Have you ever wondered what the theoretical maximum drain rate is, of sunroof drains? Every time I looked at the little drain holes in my S-Type sunroof, I felt as though they couldn’t possibly deal with a really big downpour – they look so small. So I decided to dust off my elementary physics and the result surprised me. I cheated a bit to account for friction and turbulent flow effects: I assumed that flow through the central two thirds of the hose was unrestricted but that there was no flow at all near the walls of the hose. (I was doing this at two o’clock in the morning and really didn’t feel up to re-learning Poiseuille’s Law.)

Some of you will remember potential energy and kinetic energy. When a liquid falls a vertical distance of, say, h metres through a drain hose it loses potential energy, and this is converted into kinetic energy, corresponding to a flow speed of v metres per second. The relationship between lost potential energy and gained kinetic energy is (ignoring a mass term which cancels from the two sides):

9.8 * h = 0.5 v * v

(9.8 is the acceleration due to gravity in metres per second squared)



I am going to rearrange that little equation, with the result that the velocity of the water flowing out of the drain hoses is:

v = SQRT (2 * 9.8 * h)



You can now estimate how much water will flow out of the drain hose per second. It equals the effective cross sectional area of the drain hose multiplied by the speed of flow of the water exiting the hose. The radius, r, of the hose is, of course, half the diameter and the area, A, is given by:

A = pi * r * r



Hence the volume of water flowing out of the pipe per second is given by

Volume = A * v



Now we can work out what the theoretical flow rate is for the S-Type. The diameter of each drain hose is approximately 5 mm, but let’s assume unrestricted flow only takes place in the central 3.3 mm (0.00167 metres radius), and the height difference between the entry to the drain hose and its outlet is approximately 0.9 metres. Plug those numbers into the equations above and you get a flow rate of 37 millilitres per second. This corresponds to a flow rate of 2.2 litres per minute, per hose or almost 9 litres per minute for the whole drain system!



Let’s see if we can put this into context. If we make the very conservative assumption that all of the rain landing on the sunroof ends up in its drains, then we can estimate how heavy the rainfall would need to be to generate 9 litres of drainage per minute. The area of the S-Type sunroof is about 0.4 square metres. Hence to require that much drainage, we would need to receive over two centimetres of rain in a minute. This is 32 metres in a day!

 

I'm not sure he could slalom.