Thermal boundary layer scaling

In general, the thermal boundary layer will not correspond with the velocity boundary layer. In the following treatment, the simplest non-interacting case is considered with physical properties assumed to be constant. The stream temperature is taken as constant In the first case, the wall temperature is also taken as a constant, and then by choosing the temperature scale so that the wall temperature is zero, the boundary conditions are similar to those for momentum transfer. 

Fluid flow and reaction engineering problems represent a rich spectrum of examples of multiple and disparate scales. In chemical kinetics such problems involve high values of Thiele modulus (diffusion-reaction problems), Damkohler and Peclet numbers (diffusion-convection-reaction problems). For fluid flow problems a large value of the Mach number, which represents the ratio of flow velocity to the speed of sound, indicates the possibility of shock waves a large value of the Reynolds number causes boundary layers to be formed near solid walls and a large value of the Prandtl number gives rise to thermal boundary layers. Evidently, the inherently disparate scales for fluid flow, heat transfer and chemical reaction are responsible for the presence of thin regions or "fronts in the solution. 

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. 

With the correct choice for m, this is the length scale characteristic of the inner (or boundary-layer) region. In the rescaled variables, the change in 0 from 0 = 1 to approximately the free-stream value 0 = 0 will occur over an increment A Y = 0(1) so that 30/37 = 0(1) independent of Pe. For m > 0, an increment A 7 = 0(1) clearly corresponds to a very small increment in the radial distance Ay, scaled with respect to a. In particular, the thickness of the so-called thermal boundary layer is only 0(Pe m) relative to the sphere radius a. 

We may note that the thermal boundary layer in this case is asymptotically thin relative to the boundary layer for a solid body. This is a consequence of the fact that the tangential velocity near the surface is larger, and hence convection is relatively more efficient. From a simplistic point of view, the larger velocity means that convection parallel to the surface is more efficient, and hence the time available for conduction (or diffusion) normal to the surface is reduced. Thus, the dimension of the fluid region that is heated (or within which solute resides) is also reduced. Indeed, if we define Pe by using a characteristic length scale lc and a characteristic velocity scale uc, heat can be conducted a distance... 

The conclusion to be drawn from the preceding discussion is that the potential-flow theory (10-9) [or, equivalently, (10 12) and (10 13)] does not provide a uniformly valid first approximation to the solution of the Navier Stokes and continuity equations (10-1) and (10 2) for Re 1. Furthermore, our experience in Chap. 9 with the thermal boundary-layer structure for large Peclet number would lead us to believe that this is because the velocity field near the body surface is characterized by a length scale 0(aRe n), instead of the body dimension a that was used to nondimensionalize (10-2). As a consequence, the terms V2co and u V >, in (10 6), which are nondimensionalized by use of a, are not 0(1) and independent of Re everywhere in the domain, as was assumed in deriving (10-7), but instead are increasing fimctions of Re in the region very close to the body surface. Thus in... 

We begin by considering the inner thermal boundary-layer limit for arbitrary Pr 0(1), but with Pe . The velocity distribution in the vicinity of the body surface is characterized by the length scale... 

Consider a heated vertical plate in a quiescent fluid. The plate heats the fluid in its neighborhood, which then becomes lighter and moves upward. The force resulting from the product of gravity and density difference and causing this upward motion is called buoyancy. The fluid moving under the effect of buoyancy develops a vertical boundary layer about the plate. Within the boundary layer the temperature decreases from the plate temperature to the fluid temperature, while the velocity vanishes on the plate walls and beyond the boundary layer and has a maximum in between (Fig. 5.13). Actually, in a manner similar to forced convection, the momentum boundary layer of natural convection is expected to be thicker for larger Prandtl numbers than the thermal boundary layer. However, the characteristic velocity for the enthalpy flow across should be scaled relative to Ss rather than 5,... 

This is explained by a possible higher activity of pure rhodium than supported metal catalysts. However, two other reasons are also taken into account to explain the superior performance of the micro reactor boundary-layer mass transfer limitations, which exist for the laboratory-scale monoliths with larger internal dimensions, are less significant for the micro reactor with order-of-magnitude smaller dimensions, and the use of the thermally highly conductive rhodium as construction material facilitates heat transfer from the oxidation to the reforming zone. 

The second approach assigns thermal resistance to a gaseous boundary layer at the heat transfer surface. The enhancement of heat transfer found in fluidized beds is then attributed to the scouring action of solid particles on the gas film, decreasing the effective film thickness. The early works of Leva et al. (1949), Dow and Jacob (1951), and Levenspiel and Walton (1954) utilized this approach. Models following this approach generally attempt to correlate a heat transfer Nusselt number in terms of the fluid Prandtl number and a modified Reynolds number with either the particle diameter or the tube diameter as the characteristic length scale. Examples are ... 

A schematic representation of the boundary layers for momentum, heat and mass near the air—water interface. The velocity of the water and the size of eddies in the water decrease as the air—water interface is approached. The larger eddies have greater velocity, which is indicated here by the length of the arrow in the eddy. Because random molecular motions of momentum, heat and mass are characterized by molecular diffusion coefficients of different magnitude (0.01 cm s for momentum, 0.001 cm s for heat and lO cm s for mass), there are three different distances from the wall where molecular motions become as important as eddy motions for transport. The scales are called the viscous (momentum), thermal (heat) and diffusive (molecular) boundary layers near the interface. 

C.P. Lombardo, M.C. Gregg (1989). Similarity scaling of viscous and thermal dissipation in a convecting surface boundary layer. J. Geophys. Res., 94,6273-6284. 

Abstract. Interaction between a current and surface-active material is considered. Some simple cases where the substrate motion is steady and 2D is analysed using standard boundary layer theory. Questions like how is the transversal dimension of a slick related to the film pressure and the substrate convergence and how strong substrate motion does it take to break up a surface film , are addressed. It is pointed out that the answers depend on whether the film can be considered stagnant, or develops self-organised motion. It is further pointed out how small scale thermal convection at the ocean surface is easily suppressed by a slick. 

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