Conduction Heat Transfer

Conduction = Heat Transfer

• Through matter (solids, liquids, gases)
• Without bulk motion

Important People:

Conductivity

Defintions:

• Conductivity: $k$ ($\frac{W}{mK}$)
• Thermal Diffusivity: $\alpha =\frac{k}{\rho c_p}$ ($\frac{m^2}{s}$)

Fundamental Property of Fluid:

• Independent of motion
• Depends on composition
• Depends on $T$ and $P$

Conductivity Table:

Material	Conductivity ($W/mK$)
Air	0.0235
Water	0.6
Silver, Pure	418.0
Copper 11000	388.0
Aluminum 6061 T6	167.0
Zinc, Pure	112.2
Iron, Cast	55.0
Solder, 60% Tin	50.0
Titanium	15.6
Thermal Grease, T660	0.90
Fiberglass	0.040
Air, stp	0.025

Examples:

Example 1:

How much heat transfer between two surfaces at two different Temperature?

Configuration:

• Two very large flat plates
• Temperature difference between plates
• Steady State

Key Observation:

• Heat lost by hot plate = heat gain by cold plate
• $\Rightarrow$ Same heat transfer $A{q}_H^{″}=A{q}_C^{″}$
• $\Rightarrow$ Constant heat flux $q^{″}=const$

Key Conclusions:

• Linear temperature: $T(y)=T_H+(T_C-T_H)\frac{y}{L}$
• Heat flux: $q^{″}=\frac{k}{L}(T_H-T_C)$
• Newton's Coefficient: $h=\frac{k}{L}$

Example 2:

How much heat transfer between two surfaces at two different temperatures?

Configuration:

• Two spherical surfaces
• Temperature difference between surfaces
• Steady State

Key Observation:

• Heat lost by hot oven = heat gain by cold turkey
• $\Rightarrow$ Same heat transfer $4\pi R_H^2{q}_H^{″}=4\pi R_C^2{q}_C^{″}$
• $\Rightarrow$ Heat Flux $q^{″}\propto \frac{1}{r^2}$

Key Conclusions:

• Temperature: $T(r)=\frac{\alpha }{r}+\beta$
• Heat flux: $q^{″}=\frac{k}{r^2}\frac{T_H-T_C}{\frac{1}{R_H}-\frac{1}{R_C}}$

Example 3:

Assumptions:

• No convection heat transfer
  • All doors and windows closed
  • No outside wind
• No Radiation Heat Transfer
  • All outside walls shaded

Setup:

• System =
• Uniform temperature inside the house
• Uniform temperature outside the house
• Quasi Steady State (within walls)
• Constant Pressure

First Law of Thermodynamics:

• Mass of "stuff" inside (air, furniture, ...): $m$
• Ceiling area: $A_a$
• Attic temperature: $T_a$
• Area of outside walls: $A_o$
• Outside temperature: $T_o$

Heat Transfers:

32% (attic) + 68% (walls + windows)