Dimensionless number Knudsen

The phase-density ratio is defined by i = Pp/pg and, since cpi 1, the buoyancy term is negligible. The two new dimensionless numbers generated in this process are the phase-velocity ratio 03 = t/p/t/, and the disperse-phase Knudsen number KUp = uItJL. In addition, the dimensionless form ofEqs. (1.10),... 

The analysis in this section focuses on the appropriate dimensionless numbers that are required to analyze convection, axial dispersion and first-order irreversible chemical reaction in a packed catalytic tubular reactor. The catalytic pellets are spherical. Hence, an analytical solution for the effectiveness factor is employed, based on first-order irreversible chemical kinetics in catalysts with spherical symmetry. It is assumed that the catalytic pores are larger than 1 p.m (i.e., > 10 A) and that the operating pressure is at least 1 atm. Under these conditions, ordinary molecular diffusion provides the dominant resistance to mass transfer within the pores because the Knudsen diffusivity,... 

Both microscale gas flows and rarefied-gas flows have three dimensionless numbers that characterize the flow the Reynolds number Re, the Mach number Ma, and the Knudsen number Kn. However, these three parameters are not independent in rarefied-gas flows but instead are related by... 

Knudsen number (Kn) Dimensionless number defined as the ratio of the mean free path length of gas atoms/molecules to a representative physical length scale, e.g., pore size the number is named after Danish physicist Martin Knudsen (1871-1949)... 

Another length scale of importance to the combustion of small particles is the mean free path in the surrounding gas-phase. A comparison of this length scale to the particle diameter defines whether continuum conditions exist (i.e., the particle may be distinguished separately from the gas molecules). The key dimensionless group that defines the nature of the surrounding gas to the particle is the Knudsen number,... 

When considering boundary conditions, a useful dimensionless hydrodynamic number is the Knudsen number, Kn = X/L, the ratio of the mean free path length to the characteristic dimension of the flow. In the case of a small Knudsen number, continuum mechanics will apply, and the no-slip boundary condition assumption is valid. In this formulation of classical fluid dynamics, the fluid velocity vanishes at the wall, so fluid particles directly adjacent to the wall are stationary, with respect to the wall. This also ensures that there is a continuity of stress across the boundary (i.e., the stress at the lower surface—the wall—is equal to the stress in the surface-adjacent liquid). Although this is an approximation, it is valid in many cases, and greatly simplifies the solution of the equations of motion. Additionally, it eliminates the need to include an extra parameter, which must be determined on a theoretical or experimental basis. 

Here, c, the effective particle Knudsen number, is the ratio of the relaxation time f 1 for Brownian motion of the particle to the time needed for the fictitious particle to traverse the distance Rs with its average velocity, 6 is the dimensionless surface temperature, and is the dimensionless depth of the potential well. 

For particles of unit density, in air, at a pressure of 1 atm and temperature of 298°K, the upper and lower bounds of the dimensionless coagulation coefficients are plotted as a function of Knudsen number, for different Hamaker constants, in Fig. 3. Obviously, the upper bound for the coagulation coefficient is independent of the Hamaker constant. The upper and the lower bounds tend to the Smoluchowski expression for small Knudsen numbers. For large Knudsen numbers, the upper bound coincides with the free molecular limit, as can be seen from Fig. 3. The lower bound is found to decrease dramatically with a decrease in the Hamaker constant, for large Knudsen numbers. Both the lower and the upper bounds exhibit a maximum at intermediate values of the Knudsen number. 

Particle radius Oim) Knudsen number, Kn Particle Knudsen number, Distance parameter, a No. of realizations, A No. of collisions, M No, of captures, L Dimensionless coagulation coefficient, y... 

Fig. 2. Response of the TAP reactor to an inlet pulse of a gas that is irreversibly adsorbed (or reacted) with a dimensionless rate constant k. Tp is the dimensionless time, and F p is the dimensionless flow rate. The model takes into account the number of molecules in the pulse A p A, the effective Knudsen diffusion coefficient DeA, the number of surface sites, and the dimensions of the reactor (after 55). A, = 0, standard diffusion curve B, = 3 C, U = 10.

Figure 7 Influence of Knudsen number on the transient and steady behaviors of (a) - dimensionless average temperature and (b) and (c) - local Nusselt number (parallel plates, Pe = 10, Kn =0, 0.001, 0.01 0.1 and Br = 0,...

A dilute gas mixture is assumed to behave as a continuum when the mean free path of the molecules is much smaller than the characteristic dimensions of the problem geometry. A relevant dimensionless group of variables, the Knudsen number Kn, is defined as [47, 30, 31] ... 

Values of the dimensionles.s thermophoretic velocity are. shown in Fig. 2.9 as a function of the Knudsen number with kg/kp as a parameter. For Knudsen numbers larger than unity, the dependence of the dimensionless thermophoretic velocity on particle size and chemical nature Is small. Particle sampling by theimophoresis in this range offers the advantage that particles are not selectively deposited according to size. 

The dimensionless proportionality constant A depend,s on how R, the characteristic radius of die agglomerate, is dehned it also depends on the process by which the agglomerate forms and on the Knudsen number. The value of A may also vaiy for different values of Df. If this equation holds for r 1), then A has a value of unity. However, for... 

Effects of the main governing dimensionless parameters on the momentum and heat flow transfer will be analyzed. Pure analytical correlations for Nusselt number as a function of the Brinkman number and the Knudsen number are developed for both hyrodynamically and thermally fully developed flow. In fact, this work will be a summary view of our recent studies [12-15]. 

Here Kn is Knudsen number Kn = IjD ) and designates the dimensionless radius where the maximum velocity occurs (9 / = 0). It is... 

Besides the Knudsen number, some other dimensionless parameters become important in microscale flow and heat transfer problems. The first such number is the Peclet number (Pe), which is the product of Reynolds (Re) and Prandtl (Pr) numbers (Pe = Re Pr), and signifies the ratio of rates of advection to diffusion. Peclet number enumerates the axial conduction effect in flow. In macro-sized conduits, Pe is generally large and the effect of axial conduction may be neglected. However as the channel dimensions get smaller, it may become important. 

In this lecture, the effects of the abovementioned dimensionless parameters, namely, Knudsen, Peclet, and Brinkman numbers representing rarefaction, axial conduction, and viscous dissipation, respectively, will be analyzed on forced convection heat transfer in microchannel gaseous slip flow under constant wall temperature and constant wall heat flux boundary conditions. Nusselt number will be used as the dimensionless convection heat transfer coefficient. A majority of the results will be presented as the variation of Nusselt number along the channel for various Kn, Pe, and Br values. The lecture is divided into three major sections for convective heat transfer in microscale slip flow. First, the principal results for microtubes will be presented. Then, the effect of roughness on the microchannel wall on heat transfer will be explained. Finally, the variation of the thermophysical properties of the fluid will be considered. 

For binary mixtures diffusing inside porous solids, the applicability of equation (1-100) depends upon the value of a dimensionless ratio called the Knudsen number, Kn, defined as... 

Knudsen number Kn) - A dimensionless quantity used in fluid mechanics, defined hyKn = H, where X is mean free path and I is length. [2]... 

The mass-transfer coefficients, k and k, can be estimated by analogy from available dimensionless empirical correlations for heat transfer, which are taken ftom Knudsen and Katz (1958). These analogous correlations depend on the flow regime and relate the Sherwood number to the Reynolds and Schmidt numbers. 

Overall the results show that the wall force field penetration depth is an additional length scale for gas flows in nano-channels, breaking dynamic similarity between rarefied and nanoscale gas flows solely based on the Knudsen and Mach numbers. Hence, one should define a new dimensionless parameter as the ratio of the force field penetration depth to the characteristic channel dimension, where wall effects cannot be neglected for large values of this dimensionless parameter. Additionally, the calculated tangential momentum accommodation coefficients for a specific gas-surface couple were found to be constant regardless of different base pressure, channel height, wall velocity, and Knudsen number. Results of different gas-surface couples reveal that TMAC is only dependent on the gas-surface couple properties and independent of the Knudsen number. 

In a porous medium, assuming surface diffusion is negligible, the mass transfer is limited by viscous resistance, resulting from the momentum transferred to the membrane, Knudsen diffusion resistance due to molecule-membrane collisions and ordinary diffusion due to collisions between molecules. Predominance, coexistence, or transitions between these different mechanisms are estimated by the dimensionless Knudsen number (Kn) that compares the mean free path i of diffusing molecules... 

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