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Theoretical modeling and experimental investigations for the

improvement of the mechanical efficiency in sliding vane rotary

compressors

Article  in  Applied Energy · January 2015

DOI: 10.1016/j.apenergy.2014.12.055

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Theoretical modeling and experimental investigations for the improvement of the mechanical efficiency in sliding vane rotary compressors

Giuseppe Bianchi, Roberto Cipollone∗

University of L’Aquila, Department of Industrial and Information Engineering and Economics, Via Giovanni Gronchi 18, 67100 L’Aquila, Italy

Abstract

Positive displacement compressors lead the market of compressed air production for industrial applications.

Among them, sliding vane rotary compressors represent an energetically virtuous alternative to the current

compression technologies. In the present work, the effects of compressor design parameters were investigated

through a comprehensive approach that aimed at addressing more efficient machines to promote sliding vane

compressors as the key enabling technology in compressed air systems. A comprehensive mathematical model

was developed to study the main phenomena occurring in this kind of compressors. The model provides the

cell volume evolution over a whole rotation during which filling, compression and discharge processes occur.

The first and latter phases are described by the quasi-propagatory approach that represents the inertial,

capacitive and resistive features of one-dimensional unsteady flows. The dynamics of the compressor blades

led to four different arrangements inside the rotor slots while an analysis of the hydrodynamic lubrication

established between blade tip and stator wall focused on the oil film thickness evolution to prevent dry

contacts. An extensive experimental campaign on a mid-size industrial compressor allowed the model

validation at different outlet pressure levels and revolution speeds using a direct measurement of mechanical

power and the reconstruction of the indicator diagram from piezoelectric pressure transducers. The friction

coefficient at the contact points between blades with stator and rotor was estimated in 0.065 and further

improvements of the mechanical efficiency were eventually addressed considering the roles of compressor

aspect ratio, revolution speed, and blade tilt. The first two theoretical optimizations might lead to an

increase of the compressor efficiency of 2 and 9 percentage points respectively. On the other hand, acting

on the blade tilt would not produce relevant improvements.

Keywords: sliding vane rotary compressor, positive displacement compressor, compressed air systems,

indicator diagram, piezoelectric pressure transducer, mechanical efficiency

∗Corresponding author Email addresses: [email protected] (Giuseppe Bianchi), [email protected] (Roberto

Cipollone)

Preprint submitted to Applied Energy March 5, 2015

1. Introduction1

Energy saving is today recognized as the primary action to accomplish energetic and environmental2

commitments of all the Countries in the World. CO2 concentration in atmosphere is a crucial concern,3

universally recognized as the tomorrow’s main challenge. Therefore, a new energy paradigm based on a4

sharp reduction of the energy consumptions and a renewable-based energy production is sought by many5

scientific and institutional contexts.6

Compressed air accounts for a mean 10 % of the global industrial electricity consumptions (7850 TWh in7

2012) which reaches a share close to 20 % if commercial and residential needs are included [1, 2, 3]. Facing8

this issue implies the employment of many saving measures with actions upstream and downstream of the9

compressed air production: pipeline leakages reduction, adjustable speed drives, optimization of the end use10

devices, etc. In addition, with reference to electricity consumptions in compressed air systems, the saving11

potential related to compressor technology has been estimated 25-30 % [1]. From an economic point of view,12

although energy costs for compressed air are predominant with respect to the capital ones (70-75 %) [2], an13

investment increase of 10 % in a 10 year operating period would be feasible only if the compressor efficiency14

increase was 4 % greater than the former technology [3].15

In industrial applications, rotary volumetric compressors are able to match flow rate and pressure level16

requirements with an electrical power range from a few to several hundred kilowatts. Among them, Sliding17

Vane Rotary Compressors (SVRC) revealed a better energetic behavior whether on/off load conditions were18

taken into account [3]. This operating regime is a common process to match the line pressure needs in terms19

of flow rate: when the air demand decreases, thanks to the automatic depressurization of the machine and20

some intrinsic features related to the sealing among vanes (e.g. absence of the so called ”blow hole line”),21

SVRCs accomplish in an efficient way a process that is usually energy wasteful.22

In the literature, sliding vane compressors have been investigated both theoretically and experimentally.23

The machine geometry and vane kinematics were modeled following a trigonometric approach [4, 5]. Circular24

and elliptical stator configurations to achieve higher volumetric ratios or even dual stage compressions were25

considered [6, 7] as well as slanted blade arrangements inside the rotor slots [8, 9]. Preliminary studies26

assumed suction and discharge processes as isobaric while the compression was modeled using semi-empirical27

formulations [6]. However, zero dimensional models for the cell thermodynamics were also developed based28

on the energy conservation to predict the pressure evolution over the whole cell rotation [5, 9, 10, 11].29

To detail the friction power, the contribution due to the blade dynamics was widely investigated [12, 13].30

Comprehensive analyses that involve secondary contributions such as friction at the bearings and on the31

side covers of the machine [14, 15] or the blade tip profile [16, 17] were also developed. Leakage paths32

models were presented assuming clearances as orifices [18]. As concerns the experimental activities, tests at33

different steady conditions [19, 20, 21] and on unconventional configurations like the blade tilting [22] were34

2

carried out. Experimental methodologies have been set up to measure the pressure inside the compressor35

cells [23, 24, 25] while theoretical approaches as the Helmotz’s resonator one supported the discussion of the36

results [26, 27].37

Although it deeply focused on specific aspects of the sliding vane compressor technology, the above38

mentioned literature does not strictly aim at maximizing the overall machine efficiency. This goal is of39

major concern for sliding vane rotary compressors since the benefits that would come out from an energy40

performance enhancement might make them as the key enabling technology in compressed air systems. At41

the moment, there are air flow ranges in which SVRCs are characterized by specific energy consumptions42

lower than other compressors [3]. The reinforcement of this issue and the broadening of their behavior at43

wider flow ranges appear the way to refer to these machines as premium ones concerning the energy con-44

sumptions. Furthermore, compared to other compression technologies, SVRCs have a relevant improvement45

potential whose development is referred to friction power reduction and to the internal air cooling during46

the compression phase [28, 29]. Despite similar methodologies are not currently available in literature, in47

order to identify the energy saving potential, a wide-ranging approach becomes necessary to understand how48

the different design parameters can contribute to the machine efficiency. A comprehensive model appears49

to be the best way to observe cross coupled effects among different design parameters. Once experimentally50

validated, it behaves as a SVRC virtual platform allowing the addressment of efficiency improvements and,51

at the same time, the reduction of prototype construction.52

The model presented in the current work is composed of a geometrical section that is able to provide the53

cell volume evolution for any compressor layout in terms of aspect ratio, blade tilt, intake and exhaust ports54

arrangements. A quasi-propagatory approach described the flow dynamics during the vanes filling and emp-55

tying while a lumped parameter model was adopted to predict the pressure evolution over a whole rotation56

of the compressor cells. The study of the blade dynamics coupled with an analysis of the hydrodynamic57

lubrication at the blade tip provided the oil film thickness evolution at the interface with the stator wall.58

The role of reaction forces and pressure distribution along this oil layer is of major importance to account59

for losses by friction, thus to a proper investigation of the mechanical efficiency of the machine.60

An experimental campaign on a mid-size industrial sliding vane compressor was performed at different outlet61

pressure levels and revolution speed regimes. The direct measurement of mechanical power, as well as the62

reconstruction of the indicator diagram from pressure data provided by piezoelectric transducers, allowed to63

validate the model and led to an estimation of the friction coefficient. The pressure distribution along the oil64

layer at the blade tip was further calculated. Parametric analyses on the effects of aspect ratio, revolution65

speed, blade tilt and mass on the mechanical efficiency of the compressor, were eventually carried out in66

order to identify improved configurations that would reduce the energy consumptions.67

3

2. Mathematical model68

The main phenomena occurring inside a sliding vane rotary compressor were modeled through an es-69

sential formulation: the sliding vane compressor from the suction port till the compressed air line inlet70

was represented as a sequence of ducts and capacities that exchange mass and energy according to a 1D71

quasi-propagatory formulation. The compressor core, i.e. the stator rotor and blades assembly, was modeled72

with a lumped parameter approach. Hence, the compressor cell acted as a volume of variable capacity that73

interacted with the adjacent cells and with the rest of the compressor through the suction and discharge74

ports. The solution of the momentum equation for the compressor blades provided an estimation of the75

friction power that directly affects the machine efficiency. The lubrication circuit modeling and the cal-76

culation of the load distribution at the blade tip eventually allowed to estimate the pressure distribution77

and the oil film thickness evolution along contact surface with the stator. Since the oil sealing in sliding78

vane compressors is effective, leakage flows modeling (for instance using semi-empirical correlations) was not79

pursued; this assumption allowed to keep the model platform independent from the compressor tested, thus80

able to simulate the performances of different machines.81

2.1. Compressor geometry82

The compressor vane geometry was modeled tracking the trajectories of all the relevant points in the83

compressor core represented in Figure 1.a. To investigate the effects of the blade tilt onto the mechanical84

efficiency of the machine, backward and forward slanting were also implemented (Figure 1.b). In these85

configurations, blades are tangent to an internal circumference whose radius (OrotA) is smaller than the86

rotor one (OX). As concerns the intake and exhaust ports, they are located either frontally (on the covers,87

with an axial air path) or circumferentially (with a radial air admission or delivery). Regardless of the88

configuration, the angular ports positioning has a direct influence on the volumetric compression ratio i.e.89

the ratio between the cell volume before it opens toward the exhaust and the one after closing the inlet90

port. Although the model can consider ports anyway located, in current industrial compressors the intake91

is usually axial while the discharge is radial, as reported in Figure 1.c.92

2.2. Vane filling and emptying93

In order to evaluate the flow dynamics during the intake and exhaust processes, a Quasi Propagatory94

Model (QPM) was adopted. Based on the 1D unsteady conservation equations, the method solves them in95

an analytical way taking into account the capacitive (pressure as a function of inlet and outlet mass flow96

rates), inertial (mass flow variation as a function of a pressure difference) and resistive features (losses by97

friction and heat exchange) of the flow topology [30]. QPM subdivides complex one dimensional sequence of98

pipes anyway interconnected into elementary parts, whose extremities are stimulated by boundary conditions99

(BC) modeled as functions of pressure and flow velocity. The intersection of the curves representing the100

4

Figure 1: Compressor core - the geometrical modeling is based on the trajectories of the relevant points of the assembly (a)

and may take into account forward and backward blade tilting (b) as well as axial and radial ports (c)

upstream and downstream BCs in the p−u plane gives the steady state (SS) of the flow after the transient101

(Figure 2.a).102

To model the dynamics of homentropic flows only the mass and momentum conservation equations are103

required. The homentropic steady state may be eventually corrected if heat exchange and friction occur104

[31]. Considering the mid-point of the elementary pipe, the (p,u) solution over time is represented by the105

sequence of values which result from the pressure (and speed) waves propagation calculated as intersections106

between the characteristic curves entering the BCs and the BCs themselves. The novelty of the QPM lies on107

the approximation of the boundary conditions with lines at constant slope (evaluated at the initial state):108

this allows to linearize and speed up the calculation such that the flow evolution at a given location can be109

characterized with the Equations 1:110

p = pu − pu −p∞ u∞

u = pu −Au (1a)

p = pd − p∞ −pd u∞

u = pd + B u (1b)

The flow dynamics inside the pipe starts from the initial state 0 and proceeds along the characteristic curves111

bounded by the BCs: 0-1 represents the propagation of the positive characteristic curve while 1-2 is related112

to the propagation of the negative characteristic curve until the SS, represented by the point (u∞,p∞) in113

Figure 2.a, is reached. The time base is given by the time the waves require to move between successive114

intermediate states and it is equal to the ratio of the pipe length over the local speed of sound a.115

5

Figure 2: Quasi-propagatory flow dynamics along the suction and discharge compressor ducts

With reference to the initial value of pressure, the slope of the characteristic curves C = γ p′/a and the116

linearization of boundary conditions lead to the definition of the parameter Λ:117

Λ = (C −B) (C −A) (C + B) (C + A)

(2)

Depending on the sign of Λ which is strictly related to the set of boundary conditions, the flow dynamics118

follows either an asymptotical (Λ > 0 - Eqn. 3) or an oscillatory (Λ < 0 - Eqn. 4) trend until the steady119

state is reached:120

u̇ + u

τ = u∞ τ

(3)

ü + 2

τ u̇ +

( 1

τ2 + Ω2

) u =

( 1

τ2 + Ω2

) u∞ (4)

The parameters Ω and τ depend on the boundary conditions and their slopes when linearized [30].121

Once the midpoint velocity is evaluated at a given time step, the mass flow rate in the duct with cross122

section S can be calculated according to Eqn. 5:123

ṁ = ρ∞uS = γ p∞ a∞2

uS (5)

In the compressor model, QPM decomposes the inlet and outlet sections of the compressor as a series124

of ducts and reservoirs. To model the unsteadiness of vane filling and emptying, infinitesimal ducts were125

6

considered between the cells and the suction and discharge capacities, as represented in Figure 2.b. The126

inlet (uin) and outlet (uout) flow velocities vary with time as the result of the interaction between the127

upstream and downstream BCs that characterize the intake and exhaust respectively. Both processes were128

modeled as a reservoir that feeds a throttled duct: with respect to the intake pipe, during the vane filling the129

environment behaves as a reservoir at ambient pressure while the motion of the compressor cell downstream130

becomes a flow restriction with variable crossing area. In analytical terms, the energy conservation expressed131

in terms of total pressure, leads to the energy ellipse for the upstream BC while the throttling action of132

the cell considers a subsonic (parabolic) stage followed by a sonic one whose trend is linear. Similarly, the133

emptying process occurs through a pipe that upstream has the discharging cell which acts as a variable flow134

restriction while downstream it is connected through a plenum at the line pressure. For a given contraction135

ratio ζ, that depends on the opening of the intake and exhaust ports preliminary evaluated in the geometric136

module, the steady state is analytically computable according to Eqn. 6.137

u2∞ = 2γ RTu γ − 1

ζ2

[ 1 −

( pd pu

)γ−1 γ

] (6a)

p∞ = pu

[ 1 − ζ2

[ 1 −

( pd pu

)γ−1 γ

]] γ γ−1

(6b)

Although the model scheme is the same, the boundary conditions across the suction and discharge138

ducts are greatly different. Indeed, if the compressor operated at off-design conditions, at the discharge the139

line pressure might be several bars different than the one in the discharging vane; a sudden pressure step140

(theoretically isochoric) would thus occur because of the fixed compression ratio which is imposed by the141

machine geometry. Consequently, the mismatching between pressure at the cell opening and the value of142

the compressed air line would trigger pressure unsteadiness, with a magnitude proportional to the pressure143

step. On the other hand, during the vane filling the pressure in the suction capacity is slightly greater than144

the one in the cell. Therefore, sonic flows might occur whether the cell pressure when it opens towards the145

suction port is less than 0.5 times the ambient one.146

Using an adiabatic-isentropic approach, once the inlet and outlet velocities are known, the density cal-147

culation is straightforward as well as all the thermodynamic flow properties (temperature, enthalpy, etc.).148

2.3. Cell Thermodynamics149

The model is based on a lumped parameter formulation that assumes uniform thermodynamic properties150

(temperature, pressure, composition, etc.) inside the cell. Considering the lubricant injection, the working151

fluid is a mixture of air and oil vapors. The cell behaves like an open system, characterized by input and152

output enthalpy flows that depend on the filling and emptying processes. The first law of the thermodynamics153

7

applied to an open system doing only boundary work assumes the expression in Eqn. 7:154

Ė = ṁinl Hinl − ṁoutl Houtl − q̇air−oil − q̇evap − q̇ext −pV̇ (7)

being q̇ext the thermal power exchanged with the metallic surfaces of the cells modeled according to the155

approach proposed by Tan and Ooi [32].156

The convective heat exchange between air and oil is represented by the term q̇air−oil . This contribution157

has a great energy saving potential since it would reduce the compression work pV̇ : an enhanced air cooling,158

for instance achieved by spraying the same quantity of oil currently injected to have a greater overall heat159

exchange surface would lead towards an isothermal compression [33]. Although this technique was also160

applied in screw [34] and scroll compressors [? ? ], in sliding vane machines oil injections can be performed161

along the axial length of the machine such that the residence time of the droplets for the heat exchange162

is higher, thus the expected cooling effect on the air [35]. The mathematical approach that was used to163

model the convective heat exchange between air and oil droplets relies on the Spalding low pressure film164

evaporation theory [36].165

q̇air−oil = π Ddrop Km Nu ∗ (Tair −Tdrop) (8)

Additional details on the thermal conductivity of the oil-air mixture Km and the corrected Nusselt number166

Nu∗ that are involved in Equation 8 are provided in [33, 37]. The discriminant parameter for the heat167

exchange is oil droplet diameter Ddrop. In the conventional injection technology, as the one on board of the168

compressor tested for the current study, the lubricant was supplied through calibrated holes which did not169

succeed to atomize the jet thus to accomplish the desired internal air cooling. This fact, together with the170

rapidity of the transformation, justified the adiabaticity of the compression noticed experimentally. On the171

other hand, as it was shown in [35, 37], the usage of pressure swirl nozzles led to finer oil sprays that were172

able to perform an effective air cooling so reducing the compression work.173

Even though volatility and mass diffusivity of the oil used is very low, if the oil partial pressure reached174

the saturation value at the cell temperature, the lubricant would start to evaporate and gradually lower the175

cell temperature. This phenomenon is taken into account in the term q̇evap, defined as the product of the oil176

mass flow rate evaporated and the latent heat of vaporization. In this case, the additional need to compress177

also the oil vapors might reverse the energy benefit. Hence, the oil vaporization should be taken under strict178

control.179

2.4. Lubrication Circuit180

In sliding vane compressors, the lubricant is injected inside the cells to mainly fulfill lubrication and181

sealing purposes. At the discharge, the oil is separated from the mixture with air usually through the182

8

mechanical impingement in a labyrinth chamber and a further passage through a coalescence or cyclone183

separator. The oil is eventually recovered in a tank, as shown by the grey line in Figure 4. The machine184

layout allows the oil circulation without an auxiliary device: it relies on the pressure difference between the185

cell pressure at the discharge and the one at the injection angle as driving phenomenon (αinj is reported in186

Table 1). Hence, the power required to pressurize the oil, that takes into account the compressor efficiency187

as it was an oil pump, can be calculated according to Eqn. 9.188

Poil = Qoil(pdischarge −p(αinj))

ηmech (9)

In each injector, the oil rate was calculated as through an orifice in steady conditions. This quantity differed189

from one calibrated hole to the other ones because of the sequence of pressure drops along the oil circuit,190

whose extensive modeling can be found in [5].191

2.5. Friction modeling and blade dynamics192

Friction increases the overall energy expenditure for the compressed air production and it is still an open193

issue for the energy saving in SVRC. Part of the mechanical power supplied is dissipated by friction at the194

shaft bushes, between the rotor and the side covers of the compressor and because of the blade dynamics.195

However, the first two phenomena do not produce a noticeable contribution to the overall friction power196

thanks to the bush technology and to the absence of axial loads from the rotor to the covers [14]. Except197

for the transient behavior during which oil must be pressurized, friction produced at the bushes is negligible198

because a dry contact never appears. Hence, the blade dynamics remains the only noteworthy contribution199

to the power lost by friction.200

During its motion, the blade rotates with the rotor and slides along the slot and the inner stator surface:201

consequently, there are inertial and fictitious forces (centrifugal and Coriolis) acting on it. Since blade202

thickness and rotor slot width have tight tolerances to prevent leakage flows between consecutive cells, the203

main degree of freedom of the blade is the translational one along the y direction of Figure 3.a. Hence, the204

rotational and translational inertias along the x direction were neglected. The power dissipated by friction205

depends on the reaction forces at the three contact locations, namely 1 and 2 with the side walls of the206

rotor slot and 3 at blade tip. The blade arrangement inside the rotor slot may assume four configurations:207

it can be tilted or pushed on the slot walls, either forward or backwards (Figure 3.b). The equilibrium208

configuration is the one in which all the normal forces are of compression type. From a computational point209

of view, this depends on the signs of the forces F1 and F2 with respect to the configuration shown in Figure210

3.a and modeled with the Equations 10 which state the translational equilibria along the x and y directions211

and the momentum equilibrium respectively.212

9

F1 = Fpn −Fcor + k4 Fc + F2 + k2 F3 (10a)

k1 F3 = Fin + k3 Fc + λ (F1 + F2) (10b)

F2 (λtbl − (Lbl −Lout)) = Fc(k4 (Lbl/2 −Lout) + ... (10c)

−k3 tbl/2) −Fpn Lout/2 −Fin tbl/2 + ...

−Fcor(Lbl/2 −Lout) −F3 (k2 Lout −k1 tbl/2)

Friction power is also related to the slip velocities of the contact points whose main influencing parameters213

are rotational speed and compressor aspect ratio, Eqn. 11.214

Pfr = λ(F3 U + vbl(F1 + F2)) (11)

Moreover, while the blade tip velocity U is always concordant with the angular velocity ω, the sliding velocity215

vbl inverts its direction during a complete rotation since the blades come out from the rotor slot during the216

first half of the cycle and get in during the second half. The tilt angle φ eventually affects the load along the217

directions parallel and orthogonal to the slot axis through the projection angles χ and ξ that in Eqns. 10218

are grouped inside the projection coefficients k1−4, whose mathematical formulations are reported in Eqns.219

12. Therefore, a modified friction power distribution and a different blade positioning inside the slot may220

be achieved.221

k1 = cosξ + λsign(φ) sinξ + λ (1 −|sign(φ)|) sign(sin(α)) sin(ξ) (12a)

k2 = λcosξ −sign(φ) sinξ − (1 −|sign(φ)|) sign(sin(α)) sin(ξ) (12b)

k3 = cosχ (12c)

k4 = sign(φ) sinχ (12d)

Once blade loads and slip velocities were known, a deeper analysis was performed at the contact between222

blade tip and the inner stator wall. The lubrication regime was considered hydrodynamic [38]: indeed, the223

blade tip profile and its motion with respect to the stator pressurize the oil layer as it happens in a plane (or224

curved) hydrodynamic bearing. Neglecting the effect of eccentricity on the curvature of the stator surface225

and the transversal variation of oil pressure, at a given angular location of the contact the one-dimensional226

formulation of the Reynolds equation, that embeds both the continuity and momentum equations, assumed227

the following expression:228

d

dx

( h3

12µ

dp

dx

) = w

2

dh

dx (13)

10

Figure 3: Blade dynamics - Free body diagram (a), possible blade arrangements inside the rotor slot (b) and reference scheme

for the analysis of the hydrodynamic lubrication between blade tip and stator wall (c)

being p the pressure realized inside the oil layer and µ the dynamic viscosity of the oil, a key parameter229

for the lubrication phenomena which is strongly dependent on the oil temperature. However, measurements230

revealed that the oil temperature from the injection angle until the discharge with the compressed air did not231

increase significantly (from 72◦C to 83◦C) because of the thermostatic effect of the metallic surfaces of the232

machine and the high heat capacity of the lubricant. This allowed to simplify the calculation neglecting the233

thermal analysis and using a value for the lubricant viscosity calculated at the mean operative temperature.234

The oil film thickness h(x) depends on the mutual curvature between stator wall and blade tip profile at235

a given angle which is known from the compressor geometry considering a unique radius of curvature for the236

blade tip shape. This parameter, together with blade width, influences the oil film thickness distribution237

but does not affect the blade load as long as the lubrication regime stays hydrodynamic. On the other hand,238

sharp and thin blade tip profiles would ease the penetration within the the oil layer and lead to dry contacts.239

Within the hydrodynamic assumption, the oil pressure distribution balances the blade load: the integral240

11

effect of the oil pressure distribution over the whole contact surface (whose length is the stator one Lst) is241

indeed the orthogonal component of the reaction force at the blade tip k1 F3 calculated with Eqn. 10.c.242

∫ tbl 0

p(x) Lst dx = k1 F3 (14)

The pressures of the consecutive cells separated by the blade act as boundary conditions for the Reynolds243

equation, as shown in Figure 3.c. Equation 13 was solved considering a quasi-steady approach: for each244

blade position at a given angle α̃ of the rotation cycle, the pressure boundary conditions imposed by the245

vanes were considered constant in time as well as the normal blade load and slip velocity at the tip. The246

resulting pressure distribution is reported in Eqn. 15:247

p(x,α̃) = pcell,i(α̃) +

∫ x 0

6 µU(α̃)

h2(x,α̃)

( 1 −

h0(α̃)

h(x,α̃)

) dx (15)

where h0 is the thickness at which the oil pressure reaches the maximum value. This value results from248

Equation 16 knowing the pressure boundary conditions and the geometry of the contact:249

h0(α̃) =

∫ tbl 0

h−2(x,α̃) dx− (pcell,i+1(α̃) −pcell,i(α̃))/(6 µU(α̃))∫ tbl 0

h−3(x,α̃) dx (16)

In this way, the pressure distribution along the contact surface and the minimum oil film thickness were250

calculated over the whole rotation cycle: this value must be above a minimum threshold to prevent scuffing251

or excessive wear at the stator during the machine operation.252

3. Test setup253

An experimental campaign was performed on a mid-size industrial compressor whose geometry is syn-254

thesized in Table 1. The machine was instrumented with a set of T-type thermocouples along the air and255

oil paths as shown in Figure 4.a, while low frequency transducers provided pressure data at the inlet and256

outlet sections of the compressor.257

stator diameter 136 mm intake port start 30◦

rotor diameter 111 mm intake port end 162◦

axial length 275 mm exhaust port start 325◦

blade thickness 5 mm exhaust port end 356◦

blade height 38 mm injection angle 248◦

blade mass 0.35 kg number of cells 7

Table 1: Geometrical features of the compressor tested

12

Figure 4: Experimental setup - transducers layout (a), test bench (b), piezoelectric sensors positioning (c)

The relevant feature of the experimental methodology lies on the accurate estimation of the mechanical258

efficiency of the compressor: the mechanical power (Pmech) was evaluated from the product of torque and259

revolution speed measured through a flange torque meter installed between the compressor and the electric260

motor (Figure 4.b). On the other hand, compression power was retrieved through the reconstruction of the261

indicator diagram from the pressure traces given by a set of piezoelectric transducers.262

The sensors were installed flush mounted on a side cover of the compressor, circumferentially displaced263

between stator and rotor. For each of them, the measuring range is equal to the angular extent of the264

compressor vane (∆α = 360◦/N): with reference to the same rotating cell, the transducer starts to measure265

when the first blade with respect to the direction of rotation crosses it and ends when the second blade266

leaves it. The angular position of each transducer, quoted in Figure 4.c, was carefully chosen to enhance the267

unsteady phenomena that characterize the compressor. These mainly happen during the discharge phase268

because of the difference between pressure realized by the volumetric ratio of the machine and the one of the269

compressed air line. Since the discharge port has an angular extent (31◦) lower than the measuring range270

∆α, two cells discharge at the same time. For these reasons, when the process starts, during an angular271

displacement equal to 6.5◦ two sensors (2, 3) measure the same pressure, so detailing the most important272

unsteady phenomenon. Afterwards, sensors # 3 and # 4 measure the pressure inside the discharging cells273

and inside the exhaust port. In order to reconstruct the cell pressure evolution over a whole rotation, the274

13

relative pressure signals from piezoelectric transducers were referenced to the absolute values measured at275

the inlet and outlet of the compressor core. Different pressure offsets were applied to each signal such that276

a matching of the intake and discharge pressures as well as in the overlapping ranges between consecutive277

transducers was achieved. To further refer the reconstruction procedure to a given angle, a magnetic278

incremental encoder was used following the experimental methodology proposed in [23]. This device was279

installed on the shaft of the electric motor that was directly connected to the compressor one. The aim of280

the angle measurement is twofold: it allows to discriminate which part of the signals happened during the281

same cycle and it provides the angle evolution versus time, so taking into account the little changes of the282

angular velocity during a cycle that could lead to different angular displacements in the sampling period283

(0.1 ms). In order to compensate the effects of cyclic dispersion from one vane to the other, a phase-locked284

average was performed. The complete pressure trace over the whole rotation was reconstructed assuming the285

intake process as isobaric while the angular phase between discharge end and intake start, in which the cell286

crosses the tangency between rotor and stator, was considered as an adiabatic transformation because of the287

limited extent and the velocity of the transformation. The indicator diagram was eventually reconstructed288

associating the pressure-angle trace to the volume-angle graph. The reconstruction methodology was applied289

on the averaged signals and checking the first and second derivatives while performing the overlapping.290

In order to evaluate the measurement uncertainty related to the indicated power, the concept of indicated291

mean effective pressure (IMEP) was used. This quantity is defined as the average pressure which, when292

multiplied by the vane swept volume, would require the same work out of the cycle as the real pressure of the293

indicator diagram. The uncertainty on the indicated power was defined as the ratio between the piezoelectric294

transducers uncertainty (0.125 bar) and the value of IMEP in each test. Hence, a conservative estimation295

can be safely quantified in the 5 % of the measured value. On the other hand, the direct measurement of the296

mechanical power allowed to reduce the uncertainty on this quantity to 0.2 % while the value related to the297

mass flow rate measurement through the calibrated flange (ISA 1392 nozzle) was estimated 4 % according298

to EN ISO 5167-3.299

4. Experimental Results and Model Validation300

The compressor performance was investigated at different operating points listed in Table 2. The outlet301

pressure was varied acting on the set point of the discharge valve while a change in the revolution speed was302

performed by means of an inverter.303

Figure 5 reports the comparison between experimental and simulated indicator diagrams for all the test304

cases. As it can be noticed from the solid lines, even at 1000 RPM the magnitude of the revolution speed305

narrows the cycle time for heat transfer to occur and leads to an adiabatic compression phase regardless of306

the outlet pressure level. On the other hand, the difference between the outlet pressure and the cell pressure307

14

Figure 5: Experimental and simulated indicator diagrams at the operating points listed in Table 2

at the beginning of the discharge process affects the last part of the compression phase and triggers pressure308

oscillations during the exhaust whose magnitude is proportional to the pressure gap itself. Indeed, this309

phenomenon is highly remarkable at 12.5 bar and 14.5 bar. The pressure step involved at the end of the310

compression phase deviates from the theoretical isochoric trend because of the dynamics of the discharge311

valve. The model calculations, displayed with a dashed line in Figure 5, show a satisfactory agreement with312

the experimental data although the pressure unsteadiness is not fully predicted.313

The analysis of experimental data also allowed to calculate the volumetric efficiency of the machine as314

the ratio of the actual mass flow rate measured and the theoretical value that depends on the volumetric315

capacity of the machine and the air density at the suction process, as expressed by Eqn. 17.316

ηvol = 60 ṁmeasured ρsuc N ω Vsuc

(17)

where the units of the revolution speed ω are RPM and Vsuc is the cell volume at the end of the suction317

process (αsuc = 162 ◦ + ∆α since it refers to the second blade of the cell with respect to the rotation318

sense). The values reported in Table 2 show a strong influence of the suction temperature on the volumetric319

efficiency of the compressor because it directly affects the inlet density ρsuc: in test # 2, ηvol drops of almost320

6 % compared to test # 1 that is at the same revolution speed. For similar suction temperatures (43.4◦C321

15

test 1 test 2 test 3 test 4 test 5

discharge pressure bara 8.93 12.50 14.52 8.94 12.51

revolution speed RPM 1470 1451 1455 995 981

mass flow rate kg/s 0.067 0.057 0.063 0.046 0.044

indicated power kW 18.86 21.82 23.71 12.90 14.79

mechanical power kW 21.85 25.37 27.61 14.58 17.05

suction temperature ◦C 43.4 62.8 49.5 51.8 57.0

volumetric efficiency % 94.0 88.1 91.9 96.7 96.4

Table 2: Experimental compressor performance at various discharge pressures and revolution speeds

- 49.5◦C at 1500 RPM and 51.8◦C - 57◦C at 1000 RPM), the volumetric efficiency slightly decreases at322

higher outlet pressures. This effect was expected due to the leakage flows between contiguous cells. However,323

in sliding vane compressors the effect of oil sealing is remarkable and led to these minor variations. This324

fact is in agreement with the assumption made when neglecting leakages in the compressor model. The325

experiments eventually state an increase of the volumetric efficiency at lower revolution speeds.326

Figure 6 summarizes the model validation comparing experimental and simulated values concerned to327

synthetic parameters that quantify the compressor performance, namely mass flow rate and indicated power.328

The latter quantity is calculated from the area of the indicator diagram and the blade passing frequency329

using Eqn. 18:330

Pind = N ω

60

∮ pdV (18)

Both simulated quantities falls within the uncertainty band of the measured data in all the test cases and331

state the confidence and reliability of the model.332

From the energy balance at the compressor, whose expression is reported in Eqn. 19, only part of333

the mechanical power supplied is converted into indicated power and it is so used to compress the air; the334

remaining contribution accounts for the dissipations by friction (Eqn. 11) and the power needed to pressurize335

the oil (Eqn. 9).336

Pmech = Pind + Pfr + Poil (19)

Since Pmech and Poil were available from the experiments, once the indicator diagrams were reconstructed,337

the friction coefficient λ in Eqn. 11 could be estimated in 0.065 for all the test cases.338

The model calibration and validation allowed to reproduce the compressor performance map: Figure 7339

shows the air mass flow rate delivered at different outlet pressure levels and revolution speeds. As occurred340

in the experiments reported in Table 2, the suction conditions highly affect the volumetric efficiency of the341

16

Figure 6: Model validation

compressor since they directly act on the inlet air density; for these reasons, the chart refers to ISO 1217342

conditions (1 bar, 20◦C). The volumetric nature of the machine is stated by the linear dependance of the343

mass flow rate with the revolution speed. Additionally, the effect of the delivery pressure on the volumetric344

efficiency of the machine can be noticed with reference to the slope of the performance curves at constant345

speed: the higher is the compression ratio, the lower is the mass flow that the compressor can elaborate.346

5. Parametric analysis for the performance optimization347

In order to address future performance enhancements for the sliding vane compressor technology, a348

parametric study was carried out using the simulation platform developed. The reference adopted is the349

test case #1 in Table 2, a typical operating condition of the compressor in industrial applications. The350

analysis aimed at maximizing the mechanical efficiency of the compressor, defined as the ratio of the power351

that is actually used for the air compression and the overall mechanical power supplied (Eqn. 20).352

ηmech = Pind Pmech

(20)

Figure 8 shows the effects of varying either the stator or the rotor diameter on the compressor aspect353

ratio and friction power. To keep the reference mass flow rate, axial length of the compressor was varied354

accordingly. The resulting machine layouts are of two categories: an elongated one, in which the axial355

length is predominant with respect to the radial extent and a flat one which exhibits bigger diameters and356

shorter lengths than the actual geometry reported in Table 1. A reduction of the stator diameter limits the357

17

Figure 7: Simulated compressor performance map at ISO 1217 suction conditions (1 bar, 20◦C)

Figure 8: Influence of the aspect ratio on the compressor geometry and friction power dissipated

peripheral tip speed (U) and leads to a reduction of the most significant contribution to the friction power.358

On the other hand, being the revolution speed kept constant, a decrease in the rotor diameter results in an359

increase of the sliding velocity vbl, since the displacement that the blades have to accomplish in the same360

amount of time (i.e. the revolution period) increased. Even if this contribution has a minor importance361

compared to the one at the tip, a worsening of the mechanical efficiency would result from this aspect ratio.362

A combined analysis on the effects of revolution speed and blade tilt on the mechanical efficiency of363

the compressor is reported in Figure 9 in terms of specific power dissipated by friction with forward and364

backward blade tilt varying the angular velocity from 1000 RPM to 2000 RPM. The reduction of revolution365

speed has a direct and significant effect on friction power since it reduces both the inertial and fictitious366

forces as well as all the slip velocities. This methodology cannot be applied to other rotary volumetric367

18

Figure 9: Variation of the friction power per unit air delivered with revolution speed and blade tilt

Figure 10: Blade positioning inside the rotor slot for radial and non-radial configurations

compression technologies (e.g. screw) and becomes a distinguishing feature of sliding vane machines. The368

results presented are in agreement with the experimental data of Tramschek and Mkumbwa who tested a369

circular sliding vane compressor with radial and non-radial blades [22]. The energy benefit that might be370

achieved with forward-backward blade tilting is hardly noticeable, especially at low speeds.371

However, the non-radial blade arrangement affects the load distribution on the blade since both the active372

and reaction forces involved are differently projected along the symmetry axes of the blade, as in Figure 3.373

This fact leads to another force balance, thus to a modified positioning inside the rotor slot, as displayed374

in Figure 10. With reference to the radial configuration (φ=0), at the intake the side walls of the blade375

are both exposed to the suction pressure. Therefore, inertial and fictitious forces become predominant and376

push the blade backward onto the rotor slot. As the compression starts, the pressure difference between the377

19

Figure 11: Effects of the revolution speed on the overall blade mass and friction power with respect to test case # 1 in Table 2

side walls tilts the blade towards the rotation sense and makes it to assume the configuration reported in378

Figure 3. The angular extent in which the blade assumes this position is amplified for forward blade tilt379

(φ > 0) and reduced for backward tilt (φ < 0). The pressure unsteadiness that characterize the discharge380

and the passage across the tangency between stator and rotor leads the blade to assume all the possible381

configurations. The combined examination of Figures 9 and 10 allowed to state that blade tilt does not382

effectively contribute to achieve an improvement of the mechanical efficiency of the compressor since it does383

modify the blade load distribution but without affecting the slip velocities at the friction locations.384

Being the highest slip velocity at the contact point between blade tip and stator wall, the main contribu-385

tion to the overall friction power dissipated was deeply investigated varying the mass of the compressor blade386

and revolution speed. Since the latter parameter directly affects the flow rate delivered by the machine,387

the axial length of the compressor was varied to keep the reference test conditions. Using the same linear388

density for the blade, the length adjustment led to heavier (longer) blades whether the revolution speed was389

lower than 1500 RPM and to lighter machines at revolution speeds higher than the reference value.390

Although both mass and revolution speed affect the friction power, while the mass dependence is linear, the391

effects of revolution speed are more noticeable since they have a quadratic trend on the centrifugal force as392

well as a linear influence on the slip velocities at all the contact points. However, as displayed in Figure393

11, if the volumetric capacity of the machine is kept constant, the blade mass varies with an opposite trend394

with respect to the revolution speed. Hence, the benefits of lowering the angular velocity of the machine395

onto the friction power become linear.396

Using the simulation approach of Figure 11, a further analysis was performed at the same revolution397

speeds additionally aiming at addressing the benefits of a blade mass reduction. This could be achieved398

through a mechanical removal of some material from the current blades or using unconventional materials399

able to match the lubrication issues. The results displayed in Figure 12 are reported in terms of minimum400

20

Figure 12: Effects of the blade mass reduction on the minimum thickness of the oil film established between the blade tip and

the stator wall at 1000 (a), 1500 (b) and 2000 RPM (c) 21

value of the oil film thickness to highlight the hydrodynamic capabilities of the lubricant layer in withstanding401

the blade load. Furthermore, this parameter is inversely proportional to the maximum pressure along the402

contact surface. All the simulations show that the oil film thins during the second half of the rotation since403

the inertia force becomes concordant to the centrifugal force (as in Figure 3) increasing the blade load at404

the tip. The highest stress is reached when the vane pressure is maximum, i.e. before the cell opens towards405

the exhaust port. The location of the minimum film thickness is anticipated of ∆α in Figure 12 since the406

graph refers to the second blade of the cell with respect to the rotation sense. The exhaust opening produces407

a discontinuity, thanks to which minimum thickness increases. During this phase, indeed, pressure forces408

acting on the lateral blade surfaces are equal and this reduces k1 F3 term. The blade result less loaded and409

oil thickness tends to increase. When the exhaust port is closed the remaining air is pressurized and the410

minimum thickness reassumes the values it had before the port opening. On the other hand, during the411

suction process, both sides of the blade are again exposed to the same pressure performing a reduction of412

the load at the blade tip that ends up in a thickening of the oil film.413

When the revolution speed increases, the minimum thickness decreases being the effects of the centrifugal414

force more severe. Hence, critical operating conditions get closer, opening the way to possible dry contacts.415

Furthermore, at any revolution speed, lighter blades lead to thicker oil layers thus to a decrease in the416

orthogonal load between the blade tip and the stator wall. In this way, a reduction of the friction power417

dissipated might be achieved. However, these actions need to deal with the blade stability and sealing issues418

to prevent mass leakages between adjacent vanes.419

6. Conclusions420

Sliding vane rotary compressors might represent an astonishing technology to accomplish energy saving421

commitments in compressed air systems. The current work investigated the peculiar features of these422

machines from theoretical and experimental viewpoints aiming at characterizing the compressor energy423

performance and addressing improvements of the mechanical efficiency.424

The comprehensive mathematical model developed considers all the relevant phenomena involved in a425

sliding vane rotary compressor. The geometry was modeled to explore any machine configuration: backward426

and forward blade tilt can be simulated as well as suction and exhaust ports with an axial or radial arrange-427

ments. The flow dynamics was investigated using a quasi-propagatory approach that takes into account428

the inertial, capacitive and resistive features of the flow during vane filling and emptying processes. On the429

other hand, solving the unsteady energy equation at the compressor cells provided the pressure evolution430

over the whole rotation cycle. The blade dynamics and the analysis of the hydrodynamic lubrication between431

blade tip and stator wall were eventually implemented to characterize the friction power dissipated, thus432

the mechanical efficiency of the compressor.433

22

strategy design parameter change ηmech

test #1 Table 2 reference 86.3%

aspect ratio -5% Dst, + 45% Lst 88.0%

blade tilt +10◦ 86.8%

revolution speed + blade tilt 1000 RPM, +10◦ 95.5%

revolution speed, constant blade mass 1000 RPM 95.2%

revolution speed, variable blade mass 1000 RPM, +50% mbl 91.5%

Table 3: Optimization summary

An experimental test campaign on a mid-size industrial compressor at different revolution speeds (1000434

and 1500 RPM) and discharge pressures (9, 12.5 and 14.5 bar) was further carried out using a set of435

piezoelectric transducers to retrieve the indicator diagram. The narrow time cycle led to an adiabatic436

compression phase in all the test cases while the outlet pressure value influenced the unsteadiness during the437

discharge process. Hence, the indicated power and the mass flow rate measured validated the model results438

with simulated values within the uncertainty bands and provided an estimation of the friction coefficient of439

0.065.440

The simulation platform was ultimately used to investigate geometrical and operating parameters of the441

compressor to maximize its mechanical efficiency (Table 3). Since friction power dissipated between blade442

tip and stator wall is the most significant contribution, a change in the compressor aspect ratio would result443

effective whether a reduction of the stator diameter was realized. Accordingly, a reduction of the blade mass444

would decrease the reaction force at the blade tip that balances the centrifugal effects while decreasing the445

revolution speed would also act on the slip velocities. However, these actions need to be applied without446

affecting the vane sealing and blade stability. Although the blade positioning inside the rotor slot during a447

whole rotation is affected by a forward or backward blade tilt, a noticeable effect on the overall friction power448

dissipated is not achieved: the blade tilt affects the load distribution but does not modify the slip velocity449

at the friction contact points. The oil layer at blade tip, that assures the sealing between vanes, assumes450

its most critical conditions when the blade is approaching the exhaust port. Operating conditions at lower451

revolution speed and lighter blades would favor the hydrodynamic lubrication and keep the minimum oil452

film thickness above a safety threshold to prevent dry contacts with the stator wall.453

Acknowledgement454

The Authors acknowledge Ing. E. Mattei S.p.A. and particularly its CEO, Dr. Giulio Contaldi, for455

continuous research funding and support. The work has been done also under the FP7 Project ”Complete456

Vehicle Energy-Saving CONVENIENT” founded by the European Commission.457

23

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26

List of symbols

α angular coordinate [rad] P power [W ]

γ specific heat ratio [−] Q flow rate [m3/s]

ξ contraction ratio [−] R gas constant [J/kg/K]

η efficiency [−] RY tip radius [m]

λ friction coefficient [−] S duct cross section area [m2]

µ oil dynamic viscosity [Pas] T temperature [K]

ζ force projection angle [rad] U blade tip speed [m/s]

ρ density [kg/m3] V cell volume [m3]

τ QPM parameter [s] Subscrips and superscripts

φ blade tilt angle [rad] ′ initial state

χ force projection angle [rad] 0 thickness at maximum pressure

ω revolution speed [RPM] ∞ steady state

Λ QPM parameter [−] bl blade

Ω QPM parameter [s−1] c centrifugal

a speed of sound [m/s] cor Coriolis

h film thickness [m] d downstream

k force projection coefficient [−] drop oil droplet

m mass [kg] evap evaporation

p pressure [Pa] ext external

q heat flux [J] fr friction

t thickness [m] in inertia

u flow velocity [m/s] ind indicated

v slip velocity [m/s] inj injection

w oil flow velocity [m/s] inl inlet

A upstream boundary condition slope [N s/m3] mech mechanical

B downstream boundary condition slope [N s/m3] out outside the rotor slot

C characteristic curve slope [N s/m3] outl outlet

D diameter [m] pn normal pressure

E absolute internal energy [J] st stator

F force [N] suc suction conditions

H specific enthalpy [J/kg] u upstream

Km mixture thermal conductivity [W/m/K] vol volumetric

L length [m] Mathematical operators

N number of cells [−] Υ̇ first time derivative of Υ

Nu∗ corrected Nusselt number [−] Ϋ second time derivative of Υ27

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