Thermodynamic and Heat Transfer Performance of the Organic Triangle Cycle

Abstract

Compared to the organic Rankine cycle (ORC), the organic triangle cycle (TC) is simpler in structure and is not limited by pinch point temperature differences. TC has been studied to some extent by previous researchers, such as the selection of working fluid, application, and the design of the expander. However, system optimization and parameter analysis of TC are still rare. The thermodynamic performance of TC internal circulation and TC heat recovery systems are investigated by theoretical analysis and numerical simulation, respectively. The results indicate that the expander inlet temperature T₃ and heater inlet temperature T₂ are key elements impacting the thermodynamic performance of the TC internal circulation. For the TC heat recovery system, an optimal value of the average heat-capacity flow rate of working fluid C_wf is discovered to output the maximum net power output W_net. Moreover, the total heat transfer coefficients for the heater (kA)_h and condenser (kA)_c are discussed in relation to C_wf variations. The findings will provide critical guidance for system investment and optimization.

1. Introduction

In today’s world, the environment and development are both mutually constrained and interdependent. How to balance development and environment is a hot issue today. Researchers have studied the development and utilization of many clean, renewable energy sources, such as wind [1,2], solar [3,4], and geothermal [5,6] energy. Nowadays, geothermal water is mostly used for energy generation via direct heating, flash geothermal power generation, total flow geothermal power generation, binary cycle power generation, and total flow-binary cycle combined power generation [7,8]. Additionally, the organic triangle cycle (TC) [9] can be applied to recover geothermal energy, and it has gained much attention due to its advantages, such as its simple system and high exergy efficiency. Besides, TC is suitable for lower temperature heat sources compared to ORC, and TC has the potential to generate electricity even from heat sources below 80 °C in cases where ORC is not economically feasible [10,11]. As a result, several studies have been undertaken in this sector. Smith et al. [12,13,14] investigated the construction concept, use of mixed mixture working fluids, and two-phase expansion working principle of a Lysholm double screw expander, laying a solid platform for future TC research. Bryson [15] compared TC to the traditional double cycle and discovered that the TC had significant advantages when the expander adiabatic efficiency reached certain values. Steffen et al. [16] compared TC with ORC. The exergy efficiency of TC using water as working fluid was 35 to 70% higher than that of the supercritical ORC at heat source temperatures up to 450 K. Fischer et al. [17,18] compared the thermodynamic performances of single-stage TCs with water as the working fluid, the organic Rankine cycle (ORC), and other improved TCs. It was discovered that the efficiency of TC was always greater than that of ORC under the same conditions; however, water was not a suitable working fluid. Arbab et al. [19] compared the performance of ORC and TC power generation systems. The results showed that TC power generation system could utilize up to 70% of available power and had about 50% more power generation capability than ORC. Ajimotokan [20] conducted preliminary elections and intensive studies on the appropriate working fluid for the trilateral flash cycle and discovered that light hydrocarbons were more suitable as working fluids, and n-pentane was eventually chosen as the working fluid. The comparison of the thermodynamic performance of TC and three improved TCs was reported. The effects of expander inlet temperature and expander adiabatic efficiency on cycle performance were investigated. Antonopoulou et al. [21] simulated the trilateral flash cycle with Aspen Plus and compared the net output power, thermal efficiency, and exergy efficiency for different working fluids. The thermodynamic results showed that HFO-1234yf had the best power production and thermal efficiency at a heat source temperature of 90 °C, while HFC-245fa had a stronger exergy potential. Chang et al. [22] screened the working fluid from the perspective of the heat source and expander, finding that when the heat source temperature is close to the critical temperature of the working fluid, the system performance is better.

Moreover, geothermal energy thermodynamic cycles have been extensively reported in China [23,24]. Wang et al. [25] evaluated the thermo-economic performance of a dual-pressure ORC system powered by geothermal heat. A comparison of single and dual-pressure systems was performed. The effects of the annual loan interest rate, on-grid electricity price, and carbon tax on the system’s economic performance were also discussed. Zhang et al. [26] studied the impacts of superheat and internal heat exchangers on the thermo-economic performance of ORCs based on fluid type and heat source.

As a bottom cycle, the TC is also able to recycle low-grade thermal energy from other scarce heat sources, including industrial waste heat or engine waste heat, in addition to recovering geothermal energy. Hays et al. [27] drove the TC system with waste heat from ship engines and conducted sea trials, confirming the reliability of TC applications for ships. Zeynali et al. [28] investigated the performance of modified trilateral flash cycles assisted by a solar pond and compared them with modified organic Rankine cycles. With the emergence of high-efficiency two-phase screw expanders, the application of TC has great development prospects.

In recent years, research of TC has made great progress, mainly in the selection of working fluid, application cases, the comparison between TC and traditional technology and the modeling and design of expander. However, the basic thermodynamic performance simulation and calculation methods, system optimization, and parameter analysis of TC are still rare. A deeper understanding of the fundamental thermodynamic performance of TC is required. Therefore, n-pentane is used as the working fluid and a twin-screw expander is chosen as the two-phase expander in the present work. Numerical simulations of the thermodynamic processes of TC are conducted. The influence of the system’s main parameters on the cycle’s thermodynamic performance is analyzed. The research is expected to make a significant addition to the study of TC’s thermodynamic performance and provides guidance for the design and optimization of the TC waste heat recovery system.

2. Model Description and Conditions

2.1. System Specifications

Figure 1 shows the schematic diagram and temperature-enthalpy diagram of the TC system. The system is composed of a heater, a working fluid pump, a two-phase expander, and a condenser. The ideal TC includes four reversible processes: the isentropic compression process (1–2), constant pressure heating process (2–3), isentropic expansion process (3–4), and constant pressure exothermic process (4–1). In reality, the isentropic process is impossible to achieve in the pump and expander. The working fluid in the saturated liquid state (point 1) is compressed in the pump by the condensing pressure to the heating pressure and then delivered to the heater. Point 2 indicates the state of the working fluid at the heater inlet. In the heater, the working fluid is heated to the saturated liquid condition (point 3). Afterward, the working fluid is sent to the two-phase expander and go through the expansion process. At the same time, the working fluid’s internal energy is turned into expander shaft power. The working fluid at the outlet of the expander (point 4) is in a gas-liquid two-phase state. The working fluid is then transported to the condenser, where the saturated vapor is converted to saturated liquid. Meanwhile, the latent heat of condensation of the working fluid is transferred to the condensing fluid. Finally, the working fluid returns to the saturated liquid condition (point 1). All the above processes constitute the whole cycle.

2.2. Model Establishment

2.2.1. Thermodynamic Model of TC Internal Circulation

Some hypotheses are presented below to simplify the calculations. (1) The system is in the steady state. (2) The law of conservation of energy is followed by all system components. (3) Kinetic and potential energy changes of the working fluid are neglected. (4) Heat exchanger pressure drop and heat loss are not taken into account.

For the actual process, the compression process in the pump and the expansion process in the expander are both incomplete adiabatic processes, which cannot be precisely represented in the software. As a result, the adiabatic efficiency of the pump and expander are taken as empirical values. The detailed thermodynamic description of TC is summarized as follows:

Process 1–2: the power consumed by the pump:

$$w_{\mathrm{pump}}=h_2-h_1=\frac{h_{2,s}-h_1}{{\eta }_{\mathrm{pump}}}$$

(1)

where h_2,s is the specific enthalpy after the process of the adiabatic compression of the working fluid. η_pump is the adiabatic efficiency of the pump.

Process 2–3: the heat absorbed from the heat source:

$$q_{\mathrm{h}}=h_3-h_2$$

(2)

Process 3–4: the power output of the expander:

$$w_{\exp }=h_3-h_4={\eta }_{\exp }\cdot (h_3-h_{4,\mathrm{s}})$$

(3)

where h_4,s is the specific enthalpy of the working fluid after adiabatic expansion. η_exp is the adiabatic efficiency of the expander.

Process 4–1: the heat released by the working fluid in the condenser:

$$q_{\mathrm{c}}=h_4-h_1$$

(4)

The net power output of the system:

$$w_{\mathrm{net}}=w_{\exp }-w_{\mathrm{pump}}$$

(5)

The thermal efficiency of the system:

$${\eta }_{\mathrm{th}}=w_{\mathrm{net}}/q_{\mathrm{h}}$$

(6)

The exergy of the working fluid at the state point i:

$$e_{x,i}=\left(h_i-h_0\right)-T_0(s_i-s_0)$$

(7)

where T₀ is the ambient temperature. h₀ and s₀ are the specific enthalpy and specific entropy of the working fluid at ambient temperature, respectively.

The heat exergy of the working fluid in the process 2–3:

$$e_{x,Q}=e_{x,3}-e_{x,2}$$

(8)

By substituting Equation (7) into Equation (8), the heat exergy of the working fluid can be expressed as follows:

$$e_{x,Q}=\left(h_3-h_2\right)-T_0(s_3-s_2)$$

(9)

The description of the exergy efficiency of TC is the following:

$${\eta }_{\mathrm{ex}}=w_{\mathrm{net}}/e_{x,Q}$$

(10)

2.2.2. Heat Transfer Model of the Heater

The ε-NTU method [29] is adopted in the present work since the ε-NTU method can overcome the problem that multiple iterations should be conducted for the logarithmic mean temperature difference (LMTD) method [30]. Additionally, the ε-NTU method is commonly employed in the computation of low-temperature heat exchangers. The heat transfer model of the heater is established under the conditions of a given heat source, working fluid inlet temperature, and heat source fluid average heat-capacity flow rate.

To evaluate the performance of a heat exchanger, the parameter effectiveness of heat exchanger ε is utilized. It is defined as follows:

$$\epsilon =\frac{|T_{\mathrm{out}}-T_{\mathrm{in}}{|}_{\max }}{T_5-T_2}$$

(11)

where |T_out − T_in|_max is the maximum temperature difference between the inlet and outlet of the heat source fluid and the working fluid. T₅ is the inlet temperature of the heat source fluid. T₂ is the inlet temperature of the working fluid.

According to the energy balance:

$$Q_{\mathrm{h}}=C_{\mathrm{hc}}(T_5-T_6)=C_{\mathrm{wf}}(T_3-T_2)$$

(12)

where C_hc is the average heat-capacity flow rate of the heat source fluid. C_wf is the average heat-capacity flow rate of the working fluid.

The actual heat transfer in the heater:

$$Q_{\mathrm{h}}=C_{\min }\epsilon (T_5-T_2)$$

(13)

By substituting Equation (13) into Equation (12), the outlet temperatures of the heat source fluid and working fluid in the heater can be described as the following, respectively:

$$T_6=T_5-\frac{C_{\min }\epsilon }{C_{\mathrm{hc}}}(T_5-T_2)$$

(14)

$$T_3=T_2+\frac{C_{\min }\epsilon }{C_{\mathrm{wf}}}(T_5-T_2)$$

(15)

The effectiveness of the counter-flow heat exchanger:

$$\epsilon =\frac{1-\exp [(-\mathrm{NTU})(1-R_{\mathrm{C}})]}{1-R_{\mathrm{C}}\exp [(-\mathrm{NTU})(1-R_{\mathrm{C}})]}$$

(16)

where NTU is the number of transfer units. R_C is the ratio of heat-capacity flow rate.

$$\mathrm{NTU}=\frac{{(kA)}_{\mathrm{h}}}{C_{\min }}$$

(17)

$$R_{\mathrm{C}}=C_{\min }/C_{\max }$$

(18)

where (kA)_h is the total heat transfer coefficient of the heater.

The average specific heat capacity of the working fluid during the heating process is defined as the following:

$$c_{p,\mathrm{wf}}=\frac{h_3-h_2}{T_3-T_2}$$

(19)

The mass flow rate of the working fluid:

$$m_{\mathrm{wf}}=C_{\mathrm{wf}}/c_{p,\mathrm{wf}}$$

(20)

2.2.3. Heat Transfer Model of the Condenser

Assuming that the cooling water inlet temperature T₇ is constant, the heat transfer in the condenser Q_c can be obtained from the heat balance equation [31]:

$$Q_{\mathrm{c}}=C_{\mathrm{ca}}(T_8-T_7)=m_{\mathrm{wf}}(h_4-h_1)$$

(21)

The outlet temperature of cooling water:

$$T_8=T_1-\mathsf{\Delta }{T}_{\mathrm{P}}$$

(22)

where ΔT_p is the condenser pinch temperature difference.

The average heat-capacity flow rate of cooling water:

$$C_{ca}=\frac{m_{\mathrm{wf}}(h_4-h_1)}{(T_8-T_7)}$$

(23)

When the specific heat capacity of the cooling water c_p,ca is regarded as a constant, the mass flow rate of the cooling water m_ca is depicted as follows:

$$m_{ca}=C_{\mathrm{ca}}/c_{p,\mathrm{ca}}$$

(24)

The total heat transfer coefficient in the heater and condenser is described as follows:

$$k{A}_i={\dot{Q}}_i/\mathsf{\Delta }{T}_i$$

(25)

where ΔT_i is the logarithmic mean temperature difference of heat transfer. ΔT_i can be obtained in the following form:

$$\mathsf{\Delta }{T}_i=\frac{\mathsf{\Delta }{T}_{i,\max }-\mathsf{\Delta }{T}_{i,\min }}{\ln (\mathsf{\Delta }{T}_{i,\max }/\mathsf{\Delta }{T}_{i,\min })}$$

(26)

The total heat transfer coefficient of the system:

$${\left(kA\right)}_{\mathrm{tot}}={\left(kA\right)}_{\mathrm{h}}+{\left(kA\right)}_{\mathrm{c}}$$

(27)

where (kA)_c is the total heat transfer coefficient of the condenser.

2.2.4. Model of the Second Law for the Thermodynamic Performance of the TC Heat Recovery and Power Conversion System

The heat exergy absorbed by the working fluid:

$$E_{x,Q,\mathrm{wf}}=m_{\mathrm{wf}}\left(e_{x3}-e_{x2}\right)$$

(28)

The heat exergy released by the heat source fluid:

$$E_{\mathrm{x},Q,}{}_{\mathrm{hc}}=m_{\mathrm{hc}}\left(e_{x5}-e_{x6}\right)$$

(29)

The heat exergy released by the working fluid:

$$E_{x,c,\mathrm{wf}}=m_{\mathrm{wf}}\left(e_{x4}-e_{x1}\right)$$

(30)

The heat exergy absorbed by the cooling water:

$$E_{x,c,\mathrm{ca}}=m_{\mathrm{ca}}\left(e_{x8}-e_{x7}\right)$$

(31)

Among Equations (28)–(31), e_x1–e_x8 indicate the exergy of per unit quality of working fluid, heat source fluid, or cooling water in each state point, respectively. E_x,Q,wf and E_x,c,wf represent the heat exergy absorbed by the working fluid during heating and released during condensation, respectively. E_x,Q,hc and E_x,c,ca represent the heat exergy released by the heat source fluid during the heating process and the heat exergy absorbed by the cooling water during the condensation process, respectively.

The exergy efficiency of the system:

$${\eta }_{\mathrm{ex}}=\frac{W_{\mathrm{net}}}{E_{\mathrm{x},Q,}{}_{\mathrm{hc}}}$$

(32)

The objective functions considered in the present paper are shown as follows.

For the TC internal circulation, the thermal efficiency of the system η_th, exergy efficiency η_ex, net power output W_net and heat absorption q_h are selected as the objective functions. To match the real conditions, the variation of the expansion ratio R_exp and the dryness at the outlet of the expander x₃ with the cycle parameters is calculated.

For the TC heat recovery and power conversion system, the performance of the system depends not only on the thermal efficiency of the cycle, but also greatly on the heat transfer efficiency of the heat source to the system. Furthermore, it is also related to the heat release of the system to the environment. Hence, the thermal efficiency η_th, exergy efficiency η_ex, net power output W_net and heat absorption q_h are considered as objective functions. Besides, taking the economic efficiency into consideration, the total heat transfer coefficient of the heater and condenser of the system, i.e., (kA)_h and (kA)_c, is calculated.

2.3. Model Validation

Comparisons of thermodynamic performance between TC and ORC were made by Fischer et al. [17,18], who used the exergy efficiency as the objective function. The minimum heat-capacity flow rate of heat source fluid for 1MW net power output is obtained through the optimization calculation under five preset conditions. The theoretical model established in the previous section is verified under the same conditions as the first set of parameters.

Since water is the working fluid in Refs. [17,18], the adiabatic efficiency of the pump and the expander are set at 0.65 and 0.85, respectively. The net power output is 1 MW. The goal is to obtain the minimum heat-capacity flow rate of the heat source fluid via system optimization and determine the parameters at each state point of the cycle. In the present work, the thermal efficiency is calculated based on the parameters at each state point in Refs. [17,18].

Comparisons of the thermal efficiency between the results in the reference and the present work are listed in

Table 1. Validation of the model.

Parameters	T1/K	T2/K	T3/K	T4/K	p1/kPa	p3/kPa	ηth/%
Fischer’s work	358.15	360.21	590	358.15	57.87	10,861	0.198
Present work	358.15	360.21	590	358.15	57.87	10,861	0.1947

. It is shown that the thermal efficiency of the present model is less than that of Fischer’s work. This is because the pump power consumption is considered in the present work while it is ignored in Fischer’s work. Therefore, the pump power consumption is responsible for the difference.

2.4. Calculation Conditions

Since n-pentane is the working fluid, the critical pressure and temperature of n-pentane are 3.37 MPa and 469.5 K, respectively. According to the properties of the working fluid, the boundary parameters of the cycle are listed in

Table 2. Boundary parameters of the TC.

Parameters	Value	Unit
Heat source fluid inlet temperature T5	423	K
Pump adiabatic efficiency ηpump	85	%
Expander adiabatic efficiency ηexp	85	%
Ambient temperature T0	298.15	K
Working fluid temperature range	309–496.5	K
Cooling water inlet temperature T7	298.15	K
Condenser pinch temperature difference ΔTp	5	K

.

3. Theoretical Analysis and Numerical Simulation Results of the TC Internal Circulation

3.1. Thermodynamic Performance of the TC Internal Cycle: A Theoretical Analysis

Under the condition of ignoring the power consumed by the pump, the thermal efficiency of the ideal TC is derived in Refs. [17,18] as the following form:

$${\eta }_{\mathrm{th}}=1-\frac{T_1·\ln (T_3/T_1)}{T_3-T_1}$$

(33)

In the present work, the pump power consumption is considered, and the derivation process is similar to that in Ref. [17]. Therefore, more details can be seen in Ref. [17]. Based on the established model, the thermal efficiency of the ideal TC that ignores the pump power consumption could be described as follows:

$${\eta }_{\mathrm{th}}=1-\frac{T_1·\ln (T_3/T_2)}{T_3-T_2}$$

(34)

The exergy efficiency of the ideal TC could be written as:

$${\eta }_{\mathrm{ex}}=w_{\mathrm{net}}/e_{x,Q}=\frac{T_3-T_2-T_1\cdot \ln (T_3/T_2)}{T_3-T_2-T_0\cdot \ln (T_3/T_2)}$$

(35)

It is known that the thermal efficiency is only related to T₁, T₂, and T₃ in Equation (34). Since T₁ can be determined by the parameters at state point 2 and the adiabatic efficiency of the pump, the thermal efficiency is actually related to T₂ and T₃.

The first-order derivatives of η_th to T₂ and T₃ for Equation (34) are calculated, respectively. When T₃ > T₂, the following results can be obtained.

$$\frac{\mathrm{d}{\eta }_{\mathrm{th}}}{\mathrm{d}{T}_2}=\frac{T_1[1/T_2-\ln (T_3/T_2)]}{{(T_3-T_2)}^2}<0$$

(36)

$$\frac{\mathrm{d}{\eta }_{\mathrm{th}}}{\mathrm{d}{T}_3}=\frac{T_1[\ln (T_3/T_2)-(1-T_2/T_3)]}{{(T_3-T_2)}^2}>0$$

(37)

From Equations (36) and (37), it can be predicted that the η_th decreases with the increase of T₂ but increases with the increase of T₃.

Through similar conduction to Equation (35) with that to Equation (34), Equations (38) and (39) are obtained when T₃ > T₂, the following results can be obtained.

$$\frac{\mathrm{d}{\eta }_{\mathrm{ex}}}{\mathrm{d}{T}_2}<0$$

(38)

$$\frac{\mathrm{d}{\eta }_{\mathrm{ex}}}{\mathrm{d}{T}_3}>0$$

(39)

It is predicted that η_ex decreases with the increase of T₂ and increases with the increase of T₃.

3.2. Analysis of Numerical Simulation Results of the TC Internal Circulation

In this section, the heat absorption of the cycle Q_h, the net power output W_net, the thermal efficiency η_th, and the exergy efficiency η_ex are regarded as objective functions to determine the effect of T₂ and T₃ on the thermodynamic performance of the cycle by EES. What’s more, the thermal efficiency and exergy efficiency by numerical simulation are compared with those by theoretical analysis.

3.2.1. Effect of the Heater Inlet Temperature on Thermodynamic Performance

Given that the expander inlet temperature T₃ is set at 150 °C and the expander inlet pressure P₃ is set at 1591 kPa (the saturation pressure at 150 °C), the variation trends of the thermodynamic performance are varied with the increase of the temperature T₂, as depicted in Figure 2. It is obvious that the heat absorption of the cycle and the power output of the expander show increasing trends, and the pump power consumption is reduced with the decrease of T₂ when the expander inlet temperature T₃ is constant, which means W_net, η_th, and η_ex are all increased with the decrease of T₂. However, the variation trend of T₁ behaves the opposite, and the corresponding saturation pressure (expander outlet pressure) drops. When P₁ = 101 kPa and T₁ = 35.78 °C, a smaller T₂ will result in a lower expander outlet pressure, which will increase the condenser power consumption for vacuuming. According to Figure 2b, η_th and η_ex by numerical simulations are lower than those by theoretical calculations. The reason is due to the irreversibility of the actual process and the change in specific heat capacity with temperature.

3.2.2. Effect of the Expander Inlet Temperature on Thermodynamic Performance

In this section, the heater inlet temperature T₂ is held constant at 38 °C and T₃ is varied in the range of 125–195 °C. The variation trend of the objective functions with T₃ is displayed in Figure 3. The results show that as T₃ increased, as did Q_h, W_net, η_th, and η_ex. This is because the higher T₃ is, the higher the corresponding saturation pressure P₃ is, but the lower T₁ and P₁ are. Consequently, the power output of the expander and the power consumed by the pump are increased. η_th and η_ex ranged between 10–20% and 50–90%, respectively. η_th and η_ex obtained by numerical simulation were less than those obtained by theoretical calculation. More explanations can be found in the above section.

3.2.3. Effect of the Heater Inlet Temperature and Expander Inlet Temperature on the Expansion Ratio and Dryness at the Outlet of the Expander

As shown in Figure 4a, when T₃ = 150 °C, the expansion ratio R_exp rose from 83.66 to 116.1 as T₂ reduced from 46 to 36.61 °C. As described in Figure 4b, when T₂ = 38 °C, R_exp rose from 92.59 to 117.4, with T₃ rising from 125 to 163.3 °C. As R_exp increased, the dryness at the expander outlet x₄ increased. Because the screw expander can expand with liquid, the dryness has slight damage to the expander. However, an excessive R_exp requires a large expander size, which leads to too much investment. As a result, the suitable R_exp should be determined by the actual production level before deciding the cycle parameters.

4. Simulation Results and Analysis of the TC Heat Recovery and Power Conversion System

4.1. Effect of the Average Heat-Capacity Flow Rate of the Working Fluid on Thermodynamic Performance

As depicted in Figure 5, when the effectiveness of the heater ε was 0.95, T₂ was 313 K, and the average heat-capacity flow rate of the heat source fluid C_hc was 5 kW/K, the system performance varied with the variation of the average heat-capacity flow rate of the working fluid C_wf. Under the condition of C_wf < C_hc, the heat absorption, heat release, and the net power output of the system increase with the increase of C_wf. At the same time, η_th maintained invariableness, and η_ex increased. According to Equation (15), it is known that C_min = C_wf at this time, so T₃ remained unchanged. The condensing temperature T₁ was also calculated as a constant value according to the condenser model. Thus (h₃-h₂) and (h₃-h₄) were constant and Q_h and W_net increased as C_wf (i.e., m_wf) increased. Therefore, η_th remained constant. C_wf increased the need to change the heat exchanger model, the NTU also increased, and the (kA)_h also increased, when C_wf and C_hc were equal, the W_net reached the maximum and (kA)_h also reached the maximum. When C_wf > C_hc, with the increase of C_wf, T₃ decreased, evaporation pressure P₃ decreased, T₁ increased, heat source fluid outlet temperature T₆ remained unchanged, circulating heat absorption was the maximum value and remained unchanged, circulating heat release increased, W_net decreased, η_ex decreased, NTU and (kA)_h both decreased.

Under the condition of C_wf higher than C_hc, heat absorption of the cycle keeps constant and reaches its maximum even if C_wf increased. At the same time, the heat release of the cycle increased. Contrarily, W_net, η_ex, NTU, and (kA)_h showed the opposite variety. When C_wf changes between 1 kW/K and 10 kW/K, the highest η_th and the lowest η_th of the cycle were 12.8 and 6.63%, respectively. By calculating the power output of the expander W_exp and the power consumed by the pump W_pump, it was found that with the increase of C_wf, the ratio of W_pump to W_exp rose from 0.07 to 0.08, which means the power consumed by the pump in the system cannot be ignored.

When the total heat transfer coefficient of the heater (kA)_h was kept at 20 kW/K, the performance of the system was varied with C_wf, as shown in Figure 6. It was demonstrated that with the increase of C_wf, η_th, and η_ex decreased gradually. Meanwhile, Q_h increased gradually, and W_net increased first and then decreased. In other words, there was a maximum W_net. This is because, with the increase of C_wf, T₃ decreased while T₄ was basically unchanged, which led to a reduction in W_exp by unit working fluid. When the working fluid flow rate was within a suitable range, the increase in the flow rate of the working fluid can make up for the decrease of W_exp, which leads to a rise in the overall output power. The maximum net power output was obtained when the working fluid flow rate reachedthe critical value. Subsequently, the increase in the working fluid flow rate cannot compensate for the decrease in W_exp per unit mass of the working fluid. Consequently, W_net decreased. Similarly, W_exp first increased and then decreased with the increase of C_wf.

4.2. Effect of the Average Heat-Capacity Flow Rate of the Working Fluid on the Heat Transfer Capacity of the System

Figure 7a depicts the variation trends of the total heat transfer coefficient of the heater (kA)_h and condenser (kA)_c, as well as the sum of the two coefficients (kA)_tot, with C_wf when ε = 0.95 and C_hc = 5 kW/K. It reveals that (kA)_h increased quickly and then decreased rapidly with the increase of C_wf. When C_wf = C_hc, (kA)_h reached the highest value. The increased rate of (kA)_c for C_wf < C_hc was larger than that for C_wf > C_hc. Moreover, the conversion trend of (kA)_tot showed great similarities to that of (kA)_h. As depicted in Figure 7a, when C_wf was close to C_hc, (kA)_h was greater than (kA)_c. In other cases, (kA)_c was bigger than (kA)_h. According to the simulation results in Section 4.2, when C_wf = C_hc, W_net attained its maximum value. At the same time, (kA)_h reached the maximum of 88.3 kW/K. Taking the investment of the heat exchanger into account, the system economy is not necessarily good when W_net is at its maximum.

As shown in Figure 7b, at (kA)_h = 20 kW/K, (kA)_c gradually increased and exceeded (kA)_h. However, the increased rate of (kA)_c decreased gradually. Furthermore, (kA)_tot increased with (kA)_c. As the simulation results mentioned above, W_net had a maximum value when (kA)_h = 20 kW/K and C_wf = C_hc. Nevertheless, the heater efficiency was minimum. Therefore, C_wf should be selected, reasonably based on the actual situation.

5. Conclusions

This work developed a system model for a stable-flowing triangular cycle, simulated its thermodynamic process, and investigated the effects of key operational parameters on system performance. The following are the main conclusions:

(1) For the TC internal cycle, increasing T₃ and decreasing T₂ can improve the thermal efficiency and net power output of the cycle.

(2) For the TC heat recovery and power conversion system with constant parameters of the heat source fluid, the net power output increases at first and then decreases when ε or (kA)_h is constant. The net power output can reach its maximum value at a certain average heat-capacity flow rate of the working fluid. In addition, the thermal efficiency and exergy efficiency should be taken into consideration as well; thus, a suitable objective function will be selected to optimize the thermodynamic performance of the system.

(3) For the TC heat recovery and power conversion system with constant parameters of the heat source fluid, when the total heat transfer coefficient of the heater (kA)_h is constant, the heat transfer coefficient of the condenser (kA)_c increases gradually with the increase of the average heat-capacity flow rate of the working fluid C_wf. Furthermore, (kA)_c will increase to a value that is greater than (kA)_h for a relatively small (kA)_h. Under the condition that the effectiveness of the heater ε is 0.95, (kA)_h increases first and then decreases. (kA)_h will reach its maximum when C_wf = C_hc. (kA)_c increases gradually, however, the increase rate decreases gradually for C_wf > C_hc.

(4) The performance of the TC will be improved when the heat transfer capacity between the working fluid and the heat source fluid and cooling water is enhanced. Besides, the balance of the investment and the performance of the system plays a significant role in reducing the cost of the heat exchanger and the wide application of TC. In the actual process, not only the heat transfer but also the increase in the amount of working fluid, heat source fluid, and cooling water significantly impact the net power output and thermal efficiency.