British Kinematography (1950)

2 BRITISH KINEMATOGRAPHY Vol. 16, No. 2 Picture Production. R. G. Linderman, C. W. Handley & A. Rodgers. /. Soc. Mot. Pic. Eng., 40 (June, 1943), p. 333. 2F.I.A.T. Final Report, No. 1052, by Prof. W. Finkelnburg (1949). The Carbon Arc as a Source of Artificial Sunshine, Ultra-violet and other Radiation. C. E. Greider & A. C. Downes. Trans. Ilium. Eng. Soc. (Amer.), 27 (Sept., 1932), p. 637. 3The Problems of Artificial Illumination, 4. General Electric Company, Ltd. , London, Booklet No. O.S.4000. 4Dreish. Zeit.fiir Physik 30 (1924), p. 200, and Int. Critical Tables, 5, p. 269. (McGrawHill.) 5Private communication from A. Marriage, Kodak Research Laboratories, London. "Chance Bros. Ltd., Smethwick. Data sheets Nos. 5.1315 and 5.131, and catalogue of coloured optical glasses. 7Designed by Taylor, Taylor & Hobson Ltd. Patents pending. 8Study of Radiant Energy at Motion Picture Film Aperture. R. J. Zavesky, M. R. Null & W. W. Lozier. J. Soc. Mot. Pic. Eng., 45 (August, 1945), p. 102. 9Effect of High Intensity Arcs on 35 mm. Film Projection. E. K. Carver, R. H. Talbot & H. A. Loomis. /. Soc. Mot. Pic. Eng., 41 (July, 1943), p. 69. "Private Communication from Research Laboratories, llford Ltd., London. nThe theories of cooling in this paper are based on " Heat Transmission," by W. H. McAdams. (McGraw-Hill, 1942.) i2Patents pending. APPENDIX MATHEMATICAL TREATMENT OF HEATING OF FILM IN GATE from a paper by Brian S. Kellett Fourier Method The first step is to translate the physical problem into a form suitable for mathematical treatment. Consider a piece of film as a celluloid base of thickness d, as shown in the diagram, with an emulsion coating on the left-hand face thin in comparison with the celluloid base; this is a reasonable approximation, since the celluloid base is in fact about sixteen times as thick as the emulsion, and represents a great simplification mathematically. Away from the edges of the film there will be no radial flow of heat and the temperature in the film base, 0, will satisfy the one-dimensional heat-flow equation: 56 620 k ~ = A^~2 where K= , k being the thermal conductivity, p the density and o the specific heat of the celluloid. A solution is required satisfying the boundary conditions that (i) the celluloid is initially all at the same temperature, i.e., 0 = 0 when /=0 for all x; (ii) heat is generated in the emulsion and flows into the celluloid at a steady rate of H calories per sq. cm. per sec., i.e., = = — rwhen x = 0 for all t; and (iii) no heat flows through the right-hand face, i.e., 60 £ = 0 when x = d for all t. By Fourier's method the solution can be shown to be: H\d 2d £ 1 rnx ^r™ Kt~] Hf x*\ , I |_3 ~& * Pi cos Te:m J " TV ~2d) + HK kdf The Fourier method depends on the fact that e~ K' cos mx is a solution of the differential equation. Whatever the temperature distribution in the celluloid at the end of the exposing period it can be expanded in a Fourier series of cosines of multiples ofTi/d. The principal term in the solution for the heat-flow after the exposing period will be a multiple of e &~ cos ~a and shows that the inequalities of temperature decay with a time-period of 7^' which is about 1/20 second when suitable values of the constants for ;erte ttZK celluloid are inserted CO Equally e sin ( co/ — \/ ^rj>x ) is a solution of the differential equation. id could be exp , giving co = 50' . IlK A / K celluloid which decay exponentially, falling in amplitude to 1 /em a distance \ — = 'y 25tt The exposure of 1/50 second could be expanded as a Fourier series with the principal term of period 1/25 second, giving co = 50tt, showing that heat waves are sent into the