Flow with Small Peclet Numbers

Chapter 11 consists of following Sect. 11.2 deals with the pattern of capillary flow in a heated micro-channel with phase change at the meniscus. The perturbed equations and conditions on the interface are presented in Sect. 11.3. Section 11.4 contains the results of the investigation on the stability of capillary flow at a very small Peclet number. The effect of capillary pressure and heat flux oscillations on the stability of the flow is considered in Sect. 11.5. Section 11.6 deals with the study of capillary flow at a moderate Peclet number. 

Figure 9-4. Contours of constant temperature (isotherms) for heat transfer from a sphere in a uniform flow at low Peclet numbers according to Eq. (9-51). Note that the sphere appears in this representation, in which Ic=k/Uoo, as a point source (sink) of heat (the radius of the sphere, a, is vanishingly small compared with k/Uoo in the limit as Pe - 0). In the inner region, near to the sphere, the isotherms at leading order of approximation are still spherical as illustrated in Fig. 9-2. The three contours plotted are o = 0.25, 2.75, and 5.25.

For each w satisfying condition (4.4.27), the leading terms of the asymptotic expansions for Eqs. (4.4.26) and (4.4.28) with the same boundary conditions coincide in the inner and outer regions. Therefore, as Pe - 0, in the diffusion equation one can replace the actual fluid velocity field v by w. This fact allows one to use the results presented later on in Section 4.11. Namely, as w we take the velocity field for the potential flow of ideal fluid past the cylinder. This approximation yields an error of the order of Pe in the inner expansion. By retaining only the leading terms in (4.11.15), we obtain the dimensionless diffusion flux at small Peclet numbers in the form... 

Figure 7. Longitudinal dispersion (Dl) divided by the diffusion coefficient (Df) for tracers measured in column experiments as a function of the particle scale Peclet number (Npe). It is defined as the product of the average pore fluid velocity, u, and the grain diameter, d, divided by the free fluid diffusion coefficient, D/. The magnitude of the dispersion is independent of the pore fluid velocity (Vp) for very small Peclet numbers (or fluid velocities). Note that the effective diffusion coefficient in a porous media is smaller than the diffusion coefficient in a free fluid phase due to the tortuosity. The dispersion increases linearly with increasing flow velocity (increasing Peclet number). Modified from Appelo and Postma (1999).

It is seen that mixing time scales linearly with the Peclet number. However, the mixing time is decreased dramatically since W[ /. For D = 10 m /s (typical of small molecules or ions) and Wf = 100 nm, the mixing time is only 10 ps. This mixer is a very effective tool for rapidly changing the chemical environment of species in the central focused stream at the same time consuming a smaller sample volume due to the low flow rate of the inlet stream. Hydrodynamic focusing mixers find applications in the study of fast kinetics such as protein folding. 

Dispersion In tubes, and particiilarly in packed beds, the flow pattern is disturbed by eddies diose effect is taken into account by a dispersion coefficient in Fick s diffusion law. A PFR has a dispersion coefficient of 0 and a CSTR of oo. Some rough correlations of the Peclet number uL/D in terms of Reynolds and Schmidt numbers are Eqs. (23-47) to (23-49). There is also a relation between the Peclet number and the value of n of the RTD equation, Eq. (7-111). The dispersion model is sometimes said to be an adequate representation of a reaclor with a small deviation from phig ffow, without specifying the magnitude ol small. As a point of superiority to the RTD model, the dispersion model does have the empirical correlations that have been cited and can therefore be used for design purposes within the limits of those correlations. 

Peclet number independent of Reynolds number also means that turbulent diffusion or dispersion is directly proportional to the fluid velocity. In general, reactors that are simple in construction, (tubular reactors and adiabatic reactors) approach their ideal condition much better in commercial size then on laboratory scale. On small scale and corresponding low flows, they are handicapped by significant temperature and concentration gradients that are not even well defined. In contrast, recycle reactors and CSTRs come much closer to their ideal state in laboratory sizes than in large equipment. The energy requirement for recycle reaci ors grows with the square of the volume. This limits increases in size or applicable recycle ratios. 

The onset of flow instability in a heated capillary with vaporizing meniscus is considered in Chap 11. The behavior of a vapor/liquid system undergoing small perturbations is analyzed by linear approximation, in the frame work of a onedimensional model of capillary flow with a distinct interface. The effect of the physical properties of both phases, the wall heat flux and the capillary sizes on the flow stability is studied. A scenario of a possible process at small and moderate Peclet number is considered. The boundaries of stability separating the domains of stable and unstable flow are outlined and the values of the geometrical and operating parameters corresponding to the transition are estimated. 

The capillary flow with distinct evaporative meniscus is described in the frame of the quasi-dimensional model. The effect of heat flux and capillary pressure oscillations on the stability of laminar flow at small and moderate Peclet number is estimated. It is shown that the stable stationary flow with fixed meniscus position occurs at low wall heat fluxes (Pe -Cl), whereas at high wall heat fluxes Pe > 1, the exponential increase of small disturbances takes place. The latter leads to the transition from stable stationary to an unstable regime of flow with oscillating meniscus. 

Both phases are substantially in plug flow. Dispersion measurements of the liquid phase usually report Peclet numbers, uLdp/D, less than 0.2. With the usual small particles, the wall effect is negligible in commercial vessels of a meter or so in diameter, but may be appreciable in lab units of 50 mm dia. Laboratory and commercial units usually are operated at the same space velocity, LHSV, but for practical reasons the lengths of lab units may be only 0.1 those of commercial units. 

The orientation of anisotropic particles during dip coating can be analyzed by considering the rotational diffusion of these particles in shear. Rotational diffusion in shear flow has been reviewed by Van de Ven [58]. The ratio of the shear rate, y, to the rotational diffusion coefficient, , defines the rotational Peclet number (Pe = y/ ). When the rotational Peclet number is small (i.e., near zero), the anisotropic particles are randomly oriented by diffusion. When the rotational Peclet number is large, the particles rotate but have a preferential orientation aligned with the shear. The period of rotation is given by... 

Although the model equation included the axial dispersion coefficient (Dl), plug flow was approximated by assigning a very large value to the Peclet number (uL/Dl). This is because the effect of axial dispersion is quite negligible in a small column and the model with the second derivatives can give more stable numerical results. 

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