Ok, let's try an over simplified example that will be quicker to write and easier to follow than an analysis of a real mash. It will still illustrate the basic behavior of a batch sparge vs. no sparge process. We will ignore the grain, and just use sugar and water, and so that we don't have to switch back & forth between volume and weight, we'll just do everything in volume & volume %.
For the no sparge case, let's say we start with a mash that contains 9 gallons of water and 1 gallon of sugar. The solution is then 1/(1+9) = 10 vol% sugar. If we drain 8 gallons of the solution into the BK, then the BK contains 8 * 0.1 = 0.8 gal of sugar, and 0.2 gal of sugar remains in the MLT (as grain absorption or undrainable volume, makes no difference.) Our mash efficiency is then 0.8 gal / 1 gal = 80% (BK sugar / Total sugar.)
For the sparge case, let's say we have a mash that contains 5 gal of water and 1 gal of sugar. The solution is then 1/(1+5) = 16.67 vol% sugar. If we drain 4 gal of solution into the BK, then the BK contains 4 * 0.1667 = 0.6667 gal of sugar, and 0.3333 gal of sugar remains in the MLT, along with 1.6667 gal of water. Then we sparge by adding 4 gal of water to the 2 gal of solution in the MLT. The MTL now contains 6 gal of solution, 0.3333 gal of which is sugar. So, the sugar concentration in the MLT is 0.3333/6 = 5.556 vol% sugar. We then run off another 4 gal of wort into the BK. The sugar content of this wort is 4 * 0.05556 = 0.2222 gal. Thus the total sugar in the BK is 0.6667 gal (from first runnings) + 0.2222 gal (from sparge runnings) = 0.889 gal. Thus we recovered an extra 0.089 gal of sugar by batch sparging, and our mash efficiency went from 80% to 88.9%.
Yes, that really is the simplified version.

I'll try to finish the "real" mash analysis write up in all its gory detail Thursday.
Brew on