The problem of convection

Note that flow velocities as little as 0.05 mm s can be sufficient to lead to an erroneous apparent diffiision coefficient for a species whose actual value of D is 1 x 

A more sensitive and reliable test for the presence of convection is to record two or more diffusion experiments with differing values of the diffiision period A under otherwise identical conditions. In the absence of convection (or where its influence has been suppressed), the value of D obtained should not differ between data sets. In contrast, the apparent diffusion coefficients measured in the presence of convection will vary with A, as indicated by Eq. (9.11), and produce progressively larger values of / app with longer diffusion periods. This influence is readily apparent for quinine 9.1 in CDCI3 recorded at the slightly elevated temperature of 313 K but may also be observed at a much reduced level at 298 K where probe temperature regulation is employed (Fig. 9.12). 

As indicated above, when convection can be shown to be present within a sample, there are essentially two approaches for negating this one may seek to suppress eonveetion itself or one can employ methods that are designed to compensate for it. The optimum approach will depend on a number of factors relating to the nature of the sample and the behaviour of the instrumentation, so both these approaches will be eonsidered before some general recommendations are presented. 

Before proceeding, it is instructive to consider the conditions under whieh eonveetion arises. The onset of convection, as driven by temperature gradients along the length of the NMR sample, is characterised by the Rayleigh number, R, which for a long cylindrical sample may be defined as 

It is apparent from above that one option to delay the onset of convection is to reduce the temperature gradient along the sample column. A crude approach to this is to not use active temperature regulation of the sample at all that is, have no gas flow over the sample and to ensure it is left to fully equilibrate, although this is clearly somewhat limiting and possibly impractical. Certainly, it is critical to allow samples to equilibrate under whatever conditions are used prior to diffusion measurements, and the timescales for this are considerably greater than would typically be considered sufficient for most high-resolution NMR experiments equilibration periods of 30 min or more are often appropriate. 

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The problem of convective diffusion toward the growing drop was solved in 1934 by Dionyz Ilkovic under certain simplifying assumptions. For reversible reactions (in the absence of activation polarization), the averaged cnrrent at the DME can be represented as... 

Many different mechanical arrangements have been used in the construction of rotors for centrifugation. For analytical centrifugation, these rotors must be designed to overcome the problem of convection and to make it possible to detect the component zones which are normally hard to track... 

The calculation of diffusion fluxes and the mean Sherwood number is usually carried out in three steps. First, the problem of convective mass transfer is solved and the concentration field is determined. Second, the normal derivative dC /dC) =0 on the surface is evaluated. Finally, one applies formulas (3.1.25)—... 

Solid particles. The problem of convective diffusion to a chain of solid reacting particles was studied in [168, 350], The retardation mechanism (shielding) of mass exchange in a chain of solid particles and the quantitative behavior of such a system are the same as for chains of drops. 

Since mass transfer must be rate determining, first, we consider the problem of convective diffusion of the /-species moving toward the surface. The steady-state conditions must be fulfilled so we can say that in the presence of an effective supporting electrolyte... 

It should be pointed out that in the formulation of the problem of convective diffusion one should know the velocity distribution v(z) since it appears on the lefk-hand side of Eq. (8.8). A solution of the hydrodynamic problem is possible in the limiting cases of small and large Reynolds numbers. Therefore, the theory of the boundary diffusion layer for the two limiting cases Re l, Pe l and Re l, Pe l has to be developed. 

The main distinction of the theory of a dynamic adsorption layer formed under weak and strong retardation arises when formulating the convective diffusion equation. At weak retardation the Hadamard-Rybczynski hydrodynamic velocity field is used while at strong retardation the Stokes velocity field. Different formulas for the dependence of the diffusion layer thickness on Peclet numbers are obtained. The problem of convective diffusion in the neighbourhood of a spherical particle with an immobile surface at small Reynolds numbers and condition (8.74) is solved, so that the well-known expression for the density distribution of the diffusion flow along the surface can be used. As a result, Eq. (8.10) takes the form (Dukhin, 1982),... 

The analogy between the problem of convective diffusion in a channel with semipermeable walls and the problem of mixed heterogeneous reaction makes it possible to consider these problems together. Conclusions that are true for one problem would be true for the other problem as well, even though these processes are distinct from both physical and chemical viewpoints. Besides, the analogy with chemical reactions allows us to introduce a dimensionless parameter of the problem (see (6.16)) - the Damkoler number ... 

By analogy with the problem of convective diffusion in the channel with a soluble wall (see Section 6.3) and in the channel with membrane walls (see Section 6.4), we can introduce the concepts of the region of concentration development and the region of developed concentration (see Fig. 7.5). 

Electrophoresis take place in an annulus (typically 3 mm wide) between the inner and outer cylinders of the separator A buffer electrolyte (the carrier solution) flows upward through Jthe annulus. Stable laminar flow of the carrier solution is maintained by rotation of the outer cylinder (rotor). This overcomes the problem of convective turbulence which would otherwise be caused by resistive heating, The inner cylinder (stator) is held stationary. 

However, an exact solution to the problem of convective diffusion to a solid surface requires first the solution of the hydrodynamic equations of motion of the fluid (the Navier-Stokes equations) for boundary conditions appropriate to the mainstream velocity of flow and the geometry of the system. This solution specifies the velocity of the flrrid at any point and at any time in both tube and yam assembly. It is then necessary to substitute the appropriate values for the local fluid velocities in the convective diffusion equation, which must be solved for boundary cortditiorts related to the shape of the package, the mainstream concentration of dye and the adsorptions at the solid surface. This is a very difficrrlt procedure even for steady flow through a package of simple shape. " ... 

McGregor followed Levich s approach to solve the problem of convective diffusion to a flat plate immersed in a steady flow of solution (of concentration at a mainstream velocity by considering only a two-dimensional model, hi this model the y co-ordinate is normal to the surface and the x co-ordinate is parallel to it and in the direction of flow, the origin of the co-ordinate system being located at the leading edge of the plate. He assumed that cldx ff cldy, i.e. that diffusion occurs predominantly towards or away fi om the surface along thej-axis, so Eq. 3.44 takes a much simpler form. In addition, he further assumed a steady state of convective diffusion dcldt=Qi) therefore ... 

The thermal-convection loops are limited to flow velocities up to about 6 cm s . Where higher velocities are required, the liquid must be pumped, either mechanically or electromagnetically the latter is usually preferred as it avoids the problem of leakage at the pump seal. Basically, these forced-convection systemsconsist of (c) a hot leg, where the liquid metal is... 

In many of the applications of heat transfer in process plants, one or more of the mechanisms of heat transfer may be involved. In the majority of heat exchangers heat passes through a series of different intervening layers before reaching the second fluid (Figure 9.1). These layers may be of different thicknesses and of different thermal conductivities. The problem of transferring heat to crude oil in the primary furnace before it enters the first distillation column may be considered as an example. The heat from the flames passes by radiation and convection to the pipes in the furnace, by conduction through the... 

In Section 9.3.4, consideration is given to the problem of heat transfer by conduction through a surrounding fluid to spherical particles or droplets. Relative motion between the fluid and particle or droplet causes an increase in heat transfer, much of which may be due to convection. Many investigators have correlated their data in the form ... 

The problem of burn-out prediction is a difficult one, and one on which a great deal of experimental work is being carried out, particularly in connection with nuclear-reactor development. Much of the earlier literature is rather confused, due to the fact that the mechanics of the burn-out were not carefully defined. Silvestri (S8) has discussed the definitions applicable to burn-out heat flux. It appears possible to define two distinctly different kinds of burn-out, one due to a transition from nucleate to film boiling, and one occurring at the liquid deficient point of the forced-convection region. The present discussion treats only the latter type of burn-out fluxes. The burn-out point in this instance is usually determined by the sudden rise in wall temperature and the corresponding drop in heat flux and heat-transfer coefficient which occur at high qualities. 

Here we review some of the correlations of convective mass transfer. We will find that many reactors are controlled by mass transfer processes so this topic is essential in describing many chemical reactors. This discussion will necessarily be very brief and qualitative, and we win summarize material that most students have encountered in previous courses in mass transfer. Our goal is to write down some of the simple correlations so we can work examples. The assumptions in and validity of particular expressions should of course be checked if one is interested in serious estimations for particular reactor problems. We will only consider here the mass transfer correlations for gases because for liquids the correlations are more comphcated and cannot be easily generalized. 

Crystal growth rate may be constant, which could happen if temperature is decreasing or if there is convection. Smith et al. (1956) treated the problem of diffusion for constant crystal growth rate. In the interface-fixed reference frame, the diffusion equation in the melt is... 

Because D is independently determined, and p is obtainable from initial conditions and thermod5mamic equilibrium, the problem of determining the convective dissolution rate now becomes the problem of estimating the boundary layer thickness. In fluid dynamics, the boundary layer thickness appears in a dimensionless number, the Sherwood number Sh ... 

Brenner (B6) pointed out that similar problems arise in obtaining Eq. (3-44) as in the low Re approximation for fluid flow. The neglected convection terms dominate far from the particle, since the ratio of convective to diffusive terms is 0[Pe(r/a)]. An asymptotic solution to Eq. (3-39) with Pe 0 was therefore obtained by the matching procedure of Proudman and Pearson discussed above. Brenner s result for the first term in a series expansion for Sh may be written ... 

During recent years experimental work continued actively upon the macroscopic aspects of thermal transfer. Much work has been done with fluidized beds. Jakob (D5, J2) made some progress in an attempt to correlate the thermal transport to fluidized beds with transfer to plane surfaces. This contribution supplements work by Bartholomew (B3) and Wamsley (Wl) upon fluidized beds and by Schuler (S10) upon transport in fixed-bed reactors. The influence of thermal convection upon laminar boundary layers and their transition to turbulent boundary layers was considered by Merk and Prins (M5). Monaghan (M7) made available a useful approach to the estimation of thermal transport associated with the supersonic flow of a compressible fluid. Monaghan s approximation of Crocco s more general solution (C9) of the momentum and thermal transport in laminar compressible boundary flow permits a rather satisfactory evaluation of the transport from supersonic compressible flow without the need for a detailed iterative solution of the boundary transport for each specific situation. None of these references bears directly on the problem of turbulence in thermal transport and for that reason they have not been treated in detail. 

In Section 17.8.3 we discussed the catalytic combustion of methane within a single one of the tubes in a honeycomb catalyst, illustrated in Fig. 17.18. The high velocity, and thus the dominance of convective over diffusive transport, makes the boundary layer approximations valid for this system. We will model the catalytic combustion performance in one of the honeycomb channels in this problem. 

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