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Feb., 1950.
ROSS: HEATING OF FILMS AND SLIDES
53
i.e., about -0022 cms. for celluloid. The thickness of the celluloid base is roughly 64 times this distance so that the temperature rise on the right-hand face is about e~6'4 = 1/600 times the rise at the emulsion face. The celluloid therefore behaves as if it were of infinite thickness as far as the heat-flow during the exposure period is concerned.
Due to the poor conductivity of the base and emulsion, it is also clear that during the exposing period the heat cannot spread significantly from the picture area out towards the perforated edges of the film.
Operational Calculus Solution
Having obtained this general picture of what is happening, the operational calculus readily gives a form of solution easier to use than (i) above, especially when the knowledge that the celluloid can be regarded as infinitely thick is used.
Multiplying the differential equation by e~pt and integrating from t = 0 to oo gives
— = —, 0 where 0 is the Laplace Transform of 0.
d*2 K
60 The boundary condition — =
H
for
the solution is 0 = — 7
pqk
—qx
e where q2 = p/K
whence 0
where erf
= f [ Vf
2 f« Vtt J o
AKt
i2
x (l erf
0 for all t becomes
2VKt)_\
d0 d*
H
Pk
so that
On the emulsion face x = 0 = 7-587/ or #=0-1320.
0, so that 0
2H /
When t = 1/50 sec. this gives
d x
-a
Fig. 10. Section of film with emulsion of negligible thickness.
Fig. 1 1 . Section of film with emulsion of finite thickness.
Up to this point the thermal capacity of the emulsion has been neglected. Approximate allowance can be made for this by assuming that at the end of the 1/50 sec. exposure the whole of the emulsion has risen by 0z=o. Inserting values for the constants shows the emulsion would therefore absorb heat at the equivalent of a mean rate of -0180 calories/cm. 2/sec. throughout the exposure. This has to be added to the 0-132 0 absorbed by the celluloid to give H = 0-15 0 or 0 = 6-61H. Given the thermal equivalent of the light incident on the film this formula shows the temperature rise on the emulsion face resulting from any incident light intensity.
In this calculation it is assumed that the emulsion absorbs all the incident radiation, and this is to all intents true for the darker parts of the picture area. Even at a density of 1-0, corresponding to a medium grey, the temperature rise is still 90 per cent, of that calculated. A full white in a normal release positive has a measured density of about 0-2, so that its temperature-rise is about 35 per cent, of the maximum.
More Refined Problem
As a check on the validity of the approximations involved in the above treatments
a more refined problem has been solved. It involves regarding the film as an emulsion
coating of thickness a on a celluloid base of thickness b. Light approaches from the
left in Fig. 11, and is absorbed and generates heat exponentially in crossing the
emulsion.