Abstract
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.
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Abbreviations
- a v|γκ :
-
Surface area per unit volume on the micro-scale, [1/m]
- a v|βσ :
-
Surface area per unit volume on the intermediary scale, [1/m]
- \(\varvec{b}_{\sigma}\) :
-
Vector closure variables of the σ-region, [m]
- \(\varvec{b}_{\beta}\) :
-
Vector closure variables of the β-phase, [m]
- C i :
-
Initial concentration of the η-phase, [kg/m3]
- C Aη:
-
Concentration of the η-phase, [kg/m3]
- C Aσ:
-
Concentration of the σ-region, [kg/m3]
- C Aβ:
-
Concentration of the β-phase, [kg/m3]
- \(\left\langle{C_{\rm A}}\right\rangle\) :
-
Superficial average concentration, [kg/m3]
- \(\tilde{C}_{{\rm A}\beta}\) :
-
Spatial derivation concentration of the β-phase, [kg/m3]
- \(\tilde{C}_{{\rm A}\sigma}\) :
-
Spatial derivation concentration of the σ-region, [kg/m3]
- \(\left\langle{C_{\rm A}}\right\rangle^{\gamma}\) :
-
Intrinsic average concentration of the γ-phase; [kg/m3]
- \(\left\langle{C_{{\rm A}\beta}}\right\rangle^{\beta}\) :
-
Intrinsic average concentration of the β-phase; [kg/m3]
- \(\left\langle{C_{{\rm A}\sigma}}\right\rangle^{\sigma}\) :
-
Intrinsic average concentration of the σ-region; [kg/m3]
- \(d_1^{\prime}\) :
-
Upper internal diameter of the bobbin, [m]
- \(d_2^{\prime}\) :
-
Upper external diameter of the bobbin, [m]
- \(d_3^{\prime}\) :
-
Lower internal diameter of the bobbin, [m]
- \(d_4^{\prime}\) :
-
Lower external diameter of the bobbin, [m]
- d β :
-
Velocity-like coefficient of the β-phase, [m/s]
- d σ :
-
Velocity-like coefficient of the σ-region, [m/s]
- \(\left.{\bf Deff}\right|_{\gamma\kappa}\) :
-
Effective diffusivity tensor of the γ-phase, [m2/s]
- D β :
-
Molecular diffusivity, [m2/s]
- D σ :
-
Diffusivity of the σ-region, [m2/s]
- \({D}_\beta^{\ast}\) :
-
Total dispersion tensor, [m2/s]
- \({\bf Deff}\left|_{\beta\sigma}^\beta\right.\) :
-
Effective diffusivity tensor of the β-phase, [m2/s]
- D :
-
Hydrodynamic dispersion, [m2/s]
- h :
-
Mass transfer convective coefficient, [m/s]
- H :
-
Height of the bobbin, [m]
- k r :
-
Rate constant for reaction with the fibre, [s−1]
- k h :
-
Pseudo-first-order rate constant of hydrolysis, [s−1]
- K eq :
-
Equilibrium constant, [m]
- ℓ i :
-
Lattice vectors, [m]
- ℓβ :
-
Characteristic length for the β-phase, [m]
- ℓγ :
-
Characteristic length for the γ-phase, [m]
- ℓκ :
-
Characteristic length for the κ-phase, [m]
- ℓσ :
-
Characteristic length for the σ-region, [m]
- Q :
-
Discharge rate of the dyeing bath, [m3/s]
- N :
-
Number of cotton threads
- n βσ :
-
Unit normal vector
- M bobbin :
-
Average mass of the bobbins, [kg]
- RB:
-
Liquor ratio, [kg/m3]
- \(R_{\rm th}^{\prime}\) :
-
Average radius of each cotton thread, [m]
- r σ :
-
Radius of the micro-scale, [m]
- r ω :
-
Radius of the intermediary scale, [m]
- \(r_{\rm h}^{\prime\prime\prime}\) :
-
Rate of hydrolysis, [kg/m3.s]
- s β :
-
Scalar closure variables of the β-phase
- s σ :
-
Scalar closure variables of the σ-region
- t :
-
Time, [s]
- u β :
-
Convective transport vectors of the β-phase
- u σ :
-
Convective transport vectors of the σ-region
- v β :
-
Velocity of the β-phase, [m/s]
- \(\tilde{{\bf v}}_\beta\) :
-
Spatial derivation velocity of the β-phase, [m/s]
- V :
-
Volume of the dyeing equipment, [m3]
- V σ :
-
Volume of the micro-scale, [m3]
- V ω :
-
Volume of the intermediary scale, [m3]
- V β :
-
Volume of the β-phase, [m3]
- \(V_{\rm fibre}^{\rm esp}\) :
-
Specific volume of the cotton fibre, [m3/kg]
- Greek Symbols :
-
- \(\varepsilon_\gamma\) :
-
Porosity in the γ -phase
- \(\varepsilon_\beta\) :
-
Porosity in the β-phase
- \(\varepsilon_\sigma\) :
-
Porosity in the σ-region
- Ψ:
-
Variable that defines the type of dye (reactive)
- Ω:
-
Variable that defines the type of dye (adsorptive)
References
Edwards D.A., Shapiro M., Brenner H. (1991) Dispersion and reaction in two-dimensional model porous media. Phys. Fluids 5, 837–848
Gray W.G. (1975) A derivation of the equations for multiphase transport. Chem. Eng. Sci. 30, 229–233
Guelli U. de Souza S.M.A., Whitaker S. (2003) Mass transfer in porous media with heterogenous chemical reaction. Brazil. J. Chem. Eng. 20, 191–199
Homes F.A., Whitaker S. (1985) The spatial averaging theorem revisited. Chem. Eng. Sci. 40, 1387–1392
Maliska, C.R.: Transferência de Calor e Mecânica dos Fluidos Computacional. LCT – Livros Técnicos e Científicos Editora S.A., 453 pp. Rio de Janeiro, Brazil (2004)
Paine M.A., Carbonell R.C., Whitaker S. (1983) Dispersion in pulsed systems I: heterogeneous reaction and reversible adsorption in capillary tubes. Chem. Eng. Sci. 38, 1781–1793
Plumb O.A., Whitaker S. (1990a) Diffusion, adsorption and dispersion in porous media: small-scale averaging and local volume averaging, Chap V. In: Cushman J.H. (eds) Dynamics of Fluids in Hierarchical Porous Media. Academic Press, London
Plumb O.A., Whitaker S. (1990b) Diffusion, adsorption and dispersion in heterogeneous porous media: the method of large-scale averaging, Chap VI. In: Cushman J.H. (eds) Dynamics of Fluids in Hierarchical Porous Media. Academic Press, London
Quintard M., Whitaker S. (1993) Transport in ordered and disordered porous media: volume averaged equations, closure problems, and comparison with experiment. Chem. Eng. Sci. 48, 2537–2564
Quintard M., Whitaker S. (1994a) Transport in ordered and disordered porous media I: the cellular average and use of weighting functions. Transport Porous Med. 14, 163–177
Quintard M., Whitaker S. (1994b) Convection, dispersion, and interfacial transport of contaminants: homogeneous porous media. Adv. Water Resour. 17, 221–239
Revello, J.H.P., Guelli U. de Souza, S.M.A., Ulson de Souza, A.A.: Modeling and simulation of dyeing process of threads on bobbins. XXII CILAMCE – 22. Iberian Latin American Congress on Computational Methods in Engineering, Proceedings in CD (2001)
Sherwood, T.K., Pigford, R.L., Wilke, C.R. Mass Transfer. McGraw-Hill (1975)
Schneider G.E., Zedan M. (1981) A modified strongly implicit procedure for numerical solution of field problems. Numer. Heat Transfer 4, 1–19
Ulson de Souza A.A., Whitaker S. (2003) The modelling of textile dyeing process utilizing the method of volume averaging. Brazil. J. Chem. Eng. 20(4):445–453
Whitaker, S.: A simple geometrical derivation of the spatial averaging theorem. Chem. Engr. Ed. 19, 18–21, 50–52 (1985)
Whitaker S. (1999) Theory and Applications of Transport in Porous Media: The Method of Volume Averaging. Kluwer Academic Publishers, London
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de Souza, S.M.A.G.U., de Souza, D.P., da Silva, E.A.B. et al. Modelling of the dyeing process of packed cotton threads using reactive dyes. Transp Porous Med 68, 341–363 (2007). https://doi.org/10.1007/s11242-006-9046-7
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DOI: https://doi.org/10.1007/s11242-006-9046-7