Lauter efficiency (one factor in mash efficiency) is determined solely by percentage of sugar that was created in the mash that stays in the spent grain at the end of the lauter process
Lauter Efficiency = Weight of Sugar Left in Grain / Weight of Sugar Created in Mash
Two things affect the weight of the sugar left in the grain:
- The volume of wort left in the grain
- The sugar concentration of the wort left in the grain
For equal volumes of retained wort, a higher SG retained wort contains more sugar, and thus has lower efficiency. For equal SG's of retained wort, a lower retained volume contains less sugar, and thus has higher efficiency. Sparging reduces retained wort SG by rinsing and dilution, but does not reduce retained volume. Squeezing reduces retained volume, but does not reduce the SG of the retained wort. The two effects operate independently, but when combined they have a multiplicative effect.
We can do a simplified example using gravity points, instead of the more rigorous weight, without significant loss in accuracy or generality. if you leave 1 gal of 1.050 wort in the grain, you have left 50 gravity points in the grain. If you squeezed the bag and instead left only 0.5 gal of wort in the grain then you have only left 25 points in the grain, and the efficiency will be higher. If instead, you diluted the wort by sparging to an SG of 1.025, but didn't squeeze and still left 1 gal in the grain, you would also have only left 25 points in the grain. So, equal efficiency. Now if you sparged to 1.025 and squeezed to 0.5 gal retained, then you would only leave 12.5 points in the grain for even higher efficiency. So, it all comes down to practical levels of squeezing and/or dilution of the retained wort. These can be traded off in many ways.
I have produced a simplified version of the lauter efficiency chart. For no-sparge, I have left different levels of squeezing (0.12 gal/lb = no-squeeze, 0.04 gal/lb is better than most folks can squeeze [my best is 0.05 gal/lb].) For traditional MLT sparging, I have deleted all but the no-squeeze line (0.12 gal/lb), since traditionalists don't normally squeeze. You can see that for normal beers (up to about 1.7 ratio of grain weight to pre-boil volume [about 11.4 lb of grain with 6.7 gal pre-boil volume]), it is difficult for squeezing to beat sparging for efficiency. However, as the beers get larger, it becomes easier to beat sparging alone with no-sparge squeezing.
View attachment 401202
As far as volume ratios for strike vs. sparge, it turns out that you get maximum lauter efficiency when the run-off volumes are equal for first and second runnings. But since
first runnings volume = strike volume - apparent absorbed volume, and
second runnings volume = sparge volume (since the grain already contains the absorbed volume)
you want your strike volume to be higher than your sparge volume in order to get equal runnings volume. However, it turns out that you don't have to hit exactly 50:50 run-off volumes to get essentially max efficiency. Here's a nice chart that @pricelessbrewing put together using my sparge simulator spreadsheet.
View attachment 401210
You can see that once the run-off ratio gets to about 60:40, 0.67 on the X-axis (first:sparge runnings) efficiency does not change significantly as you move towards 50:50. Moving past 50:50 ratio, things don't start to change much until you get past 40:60, 1.5 on the X-axis (first:sparge runnings.) An easy rule of thumb if you want to maximize lauter efficiency for a single batch sparge is to just use 60% of your target pre-boil volume for strike, and 40% for sparge. The grain absorption will push the actual run-off ratio towards 50:50, and you won't be in danger of going past the 40:60 ratio until you get somewhere around 27 lb of grain for a 6.7 gal pre-boil volume. If you want to do a double batch sparge, then the rule of thumb changes to 50% strike, 25% first sparge, 25% second sparge.
Hopefully this gives you additional insight. If you have more specific questions, just ask.
Brew on