Velocity variance

The first term on the LHS denotes the rate of accumulation of the velocity variance v - within the control volume. The second term on the LHS denotes the advection of the velocity variance by the mean velocity. The first term on the RHS denotes the production of velocity variance by the mean velocity shears. The momentum flux v vl is usually negative, thus it results in a positive contribution to variance when multiplied by a negative sign. The second term on the RHS denotes a turbulent transport term. It describes how variance is moved around by the turbulent eddies v. The third term on the RHS describes how variance is redistributed by pressure perturbations. This term is often associated with oscillations in the fluid (e.g., like buoyancy or gravity waves.) The fourth term on the RHS is called the pressure redistribution term. The factor in square brackets consists of the sum of three terms (i.e.,... 

In Eq. (4.116) there appear to be no explicit homogeneous fluid-phase-velocity-variance source terms. Nonetheless, the terms for mass and momentum transfer are all potential sources of fluid-phase velocity variance. Eor example, a fluid-drag term can be a source or a sink of fluid-phase velocity variance, depending on the magnitude of the mixed moments... 

Here, as an example, we have assumed that the velocity fluctuations are due to microscale turbulence in the continuous phase, and thus Z)pf will be proportional to the velocity variance of file fluid seen by the particle [Uf iflf. 

Interstitial velocity Variance operator Catalyst mass kg Observedresponse in the i experiment Calculated j response in the i experiment Matrix containing the independent variables Conversion of component A Number of collisions Position in the catalyst pellet m Axial reactor position m Greek symbols... 

If we compare the k equation (1.407) with the mean kinetic energy equation (1.459) we see that they both contain a term describing the interaction between the mean flow and turbulence. We are of course referring to the velocity variance production term, which is the second last term in (1.459). The sign of this term differ in the two equations. Thus, the energy that is mechanically produced as turbulence is lost from the mean flow, and vice versa. 

The values in these tables represent the mean values of the samples elastic parameters. However, we observed a large sound velocity variance along the line of scan direction. As this takes place, the sound velocity has the highest value near the specimen periphery and the lowest one at the central part (Table 16.2). It should be noted that the radial variation of sound velocity was not symmetric in each B scans, and varied at the angular variation of the scanning line. 

The average random force over the time step is taken from a Gaussian with a varianc 2mk T y(St). Xj is one of the 3N coordinates at time step i E and R are the relevan components of the frictional and random forces at that time n, is the velocity component. 

As already mentioned, there are two so called "dead volumes" that are important in both theoretical studies and practical chromatographic measurements, namely, the kinetic dead volume and the thermodynamic dead volume. The kinetic dead volume is used to calculate linear mobUe phase velocities and capacity ratios in studies of peak variance. The thermodynamic dead volume is relevant in the collection of retention data and, in particular, data for constructing vant Hoff curves. 

Equation (15) gives the variance per unit length of a GC column in terms of the outlet pressure (atmospheric) the outlet velocity and physical and physicochemical properties of the column, packing, and phases and is independent of the inlet pressure. However, equation (13) is the recommended form for HETP measurements as the inlet pressure of a column is usually known, and is the less complex form of the HETP expression. 

Equation (16) describes the HETP curve, or the curve that relates the variance per unit length or HETP of a column to the mobile phase linear velocity. A typical curve is shown in Figure 8. 

Figure 8. Graph of Variance per Unit Length against Linear Mobile Phase Velocity...

It is seen that by a simple curve fitting process, the individual contributions to the total variance per unit length can be easily extracted. It is also seen that there is minimum value for the HETP at a particular velocity. Thus, the maximum number of theoretical plates obtainable from a given column (the maximum efficiency) can only be obtained by operating at the optimum mobile phase velocity. 

Equation (12) indicates that the band variance is directly proportional to the square of the tube radius, very similar to that for a straight tube. At high linear velocities, Tijssen deduced that... 

Yoshida and Akita (Yl) determined volumetric mass-transfer coefficients for the absorption of oxygen by aqueous sodium sulfite solutions in counter-current-ffow bubble-columns. Columns of various diameters (from 7.7 to 60.0 cm) and liquid heights (from 90 to 350 cm) were used in order to examine the effects of equipment size. The volumetric absorption coefficient reportedly increases with increasing gas velocity over the entire range investigated (up to approximately 30 cm/sec nominal velocity), and with increasing column diameter, but is independent of liquid height. These observations are somewhat at variance with those of other workers. 

Levenspiel and Smith Chem. Eng. Sci., 6 (227), 1957] have reported the data below for a residence time experiment involving a length of 2.85 cm diameter pyrex tubing. A volume of KMn04 solution that would fill 2.54 cm of the tube was rapidly injected into a water stream with a linear velocity of 35.7 cm/sec. A photoelectric cell 2.74 m downstream from the injection point is used to monitor the local KMn04 concentration. Use slope, variance, and maximum concentration approaches to determine the dispersion parameter. What is the mean residence time of the fluid ... 

Fig. 8.3. Histogram of work values for Jarzynski s identity applied to the double-well potential, V(x) = x2(x — a)2 + x, with harmonic guide Vpun(x, t) = k(x — vt)2/2, pulled with velocity v. Using skewed momenta, we can alter the work distribution to include more low-work trajectories. Langevin dynamics on Vtot(x(t),t) = V(x(t)) + Upuii(x(t)yt) with JcbT = 1, k = 100, was run with step size At = 0.001, and friction constant 7 = 0.2 (in arbitrary units). We choose v = 4 and a = 4, so that the barrier height is many times feT and the pulling speed far from reversible. Trajectories were run for a duration t = 1000. Work histograms for 10,000 trajectories, for both equilibrium (Maxwell) initial momenta, with zero average and unit variance, and a skewed distribution with zero average and a variance of 16.0...

Figure 20. The variance of bed pressure drop versus superficial gas velocity.

In the massive range, the yields of Meynet Maeder predict some primary N production. In [2] we showed that models for the MW computed with this new set of yields lead to a plateau in log(N/0), due to massive stars with initial rotational velocities of 300 kmsec 1, at log(N/0) —4. This value is below the value of —2.2 dex observed in some DLAs and hence we suggested that in these systems both, massive and intermediate mass stars, would be responsible for the N enrichment. This is at variance with recent claims that massive stars were the only ones to enrich systems which show a log(N/0) —2.2. 

The compulsory fulfillment of conditions (4.2) and (4.3) physically follows from the fact that a one-dimensional Markov process is nondifferentiable that is, the derivative of Markov process has an infinite variance (instantaneous speed is an infinitely high). However, the particle with the probability equals unity drifts for the finite time to the finite distance. That is why the particle velocity changes its sign during the time, and the motion occurs in an opposite directions. If the particle is located at some finite distance from the boundary, it cannot reach the boundary in a trice—the condition (4.2). On the contrary, if the particle is located near a boundary, then it necessarily crosses the boundary— the condition (4.3). 

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