Velocity weighted mean

The physics and modeling of turbulent flows are affected by combustion through the production of density variations, buoyancy effects, dilation due to heat release, molecular transport, and instabiUty (1,2,3,5,8). Consequently, the conservation equations need to be modified to take these effects into account. This modification is achieved by the use of statistical quantities in the conservation equations. For example, because of the variations and fluctuations in the density that occur in turbulent combustion flows, density weighted mean values, or Favre mean values, are used for velocity components, mass fractions, enthalpy, and temperature. The turbulent diffusion flame can also be treated in terms of a probabiUty distribution function (pdf), the shape of which is assumed to be known a priori (1). 

Instead of the mean velocity weighted with the molar fractions, a velocity weiglited with the mass fractions—the mass center velocity —can be... 

The molecular weight (mean relative molecular mass) was obtained by determination of density but, in order to determine that the gas was monatomic and its atomic and molecular weights identical, it was necessary to measure the velocity of sound in the gas and to derive from this the ratio of its specific heats kinetic theory predicts that Cp/C = 1.67 for a monatomic and 1.40 for a diatomic gas. 

From MaxwelFs distribution of velocities, either in the form (2.4) or (2.6), we can easily find the moan kinetic energy of a molecule at temperature T. To find this, we multiply the kinetic energy p2/2m by the fraction of molecules in a given range dpx dpy dpgy and integrate over all values of momenta, to get the weighted mean. Thus we have... 

JVo/ M = molecular weight, average velocity, LsTP == mean free path, Z == frequency of collisions. 

Serious deviations from Eq. (3) were reported by Smith and Reddy (S2), who worked with solid materials having a wide spread of particle size (up to sixfold) and used the weight-mean diameter [dp = Xi(dp) i]. When recalculated in terms of the reciprocal mean diameter (S3), these data show much better agreement with the equation (see Table IB). Hence, Eq. (3) appears to predict the spouting velocity for materials having a wide range of particle size if the reciprocal mean diameter is used. 

In fluidized beds, bubbles form at the distribution plate or grid ports where fluidizing gas enters the bed (Figure 169).They form because the gas velocity at the interface to the bed represents an input rate that is larger than what can pass through the interstices with less frictional resistance than the bed weight. Therefore, holes are formed through whose porous surface the gas can enter the bed at the incipient fluidization velocity. This means that bubble formation is a way to increase the active interface between the gas and the particle bed. 

In order to separate the average of products into products of average, weighted averaged values are commonly introduced. The phasic - and mass averages have been defined by (3.277) and (3.278), respectively. Hence, the instantaneous velocity is decomposed into a weighted mean component and a fluctuation component in accordance with (3.279). 

In one view, the unknown velocity corrections at any node are approximated by a weighted mean of the neighbor values. 

Here, represents a weighted mean velocity of the two phases, and V, is a damping parameter due to the shear stresses. (Based on the shear stresses modelling as detailed by Brauner and Moalem Maron [43] it can be shown that V, always attains a positive value, independent of the relative velocity between the two phases). 

Equation 36 yields the dynamic wave velocity relative to the weighted mean velocity, c. ... 

In extracting the conditions for the stable or unstable modes from Equations 40, it should be emphasized that o> and (O, are expressed in terms of waves velocities relative to the weighted mean velocity and, therefore, c, c, and c may attain negative values. Thus, the condition for an unstable mode, for instance, c ... 

Co is a distribution parameter correcting the basic one-dimensional model to allow for concentration and velocity profiles across the channel Mgj is a void weighted mean drift velocity... 

The use of thermodynamics is well accepted and indeed essential in those areas of particle technology which involve heat and/or flow of gases. Properties of a dust-laden gas, for example, deviate from those of a perfect gas as a consequence of the finite particle volume. Internal energy, enthalpy and specific heats of a particulate system, needed in applications involving heat, are the suitably weighted means of the respective properties of the constituents (except that enthalpy of the particulate phase suspended in a gas also depends on the pressure). Another example is the effect of the presence of particles on both the equilibrium and frozen velocities of sound in a gas. 

Although the data in Fig. 3.10 indicate that the space velocity should be as low as possible in order to maximize NOx conversion, in practical application this may not be feasible. A very low maximum space velocity will mean a large SCR catalyst volume, which will add weight to the tmck and the SCR catalyst will occupy a larger space on the tmck, meaning that the available space on the tmck... 

The denominator that appears in eqns (11.8) and (11.9) looks strange it represents a weighted density obtained by summing the particle density multiplied by the fluid fraction and the fluid density multiplied by the particle fraction. The physical significance of this topsy-turvy combination is unclear. The same can be said for the velocity V itself, eqn (11.9), which corresponds to the weighted mean velocity defined by Wallis (1969), in which fluid velocity is weighted with particle fraction and vice versa. (Only the fluid velocity component appears in eqn (11.9) because for the case under consideration we have that m o = 0.) Mathematically, however, velocity V has an important significance, which will soon become clear. 

The significance of the weighted mean velocity V is now apparent from the forms of eqns (11.13) and (11.14) it is relative to V that system wave velocities are most naturally expressed. The only exception concerns the (V -G) term in the denominator of eqn (11.14). However, this too may be brought into line by defining a wave velocity dt such that ... 

We consider here three types of equations that convert the CMD to (1) another median diameter, (2) a weighted mean diameter, and (3) a diameter of average property. Although these equations are written for CMDs, they can be combined to convert any type of average diameter to any other type of average diameter. The examples diat follow are for the most common case, conversions between number and mass distribution. Analogous equations can be written for any moment distribution, such as the distribution of surface area or of settling velocity. 

---