flow = driving force ÷ resistance.

Per-area resistances [m²K/W] (on the formula sheet):

• Conduction through a layer: $R = \frac{d}{k}$ (thick, badly conducting → large $R$)
  • $k$ = thermal conductivity [W/(m·K)]: how well a material conducts heat (tables sometimes write $\lambda$)
• Convection at a surface (air/fluid film): $R = \frac{1}{h}$

$U = \frac{1}{R_{tot}} \quad [\text{W/(m}^2\text{K)}] \qquad \dot Q = U\,A\,\Delta T = \frac{A\,\Delta T}{R_{tot}}$

Why gas layers insulate: gas conducts badly ($k_{argon}$ = 0.02 vs $k_{glass}$ = 0.5 W/(mK)) → a 1 cm gas gap ($R$ = 0.5) beats both glass panes together ($R$ = 0.04).

Cylinder (pipe): $R = \Delta T / \dot Q$ in K/W (whole pipe, $L$ already inside), like $\Omega$ = V/A: $R_{ins} = \frac{\ln(r_o/r_i)}{2\pi k_{ins} L} \qquad R_{soil} = \frac{\ln\left(2\frac{H + r_o}{r_o}\right)}{2\pi k_{soil} L} \qquad \dot Q = \frac{\Delta T}{R_{ins} + R_{soil}}$

• $\ln$ comes from the heat spreading outward through a ring that gets bigger (not a flat slab)
• Soil: heat spreads through the ground to the nearest fixed-temperature boundary, the surface; $H$ = how far away that is

Related: Flow = driving force ÷ resistance, Advection, Sensible heat