Concentration front propagation velocity

Ca) is fixed. The velocity is variously called the concentration wave velocity of species i, the migration rate of species i or the concentration front propagation velocity of species i (Sherwood et al., 1975). A more formal method of arriving at expression (7.1.12a) is given below. 

For a system with n components (including nonad-sorbable inert species) there are n — 1 differential mass balance equations of type (17) and n — 1 rate equations [Eq. (18)]. The solution to this set of equations is a set of n — 1 concentration fronts or mass transfer zones separated by plateau regions and with each mass transfer zone propagating through the column at its characteristic velocity as determined by the equilibrium relationship. In addition, if the system is nonisothermal, there will be the differential column heat balance and the particle heat balance equations, which are coupled to the adsorption rate equation through the temperature dependence of the rate and equilibrium constants. The solution for a nonisothermal system will therefore contain an additional mass transfer zone traveling with the characteristic velocity of the temperature front, which is determined by the heat capacities of adsorbent and fluid and the heat of adsorption. A nonisothermal or adiabatic system with n components will therefore have n transitions or mass transfer zones and as such can be considered formally similar to an (n + 1)-component isothermal system. 

For high Da the column is dose to chemical equilibrium and behaves very similar to a non-RD column with n -n -l components. This is due to the fact that the chemical equilibrium conditions reduce the dynamic degrees of freedom by tip the number of reversible reactions in chemical equilibrium. In fact, a rigorous analysis [52] for a column model assuming an ideal mixture, chemical equilibrium and kinetically controlled mass transfer with a diagonal matrix of transport coefficients shows that there are n -rip- 1 constant pattern fronts connecting two pinches in the space of transformed coordinates [108]. The propagation velocity is computed as in the case of non-reactive systems if the physical concentrations are replaced by the transformed concentrations. In contrast to non-RD, the wave type will depend on the properties of the vapor-liquid and the reaction equilibrium as well as of the mass transfer law. 

The dynamic response of the column is given by solution c(i,0, (z,f)l to Eqs. (8.1) and (8.2) subject to the initial and boundary conditions imposed on the column. The response to a perturbation in the feed composition involves a mass transfer zone or concentration front Which propagates through the column with a characteristic velocity determJnttl by the equilibrium isotherm. The location of the front at any time may be found simply from an overall mass balance, but to determine the form oi the concentration front Eqs. (8.1) and (8.2) must be solved simultaneously. ... 

This is clearly an increasing function of T. Thus, for a nontrace system, the sorption effect (variation in fluid velocity due to sorption of the adsorbable species) leads to an increase in the propagation velocity with increasing concentration. The effect is thus qualitatively similar to the effect of curvature of the equilibrium isotherm and leads to the chromatographic response having a sharp front and a broadening tail as sketched in Figure 8.7. 

Another system that has been studied in this manner is the oxidation of iron(II) by nitric acid. In Chapter 3, we discussed a model that reproduces well the clock reaction experiments. Pojman et al. (1991a) numerically integrated the partial differential equations describing front propagation in that system. Figure 6.13 shows the experimental and calculated dependence of the front velocity on the initial iron(Il) and nitrate concentrations using two variations of the model. Both... 

Figure 6.13 Compari.son of the velocity of front propagation in the iron(II)-nitric acid system on (a) the initial nitrate concentration and (b) the initial Fe(ll) concentration. (Adapted from Pojman et al., 1991a.)...

Begishev et al. (1985) studied anionic polymerization fronts with e-caprolactam. There were two interesting aspects to this system. First, the polymer crystallizes as it cools, which releases heat. Thus, a front of crystallization follows behind the main front. Volpert et al. (1986) investigated this two-wave system. Second, a hot spot moved around the front as it propagated down the tube, leaving a spiral pattern in the product. The entire front propagated with a velocity on the order of 0.5 cm min which was a function of the concentrations of activator and catalyst. The hot spot circulated around the outside of the 6-cm (i.d.) front sixteen times as rapidly as the front propagated. 

If these two excess works are equal than we expect equistability and zero velocity of front propagation. If the excess work to form the front from phase 1 is less than that necessary to form the front from phase 3, then we expect phase one to be more stable than phase 3. To do that calculation we divide the inter phase region into N boxes of a specified length L, with the boundary conditions of the concentrations on the left side of the interphase... 

In this case, a reaction front proceeds from the more stable stationary state (xs3 = 6738.21) into the less stable stationary state (0 51 = 712.12). We may vary the constraints of the concentration of A and B and find front propagation in either one or the other direction. For one set of constraints there is zero velocity front propagation and the two stable stationary states are said to have equal stability [26,27]. This equal stability condition for onedimensional systems is independent of the diffusion coefficient of the species... 

Again the scenario we envisage is similar to that shown qualitatively in Fig. 11.7 we expect our best chance of such behaviour if the decay rate is small, i.e. k2 1. The reaction wave has a leading front moving with a steady velocity c1, through which most of the conversion of A to B occurs. After this front, the dimensionless concentration of A is almost zero and that of B is almost unity. At some distance, the first front is followed by a recovery wave, possibly more diffuse, in which A is completely removed and the autocatalyst also decays. The velocity of the recovery wave is c2. If ct exceeds c2, the first front will move away from the second, so the pulse will increase in width if c, = c2, the pulse will move with a constant shape if, however, c2 exceeds c, we can expect the second wave to catch the first, in which case propagation may fail. 

Constant Pattern Behavior In a real system the finite resistance to mass transfer and axial mixing in the column lead to departures from the idealized response predicted by equilibrium theory. In the case of a favorable isotherm the shock wave solution is replaced by a constant pattern solution. The concentration profile spreads in the initial region until a stable situation is reached in which the mass transferrate is the same at all points along the wave front and exactly matches the shock velocity. In this situation the fluid-phase and adsorbed-phase profiles become coincident. This represents a stable situation and the profile propagates without further change in shape—hence the term constant pattern. 

A theoretical and experimental study of multiplicity and transient axial profiles in adiabatic and non-adiabatic fixed bed tubular reactors has been performed. A classification of possible adiabatic operation is presented and is extended to the nonadiabatic case. The catalytic oxidation of CO occurring on a Pt/alumina catalyst has been used as a model reaction. Unlike the adiabatic operation the speed of the propagating temperature wave in a nonadiabatic bed depends on its axial position. For certain inlet CO concentration multiplicity of temperature fronts have been observed. For a downstream moving wave large fluctuation of the wave velocity, hot spot temperature and exit conversion have been measured. For certain operating conditions erratic behavior of temperature profiles in the reactor has been observed. 

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