Turbulence component

A dependence of both crystal and impeller material properties as well as the probability of crystal-impeller collision on fine particle generation rate has also been demonstrated. Thus the relative effects of impact, drag and shear forces responsible for crystal attrition have been identified. The contribution of shear forces to the turbulent component is predicted to be most significant when the parent particle size is smaller than a 200 pm while drag forces mainly affect larger crystals, the latter being consistent with the observations of Synowiec etal. (1993). 

The left-hand sides of Eqs. (25)-(29) have the same form as Eq. (5) and represent accumulation and convection. The terms on the right-hand side can be divided into spatial transport due to diffusion and source terms. The diffusion terms have a molecular component (i.e., /i and D), and turbulent components. We should note here that the turbulence models used in Eqs. (26) and (27) do not contain corrections for low Reynolds numbers and, hence, the molecular-diffusion components will be negligible when the model is applied to high-Reynolds-number flows. The turbulent viscosity is defined using a closure such as... 

From equation 1.41, the total shear stress varies linearly from a maximum fw at the wall to zero at the centre of the pipe. As the wall is approached, the turbulent component of the shear stress tends to zero, that is the whole of the shear stress is due to the viscous component at the wall. The turbulent contribution increases rapidly with distance from the wall and is the dominant component at all locations except in the wall region. Both components of the mean shear stress necessarily decline to zero at the centre-line. (The mean velocity gradient is zero at the centre so the mean viscous shear stress must be zero, but in addition the velocity fluctuations are uncorrelated so the turbulent component must be zero.)... 

Figure 22.9 Horizontal growth and movement of a tracer patch under the influence of turbulent currents. While the mean currents move the patch as a whole (represented by the center of mass black dots), the turbulent components increase the size of the patch. Usually, the spreading is faster in the direction of the mean current. Therefore, the patch develops approximately into an ellipse with major and minor principle axes, Omil and ami. From Peeters et al. (1996).

Now return to a view of the nature of flow in the boundary layer. It has been called laminar, and so it is for values of the Reynolds number below a critical value. But for years, beginning about the time of Osborne Reynolds experiments and revelations in the field of fluid flow, it has been known that the laminar property disappears, and the flow suddenly becomes turbulent, when the critical VUv is reached. Usually flow starts over a surface as laminar but after passing over a suitable length the boundary layer becomes turbulent, with a thin laminar sublayer thought to exist because of damping of normal turbulent components at the surface. See Fig. 6. 

In this region it can also be assumed that the molecular shearing stress and heat transfer rate are negligible compared to the turbulent components, i.e., that v and f// 2> a and that in this region therefore, because the turbulent Prandtl number is being assumed to be equal to 1, Eqs. (7.52) and (7.62) become ... 

The mass transfer coefficient, K, is defined as the ratio of the mass transport controlled reaction rate to the concentration driving force. The concentration driving force will depend on both turbulent and bulk convection. Bulk convection depends on molecular diffusivity, while the turbulent component depends on eddy diffusivity (4). The mass transfer coefficient considers the combination of the two transport mechanisms, empirically. 

Subsequent experimental validation lead to additional partitioning of the inner constant stress layer according to the relative magnitude of the viscous and turbulent components of the total shear stress. The total shear stress takes the form ... 

The mathematical formulation of these principles gives a closed system of equations governing the model. Without going into detail we have to mention that, in accordance of our foregoing discussion, small scale processes also have to be included in the model. In the governing equations the following parameters can be found horizontal and vertical components of the wind (also the random turbulent components after parameterization), the atmospheric pressure, temperature and density as well as the mass fraction of different chemical substances. 

Turbulent heat transfer is similar to turbulent momentum transfer. The heat flux in turbulent flow is thus comprised of molecular and turbulent components. 

Similarly, the turbulent component of the mixture fraction, pvjZ", is modeled as being proportional to the gradient of Z, according to... 

Fig. 5.3. Example of phase-averaged surface elevation rj) and turbulence components 1.0 m from B.P. in Case 1. (a) Free surface 77 (solid) and standard deviation rjika-q, (b)-(d) temporal variation oi(7u (thick solid), av (thin solid), and a-w (dotted) and (e) the phase-averaged void fraction (a). ...

Laminar component of fanning friction factor for a yield pseudoplastic Eanning friction factor for a pseudoplastic in a laminar regime Tomita laminar friction factor Turbulent component of farming friction factor... 

Turbulent component of fanning friction factor Fluid force tangent to the direction of flow Acceleration due to gravity (9.81 m/s )... 

These skewness factors are widely used to specify the higher-order turbulence components, because the turbulence intensity in gas-liquid two-phase flows in the baths of metallurgical processes is much higher than those of single-phase pipe flows (Tu < 15%) and single-phase jets (Tu < 30%) [21,22], It is known that 5 = 0 and F = 3 for a Gaussian (normal distribution) error curve. 

Compared with the bubble characteristics, the information on the liquid flow characteristics specified by the axial and radial mean velocities, u and v, the root-mean-square values of the axial and radial turbulence components, m and the Reynolds shear stress u V, and higher correlations of turbulence components, such as the skewness and flatness factors, are limited except for a water-air system [8-12]. 

A magnet probe, specifically the Vives probe, [21,22] was used to measure the mean velocity and turbulence components of molten Wood s metal flow at around 100°C. Helium gas was chosen to simulate the density ratio between molten steel and Ar. 

Figure 2.14 shows the axial mean velocity on the centerline of the bath, Ua, and the root-mean-square values of the axial and radial turbulence components, M ms,ci and J.J, against the axial distance, z. The measured d value remains almost unchanged in the axial direction and approaches the following empirical correlations originally proposed for a water-air system [9] ... 

Fig. 2.14 Axial mean velocity and the rms values of turbulence components on the centerline as functions of axial distance...

Fig. 2.18 Radial distributions of the tms values of axial and radial turbulence components...

Figure 2.18 compares the radial distributions of the rms values of the axial and radial turbulence components, u and measured at z = 4 cm in the Wood s metal bath with those for single-phase free jets [29,30]. Due to additional turbulence production in the wake of bubbles, the measured and... 

The measured rms values of the axial and radial turbulence components shown in Fig. 2.18 are replotted in Figs. 2.19 and 2.20, respectively. The values for a water-air system are also shown in these figures [10,11]. In Fig. 2.19, agreement between the measured values for the Wood s metal-He system and the water-air system... 

As shown in the previous section, the shape of bubbles was found to affect preferably the modulation of radial (horizontal) turbulence component [24]. However, the mechanism of turbulence production was not mentioned. In this section, the application is described of the four-quadrant classification method, [39] one of the most popular conditional sampling methods, adopted to elucidate the coherent structure of turbulence or the mechanism of turbulence production in a He-Wood s metal bubbling jet. 

Figure 2.26 presents the axial turbulence kinetic energy for each turbulent motion, ku,c, calculated from the axial turbulence component u, thus. 

The experimental results suggest that empirical correlations of the bubble characteristics and the axial mean velocity and turbulence components of liquid flow, derived from cold model experiments, are applicable to actual reflning processes stirred by bottom gas injection when the radial distributions of gas holdup and bubble frequency follow Gaussian distributions. These distributions appear to be a result of the disintegration of rising bubbles due to highly turbulent liquid motion in the bath. 

In this section, data will be given on the mean velocity and turbulence components of water flow in and near an air-water bubbling jet subjected to the Coanda effect, measured with a two-channel laser Doppler velocimeter. These quantities are closely associated with mixing in metallurgical reactors and the erosion of the side wall of the reactors [22], Particular attention is paid to whether or not the horizontal distributions of the liquid flow characteristics near the side wall are similar in the vertical region above the attachment position. 

The experimental apparatus has been described in Sect. 3.2.2. The electroresistivity probe was removed and a two-channel laser Doppler velocimeter was set up to measure the three velocity components of water flow in the bath. The origin of the cylindrical coordinates (z, r, 9) was placed at the center of the bath, as shown in Fig. 3.3. The velocity components were designated by u, v, and w, respectively. The components, u and v, were measured in the z — r plane including the centerline of the bath and the center of the nozzle exit [21,23]. Digitized velocity data were decomposed into the mean velocity and turbulence components as follows ... 

The over-bar and prime denote the mean velocity and turbulence components, respectively, N is the number of data points, and the subscript i designates the i th digitized datum. 

Velocity measurements were carried out at four fixed vertical positions (z = 0.05, 0.10, 0.15, and 0.19m) for three gas flow rates = 41.4 x 10 , 100 x 10 , and 293 x 10 m /s just like the measurements of bubble characteristics [22]. The root mean square (rms) values of the axial and radial turbulence components, u rms and i/rms. and the Reynolds shear stress V were calculated from the following equations ... 

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