Equations film flow

The problem of film flow is formulated on the assumption that the film thickness h is much smaller than the length 1 (in our case h/1 10 ). In Cartesian coordinates with transversal axis y and longitudinal one z we can write the equation for a film flow as follows ... 

Use now this equation to describe liquid film flow in conical capillary. Let us pass to spherical coordinate system with the origin coinciding with conical channel s top (fig. 3). It means that instead of longitudinal coordinate z we shall use radial one r. Using (6) we can derive the total flow rate Q, multiplying specific flow rate by the length of cross section ... 

The first part of Eq. (89), proportional to the inverse viscosity r] of the liquid film, describes a creeping motion of a thin film flow on the surface. In the (almost) dry area the contributions of both terms to the total flow and evaporation of material can basically be neglected. Inside the wet area we can, to lowest order, linearize h = hoo[ + u x,y)], where u is now a small deviation from the asymptotic equilibrium value for h p) in the liquid. Since Vh (p) = 0 the only surviving terms are linear in u and its spatial derivatives Vw and Au. Therefore, inside the wet area, the evolution equation for the variable part u of the height variable h becomes... 

Using this equation, the flow of the reaction was followed by registering the frequency shift as a function of reaction time. Typical dependencies are shown in Figure 38 (Facci et al. 1994). As one can see, the curves tend to come to saturation (saturation time depends on film thickness). On the other hand, the plateau level in all cases corresponds well to the value resulting from simple calculation of the number of cadmium atoms available for the reaction in each sample. Numerical data of such estimates are presented in Table 8. The available amount of cadmium atoms was estimated in this case, taking into ac-... 

Problems in forced convection are solved in two steps first one solves the equation of motion to obtain the velocity distribution, and then one puts the expression for v back into the diffusion equation [usually as given in Eq. (50)]. An illustration of this type of problem is that of the absorption of a gas by a liquid film flowing down a... 

The velocity distribution equation (27) indicates that in the absence of surface tension effects the maximum velocity in a film flowing in a flat channel of finite width should occur at the free surface of the film at the center of the channel. The surface velocity should then fall off to zero at the side walls. However, experimental observations have shown (BIO, H18, H19, F7) that the surface velocity does not follow this pattern but shows a marked increase as the wall is approached, falling to zero only within a very narrow zone immediately adjacent to the walls. The explanation of this behavior is simple because of surface tension forces, the liquid forms a meniscus near the side walls. Equation (12) shows that the surface velocity increases with the square of the local liquid depth, so the surface velocity increases sharply in the meniscus region until the side wall is approached so closely that the opposing viscous edge effect becomes dominant. 

In addition to the general treatments of wavy flow, a number of theories concerning the stability of film flow have been published in these the flow conditions under which waves can appear are determined. The general method of dealing with the problem is to set up the main equations of flow (usually the Navier-Stokes equations or the simplified Nusselt equations), on which small perturbations are imposed, leading to an equation of the Orr-Sommerfeld type, which is then solved by various approximate means to determine the conditions for stability to exist. The various treatments are lengthy, and only the briefest summaiy of the results can be given here. 

The stability of flow in open channels has been investigated theoretically from a more macroscopic or hydraulic point of view by several workers (Cl7, D9, DIO, Dll, 14, J4, K16, V2). Most of these stability criteria are expressed in the form of a numerical value for the critical Froude number. Unfortunately, most of these treatments refer to flow in channels of very small slope, and, under these circumstances, surface instability usually commences in the turbulent regime. Hence, the results, which are based mainly on the Ch<5zy or Manning coefficient for turbulent flow, are not directly applicable in the case of thin film flow on steep surfaces, where the instability of laminar flow is usually in question. The values of the critical Froude numbers vary from 0.58 to 2.2, depending on the resistance coefficient used. Dressier and Pohle (Dll) have used a general resistance coefficient, and Benjamin (B5) showed that the results of such analyses are not basically incompatible with those of the more exact investigations based on the differential rather than the integral ( hydraulic ) equations of motion. The hydraulic treatment of the stability of laminar flow by Ishihara et al. (12) has been mentioned already. 

For the case of two-dimensional wavy film flow, Levich (L9) has shown that Eqs. (4) and (5) reduce to the familiar form of the boundary layer equations ... 

Semenov (S7) simplified the equations of wavy flow for the case of very thin films, and this approach has also been followed by others (K20, K21). These treatments refer mainly to the case of film flow with an adjoining gas stream and will be considered in Section III, F, 3. 

For smooth laminar film flow with an interfacial shear the equations of motion remain as in Section III, B, 2, but the boundary condition du/dy = 0 at y = b must be replaced either by... 

Kapitsa (K8) extended his treatment of wavy free film flow to cover this case also. For the simplest case, in which the gas stream does not seriously affect the wavelength, it was found to a first approximation that the mean film thickness 5 could be given in terms of the flow rate per wetted perimeter Q and the mean gas velocity ugas by means of the equation... 

Turning to the turbulent regime of film flow, there are several empirical relationships to be found in the literature. The experimental data of Brauer (B14) for the zone Nne > 400 can be represented by the equation... 

As noted earlier, the equation for turbulent film flow obtained by Levich (L8) [Eq. (71)] contains an unspecified constant and therefore cannot be readily compared with the other relationships for turbulent films. 

Dukler (D12), 1959 Theoretical analysis of turbulent film flow (with and without downward cocurrent gas stream) with extension to film heat transfer. Interfacial disturbances are neglected basic equations are solved by computer giving film thicknesses, velocity profiles, local and mean heat transfer coefficients. Interfacial shear is shown to be of great importance. 

Theoretical treatment of smooth laminar film flow on vertical surface, with and without gas flow, including inertia effects. Nusselt equations (N6, N7) are shown to be special cases of the present solutions. 

Kasimov and Zigmund (Kl2), 1963 General solution is given to equations of wavy film flow (with corrected continuity equation), using simple parabolic or higher approximation to velocity profile at given plane. It is shown that the film thickness and wave amplitude should increase in direction of flow. Effect of channel slope also considered. 

Liquid-solid mass transfer has also been studied, on a limited basis. Application to systems with catalytic surfaces or electrodes would benefit from such studies. The theoretical equations have been proposed based on film-flow theory (32) and surface-renewal theory (39). Using an electrochemical cell with rotating screen disks, liquid-solid mass transfer was shown to increase with rotor speed and increased spacing between disks but to decrease with the addition of more disks (39). Water flow over naphthalene pellets provided 4-6 times higher volumetric mass transfer coefficients compared to gravity flow and similar superficial liquid velocities (17). 

Solution of the velocity profile equation in the liquid film flowing over the evaporator. 

The hydrodynamics is then described by the system of Navier-Stokes equations in the film-flow approximation (Shilkin et al., 2006) ... 

Vapor can condense on a cooled surface in two ways. Attention has mainly been given in this chapter to one of these modes of condensation, i.e.. to him condensation. The classical Nusselt-type analysis for film condensation with laminar film flow has been presented hnd the extension of this analysis to account for effects such as subcooling in the film and vapor shear stress at the outer edge of the film has been discussed. The conditions under which the flow in the film becomes turbulent have also been discussed. More advanced analysis of laminar film condensation based on the use of the boundary layer-type equations have been reviewed. 

When a co-current vapor flow is present, the basic nature of this flow does not change, but the details differ because of the thinning of the liquid film by interfacial shear stress. Dropping the convective terms we can write the Navier-Stokes equations for steady film flow as follows ... 

The real surfaces are characterized by a certain extent of roughness. It is assumed that film thickness cannot be lower than the wall roughness value. When film thickness achieves this value, the film ruptured may occur depending on rivulet width, which corresponds to film flow rate at that time. With this statement the system of equations proves to contain the relationship describing the flow rate in a rivulet in the dependence on its length and contact angle. This correlation was obtained by the same way as that for the meniscus flow rate. 

A model of gravity induced film flow is developed which can predict the shape of the interface, namely the film thickness distribution around the perimeter of rectangular minichannel. The flow is considered in terms of two components namely in the corners and on the sides of the channel. It gives the simplified equation that describes the flow interface, which is written and solved numerically. The effect of the strong capillary action that draws the liquid to the comer of the channel is noted. By comparing the model with the experimental data, it is shown that it predicts well the experimental results on interface shape and flow rate redistrihution. 

For a spinning disk, the standard model for falling film flow is complicated by the changing thickness and shear as the liquid flows over the disk. An approximation of this to conditions on a spinning disk surface can, however, be made by substitution of Eq. (9) for average liquid-solid surface shear into the above equation for mass transfer. If it is also assumed that the characteristic distance L traveled by the liquid is equal to that of the disk radius then an equation for the liquid-solid mass transfer coefficient ls can be written for an SDR as... 

H. A. Stone, Partial differential equations in thin film flows in fluid dynamics Spreading droplets and rivulets, in Nonlinear PDEs in Condensed Matter and Reactive Flows, editedby H. Berestycki and Y. Pomeau, (springer-verlag, New York, 2000). 

Apart from the trivial inclusion of the gravitational body-force terms in (6-2) and (6-3), the governing equations, and the analysis leading to them, are identical to the governing equations for the lubrication theory of the previous chapter. The primary difference in the formulation is in the boundary conditions, and the related changes in the physics of the thin-film flows, that arise because the upper surface is now a fluid interface rather than solid surface of known shape. The boundary conditions at the lower bounding surface are ... 

When the layer is thin, h/R < 1, analyze the flow in the fluid film and derive an equation for the shape h(6). Show that there is no solution for the shape when the rotation speed is less than a critical value, and determine this critical value. In film flow problems the characteristic film thickness h and the volumetric flux (per unit length of the cylinder) of fluid carried around by the rotating cylinder are related and cannot be specified independently. In this problem it is simplest if you specify the fluid flux carried around by the rotating cylinder as Q per unit length and then determine h in terms of Q. 

Figure 1.4 shows the scheme of a double-film flow and the coordinate system used. The boundary problem for the X-components Va(Y) and Vb(Y) of film the velocities consists of the equations... 

In a rectangular Cartesian coordinate system with the X-axis directed along the plane along which the film flows, the only nonzero velocity component is Vx. If we ignore the pressure gradient along the X-axis and assume that the film liquid velocity Vx and its temperature T depend only on the transverse coordinate Y, then we obtain the system of equations... 

If the wall along which the condensate film flows makes an angle 9 with the vertical, then one must replace g by g cos 8 in the original equation of motion (5.7.1) and in all subsequent relations. 

Under these assumptions we arrive at a one-dimensional velocity profile V = U(0 and pressure profile P = P( ), where f = h-Y is the coordinate measured from the wall along the normal. The corresponding equations of film flow have the form... 

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