Heat transfer boundary layer functions

Consider the locally flat description of heat transfer by convection and conduction from a hot plate to an incompressible fluid at high Peclet numbers with two-dimensional laminar flow in the heat transfer boundary layer adjacent to the hot surface. The tangential fluid velocity component Vx is only a function of position x parallel to the interface. 

We now consider bar element, and the element length is f. Two nodes are denoted by i,j. The trial function of temperature field is linear distribution. Under the convective heat transfer boundary condition, the finite element basic equation of steady heat conduction in the three-layered composite plate is [8]... 

In the previous section we discussed wall functions, which are used to reduce the number of cells. However, we must be aware that this is an approximation that, if the flow near the boundary is important, can be rather crude. In many internal flows—where all boundaries are either walls, symmetry planes, inlets, or outlets—the boundary layer may not be that important, as the flow field is often pressure determined. However, when we are predicting heat transfer, it is generally not a good idea to use wall functions, because the convective heat transfer at the walls may be inaccurately predicted. The reason is that convective heat transfer is extremely sensitive to the near-wall flow and temperature field. 

Fig. 13.2 Heat flux transferred to a flat plate in a boundary layer, plotted as In q, as a function of Ug.

In the simple case of airflow over an aerodynamically smooth surface, with a fully developed boundary layer, the velocity of deposition can be calculated as a function of the diffusivity of the vapour or particle and the air speed. Formulae, developed for mass and heat transfer (Brut-saert, 1982) have been shown to apply to both attached and unattached 212Pb in wind tunnel experiments (Chamberlain, 1966,1968 Chamber-lain etal., 1984). 

Burton has derived an equation for preferred air temperature indoors (T ), which is expressed as a function of body heat generated minus external mechanical work, heat lost by evaporation of water, and thermal resistances provided by air boundary layers on the outer surface and clothing layers (J5). His concept of the interrelationship of persons and their indoor environment by heat transfer at the skin is straightforward and useful (Figure 3). 

Using the linear-velocity profile in Prob. 5-2 and a cubic-parabola temperature distribution [Eq. (5-30)], obtain an expression for heat-transfer coefficient as a function of the Reynolds number for a laminar boundary layer on a flat plate. 

These relations are extremely valuable as they slate that for a given geometry, the friction coefficient can be expres.sed as a function of Reynolds number alone, and the Nusselt number as a function of Reynolds and Prandtl numbers alone (Fig. 6-34). Therefore, experimentalists can study a problem with a minimum number of experiments, and report their friction and heat transfer coefficient measure.ments conveniently in terms of Reynolds and Prandtl numbers. For example, a friction coefficient relation obtained with air for a given surface can also be used for water at the same Reynolds number. But it should be kepi in mind that the validity of these relations is limited by the limitations on (he boundary layer equations used in the analysis. 

The phenomena that affect drag force also affect heat transfer, and this effect appears in the Nusselt number. By nondimensionalizing the boundary layer equations, it was shown in Chapter 6 that the local and average Nusselt numbers have the functional form... 

As soon as the functional relationships between the Nusselt, Reynolds and Prandtl numbers or the Sherwood, Reynolds and Schmidt numbers have been found, be it by measurement or calculation, the heat and mass transfer laws worked out from this hold for all fluids, velocities and length scales. It is also valid for all geometrically similar bodies. This is presuming that the assumptions which lead to the boundary layer equations apply, namely negligible viscous dissipation and body forces and no chemical reactions. As the differential equations (3.123) and (3.124) basically agree with each other, the solutions must also be in agreement, presuming that the boundary conditions are of the same kind. The functions (3.126) and (3.128) as well as (3.127) and (3.129) are therefore of the same type. So, it holds that... 

While the film and surface-renewal theories are based on a simplified physical model of the flow situation at the interface, the boundary layer methods couple the heat and mass transfer equation directly with the momentum balance. These theories thus result in anal3dical solutions that may be considered more accurate in comparison to the film or surface-renewal models. However, to be able to solve the governing equations analytically, only very idealized flow situations can be considered. Alternatively, more realistic functional forms of the local velocity, species concentration and temperature profiles can be postulated while the functions themselves are specified under certain constraints on integral conservation. Prom these integral relationships models for the shear stress (momentum transfer), the conductive heat flux (heat transfer) and the species diffusive flux (mass transfer) can be obtained. 

The modeling procedure can be sketched as follows. First an approximate description of the velocity distribution in the turbulent boundary layer is required. The universal velocity profile called the Law of the wall is normally used. The local shear stress in the boundary layer is expressed in terms of the shear stress at the wall. From this relation a dimensionless velocity profile is derived. Secondly, a similar strategy can be used for heat and species mass relating the local boundary layer fluxes to the corresponding wall fluxes. From these relations dimensionless profiles for temperature and species concentration are derived. At this point the concentration and temperature distributions are not known. Therefore, based on the similarity hypothesis we assume that the functional form of the dimensionless fluxes are similar, so the heat and species concentration fluxes can be expressed in terms of the momentum transport coefficients and velocity scales. Finally, a comparison of the resulting boundary layer fluxes with the definitions of the heat and mass transfer coefficients, indiates that parameterizations for the engineering transfer coefficients can be put up in terms of the appropriate dimensionless groups. 

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).

In heat transfer, the fluid at the wall has the same temperature as the wall, different from the bulk temperature. While all fluid properties are to some degree functions of temperature, the viscosity is most strongly affected. If the viscosity at the wall is lower than the bulk value, the boundary layer is thinner than for the isothermal case, and the heat-transfer coefficient is higher than that predicted for the constant property case. Correspondingly, the coefficient is reduced if the viscosity at the wall is higher than the bulk value. The effect is usually small (5 to 10%) but can be much larger with viscous fluids or large temperature differences. 

Nonuniform Surface Temperature. Nonuniform surface temperatures affect the convective heat transfer in a turbulent boundary layer similarly as in a laminar case except that the turbulent boundary layer responds in shorter downstream distances The heat transfer to surfaces with arbitrary temperature variations is obtained by superposition of solutions for convective heating to a uniform-temperature surface preceded by a surface at the recovery temperature of the fluid (Eq. 6.65). For the superposition to be valid, it is necessary that the energy equation be linear in T or i, which imposes restrictions on the types of fluid property variations that are permitted. In the turbulent boundary layer, it is generally required that the fluid properties remain constant however, under the assumption that boundary layer velocity distributions are expressible in terms of the local stream function rather than y for ideal gases, the energy equation is also linear in T [%]. 

S. Levy, Heat Transfer to Constant-Property Laminar Boundary Layer Flows With Power-Function Free-Stream Velocity and Wall-Temperature Variation, J. Aeronaut. Sci. (19) 341-348,1952. 

The remainder of the chapter focuses on the actual spray modeling. The exposition is primarily done for the RANS method, but with the indicated modifications, the methodology also applies to LES. The liquid phase is described by means of a probability density function (PDF). The various submodels needed to determine this PDF are derived from drop-drop and drop-gas interactions. These submodels include drop collisions, drop deformation, and drop breakup, as well as drop drag, drop evaporation, and chemical reactions. Also, the interaction between gas phase, liquid phase, turbulence, and chemistry is examined in some detail. Further, a discussion of the boundary conditions is given, in particular, a description of the wall functions used for the simulations of the boundary layers and the heat transfer between the gas and its confining walls. 

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