Sharp front approach

The process concept can be described by the advanced numerical model as detailed in the previous section. However, by assuming that the fronts that are formed during the different process steps are perfectly well defined (sharp), a simplified and relatively easy-to-solve and fast model can be developed, referred to as the sharp front approach . In the first place, this approach is a very useful tool to quickly investigate the influence of process parameters on process behaviour. Furthermore, it can be used in conceptual design studies. This section describes this sharp front approach and compares the outcomes with the more advanced numerical model. Finally, the influences of several process parameters are studied. 

Figure 2.6 Axial temperature, mass deposition and gas concentration profiles used in the derivation of the sharp front approach for the capture step (a) and recovery step (b)...

All equations for the three steps for the sharp front approach have been summarized in Table 2.6. 

The outcomes of the sharp front approach are presented in this section and are compared to the simulation results of the advanced numerical model. Figure 2.7 shows axial temperature and mass profiles, which are formed after 400 seconds, when feeding a binary N2/CO2 mixture at an inlet temperature of 150 "C. Other conditions are equal to those listed in Tables 2.4 and 2.5. It can be observed that the front positions, equilibrium temperature and the amount of mass deposited per unit of bed volume match very well between the two approaches. Although heat and mass dispersion is included in the advanced model, the fronts are reasonably sharp during the capture step. Especially, the frost front is well... 

It is observed again that when assuming low axial heat and mass dispersion, the solution of the advanced model is approaching the sharp front approach. Finally, the temperature and mass deposition profiles have been computed for the recovery step after 400 seconds, as illustrated in Figure 2.8c and d. It can be observed that dispersion is playing a more prominent role during the recovery step, which is related to the higher flow rates in comparison to the capture step. For that reason, the sharp front approach is especially suited to describe the capture step. 

This section aims at giving an overview of the influences of several process parameters on the process performance, using the sharp front approach. The influences of the initial bed temperature, inlet composition, inlet temperature and packing material are analyzed on the basis of two aspects the amount of CO2 deposited per unit of bed volume and the required specific cooling duty, which is defined as follows ... 

Figure 11.7 shows the temperature history at a fixed point in the reaction tube as a front passes. The temperature at this point is ambient when the front is far away and rises rapidly as the front approaches. Hence, a polymerization front has a very sharp temperature profile (Pojman et al., 1995b). Figure 11.7 shows five temperature profiles measured during frontal free-radical polymerization of methacrylic acid with various concentrations of BPO initiator. Temperature maxima increase with increasing initiator concentration. For an adiabatic system, the conversion is directly proportional to the difference between the initial temperature of the unreacted medium and the maximum temperature attained by the front. The conversion depends not only on the type of initiator and its concentration but also on the thermodynamic characteristics of the polymer (Pojman et al., 1996b). 

To alleviate further numerical problems due to sharp changes in some of the physical variables across the front and also because of the numerical difficulties presented by the Dirac delta function in the surface tension term, the interface is defined having a fixed thickness that is proportional to the spatial mesh. This allows us to smooth the functions across the interface and replace the Dirac delta function with a smoothed delta function S. In other words, we maintain a fixed thickness of the interface within the LS approach. 

When no complexation between M and L or when adsorption of L on the support takes place, the elution volume of L (Vl) will be V(,+V-j and the front is sharp. When the association constants increase, the advancing side of the frontalogram should become more skewed and Vl approaches Vg. Slow attainment of equilibria between M and L or slow diffusions within the gel matrix should yield similar effects on the frontalo-grams. If these effects can be excluded,the slope and breakthrough volume of L also indicate the magnitude of equilibrium constants. Similar discussion holds for the trailing side. The frontalogram is conveniently expressed as its time derivative, which shows minute differences more clearly. 

Fracture toughness tests were conducted based on the linear elastic fracture mechanics (LEFM) approach. The single-edge-notch 3-point-bending (SEN-3PB) test, based on ASTM D5045, was performed to obtain the mode-I critical stress intensity factor (Kic) of the neat epoxy and epoxy/a-ZrP nanocomposites. Care was taken to ensure that the initial sharp crack, generated by tapping with a fresh razor blade, exhibited a thumbnail shape crack front... 

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