Calculate vacuum loss by using principles of physics, not physical testing?

[mhrob24]

We (company I am interning for) are meeting with various suppliers in efforts to find a new source of vacuum pumps for our brake boosters. We are looking to make a switch to an electrical pump versus the mechanically-driven pump we used previously. One of the issues we are having with this is that our team in Japan is telling us that, after conducting their "tests", they found the optimal on/off pressures for durability should be 22.3 kPa and 24.3 kPa (vacuum pressure at 100% vacuum in the booster is 22.3 kPa, so anytime the sensor detects a 2 kPa pressure increase, pump should kick on and replenish the vacuum).

Now, the supplier is coming back at these numbers, saying that this is a very small pressure increase, and that the pump would more than likely be going on and off all the time (possibly after every single brake action) which would reduce longevity as electric motors are not meant to be ran in multiple short bursts like this (not sure why that is, so if someone could touch on this as well, that would be great...but not the main question here).

Long story short, it would be helpful for us (our team here in USA), to know just how much vacuum is depleted with a single, normal brake application, in order to determine how many brake actions it would take before the 2 kPa difference is detected and the pump is turned on. We might have the tools in our garage to test this, but I wanted to see if there is some way (with the necessary background numbers) to actually determine this just by applying principals of physics (like how you would solve a HW problem) without conducting a physical test? I am interning, so I feel like this would be a good learning experience if its possible, as I am not THAT far along in school yet (just beginning statics...). This would probably expose me to some topics I'll cover down the road. It would also be pretty cool to solve this myself and help the team out.

Is this possible? If not, why? If it is possible, I don't want a step-by-step answer...just some guidance as to what background information I would need, and what topics of physics to research in order to be able to solve this myself.

 

[Baluncore]

If it is possible, I don't want a step-by-step answer...just some guidance as to what background information I would need, and what topics of physics to research in order to be able to solve this myself.

Study how a brake booster cycles. 1; Idle, 2; pressing brake pedal and 3; releasing brake pedal. How does free air enter the vacuum chamber, during which phase of the cycle? What happens if you pump the brake?
Plot the air pressure and changing volume of each chamber that plays a part through the cycle. From that you can calculate the recharge volume required per cycle.

The relative diameter of the diaphragm will decide the boost ratio. The relative volume of the vacuum chamber will decide how often re-depression is required. A bigger booster seems better, but a smaller booster weighs less and fits closer against the firewall.

 

[Tom.G]

There is often a good sized tank to act as a vacuum reservoir. That would also allow brake control in case of a vacuum pump failure, while also avoiding the pump cycling.

 

[mhrob24]

Thank you both for your responses. I appreciate the help.

Tom,

you are 100% correct. We have a 4.5 L booster, and a 7 L reservoir tank. The total vacuum capacity according to our Japan team is 13.5 L, so we are assuming the other 2 L accounts for the vacuum lines.

I think I see what you're trying to get at...which is that we should be able to utilize the vacuum in the reservoir to bring the pressure back down to 22.3 kPa instead of having the pump kick on every time the vacuum pressures moves from 22.3 to 24.3 kPa. However, the Japan team told us that we must replenish the entire vacuum system every time, so (unless the pump fails) the entire 13.5 L of vacuum must be replenished as soon as that 2 kPa pressure difference is sensed.

Baluncore,

Ok. So see the images below as references to what I am about to say. I did some additional research and this is where I am at:

So CAD data is showing me that the booster is 10.8 inches (O.D). For whatever reason, the diaphragm is not showing up in the CAD and the booster is a solid body, so I found that the typical booster thickness is between 0.125 and 0.500 inches. So I'll assume this is in between that to be safe and say its 0.375 inches thick. So the I.D should be about 10.5 inches. Since the diaphragm seals the chambers, the DIA of the diaphragm should also be 10.5 inches.

Our booster capacity is 4.5 L. Assuming the diaphragm splits the booster in half, each chamber should contain 2.25 L of vacuum.

You mentioned the ratio, and after doing some research on that, Here is where I am stuck. So the way I am thinking about it is...the PEDAL ratio will provide the multiple in terms of how the force from the driver will be transferred to the master cylinder (without power assist). I found a diagram that shows how to find this pedal ratio. Following this, I found our pedal ration to be 1.7:1 (which concerns me, because from what I've read, for power-assist brakes, the ratio should be 4:1...). I've read the average person can hit the brakes with 70 pounds of force, so this means that WITHOUT a booster, the brakes would receive 1.7 * 70 = 119 pounds of force.

Now for the BOOST assist, I found a website that said this is roughly calculated by "multiplying the atmospheric conditions by the DIA of the booster, and then multiplying that by the number of diaphragms". Assuming that by "atmospheric conditions", they mean the ATM pressure, then we have 101.3 kPa * 10.5 inches * 1 diaphragm = 1,063 kPa*in...but those units don't seem right to me.

Basically, what I am thinking is that I need a way to relate the amount of force that's being added onto the master cylinder by the booster and how much this will deplete the vacuum (increase the vacuum pressure). So for example, if I know the booster is applying 200 pounds of additional force, and I can somehow relate this to how much vacuum pressure will increase due to supplying this additional force, I should be able to get my answer. I know the atmospheric pressure being let in when the pedal is depressed and the valve is opened will increase the pressure (reduce the vacuum), but this gets pushed out when the pedal is released by a one-way check valve (from what I've read), so Idk how much of the vacuum is depleted after this entire process (atmospheric pressure enters, then exits...how much did this increase the pressure in the chamber?)Am I on the right path here?

ps: just to be clear, if you increase the pressure of a vacuum (like from 22.3 to 24.3 kPa) then you are, in theory, “depleting” the vacuum so to speak, correct?

 

[Tom.G]

For a decent explanation of how a vacuum brake booster works, see especially slides 10 thru 20 here:
https://0901.nccdn.net/4_2/000/000/...ake-Booster-System-Fundamentals---Service.pdf

Our booster capacity is 4.5 L. Assuming the diaphragm splits the booster in half, each chamber should contain 2.25 L of vacuum.

See slide 11 of the above. In the worst case, the diaphragm displaces the
full 4.5L volume.

Now for the BOOST assist, I found a website that said this is roughly calculated by "multiplying the atmospheric conditions by the DIA of the booster, and then multiplying that by the number of diaphragms". Assuming that by "atmospheric conditions", they mean the ATM pressure, then we have 101.3 kPa * 10.5 inches * 1 diaphragm = 1,063 kPa*in...but those units don't seem right to me.

The force generated by the booster is the product of the piston (diaphragm) area and the pressure difference acrossed it. However this does NOT directly determine the displaced volume. See next comment.

Basically, what I am thinking is that I need a way to relate the amount of force that's being added onto the master cylinder by the booster and how much this will deplete the vacuum (increase the vacuum pressure).

The diaphragm displacement is determined by the stroke of the master cylinder. The master cylinder stroke is mainly determined by the travel distance of the brake shoes/pads, also by the deflection of the various hardware pieces of the brake linkages and mountings, the minor expansion of the hydraulic plumbing under pressure, and of course any dissolved gases or bubbles in the hydraulic fluid.

Hope this helps some!

Cheers,
Tom

 

[Baluncore]

Am I on the right path here?

The servo-magic is mostly with the two pistons in the servo hydraulic cylinder. You must study and understand how that internal mechanism operates, which valves uncover which ports when, and how that cycle operates. In effect it “meters” air from the external filter air inlet, through the servo, then into the vacuum chamber, which must be pumped down when needed.

If you are going to model the mass of air flowing through the system, you should use absolute pressure units. Any alternative will just lead to confusion.

 

[mhrob24]

For a decent explanation of how a vacuum brake booster works, see especially slides 10 thru 20 here:
https://0901.nccdn.net/4_2/000/000/...ake-Booster-System-Fundamentals---Service.pdfSee slide 11 of the above. In the worst case, the diaphragm displaces the
full 4.5L volume.The force generated by the booster is the product of the piston (diaphragm) area and the pressure difference acrossed it. However this does NOT directly determine the displaced volume. See next comment.The diaphragm displacement is determined by the stroke of the master cylinder. The master cylinder stroke is mainly determined by the travel distance of the brake shoes/pads, also by the deflection of the various hardware pieces of the brake linkages and mountings, the minor expansion of the hydraulic plumbing under pressure, and of course any dissolved gases or bubbles in the hydraulic fluid.

Hope this helps some!

Cheers,
Tom

Wouldn’t I be able to determine the stroke by the pedal displacement? Using the diagram for the pedal ratio
shown above, wouldn’t this be the equal to the length of the horizontal red line (adjacent to the 90 deg angle)?

Also, if/when I figure out the master cylinder stroke, how would I translate this to the amount pressure that the vacuum now has? My understanding is that each brake application will deplete the vacuum to some degree (increasing the pressure), so how would I correlate the displacement volume with the pressure difference of the vacuum?

 

[mhrob24]

Wouldn’t I be able to determine the stroke by the pedal displacement? Using the diagram for the pedal ratio
shown above, wouldn’t this be the equal to the length of the horizontal red line (adjacent to the 90 deg angle)?

Also, if/when I figure out the master cylinder stroke, how would I translate this to the amount pressure that the vacuum now has? My understanding is that each brake application will deplete the vacuum to some degree (increasing the pressure), so how would I correlate the displacement volume with the pressure difference of the vacuum?

Actually, probably not, because it looks like the pedal will never reach a completely vertical position…..so I probably can’t do that to determine the stroke. But I feel like the pedal displacement has to be the main factor in determining the stroke, as the pushrod is what displaces the cylinder piston.

You said something about the travel of the brake pads…..isn’t that like, only a few millimeters away from the rotors? I mean, I realize that the cylinder will continue to travel after the the pads come in contact with the rotors (the “squeezing” down on the rotors)….idk, I’m getting confused now. I think I need to study more about the master cylinder and how that works. This problem might be a bit over my head…

 

[jrmichler]

This problem might be a bit over my head…

Trying to solve it analytically is over your head for the reasons discussed above, plus other reasons not mentioned. If you take actual measurements, do not be surprised if elastic deflection in the metal parts and bulk modulus of the brake fluid are a significant part of the system analysis.

And it's not hard to drill and tap a hole for a vacuum gauge in the vacuum reservoir. It's only sheet metal, but there is enough metal there to connect a gauge and take a few readings. Reservoirs are cheap, just throw it away when done.

The testing does not need to be done in a moving vehicle, but does need to be done in an actual vehicle. You should be able to learn everything you need to know by stepping on the brake pedal in a parked vehicle.