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where the Prandtl number, in a dimensionless coefficient, is defined as
(4)
Similarly, it was found that the mass transfer will be governed by the following correlation (which was first pronounced by Gilliland):
1 = i X 0.023 (flt)M»(&)M« (5)
where the Schmidt number is
(6)
As a result of this formulation, means are now indicated for reducing the thickness of the stagnant layer B. If a specific fluid or gas is selected for experimentation the following parameters usually remain constant within the range of experimentation:
Viscosity, /* Density, p Diffusivity, d Specific heat, Cp Conductivity, k.
As a result the Schmidt number which contains the values /i, p, and 5 will be constant. The Prandtl number containing Cp) n, and k will be constant. Consequently Eq. (3) can then be written as follows:
(7)
and Eq. (5) can be rewritten as follows:
(8)
Usually the less exact notation is preferred as follows:
(9)
This equation indicates clearly that the thickness of the stagnant layer B which is a controlling factor in the rate of diffusion (and thus the rate of develop
ment) is inversely proportional to factor Re*-*/D.
If we have a tube through which liquids or gases flow, the mass transfer coefficient cannot be indefinitely increased, as for a given diameter an increase in velocity will result in an increase in horsepower required to drive the gases or fluids through the tube. Similarly, a reduction in diameter at constant velocity will cause an increase in the pressure drop required to force the gases or liquids through the tube resulting in an increased horsepower and equipment size. Consequently, there is a practical limitation to the choice of the velocity and the diameter. Velocities near or above the sonic level are usually undesired because of other effects accompanying supersonic flow. A reasonable compromise, however, can usually be found with velocities in the vicinity of 50 to 400 miles per hour and diameters in the range of 0.1 to 7 mm. This permits the use of Reynolds numbers well in the turbulent region in the vicinity of 50,000 to 500,000 and Re°-*/D factors of considerable magnitude. The horsepower required to drive the fluids or gases under those conditions is usually quite low, and experiments can usually be performed with pumps or blowers driven by motors of reasonable size (ranging from 1/20 to 2 hp). In general, the design of the equipment is dictated to a great extent by the availability of a pump, and usually the design of a turbulent fluid chamber is based on a particular pump or blower which is available rather than on optimum theoretical considerations.
Application of the Theory
to Photographic Development
The foregoing theory indicates that improvement can be obtained from the application of turbulent fluids to photographic development and that this will be a function of Re*-*/D. The improvement, however, will only be significant if the following conditions prevail:
110
February 1953 Journal of the SMPTE Vol. 60