Institutionen för systemteknik
Department of Electrical Engineering
Examensarbete
Dynamic modeling and Model Predictive Control of a
vapor compression system
Examensarbete utfört i Reglerteknik
vid Tekniska högskolan vid Linköpings universitet av
Andreas Gustavsson LiTH-ISY-EX--12/4552--SE
Linköping 2012
Department of Electrical Engineering Link¨opings universitet
SE-581 83 Link¨oping, Sweden
Link¨opings tekniska h¨ogskola Link¨opings universitet 581 83 Link¨oping
Dynamic modeling and Model Predictive Control of a
vapor compression system
Examensarbete utfört i Reglerteknik
vid Tekniska högskolan vid Linköpings universitet av
Andreas Gustavsson LiTH-ISY-EX--12/4552--SE
Handledare: Patrik Axelsson
ISY, Linköpings Universitet
Joakim Lundkvist
Regin AB
Examinator: Daniel Axehill
ISY, Linköpings Universitet
Avdelning, Institution
Division, Department
Division of Automatic Control Department of Electrical Engineering Linköpings universitet
SE-581 83 Linköping, Sweden
Datum Date 2012-03-28 Språk Language Svenska/Swedish Engelska/English . . Rapporttyp Report category Licentiatavhandling Examensarbete C-uppsats D-uppsats Övrig rapport . . . ISBN — ISRN LiTH-ISY-EX--12/4552--SE
Serietitel och serienummer ISSN
Title of series, numbering — .
URL för elektronisk version
http://www.control.isy.liu.se http://www.ep.liu.se
Titel Dynamisk modellering och modellbaserad prediktionsreglering av kylmaskin
Title Dynamic modeling and Model Predictive Control of vapor compression system
Författare Andreas Gustavsson
Author
Sammanfattning
Abstract
The focus of this thesis was on the development of a dynamic modeling capability for a vapor compression system along with the implementation of advanced multivariable control techniques on the resulting model.
Individual component models for a typical vapor compression system were developed based on most recent and acknowledged publications within the field of thermodynamics. Parameter properties such as pressure, temperature, enthalpy etc. for each component were connected to detailed thermodynamic tables by algorithms programmed in MATLAB, thus creating a fully dynamic environment. The separate component models were then interconnected and an overall model for the complete system was implemented in SIMULINK. An advanced control technique known as Model Predictive Control (MPC) along with an open-source QP solver was then applied on the system. The MPC-controller requires the complete state information to be available for feedback and since this is often either very expensive (requires a great number of sensors) or at times even impossible (difficult to measure), a full-state observer was implemented. The MPC-controller was designed to keep certain system temperatures within tight bands while still being able to respond to varying cooling set-points. The control architecture was successful in achieving the control objective, i.e. it was shown to be adaptable in order to reflect changes in environmental conditions. Cooling demands were met and the temperatures were successfully kept within given boundaries.
Nyckelord
Keywords Model Predictive Control, MPC, Vapor Compression Systems, Modeling
v
Abstract
The focus in this work is on the development of a dynamic modeling capability for a vapor compression system along with the implementation of advanced multivariable control techniques based on the resulting model.
Individual component models for a typical vapor compression system were developed based on most recent and acknowledged publications within the field of
thermodynamics. Parameter properties such as pressure, temperature, enthalpy etc. for each component were connected to detailed thermodynamic tables by algorithms programmed in MATLAB, thus creating a fully dynamic environment. The separate
component models were then interconnected and an overall model for the complete system was implemented in SIMULINK. An advanced control technique known as Model
Predictive Control (MPC) along with an open-source QP solver was then applied on the system. The MPC-controller requires the complete state information to be available for feedback and since this is often either very expensive (requires a great number of sensors) or at times even impossible (difficult to measure), a full-state observer was implemented. The MPC-controller was designed to keep certain system temperatures within tight bands while still being able to respond to varying cooling set-points. The control architecture was successful in achieving the control objective, i.e. it was shown to be adaptable in order to reflect changes in environmental conditions. Cooling
demands were met and the temperatures were successfully kept within given boundaries.
Sammanfattning
Fokus under denna uppsats har legat på utveckling av en dynamisk modell av en kylmaskin samt implementering av modellbaserad prediktionsregling på den resulterande modellen.
Matematiska modeller för varje enhet av kylmaskinen härleds separat baserat på artiklar och rapporter publicerade inom området termodynamik. Enhetsparametrar såsom tryck, temperatur, entalpi länkas till termodynamiska tabeller via algoritmer programmerade i MATLAB för att skapa en dynamisk simuleringsmiljö. De individuella
enheterna kopplas sedan samman till ett komplett system och implementeras i SIMULINK.
En reglerstrategi känd som modellbaserad prediktionsreglering appliceras därefter tillsammans med en QP-lösare baserad på öppen källkod på systemet. Reglerstrategin kräver att den fullständiga tillståndsvektorn är tillgängligt för återkoppling och eftersom detta är oftast antingen väldigt dyrt (kräver många sensorer) eller till och med omöjligt (svårt att mäta) implementeras en observatör. Regulatorn designas att hålla vissa systemtemperaturer inom bestämda intervall och samtidigt kunna hantera förändrade referensvärden för kylkapaciteten. Regulatorstrukturen var framgångsrik för
kontrolluppgiften, det vill säga att den effektivt kunde hantera omgivande
miljöförändringar. Given kylmängd levererades samtidigt som systemtemperaturer hölls inom givna områden.
vii
Acknowledgments
First of all, I would like to thank Lennart Spiik for giving me the opportunity to conduct this thesis at Regin AB. I also would like to thank my supervisor, Joakim Lundkvist, for all his support during my time at the office in Kållered.
Likewise I would like to thank my supervisor at LiU, Patrik Axelsson, for his advice and guidance and particularly for all the time he has spent reading and responding to all my emails. Furthermore, I would like to thank my examiner, Daniel Axehill, for his interest and support throughout the thesis.
I would also like to give a special thanks to my great friend Robert Palmér for his enthusiastic support and for always being willing to assist and help despite having a busy schedule himself.
Finally, I would like to thank my family and my girlfriend for always believing in me and especially for all their support during difficult times.
Linköping, March 2012 Andreas Gustavsson
ix
Content
List of figures ... xiii
List of tables ... xvii
List of nomenclature ... xviii
Compressor ... xviii
Electronic Expansion Valve ... xviii
Evaporator and Condenser ... xviii
Subscripts ... xx Superscripts ... xx 1 Introduction ... 1 1.1 Object ... 2 1.2 Outline ... 2 2 Background ... 3 2.1 Introduction ... 3
2.2 Vapor compression system ... 3
2.2.1 Overview ... 3
2.2.2 Operation ... 6
2.2.3 System used in this thesis ... 7
3 Modeling ... 8 3.1 Introduction ... 8 3.2 Modeling assumptions ... 8 3.3 Compressor ... 10 3.3.1 Mass flow ... 10 3.3.2 Enthalpy change ... 13
3.4 Electronic expansion valve ... 14
x
3.5 Evaporator model ... 16
3.5.1 Two phase region ... 17
3.5.1.1 Conservation of refrigerant mass ... 17
3.5.1.2 Conservation of refrigerant energy ... 20
3.5.1.3 Conservation of tube wall energy ... 26
3.5.2 Superheat region ... 27
3.5.2.1 Conservation of refrigerant mass ... 27
3.5.2.2 Conservation of refrigerant energy ... 29
3.5.2.3 Conservation of tube wall energy ... 34
3.5.3 Overall mass balance of evaporator ... 35
3.5.4 State-space representation ... 36
3.6 Condenser model... 38
3.6.1 Subcool region ... 38
3.6.1.1 Conservation of refrigerant mass ... 38
3.6.1.2 Conservation of refrigerant energy ... 40
3.6.1.3 Conservation of tube wall energy ... 44
3.6.2 Superheat region ... 46
3.6.2.1 Conservation of refrigerant mass ... 46
3.6.2.2 Conservation of refrigerant energy ... 48
3.6.2.3 Conservation of tube wall energy ... 53
3.6.3 Two phase region ... 54
3.6.3.1 Conservation of refrigerant mass ... 54
3.6.3.2 Conservation of refrigerant energy ... 57
3.6.3.3 Conservation of tube wall energy ... 62
3.6.4 Overall mass balance of condenser ... 63
xi 4 Nonlinear model ... 68 4.1 Introduction ... 68 4.2 Component interconnection ... 68 4.3 Simulation ... 70 5 Model linearization ... 72 5.1 Introduction ... 72 5.2 Mathematical method ... 72
5.3 Computer software method ... 79
5.4 Linear model analysis ... 81
6 Controller Design ... 84
6.1 Introduction ... 84
6.2 Model Predictive Control ... 84
6.2.1 Overview ... 84
6.2.2 The MPC framework ... 86
6.2.3 MPC formed as a quadratic programming problem ... 89
6.2.4 MPC with reference tracking ... 90
6.2.5 MPC with integral action... 91
6.2.6 Constraints ... 94 6.2.6.1 Control signals ... 94 6.2.6.2 Controlled outputs ... 96 6.2.7 QP-solver ... 98 6.3 Observer ... 99 7 Result ...103 7.1 Introduction ...103 7.2 Choice of parameters ...103 7.2.1 Sampling time, ...103 7.2.2 Horizons, and ...104
xii
7.2.3 Weighting matrices, and ...105
7.3 Simulation ...105
7.3.1 MPC without integral action ...105
7.3.2 MPC with integral action...107
7.3.3 Controlled outputs constraints ...108
7.3.4 Observer ...110
7.3.5 qpOASES-solver ...112
7.3.6 Disturbance performance ...114
7.3.7 Three simulation scenarios ...116
7.3.7.1 Scenario 1 ...116 7.3.7.2 Scenario 2 ...118 7.3.7.3 Scenario 3 ...120 8 Conclusions ...123 9 Future work ...124 Bibliography ...125
xiii
List of figures
Figure 2.1 An oil refineriy and the Smart car - different areas in which a vapor
compression system can be found. Source [22]. ... 3
Figure 2.2 Overview of a vapor compression system. ... 5
Figure 2.3 Pressure-enthalpy diagram of the vapor compression cycle. ... 5
Figure 3.1 Pressure-Volume diagram. ... 11
Figure 3.2 Pressure-Specific volume diagram. ... 11
Figure 3.3 Compressor volumes... 12
Figure 3.4 Electronic expansion valve. Source [22]... 15
Figure 3.5 Evaporator regions. ... 16
Figure 3.6 Condenser regions. ... 38
Figure 4.1 The SIMULINK model for expansion valve mass flow. ... 68
Figure 4.2 Subsystem for calc rho_cout. ... 69
Figure 4.3 Nonlinear SIMULINK model of the vapor compression system. ... 70
Figure 4.4 Evaporator pressure, varying compressor speed. ... 71
Figure 4.5 Evaporator superheat, varying compressor speed ... 71
Figure 4.6 Evaporator pressure, varying expansion valve opening. ... 71
Figure 4.7 Evaporator superheat, varying expansion valve opening. ... 71
Figure 5.1 Overview of the vapor compression system. ... 77
Figure 5.2 Part of the evaporator model showing the To Workspace block. ... 80
Figure 5.3 Evaporator Pressure with band-limited white noise added to compressor. A zoomed in part is given in the lower right corner. ... 82
Figure 5.4 Evaporator Superheat with band-limited white noise added to compressor. A zoomed in part is given in the lower right corner. ... 82
Figure 5.5 Evaporator Pressure with band-limited white noise added to expansion valve. A zoomed in part is given in the lower right corner. ... 83
xiv
Figure 5.6 Evaporator Superheat with band-limited white noise added to expansion
valve. A zoomed in part is given in the lower right corner. ... 83
Figure 6.1 MPC scheme. ... 85
Figure 6.2 The complete system showing the observer block with its inputs and output. ...100
Figure 6.3 Observer subsystem. ...100
Figure 7.1 Pressure step response with 10% increase in compressor setting. ...104
Figure 7.2 Superheat step response with 10% increase in compressor setting. ...104
Figure 7.3 Pressure step response with 10% increase in expansion valve setting...104
Figure 7.4 Superheat step response with 10% increase in expansion valve setting. ...104
Figure 7.5 Trajectory tracking of evaporator pressure without integral action. ...106
Figure 7.6 Evaporator superheat without integral action. ...106
Figure 7.7 Compressor control signal. ...106
Figure 7.8 Expansion valve control signal. ...106
Figure 7.9 Trajectory tracking of evaporator pressure with integral action. ...107
Figure 7.10 Evaporator superheat with intergral action. ...108
Figure 7.11 Compressor control signal. ...108
Figure 7.12 Expansion valve control signal. ...108
Figure 7.13 Evaporator pressure with and without superheat constraints. ...109
Figure 7.14 Evaporator superheat with and without superheat constraints. ...109
Figure 7.15 Compressor control signal with and without superheat constraints. ...110
Figure 7.16 Expansion valve control signal with and without superheat constraints. .110 Figure 7.17 Evaporator pressure with and without observer. ...111
Figure 7.18 Evaporator superheat with and without observer. ...111
Figure 7.19 Compressor control signal with and without observer. ...112
xv
Figure 7.21 Comparison between first system state and first observer state. ...112
Figure 7.22 Difference between system states and observer states. ...112
Figure 7.23 Exitflags from the simulation. ...113
Figure 7.24 Number of iterations needed for QP solutions using either cold or warm start. ...114
Figure 7.25 Evaporator pressure with noise added to the output. ...115
Figure 7.26 Evaporator superheat with noise added to the output. ...115
Figure 7.27 Compressor control signal when noise is added to system outputs. ...115
Figure 7.28 Expansion valve control signal when noise is added to system outputs. ...115
Figure 7.29 Value of exitflag when noise is added to the output. ...116
Figure 7.30 Number of iterations needed for QP solution using either cold or warm start. ...116
Figure 7.31 Evaporator pressure and pressure reference for scenario 1. ...117
Figure 7.32 Evaporator superheat and with and without superheat constraints for scenario 1 ...117
Figure 7.33 Compressor control signal. ...118
Figure 7.34 Expansion valve control signal. ...118
Figure 7.35 Value of exitflag. ...118
Figure 7.36 Number of iterations. ...118
Figure 7.37 Evaporator pressure and pressure refrence for scenario 2. ...119
Figure 7.38 Evaporator superheat with and without superheat constraints for scenario 2. ...119
Figure 7.39 Compressor control signal. ...120
Figure 7.40 Expansion valve control signal. ...120
Figure 7.41 Value of exitflag. ...120
Figure 7.42 Number of iterations needed. Note that the number of iterations needed for qpOASES is around 60 in the beginning. ...120
xvi
Figure 7.43 Evaporator pressure with pressure reference for scenario 3. ...121
Figure 7.44 Evaporator superheat with and without superheat constraints for scenario 3. ...121
Figure 7.45 Compressor control signal. ...122
Figure 7.46 Expansion valve control signal. ...122
Figure 7.47 Value of exitflag. ...122
Figure 7.48 Number of iterations. Note the that the number of iterations needed for qpOASES is around 60 in the beginning. ...122
xvii
List of tables
Table 1.1 2006 Energy consumption per capita. Source [2]. ... 1
Table 3.1 Matrix elements of . ... 37
Table 3.2 Matrix elements of . ... 66
Table 4.1 Thermodynamic properties for Solvay Flour Solkane R134a. ... 69
xviii
List of nomenclature
Compressor
: Volume : Clearance ratio : Pressure ̇: Mass flow : Density : Compressor displacement : Compressor speed: Compressor control setting
: Polytrophic coefficient : Isentropic coefficient : Temperature
: Enthalpy : Entropy
Electronic Expansion Valve
̇: Mass flow
̇: Mass flow rate coefficient
: Area : Density : Pressure
: Valve control setting
Evaporator and Condenser
: Cross-sectional area inside of tube : Heat transfer coefficient
xix : Inside diameter of tube
: Outside diameter of tube : Void fraction
: Enthalpy : Length
̇: Mass flow rate : Pressure : Density : Temperature : Velocity : Quality
xx
Subscripts
: Evaporator : Condenser : Inlet : Outlet : Intermediate : Vapor : Liquid : Total : Wall : Region: Numbering for different regions of the evaporator and condenser
Superscripts
̅: Average value ̂: Estimated value
1
1 Introduction
Vapor compression systems are widely used all over the world for both heating and cooling. A common area of use is air-conditioning where the aim for the technique is to lower the temperature and provide comfort cooling and dehumidification. It is extensively used in offices, private residences, hotels, hospitals and automobiles but is also an important component in domestic and industrial refrigerators such as large-scale warehouses for chilled or frozen storage of food. There is a great interest both from the industry and the academic world in the development of control-oriented modeling of vapor compression systems. Only in the United States, 10.2 billion dollars was spent for household air-condition devices in 1997 [1] and the total energy demand in the United States has continued to grow ever since. The high-level of energy consumption is observed in Europe, and especially Sweden, as well. An illustration from 2006 showing the energy consumption per capita is given in Table 1.1.
Table 1.1 2006 Energy consumption per capita. Source [2].
0 5000 10000 15000 20000 25000 30000 35000 kW /c ap
2
Traditionally, chlorofluorocarbons (more known as Freons) have been extensively used as refrigerants in vapor compression systems. Their environmental impact and part in ozone depletion is widely known, therefore an increased efficiency in the operation of these systems will in other words be appealing not only from an economical but also from an environmental point of view.
1.1 Object
The purpose of this thesis can be divided into two parts; modeling and controller design. In the first part, the goal is to develop a model capable of accurately predicting the dynamics associated with a vapor compression system. Before the controller design begins, the model is implemented in MATHWORKS’SIMULINK. The controller design is then
focused on a technique called Model Predictive Control (MPC) which is a model-based control method. One of the major advantages with the method compared to for example Linear Quadratic Control is the possibility for explicit handling of input, output and state constraints. These features along with other properties of MPC are investigated and analyzed for the purpose of controlling a vapor compression system.
1.2 Outline
A background to the components of a vapor compression system and how they work is given in Chapter 2. The mathematical equations governing the different system components are presented in Chapter 3. The separate components are interconnected to create an overall, nonlinear model in Chapter 4 and a simplified, linearized version more suitable for MPC is derived in Chapter 5. The controller design is explained in Chapter 6 and to finish this thesis the result, conclusion and future work is given in Chapter 7, 8 and 9 respectively.
3
2 Background
2.1 Introduction
As mentioned in the previous chapter, vapor compression system is a popular, if not the most popular, type of refrigeration cycle when it comes to air-conditioning. It can be found in areas ranging from large-scale industries such as oil refineries and chemical processing plants to the extremely small city car Smart, see Figure 2.1. Refrigeration may be defined as a process in which the temperature of an enclosed space is lowered by removal of heat. The heat can obviously not just disappear, i.e. the heat must be transported somewhere else. How this is achieved, along with a detail explanation of the components of a vapor compression system and their functions are given in the sections to come.
2.2 Vapor compression system
2.2.1 Overview
A vapor compression system is a thermodynamic machine which uses a compressible fluid (more known as refrigerant) to transfer heat from one place to another. Often fluorocarbons are used as refrigerants and the reason for this is because of their favorable thermodynamics properties such as low boiling points, high heat of
Figure 2.1 An oil refinery and the Smart car - different areas in which a vapor
4
vaporization and a moderate density in liquid form in combination with a comparatively high density in gaseous form. The refrigerant undergoes several phase changes during an operating cycle and the different refrigerant states are subcool, liquid-vapor and superheat. In the first state, subcool, the refrigerant is cooled below its saturation point. Because of the surrounding pressure in the heat exchangers the refrigerant never freezes despite temperatures below the saturation point, thus remaining in liquid-form. The second state is a mix of liquid and vapor and it is from now on referred to as the two phase state. In the same way as the refrigerant can be subcooled the refrigerant can also be heated above its vaporization temperature. This is the third and last refrigerant state and is also known as superheat. A vapor compression system can be utilized both for heat generation and to provide cooling, however it is only the cooling application that will be discussed in this thesis. Vapor compression systems suit almost all applications with refrigeration capacities ranging from just a few watts up to megawatts. The simplest vapor compression systems operates with four components; compressor, condenser, expansion valve and evaporator. An illustration of the cycle is given in Figure 2.2 and corresponding pressure-enthalpy diagram is given in Figure 2.3. Worth noting is that the different phases in Figure 2.3 are simplifications of the phase changes in an actual system. Standard assumptions for modeling simplicity are made including adiabatic compression with isentropic efficiency, isobaric conditions in the condenser and evaporator along with isenthalpic expansion through the valve.
5
Subcool Two-phase Superheat
Condenser Evaporator Two-phase Superheat Compressor Expansion valve Heat out Heat in
Figure 2.2 Overview of a vapor compression system.
Figure 2.3 Pressure-enthalpy diagram of the vapor compression cycle.
Evaporation Compression Condensation Expansion Pressur e Enthalpy 1-2: Isentropic compression (entropy is constant). 2-3: Isobaric heat rejection
(pressure is constant). 3-4: Isenthalpic expansion
(enthalpy is constant). 4-1: Isobaric heat extraction
6
The vapor compression cycle starts with the isentropic compression where the refrigerant enters the compressor as a superheated vapor at low pressure. The refrigerant is compressed into a high-pressure refrigerant and as the pressure rises the same happens to the refrigerant temperature. The fluid leaves the compressor as superheated vapor at high pressure and enters the condenser. In the condenser the refrigerant has a higher temperature than the secondary fluid surrounding the condenser. Isobaric heat rejection then takes place where the refrigerant condenses into a subcooled liquid as heat is transferred to the secondary fluid. The refrigerant exits the condenser and enters the expansion valve, still at high pressure. As the fluid passes through the expansion valve a phase change from liquid to two-phase takes places and the pressure is gradually lowered. Since the pressure was decreased by the passage through the expansion device, the refrigerant temperature has been lowered. This results in the refrigerant having a lower temperature than the secondary fluid surrounding the evaporator. As the refrigerant flows through the evaporator, heat is transferred to it and it gradually begins to evaporate. In the isobaric heat extraction the refrigerant becomes completely vaporized and as heat continues being transferred from the secondary fluid, the refrigerant heats above its vaporization temperature and becomes superheated. The refrigerant consequently exits the evaporator as superheated vapor and then enters the compressor and the cycle is repeated.
2.2.2 Operation
In order to provide as much cooling as possible the two-phase portion of the evaporator needs to be maximized. The reason for why the two-phase region has to be maximized is because of the much higher heat transfer coefficient between the two-phase refrigerant and the secondary fluid compared to between vaporized refrigerant and the secondary fluid. The phenomenon is discussed in [3], and in [4] it was shown that for the steady state operating point the heat-transfer coefficient for the two-phase refrigerant was approximately 2.8 kW/m2K in contrast to 0.3 kW/m2K for the superheated refrigerant.
This means that virtually all the cooling capacity of the system is provided by the two-phase region. So why is not the goal to eliminate the superheat region completely and only have the much more heat-transfer efficient two-phase region? The answer is the compressor. The compressor is unable to compress liquid and since the two-phase refrigerant is a mixture of liquid and vapor it is very likely that the compressor will
7
break unless the fluid entering the compressor is completely vaporized. So, to ensure safe and reliable operation of the compressor it is necessary to have some amount of superheat present in the evaporator. A reasonable level of superheat where the evaporator performance is maximized without compromising the safety of the compressor is about 5-6 degrees Celsius [3].
2.2.3 System used in this thesis
When this project started the goal was to create a dynamic model of a vapor compression system, apply different control techniques on the model and finally validate the theoretical work on an actual system. This system was to be built for this particular project but because of a number of different reasons this was unfortunately not possible within the time-frame of this thesis. It was consequently decided that this thesis would be completely theoretical. The model in this thesis utilized thermodynamic tables to recalculate and update system parameters during simulation. So in order to be sure that the model derived in this thesis and the equations it relies on during simulation are valid approximation of an actual system, system parameters and simulation outcomes are compared to results given in [4]. More details about the model and the equations used when deriving it are presented in the chapter that follows.
8
3 Modeling
3.1 Introduction
In this chapter the vapor compression system and its components are described and mathematical models for each component are derived. The system being modeled is the system described in chapter two and the derivation follows the methods used in [5], [4], [3] and [6]. Mathematical models, based on the knowledge of underlying physics, for each of the four components are derived separately. When modeling the heat exchangers an integration rule known as Leibniz’s equation is used [7]. This rule is especially useful when removing the spatial dependence with being the spatial coordinate. In other words, the limits of integration are determined by the length of the region being integrated over. The general Leibniz’s rule may be written as
∫ ( ) ( ) ( ) [ ∫ ( ) ( ) ( ) ] ( ( ) ) ( ( )) ( ( ) ) ( ( )) (3.1)
Once equations are obtained they are rearranged and organized into a more suitable form for controller design.
3.2 Modeling assumptions
A vapor compression system, even in the simplest case with only two heat exchangers, a compressor and a valve, is a highly complex system. Difficulties when trying to derive mathematical models for the intricate dynamics associated with refrigerant phase changes are well documented in [8] and [5]. A vapor compression system is a stiff system [9], meaning there are system dynamics with time scales differing by orders of magnitude. A common technique to handle diverse time scales is to replace the fast dynamics with static (algebraic) equations. Static and dynamic analysis of time constants in a vapor compression system has been carried out in [3] and shows that the dynamics associated with the actuators (compressor and valve) react much faster to
9
changes compared to the two heat exchangers. So in order to simplify the modeling the dominant (slow) dynamics of the system are assumed to be connected to the two heat exchangers while the dynamics linked to the actuators are modeled with static relationships. As mentioned, the refrigerant undergoes several phase changes inside the heat exchangers (two in the condenser and one in the evaporator). An important assumption which significantly simplifies the modeling of these transitions is the use of lumped parameter, moving boundary between the different fluid states [10]. This assumption implies that the boundary is time-invariant and that parameters of interest in each region are “lumped” together. The method is especially useful in a control point-of-view since it generates relatively low order models. Included in the lumped parameter approach is also the concept of mean void fraction. The void fraction is defined as the ratio of vapor volume to total volume [11]. A number of experimental correlations for estimating the void fraction have been suggested, some of which are discussed in [12] and [13]. But as previously declared, a mean void fraction is used in this thesis. The concept of a mean void fraction was originally proposed in [14] and the main idea is the assumption of a time-invariant mean void fraction. This assumption greatly reduces the number of equations needed to model the void fraction and the approximation has been shown to be valid during most operating conditions [14]. Other key assumptions for modeling simplicity are:
The cross-flow heat exchanger is assumed to be a long, horizontal tube.
The flow of the refrigerant is modeled as one-dimensional flow.
Axial conduction of heat for the refrigerant is negligible.
Each region has its own mean heat transfer coefficient.
Average tube wall temperatures are used and assumed uniform throughout the thickness of the wall.
Pressure is assumed to be the same along the heat exchanger (thereby neglecting pressure drops due to viscous friction and momentum change in refrigerant).
Time invariant mean void fraction is assumed.
The two phase region in the heat exchangers can be divided into a liquid and a vapor part.10
The average wall temperature at the intermediate between the superheat/subcool and two phase region can be assumed to be the same as the average wall temperature in the two phase zone.
Air temperature remains constant throughout the cross-flow heat exchanger and is calculated as the average of the inlet and outlet temperatures.The following sections involve more assumptions than mentioned above, however, new assumptions are pointed out when being made.
3.3 Compressor
The compressor considered in this thesis is a variable speed compressor. By varying the speed of the compressor the mass flow rate of the refrigerant is changed. Also, the compression of refrigerant inside the compressor is not an isenthalpic process, i.e. there is an enthalpy difference between the inlet and outlet of the compressor. To account for these changes, two static relationships are used to model the compressor; a mass flow equation and an enthalpy change equation.
3.3.1 Mass flow
The volumetric efficiency is defined as [15]
( ( )
) (3.2)
where is the polytrophic coefficient and the clearance factor. The compression is assumed to be an isentropic process, i.e. the entropy remains unchanged. This assumption also results in that the polytrophic coefficient can be replaced by the isentropic exponent . The value of is taken from [16], thus . and
are the compressor inlet and outlet pressure respectively. A reciprocating compressor’s compression cycle can be explained in a four-part sequence that occurs with each advance and retreat of the piston (two strokes per cycle). The four parts of the cycle are intake, compression, discharge and expansion and are represented by a, b, c and d respectively in Figure 3.1. Figure 3.2 is an illustration of the same cycle as in Figure 3.1. but with specific volume instead of volume on the x-axis. and in (3.2) are volumes
11
The clearance factor is defined as the ratio of clearance volume to piston displacement volume
(3.3)
where and is the valve clearance volume and total volume respectively (see
Figure 3.3 for illustration). If for some reason, the clearance ratio is unknown or difficult
vcl c, d 4 b a 3 Specific volume Pressur e c, d out in a b Specific volume Pressur e Pin Pout
Figure 3.2 Pressure-Specific volume diagram.
b a c d Stroke Pressur e Pin Pout Stroke Volume
12
to calculate a reasonable estimation is 15% clearance [17]. This value for the clearance ratio will be used in this thesis.
Figure 3.3 Compressor volumes.
According to [15] volumetric efficiency can also be defined as
̇ ̇ ̇ (3.4)
where is the piston displacement in volume per unit of time, the compressor speed, the compressor displacement and the refrigerant density at the compressor inlet. The compressor displacement is found in the technical documentation from the compressor manufacturing company [18]. The compressor speed for this particular system is calculated based on the relationship given in [4] and can be expressed as
(3.5)
where is the maximum compressor speed and is also found in [18]. and
is the compressor control signal and compressor maximum control signal
respectively.
vtot
13
Combining (3.2) (3.5) the mass flow of the compressor may be defined as
̇ ( ) ( ( ) ) (3.6)
3.3.2 Enthalpy change
The difference between the compressor inlet and outlet enthalpy is discussed in [5] and can be expressed as
(3.7)
where is the isentropic enthalpy. Both enthalpy and entropy are functions of pressure and temperature. This means that once two or more of either enthalpy, entropy, temperature or pressure are known, remaining parameters can be interpolated using thermodynamic tables. In this thesis the thermodynamic table of choice is taken from Solvay Flour [19]. The table and its content is discussed in more detail in Chapter 4. The isentropic enthalpy is thus estimated by the following procedure:
1. let the compressor reach steady state 2. measure , and
3. interpolate from table since enthalpy is a function of pressure
and temperature, ( )
4. interpolate from table since entropy is a function of pressure and enthalpy, ( )
5. interpolate from table since isentropic enthalpy is a function of pressure and entropy, ( )
14
The isentropic efficiency coefficient may be expressed as [20]
(3.8)
where is the outlet temperature assuming an isentropic process. and are the compressor inlet and outlet temperatures respectively. may in turn be written as
(
)
(3.9)
where is the isentropic coefficient. The value of for R134a is discussed in [21] and shown to be scattered in the interval of 1.15 to 1.16 for the temperature region corresponding to the steady state condition. For this thesis the value of is chosen as the mean value of this interval, i.e. 1.155.
The enthalpy change over the compressor can thus be expressed as
(3.10)
3.4 Electronic expansion valve
The expansion device considered for this thesis is an electronic expansion valve. An illustration of a typical electronic expansion valve is given in Figure 3.4. By altering the opening of the valve, the refrigerant mass flow rate can be changed. The flow through the valve is assumed to be an isenthalpic process, i.e. the refrigerant enthalpy is unchanged from the inlet to the outlet of the valve. One static relationship is thus used to model the electronic expansion valve; a mass flow equation.
15
Figure 3.4 Electronic expansion valve. Source [22].
3.4.1 Mass flow
In order to estimate the mass flow rate through the expansion valve the model from [23] is used
̇ ̇ √ ( ) (3.11)
where ̇ is the mass flow rate coefficient, the orifice area, the refrigerant density at the expansion valve inlet and and the condenser and evaporator pressure respectively. Note that in order to obtain ̇ in kg/s, and need to be defined in Pa.
The value for the mass flow rate coefficient is discussed in [24] and determined to be scattered in the interval of 0.94 ± 0.04. The refrigerant density is assumed to remain
16 𝑥𝑒 𝑇𝑒𝑤 (𝑡) 𝑇𝑒𝑤 (𝑡) 𝑚̇𝑒 𝑖𝑛 𝑒 𝑖𝑛 𝑥𝑒 > 0 𝐿𝑒 (𝑡) 𝐿𝑒 (𝑡) 𝐿𝑒𝑇 𝑚̇𝑒 𝑜𝑢𝑡 𝑒 𝑜𝑢𝑡
Two phase Superheat
𝑃𝑒(𝑡)
unchanged by the passage through the valve, i.e. . The orifice area is calculated based on the relationship given in [4]
(3.12)
where is the maximum orifice area and can be found in the technical
documentation provided by the manufacturer [25]. and are the valve
control signal and valve maximum control signal respectively.
The mass flow rate through the electronic expansion device may then be defined as
̇ ̇ (
) √ ( ) (3.13)
3.5 Evaporator model
The derivation of the evaporator model is based on a combination of the results given in [4], [5], [6] and [26]. The structure of the evaporator is assumed to be a lumped parameter dynamic model with two types of fluid regions; a liquid-vapor section (from now on referred to as the two phase region) and a superheated vapor section. The boundary between these two regions is assumed to be a moving boundary and may thus change depending on operating conditions. An illustration of the evaporator zones is given in Figure 3.5.
17
Mass and energy conservation for the different regions of the heat exchanger are derived with help of generalized Navier-Stokes equations [27]:
Refrigerant mass balance: ( ) 0 (3.14) Refrigerant energy balance: ( ) ( ) ( ) (3.15) Wall energy balance: ( ) ( ) ( ) (3.16) These governing partial differential equations (PDEs) are in the following sections used to obtain lumped parameter ordinary differential equations (ODEs) to model fluid dynamics and relevant heat exchanger properties. In order to remove spatial dependency, each term in (3.14) to (3.16) is integrated over the length of corresponding region. After further simplifications, rearrangements and combinations of certain parameters the resulting ODEs are structured into a form more known as state-space
form.
3.5.1 Two phase region
3.5.1.1 Conservation of refrigerant mass
The first term in the equation for refrigerant mass balance (3.14) is integrated over the cross-sectional area. The left hand side of the equation below is denoted by and this
is done to make the derivation more structured and easier to follow.
∫ ∫ [ ( )] [ ( ( ) )] (3.17)
where and is the cross-sectional area of saturated liquid and vapor respectively.
18
and can therefore be replaced with . Applying the same way of thinking on and the
term may be rewritten as ( ).
In order to remove the spatial dependency, (3.17) is integrated over the length of the region and Leibniz’s rule is applied. Also note that the coefficient on the left hand side now has a new number
∫ ∫ [ ( ( ) )] ∫ [ ( ( ) )] [ ( ( ) )] 0 [ ( ( ) )] ∫ [ ( ( ) )] [ ( ( ) )] (3.18)
Before performing the integration, let us study the equation above in more detail. The last term in (3.18) is evaluated at the intermediate between the two phase and superheat region. The refrigerant can then be assumed to be completely vaporized at that location in the evaporator, i.e. the mean void fraction equals one
∫ [ ( ( ) )] [ ( ( ) )] ∫ [ ( ( ) )] (3.19)
Integrate and rearrange
∫ [ ( ( ) )] [ ( ( ̅ ) ̅ )] (3.20)
19
As the integration is carried out, the concept of mean void fraction is applied. This is the reason for being replaced by ̅ in (3.20). Now apply the product rule
( ̅ ) ̅ ( ̅ ) [ ] ̅ [ ] [ ( ̅ ) ̅ ] [ ( ̅ ) ̅ ] [ ( ̅ ) ̅ ] [ ( ̅ ) ̅ ] (3.21)
The average density in node one can be rewritten with help of the mean void fraction as
( ̅ ) ̅ . Term one in the equation for refrigerant mass balance (3.14)
thus simplifies to ( ) (3.22)
As with the first term, the second term in the equation for refrigerant mass balance (3.14) is also integrated over the cross-sectional area. A new coefficient, , is used to mark that this is a derivation of a different term in (3.14).
∫ [ ( )] [ ( ( ̅ ) ̅ )] (3.23)
20
Integrate over the length of the region and use the fact that for the vaporized part of the region ∫ ∫ [ ( ( ̅ ) ̅ )] ( ( ̅ ) ̅ )| [ ( ( ̅ ) ̅ )]| ( ( ) ) ( ( ̅ ) ̅ ) (3.24)
Since the last term inside the square brackets in (3.24) is the result of the integration over 0, i.e. at the inlet of the evaporator, that part may be denoted with the in subscript
[ ( ( ̅ ) ̅ )]
̇ ̇ ̇
(3.25)
Term two in the equation for refrigerant mass balance (3.14) thus simplifies to
̇ ̇ (3.26)
Combine the equations from the derivation of term one (3.22) and term two (3.26)
0 ⇔ ( ) ̇ ̇ 0 (3.27)
Equation (3.27) can be rewritten since is pressure dependent ( ( )). The conservation of refrigerant mass in the two phase region may thus be defined as
( ) ̇ ̇ (3.28)
3.5.1.2 Conservation of refrigerant energy
The first term in the equation for refrigerant energy balance (3.15) is integrated over the cross-sectional area
21 ∫ ( ) ∫ ∫ [ ] (3.29)
Replace the cross-sectional area ratios with the mean void fraction
[ ( ̅ ) ̅ ]
[ ( ̅ ) ̅ ]
(3.30)
Integrate over the length of the region and apply Leibniz’s rule
∫ ∫ [ ( ̅ ) ̅ ] ∫ [ ( ̅ ) ̅ ] [ ( ̅ ) ̅ ] 0 [ ( ̅ ) ̅ ] (3.31)
Since the refrigerant is assumed to be completely vaporized for the vapor region, the mean void fraction equals one
∫ [ ( ̅ ) ̅ ] [ ( ) ] ∫ [ ( ̅ ) ̅ ] (3.32)
22 Perform the integration and then rearrange
∫ [ ( ̅ ) ̅ ] [ ( ̅ ) ̅ ] ( ̅ ) [ ] ̅ [ ] (3.33)
Use the product rule
( ̅ ) [ ] ̅ [ ] [ ( ̅ ) ̅ ] [ ( ̅ ) ̅ ] (3.34)
Rearrange once again and term one in the equation for refrigerant mass balance (3.15) can be written as [ ( ̅ ) ̅ ] [ ( ̅ ) ̅ ] (3.35)
23
The second term in the equation for refrigerant mass balance (3.15) is now integrated over the cross-sectional area
∫ ( ) [ ] (3.36)
Replace the cross-sectional area ratios with the mean void fraction
[ ( ̅ ) ̅ ] (3.37)
Integrate over the length of the region and use the fact that ̅ for the vaporized part of the region ∫ ∫ [ ( ) ] [ ( ) ]| [ ( ) ]| [ ( ( ) )] [ ( ( ) )] [ ( ̅ ) ̅ ] (3.38)
The last term in (3.38) is the result of the integration over 0, i.e. at inlet of the evaporator. That part can thus be denoted with the in subscript
[ ( ̅ ) ̅ ] ̇ [ ̇ ̇ ] (3.39) Quality is defined as ̇ ̇ (3.40)
Substituting the definition of quality into (3.39) results in
̇ [( ) ̇ ̇ ]
24
Term two in the equation for refrigerant energy balance (3.15) thus simplifies to
̇ ̇ (3.42)
The last term in the equation for refrigerant energy balance (3.15) is now integrated over the cross-sectional area
∫ ( ) ( )
( )
( )
(3.43)
Integrate over the length of the region and term three in the equation for refrigerant mass balance (3.15) simplifies to
∫
∫ ( )
̅ ( ̅ ̅ ) (3.44)
As the integration is carried out, the assumptions of mean heat transfer coefficient and uniform wall temperatures are applied. This is the reason for , and being
replaced by ̅ , ̅ and ̅ respectively. Combining the resulting equations from the
derivation based on the equation for refrigerant energy balance (3.35), (3.42) and (3.44) yields ⇔ [ ( ̅ ) ̅ ] [ ( ̅ ) ̅ ] ̇ ̇ ̅ ( ̅ ̅ ) (3.45)
25 Rearrange ( ̅ )[ ] [ ( ̅ ) ̅ ] ̇ ̇ ̅ ( ̅ ̅ ) (3.46)
Substitute ̇ using the result from (3.28) and the above equation may be written as
( ̅ )[ ] [ ( ̅ ) ̅ ] ̇ [ ̇ ( ) ] ̅ ( ̅ ̅ ) ⇔ [( ̅ )( ) ( )] [ ( ̅ ) ̅ ] ̇ ( ) ̅ ( ̅ ̅ ) (3.47)
The average density in node one can be rewritten with help of the mean void fraction using ( ̅ ) ̅ . After substitution and rearrangement (3.47) is
simplified into [( ̅ ) ( )] [ ( ̅ ) ( ̅ ) ̅ ( ̅ ) ] ̇ ( ) ̅ ( ̅ ̅ ) (3.48)
26
In order to further simplify the expression, the definition of enthalpy of vaporization can be used [28]. The conservation of refrigerant energy in the two phase region may then be written as [( ̅ ) ] [ ( ̅ ) ] ̇ ( ) ̅ ( ̅ ̅ ) (3.49)
3.5.1.3 Conservation of tube wall energy
The first term in the equation for wall energy balance (3.16) is integrated over the length of the region and Leibniz’s rule is used
∫ ( ) ( ) ∫ ( ) [ ∫ | 0 | ] ( ) [ ̅ ̅ ] (3.50)
Take advantage of the product rule and term one in the equation for wall energy balance simplifies into ( ) [ ̅ ̅ ̅ ] ( ) ̅ (3.51)
The second and third term in the equation for wall energy balance (3.16) are now integrated over the length of the region
∫ ( ( ) ( ))
̅ ( ̅ ̅ ) ̅ ( ̅ )
(3.52)
Combining the resulting equations from the derivation in this section (3.51) and (3.52) and the conservation of wall energy in the two phase region may be written as
27 ⇔ ( ) ̅ ̅ ( ̅ ̅ ) ̅ ( ̅ ) (3.53)
3.5.2 Superheat region
3.5.2.1 Conservation of refrigerant mass
As for the equations associated to the two phase region, the first term in the equation for refrigerant mass balance (3.14) is integrated over the cross-sectional area
∫
(3.54)
Since the refrigerant never undergoes any phase transitions in the superheat region, there is no need to extend the cross-sectional area into a liquid and vapor part. Now integrate over the length of the region and apply Leibniz’s rule
∫ ∫ ∫ | | (3.55)
Since the total length is unchanging, the above derivative of with respect to time is equal to zero. Applying the product rule on (3.55) results in
(3.56)
Again, the total length of the evaporator is constant, thus ( ) 0 ⇔ (3.57)
28 Combining (3.56) and the result from (3.57) yields
(3.58)
After one last rearrangement, term one in the equation for refrigerant mass balance (3.14) may be written as ( ) (3.59)
As with the first term, the second term in the equation for refrigerant mass balance (3.14) is integrated over the cross-sectional area
∫
(3.60)
Integrate over the length of the region
∫ ∫ | | (3.61)
where the first part in (3.61) is the result of the integration over , i.e. at the outlet of the evaporator. Term two in the equation refrigerant mass balance (3.14) is finally rewritten as
̇ ̇ (3.62)
Combining (3.59) and (3.62) and the conservation of refrigerant mass in the superheat region may be expressed as
( 0) ⇔ ( ) ̇ ̇ 0 (3.63)
29
3.5.2.2 Conservation of refrigerant energy
The first term in the equation for refrigerant energy balance (3.15) is integrated over the cross-sectional area
∫ ( ) ( ) (3.64)
Integrate over the length of the region and apply Leibniz’s rule
∫ ∫ ( ) ∫ ( ) ( )| ( )| ∫ ( ) ( ) (3.65)
Simplifying the integration in (3.65) separately
∫ ( ) ∫ ( ) [ ( )] [ ( )] [ ( )] ( ) (3.66)
Substituting the outcome of the integration into (3.65) results in
( ) ( )
(3.67)
Use the product rule
[ ] [ ] ( ) (3.68)
30
Take advantage of the fact that the length of the evaporator is unchanging ( ) 0 ⇔ (3.69) Substitute into (3.68) [ ] [ ] ( ) [ ] ( ) (3.70)
Apply the product rule again
[ ] ( ) (3.71)
Now let us look at the equation previously derived for the conservation of refrigerant mass in (3.63). The equation can be rearranged as
( ) ̇ ̇ ⇔ ̇ ̇ [ ] (3.72)
Substitute the equivalent term in (3.71)
[ ] ̇ ̇ [ ] ( ) [ ] ̇ ̇ ( ) (3.73)
31
The enthalpy in the superheated region can be assumed to be an average of the inlet and outlet enthalpies
(3.74) Using the definition of average enthalpy in (3.73) yields
[0 ] ̇ ̇ ( 0 0 ) (3.75)
Term one in the equation for refrigerant energy balance (3.15) thus simplifies into
[0 ] 0 ̇ ̇ 0 ( ) (3.76)
Term two in the equation for refrigerant energy balance (3.15) is now integrated over the cross-sectional area
∫ ( ) ( ) (3.77)
Integrate over the length of the region and term two in (3.15) simplifies into
∫ ∫ ( ) | | ̇ ̇ (3.78)
32
The last term in the equation for refrigerant energy balance (3.15) is now integrated over the cross-sectional area
∫ ( ) ( )
( )
( )
(3.79)
Integrate over the length of the region and term three in (3.15) simplifies into
∫ ∫ ( ) ̅ ( ̅ ̅ ) (3.80)
Combining the resulting equations for term one, two and three in the equation for refrigerant energy balance (3.76), (3.78) and (3.80) results in
⇔ [0 ] 0 ̇ ̇ 0 ( ) ̇ ̇ ̅ ( ̅ ̅ ) ⇔ [0 ] 0 0 ( ) 0 ̇ ( ) 0 ̇ ( ) ̅ ( ̅ ̅ ) (3.81)
33
Use the definition of the conservation mass in (3.63) and substitute it for ̇ in (3.81)
[0 ] 0 0 ( ) 0 [ [ ] ̇ ] ( ) 0 ̇ ( ) ⇔ 0 ( ) [0 0 ( ) ] 0 ̅ ( ̅ ̅ ) ̇ ( ) (3.82)
Apply the chain rule on the time derivative of using the relation of density, enthalpy and pressure ( ( )) (3.83) Substitute (3.83) into (3.82) 0 ( ) [0 0 ( ) ] 0 [ ( ) ] ̅ ( ̅ ̅ ) ̇ ( ) (3.84)
34
The conservation of refrigerant energy in the superheat region can thus be simplified into 0 ( ) [0 0 ( ) ] 0 [ ( ) ] ̅ ( ̅ ̅ ) ̇ ( ) (3.85)
3.5.2.3 Conservation of tube wall energy
The first term in the equation for wall energy balance (3.16) is integrated over the length of the region and Leibniz’s rule is applied
∫ ( ) ( ) ∫ ( ) [ ∫ | | ] ( ) [ | | ] ( ) [ ̅ ̅ ] (3.86)
Apply the product rule and use the fact that
. Term one in (3.16) thus
simplifies into ( ) [ ̅ ̅ ̅ ] ( ) [ ̅ ( ̅ ̅ ) ] (3.87)
35
Now the spatial dependence for the second and third term in the equation for wall energy balance (3.16) are removed by integrated over the length of the region
∫ ∫ ( ( ) ( )) ̅ ( ̅ ̅ ) ̅ ( ̅ ) (3.88)
Combining (3.87) and (3.88) and the conservation of wall energy in the superheat region may be written as ⇔ ( ) [ ̅ ( ̅ ̅ ) ] ̅ ( ̅ ) ̅ ( ̅ ̅ ) (3.89)
3.5.3 Overall mass balance of evaporator
To obtain an overall mass balance equation, the previous derived mass equations are used. The equation for conservation of refrigerant mass in the two phase region, (3.28),
̇ ̇ ( ) (3.90)
is combined with the equation for conservation of refrigerant mass in the super heat region, (3.63), ̇ ̇ ( ) (3.91) which results in ( ) [ ] ̇ ̇ (3.92) The density in region two is pressure and enthalpy dependent ( ( )) and
36
( ( )) the density can be written as ( ). Use this
relationship for the time derivative of and apply the chain rule (3.93)
Now substitute this into (3.92)
( ) [ ( ) ] ̇ ̇ (3.94)
The overall mass balance of the evaporator is thus simplified into
( ) ( ) ̇ ̇ (3.95)
3.5.4 State-space representation
The governing equations for conservation of mass, refrigerant energy and wall energy (3.49), (3.53), (3.85), (3.89) and (3.95) can be rearranged into state-space form. The five time derivatives; ̇ ̇ ̇ ̇ and ̇ are organized according to the ( ) ̇ ( ) form. The states [ ] and the control
inputs [ ̇ ̇ ] are thus being used to define the evaporator model as
̇ ( ) (3.96) where [ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ] (3.97)
37 and [ ̇ ( ) ̅ ( ̅ ̅ ) ̇ ( ) ̅ ( ̅ ̅ ) ̇ ̇ ̅ ( ̅ ) ̅ ( ̅ ̅ ) ̅ ( ̅ ) ̅ ( ̅ ̅ ) ] (3.98)
The rows of correspond to ( ), ( ), ( ), ( ) and ( ) respectively. The elements of are given in Table 3.1.
Table 3.1 Matrix elements of ( ).
[( ̅ ) ] [ ( ̅ ) ] 0 ( ) [0 0 ( ) ] 0 [ ( ) ] [ ] [ ] ( ) ( ) ̅ ̅ ( )
38
3.6 Condenser model
As for the evaporator model, the derivation of the condenser model is based on a combination of the results given in [4], [5], [6] and [26]. The structure of the condenser is also assumed to be a lumped parameter dynamic model. The heat exchanger has three types of fluid regions; a superheated vapor section, a two phase section and a subcooled liquid section. The boundaries between these regions are, as for the evaporator, assumed to be moving boundaries and may thus change depending on operating conditions. An illustration of the condenser zones is given in Figure 3.6.
Equations for the conservation of mass and energy for the condenser are derived based on the generalized Navier-Stokes equations presented in Chapter 3.5, see (3.14) (3.16). The method for converting the governing PDEs to lumped parameter ODEs and removing the spatial dependence also matches that of Chapter 3.5. The resulting ODEs are finally organized into a state-space form.
3.6.1 Subcool region
3.6.1.1 Conservation of refrigerant mass
The first term in the equation for refrigerant mass balance (3.14) is integrated over the cross-sectional area 𝑥𝑐 𝑇𝑐𝑤 (𝑡) 𝑚̇𝑐 𝑜𝑢𝑡 𝑐 𝑜𝑢𝑡(𝑡) 𝑇𝑐𝑤 (𝑡) 𝑥𝑐 0 𝑇𝑐𝑤 (𝑡) 𝑚̇𝑐 𝑖𝑛 𝑐 𝑖𝑛(𝑡) 𝑃𝑐(𝑡) 𝐿𝑐 (𝑡) 𝐿𝑐 (𝑡) 𝐿𝑐 (𝑡) 𝐿𝑐𝑇 Superheat Two phase Subcool
