Thickness boundary layer

However, the transition Reynolds number depends on free-stream turbulence and may range from 3 X 10 to 3 X lO ". The laminar boundary layer thickness 8 is a function of distance from the leading edge ... 

It follows that since the addition of metal oxides has such a profound effect on the properties of liquid silicates such as the viscosity, that the Reynolds number of liquid silicates in metal-silicate liquid two-phase systems will influence the boundary layer thickness to a greater extent than in the liquid metals and alloys, mainly because of the higher viscosity of the silicate. 

The comparison of the magnitude of the two resistances clearly indicates whether tire metal or the slag mass transfer is rate-determining. A value for the ratio of the boundary layer thicknesses can be obtained from the Sherwood number, which is related to the Reynolds number and the Schmidt number, defined by... 

When the two liquid phases are in relative motion, the mass transfer coefficients in eidrer phase must be related to die dynamical properties of the liquids. The boundary layer thicknesses are related to the Reynolds number, and the diffusive Uansfer to the Schmidt number. Another complication is that such a boundaty cannot in many circumstances be regarded as a simple planar interface, but eddies of material are U ansported to the interface from the bulk of each liquid which change the concenuation profile normal to the interface. In the simple isothermal model there is no need to take account of this fact, but in most indusuial chcumstances the two liquids are not in an isothermal system, but in one in which there is a temperature gradient. The simple stationary mass U ansfer model must therefore be replaced by an eddy mass U ansfer which takes account of this surface replenishment. 

Thus the driving force for fuming is approximately equal to that for free evaporation. Using dre experimental data, and the normal expression for mass transfer across a boundary layer, it is concluded that the boundary layer thickness which would account for this rate should be about 2 x 10 cm (Turkdogan et al., 1963). 

Examples of this procedure for dilute solutions of copper, silicon and aluminium shows the widely different behaviour of these elements. The vapour pressures of the pure metals are 1.14 x 10, 8.63 x 10 and 1.51 x 10 amios at 1873 K, and the activity coefficients in solution in liquid iron are 8.0, 7 X 10 and 3 X 10 respectively. There are therefore two elements of relatively high and similar vapour pressures, Cu and Al, and two elements of approximately equal activity coefficients but widely differing vapour pressures. Si and Al. The right-hand side of the depletion equation has the values 1.89, 1.88 X 10- , and 1.44 X 10 respectively, and we may conclude that there will be depletion of copper only, widr insignificant evaporation of silicon and aluminium. The data for the boundaty layer were taken as 5 x lO cm s for the diffusion coefficient, and 10 cm for the boundary layer thickness in liquid iron. 

The majority of investigators consider it permissible and convenient to use, when calculating the boundary layer thickness, the relationships describing the concentration dependence of viscosity in the high and medium concentration range (basically Mooney s equation) [67 — 71]. 

When a fluid flowing at a uniform velocity enters a pipe, the layers of fluid adjacent to the walls are slowed down as they are on a plane surface and a boundary layer forms at the entrance. This builds up in thickness as the fluid passes into the pipe. At some distance downstream from the entrance, the boundary layer thickness equals the pipe radius, after which conditions remain constant and fully developed flow exists. If the flow in the boundary layers is streamline where they meet, laminar flow exists in the pipe. If the transition has already taken place before they meet, turbulent flow will persist in the... 

The equilibrium is considered of an element of fluid bounded by the planes 1 -2 and 3 4 at distances x and x + dx respectively from the leading edge the element is of length l in the direction of flow and is of depth w in the direction perpendicular to the plane 1 -2-3-4. The distance l is greater than the boundary layer thickness (Figure 11.5), and conditions are constant over the width w. The velocities and forces in the X-direction are now considered. 

Thus, the shear stress is expressed as a function of the boundary layer thickness S and it is therefore implicitly assumed that a certain velocity profile exists in the fluid. As a first assumption, it may be assumed that a simple power relation exists between the velocity and the distance from the surface in the boundary layer, or ... 

If the velocity profile is the same for all stream velocities, the shear stress must be defined by specifying the velocity ux at any distance y from the surface. The boundary layer thickness, determined by the velocity profile, is then no longer an independent variable so that the index of < in equation 11.25 must be zero or ... 

The velocity of the fluid may be assumed to obey the Prandtl one seventh power law, given by equation 11.26. If the boundary layer thickness S is replaced by the pipe radius r, this is then given by ... 

The procedure here is similar to that adopted previously. A heat balance, as opposed to a momentum balance, is taken over an element which extends beyond the limits of both the velocity and thermal boundary layers. In this way, any fluid entering or leaving the element through the face distant from the surface is at the stream velocity u and stream temperature 0S. A heat balance is made therefore on the element shown in Figure 11.10 in which the length l is greater than the velocity boundary layer thickness S and the thermal boundary layer thickness t. 

If equation 12.29 is applied to the outer edge of the boundary layer when y = S (boundary layer thickness) and ux = us (the stream velocity), then ... 

Obtain the momentum equation for an element of boundary layer. If the velocity profile in the laminar region may be represented approximately by a sine function, calculate the boundary-layer thickness in terms of distance from the leading edge of the surface. 

For a Prandtl number, Pr. less than unity, the ratio of the temperature to the velocity boundary layer thickness is equal to Pr 1Work out the thermal thickness in terms of the thickness of the velocity boundary layer... 

Explain why it is necessary to use concepts, such as the displacement thickness and the momentum thickness, for a boundary layer in order to obtain a boundary layer thickness which is largely independent of the approximation used for the velocity profile in the neighbourhood of the surface. 

Obtain the boundary layer thickness and its displacement thickness as a function of the distance from the leading edge of Ihe surface, when the velocity profile is expressed as a sine function. 

The value of E is insensitive to small changes in ocean temperature but is quite sensitive to wind speed over the sea surface (boundary layer thickness, wave action, and bubble formation are functions of wind speed). Therefore changes in surface wind speed accompanying a climate change could affect rates of air-sea CO2 exchange. 

Fig. 10-18 A schematic of a stagnant boundary-layer gas exchange model. Cg = gas concentration at the liquid side of the interface Q = gas concentration at the base of the stagnant boundary layer Znim = stagnant boundary layer thickness.

FIGURE 10 In vitro rates of release of progesterone from PCL films, illustrating their dependence on the film thickness and drug load. The deviation from (time)l/2 kinetics reflects the contribution of an aqueous boundary layer. The solid lines were calculated assuming an aqueous boundary layer thickness of 19 ym. (From Ref. 68.)... 

With turbulent channel flow the shear rate near the wall is even higher than with laminar flow. Thus, for example, (du/dy) ju = 0.0395 Re u/D is vaHd for turbulent pipe flow with a hydraulically smooth wall. The conditions in this case are even less favourable for uniform stress on particles, as the layer flowing near the wall (boundary layer thickness 6), in which a substantial change in velocity occurs, decreases with increasing Reynolds number according to 6/D = 25 Re", and is very small. Considering that the channel has to be large in comparison with the particles D >dp,so that there is no interference with flow, e.g. at Re = 2300 and D = 10 dp the related boundary layer thickness becomes only approx. 29% of the particle diameter. It shows that even at Re = 2300 no defined stress can be exerted and therefore channels are not suitable model reactors. 

Eor the range involved here, which is 5 x 10 < Re < 7 x 10, Eq. (22) gives an average boundary layer thickness of 6 2.8 mm, which is much greater than the... 

This can be further integrated from the wall to the boundary layer thickness y = 8, where the component is at the bulk concentration Cj,. Substituting / = - o and k = D/o, the mass-transfer coefficient yields the stagnant film model [Brian, Desalination by Reverse Osmosis, Merten (ed.), M.I.T. Press, Cambridge, Mass., 1966, pp. 161-292] ... 

The speed of agitation governs the rate. The boundary layer thickness decreases at high speeds. A physical process such as the dissolution of an ionic salt in plain water is associated with an activation energy of the order of less than about 5 kcal mol-1. The influence of temperature on physical processes is less pronounced than that from agitation. 

Karlsson, J. P. Artursson, P., A method for the determination of cellular permeability coefficients and aqueous boundary layer thickness in monolayers of intestinal epithelial (Caco-2) cells grown in permeable filter chambers, Int. J. Pharm. 7, 55-64 (1991). 

---