Dimensionless groups Peclet Number

N, IVc, IVnu IVp. K, IVh. IVs. N Proportionality coefficient, dimensionless group Grashof number, L p P Af/)U Nusselt number, hD/k or hL/k Peclet number, DGc/k Prandtl number, c A/k Reynolds number, DG/ l Stanton number, Number of sealing strips Dimensionless Dimensionless... 

Dimensionless group Dimensionless group Dimensionless concentration Peclet number for mass transfer Peclet number for heat transfer Dimensionless temperature Dimensionless length Activation energy group Dimensionless time... 

The inverse of the Bodenstein number is eD i/u dp, sometimes referred to as the intensity of dispersion. Himmelblau and Bischoff [5], Levenspiel [3], and Wen and Fan [6] have derived correlations of the Peclet number versus Reynolds number. Wen and Fan [6] have summarized the correlations for straight pipes, fixed and fluidized beds, and bubble towers. The correlations involve the following dimensionless groups ... 

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. 

For gas-liquid flows in Regime I, the Lockhart and Martinelli analysis described in Section I,B can be used to calculate the pressure drop, phase holdups, hydraulic diameters, and phase Reynolds numbers. Once these quantities are known, the liquid phase may be treated as a single-phase fluid flowing in an open channel, and the liquid-phase wall heat-transfer coefficient and Peclet number may be calculated in the same manner as in Section lI,B,l,a. The gas-phase Reynolds number is always larger than the liquid-phase Reynolds number, and it is probable that the gas phase is well mixed at any axial position therefore, Pei is assumed to be infinite. The dimensionless group M is easily evaluated from the operating conditions and physical properties. 

As in Section II,B,l,b, the gas-phase Peclet number is assumed to be infinite, and the dimensionless group M is easily evaluated. The interfacial area a can be calculated with a knowledge of the holdup of the film phase... 

If the right side of this equation is plotted versus dimensionless time for various values of the group Q)JuL (the reciprocal Peclet number), the types of curves shown in Figure 11.8 are obtained. The skewness of the curve increases with 3) JuL and, for small values of this parameter, the shape approaches that of a normal error curve. In physical terms this implies that when 3JuL is small, the shape of the axial concentration profile does not change... 

There is controvosy over the naming of Ihe dimensionless group uUDL (or its reciprocal), and, in particular, over naming it a Peclet number (see discussion by Weller, 1994)... 

Writing the model in dimensionless form, the degree of axial dispersion of the liquid phase will be found to depend on a dimensionless group vL/D or Peclet number. This is completely analogous to the case of the tubular reactor with axial dispersion (Section 4.3.6). 

Although a mechanism for stress relaxation was described in Section 1.3.2, the Deborah number is purely based on experimental measurements, i.e. an observation of a bulk material behaviour. The Peclet number, however, is determined by the diffusivity of the microstructural elements, and is the dimensionless group given by the timescale for diffusive motion relative to that for convective or flow. The diffusion coefficient, D, is given by the Stokes-Einstein equation ... 

An interesting problem arises when we consider solutions or colloidal sols where the diffusing component is much larger in size than the solute molecules. In dilute systems Equation (1.14) would give an adequate value of the Peclet number but not so when the system becomes concentrated, i.e. the system itself becomes a condensed phase. The interactions between the diffusing component slow the motion and, as we shall see in detail in Chapter 3, increase the viscosity. The appropriate dimensionless group should use the system viscosity and not that of the medium and now becomes... 

For large values of the Peclet number, Eq. (38) reduces to Eq. (11) and the exit concentration is strongly influenced by the dimensionless group BN kp/L (Fig. 7). On the other hand, at low values of the Peclet number the exit concentration is hardly influenced by this dimensionless group, which means that the exit concentration is relatively insensitive to changes in the flow rate and screw speed. We shall return to this point momentarily. 

The only two parameters appearing in eqn. (65) are the dispersion number, DjiiL, or inverse Peclet number, and the Damkohler number, or dimensionless rate group, t/jCa," - Solutions to eqn. (65) are therefore functions only of these two groups. If term (4) in eqn. (65) is absent, then... 

This characterizes the time taken for the restoration of the equilibrium microstructure after a disturbance caused, for example, by convective motion, i.e., this is the relaxation time of the microstructure. The time scale of shear flow is given by the reciprocal of the shear rate, 7. The dimensionless group formed by the ratio (tD ff/tShear) is the Peclet number... 

In addition to the Peclet number, one can also define other dimensionless groups that compare either relevant time scales or energies of interaction. Using some of the concepts previewed in Section 4.7c and Table 4.4, one can define an electrostatic group (in terms of the zeta potential f and relative permittivity cr of the liquid) as... 

Other dimensionless groups that compare the thickness of the adsorbed polymer layer to the radius of the particle or the radius of gyration of the polymer to the particle radius in polymer/colloid mixtures can also be easily defined. We are mostly concerned with the volume fraction and the Peclet number Pe in our discussions in this chapter. However, the other dimensionless groups may appear in the equations for intrinsic viscosity of dispersions when the dominant effects are electroviscous or sterically induced. 

Dispersion coefficients for packed beds are usually plotted in the form of a similar dimensionless group, the Peclet number udp/DL, which uses the diameter of the particles of the bed dp as the characteristic length rather than the bed diameter. Graphs of experimentally measured values of both axial and radial Peclet numbers plotted against Reynolds number, also based on particle diameter, are shown in Volume 2, Chapter 4. 

Examples of dimensionless groups that specify ratios of transport mechanisms are listed next in Table II and depend on the size and shape of the domain. The Peclet numbers for heat (Pet) and solute (Pes) and momentum (Re) transport are ratios of scales for convective to diffusive transport and depend on the magnitudes of the velocity field and the length scale for the diffusion gradient. Boundary layers form at large Peclet numbers (Pet or Pes) or Reynolds numbers (Re). The fonnation of a boundary layer at a large Re is particularly important in crystal growth from the melt, because the low... 

The assessment of the role of kf during protein adsorption in a fluidized bed may be performed with the help of a dimensionless transport number. Slater used the correlations provided by Rodrigues to simulate film transport limited adsorption of small ions to fluidized resins [54], In this study dimensionless groups were used to describe the influence of the system parameters particle side transport, liquid dispersion, and fluid side transport. Dispersion was accounted for by the column Peclet number analogous to Bo as introduced above and mass transport from the bulk solution to the resin was summarized in a fluid side transport number NL. 

The dimensionless group Del/uL is known as the dispersion number and is the parameter that measures the extent of axial dispersion. The degree to which axial dispersion influences the performance of a chemical reactor is determined by the value of the Peclet number (NPe). A high value of NPe corresponds to a slightly dispersed reactor. That is,... 

The capture efficiency or Sherwood number was shown to be a function of three dimensionless groups—the Peclet number, the aspect ratio (collector radius divided by par-dele radius), and the ratio of Hamaker s constant (indicating the intensity of London forces) to the thermal energy of the particles. Calculated values for the rate of deposition, expressed as Ihe Sherwood number, are plotted in Figure 6 as a function of the three dimensionless groups. 

This dimensionless group, introduced for conciseness in rate correlations, has no simple physical interpretation. It is the product of several ratios A/kT represents the ratio of the characteristic London interaction energy to the thermal energy of the particle, R is the aspect ratio, while the Peclet number may be considered as the ratio of a characteristic energy for drag losses to the thermal energy possessed by the particle. This interpretation for the Peclet number becomes evident by using the relation D = tnkT to write... 

It is always prudent to check the variables used in each dimensionless group prior to their application. This is especially true with Peclet numbers, since they can have many different characteristic lengths. 

Other dimensionless groups similar to the Deborah number are sometimes used for special cases. For example, in a steady shearing flow of a polymeric fluid at a shear rate y, the Weissenberg number is defined as Wi = yr. This group takes its name from the discoverer of some unusual effects produced by normal stress differences that exist in polymeric fluids when Wi 1, as discussed in Section 1.4.3. Use of the term Weissenberg number is usually restricted to steady flows, especially shear flows. For suspensions, the Peclet number is defined as the shear rate times a characteristic diffusion time to [see Eq. (6-12) and Section 6.2.2]. 

Predicted Deformation Mechanisms. Recent work has developed maps of the deformation mechanisms expected in films with different properties. Two dimensionless groups were found to determine which of the deformation mechanism occurs. The first is the time for particle deformation compared to the time for evaporation, captured in 1 = ERt]o/yH, where E is the evaporation rate, t]o is the polymer viscosity, and y is the water-air surface tension. The second dimensionless group is the Peclet number, which determines the vertical homogeneity in the film, Pe = 6nt] R H E/kT. The deformation regimes are shown in Fig. 9. 

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