Hydrocarbon Processing - May 2021 - 46

Maintenance and Reliability
TS
(2)
T =
k −1
× M2 + 1
2
M = V / c(3)

10.73 (ft3 × psia)/(lbmol × R)
[8.314 (m3 × Pa)/(mol × K)]
By applying the proper unit conversion
necessary for U.S. customary units, Eq. 4
can be simplified to Eq. 5.

Similar equations (Eqs. 6 and 7) can
be derived that pertain to the fluid's pressure and density within the system to account for all other variations from isentropic effects:

kRTS
c =
(4)
MW

kTS R ′
(5)
c =
MW
where:
R' = converted universal gas constant;
49,713 ft2/sec2
Since the Mach number is influenced by a
direct relationship with the velocity of the
fluid, an increase in velocity will result in
a lower fluid temperature, which was observed through FLIR imaging of the hydrogen valve. Once the velocity of the fluid slows, the system's energy is maintained
and is returned to the fluid temperature.

T k−1
k −1
PS
k−1
= ⎛ S ⎞ = ⎛⎜
× M 2 + 1⎞⎟
(6)
⎝T⎠
⎝ 2
⎠
P

where:
Ts and T = the stagnation temperature
and the high-velocity temperature;
°R (K)
V = velocity of the fluid; ft/sec (m/sec)
c = speed of sound of the fluid;
ft/sec (m/sec)
M = Mach velocity of the fluid
k = ratio of specific heats
(Cp /Cv at the Ts location)
R = the universal gas constant;

60
100

60
40

30
20

20

10

0

0.2

0.4

0.6

0.8

Mach velocity
FIG. 3. Isentropic ΔT (°F or °C) vs. Mach velocity for k-values ranging from 1.1-1.5.

46 MAY 2021 | HydrocarbonProcessing.com

1

Isentropic ΔT, °C

Isentropic ΔT, °F

50
40

1

1

where:
Ps and P = the stagnation pressure and
the current pressure; psia (Pa)
ρs and ρ = the stagnation fluid density
and the current fluid density;
lb/ft3 (kg/m3)
Hydraulic modeling can be used to determine the Mach number for the flow reading in addition to the relevant k values to
find the pressures, densities and temperatures for a fluid.
Calculation vs. FLIR imaging. For
the hydrogen valve, the FLIR camera recorded a temperature of 65.4°F. By using
hydraulic modeling for the flow reading,
the Mach number was found to be approximately 0.4, with a k-value of about 1.42.
Eq. 2 is used with a stagnation temperature
of 81.3°F to yield the following results:
81.3 + 458.67
− 458.67
1.419 − 1
× 0.4 2 + 1
2

TABLE 1. K-values for fluids at 14.7 psia
and 100°F or at fluid's dew point
if above 100°F

FIG. 2. Temperature increases approximately 3-5 pipe diameters from the reducer, as the
velocity of the fluid decreases from top left to the bottom right.

k = 1.1
k = 1.2
k = 1.3
k =1.4
k = 1.5

k

T k−1
k −1
ρS
k−1
= ⎛ S ⎞ = ⎛⎜
× M 2 + 1⎞⎟
⎝T⎠
⎝ 2
⎠
ρ
(7)

T=

80

k

Fluid

Cp /Cv

Hydrogen

1.413

Helium

1.666

Nitrogen

1.4

Carbon dioxide

1.279

Hydrogen sulfide

1.326

Methane

1.298

Ethane

1.186

Propane

1.127

i-Butane

1.097

n-Butane

1.097

i-Pentane

1.081

n-Pentane

1.081

Water

1.331

Air

1.409

Source: Proprietary softwarea simulation using
Peng-Robinson equation of state.
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Hydrocarbon Processing - May 2021

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