Download Dynamic Modeling and Control Strategies for a

Dynamic Modeling and Control Strategies for a
Micro-CSP Plant with Thermal Storage Powered
by the Organic Rankine Cycle
by
Melissa Kara Ireland
B.S., Cornell University (2008)
Submitted to the Department of Mechanical Engineering
in partial fulfillment of the requirements for the degree of
Master of Science in Mechanical Engineering
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
February 2014
c Massachusetts Institute of Technology 2014. All rights reserved.
Author . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Department of Mechanical Engineering
January 17, 2014
Certified by . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
John G. Brisson
Professor
Thesis Supervisor
Accepted by . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
David E. Hardt
Chairman, Department Committee on Graduate Theses
2
Dynamic Modeling and Control Strategies for a Micro-CSP
Plant with Thermal Storage Powered by the Organic
Rankine Cycle
by
Melissa Kara Ireland
Submitted to the Department of Mechanical Engineering
on January 17, 2014, in partial fulfillment of the
requirements for the degree of
Master of Science in Mechanical Engineering
Abstract
Organic Rankine cycle (ORC) systems are gaining ground as a means of effectively
providing sustainable energy. Coupling small-scale ORCs powered by scroll expandergenerators with solar thermal collectors and storage can provide combined heat and
power to underserved rural communities. Simulation of such systems is instrumental
in optimizing their control strategy. However, most models developed so far operate
at steady-state or focus either on ORC or on storage dynamics. In this work, a
model for the dynamics of the solar ORC system is developed to evaluate the impact
of highly transient heat sources and sinks, thermal storage, and the variable loads
associated with distributed generation.
Based on an existing micro-CSP (concentrating solar power) plant, the dynamic
model is implemented in the Modelica modeling language. Detailed steady-state component models, which are implemented in EES and validated to data where available,
form the basis for the dynamic components. The dynamic model in its current form
is used to make qualitative assessments of several control decisions based on realistic
solar irradiance input representing four reference days. Future analysis will survey
a wider range of environmental conditions to make quantitative determinations on
the efficacy of each control decision. The simulations include an approximation for
startup and shutdown, which avoids the numerical issues associated with the discontinuities in the working fluid density derivative present during such rapid phase
changes. To the author’s knowledge, this is the first model capable of continuously
simulating through startup and shutdown in addition to coupling a dynamic thermodynamic model of the power cycle with dynamic models of the solar collectors and
thermal storage tank.
Thesis Supervisor: John G. Brisson
Title: Professor
3
4
Acknowledgments
People come to help you in many ways over the course of a thesis.
Some put you on the right path. Prof. Tim Gutowski’s enthusiasm for the sustainable energy field and his encouragement to find my own way allowed me to wait
until I found the project that perfectly combined my passion for sustainable energy
and the thermodynamic modeling skills I use every day on the job. I am grateful
to Rob Stoner of the MIT Energy Initiative not only for greatly improving a related
class paper but also for introducing me to the topic that would later become this
thesis. I am greatly indebted to Dr. Sylvain Quoilin for his patience in bringing me
up to speed on all things ORC thermodynamics and dynamic modeling. His lab’s
sponsorship of my trip to University of Liège was essential in making me effective on
this project.
Others provide you with the skills and guidance to sustain you along the way.
I am grateful that Dr. Matthew Orosz gave me the chance to contribute to this
project. He devoted many hours of his time deepening my understanding of the
system and providing valuable suggestions to improve my writing. Adriano Desideri
was instrumental in keeping this project going. His willingness to delve into the
details querying the models and helping me understand and improve upon them
undoubtedly contributed to making this project a success. Prof. Harry Hemond’s
willingness to stay after hours, ask thought-provoking questions, and provide useful
suggestions assured me I was on the right track. My advisor, Prof. John Brisson,
provided invaluable guidance in crafting this work into a respectable technical paper.
His persistent encouragement to think more acutely and put myself in others’ shoes
inspired this work’s depth and clarity.
Yet others provide a sounding board when things are driving you crazy. I cannot
thank Dave Blum enough for his completely unselfish support whenever I had an
issue, from helping with class projects and concepts to providing Matlab tips and
tricks to his willingness to engage on any question no matter the strain he was under
with his own work. Phil Knodel was also a great resource for venting frustrations and
5
some useful brainstorming sessions when things weren’t making sense. Friends like
Bethany Kroese and Nina Shinday, under considerable pressure themselves, always
made time for a constructive chat.
Some come in at the very end to give you a final push. Thank you to Dave
Gutz who donated a significant amount of his time troubleshooting and improving
my understanding of control systems.
Finally, some provide constant support throughout your life no matter what your
endeavor. I am grateful to my parents for shaping me into the person I am today and
providing me with the skills, opportunities, and perseverence to complete a project
like this one. Neil is an amazing companion whose extreme patience and support,
compassionate ear, and good nature kept me going.
6
Contents
Abstract
3
Acknowledgements
5
Contents
7
List of Figures
9
List of Tables
11
Nomenclature
13
1 Introduction
17
1.1
Background and Literature Review . . . . . . . . . . . . . . . . . . .
17
1.2
Pilot System Description: Basis for Models . . . . . . . . . . . . . . .
19
2 Steady-state modeling
25
2.1
Evaporator/Recuperator . . . . . . . . . . . . . . . . . . . . . . . . .
25
2.2
Expander-generators . . . . . . . . . . . . . . . . . . . . . . . . . . .
26
2.3
Condenser . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
29
2.4
Pumps/Motors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
2.5
Solar Collectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
38
2.6
Application: Determining Optimum Set Points for Varying Working
Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
41
2.7
Potential Improvement to Cycle Optimization: Variable Condenser
Temperature Defect . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3 Dynamic modeling
46
51
3.1
Evaporator/Recuperator . . . . . . . . . . . . . . . . . . . . . . . . .
53
3.2
Condenser . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
55
3.3
Liquid Receiver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
56
3.4
Solar Collectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
57
3.5
Storage Tank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
58
3.6
Control Unit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
59
3.7
Application: Identifying Optimal Control Schemes for Daily Environmental Variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4 Conclusions and Future Work
62
71
4.1
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
71
4.2
Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
73
A Model Code and Experimental Data
75
B Sensitivity Studies on Heat Exchanger Modeling
77
B.1 Mixed vs Unmixed Cooling Air . . . . . . . . . . . . . . . . . . . . .
77
B.2 Variable vs Constant Heat Transfer Coefficients . . . . . . . . . . . .
79
B.3 Tube-to-Tube Heat Transfer . . . . . . . . . . . . . . . . . . . . . . .
80
B.4 Sensitivity to Number of Cells . . . . . . . . . . . . . . . . . . . . . .
82
C Optimized Intermediate Pressure between Two Expanders
8
85
List of Figures
1-1 Pilot micro-CSP plant. . . . . . . . . . . . . . . . . . . . . . . . . . .
20
1-2 Flow diagram for the micro-CSP plant. . . . . . . . . . . . . . . . . .
21
1-3 T-s diagram for the micro-CSP plant. . . . . . . . . . . . . . . . . . .
21
2-1 Contour plot of expander isentropic efficiency based on to Equation 2.10. 28
2-2 Finned-tube condenser geometry. . . . . . . . . . . . . . . . . . . . .
30
2-3 Modeling schematic of condenser tube bank. . . . . . . . . . . . . . .
31
2-4 Goodness of fit for condenser heat transfer correlation (Equation 2.13). 33
2-5 Goodness of fit for fan power consumption correlation (Equation 2.14). 33
2-6 Goodness of fit for WF pump isentropic efficiency correlation (Equation 2.16). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
35
2-7 Goodness of fit for WF motor efficiency correlation (Equation 2.17). .
36
2-8 Goodness of fit for HTF pump power consumption correlation (Equation 2.18). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
37
2-9 Collector modeling schematics. . . . . . . . . . . . . . . . . . . . . . .
38
2-10 Goodness of fit for collector heat loss correlation (Equation 2.19). . .
40
2-11 Goodness of fit for optimum evaporation pressure correlation (Equation 2.22). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
43
2-12 Optimum evaporation pressure correlation (Equation 2.22). . . . . . .
45
2-13 Simulation results for updated condenser heat transfer correlation (Equation 2.23). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
47
2-14 Goodness of fit for updated condenser heat transfer correlation (Equation 2.23). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
48
2-15 Results for system optimization using updated condenser correlation
for single working condition. . . . . . . . . . . . . . . . . . . . . . . .
49
3-1 Flow diagram for the micro-CSP plant. . . . . . . . . . . . . . . . . .
52
3-2 Modelica interface for the dynamic system model. . . . . . . . . . . .
52
3-3 Discretized heat exchanger model showing cell vs node parameters. .
53
3-4 Discretized condenser model showing cell vs node parameters. . . . .
56
3-5 Modeling schematic for liquid receiver. . . . . . . . . . . . . . . . . .
57
3-6 Modeling schematic for solar collectors. . . . . . . . . . . . . . . . . .
58
3-7 Modeling schematic for storage tank. . . . . . . . . . . . . . . . . . .
58
3-8 Modelica interface for the control units for the Pev,opt and Tev,const
strategies. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
61
3-9 Environmental conditions for control strategy comparison. . . . . . .
63
3-10 Controller results for Pev,opt strategy on Day 2. . . . . . . . . . . . . .
64
3-11 Effect of thermal storage for Pev,opt strategy on Day 2. . . . . . . . . .
65
3-12 Power produced or consumed by each component for Pev,opt strategy
on Day 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
67
3-13 Power consumed by condenser fans for Pev,opt strategy on Day 2. . . .
68
3-14 Control strategy comparison results.
. . . . . . . . . . . . . . . . . .
70
B-1 Modeling schematic of condenser tube bank. . . . . . . . . . . . . . .
78
B-2 Average condenser temperatures for tube-tube heat transfer analysis.
81
B-3 % error in condenser heat flow vs number of cells per tube. . . . . . .
82
10
List of Tables
1.1
Pilot system components and major modeling parameters. . . . . . .
23
2.1
Expander correlation coefficients for Equation 2.10. . . . . . . . . . .
28
2.2
WF motor correlation coefficients for Equation 2.17. . . . . . . . . . .
36
2.3
HTF pump correlation coefficients for Equation 2.18. . . . . . . . . .
37
2.4
Manufacturer parameters for solar collector optical efficiency. . . . . .
39
2.5
Collector heat loss correlation coefficients for Equation 2.19. . . . . .
40
2.6
Optimum evaporation pressure correlation coefficients for Equation 2.22. 43
2.7
Condenser heat transfer correlation coefficients for Equation 2.23. . .
11
48
12
Nomenclature
A
Area, m2
Ap
Aperture width, m
cp
Specific heat capacity, J(kgK)-1
CS
Control signal
CSP
Concentrating solar power
D
Diameter, m
EES
Engineering Equation Solver
f
Friction factor
G
Mass flux, kgm-2 s-1
h
Enthalpy, Jkg-1
HL
Heat loss, Wm-1
HT F
Heat transfer fluid
HV AC
Heating, ventilation, and air conditioning
IAM
Incidence angle modifier
Ib
Beam radiation, Wm-2
Init
Initialization
k
Thermal conductivity, W(mK)-1
L
Length, m
m
Mass, kg
ṁ
Mass flow rate, kgs-1
N
Quantity
Nrot
Rotational speed, s-1
ORC
Organic Rankine cycle
p
Pressure, Pa
P ID
Proportional-Integral-Derivative (control)
PV
Process Variable
q
Heat flux, Wm-1
Q̇
Heat flow, W
13
Re
Reynold’s number
rp
Pressure ratio
rv
Volume ratio
U
Internal energy, J, or Heat transfer coefficient, W(m2 K)-1
vw
Wind velocity, ms-1
V
Volume, m3
V̇s
Ideal volume flow rate, m3 s-1
SH
Superheat, ◦ C
SP
Set point
t
Time, s or thickness, m
T
Temperature, ◦ C
W
Width, m
Ẇ
Power, W
WF
Working fluid
∆x
Length of cell, m
Greek symbols
η
Efficiency
γ
Specific heat ratio
ν
Specific volume, m3 kg-1
φ
Expander filling factor
ρ
Density, kgm-3
θ
Incidence angle,
◦
Subscripts and superscripts
abs
Absorber
amb
Ambient
c
Center
col
Collector
cond
Condenser or conduction
const
Constant
conv
Convection
14
crit
Critical
cross
Cross-sectional
el
Electrical
em
Electromechanical
env
Environment
ev
Evaporator
ex
Exhaust
exp
Expander
f
Fluid or finside
gl
Glazing
h
Hydraulic
i
Inner
ins
Insulation
int
Intermediate
l
Liquid
lm
Log mean
m
Mechanical
nom
Nominal
N
Number of cells
o
Outer
opt
Optimum
p
Pump
r
Right
rad
Radiation
s
Swept or isentropic or surface
sf
Secondary fluid
sol
Solar
su
Supply
t
Tubeside
w
Wall
15
16
Chapter 1
Introduction
1.1
Background and Literature Review
Organic Rankine cycle (ORC) systems are gaining ground as a means of effectively
providing sustainable energy. Micro-CSP (concentrating solar power) plants based on
ORCs present a cost-effective solution to the challenge of supplying heating, cooling,
and electricity for rural health and education centers outside the range of a centralized grid [13]. Although ORCs tend to have lower efficiencies than traditional steam
Rankine cycles, the thermodynamic properties of organic fluids lead to several distinct advantages in the low to medium power range. For example, the slope of the
vapor saturation curve and the super-atmospheric saturation pressures of some organic fluids preclude the need for superheating and the removal of non-condensable
gases, respectively, used in steam Rankine cycles, reducing the complexity, cost, and
maintenance requirements [12].
Coupling micro-CSP plants with thermal storage allows for an increase in daily
operating time and also for a damping of the rapid variations that may be seen in
the solar input (e.g., due to clouds). However, input temperatures during normal
operation may still vary as much as ±20◦ C throughout the year. Furthermore, using
readily-available components, like re-purposed HVAC (heating, ventilation, and air
conditioning) scroll compressors for expansion, while cost-effective, results in the additional design objective of preventing over- or under-expansion of the working fluid
17
in the fixed-volume ratio scroll machines. The effective design of such a system depends on modeling and identifying a control strategy with the ability to adjust to
transient operating conditions.
Several authors have simulated solar thermal ORC systems in steady-state. Orosz
[13] created a physical and economic model in EES (Engineering Equation Solver)
that predicted solar ORC performance over a year for the specified location and
system components. This model condensed the hourly irradiance variation over the
year into a single average value reduced by expected cloud cover to predict net power
output and daily and annual energy production. McMahan [12] presented a validated
steady-state ORC model and an optimization methodology for solar ORCs based on
finite-time analysis but no simulation of the entire solar ORC system.
Other authors have focused on ORC or thermal storage dynamics in isolation.
Wei et al. [22] developed a dynamic ORC model and examined the efficiency of
moving boundary versus discretized heat exchanger techniques when compared to
experimental data. Casella et al. [3] developed a dynamic ORC model for normal
operation by building on the existing ThermoPower library in the Modelica modeling
language. This model was validated in steady-state and transient conditions based on
a grid-connected turbogenerator with natural gas or diesel generator exhaust as the
heat source. It was also used to simulate feedback control to match desired turbine
inlet temperature by varying pump speed. McMahan [12] examined the accuracy
and computational efficiency of several techniques for simulating packed-bed storage
dynamics in charging/discharging and idle conditions.
In addition, Twomey et al. [21] simulated grid-connected solar ORC system dynamics with the solar loop modeled as a single lumped component and solar irradiance
represented by a daily half sine curve whose amplitude was adjusted for monthly averages to estimate daily and annual power generation.
In this work, a model for the dynamics of the solar ORC system is developed to
evaluate the impact of highly transient heat sources and sinks, thermal storage, and
the variable loads associated with distributed generation. The next section describes
a micro-CSP plant that has been developed to provide combined heat and power to
18
underserved rural communities. This system is used as the basis for the detailed
steady-state component models discussed in Chapter 2, which are implemented in
EES and validated to data where available. In turn, the steady-state models provide
the basis for the dynamic models described in Chapter 3, which are developed in
the Modelica modeling language. The dynamic model in its current form is used
to make qualitative assessments of several control decisions based on realistic solar
irradiance input representing four reference days. Future analysis will survey a wider
range of environmental conditions to make quantitative determinations on the efficacy
of each control decision. The simulations include an approximation for startup and
shutdown, which avoids the numerical issues associated with the discontinuities in the
working fluid density derivative present during such rapid phase changes. Chapter 4
summarizes the conclusions of the analysis and future work.
A summary of this analysis has been submitted as a conference paper for the
ASME Turbo Expo 2014 and is currently under review.
1.2
Pilot System Description: Basis for Models
Over the past several years, researchers at MIT and University of Liège have collaborated with the non-governmental organization STG International to design a microCSP plant suitable for rural power generation. Several field prototypes have been
installed. In addition, a pilot system, pictured in Fig. 1-1, resides at Eckerd College
in St. Petersburg, Florida, and this system is used as the basis for the models.
Figure 1-2 presents a flow diagram identifying the relationships between the major components. The design of this system hinges on the use of readily-available,
commercial parts to reduce cost. The heat transfer fluid (HTF), propylene glycol,
is warmed by the sun while traveling through 20 single-axis concentrating parabolic
troughs. This fluid is the heat source for an organic Rankine cycle using HFC-245fa
as the working fluid (WF) with counterflow brazed plate heat exchangers for evaporation and recuperation plus modified HVAC scroll compressors for expansion and a
HVAC air-cooled condenser for heat rejection. The two hermetic scroll compressors
19
Figure 1-1: Pilot micro-CSP plant showing the air-cooled condenser, thermal storage, and
solar collectors with the ORC in the lower left inset.
are modified to run as expander-generators as proposed and successfully tested by
Lemort et al [11]. The counter-crossflow condenser with corrugated plain fins, which
utilizes three 61 cm diameter fans to provide cooling air, has a very complicated
geometry and will be discussed in more detail in the modeling sections. Thermal
storage is achieved with a glycol-filled, double-wall 0.79 m3 tank surrounded by 25
mm thick fiberglass wrap insulation. The target net output power is 3 kWe at an overall efficiency from solar power input to ORC power output of 3% and ORC thermal
efficiency from evaporator heat input to net power output of 8%.
Figure 1-3 is the T-s diagram for the design point of the system with a HTF supply
temperature to the ORC of 150◦ C and ambient temperature of 25◦ C. Starting at the
entrance to the pump, the WF is compressed to the high-side pressure (thermodynamic states 1-2 in the figure). The liquid is preheated in the recuperator (states 2-3)
before entering the evaporator for further preheating (states 3-4), vaporization (states
20
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+HDW7UDQVIHU)OXLG3XPS
Figure 1-2: Flow diagram for the micro-CSP plant with propylene glycol as heat transfer
fluid (shown in red) and HFC-245fa as working fluid (shown in green). Condenser cooling
air is shown in blue. The bold numbers represent the state points for the ORC T-s diagram
in Fig. 1-3 below.
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+3+
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Figure 1-3: T-s diagram for the micro-CSP plant with propylene glycol as heat transfer
fluid (HTF) and HFC-245fa as working fluid (WF). Numbers 1 through 10 indicate each
thermodynamic state of the ORC.
21
4-5), and superheating (states 5-6). The two stages of expansion through the scroll
expander-generators are represented by states 6-7 and 7-8, respectively. The vapor
then re-enters the recuperator for precooling (states 8-9) before further precooling
(states 9-10) and condensation (states 10-1) in the condenser. The liquid receiver
prevents subcooling in steady-state. The HTF temperature glide in the counterflow
evaporator is plotted against the corresponding entropy of the working fluid, so the
HTF inlet temperature aligns with the WF outlet entropy (state 6) and the HTF outlet temperature aligns with the WF inlet entropy (state 3). Similarly, the cooling air
temperature glide in the counter-crossflow finned-tube condenser is plotted against
the corresponding entropy of the working fluid, where the air inlet corresponds to the
WF outlet (state 1) and the air outlet corresponds to the WF inlet (state 9).
Figure 1-3 demonstrates that the drying behavior of an organic fluid – the negative
slope of its vapor saturation curve – allows for an evaporator exhaust with very
little superheat to remain in vapor state even with an isentropic expansion. Steam
Rankine cycles, on the other hand, exhibit a wetting behavior, or positive slope in the
vapor saturation curve. And because of turbine blade sensitivity to water droplets,
traditional Rankine cycles require a substantial amount of superheat to ensure a highquality mixture at the expander exhaust. This additional heating is detrimental to
the overall efficiency of the cycle given a fixed temperature heat source. Moreover,
organic cycles can exhibit a substantial temperature difference between the expander
exhaust and the saturation temperature at the condensing pressure (states 8-10),
again due to their drying behavior. (Since the fluid in steam cycles is often two
phase at the exit of the expander, there is zero temperature difference between these
states.) This characteristic allows for heat recovery in ORCs, as accomplished with
the recuperator in the pilot system (states 8-9), which increases their efficiency, or
for the provision of hot water, heating, or absorption cooling.
One observation apparent from the T-s diagram is that the condenser temperature pinch (between WF two-phase inlet, state 10, and air temperature at the same
location) in the counter-crossflow finned-tube arrangement can become very small.
An inherent assumption in the optimization analysis discussed in Section 2.6 and
22
Table 1.1: Pilot system components and major modeling parameters.
Component
Heat transfer fluid
Collectors
Storage Tank
Description
Propylene glycol
Single-axis SopoNova parabolic trough,
SOLEC HI/SORB II selective coating,
air-filled annulus between absorber and
glazing
Double-wall plus insulation, 3696K71
Supplier
Various
Sopogy
McMaster
HTF Pump
HTF Motor
Working fluid
Evaporator
Gear, NG11V-PH
48VDC permanent magnet
HFC-245fa
Brazed Plate, BP415-050
Dayton
Leeson
Honeywell
ITT Brazepak
Recuperator
Brazed Plate, BP410-030
ITT Brazepak
Condenser
Air-cooled finned-tube with hexagonal
tube array, Multicon CAC-28-G
Russell
Liquid receiver
Expandergenerators
WF Pump
WF Motor
Length of pipe plus sight glass
Hermetic scroll induction machine, ZR48
ZR125
Plunger, PowerLine Plus 2351B-P
48VDC permanent magnet
Various
Copeland
Hypro
Leeson
Parameter
Value
As
Lcol
Wcol
Do
V
tins
kins
Vs
Ẇnom
104 m2
73 m
1.4 m
25 mm
0.79 m3
25 mm
3.9e-2 W(mK)
24e-6 m3
373 W
As
V
Mw
As
V
Mw
At
Af
Vt
V
Vs
Vs
Vs
Ẇnom
2.4 m2
2.5e-3 m3
11 kg
0.73 m2
8.4e-4 m3
3.7 kg
13 m2
360 m2
4.0e-2 m3
1.3e-3 m3
22e-6 m3
56e-6 m3
8.7e-6 m3
746 W
-1
used in the dynamic model of Chapter 3 is a constant temperature defect of 10◦ C
between WF exhaust temperature (state 1) and inlet air temperature. The analysis
of Section 2.7 at a range of temperature defects and WF and air supply temperatures
results in pinch points as small as 0.5◦ C but not much larger than 2.8◦ C. This pinch
point in temperatures limits the performance of the heat exchanger. In addition, the
range of acceptable temperature defects at state 1 will be limited by the condition
that the pinch point remain positive (the temperature of the working fluid at state
10 must be higher than the incoming cooling air).
Table 1.1 summarizes the key modeling parameters for each component to be used
in the analysis of the next two chapters.
23
24
Chapter 2
Steady-state modeling
The following sections summarize the detailed steady-state component models, which
are developed in EES. The models have been matched to data where available, and
they are used to provide initial values, heat transfer coefficients, and pressure drops for
the dynamic simulation and for its validation when run to steady-state. In addition,
the steady-state system model is used to perform an optimization analysis for input
to a real-time control strategy discussed further in Chapter 3. The major modeling
parameters for each component are summarized in Table 1.1.
2.1
Evaporator/Recuperator
The brazed plate heat exchangers are modeled using the log mean temperature difference method, where the recuperator consists of single-phase liquid or vapor regions
and the evaporator includes liquid, two-phase, and vapor zones. For the evaporator,
this method involves solving the following system of equations for each zone:
Q̇ =ṁW F · (hW F,ex − hW F,su )
(2.1)
=(ṁ · cp )HT F · (THT F,su − THT F,ex )
(2.2)
=U A · ∆Tlm
(2.3)
25
∆Tlm =
∆T2 − ∆T1
ln(∆T2 /∆T1 )
(2.4)
∆T1 = THT F,1 − TW F,1 = THT F,su − TW F,ex
(2.5)
∆T2 = THT F,2 − TW F,2 = THT F,ex − TW F,su
(2.6)
where W F and HT F denote working fluid (HFC-245fa) and heat transfer fluid
(propylene glycol) properties, respectively; su and ex refer to supply and exhaust
conditions, respectively; Q̇ is the heat flow for that zone of the heat exchanger; ṁ
is the mass flow rate; h is specific enthalpy of the working fluid; cp is specific heat
capacity of the heat transfer fluid; T is temperature; and U A is the product of heat
transfer coefficient and heat exchange surface area. For the recuperator, Equation 2.1
refers to the cold liquid side of the heat exchanger and Equation 2.2 becomes
Q̇ = ṁW F · (hW F,su − hW F,ex )
(2.7)
referring to the hot vapor side.
Heat transfer coefficients and frictional pressure drops are derived using Thonon [20]
for single-phase flow and Hsieh and Lin [7] for two-phase flow. Knowing the inlet conditions of both fluids, the geometry of the heat exchanger, and that the total heat
transfer for all zones must also be equal between fluids, it is possible to solve for
the outlet conditions. To speed the iteration process for the full system model, the
pressure drops, while calculated, are not implemented. With pressure drops typically
below 100 mbar for these heat exchangers, the impact of internal pressure variation
on heat transfer should be negligible. For instance, a supply pressure of 30 bar and a
100 mbar pressure drop over the two-phase region of the evaporator correspond to a
saturation temperature drop of only 0.2◦ C.
2.2
Expander-generators
Neglecting ambient heat losses, the expanders are characterized by their filling factor
and isentropic efficiency. Filling factor, φ, is defined as
26
φ=
ṁ · νsu
ṁ · νsu
=
Vs · Nrot
V̇s
(2.8)
or the ratio of real to ideal mass flow rate, where νsu is the supply specific volume of
the fluid; V̇s is the ideal volume flow rate; Vs is the swept volume of the machine; and
Nrot is the rotational speed of the expander. Isentropic efficiency, ηs,exp , is the ratio
of real to ideal power generated or
ηs,exp =
Ẇel
Ẇel
,
=
ṁ · (hsu − hex,s )
Ẇs
(2.9)
where Ẇel is the electrical power generated; Ẇs is the isentropic power; and hsu
and hex,s are the supply and isentropic exhaust specific enthalpies, respectively. The
electrical power is used here as the asynchronous generator is integrated with the expander in a hermetic shell. The internal irreversibilities accounted for in the isentropic
efficiency are thus a combination of both the fluidic and electrical losses.
The expander model is based on Lemort et al. [10] who created a detailed model
accounting for multiple types of losses and validated to prototype test data. To
extend this model to expanders of various swept volumes, the validated model was
exercised over 800 different working conditions that bracket the operating envelope of
the experimental system, and polynomial fits were generated to correlate φ and ηs,exp
as a function of supply pressure, psu , and pressure ratio, rp . It is assumed that the
filling factor and isentropic efficiency remain comparable for the same supply pressure
and pressure ratio regardless of swept volume. The form of the fitting polynomials is
n−1 X
n−1
X
aij · ln(rp )i · ln(psu )j + an0 · ln(rp )n + a0n · ln(psu )n = f (rp , psu ),
(2.10)
i=0 j=0
where the coefficients, aij , are provided in Table 2.1. It should be noted that the correlations were developed for grid-synchronized, constant-speed expanders, and these
should be validated for variable load and asynchronous operation.
The pressure ratio at which a fixed volume ratio expander will achieve ideal ex27
Table 2.1: Expander correlation coefficients for Equation 2.10 [10].
ηs,exp
j
i
0
1
0
1
2
3
4
5
6.34831061E+3
-4.62226605E+3
5.18926734E+3
-2.71931292E+3
486.736446E0
53.1888731E-3
-2.07325125E+3
1.18102574E+3
-1.40315596E+3
765.497652E0
-139.912567E0
2
272.015067E0
-111.050112E0
141.445478E0
-80.6286745E0
15.0486978E0
3
4
5
-17.9964322E0
4.54486911E0
-6.30866773E0
3.77077331E0
-718.767884E-3
602.747139E-3
-67.9837592E-3
105.088614E-3
-66.0896654E-3
12.86479100E-3
-8.20388944E-3
φ
j
i
0
1
2
0
1
2
4.798
-0.06549
-0.00494
-0.6231
0.006766
0.02523
2YHUH[SDQGHG ,GHDOH[SDQVLRQ 8QGHUH[SDQGHG
7\SLFDO([SDQGHU
2SHUDWLQJ5DQJH
7\SLFDO([SDQGHU
2SHUDWLQJ5DQJH
Figure 2-1: Contour plot of expander isentropic efficiency based on Equation 2.10 with
pressure ratio on the x-axis and supply pressure on the y-axis. Red regions correspond to
high-efficiency operation, and blue regions correspond to low-efficiency operation. The black
lines represent the region of highest efficiency as predicted by the isentropic relation between
pressure and volume ratios. The gray boxes indicate the boundaries of the correlation.
28
pansion can be estimated using the isentropic relation rp = rvγ , where rv is the internal
volume ratio and γ is the specific heat ratio. Figure 2-1 illustrates the correlation for
isentropic efficiency as defined by Equation 2.10 in a contour plot. With a built-in
volume ratio for the scroll expanders of 2.85 and specific heat ratios ranging from 1.1
to 2 for HFC-245fa during operation, the region of ideal expansion should fall between
pressure ratios from 3 to 8. This range is indicated by the black lines on Fig. 2-1
and corresponds well with the red contour representing high-efficiency operation. The
dashed boxes indicate the typical operating region for each expander as determined
by the optimization study that will be discussed in Section 2.6. Expander 2 falls
within the predicted range for ideal expansion while Expander 1 sometimes operates
over-expanded.
2.3
Condenser
Figure 2-2 illustrates the complex geometry of the finned-tube condenser. Refrigerant
enters the condenser through the inlet headers with the entrance to each tube bank
represented by black circles. It then winds it way back and forth (into and out of
the page) and from top to bottom through the tube banks with its path indicated
by dotted lines until reaching the outlet headers, where the exit of each tube bank
is represented by black circles. Three 61 cm diameter fans force air flow across the
tubes and through the corrugated plain fins (parallel to the plane of the page) from
the bottom row to the top row of the tube banks. Fan operation can consist of 1, 2,
or 3 fans on with the first 2 fans controlled by fixed-speed motors and only the third
fan controlled by a variable-speed motor for economical reasons.
Because of its complex geometry, the condenser is more difficult to split into zones
as in the evaporator. Therefore, a discretized model is developed. The condenser is
approximated as a set of 12 identical parallel tube banks (as indicated in Figure 2-2)
each consisting of 12 tubes in 5 rows. Each tube is discretized into n elements for a
total of 12 x n refrigerant elements per tube bank. As the banks are assumed identical,
the set of equations for a single bank is sufficient to model the entire exchanger.
29
)DQV
'LUHFWLRQRI
DLUIORZ
6HOHFWHGWXEHEDQN
,QOHWKHDGHUV
2XWOHWKHDGHUV
Figure 2-2: End view of the condenser with the modeled tube bank outlined in blue. The
upper black rectangles represent the inlet headers, within which the black circles indicate
the refrigerant entrance to each tube bank. Similarly, the lower black rectangles represent
the outlet headers with the black circles indicating the outlet of each tube bank. The path
through each bank is indicated by dashed lines.
The modeling schematic of the tube bank, Fig. 2-3, shows refrigerant entering
as a vapor on the top right and traveling back and forth from Tube 1 to Tube 12
on the bottom left while condensing. Dashed black lines notionally represent the
discretization of fluid cells in the tubes and the hexagonal prism of air and fins (not
shown) surrounding the tube section. The discretization follows the progression of
the refrigerant as illustrated for Tubes 1 and 2 by the bold black numbers.
The air flowing over the tube bank is assumed to be well-mixed between tube rows
as this is conservative and analysis showed a negligible difference in results assuming
mixed or unmixed air (see Appendix B). Therefore, the entrance air temperature for
each row is assumed to be the average of the outlet temperatures for the cells below.
Tube-tube heat transfer due to conduction through the fins is neglected when, in
fact, the distance between the tubes suggests they would transfer some heat with
each other representing a potential refinement for future models (see Appendix B).
30
9DSRU
7ZR3KDVH
/LTXLG
$LU2XWOHW
:),QOHW
5RZ
:)2XWOHW
$LU,QOHW
Figure 2-3: Modeling schematic of condenser tube bank. Refrigerant enters as a vapor on
the top right and travels back and forth from Tube 1 to Tube 12 on the bottom left while
condensing. White tube faces indicate fluid entering at the tube face and traveling back (up
and to the left in the diagram) and gray tube faces indicate fluid that has traveled from the
back and exits at the tube face. Dashed black lines notionally represent the discretization of
the fluid cells in the tubes and the hexagonal prism of air and fins (not shown) surrounding
the tube section. Bold black numbers indicate the discretization scheme. Dashed orange
lines represent the planes of each tube row.
An energy balance is performed for each tube cell:
U Ai
· (TW F,i − Tair,su,i )
ṁW F
(2.11)
U Ai
· (TW F,i − Tair,su,i )
(ṁi · cp )air
(2.12)
hW F,i+1 = hW F,i −
Tair,ex,i = Tair,su,i +
The air is treated as a constant specific heat fluid, while the refrigerant specific heat
31
is allowed to vary. In each cell, i, the inlet properties are set to the outlet properties
of the preceding cell and used to calculate the outlet conditions, denoted by i + 1, for
the current cell – an upwind scheme.
The refrigerant-side heat transfer coefficient is determined using Gnielinski (Re <
100,000) or Dittus-Boelter (Re > 100,000) for single-phase flow [8] and Shah for twophase flow [19]. The air-side heat transfer coefficient and pressure drop are determined
using Kim et al. [9]. The fin efficiency is calculated, neglecting the corrugation,
using the Schmidt method [18] for approximating the hexagonal fins associated with
the staggered tube arrangement as circular fins of equivalent height. The model
also neglects the contribution of water in the air to heat transfer, another possible
refinement. As in the evaporator/recuperator models, to speed the iteration process,
while refrigerant pressure drop is calculated from Petukhov [8] for single-phase flow
and Choi et al. [4] for two-phase flow, it is not implemented in the heat transfer model.
For condensing temperatures below ∼15◦ C, for which pressure drop can exceed 300
mbar and therefore no longer has a negligible impact on heat transfer, a more complete
model may be necessary. For instance, a supply pressure of 1 bar and a 300 mbar
pressure drop over the two-phase region of the condenser correspond to a saturation
temperature drop of 8.5◦ C. It is chosen to neglect this effect in the current analysis.
For computational efficiency, the detailed heat transfer model is not integrated
into the system model. Instead, the calculated heat transfer, Q̇cond , is correlated as
a function of refrigerant supply temperature, TW F,su , air supply temperature, Tamb ,
and air mass flow rate, ṁair :
Q̇cond =7.48094441 × 103 + 1.25976724 × 102 TW F,su
− 1.51787477 × 102 Tamb + 6.34710507 × 103 ṁair .
(2.13)
The correlation is based on 10 working conditions. For the optimization study described further in Section 2.6, the temperature defect between ambient and condensing
temperatures is maintained at 10◦ C and WF pump volume flow rate is maintained
32
Figure 2-4: Goodness of fit for condenser heat transfer correlation (Equation 2.13) with
calculated heat flow from the detailed model on the x-axis and predicted heat flow from the
correlation on the y-axis. The R2 value of the fit is 100.00%.
Figure 2-5: Goodness of fit for fan power consumption correlation (Equation 2.14) with
measured power on the x-axis and predicted power on the y-axis. The R2 value of the fit is
86.74%.
33
at 7 LPM. In addition, the critical temperature, Tcrit , of HFC-245fa is 154◦ C. With
these constraints, the range of operation for the condenser is limited, so only a small
number of working conditions are necessary for the correlation. Figure 2-11 is a plot
of the calculated heat flow from the detailed model versus the predicted heat flow
from the correlation. The R2 value of the fit is 100.00%.
Since Modelica is much more robust than EES, the dynamic model closely matches
the detailed condenser model with the exception of constant heat transfer coefficients
and pressure drops, which are provided by the steady-state model.
The condenser fan power consumption, Ẇel,f ans , is correlated to experimental data
as a function of the volume flow rate of the air, V̇su :
Ẇel,f ans = 5.70339029 × 102 + 5.03148392 × 102 log V̇su .
(2.14)
Figure 2-5 depicts the goodness of fit, which results in an R2 of 86.74%. The measurement procedure consisted of changing the fan operation and measuring the air
velocity mid-radius above the fan grill with a Uni-T 5URG8 anemometer. Fan operation has three modes: 1) Fan 1 on/Fans 2 and 3 off, 2) Fans 1 and 2 on/Fan 3 off,
or 3) Fans 1 and 2 on/Fan 3 on with variable speed [14].
2.4
Pumps/Motors
A pump’s global isentropic efficiency, ηp , including both electromechanical and internal losses, is defined by
ηp = ηem,p · ηs,p =
Ẇs
ṁ · (hex,s − hsu )
ṁ · νsu · (pex − psu )
=
≈
Ẇel
Ẇel
Ẇel
(2.15)
assuming the liquid behaves as an incompressible fluid, where the subscript p indicates pump; the variable p represents pressure; and em is electromechanical. Since
the pump and motor are not integrated into the same shell like the expanders, the
electromechanical and internal losses can be separated.
34
Figure 2-6: Goodness of fit for WF pump isentropic efficiency correlation (Equation 2.16)
with calculated efficiency from manufacturer data on the x-axis and predicted efficiency
from the correlation on the y-axis. The R2 value of the fit is 98.75%.
For the working fluid pump, the isentropic efficiency is defined by
ηs,p,W F
pex
= 0.234247552 + 0.220591434
pnom
!
pex
− 0.0179094791
pnom
!2
,
(2.16)
which is derived from manufacturer data that is correlated as a function of normalized
outlet pressure. The nominal pressure, pnom , is set to 30 bar. Figure 2-6 depicts the
goodness of fit, which results in an R2 of 98.75%. Volumetric efficiency is neglected
in the manufacturer data.
The motor efficiency is also derived from manufacturer data and correlated with
fraction of rated mechanical power:
ηem,p,W F =
6
X
bk ·
k=0
Ẇm
Ẇnom
!k
,
(2.17)
where the rated mechanical power, Ẇnom , for the WF pump is 746 W and Table 2.2
provides the coefficients, bk . Figure 2-7 depicts the goodness of fit, which results in
an R2 of 99.83%.
35
Table 2.2: WF motor correlation coefficients for Equation 2.17.
Coefficient
b0
b1
b2
b3
b4
b5
b6
Value
2.170250E-03
4.468185E0
-9.374727E0
9.750974E0
-5.351966E0
1.474668E0
-1.608160E-01
Figure 2-7: Goodness of fit for WF motor efficiency correlation (Equation 2.17) with calculated efficiency from manufacturer data on the x-axis and predicted efficiency from the
correlation on the y-axis. The R2 value of the fit is 99.83%.
For the HTF pump, the measured electrical power output, Ẇel,p,HT F , from experimental data is correlated to the measured volume flow rate, V̇su , and supply
temperature, Tsu :
2
2
Ẇel,p,HT F = c0 + c1 · Tsu + c2 · Tsu
+ c3 · V̇su + c4 · V̇su
+ c5 · Tsu · V̇su ,
(2.18)
where Table 2.3 provides the coefficients, ck . Figure 2-8 depicts the goodness of fit,
36
Table 2.3: HTF pump correlation coefficients for Equation 2.18.
Coefficient
c0
c1
c2
c3
c4
c5
Value
1.86411220E+01
-8.91044338E-01
9.53566173E-03
5.03864468E+00
3.87583484E-01
-8.50046125E-02
Figure 2-8: Goodness of fit for HTF pump power consumption correlation (Equation 2.18)
with measured power on the x-axis and predicted power on the y-axis. The R2 value of the
fit is 99.55%.
which results in an R2 of 99.55%. Although, the HTF pump is also separated from
the motor, there was not enough information in the available data to distinguish the
electromechanical losses from the internal irreversibilities. This approximation would
result in an error in outlet enthalpy, but since enthalpy change in a pump is small
compared to the other components, the effect should be negligible. The current drawn
by the motor was measured using an Extech True RMS Ammeter 430 and the flow
rate was measured using a Blancett flow meter and B2800 flow monitor [14].
There is currently no cavitation model for the pumps, a possible future improvement.
37
Qconv,HTF
Glass
envelope
Evacuated Absorber
annulus
Qcond,abs Qcond,gl
YĐŽŶǀ͕Ĩ
2.30
Qrad,ann
Y,>
Tsky
,d&
Qrad,sky
HTF
Tgl,o
Tabs,o
Tabs,i
Qsol,abs
d,d&
Tamb
THTF
Qconv,amb
YƐŽů͕ĂďƐ
Tgl,i
Parabolic reflector
-1.7
-1.2
-0.7
-0.2
0.3
-0.20
0.8
1.3
1.8
Ap
(a) Forristall model [2]
(b) Burkholder approximation
Figure 2-9: Collector modeling schematics showing an end view of the absorber tube, glazing, and reflector, and the heat transfer processes associated with the full 1D energy balance
of Forristall and the simplified energy balance of Burkholder.
2.5
Solar Collectors
The collectors are modeled using the one-dimensional energy balance around the discretized heat collection element of Forristall [6] shown in Fig. 2-9a. To determine the
heat transfer to the fluid in the absorber considering the solar irradiance and thermal
losses, this model includes the following heat transfer processes: convection in the
heat transfer fluid, conduction in the absorber tube, convection and radiation in the
air-filled annulus, conduction in the glazing, convection and radiation to the ambient,
and solar radiation to the absorber tube and glazing. Knowing the collector geometry,
solar irradiance, ambient temperature, wind speed, incidence angle, mass flow rate,
and inlet temperature to the collector, the outlet temperature can be calculated.
Forristall’s model, available in EES, is parameterized for the collector geometry of
the pilot system and validated using the manufacturer specifications adjusted according to experimental data from commissioning the solar field. Global irradiance was
measured using a Daystar solar meter with 100 mm collimator, and the HTF flow rate
was measured using a Blancett H701A flowmeter [14]. With an accuracy of 3%, this
type of solar meter is typically better suited to benchmarking photovoltaic arrays, so
future data collection efforts will focus on acquiring a higher accuracy measurement of
38
Table 2.4: Manufacturer parameters for solar collector optical efficiency.
Parameter
Receiver Absorptivity
Mirror Reflectivity
Receiver Emittance
Glass Transmissivity
Value
0.95
0.89
0.25 at 315◦ C
0.207 at 270◦ C
0.91
direct normal irradiance through the use of a pyrheliometer. Four measurements from
the experimental data taken over five minutes were averaged to represent a steadystate reading. The irradiance, HTF mass flow rate, and collector supply temperature
were input to the Forristall model and a multiplier on optical efficiency adjusted until
the collector exhaust temperature matched that of the steady-state reading. Table 2.4
provides the manufacturer parameters for determining optical efficiency. A multiplier
of 0.55 on optical efficiency, representing unaccounted losses, is necessary to match
the experimental data.
To reduce simulation time in the dynamic model, the validated model is exercised
over 400 working conditions to determine the fitting parameters for the correlation
developed by Burkholder [2] for heat loss, HL, in Wm-1 as a function of HTF and
ambient temperatures; solar irradiance, Ib ; incidence angle modifier, IAM ; incidence
angle, θ; and wind speed, vw :
2
3
HL =d0 + d1 · (THT F − Tamb ) + d2 · THT
F + d3 · THT F +
√
2
d4 · Ib IAM cosθ · THT
vw · (d5 + d6 · (THT F − Tamb )).
F +
(2.19)
The determined coefficients, dk , are shown in Table 2.5. Figure 2-8 depicts the goodness of fit, which results in an R2 of 99.85%. The importance of several of the terms
may be explained by the heat transfer phenomena. Many of the heat transfer equations depend on THT F − Tamb . Ib IAM cosθ, or irradiance multiplied by incidence
angle modifier, appears in the equations for solar energy incident on the absorber
39
Table 2.5: Collector heat loss correlation coefficients for Equation 2.19.
Coefficient
d0
d1
d2
d3
d4
d5
d6
Value
5.34E-01
2.18E-01
-2.64E-04
4.93E-06
7.38E-08
1.06E-02
1.15E-02
Figure 2-10: Goodness of fit for collector heat loss correlation (Equation 2.19) with calculated heat loss from the Forristall model on the x-axis and predicted heat loss from the
correlation on the y-axis. The R2 value of the fit is 99.85%.
tube and glazing. Wind velocity, or vw , appears in the Reynold’s number relation for
convection from the glazing to the air.
The heat loss term, HL, participates in a simplified energy balance between the
fluid and the solar input as shown in Fig. 2-9b and described by the following equations:
Tex,i = Tsu,i +
qconv,f · ∆x
(ṁ · cp )HT F
qconv,f = qsol,abs − HL
40
(2.20)
(2.21)
where i represents the ith element of the absorber tube; qconv,f represents the heat
transferred to the fluid via convection; qsol,abs represents the solar irradiance; and ∆x
is the length of the element.
2.6
Application: Determining Optimum Set Points
for Varying Working Conditions
The effective design of a micro-CSP plant involving fixed volume ratio expanders
depends on the control system’s ability to maintain the pressure ratio across the
expanders necessary to avoid both over- and under-expansion of the working fluid
under variable ambient conditions. Considering this design objective, one application
for the steady-state system model is determining an optimization function for use in
a real-time control strategy to be analyzed with the dynamic model. With several
additional constraints discussed below, the goal of this optimization analysis is to
identify a correlation for the evaporation pressure which maximizes global efficiency
(from solar input to ORC power output) for various working conditions.
Two control strategies are analyzed with the dynamic model as discussed further
in Chapter 3. The first is based on optimizing evaporation pressure within physical
constraints, such as expander speeds, while the second aims to maintain a constant
evaporation temperature. These strategies are subsequently referred to as Pev,opt and
Tev,const , respectively. In addition, in both schemes, the superheat, SH, and condenser
temperature defect between WF exhaust and air supply, ∆Tcond , are controlled to
a constant 5◦ C and 10◦ C, respectively; and the WF pump speed is currently held
constant although further optimization may be possible in a later iteration. Since
each expander’s isentropic efficiency is modeled using a polynomial as a function
of supply pressure, psu , and pressure ratio, rp , it is possible to find an analytical
solution for the optimum intermediate pressure, Pint , by maximizing the equation for
the combined efficiency given the inlet and outlet pressures. The derivation for this
solution when the efficiency polynomial is a function of supply density, ρsu , and rp
41
was provided in [15], and the derivation for the polynomials of Equation 2.10, which
depend on psu and rp , is provided in Appendix C.
With the HTF pump, condenser fan, and Expander 2 speeds floating to control
SH, ∆Tcond , and Pint , respectively, and the WF pump speed held constant, the
steady-state system model, executed under various environmental conditions, can be
used to determine an optimum evaporation pressure to be controlled via Expander 1
speed. The input temperatures from the solar loop to the evaporator and from the
ambient are varied across the following ranges:
135◦ C ≤ THT F,su ≤ 185◦ C
0◦ C ≤ Tamb ≤ 40◦ C
For each of 30 working conditions, the evaporation pressure is varied from its maximum for that condition and below until an optimum global efficiency within physical
constraints is identified. The maximum pressure is chosen to prevent the fluid from
entering the supercritical regime where the properties are not well-known.
The resulting optimum pressure function is defined by the HTF supply temperature and ambient temperature:
2
pev =e0 + e1 · THT F,su + e2 · THT
F,su +
2
e3 · Tamb + e4 · Tamb
+ e5 · THT F,su · Tamb ,
(2.22)
where the units of pev are Pa and Table 2.6 lists the correlation coefficients, ek .
Figure 2-11 is a plot of the calculated optimum pressure from the 30 conditions
versus the predicted optimum pressure from the correlation. The R2 value of the
fit is 99.26%. When surveying potential working conditions, HTF supply temperatures below ∼140◦ C were found to require expander speeds above the hardware limit
(∼5500 rpm), so these conditions were not included in the 30 points used for the
final correlation and the relevant range for the HTF supply temperature is reduced
to 140◦ C ≤ THT F,su ≤ 185◦ C.
42
Table 2.6: Optimum evaporation pressure correlation coefficients for Equation 2.22.
Coefficient
e0
e1
e2
e3
e4
e5
Value
-1.43107571E+07
1.87886417E+05
-5.29553162E+02
-6.07260310E+04
4.43861382E+01
4.60543535E+02
Figure 2-11: Goodness of fit for optimum evaporation pressure correlation (Equation 2.22)
with calculated optimum pressure from the steady-state model on the x-axis and predicted
optimum pressure from the correlation on the y-axis. The R2 value of the fit is 99.26%.
The evaporator operating pressure that optimizes the global efficiency of the cycle
is primarily determined by a balance of the power output of the expanders with
increased inlet pressure versus the increase in the required power to drive the HTF
pump. The first phenomenon is straightforward, while the second may require more
explanation. The increase in the HTF pump power is a consequence of the fixed (and
limited) heat transfer area in the evaporator. An increase in the operating saturation
pressure of the working fluid in the evaporator increases the average temperature
throughout the WF stream. In addition, since the WF mass flow is nearly constant
(with the fixed volume flow rate of the WF pump), the heat transfer required to bring
43
the working fluid to a 5◦ C superheated state increases with increasing saturation
pressure. If the heat exchange coefficients internal to the heat exchanger do not
substantially change, the only way to achieve this increased heat transfer is to increase
the average temperature difference beween the WF and HTF streams. Since the inlet
temperature of the HTF stream is fixed by the discharge temperature from the solar
array, the only way to increase the average temperature between the streams is to
increase the HTF mass flow rate. In this way, the temperature drop of the HTF stream
is reduced as it passes through the heat exchanger so that the average temperature
difference betweem the two flows is increased. As the WF saturation temperature
approaches the inlet temperature of the heat transfer fluid, the mass flow needed to
achieve the required average temperature defect will become larger and larger to the
point where infinite mass flow would be needed to fulfill the thermal requirements
of the evaporator. It becomes apparent that as this condition is approached, the
power required by the HTF pump will also become infinite. Therefore, an optimum
evaporation pressure must exist that yields a high power output from the expanders
without requiring a significant power input to the HTF pump.
Figure 2-12 is a plot of the optimum evaporation pressure correlation (Equation 2.22) versus the HTF temperature entering the evaporator with lines of constant
ambient temperature. Considering the preceding discussion of the trade-off between
maximizing expander power and minimizing HTF pump demand, the trends exhibited
in the figure can be explained as follows:
1. The overall trend in the optimum evaporation pressure is that it increases with
increasing HTF supply temperature for all ambient temperatures. This is not
surprising as higher HTF supply temperatures allow higher WF operating temperatures and pressures that, in turn, allow for high power outputs from the
expanders without requiring large HTF mass flows.
2. Another trend evident in the figure is that a higher ambient temperature results
in a higher optimum evaporation pressure. With a constant condenser temperature defect, a higher ambient temperature corresponds to a higher condensing
pressure, which leads to a lower overall pressure ratio. The highest possible
44
/1
1
!
/1
"
1
/1
1
/
!
0
Figure 2-12: Optimum evaporation pressure correlation (Equation 2.22). HTF supply temperature is on the x-axis and optimum evaporation pressure is on the y-axis with lines of
constant ambient temperature.
evaporation pressure without a significant increase in HTF pump power is desirable to increase the output power of the expanders.
3. For low ambient temperatures, the optimum supply pressure reaches a maximum and then decreases at higher HTF supply temperatures (see 0◦ C line in
Fig. 2-12). In these circumstances, the efficiency of the expanders as a function
of the pressure ratio as discussed in Section 2.2 becomes important. High HTF
supply temperatures and low ambient temperatures correspond to the largest
potential overall pressure ratios. Under these conditions, the balance between
the increase in maximum available power associated with a higher supply pressure and the reduction in expander efficiency associated with a higher pressure
ratio is struck at a lower supply pressure than the case with higher ambient
temperature. In Fig. 2-1, the low ambient/high HTF supply temperature case
falls on the rightmost boundary of the typical operation boxes. A higher supply pressure (and pressure ratio) would push the operating points diagonally
upward and to the right in the diagram into a region of lower efficiency.
45
2.7
Potential Improvement to Cycle Optimization:
Variable Condenser Temperature Defect
Although the optimization described above is implemented in the dynamic control
scheme of the next chapter, this section analyzes a potential improvement to be considered in future work. The previous optimization maintains a constant condenser
temperature defect between the WF exhaust and air supply temperatures, ∆Tcond .
Updating the correlation characterizing the condenser heat transfer (Equation 2.13)
based on varying temperature defects allows for a potential improvement in efficiency
or net power considering trade-offs with condenser fan power consumption and variable condensing temperatures.
The updated correlation is based on varying the condenser model inputs across
the following ranges:
25◦ C ≤ TW F,su ≤ 70◦ C
0◦ C ≤ Tamb ≤ 40◦ C
5◦ C ≤ ∆Tcond ≤ 20◦ C
The WF mass flow rate is derived based on the constant 7 LPM volume flow rate
imposed at the pump inlet in addition to the imposed saturated liquid exhaust condition for the condenser. Figure 2-13 shows the simulation results used to define the
updated correlation, which is based on 122 working conditions.
Since the previous fit was based on a single temperature defect, a linear correlation
sufficed to characterize the behavior of the condenser. In fact, this can be imagined
by following the symbols beneath the dotted line representing the 10◦ C temperature
defect in Fig. 2-13; the trend of the underlying points suggests that the relationship
between heat flow and air mass flow rate is linear.
Now considering the allowance for variable temperature defects, it becomes clear
that some nonlinear terms are required, specifically to represent the relationship between heat transfer and air mass flow rate at very large and very small temperature
46
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Figure 2-13: Simulation results for updated condenser heat transfer correlation (Equation 2.23). Air mass flow rate is on the x-axis and condenser heat transfer is on the y-axis.
Different colored lines indicate different ambient temperatures and different symbols indicate different WF supply temperatures. Each symbol moving from right to left across
a single line represents condenser temperature defects from 5◦ C to 20◦ C in increments of
2.5◦ C (where the runs for all 7 defects converged).
defects (see Fig. 2-13). The resulting correlation is a function of the same parameters
as the previous:
Q̇cond = f0 + f1 · TW F,su + f2 · Tamb + f3 · ṁair + f4 · ṁ2air + f5 · ṁ3air ,
(2.23)
where Table 2.7 provides the coefficients, fk . Figure 2-14 is a plot of the calculated
heat transfer from the 122 conditions versus the predicted heat transfer from the
correlation. The R2 value of the fit is 99.67%.
Inserting the updated correlation into the system model allows for a re-evaluation
of the 30 working conditions used to define the optimum evaporation pressure and
results in several observations:
1. An optimum can be found for net power (at ∆Tcond = 7.5◦ C), but not global
efficiency, for the conditions surveyed. In other words, the maximum efficiency
47
Table 2.7: Condenser heat transfer correlation coefficients for Equation 2.23.
Coefficient
f0
f1
f2
f3
f4
f5
Value
2.49479821E+04
1.45084021E+02
-2.80330035E+02
2.46210782E+03
3.02151998E+02
1.18718021E+01
Figure 2-14: Goodness of fit for updated condenser heat transfer correlation (Equation 2.23)
with calculated heat flow from the detailed model on the x-axis and predicted heat flow
from the correlation on the y-axis. The R2 value of the fit is 99.67%.
occurs at one of the boundaries of the simulation: at the 20◦ C condenser temperature defect. Figure 2-15 is a plot of the net power and global efficiency
as a function of evaporation pressure with lines of constant condenser temperature defect for a single working condition: THT F,ev,su = 150◦ C, Tamb = 25◦ C.
For display purposes, results for ∆Tcond = 5◦ C are not shown because the net
power and global efficiency for this condition are well below optimum due to
the large amount of air flow required to maintain such a small temperature defect (see Fig. 2-13). Figure 2-15 demonstrates that global efficiency continues
to increase with increasing condenser temperature defect, primarily due to the
48
2.7
Net Power (kW)
2.6
3.5
Wdot,net,7.5
Wdot,net,15
ηglobal,7.5
ηglobal,15
Wdot,net,10
Wdot,net,17.5
ηglobal,10
ηglobal,17.5
Wdot,net,12.5
Wdot,net,20
ηglobal,12.5
ηglobal,20
3.4
3.3
2.5
3.2
2.4
3.1
2.3
3
2.2
2.9
2.1
2.8
2
2.7
1.9
20
21
22
23
24
Evaporation Pressure (bar)
25
Global Efficiency (%)
2.8
2.6
26
Figure 2-15: Results for system optimization using updated condenser correlation for single
working condition: THT F,ev,su = 150◦ C, Tamb = 25◦ C. Evaporation pressure is on the xaxis. Net power with lines of constant condenser temperature defect, ∆Tcond , (indicated
by the last value in the legend) corresponds to the left y-axis, and global efficiency with
lines of constant ∆Tcond corresponds to the right y-axis. For display purposes, results for
∆Tcond = 5◦ C are not shown because the net power and global efficiency for this condition
are well below optimum due to the large amount of air flow required to maintain such a
small temperature defect. Net power exhibits an optimum at 22 bar, ∆Tcond = 7.5◦ C.
Global efficiency fails to exhibit an optimum within the simulation range: the maximum
value shown occurs at the highest temperature defect of ∆Tcond = 20◦ C.
dramatic decrease in the condenser fan power consumption. However, the net
power output also starts to decrease with increasing ∆Tcond because the larger
temperature defect results in a smaller overall pressure ratio, which leads to
reduced power output from the expanders.
2. An optimum net power solution is not found within the range of this study for
ambient temperatures below 10◦ C. For these conditions, the simulations suggest
that very large temperature defects (in excess of 20◦ C) provide the best power
output and efficiency since these parameters continue to increase at condenser
temperature defects up to and including the boundary of the survey of 20◦ C.
This finding suggests that for such cold ambient temperatures, passive cooling
49
(with fans powered off) may be sufficient for ORC heat rejection. However,
these working conditions are in the range where condenser pressure drop, which
was neglected, will have a signficant impact on heat transfer so should be verified
with a higher fidelity model.
For the 20 working conditions where an optimum power solution is found, the
optimum power solution averages 2.5 kW net power, 84 W more than the maximum
efficiency condition at ∆Tcond = 20◦ C. Thermal efficiency averages 7.2% and global
efficiency averages 2.8% – 0.36 pts and 0.11 pts lower than the maximum efficiency
condition at 20◦ C, respectively. The optimum power solutions occur at condenser
temperature defects of 7.5◦ C or 10◦ C indicating that the previous optimization based
on ∆Tcond = 10◦ C was already very close to optimal. Indeed, on average, the net
power for the simulated working conditions only increases by 7 W between the two
optimizations with a negligible change in efficiencies. An updated optimum evaporation pressure curve with varying condenser temperature defects may result in a
negligible change in net energy production. This analysis suggests that passive cooling may be sufficient for ambient temperatures below 10◦ C, which could result in a
significant net power increase for these conditions compared to the previous analysis
where fan power consumption averaged 1.1 kW.
50
Chapter 3
Dynamic modeling
Modelica is an acausal object-oriented modeling language facilitating the deconstruction of a complicated system into its simpler component parts. In such a language,
differential-algebraic equations can be written directly for each part without regard
for order, and the models can be connected via ports, typically relating mass flow,
pressure, and enthalpy between components. The models for the solar ORC plant
either come directly from the open-source ThermoCycle library [17] or build upon the
available components, and the fluid properties are determined using the open-source
CoolProp library [1]. The Tabular Taylor Series Expansion method is used to improve
computational efficiency in calculating fluid properties. The figure depicting the flow
diagram of the plant from Chapter 1 is reproduced here as Fig. 3-1 for comparison
with Fig. 3-2, which illustrates the plant schematic in the Dymola Modelica interface. This chapter will describe the underlying thermodynamic equations for each
component in Fig. 3-2. Table 1.1 summarizes the main modeling parameters for each
component.
The model aims to capture the important dynamics of the system. With much
shorter time constants compared to the other components, the dynamic response of
the expanders and pumps is neglected. Therefore, these dynamic component models
are equivalent to their steady-state representation.
51
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Figure 3-1: Flow diagram for the micro-CSP plant with propylene glycol as heat transfer
fluid (shown in red) and HFC-245fa as working fluid (shown in green). Condenser cooling
air is shown in blue. The bold numbers represent the state points for the ORC T-s diagram
in Fig. 1-3.
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red lines, refrigerant by green lines, and cooling air by blue lines. Sensed parameters and
control signals are represented by dotted lines.
52
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Figure 3-3: Discretized heat exchanger model showing cell vs node parameters. The node
parameters are indicated with a *. Here, f represents the working fluid, HFC-245fa, or the
cold side of the recuperator and sf represents the secondary fluid, propylene glycol, or the
hot side of the recuperator. Figure is based on [15].
3.1
Evaporator/Recuperator
The brazed plate evaporator and recuperator are modeled as discretized finite-volume
counterflow heat exchangers as developed by Quoilin [15]. Figure 3-3 demonstrates
that the heat exchangers are divided into three components: working fluid, secondary
fluid, and metal wall as if the multiple plates in a brazed plate exchanger were one
long plate. The fluid components are based on dynamic mass and energy balances,
while the momentum balance is considered static.
Choosing pressure and enthalpy as state variables and using the chain rule, the
working fluid mass balance for a single element can be expressed as
dmi
δρi
δρi dhi δρi dp
=
· Vi = (
·
+
· ) · Vi = ṁ∗i−1 − ṁ∗i ,
dt
δt
δhi dt
δp dt
(3.1)
where t is time; m is mass; ṁ is mass flow rate; ρ is density; V is volume; h is specific
enthalpy; p is pressure; and a
∗
indicates a node, rather than cell, property.
Since Ui = Hi − pVi , or energy is enthalpy reduced by the product of pressure and
53
volume, and volume is constant, the working fluid energy balance becomes
dhi δρi
dp
dUi
= ρi · Vi ·
+
· Vi · hi −
· Vi = ṁ∗i−1 · h∗i−1 − ṁ∗i · h∗i + Q̇i + Ẇi , (3.2)
dt
dt
δt
dt
where U is energy; Q̇ is heat flow; and Ẇ is work generated.
With no internal work and the replacement of
δρ
δt
· Vi with the right-hand side of
Eq. 3.1, the former equation becomes
dhi
dp
dUi
= ρi · Vi ·
+ (ṁ∗i−1 − ṁ∗i ) · hi −
· Vi = ṁ∗i−1 · h∗i−1 − ṁ∗i · h∗i + Q̇i
dt
dt
dt
(3.3)
which reduces to
ρi · Vi ·
dhi
dp
= ṁ∗i−1 · (h∗i−1 − hi ) − ṁ∗i · (h∗i − hi ) + Q̇i +
· Vi .
dt
dt
(3.4)
The recuperator has a hot side and cold side of working fluid. The evaporator, on
the other hand, consists of working fluid and secondary fluid sides. For the incompressible secondary fluid side of the evaporator where pressure change is negligible,
Eq. 3.4 reduces to
ρi · V i ·
dhi
= ṁ∗i−1 · (h∗i−1 − hi ) − ṁ∗i · (h∗i − hi ) + Q̇i .
dt
(3.5)
The metal wall is assumed to have a constant specific heat capacity, cp , and to
have a negligible temperature gradient leading to the following energy balance:
Mw,i · cp,w ·
dTw,i
= Q̇HT F,i − Q̇W F,i ,
dt
(3.6)
where w indicates a wall property; T is temperature; Q̇W F is the heat flow from
the wall to the working fluid, HFC-245fa; and Q̇HT F is the heat flow from the heat
transfer fluid, propylene glycol, (or in the case of the recuperator, the hot-side of
working fluid) to the wall, respectively.
54
Constant heat transfer coefficients for each phase are input parameters and are
determined using the detailed steady-state model. At the system level and during
normal operation with small changes in mass flow rate, variations in heat transfer
coefficient can be assumed negligible as shown in Appendix B.
3.2
Condenser
The geometry and modeling strategy for the detailed steady-state condenser model,
which is the basis for the dynamic model, was discussed at length in Section 2.3. The
smallest constituents of the condenser model are the refrigerant, metal wall, and air
cells. The refrigerant and metal wall cell dynamic equations are identical to those
of the evaporator and recuperator, but the overall discretization scheme is slightly
different due to the cross-flow of the air as shown in Fig. 3-4. The air cells are assumed
to have negligible dynamics resulting in the following energy balance:
(ṁ · cp )air · (Tair,i−1 − Tair,i ) = Q̇air,i .
(3.7)
Also required are pressure drop components for the air cells. A relation for pressure
drop in off-design conditions is determined by assuming a constant friction factor:
∆p
2 · f · L · A2cross
=
= constant →
ν · ṁ2
Dh
2
ṁ
ν
∆p =∆pnom ·
·
,
νnom
ṁnom
(3.8)
where f is friction factor; L is flow length; Dh is hydraulic diameter; G is mass flux; ν
is specific volume; and Across is flow cross-sectional area. The nominal conditions are
defined using the steady-state model results for a representative working condition.
The next building block in the dynamic model is a row of tubes. Each row consists
of n tubes per row x m cells per tube such that there are n x m refrigerant, metal
wall, and air cells plus corresponding pressure drop components for the air cells.
55
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Figure 3-4: Side view for a single tube of the discretized condenser model showing cell vs
node parameters. The node parameters are indicated with a *. Here, f represents the
working fluid, HFC-245fa, and sf represents the secondary fluid, air.
Also required are splitter and joiner components to divide the incoming air from the
preceding row among the refrigerant cells and rejoin the air cells after they have
transferred heat with the metal/refrigerant.
Finally, the five rows of two or three tubes are linked to complete the model. The
condenser fan power demand is identical to that of the steady-state model.
3.3
Liquid Receiver
The liquid receiver, illustrated in Fig. 3-5, is assumed to be in thermodynamic equilibrium at all times: the vapor and liquid are saturated at the given pressure. It is
modeled by the same energy and mass conservation laws as in the discretized heat
exchangers, but the exit condition is imposed to be a saturated liquid, hl , yielding
the following energy balance:
56
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Figure 3-5: Modeling schematic for liquid receiver.
ρ·V ·
dh
dp
= ṁsu · (hsu − h) − ṁex · (hl − h) +
· V.
dt
dt
(3.9)
Since the control unit regulates condenser pressure, the pressure in the liquid receiver
is set by the saturation pressure in the condenser. Initialization sets either the initial
pressure or liquid level.
3.4
Solar Collectors
The solar collector model consists of two models connected by a thermal port: one
representing the fluid dynamics in the discretized absorber tube and the other representing the collector thermal and optical efficiency based on the fluid temperature
and environmental conditions. As discussed in Section 2.5, the efficiency, or heat loss,
model is determined using a regression of the detailed Forristall model fit to the manufacturer specification and adjusted for experimental data, where the environmental
inputs are solar irradiance, ambient temperature, wind velocity, and incidence angle.
The discretized fluid cells in the absorber tube, depicted in Fig. 3-6, are modeled in
the same way as the secondary fluid in the evaporator.
For computational efficiency, rather than being calculated in each cell, the pressure
losses are combined into lumped models after the heat exchanger and collectors as
shown in Fig. 3-2. They are modeled in the same way as those of the air cells in the
condenser model (Equation 3.8).
57
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heat loss model described in Section 2.5.
U
Figure 3-7: Modeling schematic for storage tank.
3.5
Storage Tank
Pressure is imposed in the storage tank, which is modeled as a well-mixed (single
element) control volume accounting for thermal energy losses due to conduction and
convection through the top, sides, and bottom walls and insulation as shown in Fig. 37. With constant pressure rather than constant volume, the energy balance reduces
to
ρ · Vl ·
dh
dVl
= ṁsu · (hsu − h) − Q̇env + p ·
,
dt
dt
(3.10)
where Q̇env represents the heat loss to the environment and Vl is the liquid volume.
58
3.6
Control Unit
Two control strategies are analyzed with the dynamic model. The first, referred to as
Pev,opt , aims to track the optimum evaporation pressure as discussed in Section 2.6,
and the second, referred to as Tev,const , aims to maintain a constant evaporation
temperature.
During normal operation, the control unit regulates system components to achieve
the set points by utilizing four PI controllers illustrated in Fig. 3-8. PI control is selected because of its reduced sensitivity to measurement noise compared to PID control. The sensor dynamics are currently neglected. Each controller includes steadystate initialization functions to allow any oscillations in the system at the beginning
of a simulation to settle before the controller starts tracking.
The four controllers are based on the following control signals and process variables: the speed of the HTF pump is used to regulate superheat; the speed of Expander 1 is used to regulate either Pev,opt or Tev,const ; the speed of Expander 2 is used
to regulate optimum intermediate expander pressure; and the mass flow rate of the
condenser fans (as a simplified proxy for fan speed and number of operating fans) is
used to control the condensing pressure. The leftmost column of blocks in Fig. 3-8a
correspond to the following: ∆Tev is the calculated superheat based on a correlation
for saturation temperature of HFC-245fa as a function of expander supply pressure;
Pev is the optimum evaporation/expander supply pressure as derived in Section 2.6;
and Pint is the optimum intermediate expander pressure as derived in Appendix C.
For the Tev,const strategy shown in Fig. 3-8b, the Pev block is replaced with a constant
set point and the process variable for saturation temperature is determined using the
∆Tev block.
The controllers are tuned manually with the other controllers disabled by varying
the proportional gain and integral time after a step is introduced. It is assumed that
the closed-loop system is decoupled and behaves linearly and that the same tuning
parameters can be used for the two strategies. A more robust control tuning and
design is proposed in Section 4.2.
59
Startup/Shutdown.
To simulate the daily power cycle, the control must have the ability to shut down
the ORC at the end of the day or at any time when there is insufficient irradiance to
warrant running the pumps. Similarly, it will need to simulate startup. Numerical
simulation of startup or shutdown is difficult due to flow reversals, chattering around
zero flow, and unphysical flow rate generation due to a discontinuity in the working
fluid density derivative [16]. To approximate these conditions while avoiding numerical issues, duplicate models representing operating and idling modes (see Fig. 3-2)
are simulated concurrently, transferring state variables to achieve an energy balance
between the two when startup and shutdown, which are modeled as instantaneous
events, are triggered. As experimental data indicates that the time scale of startup or
shutdown is on the order of minutes, the impact of this approximation on a full-day
simulation is expected to be negligible. However, ORC thermal inertia during these
events is also neglected under the current scheme, so future analysis is planned to
integrate this effect.
The operating model is composed of the system model discussed previously. The
idling model consists of a duplicate solar loop excluding the evaporator to represent
operation when the ORC is shut down. During normal operation, the real irradiance
signal is input to the operating model. A controller inputs a much-reduced irradiance
signal to the idling model to maintain the storage tank temperature at the shutdown
trigger temperature, where the only heat loss from the idling model is that lost to
the environment through the storage tank.
When the storage tank temperature in the operating model reaches the shutdown
trigger, several actions follow: 1) the storage tank temperature in the idling model is
re-initialized to that of the operating model (in case the controller has not settled); and
2) a controller inputs a higher irradiance signal to the operating model to maintain
the storage tank at the startup trigger temperature. If the irradiance signal falls
below 180 Wm−2 , the HTF pump, nominally set to 27 LPM in the idling model, is
shut down.
60
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3,'BH[S
,QLW
,QLW
&RQVW
'7BFRQG
3,'BPGRWBDLU
7BDPEBVX
(b) Tev,const Strategy
Figure 3-8: Modelica interface for the control units for the Pev,opt and Tev,const strategies.
The upper left arrows of the PID components represent set points; the lower left arrows
represent process variables; and the right arrows represent control signals. The Init components allow for steady-state initialization at the beginning of a simulation. The topmost
row shows the controller for superheat via HTF pump speed; the middle rows show the controllers for expander supply pressures (or constant evaporation temperature) via expander
speeds; and the bottom row shows the controller for condensing pressure via fan air flow
rate.
61
When the storage tank temperature in the idling model reaches the startup trigger,
the reverse occurs: 1) the storage tank temperature in the operating model is reinitialized to that of the idling model (in case the controller has not settled); and 2) a
controller inputs a lower irradiance signal to the idling model to maintain the storage
tank at the shutdown trigger temperature.
3.7
Application: Identifying Optimal Control Schemes
for Daily Environmental Variation
The two control strategies are compared by simulating the system response to a
solar irradiance dataset comprising four reference days that bracket a wide range of
irradiance conditions from clear sky to severe overcast.1 The current method for
evaluating the solar input consists of parsing the rapidly-varying irradiance data into
a piecewise polynomial fit to provide a continous input function to the dynamic model.
For this preliminary assessment, Tamb , approximated as a sine curve, is the same for
each of the four days and incidence angle, θ, and wind velocity, vwind , are set to zero.
Figure 3-9 is a plot of the assumed irradiance profile and ambient temperature.
At the beginning of the first day, the storage tank is initialized to 135◦ C, 10◦ C below the chosen shutdown trigger temperature of 145◦ C (approximately the overnight
temperature loss). This shutdown trigger temperature is chosen to maintain the 5◦ C
of superheat given the hardware limit of 1800 rpm imposed for the HTF pump and
is identified by executing the steady-state model under similar operating conditions.
The startup trigger is set to 155◦ C. The Tev,const strategy is analyzed with two different evaporating temperatures, 117◦ C and 127◦ C, now referred to as Tev,117 and
Tev,127 , respectively. A Tev of 127◦ C is the highest at which the 5◦ C of superheat can
1
The irradiance input is based on a global irradiance dataset measured in Lesotho in 2009 using
a SDL-1 solar data logger and converted to irradiance on a single-axis tracking surface with N-S
alignment. The measured data was compared to the respective “perfect” irradiance profile calculated
for each day using geoposition data and the solar constant resulting in a “fraction of perfect”
irradiance versus time. This perfect irradiance profile was converted to beam irradiance on a tracking
surface, then multiplied by the fraction of perfect irradiance to simulate irradiance on a tracking
surface under real atmospheric conditions (plus a perfect day) [14]. The geoposition and tracking
equations can be found in [5].
62
Irradiance (W/m2)
1000
800
600
400
200
Ambient Temperature (oC)
0
30
0
12
24
36
48
Time (hrs)
60
72
84
96
0
12
24
36
48
Time (hrs)
60
72
84
96
25
20
15
Figure 3-9: Environmental conditions for control strategy comparison. The upper plot
shows direct normal irradiance and the lower plot shows ambient temperature. In this
preliminary comparison, incidence angle and wind speed are set to zero.
be maintained within the HTF pump hardware limit under these conditions.
Figures 3-10 through 3-13 present some detailed results for a single strategy on a
single day – the Pev,opt strategy with 155◦ C startup temperature on Day 2 (between
24 and 36 hours in Fig. 3-9). The figures show the results from the idling model
when the ORC is shut down and those from the operating model when the ORC is
operating.
Figure 3-10 presents the set points and process variables for the ORC controllers.
A shutdown is indicated any time the parameters fall below the x-axis. Since the
full shutdown sequence of the ORC is not modeled as explained in Section 3.6, the
parameters are set to zero any time the simulation switches to idling mode. The
superheat and condenser temperature defect controllers use constant set points while
the evaporation pressure and intermediate expander pressure controller set points are
determined by the optimization functions described in Section 2.6 and Appendix C,
respectively. The Pev , Pint , and ∆Tcond controllers track their set points very closely.
63
6
26
Set Point
Superheat
Set Point
Evaporation Pressure
25
Pressure (bar)
24
o
Temperature ( C)
5.5
5
23
22
4.5
21
4
20
0
1
2
3
4
5 6 7 8
Time (hrs)
9 10 11 12
0
(a) Superheat Control
1
2
3
4
5 6 7 8
Time (hrs)
9 10 11 12
(b) Pev Control
9.5
11
Set Point
Mid−Expander Pressure
Set Point
∆ Tcond
9
Temperature (oC)
Pressure (bar)
10.5
8.5
8
10
9.5
7.5
7
9
0
1
2
3
4
5 6 7 8
Time (hrs)
9 10 11 12
0 1 2 3 4 5 6 7 8 9 10 11 12
Time (hrs)
(c) Pint Control
(d) ∆Tcond Control
Figure 3-10: Controller results for Pev,opt strategy on Day 2. Shutdown is indicated when
the parameters cross the x-axis.
64
Irradiance (W/m2)
1000
800
600
400
200
0
0
1
2
3
4
5
6
7
8
9
10
11
12
10
11
12
190
THTF,col,ex
Temperature (oC)
180
THTF,tank
TWF,exp,su
170
160
150
140
130
120
0
1
2
3
4
5
6
Time (hrs)
7
8
9
Figure 3-11: Effect of thermal storage for Pev,opt strategy on Day 2. The upper plot shows
the direct normal irradiance profile for Day 2, and the lower plot shows several temperatures.
The tank, whose temperature is shown in blue, reduces the variation seen by the ORC with
expander supply temperature shown in green as compared to the collectors with exhaust
temperature shown in red (see especially hours 7-9).
The superheat controller exhibits more variation although it is within ±0.5◦ C for
most of the operating time. This behavior can be attributed to the fact that a HTF
component is being used to control a WF property. One alternative is to use the WF
pump to control superheat but this would directly impact the power production of
the expanders which may not be desirable. Future analysis will examine alternatives
to the current assumption of a constant flow rate for the WF pump.
Figure 3-11 shows the effect of the thermal storage. The tank effectively dampens
the rapid oscillations in temperature seen by the collectors due to the rapidly-varying
solar input as evidenced by the smoother expander supply temperature (see especially
hours 7-9). Following the blue trace in the figure, the tank temperature takes ∼1.5
hours to reach the startup trigger of 155◦ C after the sun rises at time zero. For this
65
particular day, the ORC has to be shut down 6 times before the final shutdown of the
day as indicated by the number of times the tank temperature reaches the shutdown
trigger of 145◦ C. In the afternoon, when the solar insolation is stronger, the ORC can
operate more continuously. After the 10th hour of sunlight, the ORC is shut down
because the tank is below the shutdown trigger temperature, but there are still some
large oscillations in the collector temperature. This behavior can be explained by the
HTF pump startup trigger of 180 Wm-2 of solar irradiance. The upper plot shows that
the irradiance decreases below this trigger at 10.2 hours. At this point, the pump
is shut down and with a negligible flow rate but non-zero irradiance, the stagnant
HTF heats up very quickly. At 10.4 hours, the irradiance exceeds the trigger, so the
HTF pump is powered on and the collector exhaust temperature reduces. This cycle
repeats between 10.6 and 11.2 hours before the final shutdown at 11.4 hours.
This cycling of the HTF pump is also indicated in Fig. 3-12, which illustrates
the power requirements and outputs for the system’s turbomachinery. Under these
conditions, the expanders are optimally run with 40% of the load on the first expander and 60% on the second, and this load split is found to be similar for the other
working conditions in the optimization study of Section 2.6. The expander traces
follow the shape of the tank temperature in Fig. 3-11 as expected since their optimization functions are meant to adapt to the working conditions and a higher tank
temperature enables a higher evaporation pressure and higher power production as
discussed in Section 2.6. The WF pump power demand is also strongly a function
of the high-side or evaporation pressure since pex is in the numerator of the power
equation, Ẇel ≈ ηp · ṁ · νsu · (pex − psu ) (see Section 2.4). Therefore, the shape of
the WF pump trace mimics that of the tank temperature with a higher evaporation
pressure corresponding to a higher power demand. The converse behavior can be seen
in the HTF pump where a higher tank temperature allows for a smaller HTF pump
demand, again as discussed in Section 2.6. Finally, the condenser fans are essentially
insensitive to the tank temperature as they are being controlled to maintain a constant temperature defect between the air supply and condenser exhaust. Therefore,
they follow the trend of the ambient temperature as shown in the zoomed view in
66
4
3.5
Wdot,net
Wdot,exp2
Wdot,WF,pump
Wdot,exp1
Wdot,fans
Wdot,HTF,pump
3
2.5
Power (kW)
2
1.5
1
0.5
0
−0.5
−1
−1.5
0
1
2
3
4
5
6
7
Time (hrs)
8
9
10
11
12
Figure 3-12: Power produced or consumed by each component for Pev,opt strategy on Day
2. Nearly all the components are most heavily influenced by the HTF supply temperature
(Fig. 3-11) except the condenser fans, which react to the ambient temperature as shown in
Fig. 3-13.
Fig. 3-13 where a higher ambient temperature requires less fan power as the condenser
needs less cooling to maintain the same temperature defect.
Now that some of the details behind the simulation have been presented, the
following discussion will focus on the comparison of the various control strategies.
Figure 3-14 shows the global and ORC thermal efficiencies and the net power generated for each strategy. The vertical drops indicate that the tank temperature reached
145◦ C and a shutdown was executed. For these four reference days, the Tev,const strategy at Tev = 117◦ C, the Tev,const strategy at Tev = 127◦ C, and the Pev,opt strategy
generate 62.7 kWh, 65.5 kWh, and 70.0 kWh of energy, respectively. In other words,
as a result of its ability to flexibly adapt to varying ambient conditions, the Pev,opt
strategy generates 12% and 7% more energy than the Tev,117 and Tev,127 strategies,
respectively. It must be noted, however, that these results do not take into account
67
Ambient Temperature (oC)
Condenser Fan Power (kW)
30
25
20
15
0
1
2
3
4
5
6
Time (hrs)
7
8
9
10
11
12
0
1
2
3
4
5
6
Time (hrs)
7
8
9
10
11
12
−1.1
−1.11
−1.12
−1.13
−1.14
−1.15
Figure 3-13: Power consumed by condenser fans for Pev,opt strategy on Day 2 (lower plot).
The shape of the power consumption is a function of the shape of the ambient temperature
input shown in the upper plot and the controlled constant temperature defect.
the potential effects of repeated ORC startup/shutdown cycling (e.g., due to the heat
capacity of the hardware or the need for load following, which were neglected in this
model); a follow-on study to investigate the trade-offs between increasing thermal
storage size (and ambient losses) and decreasing operational cycling is suggested.
Another trend that can be observed is the tendency for the Tev,127 strategy net
power and efficiency to exhibit more variation during operation than that of the Tev,117
strategy under these conditions. For the Tev,127 strategy, net power and efficiency
sometimes begin higher and end lower than the Tev,117 strategy during an operational
cycle. At other times, they remain higher than the Tev,117 strategy throughout. This
behavior emphasizes the difficulty in choosing a constant evaporation temperature
which behaves optimally under many conditions, recalling the discussion of Section 2.6
on the higher HTF pump demand required to maintain the 5◦ C superheat at a reduced
HTF supply temperature. The Tev,117 strategy has more room between the constant
evaporation temperature and the shutdown trigger so the HTF pump demand stays
68
more consistent over this operational range, but it is less efficient at the beginning of
a cycle when the HTF supply temperature is higher. A more comprehensive survey
of environmental conditions and triggers would help determine an optimum although
this may be unnecessary pending more detailed studies of the Pev,opt strategy.
Having validated that an optimum pressure strategy is more effective, the role
of startup trigger temperature on cycling is further investigated. Figure 3-14 shows
that at a trigger of 170◦ C, the initial system startup in the morning is delayed, but
fewer shutdowns occur during the day once ORC operation begins. Energy generated
also increases slightly by 0.1% versus the 155◦ C startup temperature. Fewer startup
and shutdown cycles implies improved maintenance profiles and extended mean time
between failure (MTBF) of components. Further investigation of seasonal timescale
environmental influences and examination of alternative trigger temperatures and
storage tank sizes is warranted.
69
70
Efficiency (%)
0
0.5
1
1.5
2
2.5
3
3.5
4
0
1
2
3
4
5
6
7
8
9
10
11
12
0
0
6
6
12
Wdot,net,opt,170
24
Time (hrs)
18
24
Time (hrs)
Wdot,net,opt,155
18
ηglobal,opt,155
ηglobal,opt,170
12
ηthermal,ORC,opt,155
ηthermal,ORC,opt,170
30
Wdot,net,Tev127,155
30
ηglobal,Tev127,155
ηthermal,ORC,Tev127,155
36
36
42
Wdot,net,Tev117,155
42
ηglobal,Tev117,155
ηthermal,ORC,Tev117,155
48
48
Figure 3-14: Control strategy comparison results. The upper plot shows global and ORC thermal efficiency and the lower plot shows net
power generated. The diagonal hatch marks on the x-axes indicate that the overnight results, for which the plant is shut down, are not
shown for display purposes.
Net Power (kW)
Chapter 4
Conclusions and Future Work
4.1
Conclusions
A model capturing the important dynamics of a micro-CSP system with thermal
storage capable of responding and adapting to rapid environmental variations has
been developed. The dynamic model is based on detailed steady-state models validated to data where available. The steady-state models are also used to provide
initial values, heat transfer coefficients, and pressure drops for the dynamic simulation and to determine an optimum evaporation pressure correlation for input to a
control strategy analyzed with the dynamic model. Steady-state and dynamic models representing the complex geometry of the counter-crossflow finned-tube condenser
have been developed. More detailed analysis of the steady-state condenser model
indicated that the assumption of a constant 10◦ C condenser temperature defect used
in the dynamic model is close to optimal and that passive cooling may be sufficient
for ambient temperatures below 10◦ C. A method for approximating startup and shutdown in the dynamic model while avoiding the numerical issues associated with rapid
phase changes has been proposed to facilitate full- and multiple-day simulations. To
the author’s knowledge, this is the first model capable of continuously simulating
through startup and shutdown in addition to coupling a dynamic thermodynamic
model of the power cycle with dynamic models of the collectors and tank.
In this preliminary assessment based on a solar irradiance dataset spanning four
71
reference days, the Pev,opt strategy that adjusts the operating point based on the
boundary conditions generates 7% more energy than the Tev,const strategy evaporating
at 127◦ C. The Pev,const strategy, however, allows for that flexibility by reducing the
range of speeds the HTF pump experiences during operation at the expense of a wider
range of speeds experienced by the expanders, which may reduce their life and increase
maintenance costs. The relative maintenance cost of larger cycles on the expanders
versus the pump should be compared with the energy gain from the Pev,const strategy
in future analysis.
Moreover, startup trigger temperature plays an important role in system dynamics and performance, with advantages in selecting for an operational envelope at
higher temperatures of the working fluid. In this study, increasing the startup trigger
temperature from 155◦ C to 170◦ C recovers 0.1% more energy with fewer shutdowns
during the day, which should also reduce maintenance costs.
Based on the updated condenser correlation and steady-state survey of the system model, a further 0.3% energy may be gained by allowing for variable condenser
temperature defects.
The dynamic model presents several advantages over a yearly steady-state simulation. It allows for the comparison of different control strategies, which accept rapidly
varying environmental conditions as input, and for the investigation of the feasibility
of a control scheme which depends on regulating interrelated components. Dynamic
analysis provides the ability to examine the transients and cycles a component may
experience over its lifetime. The facility and intuitiveness of Modelica versus EES
software provides several advantages in itself. Since the components are designed to
be self-contained, such that the equations describing each component accept mass
flow, pressure, and enthalpy as inputs and calculate outlet mass flow, pressure, and
enthalpy, they are easy to swap with components based on a different set of equations
or of a different geometry. The Dymola Modelica interface facilitates this process by
providing for the identification of components by icons, which allows the modeler to
click-and-drag different components into a working cycle model. Parameters (e.g.,
geometries) of components are easily changed via the graphical user interface.
72
However, there are several disadvantages to a dynamic, rather than steady-state,
simulation. The dynamic model is much more complex, which results in a longer
runtime and necessitates a larger computing power. The dynamic model is also less
robust due to the interaction of the coupled controllers, which require a lengthy tuning
process. Several strategies for improving robustness are discussed below. Modelica
requires good start values to initialize all the components and begin the simulation.
This requirement indicates that a steady-state model for the components, if not the
system, is necessary involving additional effort.
4.2
Future Work
Possible improvements to this work can be split into several categories: validation,
expansion, modeling improvements, and robustness.
In future, the full system model should be validated to the pilot system in steadystate. The correlations used for the expander properties were developed for gridsynchronized, constant-speed expanders, and these should be validated for variable
load and asynchronous operation.
Expansion is meant to represent a wider survey for control strategy comparison.
The current work analyzes four days of irradiance variation with ambient temperature
variation, approximated as a sine curve, duplicated for the four days and incidence
angle and wind speed set to zero. A more comprehensive survey of environmental
conditions would allow for a quantitative comparison of control strategies.
Several improvements could increase the accuracy of the current model. The ORC
pressure drops are currently neglected in both the steady-state and dynamic models.
Adding them to the dynamic model may require better start values, which implies
including them in the steady-state model increasing its iteration time. Incorporating
pressure drop into the condenser heat transfer model becomes important for ambient
temperatures below 15◦ C, for which pressure drop can exceed 30 mbar. Including
tube-tube heat transfer could also improve the model’s accuracy as shown in Appendix B. A more realistic representation of sensor dynamics would help indicate
73
whether the proposed control schemes are viable as parameterized. In the present
work, the ORC thermal inertia is neglected during startup and shutdown transients.
In the future, a model that accounts for this effect would allow for a more accurate
comparison of the different control strategies.
However, a more realistic accounting of startup and shutdown, possibly by simulating zero flow in the ORC, leads to the issue of robustness. Several robustness
strategies for handling the numerical issues associated with rapid phase changes are
currently under development [16], but many of them rely on increasing the number of
elements in the heat exchangers which greatly increases the simulation time. A strategy for approximating shutdown that doesn’t signficantly increase run time but still
accounts for the time constants of the ORC components would increase the accuracy
of the dynamic model while retaining its usefulness for multi-day simulations.
The current method for approximating the rapidly varying solar input (piecewise
polynomials) is very computationally-intensive and allows for, at most, two days of
simulation at a time. Consecutive days must be re-initialized to the conditions from
the end of the previous simulation. A more robust way for approximating solar input
would greatly facilitate multi-day or even yearly simulations.
A more rigorous control design process should lead to a more responsive, robust
control system. The typical process would include the following steps: modeling,
linearizing, designing the controllers, and verifying by closed-loop comparison between
the linear and nonlinear models.
Future efforts focusing on any of these improvements would ensure this tool becomes an accurate, robust, and potent means of evaluating the most effective control
schemes for the proposed micro-CSP plant. In turn, underserved rural communities
and partnering government agencies, NGOs, and others would be able to make the
most of their investment to provide heating, cooling, and electricity for rural health
and education clinics by harnessing the maximum available power from the sun.
74
Appendix A
Model Code and Experimental
Data
Due to the difficulty of reading object-oriented code in print and to facilitate direct application of the models in the associated software, the models are available
electronically at the following address:
http://www.labothap.ulg.ac.be/staff/mireland/
The versions of ThermoCycle and CoolProp used are also available. See the following
website for more information on using and installing those Modelica libraries:
http://www.thermocycle.net/
The experimental and manufacturer data used to calibrate the models is also provided.
75
76
Appendix B
Sensitivity Studies on Heat
Exchanger Modeling
The following sections describe four sensitivity studies on heat exchanger modeling
based on the finned-tube condenser, namely the effect of mixed vs unmixed cooling
air, the effect of variable vs constant heat transfer coefficients, the impact of tube-tube
heat transfer, and the effect of discretization.
B.1
Mixed vs Unmixed Cooling Air
To simulate the condenser with the air between tube rows unmixed, the air cells are
futher discretized into 3 sections (left, center, right) per bank by 5 rows by n elements
per tube for a total of 3 x 5 x n air elements per tube bank. In Figure B-1a, the
modeling schematic for the mixed model from Section 2.3 is reproduced. Figure B1b is an end view indicating the additional discretization necessary for the unmixed
model with dashed lines. The left/center/right discretization model assumes a set of
adiabatic surfaces between the air cells such that tubes split by one of the surfaces
transfer half of their energy to the air parcel on the left and half to the right. The
center air parcel receives energy from the tubes on the left and right. The following
summarizes the necessary changes to the energy balances used in the mixed model of
Section 2.3.
77
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7ZR3KDVH
/LTXLG
$LU2XWOHW
5RZ
$LU)ORZ
:),QOHW
$LU,QOHW
5RZ
:)2XWOHW
(a) Modeling Schematic
(b) End View
Figure B-1: Modeling schematic of condenser tube bank. Figure B-1a shows that refrigerant
enters as a vapor on the top right and travels back and forth from Tube 1 to Tube 12 on
the bottom left while condensing. White tube faces indicate fluid entering at the tube face
and traveling back (up and to the left in the diagram) and gray tube faces indicate fluid
that has traveled from the back and exits at the tube face. Dashed black lines notionally
represent discretization of fluid cells in the tubes and the hexagonal prism of air and fins
(not shown) surrounding the tube section. Bold black numbers indicate the discretization
scheme. Dashed orange lines represent the planes of each tube row. Figure B-1b shows the
end view and the additional left/center/right discretization for the unmixed model. Dashed
black lines represent adiabatic surfaces.
The refrigerant in Tube 1 interacts with the center and right air cells in Row 5.
Equation 2.11 becomes
hW F,i+1
U Ai
1
1
= hW F,i −
· (TW F,i − Tair,c,su,i ) + (TW F,i − Tair,r,su,i ) ,
ṁW F
2
2
(B.1)
where c indicates center and r indicates right.
The center and right air cells in Row 5 interact with Tubes 1 and 2 and Tube 1,
respectively, so Equation 2.12 becomes
78
Tair,c,ex,i
Tair,r,ex,i
1
U Ai
· (TW F,i − Tair,c,su,i )
=Tair,c,su,i +
(ṁi · cp )air
2
1
+ (TW F,2N +1−i − Tair,r,su,i )
2
1
U Ai
=Tair,r,su,i +
· (TW F,i − Tair,r,su,i )
(ṁi · cp )air
2
(B.2)
(B.3)
Since the inlet air temperature is no longer the average outlet air temperature
above Row 4, two final equations are needed:
Tair,c,in,i = Tair,c,ex,4N +1−i
(B.4)
Tair,r,in,i = Tair,r,ex,4N +i
(B.5)
To compare the heat transfer between mixed and unmixed cooling air, the models
are evaluated with a WF supply temperature of 48.8◦ C, air supply temperature of
25◦ C, WF mass flow rate of 0.153 kgs-1 , air mass flow rate of 3.221 kgs-1 , and WF pressure of 2.11 bar (saturated liquid at the WF exit is not imposed). The heat transfer
coefficients and pressure drop assumptions are the same as those of Section 2.3. With
these inputs, the mixed model predicts a heat flow of 30.3 kW while the unmixed
model results in a heat flow difference of only 0.03%.
B.2
Variable vs Constant Heat Transfer Coefficients
For the set of model inputs mentioned above, the heat transfer coefficients, which
vary for each cell, can be averaged for each zone of the refrigerant. The average vapor
heat transfer coefficient is 288 Wm-2 K-1 , and the average two-phase heat transfer
coefficient is 1515 Wm-2 K-1 . (Since the outlet is on the cusp of the two-phase/liquid
boundary or fully two phase for these simulations, there is no need for the liquid heat
transfer coefficient.) The air heat transfer coefficient is 31.4 Wm-2 K-1 . Changing the
79
mixed model to use these constant heat transfer coefficients results in a heat flow
difference of only 0.76%.
With a new working condition based on the same assumptions but a 5% decrease in
air mass flow rate to 3.06 kg-1 , the average vapor heat transfer coefficient remains 288
Wm-2 K-1 , and the two-phase heat transfer coefficient becomes 1573 Wm-2 K-1 . The air
heat transfer coefficient reduces to 30.4 Wm-2 K-1 . After executing the constant heat
transfer coefficient model with the coefficients from the previous working condition,
but the new mass flow rate, the error in heat flow is only 0.66%.
Setting the air mass flow rate back to 3.221 kg-1 and reducing the WF pressure
to 1.77 bar and air supply temperature to 20◦ C (the saturation temperature at this
pressure although saturated liquid at the exit is still not imposed) results in an average
vapor heat transfer coefficient of 284 Wm-2 K-1 , two-phase heat transfer coefficient of
1609 Wm-2 K-1 , and air heat transfer coefficient of 31.2 Wm-2 K-1 . The resulting heat
flow difference relative to the constant heat transfer coefficient model with coefficients
set by the first working condition is only 0.15%.
B.3
Tube-to-Tube Heat Transfer
Tube-to-tube heat transfer is currently neglected in the condenser models. However,
its impact becomes important near the WF inlet where the fluid is still in vapor phase.
For instance, Fig. B-2, which shows the average temperature for each tube and the
surrounding air, demonstrates that Tube 1 exhibits the only significant temperature
difference relative to the rest of the tubes because it is the only tube for which
flow remains fully in single phase. The model input conditions used to generate these
average temperatures are a WF supply temperature of 48.8◦ C, air supply temperature
of 25◦ C, WF mass flow rate of 0.153 kgs-1 , air mass flow rate of 3.221 kgs-1 , and WF
pressure of 2.11 bar (saturated liquid at the WF exit is not imposed).
The tube-to-tube heat transfer, Q̇cond,tube−tube , can be estimated by modeling conduction through the fin area described on two sides by the line marked “W” in
Fig. B-2:
80
7DLU ƒ&
7:) ƒ&
;;
7XEH
;
/ PP
: PP
Figure B-2: Average condenser temperatures for tube-tube heat transfer analysis. Average
supply and exhaust air temperatures for each tube row are listed in the left column with
dotted blue lines indicating the plane of the mixed air. Average WF temperatures in each
tube are depicted on the tube faces with the progression from Tube 1 to 12 indicated to the
right of each tube. The zoomed view provides some key dimensions for the heat transfer
analysis, namely the distance between tubes, L, and the fin width used to determine the
cross-sectional area for conduction between tubes, W .
Q̇cond,tube−tube = kf in ·
Across
W · tf in · Nf ins
· ∆T = kf in ·
· ∆T,
L
L
(B.6)
where kf in is the fin conductivity; Across is the cross-sectional heat transfer area
described by the width, W , and fin thickness, tf in , multiplied by the number of fins,
Nf ins ; L is the distance between tubes; and ∆T is the temperature difference between
tubes. With kf in = 200 W(mK)-1 , W =33.0 mm, tf in = 0.13 mm, Nf ins = 1081, L =
38.1 mm, and the largest average temperature difference between Tubes 1 and 4 or 5
= 6.9◦ C, the tube-to-tube heat transfer is 168 W.
This result can be compared to the tube-to-air heat transfer, Q̇cond,tube−air , using
Q̇cond,tube−air = U A · ∆T,
(B.7)
where U A is the combination tubeside and finside product of heat transfer coefficient
81
% Error in Heat Flow vs 10 Cells
0
−0.1
−0.2
−0.3
−0.4
−0.5
−0.6
−0.7
2
4
6
8
10
Number of Cells per Tube
Figure B-3: % error in condenser heat flow vs number of cells per tube.
and heat exchange surface area for a single tube and ∆T is the average temperature
difference between the tube and air. With U A for Tube 1 = 18.3 Wm-2 K-1 (derived
from the current model) and a temperature difference between Tube 1 and its supply
air of 8.3◦ C, the tube-to-air heat transfer is 152 W, so the tube-to-tube heat transfer
actually dominates for Tube 1. This effect falls dramatically when considering tubes
in two-phase flow surrounded by other tubes in two-phase flow since they exhibit a
negligible temperature difference.
Although tube-to-tube interactions were neglected in the detailed condenser model,
based on the analysis of this working condition, they could play a significant part in
the heat transfer for up to 5 of the 12 tubes in a tube bank. This improvement will
be considered in future work.
B.4
Sensitivity to Number of Cells
Since Modelica is more robust at handling a changing number of cells, the dynamic
model run to steady-state is used for this study. Again, the model is evaluated with
a WF supply temperature of 48.8◦ C, air supply temperature of 25◦ C, WF mass flow
rate of 0.153 kgs-1 , air mass flow rate of 3.221 kgs-1 , and WF pressure of 2.11 bar
(saturated liquid at the WF exit is not imposed). The working fluid heat transfer
82
coefficients are 288 Wm-2 K-1 in vapor phase, 1520 Wm-2 K-1 in two phase, and 170
Wm-2 K-1 in liquid phase and the air heat transfer coefficient is 31.4 Wm-2 K-1 . The
nominal amount of cells per tube in the steady-state model is 10. With 12 tubes, this
results in 120 cells per tube bank. Figure B-3 shows that with only 2 cells per tube
or 24 cells per tube bank, the error in heat flow vs 10 cells per tube is less than 1%.
83
84
Appendix C
Optimized Intermediate Pressure
between Two Expanders
Since each expander’s isentropic efficiency is modeled using a polynomial as a function
of psu and rp , it is possible to find an analytical solution for the optimized intermediate
pressure, Pint , by maximizing the equation for the combined efficiency given the inlet
and outlet pressures. The derivation for this solution when the efficiency polynomial
is a function of ρsu and rp was provided in [15], and the derivation for the polynomials
used in this work (Equation 2.10), which depend on psu and rp , is provided here.
If the refrigerant is approximated as an ideal gas,
dh = cp · dT
(C.1)
and, using isentropic relations,
Tp
1−γ
γ
Tex,s
=
= constant →
Tsu
pex,s
psu
1−γ
γ
,
(C.2)
where su and ex, s refer to supply and isentropic exhaust conditions, respectively; T
is temperature; p is pressure; γ is heat capacity ratio; h is enthalpy; and cp is specific
heat capacity. Therefore,
85
1−γ
hex,s
= rp γ ,
hsu
(C.3)
when 0 K is chosen for the enthalpy reference.
Using the definition of isentropic efficiency for an expander,
η=
hsu − hex
,
hsu − hex,s
(C.4)
and defining β as
β ≡1−
1−γ
hex,s
= 1 − rp γ ,
hsu
(C.5)
the overall isentropic efficiency across both expanders can be expressed as follows:
η=
h1 − h3
h1 − h3
η1 · β1 + η2 · β2 − η1 · η2 · β1 · β2
,
=
=
h1 − h3s
β · h1
β
(C.6)
where index 1 refers to the entrance to Expander 1; index 2 refers to the exhaust of
Expander 1/supply of Expander 2; and index 3 refers to the exhaust of Expander 2.
Determining the intermediate pressure at which the combined efficiency maximizes
can be achieved by setting
dη
drp1
to zero:
dη
=(η1 β10 + η10 β1 + η2 β20 + η20 β2 − η1 η2 β1 β20 −
drp1
η1 η2 β10 β2 + η1 η20 β1 β2 − η10 η2 β1 β2 ) · β −1 = 0.
(C.7)
where a 0 indicates a derivative with respect to rp1 . Solving this differential equation
requires determination of the derivative terms.
The derivative of β1 is
dβ1
γ − 1 1−2γ
=
· rp γ ,
drp1
γ
and since rp = rp1 · rp2 , the derivative of β2 is
86
(C.8)
d
dβ2
=
drp1
drp1
rp
1−
rp1
−1
1−γ
1 − γ 1−γ
=
· rp γ · rp1γ .
γ
γ
(C.9)
The form of the expander isentropic efficiency polynomial was shown in Equation 2.10. For the first expander, the derivative of this polynomial with respect to rp1
is
n−1 n−1
dη1
1 XX
=
aij · i · ln(rp1 )i−1 · ln(psu1 )j + an0 · n · ln(rp1 )n−1 .
drp1
rp1 i=0 j=0
(C.10)
However, since the second expander parameters depend on rp1 , it is necessary to
calculate partial derivatives:
δη2 drp2
δη2 dpsu2
rp δη2
psu1 δη2
dη2
=
·
+
·
=− 2 ·
− 2 ·
.
drp1
δrp2 drp1 δpsu2 drp1
rp1 δrp2
rp1 δpsu2
(C.11)
The partial derivative of η2 with respect to rp2 is of the same form as Equation C.10
while the partial derivative of η2 with respect to psu2 is
n−1 n−1
1 XX
δη2
=
aij · j · ln(rp2 )i · ln(psu2 )j−1 + an0 · n · ln(psu2 )n−1 .
δpsu2
psu2 i=0 j=0
(C.12)
With equations for all of the derivatives, the maximum overall efficiency can be
determined by solving Equation C.7. Solving for the Expander 1 pressure ratio which
maximizes η allows for determination of the optimized intermediate supply pressure.
To approximate the behavior of a real gas, rather than the ideal gas assumed in
the previous equations, γ can be replaced with an equivalent heat capacity ratio by
solving for γ in Equation C.3:
"
γeq = 1 +
ln( hh3s1 )
ln(rp )
87
#−1
.
(C.13)
88
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