Convective dispersion convection

Effect of dispersion on a sample s flow profile at different times during a flow injection analysis (a) at injection and when the dispersion is due to (b) convection ... 

When a sample is injected into the carrier stream it has the rectangular flow profile (of width w) shown in Figure 13.17a. As the sample is carried through the mixing and reaction zone, the width of the flow profile increases as the sample disperses into the carrier stream. Dispersion results from two processes convection due to the flow of the carrier stream and diffusion due to a concentration gradient between the sample and the carrier stream. Convection of the sample occurs by laminar flow, in which the linear velocity of the sample at the tube s walls is zero, while the sample at the center of the tube moves with a linear velocity twice that of the carrier stream. The result is the parabolic flow profile shown in Figure 13.7b. Convection is the primary means of dispersion in the first 100 ms following the sample s injection. 

Convection heat transfer is dependent largely on the relative velocity between the warm gas and the drying surface. Interest in pulse combustion heat sources anticipates that high frequency reversals of gas flow direction relative to wet material in dispersed-particle dryers can maintain higher gas velocities around the particles for longer periods than possible ia simple cocurrent dryers. This technique is thus expected to enhance heat- and mass-transfer performance. This is apart from the concept that mechanical stresses iaduced ia material by rapid directional reversals of gas flow promote particle deagglomeration, dispersion, and Hquid stream breakup iato fine droplets. Commercial appHcations are needed to confirm the economic value of pulse combustion for drying. 

First order parameters affecting dispersion stem from meteorological conditions. These, as much as any other consideration, determine how a stack is to be designed for air pollution control purposes. Since the operant transport mechanisms are determined by the micro-meteorological conditions, any attempt to predict ground-level pollutant concentrations is dependent on a reasonable estimate of the convective and dispersive potential of the local air. The following are meteorological conditions which need to be determined ... 

Other factors to account for topography with regard to valley or hillside sites should include possible inversion and failure to disperse pollutants. Temperature inversion occurs when the temperature at a certain layer of the atmosphere stays constant, or even increases with height, as opposed to decreasing with height, which is the norm for the lower atmosphere. Inversions may occur on still, clear nights when the earth and adjacent air cools more rapidly than the free atmosphere. They may also occur when a layer of high turbulence causes rapid vertical convection so that the top of the turbulent layer may be cooler than the next layer above it at the interface. 

The primary boiler plant problem here is cold-end corrosion, caused by the destructive effects of sulfuric acid produced within the convection area. Further problems include acid rain, which occurs when sulfur gases are emitted and widely dispersed to eventually produce sulfuric acid in the upper atmosphere, which precipitates as rain. 

For example, for equal volumes of gas and liquid ( =0.5), Eq. (266) predicts that the Stokes velocity (which is already very small for relatively fine dispersions) should be reduced further by a factor of 38 due to hindering effects of its neighbor bubbles in the ensemble. Hence in the domain of high values and relatively fine dispersions, one can assume that the particles are completely entrained by the continuous-phase eddies, resulting in a negligible convective transfer, although this does not preclude the existence of finite relative velocities between the eddies themselves. 

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 ... 

Axial Dispersion. Rigorous models for residence time distributions require use of the convective diffusion equation. Equation (14.19). Such solutions, either analytical or numerical, are rather difficult. Example 15.4 solved the simplest possible version of the convective diffusion equation to determine the residence time distribution of a piston flow reactor. The derivation of W t) for parabolic flow was actually equivalent to solving... 

Combining hindered diffusion theory with the diffusion/convection problem in the model pore, Trinh et al. [399] showed how the effective transport coefficients depend upon the ratio of the solute to pore size. Figure 28 shows that as the ratio of solute to pore size approaches unity, the effective mobility function becomes very steep, thus indicating that the resolution in the separation will be enhanced for molecules with size close to the size of the pore. Similar results were found for the effective dispersion, and the implications for the separation of various sizes of molecules were discussed by Trinh et al. [399]. 

The form of the effective mobility tensor remains unchanged as in Eq. (125), which imphes that the fluid flow does not affect the mobility terms. This is reasonable for an uncharged medium, where there is no interaction between the electric field and the convective flow field. However, the hydrodynamic term, Eq. (128), is affected by the electric field, since electroconvective flux at the boundary between the two phases causes solute to transport from one phase to the other, which can change the mean effective velocity through the system. One can also note that even if no electric field is applied, the mean velocity is affected by the diffusive transport into the stationary phase. Paine et al. [285] developed expressions to show that reversible adsorption and heterogeneous reaction affected the effective dispersion terms for flow in a capillary tube the present problem shows how partitioning, driven both by electrophoresis and diffusion, into the second phase will affect the overall dispersion and mean velocity terms. 

Edwards, DA, Charge Transport Through a Spatially Periodic Porous Medium Electrokinetic and Convective Dispersion Phenomena, Philosophical Transactions of the Royal Society of London A 353, 205, 1995. 

Knud 0stergaard, Gas-Liquid-Particle Operations in Chemical Reaction Engineering J. M. Prausnitz, Thermodynamics of Fluid-Phase Equilibria at High Pressures Robert V. Macbeth, The Burn-Out Phenomenon in Forced-Convection Boiling William Resnick and Benjamin Gal-Or, Gas-Liquid Dispersions... 

The key analysis of hydrodynamic dispersion of a solute flowing through a tube was performed by Taylor [149] and Aris [150]. They assumed a Poiseuille flow profile in a tube of circular cross-section and were able to show that for long enough times the dispersion of a solute is governed by a one-dimensional convection-diffusion equation ... 

Aeis, R., On the dispersion of a solute by diffusion, convection and exchange between phases, Proc. R. Soc. London, A 252 (1959) 538-550. 

In the above case, D is an eddy dispersion coefficient and Z is the axial distance along the reactor length. When combined with an axial convective flow contribution, and considering D as constant, the equation takes the form... 

The development of the equations for the dynamic dispersion model starts by considering an element of tube length AZ, with a cross-sectional area of Ac, a superficial flow velocity of v and an axial dispersion coefficient, or diffusivity D. Convective and diffusive flows of component A enter and leave the element, as shown by the solid and dashed arrows respectively, in Fig. 4.12. 

In these equations the designation for dimensionless concentration c, has been dropped. Note that in the above equation, the finite differencing of the convection term has been done over two neighbouring segments. Again special relationships apply to the end segments, owing to the absence of axial dispersion, exterior to the cake. 

Strictly speaking, in this formulation the effective diffusion coefficient, is replaced by an empirical dispersion coefficient, D, to account for the effect of water flow on diffusion. However, in practice, the rate of transpirational water flow is sufficiently slow that dispersion effects are minimal and Eq. (8) can be used without error. This is because the Peclet number (see Sect. F.2) is small. For the same reason, in almost all cases diffusion is the most important process in moving nutrients to the root and the convection term can be omitted entirely. 

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