Mixing inertial

If and when projects producing energy on an industrial scale begin to appear in the middle of the twenty-first century, the theoretical advantages will be subject to confirmation. Fusion, magnetic or inertial, might then join the mix of energy resources. 

Dynamic similarity occurs in two geometrically similar units of different sizes if all corresponding forces at counterpart locations have a constant ratio. It is necessary here lo distinguish between the various types of force inertial, gravitational, viscous, surface tension and other forms, such as normal stresses in the case of viscoelastic non-Newtonian liquids. Some or all of these forms may be significant in a mixing vessel. Considering... 

The area for flow is however, A = (constant) L1, where L is the characteristic linear dimension of the system. In mixing applications, L is usually chosen as the impeller diameter D, and, likewise, the representative velocity u is taken to be the velocity at tire tip of impeller (ttDN), where N is revolutions per unit time. Therefore, the expression for inertial force may be written as ... 

For an incompressible viscous fluid (such as the atmosphere) there are two types of flow behaviour 1) Laminar, in which the flow is uniform and regular, and 2) Turbulent, which is characterized by dynamic mixing with random subflows referred to as turbulent eddies. Which of these two flow types occurs depends on the ratio of the strengths of two types of forces governing the motion lossless inertial forces and dissipative viscous forces. The ratio is characterized by the dimensionless Reynolds number Re. 

In a fully developed turbulent flow, the rate at which the size of a scalar eddy of length l,P decreases depends on its size relative to the turbulence integral scale L and the Kolmogorov scale ij. For scalar eddies in the inertial sub-range (ij < Ip, < Lu), the scalar mixing rate can be approximated by the inverse of the spectral transfer time scale defined in (2.68), p. 42 8... 

The mixing parameter Q must be chosen to yield the correct mixture-fraction-variance dissipation rate. However, inertial-range scaling arguments suggest that its value should be near unity.165... 

The Reynolds number is the ratio of inertial to viscous forces and depends on the fluid properties, bulk velocity, and boundary layer thickness. Turbulence characteristics vary with Reynolds number in boundary layers [40], Thus, variation in the contributing factors for the Reynolds number ultimately influences the turbulent mixing and plume structure. Further, the fluid environment, air or water, affects both the Reynolds number and the molecular diffusivity of the chemical compounds. 

The present analysis follows the approach taken by aU three of these authors, in which SDEs are constructed by choosing the drift and diffusivity coefficients so as to yield a desired diffusion equation. Peters [13] has pioneered an alternative approach, in which expressions for the drift and diffusivity are derived from a direct, but rather subtle, analysis of the underlying inertial equations of motion, in which (for rigid systems) he integrates the instantaneous equations of motion over time intervals much greater than the autocorrelation time of the particle velocities. Peters has expressed his results both as standard Ito SDEs and in a nonstandard interpretation that he describes heuristically as a mixture of Stratonovich and Ito interpretations. Peters mixed Ito—Stratonovich interpretation is equivalent to the kinetic interpretation discussed here. Here, we recover several of Peters results, but do not imitate his method. 

The Froude number, = vP Lg, is similar to it is a measure of the inertial stress to the gravitational force per unit area acting on a fluid. Its inclusion in Eq. (11) is justified when density differences are encountered in the absence of substantive differences in density, e.g., for emulsions more so than for suspensions, the Froude term can be neglected. Dimensionless mixing time is independent of the Reynolds number for both laminar and turbulent flow regimes, as in-... 

P 54] The mixing experiments were performed at relatively low Re. For this purpose, 85% glycerol-water solutions were used, with dynamic viscosity and density of about 100 mPas and 1.2 kg l-1, respectively [7]. Total flow rates in the range 0.2 1 h-1 Re = 0.22) and 2 1 h 1 Re = 2.2) were applied. For the given dimensions, CFD simulation showed that for Re above about 15, corresponding to a total flow rate of 13.5 1 h 1, secondary flow induced by inertial forces had a notable effect. Thus the applied flow rates were well below the critical value and pronounced secondary flow effects were not observed. 

Mixing by inertial forces has been demonstrated using a 80 zigzag micro-channel (300 pm x 600 pm x 1 mm). The mixing time achieved in this case is 1 second at the Reynolds number of 33 [17]. 

An additional advantage of using microfluidic devices, which we do not have the space to discuss in detail here, is the absence of turbulence (Koo and Kleinstreuer, 2003). In the context of nanoparticle synthesis, turbulence gives rise to unpredictable variations in physical conditions inside the reactor that can influence the nature of the chemical product and in particular affect the size, shape, and chemical composition. In microfluidic devices, turbulence is suppressed (due to the dominance of viscous over inertial forces) and fluid streams mix by diffusion only. This leads to a more reproducible reaction environment that may in principle allow for improved size and shape control. 

The dye trace thread is broken down and becomes mixed with the fluid, eventually colouring the fluid over the whole width of the pipe Inertial forces dominate Frictional losses cc fiu2 Turbulent velocity profile... 

In microscale channels, the viscous forces dominate the inertial effect resulting in a low Reynolds numbers. Hence, laminar flow behavior is dominant and mixing occurs via diffusion. However, in a liquid-liquid system, the interfacial forces acting on the interface add complexity to the laminar flow as the relationship between interfacial forces and other forces of inertia and viscous results in a variety of interface and flow patterns. Gunther and Jensen [202] illustrated this relationship as a function of the channel dimension and velocity as shown in Figure 4.12. The most regularly shaped flow pattern is achieved when interfacial forces dominate over inertia and viscous forces at low Reynolds numbers, as represented in Figure 4.12 by the area below the yellow plane [202,203]. 

Using the Grashof and the Reynolds numbers, we can indicate whether forced or free convection dominates in a particular case. Because Re equals inertial forces/viscous forces (Eq. 7.19) and Gr equals buoyant x inertial forces/(viscous forces)2 (Eq. 7.20), Re2/Gr equals inertial forces/buoyant forces. Thus Re2/Gr reflects forced convection/free convection. Experiments reveal that forced convection accounts for nearly all heat transfer when Re2/Gr is greater than 10, free convection accounts for nearly all heat transfer when Re2/Gr is less than 0.1, and the intervening region has mixed convection (i.e., both forced and free convection should then be considered, especially for Re2/Gr near 1). Using Equations 7.19 and 7.20, we obtain... 

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