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 efficiencyo Specific net outputo Exergetic efficiencyo Total UAo Surface HXso Relative cost system

Independent parameterso Maximum temperatureo Maximum pressureo 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 steady-

flow 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.

o = 6897 kWo

Hxs: S&T, counterflow, one pass each stream. Equations for performance:

o Conservation of mass and energyo Isentropic efficiency for pump and turbineo 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 variables1. Energy analysis

a. Energy balance for each componentb. Isentropic definitions

Determine states of 1-4

Specific net output

Thermal efficiency

2. Second law analysisa. 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 single-phase 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 analysisa. 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 surfacea. 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. Correlationsi. Petukhov: air, water and single-phase working fluid in condenser

ii. Krasnoshchekov-Protopopov: supercritical working fluidiii. 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 CO2The 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.

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).

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∧α ).

First parametric studyo 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°Co ηex=f (pmax , α)o For each combination of Tmax and there exists a value of pmax which

maximizes the exergetic efficiency.

o Optimising pmax is relatively independent of .o ηex ↑ if ↑ (for >0.2 and Tmax = 95°C).o 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 ↓

o 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 = 0.2o 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

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 .o For low , UAtotal, min ↑ linearly with net power output.o For higher , the increase becomes exponential and UAtotal ∞ as reached

its previous maximum.o 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 Effect of pmax on A is ± same as on UAo Tmax for Amin is different than for UAmin Amin is more importanto A temperature of 98.03°C minimises the total UA, the temperature which

minimises the total surface is 89.78°C.

o The minimum value of A is 1369 m2 and the corresponding pmax is 11.4 MPa.o 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 Amin,total linear for low and then exponential for higher .o Linear due to condenser.o 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 stepo 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 power-output 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 comparisonThe 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

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 indicatorso Thermal efficiencyo Specific net outputo Exergetic efficiency

o Total UAo Surface HXso Relative cost system

Software EESo Ready to use optimization tools

Golden Section Search (GSS): 1 degree of freedom Powell Direct Search (PDS): >1

Independent parameterso 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 optimizationo Exergy analysis and optimizationo 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/so 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 correlationo Thermal resistance tubes wall neglectedo Fouling not considered

Condenser 40 sections: 20 single-phase and 20 two-phase 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/so 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°Co Mass flow rate heat source = 314.5 kg/so Tmin cycle = 15°Co Sink temperature = 10°C

General assumptionso Each component steady-state steady-flow systemo Kinematic and potential energies neglectedo Heat and friction losses neglectedo Pump and turbine isentropic efficiencies = 0.8o Saturated liquid at condenser outleto Minimum cycle temperature (T1) is set to 15°C which fixes the minimum pressure

(p1 = p4) of the working fluido 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.

Variableso pmax: range depends on working fluido Tmax: 80-100°Co Net power output

Resultso 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.