Thermodynamic and economic investigations of transcritical CO2-cycle systems with integrated radial-inflow turbine performance predictions

Journal Pre-proofs Thermodynamic and economic investigations of transcritical CO2-cycle systems with integrated radial-inflow turbine performance predictions Jian Song, Xiaoya Li, Xiaodong Ren, Hua Tian, Gequn Shu, Chunwei Gu, Christos N. Markides PII: DOI: Reference:

S1359-4311(19)34113-4 https://doi.org/10.1016/j.applthermaleng.2019.114604 ATE 114604

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Applied Thermal Engineering

Received Date: Revised Date: Accepted Date:

14 June 2019 7 October 2019 28 October 2019

Please cite this article as: J. Song, X. Li, X. Ren, H. Tian, G. Shu, C. Gu, C.N. Markides, Thermodynamic and economic investigations of transcritical CO2-cycle systems with integrated radial-inflow turbine performance predictions, Applied Thermal Engineering (2019), doi: https://doi.org/10.1016/j.applthermaleng.2019.114604

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Thermodynamic and economic investigations of transcritical CO2-cycle systems with integrated radial-inflow turbine performance predictions Jian Song1, Xiaoya Li1,2, Xiaodong Ren3, Hua Tian2, Gequn Shu2, Chunwei Gu3, Christos N. Markides1,*

1 Clean 2 State 3

Energy Processes (CEP) Laboratory, Department of Chemical Engineering, Imperial College London, London SW7 2AZ, UK

Key Laboratory of Engines, Tianjin University, 92 Weijin Road, Nankai District, Tianjin 300072, China

Key Laboratory for Thermal Science and Power Engineering of Ministry of Education, Department of Energy and Power Engineering,

Tsinghua University, Beijing 100084, China * Corresponding author. E-mail address: [email protected]

Abstract Transcritical CO2 (TCO2) cycle systems have emerged as a promising power-generation technology in certain applications. In conventional TCO2-cycle system analyses reported in the literature, the turbine efficiency, which strongly determines the overall system performance, is generally assumed to be constant. This may lead to suboptimal designs and optimization results. In order to improve the accuracy and reliability of such system analyses and offer insight into how knowledge of these systems from earlier analyses can be interpreted, this paper presents a comprehensive model that couples TCO2-cycle calculations with preliminary turbine design based on the mean-line method. Turbine design parameters are optimized simultaneously to achieve the highest turbine efficiency, which replaces the constant turbine efficiency used in cycle calculations. A case study of heat recovery from an internal combustion engine (ICE) using a TCO2-cycle system with a radial-inflow turbine is then considered, with results revealing that the turbine efficiency is influenced by the system’s operating conditions, which in turn has a significant effect on system performance in both thermodynamic and economic terms. A more generalized heat source is then considered to explore more broadly the role of the turbine in determining TCO2-cycle power-system performance. The more detailed turbine-design modelling approach allows errors of the order of up to 10-20% in various predictions to be avoided for steady-state calculations, and potentially of an even greater magnitude at off-design operation. The model allows quick preliminary designs of radial-inflow turbines and reasonable turbine performance predictions under various operating conditions, and can be a useful tool for more accurate and reliable thermo-economic studies of TCO2-cycle systems.

Keywords: power generation; radial turbine; transcritical CO2; thermo-economic study; turbine performance; waste-heat recovery

1

Nomenclature Abbreviations A B, C, F, K c CBM cp C P0 d f F, j, R G h k Kc, K e l ṁ N P Pr Q Re rf T u U ū1 w W X Xtt Zt

area (m2) coefficients for cost calculation absolute velocity (m/s) bare module cost ($/W) specific heat capacity (J/kg·K) cost coefficient at ambient pressure and using carbon steel diameter (m) friction coefficient correction factors mass flux (kg/m2∙s) specific enthalpy (J/kg) thermal conductivity (W/m∙K) contraction and expansion loss coefficients blade height (m) mass flow rate (kg/s) number pressure (Pa) Prandtl number heat (W) Reynolds number fouling resistance temperature (K) peripheral velocity (m/s) total heat transfer coefficient, (W/m2∙K) velocity ratio relative velocity (m/s) power (W) component capacity indexes turbulent-turbulent Lockhart Martinelli parameter number of tube passes

Greek symbols α β δ η λ ρ v ζ φ ψ

absolute flow angle (°) relative flow angle (°) tip clearance (m) efficiency thermal conductivity (W/m∙K) density (kg/m3) specific volume (m3/kg) loss in radial-inflow turbine nozzle velocity coefficient rotor velocity coefficient 2

Ω

reaction degree

Subscripts 1 - 6, 3’, 5’ b bulk c cond gas HS in jw l m main n out pump r s t T tcc tcw w

state point bundle bypass flow; baffle bulk temperature of fluid baffle cut; contraction condenser engine exhaust gases heat source inlet jacket water leakage; liquid average main heater net outlet pump adverse flow; turning shell side; variable baffle spacing; isentropic tube side turbine tube rows in crossflow tube rows in baffle window wall

Superscripts *

total parameter

Acronyms BMPC CEPCI CFD EPC HTC HTX IAE ICE LMTD SIC SNL TCO2 ORC WHR

Bechtel Marine Propulsion Corporation chemical engineering plant cost index computational fluid dynamics electricity production cost heat transfer coefficient heat exchanger Institute of Applied Energy internal combustion engine log mean temperature difference specific investment cost Sandia National Laboratory transcritical CO2 organic Rankine cycle waste-heat recovery

3

1. Introduction The rate of global primary-energy consumption has been increasing rapidly with economic development and population growth, leading to increasing concerns relating to energy security, sustainability and environmental deterioration [1]. Renewable-energy technologies, and in particular distributed solar-cogeneration systems in the built environment, based on either conventional [2,3] (including with integrated thermal-energy storage [4,5]), but also more innovative/hybrid technologies [6-8], and improved efficiency energy-utilization solutions in commercial and industrial settings [9,10], have attracted interest as part of pathways aimed at fuel-use reduction and environmental impact alleviation [11]. In the context of thermal power-generation, a number of relatively mature technologies are being proposed for deployment in a wide range of applications, including systems based on a variety of power cycles. Serrano et al. [12] and Dirker et al. [13] explored the potential of using steam Rankine cycle systems for engine waste-heat recovery and solar energy utilization, respectively; Schuster et al. [14] conducted energetic and economic investigation of organic Rankine cycle (ORC) systems for different applications; Lecompte et al. [15] focused on the specific waste-heat recovery applications with various ORC architectures; while Bommbarda et al. [16] and Reddy et al. [17] studied the performance of Kalina cycle systems and conducted comparisons with other technologies. In addition, earlier-stage technologies that are still under development, such as thermoacoustic [18,19] and, more recently, thermofluidic devices, have also attracted attention due to their reduced complexity and costs. Examples of this latter category of technologies include the Non-Inertive-Feedback Thermofluidic Engine (NIFTE) [20-24] and the Up-THERM heat engine [25-28]. Transcritical CO2 (TCO2) power-cycle systems have also emerged as attractive alternatives in recent years [29,30] in various applications. Garg et al. [31] presented analyses of TCO2-cycle systems for solar power generation, including comparisons with steam Rankine cycle systems that indicated that TCO2-cycle systems may have important advantages such as improved performance and compactness. Song et al. [32] investigated a solar-driven transcritical CO2 power-cycle system with LNG as heat sink, and showed that a thermal efficiency of 6.5% may be achievable. Li and Dai [33] conducted thermo-economic analyses of TCO2-cycle systems for the exploitation of geothermal sources, and the results showed that a better thermal performance as well as a simpler system structure can make this type of technology attractive in practical applications. Ahmadi et al. [34] studied a TCO2-cycle geothermal power generation system with LNG cold energy utilisation, and investigated systematically the influence of several key thermodynamic variables on the overall system performance. Transcritical CO2-cycle systems have also appeared as an appealing option in waste-heat recovery applications, which are of direct interest to the present work. Wu et al. [35] proposed a novel type of single4

pressure, multi-stage TCO2-cycle system and the results revealed that an increase of 4-26% on the net power output could be achieved relative to an existing single-stage system. Kim et al. [36] undertook analyses of TCO2-cycle systems and compared their results with supercritical CO2-cycle systems, showing that transcritical cycles were more effective if heat was available in the low-temperature range to compensate for the difference in the specific heat of CO2 between the high- and low-pressure sides. Chen et al. [37] also conducted comparisons of transcritical CO2-cycle systems, this time relative to ORC systems for a lowtemperature waste-heat (150 °C) conversion; the results revealed that the transcritical cycles had slightly higher power outputs. Li et al. [38] performed an assessment of using TCO2-cycle system to exploit wasteheat recovery from a heavy-duty truck engine and presented the most promising layout with a preheater and a recuperator, which led to an improvement of 2.3% in brake thermal efficiency. Choosing CO2 as the working fluid is associated with benefits thanks to its non-toxic and non-flammable nature, and also given that does not suffer from temperature limitations imposed by thermal decomposition [39,40]. The high density of CO2 in the supercritical region allows smaller and more compact component designs and can facilitate device miniaturization, which is of great importance in specific applications with space restrictions such as engine WHR in vehicles [41,42]. Furthermore, a better thermal match between the heat source and CO2 as the working fluid undergoing a power cycle can be achieved, which acts to reduce the exergy losses in the heat exchangers and to improve the system’s thermodynamic performance [43]. The expander is one of the key components of power-cycle systems since its performance has a direct and significant influence on the overall system power output [44,45]. The relatively high density of CO2 in the supercritical region requires the expander in TCO2-cycle systems to be capable of dealing with small volumetric flow-rate conditions. Radial-inflow turbines offer an attractive option for employment as expanders in TCO2-cycle systems from a performance perspective, and are also widely used as turbochargers in small power-output applications [46,47]. Both numerical and experimental research has been devoted to CO2 radialinflow turbines. Zhang et al. [48] and Zhou et al. [49] conducted aerodynamic design and numerical analysis of a 1.5-MW radial-inflow turbine for CO2-cycle systems and their CFD results suggested acceptable performance from the proposed turbine designs. Qi et al. [50] investigated CO2 radial-inflow turbines over a range of power outputs from 100 to 200 kW and considered the influence of key geometric parameters on turbine performance. Comprehensive flow and head coefficient maps were generated, which have led to a range of feasible radial-inflow turbine designs. In addition to the numerical work, practical experience and experimental data from several testing facilities featuring radial-inflow turbines, with ranges from 20 to 5

200 kW, have been generated, including by the Sandia National Laboratory (SNL), the Institute of Applied Energy (IAE) and the Bechtel Marine Propulsion Corporation (BMPC) [51,52]. The design of a radial-inflow turbine in terms of its geometry and size is critically affected by the target performance expected from this component, which depends strongly on the working fluid and on overall power-cycle parameters and conditions (and vice versa) [45]. However, most research on TCO2-cycle systems to-date has involved system-level analyses or focused on optimization with a pre-specified constant value for the turbine efficiency [53-55]. Such approaches, which we refer to hear as ‘conventional approaches’, have allowed us to gain significant insight into the operation and performance of these components and to assess the potential of the overall systems within which they are embedded, but might lead to results that deviate from practical experience. In order to enhance the accuracy and reliability of such system predictions, it is necessary to couple the TCO2-cycle system models with a preliminary radial-inflow turbine design model that can be used to predict the turbine’s efficiency over a range of fluids and flow conditions. This approach has been employed in several previous studies on ORC systems. Zhang et al. [56] conducted parametric optimization and performance evaluations of a geothermal ORC system. Their results indicated deviations in the optimal working-fluid selection and system maximum exergy efficiency when correlations to predict the turbine efficiency were integrated with the thermodynamic model of the ORC system. Pan and Wang [57] presented an improved analysis method for ORC systems based on radial-inflow turbines. As with the work of Zhang et al. [56], the optimized working fluid changed when the assumption of a constant turbine efficiency was relaxed and replaced with a calculation of the turbine efficiency as part of the overall cycle model. Li et al. [58] proposed a multi-objective optimization approach for ORC system analysis coupled with a dynamic turbine efficiency model. Once again, the optimal working fluid changed from R245fa to R365mfc when a variable turbine efficiency was employed in their thermo-economic model. Rahbar et al. [59] presented a comprehensive ORC system model in which an optimization algorithm was embedded that further improved the turbine design and allowed the turbine to achieve the highest efficiency possible under various system operating conditions. As indicated by the overview of the aforementioned literature on ORC system modelling with turbine performance models, and as expected since it is known that the turbine efficiency can vary significantly over an extended operating envelope, it is necessary to employ a more detailed turbine efficiency model as part of the overall power-cycle model to achieve accurate thermodynamic and economic results. Nevertheless, there remains a research gap in transcritical CO2-cycle system assessments that consider system analysis and optimization, while accounting explicitly for turbine performance predictions with detailed turbine models. This paper presents 6

a comprehensive model that has been developed to overcome the aforementioned limitations by coupling TCO2cycle calculations with a preliminary radial-inflow turbine design model based on the 1-D mean-line method as an improved approach for the performance prediction of such systems. Several key parameters for turbine design are simultaneously optimized to achieve the highest turbine efficiency as well as the most efficient turbine design. A case study of waste-heat recovery from an internal combustion engine (ICE) using a TCO2-cycle system with a radial-inflow turbine is considered to showcase the benefits of the model developed in this work relative to the conventional approaches. Thermodynamic and economic investigations with both constant turbine efficiency values and 1-D turbine efficiency predictions from the turbine design model are conducted in order to understand the interplay between system and turbine operation and performance. In addition, the role of the turbine efficiency in determining overall TCO2-cycle system performance is considered by implementing both conventional and the methodology proposed in this work over a range of conditions on a generalized heat source. ORC technology is more mature and well-established than the TCO2-cycle systems being considered here, both technically and commercially. Nevertheless, and although ORC systems have appeared as an effective heat-to-power conversion technology especially for heat streams in the temperature range of 100400 °C [60] which, therefore make this a promising option for engine WHR [61-63], some unfavourable environmental, health and safety characteristics of certain working fluids, such as flammability and thermal decomposition at the high temperatures associated with the utilization of exhaust gases as heat sources [64,65], act to promote the consideration of alternative power-generation technologies, including the consideration of TCO2-cycle systems that are at the focus of this work.

2. System description This paper explores WHR from an ICE using a transcritical CO2-cycle system featuring a radial-inflow turbine. The detailed parameters relating to the particular ICE (E1165 500NOx L33 Natural Gas CHP Unit) considered in this work, including the (rated) conditions of the heat sources (both exhaust gases and jacket water) available for recovery, which are directly taken from manufacturer datasheets, are summarized in Table 1. Heat from both the exhaust-gases and jacket-water stream are recovered in order to improve the overall system performance and to decrease fuel consumption. To avoid corrosion in the pipes and heat exchangers, the outlet temperature of the exhaust gases from the heat recovery unit is set to be higher than 120 °C [66]. For this condition, the maximum thermal load (heat input to the CO2-cycle system) from the exhaust gases amounts to 630 kW. Jacket water is used as the coolant and the available thermal load (for heat rejection) is approximately 600 kW. 7

Table 1. ICE parameters at the investigated, rated condition. Parameter

Value

Power output (kW)

1170

Inlet temperature of jacket water (°C)

78

Outlet temperature of jacket water (°C)

89

Mass flow rate of jacket water (kg/s)

13.0

Temperature of exhaust gases (°C)

457

Mass flow rate of exhaust gases (kg/s)

1.69

A schematic diagram of the transcritical CO2-cycle system considered here is shown in Fig. 1. Liquid CO2 is firstly pressurized in the pump and absorbs heat from the low-temperature jacket water in the preheater. In the recuperator downstream of the preheater, this stream is heated further in a recuperator by the high-temperature CO2 stream that is returning from the turbine. The CO2 absorbs additional heat from the high-temperature ICE exhaust gases and then expands in the turbine to produce power. Finally, the CO2 stream flows back through the recuperator and the condenser to be cooled and condensed to the liquid state (assumed here to be saturate) required by the pump, which completes the cycle.

Figure 1. Schematic diagram of a TCO2-cycle system for WHR from an ICE.

3. Modelling methodology 3.1. Thermodynamic model A transcritical CO2-cycle for WHR from an ICEs is shown in Fig. 2.

8

Figure 2. Thermodynamic cycle (T-s) diagram of a TCO2-cycle system for WHR from an ICE taken from this work with (and showing): T1 = 25 °C, T4 (= TT,in) = 150 °C, P4 (= PT,in) = 15 MPa.

In this cycle, Process 1 to 2 in the pump can be described via a pump efficiency according to: Wpump 

m   h2s  h1 

pump

,

(1)

where h2s is the enthalpy of CO2 following (ideal) isentropic pressurization in the pump from State 1, h1 is the actual enthalpy of saturated CO2 at the pump inlet, and ηpump is the pump isentropic efficiency. Process 2 to 3 in the preheater corresponds to heat recovery from the jacket-water stream: Qpreh  m   h3  h2  =m jw  cp jw  Tjw,in  Tjw,out  ,

(2)

where ṁjw is the mass flow rate of jacket water that flows into the preheater, and which can be adjusted according to the preheating demand, such that the heat absorbed from the jacket water by the CO2 stream can vary with ṁjw. Process 3 to 3’ in the recuperator can be expressed as: Qrecup  m   h3  h3  =m   h5  h5  ,

(3)

Process 3’ to 4 in the main heater corresponds to heat recovery from the ICE exhaust gases: Qmain  m   h4  h3  =m gas  cpgas  Tgas,in  Tgas,out  ,

(4)

where Tgas,out is set to be >120 °C to avoid corrosion to the pipes and heat exchangers [66]. Similarly with the case of heat recovery from the jacket-water stream, the heat absorbed by the TCO2-cycle system from the ICE exhaust gases also varies with Tgas,out (and the mass flow rate of this stream). Process 4 to 5 in the turbine is where the mechanical power generated and can be expressed as:

9

WT  m   h4  h5s  T ,

(5)

where h5s is the enthalpy of CO2 following (ideal) isentropic expansion through the turbine from State 4, h4 is the actual enthalpy of superheated CO2 at the turbine inlet, and ηΤ is the turbine isentropic efficiency. In the conventional approach, the turbine isentropic efficiency is assumed to have a constant value in the cycle calculations even over a wide operating range. The corresponding flow procedure of this approach is shown in Fig. 3(a). Thermodynamic and economic criteria are typically taken as indicators for performance determination. The turbine’s design and efficiency are closely related to the cycle parameters, i.e., working fluid mass flow rate, turbine inlet and outlet conditions. On the other hand, the turbine efficiency also directly determines the power output and hence affects both thermodynamic and economic performance indicators. In this paper an improved approach is proposed, presented in Fig. 3(b), in which a comprehensive model that couples TCO2-cycle calculations with a radial-inflow turbine design model is presented. An assumed turbine efficiency is firstly provided to start the calculation and with the initial results, the turbine design model could deliver the preliminary design, within which the turbine efficiency is maximized by simultaneously optimizing several key aerodynamic and geometric parameters of this component. The predicted turbine efficiency, i.e., 1D turbine efficiency in this paper, is used to replace the assumed turbine efficiency and then iterations are employed to achieve the convergence between the cycle parameters and the turbine design. Finally, the net power output from the TCO2-cycle system is given by:

Wnet  WT  Wpump .

(6)

(a) Conventional approach with constant turbine efficiency.

10

(b) Improved approach coupling cycle calculation with preliminary turbine design. Figure 3. Modelling approaches for TCO2-cycle system performance assessments: (a) conventional approach, and (b) improved approach.

3.2. Preliminary design model of radial-inflow turbine The preliminary design model used to predict the performance of the radial-inflow turbine is based on the 1D mean-line method [45-47]. This model is integrated with the overall cycle calculation, which is used to optimize the turbine simultaneously with several key turbine parameters in order to maximize efficiency. The meridian flow channel of a radial-inflow turbine and the corresponding h-s diagram are shown in Fig. 4. Section 0 represents the initial state of the CO2 working fluid at the radial-inflow turbine (nozzle) inlet. In the nozzle, the working fluid flows from Section 0 (the turbine inlet) to Section 1 (the gap between the nozzle and the rotor), and expands as its enthalpy decreases and its velocity increases. Based on the turbine inlet conditions, the degree of reaction and the nozzle velocity coefficient, the parameters at the nozzle outlet can be obtained. Downstream in the rotor, the working fluid continues to expand and to flow from Section 1 (the gap) to Section 2 (the turbine outlet), causing the rotor to rotate and to produce power.

11

(a)

(b)

Figure 4. (a) Meridian channel, and (b) h-s diagram of a radial-inflow turbine.

Similar to an axial-flow turbine analysis, velocity triangles are used to express the velocity distributions and energy conversion in the rotor, with an example shown in Fig. 5.

Figure 5. Velocity triangles of a radial-inflow turbine.

The ideal velocity c0 represents the velocity after isentropic expansion through the radial-inflow turbine and is taken as the basis to define the relative velocities: C0  2  hs ,

(7)

u1 , C0

(8)

u1 

c1u  c1  cos1    1   cos1 , u2  u1 

(9)

D2  u1  D2 , D1

(10)

c2u  u2  w2  cos  2  u1  D2  w2  cos  2 .

(11)

12

The peripheral efficiency of the radial-inflow turbine is given by:

u 

u1  c1u  u2  c2u 2  u1  c1u  u2  c2u    2  u1  c1u  u2  c2u  . hs C02

(12)

Substituting Eq. (8) to (11) into Eq. (12), the peripheral efficiency can be calculated from: u  2u1  {  1    cos 1  D22  u1  D2   cos  2  1/ 2

   2  1     2  1    u1 cos 1  D22  u12  }  

.

(13)

It can be seen that the peripheral efficiency of a radial-inflow turbine is related to several parameters, including the nozzle velocity coefficient, rotor velocity coefficient, velocity ratio, reaction degree, wheel diameter ratio, absolute flow angle at rotor inlet and relative flow angle at rotor outlet. Table 2 lists the general ranges of these key parameters, which we have obtained from previous relevant publications [45,46,57-59], that have been used here for the design of the radial turbine.

Table 2. Range of key parameters for radial-inflow turbine design. Symbo l

Value

Nozzle velocity coefficient

φ

0.95

Rotor velocity coefficient

ψ

0.90

Velocity ratio

u1

0.50 – 0.70

Reaction degree

Ω

0.40 – 0.65

D2

0.35 – 0.60

Absolute flow angle at rotor inlet

α1

14 ° – 20 °

Relative flow angle at rotor outlet

β2

25 ° – 45 °

Parameter

Wheel diameter ratio

Another way of defining the peripheral efficiency of radial-inflow turbine is by capturing various losses: u  c12  c22  u12  u22  w22  w12  1   N   R   v ,

(14)

 1  where  N  1     1   2  indicates the flow loss in the nozzle,  R  w22   2  1 indicates the flow loss in  

the rotor blade passage, and  v  c22 indicates the leaving velocity loss, respectively. Friction and leakage losses, which are related to the working fluid properties and cycle operation conditions, are typically accounted for in the peripheral efficiency to obtain a practical turbine efficiency. The friction loss relates to the friction between working fluid and the wheel, and can be calculated from:

13

f 

u3 D2 f  1  1 , 6 1.36 10 m  hs v1

(15)

where f is generally taken to be equal to 4 for radial-inflow turbines. The leakage loss relates to the tip clearance between the rotor blade and shroud, and can be calculated from:   1.3  l   u   f  , m  l    0.05  0.31         ,  u f  lm   lm 

 lm

 lm

 0.05

,

(16)

 0.05

l1  l2 , 2

(17)

where δ is the tip clearance depending on the manufacturing technology, and l1 and l2 are the blade height at rotor inlet and outlet, respectively. Therefore, the efficiency of the radial-inflow turbine predicted by the preliminary design model (i.e., the 1-D turbine efficiency model) in this paper can be expressed as: T   u   f   l

(18)

The turbine efficiency is maximized by optimizing the design parameters within the range listed in Table 2. The interior-point algorithm fmincon (find minimum of constrained nonlinear multivariable function) [67] is chosen as the solver to maximize the turbine efficiency, which is the selected objective function, while a set of nonlinear constraints are also imposed that set limits to the geometric parameters of the turbine and manufacturing tolerances in practical applications. An optimal turbine design with the highest efficiency is achieved at each cycle operating condition.

3.3. Heat exchanger models Shell-and-tube heat exchangers are selected for the TCO2-cycle system under investigation, and the BellDelaware method [68] is used to calculate heat transfer coefficients (HTCs) and pressure drops. The shellside (heat source) HTC and pressure drop are given by:

 s  ji

cp,s Gs  s    Prs2/3   w 

0.14

jc jl jb js jr ,

  N Ps   N b  1 Rb Rl  2 1  tcw N tcc  

(19)  2 f s N tcc Gs2   Rb Rs  s  

 s     w 

0.14



 2  0.6 N tcw  Gw2 2 s

while the tube-side (CO2 working fluid) HTC is calculated from [69]: (1) For the single-phase region (Petukhov-Kirillov correlation): 14

N b Rl ,

(20)

t 



( f 8) Re Pr

di 12.7( f 8)0.5 (Pr 2/3  1)  1.07 

,

(21)

(2) For the supercritical region (Ptukhov-Krasnoshchekov-Protopopov correlation): t 



( f 8) Re Pr

di 12.7( f 8) (Pr 0.5

2/3

(

cp

 1)  1.07  cpbulk

)0.35 (

kbulk 0.33  bulk 0.11 ) ( ) , k wall  wall

(22)

(3) For the two-phase region (Chen correlation):  t  0.023

 di

Rel0.8 Prl0.4 F ,

(23)

and the pressure drop is calculated from: Pt 

4Z G2 1.5G2 f t LG2 Zt G2    Kc  Ke  Zt  t . 2 2di t 2 2

(24)

Finally, from the above HTCs, the overall HTC of heat exchanger is found from:

d  d 1 1 do 1    rft  o  w  o  rfs  , U  t di di w d m s

(25)

and the area of the heat exchanger is evaluated from: A

Q , U  T

(26)

where ΔT is the log mean temperature difference (LMTD) between the hot and cold side of the heat exchanger.

3.4. Cost models The module costing technique is used to calculate the bare module cost of each component [70], and the chemical engineering plant cost index (CEPCI) is used to obtain the capital cost of the system. From this, the specific investment cost (SIC) is calculated from Eq. (27) to Eq. (31): CBM  Cp0 FBM  Cp0  B1  B2 FM FP  ,

(27)

log  Cp0   K1  K 2 log  X   K 3 log  X   ,

(28)

log  Fp0   C1  C2 log  Pi   C3 log  Pi   ,

(29)

CEPCI 2017 , CEPCI 2001

(30)

2

2

Cost   CBM SIC 

Cost , Wn

(31)

where the coefficients for each component are summarized in Table 3, and we have used CEPCI2001 = 397.0 and CEPCI2017 = 567.5 [71] to convert to present cost values in Eq. (30), which are dimensionless numbers

15

employed to updating capital cost required to erect a power-cycle system from a past date to a later time.

Table 3. Coefficients used in the cost models for each ORC system component [70]. Component

K1, K 2, K 3

C1, C 2, C 3

B1, B 2

FM

FBM

Pump

K1 = 3.3892 K2 = 0.0536 K3 = 0.1538

C1 = -0.3935 C2 = 0.3957 C3 = -0.0023

B1 = 1.89 B2 = 1.35

1.0

/

Turbine

K1 = 2.2476 K2 = 1.4965 K3 = -0.1618

/

/

/

3.5

Heat exchanger

K1 = 4.3247 K2 = -0.3030 K3 = 0.1634

C1 = -0.0016 C2 = -0.0063 C3 = 0.0123

B1 = 1.63 B2 = 1.66

1.35

/

3.5. Study definition The main assumptions and other conditions employed in this paper as listed below: (1) ICE heat-source conditions are as specified in Table 1. (2) The temperature of exhaust gases cannot drop below a minimum value of 120 °C to avoid corrosion. (3) The condensation temperature of the TCO2-cycle system is set to 25 °C. (4) The pump efficiency and (constant) turbine efficiency are set to 0.80, which is a common value used for CO2 turbines in previous research [72-74]. In the case study of WHR from an ICE presented in this paper, the 1-D turbine efficiency predicted by the preliminary design model varies from 0.72 to 0.85 under various operating conditions, such that the value of 0.80 can be always achieved for different turbine inlet temperatures (see Fig. 8(a) in Section 4.1). (5) The pinch-point temperature differences for all heat exchangers are set to 6 °C. (6) All processes can be modelled as being in steady state. (7) All heat losses to the environment are neglected. (8) Mechanical (friction) and electrical (generator) losses are neglected as this paper focuses specifically on the influence of the turbine isentropic efficiency value on the overall system performance and on the performance differences between modelling approaches based on constant and 1-D turbine efficiency predictions. Identical trends are envisioned if these additional losses are adopted with a scaling factor equal to the multiple of the efficiencies, which is high (>90%) compared to the isentropic efficiency values. The isentropic efficiency results reported herein can be readily

16

multiplied by additional factors if the reader is interested in doing so, although the changes will be small and within the overall accuracy of the early-stage engineering-level predictions made here.

3.6. Model validation The TCO2-cycle model was developed using a set of in-house MATLAB codes and the properties of CO2 used in the model were acquired from NIST REFPROP [75]. Results reported in Ref. [41] and Ref. [76] were used to validate our models, with comparative results shown in Fig. 6 and Fig. 7. From these figures, it can be inferred that the results from both our thermodynamic and economic models are in good agreement with the references, which gives confidence in the validity of our models and provides an indication that these are accurate enough for the purposes of the engineering-level predictions reported in this work.

(a)

(b)

Figure 6. Validation of the present model against data from Ref. [41]: (a) net power output, and (b) electricity production cost (EPC).

17

(a)

(b)

Figure 7. Validation of the present model against data from Ref. [76]: (a) net power output, and (b) SIC.

4. Case study: Waste-heat recovery from an internal combustion engine 4.1. Turbine efficiency analysis In order to investigate the influence of key cycle parameters on the radial-inflow turbine design and its efficiency, we consider more closely the relations for the friction loss (Eq. (15)) and leakage loss (Eq. (16)). For the friction loss, we have from Eq. (15):

f 

u13 D12 u13  D12 f    ,   hs v1   hs  v1 m 1.36 106 m

(32)

 m  v  and D1  f 1/2 4 , Eq. (32) can be further simplified to: 1/ 2

and since u1  h

1/ 2 s

f 

hs

v2 , v1

(33)

which effectively states that the friction loss in a radial-inflow turbine are determined by the volume ratio of the working fluid at the turbine’s rotor outlet and inlet. Having considered the friction loss, we proceed to the leakage loss. According to Eq. (16) and Eq. (17), the leakage loss in radial-inflow turbine are directly related to the relevant tip clearance, which is defined as the absolute tip clearance divided by the average rotor blade height. The absolute tip clearance is determined by the manufacturing method used and is set to a value of 0.1 mm in this paper for all radial-inflow turbine designs. The average rotor blade height is determined by the through-flow area that is required to handle the volumetric flow rate of the working fluid. This needs to be carefully considered in turbines for CO2-cycle

18

systems since the working fluid has a relatively large density and low volumetric flow rate. The leakage loss in these components are sensitive to the system operating conditions. Figure 8 summarizes the 1-D turbine efficiencies predicted by the preliminary design model and the corresponding CO2 mass flow rate over a range of turbine inlet pressures and for different turbine inlet temperature conditions ranging from TT,in = 100 °C to TT,in = 400 °C. Except at the lowest pressures and temperatures, the turbine efficiency generally decreases at higher turbine inlet pressures due to the higher friction loss, which are directly related to the volume ratio of the working fluid. For the system at the lowest turbine inlet temperature of 100 °C, the turbine efficiency first increases as the turbine inlet pressure is increased from low values (10 MPa to ~15 MPa) and remains nearly constant in the high-pressure range (>15 MPa). The leakage loss, which is determined by the relative tip clearance, plays an important role in this situation. In can be seen from Fig. 8(b) that when TT,n = 100 °C, the mass flow rate of the working fluid (CO2) increases with the turbine inlet pressure, whereas for all other turbine inlet temperature conditions the mass flow rate decreases with the turbine inlet pressure. A larger cross-sectional flow area is required to accommodate the increased mass flow rate (and volumetric flow rate), which results in a requirement for increased rotor blade heights and, consequently, reduced leakage loss, which dominate and therefore strongly determine the turbine efficiency in the low pressure range. It can be also observed that at the high pressures, the effects of increased friction loss and decreased leakage loss are comparable thereby maintaining the turbine efficiency at a (near-) constant value. The 1-D turbine efficiency predicted by the preliminary design model varies from 0.72 to 0.85 for different conditions in Fig. 8(a), which confirms that turbine design and performance is significantly affected by wider system parameters. Note that these values bracket the turbine isentropic efficiency of 0.8, which is selected throughout this work as the fixed value employed for all system performance predictions based on a constant turbine efficiency.

19

(a)

(b)

Figure 8. (a) 1-D turbine efficiency predicted by preliminary design model, and (b) CO2 mass flow rate of TCO2-cycle system for various system conditions.

4.2. Constant and 1-D turbine efficiency performance comparison A comparison of the performance of a TCO2-cycle system with: (a) a constant turbine efficiency; and (b) a turbine efficiency predicted from the 1-D model has been performed in order to evaluate the influence of turbine efficiency variations on the performance of the whole TCO2-cycle system. Figure 9 shows net power output results for a system with a fixed turbine inlet temperature of TT,in = 100 °C. The net power output from the system is higher when using the 1-D turbine efficiency model than when using the constant turbine efficiency value of 0.8 in most of the range of turbine inlet pressures from 10 MPa to 20 MPa, except for the lowest pressure condition (10 MPa). This is linked closely to the results shown earlier concerning the turbine efficiency in Fig. 8(a) and the related discussion above this figure. The net power output increases and then decreases in both cases as the turbine inlet pressure is increased from 10 MPa to 20 MPa, with a maximum approximately in the middle of this range (~15 MPa). The TCO2-cycle system is predicted to generate a maximum net power output of 86.9 kW at a turbine inlet pressure of 14.5 MPa by the model that assumes a constant turbine efficiency of 0.80, whereas the employment of the 1-D turbine model leads to a prediction of maximum net power output of 97.5 kW, which is 12% higher, while the optimal turbine inlet pressure shifts to 15 MPa.

20

(a)

(b)

(c) Figure 9. Thermo-economic results of TCO2-cycle system with TT,in = 100 °C: (a) net power output, (b) SIC, and (c) thermal load in various HTXs associated with heat addition (with 1-D turbine efficiency). These trends are also reflected in the SIC results shown in Fig. 9(b), although the turbine inlet pressure that leads to best (lowest) SIC shifts to lower pressures due to the higher costs associated with components that operate at higher pressure conditions. Specifically, a turbine inlet pressure of 12-12.5 MPa is preferred for the system from a SIC perspective, for both turbine models. It is worth highlighting that predictions of the economic benefits of system are also dependent on the turbine model. A comparison of the minimum SIC of the TCO2-cycle system predicted when using the constant turbine efficiency is overestimated by 7% relative to the SIC predicted by the 1-D turbine design model. The maximum difference between the system SIC predicted by the conventional and improved approaches reaches up to 22% when the turbine inlet pressure is 20 MPa.

21

Variations in the thermal load of the various heat exchangers (HTXs) of the system associated with heat addition are shown in Fig. 9(c). The thermal load in the preheater (WHR from jacket water) and main heater (WHR from exhaust gases) both correspond to the maximum heat recovery that can be achieved from the two heat-source streams, i.e., jacket water and exhaust gases. Turning to the recuperator, we note that when the turbine inlet pressure is higher than 11 MPa, the outlet temperature of the CO2 flow from the turbine is lower than that at the preheater outlet after being heated from the jacket water, hence recuperation is restricted; see illustration in Fig. 10, which shows T-s diagrams of the TCO2-cycle system with turbine inlet pressures of PT,in = 10 MPa and PT,in = 15 MPa. The net power output from the system with and without a recuperator is shown in Fig. 11, which confirms that the recuperation process is active only at lower pressure ratios in the investigated TCO2-cycle system and for the conditions that are explored in the present work (as defined in Table 1, Table 2 and Section 3.5).

(a)

(b)

Figure 10. T-s diagrams of TCO2-cycle system with TT,in = 100 °C: (a) PT,in = 10 MPa, and (b) PT,in = 15 MPa.

22

Figure 11. Net power output of TCO2-cycle system with/without recuperation for TT,in = 100 °C.

We proceed now to consider equivalent results but at a higher turbine inlet temperature of TT,in = 200 °C. It can be seen from Fig. 12(a) that the net power output variation with the turbine inlet pressure is similar to the TT,in = 100 °C case, and again, the analysis with a constant turbine efficiency underestimates the system’s thermodynamic performance except at the highest pressure condition (20 MPa) where the turbine efficiency is limited by the relatively large friction loss. The maximum net power output of the TCO2-cycle system predicted with a constant turbine efficiency is 149 kW, whereas the maximum power output predicted with the 1-D turbine model is 158 kW, which is 6% higher. The net power output decreases when the turbine inlet pressure is higher than 14.5 MPa, due to the reduction in the heat that is recovered from the jacket-water stream in the preheater, which cannot be entirely utilized, as shown in Fig. 12(c); this is caused by the decreasing temperature difference inside the preheater as the CO2 temperature at the preheater inlet increases with the turbine inlet pressure.

(a)

(b) 23

(c) Figure 12. Thermo-economic results of TCO2-cycle system with TT,in = 200 °C: (a) net power output, (b) SIC, and (c) thermal load in various HTXs associated with heat addition (with 1-D turbine efficiency).

The above effect is also affected by the significant change in the heat capacity of CO2, which is highlighted here as one of the special characteristics of CO2 that establishes this as a particularly suitable working fluid for combined WHR from ICE jacket-water and exhaust-gas streams [43]. Figure 13 shows the specific heat capacity (cp) of CO2, which increases significantly at low-temperature and low-pressure conditions, but is smaller and relatively constant at higher temperatures and pressures. Because the heat gained by the jacket-water stream accounts for about 50% of the total heat addition to the cycle, but is associated with only a narrow temperature drop in temperature from 89 °C to 78 °C (relative to that experienced by the exhaust-gas stream), a larger cp in the preheater is preferable for achieving an improved thermal match and sufficient heat utilization from both the jacket water in the preheater and the exhaust gases in the main heater. (N.B.: The working-fluid flow rate is the same in these two HTXs.) The heat recovered from the exhaust gases in the main heater remains at a constant value, which amounts to the maximum amount available from this heat source for all turbine inlet pressures. The SIC results in Fig. 12(b) show that a turbine inlet pressure of 13 MPa is optimal in terms of thermoeconomic performance in both cases. The 1-D turbine model leads to a system that delivers a lower cost per unit power output, due to the higher turbine efficiency. Specifically, the lowest TCO2-cycle system SIC predicted with the constant turbine efficiency amounts to 4500 $/kW and that predicted with the 1-D turbine efficiency amounts to 4280 $/kW. An overestimation of the system SIC by of up to 5% can arise in the system economic analysis of the present application if a constant turbine efficiency value is used. The optimal pressure 24

here is lower than the 14.5 bar value that we identified in Fig. 12(a) for maximum power, which would be expected since the cost of the components in the systems under consideration increases at higher pressures, which penalises higher-pressure systems and shifts the optimum to a lower pressure.

Figure 13. Variations in the specific heat capacity of CO2 at different pressure conditions.

Figure 14 shows results for a system with a turbine inlet temperature of TT,in = 300 °C. The net power output predicted with the constant efficiency model increases with the turbine inlet pressure and then remains nearly constant for turbine inlet pressures above 13 MPa, while that predicted with the 1-D turbine model increases first but then decreases in the high pressure range (above 13 MPa) due to the reduced turbine efficiency employed in the cycle calculations, which is related to the large friction loss and also operational (cycle) parameters that limit the efficient design of the turbine. The assumption of a constant turbine efficiency underestimates the system power output in the low-pressure range (<17 MPa) but overestimates the system power output at higher turbine inlet pressures (>17 MPa). These observations in the thermodynamic performance of the system, in terms of the effects of pressure and the choice of the turbine model, are reflected when considering the SIC of the system. As before with the power output, the thermal energy in the jacket-water and exhaust-gas streams cannot be fully recovered at higher pressures, as shown in Fig. 14(c). Similarly to the above explanations, this is caused by the small cp of CO2 at high temperature and pressure conditions and the high temperature at the recuperator outlet that leads to insufficient heat recovery from the two heat sources.

25

(a)

(b)

(c) Figure 14. Thermo-economic results of TCO2-cycle system with TT,in = 300 °C: (a) net power output, (b) SIC, and (c) thermal load in various HTXs associated with heat addition (with 1-D turbine efficiency).

Figure 15 shows at with the highest turbine inlet temperature of TT,in = 400 °C, the performance differences between the two cases are the largest, suggesting that the turbine efficiency predicted by the design model deviates significantly from that with the constant value model. Based on the net power output shown in Fig. 15(a), the system model with the constant turbine efficiency delivers a maximum of 151 kW, which is 13% higher than that predicted with the 1-D turbine efficiency 134 kW, and we also note that even the turbine inlet pressure that corresponds to these two optimal systems is different. In fact, the constant efficiency model does not show a maximum in this plot. The optimal turbine inlet pressure with the constant turbine efficiency is 20MPa, but only 16 MPa with the 1-D turbine efficiency. This deviation in predictions has important

26

implications for design, as higher pressures significantly increase the sealing difficulty and, therefore, the related cost of the system, which is crucial for this technology’s uptake in most practical applications. The largest overestimation in power output at the same turbine inlet pressure reaches 14% at a pressure of 20 MPa.

(a)

(b)

(c) Figure 15. Thermo-economic results of TCO2-cycle system with TT,in = 400 °C: (a) net power output, (b) SIC, and (c) thermal load in various HTXs associated with heat addition (with 1-D turbine efficiency).

Based on the SIC of the system, a similar conclusion can be drawn that low pressure is preferable when the turbine preliminary design is considered in the system analysis. In the case of the constant turbine efficiency, the model yields a lowest SIC of 4340 $/kW at a turbine inlet pressure of 20 MPa, whereas in the case of the 1-D turbine efficiency the model reaches a lowest SIC of 4470 $/kW at a different turbine inlet

27

pressure of 14 MPa. As with the previous cases at lower TT,in, it is found here that the optimal system from a thermo-economic perspective operates at lower pressures than that identified purely from thermodynamic considerations, e.g., the generated power output here, due to the escalation of component costs at higher pressures. Overall, by comparing the results from both cases, the approach based on a constant turbine efficiency appears to simultaneously overestimate the optimal thermodynamic and economic performance of the system. Furthermore, when accounting for the preliminary design of the radial-inflow turbine, the optimal cycle (and system operating conditions) differ significantly from those obtained with the conventional approach with a constant turbine isentropic efficiency value. The variations in the thermal load of the various HTXs in Fig. 15(c) demonstrate that neither the thermal energy from the jacket-water nor the exhaust-gas stream can be entirely recovered, as was the case in the high-pressure range in Fig. 14(c).

4.3. 2-dimensional system performance maps A systematic parametric analysis of the TCO2-cycle system is conducted and results are compared here between a system model with a constant turbine efficiency and one with a 1-D turbine efficiency model, considering the full spectrum of possible turbine inlet conditions. The aim in this section is to go beyond the detailed understanding from Section 4.2 for targeted conditions towards a more complete determination of the optimal operating point based on both thermodynamic and economic indicators over a range of turbine inlet conditions that would be expected in relevant applications. The 2-dimensional system performance maps are generated and shown in Fig. 16 and Fig. 17.

(a)

(b)

Figure 16. Net power output of TCO2-cycle systems with: (a) constant turbine efficiency, and (b) 1-D turbine efficiency for the investigated case study of WHR from an ICE.

28

(a)

(b)

Figure 17. SIC of TCO2-cycle systems with: (a) constant turbine efficiency, and (b) 1-D turbine efficiency for the investigated case study of WHR from an ICE.

When assuming a constant turbine efficiency, the TCO2-cycle system model yields a maximum net power output at 166 kW within the investigated map, at conditions that correspond to a turbine inlet temperature and pressure approximately equal to 265 °C and 14 MPa. The system model with the turbine efficiency predicted from 1-D calculations, on the other hand, shows a 5% higher maximum power output of 175 kW at the operating condition of TT,in = 265 °C and PT,in = 13.5 MPa. The regions enclosed by red lines in Fig. 16(a), i.e., temperature range of 250 °C to 350 °C and pressure range of 14 MPa to 18 MPa, are optimal operating regions, in terms of the turbine inlet conditions, within which the TCO2-cycle system delivers a net power output that is within 3% of the overall optimal point. TCO2-cycle systems operating in this region could improve the ICE power output by close to 14%. Of particular interest is the observation that the model with the 1-D turbine efficiency leads to an optimal operating region, which is much more confined in its range and shifted to lower pressure conditions (from 13 MPa to 15 MPa) but similar temperature conditions relative to the model with the constant turbine efficiency. This arises due to the fact that lower pressure conditions are preferable for radial-inflow turbine design to achieve higher efficiencies due to the reduced friction loss. TCO2-cycle systems operating within the region indicated in Fig. 16(b) with the more detailed turbine model also have a power output that is approximately 5% higher than that in the region indicated in Fig. 16(a) with the constant turbine efficiency. The results in Fig. 17 demonstrate that lower-pressure conditions are also preferable for TCO2-cycle system design from a thermo-economic perspective. The lowest system SIC with a constant turbine efficiency 29

is 4070 $/kW, whereas with a 1-D turbine efficiency this decreases to 3920 $/kW. The conventional approach underestimates the system’s economic performance by up to 4%, but even more importantly the region of optimal operation again appears much smaller for the 1-D model in Fig. 17(b), with an extend in this figure that is about half that in Fig. 17(a). Coupling the turbine preliminary design and turbine performance predictions with the overall system analysis leads to a more reasonable and realistic cost expectation from the TCO2-cycle systems and this is of great importance for enabling these systems to be more competitive when comparing with other power generation technologies in the early-stage of system design.

5. Further discussion – analysis of a general heat source All previous results relate to a specific application of WHR from an ICE, with heat source conditions defined in Table 1 and Section 3.5. Here, we present results from a more generalized heat source with a temperature ranging from 300 °C to 500 °C and mass flow rate ranging from 10 kg/s to 20 kg/s, in order to further understand the role of turbine efficiency modelling in determining TCO2-cycle system performance. The turbine inlet temperature, pressure and outlet temperature of the heat source are all optimized to achieve the maximum system net power output and lowest system SIC, and results generated with a constant turbine efficiency and the 1-D turbine efficiency model are compared in Fig. 18 and Fig. 19 for power output and SIC, respectively. Since a higher turbine efficiency can be achieved by considering the design of the radial-inflow turbine, the predictions from with 1-D turbine model yields both a better thermodynamic and a better economic performance in these results. In more detail, Fig. 20 indicates an average underestimation of 7% in the net power output and SIC when a constant turbine efficiency is employed in the system analysis over the full range of heat sources considered in these maps, however, maximum deviations in prediction closer to 10% can be observed from both thermodynamic and economic perspectives, which are also of greatest practical and financial interest. In summary, coupling the turbine design model in cycle calculations and replacing a constant turbine efficiency by 1-D turbine efficiency models allows more reasonable and reliable performance predictions, and enables the system analysis to avoid errors of the order of up to 10% in these predictions even for steady-state calculations. Of note is the fact that these errors are expected to be even more severe at offdesign operation due to variations in the external ICE (or any other heat-source) conditions, since a fixed, constant efficiency is unable to capture any deterioration in the performance of the turbine and, therefore, of the wider system as the conditions change from the system’s design point.

30

(a)

(b)

Figure 18. Net power output of TCO2-cycle system with: (a) constant turbine efficiency, and (b) 1-D turbine efficiency for power generation from a generalized heat source (within a range of temperatures from 300 to 500 °C and a range of mass flow rates from 10 to 20 kg/s).

(a)

(b)

Figure 19. SIC of TCO2-cycle system with: (a) constant turbine efficiency, and (b) 1-D turbine efficiency for power generation from a generalized heat source (within a range of temperatures from 300 to 500 °C and a range of mass flow rates from 10 to 20 kg/s).

31

(a)

(b)

Figure 20. Performance deviation of TCO2-cycle system with constant turbine efficiency based on 1-D turbine efficiency for power generation from a generalized heat source (within a range of temperatures from 300 to 500 °C and a range of mass flow rates from 10 to 20 kg/s): (a) net power put, and (b) system SIC.

6. Conclusions This paper presents a comprehensive model that couples TCO2-cycle calculations with a model for the preliminary design of radial-inflow turbines, in which the turbine design parameters are optimized simultaneously with the overall system in order to achieve the best overall system performance, replacing the conventional approach of using constant turbine efficiency values in cycle calculations. It is found that the use of a constant turbine efficiency can lead to a significant underestimation or overestimation the system thermo-economic performance and also to indications of suboptimal operating conditions. Predictions with the 1-D turbine efficiency model yield a highest net power output of 175 kW under the operating condition of TT,in = 265 °C and PT,in = 13.5 MPa, which is 5% higher than that predicted with a constant turbine efficiency at TT,in = 265 °C and PT,in = 14 MPa. Lower-pressure conditions are preferable for the system since efficient turbine design can be achieved due to smaller friction loss, but also because the costs become progressively higher at elevated pressures. With respect to the system cost, the coupling of the turbine preliminary design leads to a prediction of a lowest TCO2-cycle system SIC of 3920 $/kW, which is 4% lower than that with constant turbine efficiency. This makes the TCO2-cycle system performance more attractive when comparing against alternative power generation technologies. Considering the full spectrum of possible turbine inlet conditions of the TCO2-cycle system, maximum deviations of 14% and 22% are observed between the two approaches for thermodynamic and economic performance, respectively. 32

A more generalized heat source with temperature ranging from 300 °C to 500 °C and mass flow rate ranging from 10 kg/s to 20 kg/s is selected to further explore the role of turbine design and efficiency predictions in determining overall TCO2-cycle system performance. The results show that employing a 1-D turbine efficiency model yields both better thermodynamic and economic performance since higher turbine efficiency can be achieved by considering the preliminary design of the radial-inflow turbine; underestimations close to 10% in both the net power output SIC are noted when the constant turbine efficiency is employed in the system analysis for the overall range of investigated heat sources. It can be concluded that the design of a radial-inflow turbine is influenced by the cycle parameters and that the turbine efficiency directly affects the system’s thermodynamic and economic performance. It is important to take the turbine’s preliminary design and performance predictions into consideration when performing system analysis and optimization. The model presented in this paper allows preliminary designs of radial-inflow turbines to be proposed, along with more reasonable and reliable turbine performance predictions under various operating conditions, and can be a useful tool for the more accurate and reliable thermo-economic analysis and optimization of TCO2-cycle systems. This approach allows errors of the order of up to 10-20% in various predictions to be avoided for steady-state calculations, and potentially of an even greater magnitude at off-design operation.

Acknowledgement This work was supported by the UK Engineering and Physical Sciences Research Council [grant number EP/P004709/1]. The authors would also like to thank the Zijing Scholarship from Tsinghua University that supported Jian Song. Data supporting this publication can be obtained on request from [email protected].

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Conflict of Interest The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be constructed as a potential conflict of interest.

Highlights: 

A model that couples TCO2-cycle analysis with turbine design is presented



The turbine design parameters are optimized to achieve the highest efficiency



The predicted turbine efficiency replaces a constant value to reveal differences



The coupling between system and turbine performance is investigated



Errors up to 20% in various predictions can be avoided in steady-state calculations

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