Axial convective flux

The term Ms represents the axial convective flux, while Pb a is the generative term associated to the reaction. 

The ID ideal model is a plug-flow model, which assumes that concentrations, temperature and pressure vary only in the axial direction. Moreover, no axial mixing is considered therefore the only transport mechanism is the axial convective flux. 

For the region near the attachment point, Mullis found a strong effect of axial position on flux, but no satisfactory general correlation for this effect. In addition, he found no quantitative relation for the heat-transfer characteristics of jets directed toward the propellant surface. Under most conditions studied by Mullis, the radiation contribution is approximately 10% of the convective flux. The effects of solid-particle impingement were not investigated. 

L5. Lee, D. H., and Obertelli, J. D., An experimental investigation of forced convection burn-out in high pressure water. 2. Preliminary results for round tubes with non-uniform axial heat flux distribution, AEEW-R309 (1963). 

This section derives a simple version of the convective diffusion equation, applicable to tubular reactors with a one-dimensional velocity profile V (r). The starting point is Equation (1.4) applied to the differential volume element shown in Figure 8.9. The volume element is located at point (r, z) and is in the shape of a ring. Note that 0-dependence is ignored so that the results will not be applicable to systems with significant natural convection. Also, convection due to is neglected. Component A is transported by radial and axial diffusion and by axial convection. The diffusive flux is governed by Pick s law. 

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 ... 

Lee, D. H., and J. D. Obertelli, 1963, An Experimental Investigation of Forced Convection Burnout in High Pressure Water, Part 2. Preliminary Results for Round Tubes with Non-Uniform Axial Heat Flux Distribution, UK Rep. AEEW-R-309, UK AEEW, Winfrith, England. (5)... 

PeL can be used in place of Dh as the single parameter of the axial dispersion model. The physical interpretation of PeL is that it represents the ratio of the convective flux to fee diffusive (disposed) flux ... 

One conclusion from these results is that the axial diffusion model begins to fail as Pe, - small, when an open boundary condition is used at the outlet. The case Pe, - small means increasing backmixing, or that the diffusive flux becomes increasingly significant compared with the convective flux. For an open boundary condition, it is also questionable whether the actual response C(e) can be identified with E(B). Furthermore, regardless of the boundary conditions chosen, it is difficult to envisage that cA... 

In order to obtain Xer and aw from Equations 2 and 3, the pseudohomogeneous temperature of model I-T must first be defined. Equal axial heat convection flux will be obtained if ... 

The axial dispersion terms may be required to account for the mixing phenomena created by a non-ideal flow. However, the ideal plug flow model is often appropriate for packed bed reactors because the axial mixing is negligible compared to the convective flux for many processes. 

We can also calculate the average convective flux in the axial direction, relative to the axes moving at the mean velocity U, according to... 

Neglect axial diffusion/dispersioti flux wn convective flux when summing the heat capacity times their fluxes,... 

The term on the right hand side of Eq. C5) represents the total flux of A due to Fickian diffusion and facilitated transport in the flat membrane (28). We have assumed that axial diffusion is negligible compared to axial convection and that no convection takes place across the membrane. The equilibrium facilitation fac-... 

The dimensionless group D KuL) is called the vessel dispersion number, and the reciprocal value is called the Bodenstdn number Bo, which can be used in place of Dax as the parameter for axial dispersion and represents the ratio of convective flux to diffusive (dispersed) flux. 

Axial dispersion was often ignored in early modelling of PSA separations not only because the material balance equations, e.g. equation (6.37), could be simplified substantially, but also because the diffusional fluxes were generally, but not always, small compared with the convective fluxes. This, however, may not be the case for vapour phase TSA separations and for liquid phase separations. 

In the fiber lumen, axial convection is by laminar flow with an average steady fluid velocity of u. In this region, three BCs are imposed. The first condition is a Dirichlet-type BC and accounts for the substrate concentration at the entrance of the lumen region. The second is a Neumann-type BC and represents symmetry of radial substrate concentration gradient at the center of the fiber lumen. The third BC is also a Neumann-type BC and reflects continuity of flux at the fiber wall. In the above equations, parameter D with subscripts 1,2, and 3 refers to the diffusivity values of a given species in the lumen fiber membrane wall (R, and cellular matrix R ), respectively. The fiber wall ... 

Warrier et al. (2002) conducted experiments of forced convection in small rectangular channels using FC-84 as the test fluid. The test section consisted of five parallel channels with hydraulic diameter = 0.75 mm and length-to-diameter ratio Lh/r/h = 433.5 (Fig. 4.5d and Table 4.4). The experiments were performed with uniform heat fluxes applied to the top and bottom surfaces. The wall heat flux was calculated using the total surface area of the flow channels. Variation of single-phase Nusselt number with dimensionless axial distance is shown in Fig. 4.6b. The numerical results presented by Kays and Crawford (1993) are also shown in Fig. 4.6b. The measured values agree quite well with the numerical results. 

Cross-flow filtration systems utilize high liquid axial velocities to generate shear at the liquid-membrane interface. Shear is necessary to maintain acceptable permeate fluxes, especially with concentrated catalyst slurries. The degree of catalyst deposition on the filter membrane or membrane fouling is a function of the shear stress at the surface and particle convection with the permeate flow.16 Membrane surface fouling also depends on many application-specific variables, such as particle size in the retentate, viscosity of the permeate, axial velocity, and the transmembrane pressure. All of these variables can influence the degree of deposition of particles within the filter membrane, and thus decrease the effective pore size of the membrane. 

As noted earlier, air-velocity profiles during inhalation and exhalation are approximately uniform and partially developed or fully developed, depending on the airway generation, tidal volume, and respiration rate. Similarly, the concentration profiles of the pollutant in the airway lumen may be approximated by uniform partially developed or fully developed concentration profiles in rigid cylindrical tubes. In each airway, the simultaneous action of convection, axial diffusion, and radial diffusion determines a differential mass-balance equation. The gas-concentration profiles are obtained from this equation with appropriate boundary conditions. The flux or transfer rate of the gas to the mucus boundary and axially down the airway can be calculated from these concentration gradients. In a simpler approach, fixed velocity and concentration profiles are assumed, and separate mass balances can be written directly for convection, axial diffusion, and radial diffusion. The latter technique was applied by McJilton et al. 

In this case, SN is dependent on the axial coordinate the tubular electrode is not uniformly accessible. This complicates the mathematical description of partially kinetically controlled reactions at the TE. However, for total kinetic control (irreversible reaction at the foot of the wave), the flux is uniform as radial convection is uniformly zero along the tube. 

The boundary conditions normally associated with Equation (9.14) are known as the Danckwerts or closed boundary conditions. They are obtained from mass balances across the inlet and outlet of the reactor. We suppose that the piping to and from the reactor is small and has a high Re. Thus, if we were to apply the axial dispersion model to the inlet and outlet streams, we would find Din = Dout = 0, which is the definition of a closed system. See Figure 9.8. The flux in the inlet pipe is due solely to convection and has magnitude Qi ain. The flux just inside the reactor at location z = 0+ has two components. One component, Qina(0+), is due to convection. The other component, —DAc[da/dz 0+, is due to diffusion (albeit eddy diffusion) from the relatively high concentrations at the inlet toward the lower concentrations within the reactor. The inflow to the plane at z = 0 must be matched by material leaving the plane at z = 0+ since no reaction occurs in a region that has no volume. Thus,... 

Forced Convection. In systems where the flux of A results primarily from forced conveetion, we assume that the diffusion in the direction of the flow (e.g., axial z direetion), is small in comparison with the bulk flow contribution in that direction, [/),... 

Instead of the partial differential equation model presented above, the model is developed here in dynamic difference equation form, which is suitable for solution by dynamic simulation packages, such as Madonna. Analogous to the previous development for tubular reactors and extraction columns, the development of 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 liquid phase volume of any element, n, as indicated in Fig. 4.24 below. Here j represents the diffusive flux, L the liquid flow rate and and Cla the concentration of any species A in both the solid and liquid phases, respectively. 

In this work, heat and fluid flow in some common micro geometries is analyzed analytically. At first, forced convection is examined for three different geometries microtube, microchannel between two parallel plates and microannulus between two concentric cylinders. Constant wall heat flux boundary condition is assumed. Then mixed convection in a vertical parallel-plate microchannel with symmetric wall heat fluxes is investigated. Steady and laminar internal flow of a Newtonian is analyzed. Steady, laminar flow having constant properties (i.e. the thermal conductivity and the thermal diffusivity of the fluid are considered to be independent of temperature) is considered. The axial heat conduction in the fluid and in the wall is assumed to be negligible. In this study, the usual continuum approach is coupled with the two main characteristics of the microscale phenomena, the velocity slip and the temperature jump. 

In this lecture, the effects of the abovementioned dimensionless parameters, namely, Knudsen, Peclet, and Brinkman numbers representing rarefaction, axial conduction, and viscous dissipation, respectively, will be analyzed on forced convection heat transfer in microchannel gaseous slip flow under constant wall temperature and constant wall heat flux boundary conditions. Nusselt number will be used as the dimensionless convection heat transfer coefficient. A majority of the results will be presented as the variation of Nusselt number along the channel for various Kn, Pe, and Br values. The lecture is divided into three major sections for convective heat transfer in microscale slip flow. First, the principal results for microtubes will be presented. Then, the effect of roughness on the microchannel wall on heat transfer will be explained. Finally, the variation of the thermophysical properties of the fluid will be considered. 

In this lecture, a variety of results for convective heat transfer in microtubes and microchannels in the slip flow regime under different conditions have been presented. Both constant wall temperature and constant wall heat flux cases have been analyzed in microtubes, including the effects of rarefaction, axial conduction, and viscous dissipation. In rough microchannels the abovementioned effects have also been investigated for the constant wall temperature boundary condition. Then, temperature-variable dynamic viscosity and thermal conductivity of the fluid were considered, and the results were compared with constant property results for microchannels, with the effects of rarefaction and viscous dissipation. 

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