Thermodynamic Model of a Screw Compressor

Transcription

1 L.L. van Bommel August 2016 Report number: P&E Thermodynamic Model of a Screw Compressor Master Thesis Department: Section: Process & Energy, Faculty 3mE Engineering Thermodynamics

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3 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 defed 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 3

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5 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. 5

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7 Content Abstract 5 Nomenclature 9 Abbreviations 10 Glossary 10 1 Introduction Heat Pumps Compressor Process, Arrangement and Selection Screw Compressor Motivation of the Research Research Question, Objective, Boundaries and Assumptions Chapters 19 2 Models Boundaries and Assumptions Geometry Model Theory Theory of the Thermodynamic Model Theoretical Desired Performance Outputs 30 3 Thermodynamic Model Implementation Historic Development Input Implementation Geometry Model Thermo-Physical Properties Input Values/Initial Values for the Implementation Thermodynamic Model; Factors that Influence Ideal Behaviour Mass Flows Leakage Path Areas Mass Flows of the Leakages Desired Outputs 49 4 Validation Inputs and Boundaries Model Validation: Zaytsev Adapting the model to the experimental data 54 5 Model Results and Discussion Leakages Boundary conditions van de Bor Geometry variation 58 6 Conclusions and Recommations Conclusions Recommations 66 Bibliography 67 Appices 69 Appix A: Envelope Method Rotor Element Calculation 69 Appix B: Conservation Equations of the Homogeneous Model 71 Appix C: The Thermodynamic Model in Simulink 74 Appix D: The Geometry Model in Matlab, van de Bor/Zaytsev [Matlab-Code] 79 Appix E: Calculation Pressure Difference [Matlab-Code] 98 Appix F: Calculation Efficiencies [Matlab-Code] 100 Appix G: Calculation Shaft Rotation Angle 101 7

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9 Nomenclature Symbols Units A Flow area m 2 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 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 m 3 kg -1 V Volume m 3 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 9

10 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 COP COP21 CRHP EDGAR IEA IPCC LULUCF NASA PBL UNFCCC Computational Fluid Dynamic Coefficient of Performance 21 st Conference of Parties Compression Resorption Heat Pump Emission Database for Global Atmospheric Research International Energy Agency Intergovernmental Panel on Climate Change Land Use, Land-Use Change and Forestry National Aeronautics and Space Administration Planning Board for Environment Planbureau voor Leefomgeving United Nations Framework Convention on Climate Change Glossary Leading Cavity Main Cavity Trailing Cavity The leading cavity is the cavity volume with a time and comes according to the time in front of the main cavity. The modelled volume cavity. The trailing cavity is the cavity that has a time shift and will come according to the time behind the main cavity. 10

11 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 21 st 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 11 th edition in 1997, this resulted in the well-known Kyoto Protocol. During the latest edition in Paris, the 21 st Conference of Parties (COP21) [4], the 195 atting parties adopted the firstever 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 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. Temperature Anomaly ( baseline) 1,1 1 0,9 0,8 0,7 0,6 0,5 0,4 0,3 0,2 0,1 0-0, ,2-0,3-0,4-0,5-0,6-0,7-0,8 Year Annual Mean 1 C above 19th Century Figure 1-1 Temperature increase over the years from 1880 to 2015 [5]. 11

12 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 CO 2. 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 CO 2, which comes from burning fossil fuel and from industrial processes. Reducing CO 2 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 CO 2 emission all around the world: Trs in global CO 2 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 CO 2. 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 CO 2 emission. World 6% 2% 5% 3% Eu28 CO2 fossil fuel and industrial processes 18% 4% 12% CO2 forests (representing Land Use, Land-Use Change and Forestry (LULUCF) part of UNFCCC) CH4 10% 64% 76% N2O Fluorinated gases (F-gases): HFC, PFC and SF6 Figure 1-2 Shares of greenhouse gas emission 2010 [11]. Countries all over the world are striving to emit less CO 2 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 C. Therefore, the reuse of waste heat is considered an interesting possibility to reduce CO 2 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 12

13 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].!Q out!q out Resorber (Condenser) Resorber (Condenser) Valve Solu8on Pump! W c Compressor Valve! W c Wet-Compressor Desorber (Evaporator) Desorber (Evaporator) (a)!q in 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 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 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.!q in (b) 13

14 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 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. Resorber (Condenser) Warm UAlity: 110 C Valve Waste water Stream: C! W c Wet-Compressor Cold UAlity: 5 C Desorber (Evaporator) 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 14

15 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 recomms 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 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. 15

16 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. Suc$on Compression Discharge 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]. 16

17 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. 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 17

18 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 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 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 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 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. 18

19 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. Recommations to optimise the compressor model are given. 19

20 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. Resorber (Condenser) Valve Compressor Boundary Desorber (Evaporator) 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 C, which needs to be increased to 110 C, van de Bor et al.[21]. The desorber cools the waste water stream down from C to 5 C, as illustrated in Figure

21 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 C and above 115 C respectively. In Table 2-1 these properties are listed and are depent 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. Properties Value Units Waste stream temperature range C Inlet temperature compressor C Inlet pressure compressor 0.2 bar Outlet temperature compressor 115 C Outlet pressure compressor < 5 bar Concentration H 2O 70 wt% NH 3 30 wt% The concentration range of ammonia in the mixture = 20-35wt% [21] For the model a concentration of x=30 wt% will be used. Inlet compressor Gas-liquid regime Compressor process Adiabatic to the environment 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. 21

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 Rotors generated from Envelope Method of Gearing Stosic et al. [27] The Rack Envelope Theory Sakun [30], Zaytsev [1] and You [31] Rotor elements Meshing-line Method Zaytsev and Infante Ferreira [28] Meshing line 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]. R 1h R 2h 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. 22

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 r 1h is the male rotor radius from the origin to the beginning of the male lobe and the r 2h 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 Relation between the angles in the coordinate systems: φ! φ! = ω! ω! = b! b! = i!" = 1 i!" = 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: 23

24 z = L τ τ! (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! ) (2.2.5) (2.2.6) In these equations D is the distance between the rotor axis, and k=1+i 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! x! φ! ψ y! y! φ! 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 Units Envelope radius male rotor r 1h mm Radius male rotor lobe r mm Radius male rotor is R1=r1h+r Radius tip female lobe r 0 mm Envelope radius female rotor r2h, calculated by r 2h = r 1h b 2 b 1 Radius female rotor is R2=r2h+r0 Number of lobes male rotor b 1 - Number of lobes female rotor b 2 - Wrap angle of male rotor τ w Length of the rotors L mm Clearance 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

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: D 1 C 1, C 1 A 1, A 1 I 1, I 1 L 1, L 1 F 1 and F 1 D 1, 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]. D 1 r 2h R 2 L 2 F 2 F 1 0 r 0 r 0 D 2 L 1 I 1 r 0 A 1 I 2 r 0 A 2 C 1 r O 1 D 2 D 1 O 2 r 1h R 1 Figure 2-5 The angles and segments of the male and female rotor. 25

26 The areas and segments are all calculated separately, the area D 1 C 1 is explained in detail to clarify the calculation of the areas. The D 1 C 1 segments (from the O 1 to D 1 over the curve segment D 1 C 1 and from C 1 back to the origin) is shown in Figure 2-6 and the equation for the area is given below, Area D 1 C 1 : D! C! = r 2 r!" sin θ!" sin θ!"# + r θ!" θ!"# (2.2.12) The equations of the area consist out of r, r 1h, 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 D 1, C 1 and the origin of the male rotor O 1. 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 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 Appix A. r 2h F 1 L 2 F 2 R 2 r 0 r 0 D 2 L 1 I 1 r 0 A 1 I 2 r 0 A 2 C 1 r 1h r Θ up O 1 R 1 D 1 O 2 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. 26

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 mm 3. Suction area: The suction area allows the mass flow to enter the compressor, and is calculated in mm 2. 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 mm 2 as well # Port Area in mm 2 1 Cavity Volume in m Shaft Rotation Angle in Degrees Figure 2-7 Schematic representation of the cavity volume, shown in red. The suction area and discharge area are shown in blue 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 27

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 exted 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. Inlet Upstream Outlet Downstream Compressor 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 depent. 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 appix 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 depent input values for the conservation equations will be explained in chapter 3. 28

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 Appix 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 depent of the change in mass over time!" =! n dm out l dm in k=1 k=1. The!" can!"! dt k dt k!" therefore be added to the!" relations so that the equations result in two conservation!" and!"!" 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:!", dp dt = 1 v p!,! v m!!!! dm!"# dt!!!!! dm!" dt! + 1 dv m dt v dt T!,! dt (2.3.1) The energy conservation equation becomes: T v dt dt = T!,! v m! dm!"#!!! v p!,! dt! h + T v T!,! T! dm!"!!! + 1 dv dt m! dt!!,! δq dt +!!!! m h T!,! + h!",! h mt v p!,! dm!" v T dt!!!,! (2.3.2) Equation shows the mass conservation rewritten to pressure change over time!" in units of!"!. Equation shows the energy conservation rewritten to temperature change over the time!" in units of!. The values for the physical parameters!"!", and!"!!"!,!!"!,! calculated in the homogeneous model with a combination of two fluid property programmes: Refprop [32] via Fluidprop [33].!"!!!"!,! are The change in cavity volume over time!"!" is an input together with the mass flows!"!" inlet, outlet and the leakages within the compressor. of the The implementation of the thermodynamic model will be described in chapter 3. 29

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 Equation 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 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

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. Input Model Output Desired Output Geometry Model (Matlab) Volumetric Efficiency Physical Proper6es (Fluidprop/Refprop) Thermodynamic Model (Simulink) Pressure Temperature Isentropic Efficiency Important Input Values ShaC Power Requirement 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. 31

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 depent 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 of the total compression cycle is reached. To allow the model to run, initial values and input values are required. Inputs Ini)al Values (Pressure and Temperature) Physical Proper)es (Input: Pressure and Temperature) If t < t total cycle Conserva)on Equa)ons (Thermodynamic Model) Pressure & Temperature If t = t total cycle Outputs Geometry Model Figure 3-2 Flow diagram of the thermodynamic model, time based. Screw 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 32

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 Units Envelope radius male rotor r 1h 35 mm Radius male rotor lobe r 13.8 mm Radius tip female lobe r mm Number of lobes male rotor b Number of lobes female rotor b Wrap angle of male rotor τ w 314 Length of the rotors L mm Clearance Clearance 0.1 mm Speed of rotation n 3500 rpm Concentration NH 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 time = n 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 s. The shaft rotation angle is specific for this case and will vary for different geometries, depent 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 Appix 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 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 !! s. # Cavity Volume 0.01 Cavity Volume Change in Time "dv/dt" Volume [m 3 ] Volume Change [m 3 /s] Time [s] Time [s] (a) Figure 3-3 Volumes of the compressor over time (a) the cavity volume and (b) the change in volume. (b) 33

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 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 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. 800 Suction Port Area 800 Discharge Port Area Area [mm 2 ] Area [mm 2 ] Time [s] (a) Time [s] Figure 3-4 Port areas in mm 2 (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 (b) 34

35 3.2.2 Thermo-Physical Properties Next to the geometry model, input of thermo-physical properties is required. Eight thermophysical 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 dv dp 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.!" 35

36 %% 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; Figure 3-6 Example of the Matlab code to invoke the physical properties from Refprop via Fluidprop and calculate dv dp 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. Physical property Units Selected Delta Value dv m! dp = p 10!! Pa dp T,x kg Pa dv m! dt = T 10!! K dt p,x kg K dh J dt = T 10!! K dt p,x kg K h J - kg s J - kg K v m! kg - ρ kg - m! 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 thermophysical 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 Input Values/Initial Values for the Implementation The 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. 36

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; implementation of the mass flows, leakage path areas and 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 Mass Flows The 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 dm dt = ζρau (3.3.1) For equation 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 !!!"! u = 2 vdp!!"# (3.3.2) 37

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 and the flow velocity equation 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 P high and the compressor inlet P low, P high P low. The second parameter is the pressure difference over the discharge port of the compressor. The pressure difference between the compressor outlet P high and the heat pump system inlet P low, P high -P low. Compressor Boundary P high P low P high P low Suc1on Discharge 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 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 depent 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 Prins and Infante Ferreira [37] Zaytsev [1] Leakage flow Intake port flow Discharge port flow 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 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

39 In figures 3-8(a-b) the mass flow is graphically shown as time depent 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. 0.2 "Suction Mass Flow" Mass flow through the suction port area 1.4 "Discharge Mass Flow" Mass flow through the discharge port area Mass flow [kg/s] Mass flow [kg/s] Time [s] (a) Time [s] 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 These internal leakages are based on a single cavity volume. The calculation approach of the internal leakages will be explained in the section 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. (b) Leakage Path Areas In 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 The leakages mass flows will however dep on the increased pressure over the sequenced cavities and 39

40 hence change in density. These internal flow phenomena, together with the leakage mass flow calculation, will be explained in the next section 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 clearance of the rotors. Figure 3-9 Leakage paths efficiency, based on 3000 rpm and an evaporating temperature of 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 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 In the geometry model the flow areas of the leakage paths 1, 2 and 5 are calculated with equation 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+r 0 ) for path 5, shown in Figure The contact line, sealing line and the lobe height are represented by LL in equation 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) 40