PhD Thesis
Numerical modelling of stationary vane rotary piston compressors
Author:
Bal´azsFarkas
Supervisor:
Dr. Jen˝o Mikl´osSuda
A thesis submitted in fulfillment of the requirements for the degree of Doctor of Philosophy
in the
Faculty of Mechanical Engineering
Department of Fluid Mechanics
2020
I, Bal´azs Farkas, hereby declare that this thesis titled, “Numerical modelling of sta- tionary vane rotary piston compressors” and the work presented in it are my own. I confirm that:
This work was done wholly or mainly while in candidature for a research degree at this University.
Where any part of this thesis has previously been submitted for a degree or any other qualification at this University or at any other institution, this has been clearly stated.
Where I have consulted the published work of others, this is always clearly at- tributed.
Where I have quoted from the work of others, the source is always given. With the exception of such quotations, this thesis is entirely my own work.
Where the thesis is based on work done by myself jointly with others, I have made clear exactly what was done by others and what I have contributed myself.
I have acknowledged all main sources of help.
Signed:
Date:
ii
Abstract
Numerical modelling of stationary vane rotary piston compressors by Bal´azsFarkas
The utilization of geothermal heat sources for power generation is seemingly plausible for loca- tions where significant source of thermal water (i.e. geothermally heated groundwater) is avail- able. As a proof of concept, a test bench for the experimental model of an inverse heat-pump (IHP) was designed and built by the Department of Energy Engineering (DEE) of Budapest University of Technology and Economics (BME) to investigate the feasibility of electric energy production with the use of low enthalpy geothermal sources. Along with the experimental work the numerical modelling of the novel IHP was done at the Department of Fluid Mechanics of BME. The IHP applies a Brayton cycle where the compressor and the expander are key elements of the system which were also newly designed for this recent application. The presented thesis discuses the numerical study on the novel Swinging Vane Compressor (SVC), which was designed for the new IHP, in order to estimate its performance characteristics and validate the predefined expectations. At design the two major requirements for the compressor unit were to be able to work with atmospheric air working fluid at given flow rate and pressure ratio and to be able to be converted into expander mode without significant modifications. To fulfil the initial require- ments, the rolling piston type compressor (RPC) architecture was chosen as base platform for the further developments. The RPC’s, which are generally referred as stationary vane compres- sors, are mainly applied in low capacity (<2kW) household refrigerator, air-conditioning and heat pump units. Combined with a feeding valve, an RPC unit can be effectively used as an expansion device. The redesigned SVC architecture is originated from Dr. Istv´an Magai and the most conspicuous difference to the common RPC’s is the crankshaft driven swinging vane which separates the compression and suction chambers.
To estimate the performance of the new SVC, it was evaluated based on Computational Fluid Dynamics (CFD) simulations alongside the experiments which were conducted by the DEE. The simulations were set up and solved in ANSYS Fluent. Similarly to most positive displacement compressors, the volume of the compression and suction chamber in the RPC is constantly changing, therefore the numerical domain must also be changed accordingly. Hence, to maintain adequate grid resolution for the CFD simulations, the resulted change in numerical domain has to be resolved by dynamic meshing algorithms. The position of the grid elements of the numerical domain can be controlled by deforming and re-meshing algorithms. To accomplish the mapping of the complex change in volume, the readily available dynamic meshing methods were improved by supplementing additional task specific codes, for the CFD model of the SVC.
Another challenging aspect from fluid mechanical performance viewpoint, especially in case of oil-free compressors, is the leakage flow, which appeared to have key importance on SVC per- formance. In CFD, the leakage flow paths can be directly resolved and the leakage flow can be directly computed. However, the dimensions of the sealing clearances are orders of magnitude smaller than the characteristic dimensions of the RPC’s working chambers. To avoid complica- tions connected to the different mesh size, applied in the seal clearance area and the rest of the
domain, the resolution of the leakage mesh was decreased and the restriction of the clearance gaps was modelled by the addition of porous source term. The porosity of the porous zone can be adjusted and even changed during the simulation to meet the demanded restriction.
In order to increase computational efficiency, the geometry of the compressor was reduced into a two dimensional (2D) model. However, the sudden change in flow cross sections at the inlet and the outlet ports cannot be modelled directly in 2D. Therefore, the 3D losses were evaluated and calibrated porous zones were defined both at the inlet and the outlet ports to model the additional losses. This reduction in model dimensions significantly reduced the computational cost of the simulations, but the model parameters could still not be tuned for the measurements, as the experimental results of prototype tests were significantly off from the design.
Therefore, a lumped parameter model was established based on the experimental and CFD results, as well as on the results published in the related literature. One determining factor from accuracy point of view is, how the leakage flow is predicted. The redesigned SVC investigated in this study is an oil-free unit which is a major difference compared to the common RPC’s designs. Also an oil-free compressor becomes more sensitive on clearance sizes at the seals. In case when oil is not present in the system, the flow across the narrow clearance channels can be modelled based on the Fanno theorem, while the heat exchange is neglected, but the friction and compressibility effects are taken into account. The computational cost of the applied lumped parameter simulation is low, especially in comparison to the complex CFD models. Therefore, significant amount of simulations can be completed within short period of time. With the use of a suitable optimization method, the unknown model parameters were defined by adjusting their values to minimise the difference between the model predictions and experimental data. In this study a genetic algorithm was used to find the unknown parameters which was proven to be capable to provide satisfactory results. The lumped parameter model was implemented and solved in SIEMENS Simcenter AMESim.
The comparison between the experimental and numerical results, confirmed that even care- fully built models might fail to provide adequate solutions without knowing the exact physical construction of the compressor unit. Unknown sealing clearance sizes due to material wear or inadequate precision at assembly are the usual cause of unpredictable operation. Especially, it is valid for oil-free operation, when we cannot lean on the sealing effect of the lubricant, the clearance size has more pronounced effect on performance. Slight differences in sealing gap di- mensions can significantly alter the behaviour of the investigated appliance, so that the applied models, which are validated for normal operation, would fail to resemble the experimental results outside the nominal working conditions.
To overcome the delineated issues, the indicator pressure diagrams were also resembled with the application of non-linear oscillators. First, the Van der Pol oscillators were tested but they proved to be inadequate to represent the common RPC indicator diagrams, however important findings were withdrawn which are also included in the recent thesis. To achieve better agreement with the target curves, morphed non-linear phase oscillators were established and applied. Polynomial regression, empirical regression and constructive incremental learning method was used to achieve adequate data fitting and the results of both methods were compared. The model was written
and executed in Matlab. The results show acceptable conformity and proved to be adequate to provide input data for further investigations, e.g. for parametric studies of the compressor design.
Declaration of authorship ii
Abstract iii
Contents vi
List of Figures xi
List of Tables xv
1 Introduction 1
1.1 Rolling piston compressor . . . 1
1.2 History . . . 4
1.3 The new compressor design . . . 7
2 Transient CFD simulation of a stationary vane, oil free, rotary com- pressor 11 2.1 Assessing performance by CFD . . . 11
2.1.1 Literature review . . . 12
2.2 Numerical set-up . . . 13
2.2.1 Dynamic mesh . . . 13
2.2.2 Porous media model . . . 19
2.2.3 Leakage flow . . . 21
2.2.4 Solver . . . 22
2.2.5 Boundary conditions . . . 22
2.3 A simplified modelling approach for rolling piston compressor . . . 22
2.3.1 Defining equivalent loading for 2D and 3D computations. . . 23
2.3.2 Modelling the effect of 3D cylinder-outlet transition in 2D . . . 24
2.3.3 Results . . . 28
2.4 Discharge valve . . . 32
2.4.1 Hinged discharge valve . . . 34
2.4.2 Results . . . 35
2.5 Summary . . . 38
3 Mechanistic model of the novel oil-free rotary compressor 39 3.1 Comparison between the Conventional RPC and the oil-free SVC . . . 40
3.2 Experimental study on the SVC. . . 43
3.2.1 Instrumentation . . . 44
3.2.2 Experimental results . . . 45 vii
3.3 Preliminary mathematical numerical model . . . 46
3.3.1 Suction and compression volumes. . . 46
3.3.2 Heat transfer . . . 49
3.3.3 Discharge valve . . . 50
3.3.4 Inlet . . . 51
3.3.5 Leakage . . . 51
3.3.6 Dead volume . . . 56
3.3.7 Preliminary sensitivity study . . . 57
3.4 Numerical simulation with AMESim . . . 59
3.4.1 Multi domain system. . . 59
3.4.2 Models for the suction and compression chambers . . . 60
3.4.3 Suction port . . . 61
3.4.4 Discharge valve modelling . . . 62
3.4.5 Leakage flow modelling . . . 62
3.4.6 Outlet model . . . 64
3.5 Identification of model parameters . . . 65
3.5.1 Error function . . . 65
3.5.2 Optimisation process. . . 65
3.5.3 Numerical results . . . 66
3.5.4 Summary table of the optimization study results . . . 68
3.5.5 Indicator diagrams with individually optimised parameters . . . . 69
3.5.6 Collective runs / Multi-objective optimization studies . . . 69
3.6 Sensitivity analysis . . . 74
3.6.1 Radial gap between piston and cylinder (LP I) . . . 76
3.6.2 Axial gap between piston and cylinder sidewalls (LP II) . . . 76
3.6.3 Radial gap between vane and piston (LP III) . . . 77
3.6.4 Radial gap between vane and cylinder (LP IV) . . . 77
3.6.5 Axial gap between vane and cylinder sidewalls (LP V) . . . 77
3.7 Summary . . . 78
4 Application of morphed non-linear phase oscillators for representing rolling piston compressor performance 79 4.1 Van der Pol oscillator . . . 81
4.1.1 Phase portrait and limit cycle. . . 82
4.1.2 The fitted VDP equation . . . 82
4.2 Morphed NLO . . . 83
4.2.1 Curve fitting by blended functions . . . 85
4.2.2 Learning function. . . 88
4.3 Morphed NLO with pre-defined convergence path . . . 93
4.4 Evaluation of the result oscillator with target convergence . . . 94
4.5 Summary . . . 100
5 Conclusion 101 6 Theses 105 1. Thesis: New model to predict the performance of stationary blade rotating piston compressors with radial discharge port . . . 105
2. Thesis: Dynamical meshing method developed for the Computational Fluid Dynamical model of rolling piston compressors . . . 107 3. Thesis: Development of mechanistical model to investigate the performance
of the newly designed swinging vane rotating piston compressor . . . 109 4. Thesis: Mathematical correlations to represent the change in pressure within
the cylinder of the rolling piston compressors . . . 111
7 T´ezisek 113
1. T´ezis: ´Uj modell fejleszt´ese radi´alis ki¨oml´es˝u, ´all´olap´atos, forg´odugatty´us kompresszorok m˝uk¨od´esi jellemz˝oinek
meghat´aroz´as´ahoz . . . 113 2. T´ezis: Numerikus ´araml´astani modell fejleszt´ese g¨ord¨ul˝odugatty´us kom-
presszor vizsg´alat´ahoz . . . 115 3. T´ezis: Koncentr´alt param´eter˝u matematikai modell fejleszt´ese ´uj t´ıpus´u
leng˝onyelves forg´odugatty´us kompresszor viszg´alat´ahoz . . . 117 4. T´ezis: G¨ord¨ul˝odugatty´us kompresszor nyom´asf¨uggv´eny´enek matematikai
le´ır´asa . . . 119
Bibliography 121
1.1 Schematic of a common RPC for AC units. . . 2
1.2 Representation of the compression (Vcs and suction (Vs) volumes . . . 2
1.3 Illustration of stationary vane compressor operation principle by schematic of the cross-section at different crankshaft positions. 0ocrank angle is de- fined where the center of the rotation and the center of gravity of the piston is aligned with the contact point of the vane. . . 3
1.4 Illustration of a typical RPC indication diagram [1] . . . 5
1.5 Initial prototype geometry . . . 9
1.6 Schematic of the compressor with hollow beam vane support . . . 9
1.7 Schematic of the final geometry with narrow support beam . . . 10
1.8 3D schematic of the compressor equipped with narrow support beam . . . 10
2.1 Illustration of the new meshing process applied for a conventional RPC. P is the center of the piston, O is the center of the cylinder, A is the node on the piston surface and A’ is its O centred radial projection to the cylinder surface and V and V’ are the edges of the vane (Fig.1.1) . . . 15
2.2 Schematic of the new re-meshing algorithm realized by a UDF in ANSYS Fluent . . . 16
2.3 Section of the block mesh in the narrowest gap area . . . 17
2.4 3D wedge mesh . . . 18
2.5 Hybrid mesh for the hollow beam design . . . 20
2.6 Comparison of the target and the model compressors . . . 23
2.7 The relation between the 3D (upper) and 2D (lower) outlet layouts . . . . 25
2.8 Distribution of C2 defined by Eq.2.14 at the intersection of the cylinder and the outlet joint in the 2D case . . . 27
2.9 Velocity distribution within the 3D sample model (v [m/s]) . . . 27
2.10 Velocity distribution within the simplified exit model (v [m/s]) . . . 28
2.11 Pressure upstream of the throttling valve at nominal throttling in func- tion of the crankshaft position. In ’2D wo’ case cylinder-outlet transition model was not implemented. . . 30
2.12 Pressure upstream of the throttling valve with increased throttling in function of the crankshaft position. . . 30
2.13 Outlet non-dimensional flow-rate at different rotational at nominal throt- tling as a function of the crankshaft position. In ’2D wo’ case cylinder- outlet transition model was not implemented. . . 31
2.14 Outlet non-dimensional flow-rate at different rotational speeds with in- creased throttling as a function of the crankshaft position. . . 31
2.15 The geometry of the applied check valve . . . 32
2.16 The numerical model of the discharge valve in closed and fully opened position . . . 32
xi
2.17 Representation of the different port positions. Ds I.:narrow port position,
Ds II.: wide port position . . . 33
2.18 CFD model result of ratio between the inlet and the cylinder outlet pres- sure in function of the crankshaft angle in case Ds I. and Ds II. . . 34
2.19 CFD model result of the relative valve lift in function of the crankshaft angle in case Ds I. and Ds II. . . 34
2.20 Numerical mesh of the discharge valve . . . 35
2.21 Prototype implemented valve . . . 35
2.22 CFD prediction of indicated pressure diagrams with hollow beam . . . 36
2.23 Model prediction of the effect of pressure ratio and valve pre-load on the indicator diagram, 1200 rpm. . . 36
3.1 3D model of a conventional AC compressor unit [2] . . . 40
3.2 Cross section of the rolling piston type compressor. The leakage paths are highlighted: I, II, III, IV and V leakage paths (LP) are shown . . . 41
3.3 Vane driving mechanism of the parallel compressor unit (on the left) and the simplified AMESim model of the linkage system (on the right). . . 42
3.4 3D view of the vane driving mechanism attached to one compressor unit . 42 3.5 Blueprint of the vane driving mechanism attached to one compressor unit 43 3.6 Experimental set-up. Two parallel connected compressors driven by a balance mounted electric motor instrumented with the average pressure and temperature sensors and shaft encoder installed . . . 43
3.7 Schematic of the instrumentation of the experimental set-up. . . 44
3.8 Experimental results: indicated compressor pressure at three different rotational speeds of 900 r/min (-lightgray), 1200 r/min (-gray) and 1500 r/min (-black). The throttling conditions are compared to the isentropic compression (...gray). Note that due to the location of the pressure sensor the measured data points in crank angle range between 315 deg and 360 deg are omitted in the experimental curves. . . 46
3.9 Illustration of the change of the compression volume in function of the crank angle in an RPC . . . 49
3.10 Cross-section of a stationary vane compressor highlighting the radial leak- age gap (δ)between the cylinder and the piston (LP I.) [3] . . . 53
3.11 Model channel geometry representing the radial clearance between the piston and the cylinder (LP I.) [3]. Mt:Mach-number at the throat, Me:Mach-number at the channel exit . . . 54
3.12 Simulation results of pressure change at baseline condition . . . 57
3.13 Simulation results of temperature change at baseline condition . . . 58
3.14 Wire model of the present oil-free rotary compressor developed in AMESim framework . . . 60
3.15 Positions of the piston and the vane at consecutive clockwise sequences of the compression from the upper deadlock to the end of the discharge . 61 3.16 Effect of phase delay (PHD) on clearance size between the vane and the piston (LP III). . . 64
3.17 Comparison of experimental and numerical results. Left: Indicator di- agrams. Vertical lines indicate the position of the sensor (θsens) and the outlet centreline (θout). Right: Corresponding model calculated mass flow rates at the inlet and across the leakage paths normalized to the ideal mass flow rate at different throttling conditions at 1200 r/min simulated
separately. . . 70
3.18 Comparison of experimental and numerical results. Left: Indicator di- agrams. Vertical lines indicate the position of the sensor (θsens) and the outlet centreline (θout). Right: Corresponding model calculated mass flow rates at the inlet and across the leakage paths normalized to the ideal mass flow rate at different throttling conditions at 1500 r/min simulated separately. . . 71
3.19 Comparison of experimental and numerical results. Left: Indicator di- agrams. Vertical lines indicate the position of the sensor (θsens) and the outlet centreline (θout). Right: Corresponding model calculated mass flow rates at the inlet and across the leakage paths normalized to the ideal mass flow rate at different throttling conditions at 1200 r/min simulated collectively. . . 72
3.20 Comparison of experimental and numerical results. Left: Indicator di- agrams. Vertical lines indicate the position of the sensor (θsens) and the outlet centerline (θout). Right: Corresponding model calculated mass flow rates at the inlet and across the leakage paths normalized to the ideal mass flow rate at different throttling conditions at 1500 r/min simulated collectively. . . 73
3.21 Evaluation of the effect of lowered clearance size. Left: Indicator dia- grams. Vertical lines indicate the position of the sensor (θsens) and the outlet centerline (θout). Right: Corresponding model calculated mass flow rates at the inlet and across the leakage paths normalized to the ideal mass flow rate at different throttling conditions at 1200 rpm simulated collectively. . . 74
3.22 Results of the sensitivity analysis model runs along with the regression lines . . . 75
3.23 Results of the sensitivity analysis model runs in nominal terms along with the regression lines. (Results are given specific to nominal conditions in Fig.3.23.) . . . 76
4.1 Pressurisation process within the rolling compressor cylinder . . . 80
4.2 Illustration of the course of a typical indicator diagram and its first deriva- tive. . . 81
4.3 Limit cycle fitting using combined VDP oscillator . . . 82
4.4 Intuitive way to form the desired limit cycle system [4] . . . 84
4.5 Results of polynomial fitting. . . 87
4.6 Results of empirical fitting . . . 89
4.7 Illustration of the regression quality in case of measured and CFD mod- elled Target processes . . . 90
4.8 Measured [5] and fitted indicator pressure at the discharge port,N RM SD= 0.0825 . . . 91
4.9 Measured [5] and fitted limit cycle . . . 91
4.10 The flowchart of defining the convergence function h(t) based on Schaal and Atkeson [6] . . . 93 4.11 Phase diagram of the target and the model start-up process,N RM SD=
0.0131 . . . 95 4.12 Effect of change in the initial value on the phase diagram,N RM SD0.85(p/p0)τP|0
= 0.1905 ,N RM SD1.25(p/p0)τP|0 = 0.3950 . . . 96 4.13 Effect of theδon the phase diagram,N RM SDδ=1.1δmin = 0.0159,N RM SD
δ=1.5δmin = 0.0378. . . 98 4.14 Phase diagram of the target and the model of tanh increment start-up
process, N RM SD= 0.0395 . . . 99 6.1 Representation of the meshing process: P piston centre,Ocylinder centre,
A node point on the piston, A0 radial projection of A on the surface of the cylinder and V and V0 are the edges of the vane on piston surface . . 107 6.2 The flowchart of the meshing algorithm . . . 108 6.3 Simplified flowchart of the mechanistical model . . . 109 6.4 The flowchart of defining the convergence function h(t) based on Schaal
and Atkeson [6] . . . 112 7.1 H´al´oz´asi folyamat szeml´eltet´ese: P a dugatty´u k¨oz´eppontja, O a henger
k¨oz´eppontja, A egy adott h´al´opont a dugatty´u fel¨ulet´en, amelynekA0 a henger pal´astj´ara es˝o radi´alis vet¨ulete ´esV ´es V0 a lez´ar´o elem ´elpontjai a dugatty´un. . . 115 7.2 Az ´ujonnan fejlesztett hal´oz´oalgoritmus folyamat´abr´aja . . . 116 7.3 A modell ´altal megval´os´ıtott algoritmus egyszer˝us´ıtett folyamat´abr´aja . . 117 7.4 K¨ozel´ıt˝o f¨uggv´eny meghat´aroz´asara kidolgozott elj´ar´as folyamat´abr´aja
Schaal ´es Atkeson [6] nyom´an . . . 120
1.1 Expected RPC performance data . . . 7
1.2 Main dimensions of the compressor unit . . . 8
2.1 Comparison between the estimated pressure drops calculated with the simplified 2D and 3D models . . . 27
2.2 Quality of the transition modelling (NRMSD) . . . 29
2.3 Volumetric efficiencies . . . 37
3.1 Instrumentation: list of applied sensors. . . 44
3.2 Result of the preliminary sensitivity study . . . 58
3.3 Summary of the leakage paths within the cylinder . . . 63
3.4 Summary of the optimization study results at different operating points . 67 3.5 Standardized quality parameters and the comparison between the quality of the individual and collective runs. . . 68
3.6 List of volumetric efficiency sensitivity analysis results on clearance sizes and piston-vane phase delay (PHD). . . 77
6.1 List of volumetric efficiency sensitivity analysis results on clearance sizes and piston-vane phase delay (PHD). . . 110 7.1 A volumetrikus hat´asfok ´erz´ekenys´egre a k¨ul¨onb¨oz˝o t¨om´ıt´esi r´esm´eretekre
´
es a dugatty´u ´es a lez´ar´o tolatty´u mozg´asa k¨oz¨otti f´aziselt´er´esre (PHD) . 118
xv
Acronyms
AC Air-Conditioning
ACO Amplitude Controlled Oscillator CFD Computational Fluid Dynamics CMO Compensated Morphed Oscillator CP Conversion Point
HV Hinged Vane IHP Inverse Heat Pump
LDM Layering Dynamic Meshing
MPI Message Passing Interface of Fluent MSE Mean Square Error
NFB Normalized Fractional-Bias NLH Non-Linear Harmonic NLO Non-Linear Oscillator
NMAE Normalized Mean-Average-Error
NRMSD Normalized Root-Mean-Square Deviation PHD Phase Delay
PR Pressure Ratio
RPC Rolling Piston Compressor SB Swinging Bush
TI Turbulence Intensity UDF User Defined Function VDP Van der Pol oscillator Symbols
˙
m mass flow rate, [kg·s−1]
˙
m∗ non-dimensional mass-flow rate, [−]
W diagonal matrix of the weights
X [2×n] matrix containing the location of the acquired data A cross-section, [m2]
a ratio of piston and cylinder radius (Rp/Rc), [−]
xvii
Ai polynomial regression coefficients aii linear regression coefficients b(θ) blending function
C spring stiffness, [N·m−1]
c1 coefficient of the independent variable term in the Van der Pol equation C2 inertial resistance factor, [m−1]
c2 coefficient of the first order term in the additional first order equation ck location of thek-th receptive field
cm flow parameter, [−]
cp Specific heat at constant pressure, [m2·s−2·K−1] cq flow coefficient, [−]
cv Specific heat at constant volume, [m2·s−2·K−1] D diameter, [m]
Dh hydraulic diameter, [m]
Dk distance metric of thek-th receptive field Dout discharge stub diameter, [m]
e eccentricity, [m]
Error error function, [kg2·m2·s4·r]
f laminar term,[−]
f rotational frequency, [s−1] f(θ) scaling function
H cylinder height, [m]
h specific enthalpy, [J·kg−1] h(r) local chord length, [m]
h(t) convergence function hc local channel height, [m]
ht heat transfer coefficient, [kg·s−2·K]
K number of the individual predictions (receptive fields) k thermal conductivity, [kg·m·s−2·K−1]
kF inertial permeability or Forchheimer constant, ,[−]
kv coefficient of viscous friction, [N·s·m−1] L pressurized pipe length, [m]
lf equivalent channel length, [m]
M Mach-number, [−]
m mass, [kg]
N rotational speed, [s−1]
n number of the acquired data points n∗ non-dimensional rotational speed, [−]
N u Nusselt number, [−]
p pressure [P a]
P r Prandtl number, [−]
Q heat, [kg·m2·s−2]
qV Volumetric flow-rate, [m3·s−1] R radius, [m]
r polar coordinate, radius
Rg specific gas constant, [m2·K−1·s−2] Re Reynolds-number, [−]
S Source term, [kg·m−2·s−2] s channel length, [m]
T temperature, [K]
t time, [s]
U internal energy, [kg·m2·s−2] U underlap, [m]
u specific internal energy, [kg·m2·s−2] ud discharge valve velocity, [m·s−1] V volume, [m3]
v specific volume, [m3·kg−1] v velocity, [m·s−1]
W work, [kg·m2·s−2] w cylinder width, [m]
w weight function
x streamwise coordinate, [m]
y discharge valve displacement, [m]
Greek Symbols
α permeability, [m2]
αc contraction coefficient, [−]
β smoothing coefficient
βk container ofbk andb0,k parameters of the linear regression for thek-th reception field
∆n porous zone thickness in the flow direction, [m]
δ zero crossing offset δv valve opening, [m]
δcl sealing clearance, [m]
ratio of the dead and stroke volumesVdead/Vst,p, [−]
η base cycle radius η efficiency, [%]
γ convergence ratio κ adiabatic constant, [−]
λ friction factor, [−]
µ viscosity, [kg·m−1·s−1]
µV DP coefficient of the second order term in the Van der Pol equation ω rotational speed, [s−1]
ρ density, [kg·m−3]
τ Time period, and blending function activation point, [s]
Θ limit cycle
θ crankshaft angle, [rad]
θ polar coordinate angle, [rad]
ζ pressure loss coefficient, [−]
ζs discharge valve damping, [kg·s−1] Subscripts
0 reference pressure
2D 2D (two dimensional) case 3D 3D (three dimensional) case B target function
S target function ad adiabatic b base line c cylinder
circ circular outlet stub cross section cl clearance
comp compressor cr critical cs compression
cyl cylindrical discharge port d discharge
dead dead volume disp displacement dn downstream eff effective e channel exit f viscous term h hydraulic I inertial term i sequential,i∈ <
in inlet
ind indicated value
k identifier of the individual predictions (receptive fields)
m motor me mechanical meas measured mot motor
op pressure drop from vena contracta and the outlet out outlet
p piston
pp pipe parameter pr pressurization section rect rectangular cross section
s suction
sens sensor sim simulated st stroke volume sw side wall t nozzle throat
throttle throttling imposed by porous zone up upstream
V volumetric
v valve
w wall Other Symbols
¨ second time derivative
˙ first time derivative matrix
ˆ prediction vector
f loor integer division rounding toward 0
Introduction
The utilization of geothermal heat sources for power generation is seemingly plausible for locations where significant sources of thermal water are available. Therefore, a new experimental model was designed and built by the Department of Energy Engineering (DEE) of Budapest University of Technology and Economics to investigate the feasi- bility of electricity production with the use of low enthalpy geothermal energy. The model process utilizes the Brayton thermodynamic process with the implication of a new compressor prototype unit. The major requirements for the compressor unit were to be able to work with untreated atmospheric air working fluid at given flow-rate and pressure ratio and to be able to be converted to expander mode without significant modifications. To fulfil the initial requirement, the rolling piston compressor (RPC) architecture (Fig.1.1) was used as the base of the new compressor prototype. One of themain advantages of the RPC architecture that combined with a feeding valve it can be effectively used as an expansion unit [7–9].
1.1 Rolling piston compressor
RPC’s are generally reffered to as stationary vane compressors, and are mainly applied in low capacity (<2kW) household refrigerator, air-conditioning (AC) and heat pump units [10, 11]. The RPC’s have gone through significant development since their intro- duction in the early 20th century and Ooi and Ya [12] estimated that by the end of 2015, 90% of household AC’s, typically with cooling capacity bellow 2 kW, were already equipped with RPC’s. The widespread interest in the use of the RPC’s in small AC’s from the early 1970’s is the consequence of their high efficiency, compact size, simple construction, smoother running and more quiet operation compared to the reciprocat- ing compressors at low capacity range [13–17]. The high volumetric efficiency is due to the small clearance volume, or often referred as dead volume, and correspondingly low
1
re-expansion losses inherent in their design. Common structure of a RPC, typical in AC units, is presented in Fig.1.1 and in Fig.1.2where the compression and suction volumes are also highlighted.
Figure 1.1: Schematic of a common RPC for AC units
Figure 1.2: Representation of the compression (Vcsand suction (Vs) volumes
The basic pressurization process within a stationary vane compressor is illustrated in Fig.1.3. As the compressor rotates clockwise, the volume of the compression chamber decreases. The subsequent pressurization continues until the pressure of the working fluid (pcs) reaches the discharge pressure (pd). Ideally at this point the discharge valve, which prevents the back-flow from the discharge port, opens and the pressurized working
(a) 0o (b) 90o
(c) 180o (d) 270o
Figure 1.3: Illustration of stationary vane compressor operation principle by schematic of the cross-section at different crankshaft positions. 0ocrank angle is defined where the center of the rotation and the center of gravity of the piston is aligned with
the contact point of the vane.
fluid is discharged from the compression chamber, as illustrated in Fig.1.3d. The suction takes place simultaneously with the compression, when the gas is drawn through a suction port into the suction chamber as its volume increases as it is highlighted in Fig.1.3.
The indicated pressure diagram (Fig.1.4) represents the indicated work, hence it also shows the hydraulic and heat transfer related losses during the compression process.
The corresponding volumes in Fig.1.4are the clearance volume (Vcl), the volume of the
aspirated working fluid (Vw) and the volume of the cylinder stroke (Vst). The additional work to the isentropic process comprises the following losses [1]:
• Wiredrawing loss: the work which is employed to drag the working fluid to the suction chamber from the ambient. Its value is proportional to the area of the indicator diagram which is belowp0
• Overcompression loss: the result of the excessive pressure required to overcome the resistance of the discharge port and valve to trust the working fluid into the following high pressure (pd) chamber. This work appears in the indicator diagram as the area above the discharge pressure pd.
• Re-expansion loss: arises by the pressurized working fluid in the clearance volume which re-expands into the suction chamber. The re-expansion loss is proportional to the area bounded by sections connecting the points 2s−2−30−3sin Fig.1.4
• Leakage loss: depending on the construction part of the compressed working fluid leaves the compression chamber across the sealing clearances imposing compression and heating losses. In case of hermetic compressors, the working fluid can also re- enter to the compression and suction chambers since the compressor is mounted into the high pressure vessel. The leakage into the suction chamber decreases the volumetric efficiency
• Heating loss: is the result of the heat transfer across the compressor shell and also the heat transferred by the leakage flow. This loss results in deviation from the adiabatic process, what is represented by the curve connecting the points 2−30, to the real process, which is illustrated by the section of the indicator diagram between the points 2 and 3.
1.2 History
The first mention of a stationary vane rotary machine, which was found in the literature, is a patentent of John Yule [18] form 1836. The presented steam turbine applies the same principle but implements the reversed cycle compared to the RPC’s. In a more recent study Okur and Akmador [19] and Okur and Arabaci [20] also apply a stationary vane rotary expander to realize the Brayton cycle in a cost efficient way, since the manufacturing cost of an RPC type unit is significantly lower than the cost of a dynamic compressor or turbine. Similarly, Zheng et al. [21] apply a rolling piston expander (RPE) to realize an efficient organic Rankine cycle. RPE’s can also be applied efficiently for
Figure 1.4: Illustration of a typical RPC indication diagram [1]
transcritical refrigerator cycles to regain energy from the gaseous refrigerant during the throttling process [22]. Early design of an RPC is introduced by Jaworowski in 1922.
Compared to the common RPC architecture (Fig.1.1), Jaworowski’s compressor applied a swinging vane instead of a sliding vane, and the discharge port was inserted radially and not axially, as in most hermetic RPC’s in AC units. Essential drawback of the Jaworowski design is that the positioning of the swinging vane is not supported by discharge pressure, as in the Yule engine and in most of the common RPC’s.
The stationary vane compressor went through significant development since its introduc- tion, which results in various modifications of the base design to achieve better efficiency and improve the applicability of the unit. Beside the beneficial qualities of the base- line architecture, basic RPC’s (Fig.1.1) have some considerable drawbacks. To maintain sufficient seal between the compression and suction chamber the vane has to be pressed against the piston surface at relative high force which results in increased overall friction.
The friction is also significant on the rest of the sliding surfaces as the result of the high relative velocity difference and the high extension of the sliding surfaces [12].
In order to quantify the potential in RPC design a brief summary of the published efficiency data from the literature is presented in Table 1.1. The volumetric, mechan- ical, compression and compressor efficiencies are defined by the following correlations, respectively [23]
ηV = m˙d nRp0
gT0Vdisp (1.1)
Where ˙md denotes the discharge mas flow rate, n the rotational speed, p0 the total pressure at the inlet, T0 the total temperature at the inlet, Vdisp the displacement vol- ume. Neglecting heat transfer and the leakage flow, the theoretical volumetric efficiency (ηV,T heory) can be calculated as it is shown in1.2
ηV,T heory = 1− pd
p0
κ1
−1
!
(1.2) Where the ratio of the dead volume Vdead and the piston stroke Vst,p is given as = Vdead/Vst,p.
Mechanical efficiency (Eq.1.3) gives the ratio on which the motor work can be turned into compression work.
ηme = Wind
Wmot (1.3)
The compression efficiency (ηcs, Eq.1.4) is the ratio between work done on the gas by the real compressor (Wind) and the work could be done on the same working fluid in case the process was irreversibleηcs.
ηcs = Wad
Wind (1.4)
To estimate the total efficiency, the motor efficiency ηm has to be taken into account, which is the rate the motor can transform the input electrical energy into shaft work.
The total efficiency is can be than calculated by Eq. 1.5.
ηcomp=ηmηmeηcs (1.5)
Table1.1summarizes the efficiencies presented in the literature. The results of volumet- ric efficiency scatter between 74 % and 98 % in the assessed literature. In the presented studies the low volumetric efficiencies are the consequences of operating the compressor outside the design conditions. Nevertheless, the expected volumetric efficiency lies above 95 % in case of appropriate design.
It has to be noted that the definition mechanical efficiency based on the literature is not straightforward, since the friction losses from the driving unit and the standalone compressor cannot be unambiguously differentiated. The 71 % mechanical efficiency presented by Ooi [26] is probably a more feasible prediction compared to efficiencies
Table 1.1: Expected RPC performance data
Reference
Author
Volumetriceff.ηV[%] Mechanicaleff.ηm[%] Compressioneff.ηcs[%] Compressoreff.ηcomp[%]
[24] Noh et al. (2016) 75-97 55-85
[25] Park (2010) 74.1-94.3
[26] Ooi (2004) 70.95
[27] Kim et al. (2001) 89 - 98
[23] Sakaino et al. (1984) 98.1 92 86.2 52.6 [13] Matsuzaka et al. (1982) 94 93.3 94.8 72.4
[1] Wakabayashi et al. (1982) 92.5 77.6 56
[28] Shiga et al. (1978) 85.7
above 90 %. The values of presented compression efficiencies also vary significantly between 78 and 95 %. However, all the presented compression efficiency estimations are claimed to be based on experimental data with the use of R22 refrigerant.
1.3 The new compressor design
To reach the expected volumetric efficiency (ηV), of above 95 %, RPC’s use additional lubrication, which both reduces the friction range and seals the gaps within the cylinder, thus prevent excessive leakage through the seal clearances. The lubrication requires an adequate lubrication system therefore increases the complexity of the construction and also utilizes part of the shaft work input to maintain proper lubricating oil recirculation.
In case of AC systems, the lubricant degrades the quality of the refrigerant, therefore reduction or elimination of the necessary amount of lubricant within the displacement volume is desirable. Also, some particular applications require complete elimination of any oil contamination, e.g. vacum tube manufacturing, environmental restrictions, etc..
In case of environmental requirements neutral lubricant can be considered as possible alternative. Delmot [29] applied water lubrication for a shipboard, multi-stage, high pressure air RPC system in addition to the use of non-metallic bearings and injecting water directly into the suction chamber at the compressor inlet.
The conventional RPC architecture must be modified in order to accomplish efficient compressor operation in case of the complete absence of fluid lubricant within the com- pression chamber. Stoltz [30] developed a Swinging Bush (SB) type stationary vane design for vacuum tube manufacturing, where oil contamination is inadmissible. Very
Table 1.2: Main dimensions of the compressor unit
Part name (Fig.1.1) Dimension [mm]
Cylinder radius,Rc 50
Cylinder width,w 40
Eccentricity,e 10
Piston radius,Rp 40
Design sealing gap clearance size,δ <0.01
similar architecture along with the Hinged Vane (HV) design [19] was proposed by Wu and Chen [31] for oil free operation in AC RCP’s. In case of SB and HV designs friction force and the relative velocity between the connecting parts are reduced e.g. less or no liquid lubricant needed, while still maintaining stable operation. The applied vane mounts in SB and HV compressors also reduce significantly the leakage flow.
The RPC based novel architecture (Fig.1.5) introduced in Farkas et al. [32] is also designed to minimalize leakage flow in oil-free operation. The new model compressor was designed within the framework of Research and Technological Development Fund (KMR-12-1-2012-0199) [33]. The main dimensions of the unit is listed Table1.2. In this design the vane is attached to a pivoted beam and performs swinging motion around the pivot during the operation similarly to the construction introduced by Jaworowski [34]. However, in the present case the vane is actuated by a linkage mechanism which is directly driven by the crankshaft to keep constant clearance between the vane and the piston [35], regardless the rotation speed without using extensive constrictive force and increasing the friction induced losses. The piston is solidly mounted on the crankshaft and does not roll along the inner surface of the cylinder. This design makes it similar to the trochoidal compressors (e.g. the Wankel engine) which, by theory, do not require small and expensive manufacturing tolerances because of the closed sealing border of the compression space and neither do they need oil for sealing [10].
During the test period the initial swinging vane compressor (SVC) prototype geometry was modified to improve the reliability and decrease the wire-drawing loss. The solid pivoted supporting beam (Fig.1.5) was replaced first by a hollow support beam (Fig.1.6).
In case of the hollow support beam architecture the flow is allowed to pass across opened beam structure and also the location of the inlet has been repositioned, as it is shown in Fig.1.6, compared to the original design.
Finally the hollow beam was replaced by a narrower and lighter support beam (Figs.
1.7and 1.8) which was initially applied to facilitate the manufacturing.
The modified construction resulted in lowered wiredrawing loss by the intake. It also increased the volumetric efficiency by reducing the tangential extension of the inlet slot downstream of the vane support. The mechanical efficiency is expected to be enhanced
Vane
Pivoted beam
Piston
Inlet Outlet
Figure 1.5: Initial prototype geometry
Inlet Outlet
θ
Hollow beam supported vane
(a) Cross-section of the compressor
(b) Hollow beam supported vane
Figure 1.6: Schematic of the compressor with hollow beam vane support
Narrow beam
Vane Inlet
Outlet
θ
Figure 1.7: Schematic of the final geometry with narrow support beam
Figure 1.8: 3D schematic of the compressor equipped with narrow support beam
by the lighter weight of the vane support beam, hence lower inertia, which results in lower required power for actuating the swinging motion of the vane. The inlet and outlet are kept to be radial because of installation requirements.
Transient CFD simulation of a stationary vane, oil free, rotary compressor
The substantial modification of the baseline RPC architecture necessitates a thorough investigation to validate the expected performance. Therefore, parallel to the experi- ments, conducting a CFD study was performed, which enables a more detailed view of the working process.
This chapter presents the results of the CFD simulations performed on the new SVC.
Section2.1provides an extended literate review of the previous works. Section2.2then details the basic set-up of the CFD simulation. Following Section 2.3discusses a novel method to achieve adequate preliminary performance estimations by using a reduced, hence faster 2D CFD model instead of the costly 3D simulations. Section 2.4 presents the CFD study on the effect of the discharge valve geometry and position. The results of the CFD studies are summarized in the end of the chapter in section 2.5.
2.1 Assessing performance by CFD
It is known that actualp−V diagram from a physical system differs from the ideal ther- modynamic cycle (see also in Section1.1). The deviation of each of the thermodynamic processes of suction, compression, discharge and expansion, from the ideal, can be linked to fluid mechanical reasons. For example, the most important process of compression is affected by leakage, unsteady heat transfer across the boundaries of the compression volume and viscous shear at the boundaries. To identify and quantify these losses, CFD is known to be an economical tool during evolutionary design process [36]. The use of CFD has become a common practice even in case of volumetric machines since the end of the 90’s. CFD models are used both to access performance data for design, prior to the experiments and also to supplement the experimental. Validated CFD model can be used for parametric studies in order to find the optimum design.
11
2.1.1 Literature review
There are CFD solver packages which are specifically tailored to evaluate common com- pressor and turbine designs, e.g. PumpLinx [37,38]. Other commercially available and open source CFD solvers with dynamic mesh options can be also applied to simulate volumetric machine operation, e.g. STAR-CD [39–41], ANSYS CFX [42], ANSYS Flu- ent [32, 43], etc.. While the specially tailored CFD programs allow easy set-up and faster evaluation, the more general CFD programs are more versatile, i.e. can model architectures which differ from the conventional compressor designs.
In 2004, two papers were presented at the International Compressor Engineering Con- ference in Purdue, by the same company, introducing a parametric study, using CFD simulations performed in STAR-CD, related to the effect of the design of the notch by Geng et al. [39] and suction piping by Liu and Geng [40] on the noise and efficiency for a given double discharge compressor architecture. The applied meshing techniques were not discussed in detail, but figures presented in [39] show a quadratic mesh within the cylinder with an extended region of highly skewed elements around the piston-cylinder gap and close to the vane. Despite these anomalies, the results resembled well the theory and in the work by Liu and Geng [40], where experimental tests were also conducted, the numerically predicted efficiency also aligned well with the test results and also verified the theory. The results also predicted accurately the trends in the change of performance parameters as the geometry was modified.
In a study by Shebing et al. (2010) [41], two different kinds of RPC’s were investigated which utilized pressure activated discharge valves. For the solution of the fluid dynamics and the mesh deformation, the same solvers were used as in the above mentioned studies ([40] and [39]), but no further information was detailed either. The predictions were also compared to experimental results, with an error of less than 7% for the cooling capacity and power. Thep−θdiagram of the simulation result is also claimed to be very close to the experiment, although evident discrepancies can be observed on the presented graphs.
Hui Ding et al. [37] in 2014 presented a new approach for simulating a discharge reed valve. The PumpLinx CFD package was used for the simulations which is specially targeted for simulating volumetric machines. According to the published figures in [37], the applied meshing and re-meshing algorithms result in a good quality mesh within the whole computation domain, where no highly distorted elements are evident. Unfortu- nately, no further details about the applied meshing methods were provided. Impres- sively, a whole revolution was claimed to be simulated within five hours on a general purpose quad-core Intel Xeon CPU at 2.67 GHz machine. The predictions seemed to
be aligned well with the theory but we have to resent that they were not compared to experimental test results.
A simplified porous media approach was used to model the leakage flow in the 3D CFD model for a rotary vaned expander presented by Gianluca et al. [44] in 2014. The model allowed to estimate the effect of the leakage flow on volumetric efficiency. The results claim to show encouraging agreement with the experimental data and justify the suitability of the applied real gas approach, emphasizing that the resistance coefficient used for modelling of the leakage flow must be calibrated. Also, the authors remark that despite the agreeable results, the applied mesh resolution should have been at least one order of magnitude finer than the one was used but it would have significantly increased the computational cost.
Although, the use of deforming meshing seems to be unavoidable for the CFD modelling of an RPC, in the study of Brancher et al. [45], the effect of the rotating rolling piston on the suction and discharge losses are estimated by steady 3D CFD solutions at different crankshaft angles. The predicted effective flow area and effective force area coefficients are implemented into a lumped parameter model. As a result significant improvement in the performance estimation is confirmed by experimental data.
2.2 Numerical set-up
2.2.1 Dynamic mesh
During the rotation of the piston, the volume of the suction chamber increases while the compression chamber volume decreases. Necessarily, the mesh must be also adapted to the change the volume and shape of the computational domain.
Block dynamic mesh
In case of conventional design it is enough to only deform the mesh without changing the number of mesh elements [32, 37, 39–41]. However, the related literature did not reveal any details about the applied meshing methods and some of the presented figures of the mesh show distorted elements in critical regions, which is discussed in details in Section2.1.
A new meshing method was developed to facilitate the discretization of the deforming numerical domain within the cylinder. The new discretization method is presented in Fig.2.1. The mesh contains only hexahedral elements. The position of the mesh nodes within the cylinder are controlled by an external User Defined Function (UDF) using
designated macros provided by ANSYS Fluent. The re-meshing algorithm is illustrated in Fig.2.2 and the resulted change in mesh is presented in Fig.2.1. During the rotation of the piston, the axial position of the nodes do not change, which means that the nodes do not move in the direction parallel to the crankshaft. The motion of the nodes on the piston resembles a rigid body-like motion, therefore point P, the center of the piston, point V, the bottom point of the vane on the piston and pointA which is an arbitrary node point on the piston surface retain their position with respect to each other at each instant of the piston motion.
This rigid body motion is described by the motion of the imaginary O-K axle which is part of the simpleO-P-K crank mechanism where pointO is the center of the cylinder and also defines the position of the axis of the crankshaft. The crankshaft angle is defined by the angular position of theO-Psection which rotates around pointO. The movement of pointV is restricted to the vertical direction which also means that the blade keeps its vertical position during the rotation of the cylinder. This motion resembles the motion of a piston in a hinged vane compressor [46].
The position of an arbitrary A’ node point on the cylinder surface is than defined as the projection of point A from the piston surface along the straight O-A-A’ line. The rest of the points between A and A’ are distributed equally within the straight A-A’
section, although the distribution can be changed according to the purpose. This equal distribution is optional and not restricted by the method. Also the method can be adapted to different blade thickness. With using the same settings discussed in the following section the simulation of one revolution took approximately twelve hours on the same architecture using just one core of the quad-core processor. The total number of cells equals 2×104, which remains constant throughout the simulation.
V V'
A' A
O P K
(a)θ= 0o
V V' A
A'
O P θ K
(b)θ= 135o
Figure 2.1: Illustration of the new meshing process applied for a conventional RPC.
P is the center of the piston, O is the center of the cylinder, A is the node on the piston surface and A’ is its O centred radial projection to the cylinder surface and V and V’
are the edges of the vane (Fig.1.1)
Figure 2.2: Schematic of the new re-meshing algorithm realized by a UDF in ANSYS Fluent
Figure 2.3: Section of the block mesh in the narrowest gap area
Re-meshing dynamic mesh
However, in case of the SVC (Fig.1.5,1.6and 1.7), the movement of the swinging vane mechanism and the piston result in complex change of the geometry which cannot be followed by straight forward deforming mesh algorithms. In this case deforming and re-meshing methods must be used in parallel. For the 3D simulations the triangular elements are extruded in the axial direction, forming wedge elements (Fig.2.4) which allows using the 2.5D re-meshing approach.
Figure 2.4: 3D wedge mesh
In the 2.5D model the sidewall surface mesh, which can only contain triangular elements, is modified only on one side wall and the changes are then projected to the opposite side.
The node points on the cylinder remain stationary and the nodes on the piston and vane surfaces perform rigid body like motion. This means that they keep their position on the surface with respect to each other while the piston and the vane are rotated along their axes. To control the node points on the base surface, spring/Laplace based smoothing and re-meshing algorithms are used. Both the smoothing and re-meshing algorithms are part of the Fluent’s inbuild dynamic mesh models. In case the re-meshing, as the distortion of the cells increases by the large scale movements Fluent agglomerates cells that violate the user defined skewness or size criteria and locally re-meshes the region of the agglomerated cells. If the new cells or faces satisfy the skewness criterion, the mesh is locally updated with the new mesh elements and the solution interpolated from the former cells. Otherwise, the new cells are discarded and the former cells are retained [47].
This method results in relatively easy model set-up since only the rigid body motion of the moving parts has to be predefined by the user with the use of external UDF’s and the rest of the meshing is automatically handled by Fluent’s built-in algorithms. The initial mesh is created by the use of ANSYS Workbench Mesher. The maximum size of the cells in the tangential direction on the surfaces are limited by the gap between the moving elements to gain appropriate mesh quality within the small clearances.
Hybrid mesh
To combine the advantages of the above described re-meshing and block dynamic mesh methods a hybrid mesh was set up. The stroke volume was divided into two distinct parts, the re-meshing and deforming mesh domain, as shown in Fig.2.5. The connections between these two parts are conformal. The change of geometry and node distribution on the interface between the parts has to be defined explicitly by the user via a UDF. Fig.2.5 shows the mesh in an intermediate stage. Unfortunately though, the simulations with the hybrid mesh cannot be run in parallel. The use of Fluent inbuilt re-meshing algorithm alongside with the UDF which defines the position of the nodes at the connecting region results in interference with the Fluent Message Passing Interface (abbreviated as MPI in the Fluent user interface). Although, the hybrid dynamic meshing method allows better control on mesh quality and better resolution at lower cell count, this method was currently dismissed in favour of the fully re-meshing method.
2.2.2 Porous media model
To simulate the restriction imposed by the narrow flow channels accross the sealing gaps, localy porous layers are defined within the numerical domain as discribed in details in Sections 2.2.3 and 2.3. In case of porous layer models, the momentum equation is modified by the introduction of a Darcy-Forchheimer type source term. The source term (Eq.2.1) is also implemented into Fluent [47].
Si =− µ
αvi+C2
1 2ρ|v|vi
(2.1) Whereµis the viscosity andαis the permeability of the porous media,C2 is the inertial resistance factor and |v| and vi are the absolute magnitude and the magnitude in the i-th direction of the velocity, respectively. In case of laminar flow the pressure drop across a homogeneous porous zone is proportional to the velocity. The pressure gradient can be expressed by the use of Darcy’s law as Eq.2.2[47,48].
∇pf =−µ
αv (2.2)
Deforming mesh domain Re-meshing domain Gap in the re-meshing
domain
Gap in the deforming-mesh
domain
Figure 2.5: Hybrid mesh for the hollow beam design
In order to model the effect of inertial losses at higher Reynolds number where the flow cannot be considered to be laminar any more, the Forchheimer term Eq.2.3can be added to Eq.2.2, where kF is inertial permeability or Forchheimer constant [49].
∇pI =− µ kF
v2 (2.3)
In Fluent the Forchheimer formula was modified in the form of Eq.2.5in order to get the pressure drop from the inertial loss across the porous model in function of the dynamic head. Hence, the source term (Eq.2.1) can be decomposed into the viscousSf,i(Eq.2.4) and the inertialSI,i (Eq.2.5) parts which also can be handled independently within the Fluent interface [47].
Sf,i=−µ
αvi (2.4)
SI,i =−C21
2ρ|v|vi (2.5)
2.2.3 Leakage flow
To model the sealing clearance flow, the clearance channels must be discretized at ade- quate resolution. Since the clearance channel height is several magnitudes smaller than the overall dimensions of the numerical domain (Tab.1.2) , this resolution would result in an infeasibly large cell count, if it was applied for all the mesh cells throughout the domain. Hence, the clearance channels are meshed with low resolution, allowing only one mesh element along the channel height (see Re-meshing domain in Fig.2.5). Because of meshing considerations, a gap must be kept between the connecting parts at the seal- ing, as depicted in Fig.2.5, hence direct connection between the sliding surfaces cannot be modelled. The deforming mesh-grid provides a connected domain of constant area, therefore none of the parts of the flow domain are isolated from the rest. The restriction by the friction imposed by the walls of the narrow channels are modelled by the addition of a momentum source term, whereby Darcy’s law (Eq.2.2) was applied assuming that across the narrow channel the leakage flow remains laminar. The desired pressure drop can be set for a predefined porous layer thickness (∆n) by Eq.2.6 [47].
∆pf =−µ
αvi∆n (2.6)
By adjusting theαpermeability in Eq.2.6at constantµviscosity, the desired restriction can be set. By imposing the porous term, the channel height does not have to match the exact value of the real geometry, i.e. the channel height can be increased keeping the imposed restriction at the same level by increasing the porous term. Since the channel height defines the mesh size, the size of the cells within the mesh can be also increased with the increase of the channel height which results in lower computational cost. For the current case, the highest possible porous resistance was modelled to simulate enhanced sealing, i.e. best possible compressor performance.
The center of the porous region is located at the theoretical contact point and the extension of the porous region is limited to a couple of cells. The porosity is controlled by a UDF which marks the cells located within a given radius around the contact point, and sets the porous resistance (1/α) in these cells to a predefined value. The porous resistance in the rest of the domain outside the porous regime is re-set to be zero by the governing UDF.
2.2.4 Solver
For the simulations the density based Fluent solver was selected as the expected change in density is significant. The use of double precision arithmetic is also necessary for the SVC simulations, because the description of the temporal position of the moving parts required enhanced accuracy. Rounding errors in the definition of the moving parts momentary position tend to accumulate during the solution and eventually result in collision of the parts.
The turbulence behavior was resolved by the realiziblek−εturbulence model, which was proven to be robust and reliable in similar cases [37, 50,51]. For spatial discretization second order upwind methods were applied along with computing the gradients by least square cell-based method. Because of the moving mesh, only first order implicit transient formulation was applicable in Fluent.
2.2.5 Boundary conditions
The boundary condition at the inlet and outlet were set to constant pressure. The bounding surfaces were set to be adiabatic, i.e. the heat transfer across the bounding walls was not modelled, since the transferred heat via the leakage flow expected to be significantly dominant over the heat exchange across the bounding walls. Constant ambient temperature was also fixed at the inlet boundaries.
2.3 A simplified modelling approach for rolling piston com- pressor
In order to characterize the compressor performance, experimental tests were conducted where the loading of the compressor was set by a throttling valve. To numerically describe this set-up the throttling valve was modelled by the viscous resistance part (Eq.2.6) of the porous source term assuming that along the narrow channels the flow remains laminar.
The discharge valve was not modelled at this point of the study, since in this case, no back-flow from the discharge port was expected as the pressure ratio and the volume of the cavity between the throttling valve and cylinder outlet were both moderate.
