How long to force carb: The calculator

[Lyrebird_Cycles]

A very rough way of doing it is to assume that it takes 2 weeks for any (reasonable) beer to get carbonated using a set and forget method. You can then work out the linear rate over this time, and then come up with a factor based on the volumes already dissolved and those left to dissolve to simulate the fact that an asymptotic curve is steeper at the start (i.e. faster rate of dissolution) and flatter at the end (slower rate).

Have a look at the attached sheet for an example.

I came up with an initial steepness factor of 6 times the linear rate to give a relatively stable pressure by the end of two weeks.

This sheet only mimics the asymptotic nature of the problem though, the target volume, assumed time to get there, etc are all static values.

What I'd really like is a calculator that can predict for a given time period, temperature and regulator setting, how many volumes will have dissolved in that time - i.e. not the equilibrium volume but a time dependent one.

It's a basic log curve.

The rate of change of concentration is proportional to the difference between the present concentration and the equilibrium concentration

d[CO2]/dt = k ( [CO2]eq - [CO2]t )

[CO2]eq is a constant for a given set of conditions, let k [CO2]eq = A, so the equation becomes

d[CO2]/dt = A - k[CO2]t

The solution to a differential where the rate of change varies with the output is always an exponential or a log curve (in natural logs), if the sign is negative it's a log curve.

If you can generate the data required, I can write the equation you need, Excel handles logs quite well.

BTW the model you are proposing is covered by Fick's Law, so the other factor you might want to include is viscosity, diffusion rates at liquid / gas interfaces are known to be dependent on liquid viscosity. The reason for this is that if the gas molecules move from the absorbtion surface slowly (high viscosity), this creates a higher surface concentration which reduces the effective concentration difference between gas and surface.

 

[pcqypcqy]

It's a basic log curve.

The rate of change of concentration is proportional to the difference between the present concentration and the equilibrium concentration

d[CO2]/dt = k ( [CO2]eq - [CO2]t )

[CO2]eq is a constant for a given set of conditions, let k [CO2]eq = A, so the equation becomes

d[CO2]/dt = A - k[CO2]t

The solution to a differential where the rate of change varies with the output is always an exponential or a log curve (in natural logs), if the sign is negative it's a log curve.

If you can generate the data required, I can write the equation you need, Excel handles logs quite well.

BTW the model you are proposing is covered by Fick's Law, so the other factor you might want to include is viscosity, diffusion rates at liquid / gas interfaces are known to be dependent on liquid viscosity. The reason for this is that if the gas molecules move from the absorbtion surface slowly (high viscosity), this creates a higher surface concentration which reduces the effective concentration difference between gas and surface.

Not familiar with Fick's Law (I'll read up on it) but the reading I've done so far talks about Henry's Law quite a bit.

I found this article that helps explain the process a little. Doesn't include time as a factor, but it does predict the equilibrium volume of dissolved gas as a function of regulator pressure, temperature, final gravity, alcohol content, etc.

http://www.vitalsensorstech.com/PDF's/Methods%20of%20Analysis%20for%20correcting%20CO2%20content%20for%20Specific%20Gravity%20and%20Alcohol%20November%202011.pdf

I'm a bit of an excel junky myself (engineer), so I'll have a go at generating some data and fitting a curve to it and working back to a predictive model, then share it here. Might take a while but I'll update as I test new kegs.

 

[Lyrebird_Cycles]

Henry's Law governs the [CO2] at equilibrium with a given pressure at a temperature, Fick's Law governs the route by which it gets there.

True story: I used to do some design work for a company that did alcohol reduction in wines using a membrane process. One of the principals had done as you suggest and spent some weeks developing a predictive model of membrane flux which occupied over a thousand rows of a spreadsheet.

One day he showed me the spreadsheet and the data: it took me less than half an hour to reduce the entire spreadsheet to a single equation. He was a little peeved.

 

[GABBA110360]

I use the kiss principle
300 kpa for 24 hours seems to get to the right area @about 4 deg c

 

[damoninja]

I use the kiss principle
300 kpa for 24 hours seems to get to the right area @about 4 deg c

For some, yeah... Just poured the one I kegged yesterday at about 8pm at about 2C, 40PSI... I'd say it's about 70% there, it's a bit under what I'd normally take it at... rather than leave it overnight again I've pulled it back to ~15 PSI I'll pull back to about 12-13 once it's where I want it.

But - it's a lot more carbed than I really expected... and I have left kegs at 40 PSI for 2-3 days and got good carb levels before

So, pcqypcqy is on to something... the falloff needs to be taken in to consideration for this to be go from a good tool to an awesome tool

 

[pcqypcqy]

Henry's Law governs the [CO2] at equilibrium with a given pressure at a temperature, Fick's Law governs the route by which it gets there.

True story: I used to do some design work for a company that did alcohol reduction in wines using a membrane process. One of the principals had done as you suggest and spent some weeks developing a predictive model of membrane flux which occupied over a thousand rows of a spreadsheet.

One day he showed me the spreadsheet and the data: it took me less than half an hour to reduce the entire spreadsheet to a single equation. He was a little peeved.

But I'm an engineer, I actually ENJOY playing with excel.

 

[Lyrebird_Cycles]

OK, excel away to your heart's content.

It would be interesting to see how well we agree at the end. I spent a few minutes playing with this and I came up with:

[CO2]t = [CO2]eq ( 1- e(-k * Tabs * t)).

A k value of around 0.002 seems a reasonable fit to your curve.

[CO2]eq is the equilibrium CO2 concentration, given by [CO2] = Pabs*e(2617.25/Tabs) * 0.00075/(352.5 - ABV)

[CO2] is in g/l

t is time in days

Tabs in K, just use oC + 273.15

Pabs in kPa, use Pgauge + MSLP/10



Edit: the equilibrium CO2 equation I've used is slightly different from the one I've used in the past, I modified it to take account of the lesser solubility of CO2 in the presence of remnant wort solids.

 

[Lyrebird_Cycles]

I use the kiss principle
300 kpa for 24 hours seems to get to the right area @about 4 deg c

The equation above gives 4.66 g/l CO2 for those conditions, is that about what you get?

 

[pcqypcqy]

OK, excel away to your heart's content.

It would be interesting to see how well we agree at the end. I spent a few minutes playing with this and I came up with:

[CO2]t = [CO2]eq ( 1- e(-k * Tabs * t)).

A k value of around 0.002 seems a reasonable fit to your curve.

[CO2]eq is the equilibrium CO2 concentration, given by [CO2] = Pabs*e(2617.25/Tabs) * 0.00075/(352.5 - ABV)

[CO2] is in g/l

t is time in days

Tabs in K, just use oC + 273.15

Pabs in kPa, use Pgauge + MSLP/10



Edit: the equilibrium CO2 equation I've used is slightly different from the one I've used in the past, I modified it to take account of the lesser solubility of CO2 in the presence of remnant wort solids.

not following why you divide the mean sea level pressure by 10 when calculating the absolute pressure?

 

[Lyrebird_Cycles]

Because MSLP is given in hPa by BOM.

Sorry I should have made that clear, unfortunately too late to edit.

 

[Lyrebird_Cycles]

I just realised I ignored the CO2 left from fermentation. The equation above starts from zero.

New version where [CO2] at start is [CO2]o:

[CO2]t = [CO2]o + ([CO2]eq - [CO2]o) *( 1- e(-k * Tabs * t)).

k will need to be adjusted to fit the curve, I just had a play with a remnant CO2 value around 1.8 and k = 0.0014 seems to work OK

Simplified approximation: look up the equilibium CO2 and the remnant CO2 in standard tables. Call the remnant CO2 figure "start" and the difference between equilibrium CO2 and remnant CO2 "difference"

[CO2]t = start + difference * ( 1- e(t/2))

where t is time in days.

 

[pcqypcqy]

Because MSLP is given in hPa by BOM.

Sorry I should have made that clear, unfortunately too late to edit.

Gotcha, I had already put it in as 101.3 kPa and couldn't see why I should divide by 10 when all the units were the same

For reference, I put some numbers through both your calc and the calc based on the paper I linked to earlier. For 5% beer at 12psi gauge pressure, your equation gives 2.531 volumes as the equilibrium pressure, and the other equation gives 2.51.

 

[pcqypcqy]

you updated it just AFTER I'd put it into my spreadsheet

Also I've just assumed STP for my conversion of your equation from g/l to vol/vol, I'll need to adjust this for the fridge temperature.

So my understanding of the limitations of this method so far are:

• k is based purely on our anecdotal experience that it takes 2 weeks to reach a stable carbonation of around 2.5 volumes
• the shape of the curve is based on our first principles understanding of the process involved, i.e. the rate at which the gas will dissolve in the beer is based on the difference between the

I'm just wondering how to gather the data. After applying gas to the keg for a period of time, then removing the pressure and allowing that keg to reach that equilbrium, what does the headspace pressure actually tell us? From reading through stuff on Henry's law, there's a dimensionless coefficient that is the ratio of dissolved gas pressure and the headspace pressure. For CO2 and water it's something like 80%.

So roughly speaking, if I take a reading and have a stable 10psi in the headspace at a given point in time, I'll have 8 psi in the beer? Do I then look up the standard kegging charts and for 8psi and whatever temperature we're at, I read back what the volumes should be?

(assuming all temperatures, gas pressures, etc are all at equilibrium after I've stopped applying more gas).

 

[pcqypcqy]

I can't seem to make your equations work when doing it step by step.


[CO2]t = [CO2]eq ( 1- e(-k * Tabs * t)).

This one gives a nice curve when plotted out using hourly steps, say.

[CO2]t = [CO2]o + ([CO2]eq - [CO2]o) *( 1- e(-k * Tabs * t)).

It DOES match the first one when you set [CO2]o = 0, but it doesn't match when I take [CO2]o as the value from the previous step. At small enough time steps, this should give an estimate of the CO2 that was dissolved in each time step, and when added to the previous step it should match the curve nicely, but for the life of me I can't see what I'm doing wrong.

 

[Lyrebird_Cycles]

Edit:
If you do it stepwise, the time in the exponent must be the time step, not the total time.

 

[Lyrebird_Cycles]

I'm just wondering how to gather the data.

Borrow a Zahm meter or an Orbisphere DO2/ CO2 unit.

 

[GABBA110360]

The equation above gives 4.66 g/l CO2 for those conditions, is that about what you get?

I don't know about the equation but that seems to work for my ales
the only time i'm interested in g/l is if i'm priming for bottling other than that if it's over carbed from forgetting about it or any over reason I just degas it too easy.

 

[Lyrebird_Cycles]

For Damo and / or pcqy:

I'm assuming all the data you've collected is for an almost full Corny keg or similar.

As I see it, carbonation time will be dependent on fluid column depth so you'll get different results for vessels of different heights. I can't see why the relationship wouldn't be linear but some data would be good.

 

[Zorco]

OK, excel away to your heart's content.

It would be interesting to see how well we agree at the end. I spent a few minutes playing with this and I came up with:

[CO2]t = [CO2]eq ( 1- e(-k * Tabs * t)).

A k value of around 0.002 seems a reasonable fit to your curve.

[CO2]eq is the equilibrium CO2 concentration, given by [CO2] = Pabs*e(2617.25/Tabs) * 0.00075/(352.5 - ABV)

[CO2] is in g/l

t is time in days

Tabs in K, just use oC + 273.15

Pabs in kPa, use Pgauge + MSLP/10



Edit: the equilibrium CO2 equation I've used is slightly different from the one I've used in the past, I modified it to take account of the lesser solubility of CO2 in the presence of remnant wort solids.

That's interesting, subscript on the superscript. ASCII Special? It handles like text. How did you do that LC?

 

[Lyrebird_Cycles]

I can't take any credit for it: the edit window comes up with little subscript and superscript icons.

I can write ladder logic for PLCs, that's about the extent of my coding capabilities.