# Mind Your p’s and Q’s

By Dan Helgerson, CFPS, CFPAI, CFPSD.

## I have a question:

### What is the relationship between pressure and flow?

Now, some of you may be wondering why I would even have a question about this. After all, everybody knows that “Flow means Go” and “Pressure means Push.” What could possibly be the question? For a cylinder to travel at a certain velocity, there will have to be a corresponding flow rate to the cylinder. For the cylinder to be able to push a certain load, there must be adequate pressure. We have an equation for each function: V=Q/K•d2 for velocity and F=p•A for force. Doesn’t this prove the axiom?

But here is my question: Is flow a cause or a result? Hmm…

You know I often ask that we think differently. It can be painful, but “no pain, no gain.” I am suggesting that a proper understanding of the relationship between pressure and flow will help us think more clearly about the way we transfer energy through fluids. We need to ask ourselves, “What is really going on here?”

### Acceleration

In our fluid power systems, we tend to think of acceleration and deceleration only at start up and slowdown. We do not usually consider them when we need to control velocity for either rotary or linear actuators. As a result, we may be missing an important relationship between pressure and flow.

That pedal to the right on the floor of your car is not called the “velocity controller.” It is called the “accelerator” and for a good reason. When you enter the “on-ramp” you must accelerate to increase your velocity to match that of the existing traffic. The acceleration force must overcome the inertia of your vehicle as well as the road resistance, tire deformation, and wind resistance, among other things. All of these are deceleration forces. When the vehicle arrives at the target velocity, you do not completely relax the accelerator pedal. If you did, you would not maintain your speed. Instead, you must continue to provide an acceleration force that matches the deceleration forces. When the two opposing forces are in equilibrium, velocity is maintained. Any change in deceleration (like coming to a hill or a sudden gust of wind) would require an equal change in acceleration to maintain velocity.

The equation for fluid velocity in a tube or cylinder is V=Q/K•d2 where V stands for velocity, Q stands for flow rate, K is a conversion factor, and d is the inside diameter of the tube or cylinder. For rotational velocity (rpm) the equation is N=Q•K/disp where N is rpm, Q is flow rate, K is another conversion factor, and disp stands for the displacement of the rotary actuator. In each of these equations, it appears that velocity, linear or angular, is dependent on flow rate. So, what in the world am I talking about?

There are three other equations that we can use to solve for velocity using V for velocity, F for force, p for pressure, A for area, m for mass, t for time, and a for acceleration: F=p•A, F=m•a, and a=V/t. The first two equations can be reworked to solve for acceleration: a=p•A/m. This and the third equation can be rearranged to solve for velocity: V=p•A•t/m. Notice that there is no “Q” in this equation. It is pressure, not flow that determines velocity.

Going back to the discussion about the accelerator pedal, we see that velocity is established when the acceleration force and the deceleration forces are in equilibrium.

So, in the interest of thinking correctly, the velocity of a cylinder, motor, or rotary actuator, is established when the acceleration and deceleration forces are in equilibrium. The deceleration forces include the load, the mechanical inefficiencies, and the back pressure. The acceleration force is determined by the pressure acting on the area or the displacement volume. There is no movement and no flow until the pressure is great enough to overcome the resistive forces. At breakaway pressure, the actuator begins to move, but if there is no flow, the pressure would immediately drop, and everything would stop. The flow must continue to fill the gap to maintain the pressure as the actuator moves. Flow is not the cause of velocity. It is the result of the velocity produced by acceleration.

A pressure-compensated flow control is a variable pressure control. There is more pressure in the fluid upstream from the valve than is needed for acceleration. Through a series of orifices, the pressure is squeezed off until it is just right for the necessary acceleration force. Any change in the load requires a new pressure and the orifice(s) change accordingly. The result is a constant flow rate. The pressure of the fluid determines the energy density available. It is the energy density (pressure/volume) that determines the acceleration which results in the flow.

I see by your faces that some of you are still skeptical. Suppose we have a hydraulically driven fan on an excavator. The fan is driven by a pressure-compensated pump. There is no flow control valve. The fan speed is regulated by the radiator temperature. A sensing device sends a signal to the pump to do what? Adjust the…? That’s right! adjust the pressure!

Any given fan speed will have a specific resistive (deceleration) force. That force will require a specific torque output from the hydraulic motor.

“Flow means Go” would be more accurately expressed as “Flow is the evidence of Go” but I don’t expect anyone to adopt that as a mantra. It would take too long to explain.

So, what’s the point? Why dig so deep?

For those who are interested in efficiency, this knowledge will help recognize restrictive flow controls as pressure controls that consume energy. When velocity is controlled by flow, the power density of the fluid is reduced by releasing the excess energy as heat. When velocity is controlled by pressure, the correct flow will follow.