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This is College Physics Answers
with Shaun Dychko.
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So imagine you walk into a sauna that has
a temperature of 50 degrees Celsius
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and your body has a temperature of
37.0 degrees Celsius—
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we convert both of these into Kelvin
by the way by adding 273.15—
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and what is the rate of radiation
heat transfer to your body
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assuming your body has an emissivity of
0.98 and an area of 1.50 square meters.
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So the rate of heat transfer
due to radiation is
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Stefan-Boltzmann's constant
times the emissivity
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times the area times the temperature of
the environment which is the sauna
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to the power of 4 minus the temperature of
the body to the power of 4.
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So it's 5.67 times 10 to the minus 8
times 0.98 times 1.50
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times 323.15 Kelvin to the power of 4 minus
310.15 Kelvin to the power of 4
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which is 138 watts.
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Now part (b), we are told to ignore
all the other forms of heat transfer
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so ignore conduction between your body
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and the layer of air right beside the skin
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and ignore convection which
will carry away
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the warm air besides your skin and
replace it with cooler air
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and ignore both those sayings and
assume that there's only
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radiative heat transfer and
with that assumption,
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at what speed will your body
temperature increase
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due to this net gain of 138 watts...
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well, 138 joules of energy per second?
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Okay.
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So we need to look up the specific
heat of the body which is
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3500 joules per kilogram
per Celsius degree
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and the mass of the body, we are told,
is 75.0 kilograms
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and the rate of heat transfer
then is gonna be this
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amount of energy gained,
which is *mcΔT*,
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and this is the change in temperature
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and the rate of temperature increase
will be this *ΔT* over *T*
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so we need to solve for that and
we'll divide both sides by *mc*.
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So *ΔT* over *T* is the rate of
gaining energy
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divided by the mass times
the specific heat.
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So it's 137.67 watts divided by
75.0 kilograms
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times 3500 joules per kilogram
per Celsius degree
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which is 5.244 times 10 to the minus
4 Celsius degrees per second
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but let's turn that into per minute
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so we have something that's
easier to imagine—
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this is such a small number.
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So we multiply by 60 seconds per minute
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and we get 0.0315 Celsius
degrees per minute.
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So this is a fairly... it's a slow
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change in temperature obviously
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but it does suggest that
there's some time limit
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for staying in a sauna which there is...
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I mean there's signs outside the sauna
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which will say an adult should stay
a maximum of 15 minutes typically.
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An interesting extension to
this question would be
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to calculate by how many Celsius degrees
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the body temperature would raise
in 15 minutes?
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So you multiply this by 15.