Chemical Engineering February 2014 - 37

1.2
Iso
0.2
0.4
0.6
0.8
1
1
2
3
4
fL/D
why isothermal flow is more conservative
in term of pressure-drop
calculations. Figure 7a also shows
that the temperature drop for adiabatic
flow is very small (<0.1% when
fL/D < 50).
So we may conclude that the
temperature remains constant at
a small inlet Mach number (Ma1),
even if the flow is adiabatic. Comparing
Figure 7a and Figure 4, we
can understand why the solid and
dashed lines overlap each other
when Ma1 is less than 0.3.
Critical mass flux
Critical mass flux under isothermal
flow is defined as the maximum
mass flowrate per unit duct
area at Mi1 = 1. It is reported by
Lapple [8] and referenced in API
Standard 521 [1] that the critical
mass flux of adiabatic flow (where
γ = 1.4) is 12.9% higher than that
of isothermal flow under the same
inlet conditions. However, Lappel's
model represents gas expansion
from an infinite reservoir
through a frictionless convergent
nozzle, which has an inlet velocity
equal to zero. This assumption is
not valid for gas flowing in pipelines.
With the equations derived
in this article, we can calculate
and compare the critical mass flux
for isothermal and adiabatic flow
and find that the reported value of
12.9% is not accurate. The detailed
calculations are presented below.
First, we define critical mass flux
under isothermal flow at Mi1 = 1
as the maximum mass flux (Gmax)
in Equation (28).
Gp
max = 1
M
RT
5
6
7
8
= 1.4
= 1.3
= 1.2
Figure 8. The
ratio of critical
mass flux to maximum
mass flux as
a function of pipe
length varies with
specific heat ratio
Based on Equations (14) through
(17), we arrive at Equation (34).
G
G
ca == =
2
β
max
γα
β
γα
Ma
Ma
1
2
a
a
γ
(34)
At critical flow, Ma2a is equal to
1, so Equation (34) is simplified
to Equation (35).
Gca
G
max
==1
2
Ma
Ma
a
a
γγ1Ma a
(35)
1
(28)
At any pipe length, Gmax under
isothermal flow can be expressed
as Equation (29).
Gp
ci ==
ρυ
ii
22 2i
M
RT
1
(29)
Where p2i and υ2i are outlet pressure
and velocity when the conditions
in Equation (30) are true.
Mi2
==1
υ
2i
RT
M
1
(30)
Therefore, the ratio of critical mass
flux at any pipe length to maximum
mass flux under isothermal flow is
given in Equation (31).
G
ci
G
max
=
p
p
2i
1
(31)
By solving Equation (10) and setting
Mi2 equal to one, p2i/p1 can
be determined. At any pipe length,
the critical mass flux under adiabatic
flow can be expressed as
Equation (32).
Gp
==
ca aa a
ρυ
22 2
γM
RT2a
(32)
The ratio of critical mass flux
under adiabatic flow to maximum
mass flux under isothermal flow is
shown by Equation (33).
Gca
G
max
=
p2a
p1
γ
T1
T2a
(33)
G
G
ca
ci
= γ
(37)
At γ = 1.4, Gca/Gci is equal to
1.183, so the critical mass flux
under adiabatic flow is 18.3%
higher than that under isothermal
flow, which is different from the
reported value of 12.9% from API
Standard 521.
Isothermal versus incompressible
In deciding whether to apply the
incompressible flow equation for
compressible fluid flow conditions,
engineers often rely on certain rules
of thumb. However, some of these
rules of thumb can be quite misleading
[9]. In the following section, we
examine the difference between isothermal
and incompressible equations
for pressure-drop calculations.
For an ideal gas flowing through
a horizontal pipe, Equation (38)
shows that the total pressure drop
is the summation of pressure drop
caused by friction and acceleration.
ChemiCal engineering www.Che.Com february 2014 37
In Equation (35), Ma1a can be calculated
by solving Equation (13) at
any fL/D and setting Ma2a equal to
one. Equations (31) and (35) are plotted
in Figure 8 for three fluids with
different specific heat ratios. At any
pipe length, the ratio of critical mass
flux of adiabatic flow to isothermal
is given by Equation (36).
G
G
ca
ci
G
G
==
max
ca Gmax
Gci
Ma1a γ
p1
p2a
(36)
Equation (36) can be simplified
to Equation (37) since p1/p2a and
Ma1a equal one when the square
root of fL/D equals zero.
Gc/Gmax
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Chemical Engineering February 2014

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