Working content >Heat and temperature > Heat transfer

Prerequisites:

• Heat transfer
• Heat Transfer: Example - conduction

In our first example we considered the flow of heat by conduction. Convection is a little harder to model and calculate, especially for moving organisms -- and it depends on the local situation of air or water flow. But radiation is fairly clean to calculate since the Stefan-Boltzmann equation (see Heat transfer) gives an upper limit for the electromagnetic radiation energy that a warm body emits.

Let's consider the following problem:

Suppose you are designing clothing for a polar scientist. You have material that has an R value* of 0.6 K-m²/W. If you make clothing out of this stuff do you have to worry about heat loss due to electromagnetic radiation? Or if you have handled the conduction loss will that be good enough? You want the clothing to be effective at -20 ^oC.

Even though it doesn't say so explicitly, this sounds like an estimation problem. We don't know how big the explorer is or what her temperature is. So let's treat it like one.

To solve this, we need to estimate the rates of conductive heat loss compared to the radiative heat loss. The rate of loss per unit area due to conduction expressed in terms of the R value was given in Example 1. The result was

To calculate the full heat loss due to conduction we need the area, the temperature difference, and R. R is given; we can estimate the body temperature of the scientist to be 30 ^oC (assuming she is healthy); we only have to estimate the area. Before we do that, let's look at the heat loss due to radiation.

The energy loss flux per unit area and the total flux for an area A due to radiation are given by 

We have to compare the two total fluxes. We know sigma (5.67 x 10^-8 W/m²K⁴) and the absolute temperature -- T = 303K. We only don't know the area so that's all we have to estimate.

But wait! We can estimate her surface area pretty easily -- say by approximating her by a cylinder. But the problem doesn't ask us for an absolute value of the thermal energy emitted (though that might be an interesting question). It just asks which is more important. Since both fluxes are proportional to the same A, we only have to compare ΔT/R and σT⁴. Let's do it. (Note the size of a Celsius and Kelvin degree are the same so in doing Delta T we can report it in Kelvin degrees.)

Whoa! A bit surprising! That says you definitely have to worry about the radiation properties of your clothing. You don't want your scientist to be a perfect radiator in a polar environment. She would lose almost 4 times as much heat to radiation as she does to conduction.

* See the Wikipedia article on Clothing Insulation. Note that the clothing industry uses two units for the R value. The "clo" = 0.155 K-m²/W and the "tog" = 0.1 K-m²/W. 

Joe Redish 11/24/14

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