Friday, March 26, 2021

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, wtot, is given by Eq. 14.41

wtot = S σSB (T4Ts4) ,                (14.41)

where S is the surface area, σSB 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 Ts is the temperature of your surroundings. The T4 term is the radiation you emit, and the Ts4 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; wtot is not proportional to the temperature difference TTs.

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 wtot 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 wtot = S Krad (TTs). Deduce the value of Krad at body temperature. Hint: Factor T4Ts4 =  (TTs)(…). You should get Krad = 6.76 W m−2 K−1.
The constant Krad 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 Krad, you can use either the kelvin or Celsius temperature scales for TTs, so you can write its units as W m−2 °C−1.
Air and Water, by Mark Denny, superimposed on Intermediate Physics for Medicine and Biology.
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.

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