Package dyeing dispersive flow

Ren" and Ilett" discussed the effect of dispersive flow in package dyeing. Dispersive flow is considered to reduce the concentration gradient of dye liquor flowing through the package, resulting in a more even distribution of dye in the... 

Dispersing agent for coupling bath of NAPHTOL dyes. Improves flow rate in package or beam dyeing. 

Key words convective dispersion, fluid mechanics, package dyeing geometry, modelling fluid flow in dyeing, Darcy s law, Navier-Stokes equations. 

The transport of dye liquor along the packed bed fibrous assembly is represented by an MIM model. Thus, the supply of dye from the dye bath to the surface of the fibres at any position in the package is considered to consist of additive effects of two types of transport phenomena convective flow and dispersive flow through porous media. 

Wai and Burley et developed a complex mathematical model using principles well known in chemical engineering. They considered the package as a packed bed chemical reactor. The processes of convection, dispersion, absorption and desorption of dye to and from the packed bed of fibres are modelled, while accoimt is also taken of the addition of dye to, and the possible removal of liquor from, the mixing tank. They further developed the model of a packed bed reactor and defined two equations for axial and radial flow processes. Example simulations showed that advantages inclnde a decreased time to reach a level dyeing and faster attainment of eqnilibrium. 

The above-mentioned works, which describe the influence of the dyeing conditions on the kinetics of dye uptake, either ignore the dispersion or adsorption factor, or fail to define the flow velocity within the package. The details will be discussed in the next chapter. 

One major shortcoming of these models is that the flow property during dyeing is not defined in a sonnd mathematical form, since, for both the STM and MIM approaches, an exact solntion of the problem of convective diffusion to a sohd surface first requires the solution of the hydrodynamic equations of motion of the fiuid for boundary conditions appropriate to the mainstream velocity of flow and the shape of the package. Another limitation of their work is that those workers did not consider situations like variable boundary conditions and variable dispersion coefficients, which are quite common situations in dyeing practice, in their numerical simulations. 

The derivation of the dye transfer equation within the package relies on the principle of superposition convection and dispersion can be added together if they are linearly independent. From the previous chapter, dispersion was shown to be a random process due to molecular motion. Due to dispersion, each molecule in time 6t will move either one step to the left or one step to the right (i.e. dx). Due to convection, each molecule will also move u6t in the cross-flow direction. These processes are clearly additive and independent the presence of the cross-flow does not bias the probability that the molecule will take a diffusive step to the right or the left it just adds something to that step. The net movement of the molecule is u3t 6x, and, thus, the total flux in the x-direction including the convective transport and the dispersion term, must be ... 

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