Chemical Engineering January 2015 - 49

y
Engineering Practice
to the pipe wall, and the maximum
velocity is approached in the center
of the pipe. For turbulent flow, this
boundary layer consists of a laminar
sublayer, a logarithmic outer
layer and a buffer zone of some
nominal thickness that transitions
continuously between these sublayers
[2]. The resultant velocity profile
for a constant-area pipe in fully
developed turbulent flow is shown
in Figure 1. All variables, subscripts
and superscripts presented in the
equations and figures are defined
in the Nomenclature section.
It is also useful to remember that
the majority of resistances to flow
within a pipe - the frictional losses
themselves - are actually due to
effects of this boundary layer, and
that the friction factor is used as
a means to determine these losses
as a function of the mean velocity
in the pipe. In addition, while a rigorous
analysis of the energy and
momentum balances that includes
the boundary-layer effects would
find that a slight modification of
the terms would be required for
turbulent flow (for instance, a momentum-correction
factor of 1.018
for the logarithmic-law profile in
turbulent flow), these correction
factors are commonly ignored in
engineering calculations. They tend
to cancel out from mathematical
expressions, and are minor when
compared to the loss terms [2]. As
a result, the typical energy and momentum
balances as employed in
engineering hydraulic calculations
are still valid, even when recognizing
the actual velocity profile.
For present purposes, we are concerned
with the temperature profile
that exists as a result of this actual
velocity profile. However, before
evaluating this, it is useful to also
address heat-transfer considerations.
For heat transfer within a
pipe, a temperature gradient exists
between the wall and the center of
the pipe. In general heat-transfer
calculations, it is convenient to define
a bulk temperature. The bulk
temperature is generally defined
as the effective temperature across
the gradient that drives the heat
transfer with a wall of a given
Hydraulic
boundary layer
thickness
Uw = 0
Insulated wall
U
U
Inviscid
Viscous
U(y)
Turbulent layer
Buffer zone
Thin laminar sublayer
Velocity
Figure 1. The velocity profile near the wall of a pipe shows that both turbulent and
laminar conditions are present [2]
y
T
U››/2Cp
2
T
Temperature
Profile
Thermal
boundary layer
thickness
( )


T
y
Insulated wall
Taw
= 0
w
Temperature
Figure 2. The temperature profile near the pipe wall is affected by the velocity gradient
within the pipe [5]
temperature. The normal calculations
then proceed using this bulk
temperature, and general experience
is based on the most common
heat-transfer situations - those
that deal with phase transitions (in
which case the effective temperature
is essentially fixed at the saturation
temperature) or liquid flow
(in which case the effective temperature
is taken as the average of the
flowing temperatures). Heat transfer
in gas flow is not as common,
and when it does occur, the velocity
is typically reduced. As a result, the
accumulation of experience with
heat-transfer design at petrochemical
processing facilities does not
typically include heat transfer with
high-velocity gas streams.
Adiabatic wall temperature
An area of practice that does offer
experience with this situation is
aeronautical engineering. A summary
of the cumulative theoretical
and experimental work in this
area, provided by Eckert [3, 4], indicates
that for high-speed gas flow,
the fluid temperature at the wall
is significantly greater than the
48 ChemiCal engineering www.Chemengonline.Com JanUarY 2015
static (flowing) temperature. Also
described in literature by Shapiro
[5], is the phenomenon of the temperature
profile of a gas in response
to the actual velocity profile, where
the steady-state temperature distribution
shows an adiabatic wall
temperature Taw being greater than
the free-stream temperature T», yet
less than the free-stream stagnation
temperature, T¤. This temperature
profile is shown in Figure 2.
The experimental work in this
area has confirmed that the adiabatic
wall (recovery) temperature
approaches, but does not reach, the
stagnation temperature. A recovery
factor is thus defined as the amount
of the stagnation temperature that
is recovered as the fluid decelerates
to zero velocity at the wall, based on
the actual temperatures achieved
[3]. Equation (1) shows an expression
for the recovery factor, r.
(1)
The subscripts aw, », and ¤ are
for the adiabatic wall, static and
stagnation temperatures, respectively.
The recovery factor, r, has
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Chemical Engineering January 2015

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