Thermal boundary layer uniform flow

I. THERMAL BOUNDARY-LAYER THEORY FOR SOLID BODIES OF NONSPHERICAL SHAPE IN UNIFORM STREAMING FLOW... 

It is perhaps timely to stop and reflect upon the nature of the thermal boundary-layer analysis to determine whether other generalizations of the basic result (9-230) may be possible. In particular, heat transfer from solid bodies occurs frequently when the fluid motion seen by the body cannot be approximated as a uniform streaming flow, and the reader may ask whether the correlation (9-230) can be applied in these cases with a proper choice for the characteristic velocity that appears in Pe. It is especially interesting, in this regard, to compare the present analysis with the corresponding low-Peclet-number problem that appeared earlier in this chapter. 

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... 

Processing variables that affect the properties of the thermal CVD material include the precursor vapors being used, substrate temperature, precursor vapor temperature gradient above substrate, gas flow pattern and velocity, gas composition and pressure, vapor saturation above substrate, diffusion rate through the boundary layer, substrate material, and impurities in the gases. Eor PECVD, plasma uniformity, plasma properties such as ion and electron temperature and densities, and concurrent energetic particle bombardment during deposition are also important. 

Now consider a fluid at a uniform temperature entering a circular tube whose surface is maintained at a different temperature. This time, the fluid particles in the layer in contact with the surface of the tube assume the surface temperature. Tins initiates convection heat transfer in the tube and Ihe development of a thermal hoimdaiy layer along the tube. The thickness of this boundary layer also increases in tfle flow direction until Ihe boundary layer reaches the tube center and thus fills the entire tube, as sliown in Fig. 8-7. 

But before the virtues of the results and the approach are extolled, the method must be described in detail. Let us therefore return to a systematic development of the ideas necessary to solve transport (heat or mass transfer) problems (and ultimately also fluid flow problems) in the strong-convection limit. To do this, we begin again with the already-familiar problem of heat transfer from a solid sphere in a uniform streaming flow at sufficiently low Reynolds number that the velocity field in the domain of interest can be approximated adequately by Stokes solution of the creeping-flow problem. In the present case we consider the limit Pe I. The resulting analysis will introduce us to the main ideas of thermal (or mass transfer) boundary-layer theory. 

A dilute polymer solution at 25"C flows at 2m/s over a 300 nun x 300 nun square plate which is maintained at a uniform temperature of 35°C. The average values of the power-law constants (over this temperature interval) may be taken as m = 0.3 Pa-s" and n =0.5. Estimate the thickness of the boundary layer 150 mm from the leading edge and the rate of heat transfer from one side of the plate only. The density, thermal conductivity and heat capacity of the polymer solution may be approximated as those of water at the same temperature. 

Nandeppanavar, M. M. Vajravelu, K. Abel, M. S. (2011). Heat transfer in MHD viscoelastic boundary layer flow over a stretching sheet with thermal radiation and non-uniform heat source/sink. Comm. Nonlinear. Sci and Num. Simulation, 16, pp. 3578-3590, ISSN 1007-5704. 

A boundary layer is a simplified description of a system fluctuating with time. The concept of an unstirred layer was introduced by Noyes and Whitney (Elwell and Scheel 1975). There are three terms with a simple relationship between them - the solute diffusion boundary layer , the thermal diffusion boundary layer and the hydrodynamic momentum boimdary layer, which is a layer of a solution considered as stagnant because of adhesion to the crystal surface while the remainder of the solution is flowing past this surface. The solute diffusion boundary layer has an important physical meaning in the subsequent considerations. It is common to use this concept with reference to a flat crystal surface growing uniformly in a supersaturated solution. In the following sections transport phenomena at the interface as well as in the surrounding hquid will be discussed. 

Movement of a soluble chemical throughout a water body such as a lake or river is governed by thermal, gravitational, or wind-induced convection currents that set up laminar, or nearly frictionless, flows, and also by turbulent effects caused by inhomogeneities at the boundaries of the aqueous phase. In a river, for example, convective flows transport solutes in a nearly uniform, constant-velocity manner near the center of the stream due to the mass motion of the current, but the friction between the water and the bottom also sets up eddies that move parcels of water about in more randomized and less precisely describable patterns where the instantaneous velocity of the fluid fluctuates rapidly over a relatively short spatial distance. The dissolved constituents of the water parcel move with them in a process called eddy diffusion, or eddy dispersion. Horizontal eddy diffusion is often many times faster than vertical diffusion, so that chemicals spread sideways from a point of discharge much faster than perpendicular to it (Thomas, 1990). In a temperature- and density-stratified water body such as a lake or the ocean, movement of water parcels and their associated solutes will be restricted by currents confined to the stratified layers, and rates of exchange of materials between the layers will be slow. 

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