Chapter 21 Theoretical Rates in Closed-Channel Flow

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Chapter 21 Theoretical Rates
in Closed-Channel Flow
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Many a time did I stand such a pipe and exert
myself to invent how to force these pipes so
reveal the secret of their hidden action.
Clemens Herschel (1898)
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21.1 General Remarks In closed-channel flow (
as in pipes, ducts, etc.), the system is usually
full of fluid, and consequently, the fluid is
completely bounded. For one-dimensional steady
flow, continuity can be expressed as
(21.1)
(21.2)
Under the same conditions, a general energy
accounting for a reversible thermodynamic process
(i.e., when mechanical friction and fluid
turbulence are negligible) yields, in the absence
of mechanical work and elevation change.
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(21.3)
Geometric considerations indicate the
usefulness of the definitions
(21.4)
(21.5)
It is necessary also to consider the very
important differences that exist between the
highly compressible gases and the relatively in
compressible liquids 1-4. They must be
considered separately when evaluating densities,
velocities, and flow rates.
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It is conventional in a study of flow rates
to examine theoretical relations first. In the
interest of simplicity, we also idealize the
fluids so that a liquid is taken to exhibit
constant density, whereas it is assumed that the
equation of state of a gas is given by
(21.6)
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21.2 CONSTANT-DENSITY FLUIDS Equation(21.3),
when integrated between two arbitrary positions
for the constant-density case, yields
(21.7)
Which is a form of the Bernoulli equation.
Combining equation(21.2),(21.5), and (21.7)
results in
(21.8)
Thus the theoretical rate of flow of a
constant-density fluid in a closed channel is
(21.9)
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Where mideal is in lbm/s, A2 is in2, g is in
lbm-ft/lbf-s2, ?is in lbm/f t2,and p is in
lbf/m2 .
The flow rate is to be directly proportional to
the square root of the pressure drop
p1-p2.however,the pressure drop across a constant
area section will be very small indeed, even in
the presence of frictional losses. To obtain a
measurable pressure drop, the flow is usually
obstructed in a manner similar to the way in
which open-channel flow is obstructed.
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The obstruction and the required static
pressure taps make up the closed-channel fluid
meter. The venturi, the nozzle, and the
square-edged orifice plate (and their associated
pressure taps) are the most common closed-channel
fluid meters, although porous plugs or simple
restrictions in the walls of a flow tube can
suffice to establish a suitable pressure drop
(Figure 21.1)
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21.3 Compressible Fluids When the
thermodynamic process between two arbitrary
positions in a system is isentropic (i.e, when
there is no heat transfer, no mechanical
friction, no fluid turbulence, and no
unrestrained expansion), the ideal gas of
equation(21.6) also can be characterized by
(21.10)
Where r is the static pressure ratio p2/p1 and
r is the ratio of specific heats cp/cv. The
general energy equation(21.3) under these
conditions can be integrated to yield
(21.11)
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Which with equation(21.2)can be expressed as
thus the theoretical rate of flow a compressible
fluid in a closed channel, according to
equations(21.1), (21.10), and (21.12), is
(21.12)
(21.13)
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For the same units as in equation (21.9).note
that in equations(21.10)through (21.17)the
velocities and densities are those based on an
isentropic process between the total pressure of
equation(21.23)and the thermodynamic state of
interest. equation(21.13)also can be given in the
useful form(3),(4)
(21.14)
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(No Transcript)
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• FIGURE 21.1
• Types of fluid meter for closed-channel flow.
• Herschel-type venturi tube.
• Long-radius flow nozzle.
• HEI flow nozzle.
• Square-edged orifice.
• Porous plug flow meter.
• Restrictive-type flow meter.
• (source from ASME(5))

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With the same units as given under
equation(21.9), except that R is in lbf-ft/lbm-oR
and Tc is in oR. Alternatively, if the general
energy equation(21.3) is integrated between the
actual throat static pressure and the isentropic
total pressure of equation(21.23), we have the
general relation
(21.15)
which can be expressed as
(21.16)
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Thus the theoretical rate of flow of a
compressible fluid in a closed channel is,
according to equation(21.1),(21.10), and
(21.16)3,4,
(21.17)
In terms of the generalized compressible flow
function ?, which has been defined 4,6 as
(21.18)
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Equation(21.17) also can be given in the
simplified form
(21.19)
Note that in equations(21.18) and (21.19) the
actual total pressure at meter inlet is used. The
constant in equation(21.19) is simply
(21.20)
which takes values at standard gravity
conditions of
(21.21)
(21.22)
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• For brief tablulations of the t function
  see table 21.1. For more complete tabulations
  see(4),(6). Note that p in equation(21.1) is the
  isentropic total pressure in the fluid meter,
  defined in general as (7),(8)

(21.23)
In the ideal case Cc is usually set equal to
unity.
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21.4 CRITICAL FLOW RELATIONS The flow rate of
a compressible fluid was seen equation(21.13)
to be dependent in general on the ratio of the
downstream static P2 to the upstream static
pressure P1. The variation in flow rate with
changes in the static pressure ratio is important
in studying the critical flow of gases through
nozzles. First the square of the isentropic
flow rate equation(21.13) is differentiated
with respect to r to obtain
(21.24)
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Theoretical rates in closed-channel flow The
critical static-pressure ratio (the one that
yields the maximum isentropic flow rate for given
fluid conditions at inlet and for a given
geometry) is obtained by setting equation(21.24)
equal to zero. The result is
(21.25)
When the asterisk signifies the condition of
maximum flow rate. Note that if the geometry is
such that ß-gt0,then p1-gtp0,and equation(21.25)
leads to the familiar critical point function of
thermodynamics
(21.26)
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Thus theory reveals and experiment agrees
that the flow rate of a convergent nozzle (where
CC1) attains constancy and is maximized at the
critical pressure ratio, equation(21.25). at this
critical pressure ratio, the fluid velocity
equals the local velocity of sound, and the flow
no longer responds to changes in the downstream
pressure8. Although in the case of a flow
nozzle the throat static pressure is called for
in equations(21.13)-(21.23),it is customary (and
usually preferred, see, for example, 9)to
measure the lower pressure in the larger diameter
discharge pipe. This is usually called the back
pressure Pb.If the flow is subsonic, p2 can be
taken as the back pressure.
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On the other hand, if the nozzle is choked
(i.e, if for a given inlet pressure the flow is
maximum and also independent of the back
pressure), the throat static pressure must be
greater than the back pressure. In fact,
whenever the measured static pressure ratio Pb/P1
is less than or equal to r of equation(21.13)-(21.
23),is r of equation(21.25).on the other hand, if
Pb/P1. Venturis also are operated as critical
flow meter with certain advantages noted in the
literature10. To verity that critical flow
conditions exist in the venturi, it is only
necessary to show that throat conditions are
independent of the overall pressure ratio across
the venturi.
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Contrary to the behavior of the convergent and
convergent-divergent passages of nozzles and
venturis, the square-edged orifice meter does not
exhibit a maximum flow rate. For example,
Perry11and Cunningham 12both indicate that
the flow rate (for constant upstream conditions)
continues to increase at all pressure ratios
between the critical ratio of equation(21.25)and
zero. This range is thus defined as the
supercritical range of pressure ratios. The
study of critical flowmeters for compressible
flow measurements is a complex and rapidly
changing subject for which a rapidly growing
literature is developing7, 10- 14.