DEVELOPMENT OF A DYNAMIC MODEL OF A COUNTERFLOW COMPACT HEAT EXCHANGER FOR SIMULATION OF THE GT-MHR RECUPERATOR USING MATLAB AND SIMULINK A THESIS

Transcription

1 DEVELOPMENT OF A DYNAMIC MODEL OF A COUNTERFLOW COMPACT HEAT EXCHANGER FOR SIMULATION OF THE GT-MHR RECUPERATOR USING MATLAB AND SIMULINK A THESIS Presented in Partial Fulfillment of the Requirements for the Degree of Master of Science in the Graduate School of The Ohio State University By David P. Hawn, B.S. ***** The Ohio State University 2009 Thesis Committee: Dr. Thomas Blue, Adviser Dr. Richard Christensen Approved by Adviser Nuclear Engineering Graduate Program

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3 ABSTRACT A computational model was developed to determine the dynamic behavior of counter flow compact heat exchangers. This code was written with the intention of becoming a component of a larger system dynamics model of a Brayton cycle nuclear power plant. Several configurations for the GT-MHR recuperator were analyzed, but the code can easily be modified to analyze many types of compact heat exchangers with a variety of applications. Helium was the working fluid used in this project, but the code can be modified to use other gases. This code was written in Matlab and Simulink but the methods outlined in this report could be easily reapplied in other programming languages. This code is also useful for designing counter flow compact heat exchangers in general. In this model the heat exchanger is discretized in time and in space. The resolution of the discretization is defined by the user. Helium properties are reevaluated for each volume before each time step. The dynamic inputs to the model are the inlet temperature, mass flow rate and pressure for each side of the heat exchanger. This model assumes low Mach number flows and treats the propagation of pressure and mass flow rate changes as instantaneous. The outlet temperature and pressure drop for each side is determined. The results of the simulation were successfully validated against results available in the literature. Contact the author for a copy of this code. ii

4 DEDICATION Dedicated to my family and friends iii

5 ACKNOWLEDGEMENTS First I would like to thank my friends and family who were always supportive and patient throughout this process. I am especially grateful for my fiancée who spent countless late nights and weekends working with me in the computer labs of Scott Laboratories. I am also in debt to my adviser, Dr. Blue, for his support and patience throughout this project and my time in graduate school. Of course I also owe Dr. Blue thanks for reviewing my thesis. I would also like to thank Dr. Christensen for providing encouragement, answering many questions, sitting on my committee, and reviewing my thesis. I would also like to thank Dr. Freuler and everyone I had the pleasure to work with in the FEH program. The FEH program is top-notch and I hope I can give back to it as much as it has given to me. Finally, and most importantly, is thanks due to God for the amazing world in which we live and the relentless curiosity given to me to explore it. iv

6 VITA.. February 27, Born Columbus, Ohio, USA 2005 B.S. Mechanical Engineering, The Ohio State University University Fellow, The Ohio State University Graduate Teaching Assistant, The Ohio State University 2008-Present.. NRC Fellow, The Ohio State University PUBLICATIONS 1. Theoretical and Experimental Analysis of Response of SiC in Thermal Neutron Environment, V. Krishnan, B. Khorsandi, J. Kulisek, D. Hawn, T. E. Blue, and D.W., Transactions of the American Nuclear Society, v. 96, 2007, p Major Field: Nuclear Engineering FIELDS OF STUDY v

7 TABLE OF CONTENTS Abstract... ii Dedication... iii Acknowledgements... iv Vita..... v List of Figures... ix List of Tables... xi 1. Introduction Objectives Summary Literature Review Introduction Evolution of the Design and Analysis of the GT-MHR Recuperator Introduction General Atomics Recuperator Design General Atomics and OKBM Recuperator Design Kays and London Compact Heat Exchanger Design Data Selected Heat Exchange Surfaces and Geometric Relationships Empirical Data for Selected Heat Exchanger Cores Compact Heat Exchanger Analysis Static Analysis Estimation of Average Gas Properties Determination of Outlet Temperatures Determination of Pressure Drop Transient Analysis Summary of Literature Review Recuperator modeling in Matlab & Simulink Introduction Introduction to Matlab and Simulink Differences between Matlab and Simulink Parallel Processing in Matlab and Simulink vi

8 3.3. Dynamic Model Introduction Assumptions Model Revisions and Development Heat Exchanger Discretization User Definable Parameters and Initial Conditions Initial Conditions Inlet Conditions Property Lookup Heat Transfer Model Pressure Drop Model Entrance and Exit Effects Flow Acceleration Core Friction Determination of Values at End of Time Step Error Reporting Data Output Validation Introduction Steady State Temperature Response Steady State Pressure Drop Transient Temperature Response Step Change to Inlet Temperature Non-Step Change to Inlet Temperature Mass Flow Rate Transient Stability Analysis Validation Summary Results, Conclusion and Future Work Introduction vii

9 5.2. Designs for GT-MHR Recuperator Additional Information Conclusion Future Work Bibliography viii

10 LIST OF FIGURES Figure 1: Brayton Cycle without Recuperator... 2 Figure 2: Brayton Cycle with Recuperator... 3 Figure 3: Key Components of the GT-MHR and PCU (Baxi, et al., 2006)... 5 Figure 4: Examples of Heat Exchanger Cores (London & Kays, 1998) Figure 5: Offset Strip-Fin in Counter Flow Arrangement (Etzel, Baccaglini, Schwartz, Hillman, & Mathis, 1994) Figure 6: 1994 Individual Recuperator Unit Design (Etzel, Baccaglini, Schwartz, Hillman, & Mathis, 1994) Figure 7: GT-MHR PCU Configuration in the 1994 Design (Etzel, Baccaglini, Schwartz, Hillman, & Mathis, 1994) Figure 8: 2003 Individual Recuperator Unit Design (Kostin, Kodochigov, Vasyaev, & Golovko, 2004) Figure 9: GT-MHR PCU Configuration in the 2003 Design (Kostin, Kodochigov, Vasyaev, & Golovko, 2004) Figure 10: Alternate Cross Section for GT-MHR Heat Exchanger Surface (Bikh, Golovko, Dmitrieva, Kamashev, & Krasilschikov, 2004) Figure 11: Strip-Fin Heat Exchanger Surface Parameters Figure 12: Layered Strip-Fin Surfaces Figure 13: Empirical Friction Factor and Heat Transfer Data for 1/ Figure 14: Circuit Analogy of Thermal Resistances Figure 15: Two dimensional simplification of heat exchanger surface Figure 16: Pictorial Representation of Discretized Recuperator Figure 17: Electrical Circuit Analogous to Thermal System Figure 18: Pictorial Representation of Discretized Heat Exchanger with Φ= Figure 19: Example of Transient Input Entry Figure 20: Electric Circuit Analog of Thermal System with Φ = Figure 21: Independent Heat Transfer Circuit Figure 22: Simulink Model of Coupled Equations for Heat Transfer Model Figure 23: Discretized Heat Exchanger Surface with Φ=3 (x denotes hot or cold side). 71 Figure 24: Entrance and Exit Pressure-Loss Coefficients (London & Kays, 1998) Figure 25: Example of Error Log File Figure 26: Steady-State Error in Hot and Cold Outlet Temperatures Figure 27: Influence of Steady State Error on Computational Time Figure 28: Averaged Steady State Error in Hot and Cold Outlet Pressures Figure 29: Outlet Temperature Response to a Step Inlet Temperature Change (London & Kays, 1998) ix

11 Figure 30: Hot Side Inlet and Outlet Temperatures for 100K Hot Side Inlet Step, NTU= Figure 31: Cold Side Inlet and Outlet Temperatures for 100K Hot Side Inlet Step, NTU= Figure 32: Cold Side Outlet Temperature Response to 100K Hot Side Inlet Step, NTU = Figure 33: Normalized Cold Side Outlet Temperature Response to 100K Hot Side Inlet Step, NTU= Figure 34: Hot Side Outlet Temperature Response to 100K Hot Side Inlet Step, NTU = Figure 35: Normalized Hot Side Outlet Temperature Response to 100K Hot Side Inlet Step, NTU = Figure 36: Normalized Cold Side Outlet Temperature Response to 100K Hot Side Inlet Step, NTU= Figure 37: Normalized Hot Side Outlet Temperature Response to 100K Hot Side Inlet Step, NTU = Figure 38: Averaged Hot Side and Cold Side Time Response Error Figure 39: Averaged Hot Side and Cold Side Temperature Response Error Figure 40: Non-Step Inlet Temperature Change (Cima & London, 1958) Figure 41: Hot Side Inlet Temperature for Non-Step Transient Validation Figure 42: Simulation Results for Non-Step Temperature Validation Figure 43: Mass Flow Rate Transient (Cima & London, 1958) Figure 44: Simulation Results for Mass Flow Rate Transient Validation Figure 45: Hot Side Outlet Temperatures for Stability Analysis Trials Figure 46: Cold Side Outlet Temperatures for Stability Analysis Trials Figure 47: Influence of Exchanger Volume on Pressure Drop Figure 48: GT-MHR Recuperator Cold Side Outlet Temperature Response to a 100K Hot Side Inlet Step Figure 49: GT-MHR Recuperator Hot Side Outlet Temperature Response to a 100K Hot Side Inlet Step Figure 50: Temperature in Each Cold Side Volume after a 100K Hot Side Inlet Step, NTU= Figure 51: Wall Mass Temperatures after a 100K Hot Side Inlet Step, NTU= Figure 52: Pressure in Each Cold Side Volume after a 100K Hot Side Inlet Step, NTU= x

12 LIST OF TABLES Table 1: Summary of 1994 Recuperator Design Inlet and Outlet Conditions Table 2: Summary of 2003 Recuperator Design Inlet and Outlet Conditions Table 3: Comparison of Heat Exchanger Cores Table 4: Strip-Fin Heat Exchanger Surface Geometry Table 5: Coefficients for Curve Fit Empirical Data for Heat Exchange Surfaces Table 6: Illustration of Progression of Time Steps Table 7: Units of Helium Property Data Table 8: Units of Inconel-617 Property Data Table 9: Heat Exchanger Configuration and Ideal Outlet Temperature for Steady State Validation Table 10: Heat Exchanger Configuration and Calculated Outlet Pressures for Steady State Validation Table 11: Summary of Pressure Drop Validation Analysis Table 12: Heat Exchanger Configuration for Temperature Step Validation Table 13: Temperature and Time at 63% of Response for Transient Validation Error Analysis Table 14: Summary of Transient Error Analysis of Simulation Results Table 15: Heat Exchanger Geometry and Inlet Conditions for Non-Step Inlet Temperature Validation Table 16: Calculated Heat Exchanger Properties for Non-Step Inlet Temperature Validation Table 17: Heat Exchanger Geometry and Inlet Conditions for Mass Flow Rate Transient Validation Table 18: Calculated Heat Exchanger Properties for Mass Flow Rate Transient Validation Table 19: Configuration of Heat Exchanger used for Stability Analysis Table 20: Six Trials used in Heat Exchanger Stability Analysis Table 21: Summary of 2003 Recuperator Design Inlet and Outlet Conditions Table 22: Heat Exchange Surfaces Used for GT-MHR Recuperator Model Table 23: Potential Design Variations for the GT-MHR Recuperator Table 24: Off-Peak Steady State Performance of Configuration Table 25: Off-Peak Steady State Performance of Configuration xi

13 CHAPTER 1 1. INTRODUCTION Heat exchangers are used to transfer heat energy from one fluid to another fluid, usually without allowing the two fluid streams to mix. The two fluids can be the same or different in composition and in phase. In addition, the two fluids can be at different pressures, flow rates and temperatures. The application of a heat exchanger in a system can be justified for several reasons including plant safety and cost efficiency of operation. Design and optimization of the heat exchanger is influenced by its initial justification for inclusion in the plant. Because it is important to predict the steady state and transient behavior of an entire plant in the design phase, it is important to accurately predict the steady state and transient behavior of each component in the system, including heat exchangers. The development of a computational model to determine the dynamic and steady state behavior of compact heat exchangers is the focus of this project. In a traditional pressurized water reactor, the primary loop and secondary loop are isolated from each other by a steam generator. This heat exchanger enables cooling of the primary water and heating of the secondary water without the primary and secondary water mixing. This isolation provides several benefits. First, it allows the primary 1

14 system to operate at higher pressures and temperature without requiring the entire balance of plant to operate at the same conditions. Second, it isolates the primary system from some of the potential accidents in the balance of plant. Also, it prevents radioactivity in the primary water from being piped throughout the facility. In this case, the heat exchanger is functioning as a barrier to increase the safety of the facility. This component is already justified on a safety basis and the optimization of the design serves to improve the efficiency, reliability and cost. Heat exchangers can also be used to recover heat energy that would otherwise be wasted. Figure 1 shows a schematic of a simple direct Brayton cycle using a nuclear reactor as the heat source. In this example, a single phase gas is compressed, heated in the reactor and then allowed to expand across a turbine. The turbine is generally used to power the compressor and generate electricity. Before re-entering the compressor, the hot gas is cooled in the pre-cooler with the energy being dissipated to the cold source which is often a lake, river or an ocean. reactor compressor turbine generator pre-cooler cold source Figure 1: Brayton Cycle without Recuperator 2

15 cold source pre-cooler recuperator reactor compressor turbine generator Figure 2: Brayton Cycle with Recuperator Figure 2 shows a direct Brayton cycle that includes a recuperative heat exchanger, sometimes called a recuperator or a regenerator. In this cycle the hot, low pressure gas leaving the turbine is used to preheat the cold, high pressure gas leaving the compressor. Preheating the gas entering the reactor helps achieve higher gas temperatures at the outlet of the reactor, which inherently increases the efficiency of the thermodynamic cycle. The energy used to preheat the gas in Figure 2 was being dissipated to the cold source in the cycle shown in Figure 1. Recovering this energy also improves the efficiency of the cycle and reduces the burden on the cold source. Upon first inspection it seems that a recuperator would always be included in a Brayton cycle power plant. This component, however, is justified on an energy efficiency basis and not on a safety basis. For a recuperator to be included, the efficiency increase gained by adding the recuperator must offset the capital cost of the component, the increased complexity of the system, and the change in reliability of the system. The recuperator must be optimized for efficiency, reliability and cost and prove to be economically beneficial before being a justified addition to the plant design. 3

16 The design and analysis of a complete Brayton cycle is additionally complicated because the behaviors of the individual components are non-linear functions of several parameters. A component, such as the recuperator, cannot be arbitrarily added, changed or removed from the design without reanalyzing the behavior of the system. The efficiency of turbomachinery components, for example, is non-linear and is dependent on the pressure and temperature at the inlet of the components. Linear changes to the inlet conditions are likely to cause non-linear changes to the performance of the turbomachinery components. Also, if the heat source of the Brayton cycle is a nuclear reactor, variation of the gas conditions at the inlet of the reactor will change the neutronic behavior of the reactor. The design and analysis of the entire system must be completed including dynamic models of each of the system components. The goal of this project was to develop a dynamic compact heat exchanger model for use in a larger dynamic system model of the Gas Turbine Modular Helium Reactor (GT-MHR) Objectives The objective of this project was to create a dynamic model of the GT-MHR recuperator as part of a larger goal of developing a complete dynamic system model of the GT-MHR coupled with the Power Conversion Unit (PCU). The GT-MHR PCU includes the recuperator, two compressors, one turbine, a generator, an intercooler and a pre-cooler, as shown in Figure 3 (Baxi, et al., 2006). Because an existing GT-MHR model was created using Matlab and Simulink, the recuperator model was also created using Matlab and Simulink. While a model of the recuperator was developed for this 4

17 project, the intercooler and pre-cooler are essentially simplifications of the recuperator. The recuperator model could be easily changed into dynamic models of both the intercooler and the pre-cooler. Completing the complete GT-MHR model would require development of dynamic models of the turbomachinery and integration of the separate components. Control Rod Drive Generator Recuperator Turbine & Compressor Intercooler Pre-cooler Core Vessel System Reactor Shutdown Cooling System Figure 3: Key Components of the GT-MHR and PCU (Baxi, et al., 2006) 1.2. Summary A single-phase counter flow compact heat exchanger model was developed using a variation of the methods outlined in "Compact Heat Exchangers," by Kays and London (London & Kays, 1998). Friction factor and heat transfer data used in the model were also taken from this source. The heat exchanger was approximated as one-dimensional 5

18 and was discretized in time and in space. A control volume approach was used and helium properties were evaluated for each volume at each time interval instead of using fixed property values. Continuously re-evaluate the helium properties and increasing the spatial discretization of the exchanger allows this model to be applied to heat exchangers that the previous model could not be applied to (Cima & London, 1958). The dynamic inputs and outputs to the model are helium temperature, helium pressure and helium mass flow rate. Static inputs to the model include initial conditions of the hot and cold helium volumes and the metal mass of the heat exchanger. The geometry of the heat exchanger including overall geometry and the geometry for the hot and cold cores are constants but can be easily changed to analyze different configurations. Empirical thermophysical properties of helium over a wide range of temperature and pressure are contained in look-up tables in model. These tables could easily be changed to different gas types or mixtures of gases if this type of analysis is desired. The simulation was successfully validated against analytical steady state results and published transient results. The simulation was then used to determine several possible configurations for the GT-MHR recuperator. The steady state and transient behavior of two of the possible GT-MHR recuperator configurations were investigated. Off-peak steady state performance was determined for two recuperator configurations operating at reduced mass flow rates. The dynamic response to a 100K hot-side temperature step increase was also determined. 6

19 While the primary goal of this project was to develop of a dynamic model of the GT-MHR recuperator, the resulting model could be reused with small changes to model compact heat exchangers with different geometries, heat exchange surfaces and working fluids. Both the hot and cold fluids in the GT-MHR recuperator model were helium, but the model could be modified to have different gases or single phase liquids for the hot and cold streams. By changing the cold side fluid to water, for example, the intercooler and pre-cooler can also be modeled. In addition, the model can also be used as a generic design tool for compact heat exchangers both for steady state and transient response. Kays' book provides a large number of heat exchanger surfaces and corresponding friction factor and heat transfer data than can be easily substituted into this model. Data from other sources can also be used as long as the presentation is consistent with Kays' data. 7

20 CHAPTER 2 2. LITERATURE REVIEW 2.1. Introduction Designs for several heat exchanger cores are shown in Figure 4 (London & Kays, 1998). Each of these cores was designed to maximize the heat transfer area between the hot and cold streams and to maximize heat transfer between the two fluid streams. Maximizing the heat transfer area and the heat transfer coefficient minimizes the volume required by the heat exchanger. Heat exchangers designed in this way are called compact heat exchangers. The compactness of the heat exchanger minimizes the capital cost of the component and also minimizes the volume of space consumed by the component. However, the complex shapes of some of the designs are difficult to manufacture, adding to the cost of those cores. Manufacturing capabilities have developed substantially since Kays and London's initial testing and now many of the cores they described can be manufactured economically and for use at high temperatures. An important consequence of maximizing the heat transfer area and maximizing the heat transfer coefficient is a larger pressure drop across the heat exchanger core. The energy efficiency increase gained by maximizing the heat transfer in a heat exchanger must be optimized with many 8

21 parameters including the pressure drop across the component, the capital cost of the component, the reliability impact of adding the component to a process and the effect the component has on the dynamic response of the system. Core A in Figure 4 (London & Kays, 1998) has one fluid stream contained inside the tubes with the other fluid stream in cross-flow perpendicular to the tubes. Core B is similar to Core A with the addition of fins to increase the heat transfer area of the tubes. In Core C the tubes that were initially circular have been flattened and the extended surface area has been made continuous. Core D and E are different from the others because neither fluid is contained in a tube. Instead, the fluid streams flow through stacks of plate-fin surfaces. The plate-fin surfaces in Core D have a repeating triangular cross-section that does not change in the direction of flow. Core D would be considered a plain plate-fin surface. Between each plate-fin layer is a flat metal sheet which provides a barrier between the two fluid streams in the heat exchanger. The plate-fin surfaces in Core E have a repeating rectangular cross-section that also varies in the direction of flow. This plate-fin geometry is called a strip-fin. The purpose of the geometry variation in the direction of flow is to prevent the development of a thick boundary layer which reduces heat transfer between the fluid and the heat exchanger surfaces. 9

22 Figure 4: Examples of Heat Exchanger Cores (London & Kays, 1998) This chapter includes a literature review of relevant work directly related to the evolution of the design of the GT-MHR recuperator including a summary of the recuperator parameters at full power and peak performance. In addition, a steady-state analysis method for a counter flow compact heat exchanger is summarized (London & Kays, 1998). This steady-state analysis will be referenced in later chapters when describing the development of the dynamic model. Samples of empirical data used for generalized design and analysis of compact heat exchangers is included. Finally, a description of an analog computer transient model is included that provided a basis for creating the dynamic recuperator model Evolution of the Design and Analysis of the GT-MHR Recuperator Introduction The design of the GT-MHR evolved from the design work from the Modular Helium Reactor (MHR). A significant efficiency gain was realized when the MHR was 10

23 coupled directly to a Brayton cycle instead of an indirect steam cycle and this new design was called the Gas Turbine-MHR. The publication report that outlined details of the GT- MHR including details of the recuperator was a 1994 report by General Atomics (GA) (Etzel, Baccaglini, Schwartz, Hillman, & Mathis, 1994). This report provided some details about the recuperator design and location in the GT-MHR PCU, but did not provide the inlet and outlet conditions for the helium streams or dimensions for the recuperator geometry shown in the report. Another report published by GA in 1994 included the temperature, pressure and mass flow rate of the helium at the inlet and outlet of the recuperator for both streams (Zgliczynski, Silady, & Neylan, 1994). In 2000, GA published a report indicating that a Russian partner, OKBM, had been working on design and testing of a recuperator for the GT-MHR (Schleicher, Raffray, & Wong, 2000). Several conference papers concerning the progress of the GT-MHR recuperator were presented by authors from the OKBM at the 2nd International Topical Meeting on High Temperature Reactor Technology. These papers described revised designs of GT-MHR heat exchanger components and results of testing models of those components. Inlet and outlet helium conditions for both streams were revised as was the geometry of the recuperator. Unfortunately the data presented was not remarkably useful and dimensions were not included for the revised components General Atomics Recuperator Design The GA reports published in 1994 provided some useful information about the geometry, inlet and outlet conditions, and design effectiveness of the GT-MHR 11

24 recuperator. A summary of the inlet and outlet conditions for the 1994 design is shown in Table 1 (Zgliczynski, Silady, & Neylan, 1994). An offset strip-fin heat exchanger core in a counter flow arrangement was used, as shown in Figure 5 (Etzel, Baccaglini, Schwartz, Hillman, & Mathis, 1994). The report indicated that the offset strip-fins would have fins with pitches of 1.27 and 0.91mm and fin heights of 1.27 and 1.9mm but does not assign the specified fin geometries to either side of the heat exchanger. Figure 5 shows the height of the fins on the high pressure side as being greater than the height of the fins on the low pressure side, but later modeling showed that the opposite is true. The specified strip-fin dimensions matches well with strip-fin geometries described in a well known compact heat exchanger design reference (London & Kays, 1998). Summary of 1994 GT-MHR Recuperator Inlet and Outlet Conditions at Peak Performance Temperature ( C) Pressure (MPa) Cold Side Inlet (from compressor) Cold Side Outlet (to reactor) Hot Side Inlet (from turbine) Hot Side Outlet (to precooler) Both Sides Helium Mass Flow Rate Design Effectiveness Flow Configuration 239 kg/sec 0.95 counter flow Table 1: Summary of 1994 Recuperator Design Inlet and Outlet Conditions 12

25 Figure 5: Offset Strip-Fin in Counter Flow Arrangement (Etzel, Baccaglini, Schwartz, Hillman, & Mathis, 1994) The strip-fin plates are stacked and brazed together to form a recuperator unit, as shown in Figure 6. While the majority of the flow in this design is counter-flow, near the inlets and outlets the flow geometry is changed to cross-flow to accommodate manifolds. Unfortunately, dimensions were not included with the recuperator unit shown in Figure 6 (Etzel, Baccaglini, Schwartz, Hillman, & Mathis, 1994). According to GA-A21827, six of the units shown in Figure 6 would operate in parallel and would be located in the annulus around the turbo machinery as shown in Figure 7 (Etzel, Baccaglini, Schwartz, Hillman, & Mathis, 1994). Figure 6: 1994 Individual Recuperator Unit Design (Etzel, Baccaglini, Schwartz, Hillman, & Mathis, 1994) 13

26 Figure 7: GT-MHR PCU Configuration in the 1994 Design (Etzel, Baccaglini, Schwartz, Hillman, & Mathis, 1994) General Atomics and OKBM Recuperator Design Several papers concerning the evolution of the GT-MHR design were presented in 2004 at the 2 nd International Topical Meeting on High Temperature Reactor Technology representing what GA considers to be the 2003 design. The two papers of specific interest to the GT-MHR recuperator are (Bikh, Golovko, Dmitrieva, Kamashev, & Krasilschikov, 2004) and (Kostin, Kodochigov, Vasyaev, & Golovko, 2004). Many changes were made to the GT-MHR design between 1994 and 2003 including a reactor power increase from 450MW to 600MW, a helium flow rate increase from 239 kg/sec to 320 kg/sec and a redesigned turbomachinery set operating at 4400 RPM instead of 3600 RPM. Small changes were made to the helium inlet and outlet conditions of the recuperator. A summary of the inlet and outlet conditions for the 2003 design is shown in Table 2 (Bikh, Golovko, Dmitrieva, Kamashev, & Krasilschikov, 2004). 14

27 Summary of 2003 GT-MHR Recuperator Inlet and Outlet Conditions at Peak Performance Temperature ( C) Pressure (MPa) Cold Side Inlet (from compressor) Cold Side Outlet (to reactor) Hot Side Inlet (from turbine) Hot Side Outlet (to precooler) Both Sides Helium Mass Flow Rate Design Effectiveness 320 kg/sec 0.95 Flow Configuration counter flow Table 2: Summary of 2003 Recuperator Design Inlet and Outlet Conditions The 2003 recuperator design indicates two sets of ten recuperator units, for a total of 20 units, run in parallel compared to the previous design with six units run in parallel. The recuperator was divided into a larger number of smaller units to meet the maximum pressure drop requirement, while still fitting into the available space in the PCU vessel. The recuperator unit geometry and location of the units inside the PCU was also revised. The revised design for the individual recuperator units is shown in Figure 8 (Kostin, Kodochigov, Vasyaev, & Golovko, 2004). Although the diagrams and descriptions presented in Figure 8 lack dimensions and are somewhat unclear, they are still the best available in the literature. The component on the left in Figure 8 is a complete recuperator unit. Each unit is comprised of a top header and a bottom header connected by 207 cylindrical heat exchange elements. Each of the cylindrical heat exchange elements contains a strip-fin compact heat exchanger with a cross section shown in the top right of Figure 8. The tops and bottoms of the heat exchange elements are capped with covers that isolate the high pressure helium and direct it into the correct strip-fin volume. The top and bottom 120mm of each cylindrical heat exchange element is 15

28 perforated with holes that direct and isolate the low pressure helium into the other half of the strip-fin heat exchanger volume. Part of the motivation for this design approach was the ability to repair leaking heat exchanger units by plugging individual heat exchange elements. One set of ten recuperator units occupies an annular region above the duct between the reactor and the PCU and the other set occupies an annular region below the duct, as shown in Figure 9 (Kostin, Kodochigov, Vasyaev, & Golovko, 2004). A B helium to reactor helium from turbine A B heat exchange element cover ring with holes cylindrical casing heat exchanger element surface C A A heat exchange element headers shroud C A A helium from HP compressor helium to precooler Figure 8: 2003 Individual Recuperator Unit Design (Kostin, Kodochigov, Vasyaev, & Golovko, 2004) 16

29 Exciter Generator Turbine Recuperator Duct to Reactor LP & HP Compressors Precooler Intercooler Figure 9: GT-MHR PCU Configuration in the 2003 Design (Kostin, Kodochigov, Vasyaev, & Golovko, 2004) At the same conference, another paper was presented describing a different design for the heat exchange surface used inside the cylindrical heat exchange elements. This paper indicated that the cross section of the heat exchange elements, shown in the top right of Figure 8, had been redesigned as shown in Figure 10 (Bikh, Golovko, Dmitrieva, Kamashev, & Krasilschikov, 2004). Dimensions for the configuration shown in Figure 10 are not given, but the paper indicates that the ratio of the free flow area for the high pressure side to the low pressure side is 1 to Experimental results for the heat transfer coefficient and the hydraulic resistance coefficient were also provided, but the results were not reported clearly enough and with sufficient detail to confidently utilize the data in a model (Bikh, Golovko, Dmitrieva, Kamashev, & Krasilschikov, 2004). 17

30 Figure 10: Alternate Cross Section for GT-MHR Heat Exchanger Surface (Bikh, Golovko, Dmitrieva, Kamashev, & Krasilschikov, 2004) 2.3. Kays and London Compact Heat Exchanger Design Data The accurate prediction of steady-state performance and limited dynamic response of heat exchangers, specifically compact heat exchangers, was systematically investigated by London, Kays and other researchers in the Mechanical Engineering Department at Stanford University starting in the late 1940s. A large number of compact heat exchanger cores were tested to quantify the pressure drop and heat transfer characteristics of each core. Because of their systematic and standardized approach to the testing, the data and empirical relations resulting from the studies are still relevant and useful. This data is published in the book 'Compact Heat Exchangers' which was originally published in 1955 and has been republished multiple times, most recently in Strip-fin heat exchange surfaces were originally proposed in the 1994 GT-MHR recuperator design, shown in Figure 5, and were also shown in one of the papers presented about the 2003 design, shown in the top right of Figure 8. An alternate heat 18

31 exchange surface was presented for the 2003 design as shown in Figure 10. It is unclear as to which heat exchange surfaces will be incorporated into the recuperator units, but it is a reasonable assumption that the newer design will be used. Unfortunately the data for the newer design was not presented clearly and completely making it troublesome to reuse. Data for the 1994 strip-fin design was not provided directly by GA, but data for similar, possibly identical, heat exchange surfaces is available in the literature (London & Kays, 1998). Because well documented data is available, the strip-fin surfaces were used as the heat exchange surfaces in the development of the dynamic model. The overall exchanger geometry, as shown in Figure 8, including the application of twenty individual recuperator units in parallel was utilized in the dynamic model. Also, the helium input and output conditions defined for the 2003 design, shown in Table 2, were used. The following sections provide details on the geometry of the heat exchanger surfaces and the data available for those surfaces Selected Heat Exchange Surfaces and Geometric Relationships The strip-fin heat exchange surfaces proposed in the 1994 GT-MHR recuperator design are identical to the style of the strip-fin geometry shown as Core E in Figure 4. The 1994 recuperator design indicated that the offset strip-fins would have fins with pitches of 1.27 and 0.91mm and fin heights of 1.27 and 1.9mm (Etzel, Baccaglini, Schwartz, Hillman, & Mathis, 1994). The uninterrupted flow length is not indicated but the provided geometry matches very well with the fin pitch and height of two strip-fin heat exchanger surfaces included in (London & Kays, 1998). Though speculative, it is 19

32 likely that the data in (London & Kays, 1998) was used for the design of the 1994 recuperator. The two best matching strip-fin heat exchange surfaces included in Kays and London are the 1/ and the 1/ A comparison of the heat exchange surfaces specified in the 1994 design and the two best matching geometries in (London & Kays, 1998) is shown in Table 3. Table 3 repeats each value in metric and English units, because the dimensions are defined in metric in the GA report and in inches in (London & Kays, 1998). Complete dimensions for the two heat exchange surfaces are shown in Table 4 with reference to Figure 11. The hydraulic diameter (r h ), ratio of total transfer area of one side to total exchanger volume (α), and the ratio of the free-flow area to the frontal area of one side (σ) are all derived from the other geometric quantities, but appear frequently enough to warrant explicitly defining them. Equations defining these three quantities are shown in Equation 1, Equation 2, and Equation 3. Equation 4 and provide relations for the volume used by one side of the heat exchanger and by the entire exchanger, respectively. Subscripts (1) and (2) in these equations indicate quantities that are specific to the respective heat exchange surface. These subscripts can be switched to make each equation applicable to either side of the heat exchanger. Heat Exchanger Core Selections Fin Width Fin Pitch Fin Height Uninterrupted Flow Heat Exchanger Core mm/fin inches/fin fins/cm fins/inch mm inches Length 1994 GT-MHR Recuperator Side not specified 1/ in Kays and London mm in 1994 GT-MHR Recuperator Side not specified 1/ in Kays and London mm in Table 3: Comparison of Heat Exchanger Cores 20

33 Value Description Variable Used Units 1/ / plate spacing (height) b 1.910E E-03 meters hydraulic diameter rh 3.025E E-04 meters fin thickness δ 1.020E E-05 meters flow length of uninterrupted fin l 2.800E E-03 meters fin width w 1.053E E-03 meters heat transfer area/volume between plates β 2.830E E+03 (m^2/m^3) fin area/total area 8.570E E-01 unitless ratio of total transfer area of one side to total exchanger volume Strip-Fin Heat Exchanger Surface Geometry α (m^2/m^3) ratio of free-flow area to frontal area of one side σ unitless Table 4: Strip-Fin Heat Exchanger Surface Geometry thickness (δ) Figure 11: Strip-Fin Heat Exchanger Surface Parameters Ac rh = L A where : rh hyraulic radius of one side of exchanger, (m) 2 A total heat transfer area of one side of exchanger, (m ) 2 Ac free-flow area of one side, (m ) L overall flow length of one side of heat exchanger, (m) Equation 1: Definition of Hydraulic Radius 21

34 A c Ar b1β 1r h h1 σ1 = = = A fr LA 1 fr b 1 1+ b2 + 2a where : σ ratio of free-flow to frontal area of one side of exchanger β ratio of total transfer area of one side of exchanger 2 3 to volume between plates of that side ( m /m ) 2 Afr frontal area of one side, (m ) a plate thickness (m) b plate spacing (fin height) (m) Equation 2: Definition of Sigma A σ b1β 1 α1 = = = LA fr 1 rh b 1 1+ b2 + 2a where : α ratio of total transfer area of one side to total exchanger volume m /m Equation 3: Definition of Alpha 2 3 ( ) ( ) V1 = LA fr 1 where : 3 V1 volume of one side of the exchanger (m ) Equation 4: Volume of One Side of Heat Exchanger Core σ A fr total = 1+ 2 = Ac 1 V V V where : 3 Vtotal total volume of heat exchanger (m ) Equation 5: Total Heat Exchanger Core Volume 22

35 Hot Side Strip-Fin Heat Exchange Surface Intermediate Plate Cold Side Strip-Fin Heat Exchange Surface Figure 12: Layered Strip-Fin Surfaces Strip-fin heat exchangers are created by layering individual heat exchange surfaces which are separated by an intermediate plate as shown in Figure 12. Many layers are bonded together to form large monolithic units. Inlet and outlet ducts have to be designed to direct each fluid into the appropriate heat exchange surfaces. Dimensions for the overall geometry of the individual recuperator units are not available in the literature for the 1994 or the 2003 design. This information would be useful, but provided the nominal input and output conditions shown in Table 2 with the heat exchange surfaces specified in Table 2, a heat exchanger can be designed that should behave similarly to the GA designed recuperator. In the process of this design, the overall size and shape of a GT-MHR recuperator unit could be approximated, but the determination of the physical footprint is secondary to the goal of creating a dynamic model. 23

36 Empirical Data for Selected Heat Exchanger Cores In (London & Kays, 1998), 132 different compact heat exchange surfaces were systematically tested and the friction factor and heat transfer relations were quantified for each surface. The friction factor data and the heat transfer data are both presented for each surface as a function of the Reynolds number for a range of flow conditions. The data is reported in graphical and tabular form, which prevented interpretation errors when using this data. The Reynolds number is defined as shown in Equation 6. An example of the available empirical data is shown in Figure 13 (London & Kays, 1998). 4rm h Re = Ac μ where : Re Reynolds number (unitless) Ac rh hydraulic radius (m) m mass flow rate (kg/sec) μ viscosity (kg/m/sec) 2 minimum free flow area (m ) Equation 6: Definition of Reynolds Number (Re) / Data 0.12 Value (unitless) Friction Factor Data Heat Transfer Data (StPr 2/3 ) Reynolds Number Figure 13: Empirical Friction Factor and Heat Transfer Data for 1/

37 To facilitate using this empirical data in the dynamic model, curve fits were generated of the friction factor and heat transfer data, taken from (London & Kays, 1998), for both heat exchange surfaces. The curve fitting toolbox in Matlab was used to test different forms of the curve fit equation. The equation shown in Equation 7 provided good results for the friction factor and the heat transfer data. Table 5 shows the coefficients for each of the data sets and includes the range of Reynolds numbers over which the data set is applicable. Kays and London provide conservative uncertainty values for their test data. The friction factor and heat transfer (StPr2/3) data are both indicated to have an uncertainty of ±5% (London & Kays, 1998). ( Re) ( Re) F = a + c where : b 3 ( Re ) or StPr ( Re) F f Re Reynolds Number a coefficient a from curve fit b coefficient b from curve fit c coefficient d from curve fit Equation 7: Form of Curve Fit Equation Used 2 Heat Exchange Surface Friction Factor Coefficients R 2 Value Reynolds Number Range Heat Transfer (StPr^(2/3)) Data R 2 Value Table 5: Coefficients for Curve Fit Empirical Data for Heat Exchange Surfaces Reynolds Number Range a = a = / b = Re 3000 b = Re 2000 c = c = a = 26.6 a = 1.4 1/ b = Re 3000 b = Re 3000 c = c =

38 2.4. Compact Heat Exchanger Analysis The static and dynamic behavior of a proposed heat exchanger must be analyzed and compared against the design requirements. While it is unlikely that the analysis of any proposed design will exactly match the measured behavior of the physical component, the response of the component should be predictable under conditions of interest within a reasonable margin of error. This analysis is especially important when the response of the component affects the steady-state and transient response of a large nonlinear system like a nuclear reactor. Steady state analysis of the recuperator is used to determine the helium outlet conditions, when the inlet conditions are constant with time. After the inlet conditions have remained constant for enough time, the recuperator reaches equilibrium and the outlet conditions are also constant with time. Nuclear power plants typically operate at full power between outages and the energy conversion system also operates at full power and at steady-state. The GT-MHR recuperator would be operating at steady-state for the majority of the time. The steady-state recuperator performance is optimized to maximize the heat transfer effectiveness and minimize the pressure drop. The limited volume dedicated to the recuperator must be considered in this optimization, but because this volume is not known, the volume of the recuperator was minimized. Materials cost, component reliability, near term fabrication feasibility and other aspects would also be optimized, though these topics were not considered here. Analysis of the helium outlet conditions, while one or more of the inlet conditions are changing with time, is called transient analysis. Both the inlet conditions and the 26

39 outlet conditions are functions of time. While the GT-MHR recuperator will be operating at steady-state most of the time, the reactor and PCU will have planned and unplanned transient events. Planned transient events include reactor start-up, shut-down and power changes. Unplanned transients include, but are not limited to, reactor SCRAMs, loss of coolant accidents and turbine trips. The dynamic behavior of the recuperator affects the response of the other components in the GT-MHR during transient conditions. Because of the safety requirements associated with nuclear power plants, the behavior of the recuperator during transient conditions must be considered during the design of the component. A transient model of the GT-MHR recuperator was created for this project. Methods are presented in (London & Kays, 1998) for analyzing the steady state performance of compact heat exchangers. Results for a limited set of transient conditions are also provided. A summary of their steady state analysis is described in and a summary of the transient analysis methods used in Compact Heat Exchangers and (Cima & London, 1958) is presented in Section Static Analysis London & Kays, 1998, provide a method for predicting the outlet temperature and pressure drop across a compact heat exchanger operating at steady-state. The geometry of the heat exchange surfaces, the overall geometry and the input fluid conditions are specified and the outlet temperatures, pressures and the overall effectiveness of the exchanger can be predicted. This model, however, assumes average fluid properties over the length of the exchanger. This assumption limits the application of this method, 27

40 because results become inaccurate for exchangers with large variations between inlet and outlet temperatures. In addition, the average fluid properties are calculated by making an initial, somewhat arbitrary estimate of the heat transfer effectiveness and pressure drop. Because the method presented is for steady-state conditions, the fluid properties are also constant with time. Section describes the estimates that are made in the process of determining the average fluid properties. Section outlines the method for calculating the outlet temperature for the hot and cold fluid streams in a counter-flow heat exchanger. Section outlines the method for calculating the pressure drop across the heat exchanger. It should be noted that this analysis is greatly simplified when one or both fluids are single phase liquids. Liquids can generally be treated as incompressible and the approximation of constant fluid properties can be quite appropriate. In this analysis, however, both fluids are assumed to be single phase gases Estimation of Average Gas Properties The method presented in (London & Kays, 1998) requires gas properties to be defined before the heat exchanger performance can be evaluated. The intrinsic gas properties that need to be defined are the viscosity, heat capacity, thermal conductivity and density. These properties, along with the mass flow rate of the gas, are used to determine the Reynolds number, the Prandl number, the convective heat transfer coefficient and the friction factor. The intrinsic properties of helium, for example, are functions only of the temperature and pressure of the gas. The values of the intrinsic gas 28

41 properties used in the analysis are determined by averaging the property values at the inlet temperature and pressure and at the outlet temperature and pressure. Because the outlet temperature and pressure are both unknown, the values must be estimated. This method presents two problems. First, it is awkward to be required to estimate the outlet temperature and pressure in the process of calculating the outlet temperature and pressure. The accuracy of the estimate affects the values of the intrinsic and extrinsic gas properties and the final values for the calculated outlet conditions. Second, most properties of gases are nonlinear functions of temperature and pressure. Using one averaged value for a property of a gas over the length of a heat exchanger is appropropriate for steady state analysis of heat exchangers, when the difference between the inlet and outlet conditions is not large. As the heat transfer effectivness increases, however, the properties of the gas are not well represented by this averaging scheme and this method leads to inaccuracies. As an example of an analysis of a gas-to-gas compact heat exchanger, the inlet conditions for both gases are specified and the exchanger heat transfer effectiveness is estimated to be 75%. The specific heats of the hot gas and the cold gas are assumed to be equal. These assumptions allow the outlet temperatures for the hot and cold streams to be estimated. The pressure drop is estimated to be 2% on each side of the heat exchanger. The estimated outlet temperatures and pressures are used to evaluate intrinsic gas properties at the outlet of the heat exchanger. These properties are then used to determine average extrinsic gas properties, which are used in the evaluation of the actual outlet temperatures and pressures of the heat exchanger. 29

42 Determination of Outlet Temperatures Estimated average values for the gas properties are determined for each side of the heat exchanger. The overall geometry and the exchanger surface geometry are also specified and are used to calculate the free flow area and the heat transfer area of each side using the relationships presented in The Reynolds number for each side can be calculated using Equation 6. The friction factor (f) and the StPr 2/3 factor can be determined using the empirical data for the respective heat exchange surface. A convective heat transfer coefficient is calculated for each side of the heat exchanger using Equation 8. The convective heat transfer coefficients must be scaled based on a temperature effectiveness of the heat transfer surface. The temperature effectiveness is calculated using Equation 9 and the fin effectiveness is calculated using Equation 10 (Incropera & DeWitt, 2002). Note, Equation 8, Equation 9, and Equation 10 need to be evaluated for each side of the heat exchanger. h = c μ p 2/3 Pr 4rh 1 StPr Re 2/3 ( ) where : watts h convective heat transfer coefficient 2 mk kj cp heat capacity kg K Pr Prandtl Number μ viscosity (kg/m/sec) rh hydraulic radius (m) 2/3 StPr emprical factor Re Reynolds Number Equation 8: Heat Transfer Coefficient 30

43 Af η0 = 1 ( 1 ηf ) Af + Apr where : η0 overall temperature effectiveness of the heat transfer surface η f fin efficiency 2 l b l( b 2δ ) Af fin surface area (m ), Af = + for strip-fin heat exchange surfaces 2 2 δ fin thickness (m) b plate spacing (fin height) (m) l flow length of uninterrupted fin (m) 2 Apr primary surface area (m ), Apr = l( w δ ) for strip-fin heat exchange surfaces w fin width (m) Equation 9: Temperature Effectiveness of Heat Transfer Area hp tanh H kacr η f = hp H kacr where : η f fin efficiency W h convective heat transfer coefficient 2 m K P perimeter of fin (m), P= ( 2δ + 2 l) for strip fin heat exchange surfaces b δ H adjusted height of fin (m), H = for strip-fin heat exchange surfaces 2 W k conductive heat transfer coefficient of strip-fin material m K 2 A cross sectional area of fin (m ), A = l δ for strip fin heat exchange surfaces cr cr ( ) Equation 10: Fin Effectiveness of Thin Sheet Fins Using an electrical circuit analogy, the thermal circuit for heat transfer from the hot gas to the cold gas is a function of the hot side heat transfer coefficient, the conduction resistance through the wall, and the cold side heat transfer coefficient. This thermal circuit is depicted in Figure 14. In this circuit, temperature is analogous to 31

44 voltage, thermal resistance is analogous to electrical resistance and Joules are analogous to Coulombs. Thermal current with the units Joules per second is analogous to electrical current, defined as Coulombs per second. The thermal resistances in Figure 14 are the inverse of the thermal conductivities and are defined in Equation 11 and Equation 12. Equation 12 can be used to determine the thermal resistance of the cold side convective coefficient by making the appropriate substitutions. T hot ( K) R hot R wall R cold T ( K ) cold Hot Source dq Joules dt second units for R: K watt Cold Sink Figure 14: Circuit Analogy of Thermal Resistances a Rwall = Ah + Ac k 2 where : a plate thickness (m) 2 Ah hot side heat transfer area (m ) 2 Ac cold side heat transfer area (m ) W k thermal conductivity of heat exchanger wall material mk Equation 11: Thermal Resistance of Wall 32

45 1 Rhot = η0, hha h h where : η0, h temperature effectiveness of hot side heat transfer area hh hot side convective heat transfer coefficient A h 2 hot side heat transfer area (m ) Equation 12: Thermal Resistance of Hot Side Convection Term In the analysis of an equivalent electrical circuit, the resistances would be combined into an equivalent resistance and Ohm s Law would be used to calculate the electrical current in the circuit. A similar approach is used in the analysis of this thermal circuit and the thermal resistances combine in the same way as resistances combine in an electrical circuit, as shown in Equation 13. The quantity of interest in this analysis, however, is the inverse of the combined resistance, the overall thermal conductance. Equation 13 is used to calculate the overall thermal conductance for the thermal circuit. 1 1 = = R = R + R + R U A U A h h c c total hot wall cold where : U overall conductance for heat transfer W 2 mk Equation 13: Overall Thermal Conductance The number of exchanger heat transfer units (NTU) is defined as shown in Equation 14. The minimum heat capacity used in Equation 14 is the minimum heat capacity of the two gases as calculated using Equation 15. The subscripts in Equation 15 are for the hot side, but the equation can be used to calculate the capacity rate for the cold 33

46 side with the appropriate substitutions. After calculating the NTU of the heat exchanger, Equation 16 is used to determine the heat exchanger effectiveness for a counter-flow heat exchanger. Equations for effectiveness as a function of NTU for other flow arrangements are available (London & Kays, 1998). The effectiveness calculated is then used in Equation 17, along with the given inlet temperatures, to calculate the outlet temperatures of the two gas streams. A AU h h AU c c 1 NTU = = = UdA Cmin Cmin C min 0 where : NTU number of exchanger heat transfer units C minimum of C and C min h c Equation 14: Definition of NTU Ch = m hcp, h where : kj Ch capacity rate of hot side sec K m h mass flow rate of hot side ( kg/sec) kj cph, heat capacity of hot fluid kg K Equation 15: Definition of Capacity Rate 34

47 C min 1 exp Ntu 1 Cmax ε = C min C min 1 exp Ntu 1 Cmax Cmax where : ε exchanger heat transfer effectiveness C minimum of C and C min C maximum of C and C max h h c c Equation 16: Effectiveness as a Function of NTU for Counter Flow Heat Exchangers ( ) ( ) ( ) ( ) q C t t C t t ε = = = q C t t C t t h h, in h, out c c, out c, in max min hin, cout, min hin, cin, where : ε exchanger heat transfer effectiveness q energy transfer rate realized q t max h, in/ out t c, in / out maximum energy transfer rate hot side inlet/outlet temperature (K) cold side inlet/outlet temperature (K) Equation 17: Definition of Exchanger Heat Transfer Effectiveness Determination of Pressure Drop Estimated average values for the gas properties must be determined for each side of the heat exchanger as described in Section The inlet pressures for each side of the heat exchanger are specified and the pressure drop across each side can be determined using this method. Figure 15 shows a two dimensional pictorial cross section of a heat exchange surface in one side of the heat exchanger. The cross-hatched spaces indicate wall elements where heat would be transferred to or from the gas. The white space between the cross-hatched areas indicates free flow area. The white space between (1) 35

48 and (a) in Figure 15 represents an inlet manifold or plenum region. The white space between (b) and 2 in Figure 15 represents an exit manifold or plenum region. The pressure drop from the inlet on the left to the outlet on the right can be separated into four components. These components will be described in detail in a later chapter, but a qualitative description is provided here. The four components are the entrance effect, flow acceleration, core friction and the exit effect. These four components are shown in Equation 18 which defines the steady-state pressure drop across a heat exchanger. The walls of the flow passages provide surface area for heat transfer to or from the each gas. Friction drag also results from shear stress between the gas and the walls and causes a pressure drop along the length of the heat exchanger. The friction drag is accounted for by the core friction term in Equation 18. Two effects lead to pressure changes at the entrance (a) and exit (b) as shown in Figure 15. First, the free flow area in the core is reduced because of the volume occupied by the heat exchange surfaces. This reduction in flow area causes a pressure drop as the gas enters the core at (a) with a corresponding pressure increase when the gas exits at (b). If the density of the gas does not change between (a) and (b) the pressure changes caused by the change in flow area are equal and cancel. In addition, there are irreversible entrance and exit losses (Kc and Ke) caused by the geometry of the leading and trailing edges of the core at (a) and (b). These four effects are combined into the entrance effect and the exit effect as shown in Equation 18. Finally, there is a pressure change caused by the change in density along the length of the core. This density change is a function of the temperature change of the 36

49 gas as energy is transferred to or from the fluid. This effect is accounted for by the flow acceleration term in Equation 18. FLOW FLOW 1 a b 2 Figure 15: Two dimensional simplification of heat exchanger surface 2 G 2 ρ 1 A ρm 2 ρ 1 Δ P= ( Kc + 1 σ ) f ( 1 σ Ke) 2gcρ1 ρ2 Ac ρ1 ρ2 entrance effect flow acceleration core friction exit effect where : Kc entrance loss coefficient Ke exit loss coefficient m kg G mass flux 2 Ac m sec ρ 1 ρ 2 3 density at (1) (kg/m ) 3 density at (2) (kg/m ) f friction factor Equation 18: Pressure Drop for Steady-State Analysis 37

50 Transient Analysis A paper by Cima & London present some solutions for the transient behavior of a two gas direct transfer counter-flow heat exchanger for use in a gas-turbine application (Cima & London, 1958). This is exactly the type of heat exchanger to be used as the GT- MHR recuperator. The assumptions that Cima & London used are listed below. The temperatures of each of the gases and the wall are only functions of time and distance in the direction of flow. This assumption one dimensionalizes the problem. The heat exchanger is assumed to be adiabatic and no heat is lost from the shell of the exchanger. There is no heat generation inside the heat exchanger. Heat transfer resistances are lumped between the gas volumes and the wall volumes. Thermal conduction parallel to the direction of flow is assumed to be zero. Thermophysical properties of the gases and the wall material are assumed to be constant. The properties do not vary as a function of time, temperature or position. The gases are assumed to be low velocity liquids or gases at essentially constant pressure so that enthalpy can be treated as a function of temperature. The thermal capacity of the shell of the heat exchanger is considered negligible compared to the thermal capacity of the heat transfer surfaces. The mass flow rates of both gases are constant. After making these assumptions, parameters were non-dimensionalized and the heat exchanger was discretized in time and space. A pictorial representation of the discretized heat exchanger is shown in Figure 16. An electrical circuit analogous to the thermal system was created and is shown in Figure 17. The differential equations describing this system are provided in the paper, but because a modified version of the system is used for the GT-MHR model and will be presented later, the equations for this system are not given here. 38

51 Hot helium flow direction Hot Volume 3 Hot Volume 2 Hot Volume 1 Wall Mass 1 Wall Mass 2 Wall Mass 3 Cold Volume 1 Cold Volume 2 Cold Volume 3 Cold helium flow direction Figure 16: Pictorial Representation of Discretized Recuperator C hot_3 C C hot_1 hot_2 R hot_3 R hot_2 R hot_1 ½R wall_1 C C wall_2 C wall_3 wall_1 ½R wall_2 ½R wall_3 ½R wall_1 ½R wall_2 ½R wall_3 R cold_1 R cold_2 R cold_3 C cold_1 C C cold_3 cold_2 Figure 17: Electrical Circuit Analogous to Thermal System The gas volumes shown pictorially in Figure 16 correspond to the capacitance elements C hot_n and C cold_n shown in Figure 17. The wall masses shown pictorially in Figure 16 correspond to the capacitance elements C wall_n shown in Figure 17. The resistances R hot_n and R cold_n shown in Figure 16 represent thermal resistance between the wall and the respective gas volume as defined by Equation 12. The resistances R wall_n represent the heat transfer resistance across the wall as defined by Equation

52 The indices shown on the hot and cold volumes in Figure 16 correctly describe the state of the heat exchanger for one time step. During each time step, energy is transferred from the hot gas to the wall and then to the cold gas. After the time step, the hot volumes will move to the left, the cold volumes will move to the right. Hot volume 3 and cold volume 3 will exit the heat exchanger and new volumes will enter the heat exchanger on the input side for each respective gas. Cima and London actually created an electrical circuit like the one shown in Figure 17 and used it to solve the system response to temperature transients. The circuit model had capacitors on a motorized stage moving into and out of circuit to represent the moving gas volumes. Stationary capacitances represented the thermal capacitance of the wall masses. Charging circuits at the inlets charged the incoming gas volumes to the correct voltage representing the inlet temperatures. This unique method allowed Cima and London to determine the solution to several transient conditions before digital computing resources were readily available Summary of Literature Review The data published describing the design of the GT-MHR recuperator is not complete presumably because of the continued evolution of the design and because of the proprietary nature of the GT-MHR and its components. The GT-MHR design has evolved with at least two major benchmarks, one in 1994 and the other in Published reports for either design revision do not include a transient analysis of the recuperator and do not include sufficient information to exactly model GA's design of the 40

53 recuperator. Information from each design revision was combined to create a set of parameters that represent, as well as possible, the latest design revision that could still be analyzed dynamically. These parameters were used to develop a dynamic model of the GT-MHR recuperator, as presented in later chapters. The input and output conditions used for this model are shown in Table 2. The heat exchange surfaces used in the model are indicated in Table 4. The 2003 design with two sets of ten recuperator units, all in parallel, was used in the model. Because the geometry for each recuperator unit is not specified, the model was iterated until the steady state inlet and outlet conditions shown in Table 2 were achieved. The steady state analysis method described in (London & Kays, 1998) could not be applied directly because a dynamic model was needed. The transient analysis techniques described in (London & Kays, 1998) and (Cima & London, 1958) both included too many simplifications for direct application. The assumption that the thermophysical properties of the gases were constant with time, temperature and location was not appropriate. Also, the assumption that the mass flow rates were constant was counter to the need to have a model that could accommodate mass flow rate transients. The recuperator model that was developed started from an electrical circuit analog as shown in Figure 16. Many of the methods used by (London & Kays, 1998) were modified and incorporated into the dynamic model. The details of the model that was developed will be covered in a following chapter. 41

54 CHAPTER 3 3. RECUPERATOR MODELING IN MATLAB & SIMULINK 3.1. Introduction The initial goal of this project was to create a dynamic model of the GT-MHR PCU that could be combined with an existing dynamic model of the GT-MHR core to have a complete model of the GT-MHR. The existing core model was written in Simulink, so it was appropriate to continue the PCU model with this programming language. When the goals of this project were revised so that only the GT-MHR recuperator was being modeled, work was continued using Simulink to enable future connectivity of the code. Simulink is a graphical programming language that is part of Matlab, both of which exclusively use numerical techniques to solve problems. Matlab is a proprietary language developed by MathWorks though its syntax is very similar to C. Matlab and Simulink are both popular among academic and research institutions. The final recuperator simulation developed for this project was intended to be added to a complete system model of the GT-MHR. For several reasons, the final recuperator model varies slightly from this intention, though it should still be possible to make modifications to the developed model to make it suit its original intended purpose. 42

55 The recuperator model was developed using modified versions of methods used to design and analyze compact heat exchangers, instead of a using a system dynamics approach. Because of this, the resulting simulation is useful for designing counter flow compact heat exchangers, including selecting the heat transfer surfaces, the materials, and the geometry of the overall heat exchanger. After picking the geometry of the heat exchanger, the simulation can be used to determine the steady state and transient behavior of the heat exchanger. The simulation requires the user to define the inlet pressure, temperature and mass flow rate for each side of the heat exchanger as a function of time. The outlet temperature and the pressure drop are determined, as functions of time. Although this is consistent with the design and analysis methods referenced in (London & Kays, 1998), an inconsistency exists, because the user is required to specify the inlet pressure and the inlet mass flow rate, even though the mass flow rate should be determined by specifying the inlet pressure or vica-versa. The code may need to be modified to eliminate this inconsistency, before it is used in a complete system model, but the results predicted by the current revision match well with results used for validation, as will be shown in a following chapter. Several approaches were taken during the development of this model. The final model is a combination of text-based Matlab code that uses multiple Simulink models as functions. This approach was an optimization of the conveniences and limitations of Simulink combined with the transparency of text-based code. The first section of this chapter provides an introduction to Matlab and Simulink, with an emphasis on the relative capabilities that led to the programming scheme in the final model. Section

56 details the assumptions, equations and computational methods used in the transient heat exchanger model. Section 3.4 describes the method and appearance of the data output Introduction to Matlab and Simulink Simulink is a convenient development environment because its graphical nature reduces the typical time investment needed to learn the syntax of a text-based language. Instead of writing lines of alphanumeric code, users drag and drop blocks on a screen and connect those blocks with lines. Each block is essentially a function that has specific inputs, outputs and configurable parameters. The lines that connect blocks, referred to as signals, provide data flow between functions. The blocks can represent simple operations like basic arithmetic or complex features like analog circuit components. Numerous block-sets have been included with Simulink, some of which are very general and some of which are tailored to specific disciplines. For simple models, Simulink automatically handles the data flow between blocks and the order in which blocks are invoked. As models get more complex the data flow and order of operations may need to be managed by the user Differences between Matlab and Simulink In a typical text based programming language, lines of code are organized serially top-to-bottom and it is obvious when variables are written and when data that is stored in those variables can be used. Programs are written in Simulink by arranging and connecting graphical blocks on a two dimensional layered canvas. These programs do not have the inherent top-to-bottom order of text based languages. Simulink handles 44

57 execution and dataflow by assigning an order of operation to each block based on the availability of input data required by each block. This order of operations is called the 'sorted order' in the Simulink documentation. Constants and initial conditions provide a starting location and the ordering progresses to blocks as the input data to those blocks becomes available or is updated. Occasionally the sorted order generated by Simulink needs to be modified by increasing or decreasing the priority of specific blocks or subsystems. As an embedded feature in Matlab, Simulink models can be constructed with varying degrees of interaction with text-based Matlab code. Simulink models can be written as standalone programs that are not dependent on any text-based code. In this case all constants and variables, computational logic, data output, plotting and data acquisition is handled entirely within a Simulink model using the included block-sets. While it is convenient to have an entire program constructed graphically, limitations of Simulink encourage the use of text-based Matlab code in conjunction with the graphical model. Because Matlab and Simulink share the same variable space they work together very well. Variables defined in Matlab can be read and changed by Simulink, and vice versa. If a user has a large number of variables to define, it can be more convenient to define and initialize those variables in a text-based Matlab script (M-file) instead of defining them within Simulink. Also, it is generally easier to make changes, and to conveniently track those changes, using an M-file compared to using Simulink. Sometimes writing a function with a text-based script is more time efficient than completing the same task using Simulink. In this case, the user can embed Matlab code 45

58 into a special Simulink block or the user can directly call an M-file from Simulink. In both of these examples, the program is primarily a Simulink model that takes advantage of convenient features of text-based Matlab code. Simulink models can also be used as functions that are called from a text-based Matlab program. Matlab scripts are used to handle data transfer between functions and to perform some computation. Simulink models are used when creating a function in Simulink is more convenient than writing the same function in a Matlab script. Programming a model in this way combines the convenience of graphical modeling with the transparency of traditional text-based code and overcomes several limitations in Simulink. The limitations and conveniences associated with Simulink are both caused by the abstraction of the programming environment. Because the low-level code and associated syntax are hidden, errors in programming are reduced and the language is earlier to learn. This is a convenience to users because it results in faster model development. Unfortunately, hiding the low-level code reduces the flexibility of the preprogrammed blocks limiting them to be used in the way that the Simulink developers intended. This is an inconvenience that can limit the types of systems that can be modeled and restrict particular programming methods. Variables are recalculated and re-sampled as a Simulink simulation is running. These variables affect the input and output signals connecting blocks and in some cases can be used to change parameters defining the function of blocks. In other cases, the defining parameters of a block cannot be changed dynamically while the Simulink model is running. For example, the capacitance value of the Capacitor block cannot be 46

59 changed dynamically. If the capacitance of an element is non-linear and is a function of other components in the model, Simulink is unable to correctly model the system. This limitation can be overcome by making the capacitance value a variable defined in a Matlab script and repeatedly calling the Simulink model from that script. The value of the capacitance can be updated prior to calling the Simulink model. The Simulink function can be run for a specified amount of time or until the calculated capacitance is outside an accuracy limit. Simulink would then terminate, the Matlab script would recalculate and reassign the capacitance value variable and restart the Simulink function. This allows non-linear electrical circuits to be modeled using the electrical component block-set that normally cannot accommodate non-linear components. While this method seems awkward, it is effective in overcoming several limitations in Simulink. Compared to an equivalent text-based Matlab code, a Simulink model inherently runs slower because of the computational overhead associated with interpreting and compiling the graphical model. Significant reductions in computing time can be gained by reducing the complexity of a Simulink model or by writing equivalent code in a textbased Matlab script. For example, Simulink has a block-set specifically for the analysis of electrical circuits that includes common components like resistors, capacitors, inductors, operational amplifiers, sources, etc. Two advantages of directly using the specialized Simulink blocks are a reduced probability of making mistakes in the development of the model and a reduction in model development time. Unfortunately, the computational time required to solve a Simulink model using these specialized blocks is substantially greater than the computational time required to solve an equivalent 47

60 Simulink model using the differential equations describing the system. Simulink models using the specialized block-sets can be used to validate a fundamental differential equation system model, before the equation model is used as a component in a larger computationally intensive model Parallel Processing in Matlab and Simulink Matlab R2007b includes several implicit and explicit parallel processing features that can be used to reduce computation time. These features are well described in the Matlab help resources (The MathWorks, Inc., ), but a summary is provided here. Both types of parallel processing require the user to have a computer with multiple processors and also require the user to enable multithreading in the Matlab preferences menu. Also, even with multithreading enabled, Matlab will determine if the computational overhead associated with utilizing multiple processors is outweighed by the overall runtime reduction before it will utilize multiple processors. Several matrix operations will automatically run in parallel without any additional code or special syntax. Some functions, sin and log for example, will implicitly run in parallel when performing point-wise operations on arrays or matrices. In addition, basic linear algebra commands, like matrix multiplication and addition, will implicitly run in parallel. Because matrix operations run much faster in Matlab compared to a repetition structure that performs an equivalent operation, computing time can be saved by intentionally using matrix operations whenever possible. The logic to correctly apply matrix operations, however, may be more difficult than an equivalent repetition structure. Operations performed inside a Simulink model, unfortunately, do not run in parallel. 48

61 A for-loop is a common repetition structure in many languages and is also available in Matlab. This repetition structure repeats a programmed set of operations until the for-loop is terminated. In some cases the for-loop can be replaced by one or more matrix operations that implicitly run in parallel. If the for-loop cannot be replaced, it may still be possible to parallelize the operations by explicitly calling a parallel for-loop (parfor) in place of a standard for-loop. The parfor command is part of the parallel computing toolbox that comes with Matlab starting with R2007b. The syntax for this command is the same as a standard for-loop. Unfortunately Simulink models cannot be called as functions from within a par-for loop Dynamic Model Introduction A dynamic model of the GT-MHR recuperator was created using the available design data and modified versions of the methods presented in the Literature Review. The GT-MHR recuperator is a compact heat exchanger using helium as the working fluid. The inlet temperature, pressure and mass flow rate for both inlets of the actual recuperator could vary as a function of time due to changes in other components in the PCU. The recuperator model was written so that all of the inlet parameters can be varied with time. The time dependent outlet temperatures, mass flow rates and pressure drops are determined by the model. The assumptions and simplifications used in the methods presented in the Literature Review prevented direct application of either of those methods in this model. 49

62 Two analysis techniques for compact heat exchanger performance were presented in the Literature Review. The steady-state analysis did not discretize the heat exchanger and assumed that the thermophysical properties of the gases were constant in time and space (London & Kays, 1998). The outlet conditions were arbitrarily estimated in the process of calculating the outlet conditions. The steady-state method could be improved by making a better estimate of the outlet conditions, which is difficult without substantial experience analyzing heat exchangers. Instead, a progressive approach using the calculated outlet conditions to iterate on the estimated outlet values could be used to provide a more accurate solution, without specialized knowledge or experience with heat exchangers. Also, discretizing the heat exchanger would allow the thermophysical properties to be averaged for each volume element, instead of across the entire heat exchanger. This would reduce the error for large NTU gas-to-gas heat exchangers that is caused by large variations in thermophysical properties. The transient analysis method presented in the Literature Review used an electrical circuit analog to model the thermal circuit describing the heat exchanger (Cima & London, 1958). In this analogy, the capacitors represent thermal energy storage elements, resistors represent thermal resistances and voltage represents temperature. In their work, the authors actually constructed an electrical circuit with appropriately sized capacitors and resistors and used a motorized stage to move the capacitors, representing the fluid volumes, through the circuit to represent fluid flow. Additional capacitors, representing wall masses, were statically located between the moving fluid volume capacitors. Fluid volume capacitors were charged to the correct temperature (voltage) 50

63 before entering the circuit and the voltage of the capacitors exiting the circuit were measured to determine the outlet temperature. This was an innovative solution method before digital computing was readily available. There were several drawbacks, however, to this method. First, because the authors physically constructed an electrical system, they were forced to use discrete components even though the capacitance of a gas volume varies continuously as a function of temperature, pressure and the volume size. The electrical analog used discrete capacitors that could not be conveniently changed during an experiment, meaning the thermophysical properties of the gas were fixed and assumed to be constant. Also, this experiment modeled the gas volumes as passing through the heat exchanger at a constant rate. This assumption ignores the compressibility effects of the gas. Modern computing resources can be used to make several improvements to this method. Using a numerical simulation to solve the electrical circuit allows the capacitance values of the capacitors to be recalculated based on thermophysical properties of the gas in each volume after each time step. In addition, the numeric simulation also allows the compressibility effects to be considered. The dynamic model developed for this project uses a combination of the steadystate and the transient analysis methods. The dynamic recuperator model is discretized in time and in space, similar to the transient analysis method. The heat transfer between the fluid volumes and the wall volumes is also solved in a similar way to the transient method. The pressure drop calculation from the steady-state analysis method is applied to each discrete volume at each time step. Unlike both methods in the Literature Review, the thermophysical properties of the helium are recalculated for each volume at each time 51

64 step. Also, the compressibility effects caused by the variable helium density are accounted for in the dynamic model. With this model the average helium temperature, relative pressure, mass and mass flow rate can be determined in each volume and at the outlets. In the context of this report, the hot-side refers to the recuperator volume occupied by the helium stream coming from the turbine then entering the recuperator. The cold-side refers to the recuperator volume occupied by the helium stream coming from the compressor outlet then entering the recuperator. After exiting the recuperator, the hot-side helium enters the pre-cooler and the cold-side helium enters the reactor Assumptions The recuperator was modeled in one dimension and it was assumed to be adiabatic. The thermal mass of the recuperator shell was assumed to be negligible compared to the thermal mass of the heat transfer surfaces. There is no heat generation inside the heat exchanger. It was also assumed that there was no heat conduction in the direction of flow in the wall or in either fluid. The thermophysical properties of the wall masses are assumed to be constant across each discrete wall mass. The helium thermophysical and flow conditions are assumed uniform across each discrete volume. Heat transfer resistances are lumped between helium volumes and wall masses. These assumptions are consistent with the assumptions made in the analysis methods presented in the Literature Review. In addition to these assumptions, the model assumes that the hot and cold heat transfer surfaces and the separating plate are all made from the same material. The model also assumes that the hot and cold heat transfer surfaces are stacked 52

65 in a one-to-one ratio. Finally, it is assumed that the Mach numbers of the flow on each side are small, so that the propagation of changes in inlet pressure and mass flow rate can be treated as if the fluids were incompressible (Munson, Young, & Okiishi, 2006). Several assumptions were made in the analysis methods in the Literature Review that were not made in the development of the dynamic model. In the dynamic model, all of the thermophysical properties of each helium volume are functions of time, temperature and pressure. The thermophysical properties of recuperator wall mass are also functions of temperature and time. The mass flow rates of the helium on each side of the heat exchanger are functions of time and are independent of each other Model Revisions and Development Several programming approaches were attempted in the development of the model. First, one stand-alone Simulink model was developed using the electrical system block-set. This method was initially considered because a complicated electrical circuit could be constructed without needing to directly write code for the system of coupled differential equations. The more complicated electrical analogy would have allowed modeling of heat conduction in the direction of flow. It was originally thought that the capacitance and resistance values of the capacitor and resistor blocks could be modified while Simulink was running. During the development of this model, the realization was made that the values of the electrical component blocks could not be changed programmatically, while the model was running. The values could be changed manually by the user while the Simulink model was paused or programmatically, if the model was stopped and restarted. Because dozens of values would need to be changed after each 53

66 time step, an automated approach had to be used. This limitation ended the development of a stand-alone Simulink model and forced the model to primarily be a text-based Matlab code that called Simulink models as functions. During this first major revision, the subsystems of the stand-alone Simulink model were made into separate functions called from one main M-file. The M-file included code to provide data transfer between each of the functions. The heat transfer model was still a Simulink model using the electrical system block-set. While this model was initially very convenient, it essentially hard-wired the number of volumes used to discretize the heat exchanger. Significant changes had to be made to the Simulink model and to the Matlab code to increase or decrease the number of volumes used in the discretization. It was desired to have the exchanger discretization be scalable, so that computation time for the GT-MHR specific recuperator model could be optimized by the user, and in general so that the code could be used to analyze other heat exchanger designs. In the second major revision, the heat transfer model was simplified and the M- file was revised to call the simplified circuit model from within a repetition structure. This revision allowed the user to define the number of volumes into which the heat exchanger was discretized, just by changing one constant. After revisions were made to the pressure drop and mass transfer models and individual components were verified, testing of the entire model started. In addition to discovering and fixing multiple programming mistakes, it was found that the computational time required to run the model was significant. The simulation was taking on the order of one hundred seconds to calculate response of the heat exchanger for one 54

67 second. In the model, the computational time step is defined by the minimum residence time for a volume of helium, which is dependent on the temperature and pressure of the helium in each volume. Because the residence time is a fundamental characteristic of the geometry of the heat exchanger and the thermophysical properties of the helium, the only way to increase the time step is to increase the size of the volumes used to discretize the heat exchanger. This effectively reduces the accuracy of the model and is an undesirable way to decrease the computation time. Instead, an extensive effort was made to reduce computation time by rewriting code more efficiently and parallelizing operations where possible. The heat transfer model written in Simulink using the electrical circuit blockset was the greatest contributor to the computational time. Rewriting this function using the fundamental differential equations instead of the electrical circuit block-set significantly reduced the computation time. Memory requirements were reduced by being more selective in variable storage and memory access time was reduced by conservative pre-allocation for large arrays. Matrix operations were written to replace repetition structures where possible. These efforts reduced the computational time by a factor of twenty and resulted in the third major revision of the model. Except for some debugging and the addition of several convenient data output and analysis features, the model has not been changed significantly thereafter. The third revision of the code, including the added features, is described in the remainder of this chapter Heat Exchanger Discretization The dynamic model was discretized in space and in time. The free-flow volume and the wall mass on each side of the heat exchanger were discretized into an equal 55

68 number, Φ, of elements. Note, while each side of the heat exchanger was discretized into Φ elements, the inclusion of the inlet and outlet conditions in the same numbering scheme leads to indexing with (Φ+2) elements for the hot and cold sides. Because the free-flow volume on the hot-side is different than the cold-side, the size of the discrete volume elements on the two sides is not equal. The size of the discrete fluid volume elements is determined as shown in Equation 19. The number of volumes, Φ, was left as a user definable parameter, because it controls the computational time and resources required to run the simulation and because it also affects the accuracy of the results. The heat exchanger wall material is also discretized into Φ elements. Because the wall is fixed in space and its density is assumed constant, the wall is discretized on a mass per discrete fluid volume basis. The amount of wall mass contained per discrete fluid volume is determined as shown in Equation 20. Note, Equation 19 and Equation 20 need to be calculated separately for the hot-side and the cold-side of the heat exchanger. Equation 20 was derived assuming the plates and the heat exchange surfaces were produced with the same material. σ AfrL VΦ = x Φ x where : VΦ x σ ratio of free-flow to frontal area of exchanger on side x 2 Afr frontal area on side x (m ) L overall flow length of heat exchanger on side x (m) Φ number of discrete volume elements x either hot side or cold side 3 volume of each discrete fluid volume element on side (m ) Equation 19: Definition of Discrete Fluid Volume Elements 56

69 ( δ) wa ( b) AfrLρ δ b+ w + aw M Φ = x Φ + x where : MΦ wall mass per discrete fluid volume element on side x (kg) 3 ρ density of wall material on side x (kg/m ) δ fin thickness on side x (m) a plate thickness (the same for both sides) (m) b plate spacing (fin height) on side x (m) w fin width on side x (m) x either hot side or cold side Equation 20: Definition of Discrete Wall Mass Elements Defining a discrete time step for this model was difficult, because of a particular situation in which the model would become non-physical. Because of this problem, two separate time steps are defined in this model. The first is a user controlled parameter that essentially determines how often data is stored during the simulation. Changing this parameter changes the resolution of the output data, but has little effect on the required computational time. This parameter was included so that the user could modify the memory requirements of a simulation. The second time step is a function of the calculated residence time of the helium in each fluid volume and cannot be directly defined by the user. This parameter is a function of the geometry of the heat exchanger, the number of discrete volumes Φ, and the thermophysical properties of the helium in each volume. The size of the discrete volumes is fixed (Equation 19) and the mass of helium in each volume is the product of density and volume, as shown in Equation 21. The helium density in each volume is a function of temperature and pressure, which can both vary with time. Because of this, under most operating conditions, the mass of helium in each volume will be different. The mass flow rate can be used in Equation 22 to calculate the amount of time a mass of 57

70 helium exists in a fluid volume, before it has moved downstream into another volume. This amount of time is referred to as the residence time. The helium mass flow rate is assumed to be constant in each discrete volume and Equation 22 shows that the residence time is a function of the helium density and mass flow rate in each fluid volume. Volumes with lower density helium or greater mass flow rates will become 'empty' before volumes with higher density helium and lower mass flow rates. To avoid the nonphysical situation of a fluid volume being 'empty', the minimum residence time, as defined in Equation 23, is used for the fundamental simulation step time. Because the properties of the helium can change at each time step, the minimum residence time is reevaluated at each time step. Table 6 illustrates the progression of time and reevaluation of discrete time steps in this simulation. After the heat exchanger has been discretized in space and in time, the model can be pictorially represented as shown in Figure 18. x ( φ, ) = ρ ( φ, ) for φ= { 2,3,..., Φ+1} m t V t where : Φ x x m ( φ, t) helium mass in volume φ on side x at time t (kg) V x 3 x (, t) density of helium in volume φ on side x at time t (kg/m ) ρ φ Φ 3 volume of each discrete fluid volume element on side x (m ) x either hot side or cold side Equation 21: Helium Mass per Volume 58

71 ρx( φ, tv ) Φ ρ (, ) x x φ t σ AfrL tr ( φ, t) = = for φ= 2,3,..., Φ+1 x m x( φ, t) m x( φ, t) Φ x where : { } t ( φ, t) residence time of helium in volume φ on side x at time t (sec) R x m ( φ, t) helium mass flow rate in volume φ on side x at time t (kg/sec) V x ρ φ 3 x (, t) helium density in volume φ on side x at time t (kg/m ) Φ x 3 volume of each discrete fluid volume element on side x (m ) x either hot side or cold side Equation 22: Definition of Residence Time ρhot ( φ, tv ) Φ ρ (, ) hot cold φ tvφ cold tr min ( t) = min, for φ= 2,3,..., Φ+1 m hot ( φ, t) m cold ( φ, t) where : t R min x { } ( t) minimum residence time for all volumes at time t (sec) m ( φ, t) helium mass flow rate in volume φ on side x at time t (kg/sec) ρ φ V 3 x (, t) helium density in volume φ on side x at time t (kg/m ) Φ x 3 volume of each discrete fluid volume element on side x (m ) Equation 23: Definition of Minimum Residence Time Step Model Time in Step Absolute Model Time Minimum Residence Time Evaluation 0 0 τ 0 = 0 t Rmin (τ 0 ) = t 1 1 t 1 τ 1 = t 1 t Rmin (τ 1 ) = t 2 2 t 2 τ 2 = t 1 + t 2 t Rmin (τ 2 ) = t 3 3 t 3 τ 3 = t 1 + t 2 + t 3 t Rmin (τ 3 ) = t 4 n n t n τ = t t Rmin ( τ n ) = t( n+1) n 1 Table 6: Illustration of Progression of Time Steps n 59

72 Hot helium flow direction Hot Outlet T h (5,t) P h (5,t) ṁ h (5,t) Hot Volume 3 T h (4,t) P h (4,t) ṁ h (4,t) Hot Volume 2 T h (3,t) P h (3,t) ṁ h (3,t) Hot Volume 1 T h (2,t) P h (2,t) ṁ h (2,t) Hot Inlet T h (1,t) P h (1,t) ṁ h (1,t) Wall Mass 1 T w (1,t) Wall Mass 2 T w (2,t) Wall Mass 3 T w (3,t) Cold Inlet Cold Volume 1 Cold Volume 2 Cold Volume 3 Cold Outlet T c (1,t) T c (2,t) T c (3,t) T c (4,t) T c (5,t) P c (1,t) P c (2,t) P c (3,t) P c (4,t) P c (5,t) ṁ c (1,t) ṁ c (2,t) ṁ c (3,t) ṁ c (4,t) ṁ c (5,t) Cold helium flow direction Figure 18: Pictorial Representation of Discretized Heat Exchanger with Φ= User Definable Parameters and Initial Conditions There are several parameters and initial conditions that can be specified or modified that make the simulator very flexible. Assuming the user wants to simulate the behavior of the GT-MHR recuperator, only the initial conditions, inlet conditions and the solver parameters need to be defined. While this model was created to simulate the dynamic response of the GT-MHR recuperator, model parameters can easily be redefined to allow simulation of other geometries, heat exchange surfaces, and working fluids. In addition to simulating the transient behavior of the GT-MHR, this model can also be used as a general design tool for compact heat exchangers. 60

73 Initial Conditions An initial temperature, pressure and mass flow rate need to be provided for each fluid volume and an initial temperature needs to be provided for each wall mass. To determine the initial conditions of the fluid volumes, the user must specify nominal outlet conditions for both sides of the heat exchanger. The user specified inlet conditions at t=0 are used with the nominal outlet conditions to linearly define the initial conditions for all of the fluid volumes on each side according to Equation 24. The user must also specify an initial nominal temperature for the first and last wall mass, and those two temperatures are used to linearly define initial wall temperatures using Equation 24 with the modification that φ={1, 2,, Φ}. While requiring an estimate of the outlet conditions in the process of calculating the actual outlet conditions is awkward, the values selected for the initial nominal outlet conditions do not have to be remarkably accurate. The user defined nominal outlet conditions serve as a starting point for the model to converge on the actual outlet conditions that are based only on the helium inlet conditions and the heat exchanger geometry. When defining the scenario, the inlet conditions should be kept constant, until the recuperator is operating at steady-state to allow the outlet conditions to converge to the actual initial outlet conditions. Assuming the initial nominal outlet conditions are chosen within the bounds of the thermophysical helium data, the outlet conditions should appropriately converge. 61

74 ( 1) g( Φ+ 2) g g( φ) = g(1) ( φ 1 ) Φ+ 1 for φ = 2,3,..., Φ+ 1 where : { ( )} g( φ) T( φ), P( φ), or m ( φ) Φ number of discrete volume elements Equation 24: Definition of Initial Conditions of Fluid Volumes Inlet Conditions Helium inlet conditions are provided by the user to determine the scenario to be simulated. The user enters the temperature, pressure and mass flow rate of both helium streams as a function of time for the entire scenario before the simulation is started. To make this data entry convenient, inlet conditions are entered in a Simulink model graphically. Blocks are used to produce constant, step, ramp, sine, and other common functions. Less common functions are available using Matlab and superposition of signals can be used to produce complicated inputs. Imported data from an external source can also be used to generate a transient input signal. An example of a transient entered using Simulink blocks and function superposition is shown in Figure 19. Figure 19: Example of Transient Input Entry 62

75 Property Lookup The thermophysical properties of the helium in each fluid volume are reevaluated before each time step, based on the final temperatures and pressures that are determined for each volume in the previous time step. Property data for helium was downloaded from the National Institute of Standards and Technology (NIST) website for the thermophysical properties listed in Table 7 (NIST, ). These properties were downloaded for a temperature range of 275K to 1500K with 25K increments and for a pressure range of 0.1MPa to 15MPa with an average increment of 75kPa. The heat exchanger was assumed to be constructed of Inconel-617. The heat capacity and thermal conductivity for Inconel-617 was downloaded from a vendor's website for a temperature range of 20 C to 1000 C at approximately 200 C increments (High Temp Metals, Inc.). The property data is stored in text files that are read into variables each time the program is run. The variables are embedded in Simulink look-up table blocks that provide convenient access to the data, as well as a built-in linear interpolation feature. These look-up blocks can be placed in any Simulink model that requires access to this data. 63

76 Property Units of Downloaded Data Units of Property Used in Model Temperature Kelvin Kelvin Pressure MegaPascals Pascals Density Dynamic Viscosity Heat Capacity (c p ) Thermal Conductivity kg 3 m kg Pascal Seconds= m sec Joules gram Kelvin Joules m sec Kelvin kg 3 m kg Pascal Seconds= m sec kj kg Kelvin W m Kelvin Table 7: Units of Helium Property Data Property Units of Downloaded Data Units of Property Used in Model Temperature Celsius Kelvin Density Heat Capacity Thermal Conductivity kg 3 m Joules kg Celsius Joules m sec Celsius kg 3 m J kg Kelvin W m Kelvin Table 8: Units of Inconel-617 Property Data After each time step the final temperature and pressure in each fluid volume is used to reevaluate density, dynamic viscosity, thermal conductivity and heat capacity of the helium in each volume. These properties are then used to evaluate the helium mass in each volume using Equation 21, the residence time of each volume using Equation 22, the Prandtl number of each volume using Equation 25, and the capacitance of each mass using Equation 26. The geometry of the heat exchanger and the helium mass flow rate 64

77 are used with these parameters to calculate the Reynolds number, friction factor and the overall heat transfer coefficient for each fluid volume according to definitions provided in the Literature Review. cpx, ( φ, t) μx( φ, t) Pr x( φ, t) = for φ= 2,3,..., Φ+1 k ( φ, t) x { } where : Pr x( φ, t) Prandtl number of helium in volume φ on side x at time t kg μx( φ, t) dynamic viscosity of helium in volume φ on side x at time t m sec kj cpx, ( φ, t) heat capacity of helium in volume φ on side x at time t kg K kw kx( φ, t) thermal conductivity of helium in volume φ on side x at time t m K x either hot side, cold side, or wall Equation 25: Definition of Prandtl Number { } Cx( φ, t) = mx( φ, t) cp, x( φ, t) for φ= 2,3,..., Φ+1 where : kj Cx( φ, t) thermal capacity of helium/wall in volume φ on side x at time t K mx ( φ, t) helium/wall mass in volume φ on side x at time t ( kg) kj cpx, ( φ, t) heat capacity of helium/wall in volume φ on side x at time t kg K x either hot side, cold side, or wall Equation 26: Definition of Thermal Capacitance Heat Transfer Model The heat transfer was modeled using an electric circuit analogy similar to the method described in the Literature Review (Cima & London, 1958). In this analogy, temperature is analogous to voltage, electrical current is analogous to Watts and electrical 65

78 resistance is analogous to thermal resistance. The electrical circuit analog of the discrete system shown in Figure 18 is shown in Figure 20. The resistances shown in Figure 20 are defined in the Literature Review and the thermal capacitances are defined as shown in Equation 26. The indices of the properties in Figure 18 and Figure 20 start at one at the inlet and increase in the direction of flow. Because the inlet and outlet conditions are stored in the same matrixes as the fluid volume conditions, the index of the outlet conditions is (Φ+2). Also, the indexing of the wall masses starts at one on the left and increases to Φ on the right. This indexing method reflects the matrix indexing used in the program code and in the equations in this document. Figure 20: Electric Circuit Analog of Thermal System with Φ =3 66

79 Because it was assumed that there is no heat transfer parallel to the flow direction, the thermal circuit shown in Figure 20 can be separated into Φ identical independent circuits as shown in Figure 21. By assuming that the thermophysical properties of the helium volumes and the wall masses are constant during each time step, defined by Equation 23, the solution to this circuit is shown as the coupled differential equations in Equation 27. In Equation 27, the four thermal resistances shown in Figure 21 are combined into two equivalent thermal resistances as shown in The lower integration limit for each time step starts at the end of the previous integration time, as illustrated Table 6. The integration time is the minimum residence time evaluated as shown in Equation 23. The coupled differential equations shown in Equation 27 are solved numerically using the Simulink model in Figure 22. The blue blocks in Figure 22 are all arrays of input values that are evaluated immediately before the integration and remain constant over the integration. The yellow blocks are integration operators. The orange blocks are the output temperature values. The green blocks are math operations. T h (φ,t) R h (φ,t) ½R w (φ,t) T w (φ,t) ½R w (φ,t) R c (φ,t) T c (φ,t) C h (φ,t) C w (φ,t) C c (φ,t) Figure 21: Independent Heat Transfer Circuit 67

80 ( φ, ) hot ( φ, ) R ( φτ ) n 1 Twall t T t Thot ( φ, t) = dt Thot, Chot ( φτ, n 1) + τ, n 1 n 1 hot ( ) ( ) ( ) τ n 1 Twall ( φ, t) Tcold ( φ, t) ( φ, t) = Ccold ( φτ, n 1 ) τ R (, 1) n 1 cold φτn φ= { 2,3,..., Φ+1 }, n= { 1,2,3,... } and : hot/ cold ( φ t) ( φ, t) temperature of wall mass φ ( φ τ ) n 1 ( ) ( ) ( ) n 1 Thot φ, t Twall φ, t Twall φ, t Tcold φ, t Twall ( φ, t) = dt Twall, Cwall ( φτ, n 1) + τ R, n 1, n 1 n 1 hot φτ Rcold φτ T cold where T T R T wall τ τ dt + T ( φτ, ) n 1, temperature of helium in hot/cold volume φ at time t (K) ( ) ( ) cold ( φ τ ) hot/ cold n 1 n 1 hot/ cold n 1 C C at time t (K) φτ, thermal resistance between hot/cold volume φand wall mass φat time τ φτ, temperature of helium in hot/cold volume φat time τ kj ( φτ, ) thermal capacitance of hot volume φat time τ K kj φτ, thermal capacitance of wall mass φat time τ K hot / cold n 1 n 1 ( ) wall n 1 n 1 Equation 27: Coupled Equations for Dynamic Heat Transfer Model n 1 (K) n 1 K Watt R R 1 R 2 where : ( φτ, ) = ( φτ, ) + ( φτ, ) h/ c n 1 hot/ cold n 1 wall n 1 R R ( φ τ ), combined resistance between hot/cold volume φ and center of wall mass φ at time τ hc / n 1 n 1 ( φτ ) hot / cold n 1 ( ), thermal resistance between hot/cold volume φ and wall mass φ at time τ R wall φτ, n 1 thermal conduction resistance across wall mass φat time τ n 1 Equation 28: Combined Thermal Resistance K Watt n 1 K Watt 68

81 TMP _recuperator _hot _He _lump _capacitance (2:end -1) Vh1 1 s x o tmp _recuperator _hot _He _lump _temperature TMP _recuperator _hot _He _lump _temperature (d, 2:end -1) Vh2 TMP _recuperator _wall _element _Rhot (Vh-Vw) / Rhot -1 Gain TMP _recuperator _wall _lump _capacitance Vw*Cw1 1 s x o tmp _recuperator _wall _lump _temperature Vw2 TMP _recuperator _wall _lump _temperature (d, 1:end ) TMP _recuperator _wall _element _Rcold (Vc-Vw) / Rcold 1 TMP _recuperator _cold _He_lump _capacitance (2:end -1) Vc1 1 s x o tmp _recuperator _cold _He_lump _temperature TMP _recuperator _cold _He_lump _temperature (d, 2:end -1) Vc2 Figure 22: Simulink Model of Coupled Equations for Heat Transfer Model The heat transfer model assumes the mass in each fluid volume is temporarily stationary, but with the appropriate heat transfer coefficients, as if the fluid volumes were moving. Because of this, the final temperatures of the fluid volumes that are determined by solving Equation 27 are not the actual temperatures for the fluid elements at the end of the time step. The movement of helium from upstream to downstream during the time step must be considered in the calculation of the temperatures in each volume at the end of the time step. Modeling of the mass transport process and determination of the final helium temperatures is discussed in a following section. Because wall elements cannot move, the final temperatures determined by Equation 27 are the final temperature for those masses. 69

82 Pressure Drop Model The pressure drop model uses a modified version of the pressure drop calculation presented in the Literature Review (London & Kays, 1998). This method had two main problems. First, the heat exchanger was not discretized, meaning the fluid temperature, pressure and corresponding thermophysical properties were averaged over the entire length of the heat exchanger. For large NTU heat exchangers, this approximation leads to inaccuracies. Second, the solution method required an arbitrary estimation of the outlet conditions before the fluid properties could be averaged. The averaged fluid properties were then used to calculate the actual outlet conditions. Requiring an estimate of the outlet conditions in the process of calculating the outlet conditions was awkward. The pressure drop model developed for this simulation improves on the method described in the Literature Review by discretizing the heat exchanger and by taking advantage of the inherent iterative capability of numeric computing. The discretization scheme, shown in Figure 18 applied to the one dimensional heat exchanger representation shown in the Literature Review, leads to the representation shown in Figure 23 for one side of the recuperator. In this model, the pressure drop that is associated with each fluid volume is calculated at each time step using updated fluid properties. 70

83 Figure 23: Discretized Heat Exchanger Surface with Φ=3 (x denotes hot or cold side) The pressure drop across a heat exchanger is analogous to a voltage drop across a resistor. In this analogy, pressure is analogous to voltage, mass flow rate is analogous to current and hydraulic resistance is analogous to electrical resistance. A designer typically specifies a nominal mass flow rate, an inlet pressure, and an acceptable pressure drop and then specifies the geometry and heat exchange surfaces based on an appropriate hydraulic resistance according to an Ohm's Law analogy shown in Equation 29. Iterations of this process may be required to optimize efficiency, cost and other characteristics. This analogy illustrates two important points. First, the hydraulic resistance is a function of several parameters that vary non-linearly as a function of the helium properties and flow conditions. Because of the complexity of the hydraulic resistance parameters, they are usually treated with empirical relations resulting from test data. A significant amount of test data is available in the literature (London & Kays, 1998), but extrapolating this data beyond the test conditions or making inferences on other designs may be risky. 71

84 P P = m R m T P in out (,,, geometry, etc) Equation 29: Ohm's Law Analogy to Hydraulic Resistance Second, the mass flow rate through the heat exchanger is a function of the pressure differential across the component. The outlet pressures in the GT-MHR recuperator are non-zero and the value of this backpressure is a time dependent function of other components in the system. The backpressure will affect the mass flow rates through the recuperator and an analysis of the GT-MHR PCU would not be complete without analyzing those transient effects. In the design of this model, the nominal pressure drop, mass flow rate and temperature efficiency at steady state were used as design points for determining the geometry of a heat exchanger with the appropriate hydraulic resistance, as shown in Equation 29. After the geometry of the heat exchanger was fixed, the hydraulic resistance is a determined quantity and the pressure differential and the mass flow rate cannot both be specified as this would over constrain the system. In this model, the mass flow rate, inlet temperature and inlet pressure are specified by the user defined scenario and the pressure differential and outlet temperature are determined. The specification of the inlet pressure is only useful for evaluating the thermophysical helium properties and has less of an effect on the pressure drop compared to the mass flow rate and temperature change. Also, in this model the recuperator was analyzed as an isolated component ignoring any coupling with the balance of plant. The inlet conditions are specified and the outlet conditions are determined as a function of the recuperator performance only. The rest of this section describes the pressure drop model. 72

85 The pressure drop equation presented in the Literature Review can be broken into four components as shown in Equation 30, Equation 31, Equation 32, and Equation 33. Equation 30 defines a pressure change associated with the fluid entering the heat exchanger. Equation 31 defines a pressure loss caused by flow acceleration due to the density change in the fluid. Equation 32 defines a pressure loss caused by friction between the fluid and the heat exchanger walls. Equation 33 defines a pressure loss associated with the fluid exiting the heat exchanger. The total pressure drop across each fluid volume is a sum of the terms appropriate for the volume, as defined in Equation m ( φτ, ) A c Δ Pentrance = K 1 2 c + 2 Acρφτ (, ) A fr where : kg m ( φτ, ) mass flow rate in volume φ sec kg ρφτ (, ) helium density in volume φ 3 m Kc entrance effect 2 A free flow area (m ) c Afr 2 frontal area (m ) Equation 30: Pressure Change due to Entrance 2 m ( φτ, ) ρφ ( 1, τ) Δ Pacceleration = 1 2 Ac ρφ ( 1, τ) ρφ ( + 1, τ) Equation 31: Pressure Change due to Flow Acceleration 73

86 2 m ( φτ, ) A( φτ, ) Δ Pfriction = f( φτ, ) 2 2 Ac Acρ ( φτ), where : f ( φ, τ) friction factor in volume φ A 2 friction surface area (m ) Equation 32: Pressure Change due to Friction Drag 2 2 m ( φτ, ) A c Δ Pexit = K 1 2 e + 2 Acρφτ (, ) A fr where : K e exit effect Equation 33: Pressure Change due to Exit Δ Pentrance ( τ) φ = 1 Δ P( φτ, ) = Δ Pacceleration( φτ, ) +ΔPfriction( φτ, ) 2 φ Φ Δ Pexit ( τ) φ = Φ+ 2 ( ) Equation 34: Definition of Pressure Change per Fluid Volume Entrance and Exit Effects The pressure changes caused by the entrance effect and the exit effect are very similar and they are discussed together with differences indicated where necessary. The pressure changes at the entrance and exit of the heat exchanger core are a combination of two effects. First, at the entrance there is a pressure drop caused by the reduction in free flow area and the associated contraction of the gas. At the exit there is a pressure rise caused by the increase in free flow area and an expansion of the gas. This effect is independent of friction and the transitional geometry associated with the flow area and is 74

87 only dependent on the flow conditions and the ratio of the flow area change. If the values of K c and K e in Equation 30 and Equation 33 are both set to zero, this effect is isolated and for constant density and constant mass flow rate, the pressure drop at the entrance is equal to the pressure rise at the exit. The second effect is caused by the flow disturbance associated with the inability of fluids to make sharp turns at the inlet and outlet of the heat exchanger core. At the entrance, the abrupt direction change causes flow separation and a vena contracta region can develop causing some of the kinetic energy to be dissipated (Incropera & DeWitt, 2002). Flow disturbances at the exit caused by the sudden expansion also cause some of the kinetic energy to be dissipated. The losses associated with this effect are included in the K c and K e parameters in Equation 30 and Equation 33. These parameters are a function of the geometry of the inlet and outlet regions of the exchanger core and the Reynolds number of the fluid. Figure 24 shows a plot of empirical data quantifying these parameters for a particular heat exchanger core geometry (London & Kays, 1998). This reference also indicates that for strip fin heat exchange surfaces, the K c and K e parameters should be taken for Reynolds number equal to infinity. Using Figure 24, for the 1/ heat exchange surface K c =0.35 and K e =0.37, and for the 1/ heat exchange surface K c =0.34 and K e =

88 σ = A / c A fr Figure 24: Entrance and Exit Pressure-Loss Coefficients (London & Kays, 1998) Flow Acceleration The mass flow rate across each discrete volume is assumed constant in this model. Because the temperature and pressure of the helium are changing along the length of the exchanger, the helium density also varies along the length of the exchanger. The velocity of the flow must increase or decrease to compensate for the density change and maintain constant mass flow rate, as shown in Equation 35. The pressure change associated with this term is calculated as shown in Equation 31. The helium densities in the immediate 76

89 upstream and immediate downstream volumes are used in Equation 31 to calculate the flow acceleration pressure drop each volume. x n n c n where : { } m ( φτ, ) = ρφτ (, ) A V ( φτ, ) = constant for φ= 2,3,..., Φ+1 ( ) m ( φτ, ) helium mass flow rate in volume φon side x at time τ kg/sec x n n ρφτ (, ) helium density in volume φon side x at time τ A c n 3 n ( kg/m ) 2 cross sectional area of flow channel (m ) ( ) V ( φτ, ) helium flow velocity in volume φon side x at time τ m/sec n Equation 35: Definition of Mass Flow Rate n Core Friction Core friction is a result of the shear stresses between the helium and the heat exchanger walls. The heat exchanger wall surface area provides contact area for heat to transfer to and from the fluid. Because the fluid is flowing and shear stress develops between the wall and the fluid, this same surface is referred to as friction surface area. The friction factor term, which is part of Equation 32, is a complicated function of fluid properties, flow conditions and the geometry of the heat exchanger surface. Because of the complexity, the friction factor cannot be determined analytically and experimentally determined empirical relations are used to calculate this term, as described in the Literature Review. The pressure drop associated with the core friction is a function of the mass flow rate, fluid properties and the geometry of the heat exchanger, as shown in Equation 32. The form of this function that was presented in the literature assumed the fluid was traversing the entire length of the heat exchanger. In that case, each 77

90 infinitesimal cross section of the fluid had the opportunity to contact the entire heat exchange surface area. This surface area is a product of the perimeter of the heat exchange surface and the overall flow length of the heat exchanger as shown in Equation 36. The form shown in Equation 36 has been modified to account for the spatial discretization of the heat exchanger, as shown in Equation 37. A= P L A where : 2 A friction surface area (heat transfer area) (m ) P A perimeter of flow channel cross section (m) L flow length of entire heat exchanger (m) Equation 36: Calculation of Friction Surface Area for Entire Exchanger Flow Length PA L x Ax = Φ where : 2 Ax friction surface area per discrete volume on side x (heat transfer area) (m ) P Ax perimeter of flow channel cross section on side x (m) L flow length of entire heat exchanger (m) Φ number of discrete volume elements Equation 37: Discretized Calculation of Friction Surface Area Determination of Values at End of Time Step The heat transfer and pressure drop models consider the helium mass in each discrete volume to be stationary for the duration of each time step. Fluid properties are evaluated with the correct flow conditions, but it is assumed that no mass enters or leaves 78

91 each control volume during a time step. After the pressure drop and heat transfer calculations have been performed, appropriate mass is transferred between volumes and the final helium temperatures, pressures and mass flow rates for the time step are calculated. These final values become the initial values for the next time step. The helium mass in each volume at the end of a time step is calculated as shown in Equation 38. ( φτ, + 1) = ( φτ, ) + (( φ 1 ), τ ) min( τ ) ( φτ, ) min( τ ) φ= { 2,3,..., Φ+1 } and : mx( φτ, n) helium mass in volume φon side x at time τn ( kg) m ( φτ, ) helium mass flow rate in volume φon side x at time τ ( kg/sec) m m m t m t x n x n x n R n x n R n where x n ( τ ) minimum residence time of all volumes at time τ (seconds) t R min n n Equation 38: Calculation of Mass in Discrete Volumes n The final temperature of the helium in each volume is a function of the energy stored in the helium volume after the heat transfer calculations, plus the energy entering with the upstream helium minus the energy exiting with the helium going downstream. If it is assumed that the helium quantities being transferred have equal specific heats, the final temperature in each volume is calculated as shown in Equation 39. T Tx( φτ, n) mx( φτ, n) trmin ( τn) m x( φτ, n) + Tx( ( φ 1 ), τ ) n trmin ( τn) mx( ( φ 1 ), τn) ( φτ, n+ 1) = mx( φτ, n) + m x( ( φ 1 ), τn) trmin ( τn) m x( φτ, n) trmin ( τn) φ= { 2,3,..., Φ+1 } and : T ( φτ, ) helium temperature in volume φon side x at time ( K) x where x n τ n Equation 39: Mass Averaging of Temperatures 79

92 Because it is assumed that the Mach number of the flow on both sides of the heat exchanger are small, changes in the inlet mass flow rates and pressures are assumed to propagate across the length of the heat exchanger instantly (Todreas & Kazimi, 1990). Mach number is defined as shown in Equation 40 and for this assumption to be valid, the Mach number of all of the fluid volumes should always be less than 0.3 and preferable much less than 0.3 (Munson, Young, & Okiishi, 2006). V ( φ, τ n) Max( φτ, n) = Vs( φτ, n) Max n n V ( φτ, ) helium flow velocity in volume φon side at time τ m/sec V ( φτ, ) velocity of sound in volum s ( φτ, ) Mach number of flow in volume φon side x at time τ n x n ( ) e φ on side x at time τ ( m/sec) n Equation 40: Definition of Mach Number n With this assumption, the mass flow rate in each volume is determined by the mass flow rate at the inlet as shown in Equation 41. The pressure in each volume is calculated by summing the pressure changes of all upstream volumes and subtracting the result from the inlet pressure, as shown in Equation 42. m ( φτ, ) = m ( 1, τ ) φ= { 2,3,..., Φ+1 } and : m x( φτn) φ x τn ( m ( 1, τ ) inlet helium mass flow rate on side x at time τ ( kg/sec) x n x n where, helium mass flow rate in volume on side at time kg/sec x n n Equation 41: Propagation of Mass Flow Rate Changes ) 80

93 ( ) ( ) P φτ, = P 1, τ ΔP( φτ, ) x n x n x n 2 where P φ φ= { 2,3,..., Φ+1 } and : ( φτ ) φ x τ (, helium pressure in volume on side at time Pa x n n Equation 42: Determination of Volume Pressure ) Error Reporting Several error reporting features were added to improve the convenience of the code and make the user aware of possible errors. The property data for the helium used in the model is valid between 275K to 1500K and 0.1MPa to 15MPa. Simulink look-up blocks were used to interpolate property values according to the temperature and pressure inputs. Unfortunately, these look-up blocks do not display a warning when the input values exceed the range of the table data. Instead, the look-up blocks can be set to extrapolate property values or use the last table value when the range is exceeded. Neither of these options was appropriate and an error checking and reporting feature was written to track these events. This feature was expanded to track and report other quantities that could exceed the valid range values. The temperature and pressure of the helium could go beyond the table values in two ways. First, the user could accidentally enter an inlet transient scenario that included temperatures or pressures that are outside of the helium thermophysical data. Because the simulation can take many hours to complete, it would be inconvenient to become aware of incorrect inlet conditions after the simulation had finished. To prevent this, the user-entered transient conditions are checked to ensure all of the values are within the helium property bounds, before the start of the computationally intensive part of the 81

94 simulation. If any of the values are outside the bounds, the simulation terminates and prompts the user to make them aware of the problem. The helium pressure could go below the table values, even though the inlet conditions are correct. In this case the simulation does not terminate, but an error, with an appropriate description and time stamp, is written to a variable that is included with the data output. An example of an error log is shown in Figure 25. The temperature range of the Inconel-617 wall material is similarly monitored. The empirical friction factor and heat transfer data for both heat exchange surfaces were measured for flow conditions within a specified range of Reynolds numbers. Extrapolation of this data outside of the experimental flow regime is not appropriate and the Reynolds numbers of the fluid volumes are similarly monitored. Although unlikely, a condition could exist during the mass transfer calculation where the helium mass to be transferred to a downstream volume exceeds the mass of helium available in the neighboring upstream volume. This condition could not exist in reality, but is a consequence associated with the discretization of the heat exchanger and the inability of the code to handle this situation. Finally, the maximum Mach number from all of the fluid volumes during each integration is tracked to ensure it does not exceed a value of 0.3. Errors generated when the Mach number exceeds 0.3 do not cause the code to terminate. However the errors are logged and reported in the Excel output file in the error log worksheet with appropriate error descriptions. 82

95 error description accumulated simulation time (sec) Cold He lump pressure out of bounds Cold He lump pressure out of bounds Cold He lump pressure out of bounds Cold He lump pressure out of bounds Figure 25: Example of Error Log File 3.4. Data Output A large amount of data is generated during the simulation, depending on the length of the scenario being modeled. Effective storage and presentation of this data is important for the model to be useful. The goals of the data output scheme were to display the data graphically and store the data numerically in a way that was convenient for the user and for the programmer. This included making the output automatically scale to the number of discrete fluid volumes chosen by the user. Several approaches were attempted in the process of developing a method of data output. First, the Graphical User Interface (GUI) builder tool in Matlab was used to create an interface that was similar to Windows-like menu driven interface that are commonly developed with Visual Basic. This would have been a very nice interface for the user, but the amount of time needed to learn how to write the code made this impractical from a programming perspective. Also, if this simulation is used as a component of a larger simulation in the future, development of a GUI specific to the recuperator would be wasted effort. This simulation was not intended for distribution so the plan to develop a GUI was abandoned. Plotting tools in Simulink were also explored, but the data presentation in Simulink was cumbersome, difficult to scale, and data storage was inconvenient. 83

96 Next, standard plot commands were used in the M-file to plot important quantities such as inlet and outlet temperatures, mass flow rates and pressures as a function of time. In addition, the temperatures of the discrete fluid volumes and the wall masses were also plotted. This method was sufficient during code development, but it suffered several disadvantages when it was used to analyze and store data. The plotting features in Matlab are not very user friendly and can be difficult to use especially compared to the convenience of manipulating plots in MS Excel. Making the plotting commands scalable to different numbers of discretized volumes was easy with this method. However, it was cumbersome to store and retrieve data with this method. The data was kept in memory as variables in Matlab, but if those variables were cleared the data was erased. The data could be saved to text files, but formatting and retrieving data from these files was difficult. In the process of looking for a convenient way to store data, a Matlab command was found that led to the final data output method. Matlab has a command, 'xlswrite,' that will write data to pre-existing MS Excel files. There are several quirks with this command, but overall it is remarkably useful and convenient. The command will not create an excel file, if the specified file does not exist, meaning the command can only write to existing Excel files. Any type of data can be written with this command, including strings and numeric data. When calling this command, the user specifies the worksheet and the column and row of the cell to start writing data. Variables can be stored in different worksheets making the organization of the data very easy. This also means that this data storage method is inherently scalable. Cell contents in the existing Excel file that are not specifically overwritten by the 84

97 'xlswrite' command are not disturbed by this process. In the case of this simulation, a template Excel file was created with all the appropriate worksheets that also included column and row title information, which is not overwritten when Matlab writes to the file. In addition, plots have been pre-configured in separate worksheets to use the data that Matlab writes to the file. After each simulation completes, all of the data is written to one MS Excel file and when the file is opened, plots and other analyses are automatically generated for the user. Excel is used for all graphical presentation, data manipulation, and data storage. In addition to storing the data, the entire M-file, the input scenario and the error log from the simulation are all written to the same Excel file. This method is convenient, because the file can be viewed without Matlab, all of the information necessary to repeat the simulation is available in one output file, and all of the convenient tools, which are available in Excel, can be used for data manipulation and presentation. 85

98 CHAPTER 4 4. VALIDATION 4.1. Introduction The steady state and transient behavior predicted by the simulation needed to be validated before the results of the simulation could be considered legitimate. The steady state and transient behavior predicted by the model was validated against analytical results and results available in the literature. In the process of this validation, some errors and limitations of the simulation were discovered and were either fixed or appropriately documented. In the development of the simulation it was assumed that the flow on both sides would have a small enough Mach number for the flows to be considered incompressible. Because of this assumption, changes to the inlet mass flow rates and the inlet pressures propagate instantly through the heat exchanger. Inspection of several output files indicated that the propagation of changes in mass flow rate and inlet pressure was occurring correctly and an additional formal validation of these two processes was considered unnecessary. The remaining results that needed to be validated were the steady state temperature response, transient temperature response, steady-state pressure drop, and transient pressure drop. Either analytical or published results were available to 86

99 validate all of these conditions except for the transient pressure drop response. Validation of the pressure drop during transient conditions is left for future work. In addition to validating the predicted results of the simulation, the stability of the convergence of the simulation with respect to the user determined nominal initial values was also investigated. Most of the published results that were used for validation were reported in a nondimensional form. Reporting results non-dimensionally is generally useful, because the data is more easily applied by future users. Application of the non-dimensionalized data is still difficult, because real heat exchangers have actual dimensions and the output values have actual units. The inputs to this simulation include the specific dimensions of the heat exchange surfaces, the overall dimensions of the heat exchanger and all of the inlet flow conditions, with units. During validation, heat exchangers had to be designed that met the constraints of the non-dimensionalized validation data before simulations could be run. After the heat exchanger was designed, transient conditions were applied to the exchanger and the results were compared against the corresponding reported results. The specifics of each heat exchanger used for each validation step are included for reference. In all of the results presented in this chapter, the integration time was a function of the minimum residence time of the helium masses in the discrete fluid volumes. Increasing the number of discrete volumes decreased the NTU per volume and increased the spatial resolution. Increasing the number of discrete volumes also decreased the average helium mass per volume. Because of this, the time discretization was a function 87

100 of the space discretization of the heat exchanger. Error is introduced by discretization in time and in space, but because these quantities are related in this model, the presented results cannot be used to separate the contributions. Separating the error caused by spatial discretization from error caused by time discretization is left for future work Steady State Temperature Response The steady state temperature response of a heat exchanger is a function of the NTU of the exchanger, the inlet conditions of the gas streams and the flow arrangement. The ideal steady state hot and cold outlet temperatures are calculated as shown in Equation 43 and Equation 44, respectively. This simulation was designed only to analyze counter flow heat exchangers and the exchanger effectiveness in counter flow is calculated as shown in Equation 45. C T = ε ( T T ) T where : min h, out h, in c, in h, in Ch T T T h, out hin, cin, steady state hot outlet temperature (K) steady state hot inlet temperature (K) steady state cold inlet temperature (K) ε heat exchanger effectiveness C C h c min capacity rate of hot fluid m capacity rate of cold fluid m C minimum of C and C c h ( h cp, h), ( Watts/K ) ( c cp, c), ( Watts/K) Equation 43: Ideal Steady State Hot Side Outlet Temperature 88

101 C T = ε T T + T ( ) min cout, hin, cin, cin, Cc where : T T T hin, cout, cin, steady state hot inlet temperature (K) steady state cold outlet temperature (K) steady state cold inlet temperature (K) Equation 44: Ideal Steady State Cold Side Outlet Temperature 1 e ε = C min 1 e Cmax where : min Cmin NTU 1 Cmax Cmin NTU 1 Cmax ε heat exchanger effectiveness (unitless) ( avg Cmin ) NTU number of exchanger heat transfer units AU / (unitless) C max minimum of C and C ( Watts/K) ( ) C maximum of C and C Watts/K c c h h Equation 45: Definition of Heat Exchanger Effectiveness for a Counter Flow Heat Exchanger For validation purposes, the steady state hot and cold outlet temperatures were calculated for heat exchangers with several different NTU and two capacitance rate ratios using Equations 1, 2 and 3. Next, heat exchangers were designed with appropriate geometries and flow conditions that resulted in NTUs and capacitance rate ratios that matched the analytical calculations. The heat exchanger configurations and the ideal outlet temperatures, as calculated using Equation 43, Equation 44, and Equation 45 are shown in Table 9. Heat exchangers, with the configurations shown in Table 9, were divided into 2, 4, 6, 8, 10 and 20 discrete volumes and were simulated until the output 89

102 reached steady state. The error in the hot and cold outlet temperatures was calculated for each simulation, and then averaged. The results from these twenty-four simulations are shown in Figure 26. Configuration Heat Exchanger Geometry & Materials Heat Exchange Surface Frontal Area (m2) Flow Temperature Material Length (m) (K) Helium Inlet Conditions Mass Flow Pressure Rate (kg/sec) (Mpa) Exchanger NTU Cmin/Cmax Validation Outlet Temperature (K) Hot Side 1/ Inconel Cold Side 1/ Hot Side 1/ Inconel Cold Side 1/ Hot Side 1/ Inconel Cold Side 1/ Hot Side 1/ Inconel Cold Side 1/ Table 9: Heat Exchanger Configuration and Ideal Outlet Temperature for Steady State Validation Averaged Steady-State Error in Hot and Cold Outlet Temperatures Average Error in Steady-State Outlet Temperature (%) 20% 18% 16% 14% 12% 10% 8% 6% 4% Configuration 1: NTU = 4, Cmin/Cmax = 1 Configuration 2: NTU = 8, Cmin/Cmax = 1 Configuration 3: NTU = 4, Cmin/Cmax = 0.5 Configuration 4: NTU = 8, Cmin/Cmax = 0.5 2% 0% Discrete Volumes per NTU Figure 26: Steady-State Error in Hot and Cold Outlet Temperatures 90

103 Figure 26 shows that as the discrete volume per NTU increases, the steady state error decreases. The integration time for the heat transfer calculations is determined by the minimum residence time of any of the fluid volumes in the heat exchanger. By dividing the heat exchanger into a larger number of discrete volumes, the integration time of the simulation is reduced. Increasing the discrete volumes per NTU improves the resolution in time and in space and improves the accuracy of the simulation results, as is shown in Figure 29. The steady state temperature response, for both mass flow rate conditions, behaves as expected. A predictable consequence of decreasing the integration time and increasing the spatial resolution is an increase in the computation time required for the simulation to complete. Figure 27 shows the computation time required for a desired steady state error. Influence of Steady State Error on Computational Time Average Error in Steady State Outlet Temperature (%) 20% 18% 16% 14% 12% 10% 8% 6% 4% 2% 0% Configuration 1: NTU = 4, Cmin/Cmax = 1 Configuration 2: NTU = 8, Cmin/Cmax = 1 Configuration 3: NTU = 4, Cmin/Cmax = 0.5 Configuration 4: NTU = 8, Cmin/Cmax = Computation Time (seconds) Model Run Time (seconds) Figure 27: Influence of Steady State Error on Computational Time 91

104 4.3. Steady State Pressure Drop The simulation results for the pressure drop at steady state were compared against the results of the pressure drop calculations as outlined in the Literature Review. Because of the coarseness of the analytical calculation used, it was not expected that the simulation results would match exactly. It was expected that the simulation results would match approximately and show convergence to a solution with increased numbers of discrete elements. The simulation uses the same method as the analytical calculation to determine the pressure drop, except the simulation discretizes the heat exchanger into multiple volumes and reevaluates the properties of the helium in each volume. The simulation result is probably a better calculation of the actual pressure drop, but results were not found to validate this claim. In the analytical calculations, the heat exchangers were not discretized and the outlet temperatures and pressures determined by the simulation results were used to calculate the necessary helium outlet properties in the validation calculation. While this process is awkward, this is very similar to the method presented in (London & Kays, 1998). The heat exchanger configurations and the associated steady state inlet and outlet conditions used for this validation are shown in Table

105 Exchanger NTU Side Heat Exchanger Geometry Heat Exchange Surface Frontal Area (m2) Flow Length (m) Steady State Helium Inlet Conditions Temperature (K) Pressure (Mpa) Steady State Helium Outlet Conditions Temperature (K) Pressure (Mpa) Mass Flow Rate (kg/sec) Calculated Pressure Drop (MPa) Hot 1/ Cold 1/ Hot 1/ Cold 1/ Hot 1/ Cold 1/ Hot 1/ Cold 1/ Table 10: Heat Exchanger Configuration and Calculated Outlet Pressures for Steady State Validation The steady state pressure drops, as determined by the simulations, are compared to the calculated values in Table 11. As was expected, the pressure drops determined by the simulations were only approximately equal to the calculated pressure drops. Also as expected, the simulation results converge toward a solution with increasing numbers of discrete volumes. Both of these results are shown in Table 11 and in Figure 28. The simulated results are always within 10% of the calculated results and the simulated results converge toward a value with increasing volumes per NTU, even though this value is not the calculated result. A more rigorous pressure drop validation is left to future work. 93

106 Summary of Pressure Drop Validaion Analysis Exchanger NTU Number of Discrete Volumes Side of Heat Exchanger Validation Pressure Drop (Pa) Simulation Result Pressure Drop (Pa) Difference (Pa) Error (Percent) Cold (stepped) % Hot (unstepped) % Cold (stepped) % Hot (unstepped) % Cold (stepped) % Hot (unstepped) % Cold (stepped) % Hot (unstepped) % Cold (stepped) % Hot (unstepped) % Cold (stepped) % Hot (unstepped) % Cold (unstepped) % Hot (stepped) % Cold (unstepped) % Hot (stepped) % Cold (unstepped) % Hot (stepped) % Cold (unstepped) % Hot (stepped) % Cold (unstepped) % Hot (stepped) % Cold (unstepped) % Hot (stepped) % Table 11: Summary of Pressure Drop Validation Analysis Averaged Steady State Error in Hot and Cold Outlet Pressures 10% 9% 8% 7% Error (%) 6% 5% 4% 3% 2% 1% 0% Discrete Volumes per NTU NTU = 1.5 NTU = 3 NTU = 4 NTU = 6 9 Figure 28: Averaged Steady State Error in Hot and Cold Outlet Pressures 94

107 4.4. Transient Temperature Response An outlet temperature transient can result from an inlet temperature transient or from an inlet mass flow rate transient. In general, changes to the inlet conditions of either inlet stream will affect the outlet conditions of both streams. The transient temperature response conditions were validated against several results that were available in the literature. It should be noted that while many results are available in the literature, most of the publications do not include sufficient information to make the data useful for validation. The data used for the transient temperature validation includes a temperature step increase at one arbitrary input, a non-step temperature transient at the hot inlet, and an identical and simultaneous step change in the mass flow rate at both inlets Step Change to Inlet Temperature Figure 29 shows the outlet temperature response to a step change in the inlet temperature for counter flow heat exchangers with several different NTU values (London & Kays, 1998). All four edges of this plot are used as axes. The top and left axes correspond to the un-stepped temperature response and the bottom and right axes correspond to the stepped response. The stepped response corresponds to the behavior of the fluid that had the step temperature increase applied at its inlet. The results in Figure 29 are non-dimensionalized according to the parameters shown on the axes and are redimensionalized as shown in Equation 46, Equation 47 and Equation 48. The magnitude of the step input is arbitrary, but the capacitance rate of the two gas streams must be equal (C min /C max =1) for these results to be applicable. A capacitance ratio of unity generally 95

108 means that the hot side and the cold side gases are the same and have the same or nearly the same mass flow rates, which is true for many interesting cases including the GT- MHR recuperator. Figure 29: Outlet Temperature Response to a Step Inlet Temperature Change (London & Kays, 1998) * * R 1 t = Cwθ d, C x+ 0.4 min * R 1 + where : t time, stepped response only (seconds) C θ * w wall capacitance parameter (C / C ) total residence time of the C fluid (seconds) dc, min min x the non-dimensional value read from the bottom axis w * R heat transfer resistance ratio (R on unstepped side)/(r on stepped side) min Equation 46: Re-Dimensionalization of Stepped Time Axis 96

109 * ( 1.5 ) θ, min t = x + C w d C where : t time, unstepped response only (seconds) x the non-dimensional value read from the top axis Equation 47: Re-Dimensionalization of Unstepped Time Axis [ ] Tt t T T T * () = ε f () ( ) (0 + ) + (0 + ) where : Tt ( ) temperature at time t(k) ε ( t) non-dimensional temperature parameter at time t * f T ( ) steady-state temperature reached after step input T (0 + ) steady-state temperature prior to step input Equation 48: Re-Dimensionalization of Temperature Axis for Both Fluids Heat exchange surfaces, exchanger geometry and helium inlet conditions were determined that resulted in heat exchanger configurations with a capacitance ratio very near unity and NTU values of 1.5, 3, 4 and 6. Table 12 shows the specific geometry and test conditions for the temperature step validation runs. Three different heat exchange surfaces, two different materials, and varying inlet conditions were used during the validation. In two cases, the temperature step input was applied to the hot side and in two cases the step was applied to the cold side. Three simulations, with different numbers of discrete volumes, were run for each configuration. In all of these simulations, the integration time was determined by the minimum residence time of the fluid volumes in the heat exchanger. During each simulation, the heat exchanger was run with constant inlet conditions, until steady state operation was reached. A positive temperature step was 97

110 then applied to one of the inlets, while the conditions at the other inlet were kept constant. The response of the heat exchanger was recorded, until steady state operation was reached. An example of the inlet and outlet temperatures for the 6-NTU configuration is shown in Figure 30 and Figure 31. During the first several seconds of these simulations, the helium outlet temperatures were changing despite the inlet temperatures being constant. This is caused by the user defined nominal initial conditions converging to the actual conditions, as determined by the simulation. This initial transient effect can be seen in the helium outlet temperatures during first three seconds of Figure 30 and Figure 31. Heat Exchanger Geometry & Materials Helium Inlet Conditions C max Exchanger Heat Exchange Frontal Flow Temperature (K) Mass Flow Pressure Side Material NTU Surface Area (m2) Length (m) Before Step After Step Rate (kg/sec) (Mpa) C min 1.5 Hot (unstepped) 1/ Aluminum Cold (stepped) 1/ Hot (unstepped) 1/ Aluminum Cold (stepped) 1/ Hot (stepped) 1/ Inconel 617 Cold (unstepped) 1/ Hot (stepped) 1/ Inconel 617 Cold (unstepped) 1/ Table 12: Heat Exchanger Configuration for Temperature Step Validation 98

111 900 Hot Helium Inlet and Outlet Temperatures for Step Response Validation, NTU = Temperature (K) Hot Helium Inlet Temperature (K) Hot Helium Outlet Temperature (K) 100K Temperature Step Increase Convergence of Initial Conditions Time (seconds) Figure 30: Hot Side Inlet and Outlet Temperatures for 100K Hot Side Inlet Step, NTU= Cold Helium Inlet and Outlet Temperatures for Step Response Validation, NTU = 6 Convergence of Initial Conditions Temperature (K) Cold Helium Inlet Temperature (K) Cold Helium Outlet Temperature (K) Time (seconds) Figure 31: Cold Side Inlet and Outlet Temperatures for 100K Hot Side Inlet Step, NTU=6 99

112 The results of the 6-NTU configuration validation runs are shown in Figure 32, Figure 33, Figure 34 and Figure 35. Figure 32 and Figure 34 show the cold and hot outlet temperature response, respectively, as determined by the simulation. The pre-step steady state outlet temperatures vary for the three results in each of these figures, because of the steady state response error that was described previously. Because it was desired to quantify the error associated with the transient response, the outlet temperatures were normalized to remove this steady state error, before the analysis of the transient responses. The outlet temperatures of the simulation results were shifted so the initial pre-step steady state temperature matched the ideal pre-step steady state temperature as determined using Equation 43 or Equation 44. Figure 33 and Figure 35 show the results of this normalization for the 6-NTU configuration and Figure 36 and Figure 37 show the normalized results for the 4-NTU configuration. 100

113 Cold Side Outlet Temperature Response to 100K Hot Side Inlet Step, NTU = Validation Temperature (K) Simulation Result: 16 Volumes Simulation Result: 32 Volumes 720 Simulation Result: 8 Volumes Time (seconds) Figure 32: Cold Side Outlet Temperature Response to 100K Hot Side Inlet Step, NTU = Normalized Cold Side Outlet Temperature Response to 100K Hot Side Inlet Step, NTU = 6 Validation Temperature (K) Simulation Result: 8 Volumes Simulation Result: 16 Volumes Simulation Result: 32 Volumes Time (seconds) Figure 33: Normalized Cold Side Outlet Temperature Response to 100K Hot Side Inlet Step, NTU=6 101

114 Hot Side Outlet Temperature Response to 100K Hot Side Inlet Step, NTU = 6 Temperature (K) Simulation Result: 8 Volumes Simulation Result: 16 Volumes Validation Simulation Result: 32 Volumes Time (Seconds) Figure 34: Hot Side Outlet Temperature Response to 100K Hot Side Inlet Step, NTU = Normalized Hot Side Outlet Temperature Response to 100K Hot Side Inlet Step, NTU = 6 Simulation Result: 16 Volumes 452 Temperature (K) Simulation Result: 8 Volumes Simulation Result: 32 Volumes Validation Time (seconds) Figure 35: Normalized Hot Side Outlet Temperature Response to 100K Hot Side Inlet Step, NTU = 6 102

115 Normalized Cold Side Outlet Temperature Response to 100K Hot Side Inlet Step, NTU = 4 Validation 770 Temperature (K) Simulation Result: 8 Volumes Simulation Result: 16 Volumes Simulation Result: 47 Volumes Time (seconds) 4 Figure 36: Normalized Cold Side Outlet Temperature Response to 100K Hot Side Inlet Step, NTU= Normalized Hot Side Outlet Temperature Response to 100K Hot Side Inlet Step, NTU = 4 Simulation Result: 8 Volumes Simulation Result: 16 Volumes Temperature (K) Validation Simulation Result: 47 Volumes Time (seconds) 4 Figure 37: Normalized Hot Side Outlet Temperature Response to 100K Hot Side Inlet Step, NTU = 4 103

116 The transient responses were analyzed by calculating the temperature at 63% of the ideal post-step steady state temperature and determining the ideal time required to achieve that ideal temperature according to the validation data. The temperature of each response at the ideal time was determined and compared to the ideal temperature. Also, the time required to reach the ideal temperature was determined and was compared to the ideal time. The ideal temperature and time values used to quantify the error in the transient responses are summarized in Table 13. A summary of the results of the error analysis is shown in Table 14 and averaged hot and cold side errors for the response time and the response temperature errors are shown in Figure 38 and Figure 39, respectively. Figure 38 and Figure 39 show that increasing the discrete volumes per NTU decreases the error and causes the simulation result to more accurately reproduce the validation data. These results indicate that the simulation does reproduce the expected behavior and the accuracy is a function of the resolution of the discretization in time and in space. Unfortunately, these results do not enable separation of the error contributions from the spatial and time discretization. Temperature and Time at 63% of Response for Transient Validation Error Analysis Exchanger Side NTU Hot (unstepped) 1.5 Cold (stepped) Hot (unstepped) 3 Cold (stepped) Hot (stepped) 4 Cold (unstepped) Hot (stepped) 6 Cold (unstepped) Inlet Condition Ideal Time to Reach 63% Ideal Temperature at 63% of Temperature Response (seconds) Response (K) constant temperature K step increase constant inlet temperature K step increase K step increase constant inlet temperature K step increase constant inlet temperature Table 13: Temperature and Time at 63% of Response for Transient Validation Error Analysis 104

117 Summary of Transient Error Analysis of Simulation Results Exchanger NTU Number of Discrete Volumes Side of Heat Exchanger Time to Reach Temperature at 63% of Validation Temperature Response (seconds) Error Error (%) (seconds) Temperature at 63% Validation Response Time (K) Error (K) Error (%) Cold (stepped) % % Hot (unstepped) % % Cold (stepped) % % Hot (unstepped) % % Cold (stepped) % % Hot (unstepped) % % Cold (stepped) % % Hot (unstepped) % % Cold (stepped) % % Hot (unstepped) % % Cold (stepped) % % Hot (unstepped) % % Cold (unstepped) % % Hot (stepped) % % Cold (unstepped) % % Hot (stepped) % % Cold (unstepped) % % Hot (stepped) % % Cold (unstepped) % % Hot (stepped) % % Cold (unstepped) % % Hot (stepped) % % Cold (unstepped) % % Hot (stepped) % % Table 14: Summary of Transient Error Analysis of Simulation Results 105

118 70% 60% 50% Averaged Hot Side and Cold Side Time Response Error 100K Hot Side Step, NTU = 6 100K Hot Side Step, NTU = 4 50K Cold Side Step, NTU = 3 25K Cold Side Step, NTU = 1.5 Error (%) 40% 30% 20% 10% 0% Discrete Volumes per NTU 12 Figure 38: Averaged Hot Side and Cold Side Time Response Error 0.7% 0.6% 0.5% Averaged Hot Side and Cold Side Temperature Response Error 100K Hot Side Step, NTU = 6 100K Hot Side Step, NTU = 4 50K Cold Side Step, NTU = 3 25K Cold Side Step, NTU = 1.5 Error (%) 0.4% 0.3% 0.2% 0.1% 0.0% Discrete Volumes per NTU 12 Figure 39: Averaged Hot Side and Cold Side Temperature Response Error 106

119 Non-Step Change to Inlet Temperature In addition to the validation runs involving a step change in an inlet temperature, the cold side outlet temperature response to a more complicated hot inlet temperature transient was published by (Cima & London, 1958) and is shown in Figure 40. The result of this temperature transient is not generally applicable to any heat exchanger, but only to heat exchangers that meet the parameters as shown at the bottom of Figure 40. The simulation was used to iteratively design a heat exchanger and flow conditions that very nearly met these parameters. The heat exchanger and flow condition configuration used for these validation runs is shown in Table 15. Values corresponding to the parameters shown at the bottom of Figure 40 were calculated by the simulation and are shown in Table 16. Several of the values in Table 16 are slightly different than desired values, because the interdependence between parameters made it difficult to get every value to match. Figure 40: Non-Step Inlet Temperature Change (Cima & London, 1958) 107

120 Heat Exchanger Geometry & Materials Helium Inlet Conditions Heat Exchange Frontal Flow Temperature (K) Mass Flow Pressure Material Surface Area (m 2 ) Length (m) Before Step After Step Rate (kg/sec) (Mpa) Hot Side 1/ Inconel Cold Side 1/ varied Table 15: Heat Exchanger Geometry and Inlet Conditions for Non-Step Inlet Temperature Validation NTU Calculated Heat Exchanger Parameters Dwell Time (seconds) Cmin/Cmax Cw * bar R* Hot Side Cold Side dwell time C min fluid dwell time C fluid max Table 16: Calculated Heat Exchanger Properties for Non-Step Inlet Temperature Validation The hot inlet transient shown in Figure 41 was used as the hot side inlet temperature for three simulations in which the heat exchanger was discretized into 11, 20 and 30 volumes, respectively. The resulting cold side outlet temperatures calculated by these simulations are shown in Figure 42 along with the validation response that was digitized from Figure 40. Because of the complexity of the transient response, no analysis was done on the result shown in Figure 42, however it is obvious that the simulation is predicting a valid result and the accuracy of the result is increasing along with the increased number of discrete volumes. 108

121 850 Hot Helium Inlet Temperature for Non-Step Transient Temperature (K) Time (seconds) Figure 41: Hot Side Inlet Temperature for Non-Step Transient Validation Cold Side Outlet Temperature Response to Non-Step Hot Side Inlet Transient Validation Simulation Result: 30 Volumes 770 Temperature (K) Simulation Result: 11 Volumes Simulation Result: 20 Volumes Time (seconds) Figure 42: Simulation Results for Non-Step Temperature Validation 109

122 Mass Flow Rate Transient A result was also available in (Cima & London, 1958) for the cold side temperature response to a simultaneous mass flow rate step on both sides of the heat exchanger. The response to an identical and simultaneous change in the mass flow rate is very similar to a step change in the NTU of the heat exchanger. The result shown in Figure 43 is valid for a step mass flow rate change equivalent to the NTU changing from 1 to 1.5 or 1.5 to 1. The validity of this response is not generally applicable to any heat exchanger, but only to heat exchangers that meet the non-dimensional parameters as indicated on Figure 43. The simulation was used to iteratively design a heat exchanger and flow conditions that very nearly met these parameters and resulted in a heat exchanger with an NTU of 1. Next, the simulation was used to vary the mass flow rate through the heat exchanger to determine an appropriate step change that resulted in a steady state NTU value of 1.5. The heat exchanger and flow condition configuration used for this validation run is shown in Table 17. Values corresponding to the parameters shown at the bottom of Figure 40 were calculated by the simulation and are shown in Table 18. Several of the values in Table 18 are slightly different than desired values, because the interdependence between parameters made it difficult to get every value to match. 110

123 Figure 43: Mass Flow Rate Transient (Cima & London, 1958) Heat Exchanger Geometry & Materials Heat Exchange Surface Frontal Area (m 2 ) Flow Length (m) Material Temperature (K) Helium Inlet Conditions Mass Flow Rate (kg/sec) Before Step After Step Pressure (Mpa) Hot Side 1/ Aluminum Cold Side 1/ Table 17: Heat Exchanger Geometry and Inlet Conditions for Mass Flow Rate Transient Validation Calculated Heat Exchanger Parameters Dwell Time (seconds) NTU Cmin/Cmax Cw*bar R* Hot Side Cold Side dwell time C min fluid dwell time C fluid Before Step After Step max Table 18: Calculated Heat Exchanger Properties for Mass Flow Rate Transient Validation 111