Suppose you have a pump with constant
pressure output and you have some garden
hoses.
Which would have the greatest flow rate:
20' of 5/8" connected to 20' of 5/8" connected to 30' of 5/8"
or
20' of 5/8" connected to 20' of 5/8" connected to 10' of 1/2"?

Assume that water is incompressible. That's a pretty good assumption for this
problem!
Neglect the effect of connections. There are the same number of connections in
each case anyway, so this won't affect the
comparison.
The first two parts are the same in each case (unless I've forgotten how to read
or you've fogotten how to type), so given the above
two assumptions, we can forget about those, and look only at the effect of the
third piece of hose.
I'll also assume that the numbers you gave are inside diameters.
So, we are left deciding if 30' of 5/8" will allow more flow than 10' of 1/2"
hose. When I type the numbers into my personal flow
calculator, I get *slightly* more flow in the 30' of 5/8. Let's look at why...
In turbulent flow, for reasonably small changes in D, you can consider that the
pressure loss for a given flow rate is roughly
proportional to L/D^5. (Changes in friction factor means it's not exact, but
for small change in D, that won't matter too much.)
The proof is left as an exercise for the reader...
So, going from 1/2 to 5/8 is a 1.25x increase in diameter. 1.25^5=3.05 But
you've got three times the length, so for the same flow
rate you'll have 3/3.05 =0.984x the pressure loss. But you have the same
pressure in each case, so you can get a *little bit* more
flow out of the longer but fatter hose.
I suspect it's not a coincidence that the answer is so close to "the same"...
-Paul

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