Hello there,
Newton's law of cooling looks something like this:
T3=(T1-T2)*e^(-a*t)+T2
where
T3 is the final temperature
T1 is the initial temperature
T2 is the surrounding air temperature
a is a constant
t is time.
We can not change this fundamental property of nature, but what we
can do is change the constant 'a'.
This is actually pretty easy to do too, as 'a' depends on the amount of
insulation surrounding the closed space of say a small rectangular box
built just large enough to house the cell phone and a small light bulb
or other heating device like a resistor.
With such a box, surrounded by a single layer of thick styrofoam with
the inside heated to +30 degrees C, the temperature T2 after say 1 hour
might be around +15 degrees C. Well with twice the original thickness,
the temperature after 1 hour would be +21.8 degrees, so by doubling
the insulation we effectively increase the time that a given amount of
heat keeps the inside of the box warm.
To keep that same box up to about 12 degrees C after 24 hours would
require 20 times the original thickness, but it all depends on the type
of insulation used.
To get an idea what would be required, a box could be built and lined
carefully with some insulation, then the inside heated to say +50 deg
C, and left outside or in the freezer for 1 hour, then measure the
inside temperature again. Using those two measurements and knowing
the time t and the freezer temperature the constant 'a' can be
calculated. Once this is done, the time to keep 'warm' for 24 hours
can be calculated and if the minimum temperature doesnt stay high
enough more insulation can be added. The amount to be added could
be calculated by assuming that for every additional layer of the
same thickness the constant 'a' decreases by one half. Thus, if 'a'
first calculated to 0.00001 then for two layers a would be 0.000005.
T3 can then be calculated again and if that doesnt come up high enough
add another layer, etc., etc.
All this can be calculated from that one measurement.
Alternately, try it with a thermos that has a mount big enough so that
the cell phone fits in it. Heat it up, make a measurement, calculate 'a',
calculate the new T3 for the 12 or 24 hour period. Go from there.
Note that all this is assuming that the inside of the box is heated only once.
If the heater can run off the battery, the situation gets even better. The
more insulation the less power required to keep the inside of the box above
say 10 degrees C. If there is enough room the box can be quite large meaning
lots of insulation and that would require very little power to keep warm.