Universal Journal of Applied Mathematics Vol. 14(2), pp. 39 - 46
DOI: 10.13189/ujam.2026.140202
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A Simplified Derivation of the Kubelka-Munk Equation and Its Application in the Textile Industry: A Technical Review

Elias Khalil ^1,*, Md. Saifur Rahman ², Md. Mahfujur Rahman ^2
1 Department of Wet Process Engineering, Tangail Textile Engineering College, Bangladesh
² Department of Textiles, Ministry of Textiles and Jute, Dhaka, Bangladesh

ABSTRACT

The Kubelka-Munk (K-M) theory is fundamental in the textile industry for color management, providing a mathematical framework to describe light interactions with textile substrates. This paper offers a simplified, step-by-step derivation of the K-M equation to make the underlying mathematics more accessible to students and professionals. The K-M theory, which connects absorption (K) and scattering (S) coefficients to material reflectance, plays a crucial role in modern color matching systems. This study reviews the theoretical aspects of K-M, including its assumptions and limitations, particularly in complex textile materials. The manuscript also discusses how K-M theory is applied in computer color matching (CCM), recipe prediction, and shade correction in industrial settings, reducing the need for traditional trial-and-error methods. Additionally, the paper addresses the limitations of the K-M model, such as its assumptions of homogeneity and linearity, which can lead to inaccuracies in multi-dye and multi-layered systems. The review also introduces the Saunderson correction to account for surface reflections and improve the model's accuracy. This work aims to bridge the gap between theoretical understanding and practical application, contributing to the ongoing effort to refine color matching processes in the textile industry.

KEYWORDS
Spectrophotometry, Color Matching, K-M Theory, Scattering, Absorption

Cite This Paper in IEEE or APA Citation Styles
(a). IEEE Format:
[1] Elias Khalil , Md. Saifur Rahman , Md. Mahfujur Rahman , "A Simplified Derivation of the Kubelka-Munk Equation and Its Application in the Textile Industry: A Technical Review," Universal Journal of Applied Mathematics, Vol. 14, No. 2, pp. 39 - 46, 2026. DOI: 10.13189/ujam.2026.140202.

(b). APA Format:
Elias Khalil , Md. Saifur Rahman , Md. Mahfujur Rahman (2026). A Simplified Derivation of the Kubelka-Munk Equation and Its Application in the Textile Industry: A Technical Review. Universal Journal of Applied Mathematics, 14(2), 39 - 46. DOI: 10.13189/ujam.2026.140202.