Today Damian and I got to play around with the vacuum. We were brainstorming how to increase the amount of air flow through the inhaler using the in-vitro lung and I must admit I was quite stumped when we were discussing it. Never having worked with vacuum before, I lacked the basic knowledge of how to think about it. But, thanks to the work of Danielle and her post about flow through an orifice, I think I understand what is going. Also thanks go to Damian for speaking his mind and indicating that what we were wanting to do may not be necessary. Good job! Below are my notes on what Danielle and Damian taught me.
Danielle wrote in her post that the volumetric flow rate can be written as
where is the volumetric flow rate, is the change in pressure across an orifice, is the density of the air, and are the areas of the pipe around the orifice. I think it is important to understand where this equation is coming from because it says a lot about how fluid flows through something. It of course hinges completely on Bernoulli’s equation.
where again is pressure, is the density of air and, is velocity. Excuse the horrible ascii art but below is a picture of our setup.
Tube ========================= ________ Vacuum | |Inlet _______/ | ___| |_____/ | [--] ___ A2 _____ Inhaler | | |A1 \_______ | | | \________| =========================
The inlet is what is connected to the inhaler on the right side of the tube and the left most side of the tube is where we pull the vacuum. Now, we need the air flow through the inhaler to be at least 90L/min. But, the inlet serves as an obstruction, a resistance if you will. I will get back to the idea of resistance in a moment. As we know, Bernoulli’s equation will tell us what the fluid flow through the inlet will be if we know the following: pressure inside the tube, the pressure outside the tube, and the velocity of air in the tube.
Now, if we look at Bernoulli’s equation in the tube (2) and it in the inlet (1) we obtain two sets of equations.
which can be written as
and thus setting them equal to each other gives
We can equate these together because of conservation of mass which just means that the air that flows through the inlet isn’t going to disappear in the tube because well, it can’t. Now, if we replace the velocities with a volumetric velocity, i.e. then we have that
This is readily solved for
Going through this hoopla teaches us a couple things. One important thing is that the volumetric flow rate—the quantity we need to increase—depends on the pressure difference between the tube and the orifice in a nonlinear fashion. It also tells us that the only way to increase the flow rate is to either increase the pressure differential or, make . Making the cross-sectional areas the same is impossible due to our valve constraints however, we can make them as close as possible. Danielle, can you do some calculations on comparing the difference between varying the pressure differential the areas and see which coefficient gives us the greatest boost in volumetric flow?
Inspecting the above equation also taught me an analog to vacuum and current. If I have a wire and I put current through it, the electrons will happily move along the wire with no problem. But, if I add a resistor to the wire, the rate at which the electrons move down the wire is changed. The same thing happens in our vacuum tube when air flow meets a change in diameter. The fluid flow hits a resistance and thus slows the bulk fluid motion down if the diameter is decreased.
This is good. And yes, you were right Damian. This of course means that you (Damian) are going to need to figure out a way to pull a better vacuum on the tube.