Abstract
A two-dimensional process of dyeing yarn coils was modelled in COMSOL Multiphysics to evaluate the dyeing characteristics according to a pilot-scale process. The proposed mathematical model was able to reproduce the experimental data and great importance was identified for the dyebath flow through the axial exit of the coil in relation to the radial outlet, about 2.8 times greater. To maintain the same flow, the permeability of the bobbin decreased tenfold between the radial and two-dimensional flow. Changes in the pressure gradient greatly influence the velocity field. The velocity and concentration fields found reinforce the preferential use of dyeing bobbins that do not have gaps between them, forcing the dyeing bath to follow the radial flow.
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Abbreviations
- \({C}_{ali}\) :
-
Feed concentration in the inner bobbin radius (mol m−3)
- \({C}_{i}\) :
-
Dye concentration of the dyebath (mol m−3)
- \({C}_{out}\) :
-
Average dye concentration in the fluid phase of the bobbin (mol m−3)
- \({C}_{p,j}\) :
-
Dye conccentation in the solid phase (mol m−3)
- \({D}_{h,j}\) :
-
Hydrodynamic dispersion tensor (m2 s−1)
- \({D}_{e,j}\) :
-
Effective diffusivity (m2 s−1)
- \({P}_{in}\) :
-
Inlet pressure (Pa)
- \({P}_{out}\) :
-
Outlet pressure (Pa)
- \({V}_{d}\) :
-
Volume of dyeing bath (m3)
- \({q}_{p,j}\) :
-
Adsorption isotherm (m3 kg−1)
- \({\varepsilon }_{\beta }\) :
-
Volume fraction of fluid phase (−)
- \({\varepsilon }_{\sigma }\) :
-
Volume fraction of solid phase (−)
- \({\rho }_{\beta }\) :
-
Specific mass of the fluid (kg m−3)
- \({\rho }_{\sigma }\) :
-
Specific mass of the fiber (kg m−3)
- ∇:
-
Differential operator (−)
- H :
-
Height of bobbin (m)
- Kl :
-
Langmuir constant (m3 mol−1)
- Qmax :
-
Max adsorption (mol kg−1)
- Re :
-
External radius of bobbin (m)
- Ri :
-
Internal radius of bobbin (m)
- W :
-
Average weight of bobbin (kg)
- \(F\) :
-
Gravitational force (kg m−2 s−2)
- \(I\) :
-
Unit vector (−)
- \(P\) :
-
Presure (Pa)
- \(Q\) :
-
Volumetric flow rate of the dyeing bath (m3 s−1)
- \(T\) :
-
Transposition operation (−)
- \(t\) :
-
Process time (s)
- \(u\) :
-
Velocity vector (m s−1)
- \(\beta\) :
-
Viscous force coefficient (kg m−4)
- \(\kappa\) :
-
Permeability (m2)
- \(\mu\) :
-
Dynamic viscosity (kg m−1 s−1)
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Acknowledgements
This study was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior—Brasil (CAPES)—Finance Code 001.
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Moreno, M., Valle, J.A.B., de Aguiar, C.R.L. et al. Influence of permeability and pressure on the dye concentration profile in acrylic yarn bobbins in bidirectional flow by simulation. Braz. J. Chem. Eng. 37, 515–524 (2020). https://doi.org/10.1007/s43153-020-00057-3
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DOI: https://doi.org/10.1007/s43153-020-00057-3