Leading edge conditions

Many of the earlier studies of mass transfer involved measuring the rate of vaporisation of liquids by passing a turbulent air stream over a liquid surface. In addition, some investigations have been carried out in the absence of air flow, under what have been termed still air conditions. Most of these experiments have been carried out in some form of wind tunnel where the rate of flow of air and its temperature and humidity could be controlled and measured. In these experiments it was found to be important to keep the surface of the liquid level with the rim of the pan in order to avoid the generation of eddies at the leading edge. 

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. 

Derive the momentum equation for the flow of a fluid over a plane surface for conditions where the pressure gradient along the surface is negligible. By assuming a sine function for the variation of velocity with distance from the surface (within the boundary layer) for streamline flow, obtain an expression for the boundary layer thickness as a function of distance from the leading edge of the surface. 

Where the source is limited or ceases, capillary spreading eventually slows until further migration is limited and equilibrium is reached. This stable condition is attained when the leading edge of the laterally spreading light LNAPL fails to be... 

The inverse of the tunnel experiments discussed is the propagation of a flame across a layer of a liquid fuel that has a low flash point temperature. The stratified conditions discussed previously described the layered fuel vapor-air mixture ratios. Under these conditions the propagation rates were found to be 4-5 times the laminar flame speed. This somewhat increased rate compared to the other analytical results is apparently due to diffusion of air to the flame front behind the parabolic leading edge of the propagating flame [41],... 

In electrochemical reactors, the externally imposed velocity is often low. Therefore, natural convection can exert a substantial influence. As an example, let us consider a vertical parallel plate reactor in which the electrodes are separated by a distance d and let us assume that the electrodes are sufficiently distant from the reactor inlet for the forced laminar flow to be fully developed. Since the reaction occurs only at the electrodes, the concentration profile begins to develop at the leading edges of the electrodes. The thickness of the concentration boundary layer along the length of the electrode is assumed to be much smaller than the distance d between the plates, a condition that is usually satisfied in practice. 

For relatively minor species, such as the CO mass fractions shown in Fig. 17.20, there are somewhat larger differences between the Navier-Stokes and boundary-layer models. Under these flow conditions the CO mass-fraction peaks just near the leading edge of the active catalyst. As the CO desorbs from the initial region of the catalyst, the shapes of the CO contours show less classical boundary-layer development behavior, especially at low Reynolds number. Nevertheless, the agreement between the two models is still quite good. 

Some consideration must be given to the conditions existing along the initial i = 1 line which were assumed to be known in the above discussion. The actual conditions will depend on the nature of the problem. For flow over a flat plate, because the boundary layer equations are parabolic in form, the use of these equations requires that the plate have no effect on the flow upstream of the plate. Hence, in this case, the variables will have their freestream values at all the nodal points (except at the point which lies on the surface) on the initial line which is coincident with the leading edge. At the nodal point on the surface, the known conditions at the surface must apply. This is illustrated in Fig. 3.24. 

This is actually considerably in error because V, in fact, has its maximum value at the leading edge. However, the effects of this erroneous assumption quickly die out and if the initial spacing of the i-lines is chc -en to be small, it has a negligible overall effect on the solution. These initial conditions are incorporated into the computer program discussed below. 

The program assumes the flow is turbulent from the leading edge and that 62 = 0 when x = 0. The program can easily be modified to use a laminar boundary layer equation solution procedure to provide initial conditions for the turbulent boundary layer solution which would then be started at some assumed transition point. 

---