Concentration front propagation

Fig. 51 Transport diffusivity calculated from the profiles in the y and z directions [92], Analyzed concentration profiles are at 330, 350, and 370 s, where the concentration fronts propagating from opposite faces have not yet overlapped. The thin lines represent the analytical dependence of the diffusivities which yields excellent agreement with the measured concentration integrals (standard deviation a = 0.006)...

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. 

This reaction is autocatalytic concerning the H+ ions [145]. The concentration of ions increases within the reaction front. For this reason, acid-base indicators can monitor the front propagation. 

The rate of pressure rise is indicative of the flame front propagation rate and thus of the magnitude of the explosion. The pressure rate or slope is computed at the inflection point of the pressure curve, as shown in Figure 6-15. The experiment is repeated at different concentrations. The pressure rate and maximum pressure for each run are plotted versus concentration, as shown in Figure 6-16. The maximum pressure and maximum rate of pressure rise are determined. Typically, the maximum pressure and pressure rates occur somewhere within the range of flammability (but not necessarily at the same concentration). By using this relatively simple set of experiments, the explosive characteristics can be completely established in this example the flammability limits are between 2% and 8%, the maximum pressure is 7.4 bar, and the maximum rate of pressure rise is 360 bar/s. 

Furthermore, in 3.3 we turn to reactive binary ion-exchange. An equilibrium binding reaction (adsorption) with a Langmuir-type isotherm is considered. Formation of sharp propagating concentration fronts is studied via an unconventional asymptotic procedure [1]. 

In Fig. 3 the pressure and concentration (n) variations in the wave are shown. The diffuse structure in this case may be explained qualitatively by the fact that the high-frequency waves necessary to generate a steep front propagate with a speed greater than D and, moving forward away from the wave, are damped (absorbed). As Einstein showed, waves with an oscillation period less than t decrease in amplitude by e times at a distance of order cr. 

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. 

Fig. 1.17. Schematic picture of front propagation during oxidative coke removal, (a) Oxygen concentration in the regeneration gas. e.g., air. (b) Coke loading of the solid phase.

This effect may also cause an oscillatory instability of the front propagation in the case of the isothermal mechanism of the auto wave process. However, in this case a jumplike onset of dispersion in the next layer of the solid matrix will be connected not with the critical temperature gradient of thermal fracture, but with the critical concentration of the final product accumulated at the boundary of this layer (fracture due to a local alteration of the density). 

Here, 8 = DJD is the ratio of the diffusion coefficients, = d /d z is the one-dimensional Laplacian, and % is the initial concentration the reactant. These equations describe a front propagating in the positive 4 direction, where the boundary conditions are given by... 

Fig. 10.18 Concentration wave front propagating after a step change in reflux rate to the top of...

Isothermal frontal polymerization (IFP) is a self-sustaining, directional polymerization that can be used to produce gradient refractive index materials. Accurate detection of frontal properties has been difficult due to the concentration gradient that forms from the diffusion and subsequent polymerization of the monomer solution into the polymer seed. A laser technique that detects tiny differences in refractive indices has been modified to detect the various regions in propagating fronts. Propagation distances and gradient profiles have been determined both mathematically and experimentally at various initiator concentrations and cure temperatures for IFP systems of methyl methacrylate with poly(methyl methacrylate) seeds and wilh the thermal initiator 2,2 -azobisisobutryonitrile. 

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. ... 

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