Cayer. Parametric study and optimization of a transcritical power cycle
using a low temperature source; 2009
Abstract
 Parametric study and optimization using 6 performance indicators
o Thermal efficiency
o Specific net output
o Exergetic efficiency
o Total UA
o Surface HXs
o Relative cost system
 Independent parameters
o Maximum temperature
o Maximum pressure
o Net power output
 NOT possible to simultaneously optimize all  design value is a choice
 3 working fluids: CO2, ethane and R125
 First law analysis NOT sufficient for selection
Results
 Considering variable maximum temperatures for the working fluid.
 The mass flow rate and inlet temperature of the heat source, the inlet temperature of the
heat sink and the minimum pressure of the working fluid are fixed.
 Optimization procedure is based on the evaluation of the thermal efficiency, the specific net
output, the exergetic efficiency, the UA and total surface of the heat exchangers as well as
the cost.
System description
 A regenerator not included since the previous study indicated that this component is
unattractive for low temperature heat sources because it results in substantial increases of
the total heat exchangers’ surface and only in small improvements of the thermal and
exergetic efficiencies.
 Heat source: industrial gas, 314.5 kg/s, Tinlet = 100°C.
 Coolant condenser: water at 10°C.
 General assumptions are applied: each component is considered as a steady-state steadyflow system, the kinetic and potential energies as well as the heat and friction losses are
neglected, the pump and the turbine isentropic efficiencies are set to 0.8, saturated liquid is
supposed at the condenser exit and the minimum cycle temperature (T1) is set to 15°C which
fixes the minimum pressure (p1 = p4) of the working fluid. A minimum temperature
difference of 3°C is applied in the condenser at the point where the working fluid is saturated
vapour.
 3 parameters varied for optimization: Tmax, pmax and net power output.
o Net power output normalized by the reversible power developed by a Carnot engine
operating between the specified source and sink temperatures.
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o
= 6897 kW
o 
Hxs: S&T, counterflow, one pass each stream.
Equations for performance:
o Conservation of mass and energy
o Isentropic efficiency for pump and turbine
o Standard heat transfer correlations for overall heat transfer coefficient in evaporator
and condenser.
Software EES
o Ready to use optimization tools
 Golden Section Search (GSS): 1 degree of freedom
 Powell Direct Search (PDS): >1
Methodoly
0. Specify values of fixed parameters + range of variables
1. Energy analysis
a. Energy balance for each component
b. Isentropic definitions
 Determine states of 1-4
 Specific net output
 Thermal efficiency
2. Second law analysis
a. Mass flow rate working fluid (= assumes net power output/calculated specific net
output).
b. Mass flow rate cooling water in condenser by energy balance in condenser.
In the condenser, two situations are conceivable depending on the state of the
working fluid at the turbine outlet.
First, this fluid can be superheated vapour, dividing the condenser into a singlephase region and a two-phase region for the carbon dioxide as shown in Fig. 5. In this
situation, a pinch of 3°C at the saturated vapour point (T5’) of CO2 is assumed (Tw, as =
T5’- 3). This assumption guarantees that the temperature profiles do not intersect. It
should be noted that the saturation lines in Fig. 5 only apply to the CO2 and that the
water is always in the liquid phase.
The exit temperature of the water is then calculated with the following global energy
balance for the condenser:
Second, the state of the working fluid at the exit of the turbine falls in the two-phase
region. The single-phase region is then ignored and the 3°C pinch is assumed at state
5, now situated in the liquid–vapour region.
c. Calculate: specific exergy flow (each state), exergy destruction and global exergetic
efficiency.
With all the mass flow rates known, the exergy analysis can be realised by calculating
the specific flow exergy at the different states in the cycle and at the inlet and exit of
the source and sink. With all the preceding assumptions, this quantity is given by the
following simplified expression:
The 0 subscripts are relative to the dead state: this state is assumed to be at 10°C
and at atmospheric pressure.
The exergy destruction rates can then be calculated for each component from:
Finally, the overall exergetic efficiency is expressed by:
3. Finite size thermodynamic analysis
a. UA for each HX
i. Variable properties working fluid in supercritical region and during phase
change
 Discretization:
Transcritical heater in 20 sections with identical enthalpy change for working
fluid and condenser in 40 sections (20 single phase and 20 two-phase
region).
Properties assumed constant in each section
b. UAi for each section via local energy balance + standard LMTD-expression.
c. Sum UAi = total UA for evaporator and condenser.
4. HXs surface
a. Geometric and kinematic parameters (tube diameter, number of fins, max and min
velocities) are assumed and used to determine local convection heat transfer
coefficients.
b. Correlations
i. Petukhov: air, water and single-phase working fluid in condenser
ii. Krasnoshchekov-Protopopov: supercritical working fluid
iii. Cavallini: two-phase region working fluid condenser
5. Approximate cost of major components.
All steps completed  iterative process over entire range of 3 independent parameters to determine
influence on the 6 performance indicators.
Then optimization process with the best value of each performance indicator and corresponding
values of independent parameters.
Parametric studies and optimization for CO2
The equation of state used in EES for CO2: one by Span and Wagner
The parametric studies show the qualitative influence of the 3 independent variables (Tmax, pmax
and ) on the 6 performance indicators.
Optimization technique determines the best values of the indicators with their corresponding values
of independent variables.
1. Energy analysis and optimization
 Based on 1st law.
 Depends on the states of the working fluid at different points in de cycle.
 Does not consider external flows .
 not influenced by working fluid mass flow rate or 
 Objective: determine thermal efficiency and specific net power output.

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Tmax and pmax are the independent variables
Results

𝜂𝑡ℎ and specific output ↑ if Tmax ↑
 Optimization for maximum cycle temperature NOT required, because it will
lead a value equal to T inlet heat source.
 But MAXIMUM temperature requires an infinitely large transcritical heater.
Each curve has a maximum
 Corresponding maximum pressure ↑ with Tmax.
 Optimum specific output is at a lower maximum pressure than optimum 𝜂𝑡ℎ .
For free heat sources, the focus should be in maximizing the specific output, rather than
𝜂𝑡ℎ .
Maximum values calculated with GSS optimization tool (at 100°C).
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2. Exergy analysis and optimization
 Exergy must consider the 2 external flows.
 Requires calculation of mass flow rates of working fluid and coolant condenser.
o To evaluate the Toutlet of the heat source and water in condenser.
 𝜂𝑒𝑥 = 𝑓(𝑇𝑚𝑎𝑥 , 𝑝𝑚𝑎𝑥 𝑎𝑛𝑑 𝛼).
 First parametric study
o Fixed 
o 𝜂𝑒𝑥 = 𝑓(𝑇𝑚𝑎𝑥 , 𝑝𝑚𝑎𝑥 )
o 𝜂𝑒𝑥 and pmax ↑ with Tmax (also max at 100°C  infinite HX).
o maximum 𝜂𝑒𝑥 calculated with the GSS optimization tool is 60.5% and the
corresponding Tmax and pmax are 100°C and 14.1 MPa.
o Observation: optimizing pmax is ± identical for 𝜂𝑒𝑥 and 𝜂𝑡ℎ .
 Second parametric study
o Fixed Tmax = 95°C
o 𝜂𝑒𝑥 = 𝑓(𝑝𝑚𝑎𝑥 , 𝛼)
o For each combination of Tmax and  there exists a value of pmax which
maximizes the exergetic efficiency.
o
o
o
o
Optimising pmax is relatively independent of .
𝜂𝑒𝑥 ↑ if  ↑ (for  >0.2 and Tmax = 95°C).
For  >>, for high values of  an important phenomenon occurs at low and high
values of the maximum pressure: the specific net output for such pressures being
low (see Fig. 3) the working fluid mass flow rate and the heat extracted from the
heat source increase in order to generate the high net power output
corresponding to . Since the heat source capacity is limited, extracting more
heat results in a reduction of its temperature throughout the vapour generator.
However this temperature cannot be lower than the corresponding temperature
of the carbon dioxide. This condition limits the acceptable range of values for the
maximum pressure.
 When  ↑ the acceptable range of values for pmax ↓
Eventually the pmax range is reduced to a single value which gives the maximum
𝜂𝑒𝑥 and maximum .
3. Finite size thermodynamics analysis and optimization
 Objective: determine operating conditions with MIN UA.
 Related to surface and gives a global idea of their dimensions.
 Parametric study
o Fixed  = 0.2
o UAtotal varies significantly with Tmax and pmax
 Optimizing Tmax ≠ 100°C
 Because a Tmax = Tinlet heat source  infinite HX
 Low values of Tmax (80°C) also no low values of UAtotal
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o
o
o
Because in order to obtain the net power output (corresponding
to the fixed ), requires a high mass flow rate (as specific net
output is low at low Tmax).
 If mass flow rate ↑, T between fluids ↓  UAtotal ↑ for Tmax
<90°C.
 Optimum Tmax close to 95°C for  = 0.2.
 Also optimum pmax so that UAtotal in minimized for each Tmax.
 PDS optimization tool for  = 0.2, the minimum value of UA is
4124 kW/K and that the corresponding values of pmax and Tmax
are 11.81 MPa and 98.03°C.
 Repeat procedure over entire range of .
For low , UAtotal, min ↑ linearly with net power output.
For higher , the increase becomes exponential and UAtotal  ∞ as  reached
its previous maximum.
As  approaches its maximum value the exponential augmentation of the vapour
generator UA reverses the situation. The rapid augmentation of the total UA at
high  is then due to the vapour generator; the condenser UA increases linearly
with  over the entire range of possible net power outputs.
4. Analysis and optimization of the total HXs surface
 Objective: determine operating conditions which MINIMIZE Atotal.
 Determine local heat transfer coefficients.
 Atotal = f(tmax, pmax and ).
 Parameter study
o Fixed  = 0.2
o
o
o
Effect of pmax on A is ± same as on UA
Tmax for Amin is different than for UAmin  Amin is more important
A temperature of 98.03°C minimises the total UA, the temperature which
minimises the total surface is 89.78°C.
o
o
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The minimum value of A is 1369 m2 and the corresponding pmax is 11.4 MPa.
It is to be reminded though that these results are valid for specific counter-flow
shell and tube heat exchangers and do not apply necessarily to other types of
heat exchangers.
Optimization for the entire range of .
o
o
o
Amin,total linear for low  and then exponential for higher .
Linear due to condenser.
A significant difference is noticed in the relative importance of the two heat
exchangers. Thus, the vapour generator surface is higher for a large range of a
and definitely dominant for  above 0.2 while its UA exceeds that of the
condenser only when a approaches its upper limit. This can be explained by the
significantly greater heat transfer coefficients in the condenser which reduce the
relative importance of its area despite its higher UA value.
5. Economic analysis and optimization
 First step 
o The relative total cost versus the maximum pressure for different maximum
temperatures. The minimum value of the relative cost calculated with the PDS
optimization tool is 0.899 and occurs at 8.90 MPa and 96.01°C. This value of the
maximum temperature is fairly close to the one optimising the UA. However, the
optimising maximum pressure is significantly lower than the corresponding
values determined in all the previous sections. The reason for this result is the
important dependence of the turbine and pump costs on the pressure. Indeed,
their prices increase rapidly when the pressure is augmented.
o On the other hand, at low pressures the heat exchangers’ surface increases and
consequently so does their cost.

Second step
o First step for different values of .
o The first observation is that the total relative cost increases linearly with the net
power output for low values of and exponentially for higher values of . This
behavior is similar to that observed in the two preceding sections.
o Furthermore, the total cost tends towards infinity as a approaches its maximum
value. This rapid augmentation of the cost is mainly due to the vapour generator
as illustrated in Fig. 8. Nevertheless, for most of the acceptable values of , the
relative cost of the turbine is definitely dominant. The price of this component
being highly dependant on the maximum pressure.

There is a particular value of  that minimizes the cost to power-output ratio.
o
Until now, all the optimizations have been carried out over the entire range of 
and it was not possible to determine if a particular value of  was better than
others. With the cost analysis, this is now possible by introducing the concept of
the relative cost per kW produced: the best is the one that has the lowest cost
per kW of net power output. By dividing the relative total cost of each point in
Fig. 8 by the respective net power output, Fig. 9 is obtained. This figure shows
that there is indeed a particular value of  that minimises the cost to poweroutput ratio. For the present case, the best  is near 0.21 (corresponding to a net
power output of approximately 1449 kW) with thermodynamic parameters equal
to 9.11 MPa and 96.3°C. Below that value of , the small net power output does
not overcome adequately the capital cost of the components. On the other
hand, for higher values of , the increased power output does not compensate
the additional investment. This economic optimization shows that, for any
particular source and sink conditions, there is a certain power output that is
more cost effective. It should however be noted that the economic study
presented here is not complete since it does not take into consideration
operational and maintenance costs. Therefore the quoted values of the
maximum power output and the corresponding thermodynamic parameters
should be interpreted accordingly.
Fluid comparison
The critical pressure of these fluids is 7.38 MPa for carbon dioxide, 4.87 MPa for ethane and 3.62
MPa for R125.
It should be noted that this approach would not have been acceptable for sub-critical pressures in
the vapour generator, which would be the case if ORC cycles with constant temperature evaporation
were considered. In that case the correlations used to calculate the heat transfer coefficients in the
vapour generator would have to be modified for the two-phase region.
1. Energy analysis
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The comparison of thermal efficiencies at Tmax = 95°C in Fig. 10 shows that the R125
achieves a maximum efficiency slightly above 10% which is significantly higher than the
corresponding value for the two other fluids. Moreover this maximum value for the R125
corresponds to a lower maximum pressure. The ethane and the carbon dioxide achieve
similar maximum thermal efficiencies near 8.5% but the ethane needs a lower pressure.
The comparison of the specific net output at Tmax = 95°C in Fig. 11 shows that the
highest value is 28.5 kJ/kg and is obtained with the ethane.
Maximum outputs for CO2 and R125 are significantly smaller and respectively equal to
18.8 kJ/kg and 15.9 kJ/kg. The pressure at which these outputs are obtained is lowest for
R125 and highest for CO2.
2. UA
 The minimum total UA of the heat exchangers has been calculated over a large range of
the net power output. The lowest values of this parameter are obtained with R125. It is
followed by ethane which shows a slightly lower UA than carbon dioxide except for very
high net power outputs. The comparison of the minimum surface of the heat exchangers
in Fig. 12 shows that R125 is a better choice from this point of view as well.
3. A

However, in opposition to the results for UA, carbon dioxide requires a smaller heat
exchanger surface than ethane. It can then be deduced that the CO2 has better heat
transfer properties.
4. Economic
 Finally, the economic comparison of Fig. 13 shows that the relative cost per kW with
R125 is lower, by as much as 20%, than for the other two fluids.
 Carbon dioxide comes in second position with a slightly smaller relative cost per kW than
ethane.
 The temperature and pressure conditions for the minimum cost per kW of each fluid are
the following:
o The cycle using carbon dioxide reaches the minimum cost per kW at a net power
output of 1449 kW ( = 0.21) and the conditions are a maximum pressure of 9.11
MPa and a maximum temperature of 96.31°C.
o The R125 reaches the minimum at a net power output of 1655 kW ( = 0.24) at
pmax = 3.7 MPa and Tmax = 91.6°C.
o Finally, ethane reaches its minimum at1379 kW ( = 0.20) at pmax = 6.2 MPa and
Tmax = 94.5°C.
 These comparisons indicate that, for the conditions under consideration, R125 is overall
preferable to CO2 and ethane since it leads to higher efficiencies, lower UA and heat
exchange surfaces as well as a lower relative cost per kW.
 However no fluid can simultaneously optimise all the performance indicators under
consideration.
Conclusion
 The maximum value of the specific net power output, the thermal efficiency and the
exergetic efficiency is obtained for the highest possible value of the working fluid
temperature while this is not the case for the minimum value of the UA, the heat
exchanger surfaces and the cost.
 The optimum value of each performance indicator is obtained for a particular value of
the maximum cycle pressure which is not the same for all six indicators. Maximum
thermal and exergetic efficiencies are obtained for an approximately equal maximum
pressure. Optimum values of UA, heat exchange surfaces and specific net power output
are also obtained for an approximately equal maximum pressure. This second optimum
pressure is lower than the one maximising the two efficiencies. Finally, the cost
minimization is obtained for a third optimum value of the pressure which is significantly
lower than the previous two.
 The exergetic analysis has shown that, for a limited heat source, when the net power
output increases the range of acceptable maximum pressures decreases. An upper
bound for the net power output is reached when the range of pressures reaches a single
value.
 The economic optimization has revealed the existence, for a given source and sink, of a
most profitable net power output which corresponds to the case for which the relative
total cost divided by the net power output is minimum. For the considered source and
sink this value of  is near 0.2.
 A comparison of optimum indicators for three fluids (CO2, ethane, R125) shows that
none outperforms the other two on all counts. Thus, R125 has the best thermal
efficiency, ethane the highest specific net power output and R125 the lowest UA,
surfaceand cost. The CO2 has a higher total UA but a lower  than ethane.
These results show that a simple first law analysis is not sufficient to determine the best cycle
parameters and to choose a best fluid.
SHORT

Parametric study and optimization using 6 performance indicators
o Thermal efficiency
o Specific net output

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

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


o Exergetic efficiency
o Total UA
o Surface HXs
o Relative cost system
Software EES
o Ready to use optimization tools
 Golden Section Search (GSS): 1 degree of freedom
 Powell Direct Search (PDS): >1
Independent parameters
o Tmax
o pmax
o Net power output  
NOT possible to simultaneously optimize all  design value is a choice
The parametric studies reveal the existence of a maximum value for the net power output of
the system and of another net power output minimizing the cost per kW.
3 working fluids: CO2, ethane and R125
First law analysis NOT sufficient for selection
This paper demonstrates the need to achieve a multi-point optimization and comparison in
order to study adequately a transcritical power system.
Methodology: 5 steps
o Energy analysis and optimization
o Exergy analysis and optimization
o Finite size thermodynamics and optimization: the discretisation is performed by
dividing the overall enthalpy change for one of the streams in N equal differences
h.
o Analysis and optimization of total HX surface:
 UAi standard LMTD expression, rather than approximate (Eq. 20b of previous
article).
 Vapour generator
 20 sections
 Working fluid inside the tubes, because of higher pressure
 Hot air in the shell
 Longitudinal fins on OUTside of tubes because of poor HTC of air.
 # tubes and shell diameter obtained by assuming:
o Minimum velocity working fluid = 0.5 m/s
o Maximum velocity hot air = 30 m/s
 Prandtl and Reynolds numbers + Darcy friction factor  EES
 Heat transfer coefficient HTC:
o Air side the Petukhov correlation.
o CO2 side, the Krasnoshchekov–Protopopov correlation
o Thermal resistance tubes wall neglected
o Fouling not considered
 Condenser
 40 sections: 20 single-phase and 20 two-phase
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Working fluid inside the tubes, because of higher pressure
Water in the shell
Longitudinal fins on INside of tubes because of good HTC of water +
fouling risk in water
 # tubes and shell diameter obtained by assuming:
o Minimum velocity saturated liquid CO2 = 1.5 m/s
o Maximum velocity water = 3 m/s
 Heat transfer coefficient HTC:
o Water side + single-phase CO2 the Petukhov correlation.
o Two-phase CO2 side, the Cavallini’s correlation
 Regenerator
 A regenerator is not included since the previous study indicated that
this component is unattractive for low temperature heat sources
because it results in substantial increases of the total heat
exchangers’ surface and only in small improvements of the thermal
and exergetic efficiencies.
o Economic analysis and optimization

Fixed parameters:
o Heat source: Hot air at 100°C
o Mass flow rate heat source = 314.5 kg/s
o Tmin cycle = 15°C
o Sink temperature = 10°C
General assumptions
o Each component steady-state steady-flow system
o Kinematic and potential energies neglected
o Heat and friction losses neglected
o Pump and turbine isentropic efficiencies = 0.8
o Saturated liquid at condenser outlet
o Minimum cycle temperature (T1) is set to 15°C which fixes the minimum pressure
(p1 = p4) of the working fluid
o A minimum temperature difference of 3°C is applied in the condenser at the point
where the working fluid is saturated vapour.
o Hxs: S&T, counterflow, one pass each stream.
Variables
o pmax: range depends on working fluid
o Tmax: 80-100°C
o Net power output  
Results
o The parametric studies show the qualitative influence of the 3 independent variables
(Tmax, pmax and ) on the 6 performance indicators.
o Optimization technique determines the best values of the indicators with their
corresponding values of independent variables.
o

The maximum value of the specific net power output, the thermal efficiency and the
exergetic efficiency is obtained for the highest possible value of the working fluid
temperature while this is not the case for the minimum value of the UA, the heat
exchanger surfaces and the cost.
o The optimum value of each performance indicator is obtained for a particular value
of the maximum cycle pressure which is not the same for all six indicators. Maximum
thermal and exergetic efficiencies are obtained for an approximately equal maximum
pressure. Optimum values of UA, heat exchange surfaces and specific net power
output are also obtained for an approximately equal maximum pressure. This second
optimum pressure is lower than the one maximizing the two efficiencies. Finally, the
cost minimization is obtained for a third optimum value of the pressure which is
significantly lower than the previous two.
o The exergetic analysis has shown that, for a limited heat source, when the net power
output increases the range of acceptable maximum pressures decreases. An upper
bound for the net power output is reached when the range of pressures reaches a
single value.
o The economic optimization has revealed the existence, for a given source and sink, of
a most profitable net power output which corresponds to the case for which the
relative total cost divided by the net power output is minimum. For the considered
source and sink this value of is near 0.2.
o A comparison of optimum indicators for three fluids (CO2, ethane, R125) shows that
none outperforms the other two on all counts. Thus, R125 has the best thermal
efficiency, ethane the highest specific net power output and R125 the lowest UA,
surface and cost. The CO2 has a higher total UA but a lower  than ethane
These results show that a simple first law analysis is not sufficient to determine the best cycle
parameters and to choose a best fluid.