Department: Process & Energy, Faculty 3mE Section: Engineering Thermodynamics

Thermodynamic Model of a Screw Compressor

Master Thesis

L.L. van Bommel August 2016 Report number: P&E - 2713

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Thermodynamic Model of a Screw Compressor

By

L.L. van Bommel

in partial fulfilment of the requirements for the degree of

Master of Science in Mechanical Engineering – Sustainable Process and Energy Technology

at the Delft University of Technology,

to be defended on Monday August 29, 2016 at 15:00.

Supervisor: Dr. ir. C.A. Infante Ferreira TU Delft Thesis committee: Prof. Dr. Ir. T.J.H. Vlugt, TU Delft

Dr. R. Pecnik TU Delft Ir. V. Gudjonsdottir, TU Delft PhD Candidate

An electronic version of this thesis is available at http://repository.tudelft.nl/.

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Abstract A Compressor Resorption Heat Pump (CRHP) is a potential contribution to energy reduction in applications in which waste streams are upgraded, with limited energy addition, into high value process streams for reuse in industry. Previous research concluded that a CRHP with a wet screw compressor is a suitable option for many applications. An ammonia/water mixture was found to be the most appropriate fit, in terms of thermodynamic behaviour, for such an application. Objective of this thesis was to develop an integration of a geometry model and a thermodynamic model suitable for further optimisation of the wet twin-screw compressor. The integration of the geometry model and the thermodynamic model was carried out in modelling tool Matlab/Simulink, with inclusion of the physical properties of the working fluid. The development of the integrated dynamic model was carried out based on research for a heat pump process with a pre-selected geometry and a homogeneous two-phase fluid. The existing geometry model was transformed from shaft rotation based to time based equations to achieve the dynamic model requirements and the possibility of modelling the process in Simulink. The geometry model provides inputs to the thermodynamic model that dynamically describes the wet twin-screw compressor from the suction phase through compression to the discharge phase. The thermodynamic model requires inclusion of physical properties of the fluid and these were added by importing the physical properties through Refprop via Fluidprop. Mechanical constraints of a wet twin-screw compressor inevitably lead to internal leakage paths that reduce the compressor efficiency. The leakage paths have been included together with factors for friction, flow loss, etc. to represent the process in a more realistic way. The integrated model has been validated with the calculated result by model case A and measured results from the experimental set-up by Zaytsev [1]. A number of variations have been applied to the integrated model as examples of how to evaluate options for improvements. Making use of the developed integrated model parameters can be varied to show the influence on the compressor. The evaluations used a specific set of boundary conditions from previous research, using the geometry specified by Zaytsev [1]. The effects of three input parameters on the output and efficiency were evaluated: rotor length, discharge port area and vapour quality. The main result of the evaluation is that per boundary condition, the inputs from the geometry model have to be adjusted to achieve an optimal design of the twin-screw compressor. Further research to find the optimal design can be done with the help of the model that was developed for this thesis.

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Content Abstract 5Nomenclature 9Abbreviations 10Glossary 101 Introduction 111.1 HeatPumps 131.2 CompressorProcess,ArrangementandSelection 141.3 ScrewCompressor 161.4 MotivationoftheResearch 181.5 ResearchQuestion,Objective,BoundariesandAssumptions 181.6 Chapters 19

2 Models 202.1 BoundariesandAssumptions 202.2 GeometryModelTheory 212.3 TheoryoftheThermodynamicModel 272.4 TheoreticalDesiredPerformanceOutputs 30

3 ThermodynamicModelImplementation 313.1 HistoricDevelopment 313.2 InputImplementation 323.2.1 GeometryModel 323.2.2 Thermo-PhysicalProperties 353.2.3 InputValues/InitialValuesfortheImplementation 36

3.3 ThermodynamicModel;FactorsthatInfluenceIdealBehaviour 373.3.1 MassFlows 373.3.2 LeakagePathAreas 393.3.3 MassFlowsoftheLeakages 45

3.4 DesiredOutputs 494 Validation 514.1 InputsandBoundaries 514.2 ModelValidation:Zaytsev 524.3 Adaptingthemodeltotheexperimentaldata 54

5 ModelResultsandDiscussion 575.1 Leakages 575.2 Boundaryconditions‘vandeBor’ 575.3 Geometryvariation 58

6 ConclusionsandRecommendations 656.1 Conclusions 656.2 Recommendations 66

Bibliography 67Appendices 69AppendixA:EnvelopeMethod–RotorElementCalculation 69AppendixB:ConservationEquationsoftheHomogeneousModel 71AppendixC:TheThermodynamicModelinSimulink 74AppendixD:TheGeometryModelinMatlab,vandeBor/Zaytsev[Matlab-Code] 79AppendixE:CalculationPressureDifference[Matlab-Code] 98AppendixF:CalculationEfficiencies[Matlab-Code] 100AppendixG:CalculationShaftRotationAngle 101

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Nomenclature Symbols

Units

A Flow area m2 b Number of rotor lobes - C Angle ° D Distance between rotor axes m h Specific enthalpy J∙kg-1

H Enthalpy J i Rotation speed ratio - k Ratio + 1 - L Length of the rotor mm LL Length contact line, sealing line or height of the rotors mm m Mass kg 𝑚 Mass flow rate kg∙s-1

n Speed of rotation rpm p Pressure Pa/bar Q Heat J R, r Radius m/mm t Time s T Temperature K u Flow velocity m∙s-1

v Specific volume m3∙kg-1

V Volume m3 W Work J 𝑊 Power J∙s-1 x Ammonia mole concentration mol∙mol-1

x, y, z Coordinates m Greek Symbols Units β Angle °/rad ∂ Differential - Δ Delta (angle, temperature, mass, pressure, etc.) - η Efficiency - θ Angle ° ζ Empirical flow coefficient - ρ Density kg∙m-3

∑ Summation - τ Rotation (twist) parameter ° φ Rotor turning angle ° Ψ Profile parameter - ω Rotation speed s-1

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Subscript 0 Static coordinate system attached to rotor housing 1 Male rotor 2 Female rotor c, comp Compressor (phase) discharge Discharge phase h Envelope radius high Highest value ideal Ideal process in Flow in is, s Isentropic process low Lowest value, angle out Flow out real Real process suction Suction phase up Upper angle vol Volumetric w Wrap angle

Abbreviations CFD Computational Fluid Dynamic COP Coefficient of Performance COP21 21st Conference of Parties CRHP Compression Resorption Heat Pump EDGAR Emission Database for Global Atmospheric Research IEA International Energy Agency IPCC Intergovernmental Panel on Climate Change LULUCF Land Use, Land-Use Change and Forestry NASA National Aeronautics and Space Administration PBL Planning Board for Environment ‘Planbureau voor Leefomgeving’ UNFCCC United Nations Framework Convention on Climate Change

Glossary Leading Cavity The leading cavity is the cavity volume with a time and comes

according to the time ‘in front’ of the main cavity.

Main Cavity The modelled volume cavity.

Trailing Cavity The trailing cavity is the cavity that has a time shift and will come according to the time ‘behind’ the main cavity.

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1 Introduction Sustainability is one of the words that is most commonly used when talking about global warming. In 1987 the World Commission on Environment and Development [2] wrote a report about the issue: ‘Our common future’. In this report the commission proposed a definition for sustainability, which is now widely used and is cited below.

“Sustainable development is development that meets the needs of the present without compromising the ability of future

generations to meet their own needs.” Last December the 21st climate conference was held in Paris, organised by the United Nations Framework Convention on Climate Change (UNFCCC) [3]. During this yearly event, members of the United Nations gather to discuss measures to fight global warming. During the 11th edition in 1997, this resulted in the well-known Kyoto Protocol. During the latest edition in Paris, the 21st Conference of Parties (COP21) [4], the 195 attending parties adopted the first-ever universal, legally binding global climate deal and agreed on a global warming limit of 2 degrees Celsius. Global warming is one of the largest threats to the world we live in. The highest increase in global temperature has occurred during the last 35 years [5]. The data in Figure 1-1 show that the earth’s temperature has increased by at least one degree since the average baseline of 1951-1980. Recent data show that 2015 was the warmest year on record. The temperature rise limit of 2 degrees agreed on COP21 will thus be a challenge, and drastic changes are needed to accomplish the agreement. Research in global warming is done by many different organizations. A few of these are: National Aeronautics and Space Administration (NASA), Intergovernmental Panel on Climate Change (IPCC) and International Energy Agency (IEA). Worldwide energy demand is high, and will only increase with growing population [6]. Something needs to be done to reach the agreed goal of the UNFCCC.

Figure 1-1 Temperature increase over the years from 1880 to 2015 [5].

-0,8-0,7-0,6-0,5-0,4-0,3-0,2-0,10

0,10,20,30,40,50,60,70,80,91

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1880 1895 1910 1925 1940 1955 1970 1985 2000 2015

TemperatureAnomaly(1951-1980baseline)

YearAnnualMean

1°Cabove19thCentury

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In the Netherlands, the attention to global warming and its consequences is increasing and becoming a major concern to both the government and the public as well [6, 7, 8]. The causes and effects of climate change are intensively investigated by the IPCC [10], of which the Dutch government is a member. The rising sea level is one of the most significant consequences of global warming, and even more so for a country that is partially below sea level. The Netherlands are already fighting the sea with their famous Delta works, but the question is to what extent they can strengthen them to withstand the increasing rise of the sea level. Other consequences include more extreme weather conditions and the pollution of sweet water rivers with salt water, which complicates the utilization of river water for drinking, agricultural and industrial cooling purposes. The global warming effect is mostly linked to the emission of CO2. In Figure 1-2 the shares of different greenhouse gases in the total emission of 2010 can be seen, both for the world and for Europe. This figure clearly shows that the most significant greenhouse gas is CO2, which comes from burning fossil fuel and from industrial processes. Reducing CO2 emission will thus slow down global warming and might even decrease the global average temperature. The Dutch organization Planning Board for Environment ‘Planbureau voor Leefomgeving’ (PBL) [9] released a report in 2015 concerning the CO2 emission all around the world: ‘Trends in global CO2 Emissions’ [11]. This report is based on the data from PBL and Emission Database for Global Atmospheric Research (EDGAR) [12]. Burning of fossil fuels, generating electric energy and generating heat emits most CO2. The increasing demand for energy by the growing population will prove to be a challenge for the world in the struggle to reduce the energy usage and thus decrease CO2 emission.

Figure 1-2 Shares of greenhouse gas emission 2010 [11].

Countries all over the world are striving to emit less CO2 by exploiting renewable energy sources and reducing energy losses. The IEA concluded in 2011 [13] that 47% of energy consumption in the world is thermal energy. The industry is the largest thermal energy consumer worldwide, weighing in at 44% of total thermal energy consumption. In many industrial processes a large proportion of this thermal energy is dissipated to the environment as waste heat. In the Netherlands alone, the industry is estimated to squander over 250 PJ through wasted thermal energy, of which an estimated 150 PJ/year can be reused in other processes [13,14]. This thermal energy loss to the environment has a temperature range of 40-150 °C. Therefore, the reuse of waste heat is considered an interesting possibility to reduce CO2 emissions. One of the most well-known technologies for (waste) heat recovery is the so called heat pump, which can extract thermal energy from a low caloric stream, to release it in a high caloric

64%10%

18%

6% 2%

World

76%

4%12%

5% 3%

Eu28CO2fossilfuelandindustrialprocesses

CO2forests(representingLandUse,Land-UseChangeandForestry(LULUCF)partofUNFCCC)CH4

N2O

Fluorinatedgases(F-gases):HFC,PFCandSF6

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stream. Therefore the heat pump is one of the most interesting devices to study when it comes to waste heat reduction.

1.1 Heat Pumps A heat pump, shown schematically in Figure 1-3, transfers heat from a low temperature source to a high temperature sink, with help of a working fluid, a refrigerant. Heat from the low temperature source is used to evaporate the refrigerant. The refrigerant vapour is then compressed in the compressor, increasing both its pressure and temperature. The pressurized refrigerant vapour is condensed in the condenser, releasing its heat to the high temperature sink. After throttling, the process is repeated. A heat pump can be used in many different situations; from heating up a building with geothermal energy to the reuse of waste heat in an industrial process. For more details about heat pumps, see Moran and Shapiro [16].

(a) (b)

Figure 1-3 Schematic representation of two variations, for compression, of a heat pump cycle (a) compression resorption heat pump with solution pump and (b) compression resorption heat pump with wet-compressor.

The refrigerant in a heat pump is used to facilitate the heat transfer. There are many different refrigerants to work with. One of these options is the refrigerant-absorbent combination, which consists of a mix of two elements: the refrigerant and an absorbent. Ammonia/water and water–lithium bromide are two well-known refrigerant-absorption combinations. In the combination of ammonia/water, water is the absorbent and ammonia the refrigerant. More information on refrigerants can be found in Dinçer and Kanoglu [17]. Ammonia/water is a refrigerant mixture which is suitable for high-temperature heat pump applications and allows a heat rejection temperature of 80-160 °C. According to the research of Mongey et al. [18], use of a resorption heat pump (CRHP) would be feasible for the recovery of waste heat from industry processes with waste streams in the range of 40-80°C. This temperature range is typical for waste streams in industrial process applications. Ammonia/water has another advantage compared to single component refrigerants. The concentration of ammonia/water can be adjusted in order to match the temperature glide of both the source and the sink, due to its nature as a non-azeotropic mixture, which makes it suitable for use in resorption heat pumps [19]. Another advantage of the ammonia/water mixture is that the mixture can be applied in a broad range of processes. By changing the ammonia concentration the glide of the refrigerant can be adapted to match that of the process.

CompressorValve

Desorber(Evaporator)

Resorber(Condenser)

!Qin

!Qout

!WcSolu8onPump

Wet-CompressorValve

Desorber(Evaporator)

Resorber(Condenser)

!Qin

!Qout

!Wc

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Two types of compression resorption heat pump (CRHP) can be defined: based on the Osnabrück cycle Figure 1-3(a) and wet compression Figure 1-3(b). In the Osnabrück cycle gas and liquid phases are separated before entering the compressor and the liquid is pumped parallel to the compressor; after the compression the compressed gas and liquid are mixed. The type where liquid and gas from the desorber are not separated and are compressed as ‘wet’ phase and fed into the resorber is called wet compression. According to van de Bor et al. [20], it is possible to use a wet compressor in a resorption cycle, instead of the compressor/solution pump combination of the Osnabrück cycle, Figure 1-3. Van de Bor et al. [20] have done numerical research of the performance of the heat pump, with ammonia/water mixture as refrigerant, in 50 specific industrial cases. In van de Bor et al. [21] three different heat pumps are compared for a certain situation: the recovery of heat from an industrial cooling tower stream. The temperature of this waste stream is generally around 45-60 °C. According to the research, the compression resorption heat pump (CRHP) would be most suited for the application of heat recovery from a cooling tower stream, Figure 1-4.

Figure 1-4 CRHP heat recovery from waste stream cooling tower [21].

Due to its design, the heat pump can be used both for cooling systems and for heating systems. In order to achieve the highest efficiency of the CRHP, the use of wet compression is most advisable. In the research the water flow was split into two stream, one stream to be heated above 110 °C and the other to be cooled down to 5 °C, as shown in Figure 1-4. Van de Bor et al. [21] assume that the wet compressor has an isentropic efficiency of 0.7. However in order to reach an efficiency of 0.7 or higher, further development of the wet compression is needed. The two-phase fluid in the heat pump poses a problem at the compression side; the following sections will therefore discuss the working of the compressor of the heat pump.

1.2 Compressor Process, Arrangement and Selection In a Compressor Resorption Heat Pump (CRHP) the mechanical piece of equipment that raises the pressure and temperature of the refrigerant is the compressor, which is generally driven by an electric motor. Figure 1-3 shows the two different compressor arrangement options for the CRHP that have been investigated. The option shown in Figure 1-3(a) the Osnabrück cycle uses a semi-hermetic compressor to compress the ammonia/water vapour. The vapour

Wet-CompressorValve

Desorber(Evaporator)

Resorber(Condenser)

!WcWastewaterStream:45-60°C

WarmUAlity:110°C

ColdUAlity:5°C

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compressor is combined with a pump placed parallel to the compressor, to transfer liquid stream of the ammonia/water mixture. According to the research of Mongey et al. [18] wet compression was believed not to be feasible, due to the large liquid fraction that remains after the desorber, which would enter the suction of the compressor. The other option is using a wet compressor arrangement, Figure 1-4b. Van de Bor et al. [21] make use of the wet compressor, which provides an advantage compared to dry compression with separate liquid circulation as discussed above. Information on wet compression is limited, and the aforementioned paper strongly recommends further research into optimization of compressors for this purpose. Itard [22] did research on the wet compression-resorption cycle in 1998, focusing mainly on the difference between dry and wet compression. According to her research, wet compression has an advantage, for the cases that were considered, that varies between 2.5% and 13% on the Coefficient of Performance (COP) compared to dry compression. For the compression section during the experiments a liquid ring compressor was used. In 2003 Zaytsev [1] did more research on the type of wet compressor suitable for the CRHP. A review has been done on the different compressor types available on the market and their possibilities for wet compression. In the end he concluded that a twin-screw compressor would be the best option for wet compression in CRHP. The choice for a twin-screw compressor was based on the capability of this compressor to work in the two-phase regime of an ammonia/water mixture, with a sufficiently high thermodynamic efficiency. According to data the isentropic efficiency of a twin-screw compressor can reach up to 0.75, which would fit the needed threshold as per van de Bor et al. [21]. The experiments of Zaytsev [1] showed an isentropic efficiency of the compressor still relatively low compared to the required minimum of 0.7. The twin-screw compressor can also operate over a wide range of pressure and temperature, which makes it suitable for CRHP application. The twin-screw compressor additionally has the capability to work under oil-free conditions. It also has a medium risk to hydraulic locking compared to the other compressor type options. The disadvantages of the twin-screw compressor are the built-in volume and a constraint that at the moment there are, to current knowledge, no twin-screw compressors available on the market that meet the specifications for use in the ammonia/water based CRHP. Zamfirescu et al. [23] did in 2004 further research on Zaytsev’s results on the twin-screw compressor for use as wet compressor. Zamfirescu et al. [23] developed a Computational Fluid Dynamic (CFD) simulation with a non-homogeneous model and carried out experiments on the twin-screw compressor. From the CFD model it was concluded that the available liquid-phase fluid in the compressor is spread as a layer against the compressor housing because of the centrifugal force. This liquid layer fills up the clearances between the screws and decreases the leakage between the different cavities. This has the positive effect of increasing the isentropic efficiency. Also the leakages at the bearings were researched. Several labyrinth seals were studied and it was concluded that the use of an optimal labyrinth seal would improve the thermodynamic performance of the compressor. Zamfirescu et al. [23] also did a prediction on the heat and mass transfer during compression to improve the accuracy of the model. More research has been done on the topic of liquid injection, which was proposed by Zaytsev [1] and could increase the compressor efficiency even more. Zamfirescu et al. [23] also did research on the effect of the rotational speed on the efficiency. According to their experiments the mechanical efficiency would be optimal at a rotational speed between 2500 and 3000 rpm. The mechanical losses will increase drastically at higher rotational speed.

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Lets have a deeper look into the twin-screw compressor. With the research carried out to date the type of screw, mechanical arrangement and process fluid selection provides quite a number of unknown parameters. According to the information above more research is needed to accomplish a working wet compressor suitable for the wanted specifications.

1.3 Screw Compressor Alfred Lysholm [24] from Sweden invented the first twin-screw compressor in the 1930s. The twin-screw compressor is a positive displacement compressor with two meshed helical rotors, a male rotor and a female rotor. Screw compressors create a continuous but pulsating (batch type) flow by compressing a gas between the lobes of the screws, increasing the pressure of the gas in the process. The rotors consist of a number of lobes, which can differ between the male and female rotors. The male rotor is driven electrically and in turn drives the female rotor through meshing. The helical surfaces, the meshing and the housing around the compressor rotors enclose the cavity volume of the compressor. The compression is repeated for each cavity volume in the twin-screw compressor. During the turning of the screw compressor the suction port opens and the gas enters the compressor. When the cavity volume is at it’s largest and the passage between the inlet port and the cavity volume has closed, the size of the volume decreases and the compression phase begins. When the boundary between the cavity volume and the outlet port at the discharge is opened, the compressor is in the discharge phase. The suction, compression and discharge phases are schematically shown in Figure 1-5.

Figure 1-5 Schematically shown compression in a twin-screw compressor.

Figure 1-6 shows the three phases of the twin-screw compression process (within the model these three phases will be divided into two, suction and compression/discharge). During the suction and discharge phase the temperature and pressure are assumed to be constant in this theoretical figure. The only change in these variables occurs during the compression phase. See Wennemar [25] or Arbon [26] for more info on the working of the twin-screw compressor.

Figure 1-6 Idealized pressure-volume diagram for a screw compressor with well suited built-in volume ratio [25].

Suc$on Compression Discharge

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Literature research shows that most wet screw compressors are lubricated/injected with oil. Oil is the most appropriate lubricant used. As a result there is a lot of literature about oil injected screw compressors [27]. The disadvantage of using oil for lubrication of the system is that the oil and gas need to be separated after the compression. Failing to do so results in oil staying behind in the mixture when it is transported to the heat exchangers, the resorber and the desorber. The oil forms an oil layer on the heat exchanger surface, resulting in a decreased performance of the heat exchangers. Besides decreased heat exchanger performance, the presence of oil in the mixture also requires extra equipment to separate the oil phase from the gas phase. There are a few other reasons for using oil in the compressor next to lubrication. The oil reduces the blowholes and clearances of the leakage paths between the rotors, as well as, between the rotors and the housing. Application of oil as lubricant/sealant additionally absorbs heat from the compression and keeps the temperature of the compressor low, as well as reducing wear on the lobes of the rotors. Concluding, use of a suitable lubricant/sealant is an important element in the design and use of a screw compressor. The heat pump under study uses compression with an ammonia/water mixture. Such a mixture excludes the use of oil as lubricant/sealant as it would lead to a three-phase liquid/vapour fluid, which requires further treatment downstream to separate the oil from the liquid phase [23]. The aim is to use the liquid phase of the ammonia/water mixture as a lubricant [20, 27, 28]. Using the liquid phase as lubricant requires less equipment and prevents the oil problem explained above. In order for the ammonia/water mixture to be effectively used as lubricant in the compressor, two important premises need to be met. The liquid/vapour in the suction port of the compressor needs to be ideally mixed in order for the liquid to be available on all the surface area and in all the clearances of the compressor. Flow regimes like slug flow need to be prevented by regulating the liquid intake at the suction port [23]. Screw compressors can operate either under dry or wet process conditions. Zaytsev [1] and van de Bor et al. [21] have a slightly different approach to the desired working range of the compressor. Zaytsev [1] chose for a process setup where the liquid part of the mixture is used to lubricate the compressor. As a result, the inlet and the outlet contain large portions of liquid, resulting in low efficiency. According to van de Bor et al. [21], the application of wet screw compression is the best choice in the desired range, for an average temperature lift of 85 °C. During the compression phase the temperature and pressure increase and the vapour/liquid equilibrium changes. This may result in drying up of the last section of the compression and discharge phase. Van de Bor et al. [21] researched the option to let the compressor run dry at the end. According to the study this will result in a higher efficiency and an optimal performance of the heat pump as the ideal vapour quality at the resorber inlet would be approximately 100%. There are a lot of different approaches to the problem of lubrication, but not an exact solution for the problem. For this research report a homogeneous model is assumed, and leaves the lubrication and the change in the liquid/vapour concentration open for further research. For this research report the focus is on the physical simulation of the twin-screw compressor for application in a CRHP. To make a simulation model of the twin-screw compressor two different models need to be combined. One model will describe the thermodynamics of the twin-screw compressor, while the other model describes the geometry of the compressor. The thermodynamic model will calculate the pressure and temperature increase during a whole compression cycle. Conservation equations are used to calculate the compression phenomenon

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in the compressor. Next to the thermodynamic model a geometry model will be used, which describes the geometry profile of the two rotors, where the female rotor is described as a function of the male rotor. The geometry model will be used as a source for the inputs for the thermodynamic model, such as the cavity volume. The combination of these two models allows the design of a suitable geometry of the twin-screw compressor for every application. There are some papers about these two types of models, especially on models that describe the geometry. Zaytsev made a combination of a thermodynamic and geometry model [1], using the method of Sakun [30] for the geometry. This method of modelling the geometry was also used and described by You [31] in 1994. Zaytsev and Infante Ferreira [28] developed after the use of the method of Sakun a different method for calculating the geometry of the rotors. This new method was based on the meshing line, which is used as a starting value to model the rest of the rotor profiles. Stosic et al. [27] worked on the modelling of a twin-screw compressor as well, in 2005. The method Stosic et al. use is based on a rack line, which is used as a starting position to generate the profiles. This rack line is generated by specific criteria. Modelling the geometry of the rotors will be discussed in more detail in chapter 2.

1.4 Motivation of the Research Industry produces large low caloric waste streams. Upgrade of such waste streams has significant sustainable value. The residual heat of a part of the waste stream of around 45-60 °C from a cooling tower [21] will be used to heat up the ammonia/water mixture cycle in a heat pump. This ammonia/water mixture needs to be compressed to a temperature of around 115 °C at increased pressure to achieve a water waste stream with a temperature of around 110 °C. A side stream of the waste stream will be cooled to a low temperature (desorber) in the inlet stream of the compressor of the cycle. The inlet stream of the compressor will be heated by the waste stream in the desorber and improve the compressor energy efficiency due to an increased inlet temperature. A twin-screw compressor as part of a CRHP has been defined as the most suitable compressor for such industrial application. To be able to define and design this wet compressor more knowledge about the parameters of the compression process is needed. To evaluate these parameters a model of the twin-screw compressor needs to be developed. This model will need to consist of two separate models, a geometry model and a thermodynamic model, which will be based on Zaytsev’s [1] research. Unfortunately the model Zaytsev developed in 2003 is out dated, based on C++ and not able to run anymore. This thesis will describe the new model and will be based on Zaytsev’s simulation of the geometry and thermodynamics.

1.5 Research Question, Objective, Boundaries and Assumptions How can the process of the compressor as part of a Compressor Resorption Heat Pump be described in order to determine parameters for the design of the compressor suitable for application in a CRHP system? Objective: To develop a dynamic model that describes the thermodynamic process and includes the input from a twin-screw compressor geometry model. The CRHP has the aim to transform large industry waste streams like cooling water discharge into valuable re-usable energy carriers.

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The model and variables include the following assumptions and constraints. • The model will be based and build further on the work done by Zaytsev [22].• The thermodynamic model will be based on a homogeneous model. • The model will take as boundary conditions the heat pump process as described by van

de Bor et al. [21].• The process is based on an ammonia/water mixture of given composition. • The applied compressor is a wet twin-screw compressor where the geometry is

described in ref. [22, 29, 30].• The compressor should have an isentropic efficiency of at least 0.7. • The temperature to be achieved in the discharge of the compressor should be at least

115 °C, van de Bor et al. [21].

1.6 Chapters The thesis report describes the steps carried out to develop a working model that integrates screw compressor geometry with thermodynamic behaviour of the selected ammonia/water composition and physical properties. Chapter 2 explains the models that are used to simulate the geometry and thermodynamics of the twin-screw compressor. This chapter starts with the boundaries and assumptions followed by explaining the model. The geometry model will be explained in detail and will be followed by an explanation of the theory behind the thermodynamic model. Chapter 3 explains the development of the thermodynamic model in Simulink. It includes the input from the geometry model, design inputs and physical properties. It describes the implementation of the theories in the model structure developed in Simulink. The implementation also includes empirical correction factors for reality with internal leakage paths as the most influential ones. Chapter 4 describes the validation of the developed model in Matlab/Simulink including integrated physical properties, Refprop via Fluidprop, with the results from the model and experimental results of Zaytsev [1]. The discharge port area, rotor length and the empirical flow coefficient for the leakage have been adjusted to align the model results to the measured results of Zaytsev [1]. Chapter 5 describes the model with a set of boundaries as specified by van de Bor et al. [21] and with the geometry from Zaytsev [1]. The rotor length, discharge port area and number of lobes were varied. Results show improvement of isentropic efficiency and to achieve the wanted discharge pressure and temperature. Chapter 6 includes the conclusions of the development of the thermodynamic integrated model. Recommendations to optimise the compressor model are given.

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2 Models For the dynamic modelling of the twin-screw compressor a thermodynamic model has been developed during this thesis in which existing work and application of compressor geometry have been included. In this chapter both models will be described and explained. In this chapter the boundaries and assumptions used for the development of the models for the wet twin-screw compressor are clarified first. In the next sections the thermodynamic theory and the theory of the geometry will be explained in detail. In the last section the required outputs are described. The physical properties of the ammonia/water fluid will be invoked from Refprop [32] via Fluidprop [33]. A combination of Matlab [34] and Simulink, a tool within Matlab, will be used to model the wet twin-screw compressor.

2.1 Boundaries and Assumptions The compressor is part of a heat pump system, the CRHP. The complete system of the heat pump has been explained in section 1.1 and is shown in Figure 2-1. The system determines amongst others the boundaries that are applied to the compressor. The modelling of the compressor requires determining which properties the model has to comply with. The boundaries of the compressor need to be chosen clearly and used in the dynamic model to be able to simulate the compressor as part of the CRHP.

Figure 2-1 Compressor boundary in the CRHP cycle.

As can be seen in Figure 2-1, the compressor inlet comes from the desorber. The composition of the fluid entering the compressor is a mixture of ammonia and water leaving the desorber. The stream is assumed to be a two-phase fluid consisting mainly of vapour with a dispersed liquid phase. Wet compression will be applied to attain the higher temperatures that are required for the resorber. From a thermodynamic point of view, the most efficient application of the wet compressor is when the heat of vaporization is fully used and the discharge fluid is fully vaporised and at its condensation point. The boundary conditions of the compressor are determined by the required conditions for the resorber/desorber process. The resorber inlet receives a waste water stream of 45-60 °C, which needs to be increased to 110 °C, van de Bor et al.[21]. The desorber cools the waste water stream down from 45-60 °C to 5 °C, as illustrated in Figure 1-4.

CompressorValve

Desorber(Evaporator)

Resorber(Condenser)

Boundary

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It is assumed that for industrial applications a temperature of 110 °C from the resorber can be utilised, as well as a waste water temperature of 5 °C from the desorber. To reach the required waste water conditions, the heat exchanger requires a ∆T, driving force, of approximately 5 K to occur effectively. The compressor inlet temperature and outlet temperature become 40-55 °C and above 115 °C respectively. In Table 2-1 these properties are listed and are dependent on the heat pump cycle and the specific application. The values in the table are based on van de Bor et al. [21]. The compression will ideally be an isentropic process where the entropy remains constant. The compression is assumed to be an adiabatic process, where no heat transfer through the boundaries of the compressor to the environment will occur.

Table 2-1 Assumed conditions of inlet and outlet of the twin-screw compressor boundaries.

The properties in the table above are used as starting values for the simulation model of the compressor. The concentrations are set at these values during this research. The properties in the study need to be considered as variables. The concentrations are based on the optimal of the research of van de Bor et al. [20].

2.2 Geometry Model Theory Selection of the most suitable compressor type was done during research carried out by Zaytsev [1]. A wet twin-screw compressor was selected as the most suitable type. Screw compressors have the advantage of being able to compress gas while simultaneously transporting liquid. The screw compressor process can be divided into two phases: suction (inlet) and compression, within the compression phase the discharge takes place (the outlet). The geometry of the lobes and screws influences the performance of the compressor significantly. The geometry of the lobes can be adapted to fit the requirements of the process (pressure and temperature) and gas characteristics (composition, molecular weight, density, etc.). In literature and patents different kinds of geometries for the lobes of the screw compressor have been proposed, each with their own purpose. For most cycles the compressor efficiency is important for the cycle, and needs to be as high as possible. To achieve these requirements the geometry needs to be adjusted until the efficiency meets the optimum conditions. The desired outputs will be further explained in section 2.4. The change in volume of the cavity can be assumed to function as a batch type process. This batch process can be characterised and modelled. Together with its geometry each twin-screw compressor can be characterised by its cavity volume profile.

Properties Value UnitsWastestreamtemperaturerange 45-60 °CInlettemperaturecompressor 40-55 °CInletpressurecompressor 0.2 barOutlettemperaturecompressor ≥115 °COutletpressurecompressor <5 barConcentration H2O 70 wt%

NH3 30 wt%Theconcentrationrangeofammonia in themixture=20-35wt%[21]Forthemodelaconcentrationofx=30wt%willbeused.Inletcompressor Gas-liquidregimeCompressorprocess Adiabatictotheenvironment

22

A mathematical model has been developed to simulate the geometry of the screw compressor; the different methods, which have been proposed in literature, for this model are listed in Table 2-2. Although these methods make use of different starting positions to generate the geometry of the rotors, all are essentially based on the envelope theory.

Table 2-2 Geometry method twin-screw compressor.

Method Literature RotorsgeneratedfromEnvelopeMethodofGearing Stosicetal.[27] TheRackEnvelopeTheory Sakun[30],Zaytsev[1]andYou[31] RotorelementsMeshing-lineMethod ZaytsevandInfanteFerreira[28] Meshingline The envelope theory adapted to the geometry generation was used by Deng and Shu 1988 [35] and Rinder 1979 [36]. Stosic et al. [27], Sakun [30], Zaytsev [1] and You [31] make use of the same method but have a different approach towards describing the rotor profiles. The envelope theory is used to generate the rotor profiles of the male and female rotor. In the envelope theory the male rotor makes a curve that is used to generate the female rotor curve, shown in Figure 2-2, where the rotor profiles are illustrated as circles. Sakun [30] and Zaytsev [1] applied the envelope theory in such a way that the female profile is derived from the male profile. The male rotor only spins around its centre and remains static. The female rotor spins around its centre as well as turning around the male rotor, resulting in two separate movements. The curve of the male rotor conjugates the curve of the female rotor. The female rotor profile can be derived from the conjugated curve. The way the two rotors turn around each other produces a curve for each of the rotors. You [31] inverted the envelope theory, using the female rotor to calculate the male rotor. For a detailed explanation of the envelope method see You [31].

Figure 2-2 Envelope method [31].

The difference between Stosic et al. [27] and Sakun [30], You [31] , and Zaytsev [22] is that the profiles of both the male and the female rotor are calculated from an original profile of infinite radius called the rack The rack is generated on a static coordinate system. The rack is what distinguishes the method of Stosic et al. [27] from the others, even though it is based on the envelope theory as well.

R2h

R1h

23

Zaytsev and Infante Ferreira [28] developed another methods of calculating the rotor profiles, based on the work of Sakun [30], but with a different starting view. Zaytsev and Infante Ferreira [28] further developed a method based on the meshing line, the curve where the two profiles of the rotors meet. This meshing line is used to calculate both the male and the female rotor. In the method of Stosic et al. [27] the meshing line is an output variable. The meshing line is a crucial parameter for the performance of a compressor. Zaytsev and Infante Ferreira [28] developed their own model based on the meshing line as an input for the profile generation. Zaytsev and Infante Ferreira [28] choice of using the meshing line as an input parameter results in the meshing line becoming the main variable to describe the geometry of the compressor rotor as well as the profile of the lobes. The model used in this thesis is based on the original model developed by Sakun [30] and Zaytsev [1]. The model describes the relation between the two rotor profiles through angle relations. Coordinate systems were defined to describe the angle-based relation between the rotors. For each rotor there is one static coordinate system and one rotating coordinate system. Subscript 1 represents the male rotor, 2 the female rotor and 0 the static coordinate system. The coordinate systems are shown in Figure 2-3.

Figure 2-3 Coordinate systems of the two rotors [1].

The coordinate systems can be described mathematically. The relation of change of the angles is defined as the relation between the male and the female rotor profiles. The equations used in the geometry model to calculate the male and female profiles are shown below. For more details of these geometry equations see Zaytsev [1]. The rotor turning angle φ is used to calculate the x and y coordinates of the male and female rotors. In these equations the r1h is the male rotor radius from the origin to the beginning of the male lobe and the r2h is the female rotor radius from the origin of the female rotor to the lobe tip of the female rotor minus the tip radius of the female lobe, as can be seen in Figure 2-5. 1. Relation between the angles in the coordinate systems:

𝜑!𝜑!

=𝜔!𝜔!

=𝑏!𝑏!= 𝑖!" =

1𝑖!"

=𝑟!"𝑟!"

(2.2.1)

2. The relation between the rotation and axial motion, the z-axis can be calculated with this formula to make the profile 3D:

24

𝑧 = 𝐿 ∙𝜏𝜏! (2.2.2)

3. The male rotor profile in parametrical equations. ψ is the profile parameter:

𝑥! = 𝑥! 𝜓 𝑦! = 𝑦! 𝜓

(2.2.3)(2.2.4)

4. Male to female:

𝑥! = 𝑥! 𝜑!,𝜓 = −𝐷𝑐𝑜𝑠 𝑖!",𝜑! + 𝑥! 𝜓 𝑐𝑜𝑠 𝑘𝜑! + 𝑦! 𝜓 𝑠𝑖𝑛 (𝑘𝜑!)𝑦! = 𝑦! 𝜑!,𝜓 = 𝐷𝑠𝑖𝑛 𝑖!",𝜑! − 𝑥! 𝜓 𝑠𝑖𝑛 𝑘𝜑! + 𝑦! 𝜓 𝑐𝑜𝑠 (𝑘𝜑!)

IntheseequationsDisthedistancebetweentherotoraxis,andk=1+i21.

(2.2.5)(2.2.6)

5. Female to male:

𝑥! = 𝑥! 𝜑!,𝜓 = 𝐷 𝑐𝑜𝑠 𝜑! + 𝑥! 𝜓 𝑐𝑜𝑠 𝑘𝜑! − 𝑦! 𝜓 𝑠𝑖𝑛 (𝑘𝜑!)𝑦! = 𝑦! 𝜑!,𝜓 = 𝐷 𝑠𝑖𝑛 𝜑! + 𝑥! 𝜓 𝑠𝑖𝑛 𝑘𝜑! + 𝑦! 𝜓 𝑐𝑜𝑠 (𝑘𝜑!)

(2.2.7)(2.2.8)

6. Meshing conditions: 𝜕𝑥!𝜕𝜑!

𝜕𝑥!𝜕𝜓

𝜕𝑦!𝜕𝜑!

𝜕𝑦!𝜕𝜓

= 0

(2.2.9)

7. Male to static system: 𝑥! = 𝑥! 𝜓 𝑐𝑜𝑠 𝜑! + 𝑦! 𝜓 𝑠𝑖𝑛 𝜑! 𝑦! = −𝑥! 𝜓 𝑠𝑖𝑛 𝜑! + 𝑦! 𝜓 𝑐𝑜𝑠 𝜑!

(2.2.10)(2.2.11)

The inputs for the geometry model to generate the rotor profiles are shown in Table 2-3. The model calculates the other necessary parameters itself, using these input parameters. The radii in the table can be retrieved from Figure 2-5.

Table 2-3 Input variables for the geometry model.

Symbol UnitsEnveloperadiusmalerotor r1h mmRadiusmalerotorlobe r mm

RadiusmalerotorisR1=r1h+rRadiustipfemalelobe r0 mm

Enveloperadiusfemalerotorr2h,calculatedby𝒓𝟐𝒉 = 𝒓𝟏𝒉 ∙𝒃𝟐𝒃𝟏

RadiusfemalerotorisR2=r2h+r0Numberoflobesmalerotor b1 -Numberoflobesfemalerotor b2 -Wrapangleofmalerotor τw °Lengthoftherotors L mmClearance Clearance mm

The equations above describe the relation between the geometry of the rotors. Extra calculations are necessary to generate a representation of the rotor profiles. An example of such geometry is given in Figure 2-4.

25

Figure 2-4 Example geometry of the rotor profiles: left is the male rotor with 5 lobes, right is the female rotor with 6

lobes.

Figure 2-4 shows an example of the profiles of the male and female rotor. These profiles are calculated by means of different segments that form the rotor profiles together. First the male rotor areas are calculated, then the female rotor areas are calculated with the equations to go from one coordinate system to the other, as shown above. Six areas and segments are generated in total for each rotor lobe, which form one lobe of the male rotor together, as shown in Figure 2-5. The segments of the areas for the male rotor are: D1C1, C1A1, A1I1, I1L1, L1F1 and F1D1, for the female rotor the same script is used but with subscript 2. All lobes of the male rotor are identical, after generating one; the total profile of the male rotor can be generated. The same goes for the female rotor, as shown in Figure 2-4. For more details on the areas and segments see Sakun [30] and for the English version You [31].

Figure 2-5 The angles and segments of the male and female rotor.

-10 0 10 20 30 40 50 60-10

0

10

20

30

40

50

D1

C1

A1I1

L1

R2r2h

r1h R1

r

r0

r0

r0

r0

D2

A2I2

L2 F2

D2

O2O1

-10

0

10

20

30

40

50

60

-10

0

10

20

30

40

50

F1

D1

26

The areas and segments are all calculated separately, the area D1C1 is explained in detail to clarify the calculation of the areas. The D1C1 segments (from the O1 to D1 over the curve segment D1C1 and from C1 back to the origin) is shown in Figure 2-6 and the equation for the area is given below, 2.2.12. Area D1C1:

𝐷!𝐶! = −𝑟2∙ 𝑟!" ∙ 𝑠𝑖𝑛 𝜃!" − 𝑠𝑖𝑛 𝜃!"# + 𝑟 ∙ 𝜃!" − 𝜃!"# (2.2.12)

The equations of the area consist out of r, r1h, 𝜃!"# and 𝜃!". The angles are named up and low, describing the direction the angle is going (from the lower angle to the upper angle), the angles are calculated by using the cosine rule. 𝜃!"# becomes zero, for this segment, because this is the starting line and does not make any angle with a next point. 𝜃!" as can be seen in Figure 2-6 makes an angle between D1, C1 and the origin of the male rotor O1. These angles are used together with the radii to calculate the area of the segments. This method is used for calculation of all the areas, all with their own lower and upper angles. In the end all the six segment curves are generated with the area and the upper and lower angle. The segment curves are added together and from the total curve that can be seen in Figure 2-5. The equations that are used to calculate the areas of the other segments can be found in Appendix A.

Figure 2-6 Profile segment D1C1 of the male rotor.

Parameters that define the geometry of the twin-screw compressor are calculated using the geometry model. The main parameters are the cavity volume, suction port area and discharge port area. These three parameters are used as input variables for the thermodynamic model. Cavity volume: The cavity volume is the main parameter when defining the compression part in the compressor geometry model. The total cavity volume determines the change in volume. Cavity volume is defined as the change of volume over the shaft rotation angle and is determined by the geometry of the compressor, it is the main input for the thermodynamic model.

-10 0 10 20 30 40 50 60-10

0

10

20

30

40

50

D1

C1

A1I1

L1

F1

D1

r1h

R1

r

r0

r0

r0

r0A2

I2

L2 F2

D2

O1

r2hR2

O2

Θup

27

The volume calculation is separated into two parts, the suction calculation and the compression calculation (the discharge is part of the compression). In Figure 2-7 the red line represents the cavity volume. The increasing part of the graph represents the suction phase, the decreasing part the compression phase. The two phases together give the cavity volume change for the total compressor working cycle of one cavity. The unit of the cavity volume is mm3. Suction area: The suction area allows the mass flow to enter the compressor, and is calculated in mm2. The suction port is represented by the first blue curve. The up slope of the curve represents the period during which the suction port is opening. Once the suction port is fully opened, the area remains constant for a certain amount of ‘time’, described by the shaft rotation angle, represented in Figure 2-7 by the flat part of the curve. Eventually the suction port starts closing, which is represented in the figure by the down slope of the curve. The moment the suction port is closed the compression phase starts, which is represented by the down slope of the cavity volume, represented by the red line. Discharge area: Like the suction port, the discharge port opens and closes again, represented by the second blue curve in Figure 2-7. Less port area is needed to empty the cavity volume, as the fluid in the compressor has been compressed and the volume has decreased. The discharge area is calculated in mm2 as well.

Figure 2-7 Schematic representation of the cavity volume, shown in red. The suction area and discharge area are shown

in blue.

2.3 Theory of the Thermodynamic Model The physical and thermodynamic behaviour of the twin-screw compressor are described in a thermodynamic model. A schematic representation of a twin-screw compressor is shown below in Figure 2-8. This model will be used to optimise the twin-screw compressor, which is part of

Shaft Rotation Angle in Degrees0 100 200 300 400 500 600 700

Port

Area

in m

m2

0

1000

Cav

ity V

olum

e in

m3

#10-4

0

1

2

28

the complete CRHP process. The thermodynamic model of the compressor is described as a process with two phases, each with its own conditions and parameters: suction (inlet) and compression, discharge (outlet) phases. The compression takes place between the inlet and outlet, where reduction of the cavity volume results in increase of both pressure and temperature. Besides pressure, temperature and cavity volume, the other significant variables for the compressor are specific volume, enthalpy and mass flow. Some of these variables are inputs and others are calculated in the thermodynamic model. To develop a dynamic thermodynamic model, a homogeneous model that is reproducible with current modelling tools needs to be developed first. This is done with the use of the homogeneous model developed by Zaytsev [1]. For the simulation, the modelling tool Simulink is used extended with Matlab [34] and Refprop [32] via Fluidprop [33] to calculate the required physical properties. The thermodynamic model is influenced by the geometry of the wet twin-screw compressor. The geometry of the compressor needs to be integrated into the model, in order to produce a dynamic model that simulates the compressor as realistically as possible. The model used in this thesis is based on the geometry model developed by Zaytsev [1], which has been explained in section 2.2.

Figure 2-8 Schematic compressor.

The compressor inlet is expected to be in the two-phase flow regime. This is a given property from the chosen heat pump cycle, as described in section 2.1. Within the two-phase regime the pressure and temperature are dependent. In this regime the physical state cannot be modelled using a fixed temperature and pressure only. For the outlet of the compressor saturated vapour is assumed for achieving the highest possible efficiency of the heat pump cycle. Deviating from an exact full vaporisation reduces the efficiency and this will in reality require practical process control settings to reach optimal operation. In the homogeneous pT-model the liquid and vapour are at equilibrium at any given moment in time. This allows for writing the conservation equations for the working mixture inside the cavity volume. The cavity volume is defined in the compressor geometry model, section 2.2. The equations of Zaytsev [1] were written based on the shaft rotation angle 𝜑, the derivation of the conservation equations can be found in appendix B. In order to apply the modelling tool combination Matlab/Simulink, the conservation equations need to be converted to a time basis instead of the shaft rotation angle 𝜑 basis used by Zaytsev [1]. Rewriting the conservation equations is possible because the time for each degree of rotation can be determined. From a known rotational speed, the time difference per degree rotation angle can be calculated. With the calculated time per rotation, the same conservation equation can be used replacing the change in shaft rotation angle ∆𝜑 by the change in time ∆𝑡. The changes to the time dependent input values for the conservation equations will be explained in chapter 3.

Compressor

Inlet OutletUpstream Downstream

29

The four conservation equations for the homogeneous thermodynamic model of Zaytsev [1], are written as a function of pressure, concentration, temperature and mass flow over time: !"

!",

!!!!"

, !"!"

and !"!"

. For more details of the derivation of the conservation equations see Zaytsev [1] and Appendix B. These four functions describe the thermodynamic behaviour of the mixture in the cavity volume. If it is assumed that no liquid is separately injected, the concentration change over time !!!

!" can

be assumed to be zero. The only moment that this function will have a different value than zero will be when external fluid is added during compression. In the twin-screw compressor model external injection will not be applied and hence the concentration change will stay zero during the whole compression cycle. The function !!!

!" will therefore drop out of the conservation

equations as will the conservation equation for change of concentration itself. Therefore three conservation equations remain. The remaining three conservation equations can be combined into two. The combination can be accomplished as the pressure and temperature are dependent of the change in mass over time !"

!"= !

!𝑑𝑚𝑜𝑢𝑡

𝑑𝑡 𝑘

𝑛𝑘=1 −

𝑑𝑚𝑖𝑛

𝑑𝑡 𝑘

𝑙𝑘=1 . The !"

!" can

therefore be added to the !"!"

and !"!"

relations so that the equations result in two conservation equations that are applied in the model. The rewriting leads to the following two simplified conservation equations as applied in the homogeneous model. The mass conservation of the mixture becomes:

𝑑𝑝𝑑𝑡

=1

𝜕𝑣𝜕𝑝 !,!

𝑣𝑚

𝑑𝑚!"#

𝑑𝑡 !

!

!!!

−𝑑𝑚!"

𝑑𝑡 !

!

!!!

+1𝑚𝑑𝑉𝑑𝑡

−𝜕𝑣𝜕𝑇 !,!

𝑑𝑇𝑑𝑡

(2.3.1)

The energy conservation equation becomes:

𝑑𝑇𝑑𝑡

=𝑇 𝜕𝑣𝜕𝑇 !,!

𝑣𝑚

𝑑𝑚!"#𝑑𝑡 !

!!!! − 𝑑𝑚!"

𝑑𝑡 !

!!!! + 1

𝑚𝑑𝑉𝑑𝑡

𝜕𝑣𝜕𝑝 !,!

𝜕ℎ𝜕𝑇 !,!

+ 𝑇 𝜕𝑣𝜕𝑇

!

!,!

−

𝛿𝑄𝑑𝑡 + ℎ!",! − ℎ

𝑑𝑚!"𝑑𝑡 !

!!!!

𝑚 𝜕ℎ𝜕𝑇 !,!

+ 𝑚𝑇𝜕𝑣𝜕𝑝 !,!

𝜕𝑣𝜕𝑇

!

!,!

(2.3.2)

Equation 2.3.1 shows the mass conservation rewritten to pressure change over time !"

!" in units

of !"!

. Equation 2.3.2 shows the energy conservation rewritten to temperature change over the

time !"!"

in units of !!. The values for the physical parameters !"

!" !,!, !"!" !,!

and !!!" !,!

are

calculated in the homogeneous model with a combination of two fluid property programmes: Refprop [32] via Fluidprop [33]. The change in cavity volume over time !"

!" is an input together with the mass flows !"

!" of the

inlet, outlet and the leakages within the compressor. The implementation of the thermodynamic model will be described in chapter 3.

30

2.4 Theoretical Desired Performance Outputs To identify the theoretical performance of the modelled compressor, it is necessary to determine which outputs are required to describe that performance. The main outputs are isentropic efficiency, compressor shaft power and volumetric efficiency. These outputs are required to compare the different geometrical designs of the compressors and to select the most efficient one. Isentropic efficiency and compressor shaft power: One of the calculated and used efficiencies is the isentropic efficiency. The isentropic efficiency is determined by the ratio of the work for an ideal system and the work for a real system. In the ideal situation the losses of, for example, the friction are not added and in the real situation the losses are included. The isentropic efficiency is calculated with equation 2.4.1. Equation 2.4.1 is the ratio between theoretical work and real work from Moran and Shapiro [16].

𝜂!" =𝑊!",!"#$%

𝑊!"#$=ℎ!! − ℎ!ℎ! − ℎ!

(2.4.1)

The ideal line of the compression in the equation above goes from point 1 to 2s, respectively the inlet and outlet of the compressor, and follows the isentropic line that represents constant entropy. The real work of the compression, point 1 to 2, is increased compared with the ideal work; this is according to the prediction that more energy is needed if the losses of friction and flow are added. The work is representing the compressor shaft power (ideal or real) in the isentropic efficiency calculations. Volumetric efficiency: Next to the isentropic efficiency, the volumetric efficiency must be calculated. The volumetric efficiency gives the ratio of the ‘real’ or total volume capacity of the compressor and the ‘ideal’ actual volume of the fluid in the compressor, shown in equation 2.4.2.

𝜂!"# =𝑉!"#$𝑉!"#$%

(2.4.2)

The three desired outputs as explained above are used in the model to evaluate the efficiency of the modelled screw compressor. The implementation of the two efficiency equations into the thermodynamic model is explained in section 3.4.

31

3 Thermodynamic Model Implementation The thermodynamic model was developed to include and integrate inputs derived from geometry and physical property models. To develop the model, a number of approaches and trial attempts have been made. Explanation of the historic development will support the current modelling approach. This chapter includes the implementation of the equations and the inputs of the thermodynamic model. In the model implementation diagram, Figure 3-1 the flow of information over the several models and inputs is shown. The inputs for the thermodynamic model are divided into three sections, inputs from the geometry model, physical properties and the input values/initial values. These inputs are explained in section 3.2 ‘Input Implementation’. In section 3.3 the thermodynamic model is described and includes equations and correlations to describe real life compressor energy loss parameters such as friction and leakages. In section 3.4 the implementation of the theoretical desired outputs, the isentropic en volumetric efficiencies, are explained.

Figure 3-1 Model implementation diagram.

3.1 Historic Development The CHRP system is a development to upgrade low caloric industrial streams that generate cost rather than value to high caloric streams for industrial reuse through a heat pump system. The CRHP system includes the compressor as the piece of equipment that brings the circulation fluid to its required process conditions. Modelling the compressor such that it can fit in an overall model to describe the CRHP process was a specific challenge. The fluid that was selected for the system is an ammonia/water mixture that, through its properties, had the most optimum fit of thermodynamic behaviour and allowed for the desired operating conditions of the heat pump. Matlab/Simulink [34] is the tool selected that should be able to include the integration between the geometry model and thermodynamic model. The integration also required calculated property derivatives through an integrated physical property model. In the present implementation Refprop [32] via Fluidprop [33] is accessed from Matlab to calculate the required properties and its derivatives.

ThermodynamicModel(Simulink)

GeometryModel(Matlab)

PhysicalProper6es(Fluidprop/Refprop)

ImportantInputValues

VolumetricEfficiency

IsentropicEfficiency

Input Model DesiredOutput

Pressure

Temperature

Output

ShaCPowerRequirement

32

The geometry model developed by Zaytsev has been modified to fit the requirements of the present model. It contains the conservation equations and the full geometric modelling of the compressor in the 2 phases of inlet, compression/outlet. The rotation angle based geometry model that was originally developed, has been transferred into a time-based model that provides input to the time dependent thermodynamic model. The integrated model was built in Matlab/Simulink. The combination of Refprop and Fluidprop into Matlab/Simulink gave the possibility to integrate the generation of physical properties for input, compressor internal process and outputs. The thermodynamic model is now able to describe the compressor behaviour in time (dynamic) and generate output parameters and values for compressor design and optimisation and can be included in a full CRHP model.

3.2 Input Implementation The geometry model, physical properties and input values/initial values, are the inputs for the thermodynamic model. In this section it is explained in detail how the implementation into Simulink and the use of inputs to the conservation equations in the thermodynamic model has been carried out. A flow diagram of the modelling steps is shown in Figure 3-2. This diagram gives an overview of what parameters are needed where. The pressure and temperature are iterated on time-based steps. They are obtained from the conservation equations and are used as input for the conservation equations and the physical properties of the next time step until the time for which the end of the total compression cycle is reached. To allow the model to run, initial values and input values are required.

Figure 3-2 Flow diagram of the thermodynamic model, time based.

3.2.1 GeometryModelScrew compressors have a large variation of shape, lobe size, cavity volume and rotational speed. The variations as researched in the past have resulted in the geometry model. The model output parameters provide input for the thermodynamic model. In condensing the information from geometry to cavity volume, most shapes of screw compressors can be now modelled to provide input variables for the thermodynamic model. The geometry model is simulated in

Conserva)onEqua)ons(ThermodynamicModel)

Inputs

Outputs

Ini)alValues(PressureandTemperature)

Pressure&

Temperature

PhysicalProper)es(Input:PressureandTemperature)

Ift<

t totalcy

cle

Ift=ttotalcycle

33

Matlab and needs to be combined with the thermodynamic model in Simulink. In this thesis a specific case of geometry is considered. The input variables that are used for the geometry model are shown in Table 3-1 below, the values are variable and can be changed if needed.

Table 3-1 Input values used in the geometry model, these are the inputs of the geometry model as used by Zaytsev [1].

Matlab Value UnitsEnveloperadiusmalerotor r1h 35 mmRadiusmalerotorlobe r 13.8 mmRadiustipfemalelobe r0 1.5 mmNumberoflobesmalerotor b1 5 -Numberoflobesfemalerotor b2 6 -Wrapangleofmalerotor τw 314 °Lengthoftherotors L 172.5 mmClearance Clearance 0.1 mmSpeedofrotation n 3500 rpmConcentration NH3 37.6 wt%

As already explained in chapter 2, the geometry model is based on the shaft rotation φ. The thermodynamic model requires time-based inputs. To achieve the inputs based on time the total time of one compression cycle is calculated with equation 3.2.1.

𝑡𝑖𝑚𝑒 =1

𝑛60 ∙ 360

∙ 𝜑 (3.2.1)

The time will be in seconds. The compression cycle in this case consists of a shaft rotation angle of 777 degrees; the total time of one compression cycle becomes therefore 0.03695s. The shaft rotation angle is specific for this case and will vary for different geometries, dependent on the wrap angle, the number of lobes and the geometry of the rotors. Calculation of the shaft rotation angle is shown in detail in Appendix G. To calculate the volume change !"

!" in time from the cavity volume curve, the change over time

dt needs to be calculated. To model the geometry as time based, each compression cycle has been defined as an array of individual one degree angle rotations from 0-777 degrees. The rotation angle can be transformed to time with equation 3.2.1, adding one shaft rotation degree to this equation the time per degree becomes 4.76 ∙ 10!! s.

(a) (b)

Figure 3-3 Volumes of the compressor over time (a) the cavity volume and (b) the change in volume.

Time [s]0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04

Volu

me

[m3 ]

#10-4

0

0.2

0.4

0.6

0.8

1

1.2

1.4Cavity Volume

Time [s]0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04

Volu

me

Cha

nge

[m3 /s

]

-0.01

-0.008

-0.006

-0.004

-0.002

0

0.002

0.004

0.006

0.008

0.01Cavity Volume Change in Time "dV/dt"

34

The most important property used in the thermodynamic model is the cavity volume and it is displayed in Figure 3-3(a). To decrease the run time of the thermodynamic model, the volume change !"

!" will be calculated in the geometry model and applied as input for the thermodynamic

model instead of the cavity volume curve. The curve of the volume change over time, !"!"

, is shown in Figure 3-3(b). The volume change !"

!" in Figure 3-3(b) shows the rate of increase or decrease of the volume.

The volume change changes from positive: the increase of the cavity volume, to negative: the decrease of the cavity volume. The switch from positive to negative is on the maximum value of the cavity volume, at time 0.02186 s. The other two output variables of the geometry for the homogeneous model are the suction port area and discharge port area. The two areas can be represented as in Figure 3-4. In Figure 3-4(a) the suction area is shown. In Figure 3-4(b) the opening and closing area of the discharge port are shown. As expected the discharge port area is smaller than the suction port area. Less area is needed for the discharge due to the compressed fluid leading to increased density and therefore requiring less volume per mass flow and as such less area. The discharge port area in Figure 3-4(b) shows that the port closes two times during a single cycle. This phenomenon is redundant, the discharge port area should, like the suction port area, open and close just once. The so called close-trapped volume [1] causes an extra curve at the end of the discharge. The close-trapped volume is a volume that is formed between the two rotors after the discharge port is closed. The close-trapped volume is included in the calculated discharge port area. The influence on the thermodynamic model is however negligible and has been filtered out in the thermodynamic model in Simulink.

(a) (b)

Figure 3-4 Port areas in mm2 (a) is the suction port area and (b) is the discharge port area.

The port areas of the suction and discharge are used as inputs to calculate the mass flows into and out of the compressor. The suction port area will be used to calculate the mass flow of the inlet of the compressor and the discharge port will be used to calculate the mass flow of the outlet of the compressor. This will be explained in detail in section 3.3.1.

Time [s]0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04

Area

[mm

2 ]

0

100

200

300

400

500

600

700

800Suction Port Area

Time [s]0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04

Area

[mm

2 ]

0

100

200

300

400

500

600

700

800Discharge Port Area

35

3.2.2 Thermo-PhysicalPropertiesNext to the geometry model, input of thermo-physical properties is required. Eight thermo-physical properties have been defined for the model. The eight thermo-physical properties that are used are shown in Table 3-2. These are the physical properties that are needed for the conservation equation calculations (explained in section 2.3) and mass flow calculations (which will be explained in section 3.3.1). The thermo-physical properties are invoked through equation of state calculation tools. To invoke the thermo-physical properties two inputs are needed for a given composition, the pressure and the temperature. The composition is pre-set for the used ammonia/water mixture. The thermo-physical properties of the ammonia/water mixture are invoked from Refprop via Fluidprop by a Matlab/Simulink file. The pressure and temperature inputs are calculated by the conservation equations. The change in temperature divided by the change in time is integrated over time and results in the temperature evolution during the compression cycle. For the pressure the same calculation method is applied. The Simulink blocks are shown in Figure 3-5 and the needed Matlab code in Figure 3-6. Figure 3-5 shows a block diagram where the green blocks are the inputs, pressure and temperature, the orange block calculates the thermo-physical properties through Matlab and the red block the output !"

!" ! in this case.

Figure 3-5 Example of a Simulink block diagram to invoke physical properties for 𝒅𝒗𝒅𝒑 𝑻

.

The orange block in the Simulink block diagram above calls the Matlab file to invoke the thermo-physical properties. For this specific example the Matlab code is given in Figure 3-6. The code is to invoke the physical properties and to calculate !"

!" !, which is one of the inputs

for the conservation equations. To achieve the value for !"!! !

a small variation is needed in the

pressure (dp), in this case assumed 𝑑𝑝 = 𝑝 ∙ 10!! Pa. In this code example the specific volume is invoked twice. Once with the pressure and once with the pressure added with a small difference dp, both the invoked specific volume values are at constant temperature.

36

Figure 3-6 Example of the Matlab code to invoke the physical properties from Refprop via Fluidprop and calculate 𝒅𝒗𝒅𝒑 𝑻

.

More inputs are invoked from Refprop via Fluidprop and are listed in Table 3-2. In this table the first three physical properties in the table need to be calculated as explained above, ( !"

!" !,!,

!"!" !,!

, !!!" !,!

). The Table also lists the required values for calculating these thermo-physical properties. The last four properties are invoked similarly, but without additional calculations. The only inputs needed are the pressure and temperature.

Table 3-2 Physical properties invoked from Refprop via Fluidprop.

Physicalproperty Units SelectedDeltaValue𝒅𝒗𝒅𝒑 𝑻,𝒙

m!

kg ∙ Pa𝑑𝑝 = 𝑝 ∙ 10!! Pa

𝒅𝒗𝒅𝑻 𝒑,𝒙

m!

kg ∙ K𝑑𝑇 = 𝑇 ∙ 10!! K

𝒅𝒉𝒅𝑻 𝒑,𝒙

J

kg ∙ K𝑑𝑇 = 𝑇 ∙ 10!! K

h Jkg

-

s Jkg ∙ K

-

v m!

kg -

ρ kgm!

-

As can be noted in Figure 3-2, a time-step by time-step calculation is executed in the calculation. The output parameters of the conservation equations, the pressure and temperature are needed as the input parameters to calculate the thermo-physical properties. The thermo-physical properties are needed as input for calculation of the conservation equations in the next time-step. To start the thermodynamic calculation, input values, as well as initial values are needed.

3.2.3 InputValues/InitialValuesfortheImplementationThe thermodynamic model requires input values as initial values for the equation parameters to run the model. The input values are variables specific for every application and are based on the boundary conditions. As mentioned above, initial values for the conservation equations are necessary to start the iteration. The input values are determined by the process as selected for the heat pump and are related to the optimal values for the conditions imposed by the compression resorption heat pump cycle.

%% Diff Specific volume/pressure constant T [(m3/kg)/Pa]%% % With constant Temperature and concentration % function [output] = vp(p,T) global FP % Specific volume in m3/kg, Pressure in Pa, Temperature in K dp =1; v = invoke(FP,'SpecVolume','PT',p, T); v_dp = invoke(FP,'SpecVolume','PT',p+dp, T); vp = (v_dp-v)/(dp); output(1)=vp; end

37

The two most important input values are the input values that are required for integration of the output values of two of the conservations equations !"

!" and !"

!". The integration block in

Simulink carries out the integration.

3.3 Thermodynamic Model; Factors that Influence Ideal Behaviour A thermodynamic model describes the process ideally. To achieve a realistic description of the process, factors need to be incorporated to compensate for non-idealities/irreversibility. The non-idealities describe inefficiencies in comparison with the ideal thermodynamic model that influence the system performance. The inefficiencies that are of sufficient significance are described below. The inefficiencies are: • Flow friction • Leakage through several paths. • Heat transfer within the screw compressor housing. The above inefficiencies influence the thermodynamic model. Other inefficiencies will be more related to mechanical losses and will not be dealt with in this thesis. The ODE solver that is used in the thermodynamic model is the ‘Ode23tb’. The ‘Ode23tb’ is a stiff one-step method and is based on the Runge-Kutta method with two stages. The first stage is a trapezoidal rule step while the second stage uses a backward differentiation formula of order 2. Section 3.3 is divided into three sub-sections; 3.3.1 implementation of the mass flows, 3.3.2 leakage path areas and 3.3.3 mass flows of the leakages. The heat transfer within the screw compressor housing is expected to have limited effect and will not be taken into account in the model and will not be explained in this chapter.

3.3.1 MassFlowsThe mass flows entering and leaving the cavity volume are required inputs for the conservation equations on which the homogeneous model is based. Two isentropic converging nozzle equations are used to calculate the mass flows from the calculated port areas (geometry model). The two equations are given by Zaytsev [1] and will be explained below. The equations are based on inertia and pressure force. The viscous force is neglected. The geometry model and the thermodynamic model provide the inputs for these calculations. The equation that is used in the isentropic converging nozzle is the continuity equation 3.3.1.

𝑑𝑚𝑑𝑡 = 𝜁𝜌𝐴𝑢

(3.3.1)

For equation 3.3.1 an area A is needed as input. For calculating the mass flow in and the mass flow out of the working cavity, the areas of the suction port and discharge port are needed. The ζ is an empirical flow coefficient, ρ is the density and u is the flow velocity that will be calculated with equation 3.3.2.

𝑢 = 2 ∙ 𝑣𝑑𝑝

!!!"!

!!"#

(3.3.2)

38

For the calculation of the flow velocity u, integration is done between the high pressure and the low pressure of the fluid respectively entering and leaving the specific flow areas. The specific volume v is calculated with Fluidprop whilst it is assumed that the process is ‘isentropic’. The inputs for the continuity equation 3.3.1 and the flow velocity equation 3.3.2 are generated by the geometry model (the area A) and by the thermodynamic model (the density ρ and the specific volume v). The compressor is defined from inlet port to outlet port. Two parameters (boundaries of the inlet pressure suction and outlet pressure discharge) are added to the thermodynamic model as input for the input mass flow and output mass flow calculations, see Figure 3-7. The first parameter is the pressure difference over the suction port of the compressor. The pressure difference for the inlet is defined as the difference between the heat pump system outlet Phigh and the compressor inlet Plow, Phigh – Plow. The second parameter is the pressure difference over the discharge port of the compressor. The pressure difference between the compressor outlet Phigh and the heat pump system inlet Plow, Phigh-Plow.

Figure 3-7 Pressure differences of the mass flow of the suction and discharge of the compressor boundary. The pressure

difference is needed for the calculation of the mass flows in and out of the compressor.

One of the inputs of the continuity equations 3.3.1 is an empirical flow coefficient. The empirical flow coefficient is a correction factor for the non-isentropic effects (irreversible effect), for example the friction in the flow area. For the empirical flow coefficient different values are given. The reason is that the flow coefficient is dependent on each particular application such as the working fluid and the geometry. For the application in this case, an empirical coefficient is applied that is based on a constant mass flow in and out through the cross sectional area. It has to be acknowledged that this value is a variable that can and needs to be adapted for different applications with change of fluid, geometry, and actual clearance resulting from compressor manufacturing, etc.

Table 3-3 Empirical flow coefficient values.

Path PrinsandInfanteFerreira[37] Zaytsev[1]Leakageflow 0.3 0.7Intakeportflow 1.0 0.8Dischargeportflow 0.9 0.6

The flow coefficient values applied in this thesis originate from Zaytsev’s literature review [1], Table 3-3. The empirical flow coefficient determined by Prins and Infante Ferreira [37] have been experimentally obtained. In section 3.2.1 the cross sectional port areas of the suction and the discharge have been introduced which will become the input for calculating the mass flows in and out of the compressor. The port area of the suction and the discharge are generated with the geometry model based on the input values listed in Table 3-1.

CompressorBoundary

Suc1on Discharge

Phigh Plow PlowPhigh

39

In figures 3-8(a-b) the mass flow is graphically shown as time dependent of the rotor turning a full cycle from opening of the inlet port to closing of the outlet port. The calculated mass flow for the suction within the thermodynamic model is shown in Figure 3-8(a). The mass flow increases and decreases together with the opening and closing of the suction port. In Figure 3-8(b) the mass flow leaving the working cavity volume via the discharge port is shown. The mass flow follows the curve of the port area. The curves in Figure 3-8 are exemplary for the mass flows calculated by the model.

(a) (b) Figure 3-8 (a) Mass flow into the compressor through the suction port area, calculated with the input suction port area

from the geometry model, Figure 3-4(a). (b) Mass flow out of the compressor through the discharge port area, calculated with the input discharge port area from the geometry model, Figure 3-4(b).

The flows that enter and leave the cavity volume are divided into three flows: the suction flow, discharge flow and the internal leakage flows between the different cavity volumes. The mass flows from the suction and discharge ports are explained above. The internal leakage path areas will be explained in sections 3.3.2. These internal leakages are based on a single cavity volume. The calculation approach of the internal leakages will be explained in the section 3.3.3 and will also be based on the isentropic converging nozzle equations. The empirical flow coefficients used for the internal leakages calculations are listed in Table 3-3, Zaytsev.

3.3.2 LeakagePathAreasIn the twin-screw compressor there are places where leakages occur which determine the inefficiencies compared to the ideal thermodynamic and geometry model. This means that gas and/or liquid will leak from or to the control volume. Leakage is a negative flow in relation to the defined normal flow direction of the compressor. Leakage as defined here is a leakage from or to the ideal control volumes inside the compressor other than leakage to the outside of the compressor. The leakages in a twin-screw compressor are related to clearances between the two rotors and between the rotors and the housing. The clearances induce that the gas can flow to the different cavities of the compressor. These leakages are seen as flows in and out of the cavity and reduce the efficiency of the compressor. One cavity volume has been modelled and will be applied throughout the compressor model and is defined as the main cavity volume. For the single defined main cavity volume leakage from the leading cavity volume (advanced in time) and to the trailing cavity volume (delayed in time) will be modelled. All three cavity volumes are illustrated in Figure 3-10. The leakages mass flows will however depend on the increased pressure over the sequenced cavities and

Time [s]0 0.005 0.01 0.015 0.02 0.025 0.03 0.035

Mas

s flo

w [k

g/s]

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

"Suction Mass Flow"Mass flow through the suction port area

Time [s]0 0.005 0.01 0.015 0.02 0.025 0.03 0.035

Mas

s flo

w [k

g/s]

0

0.2

0.4

0.6

0.8

1

1.2

1.4

"Discharge Mass Flow"Mass flow through the discharge port area

40

hence change in density. These internal flow phenomena, together with the leakage mass flow calculation, will be explained in the next section 3.3.3. Leakage is one of the more significant mechanisms that influence the efficiency of the compressor according to Zaytsev [1]. Zaytsev modelled five leakage paths. That is one path less when comparing to Fleming and Tang [38] where six paths were identified and defined. In Figure 3-9 the reduction of efficiency of the six leakage paths of Fleming and Tang [38] is shown. As can be seen path 5 has the lowest efficiency reduction and has been excluded by Zaytsev [1] for implementation of the leakage paths in the thermodynamic model. Path 5 of Fleming and Tang [38] represented the leakage of the suction end clearance of the rotors.

Figure 3-9 Leakage paths efficiency, based on 3000 rpm and an evaporating temperature of 253.15 K [38]

The five paths that are defined are renumbered 1-5. The five paths that are modelled in the geometric model are: 1. Leakages through the contact line between the two rotors.2. Leakages through the sealing line between the tip of the rotors and the housing.3. Leakages through the cusp blowholes at compression side with high pressure.4. Leakages through the compression start blowholes at the suction side with low pressure.5. Leakages through the discharge end clearance. The five paths summarised above are illustrated in Figure 3-10, Figure 3-11, Figure 3-15 and Figure 3-16 and will be further explained in detail. The five leakage path areas are calculated in the geometry model and will be used as inputs for the thermodynamic model to calculate the inflow and outflow of the working cavity volumes. For calculation of the leakage flows, the same calculation as for the mass flows, section 3.3.1, is used and the implementation of the leakage flows will be explained in detail in section 3.3.3. In the geometry model the flow areas of the leakage paths 1, 2 and 5 are calculated with equation 3.3.3. The clearance is multiplied by the length of the contact line for path 1, the sealing line for path 2 or the height of the lobes (r+r0) for path 5, shown in Figure 3-16. The contact line, sealing line and the lobe height are represented by LL in equation 3.3.3 and will result in the cross sectional areas for paths 1, 2 and 5. The variable clearance that is used in this application is 0.01 mm, Table 3-1.

𝐴 = 𝑐𝑙𝑒𝑎𝑟𝑎𝑛𝑐𝑒 ∙ 𝐿𝐿 (3.3.3)

41

The blowhole areas are used for path 3 and 4. For path 3 the cusp blowhole is used and for path 4 the start blowhole is used. For the calculation of the blowhole areas the definitions of Singh and Bowman [39] are used. In Figure 3-10 the first three paths are illustrated. In Figure 3-11 path 1 is shown in more detail and in Figure 3-15 the two blowholes for path 3 and path 4 are shown. The discharge end clearance of the rotors for path 5 is illustrated in Figure 3-16 and shows two different kinds of paths that are used for path 5.

Figure 3-10 Illustrated leakage paths 1, 2 and 3. View from rotors above.

Leakage path 1 Leakage path 1 shown in Figure 3-10 is the leakage path across the contact line between the male and the female rotor. The area of the leakage path is modelled and calculated in the geometry model and used as an input for the thermodynamic model. The isentropic converging nozzle equation and the leakage path 1 area are used to calculate the amount of fluid flow across the contact line between the two rotors. In Figure 3-11 a side-view of the female rotor is shown. In this figure the contact line is shown with leakage path 1 divided into five smaller leakage flows numbered from 1 to 5. In the geometry file these five cross sectional areas of the leakage flow are added together to form one cross sectional area. As can be seen, path 1 creates a leakage from the compression side to the suction side.

Figure 3-11 Path 1: in this figure the female rotor is shown from the side. The line in the middle is the contact line and the arrows crossing the line illustrate the leakage direction of path 1 through the cross sectional area which is created by the contact line and the clearance. The mass flow of leakage path 1 flows from the compression side to the suction

side [1].

This leakage path is modelled in the geometry model in two different paths. One is the leakage through the contact line during the suction phase, Figure 3-12(a), and the second is the leakage through the cross sectional area during the compression phase of the compressor, Figure

Suc$on

Compression

Discharge

LeakagePath1

LeakagePath2

LeakagePath3

1

2 3

4

5

42

3-12(b). The suction phase and compression phase are related to the cavity volume, the red line in Figure 2-7, where the increasing slope represents the suction phase and the decreasing slope represents the compression phase. Figure 3-12(a-b) show each three curves, the total curve and two curves representing different segments of the contact line. The total leakage area of path 1 is the summation of the two segment curves together. The first segment is the part of the segment line from segment 1 to 5. The second segment is the leakage path area from segment 5 to 1. In the implementation of the leakage path areas, path 1 is implemented as the two separate segment curves. This is caused by the different pressure difference for each segment during the mass flow calculations of the leakage flow and will be described in detail in the next section.

(a) (b)

Figure 3-12 The leakage area of path 1, this is the leakage area created by the contact line between the two rotors. (a) Path 1 during the suction phase. (b) Path 1 during the compression phase.

Leakage path 2 Leakage path 2 is determined by the cross sectional area of the sealing line between the rotor tips and the housing, Figure 3-10. This path is calculated in the geometry model similar to path 1, see equation 3.3.3. Figure 3-13 shows the leakage area from the leading cavity to the main cavity with the solid line and the leakage area from the main cavity to trailing cavity with the dotted line. These two paths are the summation of the leakage areas of the female rotor and male rotor.

Figure 3-13 Leakage area of path 2 is the leakage through the sealing line between the rotors and the housing.

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Leak

age

area

[mm

2 ]

0

2

4

6

8

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14Leakage Area Path 1 - Contact Line - Suction Phase

Total Path AreaSegment 1-5Segment 5-1

Time [s]0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04

Leak

age

area

[mm

2 ]

0

2

4

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8

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14Leakage Area Path 1 - Contact Line - Compression Phase

Total Path AreaSegment 1-5Segment 5-1

Time [s]0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04

Leak

age

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[mm

2 ]

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40

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60Leakage Area Path 2 - Sealing Line

From Leading CavityTo Trailing Cavity

43

Leak path 3 Leakage path 3 is the leakage flow area in and out of the cusp blowholes, Figure 3-10 and Figure 3-15. The cusp blowholes are the spaces where the contact line does not reach the housing, the space between the contact line and the sealing line. These holes are called blowholes and have a triangular shape.

(a) (b)

Figure 3-14 Both path 3 and 4 are the leakage areas through a blowhole. In figures (a) for path 3 and (b) for path 4 are the leakage areas shown. Path 3 represents the leakage through the cusp blowholes on the compression side and path 4

represents the leakage through the start blowholes on the suction side.

In Figure 3-14(a) the cross sectional area of the leading cavity to the main cavity (solid line) and the cross sectional area from the main cavity to the trailing cavity (dotted line) are displayed. Like path 2 the pressure difference needed for this leakage flow calculation is the pressure difference between the leading, main and trailing cavities.

Figure 3-15 The two different blowholes. In this figure the female rotor is shown together with the contact line. In the

compression side is the cusp blow-hole shown and in the suction side is the compression start blow-hole shown [1].

Leakage path 4 Leakage path 4 is created by leakage cross sectional areas for the inflow and outflow of the compression start blowhole and is illustrated in Figure 3-15. In this figure the side-view of the female rotor is shown together with the contact line. The compression start blowhole is like the cusp blowhole (path 3) the difference is that the compression start blowhole is located at the suction side of the rotors. The leakage area of path 4 is shown in Figure 3-14(b).

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Leak

age

area

[mm

2 ]

0

0.5

1

1.5Leakage Area Path 3 - Cusp Blowhole Compression Side

From Leading CavityTo Trailing Cavity

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age

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[mm

2 ]0

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30

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60Leakage Area Path 4 - Start Blowhole Suction Side

From Leading CavityTo Trailing Cavity

44

Figure 3-16 Leakage paths at the discharge end clearance, path 5.

Leakage path 5 Leakage path 5 is the leakage through the discharge end clearance and is shown in Figure 3-16. The leakage is defined between the end plate, the rotor end face and the discharge end. In this area there are two different kinds of paths where the fluid can go. The first path is the area between the ‘discharge’ cavity (at the compression side of the rotors) and the ‘suction’ cavity (at the suction side of the rotors). The second path is the cross sectional area between the leading cavity, main cavity and the trailing cavity. Both the two different cross sectional areas are generated by the geometry model and are shown in Figure 3-17(a-b).

(a) (b)

Figure 3-17 Cross sectional areas of leakage path 5. This is the leakage through the discharge end clearance of the rotors. (a) Is the leakage path area between the compression side and suction side. (b) Is the leakage path area between

the leading cavity, main cavity and trailing cavity.

Figure 3-17(a) corresponds with the grey line in Figure 3-16, the solid line is the area from the discharge to the discharge end clearance, the dotted line is from the discharge end clearance to the suction. Figure 3-17(b) corresponds with the orange leakage flows in Figure 3-16. The solid line is the cross sectional area from the leading cavity to the discharge end clearance and the dotted line is the cross sectional area from the discharge end clearance to the trailing cavity. This together with delays between the area curves explains why there are four different cross sectional areas needed.

FemaleRotorMaleRotor

Dischargecavitytosuc5oncavity

Leadingcavitytotrailingcavity

LeakagePaths5

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age

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1.6Leakage Area Path 5 - Discharge End Clearence - Discharge/Suction

From DischargeTo Suction

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3.5Leakage Area Path 5 - Discharge End Clearence - Leading/Trailing

From Leading CavityTo Trailing Cavity

45

The areas of the leakage flows are used for the leakage flow calculations. The leakage flows are calculated in a similar way as the mass flows of the suction and discharge of the compressor in section 3.3.1.

3.3.3 MassFlowsoftheLeakagesThe mass flows of the leakage path areas will influence the efficiencies. By adding the mass flows of the leakages in the compressor, together with the empirical friction coefficient, it will bring the model from an ideal compression to a more realistic compression. For these mass flows, the equations of section 3.3.1 will be used. These are the same equations that are used for the mass flows into the compressor through the suction port and the mass flow out of the compressor through the discharge port. The mass flows of the leakage paths depend on the pressure difference and the leakage path cross sectional areas. The leakage areas are generated by the geometry file and have been explained in the previous section, section 3.3.2. To implement the mass flows of the leakage paths the pressure difference is the last variable that needs to be defined. The thermodynamic model is running over time. The pressure difference curve for calculating the mass flow of the leakage paths is based on the pressure difference between the advance pressure curve, the main pressure curve and the delay pressure curve. The three pressure curves need to be generated by different shifts in time. The main pressure curve is generated by the conservation equations in the thermodynamic model. The main pressure curve is used to generate the advanced pressure curve and the delay pressure curve. The shift needs to be done before the model starts running. The first pressure curve (main) is generated by the homogeneous model and is used as starting value. This pressure curve will be delayed and shifted in advance for the pressure difference calculations that are needed during the run of the thermodynamic model with the leakage paths. The pressure difference calculations are performed in a separated matlab file, see Appendix E. In this Matlab file the main pressure curve is used to calculate the advanced pressure and the delayed pressure, these three curves are used to calculate the pressure difference. The generated main pressure curve will be used for the calculation of the pressure difference by shifting the main pressure curve back and forward in time. Because the calculations have to be done back in time all three pressure curves will have a delay, see Figure 3-18. By adding the difference between the advanced curve and the main curve and the same difference between the main curve and the delayed curve results in the same delay and advance to the main curve.

Figure 3-18 Illustration of the pressure curve shift and the difference between the curves.

The delay and the difference 𝑑𝜑 between the three curves are added and the pressure difference can then be calculated. The advanced curve minus the main curve and the main curve minus the delay curve generates two pressure difference curves. The last step before these curves can be

DelayedCu

rve

MainCu

rve

Advanced

Curve

Time[s]

Pressure[P

a]

DelayDelay+dφDelay+2·dφ

dφ dφ

46

used is: The delay that is added to all the pressure curves is subtracted from the beginning of the pressure difference curve. The two pressure difference curves need to become the same length as the main pressure curve. The end of these curves is subtracted till the curves have the same length. To model the leakages in the thermodynamic model three different time shifts of the pressure curve need to be calculated. Each of the three are used for different leak paths: 1. Pressure difference between the leading, main and trailing cavity. 2. Pressure difference between the compression side and the suction side. 3. Pressure difference as a combination of pressure difference 1 and 2. The three pressure differences will be next explained in detail. The data in the plots are an example of how the pressure curves and pressure difference curve will look like. The inputs of the geometry file used are based on the experimental conditions reported by Zaytsev [1], Table 3-1. The initial values used in the thermodynamic model are a suction pressure of 3.7 bar, suction temperature of 62.8°C and a discharge pressure of 9.08 bar. These initial values will likewise be used in chapter 4 during the validation, Table 4-1. The pressure difference between the leading, main and trailing cavity is calculated by 360° divided by the number of lobes of the male rotor, b1. In this case (based on Zaytsev [1]) the male rotor has 5 lobes and results in a shift of 72°, each degree has a time step of 4.76 ∙ 10!! s with a rotational speed of 3500 rpm. In Figure 3-19(a) the pressure shift is shown. Figure 3-19(b) shows the pressure difference between the advanced – main pressure curve and the main – delay pressure curve. This graph is used in the thermodynamic model for the calculation of the mass flow of the leakage paths 2, 3, 4 and part of path 5. This part of path 5 is defined as the leakage between the leading/main and main/trailing cavity at the discharge end clearance, Figure 3-16. These four paths are divided into two different areas, explained in section 3.3.2. One area of the two represents leakage flow into the main cavity; in this case the advance/main pressure difference is implemented. The second area represents the leakage flow out of the main cavity and this mass flow is connected with the pressure difference between the main - delay pressure difference curve.

(a) (b)

Figure 3-19 (a) Graph of the pressure shift of 360/b1 in this case 72°, 0.003427 s. In this graph the original pressure curve is given. (b) The pressure difference between the leading - main and trailing – main cavity.

The next pressure difference that is needed is the pressure difference between the compression side and the suction side, Figure 3-15. This pressure difference is over the contact line. To calculate this pressure difference, a pressure curve shift is made by 360°, as can be seen in Figure 3-20(a). In Figure 3-20(b) the resulting pressure difference is shown. The pressure

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difference in this case is calculated the same way as done above. This pressure difference is applied in the mass flow calculations of the leakages as part of path 1 and part of path 5. The areas of Path 1 are explained in section 3.3.2 and for this pressure difference the area of segment 1 to 5 of the contact line is used. The leakage area of path 5 from the discharge to the suction is used together with this pressure difference with pressure shift of 360° to calculate the mass flow leakage of discharge side to suction side. The pressure differences are added in the same way as explained for the pressure shift of 72°.

(a) (b)

Figure 3-20 (a) Graph of the pressure shift of 360°. (b) Pressure difference between the compression side and the suction side.

There is one leakage area that needs a different pressure shift than the two pressure differences mentioned above. This leakage path area is the area of path 1, the contact line is divided into two, segment 1 to 5 and segment 5 to 1. This last segment 5 to 1 is at the contact line between the two cavity volumes. This requires adding 72° to the 360° pressure shift, Figure 3-21(a). Segment 5 to 1 is divided into compression and suction side and needs the two pressure differences as shown in Figure 3-21(b).

(a) (b)

Figure 3-21(a) Graph of the pressure shift of 360 degrees + 360/b1 in this case 72 degrees. (b) Pressure difference between the shifted pressure curves.

With the generated pressure differences, the leakage mass flows can be calculated. Implementation of the mass flows of the leakages to the compressor model is done in Simulink equal to the mass flows for the suction and discharge as based on the leak path calculation approach as explained above. To have a total overview of the thermodynamic model in Simulink, see Appendix C.

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The leakage mass flows in and out of the main cavity volume influence the mass flow from suction to discharge. Like path 1, the leakage flow over the contact line flows out of the main cavity volume is added to the mass flow out of the conservation equation (like the discharge mass flow) in the thermodynamic model. Path 2, 3, 4 and 5 have two different flows, into the main cavity volume and out of the main cavity volume. Out of the main cavity volume is added to the mass flow out and mass calculation is done as per path 1. The leakage flows of these paths into the main cavity volume are added to the normal mass flow into the cavity volume (like the suction mass flow), and are connected to the conservation equations. The quantities of the 5 mass flows of the leakage paths are shown in Figure 3-22. In this graph first the total leakage flow is shown together with the inflow (positive) and outflow (negative) of the main cavity volume. After the total leakage flow the individual paths are shown in the graph divided into total, in and out of the main cavity.

Figure 3-22 The graph shows the mass flow in kg/s. The total mass flow and the division: into the main cavity and out of

the main cavity.

As can be seen in Figure 3-22 is the leakage mass flows of path 2 through the sealing line form the largest amount in total into the cavity volume. Path 1 has the largest leakage flow out of the main cavity. The totalized leakage flow where all the paths are summed reaches almost the 0.023 kg/s. In Figure 3-23 the total leakage paths are added together with the suction mass flow and discharge mass flow. The suction mass flow reaches over 0.085 kg/s and the leakage mass flows less than 0.023 kg/s. The calculation is done with a clearance of 0.1 mm. Changing the clearance will influence the amount of the leakage mass flow. Increasing the clearance will result in an increase of the leakage mass flow.

Figure 3-23 The suction mass flow, the discharge mass flow and the leakage paths mass flows are illustrated in this

graph.

-0,08-0,06-0,04-0,020,000,020,040,06

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The leakages calculation within the thermodynamic model will be done through iteration. First the thermodynamic model will be run without leakage paths. Second, the pressure of the thermodynamic model without leakages is used as input for the thermodynamic model with leakage, the leakages are then added as explained before. The result of the model with the leakages is used again into the model with the leakages. This iteration process will continue until the results have converged.

3.4 Desired Outputs The theoretical desired outputs are the isentropic efficiency, compressor shaft power and the volumetric efficiency. The theoretical desired outputs explained in section 2.4 will be calculated with the outputs of the thermodynamic model. In this section the implementation of the theoretical desired outputs will be described. The desired outputs will be based on Zaytsev [22]. Matlab calculates these efficiencies after running the thermodynamic model in Simulink that provides the needed inputs for the theoretical desired outputs. The Matlab script for calculation of the efficiencies can be seen in Appendix F. Isentropic Efficiency The equation below is used for the calculation of the isentropic efficiency as proposed by Zaytsev [1]. Equation 3.4.2 is substituted into equation 3.4.1. Equation 3.4.1 shows the formula for isentropic efficiency [1]. In this equation the suction and discharge enthalpies are respectively the same as stage 1 and 2 in equation 2.4.1. 𝑊!"#$ is the shaft work that is needed to turn the rotors of the compressor for compression. The equation needed to calculate the shaft work is given in 3.4.2, where ω is the shaft rotational speed in rotations per second and b1 the number of male lobes.

𝜂!",!"#$ =𝑚!"#$ ℎ!"#$% − ℎ!"#$%&'

𝑊!"#$

(3.4.1)

𝑊!"#$ = 𝑏! ∙ 𝜔 ∙ 𝑝𝑑𝑉(3.4.2)

The isentropic efficiency consists of the ideal power and the real power. The ideal power is the mass flow of the compressor multiplied with the difference between the isentropic enthalpy and the suction enthalpy. The isentropic enthalpy is extracted from Refprop with the starting values at the suction side used to calculate the ideal discharge enthalpy with the same entropy, caused by the ideal and therefore isentropic compression. The suction enthalpy is extracted from the thermodynamic model. The suction enthalpy is the enthalpy in the suction. The enthalpy is in J/kg and the mass flow in kg/s, the calculated ideal power is in J/s. The number of male lobes, b1, the rotation speed per second, 𝜔, and the change in pressure by changing cavity volume calculates the real power. The number of male lobes, rotation speed and cavity volume change are taken from the geometry model. The cavity volume is in m3 and the pressure is in Pa, the calculated real power is in J/s. The isentropic efficiency becomes dimensionless.

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The higher the isentropic efficiency the less energy is lost. According to van de Bor et al. [21] the wet screw compressor is expected to perform with an isentropic efficiency of 0.7. This is within van de Bor’s needed conditions. During the research of Zaytsev [1] the predicted isentropic efficiency by a model of the screw compressor was around 0.67. The actual efficiency from experiments was less than 0.1. This difference between the two isentropic efficiencies was caused by various of reasons; one of the reasons was that in the first model (isentropic efficiency of 0.67) there was no labyrinth seal added to the model. Other reasons were the position of injection (in the suction), leakages in the compressor and damage of the lobes of the compressor, Zamfirescu [23]. The isentropic efficiency of less than 0.1 is way too low to be a functional option for a heat pump. The leakages in the screw compressor will influence the isentropic efficiency. For instance the leakages transport energy from the main cavity to other cavity volumes; this increases the total energy loss within the compressor. Volumetric Efficiency The second efficiency that is calculated as an output is the volumetric efficiency. The volumetric efficiency will change if there is a volume change within the cavity volume. In this case the efficiency is mainly influenced by the leakage flows that are implemented into the model. Without leakage paths and friction the volumetric efficiency of the homogeneous thermodynamic model will have an ideal volume flow and the efficiency will be identical to one. If the leakage flows are added, making use of the empirical coefficients, the volumetric efficiency will decrease and result in a lower efficiency compared to the model without leakages. The volumetric efficiency decreases even further if the injection is implemented in the model, this is the case for the results by Zaytsev [1]. The volumetric efficiency is calculated with equation 3.4.3.

𝜂!"# =𝑣!"#$%&'

𝑑𝑚𝑑𝑡 𝑑𝑡

𝑉!"#$%

(3.4.3)

Implementing the volumetric efficiency into Matlab, requires the following variables. The vsuction is the specific volume at the suction side in m3/kg. The !"

!"𝑑𝑡 is calculated by taking

the maximum mass leaving the main cavity in kg. The Videal is the maximum value of the cavity volume in m3. This makes the volumetric efficiency dimensionless.

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4 Validation Validating the twin-screw compressor model is essential for knowing if the developed integrated model is working correctly. The validation of the developed model will be done with the model by Zaytsev [1] and with the measured data from his experimental set-up. Zaytsev’s model case A will be used. The inputs of geometry and boundaries need to be set equal to the inputs of the model/experiments. Both models are based on a homogeneous phase. First the developed model will be validated with the model of Zaytsev [1] and after that, the output data will be validated with the experimental data. Validation with the model from Zaytsev case A and the experimental data is assumed to be adequate.

4.1 Inputs and Boundaries Validating the developed model with the data from Zaytsev, the inputs and boundaries need to be adjusted to the inputs and boundaries of Zaytsev’s experimental compressor. The inputs listed in Table 3-1 are the used geometry inputs. Zaytsev carried out different experiments with three different cases in which the boundaries and the concentration of the ammonia/water mixture were varied. In this validation chapter case A of Zaytsev is taken. The boundaries and the concentration of the mixture are listed in Table 4-1.

Table 4-1 Thermodynamic boundaries used during the validation [1].

Symbol Value UnitsTemperatureinletcompressor TSuction 62.8 °CPressuredischarge PDischarge 9.08 barPressuresuction(intakeport) PSuction 3.7 barConcentrationAmmonia x 0.376 kg∙kg-1

Zaytsev used the empirical flow coefficient to calculate the mass flows shown in Table 3-3: the empirical flow coefficient for the mass flow at the suction (intake) port and discharge port and the leakage mass flows. Zaytsev experimented with different clearances in a range of 0.06 mm and 0.14 mm. The used clearance is 0.1 mm see Table 3-1. In Figure 4-1 two different plots are shown calculated with the integrated model. In both plots the pressure curve, with leakages and without leakages is plotted. Figure 4-1(a) shows the pressure curve plotted versus time and in Figure 4-1(b) the pV-diagram of the two different pressure curves are shown. The pV-diagram gives a view of the pressure increase compared to the volume change in the compressor. The leakage flows between the cavities have an impact on the pressure curve and on the efficiency of the compressor. The leakages flow into the cavity volume during the suction phase from the leading cavity volume in compression phase. The pressure increase starts earlier in the model with the leakages compared to the model without the leakages. The cavity volume is full before the suction port closes and this causes the compression to start earlier. The leakages leaking into the cavity volume during the suction period have an effect on the amount of fluid that can be added to the cavity volume during the suction and therefore affects the isentropic efficiency.

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(a) (b)

Figure 4-1 (a) Shows the pressure curve without and with the leakages plotted to the time. (b) pV-diagram of the pressure curve without and with leakages.

The calculated isentropic efficiency, volumetric efficiency and the needed shaft power from the integrated model are listed in Table 4-2. As expected the isentropic efficiency and the volumetric efficiency are higher for the model without the leakages and the needed shaft power is lower. This difference is only caused by the leakages. The Isentropic efficiency decreases by 21%, the volumetric efficiency by 5% and the shaft power increases by 27%. In both the models friction losses caused by mass flow are taken into account by means of the empirical flow coefficient. The mechanical losses are not taken into account in the model.

Table 4-2 The calculated outputs of the integrated model with and without leakages.

IsentropicEfficiency

VolumetricEfficiency

ShaftPower[kJ/s]

WithoutLeakages 0.84 0.80 373.10WithLeakages 0.66 0.76 474.50

The computational model that includes leakages used by Zaytsev [1] resulted in an isentropic efficiency of around 0.67. Looking at the isentropic efficiency this model and the model used by Zaytsev are in the same range. From the boundary conditions (Table 4-1) with the geometry used by Zaytsev [1] a compression ratio of 2.45 can be calculated. This value will be the pressure ratio that can be reached with this geometry. Increasing the pressure ratio would require a different geometry or reducing the discharge port area.

4.2 Model Validation: Zaytsev The computational modelling by Zaytsev produced a pressure curve that has been validated against experimental measurements see Figure 4-2. The validation between this model and experiments produced sufficient overlap for him to conclude that he had a reliable model.

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Figure 4-2 Calculated and measured result by Zaytsev [1].

Zaytsev produced a pressure and volume curve as his main outcome. Figure 4-3 shows the comparison between the pressure curve generated by Zaytsev (calculated and measured) and the pressure curves generated with the current integrated model (with and without the leakages). Figure 4-3(a) shows the pressure curve plotted against the time and in Figure 4-3(b) the pV-diagram is plotted.

(a) (b)

Figure 4-3 (a) Shows the pressure curve without leakages, with the leakages, calculated and measured pressure by Zaytsev [1] plotted to the time. (b) pV-diagram of the pressure curve without leakages, with leakages, calculated and

measured pressure by Zaytsev [1].

The maximum pressure of the integrated model increases further during a longer period of time compared to the calculated results by Zaytsev [1]. This can be caused by insufficient mass flow leaving the compressor. Adjusting the discharge port area would be an option to solve this problem. It can be observed from Figure 4-3 that the cavity volume of the integrated model is smaller compared to the cavity volume used by Zaytsev [1], while the inputs of the models are equal. More detail on adjusting the model to the experimental data is given in section 4.3. The integrated models and the calculated model case A by Zaytsev [1] are homogeneous. The difference between the integrated models and the models of case A, calculated and measured,

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are a labyrinth seal, mechanical losses and the injection of the liquid into the compressor (has a large impact on the volumetric efficiency), see section 4.3. The integrated model is validated to the calculated model of Zaytsev [1]. As can be seen in Figure 4-2 the calculated model and measured results differ, this difference will be evaluated in the next section, together with the potential causes. The effect of a too small discharge port and a smaller cavity volume will be evaluated with the integrated model of this research.

4.3 Adapting the model to the experimental data To align the integrated model better with the experimental data, see Figure 4-2, requires adjusting the inputs of the developed integrated model. It makes the results out of the integrated model more realistic and closer to the experimental data. A more realistic integrated model will result in a more precise prediction during research. Some suggestions are made in the section before and Zamfirescu et al. [23] did research on improving the calculated results of Zaytsev [1] to get closer to the measured experimental results. Figure 4-4 shows the three different plots of the results of the integrated model with leakages, the calculated results from case A and the measured experimental result also from case A in a single pV-diagram. In the figure, the three plots can be compared. It shows close overlap of the current integrated model with calculated case A of Zaytsev [22] and similarity in shape with the measured case A.

Figure 4-4 Shows the pressure results of the integrated model with leakages, the calculated results (Zaytsev [1]) and the

experimental measured results (Zaytsev [1]).

The integrated model shows a higher maximum pressure in the pV curve compared to the measured case A. To adapt the integrated model to the measured case A, as suggested above, it is required to decrease the maximum pressure and allow the flow to increase for the same amount of mass out of the compressor. The total mass balance remains unchanged. This can be achieved by for instance increasing the discharge port area. Adapting the empirical flow coefficient compensates the flow behaviour in the compressor due to leakages and irreversibilities. Adapting these inputs results in modified inputs of the integrated model compared to the inputs used by Zaytsev [1]. In Figure 4-5 the original discharge port area (a) and the adapted discharge port area (b) are shown. The adapted maximum discharge port area has been increased by 41%, which will result in the same mass through the discharge port at a lower pressure difference.

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(a) (b)

Figure 4-5 (a) Original discharge port area used in section 4.2. (b) The adapted and increased discharge port area.

What also has been observed in Figure 4-4 is that the cavity volume in case A from Zaytsev [1] is larger than the cavity volume of the integrated model in this research. This is caused by the geometry model that is used. To solve this difference the cavity volume of the integrated model needs to be increased. Changing the inputs of the geometry model can result in an increase of the cavity volume; in this case an increase of the rotor length. The compressor length will be increased from 172.5 mm to 175.45 mm to increase the maximum cavity volume from 1.3129 ∙ 10!! m3 to 1.33524 ∙ 10!! m3 which is comparable with the maximum cavity volume in case A used by Zaytsev [1]. Adapting the model with the two changes discussed above will result in the pressure curve, shown in Figure 4-6. The model is calculated first without leakages to adjust the curve to the measured results before adding the leakages. In this way calculation time is saved. The curve without leakages has been adapted with an increased discharge port area and has reached similar values for cavity volume and pressure compared to the calculated case A by Zaytsev [1].

Figure 4-6 Adapted integrated model of this research closer to the calculated pressure curve used by Zaytsev [1] (The

integrated model is without the leakages and the calculated “Case A” by Zaytsev is with leakages).

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Zamfirescu et al. [23] carried out research on improving the prediction of the calculated results to the measured result used by Zaytsev [1] see Figure 4-2. Comparing these two results it needs to be kept in mind that the calculated results are from a homogeneous model and the measured is heterogeneous (Zaytsev [1]). Zamfirescu et al. [23] stated that not all leakage paths were considered and an underestimation of clearance areas was made. Also it is mentioned that the predicted maximum pressure at the discharge end phase is higher than the maximum pressure of the experimental measurements and that a faster increase of the pressure at the beginning of the compression phase occurs by the calculated model of Zaytsev [1]. Varying the empirical flow coefficient of the leakages, Zamfirescu et al. [23] adjusted the calculated model to the measured model. The empirical coefficient for the leakages has been adapted from 0.7 to 1.2. The result after adjusting the empirical flow coefficient is shown in Figure 4-7.

Figure 4-7 The adapted integrated model to the measured results of the experiments used by Zaytsev [1].

As can be seen in Figure 4-7 the pressure curve of this research is close to the measured results of the experiments used by Zaytsev [1]. The pressure peak at the discharge phase is still too high and the pressure increases slightly faster than the measured results. This can be the explanation for the isentropic efficiency difference. The isentropic efficiencies are listed in Table 4-3. Decreasing the pressure curve peak to get the curve even closer to the measured values can be done by making use of: increasing the discharge port area and increasing the empirical flow coefficient of the discharge mass flow calculation. The empirical flow coefficient needs to be established with experiments, this would be a subject of further research.

Table 4-3 The isentropic efficiency and shaft power of the adapted model.

IsentropicEfficiency

ShaftPower[kJ/s]

WithLeakages 0.66 474.50Adaptedmodel 0.031 532.77Calculated“CaseA” 0.056 UnknownMeasured“CaseA” 0.052 Unknown

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5 Model Results and Discussion The validation of the model has been carried out in chapter 4 and the integrated model can now be used for further research. In this chapter a number of variations will be evaluated illustrative for the possibilities of the integrated model, a step-up for further development. For the evaluation of variations, the integrated model of section 4.3 will be used however without leakages. The integrated model from chapter 4.3 with the geometry of Zaytsev [1] has been adapted with the boundaries used by van de Bor et al. [21]. This has been done to be able to evaluate the integrated model consistent with the requirements of this research as explained in section 1.5. After adapting the model to the boundary conditions by van de Bor et al. [21], the rotor length, the discharge port area and the vapour quality will be varied to evaluate what the influences are of these parameters on the compressor and the isentropic efficiency. As the variations are illustrative for the use of the model, it should be realised that these subjects require further investigation.

5.1 Leakages Evaluation will be done without the leakages. Leakages are amongst others determined by the clearances and the empirical flow coefficients. The influence of the leakages is that the isentropic efficiency decreases drastically with the larger size of the clearances and the high leakage empirical flow coefficients of Zaytsev [1] and Zamfirescu et al. [23]. The clearances and the leakage empirical flow coefficients are dependent on the design and the fabrication of the screw compressor and selection of fluid. The empirical flow coefficients need to be obtained through experiments. To allow a clear evaluation of the integrated thermodynamic model the leakages have been taken out. Additional advantage is that less calculation time is needed for the evaluation without leakages.

5.2 Boundary conditions ‘van de Bor’ The validated model has been adapted to the boundary conditions by van de Bor et al. [21], listed in Table 2-1. The suction temperature is set on 45 °C, the suction pressure on 0.2 bar and an ammonia mass concentration of 0.3. The used geometry in this section is the geometry used by Zaytsev[1] see Table 3-1. The boundary conditions deviate significantly from the conditions used during the validation. With the same geometry, the compression ratio will be similar and the discharge pressure can be expected to be in the range of 0.5 bar. Additional to the boundary conditions, the geometry has been changed to the best fit of rotor length and area of discharge as explained in section 4.3. The pV-diagram and temperature curve with the boundary conditions by van de Bor et al. [21] are shown in Figure 5-1.

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(a) (b)

Figure 5-1(a) pV-diagram and (b) Temperature curve, of the compressor with the geometry used by Zaytsev[1] and the boundary conditions of van de Bor et al. [21], without leakages.

As can be observed in Figure 5-1, with the defined input values, the discharge pressure has been adjusted manually to 0.55 bar. The pressure has been established through iteration from manual adaption of the discharge pressure in the integrated model. The pressure is the highest possible discharge pressure for the selected boundary conditions where back flow from the discharge system into the compressor is close to zero. This means that the compressor reaches this pressure by compression. This pressure is in the same range when using the compression ratio as has been used with the geometry by Zaytsev see section 4.1. In Figure 5-1(b) the temperature curve is shown: the starting temperature is 318.15 K (45 °C) and the temperature reaches 340 K (66.9 °C) during compression. The 340 degrees is much lower than the required 388.15 K (115 °C, section 1.5). The calculated isentropic efficiency and shaft power are listed in Table 5-1. The shaft power is very low compared to the values in chapter 4. This means less power is needed during the compression. In this case the pressure difference is just 0.35 bar, which is a small pressure difference according to the pressure difference during the validation in chapter 4 of around 5.3 bar. This explains the lower needed shaft power in this case with the boundaries of van de Bor et al. [21].

Table 5-1 Efficiency and shaft power of the compressor using the geometry used by Zaytsev [1] and the boundaries of van de Bor et al. [21].

IsentropicEfficiency

ShaftPower[kJ/s]

WithoutLeakages 0.56 21.11 To achieve the required results of the compressor model for the case of van de Bor et al. [21], some research and adaption of the input values have to be done. For this case more pressure difference is needed to achieve the required higher temperature of 388.15 K (115 °C). This can be achieved by for instance adjusting the geometry input.

5.3 Geometry variation An increase of the efficiency of the screw compressor can be achieved by varying the inputs of the geometry of the compressor rotors to an optimal design. In this section the influence of the rotor length, the discharge port area and the vapour quality will be evaluated. The rotor length has influence on the cavity volume. The discharge port area influences the pressure increase of the compressor. Both influence the isentropic efficiency.

Cavity Volume [m3] #10-40 0.2 0.4 0.6 0.8 1 1.2 1.4

Pres

sure

[Pa]

#104

1.5

2

2.5

3

3.5

4

4.5

5

5.5

6

pV-Diagram:Boundary Conditions van de Bor et al. [21]

Time [s]0 0.005 0.01 0.015 0.02 0.025 0.03 0.035

Tem

pera

ture

[K]

315

320

325

330

335

340

345

Temperature Curve:Boundary Conditions van de Bor et al. [21]

59

Rotor Length The rotor length has been adapted during the validation in chapter 4. By varying the rotor length in the model while keeping the other inputs constant, the effect of the rotor length on the efficiency can be described. Increasing the rotor length from 130 mm to 230 mm in steps of 10 mm will have influence on the cavity volume of the compressor since the rotor length is a part of the cavity volume calculation. In Figure 5-2 the different cavity volumes related to the different rotor lengths are plotted versus time. In this figure it can be observed that by increasing the rotor length it results in an increase of the cavity volume as expected.

Figure 5-2 Cavity volume dependent on the change of rotor length.

The pV-diagram with the different rotor lengths is shown in Figure 5-3. In this pV-diagram it can be seen that the longest rotor length, 230 mm, has the highest pressure increase and the shortest rotor length, 130 mm, the lowest pressure increase. This pressure increase is related to the cavity volume. The cavity volume is the representation of the volume in the cavity of the conservation equations in section 2.3. An increase of the cavity volume results in a steeper volume curve and an increase of the mass that can be compressed by the compressor. The steeper volume curve results in more pressure increase during the compression.

Figure 5-3 pV-diagram of the different rotor lengths.

Time [s]0 0.005 0.01 0.015 0.02 0.025 0.03 0.035

Cav

ity V

olum

e [m

3 ]#10-4

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

Rotor Length:Cavity Volume

130 mm140 mm150 mm160 mm170 mm180 mm190 mm200 mm210 mm220 mm230 mm

Cavity Volume [m3] #10-40 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

Pres

sure

[Pa]

#104

1.5

2

2.5

3

3.5

4

4.5

5

5.5

6

6.5

Rotor Length:pV-Diagram

130 mm140 mm150 mm160 mm170 mm180 mm190 mm200 mm210 mm220 mm230 mm

60

In Figure 5-4 the isentropic efficiency is plotted versus the rotor length. As can be seen, increase of the rotor length results in an increase in efficiency. The isentropic efficiency increase between the rotor length of 130 mm and 230 mm is 64 %.

Figure 5-4 Isentropic efficiency plotted against the rotor length in mm.

The relation between the isentropic efficiency and the maximum pressure and maximum temperature are plotted in Figure 5-5. As can be seen in the two plots there is an ‘asymptotic’ increase between the pressure/temperature and the isentropic efficiency.

(a) (b)

Figure 5-5(a) Pressure and the isentropic efficiency, (b) Temperature and the isentropic efficiency, with the different rotor length.

These results show that increasing the rotor length will increase the isentropic efficiency; all the inputs and boundary conditions are kept constant. The length of the rotors also influences some of the leakage paths, which will be enlarged by the increasing rotor length. For example increase of the rotor length will increase the contact line and the sealing line leakage path areas and will have a negative effect on the isentropic efficiency of the compressor, as an increased leakage area will result in an increased leakage mass flow. Discharge Port Area Like the rotor length also the discharge port area is varied in section 4.3 to get closer to the measured values of Zaytsev [1]. The effect of varying the area of the discharge port on the compressor will be evaluated. While varying the discharge port area the other inputs and boundary conditions will be kept constant, to show the effects of the discharge port area clearly. The discharge port area is calculated from the cavity volume. From the cavity volume a

Rotor Length [mm]130 140 150 160 170 180 190 200 210 220 230

Isen

tropi

c Ef

ficie

ncy

0.35

0.4

0.45

0.5

0.55

0.6

0.65

Rotor Length:Rotor Length - Isentropic Efficiency

Isentropic Efficiency0.35 0.4 0.45 0.5 0.55 0.6 0.65

Pres

sure

[Pa]

#104

5.5

5.6

5.7

5.8

5.9

6

6.1

6.2

6.3

6.4

Rotor Length:Pressure - Isentropic Efficiency

130 mm140 mm150 mm160 mm170 mm180 mm190 mm200 mm210 mm220 mm230 mm

Isentropic Efficiency0.35 0.4 0.45 0.5 0.55 0.6 0.65

Tem

pera

ture

[K]

340

340.5

341

341.5

342

342.5

343

343.5

Rotor Length:Temperature - Isentropic Efficiency

130 mm140 mm150 mm160 mm170 mm180 mm190 mm200 mm210 mm220 mm230 mm

61

part is taken, for example 1/3, of the maximum cavity volume that is used in this geometry. The maximum cavity volume in this ‘case’ is 1.335 ∙ 10!! m!. In Figure 5-6 all the different discharge port areas are presented that are used during the evaluation. The discharge port areas vary from 1/3 to 1/7 of the cavity volume in steps of 1/0.5.

Figure 5-6 The varied discharge port areas.

As can be seen in Figure 5-6 with the increase of the discharge port area, the opening of the discharge port shifts earlier in time. The closing of the discharge port area is for all the discharge port areas constant. This will have effect on the pressure increase in the compressor. The pressure curves of the different discharge port areas are shown in a pV-diagram, Figure 5-7. As can be seen in this figure, the smaller the discharge port area the higher the maximum pressure in the compressor will become. The discharge port areas of 1/3 to 1/4 have a steeper pressure curve; the increase of the pressure takes place at the discharge port opening. The discharge pressure in this case is set on 0.55 bar. The early opening of the discharge port results in advanced pressure increase and is not done by the compression but caused by back flow. It can be concluded that these discharge port areas are too large and open too early for this compression case. The other smaller discharge port areas 1/4.5 to 1/7 are small enough and open later in time such that the compressor is capable of compressing the fluid to above 0.55 bar before opening of the discharge port. It is a possibility to make the discharge pressure equal to the maximum pressure in the compressor. This could lead to an optimum between the discharge port area and discharge pressure.

Figure 5-7 pV-diagram of the different discharge port areas.

Time [s]0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04

Dis

char

ge P

ort A

rea

[mm

2 ]

0

100

200

300

400

500

600

700

800

900Discharge Port Areas

1/3 * Vmax1/3.5 * Vmax1/4 * Vmax1/4.5 * Vmax1/5 * Vmax1/5.5 * Vmax1/6 * Vmax1/6.5 * Vmax1/7 * Vmax

Cavity Volume [m3] #10-40 0.2 0.4 0.6 0.8 1 1.2 1.4

Pres

sure

[Pa]

#104

1

2

3

4

5

6

7

Discharge Port Area:pV-Diagram

1/3 * Vmax1/3.5 * Vmax1/4 * Vmax1/4.5 * Vmax1/5 * Vmax1/5.5 * Vmax1/6 * Vmax1/6.5 * Vmax1/7 * Vmax

62

The relation between the different discharge port areas and the isentropic efficiency is plotted in Figure 5-8. In this figure it can be observed that there is an optimal peak in the isentropic efficiency, between the discharge port of 1/4 and 1/4.5 of the cavity volume. This is the moment that the port opens when the pressure has reached the discharge pressure of 0.55 bar, Figure 5-7.

Figure 5-8 The discharge port area plotted to the isentropic efficiency.

From this evaluation it can be concluded that per case a specific discharge port area is needed. The discharge port should open at the moment that the compressor reaches the discharge pressure. Adapting each discharge port to the required case will increase the isentropic efficiency of the compressor. In this case using the correct discharge port area the isentropic efficiency can increase by 3.7 %. This analysis is with a constant discharge pressure of 0.55 bar with different discharge port areas. Changes in the discharge pressure in relation with the discharge port area can be considered. Vapour Quality The vapour quality is a parameter that is expected to have influence on the wet twin-screw compressor in view of the 2-phase fluid used. The vapour quality, here defined as the fraction of vapour of the ammonia/water mixture, is going to be varied and will be discussed in this chapter. The discharge pressure was set at 0.55 bar, in this analysis the discharge pressure will be adapted to the pressure curve with the varying vapour quality, this will be explained further on. With the used geometry and suction pressure of 0.2 bar the discharge pressure will be in line with the compression ratio as explained in chapter 4.1 and will vary slightly. The discharge pressure has to be established manually by adapting the discharge pressure that is a constant in the integrated model. The model needs to be run several times to verify the maximum pressure at opening of the discharge. With an increase of the maximum pressure, flow from the discharge system flows back in the discharge port, this indicates that the discharge pressure is too high for this case. The discharge pressure is set at the moment that there is no back flow into the compressor; this is the ‘optimal’ discharge pressure. As can be seen Figure 5-9, the vapour quality has been varied from 0.5 to 1. Varying the vapour quality is done by adapting the suction temperature to the suction pressure of 0.2 bar.

Discharge Port Area [mm2] #1041.5 2 2.5 3 3.5 4 4.5

Isen

tropi

c Ef

ficie

ncy

0.53

0.535

0.54

0.545

0.55

0.555

0.56

0.565

Discharge Port Area:Discharge Port Area - Isentropic Efficiency

1/3 * Vmax1/3.5 * Vmax1/4 * Vmax1/4.5 * Vmax1/5 * Vmax1/5.5 * Vmax1/6 * Vmax1/6.5 * Vmax1/7 * Vmax

63

For each vapour quality a different suction temperature is used and is listed in Table 5-2. In this table also the suction pressure and the respective optimal discharge pressure to the vapour quality is added.

Table 5-2 The suction temperature, suction pressure and related discharge pressure.

VapourQuality[kg/kg]

SuctionTemperature[°C]

SuctionPressure[bar]

OptimalDischargePressure[bar]

0.5 44.1 0.2 0.570.625 47.7 0.2 0.520.75 49.9 0.2 0.490.875 51.5 0.2 0.461 52.6 0.2 0.5

The pressure curves and temperature curves of the varied vapour quality are shown in Figure 5-9. The pressure curve is plotted together with the cavity volume curve in the pV-diagram (b) and the temperature is plotted versus time (b).

(a) (b)

Figure 5-9 (a) pV-diagram of the varied vapour quality. (b) Temperature versus Time of the varied vapour quality.

From Figure 5-9(b) it can be observed that the temperature of the discharge increases less for higher vapour quality. The model shows that the optimal discharge pressure reduces with increased vapour quality, Figure 5-9(a). The results for a vapour quality of 1 deviates from the results at lower vapour qualities. The phenomena cannot be fully explained and will require further research. The various interpretations of these phenomena will be reflected below. With increase of the vapour quality, the suction temperature is in equilibrium with the suction pressure that has been set constant at 0.2 bar. This results in a higher temperature in the suction with increasing vapour quality. This temperature adjustment also results in a smaller temperature difference between suction and discharge. With a vapour quality of 1 a much higher optimal discharge temperature is observed. This can be explained by the absence of the liquid. However the optimal discharge pressure result seems lower than expected. The homogeneous model is developed based on a 2-phase system. The homogeneous model is limited by the fact that the pressure and temperature are the same for both the two phases (gas and liquid). A heterogeneous model calculates the temperature of liquid and gas phase individually with the same pressure. The difference allows a different temperature for each

Cavity Volume [m3] #10-40 0.2 0.4 0.6 0.8 1 1.2 1.4

Pres

sure

[Pa]

#104

1.5

2

2.5

3

3.5

4

4.5

5

5.5

6

Vapour Quality:pV-Diagram

0.50.6250.750.8751

Time [s]0 0.005 0.01 0.015 0.02 0.025 0.03 0.035

Tem

pera

ture

[K]

310

320

330

340

350

360

370

380

390

400

410

Vapour Quality:Temperature versus Time

0.50.6250.750.8751

64

phase throughout the compressor and can lead to a different result. The homogeneous model could be the limitation and therefore using a heterogeneous model could solve the effect of the unexpected results. Conclusion The evaluation on the rotor length, discharge port area and the vapour quality shows there is room for optimisation using this integrated model. The rotor length and the discharge port area each have a different influence on the compressor efficiency. The increase of rotor length and the optimal discharge port area related to the discharge pressure has significant effect on the isentropic efficiency. The results of these two varied parameters conclude that each parameter needs to be adjusted to the required boundary conditions to achieve/design the optimal screw compressor for the required application. For the third parameter, the vapour quality, the unexpected phenomena need to be understood before coming to a conclusion. These three analyses are examples and all three need further research to evaluate the relations between the parameters.

65

6 Conclusions and Recommendations This research describes the development of a thermodynamic model with integrated geometry model of a wet twin-screw compressor. The objective of this research as described in 1.5 is: ‘to develop a dynamic model that describes the thermodynamic process and includes the inputs from a twin-screw compressor geometry model’. In this chapter the conclusions and recommendations will be discussed.

6.1 Conclusions A thermodynamic model for a wet twin-screw compressor has been developed in Simulink and is based on a homogeneous model by Zaytsev [1]. The model has been transformed from a rotation-based model to a time-based model, integrated with the geometry model developed by Zaytsev [1], and includes the physical properties continuously calculated by Refprop via Fluidprop. Chapter 4 discusses how validation of the developed integrated model was carried out. The integrated model is compared with the homogeneous model by Zaytsev [1] and the measured data from the experiment done by Zaytsev [1]. It can be concluded that this integrated model is working properly for further use in optimising the compressor. The thermodynamic model has been extended with leakage paths based on the work by Zaytsev [1] and the work of Zamfirescu et al. [23]. Zamfirescu et al. [23] did research on the calculated and measured data used by Zaytsev [1], which resulted in the use of a higher empirical flow coefficient. During the validation of the developed integrated model it has been concluded that the leakages are dependent on the clearances and the empirical flow coefficients of the geometry model. Data and information for both these values are too scarce to provide adequate values to describe the leakages in the actual screw compressor. The thermodynamic model conditions have been set on the boundary conditions described by van de Bor et al. [21] as described by the research objectives. The geometry of the screw compressor is based on the inputs used by Zaytsev [1]. With this combination of inputs and boundaries a discharge pressure of 0.55 bar, discharge temperature of 66.9 °C and an isentropic efficiency of 0.56 have been calculated, which represents the integrated model without leakages. The isentropic efficiency decreases even further when the leakages are taken into account. As a result, it can be concluded that the required isentropic efficiency of 0.7 and the discharge temperature of 115 °C cannot be reached for this set of inputs. Additional to the development of the integrated model an evaluation has been done with the boundary conditions by van de Bor et al. [21] and the inputs from the geometry used by Zaytsev [1]. The evaluation has been done by varying the rotor length, discharge port area and vapour quality. The evaluation should be seen as an example for the possibilities of varying the parameters and use of the integrated model. This evaluation clearly shows that these three parameters can be optimized to improve (the efficiency of) the screw compressor. Increasing the rotor length can have a positive effect on the increase of the pressure, temperature and the isentropic efficiency. The discharge port area needs to be chosen such to obtain the most optimal area and opening at the correct time to increase the isentropic efficiency and the increase of the pressure and temperature during the compression. The result for the varied vapour quality will require additional thoughts to explain within constraints of the integrated model. The results of the analysis have been modelled without inclusion of the leakages. Including the leakages will likely result in a different outcome. The next step in optimising the screw compressor should be to add the correct leakages, the correct clearances, and the empirical flow coefficient.

66

The integrated model shows that variations can be evaluated both in geometry as well as in thermodynamic behaviour. As a result, it can be concluded that for every new situation of boundary conditions a compressor with different specifications and geometry is needed.

6.2 Recommendations For every CRHP situation an optimal screw compressor geometry can be defined with the help of the model developed for this research. This will require restructuring of the parameters and their influence on the compressor thermodynamics, efficiency and geometry. The efficiency as stated by van de Bor cannot be achieved within the set boundary conditions. Further research needs to be done and adaptions need to be made to the screw compressor in order to match the requirements of van de Bor for use of the screw compressor in the CRHP. In this thesis the leakages have been modelled consistent with previous work (Zaytsev [1]). Further work research is needed on the clearances within the compressor, as well as on the empirical flow coefficients. These two inputs have a large influence on the compressor-specific model and the values therefore need to be adjusted to the compressor that is going to be used. Investigation into the tightest clearance that can be achieved as well as determining the actual empirical flow coefficient from experiments is needed. Having the actual clearances and the empirical flow coefficients corresponding to the actual screw compressor the leakages can be adapted into the model and the optimisation becomes more realistic. In this research a thermodynamically homogeneous model has been developed. To develop the model closer to the actual compressor a heterogeneous model should be considered, to achieve more realistic results. This model will need to include the changing equilibrium conditions between the liquid and the gas phase throughout suction, compression and discharge, and will result in a different temperature at constant pressure for each phase. The labyrinth seal, heat exchange between the compressor and the fluid, and the mechanical friction have not been modelled. Adding these three parameters will result in a more realistic model and more realistic experiments as well. With the integrated model, working in current modelling software, the geometry and thermodynamic aspects can be further evaluated. Variations in rotor length and discharge port area have been evaluated as illustrations to show how the integrated model can be used. Adapting the rotor geometry to the boundary conditions is a real challenge and needs more insight into the dependency between the parameters. Important for this area of improvement is that the discharge pressure needs to be adjusted together with the change in geometry, and is expected to result in higher pressures than modelled in section 5.3.

67

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[24] A. Lysholm and E. Al., ‘Rotary Compressor’, 2,111,568, 1938. [25] J. Wennemar, ‘Dry Screw Compressor Performance and Application Range’, MAN

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Appendices Appendix A: Envelope Method – Rotor Element Calculation

Figure A-1 The angles and segments of the male and female rotor [30]

𝑟𝑜𝑐 = (𝑟1ℎ2 + 𝑘𝑟!) 𝑘 = 1 + 𝑖21

Area D1C1

𝜃!"# = arccos 𝑅!! − 𝑟! − 𝑟!"! /(2 ∙ 𝑟 ∙ 𝑟!") 𝜃!" = arccos 𝑟𝑜𝑐! − 𝑟! − 𝑟!"! /(2 ∙ 𝑟 ∙ 𝑟!")

𝐷!𝐶! = −𝑟2 ∙ (𝑟!" ∙ 𝑠𝑖𝑛 𝜃!" − 𝑠𝑖𝑛 𝜃!"# + 𝑟 ∙ 𝜃!" − 𝜃!"# )

Area C1A1

𝜃!"# = 𝑖!" ∙ arccos 𝐷! + 𝑟!"! − 𝑟𝑜𝑐! /(2 ∙ 𝐷 ∙ 𝑟!") 𝜃!" = 𝑖!" ∙ arccos 𝐷! + 𝑟!"! − 𝑟!"! /(2 ∙ 𝐷 ∙ 𝑟!")

𝐶!𝐴! =12 𝜃!" − 𝜃!"# ∙ 𝐷! + 𝑟!"! ∙ 𝑘 −

1+ 𝑘𝑖!"

∙ 𝐷 ∙ 𝑟!"

∙ 𝑠𝑖𝑛 𝜃!" ∙ 𝑖!" − 𝑠𝑖𝑛 𝜃!"# ∙ 𝑖!"

Area A1I1

70

𝜃!"# = arccos𝑟!

(2 ∙ 𝑟!")

𝜃!" = 0

𝐴!𝐼! =𝑟!2 ∙ (𝑟!" ∙ (sin(𝜃!")− sin(𝜃!"#))− 𝑟! ∙ (𝜃!" − 𝜃!"#)

Area I1L1

𝜃!"# = −𝜃! − 𝜃! 𝜃!" = −𝜃! − 2 ∙ 𝛼! + 𝜃!

𝐼!𝐿! =12 (𝑟!" − 𝑟!)! ∙ (𝜃!" − 𝜃!"#)

Area L1F1

𝜃!"# = 0 𝜃!" = 𝑎𝑟𝑐𝑜𝑠

𝑟!(2 ∙ 𝑟!")

𝐿!𝐹! =𝑟!2 ∙ (𝑟! ∙ 𝜃!" − 𝜃!"# − 𝑟!" ∙ (sin 𝜃!" − sin 𝜃!"# )

Area F1D1

𝜃!"# = 𝑖!" ∙ arccos 𝐷! + 𝑟!"! − 𝑟!"! /(2 ∙ 𝐷 ∙ 𝑟!") 𝜃!" = 𝑖!" ∙ arccos 𝐷! + 𝑟!"! − 𝑅!! /(2 ∙ 𝐷 ∙ 𝑟!")

𝐹!𝐷! =12

𝜃!" − 𝜃!"# ∙ −𝐷! − 𝑟!"! ∙ 𝑘 + (1+ 𝑘)𝑖!" ∙ 𝐷 ∙ 𝑟!" ∙ (sin 𝜃!" ∙ 𝑖!" − sin 𝜃!"# ∙ 𝑖!" )

71

Appendix B: Conservation Equations of the Homogeneous Model The conservation equations of the homogeneous model are based on Zaytsev [1]. In this homogeneous model the liquid/vapour phases are in equilibrium during the compression cycle. In this appendix a detailed derivation of the conservation equations that have been applied in the time dependent model is given. The following three equations B.1, B.2 and B.3 have been rewritten to time dependent conservation equations !"

!", !!!!"

and !"!"

. The conservation of the mass of the mixture:

𝑑𝑚 = 𝑑𝑚!",!

!

!!!

− 𝑑𝑚!"#,!

!

!!!

(B.1)

The conservation of the ammonia mass:

𝑑 𝑚𝑥! = 𝑥!!",!𝑑𝑚!",!

!

!!!

− 𝑥! 𝑑𝑚!"#,!

!

!!!

(B.2)

The conservation of energy:

𝛿𝑄 + ℎ!",!𝑑𝑚!",!

!

!!!

− ℎ 𝑑𝑚!"#,! = 𝑑𝐻 − 𝑉𝑑𝑝!

!!!

(B.3)

In the equations m is the mass of the fluid mixture. The overall concentration of the ammonia-water mixture is denoted by x0. 𝛿𝑄 is the heat transfer between the compressor surface and the mixture. There are two different enthalpies in the equations, h is the specific enthalpy and H is the enthalpy of the homogeneous mixture. V represents the control volume in the screw compressor. The subscripts in and out are the inflows and outflows of the cavity volume of a compression cycle. The left hand side of equation B.1 can be written as:

𝑑𝑚 = !!𝑑𝑉 + 𝑉𝑑 !

! (B.4)

In the equations above v is the specific volume and is a function of the pressure p, temperature T and the overall concentration x0 . The equation B.4 is differentiated by the time and shown in B.5.

𝑑𝑚𝑑𝑡

=1𝑣𝑑𝑉𝑑𝑡

−𝑉𝑣!

𝜕𝑣𝜕𝑝 !,!

𝑑𝑝𝑑𝑡

+𝜕𝑣𝜕𝑇 !,!

𝑑𝑇𝑑𝑡

+𝜕𝑣𝜕𝑥! !,!

𝑑𝑥!𝑑𝑡

(B.5)

Equation B.1 is differentiated by the time. It becomes the mass flow equation of the inflow and outflow of the compression:

𝑑𝑚𝑑𝑡

=𝑑𝑚!"

𝑑𝑡

!

!!!

−𝑑𝑚!"#

𝑑𝑡

!

!!!

(B.6)

The mass conservation equations B.6 and B.5 can be combined after rearranging the terms to !"!"

and becomes:

72

𝑑𝑝𝑑𝑡

=1

𝜕𝑣𝜕𝑝 !,!

𝑣𝑚

𝑑𝑚!"#

𝑑𝑡 !

!

!!!

−𝑑𝑚!"

𝑑𝑡 !

!

!!!

+1𝑚𝑑𝑉𝑑𝑡

−𝜕𝑣𝜕𝑇 !,!

𝑑𝑇𝑑𝑡

−𝜕𝑣𝜕𝑥! !,!

𝑑𝑥!𝑑𝑡

(B.7)

The differentiated conservation equation of the ammonia mass B.2 written as time dependent:

𝑚𝑑𝑥!𝑑𝑡

+ 𝑥!𝑑𝑚𝑑𝑡

= 𝑥!!",!𝑑𝑚!"

𝑑𝑡 !

!

!!!

− 𝑥!𝑑𝑚!"#

𝑑𝑡 !

!

!!!

(B.8)

The equation for the concentration change over time is obtained after adding and rearranging equations B.6 and B.8. These two equations result in the equation !"!

!" used for calculating the

change in concentration of the ammonia/water mixture, B.9.

𝑑𝑥!𝑑𝑡

=1𝑚

𝑥!!",!𝑑𝑚!"

𝑑𝑡 !

!

!!!

− 𝑥!𝑑𝑚!"

𝑑𝑡 !

!

!!!

(B.9)

The conservation equations have been rewritten to make the ammonia/water concentration dependent over time. This means that !"!

!" will change over time when the concentration of the

inflow of the compressor differs from the outflow of the compressor. That situation will for example occur if liquid of the fluid is injected downstream of the inlet stream in the compressor. The energy conservation equation B.3 is differentiated to the time and for the enthalpy change !"!"= 𝑚 !!

!"+ ℎ !"

!" is added, shown in B.10.

𝑚𝑑ℎ𝑑𝑡 + ℎ

𝑑𝑚𝑑𝑡 =

𝛿𝑄𝑑𝑡 + ℎ!",!

𝑑𝑚!"

𝑑𝑡 !

!

!!!

− ℎ𝑑𝑚!"#

𝑑𝑡 !+ 𝑉

𝑑𝑝𝑑𝑡

!

!!!

(B.10)

The change of specific enthalpy divided by the change in time, B.10, can be considered as a function of pressure, temperature and concentration: !!!"= !!

!" !,!

!"!"+ !!

!" !,!

!"!"+ !!

!!! !,!

!!!!"

. Substituting this formula in equation B.10 leads

B.11 as the result.

𝑚𝜕ℎ𝜕𝑝 !,!

𝑑𝑝𝑑𝑡

+𝜕ℎ𝜕𝑇 !,!

𝑑𝑇𝑑𝑡

+𝜕ℎ𝜕𝑥! !,!

𝑑𝑥!𝑑𝑡

+ ℎ𝑑𝑚𝑑𝑡

=𝛿𝑄𝑑𝑡

+ ℎ!",!𝑑𝑚!"

𝑑𝑡 !

!

!!!

− ℎ𝑑𝑚!"#

𝑑𝑡 !+ 𝑉

𝑑𝑝𝑑𝑡

!

!!!

(B.11)

Equation B.7 is added to equation B.11 to include !"

!". The next partial derivative for the

enthalpy by pressure is used for the thermodynamic relation: !!!" !,!

= 𝑣 − 𝑇 !"!" !,!

. The

conservation equation of energy B.3 results after substituting and rearranging the equation for the change of temperature divided by the change of time !"

!", shown in equation B.12.

73

𝑑𝑇𝑑𝑡

=𝑇 𝜕𝑣𝜕𝑇 !,!

𝑣𝑚

𝑑𝑚!"#𝑑𝑡 !

!!!! − 𝑑𝑚!"

𝑑𝑡 !

!!!! + 1

𝑚𝑑𝑉𝑑𝑡 −

𝜕𝑣𝜕𝑥! !,!

𝑑𝑥!𝑑𝑡

𝜕𝑣𝜕𝑝 !,!

𝜕ℎ𝜕𝑇 !,!

+ 𝑇 𝜕𝑣𝜕𝑇

!

!,!

−

𝛿𝑄𝑑𝑡 + ℎ!",! − ℎ

𝑑𝑚!"𝑑𝑡 !

−𝑚 𝜕ℎ𝜕𝑥! !,!

𝑑𝑥!𝑑𝑡

!!!!

𝑚 𝜕ℎ𝜕𝑇 !,!

+ 𝑚𝑇𝜕𝑣𝜕𝑝 !,!

𝜕𝑣𝜕𝑇

!

!,!

(B.12)

The equations B.7, B.9 and B.12 are the adapted conservation equations that give the information for pressure, concentration and temperature within the compressor in the homogeneous model.

74

Appendix C: The Thermodynamic Model in Simulink The thermodynamic model has been developed in Simulink. In this appendix the block-calculations in Simulink will be explained. The block calculations show the integration of physical properties in the thermodynamic model. The inputs are defined by the conservation equations. Each conservation equation is explained in the figures below:

- Figure C-1 Conservation equation, change in temperature. - Figure C-2 Conservation equation, change in pressure. - Figure C-3 Integration to pressure and temperature. - Figure C-4 Volume change input. - Figure C-5 Physical property calculation. - Figure C-6 Mass calculation by integration of the mass flows. - Figure C-7 Mass flow calculations suction, discharge and leakages. - Figure C-8 Pressure difference suction and discharge port. - Figure C-9 Pressure difference between cavities.

Figure C-1 The conservation equation of the temperature change: dT. Gets its inputs from the physical

property calculations, mass flows and other calculated values.

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Figure C-2 The conservation equation of the pressure change: dP. Gets its inputs from the physical

property calculations, mass flows and other calculated values.

Figure C-3 The integration blocks for calculating the pressure and temperature during the compression cycle. The inputs come from the conservation equations Figure C-1 and Figure C-2.

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Figure C-4 Input of the conservation equations, this is the volume change over time calculated in the

geometry model.

Figure C-5 The physical property calculation Refprop [32] via Fluidprop [33] is explained in detail in

section 3.2.2.

Figure C-6 The calculated mass is needed as input for the conservation equations. The inputs for the mass calculations are the mass flow in and out of the cavity volume. The mass flows are integrated to the mass by

the integrator block. The 0.00001 is added because the mass cannot become zero, this would result in an error at the conservation equation (divided by zero). The mass flows come from the suction/ discharge mass

flows and the leakage mass flows.

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Figure C-7 The calculation of the suction mass flow, similar to the discharge and leakage mass flows

calculation (the inputs will differ for each different mass flow). The suction port area is imported from the geometry model; the constant is the empirical flow coefficient.

Figure C-8 The calculated pressure difference for the mass flow calculation at the suction and discharge.

The constant blocks are the pressure values of the pressure outside the compressor boundaries before entering or leaving the compressor.

78

Figure C-9 The implemented calculated pressure difference that is calculated in the pressure delay Matlab file . These are the pressure differences used in the calculation for the leakage mass flows, explained in

detail in section 3.3.3.

79

Appendix D: The Geometry Model in Matlab, van de Bor/Zaytsev [Matlab-Code] format compact clear all clc tic global fdischarge global fsuction global V_dt %%% geo_in.txt r1h=24; % radius male mm r=16; % radius mm r0=1.5; % radius mm m1=4; % number of lobes male rotor m2=6; % number of lobes female rotor thau1zd = 300; % wrap angle of male rotor in degrees L = 200; % length of rotor mm epsilonv = 4; % Needed during the Discharge port area calculation clearance = 0.01; % the clearance n = 3000; % Rotation speed rpm; filename = strcat(int2str(r1h) , '_' , int2str(r), '_' , num2str(r0) , '_' , int2str(m1) , '_' , int2str(m2) , '_' , int2str(thau1zd) , '_' , int2str(epsilonv) , '_' , int2str(clearance), '_test3'); save(filename) r2h = r1h * m2 / m1; A = r1h + r2h; % distance between the two origin of the rotors i12 = r2h / r1h; i21 = 1 / i12; k = 1 + i21; M = 1 + i12; R1 = r1h + r; % male rotor radius R2 = r2h + r0; % female rotor radius %%% calculation of central angles theta2 = i12 * acos(1 + (r1h^2 - R1^2) / (2 * A * r2h)) - acos((A^2 + R1^2 - r2h^2)/(2 * A * R1)); theta3 = acos(1 - r^2 / (2 * r2h^2)); theta1 = i12 * theta3; theta4 = i21 * theta2; theta5 = acos(1 - r0^2 / (2 * r1h^2)); theta6 = acos(1 - r0^2 / (2 * r2h^2)); alpha0 = 0.5 * (2 * pi / m1 - theta1 -theta2); beta = acos((A^2 + R1^2 - r2h^2)/(2 * A * R1)); thau1z = thau1zd / 180.0 * pi; %%% conversion to degrees theta2d = theta2*180/pi; theta3d = theta3*180/pi; theta1d = theta1*180/pi; theta4d = theta4*180/pi; theta5d = theta5*180/pi; theta6d = theta6*180/pi; alpha0d = alpha0*180/pi; betad = beta*180/pi; %%% create geometry structure geometry.r1h=r1h; geometry.r=r; geometry.r0=r0; geometry.m1=m1; geometry.m2=m2; geometry.thau1zd = thau1zd; geometry.L = L; geometry.epsilonv = epsilonv; geometry.thau1z = thau1z; geometry.r2h = r2h; geometry.A = A; geometry.i12 = i12; geometry.i21 = i21; geometry.k = k; geometry.M = M; geometry.R1 = R1;

80

geometry.R2 = R2; geometry.theta2 = theta2; geometry.theta3 = theta3; geometry.theta1 = theta1; geometry.theta4 = theta4; geometry.theta5 = theta5; geometry.theta6 = theta6; geometry.alpha0 = alpha0; geometry.beta = beta; geometry.n = n; %%% segment d1c1 roc = sqrt(r1h^2 + r^2 * (1 + i21)); psidc1low = acos((R1^2 - r^2 - r1h^2)/(2 * r * r1h)); psidc1up = acos((roc^2 - r^2 - r1h^2)/(2 * r * r1h)); d1c1 = D1C1_(psidc1low, psidc1up, geometry); %%% segment c1a1 tca1low = i12 * acos((A^2 + r2h^2 - roc^2)/(2 * A * r2h)); tca1up = i12 * acos((A^2 + r2h^2 - r1h^2)/(2 * A * r2h)); c1a1 = C1A1_(tca1low, tca1up, geometry); %%% segment a1i1 psiai1low = acos(r0 / (2 * r1h)); psiai1up = 0; a1i1 = A1I1_(psiai1low, psiai1up, geometry); %%% segment i1l1 til1low = -theta1 - theta5; til1up = -theta1 - 2 * alpha0 + theta5; i1l1 = I1L1_(til1low, til1up, geometry); %%% segment l1f1 psilf1low = 0; psilf1up = acos(r0 / (2 * r1h)); l1f1 = L1F1_(psilf1low, psilf1up, geometry); %%% segment f1d1 tfd1low = i12 * acos((A^2 + r2h^2 - r1h^2)/(2 * A * r2h)); tfd1up = i12 * acos((A^2 + r2h^2 - R1^2)/(2 * A * r2h)); f1d1 = F1D1_(tfd1low, tfd1up, geometry); %%% circular arc d1d1s tdds1low = 0; tdds1up = (2 * pi) / m1; d1d1s = D1D1S_(tdds1low, tdds1up, geometry); f01 = 0.5 * (d1c1 + c1a1 + a1i1 + i1l1 + l1f1 + f1d1 + d1d1s); %%% compression for i=1:1:floor((2*pi/m1+beta)*180/pi+1) %%% changed starting value to 1 and end value to +1. Rotation of 1 male lobe phi1 = (i-1) * pi /180 - beta; %%% turning angle of the male rotor in radians, when i=0 compression begins (phi1=-beta) phi1vect_0(i) = phi1; phi1d(i) = (i-1)-betad; %%% female rotor %%% segment l2i2 tli2low = -theta3 - 2 * i21 * alpha0 + theta6; tli2up = -theta3 - theta6; l2i2 = L2I2_(tli2low, tli2up, geometry, phi1); %%% segment i2a2 psiia2low = 0; psiia2up = pi - acos(r0 / (2 * r2h)); i2a2 = I2A2_(psiia2low, psiia2up, geometry, phi1); %%% segment a2d2 psiad2low = acos((r2h^2 + r^2 - r2h^2) / (2 * r * r2h)); psiad2up = acos((r2h^2 + r^2 - (r2h - r)^2) / (2 * r * r2h)); a2d2 = A2D2_(psiad2low, psiad2up, geometry, phi1); %%% segment d2f2 tdf2low = i21 * acos((A^2 + R1^2 - (r2h - r)^2)/(2 * A * R1)); tdf2up = i21 * acos((A^2 + R1^2 - r2h^2)/(2 * A * R1)); d2f2 = D2F2_(tdf2low, tdf2up, geometry, phi1); %%% segment f2l2 psifl2low = pi - acos(r0 / (2 * r2h));

81

psifl2up = 0; f2l2 = F2L2_(psifl2low, psifl2up, geometry, phi1); %%% female rotor, second tooth l2i2_s = L2I2_S(tli2low, tli2up, geometry, phi1); i2a2_s = I2A2_S(psiia2low, psiia2up, geometry,phi1); a2d2_s = A2D2_S(psiad2low, psiad2up, geometry,phi1); d2f2_s = D2F2_S(tdf2low, tdf2up, geometry,phi1); f2l2_s = F2L2_S(psifl2low, psifl2up, geometry,phi1); %%% segment l2sh tl2shlow = -phi1 * i21 + theta4 + theta6 + 2*pi/m2; tl2shup = pi - acos((R1^2 - R2^2 - A^2) / (2 * A * R2)); l2sh = L2SH_(tl2shlow, tl2shup, geometry); %%% segment hd1s thd1slow = acos((A^2 + R1^2 - R2^2) / (2 * A * R1)); thd1sup = -phi1 + 2*pi/m1; hd1s = HD1S_(thd1slow, thd1sup, geometry); if ((phi1 >= -beta) && (phi1 < theta2)) %%% stage1 tz1(i) = fzero(@(x) G1(x,tdf2up, geometry, phi1),[tfd1low,tfd1up]); f1z1 = F1D1_(tfd1low, tz1(i), geometry); f(i) = 0.5 * (f2l2 + l2i2_s + i2a2_s + a2d2_s + d2f2_s + f2l2_s + l2sh + hd1s + d1c1 + c1a1 + a1i1 + i1l1 + l1f1 + f1z1); xa2 = -A * cos(tdf2up) + R1 * cos(M * tdf2up); ya2 = -A * sin(tdf2up) + R1 * sin(M * tdf2up); %%% coordinates of contact point f2 in the female rotor dynamic coord. system xb2 = -(r2h + r0) * cos(theta4 + theta6 + 2*pi/m2); yb2 = (r2h + r0) * sin(theta4 + theta6 + 2*pi/m2); %%% coordinates of point l2_s in the female rotor dynamic coord. system moment_slice_f(i) = MomentOfSlice(xa2, ya2, xb2, yb2, geometry); xb1 = R1 * cos(2*pi/m1); yb1 = R1 * sin(2*pi/m1); %%% coordinates of point d1_s in the male rotor dynamic coord. system xa1 = xa2; ya1 = ya2; [xa1,ya1] = FemaleDynamicToMaleDynamic(xa1, ya1, geometry, phi1); xa1vect(i) = xa1; ya1vect(i) = ya1; xa2vect(i) = xa2; ya2vect(i) = ya2; moment_slice_m(i) = MomentOfSlice(xa1, ya1, xb1, yb1, geometry); end if ((phi1 >= theta2) && (phi1 < (theta2 + theta5))) %%% stage2 psiz1 = asin(r1h / r0 * sin(theta2 + theta5 - phi1)) - (theta2 + theta5 - phi1); l1z1 = L1F1_(psilf1low, psiz1, geometry); psiz2 = (theta4 + theta6 - phi1 * i21) + asin(r2h / r0 * sin(theta4 + theta6 - phi1 * i21)); z2l2 = F2L2_(psiz2, psifl2up, geometry, phi1); f(i) = 0.5 * (z2l2 + l2i2_s + i2a2_s + a2d2_s + d2f2_s + f2l2_s + l2sh + hd1s + d1c1 + c1a1 + a1i1 + i1l1 +l1z1); xa2 = -r2h * cos(theta4 + theta6) - r0 * cos(theta4 + theta6 - psiz2); ya2 = r2h * sin(theta4 + theta6) + r0 * sin(theta4 + theta6 - psiz2); %%% coordinates of "contact" point z2 (segment f2l2) in the female rotor dynamic coord. system moment_slice_f(i) = MomentOfSlice(xa2, ya2, xb2, yb2, geometry); xa1 = xa2; ya1 = ya2; [xa1,ya1] = FemaleDynamicToMaleDynamic(xa1, ya1, geometry, phi1); moment_slice_m(i) = MomentOfSlice(xa1, ya1, xb1, yb1, geometry); xa1vect(i) = xa1; ya1vect(i) = ya1; xa2vect(i) = xa2; ya2vect(i) = ya2; end if ((phi1 >= (theta2 + theta5)) && (phi1 < (theta2 + 2*alpha0 - theta5))) %%% stage3 tz1(i) = -theta1 - theta2 - 2 * alpha0 + phi1; i1z1 = I1L1_(til1low, tz1(i), geometry); tz2 = -theta3 - theta4 - 2 * i21 * alpha0 + i21 * phi1; z2i2_s = L2I2_S(tz2, tli2up, geometry, phi1);

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f(i) = 0.5 * (z2i2_s + i2a2_s + a2d2_s + d2f2_s + f2l2_s + l2sh + hd1s + d1c1 + c1a1 + a1i1 + i1z1); xa2 = -R2 * cos(tz2 + 2*pi/m2); ya2 = R2 * sin(tz2 + 2*pi/m2); %%% coordinates of contact point z2 (segment l2i2_s) in the female rotor dynamic coord. system moment_slice_f(i) = MomentOfSlice(xa2, ya2, xb2, yb2, geometry); xa1 = xa2; ya1 = ya2; [xa1,ya1] = FemaleDynamicToMaleDynamic(xa1, ya1, geometry, phi1); moment_slice_m(i) = MomentOfSlice(xa1, ya1, xb1, yb1, geometry); xa1vect(i) = xa1; ya1vect(i) = ya1; xa2vect(i) = xa2; ya2vect(i) = ya2; end if ((phi1 >= (theta2 + 2*alpha0 - theta5)) && (phi1 < (theta2 + 2*alpha0))) %%% stage4 psiz1 = asin(r1h / r0 * sin(theta1 + theta5 - 2*pi/m1 + phi1)) - (theta1 + theta5 - 2*pi/m1 + phi1); a1z1 = A1I1_(psiai1low, psiz1, geometry); psiz2 = (theta3 + theta6 + phi1 * i21 - 2*pi/m2) + asin(r2h / r0 * sin(theta3 + theta6 + phi1 * i21 - 2*pi/m2)); z2a2_s = I2A2_S(psiz2, psiia2up, geometry, phi1); f(i) = 0.5 * (z2a2_s + a2d2_s + d2f2_s + f2l2_s + l2sh + hd1s + d1c1 + c1a1 + a1z1); xa2 = -r2h * cos(theta3 + theta6 - 2*pi/m2) - r0 * cos(theta3 + theta6 - psiz2 - 2*pi/m2); ya2 = -r2h * sin(theta3 + theta6 - 2*pi/m2) - r0 * sin(theta3 + theta6 - psiz2 - 2*pi/m2); %%% coordinates of "contact" point z2 (segment i2a2_s) in the female rotor dynamic coord. system moment_slice_f(i) = MomentOfSlice(xa2, ya2, xb2, yb2, geometry); xa1 = xa2; ya1 = ya2; [xa1,ya1] = FemaleDynamicToMaleDynamic(xa1, ya1, geometry, phi1); moment_slice_m(i) = MomentOfSlice(xa1, ya1, xb1, yb1, geometry); xa1vect(i) = xa1; ya1vect(i) = ya1; xa2vect(i) = xa2; ya2vect(i) = ya2; end if ((phi1 >= (theta2 + 2*alpha0)) && (phi1 < (2*pi/m1))) %%% stages5,6 tz1(i) = fzero(@(x) G5(x,psiad2low, geometry, phi1),[tca1low,tca1up]); c1z1 = C1A1_(tca1low, tz1(i),geometry); xa2 = -r2h * cos(theta3 + theta6 - 2*pi/m2) - r0 * cos(theta3 + theta6 - psiia2up - 2*pi/m2); ya2 = -r2h * sin(theta3 + theta6 - 2*pi/m2) - r0 * sin(theta3 + theta6 - psiia2up - 2*pi/m2); %%% coordinates of contact point a2 (segment i2a2_s) in the female rotor dynamic coord. system if (phi1 < (2*pi/m1 - beta)) %%% stage5 f(i) = 0.5 * (a2d2_s + d2f2_s + f2l2_s + l2sh + hd1s + d1c1 + c1z1); moment_slice_f(i) = MomentOfSlice(xa2, ya2, xb2, yb2, geometry); xa1 = xa2; ya1 = ya2; [xa1,ya1] = FemaleDynamicToMaleDynamic(xa1, ya1, geometry, phi1); moment_slice_m(i) = MomentOfSlice(xa1, ya1, xb1, yb1, geometry); else %stage 6 tz2 = fzero(@(x) G6(x,geometry,phi1),[tdf2low,tdf2up]); d2z2_s = D2F2_S(tdf2low, tz2, geometry, phi1); f(i) = 0.5 * (a2d2_s + d2z2_s + d1c1 + c1z1); xb2 = xb1; yb2 = yb1; %%% coordinates of contact point d1_s in the male rotor dynamic coord. system [xb2, yb2]= MaleDynamicToFemaleDynamic(xb2, yb2, geometry, phi1); moment_slice_f(i) = MomentOfSlice(xa2, ya2, xb2, yb2, geometry); xa1 = xa2; ya1 = ya2; [xa1,ya1] = FemaleDynamicToMaleDynamic(xa1, ya1, geometry, phi1); moment_slice_m(i) = MomentOfSlice(xa1, ya1, xb1, yb1, geometry); end xa1vect(i) = xa1; ya1vect(i) = ya1; xa2vect(i) = xa2; ya2vect(i) = ya2; end end xa1vect = xa1vect'; ya1vect = ya1vect'; xa2vect = xa2vect';

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ya2vect = ya2vect'; tlsl2low = theta4 + theta6; tlsl2up = -theta3 - 2 * i21 * alpha0 + theta6; ls2l2 = L2I2_(tlsl2low, tlsl2up, geometry, phi1); %%% groove cross sectional area for the female rotor f02 = 0.5 * (l2i2 + i2a2 + a2d2 + d2f2 + f2l2 + ls2l2); %%% end of compression part %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%% suction alpha = i12 * (pi - acos((R1^2 - r2h^2 - A^2) / (2 * A * r2h))) - theta1; for i=1:1:floor((2*pi/m1 + alpha + beta) * 180/pi+1) %changed starting angle from 0 to 1 and added ending angle to +1 phi1 = (i-1) * pi /180 - beta; %%% turning angle of the male rotor in radians, when i=0 formation of closed volume is begining (phi1=-beta) phi1d(i) = (i-1) - betad; %%%phi1 runs from -0.62 rad to 1.94 rad %%% female rotor segments l2i2 = L2I2_(tli2low, tli2up, geometry, phi1); i2a2 = I2A2_(psiia2low, psiia2up, geometry, phi1); a2d2 = A2D2_(psiad2low, psiad2up, geometry, phi1); d2f2 = D2F2_(tdf2low, tdf2up, geometry, phi1); f2l2 = F2L2_(psifl2low, psifl2up, geometry, phi1); %%% female rotor segments for second tooth l2i2_s = L2I2_S(tli2low, tli2up, geometry, phi1); i2a2_s = I2A2_S(psiia2low, psiia2up, geometry, phi1); a2d2_s = A2D2_S(psiad2low, psiad2up, geometry, phi1); d2f2_s = D2F2_S(tdf2low, tdf2up, geometry, phi1); f2l2_s = F2L2_S(psifl2low, psifl2up, geometry, phi1); %%% segment d1h1 td1h1low = phi1; td1h1up = thd1slow; d1h1 = -D1D1S_(td1h1low, td1h1up, geometry); %%% segment h1i2 th1i2low = tl2shup; th1i2up = phi1 * i21 + theta3 + theta6; h1i2 = -L2SH_(th1i2low, th1i2up, geometry); if ((phi1 >= -beta) && (phi1 < 0)) %%% closed volume stage z1d1 = F1D1_(tz1(i), tfd1up, geometry); tz2 = fzero(@(x) GCV(x, geometry, phi1),[tdf2low,tdf2up]); z2f2 = D2F2_(tz2, tdf2up, geometry, phi1); fs(i) = 0.5 * (z2f2 + z1d1); xa1 = R1; ya1 = 0; %%% coordinates of point D1 in the male rotor dynamic coordinate system xb2 = -A * cos(tdf2up) + R1 * cos(M * tdf2up); yb2 = -A * sin(tdf2up) + R1 * sin(M * tdf2up); %%% coordinates of contact point f2 in the female rotor dynamic coord. system xa2 = xa1; ya2 = ya1; [xa2,ya2] = MaleDynamicToFemaleDynamic(xa2, ya2, geometry, phi1); moment_slice_f_s(i) = MomentOfSlice(xa2, ya2, xb2, yb2, geometry); xb1 = xb2; yb1 = yb2; [xb1,yb1]=FemaleDynamicToMaleDynamic(xb1, yb1, geometry, phi1); moment_slice_m_s(i) = MomentOfSlice(xa1, ya1, xb1, yb1, geometry); xb1vect(i) = xb1; yb1vect(i) = yb1; xb2vect(i) = xb2; yb2vect(i) = yb2; xf(i) = xb1vect(i)-A; xfm(i) = xb1vect(i); yf(i) = yb1vect(i); end if ((phi1 >= 0) && (phi1 < alpha)) %%% stage1 psiz_2(i) = fzero(@(x) G1S(x, geometry, phi1),[psiad2low,psiad2up]); tn1(i) = asin((-A * sin(phi1) + r2h * sin(k * phi1) + r * sin(psiz_2(i) - k * phi1)) / R1) + phi1; z2d2 = A2D2_(psiz_2(i), psiad2up, geometry, phi1); z1d1 = F1D1_(tz1(i), tfd1up, geometry);

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d1n1 = -D1D1S_(phi1, tn1(i), geometry); fs(i) = 0.5 * (z2d2 + d2f2 + z1d1 + d1n1); xa2 = -r2h + r*cos(psiz_2(i)); ya2 = -r*sin(psiz_2(i)); %%% coordinates of point z2 (segment a2d2) in the female rotor dynamic coordinate system moment_slice_f_s(i) = MomentOfSlice(xa2, ya2, xb2, yb2, geometry); xb1 = xb2; yb1 = yb2; [xb1, yb1] = FemaleDynamicToMaleDynamic(xb1, yb1, geometry, phi1); moment_slice_m_s(i) = MomentOfSlice(xa1, ya1, xb1, yb1, geometry); xb1vect(i) = xb1; yb1vect(i) = yb1; xb2vect(i) = xb2; yb2vect(i) = yb2; xf(i) = xb2vect(i); xfm(i) = xb2vect(i)+A; yf(i) = yb2vect(i); end if ((phi1 >= alpha) && (phi1 < theta2)) %%% stage2 z1d1 = F1D1_(tz1(i), tfd1up, geometry); fs(i) = 0.5 * (i2a2 + a2d2 + d2f2 + z1d1 + d1h1 + h1i2); xa2 = -r2h * cos(theta3 + theta6) - r0 * cos(theta3 + theta6 - psiia2low); ya2 = -r2h * sin(theta3 + theta6) - r0 * sin(theta3 + theta6 - psiia2low); %%% coordinates of point i2 (segment i2a2) in the female rotor dynamic coordinate system moment_slice_f_s(i) = MomentOfSlice(xa2, ya2, xb2, yb2, geometry); xb1 = xb2; yb1 = yb2; [xb1, yb1] = FemaleDynamicToMaleDynamic(xb1, yb1, geometry, phi1); moment_slice_m_s(i) = MomentOfSlice(xa1, ya1, xb1, yb1, geometry); xb1vect(i) = xb1; yb1vect(i) = yb1; xb2vect(i) = xb2; yb2vect(i) = yb2; xf(i) = xb2vect(i); xfm(i) = xb2vect(i)+A; yf(i) = yb2vect(i); end if ((phi1 >= theta2) && (phi1 < theta2 + theta5)) %%% stage3 psiz1 = asin(r1h / r0 * sin(theta2 + theta5 - phi1)) - (theta2 + theta5 - phi1); z1f1 = L1F1_(psiz1, psilf1up, geometry); psiz2 = (theta4 + theta6 - phi1 * i21) + asin(r2h / r0 * sin(theta4 + theta6 - phi1 * i21)); f2z2 = F2L2_(psifl2low, psiz2, geometry, phi1); fs(i) = 0.5 * (i2a2 + a2d2 + d2f2 + f2z2 + z1f1 + f1d1 + d1h1 + h1i2); xb2 = -r2h * cos(theta4 + theta6) - r0 * cos(theta4 + theta6 - psiz2); yb2 = r2h * sin(theta4 + theta6) + r0 * sin(theta4 + theta6 - psiz2); %%% coordinates of "contact" point z2 (segment f2l2) in the female rotor dynamic coord. system moment_slice_f_s(i) = MomentOfSlice(xa2, ya2, xb2, yb2, geometry); xb1 = xb2; yb1 = yb2; [xb1, yb1] = FemaleDynamicToMaleDynamic(xb1, yb1, geometry, phi1); moment_slice_m_s(i) = MomentOfSlice(xa1, ya1, xb1, yb1, geometry); xb1vect(i) = xb1; yb1vect(i) = yb1; xb2vect(i) = xb2; yb2vect(i) = yb2; xf(i) = xb2vect(i); xfm(i) = xb2vect(i)+A; yf(i) = yb2vect(i); end if ((phi1 >= (theta2 + theta5)) && (phi1 < (theta2 + 2*alpha0 - theta5))) %%% stage4 z1l1 = I1L1_(tz1(i), til1up, geometry); tz2 = -theta3 - theta4 - 2 * i21 * alpha0 + i21 * phi1; l2z2_s = L2I2_S(tli2low, tz2, geometry, phi1); fs(i) = 0.5 * (i2a2 + a2d2 + d2f2 + f2l2 + l2z2_s + z1l1 + l1f1 + f1d1 + d1h1 + h1i2); xb2 = -R2 * cos(tz2 + 2*pi/m2); yb2 = R2 * sin(tz2 + 2*pi/m2); %%% coordinates of contact point z2 (segment l2i2_s) in the female rotor dynamic coord. system moment_slice_f_s(i) = MomentOfSlice(xa2, ya2, xb2, yb2, geometry); xb1 = xb2; yb1 = yb2; [xb1, yb1] = FemaleDynamicToMaleDynamic(xb1, yb1, geometry, phi1); moment_slice_m_s(i) = MomentOfSlice(xa1, ya1, xb1, yb1, geometry); xb1vect(i) = xb1; yb1vect(i) = yb1; xb2vect(i) = xb2; yb2vect(i) = yb2; xf(i) = xb2vect(i); xfm(i) = xb2vect(i)+A;

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yf(i) = yb2vect(i); end if ((phi1 >= (theta2 + 2*alpha0 - theta5)) && (phi1 < (theta2 + 2*alpha0))) %%% stage5 psiz1 = asin(r1h / r0 * sin(theta1 + theta5 - 2*pi/m1 + phi1)) - (theta1 + theta5 - 2*pi/m1 + phi1); z1i1 = A1I1_(psiz1, psiai1up, geometry); psiz2 = (theta3 + theta6 + phi1 * i21 - 2*pi/m2) + asin(r2h / r0 * sin(theta3 + theta6 + phi1 * i21 - 2*pi/m2)); i2z2_s = I2A2_S(psiia2low, psiz2, geometry, phi1); fs(i) = 0.5 * (i2a2 + a2d2 + d2f2 + f2l2 + l2i2_s + i2z2_s + z1i1 + i1l1 + l1f1 + f1d1 + d1h1 + h1i2); xb2 = -r2h * cos(theta3 + theta6 - 2*pi/m2) - r0 * cos(theta3 + theta6 - psiz2 - 2*pi/m2); yb2 = -r2h * sin(theta3 + theta6 - 2*pi/m2) - r0 * sin(theta3 + theta6 - psiz2 - 2*pi/m2); %%% coordinates of "contact" point z2 (segment i2a2_s) in the female rotor dynamic coord. system moment_slice_f_s(i) = MomentOfSlice(xa2, ya2, xb2, yb2, geometry); xb1 = xb2; yb1 = yb2; [xb1, yb1] = FemaleDynamicToMaleDynamic(xb1, yb1, geometry, phi1); moment_slice_m_s(i) = MomentOfSlice(xa1, ya1, xb1, yb1, geometry); xb1vect(i) = xb1; yb1vect(i) = yb1; xb2vect(i) = xb2; yb2vect(i) = yb2; xf(i) = xb2vect(i); xfm(i) = xb2vect(i)+A; yf(i) = yb2vect(i); end if ((phi1 >= (theta2 + 2*alpha0)) && (phi1 < (2*pi/m1))) %%% stage6 z1a1 = C1A1_(tz1(i), tca1up, geometry); tl2 = phi1 * i21 - theta4 -theta6; l2_i2 = -L2SH_(tl2, th1i2up, geometry); h1l2 = -L2SH_(th1i2low, tl2, geometry); fs(i) = 0.5 * (i2a2 + a2d2 + d2f2 + f2l2 + l2i2_s + i2a2_s + z1a1 + a1i1 + i1l1 + l1f1 + f1d1 + d1h1 + h1l2 +l2_i2); xb2 = -r2h * cos(theta3 + theta6 - 2*pi/m2) - r0 * cos(theta3 + theta6 - psiia2up - 2*pi/m2); yb2 = -r2h * sin(theta3 + theta6 - 2*pi/m2) - r0 * sin(theta3 + theta6 - psiia2up - 2*pi/m2); %%% coordinates of contact point a2 (segment i2a2_s) in the female rotor dynamic coord. system moment_slice_f_s(i) = MomentOfSlice(xa2, ya2, xb2, yb2, geometry); xb1 = xb2; yb1 = yb2; [xb1, yb1] = FemaleDynamicToMaleDynamic(xb1, yb1, geometry, phi1); moment_slice_m_s(i) = MomentOfSlice(xa1, ya1, xb1, yb1, geometry); xb1vect(i) = xb1; yb1vect(i) = yb1; xb2vect(i) = xb2; yb2vect(i) = yb2; xf(i) = xb2vect(i); xfm(i) = xb2vect(i)+A; yf(i) = yb2vect(i); end if ((phi1 >= (2*pi/m1)) && (phi1 < (2*pi/m1 + alpha))) %%% stage7 tn1(i) = tn1(i - 360/m1); n1d1 = -D1D1S_(tn1(i), (phi1 - 2*pi/m1), geometry); psiz_2(i) = psiz_2(i - 360/m1); a2z2_s = A2D2_S(psiad2low, psiz_2(i), geometry, phi1); fs(i) = 0.5 * (i2a2 + a2d2 + d2f2 + f2l2 + l2i2_s + i2a2_s + a2z2_s + n1d1 +d1c1 + c1a1 + a1i1 + i1l1 + l1f1 + f1d1 + d1h1 + h1i2); xb2 = -r2h * cos(2*pi/m2) + r*cos(psiz_2(i) + 2*pi/m2); yb2 = r2h * sin(2*pi/m2) - r*sin(psiz_2(i) + 2*pi/m2); %%% coordinates of point z2 (segment a2d2_s) in the female rotor dynamic coordinate system moment_slice_f_s(i) = MomentOfSlice(xa2, ya2, xb2, yb2, geometry); xb1 = xb2; yb1 = yb2; [xb1, yb1] = FemaleDynamicToMaleDynamic(xb1, yb1, geometry, phi1); moment_slice_m_s(i) = MomentOfSlice(xa1, ya1, xb1, yb1, geometry); xb_end(i) = xb1; yb_end(i) = yb1; xb2vect(i) = xb2; yb2vect(i) = yb2; xf_end(i) = xb2vect(i); xfm_end(i) = xb2vect(i)+A; yf_end(i) = yb2vect(i); end

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end

%%% volume curve building %%% calculation of dV/dphi1 and cavity geometrical moment for suction moment_m_s(1) = moment_slice_m_s(1); moment_f_s(1) = moment_slice_f_s(1); fsuck(1) = fs(1); for i=2:1:floor((thau1z + 2*pi/m1 + alpha + beta) * 180/pi+1) %changed starting angle from 1 to 2 and added ending angle to +1 phi1 = (i-1) * pi /180 - beta; %%% turning angle of the male rotor in radians, when i=0 formation of closed volume is begining (phi1=-beta) phi1d(i) = (i-1) - betad; if ((phi1 >= -beta) && (phi1 < (2*pi/m1 + alpha))) fsuck(i) = fs(i); moment_m_s(i) = moment_m_s(i-1) + moment_slice_m_s(i); moment_f_s(i) = moment_f_s(i-1) + moment_slice_f_s(i); end if ((phi1 >= (2*pi/m1 + alpha)) && (phi1 <= (thau1z - beta))) fsuck(i) = f01 + f02; moment_m_s(i) = moment_m_s(i-1); moment_f_s(i) = moment_f_s(i-1); end if ((phi1 > (thau1z - beta)) && (phi1 < (thau1z + 2*pi/m1 + alpha))) fsuck(i) = f01 + f02 - fs(i - thau1zd); moment_m_s(i) = moment_m_s(i-1) - moment_slice_m_s(i-thau1zd); moment_f_s(i) = moment_f_s(i-1) - moment_slice_f_s(i-thau1zd); end dV_dphi1suck(i) = L / thau1z * fsuck(i); end %%% volume calculation for suction Vt = L * (f01 + f02); Vsuck(1) = 0; for i = 2:1:floor((thau1z + 2*pi/m1 + alpha + beta) * 180/pi+1) phi1 = (i-1) * pi /180 - beta; Vsuck(i) = Vsuck(i-1) + L / thau1zd * 0.5 * (fsuck(i-1) + fsuck(i)); %%% volume integration end %%% calculation of dV/dphi1 and cavity geometrical moment for compression moment_m_c(1) = moment_slice_m(1); moment_f_c(1) = moment_slice_f(1); fcomp(1) = f(1); for i = 2:1:floor((thau1z + 2*pi/m1 + beta)*180/pi+1) %changed starting angle from 0 to 2 and added ending angle to +1 phi1 = (i-1) * pi /180 - beta; %%% turning angle of the male rotor in radians, when i=0 compression begins (phi1=-beta) phi1d(i) = (i-1) - betad; if ((phi1 >= -beta) && (phi1 < (2*pi/m1))) fcomp(i) = f(i); moment_m_c(i) = moment_m_c(i-1) + moment_slice_m(i); moment_f_c(i) = moment_f_c(i-1) + moment_slice_f(i); end if ((phi1 >= (2*pi/m1)) && (phi1 < (thau1z - beta))) fcomp(i) = 0; moment_m_c(i) = moment_m_c(i-1); moment_f_c(i) = moment_f_c(i-1); end if ((phi1 >= (thau1z - beta)) && (phi1 < (thau1z + 2*pi/m1+2))) fcomp(i) = f01 + f02 - f(i - thau1zd); moment_m_c(i) = moment_m_c(i-1) - moment_slice_m(i-thau1zd); moment_f_c(i) = moment_f_c(i-1) - moment_slice_f(i-thau1zd); end dV_dphi1comp(i) = -Vt / thau1z + L / thau1z * fcomp(i); end %%% volume calculation for compression Vcomp(1) = Vt; %Vt = max volume for i = 2:1:floor((thau1z + 2*pi/m1 + beta) * 180/pi+1) %changed starting angle from 1 to 2 and added ending angle to +1 phi1 = i * pi /180 - beta; Vcomp(i) = Vcomp(i-1) - Vt / thau1zd + L / thau1zd * 0.5 * (fcomp(i-1) + fcomp(i)); end %%% dV/dphi1 and cavity geometrical moment for complete working cycle for i = 1:1:floor((2*pi + thau1z + 2*pi/m1 + beta) * 180/pi+1) %changed starting angle from 0 to 1 and added ending angle to +1 phi1 = (i-1) * pi /180 - beta;

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if ((phi1 < (2*pi - beta)) && (phi1 < (thau1z + 2*pi/m1 + alpha))) %%% suction dV_dphi1(i) = dV_dphi1suck(i); moment_m(i) = moment_m_s(i); moment_f(i) = moment_f_s(i); end if ((phi1 >= (thau1z + 2*pi/m1 + alpha)) && (phi1 <= (2*pi - beta))) %%% suction is finished, but compression is not begun dV_dphi1(i) = 0; moment_m(i) = 0; moment_f(i) = 0; end if ((phi1 >= (2*pi - beta)) && (phi1 <= (thau1z + 2*pi/m1 + alpha))) %%% simultaneous suction and compression dV_dphi1(i) = dV_dphi1suck(i) + dV_dphi1comp(i - 360); moment_m(i) = moment_m_s(i) + moment_m_c(i - 360); moment_f(i) = moment_f_s(i) + moment_f_c(i - 360); end if ((phi1 > (2*pi - beta)) && (phi1 > (thau1z + 2*pi/m1 + alpha))) %%% compression dV_dphi1(i) = dV_dphi1comp(i - 360); moment_m(i) = moment_m_c(i - 360); moment_f(i) = moment_f_c(i - 360); end end %%% volume calculation for complete working cycle Vmax = 0; for i = 1:1:floor((2*pi + thau1z + 2*pi/m1 + beta) * 180/pi+1) %changed starting angle from 0 to 1 and added ending angle to +1 phi1 = (i-1) * pi /180 - beta; phi1_out(i) = (i-1)-ceil(beta*180/pi); %(i-1) * pi /180 - beta; if ((phi1 < (2*pi - beta)) && (phi1 < (thau1z + 2*pi/m1 + alpha))) %%% suction V(i) = Vsuck(i); end if ((phi1 >= (thau1z + 2*pi/m1 + alpha)) && (phi1 <= (2*pi - beta))) %%% suction is finished, but compression is not begun V(i) = Vt; end if ((phi1 >= (2*pi - beta)) && (phi1 <= (thau1z + 2*pi/m1 + alpha))) %%% simultaneous suction and compression V(i) = Vsuck(i) + Vcomp(i-360) - Vt; end if ((phi1 > (2*pi - beta)) && (phi1 > (thau1z + 2*pi/m1 + alpha))) %%% compression V(i) = Vcomp(i-360); end if (V(i) > Vmax) Vmax = V(i); %%% maximal cavity volume end if (V(i) < 0) V(i) = 0; end end %V is the complete volume curve %Vcomp is the volume curve during compression %Vsuck is the volume curve during suction %%% end of volume curve building %%% Discharge port cross sectional area Vd = Vmax / epsilonv; %%% epsilonv is built-in volume ratio. %%% Vd = discharge volume %%% transfer beta from radians (double) to degrees (integer) if ((beta * 180/pi - floor(beta * 180/pi)) < (ceil(beta * 180/pi) - beta * 180/pi)) betad = floor(beta * 180/pi); else betad = ceil(beta * 180/pi); end %%% determination of compression angle id = 360; for i = 361:1:floor((2*pi + thau1z + 2*pi/m1 + beta) * 180/pi) if abs(V(i) - Vd) < abs(V(id) - Vd) id =i; end end phi1cd = id - 360 - betad; if (phi1cd > thau1zd) phi1cd = thau1zd; end

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alphacd = thau1zd - phi1cd; alphac = alphacd * pi / 180.0; %%% transfer into radians Rc1 = sqrt(A^2 + r2h^2 - 2 * A * r2h * cos(i21 * tca1low)); %%% radius OC1 xc1 = A * cos(tca1low - theta1) - r2h * cos(k * tca1low - theta1); %%% x-coord of point C1 %%% discharge window segments on the male rotor side l1f1_w = l1f1; f1d1_w = f1d1; p0o1c1s = acos((A^2 - R2^2 + Rc1^2) / (2 * A * Rc1)); %%% angle between p0 and C1s for i = 1:1:alphacd + 360/m1+1 phi1 = (i-1) * pi /180 - alphac; %%% turning angle of the male rotor in radians (coordinate system with center at the discharge end), when i=0 phi1 = -alphac, discharge begins %%% female rotor segments l2i2 = L2I2_(tli2low, tli2up, geometry, phi1); i2a2 = I2A2_(psiia2low, psiia2up, geometry, phi1); a2d2 = A2D2_(psiad2low, psiad2up, geometry, phi1); d2f2 = D2F2_(tdf2low, tdf2up, geometry, phi1); f2l2 = F2L2_(psifl2low, psifl2up, geometry, phi1); %%% female rotor segments for second tooth l2i2_s = L2I2_S(tli2low, tli2up, geometry, phi1); i2a2_s = I2A2_S(psiia2low, psiia2up, geometry, phi1); a2d2_s = A2D2_S(psiad2low, psiad2up, geometry, phi1); d2f2_s = D2F2_S(tdf2low, tdf2up, geometry, phi1); f2l2_s = F2L2_S(psifl2low, psifl2up, geometry, phi1); %%% segment d1wd1s td1wd1slow = alphac; td1wd1sup = -phi1 + 2*pi/m1; d1wd1s = D1D1S_(td1wd1slow, td1wd1sup, geometry); %%% segment l2si2w tl2si2wlow = -phi1 * i21 + 2*pi/m2 + theta4 + theta6; tl2si2wup = alphac * i21 + theta4 + 2*alpha0 * i21 - theta6; l2si2w = L2SH_(tl2si2wlow, tl2si2wup, geometry); %%% segment i2a2_w i2a2_w = I2A2_W(psiia2low, psiia2up, geometry, alphac); %%% segment a2d2_w a2d2_w = A2D2_W(psiad2low, psiad2up, geometry, alphac); %%% segment ha2s tha2slow = acos((A^2 + r2h^2 - R1^2) / (2 * A * r2h)); tha2sup = -i21 * phi1 - theta3 + 2*pi/m2; ha2s = HA2S_(tha2slow, tha2sup, geometry); if ((phi1 >= -alphac) && (phi1 < (-alphac + 2*pi/m1))) %%% stages 1 - 6 if phi1 < (-alphac + theta1 + theta2) %%% stages 1 - 4 % numerical solution for tz2 t1 = tdf2low; tu = tdf2up; flag =0; while flag == 0 t = 0.5 * (t1 + tu); if ((G1W(t, geometry, phi1, alphac) * G1W(t1, geometry, phi1, alphac))< 0) && abs(t - t1) < 0.00001 flag = 1; elseif ((G1W(t, geometry, phi1, alphac) * G1W(t1, geometry, phi1, alphac))< 0) && abs(t - t1) > 0.00001 tu = t; elseif ((G1W(t, geometry, phi1, alphac) * G1W(t1, geometry, phi1, alphac))> 0) && abs(tu - t) < 0.00001 flag = 1; else t1 = t; end end if (phi1 == -alphac) tz2 = 0; else tz2 = t; % result of solution end psiz2w = acos((-A * cos(tz2 - 2*pi/m2) + R1 * cos(M * tz2 - 2*pi/m2) + r2h * cos(2*pi/m2 + i21*(alphac + phi1))) / r) - (2*pi/m2 + i21 * (alphac + phi1));

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z2f2_s = D2F2_S(tz2, tdf2up,geometry, phi1); a2z2_w = A2D2_W(psiad2low, psiz2w,geometry, alphac); end if (phi1 < (-alphac + 2 * alpha0 - 2 * theta5)) %%% stage1 ti1sl1wlow = -phi1 + theta2 + 2 * alpha0 - theta5; ti1sl1wup = alphac + theta2 + theta5; i1sl1w = (r1h - r0)^2 * (ti1sl1wup - ti1sl1wlow); b = 0.5 * (d1c1 + c1a1 + a1i1 + i1sl1w + l1f1_w + f1d1_w + d1wd1s + i2a2_w + a2z2_w + z2f2_s + f2l2_s + l2si2w); end if ((phi1 >= (-alphac + 2 * alpha0 - 2 * theta5)) && (phi1 < (-alphac + 2 * alpha0))) %%% stage2 psiz1 = fzero(@(x) G2W(x,geometry,phi1, alphac),[psiai1up,psiai1low]); psiz1w=psiz1; a1z1 = A1I1_(psiai1low, psiz1, geometry); z1f1_w = L1F1_(psiz1w, psilf1up, geometry); b = 0.5 * (d1c1 + c1a1 + a1z1 + z1f1_w + f1d1_w + d1wd1s + i2a2_w + a2z2_w + z2f2_s + f2l2_s + l2si2w); end if ((phi1 >= (-alphac + 2 * alpha0)) && (phi1 < (-alphac + 2 * alpha0 + 2 * (theta1 - acos(xc1 / Rc1))))) %%% stage3 tz_1 = fzero(@(x) G3W(x, geometry, phi1, alphac),[psiai1up,psiai1low]) ; tz1w = tz_1; c1z1 = C1A1_(tca1low, tz_1, geometry); z1d1_w = F1D1_(tz1w, tfd1up, geometry); b = 0.5 * (d1c1 + c1z1 + z1d1_w + d1wd1s + i2a2_w + a2z2_w + z2f2_s + f2l2_s + l2si2w); end if (phi1 >= (-alphac + 2 * alpha0 + 2 * (theta1 - acos(xc1 / Rc1)))) %%% stages 4, 5, 6 tz1w = fzero(@(x) G4W(x, geometry, phi1, alphac),[tfd1low,tfd1up]); psiz1 = acos(A^2 / (2 * r1h * r) + r2h^2 / (2 * r1h * r) - A * r2h / (r1h * r) * cos(i21 * tz1w) - r1h / (2 * r) - r / (2 * r1h)); z1d1_w = F1D1_(tz1w, tfd1up, geometry); d1z1 = D1C1_(psidc1low, psiz1, geometry); if (phi1 < (-alphac + theta1 + theta2)) %%% stage 4 b = 0.5 * (d1z1 + z1d1_w + d1wd1s + i2a2_w + a2z2_w + z2f2_s + f2l2_s + l2si2w); end if ((phi1 >= -alphac + theta1 + theta2) && (phi1 < (-alphac + theta1 + theta2 + 2 * theta5))) %%% stage 5 psiz2 = fzero(@(x) G5W(x,geometry, phi1, alphac),[psifl2up,pi/2]); psiz2w=psiz2; i2z2_w = I2A2_W(psiia2low, psiz2w, geometry, phi1); z2l2_s = F2L2_S(psiz2, psifl2up, geometry, phi1); b = 0.5 * (d1z1 + z1d1_w + d1wd1s + i2z2_w + z2l2_s + l2si2w); end if (phi1 >= (-alphac + theta1 + theta2 + 2 * theta5)) %%% stage 6.1 b = 0.5 * (d1z1 + z1d1_w + d1wd1s); end end end if (phi1 >= (-alphac + 2*pi/m1)) %%% after point D1s intersects point d1w b = 0; end if (tha2sup >= tha2slow) %%% until point A2s intersects point H c=0; end if ((tha2sup < tha2slow) && (phi1 <= (2*pi/m1))) %%% stages 6.2 - 9 %%% numerical solution for psiz2 i; t1 = psiad2up; tu = psiad2low; flag = 0; while flag == 0 t= 0.5*(t1+tu); if G6W(t,geometry,phi1)*G6W(t1,geometry,phi1)<0 && abs(t-t1)<0.00001 flag =1; elseif G6W(t,geometry,phi1)*G6W(t1,geometry,phi1)<0 && abs(t-t1)>0.00001 tu =t; elseif G6W(t,geometry,phi1)*G6W(t1,geometry,phi1)>0 && abs(tu-t)<0.00001 flag =1; else t1=t; end end psiz2 = t; % result of solution xz2 = A * cos(phi1) - r2h * cos(2*pi/m2 - k * phi1) + r * cos(psiz2 + 2*pi/m2 - k * phi1);

90

tz2w = acos(xz2 / R1) - phi1; z2wh = D1D1S_(tz2w, beta, geometry); a2sz2 = A2D2_S(psiad2low, psiz2,geometry, phi1); if (phi1 <= (2*pi/m1 - acos(xc1 / Rc1) - p0o1c1s)) %%% stage 6.2 c = 0.5 * (z2wh + ha2s + a2sz2); end if ((phi1 > (2*pi/m1 - acos(xc1 / Rc1) - p0o1c1s)) && (phi1 <= (2*pi/m1))) %%% stages 7-9 if (phi1 <= (2*pi/m1 - beta)) %%% numerical solution for psin1 on stages 7-8 psin1 = fminbnd(@(x) G7NW(x,geometry, phi1),psidc1low,psidc1up); tnw = asin((-A * sin(phi1) + r1h * sin(2*pi/m1) - r * sin(psin1 - 2*pi/m1)) / r2h) + phi1; hnw = HA2S_(tha2slow, tnw, geometry); n1c1s = D1C1_(psin1, psidc1up, geometry); end if (phi1 <= (2*pi/m1 - theta1)) %%% stage 7 %%% numerical solution for tz_1 tz_1 = fminbnd(@(x) G7ZW(x,geometry, phi1),tca1up,tca1low); tz1w = asin(-A / r2h * (sin(phi1) - sin(tz_1 - theta1 + 2*pi/m1)) - sin(k * tz_1 - theta1 + 2*pi/m1)) + phi1; c1sz1 = C1A1_(tca1low, tz_1,geometry); z1wa2s = HA2S_(tz1w, tha2sup, geometry); c = 0.5 * (z2wh + hnw + n1c1s + c1sz1 + z1wa2s + a2sz2); else %%% numerical solution for tz_1 tz_1 = fzero(@(x) G5(x,psiad2low, geometry, phi1),[tca1up,tca1low]); c1z1 = C1A1_(tca1low, tz_1,geometry); if (phi1 <= (2*pi/m1 - beta)) %%% stage 8 c = 0.5 * (z2wh + hnw + n1c1s + c1sz1 + a2sz2); else %%% stage 9 z2wd1s = D1D1S_(tz2w, (2*pi/m1 - phi1),geometry); c = 0.5 * (z2wd1s + d1c1 + c1z1 + a2sz2); end end end end fc(i) = b + c; fc(alphacd + 360/m1) = 0; end for i = 1:1:floor(thau1zd + 360/m1 - phi1cd)%%% discharge port area if ((i-1 + phi1cd - thau1zd + betad) < 0) fd(i) = f01 + f02 - fc(i); else fd(i) = f(i + phi1cd - thau1zd + betad) - fc(i); %%% equation (2) chapter "Discharge flow..." if (fd(i) < 0) fd(i) =0; end end end %%% Torque ratio phi1 = 0; d2h_ = D1D1S_(0, beta,geometry); tf2 = theta4 + theta6; h_f2 = HA2S_(tha2slow, tf2,geometry); f2d2 = -D2F2_(tdf2low, tdf2up,geometry,phi1); fh = 0.5 * (d2h_ + h_f2 + f2d2); %%% torque ratio female/male torque_ratio = i12 * (fh / (fh + f01 + f02)); %%% female rotor cavity area calculation according to Sakun equations (352), (359.2) for selfconfidence phif = i21 * acos((A^2 + R1^2 - r2h^2)/(2 * A * R1)); f02Sakun = r2h^2/2 * (0.5*r^2/r2h^2*(pi - theta3) + theta3 - sin(theta3) + A^2/r2h^2*(phif - sin(phif)) + R1^2*M/r2h^2*phif - A*R1/r2h^2*(1 + 2*i21)*sin(i12*phif) + A*R1/r2h^2*sin(M*phif) - A/r2h*sin(theta4) + theta4) + 0.5*(theta3 +theta4)*(R2^2 - r2h^2); %%% ------geometrical moment output--------- integral_moment_m = 0; integral_moment_f = 0; integral_moment_cavity = 0; for i = 1:1:floor((2*pi + thau1z + 2*pi/m1 + beta) * 180/pi+1) integral_moment_m = integral_moment_m + moment_m(i); integral_moment_f = integral_moment_f + moment_f(i); integral_moment_cavity = integral_moment_cavity + (moment_m(i) + i21 * moment_f(i)); end

91

integral_moment_m = integral_moment_m / (i + 1); integral_moment_f = integral_moment_f / (i + 1); integral_moment_cavity = integral_moment_cavity / (i + 1); %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%% code from sealline.cpp %%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%% suction for i = 1:1:floor((thau1z + 2*pi/m1 + alpha + beta) * 180/pi+1) phi1 = (i-1) * pi /180 - beta; %%% turning angle of the male rotor in radians, when i=0 formation of closed volume is begining (phi1=-beta) %%% segment 1-2" thaumin = -theta2 - theta5 + phi1; if (thaumin < 0) thaumin = 0; end if (thaumin > thau1z) thaumin = thau1z; end thaumax = beta + phi1; if (thaumax < 0) thaumax = 0; end if (thaumax > thau1z) thaumax = thau1z; end s12 = (thaumax - thaumin) * sqrt((r2h * i21)^2 + (L / thau1z)^2); %%% segment 2"-2' s22 = r0 * acos(r0 / (2 * r1h)); if ((phi1 < (theta2 + theta5)) || (phi1 > (thau1z + theta2 + theta5))) s22 = 0; end %%% segment 2'-3' thaumin = -theta2 - 2*alpha0 + theta5 + phi1; if (thaumin < 0) thaumin = 0; end if (thaumin > thau1z) thaumin = thau1z; end thaumax = -theta2 - theta5 + phi1; if (thaumax < 0) thaumax = 0; end if (thaumax > thau1z) thaumax = thau1z; end s23 = L / thau1z * (thaumax - thaumin); %%% segment 3'-3" s33 = r0 * acos(r0 / (2 * r1h)); if ((phi1 < (theta2 + 2*alpha0 - theta5)) || (phi1 > (thau1z + theta2 + 2*alpha0 - theta5))) s33 = 0; end %%% segment 3"-4 thaumin = phi1 - 2*pi/m1; if (thaumin < 0) thaumin = 0; end if (thaumin > thau1z) thaumin = thau1z; end

92

thaumax = -theta2 - 2*alpha0 + theta5 + phi1; if (thaumax < 0) thaumax = 0; end if (thaumax > thau1z) thaumax = thau1z; end s34 = (thaumax - thaumin) * sqrt((r2h * i21)^2 + (L / thau1z)^2); %%% segment 4-5 s45 = r * acos(r / (2 * r2h)); if ((phi1 < (2*pi/m1)) || (phi1 > (thau1z + 2*pi/m1))) s45 = 0; end %%% segment 5-1 thaumin = phi1; if (thaumin < 0) thaumin = 0; end if (thaumin > thau1z) thaumin = thau1z; end thaumax = beta + phi1; if (thaumax < 0) thaumax = 0; end if (thaumax > thau1z) thaumax = thau1z; end s51_s(i) = clearance * (thaumax - thaumin) * sqrt(R1^2 + (L / thau1z)^2); segments1_5(i) = clearance * (s12 + s22 + s23 + s33 + s34 + s45); path1_s(i) = segments1_5(i) + s51_s(i); end %%% compression leakages delta = pi - acos((R1^2 - R2^2 - A^2) / (2 * A * R2)); ang_hop = acos((A^2 + R1^2 - R2^2) / (2 * A * R1)); height = R1 * (ang_hop - beta); %%% height of the cusp blow hole for i = 1:1:floor((thau1z + 2*pi/m1 + beta)*180/pi)+2 phi1 = (i-1) * pi /180 - beta; %%% turning angle of the male rotor in radians, when i=0 compression begins (phi1=-beta) %%% path1 %%% segment 5-1 thaumin = phi1 - 2*pi/m1; if (thaumin < 0) thaumin = 0; end if (thaumin > thau1z) thaumin = thau1z; end thaumax = beta + phi1 - 2*pi/m1; if (thaumax < 0) thaumax = 0; end if (thaumax > thau1z) thaumax = thau1z; end s51_c(i) = clearance * (thaumax - thaumin) * sqrt(R1^2 + (L / thau1z)^2); %%% other segments are similar as for suction path1_c(i) = segments1_5(i) + s51_c(i); %%% path2, leak in

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%%% male rotor thaumin = beta + phi1; if (thaumin < 0) thaumin = 0; end if (thaumin > thau1z) thaumin = thau1z; end thaumax = thau1z; path2min_c(i) = (thaumax - thaumin) * sqrt(R1^2 + (L / thau1z)^2); %%% female rotor thaumin = delta * i12 - theta2 - 2*alpha0 + theta5 + phi1; if (thaumin < 0) thaumin = 0; end if (thaumin > thau1z) thaumin = thau1z; end thaumax = thau1z; path2fin_c(i) = (thaumax - thaumin) * sqrt((R2 * i21)^2 + (L / thau1z)^2); path2in_c(i) = clearance * (path2min_c(i) + path2fin_c(i)); %%% path2, leak out %%% male rotor thaumin = beta + phi1 - 2*pi/m1; if (thaumin < 0) thaumin = 0; end if (thaumin > thau1z) thaumin = thau1z; end thaumax = thau1z; path2mout_c(i) = (thaumax - thaumin) * sqrt(R1^2 + (L / thau1z)^2); %%% female rotor thaumin = delta * i12 - theta2 - theta5 + phi1 - 2*pi/m1; if (thaumin < 0) thaumin = 0; end if (thaumin > thau1z) thaumin = thau1z; end thaumax = thau1z; path2fout_c(i) = (thaumax - thaumin) * sqrt((R2 * i21)^2 + (L / thau1z)^2); path2out_c(i) = clearance * (path2mout_c(i) + path2fout_c(i)); %%% path3 (cusp blow hole), leak in thaumin = delta * i12 -theta2 - theta5 + phi1; if (thaumin < 0) thaumin = 0; end if (thaumin > thau1z) thaumin = thau1z; end thaumax = ang_hop + phi1; if (thaumax < 0) thaumax = 0; end if (thaumax > thau1z) thaumax = thau1z; end path3in_c(i) = 0.5 * (thaumax - thaumin) * L / thau1z * height; %%% path3 (cusp blow hole), leak out thaumin = delta * i12 -theta2 - theta5 + phi1 - 2*pi/m1; if (thaumin < 0) thaumin = 0; end if (thaumin > thau1z) thaumin = thau1z;

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end thaumax = ang_hop + phi1 - 2*pi/m1; if (thaumax < 0) thaumax = 0; end if (thaumax > thau1z) thaumax = thau1z; end path3out_c(i) = 0.5 * (thaumax - thaumin) * L / thau1z * height; end %%% complete working cycle for i = 2:1:floor((2*pi + thau1z + 2*pi/m1 + beta) * 180/pi+2); phi1 = (i-1) * pi /180 - beta; %phi1out2(i) = phi1; if i <= (thau1z + 2*pi/m1 + alpha + beta) * 180/pi path1s(i) = path1_s(i); sgmns1_5ofpath1s(i) = segments1_5(i); sgmn51ofpath1s(i) = s51_s(i); else path1s(i) = 0; sgmns1_5ofpath1s(i) = 0; sgmn51ofpath1s(i) = 0; end if (i <= 360) path1c(i) = 0; csgmns1_5ofpath1c(i) = 0; csgmn51ofpath1c(i) = 0; path2inc(i) = 0; path2outc(i) = 0; path3inc(i) = 0; path3outc(i) = 0; else path1c(i) = path1_c(i - 360); csgmns1_5ofpath1c(i) = segments1_5(i - 360); csgmn51ofpath1c(i) = s51_c(i - 360); path2inc(i) = path2in_c(i - 360); path2outc(i) = path2out_c(i - 360); path3inc(i) = path3in_c(i - 360); path3outc(i) = path3out_c(i - 360); end %%% path 4: compression start blow hole %%% leak in if (phi1 >= (2*pi - beta - 2*pi/m1)) path4inc(i) = COMP_START_BLOW_HOLE(geometry, phi1, ang_hop); else path4inc(i) = 0; end %%% leak out if (phi1 >= (2*pi - beta)) path4outc(i) = COMP_START_BLOW_HOLE(geometry,phi1 - 2*pi/m1, ang_hop); else path4outc(i) = 0; end %%% path 6: between the end plate and the rotor end face at the discharge end %%% leak in from the leading cavity if ((phi1 >= thau1z) && (phi1 < (2*pi + thau1z))) path6inl(i) = clearance * 2 * (r + r0); else path6inl(i) = 0; end %%% leak in from discharge if ((phi1 >= thau1z) && (phi1 < (thau1z + 2*pi/m1))) path6ind(i) = clearance * (r + r0); else path6ind(i) = 0; end

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%%% leak out to the trailing cavity if ((phi1 >= (thau1z + 2*pi/m1)) && (phi1 < (2*pi + 2*pi/m1 + thau1z))) path6outt(i) = clearance * 2 * (r + r0); else path6outt(i) = 0; end %%% leak out to suction if ((phi1 >= (2*pi + thau1z)) && (phi1 < (2*pi + thau1z + 2*pi/m1))) path6outs(i) = clearance * (r + r0); else path6outs(i) = 0; end end save(filename) dV_dphi1 = dV_dphi1 * 1e-9; %%% transfer into m^3 imax = length(dV_dphi1);%i - 1; i = 1; Vmax = 0; iVmax = 0; V = V * 1e-9; while (i < length(V)) if (V(i) > Vmax) Vmax = V(i); iVmax = i; %%% when i = iVmax, compression begins end i=i+1; end %%% Suction port area ftotal = fs(length(fs)); %maximum area suction port? phi1 = floor(-betad):1:length(V)-ceil(betad)-1; for i = 1:1:imax if (phi1(i) <= 0) fs(i) = 0; end if i > length(fs) fs(i) = ftotal; end if ((i > (iVmax - 360/m1)) && (i <= iVmax)) fs(i) =ftotal * (iVmax - i) / (360/m1); end if (i > iVmax) fs(i) = 0; end end %%% Discharge port area i1 = length(fd); i1 = i1 - 1; for i = imax:-1:1 if (i1 >= 1) fd(i) = fd(i1); else fd(i) = 0; end i1 = i1 - 1; end function [x,y]=FemaleDynamicToMaleDynamic(x, y, geometry, phi1) temp_x = x; temp_y = y; A = geometry.A; k = geometry.k; x = A * cos(phi1) + temp_x * cos(k*phi1) - temp_y * sin(k*phi1); y = A * sin(phi1) + temp_x * sin(k*phi1) + temp_y * cos(k*phi1); function [x,y]=MaleDynamicToFemaleDynamic(x, y, geometry, phi1) A = geometry.A; k =geometry.k; i21 = geometry.i21; temp_x = x; temp_y = y; x = -A * cos(i21*phi1) + temp_x * cos(k*phi1) + temp_y * sin(k*phi1);

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y = A * sin(i21*phi1) - temp_x * sin(k*phi1) + temp_y * cos(k*phi1); function [moment] = MomentOfSlice(xa,ya,xb,yb, geometry) L =geometry.L; thau1zd = geometry.thau1zd; dz = L / thau1zd; moment = ((xb*xb - xa*xa)/2 + (yb*yb - ya*ya)/2)*dz; %%% geometrical moment of slice dz in mm^3 function [blow_hole] = COMP_START_BLOW_HOLE(geometry, phi1, ang_hop) A = geometry.A; R1 =geometry.R1; r2h = geometry.r2h; thau1z = geometry.thau1z; r = geometry.r; i12 = geometry.i12; i21 = geometry.i21; L = geometry.L; blow_hole = 0; dthau = 0.001; olddistance = 0; thau1 = phi1 - ang_hop; while ((thau1 < phi1) && (thau1 <= thau1z)) xm = R1 * cos(phi1 - thau1); ym = -R1 * sin(phi1 - thau1); zm = L * thau1 / thau1z; %%% coordinates of current point on the male rotor tip psicos = (r2h^2 + r^2 - A^2 - R1^2 + 2*A*R1*cos(phi1 - thau1)) / (2*r*r2h); psi = acos(psicos); p = -r2h * (r2h - r*psicos) / (r*sin(psi)) + (r2h - r*psicos) / tan(psi) - r*sin(psi); q = ym + r2h * (xm - A) / (r*sin(psi)) - (xm - A) / tan(psi); phi2cos = q / p; phi2 = acos(phi2cos); thau2 = phi1 - i12 * phi2; phi2sin = ((xm - A) + (r2h - r*psicos) * phi2cos) / (r*sin(psi)); phi22 = asin(phi2sin); %%% must be equal with phi2; %%%_________________________________ %%% tangent plane will be found at point (xf, yf, zf) (psi, thau2) xf = xm; yf = ym; zf = L * thau2 / thau1z; if (zf < L) aa = -r * L / thau1z * cos(psi - i21*(phi1 - thau2)); bb = r * L / thau1z * sin(psi - i21*(phi1 - thau2)); cc = -r2h * r * i21 * sin(psi); dd = -(aa * xf + bb * yf + cc * zf); %%% tangent plane coefficients distance = (aa*xm + bb*ym + cc*zm + dd) / sqrt(aa*aa + bb*bb + cc*cc); else distance = L - zm; end blow_hole = blow_hole + (distance + olddistance)*0.5 * dthau * R1; olddistance = distance; thau1 = thau1 + dthau; end %% Properties Constants %% x0 = 0.3; %Overall ammonia concentration dx = 0.001; %Small differences in concentration %% Set fluidprop fluid NH3-H2O x0+dx global FPx FPx = actxserver ('FluidProp.FluidProp'); Msg='FluidProp: COM object created'; disp(Msg); invoke(FPx,'SetUnits','SI','','',''); %Convert to SI-units Msg = 'RefProp, Ammonia - Water'; disp(Msg); Model = 'RefProp'; nCmp = 2; Cnc=[x0+dx,0;(1-(x0+dx)),0]; Cmp1 = 'ammonia,water'; ErrorMsg = invoke(FPx,'SetFluid_M',Model,nCmp,Cmp1,Cnc);

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%% Set fluidprop fluid NH3-H2O x0=0.3 global FP FP = actxserver ('FluidProp.FluidProp'); Msg='FluidProp: COM object created'; disp(Msg); invoke(FP,'SetUnits','SI','','',''); %Convert to SI-units Msg = 'RefProp, Ammonia - Water'; disp(Msg); Model = 'RefProp'; nCmp = 2; Cnc=[x0,0;(1-x0),0]; Cmp1 = 'ammonia,water'; ErrorMsg = invoke(FP,'SetFluid_M',Model,nCmp,Cmp1,Cnc); function [f_discharge]=f_discharge(G) global fdischarge G = floor((G/(0.043722222222222/787))+1); if G<787 f_discharge=fdischarge(G); else f_discharge=fdischarge(787); end end function [f_suction]=f_suction(G) global fsuction G = floor((G/(0.043722222222222/787))+1); if G<787 f_suction = fsuction(G); else f_suction=fsuction(787); end end function [V_test]=V_test(G) global V_dt G = floor((G/(0.043722222222222/787))+1); if G<787 V_test=V_dt(G); else V_test=V_dt(787); end end

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Appendix E: Calculation Pressure Difference [Matlab-Code] %% Pressure difference calculation load('Pressure.mat') load('Time.mat') Time_H=t_h_short'; P_h_x = P_h'; global P_h_x global dPT global dPL global dPT2 global dPL2 global dPT3 global dPL3 v1 = 3.7*10^5; v2 = 9.08*10^5; n1 = 360/m1; n2 = 360; n3 = 360 - n1; n4 = 360 + 2 * n1; n5 = 360 + n1; k1_1=repmat(v1,1,n1); k1_2=repmat(v2,1,n1); k2_1=repmat(v1,1,n2); k2_2=repmat(v2,1,n2); k3_1=repmat(v1,1,n3); k3_2=repmat(v2,1,n3); k4_1=repmat(v1,1,n4); k4_2=repmat(v2,1,n4); k5_2=repmat(v2,1,n5); P_delay_1 = [k1_1,k1_1,k1_1,P_h_x]; P_main_1 = [k1_1,k1_1, P_h_x, k1_2]; P_advance_1 = [k1_1,P_h_x, k1_2, k1_2]; P_delay_2 = [k2_1,k2_1,k2_1,P_h_x]; P_main_2 = [k2_1,k2_1, P_h_x, k2_2]; P_advance_2 = [k2_1,P_h_x, k2_2, k2_2]; P_delay_3 = [k3_1,k4_1,k2_1,P_h_x]; P_main_3 = [k2_1,k2_1, P_h_x, k5_2]; P_advance_3 = [k3_1,P_h_x, k4_2, k2_2]; %% The pressure difference dP_Trailing_z1 = P_main_1-P_delay_1; dP_Leading_z1 = P_advance_1-P_main_1; dP_Trailing_z2 = P_main_2-P_delay_2; dP_Leading_z2 = P_advance_2-P_main_2; dP_Trailing_z3 = P_main_3-P_delay_3; dP_Leading_z3 = P_advance_3-P_main_3; dP_Trailing_z1(dP_Trailing_z1<100)=0; dP_Leading_z1(dP_Leading_z1<100)=0; dP_Trailing_z2(dP_Trailing_z2<100)=0; dP_Leading_z2(dP_Leading_z2<100)=0;

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dP_Trailing_z3(dP_Trailing_z3<100)=0; dP_Leading_z3(dP_Leading_z3<100)=0; dPT = dP_Trailing_z1(2*n1+1:end-n1-1); dPL = dP_Leading_z1(2*n1+1:end-n1-1); dPT2 = dP_Trailing_z2(2*n2+1:end-n2-1); dPL2 = dP_Leading_z2(2*n2+1:end-n2-1); dPT3 = dP_Trailing_z3(2*n2+n1+1:end-n2-1); dPL3 = dP_Leading_z3(2*n2+n1+1:end-n2-1);

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Appendix F: Calculation Efficiencies [Matlab-Code] clear all clc %% Efficiency Calculations %% Cavity_V = load('Cavity_volume.mat'); dV_dt = load('dV_dt.mat'); % plot(Time_G.time,Cavity_V.V); b1 = 5; n = 3500; h = load('Enthalpy.mat'); P = load('Pressure.mat'); MD = load('Discharge.mat'); Mass_out = load('Mass_in_discharge.mat'); Mass_in = load('Mass_in_suc.mat'); Mass_out_max = max(Mass_out.Mass_out_SD); Mass_in_max= max(Mass_in.Mass_in_SD); mdischarge = sum(MD.MFD_h) h_suc = h.H_h(1); h_id = 3.6606* 10^5; %% Caculate from fluidprop! (concentratie in mol/mol) W_ideal = (h_id - h_suc) x=size(P.P_h); x = x(1); Work(1)=0; for i = 2:x-1 p0(i) = P.P_h(i); p1(i) = P.P_h(i+1); V(i) = dV_dt.V_dt(i); dP(i) = p1(i)-p0(i); V_dP(i) = V(i)*dP(i); Work(i) = Work(i-1) + V_dP(i); end Work_t = sum(Work) Eff_is = W_ideal/(Work_t/mdischarge) %% Volumetric Efficiency %% S_V = load('Specific_volume.mat'); vs = S_V.Vsv_h(1); Vmax = max(Cavity_V.V); Eff_vol = ((Mass_out_max)*vs)/Vmax

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Appendix G: Calculation Shaft Rotation Angle The shaft rotation angle is based on the wrap angle of the male rotor, number of male lobes and the rotor geometry. The inputs as used by Zaytsev [1] and described in chapter 1-4 are used in this appendix as inputs, Table G-0-1.

Table G-0-1 Inputs of the shaft rotation angle calculation used by Zaytsev [1].

Symbols Description Value Units𝝉𝒘 Wrapangleofthemalerotor 314/5.48 °/rad𝜷 AnglebetweenR1andA 29/0.513 °/rad𝒎𝟏 Numberofmalerotorlobes 5 -

The shaft rotation angle is necessary in the calculations for the transfer from the shaft angle to the time. The maximum shaft rotation angle is calculated by equation G.1. The input values will be in radians and are multiplied by !"#

! to calculate to angle degrees. The one added in the

end of the equation results from the total shaft rotation that starts at 0 whilst Matlab only can start at 1, one extra step needs to be added.

𝑆ℎ𝑎𝑓𝑡 𝑅𝑜𝑡𝑎𝑡𝑖𝑜𝑛 𝐴𝑛𝑔𝑙𝑒 = 2𝜋 + 𝜏! +2𝜋𝑚!

+ 𝛽 ∙180𝜋 + 1

(G.1)

Four different angles are added to result in the shaft rotation angle, the 360 degrees of the rotor, the wrap angle in the length of the rotor, the angle representing the width of the cavity and 𝛽 (will be explained next). The shaft rotation angle in this case results in 777°. In the equation of the shaft rotation angle, angle β is used. Angle β is calculated with the cosine rule, is given in equation G.2 and illustrated in Figure G-1. Angle β represents the angle from the first touch between the male rotor and the female rotor and the ‘A-line’. The A-line is line used as start point of the rotation angle of the rotors (𝐴 = 𝑟!" + 𝑟!" in m).

𝛽 = 𝑎𝑐𝑜𝑠𝐴! + 𝑅!! − 𝑟!"!

2 ∙ 𝐴 ∙ 𝑅!

(G.2)

Figure G-1 Illustration of angle β. β is used in the shaft rotation angle calculation.

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