Suppose there is a system of pipes, and I'm trying to find the average diameter of all the pipes in that system. They have different lengths and different diameters throughout. (Assume everything is is meters.)
Pipe System
Pipe A: 1m long, 3m wide
Pipe B: 2m long, 7m wide
Pipe C: 3m long, 2m wide
Pipe D: 4m long, 1m wide
I figured that there were two ways to solve this, using just a plain average of the diameters based on the lengths as weight, or using the volumes.
Method 1: You can't just add all the diameters together and divide by four because that does take the average diameter size, but doesn't involve the lengths at all. So for example, there could be 100m of 1m diameter pipe, and 1m of 100m diameter pipe, and have an average of 50m diameter throughout the system, which is obviously wrong.
My plan was to weight it based on length, so the solution would be this:
((1m*3m)+(2m*7m)+(3m*2m)+(4m*1m))/4 which equals 6.75m as a diameter, so that doesn't seem to be right.
Method 2: Use the volumes. h*πr² is the volume of the cylinders, so adding those all up yields:
(1*π(3/2)²)+(2*π(7/2)²)+(3*π(2/2)²)+(4… m^3
Now the units are in m^3, so dividing by length would give m^2. So we do 96.60/(1+2+3+4) to get 9.66 m^2. This number represents the average cross-sectional area, so we use 2sqrt(area/pi) to get the diameter back, which turns out to be about 3.507 which seems good, but interestingly enough it's different than the previous 6.75.
So this is what I've come up with as solutions, and I was seeking enlightenment on which one of these are correct, if either of them, and if not, perhaps a possible method to find the average diameter of the pipe system. I'm just wondering because I would have thought that the numbers would be the same, due to pi and the sqrt being cancelled out through the process.
Thanks for any input. I'll edit the work if its wrong.
Pipe System
Pipe A: 1m long, 3m wide
Pipe B: 2m long, 7m wide
Pipe C: 3m long, 2m wide
Pipe D: 4m long, 1m wide
I figured that there were two ways to solve this, using just a plain average of the diameters based on the lengths as weight, or using the volumes.
Method 1: You can't just add all the diameters together and divide by four because that does take the average diameter size, but doesn't involve the lengths at all. So for example, there could be 100m of 1m diameter pipe, and 1m of 100m diameter pipe, and have an average of 50m diameter throughout the system, which is obviously wrong.
My plan was to weight it based on length, so the solution would be this:
((1m*3m)+(2m*7m)+(3m*2m)+(4m*1m))/4 which equals 6.75m as a diameter, so that doesn't seem to be right.
Method 2: Use the volumes. h*πr² is the volume of the cylinders, so adding those all up yields:
(1*π(3/2)²)+(2*π(7/2)²)+(3*π(2/2)²)+(4… m^3
Now the units are in m^3, so dividing by length would give m^2. So we do 96.60/(1+2+3+4) to get 9.66 m^2. This number represents the average cross-sectional area, so we use 2sqrt(area/pi) to get the diameter back, which turns out to be about 3.507 which seems good, but interestingly enough it's different than the previous 6.75.
So this is what I've come up with as solutions, and I was seeking enlightenment on which one of these are correct, if either of them, and if not, perhaps a possible method to find the average diameter of the pipe system. I'm just wondering because I would have thought that the numbers would be the same, due to pi and the sqrt being cancelled out through the process.
Thanks for any input. I'll edit the work if its wrong.