Knudsen number calculation

Mean Free Path and Knudsen Number Calculation... 

The nanofiber diameter for the electrospun core used in each of the test samples was obtained by SEM image analysis as described in Section 8.1.2 and listed in Table 8.20. The value of Knudsen number calculated using Eqn (8.5) and the corresponding flow regime for each of the test samples is given in Table 8.20. 

In Fig. 2, the flow rate coefficients, Qp, calculated from these different models are plotted in the range of inverse Knudsen number D from 0.01 to 100. We can see that for... 

The coagulation coefficient is a function of the radius of the particle Rp, its mass m , the effective particle Knudsen number k, the temperature of the medium T, and the depth of the interaction potential well between two particles. Using the expression for the overall interaction potential given in Appendix I, the depth of the interaction potential well can be calculated from the knowledge of the Hamaker constant for the particle. The friction coefficient f is related to the diffusion coefficient of the particle, D, through the Einstein equation... 

The Brownian coagulation coefficient for equal-sized electrically neutral particles, of unit density, for the entire range of Knudsen numbers, have been calculated through the Monte Carlo simulation. The interpartide forces due to van der Waals attraction and... 

This inaccuracy stems from their calculation of molecular transport effects, such as viscous dissipation and thermal conduction, from bulk flow quantities, such as mean flow velocity and temperature. This approximation of microscale phenomena with macroscale information fails as the characteristic length of the (gaseous) flow gradients approaches the average distance travelled by molecules between collisions - the mean path. The ratio of these quantities is referred to as Knudsen number. 

Particle behavior often depends on the ratio of particle size to some other characteristic length. The mechanisms of heat, mass, and momentum transfer between particle and carrier gas depend on the Knudsen number. 2J /d , where Ip is the mean fr. e. P ath of the gas molecules. The mean free path or mean distance traveled by a molecule between successive collisions can be calculated from the kinetic theory of gases. A good approximation for a single-component gas composed of molecules that act like rigid elastic spheres is... 

Calculate the mean free path for gases and the Knudsen number for gas flow through porous solids. 

It is desired to increase by 50% the oxygen flux in the process described in Example 1.22 by changing the pore size while maintaining all the other conditions unchanged. Calculate the required pore size. Remember to calculate the new value of the Knudsen number to corroborate that you have used the correct set of equations to relate flux to pore size. 

Before we discuss the role of the Knudsen number, we need to consider the calculation of the mean free path for a vapor. It will soon be necessary to calculate the mean free path both for a pure gas and for gases composed of mixtures of several components. Note that even though air consists of molecules of N2 and O2, it is customary to talk about the mean free path of air, X.ajr, as if air were a single chemical species. 

For Knudsen diffusion to take place, the lower limit for pore diameter has usually been set to 4>ore > 20 A. Gilron and Soffer have, however, discussed thoroughly how Knudsen diffusion may contribute to transport in even smaller pores, and from a model considering pore structure, shown that contributions to transport may both come from activated transport and Knudsen through one specific fiber. It may thus be difficult to know exactly when transport due to Knudsen diffusion is taking place. One way to approach this problem is to calculate the Knudsen number, Knudsen. for the system, which is 2/fi pore> where X is the mean free path. If Knudsen > 10, then the separation can be assumed to take place according to Knudsen diffusion. Therefore, if the preparation of the carbon membranes has been unsuccessful, one may get Knudsen diffusion. 

The main parameter determining the gas rarefaction is the Knudsen number Kn = Ha, where I is the mean free path of fluid molecules and a is a typical dimension of gas flow. If the Knudsen number is sufficiently small, say Kn < 10 , the Navier-Stokes equations are applied to calculate gas flows. For intermediate and high values of the Knudsen number, the Navier-Stokes equations break down, and the implementation of rarefied gas dynamics methods is necessary. In practical calculations usually the rarefaction parameter defined as the inverse Knudsen number, i.e.. 

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. 

A corrective coefficient has been calculated by different authors [6] who have shown that a better prediction of the flow out of the Knudsen layer would be obtained with this corrective coefficient slightly different from unity, unlike as initially proposed by Maxwell. Its value depends on the collision model for example, = 1.11 for a HS model [7]. Equation 12 is called first-order slip boundary condition, because it involves the Knudsen number (9(Kn)) and the first derivative du Jdn ) . 

A model calculation was discussed which attempted to include aspects of both finite Knudsen number transport and realistically complicated interaction potentials. The conclusion was that the oomhined effects of gas and particle properties are essential for the description of aerosol interactions. 

First, we calculate the Knudsen number for a gas in the gap between particle and wall, using Eq. (40),... 

Tunc and Bayazitoglu [3] have calculated for the T case the fully developed Nusselt numbers for microtubes through which a rarefied gas flows by taking into account the viscous dissipation but neglecting axial conduction in the fluid and the flow work. They deflned the Brinkman number (Eq. (20)) with ATref = 7 e - and used in the slip boundary conditions (Eqs. (8) and (19)) a = Ov = at = at = 1. The values of the Nusselt number for fully developed laminar flow determined by Tunc and Bayazitoglu [3] are quoted in Tab. 4 for Br = 0 and Br = 0.01 as a function of the Prandlt number and of the Knudsen number. 

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