General Case for Deriving Four Pole Coefficients for Gas Pulsation

Transcription

1 urdue University urdue e-ubs International Compressor Engineering Conerence School o Mechanical Engineering 21 General Case or Deriving Four ole Coeicients or Gas ulsation Nasir Bilal urdue University Douglas Adams urdue University Keith Novack Trane - Ingersoll Rand Jack Sauls Trane - Ingersoll Rand Follow this and additional works at: Bilal, Nasir; Adams, Douglas; Novack, Keith; and Sauls, Jack, "General Case or Deriving Four ole Coeicients or Gas ulsation" (21). International Compressor Engineering Conerence. aper This document has been made available through urdue e-ubs, a service o the urdue University Libraries. lease contact epubs@purdue.edu or additional inormation. Complete proceedings may be acquired in print and on CD-ROM directly rom the Ray W. Herrick Laboratories at Herrick/Events/orderlit.html

2 A Hybrid Approach o Calculating Gas ulsations in the Suction Maniold o a Reciprocating Compressor Nasir Bilal 1 *, ro. Douglas E. Adams 1,Keith Novak 2, Jack Sauls 2 1 urdue University, Department Mechanical Engineering, West Laayette, IN, USA urdue University Center or Systems Integrity 15 Kepner Rd., Laayette, IN 4795 hone: , bilal@ecn.purdue.edu 2 Trane - Ingersoll Rand, 36 ammel Creek Rd, La Crosse, WI 5461 USA hone: (68) , Keith.Novak@trane.com * Corresponding Author 1522, age 1 ABSTRACT A hybrid approach o calculating the gas pulsations in the suction maniold o a reciprocating compressor is developed using our pole parameters derived rom Finite Element Analysis (FEA) and acoustic analytical model. ressure response unctions are used to calculate the our pole parameters o an acoustic cavity using FEA sotware (ANSYS). The our 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 o multiple inputs, and the resulting pressure response is compared to the results obtained with a Computational Fluid Dynamics (CFD) model. The Finite Element (FE) our pole approach produces more accurate results than the 1D wave equation analytical model because the FE our pole parameters method does not require geometric simpliication and, thus, provides better estimates o the pressure response. The FE our pole approach can be easily extended to complex geometric shapes o suction and discharge maniolds. In addition, the our pole approach was preerred over the CFD method o modeling gas pulsations because the CFD model was computationally very expensive. The FE our pole parameters can be used inside a compressor simulation model to better model the low o gasses through the valve and cylinder. 1. INTRODUCTION The perormance o a reciprocating compressor is strongly inluenced by valve dynamics and mass low into the compressor, which produce pressure pulsations in the suction maniold, generate noise, and reduce the eiciency o the compressor. ressure pulsations response, or wave shapes are generated due to the intricate geometry and passages o the compressor maniold. Mathematical models developed earlier (ark, 24) were used to predict the gas pulsation in the suction maniold. However, because o the complicated shape o the compressor maniold, many simpliying assumptions were made in the models to predict the pressure response mathematically. The hybrid approach consist o using the inite element harmonic response solution using acoustic elements to calculate the our pole parameters o 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 o the system, the density o the gas, and the speed o sound. Thereore, 3D modeling o the acoustic cavity results in a more accurate calculation o the our pole than the 1D wave equation analytical model. Thus, the hybrid approach o using a inite element model together with an acoustic analytical model implemented in MATLAB produces a more accurate

3 1522, age 2 pressure response because no geometric simpliications are used. The acoustic analytical model calculates the pressure at both ends o the system rom the our pole parameters and mass low o the system. 2. FOUR OLE METHOD Four pole parameters are useul or the analysis o composite acoustic systems and are widely used to analyze gas pulsation in cavities (Soedel, 26). In essence, the our pole technique expresses the low conditions at one end o the cavity as a unction o the conditions at the other end o the cavity. The our poles o a simple tube shown in the Fig. 1 are deined as Figure 1: Coniguration o a constant area tube. Q A BQ C D 2 (1) where subscript 1 and 2 indicate the input and output ports, Q and are complex harmonic amplitudes o the volume low, velocity, and pressure, and A, B, C, and D are called the our poles and are deined as ollows: A cosh LD (2) jωs B sinh γl 2 ρ c (3) ο ο 2 ρc γ C sinh γ L j ω S (4) where ρ, c, γ, L, are luid density, speed o sound, the complex wave number and the length o the tube, respectively. The our pole parameters or any acoustic cavity are calculated using the pressure response unctions, which are described next. 3. RESSURE RESONSE FUNCTIONS A ressure Response Function (RF) ij (Singh and Soedel, 1978; and Kadam, 25) or any general location is deined as the ratio o pressure induced at point i, denoted by i, due to the low input, denoted by Q j, inward to the cavity at point j, while the other side o the cavity is blocked as shown in Fig. 2.

4 1522, age 3 Blocked end i j Acoustic Cavity Open end 1e jωt Figure 2: Illustration o generation o pressure response unctions. The RF ij can be written as ij Q i (5) j where i is the pressure measured at location i, by blocking location i while applying a unit inward volume low Q j at point j. For any acoustical cavity, two runs o the FEA simulation are needed to determine the our RFs o that cavity. Two RFs, 12 and 22, are calculated by a single run o experimental or numerical simulations. For a simple acoustic cavity as shown in Fig. 2, the RFs can be written in the ollowing orm, Q1 Q (6) Equation (1) and (6) deine the relationship between the same set o variables but are expressed dierently. It can be shown that the our pole parameter can be ormulated rom RFs o any acoustic system by extracting the ollowing relationships: A 22, 12 1 B, 21 C +, The inite element method based on simulations was used to calculate the our poles o the suction cavity as describe below. D 4. COMARISON OF FOUR OLE ARAMETERS CALCULATED BY 1D WAVE EQUATION AND THE FINITE ELEMENT METHOD The procedure to determine the our pole coeicients o the actual suction maniold is outlined below: i. Model the gas passage o a simple suction maniold, Fig. 3 below, in ANSYS using acoustic element (Fluid3) ii. Apply loading conditions as in Fig. 2 (Two loading cases) iii. erorm harmonic analysis in ANSYS through the requency range o interest iv. Determine the our pole coeicients using Improved Four ole Method (Wu and Zhang, 1998)

5 1522, age 4 Figure 3: A simple muler modeled in the ANSYS. The results o the FE our pole terms calculated rom method described above were compared with the our poles computed using the 1D analytical wave equation. Figure 4 shows the results. The FE our pole terms and the 1D analytical wave equation our pole terms are in relatively good agreement. 1 A-Term: Comparison o Analytical vs FEA Analysis Analyt FEA Frequency [Hz] 5 x Frequency [Hz] 1 x 17 B-Term: Comparison o Analytical vs FEA Analysis C-Term: Comparison o Analytical vs FEA Analysis Frequency [Hz] Figure 4: A comparison o A, B, and C terms o Analytical results with FEA results

6 1522, age 5 5. GENERAL FORMUALTION OF FOUR OLE ARAMETERS FOR MULTILE INUTS Ater comparing the results or the simple muler 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 simpliied model o a multi-cylinder compressor. First, the 1D wave equation analytical model was derived or the multiple tube coniguration, shown in Fig. 5 below, to calculate the pressure response L11 and M11 at the valve location shown in Fig. 5. Figure 5: Schematic o a multi-cylinder compressor with each section consisting o several tubes connected in series. Equation (7) and (8) show the expressions or pressure at location L 1 and M 1. The expressions or L11 and M11 have been simpliied by enorcing the pressure and volume continuity at the junction AA and BB and as explained in (Soedel, 26). C L C L AMQL 11 + ALQ M11 L 11 QM11 + CL + DLZN11 + BMZN11 AM AM AMAL + AMBLZN11 + ALBMZN11 (7) C M C M ALQM11 + AMQ L11 M11 QL 11 + CM + DMZN11 + BLZN11 AL AL AMAL + ALBMZN11 + AMBLZN11 (8) In above expressions or the A L, B L, C L and D L are the global our pole parameter or tube section L as shown in Eq. (9) AL BL AL1 BL 1AL2 BL2AL3 BL3 CL D L CL 1 D L1 CL2 D L2 CL3 D L3 (9) Similarly, or section M and N, we can write QM11 AM BM QM 32 C D M11 M M M 32 QN11 AN BN QN32 C D N11 N N N32 (1) (11)

7 1522, age 6 where, A M, B M,C M,and D M and A N,B N, C N and D N, are the global our pole parameters or section M and N. Q L11 and Q M11 are the mass low rates at the inlet at section L and M respectively and are shown in Fig. 7 and Fig. 8. Z N11 is deined in Eq. (12) N11 CN Z N11 Q 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 rom the 1D wave equation solution or multiple inputs to the FE our pole strategy. However, because the basic FE our pole ormulation allows only one input and one output, it could not be used and a new strategy was needed or multiple inputs. The basic our pole ormulation 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 perorm similar calculations as above, however real lie compressors are not this simple. Work done by Singh and Soedel in 1979, Eq (13,14 and 15) showed the pressure at L11 was aected by the mass low through point L11, the mass low through point M11 and, the mass low through section BB. The FE methods as described above was used to calculate both the Input Impedance and Transer Impedance which are then used in Equation 15 to calculate the pressure at each input. This has allowed the ormation o a new FE hybrid modeling technique that allows or multiple inputs; however the matrix is no longer a 2x2. In the case or Fig. 6, the matrix is a 3x3. This matrix can be solved by knowing the two input mass lows and a pressure boundary condition at the outlet. N11 N Input Impedance Z ~ Transer Impedance Z ~ (nω) ~ p (nω)/x ~ (nω) q 1,2,...,k (13) qq q q (nω) ~ p (nω)/x ~ (nω) q r r 1,2,..., k and q 1,2,..., k (14) qr q r ~ p (n ) Z ~ v1 ω (15) 11 (nω)[x ~ (nω)] + Z ~ (nω)[x ~ (nω)] + Z ~ (nω)[x ~ (nω)] Z ~ 1 θ θ θ 3 1m (nω)[x ~ k (nω)] θk Where, ~ p~ and X are pressure and volume velocity in the requency domain. q r

8 1522, age 7 Figure 6: Finite element model o the multiple inputs. This new FE hybrid technique was used to determine the pressure responses o the model in Fig. 6. The results in Fig. 7 and Fig. 8 show that the pressure response at location L11 and M11 obtained by using FE hybrid approach matches the pressure response obtained by 1D wave equation analytical solution and CFD methods ressure at Valve (psi) Mass Flow (lbm/s) CFD FE HYBRID 1D 4OLE MASSFLOW Time (s) Figure 7: ressure in cylinder 1 at valve location L11.

9 1522, age ressure at Valve (psi) Mass Flow (lbm/s) CFD FE HYBRID 1D 4OLE MASSFLOW Time(s) Figure 8: ressure in cylinder 2 at valve location M11. The same hybrid approach o using the pressure response unctions can be applied to the more complex suction and discharge maniolds with multiple inputs. 6. CONCLUSIONS A hybrid approach using the inite 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 o using this approach is that the pressure response in an acoustic cavity can be calculated without any geometric simpliications 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 or complex multiple input designs or multiple cylinder compressors. This approach is also more eicient compared to the CFD method, which is computationally quite expensive. The hybrid our pole parameters can also be used inside a compressor simulation model to study the eects o the geometric changes on the pressure response, in the cavity, and perormance o the compressor. Because the our poles coeicients are independent o mass low, modiications to the valve (lit or timing) and cylinder design (stroke or piston diameter) can be done inside a compressor simulation model without having to resolve new our pole parameters. The hybrid approach still needs to ollow the linear acoustic constraints o the 1D our pole method. The our pole parameters can also be used to calculate the acoustic transmission loss o the system to help in noise reduction. REFERENCES Kadam,. H., 25, Development and Comparison o Analytic, Numerical and Experimental Techniques to ormulate Four-oles Matrices o Three Dimensional Acoustic Systems, MSc. Thesis, University o Cincinnati. ark, J., 24, Mathematical Modeling and Simulation o a Multi-Cylinder Automotive Compressor, hd Thesis, urdue University. Singh, R., Soedel, W., 1979, Mathematical Modeling o Multicylinder Compressor Discharge System Interactions, Journal o Sound and Vibration, vol. 63, no. 1: p Soedel, W., 26, Sound and Vibrations o ositive Displacement Compressors, CRC ress. Wu, T. W., Zhang,., 1998, Boundary Element Analysis o Mulers with an Improved Methods or Deriving the Four ole arameters, Journal o Sound and Vibration, vol. 217, no. 4: p