Peclet number rotational

The Peclet number gauges the magnitude of the departure from equilibrium configuration of the particles. (Note that the rotational Peclet number, Pe, for a sphere has nearly the same definition only 6 is replaced by 8.) As such the Peclet number can be used in the Cross equation to determine the value of > .[= 8], giving... 

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... 

The crossover from Brownian to non-Brownian behavior in a flowing suspension is controlled by a rotational Peclet number. 

Fig. 26 Average inclination angle (0) as a function of reduced membrane viscosity for the shear rate f = 0.92 and various reduced volumes V. Results are presented for prolate (circles) and discocyte (squares) vesicles with V = 0.59, as well as prolate vesicles with V = 0.66 (triangles), 0.78 (diamonds), 0.91 (crosses), and V = 0.96 (pluses). The solid and dashed lines are calculated by K S theory, (102) and (103), for prolate (V = 0.59, 0.66, 0.78, 0.91, and 0.96) and oblate eUipsoids (P = 0.59), respectively. The dashed-dotted lines are calculated from (106) with thermal fluctuations, for V = 0.66, V = 0.78, and the rotational Peclet number f/Dg = 600 (where Dg is the rotational diffusion constant). From [180]...

As with spherical particles the Peclet number is of great importance in describing the transitions in rheological behaviour. In order for the applied flow field to overcome the diffusive motion and shear thinning to be observed a Peclet number exceeding unity is required. However, we can define both rotational and translational Peclet numbers, depending upon which of the diffusive modes we consider most important to the flow we initiate. The most rapid diffusion is the rotational component and it is this that must be overcome in order to initiate flow. We can define this in terms of a diffusive timescale relative to the applied shear rate. The characteristic Maxwell time for rotary diffusion is... 

The convective diffusion theory was developed by V.G. Levich to solve specific problems in electrochemistry encountered with the rotating disc electrode. Later, he applied the classical concept of the boundary layer to a variety of practical tasks and challenges, such as particle-liquid hydrodynamics and liquid-gas interfacial problems. The conceptual transfer of the hydrodynamic boundary layer is applicable to the hydrodynamics of dissolving particles if the Peclet number (Pe) is greater than unity (Pe > 1) (9). The dimensionless Peclet number describes the relationship between convection and diffusion-driven mass transfer ... 

However, for nonspherical particles, rotational Brownian motion effects already arise at 0(0). In the case of ellipsoidal particles, such calculations have a long history, dating back to early polymer-solution rheologists such as Simha and Kirkwood. Some of the history of early incorrect attempts to include such rotary Brownian effects is documented by Haber and Brenner (1984) in a paper addressed to calculating the 0(0) coefficient and normal stress coefficients for general triaxiai ellipsoidal particles in the case where the rotary Brownian motion is dominant over the shear (small rotary Peclet numbers)—a problem first resolved by Rallison (1978). 

A. Acrivos, Heat transfer at high Peclet number from a small sphere freely rotating in a simple shear flow, J. Fluid Mech. 46, 233-40 (1971). 

An experimental verification [405] of the fact that the leading term of the asymptotic expansion of the mean Sherwood number for Pe 1 is independent of the Peclet number for a freely rotating circular cylinder in a simple shear flow ( ff = 1) showed good qualitative and quantitative agreement with the theoretical results [132]. The measured mean Sherwood number was 2.65, which is close to the corresponding asymptotic value (4.11.4). 

Poe, G. G. and Acrivos, A., Closed streamline flows past small rotating particles heat transfer at high Peclet numbers, Int. J. Mult. Flow, Vol. 2, No. 4, pp. 365-377, 1976. 

Suspensions, even in Newtonian liquids, may show elasticity. Hinch and Leal [1972] derived relations expressing the particle stresses in dilute suspensions with small Peclet number, Pe = y/D 1 (D is the rotary diffusion coef-hcient) and small aspect ratio. The origin of elastic effect lies in the anisometry of particles or their aggregates. Rotation of asymmetric entities provides a mechanism for energy storage. Brownian motion for its recovery. Eor suspensions of spheres, this mechanism does not exist. 

This is exactly the Peclet number defined by Eq. (5.3.25), which measures the characteristic rotational Brownian diffusion time to the time scale defined by the reciprocal of the shear rate. It is the same measure found for dilute polymer solutions with the particle radius here replacing the Flory radius for the polymer. 

Todd [37] proposed an equation to describe devolatilization in co-rotating twin screw extruders based on the penetration theory discussed in Section 5.4 and Section 7.6. The equation contains the Peclet number (see Eq. 7.371), which represents the effect of longitudinal backmixing. The Peclet number must be measured or estimated to predict the devolatilizing performance of an extruder. Todd selected a Peclet number of 40 to correlate predictions to experimental results. A similar approach was followed by Werner [38], A visualization study was made by Han and Han [39], particularly to study foam devolatilization, They found substantial entrainment of the bubbles in a circulatory flow region in a partially filled screw devolatilizer. Collins, Denson, and Astarita [40] published an experimental and theoretical study of devolatilization in a co-rotating twin screw extruder. The experimentally determined mass transfer coefficients were about one-third those predicted by the mathematical model. They concluded, therefore, that the effective surface area for mass transfer is substantially less than the sum of the areas of the screws and barrel. 

Other relationships based on conventional penetration theory for packed bed have deduced that Nu = 2 i -JPe and for low Peclet numbers to the order of 10, Tscheng and Watkinson (1979) deduced an empirical correlation, where Nu = 11.6 x7e°. Any of these would suffice in estimating the wall-to-bed heat transfer coefficient as functions of the kiln s rotational speed, w, and the dynamic angle of repose, The calculated values of Nusselt numbers using Perron and Singh s... 

Before discussing theoretical models for the rheology of fiber suspensions and its connection to fiber orientation, there are three topics that must be discussed Brownian motion, concentration regimes, and fiber flexibility. Brownian motion refers to the random movement of any sufficiently small particle as a result of the momentum transfer from suspending medium molecules. The relative effect that Brownian motion may have on orientation of anisotropic particles in a dynamic system can be estimated using the rotary Peclet number, Pe s y Dm, where y is the shear rate and Ao is the rotary diffusivity, which defines the ratio of the thermal energy in the system to the resistance to rotation. Doi and Edwards (1988) estimated the rotary diffusivity, Ao, to be... 

---