Cooling by Radiation

In Chapter 14 of Intermediate Physics for Medicine and Biology, Russ Hobbie and I discuss thermal radiation. If you’re a black body, the net power you radiate, w_tot, is given by Eq. 14.41

w_tot = S σ_SB (T⁴ – T_s⁴) ,                (14.41)

where S is the surface area, σ_SB is the Stefan-Boltzmann constant (5.67 × 10⁻⁸ W m⁻² K⁻⁴), T is the absolute temperature of your body (about 310 K), and T_s is the temperature of your surroundings. The T⁴ term is the radiation you emit, and the T_s⁴ term is the radiation you absorb.

The fourth power that appears in this expression is annoying. It means we must use absolute temperature in kelvins (K); you get the wrong answer if you use temperature in degrees Celsius (°C). It also means the expression is nonlinear; w_tot is not proportional to the temperature difference T – T_s.

On the absolute temperature scale, the difference between the temperature of your body (310 K) and the temperature of your surroundings (say, 293 K at 20 °C) is only about 5%. In this case, we simplify the expression for w_tot by linearizing it. To see what I mean, try Homework Problem 14.32 in IPMB.

Section 14.9 

Problem 32. Show that an approximation to Eq. 14.41 for small temperature differences is w_tot = S K_rad (T − T_s). Deduce the value of K_rad at body temperature. Hint: Factor T⁴ – T_s⁴ =  (T − T_s)(…). You should get K_rad = 6.76 W m⁻² K⁻¹.

The constant K_rad has the same units as a convection coefficient (see Homework Problem 51 in Chapter 3 of IPMB). Think of it as an effective convection coefficient for radiative heat loss. Once you determine K_rad, you can use either the kelvin or Celsius temperature scales for T − T_s, so you can write its units as W m⁻² °C⁻¹.

 

Air and Water,
by Mark Denny.

In Air and Water, Mark Denny analyzes the convection coefficient. In a stagnant fluid, the convection coefficient depends only on the fluid’s thermal conductivity and the body’s size. For a sphere, it is inversely proportional to the diameter, meaning that small bodies are more effective at convective cooling per unit surface area than large bodies. If the body undergoes free convection or forced convection (for both cases the surrounding fluid is moving), the expression for the convection coefficient is more complicated, and depends on factors such as the Reynolds number and Prandtl number of the fluid flow. Denny gives values for the convection coefficient as a function of body size for both air and water. Usually, these values are greater than the 6.76 W m⁻² °C⁻¹ for radiation. However, for large bodies in air, radiation can compete with convection as the dominate mechanism. For people, radiation is an important mechanism for cooling. For a dolphin or mouse, it isn’t. Elephants probably make good use of radiative cooling.
 

Finally, our analysis implies that when the difference between the temperatures of the body and the surroundings is small, a body whose primary mechanism for getting rid of heat is radiation will cool exponentially following Newton’s law of cooling.