Axial dispersion, defined

Rapid Approximate Design Procedure. Several simplified approximations to the rigorous solutions have been developed over the years (57—60), but they aU. remain too compHcated for practical use. A simple method proposed in 1989 (61,62) uses a correction factor accounting for the effect of axial dispersion, which is defined as (57)... 

Chemical Reaction Measurements. Experimental studies of incineration kinetics have been described (37—39), where the waste species is generally introduced as a gas in a large excess of oxidant so that the oxidant concentration is constant, and the heat of reaction is negligible compared to the heat flux required to maintain the reacting mixture at temperature. The reaction is conducted in an externally heated reactor so that the temperature can be controlled to a known value and both oxidant concentration and temperature can be easily varied. The experimental reactor is generally a long tube of small diameter so that the residence time is well defined and axial dispersion may be neglected as a source of variation. Off-gas analysis is used to track both the disappearance of the feed material and the appearance and disappearance of any products of incomplete combustion. 

Naturally, there are two more Peclet numbers defined for the transverse direction dispersions. In these ranges of Reynolds number, the Peclet number for transverse mass transfer is 11, but the Peclet number for transverse heat transfer is not well agreed upon (121, 122). None of these dispersions numbers is known in the metal screen bed. A special problem is created in the monolith where transverse dispersion of mass must be zero, and the parallel dispersion of mass can be estimated by the Taylor axial dispersion theory (123). The dispersion of heat would depend principally on the properties of the monolith substrate. Often, these Peclet numbers for individual pellets are replaced by the Bodenstein numbers for the entire bed... 

Glaser and Lichtenstein (G3) measured the liquid residence-time distribution for cocurrent downward flow of gas and liquid in columns of -in., 2-in., and 1-ft diameter packed with porous or nonporous -pg-in. or -in. cylindrical packings. The fluid media were an aqueous calcium chloride solution and air in one series of experiments and kerosene and hydrogen in another. Pulses of radioactive tracer (carbon-12, phosphorous-32, or rubi-dium-86) were injected outside the column, and the effluent concentration measured by Geiger counter. Axial dispersion was characterized by variability (defined as the standard deviation of residence time divided by the average residence time), and corrections for end effects were included in the analysis. The experiments indicate no effect of bed diameter upon variability. For a packed bed of porous particles, variability was found to consist of three components (1) Variability due to bulk flow through the bed... 

When the axial dispersion terms are present, D > Q and E > Q, Equations (9.14) and (9.24) are second order. We will use reverse shooting and Runge-Kutta integration. The Runge-Kutta scheme (Appendix 2) applies only to first-order ODEs. To use it here. Equations (9.14) and (9.24) must be converted to an equivalent set of first-order ODEs. This can be done by defining two auxiliary variables ... 

A well-defined bed of particles does not exist in the fast-fluidization regime. Instead, the particles are distributed more or less uniformly throughout the reactor. The two-phase model does not apply. Typically, the cracking reactor is described with a pseudohomogeneous, axial dispersion model. The maximum contact time in such a reactor is quite limited because of the low catalyst densities and high gas velocities that prevail in a fast-fluidized or transport-line reactor. Thus, the reaction must be fast, or low conversions must be acceptable. Also, the catalyst must be quite robust to minimize particle attrition. 

Hint Use a version of Equation (11.49) but correct for the spherical geometry and replace the convective flux with a diffusive flux. Example 11.14 assumed piston flow when treating the moving-front phenomenon in an ion-exchange column. Expand the solution to include an axial dispersion term. How should breakthrough be defined in this case The transition from Equation (11.50) to Equation (11.51) seems to require the step that dVsIAi =d Vs/Ai] = dzs- This is not correct in general. Is the validity of Equation (11.51) hmited to situations where Ai is actually constant ... 

As shown by Wakeman, the solute material balance for the flowing liquid phase, allowing for axial dispersion and desorption of solute is given by the following defining partial differential equation... 

The Peclet number for axial dispersion is defined in a manner similar to the radial parameter... 

Equation (9.27) defines the so-called axial dispersion coefficient Dax as a model parameter of mixing. Nd is the dispersion flow rate, c the concentration of the tracer mentioned earlier, and S the cross-sectional area of the column. The complete mole flow rate of the tracer consists of an axial convection flow and the axial dispersion flow. The balance of the tracer amount at a cross section of the extractor leads to second-order partial differential equations for both phase flows at steady state. For example, for continuous liquids ... 

Mean concentration of pulse of tracer if uniformly distributed in experimental section of vessel of length L = C/C°. Dimensionless concentration = C/Cava. Dimensionless concentration = C/C Eve Dimensionless concentration Effective diameter, defined by Eq. (50) Particle diameter Tube diameter Dispersion coefficient Axial dispersion coefficient, dispersed plug flow model... 

In most adsorption processes the adsorbent is contacted with fluid in a packed bed. An understanding of the dynamic behavior of such systems is therefore needed for rational process design and optimization. What is required is a mathematical model which allows the effluent concentration to be predicted for any defined change in the feed concentration or flow rate to the bed. The flow pattern can generally be represented adequately by the axial dispersed plug-flow model, according to which a mass balance for an element of the column yields, for the basic differential equation governing llie dynamic behavior,... 

For contactors in which discrete well-mixed compartments can be identified, for example sieve-plate columns, axial mixing effects are incorporated into the stagewise model by means of the backflow ratio a which is defined as the fraction of the net interstage flow of one phase which is considered to flow in the reverse direction. For a contactor in which there are many compartments, the axial dispersion coefficient and the backflow ratio, a, are interrelated as follows ... 

Axial dispersion. An axial (longitudinal) dispersion coefficient may be defined by analogy with Boussinesq s concept of eddy viscosity ". Thus both molecular diffusion and eddy diffusion due to local turbulence contribute to the overall dispersion coefficient or effective diffusivity in the direction of flow for the bed of solid. The moles of fluid per unit area and unit time an element of length 8z entering by longitudinal diffusion will be - D L (dY/dz)t, where D L is now the dispersion coefficient in the axial direction and has units ML T- (since the concentration gradient has units NM L ). The amount leaving the element will be -D l (dY/dz)2 + S2. The material balance equation will therefore be ... 

The piston flow model in Example 9.6 showed a thermal runaway when T, = Twau =374. Will the axial dispersion model show a runaway (defined as d2Tjdz2 > 0) If so, at what value of Tin = Twalp. 

Example 11.14 assumed piston flow when treating the moving-front phenomenon in an ion-exchange column. Expand the solution to include an axial dispersion term. How should breakthrough be defined in this case ... 

The three ideal reactors form the building blocks for analysis of laboratory and commercial catalytic reactors. In practice, an actual flow reactor may be more complex than a CSTR or PFR. Such a reactor may be described by a residence time distribution function F(t) that gives the probability that a given fluid element has resided in the reactor for a time longer than t. The reactor is then defined further by specifying the origin of the observed residence time distribution function (e.g., axial dispersion in a tubular reactor or incomplete mixing in a tank reactor). 

Monodisperse silica nanoparticles of a diameter of 200-500 nm were obtained [329]. This is explained by the well-defined residence time with reduced axial dispersion of the liquid segments [328,329]. In addition, by moving the liquid segments through a tube recirculation flow sets in, which is very effective in liquid mixing. 

The parameter obtained after identification of the system as plug flow with axial dispersion, is the Peclet number defined from the packing height and the real fluid flow velocity ... 

Models with varying degrees of complexity have been employed to analyze the experimental results by a variety of techniques. The most comprehensive models include terms to account for axial dispersion in the packed bed, external mass transfer, intraparticle diffusion in both macropore and micropore regions of the pellet and a finite rate of adsorption. Of the several methods of analysis, the most popular ones are based on the moments of the response curve. The first moment of the chromatogram is defined by Equation 5.25 in which the concentration now is taken at the outlet of the column. The second central moment is calculated from equation... 

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