Although the math itself is pretty straightforward, the harder part is deciding upon the values of the many variables that go into the calculation.
For the area of the Azarah that was available for people to stand in, I used the value of 12,243 square amos based on this earlier post (with some minor adjustments). One hugely significant factor here is whether we postulate that the Ezras Yisrael was 2.5 amos lower than the Ezras Kohanim. In my models I always follow the view of Tiferes Yisrael that the Ezras Yisrael was at a lower elevation, but if so all of the blood from the main Azarah would flow into Ezras Yisrael. This would mean that the entire Ezras Yisrael would be filled with blood to a depth of 2.5 amos before the main Azarah would begin to fill, adding a significant amount of volume to the calculation! For the purposes of this post and for the sake of simplicity, I would like to assume that the Ezras Yisrael was at the same elevation as the main Azarah.
For the depth of the blood I used the value of six centimeters, the approximate height of [the lowest point of] my ankle from the floor. I am using centimeters and not amos because we will be dealing with volumes in terms of cubic centimeters and this makes the conversions easier. And why centimeters and not inches? Because the metric system is the best system (sorry, America).
Once centimeters are involved it becomes necessary to pick a value for the length of an amah. If an amah is roughly 1.5 feet this comes to 45 cm. If you like the Chazon Ish amah, you should use a value closer to 60 cm.
At this point we can calculate the volume of blood in the Azarah, which is area times depth. This comes to 148 million cc of blood.
This number must be divided by the amount of blood that spills onto the floor per animal to yield the number of animals shechted. How do you figure out how much blood comes out of a one-year old lamb when it is killed using ritual slaughter? By reading a lot of livestock websites where we learn that this is a point of great interest to the meat industry. What I extracted from my perusal of the literature is the value of roughly 2000 cc.
From these 2000 cc that exit the animal at shechitah, a certain amount must be captured for zerikah on the Mizbeyach. We will assume that this amount drains through the holes in the yesod directly to the stream of water and does not remain in the Azarah. There is no set amount needed, so I used the value of 100 cc. This leaves 1900 cc of blood to spill on the floor.
We may now calculate the number of animals slaughtered by dividing 148 million (total cc of blood) by 1900 (cc per animal) to get 78,000 animals.
If that seems like a lot, it is. If it seems like too many, it is, as we will see below. There is another factor to keep in mind, which is that there were thousands of people standing in the Azarah at this time, and all those feet take up volume on the floor. By subtracting the volume of the human feet we can get a more realistic amount for the total volume of blood on the floor. Based on the same post I referenced above (with some adjustments), we can say that there were maybe 12,000 people standing on the floor. A human foot has the volume of roughly 850 cc (says the internet), so this nets a savings of 20 million cc, leaving us with 128 million cc of blood on the floor.
We may now recalculate the number of animals slaughtered by dividing 128 million (total cc of blood) by 1900 (cc per animal) to get a reduced value of 67,000 animals.
This is still too high, because once we know that there were 12,000 people in the Azarah, and each of those people is holding one lamb for the korban Pesach [we will not count the working Kohanim here], there can only be 12,000 animals total.
If we assume that each animal was lying on the floor during slaughter, we can further subtract the volume taken up by the bodies of the lambs from the volume available for the blood. Referencing the previous post once again, we will assume that a lamb takes up about 0.5 square amos of space. The volume it takes up depends on how deep the blood is on the floor around it, but let us take 6 cm as a starting point. Thus, each lamb will reduce the volume by about 3,000 cc.
We are running into some circular reference now, because we need to know the total number of lambs present in order to calculate the number of lambs present, if you follow the logic. Let us then set an arbitrary volume of space taken up by the lambs in total just to move things along. Assuming roughly 12,000 lambs at 3,000 cc each we have 36 million cc of space taken up by lambs on the floor. Removing this volume from the 128 million gets us down to 98 million cc of blood.
We may now recalculate the number of animals slaughtered by dividing 98 million (total cc of blood) by 1900 (cc per animal) to get a reduced value of 48,000 animals.
Sadly, this is still too high, but it is the starting point of a discussion. Feel free to play with the numbers below to see if you can come up with a combination that yields better results. A gut moed!