Velocity slipping

Apart from obvious features such as laminarity, there are speculations that flows in micro channels exhibit a behavior deviating from predictions of macroscopic continuum theory. In the case of gas flows, these deviations, manifesting themselves as, e.g., velocity slip at solid surfaces, are comparatively well understood (for an overview, see [130]). However, for liquid flows on a length scale above 1 pm, there is no clear theoretical foundation for deviations from continuum behavior. Nevertheless, various unexpected phenomena such as friction factors deviating from the continuum prediction [131-133] have been reported. A more detailed discussion of this still unsettled matter is given in Section 2.2. At any rate, one has to be careful here since it may be that measurements in small systems lack precision, essentially because of the incompatibility of analysis in a confined space and with large measuring equipment... 

The physical reason for the velocity slip is the fact that close to the wall the gas is not in thermal equilibrium. For the same reason, a temperature jump is induced, and a more detailed investigation based on the kinetic theory of gases shows that heat transfer and momentum transfer are coupled. Expressions for velocity slip and temperature jump valid in the case of non-isothermal conditions are given by... 

In Figure 2.2 DSMC results of Karniadakis and Beskok [2] and results obtained with the linearized Boltzmann equation are compared for channel flow in the transition regime. The velocity profiles at two different Knudsen numbers are shown. Apparently, the two results match very well. The fact that the velocity does not reach a zero value at the channel walls (Y = 0 and Y = 1) indicates the velocity slip due to rarefaction which increases at higher Knudsen numbers. 

Isothermality in this reactor is difficult to maintain however, wall heat transfer is better than for fixed-bed reactors. Salt or sand baths may be required for isothermality. Residence-time distributions for all three phases can be measured accurately. At low velocity, slip between phases is a problem, but more... 

Terminal velocity-slip velocity theory In this theory, the steady slip velocity between solid and liquid is used in the correlation for the Sherwood number. For low particle Reynolds number, Friedlander33 gave... 

As outlined above, steady-state theories for the liquid-solid mass transfer are largely classified into two categories i.e., those based on Kolmogoroff s theory and those based on the terminal velocity-slip velocity approach. 

According to the terminal velocity-slip velocity theory, for particles less than 500 pm. 

Velocity slip and temperature jump affect the heat transfer in opposite ways a large slip on the wall increases the convection along the surface. On the other hand, a large temperature jump decreases the heat transfer by reducing the temperature gradient at the wall. Therefore, neglecting temperature jump will result in the overestimation of the heat transfer coefficient. 

Iwai, H. and Suzuki, K., Effects of Velocity Slip and Temperature Jump Conditions on Backward-Facing Step Flow in a MicroChannel, Proceedings of the 5 ASME/JSME Joint Thermal Engineering Conference, 1999, 1-8. 

Heat transfer by forced convection inside micro tube, generally referred as the Graetz problem, has been extended by Barron et al. [11] and Larrode and al. [12] to include the velocity slip described by Maxwell in 1890 [13] and the temperature jump [14] on tube surface, which are important in micro scale at ordinary pressure and in rarefied gases at low-pressure. 

Mathematica package is developed that computes the eigenvalues, the eigenfunctions, the eigenintegrals, the dimensionless temperature, the average dimensionless temperature, and the Nusselt number for steady state and periodic heat transfer in micro parallel plate channel and micro tube taking into account the velocity slip and the temperature jump. Some results in form of tables and plots are given bellow. 

The limiting Nusselt number is of great practical interest. For n=0 (parallel plate micro channel) and n=l (micro tube) the limiting Nusselt number depend on 2 parameters Kn/3v and /3. The Kn/3v control mainly the velocity slip and have influence on the temperature jump. The parameter /3 control only the temperature jump. The limiting Nusselt number is shown on Fig 6. 

The solution (89) is used to plot Fig. 9 and Fig. 10, where the vertical distances to the surface present the amplitudes, while the color of the surface present the phase angle. As the angle moves around the circle, the color of the surface will go from red to blue, green, yellow, and back to red again. Fig. 9 shows the temperature oscillations in tube without velocity slip and temperature jump. Fig. 10 shows temperature oscillations in tube... 

In the Knudsen layer, the Maxwellian velocity slip boundary condition approximates the true gas velocity at the boundary by the velocity that the molecules would have if a linear velocity gradient existed as shown in figure 1 [18-19], In other words, the magnitude of the slip is calculated from the velocity gradient evaluated at y = X,... 

Equation (3.11) gives the first-order approximation to the temperature jump if it is assumed that the temperature gradient at the wall is the same as that at y = k. To obtain the higher-order approximation, the same approaeh is applied as that whieh was used to obtain the seeond-order velocity slip equation [2]. This results in (with 0 = T/Treference) ... 

Their results for the non-slip flow ease agreed with [26], who also used the integral transform teehnique to solve for the Nusselt number for flow through a maerosized reetangular ehannel. They did not inelude viscous dissipation in the work, but they did inelude variable thermal aeeommodation eoefficients. Similar to [15], they concluded that the Knudsen number, Prandlt number, aspeet ratio, velocity slip and temperature jump can all cause the Nusselt number to deviate from the eonventional value. 

Also, data on particle-liquid mass transfer from suspended solids in gas-liquid mechanically agitated vessels are practically nonexistent (R18). However, many studies have been published on mass-transfer experiments in the absence of gas, which give an idea of the magnitude of k. Recent reviews by Nienow (N9) and Blasinski and Pyc (B17, B18) indicate two fundamentally different approaches to the prediction of A s the Kol-mogoroff theory, which implies equal at equal power input per unit volume (B17) and the terminal velocity-slip velocity theory which relates ks to the value that would apply if the solid particle moved at its terminal velocity (H2). As explained by Nienow (N9), the resulting values of A s are approximately the same. Use may be made of the graphical correlation given by Brian et al. (B29). 

At the impermeable wall, the no-slip condition is generally not appropriate for granular flows. Nevertheless, the granular phase velocity component normal to the wall is normally set to zero. However, the granular phase is usually allowed to slip along the wall. A velocity slip proportional to the velocity gradient at the wall is commonly applied ... 

In this work, heat and fluid flow in some common micro geometries is analyzed analytically. At first, forced convection is examined for three different geometries microtube, microchannel between two parallel plates and microannulus between two concentric cylinders. Constant wall heat flux boundary condition is assumed. Then mixed convection in a vertical parallel-plate microchannel with symmetric wall heat fluxes is investigated. Steady and laminar internal flow of a Newtonian is analyzed. Steady, laminar flow having constant properties (i.e. the thermal conductivity and the thermal diffusivity of the fluid are considered to be independent of temperature) is considered. The axial heat conduction in the fluid and in the wall is assumed to be negligible. In this study, the usual continuum approach is coupled with the two main characteristics of the microscale phenomena, the velocity slip and the temperature jump. 

In this context, the first goal of this lecture is thus to illustrate the results obtained from a fairly general hybrid numerical-analytical solution for temperature distributions in a fluid flowing through two- or three-dimensional micro-channels, taking into account the velocity slip and temperature jump at the walls surfaces. For this purpose, a flexible approach was employed... 

Thus, we take advantage of the accuracy, robustness and efficiency of the direct problem solution, to tackle the associated inverse heat transfer problem analysis [26, 27] towards the simultaneous estimation of momentum and thermal accommodation coefficients in micro-channel flows with velocity slip and temperature jump. A Bayesian inference approach is adopted in the solution of the identification problem, based on the Monte Carlo Markov Chain method (MCMC) and the Metropolis-Hastings algorithm [28-30]. Only simulated temperature measurements at the external faces of the channel walls, obtained for instance via infrared thermography [30], are used in the inverse analysis in order to demonstrate the capabilities of the proposed approach. A sensitivity analysis allows for the inspection of the identification problem behavior when the external wall Biot number is also included among the parameters to be estimated. 

We developed a unified flow model that can accurately predict the volumetric flowrate, velocity profile, and pressure distribution in the entire Knudsen regime for rectangular ducts. The new model is based on the hypothesis that the velocity distribution remains parabolic in the transition flow regime, which is supported by the asymptotic analysis of the Burnett equations [1]. The general velocity slip boundary condition and the rarefaction correction factor are the two primary components of this unified model. 

At low gas pressures and for small pore size, the mean free path of the gas molecules may be on the order of the pore size and therefore velocity slip occurs (Knudsen effect), resulting in higher permeabilities. However, an increase in the permeability due to an increase in gas pressure has been found in some experiments. Scheidegger [24] discusses the effect of the Knudsen slip, the internal surface roughness, surface absorption, capillary condensation, and molecular diffusion on the measured permeability. By examining these effects at the pore level, it becomes clear that the measured gas and liquid permeabilities can be noticeably different. 

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