The Method of Volume Averaging is used to model the process of dyeing textile threads on bobbins. This analysis allows one to upscale the relevant information at the micro-scale, composed of the textile fibres of the thread in contact with the dyeing bath fluid, to the macro-scale, consisting of the bobbins of threads inside the equipment. The final mathematical model consists of two equations, one for the fluid phase external to the thread and the other for the fluid phase internal to the thread. In order to solve the partial differential equations obtained in the mathematical model, the authors developed a computation code using the Method of Finite Volumes. This code utilized a system of generalized coordinates to facilitate application of the boundary conditions to different bobbin geometries. The numerical results for the kinetics of dyeing packed cotton threads with reactive dyes are compared to the experimental results obtained in Brazilian textile industries, leading to good agreement between theory and experiment. This demonstrates that the model developed in this paper is able to predict the operational conditions to be used in the textile industries, minimizing the consumption of dyes and other auxiliaries necessary for the dyeing process.