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FOREWORD
This report, "Modeling, Analysis, and Evaluation of Rankine
Cycle Propulsion System," describes work carried out under Con-
tract No. EHS-70-111 for the Office of Air Programs, Environ-
mental Protection Agency at Ann Arbor, Michigan. The work was
conducted by the Mechanical Engineering Laboratory of Corporate
Research and Development of the General Electric Company in
Schenectady, New York.
The report consists of two volumes:
Volume 1 -- Final Report
Volume II -- Users Manual
Volume I includes the derivation of the models and their
application to specific designs. Steady-state and transient re-
sults are presented. Volume II includes copies of the computer
programs, FORTRAN nomenclature, flow diagrams, and other
user information.
The Project Officer for this contract was Mr. William Zeber
of the Environmental Protection Agency. The Deputy Project
Officer was Mr. Kent Jefferies of the National Aeronautics and
Space Administration Lewis Research Center in Cleveland, Ohio.
in
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ACKNOWLEDGMENTS
The authors gratefully acknowledge assistance from
the following people:
Mr. Robert Barber and his associates of Barber-
Nichols Engineering Corporation, Arvada, Colorado, for
turbine expander data and analyses. Barber-Nichols was a
subcontractor in this project.
Mr. Dale H. Brown, Thermal Branch, General Electric
Corporate Research and Development, for advice and as-
sistance on transient thermodynamics.
Dr. Thomas Kerr of the Information Studies Branch,
Corporate Research and Development, and Mr. William
Keltz of the Specialty Fluidics Operation, General Electric
Company, for assistance in controls analysis.
Mrs. Barbara Kuhn, Contract Administrator, General
Electric Corporate Research and Development.
Mr. Peter M. Meenan and Mr. Robert C. Rustay of
the Information Studies Branch, General Electric Cor-
porate'Research and Development, for assistance in
modeling and simulation.
Dr. Dean Morgan and his associates in the Thermo
Electron Corporation, Waltham, Massachusetts, for recip-
rocating expander data and analyses. Thermo Electron
was a subcontractor on this project.
Professor Wen-Jei Yang of the University of Michigan
for consultation in the area of transient thermal analysis.
IV
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TABLE OF CONTENTS
Section Page
FOREWORD iii
ACKNOWLEDGMENTS iv
1 SUMMARY 1
Background 1
Objective ^ . . . . 1
Results 1
Recommendations 3
2 INTRODUCTION 5
Background 5
Objective 6
Approach 6
Advantages and Limitations of Modeling
and Simulation 8
3 PROPULSION SYSTEM 11
Working Fluid 11
Thermodynamic Properties 12
Transport Properties 16
Expander 19
Reciprocating Expander 19
Turbine Expander . ; 26
Feedpump 33
Nomenclature . 33
Derivation of Equations 36
Model Development 37
Results 38
Heat Exchangers 39
Nomenclature 39
Transient Thermal Analysis 42
Vapor Generator 52
Condenser 76
Regenerator ." 80
Combustor 84
Nomenclature 87
Flame Temperature Submodel 89
Thermal Transient Submodel 91
Emissions Submodel Development 94
Total Combustor Model 99
Results 100
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TABLE OF CONTENTS (Cont'd)
Section Page
3 PROPULSION SYSTEM (Cont'd)
Controls 103
Nomenclature 103
Control Definition 104
Burner Control 104
Cut-off and Feedpump Control Ill
Condenser Fan Equations .114
Discussion and Recommendations 114
4 VEHICLE SYSTEM 117
Transmission 117
Nomenclature 117
Derivation 118
Model Development 118
Results 120
Vehicle 121
Nomenclature 121
Derivation of Basic Equations 122
Model Development 123
Route 123
Model Development 123
Driver ' 125
Nomenclature 125
Development of Model 126
Results 127
5 TOTAL SYSTEM 133
Method of System Analysis 133
Steady-state Condition 134
Transient Simulation 135
System Model Structure 137
Results 137
6 DISCUSSION AND RECOMMENDATIONS 143
Heat Exchanger Dynamics • 145
Run Time Economy 147
. Control Development and System Dynamics . . . . . 149
Application to System Development 150
Appendix I -- PARAMETRIC PROPULSION SYSTEM
DESIGNS 151
VI
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TABLE OF CONTENTS (Cont'd)
Page
Appendix II -- STABILITY AND ERROR CRITERIA
FOR FINITE-DIFFERENCE SOLUTION
OF PARTIAL DIFFERENTIAL
EQUATIONS 163
Appendix III-- HEAT TRANSFER AND PRESSURE
DROP RELATIONS \ . . . 165
Appendix IV-- EVAPORATOR FLOW INSTABILITY ... 175
Appendix V -- REFERENCES 177
VII
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LIST OF ILLUSTRATIONS
Figure . Page
1 Linkage of Saturated Fluid Property Model 13
2 Linkage of Superheated Property Model 17
3 Simple Reciprocating Engine Cylinder Schematic
and Indicator Diagram 22
4 Reciprocating Expander Model-- Efficiency vs Rpm . ... 27
5 Theoretical Nozzle Performance (y = 1.4) 31
6 Effect of Gas Ratio of Specific Heat on Calculated
Nozzle Performance 31
7 Turbine Model Results 34
8 Comparison of Turbine Model with Barber-Nichols
Engineering Company Calculations 35
9 Pump Model -- Volumetric Flow Rate Versus Rpm .... 40
10 Schematic of Flow Through a Tube 46
11 Node Pattern 48
12 Information Signals for Vapor Generator Model 52
13 Node Pattern 55
14 Schematic of Phases and Interphases 62
15 Error in Approximation (vg-Vg)/vfg = (hs-h )/hfg
for CP-34 65
16 Fluid Pass Numbering Process 68
17 Thermo Electron Corporation Vapor Generator --
Cross Section Through Burner-boiler, Short Axis ..... 70
18 Vapor Generator -- Steady-state Enthalpy Distrubution
of Working Fluid. Vapor Generator Design as for
TECO System 73
19 Comparison of Steady-state Temperature Distribution
for Working Fluid in Vapor Generator . . 73
20 Vapor Generator -- Steady-state Tube Wall Temperature
Distribution. Vapor Generator Design as for TECO
System 74
21 Vapor Generator -- Steady-state Combustion-gas
Temperature Distribution 74
Vlll
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LIST OF ILLUSTRATIONS (Cont'd)
Figure
22 Vapor-generator Transient Response to a Change
in Inlet Fluid Flow 75
23 Vapor-generator Transient Response to a Change
in Combustion-gas Flow Rate 76
24 Condenser Design 78
\
25 Condenser -- Steady-state Enthalpy Distribution
Liquid Side --as Calculated by Transient Model 79
26 Condenser Transient Response to a Change in Inlet
Fluid Flow 79
27 Regenerator Design 81
28 Regenerator Liquid Enthalpy -- Derivation of Steady-
state Solution Employing Transient Model 82
29 Regenerator Tube-wall Temperature -- Derivation
of Steady-state Solution Employing Transient Model .... 83
30 Regenerator Gas Temperature -- Derivation of Steady-
state Solution Employing Transient Model 83
31 Regenerator Transient 84
32 Regenerator Transient -- Fluid Temperature 85
33 Regenerator Transient -- Tube-wall Temperature .... 86
34 Regenerator Transient -- Vapor Temperature 87
35 Combustor Schematic 91
36 Nitrogen Oxide. Measured Exhaust Concentrations .... 95
37 Carbon Monoxide. Measured Exhaust Concentrations ... 96
38 Unburned Hydrocarbons. Measured Exhaust
Concentrations 97
39 Characteristic Normalized Exhaust Concentrations
(e = 0. 59) 98
40 Combustor Model -- Linking of Combustor Submodels ... 99
41 Combustor Model Results 102
42 Schematic of Power, Working Fluid, and Air/Fuel Control . 105
43 Fuel Valve -- Simplified Schematic 106
44 Original Thermo Electron Air Valve 106
IX
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LIST OF ILLUSTRATIONS (Confd)
Figure Page
45 Burner Control -- Block Diagram 1 07
46 Fuel Flow Versus Boiler Flow -- CP-34 System;
Contoured Poppet l 09
47 Fuel Flow Versus Boiler Temperature -- CP-34
System; Contoured Poppet Ill)
48 Slope of Qf Versus Temperature Curve at Design <•
Temperature (550°F) -- CP-34 System; Contoured Poppet . Ill
49 Engine Power Level and Vapor Generator Feedpump
Control -- Functional Block Diagram 112
50 Engine Information Signal Loop 117
51 Transmission Gear-shift Sequence 119
52 Route Mission Profiles 124
53 Comparison of Vehicle Traverse and Reference
(Forcing) Conditions for a Linear Response Engine .... 129
54 Response to Wheel Slip -- Driver Releases Accelerator
and Reduces Acceleration Sensitivity 130
55 System Model Linkages • 133
56 Total Systems Model -- Initial Estimates to Derive
Cycle Design Conditions 135
57 Dynamic System Information -- Signal Flow Diagram,
Excluding Controls 136
58 Total Systems Model -- Information Flow at Cycle
Design Condition, Without Controls 138
59 Computer Output for System Steady-state Run (3 Sheets). . 139
60 Computer Cost Information for Vapor Generator
Transient Model 147
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LIST OF TABLES
Table Page
1 Results of Saturated Property Model for Water 14
2 Results of Superheated Property Model for FC-75 . ... 17
3 Methods of Transient Thermal Analysis 43
4 Features of Finite-difference Digital Method 45
5 Finite-difference Relations ^ . . . 57
6 Stability Limits for Energy Equations 58
7 Selection of Energy Equation Models 66
8 Details of Flow Paths for the TECO Vapor Generator . . 71
9 Selection of Energy Relations for Condenser 77
10 Route Mission Profile 125
11 Summary of Component Models 144
12 Cycle Design Conditions for Reciprocating Engine
with CP-34 as Working Fluid 152
13 Cycle Design Conditions for Reciprocating Engine
with Water as Working Fluid 152
14 Cycle Design Conditions for Turbine Engine
with FC-75 Working Fluid 153
15 Cycle Design Conditions for Compound Engine
with Water as Working Fluid 153
16 Combustor Designs 154
17 Reciprocating Engine System with CP-34
as a Working Fluid 154
18 Simple Reciprocating Engine System
with Water as Working Fluid 155
19 Turbine Engine System with FC-75 as Working Fluid. . . 156
20 Compound Reciprocating Engine System with Water
as Working Fluid " 157
21 Condenser Designs 158
22 Regenerator Designs 159
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Section 1
SUMMARY
BACKGROUND
The increasing concern for a cleaner environment has led to serious con-
sideration of the Rankine cycle engine for low-pollution automotive propulsion.
Typical driving cycles clearly indicate that automobiles are always in transient
operation; therefore, engine loading is continuously changing. The Rankine
cycle power plant is characterized by a number of thermal inertias which sig-
nificantly affect the ability of the vehicle to respond to varying demands. Fur-
thermore, engine transients are transmitted through the controls to the com-
bustor fuel and air supply system. High emission levels and inadequate vehi-
cle performance can result if the propulsion system dynamics are not under-
stood and properly controlled.
OBJECTIVE
The objective of the program described in this report was to develop a
generalized computer model of a Rankine-cycle automotive propulsion system
to be used for the analysis of propulsion system dynamics.
RESULTS
Digital computer models were developed for the following propulsion sys-
tem components:
Working fluid -- water and organic
Combustor
Vapor generator
Expander -- reciprocating and turbine
Condenser
Regenerator
Feedpump
Controls
The major criteria in the development of the component dynamic models
were:
• Applicability to the several alternative propulsion system designs
under development by the Office of Air Programs, Environmental
Protection Agency
• Validity over the full range of vehicle operating conditions
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Other vehicle system models were developed to permit analysis of engine dy-
namics during realistic driving transients. These included models of:
Transmission
Vehicle motion
Route mission profiles
Driver
The programming language is FORTRAN IV. The models were run on the Gen-
eral Electric 635 computer and were calibrated with experimental data when
such data were available.
A module-linkage approach was employed for the modeling. Each of the
component models is a self-contained module with several input and output in-
formation signals. Linkage of the information signals forms a total system
model which can be employed for transient analysis of the entire propulsion
system.
The computer models that have been developed are described in Volume I
of this report. The basic equations are derived, solution techniques are dis-
cussed, and preliminary results are presented. Volume II, the Users Manual,
contains copies of the computer programs, nomenclature lists, flow diagrams,
and other important user information.
The component models have been provided with input data for a propulsion
system with a reciprocating expander and organic working fluid designed by
the Thermo Electron Corporation (TECO) in Reference 1. The component
models have been linked together to form a total system model, which has been
run without controls to derive the system steady-state condition.
The. models were developed so that design modifications and different
working fluids can be easily analyzed by changing the input data. The models
are applicable to many alternative propulsion system designs, including:
• Simple reciprocating expander with water as working fluid
• Compound reciprocating expander with water as working fluid
• Turbine expander with organic working fluid
The most comprehensive models developed are for the heat exchangers
(vapor generator, condenser, and regenerator). These components play a very
significant role .in determining dynamic system performance, and their tran-
sient behavior is not well understood. A major effort was therefore directed
toward developing full-range dynamic heat exchanger models involving a mini-
mum number of assumptions and limitations.
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RECOMMENDATIONS
Although most of the effort to date has been concerned with development
of the models, and only a limited number of computer runs have been made to
obtain definitive results, several recommendations can be made. These rec-
ommendations are listed below and are fully developed and discussed in Sec-
tion 6 of this volume.
Recommendation 1: Because the thermal inertia of the heat exchanger
components will determine propulsion system response, the following
work is recommended in order to more accurately establish heat ex-
changer dynamics.
• Sensitivity analyses should be carried out for
Heat transfer coefficients
Two-phase/vapor transition point
Water-jacket resistance
These analyses would employ the models for parametric variation
of the above items to determine their effect on steady-state and
dynamic performance.
• The correlation of two-phase flow and heat transfer should be ex-
panded to account for the various two-phase flow regimes.
• The heat exchanger models should be validated with transient ex-
perimental data.
Recommendation 2: In order to improve the run-time economy of the
computer simulation it is recommended that:
• The fluid property models be utilized more efficiently.
• Fixed time steps and lump sizes be employed in the heat exchanger
models.
• A set of "parametric models" be developed.
Recommendation 3: The following control system development plan is
recommended:
• The instantaneous control models that have been developed should
be employed to bring the total system to steady state at the design
condition, and small perturbation transients around this point should
be analyzed. This will establish the basic validity of the control
scheme.
• Acceptable limits on the variation of system parameters during tran-
sients should be established.
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• The full-range dynamic models should be used to derive limited-
range parametric models as described above.
• The instantaneous control scheme should be modified to include con-
trol dynamics. The controls should be developed by means of the
parametric models.
• The final control scheme should be checked out with the full-range
dynamic models.
Recommendation 4; The models developed are highly flexible and gen-
eral. .They can be used to simulate the dynamics of many alternative
Rankine cycle propulsion-system configurations. It is recommended
that the models be employed for transient analysis of the systems under
development by the Environmental Protection Agency and the results
used to support and guide the design and experimentation.
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Section 2
INTRODUCTION
BACKGROUND
Since the exhaust emissions of internal combustion engines are a major
source of air pollution, serious consideration is being given to other auto-
motive propulsion systems with low pollution characteristics. The Rankine
cycle ei gine appears to be a most promising near-term alternative. This
engine, using components of reasonable size, weight, and cost;> is capable
of providing required vehicle performance with minimum contamination.
Automobile operation is distinctly transient; this is due to the repetitive
accelerations and decelerations that are required in most driving situations.
Propulsion systems rarely operate in steady state, and engine designs based
solely on steady-state considerations will have serious deficiencies. Propul-
sion system dynamics must be analyzed in order to determine the size and
capacity of engine components, the vehicle performance and emissions, and
the control system requirements.
In order to illustrate the importance of understanding Rankine cycle
dynamics, consider the highway passing situation. Initially, the vehicle is
cruising at a constant velocity, the engine speed is constant, and the vapor
mass flow rate is uniform throughout the cycle. The torque required for
acceleration is several times the torque at cruise. As a result the vapor
demand during the acceleration period is also several times higher than at
steady state. Therefore, the size and capacity of all the engine components
(vapor generator, expander, regenerator, condenser, pumps, and fans) must
be based on the transient operating condition.
In order to execute the passing maneuver the driver depresses the accel-
erator pedal, which opens the throttle, or increases the expander cut-off. This
produces a demand for a rapid increase in vapor flow from the vapor generator.
If the flow through the feedpump is not increased to maintain vapor generator
inventory, the pressure will drop and consequently the engine torque capability
will be reduced. Therefore, the acceleration rate of the vehicle depends upon
the dynamic response of the vapor generator and pump. Furthermore a control
linkage is required between the accelerator pedal, expander, and feedpump.
As the mass flow rate through the system increases, the vapor generator
exit temperature will drop and the cycle efficiency will be impaired, unless
there is a simultaneous increase in combustor fuel flow. However, the emission
levels are highly sensitive to air/fuel ratio. If the airflow rate is not increased
proportionately to the fuel, the air/fuel ratio will deviate from its optimum value
and the air pollution level will be high. The rate at which the air and fuel flows
change depends upon the dynamics of the air and fuel supply systems.
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Therefore, in order to maintain low emission levels during transients the
combustor dynamics must be controlled.
From the above discussion it should be apparent that understanding of
the propulsion system dynamics is essential in order to develop high-perfor-
mance low-emission Rankine cycle engines.
OBJECTIVE
The objective of this program carried out for the Office of Air Programs,
Environmental Protection Agency was:
• To develop a generalized computer model of a Rankine cycle automotive
propulsion system to be used for analysis of propulsion system dynamics.
APPROACH
A mathematical model of an engine component is a set of analytical
expressions, .equations, or algorithms which describe the component's opera-
tion. The approach taken in the present program was the development of digital
computer models of the major components of the Rankine cycle propulsion
system. The following components were modeled:
Working Fluid -- water and organics
Combustor
Vapor Generator
Expander -L reciprocating and turbine
Condenser
Regenerator
Feedpump
Controls -- power, flow rate, combustor
In addition to the propulsion system models, other models were constructed
for the analysis of dynamics in realistic driving situations. These included
models for:
« Transmission
• Vehicle -- motion resistance, traction
• Route mission profiles consisting of
Start-up
Accelerations
Cruise at various speeds
Decelerations
High-speed pass
Grades
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• Driver -- compares the vehicle performance to reference condi-
tions and adjusts the acceleration pedal or brake accordingly.
The models have been written in computer language FORTRAN IV and have
been run on the General Electric 635 high-speed digital computer. The models
can be used on any computer that employs a FORTRAN IV compiler.
The procedure used for the modeling is referred to as the module-linkage
approach. Each of the individual component models is formulated in a mod-
ular or build ing-block manner. That is, each component model is a unit in
itself, and the equations describing it are independent of thosex describing
other components. These modules are linked together by statements express-
ing the interplay or communication between components. For example, in
the expander model, the torque is expressed as a function of inlet pressure and
temperature, exit pressure, cut-off, and expander speed.
The expander model is linked to:
a. The transmission model, by rpm and torque linkages
b. The vapor generator model, by pressure temperature and mass
flow linkages
c. The control system, by the cut-off linkage
Linking together all of the component models forms a total system model
that is capable of simulating steady-state and dynamic performance over the
entire vehicle operating range.
The module-linkage approach allows rapid examination of alternative
engine and control system configurations. The component models were de-
veloped to permit simulation of the performance of four systems:
• Reciprocating engine with CP-34 as working fluid
• Reciprocating engine with water as working fluid
• Turbine engine with FC-75 as working fluid
• Compound reciprocating engine with water as working fluid
The first engine configuration, with CP-34 as working fluid, is the system
designed by the Thermo Electron Corporation under Contract CPA 22-69-162
(Ref. 1). Data for this system have been input into the models, the models
have been checked out, and component transient analyses have been run. The
total system model has been formulated, linking together the components,
and has been brought to steady state at the design condition.
The total system model can now be employed for system transient ana-
lyses. A preliminary control scheme has been developed for this purpose.
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For the remaining three systems, preliminary designs have been com-
pleted, and data generated suitable for input into the component and system
models. As of this writing the models have not been exercised for these
three alternative systems.
ADVANTAGES AND LIMITATIONS OF MODELING AND SIMULATION
Modeling and simulation are extremely powerful analytical techniques,
widely used for the investigation of dynamic systems. They are especially
suited to the analysis of complex systems containing many interacting com-
ponents -- such as Rankine cycle engines. Use of the digital Computer makes
possible the numerical solution of equations describing component dynamics.
The accuracy and validity of modeling and simulation are limited by the
theoretical basis for describing the physical processes involved. This is an
inherent limitation of all analytical techniques. The models developed in this
program have been calibrated with experimental data, whenever available, in
order to represent realistic engine designs.
The vapor generator is one of the most important components to be ac-
curately modeled, as it has a significant effect on system dynamics. This is
also the most difficult component to model, since it is necessary to simulta-
neously solve the nonlinear partial differential equations for mass, momentum,
and energy conservation. As will be brought out in later sections, vapor
generator dynamics are fairly sensitive to the assumptions made in describing
the two-phase flow during transients. Therefore, at the end' of this report,
future work is recommended which will improve the capability of the vapor gener-
ator model in predicting the dynamic behavior of this critical component.
This work consists of using the model to run sensitivity studies on the heat
transfer parameters in order to isolate those which have a strong influence
on dynamic response. The studies should be followed by experimentation to
accurately determine these parameters, and this information should be fac-
tored back into the vapor generator model.
The above recommendation provides a good example of how modeling and
simulation can be employed to complement experimentation. In any experi-
mental program there are limitations on the parameters that can be measured.
For the vapor generator, for example, pressure, temperature, and mass flow
rates are fairly easy to measure, while it is difficult to sense the motion of
the liquid two-phase interface which causes changes in pressure, temperature,
and flow. However, the vapor generator model can be used to predict the
interface motion and isolate the cause of the experimentally measured effects.
Simulation can also be used for investigations of hazardous conditions or
conditions outside the range of the experimental apparatus. Finally, simulation
is usually much less costly than actual experimentation, and should therefore
be used to direct and focus an experimental program.
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One of the requirements of the models developed in this program was
flexibility. The models readily accept design changes, different working
fluids, different component configurations, and alternative control linkages.
This requirement increased the complexity of the models, the development
time, the computer memory size, and the run time. Simpler special-pur-
pose models would have been easier and individually less expensive, but a
separate model would be required for each component of each system to be
studied. Furthermore, design changes would probably require internal
model modifications rather than just change of input data as in the present
case. The more flexible modeling is valuable in development programs where
there are many alternatives to be considered.
Throughout this program, efforts were made to minimize the limitations
and weaknesses of the digital simulation approach and increase its accuracy
and utility.
In Section 3, the propulsion system models are derived and transient
results are presented. This is followed by discussion of the other vehicle
component m.odels (Section 4) and the total system model (Section 5). Finally,
at the end of Volume I (Section 6), the results are discussed and recommendations
are made for future work.
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Section 3
PROPULSION SYSTEM
Models have been developed for the following propulsion system compo-
nents:
• Working fluid
Thermodynamic properties
Transport and metal properties
• Reciprocating expander
• Turbine expander
• Vapor generator
• Regenerator
• Condenser
• Feedpump
• Combustor
• Controls
Dimensional and geometric data have been input into these models, and steady-
state and transient analyses have been made. In most instances, the compo-
nent design has been based on the propulsion system specified in Reference 1 --
a reciprocating expander with CP-34 as working fluid.
The models were developed so that different engine designs could be read-
ily analyzed. Input data for the following three alternative propulsion systems
have been generated employing a parametric design procedure described in
Appendix I, "Parametric Propulsion System Designs, " at the end of this volume.
• Reciprocating expander with water as working fluid
• Turbine expander with FC-75 as working fluid
• Compound reciprocating expander with water as a working fluid
In the following subsections, the propulsion system models will be de-
rived and results of their application will be discussed.
WORKING FLUID
Models have been developed for the thermodynamic and transport prop-
erties of the working fluid. These models are employed throughout the
Rankine cycle and used to determine various properties of the working fluid
when other properties are known.
11
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THERMODYNAMIC PROPERTIES
Models have been developed which yield ther mo dynamic properties in the
saturated and superheated regions. These models are based upon tabular
thermodynamic data which are entered and stored in the program. These
data can then be searched to determine the properties needed. The models
apply to any working fluid for which thermodynamic data are available. Data
for CP-34, FC-75, and water are presented in the Appendix, "Working Fluid
Thermodynamic Properties for CP-34, Water, and FC-75, " of Volume II of
this report.
Saturated Properties
The saturated property model consists of three elements: a fluid prop-
erty listing, a reading program, and an interpolation program.
Saturated Fluid Property Listing. Each line of the saturated fluid property
listing contains a temperature value and the corresponding values of pressure,
specific volume of the liquid, specific volume of the vapor, enthalpy of the
liquid, and enthalpy of the vapor. The property listing can easily be extended
on both the lower and upper ends or filled in so that the intervals between tem-
perature steps are smaller. A variable temperature interval can also be
used to increase accuracy without necessitating any change in the interpola-
tion program. The amount of data in the fluid property listing is the factor
which establishes the accuracy of the interpolation.
Saturated Reading Program. The saturated reading program reads the values
in the saturated fluid properties listing and assigns subscripted labels to these
values. It stores these dimensioned arrays for subsequent use. The reading
program needs to be loaded only once during the execution of any one computer
run.
Saturated Interpolation Program. In the saturated region, specification of
one fluid property establishes the values of all remaining properties. The
saturated interpolation program was developed to determine these remaining
properties. The interpolation program is so constructed that it can be en-
tered with a value of temperature, pressure, or enthalpy of the liquid. A
logic input code is used to direct the computer to the correct portion of the
interpolation program for the given input property.
Based on the input property, the interpolation program Starts at one end
of the saturated property array and searches in sequence for the interval in
which the input-value falls. Then a linear interpolation is carried out across
this interval. This method of search works satisfactorily but can be ineffici-
ent if the interpolation program has to be entered a great many times. Alter-
native methods, employing memory or binary search, could be used so that the
entire array does not have to be searched each time.
12
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If an input value exceeds the range of that particular fluid property array,
the interpolation program extrapolates to that value.
Interrelationship Between the Reading and Interpolation Programs. Figure 1
is a diagram of the program steps for obtaining interpolated saturated prop-
erty values. The interpolation program is entered with a value of a
saturated fluid property. Before interpolation can take place, a check must
be made to ascertain whether the reading program has already been entered.
If it has not previously been entered, it is now called. The reading program,
in turn, reads and stores the saturated fluid property values in their individual
arrays. The appropriate logic variable in the interpolation program prevents
any subsequent entering of the reading program. The saturated fluid property
values are now available to the interpolation program, which uses them in its
process.
SATP
Saturated
Interpolation
Program
PROP
Saturated
Reading
Program
Saturated
Fluid Properties
Figure 1.
Linkage of Saturated Fluid Property Model
(SATP and PROP are the names of pro-
grams which are described in Volume II)
Results. Table 1 shows the results of the test case of the saturated property
model for water. Data from the American Society of Mechanical Engineers
steam table (Ref. 2) are printed next to the interpolated computer values for
comparison. A good agreement can be seen.
Superheated Properties
The superheated property model also consists of three elements: a fluid
property listing, a reading program, and an interpolation program.
13
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Table 1
RESULTS OF SATURATED PROPERTY MODEL FOR WATER
(Values in Parentheses are from Reference 2)
Temperature = 202°F
Pressure, psi
Temperature, °F
Liquid Specific Volume, ft3/lb
Vapor Specific Volume, fWlb
Liquid Enthalpy, Btu/lb
Vapor Enthalpy, Btu/lb
Temperature = 518°F
Pressure, psi
Temperature, °F
Liquid Specific Volume, ft3/lb
Vapor Specific Volume, ft3/lb
Liquid Enthalpy, Btu/lb
Vapor Enthalpy, Btu/lb
Temperature = 468°F
Pressure, psi
Temperature, °F
Liquid Specific Volume, ft3/lb
Vapor Specific Volume, ft3/lb
Liquid Enthalpy, Btu/lb
Vapor Enthalpy, Btu/lb
Pressure = 1000 psi
Pressure, psi
Temperature, UF
Liquid Specific Volume, ft3/lb
Vapor Specific Volume, ft3/lb
Liquid Enthalpy, Btu/lb
Vapor Enthalpy, Btu/lb
12.0236
202.0000
0. 0167
32.4090
170. 1020
1146.7600
798.5500
518.0000
0. 0209
0.5701
509. 6000
1199.4000
504. 8940
4681 0000
0. 0198
0. 9188
450.6500
1204.6500
1000.0000
544.5693
0. 0216
0.4461
542.5758 '
1192.9292
(12.011)
(202.0)
(0.01665)
(32.367)
s
(170.10)
(1146.7)
(798.55)
(518. 0)
(0.02086)
(0.57006)
(509.6)
(1199.4)
(504.83)
(468.0)
(0.01976)
00. 91862)
(450.7)
(1204.6)
(1000.0)
(544.58)
(0. 02159)
(0. 446)
(542.55)
(1192.9)
14
-------
Table 1 (Cont'd)
Pressure = 50 psi
Pressure, psi 50.0000 (50.0)
Temperature, UF 280.9965 (281.02)
Liquid Specific Volume, fP/lb 0.0173 (0. 01727)
Vapor Specific Volume, fts/lb 8.5199 (8. 514)
Liquid Enthalpy, Btu/lb 250. 1789 (250. 21)
Vapor Enthalpy, Btu/lb 1174.0989 (1174.1)
Liquid Enthalpy = 405. 7
Pressure, psi 336.5195 (336.463)
Temperature, °F 428.0000 (428)
Liquid Specific Volume, fp/lb 0. 0191 (0. 01906)
Vapor Specific Volume, fts/lb 1.3786 (1.3782)
Liquid Enthalpy, Btu/lb 405.7000 (405.7)
Vapor Enthalpy, Btu/lb 1203.7500 (1203.7)
Superheated Fluid Property Listings. There are three superheated fluid prop-
erty listings. The first contains the pressure and temperature steps for
which the other two listings supply the corresponding specific volume, en-
thalpy, entropy, and specific heat values. The property data can easily be
extended on both the lower and upper ends, or filled in so that the intervals
between pressure steps and temperature steps are smaller. Variable tem-
perature or pressure steps can also be used to increase interpolation accu-
racy without necessitating any change in the interpolation program. The
amount of data available, and also the amount of computer memory space,
are the limiting factors in determining the accuracy of the superheated in-
terpolation program.
Superheated Reading Pjpgram. The superheated reading program reads the
values in the superheated fluid property listings and assigns subscripted
labels to these values. It stores these dimensioned arrays in COMMON for
subsequent use. It is necessary to enter the reading program only once dur -
ing the execution of the system's program.
Superheated Interpolation Program. In the superheated region specification
of two independent fluid properties establishes the values of all remaining
properties. The superheated interpolation program was developed to deter-
mine these remaining properties. The interpolation program is constructed
15
-------
so that it can be entered with values for any of the following pairs of fluid
properties:
• Pressure and temperature
• Pressure and entropy
• Pressure and enthalpy
• Specific volume and entropy
A value of a logic input code is also entered and used to direct the computer to
the correct portion of the interpolation programforthe given input properties.
Each pressure block of the superheated table contains-superheated prop-
erties for a wide temperature range. For temperatures below the saturation
temperature (for that particular pressure block) subcooled data were employed
to fill out the pressure block.
As already mentioned in the discussion of the saturated interpolation pro-
gram, the method of search used in the interpolation programs would not be
efficient enough if the program was to be entered a great many times. Here,
as in the saturated interpolation program, appropriate messages are printed
if an input value falls outside the range of the superheated listings. This
message may be suppressed by the use of the appropriate printing logic var-
iable.
Interrelationship Between Reading and Interpolation Programs. Figure 2 is
a diagram of the program steps in obtaining interpolated superheated property
values. The interpolation program is entered with values of two superheated
fluid properties.
As in the discussion of the saturated reading and interpolation programs,
a check must be made to determine whether the reading program has already
been entered and the superheated fluid property values stored in arrays in
COMMON for use by the interpolation program. The appropriate logic vari-
able prevents any subsequent entering of the reading program.
Results. Table 2 shows the results of a test case of the superheated property
model for FC-75. Data from Reference 3 are written beside the interpolated
computer values for comparison. It is seen that agreement is good. Where
there is a discrepancy it is due to difficulty in reading the pressure enthalpy
diagram in Reference 3.
TRANSPORT PROPERTIES
Fluid transport properties such as the following were obtained as a func-
tion of pressure and temperature from many sources (Refs. 2 - 7):
Specific heat
Viscosity
16
-------
Conductivity
Density
Surface tension
These were curve-fitted with polynomials by a least squares technique.
In many cases the data available did not cover the entire operating range
and had to be extended by extrapolation. The transport properties are listed
in Volume II, and the comparison with available data is indicated. Metal,
air, and combustion gas properties are handled in a similar manner.
SUPPT
Superheated
Interpolation
Program
PROPST
Superheated
Reading
Program
Superheated Fluid
Properties I
Superheated Fluid
Properties II
Superheated Fluid
Properties III
Figure 2. Linkage of Superheated Property Model (SUPPT and PROPST
are the names of programs which are described in Volume II)
Table 2
RESULTS OF SUPERHEATED PROPERTY MODEL FOR FC-75
(Values in Parentheses are from Reference 3)
Pressure = 4 atm; Temperature = 170 °C
Pressure, atm 4. 0000
Temperature, °C 170.0000
Enthalpy, Kcal/mole 25.4000
Entropy, cal/mole °C 68. 7000
Specific Volume, liters/mole 7.8730
Pressure = 38 atm; Temperature = 294 °C
Pressure, atm 38. 0000
Temperature, °C 294.0000
(4. 00)
(170.0)
(25.40)
(68. 70)
(7.873)
(38.0)
(294.0)
17
-------
Table 2 (Cont'd)
Pressure = 38 atm; Temperature = 294°C (Cont'd)
Enthalpy, Kcal/ mole 35.5332
Entropy, cal/mole °C 85.9364
Specific Volume, liters/mole 0. 6544
Pressure = 12 atm; Entropy = 88 cal/mole °K
Pressure, atm 12.0000
Temperature, °C 272. 4402
Enthalpy, Kcal/ mole 35.8431
Entropy, cal/mole °C 88. 0000
Specific Volume, liters /mole 3. 0474
Pressure = 32 atm, Entropy = 78 cal/mole °K
Pressure, atm 32.0000
Temperature, °C 256.1174
Enthalpy, Kcal/ mole 31.0467
Entropy, cal/mole °C 78. 0000
Specific Volume, liters/ mole 0.4994
Pressure = 8 atm; Enthalpy = 32 Kcal/mole
Pressure, atm 8. 0000
Temperature, °C 235.1327
Enthalpy, Kcal/mole 32. 0000
Entropy, cal/mole °C 81.3446
Specific Volume, liters/ mole 4. 3750
Specific Volume = 39, 25 liters/ mole; Entropy = 81. 22
Pressure, atm 1.0000
Temperature, °C 210. 0000
Enthalpy, Kcal/mole 30.0500
Entropy, cal/mole °C 81.2200
Specific Volume, liters/mole 39. 2500
(35.5)
(86.0)
(.708)
(12.0)
(275.)
(35. 9)
(88.0)
(3.25)
(32.0)
(257.)
(31.0)
(78.0)
(.583)
(8.0)
(236.)
(32.0)
(81.5)
(4.79)
cal/mole °K
(1.000)
(210.0)
(30.05)
(81.22)
(39.25)
18
-------
EXPANDER
Models have been developed for a reciprocating and turbine expander.
These models are quasi-steady in that they instantaneously predict torque and
fluid properties at the outlet, as a function of inlet properties, throttle (or cut-
off) setting and rpm. The time rate of change of rpm depends upon the en-
gine loading, and is therefore determined in the vehicle model, which is dis-
cussed in Section 4 of this volume.
RECIPROCATING EXPANDER
Nomenclature
Alphabetical
Symbols
A
ei
A.
ei
C
62
C.
h
6
h.
i
J
k
m
N
Area of blowdown exhaust ports
Area of auxiliary exhaust ports
Average inlet valve area
Piston cross section area
Flow coefficient of blowdown exhaust ports
Flow coefficient of auxiliary exhaust ports
Inlet valve flow coefficient
Enthalpy after intake
Enthalpy at end of stroke
Exhaust enthalpy
Inlet enthalpy
Torque
Working fluid conductivity
Mass flow rate through engine
Number of pistons
Pressure after intake
19
-------
Alphabetical
Symbols
(Cont'd)
Pi
Pm
Q
R
RPM
s
s.
i
T
T,
T.
i
w
v.
i
V
W.
W
W
sh
x
Pressure at end of stroke
Exhaust pressure
Inlet pressure
Indicator mean effective pressure
Heat loss (Btu/lb)
Cut-off - fraction of stroke
Expander rotational speed
Exhaust entropy
Inlet entropy
Stroke
Average working fluid temperature
Temperature at end of stroke
Exhaust temperature
Inlet temperature
Average wall temperature
Specific volume after intake
Specific volume at end of stroke
Exhaust specific volume
Inlet specific volume
Piston speed
Indicator work
Isentropic work
Shaft work
Exhaust quality
20
-------
Greek
Symbols
TI Mechanical efficiency
m
TI , Thermal efficiency
th
6 Crank angle at which blowdown exhaust ports are
g
uncovered
9. Crank angle for intake opening
u Working fluid viscosity
Derivation of Equations
A schematic and indicator diagram for a simple reciprocating vapor ex-
pander is shown as Figure 3. The piston starts at the left end of the cylinder
and moves to the right, taking in vapor at inlet pressure. When the vapor
flow is cut off at point a_ by closing the inlet valve, the piston continues moving
as the vapor trapped in the cylinder expands. When the piston reaches the end
of the stroke at point b, the exhaust valve opens and the pressure drops to
the exhaust pressure. The piston then moves to the left, discharging the re-
maining vapor, the exhaust valve is closed, the inlet valve is opened, and the
cycle is repeated.
The basic equations for the reciprocating expander model are the re-
lationships for
Torque:
W
shaft m
J = RPM (1)
Mass flow:
m =
N A V
-JLJLJL
Shaft work:
Isentropic work:
W ... = TI,. TI W (3)
shaft th m s
Ws = (hi - V - Vb (pb - Pe> (4)
The properties h^, p^, and v^ in Equation 4 are determined for an isentropic
expansion from inlet conditions to point b at the end of the stroke.
21
-------
^Cylinder
Wall
Inlet
Exhaust
(inlet pressure)
(exhaust pressure)
Pressure
Close
Exhaust
Piston
Open Exhaust
V
co
V
ce
V
Volume (or Displacement)
Figure 3. Simple Reciprocating Engine Cylinder Schematic
and Indicator Diagram
22
-------
Mechanical Efficiency. The mechanical efficiency is related to piston speed
and indicator mean effective pressure by the following relation derived from
expander data by the Thermo Electron Corporation:
m
i-v
PP
0.012
(5)
m
m
The above relationship is derived for the dimensions of Vp in feet per minute
and pm in pounds per square inch. The piston speed is defined as
V
2 S RPM
P
while the mean effective pressure is defined as
P
m
Wi/vb
(6)
(7)
where v^ is the specific volume at the end of the stroke for the actual non-
isentropic expansion.
Thermal Efficiency. The thermal efficiency is defined as
'th
w./w
The indicator work is defined as
W. = W - Ap. v. - Ap v, - Q
i s, i i e b
(8)
(9)
where Ap^is the work loss during intake,
haust, and Q is the heat loss.
the work loss during ex-
For incompressible flow through the inlet valve, the following relation
can be derived for the pressure loss during intake:
(10)
(11)
1
8
A
1
2
/v \
_P_
c.
I
s
R
Vb
360
e.
i
where the crank angle for intake opening is
e.
Arcos (1 - 2R)
The ratio of piston area to average inlet valve area depends upon the valving
arrangement and schedule. The relationship employed in this model is the one
given for the Thermo Electron expander design in Reference 1. The pressure
loss during exhaust for this design can be expressed as:
Ap.
62
_E .
360
A A
p ei
(12)
The heat loss correlation is
23
-------
K v, k /Vp^p
L0.75
Q = ., £ c K— - (T-T ) (13)
-* tr /A c ' V, U I W
b
i
where K is an empirical constant.
The average working fluid temperature was taken as the arithmetic mean
of the inlet temperature, and the value at the end of the stroke (point b). assum-
ing a straight line -temperature drop during expansion,
T = T. R + 0. 5(1 - R) (T. + T, ) (14)
i i b
The average wall temperature relation is based on Thermo Electron Corpora-
tion test data for an engine operating under 550°F inlet conditions. This re-
lationship,, normalized with inlet temperature, is
T.
T = -r-r (400+ 156 R3) (15)
w 550
Model Development
The reciprocating expander model is entitled ENGINE and is listed in
Volume II of this report. Inputs to the model are:
R, RPM, p., T.. h., s., v.t and pg.
The following geometric and dimensional data must be -supplied:
N , A , S , C.f C , C , A , and A .
p p p i ei es ei es
The model was developed for fixed values of
C. = C = C = 0. 6
i ei ea
(A /A ) = 16.7
p' ei
A A
P gi -
A 'd '
62
which are based on the Thermo Electron expander design. The number of
pistons, stroke, and bore are variable.
If the equations derived above are to be employed, the specific volume
at the end of the stroke (v^) must be known. This is determined through an
iterative procedure. A first estimate of v^ is obtained by assuming an isen-
tropic intake and expansion and employing the relation
vb = v./R (16)
24
-------
Then Equations 10 and 13 are solved for the pressure drop and heat loss. The
pressure and enthalpy after intake are calculated by
pa = p.-Ap. (17)
and
6. IT. - T \
t I i tvr
(18)
hu _
— n.
a i
6.
180
IT.
0 L
«
\T
- T \
w
- T
w'
The fluid property models are then employed to determine va, and a second
approximation on v^ is obtained from
vb = va/R (19)
This is repeated until the successive iterations converge.
The flow through the expander can end in either the superheated or the
saturated region, depending on the working fluid and cut-off valve. The fluid
quality is therefore calculated and the appropriate working fluid property
model is employed.
The outputs from the reciprocating expander model are
J, m, T , h , s , v , x
e e e e e
Other alternative outputs are
Ws' V' "th- Wi' Wshaff
Results
The reciprocating expander model was run with CP-34 as a working fluid
for the following conditions:
R =0. 137
RPM = 2000
p. = 500 psi
T. = 550°F
i
h. =123 Btu/lb
s. = 0.0315 Btu/lb °F
v. =0. 1873 ft3/lb
i '
p =25 psi
N =4
P
A = 15. 3 in.3
P
S = 3 in.
P
25
-------
The results obtained are:
J
•
m
T
e
W
s
m
'th
W.
i
W
sh
m
= 365 ft-lb
= 7301 Ib/hr
= 348°F
= 77 Btu/hr
= 4.05 ft3/lb
= 1
= 48. 3 Btu/lb
= 0.915
= 0.849
= 41. 0 Btu/lb
= 37. 5 Btu/lb
= " 126. 8 psi
The fluid properties at the engine exit, flow rate, efficiencies, and mean
effective pressure are presented in Reference 1 for the same engine design
and working fluid. The model results compare very well with these values.
Figure 4 presents a parametric plot of the model results for overall ex-
pander efficiency (product of T]m and r\^) versus expander speed for several
different intake ratios. The results from Reference 1 are also plotted on
this figure for R = 0. 137. The comparison demonstrates that the model
provides a valid representation of expander performance over this range.
TURBINE EXPANDER
Nomenclature
Alphabetical
Symbols
A Nozzle exit area
e
A Nozzle throat area
C Isentropic spouting velocity
o
Ci Flow velocity relative to turbine blades
D Turbine rotor diameter
r
h. Inlet enthalpy
26
-------
1.0
0.90
0.80
c
. 0)
W
0.70
0.60
•S 0.50
CD
a
x
w
6
0.40
0.30
0.20
0. 10
R = 0.
R = 0. 137
R=0. 09
500 1000
Pi =500 psia
T =550°F
P = 25 psia
05
1=0.2^
R=0. 09
R=0. 137
x = Results from
Reference 1
for R = 0. 137
1500 2000 2500
Expander Rpm
3000
Figure 4. Reciprocating Expander Model -- Efficiency vs Rpm
27
-------
Alphabetical
Symbols
(Cont'd)
h
o
J
m
Md
M
e
M.
i
Pi
Po
R
RPM
T.
i
U
Greek
Symbols
a
Y
AM
AY
n
nd
Exit enthalpy
Torque
Mass flow
Design Mach number
Exit Mach number
Isentropic Mach number
Inlet pressure
Exhaust pressure
Gas constant
Rotational speed
Inlet temperature
Turbine tip speed
Inlet angle
Ratio of specific heats
Enthalpy change for isentropic expansion from p. to p
Design Mach number correction employed in determining
nozzle coefficient
Critical pressure ratio
Specific heat correction employed in determining nozzle
coefficient
Hydraulic efficiency
Nozzle coefficient
Design nozzle coefficient
Rotor coefficient
28
-------
Derivation of Equations
The following equations were derived for an axial impulse turbine expan-
der. The basic equations are the relationships for:
Torque:
Enthalpy change:
(h. - h ) m
i o
RPM
h. - h
i o
(20)
(21)
and the equation for mass flow rate, which is derived from compressible flow
relations.
Mass Flow Rate. The mass flow rate through the nozzle depends on the pres-
sure ratio across the turbine. The following equations apply to a converging
diverging nozzle. The critical pressure ratio is defined as (Ref. 8)
Y/Y-1
(22)
i '
For (po/Pi) greater than Apc the mass flow rate is equal to
A P-
eVR
P:
M
'-1
M
where
Y + l
2'Y-D
(23)
M
For (pe/pi) less than Ap
1/3
(24)
Y+l
2(Y-D
(25)
Hydraulic Efficiency. The hydraulic efficiency can be expressed as (Ref. 9)
UV (26)
COS Q- -
n C
The tip speed is equal to:
RPM D
U
(27)
29
-------
The spouting velocity is equal to:
C = ,/2 Ah ' (28)
O y S
The rotor coefficient is a function of the flow velocity relative to the blades:
Y = f(C)
where
f IV3
C = (CQ sin<*)2 + (CQ cosc*-U)3 (29)
The nozzle coefficient is a function of the specific heat, isentropic Mach num-
ber ratio, and design Mach number. The manner in which the nozzle and rotor
coefficients were determined will be discussed for a particular nozzle below.
Model Development
The turbine model is entitled TURBIN and is listed in Volume II of this
report.
The input to the turbine model is:
p., T., p , and RPM
The following geometric and dimensional data must be supplied:
Y, Dr, At, or. Md, *nd, andR
The main outputs are:
J, m. and h
o
The rotor coefficient relationship employed is that given in Reference 10.
The turbine model was developed for FC-75 as a working fluid with
ftlb
Y = 1.02 and R = 3.72 JiprF
m
The isentropic Mach number at design was M
-------
•a
c
c
=>-
•*-»
0)
0)
o
U
o
O
a
K
Regime 1, Isentropic Subsonic Flow
i Design Mach
5
Design Point Regime 3
Expansion
Regime 4
ormal Shocks Occu
Exit - Regime 2. 3
0.5
0.2 0.5 1.0
Isentropic Mach Number Ratio
Figure 5. Theoretical Nozzle Performance (y = 1.4)
0.2 0.5 1.0
Isentropic Mach Number Ratio
2.0
Figure 6. Effect of Gas Ratio of Specific Heats on Calculated
Nozzle Performance
31
-------
since the selected ratios of specific heats and design Mach number are not
directly represented. A new curve had to be generated for these conditions;
this was done in the following manner.
Figure 6 shows the effect of the ratio of specific heats on the nozzle co-
efficient for a design Mach number of 2. 0. Corrections for y = 1. 02 were
obtained by linear extrapolation from Y = 1.2 at each Mach number ratio. The
specific heat correction will be referred to as AY.
Figure 5 shows the effect of design Mach number on nozzle coefficient for
a ratio of specific heat of 1.4. It was assumed that the effect of the design
Mach number is independent of the ratio of specific heats. The curve for
Mj = 2.5 was considered to be close enough to the actual value of Mj = 2. 626.
Therefore, a second correction, AM^, was obtained from the difference in the
curves for M^ = 2.0 and M^ = 2. 5 in Figure 5 .
Finally, the overall correction to obtain a curve for M
-------
The inlet conditions and turbine speed at design were
p. = 220 psi
T. = 446°F
i
p = 7. 35 psi
RPM = 12,870
The pressure ratio (pt/po) was varied from 5 to 200 at fixed RPM = 12, 870.
The RPM was then varied at fixed pressure ratio. The results, which are
plotted in terms of torque coefficient and velocity ratio, are shown in Fig-
ure 7. The torque coefficient is defined as
C m D
o r
and the velocity ratio is U/CO.
These results were compared with turbine performance calculations pro-
vided by the Barber-Nichols Engineering Company. Barber-Nichols employed
test data from Reference 12 in calibrating their calculation procedure. Fig-
ure 8 shows a comparison of the torque coefficient plotted against velocity ra-
tio for the Barber-Nichols and General Electric calculations. This figure in-
dicates that for a given velocity ratio the Barber-Nichols performance averages
about ten percent higher than that predicted by the General Electric model.
The slope of the curves is also slightly different. It is felt, however, that
these differences are not significant, and the Barber-Nichols- Company recom-
mends that the General Electric model be used for the full-admission turbine.
The reason for the accuracy of the model calculations is the high specific
speed application where the parasitic losses are only about 2$. If a different
working fluid were used where the specific speed is lower, the parasitic losses
would be higher and the model predictions would be less accurate.
FEEDPUMP
NOMENCLATURE
Alphabetical
Symbols
A Piston area
P
A Valve area
v
D Maximum displacement
HP Pump power
33
-------
0. 6 -
0. 5
0.4
o
£
a;
O
O
0)
3
0.2
0.1
0. 1 0.2 0.3
0.4 0. 5 0. 6
Velocity Ratio
0.7 0.8 0. 9
Figure 7. Turbine Model Results (Single-stage Axial Turbine:
FC-75 Working Fluid)
34
-------
0.7
0.6
0.5
c
.2 0.4
o
£
8
o
0)
o
EH
0.3
0.2
0.1
Turbine Model
Barber-Nichols
Calculations
I I I I
I I
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0* 8 0.9
Velocity Ratio
Figure 8. Comparison of Turbine Model with Barber-Nichols
Engineering Company Calculations
35
-------
Alphabetical
Symbols
(Cont'd)
K
m
N
c
NPSH
P.-
s
R
RPM
T.
i
V
v
W
Greek
Symbols
V
Flow coefficient
Mass flow rate
Number of cylinders
Net positive section head
Inlet pressure
Exit pressure
Saturation pressure at inlet temperature
Variable displacement ratio
Rotational speed
Inlet temperature
Average flow velocity through inlet valve
Specific volume
Flow work
Mechanical efficiency
Volumetric efficiency
Pressure drop across inlet valve
DERIVATION OF EQUATIONS
The basic equations for the positive-displacement feedpump model are
the relationships for the following:
Power:
Mass flow:
HP
m
R D RPM r\
m
(32)
(33)
36
-------
Flow work:
W = v(po - p.)
(34)
The volumetric efficiency was derived from data supplied by the Thermo
Electron Corporation for the Hypro-Pump, Model 5420.
TI = 1-0.05
v
The mechanical efficiency was obtained as:
n
RPM
600
m
1/1+ 115/&lv(po-p.)]
(35)
(36)
Pump Cavitation
Cavitation is local vaporization at the pump inlet and can cause severe
pump .damage. The pump will cavitat e if the pressure drop across the inlet
valve is greater than the net positive suction head, defined as
Pi'Ps
NPSH
Using the incompressible pressure drop relation of the form
(37)
K Vs
2v "
(38)
where K is an empirical constant, the following approximate relation can be
derived:
3
K
2v
D
RPM
(39)
MODEL DEVELOPMENT
The feedpump model is entitled PUMP and is described in Volume II of
this report. Inputs to the model are:
R, RPM, p., p , and T.
The following geometric and dimensional data must be supplied:
Nc, D, K, Ap. and Ay
The model was developed for fixed values of
A
37
-------
K = 2.8
A = 1 in.3
P
The number of cylinders and maximum displacement are variables.
Initially, the NPSH and pressure drop across the inlet valve are calcu-
lated and compared to check for cavitation. If cavitation occurs, the model
sets the mass flow at zero. In reality, there is some mass flow during cavi-
tation, but it is very difficult to predict analytically. If the pump does not
cavitale. the mass flow and power are calculated, employing the equations
listed in the previous subsection.
RESULTS
The positive displacement pump model was run with both water and CP-34
as a working fluid. For water, the following input conditions were supplied:
R =1
RPM - 342
p. = 236 psi
p = 1000 psi
o
T. = 217°F
i
N = 1
c
D = 1. 36 in.3
The results obtained are
/p; = 1. 185 psi
NPSH = 7.43
TI = 0.971
v
ilm = 0.891
m = 0. 26 Ib/sec
HP = 1. 04 hp
For CP-34, two cases were run with different inlet temperatures to check
the cavitation calculation. In the first case the input was:
R = 1
RPM = 800
p. =250 in.3
p = 500 in.3
N =5.
c
38
-------
D
T.
= 4.78 in?
= 196°F
and the results were:
AP.
= 3. 26 psi
NPSH = 6. 75 psi
T] =0. 933
v
= 0.794
= 2. 09 Ib/sec
= 4.51
m
m
HP
In the second case the inlet temperature was changed to TI = 210°F and the
pump cavitated
Ap. = 3. 23 psi
NPSH = 2.21 psi
m =0
Figure 9 presents a comparison of the pump model results with experi-
mental data for a Hypro Pump 5530, obtained from Steam Engine Systems,
Incorporated. As can be seen, the model is within 5^ of complete agreement
with the data. It is therefore felt that the model provides a valid representa-
tion of a positive displacement pump.
HEAT EXCHANGERS
NOMENCLATURE
Alphabetical
Symbol s
A
C
Fl. F2, F3, F4
G2.G3.G4
H
h
Flow cross-section area
Energy storage capacity per unit length
Specific heat at constant pressure
Energy-transport parameter (= mass flow rate x
specific heat)
Parameters for dynamic relations
Parameters for quasi-steady relations
Heat transfer coefficient x surface area/unit length
Enthalpy
39
-------
1.6
1.4
1.2
1.0
"c
6
0>
a 0. 8
W>
V
•4-»
K
I 0.6
r-H
0.4
0.2
Rj = 1000 psig
pt = 1. 75 psig
Number of Cylinders, 4
Displacement 0. 441 in?
/
/
•
Note: Data points obtained from
experimentation on Hypro
/
/
/
/
/
y
/
/
Steam Engine Systems Inc.
Pump 5530.
100
200 300 400 500
Pump Rpm
600
700
800
Figure 9. Pump Model -- Volumetric Flow Rate Versus Rpm
40
-------
Alphabetical
Symbols
(Cont'd)
J
M
m
N
P
Q
T
t
u
V
v
X
Subscripts
con
f
fafi
Ma
fat
fg'
g
gf
gat
gs
gt
a
o
r, ref
s
Joule's constant
Captive fluid mass per unit length
Mass flow rate
Required number of iterations for gas-side energy tran-
sient
Pressure
Energy transfer rate, Btu/sec, per unit length
Temperature
Time
Internal energy
System volume
Specific volume for fluid
Flow direction
Constant
Fluid
Between f2 and fl nodes
At f2 node
Between f2 and t nodes
Saturated flu id/saturated vapor interphase
Gas
Saturated vapor/saturated fluid interphase
Between g2 and gl nodes
Between g2 and t nodes
Saturated vapor/superheated vapor .interphase
Between gas and tube
Subcooled liquid phase
At design condition
Reference condition
Superheated vapor
41
-------
Subscripts
(Cont'd)
sg Superheated vapor/saturated vapor interphase
st Stability limit
t Tube metal
tf Between tube and fluid
tg Between tube and gas
tt At tube node
1, 2 Node locations (2 at upstream location)
Greek
Symbols
At Time increment
Ax Distance step
- p Density
(Note: A dot (•) above any quantity denotes its time-derivative (d/dt))
TRANSIENT THERMAL ANALYSIS
Dynamic models of the heat exchanger consider the conservation of mass,
momentum, and energy .simultaneously on a time-dependent basis. This re-
quirement increases the model complexity, even for simple geometric config-
urations. In most practical cases, a number of simplifying assumptions are
needed to obtain manageable results.
The equations describing the physical processes must be valid over a
wide range of off-design operating conditions. This means that the lineari-
zation of process equation about the design point may not be acceptable.
The final selection of an appropriate transient model will depend upon
these and other factors described in the following pages.
Methods of Transient Analysis
For any dynamic system, the dynamic relations describing the physical proc-
esses with its boundary conditions are written first. In cases where lumped
parameter representation is adequate, the resulting relations are ordinary dif-
ferential equations. For long tubular thermal components such as vapor gen-
erators, regenerators, or condensers, both the time and distance are variable
parameters and the resulting relations are partial differential equations.
Several methods are available for solving these equations (Refs. 13-23)
and are listed in Table 3. The choice depends upon the end use and scope of
42
-------
Table 3
METHODS OF TRANSIENT THERMAL ANALYSIS
Method
Exact Solution
Method
Basis
Solution by standard
solution techniques
for differential
equations
Laplace Trans-
form Method
Transformation in
time and space
variables, then in-
verse transforma-
tion
Comments References
a. Standard solution 17, 22
available for lim-
ited cases only
b. Solutions for equa-
tions with variable-
coefficients gener-
ally not available or
extremely complex
a. .Inverse transform 14, 17,
generally very 21, 22
complex;can be
done through numer-
cal techniques
b. Variable coefficients
not admissible
State Variable
Method
Distributed system
solution technique
of Brown
Same as for La-
place Transform
Method
13, 16
Analog Method
Use of analog com-
puter
Finite-differ-
ence Digital
Method
Dusinberre method
used on digital com-
puter
a. Partial differential
equations are re-
quired to be approx-
imated as total dif-
ferential equations
b. Analog computer
size limits the prob-
lem scope
c. Variable coefficients
are admissible
though might not
prove practical
a. Lumping errors
exist but can be
made practically
insignificant
b. Numerical instabil-
ity should be
watched
c. Computing time
and memory size
depend on the selec-
tion of time and'
distance lumps
d. Can readily admit
variable coeffici-
ents
e. Applicable to any
complex geometry
or design conditions
17, 18,
19, 20.
23
15, 17
43
-------
the model. In the present case of the Rankine cycle simulation, the method
should have the ability to:
a. Accept any design changes with minimum change in program
b. Predict dynamic behavior over a much wider range, even from cold
start-up condition to full-load design operation
c. Incorporate nonlinearities and any variation of physical properties
reflected in the variable nature of the coefficients of the differen-
tial equations
d. Handle variable operating conditions (e. g. , a condenser with variable
superheat inlet condition and existing in a subcooled state, with the
location of the two-phase boundaries depending on the operating con-
ditions)
In addition to the above characteristics, the selected method should also
be:
e. Capable of giving results of acceptable accuracy'
f. Economical in run-time requirement
g. Reasonable in machine size requirement
Beyond these considerations, any other characteristics of the method do
not really pose any limitation on its usefulness. Thus, if an elegant closed
solution method or a simple numerical method are weighed almost equally on
the above considerations, neither possesses any superiority above the other
method.
Closed-form Solution Methods. The set of conservation equations are
solved in a closed-form method, satisfying appropriate boundary conditions
for each section of the heat exchanger. The first three approaches given in
Table 3 fall into this category. This method would yield explicit relations for
variation in fluid pressure, enthalpy, and mass flow for any given input dis-
turbance. Some of the customary assumptions required in this method are:
• Small perturbations (hence linear range) about any operating point
• Linear variation in fluid properties (equations-of-state linearized)
• Constant or uniform heat input rates and heat transfer coefficients
• Frictionless fluids
Some of these assumptions might be unduly restrictive. Further, a
simple change in geometric layout may void the entire solution applicable to
a previous geometry. Even then, though the final relations are in closed
form, they are usually quite complex even for a simple geometry with a
linearized range, and may require approximate numerical methods for their
evaluation (Ref. 13-17).
44
-------
Analog and hybrid methods are presented in Table 3 for completeness
but were not considered for this simulation.
Finite-difference Method. The finite-difference digital method is a simple
and most effective method, and it meets all the essential criteria with a sur-
prising simplicity and brevity. Its drawbacks -- possible numerical instabil-
ity, run time, and memory size requirements -- can be overcome. Dusin-
berre (Ref. 15) outlines a simple way of avoiding any numerical instability,
and the resulting limits on time and distance increments are neither unrea-
sonably restrictive nor overly demanding in machine-size and computing-
time requirements.
In this method, the differential equations are approximated by finite
difference equations. In Dusinberre's method of explicit solutions, the
difference equation with respect to time (which represents a time derivative)
is evaluated by using the present (or known) values of all required parameters.
This method, which does not require any of the assumptions mentioned above,
was selected for the simulation approach. Some of the very powerful features
of this method are briefly mentioned in Table 4. It will be shown later that
the method differs very little when applied to heat transfer equipments with
such diverse geometrical and functional characteristics as the vapor genera-
tor, the regenerator, or the condenser. Further, computer memory size
requirement can be significantly reduced through special programming
methods.
Table 4
FEATURES OF FINITE-DIFFERENCE DIGITAL METHOD
Flexibility • 1. Not limited to any arbitrary range of operation
2. Complex flow phenomena (e. g. , superheat,
condensing, and subcooling existing in one
pass) can be handled.
3. Stiff system (with a wide range of heat capac-
ities) can be identified, simplified (by ignor-
ing negligible heat capacities), and the pro-
gram suitably modified through very minor
changes to achieve run-time economy.
Adaptability 1. Can be used for any complex geometry (e. g. ,
counterflow, parallel flow, cross flow,
multipass) with equal ease.
2. Variable properties can be handled with the
same program.
3. Other process features -- conservation of
mass and momentum -- can be readily in-
cluded.
45
-------
Table 4 (Cont'd)
Versatility 1. Same dynamic program can be used for com-
puting the initial steady-state condition. Even
a crude guess on initial distribution is ac-
ceptable.
Basic Approach
In any transient thermal process involving fluid flow, all the physical con-
servation laws -- conservation of mass, momentum, and energy -- should be
satisfied simultaneously. However, if the primary interest is^in the dynamics
of thermal phenomena rather than the details of fluid dynamic phenomena, the
problem is simplified. This is because the flow disturbances propagate rap-
idly (at the speed of sound) compared to propagation of thermal disturbances;
hence, the fluid phenomena such as fluid inertia or liquid-phase compliance
are of secondary importance and can be neglected.
Consider the case of a fluid flowing through the tube section (Figure 10).
Assume:
1. Flow is one-dimensional.
2. Fluid inertia is neglected.
3. Geometry is uniform within a section.
4. Thermal conductivity of tube and fluid is infinite in the radial direc-
tion and zero in the longitudinal direction.
5. Fluid is radially homogeneous, and the relative velocity between liq-
uid and vapor phases is neglected.
6. Fluid pressure is uniform within a section and time increment.
7. There are no internal heat sources or heat sinks.
8. Fluid and wall properties are constant within a section and time
increment.
m
'//////////////
Figure 10. Schematic of Flow Through a Tube
46
-------
The only essential assumption is one-dimensional flow. The other assump-
tions are neither restrictive nor essential; they merely allow a simple treat-
ment of the problem.
Basic Process Equations. For the above case, the conservation equations are
Mass Balance:
^ = A ?5T (40)
OX V*. O t
Force Balance:
oP
f
•r— = f (friction and momentum (41)
pressure drop)
Qtf = Hf (Tt - Tf) (42)
Energy Balance and Heat Transfer:
= Hf
o(mh.)
Q = _! + A £ x (43)
Tube Heat Capacity:
axt
Qgt - Qtf = c -5T <44)
Equations of State:
v = f(P T )
1 (45)
hf = f(P Tf)
Application of Finite-difference Method to Energy Equation. As an example
of the application of the finite-difference method, consider a tube of length
Ax. This tube section satisfies the assumptions given earlier.
*The general energy balance relation is
- ("h) • (MU)
v/here M is the captive fluid mass per unit length ( = pA = A/v) and u is inter-
nal energy (=h-PV). Substituting these definitions and using assumption 3,
which reduces to Equation 43 when assumption 6 is used.
47
-------
Expanding Equation 43 and combining with Equation 40 yields
Q*
. f
tf
• A 9hf
m — + .— —
dx v d t
(46)
Assume that the tube section refers to an economizer. Then take v^ = vo
as average fluid specific volume. In combination with Equation 42
H
:-Tf>
m
A
(47)
Before the finite-difference approximation is applied to Equation 47, it
should be modified to base the relation in terms of either fluid temperature
or fluid enthalpy. While in the case of an economizer or superheater the choice
is not crucial, the boiling or condensing section can be represented with the
enthalpy terms only. This is a crucial observation.
To achieve this transformation, define
-------
Now, following Dusinberre's method, the forward-finite-difference approx-
imation of the differential terms of Equation 49 is
ah. h_ (t) - h, (t)
f _ fa fi
dx Ax
and
"f
at
h.
hf(t+At) - hf(t)
_fs fa
At
hfs(t)
(50)
Equation 49 now becomes
h (t+At) = Fl ' hfa(t)+ F2 ' h (t)+ F3 T(t) + F4
(51)
where
Cf Ax 2 Cf
Ax C
2 E,
f At At i
F c^ "AX " T c^
Hf
F3 = At • — ' c
At'f£
Ax C,
2 E,
Hf
F4 = At • — (h - c ' T ,)
Cf ref p ref
Equation 51 computes the fluid enthalpy at the exit node at a future time,
based on the present fluid enthalpy and tube temperature distribution; hence,
this method is called explicit. Note also that all the coefficients associated
with the distribution terms add up to a value of unity; thus the future enthalpy
is a weighted sum of the present values of the surrounding elements.
An important feature of the explicit finite-difference approximation is
that the lump size Ax and time step At should be selected for numerical sta-
bility. The upper and lower limits on Ax and At are required to minimize
errors associated with the conversion of differential to difference equations.
A simple rule to avoid the numerical instability requires that all F coeffici-
ents associated with the present distribution terms should be non-negative.
A mathematical basis for this rule is given in Appendix II, "Stability and
Error Criteria for Finite-difference Solution of Partial Differential Equations.
49
-------
Applying the stability criteria for F2:
0 < 1 - -TT —
i. e. , Ax <
for F3:
0 <
(52)
. At Er / AX Hn
TE)
Cf/Hf
2 E
Ax H
f J
(53)
Note that
1. These stability limits are time-dependent, since they require heat
transfer coefficient and mass flow rate at a given location and time.
2. These limits are from the energy equation for the fluid only. Sim-
ilar limits would arise from the conservation equations applied to
other subsystems of the equipment, such as the tube, the outer fluid,
and the shell.
3. The selected values of At and Ax should consider all the stability
limits for the system, and also external constraints such as geo-
metrical layout, maximum system time increment, etc.
4. While the stability criterion prefers smaller lump and time-step
sizes, the requirements of simulation on a digital computer are that
• Smaller lump and time-step sizes are associated with increased
memory size and run-time requirements
• The truncation errors grow in proportion to the number of lumps
and time steps.
Hence, the final choice of At and Ax should reflect all of the above
considerations.
Small Time Constant (High Frequency) Situations. In certain dynamic situ-
ations involving more than one energy subsystem (e. g. , a tube subjected
to internal and external fluid flows) conditions could arise when the energy-
storage-capacity to heat-transfer-coefficient ratio for some subsystems is
very much smaller than that for the remaining subsystems. In the mathe-
matical representation, this would require a much lower limit on allowable
time step, as can be seen from Equation 53.
50
-------
In physical terms, such a small quantity results from the fact that the
corresponding capacity to film-coefficient ratio is small; i. e. , the thermal
capacity of the fluid or metal is very small or its surface heat-transfer re-
sistance is negligible. In either case, the fluid can be treated as capable of
assuming its steady state at a much faster rate; hence, the steady-state energy
relations can be assumed applicable at any time during the transient operation
of the equipment, without any appreciable error. This procedure effectively
eliminates the term representing a fast transient -- a high-frequency term,
and allows a reasonable value of the time step At based on remaining terms.
The previous dynamic equation, 51, is here recast for its steady-state
representation. The quasi-steady behavior implies that
that is, h (t + At) = h. (t)
fa 12
Hence,
hf G2 h, + G3 T + G4 (54)
fa - fi t
where
G2 = F2/(l - Fl)
G3 = F3/(l - Fl)
G4 = F4/(l - Fl)
or
2 E
G2 = -i-
Ef
.
G3 = - - - r— ±- (55)
1 + **^
2 E
* f
Hf
At ' E; ' (href - CPf ' Tref)
G4 = - - 7~~^ -
!+**-£
2 Ef
51
-------
The stability limit on Ax associated with Equation 54 is
2 E.
Ax
f
H,
(56)
This limit is identical to that in Equation 52 for the dynamic relation. Of
course, the time-step size does not enter into the picture in this case.
VAPOR GENERATOR
The finite-difference approach outlined earlier will now be applied to simu-
late the dynamic behavior of a specific class of vapor generators. In the auto-
motive application, a once-through monotube vapor generator has significant
advantages. In the majority of such units, the hot-gas flow is arranged so
that, overall, it resembles cross-flow arrangement. The mathematical model
developed in this section refers to this specific type of unit.
General Description of Model
The overall system simulation requires a certain input-output arrange-
ment for each system component. For each component, the input-output vari-
able^ (also called information signals) indicate a preferred representation of
the dynamic situation, and do not signify the flow directions.
The complete once-through vapor generator treated as a single unit has
input-output conditions as shown in Figure 12. Note that the direction of the
signals at each end always satisfies one primary energy requirement of the
two flow quantities, pressure and mass flow rate: one is an output quantity,
and the other is an input quantity.
Note that the signal-flow arrangement in Figure 12 is selected for con-
venience in overall system simulation. If the dynamic simulation of vapor
generator alone is desired, it would be necessary to switch the signal-flow
ti
is
t3
mlf hl
(From Feed Pump)
PI
(Inlet Pressure)
i .1 i ill 111
Economizer | Evaporator I Superheater
Tl T 1 ! T ITT
(Boiler Outflow)
(Output Conditions)
Figure 12. Information Signals for Vapor Generator Model
52
-------
directions at the superheater outlet: the exit pressure would be an input signal,
and the exit mass flow would be suitably governed by the difference in boiler
pressure and exit pressure.
While Figure 12 indicates the existence of three distinct fluid zones, the
boundaries separating them need not remain at a fixed location. During the
transient operation, the last fluid zone may disappear. (Because this situ-
ation is not desirable, even transiently, the control system acts to prevent
this. ) The mathematical model should be such that it remains valid under
these dynamic conditions.
In the present approach, the decision as to the state of the" fluid (subcooled,
two-phase or superheated) is based on the average values of fluid enthalpy and
pressure at any time within a section of the tube (called a lump); the proper
relations are then used to compute the transient behavior of the fluid within
the lump. The exit fluid conditions of any lump are then transmitted as en-
trance fluid conditions for the next lump.
In the following subsection, the relations are developed for three distinct
fluid states. Note, again, that these do not necessarily coincide with the geo-
metric zones derived from a steady-state basis, but change continuously during
transient operation.
Transient Analysis
Conservation Equations for the Fluid. The basic conservation equations given
above are now applied to different fluid states.
For Subcooled Fluid: Assume that the fluid compressibility is negligible.
Then,
9v Sv
vf = constant (=VQ), — = 0, — = 0.
Consequently,
dm
f = 0 (57)
tf
For two-phase or superheated fluid:
dx
Bpf
-r— = f (friction) (53)
• f A f
Q = m-- + - (59)
t <60)
53
-------
-j:— = f (friction and momentum pressure
drop) for two-phase
= f (friction) for superheat (61)
a(mfhf) a(hf/Vf)
^tf ~ dx at
ah amf . ah Ah 3v
* -l-l-i 4- if
mf "aT f ~ax~ v"f ~aT ~vj aT x
Combining this relation with Equation 60 gives
Qtf = ™f -&T + t IT (62)
Conservation Equations fortheTube. The major dynamic effect from the
tube is that it retains a portion of the energy received from the hot fluid
stream, and transmits the remaining energy to the cold fluid stream. In
the absence of axial thermal conductivity of the tube material, the storage
capacity of tube metal can be considered on a lumped-parameter basis.
Hence,
Q - Q = C -—• (63)
Q = h ' A (T - T ) (64)
«tr ' V Atf(VTf> (65>
where Tg, T^, and Tf are bulk temperatures for the gas, tube, and fluid re-
spectively; Agt and A^f are heat transfer surfaces per unit length for the gas
side and fluid side respectively.
Conservation Equations for the Gas. It is assumed here that the total mass
flow rate of the gas is constant in both the time and space coordinates. Then
a-p
-r-^- = f (friction) (66)
O Xg
Q 4 = m -r-- + H -r (67)
gt g .Sx3 \v0jg at
Finite-difference Approximation of Energy Equations. The method already
outlined for transforming energy equations is applied to Equations 59, 62,
63, and 67, transforming them into equivalent explicit finite-difference rep-
resentation. Figure 13 defines the node pattern.
54
-------
Gas (m )
g
s s/s *
Fluid.
S / S S S / S
fl
S / S / SSS S
SS // S / / / /[
/ /// / /
Figure 13. Node Pattern
Fluid. For the two-phase fluid, the fluid temperature Tf = Tsa^, where-
as for the subcooled or superheated condition, the fluid temperature is calcu-
lated from the equation-of-state relation corresponding to the. bulk fluid en-
thalpy and pressure within a lump.
Thus from "the definition,
hf-hp
Pj.
(T, - T )
fr
(68)
where hr and Tr are fluid enthalpy and temperature respectively at a refer-
ence point, and Cpr is the mean specific heat over the range. For subcooled
fluid, Cp is almost constant; therefore the reference point is taken at the sat-
urated liquid condition. However, in the superheat case, cp varies signifi-
cantly with pressure and enthalpy; therefore, the reference enthalpy varies for
for each lump and is taken at a slightly lower value from the bulk fluid enthalpy
for that lump.
For the transient case,
fg
At) = F. f h (t) + F. , h (t) + F, .(T, - T )
fsfi fi fsfs fa fat t . x
and for the quasi-steady case,
hfa(t+At>. , h (t) - Ff
-------
where
T = T for two-phase fluid
x sat
i h \ \ <7D
/ r \
T = IT for subcooled or superheated
v I r c I
•" I A %•» I fm t j
\ Pr/ fluid
and the F coefficients are defined in Table 5. The quantities used in the
table are
and m = fluid mass flow rate at the entrance of the lump.
Tube. Since the energy transmitted to the fluid side depends on the fluid
temperature, the transient relation for tube material depends on the fluid phase.
Thus, for subcooled or superheated fluid,
Tt(t+At) = Fu Tt(t) + Ftg Tgi(t) + Tga(t)
(73)
F., h. (t)+ h, (t) + F
tf fi f3 I con
and for two-phase fluid,
Tt (t + At) = Ftt Tt(t) + F
T. (t)+ T (t) + F ' T__ (74)
gi ' g2"' *tf sat
where the F coefficients are as given in Table 5. These relations are valid
for both transient and quasi-steady situations, with proper values of F coefficients.
Gas. The energy relations are expressed here in terms of gas tempera-
ture rather than its enthalpy. Since the change-of-phase condition does not exist
here, this choice is valid. Further, the quasi-steady assumption is accepted
here, to avoid extremely small iteration time steps resulting from stability
considerations. Thus,
= Vt} = *t)+ FT(t) (75)
The F coefficients are explained in Table 5.
Stability Criteria for Energy Equations. One important feature of the ex-
plicit method is the upper limits for the lump size. Ax, and the time step,
At, necessary to avoid numerical instability. Both these limits apply to any
transient equation, whereas only the lump size limit applies to the quasi-steady
relations. These limits are obtained by requiring that all the F coefficients
associated with transient terms be non-negative and less than unity (see
Table 6).
56
-------
Table 5
FINITE-DIFFERENCE RELATIONS
(I) Fluid: Transient Case
Fluid Phase
Subcooled
or Superheat
2-phase
Ff2fl
fit Ef / Htf Ax
Ax ' Cf \ Ef ' 2
At f m
Ax Ap
Ff2f2
1 6t §1 fi + 1111 £*
ax ' Cf [' Ef ' 2
At m
AX AP
Ff2t
... V^
«-fc
(II) Fluid: Quasi-steady
Subcooled
or Superheat
2-phase
1 - B
1 +B
,,,h.-r-r n - ^ ' Htf
2 cpr m
1
0
0
28
1 +B
where n - "x Htf
2 Cpr m
Htf. Ax
Ai
(III) Tube: Transient
Fluid Phase
Subcooled
or Superheat
2-phase
Ftt
Ct ^
1 - f± (Htf + Hgt)
Ftg
At Hst
2 ' Ct
At Hat
-• -cT
Ftf
t~i Htf
2 cpr ' Ct
"•S1
F
con
Af U. T hr
1 • 'rT" Tr " c
CV CPr
0
(IV) Tube: Quasi-steady
Fluid Phase
Subcooled
of Superheat
Ftt
0
o
Ftg
H?t
2 (Hgt + Htf)
Het
2 (Hgt + Htf)
Ftf
«tr
2 Cpf (Hgt + Htf)
Htf
(Hgt + Htf)
Fcon
Htf „ hr \
(Hgt + Htf) • ' cpr|
ft
(V) Gas: Quasi-steady
1 - B
g2t - 1 - Fg2gl
where B =
57
-------
Table 6
STABILITY LIMITS FOR ENERGY EQUATIONS
(I) Fluid
2 E,
Subcooled or Superheat
Ax
At
H
tf
Cf/H
tf
2 E.
1 +
H
tf
Ax
Two-phase
At
m
(II) Tube
For any fluid phase
At
(Hgt + V
(III) Gas
No. of iterations in gas flow
direction
H
N
SL
Ax
2m c
g Pg
Note: In the two-phase situation, enthalpy change occurs
at constant temperature; hence, Ef -* °°. Consequently there
is no fluid stability limit on Ax, and an external limit should
be specified for accuracy.
Note that the stability limit on the gas side cannot impose a limit on Ax;
hence, the heat transfer area on the gas side should be subdivided to meet the
stability limit. Thus, if .
Surface area permitted by stability limit
Required number of iterations
N
then
B
h A A . Ax
gt st
2m c
58
-------
that is
2m Cp
Aqt* ,g Ag (76)
st h Ax
g*
But the total surface area on the gas side between the two gas nodes is A^.
Hence,
N =
Ast
that is h A Ax
N
2 m c
g p
S H
H tAx
N * rfrsr (77)
g Pg
Note that the. quantities on the right-hand side of Equation 76 are not affected by
by the subdivision of distance between two gas nodes. Hence, local iteration
is permissible.
Finite -difference Approximation of Continuity Equation. In this section, the ex-
plicit finite-difference form for continuity relation for two-phase or super-
heat fluid. Equation 60, is developed. It is assumed that the fluid pressure,
P, is uniform within a lump.
Writing Equation 60 in terms of density, p,
m2 - m, = -A • Ax • —
o t
But P = f (hf> P)
that is
3_P = 9_P
at dh
dh
f
P dt 3P
dp
hf dt
For simplicity, assume the pressure, P, as constant during the time in-
terval At. Then
,
ma-m1 = 'A -p" AX' ^T (78)
Equation 78 contains a time-derivative term for fluid enthalpy. To elim-
inate numerical error arising from the "high-gain" characteristics of the
59
-------
derivative term, Equation 78 is combined with the corresponding energy equa-
tion for the fluid.
Thus, taking the lump size for the continuity equation the same as that
for the energy equation,
ma-
A
-A —
lo n
hf(x+Ax, t+At) - h (x+Ax, t)
At
(79)
Note that the enthalpy change, Ahf, taken at the exit node, is consistent
with the assumption used in the finite-difference approximation of the energy
equation.
For Two-phase Flow
Ahfg = hf (x+Ax, t+At) - hf (x + Ax, t)
H.,.
mL At
Ap Ax
' At (Tt-T..t>
Substituting in Equation 79 and rearranging,
where
m2 =
1 -
+ A
(80)
(81)
(82)
(83)
For Superheated Fluid. The corresponding energy relation can be used
to derive similar mass flow relation for superheated fluid. Thus,
where
As nii + As
I» (n ~ n )
O P 1 f 1 f2
ah |p o
P PC
Pr
• . - -- r fT . T ^
2 Pr-Ut V
& ~ r t x
(84)
(85)
(86)
T = T - h / c
x r r' Pi
(87)
60
-------
Finite-difference Approximation of Momentum Equation. In the present ap-
proach, the spatial pressure distribution due to momentum relation has been
neglected during the transient calculations. This assumption would be accept-
able if the momentum pressure drop is not a very significant portion of the
system operating pressure level. The simplifications offered by this assump-
tion are significant; the major advantage is the absence of additional restric-
tions on time and lump step sizes.
Hellman and others (Ref. 24) have treated the problem of finite-difference
analysis as applied to the momentum equation. In a special case of a hori-
zontal tube with natural and forced convection, the stability criterion for the
time step is shown to be
• — ~ <88>
k! + 3k3m)
where g = acceleration, arid ka and k2 are associated with the friction-drop
relations as
Ap = f (m) = kjml+ksm3 (89)
It can be seen that this limitation is nonexistent under the assumptions
of this problem.
Pressure Transient -- Lumped-parameter Model. Earlier, explicit finite-
difference relations were developed for the enthalpy (or temperature) and flow
distribution for various flow subsystems. It was assumed in the derivation
that the fluid pressure remained constant during the integration time interval,
At. The difference between the inlet and exit mass flows would exist during
the dynamic operation; a difference would also exist between ene'rgy available
from the hot gases and that removed or absorbed by the fluid. The net re-
sult would be a fluid pressure variation.
Transient thermodynamics will be used to derive the relations for the
fluid pressure. The analysis will be based on a lumped parameter model
(this is acceptable because spatial variation of pressure (due to momentum
effects) has been ruled out during the transient operation). The derivation
is based on the extension of Brown's pioneering work in transient thermo-
dynamics (Ref. 25).
The total vapor generator volume is taken as a single unit; it is divided
into several hypothetical sections bounded by the interphases representing
the fluid change-of-phase (Figure 14). For each section, the interphases and
the system, the conservation equations are written which yield the pressure
variation information. Note that all the mass flow directions into the vapor
generator are assumed positive. The sign for the exit flow in actual compu-
tation should be watched.
61
-------
Subcooled Saturated Saturated
Liquid Liquid Vapor
Superheated
Vapor
mf
h,
/ ! / j! ^
» iTij
/ M /
/ |l ^
"•«*t|~*' ^
t v »
ms
hs
Qfg
Q
gs
Qs
(90)
Figure 14. Schematic of Phases and Interphases
Mass Balance. Assume the liquid is incompressible.
Superheated vapor:
Saturated vapor:
Saturated liquid:
Subcooled liquid:
1 A
Liquid interphase: 0 = m „ + mf_ \ (91)
Vapor interphase:
System: M + M + M
*» s
Volume Balance. Assume 1) the change of Subcooled liquid volume is
zero,and 2) the interphases have no volume.
M
s
M
g
Mf =
0 =
0 =
0 =
M
s
m
gs
m.
mf +
mf -
"gf
m
gs
>"V^ I
+ m
s
+ m
"gf
mf
+ m
+ m
sg
m
s
Superheated Vapor:
dv
M v
s s
M v + M
But vs
that is,
f (P,hs)
dv
s_
dt
s s s dt
s i dP + s
dh
Hence,
M v
s s
Sp 'hs dt 8hg p dt
B! P + B2h
M v + M [BiP + B2h ]
s s s 1 S
(92-
> Cont'd)
62
-------
Saturated Vapor:
M v
g g
Saturated liquid:
M v + M
g g g dp
dv.
—— = M v + M, -7-
Mfv f f f dp
System:
V = 0 =
M v
g g
+ M v
I s s
Mfvf
(92)
It should be noted that in each case the system equation, obtained directly,
agrees with the result obtained from combining all section relations together.
Now, combining mass and volume balance relations,
(mfg " "V Vg +
dv
+ I M —g-
dv
.dh
, + B! M + M.-— IP + B3M —jf = 0
g dp 1 s f dp | s dt
After rearranging, the equation becomes
dv dv dv
- |M -r* + M -~ , + Mr-Ti- IP
g dp s dp 'hs f dp
(m + m ) v + m (v - vj + v m +
f s f s s f fg I f g v
v - v \
s 2
m
o
gs
dh
M
s dh p dt
(93)
This equation can be used to calculate pressure transients if the terms rhfg and
m.gs can be eliminated. This will be done by using energy relations.
Energy Balance. Assume that the change of energy stored in subcooled
fluid can be neglected because of its small order of magnitude.
Superheated vapor:
Q + h m + h m
s s gs s s
M h - M v P/J
s s / s s
dh
M -rp + h M - M v P/J
s dt s s s s
63
-------
Combining with the corresponding mass balance relation,
dh v i
Q Q *
M l-rf --f P
s dt J
Saturated vapor:
Q + h m. + h m
g g fg g sg
M h - M v P/J
g g g g
That is,
Q
g
/dh v \ .
M —^ - — P
g Up j r
Saturated liquid:
Q
That is,
hfmgf = Mfhf ' Mfvfp/J
d^ v,
f I dp J
Subcooled liquid;
That is,
Interphases:
and
That is.
- hfmf
*
Q, + h, m, + h m , = 0
fg f fg g gf
Q +hm +hm =0
gs g gs s sg
Q
Q
.
fg
gs
m. h.
fg fg
m (h - h )
gs s g
System:
S Q + h- m. + h m
j? f s s
M.h. +Mh +Mh - — P
f f g g s s J
That Is,
, h -h
E Q - m. (h - h.) - h, m, + s. g m
f f A fg 1 fg h, gs
M
dh
s
s dt
M
fg
dh dh
M
f V
g dp -[ dp J
(94)
(95)
(96)
(97)
(98)
(99)
64
-------
Equations 93 and 99 should now be combined to derive an explicit rela-
tion fordP/dt. For simplicity, assume
v - v
s g
h - h
(100)
Note that both of the terms above have the same signs. Though the actual
agreement between the two quantities depends on the fluid and the operating
point in superheat zone, the error for CP-34 fluid is in the range of 10 to 30$
(Figure 15). This approximation can be relaxed as in Reference 25.
30
25
20
15
10
0
10
x 100
I
20 30
(hs - hg)
40
50
Figure 15. Error in Approximation (v "vj/v =(h~hVh for CP-34
Combining Equations 93, 99, and 100 and using the proper sign conven-
tion for m
s»
dP_
dt
/ * __
(m - m
v
fg
s)vf Ms
r dh
MS? 4-
- j '
S dP
dv
S
dh
c
M_
Ils
p \
ihf v]
dp J
an v, r T
s f g • •
dt h " mf f S. ms^ s f
r dvr dv dv 1
f a K '
i\ff~4-T\/r-S4-i\/r P
1V1 _ ~, T 1V1 , T ivl * J
f dp g dp s 5p h 1
(101)
65
-------
Analytic Procedure
For economic reasons, it might be necessary to use a quasi-steady rep-
resentation for some of the energy-storage components of a heat exchanger
unit. This is usually done at the price of eliminating some fast transients,
but it does not affect the model accuracy. The decision as to whether a quasi-
steady or a dynamic model is necessary for a particular component is not
arbitrary; it is related to the geometry, the thermal properties, and the
energy-storage capacity of surrounding streams. The selection for the par-
ticular vapor generator considered here (TECO design for CP-34 fluid, Ref. 1)
is given in Table 7.
An example of the interpretation of Table 7 would be the case of working
fluid in the boiling phase. The corresponding modes of energy equations to
be used are: quasi-steady relations for combustion gas and tube metal, and
dynamic relation for the fluid itself.
Table 7
SELECTION OF ENERGY EQUATION MODELS
Working
Fluid Phase
Subcooled
Boiling
Superheated
Combustion
Gas
Quasi-steady
Quasi-steady
Quasi-steady
Tube
Dynamic
Quasi-steady
Dynamic
Fluid
Dynamic
Dynamic
Quasi-steady
The computation procedure for the case of fluid is now explained to illus-
trate the basis of the computer model. At any given time, (t), the fluid en-
thalpy and tube temperature distribution are known. The mass flow, m, is
taken at the entrance node. From this and other known geometric and opera-
ting parameters, the dynamic parameters Ef, Cf, Hf-f, etc. , are computed
for the given lump sizes at time t. On the basis of the average fluid condi-
tions within the lump, its phase is determined. The proper stability criterion
is then applied to obtain Ax. Similarly, Ax corresponding to other energy-
subsystems (tube, gas if it is counter or parallel flow) should be calculated,
and the minimum of thes.e limiting lengths should be selected as the final lump
size. If the selected lump size is different from the original lump size the
parameter H^f is obtained for a new node pattern by linear interpolation.
The allowable time step At is calculated next. It is taken as the mini-
mum of those dictated by the stability limits for each energy subsystem. The
energy relation is then used to obtain the fluid enthalpy at the exit node at
time ( t+ At).
For each lump, the continuity equation is then used to obtain mass flow
rate at exit node. Since the fluid transients propagate very fast compared to
66
-------
thermal transients, the mass flow relation is considered an instantaneous
process, and the calculated exit flow value is used in the energy calculation
for the next lump.
The exit conditions of the previous lump are set as the inlet conditions
for the next lump, and the calculations are repeated until the vapor genera-
tor exit node is reached.
The fluid pressure change is next calculated by the use of Equation 101.
This completes the transient solution, giving the fluid-state distribution at
time (t + At).
For the next iteration, the distribution at (t + At) is now reset as the dis-
tribution at (t), and the process is repeated to obtain the transient distribution
at the next time.
At any time, the momentum relations can be used to obtain axial fluid
pressure distribution.
Computer Model
The vapor generator model is entitled VAPORG, and is listed in Volume II,
the Users Manual. The important features of this computer program are de-
scribed here. The program is designed to require minimal geometric and de-
sign data. To summarize, it first calculates a steady-state distribution of fluid
enthalpy, gas temperature, and tube-wall temperature; it then continues to cal-
culate transient behavior for a specified time period as a result of the specified
disturbance at the boundary. The details are given below.
One important convention used in the program should be stressed. Each
heat exchanger unit is subdivided into various fluid passes. Each fluid pass has
uniform geometry and is subjected to uniform flow conditions. The fluid
passes are consecutively numbered in the direction of fluid flowing through
the tube. Figure 16 illustrates various situations.
Using the basic geometric data, the program first calculates all the quan-
tities of interest (e. g., hydraulic diameter, flow cross-section area, etc. for
each fluid pass). Since-such quantities are time-independent, this calculation
section is bypassed during subsequent calculation loops.
To start the program, initial distribution of the fluid enthalpy and gas
temperature is required. The program is designed to accept estimated values
of these quantities at the inlet and exit of each fluid pass; a linear interpola-
tion is used for distribution within a fluid pass. Initially, each fluid pass has
only one lump. For each lump, a first guess on the tube temperature is ob-
tained. (From this point on, the discussion is not limited to one lump per coil,
since the calculation would restart here for each iteration step. )
67
-------
Fluid
Gas
Gas
N^
: *
1 — *
V4)
5 »
T (5)
g» ^
D
I'T fv
JT (o
D g3
D *•
^
v ™
T
<§)
1
H—1
T (5)
g»
Note: (Y)=Fluid
Pass Number 1
Fluid Pass Numbering for a Cross-flow Heat Exchanger
Fluid
V"
T (3)
S1
T (2)
S3
T (2) Gas
Fluid Pass Numbering for a Cross-flow Heat Exchanger
with a Different Fluid Flow Path
Figure 1 6. Fluid Pass Numbering Process
68
-------
For each lump, the average conditions are used to calculate the fluid- and
gas-side heat transfer coefficients and, hence, transient parameters Htf, Hgt,
d, Cf. From the stability criterion, the allowable lump size, and hence
required number of lumps for the fluid path, are obtained. The process is
repeated until all the lumps at time t for the fluid pass are covered. The maxi-
mum of all the lumps calculated from the stability criterion for each lump is
taken for time (t + At). If the new value of the number of lumps at (t + At) is
different from that at (t), the present enthalpy distribution and other transient
parameters are evaluated, through linear interpolation, to obtain all the param-
eters at the new node pattern. The fluid phase for each lump is also reestab-
lished.
If the transient condition is imposed on an initial steady state of the com-
ponent, the steady-state distribution of all dynamic parameters should be ob-
tained at a desired operating condition. This can be done by using quasi-steady
relations for all energy streams and solving the simultaneous equations through
iteration until the convergence within a specified limit is obtained. The mass
flows and pressures are held constant during the iteration process
During the transient operation, a time step At is computed from the sta-
bility criterion. Note that the selected time step should satisfy the stability
criteria for all of the energy streams for all fluid passes. The fluid enthalpy,
gas temperature, tube temperature, and fluid mass-flow distributions at time
(t + At) are obtained by the method outlined earlier. It is assumed that perfect
mixing of gases occurs; therefore temperature variation of inlet gas is ne-
glected.
At the end of each time step, the following quantities are available for
each fluid pass:
• Fluid enthalpy distribution at the entrance and exit nodes of each
lump
• Gas temperature distribution at the entrance and exit of gas nodes
at the middle of each lump
• Tube temperature at the midlump node
• Fluid mass flow rate at each fluid node
This information from the preceding lump is appropriately transmitted
to the next lump, and the computation is repeated until the exit end is reached.
With the values of parameters at the boundary known, the pressure at
the time (t + At) is calculated, using the lumped parameter pressure model.
The dynamic parameter values at (t+ At) are now reset as the 'present1
values, and the computation sequence is repeated to obtain the solution after
At. The sequence is repeated until the external time step limit is reached.
Then the axial pressure distribution is computed from the friction and mo-
69
-------
mentum pressure drop relations, and the exit pressure and enthalpy are cal-
culated. In a system model, this information at the exit plane is transmit-
ted to the next component to affect its input fluid properties.
Details of TECO Vapor Generator
The program VAPORG is presently set up for the Thermo Electron Cor-
poration vapor generator design for the CP-34 system (Ref. 1). Figure 17 is
a pictorial view of the unit. The combustion gases from the combustion cham-
bers at the top enter the central cavity of the vapor generator and flow radially
outward. The working fluid, CP-34, passes through a series of concentric
coils; it enters the outer coil in subcooled state, then passes to the inner coil
and finally through the middle coil, from which it exits. Some important de-
tails of the flow paths are given in Table 8.
23 in.
Figure 17. Thermo Electron Corporation Vapor Generator (Ref. 1)
Cross Section Through Burner-boiler, Short Axis
70
-------
Table 8
DETAILS OF FLOW PATHS FOR THE TECO VAPOR GENERATOR
Inner Tube
Outer Tube
Coil Inside
Outside
Inside Outside Length
(ft)
No. Diameter Diameter Diameter Diameter
1
0. 930
0. 930
0. 930
1.000
1.000
1.000
1. 125
1. 125
1. 125
1.315
1.315
1.315
26
17
35
Inner
Tube
Surface
Bare
Outer
Tube
Surface
Ball -
matrix
Longitudi- Circum-
nal fins ferential
fins
Bare
Bare
Note that the actual tube construction is made up of two concentric tubes
separated by a water wall. The purpose of the water wall is to limit the in-
ner tube-wall temperature so that the organic working fluid does not thermally
decompose. In the model the thermal resistance of the water wall was neg-
lected and the energy-storage capacity was lumped with the tube-wall capacities.
Important heat transfer and pressure drop relations used in the program
are given in Appendix III, "Heat Transfer and Pressure Drop Relations, " of
this volume.
Results
The VAPORG program was run with the design data of the Thermo Electron
Vapor Generator. The results are summarized below.
Steady-state Run. Before the transient behavior can be studied, a steady-
state distribution of fluid, gas, and tube-wall temperatures at desired operating
levels is required. This can be done by:
1. Using the transient program, with boundary values and fluid pres-
sure held constant.
2. Solving steady-state conservation equations simultaneously.
A combination of the two methods was used here to derive the steady-
state distribution. An approximation of the steady-state condition was ob-
tained by an iterative solution of the steady-state equations. The approxi-
mation was then input into the transient model, which was run with the boun-
dary values held constant to obtain the final steady-state condition. This
technique is much faster and more economical than driving the transient model
to steady-state from arbitrary initial conditions.
Further discussion of the manner in which steady-state distributions are
obtained is given later in this subsection under the heading "Regenerator. "
71
-------
The results for the vapor generator are summarized in Figures 18 through 21.
The steady-state fluid-enthalpy distribution obtained from the program is shown
in Figure 18. The enthalpy change across the vapor generator agrees within
S^with the steady-state results calculated by Thermo Electron (Ref. 1).
The fluid temperature distribution is shown in Figure 19. The results in
the subcooled region compare well. The boiling section predicted by the mod-
el is much shorter than that calculated by TECO. As a consequence, the super-
heat zones differ significantly in their size. Resolution of this difference is
important because the size of the superheat region has a significant effect on
the dynamic behavior. The slope of the temperature profile in the superheat
region predicted by the model compares well with the TECO calculated slope.
Therefore it seems that the boiling region is the only area in which differences
exist.
Figure 20 indicates the difference in steady-state wall temperature dis-
tribution. As can be seen, the model predicts different wall temperature
from that obtained by Thermo Electron.
The reason for this difference is believed to be that the water-jacket re-
sistance between the inner and outer tube walls has been neglected, and the
vapor generator was treated as a single-wall device with equivalent heat
capacity.
The Thermo Electron design( which includes the water jacket, results in
lower inner-wall temperature in the two-phase flow region, a lower heat
transfer rate to the working fluid, and hence a longer two-phase flow region.
The model can be easily modified and a water-jacket resistance included
so that the steady-state temperature distributions match. However, the
actual heat transfer mechanism in the 1/10-inch water jacket between the two
walls is not well known. In the Thermo Electron analysis it was assumed that
boiling occurs on the inner surface of the outer wall and condensation on the
outer surface of the inner wall. Whether this is true or not should be estab-
lished experimentally. By using the transient vapor generator model, how-
ever, the water-jacket resistance can be varied parametrically in order to
determine the sensitivity of steady-state and dynamic performance to the
value of this parameter!
Figure 21 presents the steady-state gas temperature distribution.
Transient Results. After the model was brought to steady state, it was sub-
jected to several transients. The vapor generator model was run open-loop
with different exit boundary conditions. Figure 22 shows the vapor-generator
transient response to a 17. 6^ step increase in inlet fluid flow. The combus-
tion gas flow was held constant and the exit fluid flow varied in proportion to
72
-------
30 40 JO
Tube Length (ft)
Figure 18. Vapor Generator -- Steady-state Enthalpy Distribution of
Working Fluid. Vapor Generator Design as for TECO
System (Ref. 1)
580
560 _
540 |—
520
500
480
K
£ 420
2 400
b.
380
360
340
320
300
280
Solution
Obtained
with Transient
Model
Outer Coll
t
/.
/
:oii
J-I-t-H
II
j
if
1
tl
1
Inner Coil
/"
S
\
\ TECO St
Cond
1 • Fluid
Middle C
I i I
10 20
26
30
Tube Length (feet)
Figure 19. Comparison of Steady-state Temperature Distribution for
Working Fluid in Vapor Generator
73
-------
800
700
01
2 600 —
Q.
ID
H
= 504
400
300
Model Prediction (x = Tube Metal Nodes)
TECO Prediction for Inner Wall
TECO Prediction for Outer Wall
10
20
30
40 50
Tube Length (ft)
60
70
BO
Figure 20. Vapor Generator -- Steady-state Tube Wall Temperature
Distribution. Vapor Generator Design as for TECO
System (Ref. 1)
3600t—
Model Prediction
TECO Prediction (Ref. 1)
3200
2800
£ 2400
01
1
S 2000
u
a
p
£ 1600
V
"V
-\\
\ \
\ N
\
\
,3 1200-
800-
400 Inner Coil | Middle Coil I Outer Coil
Direction of Gas Flow
Figure 21. Vapor Generator -- Steady-state
Combustion-gas Temperature
Distribution
74
-------
Boundary Conditions
.0
3
X
H
to 120 ~
ra
c
W
0 sec
2.05 Ib/sec
2.05 Ib/sec
0.51 2 Ib/sec
0. 34 sec
2.4 Ib/sec
a(Vp-50 )
0. :>12 Ib/sec
^- Steady'
'— state
Value
I
1
a 700 -
W
a
v
L,
3
Cfl.
01
(X
650
600
550
500
0
Steady
state
Value
18 20 22 24
Time (sec)
26 28 30 32 34 36 38 40 42
Figure 22. Vapor-generator Transient Response to a Change
in Inlet Fluid Flow
The steady state that must be reached at the end of the transient can be
calculated as:
670 psia
p (at exit)
h (at exit)
104 Btu/lb
where the enthalpy value is calculated assuming the heat rate remains con-
stant.
As can be seen from Figure 22, the pressure and enthalpy both tend to
level out at these steady-state values. Ninety percent of the pressure change
occurs in about 10 seconds.
A certain "noise, " due to the forward finite difference method employed,
is always present in the computer solution, irrespective of the size of the
lump and the time step. The noise is greatest (about 5^ noise on the enthalpy
trace) during the early parts of the transient and dies out as the heat ex-
changer approaches steady state. The results shown in Figure 22 are the
mean curves plotted through the noise.
75
-------
Figure 23 shows the vapor-generator transient response to a 10$ step de-
crease in combustion-gas flow rate. The inlet fluid flow was held constant
and the exit flow rate varied proportionally to p/ y T .
The steady state after the transient can be approximated as:
p (at exit) = 496 psia
h (at exit)
118 Btu/lb
where in calculating the enthalpy value the heat rate at 30 seconds is employed.
As can be seen in Figure 23, the exit pressure remains fairly constant while
the exit enthalpy drops slowly, with ninety percent of the enthalpy change oc-
curring in about 25 seconds. The noise on this set of traces was negligible.
Boundary Conditions
5 t 0 sec
3 150
M
*; 140
X '
*• 130
CO
Sf 120
OS
.c
C 110
W
•o
•3 100
* /T I |\A A ^ 1 t
mf (Jnlet) 2. U5 ID
mf (Exit) 2.05 Ib
-
-~— - rh r 0.51211
„
1 I 1 i i 1 1 1 1 1 1 I 1
••H
fc
to
c.
•5 550
K
~ 500
co
0>
L«
p 450
'/I
-
•
-
£ 40o[-
CL 1
-a
1 1 1 1 1 1 ! 1 1 1 1 I 1 1
2 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28
^ *T* • «-« n /r^l-vn^
> 0.35 sec
/f\ f\r- II / _ -.
sec i. Uo lb/ sec
/sec a (p/^rt)
j/sec 0. 460 Ib/sec
^<- Steady-
state
Value-
i.l I I I
-
-- Steady
Value
-
-
ill!!
30 32 34 36 38
Figure 23. Vapor-generator Transient Response to a Change
in Combustion-gas Flow Rate
CONDENSER
The automotive condensers are usually of the cross-flow type, with the
working fluid flowing inside the tube and cooling air outside. The fluid enters
the unit in a slightly superheated state and leaves in a subcooled condition.
76
-------
From the simulation standpoint, the condenser model is closely similar to the
vapor generator model, except for the direction of fluid enthalpy change.
Hence, the relations derived earlier for the vapor generator are directly ap-
plicable here and will not be repeated.
General Description of Model
The condenser usually has multiple fluid passes, arranged in a series
fashion. Each fluid pass consists of a number of parallel fluid paths, with
headers at both ends. If it is assumed that the only effect of parallel fluid
paths is to divide the air and fluid flows appropriately, consideration of one
fluid path is representative. Since the serial fluid passes resemble the once-
through vapor generator arrangement, the comments concerning the once-
through unit given in the "Vapor Generator" subsection are directly applicable
here.
Transient Analysis and Analytic Procedure
The conservation equations, their finite-difference approximations, and
corresponding stability criteria given for the once-through vapor generator
are also directly applicable here.
The selection of the nature of transient relation fo.r the particular conden-
ser considered is given in Table 9. For example, in the superheated working
fluid region, the air energy equations are quasi-steady, the tube equations
are dynamic, and the working fluid equations are quasi-steady.
Table 9
SELECTION OF ENERGY RELATIONS FOR CONDENSER
Fluid Phase Air Tube Fluid
Superheated Quasi-steady Dynamic Quasi-steady
Condensing Quasi-steady Quasi-steady Dynamic
Subcooled Quasi-steady Dynamic Dynamic
Computer Model and Results
The condenser model is entitled CONDENS, and is listed in Volume II.
The structure and the data requirement of this program are identical to those
for the vapor generator.
The Thermo Electron condenser design, on which CONDENS is based,
is represented in Figure 24 (Ref. 1). Briefly, it has three fluid passes, with
30 parallel flow paths in each. Each tube has louvered fins on the outside.
The flow area is not identical for all fluid passes.
77
-------
3. 0 In.
-50.0 In.-
:j
19. 9 In.
Figure 24. Condenser Design (Ref. 1)
The pressure drop and heat transfer correlations used here are mostly
similar to those given in Appendix III for the vapor generator.
The CONDENS program was run with the design data of'the Thermo Elec-
tron condenser. The steady-state temperature distribution was obtained in a
manner similar to that for the vapor generator. The results shown on Figure
25 compare with the Thermo Electron calculations within two percent at
the end points (TECO did not calculate the enthalpy distribution throughout the
condenser).
After the condenser model was brought to steady state, it was subjected
to a 10# step decrease in inlet fluid flow. The airflow rate was held constant
and the exit fluid flow was proportional to -^p - 10 . The results are shown
in Figure 26. The initial exit pressure and enthalpy are:
p (at exit)
h (at exit)
24. 55 psia
-131. 12 Btu/lb
At the end of the transient the steady-state values that should be reached are:
p (at exit) = 21. 7 psia
h (at exit)
-134. 01 Btu/lb
These values were obtained by solving the steady-state conservation equations
at the new fluid mass-flow rate. Therefore, at the end of the transient,
78
-------
50 —
-50 —
-100 —
-150
Entrance
X •= Fluid Nodes
= End Points Calculated
in Reference 1
Tube Length (inches)
Figure 25. Condenser -- Steady-state Enthalpy Distribution Liquid
Side --as Calculated by Transient Model
Boundary Conditions
£
3
ffl
X
U
rt
*
"
•••>
t
mrflnlet)
mf(Exit)
mg
X
X
s
0 sec
2. 05 Ib/sec
2.05 Ib/sec
17 Ib/sec
X
X
—
>0. 5 sec
1. 84 Ib/sec
a /p-10
17 Ib/sec
S
tea<
Jy-
|
S -1
-2
\
-3
Steady-
state
value
0 1 2 34 5 678 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 2G 27 28
Time (sec)
Figure 26. Condenser Transient Response to a Change
in Inlet Fluid Flow
79
-------
Ap (at exit) = -2. 85 psi
Ah (at exit) = -2. 89 Btu/lb
.These values are also plotted on Figure 26.
After 20 seconds the pressure is within 3# of its final value; the enthalpy
is within 0. 3"£.
An interesting point to note is that for a 10$decrease in inlet fluid flow,
the enthalpy change across the condenser increases by less than 2^. There-
fore, the heat rate of the condenser is not the same before and after the tran-
sient (remember, for a similar transient for the vapor generator, the heat
rate remained constant; the enthalpy change increase balanced the mass flow
decrease). For the condenser the decrease in fluid flow rate causes a de-
crease in interior heat-transfer coefficient; hence, the enthalpy increase is
less than it would be if the heat rate remained constant. This illustrates the
importance of the heat transfer mechanism in determining the transient be-
havior of the condenser.
REGENERATOR
The regenerator is a heat exchanger in which the thermal energy of the
superheated vapor is transferred to the subcooled fluid circulated by the feed-
pump. Ideally, the regenerator should be so sized that the vapor leaving the
regenerator still has some degree of superheat; that is, no change of phase
exists for either fluid. Generally, the subcooled fluid flows through the tube
and the vapor outside it. The particular design studied here has a cross-flow
configuration with all these features. The mathematical model of the regen-
erator is again similar to that of the vapor generator, with one major simpli-
fication arising from the absence of phase change: the dynamic fluid pressure
variation need not be considered. Since the models are similar, the relations
derived earlier are applicable here and will not be repeated.
General Description of Model
The regenerator usually has multiple fluid passes, arranged in series.
the fluid traveling through several tube lengths in each pass. If it is assumed
that perfect mixing exists on the vapor side, the cross-counterflow arrange-
ment can be safely approximated as a cross-flow arrangement. Since the sub-
cooled liquid usually will not change phase, the dynamic relations for the re-
generator are similar to those for the vapor-generator subcooled phase.
Transient Analysis and Analytic Procedure
The conservation equations, their finite-difference approximations, and
corresponding stability criteria given for the once-through vapor generator
are directly applicable here. The vapor-side relations are quasi-steady,
whereas the tube and the fluid-side relations are dynamic representations.
80
-------
Computer Model and Results
The regenerator model is entitled REGEN and is listed in Volume II. The
structure and the data requirements of this program are identical with those
for the vapor generator.
The TECO regenerator design, on which the REGEN model is based, is
represented in Figure 27 (Ref. 1). Briefly, the vapor flow to the regenerator
is divided into four sections; it enters the regenerator from the sides, flows
axially inward, and exits from the center through a common exit section. Like-
wise, the fluid-flow to the regenerator is divided into four sections. It enters at
the top of the inner tube row near the centerline, then moves through the entire
tube length and proceeds to the next tube in the same vertical fluid pass until it
3. 400 in.
j-0. 300 in.
i
r
. UU inri
-U t-0.29i
I
—11—0. 10 in.
h -2. 50 inr- 1
— i —
n. '-0.1 50 in.
Figure 27. Regenerator Design (Ref. 1)
81
-------
completes the fourth tube; it then moves to the bottom tube of the next fluid pass,
and so on, until it travels through all of the four fluid passes. Note that the fluid
exits at the same plane where the vapor enters; hence, a counterflow effect. The
tube has a ball-matrix extended surface on its outside.
The pressure-drop and heat transfer correlations applicable here are large-
ly similar to those given in Appendix III for the vapor generator.
The regenerator program was run with the design data of the Thermo
Electron Corporation regenerator.
Steady-state Runs. Since the regenerator model is simpler and more eco-
nomical to run than the condenser and vapor generator (absence of two-phase
flow) it was decided to try to obtain the steady-state solution directly. It
should be recalled that for the vapor generator and condenser this solution
was approximated by iterating the steady state conservation equations before
employing the model. For the regenerator, arbitrary liquid, gas, and tube
temperature distributions were assumed and the transient model driven to
steady state, with the inlet conditions held constant. Figure 28 gives the
variation in spatial liquid-enthalpy distribution as a function of time. The
curve marked 1 represents the initial guess, and the curve marked 105 is
the distribution at steady state. The fact that there is a wide variation in liq-
uid-enthalpy distribution between the two times is due to the initial tube tem-
perature estimation (marked 1), which is significantly different from that at
the steady state (marked 105), as shown in Figure 29. Similarly, Figure 30
gives the vapor-temperature spatial distribution as a function of time during
this process of deriving the steady-state condition.
151 Kluid Paas
Ird Fluid t'as»
200
Tube l.riglh lin. I
Figure 28. Regenerator Liquid Enthalpy -- Derivation of Steady-
state Solution Employing Transient Model
82
-------
920
200
lilt Fluid Pass [ 2nd Fluid Pass [ 3rd Fluid Pasa | 4th Fluid Pasa [
200 300
Tube Length ((n.)
Figure 29. Regenerator Tube-wall Temperature -- Derivation
of Steady-state Solution Employ ing Transient Model
Between
Fluid Pasa
1 and 2
Between
Fluid Pass
2 and 3
Gaa Nodes
Between
Fluid Pasa
3 and 4
Figure 30. Regenerator Gas Temperature -- Derivation
of Steady-state Solution Employing Transient
Model
83
-------
1 ransient Results. The steady-state distribution for the regenerator was sup-
plied as the initial distribution, and the model was subjected to 20^ drop in
fluid enthalpy at the entrance. Figure 31 gives the corresponding variation
in fluid enthalpy and vapor temperature at the exit as a function of time. The
vapor temperature shows an immediate effect of the input enthalpy disturbance;
this is due to the counterflow arrangement for the two fluid streams. The fluid
enthalpy at exit lags the input disturbance by approximately one transport delay,
which is expected. The final steady-state values also agree with a simple
energy balance.
Figure 32 shows the effect on fluid temperature distribution as a function
of time. Again the delay due to fluid transport mechanism can be noted. Fig-
ure 33 presents the tube temperature variation, and Figure 34 the vapor tem-
perature variation.
Fluid Enthalpy at Entrance
80
82
Fluid Enthalpy at Exit
Gas Temperature at Exit
84
86
88 "3
£
90 u
92 e
94
5.0 10.0 15.0 20.0 25.0 30.0 35.0 40.0
Time (sec)
Figure 31. Regenerator Transient
COMBUSTOR
The combustor model was developed as three submodels dealing with
• Flame temperature
• Thermal transients
• Emissions
Each submodel will be derived separately below. At the end of this section,
linking of the submodels to form the total combustor model will be discussed.
84
-------
290-
Time-
Step
0
5
7
14
29
62
150
Time
(sec)
0.000
0.900
1.260
2. 520
5.224
14. 511
41. 936
110
1st Fluid.Pass I 2nd Fluid Pass I 3rd Fluid Pass I 4th Fluid Pass I
0 100 200 300 400
Tube Length (in.)
Figure 32. Regenerator Transient -- Fluid Temperature
85
-------
320 -
1st Fluid Pass | 2nd Fluid Pass | 3rd Fluid Pass | 4th Fluid Pass |
100
200 300
Tube Length (in. )
400
Figure 33. Regenerator Transient -- Tube-wall Temperature
86
-------
400i_
380
360
fa 340
0)
2 320
0)
IH
(U
g 300
v
H
o 280
a
n)
260
240-
220
200-
Time
Step
0
17
28
41
73
150
Time
(sec)
0.000
3.059
5.043
8.022
17.930
41. 936
I I I ^
1st Fluid Pass | 2nd Fluid Pass | 3rd Fluid Pass | 4th Fluid Pas
100 200
. Tube Length (in.)
300
400
Figure 34. Regenerator Transient -- Vapor Temperature
NOMENCLATURE
Alphabetical
Symbols
b
c
CP..
Cp
CP,
s
cp+
D
cs
ct
Number of hydrogen moles in reaction
Percent of carbon by weight in fuel
Specific heat of air
Specific heat of gas
Specific heat of shell
Specific heat of tube
Hydraulic diameter shell - tube flow passage
Hydraulic diameter tube flow passage
Equivalence ratio
87
-------
Alphabetical
Symbols
(Cont'd)
f
f
max
f .
mm
sa
h,
ta
h(T)
L
cs .
Lct
LHV
M
m
N
f
i
g
ba
N
P,
sa
ta
tg
Fuel air ratio
Fuel air ratio maximum limit
Fuel air ratio minimum limit
Stoichiometric fuel air ratio
Heat transfer coefficient between shell and air
Heat transfer coefficient between tube and air
Heat transfer coefficient between gas and tube
Enthalpy at temperature T
Shell length
Tube length
Lower heating value of fuel
Molecular weight of carbon
Molecular weight of hydrogen
Air mass flow rate
Fuel mass flow rate
Mass flow rate of combustion gas
Equivalent turbulent friction length due to bends in air
flow path
Equivalent turbulent friction length due to bends in gas flow path
Pressure of gas at combustor exhaust
Pressure of air at combustor inlet
Shell wetted area
Tube - air wetted area
Tube - gas wetted area
88
-------
Alphabetical
Symbols
(Cont'd)
t
T
T
£
T
W
W.
Greek
Symbols
At
At
e
At
Time
Air temperature
Flame temperature
Gas temperature
Ambient temperature
Shell temperature
Tube temperature
Shell weight
Tube weight
Distance
Time step
External time step
Stability time step
FLAME TEMPERATURE SUBMODEL
Derivation of Basic Equations
Consider a reaction for the combustion of a hydrocarbon fuel containing a
fraction (c) of carbon (by weight), and a fraction (1 - c) of hydrogen. The
stoichiometric reaction -in air is
H h h v,
" ~ ' C02 +
where
and the fuel air ratio is
12
d-c)
(12
)+ (28)4(1 + )
(102)
(102a)
(103)
89
-------
For a reaction off stoichiometric conditions with a fuel air ratio (f), the equa-
tion for a lean mixture is
CH.+ (-1(1 + r)
b \el 4
- 4(1 + -) N3 - ^HaO + C03
ej * ^
-)4(1+-)N3 (104)
e/ 4 3
For a rich mixture, the reaction is
co2 + N
+ 2(1 --) (1 + 7) H (105)
e 4
where
e = j~ (105a)
s
By employing the coefficients in the above equations, the flame temperature
can be calculated. The basic equation employed is
h(T.) = h(T ) + (12 + b)LHV (106)
f . a
where h(Tf) is the enthalpy of the products of reaction at the flame temperature,
h(Ta) is the enthalpy of the products at the combustion air temperature and
LHV is the lower heating value of the fuel.
The enthalpy of the products H2O, CO3, N2, and O2 are tabulated as a func-
tion of temperature in Reference 26. These are on a per-unit weight basis
and must be multiplied by the appropriate coefficients from Equations 104 and
105 in order to be used in Equation 106.
Model Development
The flame temperature submodel has the following input:
T , m.f and m .
a i a
The following data must be supplied:
c and LHV, f , f .
max mm
The enthalpy of the products of combustion are curve-fit as a function of
temperature. Therefore, an iterative procedure is required to find Tf by
using Equation 106.
90
-------
The outputs of the model are Tf, e, and b. Tf is employed by the ther-
mal transient submodel; e andb are employed by the emission submodel. The
flame temperature submodel is entitled COMB1 and is listed in Volume II,
the Users Manual.
THERMAL TRANSIENT SUBMODEL
Derivation of Basic Equations
The schematic combustor configuration assumed for the thermal transient
submodel is shown in Figure 35. Ambient air at temperature To passes be-
tween the combustor shell and the tube and is preheated to teVnperature Ta.
Fuel is then mixed with the air, combustion takes place, and the products of
combustion reach flame temperature Tf. The combustion gas then flows through
the combustion tube, its temperature dropping to T„ as a result of heat trans-
fer to the tube wall. The gas at Tg then flows over the vapor generator coils.
Shell
Tube
Figure 35. Combustor Schematic
The basic equations which describe the transient thermal process are:
Shell: (treated in lumped manner)
dT
, s
's dt
where
+ H (T -T ) = 0
sa s a
C = W Cp
s s K
(107)
(107a)
91
-------
H
sa
h S
sa sa
(107b)
Air; (assuming quasi-steady)
dT
where:
E —r=- Ax + H (T - T ) + HA (T - TJ
a dx sa a s' ta a t
E = m Cp
a a a
H = h Sx
ta ta ta
Tube: (treated in lumped manner)
dT.
VW •
where:
- c
H.
wt cpt
tg tg "tg
Gas: (assuming quasi-steady)
dT
where:
E -j- Ax + H. (T -T.)
g dx tg g t
g
m
g
m Cp
g g
ma+ mf
(108)
(108a)
(108b)
(109)
(109a)
(109b)
(110)
(HOa)
(HOb)
The combustor is treated as a lumped model and the following finite-difference
approximations, are made:
dT T (t+At) - T (t)
S_ _ S S .- 1 -v.
dt "At ^
dT T• - T
a _ a o
dx Ax
Tt(t+At) -Tt(t)
~dT = At
dT T - T.
g_ = _g L
dx . Ax
(112)
(113)
(114)
92
-------
These approximations are substituted into the differential equations, and the
results are:
Shell:
where:
T (t+At) = F T (t) + F T (t)
s sa a ss s
sa
(115)
ss
H At
-fr -
C
115b)
Air;
where:
T (t+At) = F T (t) + F . T.(t) + F T
a as s at t ao o
F = H / (E + H + H. )
as sa ' a sa ta
F A = H. / (E + H + H. )
at ta ' a sa ta
F = E / (E + H + H. )
ao a a sa ta
Tube:
where:
T, (t+At)
t
F. T(t)+F, T (t)+ F,, T.(t).
ta a tg g tt t
ta
H
ta
At/Ct
(116)
(116a)
(116b)
(116c)
(117)
Ftg = HtgAt/Ct
H + H
At
(u?c)
Gas:
where:
T (t+At) = F T.(t) + F , T.(t)
g gt t gf f
g«
(118)
93
-------
The stability criterion requires that the coefficients F be positive. Therefore,
r Q C 1
At s Min
.
L sa ta tg J
This sets the time -step size for the integration.
Model Development
The input to the transient thermal combustor submodel is
T , T (t), T (t). T (t), T (t), T
-------
1000
900
800
700
"600
PH
-M
CQ
o
53
500
400
300
200
100
\
\
r
\
A
\
• Thermo Electron (Ref. 1)
0 Thermo Electron (Ref. 1)
A Marquardt (Ref. 28)
0 GM-SE 101 (Ref. 1)
\
^ Solar (Ref. 27)
\
\
\
1.0 0. 9 0. 8
0. 7 0. 6 0. 5
Equivalence Ratio
0. 4 0. 3
Figure 36. Nitrogen Oxide. Measured Exhaust Concentrations
95
-------
1000
900
800
700
£600
O
u
500
400
300
200
100
• Thermo Electron (Ref. 1)
0 Thermo Electron (Ref. 1)
A Marquardt (Ref. 28)
GJ GM-SE 101 (Ref. 1)
V Solar (Ref. 27)
V
I I
I I I I
1. 0 0. 9 0. 8 0. 7 0. 6 0. 5
Equivalence Ratio
0. 4 0. 3
Figure 37. Carbon Monoxide. Measured Exhaust Concentrations
96
-------
100
90
80
70
60
PH
50
40
30
•20
10
• Thermo Electron Corporation (Ref. 1)
0 Thermo Electron
(Ref. 1)
AMarquardt (Ref. 28)
0 GM-SE 101
(Ref. 1)
1.0 0.9
0. 8 0. 7 0. 6 0. 5
Equivalence Ratio
0. 4 0. 3
Figure 38. Unburned Hydrocarbons. Measured Exhaust Concentrations
97
-------
fa
o-
O
O
•o
4>
N
s
t,
o
2:
0)
O
§
U
4-*
w
3
x
W
0. 2 —
0. 1
200 400 600 800 1000
Burner Air-inlet Temperature (°F)
1200
Figure 39. Characteristic Normalized Exhaust Concentrations
(e = 0. 59)
98
-------
The information in Figures 36 to 39 has been curve-fit with polynomial
functions which are used directly in the emissions submodel. The inputs to
the model are:
T , e, and At.
a
The emissions are calculated (PPM/106 Btu/hr). This value is then multiplied
by the design heat rate of the combustor to obtain the emissions in parts per
million. The coefficients of the reaction equation determined in the flame
temperature submodel are transmitted to the emission model and the emis-
sions are calculated in two forms: 1) grams per gram fuel, and 2) the total
grams in time interval At.
The emissions submodel is entitled COMB3 and is listed in Volume II.
TOTAL COMBUSTOR MODEL
Figure 40 shows the linking of the three combustor submodels to form the
total combustor model. The .inputs are:
V" To'
' and At
The outputs are:
p , T , T , T , T , T , and the emissions after an elapsed time
6 r S a g ofAte
Flame Temperature
Submodel
At,,
At'
Emissions
Submodel
Emissions
>
Tf (t + A te)
Thermal Transient
Submodel
T (t)
Iteration
Loop for
Steady-state
Solution
T (t)
T (t)
T.(t!
Ta(t+At >
Tg(U-Ate,
v°
Vapor'
Generator
*c coefficients of reaction equation
Figure 40. Combustor Model -- Linking of Combustor Submodels
99
-------
An iterative procedure is required to derive the steady-state conditions,
since the flame temperature submodel requires a value.of Ta which is deter-
mined by the thermal transient submodel. Therefore, the following procedure
is employed:
1. Initially, Ta= To and the flame temperature submodel is used to
find Tf.
2. The thermal transient model is run with Tf, rrif, and ma held con-
stant until the transients die out and the steady-state solution is ob-
tained for Ts, Tg, Tc, and Ta.
3. The new value of Ta is used in the flame temperature submodel, and
the calculations are iterated until convergence.
Once the steady-state solution is obtained, transient cases can be run using
the value of Ta(t) to find the value of Tf(t+Ate) in the flame temperature sub-
routine, which is, in turn, used in the thermal transient subroutine to deter-
mine
T (t+At ), T (t+At ), T (t+At ), and T (t+At )
setege ae
The total combustor model is entitled COMBST and is listed in Volume II,
the Users Manual.
RESULTS
The combustor model was run to derive the steady-state solution at:
p = 14. 7 psi
o
T . = 85°F
o
m = 0. 0123 Ib/sec
ma = 0. 2435 lb/sec
LHV = 20180 Btu/lb
c = 0. 85
W = 6/81b
s
W = 3. 15 lb
S =454 in.2
sa
S =374 in.3
ta
S =374 in.3
tg
D = 4 in.
cs
D . = 7 in.
ct
100
-------
L =0. 708 ft
cs
L 4 = 1.415 ft
ct
Nu = 75
ba
N = 30
bg
The above values approximately represent a single branch of the combustor
in Reference 1 at the design condition. The following results are obtained:
T = 129°F
a
T = 3325°F
T = 125°F
s
T = 260°F
T = 3293°F
g
NO = 1.01 10~3 grams/gram of fuel
. CO = 1. 0510"3 grams/gram of fuel
HC = 7.32 10"5 grams/gram of fuel
p = 14. 699 psi
e
If a fuel economy of 10 miles per gallon is assumed, the emissions in
grams per mile are
NO = 3. 24 grams/mile
CO = 3. 37 grams/mile
HC = 0. 0235 grams/mile
The Environment Protection Agency's emission-level goals are (Ref. 29):
NO = 0.4 grams/mile
CO =4.7 grams/mile
HC = 0. 14 grams/mile
The results obtained from the model meet the goals for hydrocarbon and car-
bon monoxide emissions but are higher than the goal for nitrogen oxide. Fig-
ure 41 shows that the nitrogen oxide goals can be achieved by decreasing the
equivalence ratio to about 0. 5. However, this is accompanied by an increase
in hydrocarbon emissions and a decrease in combustion gas temperature.
This figure illustrates the sensitivity of emissions to fuel/air ratio and in-
dicates the trade-offs possible between emission levels and system efficiency.
The pressure drop calculated by the combustor model is very small and
compares to the results obtained in Reference 1. The steady-state condition
having been obtained, a fuel-air transient was run.
101
-------
4000
01
u
a
S 3000
a.
01
0>
a
O
c 2000
o
-------
Fuel Air Ratio Transient: A 10$ step decrease in fuel/air ratio is applied,
with the total flow rate held constant. The results are shown below.
t = 0" t = 0* t = 60
m = 0. 0123 Ib/sec m = 0. Ollllb/sec m = 0. 01115 Ib/sec
rh = 0.2435 Ib/sec m = 0. 245 Ib/sec m = 0. 245 Ib/sec
a a a
mr/m =0.0505 rh./rh =0.0455 m,/m =0.0455
fa fa fa
m+rh = 0. 2558 Ib/sec m ,+ rh = 0. 2561 Ib/sec m + m = 0. 2561 Ib/sec
fa fa ' fa
T = 85°F T = 85°F T = 85°F
o o o
T = 129°F T = 130°F T = S107°F
a a a
Tf = 3325°F Tf = 3061°F Tf = 3061°F
T = 125°F T = 125°F T = 123°F
s s s
T = 260°F T = 260°F T = 273°F
L L L
T = 3293°F T = 3031°F T = 3039°F
g g g
The step change in the fuel/air ratio produces an initial instantaneous step
change in both flame temperature and inlet air temperature. The air temper-
ature change is instantaneous because of the quasi-steady approximation which
neglects the air heat capacity. This is followed by a slow-transient change in
Tx, Tj., Ta, Tf, and Tg, as indicated. As can be seen from these results, the
change in gas temperature in the 60-second interval is negligible and the dom-
inant effect is caused by the initial step change in flame temperature.
CONTROLS
The controls analyzed here for a reciprocating engine and organic work-
ing fluid are the basic configuration defined in the Thermo Electron Corpor-
ation report (Reference 1). It is recognized that TECO has significantly
changed its control philosophy since this original report; before further
analysis, therefore, these changes should be considered in the model. At
this stage of the program, no attempt was made to optimize or alter the design,
and this phase of the modeling study is concerned strictly with the instanta-
neous control equations. Later phases are to consider dynamics of the con-
trol components.
NOMENCLATURE
Alphabetical
Symbols
A! through A9 Fuel valve diaphragm and bellows effective areas
(see Figure 43) in.3
AO Normalized accelerator pedal position
103
-------
Alphabetical
Symbols
(Cont'd)
C5
IR
K,
K
E
K
Q
Q
E
QF
RPM
S_
Air/fuel ratio
Cut-off, or intake ratio
Gain of temperature trim on fuel flow, Ib/hr °F
Engine flow constant, Ib/hr rpm
Pump displacement factor at full stroke Jb/hr rpm
Boiler (vapor generator) flow. Ib/hr
Air flow, Ib/hr
Engine vapor flow
Fuel flow, Ib/hr
Engine speed, rpm
Normalized pump stroke
Boiler (vapor generator) outlet temperature, °F
Pump volumetric efficiency
CONTROL DEFINITION
Figure 42 is a. simplified overall schematic diagram of the control sys-
tem. Basically, it breaks down into two distinct areas. First is the burner
fuel/air control, which serves to control the vapor generator outlet temper-
ature and maintain the fuel/air ratio. The second area comprises the cut-off
and feedpump controls. The cut-off control is directly controlled by the driver
and sets the power output of the engine. The feedpump control is linked to the
cut-off control to provide anticipation and serves to maintain vapor generator
output pressure by modulating the feedpump stroke. A third control, not shown
in Figure. 42, is the condenser fan control, which varies the ratio of the fan
drive as a function of engine speed. A more detailed discussion of the control
loops accompanies the derivations of the instantaneous equations. (Appendix I
of this volume presents the instantaneous control equations for an alternative
engine, with a reciprocating expander and steam as the working fluid. )
BURNER CONTROL
The TECO burner control consists of an airflow control and a fuel-flow
control valve. The form analyzed here uses a pressure drop across an orifice
104
-------
,Air/Fuel Control
Working
/Fluid
Control
Figure 42. Schematic of Power, Working Fluid, and Air/Fuel Control
in the organic flow line (from the regenerator) to serve as a reference for the
fuel valve (see Figure 42). This sets a fuel valve metering area so that fuel
flow is roughly proportional to organic flow rate; a quick-acting signal is pro-
vided which anticipates the need for a change in fuel flow to compensate for
changes in organic flow rate.
Pressure from a temperature sensing bulb in the vaporizer outlet line
serves as a slow-acting reset on fuel valve position so that, in effect, the fuel
valve maintains a fuel flow rate to hold boiler outlet temperature constant.
Figure 43 is a schematic diagram of the fuel valve itself; a separate regulator
maintains a relatively constant fuel supply pressure to the valve.
The fuel passing from the fuel valve goes to the fuel nozzle. The back
pressure of the fuel nozzle serves as a measure of fuel flow and is the
reference for the air valve. Air valve position is maintained as a function
of fuel nozzle back pressure in the 1970 TECO design by means of a spring-
loaded piston actuator and linkage, as shown in Figure 44. The result is the-
oretically that airflow is maintained as a direct ratio of fuel flow. This re-
lationship is highly dependent on constant combustor air-blower characteris-
tics and on low friction in the actuator. Recognizing these limitations, TECO
has altered the design. The analysis below is based on the original design, but
since it implies perfect components with instantaneous response, should yield
results similar to an analysis of the more recent system with the same as-
sumptions.
105
-------
Fuel Supply
Orifice
Temperature
Bulb Pressure
Retuno
Spring
I'uel Out
t -- Aa = A, - A,
A, = A3 - A~
Figure 43. Fuel Valve -- Simplified Schematic
Figure 44. Original Thermo Electron Air Valve
106
-------
Air Valve
As described above, the air valve is moved as a function of fuel nozzle
back pressure by means of a spring-loaded actuator and compensating link-
age. The net effect, assuming no friction, constant blower characteristics,
and instantaneous response, is to maintain a constant air/fuel ratio, or
Q
A
CQ
f
(120)
where C = 19.8.
Fuel Valve
The fuel-valve instantaneous equations depend on inputs from the organic-
flow sensing orifice, the temperature sensing bulb, the fuel supply pressure,
and the fuel nozzle back pressure. Figure 43 shows the fuel valve configura-
tion; Figure 45 is the interacting block diagram showing the instantaneous re-
lationship of the fuel valve position to the above parameters.
The relationship implies negligible mass and damping of the moving parts
and neglects inertance and compressibility of the fuel and the organic fluid.
It also assumes a negligible time constant of the temperature sensing bulb.
These dynamic terms must, of course, be accounted for in the next phase of
the analysis but are neglected at this point.
Organic
Flow from
Vapor-generator
Outlet Temperature
tf-^^ Temperature J
Reference I
Dowthernv A
L _ _ j „ _ Prcsaure
Equivalent Spring
Preload X0
Poppet
Contour
L£
x
t
Fuel Suppl_y_
Pressure
Link and Valve
Figure 45. Burner Control -- Block Diagram
107
-------
Thermo Electron has made provision for a contoured metering poppet in
the fuel valve in order to linearize the relationship between fuel flow and or-
ganic flow. The relationships of Figure 45 were solved by an iterative com-
puter solution to determine the fuel flow as a function of vapor temperature
and organic flow rate. Areas used were supplied by the Thermo Electron
Corporation, and the characteristics of Dowtherm A® fluid were used in es-
tablishing the temperature bulb pressure. The result is shown in Figure 46,
where the valve characteristic of fuel flow versus organic flow is indeed
linear at the outlet temperature of 550°F for the design vapor generator. As
the temperature deviates from 500°F, the characteristic loses its linearity.
A model incorporating an exact solution must continuously solve the
equations represented by the block diagram of Figure 45. This will be ex-
pensive in terms of computer time, since it will involve algebraic loops.
For this first cut at the analysis, it was decided to develop a simpler model,
which approximates the characteristics of Figure 4 6. This was done by con-
sidering that the normal operation will be at 550°F organic temperature and
that deviations from this temperature will not be overly large. With that as-
sumption, the fuel flow could be characterized by the equation
Qc, = 0. 01208 Q + K.[T-550] (121)
-b b
where, by inspection of Figure 46, it can be seen that K^, the sensitivity of
flow to vapor generator temperature, will vary as a function of vapor-gener-
ator flow rate.
Kb was determined by first cross-plotting the curves of Figure 46 into
the format of Figure 47 and finding the slope of the resulting Qf versus T lines
at the 550°F point. The resulting values of K^ are plotted in Figure 48 as a
function of organic flow rate Q. A curve fit of Figure 48 shows that it can be
characterized to better than 2% accuracy by the equation
K = -8. 1344 + 3. 5389 x 10~3 Q - 7. 7945 x 10~7 Q3
+ 8.6304 X 10~u Q3 - 3.7709 x 10~15 Q4 (122)
Q_ = Fuel flow,. Ib/hr
b
Q = Organic flow, Ib/hr
T = Vapor-generator outlet temperature, °F
In addition, the following absolute limits are provided by the control
MAX Q^ = 47. 02 Ib/hr (123)
-T
MIN Q_ = 4.851 Ib/hr (124)
r
®Registered trademark of the Dow Chemical Company
108
-------
100
1000
2000
3000 4000 5000
Q -- Boiler Flow (Ib/hr)
6000 7000 8000
Figure 46. Fuel Flow Versus Boiler Flow -- CP-34 System;
Contoured Poppet
109
-------
100
Design Temperature
550°F
510
520 530 540 550 560 570
T -- Boiler Temperature (°F)
580
Figure 47. Fuel Flow Versus Boiler Temperature -- CP-34
System; Contoured Poppet
110
-------
Boiler Flow (lb/hr)
-1
-2
Kb-3
-4
-5
1000 2000 3000 4000 5000 6000 7000 8000 9000 10.000
-6U
Figure 48. Slope of Qf Versus Temperature Curve at Design Temperature
(550°F) -- CP-34 System; Contoured Poppet
CUT-OFF AND FEEDPUMP CONTROL
The cut-off will be varied directly as a function of driver demand, with
provision for a minimum cut-off limit to maintain idle speed under variable
accessory load.and a maximum cut-off to assure that flow demand does not
exceed boiler capacity. The feedpump is a variable-stroke positive-displace-
ment pump running at a fixed ratio to engine speed in normal operation and is
linked directly to the valve cut-off, as will be explained below. A pressure
bias is applied on the feedpump stroke control so that the feedpump control
basically maintains boiler pressure. The simplified block diagram for the
cut-off and feedpump control is given in Figure 49.
Cut-off Control
TECO limits the maximum intake ratio, or cut-off, to 0. 29 for several
reasons, feedpump size limits being probably the most overriding. For this
reason, a fixed maximum of 0. 29 is set; it is presumed tnis will occur when
the driver input (accelerator pedal position) is at full scale (AO = 1). There-
fore,
IR = 0.29(AO) (125)
However, it is necessary to place a maximum limit on engine flow demand in
order that it not normally exceed boiler capacity. The design flow of the
111
-------
Vapor
Generator
Output Pressure
Feedpump Stroke
Maximum Limit
Cam
Linkage
Accelerator
Pedal Input
Figure 49. Engine Power Level and Vapor Generator Feedpump
Control -- Functional Block Diagram
boiler is 7301 pounds per hour, this flow occurring at a cut-off of 0. 137 and
2000 rpm engine speed. Since engine flow is approximately
RPM • IR
(126)
it follows that KE = 26. 647 Ib/hr rpm and, to maintain maximum engine flow
constant,
IR
MAX
E design
K ' RPM
274
RPM
(127)
It can be seen that IR of Equation 120 reaches the maximum level of 0. 29
at 944. 83 rpm. The maximum cut-off limit can thus be defined as
MAX IR - 0.29 for RPM ^ 944.83 (128)
274
RPM
for RPM >944. 83
(129)
The minimum cut-off must be maintained to hold an idle speed high
enough to drive accessories under a wide range of accessory loads. The
nominal idle speed will be 300 rpm, and it will be assumed that a 5$ droop
control will be maintained. In other words, a 5$ drop in speed below 300
rpm would be. enough to increase cut-off from zero to the maximum limit of
0. 29. In equation form this works out to
112
-------
MIN IR = 0.29 for RPM < 285 (130)
= 0.01933 (300-RPM) for 285 < RPM < 300 (131)
= 0 for RPM > 300 (132)
The minimum limit of zero for speeds above 300 rpm may be modified in later
models, since system requirements may demand at least a very small mini-
mum flow under most conditions.
Pump Stroke Control
The pump stroke is varied in order to maintain vapor generator output
pressure in the TECO design. This is done by two inputs. The first input is
from the cut-off control and roughly holds pump flow equal to engine demand
flow.
This can best be understood by comparing the engine and pump flow equa-
tions. The pump flow can be described as
Qp = Kp • RPM • Sp • T (133)
where
Kp = Pump displacement factor at full stroke,based on
engine speed, Ib/hr-rpm
RPM = Engine speed in revolutions per minute
Sp = Pump stroke (unitless) where Sp = 1 = full displacement
r\ •= Volumetric efficiency
Referring back to Equation 126,
-------
The pressure trim on the pump stroke assures a pressure control and
compensates for variations in pump and engine efficiencies which are not ac-
counted for in Equation 134. Here a 5$pressure droop control will be used.
A decrease of 5$ from set point pressure of 500 psi will cause the pump to
go from zero to full stroke. To maintain a 500-psig vapor generator output
pressure at design point flow and pump speed will require a pressure set
point of 501. 25 psi, based on a volumetric efficiency of
\ - 1 -
The resulting instantaneous pump stroke equation is
P - 501.25
Sp = 3.448 (IR) ^5
CONDENSER FAN EQUATIONS
(135)
(136)
The condenser fan will be engine-driven and clutched between different
speed ratios as a function of engine speed in order to provide sufficient cool-
ing air at minimum power penalty. Equations supplied by TECO are
N = 3 X RPM for 0 < RPM < 800 (137)
N = 2 x RPM for 800 < RPM < 1400 (138)
N = 1 x RPM for RPM > 1400 (139)
where
N = Fan speed, rpm
RPM = Engine speed, rpm
DISCUSSION AND RECOMMENDATIONS
The control modeling reported here is almost the simplest possible, in
that it totally neglects the dynamics of the control elements. Yet, consider-
ing the fact that most of the control elements will react in fractions of a sec-
ond compared to many seconds response time for most of the systems ele-
ments, this assumption tis not unreasonable for a first cut at system modeling.
Modeling of the entire system using the instantaneous control model will
give a valuable insight into overall system response and will immediately point
up deficiencies in both control mode and component sizing. In fact, the one
significant lag to be expected in the control as it now stands is the time lag
of the freon-filled temperature sensing bulb used to sense boiler outlet tem-
perature. These bulbs frequently have equivalent time constants of several
seconds, and this lag should be incorporated into the model as soon as .tests
or analyses can be performed to determine its magnitude.
114
-------
As the system model is run through transients, the need for additional
compensating dynamics in the control system may become apparent. One area
of concern involves the dynamic effects in the vapor generator during tran-
sients. It is entirely possible that an initial increase in flow could conceiv-
ably produce an initial decrease in pressure and a longer-term increase. A
simple pressure control could be driven into instability (either sustained os-
cillations or a "hard over" condition) by such a situation.
In the event that the feedpump control were to react in this manner on the
system model, two possible cures immediately come to mind. The first would
be to provide limited authority to the pressure trim. The implications of this
on system safety would, of course, have to be considered aj: the same time.
The second approach would be to consider dynamic compensation (dashpot, etc. )
of the pressure trim, or else provision for dynamic compensation in the link .
between the cut-off valve and the feedpump stroke. The point to be observed
here is that the present system model is complete enough to show up such po-
tential problems and will allow realistic testing of the possible cures.
A useful tool in synthesizing controls will also be parametric models of
the system components. These will be simplified models whose dynamics
will be empirically matched to those of the more complete model. The para-
metric models will undoubtedly have to be altered for different operating
points, but will offer the advantage of reductions in computer time plus the
capability of closed-form analytical representation where possible. The lat-
ter will be valuable in providing analytical insight into the control problems
and will help to avoid the "cut and try" approach.
Using parametric models, the detailed control dynamics can be investi-
gated at reasonable cost. Here the individual loops can be checked out ana-
lytically first, for stability and then response. Following that, the complete
system can be checked for interaction effects due to control dynamics, using
the parametric models. A subsequent check of the controls with dynamics on
the complete detailed model would then be desirable.
The scope of this program did not permit consideration of alternate con-
trols or different mechanization of the TECO control system. However, the
general TECO control concept appears attractive in that it attempts to maintain
quasi-steady-state conditions during transients, thereby avoiding major un-
balances and upsets. Linking the feedpump stroke to valve cut-off is a form
of feed-forward that should make a much tighter pressure loop than a straight
pressure feedback control. Tying fuel flow to organic flow is also desirable
because of the long time constants to be expected in the temperature sens-
ing bulb.
The air/fuel mixture control will require additional design to assure
adequate control over expected operating and ambient ranges. It might be
desirable to control the air rather than the fuel and link the fuel flow to the
115
-------
air flow. Since the control of the combustor is crucial to emissions of the
engine, future modeling should be focused on this area to assure low emis-
sions during both steady-state and transient operation.
116
-------
Section 4
VEHICLE SYSTEM
In order to simulate propulsion system dynamics during transients that
occur in realistic driving situations the following vehicle-system models
were prepared.
e Transmission
• Vehicle
• R6ute
• Driver
Figure 50 shows the linkage of these models with the expander model.
In summary, the expander torque is conveyed through the transmission to
the vehicle wheels, where it acts to overcome motion resistance and accel-
erate the vehicle. The speed change is conveyed back through the trans-
mission to the expander. The driver model senses the vehicle speed, location,
and acceleration, compares them to reference values provided by the route-
mission profile, and makes an appropriate correction to the accelerator
pedal setting. This correction is transmitted through the controls to the
propulsion system. This section describes the vehicle system models.
Cut-off
or Throttle
Accelerator Pedal
Displacement
Acceleration
Transmission
Torque
Rpm
Driver
Vehicle
Wheel
Slip
Reference
Acceleration
Speed
Limit
Grade
Traction
Coefficient
TRANSMISSION
Figure 50. Engine Information Signal Loop
NOMENCLATURE
HR
Power required for driving auxiliaries and overcoming
drive train losses
117
-------
Alphabetical
Symbols
(Cont'd)
r
Rue
RIU
RPMe
RPM
x
RPM^g
RPM1U
Engine torque
Axle torque
Torque for driving auxiliaries and overcoming drive
train losses
Gear ratio
Cut-off points lu, xjg, 2u, s£ in Figure 51
Engine rpm
Axle rpm
Axle rpm at points iu, v£, su, a£ in Figu're 51
DERIVATION
The basic equations for the transmission model are:
RPM = r (RPM )
e x
(140)
(141)
(142)
MODEL DEVELOPMENT
The transmission model is entitled TRANSM and is listed in Volume II of
this report. The specific model has been developed for a transmission designed
for the Thermo Electron Corporation (TECO) engine. Figure 51 shows the
gear-shift sequence as a function of cut-off and RPM. There are three dis-
tinct regions.
118
-------
0.3 h-
0.275
0.25
0.225
0.2
0. 175
o
~ 0. 15
U
0.125
0.1
0. 075
0.05
0. 025
Downshift
Region I
I
Notes: For Transmission designed for
Thermo Electron Corporation
Engine
• Gear Ratio after Upshift,
0.584
Rear-end Ratio, 2.79
Upshift
Region III
200
800
400 600
Rpm at Axle
Figure 51. Transmission Gear-shift Sequence
1000
• In Region I the gear ratio is 1.
e The ratio remains 1 as the vehicle accelerates through Region II to
the upshift.
• In Region III the gear ratio is 0. 584 and remains 0. 584 as the vehicle
decelerates through Region II to the downshift.-
• There is also a rear-end ratio of 2. 79; therefore, the overall ratio
is the product of two values.
The power required for the transmission depends on speed and load, and
in most cases is very small. The maximum power requirement observed in
119
-------
the transmission data was approximateiy 3 hp. For simplicity, this maximum
power penalty was employed over the entire range. Therefore, HPj> = 3 hp.
The input to the model is r, RPMX, and Je, while the output is RPM
and Jx.
The upshift and downshift lines are automatically determined when r and
RPMX are supplied at points m, -JL, su, *$. (see Figure 51). In this instance,
R =0. 065
u?
R = 0.26
iu
R „ = 0.065
zS.
R =0. 155
su
RPM
RPM
i
RPM.
t
RPM
\JL
iu
370
550
430
660
RESULTS
The main point to be checked out in this model is the correct handling of the
upshifts and downshifts. Two cases were run, one case at r = 0. 035 and the other
case at r = 0. 15, both cases with increase and then decrease in axle speed. The
results are summarized below.
Case 1:
Input
r = 0. 035, J = 1000
e
Output
Initial
After Upshift
RPM
X
200
40Q
600
400 .
RPM -
e
558
1116
977
651
Jv
X
2790
2790
1629
1629
After Downshift 200
558
2790
120
-------
Case 2: r = 0. 15, J = 1000
e
After
Input
I nitial
Upshift
RPM
X
200
500
700
500
Output
RPMP
c
558
1395
1140
815
Jv
X
2790
2790
1629
1629
After Downshift 200 558 2790
VEHICLE
NOMENCLATURE
Alphabetical
Symbols
A Acceleration
Af Frontal area
C^ Aerodynamic drag coefficient based
on frontal area
D Drag
D Aerodynamic drag
Dff Grade drag
o
Dm Rolling and mechanical resistance
F Tractive effort
F Tractive limit
m
G Grade %
J Torque at axle
X
K/> Traction coefficient
M Rotational inertia
R^ Wheel radius
V Vehicle velocity
W Weight of vehicle
W Normal force on road
n
121
-------
Greek
Symbols
a Rotational acceleration
p Air density
DERIVATION OF BASIC EQUATIONS
The basic equation describing vehicle motion is Newton's Law:
A=-Sr 043)
Motion Resistance
The motion resistance is made up of aerodynamic drag, rolling and
mechanical resistance, and grade drag:
D = Do + D^ + D^ (144)
^^ cL m ^^ P
The aerodynamic drag is
Da= 1/2 p V3 Cd Af (145)
The rolling resistance is calculated by the method specified in Reference
29.
W I -a -5
Dm = JTJT 1 + 1.410 3V + 1. 2110 5 (V)2 (146)
\
where
Dm is in pounds
W is in pounds
V is in feet per second
The grade resistance is calculated as
D = W sin
arctan '0. 01 G| (147)
Tractive Effort
The torque applied at the rear axle must accelerate the vehicle and all
the rotating parts (drive train, transmission, and expander). Therefore the
tractive effort available is
Jv Ma
F = -2L - (148)
Rw Rw
122
-------
The maximum tractive effort that can be applied (or the tractive limit)
is
Fm = Kf W^ (149)
where
Wn = W cos
arctan (o. 01 G] (150)
If F is greater than F , the wheels will break away from the road
surface and start to slip. This is an important consideration for vapor
engines because the starting torques can be high. If the wheels are slip-
ping they accelerate at
J - F R
_ x m w nsi)
a - M ,(151)
If the wheels are not slipping:
a = ^- (152)
w
F is always less than or equal to Fm.
In both cases (wheel slip and no wheel slip) the linear and rotational ac-
celerations are integrated to determine the vehicle and rotational velocities.
The vehicle velocity, in turn, is integrated to determine the vehicle position.
MODEL DEVELOPMENT
The equations for vehicle motion are included in the total system model
MAINSYS, listed in Volume II of this report.
The following input data are specified by the user:
Cd, Af, Rw, M, W
The grade G and friction coefficient Kr are provided from the route profile.
The acceleration A, velocity V, and vehicle position are transmitted to the
driver model along with'the peripheral velocity, to check for wheel slip.
ROUTE
MODEL DEVELOPMENT
The route mission profile has been prepared as a data file, entitled
ROUTE, which is read by the total system model, MAINSYS. Each line of
the data file indicates:
1. The next marker location (LR)
2. The grade (G)
123
-------
3. Idle time (TI)
4. Reference acceleration (AR)
5. Reference velocity (speed limit) (VR)
6. Traction coefficient (KF)
7. A logic variable, to tell the driver whether to accelerate (CR = +1),
decelerate (CR = -1), or cruise (CR = 0).
The driver model compares the vehicle performance with the route ref-
erence conditions and adjusts the accelerator pedal accordingly. A new line
of route data is read whenever the vehicle velocity reaches the speed limit,
the idle time is exceeded, or the vehicle reaches the next marker location.
The data file ROUTE, which is listed in Volume II, is repeated here in
Table 10. The route profile corresponding to Table 10 is plotted in Figure
52 and includes:
1. Initial idle
2. Acceleration to 60 mph in 13. 5 seconds
3. Acceleration in merging traffic
4. Cruise at various speeds
5. High-speed pass maneuver
6. Acceleration on a 5% grade
This route mission profile allows testings of some of the important vehicle
performance goals in Reference 29.
It should be noted that the route profile is a forcing function but that the
vehicle might not meet the required or designated performance level. That
would depend upon the capability of the propulsion system.
80
70
60
50
High-speed
Pass Maneuver
_ 13. 5 sec acceleration
to 60 mph . .
K Acceleration
iri Merging
Traffic
0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2. 8 3. 0 3.2 3. 4 3. 6 3.8
sec Idle
Distance (miles)
Figure 52. Route Mission Profiles
124
-------
Table 10
ROUTE MISSION PROFILE
LR
(miles) G
0. 65 0
0. 65 0
0. 65 0
0. 65 0
0. 65 0
1.2 0
1. 2 0
1. 93 0
1. 93 0
1. 93 0
1. 93 0
2. 3 0
2. 3 0
2. 98 0
2. 90 0
2. 98 0
3. 65 0
3. 65 0
3. 65 0.
3. 65 0.
DRIVER
NOMENCLATURE
Alphabetical
Symbols
Af
AR
AS
TI
(sec)
1
0
0
0
0
0
0
0,
0
0
0
0
0
0
0
0
0
0
05 0
05 0
Frontal area
AR
(ft/ sec3)
0
10
6
3
3
-5
3
10
6
3
3
-10
3
2. 23
-10
3
-10
8.8
3
-10
VR
(mph)
0
40
50
60
60
25
25
40
50
70
70
50
50
80
50
50
0
65
65
0
KF
0.5
0. 5
0. 5
0,5
0. 5
0. 5
0. 5
0. 5
0. 5
0. 5
0. 5
0. 5
0. 5
•0. 5
0. 5
0. 5
0. 5
0. 5
0. 5 .
0. 5
CR
0
1
1
1
0
-1
0
1
1
1
0
-1
0
1
-1
0
-1
1
0
-1
Reference acceleration
Accelerator si
stting
125
-------
Alphabetical
Symbols
(Confd)
C Torque-accelerator constant for "linear engine"
Cd Drag coefficient based on frontal area
OR Logic variable -- acceleration (CR = +1), cruise (CR = 0),
deceleration (CR = -1)
G Grade
J Axle torque
J\.
Kf Traction coefficient
KA Accelerator sensitivity
LR Next marker location
M Rotational inertia
R, Wheel radius
TI Idle time
VR Reference velocity
W Vehicle weight
DEVELOPMENT OF MODEL
The driver model is entitled DRIVER and is listed in Volume II of this
report. The driver model closes the loop between the vehicle, propulsion
system, and route. The driver receives the following information signals:
1. Maximum idle time and elapsed idle time
2. Vehicle position and the next marker location
3. Vehicle velocity and reference velocity (speed limit)
4. Vehicle acceleration and reference acceleration
5. Wheel-slip signal
After analyzing this information, the driver model regulates either the ac-
celerator setting or the deceleration rate. If the elapsed idle time is less
than the maximum idle time, the accelerator setting is maintained at zero.
If the vehicle has not yet reached the next marker location,the driver attempts
to follow the current acceleration and speed limit instructions. When the
.vehicle reaches the next marker location, a new set of instructions are relayed
to the driver.
The acceleration and speed-limit instructions are obeyed as follows. A
logic variable tells the driver whether he is supposed to be accelerating,
126
-------
cruising, or decelerating. If the vehicle is to accelerate, the driver attempts
to reach the speed limit by adjusting the accelerator setting according to the
following schedule:
AS = ASO + KA (AR - A) (153)
where
AS = New accelerator setting
ASO = Previous accelerator setting
KA = Accelerator sensitivity
AR = Reference acceleration
A = Vehicle acceleration
If the vehicle is to cruise at the speed limit but the instantaneous velocity
is less than the speed limit, the driver adjusts the accelerator setting ac-
cording to Equation 153. If the vehicle velocity is greater than the speed
limit, the accelerator setting is adjusted by
AS= ASO+ KA (-A - AR), (154)
and if the vehicle velocity is equivalent to the speed limit, the setting is ad-
justed by
AS = ASO + KA(-A) (155)
If the vehicle is to decelerate, the driver sets the deceleration rate equal to
the reference acceleration, AR. There is no brake-force model. It is as-
sumed that the vehicle can decelerate at any selected rate.
All of the above considerations are superseded if the wheels are slipping. In
this instance, the driver model has two options. It can either hold the accelerator
setting constant or reduce it to zero. In each instance the acceleration sen-
sitivity, KA, is decreased so that the chances of future wheel slips are reduced.
It is felt that the above driver model presents a reasonable approximation
to the actual behavior of a human driver and will permit the various propulsion
systems to be evaluated and compared in a consistent manner.
RESULTS
The driver, vehicle, and route models were linked together and test
cases were run. The purpose of these tests was to determine if the linkages,
logic, and programming were correct.
An imaginary "linear engine" was used to provide the torque for acceler-
ation according to the schedule:
J = C AS
X
127
-------
If the C is large the torque will be high and wheel slip will occur. Two
cases were run to investigate
• Vehicle route traverse
• Wheel-slip conditions
Case 1: Vehicle Route Traverse
The purpose of this test was to determine if the driver model can hold the
vehicle on a given route model.
The following data were input:
C = 1200 ft-lb
M = 383 Ib ft3
Af = 25 ft3
Cd = 0. 5
W = 4000 Ib
R = 1 ft
w
KA = 0. 03
The route profile consisted of an acceleration to 30 mph, cruise at 30 mph,
followed by deceleration to rest (Figure 53). The route data are as follows: .
LR
(miles)
2.4
2.4
2.4
6.1
G
0
0
0
0
TI
(sec)
0
0
0
0
AR
(ft/ sec3)
7
5
5
-7
VR
(mph)
15
30
30
0
KF
0. 5
0. 5
0. 5
0. 5
CR
1
1
0
-1
The integration was first selected as one second; the result is shown as
Vehicle Traverse 1 in Figure 53. The vehicle tracks the reference conditions
well during the acceleration period, but overshoots and oscillates about the
speed limit. Reducing the integration step size to 0. 1 second significantly
reduces the overshoot and oscillation. These results indicate that the driver
model is capable of holding the vehicle on a given route profile if the engine
can provide the torque.
Case 2: Wheel Slip
The purpose of this test was to check out the wheel-slip logic of the driver
and vehicle model.
128
-------
Vehicle
Traverse No. 2
Vehicle
Traverse No. 1
IJ. 2
Distance (miles)
IT. 3
Figure 53. Comparison of Vehicle Traverse and Reference (Forcing)
Conditions for a Linear Response Engine
The data were the same as in Case 1 except that C was made to equal
12, 000 foot-pounds so that the wheels would break away.
One line of data was provided for the route:
LR
(miles)
0. 61
TI
(sec)
AR
(ft/sec2)
0
VR
(mph)
30
CR
1
Figure 54 shows the results. The wheels initially break away and accelerate
to a high speed (the speed would not be as high in a real engine, but the fictional
linear engine employed for this test is lossless). The driver model senses
the wheel slip, releases the accelerator, and the engine slows down. The
vehicle continues to accelerate.
In this instance the wheel speed undershoots the vehicle speed and the
driver depresses the accelerator at a reduced sensitivity (KA is cut in half).
129
-------
Wheel Periphery Velocity
— — — Vehicle Velocity
0.4
0.8
1. 2
Time (sec)
1. 6
2.0
2.4
Figure 54. Response to Wheel Slip -- Driver Releases Accelerator
and Reduces Acceleration Sensitivity
130
-------
The wheels break away again, the accelerator sensitivity is reduced again
until the wheel slip is under control. The above results illustrate the tran-
sients that the driver model is able to induce in the propulsion system.
The driver model is entitled DRIVER and is listed in Volume II of this
report.
131
-------
Section 5
TOTAL SYSTEM
The total system dynamic behavior depends upon the component dynamics,
the control system and its dynamics, and the interaction between components
and controls. The total system model therefore links the components and con-
trols, and simulates the dynamic operation of the entire propulsion system.
In order to analyze the vehicle performance in a driving situation, the propul-
sion system is linked to the transmission and vehicle models, and the route
mission model is employed to generate system transients through the action
of the driver model. The system model developed here is entitled MAINSYS,
and is listed in Volume II (Users Manual) of this report.
METHOD OF SYSTEM ANALYSIS
The arrangement of component models in the system model is shown in
Figure 55. The expander, the transmission, and the vehicle are joined by
torque and speed linkages. The working fluid flow and properties link the ex-
pander, regenerator, condenser, pump, and vapor generator. The combus-
Air and Fuel
Speed and Torque
Linkages
Displacement
Transmission
Input
to Controls
Working fluid
Linkages
Figure 55. System Model Linkages
133
-------
tion gas flow and temperature link the combustor and vapor generator. Speed,
acceleration, and route conditions link the vehicle, route, and driver. The ex-
pander cut-off, pump displacement, condenser airflow, and fuel and combus-
tor airflow are set by the controls which close the loop linking the driver to
.the rest of the system. The control system linkage is not shown in Figure 55.
The above arrangement provides a means of information transfer between
models. The dynamic models of the components, derived earlier, predict
transient behavior in response to input disturbances. The system model pro-
vides the necessary information link to change the input signals of a compo-
nent in accordance with the variation in output signals of a neighboring com-
ponent. Thus component interaction is maintained.
The direction of information flow depends upon the particular use of the
system model; as will be shown later, the direction for deriving the total
system steady-state condition differs from the direction for system transient
analyses.
If the path connecting the components (duct, shaft, etc. ) possesses a sig-
nificant static or dynamic behavior, this should be considered when informa-
tion is transferred between the components in the system model.
STEADY-STATE CONDITION
The usual first step in total system transient analysis is to derive steady-
state cycle conditions and detailed distributions for each heat exchanger com-
ponent at a desired power level. System transients are then superimposed
on this steady-state condition.* The procedure to derive the steady-state
distributions for the individual heat exchanger models was explained in the
subsection "Heat Exchangers" of Section 3.
The system steady-state condition is derived as follows. Initial esti-
mates are made for the expander cut-off, pump displacement, fuel and air-
flows, and some working fluid properties around the cycle (see Figure 56).
The expander and pump speed are fixed.
Starting with the combustor model, the steady-state value of combustion
gas temperature is obtained and transmitted to the vapor generator model.
By application of the steady-state solution procedure, the pressure and enthalpy
of the fluid stream at exit are obtained, as well as the detailed nodal distri-
bution. These values are transmitted to the expander model to find the fluid
flow rate and exit fluid enthalpy.
*It is not necessary to first bring the system to steady state. For example,
a cold-start situation can be analyzed where the heat exchanger walls are
not initially in steady state. All that is required is that the initial tempera-
ture distribution be prescribed.
134
-------
In a similar manner each component is individually brought to a steady-
state condition based upon the exit conditions of the previous component.
The calculations proceed sequentially around the cycle in the direction of the
working fluid until the vapor generator inlet is reached. The engine speed
can be input to the transmission and the vehicle velocity determined.
p -14.7
T -85
m«0. 2435
Cut-off « 0.147
Rpm • 2000.
Route '
I
I
_*_
JW3U*id3
Trinsmission I Jfe Vehicle f
'" -"-•-•'- '*»-r.** FraEBip••"»'
_
Input to
Control*
^ -oK".*..'.i,.V '5T71
, Regenerator ! 1
*~.:\T£fff7r. I
p • 550. 0
h - -124.
m- 2. 05
Rpm < 2000
Stroke - 0. 434
-126
h • 43.
m- 2. 05
Condenser
175.
P • 14. 7
T- 85.
fc- 17
Figure 56. Total Systems Model -- Initial Estimates to Derive
Cycle Design Conditions
At this point each component will be in a steady-state condition. The
mass flows and fluid properties around the cycle may not match, however,
because of the initial estimates on pump stroke, engine displacement, and
fluid conditions. The control system can then be linked to the models so
that these parameters are varied according to the control laws until the
final steady-state condition is obtained.
TRANSIENT SIMULATION
After the total system model is brought to steady state at a particular
operating condition corresponding to engine idle or vehicle cruise, propul-
sion-system transients can be analyzed. This requires a different linking of
135
-------
components, which was employed in deriving the steady-state condition. This
linkage is shown in Figure 57.
Reference
Figure 57. Dynamic System Information -- Signal Flow Dia-
gram, Excluding Controls
The route mission information is transmitted to the driver, which acts
by changing the accelerator pedal displacement. This changes other system
parameters (fuel flow, feedpump stroke, expander cut-off, etc. ) in accor-
dance with the control strategy.
Initially, the cycle conditions and mass flow rates are at the steady-state
condition previously derived. As the pump stroke and cut-off are adjusted by
the control system, the mass flow through the pump and expander varies.
This mass flow imbalance is transmitted to the vapor generator, regenerator,
and condenser, causing a change in pressure in these heat exchanger compo-
nents.
The new pressure levels are transmitted back to the expander and pump,
and the power and torque are determined. The expander torque is conveyed
through the transmission to the vehicle; this results in a change in vehicle and
engine speed. The driver senses the acceleration rate and velocity of the ve-
hicle, makes a comparison with the route mission demands, adjusts the accel-
erator pedal accordingly, and the transients continue.
136
-------
The procedure is repeated to obtain new cycle conditions at selected time
intervals. At each time interval, the new cycle conditions are stored and the
effect of the next system disturbance is evaluated, starting from the cycle
conditions at that time. This is continued until the route-mission traverse
is completed. The cycle conditions plotted as a function of time represent
the system dynamic behavior.
SYSTEM MODEL STRUCTURE
The system program, MAINSYS, is designed with a major emphasis on
simplicity and flexibility. The program combines the components as shown
in Figure 55. However, its modular structure and special data input arrange-
ment makes the addition or elimination of any component (for example, regen-
erator) a simple task. Of course, the information signals rerouted as a re-
sult of such changes should be properly considered in the system model.
The input component design data and the initial values of various cycle
parameters are organized on a component basis; hence, modification of data
files is also easy. Similarly, the program output is printed on a component
basis, with proper identification and clearly defined boundary values. A
logic variable is available to specify the printing of additional details of the
component model if these are needed. At the termination of the program,
a detailed list of additional information is printed out for each fluid pass of
the thermal components.
The system program can also be used,with minor changes, to study in-
dividual component transients. The changes are primarily due to the need
to rearrange the direction of informational flow signals at the exit plane, as
explained in Section 3 (subsection "Heat Exchanger") for the vapor generator.
RESULTS
The system program was run to derive steady-state cycle conditions un-
der full-power design conditions. The program does not include any static or
dynamic losses between the components, and the controls are not included.
For the present purpose, the route and the driver models are bypassed.
Figure 56 represents the initial cycle conditions required to start the
program. Figure 58 gives the cycle conditions obtained after one iteration,
and Figure 59 is the actual computer print-out for this case. Since the mass
flows through the expander, feedpump, and vapor generator and the fluid en-
thalpy values at the regenerator exit and vapor generator inlet do not match,
further iterations with control interaction are required to bring the system
to final steady state. After this has been accomplished, system transients
can be run.
137
-------
CO * 0.01056
00
oo
NO - 0. 01009
0. 00007
T - 85
ED> 0.2435
tn> 0.0123
Cut-off * 0. 147
R pen * 2000
- —•-•-t-1 Torque
Expander
304.04
Rpm = 1228.0
Torque » 474. 48
p * 546.24
h « -81.069
m - 2. 050
' 25.0
' 88.413
m * 2.0936
Input to
Controls
I Regenerator I
*' ~ ~~ '
P '
h >
• 550. 0 f[
: _
m= 2
Rpm • 2000
Stroke * 0. 434
i
i
p • 550.
m - 2.
Feedpump
124.01 1
.050 j
P •
rh -
in *
° p « 24. 553
h « -128. 9(
m- 2. 0936
22. 355
46. 666
2. 0936
' ^
r
J^Condenscri
p * 14.571
T * 175. 78
J « 14.7 f
r * as. oo I
i» 17.0 '
Mph - 90. 84
Figure 58. Total Systems Model -- Information Flow at Cycle Design Condition, Without Controls
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140
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141
-------
Section 6
DISCUSSION AND RECOMMENDATIONS
Mathematical models required for digital simulation of the dynamics of
Rankine cycle automotive propulsion systems have been developed. The fol-
lowing propulsion system components have been modeled:
Working fluid -- water and organic
Combustor
Vapor generator
Expander -- reciprocating and turbine
Regenerator
Condenser
Feedpump
Controls
Transmission, vehicle, driver, and route models have been developed to
simulate transients produced in actual driving situations.
The dynamic models are valid over any operatingrange, and they have been
calibrated with experimental results when such results were available. Work-
ing fluid properties and geometric and dimensional data are input quantities;
hence, change of working fluid and design modifications are easily accounted
for. The modular structure of the system 'program allows change in compo-
nent arrangement or addition or elimination of any component (for example,
regenerator). The programming language is FORTRAN IV and the models
have been run on the General Electric 635 digital computer.
Additional important features, along with the strengths and limitations of
each component model, are summarized in Table 11.
Data for a propulsion system with a reciprocating expander and organic
working fluid (Ref. 1) were provided as input, and the component models were
subjected to some representative open-loop transients. The component models
were linked together to form a total system model, which was exercised with-
out controls to derive the steady-state condition.
Although the main emphasis to date has been on model development
rather than analysis of results, several recommendations can still be
made. These are based on the experience obtained from the development
of the models and the limited number of computer runs made for calibra-
tion and verification.
143
-------
Table 11
SUMMARY OF COMPONENT MODELS
Component
Working Fluid
Features
Limitations
1. Thermodynamic properties in 1. Thermodynamic models are Computer memory requlre-
superheatcd or saturated re-
gion determined through tabu-
lar interpolation.
2. Transport properties curve-
Tit with polynomial functions.
3. Input data for water. CP-34,
and FC-75.
valid for any fluid for which
tabular data are available.
2. Model results in excellent
agreement with standard
thermodyramie tables.
ment for tabular data.
Time required for full
table search
Combustor
Vapor Generator
1. Calculates emissions and com-
bustion gas temperature.
2. Thermal transients calculated
employing lumped-parameter
method.
3. Air preheated before entering
combustion zone.
1. Once-through monotube cross-
flow configuration, with fluid
passes arranged as in Refer-
ence 1.
2. Distributed parameter model. 2.
3. Fluid change of phase (sub-
cooled, boiling, and super-
• heated) accounted for.
1. Vapor-generator dynamic
behavior is simulated over
wide nonlinear operating
range.
Distributed parameter ap-
proach accounts for variation
in properties along length
of vapor generator.
Design modifications are
easily accounted for by
changing input data.
Forward finite-difference
method gives explicit rela-
tions. Associated stability
limits on distance and time
step are computed in the pro-
gram so that integration is al-
ways stable.
Emissions are based on
steady-state emission
data
Complexity
-------
Table 11 (Cont'd)
Expander-turbine
Feedpump
Transmission
1. Quasi-steady model: speed
change determined by ve-
hicle dynamics.
2. Single-stage axial Impulse
turbine.
3. Calculates off-design per-
formance.
4. Nozzle and rotor losses ac-
counted for.
1. Variable stroke single-stage
reciprocating pump.
2. Cavitation effects Included.
Specific model for two-speed
transmission from Reference 1.
Good correlation with exper-
mental data
Losses would be under-
estimated for low specific-
speed applications.
Vehicle
Route
Driver
Controls
1. Motion is calculated on the
basis of excess torque and
road condition.
2. Air drag, grade, and trac-
tion loads are considered.
3. Wheel-slip conditions are
predicted.
1. Input data in tabular form.
2. Reference velocity, acceler-
ation, grade and Idle time
specified as a function of
marker location.
Instantaneous model: physi-
ological effects on gain not
considered.
1. Instantaneous control mod-
els.
2. Strategy generally based on
that described In Reference 1.
3. Power, working fluid, and
air/fuel control relations.
Table can be extended to pro-
duce any driving cycle.
Perception-execution de-
lay or physiological ef-
fects on gain not consid-
ered.
HEAT EXCHANGER DYNAMICS
The preliminary runs of the heat exchanger models (vapor generator,
condenser, and regenerator) indicate that the heat exchanger transients have
time constants of the order of 10 to 20 seconds. The other components (ex-
pander, pump, and controls) can be considered quasi-steady relative to the
heat exchangers. Therefore the thermal inertials will determine propulsion
system response. It is important to be able to accuratelypredict heat ex-
changer transients so that controls can be developed that will anticipate sys-
tem changes and make appropriate adjustments.
Since heat exchanger dynamics are fairly sensitive to heat transfer coef-
ficients, several assumptions made concerning the details of the heat transfer
should be verified.
145
-------
For example, the fluid capacitance of the vapor generator is related to the
percentage volume occupied by the superheat region. The preliminary model
results indicate that neglect of the water-jacket resistance in the TECO design
resulted in a superheat region more than twice as long as calculated in Ref-
erence 1. However, the actual heat transfer mechanism in the 1/10-inch water
jacket between the heat exchanger walls is not well known. By the use of the
transient model this resistance can be varied parametrically in order to deter-
mine the sensitivity of the vapor generator performance to its value.
A second area where the vapor generator model can be expanded is the
prediction of the heat transfer coefficient in the region wherexthe quality is
between 0. 8 and 1. 0. The two-phase heat transfer coefficient in the present
model is based on the film boiling correlation below quality 0. 8, and a linear
interpolation between film boiling and convective heat transfer between qual-
ities 0. 8 and 1. 0. This cut-off point is purely arbitrary and can be improved
by using a better criterion, like the critical heat flux condition if such data
are available.
If dynamics are found to be sensitive to the boiling-heat transfer correl-
ation, the existence of various flow regimes -- stratified, annular, dispersed,
bubble, or plug flow -- and their effect on boiling mechanisms can be included
(Ref. 30).
All of the heat transfer mechanisms have been treated as quasi-steady;
that is, steady-state correlations have been employed with instantaneous flow
rates and fluid properties. Furthermore, standard correlations have been
used which can vary from 10$ to 20$ when applied to different designs. There-
fore it is essential to calibrate the heat exchanger models with transient ex-
perimental data for the particular designs to be analyzed.
Finally, in addition to the control system instability problems that can be
caused by heat-exchanger dynamic response, the vapor generator itself may
be subject to flow instabilities. Analyses of the latter were outside the scope
of the present study, but they should not be ignored. Appendix IV, "Evapor-
ator Flow Instability, " of this volume gives a brief description of the effects
and outlines the recommended steps for predicting stability limits.
On the basis of the above discussion, the following, Recommendation 1,
is made.
Recommendation 1:
• Sensitivity analyses should be carried out for
Heat transfer coefficients
Two-phase/vapor transition point
Water-jacket resistance
146
-------
These analyses would employ the models for parametric variation of
the above items to determine their effect on steady-state and dynamic
performance.
• The correlation of two-phase flow and heat transfer should be ex-
panded to account for the various two-phase flow regimes.
• The heat exchanger models should be validated with transient ex-
perimental data.
RUN TIME ECONOMY
Figure 60 shows computer cost information for the vapor generator tran-
sient model. The data points used to construct this curve were obtained from
actual computer runs with the model on the GE 635 computer.
20
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-------
a. More efficient utilization of fluid property model
b. Using fixed time steps and lump sizes in the heat exchanger models
c. Development of "parametric models"
The heat exchanger models employ the fluid property models a large num-
ber of times. The heat exchanger models are incremented in approximately
I/100-second time steps as dictated by the stability criteria, and the fluid
property models are called for each time step. Since the fluid properties do
not change significantly for several time steps, the fluid property models could
be used less frequently with little or no reduction in accuracy.
Furthermore, each time a fluid property model is used the entire satu-
rated or superheated table is searched to find the appropriate fluid properties.
For many transient situations the fluid properties are not expected to vary
over such a wide range. An alternative search routine could therefore be
employed where a selected region of the fluid property tables is searched
first.
Another alternative would be to employ a binary search technique. Here
the table is initially divided in half. It is then determined which half of the
table contains the required fluid property; this half is, in turn, divided in half.
The process is continued until the property is converged upon. More efficient
utilization of the fluid property models should improve run-time economy.
The manner in which time steps and lump sizes are determined in the heat
exchanger model is explained in Section 3. In summary, at a particular time
the minimum lump size is determined from the stability criteria, and a node
pattern is set up along the length of the heat exchanger. If the node pattern is
different from that of a previous time, fluid and transport properties are de-
termined at the new nodes by linear interpolation. The minimum time step is
then determined, employing the stability criteria at each node. This proce-
dure is repeated at each time step.
Considerable run-time economy can be achieved if a fixed time step and
lump size are employed. The only requirement is that the values used to be
less than as specified by the stability criteria. This can be assured by cal-
culating the stability limits over the anticipated operating range and selecting
smaller values for the fixed time step and lump size. The sections of the
model which calculate the stability limits can then be bypassed when transient
runs are made. If by chance the stability limits are exceeded during a run it
will be immediately obvious, as the solutions will rapidly diverge. A simple
limit check can be employed to stop the program if this happens.
A final method that can be employed to reduce run time and costs is to
develop a set of "parametric models. "
148
-------
The dynamic component models are exercised for several transients at
different power levels. The resulting open-loop performance is plotted para-
metrically and curve-fit with appropriate functions. The curve fits are pro-
grammed and then employed as models. It should be noted that the paramet-
ric models are not simple or approximate, involving a number of restrictions.
They pro.vide a valid representation of process dynamics over a limited range.
Parametric models are more economical than the full-range dynamic models
in situations requiring continual repeated use.
In summary of the above discussion the following. Recommendation 2,
is made.
Recommendation 2:
In order to improve run-time economy it is recommended that
• Fluid property models be utilized more efficiently.
• Fixed time steps and lump sizes be employed in the heat exchanger
models.
• Parametric models be developed.
CONTROL DEVELOPMENT AND SYSTEM DYNAMICS
Since the control performance depends directly on the process dynamics,
a valid dynamic model of the inherently complex behavior of the Rankine cycle
is required. The nature of system dynamics imposes the constraint on con-
trol dynamics. The optimum control approach would utilize the major fea-
tures of system dynamics, would maintain desired operating conditions dur-
ing any normal working range, and would be fail-safe under unusual working
situations.
Over the wide operating range inherent in route mission situations, the
process dynamics may differ significantly. The detailed, nonlinear models
developed maintain all important dynamic features of the system at any power
level. Hence, the models developed are an extremely powerful and useful
tool which should be employed for the control system development.
By the use of paranretric models, the detailed control dynamics can be
investigated at reasonable cost. Here the individual loops can first be checked
out analytically for stability and then for response. Following that, the com-
plete system can be checked for interaction effects due to control dynamics us-
ing the parametric models. A subsequent check of the controls with dynamics
on the complete detailed model would then be desirable.
Recommendation 3:
The following control systems development plan is recommended.
149
-------
• The instantaneous control models that have been developed should
be employed to bring the total system to steady state at the design
condition, and small perturbation transients around this point should
be analyzed. This will establish the basic validity of the control
scheme.
• Acceptable limits on the variation of system parameters during trans-
sients should be established (e. g. , the variation in expander mass
flow demand during driving situations, assuming a droop in boiler
pressure).
• The full-range dynamic models should be used to derive limited -
range parametric models as described above.
• The instantaneous control scheme should be modified to include con-
trol dynamics. The controls should be developed by means of the
parametric models.
• The final control scheme should be checked out with the full-range
dynamic models.
Following this, route mission profiles can be traversed in order to de-
termine fuel economy and emissions in grams per mile. The main advantage
of this approach is in the minimal use of the full-range dynamic model and the
economies achieved thereby. Furthermore, the parametric models will visu-
ally display the essential features of component dynamics and will help guide
the control development.
APPLICATION TO SYSTEM DEVELOPMENT
Automobiles are always in transient operation; hence, development of pro-
pulsion systems requires thorough understanding of process dynamics so that
control systems can be developed to insure desired performance over the to-
tal operating range. This is especially critical for maintaining low emission
levels in spite of rapidly changing load conditions. To avoid costly and time-
consuming development cycles, system simulation can and should be used in
the early stages to uncover major problem areas in operation and control.
The models developed under the present program are highly flexible and
general. They can be used to simulate the dynamics of (a) the water-based
system with reciprocating expander, and (b) the organic-working-fluid sys-
tem with turbine expander as well as (c) the organic fluid system with recip-
rocating expander for which they were checked out. The models can be easily
modified and used to study the effect of transients typical of each system.
Recommendation 4:
The models should be employed for transient analysis of the systems and
components under development by the Environmental Protection Agency,
and the results used to support and guide the design and experimentation.
150
-------
APPENDIX I
-------
Appendix I
PARAMETRIC PROPULSION SYSTEM DESIGNS
In order to employ the dynamic model for simulation of Rankine cycle
performance, dimensional and geometric data for the propulsion systems
must be provided as input. Such data can come from existing system designs
or can be generated.by the use of the parametric design programs discussed
in this Appendix.
These programs were calibrated with the Thermo Electron Corporation
engine design -- simple reciprocating expander with CP-34 as working fluid
(Ref. 1) -- and then employed to prepare preliminary designs for three other
systems:
• Simple reciprocating engine with water as the working flud
• Turbine engine with FC-75 as the working fluid
• Compound reciprocating engine with water as the working fluid
Each system was designed for approximately 105 horsepower. The boiler
conditions (pressure and temperature) at design were chosen fairly arbitrar-
ily (reasonable values without exceeding the stability limits of the working
fluid). Similarly, the condenser conditions.at design were selected to give
reasonable condensing temperatures.
These values could be optimized for high efficiency, but that was not the
purpose here. The fluid properties around the cycle at design and the expander
sizes were determined by the use of the programs EEFF (for the simple recip-
rocating expander), ECOMP (for the compound expander), and TSIZE (for the
rotating expander). These programs are listed in Volume II, the Users Manual.
The results are listed in Tables 12 to 15.
The components were sized on the basis of the design cycle conditions.
The combustor, vapor generator, condenser, and regenerator dimensions
are presented as Tables 16 to 22. These were determined by the use of the
programs BLSIZ1 (for single-phase flow regimes of the vapor generator),
BLS1Z2 (for two-phase flow regime of the vapor generator), CONDSZ (for
the condenser), and RGSIZE (for the regenerator). These programs are
listed in Volume II.
The heat exchanger components were sized using a computerized NTU
(number of thermal units) method. In order to check the NTU method, the
programs were run for the Thermo Electron system and the results compared
to those given in the TECO report. This comparison is also shown in the
tables.
151
-------
Table 12
CYCLE DESIGN CONDITIONS FOR RECIPROCATING ENGINE
WITH CP-34 AS WORKING FLUID
Location
Boiler Exit
Engine Exit
Regenerator Vapor Exit
Condenser Exit
Pump Exit
Regenerator Liquid Exit
Mass Flow Rate: 7301 Ib/hr
Engine:
Pressure
(psia)
500
25
25
25
500
500
Temperature
CF)
550
348
230
196
199
285
Enthalpy
(Btu/lb)
123
77
43
-126
-124
-90
Specific
Volume
(ft3/lb)
0. 187
4.05
3!4
0. 016
0. 016
0. 018
Boiler:
Regenerator:
Condenser:
Pump:
Cycle Efficiency:
4 cylinders. 4.42-in. bore, 3-in. stroke
Design conditions: 2000 rpm, 127 indicated mean effective pressure.
0. 137 intake ratio. 107. 6 hp
Heat rate 1. 56 108 Btu/hr, efficiency (higher heating value) 82. 5<
Heat rate 2.468 106 Btu/hr
Heat rate 1.236 106 Btu/hr
5. 2 hp at design
16.8<
Table 13
CYCLE DESIGN CONDITIONS FOR RECIPROCATING ENGINE
WITH WATER AS WORKING FLUID
Location
Boiler Exit
Engine Exit
Condenser Exit
Pump Exit
Mass Flow Rate:
Engine:
Boiler:
Condenser:
Pump;
Cycle Efficiency:
Pressure
(psia)
1000
24
24
1000
Temperature
820
237
217
219
Enthalpy
(Btu/lb)
1401
1098
185
189
Specific
Volume
(ft3/lb)
0. 702
16. 1
0.0167
0.0167
939 Ib/hr
4 cylinders, 2. 78-in. bore, 3-in. stroke
Design conditions: 2000 rpm, 306 indicated mean effective pressure.
0. 137 intake ratio, 105. 3 hp
Heat rate 1.139 10s Btu/hr, efficiency (higher heating value) 82. 5<
Heat rate 8.575 10s Btu/hr
1. 26 hp at design
23. 3«
152
-------
Table 14
CYCLE DESIGN CONDITIONS FOR TURBINE ENGINE
WITH FC-75 WORKING FLUID
Location
Boiler Exit
Turbine Exit
Regenerator Vapor Exit
Condenser Exit
Pump Exit
Regenerator Liquid Exit
Pressure
(psia)
Temperature
446
374
230
177
177
329
Enthalpy
(Btu/lb)
130
121
78. 6
36. 7
37. 2
79. 4
Sperilir
Volume
(fr'Vlh)
0. 0553
2 49
0 01
0 01
x
Mass Flow Rate:
Engine:
Boiler:
Regenerator:
Condenser:
Pump:
Cycle Efficiency:
30.770 Ib/hr
Single-stage impulse turbine. 7. G5-in. diameter. 0. 6-in. blade height,
0. 012-in. tip clearance, 10" nozzle angle, 0. 392-inf throat area. 2. 62-in'.
exit area
Design conditions: 12, 872 rpm, 0. 548 speed ratio. 2. 63 Mach number.
105 hp
Heat rate 1. 55 108 Btu/hr, efficiency (higher heating value) 82. 5<
Heat rate 1.3 108 Btu/hr
Heat rate 1. 285 106 Btu/hr
6. 6 hp at design
16. 1*
Table 15
CYCLE DESIGN CONDITIONS FOR COMPOUND ENGINE
WITH WATER AS WORKING FLUID
Location
Boiler Exit
First-stage Engine Exit
Second-stage Engine Exit
Condenser Exit
Condenser-Pump Exit
Feedwater Heater Exit
Boiler-Pump Exit
Pressure
(psia)
Temperature
820
475
235
215
216
442
444
Specific
Volume
Mass Flow Rate:
Engine:
First stage:
Second stage:
1240 Ib/hr
062 Ib/hr
Boiler:
Condenser:
Pumps:
First stage: 2 cylinders, 2. 68-in. bore. 3-in. stroke
Design conditions: 2000 rpm. 361 indicated mean effective pressure.
0. 37 intake ratio
Second staye: 2 cylinders, 4.05-in. bore, 3-in. stroke
Design conditions: 2000 rpm. 132 indicated mean effective pressure.
0. 37 intake ratio
Hp at design: 106. 6
Heat rate 1.21 10B Btu/hr, efficiency (higher heating value) 82.5''
Heat rate 8. 76 106 Btu/hr
Condenser pump. 0. 4 hp at design
Boiler pump, 1.31 hp at design
Cycle Efficiency: 24.
153
-------
Table 16
COMBUSTOR DESIGNS
Design Conditions
82. 7 £ boiler-burner effectiveness
21, 600 Btu/lb, higher-heating-value
3330*F combustion-gas temperature
18. 8 air/fuel ratio
As In TECO design, two combustion
Heat Release at Design (Btu/hr)
Heat Release Maximum (Btu/hr)
Fuel Rate at Design (Ib/sec)
Fuel Rate Maximum (Ib/sec)
Air Rate at Design (Ib/sec)
Air Rate Maximum (Ib/sec)
Burner Length (In. )
Burner Diameter (In. )
Burner Weight (Ib)
Shell Length (In. )
Shell-Burner Hydraulic Diameter (in. )
Shell Weight (Ib)
Combustion Gas -Burner Wetted Area (In?)
Shell-Air Wetted Area (In?)
Air-Burner Wetted Area (In?)
fuel
chambers with equal
Reciprocating
Engine with
CP-34
1.G1 10'
2. 1 10*
0. 0245
0. 0275
0.485
0. 534
17
7
3.15
17
4
6.8
374
454
374
heat release rates are employed.
Reciprocating
Engine with
Water
1.38 10"
1. 51 10*
0.0178
0. 0194
0.362
0.385
12
7
2.22
12
4
4.8
263
320
263
Compound
Reciprocating
Water
1.46 10'
1.60 10*
0.0187 v
0. 0205
0.371
0.466
13
7
2.34
13
4
5.16
284
346
284
Turbine
Engine with
FC-75
1.88 10*
2. 06 10*
' 0. 0242
0. 0267
0.479
0. 529
16.7
7
3. 1
16.7
4
6.7
368
446
368
Table 17
RECIPROCATING ENGINE SYSTEM
WITH CP-34 AS A WORKING FLUID
Cnil 1
Working Fluid
Klow rate (Ih/sec) 2-05
Inlet temperature CF> 200
Exit temperature ('¥) 390
Pressure (psi) 550
Combustion Gas
Klow rate
-------
Table 18
SIMPLE RECIPROCATING ENGINE SYSTEM
WITH WATER AS WORKING FLUID
Vapor Generator Design
Working Fluid
Flow rate (Ib/sec)
Inlet temperature (°F)
Exit temperature (°F)
Pressure (psi)
Combustion Gas
Flow rate (Ib/sec)
Inlet temperature (°F)
Exit temperature (°F)
Tube Diameter
Outer (in. )
Inner (in. )
Extended Surface, Outer
Ball diameter (in.)
Matrix thickness (in. )
Matrix porosity
Outer fin height (in. )
Outer fin thickness (in. )
Outer fin spacing
Extended Surface, Inner
Inner fin height (in. )
Inner fin thickness (in. )
Inner fin number
Tube spacing (pitch) (in. )
Tube length (ft)
Coil 1
0. 253
220
545
1000
0. 369
1274
330
1
0. 9
Ball Matrix
3/32
0. 5
0.39
--
--
—
None
--
--
--
2
61
Coil 2
0. 253
545 (0 qual. )
545 (1 qual. )
1000
0. 369
3330
1776
1
0. 9
Finned
--
--
--
0. 356
0. 012
10
Finned
0. 120
0.0312
16
2. 0
8.3
Coil 3
0.253
545
820
1000
0. 369
1776
1274
1
0. 9
None
™" ~
--
--
--
--
--
None
--
--
--
1. 1
19
155
-------
Table 19
TURBINE ENGINE SYSTEM WITH FC-75 AS WORKING FLUID
Working Fluid
Flow rate (Ib/sec)
Inlet temperature (°F)
Exit temperature (°F)
Pressure (psi)
Combustion Gas
Flow rate (Ib/sec)
Inlet temperature (°F)
Exit temperature (°F)
Tube Diameter
Outer (in. )
Inner (in. )
Extended Surface, Outer
Ball diameter (in. )
Matrix thickness (in. )
Matrix porosity
Outer fin height (in. )
Outer fin thickness (in.)
Outer fin spacing
Extended Surface, Inner
Inner fin height (in. )
Inner fin thickness (in. )
Inner fin number
Tube spacing (pitch) (in. )
Tube length (ft)
Vapor Generator Design
Coil 1
Coil 2
8. 55
362
402
220
0. 503
1850
400
1
0. 9
Ball Matrix
3/32
0. 5
0. 39
8. 55
402
446
None
2
15
0. 503
3330
1850
1
0. 9
Ball Matrix
3/32
0. 5
0.39
None
2
4. 5
156
-------
Table 20
COMPOUND RECIPROCATING ENGINE SYSTEM
WITH WATER AS WORKING FLUID
Vapor Generator Design
Working Fluid
Flow rate (Ib/sec)
Inlet temperature (°F)
Exit temperature (°F)
Pressure (psi)
Combustion Gas
Flow rate (Ib/sec)
Inlet temperature (°F)
Exit temperature (°F)
Tube Diameter
Outer (in. )
Inner (in. )
Extended Surface, Outer
Ball diameter (in. )
Matrix thickness (in. )
Matrix porosity
Outer fin height (in. )
Outer fin thickness (in. )
Outer fin spacing
Extended Surface. Inner
Inner fin height (in. )
Inner fin thickness (in.)
Inner fin number
Tube spacing (pitch") (in. )
Tube length (ft)
Coil 1
0. 345
445
545
1000
0.39
1102
702
1
0.9
Ball Matrix
3/32
0. 5
0. 39
--
--
--
None
--
--
--
2
7. 75
Coil 2
0.345
545 (0 qual. )
545 (1 qual. )
1000
0. 39
3330
1755
1
0. 9
Finned
--
'
0. 356
0. 012
10
Finned
0. 120
0. 0312
16
2
9
Coil 3
0.345
545
820
1000
0. 39
1755
1102
1
0.9
None
--
--
--
--
--
None
--
--
1-. 1
49. 5
157
-------
Table 21
CONDENSER DESIGNS
Flat Tubes with Louvered Fins
Number of Parallel Tubes 30
Frontal Height 19.9 in.
Tube Width. Outer 0. 75 in.
Tube Width, Inner 0. 69 in.
Tube Height, Outer 0. 206 in.
Tube Height, Inner 0. 146 in.
Tube Spacing (Pitch) 0. 664 in.
Fin Number 14/in.
Fin Height 0.465 in.
Fin Thickness 0. 0025 in.
Metal - Tubes Steel
Metal - Fins Copper
Frontal Width 50 in.
en
oo
Design Heat Rate (Btu/hr)
Working Fluid
Flow rate (Ib/sec)
Quality, in
Quality, out
Condenser Temperature (°F)
Pressure (psi)
Reciprocating
Engine with
CP-34
1.23 10e
2. 03
1
0
216
25
Reciprocating
Engine with
Water
0.857 10s
0. 261
1
0
237
24
Compound
Reciprocating
Engine - Water
0. 876 108
0. 267
1
0
235
23
Turbine
Engine with
FC-75
1. 51 108
8. 54
1
0
177
7.35
Air
Flow rate (Ib/sec)
Inlet temperature (°F)
Tube Length (in. )
Calculated
TECO design
Condenser thickness
17
95
240
200
3. 75
11.8
95
167
3
12
95
174
3
20. 7
95
384
6
-------
Table 22
REGENERATOR DESIGNS
Designs Based Upon TECO Regenerator
Liquid Flow Inside Tubes. Gas Flow Across Tube Banks
Number of Flow Sections
Tube Banks
Tube Diameter, Outer
Tube Diameter, Inner
Tube Spacing (Pitch)
Outer Surface
Matrix Porosity
Ball Diameter
Matrix Thickness
Matrix Height
Metal
Flow Rate (Ib/sec)
Gas Side
Inlet temperature (°F)
Exit temperature (°F)
Pressure (psi)
Liquid Side
Inlet temperature (°F)
Exit temperature (°F)
Pressure (psi)
Tube Length (in. )
Calculated
TECO design
25 in. long, 4 tubes high
0. 550 in.
0. 5 in.
0. 85 in.
Ball Matrix
0.35
0. 0625 in.
0.29in.(CP-34), 0.55 in. (FC-75)
0. 3 in.
Steel
CP-34 System FC-75 System
2.05 8.54
348
230
25
199
285
540
456
400
374
230
7. 35
177
322
220
1334
159
-------
It can be seen that the NTU method does not accurately duplicate the TECO
results in all cases. In fact, in some instances it differs by about 25 percent.
This is probably due to the fact that the heat exchanger effectiveness corre-
lations employed in these programs did not properly represent extended-sur-
face heat exchangers (fins and ball-matrix). It should be noted that in the
case of the bare surfaces the agreement is much better.
The NTU method is probably adequate for the purpose --to roughly size
the heat exchanger components so that the dynamic models can be checked out
for the various systems.
In any case, if the dimensions derived for these components &re grossly
incorrect it will be immediately apparent in the dynamic modeling and modifi-
cations can be made.
The instantaneous control equations have been derived for the reciprocating.
expander with CP-34 as working fluid in Section 3 (subsection "Controls") of
this volume. Employing the same techniques, the following, instantaneous con-
trol equations apply for the reciprocating expander with water as the working
fluid specified as the preceding paragraphs.
For Intake Ratio and Feed Pump
IR = 0. 8 AO
where
IR = Intake ratio
AO = Operator's input
The following limits apply:
MAX IR = 0. 8 for 0 s RPM * 342. 5 (157)
274
MAX IR = _ * for RPM > 342. 5 (158)
KlrM
MIN IR = 0.8 for 0 <: RPM s 285 (159)
MIN IR = 0. 05333 (300-RPM) for 285 SRPM < 300 (160)
MIN IR = OforRPM>300 (161)
The pump-stroke equation, based on 5$ pressure drop, is
S = 1 25 (IR) {Pb ' 1001-42)
Sp 1.25(IR) __D _ (162)
160
-------
APPENDIX II
-------
where
S_ = Normalized pump stroke
IR = Intake ratio
= Boiler outlet pressure psi
For Fuel Air Control
Qf = 3.42836 + 0.00070 Qb
+ 0. 00011 Q2 - 4.09780 I0~e Qg (163)
where ^
Q = fuel flow (Ib/hr) at 820°F boiler outlet temperature
and
Q = boiler inlet flow (Ib/hr)
Qf = Q* + K (T - 820«) (164)
where
K = -2.125 + 0. 00118 OX
b • b
Q. = fuel flow (Ib/hr) for small deviations
(± 10°F) off 820°F boiler outlet temperature (°F)
T = boiler outlet temperature
The maximum fuel flow limit is 77 pounds per hour, and Qf can never go
negative. The air fuel relationship is
= 19.8Qf (165)
where Q. is airflow rate (Ib/hr).
161
-------
Appendix II
STABILITY AND ERROR CRITERIA FOR FINITE-DIFFERENCE SOLUTION
OF PARTIAL DIFFERENTIAL EQUATIONS
NOMENCLATURE
Alphabetical
Symbols
A
H
h
m
n
Q
t
v
w
X
\
Flow cross-section area
Enthalpy by finite-difference solution
Enthalpy by actual solution
Mass flow rate
Node position n
Heat addition
Time
Specific volume
Discretization error
Distance
mv At 1
A Ax
The finite-difference approximation of a partial differential equation
(PDE) should satisfy convergence criteria. The difference between the finite-
difference solution (H) and the solution of PDE (h) at any grid point is known
as the local discretization error,w, defined as
w = h-H
The convergence requires that w-« 0 as the grid spacing Ax and At tend to
zero; this requirement will give stability criteria. During computation, since
only a finite number of digits can be retained by the computer, the round-off
error is introduced. Generally, this error grows in direct proportion to the
grid refinement.
To obtain the stability criteria, examine the following energy equation:
A_ ah _ •
v St
m
(166)
For simplicity assume m and Q constants for the given grid and for a
given time interval At. From Taylor's series expansion, supposing that h
possesses a sufficient number of derivatives,
163
-------
h (n+1, t) = h (n, t) + Ax
-«f
+ 0
h (n+1, t+At) = h (n+l,t) + At . +
2!
[(Ax)3]
0 [(At)3]
Then
. ah m
,. u / 4.\ (Ax)2 a2h
, t) - h ° KJj
where the derivatives are evaluated at ((n+1) AX, t)
From Equations 166 and 167, the PDE can be represented as
m
m v At
A Ax
-h
a2h
ax2"
(168)
From Dusinberre's explicit finite-difference approximation of Equation
166 (using H to denote the approximate solution),
JKn+l.t+AtM1^ |H H(n't)
H (n+l,t) +^^ (169)
From the definition of discretization error w, and from Equations 168
and 169, and using X = (mv/A)(At/Ax).
w (n+1, t+At) = X w (n, t) + (l-X)w (n+1, t) +
+ Z (n+l.t) (170)
where
Z (n+1, t) =
0 [(Atf j + 0
If 0 < X £ 1, the coefficients X and (1 - X) are non -negative, and the inequality
is
w (n+1, t+At) * X w (n, t) + (1-X) w (n+1, t)
|Z (n+l,t) (171)
It can be proved that, provided 0< X^l, the discretization error is
0[(A*)2] and' 0[(At)a], and thus the explicit finite-difference representation
converges as Ax -0 and At-0.
Note that Dusjinberre's stability criterion is equivalent to the convergence
criteria developed above. It can be shown that, fora linear partial differential
equation, stability is a necessary and sufficient condition for convergence.
164
-------
APPENDIX III
-------
Appendix III
HEAT TRANSFER AND PRESSURE DROP RELATIONS
NOMENCLATURE
Alphabetical
Symbols
A
B
b
Cf, f
D
d
E
G
h
k
m
n
P
Pr
R
Re
S
T
t
w
x
y
Flow area
Used in fin efficiency equation
Fin height
Friction factor
Specific heat
Hydraulic diameter
Tube diameter
Fin (or ball matrix) effectiveness
Volumetric flow rate
Heat transfer coefficient
Conductivity
Entrance expansion coefficient
Exit expansion coefficient
Mass flow rate
Number of fins per inch
Pressure
Prandtl number
Matrix porosity (based on flow area)
Reynolds number
Wetted area
Temperature
Tube spacing
Fin width
Quality
Length
Fin efficiency
165
-------
Alphabetical
Symbols
Cont'd
Subscripts
bTi
bTo
e
eTi
eTo
f
g
gTo
k
L
m
Ti
Tif
v
Viscosity
Density
Bare inner tube wall
Bare outer tube wall
Fins above
Finned inner wall
Finned outer wall
Working fluid
Combustion gas
Combustion gas and tube outer wall
Fuel
Liquid phase
Matrix
Tube inner wall
Tube inner wall and working fluid
Vapor phase
The heat transfer and pressure drop relations used for the Thermo Elec-
tron vapor-generator case are listed below.
HEAT TRANSFER COEFFICIENTS
BETWEEN WORKING FLUID AND INNER TUBE WALL (hTif)
Single-phase Flow
hTif = (0.023)
Gf
Uf
kf
(172)
where the fluid properties are evaluated at the average fluid bulk temperature
at a particular location.
Two-phase Flow
The heat transfer coefficient is broken up into two components (Ref. 31):
166
-------
hTif =
The convective heat transfer coefficient, h , is defined as:
= (0.023)
where
GfDTi
Ti
(173)
Re
C T ^1
pL ]
(173a)
F = Function (XT)
XT =
1 - x
o.g
'V
OS
u
V
u
0 1
and
XT &0.1
0.1 < XT * 0. 4
0. 4 < XT * 2
2 < XT
Function
Function
Function
Function
= 1
1.996 (XT>°-3
2. 730
2. 584
(I73b)
The boiling heat transfer coefficient, ha, is defined as:
0.79 0.45 0.49
(0.00122) kr C T p. (32.2)
j_i
OJ3* 0.75
(AT) (Ap) s
0.5 0.29
^ UT
1V
0.34 0.34
) PV
(174)
wnere
AT
T - T
Ti xf
*» = Pv(TTi> - PV(V
1.35
s = Function Re % (F )
where F is as defined in Equation 173a and
(174a)
167
-------
l-a.6
< 20,000; s = 2. 282 - 0. 15 in ( Re^ F )
. 1'3B
Re^F < 200,000; s = 3. 343 - 0. 257 £n (Re A F )
1.36
10,000 <
20,000 *
1.Z 5 1.8
200,000 s Re^F < 4 10 ; s = 2. 032 - 0. 150 j?n ( Re 4 F )
4 10
s = 0. 1
(I74b)
Further,
o(T) = 0. 2317 10-
1 -
T
584
0. 818
L318
(175)
The above two-phase heat transfer correlation for hTif is valid up to a
quality of 0. 8-, for qualities between 0. 8 and 1. 0>
,o.e
h3 = 0. 023
and 174)
,0-4
PV
k
DTi
x = °- 8' from Equations 173
(176)
and
hTif =
(176a)
Fins In Tube
If the inside of the tube is finned, the heat transfer coefficient becomes:
, „
hTif " ETi
SeTi
(177)
where
SeTi
SbTi
nT. (2bT.)
(178)
E
Ti
SeTi SeTi
(179)
168
-------
"Ti (2bTi * WTi>
tanh
keTi WTi
BETWEEN COMBUSTION GAS AND OUTER TUBE WALL (h 9To)*
Bare Tube
where
h _ = 0.237
gTo
Re
Pr
Pr
g g
G D
C G
pg t
.
"
g
Finned Tube
r -0.388 -2/
0. 1632 [Re^ Pr_ / j C. _ G
g
g
pg g
Then
E
To
bTo
where
(183)
(184)
(I84a)
(185)
m
eTo
To
^ ^
. To To To
bTo
eTo
*Only convective heat transfer relations are given here. The radiative effect,
if important, should be included.
169
-------
To ITo
To
eTo
2nTo (bTo
WTo bTo/dTo>
(186c)
'To
tanhlBTo bTol
BTo bTo
(186d)
/Vr
o 2
keTo WTo
(186e)
Ball Matrix Between Tubes
-0.3 - S/S
h. _, = 0. 23 (Re Pr ' ) G C
bTo g g g FW
h _ = E hum 5. 249
gTo m bTo
(187)
(188)
E
m
tanh (z)
= 3.63 (1.94 10 ) B
m
(189)
(190)
B
bTo
m
k D
m m
(191)
PRESSURE DROP
WORKING FLUID
Single-phase Flow
AP
2f Gfa
°Tipf
(192)
Gf°Ti
If (Re,) = — ^- < 3000
^ ^
f = 16/Ref
If 3000 s Ref < 20,000
f = 0. 0791/Ref
0.85
170
-------
o.a
If 20,000 * Ref
f = 0. 046/Ref"
Two-phase Flow -- Martinelli-Nelson correlation (Ref. 30, p. 79)
Ap = Ap
where Ap = friction pressure drop
Apa = acceleration pressure drop
Ap
(193)
AP,
Ap,
where Ap^ is the pressure drop as if all the fluid is liquid at the saturation
temperature, and
Ap. 3 !.a
-7-^- = (C» (1 - X)
ApL
where x is the average quality in length Ay
0 = Function (XT)
(194)
(195)
where XT is defined in the section on two-phase-flow heat transfer. The
functional relationship between Q) and XT is:
In Q) = 1.4516 - 8.688 10~* in (XT) +
5.463 lO* [j£n (XT)]
The acceleration pressure drop is:
- 0.4784 [in (XT)]3 (196)
A r-\
Aps
1 - x9
( PL2
_x^\ (197)
V2
where the subscripts 1 and 2 refer to locations in the tube.
COMBUSTION GAS (Ref. 32)
G..«
Ap =
- 1
pv p*
171
- U-a2-K3)
(198)
-------
KX, K2, and a are defined below; subscripts 1 and 2 refer to axial locations.
Bare Tube
Finned Tube
0 = (t
To
To
(199)
(200)
f (Cf, Reg)
Cf (Reg)
T5.18
(201)
Cf (z) = 0.3906 z - 0. 3321
= WTO
(202)
(203)
Re_
GgDTo
Kj = 0
Ka = 0
> (204)
o -
- dTo " 2nTo bTo WTo
To
(205)
S . 4(dTo
A
D
To
(206)
f(Re
g
(a)
Ka(a)
-O.346
0. 151 (Re )
g
3 Z
0.395 a + 0. 685 a + 0. 065 a+0.
-0. 87 a + 1. 573 a - 2. 46 a + 1
.(207)
Ball-matrix Tube
a = R
m
(208)
172
-------
S = 4 (0. 0417)
A DTo
-1.179 -1.4B2
f (Re ) = 78. 63 Re + 1.397 Re
g g g
K = 0
(2U9)
\ (210)
K = 0
HYDRAULIC DIAMETERS
Hydraulic diameters are defined as
4A .
INNER TUBE
Bare Tube
Finned Tube
Ti
D
TTd
Ti
(211)
OUTER TUBE
Bare Tube
D = 4(tTo - dTo'
TT
Finned Tube
(212)
D
g
(tTo- - dTo
WTo bTo) (dTo + 2bTo)
(dTo bTo
2WTO bTo) + nCTo
(213)
Ball-matrix Tube
4 R
D
m
g 6(l--Rm)
(214)
173
-------
FLOW RATES
G = m/A
(215)
WORKING FLUID
Bare Tube
G =
(216)
Finned Tube
Gf = mf
1 nTiWTibTi
COMBUSTION GAS
(217)
Bare Tube
(m
(tTo - dTo) LTo
(218)
Finned Tube
g (tTo * dTo - 2nTo WTo ^ LTo
(219)
Ball-matrix Tube
mk)
W
To
Rm LTo
(220)
174
-------
APPENDIX IV
-------
Appendix IV
EVAPORATOR FLOW INSTABILITY
Under various conditions, some of which have been relatively well defined
and analyzed, static, dynamic, or compound flow instabilities can exist in
evaporators. These instabilities can lead to either high-frequency (acous-
tic) pressure oscillations or low-frequency pressure and flow oscillations
under subcritical or near-critical thermodynamic conditions. They have
caused premature burnout or mechanical evaporator failure when various
means now available for avoiding unstable operation have not Ueen introduced.
Static instabilities, such as the chugging oscillation, occur mostly with
alkali liquid metals and with fluorocarbons, where large superheat is required
for nucleation and where the boiling curve hysteresis sustains the oscillation.
Dynamic instabilities, such as acoustic or density wave instabilities, can
occur at low (5 - 30 Hz) or high (>1000 Hz) frequencies; the low frequencies
are usually associated with long evaporators. The dynamic instabilities can
be associated with large-amplitude (100-psi amplitude in a 500-psia system)
pressure oscillations.
The primary phenomena leading to static instabilities can be predicted by
using steady-state criteria or correlations. The threshold of static instabil-
ity can thus be predicted. For dynamic instabilities linearized solutions of
the constitutive equations, together with carefully taken experimental data,
have permitted the correct definition of several stable and unstable operating
regions. These methods of analysis (Refs. 33-36) at least allow the de-
signer to predict whether he is clearly stable or unstable or if he is in a
"grey zone".
The following steps are therefore recommended for predicting evaporator
stability or alleviating existing instabilities:
1. Determine system or loop instability by using Ledinegg criterion.
Check for static instability, using steady-state correlations, to
avoid or alleviate instabilities caused by boiling crises (dry wall),
flow pattern transition, etc.
2. Check for the onset of dynamic (density wave) instabilities by using
a simplified analytical model (Ref. 35) or empirical correlations
(Ref. 37).
3. Validate the prediction from step 2, above, by a dynamically instru-
mented and controlled (single-tube) experiment carried out at the
phase density ratio of interest. Since the amplitude of oscillations
in a multitube evaporator will be less than that in a single-tube unit,
this procedure should result in a conservative design.
175
-------
4. Use the analytical model in step 2 and the instrumented experimental
test section in step 3 to define system or control changes required to
alleviate existing instabilities.
176
-------
APPENDIX V
-------
Appendix V
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-------
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