Concentration front

Under constant pattern conditions the LUB is independent of column length although, of course, it depends on other process variables. The procedure is therefore to determine the LUB in a small laboratory or pilot-scale column packed with the same adsorbent and operated under the same flow conditions. The length of column needed can then be found simply by adding the LUB to the length calculated from equiUbrium considerations, assuming a shock concentration front. 

If the feed rate is decreased, the trends of curves in Fig. 13-109 are reversed. The disturbance of other variables such as feed composition, boil-up ratio, and recycle of water-rich effluent from the decanter produces similar shifts in the steep concentration fronts, indicating that azeotropic towers are among the most sensitive separation operations, for which dynamic studies are essential if reli-... 

Figure 3.2 Three major methods in chromatography. The commonest form of chromatography involves the introduction of a small volume of sample onto a column and is known as zonal chromatography. Movement down the column is effected by the mobile phase, which may be simply a solvent (A) allowing partition of the test molecules between the stationary and mobile phases. Alternatively, the mobile phase may be a solvent containing solute molecules (B), which actively displace test molecules from the stationary phase. A less frequently used method known as frontal separation (C) does not involve a separate mobile phase but a large volume of the sample is allowed to pass through the column and as the various components separate, concentration fronts develop and their movement can be monitored.

Sometimes the boundary conditions can be approximated as a step in concentration instead of a step in mass released. This subtle difference in boundary conditions changes the solution from one that is related to pulse boundaries (known mass release) to one resulting from a concentration front with a known concentration at one boundary. 

EXAMPLE 2.7 Dichlowbenzene concentration in lake sediments due to a plating facility discharge (solution to a concentration front)... 

Figure E2.7.1. Illustration of a concentration front moving down into the sediments of a lake.

EXAMPLE 5.4 The mixing of two Streams (concentration front withmoving coordinates and first order and zero-order source terms)... 

These boundary conditions, illustrated in Figure E5.5.2, will give us a concentration front, but in two dimensions. In addition, we have a zero flux condition that will require an image solution. We will use the solution of Example 2.7 to develop a solution for this problem. The solution, before applying boundary conditions, was... 

Now, we need an image to the concentration front about the z = 0 plane. In the y-direction, we have a step up at y = -Ay and a step down at y = Ay. We will also use the product rule (Example 2.3) to indicate that the solution to our governing equation (E5.5.2) for the y-direction should be multiplied times the solution in the z-direction. Then, the solution can be given as... 

A reactor modeled as a complete mix reactor with a leaky dead zone will have a concentration front from Q =0to Q = Co applied. Develop an equation for the concentration in the outflow of the reactor. 

Figure E6.8.2. Solution of a complete mix reactor with a leaky dead zone to a step-down concentration front applied at f = 0 with ty = 1.2 hrs and h = 1 hr.

Industrially relevant consecutive-competitive reaction schemes on metal catalysts were considered hydrogenation of citral, xylose and lactose. The first case study is relevant for perfumery industry, while the latter ones are used for the production of sweeteners. The catalysts deactivate during the process. The yields of the desired products are steered by mass transfer conditions and the concentration fronts move inside the particles due to catalyst deactivation. The reaction-deactivation-diffusion model was solved and the model was used to predict the behaviours of semi-batch reactors. Depending on the hydrogen concentration level on the catalyst surface, the product distribution can be steered towards isomerization or hydrogenation products. The tool developed in this work can be used for simulation and optimization of stirred tanks in laboratory and industrial scale. 

Some simulation results for trilobic particles (citral hydrogenation) are provided by Fig. 2. As the figure reveals, the process is heavily diffusion-limited, not only by hydrogen diffusion but also that of the organic educts and products. The effectiviness factor is typically within the range 0.03-1. In case of lower stirrer rates, the role of external diffusion limitation becomes more profound. Furthermore, the quasi-stationary concentration fronts move inside the catalyst pellet, as the catalyst deactivation proceeds. 

This partial derivative is the velocity of the concentration front hi the bed. The constant pattern assumption presupposes that this velocity is constant, or in other words, is independent of the solution concentration. This means that all points on the breakthrough curve are traveling in the bed under the same velocity, and thus a constant shape of this curve is established (Wevers, 1959). According to the above equation, this could happen only if (Perry and Green, 1984)... 

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]. 

As illustrated by (3.2.11), for m > 2 the first derivative of concentration at the boundary of support is discontinuous that is, a weak shock is formed at the zero concentration front. This stands in accord with the classical Rankine-Hugoniot condition that prescribes for any moving interface Xi(t)... 

We now observe how the concentration front penetrates into system A when time increases from tx to t3 (Fig. 18.5c). In order to measure the penetration speed we can look, for instance, at the movement away from the boundary of the location where CA reaches Cg/2, that is, half the concentration in system B. We call this point xil2(t). 

The concentration front penetrates the system A with a speed which is proportional to tm. Except for the numerical factor, we again find the law by Einstein and Smolu-chowski (Eq. 18-8). Note that if we had chosen another criterion to define the front, such as 0.9 Cg, yV2 would be replaced by yo g for which erfc(y0 9) = 0.1. From Table A.2 of Appendix A we gety0 9 = 0.086. Thus the prefactor in Eq. 18-23 would change (from 1 to 0.17), but not the law itself. 

Yet, this is not the only, and usually not even the most important reason why the concentration measured at a fixed location is asymmetric in time. In many cases chemicals enter the river from outfalls (see Fig. 24.4). Remember that vertical mixing usually occurs over a short distance, whereas lateral mixing may need more time (or distance). As discussed before, it is mostly the lateral mixing (or rather its slowness ) which allows longitudinal dispersion. As long as not all streamlines are occupied , the dispersion coefficient is small and the concentration front steep. 

At sufficiently large values of X the saturation curves approach a constant pattern form, and thereafter the concentration front progress through the columii at a steady velocity, governed by the capacity of the adsorbent and the feed concentration, with no further change in the shape of the curve. Such behavior is characteristic of systems with a favorable equilibrium isotherm (12). The constant pattern limit is reached when the dimensionless concentration profile in fluid phase and adsorbed phase become practically coincident, and the asymptotic form of the break-... 

Figure 1. Case I (fully stable) profiles (a) progression of concentration front, (b) progression of temperature front, (c) breakthrough curve, (d) equilibrium (Y ) and operating (Y) lines for steady-state front, (e) graphical representation of the integral used in the prediction of steady-state LUB...

In the case of an unfavorable isotherm (or equally for desorption with a favorable isotherm) a different type of behavior is observed. The concentration front or mass transfer zone, as it is sometimes called, broadens continuously as it progresses through the column, and in a sufficiently long column the spread of the profile becomes directly proportional to column length (proportionate pattern behavior). The difference between these two limiting types of behavior can be understood in terms of the relative positions of the gas, solid, and equilibrium profiles for favorable and unfavorable isotherms (Fig. 7). 

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. 

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