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simpler expressions can be deduced for systems common in motion-picture developing machines. In some tanks, the leader is dry when it enters, so that the carry-in rate, A, is zero. It is evident that in such tanks the concentration remains constantly at C0. In other tanks of processing machines, the rate at which water is introduced into the tank by carry-over is equal to the rate at which solution is removed from the tank (i.e., A is equal to B). A differential equation may be set up describing this condition, and solved as follows:
dW
W
dW
W
w_
W0
-WB T
-Bdf T
-Bf T
-Bf
-Af
(6) (7) (8)
(9) (10)
This equation applies as long as the carry-over rate at the entrance is at least equal to the carry-over rate at the exit. The carry-in rate and carry-out rate are equal, for example, when rollers at the entrance and exit ends of a tank are positioned at the same height, and no hardening action takes place in the tank. The carry-in rate exceeds the carry-out rate when, for example, a tank is equipped with a squeegee at the exit end, but not at the entrance. Liquid then overflows from the tank, and despite the difference in carry-over rates, the rates of loss and gain of liquid nevertheless remain equal, so that the equation still applies. Figure 1 summarizes the relations between concentration, carry-over rates, tank volume, and footage for the different tanks. Sodium sulfate, sodium acetate and glycerol may be cited as examples of
chemicals depleted by film primarily because of carry-over.
Before replenisher formulas and rates can be calculated, values of carry-in (A) and carry-out (B) rates must be obtained. Only the simplest methods of determining carry-over rates are mentioned here, but more involved methods may prove more precise. A measured quantity of film is first passed through the machine. If the volume of solution in the tank changes, measurement of the drop in level of the solution or the amount of liquid needed to maintain a constant level will indicate the net liquid loss. Also, both before and after the film is passed through, the solution is analyzed for one of the nonreactive constituents. The value of A, for the general type of tank, may then be calculated when these figures are substituted in Eq. (5), expressed in the form:
_ Volume Change
•"• — 7 X
log Co log C
log T — log ( T — Volume Change)
The volume reduction per unit length of film represents the difference between carry-in (A) and carry-out (B) rates. In tanks where the film is dry at the entrance, A is zero, and B therefore is equal to the volume loss divided by the quantity of film. In those tanks where the liquid level remains constant, values of A and B can be calculated from Eq. (10) expressed in the form:
(ID
(12)
If the solution overflows, then the liquid loss rate (B) calculated from this equation includes both the carry-over rate at the exit (Be) and the overflow rate (B0). Since the overflow can be readily collected and measured, the value of overflow per foot of film can be calculated. The difference between carryin and overflow rates indicates the carryout rate. Most machines are operated over a range of speeds, and so values of
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January 1954 Journal of the SMPTE Vol. 62