Momentum diffusivity

A fruitful approach for velocity computation in the first three zones of jets supplied from outlets with finite size was developed based on the hypothesis that momentum diffuses with distance from the source in the same manner as heat energy." 40 approach, developed by Elrod,Shepelev and Gelman, - and Regenscheit, utilizes the method of superposition of jet momentum from the multiple-jet system. These jets originate from the points with supply air veloc-... 

Along with a constant velocity zone (Zone 1), there is a constant temperature zone in a jet. Heat diffusion in a jet is more intense than momentum diffusion therefore the core of constant temperatures fades away faster than that of constant velocities and the temperature profile is flatter than the velocity profile. Thus the length of the zone with constant temperature (Fig. 7.23) is shorter than the length of the constant velocity zone (Zone I... 

These groups have a definite, important, physical meaning. The Reynolds number is the ratio of inertial forces to viscous forces, the Sherwood number the ratio of mass transfer resistance in fluid film to mass transfer in bulk fluid, and Schmidt number the ratio of momentum diffusivity to mass diffusivity. 

Consider a long cylindrical shell whose interior is filled with an incompressible fluid. If the fluid is initially at rest when the cylinder begins to rotate, a boundary layer develops as the momentum diffuses inward toward the center of the cylinder. The fluid s circumferential velocity vu comes to the cylinder-wall velocity immediately, owing to the no-slip condition. At very early time, however, the interior fluid will be only weakly affected by the rotation, with the influence increasing as the boundary layer diffuses inward. If the shell continues to rotate at a constant angular velocity, the fluid inside will eventually come to rotate as a solid body. 

The Rayleigh criterion can also be written as a function of the Grashof criterion, which compares convective with conductive heat transfer and the Prandtl criterion, which compares the momentum diffusivity (kinematic viscosity) with the thermal diffusivity ... 

Pr Prandtl Cp/X momentum diffusivity and pumps Heat transfer and... 

The mass diffusivity Dt], the thermal diffusivity a = k/pCp, and the momentum diffusivity or kinematic viscosity v = fi/p, all have dimensions of (length)2/time, and are called the transport coefficients. The ratios of these quantities yield the dimensionless groups of the Prandtl number, Pr, the Schmidt number, Sc, and the Lewis number, Le... 

Remember that the constant of proportionality in Fourier s law was defined as the transport propert> thermal conductivity. Similarly, lire constant of proportionality in Pick s law is defined as another transport property called the binary diffusion coefficient or mass diffusivity, D g. The unit of mass diffu-sivity is m /s, which is the same as the units of thermal diffusivity ov momentum diffusivity (also called kinematic viscosity) (Fig, 14-11). 

Viscosity 11, r/R p. Density p Slip radius V, Momentum diffusivity 9, Dimensionless temperature Dimensionless axial coordinate... 

The Prandtl number is the ratio of the kinematic viscosity (i.e., the momentum diffusivity) to the thermal diflusivity. Because the Schmidt number is analogous to the Prandtl number, one would expect that Sc is the ratio of the momentum diffusivity (i.e., the kinematic viscosity), v, tothe mass diffusivity Dab- Indeed, this is true ... 

For integrating Eq. (4-9), vji= ei Er) should be known as a function of and operating variables. However, the momentum diffusivity is the only term we know, with essentially no systematic data for In the case of free turbulence of a homogeneous fluid, the diffusivity of a scalar quantity like heat and mass is estimated to be about two times that of momentum (S4) and the two diffusivities are not far apart for turbulent pipe flow (S8). However, such a relation is not available yet for gas-liquid bubble flow in bubble columns. Generally the local radial mass diffusivity may be expressed by a, with a being a numerical coefficient of order unity. 

If these Peclet numbers are divided by the Reynolds number, the resulting dimensionless numbers are called the Pradtl number, Pr and Schmidt number. Sc, respectively. The Prandtl number (Pr) is the ratio of momentum diffusivity and thermal diffusivity. The Schmidt number (Sc) is the ratio of momentum diffusivity and mass diffusivity. These five dimensionless numbers can convey very useful information about the relative contributions of convective and molecular transport and relative magnitudes of momentum, heat and mass transfer. 

Note that for clarity, gravity and additional force terms are excluded from the above equation. The last term, pu u", is a result of the averaging procedure. It represents the turbulent stresses, an additional source of momentum diffusion due to turbulence. A constitutive relationship, or modeling, must be introduced to relate the turbulent stresses to mean flow quantities, an area known as turbulence modeling. 

The Prandtl number is a dimensionless number named after Ludwig Prandtl. It is defined as the ratio of momentum diffusivity (kinematic viscosity) to the thermal diffusivity, as well as the ratio of viscous diffusion rate to thermal diffusion rate ... 

The Prandtl values for some fluids are listed in Table 2.8. For mercury, heat conduction is more effective compared to convection when thermal diffusivity is dominant. For engine oil, convection is very effective in transferring energy from an area, compared to pure conduction, where the momentum diffusivity is dominant. In heat transfer problems, the Prandtl number controls the relative... 

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