# Pipe Flow Rate Measurement

Hello,

I am stuck on a pipe flow measurement problem and I would love if some excellent minds would help me out.
I have a robot that sprays paint, and there is a motor pump providing the paint to it. I need to be able to measure the flow rate (velocity) of the paint going through the nozzle.
Would someone be able to give me a very elegant and simple way to theoretically explain why I can't use one pressure sensor to get flow rate?
I know that Bernoulli's principle relies on a pressure difference, and that there doesn't exist a commercial solution that measures flow rate with just one sensor.
But what about these points that seems to counter the above?
- If I have a vacuum cleaner, and there is a hole in the side of the tube, I can feel the air blowing into that hole. If I were to increase the motor speed of the vacuum, then this air blowing into the side hole is increased. If I were to put just one pressure sensor in that hole, wouldn't I get a correlation between that reading and the flow rate of the air? What am I missing in this analogy?
- If we need a pressure difference to get the flow rate, why can't I just use the outside air (that we are spraying into) as the other pressure value? Since it is about 0 psi environment that we are spraying to, can't we use that as the other pressure value, thereby calculating flow?
- I know that if we close the nozzle valve and run the pump, then the sensor would sense increased pressure while speed remains at zero, giving us a faulty value. But what if we just choose to measure the pressure only when we know the valve is open? Could we get a correlation between the one pressure reading and flow?

The problem is, I intuitively know that it isn't possible to use just one pressure sensor. I know I need two, but I can't come up with a really elegant theoretical explanation/analogy to really convince myself. I also need to disprove the above counterpoints to really convince myself. Your wisdom and knowledge is greatly appreciated!

Sangwoo