**Recipe Design and Required Math**

**Building a Grist Bill Part 1:**

Da Shugah

Da Shugah

Extract:

**Recipe Variables**

**Plato vs. Specific Gravity**

Calculating Strike Water Temperature

Basic theory: We are looking at understanding how malt, strike water, and our brew system come into thermal equilibrium.

Building a Grist Bill Part 1:

Color Me Shocked!

**Algebra Basics**

**Algebraic Properties**

Associative

Cumulative

Distributive

Associative

Cumulative

Distributive

Associative

Addition:

a+(b+c)=(a+b)+c

Multiplication:

a(bc) = (ab)c

Commutative

Addition:

a+b=b+a

Multiplication:

ab = ba

Distributive

Addition & Multiplication:

a(b+c) = ab+ac

Multiplication

ab = c => a = c/b, b = c/a c/(ab) = 1, 1*c = c

Division is the inverse of multiplication

1 is the multiplicative identity

Addition

a + b = c => a = c - b, b = c - a, 0 = c - ( a + b ), a + 0 = a

Subtraction is the inverse of addition

0 is the additive identity

Multiplication

A negative times a positive equals a negative:

a(-b) = -ab = -c

A negative times a negative equals a positive:

-a(-b) = ab = c

Addition

Adding a negative number -

a

to a number

b

, moves to the left from

b

by a distance

a

on the number line:

b + (-a) = b - a

Subtracting a negative number -

a

from a number

b

, moves to the right from

b

by a distance

a

on the number line:

b - (-a) = b + a

**Inverse Relationships**

Multiplication

Addition

Multiplication

Addition

**Negative Numbers**

Addition/Subtraction

Multiplication/Division

Addition/Subtraction

Multiplication/Division

**Order of Operation**

PEMDAS

PEMDAS

**The Malt Spec Sheet**

Hot Water Extract (HWE)

Fine Course Differential (FCD)

Moisture Content (MC)

Hot Water Extract (HWE)

Fine Course Differential (FCD)

Moisture Content (MC)

**Brew House Yield**

Specific Gravity:

A measurement of the density of a sample relative to the density of a reference material.

Density of sample can be the result of anything.

Plato:

Empirically defined as the mass of sucrose by weight in solution with water.

Expressed as a percentage, grams sucrose

Conversion:

Specific gravity is converted to Plato via the following:

But this can be approximated with the following:

Hot Water Extract is a specification of malt that describes how much soluble material is available in the product. It can be described in terms of two different variables:

Grind:

Coarse Grind vs. Fine Grind

Moisture:

Dry Basis vs. As-is

This gives us four options to characterize the extract in our malt:

Fine Grind, Dry Basis: A theoretical maximum in a lab

Fine Grind, As-is: A batch specific, lab-standardized measurement

Coarse Grind, Dry Basis: A theoretical maximum in a brewery

Coarse Grind, As-is: A realistic measurement of extract potential

Luckily, we can translate from one to the other, with a few other pieces from the malt spec sheet:

Fine Course Difference (FCD): a metric describing the numerical difference between the extract from a fine grind congress mash (

a

) and a course grind congress mash (

b

). Described by

a-b.

Moisture Content (MC): A description of the water still bound up in the malt.

Dry Basis -> As-is:

Fine Grind -> Coarse Grind:

Fine Grind, Dry Basis -> Course Grind, As-is

Brewhouse Yield: a percentage (always less than 100%) that describes the efficiency of any specific brewhouse.

Usually somewhere between 80% and 90% for micro breweries.

May vary by brewer and/or environmental conditions.

Ground water temp, outside temp, barometric pressure, etc.

Calculating brewhouse yield:

Working with multiple malts

What exactly are we looking for?

The mass of our extract! In terms of information we know or can measure.

Lets go through the derivation...

Measuring the mass of our wort would require us to measure the density of out wort.

We can get to where we want to go by multiplying by unity

The conclusion is something that can be measured or known:

We know how Gravity and Plato are related, let's simplify...

So this is where we left off:

And remember, S.G. can be defined in terms of Plato:

The simplification from here is pretty straight forward...

We have to make an estimation:

And so this is our final solution for determining the amount of extract in a known volume of wort at a known gravity. This will give us the Total Extract Mass (TEM):

Now we have all the information we need to build our grist bill:

TEM, our chosen Total Extract Mass (aka Total Extract Weight, TEW), determined by the volume and gravity of the brew.

HWE: Hot Water Extract for Course Grind, As-is malt

BY: at the worst, a guess at our brewhouse yield

Malt %: the percentage of each malt type chosen by the brewer.

Lets take a breath. The math gets (a little) easier at this point. Take an inventory of what we have, and what we don't:

Whew!

We gots:

A grist bill! Hooray!

Our brewhouse yield.

A real understanding of what happens in our mash, mathematically speaking.

We need:

A calculation for color.

A way of determining our strike water temperature

An understanding of bitterness and hopping

Color measurement and variables

Metrics of color:

Degrees Lovibond: The original measurement of beer and wort color. Small tinted glass slides compared to beer, each assigned a number.

SRM (Standard Reference Method): A scale that was designed to be normalized to match lovibond, with a scientific explanation:

The relationship between Lovibond and SRM is not simple. Lovibond is linear, SRM is logarithmic, described by:

EBC (European Brewing Convention) is simply 2x SRM.

Loss

Color Loss has multiple sources:

Fermentation in general, with some correlation from temperature

Yeast type (flocculation)

Filtration/fining

Oxidation

Trub removal (aka haze)

Boiling of wort (temp and time)

Loss is typically between 20% & 30% for most microbreweries

Reliably characterizing each step in the process would be costly and time consuming, easier to characterize entire process.

So let's model this, via the following:

And so, calculating color

Similar to building a grist bill, we must use a sum:

Now an example:

In layman's terms, this is expressed mathematically as such:

The "thermal state" of any system is described by three characteristics

It's Mass, M

It's Heat Capacity, C

It's Temperature, T

Objects in thermal contact come into equilibrium by changing one of the characteristics above. Mass and Heat Capacity can't change, so temperature must. Thus we have:

For the particulars of our brew, adding the three components of a mash that come into equilibrium, this becomes definite:

Setting up the Physics

The Details...

Sorting out the Math:

The temperature is the only thing that can change, the deltaT indicates that:

We know some equations that we can use to make things a little easier.

And if we keep things metric (KG and Celcius), the basis for Heat Capacity is that for water, it's 1

A little rearrangement...

And voila! We can calculate our Strike Temp

As a side note:

Hopping

The International Bittering Unit (IBU) is defined as the ppm of isomerized alpha acids in beer, symbolically known as:

From our definition of IBU:

The mass of isomerized alpha acids can be determined:

Utilization is is primarily dependent upon boil time

Now we can convert this into any unit we want:

kgs of hops in liters of beer

lbs of hops in barrels of beer

kgs of hops in hectoliters of beer

Milligrams are not convenient for professional brew scales:

I think in terms of kilograms and hectoliters...

So how do we use this?

There are as many theories on how to hop beer as there are brewing traditions...

The Germans: low alpha, high amount, aroma is important

The English: high quality, low alpha for flavor. High alpha, low quality for bitterness

The Americans: High alpha, high IBU throughout

The Belgians: Hops are really just microbio control...

Whatever the case, a brewer should start by deciding on a bitterness level: choose how many IBUs you want your beer to be.

Next determine what you want the weight of your late contributions to be. Determine how much bitterness this will add:

In any case you will need a hop utilization chart, note the gravity dependence:

Use the equation we derived on the previous page to determine how much bittering hop you need at the beginning of your boil:

Lets go through an example:

Lets continue with the theme of a Brown Porter that we've started...

I know that I want to use 1 kg of Kent Goldings in the last 15 minutes of the boil for some hop flavor. How much bitterness does that add? We've got some maths for that!

Using our hop utililization chart, and the alpha acid percentage from our hops (5%), we can calculate:

Knowing that we want our beer to be right near the middle of the style at 25 IBUs, we have to make up 25-5=20 IBUs with a bittering addition 1 hour before the end of boil. How much Magnum (at 11% A.A.) should we add? Again, we know how to calculate that:

Plug in the numbers (ref. util. chart)...

So there you have it, to achieve 25 IBUs, with a flavor hop addition of 1 kg, our hop charge would be:

60min: 0.9 kg Magnum at 11% A.A.

15min: 1 kg Kent Goldings at 5% A.A.