Equation thermal boundary layer

It is assumed that the velocity boundary layer is everywhere thicker than the thermal boundary layer, so that 8 > 8, (Figure 11.11). Thus the velocity distribution everywhere within the thermal boundary layer is given by equation 11.12. The implications of this assumption are discussed later. 

The integral in equation 11.55 clearly has a finite value within the thermal boundary layer, although it is zero outside it. When the expression for the temperature distribution in the boundary layer is inserted, the upper limit of integration must be altered from /... 

It is seen from equation 11.66 that the heat transfer coefficient theoretically has an infinite value at the leading edge, where the thickness of the thermal boundary layer is zero, and that it decreases progressively as the boundary layer thickens. Equation 11.66 gives the point value of the heat transfer coefficient at a distance x from the leading edge. The mean value between. v = 0 and x = x is given by ... 

Another important case is where the heat flux, as opposed to the temperature at the surface, is constant this may occur where the surface is electrically heated. Then, the temperature difference 9S — o will increase in the direction of flow (x-direction) as the value of the heat transfer coefficient decreases due to the thickening of the thermal boundary layer. The equation for the temperature profile in the boundary layer becomes ... 

If a concentration gradient exists within a fluid flowing over a surface, mass transfer will take place, and the whole of the resistance to transfer can be regarded as lying within a diffusion boundary layer in the vicinity of the surface. If the concentration gradients, and hence the mass transfer rates, are small, variations in physical properties may be neglected and it can be shown that the velocity and thermal boundary layers are unaffected 55. For low concentrations of the diffusing component, the effects of bulk flow will be small and the mass balance equation for component A is ... 

Again, the form of the concentration profile in the diffusion boundary layer depends on the conditions which are assumed to exist at the surface and in the fluid stream. For the conditions corresponding to those used in consideration of the thermal boundary layer, that is constant concentrations both in the stream outside the boundary layer and at the surface, the concentration profile is of similar form to that given by equation 11.70 ... 

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... 

Equation 9.1-15 equates the rate of heat transfer by conduction at the surface to the rate of heat transfer by conduction/convection across a thermal boundary layer exterior to the particle (corresponding to the gas film for mass transfer), expressed in terms of a film coefficient, h, and the difference in temperature between bulk gas at Tg and particle surface at Ts ... 

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... 

The velocity field is caused in free convection by the temperature field. Therefore, the thickness 8 of the thermal boundary layer can be used as the single length scale that characterizes both the temperature and velocity fields. Denoting the velocity scale in the x direction by u0, the continuity equation [Eq. (39)] shows that the velocity scale v0 in the y direction is of the order of u08/x. 

The equivalent equation for the thermal boundary layer will be ... 

The coefficients in this equation, i.e., e,f, g, and h, are determined by applying the boundary conditions on temperature at the inner and outer edges of the thermal boundary layer. Three such boundary conditions, which are analogous to those given for the velocity in Eq. (3.123), are ... 

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... 

If the same procedure were followed at this point as in Sec. 5-6, the temperature and velocity functions given by Eqs. (12-19) and (12-29) would be inserted in (12-27) in order to arrive at a differential equation to be solved for 5 the thermal-boundary-layer thickness in the presence of the magnetic field. The problem with this approach is that a nonlinear equation results which must be solved by numerical methods. 

The above definition of the lithosphere equates the lithosphere with a thermal boundary layer... 

Liquid metals such as mercury have high thermal conductivities, and are commonly used in applications that require high heat transfer rates. However, they have very small Prandtl numbers, and thus the thermal boundary layer develops much faster than the velocity boundary layer. Then we can assume the velocity in the thermal boundary layer to be constant at the free stream value and solve the energy equation. It gives... 

Equations (2.1) and (2.2) are the velocity boundary layer approximations and (2.3) is the thermal boundary layer approximation. Thus, for the steady, two-dimensional flow of an incompressible fluid with constant properties, flow continuity can be expressed as ... 

Dimensional analysis of the energy equation then gives a criterion for when convective terms become comparable in magnitude to conductive terms favoring the formation of thermal boundary layers. 

The influence of a wall on the turbulent transport of scalar (species or enthalpy) at the wall can also be modeled using the wall function approach, similar to that described earlier for modeling momentum transport at the wall. It must be noted that the thermal or mass transfer boundary layer will, in general, be of different thickness than the momentum boundary layer and may change from fluid to fluid. For example, the thermal boundary layer of a high Prandtl number fluid (e.g. oil) is much less than its momentum boundary layer. The wall functions for the enthalpy equations in the form of temperature T can be written as ... 

By this normalisation the temperature in the thermal boundary layer varies between the values 0 < d+ < 1. With these dimensionless quantities the energy equation (3.115) can be rearranged into... 

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-... 

The concentration boundary layer and the mass transfer coefficient can immediately be found from the equations given previously, as the integral condition (3.167) for mass transfer corresponds to that for heat transfer. The temperature is replaced by the mass fraction A, the thermal diffusivity a by the diffusion coefficient D, and instead of the thermal boundary layer ST the concentration boundary layer Sc is used. This then gives us the concentration profile corresponding to (3.172) ... 

I. Governing Equations and Rescaling in the Thermal Boundary-Layer Region... 

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. 

The asymptotic formulation of the previous subsection has led not only to the important result given by (9-230) but also to a very considerable simplification in the structure of the governing equation in the thermal boundary-layer region. As a consequence, it is now possible to obtain an analytic approximation for 0. 

In the limit Pe p> 1, we have seen that the thermal boundary layer equations hold for a region very close to the surface of the body, and thus that we require only the limiting form of the velocity components as we approach the body surface. We can again use a Taylor series approximation to deduce the appropriate form for the velocity components,... 

We can now find the appropriate form for the thermal boundary-layer equation by substituting the approximate forms for the two velocity components into the thermal equation (9-247) and rescaling according to... 

This is the generalization of the thermal boundary-layer equation (9-225). 

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