The velocity boundary layer

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. 

We will introduce so-called boundary layer coordinates, Fig. 3.13, in which the coordinate x1 = x is chosen to be along the surface of the body and x.2 = y as perpendicular to it. We will presume an initial velocity wa(y) its integral mean value will be wm. 

3 Convective heat and mass transfer. Single phase flow 

Therefore, gases at moderate velocities, without large changes in their temperature, may be assumed to be incompressible, dg/dt — 0, and their flow can be approximately described by the continuity and momentum equations for incompressible flow. 

Under these presumptions, the continuity and Navier-Stokes equations (3.93) 

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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... 

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. 

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. 

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 ... 

Consider the parallel flow of a fluid over a flat plate of length L in the flow direction, as shown in Fig. 7-6. The. t-coordinate is measured along the plate surface from the leading edge in the direction of the flow. The fluid approaches the plate in the, T-direction with a uniform velocity V and temperature T. The flow in the velocity boundary layers starts out as laminar, but if the plate is sufficiently long, the flow become.s turbulent at a distance. r r from the leading edge where the Reynolds number reaches its criiical value for transition. 

Liquid metals such as mercury have high thermal conductivities, and are commonly used in applications that require high heat transfer rates. However, they have very small Prandtl numbers, and thus the thermal boundary layer develops much faster than the velocity boundary layer. Then we can assume the velocity in the thermal boundary layer to be constant at the free stream value and solve the energy equation. It gives... 

So far we have limited our consideration to situations for which the entire plate is heated from the leading edge. But many practical applications involve surfaces with an unheated stalling section of length shown in Fig. 7-11, and thus there is no heat transfer for 0 < a <. In such cases, the velocity boundary layer starts to develop at llie leading edge. v = 0), but the thermal boundary layer starts to develop where heating starts (.v = ). 

During laminar flow in a tube, the magnitude of the dimensionless Prandtl number Pr is a measure of the relative growth of the velocity and thermal boundary layers. For fluids with Pr = I, such as gases, the two boundary layers essentially coincide with each other. For fluids with Pr > I, such as oils, the velocity boundary layer outgrows the thermal boundary layer. As a result, the hydrodynamic entry length is smaller than the thermal entry length. The opposite is tnie for fluids with Pr < 1 such as liquid metals. 

The velocity and temperature profiles for natural convection over a vertical hot plate are also shown in Fig. 9 -6. Note lhat as in forced convection, the thickness df the boundary layer increases in the flow direction. Unlike forced convection, however, the fluid velocity is zero at the outer edge of the velocity boundary layer as well as at the surface of the plate. This is expected since the fluid beyond the boundary layer is motionless. Thus, the fluid velocity increases with distance from the surface, reaches a maximum, and gradually decreases to zero at a distance sufflciently far from (be surface. At the. surface, the fluid temperature is equal to the plate temperature, and gradually decreases to the temperature of the surrounding fluid at a distance sufficiently far from the surface, as shown in the figure. In the case of cold surfaces, the shape of the velocity and temperature profiles remains the same but their direction is reversed. 

Equations (2.1) and (2.2) are the velocity boundary layer approximations and (2.3) is the thermal boundary layer approximation. Thus, for the steady, two-dimensional flow of an incompressible fluid with constant properties, flow continuity can be expressed as ... 

The Prandtl numbers of ideal gases lie between around 0.6 and 0.9, so that their thermal boundary layer is only slightly thicker than their velocity boundary layer. Liquids have Prandtl numbers above one and viscous oils greater than 1000. The thermal boundary layer is therefore thinner than the velocity boundary layer. By the presumptions made, the solution is only valid if 5T/6 < 1. This means that the solution is good for liquids, approximate for gases but cannot be applied to fluids with Prandtl numbers Pr [Pg.318]

Putting the thickness of the thermal boundary layer according to (3.175), along with that of the velocity boundary layer (3.170) yields, after a slight rearrangement, the Nusselt number... 

As according to (3.170), S (vxjwoo) /2 holds for the velocity boundary layer, the distance y from the wall can usefully be related to the boundary layer thickness <5, so introducing a dimensionless variable... 

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