Effect of Peclet Number

Effect of Peclet number and the necessity for mesh refinement Viscosity depending upon concentration or temperature ... 

To demonstrate the effect of Peclet number, Krieger (1972), in a series of classic experiments, measured the relative viscosity of suspensions of monomod-al spheres with sizes from about 0.1 to 0.5 (im. By adjusting the solution... 

Figure 15.13 The effect of Peclet number on stearic acid consumption (catalyst (Pd/C (Aldrich))).

The forced fluid flow in heated micro-channels with a distinct evaporation front is considered. The effect of a number of dimensionless parameters such as the Peclet, Jacob numbers, and dimensionless heat flux, on the velocity, temperature and pressure within the liquid and vapor domains has been studied, and the parameters corresponding to the steady flow regime, as well as the domains of flow instability are delineated. An experiment was conducted and demonstrated that the flow in microchannels appear to have to distinct phase domains one for the liquid and the other for the vapor, with a short section of two-phase mixture between them. 

Two-phase flows in micro-channels with an evaporating meniscus, which separates the liquid and vapor regions, have been considered by Khrustalev and Faghri (1996) and Peles et al. (1998, 2000). In the latter a quasi-one-dimensional model was used to analyze the thermohydrodynamic characteristics of the flow in a heated capillary, with a distinct interface. This model takes into account the multi-stage character of the process, as well as the effect of capillary, friction and gravity forces on the flow development. The theoretical and experimental studies of the steady forced flow in a micro-channel with evaporating meniscus were carried out by Peles et al. (2001). These studies revealed the effect of a number of dimensionless parameters such as the Peclet and Jacob numbers, dimensionless heat transfer flux, etc., on the velocity, temperature and pressure distributions in the liquid and vapor regions. The structure of flow in heated micro-channels is determined by a number of factors the physical properties of fluid, its velocity, heat flux on... 

Use the program to assess the effects of differing degrees of axial dispersion, for values of Peclet number ranging from 0.005 to 0.25. Modify the program to account for zero axial dispersion. 

Before leaving this discussion, it is important to note that other forms of Peclet numbers are also possible and may be more appropriate depending on the type of convective influence studied. For example, in the case of oscillatory flows (as in oscillatory viscometers), it is more useful to define the Peclet number as (Rfa/D), where co is the frequency of oscillation. Regardless of the particular definition, the general significance of the Peclet number remains the same, i.e., it compares the effect of convection relative to diffusion. 

Fig. 6. Dispersion curves predicted by the hyperbolic model [Eq. (57)] for various values of the effective local Peclet number, jj.

The effect of Peclet on the Sherwood number in high porosity granular media... 

A noticeable deviation of sedimentation potentials from Smoluchowski s formula takes place at large siuface concentration variation along the bubble surface. Before considering experimental data, it has to be pointed out that the validity of Smoluchowski s formula for the description of the Dorn effect at large Peclet numbers applies only to solid spherical particles. In particular, the correctness of conclusions of some papers (Dukhin, 1964 Dukhin Buikov, 1965 Derjaguin Dukhin, 1967, 1971) is experimentally confirmed by Usui et al. (1980). Sedimentation potential for four sizes of glass balls appears to be the same. Since the radii of the particles under consideration are approximately 50, 150, 250, and 350 pm, the absence of any effect of Peclet and Reynolds numbers on the sedimentation potential could be demonstrated. 

By including the effect of axial conduction in the fluid, the fully developed average Nusselt number becomes a function of the Peclet number. An approximate relation by means of which Nux can be calculated as a function of Peclet number is the following ... 

The reactor was modeled as an axial dispersion unit The most common single criterion for the extent of axial dispersion is the Peclet (Pe) number. A study highlighting the importance of the effect of Pe number and the effect of the extent of axial dispersion on the productivity was included. Sensitivity of the model parameters to possible inaccuracy in the value of Pe number was also studied. The value of Pe number that was used in simulation studies was obtained from empirical correlations presented in the literature [24-26]. 

Dimensionless effective radial thermal conductivity X and effective wall heat transfer coefficient h as function of Peclet-number (a measured under reacting conditions, b calculated according to /42/for nonreacting conditions)... 

Film Heat Transfer Coefficient Value Most of the experimental data for liquid metals in forced convection have been obtained for round tubes. Since a large fraction of heat transfer to liquid metals in forced convection is by molecular and electronic conduction, the velocity and temperature distribution of the fluid in the channel is expected to have a noticeable effect. Until data are obtained for the reference channel, however, the data for round tubes is used with the equivalent diameter of the channel replacing the diameter of the tube. Most of the round tube data fall below the L.yon-Martinelli theoretical prediction, and therefore 85% of the Lyon-Martinelli Nusselt Number is used as the best average value in the range of Peclet Number of interest (500-1000). The factor shown in Table X represents the expected accuracy of experimental data. 

The variations of Peclet number with Reynolds number for gas and liquid phase systems are compared in Figure 6.12. At high Re, the asymptotic value of Pe = 2 is reached for liquids, but at lower values of Re, the axial dispersion is greater than that for gases. The increased dispersion with liquids is believed to be due to the effect of greater liquid hold-up in the laminar boundary layer surrounding particles, together with small random fluctuations in the flow (Ruthven 1984). 

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. 

Both phases are siibstantiaUy in plug flow. Dispersion measurements of the hquid phase usuaUy report Peclet numbers, Uid /D, less than 0.2. With the usual smaU particles, the waU effect is negligible in commercial vessels of a meter or so in diameter, but may be appreciable in lab units of 50 mm (1.97 in) diameter. Laboratory and commercial units usuaUy are operated at the same space velocity, LHSy but for practical reasons the lengths of lab units may be only 0.1 those of commercial units. 

Figure 3.2.1 illustrates the mixing in packed beds (Wilhelm 1962). As Reynolds number approaches the industrial range Rep > 100, the Peclet numbers approach a constant value. This means that dispersion is influenced by turbulence and the effect of molecular diffusion is negligible. 

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 dependence of the local Nusselt number on non-dimensional axial distance is shown in Fig. 4.3a. The dependence of the average Nusselt number on the Reynolds number is presented in Fig. 4.3b. The Nusselt number increased drastically with increasing Re at very low Reynolds numbers, 10 < Re < 100, but this increase became smaller for 100 < Re < 450. Such a behavior was attributed to the effect of axial heat conduction along the tube wall. Figure 4.3c shows the dependence of the relation N /N on the Peclet number Pe, where N- is the power conducted axially in the tube wall, and N is total electrical power supplied to the tube. Comparison between the results presented in Fig. 4.3b and those presented in Fig. 4.3c allows one to conclude that the effect of thermal conduction in the solid wall leads to a decrease in the Nusselt number. This effect decreases with an increase in the... 

The problem of axial conduction in the wall was considered by Petukhov (1967). The parameter used to characterize the effect of axial conduction is P = (l - dyd k2/k ). The numerical calculations performed for q = const, and neglecting the wall thermal resistance in radial direction, showed that axial thermal conduction in the wall does not affect the Nusselt number Nuco. Davis and Gill (1970) considered the problem of axial conduction in the wall with reference to laminar flow between parallel plates with finite conductivity. It was found that the Peclet number, the ratio of thickness of the plates to their length are important dimensionless groups that determine the process of heat transfer. 

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. 

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