3. Numerical approach on D using the concept of area exchange

When the practical dyeing conditions are set by using equation (5’), Vs can be treated as a known value for each individual dye/fiber system because flux (B) and liquor ratio (A) can be determined as the operating conditions.

On the other hand, the leveling capacity of dyeing machine (D) has not been analyzed theoretically, but obtained experimentally for an individual dyeing machine.

The concept of area exchange (see “Theory of area exchange”) was introduced to make a theoretical approach to determining D value.

In the concept of area exchange, the following relations are known;

Ratio of effective area: R=P/Q (14)
Ratio of area exchange: K=P/Q×100×A/B=RC (15)

Where

P: effective area (L)
Q: gross volume (L)
A: flux (L)
B: liquor ratio (L/kg)
C: number of circulations of dye liquor or fabric (times/min.)

According to the theory of area exchange, the leveling capacity of a dyeing machine relates to the ratio of effective area of the dyeing machine. The critical D value (Dcrit) for leveling is given in following equation (16).

Dcrit=kR×100（%/circulation） (16)

Dcrit means maximum D value for level dyeing, which is obtained when the D value is below Dcrit .

Here, k is a coefficient relating to the leveling capacity, and it is considered to depend mainly on the uniformity in the dyeing machine, such as temperature variation, flux variation or condition of circulation of fabric, but not on the type of dyeing machine (jet or cheese).

Therefore, equation (5’) can be changed to equation (17)

T=kRA/BVs(1)=kRC/Vs(1)=kK/Vs(1) (17)

A test dyeing is carried out to obtain k value using a high performance test dyeing machine, and the result is shown in Figure 4, where the relation between the levelness (strength difference between inner layer and outer layer) and the ratio of area exchange is presented. A detailed discussion is given in the next section.

The ratio of area exchange corresponding to 100% levelness can be obtained by extrapolating the plotted line, and the value is 36%.

Then, k=0.143 is obtained by substituting the conditions (Vs(1)=2.58(%/min), T=2°C/min. and k=36) for the parameters in equation (17)

D in the concept of area exchange is obtained from this result.

Dcrit=0.143R×100 (%/circulation) (18)

Equation (17) can be modified as shown in equation (19) to obtain the rate of temperature rise for level dyeing.

T=0.143AR/BVs(1)=0.143K/Vs(1) (19)

In this experiment, the rate of temperature rise for level dyeing was 0.46°C/min. and the result of levelness in this experiment was 98.5%.

In jet dyeing, the number of fabric circulations is applied instead of liquor circulation;

C’=M/N (3’)

Where

C: number of circulations of fiber
M: fabric speed(m/min.)
N: loop length per one flow(m)
And T=kRM/Vs(1)L=kRC’/Vs(1) (17’)

k value is independent of the kind of dyeing machine, therefore, the following equation can be applied for jet dyeing.

T=0.143RM/Vs(1)L=0.143RC’/Vs(1) (20)