John Palmer Batch Sparge Equations

[PThatcher]

I have been rereading How to Brew, and am having problems with His equations on pg184.
His input values are:
OG – 49 Points (Original Gravity, obviously)
Gr – 8.25 Pounds (Total recipe weight)
Vr – 5 Gallons (Recipe Volume)
Vb – 6 Gallons (Boil Volume)
k – 4.125 quarts (Water-retention Coefficient, .5 quart/pound)

I am having trouble with the equations for The Sparge Mash ratio (Rb) and the Scale up factor (S)

Rb = (Vb + (Vb2 + 2k x Vb x Gr)^.5 ) / Gr -or- ( 6 + (12 + 8.25 x 6 x 8.25)^.5 ) / 8.25
His Value, 1.85... Mine, 3.21
I get a closer number if I convert k to gallons (1.03), 2.02

S = 1/(1-k2/Rb2)
This is where I started to suspect typos, because there is no need to double the values.
But anyway It is (in Gallons):
1/(1-1.03/1.85) -or- 1/(1-1.03/3.21)
I used gallons because in quarts the values are negative.
He calculates 1.08, I got 2.25 (with his Rb) and 1.47 (my Rb)

These values are used in all the other equations, and they are (for the most part) not adding up. I haven’t ruled out operator error, and it wouldn’t surprise me. My numbers are consistent though, so I don’t think I am entering anything incorrectly. If I’m wrong I would like to figure out why before I start my first AG batch.
PLEASE HELP

 

[bartyen82]

There are typos in both equations that he corrects here.

In the sparge mash ratio equation, Vb2 should read Vb^2. However, k should be in quarts per pound, and NOT total quarts as you've calculated it (replace your 4.125 with 0.5). Make those corrections, and you'll get:

Rb = (Vb + (Vb^2+2k*Vb*Gr)^.5)/Gr = (6 + (36+2*0.5*6*8.25)^.5)/8.25 = 1.84xx

In the grain scale-up factor equation, k2/Rb2 should read k^2/Rb^2, and you'll get:

S = 1 / (1-k^2/Rb^2) = 1 / (1-(0.5^2)/(1.84^2)) = 1.07xx

Good luck on your first AG batch!

 

[Pierce]

THANK YOU. Thank You. Thank you.

I spent a couple of hours the other night trying to get this into an excel file and couldn't figure it out. Needless to say it was starting to frustrate me very much.

 

[bartyen82]

On a slightly different note, a guy named Ken Schwartz discusses the calculations at his website here. I expect JP referenced his work for the calculations in How to Brew, but what I like about the original is that KS actually explains the derivation of the equations that JP just gives you.

For Pierce - KS has a spreadsheet available as well.

The only thing I don't like about Schwartz's calculations is that for big beers (>1.080 or so) with reasonable boil volumes (less than 7 gallons), the equations return negative values for sparge water volume additions. Since there's no such thing as a negative water addition, this implies that your grain scale-up factor is too large and that you're actually leaving water (and sugar!) in the mash-tun. You are advised to choose a larger boil volume, but it seems to me that you could just go with a smaller grain scale up factor such that your first (or only) sparge water addition would be 0gal (and not negative). Nothing in the equations accounts for it, leaving you to guess about how much to scale your grain weight. I've never seen a discussion about what to do in that scenario.

Anyway, I'm glad I helped!

 

[Pierce]

Thanks for that. Unfortunately I'm minutes away from finishing my spreadsheet, but I think its been a good exercise to really understand what's going on.

 

[PThatcher]

Thank You! It is so much clearer now. Ken Schwartz's explanations should be just what I am looking for.

Pierce, I know just how you feel. I think at one point the right side of my brain looked at the left and said "Gee it's dark in here."

 

[bartyen82]

Does anyone else looking at JP's sparge water equation think he's got his example wrong? I don't think the units work out...

Rb has units of vol/lb. If you assume Vb is in gallons, but k is in quarts/lb, then the part inside the square root has units of:

(gal^2+qt/lb*gal*lb)^.5 = (gal^2+qt*gal)^.5

k should have the same volume unit that Vb does for the radical to work out, but JP doesn't do that in his example...

 

[Pierce]

Ya I definitely saw that and immediately heard my physics teaching yelling at the class about keeping the units straight.

To be honest. I figured he probably got the math right with or without the units. So I just rolled with it. After, of course, I tried to change the formula to have appropriate units.

 

[PThatcher]

Yeah that occured to me too. I think that was my big Hang-up, to the point that I was ignoring what appears obvious now. I heard my prof.s yelling at me too.

I have to laugh, I have the 2006 printing and the typos aren't fixed. Or if they were, the printer forgot and put them back.

 

[TrailRunner]

So glad I found this thread...been twisting those equations every which way and finally decided something's rotten in Denmark. Thank you!

Now, I've read his update and see where he mentions grain bills being listed for "standard" sparking (continuous or fly) methods and batch spargers having to increase the grain bill. I'm new to AG...is this true? Should I be scaling up a recipe I see listed on here or magazines assuming it was fly sparged unless stated otherwise (just like it's assumed that gravity has been temperature corrected, etc.)?

Thanks!

 

[nexy_sm]

Let's ressurect this topic a little bit again
So, this is kind of frustrating.

So, Rb=(Vb+(Vb^2+2k*Vb*Gr)^0.5)/Gr doesn't lead to 1.85 q/lb.

First case (assumption that k is correct):
Vb should be converted to quats, which yealds to 5.94 q/lb

Second case (assumption: palmer wanted to say k=0.5 gal/lb, which is probably unreal):

Result: 1.85 gal/lb

So my question is what is wrong here. In the suggested update, just the equation is rewritten in order to look much prettier.