PX154 - B4 - heat transfer mechanisms

[YF 17.6]

conduction

• important one for solids, occurs in liquids too
  • no mass transport for solids
• consider heat along a 1D rod
  • heat flow/flux:
  $$\frac{dQ}{dt}=-\kappa A\frac{dT}{dx}$$- where,
  - $A=$ cross-sectional area
  - $\frac{dT}{dx}=$ temperature gradient
  - $\kappa =$ thermal conductivity
  - for copper/silver, $\kappa =400W{m}^{-1}{K}^{-1}$
  - for glass, $\kappa =0.8W{m}^{-1}{K}^{-1}$
  - for diamond, $\kappa =1800W{m}^{-1}{K}^{-1}$
  - for expanded polystyrene, $\kappa ={0.08}^{-1}{K}^{-1}$
  - need the negative sign as heat flows from hot to cold
• heat conduction can be via electrons in an electrical conductor and via vibrations of the atoms

examples of conduction

• case 1
  
  • in equilibrium, $\frac{dQ}{dt}$ is the same in both materials, ie: $\frac{d{Q}_1}{dt}=\frac{d{Q}_2}{dt}$
    • [YF 17.12 example]
• case 2
  
  • $\frac{dT}{dx}$ is the same for both rods
    $\therefore$ total $\frac{dQ}{dt}=\frac{d{Q}_1}{dt}+\frac{d{Q}_2}{dt}$
• eg:
  $$\frac{d{Q}_1}{dt}=\frac{d{Q}_2}{dt}+\frac{d{Q}_3}{dt}$$

convection

• heat flow arising from motion of a fluid

radiation

• a surface at temperature $T$ will radiate at a power $P$ proportional to the surface area $A$ and $T^4$:

$e=$ surface emissivity
$\sigma =5.67\times {10}^{-8}W{m}^2{K}^{-1}$ (stefan boltzmann constant)
- all bodies emit and absorb radiation
- we will need a quantum theory to describe it properly - stefan-boltzmann law for blackbodies