*This post is intended to serve primarily as documentation for the Batch Sparging Calculator. If you’re looking to skip to the end of the page, head over that way instead.*

One great thing about batch- or no-sparge brewing is that it’s fairly easy to predict lauter efficiency, and with good results. This is due to the fact that in batch sparging, the actual mechanics of the lauter tun – the way that wort flows through the grain bed – are neglected. Some of the wort is drained, and a set fraction is left behind in the tun. We begin with a few assumptions:

- Conversion efficiency is 100%.
- Conversion is complete before lautering begins.
- No additional grain is added during lautering.
- Each infusion is fully drained, less any deadspace.

The lauter efficiency is then a simple ratio of the two wort fractions:

E = V_{1}/V_{0}

Where E is the lauter efficiency, V_{1} is the volume run off to the kettle, and V_{0} is the total strike volume in the lauter tun. As a practical matter, efficiency is maximized when the tun is drained as completely as possible, and so the volume run off is equal to the infusion volume minus the volume absorbed by the grain and any deadspace that can’t be drained:

V_{1} = V_{0} – V_{a} – V_{d}

There is one additional factor that must be considered, and that is the expansion of the wort due to dissolved sugars. When measuring the volume and gravity of the wort, the apparent extract will be less than the estimated efficiency. We can compensate by defining an expansion coefficient, C, which is the inverse of the wort specific gravity:

C = 1/SG

When it comes to analytically determining the expansion coefficient, however, we encounter a Catch-22: the efficiency determines how much extract is in solution, but the amount of sugar extracted also depends on efficiency. In practice, the calculator just takes a brute-force numerical approach of iterating the efficiency calculation three times, approximating the expansion coefficients. The full expression for lauter efficiency is therefore:

E = CV_{1}/(V_{1} + V_{a} + V_{d})

For a no-sparge beer, this is all that is needed. When considering one or more sparges, however, we have to add the extract contributions from the additional infusions. The extract available for sparging is, by definition, whatever remains after draining the previous infusion(s). This is the complement of the lauter efficiency:

(V_{a} + V_{d})/(V_{1} + V_{a} + V_{d})

And so the overall efficiency contribution of the n-th sparge is:

E_{n} = C_{n}(V_{n}/(V_{n} + V_{a} + V_{d}))*((V_{a} + V_{d})/(V_{n-1} + V_{a} + V_{d}))

Summing the individual extract efficiencies gives the overall lauter efficiency. Expanded to four terms (three sparges), which is about the most that would ever be reasonable, the expression becomes:

E = C_{1}((V_{0} – V_{a} – V_{d})/V_{0}) + C_{2}((V_{a} + V_{d})/V_{0})(V_{2}/(V_{2} + V_{a} + V_{d})) + C_{3}((V_{a} + V_{d})/V_{0})((V_{a} + V_{d})/V_{2})(V_{3}/(V_{3} + V_{a} + V_{d})) + C_{4}((V_{a} + V_{d})/V_{0})((V_{a} + V_{d})/V_{2})((V_{a} + V_{d})/V_{3})(V_{4}/(V_{4} + V_{a} + V_{d}))

Which I realize looks ridiculous written out like that but is computationally really straightforward.

With that done, all that remains is to multiply the efficiency by the (apparent) total extract, and divide by volume to get gravity. If you look at the source code for the calculator you’ll see that everything else is just parsing input and prettying up the results for output.

By the way, this is nothing new; batch sparging analysis has previously been taken up by Ken Schwartz and Kai Troester, among others.