Chapter 21
Theoretical Rates
in Closed-Channel Flow
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)
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
m  1 AV
1 1  2 A2V2
  2   A2 
V1      V2
 1   A1 
(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.
dp VdV
0


gc
(21.3)
Geometric considerations indicate the usefulness of the
definitions
D2
(21.4)

D1
A2
 
A1
2
(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.
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
p   RT
(21.6)
21.2 CONSTANT-DENSITY FLUIDS
Equation(21.3), when integrated between two arbitrary
positions for the constant-density case, yields
V22  V12 P1 P2
 
2 gc
 
(21.7)
Which is a form of the Bernoulli equation. Combining
equation(21.2),(21.5), and (21.7) results in
12
 2g  P  P  
V2   c 1 4 2 
  1    
(21.8)
Thus the theoretical rate of flow of a constant-density fluid
in a closed channel is
mideal
 2g  P  P  
 A2  c 1 42 
 144 1    
12
(21.9)
Where mideal is in lbm/s, A2 is in2, g is in lbm-ft/lbf-s2,
ρis 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 openchannel flow is obstructed.
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)
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
2  1r1 r
(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
V22  V12    P1 P2 

  
2 gc
   1    
(21.11)
Which with equation(21.2)can be expressed as


12
 2 gc P1 1  r


V2  
    1 1 1  r 2 r  4  


 1 
(21.12)
thus the theoretical rate of flow a compressible fluid in a
closed channel, according to equations(21.1), (21.10), and
(21.12), is
mideal

 2 gc P11 r  r
 A2 
    1 1  r 2 r  4 144

2
 1 
 
12


(21.13)
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)
mideal 

A2 P1
 RT1
gc 
12

 2 r  r
1 r

2r 4 2

  1 1  r  

2
 1 
 1 

12
 
4


(21.14)
FIGURE 21.1
Types of fluid meter for closed-channel flow.
(a) Herschel-type venturi tube.
(b) Long-radius flow nozzle.
(c) HEI flow nozzle.
(d) Square-edged orifice.
(e) Porous plug flow meter.
(f) Restrictive-type flow meter.
(source from ASME(5))
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
    P1 P2 
V22




2 gc


1





(21.15)
which can be expressed as
 2 g P   P 

c 1
1   2 
V2  
  1 1   P1 




 1 
12



 

(21.16)
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],
mideal
2

A P  P 
 21 21  2 
T1  P1 

 P 
1   2 
  P1 
 1 
12
12



  g   2 
   c  

   R     1  

(21.17)
In terms of the generalized compressible flow function Г,
which has been defined [4],[6] as
12
 P    P 

  2  1   2 

  P1    P1 
 



2   1
  1  
  2 
  1 
   1


 
 

2
 1 
(21.18)
Equation(21.17) also can be given in the simplified form
mideal
 A2 P1 
  1 2  2 K
 T1 
(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
 g   2  2 
K   c  


 R     1    1 
2  1
12
  1 


   1  
(21.20)
which takes values at standard gravity conditions of
Kair  0.531748
(21.21)
Ksteam  0.408650
(21.22)
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)
 1 
1  r
 C 
Pt  P1 
2
4 2 
 1  r  Cc 
4
2
c
  1
In the ideal case Cc is usually set equal to unity.
(21.23)
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
2
d  mideal

dr
 2  2 

  A2 
 P1 1 g c  
   1 



  2    4 r  2   r 2   r  1 
2   r  2       1   r1 r 




2
2
2
4
2
4


1

r

1

r







(21.24)
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
r
1  
  1  4 2    1

 r 
2
 2 
(21.25)
When the asterisk signifies the condition of maximum flow
rate. Note that if the geometry is such that β->0,then p1>p0,and equation(21.25) leads to the familiar critical point
function of thermodynamics
   1
 P2   2 
 

P


1

 1 
(21.26)
Thus theory reveals and experiment agrees that the flow rate
of a convergent nozzle (where CC=1) 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 pressure[8].
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.
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 literature[10].
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.
Contrary to the behavior of the convergent and convergentdivergent passages of nozzles and venturis, the square-edged
orifice meter does not exhibit a maximum flow rate.
For example, Perry[11]and Cunningham [12]both 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 developing[7], [10]- [14].