At the south-western corner of the base of the altar there were two holes like two fine nostrils through which the blood which was poured on the western side of the foundation and on the southern side flowed down till the two streams became mingled in the channel, through which they made their way out to the brook of Kidron.
On the pavement beneath at that corner there was a place a cubit square on which was a marble slab with a ring fixed in it, and through this they used to go down to the pit to clean it out.
After the blood of the sacrifices was sprinkled according to their requirements, the remainder of the blood went down a drain that was built into the foundation of the altar itself. This blood mixed in the plumbing pipes of the Temple with the remainder of other blood that had been poured into a similar drain in the outer altar. From there they emptied into the Kidron Valley, where their remains were sold as fertilizer.
The Kidron Valley, running to the east of the Temple Mount, is the natural run-off point for sewage from the Temple. The walls of the Temple Mount actually stand at the very edge of the banks of the dry river, in which the Shiloaḥ spring flows.
We find in the Gemara in Massekhet Sukka (daf 49a) that Rabbi Elazar bar Zadok is quoted in a baraita as describing that through the opening in the Temple floor young kohanim were sent down once every seventy years to clean out the congealed wine and blood, after which it was burned in the Temple (based on 28:7). The Meiri explains this to mean that the kohanim did not actually go down to the very bottom, but that they would clean as deep as they could using implements that were available to them.