Thickness thermal boundary layer

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. 

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... 

The thermal boundary-layer thicknesses in the liquid before bubble nucleation are much greater. 

Hsu and Graham (1961) took into consideration the bubble shape and incorporated the thermal boundary-layer thickness, 8, into their equation, thus making the bubble growth rate a function of 8. Han and Griffith (1965b) took an approach similar to that of Hsu and Graham with more elaboration, and dealt with the constant-wall-temperature case. Their equation is... 

Other factors do intervene. Significant solar heating of the soil surface, so that the soil becomes warmer than the air, causes vertical thermal convection currents to develop within the boundary layers. This introduces turbulence or instability that acts to move the chemical signature up into the free air. When the molecules are moved into the free flow of the air, the effect is to reduce the concentration by dilution. Conversely, when the soil surface is cooler than the air, thermal convection is inhibited, with the result that the molecules are effectively trapped in the boundary layer. This effect is strengthened by the cooling of the air adjacent to the surface, which increases its viscosity. Higher viscosity lowers the Reynold s number, thus decreasing boundary layer thickness. 

One example would be ice melting or methane hydrate dissociation when rising in seawater. Convective melting rate may be obtained by analogy to convective dissolution rate. Heat diffusivity k would play the role of mass diffusivity. The thermal Peclet number (defined as Pet = 2aw/K) would play the role of the compositional Peclet number. The Nusselt number (defined as Nu = 2u/5t, where 8t is the thermal boundary layer thickness) would play the role of Sherwood number. The thermal boundary layer (thickness 8t) would play the role of compositional boundary layer. The melting equation may be written as... 

Diffusion-layer thickness Thermal-boundary layer... 

Fig. 7.93. Influence of location on boundary layer thickness in laminar flow along an electrode 8,8p and 8V are the thicknesses of the diffusion, thermal, and hydrodynamic boundary layers, respectively.

Determine the effect of Prandtl number on the thermal boundary-layer thickness. Consider the range 1 < Pr < 100. 

The boundary layer equations may be obtained from the equations provided in Tables 6.1-6.3, with simplification and by an order-of-magnitude study of each term in the equations. It is assumed that the main flow is in the x direction. The terms that are too small are neglected. Consider the momentum and energy equations for the two-dimensional, steady flow of an incompressible fluid with constant properties. The dimensionless equations are given by Eqs. (6.46) to (6.48). The principal assumption made in the boundary layer is that the hydrodynamic boundary layer thickness 8 and the thermal boundaiy layer thickness 8t are small compared to a characteristic dimension L of the body. In mathematical terms,... 

Use these assumptions to find a relation between the hydrodynamic boundary layer thickness 8 and the thermal boundary layer thickness 8,. For gases, Pr is of the order of unity. For liquids, Pr ranges from about 10 to 1000. For liquid metals, Pr ranges from about 0.003 to about 0.03. Deduce the relative thicknesses of 8 and 8, for gases, liquids and liquid metals. 

The thermal boundary layer thickness is zero at the entrance where x = 0. 

The momentum boundary layer thickness is represented by 8, and the thermal boundary layer thickness is represented by 8,. 

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 ... 

In the above procedure, it was assumed that the outermost grid point, i.e., the N-point, was always outside both the velocity and thermal boundary layers. One way of ensuring this is the case is, of course, to simply estimate the maximum boundary layer thickness expected and then select the number of grid points and their positioning such that the outermost grid line is at a greater distance from the surface than this maximum boundary layer thickness. This is illustrated in Fig. 3.22. 

Air at 300 K Ad 1 atm flows at a velocity of 2 m/s along a flat plate which has a length of 0.2 m. The plate is kept at a temperature of 330 K. Plot the variations of the velocity and thermal boundary layer thicknesses along the plate. 

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]. 

To calculate the heat transfer at the wall, we need to derive an expression for the thermal-boundary-layer thickness which may be used in conjunction... 

Calculate the ratio of thermal-boundary-layer thickness to hydrodynamic-bound-ary-layer thickness for the following fluids air at 1 atm and 20°C, water at 20°C, helium at 1 atm and 20°C, liquid ammonia at 20°C, glycerine at 20°C. 

Let us first consider the simple flat plate with a liquid metal flowing across it. The Prandtl number for liquid metals is very low, of the order of 0.01. so that the thermal-boundary-layer thickness should be substantially larger than the hydrodynamic-boundary-layer-thickness. The situation results from the high values of thermal conductivity for liquid metals and is depicted in Fig. 6-15. Since the ratio of 8/8, is small, the velocity profile has a very blunt shape over most of the thermal boundary layer. As a first approximation, then, we might assume a slug-flow model for calculation of the heat transfer i.e., we take... 

Now consider the flat plate shown in Fig. 12-3. The plate surface is maintained at the constant temperature Tw, the free-stream temperature is 7U, and the thermal-boundary-layer thickness is designated by the conventional symbol 5,. To simplify the analysis, we consider low-speed incompressible flow so that the viscous-heating effects are negligible. The integral energy equation then becomes... 

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