Convection plugs

When a fluid passes through a packed column, the flow is divided due to the packing. Modelling of these phenomena is carried out by superimposing a dispersion, characterized by a coefficient D on the convective plug flow of velocity U. This is the model for an axial dispersion reactor. This model allows characterisation of a flow with intermediate properties between those of the plug flow reactor and those of a continuous stirred reactor. 

Heat Transfer in Rotary Kilns. Heat transfer in rotary kilns occurs by conduction, convection, and radiation. In a highly simplified model, the treatment of radiation can be explained by applying a one-dimensional furnace approximation (19). The gas is assumed to be in plug flow the absorptivity, a, and emissivity, S, of the gas are assumed equal (a = e ) and the presence of water in the soHds is taken into account. Energy balances are performed on both the gas and soHd streams. Parallel or countercurrent kilns can be specified. 

Macromixing is estabflshed by the mean convective flow pattern. The flow is divided into different circulation loops or zones created by the mean flow field. The material is exchanged between zones, increasing homogeneity. Micromixing, on the other hand, occurs by turbulent diffusion. Each circulation zone is further divided into a series of back-mixed or plug flow cells between which complete intermingling of molecules takes place. 

The effectiveness of a fluidized bed as a ehemical reactor depends to a large extent on the amount of convective and diffusive transfer between bubble gas and emulsion phase, since reaction usually occurs only when gas and solids are in contact. Often gas in the bubble cloud complex passes through the reactor in plug flow with little back mixing, while the solids are assumed to be well mixed. Actual reactor models depend greatly on kinetics and fluidization characteristics and become too complex to treat here. 

Separation layer mixers use either a miscible or non-miscible layer between the reacting solutions, in the first case most often identical with the solvent used [48]. By this measure, mixing is postponed to a further stage of process equipment. Accordingly, reactants are only fed to the reaction device, but in a defined, e.g. multi-lamination-pattem like, fluid-compartment architecture. A separation layer technique inevitably demands micro mixers, as it is only feasible in a laminar flow regime, otherwise turbulent convective flow will result in plugging close to the entrance of the mixer chamber. 

Convection is mass transfer that is driven by a spatial gradient in pressure. This section presents two simple models for convective mass transfer the stirred tank model (Section II.A) and the plug flow model (Section n.B). In these models, the pressure gradient appears implicitly as a spatially invariant fluid velocity or volumetric flow rate. However, in more complex problems, it is sometimes necessary to develop an explicit relationship between fluid velocity and pressure gradients. Section II.C describes the methods that are used to develop these relationships. 

Figure 3 The plug flow model for convective mass transfer. The drawing shows a tube of diameter D and length L. Discrete fluid plugs (shaded rectangles) move down the tube fluid within each plug is completely mixed, while there is no mixing between adjacent plugs. The system is a cylindrical section of the tube between z + z and Az and is fixed in space. Fluid enters the system at z with density p(z) and volumetric flow rate q(z) fluid exits the system at z + Az with density p(z + Az) and volumetric flow rate q(z, + Az).

While the simple stirred tank and plug flow models are adequate to describe convective transport in many cases, a more complete description of fluid flow is sometimes needed. For example, an accurate description of tablet dissolution in a stirred vessel may require information about the changing fluid velocity near the tablet surface. Neither the stirred tank nor the plug flow models can address these velocity changes, since both assume that velocity is independent of position and time. In such cases, a more detailed description of fluid flow can be developed using the Navier-Stokes equations, which describe the effects of pressure, viscos-... 

Note that when the fluid velocity (v) is constant, the description of convection given by the second term on the right-hand side of this equation is identical to that of the plug flow model [Eq. (8)]. In more complex systems, a spatially varying fluid velocity may by incorporated by using the Navier-Stokes equations [Eqs. (10)—(12)] to describe velocity profiles. 

Plug flow A simple convective flow pattern in pipes and tubes that is characterized by a fluid velocity independent of radial position, complete mixing in the radial direction, and no mixing in the axial direction. Also called the parallel tube model or tubular flow. See Eqs. (7) and (8) and Figure 3. 

In PF, the transport of material through a vessel is by convective or bulk flow. All elements of fluid, at a particular axial position in the direction of flow, have the same concentration and axial velocity (no radial variation). We can imagine this ideal flow being blurred or dispersed by backmixing of material as a result of local disturbances (eddies, vortices, etc.). This can be treated as a diffusive flow superimposed on the convective flow. If the disturbances are essentially axial in direction and not radial, we refer to this as axial dispersion, and the flow as dispersed plug flow (DPF). (Radial dispersion may also be significant, but we consider only axial dispersion here.)... 

Figure 6. A schematic of the electroosmotic flow of the medium (e.g., an electrolyte) in a capillary caused by the flow of counter ions as a plug, under the influence of the applied electric field, E UL, is the convective liquid velocity from electroosmosis. Adapted from Everett.48...

For laminar flow in short tubes or laminar flow of viscous materials these models may not apply, and it may be that the parabolic velocity profile is the main cause of deviation from plug flow. We treat this situation, called the pure convection model, in Chapter 15. 

Figure 15.10 Convective flow lowers conversion compared to plug flow.

A viscous liquid is to react while passing through a tubular reactor in which flow is expected to follow the convection model. What conversion can we expect in this reactor if plug flow in the reactor will give 80% conversion ... 

Note the mean residence time for this distribution is the same as for the pure plug flow. However, in this case, only an infinitesimal element of the flow has this precise residence time. Much of the fluid passes through the vessel by mixing rather than convective forces so that it spends too short a time in the vessel. This is offset by elements of fluid which spend too long a time in the vessel, and thus reduce Its effective capacity. 

Fig. 5 a. The development of the concentration profile due to a plug of protein solution entering a buffer-primed, thin plate flow channel. Note that a bullet-shaped concentration profile develops with time (assuming no diffusion) b. the geometry and coordinate system used in the convection-diffusion treatment... 

In order to understand these boundary conditions, let us consider that the inlet pipe in which ideal plug flow occurs has the same diameter (shown by broken lines) as the reactor itself (Fig. 2.21). Inside the reactor, across any section perpendicular to the z-direction, the flux of the reactant, i.e. the rate of transfer is made up of two contributions, the convective flow uC and the diffusion-like dispersive flow... 

Although the mixing patterns in bubble columns do not obviously correspond to simple axial dispersion, the dispersed plug flow model has been found to hold reasonably well in practice. For a two-phase gas-liquid system, the equation for gas-phase convection and dispersion (Chapter 2, equation 2.14) becomes ... 

The Peclet numbers are useful for estimating the relative contributions of convection and diffusion to mass and heat transfer. If Pe is large (>10), convection dominates, and a plug-flow model may be appropriate for simple reactor computations. When Pe is small (<<1), diffusion dominates, and the system behaves like a well-stirred reactor. Thus, Pe may be used to estimate whether downstream impurities can diffuse into the deposition zone. 

Most of the models assume that neutral-species transport can be represented with either a well-mixed model or a plug flow model. The major drawback to these assumptions is that important inelastic rate processes such as molecular dissociation are usually localized in space in the reactor and are often fast compared with rates of diffusion or convection. As a result, the spatial variation of fluid flow in the reactor must be accounted for. This variation introduces a major complication in the model, because the solution of the nonisothermal Navier-Stokes equations in multidimensional geometries is expensive and difficult. 

For plug flow, only the flow and the processes other than mixing, diffusion, and conduction are considered. These have been studied in Chapter 4. In a plug flow tubular reactor model we consider only the convective one-dimensional flow and the chemical reaction as shown in Figure 5.1, where n is the convective molar flow rate for the constant volumetric flow rate g of component i. These two rates are connected by the equation rq = q Ci for the concentration Cj. 

In the nonisothermal case, the plug flow model accounts only for the convective heat flow as shown in Figure 5.3. 

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