Adams, Douglas

Bilal, Nasir

Novack, Keith

Sauls, Jack

English

Purdue E-Pubs

Purdue University Purdue e-Pubs International Compressor Engineering Conference School of Mechanical Engineering 2010 General Case for Deriving Four Pole Coefficients for Gas Pulsation Nasir Bilal Purdue University Douglas Adams Purdue University Keith Novack Trane - Ingersoll Rand Jack Sauls Trane - Ingersoll Rand Follow this and additional works at: https://docs.lib.purdue.edu/icec Bilal, Nasir; Adams, Douglas; Novack, Keith; and Sauls, Jack, "General Case for Deriving Four Pole Coefficients for Gas Pulsation" (2010). International Compressor Engineering Conference. Paper 2036. https://docs.lib.purdue.edu/icec/2036 This document has been made available through Purdue e-Pubs, a service of the Purdue University Libraries. Please contact epubs@purdue.edu for additional information. Complete proceedings may be acquired in print and on CD-ROM directly from the Ray W. Herrick Laboratories at https://engineering.purdue.edu/Herrick/Events/orderlit.html  1522, Page 1 International Compressor Engineering Conference at Purdue, July 12-15, 2010      ??A Hybrid Approach of Calculating Gas Pulsations in the Suction Manifold of a Reciprocating Compressor Nasir Bilal1*, Prof. Douglas E. Adams1, Keith Novak2, Jack Sauls21Purdue University, Department Mechanical Engineering, West Lafayette, IN, USA Purdue University Center for Systems Integrity 1500 Kepner Rd., Lafayette, IN 47905 Phone: 765 446 1709, bilal@ecn.purdue.edu2 Trane - Ingersoll Rand, 3600 Pammel Creek Rd, La Crosse, WI 54601 USA Phone: (608)787-4105, Keith.Novak@trane.com* Corresponding Author ABSTRACT A hybrid approach of calculating the gas pulsations in the suction manifold of a reciprocating compressor is developed using four pole parameters derived from Finite Element Analysis (FEA) and acoustic analytical model.   Pressure response functions are used to calculate the four pole parameters of an acoustic cavity using FEA software (ANSYS).  The four pole parameters are then used with an analytical model to calculate the pressure response in the acoustic cavity.  The same scheme is then extended to a case of multiple inputs, and the resulting pressure response is compared to the results obtained with a Computational Fluid Dynamics (CFD) model.  The Finite Element (FE) four pole approach produces more accurate results than the 1D wave equation analytical model because the FE four pole parameters method does not require geometric simplification and, thus, provides better estimates of the pressure response. The FE four pole approach can be easily extended to complex geometric shapes of suction and discharge manifolds.  In addition, the four pole approach was preferred over the CFD method of modeling gas pulsations because the CFD model was computationally very expensive.  The FE four pole parameters can be used inside a compressor simulation model to better model the flow of gasses through the valve and cylinder. 1. INTRODUCTION The performance of a reciprocating compressor is strongly influenced by valve dynamics and mass flow into the compressor, which produce pressure pulsations in the suction manifold, generate noise, and reduce the efficiency of the compressor.  Pressure pulsations response, or wave shapes are generated due to the intricate geometry and passages of the compressor manifold.  Mathematical models developed earlier (Park, 2004) were used to predict the gas pulsation in the suction manifold.  However, because of the complicated shape of the compressor manifold, many simplifying assumptions were made in the models to predict the pressure response mathematically. The hybrid approach consist of using the finite element harmonic response solution using acoustic elements to calculate the four pole parameters of the acoustic cavity which is then used in an analytical model to calculate the pressure response in the cavity.  Four pole parameters are based on the geometric characteristics of the system, the density of the gas, and the speed of sound.  Therefore, 3D modeling of the acoustic cavity results in a more accurate calculation of the four pole than the 1D wave equation analytical model.  Thus, the hybrid approach of using a finite element model together with an acoustic analytical model implemented in MATLAB produces a more accurate  1522, Page 2 International Compressor Engineering Conference at Purdue, July 12-15, 2010 pressure response because no geometric simplifications are used.  The acoustic analytical model calculates the pressure at both ends of the system from the four pole parameters and mass flow of the system. 2. FOUR POLE METHOD  Four pole parameters are useful for the analysis of composite acoustic systems and are widely used to analyze gas pulsation in cavities (Soedel, 2006).  In essence, the four pole technique expresses the flow conditions at one end of the cavity as a function of the conditions at the other end of the cavity. The four poles of a simple tube shown in the Fig. 1 are defined as Figure 1: Configuration of a constant area tube. 1 21 2Q QA BP PC D? ? ? ?? ?=? ? ? ?? ?? ?? ? ? ?      (1) where subscript 1 and 2 indicate the input and output ports, Q and P are complex harmonic amplitudes of the volume flow, velocity, and pressure, and A, B, C, and D are called the four poles and are defined as follows: A= cosh ?L=D      (2) 2 sinhj SB Lcο οω= γρ     (3) 20 0 sinhcC Lj Sγρ γ=ω      (4) where ρ , c  , γ , L ,  are fluid density, speed of sound, the complex wave number and the length of the tube, respectively. The four pole parameters for any acoustic cavity are calculated using the pressure response functions, which are described next. 3. PRESSURE RESPONSE FUNCTIONS  A Pressure Response Function (PRF)  ijf  (Singh and Soedel, 1978; and Kadam, 2005) for any general location is defined as the ratio of pressure induced at point i, denoted by iP , due to the flow input, denoted by jQ  , inward to the cavity at point j, while the other side of the cavity is blocked as shown in Fig. 2. ?????? ????????? ??????? 1522, Page 3 International Compressor Engineering Conference at Purdue, July 12-15, 2010 Figure 2: Illustration of generation of pressure response functions. The PRF ijf can be written as iijjPfQ=     (5) where iP is the pressure measured at location i , by blocking location i  while applying a unit inward volume flow jQ  at point j .  For any acoustical cavity, two runs of the FEA simulation are needed to determine the four PRFs of that cavity. Two PRFs,  12f  and 22f , are calculated by a single run of experimental or numerical simulations.  For a simple acoustic cavity as shown in Fig. 2, the PRFs can be written in the following form, 1 11 12 12 21 22 2P f f QP f f Q? ? ? ? ? ?=? ? ? ?? ?? ? ? ? ? ?      (6) Equation (1) and (6) define the relationship between the same set of variables but are expressed differently. It can be shown that the four pole parameter can be formulated from PRFs of any acoustic system by extracting the following relationships: 2212fAf= ,   211Bf= ,  1112 2212fC f ff= − + ,  1112fDf=The finite element method based on simulations was used to calculate the four poles of the suction cavity as describe below. 4. COMPARISON OF FOUR POLE PARAMETERS CALCULATED BY 1D WAVE EQUATION AND THE FINITE ELEMENT METHOD  The procedure to determine the four pole coefficients of the actual suction manifold is outlined below: i. Model the gas passage of a simple suction manifold, Fig. 3 below, in ANSYS using acoustic element (Fluid30) ii. Apply loading conditions as in Fig. 2 (Two loading cases) iii. Perform harmonic analysis in ANSYS through the frequency range of interest iv. Determine the four pole coefficients using Improved Four Pole Method (Wu and Zhang, 1998). Blocked end1 j te ωi jOpen endAcoustic Cavity 1522, Page 4 International Compressor Engineering Conference at Purdue, July 12-15, 2010 Figure 3: A simple muffler modeled in the ANSYS. The results of the FE four pole terms calculated from method described above were compared with the four poles computed using the 1D analytical wave equation.  Figure 4 shows the results.  The FE four pole terms and the 1D analytical wave equation four pole terms are in relatively good agreement.  Figure 4: A comparison of A, B, and C terms of Analytical results with FEA results 0 200 400 600 800 1000 1200 1400 1600 1800 2000-10010Frequency [Hz]A-Term: Comparison of Analytical vs FEA AnalysisAnalytFEA0 200 400 600 800 1000 1200 1400 1600 1800 2000-505x 10-5Frequency [Hz]B-Term: Comparison of Analytical vs FEA Analysis0 200 400 600 800 1000 1200 1400 1600 1800 2000-101x 107Frequency [Hz]C-Term: Comparison of Analytical vs FEA Analysis 1522, Page 5 International Compressor Engineering Conference at Purdue, July 12-15, 2010 5. GENERAL FORMUALTION OF FOUR POLE PARAMETERS FOR MULTIPLE INPUTS  After comparing the results for the simple muffler model, shown in Fig. 3, the 1D wave equation model was extended to multiple inputs, shown in Fig. 5.  This model can be considered a simplified model of a multi-cylinder compressor. First, the 1D wave equation analytical model was derived for the multiple tube configuration, shown in Fig. 5 below, to calculate the pressure response PL11 and PM11 at the valve location shown in Fig. 5. Figure 5: Schematic of a multi-cylinder compressor with each section consisting of several tubes connected in series. Equation (7) and (8) show the expressions for pressure at location L1 and M1.  The expressions for PL11 and PM11have been simplified by enforcing the pressure and volume continuity at the junction AA and BB and as explained in (Soedel, 2006). 11 1111 11 11 1111 11L L M L L ML M L L N M NM M M L M L N L M NC C A Q A QP Q C D Z B ZA A A A A B Z A B Z? ?? ? += − + + + ? ?? ?+ +? ?? ?   (7) 11 1111 11 11 1111 11M M L M M LM L M M N L NL L M L L M N M L NC C A Q A QP Q C D Z B ZA A A A A B Z A B Z? ?? ? += − + + + ? ?? ?+ +? ?? ?   (8) In above expressions for the AL, BL, CL and DL are the global four pole parameter for tube section L as shown in Eq. (9) 3 31 1 2 23 31 1 2 2L LL L L L L LL LL L L L L LA BA B A B A BC DC D C D C D? ?? ? ? ? ? ?= ? ?? ? ? ? ? ?? ? ? ? ? ? ? ?   (9) Similarly, for section M and N, we can write 32113211MM M MMM M MQQ A BPP C D? ?? ? ? ?=? ? ? ?? ?? ? ? ? ? ?      (10) 11 3211 32N N N NN N N NQ A B QP C D P? ? ? ? ? ?=? ? ? ?? ?? ? ? ? ? ?      (11) ????????????????????????????????????????????????????????????????????? 1522, Page 6 International Compressor Engineering Conference at Purdue, July 12-15, 2010 where, AM , BM ,CM ,and DM and AN ,BN, CN and DN , are the global four pole parameters for section M and N.  QL11and QM11 are the mass flow rates at the inlet at section L and M respectively and are shown in Fig. 7 and Fig. 8.  ZN11is defined in Eq. (12) 111111N NNN NP CZQ A= =      (12) Equations (7) and (8) were used to calculate the pressure response in the cavities as shown in Fig. 5.  An FE model with multiple inputs was created in ANSYS, see Fig. 6, to compare the results from the 1D wave equation solution for multiple inputs to the FE four pole strategy.  However, because the basic FE four pole formulation allows only one input and one output, it could not be used and a new strategy was needed for multiple inputs.  The basic four pole formulation required only one input and one output.  This new design had two inputs and one output.   One option was to cut the design into pieces at Section AA in Fig. 5 and perform similar calculations as above, however real life compressors are not this simple.  Work done by Singh and Soedel in 1979, Eq (13,14 and 15) showed the pressure at PL11  was affected by the mass flow through point PL11, the mass flow through point PM11 and, the mass flow through section BB.  The FE methods as described above was used to calculate both the Input Impedance and Transfer Impedance which are then used in Equation 15 to calculate the pressure at each input.  This has allowed the formation of a new FE hybrid modeling technique that allows for multiple inputs; however the matrix is no longer a 2x2.  In the case for Fig. 6, the matrix is a 3x3. This matrix can be solved by knowing the two input mass flows and a pressure boundary condition at the outlet.   Input Impedance k    1,2,...,q      )n(X~)/n(p~)n(Z~ qqqq == ωωω ?           (13) Transfer Impedance k   1,2,...,q  andk   1,2,...,rr  q     )n(X~)/n(p~)n(Z~ rqqr ==≠= ωωω ?  (14)      )]n(X~)[n(Z~...)]n(X~)[n(Z~    )]n(X~)[n(Z~)]n(X~)[n(Z~)n(p~ kk1m331322121111v1 θθθθ ωωωωωωωωω ???? ++++=                    (15) Where, qp~ and X~r?  are pressure and volume velocity in the frequency domain.  1522, Page 7 International Compressor Engineering Conference at Purdue, July 12-15, 2010 Figure 6: Finite element model of the multiple inputs. This new FE hybrid technique was used to determine the pressure responses of the model in Fig. 6.  The results in Fig. 7 and Fig. 8 show that the pressure response at location PL11 and PM11 obtained by using FE hybrid approach matches the pressure response obtained by 1D wave equation analytical solution and CFD methods. Figure 7:  Pressure in cylinder 1 at valve location PL11. 1341361381401421441461481501521540 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018Time (s)Pressure at Valve (psi)00.050.10.150.20.250.30.350.4Mass Flow (lbm/s)CFDFE HYBRID1D 4POLEMASSFLOW 1522, Page 8 International Compressor Engineering Conference at Purdue, July 12-15, 2010 Figure 8: Pressure in cylinder 2 at valve location PM11. The same hybrid approach of using the pressure response functions can be applied to the more complex suction and discharge manifolds with multiple inputs. 6. CONCLUSIONS A hybrid approach using the finite element method in combination with an analytical model can be used to calculate the pressure response in the suction or a discharge system.  The advantage of using this approach is that the pressure response in an acoustic cavity can be calculated without any geometric simplifications on actual 3D CAD designs. This approach has been used to model a complete intake system including the internal shell cavity.  The hybrid approach allows for complex multiple input designs or multiple cylinder compressors.  This approach is also more efficient compared to the CFD method, which is computationally quite expensive. The hybrid four pole parameters  can also be used inside a compressor simulation model to study the effects of the geometric changes on the pressure response, in the cavity, and performance of the compressor.  Because the four poles coefficients are independent of mass flow, modifications to the valve (lift or timing) and cylinder design (stroke or piston diameter) can be done inside a compressor simulation model without having to resolve new four pole parameters.  The hybrid approach still needs to follow the linear acoustic constraints of the 1D four pole method.  The four pole parameters can also be used to calculate the acoustic transmission loss of the system to help in noise reduction. REFERENCES  Kadam, P. H., 2005, Development and Comparison of Analytic, Numerical and Experimental Techniques to formulate Four-Poles Matrices of Three Dimensional Acoustic Systems, MSc. Thesis, University of Cincinnati. Park, J., 2004, Mathematical Modeling and Simulation of a Multi-Cylinder Automotive Compressor, PhD Thesis, Purdue University. Singh, R., Soedel, W., 1979, Mathematical Modeling of Multicylinder Compressor Discharge System Interactions, Journal of Sound and Vibration, vol. 63, no. 1: p. 125-143. Soedel, W., 2006, Sound and Vibrations of Positive Displacement Compressors, CRC Press. Wu, T. W.,  Zhang, P., 1998, Boundary Element Analysis of Mufflers with an Improved Methods for Deriving the Four Pole Parameters, Journal of Sound and Vibration, vol. 217, no. 4: p. 767-779.1321341361381401421441461481501521540 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018Time(s)Pressure at Valve (psi)00.050.10.150.20.250.30.350.4Mass Flow (lbm/s)CFDFE HYBRID1D 4POLEMASSFLOW

General Case for Deriving Four Pole Coefficients for Gas Pulsation

https://docs.lib.purdue.edu/cgi/viewcontent.cgi?article=3035&amp;context=icec

General Case for Deriving Four Pole Coefficients for Gas Pulsation

Abstract

Similar works

Full text

Available Versions

Purdue E-Pubs