A normal practice when slaughtering animals is to drain all their blood from the body while it is still hot. So most of the blood which would have been in the blood vessels in the muscle would have drained out at the slaughter house. Any remaining dropps of blood would become thicker as the meat cooled down from body temperature and therefore does not flow as well. And over time it would coagulate as well which makes it solid. This means that when you cut open a piece of meat there will not be much blood in it at all and the little blood there is will be a solid and not flow out all over the place.
Packaged meat is washed and drained of blood before it is packaged.
Muscles are not sealed systems. All the parts of your body that require blood are threaded with tiny blood vessels that help carry it there. A standard part of butchering any animal is hanging the animal up to let all of the blood drain.
If you were to kill a chicken and extract the chicken breast yourself, it would be plenty bloody.
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