Equations and boiler pressure

Want to talk espresso but not sure which forum? If so, this is the right one.
jedovaty
Posts: 537
Joined: 13 years ago

#1: Post by jedovaty »

Hi!

I'm building a home-brew PID controller with an old arduino for fun, and am getting schooled in math and physics :roll: I'd like to double check I'm on the right track here, and hopefully someone can offer a little help.

1. Assume sea level (indeed, I am at sea level).

2. At sea level, is the following approximately correct?
1 bar atmospheric pressure, 0 bar gauge pressure, and 1 bar absolute pressure

3. Is it correct that typical temperature range for 4L espresso boilers will be 212-270F ? For my specific case, I have read a 50-60F difference in boiler temp vs brew temp, which would mean, if I want to brew with 198F water, I need boiler temp between 248 and 258F.

4. Looking up saturated steam tables, this translates to roughly 2 bar gauge pressure at the upper end, and a 3 bar absolute pressure gauge

5. The pressure transducers I am looking at are 0-30psi and 0-50psi, and apparently measured in absolute - according to the data sheet. 30 psi converts to just under 2 bar.

6. If all the above is correct, I would want the 0-50psi measuring device? If the above is incorrect, please help me figure out which measurement range would best suite a 4L boiler with a 50-60F water temp offset.

What extra confuses me: the mater xp110 p-stat I have, has the labels, 0,5-1,2 bar. Huh?

Next, I'm also trying to find some kind of approximate formula for calculating temperature from pressure. Can someone provide a rough formula that could work?

In searching for this online, all I've found are calculators to do so, or the actual saturated steam pressure tables! That's not really helpful when trying to code the math. I did find some forum post where is was suggested to take the square root of the absolute pressure in bar twice, then multiple by 100. So for 3 bar, that would be:
(sqrt ( sqrt 3 )) * 100 = 131.6 C, which is 269F.... steam tables tell me this is 133C / 272F. I checked the formula at other pressures, and I think I just need to add about 2C to it (I'll do the actual math later) so.. hmmm.

Close enough, or is there something a little better?

Thanks for help :)

jpender
Posts: 3917
Joined: 12 years ago

#2: Post by jpender »

jedovaty wrote:Next, I'm also trying to find some kind of approximate formula for calculating temperature from pressure. Can someone provide a rough formula that could work?
I haven't looked at this closely but it might work for you:

Simplified Equations for Saturated Steam Properties for Simulation Purpose

kirby
Posts: 33
Joined: 4 months ago

#3: Post by kirby »

You're looking for the Clausius-Clapeyron Equation. See the link for a more detailed explanation.

https://chem.libretexts.org/Bookshelves ... n_Equation

jedovaty (original poster)
Posts: 537
Joined: 13 years ago

#4: Post by jedovaty (original poster) »

jpender wrote:I haven't looked at this closely but it might work for you:

Simplified Equations for Saturated Steam Properties for Simulation Purpose
Thanks for the response. I came across this in my search and the first word "Simplified" is a tiny little understatement. Actually, it's a gross understatement. I did give it multiple goes, and it seems like I need to include other kinds of pressure (e.g. critical and reduced pressures), and it's all assuming a third unit of measurement so I would need to do a three way calculation and that's.. no.
kirby wrote:You're looking for the Clausius-Clapeyron Equation. See the link for a more detailed explanation.
Hmm, this is new. I took a few minutes and looks like that needs more than just basic bath, including derivates and such.. that's too much beyond my abilities. UNCLE.

I like the double sqrt equation better so far, adding 2C to it seems to get me close. Any other thoughts?

Think about pi. You can use symbols and such, or you can use 22/7 or 3.14 to approximate. That's what I'm looking for here.

Also, the correct device measurement. Oi, this is way harder than expected :(

jpender
Posts: 3917
Joined: 12 years ago

#5: Post by jpender »

Ignore my response. The Clausius-Clapeyron equation isn't as exact but it's close enough at the relatively low temperature range you are interested in.

Simplified:

P = exp(4898 * (1/373 - 1/(T+273)))

where T is in °C and P is in bar

User avatar
Jeff
Team HB
Posts: 6913
Joined: 19 years ago

#6: Post by Jeff »

You could also just do a table lookup with interpolation.

jedovaty (original poster)
Posts: 537
Joined: 13 years ago

#7: Post by jedovaty (original poster) »

jpender wrote: P = exp(4898 * (1/373 - 1/(T+273)))

where T is in °C and P is in bar
Does the attached look correct to solve for T? I ran out of paper.


jpender
Posts: 3917
Joined: 12 years ago

#8: Post by jpender »

Oh, you wanted T. Yeah, that looks good.

I like the coffee filter as scratch paper.


Also I agree with Jeff. You could just do a look up table with an entry for each 0.1bar or whatever pretty easily. You can generate a table here:

https://webbook.nist.gov/chemistry/fluid/

jedovaty (original poster)
Posts: 537
Joined: 13 years ago

#9: Post by jedovaty (original poster) »

Hmmm, I guess that's another way to do it, sure. A lookup table means something called arrays/matrices, right? Eitherway more research, whether to look up how to do arrays, or how to do log base e math in arduino code. Just mumbling now.

I think we have the second question addressed then, yay!

Back to the first question then: ultimately would like to know whether to get a 0-30PSI or 0-50PSI (absolute) measuring device for a 4L espresso boiler? I can provide datasheet for the device if anyone is interested to read the specs.

I bought the 0-30psi last week, before I knew that pressure is measured different ways. So after reading up on it, I think the 0-50 is the correct one. But the mater p-stat details are throwing a monkey wrench in this. I would test the whole thing, but the mater died a couple weeks ago, I didn't head the warnings and didn't use an intermediary relay between it and the heating element..

kirby
Posts: 33
Joined: 4 months ago

#10: Post by kirby »

First, I love that this conversation ended up with some equations written on a coffee filter.

Yeah if you could send some details on the measurement devices, that would be great. Most manometers measure difference relative to atmosphere. In that case, your 30 psi one would be close, but likely fine. However, if its reading is absolute, meaning relative to vacuum, you'll be out of its range.