Heat transfer boundary layer thickness

J9A,mix in the expressions for 5c and Sc represents a diffusivity instead of a molecular transport property, one must replace a, mix by the thermal diffusivity 0 (= kidpCp, where p = density, Cp = specific heat, and kjc = thermal conductivity) to calculate the analogous heat transfer boundary layer thickness Sj and the Prandtl number [i.e., Pr = d/p)ja. In the creeping flow regime, where g 9) = I sine. 

Since a, mix appears everywhere in part (a) as a diffusivity, not a molecular transport properly, the corresponding heat transfer boundary layer thickness is calculated from the preceding equation via replacement of A,mix by a. 

This dimensionless group is recognized as the Prandtl number, which is currently used in heat transfer processes. This number is very important when the boundary layer theory is applied because it shows the relationship between the corresponding thickness of the heat transfer boundary layer and the hydrodynamic boundary layer [6.12]. 

Now let us consider the mixing time, t. This will be estimated by an order of magnitude estimate for diffusion to occur across the boundary layer thickness, <5Bl- If we have turbulent natural conditions, it is common to represent the heat transfer in terms of the Nusselt number for a vertical plate of height, , as... 

Although the correlations given by Eq. (6.48) are useful for practical evaluation of heat transfer to a wall, one must not lose sight of the fact that the temperature gradient at the wall actually determines the heat flux there. In transpiration cooling problems, it is not so much that the injection of the transpiring fluid increases the boundary layer thickness, thereby decreasing the... 

Once the system of equations has been solved, the nondimensional temperature gradient can be easily evaluated at the surface, providing the Nusselt number. It should be expected that the heat transfer depends on the boundary-layer thickness, which in turn depends on the flow field, which is principally governed by the Reynolds number. Figure 6.9 shows a correlation between the Nusselt number and the Reynolds number that was obtained by solving the nondimensional system for several Reynolds numbers. 

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. 

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]. 

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... 

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... 

For vertical surfaces, the Nusselt and Grashof numbers are formed with L, the height of the surface as the characteristic dimension. If the boundary-layer thickness is not large compared with the diameter of the cylinder, the heat transfer may be calculated with the same relations used for vertical plates. The general criterion is that a vertical cylinder may be treated as a vertical flat plate [13] when... 

Dimensionless numbers have proved useful for analyzing relationships between heat transfer and boundary layer thickness for leaves. In particular, the Nusselt number increases as the Reynolds number increases for example, Nu experimentally equals 0.97 Re0-5 for flat leaves (Fig. 7-9). By Equations 7.18 and 7.19, d/8bl is then equal to 0.97 (vd/v)V2y so for air temperatures in the boundary layer of 20 to 25°C, we have... 

Equation 8.2 shows how the net flux density of substance depends on its diffusion coefficient, Dj, and on the difference in its concentration, Ac] 1, across a distance Sbl of the air. The net flux density Jj is toward regions of lower Cj, which requires the negative sign associated with the concentration gradient and otherwise is incorporated into the definition of Acyin Equation 8.2. We will specifically consider the diffusion of water vapor and C02 toward lower concentrations in this chapter. Also, we will assume that the same boundary layer thickness (Sbl) derived for heat transfer (Eqs. 7.10-7.16) applies for mass transfer, an example of the similarity principle. Outside Sbl is a region of air turbulence, where we will assume that the concentrations of gases are the same as in the bulk atmosphere (an assumption that we will remove in Chapter 9, Section 9.IB). Equation 8.2 indicates that Jj equals Acbl multiplied by a conductance, gbl, or divided by a resistance, rbl. 

This overlapping will in fact reduce the available area for heat and mass transfer. During the present work, some boundary layer thicknesses were estimated for the experimental conditions of this work. As a result, the boundary layers only overlap for Reynolds numbers below 0.826. For the case of Reynolds numbers of 1.74 and 3.05 using the particle diameter of 0.035 cm., the boundary layers do not overlap.Table III shows some of the values obtained.Clearly, this effect cannot explain completely the low heat and mass transfer coefficients at low Reynolds numbers. 

Note that /i, is proportional to Re and thus to. v- - for laminar flow. Therefore, is infinite at the leading edge (jc = 0) and decreases by a factor of.r in the flow direction. The variation of the boundary layer thickness 5 and the friction and heat transfer coefficients along an isothermal flat plate are shown in Fig. 7-9. The local friction and heat transfer coefficients are higher in... 

A simple graphical illustration of a follows from (1.25). As shown in Fig. 1.7 the ratio X/a is the distance from the wall at which the tangent to the temperature profile crosses the = t F line. The length of X/a is of the magnitude of the (thermal) boundary layer thickness which will be calculated in sections 3.5 and 3.7.1 and which is normally a bit larger than X/a. A thin boundary layer indicates good heat transfer whilst a thick layer leads to small values of a. 

The Schmidt number for the mass transfer is analogous to the Prandtl number for heat transfer. Its physical implication means the relative thickness of the hydrodynamic layer and mass-transfer boundary layer. The ratio of the velocity boundary layer (S) to concentration boundary layer (Sc) is governed by the Schmidt number. The relationship is given by... 

The temperature dependence of the viscosity of the liquid and thereby the boundary layer thickness upon cooling and heating is taken into consideration with the viscosity term Vis = Following the suggestion of Sieder and Tate [505], that experimental data for heat transfer in pipes upon heating and cooling correlated upon inclusion of Vis , this expression was also accepted in most research studies over heat transfer in mixing. 

It has been shown that there exists a continuous change in the physical behavior of the turbulent momentum boundary layer with the distance from the wall. The turbulent boundary layer is normally divided into several regions and sub-layers. It is noted that the most important region for heat and mass transfer is the inner region of the boundary layer, since it constitutes the major part of the resistance to the transfer rates. This inner region determines approximately 10 — 20% of the total boundary layer thickness, and the velocity distribution in this region follows simple relationships expressed in the inner variables as defined in sect 1.3.4. 

We have seen how heat transfer and thus dry deposition of SO2 is reduced on large surfaces, due to the buildup of boundary layer thickness (which reduces the local gradients). However, there are economically important structural objects composed of many elements of small dimension which show the opposite effect. These include fence wire and fittings, towers made of structural shapes (pipe, angle iron, etc.), flagpoles, columns and the like. Haynie (11) considered different damage functions for different structural elements such as these, but only from the standpoint of their effect on the potential flow in the atmospheric boundary layer. The influence of shape and size act in addition to these effects, and could also change the velocity coefficients developed by Haynie (11), which were for turbulent flow. Fence wire, for example, as shown below, is more likely to have a laminar boundary layer. 

Figure 9-12. The self-similar temperature profile given by Eq. (9-240) for forced convection heat transfer from a heated (or cooled) solid sphere in a uniform velocity field at small Re and large Pe. The function g( i]) represents the dependence of the thermal boundary-layer thickness on // and is given by (9-237).

---