Flowrate and pressure after a size reduction

[TSN79]

I work for a plumbing company, and I'm trying to understand things like pressure loss etc. Right now I'm trying to figure out what's shown in the attachment. It is a pipe reduction with water flowing trough it.

First, is it so that the flowrate at the left end must be equal to the flowrate at the right end? I believe it must be, otherwise water must be lost or gained as it moves through.

If I'm right, does that mean that a house's need for water can be supplied through a whatever small pipe? If x liters per second go in on the left, won't that result in x liters per second on the right, only with much greater pressure? This is probably where pressure loss comes in. I just don't get how exactly...

Last question; what equation links pressure and flowrate on the two sides? Bernoulli?

 

[Q_Goest]

Hi TSN,
Regarding Bernoulli's equation - without modification to add permenant pressure loss, Bernoulli's would predict that the stagnation pressure along any pipe at constant elevation remains constant. If this were true, then you'd be correct in saying any size pipe could supply a house. Bernoulli's equation doesn't predict permenant pressure loss. It's an idealized equation which ignores frictional flow.

Imagine a very long pipe of constant diameter with water flowing through it. In reality, there is a pressure drop as water flows through the pipe. This is a permenant pressure drop which can not be recovered.

To account for this permenant pressure drop, Bernoulli's equation must be modified by adding an equation which accounts for this irreversible pressure drop. That equation is called the Darcy-Weisbach equation. By adding this, we can accurately predict the permenant pressure loss. This modification is shown in equation 16 of the attached. The Darcy-Weisbach equation is shown in equation 1.

 

[russ_watters]

For sizing plumbing systems, you have a certain pressue available at the street, so you need to size the piping big enough to get your desired flow rate out of that pressure. If you size the pipes too small, your pressure drop will make it impossible to get your desired flow rate. I use a pipe sizing book at work that has pressure losses listed for straight lengths of pipes and certain fittings. You should get one. (Not sure the name, though)