Noncontinuum effects

The linear relationship between Nu and /r at a fixed Re proposed in Fig. 10.13 has also been found experimentally for stagnation point transfer from eylinders (K4) and spheres (G6). In addition, it has received some theoretical confirmation from predictions of turbulence models for stagnation point transfer (Gl, S29, T13, W2). 

Few reliable data are available on the effect of turbulence on transfer at high Pr or Sc. The data of Henry and Epstein (H8) for transfer from spheres in turbulent gas streams with Sc 5 and the data of Mizushina et al. (MIO) for cylinders with Sc = 10 suggest that the representation in Fig. 10.13 should be independent of Pr or Sc. 

Elsewhere in this book attention is focused on particles whose Mach and Knudsen numbers are small. The Mach number is defined as the ratio of the relative velocity between the particle and the fluid to the speed of sound in the fluid  

For all practical purposes, isothermal flows with Ma 0.2 can be treated as incompressible, i.e., density variations in the fluid around the particle are negligible. Compressibility effects become important as Ma is increased, especially for Ma approaching and in excess of unity. The Knudsen number is defined as the ratio of the molecular mean free path in the fluid to some characteristic particle dimension. For a spherical particle 

For Kn less the scale of treated as a liquids, c is tually never centimeters (BIO) from 

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Table 5.2 gives a new correlation, based on a critical examination of available data for spheres (N6). Results in which wall effects, compressibility effects, noncontinuum effects, support interference, etc. are significant have been excluded. The whole range of Re has been divided into 10 sub intervals, with a distinct correlation for each interval. Adjacent equations for match within 1% at the boundaries between sub intervals, but the piecewise fit shows slight gradient discontinuities there. The Re = 20 boundary corresponds to onset of wake formation as discussed above, the remaining boundaries being chosen for convenience. 

Compressibility and noncontinuum effects are related. For an ideal gas, kinetic theory leads to the relationship (Sll) ... 

Analogous to the slip velocity between gas and particle at Kn above the continuum flow range discussed in Section A above, a temperature discontinuity exists close to the surface at high Kn. Such a discontinuity represents an additional resistance to transfer. Hence, transfer rates are generally lowered by compressibility and noncontinuum effects. The temperature jump occurs over a distance 1.996kl 2 — a )/Fva k + 1) (K2, Sll) where is the thermal accommodation coefficient, interpreted as the extent to which the thermal energy of reflected molecules has adjusted to the surface temperature. 

For small particles, subject to noncontinuum effects but not to compressibility, Re is very low see Eq. (10-52). In this case, nonradiative heat transfer occurs purely by conduction. This situation has been examined theoretically in the near-free-molecule limit (SI4) and in the near-continuum limit (T8). The following equation interpolates between these limits for a sphere in a motionless gas ... 

Actual construction of such a device presents a number of technical challenges. When electrodes are synthesized at 10- to 100-nm diameters, with the anode and cathode separated by similar distances, problems in hard wiring and assembly are to be expected. In addition, there currently is no three-dimensional architecture for an electrochemical cell that would achieve uniform current density. Also, at the nanometer scale, noncontinuum effects, especially mass transport, become a concern. Other issues of concern include ensuring that there is enough territory for phase nucleation to occur and quantized charging when the electrode material approaches nanoscale dimensions. 

Stokes Law and Noncontinuum Effects Slip Correction Factor... 

Equation (17.60) neglects noncontinuum effects that may influence very small cloud droplets. These effects can be included in this equation by introducing a modified diffusivity D v, where (Fukuta and Walter 1970)... 

FIGURE 17.13 Water vapor diffusivity corrected for noncontinuum effects and imperfect accommodation as a function of the droplet diameter at T = 283 K and p = 1 atm. 

C Repeat Problem 17.7 neglecting the correction to the diffusion coefficients and thermal conductivities for noncontinuum effects (assume that D v = Dv, k a = ka). Compare with Problem 17.7 and discuss your observations. 

Stokes law is based on the solution of equations of continuum fluid mechanics and therefore is applicable to the limit Kn -> 0. The nonslip conditions used as a boundary condition are not applicable for high Kn values. When the particle diameter Dp approaches the same magnitude as the mean free path X of the suspending fluid (e.g., air), the drag force exerted by the fluid is smaller than predicted by Stokes law. To account for noncontinuum effects that become important as Dp becomes smaller and smaller, the slip correction fac-... 

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