University of Liège FACULTY OF APPLIED SCIENCES AEROSPACE AND MECHANICAL ENGINEERING DEPARTMENT THERMODYNAMIC LABORATORY Development of an experimentally validated dynamic model of a micro scale solar organic Rankine cycle Laura PEIGNEUX Thesis Submitted in Partial Fulfillment of the Requirements for the Degree of Electro-Mechanical Engineer Academic year 2012-2013 President of the jury: O. Leonard Composition of the jury: V. Lemort (Main adviser) P. Dewallef S. Quoilin E. Winandy
Transcript
University of Liège
FACULTY OF APPLIED SCIENCES
AEROSPACE AND MECHANICAL ENGINEERING
DEPARTMENT
THERMODYNAMIC LABORATORY
Development of an experimentally validated dynamic model of a micro scale solar organic
Rankine cycle
Laura PEIGNEUX
Thesis Submitted in Partial Fulfillment of the Requirements for the Degree of
Electro-Mechanical Engineer
Academic year 2012-2013
President of the jury: O. Leonard
Composition of the jury:
V. Lemort (Main adviser)
P. Dewallef
S. Quoilin
E. Winandy
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Acknowledgements I would like to thank Professor Lemort for entrusting me with this work and giving me the opportunity to study such an interesting and topical subject. A special thanks goes to Sébastien Declaye who guided and advised me all along the year. His counsels and remarks were always judicious. I am very grateful to Adriano Desideri and Sylvain Quoilin for helping me deal with the subtleties of modelica. This work owes them a lot. I would also like to thank all the members of the laboratory for providing agreeable working conditions. Finally, I want to thank my family and friends for being such a great support in bad and good times.
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Development of an experimentally validated dynamic model of a micro scale solar
organic Rankine cycle
Laura Peigneux
Thesis Submitted in Partial Fulfillment of the Requirements for the Degree of Electromechanical Engineer
(Academic year 2012-2013)
Abstract
The present work studies a micro scale solar organic Rankine cycle (ORC).
The representative test bench is made up of two scroll expanders, a recuperator, pumps and an air-
condenser. This test bench has been elaborated by the thermodynamics laboratory of the University of Liège
as part of the sun2power project.
The first part of this work focuses on measurements which have been previously done on the test bench.
These will be corrected using a reconciliation method. The method modifies the raw values of temperatures,
pressures and mass flows through the minimization of an error function defined according to the weighted
least-squares method. The results are then studied and criticized in order to judge the precision and the
reliability of the method.
Then, a model of the test bench has been developed using the «Dymola» software which uses «Modelica» as
modeling language. First, a simple model of the ORC was performed. Then, this model was gradually made
more complex by adding PID controllers and solar collectors.
This enhancement made it possible to run the model for four typical days based on the direct normal
irradiance (DNI) of Marseille. The performances of the cycle could then be apprehended in terms of net
production and spread over the day. The simulation also showed the importance of the ambient conditions.
Finally, some possible enhancements of the model were brought into light. These mainly concern the
storage along with the clouding and a more representative modeling of the expanders and of the condenser.
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Résumé
Le présent travail base son étude sur une microcentrale solaire utilisant un cycle de Rankine organique
(ORC). Le banc d’essais de cette centrale est constitué de deux expanseurs scroll, d’un régénérateur, de
pompes et d’un condenseur à air réalisé par le laboratoire de thermodynamique de l’Université de Liège
dans le cadre du projet sun2power.
Dans un premier temps, des mesures précédemment réalisées sur le banc d’essai ont été étudiées et
corrigées à l’aide d’une méthode de réconciliation. Cette méthode ajuste les variables brutes de
températures, pressions et débits qui lui sont fournies afin de minimiser une fonction d’erreur obtenue via la
méthode des moindres carrés.
Les résultats obtenus ont été analysés et critiqués afin de juger au mieux de la précision et la fiabilité de la
méthode.
Un modèle du banc a ensuite été conçu via le logiciel « Dymola » qui se base sur le langage de
programmation « Modelica ». Un premier modèle simple de l’ORC a été réalisé avant d’être affiné grâce à
l’introduction de contrôleurs PID et de collecteurs solaires.
Ceux-ci ont permis la modélisation et l’étude de quatre journées type sur base de l’irradiation à Marseille.
L’analyse de ces journées donne une idée d’une part de l’ordre de grandeur de la production nette du cycle
et d’autre part de son étalement sur une journée. Elle met également en avant l’importance des conditions
extérieures.
Enfin, des améliorations possibles du modèle ont été suggérées. Celles-ci reprennent principalement le
stockage, allant de pair avec la prise en compte de la nébulosité, ainsi qu’une meilleure modélisation des
Table of figures .................................................................................................................................................... 8
2. State of the art........................................................................................................................................... 11
2.1 The Rankine Cycle .................................................................................................................................... 11
2.1.1. Real Rankine Cycle ........................................................................................................................... 12
3.1.1.1 Working fluid ................................................................................................................................. 14
4. Data Reconciliation .................................................................................................................................... 18
4.1 The Method ............................................................................................................................................. 18
4.2.1 Standard deviation ........................................................................................................................... 19
4.2.2 Differentiation of the measurement points ..................................................................................... 20
4.2.3 Application of the method ............................................................................................................... 22
5. Establishment of the model ...................................................................................................................... 30
5.3 Establishment of a simple ORC model .................................................................................................... 42
6. Improvement of the cycle ......................................................................................................................... 47
6.1.1 Implementation of the PID controller to the cycle .......................................................................... 48
6.2 Solar collectors ........................................................................................................................................ 53
6.3 Direct normal irradiance (DNI) ................................................................................................................ 54
6.4 Air mass flow rate and fan consumption at the condenser .................................................................... 55
7. Simulations of the model .......................................................................................................................... 57
7.1 Adjustment of the frequency at the condenser ...................................................................................... 59
8. Possible enhancement of the cycle ........................................................................................................... 63
Appendix I .......................................................................................................................................................... 70
Appendix II ......................................................................................................................................................... 71
Appendix III ........................................................................................................................................................ 73
Appendix IV ....................................................................................................................................................... 78
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Nomenclature
Bracket root used in the bisection method [J/kg] Bracket root used in the bisection method [J/kg] Generic expression of the initial measurements Specific heat capacity [J/(kg.K)] Frequency [Hz] Differential pressure at the first expander [bar]
Differential pressure at the second expander [bar]
Outlet enthalpy used in the determination of the transfer coefficients
[J/kg]
Generic expression of the reconciled measurements Specific volume [m³/kg] Oil volume [m³] Mass flow rate [kg/s] Pressure [bar] Pinch point at the evaporator [°C]
Generic expression of the internal volume ratio Entropy [J/(kg.K)] Temperature [°C] Oil fraction contained in the working fluid [%] Independant variable of the reconciliation method
Subscripts Before correction Secondary fluid of the condenser Calculated Condenser Corrected Exhaust Expander Evaporator Isentropic Measured Secondary fluid of the evaporator Post correction Working fluid of the cycle Saturation Supply
Greek symbols Efficiency Lagrange multiplier Standard deviation
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Table of figures Figure 1 : Simple working cycle with superheating [4]...................................................................................... 11
Figure 3: Working principle of a gear pump [1] ................................................................................................ 16
Figure 4: Final configuration of the test bench ................................................................................................. 17
Figure 5 : Differentiation of the measurement points ...................................................................................... 21
Figure 6 : locating of the sensors ....................................................................................................................... 21
Figure 22: Construction of the model ............................................................................................................... 43
Figure 23 : Simple ORC model ........................................................................................................................... 43
Figure 24: Pressure and temperature at the evaporator’s outlet ..................................................................... 44
Figure 25: Temperature at the condenser's exhaust ........................................................................................ 44
Figure 35: Temperature of the oil at the inlet of the evaporator ..................................................................... 52
Figure 36 : Variation of the PID parameters ...................................................................................................... 52
Figure 37: Solar Collector .................................................................................................................................. 53
Figure 38: Solar Collector and oil circuit ............................................................................................................ 54
Figure 39: Direct Normal Irradiance .................................................................................................................. 55
Figure 40: Final aspect of the ORC .................................................................................................................... 56
Figure 41 : Available power and net production (4 typical days) ...................................................................... 57
Figure 42 : Productions and consumptions for the summer solstice and the summer equinox ...................... 58
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Figure 43: Variable frequency at the condenser ............................................................................................... 60
Figure 44: Consumption of the condenser ........................................................................................................ 60
Figure 45: Comparison of the net production of the cycle when varying the condenser's frequency ............. 61
Figure 46 : Outlet pressure of the second expander ......................................................................................... 62
On the oil side, the secondary fluid flow rate is insured by a vane pump.
3.1.4 Condenser
The choice of an aero-condenser was mainly motivated by the fact that no source of cold water is available
near the power plant, which disables the use of a classical open circuit condenser. Moreover, a secondary
circuit would have induced higher costs and an unnecessary complexity.
3.1.5 Recuperator
A recuperator is a heat exchanger which increases the cycle efficiency through energy recovery.
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In the case of the Sun2power test bench, a counter flow exchanger equipped with 40 brazed plates was used
(CB 30 Alfa Laval).
3.1.6 Evaporator
The evaporator mounted on the test bench is a typical counterflow heat exchanger equipped with 100
plates.
Figure 4 shows the final configuration of the test bench.
Figure 4: Final configuration of the test bench
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4. Data Reconciliation
4.1 The Method
The first step towards the establishment of a model of the sun2power ORC is finding reliable data to
incorporate in the model-to-be. In order to do so, it is possible to use measurements which have been
previously done on the test bench [1].
However, one has to be cautious when using this kind of approach: random errors in the measurements may
cause discrepancies in the mass and energy balances. Moreover, calculations which have been carried out
based on different measurement sets may give different results.
To overcome these problems, a data reconciliation methodology will be used [14]. The latter stipulates a
mathematical representation of the constrained minimization problem. The goal of the methodology is to
rectify the measured variables by assigning to them a value which is as close as possible to the initial value
weighted by its standard deviation.
This formulation uses the weighted least-squares problem:
( ) ∑( )
(1)
In this equation, represents the reconciled measurements, represents the initial measurements and
represents the standard deviations chosen according to the studied measured variable.
The Lagrange multipliers method is then used in order to reconcile the measures of flow rate, temperature
and pressure:
( ) ∑ ( )
(2)
Where are the Lagrange multipliers and ( ) is the constraint function derived from the heat
balance calculations over each component of the cycle. This function depends on which has been
previously defined and on which represents the independent variables. Unlike the variables, these
independent variables haven’t been measured.
The derivative of (2) for each measured variable gives us a certain amount of equations (see Table 3). To
these equations, we can add another equation obtained by calculating the derivative of (2) according to the
sole independent variable: the oil fraction stirred in the working fluid.
Once the different derivatives have been calculated, the system consisting of the following equations is
implemented in the software EES (Engineering Equation Solver):
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(3)
As said earlier, this method will first be applied to previously done measurements. A reliable set of
measurement will then be extracted and introduced into an ORC model developed under EES.
The data collected from the EES file will then be used as initialization for the modelica model of the
test bench.
4.2 Application
4.2.1 Standard deviation
The application of (1) to our case requires the evaluation of the standard deviation for each measurement
instrument installed on the test bench.
The standard deviation depends on the admissible error, also called the confidence level. The standard
deviation can be estimated by equation (4) in which represents the error and k is a constant which is
representative of the confidence level. For a confidence level of 95%, k equals 2.
(4)
The standard deviations used in the reconciliation method are listed below and have been determined based
on the datasheets given by the constructor.
- Temperature
Measurement device: Thermocouple T class1
Standard deviation [°C] :
- Absolute pressure
Measurement device : Sensor Keller 21Y
Standard deviation [bar] :
- Differential pressure
Measurement device: Sensor Siemens P250
Standard deviation [bar] :
- Mass flow
Measurement device: Coriolis sensor, Krohne Optimass 7300C S06 for the R245fa
Standard deviation [kg/s]
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In order to establish the standard deviation of the mass flow rate on the oil side, some hypothesis had to be
made. The mass flow rate was initially obtained through the measurement of the volume of oil going
through a given section during a time interval of 120 seconds.
By considering a 1 percent error on the measured volume and a 1 second error on the time interval, and by
considering the error one does on a quotient, the following standard deviation was set for the mass flow
rate on the oil side:
(
) (
)
As for the mass flow rate of the air, a 5% error on the measurement was imposed. This led to the following
value of the standard deviation in kg/s :
4.2.2 Differentiation of the measurement points
The measurement points have been divided into two groups. The first group contains the nineteen first
essays whereas the second group contains the seven last ones.
This differentiation is based on the opening of the red circled valve in
Figure 5.
For the measurement points of the first group, this valve was opened leading to the theoretical equivalence
given by equation (5).
(5)
The other valves around the expanders which haven’t been circled were closed at all time. Both expanders were therefore inoperative and their action was simulated by the use of the valve encircled in blue.
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Figure 5 : Differentiation of the measurement points
However, for the second group of measurement points the valve encircled in red was closed. By closing this
valve, and since all the other valves stayed closed, a certain pressure state was trapped in between the two
expanders.
Therefore, the pressure in the region between the expanders was constant while the pressure at the
regenerator’s inlet on the hot side was varying.
This created a situation in which the pressure measured by the sensor at the expander’s exhaust (« »)
became superior to the pressure measured at its inlet (« »).
To ease the comprehension of the phenomenon, Figure 6 shows the exact location of the differential
pressure sensors.
Equation (5) therefore became meaningless for the last set of measurement points.
Figure 6 : locating of the sensors
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4.2.3 Application of the method
The measures to reconcile are given in Table 2. Following the explanations given in section 4.2.2, the variable
is only taken into account for the first set of measurements which includes assays 1 to 19.
Table 2 : Measures to reconcile
Temperatures Mass flows Pressures Independent variable
The formulation of (1) applied to our case is given by equation (6). Again, this equation is valid for the first
set of measurements. The second set responds to a similair equation, only the last term is missing.
( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
(
)
( )
( )
(
)
( )
( )
( )
( )
(6)
The three common constraints to each set of measurements are the energy balances on the heat
exchangers:
- Constraint on the evaporator
( ) ( )
(7)
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Where
(8)
(9)
(10)
(11)
As the derivative of this constraint has to be evaluated, it is necessary to express as a third order
polynomial of the pressure at the evaporator. This expression can be seen in (12).
(12)
- Constraint on the condenser
( ) ( )
(13)
Where
(14)
(15)
(16)
(17)
Again, the expression of has to be expressed in a way that allows the evaluation of the constraint’s
derivative.
This expression is given by (18)
(18)
Where is expressed as the average pressure between the inlet and the outlet of the condenser.
(19)
- Constraint on the recuperator
( ) ( )
(20)
Where
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The first set of measurement also has to respect a fourth constraint which expresses the equality of
pressures between two points of the cycle.
- Constraint on the pressure (Measurement points 1 to 19)
(21)
All in all, the application of the method gives us the following distribution of equations and unknowns:
Table 3: Application of the reconciliation method
Measurement points 1 to 19 20 to 26
Equations - Derived from (2) - Constraints
19 4
18 3
Unknowns - - Lagrange multipliers
18 4
17 3
4.2.4 Results
The results obtained by using this method have been analyzed using two types of indicators:
1. The new variable defined by equation (22)
| |
(22)
2. Heat balances on the exchangers before and after correction
For a confidence level of 95%, the first indicator has to be lower than 1,96. If not, one can conclude to an
oversized correction.
Before studying the final reconciled measures, it may be useful to look into the energy balances on each side
of the heat exchangers before application of the method. These are shown in Table 4 below.
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Table 4 : Heat transfer before the application of the reconciliation method
1 15,01 15,42 17,52 15,48 3,092 3,072
2 13,86 11,32 11,57 3,059 10,59 2,2
3 19,08 16,84 16,87 2,914 17,24 3,159
4 18,72 15,89 16,97 16,16 3,335 3,517
5 17,6 15,23 16,38 15,53 3,366 3,573
6 17,32 14,94 15,59 15,22 3,363 3,546
7 16,46 14,12 14,94 14,38 3,181 3,372
8 15,58 13,31 14,28 13,45 2,923 3,08
9 16,41 13,01 13,85 13,13 3,155 3,261
10 15,35 12,39 13,4 12,51 2,996 3,114
11 26,8 23,06 24,51 23,37 7,201 7,541
12 24,62 1,962 23,02 21,79 5,481 25,29
13 25,37 22,3 23,72 22,58 6 26,4
14 24,15 1,933 20,59 23,17 2,657 21,8
15 24,26 2,015 21,05 23,47 2,755 22,2
16 25,67 2,296 23,31 23,01 3,492 23,45
17 30,7 1,83 22,19 22,74 3,547 24,14
18 16,46 14,12 14,94 14,38 3,181 3,372
19 23,43 22,58 22 5,259 23,01 5,561
20 23,94 2,37 21,45 22,03 5,671 25,27
21 22,28 2,39 21,27 21,97 5,473 24,93
22 21,82 2,271 21,14 21,95 5,243 24,63
23 19,76 2,249 21,02 21,93 5,084 24,43
24 18,41 2,169 20,98 21,66 4,659 23,63
25 17,74 2,216 20,84 21,66 4,659 23,63
26 16,76 2,294 20,66 21,25 4,674 23,25
The grey cells point out the presence of incoherent results. In the case of the evaporator, these can be
explained by the uncertainties in the measures and by the fact that the temperature at the regenerator’s
outlet is close, even sometimes equal, to the saturation temperature.
Because of this, the measured temperature at the outlet of the regenerator may be superior to the
saturation temperature. In this case, the evaporation is considered to take place in the regenerator even
though it might not be so in reality. The evaporator’s use is then only to increase slightly the fluid’s
temperature. This phenomenon has been graphically represented in Figure 7 for the second measurement
point. The left part of the figure shows three key points, starting from the left: the inlet and outlet of the
regenerator on the cold side and the outlet of the evaporator. As said previously, the temperature at the
outlet of the regenerator is very close to the saturation temperature, this is shown in the right part of the
figure which zooms on the critical zone.
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Figure 7: Evaporation
When considering the condenser, the same logic can be applied backwards: the measure uncertainties may
lead to an outlet temperature at the regenerator which is lower than the saturation temperature.
Because of this measurement issue, it will be considered that the condensation has taken place into the
regenerator. Since the change of phase doesn’t take place in the condenser, the latter will only subcool the
fluid.
This is shown in Figure 8 for the fifteenth measurement point. The left part of the figure shows three points.
The first point on the left of the temperature-entropy diagram represents the outlet conditions of the
condenser, followed by the second which gives the outlet conditions of the regenerator’s hot side. The last
point on the right of the diagram for its part shows the inlet condition at the recuperator.
The second part of the figure shows a zoom on the outlet conditions of the regenerator which, as expected,
is below the saturation temperature.
Because of these discrepancies in the heat balances, the measurement points 20 to 26 have been rejected.
Indeed, The conditions in which the measures were done lead to the situation described above in every case,
making the reconciled data unusable for further operations.
Figure 8: Condensation
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A first version of the reconciliation method containing the fraction of lubricating oil trapped in the
refrigerant was computed. A special attention was then brought to the heat balances and the final value of
the oil fraction. The results are shown in Table 5.
The first two columns shows the heat exchanged on the thermal oil side of the evaporator before (AC) and
after (PC) the application of the reconciliation method. The two following columns follow the same logic, this
time for the working fluid. Finally, the last column shows the lubricating oil fraction trapped in the R245fa.
These results immediately trigger the attention: the heat balance on the thermal oil side is untouched
whereas that of the refrigerant is systematically corrected. Moreover, one can also notice that the
lubricating oil fraction is corrected in an extreme way, sometimes giving negative results. The other variables
such as pressure, temperature or mass flow have only been slightly corrected and cannot account for the
results of Table 5.
The correction of y explains it all. Since no standard deviation has been attached to the lubricating oil
fraction, the program is free to modify it at its will without introducing any dissonance in equation (6). That
is why the lubricating oil fraction has been modified in priority, leading to small corrections in the other
variables.
Moreover, when applying the reconciliation method on a single heat exchanger, the program adapts the
lubricating oil fraction to suit the heat balance over the studied element.
The problem arises when a second heat exchanger is added to the method. In this case, the correction
needed for the lubricating oil fraction might vary from one exchanger to the other. Therefore, satisfying one
heat balance may lead to the degradation of the other. The program will then keep the most befitting value
found for the oil fraction, i.e. the value that allows a modification of the other variables leading to a
minimization of equation (6).
Considering all this, the negative values found by the program can be explained as follows: at least one heat
exchanger requires these values to satisfy its heat balance and this configuration leads to a minimization of
the constrained equation given by (6).
The question that now has to be answered is why would a heat exchanger call for such values of the variable
y?
The solution which has been brought into light suggests that these are due to the lack of thermal insulation
of the exchanger. Therefore, if we consider the case of the evaporator, the thermal oil gives a certain
amount of heat to the refrigerant and to the ambiance. In the model, only the exchange from thermal oil to
refrigerant has been considered. During the calculation, this simplification may lead to a situation where the
quantity of heat held in the lubricating oil trapped in the refrigerant becomes negative. Hence, the program
will identify a negative value of the lubricating oil fraction to satisfy the heat balance over the exchanger.
A solution to this would be to improve the insulation of the heat exchangers.
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Table 5 : Heat transfer before and after the application of the reconciliation method
1 17,03 17,03 15,48 17,02 -0,1583
2 14,88 14,88 3,059 1,712 -0,4729
3 18,79 18,79 2,914 2,575 -0,1335
4 17,66 17,66 16,16 17,62 -0,1425
5 17,04 17,04 15,53 17 -0,1595
6 16,58 16,58 15,22 16,56 -0,1576
7 15,72 15,72 14,38 15,7 -0,164
8 14,6 14,6 13,45 14,59 -0,1516
9 14,01 14,01 13,13 14,01 -0,1259
10 13,42 13,42 12,51 13,4 -0,1425
11 25,22 25,22 23,37 25,22 -0,1627
12 22,73 22,73 21,79 22,72 -0,06968
13 24,16 24,16 22,58 24,14 -0,1152
14 20,22 20,22 23,17 20,21 0,1804
15 19,92 19,92 23,47 19,89 0,2015
16 19,39 19,39 23,01 19,34 0,193
17 21,94 21,94 22,74 21,87 0,02196
18 15,72 15,72 14,38 15,7 -0,164
19 22,05 22,05 5,259 5,382 0,03735
Because of this and since the oil fraction is in reality supposedly rather small, the method has been applied
considering that no oil was mixed in the refrigerant.
Now that the energy balances have been studied before the application of the method, it is interesting to
evaluate those same heat balances with the reconciled measurements.
In the case of the condenser, Figure 9 shows the correction in energy balance on both sides. The red dots
show the energy balance before correction whereas the blue ones show the results of the reconciliation
method.
The correction obtained thanks to the method is more than satisfactory for the majority of the reconciled
data. It is however noticeable that five points haven’t been corrected enough to align on the bisector. These
points correspond to the measurement points 12 and 14 to 17 which had been pinned in Table 4.
When studying the corrected data, it can be seen that the mass flow rate of the refrigerant is untouched.
This is due to the extremely small value of its standard deviation.
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Figure 9: Result of the reconciliation method on the condenser
The same analysis can be done for the evaporator and the regenerator. This is shown in Figure 10. Again, it
can be seen that the majority of the reconciled measure now respect the heat balance of the concerned heat
exchanger. Those who do not respect this heat balance can easily be spotted either numerically through the
parameter w defined by equation (22) or graphically.
Therefore, the method not only provides a way of correcting the measures, it also highlights incoherent
results.
Figure 10: Result of the reconciliation method on the evaporator and on the regenerator
Figure 9 and Figure 10 show the results of the reconciliation method through the evaluation of the heat
exchanges. The same type of chart has been established for random pressure, temperature and flow rate
and can be studied in Appendix I.
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5. Establishment of the model
Using the characteristics of each components of the cycle, a dynamic model of the test bench can be
developed using the program «Dymola» which runs with the open Modelica modeling language.
5.1 Dymola
Dymola, short for Dynamic Modeling Laboratory, is an engineering tool which allows the modeling and the
simulation of complex systems. The program covers a large range of engineering applications in different
fields such as robotics, automotive, aerospace and thermodynamics.
Therefore, Dymola offers five main libraries: a first library has been elaborated for electrical, electronic and
magnetic components, another one focuses on the mechanical components. The third library concerns the
fluid component, giving access to specific types of fluid, sensors, valves, thermal conductor, etc.
The fourth library concerns the control systems, mainly introduced as blocks representing either
input/output blocs or controller blocks.
Finally, the last library concentrates on functions used for vectors, matrices or linear systems.
Even though the in-built libraries allow a great operating liberty, Dymola also offers the user the possibility
to create its own model libraries or even to modify the existing ones, making it possible to create unique and
precise models.
The present work uses the «ThermoCycle» library which has been developed by the Thermodynamics
Laboratory of the University of Liège.
Just like other libraries, «ThermoCycle» considers three basic equations to model a given system: the mass
balance, the energy balance and the momentum balance.
5.2 Description of the components
A full model is created by assembling elementary components together. Each elementary component has to
be tuned to better represent the reality. This tuning can be separated into two categories: a first category
focuses on the parameters which define the item whereas the second category concentrates on the initial
values given to the thermodynamic variables.
These initial values were found by running a simplified EES model of an organic Rankine cycle.
This section will give an overview of the components used to describe the cycle and of the way in which they
have been initialized.
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5.2.1 Pump and liquid receiver
The pump model implemented in modelica requires the knowledge of different parameters.
The first parameter, i.e. the working fluid, had been set to R245faCool which uses a medium library named
«Coolprop». This working fluid is, of course, chosen in the other components as well.
Then, the swept volume has to be defined. In the case of the investigated pump, it is of . This
value has been calculated based on the volumetric flow rate going through the pump at a frequency of 50Hz.
Finally, the model was updated with an equation representing the internal isentropic efficiency of the pump
as a function of the pressure ratio (
) and of the frequency ( ) .
Using the dotted data represented in Figure 11, the following equation was obtained:
(
)
(23)
The representative curve of this equation can also be seen in Figure 11 for the frequencies of 33, 50 and 52
Hz.
Figure 11: Efficiency of the pump versus pressure ratio
Concerning the liquid receiver, very few information was necessary. The volume of the tank was set to
and the presence of non-condensable gases was not taken into account, their pressure was
therefore set to zero.
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5.2.2 Evaporator
The secondary fluid used in the evaporator is the organic oil «Slytherm XLT» which can be referred to as
«therminoil 66» in Dymola. The mass flow rate of this fluid is initialized at 0,21 kg/s whereas the mass flow
rate of the secondary fluid is of 0,108 kg/s.
Figure 12: Evaporator
The geometric parameters of the evaporator are given in the datasheet either explicitly or through
theoretical formulas depending on the number of plates.
On the investigated test bench, the evaporator is a “Swep B35” with 100 brazed plates. Knowing that, the
exchange area and the hold-up volume can be evaluated as shown in equations (24) and (25).
( )
(24)
(
)
(25)
Equation (24) considers the area of one plate, , calculated via the data of Figure 12. Since the extreme
plates of the heat exchanger do not take part in the heat exchange as efficiently as the others, they have
been taken out of the calculus of the total exchange area.
Since the exchanger is symmetric, it is possible to derive the volume of the working fluid and the volume of
the secondary fluid simply by considering half of the value found in (25).
The mass of metal between the two fluids also has to be implemented in the evaporator model. Again, this
can be found in the datasheet where the total mass is given by (26) in kg . In this equation, the second term
refers to the mass of the plates and therefore gives us the mass of the wall.
(26)
In the end, the mass of the wall is 33,32 kg.
33
Another important parameter to implement in the heat exchanger model is the specific capacity of the metal
wall: the plates are made of stainless steel whose specific capacity is of 500 J/kgK.
Finally, the last parameters required by modelica are the heat transfer coefficients. These have been
determined by a calculation which is the object of the following subsection.
The evaporator, as all of the heat exchangers models under modelica, will then use a finite volume
approach. In other words, the exchanger is split into smaller elements of equal volume. The energy and mass
balances are then applied for the medium fluid chosen during the initialization.
Since the fluid properties are assumed to vary only along the flow direction, this type of model is referred to
as a 1D model.
The user can select the number of elements necessary to represent his exchanger while still taking into
account that as the elements get smaller and smaller, the program gets closer and closer to the real solution
but the time calculation is also increased.
5.2.2.1 Evaluation of the heat transfer coefficients
The evaluation of the heat transfer coefficients in an exchanger where a phase change occurs is complex
because the exchange area for each zone is unknown.
To override this difficulty, the heat transfer coefficients have been calculated through an iterative method
using the Engineering Equation Solver (EES) program.
The first step of the calculation is to select the necessary reconciled data. Concerning the evaporator, these
data are the following:
- mass flow rate of both the working and the secondary fluid
- pressure at the inlet and at the outlet of the evaporator
- supply temperature on the oil side
These will allow the calculation of the saturation temperature and the specific heats.
Of course, the points corresponding to the greyed cells of Table 4 haven’t been taken into account for the
evaluation of the heat transfer coefficients.
34
Figure 13 : Determination of the heat transfer coefficients
At first, the heat transfer coefficients were set to a default value. They were then introduced into a
procedure along with the forementioned data.
The primary aim of the procedure is to determine the vapor area, the liquid area and the two-phase area in
the evaporator. To do this, the Log Mean Temperature Difference (LMTD) Method has been applied (Figure
13).
The method terms a logarithmic mean temperature difference and applies it to the evaluation of the heat
transfer.
Equation (27) shows the definition of the log-mean temperature difference. In this equation and
are the temperature differences between the two involved fluids at each end of the exchanger ((28),(29)).
( )
(27)
(28)
(29)
These equations have been applied thrice in order to consider the presence of three distinct phases in the
heat exchanger. Maintaining the previous notations, this gives us equations (30) to (38). Some of these
equations introduce and which are represented in Figure 15.
Note that the complete code can be found in Appendix III.
(30)
(31)
35
(
)
(32)
(33)
(34)
(
)
(35)
(36)
(37)
(
)
(38)
The outlet temperature of each fluid is determined in the EES procedure.
The outlet temperature of the refrigerant is updated via the bisection method. This method is a typical root-
finding method for a continuous function defined in a given interval. Its principle is simple: if we can find two
values, and , of the variable for which the studied function ( ) has opposite signs, these two variables
are considered as being bracket roots since they frame at least one of the function’s root.
Considering that the function has a unique root between and , the method will divide the initial interval
by two and memorize a midpoint given by ( ) .
If this midpoint is the root, the method ends there. Otherwise, one has to take into account the sign of ( ):
- If ( ) and ( ) have opposite signs, they become the new bracket variables
- If ( ) and ( ) have opposite signs, they become the new bracket variables
The interval surrounding the root of the function is thus decreased until the subinterval is sufficiently small,
as shown in Figure 14.
Figure 14 : the bisection method [15]
36
In our case, the function of which we wish to find the root is a residue defined as the normalized difference
between the calculated area and the real area (equation (39)):
(39)
The residue is a function of the outlet enthalpy of the working fluid which is the variable used in the
bisection method.
Hence, the first step of the procedure is finding two values of the outlet enthalpy which give opposite signs
for the residue function (i.e. finding the bracket roots).
Once this is done, the outlet enthalpy is determined as the average of the two bracket roots. This is shown in
equation (40). In this equation, and represent the bracket roots.
(40)
At this stage, the outlet enthalpy has been determined and the outlet pressure of the R245fa is known
(reconciled data).
These informations allow the evaluation of the outlet temperature of the R245fa.
Figure 15 : - scheme at the evaporator
Knowing the outlet temperature, the exchanged heat of the vapor phase can be calculated as shown in
equation (41).
As said earlier, the saturation temperature has been determined based on the reconciled pressure
measurements.
37
Then, considering that there are no heat losses, it is possible to determine the temperature on the oil side
which marks the limit of the vapor area of the refrigerant on the oil side as shown in Figure 15.
( )
(41)
(42)
At this point, everything is known and equations (30) to (32) can be evaluated. Once this is done, the vapor
area is determined as shown in equation (43).
(43)
In the same line of thinking, we can write the following equations for the two phase zone.
The enthalpy at the inlet and at the outlet of the two-phase zone is determined based on the saturation
temperature and the quality.
(44)
(45)
Which allows the evaluation of the two –phase area in the evaporator (equation (46))
(46)
Again, this procedure can be applied to the liquid zone for which we obtain the following equations
( )
(47)
(48)
And
(49)
As foresaid, the outlet temperature on the oil side is not known at the beginning of the procedure. It is
evaluated through equation (48).
Thanks to equations (43),(46) and (49) it is possible to determine the total exchange area calculated by the
program. This value will then be compared to the real transfer area via the residue (39).
If this residue is too big, then the bracket roots of the bisection method are updated. The outlet enthalpy is
therefore updated as well leading to a new outlet temperature for the R245fa.
38
This iterative process goes on until a satisfactory value of the residue is reached.
Throughout this procedure, the exchange coefficient and have been considered as already
known. In fact, they have been initialized to arbitrary values in order to run the procedure.
Once the program was capable of running on its own and giving viable results, these arbitrary values were
commented off the code.
The final values of the heat transfer coefficients were determined by minimizing the error given by the
absolute difference of the calculated pinch and the measured one.
| | (50)
The result of the minimization gives the final results of Table 6 for the heat transfer coefficient.
These were obtained based on very few measurement points done mostly in the same experimental
conditions. More accurate values could have been found with a wider range of measures considering
different phase-configurations in the evaporator.
Table 6 : Heat transfer coefficient in the evaporator
Heat transfer Coefficients
Working fluid: Liquid phase 1004 W/m²K
Working fluid: Two-Phase 1 182 W/m²K
Working fluid: Vapor phase 1 102 W/m²K
Secondary fluid 99 W/m²K
Figure 16 shows the result of the method by considering the calculated pinch at the evaporator using the
heat transfer coefficients of Table 6 ( ) and the measured pinch ( ).
It can be seen that the majority of the points tend to concentrate around the bisector. The greatest error
committed is of 4,5 K and concerns an isolated point.
Figure 16 : Comparison of the calculated and the measure pinch point (evaporator)
39
5.2.3 Expanders
The measures previously done on the test bench were not sufficient to characterize completely the
expanders.
Therefore, some of the initialization parameters have been left to the default values. This is the case of the
filling factor which has been set to 1.
On the other hand, some parameters could easily be found. This concerns the swept volume which has been
found in the constructor’s database 69[5].
During the construction of the cycle, the isentropic efficiency was set to the default value of 70%.
5.2.4 Recuperator
As said earlier, the recuperator is a counter flow brazed plate heat exchanger (CB 30 Alfa Laval).
Just like in the case of the evaporator, the volume of both fluids needs to be specified along with the
exchange area, the mass of the wall and the specific heat capacity of the metal wall.
Based on Figure 48 in Appendix IV we can establish the exchange area similarly to equation (24). The
procedure is reminded in equation (51).
( ) (51)
This figure also shows that the volume of both cold and hot fluids is given by and
respectively.
As for the mass of the wall, it is given in the datasheet and is equal to 6,31 kg. The heat capacity of this same
metal wall is of 502 J/(kg.K).
Of course, the heat transfer coefficients also have to be determined. The main difference that we can
observe when comparing their method of calculation with that of the evaporator comes from the absence of
phase change in the recuperator. This largely simplifies the calculation a lot since we do not need to
determine the different exchange areas.
The program given in Appendix III will thus minimize the error given by the difference between the
exchanged heat obtained by calculation and the exchanged heat obtained with the reconciled measures.
The resulting heat transfer coefficients are given in Table 7.
Table 7 : Heat Transfer Coefficients in the recuperator
Heat transfer Coefficients
Working fluid: cold side 246 W/m²K
Working fluid: hot side 246 W/m²K
40
Again, the accuracy of the method can be judged by studying the difference between the variables present
in equation (51). These are shown in Figure 17.
Figure 17 : Comparison of the calculated and measured heat exchange
5.2.5 Condenser
The thermodynamic libraries implemented in modelica offer different types of heat exchangers. The
problem to which we are confronted is that none of these models can represent the condenser used on the
test bench. This is why, even though the real condenser is a cross-flow multi-pass heat exchanger, it will be
modeled using a counter flow heat exchanger.
Regarding to the results, this won’t be too detrimental. Indeed, when looking at the -NTU curves for cross-
flow multiple passes and for counter-flow heat exchanger, it can be seen that they tend to converge at a
high number of passes [25].
The initialization of this last heat exchanger requires the knowledge of the volume of both the primary and
the secondary fluid along with the exchange areas, the mass of the wall between the two fluids and the
specific heat capacity of the metal wall.
The volume of the working fluid can easily be found in the datasheet and is given by . Concerning
the secondary fluid, a very small volume has been chosen to represent the air passing through.
Still in the datasheet, the mass of the wall which is of 75 kg and the constitutive material used for the fins,
that is to say aluminum, can be found.
The constitutive material of the tube is copper, which means that the specific heat capacity has to be
determined based on the specific heat capacities of both materials.
This has been done by considering the tube and the fins as a hole and evaluating the proportions of
aluminum and of copper that this entity contained.
Doing so let to a specific heat capacity of .
Concerning the exchange area, we now need to distinguish the exchange area of the working fluid from the
exchange area of the secondary fluid because of the presence of fins.
41
The datasheet gives us the exchange area of the secondary fluid which takes the fins into account. This area
is of . Since the exchange area of the unfinned tubes is not specified, it has to be calculated through
equation (52). In this equation, R represents the radius of the tubes which has been measured with a caliper
rule on a tube sample. To evaluate the number of tubes, n, it was considered that instead of having a unique
winding tube, the condenser was made of a series of straight tubes as shown in Figure 18. By doing so, the
corners in the redden area were neglected and only the central part was taken into account. The impact on
the exchanged heat is limited since these corners actually stitch out of the condenser and therefore are not
into the air draft.
(52)
Figure 18 : Winding tube at the condenser
All the necessary data could easily be found and equation (52) brought to the conclusion that the exchange
area on the refrigerant side was of .
5.2.5.1 Evaluation of the heat transfer coefficients
The evaluation of the heat transfer coefficients has been done using the same method as described in
section 5.2.2.1 for the evaporator.
The results are given in Table 8.
Table 8: heat transfer coefficients in the condenser
Heat transfer Coefficients
Working fluid: Liquid phase 325 W/m²K
Working fluid: Two-Phase 475 W/m²K
Working fluid: Vapor phase 380 W/m²K
Secondary fluid 250 W/m²K
Again, the precision of the method can be assessed by considering Figure 19 were it can be seen that the
difference between the measured pinch ( ) and the calculated pinch ( ) is small for the
majority of the points. The greatest error committed with the use of the heat transfer coefficients from
Table 8 is of 4 K.
42
Figure 19 : Comparison of the measured and calculated pinch point (condenser)
5.3 Establishment of a simple ORC model
The realization of the model had to be taken smoothly to avoid mismatches between the components.
Therefore, the model was built step by step using sources and sinks to cope for the missing items.
The source is an element which allows the user to impose different inlet conditions to a component
according to the needs. There are three main types of sources in the ThermoCycle library. These are shown
in Figure 20.
The first source is called a «sourceP» as one can specify, in addition to the medium and the specific nominal
enthalpy, the nominal pressure. This imposition can be done either by specifying the values as constant
parameters or by connecting blocks in the triangular entries. This last case scenario allows the user to
impose variable pressure or enthalpy.
For the second type of source, the specified parameters are the pressure and the temperature along with
the medium. Based on these, the «source_pT» is capable of determining the inlet enthalpy.
The last source is parameterized with the specific heat capacity and the fluid density.
Figure 20: ThermoCycle sources
The sink is a similar element except that instead of specifying the inlet conditions of an element, it imposes
the outlet conditions.
Figure 21: ThermoCycle sinks
43
As previously said, these components will allow the construction of the cycle step by step as shown in Figure
22 .Each element is tested separately at first. Once the model of the element is validated, it can be added to
the cycle.
It is very important to check that at each stage the element presents properties whose order of magnitude is
physically acceptable. The outlet conditions of an item also have to match the inlet conditions of the
following item.
These measures may seem obvious but if they are not applied, an inconsistency could go unnoticed until at
some point the model stops simulating. Finding the error to get back on track can then be complex.
Figure 22: Construction of the model
The final model of the simple organic Rankine cycle is represented in Figure 23.
Figure 23 : Simple ORC model
The simple model is then simulated in steady state. This allows a verification of the plausibility of the results
given by the program.
The result of the simulation is shown in Figure 24 for the temperature and the pressure at the outlet of the
evaporator. For the time being, this outlet temperature shows a superheating of about 3 K.
44
Figure 24: Pressure and temperature at the evaporator’s outlet
At the exhaust of the condenser, the fluid’s temperature is equal to the saturation temperature as shown in
Figure 25.
Figure 25: Temperature at the condenser's exhaust
As for the expanders, Figure 26 shows in blue this inlet enthalpy of the first expander, in red its outlet
enthalpy and finally in green the outlet enthalpy of the second expander.
In these conditions, the expanders are able to provide about .
45
Figure 26: Expander's enthalpy
Another interesting way to check the model would be to impose a condition that varies over time. This has
been done by imposing a step in temperature at the inlet of the oil side of the evaporator.
After 500 seconds of simulation, the temperature of the oil suddenly rises as shown in Figure 27.
Figure 27 : Temperature of the oil at the evaporator's inlet
The consequences of this rise in temperature can immediately be seen in the cycle. Figure 28 shows the
example of the outlet temperature of the R245fa at the evaporator. In response to the raise of temperature
on the oil side, it can be seen that the temperature of the working fluid is also gradually increased.
46
Figure 28 : Temperature of the R245fa at the evaporator's outlet
47
6. Improvement of the cycle The previous simulations were useful to judge the reactions of the model in different basic situations.
However, now that the model works and that its functioning is understood, it would be much more relevant
to complexify the model.
6.1 PID controller
PID control is a widespread way of using feedback in engineering systems. It uses the difference between the
desired value of a quantity, called the set point (SP), and its actual value, called the process variable (PV), to
create a corrective action. This action is done through the adjustment of a third variable called the control
signal (CS).
The PID controller bases its action on three levels:
- The proportional control can be related to the present error. This action can be found alone is some
controllers which are then simply called P controller. These respond to the control law given by (53)
where represents the proportional gain and e the error.
(53)
- The integral control considers the accumulation of the past errors. The control error is thus
expressed through the integral of the error as shown in equation (54) where us the integral gain,
sometimes replaced by the time constant which is its inverse.
( ) ∫ ( )
(54)
- Finally, the derivative control will provide the controller with an anticipative ability through its
predictive action based on a linear extrapolation as shown in equation (55). In this equation,
represents the time constant of the derivative control. It is sometimes replaced by its inverse, the
derivative gain .
( ) ( )
(55)
Finally, the control signal can be expressed as the sum of these three actions. Its mathematical expression is
given in equation (56).
( ) ∫ ( )
(56)
In the case of the sun2power model, a first PID is needed to modify the pump frequency in order to obtain a
superheating of 5°C at the evaporator’s exhaust.
48
The reminder which just has been developed shows that three constants have to be determined: , and
.
To do so, the method known as the step response method has been applied. This method identifies two
parameters by studying how the output reacts after a step input.
The two parameters that characterize the response are listed below:
- represents the intersection point of the steepest tangent of the step response with the vertical
axis.
- represents this same intersection on the horizontal axis.
A graphical representation is given in Figure 29.
Figure 29: The step response method [6]
However, when applied to the cycle, the method gave parameters that were slightly overestimated leading
to a saturation of the control signal. Therefore, the parameters obtained through this method have been
used as starting point and have been corrected manually.
6.1.1 Implementation of the PID controller to the cycle
At first, a single PID that modifies the pump frequency in order for the temperature at the exhaust of the
evaporator to match that of the saturation temperature increased of 5 K was implemented.
In order to do so, a pressure and temperature sensor was placed between the evaporator and the expander
as shown in Figure 30. This element extracts the value of the two aforementioned thermodynamic variables
at the location were it has been placed.
These data can then be processed as follows:
- the temperature is directly injected at the PID inlet for the process variable
- the pressure goes through an element which evaluates the saturation temperature for the given
pressure and adds 5 K to it. This temperature is then used as set point for the PID.
The problem with this configuration is that it leads to a modification of the pressure at the evaporator’s inlet
resulting in a saturation of the controller. Therefore, two controllers have to be implemented at once.
49
Figure 30: ORC cycle with PID for temperature
The second PID controller will modify the frequency of the generator connected to the first expander in
order to control the expander’s inlet pressure. Its parameters have been set in the same way than before,
that is to say using the step response and eventually applying a slight correction based on the measurement
points.
Table 9: PID parameters
Temperature PID Pressure PID
The sensor that has been previously placed for the temperature PID controller can be reused. Its measure of
the pressure will be used as the process variable of the second PID controller. The set point is set to a
constant value of 28,1 bars which is a coherent order of magnitude for this type of cycle [24].
The final configuration of the enhanced cycle can be seen in Figure 31.
50
Figure 31: ORC Cycle with PID for temperature and pressure
To test the reliability of the cycle, some of the constant inputs have been changed. For example, the inlet
temperature of the secondary fluid in the evaporator has been decreased of ten degrees over a time period
of 50 seconds.
This abrupt change and its effect on the outlet temperature of the working fluid at the evaporator are shown
in Figure 32. It shows that the PID controller has done is part perfectly well since the effect of the ramp in
temperature on the oil side is barely perceptible at the outlet of the evaporator on the R245fa side.
Figure 32: Ramp in temperature on the oil side
To achieve this result, the temperature PID controller had to readjust its control signal, aka the pump
frequency. Since the temperature on the oil side had dropped, there was a risk for the temperature at the
exhaust of the evaporator to drop as well. To deal with this, the PID controller had to decrease the frequency
of the pump. This can be seen in Figure 33.
51
Figure 33: Control signal of the temperature PID
As for the pressure controller, its action can be seen in Figure 34. The first graphic represents the set point
signal and the pressure at the outlet of the evaporator. It shows that the outlet pressure at the evaporator
decreases after 550 seconds, that is to say when the ramp starts. Figure 34 also shows the immediate
corrective action of the PID controller which, as a reaction to the pressure drops, diminishes the frequency
of the expander’s generator.
Figure 34 : Pressure controller
Another test which was smoother and perhaps closer to the reality of things was conducted. This test
applied another temperature variation on the oil side. Only this time, instead of having an abrupt change to
it, the temperature was describing a sine curve as depicted in Figure 35.
52
Figure 35: Temperature of the oil at the inlet of the evaporator
Since the pressure at the outlet of the evaporator was fixed to 28,1 bars, the saturation temperature was
also fixed. In these conditions, the PID had to bring the temperature of the R245fa at the outlet of the
evaporator at 145°C regardless of the variations to which the state of the oil was submitted.
Figure 36 shows the results of this simulation. The first graphic shows the variations of the process variable
(the temperature at the evaporator’s outlet) and the set point (the saturation temperature incremented of
five degrees). It can be seen that these two temperatures may differ of a tenth of degree at most.
The second graphic shows the pump frequency which decreases to counter balance the drop in temperature
that occurs after 500 seconds (Figure 35).
Figure 36 : Variation of the PID parameters
53
6.2 Solar collectors The model of the solar collector comes from the “ThermoCycle” library. This model is represented in Figure
37 were it can be seen that the following inputs have to be defined:
- the wind speed
- the inclination of the sunbeam
- the ambient temperature
- the direct normal irradiation (DNI)
These inputs are then processed by the “absorberSchott” component in order to solve the 1D radial energy
balance based on the Schott test analysis [9].
Figure 37: Solar Collector
The parameters of the solar collector have been set to the following values based on the researches done by
[10]:
- Shadowing - Mirror reflectivity - Number of tubes - Tube thickness
- Aperture of the parabola - External diameter of the
tube
The length of the tubes however hasn’t been set to the value calculated by [10], that is to say 3,3m per tube.
Instead a total length of 22m (2,75 m per tube) was used.
The reason for that comes from the fact that the test bench has been studied to accept 25 kW at the solar
collector. When using a total tube length of 22m, the maximal power gained from the sun in June is of 25,04
kW.
54
As usual, the component has first been tested alone. Once its functioning was fully understood and the
results judged satisfactory, the oil circuit was added (Figure 38).
Since little is known concerning the tank, some hypothesis had to be made: a very small volume was
considered to avoid conditions in which hot fluid gets stored at the outlet of the evaporator. Indeed, such a
situation could lead to a stratification of the temperature inside the tank and therefore an incoherent inlet
temperature at the solar collector.
As for the pump, an isentropic efficiency of 75% has been used. Finally, the pressure drop illustrated by the
sky blue component has been set to 1bar [12]
Figure 38: Solar Collector and oil circuit
6.3 Direct normal irradiance (DNI)
The direct normal irradiance is the quantity of solar radiation which is received per unit area by a surface
held perpendicular to the sunbeams.
Figure 39 shows the evolution of this irradiation for four typical days of the year 2012: the March and
September equinoxes and the spring and winter solstices which take place respectively in June and
December.
These data have been evaluated for the longitude and latitude of Marseille where the test bench will be
installed [11].
55
Figure 39: Direct Normal Irradiance
6.4 Air mass flow rate and fan consumption at the condenser
The air mass flow rate at the condenser was previously imposed to a constant value. However, it is possible
to determine a linear expression giving its evolution according to the frequency.
To do so, the nominal mass flow rate at the frequency of 50 Hz was considered as well as a null mass flow
rate for frequency 0 Hz.
The linear interpolation between these two points gave the following expression (57)
(57) The condenser’s consumption for its part can also be expressed as a function of the frequency [1]. The resulting expression is showed in equation (58)
(58)
Given these expressions, it is simple to create a modelica block which takes as input the frequency and gives
back the mass flow rate and the consumption.
The final aspect of the model is shown in Figure 40.
0
100
200
300
400
500
600
700
800
900
0 5 10 15 20 25
DN
I [W
/m²]
Time [h]
December
September
March
June
56
Figure 40: Final aspect of the ORC
6.1
6.2 & 6.3
6.4
57
7. Simulations of the model
The model presented in Figure 40 has been simulated for the four typical days described in section 6.3.
Along with the DNI, the ambient temperature has also been changed to match the considered day of the
year. The following values have been imposed:
- June : 25°C
- September : 20°C
- December: 8 °C
- Mars: 15°C
The results of the four simulations have been studied in terms of available power at the solar collector and
net production.
The net production takes into account the share of the expanders from which the consumption of the two
pumps and of the condenser has been cut off.
Figure 41 : Available power and net production (4 typical days)
0
5000
10000
15000
20000
25000
30000
0 5 10 15 20 25
Po
we
r [W
]
Time [h]
Solar power at the collector
March
June
December
September
0
1000
2000
3000
4000
0 5 10 15 20 25
Pro
du
ctio
n [
W]
Time [h]
Net production
March
June
December
September
58
Some results shown in Figure 41 draw the attention. It is the case, for example, of the spring equinox in
March. The figure shows that the power coming from the sun and captured by the solar collector is at all-
times inferior to that of the summer solstice (June). However, when looking at the corresponding
production, it can be seen that the representative curve of the equinox goes beyond that of the solstice. At
its maximum, the net production in March is almost equal to the net production in June.
To better understand this phenomenon, the different productions and consumptions of the cycle have been
represented in Figure 42. This figure allows a global analysis of the two considered days.
Yet, a more throughout study could be carried out by considering the numerical data relative to the peak
production in March. This specific time of the day is featured by the dotted line in Figure 42.
The corresponding production and consumptions is given in Table 10.
Figure 42 : Productions and consumptions for the summer solstice and the summer equinox
0
500
1000
1500
2000
2500
3000
0 5 10 15 20 25
W_d
ot
[W]
Time [h]
June
Pump (R245fa)
Pump (Oil)
Condenser
First Expander
Second expander
0
500
1000
1500
2000
2500
3000
0 5 10 15 20 25
W_d
ot
Time [h]
March
Pump (R245fa)
Pump (Oil)
Condenser
First expander
Second expander
59
Table 10: Numerical data (March & June) at 11:51
March Pump (R245fa) Pump (Oil) Condenser First expander Second expander
207,22 W 23,8 W 165,37 W 1720,1 W 2349,3 W
June Pump (R245fa) Pump (Oil) Condenser First expander Second expander
286,88 W 23,6 W 165,67 W 1755,9 W 2276,41 W
Table 10 shows that the consumption of the R245fa pump is somewhat inferior in March.
This can be explained by the fact that the direct normal irradiance is slightly lower in March at 11:51 than it
is in June at the same time (respectively 721,6 W/m² and 785 W/m²). In general, the higher the DNI, the
faster will the pump work to increase the speed in order to match the outlet conditions at the evaporator.
On the other hand, Table 10 also shows a much more interesting result: the production of the expanders.
In the case of the second expander, this production is increased by 73 W on the spring equinox in
comparison with the summer solstice at the same time.
The main reason for this comes from the ambient temperature. As said before, it has been set to 25°C in
June and to 15 °C in March. This difference impacts the outlet conditions of the second expander.
Therefore, a lower ambient temperature used for the air mass flow rate at the condenser will cause a lower
outlet pressure at the second expander. As a consequence, the production of the second expander is
increased.
When considering the first expander, it can be seen that the productions around 11:51 are very similar in
June and in March. This is due to the fact that the DNI at this moment of the day are also very close.
7.1 Adjustment of the frequency at the condenser
The simulations carried out in the previous section were done under the following regulation:
- Variable speed at the pump
- Variable speed at the first expander
- Constant speed at the second expander
- Constant speed a the condenser
The frequency of the condenser has been set to the arbitrary value of 48Hz. However, this could be
improved.
The frequency of the fans can be linked to the direct normal irradiance. That is to say, when the irradiance
increases the frequency at the condenser should increase as well to disperse the surplus of heat gained by
the cycle.
Therefore, a multiplicative coefficient can be used to convert the direct normal irradiance into a frequency.
To find this coefficient, it has been considered that the maximum frequency of the condenser should not
exceed 52 Hz. logically, this maximum frequency ought to be reached for the maximum DNI, namely 892
W/m² (June).
60
This reasoning led to a multiplicative coefficient of 0,0583. It has been applied to the cycle using a constant
and a multiplicative block from the modelica library (Figure 43).
Figure 43: Variable frequency at the condenser
The consumption of the condenser over time has been represented in Figure 44 before and after
implementing a variable frequency to the fans.
Figure 44: Consumption of the condenser
It is possible to compare quantitatively the consumption of the condenser with and without a variable
frequency at the condenser.
Still while considering the typical day of June, a gain of 1,1 kWh can be achieved at the condenser’s
consumption (Table 11).
0
20
40
60
80
100
120
140
160
180
200
0 5 10 15 20 25
Co
nsu
mp
tio
n [
W]
Time [h]
Consumption of the condenser
Variable frequency
Constant frequency
61
Table 11: Consumption of the condenser
Consumption at the condenser
Basic Case 2,7 kWh
Variable frequency 1,6 kWh
However, changing the frequency of the condenser will also impact the whole cycle. Therefore, the 1,1 kWh
that have been gained at the condenser won’t be the final power gained on the hole cycle.
The actual saving achieved on the whole cycle is of 0,32 kWh. Again, this has been calculated while
considering the summer solstice.
The difference in the net productions for the summer solstice is represented in Figure 45.
As expected according to the numerical study, the difference between the two productions is marginal.
Figure 45: Comparison of the net production of the cycle when varying the condenser's frequency
This difference between the 1,1 kWh gained at the condenser and the 0,32 kWh gained on the whole cycle
can be related to two factors: a reduction of the production at the second expander and an increase in the
consumption of the pump on the refrigerant side.
The reduction in the production at the second expander can be explained just like when comparing typical
days of June and March. With a variable frequency at the condenser, the outlet temperature of the
expander is most of the time higher than with a constant frequency.
Because of that, the corresponding pressure also rises causing the production of the expander to drop. The
outlet pressure at the second expander is depicted in Figure 46
0
500
1000
1500
2000
2500
3000
3500
4000
0 5 10 15 20 25
Pro
du
ctio
n [
W]
Time [h]
Net production of the cycle
Variable frequency at thecondenser
Constant frequency at thecondenser
62
The increase in the pump’s consumption can also be linked to the fluid’s temperature. As seen before, as the
fluid’s temperature increases, the mass flow increases as well to meet the conditions fixed by the PID
controllers.
Figure 46 : Outlet pressure of the second expander
63
8. Possible enhancement of the cycle
Two main aspects of the solar application of the organic Rankine cycle haven’t been taken into account
in the model. This concern the storage and the cloud covering.
They are often combined since the storage allows to cope with the deficit created by clouding.
Different configurations regarding the two expanders are also possible and haven’t been taken into account.
As for the condenser, a cross-flow multi-pass model is currently under development at the University of
Liège. Using it would bring the model even closer to reality.
Finally, the expanders have been considered to have a constant efficiency of 0,7 at all time. However, it
would be interesting to study the cycle’s behavior with a varying efficiency at the expanders.
8.1 Storage
The storage of energy in any type of power plant is a very important problematic. Not only will it improve
the performance and reliability of the system, it will also make the system more cost effective.
Storage of energy can also solve the imbalance that sometimes occurs between the energy demand and
supply.
Furthermore, in the case of solar energy one has to keep in mind that an intermittent energy source is being
used. Therefore, the absence of storage leads to the use of back-up or auxiliary energy. Most of the time,
these other sources of energy run on fuel or use expensive systems.
Some tests have been done to store the solar energy using techniques that have already proven themselves
in other fields:
- Storing the exceeding energy by moving water uphill
- Compressing air and un-compress it to regain the energy
- Battery storage
- …
Unfortunately, these methods are either too expensive of inefficient.
Because of this, ways have been elaborated to store the energy as heat.
8.1.1 Thermal energy storage methods
Over the years, three means of thermal energy storage have distinguished themselves:
1) Sensible heat storage: A liquid or a solid is heated without any phase change. The energy stored by
this means depends on the change in temperature of the material.
An example of such storage can be seen in Andasol(Spain) were molten salts were used.
2) Latent heat storage: A material is heated up and undergoes a phase shift. In this case, the amount of
energy that can be stored depends on the mass and on the latent heat of fusion of the material.
64
3) Thermochemical energy storage: This final type of storage bases its action on reversible chemical
reactions. During these, the energy is used to break chemical bonds.
A study has been carried out on the sun2power project. The goal of this study was to determine which type
of storage would be best suited for the current application 69[10].
8.2 Cloud covering
The model developed in the previous section considers a clear sky. In other words, the DNI chosen for the
simulations doesn’t take into account the presence of clouds.
This presence increases the diffuse radiation and reduces the global radiation. Indeed, clouds act as a
blanket protecting the earth from the incoming radiation. Therefore, the temperature at the surface of the
earth may be lower than expected during cloudy days.
Concerning the solar collectors, this means that less energy coming from the sun can be brought to the
secondary fluid.
8.3 Optimization of the use of the expanders
All along this work, both expanders have been working in series. However, given certain conditions, it might
be interesting to work differently.
The following configuration can be applied to the expanders:
1. Expanders are in series
2. Expanders are in parallel
3. Only the first expander is working
4. Only the second expander is working
8.4 Isentropic efficiency at the expanders
As said earlier, the isentropic efficiency of both expanders was set to 0,7. The following reasoning could
parry to that [8].
The expansion can be divided in two
- First, an isentropic expansion taking place between the inlet pressure and the adapted pressure. This
last pressure is imposed by the internal volume ratio and the inlet entropy.
(59)
( )
(60)
- Then, a constant volume expansion can be considered.
65
These hypothesis lead to the following estimation of the isentropic efficiency stated in equation (61). In this
equation, , and are defined according to equations (62) to (64).
(61)
(62)
( ) (63)
(64)
In equation (64), the subscript “s” in the expression of the outlet enthalpy refers to the fact that is
calculated for an isentropic process (i.e is a function of the inlet entropy).
The difficulty comes from the implementation of this reasoning under modelica: In order to evaluate the
adapted pressure, one needs to find the thermodynamic state based on the density and the entropy.
Yet, modelica only allows the use of the following combinations:
- Pressure and temperature
- Pressure and enthalpy
- Pressure and entropy
- Density and temperature
None of these allow the direct determination of the pressure knowing the density and the entropy. It is
however possible to determine the adapted pressure when taking into account that modelica is an acausal
language.
The density, which is known, could then be described as a function of the entropy, which is also known and
the adapted pressure.
Sadly, this only works in a simple system and gave results when testing the expander alone. Once in the
cycle, the calculation of the adapted pressure became too complicated. This led to an interruption of the
simulation.
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9. Conclusion
The first part of this report settles the context of the work: the test bench is described along
with key results obtained from previous studies.
Then, experimental data is analyzed and corrected using a reconciliation method. This method bases its
action on the minimization of a constraint equation while verifying the energy balances on each heat
exchanger. The results show that in order to respect the energy balances, some variables have to be greatly
modified. This is mainly the case for the temperature, which implies that the measurement results could be
improved by increasing the insulation of the heat exchangers.
The motivation behind the introduction of this method is double: the reconciled data is then available for
further use and the method can be applied to future sets of measurement.
A first model was then developed using the modeling language modelica. It contained the main components
of the organic Rankine cycle. These were tuned and updated to represent as best as possible the element
mounted on the test bench.
One of the necessary data for this tuning was the heat transfer coefficients of the heat exchangers.
These were evaluated under the Engineering Equation Solver (EES) using an iterative method. This method
updated the calculated value of the heat transfer coefficients by diminishing a given error. For the condenser
and the evaporator, this error was the difference between the measured and the calculated pinch point. In
the case of the recuperator, the error to minimize was the difference between the measured and the
calculated heat exchange.
The accuracy of the method has been judged graphically by comparing the terms of the error in each case
(i.e. the measured and calculated pinch or the measured and calculated heat exchange). This showed that in
general most of the depicted points tended to evolve around the bisector. However, more accurate results
should be expected if using a wider range of measurements.
Afterwards, the simple model of the ORC was gradually made more complex. First, two PID controllers were
implemented. The first one adapted the frequency of the working cycle’s pump in order to maintain a
superheating of 5°C at the evaporator’s outlet. The second PID imposed a given pressure at the inlet of the
first expander.
This first enhancement was followed by the implementation of solar collectors.
This final model was then run using a variable daily irradiance. Since the test bench is meant to be installed
in Marseille, the latitude and longitude of this town was used to find the aforesaid daily irradiance. Four
typical days were then simulated.
The graphical results showed the difference in daily production in terms of spread and peak production. The
impact of the ambient conditions was also highlighted by studying the production of two different days
subject to a similar DNI.
Finally, possible enhancements of the cycle were suggested. These cover the storage and clouding along with
a more realistic modeling of the condenser and the expanders.
67
General conclusion
The use of micro scale solar plants is particularly interesting for sunny and isolated regions which do not
have an easy access to electricity. This is the case for example of villages in Africa or India.
In such countries, the demand for electricity is not as oversized as ours. Therefore, only a few kilowatts of
electricity could radically change the way of living of the inhabitants. Basic needs, such as water supply or
medical device fueling could be satisfied.
Several prototypes have already been installed in Africa and are capable of producing 3 to 4 kilowatts of
electricity along with hundreds of liters of hot water per day.
On a larger scale, solar power plants with a capacity up to the megawatt have also seen the day.
In this domain, Spain and the United-States possess about 90% of the current installed concentrated solar
power. In 2012, Spain held around 1,7 GW of installed capacity and the United-States 510 MW. This can be
explained by the fact that these two countries are subject to a high DNI and benefit from government
support (tax incentives, renewable portfolio or feed-in tariffs).
Other countries such as those presented in Figure 47 also tend to develop the use of solar power.
Figure 47 : Estela forcast
68
Hence, solar energy provides a source of renewable and non-polluting energy which fits in our initiative to
reduce our consumption of fossil fuel. Moreover, it allows the developing countries to assert themselves and
access a basic comfort which was hitherto unknown to them.
69
Bibliography
[1] O. Dumont – Design, Modeling and optimization of a solar Organic Rankine Cycle, University Thesis
(2012)
[2] S. Declaye – Design, optimization and modeling of an organic Rankine cycle for waste heat recovery
University Thesis (2009)
[3] M. Bauduin & E. Georges – Projet intégré en énergétique, University project (2011-2012)
[4] Power from the sun – Chapter 12 : Power Cycles for Electricity Generation
Evaluation of the heat transfer coefficients in the recuperator
PROCEDURE STAT(start#;stop#:x_bar) x = start# y= stop# $common error[start#..stop#] "!Reading of data" N = (y-x)+1 "Total number of readings" i = 1 repeat x[i] = error[x] i = i+1 x = x+1 until (x>y) "!Mean value" x_bar = 0 i = 1 repeat x_bar = x_bar+x[i]/N i = i+1
