Energy Analysis of Waste Heat Recovery Using Supercritical CO₂ Brayton Cycle for Series Hybrid Electric Vehicles

Abstract

Waste heat recovery from exhaust gas is one of the most convenient methods to save energy in internal combustion engine-driven vehicles. This paper aims to investigate a reduction in waste heat from the exhaust gas of an internal combustion engine of a serial Diesel–electric hybrid bus by recovering part of the heat and converting it into useful power with the help of a split-flow supercritical CO₂ (sCO₂) recompression Brayton cycle. It can recover 17.01 kW of the total 33.47 kW of waste heat contained in exhaust gas from a 151 kW internal combustion engine. The thermal efficiency of the cycle is 38.51%, and the net power of the cycle is 6.55 kW. The variation in the sCO₂ temperature at the shutdown of the internal combustion engine is analyzed, and a slow drop followed by a sudden and then a slow drop is observed. After 80 s from stopping the engine, the temperature drops by (23–33)% depending on the tube thickness of the recovery heat exchanger. The performances (net power, thermal efficiency, and waste heat recovery efficiency) of the split-flow sCO₂ recompression Brayton cycle are clearly superior to those of the steam Rankine cycle and the organic Rankine cycle (ORC) with cyclopentane as a working fluid.

1. Introduction

The road transport sector is responsible for 1/3 of the final energy consumed in the EU and for 1/5 of the EU’s greenhouse gas emissions. Trucks and buses (heavy-duty vehicles) contribute 25% to road transport greenhouse gas emissions in the EU and 6% to total EU greenhouse gas emissions [1].

As emissions from road transport are increasing, the European Commission has proposed new and ambitious targets for new heavy-duty vehicles from 2030 in order for the EU to achieve climate neutrality in 2050. Currently, heavy vehicles in the EU are powered by internal combustion engines. The new CO₂ standards will have the effect of both reducing the consumption of fossil fuels through energy efficiency increase and the electrification of vehicles, as well as increasing air quality and the health of EU citizens since heavy vehicles travel in urban areas [1].

Although the energy efficiency of heavy vehicles has increased significantly lately, there are many concerns about increasing the efficiency of internal combustion engines. One of these concerns relates to the recovery of waste heat discharged from the engine. The main technologies available for waste heat recovery are as follows: thermoelectric generators [2]; thermoacoustic generators [3]; turbocompound systems [4]; technologies using thermal cycles (steam Rankine [5]; organic Rankine [6]; Stirling [7] and Brayton with air [8] or with CO₂ [9]) and refrigeration systems [10]). The purpose of waste heat recovery systems is to recover as much heat as possible to convert it into electricity or cold. The performance of these systems depends on the choice of the heat source and the working fluid, the cycle configuration, the performance of the components (heat exchangers, pumps, compressors, turbines), and the operating mode of internal combustion engines (variation in the operating regime as a result of the variation in road conditions determines the variation in temperature and the exhaust gas flow rate). Two solutions have been proposed to reduce the variation in engine exhaust gas characteristics: (i) the use of control strategies that monitor the engine’s operating status and adjust the waste heat recovery system and (ii) the use of heat storage technology [11]. The only internal combustion engines that operate at almost constant speed, regardless of road conditions, are those on series hybrid vehicles. In these vehicles, the internal combustion engine is completely mechanically decoupled from the driving wheels, and all the energy produced by it is converted into electricity by the generator, which supplies one or more electric traction motors or recharges the electric accumulator. Because the internal combustion engine is not directly connected to the wheels, it can be smaller than the engine should be for a vehicle of the same size; in addition, it runs at a constant speed when optimized for the lowest fuel consumption and the lowest CO₂ emission for a certain period of time, followed by a period of inactivity (standstill). The average frequency of stops and starts is (60–70) stop/h, with an average duration of (15–50) min, which represents an average downtime of (25–40)% [12]. Thus, the fuel consumption of Diesel hybrid vehicles can be lower by about (18–29)% than that of conventional standard Diesel vehicles [13].

The efficiency of series hybrid vehicle engines can be further increased by applying a waste heat recovery system. Several studies [14,15,16] show that operation at a constant engine load is an advantage in the implementation of waste heat recovery systems. This paper thermodynamically analyses waste heat recovery from the Diesel engine exhaust gas of a series hybrid bus to increase engine power using the split-flow sCO₂ recompression Brayton cycle. This thermodynamic cycle was chosen for engine exhaust temperatures of about 400 °C (after catalytic treatment and particulate filter) and power less than 1MW, where the most suitable cycles are the Brayton cycle with supercritical CO₂ [17], the steam Rankine cycle [18,19] and the ORC with ethanol or cyclopentane [20,21].

All investigations related to the application of waste heat recovery technology using a thermal cycle for heavy-duty Diesel engine vehicles aimed at obtaining maximum thermal efficiency, reducing CO₂ emissions, maximum heat recovery, minimizing the heat transfer area/net power output ratio or minimizing the period of investment recovery [22].

Depending on the objectives pursued, a cycle with a certain working fluid, a certain configuration, and a certain type of heat exchangers and expanders resulted. Because organic working fluids tend to decompose at high temperatures, limiting the inlet temperature to the expander, and the constant temperature evaporation process creates a large temperature difference between the working fluid and the heat source, resulting in reduced utilization of the heat source and an increase in the conversion efficiency of ORC systems is limited [23].

In the sCO₂ Brayton cycle, the weak points of ORC are not encountered due to the thermophysical properties of CO₂ (a non-toxic, stable, and non-flammable fluid) around the critical point that allow the introduction of heat into the cycle and its removal from the cycle with the avoidance of constant temperature evaporation and condensation with better coupling of the working fluid to the heat sources. In other words, all the processes undergone by the working fluid occur only in the supercritical phase, which combines the properties of gas with those of a liquid. In addition, the high density of CO₂ leads to a relatively high-power density, which allows reducing the dimensions of the compressor and the expander, as well as reducing the energy consumed by the compressor [24].

Paper [24] states that there is no optimal sCO₂ Brayton cycle configuration that covers all operating conditions, but split-flow, recompression, and double-expansion cycles are the best performers. Also, paper [25] states that the split-flow cycle with recompression has the highest efficiency over the cycle with a single cycle having a double expansion.

For real closed Brayton cycles, the thermodynamic efficiency depends on the pressure ratio but also on the minimum pressure in the cycle, with a maximum value of around 80 bar [26]. The recuperated cycle has the maximum efficiency at a much lower value of the pressure ratio than the simple cycle, and, in addition, the efficiency variation curve is relatively flatter, which allows a more stable operation [26]. For the recuperated Brayton cycle, the efficiency increases with the increasing turbine inlet temperature and turbine efficiency and decreases with increasing pressure drop, heat loss, and minimum recuperator temperature [26].

The Brayton cycle with split flow and recompression has a higher efficiency at moderate and high values of the turbine inlet temperature than the simple cycle [26]. For the hot source temperature (liquid Na from a nuclear reactor) of 545 °C, the cycle with recompression has the highest thermal efficiency (43.83%) compared to the other cycle configurations [25].

The higher efficiency of the recompression cycle compared to the simple recuperated cycle is due to the two-stage heat recovery and the use of the additional compressor to divert part of the working fluid from the main compressor. The recompressed fraction of the working fluid significantly influences the thermal efficiency of the cycle. The optimum value of the recompressed fraction (influenced by the high-pressure and low-pressure values of the cycle and the efficiency of the heat recuperators) is around 0.4 [27]. The only disadvantage of the sCO₂ Brayton cycle is that the lack of phase change in the cycle leads to lower heat transfer coefficients than in the classic cycle, which increases the size of the heat exchange surfaces [28].

Thermodynamic cycles with CO₂ as the working fluid are increasingly attracting the attention of specialists, especially in the field of electricity production with nuclear reactors, electricity production using geothermal and solar energy, and in the field of waste heat recovery from gas turbines and naval internal combustion engines and more recently in the field of waste heat recovery from road vehicle internal combustion engines.

A Brayton cycle with transcritical CO₂ has the following turbine inlet parameters: pressure of 130 bar and temperature of 200 °C, which can convert about 20% of the waste heat of the gas discharged by an engine into electrical energy [29].

In paper [30], it is shown that a transcritical CO₂ Rankine cycle system with a preheater and regenerator can increase the power of a 43.8 kW internal combustion engine by 9 kW (by about 20%) via recovering waste heat from the cooling liquid and exhaust gas from the engine. This system proved to be significantly more efficient in terms of the simultaneous recovery of heat from the coolant and exhaust gas from the engine than the organic Rankine cycle with R123 as the working fluid.

As the CO₂ Brayton cycle can increase by (10–11)%, the efficiency of a naval gas turbine can increase by recovering the waste heat of the discharged gas [31]. The study of a split-flow sCO₂ Brayton cycle with recompression and regeneration coupled to a marine gas turbine demonstrated an increase in turbine thermal efficiency of 12.38% [32].

The preheated transcritical CO₂ Rankine cycle can improve the power and efficiency of an internal combustion engine through the combined recovery of waste heat from the coolant and exhaust gas, with improved dynamic performance at various engine operating conditions [33].

The analysis results of a sCO₂ Brayton/Rankine cycle system with regeneration and recompression for waste heat recovery from a naval turbine showed that the thermal efficiency of the turbine increased by 10%, the power increased by 25%, and the efficiency of the system was highly dependent on the composition and the temperature of the exhaust gas [34].

When the cooling source cannot cool the CO₂ to condensation, as is the case with the waste heat recovery system from the exhaust gas of the internal combustion engines of vehicles, the transcritical CO₂ Rankine cycle becomes the sCO₂ Brayton cycle.

The current study aims at to analyze the split-flow sCO₂ recompression Brayton cycle for waste heat recovery from the exhaust gas of a Diesel engine for a series-hybrid bus. The cycle performance (thermal efficiency, efficiency of heat recovery, net power) is compared to those of the steam Rankine cycle and ORC with cyclopentane. Unlike other studies, here, the variation in the system performance to the variation in the operating conditions is evaluated (exhaust gas and cold source temperature). The variation in sCO₂ temperature at the turbine inlet when the exhaust gas flow is interrupted as a result of stopping the engine is also analyzed.

The contribution of the paper is to analyze the advantage of the split-off sCO₂ recompression Brayton cycle in terms of thermal efficiency over more mature technologies used to recover waste heat from the exhaust gas of an internal combustion engine.

2. System Analysis

The configuration of the heat recovery system based on the split-flow sCO₂ recompression Brayton cycle and the cycle thermodynamic processes are shown in Figure 1. The system consists of a heater (HE) for sCO₂ heating, a gas turbine (GT), a high-temperature heat recuperator (HTR), a low-temperature heat recuperator (LTR), a flow splitter, a cooler (C), a compressor for the flow passed through the cooler (MC), a compressor for the rest of the flow (RC) and an alternator (A). The distribution of the sCO₂ flow between the two compressors depends on the operating conditions of the system (heat addition and environmental temperature) [35].

3. Thermodynamic Model

The thermodynamic model of the split-flow sCO₂ recompression Brayton cycle is given in

Table 1. Thermodynamic model of the split-flow sCO₂ recompression Brayton cycle.

Component	Energy Equations
Gas turbine	Gas turbine power: ${\dot{W}}_{\mathit{GT}}={\mathsf{\eta }}_{GT}\cdot {\dot{m}}_{C{O}_2}\cdot (h_4-{ h}_5)$  ${\mathsf{\eta }}_{GT}$—gas turbine isentropic efficiency ${\dot{m}}_{C{O}_2}$—CO2 mass flow rate, kg/s h—CO2 enthalpy, kJ/kg
High-temperature recuperator	Effectiveness: ${\mathsf{\epsilon }}_{HTR}=\frac{h_3-h_{2a}}{h_5-h_{2a}}$
Low-temperature recuperator (LTR)	$\mathrm{Effectiveness}: {\mathsf{\epsilon }}_{LTR}=\frac{\left(1-SR\right)\left(h_{2a}-h_2\right)}{h_{5a}-h_2}$ $\mathrm{Split} \mathrm{ratio}: \mathit{SR}=\frac{C{O}_2 flow through recompressor}{C{O}_2 flow through turbine}$
CO2 cooler (C)	Heat rejected from the cycle: ${\dot{Q}}_{out}=\left(1-SR\right){\dot{m}}_{C{O}_2}\cdot (h_6{- h}_1)={\dot{m}}_w\cdot c_{p,w}(t_{wo}-{ t}_{wi})$ ${\dot{m}}_w$—cooling water mass flow rate, kg/s; $c_{p,w}$—specific heat of cooling water, kJ/(kg·K); twi, two—temperatures of cooling water at the entrance and exit of cooler, °C
Main compressor	$\mathrm{Power}: {\dot{W}}_{\mathit{MC}}={\mathsf{\eta }}_{MC}\left(1-SR\right){\dot{m}}_{C{O}_2}\cdot (h_2{- h}_1)$ ${\mathsf{\eta }}_{MC}$—main compressor isentropic efficiency
Recompressor	$\mathrm{Power}: {\dot{W}}_{\mathit{RC}}={\mathsf{\eta }}_{RC}\cdot \mathit{SR}\cdot {\dot{m}}_{C{O}_2}\cdot (h_{2a}{- h}_6)$ ${\mathsf{\eta }}_{RC}$—recompressor isentropic efficiency
CO2 heater (HE)	Heat introduced in the cycle: ${\dot{Q}}_{in}={\dot{m}}_{C{O}_2}\cdot (h_4{- h}_3)={\dot{m}}_{exh}\cdot \left(h_{exhi}-h_{exho}\right)$ ${\dot{m}}_{exh}$—mass flow rate of exhaust gas, kg/s; hexhi, hexho—exhaust gas enthalpy at the heater inlet and outlet, respectively, kJ/kg
Cycle net power	Cycle net power: ${\dot{W}}_{net}={\dot{W}}_{GT}-\left({\dot{W}}_{MC}+{\dot{W}}_{RC}\right)$
Cycle thermal efficiency	${\mathsf{\eta }}_t=\frac{{\dot{W}}_{net}}{{\dot{Q}}_{in}}$
Heat recovery efficiency	${\mathsf{\eta }}_{HR}=\frac{{\dot{Q}}_{in}}{{\dot{Q}}_{exh}}=\frac{{\dot{Q}}_{in}}{{\dot{m}}_{exh}\left(h_{exh}-h_{env}\right)}$; henv—exhaust gas enthalpy at the environmental temperature, °C.

.

The enthalpy of the engine exhaust gas is calculated using the following equation:

$$h_{exh}(T)=x_{C{O}_2}{h}_{C{O}_2}(T)+x_{N_2}{h}_{N_2}(T)+x_{O_2}{h}_{O_2}(T)+x_{H_{2}O}\left[h_{H_{2}O}(T)+h_l\right], \mathrm{kJ}/\mathrm{kg} \mathrm{flue} \mathrm{gas}$$

(1)

where $x_{C{O}_2}$, $x_{N_2}$, $x_{O_2}$, $x_{H_{2}O}$—mass fraction of carbon dioxide, water vapor, and nitrogen and oxygen, respectively, in the engine exhaust gas;

h_CO2(T), h_N2(T), h_O2(T), h_H2O(T)—the specific enthalpy of carbon dioxide, water vapor, and nitrogen and oxygen corresponding to temperature T, kJ/kg (

Table 2. Specific enthalpy [36].

$\mathit{h}(\mathit{T})={\mathit{c}}_{\mathit{p}}(\mathit{T})\cdot \mathit{T}=\left({\mathit{A}}_0+{\mathit{B}}_0\mathit{T}+{\mathit{C}}_0{\mathit{T}}^2+{\mathit{D}}_0{\mathit{T}}^3\right)\mathit{T}$, kJ/kg (T in K)
Gas	A0	B0	C0	D0
Carbon dioxide	0.505	1.359 × 10−3	−7.955 × 10−7	−1.697 × 10−10
Water vapor	1.789	0.106 × 10−4	5.856 × 10−7	1.995 × 10−10
Nitrogen	1.0316	−0.5608 × 10−4	2.884 × 10−7	−1.0256 × 10−10
Oxygen	0.7962	4.75 × 10−4	−2.235 × 10−7	4.1 × 10−11

);

h_l—latent heat of water vapor, h_l = 2502 kJ/kg.

3.1. Model Validation

The thermodynamic model was implemented in a Matlab 9.10 program based on the CoolProp v6.4.1 [37] to simulate the operation of the waste heat recovery system in a steady state. To verify the proposed model, the split-flow sCO₂ recompression Brayton cycle was simulated with the input data presented in [17], and the simulation results were compared. The cycle parameters and the results of the two simulations are given in

Table 3. Split-flow sCO₂ recompression Brayton cycle.

Parameter	Results
	[17]	Present Study
p1, MPa	7.8	7.8
p2, MPa	20	20
texh, °C	660	660
t4, °C	650	650
${\mathrm{CO}}_2 \mathrm{mass} \mathrm{flow} \mathrm{rate}, {\dot{m}}_{C{O}_2}$, kg/s	900	900
Cooling liquid temperature, tw, °C	22	22
Split-flow ratio	0.4	0.4
Turbine isentropic efficiency, ηGT		0.93
Compressor isentropic efficiency, ηC		0.89
Effectiveness of high-temperature recuperator, εHTR		0.83
Effectiveness of low-temperature recuperator, εLTR		0.74
Heater average temperature difference, Δta, °C	10	10
Cooler average temperature difference, Δtw, °C	5	5
$\mathrm{Heat} \mathrm{flow} \mathrm{rate} \mathrm{input}, {\dot{Q}}_{in}$, MW	200	209.85
$\mathrm{Heat} \mathrm{flow} \mathrm{rate} \mathrm{output}, {\dot{Q}}_{out}$, MW	100	103.37
$\mathrm{Main} \mathrm{compressor} \mathrm{power}, {\dot{W}}_{MC}$, MW	10.1	9.91
$\mathrm{Recompressor} \mathrm{power}, {\dot{W}}_{RC}$, MW	21	20,37
$\mathrm{Gas} \mathrm{turbine} \mathrm{power}, {\dot{W}}_{TG}$, MW	131.1	136.7
$\mathrm{Net} \mathrm{power}, {\dot{W}}_{net}$, MW	100	106.42
Thermal efficiency, ηt, %	50	50.74

.

It can be seen from Table 3 that there are small differences between the results of the two simulations (the highest at 4.925% in the case of the input heat flow and the lowest at 1.88% in the case of the power of the main compressor), which proves that the developed model is correct. These differences are due to the different values chosen for the isentropic efficiency of the turbine and compressors and for the effectiveness of the heat recuperators.

3.2. Simulation of Split-Flow sCO₂ Recompression Brayton Cycle Applied to an Internal Combustion Engine of a Serial Diesel-Electric Hybrid Bus

The developed numerical model was applied to the split-flow sCO₂ recompression Brayton cycle, which recovers heat from the exhaust gas of a Diesel engine from a series hybrid electric bus. The characteristics of a Diesel engine are given in

Table 4. Diesel engine characteristics.

Parameter	Value
Displacement	4.5 L
Number of cylinders	4
Speed	2500 rpm
Max. power	149 kW
Min. power	90 kW
Temperature of exhaust gas, texhi	410 °C
$\mathrm{Mass} \mathrm{flow} \mathrm{rate} \mathrm{of} \mathrm{exhaust} \mathrm{gas}, {\dot{m}}_{exh}$	0.0738 kg/s
Composition (mass fraction) of exhaust gas	$x_{C{O}_2}$ = 11.93; $x_{H_{2}O}$ = 4.67; $x_{O_2}$ = 3.86; $x_{N_2}$ = 79.54

.

Respecting the limits imposed by the thermal level of the heat source and the cooling source, the minimum temperature difference in the heat exchangers (especially in the heater and the cooler), and the optimal configuration of the heat recovery system based on split-flow sCO₂ recompression Brayton cycle was chosen.

The data used for heat recovery system simulation are given in

Table 5. Input data for simulation of split-flow sCO₂ recompression Brayton cycle.

Parameter	Value
p1, MPa	7.8
p2, MPa	20
t4, °C	400
${\mathrm{CO}}_2 \mathrm{mass} \mathrm{flow} \mathrm{rate}, {\dot{m}}_{C{O}_2}$, kg/s	0.09
Cooling fluid temperature, tw, °C	22
Split-flow ratio	0.4
Turbine isentropic efficiency, ηGT	0.93
Compressor isentropic efficiency, ηC	0.89
Effectiveness of high-temperature recuperator, εHTR	0.64
Effectiveness of low-temperature recuperator, εLTR	0.74
Heater average temperature difference, Δta, °C	10
Cooler average temperature difference, Δtw, °C	5

, and the simulation results for steady-state operation are presented in

Table 6. Simulation results of split-flow sCO₂ recompression Brayton cycle.

Parameter	Value
$\mathrm{Input} \mathrm{heat} \mathrm{flow} \mathrm{rate}, {\dot{Q}}_{in}$, kW	17.01
$\mathrm{Output} \mathrm{heat} \mathrm{flow} \mathrm{rate}, {\dot{Q}}_{out}$, kW	10.46
$\mathrm{Heat} \mathrm{flow} \mathrm{rate} \mathrm{of} \mathrm{high}-\mathrm{temperature} \mathrm{recuperator}, {\dot{Q}}_{HTR}$, kW	12.77
$\mathrm{Heat} \mathrm{flow} \mathrm{rate} \mathrm{of} \mathrm{low}-\mathrm{temperature} \mathrm{recuperator}, {\dot{Q}}_{LTR}$, kW	12.56
$\mathrm{Power} \mathrm{of} \mathrm{main} \mathrm{compressor}, {\dot{W}}_{MC}$, kW	0.99
$\mathrm{Power} \mathrm{of} \mathrm{recompressor}, {\dot{W}}_{RC}$, kW	2.06
$\mathrm{Gas} \mathrm{turbine} \mathrm{power}, {\dot{W}}_{TG}$, kW	9.60
$\mathrm{Net} \mathrm{power}, {\dot{W}}_{net}$, kW	6.55
Thermal efficiency, ηt, %	38.51
Efficiency of waste heat recovery, ηHR, %	50.81

and

Table 7. Parameters of simulated cycle.

Point	Pressure (MPa)	Temperature (°C)	Enthalpy (kJ/kg)
1	7.4	27	−231.8
2	20	47.09	213.5
2a	20	151.6	19.03
3	20	252.6	160.9
4	20	400	349.9
5	7.4	296.3	296.3
5a	7.4	170.2	101.3
6	7.4	61.29	−38.2

.

The system can recover 17.01 kW of the total 33.47 kW contained in hot exhaust gases from the 151 kW Diesel engine and generates a net mechanical power of 9.6 kW, increasing the power of the engine by 6.33%.

As the Diesel engine of the series hybrid bus has periods of operation at constant load followed by periods of non-operation, it is desired to find out how the sCO₂ temperature varies at the exit of the heater. When the engine is turned off, no more exhaust gas flows through the heater, and the CO₂ heats up less, taking only the heat stored in the metal of the tubes through which it flows.

The variation over time in the average temperature of the tube T_t as a result of heat loss to the fluid flowing through it is described by the following equation:

$$m_t\cdot c_t\frac{\partial T_t}{\partial \mathsf{\tau }}=U\cdot A\left(T_t-T_{C{O}_2}\right)$$

(2)

where m_t is tube mass, kg; c_t is the specific heat of tube material, kJ/kg·K; $T_{C{O}_2}$ is CO₂ temperature, °C; T_t is the average tube temperature, °C; U is overall heat transfer coefficient, W/(m²·K):

$$U=\frac{1}{\frac{1}{{\mathsf{\alpha }}_i}+\frac{d_i}{2{\mathsf{\lambda }}_t}\mathit{ln}\left(\frac{d_0}{d_i}\right)}$$

(3)

in which d_i is the inner tube diameter, m; d₀ is the outer tube diameter, m; λ_t is the tube thermal conductivity, W/(m·K); A is the heat transfer surface area, m²; α_i is the heat transfer coefficient inside the tube, W/(m²·K).

To calculate the convection coefficient in the case of sCO₂, it was assumed that the tube was horizontal and that the pressure in the tube was high. For these conditions, the following equation was adopted at the temperature of the wall (T_t) and the fluid (T_f) [38]:

$$N{u}_w=0.01{\mathit{Re}}_w^{0.9}{\mathit{Pr}}_w^{0.5}{\left(\frac{{\mathsf{\rho }}_w}{{\mathsf{\rho }}_f}\right)}^{0.906}{\left(\frac{c_{p,w}}{c_{p,f}}\right)}^{-0.585}$$

(4)

where $N{u}_w=\frac{{\mathsf{\alpha }}_i\cdot d_0}{k_w}$; ${\mathit{Re}}_w=\frac{{\mathsf{\rho }}_w\cdot v\cdot d_0}{{\mathsf{\mu }}_w}$; ${\mathit{Pr}}_w=\frac{{\mathsf{\mu }}_w\cdot c_{p,w}}{k_w}$, k_w is CO₂ thermal conductivity at wall temperature is W/(m·K); ρ_w is CO₂ density at wall temperature, kg/m³; ρ_f is CO₂ density at fluid temperature, kg/m³; ρ_w is CO₂ density at wall temperature, kg/m³; v is CO₂ velocity inside the tube, m/s; c_p,w is CO₂-specific heat at wall temperature, J/(kg·K); c_p,f is CO₂-specific heat at fluid temperature, J/(kg·K); and μ_w is CO₂ dynamic viscosity at fluid temperature, kg/(m·s).

The CO₂ temperature variation as it flows through the tube is described by the following equation:

$${\dot{m}}_{C{O}_2}\cdot c_{p,C{O}_2}\cdot L\frac{\partial T_{C{O}_2}}{\partial x}+M_{C{O}_2}\cdot c_{p,C{O}_2}\frac{\partial T_{C{O}_2}}{\partial \mathsf{\tau }}=U\cdot A\left(T_t-T_{C{O}_2}\right)$$

(5)

where $M_{C{O}_2}$ is the mass of CO₂ inside the tube, kg; $c_{p,C{O}_2}$ is specific heat at a constant pressure of CO₂, kJ/kg·K; and L is the tube length, m.

The two Equations (2) and (5) are linked by the following initial conditions:

$$\frac{\partial T_{C{O}_2}}{\partial \mathsf{\tau }}=0 \mathrm{at} x=0; \frac{\partial T_t}{\partial \mathsf{\tau }}=0 \mathrm{at} x=0;$$

(6)

A heat exchanger with tubes made of stainless steel (S34709 material) was chosen as a heat recovery device for CO₂ heating. The characteristics of the heat exchanger are as follows: specific heat c_t = 662 J/(kg·°C); thermal conductivity λ_t = 28.7 W/(m·°C); inner diameter of 21.7 mm, length of 2086.7 mm, and thickness of 4.9 mm [39]. The thermophysical properties (density, thermal conductivity, specific heat, and dynamic viscosity) of sCO₂ were calculated using the data available in [40].

In Figure 2, a smooth decrease, followed by a sharp decrease, and again a smooth decrease can be seen in the sCO₂ temperature at the exit of the heater. A total of 80 s after the engine stops, the temperature drops by 32.5% when the heat recovery tube thickness is 3 mm, by 27.5% when the thickness is 4 mm, and by 23.5% when the thickness is 5 mm.

In addition to the variation in the characteristics of the heat source (engine exhaust gas), the cooling source can change its thermal level determined by the ambient temperature. Next, the variation in cycle thermal efficiency and heat recovery efficiency with the variation in the cooling source temperature, turbine inlet pressure, and temperature were analyzed.

Keeping the turbine inlet temperature and turbine outlet pressure unchanged, the variation in the thermal efficiency and the heat recovery efficiency with variation in turbine inlet pressure and the cooling source temperature was sought (Figure 3 and Figure 4).

Keeping the cooling source temperature and turbine outlet pressure unchanged, the variation in thermal efficiency and heat recovery efficiency was sought with the variation in turbine inlet pressure and temperature (Figure 5 and Figure 6).

It can be seen in Figure 3 that the cycle thermal efficiency has a greater variation with the turbine inlet pressure variation at lower cooling source temperatures. For high cooling source temperatures, the cycle efficiency increases with turbine inlet pressure up to a certain value, after which it remains constant.

The efficiency of waste heat recovery has similar variations to the thermal efficiency of the cycle (Figure 4). The thermal efficiency decreases with the decrease in the turbine inlet temperature, with this decrease being greater at higher values of the turbine inlet pressure (Figure 5). A decrease in the turbine inlet temperature of 60 °C for the turbine inlet pressure of 20 MPa causes a decrease in thermal efficiency by 15%. Considering the high rate of the temperature drop at the turbine inlet, when the internal combustion engine turns off (Figure 2), it is necessary to stop the heat recovery system when the internal combustion engine turns off. The waste heat recovery efficiency varies similarly to the thermal efficiency, with the difference that it decreases more at lower turbine inlet pressure values (Figure 6).

4. Performance Comparison of Split-Flow sCO₂ Recompression Brayton Cycle with Other Cycles

To see which cycle performs better among those studied and presented in the literature, the performance of the split-flow sCO₂ recompression Brayton cycle was compared with that of the steam Rankine cycle and the ORC with cyclopentane for the same characteristics of the hot source and the cold source.

The parameters of the steam Rankine cycle (Figure 7) were chosen in such a way that the steam quality at the turbine outlet was greater than 0.88, and the minimum temperature difference between the engine exhaust gas and the water in the Rankine cycle at the pinch point was at least 10 °C.

The results of the cycle simulation using Matlab 9.10 and CoolProp v6.4.1 are presented in

Table 8. Simulation results of steam Rankine cycle.

Point	Temperature (°C)	Pressure (bar)	Mass Flow Rate (kg/s)	Enthalpy (kJ/kg)	Steam Quality
1	99.51	0.1	0.005356	417.5	0
2	100	30	0.005356	421.3	0
3	235.7	30	0.005356	798.9	0
4	235.7	30	0.005356	1016.6	1
5	250	30	0.005356	2850.8	0.88
6	99.51	0.1	0.005356	2392	0

.

Because cyclopentane decomposes at temperatures higher than 275 °C [41], was considered an intermediate circuit with thermal oil between the engine exhaust gas and the cyclopentane ORC, thus ensuring a safe operating temperature of 250 °C. The scheme and the T-s diagram of the cyclopentane ORC are given in Figure 8 and Figure 9.

The simulation results of cyclopentane ORC achieved using the Matlab 9.10 and CoolProp v6.4.1 are given in

Table 9. Simulation results of cyclopentane ORC.

Point	Temperature (°C)	Pressure (bar)	Mass Flow Rate (kg/s)	Enthalpy (kJ/kg)
1	48.86	0.1	0.002639	−0.3636
2	50.57	30	0.002639	4.704
3	116.6	30	0.002639	144.6
4	250	30	0.002639	696.3
5	156.8	0.1	0.002639	560.5
6	71.8	0.1	0.002639	420.6

.

In

Table 10. Performance of split-flow sCO₂ recompression Brayton cycle, steam Rankine cycle and cyclopentane ORC.

Cycle	${\dot{\mathit{W}}}_{\mathit{n}\mathit{e}\mathit{t}}$ (kW)	${\dot{\mathit{Q}}}_{\mathit{i}\mathit{n}}$ (kW)	${\dot{\mathit{Q}}}_{\mathit{o}\mathit{u}\mathit{t}}$ (kW)	ηt	ηHR
Split-flow sCO2 recompression Brayton cycle	6.55	17.01	10.46	38.52	50.81
Steam Rankine cycle	2.46	14.52	12.09	16.94	43.27
Cyclopentane ORC	3.45	14.56	11.11	23.69	43.49

and Figure 10, the performance of the three cycles for waste heat recovery of the engine exhaust gas is presented comparatively. It can be seen that the split-flow sCO₂ recompression Brayton cycle has the highest net power, the highest thermal efficiency, and the highest heat recovery efficiency. This is explained by the fact that, in this cycle, the temperature variation curves of the heat source and the cycle working fluid are the closest compared to the other cycles. In the cyclopentane ORC and the steam Rankine cycle, with respect to the minimum pinch point temperature difference of 10 °C, the two curves were further apart.

The cyclopentane ORC cycle follows the split-flow sCO₂ recompression Brayton cycle in terms of net power, thermal efficiency, and heat recovery efficiency. The higher performance compared to the steam Rankine cycle was due to the internal heat recovery.

Compared to technologies with a higher readiness level, only the turbocompound and combined cycles have a superior specific recoverable power (ratio between the electrical power produced by the recovery unit and the engine brake power) (3–20% and 3–7%, respectively [29]) and thermal efficiency (ratio between the output power and the inlet thermal power of the recovery unit) (55–70% and 8–40%, respectively [29]). The thermoelectric, ORC, Stirling engine, Inverted Brayton Cycle, and Trilateral Flash Cycle have lower performance (specific parameters) [29].

5. Conclusions

There are multiple studies related to waste heat recovery from the exhaust gas of internal combustion engines. This paper addresses the recovery of waste heat from the exhaust gas of the internal combustion engine of a series of hybrid electric buses using the split-flow sCO₂ recompression Brayton cycle. A mathematical model of the heat exchange in the heat recuperator was developed, based on which the temperature variation in the CO₂ was determined when the internal combustion engine was turned off. A mathematical model was developed, and the simulation of the thermal cycle was carried out for different operating conditions (turbine inlet temperature and pressure, cooling source temperature). It was observed that the performance of the cycle decreased greatly when the temperature at the turbine inlet was reduced, which led to the idea of stopping the cycle with the shutdown of the internal combustion engine or the use of a heat storage tank, which mitigated large variations in the temperature of the turbine inlet. For the same waste heat source characteristics, the net power, thermal efficiency, and heat recovery efficiency of the split-flow sCO₂ recompression Brayton cycle are clearly superior to those of the steam Rankine cycle and cyclopentane ORC.