Treat the blue clay (mass 1, temperature T) as a single lump. Feed it infinitesimal lumps of red clay (mass dX, temperature TR). After each infinitesimal feeding, the temperature of the blue clay changes by
dT=T+TRdX1+dX−T=−TdX+TRdX
(Final equality comes from series expansion).
Then you can integrate: ∫Tfinal0dTTR−T=∫10dX.
i.e. log(TRTR−Tfinal)=1
The final solution is:
Tfinal=TR⋅(1−1/e).
It’s worth saying that this is the most efficient way to transfer energy from red to blue because each feeding step is thermodynamically reversible.
Can’t you do better by splitting both blocks up, and doing equilibrating each piece of blue with each piece of red, in turn? By not dividing blue, you are leaving the red pieces a lot warmer than necessary
When T is 1000 you can get the blue to 0.98 of what red was initially… so I imagine in the infinite limit you can get all the way there.
This seemed weird to me, so I looked at the final temperature distribution of all of the n blue blobs, as n increases. I’ve plotted them here (blue lines have n being small; red lines have n being big). You can see that as the number of blobs becomes very big, the fraction which have temperature less than 1 falls… (i.e. knee of graph moves to the right...)
Treat the blue clay (mass 1, temperature T) as a single lump. Feed it infinitesimal lumps of red clay (mass dX, temperature TR). After each infinitesimal feeding, the temperature of the blue clay changes by
dT=T+TRdX1+dX−T=−TdX+TRdX
(Final equality comes from series expansion).
Then you can integrate: ∫Tfinal0dTTR−T=∫10dX.
i.e. log(TRTR−Tfinal)=1
The final solution is:
Tfinal=TR⋅(1−1/e).
It’s worth saying that this is the most efficient way to transfer energy from red to blue because each feeding step is thermodynamically reversible.
Can’t you do better by splitting both blocks up, and doing equilibrating each piece of blue with each piece of red, in turn? By not dividing blue, you are leaving the red pieces a lot warmer than necessary
Having checked this, you’re totally correct.
Here’s a short Python program that checks this empirically.
A graph of the scaling of the max blue temperature with respect to number of divisions is here:
https://imgur.com/a/r4hdaJD
When T is 1000 you can get the blue to 0.98 of what red was initially… so I imagine in the infinite limit you can get all the way there.
This seemed weird to me, so I looked at the final temperature distribution of all of the n blue blobs, as n increases. I’ve plotted them here (blue lines have n being small; red lines have n being big). You can see that as the number of blobs becomes very big, the fraction which have temperature less than 1 falls… (i.e. knee of graph moves to the right...)