18-6 Heat Transfer Mechanisms#

Prompts

Name the three heat transfer mechanisms. For each, describe how energy is transferred.
Write the conduction rate equation \(P = kA\,\Delta T/L\). What does \(k\) depend on? Why do metals conduct heat well?
What is convection? How does natural convection differ from forced convection?
Write the Stefan–Boltzmann law for radiation. Why does \(P \propto T^4\)? What is emissivity \(\varepsilon\)?
An object at temperature \(T\) is in an environment at \(T_{\text{env}}\). When does it absorb net radiation? When does it emit net radiation?

Lecture Notes#

Overview#

Heat can be transferred by conduction (through a material), convection (by fluid motion), and radiation (by electromagnetic waves).
Conduction: energy passed by collisions; rate \(\propto\) area, \(\Delta T\), and \(1/L\); depends on thermal conductivity \(k\).
Convection: warmer fluid rises, cooler sinks; buoyancy-driven (natural) or forced.
Radiation: no medium needed; \(P \propto T^4\); works in vacuum (e.g., Sun to Earth).

Conduction#

Energy flows through a material by collisions between atoms and electrons. The conduction rate (power) through a slab of area \(A\), thickness \(L\), with temperature difference \(\Delta T = T_H - T_C\):

(159)#\[ P_{\text{cond}} = kA\,\frac{\Delta T}{L} \]

\(k\): thermal conductivity (W/(m·K)); material property.
High \(k\): good conductor (metals). Low \(k\): good insulator (foam, air, fiberglass).

Thermal resistance: \(R = L/k\). Higher \(R\) → better insulator.

Material	\(k\) (W/(m·K))
Copper	401
Aluminum	235
Iron	67
Glass	~1
Air	0.026
Foam	~0.02

Poll: Conduction through a window

Estimate the power of heat loss via thermal conduction through a single-pane window on a chilly night in San Diego. Use \(k_{\text{glass}} = 0.8\) W/(K·m); estimate area, thickness, and \(\Delta T\) yourself.

(A) 2000 W
(B) 300 W
(C) 40 W
(D) 5 W

Poll: Composite slab—steady-state conduction

A composite slab is made of two materials (gold and iron) of the same dimensions, with hot reservoir on the left and cold reservoir on the right. In steady state, which is/are true?

(A) The rate of energy flowing through the gold slab equals that through the iron slab
(B) \(T_H - T_M = T_M - T_C\) (linear drop in each slab)
(C) Both of the above
(D) Neither

Convection#

Heat is carried by bulk motion of a fluid (gas or liquid). Warmer fluid near a hot surface expands, becomes less dense, and rises; cooler fluid replaces it.

Natural convection: driven by buoyancy (density differences). Examples: candle flame, room heating, atmospheric circulation.
Forced convection: fluid is pumped or blown (e.g., fan, wind).

Radiation#

Energy transferred by electromagnetic waves. No medium required—works in vacuum.

Stefan–Boltzmann law (emission rate):

(160)#\[ P_{\text{rad}} = \sigma \varepsilon A T^4 \]

\(\sigma = 5.67 \times 10^{-8}\) W/(m²·K⁴)—Stefan–Boltzmann constant.
\(\varepsilon\): emissivity (\(0 \le \varepsilon \le 1\)); \(\varepsilon = 1\) for ideal blackbody.
\(T\) in kelvins. Any object with \(T > 0\) K emits radiation.

Poll: Radiation scaling

One copper rod of dimensions \(L \times w \times h\) radiates at temperature \(T\) with net power \(P\). A second rod has dimensions \(2L \times 2w \times 2h\) and is also at \(T\). What is the net power radiated by the second rod?

(A) \(P\)
(B) \(2P\)
(C) \(4P\)
(D) \(8P\)
(E) \(16P\)

Net radiation (object at \(T\) in environment at \(T_{\text{env}}\)):

(161)#\[ P_{\text{net}} = \sigma \varepsilon A (T_{\text{env}}^4 - T^4) \]

\(T_{\text{env}} > T\) → net absorption. \(T_{\text{env}} < T\) → net emission.

Poll: Radiation

Two identical objects at the same temperature, one black (\(\varepsilon = 1\)) and one shiny (\(\varepsilon \approx 0\)). Which emits more radiation?

(A) The black one
(B) The shiny one
(C) Same

Summary#

Conduction: \(P = kA\,\Delta T/L\); energy by collisions; \(k\) = thermal conductivity.
Convection: heat by fluid motion; natural (buoyancy) or forced.
Radiation: \(P = \sigma \varepsilon A T^4\); no medium; \(P \propto T^4\).