The laminar sub-layer

If at a distance a from the leading edge the laminar sub-layer is of thickness 5 and the total thickness of the boundary layer is 8, the properties of the laminar sub-layer can be found by equating the shear stress at the surface as given by the Blasius equation (11.23) to that obtained from the velocity gradient near the surface. 

It has been noted that the shear stress and hence the velocity gradient are almost constant near the surface. Since the laminar sub-layer is very thin, the velocity gradient within it may therefore be taken as constant. 

Equating this to the value obtained from equation 11.23 gives  

The velocity at the inner edge of the turbulent region must also be given by the equation for the velocity distribution in the turbulent region. 

Thus Sh x x(K 1 that is, b it increases very slowly as x increases. Further. /, a u 0 and therefore decreases rapidly as the velocity is increased, and heat and mass transfer coefficients are therefore considerably influenced by the velocity. 

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This relation holds provided that the one-seventh power law may be assumed to apply over the whole of the cross-section of the pipe. This is strictly the case only at high Reynolds numbers when the thickness of the laminar sub-layer is small. By combining equations 3.59 and 3.63, the velocity profile is given by ... 

Thus for turbulent flow at high Reynolds numbers, where the thickness of the laminar sub-layer may be neglected, a 1. 

When the flow in the boundary layer is turbulent, streamline flow persists in a thin region close to the surface called the laminar sub-layer. This region is of particular importance because, in heat or mass transfer, it is where the greater part of the resistance to transfer lies. High heat and mass transfer rates therefore depend on the laminar sublayer being thin. Separating the laminar sub-layer from the turbulent part of the boundary... 

The relation between the mean velocity and the velocity at the axis is derived using this expression in Chapter 3. There, the mean velocity u is shown to be 0.82 times the velocity us at the axis, although in this calculation the thickness of the laminar sub-layer was neglected and the Prandtl velocity distribution assumed to apply over the whole cross-section. The result therefore is strictly applicable only at very high Reynolds numbers where the thickness of the laminar sub-layer is vety small. At lower Reynolds numbers the mean velocity will be rather less than 0.82 times the velocity at the axis. 

The expressions for the shear stress at the walls, the thickness of the laminar sub-layer, and the velocity at the outer edge of the laminar sub-layer may be applied to the turbulent flow of a fluid in a pipe. It is convenient to express these relations in terms of the mean velocity in the pipe, the pipe diameter, and the Reynolds group with respect to the mean velocity and diameter. 

The discrepancy between the coefficients in equations 11.45 and 11,46 is attributable to the fact that the effect of the curvature of the pipe wall has not been taken into account in applying the equation for flow over a plane surface to flow through a pipe. In addition, it takes no account of the existence of the laminar sub-layer at the walls. 

The thickness of the laminar sub-layer is therefore almost inversely proportional to the Reynolds number, and hence to the velocity. 

Calculate the thickness of the laminar sub-layer when benzene flows through a pipe 50 mm in diameter at 2 1/s. What is the velocity of the benzene at the edge of the laminar sub-layer Assume that fully developed flow exists within the pipe and that for benzene, p — 870 kg/m3 and p = 0.7 mN s/m2. 

At the surface, the laminar sub-layer, in which the only motion at right angles to the surface is due to molecular diffusion. 

In addition to momentum, both heat and mass can be transferred either by molecular diffusion alone or by molecular diffusion combined with eddy diffusion. Because the effects of eddy diffusion are generally far greater than those of the molecular diffusion, the main resistance to transfer will lie in the regions where only molecular diffusion is occurring. Thus the main resistance to the flow of heat or mass to a surface lies within the laminar sub-layer. It is shown in Chapter 11 that the thickness of the laminar sub-layer is almost inversely proportional to the Reynolds number for fully developed turbulent flow in a pipe. Thus the heat and mass transfer coefficients are much higher at high Reynolds numbers. 

Equation 12.35 applies only in those regions where eddy transfer dominates, i.e. outside both the laminar sub-layer and the buffer layer (see below). 

In the laminar sub-layer, turbulence has died out and momentum transfer is attributable solely to viscous shear. Because the layer is thin, the velocity gradient is approximately linear and equal to Uj,/Sb where m is the velocity at the outer edge of a laminar sub-layer of thickness <5 (see Chapter ll). 

The buffer layer covers the intermediate range 5 < >,+ < 30. A straight line may be drawn to connect the curve for the laminar sub-layer (equation 12.40) at y+ = 5 with the line... 

Similarly, the velocity ub at the edge of the laminar sub-layer is given by ... 

When the thickness of the laminar sub-layer is large compared with the height of the obstructions, the pipe behaves as a smooth pipe (when e < <5 /3). Since the thickness of the laminar sub-layer decreases as the Reynolds number is increased, a surface which is hydrodynamically smooth at low Reynolds numbers may behave as a rough surface at higher values. This explains the shapes of the curves obtained for plotted against Reynolds number (Figure 3.7). The curves, for all but the roughest of pipes, follow the... 

The original Reynolds analogy involves a number of simplifying assumptions which are justifiable only in a limited range of conditions. Thus it was assumed that fluid was transferred from outside the boundary layer to the surface without mixing with the intervening fluid, that it was brought to rest at the surface, and that thermal equilibrium was established. Various modifications have been made to this simple theory to take account of the existence of the laminar sub-layer and the buffer layer close to the surface. 

Taylor(4) and Prandtl(8 9) allowed for the existence of the laminar sub-layer but ignored the existence of the buffer layer in their treatment and assumed that the simple Reynolds analogy was applicable to the transfer of heal and momentum from the main stream to the edge of the laminar sub-layer of thickness <5. Transfer through the laminar sub-layer was then presumed to be attributable solely to molecular motion. 

Tf auH and b8s are the velocity and temperature, respectively, at the edge of the laminar sub-layer (see Figure 12.5), applying the Reynolds analogy (equation 12,99) for transfer across the turbulent region ... 

The quantity a, which is the ratio of the velocity at the edge of the laminar sub-layer to the stream velocity, was evaluated in Chapter 11 in terms of the Reynolds number for flow over the surface. For flow over a plane surface, from Chapter 11 ... 

It is thus seen that by taking account of the existence of the laminar sub-layer, correction factors are introduced into the simple Reynolds analogy. 

Equation 12.121 is a modified form of the Lewis Relation, which takes into account the resistance to heat and mass transfer in the laminar sub-layer. 

The method is based on the calculation of the total temperature difference between the fluid and the surface, by adding the components attributable to the laminar sub-layer, the buffer layer and the turbulent region. In the steady state, the heat flux (<70) normal to the surface will be constant if the effects of curvature are neglected. 

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