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["WILLIAM JONES" <billj AT harborside.com>]

Now let's turn to the question of how long it takes for a blackbottle to 
complete one full cycle of heating and cooling.  Before discussing the very 
simple differential equations that govern this, lets establish a few 
parameters for them
Tw = water temperature
Ta = air temperature
s = solar brightness factor
Ca = cooling coefficient in air
Cw = cooling coefficient in water
Cooling is easy.  Newton's Law of Cooling says that the rate of change of 
temperature is a negative mutliple of the difference between the current 
blackbottle temperature and the water temperature Tw:
            T' = -Cw (T-Tw) = -CwT + CwTw
This is a first-order inhomogeneous differential equation with a constant 
term.
During my last night at OHSU hospital I figure out the law that governs 
heating.  There is no heat reservoir at a fixed temperature, equivalent to 
the cold water.  Instead, the temperature increases constantly from the sun's 
heat, but the blackbottle simultaneous tends to cool toward the air 
temperature.  So the equation is
            T' = s - Ca(T - Ta) = (s + CaTa) - CaT
This is also a first-order inhomogeneous differential equation with a 
constant term.
While cooling causes the blackbottle to approach a temperature asymptote, we 
should think of the final temperature in the sun as the equilibrium point at 
which energy entering equals energy leaving.