Chemical Engineering May 2012 - 66

0.04
Displacement plot
Engineering
programmed using macros
available in a standard
spreadsheet.
0.035
0.03
0.025
0.015
0.02
0.005
0.01
Example problem
The method has been applied
to vibrations in the discharge
piping (8 in. nominal
bore) leading from a refinery
fuel gas (RFG) screw compressor up to
the oil separator. Figure 2 shows the
model for numerical simulation. The
rotor frequency is around 3,000 rpm.
Heavy vibrations, along with failures,
in the small-bore connections have
been reported. The goal was to study
the problem and provide a solution for
reducing vibration levels and preventing
such failures in the future.
Vibration measurements were
180 184 188 192
Frequency, Hz
196 200 204 208
FIGURE 7. Results of the proposed method
match those using inite element analysis
As a check, a direct solution
(benchmark) based on calculated
end moments was obtained
through finite element
analysis (FEA) by standard
commercial software. The results
show a close match with
those of the proposed method
(Figures 7 and 8).
Figures 5 and 6 show the
taken at the points of failure. An FFT
displacement plot of a point in the
mid-span is shown in Figure 3. There
is a peak at 200 Hz (that is, four times
the running speed), which is typical
of screw compressor pulsations. The
vibration velocity is around 62 mm/s,
which is much higher than the ASME
limit of 12.7 mm/s [5]. Hence for a comprehensive
design check, the actual
stresses and the support reactions are
required. Also, there is no excitation
source of forces in the span. The excitations
are by the end moments.
Numerical simulation
FFT plots of displacements at points
0.25 and 0.5 of the pipe span have
been considered as inputs. As the
quantities are complex, both modulus
and phase were required. From Equation
(14), the coefficients are solved.
The plots of coefficients A to D are
shown in Figure 4. On their basis,
the response (the stress and end reactions)
were calculated (Figures 5 and
6). As a part of the error analysis, a
random error with a peak magnitude
of 1% was introduced into the measurements.
The exercise was repeated
and the resultant plots are also shown
in Figures 5 and 6 for comparison.
Reduction of vibration stress
The plots (Figures 4-6) show peaks
at 200 Hz, which is the excitation frequency
due to pulsations generated
by the compressor. The stresses are
high and exceed the endurance limit.
results after the introduction
of the error. The variation in
the solution is about the same
order of magnitude of the
maximum error, which is also
in agreement with the theory.
A distinguishing feature
FIGURE 9. A view of the inal coniguration of the
piping shows additional pipe supports
of this method is that no information
is required on the
natural boundary conditions
(BCs). This is remarkable
since in the direct theory, the
solution depends on the BCs,
whereas in this inverse problem,
the BCs do not play a
role. This is also significant in
the sense that practically, it
is almost impossible to assess
the true support conditions.
In order to reduce the
stresses, the modes around
the observed frequency of
200 Hz were identified. The modes
were then iteratively shifted by means
of additional restraints. The end moments
were applied to determine the
stresses and the reactions. The final
configuration was achieved by further
fine-tuning considering practical constraints.
Figures 9 and 10 show the
final configuration of the piping.
Vibration readings were again
180 184 188 192
Frequency, Hz
196 200 204 208
End Reaction)
FIGURE 8. The proposed method requires no
information on natural boundary conditions
Benchmark
Present
method
4,000
2,000
6,000
Present
method
Benchmark
Reaction
at x = 0
New support
FIGURE 10. New supports can be added to reduce
vibration stresses
taken after the implementation of the
recommendations (Figure 11). The
maximum reported vibration velocity
is around 5 mm/s. The results show
a drastic reduction in the vibration
levels, which proves the success of
the resolution and vindicates the proposed
method.
48 CHEMICAL ENGINEERING WWW.CHE.COM MAY 2012
Final assessment
Vibration failure in operational piping
is a serious problem that requires
comprehensive study and analysis
to solve. In this sense, the proposed
method has tremendous practical
value. A quantitative method with
proper mathematical basis has been
provided as an alternative to the
cookbook approach.
The method provides a basis for a
proper engineering design, and can be
easily adopted by engineers involved
in troubleshooting. It should be acknowledged,
however that troubleshooting
vibrations in plant piping is
the job of a specialist with experience
in this field.
Displacement, mm
Reaction, N
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Chemical Engineering May 2012

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