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David C. Silverman |
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The Momentum Boundary Layer and Related Friction Factor Many corrosion situations under the influence of heat transfer also occur when fluid is in motion, whether by forced or natural convection. An understanding of fluid mechanics and boundary layer theory is a necessary underpinning for any discussion of heat transfer and its effect on corrosion. The hydrodynamic boundary layer affects the thermal (heat transfer) boundary layer. The latter boundary layer controls heat transfer in flowing sytems. As shown by the Reynolds analogy, knowledge of the friction factor enables estimation of the heat transfer coefficient. The Reynolds analogy provides a way to estimate the heat transfer coefficient from the hydrodynamic friction factor. Thus, an understanding or at least awareness of the characteristics of the hydrodynamic boundary layer is fundamental to understanding heat transfer and ultimately corrosion under heat transfer conditions in flowing systems. This awareness is also critical for assessing the applicability of laboratory simulation tools for predicting performance under heat transfer conditions in dynamic systems. The purpose of this section is to provide an introduction and summary to serve as background for the other parts of this tutorial. Numerous textbooks provide more detail.Under normal conditions, when a fluid moves past a stationary, hydraulically smooth surface an implicit assumption is that the fluid touching that surface is stationary or, as in the case of the rotating cylinder electrode moves at the velocity of the wall. That assumption is often called "no slip at the wall". In terms of a staionery wall, the distance over which the fluid velocity changes from 0 to the free stream velocity is called the hydrodynamic boundary layer. Under laminar flow conditions, the boundary layer is laminar at all positions. When the flow is turbulent, the hydrodynamic boundary layer may be represented in its simplest form as having two sections, a viscous sublayer and a turbulent outer layer with a transition region between the two. When the flow is turbulent, the velocity profiles in those regions are time-averaged profiles. A full discussion of the meaning of the boundary layer and the differences between laminar and turbulent flow can be found in the literature one of the best of which is H. Schlichting, Boundary-Layer Theory, 7th edition, McGraw-Hill Book Co., New York, 1979. Internal Flow (Pipe or Equivalent) - Turbulent Flow An example of internal flow is fluid flowing in a pipe or annulus. In pipes, turbulent flow has been traditionally assumed to occur at values of the Reynolds number above 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. This figure ) .
shows the approximate velocity profile for a Reynolds number of about 105.
The profile in this figure is time averaged. Turbulent flow creates
eddies that blur the profile demarcation. The profile would be sharper if the flow were laminar.
The velocity profile at this Reynolds number follows an approximately 1/7 power
law in that the velocity profile is proportional to the distance from the wall
raised to about the 1/7 power. This power is a function of Reynolds number. The
fluid velocity increases from 0 at the solid-fluid interface to a value close
to the maximum value across a relatively short distance into the fluid.
The velocity actually increases all of the way to the centerline which can be
considered the point at which the boundary layers from all radial positions
meet. But, in reality, most of the change in velocity occurs over a relatively
short distance. This change in velocity is a result of momentum transfer
between the fluid and wall. The boundary layer thickness is independent of
position on the surface once that position is free of entrance effects. The
result is that the velocity profile, momentum transfer,
and wall shear stress are independent of position.
The velocity profile that exists between the wall and centerline may be modeled as three distinct regions. Very close to the wall, the flow is purely laminar (the laminar sublayer). The next region is a transition region between laminar and turbulent flow. The final region which comprises most of the flow is turbulent flow. The momentum transfer of the flowing fluid to the wall creates a shear stress at the wall that is felt as the pressure drop in the fluid. This shear stress describes the momentum transfer between the fluid and the wall. The wall shear stress is often expressed in terms of a dimensionless number called the friction coefficient "f". The relationship between shear stress and friction factor for flow in a pipe or comparable geometry is given by 
(1)
where τ is the shear stress at the wall, f is the friction coefficient, ρ is the fluid density, and V is the mean velocity of the fluid (volumetric flow divided by cross sectional area). The friction factor can be defined in terms of the Reynolds number for a hydraulically smooth pipe as 
(2)
where f is the friction factor and Re is the Reynolds number in which V is the characteristic velocity and the diameter of the pipe is the characteristic dimension. Equation (2) is a bit unwieldy. Between Reynolds numbers of 30000 and 1000000, equation (2) can be approximated by 
(3)
and between Reynolds numbers between about 5000 and 30000, the friction factor can be approximated by 
(4)
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. The region over which the velocity increases from zero to the free stream velocity (Vfs) is the hydrodynamic boundary layer. This boundary layer increases in thickness from the leading edge of the external flow surface. This figure shows an example of this type of momentum boundary layer ) .
The flow in this geometry is laminar until large values of Reynolds numbers are reached. In this case, the characteristic dimension used in the Reynolds number is the distance from the leading edge of the flat plate. Since the boundary layer increases in thickness with distance from the leading edge, the friction factor is a function of position. The mean friction factor from the leading edge to the position x along the plate is given by 
(5)
Equation (5) was developed on the assumption that the boundary layer remains "thin" so that the shear parallel to the plate 
where y is perpendicular to
the plate and the velocity gradient in this direction is always very much greater
than any other velocity gradient. This
assumption means that the displacement boundary layer δ is much less than
x. The relationship between δ and x is
(6)Thus, Rex>>1 for the boundary layer approximation that led to equation (5) to be valid,. The assumption breaks down close to the leading edge. Solutions similar to those above can be developed for surfaces which are not flat plates but are wedges shown in this figure ) .
The velocity profile is proportional to the position on the surface raised to
the power m. The friction factors as a function of x in the figure are given by
(7)where K is a function of β shown in the above figure. The free stream velocity (velocity at infinite distance) is proportional to the distance raised to a power m. Values for K in equation 7 and m are shown in the following table for selected values of β
As with flow in a pipe, the laminar boundary layer becomes unstable if the Reynolds number becomes sufficiently large. For a flat plate in which the free stream velocity is constant, the laminar boundary layer is present for Reynolds numbers up to about 80000 in which the characteristic distance is the distance measured from the leading edge. It can persist to Reynolds numbers of several million if the free stream turbulence is very low. A reasonable assumption is that for smooth surfaces the transition occurs for Reynolds numbers between about 200000 and 500000 but those numbers must be used with caution. Two expressions that have been developed for the friction factor under turbulent conditions are the somewhat theoretical equation 
(8)and the more empirical 
(9)Equation (9) seems to fit the data better at higher Reynolds numbers. Previous Page: Introduction and Overview Next Page: The Thermal (Heat Transfer) Boundary Layer and Heat Transfer Coefficient |
David C. Silverman, Ph.D. - Primary Consultant |