Mathematics of Dyeing

For very long, I have been working to find out mathematically, the Material to Liquor Ratio for Optimum Dyeing. Finally, I crack the code.  Sharing some insight of my New Book
 " Mathematics of Dyeing "
 And as in my one of the earliest post, where I published " Yarn Dyeing Machines, Fraud of the Century"  Herein the same gets proven mathematically

Pictorial View Of Your Favorite Dye In Package Dyeing

Where Br : Brownian Motion. F( A) : Adhesion to Yarn. F(D) : Drag From The cross flow 

F(F) : Frictional Force,  F(L) ;Hydrodyamanic Lift Force, F(y) : Drag from dye liquor

For small particles
F(L) < F(Y) and hence are retained by the surface of the yarn by Van Der Waals Adhesion Forces F(A) and are not swept away.


Study this picture very carefully. 
The Dye molecule is trying to enter the yarn and the various forces acting on the dye particle

All these forces can be mathematically calculated.
Eg

Resistance Coefficient of Boundary Layer of Yarn

Pressure Differential = Beta x Flow Rate x Viscosity of dye liquor

Where Beta is the Resistance Coefficient

Similarly all the other Forces can be calculated for the Dye Particle to find its way into the yarn capillaries

The Diffusion Coefficient of the Dye which can be calculated from


 Weisz Zollinger Model

D = Do PT ( sigma0/ sigma)

Where P is porosity
Sigma0 is the concentration in the external solution ,when equilibrium is achieved
Sigma is the initial concentration
Do is Diffusion coefficient in free water calculated as below

Do = 3.6 Sq Rt 76/M
 ( M is the mass of the dye particle)

In order to calculate the final uptake of the Dye Into Yarn

Porosity of package can be  calculated from the Packing Density of yarn

The Porosity will then defines the uptake of Dye Into the Yarn

The Intake of Dyeliquor Into Yarn can be calculated from the equation


L= a T kh

Where a is the relative surface area
T is the count of the yarn in Tex
Kh is the constant ( depending upon wetting energy ) Square Root fibre volume density x fibre fineness

The saturation point of equilibrium is calculated from the partition coefficient.

With all the above equations and one dozen more, one can now use mathematics to find out exactly, how much of water is required to exhaust the dye into the capillaries of the yarn.

The Rate of Reaction is given by
Rate = - Delta A/ Delta T
Since concentration of the dye is dropping with the reaction time, the rate of reaction is negative for the dye. But positive for the Product

Rate of Dyeing : k x Df x Ds

where k is the coefficient of dyeing
Df : Adsorption Isotherm of dye uptake in yarn
Ds : Adsorption Isotherm of dye left in liquor

Essentially, one can calculate exactly the amount of water required for the dyeing of the yarn and exactly the amount of dye, which will diffuse into the yarn core and not leave any trace of effluent .

Which means, one can dye yarn package or for that matter any textiles in a bio reactor like rice cooker, with the specified water and Ph and Temperature and Dyestuff. And the water with dye will be adsorbed into the yarn without a trace of effluent .


Dyeing is therefore Diffusion in an Asymptotic Process and depends upon Time Elapsed and Not on Quantity Of Liquor

This Material to Liquor Ratio theory is applicable only when the partition coefficient of dyestuff is poor and undissolved dyes cannot migrate into the package.

Therefore, Material to Liquor Ratio is not a function of Machine. But is a function of Partition Coefficient of Dyes.

Further, study carefully that if pump should actually bring the dye liquor to a steady state and flow should be very weak or extremely slow for the molecules to be not forced away from the vander waal forces which are trying to move the dye into the yarn.

All this bull shit of high powered pump and centrifugal pump and axial flow pump does not matter. What matters is that the liquor should reach each element of the package and be there in a steady state. Finally displacing the depleted liquor with fresh liquor and so on, if required. That is why the best results are achieved with pump speeds of only 5 liters per minute of flow.

At Mantex Technologies, I am now developing a system to actually capture the colors as they get matured in the machine and give an online Delta E value. Thereby providing an opportunity for the dyer to make corrections while dyeing itself

But of course, the best is yet to come ,when machine will be a bio reactor with dyes getting fully adsorbed.