heat

Computing the Surface Area of a Person – Mosteller Formula

While doing research for my new book “More Great Formulas Explained”, I came across a neat formula that can be used to calculate the surface area of a person. It goes by the name Mosteller formula and requires two inputs: the mass m (in kg) and the height h (in cm). The surface area S (in m²) is proportional to the square root of m times h:

S = sqrt (m * h / 3600)

For example, a person with the weight m = 75 kg and height h = 175 cm can be expected to have the body surface area S = 1.91 m². A note for American readers: you can use this table to easily convert the height in feet / inches to centimeters.

What’s the use of this? In my book I needed to know this quantity to compute heat loss. According to Newton’s law of cooling, the heat loss rate P (in Watt = Joules per second) is proportional to the surface area S and the temperature difference ΔT (in °C or K):

P = a * S *ΔT

with a being the so called heat transfer coefficient. For calm air it has the value a = 10 W/(m² * K). A person’s body temperature is around 37 °C. So the m = 75 kg and h = 175 cm person from above would lose this amount of heat every second at an air temperature of 20 °C:

P = 10 W/(m² * K) * 1.91 m² * 17 °C = 325 Watt

That is of course assuming the person is naked, clothing will reduce this value significantly. So the surface area formula indeed is useful.

Physics (And The Formula That Got Me Hooked)

A long time ago, in my teen years, this was the formula that got me hooked on physics. Why? I can’t say for sure. I guess I was very surprised that you could calculate something like this so easily. So with some nostalgia, I present another great formula from the field of physics. It will be a continuation of and a last section on energy.

To heat something, you need a certain amount of energy E (in J). How much exactly? To compute this we require three inputs: the mass m (in kg) of the object we want to heat, the temperature difference T (in °C) between initial and final state and the so called specific heat c (in J per kg °C) of the material that is heated. The relationship is quite simple:

E = c · m · T

If you double any of the input quantities, the energy required for heating will double as well. A very helpful addition to problems involving heating is this formula:

E = P · t

with P (in watt = W = J/s) being the power of the device that delivers heat and t (in s) the duration of the heat delivery.

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The specific heat of water is c = 4200 J per kg °C. How much energy do you need to heat m = 1 kg of water from room temperature (20 °C) to its boiling point (100 °C)? Note that the temperature difference between initial and final state is T = 80 °C. So we have all the quantities we need.

E = 4200 · 1 · 80 = 336,000 J

Additional question: How long will it take a water heater with an output of 2000 W to accomplish this? Let’s set up an equation for this using the second formula:

336,000 = 2000 · t

t ≈ 168 s ≈ 3 minutes

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We put m = 1 kg of water (c = 4200 J per kg °C) in one container and m = 1 kg of sand (c = 290 J per kg °C) in another next to it. This will serve as an artificial beach. Using a heater we add 10,000 J of heat to each container. By what temperature will the water and the sand be raised?

Let’s turn to the water. From the given data and the great formula we can set up this equation:

10,000 = 4200 · 1 · T

T ≈ 2.4 °C

So the water temperature will be raised by 2.4 °C. What about the sand? It also receives 10,000 J.

10,000 = 290 · 1 · T

T ≈ 34.5 °C

So sand (or any ground in general) will heat up much stronger than water. In other words: the temperature of ground reacts quite strongly to changes in energy input while water is rather sluggish. This explains why the climate near oceans is milder than inland, that is, why the summers are less hot and the winters less cold. The water efficiently dampens the changes in temperature.

It also explains the land-sea-breeze phenomenon (seen in the image below). During the day, the sun’s energy will cause the ground to be hotter than the water. The air above the ground rises, leading to cooler air flowing from the ocean to the land. At night, due to the lack of the sun’s power, the situation reverses. The ground cools off quickly and now it’s the air above the water that rises.

Image
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I hope this formula got you hooked as well. It’s simple, useful and can explain quite a lot of physics at the same time. It doesn’t get any better than this. Now it’s time to leave the concept of energy and turn to other topics.

This was an excerpt from my Kindle ebook: Great Formulas Explained – Physics, Mathematics, Economics. For another interesting physics quicky, check out: Intensity (or: How Much Power Will Burst Your Eardrums?).