[/I]What you will determine is the following:

"S", the grain scale-up factor (the weight of each grain in the standard recipe is multiplied by "S" to obtain the no-sparge or batch-sparge recipe),

"Wg", the total weight of grain needed for the no-sparge or batch-sparge version,

"R", the required mash thickness at runoff,

"Vm", the total volume of mash water that has been put into the mash tun to achieve "R" at the first runoff,

"V1", the first runoff volume (equal to your "Vb" if no-sparge brewing),

"G1", the gravity of the first runoff,

"Vs", the required volume of "sparge" water to be added after the first runoff (if batch-sparge brewing),

"V2", the second runoff volume ( if batch-sparge brewing),

"G2", the second runoff gravity (if batch-sparge brewing),

"Vt", the total mash-tun capacity required to hold all the grain and water.

A note on "gravity" figures. Ideally you should use a gravity scale that is based on percentage of sugar, such as the Plato scale. You can use specific gravity "points", which are the digits following the decimal place in specific gravity (1.045 SG = 45 "points"), but there will be a

*slight* error since SG does not correspond exactly linearly with sugar content. In any case, do NOT use specific gravity in the form 1.XXX, it will NOT work in these formulas!

You can use any units in these equations, as long as you use them throughout all the calculations. Please note that if you use gallons and pounds, the mash thickness "R" will also be in units of gallons per pound, not the more often used quarts per pound. Multiply gallons per pound by four to convert to quarts per pound if desired.

Grain occupies about 0.08 gal/lb (0.67 l/lb) when mixed with water. Call this figure "Q" and use it to find Vt, the total mash tun volume (capacity) required for the scaled-up recipe.

** The Equations**
For a no-sparge recipe:

S = Vb / (Vb - (Ra x Wn))

G1 = Vr x Gr / Vb

R1 = Ra x S / (S - 1)

Wg = S x Wn

Vm = (Ra + R1) x Wg

Vt = Vm + (Q x Wg) To minimize the extra grain required, collect as much wort as possible by establishing a thin mash just before runoff. Again, it's probably better to mash using a "normal" mash thickness, and thin the mash to "R" only after the conversion is complete.

When batch-sparge brewing, it turns out that the best extraction efficiency is obtained when the two runoffs are of equal volume (V1 = V2 = Vb/2):

R = (Vb + SQRT

{Vb2

+ (8 x Wn x Vb x Ra)}) / (4 x Wn)

S = 1 / (1 - (Ra2/R2))

Wg = S x Wn

Vm = R x Wg

V1 = Vb /2

G1 = S x Vr x Gr / (V1 + (Ra x S x Wn))

Vs = V1

V2 = V1

G2 = Vr x Gr x (Ra/R) x (1 - Ra/R) / (Wn x (R - Ra))

Vt = Vm + (Q x Wg) "SQRT{}" means take the square root of the expression inside the curly brackets {}

For a full explanation of the math behind these equations, see the

sidebar.

Definition of Variables Used in the Analysis -- Mash Conditions

Vr = recipe volume (can be more or less than available boiler capacity)

Gr = recipe gravity (expressed as "points/unit volume"; see below)

Vb = total volume to be sparged to boiler (there will be one runoff for no-sparge, two runoffs for batch-sparge)

E = standard continuous-sparge extraction efficiency (range 0 to 1, or percent ÷ 100)

Wn = weight of grain of the standard recipe

Wg = weight of grain of the no-sparge or batch-sparge recipe

Ra = absorption rate of the grain

R1 = final mash water volume to grain weight ratio (just before first runoff)

R2 = desired sparge water to grain ratio (just before second runoff, after sparge infusion and rest, batch-sparge only)

Ptn = total potential extract points of the standard recipe (see below)

Pt = total potential extract points of the no-sparge or batch-sparge recipe (see below)

** Analyzing the Mash**
Now we can analyze the process to find all the other relevant factors:

Definition of Variables Used in the Analysis -- Recipe Results

Pn = predicted total extracted points for the standard recipe

Pa = predicted total extracted points for the no-sparge or batch-sparge recipe

Vm = total volume of mash water (at time of runoff)

Va = total volume of liquid absorbed by the grain (or not otherwise removed from the tun)

V1 = volume of first runoff (it's the only runoff if no-sparge)

G1 = gravity (points per unit volume) of first runoff

P1 = total points in first runoff

Pm = total points remaining in mash tun (in the liquid absorbed by the grain) after first runoff

Vs = volume of sparge water added to mash after first runoff (batch-sparge only)

V2 = volume of second runoff (batch-sparge only)

G2 = gravity of second runoff (batch-sparge only)

P2 = total points in second runoff (batch-sparge only)

Ps = total points remaining in mash tun (in the liquid absorbed by the grain, or not otherwise removed from the tun) after second runoff (batch-sparge only)

Gb = gravity (points) of boiler contents after both runoffs (batch-sparge only)

Pb = total points in boiler kettle after both runoffs (batch-sparge only)

Just before recirculation and runoff, there will exist a certain water to grain ratio (mash thickness) R1 which is directly related to the total quantity of mash water used(Vm):

Vm = R1 x Wg

The gravity of the first runnings (in points per volume) is by definition the total actual points divided by the mash water volume:

G1 = Pa / Vm

The portion of the total mash water volume which is absorbed by the grain (more accurately, that which is left behind after the runoff) is

Va = Ra x Wg

For the first runoff volume V1, we will drain off all the liquid except that which is absorbed by the grain:

V1 = Vm - Va

= (R1 x Wg) - (Ra x Wg)

= (R1 - Ra) x Wg

P1 / Pa = V1 / Vm

P1 = Pa x V1 / Vm

= Pa x ((R1 - Ra) x Wg) / (R1 x Wg)

= (1 - Ra / R1) x Pa

The number of points remaining in the mashtun after the first runoff is

Pm = Pa - P1

= Pa - (1 - Ra / R1) x Pa

= (Ra / R1 ) x Pa

I

As with the first runoff, the number of points in V2 is the same as the proportion drained versus the total volume so

P2 / Pm = V2 / (V2 + Va)

P2 = Pm x Vs / (Vs + Va)

= Pm x (R2 - Ra) x Wg / ((R2 - Ra) x Wg + Ra x Wg)

= Pm x (1 - Ra / R2)

and the runoff gravity is

G2 = P2 / V2

The proportion of sugar left in the mash tun after sparging is the same proportion as the remaining (absorbed) liquid:

Ps / Pm = Ra / R2

Ps = Pm x (Ra / R2)

= Pm x (Ra / R2) x (Ra / R1)

= Pm x (Ra2 / (R1 x R2))

In the boiler, then, we now have a volume

Vb = V1 + V2

with a total number of points

Pb = P1 + P2

and therefore a gravity of

Gb = Pb / Vb

** Optimizing the Batch-Sparge Process**
Let's digress for a moment and see if we can optimize the extraction of sugar into the boiler for the batch-sparge case. The total number of points we put into the boiler was Pb = P1 + P2

= (1 - Ra / R1) x Pa + (1 - Ra / R2) x Pm

= (1 - Ra / R1) x Pa + (1 - Ra / R2) x (Ra / R1) x Pa

= Pa x [(1 - Ra / R1) + (1 - Ra / R2) x (Ra / R1)]

= Pa x (1 - Ra2 / (R1 x R2))

so from the above relationship,

S = Pa / Pb

= 1 / (1 - Ra2 / (R1 x R2))

Unfortunately at this point, we don't yet know the values of R1 and R2, so we can't evaluate S numerically. But we'll get there!

Let's take another look at

Pb / Pa = 1 - (Ra2 / (R1 x R2))

Since this represents the "batch-sparge efficiency", it is the expression we want to maximize. If you took calculus you might remember that to minimize or maximize a function, you must differentiate it with respect to the independent variable of interest, then set that expression equal to zero. However, we have two variables, R1 and R2. But, note that these variables are related in that we are obtaining a fixed volume of wort Vb, and therefore R2 depends on R1. Rewriting the expression for Vb in terms of R1 and R2, we see that

Vb = V1 + V2

= (R1 - Ra) x Wg + (R2 - Ra) x Wg

= (R1 + R2 - 2Ra) x Wg

Rearranging, we can write R2 in terms of R1:

R2 = (Vb / Wg) + 2Ra - R1

To maximize the batch-sparge efficiency expression, it is sufficient to maximize just the product R1 x R2. Rewriting R1 x R2 using the last equation,

R1 x R2 = R1 x ((Vb / Wg) + 2Ra - R1)

or

R1 x R2 = ((Vb / Wg) + 2Ra) x R1 - R12

We differentiate this expression with respect to R1 and set it equal to zero to maximize:

Vb/2 = Wg x (R - Ra)

Since Wg = S x Wn and S = 1 / (1 - (Ra2 / R2)),

Vb/2 = S x Wn x (R - Ra)

= Wn x (R - Ra) / (1 - (Ra2 / R2))

= Wn x (R - Ra) x R2 / (R2 - Ra2)

Rearranging,

Vb/2 x (R2 - Ra2) = Wn x (R - Ra) x R2

The difference of two squares on the left side is rewritten:

Vb/2 x (R - Ra) x (R + Ra) = Wn x (R - Ra) x R2

Dividing through by (R - Ra),

Vb/2 x (R + Ra) = Wn x R2

So

0 = Wn x R2 - Vb/2 x R - Vb/2 x Ra

= 2 x Wn x R2 - Vb x R - Vb x Ra

which is a quadratic in R with two solutions

R = (Vb +/- SQRT{Vb2 + 8 x Wn x Vb x Ra}) / (4 x Wn)

where everything inside the curly brackets after "SQRT" is considered to be under a square-root radical.

Note that the quantity under the radical is always greater than Vb. Since we need R > 0 (negative values for R don't make real-world sense!), the only root that "works" is

R = (Vb + SQRT{Vb2 + 8 x Wn x Vb x Ra}) / (4 x Wn)

This gives us all we need to design an optimum

batch-sparge session.