Rarefaction coefficient

In general, the increased rarefaction effects in our flowrate model can be taken into account by introducing a correction expressed as rarefaction coefficient CfKn), which is a function of the Knudsen number. The flowrate is then obtained as... 

The asymptotic values of the flowrate for duct flows at high Knudsen number are constants depending on the duct aspect ratio. This offers the possibility of obtaining a model for the rarefaction coefficient Cr Kn) and in particular the coefficient a. The objective is to construct a unified expression for a Kn) that represents the transition of a from zero in the slip flow regime to its asymptotic constant value in the free-molecular flow regime. 

Iwai and Suzuki [15] numerically investigated the effects of rarefaction and compressibility on heat transfer for a flow over a backward-facing step in a microchannel duct. They applied the velocity shp boundary condition to all the walls and considered tem-peratme jump at the heated wall. Skin friction was seen to reduce when the velocity shp was taken into account. It was further reduced if the accommodation coefficient takes smaller values, which results in larger slip velocities. They found that the compressibil-... 

Heat convection for gaseous flow in a circular tube in the slip flow regime with uniform temperature boundary condition was solved in [23]. The effects of the rarefaction and surface accommodation coefficients were considered. They defined a fictitious extrapolated boundary where the fluid velocity does not slip by scaling the velocity profile with a new variable, the shp radius, pj = l/(l + 4p.,Kn), where is a function of the momentum accommodation coefficient, and defined as p, =(2-F,j,)/F,j,. Therefore, the velocity profile is converted to the one used for the... 

The Knudsen number (Kn) is used to represent the rarefaction effects. It is the ratio of the molecular mean free path to the characteristic dimension of the flow. For Knudsen numbers close to zero, flow is still assumed to be continuous. As the Knudsen number takes higher values, due to a higher molecular mean free path by reduced pressure or a smaller flow dimension, rarefaction effects become more significant and play an important role in determining the heat transfer coefficient. 

It follows from [378, 383] that the formula for the osmotic pressure differs from the corresponding formula for the capillary rarefaction only by the coefficient g(< ) = (1 - 1.83 1 - )2 which characterizes the fraction of the membrane area adjacent to flat faces of foam cells. In the case of polydisperse foam, it is also expedient to use formula (7.1.16) with a replaced by the Sauter mean radius (7.1.9). 

Here p and p are the liquid and gas densities, respectively, g is the vector of the gravitational acceleration, and AP is the capillary rarefaction given by (7.1.10) and (7.1.15). The kinetic coefficient H was called the coefficient of hydroconductivity and calculated for polyhedral foam models [245, 246]. Generally speaking, the variable H is a tensor, but usually the isotropic approximation is used, where this parameter is a scalar. Various expressions for the coefficient H were proposed and made more precise in [125, 214, 245]. Thus, different approaches used to calculate the coefficient of hydroconductivity were analyzed in [488]. For example, the structure of spherical and cellular foam was studied under the assumption that liquid flows through a porous layer according... 

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. 

In order to model the flowrate variations with respect to the Knudsen number Kn, we introduced the rarefaction correction factor as Cr =l+ Kn. This form of the correction factor was justified using two independent arguments first, the apparent diffusion coefficient and second, the ratio of intermolecular collisions to the total molecular collisions. We must note that a cannot be a constant. Physical considerations to match the slip flowrate require 0 fox Kn < 0.1, while in the free molecular flow... 

However, in the transitional regime, i.e., when the rarefaction parameter is intermediate (8 1), the slip solution is not valid. Such a situation is realized when the channel/tube size a is less than 2 X 10 m. In this case, the kinetic Boltzmaiui equation is applied to calculate the coefficients Gp and Gt- hi the free-molecular regime, i.e., when the rarefaction parameter is... 

Gas Flow in Nanochannels, Fig. 2 Poiseuille coefficient versus rarefaction parameter 5 solid lines, kinetic equation solution [11] pointed line, free-molecular value based on Eq. 12 dashed line, Navier-Stokes solution Eq. 16... 

Cy T — Tc) ", where a 0.09. This implies that the sound speed will approach zero and that the nonlinearity coefficient will change sign from positive to negative sufficiently near the critical point. Thus, the conventional rules for wave steepening will reverse, with the attendant possibility that the rarefaction wave will steepen to form a rarefaction shock, which are discussed in detail in the following section. 

The reflection coefficient R=Piaax/P20 s M, the Mach number of a blast wave, which is just in front of the surface of the PUR foam (P20 denotes here the normal reflection pressure of air shock with Mg=Mi), is presented on Fig. 6 by a solid line. The dotted line represents computed pressure values. One can mention that on this curve there is no amplification effect (R<1) up to Mi l.2. The least effect is caused by two forces first, the reflection of the rarefaction wave from a boundary of the foam and second, inertia losses due to filtration effects. Even in the case of a blast wave the second force seems to be negligible with Mi>1.3. 

Note that if the super-index ch or tb is omitted the quantity is referred to both channel and tube. The coefficient Gj describes a gas flow due to a temperature gradient and it is called the thermal creep coefficient. The coefficients Gp and Gt are introduced so that they are always positive. They are calculated from the kinetic equation and determined by the rarefaction parameter... 

Gas Row in Nanochannels, Rgure 3 Thermal creep coefficient 6 vs rarefaction parameter solid lines - kinetic equation solution [ ], pointed line - free molecular value based on Eqs. (12) and (15), dashed line -Navier- okes solution Eq. (19)... 

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