Convection, natural equations

Overall, the RDE provides an efficient and reproducible mass transport and hence the analytical measurement can be made with high sensitivity and precision. Such well-defined behavior greatly simplifies the interpretation of the measurement. The convective nature of the electrode results also in very short response tunes. The detection limits can be lowered via periodic changes in the rotation speed and isolation of small mass transport-dependent currents from simultaneously flowing surface-controlled background currents. Sinusoidal or square-wave modulations of the rotation speed are particularly attractive for this task. The rotation-speed dependence of the limiting current (equation 4-5) can also be used for calculating the diffusion coefficient or the surface area. Further details on the RDE can be found in Adam s book (17). 

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. 

The strategies discussed in the previous chapter are generally applicable to convection-diffusion equations such as Eq. (32). If the function O is a component of the velocity field, the incompressible Navier-Stokes equation, a non-linear partial differential equation, is obtained. This stands in contrast to O representing a temperature or concentration field. In these cases the velocity field is assumed as given, and only a linear partial differential equation has to be solved. The non-linear nature of the Navier-Stokes equation introduces some additional problems, for which special solution strategies exist. Corresponding numerical techniques are the subject of this section. 

Table 1 shows the particular forms of the convective diffusion equation for different geometries. It is fortunate that, due to the symmetrical nature of hydrodynamic electrodes, some of these terms may be neglected. Also, the major part of investigations conducted are under conditions of steady-state flow where dc/dt = 0. The exception to this is, of course, the cyclic operation of the DME. 

In stagnant environments, if natural convection can be ignored, the convective-diffusion equation is reduced to... 

The latter interpretation was successfully used to analyze natural convection at vertical plate electrodes by including a magnetic field-related term in the classical convective diffusion equation. " The beneficial effect of the magnetic field on mass transport may be estimated from the ratio of the limiting current density in a magnetic field to that in its absence, called the augmentation factor ... 

As discussed in the introduction, many current-distrihution problems are not described by a simple, convection-diffusion equation or by Laplace s equations. Alavyoon era/. provided an example in which the coupled concentration and potential fields were solved throughout the entire computational domain, along with natural convection flow fields. The equations were solved by evaluating the nonlinear terms at the previous time step. Gu etal. provided an additional study that coupled charge and mass transfer to natural convection. This work is related to... 

We can see that Eqs. (2 101) (2-104) are sufficient to calculate the continuum-level stress a given the strain-rate and vorticity tensors E and SI. As such, this is a complete constitutive model for the dilute solution/suspension. The rheological properties predicted for steady and time-dependent linear flows of the type (2-99), with T = I t), have been studied quite thoroughly (see, e g., Larson34). Of course, we should note that the contribution of the particles/macromolecules to the stress is actually quite small. Because the solution/suspension is assumed to be dilute, the volume fraction is very small, (p 1. Nevertheless, the qualitative nature of the particle contribution to the stress is found to be quite similar to that measured (at larger concentrations) for many polymeric liquids and other complex fluids. For example, the apparent viscosity in a simple shear flow is found to shear thin (i.e., to decrease with increase of shear rate). These qualitative similarities are indicative of the generic nature of viscoelasticity in a variety of complex fluids. So far as we are aware, however, the full model has not been used for flow predictions in a fluid mechanics context. This is because the model is too complex, even for this simplest of viscoelastic fluids. The primary problem is that calculation of the stress requires solution of the full two-dimensional (2D) convection-diffusion equation, (2-102), at each point in the flow domain where we want to know the stress. 

The high sensitivity of the Allendoerfer cell makes it of great value in the detection of unstable radicals but, for the study of the kinetics and mechanism of radical decay, the use of a hydrodynamic flow is required. The use of a controlled, defined, and laminar flow of solution past the electrode allows the criteria of mechanism to be established from the solution of the appropriate convective diffusion equation. The uncertain hydrodynamics of earlier in-situ cells employing flow, e.g. Dohrmann [42-45] and Kastening [40, 41], makes such a computational process uncertain and difficult. Similarly, the complex flow between helical electrode surface and internal wall of the quartz cell in the Allendoerfer cell [54, 55] means that the nature of the flow cannot be predicted and so the convective diffusion equation cannot be readily written down, let alone solved Such problems are not experienced by the channel electrode [59], which has well-defined hydrodynamic properties. Compton and Coles [60] adopted the channel electrode as an in-situ ESR cell. 

The surface electric current is approximately zero since electric currents of the anions and cations are of a convective nature and roughly compensate each other. The electric current density within the diffuse layer of ionic surfactants obeys the equation... 

Assuming that the active layer of the nanofiltration membrane is porous in nature, the extended Nernst-Planck equation is apphcable to describe the transport of multieomponent systems in nanofiltration membranes. It represents transport due to diffusion, eleetrieal potential gradient and convection. The equations can be written as ... 

We next consider cells designed with the intention of studying the kinetics and mechanism of radical decay, as well as identifying the presence of particular radicals through their ESR spectra. We have noted (see above) that for this end, it is desirable not only to have a hydrodynamic flow over the electrode surface, but also that this flow be well defined and calculable so that the distribution of radicals in space and time may be calculated by solving the relevant convective-diffusion equation. We suggested above that this process could be expected to be difficult for the flow cells of Dohrmann and of Kastening because of the uncertain hydrodynamics of those cells. Likewise, the nature of the flow in Allendoerfer s cell cannot be confidently predicted, since it involves a complex flow between the inside of a silica tube and the surface of a coiled helix. We now describe the flow cell due to Compton and Coles, which was shown to display predictable and calculable hydrodynamics. 

We see from our discussion of the 1-D convection/diffusion equation that the qualitative nature of an equation, and of its numerical solution, can change as we vary the parameters. These changes are related to the nature of how information about the field is propagated by the differential equation in space and time. The importance of information flow is reflected in a common naming convention for second-order PDEs. As this nomenclature is employed often in the literature, we briefly review it here. 

Natural convection occurs when a solid surface is in contact with a fluid of different temperature from the surface. Density differences provide the body force required to move the flmd. Theoretical analyses of natural convection require the simultaneous solution of the coupled equations of motion and energy. Details of theoretical studies are available in several general references (Brown and Marco, Introduction to Heat Transfer, 3d ed., McGraw-HiU, New York, 1958 and Jakob, Heat Transfer, Wiley, New York, vol. 1, 1949 vol. 2, 1957) but have generally been applied successfully to the simple case of a vertical plate. Solution of the motion and energy equations gives temperature and velocity fields from which heat-transfer coefficients may be derived. The general type of equation obtained is the so-called Nusselt equation hL I L p gp At cjl... 

Nusselt Equation for Various Geometries Natural-convection coefficients for various bodies may be predicted from Eq. (5-32). The various numerical values of 7 andm have been determined experimen-... 

The lower Emit of applicability of the nucleate-boiling equations is from 0.1 to 0.2 of the maximum limit and depends upon the magnitude of natural-convection heat transfer for the liquid. The best method of determining the lower limit is to plot two curves one of h versus At for natural convection, the other ofh versus At for nucleate boiling. The intersection of these two cui ves may be considered the lower limit of apphcability of the equations. 

Experimental gas-solid mass-transfer data have been obtained for naphthalene in CO9 to develop correlations for mass-transfer coefficients [Lim et al., Am. Chem. Soc. Symp. Ser, 406, 379 (1989)]. The data were correlated over a wide range of conditions with the following equation for combined natural and forced convection ... 

Evidently, from the low value of the exponent in Equation 7-80, the contribution from natural convection and, hence, its practical significance is small. 

Flemeon is the first standard reference book that presents the equations for calculating thermal updrafts. These equations are repeated and expanded in other standard reference books, including Heinsohn, Goodfellow, and the ACGIFl Industrial Ventilation Manual.These equations are derived from the more accurate formulas for heat transfer (Nusselt number) at natural convection (where density differences, due to temperature differences, provide the body force required to move the fluid) and both the detailed and the simplified formulas can be found in handbooks on thermodynamics (e.g., Perry--, and ASHRAE -). 

Convection is heat transfer between portions of a fluid existing under a thermal gradient. The rate of convection heat transfer is often slow for natural or free convection to rapid for forced convection when artificial means are used to mix or agitate the fluid. The basic equation for designing heat exchangers is... 

For organic liquids, evaluate the natural convection film coefficient from Figure 10-103. Equation 10-29 may be used for the inside horizontal tube by multiplying the right side of the equation by 2.25 (1 + 0.010 Gr,i/")/logRe. 

Equation 9.215 is valid for Reynolds Numbers in excess of 10,000. Where the Reynolds Number is less than 2000, the flow will be laminar and, provided natural convection effects... 

Garner and Keey(52 53) dissolved pelleted spheres of organic acids in water in a low-speed water tunnel at particle Reynolds numbers between 2.3 and 255 and compared their results with other data available at Reynolds numbers up to 900. Natural convection was found to exert some influence at Reynolds numbers up to 750. At Reynolds numbers greater than 250, the results are correlated by equation 10.230 ... 

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