Boundary layer equations energy

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,... 

The boundary layer equations were derived in a previous chapter, or may be deduced from the general convection equations in the early part of this chapter. For two-dimensional, steady flow over a flat plate of an incompressible, constant-property fluid, the continuity, x-momentum and the energy equations are as follows ... 

Because they do not contain the pressure as a variable. Eqs. (2.76) and (2.77) have been used quite extensively in solving problems for which the boundary layer equations (see later) cannot be used. For this purpose, instead of solving the Navier-Stokes and energy, simultaneously with the continuity equation, it is convenient to introduce the stream function, ip, which is defined such that... 

In all the solutions given in the present chapter, the fluid properties will be assumed to be constant and the flow will be assumed to be two-dimensional. In addition, dissipation effects in the energy equation will be neglected in most of this chapter, these effects being briefly considered in a last section of this chapter. Also, solutions to the full Navier-Stokes and energy equations will be dealt with only relatively briefly, the majority of the solutions considered being based on the use of the boundary layer equations. 

It is next noted that because the velocities are very low in the inner, i.e., nearwall, region, the convective terms in the boundary layer equations can be neglected in this region, i.e., in this region the momentum and energy equations can be assumed to have the form ... 

There are many situations in which similarity-type solutions to the boundary layer equations cannot be obtained. Numerical solutions to these equations can be obtained in such cases. In general, such solutions first involve numerically solving for the surface Velocity distribution and then using the energy equation to obtain the temperature distribution. Here, in order to illustrate how the energy equation can be numerically solved, it will be assumed that the variation of the surface velocity with... 

Theoretical approaches to the prediction of H x,y,t) would involve the solution of the boundary layer equations for coupled energy and momentum transport or, more simply, the solution of the energy equations in conjunction with a constructed wind field. The application of such approaches to the prediction of inversion height has not yet been reported. Now, empirical models offer the only available means to estimate H. For those areas where it is necessary only to account for temporal variations in H, interpolation and extrapolation of measured mixing heights may be sufficient. When it is important to estimate // as a function of x,y, and t, a detailed knowledge of local meteorology is essential. 

Corresponding considerations are also valid for the thermal boundary layer in multicomponent mixtures. The energy transport through conduction and diffusion in the direction of the transverse coordinate x is negligible in comparison to that through the boundary layer. The energy equation for the boundary layer follows from (3.97), in which we will presuppose vanishing mass forces k Ki-... 

This is known as the thermal boundary-layer equation for this problem. Because we have obtained it by taking the limit Pe -> oo in the full thermal energy equation (9-222) with m = 1/3, we recognize that it governs only the first term in an asymptotic expansion similar to (9-202) for this inner region. 

Uniform Surface Temperature, Foreign Gas as Coolant. The effectiveness of air injection in reducing convective heat flux stimulated investigations into the use of other coolants With the introduction of a foreign species into the boundary layer, the boundary layer equations reduce to the continuity equation (Eq. 6.6), the momentum equation (Eq. 6.7), the energy equation... 

Using the same assumptions that were made in the vapor-layer model, the energy-conservation equation for the incompressible 2-D vapor phase can be simplified to a 1-D equation in boundary layer coordinates ... 

The temperature of a liquid metal stream discharged from the delivery tube prior to primary breakup can be calculated by integrating the energy equation in time. The cooling rate can be estimated from a cylinder cooling relation for the liquid jet-ligament breakup mechanism (with free-fall atomizers), or from a laminar flat plate boundary layer relation for the liquid film-sheet breakup mechanism (with close-coupled atomizers). 

There is a natural draw rate for a rotating disk that depends on the rotation rate. Both the radial velocity and the circumferential velocity vanish outside the viscous boundary layer. The only parameter in the equations is the Prandtl number in the energy equation. Clearly, there is a very large effect of Prandtl number on the temperature profile and heat transfer at the surface. For constant properties, however, the energy-equation solution does not affect the velocity distributions. For problems including chemistry and complex transport, there is still a natural draw rate for a given rotation rate. However, the actual inlet velocity depends on the particular flow circumstances—there is no universal correlation. 

The pressure term has been retained in the thermal-energy equation, although in low-speed flow there is very little energy content in this term. Indeed, the term is very often neglected in boundary-layer analysis. For high-speed flow (i.e., Ma > 0.3), however, the thermal energy can be substantially affected by pressure variations. 

In this equation, if the rate of diffusion is faster than that of the catalytic reaction at the surface (ko kc), the Arrhenius plot of rr gives the apparent activation energy Ec of kc. This is the reaction-controlled condition. On the other hand, if the rate of the catalytic reaction is faster than that of diffusion (kc 2> kid, the Arrhenius plot of rr gives the characteristics of temperature dependence of ko. This is the diffusion-controlled condition. Under diffusion-controlled conditions, the transferred reactant decreases at once at the surface (Cs = 0) because of the fast catalytic reaction rate. The gas flow along the catalyst surface forms a boundary layer above the surface, and gas molecules diffuse due to the concentration gradient inside the layer in the thickness direction. As the total reaction... 

When the Bom, double-layer, and van der Waals forces act over distances that are short compared to the diffusion boundary-layer thickness, and when the e forces form an energy hairier, the adsorption and desorption rates may be calculated by lumping the effect of the interactions into a boundary condition on the usual ccm-vective-diffusion equation. This condition takes the form of a first-order, reversible reaction on the collector s surface. The apparent rate constants and equilibrium collector capacity are explicitly related to the interaction profile and are shown to have the Arrhenius form. They do not depend on the collector geometry or flow pattern. 

Take into consideration two-dimensional, rectilinear, steady, incompressible, constant-property, laminar boundary layer flow in the x direction along a flat plate. Assume that viscous energy dissipation may be neglected. Write the continuity, momentum and energy equations. 

If terms of the order (8/L)2 and less are again neglected, it will be seen that the energy equation for laminar two-dimensional boundary layer flow becomes ... 

The same line of reasoning can be applied to the energy equation and if it is assumed that the turbulence terms (kT /M wt - T ) and (v T )/ui(Twr - T ) in the resultant equation have the same order of magnitude as the turbulence terms in the momentum equation, i.e., (8/L), then the energy equation for turbulent boundary layer flow becomes... 

As was the case with the full equations, these contain beside the three mean flow variables u, v, and T (the pressure is, of course, by virtue of Eq. (2.157) again determined by the external in viscid flow) additional terms arising as a result of the turbulence. Therefore, as previously discussed, in order to solve this set of equations, there must be an additional input of information, i.e., a turbulence model must be used. Many turbulence models are based on the turbulence kinetic energy equation that was previously derived. When the boundary layer assumptions are applied to this equation, it becomes ... 

The limiting forms of the equations that result from the application of these conservation principles to this control volume as dx -+ 0 give a set of equations governing the average conditions across the boundary layer. These resultant equations are termed the boundary layer momentum integral and energy integral equations. 

Consider next the application of the conservation of energy principle to the control volume that was used above in the derivation of the momentum integral equation. The height, , of this control volume is taken to be greater than both the velocity and temperature boundary layer thicknesses as shown in Fig. 2.21. 

This is termed the boundary layer energy integral equation. 

The energy integral equation is applied in basically the same way as the momentum integral equation. The form of the boundary layer temperature profile, i.e., of the variation of (T - T ) with y, is assumed. In the case of laminar flow, for example, a polynomial form is again often used. The unknown coefficients in this assumed temperature profile are then determined by applying known boundary conditions on temperature at the inner and outer edges of the boundary layer. For example, the variation of the wall temperature Tw with x may be specified. Therefore, because at the outer edge of the boundary layer the temperature must become equal to the freestream temperature T, two boundary conditions on the assumed temperature profile in this case are ... 

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