Boundary-layer flow curved

The responses of this system to ideal step and pulse inputs are shown in Figure 11.3. Because the flow patterns in real tubular reactors will always involve some axial mixing and boundary layer flow near the walls of the vessels, they will distort the response curves for the ideal plug flow reactor. Consequently, the responses of a real tubular reactor to these inputs may look like those shown in Figure 11.3. 

As Re increases further and vortices are shed, the local rate of mass transfer aft of separation should oscillate. Although no measurements have been made for spheres, mass transfer oscillations at the shedding frequency have been observed for cylinders (B9, D6, SI2). At higher Re the forward portion of the sphere approaches boundary layer flow while aft of separation the flow is complex as discussed above. Figure 5.17 shows experimental values of the local Nusselt number Nuj c for heat transfer to air at high Re. The vertical lines on each curve indicate the values of the separation angle. It is clear that the transfer rate at the rear of the sphere increases more rapidly than that at the front and that even at very high Re the minimum Nuj. occurs aft of separation. Also shown in Fig. 5.17 is the thin concentration boundary layer... 

Coordinate system used for boundary layer flow over a curved surface. 

The separation efficiency of a hydrocyclone has a character of probability. This is to do with the probability of the position of the different particles in the entrance to the cyclone, their chances of separation into the boundary layer flow and the general probability character of turbulent flow. Coarse particles are always more likely to be separated than fine particles. Effectively, the hydrocyclone processes the feed solids by an efficiency curve called grade efficiency , which is a percentage increasing with particle size (see chapter 3 for more details about grade efficiency). Figure 6.6 shows the process schematically the solids in the feed enter the cyclone and are... 

Similarly, in a spherical tank, the primary convection currents are believed to be boundary layer flows at and round the inner curved walls. 

For LNG and LPG, the boundary layer flow at the wall absorbs all the heat flow entering the liquid. Furthermore, for a container or tank which has a liquid depth/ diameter ratio greater than about 0.5, the heat flow through the base is absorbed convectively by a boundary layer flow across the base which is continuous with the vertical wall boundary layer flow via a boundary layer suction process. The same is believed to be true for spherical containers and tanks, i.e. The boundary layer flow follows the vertical, inner curve of the tank wall. 

Equations (5,61) and (5.62) can be used to derive a pressure potential equation applicable to thin-layer flow between curved surfaces using the following procedure. In a thin-layer flow, the following velocity boundary conditions are prescribed ... 

These equations were derived for flow over a plane surface. They may be applied to flow over a curved surface provided that the boundary layer thickness remains small compared to the radius of curvature of the surface. When applied to flow over a curved surface, x is measured along the surface and y is measured normal to it at all points as shown in Fig. 2.15, i.e., body-fitted coordinates are used. 

Figure 7-6. Schematic illustration of originally nonturbulent air (straight anrows in upwind side on left) flowing over the top of a flat leaf, indicating the laminar sublayer (shorter straight anrows), the turbulent region (curved arrows), and the effective boundary layer thickness, 5bl. The length of an arrow indicates the relative speed, and the curvature indicates the local direction of air movement. A similar airflow pattern occurs on the lower leaf surface.

Chaplcr 6, we considered the general and theoretical aspects ot forced convection, with emphasis on differential formulation and analytical solutions. In this chapter we consider the practical-aspects of forced convection to or from flat or curved surfaces subjected to external flow, characterized by the freely growing boundary layers surrounded by a free How region that involves no velocity and temperature gradients. 

The range of liquid flow rates used to generate the date in Figure 23.10 is similar to the blood flow rates used in clinical practice. Figure 23.10 shows that for Reynolds numbers between 5 and 10, the slope of the friction factor versus Reynolds number curve changes suggesting the onset of boundary layer separation. Boundary layer separation will lead to mixing of the blood and a decrease in the blood side mass-transfer resistance. 

Standard k-s Simplest model to represent variation of turbulence length and velocity scales Robust and economical Excellent performance for many industrial flows The most widely validated model More expensive than zero equation models Assumes isotropic eddy viscosity Performs poorly for — some unconfined flows — rotating flows — non-circular ducts — curved boundary layers... 

In the following we will assume the velocities, temperatures and concentrations in the outer region to be known, and consider a steady-state, two-dimensional flow. The body forces are negligible. Flow along a curved wall can be taken to be two-dimensional as long as the radius of curvature of the wall is much bigger compared to the thickness of the boundary layer. The curvature is then insignificant for the thin boundary layer, and it develops just as if it was on a flat wall. The curvature of the wall is merely of influence on the outer flow and its pressure distribution. 

As the previous illustrations showed, the heat and mass transfer coefficients for simple flows over a body, such as those over flat or slightly curved plates, can be calculated exactly using the boundary layer equations. In flows where detachment occurs, for example around cylinders, spheres or other bodies, the heat and mass transfer coefficients are very difficult if not impossible to calculate and so can only be determined by experiments. In terms of practical applications the calculated or measured results have been described by empirical correlations of the type Nu = f(Re,Pr), some of which have already been discussed. These are summarised in the following along with some of the more frequently used correlations. All the correlations are also valid for mass transfer. This merely requires the Nusselt to be replaced by the Sherwood number and the Prandtl by the Schmidt number. 

The results of the numerical solution of the conjugation boundary-value problem for the mass exchange (3.106), (3.107) are shown in Fig. 3.20. The initially homogeneously cold air gets the latent heat from the droplet layer and becomes warmer and warmer from cross section 1 to 5, etc. (family (I) of curves in Fig. 3.20,A). In turn, the droplets become cooler their temperature is reduced especially intensively at the EPR entrance (the dashed curve 0) but less and less intensively in subsequent cross sections (the family of curves (II)). The curves TE(z) and t(z) attract each other and meet at the theoretical infinity z —> -oo. The complex relation between the two profiles is testified by the local maxima on TE(z) at early cross sections (curves 1 and 2) and by negative values of the mass flow (3.97) presented in Fig. 3.20,B. A dilative thermal boundary layer grows over the droplet EPR. Its width dE(x) grows theoretically to infinity over the EPR, but the internal portions of all the variables tend to a certain final position within the EPR, 0 < z < 1. 

In the case where a gas flows over the front portion of a curved surface or a convex curve surface, the flow outside the boundary layer accelerates. The boundary layer over most of the front portion remains fairly thin and has a uniform streamline pattern over this portion. According to the Bernoulli equation [48], accelerated flow results in a pressure decrease in the vicinity of the front portion area. In this zone, the pressure gradient is positive, i.e. the direction of pressure gradient is the same as the flow direction. The positive pressure gradient is helpful to push the flow within the boundary layer forwards. The variations of velocity and pressure are expressed as... 

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