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All content in this area was uploaded by Martin White on Mar 30, 2021
Content may be subject to copyright.
*corresponding author(s)
Email address: Omar.Aqel@city.ac.uk 1
The 4th European sCO2 Conference for Energy Systems
March 23-24, 2021, Online Conference
2021-sCO2.eu-126
BINARY INTERACTION UNCERTAINTY IN THE OPTIMISATION OF A
TRANSCRITICAL CYCLE: CONSEQUENCES ON CYCLE AND TURBINE DESIGN
Omar Aqel*
Department of Mechanical
Engineering and Aeronautics, City
University of London
London, UK
Email: Omar.Aqel@city.ac.uk
Martin White
Department of Mechanical
Engineering and Aeronautics, City
University of London
London, UK
Email: Martin.White@city.ac.uk
Abdulnaser Sayma
Department of Mechanical
Engineering and Aeronautics, City
University of London
London, UK
Email: A.Sayma@city.ac.uk
ABSTRACT
Doping CO2 with an additional fluid to produce a CO2-based
mixture is predicted to enhance the performance of the super
critical CO2 power cycle and lower its cost when adapted to
Concentrated Solar Power plants. A consistent fluid mixture
modelling process is necessary to reliably design and predict the
performance of turbines operating with CO2-based working
fluids. This paper aims to quantify the significance of the choice
of an Equation of State (EoS) and the uncertainty in the binary
interaction parameter () on the cycle and turbine design.
To evaluate the influence of the thermodynamic model, an
optimisation study of a 100 MWe simple recuperated
transcritical CO2 cycle is conducted for a combination of three
mixtures, four equations of state, and three possible values of the
binary interaction parameter. Corresponding multi-stage axial
turbines are then designed and compared based on the optimal
cycle conditions.
Results show that the choice of the dopant fraction which yields
maximum cycle thermal efficiency is independent from the fluid
model used. However, the predicted thermal efficiency of the
mixtures is reliant on the fluid model. Absolute thermal
efficiency may vary by a maximum of 1% due to the choice of
the EoS, and by up to 2% due to uncertainty. The maximum
difference in the turbine geometry due to EoS selection
corresponded to a 6.3% (6.6 cm) difference in the mean diameter
and a 18.8% (1.04 cm) difference in the blade height of the final
stage. On the other hand, the maximum difference in turbine
geometry because of uncertainty amounted to 6.7% (5.6 cm)
in mean diameter and 27.3% (2.73 cm) in blade height of the last
stage.
INTRODUCTION
Several studies have identified the potential of supercritical
carbon dioxide (sCO2) cycles to outperform traditional steam
cycles in concentrated solar power (CSP) plants [1]–[6].
However, the lack of cooling water hinders the performance of
CSP plants and reduces its thermal efficiency. This is because the
use of air-cooled condensers prevents condensing cycles,
increases the cycle's compression work, and limits its efficiency.
Doping CO2 with an additional fluid to produce a CO2-based
mixture could alleviate the limitations of dry cooling. It does so
by increasing the critical temperature of the working fluid and
expanding the operation of transcritical carbon dioxide (tCO2)
cycles, which compress the fluid in its liquid state and expand it
in its supercritical state, into arid environments [7].
A variety of dopants have been considered in the past. Xia et al.
[8] identified organic dopants that might improve cycle thermal
efficiency. However, Invernizzi et al. [9] concluded that organic
dopants, such as hydrocarbon mixtures, are not stable enough for
temperatures above , which is below the expected
temperature range of CSP, and hence alternatives are needed.
Inorganic dopants with critical temperatures higher than that of
CO2, such as dinitrogen tetroxide (N2O4) or titanium
tetrachloride (TiCl4), were proposed by Bonalumi et al. [10] and
further studied by Manzolini et al. [11]. Results showed that they
may achieve cycle efficiencies of up to 50%, reduce the specific
DOI: 10.17185/duepublico/73959
2
cost of the power block by 50%, and reduce the levelised cost of
electricity (LCoE) by 11 to 13% with respect to a conventional
steam cycle. Moreover, the power block cost may be reduced by
20% compared to pure sCO2.
Regardless of the working fluid, the choice of the
thermodynamic equation that describes the fluid’s state
properties (fluid model) affects cycle performance prediction
and equipment sizing. Specifically, the thermodynamic
properties determine the cycle thermal efficiency and equipment
sizing, while the transport properties affect equipment sizing.
However, transport properties are not considered in this study
because they are not directly calculated by an equation of state
(EoS). It is also worth noting that the choice of the fluid model
does not alter the actual behavior of the fluid or the cycle, but
only effects the ability to predict their behavior.
The influence of the fluid model on the estimated cycle
performance and equipment sizing has been investigated in the
past. Zhao et al. [12] conducted a selection procedure which
compared six EoS to identify the best option for the modelling
of a pure CO2 working fluid in a recompression cycle. The six
EoS compared were of three types: (1) Cubic-type, including the
Peng-Robinson (PR), the Peng-Robinson combined with
Boston-Mathias alpha function (PR-BM), and the Soave-
Redlich-Kwong (SRK); (2) Virial-type, including the Lee-
Kesler-Plocker (LKP) and the Benedict-Webb-Rubin modified
by Starling and Nishiumi (BWRS); and (3) Helmholtz-type in
the form of the Span-Wagner (SW) EoS. It was concluded that
the SW EoS provided the most accurate predictions of CO2
properties in the near-critical and supercritical regions.
In the study by Zhao et al. [12], the Mean Absolute Percentage
Error (MAPE) in the specific heat calculated by SW EoS was
0.5% compared to experimental data. Other EoS resulted in
MAPE of about 2% in the calculated specific heat values. At
most the variation in thermal efficiency was within 2%
depending on the EoS. In terms of equipment sizing, they noted
that a deviation of 10% in recuperator size (specified by the
product of the overall heat transfer coefficient and heat
exchange area ) and compressor diameter is possible depending
on the choice of EoS. The variation in the compressor size was
attributed to its operation near the critical point where evaluation
of the specific heat capacity becomes less precise. Conversely,
the influence on turbine diameter was found to be more limited
(from 0.2% to 3.0%), which is expected since equations of state
converge to the ideal gas law at high temperatures above the
critical dense-gas region.
The study of mixtures adds another uncertainty in
thermodynamic property predictions because of the use of the
Binary Interaction Parameter (), which is a correction factor
applied to an EoS to account for intermolecular interactions
between mixture components. A value for may be obtained
by regressions based on experimental Vapor-Liquid Equilibrium
(VLE) data, where is calibrated to fit the EoS predictions with
empirical results. It is also possible to predict the value of
using models such as the predictive-Peng Robinson or the
Enhanced-Predictive-Peng-Robinson-78 equation of state [13].
However, predictive models will not be used in this study since
experimental data is available for all the mixtures involved.
Di Marcoberardinoa et al. [14] compared the cycle performance
of a CO2/C6F6 mixture using five different EoS. The choice of
EoS resulted in an inconsistent cycle thermal efficiency which
ranged from 40.5% to 42.5%. They also noted that the choice of
the EoS slightly effects the identification of the optimal dopant
molar fraction. In the same study, they varied by +/-50% and
found that it had a limited effect on the cycle efficiency (+/-
0.2%). However, they did not study the effect of on
equipment sizing, nor did they investigate its influence in other
mixtures.
Previous studies have indicated that thermodynamic property
prediction is most consistent near the turbine operating
conditions [12], [14]. Therefore, it follows that the turbine should
be the component least affected by the fluid model. However, it
has not yet been shown to what extent any small variation will
impact the final turbine geometry or performance predictions.
Answering this question will help with future design efforts by
guiding the most suitable choice for the EoS to be used during
the mean-line design and numerical computational-fluid
dynamic simulation of the turbine, which is a critical component
of the cycle.
The aim of the current work is to investigate the sensitivity of
key cycle and turbine design parameters to the choice of EoS and
uncertainty within a simple recuperated transcritical cycle
layout using CO2-based mixtures as working fluids. Ultimately,
the aim is to quantify the effect of EoS and on turbine design.
A large scale 100 MWe CSP power plant is considered as a test
case because it is the target scale of the SCARABEUS project
[15].
METHODOLOGY
Working fluid modelling
This work is part of a research effort that aims to explore the use
of CO2-based working fluids in CSP plants. Therefore, the choice
of dopants is focused on those that increase the critical
temperature of CO2 to enable the operation of transcritical cycles
in CSP plants. Although the list of chemical compounds is
virtually endless, the choice of dopant can be focused by a set of
desirable dopant properties: (1) critical temperature above ;
(2) thermal stability above ; and (3) solubility in CO2 in
all cycle conditions. The minimum critical temperature is set to
ensure compression occurs far enough from the critical point that
the liquid’s properties are not drastically affected by small
changes in temperature. A critical temperature of is a safe
3
distance away from the design pump inlet temperature of
that liquid compression is ensured.
Based on the above criteria, the chosen dopants are: H2S, C6F6,
and an unnamed Non-Organic Dopant (NOD). The latter dopant
will not be named as it remains confidential within the project
consortium. The former two of these dopants have been
considered for CO2 power cycles in previous publications [14],
[17]. The main dopant thermophysical parameters of interest are
shown in Table 1.
Table 1. Select properties of CO2 and dopants
Compound
Molecular
Weight
(g/mol)
Critical
Temperature
(K)
Critical
Pressure
(MPa)
CO2
44.01
31.0
7.382
H2S
34.08
100.4
8.963
C6F6
186.1
242.8
3.273
NOD
>60
<125
<7.000
To calculate the thermophysical properties of the working fluids
the thermodynamic models available within Simulis
Thermodynamics were used [18]. Validation details were
described in the authors’ earlier work [19].
The four candidate EoS that were selected for the study are
shown in Table 2. The EoS were chosen as to cover three
different types: Cubic, Virial, and SAFT. Among these, the cubic
types are the most popular owing to their accuracy in the
estimation of VLE properties for most fluids. They also require
little computational overhead because of their simplicity.
However, the accuracy of cubic EoS are limited with highly polar
compounds. Although they have the ability to describe mixtures
accurately, the application of virial type EoS is limited to low
and moderate density fluids. SAFT equations of state are known
to produce accurate property estimations away from the critical
point and are suitable for systems in which the strength of
association varies from weak hydrogen bonds to strong covalent
bonds. However, their accuracy comes at a high computational
cost.
Table 2. Equations of State used to model the mixtures.
Equation of State
Type
Reference
Peng-Robinson (PR)
Cubic
[20]
Benedict-Webb-Rubbin modified
Starling-Nishiumi (BWRS)
Virial
[21]
Soave-Redlich-Kwong (SRK)
Cubic
[22]
Peturbed Chain Statistical Associating
Fluid Theory (PC-SAFT)
SAFT
[23]
The cubic EoS requires the definition of the following fluid-
specific parameters: acentric factor, critical temperature, and
critical pressure. In addition to the parameters required to solve
a cubic EoS, the PC-SAFT model requires the following
parameters for each pure component of the mixture: (i) the
characteristic segment number , (ii) the characteristic segment
size parameter , and (iii) the characteristic segment energy
parameter . These parameters are listed in Table 3.
Table 3. SAFT parameters for the pure components
Dopant
(Å)
Reference
CO2
1.8464
2.98388
140.00
Simulis preset
NOD
>2
>2
>200
Undisclosed
H2S
1.6686
3.0349
229
[24]
C6F6
3.779
3.396
221.65
[14]
Along with the EoS, a value must be specified for each
mixture. In this study, was calculated against regressed
Vapor-Liquid Equilibrium (VLE) experimental data and used to
tune the mixing models for each mixture and EoS pair.
Determining the value of required an optimisation problem.
By tuning , the calculated VLE lines were manipulated and
compared with experimental data to find the best-fit value.
An unconstrained gradient-based optimisation approach was
used. The weighted least mean square method was used as the
objective function. Like the simple least square method, it
minimises the residuals between experimental and calculated
data, but it also weighs each residual with the experimental
uncertainty of the experimental data. The objective function is
reduced or expanded depending on the availability of
experimental data. The objective function for the optimisation is
defined as:
Where is the temperature, is the pressure, and and are
the liquid and vapour molar fractions of CO2, respectively. The
accents () and () indicate the measured and calculated values,
respectively. Experimental uncertainty is represented by the term
. The number of experiments is denoted by .
The Mean Absolute Percentage Error (MAPE) is a measure of
the accuracy of the thermodynamic model. The lower it is, the
more accurate is the model. The MAPE is calculated as follows:
4
where corresponds to either the temperature or pressure. The
MAPE may be used to compare the accuracy of the models to
determine their suitability. Based on the MAPE values presented
in Table 4, the two cubic equations of state (PR and SRK) are
more suitable than the virial equation of state (BWRS) for all
mixtures.
Table 4. Binary interaction coefficient and its associated MAPE
for each CO2-based mixture and EoS combination
Binary Interaction Parameter ()
PR
BWRS
SRK
PC-
SAFT
NOD
0.0214
0.0182
0.0249
-0.0939
C6F6
0.0332
0.0626
0.0394
-0.0571
H2S
0.0871
0.0453
0.0871
-0.0393
Mean Absolute Percentage Error (MAPE %)
PR
BWRS
SRK
PC-
SAFT
NOD
2.089
1.938
2.068
4.722
C6F6
2.619
5.028
2.374
2.227
H2S
0.3862
0.4901
0.4025
0.275
No. of exp. pts
Source of data
NOD
48
Undisclosed
C6F6
64
[25]
H2S
122
[26]
The increase in fidelity with the availability of experimental
data is noticeable from Table 4. Among the three mixtures, the
experimental data for CO2/H2S is the most abundant, thus it has
the lowest MAPE in property estimation.
Since the uncertainty in depends on the available VLE data,
each mixture has a different range of uncertainty. However, to
properly compare the influence of uncertainty in each
mixture, a uniform uncertainty of 50% is applied to all
estimates. This negates the effect of VLE data availability when
comparing mixtures, which can always be collected through
experiments to narrow the uncertainty margins and improve
model fidelity.
Thermodynamic cycle model
A simple recuperated tCO2 cycle is suitable for the purposes of
this study because it is a viable option for CSP applications with
CO2-based mixtures. A schematic of the tCO2 cycle and its
Temperature–Entropy diagram are shown in Figure 1.
Figure 1. Temperature-Entropy diagram and cycle layout of a
simple recuperated tCO2 cycle operating with a CO2/C6F6
mixture.
The cycle is modelled by applying the first law of
thermodynamics to all equipment. Cycle thermal efficiency is
expressed as the ratio of the net work produced to the heat
consumed by the cycle in Eq.3:
The losses within the pump and turbine are approximated by
assuming isentropic efficiencies for each component, as
expressed by Eq.4 and Eq.5:
where the subscript ‘s’ denotes the outlet conditions assuming
isentropic compression and expansion.
The recuperator effectiveness determines the ratio of the actual
heat load to the maximum attainable heat load from the stream
with the lowest heat-capacity rate, as expressed in Eq.13:
The cycle state points are determined by setting the pump inlet
temperature (), the turbine inlet temperature (), pressure
ratio, component efficiencies, and pressure drops. Within this
study, and will be set according to the values expected in
state-of-the-art dry-cooled CSP plants.
Turbine model
5
For a large 100 MW turbine, a multi-stage axial architecture is
recommended [27]. A 1-D mean line turbine design approach
was used to model the turbine. The main parameters used to
inform turbine design are shown in Eq. 7 to Eq. 9. The blade-
loading coefficient , turbine flow coefficient , and degree
of reaction () control the blade speed, fluid axial velocity, and
the expansion in the stator and rotor:
where is the total enthalpy drop across the stage, is the
blade speed of the rotor at the mean radius, is the axial flow
velocity at the rotor outlet of the stage, and is the enthalpy
drop across the rotor. Further details on this design approach are
reported in Salah et al. [28]. The number of stages is based on
preliminary stress calculations of a previous publication [19].
The specific speed in Eq. 10 is a ratio used to indicate a turbine’s
size and shape [29].
where are the specific speed and nominal speed,
respectively. The volume flow rate out of the turbine is
represented by (in m3/s).
Optimisation program
A MATLAB program was developed to study the effect of the
EoS and on optimal cycle and turbine design. The flowchart
in Fig A.1 illustrates the calculation processes for a single CO2
mixture. The flowchart shows four layers, three of which are
parametric studies which change the EoS, , and dopant molar
fraction. The inner most layer identifies the optimal cycle
condition for the given EoS, , and dopant molar fraction
combination. Once optimum cycle conditions are found, the
program then produces a turbine geometry using the turbine
boundary conditions resulting from the optimal cycle.
Cycle conditions are chosen to simulate those of a CSP plant with
dry cooling. Assuming an ambient dry-bulb temperature of
and a minimum temperature difference of in the condenser,
the pump inlet temperature () is fixed to . Liquid flow
into the pump is assumed to be subcooled by below the
saturation pressure. Therefore, the pump inlet pressure () is set
equal to the saturation pressure at . The turbine inlet
temperature () is fixed to , which is the targeted
temperature of advanced CSP receiver employing sodium salt as
its Heat Transfer Medium (HTM). Finally, the turbine inlet
pressure () is limited to as recommended by Dostal et
al. [30]. The turbine design parameters were set based on authors
experience. Both the cycle and turbine design inputs are shown
in Table 5 and Table 6, respectively.
Table 5. Inputs required for cycle solution
Table 6. Inputs required for turbine design
RESULTS AND DISCUSSION
Attention is first given to the choice of the optimal dopant molar
fraction. This is important because it defines all subsequent fluid
properties. In essence, a variation in the dopant molar fraction of
Fixed Parameters
Parameter
Range
Unit
Dopant Molar Fraction
Max(65)
%
Turbine inlet temperature
700
Pump inlet temperature
50
Pump efficiency
85
%
Turbine efficiency
90
%
Generator efficiency
99
%
Net electrical power
100
MW
Pressure drop primary
heat exchanger
0.015
-
Pressure drop in
recuperator high and low-
pressure sides
0.01 and
0.015
-
Pressure drop in
condenser
0.02
-
Dependant Parameters
Pump inlet pressure
MPa
Turbine inlet pressure
Max (25)
MPa
Optimised Parameters
Pressure Ratio (PR)
2 – 4
-
Recuperator Effectiveness
70 – 98
%
Parameter
Value
Unit
Rotational speed
3000
RPM
Number of stages
4
-
Turbine efficiency
90
%
Loading coefficient
1.65
-
Flow coefficient
0.23
-
Degree of reaction
0.5
-
6
a mixture produces different fluids. As seen in Figure 2, the
thermal efficiency of the cycle is affected by the dopant molar
fraction, the EoS, and the value of , but this effect differs
depending on the mixture. Generally, thermal efficiency
fluctuates around 0.1% to 0.3% of the baseline value for both
variations in the EoS and . The largest variation due to is
0.7% and it is observed in BWRS when used with CO2/C6F6.
Among the four EoS, the two cubic EoS, PR and SRK, are the
least sensitive to variations in . This is confirmed by the
averaged MAPE values shown in the Figure 2 for each
combination. The MAPE was calculated by comparing the
results obtained with variations against the baseline of no
variation in .
Moreover, the cycle thermal efficiency for all mixtures and EoS
shows a positive correlation with , where higher values of
produce cycles of higher thermal efficiencies. Also common
among mixtures is that the thermal efficiency exhibits the same
trend with dopant fraction regardless of the EoS or , which
suggests that the dopant fraction that yields the highest thermal
efficiency is independent of the fluid model used. This is further
confirmed through Figure 3, which shows that neither EoS nor
affect the choice of the optimal dopant fraction. Furthermore,
during the optimisation it was found that all optimal cycles lead
to the maximum permissible pressure at turbine inlet (i.e., 25
MPa).
Taking a closer look at the thermal efficiency for the same
optimal dopant fractions reveals some differences in the
prediction of cycle performance. Figure 4 shows the maximum
percentage difference resulting from the choice of the EoS and
the variation in . The maximum percentage difference
between the EoS was calculated based on the difference between
the EoS that yields the lowest efficiency and the EoS that yields
the highest. The maximum percentage difference resulting from
variation was calculated by comparing the efficiency change
due to variation with the baseline case of no variation.
Figure 3. Effect of choice of EoS and variation in on the
optimal dopant molar fraction.
As observed from Figure 4, different mixtures respond
differently to the EoS. Cycles operating with CO2/C6F6 are
affected the most, with a maximum percentage change of 2.3%,
which equates to a 1% nominal change in efficiency between
SRK and PC-SAFT. The other two dopants are affected half as
Figure 2. The effect of dopant molar fraction on cycle thermal efficiency depending on the choice of EoS and the variation in .
7
much. Overall, the effect of variation is less pronounced,
except when modelling a CO2/C6F6 mixture using the BWRS
EoS where a decrease in the value of results in cycle
efficiency estimates lower by more than 2% nominal efficiency
(i.e. 44.77% compared to 46.90%).
Figure 4. Effect of choice of EoS and variation in on cycle
thermal efficiency.
Before investigating the effect on turbine design, the change in
pump design will be addressed. Because of real-gas effects any
deviation in the EoS leads to a larger variation in properties near
the saturation region. This variation is apparent in the design
point parameters of the pump, indicated by the differential head
and volume flow rate. As observed in Figure 5, both head and
flow rate change with EoS and . For all cases, the change in
flow rate is greater than the change in head. In the most extreme
case, the SRK predicts almost twice the flow rate for CO2/H2S
and CO2/C6F6 than the PC-SAFT does. Consequently, half as
many pumps may be predicted to be required if the PC-SAFT
EoS is used to model the cycle. The change in head and flow rate
is also significant due to variation. The maximum change in
flow rate and pump head is 34.2% and 14.7%, respectively, if the
BWRS is used to model CO2/C6F6. These findings agree with
previous studies which identified the significant influence of the
fluid model on the pump in particular.
Figure 5. Effect of choice of EoS and variation in on the
pump specific speed.
As seen in the following figures, the fluid properties at turbine
inlet are less affected by the change in the fluid model than the
properties at pump inlet. This is partly because of the
aforementioned real gas effects at pump inlet, but also because
Figure 6 The effect of dopant molar fraction on the fluid’s density at turbine inlet depending on the choice of EoS and the variation in .
8
the turbine inlet conditions are identical for all cases (700 and
25 MPa), while the pump inlet pressure varies depending on the
EoS and . However, there is an observable variation in turbine
inlet density, as shown in Figure 6. The maximum MAPE
between the baseline models of the EoS are 8.3%, 2.4%, and
4.5% for H2S, NOD, and C6F6 mixtures, respectively. Therefore,
the turbine inlet density of cycles operating with CO2/NOD is
generally less sensitive than those operating with the other two.
Moreover, The variation in density for the optimal blend fraction,
described in Figure 7, is less severe than the general trend shown
in Figure 6, yet the MAPE is still the highest for CO2/C6F6.
Similar to the trend observed in pump design variation, the two
cubic EoS exhibit the lowest sensitivity to variation even
away from the critical point. This consistency suggests that cubic
EoS are more robust and may be a good option in the absence of
quality experimental data from which can be calibrated.
Among the four EoS, the BWRS is the most sensitive to
variation, therefore precise values are recommended when
employing the BWRS EoS.
The density of the fluid directly affects the expansion work of
the turbine. Therefore, its variation is reflected in the variation in
the turbine specific work. As seen in Figure 8 and Figure 9, the
trend in specific work variation is similar to that in density, but
in the opposite direction. Turbine loading, mechanical stresses,
and number of stages are partly determined by the specific work,
since the specific work is related to blade speed through the
loading coefficient and the blade speed is constrained based on
mechanical design constraints. A difference of 13.7% in density
like that observed between SRK and PC-SAFT for CO2/H2S
could produce slightly different turbine designs, as seen later.
Figure 7 Effect of choice of EoS and variation in on the
density of the fluid at turbine inlet.
The implications of density dissimilarity and the subsequent
dissimilarity in turbine specific work are reflected in the specific
speed of the turbine, shown in Figure 10. The specific speed was
calculated for the entire turbine (across the four stages) by
substituting the total enthalpy drop across the turbine in Eq.10.
Although the percentage change is considerable for some cases,
such as 21% for BWRS with CO2/C6F6, the nominal change in
specific speed is miniscule; no larger than 0.3 rad/s for the same
case. This indicates that the resulting turbine designs will be
comparable in shape and size for all EoS and values.
Figure 8 The effect of dopant molar fraction on the turbine specific work depending on the choice of EoS and the variation in .
9
Figure 9 Effect of choice of EoS and variation in on the
turbine specific work
The changes in turbine design parameters culminate in the
resulting turbine geometry represented by its mean diameter and
blade height. The turbine mean diameter is measured at the
meridional profile midspan of the turbine blades and is assumed
to be constant across all turbine stages. The meridional blade
profile of the turbine geometry corresponding to the optimal
dopant fraction for each mixture, EoS, and variation
combinations are presented in Figure 11. Consistent with the
trends observed in the previous figures, the effect of EoS is more
pronounced in CO2/H2S mixtures. The largest difference is
between SRK and PC-SAFT, which amounts to 6.3% (6.6 cm)
difference in the mean diameter and a 18.8% (1.04 cm)
difference in the blade height at the final stage. The EoS effects
the other two mixtures half as much.
Figure 10 Effect of choice of EoS and variation in on the
turbine specific speed
The effect of variation on turbine geometry is smaller than
that of the EoS, except in the case of BWRS with CO2/C6F6
which yields a difference of 6.7% (5.6 cm) in mean diameter and
27.3% (2.73 cm) in blade height of the last stage when is set
to 0.5 of its baseline values. The mixture and EoS combinations
that are practically insensitive to variations are: CO2/H2S
with PC-SAFT, CO2/NOD with BWRS or SRK, and CO2/C6F6
with PR or SRK. Moreover, the PR EoS with CO2/H2S or
CO2/NOD mixtures or PC-SAFT with CO2/C6F6 are also
Figure 11 The meridional flow path for the four-stage turbine corresponding to the optimal dopant fractions.
10
relatively insensitive to variations as they exhibit only small
variations. Even though the turbine geometry is affected by fluid
model variations, that effect does not change the turbine design
fundamentally as seen in Figure 11. The same was revealed
through the small change in turbine specific speed.
The Mach number of a fluid is important to determine whether
subsonic or supersonic flow is present. By assuming similar
velocities for all cases, the Mach number may be compared by
comparing the speed of sound at turbine inlet. As shown in
Figure 12, the speed of sound remains fairly constant regardless
of the fluid model. The MAPE due to variation for all EoS
and mixture combinations were less than 0.2%. Moreover, the
MAPE due to EoS selection is 2.0% at its highest for CO2/C6F6.
Overall, it seems that Mach number prediction will unlikely to
be affected by the fluid model.
Figure 12 Effect of choice of EoS and variation in on the
speed of sound at turbine inlet.
CONCLUSION
In this paper, the sensitivity of cycle and turbine design to the
fluid property model was investigated. The study included three
CO2-based mixtures (CO2/H2S, CO2/NOD, and CO2/C6F6) in
combination with four equations of state (PR, BWRS, SRK, and
PC-SAFT), each modelled under scenarios of uncertainty
(50%).
It was found that the choice of the dopant fraction which yields
maximum cycle thermal efficiency for each mixture is
independent from the fluid model used. However, the predicted
optimal thermal efficiency of the mixtures is reliant on the fluid
model. Absolute thermal efficiency may vary by a maximum of
1% due to the choice of the EoS when modelling CO2/C6F6, and
by up to 2% due to uncertainty when the BWRS EoS is used
to model CO2/C6F6. Moreover, cycle thermal efficiency was
observed to have a positive correlation with the value for all
mixture and EoS combinations.
In terms of turbine design, among the three mixtures, CO2/NOD
is the least sensitive to the fluid model, while the other two are
nearly equally sensitive. Furthermore, the two cubic equations of
state, PR and SRK, are generally less sensitive to variation
the other two EoS, except for the case of CO2/H2S. This suggests
that they offer more robust property prediction in the absence of
quality experimental data. On the other hand, the BWRS EoS is
especially sensitive to uncertainty, thus requires precise
model calibration before it can be used reliably. Lastly, the
maximum difference in the turbine geometry due to the choice
of the EoS amounted to 6.3% (6.6 cm) difference in the mean
diameter and a 18.8% (1.04 cm) difference in the blade height at
the final stage. On the other hand, the maximum difference in
turbine geometry as a result of uncertainty amounted to 6.7%
(5.6 cm) in mean diameter and 27.3% (2.73 cm) in blade height
of the last stage.
NOMENCLATURE
Acronyms
BWRS
Benedict-Webb-Rubin-Starling
CSP
Concentrated Solar Power
EoS
Equation of State
HTM
Heat Transfer Medium
MAPE
Mean Absolute Percentage Error
MITA
Minimum Internal Temperature Approach
PC-SAFT
Perturbed Chain Statistical Associating Fluid Theory
PR
Peng-Robinson
sCO2
Supercritical Carbon Dixoide
SRK
Soave-Redlich-Kwong
tCO2
Transcritical Carbon Dioxide
VLE
Vapor-Liquid Equilibrium
Symbols
Efficiency
Effectiveness
Binary interaction coefficient
Loading coefficient
Flow coefficient
Degree of reaction
Specific enthalpy,
Density,
Head,
Rotational speed,
Pressure,
Pressure ratio
11 DOI: insert your DOI here once received
Temperature,
ACKNOWLEDGEMENTS
This project has received funding from the European Union’s
Horizon 2020 research and innovation programme under grant
agreement No. 814985.
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13
Appendix A
Fig A1. Flowchart of the sensitivity study for a single CO2 mixture.
This text is made available via DuEPublico, the institutional repository of the University of
Duisburg-Essen. This version may eventually differ from another version distributed by a
commercial publisher.
DOI:
URN: 10.17185/duepublico/73959
urn:nbn:de:hbz:464-20210330-104458-0
This work may be used under a Creative Commons Attribution 4.0
License (CC BY 4.0).
Published in: 4th European sCO2 Conference for Energy Systems, 2021
