First let me say these are all ballpark calculations because as you pointed out we really don't have a solid understanding of the pump performance.
What you are putting together is a system. Each element in the system has a transfer function, meaning it converts something to something else in a predictable way. For example, the motor converts current and voltage into rotation and torque. The pump converts rotation and torque to pressure and flow. The cylinder converts pressure and flow to linear force and speed. The mechanics of the bender and cylinder arrangement convert linear force and speed to rotary torque and speed. Eventually that energy is absorbed by the deformation of the tubing. Energy is also lost along the way because each component has <100% efficiency.
I think about systems in this way partially because I've been trained to do so and also because I find the system-based approach gives me the clearest picture of power in -> power out.
As a designer of a system with many possible variants, you have about a million places you could start and lots of things to iterate on, constraints to establish, and components to select. I'll start this out kinda how I started my bender project by considering power and output torque.
Knowing the maximum tube size you intend to bend and knowing the power rating of your motor, the fastest bending speed is already determined. Let's assume you're designing for 5,000ft-lb of torque and you have a 3hp motor. Knowing torque and power, we can calculate maximum possible speed:
3hp x 550 ft-lb/sec-hp / 5,000ft-lb = 0.33radian/sec = 19deg/sec.
So if you had a full 3hp of output and you optimized the mechanics of the system, you could theoretically bend 90 degrees in just 3.5seconds - plenty, if not stupid- fast for a hobby bender that has no CNC control (see videos of Zag's bender for an example; he also used a 3hp motor).
Now let's work our way back from the tube to the motor and try to fill in some gaps. If the cylinder is mounted to the bender at 45* 18" from the center pivot (IMPORTANT ASSUMPTION) and angle changes are small, to rotate at 19deg/second, the speed the cylinder has to travel can be calculated as follows:
19deg/sec x 18in x pi/180deg x cos(45) = 4.2in/sec
Since you’ve selected a 3” cylinder, the flow rate required to move at this speed can be calculated based on the swept volume of the piston as follows:
4.2in/sec x pi x 3in^2 / 4 = 29.8cuin/sec
Converted to gallons per minute:
(29.8 cuin/sec) x (60sec/min) x (1gal/231cuin) = 7.7 gpm
So, based on the above calculations, you should set your pulley ratio for a flow no more than ~7gpm if you want to respect the 3hp limit of your motor. Once you get your motor and pump connected, you can measure the flow rate as I described before and find out where you are at and whether or not you can use a different pulley ratio to speed things up.
We also need to consider the operating pressure of the system for the above conditions. We have many options for calculation this, but I am going to do it using the power delivered by the cylinder. We know the power is approximately 3hp and the cylinder speed is 4.2in/sec. Let’s calculate force:
3hp x (550ft-lb/sec-hp) x (12in/ft) x (sec/4.2in) = 4,714lb
We could also have made the same calculation using geometry and output torque as follows:
(5,000ft-lb x 12in/ft x 1/cos45) / 18in = 4,714lb
Okay so now we know the force on the cylinder, let’s calculate the pressure based on the size of the piston 3”:
4714lb / (pi x 3in^2 / 4) = 667psi.
Whatdya know – very little pressure is required. This of course assumes a nice frictionless lessless system where the cylinder maintains no more than 45 degrees from perpendicular to the bender arms, so you should definitely err to the side of conservative when choosing your pulley ratio / flow rate.
FYI I reposted this after making some important corrections to my calculations and I give no guarantee of accuracy or correctness. Be sure to check it over. You really should get the dimensions of your bender to refine the calculations above.
So in conclusion – hook the damn thing up and see what it does!