Excretion-related thermodynamics
[Slightly vulgar shower thought]
How does excretion (defection or urination) affect the thermal energy of the body? The substance retains some thermal energy from the body, but does losing that substance cool the body in any way?
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I'm not sure about number 2, but your bladder actually acts as a heat sink since piss is mostly water. Water has a crazy high heat capacity, so it drains heat from your body. It's a common tip for camping in extreme cold temperatures to not go to bed with a full bladder since your body has to spend some amount of calories warming up the piss.
I don't see how it makes a difference in the cold. It's already body temperature, so there would be a thermal equilibrium. Heat loss is a function of the body's surface area, which is unaffected.
The body is not a perfect thermal insulator so you must note that the liquid in the bladder is constantly losing heat due to dissipation into the surrounding tissues then the environment around the body. The greater the temperature differential between the body and the environment, the faster the rate of transfer. Your body won’t (or at least it’ll try its damndest) to not let that internal temp drop, which will take more and more energy to maintain as the external temp drops.
This is the right answer. To add to this; you produce more urine when you're cold for this very reason. When you have to go, you have to go. But try to stay in your tent and piss in an empty bottle, then cuddle up to that bottle when it's still warm. You get the best of both worlds.
But the piss will eventually leave your body, so all of the heat energy that went into it is going to be dumped out.
Well it starts at body temperature, and if it stays in thermal equilibrium, then it doesn't require energy to maintain. It wouldn't make a difference, you'd be losing the same amount of heat energy either way.
Based on what others have said, I think the answer is that it isn't really in equilibrium. If the bladder is in contact with tissues that are warmer on one side than the other, then it will basically act as a thermal conductor to move heat away from your core.
Your body isn't a closed system. You are constantly radiating heat away, and you are constantly burning calories to maintain your body's core temperature. So the bladder is a tank full of mostly water inside that is draining those valuable calories and then will just be pissed away.
It's not like the bladder is radiating heat away when it's inside the body. Heat doesn't just get used up, it has to go somewhere. If it was in equilibrium it would just increase your thermal mass and have no effect.
Like I said though, I think the issue is that it's constantly moving heat away from the core to the lower abdomen where it can be radiated or conducted away.