Velocity mean-mass

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). 

Mass flux of A by diffusion with respect to the mean mass velocity kmoP(m -s) or moP(cm -s) lbmol/(fF-h)... 

Panicles entrained in the airstream deposit along the airway as a function of size, density, airstream velocity, and breathing frequency. Sizes of rougjily spherical or irregularly shaped particles arc commonly characterized by relating the settling velociiy of the particle to that of an idealized spherical particle. For example, an irregular particle which settles at the same rate as a 5 pm spherical particle has a mean mass aerodynamic diameter (MMAD) of. 5 pm. Since spherical particle mass, is a function of particle diameter, J... 

Comparing this result with Eq. (14.48) and Fig. 14.5, we see that the mean size of the particle in the sense of linear momentum, and therefore also in the sense of free-falling velocity, is always greater than the mean mass size... 

Determine geometric mean mass velocity, G, using Figure 10-60. 

Gg = geometric mean mass velocity through shell side, lb/hr(fti). 

In order to use Eqs. (3) and (4) or the data given in Fig. 1, for the calculation of maximum turbulent fluctuation velocity the maximum energy dissipation e , must be known. With fully developed turbulence and defined reactor geometry, this is a fixed value and directly proportional to the mean mass-related power input = P/pV, so that the ratio ,/ can be described as an exclusive function of reactor geometry. In the following, therefore details will be provided on the calculation of power P and where available the geometric function ,/ . 

In the first place, we shall find that the four quantities Ty px0y py0y pz0 must be constant at all points of space, for equilibrium. By comparison with Eq. (2.4) of Chap. IV, the formula for the Maxwell distribution of velocities, we see that T must be identified with the temperature, which must not vary from point to point in thermal equilibrium. The quantities pxo, pyo, p 0 are the components of a vector representing the mean momentum of all the molecules. If they are zero, the distribution (2.15) agrees exactly with Eq. (2.4) of Chap. IV. If they are not zero, however, Eq. (2.15) represents the distribution of velocities in a gas with a certain velocity of mass motion, of components pxo/my pyQ/my pzo/m. The quantities px — pxo, etc., represent components of momentum relative to this momentum of mass motion, and the relative distribution of velocities is as... 

This equation appears to predict that Z i2 will be a function of the composition of the gas, although experimentally the diffusion constant is almost independent of composition. However, we must be careful in our definition of Aj and A2. We must take account of the fact that collisions of molecules of one species with one another can have no significant effect on the diffusion such collisions do not affect the total momentum possessed by all the molecules of that species and thus do not affect the mean mass velocity of the species in its diffusion. Thus the total number of molecules of that species crossing the reference plane in a given period of time is not affected by such collisions and is the same as if such... 

From an analysis of 6000 meteors by radio echo techniques, Verniani concluded that the mean velocity was 34 km/s, and that their mean mass was 10" g. Verniani also found that the velocity distribution of the meteors shifted with meteor mass distribution. The physics of radar scatter from meteor trails extracts the ionization coefficient, Eq. (5), from the observed echo intensities and associated electron densities. More recent analysis of the ionization trails of meteors has indicated a slightly different expression than Eq. (6) for the ionization coefficient associated with faint, low velocity (< 35 km s ), radio meteors, namely (3 = 9.4 x 10" (u - 10) u , where v is in km s". The magnitude and velocity dependence of (3 remains a major puzzle. It must be noted that the calculations assume no chemical reactions in the trail this assumption is needed to make the calculations tractable and because many of the reaction cross sections are not known. Collision processes leading to ionization are discussed in Secs. 3.2.2 and 3.4. 

Boundary layer theory, just like film theory, is also based on the concept that mass transfer takes place in a thin him next to the wall as shown in Fig. 1.48. It differs from the him theory in that the concentration and velocity can vary not only in the y-direction but also along the other coordinate axes. However, as the change in the concentration prohle in this thin him is larger in the y-direction than any of the other coordinates, it is sufficient to just consider diffusion in the direction of the y-axis. This simplihes the differential equations for the concentration signihcantly. The concentration prohle is obtained as a result of this simplihcation, and from this the mass transfer coefficient [3 can be calculated according to the dehnition in (1.179). In practice it is normally enough to use the mean mass transfer coefficient... 

Mean mass temperature. Heat flux. Taking into account the fact that the dimensionless distribution of fluid velocity in a tube is u(g) = 1 - g2, we have the mean mass temperature in an arbitrary cross-section... 

But an accurate calculation of the performance of the nozzle requires a knowledge of both the stagnation and the local values of pressure. For example, the equation for mass flow for an impulse stage (e.g. equation (14.63)) will depend on the ratio pi.i/por.z. where the absence of a pressure drop over the blade implies p j = p2j = po.i+il unfortunately we will not have po.i+i available, only the value p oj+i, which will have to stand for bothpo.i+i and Por./+i- In fact, the assumption of zero interstage velocities means that we cannot distinguish between the stagnation and local values of any of the thermodynamic variables, and must be content with the approximations ... 

Ge = (CcGb) 2, geometric mean mass velocity, hQ= Shell-side heat transfer coefficient,... 

Ge = (GcG b) 2,geometric mean mass velocity, hp= Shell-side heot tronsfer coefficient, Viscosity at average temperature. Viscosity at tube-wall temperoture. 

EXAMPLE A 52 API gasoline which hos a viscosity of 0.5 centi-poise at its average temperature flows through the segmentally baffled shell of an exchanger with Va O.D. tubes. The geometric mean mass velocity is 77.8 lb./(sec.) (sq. ft.) and the viscosity gradient is very close to 1. What is the shell-side heat transfer coefficient 1—Connect G<. = 77.8 and /u = 0.5 with Line 1. 2—Pivot Line 1 about its intercept on uncalibrated line until it intersects 52° API. 3—Read h.. = 184. 

The following hold for calculation of molecular velocity, mean free path, mean time between collisions, and collision frequency for molecules with a diameter of 3 10" ° m, of mass 28 Da, at pressure of 760 torr, and temperature of 60°C ... 

Note that except for the mean mass m(f), the actual size distribution of the product droplet has not been specified yet. For high-velocity sprays, where drops... 

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