You are given two containers A and B with equal amounts of liquid in each. The l
ID: 2261538 • Letter: Y
Question
You are given two containers A and B with equal amounts of liquid in each. The liquid in A and B is identical in every way except that the one in B is at a higher temperature than the one in A. Your task is to raise the temperature of the liquid in A as much as possible --- at the expense of the temperature of the liquid in B --- by applying the following operations any number of times:
>> You are allowed to partition any parcel of A or B into parcels of any number and size.
>> You are also allowed to put any two parcels into contact with each other thereby bringing their contents into thermal equilibrium --- without mixing their contents.
>> You may combine (mix) the contents of any set parcels of A or any set of parcels of B into one parcel --- but you are never to combine a parcel of A with a parcel of B. i.e. you must keep liquids A and B from mixing!
At the end you will recombine all parcels of A and all parcels of B. Obviously the final temperature of A depends on what you did.
Q: What is the supremum of the set of possible final temperatures of A and why?
_________
Assume ideal conditions: the combined thermal energy of the two liquids stays constant; thermal equilibrium is attained instantaneously; specific heat, density, etc. stay fixed, so when parcels x and y are brought into thermal equilibrium, their final temperature will be
T = (T_x?V_x + T_y?V_y) / (V_x + V_y),
where T_i and V_i is the temperature and volume of parcel i, respectively.
Explanation / Answer
Some preliminary thoughts....
My first approach was to partition each container's liquid
into 2N equal portions. I then brought in turn each of the portions
from container A into contact with one portion of container B
and then with a second portion, (the same 2 portions of container
B each time). I then repeated this process with another two
portions of container B, and then another two, etc., until I've
'extracted' all the thermal energy from the portions of
container B that I could. I then re-combined the portions from
container A and calculated the final temperature for all of the
liquid from container A.
For N = 2 I end up with a final temperature for A of
T_A + R*(T_B - T_A) = T_A + (93/128)*(T_B - T_A)
I'll try higher values of N and see if I can establish a sequence
of values for R as N -> infinity.
I'm guessing that lim(N->infinity)(R(N)) might be 3/4, since the
R-value for the 'lead' portion of container A will go to 1 and the
R-value for the 'last' portion of container A will go to 1/2.
However, there may be a more effective sequencing of
exposing portions to one another to produce a higher final R-value.
Is this what you had in mind? When I first read the question
my sense was that we wouldn't be able to get an R-value higher
than 1/2, so I'm a bit surprised that we can (potentially) achieve
an R-value of 3/4, and perhaps higher.
@Zsolt. I wouldn't be at all surprised if my guess of (sup)R = 0.75
is wrong; I just wanted to get the conversation started on this
interesting problem. I don't really have a 'gut feeling' for the
problem yet, although I am uneasy about the notion that
sup(R) = 1 in the limit; if the math says it is then that would
be fascinating, but I'll have to see it to believe it. :D
We did take slightly different approaches but the general idea
was similar. I like your approach more in theory; I took mine
because I wanted to be able to do a fairly quick calculation
to get a feel for what was going on. I'll spend more time on it
shortly, taking into account your ideas as well as more of my
own and see what happens. I wonder, too, if there is some
elegant argument using statistical mechanics that would avoid
all the messy calculations.
Edit: Some more results; partitioning each container into 2N equal
portions and then proceeding as discussed above, I find the
following for R(N):
R(1) = 0.625, R(2) = 0.7383, R(3) = 0.7886, R(4) = 0.8165,
each to 4 decimal places. I can't see R(N) going to 1 as N goes
to infinity using this method, but it may come close. I now need
to form an actual expression for R(N) so that I can get a firm
value for sup(R(N)). Zsolt's approach may be more promising,
but I'm finding mine to be easier with regards to making
calculations. Hopefully I'll have some more definitive results later.