2. Numerical approach on the rate of dyeing curve

Till now, the rate of dyeing is discussed as a constant value V, but the rate of dyeing changes during dyeing as a matter of fact, therefore, it should not be treated as a constant value.

Several numerical approaches have been made to present the rate of dyeing.

The simplest method is the use of a maximum rate of dyeing (Vmax) corresponding to the steepest slope in the rate of dyeing curve, but the practical rate of dyeing is much lower than Vmax, so this is found to be impractical for setting reasonable dyeing conditions.

Alternative methods are the significant rate of dyeing (Vsig) and the average rate of dyeing (Vs), and these have proved to be useful in practice.

1) The significant rate of dyeing (Vsig)

The significant rate of dyeing (Vsig) is presented as equation (7)

 (7)

where

c: concentration of dye on the fiber
dc: differential change of concentration of dye on the fiber
Vt =dc/dt(rate of dyeing at time=t) (8)

Vsig can be obtained approximately from the rate of the dyeing curve using equation (9)

(%/min.) (9)

where

 (10)
CR: relative dye uptake (CR=100% at the end of dyeing)
△CR：relative dye uptake at a part of linear line in the rate of dyeing curve

Vsig can be obtained approximately from Figure 1 using equation (11) derived from equations (9) and (10).

 (11)

Figure 1 Method for obtaining the significant rate of dyeing

2) The average rate of dyeing (VS)

The differential curve of rate of dyeing with temperature rise in polyester dyeing with disperse dye is close to a frequency curve, which can be regarded as the approximate normal distribution.

In the case of normal distribution, 68.3% of dye is absorbed in the range of average value±standard deviation (m±s) as shown in Figure 2.

Therefore, Vs can be obtained from equation  (12)

 (12)

Figure 2 The average rate of dyeing(Vs)

In the dyeing of polyester with disperse dye, the following relation is observed in practical dyeing;

 (13)

where

T: rate of temperature rise (°C/min.)
Vs(1): average rate of dyeing at the rate of temperature rise is 1°C/min.

Then, the average rate of dyeing with optional rate of temperature rise can be calculated from Vs(1).

Although equation (13) can not always be applied to dyeing systems other than disperse dye/polyester ones, Vs can be obtained from the rate of dyeing curve in the practical dyeing conditions as shown in Figure 2.

Both Vsig and Vs were measured in one disperse dye/polyester dyeing system, and both values were found to be almost the same.

Although either parameter can be used in practice, Vs is used in the following discussion, but Vsig can be used alternatively.

3) Average rate of dyeing (Vs) and the condition of temperature rise

By replacing V in equations (5) and (6) with Vs, equations (5’) and (6’) are given.

T=CD/Vs=AD/BVs (5’)
T=MD/Vs L (6’)

The average rate of dyeing can refer to the rate of temperature rise through these equations.