The Relation between Dye Concentration and Reflectance–Amendments to the Kubelka–Munk Equation

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Abstract
The basic Kubelka–Munk formula used in our recipe–prediction work (Eqn 1) relates the reflectance and concentration at every wavelength for every dye. In practice, the equation is satisfactory in most instances, but for some dyes at certain wavelengths the agreement between prediction and actual dyeings is very poor. To use the equation, one parameter, the absorption coefficient, is required for every wavelength for every dye. The discrepancy cannot be overcome by changing the value of the absorption coefficients for the particular dye. A more complicated equation (Eqn 3) is shown to be of little help in overcoming the difficulties. A more general relation (Eqn 7), including two parameters at each wavelength, is found to hold extremely well for dyeings of single dyes. The extension of this equation to mixtures of dyes is considered and alternative equations are suggested. A computer programme has been written to predict recipes with the suggested alternative formulae for mixtures of dyes. This has shown that such amendments to the theory are likely to slow down the rate of recipe prediction on a digital computer by a factor of about four.

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