Velocity capillary flow

Figure 9.5a shows a portion of a cylindrical capillary of radius R and length 1. We measure the general distance from the center axis of the liquid in the capillary in terms of the variable r and consider specifically the cylindrical shell of thickness dr designated by the broken line in Fig. 9.5a. In general, gravitational, pressure, and viscous forces act on such a volume element, with the viscous forces depending on the velocity gradient in the liquid. Our first task, then, is to examine how the velocity of flow in a cylindrical shell such as this varies with the radius of the shell.

Chapter 9 is devoted to regimes of capillary flow with a distinct interface. The effect of certain dimensionless parameters on the velocity, temperature and pressure within the liquid and vapor domains are considered. The parameters corresponding to the steady flow regimes, as well as the domains of flow instability are defined. 

The effect of wall heat flux on the length of the heating and evaporation regions, vapor velocity, temperature and pressure in the outlet cross-section is shown in Figs. 8.13, 8.14, and 8.15. These data illustrate some important features of capillary flow at large Euler numbers. 

The quasi-one-dimensional model of flow in a heated micro-channel makes it possible to describe the fundamental features of two-phase capillary flow due to the heating and evaporation of the liquid. The approach developed allows one to estimate the effects of capillary, inertia, frictional and gravity forces on the shape of the interface surface, as well as the on velocity and temperature distributions. The results of the numerical solution of the system of one-dimensional mass, momentum, and energy conservation equations, and a detailed analysis of the hydrodynamic and thermal characteristic of the flow in heated capillary with evaporative interface surface have been carried out. 

OS 90] [R 31] [P 70] At weak electrical field, the propagation velocity of a reaction front in a capillary-flow reactor was increased or decreased depending on the mutual orientation of the electrical field and the reaction zone propagation [68]. The movement of two reaction fronts was given by optical images in [68]. 

Resolution in forced-flow development is not restricted by the same limitations that apply to capillary flow controlled systems. The maximum resolution achieved usually corresponds to the optimum mobile phase velocity and R, increases approximately linearly with the solven)t migration distance (48). Thus there is... 

For axial capillary flow in the z direction the Reynolds number, Re = vzmaxI/v = inertial force/viscous force , characterizes the flow in terms of the kinematic viscosity v the average axial velocity, vzmax, and capillary cross sectional length scale l by indicating the magnitude of the inertial terms on the left-hand side of Eq. (5.1.5). In capillary systems for Re < 2000, flow is laminar, only the axial component of the velocity vector is present and the velocity is rectilinear, i.e., depends only on the cross sectional coordinates not the axial position, v= [0,0, vz(x,y). In turbulent flow with Re > 2000 or flows which exhibit hydrodynamic instabilities, the non-linear inertial term generates complexity in the flow such that in a steady state v= [vx(x,y,z), vy(x,y,z), vz(x,y,z). ... 

Fig. 5.1.2 Non-ideal capillary flow reactor (a) propagators [13] and (b) corresponding RTDs calculated from the propagator data, (a) The propagators indicate the distribution of average velocities over each observation time (A) ranging from 50 ms to 1 s. As the observation time increases the spins exhibit a narrowing distribution of average velocities due to the motional narrowing effect of molecular diffusion across the streamlines. The dashed vertical line represents the maximum velocity that would be present in the absence of molecular...

It is probable that capillary flow of water contributes to transport in the soil. For example, a rate of 7 cm/year would yield an equivalent water velocity of 8 x 10-6 m/h, which exceeds the water diffusion rate by a factor of four. For illustrative purposes we thus select a water transport velocity or coefficient U6 in the soil of 10 x 10 6 m/h, recognizing that this will vary with rainfall characteristics and soil type. These soil processes are in parallel with boundary layer diffusion in series, so the final equations are... 

Methane is commonly used as a marker for measuring the gas holdup time (tm), which was done on a capillary column 25 m long by 0.25 mm ID by 0.25 pm film thickness. A retention time for methane of 1.76 min was obtained. Determine the average linear gas velocity (v) and the average volumetric flow rate (Fc). Explain how these values differ from the actual velocity and flows at the column inlet and outlet. 

The analysis of this effect in a closed cylindrical cell is obtained by subtracting from the electroosmotic velocity vEO the velocity of flow vP through a capillary given by Poiseuille s equation (Equation (4.18)) ... 

If one considers fluid flowing in a pipe, the situation is highly illustrative of the distinction between shear rate and flow rate. The flow rate is the volume of liquid discharged from the pipe over a period of time. The velocity of a Newtonian fluid in a pipe is a parabolic function of position. At the centerline the velocity is a maximum, while at the wall it is a minimum. The shear rate is effectively the slope of the parabolic function line, so it is a minimum at the centerline and a maximum at the wall. Because the shear rate in a pipe or capillary is a function of position, viscometers based around capillary flow are less useful for non-Newtonian materials. For this reason, rotational devices are often used in preference to capillary or tube viscometers. 

Figure 5. Variation of Xanthan slip velocity with tube length in capillary flows (Reproduced with permission, ref. 27).

Another important ramification of shear-thinning behavior in capillary or tube flow, relevant to polymer processing, relates to the shape of the velocity profiles. Newtonian and shear-thinning fluids are very different, and these differences have profound effects on the processing of polymer melts. The former is parabolic, whereas the latter is flatter and pluglike. The reason for such differences emerges directly from the equation of motion. The only nonvanishing component for steady, incompressible, fully developed, isothermal capillary flow, from Table 2.2, is... 

Estimation of Entrance Pressure-Pressure Losses from the Entrance Flow Field17 Consider the entrance flow pattern observed with polymer melts and solutions in Fig. 12.16(a). The flow can be modeled, for small values of a, as follows for 0 < a/2 the fluid is flowing in simple extensional flow and for a/2 < 0 < rc/2 the flow is that between two coaxial cylinders of which the inner is moving with axial velocity V. The flow in the outer region is a combined drag-pressure flow and, since it is circulatory, the net flow rate is equal to 0. The velocity V can be calculated at any upstream location knowing a and the capillary flow rate. Use this model for the entrance flow field to get an estimate for the entrance pressure drop. 

The influence of interphase mass transfer between liquid-liquid slugs was investigated for nitration of aromatic compounds in a capillary-flow reactor (see Figure 5.2) [22]. This was achieved by changing flow velocity via volume flow setting, while residence time was kept constant by increasing the capillary length. 

It is instructive to consider steady fluid flow (sometimes called Poiseuille flow) in a thin capillary tube. This example has many purposes it provides (1) a model flow calculation, (2) an illustration of how velocity profiles arise, (3) an explanation of the nature of flow in capillary chromatography, and (4) a foundation for capillary flow models of packed beds. 

This equation shows that the capillary flow velocity increases with capillary radius rc and surface tension y it decreases, as do all flow velocities, with viscosity 17. The equation also shows that (v) decreases with the length Xf of penetration of liquid into the capillary space that is, flow diminishes as the liquid progresses further and further into the pore space. This diminution can be calculated as a function of time as follows. 

Since chromatographic migration rates depend on the amount of liquid phase and on the liquid velocity, the above noted gradients, established by the actions of capillarity, have important chromatograhic effects [14]. More details on the chromatographic consequences of capillary flow can be found in the literature (12) and in subsequent chapters. 

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