How would I work this continuity problem? I am trying to design a system for a project and I need some input on if this is right or not.
I have $10 \frac{kg}{s}$ of fluid flowing through a pipe. It breaks up into $10$ 1 inch diameter pipes. I am giving a picture below. Does this mean that the feed pipe, the one supplying the $10 \frac{kg}{s}$, should be just 10 inches in diameter instead of 1 inch?
In case your wondering, the fluid is ethylene glycol, I dont think that matters

 A: I assume your objective is to minimize resistance to flow, in which case the dominating influence would be the length of the 1-inch pipes and the shape of their entry points where they enter the plenum.
If you want the fluid velocity at the entrance to the plenum to be the same as it is in the 1-inch pipes, then the plenum should have the same cross-sectional area as the small pipes, and a diameter of sqrt(10) = 3.16 inches would be sufficient.
However, you could use a smaller diameter plenum, with a higher fluid velocity, without introducing too much drag since its ratio of volume to surface area is larger.
On the other hand, a larger plenum would have less drag, at the cost of its containing a larger amount of fluid.
A: To conserve momentum of the fluid, it's the cross sectional area that should be conserved.  The area is in a square relationship to the diameter, so simply multiplying the diameter of the small pipes by 10 won't be correct.
Here's how I figure the calculation should go:
$A_{in} = A_{out}$
$\pi(D_{in}/2)^2 = \pi(D_{out}/2)^2 * 10$
$D_{in}/2 = \sqrt{(D_{out}/2)^2 * 10}$
$D_{in} = \sqrt{10} D_{out}$
