Velocity boundary layers

MAISEL and SHERWOOD(46) also carried out experiments in a wind tunnel in which water was evaporated from a wet porous surface preceded by a dry surface of length Lq. Thus, a velocity boundary layer had become established in the air before it came into... 

Thus, a velocity boundary layer and a thermal boundary layer may develop simultaneously. If the physical properties of the fluid do not change significantly over the temperature range to which the fluid is subjected, the velocity boundary layer will not be affected by die heat transfer process. If physical properties are altered, there will be an interactive effect between the momentum and heat transfer processes, leading to a comparatively complex situation in which numerical methods of solution will be necessary. 

In general, the thermal boundary layer will not correspond with the velocity boundary layer. In the following treatment, the simplest non-interacting case is considered with physical properties assumed to be constant. The stream temperature is taken as constant In the first case, the wall temperature is also taken as a constant, and then by choosing the temperature scale so that the wall temperature is zero, the boundary conditions are similar to those for momentum transfer. 

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. 

The flow of fluid over a plane surface, heated at distances greater than. to from the leading edge, is now considered. As shown in Figure 11.11 the velocity boundary layer starts at the leading edge and the thermal boundary layer at a distance o from it. If the temperature of the heated portion of the plate remains constant, this may be taken as the datum temperature. It is assumed that the temperature at a distance y from the surface may be represented by a polynomial of the form ... 

Thus the conditions for the thermal boundary layer, with respect to temperature, are the same as those for the velocity boundary layer with respect to velocity. Then, if the thickness of the thermal boundary layer is 5 the temperature distribution is given by ... 

It is assumed that the velocity boundary layer is everywhere thicker than the thermal boundary layer, so that 8 > 8, (Figure 11.11). Thus the velocity distribution everywhere within the thermal boundary layer is given by equation 11.12. The implications of this assumption are discussed later. 

Explain the concepts of momentum thickness" and displacement thickness for the boundary layer formed during flow over a plane surface. Develop a similar concept to displacement thickness in relation to heat flux across the surface for laminar flow and heat transfer by thermal conduction, for the case where the surface has a constant temperature and the thermal boundary layer is always thinner than the velocity boundary layer. Obtain an expression for this thermal thickness in terms of the thicknesses of the velocity and temperature boundary layers. 

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... 

Figure 17.2 shows SiFLj and SiH2 species profiles for three different surface temperatures. In all cases there is a boundary layer near the surface, which is about 0.75 cm thick. The boundary becomes a bit thicker at the higher temperatures, owing to the temperature-dependent increases in viscosity, thermal conductivity, and diffusion coefficients. The temperature and velocity boundary layers (not illustrated) are approximately the same thickness as the species boundary layers. 

It can be seen that the expression for the average Nusselt number for Pr 1 is closer in form to the case where Pr — oo, than the case where Pr —> 0. The reason for this is that in natural convection, the driving force is caused by the temperature gradients, and thus defined by the thermal boundary layer. When Pr 1 and when Pr — co, the thermal boundary layer is thicker than the velocity boundary layer. Hence, the behavior of the Nusselt number would be similar in form for both cases. When Pr — 0, the behavior of the kinematic viscosity relative to the thermal diffusivity is going to be different from that of the other two cases. In addition, the right-hand side of the expression for Pr — 0 is independent of o, as one would expect for this case where the effects of the kinematic viscosity are very small or negligible. 

Now, in general, the effects of viscosity and heat transfer do not extend to the same distance from the surface. For this reason, it is convenient to define both a velocity boundary layer thickness and a thermal or temperature boundary layer thickness as shown in Fig. 2.14. The velocity boundary layer thickness is a measure of the distance from the surface at which viscous effects cease to be important while the thermal boundary layer thickness is a measure of the distance from the wall at which heat transfer effects cease to be important. 

As with the velocity boundary layer, the thermal boundary layer is assumed to have a definite thickness, dr, and outside this boundary layer the temperature is assumed to be constant. 

It will be seen from the results given in Fig. 3.5 that, if the thermal boundary lay r thickness, is defined in a similar way to the velocity boundary layer thickness as the distance from the wall at which 0 becomes equal to 0.99, i.e.. reaches to within 1% of its free stream value, then ... 

Now the analysis of the velocity boundary layer on a flat plate presented above... 

Air at a temperature of 10°C flows upward over a 0.25 m high vertical plate which is kept at a uniform surface temperature of 40°C. Plot the variation of the velocity boundary layer thickness and local heat transfer rate along the plate for air velocities of between 0.2 and 1.5 m/s. Assume two-dimensional flow. 

If the Darcy assumptions are used then with forced convective flow over a surface in a porous medium, because the velocity is not assumed to be 0 at the surface, there is no velocity change induced by viscosity near the surface and there is therefore no velocity boundary layer in the flow over the surface. There will, however, be a region adjacent to the surface in which heat transfer is important and in which there are significant temperature changes in the direction normal to the surface. Under many circumstances, the normal distance over which such significant temperature changes occur is relatively small, i.e., a thermal boundary layer can be assumed to exist around the surface as shown in Fig. 10.9, the ratio of the boundary layer thickness, 67, to the size of the body as measured by some dimension, L, being small [15],[16]. 

As discussed in Chapters 2 and 3, in the integral method it is assumed that the boundary layer has a definite thickness and the overall or integrated momentum and thermal energy balances across the boundary layer are considered. In the case of flow over a body in a porous medium, if the Darcy assumptions are used, there is, as discussed before, no velocity boundary layer, the velocity parallel to the surface near the surface being essentially equal to the surface velocity given by the potential flow solution. For flow over a body in a porous medium, therefore, only the energy integral equation need be considered. This equation was shown in Chapter 2 to be ... 

In high-velocity boundary layers substantial temperature gradients may occur, and there will be correspondingly large property variations across the boundary layer. The constant-property heat-transfer equations may still be used if the properties are introduced at a reference temperature T as recommended by Eckert ... 

The region of the flow above the plate bounded by 5 in which the effects of the viscous shearing forces caused by fluid viscosity are fell is called the velocity boundary layer. The boundary layer iliickiiess, 8, is typically defined as the distance) from the. surface at which u = 0.99F. 

We have seen that a velocity boundary layer develops when a fluid flows over a surface as a result of the fluid layer adjacent to the surface assuming the surface velocity (i.e., zero velocity relative to the surface). Also, we defined the velocity boundary layer as the region in which the fluid velocity varies from zero to 0.99V. Likewise, a thermal boundary layer develops when a fluid at a specified temperature flows over a surface that is at a different temperature, as shown in Fig. 6-15. 

The Prandtl numbers of gases are about 1, which indicates that both momentum and heat dissipate tlirough tine fluid at about the same rale. Heat diffuses very quickly in liquid metals (Pr < 1) and very slowly in oils (Pr > 1) relative to momentum. Consequently the thermal boundary layer i.s much thicker for liquid melals and much thinner for oils relative to the velocity boundary layer. 

Differential control volume used in the derivation of mass balance in velocity boundary layer in two-dimensional flow over a surface. 

The differential forms of the equations of motion in the velocity boundary layer are obtained by applying Newton s second law of motion to a differential conlrol volume element in the boundary layer. Newton s second law is an expression for momenluin balance and can be slated as the net force acting on the control volume is equal to the mass times the acceleration of the fluid element wilhht the control volume, which is also equal to the net rate of momentum outflow from the control volume. 

Recall that we defined the boundary layer thickness as the distance from the surface for which utV 0.99. We observe from Table 6-3 that the value of tj corresponding to ulV = 0.99 is = 4.91. Substituting rj = 4.9J andy = B into the definition of the similarity variable (Eq. 6-43) gives 4.91 = 5 Vv/vx. Then the velocity boundary layer thickness becomes... 

Temperature profiles for flow over an isothermal flat plate are similar, just like the velocity profiles, and thus we expect a similarity solution for temperature to exist. Further, the thickness of the thermal boundar y layer is proportional to /i. T/V,just like the thickness of the velocity boundary layer, and thus the similarity variable is also t), and 0 = 6(ri). Using thechain rule and substituting the It and tt e.xpres ions from Eqs. 6-46 and 6—47 into the energy equation gives... 

C What fluid property is responsible for the development of the velocity boundary layer For what kind of fluids will there be no velocity boundary layei on a flat plate ... 

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