Will changing the pipe diameter reduce the pressure head loss? I have this system installed in my house (fig 1), and I'm having problems with the shower. I want to know whether, if I change the PEX tubes shown in red from 1/2" to 1" as shown in fig 2, I will have more pressure. Will I enjoy the shower?
The head loss due to friction is calculated as   $\frac{fL}D \frac{v^2}{2g}$, where


*

*$f$ is the friction factor,

*$L$ is the length of the pipe,

*$D$ is the pipe diameter,

*$v$ is the fluid velocity, and

*$g$ is the gravitational acceleration.


I understand this to mean that when I increase the diameter, the head loss of friction decreases, so I will get more pressure. Is this correct?
Finally, what about the flow rate? How this will change affect it? How does the flow rate depend on these variables?

 A: Think about it like this: friction happens at the boundary layer (in the vicinity of where the water touches the pipe). So doubling the diameter will double the circumference (double the friction) but the quadruple the cross sectional area (quadruple the flow).
Will you get a lot more pressure by installing a bigger pipe in that one section? No.
Using an online Hazen-Williams calculator (source) with l = 6 ft, c = 140, q = 2 gal/min and d = 1/2" (conservative, the ID's probably a little higher) gives a pressure drop of about 0.3 psi with the 1/2" pipe.
To put 0.3 psi into perspective, your house water pressure should be somewhere between 40-60 psi. Check your water pressure, and the water pressure at your water heater. Plenty of videos on YouTube show how... note that before you connect the meter you really want to let some water flow, odds are there's gunk in the bottom of the tank that will foul your meter and give you a bad reading.
Most of your friction will be not in the straight run but at the 90 degree turn, although in this case I would guess you've got a bad connection, maybe a leak.
