Analysis of pipeline networks using two-ports

Applied Acoustics 109 (2016) 44–53

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Applied Acoustics journal homepage: www.elsevier.com/locate/apacoust

Analysis of pipeline networks using two-ports Ahmed Okasha a,⇑, Tamer Elnady a, Mats Åbom b a b

Group for Advanced Research in Dynamic Systems (ASU-GARDS), Ain Shams University, 1 Elsarayat St., Abbaseya, 11517 Cairo, Egypt Marcus Wallenberg Laboratory for Sound and Vibration Research, The Royal Institute of Technology (KTH), Teknikringen 8, 10044 Stockholm, Sweden

a r t i c l e

i n f o

Article history: Received 15 May 2015 Received in revised form 3 December 2015 Accepted 29 February 2016

Keywords: Two-port theory Pulsations in pipeline networks Pulsation suppression device

a b s t r a c t Majority of vibration problems arise in pipeline networks are attributed to the high-pressure pulsations. Pulsations are generated by fluid machines such as compressors and pumps. These pulsations turn into shaking forces at elements such as pipe bends and pipe reducers, which in turn excite vibrations in the connected piping network. High vibrations beyond the endurance limit of the pipe material may cause damage to pipes, supports, and equipment. In addition, if the source pulsation frequency coincides with one of the natural frequencies of the piping network, resonance will take place and the vibrations will be magnified to a large scale. Obviously, if these vibrations are not well controlled, they might cause damage to the whole system and foundation, and might lead to substantial financial losses. Thus, prediction of pulsations is important for safe and proper operation. In this paper, a pilot plant equipped with a reciprocating compressor, pipes, bends, and terminated by a vessel is built. The network is modeled using the two-port theory that splits the network into several cascaded elements, and predicts the response of the network. The prediction model uses the measured compressor source data as an input, which is determined by the indirect multi-load method that is usually used to characterize internal combustion engines. A pulsation suppression device is designed, modeled, manufactured and inserted into the pilot network. The pressure pulsations are measured with and without the pulsation suppression device, and compared to the predictions using the two-port theory. Ó 2016 Elsevier Ltd. All rights reserved.

1. Introduction In Oil and Gas Industry, there are usually complex installations from the well to the storage vessels. This configuration can be found onshore and/or offshore installations. Common vibration problems arise in pipelines are associated to pressure pulsations. Fluid Machines – devices that exchange work and heat with a fluid – such as compressors and pumps produce acoustical, pressure, or dynamic pulsations. Elements such as pipe bends and pipe reducers act as a coupling mechanism. At these elements, reflection take place and standing waves might be formed. The pulsation standing wave amplitude is dependent on the network geometry as well as on the frequency, amplitude, and phase of the initial pulsation wave. Shaking forces are then generated from the pulsation standing waves. The generated shaking forces excite vibrations at the piping network. If these vibration levels exceed the permissible limits, they can lead to various problems such as machinery downtime, damage to pipes, supports, foundations, and may cause explosions and fires. The problem might be even worse if one of ⇑ Corresponding author. E-mail addresses: [email protected] (A. Okasha), [email protected]. edu.eg (T. Elnady), [email protected] (M. Åbom). http://dx.doi.org/10.1016/j.apacoust.2016.02.008 0003-682X/Ó 2016 Elsevier Ltd. All rights reserved.

the pipeline network’s Mechanical Natural Frequencies (MNFs) coincides with the source excitation frequency. In this case, the resulting excitations are magnified over 30 times [1] creating dangerous impact on the piping foundation. Thus, it is important to predict the pressure pulsations to assure that they are below allowable limits. In the literature, pressure pulsations can be predicted by modeling the acoustic characteristics of the piping, the pulsations generated by the compressor, and the interaction of the two [2]. Digital and analog simulation techniques have been developed to model the pulsation generating characteristics of compressor and pump systems. While, the analog technique, developed in the 1950’, solves the differential equations by building electrical models of the piping and the compressors, the digital technique uses advanced computers to solve the differential equations with complex matrix algorithms. In 1973, the commercial software ‘‘MAPAK”, provided by ‘‘Beta Machinery” was introduced as a digital frequency based simulation software. In the late 1990’s, the software was expanded to include nonlinear time domain algorithms [1]. Another convenient technique to predict the pressure pulsations in pipeline networks is to use the building block or matrix methods [3]. The network is divided into a number of two-port elements, each described by a transfer matrix. Since plane acoustic

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waves propagate through different elements, the sound field can be characterized by two state variables, the acoustic pressure and the volume velocity. The reciprocating compressor or the pump can be described by a one-port source connected to the network through a node. The one-port can only affect the network through one opening (port). In the frequency domain, the one-port source can be characterized by two complex parameters, the source strength and the source impedance. The source data can be determined experimentally either by direct or indirect methods. The source characteristics can also be determined based on nonlinear time domain simulation of the source [4]. The building block or the two-port technique is implemented in SIDLAB software [5]. Several passive control schemes are suggested to limit the vibration levels in the piping network. First, reducing vibrations can be accomplished by modifying the source internals [2] or in other words by conducting a pulsation analysis that deals with controlling the level of pulsation forces entering the system, while not taking into consideration other dynamic forces. Another effective approach is by carrying out a mechanical analysis that deals with controlling the responsiveness of the system. This method will increase the piping network stiffness to reduce pulsationinduced vibrations [1]. This is recommended if the source excitation frequency matches one of the system’s resonance frequencies. The standard API-618 [6], presented a detailed description for three-detailed design approaches to deal with vibrations in reciprocating compressors networks. For relatively low power and low pressure piping networks, design approach one is recommended which proposes adding of a pulsation suppression device to the network. In the work discussed here, a prototype gas pipeline network that consists of a 3-HP reciprocating compressor, pipes, bends, and terminated by a vessel is constructed. The reciprocating compressor source data is determined experimentally using the indirect multi-load method. The network is modeled using the two-port theory. The pressure pulsations are measured in two different locations and compared to simulations. A pulsation suppression device is designed, manufactured, and coupled to the network. To investigate its effect in attenuating the gas pulsations, the pressure pulsations are measured after adding the suppression device. It is worthy to note that in reciprocating compressor, there are other forces generated beside the pulsation forces, such as gas forces and crosshead forces. However, the focus in this paper is on the pulsation forces only.

2. Theoretical analysis

For low frequencies, f < c/2d, only plane waves can exist in a straight pipe.

Fig. 1. The definition of a two-port element (courtesy of SIDLAB software).

is considered, and plane wave propagation can be assumed, this approach leads to the two-port transfer matrix method, as described in Ref. [3]. This method splits the system into several smaller duct parts, acoustic elements, in which the sound propagation is well defined. Since plane acoustic waves propagate through different elements, the sound field can be characterized by two state variables. One convenient choice is to use acoustic pressure (p) and volume velocity (q). The sound propagation inside each element is analyzed separately and higher order modes can exist inside the element. The definition of a two-port element is shown in Fig. 1. There exists a complex frequency dependent 2
2 matrix, (T) the two-port transfer matrix, which describes the sound transmission within a certain two-port element. The acoustic pressure and volume velocity on each side of the element can be related with the following expression:




p1 q1




¼

T 11 T 21



 p2 ps  þ T 22 q2 qs T 12

ð1Þ

where (p) and (q) are the acoustic pressure and volume velocity respectively, the subscript 1 refers to the inlet side (or node) and 2 refers to the outlet side (or node), and (Tij) are the elements of the two-port transfer matrix, (ps) is the source strength and (qs) is the source volume velocity. The definition of a one-port element is that it can only affect the network through one opening (port). It is usually connected to a single node, i.e., a connection point between one or several twoports in a duct network. A one-port connected to a node can be completely described by the source strength (either pressure source or volume velocity source) and the source impedance. If no one-port element is connected to a node, it is supposed to be an ideal node with zero source strength and infinite impedance. Fig. 2 shows the electric circuit analogies for the one-port element. The relation between pressure and volume flow for an active one-port is found as:

2.1. Two-port theory

p ¼ ps  Z s q

Duct systems or networks acting as acoustic wave-guides assume linear wave propagation suitable for analysis using the building block or matrix methods. When the low frequency regime

When all the elements (one-ports and two-ports) in a network are defined, they are connected to each other as defined by the network structure, and the sound propagation in the complete duct

ð2Þ

Fig. 2. One-port element analogy: (a) pressure source and (b) volume velocity source. Where (ps) is the pressure source strength, (qs) is the volume velocity source strength, (Zs) is the source impedance, (pL) is the load pressure, and (ZL) is the load impedance.

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system can be analyzed. If the elements are connected in cascade (‘‘a chain”), the problem is simple and the transfer matrix for the system is the result of the successive multiplication of the transfer matrices of all elements. If the elements are connected in an arbitrary fashion, the analysis developed by Glav and Åbom [7] is used to set up a global network matrix. The two-port technique is implemented in SIDLAB software, which is based on the representation of a duct network as a network of arbitrary connected two-ports. As for the flow and temperature calculation, the same SIDLAB algorithm used to calculate the sound propagation could be used to calculate the pressure drop [8] and temperature gradients [9]. 2.2. Two-port elements models Oil and Gas pipeline networks contain different set of elements such as pipes, bends, diffusers, area expansion, area contraction, and vessels plus many other so-called two-port elements. As SIDLAB software includes typical two-port elements for onedimensional acoustic, flow, and temperature calculations, it is used to predict the pressure pulsations inside a pipeline network by carrying out an acoustic modeling of the receiving system elements as well as the interaction between the source and the piping network. Passive part models (that depends on the duct geometry and speed of sound) of the elements such as pipes, diffusers, area expansion and contraction, and bends will be presented. In addition, the active part model (that describes the sound generation mechanism inside the element) for bends developed by Nygård [10] is reported. 2.2.1. Pipe The hard-walled pipe can be found in many applications. It is defined by its length and cross-sectional area. The transfer matrix can be easily deduced for this element using basic equations. The effects of viscous and turbulent damping are taken according to Allam and Åbom [11]. The transfer matrix describing the pipe is found as:

" T¼

1 2 S

2q0

 ik L  e þ þ eik L  ik L  e þ  eik L c0

q0 c0  ikþ L e 2S  ik L 1 e þ 2

 eik L  þ eik L

# ð3Þ

where (L) is the pipe length, and (S) is the cross sectional area, (q0) and (c0) are the density and speed of sound at reference temperature respectively, (k+) and (k) are the wave numbers in the positive and negative directions respectively.

whose cross-sectional areas vary from the inlet area to the outlet area as shown in Fig. 3. 2.2.3. Area expansion/area contraction These elements represent a sudden change in the cross section area of a flow duct. They are defined by the inlet area, outlet area, and the length of the extended smaller pipe (whether at the inlet or at the outlet). The analysis of sound propagation through sudden changes in the cross section area of a flow duct is complicated in the case of superimposed mean flow. This is due to the interaction between the mean flow field and the acoustic field which appears in these regions. From a mathematical point of view, the problem is that the actual mean flow velocity profile near the expansion or the contraction is too complicated to suggest any exact analytical approach. The original assumption made by Ronneberger in [12], is that the distance over which the mean flow is expanded or contracted is negligible compared to the acoustic wave length. This assumption will be used here which is based on Ref. [13]. The effects of higher modes are taken care of by using Karal’s end correction [14]. 2.2.4. Expansion chamber This is a conventional expansion chamber with concentric extended inlet and outlet. All walls but the end plates, which can be given a wall impedance, are assumed to be hard. The effects of mean flow are neglected whereas higher order modes are included. This analysis using the mode-matching technique is explained in detail by Åbom [15]. 2.2.5. Bend Nygård [10] presented a derivation for the passive and active parts models for different types of bends (mitre, normal, and smooth bend) and constrictions. In our network, there are only several 90-degree smooth bend (with R/D = 2) in the prototype network, that is shown in Fig. 4. The passive and active parts can be related using the scattering matrix Sm rather than the transfer matrix, using the following formula:




p2

Fig. 3. (a) Diffuser general layout and (b) diffuser model in SIDLAB where the element is divided cascaded pipes whose cross-sectional areas vary from the inlet area to the outlet area.




¼

S11 S21



s  p1þ p  þ 1 p2þ ps2 S22 S12

ð4Þ

where (p1+), (p1) and, (p2+), (p2) are the traveling pressure amplitudes at the inlet and outlet port respectively, and the scattering matrix is found as:


2.2.2. Diffuser This is a smooth expansion or contraction in a flow duct. A diffuser is often used to decrease the pressure drop in a flow duct network. Any effects of flow separation are assumed negligible. The diffuser is modeled by considering the element as cascaded pipes

p1

S11 S21

S12 S22

 ¼


MC L 1 2 þ MC L 2

2 MC L

 ð5Þ

where (M) is the Mach number and (CL) is the loss coefficient for the element under test. According to [16], the loss coefficient for different bends types can be calculated as a function of the bend degree, the ratio R/D, and Reynolds number.

Fig. 4. 90-Degree smooth bend with R/D = 2.

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LW ¼ Ls þ 10 log

Aq0 c30 C 2L M 4 16W ref

! þD

ð6Þ

where (LW) is sound power level traveling in one direction in the duct, (LS) is the non-dimensional scaled spectrum as function of Strouhal number (St) and bandwidth (Df), (A) is the element cross-sectional area, (Wref) is sound power reference, and (D) is a correction factor of 3 dB for bends. The sound power has a frequency width (Df) for the current analyzed frequency point (fi). Hence, the bend’s source amplitude can be written as:

ps1;2

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 10LW =10 W ref Df i ¼ Z0 Df

ð7Þ

where (Z0) is the characteristic acoustic impedance of the pipe. Because of the assumption that the noise is described as a dipole source, these pressure amplitudes are 180 degree out of phase. This model which is valid at low Mach numbers and for low frequencies. 2.3. Pulsation attenuation In Ref. [17], a review of various Pulsation Suppression Devices (PSDs) such as surge volumes, pressure drop elements and acoustic filters that have been developed in the literature are presented. In Ref. [18], a parametric study of different PSDs including the one bottle, and the two-bottle design, with different sizes, were simulated and analyzed to study the performance of each of them. The study revealed that the two-bottle design with a cut-off frequency below the first excitation frequency of the source (compressor or pump) has a significant pulsation attenuation. In addition, a study has been carried out to show that one bottle design is not cost effective as the two-bottle design. The two-bottle design or the so called volume-choke-volume filter consists of two volumes, and a choke tube in between. The two volumes act as two compliance components, and the choke tube acts as an acoustical inertance to resist changes in velocity

1

Helmohltz frequecny (fH)

Transmission coefficient [-]

In order to describe the active part, the source pressure ampli    tudes, ps1 and ps2 have to be estimated. These are the outgoing pressure amplitudes, describing the active part of the two-port, assuming reflection free terminations. The pressure amplitudes are independent of the passive part of the two-port scattering matrix. In order to determine the source pressure amplitude, the sound power level of the element under test should be determined. Nygård [10] carried out a collapse of sound power measurement for different bends and constrictions. By carrying out measurements for a specific geometry, at different flow conditions and sometimes with different duct dimensions, the generated sound pressure level is determined. By measuring the sound power level, a non-dimensional reference spectrum (Ls) can be obtained. These spectrums can be used in order to predict the sound power level and consequently the element source strength when having other similar flow situations for the same element from the relation:

0.8

0.6

0.4

0.2

0 0

50

100

150

200

250

300

Frequency [Hz] Fig. 6. Typical frequency response of volume-choke-volume filter system.

of the fluid. The combination of these acoustic elements in this manner produces a ‘‘low pass filter” which attenuates pulsation at frequencies above its ‘‘cutoff” or Helmholtz frequency [18]. Fig. 5 shows the configuration of a non-symmetrical two-bottle PSD according to API-618 [6], and its pulsation response characteristics are shown in Fig. 6. As described in Ref. [18], the internal resonances of the acoustic filter elements can have the effect of passing particular frequencies as the choke tube acts as an open-open pipe, such that a pass band occurs at the half-wave length of the choke, which can be calculated from the relation:

f ¼

c n 2L

ð8Þ

where (f) is the frequency in Hz, (c) is the speed of sound in m/s, (L) is the equivalent length of the pipe in m, and (n) is the mode number. It is therefore recommended to design the PSD such that the first pass band frequency is located at higher frequencies where the pressure pulsations become of relatively low values. A parameterization study has been carried out to study the effect of varying the length and the diameter of both the choke tube and the bottles of the PSD. A single parameter is studied while other parameters are kept fixed. Fig. 7a shows the effect of varying the choke tube diameter on the PSD transmission loss. As shown, decreasing the diameter increases the transmission loss, however, this is accompanied by an increase in the pressure drop over the PSD. On the other hand, increasing the length of the choke tube gets the Helmholtz frequency to lower frequency values, which means better attenuation performance for the PSD. However, increasing the choke tube length also moves the pass band frequencies region of the filter to the lower frequencies as shown in Fig. 7b. For the bottles, increasing the diameter (D1 or D2) increases the transmission loss as shown in Fig. 8. The effect of varying the bottle length has the same effect of varying the choke tube length.

Fig. 5. Non-symmetrical volume-choke-volume PSD schematic layout according to Annex-O in API-618.

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A. Okasha et al. / Applied Acoustics 109 (2016) 44–53 80

Transmission Loss [dB]

Transmission Loss [dB]

80

60

40

Dc = 15 mm Dc = 25 mm 20

Dc = 35 mm

0

60

40

Lc = 300 mm Lc = 400 mm 20

Lc = 500 mm

0 0

100

200

300

400

500

0

100

200

300

Frequency [Hz]

Frequency [Hz]

(a)

(b)

400

500

Fig. 7. The effect of varying the choke tube diameter (a), and length (b), on the PSD transmission loss.

Transmission Loss [dB]

80

60

D1 or D2 = 300 mm

40

D1 or D2 = 400 mm D1 or D2 = 500 mm 20

0 0

100

200

300

400

500

Frequency [Hz] Fig. 8. The effect of varying the bottle diameters (D1 or D2) on the PSD transmission loss.

3. Experimental setup The test rig consists of a two-cylinder single-stroke 3-HP reciprocating compressor that supplies a pipeline system terminated by a 200 L vessel. The test rig has several test points that are utilized to mount pressure transducers to measure the pressure pulsations. As indicated by Elnady in [19], the pressure transducers should be mounted subsequent to the source discharge with a minimum distance of three tube diameter to make sure that the non-plane waves die out before they reach the pressure transducers positions. A piezo-resistive pressure transducer (Kulite WCT-312) is used to measure the dynamic pressure. It has both dynamic and static pressure capabilities at high temperature environments. The transducers are mounted flush to the inlet surface of the pipe and are 0.47 m apart to be able to catch low frequency excitations by the compressor. The flow is measured by VA 420 thermal mass flow sensor (in-line type). The temperature is measured via thermocouple K-type probe. The rpm signal is captured by B&K Photoelectric tachometer probe MM0024 that has a useful bandwidth up to 20,000 rpm. The rpm signal is important for time synchronous averaging. The measurements are implemented through LMS SCADAS Mobile package and its associated software LMS Test.Xpress. The first step to analyze the acoustic field inside the pipeline network is to determine the characteristics of the source at the operating conditions. Any fluid machine source generates

broadband noise (flow noise) plus a few tones (fundamental mechanism). The source character (amplitude and frequency) depends on the operating state of the machine (i.e. RPM, flow rates, and gas temperatures). It is difficult to keep the rotational speed of the compressor constant; therefore, a tachometer is used to track the rotation of the compressor shaft. Its signal is recorded in the time domain together with the time domain data of the dynamic pressure sensors. The tachometer signal is used to perform a time domain synchronous averaging, and the pressure signals can be shown in the angle domain. Fig. 9 presents the procedure followed from the start of measuring the pressure pulsations up to determining the source strength and the source impedance of the reciprocating compressor. In the cases studied here, the compressor speed is 990 rpm. Since the compressor is single stroke, the first firing frequency is at 33 Hz (2nd Order). The suction pressure is the atmospheric pressure and the average discharge pressure is kept at 200 kPa. The dynamic pressure measurements are on the discharge side. This measurement is repeated for a seven different acoustic loads shown in Table 1, along with the operating conditions for each load. Obviously, these conditions should be kept as constant as possible. The first six loads are used for the least square fit to determine the source strength and source impedance, whereas the seventh load is used for validation. The source strength and the source impedance are calculated based on the analysis in Ref. [20]. The source plane is defined at the plane of the measurement point 1 (the pressure transducer away from the source). The results of Ps and Zs are shown in Fig. 10. Two cases are considered. Both are driven by the reciprocating compressor, connected to a small pipe network, which is terminated by a vessel and a valve used to regulate the pressure inside the network. The network in case 1 consists of a straight pipe and the two measurement points that are 2.1 m apart (Fig. 11a). The network in case 2 shown in Fig. 11b involves the same elements of case 1, but the long pipe has been replaced by the pulsation suppression device, that is designed according to the API-618 Standard. The designed PSD has a Helmholtz frequency of 20 Hz, a pressure drop of 254 Pa, and the first pass band frequency is at 600 Hz. The measured pressure is compared to the predicted pressure for the two cases. For each case, a comparison is done at two different points, one at the source plane, and the other is at the end of the network (before the vessel). Fig. 12 shows a schematic diagram detailing the components of the rig for the two cases.

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A. Okasha et al. / Applied Acoustics 109 (2016) 44–53

Measurement Pipe RPM Measurement

Reciprocating

Flow Measurement

Compressor

Source

Temperature Pressure

Measurement

Measurement

Pressure Signals Resampling Angle Domain Data

4

SPL [PA ]

x 10 1

Time Synchronous Averaging (TSA)

0 -1 0

50

100

150

200

250

300

350

Angle [deg]

Order Domain Analysis

100 50 5

10

15

Order [-]

Transfer Function and Load Impedance Calculation SIDLAB

Yes Load no. <=No. of Loads

No Calculate (ps) and (Zs) Source Strength (ps) 180 160 140 120 100 80 60 40 20 0

Source Impedance (Zs) Real (Zs)

0

5 0 -5

0

5

10

15

20

25

20

25

Order [-]

0

5

10

15

Order [-]

20

25

Imag (Zs)

0

Source Strength Level (dB)

PA [dB]

Order Domain Data 150

5 0 -5

0

5

10

15

Order [-]

Fig. 9. Measurement and post processing procedure for the multi-load indirect source characterization method.

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A. Okasha et al. / Applied Acoustics 109 (2016) 44–53

Table 1 Process variables recorded during measurements.

4. Results and discussion

Temperature (°C)

Flow speed (m/s)

Mach no. (–)

4.1. Case 1

1 2 3 4 5 6 7

200 200 200 200 200 200 200

45 48 49 47 48 49 51

14.5 14.3 12.5 13.9 12 13.7 12.3

0.043 0.042 0.037 0.041 0.035 0.04 0.036

The two pressure transducers used for the source characterization, are also used to calculate the load impedance using the two-microphone technique [19]. This load impedance is what the system feels looking through point 1 into the downstream direction. The load impedance can be moved from the measurement point 1 to point 2 using the following relationship of moving the reflection coefficient along a hard pipe:

Real (Z s ) [-]

Pressure (kPa)

150

2 0 -2 0

100

5

10

15

10

15

Order [-]

Imag (Zs ) [-]

Source Strength Level [dB]

Load case

50

2 0 -2

0 0

5

10

15

0

5

Order [-]

Order [-]

(a) Source strength.

(b) Normalized source impedance.

Fig. 10. The source characteristics of the reciprocating compressor.

Pulsation suppression device

(a)

(b)

Fig. 11. (a) Photo of the test rig (case 1: compressor–pipe–vessel) and (b) photo of the test rig (case 2: compressor–pipe–PSD–pipe–vessel).

Fig. 12. Layout of the test rig, (a) case 1, and (b) case 2.

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A. Okasha et al. / Applied Acoustics 109 (2016) 44–53

both points, especially for the first two excitation orders, which are the highest and consequently of more interest. 4.2. Case 2

Fig. 13. SIDLAB network for case 1, points 1 and 2.

R2 ¼ R1 ejðkþ þk ÞL

ð9Þ

150

Sound Pressure Level [dB]

Sound Pressure Level [dB]

where (R2) and (R1) are the translated and original reflection coefficient respectively, and (L) is the distance between the two points. Using this relation, the network can be considered a simple network. The one-port source can be considered directly connected to the one-port load for point 1. These configurations can be easily simulated in SIDLAB using one and two-port elements as shown in Fig. 13. The source is defined as a user defined impedance element and the measured source strength and source impedance are loaded for this element. The whole network is defined as a user defined impedance element and the measured load impedance is used for this element. In the model of point 1, the pipe length is approximately zero, and in the model of point 2, the pipe length is 2.1 m. Fig. 14 shows the comparison between the measured pressure pulsations and the predicted pressure pulsations using SIDLAB at points 1 and 2. The prediction input data is the measured source characteristics, and the measured load impedances for points 1 and 2. The results are shown up to the 18th order, which is equivalent to approximately 300 Hz, which is the limit, based on the used microphone spacing. The figure shows good agreement in

100

Measurement SIDLAB

50

For Point 1, the load impedance is measured as case 1. For point 2, Eq. (9) can no longer be used since the system in between is not a simple straight hard pipe. Therefore, the full network is modeled in SIDLAB, including the vessel and the opening. No load impedance was measured to predict the dynamic pressure at point 2 in this case. The SIDLAB elements for the full network are shown in Fig. 15. In this network, all the bends active part models have been included. Since the vessel dimensions are quite big compared to the cross dimensions of the network pipes, it was interesting to investigate the validity of the 1D model for this vessel. A separate 1D model for the vessel only was constructed and compared to FEM model using COMSOL Multiphysics. Fig. 16 shows the details of both models, and Fig. 17 shows the comparison between the Transmission Loss results comparison. The Transmission Loss demonstrates very good correlation in the plane wave region indicated by the vertical dotted line. This is the region covered by the important excitation frequencies of the compressor. One trick part to model in 1D is the rounded ends of the vessel. This was handled by dividing each end to two conical parts (diffusers) as shown in Fig. 16c. Fig. 18 shows the comparison between the measured dynamic pressure at points 1 and 2 for case 2, and the predicted dynamic pressure at the same points using SIDLAB. The predicted dynamic pressure at point 1 shows good agreement with the measurements. For point 2, the reason for having lower measured pressure values

150

100

Measurement SIDLAB

50

0

0 0

5

10

15

0

5

10

Order [-]

Order [-]

(a)

(b)

Fig. 14. Comparison between measured and predicted sound pressure level, (a) case 1, point 1 and (b) case 1, point 2.

Fig. 15. Case 2: network elements in SIDLAB.

15

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A. Okasha et al. / Applied Acoustics 109 (2016) 44–53

(a) The vessel at the end of the network.

(b) The FEM mesh in COMSOL Multiphysics

(c) The 1D network elements in SIDLAB

Fig. 16. Details of the vessel used in case 2 to regulate the pressure inside the network.

150 Measurement

3D COMSOL Plane wave limit 100

50

SIDLAB (with bends)

150

Sound Pressure Level [dB]

Transmission Loss [dB]

1D SIDLAB

SIDLAB (without bends)

100

50

0 0

500

1000

1500

0 0

Frequency [Hz]

15

Measurement SIDLAB

100

50

0

Fig. 19. Comparison between predicted sound pressure level for case 2, point 2, with and without the effect of flow generated noise by bends.

two for the same test-rig but with different PSD design, shows a pass band frequency at the 14th order, which affects the pulsation attenuation at this region. Fig. 19 shows the comparison of case 2, point 2 measurement with and without taking into consideration the effect of the flowgenerated noise or active part models inside all bends. As shown, the sound pressure level added by the flow-generated noise is low compared to the pulsation levels of the reciprocating compressor, therefore they can be neglected in this case.

Sound Pressure Level [dB]

than those predicted by SIDLAB is attributed to the used pressure transducers. The used pressure transducers do not give good readings at lower dynamic pressures because of its low sensitivity. Thus, when the level of the predicted pressure is low, a shift between the two spectrums begin to appear. The first and second orders have high level and consequently agree well with the measurements. Other orders with low levels, as predicted by SIDLAB, do not match the measurements well, because of the noise floor of the pressure transducers. In Ref. [20], the response of point

Sound Pressure Level [dB]

10

Order [-]

Fig. 17. Comparison between the Transmission Loss of the vessel calculated using 3D finite elements in COMSOL Multiphysics, and the 1D Model in SIDLAB.

150

5

Measurement SIDLAB

150

100

50

0 0

5

10

15

0

5

10

Order [-]

Order [-]

(a)

(b)

Fig. 18. Comparison between measured and predicted sound pressure level, (a) case 2, point 1 and (b) case 2, point 2.

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5. Conclusion In this paper, the two-port theory is used to model a pipeline network, where the network can be divided into several cascaded elements, each described by a transfer matrix. Passive models for the elements used have been introduced. And the active part model for the bends have been presented. A pilot plane equipped with a reciprocating compressor, pipes, bends, and a vessel were constructed. Two cases are investigated; one with a long pipe, the other with a pulsation suppression device designed according to the API-618 standard. The comparison shows good agreement between the measured dynamic pressures and the predicted ones using 1D propagation models. This linear analysis for dynamic pulsations of reciprocating compressors is still possible when the ratio of the dynamic to the static pressures is 4%. Acknowledgment The work described in this paper was funded by Research, Development and Innovative Programme (RDI), contract number C2/S1/129. The authors would like to thank Mohammed Harb and Mostafa Arafa for their help with the measurements. References [1] Beta Machinery Analysis. Primer vibration control strategies for reciprocating compressors; 2009. [2] Wachel J. Acoustic pulsation problems in compressors and pumps. Dara childs lecture series. San Antonio: Texas A&M University; 1992. [3] Munjal M. Acoustics of ducts and mufflers with applications to exhaust and ventilation system design. New York: John Wiley; 1987.

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