Linear exhaustion profile

Carbonell et al. developed a mathematical representation of various exhaustion profiles and went on to calculate practical time/temperature profiles that would result in linear exhaustion profiles. Cegarra et al. carried out more work aimed at establishing detailed kinetic relationships in order to carry out isoreactive dyeings, in which the dyeings have a linear exhaustion profile. ... 

The final correction to the model is based on adjusting the exhaustion rate at the end of the non-linear exhaustion profiles, when it drops to very low values which would result in the control program trying to cool off the machine. The controller may be adjusted to accommodate a pre-set final ramp rate when the target exhaustion rate drops below 0.2%. 

Simulation results demonstrating a comparison of set and achieved exhaustion values using a linear exhaustion profile. 

Brooks showed that, for a given flow direction within a package, the difference in concentration across the radios was found to be proportional to the overall rate of exhaustion of the dye nsed. For a constant rate of depletion of dye, this difference must be constant thronghont the dyeing. This has cast serious doubts on the linear function being the optimnm exhaustion profile. He stated that neither linear exhaustion curves nor the time-temperature profile recotmnended by dye manufacturers result in best level dyeing. He showed that exponential curves and exhaustion curves proportional to the square root of time t give superior levelness compared with previous models. 

The woik of Nobbs and Ren, unlike that of Hoffmann and Mueller, does not produce solutions which provide the concentration of dye at aity positions in the package at aity time. It gives a relationship between a number of variables and the unlevelness of dyeing. The control scheme developed and described subsequently by the authors was aimed at achievement of any of the three exhaustion profiles -linear, exponential or quadratic - by on-hne measurement of the dye bath conditions and manipulation of the temperature. 

The control parameter for different exhaustion profiles is then defined as follows Linear ... 

As was shown earlier, three exhaustion profiles were used in this model linear, quadratic and exponential. The general equation defining the exhaustion rate can be shown as ... 

In a similar manner to the chemical relaxation case (see Section 6.7) the time-dependence reduces to a /t law in the short-term approximation and, for longer periods of time, to an exponential law. When t —> oo (in practice t > 5r ), that is in the stationary state, the function disappears and a linear concentration profile is the result, as it should be on account of dc/dt — l dc /dy — 0 (Fig. 7.26). The behaviour on switching off the current is found analogously, if the stationary profile is taken as the initial condition. The reader is referred to Refs. [324,431] for an exhaustive discussion. 

The mass balance was integrated using Gear s method, taking into account the temperature profile by linear interpolation between the experimental points. The significance of the model was judged by variance analysis [18]. The F-test revealed that the model cannot be rejected at a significance level of a = 0.05 for both the CO/O mixture and the simulated exhaust gas (fig. 3). 

Experimental work on dyebath exhaustion control was also carried out using a pilot-scale radial flow package dyeing machine, and the results supported the flndings of the theoretical model. In particular, it was found that a quadratic profile was preferable to an exponential profile, which in turn was preferable to a linear profile. All controlled exhaustion dyeings gave better levelness than a standard (constant temperature ramp) dyeing method. This work was continued by Illett, who improved the mathematical solution of the Nobbs-Ren model and applied it to axial flow machines. 

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