[ELI5] Can someone explain Square Cube Law and how would applying it to HVAC/Heater units save electricity?


Suppose a much larger house using less electricity using these methods?

In: Engineering

Cube square law is like, if you have a square and then make it twice as long you have twice as much space, if you make it twice as wide you have twice as much space, but if you make the box twice as long and wide (make it twice as big entirely) that means you have 4x as much space in the box, not 2x.

It’s saying if you spread out an area the amount you need to use to fill it grows really fast. So a room that is twice as big has 4 times as much air in it.

This works nice if you have some graph paper.

Draw a line one line segment long.

Now make a square in which the length of each side of the square is equal to the length of the line you drew.

How many square boxes does it make? Just 1.

Now make your line 2 line segments long. Make your square out of it.

How many square boxes this time? 4.

By doubling the length of the line segment, we scale our boxes by 4.

If we were to extend this analogy, then we can imagine how a cube is constructed of six 2D squares.

If each square has side length 1, then the cube will have a volume of 1x1x1 = 1

If each square has side length 2, then the 2D boxes will have 4 squares each, and there will be a total of 2x2x2 of those small cubes.

So x2 line length = x4 box size = x8 cube size

This is what the square-cube law says – As we increase the length of lower dimensions (length vs cubes), the higher dimension (cubes) will grow significantly quicker than the lower dimension (length)

With regards to heating, this says that the volume of your house (the inside) will grow quicker than the walls of your house (the surface). Since heat can only be exchanged at a surface, then a bigger house will lose less heat than two individual houses whose total size is the same as the big house. More surface area = more heat loss.

You’ll still have to spend the energy to heat up all of that space of course, but the bigger the house is, the more efficient it is at preventing heat loss for higher volumes.

The Square-Cube Law says that when you make an object larger by some amount, the surface area increases by the *square* of that amount, and the volume increases by the *cube* of that amount.

If you have a 1ft x 1ft x 1ft box, its surface area is 6 ft^(2) and its volume is 1 ft^(3). Now let’s make the cube twice as big, so now it’s 2 ft x 2 ft x 2 ft. The surface area is now 24 ft^(2), while the volume is now 8 ft^(3). When you made it 2x bigger, the surface area became 4x bigger (your 2x increase *squared* == 2^(2)x bigger == 4x bigger), while the volume became 8x bigger (your 2x increase *cubed* == 2^(3)x bigger == 8x bigger).

I can’t imagine how this could possibly be applied to making HVAC cheaper – it does the opposite, in fact. HVAC systems affect some volume of air, so a larger house has a massively larger volume of air and needs a stronger HVAC system which uses more electricity.