I really appreciate your interest in this, so let me explain in detail:
Strike water was 9L, this was measured.
Grain was 2.7kg and it should have absorbed 1.8L and it then should be 7.2L (estimate)
I sparged with 6.8L (measured) and it should then be up to 13.8L (estimate)
I took a hydrometer reading at the time, it was 1.034 @ 55.5 C, which ends up in a calibrated 1.045 pre boil gravity. This was also measured once it was cooler and it was precise.
So pre-boil volume = 13.8L (estimate) and pre-boil gravity 1.045 (precise). So far, all according to plan.
After 108 minutes of boil, my kettle should have lost 6.8L. So post boil volume should be 7L (estimate)
I took hydrometer reading at that time and it was 1.072 @ 26 C, which ends up in a calibrated 1.073 post boil gravity. This was also measured once it was cooler and it was precise.
I then diluted my wort with 5L (measured) of water, and this should bring the volume up to 12L (estimate). Once I transferred the wort into the fermenter, I saw it was indeed 12L. Because the 5L addition was measured, this means the estimate of my post volume was actually precise as well!
I took a final hydrometer reading at that time and it was 1.045 (calibrated).
So, post-boil volume 7L (precise) and post-boil gravity 1.073 (precise) and
post-dilution volume 12L (precise) and post-dilution gravity 1.045 (precise)
The only volume for which I can't be sure is the pre-boil volume. This leaves room for:
- grain absorption miscalculation
- boil off losses miscalculation
But are these enough to explain the odd numbers?
I guess I enjoy problem solving. So I have completed a lot more number crunching to try to determine the most likely source of error, as these numbers cannot possibly all be accurate. Here is what I have come up with.
The most likely error might be your preboil temperature. If preboil temp was actually 35 C instead of 55.5 C, you would have preboil gravity of 1.038 instead of 1.045 (corrected for room temperature from 1.034 to 1.038). From this point, all of the math works out pretty well, with minor adjustments. I was able to figure out the most likely actual numbers, allowing for minor margins of error for each data point. Given the preboil gravity of 1.038 instead of 1.045, if the following data were most accurate, then everything works out:
Grain absorption was not 1.8L but rather 2.3L. This leaves (9 - 2.3) = 6.7L from the first strike.
Sparging with 6.8L gave you a total 13.5L preboil volume (not 13.8L). The SG here then was 1.038 @ 13.5L as corrected for room temperature.
Boiloff rate was about 25%, taking the volume down to 7.4L.
Important: At this point, shrinkage occurs due to the boil temperature cooling down to room temperature. This results in a loss of about 0.3L, taking the volume down to 7.1L. Gravity at this point was measured to be 1.073 which appears to be pretty accurate, or perhaps was 1.074.
From here, you then diluted with 5L more to about 12L, with a gravity of about 1.044 (perhaps a point lower than your 1.045). Tiny errors of 0.001 like this seem insignificant but can make some difference in the end. More significant perhaps might be any volume losses, if you lose a cup or two of volume here & there, it can throw off some of the readings.
So now... what does all of this mean? Well... honestly..... not a whole lot maybe! It just looks like something was messed up with the preboil gravity or temperature measurements, causing additional confusion. With some number crunching, it shows how tiny errors can make a difference.
But the real bottom line, as far as efficiency goes.....
Yes, you only got about 65% brewhouse efficiency. But so what! To some extent, this should be expected for high gravity beers and/or concentrated boils. You had a high gravity mash and boil that went to 1.073-1.074 by the end of the boil. That will tend to hurt efficiency unless you sparge more to collect more sugars. Otherwise, higher gravity beers will tend to give you lower efficiency. It's just part of the nature of high gravity brewing. If you want to improve your efficiency, you can try a few different things:
1) Mill the grains harder. This can improve efficiency by as much as 10-15%. It really helps a lot.
2) Sparge more, and boil longer. The more you sparge, the more sugars you collect. And you can always boil longer to achieve a higher gravity. Combining a bigger sparge and longer boil, you can easily have 75-80% or more brewhouse efficiency with higher gravity brews. Something to keep in mind for future batches.
3) Don't concentrate the boil and dilute later. The water you added at the end of the process, if added to the sparge instead, would have collected more sugars.
Hope this helps. If not, oh well, I just enjoy mathematics a lot. Yeah I'm kind of weird. Cheers.