With respect to water flowing through a hose or pipe, if increasing pressure increases flow rate, would increasing the flow rate also increase the pressure? In other words, if you had a cross sectional area of 3 ft on a pipe with x flow rate, would 2x increase the pressure, p? I don't remember these formulas from school... Thanks in advance for the refresher!
You are confusing cause and effect. Pressure is the cause. Flow rate is the effect. Higher pressure causes increased flow rate. If the flow rate increases, it is caused by increased pressure.
I have a different take on this. This is not a question about cause and effect, but is a question about relationships. In electronics we deal with current sources and voltage sources and don't dismiss questions about current sources by insisting that voltage is electromotive force. The question about cross sectional area of a pipe is similar to a question about resistance in an electrical circuit driven by a non-ideal source. If a certain amount of water is flowing downhill in a pipe, then the existance of gravity and water and the hill are the causes, but the pipe area controls the pressure and flow rate in the pipe. Smaller cross section implies greater velocity and higher pressure, in that situation.
This subject can get quite complex, but it is instructional to look at a simple case of a circular pipe with laminar flow with average velocity . The change in pressure is approximately given as follows. where ρ is density, g is gravitational accelleration, z is change in vertical height, μ is the viscocity, L is the length and A is the cross sectional area of the pipe. The first term is the gravitational component related to changes in height. The second term is the "resistive" drop of the pipe due to friction. Here you can see that pressure drop is proportional to average velocity. It is instructive to compare this second term to Ohm's Law in electricity; V=RI, where V is voltage drop, I is current (charge flow rate) and R is resistance. The voltage is similar to pressure and current is similar to flow velocity. The electrical resistance is given by ... , where ρe is resistivity of the material. By analogy, the pipe resistance is ... Hence, you can think of fluid viscocity as analogous to electrical resistivity. Also, keep in mind that flow rate is a little different than average velocity. You might have volumetric flow rate Q or mass flow rate Qm, and you would consider area and velocity (and density for Qm, since Qm=ρQ) in flow rate estimation. There is also a difference between pressure and force. Pressure is force per unit area, so for a given pressure and pipe diameter, there is a particular total force. For this reason, you might want to recast the pressure equation ... Instead of , multiply through by the area A, with F=PA and Q=vaA, to get.
Please note: Cross sectional (or any other area) is measured in square feet not feet. You are unlikely to get laminar flow with water in an ordinary hose pipe. BTW I've never seen a hose pipe with a 3 sq feet Xsection. Most piped flow is controlled as much by the pipe resistance (fanning friction factor) as the pressure. This is experimentally determined. Unlike electric flow, if you have a pump in your system you cannot apply the Bernoulli equation across the pump.