Thermal Exchanges

View Factors:

View Factor $F_{1 \to 2}$ :

Fraction of radiated energy from surface 1 arriving at surface 2.

Examples:

Example: Sun $⟶$ Earth

Sun:


Surface Temperature	$T_{s u n} \approx 5800 K$
Blackbody Radiation	$E_{b} \approx 6.42 * 10^{7} \frac{W}{m^{2}}$
Sun Radius	$R_{s u n} \approx 695, 000 k m$
Radiated Heat	${\dot{Q}}_{s u n} \approx 3.86 * 10^{26} W$

Earth:


Earth Radius	$R_{e a r t h} \approx 6, 400 k m$
Distance to sun	$R_{o r b i t} \approx 150 * 10^{6} k m$

View Factor:

F_{S \to E} = \frac{π R_{e a r t h}^{2}}{4 π R_{o r b i t}^{2}} \approx 4.6 * 10^{- 10}

Incoming Sun Power:


At Sun Surface	${\dot{Q}}_{s u n} \approx 3.86 * 10^{2} 6 W$
Total hitting Earth	${\dot{Q}}_{s u n} F_{S \to E} \approx 1.8 * 10^{17} W$
Per unit Earth area	$\frac{{\dot{Q}}_{s u n} F_{S \to E}}{4 π R_{e a r t h}^{2}} \approx 340 \frac{W}{m^{2}}$

Earth Radiated Power:


Surface temperature	$T_{e a r t h} \approx 288 K$
Blackbody radiation	$E_{b} \approx 390 \frac{W}{m^{2}}$
Earth Radius	$R_{e a r t h} \approx 6, 400 k m$
Heat radiated away	${\dot{Q}}_{e a r t h} \approx 2.01 * 10^{17} W$

Net Budget:

Setup:

System =
Dorm room at constant and uniform Temperature

Energy Budget:

Heat radiated away: $A σ T^{4}$
- Where $A$ is the area of human body $\approx 1.7 m^{2}$
Incoming heat: $A_{r o o m} σ T_{\infty}^{4} F_{r o o m \to p e r s}$

View Factors:

Key property: $A_{1} F_{1 \to 2} = A_{2} F_{2 \to 1}$ (always true)
View factor: $F_{p e r s \to r o o m} = 1$

(Back to) Energy Budget:

Net heat radiated: $A σ (T^{4} - T_{\infty}^{4})$

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