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.

Nano Dentures

Last week, I was  browsing for venture capitalist, who would be willing to invest into my new product line.

The way, the VC work is that though they call themselves Risk Investors, however, generally, they invest into water tight already invested biz, which is making cash profits and has only issue of funding to scale up the supplies. Then they jump and take the opportunity. Rarely, does a VC support the biz model from a drawing  board stage to Global scale ability stage.

Interestingly, despite the diligence into biz models and funding, most of the time biz fail as some of the  simplest issues are ignored.

Here is an interesting case : A German Nano venture invested into a Denture Biz as below

Where Students of Singapore University designed a special bio polymer to replace the steel wires conventionally  used on the dentures.

Well, the students certainly need to be applauded to create this unique piece . Super Invention Indeed.

However, for a nano VC, whether this makes biz sense, one has to really think "n" times.

The denture USP is the white biopolymer which replaces the steel wire and hence the color merges with the teeth. However, the simplest of the solution is to take Silk yarn, twist it and then take calcium substrates and from within the calcium block run the twisted silk yarn .

Silk is a protein yarn, hence no issues of foreign body , again calcium is the same material as that the teeth are formed, hence again no rejection issues. Plus the color merges with the teeth color.

Silk has same strength as that of steel and besides that it is highly elastic. This gives ductility to give a form as required.

The Silky Denture will cost not more then USD10 per piece and will have a global market, with no issues of cyclic sales or cash flow constraints and strong positioning in the market against steel dentures or biopolymer dentures.

But surely, no VC supports such a simple solution. But, if you show enough complexity, which is barrier to entry for others, then without looking at the simple competition, which can displace the investment, the VC justifies all its investment with supporting high tech arguments and global sales and unique positioning in the market etc etc till it finds out that an ingenious Chinese has come out with a simple solution of the same copied product.

Well the same VC, did it not even bother to reply my email , because my solutions are Intelligent , Simple, Low Cost and Scale able and does not pass VC funding Criterion.

End Of Overhead Cleaners

As per my memory goes, the only machine which I recalled from my first school trip to a textile mill as a kid was a Barber Coleman winder, with an overhead cleaner.

More then then the winder, the overhead cleaner was very amusing for me, because it moved on a track and had two arms and looked more like an octopus with two arms sucking fibers out of the machine.

As luck would have had it, I did my specialization in Polymer based Textiles, yet worked most of my life with machines and polymers. And the Octopus on the Coleman winder was always flashed into my mind, whenever I saw the overhead cleaner on the machines.

Well, the lazy thinking Textile Industry , it would take time to understand that a Robot, which can clean the floor can do the same job for the machine. It can also vacuum the fibres out of the machine , including floor cleaning . Job, which is currently being done by the OHTC.

At Mantex Technologies , now I am working out with a Robot Vacuum Cleaner to make a Vacuum cleaner, which vacuums in X axis and also in Y axis. Which means, the neck of cleaner will extend right till the area of operation, where the vacuum needs to be done.

The best thing about the Robot Cleaner is that it simply does not keep running till it has already " All Cleared" the dust and fibre from the area of operation. So, it is miles ahead of the old system of travelling OHTC.

The Robot is intelligent, so it also keeps taking pictures in real time and if you do wish to monitor, it can flash the pictures to a receiver.

The Second generation of robot will also calculate, the fly generation of the machine per day or per hour by using a small pizo sensor and alarm the winder or spinner on the short fibres or dust in the system. It will then report this via wireless to your hand phone.

The line of operation of the robot is very easy to make. A simple color sensor will follow the color line drawn besides the machine.  The robot will not leave this line and at the end of the machine, will turn back with the inbuilt systems of sensing obstacles etc. Robot will also stop on its own, when it senses that the line of operation is blocked by the operator.

No more Over Head Cleaners. It will now be Robot Cleaners . I only hope my design is not quickly copied away by some jerk.

Wait for the New Robot Cleaner against the Over Head Cleaner.

Low power, no messy wires, no overhead railings etc etc. Nice and clean machines. And interestingly lower priced then the OHTC.