internal flow and valve dynamics in a reciprocating compressor
TRANSCRIPT
DISSERTATION
Internal Flow and Valve Dynamics
in a Reciprocating Compressor
ausgefuhrt zum Zwecke der Erlangung des akademischen Grades eines Doktors der
technischen Wissenschaften unter der Leitung von
Ao.Univ.Prof. Dipl.-Ing. Dr. techn. Herbert Steinruck
am Institut fur Stromungsmechanik und Warmeubertragung (E 322)
eingereicht an der Technischen Universitat Wien
Fakultat fur Maschinenwesen und Betriebswissenschaften
von
Dipl.-Ing. Roland Aigner
Matrikelnummer 9625691
1230 Wien, Triester Straße 221/3/9
Wien, im Juni 2007
Kurzfassung
Die Stromung in einem Koblenkompressor wird durch die Bewegung des Kolbens unddurch das Offnen und Schließen der Ventile hervorgerufen. Vorallem die Wechselwirkungvon Druckwellen mit der Ventilbewegung bestimmt die auftretenden Stromungsformen.
Die vorliegende Dissertation liefert ein genaues aber einfaches numerisches Verfahrenum die Innenstromung und die Ventildynamik zu berechnen. Dabei erstreckt sich dasbetrachtete Rechengebiet von den Ventilen der Saugseite bis zur Druckkammer des Kom-pressors auf der Auslassseite. Fur die mathematische Beschreibung des Stromungsfeldeswerden quasi eindimensionale Eulergleichungen verwendet. Bei Kompressoren mit mehrals zwei Ventilen wird hingegen ein zweidimensionales Modell eingefuhrt um den Ein-fluss mehrerer Ventiltaschen zu berucksichtigen. Obwohl die Energiegleichungen bei denEulergleichungen durch die Isentropenbeziehung ersetzt wird, werden Druckverluste anSprungen des Stromungsquerschnittes mit Hilfe von Verlustkoeffizienten eingerechnet. Ver-schiedene Randbedingungen vervollstandigen das mathematische Stromungsmodell. Furdie Ventilbewegung wird das Modell von Costagliola dahingehend modifiziert, dass auchDruckwellen im Zylinder berucksichtigt werden konnen.
Die resultierenden Gleichungssysteme werden mit Hilfe finiter Volumenverfahren vonLe Veque, Mac-Cormack, Lax, Wendroff und Friedrichs gelost. Ein besonderes Augenmerkwird dabei auf die Vor- und Nachteile der einzelnen Verfahren gerichtet.
Schließlich wurden Messungen an einem Testkompressor mit zwei Ventilen von Burck-hardt Compression (Schweiz) durchgefuhrt. Vergleiche mit den numerischen Losungenzeigen, dass die wichtigsten physikalischen Effekte genau beschrieben werden und dahersowohl die Stromungsform als auch die Ventilbewegung gut bestimmbar sind. Zusatzlichwurden die Messergebnisse eines Kompressors mit acht Ventilen von Ariel (USA) mit derLosung des zweidimensionalen Modells verglichen. Hier werden die von der Simulationvorhergesagten komplizierten Wellensystem von der Messung bestatigt.
Abstract
The gas flow inside a reciprocating compressor is the result of the oscillating piston motionand the opening and closing of the valves. Above all, the interaction of pressure waveswith the valve motion determines the present flow patterns.
This thesis provides an accurate but simple numerical model capable of calculatinginternal flows and valve dynamics in a reciprocating compressor. The considered compu-tational domain extends from the valves of the suction side to the pressure chamber ofthe discharge side. Quasi one-dimensional Euler equations are employed to describe thegas flow. However, in case of cylinders with more than two valves two-dimensional modelsare introduced to account for the effects of multiple valve pockets. Although the energyequation is replaced by the isentropic condition, pressure losses at sudden changes of flowcross-section are taken into account by means of loss coefficients. Different kinds of bound-ary conditions complete the mathematical model of the gas flow. The valve dynamics aredescribed by a modified Costagliola-model which takes pressure waves in the cylinder intoaccount.
In order to solve the governing systems of equations finite volume methods from LeV-eque, Mac-Cormack, Lax, Wendroff and Friedrich are employed and their advantageshighlighted.
Finally measurements have been carried out on a test compressor with two valvesfrom Burckhardt Compression (Switzerland). Comparisons with the results of the quasi-one-dimensional model show that main physical effects are described accurately and thusboth, gas flow in the compressor and valve motion can be predicted well. In addition themeasurement results of a compressor from Ariel (USA) with eight valves are compared tothe solution of the quasi two-dimensional model. Here the complicated two-dimensionalwave pattern inside the cylinder and valve motion agrees well with the the measurementdata.
Acknowledgement / Danksagung
Herrn Ao. Prof. Dr. Herbert Steinruck sei an dieser Stelle mein besonderer Dank ausge-sprochen. Nur durch seine Unterstutzung, Betreuung, und Hilfe wurde diese Dissertationmoglich.
Ich bedanke mich sehr herzlich bei Prof. Gotthard Will fur die Ubernahme des Koreferats.
Herrn Dr. Peter Steinruck, Herrn Dr. Georg Samland und Herrn Dr. Gunther Machudanke ich sehr herzlich fur die zahlreichen fachlichen Diskussionen.
I am very grateful to Fred Newman who provided me with valuable suggestions and mea-surement data from Ariel Corporation.
Dank gebuhrt auch Dr. Daniel Sauter fur die reibungslos verlaufenden Messungen beiBurckhardt Compression AG.
Herzlich bedanken mochte ich mich bei meiner Freundin Ines, meinen Freunden und meinerFamilie.
Des Weiteren mochte ich mich bei den Mitarbeitern des Instituts bedanken. Vor allemRichard, Uli, Guido und Thomas haben den Arbeitsalltag erleichtert und in so manchenunproduktiven Stunden fur Ausgleich gesorgt. Besonders dankbar bin ich Herrn Prof. Wil-helm Schneider fur die zahlreichen wissenschaftlichen und facherubergreifenden Ratschlage.
I acknowledge the European Forum for Reciprocating Compressors that has financed thiswork.
1
Contents
Content . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . III
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IV
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VII
List of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IX
1 Introduction 1
2 Modelling of the Reciprocating Compressor 5
2.1 Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.1.1 Simulation domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.1.2 One-dimensional model . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.1.3 Two-dimensional model . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 Characteristic Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2.1 Gas Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2.2 Heat Transfer and Dissipation . . . . . . . . . . . . . . . . . . . . . . 9
2.2.3 Classification of Compressors . . . . . . . . . . . . . . . . . . . . . . 9
2.3 Piston Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.4 Gas Flow inside the Compressor . . . . . . . . . . . . . . . . . . . . . . . . 10
2.4.1 Quasi one-dimensional Model . . . . . . . . . . . . . . . . . . . . . . 10
2.4.2 Quasi two-dimensional Model . . . . . . . . . . . . . . . . . . . . . . 13
2.4.3 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.4.4 T-piece . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.4.5 Interface Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.4.6 Pressure Losses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.5 Valve Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.5.1 Valve Dynamic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.5.2 Flow through the valve . . . . . . . . . . . . . . . . . . . . . . . . . 19
I
2.5.3 Asymmetric Flow in Valve Pockets, Valve Masking and Slotted ValvePockets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3 Numerical Method 22
3.1 Finite Volume Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.1.1 One-dimensional Finite Volume Schemes . . . . . . . . . . . . . . . . 22
3.1.2 Two-dimensional Finite Volume Schemes . . . . . . . . . . . . . . . 30
3.2 Numerical Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . 34
3.2.1 Walls one-dimensional . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.2.2 Mass-flow through the Boundary . . . . . . . . . . . . . . . . . . . . 35
3.2.3 Pressure Outlet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.2.4 Boundary Problem at Sudden Changes of Cross-section . . . . . . . 37
3.2.5 Boundary Conditions at the Valve . . . . . . . . . . . . . . . . . . . 38
3.2.6 Boundary Conditions at the T-piece . . . . . . . . . . . . . . . . . . 39
3.2.7 Wall two-dimensional . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.2.8 Interface Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.3 Equation of motion for the valve plate . . . . . . . . . . . . . . . . . . . . . 42
4 Measurements 44
4.1 Compressor with two valves . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
4.1.1 Test Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
4.1.2 Results of Measurement . . . . . . . . . . . . . . . . . . . . . . . . . 46
4.2 Compressor with eight valves . . . . . . . . . . . . . . . . . . . . . . . . . . 56
5 Comparison of Measurement and Numerical data 59
5.1 One-dimensional Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
5.1.1 Comparison with Measurement Results of the Burckhardt test com-pressor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
5.1.2 Super elevation of the pressure due to initial sticking . . . . . . . . . 63
5.1.3 Pressure loss at sudden changes of cross section . . . . . . . . . . . . 63
5.1.4 Different locations of outflow boundary . . . . . . . . . . . . . . . . 64
5.1.5 Impact velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
5.1.6 Valve losses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
5.2 Two-dimensional model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
5.2.1 Comparison with measurement results of the Ariel test compressor . 65
5.2.2 Variation of discharge pressure . . . . . . . . . . . . . . . . . . . . . 69
II
5.2.3 Pressure distribution inside the cylinder . . . . . . . . . . . . . . . . 70
5.2.4 Moment onto the piston . . . . . . . . . . . . . . . . . . . . . . . . . 71
5.2.5 Valve Opening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
5.2.6 Valve Masking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
6 Remarks on Numerical Schemes 75
6.1 Comparison of full three-dimensional, quasi one-dimensional and quasi two-dimensional numerical methods . . . . . . . . . . . . . . . . . . . . . . . . . 75
6.1.1 Pressure waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
6.1.2 Computational Time . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
6.2 Limitations of the Numerical Schemes . . . . . . . . . . . . . . . . . . . . . 78
6.2.1 Limitations of the one-dimensional model . . . . . . . . . . . . . . . 78
6.2.2 Limitations of the quasi two-dimensional model . . . . . . . . . . . . 79
7 Summary 80
A Tables 82
A.1 Sensors of Burckhardt Test Compressor . . . . . . . . . . . . . . . . . . . . 83
A.2 Pressure loss coefficients for the T-piece . . . . . . . . . . . . . . . . . . . . 83
Reference 85
Curriculum Vitae 87
III
List of Figures
1.1 Double-acting compressor of barrel-design. . . . . . . . . . . . . . . . . . . . 1
1.2 Results of a full three-dimensional simulation: pressure distribution at thecylinder head and in the symmetry plane [22]. . . . . . . . . . . . . . . . . 2
2.1 Flow geometry inside the compressor. . . . . . . . . . . . . . . . . . . . . . 6
2.2 Schema of the one-dimensional model. . . . . . . . . . . . . . . . . . . . . . 6
2.3 Cross-section areas of the one-dimensional model. . . . . . . . . . . . . . . . 6
2.4 Model of discharge system VR1: Pressure chamber is modelled at the endof the valve retainer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.5 Model of discharge system VR2: Pressure chamber is connected to the valveretainer by a T-piece. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.6 Model with discharge pipe and damper. Crank end working chamber isoptional (dashed lines). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.7 Schema of the two-dimensional model. . . . . . . . . . . . . . . . . . . . . . 9
2.8 Schema of the two-dimensional model with eight valves. . . . . . . . . . . . 9
2.9 Right boundary of computational domain: Characteristics variables enter-ing and leaving at the right end of the computational domain. . . . . . . . . 14
2.10 T-piece consisting of tube 1, tube 2 and tube 3: The shaded square displaysthe junction area. Its appropriate in-going and out-going characteristics w1,i
and w2,i are indexed by the adjacent tube i = 1, 2, 3. . . . . . . . . . . . . . 15
2.11 Interface between one-dimensional and two-dimensional domain. . . . . . . 16
2.12 Modelling pressure losses . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.13 Sudden expansion of cross-section . . . . . . . . . . . . . . . . . . . . . . . . 17
2.14 Pressure losses at the T-piece . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.15 Schema of a plate valve. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.16 Pressure distribution and velocity vectors in the symmetry plane duringdischarge (full three-dimensional simulation). . . . . . . . . . . . . . . . . . 21
2.17 Valve Masking. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.18 Slotted Valve Pockets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
IV
3.1 Subdivision of the computational domain into grid cells Ci. . . . . . . . . . 23
3.2 Riemann problems at each cell boundary. . . . . . . . . . . . . . . . . . . . 25
3.3 Riemann problem at the left boundary xi− 12
of the ith cell . . . . . . . . . . 26
3.4 Riemann problem of a non-autonomous system at the left boundary xi− 12
of the i-th cell. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.5 Riemann problem of a non-autonomous system at the left boundary xi− 12
of the i-th cell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.6 Block-structured grid. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.7 Typical control volume in a non-orthogonal grid. . . . . . . . . . . . . . . . 30
3.8 Ghost cell Ck+1 at the boundary . . . . . . . . . . . . . . . . . . . . . . . . 35
3.9 Mass flow through the boundary . . . . . . . . . . . . . . . . . . . . . . . . 35
3.10 Numerical treatment of sudden changes in flow cross-sections: Computa-tional domain is separated and pressure loss pv is introduced . . . . . . . . 37
3.11 Valve region and the adjacent finite volume cells: Ghost cells Ck+1,1 andC0,2 are introduced for the numerical boundary conditions. . . . . . . . . . . 38
3.12 Finite Cells at the T-piece. Junction area is shaded . . . . . . . . . . . . . . 39
3.13 Boundary cell in the two-dimensional domain . . . . . . . . . . . . . . . . . 40
3.14 Interface between two-dimensional and one-dimensional domain. . . . . . . 41
3.15 Mass correction: Mass inside the artifical cells and mass fluxes during onetime step . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4.1 Shema of the Burckhardt experimental compressor . . . . . . . . . . . . . . 45
4.2 Location of the sensors inside the cylinder 1, the suction and the dischargevalve retainer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4.3 Comparison of relative pressure at pc5 and valve lift at different times ta,tb and tc. Case: pout = 3 bar, cs =48N/mm , xv,max=1.35 mm. . . . . . . . 47
4.4 Comparison of relative pressure at pc5 and valve lift with 2 cylinders,1 cylinder and 1 chamber working. Case: pout = 3 bar, cs=48 N/mm,xv,max=1.35 mm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.5 Comparison of relative pressure in the valve retainer: 2 cylinders, 1 cylinderand 1 working chamber. Case: pout = 4 bar, cs=48 N/mm, xv,max=1.35 mm. 48
4.6 Relative Pressure at pressure sensors pc5, pc6 and pout and valve lift. Case:pout = 4 bar, cs=48 N/mm, xv,max=1.35 mm. . . . . . . . . . . . . . . . . . 49
4.7 Pressure difference pc5-pc1. Case: pout = 4 bar, cs=48 N/mm, xv,max=1.35mm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.8 Relative Pressure at pressure sensors pc1, pc3, pc4, pc5 and pc6. Case:pout = 4 bar, cs=48 N/mm, xv,max=1.35 mm. . . . . . . . . . . . . . . . . . 51
V
4.9 Relative pressure in front (pc5) and behind pressure valve (pout) . Cases:1 chamber, pout = 4 bar, cs=48 N/mm, xv,max = 1.35 mm and 1 chamber,pout = 1 bar, cs=48 N/mm, xvmax=1.35 mm. . . . . . . . . . . . . . . . . . 51
4.10 Difference pressure pc5-pout over valve lift for different discharge pressures. Cases: 1 chamber, pout = 1/2/3/4 bar, cs = 48 N/mm, xv,max = 1.35 mm. 52
4.11 Comparison of valve plate motion at sensor VP1, VP2 and VP3. Case: 1chamber, pout = 4 bar, cs =48 N/mm, xv,max=1.35 mm. . . . . . . . . . . . 53
4.12 Measurements of valve plate motion with sensor VP1 and VP2. . . . . . . . 54
4.13 Askew valve plate: Different contributions. Case: 1 chamber, pout = 4 bar,cs =48 N/mm, xv,max=1.35 mm. . . . . . . . . . . . . . . . . . . . . . . . . 54
4.14 Comparison of relative pressure at pc5 and valve lift. Cases: pout = 4 bar,cs=48 N/mm and cs=80 N/mm, xv,max=1.35 mm. . . . . . . . . . . . . . . 55
4.15 Comparison of relative pressure at pc5 and valve lift. Cases: pout = 4 bar,cs=48 N/mm, xv,max = 1.05 / 1.35 / 1.65 mm . . . . . . . . . . . . . . . . 56
4.16 Location of sensors inside cylinder . . . . . . . . . . . . . . . . . . . . . . . 57
4.17 Measurement data of absolute pressure at different sensors. . . . . . . . . . 58
5.1 Burckhardt test compressor: Comparison of absolute pressure at pc5 andvalve lift. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
5.2 Burckhardt test compressor: Comparison of absolute pressure at pc6 andvalve lift. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
5.3 Burckhardt test compressor: Comparison of absolute pressure at pout. . . . 61
5.4 Burckhardt test compressor: Comparison of difference pressure pc5-pc1. . . 61
5.5 Initiation of waves: Pressure distribution inside the cylinder and in thevalve retainer at the beginning of the discharge . . . . . . . . . . . . . . . . 63
5.6 Delay of valve opening due to viscosity in valve gap . . . . . . . . . . . . . . 64
5.7 Effect of pressure loss at sudden change of cross-section . . . . . . . . . . . 64
5.8 Comparison of absolute pressure at pout for different models of the pressurechamber. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
5.9 Comparison of absolute pressure at pc5 and valve lift for different modelsof the pressure chamber. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
5.10 Valve Losses at discharge valve . . . . . . . . . . . . . . . . . . . . . . . . . 66
5.11 Impact velocity of discharge valve depending on discharge pressure . . . . . 66
5.12 Ariel test compressor: Comparison of pressure distribution at pc2 and pout 67
5.13 Ariel test compressor: Comparison of pressure distribution at pc4 and pin . 68
5.14 Ariel test compressor: Comparison of difference pressure pc2-pc4. . . . . . . 68
5.15 Variation of discharge pressure: pout = 640000 / 610000 Pa . . . . . . . . . 70
5.16 Ariel test compressor: Pressure distribution in the cylinder at various times. 72
VI
5.17 Ariel test compressor: Velocity magnitudes in the cylinder at various times. 73
5.18 Moment onto Piston . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
5.19 Calculating the moment onto the piston: Pressure distribution inside cylinder 73
5.20 Valve motion for different discharge valves . . . . . . . . . . . . . . . . . . . 74
5.21 Mass flux through different discharge valves . . . . . . . . . . . . . . . . . . 74
5.22 Effect of Valve Masking: Comaparison of pressure at suction side pc2 . . . 74
6.1 Comparison of different finite volume schemes: Pressure pc5 inside thecylinder and valve motion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
6.2 Comparison of different finite volume schemes: Pressure difference pc6-pc5. 77
6.3 Comparison of absolute pressure at pc5 and valve lift with discharge pres-sure of 2 bar. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
VII
List of Tables
3.1 Different one-dimensional finite volume schemes . . . . . . . . . . . . . . . . 30
4.1 Main Specifications of the Burckhardt test compressor. . . . . . . . . . . . . 45
4.2 Specifications of the valves. . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4.3 Pressure difference at closed valve (valve starts to open) for different dis-charge pressures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.4 Impact velocities for different maximum valve plate lifts. . . . . . . . . . . . 55
4.5 Main Specifications of the Ariel test compressor . . . . . . . . . . . . . . . . 56
5.1 Specifications of the test case . . . . . . . . . . . . . . . . . . . . . . . . . . 59
6.1 Calculation Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
A.1 Sensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
A.2 Pressure loss coefficients for the T-piece [14] . . . . . . . . . . . . . . . . . . 84
A.3 Values for fbr [14] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
VIII
List of Symbols
A area
Av force area of the valve
c speed of sound
cF force coefficient
cm1 valve masking constant
cp specific heat capacity at constant pressure
cv specific heat capacity at constant volume
cs spring constant
C area of two-dimensional cell
d diameter of cylinder
fe1mm valve flow area when xv = 1mm
F numerical flux vector
FLF Lax-Friedrich’s numerical flux vector
FR Richtmyer numerical flux vector
f flux vector in x-direction
Fadh valve plate force due to adhesion effects
Fv total valve plate force
F second order correction of F-wave
g flux vector in y-direction
h specific enthalpy
h tensor
ht total specific enthalpy
i unit vector in x-direction
j unit vector in y-direction
J Jacobian matrix
K pressure loss coefficient
l length of piston rod
l1 initial deflection of valve spring
m mass
m mass flux
IX
mv mass of valve plate
M mass per area
n speed of the crankshaft
n unit normal vector
p pressure
pin suction pressure
pout discharge pressure
pt total pressure
p0 reference pressure
Q heat flux
r length of the cranklever
r eigenvector of the Jacobian matrix
R mass per length
Re Reynolds number
s source vector
S surface
T temperature
Ty moment onto piston
T matrix of right eigenvectors of the Jacobian matrix
t time
t dimonsionless time composed with wave speed and diameter of cylinder
t dimonsionless time composed with piston motion
u state vector
U averaged state vector
u velocity in x-direction
v velocity in y-direction
vv valve plate velocity
V volume
Vch volume of cylinder head recess
w characteristic variable
ws width of valve pocket opening
W shock-wave discontinuities
x coordinate in lateral direction, coordinate in plane parallel to cylinder head
xm0 characteristic length of valve masking
xm1 characteristic length of valve masking
xv distance between valve plate and valve seating
xv,min minimum distance between valve plate and valve seating
xv,max maximum distance between valve plate and valve seating
y coordinate in plane parallel to cylinder head
z coordinate in direction of cylinder axis
Z distance between cylinder head and piston
X
Z0 cylinder clearance, Z at dead centner
Z F-wave, numerical flux increments
Greek symbols
α coefficient for the wave decomposition
αnum numerical viscosity in x-direction
αv valve constant describing effective cross-section
β coefficient for the F-wave decomposition
βnum numerical viscosity in y-direction
βv valve constant describing effective cross-section
γ ratio of specific heats
∆t time step
∆x width of cell in x-direction
∆y width of cell in y-direction
∆pmax maximum pressure difference inside cylinder
ǫ compressor classification number
Θ quotient of pressure gradients
λ eigenvalue of the Jacobian matrix
µ dynamic viscosity
Ψ flux limiter
ρ density
ρ0 reference density
σ eigenvalue of the Jacobian matrix
φ effective flow area of the valve
ϕ angle of the crankshaft
XI
Chapter 1
Introduction
The task of a reciprocating compressor is to increase the pressure of a gas. The mainparts are a cylinder, a driven piston and passive unidirectional restrictor valves (figure1.1). During expansion the piston movement increases the volume of the compressionchamber. When the pressure inside the cylinder drops below the suction pressure gasenters the cylinder via the suction valves. The piston reaches the lower dead centre,reverses its movement and therefore starts to compress the fresh gas. Now the suctionvalves close and the gas undergoes an almost adiabatic compression. When the pistonapproaches the top dead centre the pressure inside the compression chamber forces thedischarge valves to open and the gas is delivered to the discharge pipe. Finally, the pistonreaches the top dead centre, discharge valves close and the cycle starts again.
Smaller reciprocating compressors can be found in everyday life applications such asrefrigerators and trucks whereas this thesis addresses rather large barrel design compressorsused in the gas industry, steel industry, chemical industry, oil refineries and low densitypolyethylene productions. In a barrel design compressor the valves are located alongsidethe cylinder (figure 1.1).
Figure 1.1: Double-acting compressor of barrel-design.
Reciprocating compressors are often crucial and expensive systems in this fields ofapplication and reliability is very important. Since most of the machines are custom
1
designed or produced in small batches designing by means of prototypes can not be used.Therefore accurate design tools play an important role. Not only power consumption andoverall flow rate are important design criteria but also valve motion and moments on thepiston, which influence the life-cycle of the the machines considerably, must be determinedaccurately.
Full three-dimensional simulations of reciprocating compressors with present commer-cial CFD-programs are the state of the art. They produce accurate and reliable resultsregarding valve motion, gas flow and heat transfer [22]. However, they cannot meet thedemands of engineers for usability and short computation times [22].
Common simplified approaches with short computation times are based on the worksdone by Costagliola [5]. In this model the change of state inside the cylinder is assumedto be quasi static and isentropic. Thus the thermodynamic state (pressure, temperature,density) is taken to be uniform within the cylinder. The valve motion is determined bythe pressure difference across the valve. Since the valve is responsible for most of thebreakdowns during a lifecylce of a compressor [25] special attention has to be paid to thevalve motion. Moreover valve manufacturers have identified the impact velocity of thevalve plate onto the catcher as one of the most important design criteria. Due to thefact that wave phenomena within the cylinder are neglected the predicted valve motiondeviates considerably from the measured one and thus these simplified models fail to be asuitable base of a design tool. In addition, moments on the piston which are caused by thepressure waves inside the cylinder cannot be determined. Therefore critical excitations ofthe piston rod cannot be predicted.
Calculating the gas flow in a compressor taking waves into account seems to be acomplicated task and a full three-dimensional simulation unavoidable. However, the resultsof [22] for the cylinder show that plane waves in lateral direction dominate (figure 1.2).Near the centre line the curvatures of the pressure surfaces are small. It turns out that aquasi one-dimensional model of the gas flow in the compressor, coupled to a simple valvemodel is sufficient to describe the main effects in a compressor with two valves. Namelythe interaction of the plane waves with the valve dynamics. Compressors with more thantwo valves require a quasi two-dimensional approach in order to account for the multiplevalve pockets, where the gas enters and leaves the cylinder.
Figure 1.2: Results of a full three-dimensional simulation: pressure distribution at thecylinder head and in the symmetry plane [22].
Studies of E. Machu [19] give a first insight of the effect of waves and unsteady flow
2
inside the cylinder. He calculates simple waves by using the method of characteristics.In his paper it is shown that calculated impact speed is considerably reduced, comparedto Costagliola models, by taking the waves into account. However, only the initial stagesof the outflow phase can be captured well. Moreover this model does not account forgeometrical effects. Later G. Machu [20], [21] introduced a model based on the one di-mensional Euler equations, which is capable of describing laterally running waves in thecompressor and their interaction with the valve dynamics. This model provides the basisfor the present work.
In Chapter 2 different models for the interaction of the gas flow and the valve dy-namics are described. Depending on the description of the geometry three-, two- andone-dimensional models are considered. It will be discussed how a complicated three-dimensional geometry is mapped onto a quasi one- or two-dimensional geometry. Specialcare is taken selecting the computational domain and appropriate boundary conditions.The main advantage of quasi one- and two- dimensional models is that the computationaldomains can be kept constant although the flow domain inside the cylinder changes due tothe piston motion. Furthermore we derive the governing equations for the gas flow fromthe Euler equations by integrating over a cross-section or cell. The resulting system ofhyperbolic equations with source terms can be written in the form
∂u
∂t+
∂f(u, x, y)
∂x+
∂g(u, x, y)
∂y= s(u, x, y) . (1.1)
where u stands for the state vector. The flux vector f and g and the source vectors depend not only on u, but also on the space coordinates x and y. In case of one-dimensional models the dependencies on y and the source terms s can be eliminated. Thesystem of equations (1.1) is supplemented by adequate boundary conditions at the valves,the walls, the inlets and the outlets. T-pieces and interfaces between one-dimensional andtwo-dimensional computational domains are treated as special boundaries with additionalequations. Finally the description of the valve dynamics and the flow through the valvescomplete the mathematical model.
In Chapter 3 we address the numerical solution of the governing equations. The focuslies on the discussion of finite volume schemes for the one-dimensional and two-dimensionalgas flow equations. Here difficulties arise from the space varying flux vectors and due tothe fact that in (1.1) the source terms s become infinite in case of jumps in cross-sections orcell heights. Different approaches for this problem and their advantages and disadvantagesare discussed. In addition the boundary conditions as well as the T-piece and interfaceconditions are implemented into the numerical scheme. Finally the numerical treatmentof the motion equation for the valve plate is presented.
Chapter 4 is concerned with the measurements of two test compressors at BurckhardtCompression AG, Switzerland and Ariel Corporation, USA. Both compressors are double-acting barrel design compressors. However, the Burckhardt compressor has two valvesand is about three times smaller than the compressor from Ariel with eight valves. Theresults of these measurements give good hints about the interaction of pressure waves andvalve motion in a reciprocating compressor.
In Chapter 5 the quasi one-dimensional and two-dimensional simulations are comparedto selected test cases of the measurements. It will be shown that the models are capable ofcapturing the most important physical effects. Moreover they predict accurately not onlygas flow and pressure distribution inside the cylinder but also important design criteria of
3
the valves such as impact velocity of the valve plate and valve losses.
Chapter 6 is concerned with comparison of different numerical schemes and their so-lution. Computational time, limitations and accuracy of the different approaches arediscussed.
4
Chapter 2
Modelling of the ReciprocatingCompressor
In this chapter the mathematical model of a reciprocating compressor is discussed. Start-ing from a complicated three-dimensional geometry of a compressor the one-dimensionalmodel is derived. In order to include the effects of multiple discharge and suction valvesthe gas flow inside the cylinder is also modelled quasi two-dimensionally. Integrated Eulerequations form the governing equations for the gas flow and different boundary condi-tions complete the mathematical model. A major part of this chapter is dedicated to thevalve description which follows and extends the ideas of Costagliola [5]. In addition pistonmotion and heat transfer are addressed.
2.1 Geometry
2.1.1 Simulation domain
The simulation domain covers not only the working chamber (consitsting of the cylinder,the valve pockets and the cylinder head recesses), but also the valve retainers and pressurechambers (figure 2.1). Since measurements show that the pressure variations in the suctionvalve retainers are negligible small and thus pressure and temperature are constant, theinlet boundary of the simulation domain is set right before the suction valve. However, atthe discharge side the conditions are much more complicated. Due to high mass outflowrates during discharge, pressure waves with high amplitudes are running back and forththrough the valve retainer, the pressure chamber and the discharge pipes and thus influencethe valve motion. Therefore the outlet boundary must be set at a sufficient distance tothe valve retainer, where the pressure is almost constant. This distance depends stronglyon the geometry of the pressure chamber and the discharge pipe.
2.1.2 One-dimensional model
In the quasi one-dimensional model the wave propagation along the diameter (x-axis) ofthe cylinder from the suction to the pressure side is considered (figure 2.2). The Eulerequations are integrated over a cross section A(x, t) perpendicular to the x-axis. The
5
Cylinder
Suction
Discharge
xy
z
Valve
Valve
Valve
Valve
Retainer
Piston
Pressure
Chamber
Boundary
Boundary
Outlet
Inlet
Cylinder Head
Recess
Figure 2.1: Flow geometry inside the compressor.
effective cross sections of the quasi one-dimensional model are displayed schematicallyin figure 2.2. Starting from left we have the suction valve followed by suction valvepocket, cylinder, discharge valve pocket, discharge valve and valve retainer. In the case ofcompressors with more than two valves the suction valves and discharge valves are replacedby one adequate suction and discharge valve, respectively. Due to the piston motion thecross-section A(x, t) inside the cylinder varies with time. Figure 2.3 shows the appropriatearea functions depending on x at different times t1 and t2. At t2 the piston masks thevalve pocket in a way that the effective cross section area A(x, t) is discontinuous there.
suction discharge
cylinder
A(x, t)
Z
valve pocketvalve
valve valve
retainer
Figure 2.2: Schema of the one-dimensionalmodel.
x 0
suction dischargecylinder
A(x
,t)
t2
t1
valvevalvevalve retainer
Figure 2.3: Cross-section areas of the one-dimensional model.
Several approaches are used to model the discharge side of the compressor.
6
• The first model has been used by Machu [20] (figure 2.4). Here, the pressure chamberis added to the subdomain of the valve retainer and the outlet boundary is set at theend of the valve retainer. A pressure loss at the outlet is specified to simulate thehindered flow of the valve retainer to the pressure chamber. Moreover, a reducedoutflow area causes reflections of the pressure waves at the end of the valve retainer.
• In the second approach the complicated geometry of the pressure chamber is replacedby a cylinder connected to the valve retainer using a T-piece (figure 2.5). Again thepressure is assumed to be constant at the end of the pressure chamber.
• The third approach is used when effects of the discharge pipe and damper are takeninto account. Here the outlet boundary is located at the end of the damper (figure2.6). In case of double-acting compressors the pressure chamber links the valveretainer of the crank end (CE) to the one of the head end (HE). Again we use aT-piece to connect the discharge pipe to the pressure chamber.
It must be pointed out that reducing the pressure chamber to a simple cylinder may bean oversimplification in some cases.
Cylinder
Piston
Outlet BoundaryValve
Valve Retainer
Valve Pocket
Figure 2.4: Model of discharge systemVR1: Pressure chamber is modelled at theend of the valve retainer.
Cylinder
Pressure Chamber
Piston
Outlet Boundary
Valve
T-piece
Valve RetainerValve Pocket
Figure 2.5: Model of discharge systemVR2: Pressure chamber is connected tothe valve retainer by a T-piece.
2.1.3 Two-dimensional model
The two-dimensional model takes the wave propagation in a plane parallel to the cylinderhead (x,y-plane) into account. The equations of motion (Euler equations) are integratedover the height z(x, y, t) of the cylinder. In terms of the compressor the height is thedistance between the piston and the cylinder head. The two-dimensional computationaldomain is displayed schematically in figure 2.7.
In order to keep the model as simple as possible and hence least time consuming (interms of computational time) a one-dimensional approach is used for the valve pocketsand valve retainers. The coupling between the quasi one-dimensional computational sub-domain and the quasi two-dimensional one is described by the interface conditions, see
7
Cylinder
Cylinder
PressureChamber
Discharge Pipe
Piston
BoundaryOutlet
crank end
head end
Valve
Damper
T-piece
T-piece
T-piece
Valve RetainerValve Pocket
Figure 2.6: Model with discharge pipe and damper. Crank end working chamber is op-tional (dashed lines).
section 2.4.5. A typical arrangement for a compressor with eight valves is shown in figure2.8.
2.2 Characteristic Data
In this section the basic assumptions about the gas flow and heat transfer in a compressorare stated. They are based on measurements, full three-dimensional CFD-simulations andexperiences of compressor manufacturers.
2.2.1 Gas Flow
For each phase of the compression cycle a Reynolds number characterises the gas flow.During discharge and intake the maximum gas velocity, the actual distance between cylin-der head and piston and the actual viscosity are used as the reference values. Accordingto full three-dimensional CFD-simulations [3], [22] the Reynolds numbers range from 104
to 105. During compression and expansion the mean piston velocity, the diameter of thecylinder and the actual viscosity yields a Reynolds number in the order of magnitude of104. Comparisons with critical Reynolds numbers leads to the assumption that in thecylinder turbulent flow occurs during the whole cycle. Furthermore the high Reynoldsnumbers allow the use of inviscid gas flow models. Although these results are true for thetest cases calculated (see tables 4.1 and 4.5, the assumptions can be extended to most ofthe gases compressed.
8
Suction Discharge
Z
valve pocket
retainer
Cylinder
Valve
Valve Valve
cell
Figure 2.7: Schema of the two-dimensional model.
Interface
valve pocket
Cylinder
Valve
x
y
one-dimensional
two-dimensional
Figure 2.8: Schema of the two-dimensional model with eight valves.
2.2.2 Heat Transfer and Dissipation
In turbulent flow the heat transfer from the gas to the surrounding structure is determinedby local flow properties only [4], [9]. However, if we assume inviscid flow and Prandtl num-bers in the order of 1 then an almost adiabatic change of condition can be expected. Inother words, the gas flow is not influenced by the heat transfer from and to the surround-ing material. In addition for small Eckert numbers compared to the Reynolds number(EC ≪ Re) the dissipation can be neglected [4]. During compression and expansion themean piston velocity, the specific heat capacity for constant pressure and the temperaturedifference between suction pipe and discharge pipe yields a Eckert number in the order ofmagnitude of 10−4. During discharge and intake the maximum gas velocity, the specificheat capacity for constant pressure and the temperature difference across the cylinder areused as the reference values. According to full three-dimensional CFD-simulations [3], [22]the Eckert numbers of the test compressors are in the order of magnitude of 1. Hence, heattransfer and dissipation are neglected. In fact the essential part of temperature changesof the gas are due to compression and expansion.
When the dissipation and heat transfer are very small, the Euler equations [9] can beused. However, at sudden changes of flow cross-sections pressure losses occur and thisproblem is addressed in section 2.4.6.
2.2.3 Classification of Compressors
The sudden opening of the discharge valves excites a rarefaction wave. Considering differ-ent time scales allows to estimate the response to this excitation. Moreover, this allows aclassification of compressors whether or not pressure waves play an important role. Dur-ing discharge two different characteristic time scales can be identified. On one hand wehave the time t a wave needs to travel through the cylinder, defined by the diameter ofthe cylinder D and the wave speed at reference state c0. On the other hand we have thetime t specified by the parameters of the piston motion, more precisely the distance of the
9
piston to the cylinder head Z0 and the acceleration of the piston Z0 in dead centre.
t =tDc0
, t =t
√
Z0
Z0
(2.1)
Now for the non-dimensional characteristic parameter (classification number) ǫ = t/t/ wecan distinguish three different orders of magnitude. If ǫ ≪ 1 then the speed of the wavesis much faster than the piston motion and an incompressible outflow of the compressoroccurs. However, ǫ of typical compressors are of order 1. Therefore compressibility effectsmust be taken into account. ǫ ≫ 1 holds true in the case of heavy gases and very fastrunning compressors.
2.3 Piston Motion
The distance between the piston and the cylinder head Z depending on time t can bedetermined using the properties of the crank mechanism. We use the length of the crank-lever r, the length of the piston rod l, the smallest distance between piston and cylindertop Z0 and the present angle of the crankshaft ϕ to obtain
Z(t) = r + l + Z0 − r cos ϕ(t) − l
√
1 −(r
l
)2sin2 ϕ(t) , (2.2)
where φ(t) = 2πnt. Here, n denotes the speed of the crankshaft. As a first approximationfor r
l ≪ 1 (2.2) reduces toZ(t) = r + Z0 − r cos φ(t) . (2.3)
2.4 Gas Flow inside the Compressor
2.4.1 Quasi one-dimensional Model
The governing equations for the gas flow depending on time t and the space coordinatex are obtained by taking the mass, momentum and energy balance over a cross section.The system of equations is written as
∂
∂t[ρA] +
∂
∂x[ρuA] = 0 , (2.4)
∂
∂t[ρuA] +
∂
∂x
[
ρu2A + pA]
= p∂A
∂x, (2.5)
∂
∂t
[
ρA
(
cvT +u2
2
)]
+∂
∂x
[
ρuA
(
cvT +u2
2
)
+ uAp
]
= 0 , (2.6)
where the variables p, ρ, T and u have their usual meaning, pressure, density, temperatureand velocity in x-direction, respectively. The specific heat capacity for constant volumeand for constant pressure are denoted by cv and cp respectively. In case of smooth solutionsthe energy equations (2.6) can be replaced by
∂
∂t(ρs) +
∂
∂x(ρus) = 0 , (2.7)
10
thus allowing isentropic solutions (s = const.). Assuming isentropic solutions we canexpress the pressure p in terms of the density and entropy
p = p(ρ, s) . (2.8)
In case of constant heat capacities following relation is obtained
pρ−γ = const. , (2.9)
with γ = cp/cv . Smooth solutions are the case if no shocks or flow separations occur.Measurements show that the first condition is satisfied. However the second one is violatedat sudden changes of cross-sections, sharp turns of the flow direction, and T-pieces. Atthis point we accept this violation and assume that the associtated entropy productionsare sufficiently small. In section 2.4.6 we present a correction of this problem.
With an adequate reference state, an associated reference pressure p0 and referencedensity ρ0, (2.9) can be rewritten as
pρ−γ = p0ρ0−γ . (2.10)
We observe that (2.5) is not a homogenous conservation law [28]. The right hand sideof (2.5) constitutes a momentum source of the flow due to a variation of the cross section.Since the cross section A may vary rapidly at the transition from the cylinder to the valvepocket, the derivative dA/dx may become large causing numerical problems. For smoothsolutions these problems can be avoided by transforming the momentum equation (2.5) to
∂u
∂t+
∂
∂x
(
u2
2
)
+1
ρ
∂p
∂x= 0 . (2.11)
Furthermore, using the definition (2.8) and assuming constant entropy yields
∂u
∂t+
∂
∂x
(
u2
2
)
+1
ρ
∂p
∂ρ
∂ρ
∂x= 0 . (2.12)
With the definition of the speed of sound c =√
(∂p∂ρ )s we obtain
∂u
∂t+
∂
∂x
u2
2+
ρ∫
ρ0
c2(ρ, s0)
ρdρ
= 0 . (2.13)
In case of constant specific heat capacities equation (2.13) reduces to
∂u
∂t+
∂
∂x
(
1
2u2 +
γ
γ − 1
p
ρ
)
= 0 . (2.14)
From now on we will consider constant specific heat capacities only, since the extensionto the general case with heat capacities depending on the temperature of the gas causesno difficulties.
Finally the system of governing equations can be written in conservative form:
∂u
∂t+
∂f(u, x)
∂x= 0 . (2.15)
11
Here the state vector u and the flux function f(u) can be given by
u =
(
R
u
)
, f(u) =
(
Ruu2
2 + p0
ρ0γ
(
RA
)γ−1
)
, (2.16)
where R = ρA. Note that 2.16 does not correspond to a physical conservation law since uis no conservation variable.
Local Properties of the system equations In case of constant cross-sections A theflux vector in (2.15) depends on the state vector only and the system of equations reducesto
∂u
∂t+
∂f(u)
∂x= 0 . (2.17)
Furthermore, the linearisation around an arbitrary fixed state u0 yields
∂u
∂t+ J(u0)
∂u
∂x= 0 , (2.18)
where J(u0) is the Jacobian matrix of the flux function f(u) and given by
J(u) =
(
u Rc2
R u
)
. (2.19)
Here, c denotes the speed of sound. The eigenvalues λ1, λ2 of J are
λ1 = u − c ,
λ2 = u + c . (2.20)
The matrix T of right eigenvectors of the Jacobian J is
T = (r1, r2) =
(
−Rc
Rc
1 1
)
. (2.21)
We introduce the characteristic variables w = (w1(u(x, t)), w2(u(x, t)))T defined by
w := T−10 u, T−1
0 = T−1(u0) . (2.22)
Here w1 and w2 are given by
w1 =1
2(u − c) , (2.23)
w2 =1
2(u + c) . (2.24)
Finally 2.18 can be transformed to a decoupled system of linear partial differential equa-tions in terms of the characteristic variables
∂w1
∂t+ λ1(u0)
∂w1
∂x= 0 , (2.25)
∂w2
∂t+ λ2(u0)
∂w2
∂x= 0 . (2.26)
In section 3.2 we will use these results, especially the characteristic variables and theircorresponding partial differential equations, in order to derive the numerical formulationsof different boundary conditions. Moreover the wave propagation structure described bythe characteristics will be incorporated in the finite volume scheme F-wave (see section3.1.1).
12
2.4.2 Quasi two-dimensional Model
The governing equations for the flow of gas are obtained by taking the mass and momentumbalance over a cell. In addition to the one-dimensional approach v denotes the velocityin y-direction and the momentum equation is written for the x-direction and y-direction,respectively.
∂
∂t[ρZ] +
∂
∂x[ρZu] +
∂
∂y[ρZv] = 0 , (2.27)
∂
∂t[ρZu] +
∂
∂x
[
ρZu2 + pZ]
+∂
∂y[ρZuv] = p
∂Z
∂x, (2.28)
∂
∂t[ρZv] +
∂
∂y[ρZvv + pZ] +
∂
∂x[ρZuv] = p
∂Z
∂y, (2.29)
where Z denotes the distance between the piston and the cylinder head.
Assuming isentropic flow conditions we replace the energy equation by the isetropicequation of state (2.9). Note that equations (2.28) and (2.29) are not of conservation form.The right hand terms constitute a momentum source of the flow due to the variation of theheight inside the cylinder. However, only smooth cylinder head recesses can be consideredsince the gradients of Z become very large in case of sharp changes.
The governing equations (2.27)-(2.29) reduce to the one-dimensional model equationsby replacing the height Z with the cross-section A and setting the velocity in y-directionv to zero.
For the numerical analysis it is useful to write the continuity equation (2.27) and bothequations of motion (2.28) and (2.29) in the following form:
∂u
∂t+
∂f(u)
∂x+
∂g(u)
∂y= s . (2.30)
Here the the state vector u and the analytical flux vectors f and g are given by
u =
M
Ix
Iy
, f(u) =
Ix
Ix2
M + p0
ργ0
Mγ
Zγ−1
IxIy
M
, g(u) =
IyIxIy
MIy
2
M + p0
ργ0
Mγ
Zγ−1
, (2.31)
where M = ρZ, Ix = ρZu and Iy = ρZv. The source vector s can be written as
s =
0
p∂Z∂x
p∂Z∂y
. (2.32)
When Z is constant 2.32 becomes s = (0, 0, 0)T . This is true for compressors with planecylinder heads and pistons.
2.4.3 Boundary Conditions
The computational domain of the compressor is divided into the subdomains of the cylin-der, the valve retainers, and the pressure chamber. In addition, if the flow cross section
13
in the cylinder changes abruptly the cylinder is divided into two further computationalsubdomains. On each subdomain the governing equations for the gas flow have to besupplemented with boundary conditions or interface conditions to adjacent subdomains.In general we can distinguish two types of boundary conditions needed: on the one handimpermeable walls such as closed valves, or the cylinder wall in the two-dimensional ap-proach and on the other hand permeable boundaries where mass flows enter and leave thecomputational domain (flow boundaries).
Walls On walls we require that the velocity and the momentum normal to the boundaryis zero.
Flow Boundaries In order to obtain adequate boundary conditions we must considercharacteristics leaving and entering the computational domain. A necessary conditionfor a well posed problem is that the boundary condition in terms of physical variablesdo not lead to prescription of a characteristic variable associated with a characteristicleaving the computational subdomain under consideration. In other words, the numberof boundary conditions should be equal to the number of characteristics pointing into theregion at a boundary. Considering linearised system (2.18) and assuming subsonic flow thesigns of the eigenvalues λ1, λ2 defined in (2.20) remain constant. Hence the characteristiccorresponding to the negative eigenvalue λ1 always points to the left whereas the secondone always points to the right. Figure 2.9 shows the boundary-value problem on the rightside of a computational domain. Wibmer [31] has shown, that a well posed problem isobtained, if one and only one of the variables pressure, density, mass flow or velocity isprescribed (due to one entering characteristic).
x
computational domain
right boundary
w1
w2
Figure 2.9: Right boundary of computational domain: Characteristics variables enteringand leaving at the right end of the computational domain.
2.4.4 T-piece
In the valve retainer the mass flow coming from the cylinder is divided in the one thatruns further downstream and the one that enters the pressure chamber of the compressor.This junction, or so called T-piece, can be described by a simple one-dimensional model.However at this point we have to neglect effects from different branching angles, pressurelosses, and flow separations. These issues are subject to section 2.4.6 where pressure losses
14
for T-pieces are introduced. Furthermore, in case of double-acting cylinders this modelcan be applied to the T-piece in the pressure chamber, where the mass flows from thecrank end side and head end side merge and enter the discharge pipe.
1 2
3
w1,1
w2,1
w2,2
w1,2
w2,3
w1,3
x1 x2
x3
h1 = h2 = h3
m1 + m2 + m3 = 0
Figure 2.10: T-piece consisting of tube 1, tube 2 and tube 3: The shaded square displaysthe junction area. Its appropriate in-going and out-going characteristics w1,i and w2,i areindexed by the adjacent tube i = 1, 2, 3.
Now the T-piece itself is reduced to a boundary condition problem for the three pipes.Similar to section 2.4.3 the number of conditions must be equal to the number of char-acteristics pointing into the computational domains of the pipes. Figure 2.10 shows thecharacteristics entering and leaving the shaded junction area. We assume that the junctionitself has no volume, therefore the net mass flow m entering and leaving the junction iszero. Furthermore we require that the net enthalpy flux at the T-piece is also zero. Nowusing the mass balance for the T-piece and assuming that the entropy is constant in allthree pipes yield that the total specific enthalpy ht on each boundary must be equal. Thethree analytical boundary conditions can be written as
m1 + m2 + m3 = 0 , (2.33)
ht1 = ht2 = ht3 , (2.34)
where
ht = h +u2
2. (2.35)
The numerical treatment of the T-piece is subject of section 3.2.
2.4.5 Interface Conditions
The connection between one-dimensional and two-dimensional areas are called interfaces(figure 2.4.5). We assume that the interface has no volume and thus the mass flux leavingthe cylinder via the interface must enter the computational domain of the valve pocket andvice versa. Furthermore conservation of energy across the interface is required resultingin a parity of enthalpies on each face. Similar to a T-piece with only two branches theequations for the interfaces can be written as
m1d + m2d = 0 , (2.36)
ht1 = ht2 , (2.37)
15
where m1d and m2d denotes the mass flux from cylinder and valve pocket to the interface,respectively.
Cylinder
Interface
Valve pocket
Governing equations (2.15) Governing equations (2.30)
Face 1dFace 2d
Figure 2.11: Interface between one-dimensional and two-dimensional domain.
2.4.6 Pressure Losses
At sudden changes in cross-section and in T-pieces pressure losses occur. Although inthe quasi one-dimensional gas flow model the energy equation is not solved and isentropicflow is assumed, the pressure losses can be taken into account by means of simplifiedmodels. However, the governing equations for the gas flow (2.15) are not valid any more.Moreover at this locations the computational domain must be split up and new conditionsconnecting the resulting subdomains must be chosen. Similar to section 2.4.3 the numberof connecting conditions must be equal to the number of characteristics pointing into thecomputational subdomains.
Usually pressure losses in adiabatic flows are associated with conservation of energyand an entropy production. However in the present model constant entropy is assumed.Therefore we have to drop the requirement for adiabatic flow and compensate the entropyproduction by an artificial heat flux Q12 across the side walls.
For the transition from state 1 in front of the pressure loss (p1, ρ1, u1) to state 2 (p2,ρ2, u2) we require conservation of mass and entropy (figure 2.12):
m = m1 = m2 , (2.38)
s = m1s1 = m2s2 , (2.39)
and thus 2.39 can be replaced by s1 = s2. In addition the losses pv in total pressure pt aregiven by
pt1 = pt2 + pv . (2.40)
The total pressure is the pressure at the thermodynamic state that would exist if the gaswas brought to zero velocity at constant entropy. For a compressible gas with constantspecific heat capacities using the isentropic equation of state (2.9) we can write
pt = p
(
1 +γ − 1
γ
ρ
p
u2
2
)
γγ−1
. (2.41)
16
In case of incompressible flows losses in total pressure are described by loss coefficientsK, which can be written as
K =pv
ρu2/2, (2.42)
where ρu2/2 denotes the velocity pressure. According to Miller [23] the incompressibleloss coefficients are suitable for compressible flows as well if the losses are related to thedynamic pressure pt − p instead of the velocity pressure. For compressible flows equation(2.42) takes the form
K =pt1 − pt2
pt1 − p1, (2.43)
However, Miller points out that in case of separated flows the application of these coeffi-cients may not be very satisfactorily, since choked flow is neglected.
Sudden Expansions in Cross-section In consequence of a sudden expansion in of theflow cross-section A(x, t) separation zones arise (figure 2.13). In this area the velocitiesare rather small and the pressure is equal to p1. Further downstream the inflowing streammixes with the surrounding gas. The incompressible pressure loss coefficient for a suddenexpansions is given by
K = (1 − A1/A2)2 , (2.44)
where A1 is the inlet area and A2 the outlet area. In case of infinite outlet areas K becomes1 and thus (2.43) reduces to
p1 = pt2 = p2. (2.45)
p1
u1
ρ1
p2
u2
ρ2
Q12
Pressure Loss
K
Figure 2.12: Modelling pressure losses
p1
u1ρ1
A1 p2
u2
ρ2
A2
Dead water zones
Figure 2.13: Sudden expansion of cross-section
In order to implement the pressure losses at sudden expansions in cross-section a modelas described above can be employed resulting in computing the quasi-stationary systemof equations (2.38) and (2.40), see section 3.2.4.
Pressure losses at T-pieces Starting from the loss-free T-piece (section 2.4.4) eachpipe is provided with an pressure loss (figure 2.14). The thermodynamic states adjacent tothe junction are marked with stars. The pressure loss coefficients K1, K2 and K3 dependon the flow direction of the single branches, the cross-section areas A and the mass fluxes.They are derived from the incompressible loss coefficients of Idelchik [14] and given in the
17
appendix for important cases. Even the effects of branching angles and separation zonescan be easily incorporated by adjusting the loss coefficients.
We employ the equations of the T-piece (2.33) and (2.34) to connect each branch andthe pressure loss equations (2.38) and (2.43) to obtain
3∑
i
m∗i = 0 ,
h∗t1 = h∗
t2 ,
h∗t2 = h∗
t3 ,
Ki =pti − p∗tipti − pi
,
uiρi = u∗i ρ
∗i ,
piρ−γi = p∗i ρ
∗i−γ . (2.46)
where i is the subscript of the branch running from 1 to 3.
p1 p2p∗1 p∗2
A1 A2
A3
p∗3, u∗3, ρ∗3
p3, u3, ρ3
u1 u2u∗1 u∗
2
ρ1 ρ2ρ∗1 ρ∗2K1 K2
K3
Figure 2.14: Pressure losses at the T-piece
2.5 Valve Model
2.5.1 Valve Dynamic
The discharge and the suction valves are located adjacent to the circular lateral surfacesof the valve pockets. Both valves are unidirectional restrictor valves, plate valves or ringvalves are usually used. In this work we concentrate on plate valves but most of theequations can be easily adopted to any kind of restrictor valve. Figure (2.15) shows thebasic function of a plate valve. The state of a valve is specified by the distance betweenthe valve plate and the seating xv. The motion of the valve plate is determined by theforces acting on it. We consider the following three contributions to the resulting force:the pressure difference across the valve acting on an effective force area Av of the valve
18
p1
p2
Seating
Valve Platexv
sv
Catcher
Valve Springing
Figure 2.15: Schema of a plate valve.
plate, the springing and thirdly a contribution due to viscous forces in the initial stagesof valve opening. Denoting the pressure in front of the valve p1 and behind the valve p2
we obtain the equation of motion for the valve plate
mvxv = (p1 − p2)Av − cs (xv + l1) − Fadh , (2.47)
Here, mv stands for the mass of the valve plate. The constants cs and l1 denote springconstant and initial deflection of the spring, respectively. An initial sticking effect ismodelled by the force Fadh. It is caused by the viscosity µ of the gas or oil in the valve gapresulting in a small time delay when the valve is opening. We follow the ideas of Reynoldslubrication theory to obtain
Fadh = µsv3dv
1
xv3
dxv
dt. (2.48)
Here sv and dv stand for the length and the total depth of the valve gap. A detaileddescription and derivation can be found in [7] and [8]. The motion of the valve plateis limited by the seating on one side and by the catcher on the other side. Thereforethe deflection of the plate xv ranges from xv,min to xv,max. Because of the roughness ofthe seating and valve plate it is necessary that xv,min > 0 [8]. Due to the fact that theflow within the valve changes with xv, and thus the effective force area Av changes, weintroduce a force coefficient cF . The force area Av can be written as
Av = Av,0cF , (2.49)
where Av,0 stands for the force area at closed valve. For common plate valves the relationbetween the force coefficient and valve lift takes the form
cF =
(
1 − 0.2xv
xv,max
)
. (2.50)
2.5.2 Flow through the valve
The flow through the valves is considered as (quasi stationary) outflow of gas from apressurised vessel through a convergent nozzle. In case of constant heat capacities it is
19
given by St.Venant and Wantzell [32]
m = φρ1
(
p2
pt1
)1γ
√
√
√
√
2γ
γ − 1
pt1
ρ1
(
1 −(
p2
pt1
)γ−1
γ
)
, (2.51)
where pt1 is the total pressure before and p2 is the pressures after the valve, respectively.The effective flow cross section φ of the valve is assumed to be a function of the positionof the valve plate xv only. This is not true if effects of asymmetric flow conditions in thevalve pocket are taken into account (section 2.5.3). For common valves the equation forthe effective cross section takes the form
φ(xv) =
√
(fe1mmxv)2
αv + βvxv2
. (2.52)
The constant fe1mm denotes the valve flow cross-section when the valve plate is 1 mmabove the valve seating. The constants αv and βv describe non-linear dependencies of thevalve plate lift and have to be determined empirically.
2.5.3 Asymmetric Flow in Valve Pockets, Valve Masking and SlottedValve Pockets
Valve masking occurs when the piston approaches top dead centre (the distance betweencylinder head and piston becomes very small) and a part of the valve pocket is maskedby the piston itself, see figure 2.17. Two effects take place. Firstly, the sudden change ofcross-section areas produce pressure losses due to separation. This problem is discussed indetail in section 2.4.6. Secondly, if the distance between piston edge and valve is so smallthat the separation zone extends across the whole valve pocket length then the outflowinggas will use only a part of the valve flow cross-section. Similar effects take place in slottedvalve pocket entries where the effective flow cross-section of the valve is reduced due tothe separation zones (figure 2.18). However, full three-dimensional simulations show thateven in open valve pockets without valve masking asymmetric flow and separation zonescan occur. In figure 2.16 the simulation results of an academic compressor are displayed[24]. In order to take above effects into account we assume that the valve flow area is notonly a function of valve plate lift xv but also of the piston position Z,
φ(xv, h) = fmask(h)
√
(fe1mmxv)2
αv + βvxv2
. (2.53)
Here fmask denotes a empirical function dependent on Z. In case of valve masking thefollowing approximation is used:
fmask(h) = 1 − cm1e−(Z−xm0)
xm1 . (2.54)
where the constants cm1, xm0, and xm1 must be matched to measurement data.
20
Crank Angle=654.00(deg)Contours of Total Pressure (pascal) (Time=7.8333e-02)
FLUENT 6.2 (3d, dp, segregated, dynamesh, ske, unsteady)Mar 14, 2007
5.80e+065.79e+065.78e+065.76e+065.75e+065.74e+065.72e+065.71e+065.70e+065.69e+065.68e+065.66e+065.65e+065.64e+065.62e+065.61e+065.60e+065.59e+065.58e+065.56e+065.55e+06
Z
Y X
valve
valve pocket
cylinder
piston
discharge
Figure 2.16: Pressure distribution and velocity vectors in the symmetry plane duringdischarge (full three-dimensional simulation).
valve
valve
retainer
valve pocketcylinder
piston
separationzone
Figure 2.17: Valve Masking.
valve
valveretainer
valve pocket
cylinder
piston
separationzone
Figure 2.18: Slotted Valve Pockets.
21
Chapter 3
Numerical Method
In the first part of this chapter we present the treatment of the hyperbolic Euler equa-tions by finite volume schemes, following [16], [17], [13], [29], [6] and [30] . Here attentionhas to be turned to the fact that the flux functions in the governing systems of equa-tions (2.15) and (2.30) are not only a function of the state vector but also a function ofspace coordinates x and y. Therefore not every finite volume method is suitable. Forthe one-dimensional conservation law the finite volume methods by LeVeque [16],[17] andthe approach of MacCormack [13],[17], [29] are employed. Both methods are second orderaccurate in time and space. However LeVeques flux splitting scheme has the advantagethat boundary conditions, where calculating characteristics is required, can be easily in-corporated in the scheme. The two-dimensional Euler equations in the cylinder are solvedby Lax-Friedrich’s [13] finite volume scheme of first order and the second order method ofLax-Wendroff [15]. In case of discontinuities the performance of the second order approachis rather poor and thus a flux limiter is employed.
The boundary conditions are subject of the second part of this chapter. We distinguishbetween physical boundary conditions derived in section 2.4.3 and numerical boundaryconditions required for computation. The physical boundary conditions almost never fullydetermine the set of dependent variables of the underlying system of differential equations,whereas the numerical methods used require all dependent variables at the boundaries.In this work two different approaches are discussed and employed. The first methodextrapolates the interior state to an additional cell (ghost cell) outside the computationaldomain. The second method adds the characteristic equation (2.25) corresponding to theoutgoing characteristic to the imposed physical boundary conditions.
Finally, the integration of the motion equation of the valve plate (2.47) is discussed.
3.1 Finite Volume Schemes
3.1.1 One-dimensional Finite Volume Schemes
Each spatial subdomain is divided into intervals Ci (finite volumes, cells) of constant length∆x with midpoint xi (figure 3.1). Denote the i-th cell by
Ci =(
xi− 12, xi+ 1
2
)
, (3.1)
22
CellCell
CellCellCell
C0
x0
Ci−1 Ci Ci+1Ck
xi−1 xi xi+1 xk
∆x
Figure 3.1: Subdivision of the computational domain into grid cells Ci.
where the cell boundary on the left side is located at xi− 12
= 12 (xi + xi−1) and on the
right side at xi+ 12
= 12 (xi + xi+1). Note that in a quasi one-dimensional problem, the
cross-section areas and thus the ”physical” volume of the cells can vary along x.
The actual derivation of the governing equations (quasi one-dimensional Euler equa-tions) is based on balances over control volumes. It seems natural to return to this formu-lation, since subdomains are divided into finite control volumes. Now the Euler equationsare satisfied in each cell, above all the conservation of the state quantities is ensured.Moreover, the integral formulation allows discontinuous solutions (shocks), whereas thedifferential formulation requires smooth solutions. Integrating the differential form of thequasi one-dimensional Euler equation
∂u
∂t+
∂f(u, A)
∂x= 0 ,
over the cell Ci and one time step,[
xi− 12, xi+ 1
2
]
× [tn, tn+1], yields the integral formulation:
xi+1/2∫
xi−1/2
u(x, tn+1)dx−xi+1/2∫
xi−1/2
u(x, tn)dx+
tn+1∫
tn
f(u(xi+1/2, t), A)dt−tn+1∫
tn
f(u(xi−1/2, t), A)dt = 0 .
(3.2)Note that the cross-section area A, and thus the analytical flux f , are a function of timet and coordinate x. The first two terms in (3.2) correspond to the total change of thestate quantities inside the cell during the time step ∆t = tn+1 − tn. The third and fourthintegral represent the total fluxes of the state quantities across the boundaries xi+ 1
2and
xi− 12. Therefore the system of equations (3.2) simply states that every gain or loss of
state quantities inside the cell is due to the fluxes across the boundaries only. Moreoverthe conservation of the state quantities within the whole computational subdomain willbe ensured [16].
Let Uni denote the cell average of the state quantities over the ith cell at time tn:
Uni ∼ 1
∆x
xi+1/2∫
xi−1/2
u(x, tn)dx . (3.3)
23
The flux integral of (3.2) at the boundary xi− 12
is approximated by the numerical flux
Fni− 1
2
:
Fni− 1
2(Ui−q1 ..Ui+q2 , A) ∼ 1
∆t
tn+1∫
tn
f(u(xi−1/2, A, t))dt . (3.4)
The numerical flux depends on the cell averages of the state quantities Ui−q1 ..Ui+q2 andthe cross-section area A. Whereas the actual number of involved cell averages given by q1
and q2 is determined by the numerical scheme used. Finally, when employing the aboveapproximations the system of equations for Un+1
i of cell Ci at the time tn+1 can be writtenas:
Un+1i = Un
i − ∆t
∆x
(
Fni+ 1
2−Fn
i− 12
)
(3.5)
A numerical scheme which has the form (3.5) is called conservative since it mimics theproperties of (3.2).
Generally we require that a numerical scheme is convergent which means that thenumerical solution converges to the true solution of the differential equations as the gridis refined. According to Leveque [16] this is true if following two conditions are satisfied:
• Firstly the method must be stable, meaning that small errors made in each timestep do not grow too fast in later time steps.
• Secondly the method must be consistent, meaning that the numerical flux ap-proximates the analytical flux well. More precisely, for constant state quantitiesU = U = const. and constant cross-sections A = A the numerical flux must beequal to the analytical flux:
F(U, .., U, A) = f(U, A) . (3.6)
A detailed description and mathematical formulation of these conditions can be found in[16], [13], [29], [6] and [30]. Later we will state the conditions under which the numericalschemes used in this work are convergent. Now we introduce different numerical fluxfunctions of (3.4) resulting in different finite volume methods.
Godunov method First we consider autonomous systems, where the flux vector f isa function of the state vector u only. In case of quasi onedimensional Euler equationsthis means that the cross-sections are constant through the entire computational domain(A = A(t)). The resulting system of equations can be written as
∂u
∂t+
∂f(u)
∂x= 0 . (3.7)
Following the ideas of Godunov [16] we decompose the original initial value problem intoa set of Riemann problems, by replacing the initial data by the cell average on every cell.Thus the resulting initial data has discontinuities at every cell boundary and we have tosolve a Riemann problem for every cell boundary. In particular the Riemann problemat the left boundary xi− 1
2of the i-th cell consists of the conservation law (3.7) and the
piecewise constant initial data at the time tn
u(x, tn) =
{
Uni−1 : x < xi− 1
2
Uni : x > xi− 1
2
(3.8)
24
with the similarity solution
u(x, t) = Ui− 12
(x − xi− 12
t
)
. (3.9)
We note that u(x, t) = U(0) is independent of the time t. Hence the numerical flux at theleft cell boundary takes the form
Fni− 1
2= f
(
Uni− 1
2(0))
, (3.10)
and the balance over the control volumes 3.5 can be written as
Un+1i = Un
i − ∆t
∆x
[
f(
Uni+ 1
2
)
− f(
Uni− 1
2
)]
. (3.11)
Thus we only have to determine for every Riemann problem the value of the solutionalong the line xi− 1
2. If we choose the time step ∆t sufficient small, so that the propagating
discontinuities of the neighbouring Riemann problems do not reach the cell edge xi− 12
(figure 3.2) the intermediate state Ui− 12
is constant along the cell edge xi− 12
in the time
step. Note that the intermediate state Ui− 12
and thus the numerical flux Fi− 12
depends
xtn
tn+1
xi− 12
xi− 32
xi+ 12
xi+ 32
Uni−1 Un
i Uni+1
Figure 3.2: Riemann problems at each cell boundary.
on the state quantities Ui and Ui−1 only. Finally the solution for the state quantitiesUn+1 at time tn+1 is obtained by piecing together the Riemann problems of each cellinterface. However, this method involves the solution of nonlinear problems and can bevery time consuming. Therefore the Riemann problems will be replaced by appropriatelinear Riemann problems in the following method.
Roe method At every cell boundary xi− 12
the associated Riemann problem is replaced
by a linear Riemann problem∂U
∂t+ An
i− 12
∂U
∂x= 0 , (3.12)
where the locally defined flux matrix (so called Roe matrix) Ai−1
2
(Ui−1,Ui) of the cell
edge xi− 12
must satisfy following conditions in order to obtain a conservative, consistent
finite volume method:
Ai−1
2
is diagonalisable with real eigenvalues , (3.13)
Ai−1
2
(Ui−1 − Ui) = f (Ui−1) − f (Ui) , (3.14)
Ai−1
2
(U,U) = J(U) . (3.15)
25
Here J(U) is the Jacobian matrix of the flux function f(U) of (3.7). Autonomous systems(3.7) have been studied extensively and a variety of Roe-type linearisations have beendeveloped. For an overview we refer to [16], [10].
x
t
xi− 12
Ui−1
Ui− 12
Ui
W1i− 1
2
W2i− 1
2
Figure 3.3: Riemann problem at the left boundary xi− 12
of the ith cell
For a linear system consisting of two equations (conservation of mass and equation ofmotion) the discontinuous initial condition at time tn decomposes in two shock disconti-nuities (so called waves) of size Wk
i− 12
for k = 1, 2, propagating along the characteristics
with speed σki− 1
2
(figure 3.3). The discontinuities Wki− 1
2
separate three distinct regions. On
the one hand two undisturbed zones, where the uniform initial conditions Uni and Un
i−1
are present, and on the other hand an intermediate region, where a constant state Uni− 1
2
can be found. Now the discontinuities can be expressed in terms of the state quantities:
W1i− 1
2= Un
i− 12− Un
i−1 , W2i− 1
2= Un
i − Uni− 1
2, (3.16)
and the sum of the equations (3.16) yields
Uni − Un
i−1 = W1i− 1
2+ W2
i− 12
. (3.17)
The two discontinuities are proportional to the eigenvectors rki− 1
2
of Ai−1
2
and propagate
with speeds σki− 1
2
= λki− 1
2
given by the eigenvalues of the Roe-matrix. More precisely the
solution is obtained by solving the linear system
Ui −Ui−1 =2∑
k=1
αki− 1
2rki− 1
2(3.18)
for the coefficients αki− 1
2
and then setting for the left- and rightgoing discontinuities
W1i− 1
2= α1
i− 12r1i− 1
2, W2
i− 12
= α2i− 1
2r2i− 1
2. (3.19)
Using (3.16) yields the intermediate state vector Ui− 12
at the cell boundary interface xi− 12:
Ui− 12
= Ui−1 + α1i− 1
2r1i− 1
2= Ui − α2
i− 12r2i− 1
2(3.20)
Furthermore employing (3.10) we can write for the numerical flux at the cell edge xi− 12
Fi− 12
= f(Ui−1) + λ1i− 1
2α1
i− 12r1i− 1
2= f(Ui) − λ2
i− 12α2
i− 12r2i− 1
2. (3.21)
26
xxi− 12
Ui−1
Uli− 1
2
Uri− 1
2
Ui
W1i− 1
2
W2i− 1
2
Figure 3.4: Riemann problem of a non-autonomous system at the left boundary xi− 12
of
the i-th cell.
F-wave algorithm Our aim is to solve quasi one-dimensional Euler equations which arenon-autonomous conservation laws since the flux function f depends not only on the statevector u but also on the space-varying cross-section A(x, t). Here we extend the ideasof the approximated Riemann solvers. First the space dependent flux function f(u, x) isdiscretised to yield a flux function fi(u) that holds throughout the i-th cell. This approachis called cell-centred flux function and in case of quasi one-dimensional Euler equations itsimply means that the cross section areas are constant in each cell. We denote the cross-section area of the i-th cell at time tn by An
i . The resulting Riemann problem is muchmore complicated since the intermediate state Ui− 1
2is split up into the intermediate state
Uri− 1
2
at the right side of the cell boundary and the one on the left side Uli− 1
2
(figure 3.4).
Note that attempting to solve this Riemann problem by a decomposition of the form (3.17)would fail in this case, since the jump in U at xi− 1
2would have been neglected. However
taking this jump into account and computing the two intermediate states require solvinga nonlinear system of equations in each time step. Moreover a Roe-type linearisationdescribed above cannot be found in this case. Therefore the F-wave algorithm developedby Leveque [17] is employed. In order to avoid these difficulties this scheme does notattempt to perform a classical decomposition of the jump in U. Instead it uses a flux-based wave decomposition, in which the flux difference fi(Ui)− fi−1(Ui−1) is decomposed.
x
t
xi− 12
fi−1(Ui−1, Ai−1)
Fi− 12
fi(Ui, Ai)
Z1i− 1
2
Z2i− 1
2
Figure 3.5: Riemann problem of a non-autonomous system at the left boundary xi− 12
of
the i-th cell
Again we consider the the Riemann problem at the cell edge xi− 12
(figure 3.5). Here,
27
linearisation of the flux vector about the state vector Ui−1 on the left side of the cell edgeyields
fi−1(U) = fi−1(Ui−1, Ai−1) + Ji−1 (U −Ui−1) . (3.22)
Similar, on the right side of the cell edge we obtain
fi(U) = fi(Ui, Ai) + Ji (U− Ui) . (3.23)
The Jacobians Ji−1 and Ji of the flux function are evaluated with the quantities of the leftside Ui−1, Ai−1 and the quantities of the right side Ui, Ai, respectively. The appropriateeigenvectors rk
i− 12
and eigenvalues σki− 1
2
are given by
σ1i− 1
2= λ1(U
ni−1, A
ni−1) σ2
i− 12
= λ2(Uni , An
i ) (3.24)
andr1i− 1
2= r1(U
ni−1, An
i−1) r2i− 1
2= r2(U
ni , An
i ) . (3.25)
In order to ensure a conservative finite volume method the fluxes at the cell edge xi− 12
must be constant. So we set
Fi− 12
= fi−1(Ui−1, Ai−1) + Ji−1
(
ULi− 1
2− Ui−1
)
= fi(Ui, Ai) + Ji
(
URi− 1
2− Ui
)
, (3.26)
where the resulting intermediate flux is denoted by Fi− 12. Similar to the decomposition
(3.16) two flux discontinuities of size Zki− 1
2
for k = 1, 2, propagate along the characteristics
with speed σki− 1
2
(figure 3.5):
Z1i− 1
2= Fn
i− 12− fn
i−1 , Z2i− 1
2= fn
i − Fni− 1
2, (3.27)
and the sum of the equations (3.27) yields
fni − fn
i−1 = Z1i− 1
2
+ Z2i− 1
2
. (3.28)
Here Z is called F-wave which bears analogy to waves W of (3.19) but carries flux incre-ments instead of increments in U. Now instead of solving the system (3.18), we solve
fi(Ui, Ai) − fi−1(Ui−1, Ai−1) = β1i− 1
2r1i− 1
2+ β2
i− 12r2i− 1
2(3.29)
for the coefficients βki− 1
2
and then set
Zki− 1
2= βk
i− 12rki− 1
2. (3.30)
Note that we can recover the waves Wki− 1
2
by setting
Wki− 1
2=
1
σki− 1
2
Zki− 1
2. (3.31)
Finally we can write for the numerical flux at the cell interface xi− 12
Fi− 12
= f(Ui−1, Ai−1) + Z1i− 1
2= f(Ui, Ai) −Z2
i− 12
(3.32)
28
The finite volume scheme for Un+1i of cell Ci at the time tn+1 is given by:
Un+1j = Un
j − ∆t
∆x
(
Z1i+ 1
2+ Z2
i− 12
)
(3.33)
It can be easily extended to second order for smooth solutions by adding a correctionterm
Un+1j = Un
j − ∆t
∆x
(
Z1i+ 1
2+ Z2
i− 12
)
− ∆t
∆x
(
Fni+ 1
2−Fn
i− 12
)
(3.34)
where
Fni− 1
2
=1
2
2∑
p=1
|σp12
|(
1 − ∆t
∆x|σp
12
|)
Zp
i− 12
(3.35)
Mac Cormack Scheme It is worth noting that there are finite volume methods whichdo not require the computation of the wave structure. One of the most popular is theMac Cormack scheme [16], [29], [13], which is based on Lax-Wendroff methods. Herewe employ the two-step approach which can handle the problems arising from the space-varying flux functions. The numerical flux Fi− 1
2at the cell boundary xi− 1
2for the quasi
one-dimensional Euler equations is given by
Fni− 1
2=
1
2
(
f(Uni+1, A
ni+1) + f(U∗
i , A∗i ))
, (3.36)
where
U∗i = Un
i − ∆t
∆x
(
f(Uni+1, A
ni+1) − f(Un
i , Ani ))
. (3.37)
This finite volume method is also second order accurate.
Stability conditions of the finite volume schemes The time step ∆t of the numer-ical integration has to be chosen in a way that the stability conditions of the numericalschemes are satisfied. For all finite volume methods mentioned above the CFL conditionprovides a necessary and sufficient condition [13], [16], [29]. It simply states that the ana-lytical domain of influence lies within the numerical domain of influence. In case of quasione-dimensional Euler equations the CFL-condition can be written as
∆t ≤ ∆x
c + |u| . (3.38)
Comparison of schemes Both finite volume schemes for non-autonomous systems arelisted in table 3.1. The advantage of the Mac Cormack approach is that it is very easy toimplement since it does not require the computation of the wave structure of the systemof differential equations. On the other side employing the F-wave algorithm makes theimplementation of boundary conditions, where the characteristics are calculated, easier.
29
MacCormack: Un+1j = 1
2
(
Unj + U∗
i
)
− ∆t2∆x
[
f(U∗i , A
∗i ) − f(U∗
i−1, A∗i−1)
]
with
U∗i = Un
i − ∆t∆x
(
f(Uni+1, A
ni+1) − f(Un
i , Ani ))
f-wave : Un+1j = Un
j − ∆t∆x
(
Z1i+ 1
2
+ Z2i− 1
2
)
− ∆t∆x
(
Fni+ 1
2
−Fni− 1
2
)
Table 3.1: Different one-dimensional finite volume schemes
3.1.2 Two-dimensional Finite Volume Schemes
Figure 3.6: Block-structured grid.
P
N
S
W
E
Ci,j
Ci,j+1
Ci,j−1
Ci−1,j
Ci+1,j
x, i
y,j sw
se
nwne
n
Figure 3.7: Typical control volume in anon-orthogonal grid.
The domain of the cylinder is divided into a finite number of small control volumes(areas). To simplify matters we only employ non-orthogonal, block-structured grids (figure(3.6). A finite volume Ci,j is defined by its corners sw, se, nw and ne (see figure 3.7). Thecomputational node and therefore all dependent variables are assigned to the centre P ofthe control volume. At this point we assume that the distance between piston and cylinderhead is only a function of time t and later on we will discuss the effects of space-varyingheights Z(x, y, t). A detailed derivation of the finite volume methods for two-dimensionalgrids can be found in [29]. Similar to the one-dimensional methods we transform thegoverning differential equations into integral formulation. Integrating
∂u
∂t+
∂f(u)
∂x+
∂g(u)
∂y= 0 (3.39)
over the cell Ci,j gives∫
V
(
∂u
∂t+
∂f(u)
∂x+
∂g(u)
∂y
)
dV = 0 . (3.40)
30
Using Green’s theorem , this equation becomes
∂
∂t
∫
V
udV +
∫
S
h · ndS = 0. (3.41)
Here, n denotes the unit vector normal to the surface S of the finite volume and the tensorh can be expressed in Cartesian coordinates as
h = f(u)i + g(u)j ,
where i and j denotes the unit vector in x and y direction, respectively. In case of two-dimensional non-orthogonal grids one obtains
h · n dS = (f(u)dy − g(u)dx) , (3.42)
which can be substituted in 3.41 to yield
∂
∂t
∫
V
u dx dy +
∫
S
(f(u)dy − g(u)dx) = 0. (3.43)
The surface S consists of the four sections nw− sw, sw− se, se−ne and ne−nw. Nowintegrating over one time step tn+1 − tn = ∆t one obtains
∫
V
u(x, y, tn+1) dx dy −∫
V
u(x, y, tn) dx dy +
tn+1∫
tn
∫
S
(f(u)dy − g(u)dx) dt = 0. (3.44)
Again, the first two terms in (3.44) correspond to the total change of the state quantitiesinside the cell during the time step ∆t = tn+1 − tn. The third integral represent the totalfluxes of the state quantities across the cell boundaries. Let Un
i,j denote the cell averageof the state quantities over the cell (i, j) at time tn:
Uni,j ∼
1
|Ci,j|
∫
V
u(x, y, tn) dx dy . (3.45)
where the cell area |Ci,j| can be written as
|Ci,j| =1
2|(ne − sw) × (nw − se)| , (3.46)
The flux integral of (3.44) at the section sw− se is approximated by the numerical fluxesFn
i,j− 12
and Gni,j− 1
2
:
Fni,j− 1
2∼ 1
∆t
tn+1∫
tn
∫
S
f(u)dy , (3.47)
Gni,j− 1
2∼ 1
∆t
tn+1∫
tn
∫
S
g(u)dy . (3.48)
31
Finally, when employing the above approximations the system of equations for Un+1i,j of
cell Ci,j at the time tn+1 can be written as:
Un+1i,j = Un
i,j− ∆t|Ci,j |
[(
Fi,j− 12(yse − ysw) + Fi+ 1
2,j (yne − yse) +
Fi,j+ 12(ynw − yne) + Fi− 1
2,j (ysw − ynw)
)
−(
Gi,j− 12(xse − xsw) + Gi+ 1
2,j (xne − xse) +
Gi,j+ 12(xnw − xne) + Gi− 1
2,j (xsw − xnw)
)]
, (3.49)
Similar to the one-dimensional finite volume schemes, a numerical method written in theform (3.49) is called conservative. The boundary problems in the two-dimensional domain(cylinder) do not require the calculation of characteristics and therefore we employ simpleand robust finite volume methods, where calculating the wave structure is not necessary.
Lax-Friedrich’s scheme The space centred explicit Lax-Friedrich’s scheme was one ofthe first schemes used to solve Euler equations [13]. Due to the poor first order accuracy thesolutions must be questioned. However it can be used as intermediate step in high orderschemes, as in the two-step Lax-Wendroff method (Richtmyer). A detailed descriptionof this method can be found in [13] and [29]. For the numerical flux Fi+ 1
2,j of the face
ne − nw we can write
FLFi,j+ 12
=1
2(fi,j+1 + fi,j) +
1
16
|Cij| + |Ci,j+1|∆t (ynw − yne)
(
Uni,j −Un
i,j+1
)
, (3.50)
wherefi,j+1 = f
(
Uni,j+1
)
. (3.51)
The numerical fluxes for the other faces can be easily derived. The second term on theright side of equation (3.50) is used to stabilise the method. It can be interpreted as andissipative term proportional to the second derivative of u. Therefore the discretisationscheme (3.50) is consistent with
∂u
∂t+
∂f(u)
∂x+
∂g(u)
∂y= αd
∂2u
∂x2+ βd
∂2u
∂y2, (3.52)
which is a dissipative system of equations with the numerical viscosities αnum and βnum
of order O(∆x). Again, the time step ∆t of the numerical integration has to be chosensuch that the stability conditions of the numerical schemes are satisfied. Unfortunately,a sufficient stability criterion can not be given for this finite volume scheme on non-orthoganal quadrilateral grids. However, it turns out that using following conditions leadto stable methods in the tested cases:
∆t ≤ ∆x
2 (c + |u|) , ∆t ≤ ∆y
2 (c + |v|) . (3.53)
Lax-Wendroff schemes, Richtmyer scheme In this work we discuss the two-stepLax-Wendroff schemes introduced by Richtmyer [13], [29]. There are also one-step Lax-Wendroff schemes but they suffer from the difficulty of requiring calculations of Jacobianmatrices. The Richmyer scheme consits of two steps: The predictor step, in which an
32
estimated solution is calculated, and a corrector step, where the actual solution is found.The numerical fluxes from the Lax-Friedrich’s scheme are used as first guess. In the secondstep the same fluxes but without the dissipative terms and evaluated at the intermediatestep are employed. As an example we approximate the analytical flux fi+ 1
2,j of the face
ne/nw by the numerical flux FRi+ 12,j. First the predictor flux at the time n + 1
2 is given
by
Fn+ 1
2
i,j+ 12
=1
2(fi,j+1 + fi,j) +
1
16
Ai,j + Ai,j+1
∆t (ynw − yne)
(
uni,j − un
i,j+1
)
. (3.54)
Similarly, we obtain the numerical fluxes for the other faces. Thus we can calculate thepredictor state vectors Un+ 1
2 by means of equation (3.49). Second the corrector step canbe written as
FRi,j+ 12
=1
2
(
fn+ 1
2i,j+1 + f
n+ 12
i,j
)
, (3.55)
where
fn+ 1
2i,j+1 = f
(
Un+ 1
2i,j+1
)
. (3.56)
The necessary and sufficient stability condition for the case of orthogonal grids where∆x = ∆y was given by Richtmyer [13] and can be written as
∆t ≤ ∆x√
2(
c +√
u2 + v2) . (3.57)
Although, in case of non-orthogonal grids this relation is not sufficient, satisfying resultshave been obtained using it.
Flux Limiter In the vicinity of discontinuities second order methods performs ratherpoor. Precisely, non-physical oscillations can occur. Now we want a method which usesan accurate second order scheme in regions of smooth solutions and non-oscillating firstorder methods in all other regions. Therefore the numerical flux is computed by a linearcombination of first order and second order fluxes. A detailed description can be found in[16]. Here we use the first order Lax-Friedrich’s flux FLF and the second order Richtmyerflux FR to obtain
F(Un) ≈ FLF(Un) + Ψ(Un) [FR(Un) − FLF(Un)] , (3.58)
where the flux limiter Ψ(Un) describes the smoothness of the solution. The quotient ofthe pressure gradients in two adjacent cells Θ represents a measure of smoothness. Forinstance, at the face between nw − ne it takes following form:
Θi,j+ 12
=pi,j+2 − pi,j+1
pi,j+1 − pi,j, (3.59)
In order to avoid biased weighting 3.59 is replaced by the following formula in every othertime step:
Θi,j+ 12
=pi,j+1 − pi,j
pi,j − pi,j−1, (3.60)
Note that for smooth solutions Θ becomes 1. A wide variety of flux limiters have beendeveloped and we refer to [13] for an overview. In this work we employ the Minmod limiter
33
where the flux limiter Ψ as function of Θ is given by
Ψ(Θ) =
0 : Θ < 0
Θ : 0 ≤ Θ ≤ 1
1 : Θ > 1
(3.61)
Treatment of Source Terms Cylinder head recesses are often used in order to channelthe gas flow into the valve pockets. Thus the distance between cylinder head and pistonZ varies not only with time but also with the space coordinates x and y, resulting in asource vector s in the two-dimensional Euler equations
∂U
∂t+
∂f(u, Z)
∂x+
∂g(u, Z)
∂y= s . (3.62)
Here, the source vector takes the form
s =
0
p∂Z∂x
p∂Z∂y
. (3.63)
In order to solve this problem we employ a fractional step method in which we split thegeneral problem (3.62) into a homogeneous conservation law and an ODE given by
∂u
∂t= s . (3.64)
Now for both subproblems standard methods can be employed. First we use one of thefinite volume methods above to solve ∂u
∂t + ∂f(u,Z)∂x + ∂g(u,Z)
∂y = 0 and second (3.64) isintegrated by a forward Euler approach which is written as
Un+1i,j = U∗
i,j + ∆t
0
p∗i,j∂h∂x
p∗i,j∂h∂y
. (3.65)
The vector U∗i,j denotes the intermediate state vector derived from a finite volume scheme.
In case of second order two-step methods, in the predictor step and in the corrector step(3.65) applies.
3.2 Numerical Boundary Conditions
In terms of numerical treatment of the boundary conditions, we have to distinguish be-tween the physical boundary conditions that follow from Section 2.4.3, 2.4.5, 2.4.6 and 2.4.4and the numerical boundary values required for computation. The numerical algorithmsrequire the information of all state quantities at the boundaries. The physical boundaryconditions only provide the boundary value corresponding to the incoming characteristic.In this section we discuss the various possibilities to determine the missing conditions fordifferent boundary problems.
34
3.2.1 Walls one-dimensional
The mass flux m through the wall is equal to zero and thus the velocity u at the boundaryxk+ 1
2is also equal to zero. This boundary problem can be modelled by simply using a
ghost cell (boundary cell) Ck+1 as mirror element of Ck (figure 3.8. The state quantitiesUk+1 of the ghost cell are set to the ones in the adjacent cell, except that the sign ofvelocity is changed.
CellCellCell
Ck−1 Ck Ck+1
xk−1 xk xk+1
Wall
Figure 3.8: Ghost cell Ck+1 at the bound-ary
CellCellCell
Ck−1 Ck Ck+1
xk−1 xk xk+1
m
Boundary
Figure 3.9: Mass flow through the bound-ary
3.2.2 Mass-flow through the Boundary
Again, the ghost cell Ck+1 outside the subdomain is added (figure 3.9). The mass flux mthrough the boundary provides the necessary first equation for the two state quantities ofthe ghost cell Ck+1. Now a wide variety of methods for the missing condition have beenfound and we refer to [13] for an overview. Here we introduce three different approaches:
Leaving characteristic The additional relationship for the missing state quantity isgiven by the linear partial differential equations (2.25) for the characteristic leaving thesubdomain and entering the ghost cell. These linear PDEs are discretised by a simpleupwind scheme and the outgoing characteristic can be calculated at the desired boundaryat the new time. According to (2.25) the equation for the leaving characteristic on theright boundary takes the form:
∂w2
∂t+ λ2
∂w2
∂x= 0 . (3.66)
We assume, that the flow situation at time tn is known. Then first order upwinding of3.66 yields
wn+12,k+1 = wn
2,k+1 + λ2(Unk+1, A
nk+1)
∆t
∆x
(
wn2,k − wn
2,k+1
)
(3.67)
Together with the definition of the characteristic variables (2.23) we are able to calculatethe full boundary data of Ck+1 to apply to the numerical scheme. Equation (3.67) and theprescription of m form a system of equations which is solved by means of a well-knownnumerical Newton-Raphson iteration [26]. Similarly, solutions for this boundary problem
35
at the left side of the subdomain can be found. Let C0 denote the ghost cell on the leftside. Here the equation for the leaving characteristic takes the form
wn+11,0 = wn
1,0 + λ1(Un0 , An
0 )∆t
∆x
(
wn1,0 − wn
1,1
)
. (3.68)
Note that the ideas of this approach can be used for all kind of boundary problems(e.g. pressure, mass flow, velocity, density).
Extrapolation The second method to determine the missing equation is extrapolation.Instead of considering characteristics leaving the state quantities for the ghost cell Ck+1
are extrapolated from the interior points. According to Hirsch [13] the extrapolationused can be one order lower than the finite volume scheme without altering the accuracyand stability of the interior numerical method. For this boundary problem one possibleassumption is, that the mass per unit length of the boundary cell Rk+1 can be extrapolatedfrom the adjacent ones. It is given by
Rk+1 = 2Rk − Rk−1 . (3.69)
Together with the given mass flux m one is able to determine both state quantities. Thestate vector for the ghost cell Ck+1 takes following form
Uk+1 =
(
Rk+1
uk+1
)
=
(
2Rk − Rk−1
mRk+1
)
. (3.70)
Source Term The third approach also models this boundary problem without consid-ering the leaving characteristics. As in the case of walls the ghost cell is used as a mirrorelement. The loss or gain of mass and impulse inside the computational domain due tomass flux m through the valve are included by a source term in the cells adjacent to themirror element. The state vector Uk of cell Ck at the time tn+1 is given by:
Un+1k = Un
k +∆t
∆x
(
Fnk+ 1
2
) − Fnk− 1
2
))
− ∆tS , (3.71)
where
S =1
∆x
(
m,m2
2R1dnk2
)T
. (3.72)
3.2.3 Pressure Outlet
At the end of the valve retainer or at the end of the pressure chamber a pressure outlet isdefined. Either the characteristic method or the extrapolation method, both described insection 3.2.2, can be employed. However, instead of the mass flux the pressure pout is pre-scribed at the boundary. In addition, the pressure losses due to sudden changes pv, whichhave been derived in section 2.4.6, can be easily incorporated. Using the extrapolationmethod the state vector Uk+1 of the ghost cell Ck+1 is given by
Uk+1 =
(
R1dk+1
uk+1
)
=
Ak+1pres
1γ ρ0
p0
1γ
2unk − un
k−1
, (3.73)
36
where the resulting pressure pres is written as
pres = pout + pv . (3.74)
Note that pv and thus pres depend on the state quantities Uk at time tn.
3.2.4 Boundary Problem at Sudden Changes of Cross-section
Separation
CellCellCell
CellCellCell
Cell
Cellpv
Ck−2
Ck−2
xk−2
xk−2
Ck−1
Ck−1
Ck
Ck
Ck+1
Ck+1
xk−1
xk−1
xk
xk
xk+1
xk+1
Figure 3.10: Numerical treatment of sudden changes in flow cross-sections: Computationaldomain is separated and pressure loss pv is introduced
At the location of sudden changes in flow cross-section the computational domain issplit up (figure 3.10. Then the reconnection of the resulting subdomains is consideredas as a boundary problem for the two cells Ck and Ck−1 adjacent to the splitting. Asmentioned above the finite volume methods used require the full set of state quantities atthese cells. First we make use of the equations of the pressure losses (2.40),(2.38) derivedin section 2.4.6. Second, the characteristics pointing into the cells Ck and Ck−1 providethe equations for the missing conditions. Similar to the mass flux boundary conditions weobtain equations for the characteristic variables by solving the partial differential equationsfor the characteristic variables (2.25) with and Upwind scheme. Finally the system ofequations for the state quantities of the cells Ck and Ck−1 are given by
mn+1k−1 = mn+1
k
ptn+1k−1 = pt
n+1k + pv
wn+12,k−1 = wn
2,k−1 + λ2(unk−1)
∆t
∆x
(
wn2,k−2 − wn
2,k−1
)
(3.75)
wn+11,k = wn
1,k + λ1(unk)
∆t
∆x
(
wn1,k − wn
1,k+1
)
.
37
A drawback of this method is that it imposes unwanted mass sinks or mass sources onthe location of the separation. However, this problem can be easily solved using followingprocedure. First the state vectors of the boundary cells un+1
k−1 and un+1k are calculated
without the pressure losses by means of finite volume methods. Then the system ofequations (3.75) is solved by means of Newton-Raphson iterations. Finally the lost mass∆m is added to the cells Ck and Ck−1 according to their volumes. We obtain
∆m = Rk − 1n+1 |WL +Rn+1k |WL −(Rn+1
k−1 + Rn+1k ) (3.76)
and
Rn+1k−1 = Rn+1
k−1 |WL −∆m
∆x
An+1k−1
An+1k−1 + An+1
k
(3.77)
Rn+1k = Rn+1
k |WL −∆m
∆x
An+1k
An+1k−1 + An+1
k
, (3.78)
where |WL denotes solutions of (3.75).
3.2.5 Boundary Conditions at the Valve
1 2ValveCylinder Chamber
CellCellCell
Cell
Cell
Cell
CellCell
Cell
Cell
m
analytical
numerical
Ck−1,1
Ck−1,1
xk−1
xk−1
Ck,1
Ck,1
C1,2
C1,2
C2,2
C2,2
Ck+1,1
C0,2
xk,1
xk,1
x1,2
x1,2
x2,2
x2,2
xk+1,1
x0,2
Figure 3.11: Valve region and the adjacent finite volume cells: Ghost cells Ck+1,1 and C0,2
are introduced for the numerical boundary conditions.
A valve separates the computational subdomain of the cylinder from the adjacent valveretainer subdomain. Figure 3.11 shows the valve region and its adjacent boundary cells.Again we introduce two ghost cells Ck+1,1 and C0,2 and define the boundary conditions forthese cells in a way that the equation of the valve connecting this subdomains is satisfied.Now we distinguish two different states of the valve:
38
• Closed Valve (xv = xv,min): Here the mass flux m through the valve is equal to zero.Therefore the method for the one-dimensional wall (section 3.2.1) can be applied tothe ghost cells Ck+1,1 and C0,2.
• Open Valve (xv,min < xv ≤ xv,max): The mass flux m through the valve is given by(2.51). Thus the methods for the mass-flow boundary in section 3.2.2 can be appliedto the ghost cells Ck+1,1 and C0,2.
3.2.6 Boundary Conditions at the T-piece
Following the solution of the boundary problem at sudden changes of cross-section weexpand the method of characteristics to model the T-piece. Now there are three compu-tational domains with their appropriate boundary elements Ck,1, Ck,2 and C1,3. Again theghost cells Ck+1,1, Ck+1,2 and C0,3 were added. Figure 3.12 shows the T-piece region andadjacent cells.
junction area
Ck−1,1 Ck,1 Ck+1,1 Ck+1,2 Ck,2 Ck−1,2
C0,3
C1,3
C2,3
Figure 3.12: Finite Cells at the T-piece. Junction area is shaded
The equations for the leaving characteristics and the equations for the T-piece (2.33),(2.34) form a system of equations for the required state quantities in the boundary cells:
mn+1k+1,1 + mn+1
k+1,2 = mn+10,3 (3.79)
htotn+1k+1,1 = htot
n+1k+1,2 = htot
n+10,3 (3.80)
wn+12,k+1 = wn
2,k+1 + λ2(unk+1)
∆t
∆x
(
wn2,k − wn
2,k+1
)
|1 (3.81)
wn+12,k+1 = wn
2,k+1 + λ2(unk+1)
∆t
∆x
(
wn2,k − wn
2,k+1
)
|2 (3.82)
wn+11,0 = wn
1,0 + λ1(un0 )
∆t
∆x
(
wn1,0 − wn
1,1
)
|3 . (3.83)
Again, a numerical Newton-Raphson iteration [26] is employed to solve this system. Ifthe pressure losses in the T-piece are taken into account the above system of equations isextended by the six pressure loss equations (2.47).
39
3.2.7 Wall two-dimensional
At walls we require that the velocity normal to the boundary vn vanishes (figure 3.13).Therefore we introduce a ghost cell C∗ and set its computational variables to the ones ofthe adjacent cell C except for the velocity v∗ . Simple vector analysis yields
v∗ = v − 2 |nbv|nb , (3.84)
where nb is the normal vector to the boundary.
C
C∗
v
v∗
vt
vnnb
wall
ghostcell
Figure 3.13: Boundary cell in the two-dimensional domain
3.2.8 Interface Condition
The interface connects the two-dimensional computational domain representing the cylin-der with the one-dimensional domain representing the valve pockets (figure 3.14). Sincewe are linking quantities of different dimensions we have to reduce the two-dimensionalones to suitable one-dimensional information by introducing artificial cells C0,2d and C1,2d.These cells are two-dimensional cells but extend across the whole cross-section. Similarly,at the side of the one-dimensional domain two artifical cells, C0,1d and C1,1d, are introduced.The volume of each artificial cell is the sum of all volumes of the cells inside the cylinderadjacent to the interface under consideration. Moreover we require, that the artificial cellsC0,2d and C1,2d are identified with C0,1d and C1,1d, respectively. Without loosing generalitywe assume that the valve pocket is parallel to the x-axis.
Now we employ the following calculation procedure in each timestep: First a two-dimensional finite volume scheme is applied to determine the state vector U0,2d at the timetn+1. In doing so the velocity normal to the valve pocket axis (x-axis) is set to zero andthus impulse Iy0,2d is zero. Similarly, the state quantities Un+1
1,1d of cell C1,1d are calculatedusing one-dimensional finite volume methods. However we still have to determine theboundary cells C0,1d and C1,2d. Here we follow a different approach compared to theprevious boundary condition problems. The state quantities of the cell C0,1d are obtainedby projecting the state vector Un+1
0,2d of the two-dimensional computational domain on
40
Cylinder
Valve pocket
One-dimensional
Two-dimensional
Interface
Interface
Artificial
ArtificialCells
Cells
Projection
Extension
ws
C0,2d C1,2d
C0,1d C1,1d C2,1d
Figure 3.14: Interface between two-dimensional and one-dimensional domain.
the state vector Un+10,1d of the one-dimensional domain. Furthermore, we extend the state
vector Un+11,1d of C1,1d in order to derive the state vector Un+1
1,2d . So we can write
U1,2d =
M1,2d
Ix1,2d
Iy1,2d
Ext.:=
R1,1dws
u1,1dR1,1dws0
, (3.85)
U0,1d =
(
R0,1d
u0,1d
)
Proj.:=
M0,2dws
Ix0,2d
M0,2d
0
. (3.86)
Here ws denotes the width of the interface.
The drawback of this procedure is that the requirement for conservation of state quan-tities can not be satisfied, which is seen from following considerations (figure 3.15). Letmn
0 and mn1 denote the mass inside the artificial cells C0,2d, C0,1d and C1,2d, C1,1d, respec-
tively. In addition, ∆mc and ∆mv stand for the mass fluxes through the interface at thecylinder and valve pocket side during one time step. The mass fluxes between the artificalcells in the one-dimensional and two-dimensional domain are denoted by ∆m1 and ∆m2,respectively. All these mass fluxes can be easily derived from the numerical fluxes of thefinite volume methods used. Now the masses mn+1
0 and mn+11 inside the cells C0,2d and
41
C1,1d at time tn+1 are given by
mn+10 = mn
0 + ∆mc − ∆m2 , (3.87)
mn+11 = mn
1 + ∆m1 − ∆mv . (3.88)
(3.89)
Then the projection and extension described above assigns the new masses to the appro-priate cells in the other computational domain. Finally the mass inside the artifical cellsin the one-dimensional domain and two-dimensional domain at the time tn+1 is given by:
(mn+10 +mn+1
1 )1d = (mn+10 +mn+1
1 )2d = (mn0 +mn
1 )+(∆mc−∆mv)+(∆m1−∆m2) . (3.90)
If the last term of (3.90) is zero the method used would be conservative, since every gainor loss of mass inside the artifical cells is due to the fluxes across the interfaces. However,the numerical fluxes and thus the mass fluxes of the one-dimensional scheme differ fromthe fluxes of the two-dimensional scheme, leading to unwanted mass sources and sinks.
Since we can quantify the mass gains produced by the procedure described above theerrors made can be easily corrected. In detail, the gained mass ∆m1 −∆m2 is subtractedfrom the artifical cells Ci,1d and Ci,2d for i = 0, 1. We obtain
Rn+1i,1d = Rn+1
i,1d |B −∆m1 − ∆m2
2∆x,Mn+1
i,2d = Mn+1i,2d |B −∆m1 − ∆m2
2∆xws, (3.91)
where |B denotes the state quantities before the correction.
Cylinder
Valve pocketto
to
Interface
Interface
Projection Extension
C0,2d C1,2d
C0,1d C1,1d
m0
m0
m1
m1
∆mc ∆m2
∆m1 ∆mv
Figure 3.15: Mass correction: Mass inside the artifical cells and mass fluxes during onetime step
3.3 Equation of motion for the valve plate
The second order ODE of the valve motion (2.47) can be solved analytically if a constantvalve force Fv is assumed during the time step ∆t. The solution for the valve lift xv andthe valve plate velocity vv at the time tn+1 takes the form
42
vvn+1 = vv
n cos
√
cs
mv∆t +
√
1
mvcs
[
Fv − cs (l1 + xvn) sin
√
cs
mv∆t
]
(3.92)
xvn+1 =
Fv
cs− l1 +
(
l1 + xvn − Fv
cs
)
cos
√
cs
mv∆t +
√
mv
csvv
n sin
√
cs
mv∆t , (3.93)
whereFv = (p1 − p2)Av − Fadh . (3.94)
43
Chapter 4
Measurements
In this chapter measurements for two test compressors from Burckhardt Compression,Switzerland and Ariel, USA are discussed. Both compressors are double-acting barreldesign compressors. However, Burckhardt’s compressor has two valves and is about threetimes smaller than Ariel’s compressor with eight valves.
4.1 Compressor with two valves
A double-acting, 2 cylinder, barrel design reciprocating compressor was tested. The exper-iment was carried out by Daniel Sauter, Markus Lehmann and Roland Aigner in November2004 at Burckhardt Compression, Switzerland. The main specifications of the compressorcan be found in table 4.1. Various sensors were used to measure the
• pressure inside the cylinder and inside the suction and discharge chamber,
• temperature inside the cylinder and in the pipes,
• valve motion of the discharge valve,
• volume flow,
• speed of the crankshaft.
Figure 4.1 shows the experimental compressor and its main components. Figure 4.2 showsthe location of each sensor in cylinder 1. A detailed description of the sensors used canbe found in the appendix A.1.
4.1.1 Test Cases
In order to get a wide variety of test cases the valve configuration (see table 4.2) of thepressure valve of the first cylinder and the discharge pressure have been varied. In addition,the influence of the number of working cylinders has been examined. For the comparisonof the measurements with numerical simulations three of four working chambers have beensealed off during measurements by covering the valve pockets of these chambers. All testconfigurations were mesasured at 4 different relative discharge pressures: 1, 2, 3 and 4 bar.The ambient pressure at the days of measurement was 0.97 bar. Therefore in this section
44
Type of compressor: Burckhardt compression 2K90-1A
2 cylinders, double-acting, barrel design
Diameter of cylinder: 0.22 m
Stroke: 0.09 m
Speed of crankshaft: 980-990 rpm
Suction pressure: ambient pressure
Maximum pressure ratio: 1/5
Clearance: 1.6-2 mm
Gas: Air
Length of connecting rod: 0.25 m
Type of valves: 110K12
Classification ǫ: 0.22
Table 4.1: Main Specifications of the Burckhardt test compressor.
������������������������������������������������������
������������������������������������������������������
������������������������������
������������������������������
cylinder 1cylinder 2
flywheel with trigger
engine
heat exchanger
orifice
damper
outlet restriction valve
suction pipe
discharge pipe
pressure and temperature sensors
figure 4.2
Figure 4.1: Shema of the Burckhardt experimental compressor
45
pressure is displayed relative to this ambient pressure. Six different valve configurationshave been tested. First, the maximum valve lift has been set to 1.05, 1.35 and 1.65 mm.Second, a total spring constant of the valve of 48 and 80 N/mm has been used. The signalsof the sensors were recorded with a sampling rate of 10000 Hz. For every test case threeindependent measurements were taken for a complete compressor cycle at three differenttimes ta, tb and tc, which were approximately one minute apart.
Discharge valve mass of valve plate 0.076 kg
Measurement valve number of springs 8
spring deflection 0.0009 m
spring constant cs 48/80 N/mm
maximum valve lift xv,max 1.05/1.35/1.65 mm
valve force area 0.0056 m2
parameter of valve αv/βv 2.042 [1]/ 80000 1/m2
parameter of valve fe1mm 1.86 m2/m
Discharge valve number of springs 12
spring deflection 0.9 mm
spring constant cs 48 N/mm
maximum valve lift xvmax 1.35 mm
Suction valve number of springs 12
spring deflection 0.594 mm
spring constant cs 21.3 N/mm
maximum valve lift xv,max 1.35 mm
Table 4.2: Specifications of the valves.
4.1.2 Results of Measurement
Periodic Process After a warm-up of several minutes the process inside the compressoris periodic (figure 4.3). Then comparisons of valve motion and pressure distribution overone period at different times show only small deviations.
Variation of number of Working Chambers The difference between one cylinderand two cylinders working are very small. Whereas, if only one working chamber is active,pressure distribution and valve motion show slightly different behaviour (figure 4.4 andfigure 4.5). This is due to the fact that in case of double-acting cylinders pressure wavesfrom the crank end side and the head end side interact, since both sides are connectedby the pressure chamber. Hence the outflow condition for the discharge valve changes,resulting in different pressure distributions inside the working chamber. We observe thatthe pressure in the pressure chamber has one or two peaks depending on whether thecompressor is double-acting or not. However, the differences in the valve retainer duringdischarge (between 130◦ CA and 180◦ CA) are rather small and therefore differences ofthe pressure inside the cylinder and differences of the valve motion are small.
46
6
7
1 5 9
4
8
2
3
1 . . . . . . pc12 . . . . . . pc23 . . . . . . pc3
4 . . . . . . pc45 . . . . . . pc56 . . . . . . pc6
7 . . . . . . pc78 . . . . . . pin and Tsc9 . . . . . . pout and Tpc
suction valve discharge valve
Figure 4.2: Location of the sensors inside the cylinder 1, the suction and the dischargevalve retainer.
−50000
0
50000
100000
150000
200000
250000
300000
350000
400000
0 50 100 150 200 250 300 350−0.0002
0
0.0002
0.0004
0.0006
0.0008
0.001
0.0012
0.0014
0.0016pressure, time apressure, time bpressure, time cvalve lift, time avalve lift, time bvalve lift, time c
Rel
ativ
epre
ssure
atpc5
[Pa]
Val
velift
[m]
Crank angle CA[◦]
Figure 4.3: Comparison of relative pressure at pc5 and valve lift at different times ta, tband tc. Case: pout = 3 bar, cs =48N/mm , xv,max=1.35 mm.
47
−50000
0
50000
100000
150000
200000
250000
300000
350000
400000
0 50 100 150 200 250 300 350−0.0002
0
0.0002
0.0004
0.0006
0.0008
0.001
0.0012
0.0014
0.0016pressure, 2 cylinderpressure, 1 cylinderpressure, 1 chambervalve lift, 2 cylindervalve lift, 1 cylindervalve lift, 1 chamber
Rel
ativ
epre
ssure
atpc5
[Pa]
Val
velift
[m]
Crank angle [◦]
Figure 4.4: Comparison of relative pressure at pc5 and valve lift with 2 cylinders, 1 cylinderand 1 chamber working. Case: pout = 3 bar, cs=48 N/mm, xv,max=1.35 mm.
260000
270000
280000
290000
300000
310000
320000
330000
340000
0 50 100 150 200 250 300 350
pout, 2 cylinderpout, 1 cylinderpout, 1 chamber
Rel
ativ
epre
ssure
atpou
t[P
a]
Crank angle [◦]
Figure 4.5: Comparison of relative pressure in the valve retainer: 2 cylinders, 1 cylinderand 1 working chamber. Case: pout = 4 bar, cs=48 N/mm, xv,max=1.35 mm.
48
Initiation of Waves In figure 4.6 the pressure readings at pc5 and pc6 close to thedischarge valve and the suction valve, respectively, and the position of the valve plate asa function of the crank angle CA are shown. During compression, when both valves areclosed the pressure is uniform in the cylinder (CA = 140◦). When the pressure in thecylinder exceeds the pressure in the valve retainer plus the springing the valve starts toopen. The pressure near the outflow valve ceases to increase and reaches a maximum,while the pressure close to the suction valve still increases. The opening of the valve hasinitiated a wave which reaches the opposite side of the cylinder at a crank angle CA=146◦,just before the valve has completely opened. Then the rarefaction wave is reflected at thesuction side. Thus the pressure at the suction side (pc6) attains also a maximum. But it ismarkedly larger than that on the discharge side (pc5). When the valve plate hits the valveseat the pressure in front of the pressure valve is large enough to keep the valve open untilCA = 155◦. Then the pressure has dropped and the valve begins to close again reducingthe mass outflow. Again this leads to an increase of the pressure inside the cylinder. Nowincreasing of pressure forces the valve to open and a complicated interaction of the valvemotion with the pressure waves in the cylinder takes place. Shortly after the lowest volumepoint the valve closes. At that time a complicated system of waves travelling back andforth is left in the cylinder. In figure 4.7 the difference of the pressure readings betweensuction side and cylinder centre (pc5 - pc1) is shown. We observe that during expansionthe amplitudes of the waves are proportional to the mean pressure inside the cylinderand their periodic time tends to the travelling time of a wave across the cylinder 2d/ccorresponding to the lowest eigen mode in lateral direction. We want to point out thatat the beginning of the expansion phase a mix of modes with several eigen frequenciesis present. These modes are initiated by the valve motion. As said above the zerotheigenmode has the smallest damping and remains visible until the suction phase starts.
0
100000
200000
300000
400000
500000
600000
100 120 140 160 180 200 220 240
0
0.0005
0.001
0.0015
0.002
0.0025
0.003pc5pc6
poutvalve lift
Rel
ativ
epre
ssure
[Pa]
Val
velift
[m]
Crank angle [◦]
Figure 4.6: Relative Pressure at pressure sensors pc5, pc6 and pout and valve lift. Case:pout = 4 bar, cs=48 N/mm, xv,max=1.35 mm.
49
−60000
−40000
−20000
0
20000
40000
60000
0 50 100 150 200 250 300 350
diff. pressure pc5−pc1
Pre
ssure
diff
eren
cepc5
-pc1
[Pa]
Crank angle [◦]
Figure 4.7: Pressure difference pc5-pc1. Case: pout = 4 bar, cs=48 N/mm, xv,max=1.35mm.
Plane Waves As described above, pressure waves run through the cylinder in lateraldirection (x-direction). For example the opening of the valve initiates a wave. This waveruns at the speed of sound and reaches the sensor next to the discharge valve (pc5) first(figure 4.8). Then sensors pc1, pc2, pc3, pc4 at the y-axis notice the rarefraction wave.Since the differences between the pressure readings at these sensors are rather small, aplane wave can be assumed. Finally, the information of the opening of the valve hastravelled to the sensors pc6 and pc7.
Pressure differences across the discharge valve When the discharge valve startsopening, the gas velocity is almost equal to zero and the pressure is constant inside thecylinder except from very small disturbances. Therefore the pressure difference (pressureloss) across the valve can be determined exactly from the pressure readings at pc5 andpout (figure 4.9). Both, Figure 4.10 and table 4.1.2 show that higher pressure differencesoccur at higher discharge pressures. Since pressure forces due to the valve springing areconstant and flow inside the cylinder is negligible small we assume that faster increasingpressure inside the cylinder at higher discharge pressures, valve plate sticking, and theinertia of the valve plate contribute to this behaviour. Note that in case of open valves,the time, waves need to cover the distance between the valve and the sensors, must betaken into account and thus the difference of the pressure readings at pc5 and pout do notgive the exact pressure difference across the valve.
Valve Plate Motion The motion of the discharge valve plate has been recorded usingthree distance sensors. They were mounted on the circumference of a mean diameter of thevalve each with a distance of 90◦ to the next one. While at the first opening of the valve theplate keeps plane, at the end of the discharge phase the valve plate becomes slightly askew(figure 4.11). We have identified that asymmetric flow and variations in the springing
50
300000
350000
400000
450000
500000
120 130 140 150 160 170 180 190 200
pc1pc5pc6pc4pc3
Rel
ativ
epre
ssure
[Pa]
Crank angle [◦]
Figure 4.8: Relative Pressure at pressure sensors pc1, pc3, pc4, pc5 and pc6. Case:pout = 4 bar, cs=48 N/mm, xv,max=1.35 mm.
0
50000
100000
150000
200000
250000
300000
350000
400000
450000
500000
0 50 100 150 200 250 300 350
pout, 4 barpout, 1 barpc5, 4 barpc5, 1 bar
Rel
ativ
epre
ssure
[Pa]
Crank angle [◦]
Figure 4.9: Relative pressure in front (pc5) and behind pressure valve (pout) . Cases: 1chamber, pout = 4 bar, cs=48 N/mm, xv,max = 1.35 mm and 1 chamber, pout = 1 bar,cs=48 N/mm, xvmax=1.35 mm.
51
0
20000
40000
60000
80000
100000
120000
0 0.0002 0.0004 0.0006 0.0008 0.001 0.0012 0.0014
pc5−pout, 4 barpc5−pout, 3 barpc5−pout, 2 barpc5−pout, 1 bar
Diff
eren
cepre
ssure
pc5
-pou
t[P
a]
Valve Lift [m]
Figure 4.10: Difference pressure pc5-pout over valve lift for different discharge pressures .Cases: 1 chamber, pout = 1/2/3/4 bar, cs = 48 N/mm, xv,max = 1.35 mm.
Discharge Pressure Pressure Difference (pc5-pout) Pressure Gradient
×105 [Pa] ×105 [Pa] ×105[Pa/s]
2 0.08 260
3 0.16 480
4 0.30 690
5 0.40 842
Table 4.3: Pressure difference at closed valve (valve starts to open) for different dischargepressures.
52
are responsible for this. In order to quantify the contribution of both effects, followingmeasurement procedure has been conducted (figure 4.12). In the first measurement thecompressor was tested with a certain position of the valve plate. In detail, the sensorsVP1 and VP2 were located at Position A and Position B, respectively. Then the dischargevalve was rotated at 180◦ and again the valve motion recorded (sensor VP1 at Postion Band sensor VP2 at position A). The influence of the asymmetric springing is representedby the line AS which is given by
AS = xv,VP1, Pos. B − xv,VP2, Pos. B , (4.1)
and the line for the influence of the asymmetric flow AF is obtained by
AF = 0.5((
xv,VP1, Pos.B + xv,VP2, Pos.B
)
−(
xv,VP1, Pos.A + xv,VP2, Pos.A
))
. (4.2)
It turns out that the effect of springing dominates (figure 4.13).
0
0.0002
0.0004
0.0006
0.0008
0.001
0.0012
0.0014
100 120 140 160 180 200
VP1VP2VP3
VP average
Val
velift
[m]
Crank angle [◦]
Figure 4.11: Comparison of valve plate motion at sensor VP1, VP2 and VP3. Case: 1chamber, pout = 4 bar, cs =48 N/mm, xv,max=1.35 mm.
Variation of springing In figure 4.14 the comparison of measurement data with dif-ferent springing configurations (cs=48 N/mm and cs=80 N/mm) is shown. It turns outthat stiffer springing leads in this case to following results:
• Lower impact velocity of the valve plate: In this case the impact velocity is reducedfrom 1.67 m/s for cs = 48 N/mm to 1.48 m/s in case of cs = 80 N/mm. This is truefor most compressors.
• The valve closes faster
• The valve plate tends to open and close more often and therefore excite more oftenpressure waves.
53
2
1
1
2
1. Measurement 2. Measurement
1 . . . VP1 (distance sensor)
2 . . . VP2 (distance sensor)
piston
discharge valve
Pos. A
Pos. B
Figure 4.12: Measurements of valve plate motion with sensor VP1 and VP2.
−0.5
0
0.5
1
1.5
120 140 160 180 200 220
VP1 Pos. AVP2 Pos. BVP1 Pos. BVP2 Pos. A
asymetric springingasymetric flow
Val
velift
[m]
Crank angle [◦]
Figure 4.13: Askew valve plate: Different contributions. Case: 1 chamber, pout = 4 bar,cs =48 N/mm, xv,max=1.35 mm.
54
−100000
0
100000
200000
300000
400000
500000
0 50 100 150 200 250 300 350−0.0005
0
0.0005
0.001
0.0015
0.002
0.0025pc5 c=48 N/mmpc5 c=80 N/mm
valve lift, c=48 N/mmvalve lift, c=80 N/mm
Rel
ativ
epre
ssure
atpc5
[Pa]
Val
velift
[m]
Crank angle [◦]
Figure 4.14: Comparison of relative pressure at pc5 and valve lift. Cases: pout = 4 bar,cs=48 N/mm and cs=80 N/mm, xv,max=1.35 mm.
Variation of maximum valve plate lift xv,max The variation of the valve plate liftxv,max from 1.05 mm to 1.65 mm has very little effect on the pressure distribution inside thecylinder (figure 4.15). Whereas the impact velocity of the valve plate increases markedlywith higher maximum valve plate lifts xv,max since the plate is accelerated over a largerdistance (see table 4.1.2).
Maximum valve plate lift xv,max [m] Impact velocity [m/s]
1.05 1.36
1.35 1.67
1.65 1.91
Table 4.4: Impact velocities for different maximum valve plate lifts.
55
−100000
0
100000
200000
300000
400000
500000
0 50 100 150 200 250 300 350−0.0005
0
0.0005
0.001
0.0015
0.002
0.0025
pc5 xvmax=1.35 mmpc5 xvmax=1.65 mm
valve lift, 1.05 valve lift, 1.35
pc5 xvmax=1.05 mm
valve lift, 1.65 R
elat
ive
pre
ssure
atpc5
[Pa]
Val
velift
[m]
Crank angle [◦]
Figure 4.15: Comparison of relative pressure at pc5 and valve lift. Cases: pout = 4 bar,cs=48 N/mm, xv,max = 1.05 / 1.35 / 1.65 mm
4.2 Compressor with eight valves
A double-acting, 1 cylinder, barrel design reciprocating compressor with eight valves wastested. The experiment was carried out by Ariel Compression Ltd. The main specificationsof the compressor can be found in table 4.5. Various sensors were used to measure pressureinside the cylinder and at the suction and discharge flange (figure 4.16).
Type of compressor: Ariel Compression JGD26.5
1 cylinder, double-acting, barrel design
Diameter of cylinder: 0.6731 m
Stroke: 0.1397 m
Speed of crankshaft: 1182 rpm
Gas: Nitrogen
Suction pressure: 2.2 · 105 Pa
Discharge pressure: 6.4 · 105 Pa
Valves: 8
Classification ǫ: 0.65
Table 4.5: Main Specifications of the Ariel test compressor
The pressure readings of the various sensors can be found in figure 4.17. Note thatthe locations of the pressure sensors differ from the Burckhardt test compressor. Mostof the statements made above for the Burckhardt test compressor can be applied to thiscompressor as well. However, since this compressor is almost three times bigger and hashigher overall mass flow rates some significant differences compared to the Burckhardt
56
2 4
3
5
1
1 . . . . . . pc12 . . . . . . pc23 . . . . . . pc34 . . . . . . pc45 . . . . . . pc5pout at discharge flangepin at suction flange
suction valves discharge valvesandvalve retainers
Figure 4.16: Location of sensors inside cylinder
compressor can be identified. Therefore the measurements of the Ariel compressor arediscussed in the following. Although the gas can leave the cylinder through four valves,the maximum measured pressure inside the cylinder exceeds the discharge pressure by morethan 1.4 bar. This is due to the fact that high mass fluxes through the valves increase thepressure in the valve retainers and discharge pipe markedly resulting in hindered outflow ofgas. In addition, the pressure differences between suction and discharge side in the cylinderare very high since the diameter of the cylinder and thus the distance from the suctionside to the discharge valves is large. Note that these differences of the test compressorscan be well described by the classification number ǫ. In case of the Ariel test compressorǫ is 0.65, which is almost three times bigger than the ǫ of the Burckhardt test compressor.
Although the pressure sensor pc1 is located in the vicinity of pc3, the pressure read-ings differ during discharge. We assume that the sensor pc1 in the port records totalpressure whereas the sensor pc3 registers the static pressure only. Therefore the differencecorresponds to the dynamic pressure.
Another difference to the Burckhardt test compressor is that the pressure waves in thedischarge pipe do not fade away. This behaviour can not be described by modelling thevalve retainer and pressure chamber only. Instead the whole discharge system, consistingof discharge valve retainer, pressure chamber, discharge pipe, and damper, must be con-sidered. However for calculating the gas flow in the cylinder it is sufficient to give a goodapproximation of the pressure at the downstream side of the discharge valves. This issueis discussed in detail in section 5.2.
57
0
100000
200000
300000
400000
500000
600000
700000
800000
900000
−150 −100 −50 0 50 100 150
pc3
pinpout
pc1pc2
pc4
Abso
lute
pre
ssure
[Pa]
Crank angle [◦]
Figure 4.17: Measurement data of absolute pressure at different sensors.
58
Chapter 5
Comparison of Measurement andNumerical data
In this chapter the quasi one-dimensional and two-dimensional simulation results are com-pared to measurement data of the test compressors. The deviations are discussed andimportant physical effects highlighted.
5.1 One-dimensional Model
5.1.1 Comparison with Measurement Results of the Burckhardt testcompressor
The quasi-onedimensional model is employed in order to obtain the numerical results.For modelling the pressure chamber it is assumed that the pressure variations inside thedischarge pipe are negligible small and thus the second approach VR2 is chosen where thepressure chamber is connected to the valve retainer by a T-piece and the outflow boundaryis located at the end of the pressure chamber. The specifications of the test case calculatedcan be found in table 5.1.
Note that from now on all pressure results are displayed in absolute pressure. Thedead centre with the smallest cylinder volume is at 180◦ CA.
Number of active Discharge Pressure Valve springing Maximum valve lift
Working Chambers pout [Pa] cs [N/m] xv,max [m]
1 5 · 105 48 · 103 1.35 · 10−3
Table 5.1: Specifications of the test case
In Figure 5.1, 5.2 and 5.3 the comparisons of the measured pressures at the dischargeside (pc5), the suction side (pc6) and in the valve retainer (pout) with the numericalsolution of the quasi one dimensional model are given. In addition, the motion of thedischarge valve obtained by numerical simulation and measurements can be found in thesame figures. The pressure difference between discharge side (pc5) and cylinder centre(pc1) is displayed in figure 5.4. In the following we will discuss the solutions in detail.
59
0
100000
200000
300000
400000
500000
600000
0 50 100 150 200 250 300 350 0
0.0005
0.001
0.0015
0.002
0.0025
0.003pressure pc5, Num.pressure pc5, Exp.
valve lift, Num.valve lift, Exp.
Abso
lute
pre
ssure
atpc5
[Pa]
Val
velift
[m]
Crank angle [◦]
Figure 5.1: Burckhardt test compressor: Comparison of absolute pressure at pc5 and valvelift.
0
100000
200000
300000
400000
500000
600000
700000
0 50 100 150 200 250 300 350 0
0.0005
0.001
0.0015
0.002
0.0025
0.003
0.0035pressure pc6, Num.pressure pc6, Exp.
valve lift, Num.valve lift, Exp.
Abso
lute
pre
ssure
atpc6
[Pa]
Val
velift
[m]
Crank angle [◦]
Figure 5.2: Burckhardt test compressor: Comparison of absolute pressure at pc6 and valvelift.
60
460000
480000
500000
520000
540000
0 50 100 150 200 250 300 350
pout numericalpout experimental
Abso
lute
pre
ssure
atpou
t[P
a]
Crank angle [◦]
Figure 5.3: Burckhardt test compressor: Comparison of absolute pressure at pout.
−60000
−40000
−20000
0
20000
40000
60000
0 50 100 150 200 250 300 350
diff. pressure pc5−pc1, num.diff. pressure pc5−pc1, meas.
Val
velift
[m]
Crank angle [◦]
Figure 5.4: Burckhardt test compressor: Comparison of difference pressure pc5-pc1.
61
Compression The gas velocities in the cylinder are very small and an isentropic com-pression takes places. The almost uniform pressure inside the cylinder (figure 5.5) is afunction of the actual cylinder volume and the initial pressure and volume at the begin-ning of the compression only. In other words differences in pressure between simulationand measurement during compression can be traced back to the fact that the compressionvolume is not specified correctly.
Discharge When the pressure in the cylinder exceeds the pressure in the valve retainerplus the springing the valve starts to open and a wave is initiated (figure 5.5). Therarefaction wave, which is initiated at CA = 142◦, reaches the suction side at CA = 146◦.Note that in undisturbed regions the isentropic compression still takes place. However,the following complicated interaction of the valve motion and the pressure waves is wellrepresented by the numerical model. Most notably the vale motion is given correctly. Alsothe calculated pressure differences in the cylinder, which are the cause for the momentonto the piston, matches the measured ones.
Due to the outflow of gas through the valve, the pressure in the valve retainer rises.Again this increase is represented well by the model.
We observe that at the end of the discharge the calculated amplitudes of the pressurewaves are bigger than the measured ones. Resulting in higher pressure differences acrossthe cylinder in the simulation. In other words, the one-dimensional model overestimatesthe pressure waves and thus the moment on the piston cannot be determined accuratelyin this stage of the discharge phase.
Expansion We observe that the calculated amplitudes of the pressure waves are higherthan the measured ones. Whereas the frequencies agree. Furthermore, an agreement ofthe decay of the amplitudes, due to the increase of volume, can be observed.
The decrease of mean pressure inside the cylinder is not well represented by the numer-ical model. More precisely, the pressure decreases slower in the model. In order to find thecause of the deviations full three dimensional simulations have been carried out. It turnsout that the remaining gas runs through the cylinder at higher velocities as compared tothe compression phase. However heat transfer from the gas to the surrounding and dissi-pation can still be neglected. Hence, similar to the compression phase the mean pressureis only a function of volume (for given initial pressure and volume at the beginning ofthe expansion), and thus the mass inside the cylinder must be higher at the time whenthe discharge valve closes. Two different causes can be identified. Firstly the minimumclearance Z0 is smaller than specified for the calculation resulting in smaller volumes atthe start of the expansion. Secondly higher amplitudes of the pressure waves lead to moremass inside the cylinder. Since the calculated pressure during compression matches themeasured one perfectly the second cause is most likely responsible for different slopes ofthe pressure during expansion.
Suction All statements made for the discharge phase concerning pressure waves are truefor the suction phase as well. However, since the overall pressure level is low comparedto the discharge phase, the resulting pressure waves have very low amplitudes. Thereforeproblems arising from the pressure difference across the cylinder cannot be found duringthe suction phase. The gas velocities inside the cylinder are rather small during the suction
62
phase.
5
5.5
6
6.5
0 0.1 0.2 0.3
140° CA142° CA144° CA146° CA
x [m]
Abso
lute
pre
ssure
×10
5[P
a]
cylinder
cylinder head recess
valve pocket
valve retainer
Figure 5.5: Initiation of waves: Pressure distribution inside the cylinder and in the valveretainer at the beginning of the discharge
5.1.2 Super elevation of the pressure due to initial sticking
Comparing the pressure reading pc5 close to the discharge valve with the correspondingnumerical solution, where the initial sticking is not taken into account, shows that theopening of the valve is predicted too early (figure 5.6). As a consequence the pressuremaximum at pc5 is about 0.3 bar to small. With other words in reality the opening ofthe valve does not start when the pressure in the cylinder balances the pressure in theretainer and the springing. An additional delay occurs. In [8] the dynamic behaviour of avalve plate is analysed. If the initial gap is very small, say 1-5 µm, viscosity hinders thegas to flow into the valve gap resulting in a local pressure drop in the valve gap. ApplyingReynolds lubrication theory, an additional force Fadh holding back the valve plate can beidentified. It turns out that the result depends on the initial gap width, which cannot bemeasured easily. In this case a initial gap width of 2 µm has been chosen to fit the data.
5.1.3 Pressure loss at sudden changes of cross section
The pressure at pc5 predicted by the numerical solution, wherein the pressure loss has beenneglected, drops after the first maximum at a crank angle of 144 degrees to a minimum atCA 156◦ (figure 5.7). This minimum is about 5 ·104 Pa smaller than that of the measureddata. At the second minimum at a crank angle of 180◦ (dead centre) the difference is evenworse (8 · 104 Pa). An explanation for this difference is the following. During outflow thegas is compressed under the piston. There it flows through a narrow channel towards thevalve pocket. At the entrance of the valve pocket the gas faces a sudden increase of the
63
ExperimentNumerical with adhesionNumerical without adhesion
400000
450000
500000
550000
600000
120 130 140 150 160 170 180 190 200
Crank angle [◦]
Pre
ssure
pc5
[Pa]
Figure 5.6: Delay of valve opening due toviscosity in valve gap
ExperimentNumerical with pressure lossesNumerical without pressure losses
400000
450000
500000
550000
600000
120 130 140 150 160 170 180 190 200
Crank angle [◦]
Pre
ssure
pc5
[Pa]
Figure 5.7: Effect of pressure loss at sud-den change of cross-section
effective cross section of the flow channel when the piston is close to the dead centre. Thissudden increase of the cross section is physically associated with a pressure loss.
Although the modelling of the pressure losses at sudden changes of cross-section in-creases the quality of the numerical solution in the vicinity of the dead centre (180◦)markedly, the deviations of pressure at the first minimum remain.
5.1.4 Different locations of outflow boundary
Different approaches for the valve retainer and pressure chamber have been introducedin section 2.1.2. The comparison of the measured pressure in the valve retainer with thesolutions of the two different numerical models is given in figure 5.8. In the following wewill discuss the differences in detail.
• First approach VR1 (figure 2.4): The pressure chamber is added to the subdomainof the valve retainer and the outlet boundary is set at the end of the valve retainer.This model describes the first peak of the pressure in the valve retainer (pout) well,although the amplitude is too high. However, the gas leaves the valve retainer almostimmediately and thus the pressure oscillates about the discharge pressure. Duringexpansion the pressure is damped due to the pressure loss at the outflow.
• Second approach VR2 (figure 2.5): Here the pressure chamber is replaced by acylinder connected to the valve retainer using a T-piece. The calculated pressure inthe valve retainer follows the measured one since the gas is hindered to flow out of thevalve retainer. However, the small oscillations of the pressure cannot be describedwell by the simulation. These small variations have only a minor influence on thepressure distribution inside the cylinder.
Observing the pressure at the discharge side of the cylinder (pc5) shows that bothapproaches give good results and differences are rather small (figure 5.9). Whereas the
64
valve motion shows different behaviour at the second opening of the valve. However interms of pressure distribution inside the cylinder these differences play a minor role, sinceduring this stage of the discharge phase only a small amount of gas flows through thevalve.
460000
480000
500000
520000
540000
0 50 100 150 200 250 300 350
pout, Num., VR2
pout experimentalpout, Num., VR1
Abso
lute
pre
ssure
atpou
t[P
a]
Crank angle [◦]
Figure 5.8: Comparison of absolute pressure at pout for different models of the pressurechamber.
5.1.5 Impact velocity
The impact velocity of the valve plate onto the valve seat can be extracted from the valvemotion. In figure 5.11 the calculated and measured impact velocities for different dischargepressures are compared. The difference between measurement and simulation is less than15%.
5.1.6 Valve losses
In figure 5.10 the computed mean pressure p is shown in the p,V -diagram. The shadedarea corresponds to losses at the discharge valve. In the standard case defined above thevalve losses amount 5.3% of the total input energy.
5.2 Two-dimensional model
5.2.1 Comparison with measurement results of the Ariel test compressor
The two-dimensional model is employed to simulate the Ariel test compressor specified insection 4.2. The pressure chamber of this compressor, where the gas flows from four valve
65
0
100000
200000
300000
400000
500000
600000
0 50 100 150 200 250 300 350 0
0.0005
0.001
0.0015
0.002
0.0025
0.003pc5, Num., VR2pc5, Num., VR1
pc5, Experimentalvalve lift, Num., VR2valve lift, Num., VR1
valve lift, Exp.
Abso
lute
pre
ssure
atpc5
[Pa]
Val
velift
[m]
Crank angle [◦]
Figure 5.9: Comparison of absolute pressure at pc5 and valve lift for different models ofthe pressure chamber.
0
1
2
3
4
5
6
7
0 1 2 3 4 5
valve losses
aver
aged
pre
ssure
×10
5[P
a]
volume [dm3]
Figure 5.10: Valve Losses at dischargevalve
0
0.5
1
1.5
2
2.5
0 1 2 3 4 5
numericalmeasurement
relative discharge pressure ×105 [Pa]
impac
tve
loci
ty[m
/s]
Figure 5.11: Impact velocity of dischargevalve depending on discharge pressure
66
retainers merge, is far too complicated to be modelled as a simple cylinder. Therefore,the first approach VR1 for the pressure chamber is chosen. The measured pressures atdifferent locations inside the cylinder are compared to simulated ones (figures 5.12 and5.13). Furthermore the pressure difference between suction and discharge side (pc2-pc4)is displayed in figure 5.14. Note that in the following diagrams the dead centre with thesmallest volume is at 0◦ CA.
0
100000
200000
300000
400000
500000
600000
700000
800000
900000
−150 −100 −50 0 50 100 150
pressure pc4, Num.pressure pc4, Exp.pressure pin, Num.pressure pin, Exp.
Pre
ssure
[Pa]
Crank angle [◦]
Figure 5.12: Ariel test compressor: Comparison of pressure distribution at pc2 and pout
In the following the four stages of the compression cycle will be discussed:
Compression The computed pressure inside the cylinder increases slightly faster thanthe measured one. As pointed out in section 5.1.1 this can be attributed to errors inthe specification of the volume. Since the calculated slope of the pressure is steeper, theclearance volume, which has been derived from drawings provided by Ariel and specifiedin the numerical simulation, must be smaller than the real one.
Discharge The discharge valves start to open simultaneously at −73◦ CA (figure 5.20).
During discharge the calculated pressures inside the cylinder are about 0.7 bar higherthan the measured ones. In addition we observe that this difference varies only slightly.This issue will be discussed in detail in section 5.2.2.
The pressure difference between suction side and discharge side is well represented bythe model. Not only the frequency but also the amplitudes are calculated almost correctly.However at the end of the discharge phase the calculated second peak is lower than themeasured one. Moreover from this point on the calculated pressure difference is about 8◦
CA ahead. The moment onto piston is derived from the pressure acting on the piston.Due to the good approximation of the pressure difference we expect the computed moment
67
0
100000
200000
300000
400000
500000
600000
700000
800000
900000
−150 −100 −50 0 50 100 150
pressure pc2, Num.pressure pc2, Exp.
pressure pout, Num.pressure pout, Exp.
Pre
ssure
[Pa]
Crank angle [◦]
Figure 5.13: Ariel test compressor: Comparison of pressure distribution at pc4 and pin
−200000
−150000
−100000
−50000
0
50000
100000
150000
200000
−150 −100 −50 0 50 100 150
diff. pressure pc2−pc4, Num.diff. pressure pc2−pc4, Exp.
Pre
ssure
diff
eren
ce[P
a]
crank angle [◦]
Figure 5.14: Ariel test compressor: Comparison of difference pressure pc2-pc4.
68
to approximate the real one also well (section 5.2.4).
Expansion The computed pressures inside the cylinder decrease faster than the mea-sured ones. Similar to the compression phase we identify deviations in the clearancevolume as cause for the different slopes. However in this case different initial pressuredistribution at the beginning of the expansion phase leads to deviations in the pressurereadings during expansion too.
The calculated pressure difference between discharge side and suction side matches themeasured one with respect to frequency, amplitude, and damping. However, it is still 8◦
CA ahead.
Suction During suction the small pressure variations inside the cylinder are underes-timated by the simulation model. It turns out that at the end of the suction phase themeasured mean pressure is slightly higher (0.1 bar) than the computed one. Again, thepressure oscillations inside the suction pipe, which are neglected in the model, are respon-sible for this.
5.2.2 Variation of discharge pressure
The results of the Ariel test compressor show that during discharge the calculated pressureinside the cylinder is too high compared to the measured one. This can be explained asfollows. Firstly, the computed pressure inside the cylinder increases faster at the beginningof the discharge phase than the measured one resulting in an overestimation of the pressureduring discharge, and secondly the actual discharge pressure is lower than the specified one.In order to determine the influence of the first effect simulations have been performed wherethe clearance volume has been adjusted in a way so that the computed pressure increaseduring compression fits the measured one. However, it turns out that the difference in thepressure rise at the beginning of the discharge can be neglected. In order to evaluate thesecond explanation we take a closer look at the discharge system and its numerical model.From the measured pressure readings at pout we can see that in this case the amplitudesof the pressure waves inside the discharge system do not decay. Moreover the differencebetween pressure maximum and minimum at the discharge flange is 1 bar. It can beassumed that the pressure variations are of the same order in the valve retainer. So if ithappens that the discharge valves open during a pressure valley at the downstream sideof the valves then the actual discharge pressure would be lower than the nominal one. Onthe other hand the numerical simulation uses a model of the discharge system, where thepressure waves are damped. Thus at the beginning of the discharge phase the numericalsimulation cannot provide a good approximation of the pressure at the downstream sideof the valve. However, by reducing the discharge pressure specified for the numericalsimulation, the model can mimic lower pressures at the downstream side of the valves. Infigure 5.15 the computed pressure for a reduced discharge pressure of 6.1 bar is comparedto measurement data and numerical results where the discharge pressure is 6.4 bar. Itturns out that the pressure inside the cylinder is reduced and the trend of the pressuredistribution is not changed. Therefore following conclusions may be drawn:
• The pressure variations in the discharge system, which cannot be represented by thesimplified numerical model, are responsible for the overestimation of pressure inside
69
the cylinder during discharge.
• The trend of the pressure distributions inside the cylinder is not influenced by smallvariations of the discharge pressure. In other words the resulting pressure level inthe cylinder is only shifted by the variation of the discharge pressure.
• The simplified one-dimensional models of the valve retainer and pressure chamberare not sufficient to compute the pressure distribution inside the discharge systemfor the whole working period. However they are sufficient to describe the pressurerise in the valve retainer during discharge.
• During discharge the pressure level inside the cylinder is determined by the pressureat the downstream side of the cylinder.
In order to provide a good approximation of the pressure at the downstream side of thevalve the whole discharge system must be modelled. Furthermore the model must be validthroughout the whole working periode. However the gas flow in the pressure chamber istoo complicated to be modelled by an one-dimensional approach, at least two-dimensionalapproaches must be employed.
0
100000
200000
300000
400000
500000
600000
700000
800000
900000
−100 −50 0 50 100
press. pc4, Num. pout=6.4 barpress. pc4, Num. pout=6.1 bar
pressure pc4, Exp.
Pre
ssure
pc4
[Pa]
Crank angle [◦]
Figure 5.15: Variation of discharge pressure: pout = 640000 / 610000 Pa
5.2.3 Pressure distribution inside the cylinder
The Ariel test compressor has 8 valves located at the circumference of the cylinder. The en-tering and leaving of gas through the valves initiates pressure disturbances which stronglydeviate from plane waves. In fact a two-dimensional gas flow in the x, y-plane of the cylin-der can be observed (figure 5.16). At four selected crank angles the pressure distributionsinside the cylinder are discussed in the following.
70
At -64◦ CA the discharge valves are already open and the pressure in the vicinity ofthe discharge valves decreases due to the outflow of gas. On the other hand the gas in thesuction side and in the centre of the cylinder still undergoes isentropic compression. Inthis regions the velocity is zero (figure 5.17). The pressure difference between suction anddischarge side is about 0.9 bar resulting in a turning moment onto the piston relative tothe cylinder centre of 2350 Nm (figure 5.18).
At -32◦ CA a maximum velocity of about 130 m/s occurs at the valve pocket entriesleading to the discharge valves. We observe that due to these high gas velocities thepressure in the vicinity of the discharge valve pockets drop to 6.95 bar. Whereas thehighest pressure can be found at the stagnation point between the two rightmost dischargevalves.
Although the piston has reached the dead centre and reversed its movement, the valvesare still open at 3◦ CA. This is due to the fact that at this time the pressure peak is atthe discharge side of the cylinder. Moreover the expansion of the gas has not yet resultedin a sufficient pressure drop.
At 26◦ CA the valves have already closed and the remaining pressure difference acrossthe cylinder of 0.8 bar drives the pressure waves back and forth in the cylinder. Howeverthe amplitudes will be damped due to the decrease of mean pressure in the cylinder.
5.2.4 Moment onto the piston
The moment onto the piston is derived from the pressure distribution inside the cylinder.The pressure difference results in a moment onto the piston, which is given by
Ty =
∫
S
p(x, y)x dS . (5.1)
Here S denotes the area of the piston. In figure 5.18 the y-component of the turningmoment relative to the centre of the cylinder is displayed. There the trend of the momentmust follow the difference pressure. In case of special pressure distributions inside thecylinder a relationship between turning moment Ty and maximum pressure difference∆pmax can be found. For example, if the pressure is constant along y and linear along x(figure 5.19), then the equation 5.1 yields
Ty =1
8π∆pmax
(
d
2
)3
. (5.2)
Note that all other components of the moment must vanish due to symmetry.
5.2.5 Valve Opening
Let us denote the discharge valve which is located closer to the suction side with Vp1and one of the rightmost discharge valves with Vp2. Figures 5.20 and 5.21 show the valvemotion and mass flux of Vp1 and Vp2. Since the pressure in the cylinder is almost uniformduring compression, the discharge valves start to open simultaneously at −73◦ CA. Theystay open until the piston reaches the dead centre. It turns out that the mass flux throughthe different valves differ only slightly. However, the discharge valve Vp1 located closer tothe suction side closes earlier due to the pressure distribution inside the cylinder.
71
(a) Shortly after opening of discharge valves(CA = -64◦)
(b) During discharge (CA = -32◦)
(c) Closing of the discharge valves (CA = 3◦) (d) During expansion (CA = 26◦)
Figure 5.16: Ariel test compressor: Pressure distribution in the cylinder at various times.
72
(a) Shortly after opening of discharge valves(CA = -64◦)
(b) During discharge (CA = -32◦)
Figure 5.17: Ariel test compressor: Velocity magnitudes in the cylinder at various times.
−4000
−2000
0
2000
4000
−60 −30 0 30 60 90−2
−1
0
1
2
Moment onto pistonPressure difference pc2−pc4
Diff
eren
cepre
ssure
×10
5[P
a]
Crank angle [◦]
Mom
ent
[Nm
]
Figure 5.18: Moment onto Piston
p
−d2
d2 x
∆p
max
Figure 5.19: Calculating the moment ontothe piston: Pressure distribution insidecylinder
73
0
0.0005
0.001
0.0015
0.002
0.0025
0.003
0.0035
0.004
−100 −80 −60 −40 −20 0 20 40
Valve lift Vp1Valve lift Vp2
crank angle [◦]
valv
elift
[m]
Figure 5.20: Valve motion for differentdischarge valves
0
1
2
3
4
5
6
−100 −80 −60 −40 −20 0 20 40
mass flux through Vp1mass flux through Vp2
crank angle [◦]
mas
sflux
m[k
g/s]
Figure 5.21: Mass flux through differentdischarge valves
5.2.6 Valve Masking
Measurements at the Ariel test compressor have shown that valve masking leads to sep-aration zones resulting in reduced effective flow cross-sections of the valve and pressurelosses at the valve pocket entries. In the two-dimensional model the pressure losses areneglected. Only the valve flow area is reduced according to the distance between pistonand cylinder head. We observe that taking this effect into account hinders the gas to flowout. Therefore the calculated pressure inside the cylinder is higher (figure 5.22).
0
100000
200000
300000
400000
500000
600000
700000
800000
900000
−100 −50 0 50 100
pc2, Num. with valve mask.pc2, Num. without valve mask.
pressure pc2, Exp.
Pre
ssure
[Pa]
Crank angle [◦]
Figure 5.22: Effect of Valve Masking: Comaparison of pressure at suction side pc2
74
Chapter 6
Remarks on Numerical Schemes
This chapter is concerned with comparison of different numerical schemes and their so-lution. Not only computational time but also accuracy of the different approaches arediscussed. In addition the limitations of the one-dimensional and two-dimensional ap-proach is considered.
6.1 Comparison of full three-dimensional, quasi one-dimensional
and quasi two-dimensional numerical methods
In order to evaluate the simplified models the test case of section 5.1.1 has been simulatedusing full three-dimensional, quasi one-dimensional and quasi two-dimensional numericalmethods. However to simplify matters the outflow boundary has been placed at thedownstream side of the discharge valve. That means that the effects of the dischargesystem have been neglected.
For the three dimensional simulation the commercial CFD-code of Fluent Inc. isemployed. Here the three-dimensional Reynolds-averaged Navier-Stokes equations, thecontinuity equation and the energy equation are solved by means of first order methodsin the flow domain of the cylinder. In addition an appropriate turbulence model mustbe chosen. The simple valve models presented in section 2.5 have formed the in- andoutflow boundaries. The piston motion and thus the variation of the computational do-main with time has been modelled using dynamic meshing. A detailed description of theimplementation can be found in [22].
In figure 6.1 the pressure inside the cylinder and the valve motion are compared. Itturns out that the simple models are capable of capturing the most important physicaleffects, resulting in a very good agreement with the three dimensional model. Most notablythe valve motion is predicted accurately by the simplified models although differences canbe observed at the end of the discharge phase. However the differences of the models arediscussed in the following.
75
0
100000
200000
300000
400000
500000
600000
−150 −100 −50 0 50 100 150 0
0.0005
0.001
0.0015
0.002
0.0025
0.003pressure pc5, Num. 1Dpressure pc5, Num. 2Dpressure pc5, Num. 3D
valve lift, Num. 2Dvalve lift, Num.3D
valve lift, Num. 1DA
bso
lute
pre
ssure
atpc5
[Pa]
Val
velift
[m]
Crank angle [◦]
Figure 6.1: Comparison of different finite volume schemes: Pressure pc5 inside the cylinderand valve motion.
6.1.1 Pressure waves
During discharge the interaction of the pressure waves and the valve motion is the domi-nant physical effect inside the cylinder. We use the pressure difference across the cylinder(pc6-pc5) to describe the properties of the pressure waves (figure 6.1). It turns out thatboth simple models are capable of describing the first excitation of the rarefraction wavedue to the opening of the discharge valve. Most notably the amplitudes and the frequen-cies match. However in the course of the discharge small deviations can be observed. Mostnotably, at the end of the discharge phase the one-dimensional and two-dimensional modeldiffer from the three-dimensional model. This is due to the fact, that the distance betweencylinder head and piston is so small, that the resulting flow in the discharge valve pocketand suction valve pocket is three-dimensional. This behaviour cannot be described by thesimple models. In case of the one-dimensional model the differences at the dead centre(0◦CA) causes the discharge valve to open a third time, resulting in another wave exci-tation with high amplitudes. Finally, when the discharge valve stays close, the remainingpressure waves run back and forth in the cylinder. However the amplitudes of the wavesdecay on the one hand due to dissipation and on the other hand due to the increase ofthe volume, whereas the effects of the dissipation are rather small. In numerical simula-tions another effect takes place. The numerical dissipation of the used method leads toan artificial damping. However the magnitude of the numerical dissipation decreases withdecreasing cell size ∆x. In other words with a sufficient fine grid the numerical dissipationcan be neglected. For a detailed description of this issue we refer to [13] and [30]. In table6.1 the number of cells per diameter are listed for each numerical model. Since a veryfine grid has been used for the one-dimensional simulation the numerical dissipation canbe neglected and the decrease of the amplitudes correspond to the increase of the cylin-der volume only. Although the two-dimensional mesh is finer than the three-dimensional
76
one, the numerical dissipation is higher. Hence the amplitudes of the waves decay faster.The phase shift between the different models during expansion can be attributed to thedifferent behaviour at the end of the discharge phase.
−100000
−50000
0
50000
100000
−100 −50 0 50 100
pc6−pc5, Num. 2Ddiff. press., Num.3D
pc6−pc5, Num. 1D
Pre
ssure
Diff
eren
cepc6
-pc5
[Pa]
Crank angle [◦]
Figure 6.2: Comparison of different finite volume schemes: Pressure difference pc6-pc5.
6.1.2 Computational Time
The computational time for the one-dimensional and two-dimensional simulation was mea-sured on a Pentium 4 Mobile processor with 512 MB RAM. The simulation of one crankshaft revolution took about two minutes and about four hours for the one-dimensional andtwo-dimensional model, respectively. The three-dimensional calculations were performedon an Athlon 64 X2 Dual Core 4800+ processor with 2 GB RAM. Here the computationaltime of one crank shaft revolution was 2.5 days. In table 6.1 the calculation time and thegrid size are listed for each model.
Method Total number of cells Cells per Diameter Calculation Time per periode
1D 400 330 2 min
2D 6272 104 4 hours
3D 22035-102297 70 2.5 days
Table 6.1: Calculation Time
77
6.2 Limitations of the Numerical Schemes
6.2.1 Limitations of the one-dimensional model
The quasi one-dimensional model can not be applied on all of the test cases success-fully. Although, for most of the compressor specifications a solution can be obtained,the results must be questioned. In the following we will discuss different cases where theone-dimensional model is only limited applicable.
Two-dimensional flow in the x,y-plane One main assumption of the quasi one-dimensional model is that the thermodynamic state in each flow cross-section is uniform.This is violated in case of compressors with more than two valves since multiple valve pock-ets leads to a two-dimensional flow in the x, y-plane (figure 5.16). Hence the results ofthe one-dimensional model are not satisfying. Also another problem arises when multiplevalve pockets have to be calculated: The two equivalent valve pockets which are derivedby adding up all discharge and suction valve pockets, have very large cross-sections. Es-pecially, when the piston approaches dead centre the cross-sections of the cylinder volumeare rather small compared to the valve pockets. This may lead to supersonic flow inthe cylinder cross-sections and furthermore in connection with large jumps in flow areasto the breakdown of the numerical calculation. It must be pointed out that supersonicflow is not likely to occur in common compressors. For example, when employing theone-dimensional model to the Ariel test compressor with eight valves the results showsupersonic flow before the simulation breaks down. Whereas the gas velocity computedby the two-dimensional model never exceeds sonic speed.
Two-dimensional flow in the x, z-plane Again the thermodynamic state is not uni-form in the cross-section at a time t instead it is a function of x and z. This flow can befound in compressors with low compression ratios and in cylinders, where the flow cannotfollow the form of the cross-sections and separation zones are present (figure 2.16). In thefirst case the discharge valve opens when the piston is far away from the dead centre andthe pressure disturbance coming from the valve pocket can expand in x and z direction,resulting in a complicated wave pattern. In the second case the height Z may be smallduring discharge but the geometry of the cylinder head recesses or the valve pockets leadsto separation zones or at least to cross-section areas where the thermodynamic state andthe gas velocity is not uniform. Here a two-dimensional model in the x,z-plane mustbe employed. We refer to Meyer who used a commercial CFD code to modell this two-dimensional problem [22]. It is worth mentioning that with above finite volume schemesa two dimensional model in the x,z plane can be easily derived.
Figure 6.3 shows the comparison of the measured pressure and valve lift of the Burck-hardt test compressor with simulation results. Here a discharge pressure of 2 bar is chosenand thus the compression ratio is about 2. Although the initial stage of the discharge iscalculated correctly, later the calculated pressure distribution and valve motion show largedeviations from the measured one. Again the one-dimensional model fails to represent thewave structure and at least two-dimensional approaches must be employed in order tocalculate this case.
78
0
50000
100000
150000
200000
250000
300000
0 50 100 150 200 250 300 350 0
0.0005
0.001
0.0015
0.002
0.0025
0.003pressure pc5, Num.pressure pc5, Exp.
valve lift, Num.valve lift, Exp.
Abso
lute
pre
ssure
atpc5
[Pa]
Val
velift
[m]
Crank angle [◦]
Figure 6.3: Comparison of absolute pressure at pc5 and valve lift with discharge pressureof 2 bar.
Three dimensional flow If a compressor features more than one different characteristicstated above the cylinder flow will be three dimensional. Therefore the thermodynamicstate at a time t depends on x, y, z and thus three-dimensional models must be employed.
6.2.2 Limitations of the quasi two-dimensional model
Also the quasi two-dimensional model can not be applied to all compressor specifications.In the following most problem cases will be described.
Gas flow depending on z If the thermodynamic state and gas velocity depends onthe z-coordinate, one main requirement for the two-dimensional model is violated andthree-dimensional model must be employed in order to calculate this flow accurately.
Sudden changes of height inside the cylinder Certain cylinder head recess lead tosudden changes of height inside the cylinder and following problems may occur. Firstly thethermodynamic state and the gas velocity depends on z and three-dimensional gas flowsare present. Secondly, as pointed out in section 2.4.2, the source terms in the momentumequations (2.28) and (2.29) become very large, resulting in the break-down of the numericalsimulation. However, in this case following simplification of the cylinder geometry can beused. The volume Vch of the cylinder head recess is included in the cylinder clearance.The resulting cylinder clearance Z0 = Z0 + Vch/(d2π/4) is only a function of t.
79
Chapter 7
Summary
Measurements conducted on two different test compressors showed that special attentionhas to be paid for the interaction of the pressure waves and the valve motion in order tosimulate the gas flow and valve motion of barrel design reciprocating compressors accu-rately. Most notably important design criteria such as impact velocity of the valve, thevalve losses, and the moments onto the piston are governed by these interactions.
In the case of compressors with two valves it turned out that considering plane wavesonly is sufficient to capture the wave structure inside the cylinder. The resulting governingequations were solved by means of the finite volume scheme F-wave which offers two ben-efits: Firstly this numerical scheme directly incorporates the underlying wave propagationstructure and secondly the boundary conditions can be easily implemented. Finally sim-ple quasi-static valve models complete the numerical model of a reciprocating compressor.Since the model is one-dimensional, only very short computational times are required, evenwith very fine grids. The comparison of the measurements with the simulation show, thatthe one-dimensional model is capable of capturing the most important physical effects,resulting in a very good agreement. Most notably the valve motion and the pressure dis-tribution inside the cylinder and the valve retainer can be computed accurately. However,the amplitudes of the pressure waves at the end of the discharge phase and during theexpansion are overestimated, since three-dimensional effects at the end of the dischargecannot be described by this simplified model.
In the case of compressors with more than two valves a two-dimensional approachwas chosen in order to account for the multiple valve pocket entries, which lead to two-dimensional flows in the cylinder. The valve pockets and the valve retainers were mod-elled one-dimensionally. The resulting governing equations for the gas flow were solvedby means of the robust and simple finite volume scheme of Richtmyer. Again the com-parison of the measurements with the simulation show a good agreement in terms of gasflow and valve motion. Even the pressure difference across the cylinder is computed accu-rately, resulting in an accurate solution for the moment onto the piston. However duringdischarge the pressure inside the cylinder is overestimated, which is attributed to the pres-sure variations in the valve retainers. In fact the simplified one-dimensional models of thevalve retainer and pressure chamber are not sufficient to compute the pressure distribu-tion inside the discharge system throughout the whole working period. However they aresufficient to describe the pressure rise in the valve retainer during discharge. In orderto correct this shortcoming we suggest to calculate the gas flow in the pressure chamber
80
two-dimensionally.
Due to the low computation time and accurate results the simplified models presentedin this work, namely the one-dimensional and two-dimensional model, can form the basisfor a engineering design tool, which predicts gas flow and valve motion in a reciprocatingcompressor.
81
Appendix A
Tables
82
A.1 Sensors of Burckhardt Test Compressor
The Table A.1 lists all sensors and their appropirate coordinates in the Burckhardt testcompressor. The point of origin of the coordinate system is the top dead center of thefirst cylinder.
Identifier Type of sensors Name Coordinates or Position
[mm/mm/mm]
TRIG light trigger flywheel
VP1 distance valve plate clearance 1 valve plate
VP2 distance valve plate clearance 2 valve plate
VP3 distance valve plate clearance 3 valve plate
Ta temperature ambient air temperature -
Tst temperature suction tube 30 cm behind flange
Tsc temperature suction chamber -265/-30/0
Tpc temperature pressure chamber 265/0/-30
Tpt temperature pressure tube 30 cm behind flange
pc1 pressure cylinder pos. 1 0/0/0
pc2 pressure cylinder pos. 2 0/-33/0
pc3 pressure cylinder pos. 3 0/-66/0
pc4 pressure cylinder pos. 4 0/66/0
pc5 pressure cylinder pos. 5 75/0/0
pc6 pressure cylinder pos. 6 -75/0/0
pc7 pressure cylinder pos. 7 -75/-33/0
pin pressure suction chamber -265/0/-20
pout pressure pressure chamber 265/-30/0
po pressure pressure orifice orifice
dpo pressure pressure difference orifice orifice
To temperature temperature orifice orifice
Table A.1: Sensors
A.2 Pressure loss coefficients for the T-piece
83
A1 A2
A3
m1 m2
m3
K1 =
{
1 +
(
A3
A1
)2
+ 3
(
A3
A1
)2[
(
m1
m3
)2
−(
m1
m3
)
]}
∗
∗fbrpt3 − p3
pt1 − p1
K2 = K1but subscript 1 is to be replaced by 2
K3 = 0
fbr see table A.3
A1 A2
A3
m1 m2
m3
K1 = 0
K2 = −0.6pt1 − p1
pt2 − p2
K3 = −(
1 − 0.5m3
m1+ 2
(
m3
m1
)2)
pt1 − p1
pt3 − p3
A1 A2
A3
m1 m2
m3
K1 = 1.55m3
m2−(
m3
m2
)2 pt2 − p2
pt1 − p1
K2 = 0
K3 =
[
1 +
(
m3
m2
A2
A3
)2
− 2
(
1 − m3
m2
)
]
fbrpt2 − p2
pt3 − p3
fbr see table A.3 but subscript 3 is to be replaced by 2
A1 A2
A3
m1 m2
m3
K1 = −[
1 + 1.5
(
u1
u3
)2]
pt3 − p3
pt1 − p1
K2 = K1but subscript 1 is to be replaced by 2
K3 = 0
Table A.2: Pressure loss coefficients for the T-piece [14]
fbr
A1/A3 ≤ 0.35 1
A1/A3 > 0.35 m1/m3 ≤ 0.4 0.9 (1 − m1/m3)
m1/m3 > 0.4 0.55
Table A.3: Values for fbr [14]
84
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86
Curriculum Vitae
Personal Data
Name: Roland Aigner
Nationality: Austria
Date of Birth: November 13th, 1976
E-mail: roland [email protected]
Education
06/2004 - present Vienna University of Technology, Austria
Institut of Fluid Mechanics and Heat Transfer
• Doctorate in Mechanical Engineering
10/1996 - 01/2003 Vienna University of Technology, Austria
Department of Mechanical Engineering
• Diploma Thesis ”Formation of the strand for continuouscasting of steel”
• Graduation with honours as Master of Science (Diplomingenieur)
09/2000 - 01/2001 University of Salford, Salford (Greater Manchester), UK
Department of Mechanical Engineering
• Exchange semester
09/1991 - 06/1996 Hohere Technische Bundeslehranstalt Wels, Austria
Department of Mechanical Engineering
• Graduation with distinction (Matura)
Professional Experience
06/2003 - 06/2007 Vienna University of Technology, Austria
Institut of Fluid Dynamics and Heat Transfer
Teaching, Supervisor of exercise courses:
• Heat Transfer
• Numerical Methods for Fluid Mechanics and Heat Transfer
Research / Industrial Projects:
• Hoerbiger Valve Tec GmbH: ”Numerical Simulation ofa pipe branching wye”
• Berndorf AG: ”Heat Transfer of a rotating cylinder”
• European Forum for Reciprocating Compressors: ”HeatTransfer in a Reciprocating Compressor”
08/1999 - 09/1999 General Motors Austria GmbH, Vienna, Austria
• Intern
07/1995 - 08/1995 BRP-Rotax GmbH & Co. KG, Gunskirchen, Austria
• Intern
87