Boundary layer thickness velocity

At 700°C and 1 atm this leads to a diffusion constant of 0.81 cm /s The flow field around a superheater tube is very complex involving both laminar and turbulent boundary layers and the estimation of the local boundary layer thicknesses (velocity, diffusion and thermal boundary layers) around the tube requires computer simulations with computational fluid dynamic (CFD) software packages. However, for this rough analysis an average value of the thermal boundary layer thickness around the tube is enough and can be estimated if the average Nusselt number around the tube is known... 

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. 

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. 

Figure 5. Exact (numerical solution, continuous line) and linearised (equation (24), dotted line) velocity profile (i.e. vy of the fluid at different distances x from the surface) at y = 10-5 m in the case of laminar flow parallel to an active plane (Section 4.1). Parameters Dt = 10 9m2 s-1, v = 10-3ms-1, and v = 10-6m2s-1. The hydrodynamic boundary layer thickness (<50 = 5 x 10 4 m), equation (26), where 99% of v is reached is shown with a horizontal double arrow line. For comparison, the normalised concentration profile of species i, ct/ithe linear profile of the diffusion layer approach (continuous line) and its thickness (<5, = 3 x 10 5m, equation (34)) have been added. Notice that the linearisation of the exact velocity profile requires that <5,

Figure 1.25 shows the boundary layer that develops over a flat plate placed in, and aligned parallel to, the fluid having a uniform velocity upstream of the plate. Flow over the wall of a pipe or tube is similar but eventually the boundary layer reaches the centre-line. Although most of the change in the velocity component vx parallel to the wall takes place over a short distance from the wall, it does continue to rise and tends gradually to the value vx in the fluid distant from the wall (the free stream). Consequently, if a boundary layer thickness is to be defined it has to be done in some arbitrary but useful way. The normal definition of the boundary layer thickness is that it is the distance from the solid boundary to the location where vx has risen to 99 per cent of the free stream velocity v . The locus of such points is shown in Figure 1.25. It should be appreciated that this is a time averaged distance the thickness of the boundary layer fluctuates owing to the velocity fluctuations.

In terms of hydrodynamics, the boundary layer thickness is measured from the solid surface (in the direction perpendicular to a particle s surface, for instance) to an arbitrarily chosen point, e.g., where the velocity is 90-99% of the stream velocity or the bulk flow ((590 or (599, respectively). Thus, the breadth of the boundary layer depends ad definitionem on the selection of the reference point and includes the laminar boundary layer as well as possibly a portion of a turbulent boundary layer. 

The Reynolds number is the ratio of inertial to viscous forces and depends on the fluid properties, bulk velocity, and boundary layer thickness. Turbulence characteristics vary with Reynolds number in boundary layers [40], Thus, variation in the contributing factors for the Reynolds number ultimately influences the turbulent mixing and plume structure. Further, the fluid environment, air or water, affects both the Reynolds number and the molecular diffusivity of the chemical compounds. 

Figure 4-19 Plagioclase phase diagram and plagioclase melting Figure 4-20 Free falling velocity of a mantle xenolith in a basalt Figure 4-21 Sketch of boundary layer, and boundary layer thickness Figure 4-22 MgO diffusion profile in olivine and in melt during olivine growth...

Momentum boundary layer calculations are useful to estimate the skin friction on a number of objects, such as on a ship hull, airplane fuselage and wings, a water surface, and a terrestrial surface. Once we know the boundary layer thickness, occurring where the velocity is 99% of the free-stream velocity, skin friction coefficient and the skin friction drag on the solid surface can be calculated. Estimate the laminar boundary layer thickness of a 1-m-long, thin flat plate moving through a calm atmosphere at 20 m/s. 

Equation (E4.4.1) is a nonlinear partial differential equation, because of the velocity u that appears in front of the velocity gradient du/dx. The boundary layer thickness is generally defined as the distance from the plate where the momentum reaches 99% of the free-stream momentum. We will assign (Blasius, 1908)... 

Laminar boundary layer theory assumes that a uniform flow (V = constant) approaches a flat plate. A laminar flow region develops near the plate where the thickness of the laminar boundary layer increases with thickness along the plate, as developed in Example 4.2. If we assign 5 to be the boundary layer thickness, or the distance from the plate where the velocity is equal to 0.99 times the velocity that approached the plate, and 5c to be the concentration boundary layer thickness, then we can see that both 5 and 5c are functions of distance, x, from the leading edge, as shown in Figure 8.11. 

The increase of the wind velocity to mi0 = 10 m s-1 reduces vjr of formaldehyde by about 8. This results from the significantly reduced boundary layer thickness which makes diffusion time tw small. In contrast, wind velocity does not affect vjr of acetaldehyde very much since it is already close to 1. 

A principal assumption for similarity is that there exists a viscous boundary layer in which the temperature and species composition depend on only one independent variable. The velocity distribution, however, may be two- or even three-dimensional, although in a very special way that requires some scaled velocities to have only one-dimensional content. The fact that there is only one independent variable implies an infinite domain in directions orthogonal to the remaining independent variable. Of course, no real problems have infinite extent. Therefore to be of practical value, it is important that there be real situations for which the assumptions are sufficiently valid. Essentially the assumptions are valid in situations where the viscous boundary-layer thickness is small relative to the lateral extent of the problem. There will always be regions where edge effects interrupt the similarity. The following section provides some physical evidence that supports the notion that there are situations in which the stagnation-flow assumptions are valid. 

Solution to the nondimensional axisymmetric stagnation-flow problem is plotted in Fig. 6.3. Since the viscous boundary layer merges asymptotically into the inviscid potential flow, there is not a distinct edge of the boundary layer. By convention, the boundary-layer thickness is defined as the point at which the radial velocity comes to 99% of its potential-flow value. From Fig. 6.3 it is apparent that the boundary-layer thickness S is approximately z 2. In addition to the boundary-layer thickness, a displacement thickness can be defined. The displacement thickness is the distance that the potential-flow field appears to be displaced from the surface due to the viscous boundary layer. If there were no viscous boundary layer (i.e., the inviscid flow persisted right to the surface), then the axial velocity profile would have a constant slope du/dz = —2. As shown in Fig. 6.3, projecting the constant axial-velocity slope to the surface obtains an intercept of u = 0 at approximately z = 0.55. Since the inviscid flow would have to come to zero velocity at the surface, z = 0.55 is the distance that the potential flow is displaced due to the viscous boundary layer. Otherwise, the potential flow is unaltered by the boundary layer. 

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