Tools for Supercritical Carbon Dioxide Cycle Analysis and the
Cycle’s Applicability to Sodium Fast Reactors
By
Alexander R. Ludington
B.S. Physics
United States Naval Academy, 2007
SUBMITTED TO THE DEPARTMENT OF NUCLEAR SCIENCE AND ENGINEERING IN
PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF
MASTER OF SCIENCE IN NUCLEAR SCIENCE AND ENGINEERING
AT THE
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
JUNE 2009
Copyright © Massachusetts Institute of Technology (MIT)
All rights reserved
Signature of Author: ____________________________________________________________
Department of Nuclear Science and Engineering
May 8, 2009
Certified by:___________________________________________________________________
Dr. Pavel Hejzlar – Thesis Co-Supervisor
Principal Research Scientist
Certified by: ___________________________________________________________________
Prof. Michael J. Driscoll – Thesis Co-Supervisor
Professor Emeritus of Nuclear Science and Engineering
Certified by: ___________________________________________________________________
Prof. Neil E. Todreas – Thesis Co-Supervisor
Professor Emeritus of Nuclear Science and Engineering
Accepted by: __________________________________________________________________
Prof. Jacquelyn Yanch
Chairman, Department Committee on Graduate Students
1
2
Tools for Supercritical Carbon Dioxide Cycle Analysis and the Cycle’s
Applicability to Sodium Fast Reactors
By
Alexander R. Ludington
Submitted to the Department of Nuclear Science and Engineering on May 8, 2009 in partial
fulfillment of the requirements for the degree of Master of Science in
Nuclear Science and Engineering
Abstract
The Sodium-Cooled Fast Reactor (SFR) and the Supercritical Carbon Dioxide (S-CO2)
Recompression cycle are two technologies that have the potential to impact the power generation
landscape of the future. In order for their implementation to be successful, they must compete
economically with existing light water reactors and the conventional Rankine cycle.
Improvements in efficiency, while maintaining safety and proliferation goals, will allow the SFR
to better compete in the electricity generation market. These improvements will depend on core
design as well as the balance of plant, including the choice of steam or CO2 as the working fluid.
This work has developed some of the tools necessary for evaluating different design core and
balance of plant options. Much of it has concentrated on the S-CO2 Recompression cycle.
S-CO2 promises to be useful as a working fluid in high-efficiency power conversion
systems for SFRs because it achieves higher efficiencies at the high temperatures associated with
SFRs. The recompression cycle is capable of operating with very high efficiencies due to the
low compressor work needed when CO2 approaches its critical point at the compressor inlet.
The potential of this cycle to meet the needs of next-generation plants must be investigated
across the entire range of operations and within each component of the system. A steady-state
code for analysis of the recompression cycle was previously developed at MIT in the form of
CYCLES II, but the present work has made significant improvements to this code that make the
new version, CYCLES III, more versatile. This code can help to size components of the system
and predict the costs and performance of the system at steady-state. Coupling of the primary and
secondary loops is a major concern, the construction of the intermediate loop and associated heat
exchangers (IHX) being critical to cost, efficiency, and safety. Furthermore, there is little
experience in industry with large-scale compressors for S-CO2. The experience that has been
gained is typically proprietary. Most existing CO2 compressors do not operate near the critical
point and therefore, perform much like any other semi-ideal gas compressor. Accordingly,
consistent, usable models of non-ideal gas compressors have been developed in the present work
to produce preliminary designs and performance maps for the compressors in S-CO2
3
recompression cycles. Compressor designs were developed for a 500 MWth S-CO2
recompression cycle. The main compressor achieves an operating point total-to-static efficiency
of 90.4 % and the recompressing compressor achieves 91.4 %. Further work can continue once
these areas have been developed, including transient analysis, the effects of impurities on the
system, and investigation of cycles which operate on other working fluids. Additionally,
changes in the intermediate loop, the arrangement of the reactor vessel, and in-core changes will
affect the efficiency of the SFR. These include the option of diluent grading in the fuel,
flattening of the core outlet temperature profile, choosing Rankine or S-CO2 for the balance of
plant, and heat exchanger design. All these have been evaluated for their impact on plant
efficiency. It has been determined that the S-CO2 recompression cycle can provide efficiency
benefits over conventional Rankine cycles for SFRs with core outlet temperatures at or above
510 oC. With the S-CO2 cycle, SFRs can achieve thermal efficiencies of ~42 %.
Thesis Co-supervisor: Prof. Michael J. Driscoll
Title:
Professor Emeritus of Nuclear Science and Engineering
Thesis Co-supervisor: Prof. Neil E. Todreas
Title:
Professor Emeritus of Nuclear Science and Engineering
Thesis Co-supervisor: Dr. Pavel Hejzlar
Title:
Principal Research Scientist
4
Acknowledgments
First, I would like to thank Dr. Pavel Hejzlar for his patience, encouragement, and
continual assistance in completing my thesis. His guidance was crucial to every part of my
research, from the very beginning. Professors Michael J. Driscoll and Neil E. Todreas were
ever-present, and their vast knowledge and experience kept me from falling victim to
innumerable blunders. Jonathan P. Gibbs assisted me with my initial familiarization with the SCO2 cycle and CYCLES II and Anna Nikiforova helped to introduce me to heat exchanger
modeling. Matthew Denman and Matthew Memmott have been great sources of background
information on SFRs, having devoted many months to the DOE/NERI project on SFRs. Tri
Trinh provided constant feedback on the compressor models and insight into the transient
behavior of the recompression cycle. Thermoflow, Inc. has my appreciation for furnishing a free
academic license for their software to the Department of Nuclear Science and Engineering. The
steam cycle calculations were much simplified with this tool.
My thesis spans several aspects of the work done in the Nuclear Science and Engineering
Department and at MIT at large for that matter. Compressors, heat exchangers, and SFRs have
brought me in touch with many students and faculty at MIT to whom I am grateful for their
expertise and assistance. I have received support for this research from a NERI/DOE
investigation of SFRs and a Sandia National Laboratories project on the S-CO2 power cycle.
5
Table of Contents
ABSTRACT………………………………………………………………………………………………..3
ACKNOWLEDGEMENTS………………………………………………………………………. ……...5
TABLE OF CONTENTS………………………………………………………………………………….6
LIST OF FIGURES……………………………………………………………………………………...10
LIST OF TABLES……………………………………………………………………………………….12
ACRONYMS……………………………………………………………………………………………..13
1
INTRODUCTION………………………………………………………………………………14
1.1 Motivation………………………………………………………………………………………..........14
1.2 Sodium-Cooled Fast Reactor Background…………………………………………………………….14
1.3 S-CO2 Recompression Cycle Background……………………………………………………………16
1.4 HEATRICTM Printed Circuit Heat Exchanger Background…………………………………………18
1.5 Thesis Outline…………………………………………………………………………………………20
2
RECOMPRESSION CYCLE DEVELOPMENT……………………………………………..22
2.1 Introduction ……………………………………………………………………………………………22
2.2 CYCLES to CYCLES III……………………………………………………………………………...25
2.2.1 Extended Use of the Legault Nomenclature ………………………………………………..27
2.2.2 Optimization and Single Point Calculations………………………………………………..28
2.2.3 The Simple Recuperative Brayton Cycle…………………………………………………...29
2.2.4 Inclusion of other NIST Fluids……………………………………………………………..30
2.2.5 Interfacing with TSCYCO …………………………………………………………………32
2.3 The Ethane Cycle……………………………………………………………………………………...34
2.4 Fluid Impurities in the S-CO2 Recompression Cycle…………………………………………………36
2.4.1 Helium Additions for Leak Detection………………………………………………………37
2.4.2 Air Impurities……………………………………………………………………………….40
2.5 Chapter Summary……………………………………………………………………………………..41
2.7 Nomenclature for Chapter 2…………………………………………………………………………..42
3
S-CO2 COMPRESSOR DESIGN………………………………………………………………43
3.1 Introduction ……………………………………………………………………………………………43
3.2 Developing Compressors for the S-CO2 Cycle……………………………………………………….43
3.2.1 The Need for a Compressor Model ………………………………………………………...43
3.2.2 Compressor Background……………………………………………………………………43
3.3 Real Gas Radial Compressor (RGRC) Code………………………………………………………….50
3.3.1 Issues with Earlier Codes CCDS/CCODS and Motivation for RGRC Development……...50
3.3.2 Basic Outline of the RGRC Code…………………………………………………………..54
3.3.3 Variable Nomenclature……………………………………………………………………..55
3.3.4 Impeller Calculations……………………………………………………………………….55
3.3.5 Loss Calculations…………………………………………………………………………...60
3.3.6 Vaneless Space and Diffuser Calculations…………………………………………………67
6
3.3.7 Off-Design Compressor Performance………………………………………………………71
3.4 The Multi-Stage Code RGRCMS……………………………………………………………………..72
3.5 S-CO2 Compressor Designs…………………………………………………………………………...73
3.5.1 The Main Compressor Design……………………………………………………………...73
3.5.2 The Recompressing Compressor Design…………………………………………………...75
3.5.3 Benchmarking RGRC and RGRCMS………………………………………………………76
3.6 Chapter Summary……………………………………………………………………………………..78
3.7 Nomenclature for Chapter 3…………………………………………………………………………...80
4
BALANCE OF PLANT OPTIONS…………………………………………………………….82
4.1 Introduction……………………………………………………………………………………... …….82
4.1.1 Necessary Tools for Analysis………………………………………………………………82
4.1.2 Heat Losses in Intermediate Piping and Pumping Power of an SFR……………………….83
4.2 Alternate Fluids in the Intermediate Loop…………………………………………………………….88
4.3 Eliminating the Intermediate Loop……………………………………………………………………89
4.4 PCHE versus Shell-and-Tube Heat Exchangers………………………………………………………90
4.4.1 The Shell-and-Tube Code SoSaT…………………………………………………………..91
4.4.2 Expanding the Capabilities of the PCHE Codes…………………………………………..100
4.5 Benchmarking Heat Exchanger Codes………………………………………………………………100
4.6 S-CO2, Traditional Rankine, or Supercritical Steam PCS…………………………………………...101
4.7 Chapter Summary……………………………………………………………………………………103
4.8 Nomenclature for Chapter 4………………………………………………………………………….104
5
INCREASING THE EFFICIENCY OF THE SFR………………………………………….106
5.1 Introduction ………………………………………………………………………………………….106
5.2 Options for Increasing Core Outlet Temperature……………………………………………………106
5.3 A Reference Design: The ABR-1000 ………………………………………………………………..107
5.4 Option Space…………………………………………………………………………………………108
5.4.1 Methodology………………………………………………………………………………110
5.5 Results of Efficiency Comparisons…………………………………………………………………..112
5.6 The Choice of Cycle Pressure for Rankine Cycles…………………………………………………..118
5.7 Chapter Summary……………………………………………………………………………………119
6
SUMMARY, CONCLUSIONS AND RECCOMENDATIONS…………………………….121
6.1 Summary……………………………………………………………………………………………..121
6.2 Conclusions …………………………………………………………………………………………..121
6.2.1 Conclusions on the S-CO2 Cycle and its Compressors…………………………………....121
6.2.2 Conclusions on Sodium Fast Reactors…………………………………………………….122
6.3 Recommended Future Work ………………………………………………………………………...123
6.3.1 S-CO2 Compressors……………………………………………………………………….123
6.3.2 The S-CO2 Cycle as a Whole……………………………………………………………...124
6.3.3 SFR Heat Exchangers……………………………………………………………………..124
6.3.4 Safety and Availability Consequences of SFR Design Options…………………………..125
6.4 Where to Obtain Codes used in this thesis…………………………………………………………..126
REFERENCES………………………………………………………………………………………….127
7
APPENDIX A
CYCLES III CODE MANUAL……………………………………………...132
A.1 Introduction………………………………………………………………………………………….132
A.2 Inputs and Outputs…………………………………………………………………………………..132
A.3 Troubleshooting……………………………………………………………………………………..134
A.4 Conclusion…………………………………………………………………………………………...135
APPENDIX B
RGRC AND RGRCMS CODE MANUAL………………………………….136
B.1 Introduction………………………………………………………………………………………….136
B.2 Inputs and Outputs…………………………………………………………………………………...136
B.3 Recommendations and Trouble-Shooting…………………………………………………………...138
B.4 Fluid Properties……………………………………………………………………………………...140
B.5 Conclusion…………………………………………………………………………………………...141
APPENDIX C
PCHE CODES MANUAL……………………………………………………143
C1. Introduction………………………………………………………………………………………….143
C.2 Inputs and Outputs…………………………………………………………………………………..143
C.3 Cautions with the PCHE codes……………………………………………………………………...145
C.4 Improvements to the PCHE Codes…………………………………………………………………..145
C.5 Cost Estimates for PCHEs……………………………………………………………………….…..147
C.6 Conclusion…………………………………………………………………………………………...147
APPENDIX D
SoSaT CODE MANUAL……………………………………………………..148
D.1 Introduction………………………………………………………………………………………….148
D.2 Inputs and Outputs…………………………………………………………………………………..148
D.3 Cautions and Considerations for SoSaT…………………………………………………………….151
D.4 Correlations and Supporting Information…………………………………………………………...152
D.5 Conclusion…………………………………………………………………………………………..153
8
9
List of Figures
Figure 1.1: Arrangement of the primary IHX in a pool-type SFR
Figure 1.2: The density spike near the critical point of CO2
Figure 1.3: Heatric TM Printed Circuit Heat Exchangers
Figure 1.4: PCHE with two S-CO2 plates to each Na plate and a helium fill-gas plate
Figure 2.1: The Simple Recuperative Brayton Cycle
Figure 2.2: The S-CO2 Recompression Cycle
Figure 2.3: Turbine size comparison for different fluids
Figure 2.4: Efficiency Comparison of S-CO2 Recompression Cycle to other cycles
Figure 2.5: Pipe paths in the Recompression Cycle of CYCLES III
Figure 2.6: The original output “res.txt” from the original CYCLES
Figure 2.7: The effect of impurities on the critical temperature of CO2
Figure 2.8: The efficiency of a pure ethane simple cycle
Figure 2.9: The effect of dissociation on the ethane cycle
Figure 2.10: The effect of turbine inlet temperature on cycle efficiency
Figure 2.11: The effect of a helium leak detection gas on the S-CO2 cycle efficiency
Figure 2.12: The compressor work as the helium mole fraction is changed
Figure 2.13: The effect of air impurity on the S-CO2 cycle
Figure 3.1: Design ranges for different compressor types
Figure 3.2: The meridional plane view of a typical centrifugal compressor
Figure 3.3: A straight vaned diffuser
Figure 3.4: Example compressor performance map showing the surge line
Figure 3.5: Fast-running fluid property subroutines for S-CO2 produced by Gong
Figure 3.6: Range of Applicability for Gong’s S-CO2 property subroutines
Figure 3.7: A single-stage main compressor map from CCODS showing unusual features
Figure 3.8: The velocity triangles at impeller inlet and outlet
Figure 3.9: A simplified schematic of an impeller showing the definition of backsweep angle
Figure 3.10: The process for determining impeller performance
Figure 3.11: The flow path in the vaneless space
Figure 3.12: The pressure ratio of the main compressor for varying speeds
Figure 3.13: The total-to-static efficiency of the main compressor for varying speeds
Figure 3.14: The pressure ratio of the recompressing compressor for varying speeds
Figure 3.15: The total-to-static efficiency of the two-stage recompressing compressor for varying speeds
Figure 3.16: The pressure ratio of the modeled test compressor for varying speeds
Figure 3.17: The total-to-static efficiency of the modeled test compressor for varying speeds
Figure 4.1: Schematic of the intermediate loop
Figure 4.2: The temperature drop in a 24 m long intermediate pipe
Figure 4.3: Heat Lost in a 24 m long intermediate pipe
Figure 4.4: The division of the tube length according to heat transfer regime in SoSaT
Figure 4.6: Rankine Cycle efficiency as a function of temperature and pressure
10
Figure 5.1: The arrangement of components in the SFR balance of plant
Figure 5.2: The design choices affecting efficiency that are considered in this study
Figure 5.3: STEAM PRO 16 diagram of the Rankine Cycle
Figure 5.4: Efficiency comparison with PCHEs for both the P-IHX and S-IHX
Figure 5.5: Efficiency comparison with a PCHE for the P-IHX and a shell-and-tube S-IHX.
Figure 5.6: Efficiency comparison with a shell-and-tube P-IHX and a PCHE for the S-IHX.
Figure 5.7: Efficiency comparison with shell-and-tube heat exchangers for both the P-IHX and S-IHX.
Figure 5.8: Efficiency comparison with no intermediate loop and a PCHE IHX
Figure 5.9: Efficiency comparison with no intermediate loop and a shell-and-tube IHX
Figure 5.10: Cost of JSFR steam generator for varying steam pressures
Figure B.1: The RGRC input file
Figure B.2: The beginning of an RGRC output file
Figure C.1: The input file ihxNa_hyb.in with representative values
Figure C.2: The output file ihxNa_hyb.out with representative values
Figure D.1: The SoSaT input file, matching the steam generator of the JSFR
Figure D.2: The SoSaT output file
Figure D.3: The placement of the central downcomer in SoSaT
11
List of Tables
Table 2.1: Critical points of fluids important to the Ethane Simple Cycle
Table 2.2: Properties of selected fluids at their critical points
Table 3.1: The operating points of the compressors in the recompression cycle
Table 3.2: The non-dimensional parameters of the compressors in the recompression cycle
Table 3.3: Selected Sandia Test Compressor Parameters
Table 4.1: Representative Values used in Eqn. 4-1
Table 4.2: Intermediate Loop Pumping Power Requirements for SFRs
Table 4.3: Characteristics of CRBR and ABR-1000 Intermediate Loops
Table 4.4: Correction factors for the cross-flow Nusselt number
Table 4.5: Conditions for each boiling regime in SoSaT
Table 4.6: Benchmarking of the SoSaT Code
Table 5.1: The reference balance of plant
Table 5.2: Core Outlet Temperatures of Selected SFRs
Table 5.3: Standard channel dimensions used for PCHEs in this study
Table 5.4: Options Considered in the Efficiency study
Table 5.5: Efficiency Comparison of SFR options for a core outlet temperature of 510 oC
Table 5.6: Efficiency Comparison of SFR options for a core outlet temperature of 530 oC
Table B.1: RGRC inputs and their suggested ranges
Table C.1: Files needed for each PCHE code
12
Acronyms
P-IHX
S-IHX
IHX
PCHE
S-CO2
SFR
PCS
HTR
LTR
NIST
ANL
INL
GTL
RMS
DNB
CHF
ASME
ONB
NACA
IAEA
NRC
BOP
BOL
EOL
MC
RC
TNF
Primary Intermediate Heat Exchanger
Secondary Intermediate Heat Exchanger
Intermediate Heat Exchanger
Printed Circuit Heat Exchanger
Supercritical Carbon Dioxide
Sodium-Cooled Fast Reactor
Power Conversion System
High Temperature Recuperator
Low Temperature Recuperator
National Institute of Standards and Technology
Argonne National Laboratory
Idaho National Laboratory
Gas Turbine Laboratory (MIT)
Root-Mean-Square
Departure from Nucleate Boiling Regime
Critical Heat Flux
American Society of Mechanical Engineers
Onset of Nucleate Boiling
National Advisory Committee for Aeronautics
International Atomic Energy Agency
Nuclear Regulatory Commission
Balance of Plant
Beginning of Life
End of Life
Main Compressor
Recompressing Compressor
Technology Neutral Framework
13
1 Introduction
1.1 Motivation
There are four primary objectives in this research. The first objective is to update the
CYCLES II code with the abilities to model a system running on any fluid available in the NIST
database and to model either simple or recompression cycles. The code should also be as userfriendly as possible owing to the fact that it may be used by many students who will have to
become familiar with it in a short period of time. Its applicability as a research tool will be
enhanced to include any simple Brayton cycle and more flexible investigations of the
recompression cycle. The second objective is to develop models for single and multi-stage
centrifugal CO2 compressors using a user-friendly computer code which will be developed as
part of this research. The code will fill in gaps in MIT’s capability to model turbomachinery
performance and will enable transient models of the recompression cycle to more accurately
incorporate compressor performance. The third objective is to perform a preliminary
investigation of the intermediate loop and heat exchangers (IHX) that might be used in an SFR.
This investigation will focus on cost, size, and efficiency. Fourth, the investigation will look into
the range of options for increasing the efficiency of the SFR. All of these objectives will
enhance the work being done on Sodium-Cooled Fast Reactors and the S-CO2 Power Conversion
Systems that might be used as a balance of plant in a number of next-generation power reactors
including the SFR.
This study will compare heat exchangers, and power cycles based on an assumed thermal
power of 250 MW, modeled after the thermal power of a single loop in the ABR-1000. S-CO2
recompression cycles and their associated compressors are modeled on an assumed cycle thermal
power of 500 MW. This value was selected because compressors become difficult to design for
an operating speed of 3600 RPM in smaller power systems and two heat exchanger loops could
be coupled to a single turbomachinery train. The operating speed of 3600 RPM is selected in
order to synchronize the cycle to the electric grid. These restrictions on compressor design are
discussed further in Chapter 3.
1.2 Sodium-Cooled Fast Reactor Background
Sodium-Cooled Fast Reactors have been operated in the United States since EBR-1 in
1951 [IAEA, 2006]. They are currently of interest as a means to manage actinides from LWR
spent fuel. The SFR is assumed to be one of two types: a pool design in which the primary
sodium flows from a lower cold pool up through the core and into an upper hot pool, or a loop
design in which the primary sodium exits the reactor vessel and flows through a heat exchanger,
14
returning in a cold leg to the core inlet. The primary intermediate heat exchanger, which
transfers heat from radioactive primary sodium to clean intermediate sodium, is located within
the reactor vessel in a pool design, and outside the vessel in the loop design. The pool design
essentially eliminates the loss of coolant accident (LOCA) sequence, while the loop design
reduces the amount of primary coolant and saves capital cost by reducing the size of the vessel.
Figure 1.1 schematically shows the pool-type design option. The P-IHX is located within the
vessel, whereas a loop-type design would require pumping primary sodium through primary
piping to a P-IHX outside the vessel. Both designs employ an intermediate sodium loop which
serves as a buffer between the radioactive primary coolant and the power conversion system
(PCS).
Figure 1.1: Arrangement of the primary IHX in a pool-type SFR. Primary sodium flows
downward through the P-IHX and into the cold pool. Primary pumps located within the cold
pool pump the sodium back up through the core. In this way the hot and cold pools are separated
by what is called a redan.
With the intermediate loop present, steam generator leaks will not release any activated
sodium and will constitute less of a safety risk.
SFRs operate with core outlet temperatures up to 575 oC, as in the BN-1800 design
[IAEA, 2006]. They typically have a temperature rise across the core of less than 200 oC which
makes them well suited for the S-CO2 recompression cycle because it is so highly recuperative.
Fuels can be metal or oxide and SFR cores can be designed with conversion ratios from 0 to 1, or
above. The advantages of SFRs are the excellent heat transfer characteristics of sodium,
excellent material performance in a sodium environment, and high temperatures. Large-scale
SFRs have been operated around the world with varying success.
15
1.3 S-CO2 Recompression Cycle Background
Increasing cycle efficiency and reducing capital costs are the best ways to reduce
electricity costs in the nuclear power industry. There is great interest in new balance of plant
options that maximize efficiency while reducing plant capital costs. Closed Brayton cycles are
simple and compact and can achieve very high efficiencies at the proper conditions. The most
interesting of these is the supercritical CO2 recompression cycle. Much of this thesis is devoted
to the S-CO2 recompression cycle because it is so promising for applications with core outlet
temperatures above 500 oC. Other Brayton cycles, like the helium Brayton cycle, achieve very
high efficiencies at much higher temperatures. These high temperatures, however, are much
more challenging to materials than the S-CO2 recompression cycle. S-CO2 recompression cycles
have been investigated at MIT for several years, beginning in 2000 [Dostal, 2004]. The use of
CO2 as a working fluid in power conversion systems has enjoyed success in British gas-cooled
reactors (GCR) and has been studied by since the 1960’s [Dostal, 2004], but the operating range
of industry experience has not produced much data in the supercritical regime. The
recompression cycle has significant advantages over other cycles, and especially over other
Brayton cycles for turbine inlet temperatures above 490 oC.
The ability of the S-CO2 cycle to reach high efficiency comes from the reduced
compressor work as the compressor inlet conditions approach the critical point of CO2. The
density of the fluid increases dramatically, as shown in Figure 1.2. The increased density close
to the critical point reduces the compressor work.
16
Figure 1.2: The density spike near the critical point of CO2. As the temperature decreases and
approaches the critical temperature (indicated with the red line),
the density rises more rapidly [NIST, 2007].
Earlier research on the S-CO2 cycle by Vaclav Dostal sized components and calculated
efficiencies of the S-CO2 recompression cycle. Dostal’s research optimized heat exchanger
sizes, roughly sized turbomachinery, and showed that the S-CO2 cycle could be economically
competitive, especially at higher turbine inlet temperatures [Dostal, 2004] through the use of a
steady state code called CYCLES. This work has been continued and studies of the steady-state
and transient cycle performance have been conducted through the use of computer codes at MIT
and in an experimental compression loop operated by Sandia National Laboratory in
collaboration with Barber-Nichols Inc. [Wright et al., 2008].
CYCLES III is the result of updates to Dostal’s CYCLES and its operation is detailed in
Chapter 2. It is a steady-state code that models the recompression cycle as well as the simplerecuperative Brayton cycle. It models compressors simply, because it is operating at steady state
and the user inputs a value for the compressor efficiency. More information is needed about how
compressors will operate in the recompression cycle, so the development of a mean-line
compressor design and performance code (RGRC) was undertaken in this research. RGRC is
detailed in Chapter 3 and has produced compressor performance maps which are useful to
17
transient analyses as well as to steady-state studies which require a value for compressor
efficiency.
The recompression cycle is interesting to next generation nuclear reactors because it
achieves high efficiencies, especially at high turbine inlet temperatures. The turbomachinery is
compact and sodium reactions with CO2 are less exothermic than sodium reactions with water.
The working fluid, CO2, is abundant, non-toxic, and cheap. It appears that at higher
temperatures (500 oC and up), the S-CO2 cycle can outperform the Rankine cycle, but even if the
advantage is moderate, the compactness of the S-CO2 cycle may be a very significant savings in
cost and space. This compactness has made the cycle a consideration for such applications as
space and naval power cycles.
1.4 HEATRICTM Printed Circuit Heat Exchanger Background
HEATRICTM Printed Circuit Heat Exchangers (PCHEs) are discussed often in this
research. They are the heat exchanger of choice for compactness and ruggedness. They are
formed by diffusion bonding plates with etched channels. Alternating plates for hot and cold
fluids allow for very high heat transfer area in a relatively compact volume. Figure 1.3 shows
the stacking of hot and cold PCHE plates and the cross section of semi-circular etched channels.
PCHEs have been modeled at MIT with Fortran codes written by Pavel Hejzlar [Hejzlar et al.,
2007] and updated by several others [Shirvan, 2009].
Figure 1.3: HeatricTM Printed Circuit Heat Exchangers are formed by diffusion bonding stacked
plates (left) of alternating hot and cold fluid channels. This creates a monolithic block of parallel
channels (right). [Heatric, 2009]
PCHEs can be built with a varied range of channel diameters, but they are, in general,
very small. The PCHEs used in this models discussed here have channel diameters of 2.5 mm.
They can be constructed with alternating hot and cold plates, one hot plate for two cold plates,
etc. They have the option of producing “hybrid” designs which include a separating plate with
fill gas channels for leak detection, as shown in Figure 1-5.
18
Unit cell
S-CO2 plate
S-CO2 plate
Helium plate
Sodium plate
Helium plate
S-CO2 plate
S-CO2 plate
Figure 1.4: PCHE with two S-CO2 plates to each Na plate and a helium fill-gas plate in between
[Ludington et al., 2007].
The channels can be either straight or zig-zag channels, but are usually straight for sodium. Zigzag channels improve the heat transfer coefficient by a factor of ~2.3, but have a pressure drop
penalty. The PCHE codes at MIT model the heat transfer by nodalizing the heat exchanger core
along the flow path and iterating along the heat exchanger length. A heat balance is solved for
each node until the desired power is reached.
Simplifying assumptions used in the PCHE models are very similar to those made in
modeling shell-and-tube heat exchangers in Chapter 4. For the PCHE, they are:
1.
2.
3.
4.
5.
6.
Mass flow of each fluid is uniformly distributed among the channels.
Every unit cell has the same temperature within the heat exchanger core.
The wall channel is uniform around the channel periphery.
Zero heat is lost and axial conduction is negligible.
Kinetic and potential energy are neglected
Fluid properties are constant along a node.
The total heat transfer coefficient is determined using correlations appropriate to the
fluids being used and, in the case of PCHEs with a helium plate, by adjusting the conduction
length of the plates to account for the thermal resistance of the helium channels. Detailed
discussion of the solution methodology is available from Dostal [2004].
PCHEs, in general, have high power density. Leaks can be controlled by using the
helium hybrid plate, but the small channels also mean that a single leak will not be likely to pose
serious problems. Pumping sodium through narrow channels has been avoided in the past, but
recent research shows that clean sodium can be pumped through narrow channels without
19
problems [Hejzlar, 2008]. These benefits make PCHEs very appealing for applications where
compact heat exchangers are desirable.
1.5 Thesis Outline
Chapter 2: Recompression Cycle Optimization
Chapter 2 details the changes made to the CYCLES II code for modeling the
Recompression cycle, in addition to the history of the code and its capabilities. Also in this
chapter are discussions of fluid impurities and detection gases, heat exchanger sizing, and the
affects all these have on cycle efficiency. A discussion of the recompression cycle as compared
to other cycles is also included. In addition, a brief study of an Ethane simple recuperative
Brayton cycle is discussed.
Chapter 3: S-CO2 Compressor Design
This chapter details the history of S-CO2 compressors and the code, RGRC, developed in
this work as a mean-line design code for real gases. For the reader unfamiliar with compressor
design and operation, an overview of these topics is included as well. Detailed discussion of the
code’s operation and results are presented. The compressors are developed to run with a 500
MWth recompression cycle to scale with the 250 MWth heat exchangers considered in Chapters
4 and 5. The compressor designs are chosen to produce both efficient and versatile compressors.
An attempt at benchmarking has been made by using estimated geometry for an existing test
compressor [Wright et al., 2008].
Chapter 4: Balance of Plant Options for the SFR
Chapter 4 discusses the ex-core options for the SFR, from an efficiency focused
perspective. Tools developed for modeling the S-CO2 recompression cycle are used to compare
its performance to that of supercritical steam and conventional Rankine plants. Different
configurations of the plant, including elimination of the intermediate loop, are considered.
Computer codes for modeling heat exchangers are discussed, as well as the impact of the
intermediate loop on efficiency.
Chapter 5: Increasing SFR Efficiencies
Chapter 5 details the changes that can be made within the core and choices in the balance
of plant that improve the efficiency of the SFR. Any such effort is primarily aimed at improving
the turbine inlet temperature. At higher temperatures differences between power cycles become
important. Comparisons are made between several different configuration options and
discussions of safety and cost are included with the efficiency results. The design configurations
20
considered span every combination of PCHEs and shell-and-tube heat exchangers within sizing
constraints.
Unless otherwise noted, all fluid properties have been computed using REFPROP 8.0,
available from the National Institutes of Standards and Technology (NIST) [NIST, 2007]. For
compressor development fluid properties in Chapter 3 have been calculated using faster
polynomial subroutines for CO2. These are discussed in Chapter 3. Sodium properties come
from experimental temperature-dependent relationships developed at Argonne National
Laboratory [Fink and Leibowitz, 1995].
21
2 Recompression Cycle Development
2.1 Introduction
The S-CO2 recompression cycle is a variation of the well-known simple recuperative
Brayton cycle. The open Brayton cycle is familiar to anyone who has flown on a commercial
jetliner. Closed Brayton cycles are used in power conversion systems (PCS) with a variety of
working fluids. The simple closed cycle is used with helium, CO2, or gas mixtures in a variety
of engineering applications. Figure 2.1 shows the simple recuperative Brayton cycle schematic
and its T-s diagram for ideal components.
Figure 2.1: The Simple Recuperative Brayton Cycle
The simple recuperative Brayton cycle includes a heat exchanger (recuperator) as shown
in Figure 2-1. This recuperator preheats the fluid entering the reactor (IHX) with energy from
22
the turbine exhaust. The precooler is the mechanism for heat rejection to the environment from
the S-CO2 cycle. The precooler outlet (state 1) is just above the critical point of the fluid,
reducing the work of the compressor and thus enhancing efficiency. With CO2 as the working
fluid, the simple recuperative cycle does not achieve attractive efficiencies because heat transfer
is not effective in the recuperator. The reason is the development of a pinch-point in the
recuperator. Pinch -point refers to the location along the heat exchanger width at which the
temperature difference between the hot stream and the cold stream reaches zero. At that point no
more heat transfer will occur, resulting in very poor recuperator effectiveness. The underlying
cause of the pinch-point in a simple recuperative cycle is the mismatch in specific heats at
different pressures of the two streams. The solution to this problem is the recompression cycle,
shown in Figure 2.2. By splitting the recuperator into a low temperature recuperator (LTR) and a
high temperature recuperator (HTR), the pinch-point problem is avoided. The flow is split and
two compressors are needed [Dostal, 2004].
Figure 2.2: The S-CO2 Recompression Cycle
23
The recompression cycle is designed with two recuperators and two compressors because
of the pinch-point problem that arises if a simple recuperative cycle is used [Dostal, 2004]. Flow
is split at the inlet to the precooler (point 6 in Figure 2.2) and then merges again at the inlet to the
cold side of the HTR (point 2’ in Figure 2.2). Typically about 38% to 40 % of the flow will be
directed to the recompressing compressor. Without using the recompression cycle, large
differences in specific heat capacity cause the temperature difference in the recuperator of the
simple Brayton cycle to reach zero; a pinch-point. The result of the pinch-point problem is poor
effectiveness in the recuperator and lower cycle efficiency.
The recompression cycle can achieve thermal efficiencies far superior to those of any
existing LWR if the turbine inlet temperature is sufficiently high. S-CO2 cycles become
competitive at turbine inlet temperatures of about 490 oC. This temperature threshold makes the
S-CO2 recompression cycle attractive to next-generation plants, like SFRs, because core outlet
temperatures will likely be in the range of 500 oC to 550 oC. Besides efficiency, another benefit
of the S-CO2 cycle is its compactness. The turbine, for example, is an order of magnitude
smaller in an S-CO2 plant than in most other power cycles. Figure 2.3 shows the size
comparison of the CO2 turbine with those of helium and steam cycles as developed by Dostal
[2004].
Figure 2.3: Turbine size comparison for different fluids, from Dostal, [2004]
Small turbomachinery and relatively small heat exchangers mean that capital costs and
plant footprint can be reduced. The ability to model this cycle and predict efficiency is critical to
evaluating its competitiveness with other cycles.
24
The compressors and the turbine are assumed to run on a single shaft in the
recompression cycle in order to simplify the design and reduce capital costs. This method
introduces controllability issues, however, and the transient operation of the cycle must use
finely tuned controllers. The main compressor inlet runs very near the critical point of CO2,
where fluid properties are highly variable. Therefore, the performance of the cycle is very
sensitive to control of the main compressor inlet conditions. Aspects of cycle control are
discussed in Tri Trinh’s SM thesis and he has developed a code (TSCYCO) that models the cycle
under a variety of transients [Trinh, 2009], [Kao, 1984]. TSCYCO models the response of the
entire PCS and allows the user to adjust parameters which operate a set of control valves.
Dostal’s main goal in CYCLES was to size heat exchangers and to optimize the
configuration of the recompression cycle for maximum efficiency. Optimizing the S-CO2 cycle
allows for a comparison with the traditional Rankine cycle, to determine whether the cycle can
achieve economic competitiveness beyond that of Gen III+ reactors when coupled to Gen IV
reactor designs like the SFR. Dostal’s results showed that the S-CO2 cycle can be very
competitive in efficiency, especially as turbine inlet conditions are increased, as shown in Figure
2.4.
Figure 2.4: Efficiency Comparison of S-CO2 Recompression Cycle
and other Cycles [Dostal, 2004]
2.2 CYCLES to CYCLES III
CYCLES III is the latest edition of a code that models the performance of an S-CO2
power conversion system, for either a simple recuperative Brayton cycle or a Recompression
cycle. It was originally written as CYCLES by Vaclav Dostal and was used as the main analysis
25
tool in his Ph.D. work [Dostal, 2004]. The original CYCLES included no piping losses and only
performed calculations for the recompression cycle. Subsequent improvements by Pavel Hejzlar
and David Legault [Legault, 2006] improved the code to include pipe modeling, and a much
more readable, user-friendly structure and variable nomenclature. Legault’s improvements made
the code much more accessible to new users because the variable names are very logically
constructed. Legault named his code CYCLES II, but the code still only performed
recompression cycle calculations. The simple, recuperative Brayton cycle is typically used in the
nuclear power industry with helium as the working fluid, and can reach efficiencies superior to
LWRs. High Temperature Gas-cooled Reactors (HTGRs) can reach efficiencies of 43-48 % and
the Modular Pebble Bed Reactor (MPBR) is expected to reach efficiencies approaching 45 %
[Wang, 2003]. The ability to model the simple cycle is included in CYCLES III, as a
comparison with the recompression cycle and so that the code is useful for more applications.
The main improvements in CYCLES II were the inclusion of a detailed pipe model and
the consolidation of the input and output files. Figure 2.5 shows the numbering scheme applied
to the pipe paths in CYCLES III for the recompression cycle.
9-------->-------9
|
|
|
1--<--PRE-10-|
|
|
11
7--->----7
|
\/
\/
^
|
|
|
|
|
|
| MCOMP=========RECOMP===========TURBINE
|
|
|
12
|
|
^
|
\/
^
|
| \/
|
6
|
|
|
3--->-3----4->---4
|
|
|
|
|
|
^
|
|
2-->-LTR<--<---8---<--HTR->-5>-IHX
|
|
|
|
|
|
|
^
|
|
|
|
|
|
|
7------<----------7
9----<----9
Figure 2.5: Pipe paths in the Recompression Cycle of CYCLES III
Each of the twelve paths shown in Figure 2.5 has detailed information about the
dimensions of the pipes, to allow for the effect of pressure drop on efficiency. Legault was able
to show, with Hejzlar’s detailed pipe model, that the pressure drops within pipes and plena were
important to the efficiency of the cycle [Legault, 2006]. Piping losses can reduce the efficiency
of the plant by as much as 2.0 %, depending on the design of the piping. Consolidating the
26
input and output files made the code much easier to use. For the user, these input and output
files made the information very readable, and the files could be used in papers and presentations
without significantly altering their format.
CYCLES II eliminated Dostal’s option of optimizing heat exchanger volumes. CYCLES
II, therefore could only produce results for the cycle defined by the user and the user would have
to make judgments about how to size the heat exchangers. The goal with CYCLES III was to
include all of the best features of the previous editions, along with some new features. After
completion, the new developments in CYCLES III are:
1.
2.
3.
4.
5.
6.
Extended use of the easy to read Legault code structure and nomenclature
Inclusion of both optimization and single point calculations
Inclusion of both Simple and Recompression Cycles
The ability to model impure working fluids (or a variety of different working fluids)
An interface for the headers in TSCYCO
A convenient code manual (see Appendix A) for troubleshooting
2.2.1 Extended Use of the Legault Nomenclature
CYCLES II incorporated an easy to read nomenclature, but Legault recommended that
the use of it be extended further into some of the subroutines and that the structure should be
simplified further. Though it may be contrary to some programmers’ preference, the use of a
large variable module has been incorporated in CYCLES III so that the program code is easily
readable and understandable for new users. Variable names are more intuitive than in the
original CYCLES. For example, Legault changed temperature variables in the LTR from
Dostal’s structure:
trl(1)
inlet temperature of the cold side (°C)
trl(2)
outlet temperature of the cold side (°C)
trl(3)
(not used)
trl(4)
inlet temperature of the hot side (°C)
trl(5)
real outlet temperature of the hot side (°C)
trl(6)
ideal outlet temperature of the hot side (°C)
to a derived type structure that looks like:
ltr.TinCold
inlet temperature of the cold side (K)
ltr.ToutCold
outlet temperature of the cold side (K)
ltr.TinHot
inlet temperature of the hot side (K)
ltr.ToutHot
real outlet temperature of the hot side (K)
ltr.ToutHoti
ideal outlet temperature of the hot side (K)
27
Legault used similar derived-type structures for enthalpy, pressure, entropy, density, and
many other characteristics of the LTR. Likewise, the HTR, pre-cooler, turbine, and compressors
have their own variable names with derived-type structures. These are contained in the file
modGlobalVariables.f90. This file has undergone only minor changes from CYCLES II to III.
A new data type, optdata has been added. It consists of dimensions for the heat exchangers so
the code can read through optimization results and select the dimensions consistent with the
highest efficiency while optimizing the cycle. Other small changes include the addition of a
throttle on the compressors. An input has been added so that the user can define a form loss
coefficient to represent the throttle. These inputs are explained more in Appendix A. The form
loss is necessary because, as TSCYCO shows, compressor throttles must be in the partially
closed position at steady state operation in order to allow for controllability during transients.
Form loss coefficients can be based upon experience from operating TSCYCO.
In CYCLES III, Legault’s nomenclature was expanded to the subroutines that compute
the performance of each component of the cycle. The variable module modGlobalVariables has
been expanded to:
EXPAND
COMPRESS
PCHEVOL
PRECOOLER
The subroutine for the turbine
The subroutine for compressors
The recuperator subroutine
The precooler subroutine
This increases the general readability of the code.
2.2.2 Optimization and Single Point Calculations
The optimization function available in Dostal’s original CYCLES has been returned in
CYCLES III. The code is easier to use than the original because it simplifies the outputs for the
user and minimizes the manipulation of data required on the part of the user. The optimization
approach is “brute force”, iterating from an initial guess of recuperator volumes until the
efficiencies of many different configurations have been examined. The heat exchangers in the SCO2 cycle are assumed to be counter-flow PCHEs, without a detection gas plate. Optimization is
performed by selecting a total volume of heat exchangers. This total volume will then be divided
among the pre-cooler, High Temperature Recuperator (HTR), and Low Temperature Recuperator
(LTR) until the configuration of highest efficiency is found. In the original CYCLES, an output
file called res.txt listed heat exchanger volumes and the resultant cycle efficiency, as shown in
Figure 2.6.
28
Figure 2.6: The original output res.txt from the original CYCLES.
The code user then had to pick through the output to find the configuration which yielded
the highest efficiency. The optimization output included forty parameters of the cycle, far more
than necessary to perform an optimization. Heat exchanger volumes are not even visible when
the file is opened, because they are so far to the right in the text. In subroutines named
OPTIMIZE_R and OPTIMIZE_S, for the recompression and simple cycles respectively,
CYCLES III performs this calculation and then steps through all of the results in res.txt to locate
the optimum configuration without burdening the user with the task. As the code steps through
the optimization results, the heat exchanger dimensions are stored in optdata, replacing the
dimensions each time a configuration of higher efficiency is found. Once res.txt has been read,
the code calculates the performance of the cycle having heat exchanger dimensions defined by
optdata and then outputs the results.
2.2.3 The Simple Recuperative Brayton Cycle
Including the Simple Recuperative Brayton cycle in CYCLES III allows the user to
quantify the efficiency gain that is achieved by using the Recompression cycle. It also makes
CYCLES III a useful tool for designers in power conversion systems using other NIST fluids, a
new capability discussed in Section 2.2.4. Coding of the cycle calculations for the simple cycle
in CYCLES III was based on the structure of the code for the recompression cycle in CYCLES
II. If the user looks into the code, he or she will find that the file simple.f90 looks almost
identical to the file recompress.f90. It has been modeled after Legault’s structure so that the user
29
can easily transfer knowledge and experience between the two. Choosing the simple cycle or the
recompression cycle is as easy as changing the value of itype in the input file HXdata.txt. The
simple cycle can be used to model typical helium Brayton cycles. Investigation will show that a
simple cycle will run with S-CO2, but cycle efficiencies will be too low to ever be viable in
industrial use.
2.2.4 Inclusion of other NIST Fluids
The most drastic change in CYCLES III, in terms of functionality, is the ability to model
the cycle with any NIST fluid and mixtures of NIST fluids. CYCLES III determines fluid
properties from highly developed polynomials, available from the National Institute of Standards
and Technology (NIST). These polynomials are available through Fortran subroutines, but can
be slow to use if many calculations are needed [NIST, 2007]. To speed up calculations,
CYCLES has always included the ability to develop tabulated fluid properties from these
polynomials. Once the tables are complete, the code can run much faster, however, the original
table creation subroutines were written only for pure fluids. It is certain that no working fluid
can be entirely pure, and that impurities will change the critical point of the mixture. Figure 2.7
shows the effect of a few impurities on the critical point of CO2. The critical point can be
identified by the spike in heat capacity at constant pressure.
30
Figure 2.7: The effect of impurities on the critical temperature of CO2. The molar specific heat
capacities of CO2 mixtures are shown for lines of constant pressure. a.) pure, b.) 1% H2 by mole
fraction, c.) 1% Propane by mole fraction
The modifications in CYCLES III allow the user to include any fluid available in the
NIST database as an impurity, including an approximation for air. The input file HXdata.txt
includes a portion near the top that reads:
31
0
1
2
0.010d0
!ifluid This is the first fluid, 0-CO2, 1-Ethane, 2-Helium
!mix IDs if there is a 2nd fluid, 0-pure, 1-2nd fluid exists
!ifltwo IDs 2nd fluid, 0-He, 1-Air,2-Hydrogen,3-Nitrogen,4-Methane
!fracgas This is the mole fraction of the 2nd fluid(if applicable)
The inputs listed above would correspond to the mixture shown in Figure 2.6b. Air is
included in CYCLES III as a mixture of 78.12 % N2, 20.96 % O2, and 0.92 % Ar by mole
fraction, as approximated in NIST’s REFPROP program. The computing time required to
develop tables is much longer for mixtures, but it is necessary because the effect of impurities
can be substantial, as discussed in Section 2.4. The user could choose to modify the code and
change the available NIST fluids, simply by changing the reference fluid property file called by
CYCLES III. This is discussed in Appendix A.
The ability to model the cycle with other fluids is especially useful in the simple cycle,
because the simple cycle has been used in many industries, with several different working fluids.
Research into new power cycles can take advantage of this capability, as discussed further in
Section 2.3.
2.2.5 Interfacing with TSCYCO
The Transient S-CO2 Cycles Code (TSCYCO) is Tri Trinh’s updated version of Shih
Ping Kao’s S-CO2 Power Systems (SCPS) code [Trinh, 2009], [Kao, 1984]. TSCYCO performs
transient analysis of the S-CO2 recompression cycle for several different transients based on
energy, mass, and momentum conservation laws. Its function is very different from that of
CYCLES III, but it does require information about the structure of the piping that is in a different
format than CYCLES III. The headers in TSCYCO are modeled as single equivalent pipes, one
for each of the twelve paths in Figure 2-5. The required dimensions are:
The total internal volume of the header
The thickness of the equivalent pipe
The length and diameter of the header
The total heat transfer area of the header
Because each path in Figure 2.5 has a single header in TSCYCO, CYCLES III lumps all
the passages in each of the twelve paths into one header, preserving total volume of steel, total
internal volume, and pressure drop. Because transient effects depend on mass flow rates, and
therefore accumulations of mass, the volumes of the headers are important. These are preserved
from CYCLES III in a rough approximation based on the hydraulic diameter of each passage in
32
the detailed pipe model of CYCLES III. Because of the thermal inertia of the pipes’ steel, an
approximation of the volume of steel is made for each header based on the ASME required
thickness of an equivalent pipe. The ASME minimum required thickness for a circular pipe is
given by
𝑃𝐷
𝑡≥
Eqn. 2-1
2 𝑆 + 𝑃𝑦
where t is the pipe’s thickness, P is the internal pressure, S is the maximum allowable stress
intensity for the material, y is a safety factor equal to 0.4, and D is the outside diameter of the
pipe [ASME, 2007]. In CYCLES III, the subroutine HEADERS calculates the header
dimensions for TSCYCO by determining approximate:
Heat transfer areas,
Volumes of steel, and
Internal pipe volumes
for each of the twelve paths. Then each path is recreated as a single pipe by increasing the length
of the pipe and adjusting dimensions to preserve the three quantities listed above. The length is
increased until the pressure drop of the header matches that calculated in CYCLES III for the
path. Pressure drops are calculated as
∆𝑃 =
𝑓 2𝐿 𝑘 2
𝜌𝑣
+ 𝜌𝑣
2
𝐷 2
Eqn. 2-2
where ρ is the fluid density, v is the velocity, L is the header length, D is the header diameter, k
is a form loss coefficient, and the friction factor, f, is determined from the Blasius correlation for
low Reynolds numbers and the McAdams correlation for higher Reynolds numbers [Todreas and
Kazimi 1993].
𝑓=
0.316𝑅𝑒 −0.25 , 𝐵𝑙𝑎𝑠𝑖𝑢𝑠 𝑐𝑜𝑟𝑟𝑒𝑙𝑎𝑖𝑜𝑛: 𝑅𝑒 < 30,000
0.184𝑅𝑒
−0.20
Eqn. 2-3
, 𝑀𝑐𝐴𝑑𝑎𝑚𝑠 𝑐𝑜𝑟𝑟𝑒𝑙𝑎𝑡𝑖𝑜𝑛: 𝑅𝑒 ≥ 30,000
For every pipe in CYCLES III, the form loss factor k was assumed to be 1.50. This
comes from a contribution of 1.0 for fluid expansion and 0.5 for fluid contraction at the pipe
outlets and inlets, respectively. The dimensions and pressure drops are presented as output in
headers.txt. CYCLES III and TSCYCO showed good agreement in the header calculations,
producing pressure drops that matched by + 5 % at TSCYCO’s steady state [Trinh, 2009].
33
Another change to CYCLES III was the inclusion of form losses at each compressor
outlet. These losses are due to the throttles required in a real design for controllability. In
TSCYCO, the controllers can be tuned to provide adequate controllability, but throttles must be
partially closed in steady state operation. The losses associated with partially closed throttles can
be represented in CYCLES III by the form loss coefficients Kmc and Krc for the main and
recompressing compressors respectively.
2.3 The Ethane Cycle
The increased capabilities of CYCLES III allowed modeling of a simple recuperative
ethane cycle, to determine if it could be cost competitive for electricity production. Hejzlar and
Driscoll noted that the critical point of ethane makes it appealing for use in power cycles, due to
the advantage of the low compressor work as in the S-CO2 cycle [Driscoll and Hejzlar, 2007].
Ethane dissociates at high temperatures into myriad hydrocarbons. The dominant
dissociation reactions are
𝐶2 𝐻6 → 2𝐶𝐻3
Eqn. 2-4
𝐶𝐻3 + 𝐶2 𝐻6 → 𝐶𝐻4 + 𝐶2 𝐻5
Eqn. 2-5
2𝐶2 𝐻5 → 2𝐶2 𝐻4 + 𝐻2
Eqn. 2-6
producing methane (CH4), ethylene (C2H4), and hydrogen as major constituents of the
dissociated mixture [Perez, 2008]. Other reactions result in the presence of propylene (C3H6),
propane (C3H8), and butane (C4H10), though to lesser, but still not well known, degrees. The
critical points of methane and ethylene are such that they hurt the efficiency of the ethane cycle.
High pressures suppress the dissociation, so it is difficult to determine the extent to which
dissociation will occur. Table 2-1 shows the critical points of pure ethane (C2H6), methane
(CH4), ethylene (C2H4), and the dissociated mixture. The mixture is expected to undergo very
high dissociation at PCS temperatures. For example, the equilibrium concentration of ethane
after temperature has been increased to ~700 K is only ~55 % at 20 MPa [Perez, 2008].
34
Table 2.1: Critical points of fluids important to the Ethane Simple Cycle
Critical Temp. (K)
305.32
190.56
282.35
33.15
Ethane
Methane
Ethylene
Hydrogen
Critical Press. (MPa)
4.872
4.592
5.042
1.296
Critical Dens. (kg/m3)
206.18
162.66
214.25
31.26
Modeling the simple Brayton cycle in CYCLES III, pure ethane can achieve high
efficiencies as shown in Figure 2.8 for a range of turbine inlet temperatures.
46
Cycle Efficiency (%)
45
44
43
42
41
40
400
450
500
550
600
Turbine Inlet Temperature (oC)
Figure 2.8: The efficiency of the pure ethane simple cycle
The simplicity of the cycle, high efficiency, and the easy availability of ethane made this
concept very appealing. In a real system, some of the ethane must dissociate, but if the critical
point was not impacted too greatly high efficiencies might still be achievable. Ethane/methane
mixtures were used in CYCLES III to model the behavior of the simple cycle under different
dissociation conditions since the real dissociation effects are not well known. Ethylene was not
included because the REFPROP data for ethylene only extends to a temperature of 450 K. The
critical point of ethylene is not as damaging to the cycle as that of methane is, so the use of
methane impurities only should still give a good assessment of how dissociation affects cycle
efficiency. Hydrogen is not expected to appear in large portions, but the critical point of
hydrogen is so low that its presence will be very detrimental to cycle efficiency. Figure 2.9
35
shows the effects of methane concentration on the ethane cycle operating with a turbine inlet
temperature of 470 oC. This turbine inlet temperature was chosen because the applicable range
of methane data in REFPROP is also limited by temperature.
42
Cycle Efficiency (%)
41
40
39
38
37
36
0
5
10
15
20
Methane Content (mole %)
Figure 2.9: The effect of dissociation on the ethane cycle
After a methane concentration of 20 % by mole, the efficiency loss is approximately
linear with methane percentage. Equilibrium concentrations of ethane at the high temperatures
of the cycle are less than 60 %. At the expected level of dissociation, the simple ethane cycle
fails to perform beyond the efficiencies achieved in current LWRs. As mentioned in Chapter 1,
other simple gas cycles running on helium have achieved very high efficiencies. Unless
experiments show that dissociation does not occur nearly to the predicted levels or that
recombination mitigates reverses the process at low temperature, the ethane simple cycle cannot
be viable for economic power conversion systems. Also, the flammability of ethane and
methane introduces problems for safety.
2.4 Fluid Impurities in the S-CO2 Recompression Cycle
The performance of the recompression cycle is attractive because the main compressor
operates just above the critical point, reducing compressor work a great deal. The critical point
of CO2 is at 7.377 MPa and 30.98 oC. The low dissociation and low corrosion rates of CO2, and
temperature of commonly available cooling water mean that CO2 is a good choice for working
fluid. Inevitably present impurities in the fluid will change the critical point. Compressor work
will rise if the critical point is lowered from that of pure CO2 as the compressor inlet conditions
become further from the critical point. Cooling water puts a lower limit on the temperature of
36
the fluid at the inlet to the main compressor. In order to predict the effect of expected impurities
or a detection gas on cycle performance, optimized cycles were run on a series of different fluids.
A detection gas is desirable in many cases because the S-CO2 cycle could be used as a
direct cycle. In a direct cycle, detecting CO2 leaks would warn of a primary system rupture
before the problem became dire. For SFRs, sodium reacts exothermically with CO2, so detecting
leaks early can prevent major problems. Helium is frequently used as a detection gas because it
reveals leaks sooner than almost any other gas (due to its small atomic size). Also, it usually has
minimal detrimental effect on engineering systems because it is chemically inert. Other options
for a detection gas could be any chemical lighter than CO2 that doesn’t significantly lower the
critical temperature.
Air is inevitably present as an impurity in any gas purchased for industrial use. The cost
of gases depends strongly on their purity as well. Studying the amount of air tolerable in an SCO2 cycle will help to reduce the operating cost of the cycle by reducing the required purity of
the working fluid. It will also let designers know what level of air impurity can be tolerated
before they can expect the efficiency of the system to be unacceptably low.
2.4.1 Helium Additions for Leak Detection
Based on a 2400 MWth, 4 loop design, it is estimated that 0.5 mole percent helium is
needed to detect leaks in the CO2 recompression cycle [Freas, 2007]. At 600 MWth per loop,
this estimate can be applied to the recompression loop studied here. Recompression cycles will
likely be built for 400 MWth or larger systems (per loop) because of the constraints on
compressor design, as discussed in Chapter 3.
Shifting the critical temperature to too low a value will cause the cycle to lose efficiency
because cooling water temperature cannot be drastically changed. Shifting it to too high a value
will cause the main compressor inlet state-point to fall below the vapor dome, a consequence to
be avoided. Some test results, however, show that operation below the vapor dome is not
necessarily damaging to the system [Hejzlar, 2008b]. The critical points of gases discussed in
this chapter are shown in Table 2.2.
37
Table 2.2: Properties of selected fluids at their critical points
Critical
Critical
Added
Pressure
Temperature
Constituent
(MPa)
(K)
Carbon Dioxide(CO2)
Pure
7.377
304.13
1 % Helium
7.312
301.14
1 % Propane
7.316
304.10
0.2% Air
7.404
303.96
Ethane (C2H6)
Pure
4.872
305.33
Helium
Pure
0.227
5.195
Propane (C3H8)
Pure
4.251
369.89
Air (78%N2, 21%O2, 1%Ar)
3.786
132.53
Critical
Density
(kg/m3)
467.6
465.2
461.8
467.0
206.6
69.6
220.5
342.6
CYCLES III predicts an almost linear relationship between turbine inlet temperature and
net cycle efficiency. Net cycle efficiency refers to the efficiency of the cycle once all losses,
including the pumping power of water in the precooler, have been accounted for. The behavior
is similar regardless of the pressure drop in the IHX, i.e. the linear relationship between turbine
inlet temperature and efficiency holds, but the efficiency falls for higher pressure drops. Figure
2.10 shows the net cycle efficiency of the recompression cycle for an IHX pressure drop of 300
kPa and turbine inlet temperature of 510 oC.
Net Cycle Efficiency (%)
43
42.5
42
41.5
41
40.5
40
480
490
500
510
520
530
540
550
Turbine Inlet Temperature (oC)
Figure 2.10: The effect of turbine inlet temperature on the cycle operating with pure CO2,
assuming an IHX pressure drop of 300 kPa.
The behavior of the cycle at varying turbine inlet temperatures is similar for changing
impurity concentrations. For the impurities discussed next, 510 oC was chosen as the constant
38
turbine inlet temperature, but the efficiency lost for a given impurity concentration, as compared
to the efficiency when running on pure fluid, will be the same regardless of turbine inlet
temperature. Figure 2.11 shows the net cycle efficiency when helium impurities are added to the
working fluid. No real change is observed until the mole fraction of helium reaches 0.002. The
change in the critical point causes a rise in the main compressor work, as shown in Figure 2.12.
Net Cycle Efficiency
42
41.5
41
40.5
40
39.5
0
0.002
0.004
0.006
0.008
0.01
0.012
Mole Fraction of He
70
65
60
55
50
45
40
35
30
RC Work (MW)
MC Work (MW)
Figure 2.11: The effect of a helium leak detection gas on the S-CO2 cycle efficiency. About 0.5
mole % He is expected to be needed for leak detection in a large plant.
0
0.005
0.01
0.015
70
65
60
55
50
45
40
35
30
0
Mole Fraction of He
0.005
0.01
0.015
Mole Fraction of He
Figure 2.12: The compressor work as the helium mole fraction is changed. The main compressor
(left) is affected more than the recompressing compressor (right).
As the critical point shifts, the main compressor becomes closer and closer to an ideal gas
compressor, requiring more work. The recompressing compressor, already further from the
critical point, shows less of a change in the work required. It actually requires less work for
39
compression as the mole fraction of helium is increased, but the decrease in the work of the
recompressing compressor is more than offset by the increase in work of the main compressor.
It is evident that helium has a detrimental effect on the efficiency of the S-CO2
recompression cycle. A level of 0.5 mole % helium degrades the efficiency of the representative
plant by about 1.0 %. If the cooling water temperature could be decreased commensurate with
the critical point, efficiency could be maintained, but that is only realistic to a point and only in
colder latitudes.
2.4.2 Air Impurities
Other inevitable impurities will exist, air being the most likely. However, the critical
point of helium is so low that if it is present, helium’s effects will dominate and any efficiency
penalty due to air will be negligible in comparison. It can be assumed that CO2 used in any
operating cycle will be relatively pure, but some air will be present. Unless the fraction of air
impurities becomes high, it is expected that little to no effect on efficiency will be observed.
Figure 2.13 shows the effect of air impurities on the net cycle efficiency.
Net Cycle Efficiency (%)
42
41.5
41
40.5
40
39.5
0
0.001
0.002
0.003
0.004
0.005
0.006
0.007
Mole Fraction of Air
Figure 2.13: The effect of air impurity on the S-CO2 cycle. The mole fraction of air in
commercially available CO2 can be 0.001 or even lower. The effect of air on the cycle is not
detrimental at the concentrations expected.
Air has no effect on the cycle performance, up to 0.0035 mole fraction and a small
negative impact up to mole fractions of about 0.006. These air concentrations are well above the
expected range of air impurities because commercially available CO2 has purities of 99.8 % and
above [Freas, 2007]. Even at an air mole fraction of 0.010, the efficiency penalty is still just 1.05
40
%. Therefore, likely levels of air impurities are expected to contribute negligibly to losses in
cycle efficiency.
2.5 Chapter Summary
The S-CO2 recompression cycle outperforms other cycles in efficiency terms if operated
within certain temperature ranges. Its turbomachinery is markedly more compact, as evidenced
in the compressor designs discussed next in Chapter 3. Test recompression cycles have not been
built and further work is needed to model these systems. CYCLES III is designed to give
accurate calculations for the operation of simple and recompression Brayton cycles at steady
state. Its major improvements are the inclusion of the simple cycle and the ability to model fluid
impurities, or cycles using any fluid.
CYCLES III runs showed that a helium leak detection gas was detrimental to the
efficiency of the S-CO2 cycle, causing a loss of 1.0 % efficiency at a helium mole fraction of
0.005. If leak detection capability can be proven effective in supercritical CO2 with a helium
mole fraction of less than 0.005, then helium leak detection could be a very appealing feature of
the recompression cycle. Air impurities are not as harmful to the cycle and can be tolerated at
mole fractions up to 0.0035 with essentially zero loss in efficiency. Expected air impurities have
no real effect on cycle efficiency.
The ethane simple recuperative cycle shows little promise unless ethane cracking can be
maintained at a low mole fraction. This is likely not possible at the high temperatures needed to
produce an attractive efficiency. The rate of recombination, and the equilibrium concentration of
different hydrocarbons are not known for an ethane cycle, though. Therefore, conclusions about
the attractiveness of the cycle cannot be complete until more experimental data is available in the
high pressure and high temperature ranges that would be necessary for a PCS. From analysis of
the cycle performance with expected dissociation, however, the ethane recuperative cycle
appears to have very little real promise.
41
2.7 Nomenclature for Chapter 2
D
f
k
L
𝑚
P
Re
S
t
v
y
Diameter (m)
Friction factor, main compressor flow fraction
form loss coefficient
Length of pipe (m)
mass flow rate (kg/s)
Pressure (Pa)
Reynolds number
Maximum allowable stress (Pa)
pipe thickness
flow velocity (m/s)
Safety factor of 0.4
Greek Letters
𝜌
Density (kg/m3)
42
3 S-CO2 Compressor Design
3.1 Introduction
CO2 compressors are used in many industries, but few applications require operation
close to the critical point. Carbon sequestration is one area in the electric power production
industry that may spawn interest in S-CO2 compressor design. Presently, however, there is little
available information on large-scale S-CO2 compressors operating near the critical point.
Modeling of CO2 compressors, especially those operating near the critical point, is important to
further development of the recompression cycle.
3.2 Developing Compressors for the S-CO2 Cycle
3.2.1 The Need for a Compressor Model
Efforts on the part of Tri Trinh have been directed toward modeling of S-CO2 power
conversion systems in transients and tuning controls for the system [Trinh, 2009], but
compressor performance in his work was originally based on ideal-gas models adjusted to CO2
densities. The adjustment consists of scaling compressor work inversely with density. The
profound changes in fluid properties near the critical point make this approach less than certain.
Also, the reductions in compressor work and in turbomachinery size are some of the main
advantages of an S-CO2 recompression cycle. These factors demand the development of models
to design the compressor for the steady state design point and to model the off-design
performance of S-CO2 compressors. Centrifugal compressors are the first choice for the
recompression cycle because they are durable, have larger margin to surge than axial
compressors, and are expected to perform well based on the operating parameters of the S-CO2
cycle. The Real Gas Radial Compressor (RGRC) code was developed in this work for the
purpose of sizing and modeling the performance of S-CO2 centrifugal compressors.
3.2.2 Compressor Background
There are many types of industrial compressors, including axial, positive displacement,
and centrifugal (a.k.a. radial) compressors. All compressors can be described by a set of nondimensional numbers. These are specific speed, flow coefficient, and head coefficient [Japikse,
1994]. They are defined as
43
Specific Speed,
𝛺 𝑉
𝑁𝑠 =
3
(𝑔𝐻)4
Eqn. 3-1
Flow coefficient,
𝜙=
𝑚
𝜌Ω𝐷3
Eqn. 3-2
𝜓=
Δ𝑃
𝜌Ω2 D2
Eqn. 3-3
Head coefficient,
where D is the outlet diameter of the impeller, Ω is the rotational speed in revolutions per
second, 𝑉 is the volumetric flow rate at the inlet, g is the acceleration of gravity, H is the
adiabatic head (in units of length), 𝑚 is the mass flow rate, and ρ is the density at the impeller
inlet in consistent units. Using these parameters, a designer can select the type of compressor
appropriate for a given application. Figure 3.1 shows the range of applicability for a few
different compressor types. In industry practice it is common to encounter “non-dimensional”
parameters that retain many Imperial units. For these cases, volumetric flow is in gallons per
minute, rotational speed is in RPM, and head is in feet. The result is that a truly non-dimensional
specific speed will be equal to 1/129 the value obtained with Imperial units. Care should be
taken to examine the unit system for specific speed calculations, as the Imperial unit system is
not the only one used in industry.
44
Figure 3.1: Design ranges for different compressor types. Lines of constant specific
speed are labeled. Centrifugal compressors are appropriate in Region A, Axial compressors in
Region B, and Reciprocating compressors in Region C. Other compressor designs are used in
other regions of this plot. Adapted from Japikse [1994].
In the S-CO2 recompression cycle, centrifugal compressors are the most appropriate,
based on experience with the non-dimensional parameters. Every design discussed here will
assume a rotational speed of 3600 RPM at the design point, in order to synchronize the shaft with
the electric grid. Looking at the equation for specific speed, it becomes clear that, to achieve
good performance, smaller machines should be built with higher rotational speeds or lower stage
pressure ratios. For example, the small test compressor operated by Sandia National Laboratory
has a mass flow rate of only 3.53 kg/s at the design point. With a pressure ratio of about 1.8, this
requires that the compressor be operated at speeds of 65,000 to 75,000 RPM [Wright et al.,
2008]. Tables 3.1 and 3.2 show the operating points and non-dimensional parameters
characteristic of a recompression cycle design of 500 MWth. This cycle is identical to the
recompression cycle discussed in Chapter 2.
45
Table 3.1: The operating points of the compressors in the recompression cycle. The
recompressing compressor is assumed to be a two-stage design.
Inlet
Outlet
Mass Flow
Pressure
Temperature
Pressure
Temperature
Rate (kg/s)
(MPa)
(oC)
(MPa)
(oC)
1589.1
7.67
31.89
20.00
60.97
Stage 1
989.6
7.71
66.99
12.42
107.44
Stage 2
989.6
12.42
107.44
19.92
157.34
Stage
Main
Recomp
Table 3.2: The non-dimensional parameters of the compressors in the recompression
cycle. The two stages of the recompressing compressor are shown separately.
Mcomp
Specific Speed Ns
Recomp 1
Recomp 2
Sandia Test
3.42
4.04
2.83
3.39
Head Coefficient Ψ
1.917E-3
1.640E-3
2.885E-3
3.689E-3
Flow Coefficient φ
9.80E-04
1.08E-03
1.24E-03
2.58E-03
Based on the non-dimensional parameters given in Table 3.2, the centrifugal compressor
is the best choice for the S-CO2 recompressing compressor. The values of the non-dimensional
parameters given in Table 3.2 are extreme values in the centrifugal compressor range. They do
not match Figure 3.1 but the centrifugal compressor is a better option than any other based on
Figure 3.1. The values for the Sandia test compressor are based on a rotational speed of 45,000
RPM. The Sandia test compressor comes closer to the centrifugal compressor region in Figure
3.1. The recompression cycle’s main compressor can be designed as a single stage for a PCS
rated above ~400 MWth and a two-stage design is best suited for the recompressing compressor.
If the PCS power is reduced below 400 MWth, the mass flow rates of the compressors are
reduced. In order for the specific speed of each stage to remain within the range of centrifugal
compressors, the pressure ratio of each stage must be reduced, and therefore the number of
stages must be increased. In order to produce a design of only one stage for the main
compressor, the cycle power must be above 400 MWth.
To understand the operation of a compressor design code a few centrifugal compressor
concepts will be necessary. First, terminology is necessary to understanding. Figure 3.2 shows
several parts of a centrifugal compressor. The view given in Figure 3.2 is termed the meridional
plane. It is defined as the plane that contains the axis of rotation.
46
Figure 3.2: The meridional plane view of a typical centrifugal compressor. This exaggerated
view is probably not to scale for any real compressor design.
Flow enters at the impeller inlet, sometimes called the inducer. The term inducer
identifies the portion of the rotor whose blade passage is at the inlet, before radius has begun to
increase. The impeller turns along the axis, and flow exits the impeller into a vaneless space.
Sometimes the vaneless space is formed with two parallel plates and sometimes the plates
diverge. In the Real Gas Radial Compressor (RGRC) code, the vaneless space is always of
constant depth in the axial direction. Beyond the vaneless space is typically a vaned diffuser,
which serves to slow down the flow and raise the static pressure of the fluid. It is a circular plate
of metal that has vanes inside it which direct the flow, such that the flow area of the passage
increases as radius increases. Figure 3.3 shows a view of a straight-vaned diffuser looking along
the axis of rotation. In RGRC the vanes can be curved by the user up to 25o. This just means
that the angle of the vane with respect to the radial direction will be 25o greater at the diffuser
outlet than it is at the diffuser inlet. The area ratio of a diffuser refers to the ratio of the outlet
flow area to the throat flow area.
47
Figure 3.3: A straight vaned diffuser. These vanes are of width 8o and there are twenty of them.
They are inclined at 70o to the radial direction.
After the vaned diffuser is a scroll, or volute, which collects the flow and directs it to the
next component of the system. In a multi-stage compressor the vaned diffuser of upstream
stages will be followed instead by a return channel which collects the flow and directs it to the
next stage.
The volute is neglected in RGRC because its design will depend on available space and
material. In the multi-stage performance code, the return channel is treated as a form loss with a
form loss coefficient chosen by the user. The casing refers to the static portion of the compressor
which surrounds the impeller. In shrouded impellers, the blade passage is completely enclosed
from hub to shroud. Unshrouded impellers, like those in RGRC, have blade passages which are
open at the tips of the blades. This produces some losses which will be discussed later.
Compressors operate between two limits: surge and choke. At low mass flow rates,
compressors stall or surge depending on the phenomenon exhibited. Both are limiting
conditions. Stall occurs when the flow experiences separation of the boundary layer on blades or
in narrow passages. Stalled portions of the flow develop and form stall cells. Sometimes the
cells will rotate through the compressor in what is aptly called rotating stall. RGRC neglects
these phenomena and simply states that stall is likely to have occurred. Surge, the other
phenomenon which occurs at low mass flow rates, results in reversal of the flow and can cause
violent vibrations. The operating range of a compressor is bounded at low mass flow rates by
what is termed the surge line, as shown in the example compressor map of Figure 3.4. The
operating limit at low mass flow rates is termed the surge line by convention, whether the actual
phenomenon occurring is surge or stall. In practice, the surge line occurs where the slope of the
48
pressure ratio curve is zero. An operator will likely find that the surge line in RGRC occurs at
higher mass flows than the zero slope condition, due to the many conditions that the code will
identify as surge/stall. These conditions are believed to be conservative estimates because none
of these criteria has been seen to be sufficient to predict surge in experiments.
Figure 3.4: Example compressor performance map showing the relationship between the surge
points for different speeds (in % of design speed) in a typical centrifugal compressor.
At high mass flow rates, regions of the flow will become supersonic, resulting in severely
deteriorating performance at high mass flow rates. Choke is assumed to occur when the ratio of
the core flow velocity is greater than 90 % of the critical velocity. A note should be made about
calculating the critical velocity. The critical velocity will always be different than the speed of
sound and will always be smaller. Critical velocity is defined as the velocity that must be chosen
to achieve supersonic flow for a constant value of stagnation enthalpy. Critical velocity is
therefore a function of stagnation enthalpy whereas the speed of sound is a function of static
enthalpy. Appendix B discusses the development of the critical velocity database. At each point
in the calculation of compressor performance, the core flow velocity is compared to the critical
velocity to test for choke.
The Euler turbomachinery equation describes the fundamental cause of pressure rise in a
compressor. By imparting rotation to the fluid, the total enthalpy of the fluid is increased. The
rise in static enthalpy through the impeller is given by [Cumpsty, 2004]
𝑕2,𝑠𝑡 − 𝑕1,𝑠𝑡 =
1 2
𝑈 − 𝑈12 + 𝑊12 − 𝑊22
2 2
49
Eqn. 3-4
where h2,st and h1,st represent the static enthalpies of the fluid at the impeller inlet and outlet
respectively. The letter U represents blade speed and W is the relative speed of the fluid.
Subscripts indicate inlet at the Root Mean Square (RMS) radius (1) or outlet (2). The Euler
turbomachinery equation is used directly in RGRC to calculate the static enthalpy rise produced
by the impeller. Further rise in static enthalpy is achieved by pressure recovery in diffusers,
where the dynamic pressure, ½ρv2, is reduced and static pressure rises as the flow area expands.
Mean-line codes treat the flow in a compressor by modeling the behavior of the fluid at
the mean streamline, the surface along the RMS radius of the blade passage. Mean-line
compressor codes can be used to develop initial sizing, design, and performance estimates for
compressors. In fact, these methods are typically more accurate for determining off-design
performance than more advanced computational fluid dynamics (CFD) codes due to their
empirical nature [Aungier, 2000]. The mean stream line method was used to improve on
existing compressor codes to achieve some understanding of how compressors will perform in an
S-CO2 recompression cycle.
Existing codes available to MIT did not produce results consistent with themselves and
they were very difficult to use and manipulate. These codes were adjusted NASA codes known
as CCD and CCODP, which had been altered to work for supercritical CO2. The code structure
made editing very tricky. Several remaining ideal-gas assumptions were not appropriate for a
real gas compressor. These assumptions persisted despite the fact that real gas property
subroutines were included in the codes. Therefore, effort was needed to produce an improved
mean-line real gas compressor code.
3.3 Real Gas Radial Compressor (RGRC) Code
3.3.1 Issues with Earlier Codes CCDS/CCODS and Motivation for RGRC Development
The NASA mean line centrifugal compressor codes were adjusted by Yifang Gong of the
MIT Gas Turbine Laboratory (GTL) to operate with CO2 near the critical point [Hejzlar et al.,
2007] These codes consisted of a design code, CCD, and an off-design code, CCODP. They
had originally been developed at the NACA Lewis Flight Laboratory, primarily by Jerry Wood
[Wood, 1995]. After updates for S-CO2 by Gong, they were renamed CCDS and CCODS, with
the “S” added for S-CO2.
The CCD/CCODP input data requires the user to choose some basic parameters, but most
of the geometry of the compressor is determined by the code in a complicated iterative process.
The NASA codes rely on correlations and experimental diffuser data for air and its calculations
50
are based on a constant value for the ratio of specific heats, Cp/Cv. For diffuser performance the
NASA codes used Runstadler’s database of diffuser performance [Runstadler, 1969]. This
database is one of the most thorough available, but it was determined that the data were not
realistic for S-CO2. For Supercritical CO2 the ratio of specific heats and other properties change
dramatically near the critical point, so Gong’s changes were focused on developing a set of
polynomials for determining fluid properties for S-CO2 and implementing those subroutines
within the existing NASA codes. He succeeded in producing a set of polynomials that run very
efficiently for the range of fluid properties that is of interest to recompression cycles. Figure 3.5
shows Gong’s subroutines compared with REFPROP results. In the range of interest for the
recompression cycle, Gong’s polynomials produce very good results and are able to do so
quickly.
1000
Density (kg/m3)
900
1350 J/kgK : Gong
800
1350 J/kgK : NIST
1450 J/kgK : Gong
700
1450 J/kgK : NIST
1550 J/kgK : Gong
600
1550 J/kgK : NIST
500
400
300
325
350
375
400
425
450
Enthalpy (kJ/kg)
Figure 3.5: Fast-running fluid property subroutines for S-CO2 produced by Gong, as they
compare to NIST REFPROP values. Density is plotted against enthalpy for three lines of
constant entropy: 1350 J/kgK, 1450 J/kgK, and 1550 J/kgK as indicated in the legend to the
right. Each entropy line has a value of enthalpy above which Gong’s subroutines are not
applicable.
It is evident in Figure 3.5 that both property subroutines have their limitations. At very
low enthalpy, the REFPROP subroutines return errors, and at higher enthalpies, Gong’s
subroutines deviate from the REFPROP subroutines. For recompression cycles, however,
Gong’s subroutines perform well. Figure 3.6 shows the range of applicability for Gong’s
subroutines.
51
Density (kg/m3)
1000
750
500
250
0
300
325
350
375
400
425
450
Enthalpy (kJ/kg)
Figure 3.6: Range of Applicability for Gong’s S-CO2 property subroutines
Figure 3-5 shows the density as a function of enthalpy for three lines of constant entropy.
By drawing a line through the points where the property routines diverge, an expression for the
range of validity for Gong’s subroutines was developed. If the density at the impeller outlet falls
outside the range bounded by the two lines in Figure 3.6, RGRC will return a warning to the user
that the fluid property subroutines cannot be trusted in that range. On the whole, Gong’s fluid
property subroutines are quite successful. They reduce the computing time for many calculations
by orders of magnitude.
Subroutines calling Gong’s polynomials were substituted for the calculations already
present in CCD and CCODP. In principle, these changes should have been enough to produce
basic compressor designs and performance maps. The structure of the code and its variable
naming scheme were left unchanged by Gong.
CCD and CCODP are written in Fortran 77, using almost no subroutines and scores of
GO TO statements. The code is very difficult to understand for these and other reasons.
Furthermore, there were large disagreements between the design code and the off-design code
when run on CO2. Part of the problem was inconsistency in Gong’s application of his new
property subroutines. The performance maps did not agree with the design code, sometimes
showing as much as 20% difference in the stage pressure ratio for the design point. An example
of the discrepancies is shown in Figure 3.7. The pressure ratio graph shows a vertical surge line,
a feature never observed in real compressors. The effect is produced by CCODS because no
surge criterion has been met. Instead, the surge line in Figure 3.7 only serves to indicate the
mass flow rate at which calculations begin.
52
Total-to-Static Pressure Ratio
3
100%
2.5
90%
2
80%
70%
1.5
60%
1
0
500
1000
1500
2000
Total-to-Static Efficiency
Mass Flow Rate (kg/s)
0.9
0.88
0.86
100%
0.84
90%
0.82
0.8
80%
0.78
70%
0.76
60%
0
500
1000
1500
2000
Mass Flow Rate (kg/s)
Figure 3.7: A single-stage main compressor map from CCODS showing unusual features.
The pressure ratio (top) shows that surge occurs at the same mass flow rate regardless of speed
and the design point does not appear at all. The design point is 3600 RPM and 1580.9 kg/s.
Speeds are in percent of design speed
Also, the off-design code was not self-consistent if the speeds for the compressor map
were changed by the user. The source of this error was never discovered. Evidently, there were
some data being preserved as calculations proceeded from one compressor speed to the next, but
the complexity of the code made it impossible to diagnose the problem.
CCDS and CCODS develop a compressor design based on some methods which were not
advantageous because of inconsistencies in the results and difficulties involved in using the code
for power producing S-CO2 cycles. First, CCDS develops a compressor design based on an
input value of specific speed. The choice of specific speed as an input unnecessarily complicates
the code when it is used for compressors in electricity production. Presumably the choice of
input specific speed was made because specific speed is used to characterize compressors based
on their application. The code used the input value of specific speed to calculate a value for
53
RPM. In the case of electricity production, the compressor will be operating at 3600 RPM or
some fraction of 3600 RPM, so it was desirable to input RPM and avoid the complications of
specific speed. Choosing a specific speed meant that the designer would have a more difficult
process than if the rotational speed was an input.
CCDS also required an input value of the de Haller number, the ratio of relative velocity
at impeller inlet to that at the outlet, W2OW1T in CCDS/CCODS. This limit was intended to
allow the user to guarantee a design that would meet the criterion of W2/W1 > 0.75. de Haller
introduced this criterion as a way of guaranteeing that the stage would avoid stall, but this
method is not entirely sufficient [Cumpsty, 2004]. The de Haller number is a limit on the head
loading of each stage in a multi-stage compressor. In other words, it is a limit on the pressure
ratio for a single stage. Therefore, a low de Haller number should be an indication to the
designer that a compressor design needs to be reevaluated in terms of stage number. The
constraint resulted in some unphysical results. By restricting the relative velocities to be
proportioned by the designer, the mass flow rate at impeller outlet did not match that at impeller
inlet in every design. The difference could be as much at 5.6 % in CCDS. Furthermore, the
constraint in the outlet velocity meant that the calculated enthalpy and pressure rise were not
reliable. The outlet velocity triangle was calculated based on an assumed relationship to the inlet
triangle and the proportional relationship between outlet diameter and blade speed.
The above shortcomings prompted an effort to produce a new code that would take
advantage of Fortran 90’s improvements and would be accurate, user-friendly and more readable
for new users. The new code’s compressor maps agree with the design point and there is a
supplement that allows stages to be coupled into a multistage design (up to three stages). It has
been named RGRC, for Real Gas Radial Compressor, and the multistage code has been called
RGRCMS, with the MS added for Multi-Stage.
3.3.2 Basic Outline of the RGRC Code
Designing compressors is sometimes described as a combined art and science. Some
parameters of the design must be “known” by the designer in order to achieve acceptable
performance. Some geometrical parameters are determined within RGRC in order to alleviate
some decision-making on the part of a code user who may be inexperienced with compressors.
However, the designer must make many important choices about compressor parameters in order
to achieve a successful design. The code will return warnings to the designer to indicate what
parameters can be changed in order to avoid problems. The code calculates all quantities based
on physical laws, experimental correlations, and experimentally determined fluid properties. The
code can be broken into several portions. They are:
54
1.
2.
3.
4.
5.
6.
The Impeller Inlet (Inducer)
Dimensioning the impeller
The Impeller Outlet
Losses
The Vaneless Space
The Vaned Diffuser
After the design portion is complete, the off-design performance is computed by repeating the
above list (with the exception of item 2) for a range of impeller speeds and mass flow rates.
3.3.3 Variable Nomenclature
There is a consistent naming scheme to the variables in the code. Numbers 0 through 4 are
used to denote positions in the compressor beginning from inlet and going to outlet.
01234-
Conditions upstream of the compressor
Inducer inlet
Impeller outlet
Vaned Diffuser Inlet
Vaned Diffuser Outlet
Absolute velocities are designated with a “V”, relative velocities with a “W”, and blade
speeds with a “U”. At the inducer inlet a “T” denotes the blade tip, “H” the hub, and “MF” the
root mean square radius. A “U” or “M” for tangential and meridional, respectively, will appear
as the second letter in velocity components. For example, VU1MF is the tangential component
of the absolute velocity at the RMS radius of the inducer inlet. Simpler examples include U1T
and U2: the blade tip speeds at inducer inlet and impeller outlet, respectively. The blade tip
speed at the impeller outlet (2) does not have a tip and a hub speed because they are equivalent
when the blade passage is oriented perpendicular to the rotating axis as at the impeller outlet.
Enthalpies, densities, and pressures are saved as structured data types of “total”, “stat”,
and “rel”, such that P3.stat is used for the static pressure at position “3”, H1.rel is the relative
enthalpy at position “1”, and H2.total denotes the total enthalpy at position “2”.
3.3.4 Impeller Calculations
The impeller inlet dimensions are determined from a few inputs from the user. These include
hub diameter and the fluid properties upstream of the compressor, including mass flow rate,
static pressure, entropy and velocity. RGRC always assumes a zero prewhirl condition, because
the design of the inlet guide vanes will depend on many things external to the compressor and
will be important for transient control of the cycle as a whole. Prewhirl, however, can be very
55
important to the performance of a centrifugal compressor. Hub diameter will be determined by
the diameter of the shaft and whether or not the shaft must protrude through the hub (as in a
multistage design, for example). This determination will have to be made by the user,
considering torque, materials and the configuration of the system. The code uses the input
parameters to determine the diameter of the blade tips that the impeller must have in order to
accommodate the design mass flow rate at the inlet, given the velocity and density of the fluid.
The calculation is among the most simple in the entire code, basing inducer size on the mass
flow rate, the designer’s chosen number of blades and the dimension for blade thickness.
Inducer blade tip diameter is determined by
𝜋
𝐷
4 1𝑇
2
−
𝜋
𝐷
4 1𝐻
2
− 𝑍1 (𝐷1𝑇 − 𝐷1𝐻 )𝑡𝐿𝐸 =
𝑚
𝜌𝑉1
Eqn. 3-5
where D1T is the tip diameter, D1H is the hub diameter, Z1 is the number of blades, tLE is the
leading edge thickness of the blades and V1 is the inlet velocity at the RMS radius. This is just a
statement of conservation of mass. These calculations are contained in the subroutine INLET
and the velocity triangle is computed in the subroutine INLET_TRI.
To calculate the velocity triangle at the outlet, a phenomenon known as “slip” must be
understood. The meridional plane is defined in centrifugal compressor design as the plane that
contains the axis of rotation. At the inducer, the meridional direction is the same as the axial
direction and at the impeller outlet, it is equivalent to the radial direction. Based on conservation
of mass as the flow progresses in the meridional direction, the velocity triangle at impeller outlet
can be determined. Observed impeller performance, however, reveals that the flow “slips” as
shown in Figure 3.8. Slip is the result of pressure gradients at the blade passage outlet and is
explained in more detail in [Cumpsty, 2004].
56
Figure 3.8: The velocity triangles at impeller inlet (above) and outlet (below).
The Weisner slip correlation [Cumpsty, 2004] is used to determine the slip velocity and from
that, the impeller outlet velocity triangle. The slip velocity is given by
𝑉𝑠𝑙𝑖𝑝 =
𝑈2 𝜒2
,
(𝑍2 )0.7
Eqn. 3-6
where χ2 is the blade angle at outlet, Z2 is the number of blades at outlet, and U2 is the blade
speed at outlet. The “ideal” case assumes that the relative flow at impeller outlet will be at an
angle of zero to the blades. In other words, the ideal relative flow angle (angle from meridional)
is equal to the blade backsweep angle χ2. Figure 3.9 shows the backsweep angle on a schematic
impeller.
57
Figure 3.9: A simplified schematic of an impeller showing the definition of backsweep angle, χ2.
Backsweep angle in RGRC is measured with respect to the radial direction.
With the ability to calculate the impeller outlet velocity triangle, the code first assumes an
outlet diameter based on a ratio of inlet tip to outlet diameter, which is an input from the user.
Within the code, this ratio is called LAMX. This ratio is typically between 0.3 and 0.5 for a
centrifugal compressor impeller.
𝐷1𝑇
𝐿𝐴𝑀𝑋 =
Eqn. 3-7
𝐷2
If this ratio is too high, the impeller will begin to resemble an axial machine, but the flow
through it will not benefit from the blade shape. Therefore, there will be a decrease in fluid
density through the impeller. In other words, if the diameter ratio is initially too high, then the
machine will not compress the fluid at all. In this case, the code will alert the user to choose a
lower starting value for the diameter ratio. With the first value of the impeller outlet diameter
chosen, the code begins by assuming a density value slightly higher than the impeller inlet
density. Using this value, the slip correlation, and the conservation of mass requirement, the
code will calculate the velocity triangle at outlet. The key relationships are determined as
described in Figure 3.8. The mass flow rate relation is defined by
𝑊1 𝜌2 𝐴2
=
𝑊2 𝜌1 𝐴1
𝐴2 = 𝜋𝐷2 𝑏2 − 𝑍2 𝑏2 𝑡𝑇𝐸 𝑐𝑜𝑠𝜒2
58
Eqn. 3-8
Eqn. 3-9
𝐴1 =
𝜋 2
2
𝐷 − 𝐷1𝐻
𝑐𝑜𝑠𝛽1
4 1𝑇
Eqn. 3-10
Here, W2 is the ideal relative velocity at impeller outlet, because outlet flow area A2 is
defined by the backsweep angle. The relative velocity at the inlet, W1, represents the value at the
RMS radius. The true relative velocity at the outlet is determined with the slip correlation
described previously. Once the velocity triangle is determined, the enthalpy rise is calculated
using the Euler turbomachinery equation. This enthalpy rise is then used to determine a new
value for the fluid density at the outlet by calling the property subroutines with the calculated
enthalpy as an input. The new value of density is then used to repeat the calculation until the
density of the fluid converges. These calculations are contained in the subroutine OUTLET.
Once the density converges, the velocity triangle and enthalpy rise are known for the current
impeller outlet diameter. The process of determining the impeller’s performance is shown in
Figure 3.10. The impeller will not be resized in the design portion of the code until the diffuser
outlet static pressure is known, and therefore the entire compressor performance can be
compared to the desired performance.
Figure 3.10: The process for determining impeller performance. The outlet conditions of the
impeller rely on the convergence of fluid density. Mass conservation determines the outlet
triangle for each density iteration.
59
3.3.5 Loss Calculations
Within the impeller losses will occur. They are represented by both pressure losses and
additional work terms. The loss calculations depend on some parameters that describe the shape
of the impeller. The impeller shape is determined by the hub and tip diameters at inlet, the outlet
diameters, blade height, and blade angles. The shape will affect the losses as the fluid flows
through the passages. The subroutine IMP_SHAPE calculates the blade length based on the
impeller diameters and the blade angles. The blade passage is assumed to follow an elliptical
shape in the meridional plane. The impeller losses are then calculated using subroutines that are
based on empirical or semi-empirical formulas. There is no hard and fast convention for how
losses are calculated. When more experimental data for S-CO2 compressors is available in the
literature, the loss calculations may be improved by adding appropriate correction factors to the
present loss correlations.
Losses in the impeller are subdivided into incidence, blade loading, skin friction, hub-toshroud loading, mixing, diffusion, expansion, and tip clearance loss. These are expressed as
fractional pressure losses and are defined as described by Aungier [2000]. Disk friction, leakage,
and recirculation losses are expressed as additional work terms, per unit mass. The total
fractional pressure loss, 𝜔𝑡𝑜𝑡 , will be converted into an absolute pressure loss by
∆𝑃𝑖𝑚𝑝
𝜌1 𝑉12 𝜔𝑡𝑜𝑡
=
2
Eqn. 3-11
so that 𝜔𝑡𝑜𝑡 represents a portion of the dynamic pressure at the impeller inlet.
Clearance loss is the loss associated with Coriolis forces on the fluid between the
impeller blades and the stationary casing. It is defined as a fractional pressure loss and is given
by
𝜔𝐶𝐿 =
2𝑚𝐶𝐿 ∆𝑃𝐶𝐿
𝑚𝜌1 𝑊1
Eqn. 3-12
where ∆𝑃𝐶𝐿 is the pressure gradient along the clearance gap and is determined by the geometry
of the impeller blades, the blade speeds, and the mass flow rate through the impeller, as
∆𝑃𝐶𝐿 =
𝑚 𝐷2 𝑉𝑈2 − 𝐷1 𝑉𝑈1
2 𝑍𝑒𝑓𝑓 𝑟 𝑏 𝐿𝑏
60
Eqn. 3-13
where 𝑟 and 𝑏 are the average passage radius from the axis of rotation and the average passage
height from hub to shroud, respectively. The clearance mass flow rate is a function of the blade
geometry and flow rate.
𝑚𝐶𝐿 = 𝜌2 𝑈𝐶𝐿 𝑍𝑒𝑓𝑓 𝐿𝑏 𝑏𝐶𝐿
Eqn. 3-14
where bCL is the blade to shroud clearance and UCL is an empirical clearance flow velocity, given
by
𝑈𝐶𝐿 = 0.816 𝐴𝐵𝑆(2∆𝑃𝐶𝐿 /𝜌2 )
Eqn. 3-15
Skin Friction is the loss associated with the shear forces of the flow against the surfaces of the
impeller and the casing. It is defined as a fractional pressure loss and is given by
𝑊𝑟𝑚𝑠
𝑊1,𝑚𝑓
𝜔𝑆𝐹 = 4𝐶𝑓𝑡
2
𝐿𝑏
𝐷𝑕𝑦𝑑
Eqn. 3-16
where 𝑊𝑟𝑚𝑠 is the RMS value of relative velocity from impeller inlet to impeller outlet, given by
𝑊𝑟𝑚𝑠 =
1
𝑊 2 + 𝑊22
2 1
1/2
Eqn. 3-17
The turbulent skin friction factor, Cft, is given by
𝐶𝑓𝑡 =
𝐶𝑓𝑡𝑠
𝐶𝑓𝑡𝑠 + 𝐶𝑓𝑡𝑟 − 𝐶𝑓𝑡𝑠
𝑅𝑒𝑒 < 60
1 − 60/𝑅𝑒𝑒
𝑅𝑒𝑒 ≥ 60
Eqn. 3-18
where 𝑅𝑒𝑒 is a Reynolds number which takes into account the roughness of the surface. It is
used only to define the friction factors and is defined as
𝑅𝑒𝑒 = 𝑅𝑒𝑑 − 2,000
𝑒
𝐷𝑕𝑦𝑑
Eqn. 3-19
where e is the surface roughness, Red is the Reynolds number based on average hydraulic
diameter, and Dhyd is the average hydraulic diameter of the impeller passage. The additional
subscripts “s” and “r” refer to the smooth surface and rough surface friction factors. The
roughness effect becomes important when Ree is greater than 60. The friction factors are given
61
in Eqn. 3-20 and 3-21. The smooth surface friction factor must be solved iteratively. It depends
only on the Reynolds number based on hydraulic diameter.
1
= −2𝑙𝑜𝑔10
4𝐶𝑓𝑡𝑠
2.51
𝑅𝑒𝑑 4𝐶𝑓𝑡𝑠
Eqn. 3-20
The rough surface friction factor depends on the hydraulic diameter and the average surface
roughness, e.
1
4𝐶𝑓𝑡𝑟
= −2𝑙𝑜𝑔10
𝑒
3.71𝐷𝑕𝑦𝑑
Eqn. 3-21
Blade loading loss, or diffusion blading loss, is the loss associated with buildup of the
boundary layer inside the blade passage. It reflects the increase of the boundary layer thickness
and the separation of the flow caused by pressure gradients within the passage. Blade loading
loss is defined as a fractional pressure loss and can be calculated by
𝜔𝐵𝐿
Δ𝑊/𝑊1
=
24
2
Eqn. 3-22
where Δ𝑊 is a velocity representing the increase in loading from the impeller inlet to the outlet.
It is given by
Δ𝑊 =
2𝜋𝐷2 (𝑑𝐻𝑎𝑒𝑟𝑜 )
𝑍𝑒𝑓𝑓 𝐿𝑏 𝑈2
Eqn. 3-23
where 𝑍𝑒𝑓𝑓 is the effective number of blades, taking into account that splitter blades of half the
length of the full blades may be used in some impellers. The effective blade number is given by
𝑍𝑒𝑓𝑓 = 𝑍1 + 𝑍2 − 𝑍1
𝐿𝑠𝑝𝑙𝑖𝑡
𝐿𝑏
Eqn. 3-24
where Lsplit is the length of splitter blades and dHaero is the total aerodynamic enthalpy rise in the
impeller. The total aerodynamic enthalpy rise is defined by the velocity triangles calculated in
OUTLET. The ideal aerodynamic head is equal to 𝑑𝐻𝑎𝑒𝑟𝑜 less the enthalpy change attributable
to losses.
62
The incidence loss is the loss associated with redirection of the flow when it makes
contact with the inducer blades. The incidence loss should be a minimum at the design point.
Negative or positive incidence upon the inducer will result in a greater loss. Incidence loss is
defined as a fractional pressure loss and is given by
𝜔𝐼𝑁𝐶 = 0.8 1 − 𝑉1 / 𝑊1 𝑠𝑖𝑛𝛽1
2
+ 𝑍1 𝑡𝐿𝐸 / 𝜋𝐷1 𝑠𝑖𝑛𝛽1
2
Eqn. 3-25
Recirculation loss is caused by recirculation of the flow at the impeller outlet. It is
caused by mixing of the wake at the impeller blade tip and is greater when there is backflow to
the impeller. Recirculation loss is defined as a total work addition and can be calculated by
𝐼𝑅𝐶 =
𝐷𝑒𝑞
−1
2
𝑊𝑈2
− 2𝑐𝑜𝑡𝛽2
𝑉𝑀2
Eqn. 3-26
𝐷𝑒𝑞 = 𝑊𝑚𝑎𝑥 /𝑊2
Eqn. 3-27
where Deq is the equivalent diffusion factor. The equivalent diffusion factor takes into account
the change in relative velocity from the impeller inlet to the outlet. The relative velocity 𝑊𝑚𝑎𝑥 is
a function of the inlet and outlet velocity triangles and Δ𝑊 which was defined earlier in Eqn. 323.
𝑊𝑚𝑎𝑥 = 𝑊1 + 𝑊2 + Δ𝑊 /2
Eqn. 3-28
The mixing loss is caused by mixing of the flow at the impeller outlet and is modeled similar to
an abrupt expansion loss. It depends on a wake velocity, 𝐶𝑚 ,𝑤𝑎𝑘𝑒 , and a mixing velocity, 𝐶𝑚 ,𝑚𝑖𝑥 .
The mixing loss is defined as a fractional pressure loss.
𝜔𝑀𝐼𝑋 =
𝐶𝑚 ,𝑤𝑎𝑘𝑒 − 𝐶𝑚 ,𝑚𝑖𝑥 /𝑊1
2
Eqn. 3-29
The wake velocity is a function of the blades speed and a separation velocity, 𝑊𝑆𝐸𝑃 , which
depends upon the recirculation of the flow.
𝐶𝑚 ,𝑤𝑎𝑘𝑒 =
63
2
2
𝑊𝑆𝐸𝑃
− 𝑊𝑈,1
Eqn. 3-30
0.5𝑊2 𝐷𝑒𝑞 𝑖𝑓 𝐷𝑒𝑞 > 2.0
𝑊𝑆𝐸𝑃 =
Eqn. 3-31
𝑊2 𝑖𝑓 𝐷𝑒𝑞 < 2.0
Where 𝐷𝑒𝑞 is the equivalent diffusion factor as defined earlier in Eqn. 3-27. The mixing velocity
simply takes into account the flow expansion at the impeller outlet, and is therefore a function of
the flow velocity and the blade passage geometry at the outlet.
𝐶𝑚 ,𝑚𝑖𝑥 = 𝑉2 𝐴2 / 𝜋𝐷2 𝑏2
Eqn. 3-32
Diffusion losses are those losses which occur between the leading edge of the impeller
blade and the blade passage throat. They are negligible for most impellers and are rarely greater
than the blade incidence loss. The diffusion loss is defined as a fractional pressure loss.
𝜔𝐷𝐼𝐹 = 0.8 1 − 𝑊𝑡𝑕 /𝑊1
2
− 𝜔𝐼𝑁𝐶
Eqn. 3-33
where 𝑊𝑡𝑕 is the relative velocity without the inlet blockage considered. The diffusion
loss must be greater than zero. If the calculated value is negative it will be set to zero.
Hub-to-shroud mixing losses are caused by the pressure gradient between the hub and the
shroud, within the blade passage. The hub-to-shroud mixing loss is defined as a fractional
pressure loss.
2
𝜔𝐻𝑆 = 𝜅𝑚 𝑏𝑊 /𝑊1 /6
Eqn. 3-34
𝜅𝑚 = 𝛼2 − 𝛼1 /𝐿𝑏
Eqn. 3-35
and 𝑏 and 𝑊 are the average blade passage height and relative velocities, respectively.
Within the blade passage there is an expansion loss caused by imperfect diffusion through the
expanding blade passage. The result depends on the blockage at the impeller outlet which is
determined by correlation to be
𝐵2 = 𝜔𝑆𝐹
𝜌1 𝑉12
𝜌2 𝑉22
𝑊1 𝐷𝑕𝑦𝑑
𝑏22
+ 0.3 + 2
𝑊2 𝑏2
𝐿𝑏
𝐴2
𝐴1
2
𝜌2 𝑏2
𝑏𝑐𝑙𝑒𝑎𝑟
+
𝜌1 𝐿𝑏
2𝑏2
and the expansion loss is defined as a fractional pressure loss given by
64
Eqn. 3-36
𝜔𝐸𝑋𝑃 =
1
𝑉𝑀2
−1
1 − 𝐵2
𝑊1
2
Eqn. 3-37
The disk friction loss is caused by frictional torque on the back surface of the disk. Other
losses, in bearings, seals, and gear boxes, are combined with the disk friction loss. Disk Friction
and windage are defined as a total work addition, given by
𝐼𝐷𝐹 = 𝐶𝑀𝐷 + 𝐶𝑀𝐶 𝜌2 𝑈2 𝑅22 / 2𝑚
Eqn. 3-38
where 𝐶𝑀𝐷 and 𝐶𝑀𝐶 are experimentally determined constants related to the blade passage
geometry, the velocities, and the Reynolds number of the flow.
𝐶𝑀𝐷 = 0.75𝐶𝑀
0.75𝐿𝑏 𝐶𝑀
𝐶𝑀𝐶 =
Eqn. 3-39
𝐷1𝑇
1 − 𝐷2
5
Eqn. 3-40
𝑅2 − 𝑅1𝑇
𝑉
𝐶𝑀0 1 − 𝑈𝑈2
2
𝐶𝑀 =
𝑉𝑈1 2
1−
𝑈1𝑇
𝐶𝑀0 = 𝐶𝑀𝑠 + 𝐶𝑀𝑟 −𝐶𝑀𝑠
2
Eqn. 3-41
𝑙𝑜𝑔 𝑅𝑒/𝑅𝑒𝑠
𝑙𝑜𝑔 𝑅𝑒𝑟 /𝑅𝑒𝑠
Eqn. 3-42
2𝜋
𝑠/𝑟 𝑅𝑒
3.7 𝑠/𝑟
𝐶𝑀𝑠 = 𝑀𝐴𝑋
0.1
𝑅𝑒
Eqn. 3-43
0.08
𝑠/𝑟 1/6 𝑅𝑒1/4
0.102 𝑠/𝑟
𝑅𝑒 0.2
0.1
𝐶𝑀𝑟 = 3.8𝑙𝑜𝑔10 𝑟/𝑒 − 2.4 𝑠/𝑟
65
0.25 −2
Eqn. 3-44
Roughness becomes important at Reynolds numbers greater than Res, which is defined in terms
of the constant 𝐶𝑀 and the surface roughness.
𝑅𝑒𝑠 =
1,100 𝑒/𝑟
−0.4
𝐶𝑀
𝑅𝑒𝑟 = 1,100 𝑟/𝑒 − 6𝑥106
Eqn. 3-45
Eqn. 3-46
Leakage Loss is defined as a total work addition, given by
𝐼𝐿𝑒𝑎𝑘 =
𝑚𝐶𝐿 𝑈𝐶𝐿
2𝑈2 𝑚
Eqn. 3-47
where 𝑈𝐶𝐿 is the same clearance flow velocity used in the clearance loss calculation and 𝑚𝐶𝐿 is
the leakage mass flow rate also required in the clearance loss calculation.
Disk friction, recirculation, and leakage loss are calculated as additional work inputs to
the system. All other losses are calculated as fractional pressure drops. Once all the pressure
drops are known, they are summed and the fluid pressure at the impeller outlet is calculated. The
loss of pressure corresponds to a loss in fluid enthalpy, which is calculated using the fluid
property subroutines.
Once the losses are calculated, the entropy of the fluid is updated, by making the
approximation that the increase in entropy is given by
𝑑𝑠 =
𝑑𝑕
𝑇
Eqn. 3-48
where dh is the loss in enthalpy and T is the static temperature of the fluid at the impeller inlet.
This approximation results is based on only about a 10 % change in temperature through the
compressor stage, so as long as the losses are accurate, the change in entropy should be within 10
% of the correct value. Since the increase in entropy is small compared to the absolute value of
the entropy (less than 3 %) the approximation produces very small errors in the state-point of the
compressor outlet.
The impeller efficiency is determined from all of the actual energy input to the fluid over the
total energy input. Impeller efficiency is given by
66
𝜂𝑅 =
𝑑𝐻𝑎𝑒𝑟𝑜 − 𝑑𝑕
𝑑𝐻𝑎𝑒𝑟𝑜 + 𝐼𝐿𝑒𝑎𝑘 + 𝐼𝑅𝐶 + 𝐼𝐷𝐹
Eqn. 3-49
3.3.6 Vaneless Space and Diffuser Calculations
The vaneless space is modeled by assuming conservation of angular momentum and the
expansion of the turbulent boundary layer along the walls of the vaneless space. By conservation
of angular momentum, rVθ is a constant and rVr is a constant. Thus, the flow angle with respect
to the radial direction remains constant with the exception of changes in radial velocity resulting
from narrowing or widening of the space. Figure 3.11 shows the flow path through the vaneless
space. Due to the conservation requirements, the flow follows a curved path through the space.
Figure 3.11: The flow path in the vaneless space (not to scale). The flow angle with respect
to the radial direction, α, is a constant.
In real compressors the velocity profile at the impeller outlet is highly non-uniform. For that
reason the core flow and the flow near the walls will follow very different flow paths. Near the
wall, flow can re-enter the impeller and therefore an additional surge condition is used. If the
tangent of the absolute flow angle is above 4.0, then the flow is said to be surged.
The subroutine VANELESS begins by setting a time step such that the flow will travel
1/500 of the distance to the vaned diffuser in the first time step. That same time step is then used
throughout the vaneless space. The velocity and flow direction at the beginning of the time step
are assigned to the entire time step. The position of a fluid particle at the end of that time step is
then computed.
The velocity is computed at the new position, based on conservation of angular
momentum, and then the code continues until the position has reached the entrance to the vaned
diffuser. Figure 3.11 is not accurate in that it shows a constant angle θ between time steps. In
RGRC the angular distance traveled each step will actually decrease, as the velocity decreases.
67
For CO2, the kinematic viscosity is so low that the buildup of the boundary layer in the vaneless
space is negligible. The blockage calculated at the impeller outlet is assumed to represent the
blockage at the diffuser inlet.
The diffuser is modeled very simply as an increase in area, with corresponding losses due
to friction, incidence, and mixing of the wake at the outlet. A database of diffuser performance
by Runstadler contains the most thorough data on diffuser performance and was used as the
diffuser model in the NASA codes [Runstadler, 1969]. However, its use for a real gas resulted in
unphysical results. Thus, it was abandoned in favor of a simple model. Diffusers are always
vaned diffusers in RGRC. Pipe diffusers are more expensive to manufacture and are covered by
a Pratt and Whitney patent [Cumpsty, 2004]. The subroutine AUNGIER will create a vaned
diffuser according to the area ratio input by the user and it will orient the vanes such that the
incidence angle for the core flow is zero at the design point. It has been named for the source of
the diffuser model [Aungier, 2000]. The vanes can be straight or curved. They will be in line
with the flow at the design point at position 3 and can curve up to 25 o further at position 4.
The vaned diffuser performance is based on the area ratio, which is described by diffuser
inlet and outlet passage widths. The axial dimension of the vaned diffuser does not change in
RGRC, so the area ratio is the same as the ratio of the arc lengths at the diffuser inlet and outlet.
The diffuser performance is calculated by assuming that the flow follows the blade angle at
position 4. By calculating the blockage at position 4 and the losses within the diffuser passage,
the code iterates the velocity at position 4 until it achieves conservation of mass from position 3
to position 4. The initial guess of the velocity at position 4 is based simply on the area ratio of
the vane, but changes in density and blockage cause this value to be incorrect. The blockage at
position 4 is calculated as a portion of the flow area by
2
𝐵4 = 𝐾1 + 𝐾2 𝑉𝑅 − 1
𝐿𝑣𝑎𝑛𝑒
𝑤4
Eqn. 3-50
where 𝑉𝑅 is given by
𝑉𝑅 =
1 𝑉𝑀3 𝑠𝑖𝑛𝛽4
+1
2 𝑉𝑀4 𝑠𝑖𝑛𝛽3
Eqn. 3-51
and 𝐾1 and 𝐾2 are empirical constants determined from the diffuser divergence angle, 𝜃𝐶 , and the
blade loading parameter, 𝐿. These are defined in terms of the average blade-to-blade velocity
difference, Δ𝑉.
𝐾1 = 0.2 1 − 1/𝐶𝐿 𝐶𝜃
68
Eqn. 3-52
𝐾2 =
2𝜃𝐶
2𝜃𝐶
1−
125𝐶𝜃
22𝐶𝜃
Eqn. 3-53
The average blade-to-blade velocity difference is a function of the number of vanes, Z, the length
of the vanes, the radius of the diffuser at inlet and outlet and the velocity of the core flow at each
position. It is defined as
2𝜋 𝑅3 𝑉𝑈3 − 𝑅4 𝑉𝑈4
Δ𝑉 =
Eqn. 3-54
𝑍𝐿𝑣𝑎𝑛𝑒
and the defining correction coefficients for blade loading and divergence angle are given by
1 ≤ 𝐶𝐿 = 2𝜃𝐶 /11
Eqn. 3-55
1 ≤ 𝐶𝜃 = 3𝐿
Eqn. 3-56
where the diffuser divergence angle is defined as
𝜃𝐶 = 𝑡𝑎𝑛−1 (𝑤4 − 𝑡𝑏4 − 𝑤3 + 𝑡𝑏3 )/(2𝐿𝑣𝑎𝑛𝑒 )
Eqn. 3-57
and the blade loading parameter is defined as
𝐿=
Δ𝑉
V3 − V4
Eqn. 3-58
Losses within the vaned diffuser are divided into a skin friction loss, an incidence loss, and a
wake mixing loss. The skin friction loss is defined as a fractional pressure loss and is given by
𝜔𝑆𝐹
4𝑐𝑓 𝑉 /𝑉3 2 𝐿𝑣𝑎𝑛𝑒
=
,
𝑑𝐻 (2𝛿/𝑑𝐻 )0.25
Eqn. 3-59
where the skin friction coefficient is calculated just as it is in the impeller. The hydraulic
diameter is the average of the hydraulic diameters from the diffuser inlet and outlet. The
boundary layer thickness, 𝛿, is given by
2𝛿 5.142𝐶𝑓𝑡 𝐿𝑣𝑎𝑛𝑒
=
,
𝑑𝐻
𝑑𝐻
69
Eqn. 3-60
which is a flat plate approximation. The boundary layer thickness is restricted to be less than or
equal to one half the hydraulic diameter. Incidence loss upon the diffuser vane depends upon the
incidence angle relative to the vane angle.
𝜔𝐼𝑁𝐶 = 0.8 (𝑉3 − 𝑉3∗ )/𝑉3
2
Eqn. 3-61
where 𝑉3∗ is the throat velocity at the design point. The diffuser wake will experience a mixing
loss as the flow slows from the wake velocity to the mixed velocity. The mixing loss is
calculated similar to that of the impeller. It depends on a mixing velocity and a wake velocity at
the diffuser outlet.
𝜔𝑀𝐼𝑋 =
𝑉𝑀,𝑤𝑎𝑘𝑒 − 𝑉𝑀,𝑚𝑖𝑥 /𝑉3
2
Eqn. 3-62
The wake velocity again depends on the separation velocity and the tangential component of the
flow velocity at the diffuser outlet.
𝑉𝑠𝑒𝑝 2 − 𝑉𝑈4 2
𝑉𝑀,𝑤𝑎𝑘𝑒 =
Eqn. 3-63
The mixing velocity is again the result of the abrupt flow expansion at the blade passage outlet.
It therefore depends on the meridional flow velocity and the geometry of the diffuser blade
passage at the outlet.
𝐴𝑍𝑉𝑀 4
𝑉𝑀,𝑚𝑖𝑥 =
Eqn. 3-64
2𝜋𝑅4 𝑏4
The separation velocity depends on the diffuser divergence angle and the inlet velocity.
𝑉𝑠𝑒𝑝 =
𝑉3
1 + 2𝐶𝜃
Eqn. 3-65
Going into the diffuser calculation, the code determines the length of the vane based on R3 and
the input area ratio, the throat area, and the incidence angle on the diffuser vane. It then begins
the velocity iteration for station 4. After calculating the losses and blockage, a new value for
velocity is determined. The process continues until the flow rate at station 4 matches that at
station 3, thus conserving mass.
The pressure recovery of the diffuser is defined by the portion of the dynamic pressure at
station 3 that is recovered as static pressure at station 4.
70
𝐶𝑝∗ =
𝑃4,𝑠𝑡 − 𝑃3,𝑠𝑡
𝑃3,𝑡 − 𝑃3,𝑠𝑡
Eqn. 3-66
Once the conditions are known at station 4, calculations are mostly complete. During the
design portion of RGRC, the static pressure ratio from station 0 to 4 will be compared with the
desired pressure ratio, and the impeller diameter (along with the diffuser size) will be adjusted
accordingly until the design matches the desired performance. In the off-design portion,
efficiencies are calculated and then the next operating condition is run. The total-to-static
efficiency is defined as
𝜂 𝑇𝑆 =
𝑑𝐻𝑎𝑒𝑟𝑜
𝑕4,𝑠𝑡 − 𝑕0,𝑡
+ 𝐼𝐷𝐹 + 𝐼𝑅𝐶 + 𝐼𝐿𝑒𝑎𝑘
Eqn. 3-67
The less frequently used total-to-total efficiency is defined as
𝜂 𝑇𝑇 =
𝑑𝐻𝑎𝑒𝑟𝑜
𝑕4,𝑡 − 𝑕0,𝑡
+ 𝐼𝐷𝐹 + 𝐼𝑅𝐶 + 𝐼𝐿𝑒𝑎𝑘
Eqn. 3-68
It is clear that the total-to-static efficiency must be less than the total-to-total efficiency.
3.3.7 Off-Design Compressor Performance
When the code performs a calculation for the performance map, it uses the geometry
developed in the design stage and changes the compressor’s speed and mass flow rate through a
range of values. It performs all the calculations discussed previously; the velocity triangles,
losses, and diffuser performance for a range of mass flow rates from 10 % to 300 % of the design
flow rate. RGRC performs the calculations for six speeds, selected by the user in the input file.
Whether or not the pressure ratio and efficiency of the compressor are displayed in the output for
each combination of speed and mass flow rate, is decided by choke and surge. If the compressor
has encountered either, the code will not display the output, and if choke has occurred, the code
will go on to the next operating speed to be included in the output. In RGRC and RGRCMS,
surge is defined as any of the following conditions:
1. A de Haller number (W1MF/W2) of less than 0.7
2. An impeller outlet flow angle with tangent greater than 4.0
3. Blockage at the diffuser inlet exceeds 15 %
71
Stall could also be experienced by an extreme flow incidence at the diffuser, but experimental
data determining where that flow angle lies for CO2 is not available. In practice, the first two
conditions above are sufficiently limiting that severe negative incidence does not occur at the
diffuser inlet.
Choke occurs when the flow velocity at any point rises above the critical velocity or
when losses in the impeller cause the density at the diffuser inlet to fall below the impeller inlet
density. This occurs due to excessively high flow rates, since the impeller losses are calculated
as pressure losses. The critical velocity database applies to CO2 and no other fluids have had
databases created for them yet. The process of creating a critical velocity database is discussed
in Appendix B.
RGRC will output the location of choke or surge so that the user can design around the
limiting conditions. In RGRCMS the output includes the stage number of the choke or surge
also, so the user knows which stage design needs to be improved.
3.4 The Multi-Stage Code RGRCMS
The multi-stage code reads in geometric parameters and first stage inlet conditions
formed from output of the single stage code. The geometry data required by the multi-stage code
is written by a subroutine called MULTISTAGE in RGRC. The multi-stage code assigns the
geometrical parameters to vectors with an index identifying the stage number. The code then
begins looping through the range of shaft speeds and mass flow rates. Within each combination
of shaft speed and mass flow rate, the code proceeds through each stage. After calculating the
first stage, the outlet conditions of the first stage are assigned as the inlet conditions of the
second stage, with a form loss for the return channel in between. The losses expressed as
additional work terms are progressively summed as the code steps through stages. Otherwise,
the multi-stage code proceeds almost exactly as does the off-design portion of RGRC. It
includes the exact same calculations for impeller and diffuser flows and the losses are calculated
with exactly the same subroutines.
Stage matching can be difficult and the designer must consider the effects of the return
channel on downstream stages. Oftentimes, stall and choke will be a serious issue in
downstream stages at mass flow rates that were well within the operating range of the stage when
designed independently. This is due to the changing inlet conditions of the second stage as the
first stage performance changes. Designing a multi-stage compressor with RGRCMS will
inevitably be an iterative process. Multi-stage compressors are usually designed so that each
stage has the same static pressure ratio at the design point.
The form loss model of the return channel is really not sufficient to describe the effect of
the return channel on the flow. The rotation of the fluid will still affect the second stage as the
72
velocity changes. This is an aspect of RGRCMS that could use some improvement. The current
model does predict the appropriate pressure ratio at the design point and clearly shows that surge
and choke are more limiting in the multi-stage design. Also, return channel design may depend a
great deal on available space and other factors independent of compressor performance.
3.5 S-CO2 Compressor Designs
Preliminary design estimates are frequently based on specific speed as a way of
determining the optimum compressor type, or number of stages. Centrifugal compressors were
chosen as the design option for the recompression cycle based on specific speed, and the
assumption that the compressors would be synchronized with the electricity grid at 3600 RPM.
Testing different approaches to the geometrical design of recompression cycle compressors
resulted in the designs detailed in this section. Designs were developed for the compressors
needed to operate the 500 MWth PCS described in Chapter 2.
3.5.1 The Main Compressor Design
The main compressor operates at steady state with the inlet conditions just above the
fluid’s critical point. The density is very high and therefore the compressor behaves more like a
pump. At the design point the main compressor achieves a total-to-static efficiency of 90.4 %.
Figures 3.12 and 3.13 show the pressure ratio and efficiency as a function of mass flow rate and
operating speed for the main compressor in a 500 MWth representative cycle. At the design
point the main compressor work is 32.3 MW. In the reference cycle, the main compressor
receives 62 % of the flow, which is a typical value for recompression cycles. Despite the much
larger share of the flow, the main compressor work is much less than the recompressing
compressor, due to the proximity of the main compressor inlet to the critical point of the fluid.
73
Static-to-Static Pressure Ratio
4
3.5
3
2.5
120%
2
110%
1.5
100%
1
90%
0.5
80%
0
70%
Mass Flow Rate (kg/s)
Figure 3.12: The pressure ratio of the main compressor for varying speeds in percentage of the
design speed (3600 RPM). The design mass flow rate is 1580.9 kg/s.
Total-to-Static Efficiency
1
0.9
120%
110%
0.8
100%
90%
0.7
80%
70%
0.6
0.5
0.4
Mass Flow Rate (kg/s)
Figure 3.13: The total-to-static efficiency of the main compressor for varying speeds in
percentage of the design speed (3600 RPM).
The surge margin in the main compressor is not as large as power producing cycles may
require. This is due to the very high density of the fluid necessitating a small blade height at the
impeller outlet in order to avoid surge. By reducing the blade height, losses are incurred and the
blade height can only be reduced so far in practice. To reduce this problem, a designer could use
RGRC to develop a design for a lower mass flow rate, and then just apply that design to the main
compressor of the 500 MWth power cycle.
74
3.5.2 The Recompressing Compressor Design
The recompressing compressor was attempted as single-stage, two-stage, and three-stage
designs. The best results were achievable for the two-stage design, in terms of both efficiency
and operating range. The single-stage recompressing compressor design achieved undesirable
efficiencies (79.8 %) and had a very small surge margin. This is attributable to excessive losses
when the pressure ratio is too high for a lower density fluid. The lower density in the
recompressing compressor means that a lower stage pressure ratio is needed. At the design
point, the two-stage recompressing compressor achieves a total-to-static efficiency of 91.4 %.
Figures 3.14 and 3.15 show the pressure ratio and efficiency of the two-stage recompressing
compressor at various operating speeds. The inlet conditions of the recompressing compressor
are far enough away from the critical point that, although compressing only 38 % of the flow, the
recompressing compressor requires 57.0 MW at steady state, about 1.76 times the requirement of
the main compressor, despite the fact that the recompressing compressor only receives 38% of
the flow.
Static-to-Static Pressure Ratio
4.5
4
3.5
3
120%
2.5
110%
2
100%
1.5
1
90%
0.5
80%
0
70%
Mass Flow Rate (kg/s)
Figure 3.14: The pressure ratio of the recompressing compressor for varying speeds in
percentage of the design speed (3600 RPM). The design mass flow rate is 989.6 kg/s.
75
Total-to-Static Efficiency
1
0.9
120%
0.8
110%
0.7
100%
0.6
90%
0.5
80%
70%
0.4
Mass Flow Rate (kg/s)
Figure 3.15: The total-to-static efficiency of the two-stage recompressing compressor for varying
speeds in percentage of the design speed (3600 RPM).
The design point efficiencies of the recompression cycle designs were incorporated in cycle
calculations in Chapter 2. Performance maps similar to these designs have been incorporated in
the transient analysis of TSCYCO performed by Trinh [2009]. These compressors display many
of the features of ideal gas compressors. They achieve their highest efficiencies at the design
point. Pressure recovery in the vaned diffuser is usually achievable up to 0.80 at the design point
and efficiencies are rather high overall. As speed increases, the compressor achieves the same
peak efficiency, but at a higher mass flow rate. The surge line occurs prior to the flattening of
the pressure ratio curves as mass flow rate is decreased. The flattening of the pressure ratio
curve can be considered the definition of stall, but the early surge in these performance maps is
believed to be indicative of the conservative assumptions in the code.
3.5.3 Benchmarking RGRC and RGRCMS
Benchmarking RGRC is important to validate the code. Few compressors have operated
close to the critical point and benchmarking is difficult given the lack of data. A test S-CO2
compression loop has been operated by Sandia National Laboratory close to the critical point.
Some of its parameters are given in Table 3.3.
76
Table 3.3: Selected Sandia Test Compressor Parameters
Impeller Diameter
Diffuser Area Ratio
Blade Thickness
Blade Height at Impeller Outlet
Number of Inlet Blades
Number of Outlet Blades
Blade Backsweep Angle
Operating Mass Flow Rate
Design Speed
3.7 cm
1.90
0.40 mm
0.62 mm
6
12
40 o
3.53 kg/s
75,000 RPM
Estimates of the test compressor’s geometry have enabled use of RGRCMS to develop a
compressor map, but until more experimental data are available, benchmarking is incomplete.
The performance maps produced for the design described in Table 3.3 are shown in Figures 3.16
and 3.17.
2.5
Pressure Ratio
2
120%
110%
1.5
100%
1
90%
80%
0.5
70%
0
Mass Flow Rate (kg/s)
Figure 3.16: The pressure ratio of the modeled test compressor for varying speeds in percentage
of design speed (75,000 RPM). The design mass flow rate is 3.53 kg/s.
77
0.9
Total-to-Static Efficiency
0.8
0.7
120%
0.6
110%
0.5
100%
0.4
90%
0.3
80%
0.2
70%
0.1
0
Mass Flow Rate (kg/s)
Figure 3.17: The total-to-static efficiency of the modeled test compressor for varying speeds in
percentage of design speed (75,000 RPM). The design mass flow rate is 3.53 kg/s.
The Sandia test compressor, operating at 75,000 RPM and 3.53 kg/s with inlet conditions
of 7.69 MPa and 305 K achieved a pressure ratio of 1.8 and a total-to-static efficiency of 66.0 %
[Wright et al., 2008]. The steady-state operating speed is really a lower value than 75,000 RPM.
It is actually closer to 45,000 RPM. The performance maps produced by RGRCMS are
reasonably accurate considering that dimensions were determined by looking at a photograph of
the test compressor. Continued operation of this test compressor and input of more precise
geometry will yield more data and eventually RGRC can be effectively benchmarked. Currently,
the exact dimensions of the impeller are not available for release, nor are tabulated data.
3.6 Chapter Summary
A mean-line compressor code was developed using the NIST fluid property polynomials
as a means of sizing and estimating performance for S-CO2 compressors to be used in the
recompression cycle. This code produces results for single and multi-stage compressors,
estimating losses based on experimental correlations. It is a first step in designing actual
compressors.
The geometry of the compressor impeller can be manipulated by the user in a single input
file and the code performs sizing of the impeller and related diffuser to achieve the desired
pressure ratio at the design point. Off-design performance estimation is automatic for the
resulting design. RGRC produces a performance map for the stage in a single output file with
the important geometrical parameters. RGRC also produces an output that can be used with
other stage outputs in the multi-stage code, RGRCMS.
78
The codes incorporate fluid property polynomials produced by Yifang Gong which speed
up calculations for CO2. These are suitable for the range of applications that recompression
cycles require, but can be backed up with NIST fluid property subroutines as well. This
compressor code has been used to estimate the performance of Sandia’s test compressor, as a
start on benchmarking the code. Results of the benchmarking attempt show that the codes
operate without any problems at high relative velocities and that the codes are consistent with
expectations. The much smaller Sandia compressor has a similar specific speed to the larger
compressor designs and its performance was shown to be similar as well.
S-CO2 recompression cycle compressors achieve total-to-static efficiencies greater than
90 % at the design point and they display none of the anomalous features that plagued CCODS
in their compressor maps. Multi-stage designs are now possible, up to three stages, and the use
of multiple stages becomes necessary for the recompressing compressor of the 500 MWth cycle.
79
3.7 Nomenclature for Chapter 3
a
A
b
B
D
Dhyd
g
h
H
I
Lb
Ns
P
R
Re
s
t
T
U
V
Vcr
W
Z
speed of sound (m/s)
Flow area (m2)
blade height (m)
Blockage (dimensionless)
Impeller diameter (m)
hydraulic diameter of a passage (m)
acceleration due to gravity (m/s2)
enthalpy (J/kg)
pressure head (m)
Loss (J/kgK)
blade length (m)
Specific Speed (dimensionless)
Pressure (kPa)
radius (m)
Reynolds number
entropy (J/kgK)
thickness (m)
Temperature (oC)
Blade velocity (m/s)
Absolute flow velocity (m/s)
critical velocity (m/s)
Relative flow velocity (m/s)
Number of Blades
Greek Letters (for Compressors)
α
𝛽
δ
η
μ
ω
Ω
φ
ρ
𝜃𝐶
χ
Absolute flow angle
Relative flow angle
Boundary layer thickness (m)
Efficiency
Viscosity (Pas)
fractional pressure drop
Rotational speed (rad/s)
Flow Coefficient
Density (kg/m3)
Diffuser divergence angle
Blade angle (rad)
80
Subscripts (for Compressors)
1
2
3
4
Inducer inlet
Impeller outlet
Diffuser Inlet
Diffuser outlet
b
BL
CL
DF
e
eff
eq
EXP
HS
INC
LE
Leak
M
MIX
MF
RC
rel
rms
SEP
SF
split
st
SS
TE
th
TS
TT
t, tot
U
vane
wake
full length blade
Blade loading
Tip Clearance
Disk Friction
roughness
effective
equivalent
Expansion
Hub-to-Shroud
Incidence
Leading Edge
leakage losses
meridional
Mixing
At the RMS radius
Recirculation
relative
root mean square
Separation
Skin Friction
splitter blade
static
Static-to-Static
Trailing Edge
throat
Total-to-Static
Total-to-Total
total
tangential
diffuser vane
diffuser wake flow
81
4 Balance of Plant Options
4.1 Introduction
4.1.1 Necessary Tools for Analysis
SFRs are among the reactor concepts included in the Generation IV Roadmap [GIF,
2002]. They have the potential to be built as, burners, break-even or breeder cores. Their
relatively high core outlet temperature means that significant efficiency gains could be achieved
over current LWR technology. The choice of balance of plant, heat exchanger type and
dimensions, and intermediate loop will affect the overall efficiency of the SFR a great deal. The
ABR-1000 design claims a Rankine cycle efficiency of ~38 %, but the cycle efficiency alone
doesn’t tell the whole story. The pumping power of primary and intermediate pumps is
important, and the net cycle efficiency can be improved or degraded by design changes from the
reference case. As shown in Figure 2.4, the cycle efficiency is strongly dependent on the turbine
inlet temperature and the choice of cycle. In order to understand where efficiency improvements
could be made, many design configurations needed to be investigated. A principal such
configuration is the use of Printed Circuit Heat Exchangers (PCHEs) for the secondary
intermediate heat exchanger (S-IHX) or primary intermediate heat exchanger (P-IHX). PCHEs
are compact and rugged. They could replace shell-and-tube designs in the SFR plant if sodium
plugging is shown to be avoidable in small channels. The S-CO2 cycle will make higher plant
efficiencies a possibility for higher turbine inlet temperatures. Additionally, elimination of the
intermediate loop will reduce the temperature difference between the primary fluid and the PCS
working fluid, thus increasing turbine inlet temperature. All these changes need to be evaluated
for their effect on efficiency.
Tools already available at MIT for this analysis included CYCLES III for the S-CO2 PCS
and a number of PCHE codes written by Pavel Hejzlar [Hejzlar et al., 2007]. There was no
available model at MIT for shell-and-tube heat exchangers, nor do the PCHE codes cover all of
the cases of interest to this study. These existing tools also do not take into account efficiency
losses due to heat conduction from intermediate piping or pumping power losses in either the
primary or intermediate loops. Heat losses in intermediate piping are negligible and do not
produce any real change in the plant efficiency. Pumping power for the entire plant can reach
almost 1 % of the core thermal power, so it is important for efficiency considerations.
82
The most important factors in the efficiency of SFRs are therefore:
1.
2.
3.
4.
Core outlet temperature
Heat Exchanger Design
PCS efficiency
Total System Pump work
These four items will be evaluated in Chapter 5, but in order to do so, some tools need to
be developed. These include heat exchanger models for both the P-IHX and the S-IHX, models
of the achievable efficiencies for Rankine, supercritical steam, and S-CO2 cycles, and a
discussion of the likely range of achievable core outlet temperatures.
With these important considerations in mind, the task of expanding the modeling
capability was undertaken. The major goals were to:
1. Develop a shell-and-tube code to model sodium-sodium or sodium-CO2 heat
exchangers
2. Develop a straight tube shell-and-tube steam generator model
3. Develop a PCHE model for sodium-sodium heat exchangers
4. Develop a PCHE model for a sodium-water steam generator
5. Model the intermediate heat exchanger pressure drops to determine pumping power
6. Show that heat lost through insulated intermediate loop piping is negligible
First, simple calculations of pumping power and heat lost through intermediate piping are
performed. Then, descriptions of the more involved calculations are included in later sections.
Sodium properties are calculated using correlations from a study of sodium at Argonne National
Laboratory [Fink and Leibowitz, 1995].
4.1.2 Heat Losses in Intermediate Piping and Pumping Power of an SFR
The heat lost in the intermediate loop piping is negligible in an SFR. This can be shown
by performing some simple calculations based on convective heat transfer from the sodium flow
to the air outside. The temperature changes in the sodium flow are negligible in the intermediate
loop, allowing the system to be modeled as though temperatures are identical at positions 2 and 3
and at positions 1 and 4 shown in Figure 4.1. This can be shown by assuming a constant linear
heat rate in the sodium piping from 2 to 3. If the ambient air is at a temperature of around 300 K
and the sodium is in the range of typical SFR temperatures, then the temperature difference from
the sodium to the air will be ~450 oC. The linear heat rate will be given by
83
−1
1
ln 𝑑2 𝑑1
ln 𝑑3 𝑑2
1
𝑞 =
+
+
+
𝜋𝑑1 𝑕𝑁𝑎
2𝜋𝑘𝑝𝑖𝑝𝑒
2𝜋𝑘𝑖𝑛𝑠
𝜋𝑑3 𝑕𝑎𝑖𝑟
(450 o C)
′
Eqn. 4-1
where 𝑑1 , 𝑑2 , and 𝑑3 are the pipe inside diameter, pipe outside diameter, and the insulation
outside diameter, respectively. The thermal resistance of the insulation is the dominant term in
Eqn. 4-1, contributing about 91 % of the resistance even for the thinnest insulation of just 1 cm.
Using the Clinch River Breeder Reactor (CRBR) as an example shows that the
temperature changes in the intermediate loop are negligible. CRBR was chosen as an example
because it is matches the ABR-1000 closely in intermediate loop mass flow rate and
temperatures. It also has the highest intermediate loop pumping power of any SFR, so it is a
conservative example. The temperature drop in the intermediate loop from points 2 to 3 is
determined by calculating the heat transfer to the atmosphere. Representative values used in
Eqn. 4-1 are given in Table 4.1.
From Core
To PCS
2
3
P-IHX
To Core
S-IHX
1
4
From PCS
Figure 4.1: Schematic of the intermediate loop. The P-IHX could be incorporated in a loop
design (as pictured) or submerged within a pool of primary coolant.
84
Table 4.1: Representative Values used in Eqn. 4-1
Pipe inside diameter
Pipe outside diameter
Insulation outside diameter
Mass flow rate
Conductivity of the pipe
Conductivity of Insulation
Temp. of Sodium
Conductivity of Sodium
Heat Capacity of Sodium
Density of Sodium
Viscocity of Sodium
Heat transfer coeff. of air
d1
d2
d3
𝑚
kpipe
kins
T
kNa
Cp
ρ
μ
hair
0.587 m
0.610 m
Variable
1256 kg/s
21.0 W/mK
0.04 W/mK
510 oC
63.71 W/Mk
1262.47 J/kgK
832.29 kg/m3
2.324 E-4 Pas
40 W/m2K [Mills, 1995]
The value of the heat transfer coefficient for the ambient air is conservatively chosen to be high.
The results show that the insulation provides the most significant thermal resistance, and that
even relatively high heat transfer coefficients on the air side do not result in significant heat
losses in the intermediate piping. The sodium Nusselt number was calculated using a correlation
by Notter and Sleicher [Todreas and Kazimi, 1993]
𝑁𝑢 = 4.8 + 0.0156𝑅𝑒 0.85 𝑃𝑟 0.93
Eqn. 4-2
Based on an assumed intermediate pipe length of 24 m from point 2 to 3, the coolant
temperature difference between points 2 and 3 can be plotted as a function of insulation
thickness, as shown in Figure 4.2 for a 250 MW loop. Even for very thin insulation, the
temperature difference is very small. Figure 4.3 shows the heat lost to the environment by this
example loop, also as a function of insulation thickness. Both the hot and cold legs of the
intermediate loop will experience similar losses because the fluid properties of sodium do not
change much with temperature. The cold leg will lose about 33 % less heat than the hot leg
because the temperature difference between the cold leg and ambient air is about 155 oC less
than that of the hot leg.
85
0.06
Temperature Drop (oC)
0.05
0.04
0.03
0.02
0.01
0
0.01
0.1
1
Thickness of Insulation (m)
Figure 4.2: The temperature drop through the 24 m long intermediate pipe, as a function of
insulation thickness
The heat lost by the intermediate coolant, corresponding to this temperature drop, is shown in
Figure 4.3. Despite the high heat capacity of sodium, the temperature drop is so small that very
little heat is lost through the intermediate piping walls.
90
80
Heat Lost (kW)
70
60
50
40
30
20
10
0
0.01
0.1
1
Thickness of Insulation (m)
Figure 4.3: Heat Lost in through the 24 m long intermediate pipe, as a function of insulation
thickness
86
The thermal resistance of the insulation is so high that changes in the insulation’s thermal
conductivity result in almost perfectly linear responses in heat lost. Thus, if the conductivity of
the insulation is underestimated by a factor of three, then the corresponding results in Figures 4.2
and 4.3 are just a multiple of three higher, which is still very little heat lost.
In the intermediate loop sodium pump, the temperature rise is also very small. The
pumping power for an incompressible fluid is given by
𝑊𝑝𝑢𝑚𝑝 =
Δ𝑃𝑚
𝜌
Eqn. 4-3
The CRBR, whose intermediate pipes served as a model for the calculation above, has an
intermediate pump head of 0.86 MPa. The CRBR is a conservative test case because it has one
of the highest pumping powers for SFRs, as shown in Table 4.2, and the CRBR matches the
ABR-1000 closely in intermediate loop mass flow rate and temperature [IAEA, 2006], as shown
in Table 4.3.
Table 4.2 Intermediate Loop Pumping Power Requirements for SFRs
Reactor
CRBR (USA)
Super-Phenix 1 (France)
ALMR (USA)
BN-1600 (Russian Federation)
JSFR-1500 (Japan)
Intermediate Pump Head (MPa)
0.86
0.25
0.34
0.331
0.335
Table 4.3 Characteristics of CRBR and ABR-1000 Intermediate Loops
CRBR
502
344
1629
325
3
o
Hot Temperature ( C)
Cold Temperature (oC)
Mass Flow Rate (kg/s)
Intermediate Loop Power (MWth)
Number of Loops
ABR-1000
488
333
1256
250
4
With the head requirement of 0.86 MPa, the pump work comes out to 1.30 MW for the
intermediate pump. The sodium temperature rise through the pump is given by
87
∆𝑇 = 𝑊𝑝𝑢𝑚𝑝 1 −
1
𝜂
1
𝑚𝑐𝑝
Eqn. 4-4
for a fluid with constant specific heat capacity, where 𝜂 is the polytropic efficiency of the pump,
cp is the specific heat capacity of the fluid, and the pump work is calculated as before. Constant
specific heat capacity is a good assumption for liquid sodium within the intermediate loop. If the
pump is 75 % efficient, this results in a temperature rise of 0.27 oC, which is negligible.
Basically, the intermediate piping can be ignored for efficiency studies, with the exception of the
pump work’s effect on overall plant efficiency.
The pumping power of the intermediate loop has already been shown to have very little
effect on the temperature of the fluid. Pumping power in the intermediate loop will depend a
great deal upon the size and type of heat exchangers used. In the primary fluid, pumping power
is greater due to pressure drops in the core and in primary piping (for a loop design). In the
ABR-1000 reference case each of the primary pumps in ABR-1000 requires 1.15 MW of power
if perfectly efficient, based on a pressure head of 0.76 MPa and a flow rate of 90.9 m3/min. To
overcome the friction pressure drops within intermediate piping and heat exchangers, the pump
work of each intermediate loop pump is about 340 kW if perfectly efficient. Because centrifugal
sodium pumps can be expected to have efficiencies between 70-85 % [Grandy and Seidensticker,
2007], the total pumping power required for the ABR-1000 plant is from about 7.0 MW to 8.5
MW for a 1000 MWth plant, including both primary and intermediate sodium pumps.
Therefore, reducing pressure drops and quantifying pump work are certainly important from a
plant efficiency standpoint.
4.2 Alternate Fluids in the Intermediate Loop
Sodium has been used as the working fluid in the intermediate loop in SFRs due to its
extremely good thermodynamic properties. Using other fluids has been considered as an option
in previous research. Here, these results are reported so the reader is familiar with work that has
already been completed. Elemental liquids, organics, inorganic salts, gases, and gas-solid
suspensions were investigated in a study of thirty different heat transfer media by Cooper and
Lee [1975]. They were compared on the basis of required heat transfer area, pumping power,
melting points, and chemical reactivity. Their conclusions are summarized as follows:
Liquid Metals: Heat transfer areas and pumping powers for liquid metals are all comparable to
sodium, however, many react nearly as strongly with water. Work has been done on leadbismuth eutectic reactors in numerous studies. These have been interesting since the solution of
some practical problems in a “comprehensive Russian development program” [Hejzlar et al.,
88
2000], but lead-bismuth eutectic fails to perform better than sodium due to excessive pumping
power. On the whole, no liquid metals perform better than sodium from a heat transfer
comparison, though some are comparable. The lead-bismuth eutectic is attractive for its low
reaction rate with water.
Molten Salts: All of the salts studied require a higher heat transfer area than sodium. Some are
explosively reactive with sodium, and excessively high melting point is a problem for others.
Organics: Organics thermally decompose at temperatures above 427 oC, which is well below the
temperatures of interest to SFRs.
Simple gases: Gases don’t perform as well as a heat transfer medium, and their pumping power
is often orders of magnitude greater than that of sodium.
Gas-solid suspensions: Gas-solid suspensions require 40-100 times the heat transfer area of
sodium, making them impractical.
As a heat transfer medium sodium is the best available and it enjoys lower pumping
powers and a greater useful temperature range than any other option. Consequently, for the
intermediate loop sodium is still the best option. The lead-bismuth eutectic coolant is also
appealing due to its inertness in terms of reactions with water. Bismuth does not react with water
and lead releases only 1.5 % of the heat that sodium does in a reaction with water, per unit mass
of reactant. The pumping power requirements for lead and bismuth are factors of 4.18 and 3.67
higher, respectively, than that of sodium. The pumping power of the intermediate loop is low
compared to that of the primary loop, but substituting a lead-bismuth eutectic for sodium would
make the intermediate pumping power comparable to the primary pumping power.
4.3 Eliminating the Intermediate Loop
The intermediate loop in operating SFRs is a safety measure to reduce the radiological
consequences of a sodium-water or sodium-air interaction. By including the intermediate loop,
primary sodium is isolated and the danger of a radiation release is mitigated in accidents
involving leaks in the P-IHX. Eliminating the intermediate loop will have substantial efficiency
gains, however, and the safety consequences may not be severe enough to warrant the
requirement of an intermediate loop. The effects of eliminating the intermediate loop on core
damage frequency (CDF) or large early release frequency (LERF) are not examined here.
Instead, the calculations performed were aimed only at determining the efficiency benefit for the
system if the intermediate loop were eliminated. Probabilistic Risk Assessments (PRA) of the
89
effect on plant damage states is important to determining whether a design configuration is
acceptable. Efficiency improvements may be outweighed by increased risk.
Eliminating the intermediate loop is a much more feasible option in loop designs for
SFRs. One reason is the required heat transfer area of intermediate heat exchangers and the core
dimensions of SFRs. SFRs are generally short, squat cores due to the requirement to avoid
designing a core with too high of a coolant void worth [Wigeland et al., 1994]. For that reason,
the vertical temperature change from core inlet to core outlet in a pool reactor will only span
about 5.0 m, the height of the core. Large heat transfer areas would require tall heat exchangers
unless the reactor vessel can be substantially enlarged radially. Either way, inclusion of a steam
generator inside a pool reactor vessel would require substantial enlargement of the vessel in
height or diameter or both. Also, the safety consequences of eliminating the intermediate loop in
a pool design are such that the design is not likely to be licensable. Loop designs allow for the
use of a taller P-IHX and do not include the hazard of a large secondary coolant blowdown
within the reactor vessel. Further discussion of this option is included in Chapter 5, as the
different configuration options are examined.
4.4 PCHE versus Shell-and-Tube Heat Exchangers
In order to compare design options with different combinations of heat exchanger
designs, a method for modeling heat exchanger performance was necessary. MIT’s in-house
PCHE code models printed circuit heat exchangers. It was written by Pavel Hejzlar [Hejzlar et
al., 2007]. A flexible model of shell-and-tube heat exchangers was needed to provide a first
estimate of the size of these heat exchangers. A Fortran code named SoSaT (Sodium Shell and
Tube) was written to accomplish this, so that different types of shell-and-tube heat exchangers
could be analyzed in a short period of time. When this project was undertaken, however, the
PCHE codes had the following capabilities:
Sodium to gas hybrid heat exchangers
Sodium to gas direct PCHE
Gas to gas direct PCHE
Water to Water Two Phase PCHE [Shirvan, 2009]
Additions were made to the list of capabilities such that these codes can now model:
Sodium to Sodium direct PCHE
Sodium to Water Two Phase Hybrid PCHE
90
In optimizing the recompression cycle, CYCLES III treats the IHX very simply, requiring a
pressure drop, power rating, and turbine inlet temperature as the only inputs related to the IHX.
Mass flow rate is scaled with the power as
3000 𝑘𝑔/𝑠
𝑚=𝑄
Eqn. 4-5
600 𝑀𝑊
where Q is the thermal power input in the IHX. This scaling of mass flow rate with power has
been a feature of the CYCLES code since its original version. The only PCHE type that needs to
be modeled within CYCLES III is the gas-to-gas recuperator because CYCLES III only requires
pressure drop, power, and temperature for the IHX. Therefore, the existing codes proved
sufficient for optimizing the S-CO2 balance of plant. In order to model the heat exchangers of
the entire system, the PCHE modeling capabilities required expansion and a shell-and-tube code
needed to be written.
4.4.1 The Shell-and-Tube Code SoSaT
In order to size shell-and-tube heat exchangers, a Fortran code called SoSaT (Sodium
Shell and Tube) was written. The code has the capability to size sodium-sodium, sodium-CO2,
or sodium-water heat exchangers. The code can perform calculations for single phase, boiling,
or supercritical water. For sodium-sodium and sodium-CO2 heat exchangers, the method of the
number of transfer units (NTU) is used. NTU is an expression of the heat exchanger
effectiveness relative to the overall heat transfer coefficient and fluid heat capacities [Shah,
2003]. The relationships between heat transfer coefficient, NTU and effectiveness are
summarized as follows:
𝑈𝐴
−1
=
1
𝐿
𝑙𝑜𝑔 𝑑𝑜,𝑜 /𝑑𝑖,𝑖
1
1
+
+
𝜋𝑑𝑖,𝑖 𝑕𝑡𝑢𝑏𝑒
2𝜋𝑘𝑡𝑢𝑏𝑒
𝜋𝑑𝑜,𝑜 𝑕𝑁𝑎
Eqn. 4-6
where UA is the product of the overall heat transfer coefficient and the heat transfer area. The
number of transfer units is based on the overall heat transfer coefficient and the mass flow rates
and heat capacities of the two fluids.
𝑁𝑇𝑈 = 𝑈𝐴/𝑀𝐼𝑁(𝐶1, 𝐶2)
Eqn. 4-7
where C1 and C2 are the products of mass flow rate and heat capacity for each fluid. The
effectiveness is related to NTU by
91
𝜀=
𝑒𝑥𝑝 𝑁𝑇𝑈 1 − 𝐶 ∗
−1
𝐶∗
− 𝐶∗
𝑒𝑥𝑝 𝑁𝑇𝑈 1 −
Eqn. 4-8
where C* is given by
𝐶 ∗ = 𝑀𝐼𝑁(𝐶1, 𝐶2)/𝑀𝐴𝑋(𝐶1, 𝐶2)
Eqn. 4-9
The sodium heat transfer coefficient, in un-baffled counter flow along a tube bank, is given by
the Westinghouse correlation [Todreas and Kazimi, 1993]
𝑁𝑢𝑁𝑎 ,𝑐𝑓𝑙𝑜𝑤 = 4 + 0.33 𝑃/𝑑
3.8
+ 𝑅𝑒𝑃𝑟/100
𝑕𝑁𝑎 =
0.86
+ 0.16 𝑃/𝑑
𝑁𝑢𝑘𝑁𝑎
𝐷𝑕𝑦𝑑
5.0
Eqn. 4-10
Eqn. 4-11
where the Reynolds number is calculated as the average value from the sodium inlet to the outlet,
based on hydraulic diameter.
When baffles are used on the shell side, the Bell-Delaware method is used to determine
the heat transfer coefficient [Shah, 2003]. Baffles cause the flow to be more mixed and to follow
an almost cross-flow pattern through the tube banks. This enhances the shell side heat transfer
coefficient.
The Bell-Delaware method correlates correction factors to a cross-flow heat transfer
coefficient based on the baffle geometry. The cross-flow Nusselt number for sodium is given by
the Kalish-Dwyer correlation [Foust, 1976].
𝑁𝑢𝑁𝑎 ,𝑥𝑓𝑙𝑜𝑤 =
𝜙1
𝑑
1−
𝑑
𝑃
𝑕𝑥𝑓𝑙𝑜𝑤 =
5.25 + 0.225𝑃𝑒 0.653
𝑁𝑢𝑁𝑎 ,𝑥𝑓𝑙𝑜𝑤 𝑘𝑁𝑎
𝐷𝑕𝑦𝑑
Eqn. 4-12
Eqn. 4-13
where φ1 is an empirical constant which is determined from tabulated values by Hsu [1964]. The
resulting heat transfer coefficient is given by
𝑕𝑠 = 𝑕𝑥𝑓𝑙𝑜𝑤 𝐽𝑐 𝐽𝑟 𝐽𝑏 𝐽𝑠 𝐽𝑙
where the correction factors, J, are given according to Table 4.4.
92
Eqn. 4-14
Table 4.4: Correction factors for the cross-flow Nusselt number from the Bell-Delaware Method
Correction
For
Correction Factor J
The cut of
the baffles
Leakage
around the
baffles
Bundle
bypass flow
Temperature
Gradient
Buildup
Baffle
Spacing at
inlet and
outlet
Notes
𝐽𝑐 = 0.55 + 0.72𝐹𝑐
Definitions
𝜃𝑐𝑡𝑙 + 𝑠𝑖𝑛𝜃𝑐𝑡𝑙
Eqn. 4-16
𝜋
𝐷𝑠 − 2𝑙𝑐
= 2𝑐𝑜𝑠 −1
Eqn. 4-17
𝐷𝑐𝑡𝑙
𝐹𝑐 = 1 −
Eqn. 4-15
𝜃𝑐𝑡𝑙
𝑟𝑠 =
𝐽𝑙 = 0.44 1 − 𝑟𝑠 + 0.44 1 − 𝑟𝑠 𝑒𝑥𝑝−2.2𝑟 𝑙𝑚
𝐴𝑜,𝑠𝑏
𝐴𝑜,𝑠𝑏 + 𝐴𝑜,𝑡𝑏
Eqn. 4-18
𝑟𝑙𝑚 =
𝐽𝑏 =
𝑒𝑥𝑝−𝐶𝑟 𝑏
+ )1/3
1−(2𝑁𝑠𝑠
𝐴𝑜,𝑠𝑏 + 𝐴𝑜,𝑡𝑏
𝐴𝑜,𝑐𝑟
𝑁𝑠𝑠
𝑁𝑟,𝑐𝑐
𝐴𝑜,𝑏𝑝
𝑟𝑏 =
𝐴𝑜,𝑐𝑟
+
𝑁𝑠𝑠
=
+
1 𝑓𝑜𝑟 𝑁𝑠𝑠
≥ 1/2
Eqn. 4-21
+
𝑓𝑜𝑟 𝑁𝑠𝑠
≥ 1/2
Eqn. 4-19
Eqn. 4-20
Eqn. 4-22
Eqn. 4-23
1 𝑓𝑜𝑟 𝑅𝑒 ≥ 100
𝐽𝑟 =
10
𝑁𝑟,𝑐𝑐
leakage area
𝐴𝑜,𝑡𝑏 = tube-to-baffle leakage
area
𝐴𝑜,𝑐𝑟 = cross flow area
𝑁𝑠𝑠 = the number of sealing
strips or obstructions to
bypass flow
in cross flow zone
1.35 𝑓𝑜𝑟 𝑅𝑒 ≤ 100
𝑓𝑜𝑟 𝑅𝑒 ≤ 20
𝑁𝑏 − 1 + (𝐿+𝑖 )(1−𝑛) + (𝐿+𝑜 )(1−𝑛)
𝐽𝑠 =
𝑁𝑏 − 1 + 𝐿+𝑖 + 𝐿+𝑜
𝐴𝑜,𝑠𝑏 = shell-to-baffle
𝑁𝑟,𝑐𝑐 = number of tube rows
Eqn. 4-24
0.18
𝐷𝑠 = shell diameter
𝑙𝑐 = dist. baffle edge to shell
𝐷𝑐𝑡𝑙 = tube array diam. - tube
diam.
𝐶=
1.25 𝑓𝑜𝑟 𝑅𝑒 > 100
Eqn. 4-25
0.6 𝑡𝑢𝑟𝑏𝑢𝑙𝑒𝑛𝑡 𝑓𝑙𝑜𝑤
Eqn. 4-26
𝑛=
93
1
𝑙𝑎𝑚𝑖𝑛𝑎𝑟 𝑓𝑙𝑜𝑤
3
Eqn. 4-27
𝑁𝑏 = number of baffles
𝐿+𝑖 = entrance baffle spacing
𝐿+𝑜 = outlet baffle spacing
For a P-IHX, with sodium on the tube side, the heat transfer coefficient is given by
𝑁𝑢𝑁𝑎 ,𝑡𝑢𝑏𝑒 = 6.3 + 0.0167𝑅𝑒 0.85 𝑃𝑟 0.93
Eqn. 4-28
for upward flow of sodium in tubes [Mills, 1995]. For S-CO2 flow on the tube side, Kim et al.
[2008] have modified the Jackson and Fewster correlation for supercritical pressures in smooth
tubes. Given that the operating pressure of S-IHXs will be much higher than the critical pressure
of 7.377 MPa, the correlation can be used to produce a Nusselt number that applies to the entire
heat exchanger once the Reynolds and Prandtl numbers are obtained from average properties at
the inlet and outlet. Kim et al. [2008] determine the S-CO2 Nusselt number to be
𝑁𝑢 = 0.0182𝑅𝑒
0.824
𝑃𝑟
0.515
𝜌𝑤𝑎𝑙𝑙
𝜌𝑏𝑢𝑙𝑘
0.299
Eqn. 4-29
For enhanced tubes, the Ravigurarajan and Bergles correlation is used to determine the
heat transfer coefficient [Ravigurarajan and Bergles, 1996]. The enhancements consist of spiral
ribs on the interior of the tubes. Pressure drops are increased, but the heat transfer can be vastly
improved, saving a great deal in terms of the size of the heat exchanger. The Ravigurarajan and
Bergles correlations for heat transfer and pressure drop are excluded here due to their length, but
they can be found in Appendix D. In all designs discussed hence, enhanced tubes are assumed to
be used for CO2 only. The experiments used to develop enhanced tube correlations were
conducted with air and single-phase water, so other fluids are assumed to be outside the range of
validity of their correlations. With the average heat transfer coefficients determined for the two
fluids, the overall heat transfer coefficient can be calculated.
For sodium-sodium heat exchangers, single-wall tubes are appropriate, but to ensure a
leak detection capability and increase the robustness of S-IHXs, double-walled tubes are
assumed to be required in a sodium-CO2 or a sodium-water heat exchanger. The work of Kubo
et al. [1997] produced results for the effective thermal conductivity of a 9Cr-1Mo double-walled
steam generator tube. Using this data, and a thermal conductivity of 9Cr-1Mo steel of 27.9
W/mK [Williams et al., 1984] the value of the effective gap conductivity was determined to be
293.0 mW/mK.
This value for effective helium gap conductivity has been used in SoSaT so that the user
may define a double-walled tube of any wall thickness and any material. It should be noted that
the experiments of Kubo et al. were conducted for a beginning of life (BOL) gap thicknesses of 3
μm and the end of life (EOL) gap thickness was measured to be ~7 μm. Comparison of the
calculated value of gap conductivity with the conductivity of helium at 400 oC and 0.90 MPa, the
94
condition of the experiment performed by Kubo et al., reveals that the gap is behaving almost
exactly as a stagnant gas thermal resistance. REFPROP determines the conductivity to be 274.01
mW/mK.
SoSaT will calculate the ASME required thickness of tubes based on the material chosen,
the operating temperature, and the internal pressure based on Eqn. 2-1. The allowable stress
intensity for each material is interpolated from ASME Code tabulated values assuming a
temperature equal to the hot side inlet temperature [ASME, 2007]. These calculations are
contained in the subroutines MATPROP and GEOMETRY.
For single phase fluids, the heat transfer coefficients, tube conductivity and dimensions
are then used in the subroutine EFFECTIVE to determine the overall heat transfer coefficient
and NTU. Then the effectiveness is calculated and outlet temperatures are based on the
effectiveness. The process is continued by iterating the outlet temperatures of each fluid until
convergence.
The NTU method is not used in steam generators because the heat transfer coefficients
and fluid properties vary widely through different boiling regimes. Therefore, SoSaT is really
two codes which share common inputs and outputs. The calculation method is totally different
for boiling water than it is when both fluids are single phase.
Modern steam cycles employ ever-increasing system pressures as a means of increasing
cycle efficiency. Substantial efficiency gains can be achieved by raising steam pressures, even
above the critical pressure of 22.06 MPa [MIT, 2007]. Therefore, it is assumed that Generation
IV reactors operating on a steam cycle will have system pressures considerably higher than those
of current LWR steam cycles. Modeling the heat transfer within high pressure steam generators
requires the use of appropriate experimental correlations for the heat transfer coefficients within
different boiling regimes. The boiling process, as modeled by SoSaT is summarized in Figure
4.4.
95
Figure 4.4: The division of the tube length according to heat transfer regime in SoSaT
.
The shell-and-tube steam generator model assumes that sodium will flow in a uniform
manner downward through a cylindrical shell. There is no option for baffles in the current steam
generator model. Upward flow of water is contained in a hexagonally pitched bank of smooth
surfaced tubes. The tubes are assumed to be double wall tubes and can be constructed of several
materials. These are 9Cr-1Mo steel, 304 SS, and 316 SS. The code begins by calculating what
the sodium outlet temperature must be, assuming that all of the designated power is transferred
as heat to the water. Based on the sodium inlet conditions, the heat transferred is given by
𝑄 = 𝑚𝑁𝑎 𝑐𝑝
𝑎𝑣𝑔
∆𝑇
Eqn. 4-30
By assuming that the specific heat capacity of sodium does not change appreciably with
temperature, the outlet temperature is calculated from Eqn. 4-30, using the user’s input of total
heat to be transferred. The specific heat of sodium at each temperature is calculated based on
known fluid properties from an Argonne National Laboratory study [Fink and Leibowitz, 1995]
96
and the process is iterated until the sodium outlet temperature converges. This outlet
temperature is necessary for determining the fluid properties (and then the heat transfer
coefficient) of sodium at the bottom of the shell, where the first node of the heat transfer
calculation lies. The calculation will proceed upward in nodes of 1 cm length.
The critical heat flux is determined for each node from interpolation of Groeneveld’s
2006 CHF Lookup table [Groeneveld, 2007], which requires tube diameter, mass flow rate,
pressure, and quality as inputs. This calculation occurs at each node because many of these
parameters will change at each node.
The sodium heat transfer coefficient is calculated based on properties obtained from the
outlet temperature and the Westinghouse correlation for liquid-metal, counterflow heat transfer
in a tube bundle.
𝑁𝑢 = 4 + 0.33 𝑃/𝑑
3.8
+ 𝑅𝑒𝑃𝑟/100
0.86
+ 0.16 𝑃/𝑑
5.0
Eqn. 4-31
Based on conditions on the tube side, an appropriate heat transfer correlation will be chosen.
When the water is still subcooled, the Dittus-Boelter correlation is used. The code identifies
subcooled water by the wall temperature and the bulk fluid temperature.
𝑕𝐻2𝑂 = 0.023𝑅𝑒 0.8 𝑃𝑟 0.4 𝑘/𝑑
Eqn. 4-32
and friction factors are given by the Blasius or McAdams correlation, according to the Reynolds
number.
𝑓=
0.316𝑅𝑒 −0.25 , 𝐵𝑙𝑎𝑠𝑖𝑢𝑠 𝑐𝑜𝑟𝑟𝑒𝑙𝑎𝑖𝑜𝑛: 𝑅𝑒 < 30,000
0.184𝑅𝑒
−0.20
Eqn. 4-33
, 𝑀𝑐𝐴𝑑𝑎𝑚𝑠 𝑐𝑜𝑟𝑟𝑒𝑙𝑎𝑡𝑖𝑜𝑛: 𝑅𝑒 ≥ 30,000
The code assumes that, until saturation, all heat transfer to the water manifests as a temperature
rise. This is valid on a bulk fluid basis, and since the total length of pipe is more interesting than
the details of the flow, the assumption is used in SoSaT. Therefore, the temperature increase of
the subcooled fluid is given by
𝑑𝑞
∆𝑇𝑏𝑢𝑙𝑘 =
Eqn. 4-34
𝑚𝐻2𝑂 𝑐𝑝
where dq represents the heat transferred in the node. The fluid reaches subcooled boiling when
the wall temperature reaches a minimum value above the saturation temperature. This point is
termed the onset of nucleate boiling. The temperature above which the wall must rise is given by
97
Twall = Tsat
8σq′′ONB Tsat
+
hfg k f ρg
0.5
Eqn. 4-35
which was developed by Davis and Anderson [1966]. Once the wall temperature reaches this
value, the flow enters the subcooled boiling regime. The heat transfer coefficient of the
subcooled boiling region is given by
𝑕𝑠𝑢𝑏 = 𝑕 𝑇𝑃
𝑇𝑤𝑎𝑙𝑙 − 𝑇𝑠𝑎𝑡
+ 𝑕𝑙𝑜
𝑇𝑤𝑎𝑙𝑙 − 𝑇𝑓
Eqn. 4-36
where hlo indicates the liquid only heat transfer coefficient given by the Dittus-Boelter
correlation and hTP is the two-phase nucleate boiling heat transfer coefficient given by Kandlikar
[1990].
𝑕 𝑇𝑃
= 𝐶1 𝐶𝑜𝐶2 (25𝐹𝑟𝑙𝑜 )𝐶5 + 𝐶3 𝐵𝑜𝐶4 𝐹𝑓𝑙
Eqn. 4-37
𝑕𝑙𝑜
Once the bulk temperature has reached saturation, the Kandlikar correlation predicts the rise in
quality for each node according to the heat of vaporization. This regime is saturated nucleate
boiling. Critical heat flux is still tested for in each node and once it is surpassed, the code
switches to the Bishop correlation for post-dryout heat transfer [Groeneveld, 1975]. The heat
transfer coefficient after dryout is given by
𝑁𝑢 = 0.033𝑅𝑒𝑤0.80 𝑃𝑟𝑤1.25 𝑥 +
𝜌𝑔
1−𝑥
𝜌𝑙
0.738
𝜌𝑔
𝜌𝑙
0.197
Eqn. 4-38
which is valid up to quality of 1.0 and pressures up to 21.5 MPa. Once the equilibrium quality
reaches 1.0, the Gnielinski correlation for vapor heat transfer coefficient is used. It gives the
Nusselt number as
𝑁𝑢 =
𝑓
𝑅𝑒𝑔 − 1000 𝑃𝑟𝑔
2
1 + 12.7 𝑃𝑟𝑔0.66 − 1.0
𝑓
2
𝑓 = 0.00128 + 0.1143𝑅𝑒𝑔−0.311
Eqn. 4-39
Eqn. 4-40
The calculation concludes once the prescribed power has been transferred to the fluid. The test
for heat transferred is the condition for continuation of the loop, so even if the flow never enters
98
the superheated vapor regime, the calculations will stop once the total desired power is
transferred.
For supercritical water, modeling the heat transfer is much easier since the water does not
undergo a distinct phase change. In the case of supercritical pressures, the water side heat
transfer coefficient is determined by the correlation of Cho, Chou, and Cox [Herron, 2002].
𝑁𝑢𝑠−𝐻2𝑂
𝐺𝑑
= 0.00459
𝜇𝑏𝑢𝑙𝑘
0.923
𝑕𝑤𝑎𝑙𝑙 − 𝑕𝑏𝑢𝑙𝑘 𝜇𝑤𝑎𝑙𝑙
𝑇𝑤𝑎𝑙 𝑙−𝑇𝑏𝑢𝑙𝑘
𝑘
0.613
𝜌𝑤𝑎𝑙𝑙
𝜌𝑏𝑢𝑙𝑘
0.231
Eqn. 4-41
For supercritical pressures, the Cho, Chou, and Cox correlation will be acceptable, assuming that
pressure drops never cause the fluid to fall below the saturation dome. The critical pressure for
water is 22.09 MPa.
The conditions tested at each node determine which boiling regime the water has entered,
and therefore which correlations to use for the heat transfer coefficient and pressure drop. Table
4.5 summarizes the criteria for each boiling regime based on bulk temperature, wall temperature
and equilibrium quality, x.
Table 4.5: Conditions for each boiling regime in SoSaT
Heat Transfer Regime
Subcooled Liquid
Subcooled Boiling
Saturated Nucleate Boiling
Post CHF heat transfer
Vapor
Heat Transfer
Correlation
Dittus-Boelter
Kandlikar
Kandlikar
Bishop
Gnielinski
References
Conditions*
[Todreas and Kazimi, 1993]
[Kandlikar, 1990]
[Kandlikar, 1990]
[Groeneveld, 1975]
[Rohsenow et al., 1998]
Tbulk<Tsat, Twall<Tsat+ ΔT
Tbulk<Tsat,Twall>Tsat+ΔT
x>0.0,x<1.0, q’’,q’’crit
x<1.0, q’’>q’’crit
x>1.0
* ΔT is represented by the second term in Eqn. 4-35
To aid in the operation of SoSaT, the code outputs the difference between the bulk
temperature and saturation temperature, along with the boiling regime to the screen every 10 cm
in the upward flow calculation. That way, the user can see if the prescribed geometry has
resulted in a pinch-point or if an excessively high flow rate has resulted in poor steam conditions
without ever opening the output file.
99
4.4.2 Expanding the Capabilities of the PCHE Codes
The PCHE codes needed to have a hybrid steam generator model and a sodium-tosodium model added. Creating a sodium-to-sodium PCHE model was very simple. The existing
PCHE code with no helium plate was altered by taking the primary sodium properties and
applying them to the secondary side as well. The task was almost trivial. Creating a steam
generator model for the PCHE was more difficult. The original hybrid code was altered by
including a new heat transfer subroutine and CHF calculations. The heat transfer subroutine was
modified from Shirvan [2009] to include the same high pressure correlations as SoSaT. One
challenge with the PCHE code was dealing with convergence. The code operates by calculating
the heat transfer coefficient of each fluid at its average temperature. This heat transfer
coefficient is then used to make a rough judgment of the size of the heat exchanger, and therefore
the size of each node. The wide variation in boiling water heat transfer coefficient meant that the
initial guess was far from correct and the code’s node length was too long. The result was that
the code stepped beyond the desired power of the heat exchanger and crashed. Resolving this
error by stopping calculations once the desired power is reached allowed the code to run
smoothly.
4.5 Benchmarking Heat Exchanger Codes
There are not enough data available to benchmark PCHE code results, but the shell-andtube design has been used in many reactor designs. For S-CO2 heat exchangers, the Flexible
Conversion ratio Reactor (FCR) IHX was used as a case for comparison. The FCR is a leadcooled reactor with S-CO2 power conversion system. It uses kidney shaped IHXs, so for
comparison, the geometry of the tube array and the total number of tubes was retained from the
FCR analysis. The comparison between the FCR IHX and that produced by SoSaT is included
in Table 4.6. Also included are benchmark designs for a shell-and-tube P-IHX from the ABR1000 design, and a shell-and-tube steam generator from the JSFR design.
100
Table 4.6: Benchmarking of SoSaT code**
Reference Design
Reactor Design
Number of Tubes
S-CO2 heat transfer coeff. (W/m2K)
Lead Heat transfer coeff. (W/m2K)
Tube length (m)
Reactor Design
Number of Tubes
Primary heat transfer coeff. (W/m2K)
Secondary heat transfer coeff. (W/m2K)
Tube length (m)
Reactor Design
Number of Tubes
Sodium heat transfer coeff. (W/m2K)
Water heat transfer coeff. (W/m2K)
Heat transfer area (m2)
19173
6812
22500
5.64
SoSaT Results
FCR IHX
19181
6524
25100
5.70
4500
N/A
N/A
4.78
ABR-1000 P-IHX
4550
30219
41266
4.90
7200
N/A
N/A
12500
JSFR Steam Generator
7220
32765
variable
10140*
*Including a 20% margin on the heat transfer area, as is common practice in steam generator
design would yield a value closer to the reference JSFR design.
**[Grandy and Seidensticker, 2007], [IAEA, 2006], [Todreas and Hejzlar, 2008]
For the benchmark designs, tube dimensions and pitch were identical to the reference
design. The number of tubes for each design is a result of the shell geometry, so it does not
match the reference designs perfectly, but it comes close for each design. Heat transfer
coefficients are averages that where given in the FCR reports or calculated in SoSaT and are not
available for the ABR-1000 or JSFR reference designs. The FCR IHX includes enhanced tubes
which were replicated in the SoSaT design. The improvements over smooth tubes are similar for
the FCR design and the SoSaT model of the same heat exchanger. The results of SoSaT are in
very good agreement with reference designs.
4.6 S-CO2, Traditional Rankine, or Supercritical Steam PCS
Commercial power SFRs have used the traditional Rankine cycle, though at temperatures
and pressures higher than those of LWRs. Supercritical steam cycles have been operated in
some coal fired power plants [MIT, 2007] with pressures up to 24.3 MPa and temperatures of
565 oC. Achievable generating efficiency of these plants is about 38 %. Target steam conditions
101
are up to 38.5 MPa and 700-720 oC. Supercritical CO2 cycles are promising for any high
temperature application. All of these technologies could be options for the SFRs of the future,
but they will inevitably produce different cycle efficiencies. The trend in conventional power
technology is to use steam cycles of ever-increasing pressures because cycle efficiency improves
with increased pressure, as shown in Figure 4.6. These data were developed using a standard
commercial steam cycle model with a high and a low pressure turbine and four reheat stages.
The feedwater temperature is maintained constant at 216 oC, as per the ABR-1000 reference
case. Cooling water is at 20 oC. The data were obtained using STEAM PRO 16, steam cycle
modeling software from Thermoflow, Inc. [Thermoflow, 2008].
41.5
Cycle Efficiency (%)
41
14.5 MPa
40.5
15.5 MPa
16.5 MPa
40
39.5
39
440
460
480
500
520
540
Turbine Inlet Temperature (oC)
Figure 4.6: Rankine Cycle efficiency as a function of temperature and pressure
The S-CO2 cycle has already been discussed in detail in Chapter 2. It can achieve very
high efficiencies and is compact. The S-CO2 recompression cycle will be one of the options
considered in Chapter 5.
The supercritical water cycle shows promise for the future. Corrosion is an issue in a
supercritical water environment, but efficiency gains can be realized. A good model of the
supercritical water cycle was not available in this research, but gains over conventional Rankine
cycles can be estimated based on running the Thermoflow software with a pressure just below
critical. STEAM PRO 16 can run at pressures up to 22.0 MPa, and a 1 MPa increase over this
value will not result in a very large efficiency increase.
Using SoSaT and the PCHE codes, the ultimate turbine inlet temperature can be
determined, based on limitations on heat exchanger size which are discussed in Chapter 5.
Chapter 5 details the calculation of cycle efficiencies for a wide range of balance of plant
configurations.
102
4.7 Chapter Summary
The development of Fortran codes for heat exchanger modeling has been a major task in
examining the possible range of efficiencies for SFRs. These codes have been expanded to
include new capabilities needed for modeling different SFR design configurations. Coupled with
CYCLES III and STEAM PRO 16, the heat exchanger codes can be used to determine the
achievable efficiency of an SFR. Printed Circuit Heat Exchanger (PCHE) codes have been
expanded to include sodium-to-sodium and sodium-to-water heat exchangers. They are compact
and highly effective. The SoSaT code was written from scratch to model shell-and-tube heat
exchangers. It performs calculations for sodium-to-sodium, sodium-to-CO2, and sodium-towater heat exchangers. Water can be either boiling or supercritical. Tubes can be single or
double-walled and can include enhanced heat transfer surfaces for CO2. Baffles can be included
on the shell side. Eliminating the intermediate loop is only a practical option for loop-type SFR
designs. This idea is elaborated in Chapter 5. Using an alternate fluid in the intermediate loop
was shown by Cooper and Lee to have no advantage for any of the fluids studied. Most
efficiency gains therefore, will come from measures to raise the average core outlet temperature
or changes in the balance of plant. The tools developed here are used in Chapter 5 to quantify
the achievable efficiency of an SFR and to compare the performance of heat exchangers and
power cycles from an efficiency perspective. In Chapter 5, the range of options for SFR design
will be discussed and results from the heat exchanger codes will be used to show the achievable
efficiencies of SFRs.
103
4.8 Nomenclature for Chapter 4
A
cp
d
D
e
f
g
G
h
h
J
K
k
L
𝑚
N
Nu
NTU
Pw
p
flow area or heat transfer area (m2)
specific heat at constant pressure (J/kgK)
tube or channel diameter (m)
shell diameter (m)
height of axial ribs in enhanced tubes (m)
friction factor
acceleration due to gravity (9.80 m/s2)
mass flux (kg/m2s)
heat transfer coefficient (W/m2K)
enthalpy (J/kg)
adjustment factor for the Bell-Delaware method (dimensionless)
form loss coefficient
thermal conductivity (W/mK)
length (m)
mass flow rate (kg/s)
integer number of tubes, ribs, channels, etc.
Nusselt number
Number of Transfer Units
Wetted Perimeter (m)
Pressure (Pa)
𝑄
q’’
q’
Re
T
U
v
W
x
thermal power (W)
heat flux (W/m2)
linear heat rate (W/m)
Reynolds number
Temperature (oC)
Overall heat transfer coefficient (W/m2K)
velocity (m/s)
work (W)
equilibrium steam quality
Greek Letters
ε
η
μ
ρ
ζ
Effectiveness
efficiency
viscosity (Pa•s)
Density
Liquid surface tension
104
Subscripts
b
c
f
fin
g
i
in
l
lo
Na
o
out
r
s
baffle bundle bypass flow
baffle cut
liquid
characteristic of the tube enhanced surface
vapor
inside dimension
Inlet condition
baffle leakage
liquid only
Sodium property
outside dimension
Outlet condition
baffle temperature gradient
baffle spacing
105
5 Increasing the Efficiency of the SFR
5.1 Introduction
As stated in Chapter 4, increasing average core outlet temperature, improving heat
transfer to the PCS, and improving PCS performance are likely to produce the greatest efficiency
increases for an SFR. Previous research has indicated that several methods of increasing average
core outlet temperature can be employed in the SFR. It is feasible that SFR core outlet
temperatures can be up to 575 oC, as in the BN-1800 design [IAEA, 2006]. With such high
temperatures, and effective heat transfer to the PCS, efficiencies can be very high indeed.
Determining the efficiency benefit of these temperature increases will allow financial
considerations to take their long term effect into account more accurately. The tools developed
in this research and described in Chapter 4 allow a comparison between different heat exchanger
options. CYCLES III and STEAM PRO 16 allow for a comparison between the S-CO2
recompression cycle and the traditional Rankine cycle. All of these tools combined produce a
wide range of BOP options that can be considered. A small increase in plant efficiency could
result in very large economic benefits, and therefore a difference of less than 1% in efficiency
may be economically significant.
5.2 A Reference Design: The ABR-1000
In order to make meaningful comparisons of design options, a reference design is
required. The ABR-1000 is a concept reactor designed by Argonne National Laboratory
[Grandy and Seidensticker, 2007]. It is a four-loop, pool type, 1000 MWth reactor with a high
pressure traditional Rankine PCS. The ABR-1000 design reports that the PCS achieves a
thermal efficiency of 38 % with a steam temperature of 454 oC and a pressure of 15.5 MPa at the
turbine inlet. Comparisons will be made with this reference design in regard to heat exchanger
options, the effect of increased core outlet temperature, and the choice of PCS.
For ease of comparison, the steam generator for the reference temperature and flow rate
conditions was assumed to be a straight shell-and-tube heat exchanger that matched the reference
design in flow rates, temperatures, and overall heat transfer area. This approach was taken
because heat transfer coefficients will not vary appreciably between a helical coil steam
generator and a straight tube steam generator, and SoSaT has been written for straight tubes.
Both straight tube and helical coil steam generators are considered the best options for SFRs of
the future and both have been included in SFR designs [IAEA, 2006], [Chikazawa et al., 2008].
The ABR-1000 steam generator has a helical coil bundle height of 11.6 m and this dimension
was preserved in the comparisons performed. The heat transfer in a straight-tube steam
generator of 11.6 m in height is much better than that of a helical coil steam generator of 11.6 m
106
because the number of tubes is much larger in a straight tube steam generator. For example, the
total heat transfer area of the ABR-1000 helical coil steam generator is 1806 m2, as compared to
a heat transfer surface of ~5000 m2 for a straight tube design with equal shell dimensions. The
additional heat transfer area makes a marked difference. The ABR-1000 steam generator
achieves a steam temperature of 454 oC at outlet, while an 11.6 m tall straight tube steam
generator achieves a steam temperature of 486 oC. This translates to an efficiency gain of 0.8 %
in a Rankine cycle. Because the straight-tube steam generator is modeled in SoSaT, it has taken
the place of the helical coil steam generator in the reference balance of plant while preserving the
shell dimensions. Thus, the efficiency of the reference design is higher than that of the ABR1000. The total heat transfer area of a straight-tube design and a helical coil design will not be
very different, but some studies have shown that the reliability of helical coil designs could be
significantly higher than that of straight tube steam generators. For example, helical coil designs
have not experienced fretting or high cycle fatigue due to the different welds used at the tube
sheet [Chikazawa et al., 2008]. For efficiency comparisons, this study only considers straighttube designs, but the helical coil steam generator design could be important for reducing the cost
of steam generators. The reference balance of plant is summarized in Table 5.1.
Table 5.1: The reference balance of plant
o
Core Outlet Temperature ( C)
Core Thermal Power (MW)
P-IHX
Height (m)
Shell Diameter (m)
S-IHX
Height (m)
Shell Diameter (m)
Steam Pressure (MPa)
Steam Temperature (oC)
Net Cycle Efficiency (%)
Intermediate Pumping Power (MW)
Primary Sodium Flow Rate (kg/s)
Intermediate Sodium Flow Rate (kg/s)
Primary Pumping Power (MW)
510
1000 (250 MWth per loop)
5.20
1.72
11.6
2.81
15.5
454
38
~1.6 (~0.4 MW per loop)
1256
1256
4.60
5.3 Options for Increasing Core Outlet Temperature
Placing ribs, or “long, semi-circular protrusions” within the hexagonal assembly cans can
flatten the temperature profile of the core outlet flow by reducing flow in non-heated edge
107
subchannels [Memmott, 2009]. During transients, the peak cladding temperature limit will be
met in the hottest channel, but if all the channels are at about the same temperature, the average
core outlet temperature can be much higher without increasing the temperature of the hot
channel. Thus, the average core outlet temperature can be increased without endangering any
margins to the peak clad temperature limit. Because the efficiency of the PCS is driven
primarily by temperature, this modification is particularly appealing. RELAP-5 models of the
SFR have shown that core outlet temperature profile has a difference of 60 oC for cold assemby
dimensions and 30 oC for hot assembly dimensions from the interior to exterior subchannels.
With ribs in place, this temperature difference can be reduced to less than ~2 oC [Memmott,
2009]. This result means that the average core outlet temperature could be increased by almost
15 oC from the reference case of 510 oC without endangering any peak temperature limits. This
method appears to be the most effective available.
Diluent grading in the fuel can flatten the core power profile as well. Power is reduced in
the high power region by increasing the fraction of Zr in the fuel, or by placing “dummy rods”
into these regions to flatten the core power profile. This option greatly affects the refueling cycle
and will probably produce cycle lengths of too short a period to be economically attractive
[Denman, 2009]. The opposite approach is to create enrichment zones in the lower power
regions to increase the power there.
These and other design options create a range of core outlet temperatures in SFRs. Table
5.2 includes the core outlet temperatures of some power-producing SFRs. The BN-1800 design
still requires many decisions to be made about the construction of the core, so the outlet
temperature for the BN-1800 reflects an estimate of what the designers believe to be achievable.
Table 5.2: Core Outlet Temperatures of Selected SFRs
Core Outlet Temperature (oC)
545
498
550
547
575
510
Reactor
Super-Phenix 1(France)
ALMR (USA)
JSFR-1500 (Japan)
BN-800 (Russian Federation)
BN-1800 (Russian Federation)
ABR-1000 (USA)
5.4 Option Space
The option space consists of variation in core outlet temperature, heat exchanger type,
elimination of the intermediate loop, and choice of PCS as shown in Figure 5.1.
108
Figure 5.1: The arrangement of components in the SFR balance of plant and the options available
for each component.
These choices are summarized in analogy to a fault tree in accident space, as shown in Figure
5.2. The selection of each component will lead to changes in the efficiency of the plant.
Figure 5.2: The design choices affecting efficiency that are considered in this study
Figure 5.2 is meant to reflect the range of options in developing a balance of plant for an
SFR. Core outlet temperature could very well reach up to 575 oC. The primary IHX could be a
PCHE or a shell-and-tube design, or the intermediate loop could be eliminated. The secondary
IHX could be either PCHE or shell-and-tube, and the PCS could be an S-CO2, conventional
109
Rankine, or supercritical water cycle. The range of system pressures in the steam cycle can vary
substantially as well. In this study, two core outlet temperatures will be compared: 510 oC and
530 oC and the steam cycles will be assumed to operate at 15.5 MPa and 23 MPa (supercritical).
5.4.1 Methodology
By assuming a core outlet temperature, a primary mass flow rate of 1256 kg/s, and a
thermal power of 250 MW, the primary fluid cold leg temperature is known. With this
information, the P-IHX, then the S-IHX, and finally the PCS are examined to determine the
efficiency of the cycle. In order to compare different options, the dimensions of the ABR-1000
heat exchangers have not been exceeded. The much higher flow rate in CO2 heat exchangers
causes pressure drops to become excessively large if the flow length is allowed to be longer than
about 2 m. Therefore, the CO2 PCHE has the same dimensions as the water PCHE, but the flow
direction is changed to create a shorter flow path through the monolithic block.
Table 5.3 lists the standard dimensions of etched channels in the PCHEs and the standard
tube dimensions for shell-and-tube heat exchangers. For comparison, they have been kept
constant in every design, though in reality the designer could have considerable leeway. For
PCHEs, the dimensions do not correspond exactly due to their rectangular shape.
Table 5.3: Standard channel dimensions used for PCHEs in this study
Plate Thickness
Channel Diameter
Channel Pitch
Helium Plate Thickness
Helium Plate Channel Diameter
Formed Plate Thickness
Formed Plate Channel Height
Formed Plate Channel Width
1.50 mm
2.50 mm
3.00 mm
0.50 mm
0.30 mm
2.00 mm
2.50 mm
3.00 mm
The PCSs are standardized as well. The CO2 cycle modeled in CYCLES III is a 500
MWth cycle, operating with a maximum pressure of 20 MPa and 70 m3 of heat exchanger
volume. Though the heat exchangers are designed for 250 MWth, the 500 MWth cycle is used
to determine efficiency because the compressor efficiencies determined in Chapter 3 applied to
the 500 MWth cycle. Two intermediate loops of 250 MWth could be coupled to each of the 500
MWth power cycles. CYCLES III is used to determine the efficiency of the recompression cycle
based on pressure drops and temperatures calculated in the S-IHX. The mass flow rate through
the S-IHX is held constant for the CO2 cycle at 1250 kg/s (for a 250 MWth cycle, doubled in
CYCLES III for efficiency calculations). The Rankine cycle is modeled by STEAM PRO 16,
110
which assumes a constant feedwater temperature as a function of system pressure. The mass
flow rate used in the PCS then depends upon the design of the S-IHX. STEAM PRO models the
Rankine cycle with low and high pressure turbines and four reheat stages. The feedwater
temperature is set to a constant 216 oC and the pressure is set to 15.5 MPa. Figure 5.3 shows the
layout of the STEAM PRO 16 Rankine cycle model.
Figure 5.3: STEAM PRO 16 diagram of the Rankine Cycle
STEAM PRO 16 cannot model a supercritical water cycle, but the cycle efficiency at
pressures of 23 MPa should be less than 0.2 % higher than the results at just below critical
pressure (22.0 MPa). Therefore, efficiency results for the supercritical steam cycle are based on
the STEAMPRO 16 results for 22.0 MPa.
The efficiency for each configuration was determined by calculating temperatures
achievable in the intermediate loop by using the PCHE codes or SoSaT as appropriate. An
intermediate loop flow rate of 1256 kg/s was assumed for all cases, in accordance with the ABR1000 report. This is an appropriate choice of flow rate because it matches the primary sodium
flow rate and the heat transfer coefficients and heat capacities are almost identical on either side
of the P-IHX. The intermediate fluid temperature at the P-IHX inlet was reduced until the PIHX adhered to the size restrictions set forth in Table 5.3. Once the intermediate loop
temperatures were known, these temperatures could be applied to the S-IHX, neglecting the
small losses in the intermediate piping, as discussed in Section 4.1.1. The S-IHX was modeled,
111
again using the PCHE codes or SoSaT as appropriate, and the PCS fluid temperatures were
determined from the S-IHX model. With pressure drops and temperatures determined, STEAM
PRO 16 and CYCLES III were used to determine the cycle efficiency.
It is important to keep in mind that the option space does not include every configuration.
For example, every option could also include variation in the steam temperature, or even in the
type of Rankine cycle used (turbine stages, reheat, etc.). This study aims to show a broad
comparison between options and inform the SFR designer about the general advantages of
certain options. Furthermore, a principle goal of this research was to develop tools for future
analysis. The PCHE codes, SoSaT, and CYCLES III are all useful for future researchers in this
field.
5.5 Results of Efficiency Comparisons
The option space described in Figure 5.2 has been tabulated in Table 5.4 along with the
efficiency results for each case in Tables 5.5 and 5.6. Twenty-four cases were examined, but
only 18 are displayed in Table 5.4. The last six cases (numbers 19-24) corresponded to the
elimination of the intermediate loop for a pool design, which was determined to be impractical
for safety and efficiency reasons.
Table 5.4: Options Considered in the Efficiency study
Option Number
P-IHX
S-IHX
Type
PCS
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
PCHE
PCHE
PCHE
PCHE
PCHE
PCHE
S-and-T
S-and-T
S-and-T
S-and-T
S-and-T
S-and-T
-------------------------------------------------------
PCHE
PCHE
PCHE
S-and-T
S-and-T
S-and-T
PCHE
PCHE
PCHE
S-and-T
S-and-T
S-and-T
PCHE
PCHE
PCHE
S-and-T
S-and-T
S-and-T
Pool/Loop
Pool/Loop
Pool/Loop
Pool/Loop
Pool/Loop
Pool/Loop
Pool/Loop
Pool/Loop
Pool/Loop
Pool/Loop
Pool/Loop
Pool/Loop
Loop Only
Loop Only
Loop Only
Loop Only
Loop Only
Loop Only
Water (15.5 MPa)
CO2
Water (23 MPa)
Water (15.5 MPa)
CO2
Water (23 MPa)
Water (15.5 MPa)
CO2
Water (23 MPa)
Water (15.5 MPa)
CO2
Water (23 MPa)
Water (15.5 MPa)
CO2
Water (23 MPa)
Water (15.5 MPa)
CO2
Water (23 MPa)
112
Designers are commonly nervous about placing a steam generator or large CO2 plenum
inside a pool of sodium, even if they are double-walled, so the inclusion of a secondary coolant
plenum inside the pool is a safety issue. The failure of the plenum could constitute a single
failure leading to core damage due to the positive reactivity insertion resulting from core
voiding. Some designs have created double pool levels which prevent such a gas bubble from
entering the core [Hejzlar et al., 2000]. The impracticality of eliminating the intermediate loop
in a pool design was also the result of the very tight space for the S-IHX within the reactor
vessel. By using the dimensions of the ABR-1000 P-IHX as the maximum dimensions of the
single IHX for this case, the heat transfer to the PCS was severely limited. Steam generators and
S-CO2 heat exchangers require more heat transfer area than this small heat exchanger could
provide. As an example, a steam generator inside the pool of an SFR with a core outlet
temperature of 530 oC achieves a Rankine cycle efficiency of35.85 %, which is much lower than
any of the efficiencies found for the eighteen other cases examined. In addition, pressure drops
were higher for this case, further hurting plant efficiency. All in all, the option of eliminating the
intermediate loop is considered to be practical only for loop-type SFRs.
Options 13 through 18 have no P-IHX listed because the elimination of the intermediate loop
means that the S-IHX is the only heat exchanger linking the primary sodium to the PCS.
Table 5.5 shows the results of the efficiency comparison for a constant core outlet
temperature of 510 oC. The heat exchanger and PCS types have been eliminated from Table 5.5,
so as to display more information about the fluid temperatures and pump work. Cross
referencing with Table 5.4 will reveal the efficiency consequences of each design choice. Table
5.6 is produced just like Table 5.5, but for a core outlet temperature of 530 oC. The temperature
listed in Tables 5.5 and 5.6 is the turbine inlet temperature. Intermediate pump work is low for
heat exchanger pressure drops, but the design of intermediate piping will have a substantial
effect on the total intermediate pump work. Pump work accounting for the heat exchanger
pressure drops is between 100 kW and 500 kW for each design. On a 250 MWth loop, this
difference cannot account for more than 0.16 % in the efficiency of the entire system. Given the
uncertainty of the intermediate piping design, it can be safely stated that the intermediate loop
pumping power will not be a deciding factor for the SFR. Cycle efficiencies for supercritical
water are based on the results of STEAM PRO 16 for a pressure of 22.0 MPa because this is the
highest pressure at which STEAM PRO 16 can produce results.
113
Table 5.5: Efficiency Comparison of SFR options for a core outlet temperature of 510 oC
Option Number
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
T (turbine inlet) (oC)
504.4
501.9
503.1
505.2
500.4
475.9
486.4
483.1
487.5
486.2
481.2
433.5
507.4
504.2
509.8
508.4
502.6
473.6
Efficiency (%)
40.68
41.05
~41.20
40.70
41.09
~40.40
40.27
40.13
~40.75
40.27
40.14
~38.75
40.75
41.15
~41.38
40.78
41.19
~40.33
Table 5.6: Efficiency Comparison of SFR options for a core outlet temperature of 530 oC
Option Number
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
T (turbine inlet) (oC)
525.6
522.0
526.1
526.0
520.3
505.9
505.0
502.5
507.1
506.6
500.9
477.6
525.6
524.1
529.7
527.6
522.4
507.5
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Efficiency (%)
41.19
41.94
~41.80
41.20
42.00
~41.28
40.70
41.08
~41.30
40.73
41.10
40.46
41.19
42.03
~41.88
41.22
42.09
~41.31
By increasing core outlet temperature 20 oC, the SFR achieves an increase of about 0.9 %
for most of the cases using S-CO2 and supercritical water cycles, and about 0.5 % for traditional
Rankine cycles. Supercritical water cycles are sometimes beneficial and sometimes not,
depending on the performance of the S-IHX. The S-CO2 cycle produces efficiencies that are, in
general, comparable to, but slightly higher than, a traditional Rankine cycle when the core outlet
temperature is 510 oC. After raising the core outlet temperature to 530 oC the S-CO2 cycle has a
clear advantage. The results are shown in graphical form in Figures 5.4 through 5.9.
42.5
Cycle Efficiency (%)
42
41.5
41
40.5
S-CO2
40
Case 2
SC Steam Case 3
Rankine
39.5
Case 1
39
38.5
505
510
515
520
525
530
535
Core Outlet Temperature (oC)
Figure 5.4: Efficiency comparison with PCHEs for both the P-IHX and S-IHX
42.5
Cycle Efficiency (%)
42
41.5
41
40.5
40
39.5
S-CO2
Case 5
SC Steam
Case 6
Rankine
Case 4
39
38.5
505
510
515
520
525
530
535
Core Outlet Temperature (oC)
Figure 5.5: Efficiency comparison with a PCHE for the P-IHX and a shell-and-tube heat
exchanger for the S-IHX
115
42.5
Cycle Efficiency (%)
42
41.5
41
40.5
40
39.5
S-CO2
Case 8
SC Steam
Case 9
Rankine
Case 7
39
38.5
505
510
515
520
525
530
535
Core Outlet Temperature (oC)
Figure 5.6: Efficiency comparison with a shell-and-tube heat exchanger for the P-IHX and a
PCHE for the S-IHX.
42.5
Cycle Efficiency (%)
42
41.5
41
40.5
S-CO2
Case 11
SC Steam Case 12
40
Rankine
39.5
Case 10
39
38.5
500
510
520
530
540
Core Outlet Temperature (oC)
Figure 5.7: Efficiency comparison with shell-and-tube heat exchangers for both the P-IHX and
the S-IHX.
116
42.5
Cycle Efficiency (%)
42
41.5
41
40.5
S-CO2
40
Case 14
SC Steam Case 15
Rankine
39.5
Case 13
39
38.5
505
510
515
520
525
530
535
Core Outlet Temperature (oC)
Figure 5.8: Efficiency comparison with no intermediate loop and a PCHE IHX
42.5
Cycle Efficiency (%)
42
41.5
41
40.5
S-CO2
40
Case 17
SC Steam Case 18
Rankine
39.5
Case 16
39
38.5
505
510
515
520
525
530
535
Core Outlet Temperature (oC)
Figure 5.9: Efficiency comparison with no intermediate loop and a shell-and-tube IHX
The reference case (option 10) is an efficiency improvement over the ABR-1000 design
due to the enlarged heat transfer surface of a straight tube steam generator. Beyond this increase
in efficiency is a possible increase of about 2.0 % which is achievable by using the S-CO2 cycle
and increasing the core outlet temperature to 530 oC. If core outlet temperature is restricted to
510 oC, the efficiency can be improved by about 0.9 % if S-CO2 or supercritical water cycles are
used and the P-IHX is changed to a PCHE design or eliminated altogether.
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5.6 The Choice of Cycle Pressure for Rankine Cycles
In the Rankine cycle, increasing steam pressure yields increasing cycle efficiency.
However, higher pressure steam requires thicker tubes in a steam generator, and the heat transfer
area required is generally higher as steam pressure increases. Quantifying the efficiency gain
versus the cost of producing larger, more robust steam generators could allow designers to select
an economically optimal steam pressure. Though the choice may not yield the highest thermal
efficiency, savings in steam generator cost could make up for it. An example of this effect is the
model of the JSFR steam generator, run in SoSaT. The JSFR steam pressure is 19.2 MPa, but if
the cross-sectional geometry is run in SoSaT for varying pressures, the heat transfer area varies
due to changes in the heat transfer coefficients and CHF. Figure 5.10 shows the total volume of
steel (a good surrogate for cost) in steam generator tubing for the JSFR design at different
pressures. The tube thickness was maintained at the design thickness. As pressure increases
above about 15.5 MPa, the heat transfer area increases.
0.0025
Cost/kWe (m3/kWe)
0.002
0.0015
0.001
0.0005
0
14
15
16
17
18
19
20
Steam Pressure (MPa)
Figure 5.10: Increasing volume of steel in tubes of the JSFR steam generator
with increasing steam pressure
The cost in Figure 5.10 is described in terms of volume of steel used in the steam
generator tubing. The cost is expressed in m3 of tube material per kWe based on the increased
tube area and cycle efficiency when compared to a 14 MPa base case. The cost does not vary
smoothly with pressure because of the different effects of pressure on heat transfer coefficient
and CHF. Designers should keep in mind that this exact trend will not hold for every steam
generator geometry. The results in Figure 5.10 are presented to inform the reader that steam
118
generator designs should consider the effect of steam pressure on the overall heat transfer area
(and therefore cost) of the steam generator tubing. This cost is significant because, although
busbar costs will be reduced as steam pressure and efficiency increase, the additional capital cost
of a larger steam generator may be a outweigh any gains achieved through the improvement of
efficiency.
5.7 Chapter Summary
It is clear that the S-CO2 cycle has an efficiency benefit over the Rankine cycle in the
temperature range considered here. The S-CO2 cycle achieves efficiencies about 1.0 % higher
than the traditional Rankine cycle for comparably sized heat exchangers, either PCHE or Shelland-tube. PCHEs suffer from higher intermediate and primary fluid pump work requirements,
but are very compact. The intermediate loop pump work for all of these designs is small and the
only pumping requirements that will be significant to overall efficiency are those of the primary
pumps and that associated with intermediate piping pressure drops. The PCHE S-IHXs modeled
here did not require the whole available space. In other words, the PCHE designs always
approached a pinch point prior to the size reaching the maximum allowed dimensions. However,
the heat transfer coefficients have not been thoroughly studied for PCHE geometries and
improved PCHE codes should be developed to model them more accurately if more data become
available. The limiting factor for most of the design options was the cycle choice, rather than the
S-IHX.
Consistent with Dostal’s findings [Dostal, 2004], the S-CO2 cycle is slightly better than
the Rankine cycle above a turbine inlet temperature of about 485 oC. For the lowest turbine inlet
temperatures achieved, the cycles’ efficiencies are within 0.15 % of each other. When heat
exchanger options raise the turbine inlet temperature higher, however, the S-CO2 cycle efficiency
exceeds that of the Rankine cycle. The highest efficiencies are achieved with a S-CO2 cycle
coupled to a shell-and-tube heat exchanger linked to primary sodium in a loop-type design, i.e.
there is no intermediate loop present. The CO2 shell-and-tube S-IHX must incorporate enhanced
tubes, however, as smooth tubes do not provide the heat transfer coefficient required.
Rankine cycles and S-CO2 recompression cycles perform comparably at a core outlet
temperature below 510 oC. At 510 oC, the S-CO2 cycle and the supercritical water cycle achieve
comparable efficiencies which are, on the whole, a slight improvement over the traditional
Rankine cycle. Clear differences in cycle efficiency become evident if the core outlet
temperature reaches 530 oC when the S-CO2 cycle has a clear efficiency advantage.
Eliminating the intermediate loop is only practical for loop-type designs. Eliminating the
intermediate loop, though effective in increasing efficiency, has safety implications, and there
are other options for increasing efficiency almost as much without the safety issues. If the loop
119
were eliminated in the pool design, the SFR would be in violation of the Technology Neutral
Framework (TNF) because rupture of the IHX plenum could constitute a single failure that could
lead to core damage when a gas bubble voids the core and causes positive reactivity insertion
[NRC, 2007]. This can be prevented by the design of the vessel to include an appropriate redan
separating the hot and cold sodium pools. In the loop design, primary sodium must be pumped
out of the vessel and into the IHX via primary sodium piping. Switching to the loop design
introduces the large LOCA as a credible accident sequence, and therefore must be evaluated for
its safety consequences in the PRA.
Intermediate loop pumping power ranged between 100 kW and 500 kW for each 250
MWth loop. The ABR-1000 reference design requires an intermediate pumping power of about
340 kW, so these numbers are in fair agreement. The overall pumping power of the intermediate
loop is therefore about 30 % that of the primary pumps. From an overall plant efficiency
standpoint, differences in intermediate pumping power will not result in efficiency differences of
more than 0.16 %.
High turbine inlet temperatures are possible with supercritical water cycles as well.
Modeling the performance of the cycle was not achieved here, but water cycles operating at 23
MPa can achieve efficiencies about 0.4 % higher than those of 15.5 MPa Rankine cycles for
comparable temperatures. In Tables 5.6 and 5.7 STEAM PRO 16 was used to determine an
estimated efficiency for the supercritical water cycle by running at a steam pressure of 22.0 MPa.
Increasing the steam pressure can produce benefits. Achieving a steam pressure of 17.5
MPa will increase the efficiency of the Rankine cycle about 0.21 % over that of the 15.5 MPa
cycle. Heat transfer to boiling water at 17.5 MPa is not as effective as it is at 15.5 MPa, so a
larger steam generator is required for higher pressures. The extra steam generator cost must be
weighed against the gain in cycle efficiency.
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6 Summary, Conclusions, and Recommendations for Future Work
6.1 Summary
Sodium-Cooled Fast Reactors have the potential to achieve efficiencies of up to 42% and
they have unique benefits for the fuel cycle and operations. They are an important technology
for the future, but the development of the S-CO2 cycle and supercritical water cycles will be
critical for achieving very high efficiencies with these designs.
Development of the S-CO2 recompression cycle holds promise for the SFR and other
high temperature applications in the future. High efficiencies and relatively small components
make the S-CO2 cycle an economically attractive alternative to traditional Rankine cycles. The
development of turbomachinery and the detailed modeling of cycle control are important for the
cycle to actually be introduced into industrial use.
The compressors, heat exchangers, and SFR design options investigated here show
promise for S-CO2 cycles and for SFRs, but will require further work. The developments made
here also introduce new questions about the economic impacts of design choices. Tools
developed in this research can be used in future investigations to continue this work.
This work has resulted in several codes that are available for use in the future, for
compressor design (RGRC and RGRCMS), heat exchanger sizing (SoSaT and improved PCHE
codes), and steady-state S-CO2 cycle analysis (CYCLES III).
6.2 Conclusions
6.2.1 Conclusions on the S-CO2 Cycle and its Compressors
The S-CO2 recompression cycle is expected to perform well with realistic levels of air
impurities in the working fluid and can tolerate a helium mole fraction of 0.005 with an
efficiency loss of about 1.0%, based on runs of the updated CYCLES III. With the new ability to
run with fluid mixtures, CYCLES III has shown that the recompression cycle need not run on
extremely pure CO2 and that helium may be used as a leak detection gas, if the efficiency penalty
is tolerable. Producing leak detection systems that require a lower concentration of helium in the
CO2 would alleviate the efficiency penalty. Also, the ethane simple Brayton cycle was shown to
have dubious hopes for the future, based on expected levels of ethane dissociation at high
temperatures. If dissociation is shown to be low, then the cycle could be successful. When
coupled to an SFR, the S-CO2 cycle performs comparably to the Rankine cycle for core outlet
121
temperatures of about 500 oC and can produce higher efficiencies than the Rankine cycle for
higher core outlet temperatures. Even if efficiencies are not shown to be markedly different from
Rankine cycle efficiencies, the compactness of the cycle may prove to be a deciding advantage
for the S-CO2 recompression cycle.
S-CO2 compressors appear to be operable over appreciable ranges of speeds and mass
flow rates. They achieve good efficiencies in a 500 MWth balance of plant, with single stage
and two stage designs for the main and recompressing compressors, respectively. The Real Gas
Radial Compressor (RGRC) code has been developed as a useful tool for mean-line, real gas
centrifugal compressor design in the future. Gong’s CO2 property subroutines have been
incorporated to make the code run quickly. NIST fluid properties are also available, in order to
get more accurate results when computation time is not as much of an issue.
Producing centrifugal compressor designs for the S-CO2 cycle has shown that a singlestage design can be used with the main compressor for systems above about 400 MWth. If the
system power is reduced below this level, either the compressor design speed or the number of
stages must be increased. For a 500 MWth system with a single-stage main compressor, the
recompressing compressor was designed as a two-stage design. These designs are compact,
having diffuser outlet diameters of 2.52 m or less. Efficiencies are at or slightly above 90 % at
the design point for the recompression cycle designs.
Benchmarking the RGRC code was attempted with estimated parameters from a test
compressor operated by Sandia National Laboratory, showing a rough agreement with expected
compressor performance. Further benchmarking can be performed when more data become
available.
6.2.2 Conclusions on Sodium Fast Reactors
Increasing the plant efficiency is the most effective method of reducing busbar costs if
the primary or even secondary mission of the SFR is to produce electricity [Nitta, 2009]. This
study has shown that the S-CO2 cycle, and the use of Printed Circuit Heat Exchangers (PCHEs)
can significantly improve the efficiency of the SFR, translating to a reduced cost of electricity.
Eliminating the intermediate loop is only practical, from an efficiency perspective, in a
loop design for the SFR because the large S-IHX required for effective heat transfer would be
too tall to fit within a pool design. The height could be reduced if the designer is willing to use
much wider reactor vessels. Forging and transportation of larger vessels become issues for the
construction of an SFR. The loop design would probably be required from a safety standpoint
anyway, since the potential for a vapor or gas release from a large, high pressure water or CO2
plenum within the primary sodium pool, which could pass through the core, would exist in a pool
design.
122
The Sodium Shell and Tube (SoSaT) code was developed to model heat exchangers with
sodium on the shell side and either sodium, CO2, or water on the tube side. It uses straight tubes
and can produce results for water at pressures up to and beyond critical. Enhanced heat transfer
surfaces can be used on the tube side with CO2 and the shell side can incorporate baffles also.
Results show that the code sizes heat exchangers to within a few percent of benchmark designs.
This tool will be useful for future studies of SFRs and is easily adjusted to different design
parameters.
MIT’s Printed Circuit Heat Exchanger (PCHE) codes have been expanded to perform the
same functions as SoSaT. These codes show that PCHEs are very compact and still do not suffer
from severe pressure drops. The codes can still be improved, however, for operability and to
incorporate better heat transfer correlations.
For core outlet temperatures of 510 oC and below, the Rankine cycle and the S-CO2 cycle
produce comparable efficiencies for most configurations. At higher core outlet temperatures, the
S-CO2 cycle achieves the higher efficiency. Efficiency can be increased by changing the P-IHX
to a PCHE design, switching the PCS to S-CO2, removing the intermediate loop, or increasing
the core outlet temperature. Achievable efficiencies are around 42 %.
6.3 Recommended Future Work
6.3.1 S-CO2 Compressors
S-CO2 compressors and the cycle’s transient response require more study before they can
be ready for full scale development. In general, compressor performance predictions are based
on empirical relationships, so experimental work with S-CO2 compressors will be required in
order to learn more about their performance. The surge and choke limits are identified by
conditions that have been used in air compressors, so the operating range of S-CO2 compressors
may differ. Adjusting coefficients in the loss calculations and the criteria for surge and choke
can make RGRC match experimental data more closely, once they are available. Continued
operation of test compressors will help to develop the state of the art. Because these
compressors are relatively compact, when compared to other fluids, research may be more
affordable than basic compressor research has been in the past. Also, existing compressor
designs can be tested with CO2 in order to develop a database for correlating the performance. In
addition, RGRC could be enhanced to include some other calculations. For example, there is no
calculation of the stresses on the impeller blades. Narrower blades in dense fluids will need to be
engineered to withstand the stresses placed on them during operation. Also, RGRC does not
include a model of the Inlet Guide Vanes (IGVs) which are used to control compressor
123
performance, nor does it include the possibility of prewhirl. For multi-stage compressors, the
return channel is an oversimplified model and could be improved. As more research is
conducted, some work with computational fluid dynamics (CFD) codes may enhance the
development of S-CO2 compressors as well.
6.3.2 The S-CO2 Cycle as a Whole
The S-CO2 cycle requires validation, especially in the development of control techniques
for transient operation. Trinh [2009] has made significant progress in furthering the transient
model, but additional work will have to enter the experimental realm. A full recompression
cycle test loop would probably be required in order to test the performance of the cycle under
system transients.
Compact turbomachinery is a benefit of the S-CO2 recompression cycle, but more
detailed studies of the expected costs must be performed in order to quantify the economic
advantage of this cycle beyond that achieved from efficiency improvements alone. Cost
estimates for turbomachinery and heat exchangers will allow for comparisons with Rankine and
supercritical water cycles. Then the entire economic benefit of the S-CO2 cycle can be
understood.
The corrosion of stainless steels in an S-CO2 environment must be investigated further.
Corrosion will affect the lifetime and reliability of heat exchangers and the failure modes of
every component. Of interest are high-alloy steels and austenitic stainless steels. Experimental
work at MIT is building on the low-pressure experience of the British AGR program [Medina,
2008].
6.3.3 SFR Heat Exchangers
The relative costs of PCHEs and shell-and-tube heat exchangers will need to be
thoroughly studied, in order to understand the long term benefit of choosing one design over the
other. The performance of PCHEs relative to their cost is important as well, and information on
the failure modes and repair costs for all components will need to be gathered, or at least
estimated in the absence of experience.
Printed Circuit Heat Exchangers (PCHEs) are very rugged and have high heat transfer
areas, producing very highly effective heat exchangers in a relatively small volume. More
experience with sodium flow in narrow channels will be needed to assess the operational
readiness of PCHEs for use in SFRs. Also, the forced convective boiling in a horizontal, zig-zag,
semi-circular channel is not well documented. Studies of the nucleate boiling heat transfer and
the CHF of this geometry will be important for detailed design. For now, PCHEs appear to be an
124
excellent way to improve efficiency without seriously increasing capital costs. The PCHE codes
at MIT could be consolidated and simplified into a single PCHE code with integer flags in the
input file which would allow the user to specify different types of PCHE. This would simplify
work for the user and reduce the chance of error in running the codes. The experimental
performance of PCHEs will have to be used to benchmark and adjust the existing PCHE codes.
SFR steam generators have been constructed as both straight and helical coil shell-andtube heat exchangers. The straight-tube design was studied here because of its relative ease in
modeling, but the helical coil steam generator should not be discounted. If an effective helical
coil model is developed to compliment SoSaT, comparisons could be made for the purpose of
choosing a design. The helical coil design increases the heat transfer coefficient, thereby
decreasing the total heat transfer area. The size of the shell is much larger in a helical coil steam
generator though, so if restrictions on size within the containment are important, the straight tube
steam generator may be a better option. Both have been used in operating SFRs.
6.3.4 Safety and Availability Consequences of SFR Design Options
Current efforts at MIT on the part of Brian C. Johnson are aimed at analyzing safety
consequences of proposed design choices using the ALMR and PRISM Probabilistic Risk
Assessment (PRA) models. The consequences of BOP configuration changes need to be
analyzed in the PRA of an SFR to determine if they are acceptable under the NRC’s Technology
Neutral Framework (TNF). The TNF is expected to be the regulatory framework under which
new SFRs would be licensed.
The major considerations in the PRA, stemming from this investigation, would be the
consequences of removing the intermediate loop and therefore switching to a loop design.
Primary coolant would then travel outside the vessel in piping and the LOCA would be a
significant contributor to risk, if not the most significant. The elimination of the intermediate
loop in a pool design is not feasible due to the size restrictions on heat exchangers within the
vessel and the fact that core damage could be caused by a gas bubble resulting from the failure of
an IHX plenum in the vessel.
The overall performance of PCHEs is of interest because steam generator tube leaks have
been problems for several existing SFRs. The small channel diameter and the robust design of
PCHEs should reduce the frequency and severity of steam generator leaks. Also, every shelland-tube design used in this study was modeled as a double-walled tube with a helium gap. The
use of double walled tubes or hybrid PCHEs will improve the availability of the entire system.
125
6.4 Where to Obtain Codes used in this thesis
The codes described in this work are available at MIT for future use. They can be
obtained through the computer facilities manager in the Department of Nuclear Science and
Engineering (Rachel Morton at time of writing). These include:
CYCLES III
RGRC
RGRCMS
SoSaT
PCHE
The Steady-State Recompression and Simple Cycle Code
The Real Gas Radial Compressor Code
The Multi-Stage Real Gas Radial Compressor Code
The Sodium Shell-and-Tube Heat Exchanger Code
The Printed Circuit Heat Exchanger Codes
Information on the operation of these codes is included in the following appendices.
These appendices are geared toward operating the codes rather than the specifics of the codes’
function.
126
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131
Appendix A. CYCLES III Code Manual
A.1 Introduction
This Appendix is intended to expand upon David Legault’s SB thesis, which explains the
operation of CYCLES II. Legault’s thesis is an excellent companion to CYCLES II, however,
the changes in CYCLES III demand a short manual to assist in its use.
A.2 Inputs and Outputs
For the new user, the input is mostly self-explanatory. Comments identify what each
input parameter is. The comments from the beginning of the input file, called HXdata.txt, are
shown below:
Main cycle input data for CYCLES III
itables
Table trigger, if 0 old tables are used, if 1 new tables created
iprop
Property trigger, if 0 tables are used, if 1 polynomials are used
icase
Case trigger: 0 runs a single operating point, 1 optimizes
itype
Cycle you are running: 0 for simple, 1 for recompression
The following are data concerned with the working fluid
ifluid
mix
ifltwo
fracgas
This is the first fluid, 0-CO2, 1-Ethane, 2-Helium
IDs if there is a 2nd fluid, 0-pure, 1-2nd fluid exists
IDs 2nd fluid, 0-He, 1-Air,2-Hydrogen,3-Nitrogen,4-Methane
This is the mole fraction of the 2nd fluid(if applicable)
It is recommended that tables are always used, as the same fluid will likely be used
several times, and the calculations will be much faster than if the polynomials are used.
CYCLES III should include tables with it. If not, or if a new fluid is needed, new tables are
created by changing itables to 1 and running the code. The table generation subroutine will take
considerably longer when mixtures are involved (especially air, which is a mixture of Nitrogen,
Oxygen, and Argon). The tables consist of values for the thermal conductivity, density,
enthalpy, viscosity, and specific heat over a temperature and pressure range that is defined by the
user. The range and spacing of data point in the table are determined in createtables.txt, which
looks like:
132
600
800
6000.0
21000.0
20.0
700.0
!nTpoints
!nPpoints
!tabPmin
!tabPmax
!tabTmin
!tabTmax
number of temperature points
number of pressure points
table minimum pressure (kPa)
table maximum pressure (kPa)
table minimum temperature (C)
table maximum temperature (C)
Experience has shown that one of the most common errors in operating CYCLES III is forgetting
to change the table limits when operating the cycle on a new fluid which requires a new pressure
range.
If the user should desire to add different fluids than those that are listed in HXdata.txt,
the fluid files are called in the subroutine readHX. The fluid files must be located in the path:
C:\Program Files\REFPROP\fluids\
or the subroutine READHX must be changed to reflect a different position.
In HXdata.txt, the pipe data appears below the heat exchanger data. It is read in as follows:
IP:
Nsec:
Npipe:
Tells the code which path is being described. Don't change these.
Tells the code how many sections are in that path so it can read the data correctly
How many pipes are described by the data for a certain section. For example,
Npipe might be 48 for the first section of IP=1. This means that the precooler will
be divided into eight modules, each with six outlet plena for a total of 48 outlet
plena. The number six comes from the construction of the recuperators and the
number of modules is up to you (within reason of course).
Dpipe:
pipe inner diameter
Apipe:
pipe flow area. Obviously this must be compatible with the pipe inner diameter
Elpipe:
length of the pipe
xsi_pipe:
The K factor describing the pipe. Inlet and outlet factors are included here, as
well as bends.
rough_pipe: The roughness of the pipe. You will likely never change these values.
The new user will notice that pipe areas do not always correspond to the pipe diameter in
sample input files. This is due to non-circular shapes for some of the plena. The user should
keep in mind that the pipe data is used to try to create a realistic model of system pressure drops,
so the exact shapes of the pipes are not extremely important. Realistic pressure drops are. Each
system design will require a virtually complete re-working of the pipe inputs. The user should
also keep in mind the maximum dimensions of manufactured pipes, with consideration of the
required thickness for high internal pressure (20.0 MPa).
133
The output has been consolidated into a single file output.txt. This file has information
on the heat exchanger dimensions, fluid statepoints at every location in the cycle, the work
requirements of compressors, and the efficiency of the cycle.
A.3 Troubleshooting
CYCLES III usually runs without any problem, but if problems are encountered they will
usually be due to excessive pressure drops or failure of the precooler to converge. Pressure
drops can be reduced simply by expanding the pipes in the pipe model to increase flow area.
The precooler will fail to converge if its volume is too small for the power rating of the system.
The message on the screen output will read:
function tp2-not converged
Increasing the precooler volume, or the initial guess of the precooler volume for optimizations,
will resolve the problem.
In CYCLES III, the efficiency of the PCS is calculated using two methods. The first is
based upon the heat added and heat rejected in the precooler.
 th 
Qadded  Qrejected
Qadded
Eqn. A-1
The second calculation is based on the work of the turbomachinery.
 th 
WT  WC
Qadded
Eqn. A-2
Both calculations should produce the same result, but slight differences will result
depending on how finely spaced the fluid property tables are. Also, pressure drops in piping and
kinetic energy changes affect the static enthalpy in pipe flows, which translates to changes in
turbomachinery work, contributing to differences in the two efficiency calculations. In addition,
the coolant will lose heat through the pipe walls to the atmosphere. This is generally negligible.
The two calculation methods for thermal efficiency are displayed as eta1 and eta2 in the file
output.txt. If they are substantially different, the tables should be regenerated with finer spacing.
The other way to increase the accuracy is to reduce the number labeled as tolerance in the input
file. Otherwise, their difference can be viewed as an uncertainty in the calculation.
134
For S-CO2 cycles the optimum pressure ratio is about 2.60 exactly, with a maximum cycle
pressure of 20.0 MPa. For all configurations this value produces the highest efficiencies. In the
ethane cycle with a maximum pressure of 20.0 MPa, the corresponding pressure ratio should be
4.00.
A.4 Conclusion
CYCLES III can accurately model the heat exchangers and piping of the S-CO2
recompression cycle or the simple recuperative Brayton cycle with a variety of fluids and fluid
impurities. The user should be careful to model the piping database in such a way as to produce
realistic pressure drops. Optimizations can be performed to determine the best division of heat
exchanger volume between pre-cooler, HTR, and LTR. The code incorporates tabulated fluid
properties for faster calculations and consolidated input and output files for easy use.
135
Appendix B. RGRC and RGRCMS Code Manual
B.1 Introduction
This Appendix is intended to be a user’s manual for the RGRC and RGRCMS codes. For
further insights into compressor design the manual for CCD and CCOD, by Jerry Wood, can be
useful. However, the operation of CCD/CCOD is very different from RGRC and RGRCMS and
they will not produce good results for S-CO2.
B.2 Inputs and Outputs
The input file, input.txt, for RGRC is shown in Figure B.1.
Input File for Centrifugal Compressor Design Code RGRC
1
!Fluid indicator, 1-CO2
1858.7
!Mass Flow rate of the Compressor Stage (kg/s)
2.608
!Desired Static-to-Static Pressure Ratio
3600.00
!Operating RPM
30.0
!Velocity of fluid at inlet (m/s)
0.100
!Hub diameter at inlet (m)
7667300.0
!Static Pressure at inlet (Pa)
1285.6
!Entropy in inlet pipe away from compressor (J/kgK)
19
!Number of blades at inlet
0
!Splitter Blades? 1-Yes, 0-No
0.0030
!Thickness of blades at inlet (m)
0.0025
!Thickness of blades at outlet (m)
2.00
!Axial height of blades at outlet in % of D2
2.00
!Blade clearance as a % of blade height
40.0
!Impeller Backsweep (degrees)
0.00
!Abs. flow angle at impeller inlet(deg) (normally zero)
0.00
!Slope of hub at inlet (degrees)
0.00
!Slope of casing at inlet (degrees)
2.00
!Area ratio of vaned diffuser
0
!0-vaned diffuser, 1-pipe diffuser
1.10d0
!The ratio of R3 to R2
35
!number of diffuser vanes
15.0
!Curvature of diffuser vanes
0.40
!Initial Guess of Ratio D1T to D2
136
Off-design speeds (choose 6)(ratio to design speed)
1.2
1.1
1.0
0.9
0.8
0.7
Figure B.1: The input file for RGRC
Currently, RGRC does not include other fluid options, but they could be added. This will
require the development of a critical velocity database for any new fluid and polynomials for its
fluid properties in the range of interest would greatly help to reduce computing time. For the
time being, only CO2 can be used in RGRC.
Mass flow rate, pressure ratio, inlet fluid conditions, and velocity will be determined by
the application of the compressor. The hub diameter will be determined by the dimensions of the
shaft required for the torque. The other inputs are flexible and can be chosen by the user. Some
guidance for each input is given in Table B.1.
Table B.1: RGRC inputs and their suggested ranges
Number of blades at input
Splitter Blades
Thickness of blades (inlet)
Thickness of blades (outlet)
Axial height of blades (outlet)
Blade clearance
Impeller backsweep
17 to 35 recommended
will help to avoid stall, but there is an efficiency penalty
should be a few percent of the hub diameter
See discussion to follow*
About 2 % of the outlet diameter, but could be 1 % to 4 %
2 % to 4 % of blade height
Anywhere from 20 o to 60 o
*The thickness of blades at the impeller outlet will depend on the impeller diameter, the blade
height, and the rotational speed. The designer should ensure that there are no undue stresses on
the blades and thicker blades will be needed when the stresses are high. This is one area where
the calculations in RGRC are lacking. There is no determination of the stresses within the
blades, and therefore it can be difficult to judge whether or not the blade thickness is appropriate.
The output from RGRC will be in the form of a text file, output.txt. This file contains all
the information needed to produce a compressor map. Figure B.2 shows the top portion of an
RGRC output.
137
10.00
53.12
90.83
1.82
99.92
199.83
35
0.05
77.33
40.00
2.00
28.70
19980.34
1337.66
Hub Diameter (cm)
Inlet Tip Diameter (cm)
Impeller Outlet Diameter (cm)
Axial Blade Height (cm)
Diffuser Inlet Diameter (cm)
Diffuser Outlet Diameter (cm)
Number of Diffuser Vanes
Axial blade clearance (cm)
Blade angle at inlet
Blade angle at impeller outlet
Diffuser Area Ratio
Velocity at outlet (m/s)
Static Pressure at outlet (kPa)
Entropy at outlet (kJ/kgK)
Speed
M_dot
Press. Rat.
1.200
158.090
Stall at
1.200
316.180
Stall at
1.200
474.270
Stall at
1.200
632.360
Stall at
1.200
790.450
Stall at
1.200
948.540
Stall at
1.200 1106.630
Stall at
1.200 1264.720
Stall at
1.200 1422.810
Stall at
1.200 1580.900
3.469
1.200 1738.990
3.437
1.200 1897.080
3.393
1.200 2055.170
3.339
1.200 2213.260
3.273
etaR
impeller
impeller
impeller
impeller
impeller
impeller
impeller
impeller
impeller
0.997
0.997
0.996
0.996
0.996
etaTT
outlet.
outlet.
outlet.
outlet.
outlet.
outlet.
outlet.
outlet.
outlet.
0.899
0.912
0.922
0.929
0.933
etaTS
0.887
0.897
0.903
0.907
0.906
Cpstar
0.721
0.755
0.779
0.792
0.793
Figure B.2: The beginning of an RGRC output file. The performance for
each speed is listed in order.
The top portion of the output has important geometrical parameters for the user to get an
idea of the size of the compressor. Then the off-design performance is listed. The columns,
from left to right are: speed, mass flow rate (kg/s), static-to-static pressure ratio, impeller
efficiency, total-to-total efficiency, total-to-static efficiency, and diffuser pressure recovery
coefficient. Notice that at low mass flow rates the impeller is stalled so RGRC does not output
any performance calculations.
138
B.3 Recommendations and Trouble-Shooting
The faster fluid property subroutines should be used for stage design unless time is not a
concern. Running RGRC on the NIST subroutines can take a few days. The best approach is to
use the Gong subroutines for RGRC and then develop the performance map in RGRCMS with
the NIST subroutines. RGRC will return a warning when the code operates out of the range of
agreement between Gong’s subroutines and NIST values. The user should realize that this
usually only occurs at very low mass flow rates, when the compressor is stalled anyway. A few
data points outside the range of applicability will not be a problem if they occur within the range
of choke and surge.
There are common problems that arise in the use of RGRC and an inexperienced user
may not have the compressor design intuition to develop a solution. These common problems,
and their typical solutions, are listed here. Oftentimes, a designer will find that the solution to
one problem is the cause of another. Below are some issues commonly encountered and some
suggestions for new users to find solutions.
Problem:
Velocity at the diffuser outlet (V4) is too small.
Solution:
This problem will arise if the diffuser pressure recovery is too high. If the
diffuser area ratio is very high, then the velocity at the outlet will necessarily be low. Lowering
the diffuser area ratio will alleviate this problem. The velocity at the outlet must be maintained
above some minimum, especially for multi-stage machines. If the outlet velocity from the first
stage is very small, then the second stage is likely to stall.
Problem:
The stall margin for the operating point is too small.
Solution:
Stall will usually occur at the impeller outlet due to excessively large flow angles.
This will result in incidence stall at the diffuser inlet as well. To eliminate this problem, the axial
height of the blades at the impeller outlet will have to be decreased. One could also increase the
number of blades at the outlet, increase the blade thickness, or divide the compressor into stages.
If the diffuser stalls before the impeller stalls, the axial height of the blades can be increased, but
the diffuser angle may have to be changed. Decreasing the axial height of the blades will have
penalties in the losses however, and will reduce the margin to choking. Another option for
avoiding stall is to design the system with an increased flow velocity at the impeller inlet.
139
Problem:
Choke occurs too soon.
Solution:
Choke is likely to occur at the vaned diffuser inlet. At this location, the flow must
not encounter an excessive area reduction, or the velocity will approach the critical velocity. To
solve this problem, the vaneless space could be made longer or wider. Widening of the vaneless
space could be accomplished by increasing the clearance or the axial blade height. Increasing
the input variable R3/R2 will increase the pressure recovery of the vaneless space, decreasing the
flow velocity and avoiding choke. The problem can be avoided by increasing the diffuser area
ratio also. By increasing the area ratio, pressure recovery is improved. Thus, the code will not
design the impeller quite as large and flow velocities will be reduced as a result.
Problem:
The operating range at high speeds is small or non-existent.
Solution:
This problem can occur due to impeller or diffuser dimensions. Decreasing the
diffuser area ratio or increasing the backsweep angle will expand the operating range. Typically,
backsweep angle will only achieve so much. Beyond a certain point, compressor performance is
not changed significantly. Experience has shown this angle to be about 50 degrees for most
designs.
Problem:
The efficiency is unappealingly low.
Solution:
The compressor could be divided into multiple stages or the backsweep of the
impeller could be decreased. Decreasing the backsweep angle will typically result in rapidly
shrinking operating range. It is often best to attempt a design that will give an acceptable
operating range and then tweak parameters to achieve a decent efficiency without compromising
the operating range much. Efficiency will depend strongly on the size of the impeller and flow
velocity.
Problem:
A multi-stage design has an extremely narrow operating range
Solution:
Usually the problem is caused by surge in the second or subsequent stage. This is
due to off-design losses in the first stage magnifying the effects of reduced flow velocity in the
down-stream stages. Second and subsequent stages should be designed with special
consideration for the margin to stall. Usually, improving on a stage design will have a direct
effect on the entire compressor’s operation. In other words, if the second stage is improved such
that it has a higher stall margin alone, the multi-stage compressor will be less likely to exhibit
stall in the second stage. Inevitably, the operating range of the multi-stage compressor will be
narrower than that of any one stage. The designer must make a trade-off between operating
range and total compressor efficiency.
140
B.4 Fluid Properties
The critical velocity database developed for CO2 by Gong could be produced for any other fluid,
allowing the operation of RGRC for any gas. The database is produced by successive iteration
of velocity and enthalpy until convergence is achieved. By choosing a total enthalpy and then
increasing velocity, the database can be produced. The steps are:
1.
2.
3.
4.
5.
Choose an entropy value
Begin at a total enthalpy value
Increase velocity and calculate static enthalpy
Find the speed of sound and compare to the velocity
If the velocity is equal to the static enthalpy’s speed of sound it is the critical velocity for
the present value of total enthalpy. If not, increase velocity again.
Gong’s polynomials operate by using coefficients which have been fit to the NIST data in
the range of interest. These coefficients are contained in the file co2p4s.dat. Fluid properties, in
this example the pressure, are then calculated as polynomial functions of the enthalpy by
5
𝑃=
𝑝𝑜𝑙𝑦𝑐(𝑖) 𝐻
5−𝑖
Eqn. B-1
𝑖=1
where polyc is a vector of five coefficients which are interpolated between 67 values of entropy.
Therefore, each fluid property is determined from enthalpy and entropy. These polynomial fluid
property subroutines are contained in the file property.f90. To use the NIST subroutines,
property.f90 can be replaced with nist_property.f90. The subroutines are all named the same, so
the rest of the code will run exactly as it had with the Gong polynomials.
RGRC will return a warning when Gong’s polynomials are approaching the unreliable
region. The user should not automatically discard any compressor map for which some points
return this error. It is likely that the error will be returned in the choke or surge region and
therefore the entire operating range will have been produced with valid property subroutines.
B.5 Conclusion
Compromise will govern the design of any compressor stage, with a careful regard for the
aspects of the compressor performance which are most important for its application.
Determining whether or not a centrifugal compressor is practical for a particular application is
the first step. Based on the non-dimensional specific speed, applications for centrifugal
141
compressors can be identified. A great deal of guess-and-test operation of RGRC will lead to a
better intuition and a faster design process.
142
Appendix C. PCHE Codes Manual
C1. Introduction
The PCHE models developed by Hejzlar [Hejzlar et al., 2007] are based on a simple
convective heat transfer model which uses fluid properties from NIST to calculate the heat
transfer inside the semi-circular channel of a PCHE. The codes included the option of a formed
plate containing detection gas channels for heat exchangers like the S-IHX where exothermic
reactions occur between the two fluids. There is considerable room for improvement in the
PCHE codes, such as combining them into one user-friendly code with a common input and
output file. Also, heat transfer coefficients for boiling water in zig-zag semi-circular channels
are not developed well enough to use in heat exchanger design with confidence. Experimental
work with this geometry would advance the use of PCHEs as steam generators in SFRs.
C.2 Inputs and Outputs
The PCHE codes are easy to run and the outputs are easy to read. The main caution for a
new user is to ensure that none of the inputs are unphysical. For instance, the plate thickness
cannot be narrower than ½ the channel diameter. Also, water pressures must remain below 21.0
MPa for the heat transfer correlations to be appropriate. The user need not remember to use
straight channels for sodium; the code is hardwired to do so.
Each PCHE code is wholly contained in its own file. The names of the files, and their
individual purposes for calculations, are listed in Table C.1. Each code can be run by replacing
the file in the workspace with the desired code and compiling again.
Table C.1: Files needed for each PCHE code
Program File
Input File
Output File
Sodium-Sodium Heat Exchangers
ihxNa.f90
ihxNa.in
ihxNa.out
Sodium-CO2 Heat Exchangers
ihxNa_Hyb.f90
IhxNa_Hyb.in
ihxNa_Hyb.out
Sodium-Water Heat Exchangers
ihxNa_Hyb2.f90
ihxNa_Hyb2.in
IhxNa_Hyb2.out
The input and output files require the user to specify the geometry of the plates at the
entrance to the channels. Then the code will determine the necessary length of channels to
achieve the desired heat transfer. As an example, the input file for a hybrid Na-CO2 heat
exchanger is given in Figure C.1 and the output file is in Figure C.2.
143
zig
! Na/CO2 IHX
-Na inlet for SFR design - 250MWt
1h2c
250000.0
! power (kW)
200.0
! pressure - hot fluid - hot end (kPa)
19920.0
! pressure - cold fluid - cold end (kPa)
530.0
! temperature - hot fluid - hot end (oC)
361.3
! temperature - cold fluid - cold end (oC)
1256.0
! mass flow rate on hot side (kg/s)
1250.0
! mass flow rate on cold side (kg/s)
0.0025 0.003 !hot channel passage width and height (m)
0.0005
! separating plate thickness (m)
1 0.0005 0.0003 0.0024 ! helium plate indicator (0 if no helium plate), He plate thickness and channel D
(m)
0.0035
!cold channel diameter (m)
0.0020
!hot formed plate thickness (m)
0.0025
!cold plate thickness (m)
5.0
!cold pitch divider (Pitch=channel diameter + (channel diameter)/(pitch divider)
2.90
!HX height (m) - this is total for 4 submodules (actual module H=height/4=5.5)
2.80
!HX width (m) - this is maximum size of photo etching (2x0.6 submodules separated by chamber)
4.0
! # of modules - how many modules stacked, hence total H=H/4* # of mudules
21.4
!thermal conductivity of the plate (W/mK) (ss 304 at 500C)
40.0
! # of HX longitudal cells
15.0026
!wmh - molar mass
44.0026
!wmc - molar mass
Na !hot fluid Na
co2.fld
!cold fluid id (0 co2 else helium)
1.0d-5
!pressure iteration precision
40
!nodes
Figure C.1: The input file ihxNa_hyb.in with representative values
144
OPERATING CONDITIONS
250000.00 power (kW)
200.00 pressure - hot fluid - hot end (kPa)
19920.00 pressure - cold fluid - cold end (kPa)
2227.5311 hot side pressure drop (Pa)
-119247.0789 cold side pressure drop (Pa)
1256.0000 mass flow rate on hot side (kg/s)
1250.0000 mass flow rate on cold side (kg/s)
1.00 cold mass flow rate over hot mass flow rate
373.53 temperature - hot fluid cold end (oC)
361.22 temperature - cold fluid cold end (oC)
530.00 temperature - hot fluid hot end (oC)
524.10 temperature - cold fluid hot end (oC)
773.19 enthalpy - hot fluid - hot end (kJ/kg)
574.04 enthalpy - hot fluid - cold end (kJ/kg)
1003.36 enthalpy - cold fluid - hot end (kJ/kg)
803.26 enthalpy - cold fluid - cold end (kJ/kg)
1.2195 cold side inlet velocity (m/s)
0.3246 hot side outlet velocity (m/s)
3085.44 Reynolds number - hot fluid average (-)
12471.60 Reynolds number - cold fluid average (-)
120738.14 Heat transfer coef. - hot fluid average (W/m2/K)
2200.20 Heat transfer coef. - cold fluid average (W/m2/K)
3637.79 Total heat transfer coeff. - average (W/m2/K)
HX GEOMETRY
90.22050
Total HX volume - all modules (m3)
22.55512
1 Module HX volume (m3)
2.90000
HX height (m)
2.80000
HX width (m)
2.77772
HX length (m)
3.62606
Cold channel length (m)
2.50
3.00
Channel width and height on hot side (mm)
3.50
Channel diameter on cold side (mm)
4.50
Channel pitch on hot side (mm)
4.20
Channel pitch on cold side (mm)
149420.
Number of channels on hot side per 1 module
320048.
Number of channels on cold sideper 1 module
2.00
Plate thickness on hot side (mm)
2.50
Plate thickness on cold side (mm)
0.50
Separator plate thickness (mm)
0.50
Helium plate thickness (mm)
2.40
Helium channel pitch (mm)
0.30
Helium channel diamater (mm)
241.00
Number of hot plates
482.00
Number of cold plates
4.00
Number of HX modules
OTHERS
21.40
HX material conductivity (W/mK)
3.31
Conduction length (mm)
40
Number of nodes
0.10D-04
Pressure iteration precision
Figure C.2: The output file ihxNa_hyb.out
145
The number of modules can be selected by the user and will just be used by the code to create as
many identical units which evenly split the flow. The number of nodes can be increased if the
code returns warnings about the accuracy of convergence. Pressure drops in the output file will
be calculated from the hot fluid inlet, so they will appear as negative values for the cold fluid.
This is not an indication of an error. The magnitude of the pressure drop is correct.
C.3 Cautions with the PCHE codes
When running any of the PCHE codes, if a runtime error occurs, the input value of cold
fluid temperature is probably too high. The code is calculating the output temperature of the
cold fluid based on the power and the inlet temperature and mass flow rate. If the outlet
temperature it computes is higher than the inlet temperature of the hot side, the code runs into an
error. Regrettably, this has not yet been corrected. Increasing cold side mass flow rate or
increasing the cold side inlet temperature will be required in this case.
The steam generator code does not converge as well as the others due to the wider range
of heat transfer coefficients on the water side. The code returns warnings to the program
window each time a node does not converge. Before producing the final output, the code will
also warn if the entire heat transferred has not converged or if the pressure drop has not
converged. For these cases, increasing the number of nodes is an option, or the user could
increase the number of iteration loops within the code.
If the flow path is too long, the PCHE codes will deliver an extremely high pressure drop.
This is an indication to the user that the configuration of the PCHE is not a good one and the
dimensions should be rearranged in order to produce a greater number of channels, and therefore
a shorter flow path. In practice, this might not be possible because a greater number of channels
results in larger inlet and outlet plena, which have their own associated pressure drops and costs.
Usually, a flow path greater than about 1.5 m is not acceptable. Pressure drops within the
channels are underestimates since form losses at the channel inlets and outlets, as well as in the
plena, will raise the overall pressure drop of the IHX.
C.4 Improvements to the PCHE Codes
The PCHE codes can be improved in a number of ways. Convergence is still an issue for
the hybrid steam generator code. Oftentimes, a single node will not converge at some point in
the calculation. This is not a serious problem for the overall sizing of the heat exchanger, but
when several nodes repeatedly fail to converge the code sometimes results in an error of several
percent for the overall heat transferred. The code fails when the inlet temperature of the cold
side is too high, causing a pinch point. It should provide a warning to the user to explain the
failure. Correlations used in the PCHE codes for two phase flow and for sodium are best
146
estimates based on other geometries. Experimental data will need to be developed to understand
how heat transfer and CHF are affected by the semi-circular, zig-zag channels of a PCHE.
C.5 Cost Estimates for PCHEs
Dostal developed estimates for the cost of PCHE cores based on the mass of steel within
them. The estimated volume of steel was calculated by
𝑓𝑚 = 1 −
𝜋𝑑 2
8𝑃𝑡
Eqn. C-1
where 𝑓𝑚 is the volume fraction of metal, d is the channel diameter, P is the channel pitch, and t
is the plate thickness. He used this volume fraction to estimate the cost based on a quoted price
of $30/kg for stainless steel units and $120/kg for titanium units in 2003 [Dostal, 2004]. Dostal
also recommends determining the minimum required wall thickness based on an expression from
Hesselgraves relating the walls to fins. The minimum wall thickness is given by
𝑡𝑓 =
1
𝜍
𝑁
∆𝑃 𝑓
Eqn. C-2
where 𝑡𝑓 is the wall (fin) thickness, 𝑁𝑓 is the number of walls (fins) per meter, ∆𝑃 is the pressure
difference from the hot side to the cold side, and 𝜍 is the maximum allowable stress of the
material [Dostal, 2004]. These calculations will have to be verified by experiment and detailed
modeling of the thermo-mechanical stresses in PCHEs.
C.6 Conclusion
The PCHE codes give the user a first estimate of PCHE volume and achievable
temperatures. The user will have to make decisions about the channel geometry based on system
pressures and materials. Pressure drops are a major consideration for PCHEs, as they usually
limit the flow path length to less than 2 m. The designer must be cognizant of how plenum
construction, pressure drop, and volume limit the heat exchanger design. Improvements in heat
transfer correlations and the overall readability/user-interface of the PCHE codes can and should
be made in the future. In addition, studies of the thermo-mechanical stresses and the
construction of PCHE plena should be performed and included with the code suite.
147
Appendix D. SoSaT Code Manual
D.1 Introduction
SoSaT is the Sodium Shell-and-Tube heat exchanger code. It performs calculations for
sodium-sodium, sodium-CO2, and sodium-water heat exchangers. Water can be single phase,
two phase, or supercritical. The code has a single input file and a single output file and can be
used for a variety of tube materials, for double-walled tubes, and with enhanced heat transfer
surfaces in CO2 tubes.
D.2 Inputs and Outputs
Operating SoSaT is accomplished with a single input file, input.txt. The user must know
mass flow rates, pressures and inlet temperatures for both fluids. The outside diameter of the
shell, the total power, and some parameters describing the geometry of the tube array are also
required. The input file, matching the steam generator of the JSFR, is shown in Figure D.1.
JSFR IHX Geometry
0
0
1
0
3
0
1765.0
7500.0
799.0
2.79
0.00
19200.0
200.0
520.0
355.0
240.0
488.0
1.60
0.019d0
0.000007d0
0
!Zero
0.0015d0
!0-single wall, 1-double wall
!shell material, 0-9Cr1Mo, 1-304 SS, 2-316 SS
!tube material, same thing
!Primary fluid, 0-Na, 1-He, 2-CO2, 3-H2O
!Secondary fluid, 0-Na, 1-He, 2-CO2, 3-H2O
!0-smooth tube, 1-enhanced tube
!thermal power (MW)
!mass flow rate on primary side (kg/s)
!mass flow rate on secondary side (kg/s)
!Outside diameter of shell (m)
!Outside diameter of inlet pipe (m)
!Inlet pressure of secondary side (kPa)
!Inlet pressure of primary side (kPa)
!Inlet temperature of primary side (deg C)
!GUESS Outlet temperature of primary side (deg C)
!Inlet temperature of secondary side (deg C)
!GUESS Outlet temperature of secondary side (deg C)
!Pitch to diameter ratio of tube array
!Outside diameter of outside tube (m)
!Double tube gap thickness (m)
indicates that the following 4 thicknesses are used
!Outside tube (m)
148
0.0011d0
0.008d0
0.000d0
0
!Inside tube (m)
!Shell (m)
!Pipe (m)
!0 indicates no baffles, 1 indicates baffles
Figure D.1: The SoSaT input file, matching the steam generator of the JSFR. The data
for the enhanced tubes is never read, since enhanced tubes cannot be used for steam.
The code outputs information about the length, heat transfer area, heat exchanger
effectiveness, and the fluid temperatures at inlet and outlet in output.txt. The output file appears
as shown in Figure D.2.
Shell and Tube Design Data
Number of Baffles
Number of Tubes
0
7220
Heat Exchanger Effectiveness
Heat Exchanger length (m)
Hot fluid Inlet Temperature (deg C)
Hot fluid Outlet Temperature (deg C)
Cold fluid Inlet Temperature (deg C)
Cold fluid Outlet Temperature (deg C)
Hot side delta P (kPa)
Cold side delata P (kPa)
Heat transferred (MW)
Heat transfer area (m^2)
Outlet Steam Superheat (deg C)
0.324
23.530
520.350
336.145
240.000
498.002
10.280
14.292
1765.168
10140.470
135.719
Figure D.2: The SoSaT output file
The results produced by SoSaT for the JSFR steam generator are in agreement with the
published design. The heat transfer area of the JSFR steam generator is 12,500 m3, which agrees
with the SoSaT calculated value of 10,140 m3 very well after the customary 20 % margin is
included.
For a new user designing a heat exchanger, probably the most interesting outputs will be
the fluid temperatures and the dimensions of the heat exchanger. Pressure drops, if high, will
eliminate some designs from consideration. A designer should keep in mind that pressure drops
calculated in SoSaT are only for the tubes. Plena and piping are not considered and these
pressure drops will be considerable for most situations. For comparison purposes, the total heat
149
transfer area is included as an output. This value is the heat transfer area on the shell side (i.e. it
is calculated using the outside diameter of the tubes). The total heat transfer area will give the
user an idea of how effective the heat exchanger is. Effectiveness is also given in the output.
The user can define thicknesses for the tubes or have the code calculate thicknesses (as
stated in the input file). If the code is allowed to calculate thicknesses it will base the calculation
on the minimum required thickness to meet ASME requirements (for both inner and outer tubes
in double-walled tubes) due to the internal pressure.
A central downcomer pipe is present in the SoSaT model. This is used for heat
exchangers which can only have pipe inlets at the top of the shell, for instance in a pool-type
SFR P-IHX. The downcomer pipe lies along the axis of the shell and distributes secondary
coolant at the bottom of the shell to a plenum which then leads to the upward flowing secondary
tubes, as shown in Figure D.3. Primary coolant flows within the shell around the tube bank.
Figure D.3: The placement of the central downcomer (dashed line) in SoSaT. Such an
arrangement is required when the heat exchanger cannot have penetrations in the lower
secondary coolant plenum. Secondary coolant is shown as dark arrows and primary coolant is
shown as light arrows.
150
D.3 Cautions and Considerations for SoSaT
Operation of SoSaT is relatively straight-forward, but a few cautions are included here to
simplify the process. They are arranged here much as the troubleshooting section of Appendix B
is arranged for RGRC and RGRCMS. They are titled “issues” rather than “problems” because it
may be difficult to tell whether there is a problem with the results. Rather, these are things to
keep in mind before developing a heat exchanger.
Issue:
Water pressure in a Steam Generator
Cautions:
The size of a steam generator will depend greatly upon the location of the critical
heat flux. Once the critical heat flux is reached, the heat transfer becomes poor and more heat
transfer area will be needed, owing to the longer length of the heat exchanger and rapidly
expanding capital costs. Changing the steam pressure can have a marked impact on the value of
the critical heat flux and therefore a number of steam pressures should be attempted before
settling on a design. Increasing the pressure will increase the efficiency of the Rankine cycle,
but it may be at the cost of a larger steam generator.
Issue:
Size and Pitch of secondary tubes
Cautions:
The correlations used in SoSaT (for water) are only applicable to tubes of inside
diameter less than 4.0 cm, so results for any tube larger than that are suspect. The tube size and
pitch affect the mass flux in each tube, and this will have a strong effect on the critical heat flux
and all heat transfer coefficients. Changing the shell diameter and the pitch will give the user a
feel for how the effectiveness is changed. This is an issue in CO2 and sodium secondary coolants
also, because Nusselt number is a function of Reynolds number for all fluids. Especially for
water, changing the dimensions of the tubes will have sometimes unexpected results because so
many characteristics of the flow change (perhaps most important is CHF).
Issue:
Mass Flow rates
Cautions:
Increasing the mass flow rate on the secondary side can improve heat transfer
because the secondary coolant will not undergo as much of a temperature rise, keeping ΔT large.
However, the efficiency of the PCS will be reduced by a reduction in the outlet temperature.
Again, there is a tradeoff between size and efficiency. For boiling water the relationship is very
complicated and it is difficult to predict how a change in mass flow rate will affect outlet
temperature, due to dependence of heat transfer coefficient and critical heat flux on the mass
flux.
151
Issue:
Enhanced Tubes
Cautions:
The Ravigurarajan and Bergles correlations for pressure drop and Nusselt number
in a tube with enhanced heat transfer surfaces are sufficiently complicated that results are
difficult to predict a priori. The user is cautioned to have a maximum acceptable pressure drop
in mind when designing a CO2 heat exchanger with enhanced tubes. Changes to the tube
geometry should be geared toward achieving maximum heat exchanger effectiveness while
remaining below the user’s pre-determined maximum pressure drop. Enhanced tubes are not a
current option for steam generators in SoSaT.
Issue:
Pinch Points
Cautions:
SoSaT will simply stop running if a pinch point exists. The code identifies a
pinch point by comparing the temperature at the inlet of a node to the outlet of the same node. If
the temperature rise is less than 0.001 K, there is a pinch point. This restriction doesn’t
necessarily catch all cases, but if the code can be seen to run in the superheated steam regime for
some time, then the user will know that the parameters used in the design will not work.
D.4 Correlations and Supporting Information
The Ravigurarajan and Bergles Correlation for enhanced tubes is written as a correction to
smooth tube Nusselt numbers and pressure drops. The Nusselt number for enhanced tubes is
given by
1/ 7
7
0.212
0.21
0.29
 
 


 p
 
0.036  e 
0.024
Eqn. D-1
 1  2.64 Re
   
  Pr
 
Nu sm
d
d
90









 
 

where e is the height of the ribs, p is the pitch of the ribs, α is the helix angle, and Nusm is the
smooth tube Nusselt number given by the Gnielinski correlation. The pressure drop friction
factor in enhanced tubes is given by
Nu Bergles
𝑓𝑎
𝑓𝑠𝑚
= 1 + 29.1𝑅𝑒
𝑝
𝛼
0.67−0.06 −0.49
𝑑
90
𝑒
𝑑
1.37−0.157
𝑝
𝑑
𝑝
−1.66𝐸−6𝑅𝑒 −0.33
𝑑
𝛼
90
𝛼
90
4.59+4.11𝐸−6𝑅𝑒 −0.15
𝑝
𝑑
1+
2.94𝑠𝑖𝑛𝛽
15
16
16
15
𝑛
Eqn. D-2
152
where n is the number of ribs and β is the contact angle of the rib with the inside surface of the
tube.
Operation of SoSaT requires the supplementary files KandD.txt and chf_lookup.txt to be
present in the program file. The former is a table of values used in the sodium heat transfer
calculation, named for the Nusselt number correlation of Kalish and Dwyer [Foust, 1976]. The
latter contains a formatted table of CHF values from which the code interpolates the value at
each node [Groeneveld, 2007]. The CHF values include pressures above 14 MPa and mass
fluxes up to 1000 kg/m3. The table will have to be expanded from the reference if values for
lower pressures or higher mass fluxes are required.
D.5 Conclusion
SoSaT can be used for SFR heat exchangers from the vessel on outward. The user must
be cognizant of pinch-point problems, as they will result in a heat exchanger that continues
increasing in length without continued heat transfer. Otherwise, the code is very user friendly
and provides valuable insight into the way design choices affect the heat transfer in fluids of
interest to SFRs.
153