# Suck back calculation -- Need physics help

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#### DavidWood2115

##### Well-Known Member
I have a conundrum regarding how much suck back volume occurs when cold crashing.

Background: I accidentally created a suck back situation when I soft crashed my CF10 fermenter and forgot that I hadn't used my spunding valve on this batch (more precisely I didn't close off the blow off valve, so the spunding valve did nothing). The fermenter was holding a 7 gallon batch, leaving perhaps 8 gallons of headspace (including the domed lid). @doug293cz's analysis suggests this should result in approximately 35 oz of suck back due to gas contraction (head space * 10/293.15) and neglecting CO2 absorption which would increase the suck back over time. However, in actuality the sanitizer only rose 35" up the blow off hose (~40 hours after reducing the temperature). The photo below shows that the sanitizer is an inch or so below the TC connection to the blow off cane. (Note that this picture was taken after closing the valve to the blow off hose, which was done after the beer had cooled to 50F). The hose is 1/2" i.d. vinyl, so 35" is only about 4 oz or less than 1/8th the ideal gas law calculation.

So, this was a puzzle. I woke up this morning with the Ahh! that this calculation assumes atmospheric pressure both before and after the cold crash. But in actuality, the ~4" of sanitizer in flask creates a small positive pressure during fermentation and it should require a larger negative pressure to suck the sanitizer 35" up the hose.

So what should those pressures be? I assumed that since StarSan is mostly water, the pre-crash absolute pressure should be 1 atm + 4"/407 = 1.00982801 atm and the post-crash pressure should be 1 atm - 35/407 = 0.914004914 atm (using an approximation I found on the web to convert inches of H20 to atm). However, when I plug those numbers into my spreadsheet, it gives the nonsensical result that the gas actually expands when it's colder.

PV = nRT = kT
k = PV/T = 1.00982801 * 8 gallons/293.15 = .027558
V = kT/P = .027558 * 283.15 / 0.914004914 = 8.54 gallons

I'm guessing that the problem is that 35" of water in a vinyl hose is not actually equivalent to 35" of a water column, but why? Is it capillary effect? Something else? Some other problem with my analysis? I clearly need some help here.

#### Beholder

##### Well-Known Member
You may want to check your units conversion on the ideal gas, as you are mixing atm, gallons, and Kelvin, which is ok if you do ratios, but doesn’t fly for absolutes.

I would calculate the pressure change based on temperature ratio at constant volume and mols. Going from 70 F to 35 F would drop the pressure around 7000 Pa or ~ 1 psi.

From a manometer, 1 psi provides around 27in H2O, so pretty close to your observed 35” using the numbers you provided. If your volume estimate is off, then the atmospheric mol calculation would be different (that is constant when calculating the pressure effect of changing temp of a fixed volume), which could explain the inaccuracy.

You can play around with the numbers using Ideal Gas Law Calculator, which at least will eliminate unit conversion errors.

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#### DavidWood2115

##### Well-Known Member
You may want to check your units conversion on the ideal gas, as you are mixing atm, gallons, and Kelvin, which is ok if you do ratios, but doesn’t fly for absolutes.

I would calculate the pressure change based on temperature ratio at constant volume and mols. Going from 70 F to 35 F would drop the pressure around 7000 Pa or ~ 1 psi.

From a manometer, 1 psi provides around 27in H2O, so pretty close to your observed 35” using the numbers you provided. If your volume estimate is off, then the atmospheric mol calculation would be different (that is constant when calculating the pressure effect of changing temp of a fixed volume), which could explain the inaccuracy.

You can play around with the numbers using Ideal Gas Law Calculator, which at least will eliminate unit conversion errors.
Thanks for your reply. I believe the units conversion is fine, as I am consistent in how I compute k and use it. Thus all the other unit conversion constants just get lumped into k. k does have a Frankenstein set of units (atm * gallons / K), but that shouldn't matter so long as one ensures the units cancel out correctly when you use it (as I do).

I only did a soft crash from 68F to 50F, so that should drop the pressure by ~3500 Pa or ~0.5psi. This should result in only 14" in a manometer, not the 35" I observed in the hose. My estimates may be off by a bit (half gallon of beer, maybe a gallon of headspace, starting temperature +2F), but not enough to account for a factor of two in the final pressure.

Once I keg this batch I can try a controlled experiment. I should be able to more accurately estimate the true volume of the fermenter and put a precise amount of water in it for the test. But I don't see how this is going to account for a factor of two. I think there must be some capillary effect or tube friction effect that makes the vinyl tube not act entirely like a manometer.

#### tracer bullet

##### Supporting Member
HBT Supporter
Maybe use this: Water - Specific Volume vs. Temperature

I used it to check volume changes from cold to mash to boil temps. I'd think it'd work for changes from fermenting to cold crash temps as well. Of course you aren't exactly working with water, but it is close-ish. If nothing else it gives you a theoretical point to compare to.

I think you're right about checking your own setup in a controlled manner to know what's really real for it.

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#### DavidWood2115

##### Well-Known Member
Thanks for the pointer. I did a similar calculation for the beer, but that shows that the liquid volume only decreases by about an ounce with the temperature change, so the dominate effect is due to the gas.

I need to crunch the numbers, but I think the mystery is actually explained by @doug293cz's analysis linked above. I took the picture (and shut the blowoff valve) about 40 hours after turning down the temperature. @doug293cz's chart suggests that about 10% of the CO2 would be absorbed into the beer in that time. P=nRT/V so if n (the number of gas molecules) decreases by 10%, so should the pressure. This is roughly consistent with my estimate of the pressure inside the fermenter when I closed off the valve: 0.914004914 atm / 1.00982801 = .9051 or about 10% less.

#### tracer bullet

##### Supporting Member
HBT Supporter
... the dominate effect is due to the gas.
Argh, yeah - now that you mention it, it should have been obvious. And I should have known it!

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#### DavidWood2115

##### Well-Known Member
Haven’t had time to crank the math, but gas absorption is definitely what’s going on here. Looking at my photos more carefully, the top of the sanitizer was at least three inches below the TC clamp Thursday evening, 1” below it when I took the picture on Friday (in the original post), and it is now at/above the TC clamp. And the valve at the top has remained closed the entire time.

#### tracer bullet

##### Supporting Member
HBT Supporter
The diameter of the tubing would play a role, 35" in a 1/4" tube is a lot different of course than 35" in a 1" tube (for a vacuum pump I don't think it matters, for a fermentation vessel there's a limited supply of said vacuum). Also the height of the tubing, it's easier to suck an inch of water from 0" to 2" than it is to take it from 33" to 35". So this probably won't change linearly. And of course the liquid level in the flask is the origin of the measurement (I think the level before any suction occurs) which you seem to have considered. Your location could also play a role, a vacuum generated in Florida may do more lifting than say Denver.

It would be interesting to have a vacuum gauge attached.

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#### doug293cz

##### BIABer, Beer Math Nerd, ePanel Designer, Pilot
Staff member
Mod
HBT Supporter
I have a conundrum regarding how much suck back volume occurs when cold crashing.

Background: I accidentally created a suck back situation when I soft crashed my CF10 fermenter and forgot that I hadn't used my spunding valve on this batch (more precisely I didn't close off the blow off valve, so the spunding valve did nothing). The fermenter was holding a 7 gallon batch, leaving perhaps 8 gallons of headspace (including the domed lid). @doug293cz's analysis suggests this should result in approximately 35 oz of suck back due to gas contraction (head space * 10/293.15) and neglecting CO2 absorption which would increase the suck back over time. However, in actuality the sanitizer only rose 35" up the blow off hose (~40 hours after reducing the temperature). The photo below shows that the sanitizer is an inch or so below the TC connection to the blow off cane. (Note that this picture was taken after closing the valve to the blow off hose, which was done after the beer had cooled to 50F). The hose is 1/2" i.d. vinyl, so 35" is only about 4 oz or less than 1/8th the ideal gas law calculation.

View attachment 771396
So, this was a puzzle. I woke up this morning with the Ahh! that this calculation assumes atmospheric pressure both before and after the cold crash. But in actuality, the ~4" of sanitizer in flask creates a small positive pressure during fermentation and it should require a larger negative pressure to suck the sanitizer 35" up the hose.

So what should those pressures be? I assumed that since StarSan is mostly water, the pre-crash absolute pressure should be 1 atm + 4"/407 = 1.00982801 atm and the post-crash pressure should be 1 atm - 35/407 = 0.914004914 atm (using an approximation I found on the web to convert inches of H20 to atm). However, when I plug those numbers into my spreadsheet, it gives the nonsensical result that the gas actually expands when it's colder.

PV = nRT = kT
k = PV/T = 1.00982801 * 8 gallons/293.15 = .027558
V = kT/P = .027558 * 283.15 / 0.914004914 = 8.54 gallons

I'm guessing that the problem is that 35" of water in a vinyl hose is not actually equivalent to 35" of a water column, but why? Is it capillary effect? Something else? Some other problem with my analysis? I clearly need some help here.
The basic problem with any such calculation is that you have to assume that CO2 is no longer being off gassed from the beer. This would only be true if fermentation had completely finished, and the CO2 content of the beer had come into equilibrium with the headspace CO2. My analysis that you linked made this assumption. This assumption is okay if you are interested in the "worst" case suck back, but in a real situation, the assumption is not likely to be valid. Without knowing how much CO2 needs to come out of the beer to create equilibrium with the headspace, you cannot do the calculation you are attempting.

Brew on

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#### DavidWood2115

##### Well-Known Member
The basic problem with any such calculation is that you have to assume that CO2 is no longer being off gassed from the beer. This would only be true if fermentation had completely finished, and the CO2 content of the beer had come into equilibrium with the headspace CO2. My analysis that you linked made this assumption. This assumption is okay if you are interested in the "worst" case suck back, but in a real situation, the assumption is not likely to be valid. Without knowing how much CO2 needs to come out of the beer to create equilibrium with the headspace, you cannot do the calculation you are attempting.
I neglected to say that this beer is an IPA and I did the soft crash on Day 11, in preparation for dry hopping. I think it is safe to assume that fermentation was complete, the headspace was 100% CO2, and the system was at equilibrium. I believe I know how to do the calculation of pressure/volume change due to an instantaneous temperature drop given these assumptions. However, it is clear from the empirical evidence that CO2 is being absorbed into the lower temperature beer over time, as your post linked above suggests will happen. This has the larger effect in this system. Can you share your model for CO2 (re-)absorption? I don't know how to do that calculation. Thanks!

#### ScrewyBrewer

##### ezRecipe - The Easy Way To Awesome Beer!
HBT Supporter
You may need to include the volumes of residual Co2 in the beer, produced during fermentation.

#### doug293cz

##### BIABer, Beer Math Nerd, ePanel Designer, Pilot
Staff member
Mod
HBT Supporter
I neglected to say that this beer is an IPA and I did the soft crash on Day 11, in preparation for dry hopping. I think it is safe to assume that fermentation was complete, the headspace was 100% CO2, and the system was at equilibrium. I believe I know how to do the calculation of pressure/volume change due to an instantaneous temperature drop given these assumptions. However, it is clear from the empirical evidence that CO2 is being absorbed into the lower temperature beer over time, as your post linked above suggests will happen. This has the larger effect in this system. Can you share your model for CO2 (re-)absorption? I don't know how to do that calculation. Thanks!
I don't have a dynamic model for CO2 absorption/desorption. I just use equation 2.1 provided in the attached .pdf (by A. J. deLange, I believe) that relates CO2 volumes to temperature and pressure under equilibrium conditions.

At the end of fermentation, and assuming equilibrium of CO2 in the beer vs. headspace, if you know the temp, CO2 partial pressure, headspace volume and beer volume, you can calculate the mass of CO2 in the headspace, the beer, and the total CO2 in the fermenter.

Total CO2 won't change during the cold crash (any suck back is assumed to be air), so you can solve for the new equilibrium conditions (CO2 partial pressure, and beer CO2 volumes) at the crashed temperature using "Goal Seek" in a spreadsheet (which is easier than trying to solve the simultaneous non-linear equations.)

Brew on

#### Attachments

• CO2 Volumes.pdf
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#### DavidWood2115

##### Well-Known Member
Thank you for all the replies and sorry for not getting back to this sooner, but I claim a parental pass. My son came up for a few days and we kegged the batch in question (a Belgian IPA that turned out great) and brewed a new batch (a Belgian Wit, which is my wife's go to beer), which in my book takes priority over this reply.

In any case, I cranked the math. There are basically three subproblems (plus a fourth, "well that's interesting").

1. Beer volume. Beer becomes denser as it cools. The 7 gallon (896 oz) contracts about 1.4 oz when soft crashing from 68F to 50F. This is assumed to be independent of the amount of CO2 absorbed.

2. (Instantaneous) Headspace pressure and volume. The gas in the headspace, which we assume is 100% CO2 after fermentation has completed, will also become less energetic as it cools. If the volume were fixed, this would just decrease the pressure. But in my setup, the volume can also change as sanitizer "sucks back" up the blow off hose. If we assume this is instantaneous (so the volumes of CO2 in the beer remains constant, which we address next) then this turns out to be a straight forward quadratic equation.

For my parameters, laid out above, this translates to 13.5" of suck back in the blow off hose (specifically, the sanitizer level is 13.5" above the end of the hose). I can share this derivation if anyone cares.

3. "Equilibrium" headspace pressure and volume. As @doug293cz points out above and earlier, the volumes of CO2 increase as temperature decreases, so the headspace will have fewer CO2 molecules, less pressure, and thus less volume due to increased suck back. I did not attempt a closed form solution, but goal seek says that at equilibrium the sanitizer would suck back 46". But recall that I was doing a soft crash to 50F. If I had done a full cold crash to 32F, my model says the suck back would have been 92 inches which means that it would have started sucking sanitizer back into the beer. I did not compute whether it would have been enough drain the flask I was using, but that is theoretically possible.

4. Finally, in thinking about this I had a "well that's interesting" moment (hence my putting "Equilibrium" in quotes). We like to think about blow off hoses as preventing air (and thus oxygen) from making their way into our fermenters. However, the really only slow it down. Post fermentation, our CO2 headspace contains essentially 100% CO2. This will tend to diffuse into the sanitizer through the blow off hose just as it does with our beer (albeit with a much smaller surface area). But the bucket of sanitizer is also open to air, so the excess CO2 that diffuses in via the blow off hose will diffuse out to the atmosphere. O2 is also soluble in H20, with a partial pressure of 0.2 atm, so there must also be some reverse transport of O2 from outside the fermenter to inside. How much? Possibly not a lot since the blow off hose has a small diameter. But over time this must add up.

#### VikeMan

##### It ain't all burritos and strippers, my friend.
But the bucket of sanitizer is also open to air, so the excess CO2 that diffuses in via the blow off hose will diffuse out to the atmosphere. O2 is also soluble in H20, with a partial pressure of 0.2 atm, so there must also be some reverse transport of O2 from outside the fermenter to inside.

Not to mention the O2 that permeates the walls of the blowoff hose itself. I cringe whenever I see someone using silicon (in particular) blowoff hoses all the way through to packaging day. Silicon is so O2 permeable that it's used in applications like blood oxygenation membranes.

#### ScrewyBrewer

##### ezRecipe - The Easy Way To Awesome Beer!
HBT Supporter
If the volume of boiling wort increases by 4% from its room temperature volume. Is it safe to say the volume of wort will decrease by 4% when cooled to 36F?

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#### DavidWood2115

##### Well-Known Member
If the volume of boiling wort increases by 4% from its room temperature volume. Is it safe to say the volume of wort will decrease by 4% when cooled to 36F?
The specific volume of water at:
211F is 1.04304 cm3/g
68F is 1.0018 cm3/g
36F is 1.00005 cm3/g

So wort shrinks about 4% when shrinking from just off a boil to room temperature. Beer will shrink less than 2% when cold crashed from 68F to 36F.

#### tracer bullet

##### Supporting Member
HBT Supporter
So wort shrinks about 4% when shrinking from just off a boil to room temperature. Beer will shrink less than 2% when cold crashed from 68F to 36F.
That's what I cam up with a long time ago as well. Some other info if interested, though it doesn't relate to suckback it's on the thermal expansion idea -

* Mash temps are almost right in the middle, 2% away from each of those temps above
* For 6 gallons, 2% is about 1/8 gallon and 4% is about 1/4 gallon

So if you have 6.25 gallons at boil temps you'll have 6 gallons at chilled temps.

Sorry for the off topic but it seemed related and possibly (hopefully) helpful for other reasons