Boundary layer equations natural convection

BOUNDARY LAYER EQUATIONS FOR NATURAL CONVECTIVE FLOWS... 

NUMERICAL SOLUTION OF THE NATURAL CONVECTIVE BOUNDARY LAYER EQUATIONS... 

A numerical solution to the laminar boundary layer equations for natural convection can be obtained using basically the same method as applied to forced convection in Chapter 3. Because the details are similar to those given in Chapter 3, they will not be repeated here. 

Equation (8.166) cannot be directly applied to natural convective boundary layer flows because in such flows the velocity is zero at the outer edge of the boundary layer. However, Eq. (8.166) should give a good description of the velocity distribution near the wall. It is therefore assumed that in a turbulent natural convective boundary layer ... 

Some of the more commonly used methods of obtaining solutions to problems involving natural convective flow have been discussed in this chapter. Attention has been given to laminar natural convective flows over the outside of bodies, to laminar natural convection through vertical open-ended channels, to laminar natural convection in a rectangular enclosure, and to turbulent natural convective boundary layer flow. Solutions to the boundary layer forms of the governing equations and to the full governing equations have been discussed. 

It will be seen from the results given by the similarity solution that the velocities are very low in natural convective boundary layers in fluids with high Prandtl numbers. In such circumstances, the inertia terms (i.e., the convective terms) in the momentum equation are negligible and the boundary layer momentum equation for a vertical surface effectively is ... 

In this section we derive the equation of motion that governs the natural convection flow in laminar boundary layer. The conservation of mass and energy equations derived in Chapter 6 for forced convection are also applicable for natural convection, but tlie momentum equation needs to be modified to incorporate buoyancy. 

Introductory note Most transport and/or fluids problems are not amenable to analysis by classical methods for linear differential equations, either because the equations are nonlinear (or simply too comphcated in the case of the thermal energy equation, which is linear in temperature if natural convection effects can be neglected), or because the solution domain is complicated in shape (or in the case of problems involving a fluid interface having a shape that is a priori unknown). Analytic results can then be achieved only by means of approximations. One approach is to simply discretize the equations in some way and turn on the computer. Another is to use the family of approximations methods known as asymptotic approximations that lead to useful concepts such as boundary layers, etc. This course is about the latter approach. However, it is not just a... 

In Chapter 5, we learned the foundations of convection. Integrating the governing equations for laminar boundary layers, we obtained expressions for the heat transfer associated with forced convection over a horizontal plate and natural convection about a vertical plate. We also found analytically, as well as by the analogy between heat and momentum, that the thermal and momentum characteristics of laminar flow over a flat plate are related by... 

The thin-layer approximation fails because natural convective boundary layers are not thin. From the interferometric fringes in Fig. 4.2ft (which are essentially isotherms), the thermal boundary layer around a circular cylinder is seen to be nearly 30 percent of the cylinder diameter. For such thick boundary layers, curvature effects are important. Despite this failure, thin-layer solutions provide an important foundation for the development of correlation equations, as explained in the section on heat transfer correlation method. 

The equation for the laminar Nusselt number Nut is obtained in a two-step procedure. In the first step, not only is the flow idealized as everywhere laminar, but the boundary layer is treated as thin. There results from this idealization the equation for the laminar thin-layer Nusselt number Nur. As already explained, natural convection boundary layers are generally not thin, so the second step is to correct Nur to account for thick boundary layers. This correction uses the method of Langmuir [175]. The corrected Nusselt number is the laminar Nusselt number Nuc. 

The value of E in relation to like the value of gj, indicates the evaporative nature of the surface so E < E q or E/Ee, < 1 reflects surface dryness or stomatal closure as well as the balance of energy exchange between the atmosphere and the underlying surface. By definition, E > Ef, can be caused only by advection. As implied above with respect to the partitioning of temperature, this may also result from the entrainment of dry air from above the convective boundary layer that develops daily over the earth surface. To further illustrate the relation between E and Ef, in terms of surface characteristics, it is helpful to write the Penman-Monteith equation (Monteith and Unsworth, 1990),... 

At the top surface of the Si02 layer in the device, natural convection and radiation occur due to heat being conducted to the surface of the device from the sensor heater. The boundary equation there can thus be formulated by equating the conduction of heat and the effective convection on the top surface, i.e.,... 

Heat transfer by convection occurs in liquids and gases where there is a velocity field caused by extorted fluid motion or by natural fluid motion caused by a difference in density. The former case involves forced convection, and the latter case free convection. Combined convection occurs when both forced and free convection are present. The convection coefficient of surface heat transfer, a, defining the heat exchange in the contact boundary layer between fluid and soUd, is determined. Coefficient or is often expressed by equations containing criteria numbers, such as those of Nusselt (Nu), Prandtl (Pr), Reynolds (Re) and Grashof(Gr) ... 

Simple integration of Equations (A.4) or (A.5) and substitution of the boundary conditions (A.l2) and (A.9) shows that as stated in Chapter 1, the solutions of the equations are indeterminate. It can be shown by solving the time dependent equations (see later) and letting r that the only steady state solution is Co = 0 and 7=0. This is because with the model used, there are no steady concentration profiles until all the species 0 is removed from solution. In practice we know that a steady state current is easily obtained, and the experimental situation is readily predicted if we define a boundary layer, thickness 6, and we assume that outside this layer the concentrations of O and R are maintained constant by convection, either natural or forced. The boundary conditions to (A.4) and (A. 5) are then... 

It is worthwhile to note that Equations (A.22) and (A. 23) have genuine steady state solutions without the need to introduce a boundary layer. This is because the chemical reaction (A. 19) causes the formation of a steady state kinetic layer. Only within this boundary layer is R present in solution, and the concentration of 0 perturbed from its initial concentration. The thickness of the kinetic layer depends on k the larger k the thinner the layer. Certainly for high values of k, the kinetic layer will lie well within the normal steady state diffusion layer defined by natural convection. The time required to reach the steady state (form the kinetic layer) also depends inversely on k. 

In order to determine the heat transfer rate by convection, the temperature distribution in the thermal boundary layer needs to be known. This temperature distribution depends on the nature of the fluid motion or the velocity field, and this is determined by solving the energy equation along with the mass and momentum equations for specific flow geometry. 

The fluid dynamic structure within the boundary layers adjacent to natural solid surfaces such as soil, sediment, snow, and ice, is complex. Typically, the flows fields have both laminar and turbulent regions. The flows magnitudes and directions respond according to the angle of incidence to the surface and the overall shape of the object as well as thermal-induced fluid density differences (i.e., stratification) and so on. All these factors operate into shaping the mass transfer boundary layer which controls the chemical flux. Ironically, the traditional approach to handling such complex flow situations has been to use a simple flux equation. The so-called convective mass flux equation is... 

The dishes were 5, 7, and 10 cm in diameter (Thibodeaux et al., 1980). Using heat transfer data for a cold plate facing upward. Equation 2.34 in Table 2.3 can be applied to assess natural convection as well. In the above cases, the correlations relate chemical dissolution at the sediment-water interface, which forms a boundary layer with fluid density slightly greater than that of pure water. This particular mass transfer process is very slow since a high-density fluid accumulates on the bottom surface and forms a stable layer, which resist the generation of BL turbulence. The resulting estimated MTCs should be the lowest for the water-side bed sediment surfaces, and appropriate for waterbodies in the absence of bottom of currents. 

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