A cup of coffee cools off in a room at temperature Tenv.
The temperature T of the coffee
satisfies the differential equation (with respect to time)
T' = - (T - Tenv)/tau
where tau is the relaxation time,
the amount of time the temperature difference in
a constant environment
scales by 1/e.
The temperaure can be at least approximately
calculated by simple steps
T(t + dt) = T(t) - (T - Tenv).dt / tau
The blue graph plots the approximate solution with steps
dt = 1 second. The red graph plots the room temperature.
With a constant room temperature
Trm, the formula
for the temperature T(t) of the coffee is
T = T(0) e- (T - Trm)/tau
In this graph, the room temperature varies in a simple periodic
fashion.
Click on the black node to suspend the process.
Move it to change its speed. Set initial temperature with the blue
node, frequency of environment change with red one.
[Where the room temperature varies
in discontinuous
steps|
Where the room temperature increases
linearly with time|
Back to the Mathematics 256 page]