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David C. Silverman |
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The Thermal (Heat Transfer) Boundary Layer and Heat Transfer Coefficient Heat will flow between a wall and the fluid adjacent to it when a temperature gradient is established between the wall and the fluid. Near the wall the fluid velocity increases from zero at the wall to the bulk velocity, sometimes not too distant from the wall relative to the radius of curvature. Likewise, the temperature changes from that at the wall to that in the free stream. The result is that the fluid temperature adjacent to the wall is assumed to be equal to the surface temperature of the wall at the interface and is equal to the bulk fluid temperature at some point in the fluid. The distance over which the temperature change occurs is called the thermal boundary layer. A momentum boundary layer also is present if the fluid is flowing past the wall. The momentum (hydrodynamic) boundary layer and the thermal boundary layer can affect each other. The distances over which the velocity changes from zero to the free stream velocity and the temperature changes from the wall temperature to the free stream temperature are often different.From a corrosion standpoint, the wall temperature or more specifically, the temperature at the wall-fluid boundary is the important parameter driving corrosion. The fluid temperature and even the average temperature in the wall could be vastly different. The two temperatures are related through an equation of the form 
where Q is the heat transferred through the wall, A is the area through which the heat is transferred,
Tfs is the free stream or bulk temperature, Tw is the wall temperature
at the fluid-wall boundary, and h is the heat transfer coefficient. The science of heat
transfer enables "h", the
heat transfer coefficient, to be estimated from the fluid properties and fluid dynamics. Once
the value of h is estimated, the interfacial temperature can be estimated (at least in principle).
The discussion below is the underpinning theory for the estimation of the heat
transfer coefficient. The next section Heat Exchangers-Analysis in the Absence of Fouling
is an example of how the heat transfer coefficient can be used in a practical situation.
The Prandtl number helps to define the relative thickness of the hydrodynamic and thermal boundary layers. This figure ) .
shows the relative thickness of these boundary layers for liquids and oils
(high Prandtl number) and liquid metals (low Prandtl number). The ultimate
relationship among the
Nusselt number ,
Reynolds number ,
and Prandtl number
depends on the relative magnitude of the Prandtl number, i.e. the
relative magnitudes of the hydrodynamic and thermal boundary layers. Large Prandtl numbers
(e.g. much greater than 1) mean that the thermal boundary layer lies well within
the hydrodynamic boundary layer. Small Prandtl numbers (e.g. much less than 1) mean that
the thermal boundary layer is thicker than the hydrodynamic boundary layer.
Internal Flow - Turbulent Flow Conditions An example of internal flow is fluid flowing in a pipe. In pipes, turbulent flow has been traditionally assumed to occur at values of the Reynolds number of about 2300. But that value depends on the “smoothness” of the pipe. For very smooth pipes, laminar flow can persist to Reynolds numbers of 10000 or more. The fluid velocity is zero at the wall and maximum at the center line. See the section entitled The Momentum Boundary Layer and Friction Factor for more details. Since the Prandtl number can vary widely depending of the fluid, much effort has been expended to develop appropriate equations as a function of Prantdl number. Whether the surface is at constant temperature or constant heat rate can affect the relationship among the Nusselt, Reynolds, and Prandtl numbers especially when the Prandtl number is less than 1. The following table gives some reasonable estimates of the relationship.
Alternative relationships exist ("Perry’s Chemical Engineers Handbook", 7th, McGraw-Hill, 1997). But, most enable the user to get a reasonable estimate of the Nusselt number, Nu, and the convective heat transfer coefficient, h, if used within the stated limits. Surface roughness can affect heat transfer in the same way it can affect hydrodynamics. The section of the tutorial in the rotating cylinder electrode entitled Effect of Surface Roughness provides a graphic example of the effect of roughness on mass transfer. One way to estimate the Nusselt number for a rough surface is 
10where Nu is the Nusselt number and f is the friction factor estimated for the smooth and rough pipe. Values of "n" of 0.5 and 1 have been reported. Data for gases seem to conform best to a value of "n" of 0.5. One complication is that estimation of the friction factor for the rough surface requires knowledge of the roughness factor, a quantity difficult to estimate under field conditions. This factor reflects the size of the protrusions above the surface. Usually, the greater the roughness factor, the greater the heat transfer through the surface to or from a fluid flowing over that surface. External Flow An example of external flow is fluid flowing along a flat surface in which the boundary layer is small relative to the distance to any other surface. The flow region is bounded by a solid surface on one side and viscous flow on the other. Though in theory the fluid velocity increases from 0 at the wall to a free stream value at infinity, in reality it reaches approximately the free stream velocity relatively close to the wall. As discussed in the section on The Momentum Boundary Layer and Friction Factor laminar flow can be present up to large Reynolds numbers based on distance from the leading edge. The relationship among the thermal and hydrodynamic boundary layers is shown in this figure for a Prandtl number greater than 1 ) .
In this figure, the subscript "fs" means "free stream".
Under laminar flow conditions and constant wall temperature, the relationship
among the local Nusselt number, the 
(11)For the case of very small Prandtl numbers: 
(12)For the case of large Prandtl numbers (e.g. Pr > 10) 
(13)The equations are a bit more approximate for intermediate values of the Prandtl number. Note that x is the position from the leading edge. The local convective heat transfer coefficient, hx, decreases with distance from the leading edge because the boundary layer thickness increases with distance from the leading edge. At the leading edge, the heat transfer coefficient becomes very large. To obtain the average heat transfer coefficient from the leading edge to the desired point on the external surface, one must integrate any of the equations (11)-(13) from zero to the desired position and divide by the distance. The result is that the average heat transfer coefficient to the point "x" on the surface is twice the local heat transfer coefficient at that point "x". Solutions similar to those for the flat plate can be developed for surfaces which are not flat plates but are wedges, the velocity profiles of which are shown in this figure ) .
The free stream velocity is proportional to the position on the surface raised
to the power m (free stream velocity along edge is proportional to the distance from the entry point
raised to the power m). The relationship takes the form of
(14)where the constant is function of the Prandtl number and exponent m. A two dimensional stagnation point exists when m = 1. In that case, the thermal boundary layer is of constant thickness along the surface. Further information on correlations for these more complex situations as well as a blunt nose axisymmetric body can be found in "Perry’s Chemical Engineers Handbook", 7th, McGraw-Hill, 1997 and W. M. Kayes,"Convective Heat and Mass Transfer", McGraw-Hill Book Company, 1966 and later editions. The important point is that the Prandtl number has a significant influence on the heat transfer relationship to hydrodynamics and practical situations in corrosion can arise in which the Prandtl number is greater than or less than 1. Previous Page: The Momentum Boundary Layer and the Friction Factor Next Page: Heat Exchangers-Analysis in the Absence of Fouling |
David C. Silverman, Ph.D. - Primary Consultant |
