Supercritical CO2 Power Cycle Symposium May 24-25, 2011 Boulder, Colorado

Modeling Off-Design Operation of a Supercritical Carbon Dioxide Brayton Cycle John J. Dyreby, Sanford A. Klein, Gregory F. Nellis, and Douglas T. Reindl University of Wisconsin-Madison, Solar Energy Laboratory 1343 Engineering Research Building, 1500 Engineering Drive, Madison, WI 53706 Email: [email protected]

Abstract In the search for increased efficiency of utility-scale electricity generation, Brayton cycles operating with carbon dioxide have found considerable interest. Due to the unique properties of carbon dioxide, high design-point efficiencies can be realized by operating the compressor near the critical point (7.4 MPa and 31°C). In this paper, we show that the thermal efficiency and power production of a cycle using fixed turbomachinery designed to provide optimal performance under these conditions will decrease as the compressor inlet temperature increases. Conversely, turbomachinery designed to provide optimal performance at a higher compressor inlet temperature (e.g., 60°C), which results in lower efficiency at the design point, exhibits an increase in both efficiency and power production under off-design conditions as the inlet temperature is reduced. The findings of this research are significant in that they suggest the optimal design for the turbomachinery in a supercritical carbon dioxide (S-CO2) Brayton power cycle, considering its overall performance, may not coincide with the optimal design suggested by a simple on-design thermodynamic analysis. Initial results suggest that designing for a higher compressor inlet temperature will not significantly degrade plant efficiency and it can yield better off-design power production, possibly increasing the overall power plant performance evaluated on an annual basis. These results are particularly relevant to renewable energy applications that are inherently transient, such as concentrating solar power (CSP) systems.

1. Introduction This paper reports on the development of models that allow the characterization and evaluation of supercritical carbon dioxide (S-CO2) power cycles for concentrating solar power (CSP) applications. Specifically, the models developed are used to predict the design point and off-design operation of the cycle in order to characterize the performance of different cycle configurations for a CSP plant operating on an annual basis. High side cycle conditions are assumed to be compatible with power tower operating temperatures in the 500°C to 650°C range and low side conditions are consistent with dry or hybrid cooling heat rejection systems in an arid climate. Dry cooling in this context involves the use of ambient air as the sole heat rejection medium, which has potential policy and water conservation advantages that are moderated by a decrease in cycle efficiency (compared to the more traditional use of a water-cooled system that uses a cooling tower for heat rejection). Initial model development focuses on a recuperated Brayton cycle (shown in Figure 1) that consists of a compressor, a turbine, and three heat exchangers: the precooler, the recuperator, and the primary heat exchanger.

Supercritical CO2 Power Cycle Symposium May 24-25, 2011 Boulder, Colorado

Compressor

Turbine Generator

Primary Heat Exchanger Heat Rejection

Precooler

Heat Addition

Recuperator

Figure 1 – Diagram of a recuperated Brayton cycle.

The model is flexible with respect to compressor and turbine design but the parameters are currently based on the radial turbomachinery under investigation by Sandia National Laboratory (SNL) [1]. Specifically, turbomachinery performance is characterized by a dimensionless head-flow curve based on experimental data from SNL. Heat exchangers are modeled in a counter-flow configuration using discrete sections that are connected in series; this representation allows the effect of the rapidly changing properties of carbon dioxide near the critical point to be accurately captured. Given a particular physical plant (i.e., fixed turbomachinery and heat exchanger parameters) that is designed for a specific operating point, the model is capable of simulating its performance under off-design conditions. Further details on component modeling are provided in the section that follows.

2. Modeling Methodology The three major components of a closed-loop Brayton cycle and variations thereof are the compressor(s), turbine(s), and heat exchangers. The simple Brayton cycle with recuperation consists of one compressor, one turbine, and three heat exchangers: the precooler, the primary heat exchanger, and the recuperator. Another cycle variation that shows promise for S-CO2 applications is the recompression Brayton cycle [2, 3], which adds a second compressor and a second recuperator. Although the recompression cycle is more complex than the simple Brayton cycle, the fundamental components required to analyze the cycle are the same. The approach followed in the present analysis is to model each component separately and then integrate those component models into a system-level model; thereby allowing analyses of multiple Brayton cycle variations under both design and off-design conditions to be performed. To this end, a semi-empirical model for each of the components was developed in the Fortran programming language with fluid properties provided by REFPROP [4, 5]. The semi-empirical model uses performance parameters that are based on the underlying physics of the component and therefore allows off-design point operation to be estimated. The advantage of a semi-empirical model as compared to a completely physics-based model is that it is computationally fast. The disadvantage is that it is limited to a more narrow range of conditions that correspond to the region where the performance parameters used in the model remain valid. 2.1 Heat Exchanger Model Heat exchangers are modeled assuming a counter-flow configuration using discrete sections, or sub-heat exchangers, connected in series; this arrangement allows the effect of the rapidly changing carbon dioxide properties near the critical point to be accurately captured. A concise description of the sub-heat exchanger analysis is available in Nellis and Klein [6]. The design heat exchanger conductance (also referred to as the UA value) is specified and an iterative approach is used to determine the corresponding outlet temperatures, assuming that there is no heat loss through the heat exchanger jacket to the surroundings. The UA value is strongly dependent on heat exchanger geometry and relatively constant over a range of conditions and is therefore useful for off-design calculations. The effect of mass

Supercritical CO2 Power Cycle Symposium May 24-25, 2011 Boulder, Colorado

flow rate on heat transfer coefficients, and hence the conductance, is accounted for by scaling the UA value under off-design conditions according to:

 m UA  UAdesign   m  design

  

0.8

(1)

 design. The derivation where UAdesign is the conductance of the heat exchanger at the design-point mass flow rate of m of Eq. (1) uses the Dittus-Boelter heat transfer correlation and is presented in Patnode [7]. A similar analysis using the Darcy friction factor and Blasius correlation is used to scale pressure drop (ΔP) through the heat exchanger under off-design conditions:  m P  Pdesign   m  design

  

7/ 4

(2)

Specifying the design heat exchanger conductance as the performance parameter for the model allows for a fair comparison between cycles because a larger UA value typically corresponds to a larger and more expensive heat exchanger. 2.2 Compressor Model The compressor model uses dimensionless flow and ideal head coefficients, which commonly describe compressor performance and are derived by applying the Buckingham Pi theorem [8]. The ideal head coefficient and the compressor efficiency are both functions of the flow coefficient, and this functional relation can vary significantly between different compressor designs. The functional relation used within the model can be based on experimental data or numerical predictions and therefore the semi-empirical compressor model is flexible with respect to how the ideal head and efficiency curves are specified. To facilitate development of the present turbomachine model, the compressor currently being studied for use with carbon dioxide at Sandia National Laboratory [1] is used to generate the necessary relationships between the ideal head coefficient, efficiency, and flow coefficient. An additional correction for shaft speed is proposed and used to collapse the performance map into a single dimensionless head-flow curve. The modified head coefficient (ϕ*), flow coefficient (ψ*), and efficiency (η*) are defined as:

m    U c Dc 2 *

 N   N design

 h  N  i*  2i  design  Uc  N  N   *    design   N 

1/5

  

(3)

 20 

* 3

(4)

 20 

* 5

(5)

 is the mass flow rate through the compressor, ρ is the density of the fluid at the compressor inlet, Uc is the where m tip speed of the rotor, Dc is the diameter of the rotor, N is the shaft speed, Ndesign is the design shaft speed (75,000 rpm for the SNL compressor), and Δhi is the isentropic enthalpy rise of the fluid through the compressor. Figure

Supercriticaal CO2 Power Cycle Sympoosium May 24-25,, 2011 Boulder, Collorado

2(a) is the performance map m for the com mpressor overlaaid onto experiimental data, aand Figure 2(b)) shows the ressult of E (3-5) for sh haft speeds greeater than 35,00 00 rpm. applying Eqs.

(a)

(b)

Figure 2 – (a) Perform mance map for th he main comprressor in the San ndia National L Laboratory supeercritical CO2 ttest loop [1] an nd (b) result of applying a the mo odified efficienccy, ideal head, aand flow coefficcients.  

Curve fits to the modified d ideal head co oefficient and efficiency, e bothh as a function of modified fllow coefficientt, provide thee relations requ uired by the sem mi-empirical compressor c moodel. For this pparticular comppressor, the low wer limit for th he unmodified flow f coefficien nt is roughly 0..021; operationn below this caan result in surgge conditions, w which are charactterized by aero odynamic instab bility and rapid d flow reversalls. The upper llimit of the flow w coefficient iis 0.05, which h corresponds to an ideal heaad coefficient of o zero (i.e., noo enthalpy rise through the coompressor). The comprressor model iss explicit: given n the inlet cond ditions and maass flow rate, thhe outlet condittions and poweer are calculated without requirring iteration. Although A this model m is expliccit, the system model that usees the semiempirical compressor c mo odel typically is not. The inleet conditions annd mass flow rrate for a closed Brayton cyclle operating under u off-desig gn conditions are a typically no ot known and m must be solved for by integratting various componentt models. The result is an im mplicit system model m that requuires iteration. 2.3 Turbine Modell The initial model develop pment is appliccable to radial turbines, t whichh are appropriaate for applicattions up to 50 M MWe [9]. The mass m flow rate through t a turbiine is strongly dependent on iits pressure rattio and inlet conditions and w weakly dependent on shaft speed d. As a first ord der approximattion, this relatiionship is:

m  Cs Anozzle 

(6)

where Anozzzle is an effectiv ve nozzle area that is based on o the geometryy of the turbinee, ρ is the denssity of the fluidd at the turbine inleet, and Cs (refeerred to as the spouting s velociity) is the veloccity that wouldd be achieved iif the fluid wass expanded isentropically i to t the outlet preessure through h an ideal nozzlle. B [10] hav ve proposed a general relationship betweenn the aerodynam mic efficiency (ηturbine,aero, thee Chen and Baines efficiency of an ideal turb bine with no in nternal losses) and a velocity raatio (ν, the ratioo of rotor tip sppeed to spoutinng

Supercritical CO2 Power Cycle Symposium May 24-25, 2011 Boulder, Colorado

velocity) for a turbine based on blade geometry and loading. For a well-designed turbine with a low loading coefficient, the relationship simplifies to:

turbine, aero  2 1  2

(7)

Plotting Eq. (7) results in the familiar relationship between aerodynamic efficiency and velocity ratio for a radial turbine, shown in Figure 3. Note that the maximum efficiency occurs at a velocity ratio of 0.707, as expected [11]. 1.0 0.9 0.8

Efciency

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

U Cs

Figure 3 – Aerodynamic efficiency of a radial turbine as a function of velocity ratio.

The aerodynamic efficiency assumes an ideal turbine and does not take into account internal losses (e.g., recirculation, viscous effects, etc.). To account for these losses in the semi-empirical model, the efficiency of the turbine is calculated by scaling the aerodynamic efficiency predicted by Eq. (7) by the efficiency of the turbine at the design point. The semi-empirical turbine model and the semi-empirical compressor model are complementary in that the compressor model uses the mass flow rate as an input and returns the compressor outlet pressure (i.e., there is a head-flow curve that relates pressure rise to mass flow rate), while the turbine model uses the inlet conditions and outlet pressure as inputs and returns the mass flow rate (i.e., there is a flow resistance afforded by the fixed area restriction in the turbine). Matching the head-flow curve of the compressor with the flow resistance of the turbine (as well as the minor pressure drops through the heat exchangers) allows the operating point of the system to be determined. Figure 4 illustrates this concept, showing the effects of changing shaft speed and turbine inlet temperature on the operating point of the cycle. Note the operating characteristics shown are specific to the geometry of the compressor and turbine used in the analysis, which were chosen arbitrarily for the purpose of model illustration.

Supercritical CO2 Power Cycle Symposium May 24-25, 2011 Boulder, Colorado

5.0 650°C 550°C

4.5

450°C 35,000 rpm

Pressure Ratio

4.0

3.5 30,000 rpm

3.0

2.5

25,000 rpm

2.0

1.5 Turbine 1.0 0

20

40

60

80

100

120

140

160

180

Mass Flow Rate (kg/s)

Figure 4 – Head-flow and flow resistance curves for different shaft speeds and turbine inlet temperatures; the curves intersect at the corresponding operating point of the cycle.

The intersection of the compressor and turbine curves corresponds to the operating point of the system. Note that changing the speed of the compressor has a significant impact on the operating point, whereas a change in turbine inlet temperature has a much smaller effect.

3. Recuperated Brayton Cycle Modeling The compressor and turbine inlet temperatures are specified as inputs for each respective model. These specifications remove the need for modeling the precooler and primary heat exchanger explicitly using a sub-heat exchanger analysis and allow the effect of compressor or turbine inlet temperatures to be investigated more directly. The design-point efficiency of the cycle shown in Fig. 1 is determined for a range of compressor inlet temperatures and pressures. A 10 MW power plant is modeled while holding constant the compressor and turbine isentropic efficiency (0.9), the recuperator conductance (2,000 kW/°C), and the turbine inlet temperature (550°C). The compressor outlet pressure is set to either 25 MPa or the maximum possible outlet pressure that could be achieved while maintaining the tip speed of the compressor and turbine at Mach 0.9. The relative pressure drop through the heat exchangers (ΔP/P) is assumed to be 0.5% , which was chosen based on the average values presented by Dostal [12]. The results of this analysis are shown in Figure 5; note that every point in Fig. 5 corresponds to a unique set of system hardware designed specifically to optimize the plant efficiency at that condition.

Supercritical CO2 Power Cycle Symposium May 24-25, 2011 Boulder, Colorado

0.43 7.7 MPa

Design 1

0.42 9 MPa Optimal Inlet Pressure

0.41

Thermal Ef)ciency

10 MPa

0.40 0.39

Design 2

0.38 0.37 0.36 0.35 10

15

20

25

30

35

40

45

50

55

60

Compressor Inlet Temperature (°C)

Figure 5 – Parametric design point efficiency for three compressor inlet pressures (solid lines) and the optimal compressor inlet pressure (dotted black line) as a function of inlet temperature.

The abrupt decrease in thermal efficiencies shown in Fig. 5 for the 7.7 and 9 MPa cases with increasing inlet temperature (corresponding to the transition from a solid to a dashed line) occur when the compressor tip speed reaches a Mach number of 0.9; this constraint reduces the compressor outlet pressure and overall pressure ratio of the cycle. As expected from literature, the highest design point efficiency is realized at pressures approaching the critical pressure [2]. However, as the design-point compressor inlet temperature increases, the optimal inlet pressure increases and the cycle operation moves away from the critical point. Allowing the inlet pressure to vary in order to maximize the thermal efficiency of the cycle for a given inlet temperature results in the dotted black line in Fig. 5. The thermal efficiency shown in Fig. 5 does not include any pumping or fan energy associated with heat rejection from the cycle. The two designs identified in Fig. 5 (Design 1 and Design 2) were selected in order to investigate the effect of design-point compressor inlet temperature on the off-design performance of the cycle. At the design point, the two cycles are identical with the exception of the turbomachinery parameters; both cycles have the same recuperator conductance and operate at a high side pressure of 25 MPa. The fixed turbomachinery parameters are calculated by setting the flow coefficient to the value that maximizes compressor efficiency and the turbine velocity ratio to the value that maximizes turbine efficiency. Due to page space limitations here, the parameters for these two designs will not be listed, but in general designing for a higher compressor inlet temperature (Design 2) results in slightly larger turbomachinery operating at a lower shaft speed at the design point. An off-design analysis was run for each design over a range of compressor inlet temperatures and pressures. The shaft speed was allowed to vary such that thermal efficiency was maximized. Cycle efficiency as a function of temperature is plotted in Figure 6(a) for Design 1 and Figure 6(b) for Design 2.

Supercritical CO2 Power Cycle Symposium May 24-25, 2011 Boulder, Colorado

0.43

0.43 Design Point

0.42

0.42 Optimal Inlet Pressure

0.41

0.41 0.40

0.39 0.38 0.37

11 MPa

0.36 10 MPa

0.35 0.34

Thermal Ef)ciency

Thermal Ef)ciency

0.40

Design Point

0.38

7.7 MPa

8 MPa 9 MPa

0.37 0.36

10 MPa 11 MPa

0.35 0.34

9 MPa 8 MPa

0.33

Optimal Inlet Pressure

0.39

0.33

7.7 MPa

0.32

0.32 25

30

35

40

45

50

55

60

25

30

Compressor Inlet Temperature (°C)

35

40

45

50

55

60

Compressor Inlet Temperature (°C)

(a)

(b)

Figure 6 – Off-design thermal efficiency for (a) Design 1 and (b) Design 2.

Actively controlling compressor inlet pressure (also referred to as inventory control) is clearly advantageous when operating the cycle at an off-design inlet temperature. This is especially true for the low temperature Design 1, which, for a fixed design compressor inlet pressure of 7.7 MPa, shows a large decrease in thermal efficiency as the inlet temperature increases. Using both inventory and shaft speed control in order to maximize thermal efficiency results in the thermal performance curve mapped by the dotted black lines shown in Fig. 6. Note that while Design 2 exhibits a larger efficiency at its design point of 60°C, Design 1 has a slightly greater efficiency over a larger range of temperatures (Figure 7(a) overlays the two dotted black lines from Fig. 6 for easier comparison). However, increasing the compressor inlet temperature decreases the net power output of the cycle, which results in Design 2 possibly being able to take advantage of cooler conditions in order to produce more power, as shown in Figure 7(b). 16

0.43 Design Point

Design 2

0.42

14

0.41 12

Design 1

Design Point

0.39 0.38

Design 2

0.37 0.36 0.35

Net Power (MW)

Thermal Ef'ciency

0.40

Design 1

Design Point

10 Design Point

8 6 4

0.34 2

0.33 0.32

0 25

30

35

40

45

50

Compressor Inlet Temperature (°C)

(a)

55

60

25

30

35

40

45

50

55

Compressor Inlet Temperature (°C)

(b)

Figure 7 – Off-design thermal efficiency (a) and net power production (b) for Design 1 (blue) and Design 2 (red).

60

Supercritical CO2 Power Cycle Symposium May 24-25, 2011 Boulder, Colorado

4. Summary & Conclusions The off-design modeling efforts presented in this paper are ongoing, with initial results indicating that the selection of the design point for the cycle has a relatively small effect on thermal efficiency in off-design operation provided that both shaft speed and pressure can be controlled, but it can have a large effect on net power production. Designing for a warmer compressor inlet temperature (e.g., 60°C), which would be advantageous for a dry-cooled power plant, results in only a slight decrease in thermal efficiency over a range of ambient temperatures compared to a lower temperature design (e.g., 33°C). However, the high temperature design may able to take advantage of cooler conditions and, depending on available solar radiation, produce more power over a range of ambient temperatures. While the current results highlight the importance of design point selection on off-design operation, ongoing work is focused on modeling the off-design performance of power plants on an annual basis. Specifically, weather information for various locations will be used in conjunction with the developed off-design models in order to characterize and evaluate various cycles and designs on an annual basis.

Acknowledgements The authors would like to thank the Concentrating Solar Power Program at the National Renewable Energy Laboratory for their sponsorship of this work. The collaboration with Steven Wright at the Sandia National Laboratory is also greatly appreciated.

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