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Dynamic Modeling and Control Strategies for a Micro-CSP Plant with Thermal Storage Powered by the Organic Rankine Cycle by Melissa Kara Ireland B.S., Cornell University (2008) Submitted to the Department of Mechanical Engineering in partial fulfillment of the requirements for the degree of Master of Science in Mechanical Engineering at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY February 2014 c Massachusetts Institute of Technology 2014. All rights reserved. Author . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Department of Mechanical Engineering January 17, 2014 Certified by . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . John G. Brisson Professor Thesis Supervisor Accepted by . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . David E. Hardt Chairman, Department Committee on Graduate Theses 2 Dynamic Modeling and Control Strategies for a Micro-CSP Plant with Thermal Storage Powered by the Organic Rankine Cycle by Melissa Kara Ireland Submitted to the Department of Mechanical Engineering on January 17, 2014, in partial fulfillment of the requirements for the degree of Master of Science in Mechanical Engineering Abstract Organic Rankine cycle (ORC) systems are gaining ground as a means of effectively providing sustainable energy. Coupling small-scale ORCs powered by scroll expandergenerators with solar thermal collectors and storage can provide combined heat and power to underserved rural communities. Simulation of such systems is instrumental in optimizing their control strategy. However, most models developed so far operate at steady-state or focus either on ORC or on storage dynamics. In this work, a model for the dynamics of the solar ORC system is developed to evaluate the impact of highly transient heat sources and sinks, thermal storage, and the variable loads associated with distributed generation. Based on an existing micro-CSP (concentrating solar power) plant, the dynamic model is implemented in the Modelica modeling language. Detailed steady-state component models, which are implemented in EES and validated to data where available, form the basis for the dynamic components. The dynamic model in its current form is used to make qualitative assessments of several control decisions based on realistic solar irradiance input representing four reference days. Future analysis will survey a wider range of environmental conditions to make quantitative determinations on the efficacy of each control decision. The simulations include an approximation for startup and shutdown, which avoids the numerical issues associated with the discontinuities in the working fluid density derivative present during such rapid phase changes. To the author’s knowledge, this is the first model capable of continuously simulating through startup and shutdown in addition to coupling a dynamic thermodynamic model of the power cycle with dynamic models of the solar collectors and thermal storage tank. Thesis Supervisor: John G. Brisson Title: Professor 3 4 Acknowledgments People come to help you in many ways over the course of a thesis. Some put you on the right path. Prof. Tim Gutowski’s enthusiasm for the sustainable energy field and his encouragement to find my own way allowed me to wait until I found the project that perfectly combined my passion for sustainable energy and the thermodynamic modeling skills I use every day on the job. I am grateful to Rob Stoner of the MIT Energy Initiative not only for greatly improving a related class paper but also for introducing me to the topic that would later become this thesis. I am greatly indebted to Dr. Sylvain Quoilin for his patience in bringing me up to speed on all things ORC thermodynamics and dynamic modeling. His lab’s sponsorship of my trip to University of Liège was essential in making me effective on this project. Others provide you with the skills and guidance to sustain you along the way. I am grateful that Dr. Matthew Orosz gave me the chance to contribute to this project. He devoted many hours of his time deepening my understanding of the system and providing valuable suggestions to improve my writing. Adriano Desideri was instrumental in keeping this project going. His willingness to delve into the details querying the models and helping me understand and improve upon them undoubtedly contributed to making this project a success. Prof. Harry Hemond’s willingness to stay after hours, ask thought-provoking questions, and provide useful suggestions assured me I was on the right track. My advisor, Prof. John Brisson, provided invaluable guidance in crafting this work into a respectable technical paper. His persistent encouragement to think more acutely and put myself in others’ shoes inspired this work’s depth and clarity. Yet others provide a sounding board when things are driving you crazy. I cannot thank Dave Blum enough for his completely unselfish support whenever I had an issue, from helping with class projects and concepts to providing Matlab tips and tricks to his willingness to engage on any question no matter the strain he was under with his own work. Phil Knodel was also a great resource for venting frustrations and 5 some useful brainstorming sessions when things weren’t making sense. Friends like Bethany Kroese and Nina Shinday, under considerable pressure themselves, always made time for a constructive chat. Some come in at the very end to give you a final push. Thank you to Dave Gutz who donated a significant amount of his time troubleshooting and improving my understanding of control systems. Finally, some provide constant support throughout your life no matter what your endeavor. I am grateful to my parents for shaping me into the person I am today and providing me with the skills, opportunities, and perseverence to complete a project like this one. Neil is an amazing companion whose extreme patience and support, compassionate ear, and good nature kept me going. 6 Contents Abstract 3 Acknowledgements 5 Contents 7 List of Figures 9 List of Tables 11 Nomenclature 13 1 Introduction 17 1.1 Background and Literature Review . . . . . . . . . . . . . . . . . . . 17 1.2 Pilot System Description: Basis for Models . . . . . . . . . . . . . . . 19 2 Steady-state modeling 25 2.1 Evaporator/Recuperator . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.2 Expander-generators . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.3 Condenser . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.4 Pumps/Motors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 2.5 Solar Collectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 2.6 Application: Determining Optimum Set Points for Varying Working Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 41 2.7 Potential Improvement to Cycle Optimization: Variable Condenser Temperature Defect . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Dynamic modeling 46 51 3.1 Evaporator/Recuperator . . . . . . . . . . . . . . . . . . . . . . . . . 53 3.2 Condenser . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 3.3 Liquid Receiver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 3.4 Solar Collectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 3.5 Storage Tank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 3.6 Control Unit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 3.7 Application: Identifying Optimal Control Schemes for Daily Environmental Variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Conclusions and Future Work 62 71 4.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 4.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 A Model Code and Experimental Data 75 B Sensitivity Studies on Heat Exchanger Modeling 77 B.1 Mixed vs Unmixed Cooling Air . . . . . . . . . . . . . . . . . . . . . 77 B.2 Variable vs Constant Heat Transfer Coefficients . . . . . . . . . . . . 79 B.3 Tube-to-Tube Heat Transfer . . . . . . . . . . . . . . . . . . . . . . . 80 B.4 Sensitivity to Number of Cells . . . . . . . . . . . . . . . . . . . . . . 82 C Optimized Intermediate Pressure between Two Expanders 8 85 List of Figures 1-1 Pilot micro-CSP plant. . . . . . . . . . . . . . . . . . . . . . . . . . . 20 1-2 Flow diagram for the micro-CSP plant. . . . . . . . . . . . . . . . . . 21 1-3 T-s diagram for the micro-CSP plant. . . . . . . . . . . . . . . . . . . 21 2-1 Contour plot of expander isentropic efficiency based on to Equation 2.10. 28 2-2 Finned-tube condenser geometry. . . . . . . . . . . . . . . . . . . . . 30 2-3 Modeling schematic of condenser tube bank. . . . . . . . . . . . . . . 31 2-4 Goodness of fit for condenser heat transfer correlation (Equation 2.13). 33 2-5 Goodness of fit for fan power consumption correlation (Equation 2.14). 33 2-6 Goodness of fit for WF pump isentropic efficiency correlation (Equation 2.16). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 2-7 Goodness of fit for WF motor efficiency correlation (Equation 2.17). . 36 2-8 Goodness of fit for HTF pump power consumption correlation (Equation 2.18). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 2-9 Collector modeling schematics. . . . . . . . . . . . . . . . . . . . . . . 38 2-10 Goodness of fit for collector heat loss correlation (Equation 2.19). . . 40 2-11 Goodness of fit for optimum evaporation pressure correlation (Equation 2.22). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 2-12 Optimum evaporation pressure correlation (Equation 2.22). . . . . . . 45 2-13 Simulation results for updated condenser heat transfer correlation (Equation 2.23). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 2-14 Goodness of fit for updated condenser heat transfer correlation (Equation 2.23). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 48 2-15 Results for system optimization using updated condenser correlation for single working condition. . . . . . . . . . . . . . . . . . . . . . . . 49 3-1 Flow diagram for the micro-CSP plant. . . . . . . . . . . . . . . . . . 52 3-2 Modelica interface for the dynamic system model. . . . . . . . . . . . 52 3-3 Discretized heat exchanger model showing cell vs node parameters. . 53 3-4 Discretized condenser model showing cell vs node parameters. . . . . 56 3-5 Modeling schematic for liquid receiver. . . . . . . . . . . . . . . . . . 57 3-6 Modeling schematic for solar collectors. . . . . . . . . . . . . . . . . . 58 3-7 Modeling schematic for storage tank. . . . . . . . . . . . . . . . . . . 58 3-8 Modelica interface for the control units for the Pev,opt and Tev,const strategies. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 3-9 Environmental conditions for control strategy comparison. . . . . . . 63 3-10 Controller results for Pev,opt strategy on Day 2. . . . . . . . . . . . . . 64 3-11 Effect of thermal storage for Pev,opt strategy on Day 2. . . . . . . . . . 65 3-12 Power produced or consumed by each component for Pev,opt strategy on Day 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 3-13 Power consumed by condenser fans for Pev,opt strategy on Day 2. . . . 68 3-14 Control strategy comparison results. . . . . . . . . . . . . . . . . . . 70 B-1 Modeling schematic of condenser tube bank. . . . . . . . . . . . . . . 78 B-2 Average condenser temperatures for tube-tube heat transfer analysis. 81 B-3 % error in condenser heat flow vs number of cells per tube. . . . . . . 82 10 List of Tables 1.1 Pilot system components and major modeling parameters. . . . . . . 23 2.1 Expander correlation coefficients for Equation 2.10. . . . . . . . . . . 28 2.2 WF motor correlation coefficients for Equation 2.17. . . . . . . . . . . 36 2.3 HTF pump correlation coefficients for Equation 2.18. . . . . . . . . . 37 2.4 Manufacturer parameters for solar collector optical efficiency. . . . . . 39 2.5 Collector heat loss correlation coefficients for Equation 2.19. . . . . . 40 2.6 Optimum evaporation pressure correlation coefficients for Equation 2.22. 43 2.7 Condenser heat transfer correlation coefficients for Equation 2.23. . . 11 48 12 Nomenclature A Area, m2 Ap Aperture width, m cp Specific heat capacity, J(kgK)-1 CS Control signal CSP Concentrating solar power D Diameter, m EES Engineering Equation Solver f Friction factor G Mass flux, kgm-2 s-1 h Enthalpy, Jkg-1 HL Heat loss, Wm-1 HT F Heat transfer fluid HV AC Heating, ventilation, and air conditioning IAM Incidence angle modifier Ib Beam radiation, Wm-2 Init Initialization k Thermal conductivity, W(mK)-1 L Length, m m Mass, kg ṁ Mass flow rate, kgs-1 N Quantity Nrot Rotational speed, s-1 ORC Organic Rankine cycle p Pressure, Pa P ID Proportional-Integral-Derivative (control) PV Process Variable q Heat flux, Wm-1 Q̇ Heat flow, W 13 Re Reynold’s number rp Pressure ratio rv Volume ratio U Internal energy, J, or Heat transfer coefficient, W(m2 K)-1 vw Wind velocity, ms-1 V Volume, m3 V̇s Ideal volume flow rate, m3 s-1 SH Superheat, ◦ C SP Set point t Time, s or thickness, m T Temperature, ◦ C W Width, m Ẇ Power, W WF Working fluid ∆x Length of cell, m Greek symbols η Efficiency γ Specific heat ratio ν Specific volume, m3 kg-1 φ Expander filling factor ρ Density, kgm-3 θ Incidence angle, ◦ Subscripts and superscripts abs Absorber amb Ambient c Center col Collector cond Condenser or conduction const Constant conv Convection 14 crit Critical cross Cross-sectional el Electrical em Electromechanical env Environment ev Evaporator ex Exhaust exp Expander f Fluid or finside gl Glazing h Hydraulic i Inner ins Insulation int Intermediate l Liquid lm Log mean m Mechanical nom Nominal N Number of cells o Outer opt Optimum p Pump r Right rad Radiation s Swept or isentropic or surface sf Secondary fluid sol Solar su Supply t Tubeside w Wall 15 16 Chapter 1 Introduction 1.1 Background and Literature Review Organic Rankine cycle (ORC) systems are gaining ground as a means of effectively providing sustainable energy. Micro-CSP (concentrating solar power) plants based on ORCs present a cost-effective solution to the challenge of supplying heating, cooling, and electricity for rural health and education centers outside the range of a centralized grid [13]. Although ORCs tend to have lower efficiencies than traditional steam Rankine cycles, the thermodynamic properties of organic fluids lead to several distinct advantages in the low to medium power range. For example, the slope of the vapor saturation curve and the super-atmospheric saturation pressures of some organic fluids preclude the need for superheating and the removal of non-condensable gases, respectively, used in steam Rankine cycles, reducing the complexity, cost, and maintenance requirements [12]. Coupling micro-CSP plants with thermal storage allows for an increase in daily operating time and also for a damping of the rapid variations that may be seen in the solar input (e.g., due to clouds). However, input temperatures during normal operation may still vary as much as ±20◦ C throughout the year. Furthermore, using readily-available components, like re-purposed HVAC (heating, ventilation, and air conditioning) scroll compressors for expansion, while cost-effective, results in the additional design objective of preventing over- or under-expansion of the working fluid 17 in the fixed-volume ratio scroll machines. The effective design of such a system depends on modeling and identifying a control strategy with the ability to adjust to transient operating conditions. Several authors have simulated solar thermal ORC systems in steady-state. Orosz [13] created a physical and economic model in EES (Engineering Equation Solver) that predicted solar ORC performance over a year for the specified location and system components. This model condensed the hourly irradiance variation over the year into a single average value reduced by expected cloud cover to predict net power output and daily and annual energy production. McMahan [12] presented a validated steady-state ORC model and an optimization methodology for solar ORCs based on finite-time analysis but no simulation of the entire solar ORC system. Other authors have focused on ORC or thermal storage dynamics in isolation. Wei et al. [22] developed a dynamic ORC model and examined the efficiency of moving boundary versus discretized heat exchanger techniques when compared to experimental data. Casella et al. [3] developed a dynamic ORC model for normal operation by building on the existing ThermoPower library in the Modelica modeling language. This model was validated in steady-state and transient conditions based on a grid-connected turbogenerator with natural gas or diesel generator exhaust as the heat source. It was also used to simulate feedback control to match desired turbine inlet temperature by varying pump speed. McMahan [12] examined the accuracy and computational efficiency of several techniques for simulating packed-bed storage dynamics in charging/discharging and idle conditions. In addition, Twomey et al. [21] simulated grid-connected solar ORC system dynamics with the solar loop modeled as a single lumped component and solar irradiance represented by a daily half sine curve whose amplitude was adjusted for monthly averages to estimate daily and annual power generation. In this work, a model for the dynamics of the solar ORC system is developed to evaluate the impact of highly transient heat sources and sinks, thermal storage, and the variable loads associated with distributed generation. The next section describes a micro-CSP plant that has been developed to provide combined heat and power to 18 underserved rural communities. This system is used as the basis for the detailed steady-state component models discussed in Chapter 2, which are implemented in EES and validated to data where available. In turn, the steady-state models provide the basis for the dynamic models described in Chapter 3, which are developed in the Modelica modeling language. The dynamic model in its current form is used to make qualitative assessments of several control decisions based on realistic solar irradiance input representing four reference days. Future analysis will survey a wider range of environmental conditions to make quantitative determinations on the efficacy of each control decision. The simulations include an approximation for startup and shutdown, which avoids the numerical issues associated with the discontinuities in the working fluid density derivative present during such rapid phase changes. Chapter 4 summarizes the conclusions of the analysis and future work. A summary of this analysis has been submitted as a conference paper for the ASME Turbo Expo 2014 and is currently under review. 1.2 Pilot System Description: Basis for Models Over the past several years, researchers at MIT and University of Liège have collaborated with the non-governmental organization STG International to design a microCSP plant suitable for rural power generation. Several field prototypes have been installed. In addition, a pilot system, pictured in Fig. 1-1, resides at Eckerd College in St. Petersburg, Florida, and this system is used as the basis for the models. Figure 1-2 presents a flow diagram identifying the relationships between the major components. The design of this system hinges on the use of readily-available, commercial parts to reduce cost. The heat transfer fluid (HTF), propylene glycol, is warmed by the sun while traveling through 20 single-axis concentrating parabolic troughs. This fluid is the heat source for an organic Rankine cycle using HFC-245fa as the working fluid (WF) with counterflow brazed plate heat exchangers for evaporation and recuperation plus modified HVAC scroll compressors for expansion and a HVAC air-cooled condenser for heat rejection. The two hermetic scroll compressors 19 Figure 1-1: Pilot micro-CSP plant showing the air-cooled condenser, thermal storage, and solar collectors with the ORC in the lower left inset. are modified to run as expander-generators as proposed and successfully tested by Lemort et al [11]. The counter-crossflow condenser with corrugated plain fins, which utilizes three 61 cm diameter fans to provide cooling air, has a very complicated geometry and will be discussed in more detail in the modeling sections. Thermal storage is achieved with a glycol-filled, double-wall 0.79 m3 tank surrounded by 25 mm thick fiberglass wrap insulation. The target net output power is 3 kWe at an overall efficiency from solar power input to ORC power output of 3% and ORC thermal efficiency from evaporator heat input to net power output of 8%. Figure 1-3 is the T-s diagram for the design point of the system with a HTF supply temperature to the ORC of 150◦ C and ambient temperature of 25◦ C. Starting at the entrance to the pump, the WF is compressed to the high-side pressure (thermodynamic states 1-2 in the figure). The liquid is preheated in the recuperator (states 2-3) before entering the evaporator for further preheating (states 3-4), vaporization (states 20 6FUROO([SDQGHU*HQHUDWRUV 6RODU&ROOHFWRUV 7KHUPDO 6WRUDJH (YDSRUDWRU 5HFXSHUDWRU &RQGHQVHU :RUNLQJ)OXLG3XPS +HDW7UDQVIHU)OXLG3XPS Figure 1-2: Flow diagram for the micro-CSP plant with propylene glycol as heat transfer fluid (shown in red) and HFC-245fa as working fluid (shown in green). Condenser cooling air is shown in blue. The bold numbers represent the state points for the ORC T-s diagram in Fig. 1-3 below. 5ID +3+ $- $/ ,,5%%$%( 1 7>&@ + +3+/ 2 " 2 +3+/$ 6( +4 6( $/ 75%%$%( (% " ! +/$ +/ " 2 V>-NJ.@ Figure 1-3: T-s diagram for the micro-CSP plant with propylene glycol as heat transfer fluid (HTF) and HFC-245fa as working fluid (WF). Numbers 1 through 10 indicate each thermodynamic state of the ORC. 21 4-5), and superheating (states 5-6). The two stages of expansion through the scroll expander-generators are represented by states 6-7 and 7-8, respectively. The vapor then re-enters the recuperator for precooling (states 8-9) before further precooling (states 9-10) and condensation (states 10-1) in the condenser. The liquid receiver prevents subcooling in steady-state. The HTF temperature glide in the counterflow evaporator is plotted against the corresponding entropy of the working fluid, so the HTF inlet temperature aligns with the WF outlet entropy (state 6) and the HTF outlet temperature aligns with the WF inlet entropy (state 3). Similarly, the cooling air temperature glide in the counter-crossflow finned-tube condenser is plotted against the corresponding entropy of the working fluid, where the air inlet corresponds to the WF outlet (state 1) and the air outlet corresponds to the WF inlet (state 9). Figure 1-3 demonstrates that the drying behavior of an organic fluid – the negative slope of its vapor saturation curve – allows for an evaporator exhaust with very little superheat to remain in vapor state even with an isentropic expansion. Steam Rankine cycles, on the other hand, exhibit a wetting behavior, or positive slope in the vapor saturation curve. And because of turbine blade sensitivity to water droplets, traditional Rankine cycles require a substantial amount of superheat to ensure a highquality mixture at the expander exhaust. This additional heating is detrimental to the overall efficiency of the cycle given a fixed temperature heat source. Moreover, organic cycles can exhibit a substantial temperature difference between the expander exhaust and the saturation temperature at the condensing pressure (states 8-10), again due to their drying behavior. (Since the fluid in steam cycles is often two phase at the exit of the expander, there is zero temperature difference between these states.) This characteristic allows for heat recovery in ORCs, as accomplished with the recuperator in the pilot system (states 8-9), which increases their efficiency, or for the provision of hot water, heating, or absorption cooling. One observation apparent from the T-s diagram is that the condenser temperature pinch (between WF two-phase inlet, state 10, and air temperature at the same location) in the counter-crossflow finned-tube arrangement can become very small. An inherent assumption in the optimization analysis discussed in Section 2.6 and 22 Table 1.1: Pilot system components and major modeling parameters. Component Heat transfer fluid Collectors Storage Tank Description Propylene glycol Single-axis SopoNova parabolic trough, SOLEC HI/SORB II selective coating, air-filled annulus between absorber and glazing Double-wall plus insulation, 3696K71 Supplier Various Sopogy McMaster HTF Pump HTF Motor Working fluid Evaporator Gear, NG11V-PH 48VDC permanent magnet HFC-245fa Brazed Plate, BP415-050 Dayton Leeson Honeywell ITT Brazepak Recuperator Brazed Plate, BP410-030 ITT Brazepak Condenser Air-cooled finned-tube with hexagonal tube array, Multicon CAC-28-G Russell Liquid receiver Expandergenerators WF Pump WF Motor Length of pipe plus sight glass Hermetic scroll induction machine, ZR48 ZR125 Plunger, PowerLine Plus 2351B-P 48VDC permanent magnet Various Copeland Hypro Leeson Parameter Value As Lcol Wcol Do V tins kins Vs Ẇnom 104 m2 73 m 1.4 m 25 mm 0.79 m3 25 mm 3.9e-2 W(mK) 24e-6 m3 373 W As V Mw As V Mw At Af Vt V Vs Vs Vs Ẇnom 2.4 m2 2.5e-3 m3 11 kg 0.73 m2 8.4e-4 m3 3.7 kg 13 m2 360 m2 4.0e-2 m3 1.3e-3 m3 22e-6 m3 56e-6 m3 8.7e-6 m3 746 W -1 used in the dynamic model of Chapter 3 is a constant temperature defect of 10◦ C between WF exhaust temperature (state 1) and inlet air temperature. The analysis of Section 2.7 at a range of temperature defects and WF and air supply temperatures results in pinch points as small as 0.5◦ C but not much larger than 2.8◦ C. This pinch point in temperatures limits the performance of the heat exchanger. In addition, the range of acceptable temperature defects at state 1 will be limited by the condition that the pinch point remain positive (the temperature of the working fluid at state 10 must be higher than the incoming cooling air). Table 1.1 summarizes the key modeling parameters for each component to be used in the analysis of the next two chapters. 23 24 Chapter 2 Steady-state modeling The following sections summarize the detailed steady-state component models, which are developed in EES. The models have been matched to data where available, and they are used to provide initial values, heat transfer coefficients, and pressure drops for the dynamic simulation and for its validation when run to steady-state. In addition, the steady-state system model is used to perform an optimization analysis for input to a real-time control strategy discussed further in Chapter 3. The major modeling parameters for each component are summarized in Table 1.1. 2.1 Evaporator/Recuperator The brazed plate heat exchangers are modeled using the log mean temperature difference method, where the recuperator consists of single-phase liquid or vapor regions and the evaporator includes liquid, two-phase, and vapor zones. For the evaporator, this method involves solving the following system of equations for each zone: Q̇ =ṁW F · (hW F,ex − hW F,su ) (2.1) =(ṁ · cp )HT F · (THT F,su − THT F,ex ) (2.2) =U A · ∆Tlm (2.3) 25 ∆Tlm = ∆T2 − ∆T1 ln(∆T2 /∆T1 ) (2.4) ∆T1 = THT F,1 − TW F,1 = THT F,su − TW F,ex (2.5) ∆T2 = THT F,2 − TW F,2 = THT F,ex − TW F,su (2.6) where W F and HT F denote working fluid (HFC-245fa) and heat transfer fluid (propylene glycol) properties, respectively; su and ex refer to supply and exhaust conditions, respectively; Q̇ is the heat flow for that zone of the heat exchanger; ṁ is the mass flow rate; h is specific enthalpy of the working fluid; cp is specific heat capacity of the heat transfer fluid; T is temperature; and U A is the product of heat transfer coefficient and heat exchange surface area. For the recuperator, Equation 2.1 refers to the cold liquid side of the heat exchanger and Equation 2.2 becomes Q̇ = ṁW F · (hW F,su − hW F,ex ) (2.7) referring to the hot vapor side. Heat transfer coefficients and frictional pressure drops are derived using Thonon [20] for single-phase flow and Hsieh and Lin [7] for two-phase flow. Knowing the inlet conditions of both fluids, the geometry of the heat exchanger, and that the total heat transfer for all zones must also be equal between fluids, it is possible to solve for the outlet conditions. To speed the iteration process for the full system model, the pressure drops, while calculated, are not implemented. With pressure drops typically below 100 mbar for these heat exchangers, the impact of internal pressure variation on heat transfer should be negligible. For instance, a supply pressure of 30 bar and a 100 mbar pressure drop over the two-phase region of the evaporator correspond to a saturation temperature drop of only 0.2◦ C. 2.2 Expander-generators Neglecting ambient heat losses, the expanders are characterized by their filling factor and isentropic efficiency. Filling factor, φ, is defined as 26 φ= ṁ · νsu ṁ · νsu = Vs · Nrot V̇s (2.8) or the ratio of real to ideal mass flow rate, where νsu is the supply specific volume of the fluid; V̇s is the ideal volume flow rate; Vs is the swept volume of the machine; and Nrot is the rotational speed of the expander. Isentropic efficiency, ηs,exp , is the ratio of real to ideal power generated or ηs,exp = Ẇel Ẇel , = ṁ · (hsu − hex,s ) Ẇs (2.9) where Ẇel is the electrical power generated; Ẇs is the isentropic power; and hsu and hex,s are the supply and isentropic exhaust specific enthalpies, respectively. The electrical power is used here as the asynchronous generator is integrated with the expander in a hermetic shell. The internal irreversibilities accounted for in the isentropic efficiency are thus a combination of both the fluidic and electrical losses. The expander model is based on Lemort et al. [10] who created a detailed model accounting for multiple types of losses and validated to prototype test data. To extend this model to expanders of various swept volumes, the validated model was exercised over 800 different working conditions that bracket the operating envelope of the experimental system, and polynomial fits were generated to correlate φ and ηs,exp as a function of supply pressure, psu , and pressure ratio, rp . It is assumed that the filling factor and isentropic efficiency remain comparable for the same supply pressure and pressure ratio regardless of swept volume. The form of the fitting polynomials is n−1 X n−1 X aij · ln(rp )i · ln(psu )j + an0 · ln(rp )n + a0n · ln(psu )n = f (rp , psu ), (2.10) i=0 j=0 where the coefficients, aij , are provided in Table 2.1. It should be noted that the correlations were developed for grid-synchronized, constant-speed expanders, and these should be validated for variable load and asynchronous operation. The pressure ratio at which a fixed volume ratio expander will achieve ideal ex27 Table 2.1: Expander correlation coefficients for Equation 2.10 [10]. ηs,exp j i 0 1 0 1 2 3 4 5 6.34831061E+3 -4.62226605E+3 5.18926734E+3 -2.71931292E+3 486.736446E0 53.1888731E-3 -2.07325125E+3 1.18102574E+3 -1.40315596E+3 765.497652E0 -139.912567E0 2 272.015067E0 -111.050112E0 141.445478E0 -80.6286745E0 15.0486978E0 3 4 5 -17.9964322E0 4.54486911E0 -6.30866773E0 3.77077331E0 -718.767884E-3 602.747139E-3 -67.9837592E-3 105.088614E-3 -66.0896654E-3 12.86479100E-3 -8.20388944E-3 φ j i 0 1 2 0 1 2 4.798 -0.06549 -0.00494 -0.6231 0.006766 0.02523 2YHUH[SDQGHG ,GHDOH[SDQVLRQ 8QGHUH[SDQGHG 7\SLFDO([SDQGHU 2SHUDWLQJ5DQJH 7\SLFDO([SDQGHU 2SHUDWLQJ5DQJH Figure 2-1: Contour plot of expander isentropic efficiency based on Equation 2.10 with pressure ratio on the x-axis and supply pressure on the y-axis. Red regions correspond to high-efficiency operation, and blue regions correspond to low-efficiency operation. The black lines represent the region of highest efficiency as predicted by the isentropic relation between pressure and volume ratios. The gray boxes indicate the boundaries of the correlation. 28 pansion can be estimated using the isentropic relation rp = rvγ , where rv is the internal volume ratio and γ is the specific heat ratio. Figure 2-1 illustrates the correlation for isentropic efficiency as defined by Equation 2.10 in a contour plot. With a built-in volume ratio for the scroll expanders of 2.85 and specific heat ratios ranging from 1.1 to 2 for HFC-245fa during operation, the region of ideal expansion should fall between pressure ratios from 3 to 8. This range is indicated by the black lines on Fig. 2-1 and corresponds well with the red contour representing high-efficiency operation. The dashed boxes indicate the typical operating region for each expander as determined by the optimization study that will be discussed in Section 2.6. Expander 2 falls within the predicted range for ideal expansion while Expander 1 sometimes operates over-expanded. 2.3 Condenser Figure 2-2 illustrates the complex geometry of the finned-tube condenser. Refrigerant enters the condenser through the inlet headers with the entrance to each tube bank represented by black circles. It then winds it way back and forth (into and out of the page) and from top to bottom through the tube banks with its path indicated by dotted lines until reaching the outlet headers, where the exit of each tube bank is represented by black circles. Three 61 cm diameter fans force air flow across the tubes and through the corrugated plain fins (parallel to the plane of the page) from the bottom row to the top row of the tube banks. Fan operation can consist of 1, 2, or 3 fans on with the first 2 fans controlled by fixed-speed motors and only the third fan controlled by a variable-speed motor for economical reasons. Because of its complex geometry, the condenser is more difficult to split into zones as in the evaporator. Therefore, a discretized model is developed. The condenser is approximated as a set of 12 identical parallel tube banks (as indicated in Figure 2-2) each consisting of 12 tubes in 5 rows. Each tube is discretized into n elements for a total of 12 x n refrigerant elements per tube bank. As the banks are assumed identical, the set of equations for a single bank is sufficient to model the entire exchanger. 29 )DQV 'LUHFWLRQRI DLUIORZ 6HOHFWHGWXEHEDQN ,QOHWKHDGHUV 2XWOHWKHDGHUV Figure 2-2: End view of the condenser with the modeled tube bank outlined in blue. The upper black rectangles represent the inlet headers, within which the black circles indicate the refrigerant entrance to each tube bank. Similarly, the lower black rectangles represent the outlet headers with the black circles indicating the outlet of each tube bank. The path through each bank is indicated by dashed lines. The modeling schematic of the tube bank, Fig. 2-3, shows refrigerant entering as a vapor on the top right and traveling back and forth from Tube 1 to Tube 12 on the bottom left while condensing. Dashed black lines notionally represent the discretization of fluid cells in the tubes and the hexagonal prism of air and fins (not shown) surrounding the tube section. The discretization follows the progression of the refrigerant as illustrated for Tubes 1 and 2 by the bold black numbers. The air flowing over the tube bank is assumed to be well-mixed between tube rows as this is conservative and analysis showed a negligible difference in results assuming mixed or unmixed air (see Appendix B). Therefore, the entrance air temperature for each row is assumed to be the average of the outlet temperatures for the cells below. Tube-tube heat transfer due to conduction through the fins is neglected when, in fact, the distance between the tubes suggests they would transfer some heat with each other representing a potential refinement for future models (see Appendix B). 30 9DSRU 7ZR3KDVH /LTXLG $LU2XWOHW :),QOHW 5RZ :)2XWOHW $LU,QOHW Figure 2-3: Modeling schematic of condenser tube bank. Refrigerant enters as a vapor on the top right and travels back and forth from Tube 1 to Tube 12 on the bottom left while condensing. White tube faces indicate fluid entering at the tube face and traveling back (up and to the left in the diagram) and gray tube faces indicate fluid that has traveled from the back and exits at the tube face. Dashed black lines notionally represent the discretization of the fluid cells in the tubes and the hexagonal prism of air and fins (not shown) surrounding the tube section. Bold black numbers indicate the discretization scheme. Dashed orange lines represent the planes of each tube row. An energy balance is performed for each tube cell: U Ai · (TW F,i − Tair,su,i ) ṁW F (2.11) U Ai · (TW F,i − Tair,su,i ) (ṁi · cp )air (2.12) hW F,i+1 = hW F,i − Tair,ex,i = Tair,su,i + The air is treated as a constant specific heat fluid, while the refrigerant specific heat 31 is allowed to vary. In each cell, i, the inlet properties are set to the outlet properties of the preceding cell and used to calculate the outlet conditions, denoted by i + 1, for the current cell – an upwind scheme. The refrigerant-side heat transfer coefficient is determined using Gnielinski (Re < 100,000) or Dittus-Boelter (Re > 100,000) for single-phase flow [8] and Shah for twophase flow [19]. The air-side heat transfer coefficient and pressure drop are determined using Kim et al. [9]. The fin efficiency is calculated, neglecting the corrugation, using the Schmidt method [18] for approximating the hexagonal fins associated with the staggered tube arrangement as circular fins of equivalent height. The model also neglects the contribution of water in the air to heat transfer, another possible refinement. As in the evaporator/recuperator models, to speed the iteration process, while refrigerant pressure drop is calculated from Petukhov [8] for single-phase flow and Choi et al. [4] for two-phase flow, it is not implemented in the heat transfer model. For condensing temperatures below ∼15◦ C, for which pressure drop can exceed 300 mbar and therefore no longer has a negligible impact on heat transfer, a more complete model may be necessary. For instance, a supply pressure of 1 bar and a 300 mbar pressure drop over the two-phase region of the condenser correspond to a saturation temperature drop of 8.5◦ C. It is chosen to neglect this effect in the current analysis. For computational efficiency, the detailed heat transfer model is not integrated into the system model. Instead, the calculated heat transfer, Q̇cond , is correlated as a function of refrigerant supply temperature, TW F,su , air supply temperature, Tamb , and air mass flow rate, ṁair : Q̇cond =7.48094441 × 103 + 1.25976724 × 102 TW F,su − 1.51787477 × 102 Tamb + 6.34710507 × 103 ṁair . (2.13) The correlation is based on 10 working conditions. For the optimization study described further in Section 2.6, the temperature defect between ambient and condensing temperatures is maintained at 10◦ C and WF pump volume flow rate is maintained 32 Figure 2-4: Goodness of fit for condenser heat transfer correlation (Equation 2.13) with calculated heat flow from the detailed model on the x-axis and predicted heat flow from the correlation on the y-axis. The R2 value of the fit is 100.00%. Figure 2-5: Goodness of fit for fan power consumption correlation (Equation 2.14) with measured power on the x-axis and predicted power on the y-axis. The R2 value of the fit is 86.74%. 33 at 7 LPM. In addition, the critical temperature, Tcrit , of HFC-245fa is 154◦ C. With these constraints, the range of operation for the condenser is limited, so only a small number of working conditions are necessary for the correlation. Figure 2-11 is a plot of the calculated heat flow from the detailed model versus the predicted heat flow from the correlation. The R2 value of the fit is 100.00%. Since Modelica is much more robust than EES, the dynamic model closely matches the detailed condenser model with the exception of constant heat transfer coefficients and pressure drops, which are provided by the steady-state model. The condenser fan power consumption, Ẇel,f ans , is correlated to experimental data as a function of the volume flow rate of the air, V̇su : Ẇel,f ans = 5.70339029 × 102 + 5.03148392 × 102 log V̇su . (2.14) Figure 2-5 depicts the goodness of fit, which results in an R2 of 86.74%. The measurement procedure consisted of changing the fan operation and measuring the air velocity mid-radius above the fan grill with a Uni-T 5URG8 anemometer. Fan operation has three modes: 1) Fan 1 on/Fans 2 and 3 off, 2) Fans 1 and 2 on/Fan 3 off, or 3) Fans 1 and 2 on/Fan 3 on with variable speed [14]. 2.4 Pumps/Motors A pump’s global isentropic efficiency, ηp , including both electromechanical and internal losses, is defined by ηp = ηem,p · ηs,p = Ẇs ṁ · (hex,s − hsu ) ṁ · νsu · (pex − psu ) = ≈ Ẇel Ẇel Ẇel (2.15) assuming the liquid behaves as an incompressible fluid, where the subscript p indicates pump; the variable p represents pressure; and em is electromechanical. Since the pump and motor are not integrated into the same shell like the expanders, the electromechanical and internal losses can be separated. 34 Figure 2-6: Goodness of fit for WF pump isentropic efficiency correlation (Equation 2.16) with calculated efficiency from manufacturer data on the x-axis and predicted efficiency from the correlation on the y-axis. The R2 value of the fit is 98.75%. For the working fluid pump, the isentropic efficiency is defined by ηs,p,W F pex = 0.234247552 + 0.220591434 pnom ! pex − 0.0179094791 pnom !2 , (2.16) which is derived from manufacturer data that is correlated as a function of normalized outlet pressure. The nominal pressure, pnom , is set to 30 bar. Figure 2-6 depicts the goodness of fit, which results in an R2 of 98.75%. Volumetric efficiency is neglected in the manufacturer data. The motor efficiency is also derived from manufacturer data and correlated with fraction of rated mechanical power: ηem,p,W F = 6 X bk · k=0 Ẇm Ẇnom !k , (2.17) where the rated mechanical power, Ẇnom , for the WF pump is 746 W and Table 2.2 provides the coefficients, bk . Figure 2-7 depicts the goodness of fit, which results in an R2 of 99.83%. 35 Table 2.2: WF motor correlation coefficients for Equation 2.17. Coefficient b0 b1 b2 b3 b4 b5 b6 Value 2.170250E-03 4.468185E0 -9.374727E0 9.750974E0 -5.351966E0 1.474668E0 -1.608160E-01 Figure 2-7: Goodness of fit for WF motor efficiency correlation (Equation 2.17) with calculated efficiency from manufacturer data on the x-axis and predicted efficiency from the correlation on the y-axis. The R2 value of the fit is 99.83%. For the HTF pump, the measured electrical power output, Ẇel,p,HT F , from experimental data is correlated to the measured volume flow rate, V̇su , and supply temperature, Tsu : 2 2 Ẇel,p,HT F = c0 + c1 · Tsu + c2 · Tsu + c3 · V̇su + c4 · V̇su + c5 · Tsu · V̇su , (2.18) where Table 2.3 provides the coefficients, ck . Figure 2-8 depicts the goodness of fit, 36 Table 2.3: HTF pump correlation coefficients for Equation 2.18. Coefficient c0 c1 c2 c3 c4 c5 Value 1.86411220E+01 -8.91044338E-01 9.53566173E-03 5.03864468E+00 3.87583484E-01 -8.50046125E-02 Figure 2-8: Goodness of fit for HTF pump power consumption correlation (Equation 2.18) with measured power on the x-axis and predicted power on the y-axis. The R2 value of the fit is 99.55%. which results in an R2 of 99.55%. Although, the HTF pump is also separated from the motor, there was not enough information in the available data to distinguish the electromechanical losses from the internal irreversibilities. This approximation would result in an error in outlet enthalpy, but since enthalpy change in a pump is small compared to the other components, the effect should be negligible. The current drawn by the motor was measured using an Extech True RMS Ammeter 430 and the flow rate was measured using a Blancett flow meter and B2800 flow monitor [14]. There is currently no cavitation model for the pumps, a possible future improvement. 37 Qconv,HTF Glass envelope Evacuated Absorber annulus Qcond,abs Qcond,gl YĐŽŶǀ͕Ĩ 2.30 Qrad,ann Y,> Tsky ,d& Qrad,sky HTF Tgl,o Tabs,o Tabs,i Qsol,abs d,d& Tamb THTF Qconv,amb YƐŽů͕ĂďƐ Tgl,i Parabolic reflector -1.7 -1.2 -0.7 -0.2 0.3 -0.20 0.8 1.3 1.8 Ap (a) Forristall model [2] (b) Burkholder approximation Figure 2-9: Collector modeling schematics showing an end view of the absorber tube, glazing, and reflector, and the heat transfer processes associated with the full 1D energy balance of Forristall and the simplified energy balance of Burkholder. 2.5 Solar Collectors The collectors are modeled using the one-dimensional energy balance around the discretized heat collection element of Forristall [6] shown in Fig. 2-9a. To determine the heat transfer to the fluid in the absorber considering the solar irradiance and thermal losses, this model includes the following heat transfer processes: convection in the heat transfer fluid, conduction in the absorber tube, convection and radiation in the air-filled annulus, conduction in the glazing, convection and radiation to the ambient, and solar radiation to the absorber tube and glazing. Knowing the collector geometry, solar irradiance, ambient temperature, wind speed, incidence angle, mass flow rate, and inlet temperature to the collector, the outlet temperature can be calculated. Forristall’s model, available in EES, is parameterized for the collector geometry of the pilot system and validated using the manufacturer specifications adjusted according to experimental data from commissioning the solar field. Global irradiance was measured using a Daystar solar meter with 100 mm collimator, and the HTF flow rate was measured using a Blancett H701A flowmeter [14]. With an accuracy of 3%, this type of solar meter is typically better suited to benchmarking photovoltaic arrays, so future data collection efforts will focus on acquiring a higher accuracy measurement of 38 Table 2.4: Manufacturer parameters for solar collector optical efficiency. Parameter Receiver Absorptivity Mirror Reflectivity Receiver Emittance Glass Transmissivity Value 0.95 0.89 0.25 at 315◦ C 0.207 at 270◦ C 0.91 direct normal irradiance through the use of a pyrheliometer. Four measurements from the experimental data taken over five minutes were averaged to represent a steadystate reading. The irradiance, HTF mass flow rate, and collector supply temperature were input to the Forristall model and a multiplier on optical efficiency adjusted until the collector exhaust temperature matched that of the steady-state reading. Table 2.4 provides the manufacturer parameters for determining optical efficiency. A multiplier of 0.55 on optical efficiency, representing unaccounted losses, is necessary to match the experimental data. To reduce simulation time in the dynamic model, the validated model is exercised over 400 working conditions to determine the fitting parameters for the correlation developed by Burkholder [2] for heat loss, HL, in Wm-1 as a function of HTF and ambient temperatures; solar irradiance, Ib ; incidence angle modifier, IAM ; incidence angle, θ; and wind speed, vw : 2 3 HL =d0 + d1 · (THT F − Tamb ) + d2 · THT F + d3 · THT F + √ 2 d4 · Ib IAM cosθ · THT vw · (d5 + d6 · (THT F − Tamb )). F + (2.19) The determined coefficients, dk , are shown in Table 2.5. Figure 2-8 depicts the goodness of fit, which results in an R2 of 99.85%. The importance of several of the terms may be explained by the heat transfer phenomena. Many of the heat transfer equations depend on THT F − Tamb . Ib IAM cosθ, or irradiance multiplied by incidence angle modifier, appears in the equations for solar energy incident on the absorber 39 Table 2.5: Collector heat loss correlation coefficients for Equation 2.19. Coefficient d0 d1 d2 d3 d4 d5 d6 Value 5.34E-01 2.18E-01 -2.64E-04 4.93E-06 7.38E-08 1.06E-02 1.15E-02 Figure 2-10: Goodness of fit for collector heat loss correlation (Equation 2.19) with calculated heat loss from the Forristall model on the x-axis and predicted heat loss from the correlation on the y-axis. The R2 value of the fit is 99.85%. tube and glazing. Wind velocity, or vw , appears in the Reynold’s number relation for convection from the glazing to the air. The heat loss term, HL, participates in a simplified energy balance between the fluid and the solar input as shown in Fig. 2-9b and described by the following equations: Tex,i = Tsu,i + qconv,f · ∆x (ṁ · cp )HT F qconv,f = qsol,abs − HL 40 (2.20) (2.21) where i represents the ith element of the absorber tube; qconv,f represents the heat transferred to the fluid via convection; qsol,abs represents the solar irradiance; and ∆x is the length of the element. 2.6 Application: Determining Optimum Set Points for Varying Working Conditions The effective design of a micro-CSP plant involving fixed volume ratio expanders depends on the control system’s ability to maintain the pressure ratio across the expanders necessary to avoid both over- and under-expansion of the working fluid under variable ambient conditions. Considering this design objective, one application for the steady-state system model is determining an optimization function for use in a real-time control strategy to be analyzed with the dynamic model. With several additional constraints discussed below, the goal of this optimization analysis is to identify a correlation for the evaporation pressure which maximizes global efficiency (from solar input to ORC power output) for various working conditions. Two control strategies are analyzed with the dynamic model as discussed further in Chapter 3. The first is based on optimizing evaporation pressure within physical constraints, such as expander speeds, while the second aims to maintain a constant evaporation temperature. These strategies are subsequently referred to as Pev,opt and Tev,const , respectively. In addition, in both schemes, the superheat, SH, and condenser temperature defect between WF exhaust and air supply, ∆Tcond , are controlled to a constant 5◦ C and 10◦ C, respectively; and the WF pump speed is currently held constant although further optimization may be possible in a later iteration. Since each expander’s isentropic efficiency is modeled using a polynomial as a function of supply pressure, psu , and pressure ratio, rp , it is possible to find an analytical solution for the optimum intermediate pressure, Pint , by maximizing the equation for the combined efficiency given the inlet and outlet pressures. The derivation for this solution when the efficiency polynomial is a function of supply density, ρsu , and rp 41 was provided in [15], and the derivation for the polynomials of Equation 2.10, which depend on psu and rp , is provided in Appendix C. With the HTF pump, condenser fan, and Expander 2 speeds floating to control SH, ∆Tcond , and Pint , respectively, and the WF pump speed held constant, the steady-state system model, executed under various environmental conditions, can be used to determine an optimum evaporation pressure to be controlled via Expander 1 speed. The input temperatures from the solar loop to the evaporator and from the ambient are varied across the following ranges: 135◦ C ≤ THT F,su ≤ 185◦ C 0◦ C ≤ Tamb ≤ 40◦ C For each of 30 working conditions, the evaporation pressure is varied from its maximum for that condition and below until an optimum global efficiency within physical constraints is identified. The maximum pressure is chosen to prevent the fluid from entering the supercritical regime where the properties are not well-known. The resulting optimum pressure function is defined by the HTF supply temperature and ambient temperature: 2 pev =e0 + e1 · THT F,su + e2 · THT F,su + 2 e3 · Tamb + e4 · Tamb + e5 · THT F,su · Tamb , (2.22) where the units of pev are Pa and Table 2.6 lists the correlation coefficients, ek . Figure 2-11 is a plot of the calculated optimum pressure from the 30 conditions versus the predicted optimum pressure from the correlation. The R2 value of the fit is 99.26%. When surveying potential working conditions, HTF supply temperatures below ∼140◦ C were found to require expander speeds above the hardware limit (∼5500 rpm), so these conditions were not included in the 30 points used for the final correlation and the relevant range for the HTF supply temperature is reduced to 140◦ C ≤ THT F,su ≤ 185◦ C. 42 Table 2.6: Optimum evaporation pressure correlation coefficients for Equation 2.22. Coefficient e0 e1 e2 e3 e4 e5 Value -1.43107571E+07 1.87886417E+05 -5.29553162E+02 -6.07260310E+04 4.43861382E+01 4.60543535E+02 Figure 2-11: Goodness of fit for optimum evaporation pressure correlation (Equation 2.22) with calculated optimum pressure from the steady-state model on the x-axis and predicted optimum pressure from the correlation on the y-axis. The R2 value of the fit is 99.26%. The evaporator operating pressure that optimizes the global efficiency of the cycle is primarily determined by a balance of the power output of the expanders with increased inlet pressure versus the increase in the required power to drive the HTF pump. The first phenomenon is straightforward, while the second may require more explanation. The increase in the HTF pump power is a consequence of the fixed (and limited) heat transfer area in the evaporator. An increase in the operating saturation pressure of the working fluid in the evaporator increases the average temperature throughout the WF stream. In addition, since the WF mass flow is nearly constant (with the fixed volume flow rate of the WF pump), the heat transfer required to bring 43 the working fluid to a 5◦ C superheated state increases with increasing saturation pressure. If the heat exchange coefficients internal to the heat exchanger do not substantially change, the only way to achieve this increased heat transfer is to increase the average temperature difference beween the WF and HTF streams. Since the inlet temperature of the HTF stream is fixed by the discharge temperature from the solar array, the only way to increase the average temperature between the streams is to increase the HTF mass flow rate. In this way, the temperature drop of the HTF stream is reduced as it passes through the heat exchanger so that the average temperature difference betweem the two flows is increased. As the WF saturation temperature approaches the inlet temperature of the heat transfer fluid, the mass flow needed to achieve the required average temperature defect will become larger and larger to the point where infinite mass flow would be needed to fulfill the thermal requirements of the evaporator. It becomes apparent that as this condition is approached, the power required by the HTF pump will also become infinite. Therefore, an optimum evaporation pressure must exist that yields a high power output from the expanders without requiring a significant power input to the HTF pump. Figure 2-12 is a plot of the optimum evaporation pressure correlation (Equation 2.22) versus the HTF temperature entering the evaporator with lines of constant ambient temperature. Considering the preceding discussion of the trade-off between maximizing expander power and minimizing HTF pump demand, the trends exhibited in the figure can be explained as follows: 1. The overall trend in the optimum evaporation pressure is that it increases with increasing HTF supply temperature for all ambient temperatures. This is not surprising as higher HTF supply temperatures allow higher WF operating temperatures and pressures that, in turn, allow for high power outputs from the expanders without requiring large HTF mass flows. 2. Another trend evident in the figure is that a higher ambient temperature results in a higher optimum evaporation pressure. With a constant condenser temperature defect, a higher ambient temperature corresponds to a higher condensing pressure, which leads to a lower overall pressure ratio. The highest possible 44 /1 1 ! /1 " 1 /1 1 / ! 0 Figure 2-12: Optimum evaporation pressure correlation (Equation 2.22). HTF supply temperature is on the x-axis and optimum evaporation pressure is on the y-axis with lines of constant ambient temperature. evaporation pressure without a significant increase in HTF pump power is desirable to increase the output power of the expanders. 3. For low ambient temperatures, the optimum supply pressure reaches a maximum and then decreases at higher HTF supply temperatures (see 0◦ C line in Fig. 2-12). In these circumstances, the efficiency of the expanders as a function of the pressure ratio as discussed in Section 2.2 becomes important. High HTF supply temperatures and low ambient temperatures correspond to the largest potential overall pressure ratios. Under these conditions, the balance between the increase in maximum available power associated with a higher supply pressure and the reduction in expander efficiency associated with a higher pressure ratio is struck at a lower supply pressure than the case with higher ambient temperature. In Fig. 2-1, the low ambient/high HTF supply temperature case falls on the rightmost boundary of the typical operation boxes. A higher supply pressure (and pressure ratio) would push the operating points diagonally upward and to the right in the diagram into a region of lower efficiency. 45 2.7 Potential Improvement to Cycle Optimization: Variable Condenser Temperature Defect Although the optimization described above is implemented in the dynamic control scheme of the next chapter, this section analyzes a potential improvement to be considered in future work. The previous optimization maintains a constant condenser temperature defect between the WF exhaust and air supply temperatures, ∆Tcond . Updating the correlation characterizing the condenser heat transfer (Equation 2.13) based on varying temperature defects allows for a potential improvement in efficiency or net power considering trade-offs with condenser fan power consumption and variable condensing temperatures. The updated correlation is based on varying the condenser model inputs across the following ranges: 25◦ C ≤ TW F,su ≤ 70◦ C 0◦ C ≤ Tamb ≤ 40◦ C 5◦ C ≤ ∆Tcond ≤ 20◦ C The WF mass flow rate is derived based on the constant 7 LPM volume flow rate imposed at the pump inlet in addition to the imposed saturated liquid exhaust condition for the condenser. Figure 2-13 shows the simulation results used to define the updated correlation, which is based on 122 working conditions. Since the previous fit was based on a single temperature defect, a linear correlation sufficed to characterize the behavior of the condenser. In fact, this can be imagined by following the symbols beneath the dotted line representing the 10◦ C temperature defect in Fig. 2-13; the trend of the underlying points suggests that the relationship between heat flow and air mass flow rate is linear. Now considering the allowance for variable temperature defects, it becomes clear that some nonlinear terms are required, specifically to represent the relationship between heat transfer and air mass flow rate at very large and very small temperature 46 '7FRQG & '7FRQG & '7FRQG & 4FRQG>:@ 7:)VX & 7:)VX & 7:)VX & 7:)VX & 7:)VX & 7:)VX & 7:)VX & 7:)VX & 7:)VX & 7:)VX & 7DPE & '7FRQG & '7FRQG & '7FRQG & '7FRQG & 7DPE & 7DPE & 7DPE & 7DPE & - PDLU>NJV@ Figure 2-13: Simulation results for updated condenser heat transfer correlation (Equation 2.23). Air mass flow rate is on the x-axis and condenser heat transfer is on the y-axis. Different colored lines indicate different ambient temperatures and different symbols indicate different WF supply temperatures. Each symbol moving from right to left across a single line represents condenser temperature defects from 5◦ C to 20◦ C in increments of 2.5◦ C (where the runs for all 7 defects converged). defects (see Fig. 2-13). The resulting correlation is a function of the same parameters as the previous: Q̇cond = f0 + f1 · TW F,su + f2 · Tamb + f3 · ṁair + f4 · ṁ2air + f5 · ṁ3air , (2.23) where Table 2.7 provides the coefficients, fk . Figure 2-14 is a plot of the calculated heat transfer from the 122 conditions versus the predicted heat transfer from the correlation. The R2 value of the fit is 99.67%. Inserting the updated correlation into the system model allows for a re-evaluation of the 30 working conditions used to define the optimum evaporation pressure and results in several observations: 1. An optimum can be found for net power (at ∆Tcond = 7.5◦ C), but not global efficiency, for the conditions surveyed. In other words, the maximum efficiency 47 Table 2.7: Condenser heat transfer correlation coefficients for Equation 2.23. Coefficient f0 f1 f2 f3 f4 f5 Value 2.49479821E+04 1.45084021E+02 -2.80330035E+02 2.46210782E+03 3.02151998E+02 1.18718021E+01 Figure 2-14: Goodness of fit for updated condenser heat transfer correlation (Equation 2.23) with calculated heat flow from the detailed model on the x-axis and predicted heat flow from the correlation on the y-axis. The R2 value of the fit is 99.67%. occurs at one of the boundaries of the simulation: at the 20◦ C condenser temperature defect. Figure 2-15 is a plot of the net power and global efficiency as a function of evaporation pressure with lines of constant condenser temperature defect for a single working condition: THT F,ev,su = 150◦ C, Tamb = 25◦ C. For display purposes, results for ∆Tcond = 5◦ C are not shown because the net power and global efficiency for this condition are well below optimum due to the large amount of air flow required to maintain such a small temperature defect (see Fig. 2-13). Figure 2-15 demonstrates that global efficiency continues to increase with increasing condenser temperature defect, primarily due to the 48 2.7 Net Power (kW) 2.6 3.5 Wdot,net,7.5 Wdot,net,15 ηglobal,7.5 ηglobal,15 Wdot,net,10 Wdot,net,17.5 ηglobal,10 ηglobal,17.5 Wdot,net,12.5 Wdot,net,20 ηglobal,12.5 ηglobal,20 3.4 3.3 2.5 3.2 2.4 3.1 2.3 3 2.2 2.9 2.1 2.8 2 2.7 1.9 20 21 22 23 24 Evaporation Pressure (bar) 25 Global Efficiency (%) 2.8 2.6 26 Figure 2-15: Results for system optimization using updated condenser correlation for single working condition: THT F,ev,su = 150◦ C, Tamb = 25◦ C. Evaporation pressure is on the xaxis. Net power with lines of constant condenser temperature defect, ∆Tcond , (indicated by the last value in the legend) corresponds to the left y-axis, and global efficiency with lines of constant ∆Tcond corresponds to the right y-axis. For display purposes, results for ∆Tcond = 5◦ C are not shown because the net power and global efficiency for this condition are well below optimum due to the large amount of air flow required to maintain such a small temperature defect. Net power exhibits an optimum at 22 bar, ∆Tcond = 7.5◦ C. Global efficiency fails to exhibit an optimum within the simulation range: the maximum value shown occurs at the highest temperature defect of ∆Tcond = 20◦ C. dramatic decrease in the condenser fan power consumption. However, the net power output also starts to decrease with increasing ∆Tcond because the larger temperature defect results in a smaller overall pressure ratio, which leads to reduced power output from the expanders. 2. An optimum net power solution is not found within the range of this study for ambient temperatures below 10◦ C. For these conditions, the simulations suggest that very large temperature defects (in excess of 20◦ C) provide the best power output and efficiency since these parameters continue to increase at condenser temperature defects up to and including the boundary of the survey of 20◦ C. This finding suggests that for such cold ambient temperatures, passive cooling 49 (with fans powered off) may be sufficient for ORC heat rejection. However, these working conditions are in the range where condenser pressure drop, which was neglected, will have a signficant impact on heat transfer so should be verified with a higher fidelity model. For the 20 working conditions where an optimum power solution is found, the optimum power solution averages 2.5 kW net power, 84 W more than the maximum efficiency condition at ∆Tcond = 20◦ C. Thermal efficiency averages 7.2% and global efficiency averages 2.8% – 0.36 pts and 0.11 pts lower than the maximum efficiency condition at 20◦ C, respectively. The optimum power solutions occur at condenser temperature defects of 7.5◦ C or 10◦ C indicating that the previous optimization based on ∆Tcond = 10◦ C was already very close to optimal. Indeed, on average, the net power for the simulated working conditions only increases by 7 W between the two optimizations with a negligible change in efficiencies. An updated optimum evaporation pressure curve with varying condenser temperature defects may result in a negligible change in net energy production. This analysis suggests that passive cooling may be sufficient for ambient temperatures below 10◦ C, which could result in a significant net power increase for these conditions compared to the previous analysis where fan power consumption averaged 1.1 kW. 50 Chapter 3 Dynamic modeling Modelica is an acausal object-oriented modeling language facilitating the deconstruction of a complicated system into its simpler component parts. In such a language, differential-algebraic equations can be written directly for each part without regard for order, and the models can be connected via ports, typically relating mass flow, pressure, and enthalpy between components. The models for the solar ORC plant either come directly from the open-source ThermoCycle library [17] or build upon the available components, and the fluid properties are determined using the open-source CoolProp library [1]. The Tabular Taylor Series Expansion method is used to improve computational efficiency in calculating fluid properties. The figure depicting the flow diagram of the plant from Chapter 1 is reproduced here as Fig. 3-1 for comparison with Fig. 3-2, which illustrates the plant schematic in the Dymola Modelica interface. This chapter will describe the underlying thermodynamic equations for each component in Fig. 3-2. Table 1.1 summarizes the main modeling parameters for each component. The model aims to capture the important dynamics of the system. With much shorter time constants compared to the other components, the dynamic response of the expanders and pumps is neglected. Therefore, these dynamic component models are equivalent to their steady-state representation. 51 6FUROO([SDQGHU*HQHUDWRUV 6RODU&ROOHFWRUV 7KHUPDO 6WRUDJH (YDSRUDWRU 5HFXSHUDWRU &RQGHQVHU :RUNLQJ)OXLG3XPS +HDW7UDQVIHU)OXLG3XPS Figure 3-1: Flow diagram for the micro-CSP plant with propylene glycol as heat transfer fluid (shown in red) and HFC-245fa as working fluid (shown in green). Condenser cooling air is shown in blue. The bold numbers represent the state points for the ORC T-s diagram in Fig. 1-3. 2SHUDWLQJ0RGHO ,GOLQJ0RGHO &RQWURO 8QLW 3UHVVXUH'URS 6WRUDJH 7DQN 6HQVRU ,QVRODWLRQ 3UHVVXUH 'URS 6WRUDJH 7DQN &RQWURO 8QLW 9BZLQG ([SDQGHU 7KHWD 7BDPE 5HFXSHUDWRU ([SDQGHU &ROOHFWRUV & (YDSRUDWRU &ROOHFWRUV V 3UHVVXUH 'URS 6LQN &RQGHQVHU :)3XPS +7) 3XPS /LTXLG 5HFHLYHU +7) 3XPS 6SHHG 6LJQDO )DQV Figure 3-2: Modelica interface for the dynamic system model with the HTF represented by red lines, refrigerant by green lines, and cooling air by blue lines. Sensed parameters and control signals are represented by dotted lines. 52 KI 0I KI1 0I1 KI 0I 4I KI 4I KI 7Z 7Z :DOO KI1 0I1 4I1 KI1 4I1 KI1 7Z1 7Z1 :RUNLQJ )OXLGLQ 6HFRQGDU\ )OXLG LQ 4VI KVI KVI 0VI 4VI KVI 4VI1 KVI1 KVI 0VI 4VI1 KVI1 KVI1 0VI1 KVI1 0VI1 Figure 3-3: Discretized heat exchanger model showing cell vs node parameters. The node parameters are indicated with a *. Here, f represents the working fluid, HFC-245fa, or the cold side of the recuperator and sf represents the secondary fluid, propylene glycol, or the hot side of the recuperator. Figure is based on [15]. 3.1 Evaporator/Recuperator The brazed plate evaporator and recuperator are modeled as discretized finite-volume counterflow heat exchangers as developed by Quoilin [15]. Figure 3-3 demonstrates that the heat exchangers are divided into three components: working fluid, secondary fluid, and metal wall as if the multiple plates in a brazed plate exchanger were one long plate. The fluid components are based on dynamic mass and energy balances, while the momentum balance is considered static. Choosing pressure and enthalpy as state variables and using the chain rule, the working fluid mass balance for a single element can be expressed as dmi δρi δρi dhi δρi dp = · Vi = ( · + · ) · Vi = ṁ∗i−1 − ṁ∗i , dt δt δhi dt δp dt (3.1) where t is time; m is mass; ṁ is mass flow rate; ρ is density; V is volume; h is specific enthalpy; p is pressure; and a ∗ indicates a node, rather than cell, property. Since Ui = Hi − pVi , or energy is enthalpy reduced by the product of pressure and 53 volume, and volume is constant, the working fluid energy balance becomes dhi δρi dp dUi = ρi · Vi · + · Vi · hi − · Vi = ṁ∗i−1 · h∗i−1 − ṁ∗i · h∗i + Q̇i + Ẇi , (3.2) dt dt δt dt where U is energy; Q̇ is heat flow; and Ẇ is work generated. With no internal work and the replacement of δρ δt · Vi with the right-hand side of Eq. 3.1, the former equation becomes dhi dp dUi = ρi · Vi · + (ṁ∗i−1 − ṁ∗i ) · hi − · Vi = ṁ∗i−1 · h∗i−1 − ṁ∗i · h∗i + Q̇i dt dt dt (3.3) which reduces to ρi · Vi · dhi dp = ṁ∗i−1 · (h∗i−1 − hi ) − ṁ∗i · (h∗i − hi ) + Q̇i + · Vi . dt dt (3.4) The recuperator has a hot side and cold side of working fluid. The evaporator, on the other hand, consists of working fluid and secondary fluid sides. For the incompressible secondary fluid side of the evaporator where pressure change is negligible, Eq. 3.4 reduces to ρi · V i · dhi = ṁ∗i−1 · (h∗i−1 − hi ) − ṁ∗i · (h∗i − hi ) + Q̇i . dt (3.5) The metal wall is assumed to have a constant specific heat capacity, cp , and to have a negligible temperature gradient leading to the following energy balance: Mw,i · cp,w · dTw,i = Q̇HT F,i − Q̇W F,i , dt (3.6) where w indicates a wall property; T is temperature; Q̇W F is the heat flow from the wall to the working fluid, HFC-245fa; and Q̇HT F is the heat flow from the heat transfer fluid, propylene glycol, (or in the case of the recuperator, the hot-side of working fluid) to the wall, respectively. 54 Constant heat transfer coefficients for each phase are input parameters and are determined using the detailed steady-state model. At the system level and during normal operation with small changes in mass flow rate, variations in heat transfer coefficient can be assumed negligible as shown in Appendix B. 3.2 Condenser The geometry and modeling strategy for the detailed steady-state condenser model, which is the basis for the dynamic model, was discussed at length in Section 2.3. The smallest constituents of the condenser model are the refrigerant, metal wall, and air cells. The refrigerant and metal wall cell dynamic equations are identical to those of the evaporator and recuperator, but the overall discretization scheme is slightly different due to the cross-flow of the air as shown in Fig. 3-4. The air cells are assumed to have negligible dynamics resulting in the following energy balance: (ṁ · cp )air · (Tair,i−1 − Tair,i ) = Q̇air,i . (3.7) Also required are pressure drop components for the air cells. A relation for pressure drop in off-design conditions is determined by assuming a constant friction factor: ∆p 2 · f · L · A2cross = = constant → ν · ṁ2 Dh 2 ṁ ν ∆p =∆pnom · · , νnom ṁnom (3.8) where f is friction factor; L is flow length; Dh is hydraulic diameter; G is mass flux; ν is specific volume; and Across is flow cross-sectional area. The nominal conditions are defined using the steady-state model results for a representative working condition. The next building block in the dynamic model is a row of tubes. Each row consists of n tubes per row x m cells per tube such that there are n x m refrigerant, metal wall, and air cells plus corresponding pressure drop components for the air cells. 55 KI 0I KI 0I 7VIH[ 4I KI 7VIH[ 4I KI 7Z 4VI KVI 0VI 7VIVX 7Z 4VI KVI 0VI 7VIVX KI1 0I1 :DOO KI1 0I1 7VIH[1 4I1 KI1 7VIH[1 4I1 KI1 7 Z1 4VI1 KVI1 0VI1 7VIVX1 7Z1 4VI1 KVI1 0VI1 7VIVX1 :RUNLQJ )OXLGLQ $LULQ Figure 3-4: Side view for a single tube of the discretized condenser model showing cell vs node parameters. The node parameters are indicated with a *. Here, f represents the working fluid, HFC-245fa, and sf represents the secondary fluid, air. Also required are splitter and joiner components to divide the incoming air from the preceding row among the refrigerant cells and rejoin the air cells after they have transferred heat with the metal/refrigerant. Finally, the five rows of two or three tubes are linked to complete the model. The condenser fan power demand is identical to that of the steady-state model. 3.3 Liquid Receiver The liquid receiver, illustrated in Fig. 3-5, is assumed to be in thermodynamic equilibrium at all times: the vapor and liquid are saturated at the given pressure. It is modeled by the same energy and mass conservation laws as in the discretized heat exchangers, but the exit condition is imposed to be a saturated liquid, hl , yielding the following energy balance: 56 KVX 0VX S9UK KH[ KO 0H[ Figure 3-5: Modeling schematic for liquid receiver. ρ·V · dh dp = ṁsu · (hsu − h) − ṁex · (hl − h) + · V. dt dt (3.9) Since the control unit regulates condenser pressure, the pressure in the liquid receiver is set by the saturation pressure in the condenser. Initialization sets either the initial pressure or liquid level. 3.4 Solar Collectors The solar collector model consists of two models connected by a thermal port: one representing the fluid dynamics in the discretized absorber tube and the other representing the collector thermal and optical efficiency based on the fluid temperature and environmental conditions. As discussed in Section 2.5, the efficiency, or heat loss, model is determined using a regression of the detailed Forristall model fit to the manufacturer specification and adjusted for experimental data, where the environmental inputs are solar irradiance, ambient temperature, wind velocity, and incidence angle. The discretized fluid cells in the absorber tube, depicted in Fig. 3-6, are modeled in the same way as the secondary fluid in the evaporator. For computational efficiency, rather than being calculated in each cell, the pressure losses are combined into lumped models after the heat exchanger and collectors as shown in Fig. 3-2. They are modeled in the same way as those of the air cells in the condenser model (Equation 3.8). 57 KI 0I 6HFRQGDU\ )OXLGLQ KI 0I 4FRQYI 4FRQYI KI KI KI1 KI1 0I1 0I1 4FRQYI1 4FRQYI1 KI1 KI1 Figure 3-6: Modeling schematic for solar collectors, where Q̇conv,f,i is determined using the heat loss model described in Section 2.5. U Figure 3-7: Modeling schematic for storage tank. 3.5 Storage Tank Pressure is imposed in the storage tank, which is modeled as a well-mixed (single element) control volume accounting for thermal energy losses due to conduction and convection through the top, sides, and bottom walls and insulation as shown in Fig. 37. With constant pressure rather than constant volume, the energy balance reduces to ρ · Vl · dh dVl = ṁsu · (hsu − h) − Q̇env + p · , dt dt (3.10) where Q̇env represents the heat loss to the environment and Vl is the liquid volume. 58 3.6 Control Unit Two control strategies are analyzed with the dynamic model. The first, referred to as Pev,opt , aims to track the optimum evaporation pressure as discussed in Section 2.6, and the second, referred to as Tev,const , aims to maintain a constant evaporation temperature. During normal operation, the control unit regulates system components to achieve the set points by utilizing four PI controllers illustrated in Fig. 3-8. PI control is selected because of its reduced sensitivity to measurement noise compared to PID control. The sensor dynamics are currently neglected. Each controller includes steadystate initialization functions to allow any oscillations in the system at the beginning of a simulation to settle before the controller starts tracking. The four controllers are based on the following control signals and process variables: the speed of the HTF pump is used to regulate superheat; the speed of Expander 1 is used to regulate either Pev,opt or Tev,const ; the speed of Expander 2 is used to regulate optimum intermediate expander pressure; and the mass flow rate of the condenser fans (as a simplified proxy for fan speed and number of operating fans) is used to control the condensing pressure. The leftmost column of blocks in Fig. 3-8a correspond to the following: ∆Tev is the calculated superheat based on a correlation for saturation temperature of HFC-245fa as a function of expander supply pressure; Pev is the optimum evaporation/expander supply pressure as derived in Section 2.6; and Pint is the optimum intermediate expander pressure as derived in Appendix C. For the Tev,const strategy shown in Fig. 3-8b, the Pev block is replaced with a constant set point and the process variable for saturation temperature is determined using the ∆Tev block. The controllers are tuned manually with the other controllers disabled by varying the proportional gain and integral time after a step is introduced. It is assumed that the closed-loop system is decoupled and behaves linearly and that the same tuning parameters can be used for the two strategies. A more robust control tuning and design is proposed in Section 4.2. 59 Startup/Shutdown. To simulate the daily power cycle, the control must have the ability to shut down the ORC at the end of the day or at any time when there is insufficient irradiance to warrant running the pumps. Similarly, it will need to simulate startup. Numerical simulation of startup or shutdown is difficult due to flow reversals, chattering around zero flow, and unphysical flow rate generation due to a discontinuity in the working fluid density derivative [16]. To approximate these conditions while avoiding numerical issues, duplicate models representing operating and idling modes (see Fig. 3-2) are simulated concurrently, transferring state variables to achieve an energy balance between the two when startup and shutdown, which are modeled as instantaneous events, are triggered. As experimental data indicates that the time scale of startup or shutdown is on the order of minutes, the impact of this approximation on a full-day simulation is expected to be negligible. However, ORC thermal inertia during these events is also neglected under the current scheme, so future analysis is planned to integrate this effect. The operating model is composed of the system model discussed previously. The idling model consists of a duplicate solar loop excluding the evaporator to represent operation when the ORC is shut down. During normal operation, the real irradiance signal is input to the operating model. A controller inputs a much-reduced irradiance signal to the idling model to maintain the storage tank temperature at the shutdown trigger temperature, where the only heat loss from the idling model is that lost to the environment through the storage tank. When the storage tank temperature in the operating model reaches the shutdown trigger, several actions follow: 1) the storage tank temperature in the idling model is re-initialized to that of the operating model (in case the controller has not settled); and 2) a controller inputs a higher irradiance signal to the operating model to maintain the storage tank at the startup trigger temperature. If the irradiance signal falls below 180 Wm−2 , the HTF pump, nominally set to 27 LPM in the idling model, is shut down. 60 6+B63 ,QLW ,QLW 3BVXBH[S &RQVW 7BKWIBVX '7B HY 3,'BSXPS ,QLW 3B HY ,QLW 7BVXBH[S 3BVXBH[S &RQVW ,QLW 3B LQW 3,'BH[S ,QLW &RQVW 3BH[BH[S 7BH[BFRQG 3,'BH[S '7BFRQGB63 ,QLW ,QLW &RQVW '7BFRQG 3,'BPGRWBDLU 7BDPEBVX (a) Pev,opt Strategy 6+B63 ,QLW ,QLW 3BVXBH[S &RQVW '7B HY 7BHY 3,'BSXPS ,QLW ,QLW 7BVXBH[S &RQVW ,QLW 3B LQW 3BVXBH[S 3,'BH[S ,QLW &RQVW 3BH[BH[S 7BH[BFRQG '7BFRQGB63 3,'BH[S ,QLW ,QLW &RQVW '7BFRQG 3,'BPGRWBDLU 7BDPEBVX (b) Tev,const Strategy Figure 3-8: Modelica interface for the control units for the Pev,opt and Tev,const strategies. The upper left arrows of the PID components represent set points; the lower left arrows represent process variables; and the right arrows represent control signals. The Init components allow for steady-state initialization at the beginning of a simulation. The topmost row shows the controller for superheat via HTF pump speed; the middle rows show the controllers for expander supply pressures (or constant evaporation temperature) via expander speeds; and the bottom row shows the controller for condensing pressure via fan air flow rate. 61 When the storage tank temperature in the idling model reaches the startup trigger, the reverse occurs: 1) the storage tank temperature in the operating model is reinitialized to that of the idling model (in case the controller has not settled); and 2) a controller inputs a lower irradiance signal to the idling model to maintain the storage tank at the shutdown trigger temperature. 3.7 Application: Identifying Optimal Control Schemes for Daily Environmental Variation The two control strategies are compared by simulating the system response to a solar irradiance dataset comprising four reference days that bracket a wide range of irradiance conditions from clear sky to severe overcast.1 The current method for evaluating the solar input consists of parsing the rapidly-varying irradiance data into a piecewise polynomial fit to provide a continous input function to the dynamic model. For this preliminary assessment, Tamb , approximated as a sine curve, is the same for each of the four days and incidence angle, θ, and wind velocity, vwind , are set to zero. Figure 3-9 is a plot of the assumed irradiance profile and ambient temperature. At the beginning of the first day, the storage tank is initialized to 135◦ C, 10◦ C below the chosen shutdown trigger temperature of 145◦ C (approximately the overnight temperature loss). This shutdown trigger temperature is chosen to maintain the 5◦ C of superheat given the hardware limit of 1800 rpm imposed for the HTF pump and is identified by executing the steady-state model under similar operating conditions. The startup trigger is set to 155◦ C. The Tev,const strategy is analyzed with two different evaporating temperatures, 117◦ C and 127◦ C, now referred to as Tev,117 and Tev,127 , respectively. A Tev of 127◦ C is the highest at which the 5◦ C of superheat can 1 The irradiance input is based on a global irradiance dataset measured in Lesotho in 2009 using a SDL-1 solar data logger and converted to irradiance on a single-axis tracking surface with N-S alignment. The measured data was compared to the respective “perfect” irradiance profile calculated for each day using geoposition data and the solar constant resulting in a “fraction of perfect” irradiance versus time. This perfect irradiance profile was converted to beam irradiance on a tracking surface, then multiplied by the fraction of perfect irradiance to simulate irradiance on a tracking surface under real atmospheric conditions (plus a perfect day) [14]. The geoposition and tracking equations can be found in [5]. 62 Irradiance (W/m2) 1000 800 600 400 200 Ambient Temperature (oC) 0 30 0 12 24 36 48 Time (hrs) 60 72 84 96 0 12 24 36 48 Time (hrs) 60 72 84 96 25 20 15 Figure 3-9: Environmental conditions for control strategy comparison. The upper plot shows direct normal irradiance and the lower plot shows ambient temperature. In this preliminary comparison, incidence angle and wind speed are set to zero. be maintained within the HTF pump hardware limit under these conditions. Figures 3-10 through 3-13 present some detailed results for a single strategy on a single day – the Pev,opt strategy with 155◦ C startup temperature on Day 2 (between 24 and 36 hours in Fig. 3-9). The figures show the results from the idling model when the ORC is shut down and those from the operating model when the ORC is operating. Figure 3-10 presents the set points and process variables for the ORC controllers. A shutdown is indicated any time the parameters fall below the x-axis. Since the full shutdown sequence of the ORC is not modeled as explained in Section 3.6, the parameters are set to zero any time the simulation switches to idling mode. The superheat and condenser temperature defect controllers use constant set points while the evaporation pressure and intermediate expander pressure controller set points are determined by the optimization functions described in Section 2.6 and Appendix C, respectively. The Pev , Pint , and ∆Tcond controllers track their set points very closely. 63 6 26 Set Point Superheat Set Point Evaporation Pressure 25 Pressure (bar) 24 o Temperature ( C) 5.5 5 23 22 4.5 21 4 20 0 1 2 3 4 5 6 7 8 Time (hrs) 9 10 11 12 0 (a) Superheat Control 1 2 3 4 5 6 7 8 Time (hrs) 9 10 11 12 (b) Pev Control 9.5 11 Set Point Mid−Expander Pressure Set Point ∆ Tcond 9 Temperature (oC) Pressure (bar) 10.5 8.5 8 10 9.5 7.5 7 9 0 1 2 3 4 5 6 7 8 Time (hrs) 9 10 11 12 0 1 2 3 4 5 6 7 8 9 10 11 12 Time (hrs) (c) Pint Control (d) ∆Tcond Control Figure 3-10: Controller results for Pev,opt strategy on Day 2. Shutdown is indicated when the parameters cross the x-axis. 64 Irradiance (W/m2) 1000 800 600 400 200 0 0 1 2 3 4 5 6 7 8 9 10 11 12 10 11 12 190 THTF,col,ex Temperature (oC) 180 THTF,tank TWF,exp,su 170 160 150 140 130 120 0 1 2 3 4 5 6 Time (hrs) 7 8 9 Figure 3-11: Effect of thermal storage for Pev,opt strategy on Day 2. The upper plot shows the direct normal irradiance profile for Day 2, and the lower plot shows several temperatures. The tank, whose temperature is shown in blue, reduces the variation seen by the ORC with expander supply temperature shown in green as compared to the collectors with exhaust temperature shown in red (see especially hours 7-9). The superheat controller exhibits more variation although it is within ±0.5◦ C for most of the operating time. This behavior can be attributed to the fact that a HTF component is being used to control a WF property. One alternative is to use the WF pump to control superheat but this would directly impact the power production of the expanders which may not be desirable. Future analysis will examine alternatives to the current assumption of a constant flow rate for the WF pump. Figure 3-11 shows the effect of the thermal storage. The tank effectively dampens the rapid oscillations in temperature seen by the collectors due to the rapidly-varying solar input as evidenced by the smoother expander supply temperature (see especially hours 7-9). Following the blue trace in the figure, the tank temperature takes ∼1.5 hours to reach the startup trigger of 155◦ C after the sun rises at time zero. For this 65 particular day, the ORC has to be shut down 6 times before the final shutdown of the day as indicated by the number of times the tank temperature reaches the shutdown trigger of 145◦ C. In the afternoon, when the solar insolation is stronger, the ORC can operate more continuously. After the 10th hour of sunlight, the ORC is shut down because the tank is below the shutdown trigger temperature, but there are still some large oscillations in the collector temperature. This behavior can be explained by the HTF pump startup trigger of 180 Wm-2 of solar irradiance. The upper plot shows that the irradiance decreases below this trigger at 10.2 hours. At this point, the pump is shut down and with a negligible flow rate but non-zero irradiance, the stagnant HTF heats up very quickly. At 10.4 hours, the irradiance exceeds the trigger, so the HTF pump is powered on and the collector exhaust temperature reduces. This cycle repeats between 10.6 and 11.2 hours before the final shutdown at 11.4 hours. This cycling of the HTF pump is also indicated in Fig. 3-12, which illustrates the power requirements and outputs for the system’s turbomachinery. Under these conditions, the expanders are optimally run with 40% of the load on the first expander and 60% on the second, and this load split is found to be similar for the other working conditions in the optimization study of Section 2.6. The expander traces follow the shape of the tank temperature in Fig. 3-11 as expected since their optimization functions are meant to adapt to the working conditions and a higher tank temperature enables a higher evaporation pressure and higher power production as discussed in Section 2.6. The WF pump power demand is also strongly a function of the high-side or evaporation pressure since pex is in the numerator of the power equation, Ẇel ≈ ηp · ṁ · νsu · (pex − psu ) (see Section 2.4). Therefore, the shape of the WF pump trace mimics that of the tank temperature with a higher evaporation pressure corresponding to a higher power demand. The converse behavior can be seen in the HTF pump where a higher tank temperature allows for a smaller HTF pump demand, again as discussed in Section 2.6. Finally, the condenser fans are essentially insensitive to the tank temperature as they are being controlled to maintain a constant temperature defect between the air supply and condenser exhaust. Therefore, they follow the trend of the ambient temperature as shown in the zoomed view in 66 4 3.5 Wdot,net Wdot,exp2 Wdot,WF,pump Wdot,exp1 Wdot,fans Wdot,HTF,pump 3 2.5 Power (kW) 2 1.5 1 0.5 0 −0.5 −1 −1.5 0 1 2 3 4 5 6 7 Time (hrs) 8 9 10 11 12 Figure 3-12: Power produced or consumed by each component for Pev,opt strategy on Day 2. Nearly all the components are most heavily influenced by the HTF supply temperature (Fig. 3-11) except the condenser fans, which react to the ambient temperature as shown in Fig. 3-13. Fig. 3-13 where a higher ambient temperature requires less fan power as the condenser needs less cooling to maintain the same temperature defect. Now that some of the details behind the simulation have been presented, the following discussion will focus on the comparison of the various control strategies. Figure 3-14 shows the global and ORC thermal efficiencies and the net power generated for each strategy. The vertical drops indicate that the tank temperature reached 145◦ C and a shutdown was executed. For these four reference days, the Tev,const strategy at Tev = 117◦ C, the Tev,const strategy at Tev = 127◦ C, and the Pev,opt strategy generate 62.7 kWh, 65.5 kWh, and 70.0 kWh of energy, respectively. In other words, as a result of its ability to flexibly adapt to varying ambient conditions, the Pev,opt strategy generates 12% and 7% more energy than the Tev,117 and Tev,127 strategies, respectively. It must be noted, however, that these results do not take into account 67 Ambient Temperature (oC) Condenser Fan Power (kW) 30 25 20 15 0 1 2 3 4 5 6 Time (hrs) 7 8 9 10 11 12 0 1 2 3 4 5 6 Time (hrs) 7 8 9 10 11 12 −1.1 −1.11 −1.12 −1.13 −1.14 −1.15 Figure 3-13: Power consumed by condenser fans for Pev,opt strategy on Day 2 (lower plot). The shape of the power consumption is a function of the shape of the ambient temperature input shown in the upper plot and the controlled constant temperature defect. the potential effects of repeated ORC startup/shutdown cycling (e.g., due to the heat capacity of the hardware or the need for load following, which were neglected in this model); a follow-on study to investigate the trade-offs between increasing thermal storage size (and ambient losses) and decreasing operational cycling is suggested. Another trend that can be observed is the tendency for the Tev,127 strategy net power and efficiency to exhibit more variation during operation than that of the Tev,117 strategy under these conditions. For the Tev,127 strategy, net power and efficiency sometimes begin higher and end lower than the Tev,117 strategy during an operational cycle. At other times, they remain higher than the Tev,117 strategy throughout. This behavior emphasizes the difficulty in choosing a constant evaporation temperature which behaves optimally under many conditions, recalling the discussion of Section 2.6 on the higher HTF pump demand required to maintain the 5◦ C superheat at a reduced HTF supply temperature. The Tev,117 strategy has more room between the constant evaporation temperature and the shutdown trigger so the HTF pump demand stays 68 more consistent over this operational range, but it is less efficient at the beginning of a cycle when the HTF supply temperature is higher. A more comprehensive survey of environmental conditions and triggers would help determine an optimum although this may be unnecessary pending more detailed studies of the Pev,opt strategy. Having validated that an optimum pressure strategy is more effective, the role of startup trigger temperature on cycling is further investigated. Figure 3-14 shows that at a trigger of 170◦ C, the initial system startup in the morning is delayed, but fewer shutdowns occur during the day once ORC operation begins. Energy generated also increases slightly by 0.1% versus the 155◦ C startup temperature. Fewer startup and shutdown cycles implies improved maintenance profiles and extended mean time between failure (MTBF) of components. Further investigation of seasonal timescale environmental influences and examination of alternative trigger temperatures and storage tank sizes is warranted. 69 70 Efficiency (%) 0 0.5 1 1.5 2 2.5 3 3.5 4 0 1 2 3 4 5 6 7 8 9 10 11 12 0 0 6 6 12 Wdot,net,opt,170 24 Time (hrs) 18 24 Time (hrs) Wdot,net,opt,155 18 ηglobal,opt,155 ηglobal,opt,170 12 ηthermal,ORC,opt,155 ηthermal,ORC,opt,170 30 Wdot,net,Tev127,155 30 ηglobal,Tev127,155 ηthermal,ORC,Tev127,155 36 36 42 Wdot,net,Tev117,155 42 ηglobal,Tev117,155 ηthermal,ORC,Tev117,155 48 48 Figure 3-14: Control strategy comparison results. The upper plot shows global and ORC thermal efficiency and the lower plot shows net power generated. The diagonal hatch marks on the x-axes indicate that the overnight results, for which the plant is shut down, are not shown for display purposes. Net Power (kW) Chapter 4 Conclusions and Future Work 4.1 Conclusions A model capturing the important dynamics of a micro-CSP system with thermal storage capable of responding and adapting to rapid environmental variations has been developed. The dynamic model is based on detailed steady-state models validated to data where available. The steady-state models are also used to provide initial values, heat transfer coefficients, and pressure drops for the dynamic simulation and to determine an optimum evaporation pressure correlation for input to a control strategy analyzed with the dynamic model. Steady-state and dynamic models representing the complex geometry of the counter-crossflow finned-tube condenser have been developed. More detailed analysis of the steady-state condenser model indicated that the assumption of a constant 10◦ C condenser temperature defect used in the dynamic model is close to optimal and that passive cooling may be sufficient for ambient temperatures below 10◦ C. A method for approximating startup and shutdown in the dynamic model while avoiding the numerical issues associated with rapid phase changes has been proposed to facilitate full- and multiple-day simulations. To the author’s knowledge, this is the first model capable of continuously simulating through startup and shutdown in addition to coupling a dynamic thermodynamic model of the power cycle with dynamic models of the collectors and tank. In this preliminary assessment based on a solar irradiance dataset spanning four 71 reference days, the Pev,opt strategy that adjusts the operating point based on the boundary conditions generates 7% more energy than the Tev,const strategy evaporating at 127◦ C. The Pev,const strategy, however, allows for that flexibility by reducing the range of speeds the HTF pump experiences during operation at the expense of a wider range of speeds experienced by the expanders, which may reduce their life and increase maintenance costs. The relative maintenance cost of larger cycles on the expanders versus the pump should be compared with the energy gain from the Pev,const strategy in future analysis. Moreover, startup trigger temperature plays an important role in system dynamics and performance, with advantages in selecting for an operational envelope at higher temperatures of the working fluid. In this study, increasing the startup trigger temperature from 155◦ C to 170◦ C recovers 0.1% more energy with fewer shutdowns during the day, which should also reduce maintenance costs. Based on the updated condenser correlation and steady-state survey of the system model, a further 0.3% energy may be gained by allowing for variable condenser temperature defects. The dynamic model presents several advantages over a yearly steady-state simulation. It allows for the comparison of different control strategies, which accept rapidly varying environmental conditions as input, and for the investigation of the feasibility of a control scheme which depends on regulating interrelated components. Dynamic analysis provides the ability to examine the transients and cycles a component may experience over its lifetime. The facility and intuitiveness of Modelica versus EES software provides several advantages in itself. Since the components are designed to be self-contained, such that the equations describing each component accept mass flow, pressure, and enthalpy as inputs and calculate outlet mass flow, pressure, and enthalpy, they are easy to swap with components based on a different set of equations or of a different geometry. The Dymola Modelica interface facilitates this process by providing for the identification of components by icons, which allows the modeler to click-and-drag different components into a working cycle model. Parameters (e.g., geometries) of components are easily changed via the graphical user interface. 72 However, there are several disadvantages to a dynamic, rather than steady-state, simulation. The dynamic model is much more complex, which results in a longer runtime and necessitates a larger computing power. The dynamic model is also less robust due to the interaction of the coupled controllers, which require a lengthy tuning process. Several strategies for improving robustness are discussed below. Modelica requires good start values to initialize all the components and begin the simulation. This requirement indicates that a steady-state model for the components, if not the system, is necessary involving additional effort. 4.2 Future Work Possible improvements to this work can be split into several categories: validation, expansion, modeling improvements, and robustness. In future, the full system model should be validated to the pilot system in steadystate. The correlations used for the expander properties were developed for gridsynchronized, constant-speed expanders, and these should be validated for variable load and asynchronous operation. Expansion is meant to represent a wider survey for control strategy comparison. The current work analyzes four days of irradiance variation with ambient temperature variation, approximated as a sine curve, duplicated for the four days and incidence angle and wind speed set to zero. A more comprehensive survey of environmental conditions would allow for a quantitative comparison of control strategies. Several improvements could increase the accuracy of the current model. The ORC pressure drops are currently neglected in both the steady-state and dynamic models. Adding them to the dynamic model may require better start values, which implies including them in the steady-state model increasing its iteration time. Incorporating pressure drop into the condenser heat transfer model becomes important for ambient temperatures below 15◦ C, for which pressure drop can exceed 30 mbar. Including tube-tube heat transfer could also improve the model’s accuracy as shown in Appendix B. A more realistic representation of sensor dynamics would help indicate 73 whether the proposed control schemes are viable as parameterized. In the present work, the ORC thermal inertia is neglected during startup and shutdown transients. In the future, a model that accounts for this effect would allow for a more accurate comparison of the different control strategies. However, a more realistic accounting of startup and shutdown, possibly by simulating zero flow in the ORC, leads to the issue of robustness. Several robustness strategies for handling the numerical issues associated with rapid phase changes are currently under development [16], but many of them rely on increasing the number of elements in the heat exchangers which greatly increases the simulation time. A strategy for approximating shutdown that doesn’t signficantly increase run time but still accounts for the time constants of the ORC components would increase the accuracy of the dynamic model while retaining its usefulness for multi-day simulations. The current method for approximating the rapidly varying solar input (piecewise polynomials) is very computationally-intensive and allows for, at most, two days of simulation at a time. Consecutive days must be re-initialized to the conditions from the end of the previous simulation. A more robust way for approximating solar input would greatly facilitate multi-day or even yearly simulations. A more rigorous control design process should lead to a more responsive, robust control system. The typical process would include the following steps: modeling, linearizing, designing the controllers, and verifying by closed-loop comparison between the linear and nonlinear models. Future efforts focusing on any of these improvements would ensure this tool becomes an accurate, robust, and potent means of evaluating the most effective control schemes for the proposed micro-CSP plant. In turn, underserved rural communities and partnering government agencies, NGOs, and others would be able to make the most of their investment to provide heating, cooling, and electricity for rural health and education clinics by harnessing the maximum available power from the sun. 74 Appendix A Model Code and Experimental Data Due to the difficulty of reading object-oriented code in print and to facilitate direct application of the models in the associated software, the models are available electronically at the following address: http://www.labothap.ulg.ac.be/staff/mireland/ The versions of ThermoCycle and CoolProp used are also available. See the following website for more information on using and installing those Modelica libraries: http://www.thermocycle.net/ The experimental and manufacturer data used to calibrate the models is also provided. 75 76 Appendix B Sensitivity Studies on Heat Exchanger Modeling The following sections describe four sensitivity studies on heat exchanger modeling based on the finned-tube condenser, namely the effect of mixed vs unmixed cooling air, the effect of variable vs constant heat transfer coefficients, the impact of tube-tube heat transfer, and the effect of discretization. B.1 Mixed vs Unmixed Cooling Air To simulate the condenser with the air between tube rows unmixed, the air cells are futher discretized into 3 sections (left, center, right) per bank by 5 rows by n elements per tube for a total of 3 x 5 x n air elements per tube bank. In Figure B-1a, the modeling schematic for the mixed model from Section 2.3 is reproduced. Figure B1b is an end view indicating the additional discretization necessary for the unmixed model with dashed lines. The left/center/right discretization model assumes a set of adiabatic surfaces between the air cells such that tubes split by one of the surfaces transfer half of their energy to the air parcel on the left and half to the right. The center air parcel receives energy from the tubes on the left and right. The following summarizes the necessary changes to the energy balances used in the mixed model of Section 2.3. 77 9DSRU 7ZR3KDVH /LTXLG $LU2XWOHW 5RZ $LU)ORZ :),QOHW $LU,QOHW 5RZ :)2XWOHW (a) Modeling Schematic (b) End View Figure B-1: Modeling schematic of condenser tube bank. Figure B-1a shows that refrigerant enters as a vapor on the top right and travels back and forth from Tube 1 to Tube 12 on the bottom left while condensing. White tube faces indicate fluid entering at the tube face and traveling back (up and to the left in the diagram) and gray tube faces indicate fluid that has traveled from the back and exits at the tube face. Dashed black lines notionally represent discretization of fluid cells in the tubes and the hexagonal prism of air and fins (not shown) surrounding the tube section. Bold black numbers indicate the discretization scheme. Dashed orange lines represent the planes of each tube row. Figure B-1b shows the end view and the additional left/center/right discretization for the unmixed model. Dashed black lines represent adiabatic surfaces. The refrigerant in Tube 1 interacts with the center and right air cells in Row 5. Equation 2.11 becomes hW F,i+1 U Ai 1 1 = hW F,i − · (TW F,i − Tair,c,su,i ) + (TW F,i − Tair,r,su,i ) , ṁW F 2 2 (B.1) where c indicates center and r indicates right. The center and right air cells in Row 5 interact with Tubes 1 and 2 and Tube 1, respectively, so Equation 2.12 becomes 78 Tair,c,ex,i Tair,r,ex,i 1 U Ai · (TW F,i − Tair,c,su,i ) =Tair,c,su,i + (ṁi · cp )air 2 1 + (TW F,2N +1−i − Tair,r,su,i ) 2 1 U Ai =Tair,r,su,i + · (TW F,i − Tair,r,su,i ) (ṁi · cp )air 2 (B.2) (B.3) Since the inlet air temperature is no longer the average outlet air temperature above Row 4, two final equations are needed: Tair,c,in,i = Tair,c,ex,4N +1−i (B.4) Tair,r,in,i = Tair,r,ex,4N +i (B.5) To compare the heat transfer between mixed and unmixed cooling air, the models are evaluated with a WF supply temperature of 48.8◦ C, air supply temperature of 25◦ C, WF mass flow rate of 0.153 kgs-1 , air mass flow rate of 3.221 kgs-1 , and WF pressure of 2.11 bar (saturated liquid at the WF exit is not imposed). The heat transfer coefficients and pressure drop assumptions are the same as those of Section 2.3. With these inputs, the mixed model predicts a heat flow of 30.3 kW while the unmixed model results in a heat flow difference of only 0.03%. B.2 Variable vs Constant Heat Transfer Coefficients For the set of model inputs mentioned above, the heat transfer coefficients, which vary for each cell, can be averaged for each zone of the refrigerant. The average vapor heat transfer coefficient is 288 Wm-2 K-1 , and the average two-phase heat transfer coefficient is 1515 Wm-2 K-1 . (Since the outlet is on the cusp of the two-phase/liquid boundary or fully two phase for these simulations, there is no need for the liquid heat transfer coefficient.) The air heat transfer coefficient is 31.4 Wm-2 K-1 . Changing the 79 mixed model to use these constant heat transfer coefficients results in a heat flow difference of only 0.76%. With a new working condition based on the same assumptions but a 5% decrease in air mass flow rate to 3.06 kg-1 , the average vapor heat transfer coefficient remains 288 Wm-2 K-1 , and the two-phase heat transfer coefficient becomes 1573 Wm-2 K-1 . The air heat transfer coefficient reduces to 30.4 Wm-2 K-1 . After executing the constant heat transfer coefficient model with the coefficients from the previous working condition, but the new mass flow rate, the error in heat flow is only 0.66%. Setting the air mass flow rate back to 3.221 kg-1 and reducing the WF pressure to 1.77 bar and air supply temperature to 20◦ C (the saturation temperature at this pressure although saturated liquid at the exit is still not imposed) results in an average vapor heat transfer coefficient of 284 Wm-2 K-1 , two-phase heat transfer coefficient of 1609 Wm-2 K-1 , and air heat transfer coefficient of 31.2 Wm-2 K-1 . The resulting heat flow difference relative to the constant heat transfer coefficient model with coefficients set by the first working condition is only 0.15%. B.3 Tube-to-Tube Heat Transfer Tube-to-tube heat transfer is currently neglected in the condenser models. However, its impact becomes important near the WF inlet where the fluid is still in vapor phase. For instance, Fig. B-2, which shows the average temperature for each tube and the surrounding air, demonstrates that Tube 1 exhibits the only significant temperature difference relative to the rest of the tubes because it is the only tube for which flow remains fully in single phase. The model input conditions used to generate these average temperatures are a WF supply temperature of 48.8◦ C, air supply temperature of 25◦ C, WF mass flow rate of 0.153 kgs-1 , air mass flow rate of 3.221 kgs-1 , and WF pressure of 2.11 bar (saturated liquid at the WF exit is not imposed). The tube-to-tube heat transfer, Q̇cond,tube−tube , can be estimated by modeling conduction through the fin area described on two sides by the line marked “W” in Fig. B-2: 80 7DLU & 7:) & ;; 7XEH ; / PP : PP Figure B-2: Average condenser temperatures for tube-tube heat transfer analysis. Average supply and exhaust air temperatures for each tube row are listed in the left column with dotted blue lines indicating the plane of the mixed air. Average WF temperatures in each tube are depicted on the tube faces with the progression from Tube 1 to 12 indicated to the right of each tube. The zoomed view provides some key dimensions for the heat transfer analysis, namely the distance between tubes, L, and the fin width used to determine the cross-sectional area for conduction between tubes, W . Q̇cond,tube−tube = kf in · Across W · tf in · Nf ins · ∆T = kf in · · ∆T, L L (B.6) where kf in is the fin conductivity; Across is the cross-sectional heat transfer area described by the width, W , and fin thickness, tf in , multiplied by the number of fins, Nf ins ; L is the distance between tubes; and ∆T is the temperature difference between tubes. With kf in = 200 W(mK)-1 , W =33.0 mm, tf in = 0.13 mm, Nf ins = 1081, L = 38.1 mm, and the largest average temperature difference between Tubes 1 and 4 or 5 = 6.9◦ C, the tube-to-tube heat transfer is 168 W. This result can be compared to the tube-to-air heat transfer, Q̇cond,tube−air , using Q̇cond,tube−air = U A · ∆T, (B.7) where U A is the combination tubeside and finside product of heat transfer coefficient 81 % Error in Heat Flow vs 10 Cells 0 −0.1 −0.2 −0.3 −0.4 −0.5 −0.6 −0.7 2 4 6 8 10 Number of Cells per Tube Figure B-3: % error in condenser heat flow vs number of cells per tube. and heat exchange surface area for a single tube and ∆T is the average temperature difference between the tube and air. With U A for Tube 1 = 18.3 Wm-2 K-1 (derived from the current model) and a temperature difference between Tube 1 and its supply air of 8.3◦ C, the tube-to-air heat transfer is 152 W, so the tube-to-tube heat transfer actually dominates for Tube 1. This effect falls dramatically when considering tubes in two-phase flow surrounded by other tubes in two-phase flow since they exhibit a negligible temperature difference. Although tube-to-tube interactions were neglected in the detailed condenser model, based on the analysis of this working condition, they could play a significant part in the heat transfer for up to 5 of the 12 tubes in a tube bank. This improvement will be considered in future work. B.4 Sensitivity to Number of Cells Since Modelica is more robust at handling a changing number of cells, the dynamic model run to steady-state is used for this study. Again, the model is evaluated with a WF supply temperature of 48.8◦ C, air supply temperature of 25◦ C, WF mass flow rate of 0.153 kgs-1 , air mass flow rate of 3.221 kgs-1 , and WF pressure of 2.11 bar (saturated liquid at the WF exit is not imposed). The working fluid heat transfer 82 coefficients are 288 Wm-2 K-1 in vapor phase, 1520 Wm-2 K-1 in two phase, and 170 Wm-2 K-1 in liquid phase and the air heat transfer coefficient is 31.4 Wm-2 K-1 . The nominal amount of cells per tube in the steady-state model is 10. With 12 tubes, this results in 120 cells per tube bank. Figure B-3 shows that with only 2 cells per tube or 24 cells per tube bank, the error in heat flow vs 10 cells per tube is less than 1%. 83 84 Appendix C Optimized Intermediate Pressure between Two Expanders Since each expander’s isentropic efficiency is modeled using a polynomial as a function of psu and rp , it is possible to find an analytical solution for the optimized intermediate pressure, Pint , by maximizing the equation for the combined efficiency given the inlet and outlet pressures. The derivation for this solution when the efficiency polynomial is a function of ρsu and rp was provided in [15], and the derivation for the polynomials used in this work (Equation 2.10), which depend on psu and rp , is provided here. If the refrigerant is approximated as an ideal gas, dh = cp · dT (C.1) and, using isentropic relations, Tp 1−γ γ Tex,s = = constant → Tsu pex,s psu 1−γ γ , (C.2) where su and ex, s refer to supply and isentropic exhaust conditions, respectively; T is temperature; p is pressure; γ is heat capacity ratio; h is enthalpy; and cp is specific heat capacity. Therefore, 85 1−γ hex,s = rp γ , hsu (C.3) when 0 K is chosen for the enthalpy reference. Using the definition of isentropic efficiency for an expander, η= hsu − hex , hsu − hex,s (C.4) and defining β as β ≡1− 1−γ hex,s = 1 − rp γ , hsu (C.5) the overall isentropic efficiency across both expanders can be expressed as follows: η= h1 − h3 h1 − h3 η1 · β1 + η2 · β2 − η1 · η2 · β1 · β2 , = = h1 − h3s β · h1 β (C.6) where index 1 refers to the entrance to Expander 1; index 2 refers to the exhaust of Expander 1/supply of Expander 2; and index 3 refers to the exhaust of Expander 2. Determining the intermediate pressure at which the combined efficiency maximizes can be achieved by setting dη drp1 to zero: dη =(η1 β10 + η10 β1 + η2 β20 + η20 β2 − η1 η2 β1 β20 − drp1 η1 η2 β10 β2 + η1 η20 β1 β2 − η10 η2 β1 β2 ) · β −1 = 0. (C.7) where a 0 indicates a derivative with respect to rp1 . Solving this differential equation requires determination of the derivative terms. The derivative of β1 is dβ1 γ − 1 1−2γ = · rp γ , drp1 γ and since rp = rp1 · rp2 , the derivative of β2 is 86 (C.8) d dβ2 = drp1 drp1 rp 1− rp1 −1 1−γ 1 − γ 1−γ = · rp γ · rp1γ . γ γ (C.9) The form of the expander isentropic efficiency polynomial was shown in Equation 2.10. For the first expander, the derivative of this polynomial with respect to rp1 is n−1 n−1 dη1 1 XX = aij · i · ln(rp1 )i−1 · ln(psu1 )j + an0 · n · ln(rp1 )n−1 . drp1 rp1 i=0 j=0 (C.10) However, since the second expander parameters depend on rp1 , it is necessary to calculate partial derivatives: δη2 drp2 δη2 dpsu2 rp δη2 psu1 δη2 dη2 = · + · =− 2 · − 2 · . drp1 δrp2 drp1 δpsu2 drp1 rp1 δrp2 rp1 δpsu2 (C.11) The partial derivative of η2 with respect to rp2 is of the same form as Equation C.10 while the partial derivative of η2 with respect to psu2 is n−1 n−1 1 XX δη2 = aij · j · ln(rp2 )i · ln(psu2 )j−1 + an0 · n · ln(psu2 )n−1 . δpsu2 psu2 i=0 j=0 (C.12) With equations for all of the derivatives, the maximum overall efficiency can be determined by solving Equation C.7. Solving for the Expander 1 pressure ratio which maximizes η allows for determination of the optimized intermediate supply pressure. 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