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Unformatted text preview: practical conditions, the flow in a
tube is laminar for Re 2300, turbulent for Re 10,000, and transitional in
between. That is,
2300 Die trace (86) where Ac is the cross sectional area of the tube and p is its perimeter. The
hydraulic diameter is defined such that it reduces to ordinary diameter D for
circular tubes since a Laminar 4Ac
p Dh a 4a2
Dh =
=a
4a (85) where m is the mean fluid velocity, D is the diameter of the tube, and
/ is the kinematic viscosity of the fluid.
For flow through noncircular tubes, the Reynolds number as well as the
Nusselt number and the friction factor are based on the hydraulic diameter
Dh defined as (Fig. 8–4) D Square duct: mD Re
Re
Re 2300
10,000
10,000 laminar flow
transitional flow
turbulent flow In transitional flow, the flow switches between laminar and turbulent
randomly (Fig. 8–5). It should be kept in mind that laminar flow can be
maintained at much higher Reynolds numbers in very smooth pipes by
avoiding flow disturbances and tube vibrations. In such carefully controlled cen58933_ch08.qxd 9/4/2002 11:29 AM Page 423 423
CHAPTER 8 experiments, laminar flow has been maintained at Reynolds numbers of up to
100,000. 8–3 I THE ENTRANCE REGION Consider a fluid entering a circular tube at a uniform velocity. As in external
flow, the fluid particles in the layer in contact with the surface of the tube will
come to a complete stop. This layer will also cause the fluid particles in the
adjacent layers to slow down gradually as a result of friction. To make up for
this velocity reduction, the velocity of the fluid at the midsection of the tube
will have to increase to keep the mass flow rate through the tube constant. As
a result, a velocity boundary layer develops along the tube. The thickness of
this boundary layer increases in the flow direction until the boundary layer
reaches the tube center and thus fills the entire tube, as shown in Figure 8–6.
The region from the tube inlet to the point at which the boundary layer
merges at the centerline is called the hydrodynamic entrance region, and the
length of this region is called the hydrodynamic entry length Lh. Flow in the
entrance region is called hydrodynamically developing flow since this is the
region where the velocity profile develops. The region beyond the entrance region in which the velocity profile is fully developed and remains unchanged
is called the hydrodynamically fully developed region. The velocity profile
in the fully developed region is parabolic in laminar flow and somewhat flatter in turbulent flow due to eddy motion in radial direction.
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 will assume the
surface temperature. This will initiate convection heat transfer in the tube and
the development of a thermal boundary layer along the tube. The thickness of
this boundary layer also increases in the flow direction until the boundary
layer reaches the tube center and thus fills the entire tube, as shown in
Figure 8–7.
The region of flow over which the thermal boundary layer develops and
reaches the tube center is called the thermal entrance region, and the length
of this region is called the thermal entry length Lt. Flow in the thermal entrance region is called thermally developing flow since this is the region where
the temperature profile develops. The region beyond the thermal entrance region in which the dimensionless temperature profile expressed as (Ts T)/
(Ts Tm) remains unchanged is called the thermally fully developed region.
The region in which the flow is both hydrodynamically and thermally developed and thus both the velocity and dimensionless temperature profiles remain unchanged is called fully developed flow. That is,
Velocity boundary layer Velocity profile r x
Hydrodynamic entrance region Hydrodynamically
fully developed region FIGURE 8–6
The development of the
velocity boundary layer in a
tube. (The developed mean velocity
profile will be parabolic in laminar
flow, as shown, but somewhat
blunt in turbulent flow.) cen58933_ch08.qxd 9/4/2002 11:29 AM Page 424 424
HEAT TRANSFER
Thermal
boundary layer
Ti FIGURE 8–7
The development of the
thermal boundary layer in a tube.
(The fluid in the tube is being cooled.) Temperature profile Ts x
Thermal
entrance region Thermally
fully developed region (r, x)
0 →
x
Ts(x) T(r, x)
0
x Ts(x) Tm(x) Hydrodynamically fully developed:
Thermally fully developed: (r) (87)
(88) The friction factor is related to the shear stress at the surface, which is related to the slope of the velocity profile at the surface. Noting that the velocity profile remains unchanged in the hydrodynamically fully developed
region, the friction factor also remains constant in that region. A similar argument can be given for the heat transfer coefficient in the thermally fully developed region.
In a thermally fully developed reg...
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This note was uploaded on 01/28/2010 for the course HEAT ENG taught by Professor Ghaz during the Spring '10 term at University of Guelph.
 Spring '10
 Ghaz

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