Chemical Engineering February 2014 - 36

Cover Story
10
12
2
4
6
8
(a) Ma1 = 0.1
Solid line
- Isothermal
Dashed line - Adiabatic
0.995
0.996
0.997
0.998
0.999
1.000
1.001
20
40
fL/D
(b) Ma1 = 0.5
Figure 7. Velocity and temperature profiles of isothermal and adiabatic flow with known inlet conditions for two different Ma1
values show that isothermal flow assumptions are generally more conservative in terms of pressure-drop calculations
thermal and adiabatic conditions
for the pressure-drop calculation.
The results are then compared. Two
scenarios are considered in the calculation,
one with known inlet conditions,
and one with known outlet
conditions. For both isothermal
and adiabatic flow, the Mach number
(Ma) based on sonic velocity, as
defined in Equation (8), is used in
the comparison.
Scenario 1: known inlet conditions.
With inlet conditions p1, υ1,
T1 and Ma1 known, outlet conditions
p2, υ2, T2 (which is the same
as T1 for isothermal flow) and Ma2
must be calculated. Using Equations
(9) and (26), adiabatic and isothermal
curves are plotted for p2/
p1 versus fL/D in Figure 4, where
the red and blue lines represent the
boundary between subsonic and supersonic
region. In Figure 4, it is notable
that, at large Ma1 values, the
solid line is below the dashed line,
which means that calculated p2 values
for isothermal flow are smaller
than that for adiabatic flow at
the same fL/D.
In other words, isothermal flow
provides more conservative results
regarding pressure-drop calculations.
Also, it should be noted that
the solid and dashed lines nearly
overlap with each other at small
values of Ma1, implying that the
difference between isothermal and
adiabatic flow assumptions is insignificant.
The discrepancy only
becomes obvious when the Mach
number is very large (>0.3). Additionally,
the red line is above the
blue line, which shows that the
critical pressure at the pipe's outlet
(p2), for a given flowrate and pipe
length, is lower for adiabatic flow
than for isothermal flow.
Scenario 2: known outlet conditions.
In the next scenario, p1, υ1,
T1 and Ma1 under isothermal and
adiabatic flow are to be calculated
and compared using known outlet
parameters (p2, υ2, T2 and Ma2).
The result is shown in Figure 5.
Similar to the previous scenario
with known inlet conditions, the
solid line and dashed line almost
overlap with each other at small
Mach numbers (Ma2 < 0.3), implying
that the difference between the
two methods is negligible. When
Ma2 becomes larger (Ma2 ≥ 0.4), the
two lines separate and the solid line
is below the dashed line, suggesting
that isothermal flow gives a larger
p1 value than predicted by adiabatic
flow. It is also worth mentioning
that the difference between isothermal
and adiabatic flow is small
at large fL/D values (greater than
10) regardless of the Mach number.
Thus, we can apply either isothermal
or adiabatic flow equations in
this scenario.
Overall, the isothermal flow equation
is more conservative than the
adiabatic equation in terms of pressure-drop
calculations, because the
solid line is below the dashed line in
Figures 4 and 5 at most conditions.
36 ChemiCal engineering www.Che.Com february 2014
However, the question arises as to
whether isothermal flow is always
conservative, even when the solid
and dashed lines are visually inseparable.
A careful look at Figures
4 and 5 finds that the answer is
" yes " for known inlet conditions but
" no " for known outlet conditions,
in which the dashed line turns
out to be above the solid line when
p2/p1 decreases to a critical value.
This critical p2/p1 value is found to
be greatly dependent on the fluid's
molecular weight, but not on specific
heat ratio. As shown in Figure
6, critical p2/p1 continues to decrease
with molecular weight. For
hydrogen with a molecular weight
of 2 and a specific heat ratio of 1.3,
the adiabatic equation becomes
more conservative (larger p1) as p2/
p1 drops below 0.53 when the outlet
Mach number Ma2 is 0.04.
Temperature and velocity
The pressure drop and variation in
temperature and velocity along a
pipe are always interrelated. With
known inlet conditions, the dimensionless
temperature (T2/T1) and
velocity profiles (υ2/υ1) for both
isothermal and adiabatic flow are
depicted in Figure 7. At both small
and large Mach number values, it
can be clearly seen that υ2/υ1 under
isothermal flow is larger than that
under adiabatic flow at the same
pipe length (fL/D). Since higher velocity
will result in larger pressure
drop, Figure 7 indirectly explains
60
80
1.2
1.4
1.6
1.8
0.8
1
0.2
0.4
fL/D
0.6
0.8
1
Solid line
- Isothermal
Dashed line - Adiabatic
0.994
0.995
0.996
0.997
0.998
0.999
1.000
1.001
2 / 1
T2 / T1
2 / 1
T2 / T1

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Chemical Engineering February 2014

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