keithe@teklabs.UUCP (Keith Ericson) (01/18/84)
OK, here's the problem statement: you pour a cup of coffee, to which
you are going to add some milk, when suddenly (would it be any other
way?) the door-bell rings. You want to maintain the coffee at the
highest temperature possible, so you have to decide whether to add
the milk before or after answering the door. Which do you choose?
And, does the initital temperature of the milk affect the decision?
Well, let's use an analog of the problem, couched in EE terms:
we'll use charged capacitors, resistors and a switch...
R R
/ 1 2
|--/ --/\/\/---------|-------/\/\/\------|
| | |
+ | + | + |
V ----- C V ----- C V ----- C
milk ----- milk coffee----- coffee room----- room
- | - | - |
| | |
| | |
|---------------------|-------------------|
The (initial) temperature of each is analogous to the (initial)
voltage on the corresponding capacitor. When the coffee, of
temperature T is poured into the cup, heat (energy) transfers
coffee
to the room, which is at T . At some time later the milk, at
room
T is added, which is accomplished in the analog by closing
milk
the switch. R is very small, while R is considerably larger.
1 2
(The "capacitance" of the room is nearly infinite
compared to the rest of the system, so C could be replaced
room
with a constant voltage source.)
So now what happens? The expected exponential decay of
voltage/temperature from the initial, to the final, with a
discontinuity in the curve when the switch is closed.
Use either conservation of charge or conservation of energy
determine the solution.
Qualitatively, the effect is as follows:
If the milk is never added, you will get a temperature
curve something like...
T | *
coffee | *
| *
| *
| *
| *
| *
| *
| * (That's supposed to look like
| * an exponential decay, folks...)
| *
| *
| *
| *
| *
| *
| *
| *
| *
| *
| (asymptotically approaching *
| the final value...) *
T |------------------------------------------------------------------
room
By adding the milk, you chop out an internal section of this curve and
glue the resulting pieces together. The size of the piece you cut out
depends on the difference between the temperature of the milk and the
temperature of the coffee at the time the milk is added.
The math is left for the student (and others who still remember enough
of it to perform it)...
keithe at teklabs