caa a22 4500 463163702 CHVBK 20180406164755.0 cr unu---uuuuu 170326e20070701xx s 000 0 eng 10.1007/s11242-006-9046-7 doi (NATIONALLICENCE)springer-10.1007/s11242-006-9046-7 Modelling of the dyeing process of packed cotton threads using reactive dyes [Elektronische Daten] [Selene de Souza, Dilênio de Souza, Eduardo da Silva, Antônio de Souza] 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. Springer Science+Business Media, Inc., 2006 Mathematical modelling nationallicence Mass transfer nationallicence Method of volume averaging nationallicence Textile nationallicence Multiphase nationallicence Dyeing nationallicence Diffusion nationallicence Kinetics nationallicence a v|γκ : Surface area per unit volume on the micro-scale, [1/m] nationallicence a v|βσ : Surface area per unit volume on the intermediary scale, [1/m] nationallicence $$\varvec{b}_{\sigma}$$ : Vector closure variables of the σ-region, [m] nationallicence $$\varvec{b}_{\beta}$$ : Vector closure variables of the β-phase, [m] nationallicence C i : Initial concentration of the η-phase, [kg/m3] nationallicence C Aη : Concentration of the η-phase, [kg/m3] nationallicence C Aσ : Concentration of the σ-region, [kg/m3] nationallicence C Aβ : Concentration of the β-phase, [kg/m3] nationallicence $$\left\langle{C_{\rm A}}\right\rangle$$ : Superficial average concentration, [kg/m3] nationallicence $$\tilde{C}_{{\rm A}\beta}$$ : Spatial derivation concentration of the β-phase, [kg/m3] nationallicence $$\tilde{C}_{{\rm A}\sigma}$$ : Spatial derivation concentration of the σ-region, [kg/m3] nationallicence $$\left\langle{C_{\rm A}}\right\rangle^{\gamma}$$ : Intrinsic average concentration of the γ-phase nationallicence [kg/m3] nationallicence $$\left\langle{C_{{\rm A}\beta}}\right\rangle^{\beta}$$ : Intrinsic average concentration of the β-phase nationallicence [kg/m3] nationallicence $$\left\langle{C_{{\rm A}\sigma}}\right\rangle^{\sigma}$$ : Intrinsic average concentration of the σ-region nationallicence [kg/m3] nationallicence $$d_1^{\prime}$$ : Upper internal diameter of the bobbin, [m] nationallicence $$d_2^{\prime}$$ : Upper external diameter of the bobbin, [m] nationallicence $$d_3^{\prime}$$ : Lower internal diameter of the bobbin, [m] nationallicence $$d_4^{\prime}$$ : Lower external diameter of the bobbin, [m] nationallicence d β : Velocity-like coefficient of the β-phase, [m/s] nationallicence d σ : Velocity-like coefficient of the σ-region, [m/s] nationallicence $$\left.{\bf Deff}\right|_{\gamma\kappa}$$ : Effective diffusivity tensor of the γ-phase, [m2/s] nationallicence D β : Molecular diffusivity, [m2/s] nationallicence D σ : Diffusivity of the σ-region, [m2/s] nationallicence $${D}_\beta^{\ast}$$ : Total dispersion tensor, [m2/s] nationallicence $${\bf Deff}\left|_{\beta\sigma}^\beta\right.$$ : Effective diffusivity tensor of the β-phase, [m2/s] nationallicence D : Hydrodynamic dispersion, [m2/s] nationallicence h : Mass transfer convective coefficient, [m/s] nationallicence H : Height of the bobbin, [m] nationallicence k r : Rate constant for reaction with the fibre, [s−1] nationallicence k h : Pseudo-first-order rate constant of hydrolysis, [s−1] nationallicence K eq : Equilibrium constant, [m] nationallicence ℓ i : Lattice vectors, [m] nationallicence ℓβ : Characteristic length for the β-phase, [m] nationallicence ℓγ : Characteristic length for the γ-phase, [m] nationallicence ℓκ : Characteristic length for the κ-phase, [m] nationallicence ℓσ : Characteristic length for the σ-region, [m] nationallicence Q : Discharge rate of the dyeing bath, [m3/s] nationallicence N : Number of cotton threads nationallicence n βσ : Unit normal vector nationallicence M bobbin : Average mass of the bobbins, [kg] nationallicence RB : Liquor ratio, [kg/m3] nationallicence $$R_{\rm th}^{\prime}$$ : Average radius of each cotton thread, [m] nationallicence r σ : Radius of the micro-scale, [m] nationallicence r ω : Radius of the intermediary scale, [m] nationallicence $$r_{\rm h}^{\prime\prime\prime}$$ : Rate of hydrolysis, [kg/m3.s] nationallicence s β : Scalar closure variables of the β-phase nationallicence s σ : Scalar closure variables of the σ-region nationallicence t : Time, [s] nationallicence u β : Convective transport vectors of the β-phase nationallicence u σ : Convective transport vectors of the σ-region nationallicence v β : Velocity of the β-phase, [m/s] nationallicence $$\tilde{{\bf v}}_\beta$$ : Spatial derivation velocity of the β-phase, [m/s] nationallicence V : Volume of the dyeing equipment, [m3] nationallicence V σ : Volume of the micro-scale, [m3] nationallicence V ω : Volume of the intermediary scale, [m3] nationallicence V β : Volume of the β-phase, [m3] nationallicence $$V_{\rm fibre}^{\rm esp}$$ : Specific volume of the cotton fibre, [m3/kg] nationallicence Greek Symbols nationallicence $$\varepsilon_\gamma$$ : Porosity in the γ -phase nationallicence $$\varepsilon_\beta$$ : Porosity in the β-phase nationallicence $$\varepsilon_\sigma$$ : Porosity in the σ-region nationallicence Ψ : Variable that defines the type of dye (reactive) nationallicence Ω : Variable that defines the type of dye (adsorptive) nationallicence de Souza Selene Chemical Engineering Department, Federal University of Santa Catarina, P.O. Box 476, CEP 88040-900, Florianópolis, SC, Brazil aut de Souza Dilênio Chemical Engineering Department, Federal University of Santa Catarina, P.O. Box 476, CEP 88040-900, Florianópolis, SC, Brazil aut da Silva Eduardo Chemical Engineering Department, Federal University of Santa Catarina, P.O. Box 476, CEP 88040-900, Florianópolis, SC, Brazil aut de Souza Antônio Chemical Engineering Department, Federal University of Santa Catarina, P.O. Box 476, CEP 88040-900, Florianópolis, SC, Brazil aut Transport in Porous Media Kluwer Academic Publishers 68/3(2007-07-01), 341-363 0169-3913 68:3<341 2007 68 11242 https://doi.org/10.1007/s11242-006-9046-7 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/s11242-006-9046-7 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- de Souza Selene Chemical Engineering Department, Federal University of Santa Catarina, P.O. Box 476, CEP 88040-900, Florianópolis, SC, Brazil aut NATIONALLICENCE 700 1- de Souza Dilênio Chemical Engineering Department, Federal University of Santa Catarina, P.O. Box 476, CEP 88040-900, Florianópolis, SC, Brazil aut NATIONALLICENCE 700 1- da Silva Eduardo Chemical Engineering Department, Federal University of Santa Catarina, P.O. Box 476, CEP 88040-900, Florianópolis, SC, Brazil aut NATIONALLICENCE 700 1- de Souza Antônio Chemical Engineering Department, Federal University of Santa Catarina, P.O. Box 476, CEP 88040-900, Florianópolis, SC, Brazil aut NATIONALLICENCE 773 0- Transport in Porous Media Kluwer Academic Publishers 68/3(2007-07-01), 341-363 0169-3913 68:3<341 2007 68 11242 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer