The Effect of Scale
![[url=https://pixabay.com/en/giant-small-large-crush-fear-1013731/]"Size"[/url] by 3dman_eu is in the [url=http://creativecommons.org/publicdomain/zero/1.0/]Public Domain, CC0[/url]](https://www.geogebra.org/resource/wK5CFCxW/36F3Rr5p1u4LRyYX/material-wK5CFCxW.png)
Can you see from our previous sections how the size of an object affects its tendency to heat or cool? Let's look closely at the equations and figure it out. It is important for practical reasons as well as for technological ones.
There is a mass term in the denominator of all of the heat transfer equations. The larger the mass, the slower the rate of temperature change. But, there is also an area in the numerator of all the equations. Bigger objects have bigger areas, which would cause the temperature to change faster for bigger objects. Which term wins - the mass or the area - or do their effects cancel out?
First off, the mass can be written as the volume times the density of the material of which the object is made, or . So really we have an area over a volume, or a surface area to volume ratio on the right side of each heat transfer equation.
If you take any shape and scale it larger or smaller, you can see that the larger the object is, the smaller the surface area to volume ratio is. Take for example a sphere. The surface area is and the volume is . Thus the surface area to volume ratio is 3/r. The bigger the radius, the smaller the ratio. This means that a big sphere will cool slower than a small sphere, and also heat slower if submerged in a hot pot of water.
While babies are not spherical, the principle is true regardless of shape. A small child will overheat in hot weather (or a hot tub) and be subject to hypothermia in cold weather (or the ocean) much sooner than an adult. A simple rule is to realize that the dependence on the radius r above means that these effects are dependent on the linear dimension and not area or volume. Since a baby is usually around 20 inches long and an adult is 3 to 4 times that, a baby will heat or cool 3-4 times faster than an adult in the same environment. While the proportions of a baby are not exactly the same as an adult, this is still a good estimate.
While this is true of babies, it also speaks to the much faster metabolisms required of smaller animals to maintain body temperature. You've probably heard that some small creatures in nature need to consume their body weight in food each day. This is largely to keep themselves warm due to a very large surface to volume ratio.
As a final example, these effects are also true of homes. A small home has more wall area per internal volume than a bigger home. If we assume the shape of both homes is comparable and materials are the same, the smaller home will heat up faster on a hot day and cool off faster on a cold day or night than the larger home.
Since we have not discussed shape yet, what shape of object do you suppose would heat or cool slowest for a given mass or volume? Obviously the goal is to maximize volume for a given surface area, or minimize the surface area to volume ratio. One common shape does just that - a sphere. No other shape has as good a ratio.
Surface Area to Volume Examples
Since the surface area to volume affects the rate of temperature change, let's see how it compares for different shapes. We looked at a sphere above. The ratio A/V =3/r. How does this look for a cube? Well a cube of side length 'a' has a surface area of and a volume So
At first inspection this might suggest that cubes cool off twice as fast as spheres since one has a 6 in the numerator and the other a 3. But the constants are not the same. One is the radius and the other is the side length. So to make a fair comparison, let's consider a fixed volume of some material that is formed in either the shape of a sphere or a cube. How will their cooling rates compare?
Let's start with equating their volumes: So naturally I can write With this I can write the surface area to volume of the sphere as
Now we have a fair comparison. A sphere cools at a rate where , and a cube of the same volume cools at a rate where The ratio of cooling rates is 3/3.7221 is 0.8060. This means that a sphere loses heat in a cold environment only 80.6% as fast as a cube of equal volume. Of all shapes in nature, the sphere is the most efficient in this regard. In fact, one way to define the sphere is the shape that has the lowest surface area to volume ratio in 3d space.
What if I can't calculate based on shape?
If you wanted to compare cooling or heating rates of objects for which you can't easily compute the surface area to volume ratios... like for eggs or baking of poultry, how might this be done? The problem with eggs and poultry is that they have odd enough shapes that there is no standard geometrical expression that accounts for their volumes or surface areas. Of course the egg is close to an ellipsoid, but it isn't.
If all you're after is ratios, then we don't need details of the shape. If we assume that the egg shape is consistent across eggs regardless of volume or size of the egg, then while the ratio of surface area to volume for a single egg contains an unknown parameter The unknown in the right side is an egg-shape-dependent parameter like the 3 or the 3.7221 in the above problem. But if my goal is to figure out how the time to hard boil a small egg compares with the time to hard boil a larger one, then I only need to know the linear dimension ratio. So maybe stand the egg tall on a table and measure its height. Call that a. Then the rate at which the two eggs grow hotter inside will be inversely proportional to their height ratio, or What does this mean? That if, for instance, you could find one egg that is twice as tall as another, that it will heat up half as fast as the smaller one. So the ratio of the linear dimensions relates to the rate of heating. Of course, another way of saying this is that the egg that's twice as tall will take twice as long to become hard boiled. Easy:)