Sensitivity Analysis of Transcritical CO₂ Cycle Performance Regarding Isentropic Efficiencies of Turbomachinery for Low Temperature Heat Sources

Abstract

The transcritical CO₂ (T-CO₂) power cycle using low temperature waste heat is a promising technique for energy saving and environmental protection. However, according to the literature, there is no commercialized unit in service yet. This study provides developers a reference to shorten the design phase of the T-CO₂ cycle commercialization process. A sensitivity analysis of the system performance, i.e., thermal efficiency and net power output, regarding the isentropic efficiencies of pump (${\eta }_p$) and expander (${\eta }_e$) and the heat source temperature ($T_{h,in}$) has been carried out using MATLAB and NIST REFPROP database. Simple and recuperative configurations are investigated based on their own optimal working pressures. The results show that the enhancement of ${\eta }_e$ has a greater influence on improving the system performance, but the improvement will diminish as ${\eta }_p$, ${\eta }_e$, and $T_{h,in}$ increase. Although better system performance can be achieved with higher ${\eta }_p$, ${\eta }_e$, and $T_{h,in}$, the cost of the system equipment will also increase due to the higher optimal working pressure. In addition, increasing ${\eta }_p$ and ${\eta }_e$ will negatively affect the effectiveness of the recuperator. Therefore, the turbomachinery efficiencies and the heat source temperature should be considered simultaneously for the most cost-effective system design.

1. Introduction

The global energy demand has grown considerably in recent years due to the population explosion and increasing industrial activities. Fossil fuels, the primary energy source nowadays, are no longer sustainable because of their limited amount. Burning fossil fuels also leads to undesirable effects, such as air pollution and global warming, etc. [1,2]. Therefore, seeking alternatives for power generation to mitigate the dependence on fossil fuels and relieve the environmental impacts is an urgent task around the globe. As such, much attention has been given to renewable energies. However, renewable energies are usually limited by their availability and cannot be used for providing base load power [3,4]. Using a heat-to-power system to recover the industrial waste heats, especially at low temperature levels, is another feasible way to reduce the fossil fuel consumption.

It is reported that approximately 50% of energy in industrial processes is released to the environment in the form of low grade heat without further usage, causing thermal pollution [5]. In Europe, approximately 100 TWh of potentially recoverable thermal energy, with the temperature range between 100 and 200 °C, has been wasted annually through industrial processes [6]. Generally, the waste heat rejected by conventional fossil fuel power plants also lies within this temperature range [7]. If these industrial waste heats can be properly re-utilized, it not only fulfills the goal of sustainable energy supply and the reduction of fossil fuel consumption, but also possesses environmentally friendly merits at the same time [8,9,10,11].

Steam-based power plants cannot efficiently utilize such low temperature heat sources due to the high evaporation temperature of water [8,12]. Organic Rankine cycles (ORC) have been widely adopted in geothermal fields and industrial waste heat recovery systems with low-to-medium temperature heat sources [11,12,13,14,15,16,17]. However, the inherent high global warming potential (GWP) of the organic fluids that are commonly used in existing ORCs (i.e., R134a, R245fa) have imposed uncertainties on their future development [2,18]. Some of the organic fluids are flammable, toxic, or chemically unstable at a certain temperature level [15], which are not favorable features for the working fluid in power units. In addition, most of the ORCs are operated as subcritical cycles [15], the performances of which are limited by the constant temperature evaporation process. The unmatched temperature profile between the working fluid and the heat source in subcritical ORCs results in a so-called pinching problem and the irreversibility in the evaporator is consequently increased [19,20,21,22].

Transcritical CO₂ (T-CO₂) power cycles have been drawing attention in low temperature heat recovery in the past decade (the term “transcritical” means the high-pressure side of the cycle is above the critical point while the low-pressure side is below it). In a transcritical cycle, the working fluid temperature during the heating process is continuously rising, thus providing a well-matched temperature profile with the sensible heat sources [19,20,21,22,23]. Therefore, the T-CO₂ cycles can make better use of the thermal energy from the heat sources [1,2]. Compared to the fully supercritical CO₂ cycles, less power consumption for pressurizing is needed in T-CO₂ cycles attributed to the high density of the condensed CO₂ [4,15,24,25]. Studies show that the T-CO₂ cycles have the potential to outperform ORCs and supercritical CO₂ cycles in recovering low temperature heat sources [23,25].

As for the working fluid, CO₂ possesses low environmental impact and potentially high energy performance [4,22,26]. CO₂ has zero ozone depletion potential (ODP) and the GWP of CO₂ is 1 over 100 years [18]. The low critical temperature of CO₂ (31.0 °C) make it easily surpass the critical point by low temperature heat sources, and it has been verified to work stably in the transcritical region [24]. Velez et al. [21] showed that the T-CO₂ cycle can convert heat into power with the expander inlet temperature as low as 60 °C. CO₂ being a natural fluid is abundant in the nature, easily accessible and less expensive than organic fluids [7,9,10,23]. In addition, CO₂ is non-toxic, non-flammable, non-explosive, less corrosive than steam, and compatible with most of the materials [10,23,27,28]. Furthermore, the thermodynamic properties of CO₂ in the working range of interest are well known, even in supercritical state [27]. These characteristics make CO₂ an attractive working fluid for low temperature heat-to-power units.

Chen et al. [1] indicated that the T-CO₂ cycle performs slightly better than the R123-ORC based on the same heat rejection condition. In another study that aimed to recover the 200–350 °C exhaust from vehicles [29], the authors found that the recuperative T-CO₂ cycle with low working pressure has higher thermal efficiency compared to the CO₂ Brayton cycles. Li et al. [2] compared a T-CO₂ cycle and a R245fa-ORC in recovering 120–260 °C low grade heat. Although R245fa-ORC presents higher energy and exergy efficiencies, T-CO₂ cycle is more promising in the future because of its negligible environmental impacts. They also pointed out that the installation of a recuperator is recommended for both cycles in most cases. Amini et al. [7] conducted a practical study of using T-CO₂ cycle to recover the 150 °C residual heat from a power plant in Iran. Taking the economic considerations into account, the T-CO₂ cycle can generate 0.9% of the power plant capacity (4.04 MW) and the overall efficiency can be increased by 0.4%. Lecompte et al. [10] reviewed the experimental results of supercritical and transcritical heat recovery systems. CO₂ is a promising working fluid theoretically for close-loop transcritical cycles. However, poor test results were obtained owing to the non-optimized turbomachinery. Cayer et al. [27,30] performed two consecutive studies on T-CO₂ cycle with an industrial waste heat also at 100 °C. By varying the working pressure while the maximal and minimal temperatures in the cycle are fixed, optimal working pressures can be found with respect to each objective [27]. T-CO₂ cycle has advantages in heat transfer characteristics but falls short in energy efficiency and power output compared to R125 and ethane ORCs, respectively. However, it is also suggested that no fluid can outperform the others in all aspects [30]. Baik et al. [20] compared two transcritical cycles using CO₂ and R125, respectively, in terms of power output with heat source temperature at 100 °C. Results showed that higher expander inlet temperature can be obtained with CO₂ due to its better heat transfer characteristic. Even though the relatively high working pressure of the T-CO₂ cycle results in larger pumping power consumption and its net power output is 14% lower than that of the R125 cycle consequently, R125 is no longer acceptable owing to its high GWP [23]. Yoon et al. [31] compared a T-CO₂ cycle and a toluene-ORC as the bottoming cycle of a Capstone C30 micro gas turbine with exhaust gas at 276 °C. The T-CO₂ cycle performs better than the toluene-ORC in certain situations since it has no phase change and no pinching problem during the heating process. H. Yamaguchi et al. [24] established and tested a prototype of solar-powered T-CO₂ cycle, where 11% of thermal efficiency was acclaimed with the expander inlet temperature at 165 °C and pressure ratio of about 1.4. However, the expansion process was via an expansion valve instead of a real expander.

Despite the aforementioned advantages of T-CO₂ cycle in recovering low temperature heat sources, there is no commercialized unit in service yet according to literature. Among the main components in T-CO₂ cycles, the performances of pump and expander can greatly influence the overall system performance [7,29]. Both of them are still in developing phase and neither has been commercially manufactured yet [26,32,33]. However, most of the previous studies on T-CO₂ cycles were based on constantly fixed isentropic efficiencies of pump (${\eta }_p$) and expander (${\eta }_e$) (the commonly assumed values of ${\eta }_p$ and ${\eta }_e$ in literatures are collected in

Table 1. Commonly assumed values of ${\eta }_p$ and ${\eta }_e$ in literatures.

Literature	${\mathit{\eta }}_{\mathit{p}}$ (%)	${\mathit{\eta }}_{\mathit{e}}$ (%)	Literature	${\mathit{\eta }}_{\mathit{p}}$ (%)	${\mathit{\eta }}_{\mathit{e}}$ (%)
Wang et al. [9]	70	75	Salah et al. [33]	70	80
Velez et al. [21]	75	75	Chen et al. [1] Amini et al. [7] Chen et al. [29]	80	70
Cayer et al. [27,30] Baik et al. [20] Tuo [22] Kim et al. [34,35]	80	80	Li et al. [2] Bellos et al. [4] Yoon et al. [31]	85	85
Kim et al. [25]	90	90

). Those results are only applicable for specific ${\eta }_p$ and ${\eta }_e$, but they may not have the expected values in real cases. Therefore, a more comprehensive analysis of the T-CO₂ cycle performance regarding ${\eta }_p$ and ${\eta }_e$ was conducted in this study.

This study aims at providing a clear view of how T-CO₂ cycles will perform under various scenarios of ${\eta }_p$ and ${\eta }_e$ with low temperature heat sources. Developers can take it as a reference in designing the T-CO₂ cycles to shorten the design phase of the commercialization process. To this end, a sensitivity analysis of T-CO₂ cycles regarding ${\eta }_p$ and ${\eta }_e$ (65–95%) has been carried out at low temperature levels (120–180 °C). A similar study has been conducted by D. Novales et al. [32] on supercritical CO₂ cycles with a high heat source temperature of 700 °C, but very few studies can be found focusing on the current study. The configurations investigated in this study are simple and recuperative cycles, since they are more preferable for low temperature heat sources according to the literature [26].

2. Cycle Configurations

The working principle of the T-CO₂ cycles is illustrated by the temperature-entropy (T-s) and pressure-volume (P-v) diagrams shown in Figure 1 and Figure 2, respectively. The cycle mainly consists of turbomachinery (Pump, Expander) and heat exchangers (Heater, Condenser, Recuperator). The subcritical working fluid is pressurized over critical pressure through the Pump (1-2) and isobarically heated by heat source in the Heater (2-3) to reach supercritical state. The supercritical working fluid is then expanded through the Expander (3-4) and the high pressure and high temperature energy could be converted into electricity via a coupled generator (the generator will not be discussed in this study). Finally, the exhaust from the expander is isobarically cooled down by heat sink in the Condenser (4-1). To this point, the cycle is completed and then recommences. The red and blue segments in the figures display the hot side and the cold side of the system, respectively. The yellow marks in Figure 1 indicate the locations of minimal temperature difference in heat exchangers (the pinch points).

A recuperator can be added to preheat the working fluid before entering the heater by recovering the residual heat from the expander exhaust. The recuperating process is presented by the green segments in the figures and the subscripts for the inlet of heater and the inlet of condenser in recuperative cycle are re-denoted as 2r and 4r, respectively. It should be mentioned here that, in Figure 1, the temperature profiles of the heat source and the heat sink are intentionally aligned with that of the working fluid in heater and condenser, respectively, so the entropy values on the x-axis refer to CO₂ only.

3. Methods

For simplicity, this study was carried out based on the following assumptions: (i) Saturated liquid state CO₂ was assumed at the pump inlet. (ii) The energy losses (heat loss, pressure loss, etc.) within piping and heat exchangers were neglected. (iii) All the fluids were steady-state flow throughout the process.

3.1. Energy Analysis

The correlations of energy analysis and the simulation process were established in MATLAB (R2019b) environment based on the first law of thermodynamics. The fluid properties were read from the NIST REFPROP database (v.9.1) [36] with ready-to-use formulas in MATLAB (Mathworks, Natick, MA, USA).

For pump and expander:

$${\eta }_p=\frac{h_{2,s}-h_1}{h_2-h_1}$$

(1)

$${\eta }_e=\frac{h_3-h_4}{h_3-h_{4,s}}$$

(2)

where ${\eta }_p$ and ${\eta }_e$ denote the isentropic efficiencies as mentioned. h is the enthalpy at each state point, where the subscript s suggests the isentropic process.

For heater and condenser:

$$q_h=\left(h_3-h_2\right)=\frac{{\dot{m}}_{air}}{{\dot{m}}_{{CO}_2}}\left(h_{h,in}-h_{h,out}\right)$$

(3)

$$q_c=\left(h_4-h_1\right)=\frac{{\dot{m}}_{water}}{{\dot{m}}_{{CO}_2}}\left(h_{c,in}-h_{c,out}\right)$$

(4)

where q is the heat transfer rate in each heat exchanger, where the subscripts h and c represent the heater and the condenser, respectively. $\dot{m}$ is the mass flow rate of the fluid. In the recuperative cycle, $h_2$ and $h_4$ are replaced by $h_{2r}$ and $h_{4r}$, respectively.

For recuperator:

$$q_{rec}=\left(h_{2r}-h_2\right)=\left(h_4-h_{4r}\right)$$

(5)

$${\epsilon }_{rec}=\frac{q_{rec}}{(h_4-h_{4r,min})}$$

(6)

where ${\epsilon }_{rec}$ is the effectiveness of the recuperator. $h_{4r,min}$ is defined as the theoretical minimal enthalpy of CO₂ at the recuperator hot side outlet when $T_{4r}$ equals to $T_2$ [2,25,34].

For the overall system:

$$w_{net}=\left(h_3-h_4\right)-\left(h_2-h_1\right)$$

(7)

$${\eta }_{th}=\frac{w_{net}}{q_h}$$

(8)

where $w_{net}$ is the specific net power output and ${\eta }_{th}$ represents the thermal efficiency of the cycle.

3.2. Simulation Process

The schematic of the simulation process is shown in Figure 3. Three cases of heat source temperatures ($T_{h,in}$), 120 °C, 150 °C, and 180 °C, were adopted to cover possible applications. Firstly, the fixed parameters shown in

Table 2. Fixed parameters.

Parameter	Value	Parameter	Value
$T_{h,in}$ (°C)	120/150/180	$\Delta T_{pp}$ (K)	5
$T_{h,out}$ (°C)	70	$P_0$ (MPa)	0.1
$T_{c,in}$ (°C)	15	$P_2$ (MPa)	8–30
$T_0$ (°C)	15	${\eta }_p$ (%)	65–95
$T_1$ (°C)	25	${\eta }_e$ (%)	65–95

were imported into the mathematical model established in MATLAB. The pump inlet temperature ($T_1$) was fixed at 25 °C and the corresponding saturated pressure ($P_1$) was 6.43 MPa. The ambient pressure ($P_0$) and temperature ($T_0$) were assumed at 0.1 MPa and 15 °C, respectively. The pump outlet pressure ($P_2$), which is also known as the working pressure in this study, was varied from 8 MPa to 30 MPa with a step increase of 0.02 MPa. The pinch point temperature differences ($\Delta T_{pp}$) in heat exchangers were all assumed at 5K [30,37]. The location of the pinch point in the heater was kept at its outlet or equivalently, the expander inlet, to obtain the theoretically highest expander inlet temperature ($T_3$) under the $\Delta T_{pp}$ restriction [11,17,21,31]. ${\eta }_p$ and ${\eta }_e$ were initially set at 65% and will be gradually increased to 95% afterwards. Secondly, as the CO₂ conditions at state point 1 and 3 are known, the pump outlet temperature ($T_2$) and the expander outlet temperature ($T_4$) can be derived using Equations (1) and (2). For the recuperative cycle, the pinch point in recuperator will occur at its hot side outlet due to the smaller specific heat capacity of CO₂ in low pressure region [25,34]. Therefore, the hot side outlet temperature of the recuperator ($T_{4r}$) was constantly 5K above $T_2$ and then, its cold side outlet temperature ($T_{2r}$) can be calculated using Equation (5). To this point, the conditions of CO₂ at each state point in both cycles under specific ${\eta }_p$, ${\eta }_e$, and $T_{h,in}$ were all acquired. Then, the optimal $P_2$ for both cycles can be found, respectively [2,27,37]. Finally, ${\eta }_p$ and ${\eta }_e$ were varied from 65% to 95% with a 5% step increase of one component at a time and the whole process above was repeated with different $T_{h,in}$. A system performance map related to ${\eta }_p$, ${\eta }_e$, and $T_{h,in}$ was established for the sensitivity analyses.

For the surroundings, hot air was selected as the heat source in this study and a lower limit of 70 °C was imposed on its outlet temperature ($T_{h,out}$) to prevent acid formation or fouling in the component and piping [7]. For convenience, $T_{h,out}$ was fixed at 70 °C since the heat recovery efficiency is not the focus in this study. Then, the mass flowrate of hot air (${\dot{m}}_{air}$) can be derived by energy balance in the heater. In order to provide readers a sense of the amount of heat source needed per unit mass of CO₂ (or vice versa), ${\dot{m}}_{air}$/${\dot{m}}_{{CO}_2}$ will be presented. The heat sink was cold water at the ambient temperature of 15 °C [9,15], and its mass flowrate (${\dot{m}}_{water}$) and outlet temperature ($T_{c,out}$) were calculated to meet the $\Delta T_{pp}$ restriction within 0.1 °C difference. However, as the water-cooling mechanism is relatively mature, only a brief description will be given to the condenser side.

3.3. Verification

The mathematical models of the simple cycle and the recuperative cycle in this study have been verified with references [20,34], respectively. It can be seen in

Table 3. Verification with references.

Parameter	Model	Ref [20]	Deviation (%)
${\eta }_{th}$ (%)	5.6	5.6	0
$w_{net}$ (kj/kg)	9.74	9.66	0.8
${\dot{m}}_{air}$/${\dot{m}}_{{CO}_2}$	0.848	0.833	1.8
Parameter	Model	Ref [34]	Deviation (%)
${\eta }_{th}$ (%)	29.7	28.7	3.5
$w_{net}$ (kj/kg)	94.0	96.9	−3
${\dot{m}}_{air}$/${\dot{m}}_{{CO}_2}$	0.847	0.832	1.8
${\epsilon }_{rec}$ (%)	94	90	4.4

that the main parameters derived from the mathematical models well agree with the literature, considering that not all the information required was revealed. Therefore, the accuracy of the simulation in this study is considered acceptable.

4. Results and Discussions

In this section, the sensitivity analyses of simple and recuperative T-CO₂ cycle performances regarding ${\eta }_p$ and ${\eta }_e$ at low temperature levels will be presented. The main performance indicators are ${\eta }_{th}$ and $w_{net}$ at the design point. The effectiveness of recuperator will be investigated as well. In addition, two mass flow rate ratios, i.e., the heat source to CO₂ and the heat sink to CO₂, under various operating conditions are provided for estimating the size of the heat exchangers.

4.1. Simple Cycle

4.1.1. Design Point Setting for the Simple Cycle

The system performance of the simple cycle regarding $P_2$ is demonstrated using a standard case with 80% of both ${\eta }_p$ and ${\eta }_e$ and 150 °C of $T_{h,in}$, since the other sets would show similar trend of results. It is clearly seen from Figure 4 that both ${\eta }_{th}$ and $w_{net}$ will first increase and then decrease as $P_2$ varies from 8 MPa to 30 MPa (the corresponding pressure ratio is about 1.2 to 4.7). This can be explained by the enthalpy variations at each sate point as shown in Figure 5. It is observed that the increment of enthalpy difference in the expander ($h_3-h_4$) becomes gradually smaller, while that in the pump ($h_2-h_1$) grows steadily as $P_2$ increases. Therefore, there exist peak values for $w_{net}$ and ${\eta }_{th}$ over the investigated range of $P_2$. In addition, since the enthalpy difference in the heater ($h_3-h_2$) is gradually decreased with the increase of $P_2$, the maximal thermal efficiency (${\eta }_{th,max}$) is found to occur at higher pressure region with regard to the maximal net power output ($w_{net,max}$) in the simple cycle [23].

Considering the ${\eta }_{th}$ and $w_{net}$ values and the corresponding $P_2$, the $w_{net,max}$ design criterion is selected for the simple cycle in the following analysis. In this way, more $w_{net}$ can be extracted from the heat source with lower $P_2$ and just a small penalty of ${\eta }_{th}$ loss compared to the ${\eta }_{th,max}$ design criterion. Studies also suggested that in such heat recovery systems with nearly-free heat sources, the focus should be on maximizing the power output [22,27,30,35].

4.1.2. Sensitivity Analysis of the Simple Cycle Performance Regarding ${\eta }_p$ and ${\eta }_e$

Figure 6 shows the ${\eta }_{th}$ and $w_{net}$ of the simple cycle at the design point under different circumstances of ${\eta }_p$, ${\eta }_e$, and $T_{h,in}$. It can be clearly noticed that both ${\eta }_{th}$ and $w_{net}$ will increase with ${\eta }_p$ and ${\eta }_e$ as expected. Furthermore, they are more sensitive to ${\eta }_e$ rather than ${\eta }_p$ as can be inferred from Figure 6. Taking the standard case for example, if ${\eta }_e$ is enhanced from 80% to 85 % and the other parameters are fixed, then ${\eta }_{th}$ and $w_{net}$ can be increased by 12% and 10.7%, respectively. On the other hand, if the variable becomes ${\eta }_p$ instead, then the increments will decrease to 5% for ${\eta }_{th}$ and 4.2% for $w_{net}$, respectively. It can also be found that $w_{net}$ is less sensitive to ${\eta }_p$ and ${\eta }_e$ compared to ${\eta }_{th}$. Thus, by setting the design point based on $w_{net}$, the system performance will be more stable as the ${\eta }_p$ and ${\eta }_e$ fluctuate during operation. The developers can also gain more flexibility in designing the components. A more detailed sensitivity of the system performance per 5% variation of ${\eta }_e$ is shown in Figure 7, since ${\eta }_e$ is more influential than ${\eta }_p$.

It is clearly seen from Figure 7 that the contribution of enhancing ${\eta }_e$ to the system performance is gradually reduced as ${\eta }_e$ increases. In addition, the benefits gained from enhancing ${\eta }_e$ are slightly decreased with higher ${\eta }_p$ and $T_{h,in}$ as well. Thus, compromises should be made between the investment for more refined turbomachinery and the profits gained from the system performance improvement. Based on the assumptions made in this study, the largest increments of ${\eta }_{th}$ and $w_{net}$ derived from a 5% enhancement of ${\eta }_e$ are 19.2% and 17.5% over the investigated range, respectively, both of which occur at the point when ${\eta }_e$ is increased from 65% to 70% with 65% of ${\eta }_p$ and 120 °C of $T_{h,in}$.

It is noteworthy that $P_2$ will increase with ${\eta }_p$ and ${\eta }_e$ and the increase becomes more pronounced with higher $T_{h,in}$, as shown in Figure 8. Since $P_2$ is strongly related to the total cost [7,30], the extra cost for the high-pressure equipment should be concerned when attempting to increase the system performance by utilizing higher temperature heat sources or pursuing higher turbomachinery efficiencies. It is also suggested that enhancing the ${\eta }_e$ in simple cycle is most profitable in recovering lower temperature heat sources, due to the larger increments of the system performance and the lower penalty of the additional cost resulting from the pressure increase.

4.2. Recuperative Cycle

4.2.1. Design Point Setting for the Recuperative Cycle

In contrast to the simple cycle, ${\eta }_{th,max}$ is achieved at a lower $P_2$ with respect to $w_{net,max}$ in the recuperative cycle as shown in Figure 9. This is because the enthalpy difference in the recuperator ($h_4-h_{4r}$ and $h_{2r}-h_2$, or equivalently, $q_{rec}$) is gradually reduced as $P_2$ increases, which is displayed in Figure 10. The lower performance of the recuperator at high pressure leads to a larger amount of heat energy required from the heat source. Consequently, ${\eta }_{net,max}$ occurs at lower pressure region (the vertical lines in Figure 9 and Figure 10 represent the point where $q_{rec}$ is reduced to zero).

It should be noted that the recuperating process mainly affects the heat transfer rate in the heat exchangers. The behavior of $w_{net}$ regarding $P_2$ in the recuperative cycle is basically the same as that in the simple cycle, while ${\eta }_{th}$ is generally higher since the heat energy required from the heat source is reduced. Thus, the ${\eta }_{th,max}$ design criterion is selected for the following analysis of the recuperative cycle, considering that higher ${\eta }_{net}$ is achievable with a lower $P_2$ and a negligible expense of $w_{net}$.

4.2.2. Sensitivity Analysis of the Recuperative Cycle Performance Regarding ${\eta }_p$ and ${\eta }_e$

Similar to Section 4.1.2, Figure 11 shows ${\eta }_{th}$ and $w_{net}$ of the recuperative cycle at the design point under different circumstances of ${\eta }_p$, ${\eta }_e$, and $T_{h,in}$. It is clear that the ${\eta }_{th}$ and $w_{net}$ of the recuperative are also more sensitive to ${\eta }_e$ rather than ${\eta }_p$. Based on the standard case, ${\eta }_{th}$ and $w_{net}$ can be increased by 8.9% and 10.7%, respectively, when ${\eta }_e$ varies from 80% to 85 %. While the increments are only 3.5% and 4.2% for ${\eta }_{th}$ and $w_{net}$, respectively, if the variable becomes ${\eta }_p$ instead. As such, the focus in the following paragraph will be on ${\eta }_e$ as well. It can also be found that, unlike in the simple cycle, ${\eta }_{th}$ responds less sensitively to the ${\eta }_p$ and ${\eta }_e$ variations than $w_{net}$ does. For the same reason as in Section 4.1.2, adopting the ${\eta }_{th,max}$ design criterion will be more beneficial for the recuperative cycle.

From Figure 12, the behaviors of ${\eta }_{th}$ and $w_{net}$ in recuperative cycle regarding ${\eta }_e$ show the same trend as that in the simple cycle. The variation of $w_{net}$ has no significant difference from the simple cycle despite the fact a different design criterion is adopted. Although the increment of ${\eta }_{th}$ contributed by enhancing ${\eta }_e$ is smaller compared to the counterparts in simple cycle, especially with higher $T_{h,in}$, the recuperative cycle generally performs better with higher $T_{h,in}$. In addition, it can be seen from Figure 13 that the advantage of lower $P_2$ of the recuperative cycle is more pronounced as $T_{h,in}$ increases.

In fact, it is not worthwhile to increase the ${\eta }_p$ and ${\eta }_e$ in the recuperative cycle if $T_{h,in}$ is too low. For example, at 120 °C of $T_{h,in}$ and 95% of both ${\eta }_p$ and ${\eta }_e$, the ${\eta }_{th}$ of recuperative cycle is 11.99%, which is even slightly lower than the 12.16% obtained in the simple cycle under the same conditions. This is because the low $T_{h,in}$ and high ${\eta }_p$ and ${\eta }_e$ will result in the low performance of the recuperator. In this situation, the recuperator does not help improving the system performance, so the system performance is inevitably inferior to the simple cycle if the design point is still set at a lower $P_2$.

4.2.3. Effectiveness of the Recuperator Regarding ${\eta }_p$ and ${\eta }_e$

From Figure 14, it can be seen that the sensitivity of ${\epsilon }_{rec}$ regarding ${\eta }_p$ and ${\eta }_e$ will be significantly affected by $T_{h,in}$ and the influence of ${\eta }_e$ on ${\epsilon }_{rec}$ is larger than that of ${\eta }_p$. In lower temperature environments, ${\epsilon }_{rec}$ varies more drastically with ${\eta }_p$ and ${\eta }_e$. As ${\eta }_p$ and ${\eta }_e$ increase, ${\epsilon }_{rec}$ will decrease due to the higher $T_{4r}$ and lower $T_4$ resulting from higher ${\eta }_p$ and ${\eta }_e$, respectively. In other words, pursuing the high turbomachinery efficiencies will adversely affect the recuperator performance, especially with lower $T_{h,in}$. Therefore, the influences of ${\eta }_p$, ${\eta }_e$, and $T_{h,in}$ should be considered simultaneously when evaluating the cost-effectiveness of the recuperative cycle.

4.3. Surrounding Conditions

4.3.1. Mass Flow Rate Ratio of Heat Source to CO₂

For the heat source, the average values of ${\dot{m}}_{air}$/${\dot{m}}_{{CO}_2}$ regarding ${\eta }_e$ are plotted in Figure 15 with the range covering all the conditions of ${\eta }_p$. The influences of ${\eta }_p$ and ${\eta }_e$ on ${\dot{m}}_{air}$/${\dot{m}}_{{CO}_2}$ are insignificant, while $T_{h,in}$ plays a decisive role on that in both cycles. In general, the pump has less effect, especially in the recuperative cycle.

In the simple cycle, ${\dot{m}}_{air}$/${\dot{m}}_{{CO}_2}$ is slightly decreased as ${\eta }_e$ increases. At all temperature levels investigated, ${\dot{m}}_{air}$/${\dot{m}}_{{CO}_2}$ decreases around 7% as ${\eta }_e$ increases from 65% to 95%. On the other hand, ${\dot{m}}_{air}$/${\dot{m}}_{{CO}_2}$ will increase significantly as $T_{h,in}$ decreases. As $T_{h,in}$ decreases from 180 °C to 150 °C, ${\dot{m}}_{air}$/${\dot{m}}_{{CO}_2}$ will increase around 23%, and an additional 41% will occur as $T_{h,in}$ further decreases to 120 °C.

In the recuperative cycle, it can be seen from Figure 15 that the ${\dot{m}}_{air}$/${\dot{m}}_{{CO}_2}$ is generally lower compared to the simple cycle, which means the size and the cost of the heater can be potentially reduced. It can also be noticed that, in contrary to the simple cycle, the ${\dot{m}}_{air}$/${\dot{m}}_{{CO}_2}$ in recuperative cycle will increase with ${\eta }_e$. As ${\eta }_e$ increases from 65% to 95%, ${\dot{m}}_{air}$/${\dot{m}}_{{CO}_2}$ increases around 8–12% due to the lower ${\epsilon }_{rec}$. The temperature influence on ${\dot{m}}_{air}$/${\dot{m}}_{{CO}_2}$ is more significant in the recuperative cycle. When $T_{h,in}$ decreases from 180 °C to 150 °C, ${\dot{m}}_{air}$/${\dot{m}}_{{CO}_2}$ increases around 30%, and another 50% increment occurs when $T_{h,in}$ further reduces to 120 °C. Therefore, the advantage of smaller heater size of recuperative cycle will be more pronounced with lower ${\eta }_e$ and higher $T_{h,in}$.

Since in general the mass flow rate can be directly related to the size of the heat exchangers, the heater size will almost double as $T_{h,in}$ decreases from 180 °C to 120 °C in both cycles. Although T-CO₂ cycle is promising in recovering the low temperature heat sources, the larger physical footprint and more material used will bring higher costs and should be considered when exploiting the low temperature heat sources by T-CO₂ cycles.

4.3.2. Mass Flow Rate Ratio of the Heat Sink to CO₂

For the heat sink, ${\dot{m}}_{water}$/${\dot{m}}_{{CO}_2}$ is almost not affected by ${\eta }_p$, ${\eta }_e$, and $T_{h,in}$. In the simple cycle, the required ${\dot{m}}_{water}$ is about six times ${\dot{m}}_{{CO}_2}$ and $T_{c,out}$ is at around 23.7 °C for most cases. In the recuperative cycle, the cooling demand is slightly lower. The required ${\dot{m}}_{water}$ is roughly 5.8 times ${\dot{m}}_{{CO}_2}$ and $T_{c,out}$ is at around 22.5 °C for most cases in the recuperative cycle.

5. Conclusions

The T-CO₂ cycle using low temperature heat sources is a promising alternative for power generation to mitigate fossil fuel consumption. However, the turbomachinery components for CO₂ in the working conditions of interest are still in developing phase. Therefore, a sensitivity analysis of T-CO₂ cycles regarding the turbomachinery efficiencies has been carried out.

Considering the optimal working pressure, the design point of the simple cycle is set to the value at which the maximal net power output occurs, while the maximal thermal efficiency is chosen as the design point for the recuperative cycle. In this way, the system performances will also be less sensitive to the turbomachinery efficiencies and the developers can gain more flexibility in designing the components.

In both cycles, the expander has greater influence on the system performances. As ${\eta }_p$, ${\eta }_e$, and $T_{h,in}$ increase, the benefits of enhancing ${\eta }_e$ will diminish. Besides, the cost may be higher because of the higher optimal working pressure. Therefore, compromises should be made between the investment for more refined turbomachinery and the resulted profits gained from the system performance improvement. In general, the simple cycle is more suitable for lower $T_{h,in}$, while the recuperative cycle performs better with higher $T_{h,in}$.

In the simple cycle, enhancing ${\eta }_e$ is more profitable with lower $T_{h,in}$, since its contribution to the system performance will be more pronounced and the cost resulting from the pressure increment is potentially lower.

In the recuperative cycle, the working pressure at the design point is lower than the simple cycle, especially with higher $T_{h,in}$. In addition, ${\epsilon }_{rec}$ is generally higher and less sensitive to the variations of ${\eta }_p$ and ${\eta }_e$ with higher $T_{h,in}$ as well. However, ${\epsilon }_{rec}$ will gradually decrease as ${\eta }_p$ and ${\eta }_e$ increase, so the influences of ${\eta }_p$, ${\eta }_e$, and $T_{h,in}$ should be considered simultaneously when evaluating the cost-effectiveness of the recuperative cycle.

For the surroundings, ${\dot{m}}_{air}$/${\dot{m}}_{{CO}_2}$ is slightly reduced as ${\eta }_e$ increases in the simple cycle, while it increases with ${\eta }_e$ in the recuperative cycle. The advantage of smaller heater size in recuperative cycle, due to the smaller ${\dot{m}}_{air}$/${\dot{m}}_{{CO}_2}$, is more pronounced with lower ${\eta }_e$. As $T_{h,in}$ decreases, ${\dot{m}}_{air}$/${\dot{m}}_{{CO}_2}$ will increase dramatically in both cycles. Therefore, although the T-CO₂ cycles are promising in recovering low temperature heat sources, one should be aware that the heater size will increase significantly if the target $T_{h,in}$ is low. On the other hand, ${\dot{m}}_{water}$/${\dot{m}}_{{CO}_2}$ is almost not affected by ${\eta }_p$, ${\eta }_e$, and $T_{h,in}$.

The importance of this study lies in providing future developers the interactions between system performance, turbomachinery efficiencies, and heat source temperatures in the T-CO₂ cycles. The developers can take it as a reference in designing the T-CO₂ cycles, and for shortening the design phase during its commercialization process.