Design condition and operating strategy analysis of CO2 transcritical waste heat recovery system for engine with variable operating conditions

Energy Conversion and Management 142 (2017) 188–199

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Energy Conversion and Management journal homepage: www.elsevier.com/locate/enconman

Design condition and operating strategy analysis of CO2 transcritical waste heat recovery system for engine with variable operating conditions Gequn Shu, Xiaoya Li, Hua Tian ⇑, Lingfeng Shi, Xuan Wang, Guopeng Yu State Key Laboratory of Engines, Tianjin University, 92 Weijin Road, Nankai District, Tianjin 300072, China

a r t i c l e

i n f o

Article history: Received 29 December 2016 Received in revised form 21 February 2017 Accepted 23 February 2017

Keywords: Design condition analysis CO2 transcritical power cycle Variable heat source Waste heat recovery Gasoline engine

a b s t r a c t Waste heat recovery by means of a CO2 transcritical power cycle (CTPC) is suitable for dealing with hightemperature heat sources and achieving miniaturization. Considering the variable operating conditions of engines, the object of current work is to reveal the influence of design condition selection on CTPC systems. Two different engine operating conditions are chosen for system design. System performance has been predicted by a dynamic model and compared by net power output at off-design conditions. Constraints on temperatures, pressures and pump rotational speed have been taken into account. The results show that system designed under a partial load condition possesses a broad range of operation which will be beneficial to operate continuously when engine condition varies. The operating condition determined by driving cycles is recommended for system design of waste heat recovery for gasoline engines. Optimal performance can be obtained by adopting the mass flow rate guided operation strategy. Moreover, the average fuel consumption reduction during the Highway Fuel Economy Test Cycle over the original is 2.84% if system is designed under a partial condition. These preliminary results give reference to system design and optimization for waste heat recovery of engines based on thermodynamic cycles. Ó 2017 Elsevier Ltd. All rights reserved.

1. Introduction Driven by escalating fuel prices and future carbon dioxide emission legislations, the automotive industry is strongly focusing on improving the thermal efficiency of internal combustion engines (ICEs). However, only about one third of the fuel energy is transformed into useful power at crankshaft. The most part of the fuel energy is lost by coolant and exhaust gas. The potentially available energy to be converted into usable power in exhaust gas and coolant is quite significant. Therefore, waste heat recovery (WHR) technology has recently attracted a lot of interest beyond the limit of in-cylinder techniques. A wide range of research work has shown that organic Rankine cycle (ORC) has been accepted as a viable technology to convert waste heat into electrical or mechanical energy [1]. Research mainly includes system configurations design [2–4], working fluids selection [5,6], thermodynamic parameters optimization [7] and performance evaluation [8,9]. These investigations have made excellent contributions to improving cycle performance and developing the potential for application. Yet before

⇑ Corresponding author. E-mail address: [email protected] (H. Tian). http://dx.doi.org/10.1016/j.enconman.2017.02.067 0196-8904/Ó 2017 Elsevier Ltd. All rights reserved.

an ORC system being applied to commercial vehicles, several specific challenges of its integration have to be faced. The first challenge deals with the high temperature of engine exhaust gas as well as system miniaturization requirement. In view of the relatively low thermal decomposition temperature of organic working fluids, a traditional solution is to add a thermaloil circuit between exhaust gas and ORCs [10] or use a dual-loop ORC [2]. However, a thermal-oil circuit not only leads to extra energy and exergy losses but also forms a bulky system. A dualloop ORC also makes a complicated and costly system. It is difficult to integrate these schematics with vehicles due to space limitation. To overcome this challenge, CO2 transcritical power cycle (CTPC) has been put into perspective for engine WHR [11–15]. The decomposition temperature of CO2 reaches as high as 2000 K which shows a high thermal stability. Carstens [16] investigated a supercritical CO2 (sCO2) power conversion system and showed that CO2 could be heated up to 650 °C. It means CO2 could be directly heated by the high temperature of engine exhaust gas, eliminating the cost and complexity of an intermediate heat transfer loop typically used in ORC applications. Besides, sCO2 enables extremely compact turbomachinery designs for high fluid density and permits the use of compact and microchannel-based heat exchanger technology for single-phase heat transfer process [11].

G. Shu et al. / Energy Conversion and Management 142 (2017) 188–199

189

Nomenclature A area [m2] a0, a1, a2, a3 parameters of pump B fuel consumption [kg/h] b0, b1, b2, b3, c0, c1, c2, c3 parameters of expander cp specific heat capacity [J/(kg K)] Cv coefficient of expander [m2] leakage coefficients Csv/Cst D diameter [m] f friction factor G mass flux [kg/(m2 s)] h specific enthalpy [J/kg] Hu low heating value [J/kg] k thermal conductivity [W/(m K)] L length [m] m mass flow rate [kg/s] Nu Nusselt number p pressure [Pa] Pr Prandtl number Q volume flow rate [m3/s] Re Reynolds number S slip ratio T temperature [K] u specific internal energy [J/kg] U heat transfer coefficient [W/(m2 K)] V volume [m3] W power [W] x vapor quality z direction along tube length Greek letters a heat transfer coefficient [W/(m2 K)] q density [kg/m3] x rotational speed [rpm/s] g efficiency

Gas-like viscosity and diffusion coefficient of sCO2 allow good flow ability and transmission characteristics. Echogen Power Systems Company [13] compared a 10 MW CO2 turbine with a 10 MW steam turbine, showing an extremely compact and highly efficient CO2 turbine design with simpler and single stage. Also, their comparison between a shell-tube and a highly compact heat exchanger of comparable overall heat duty indicated that microchannel heat exchangers had a smaller dimension and 87.4% weight reduction. Shu et al. [15] compared CTPC with R123-based ORC and proposed that the total heat exchange area of basic CTPC is smaller than that of basic transcritical ORC due to a larger heat transfer coefficient. Also, the turbine size of CO2 system (0.010–0.020 m) is rather smaller than that of R123 system (0.055–0.070 m). Therefore, CTPC ought to outperform ORCs because it has an outstanding potential in miniaturizing. The second challenge is the transient and variable behavior of the heat source depending on engine operating conditions. Exhaust energy under mapping characteristics changes dramatically [17], which may cause safety and operation problems to CTPC systems. Xie and Yang [18] analyzed an ORC system under actual driving cycle and concluded that energy fluctuation caused on-road thermal inefficiency (3.63%) of Rankine cycle which was less than half of the design point (7.77%). The experimental investigation on thermal oil storage/ORC conducted by Shu et al. [19] demonstrated that even with a significant inertia of thermal oil, the superheat degree of R245fa after evaporation easily went below zero which would cause safety issues to the expander. Therefore, variable

l c

viscosity [Pa∙s] void fraction

Subscripts 1,2 first, second region 12 intermediate of the first and second region amb ambient c condenser cs cross section expander exp, v f saturated liquid g saturated vapor, exhaust gas i, in inlet o, out outlet p pump r working fluid rec receiver s isentropic sv, st laminar/turbulent leakage total total length w tube wall water cooling water Abbreviations CTPC CO2 transcritical power cycle EIT expander inlet temperature FV finite volume HWY Highway Fuel Economy Test Cycle ICE internal combustion engine MB moving boundary ORC organic Rankine cycle PDE partial differential equation sCO2 supercritical CO2 WHR waste heat recovery

engine operating conditions greatly influence the performance of WHR systems. Moreover, engine operating condition selection is a prerequisite for CTPC system design. Systems designed under different engine operating conditions will have differences in size, volume, weight and performance at off-design conditions. CTPC systems are easily forced to operate at their partial load conditions for any change in engine operating conditions. Operating beyond or below the design points will lead to degradation of the isentropic efficiencies of pumps [20] and expanders [21] as well as variations in the effectiveness of heat exchangers [22]. Usman [23] presented the impact of ORC system installation on vehicles and recommended that the heat exchanger should not be designed for maximum heat recovery due to its unsuitable at partial load operation. However, many waste heat recovery systems were designed under the maximum condition to get a considerable energy saving potential, little research concerns on comparing design condition selection. For vehicle engines, it is meaningful to combine the mapping characteristics with driving cycles when designing a CTPC system. Based on the analysis above, CTPC is adopted for extracting the energy of exhaust gas from a gasoline engine. This current work is focused on providing an insight into the effects of design condition selection on CTPC systems. Two different engine operating conditions are selected for system design, one is the rated condition and the other is chosen based on driving cycles. For safe operation, constraints on temperatures, pressures and pump rotational speed are considered and the performance of CTPC systems is predicted

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and compared at off-design conditions (engine mapping conditions and Highway Fuel Economy Test Cycle). The corresponding control strategies are given in the last part. The results will be helpful for decision-making of WHR system design for gasoline engines and provide guidance for future controller design.

3.1. Heat exchanger models Both the gas heater and the condenser are modeled by applying the moving boundary (MB) approach which has been widely used for dynamic modeling [24,25]. The following assumptions are made in order to simplify the MB models.

2. System description In this study, a water-cooled 4-stroke 4 cylinder naturally aspirated gasoline engine in a Toyota vehicle is used as a topping system. Details of the gasoline engine specifications are listed in Table 1. Fig. 1 describes the system configuration and the overview cycle model of the CTPC. The simple cycle operates with four main components including a gas heater, an expander, a condenser and a pump. The working fluid is compressed to a supercritical pressure in the pump. At this high pressure, the working fluid is heated by exhaust gas in the gas heater. Next, the working fluid expands through an expander and produces mechanical work. This shaft power can then be converted to electricity by a generator. Finally, the working fluid enters the condenser and discharges heat to the cooling water. A receiver is placed at the exit of the condenser as an accessory. In this case, the working fluid leaves from the bottom of the receiver and always enters the pump in a saturated liquid to avoid pump surge.

3. Mathematic model The CTPC system is modeled by a dynamic model for performance prediction. Matlab/Simulink software is used to establish the mathematic models and Refprop 9.0 is used to calculate the thermodynamic properties of fluids. Each of the component models is compiled using sequential code s-function. Matrix calculations and inversions are included.

Table 1 Basic parameters of gasoline engine. Parameter

Unit

Content

Engine type Bore Stroke Displacement Compression ratio Maximum power Maximum torque Ignition mode Intake mode

– mm mm L – kW Nm – –

Inline 4 cylinder 86 86 2.0 9.8 110 (@6000 rpm) 192 (@4000 rpm) 1-3-4-2 Naturally aspirated

(1) The heat exchanger is a long, thin, horizontal tube-in-tube heat exchanger. (2) The fluid is mixed adequately and can be modeled as a onedimensional fluid flow through a tube with effective diameter, length and surface areas. (3) The pressure drop along the tube caused by momentum change and viscous friction is negligible. Therefore, fluid pressure can be assumed uniform along the entire tube. It should be noted that the pressure of the working fluid is still time dependent. (4) Axial heat conduction in the working fluid and the secondary fluids as well as in the pipe wall is negligible. (5) The external pipe is assumed to be ideally insulated. For the one-dimensional case, the differential conservations of mass and energy equation for the fluid are given in Eqs. (1) and (2), respectively. The differential conservation of energy equation for the heat exchanger wall energy is given in Eq. (3).

_ @ðAqÞ @ m þ ¼0 @t @z

ð1Þ

_ @ðAqh  ApÞ @ mh þ ¼ ai pDi ðT w  T r Þ @t @z

ð2Þ

cpw qw Aw

dT w ¼ ai pDi ðT r  T w Þ þ ao pDo ðT a  T w Þ dt

Wherein, A means cross-sectional area of the inside of tube; q, h, p, _ are density, specific enthalpy, pressure, temperature and Tr and m mass flow rate of fluid, respectively; cpw, qw, Aw and Tw are specific heat capacity, density, cross-sectional area and temperature of tube wall, respectively; ai means heat transfer coefficient between tube wall and internal fluid; ao means heat transfer coefficient between tube wall and external fluid; Di means inner diameter of tube; Do means outer diameter of tube. The governing partial differential equations (PDEs) are integrated using the Leibniz integration rule given in Eq. (4) along the length of the heat exchanger to remove spatial dependence.

Z

z2

z1

@f ðz; tÞ d dz ¼ @t dt

[me,in; he,in]

ωp

Gas heater

Condenser (a)

me,out

Pe

Generator

dz2 dz1 þ f ðz1 ; tÞ dt dt

[Pe; he,out]

Pump

[Pc; hc,out] Receiver

f ðz; tÞdz  f ðz2 ; tÞ

z1

Expander/Valve

mc,out Expander

Pump

z2

Gas heater

Exhaust gas Cooling water

Z

mg,in Tg,in

Engine CO2

ð3Þ

Pc

Condenser with receiver mwater,in Twater,in (b)

Fig. 1. System schematic and the overview cycle model of the CTPC.

[mc,in; hc,in]

ð4Þ

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3.1.1. Gas heater model For transcritical cycles, a gas heater replaces an evaporator which is a traditional component in subcritical cycles. The working fluid is assumed to be supercritical in the whole gas heater, thus a single fluid region is assumed, as shown in Fig. 2. For the gas heater, the limits of integration are simply z1 = 0 and z2 = Ltotal. Selecting pressure and enthalpy as the independent variables for calculating thermodynamic properties of working fluid, and assuming an average enthalpy and an average density, the conservations of working fluid mass, working fluid energy and inner tube wall energy are derived as shown in Eqs. (5)–(7), respectively. The exhaust gas side is also considered here. And since the mass is assumed constant, only Eq. (2) is integrated. Temperature is chosen as the independent variable because it is more concerned than enthalpy. The final conservation of exhaust gas energy is shown in Eq. (8). Mass conservation of working fluid:

      dp 1   dhout dhin @q @q _ in ¼ 0 _ out  m þ Acs Ltotal   Acs Ltotal  þ þm @p h dt 2 dt @ h p dt

ð5Þ

3.1.2. Condenser with receiver model The condenser is coupled with the receiver and modeled together as shown in Fig. 3. Superheated region (SH) and twophase region (TP) are assumed in the condenser. The combined model can be expressed in matrix form as shown in Eq. (9), Z c ðxc ; uc Þ  x_ c ¼ f c ðxc ; uc Þ, with the elements of Z c ðxc ; uc Þ and f c ðxc ; uc Þ matrixes found in Tables A.1 and A.2, respectively. Detailed modeling processes are given in Appendix B.

2

z11

6 6 z21 6 6 6 z31 6 6 6 0 6 6 6 z51 6 6 0 6 6 6z 4 71 z81

z12

z13

z14

0

0

0

z22

0

0

0

0

0

z32

z33

z34

0

0

0

z42

0

z44

0

0

0

0

0

0

z55

0

0

0

0

0

0

z66

0

0

0

0

0

0

z77

0

0

0

0

0

z87

3 2 3 32 f1 L_ 1 7 6 7 6 7 7 6f 7 _ 0 76 p c 7 6 27 76 7 6 7 76 7 6f 7 _ 0 76 m rec 7 6 37 76 7 6 7 76 6 _  0 76 f4 7 c 7 7 6 76 7¼6 7 7 7 _ w1 7 6 0 76 T 6 7 6f5 7 76 7 7 6 6f 7 0 7 T_ w2 7 76 67 7 6 76 6 7 6 7 6 _ 7 4f7 7 z78 7 T 56 5 water;12 5 4 _T water;out 0 f8 0

ð9Þ

Energy conservation of working fluid:



Acs Ltotal

        dhout dhin  @q  @q    1 dp þ 1 Acs Ltotal q  h þ h þ  dt 2 @p h dt dt @h p

_ out hout  m _ in hin ¼ ai pDi Ltotal ðT w  T r Þ þm

ð6Þ

3.1.3. Heat transfer coefficients Numerous correlations to calculate the heat transfer coefficients (HTCs) can be found in the literature. In single phase, the Petukhov–Kirillov correlation [26] is implemented:

ð7Þ

Nu ¼

Energy conservation of inner tube wall:

cpw qw Aw

dT w ¼ ai pDi ðT r  T w Þ þ ao pDo ðT g  T w Þ dt

Energy conservation of exhaust gas:

  dT g;out dT  g 1 @h in g @ q _ g hg;in _ g hg;out  m g g  Acsg Ltotal h q þ þ þm 2 dt dt @ T @ T ¼ ao pDo Ltotal ðT w  T g Þ ð8Þ

Tw(t)

CO2 minhin(t)

Pe(t)

CO2 mouthout(t)

Exhaust gas Ltotal Supercritical

a¼

ðf =8Þ  Re  Pr ½12:7  ðf =8Þ

0:5

 ðPr2=3  1Þ þ 1:07

Nu  k Deq

ð11Þ

Wherein, Re ¼ G  Deq =l, Pr ¼ cp  l=k,f ¼ ½0:79 lnðReÞ  1:64 . Note that in supercritical phase, the Ptukhov-Krasnoshchekov-Proto popov correlation [27] is employed for the Nu calculation:       cp 0:35 kbulk 0:33 lbulk 0:11 ðf =8Þ  Re  Pr Nu ¼ 0:5 kwall lwall ½12:7  ðf =8Þ  ðPr 2=3  1Þ þ 1:07 cpbulk ð12Þ 2

Wherein, the subscript ‘wall’ and ‘bulk’ represent the property of the working fluid at wall-temperature and bulk-temperature, respectively. cp is the averaged over cross-section specific heat under constant pressure, cp ¼ ðhbulk  hwall Þ=ðT bulk  T wall Þ. In the two-phase region, the heat transfer coefficient is calculated by the Chen correlation with a modified form of the Dittus-Boelter equation [28].

a ¼ 0:023  Re0:8  Pr 0:4  f f

Fig. 2. Gas heater model.

Pc(t)

CO2

kf F Deq

Tw2(t)

Tw1(t)

CO2

minhin(t) Cooling water

mouthout(t) (mrec,inhin(t))

L2(t)

L1(t)

Superheated

ð10Þ

Ltotal Two-phase

Fig. 3. Condenser with receiver model.

mghg(t) mfhf(t) mrec,outhf (t)

ð13Þ

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G. Shu et al. / Energy Conversion and Management 142 (2017) 188–199

Wherein, the subscript ‘f’ represents liquid fraction. F is the ratio of the two-phase Reynolds number to the liquid Reynolds number, based on the liquid fraction of the flow. And it is a function of the turbulent-turbulent Lockhart-Martinelli parameter, Xtt [29]. They are expressed as follows, respectively.

Ref ¼

Prf ¼

G  Deq  ð1  xÞ cpf  lf kf

( F¼

ð14Þ

lf

ð15Þ

CF 1 ¼ b0 þ b1

1:0

1=X tt 6 0:1

2:35  ð1=X tt þ 0:213Þ0:736

1=X tt > 0:1

!0:5  0:9 qg 1x X tt ¼ x qf

lf lg

ð16Þ

ð17Þ

The dynamics of this system is assumed to be dominated by the dynamics of the heat exchangers. The dynamics of the pump and expander are considered to be fast relative to the dynamics of the heat exchangers [25,30,31]. Thus the pump and expander are modeled with static (algebraic) relationships. A piston pump is used in this work. The pump mass flow rate corresponding to the rotational speed can be expressed by:

_ p ¼ gv  qp;in  V p  xp m

ð18Þ

Wherein, qp,in is the working fluid density at the inlet of the pump, Vp is the pump displacement, xp is the pump rotational speed and gv is the volumetric efficiency. Ignoring the losses caused by clearance volume, gv can be expressed as follows, where Dgsv and Dgst are the volumetric efficiency losses of laminar leakage and turbulent leakage, respectively. Csv and Cst are the dimensionless flow leakage coefficients [32].

gv ¼ 1  Dgsv  Dgst

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 15  C sv  pp;out 30  C st  2  pp;out =qp;in ¼1  p2  l  xp p  xp  ðV p =2pÞ1=3

ð19Þ

The consumed power of pump can be described by:

_ p ðhp;out  hp;in Þ Wp ¼ m

ð20Þ

hp;out ¼ hp;in þ ðhp;out;s  hp;in Þ=gp;s

ð21Þ

Wherein, gp,s is the isentropic efficiency of the pump. The off-design efficiency varies and it is expressed by the following equation in terms of volume flow rate variation and the design point efficiency [33]. a0, a1, a2 and a3 depend on the specific case of pump.

gp;s;design

¼ a0 þ a1

Q_ p Q_ p;design

! þ a2

Q_ p Q_ p;design

!2 þ a3

Q_ p

!3

Q_ p;design ð22Þ

The model of the turbine is simplified and computed as follows:

_ exp;in ¼ C v m

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qexp;in ðpexp;in  pexp;out Þ

ð23Þ

_ exp;in ðhexp;out  hexp;in Þ W exp ¼ m

ð24Þ

hexp;out ¼ hexp;in  ðhexp;in  hexp;out;s Þ  gexp;s

ð25Þ

   2  3 u u u þ b2 þ b3 cs cs cs

 CF 2 ¼ c0 þ c1

!0:1

3.2. Pump and expander models

gp;s

Wherein, gexp,s is the isentropic efficiency of the turbine. It is calculated by the design value (0.7) multiplying two correction factors [20]. The first correction factor (CF1) is related to the variation of u/cs that results from the variation of the isentropic enthalpy drop at off-design conditions, where u is the wheel tip speed and cs is pffiffiffiffiffiffiffiffiffiffiffiffiffiffi spouting velocity (cs ¼ 2  Dhs ). The second correction factor (CF2) is related to the variation of the mass flow rate from the design value. b0, b1, b2, b3 and c0, c1, c2, c3 depend on the specific case of turbine design.

_ exp m

_ exp;design m



 þ c2

_ exp m

_ exp;design m

ð26Þ 2

 þ c3

_ exp m

3

_ exp;design m ð27Þ

gexp;s ¼ gexp;s;design  CF 1  CF 2

ð28Þ

The global model of the system is obtained by interconnecting each subcomponent model according to the system configuration schematic described in Fig. 1. Net power output and overall effective thermal efficiency are chosen to evaluate system performance. They can be expressed as follows considering the efficiency of the generator gG:

W net ¼ W G  W p ¼ W exp  gG  W p

ge ¼

W e;ICE þ W net B=3600  Hu

ð29Þ ð30Þ

3.3. Optimization model Based on the system model of the CTPC built above, there are five external inputs including the temperature of exhaust gas, the mass flow rate of exhaust gas, the pump rotational speed, the temperature of cooling water and the mass flow rate of cooling water. Once the parameters are given, system performance can be predicted. For each of the engine operating conditions, we can know whether the CTPC system is able to work normally and furthermore obtain the optimal performance if it is able to work normally. Several factors must be considered as constraints to reflect the real operation of the CTPC system, they are listed as follows: (1) To avoid low-temperature corrosion, the outlet temperature of exhaust gas should be maintained over 120 °C which is generally assumed as the temperature at the acid dew point for exhaust gas [2]. (2) The pressure in the CTPC must be kept below 20 MPa due to structural and material limitations [16]. (3) The upper boundary of the expander inlet temperature (EIT) is 400 °C [34]. Temperature beyond this boundary can cause damage to the system due to material temperature constraints being exceeded. (4) The adjustable range of the pump rotational speed is restricted. (5) A further operational requirement for the CTPC is that supercritical conditions are maintained in the gas heater and subcritical conditions are kept in the condenser. Net power output is chosen as the optimization objective rather than cycle efficiency because net power output directly reflects the contribution to the original engine. For a given operating condition which means the mass flow rate and the temperature of exhaust gas are known, the optimization objective function can be expressed as follows:

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G. Shu et al. / Energy Conversion and Management 142 (2017) 188–199

max

_ g; Tg; m _ water ; T water;in Þ W net ¼ max f ðx; m 8 > T T > gout acid > > > > > < T exp;in < T max s:t: pcrit < p < pmax > > > pc > pcrit > > > : xmin 6 x 6 xmax

ð31Þ

4.1. Design condition determination

3.4. Model validation Heat exchangers play an important role in the system. The validation of system model was done by using the equations to calculate an R134a vapor compression cycle system in Ref. [35], because both consist of a condenser with receiver model. Also a subcritical evaporator model has been built using the mathematical approach in this paper. Fig. 4 shows the simulated results in present study and the experimental results in Ref. [35]. At 1000 s, the valve opening is increased by 9.4%; at 1400 s, the mass flow rate of working fluid is increased by 14.3%; at 1800 s, the mass flow rate of secondary fluid in condenser is decreased by 2.7% and finally the mass flow rate of secondary fluid in evaporator is decreased by 13.1% at 2200 s. The transient changes of condensing pressure match well with the experimental ones except for a constant steady state offset. This offset is due to a gap in the initial conditions during 0–1000 s. Also, the variation of evaporator superheat is almost consistent except the initial offset. The average relative errors of condensing pressure and evaporator superheat are 3.49% and 6.22%, respectively. It gives confidence in heat exchanger modeling, especially in the condenser with receiver modeling approach presented before. 4. Results and discussion For the engine in this study, the energy balance test has been conducted under gasoline engine mapping characteristics which covered the engine speed from 1000 rpm to 6000 rpm and the engine load from 0.1 MPa to full load. Only exhaust energy is considered and recovered in this investigation. Figs. 5 and 6 depict the distributions of the temperature downstream the three-way cat-

Pressure (kPa)

alytic converter and the available energy of exhaust gas (above 120 °C), respectively. As it can be seen, the temperature and energy of exhaust gas vary greatly with engine speed and engine load. Under low-speed conditions, they are rather less than those under the rated condition.

In this investigation, two different engine operating conditions are selected to design the CTPC systems. The first one is the rated condition which is commonly chosen for system design, named as operating condition A, where engine speed is 6000 rpm and engine load is the maximum. The second one is a representative for partial load operations based on driving cycles, named as operating condition B, where the engine speed is 2500 rpm and the engine torque is 112.7 N m. Fig. 7 gives the reason for the selection of operating condition B. Three standard cycles for passenger cars are used to analyze the real operating points of the engine: New European Driving Cycle (NEDC), Highway Fuel Economy Test Cycle (HWY), and the WLTC driving cycle for a Class 3 vehicle. All the operating points of each second have been depicted in Fig. 7 except the idle condition. It could be concluded that these scattered operating points are representative for common conditions of the engine. And operating condition B is selected as a design point due to the major operating points of the engine concentrate around it (2500 rpm, 70% load). Table 2 gives the main engine parameters under the two operating conditions, including the mass flow rate and temperature of exhaust gas. And two CTPC systems are designed based on the first law of thermodynamics, thus the specific cycle operating condition parameters and component physical parameters are determined. In order to be simple, system A means the CTPC system designed under operating condition A and system B means the CTPC system designed under operating condition B in the subsequent context. Table 3 shows the designed system parameters. The main operating parameters are designed to be the same, for example, the high pressure in the gas heater is 15 MPa; the expander inlet temperature is 370 °C, exhaust outlet temperature is 200 °C, etc. Table 3 also reveals that system B is much more compact than system A. Both the gas heater and the condenser of system B are smaller than

1090

Pc Pc [Ref]

1050 1010 970 930 1000

1400

1800

2200

2600

Time (s)

Temperature (C)

(a) SH SH [Ref]

25 22 19 16 13 1000

1400

1800

2200

Time (s)

(b) Fig. 4. Model validation: (a) condensing pressure; (b) evaporator superheat.

2600

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G. Shu et al. / Energy Conversion and Management 142 (2017) 188–199

200

Temperature of exhaust gas (ºC ) 180

Engine torque (N.m)

160

770

730 752

140

730 120 100

752 391 519 583

80

690

60 262 40

134 198 262 327 391 455 519 583 648 690 730 752 770

648

2000

3000

4000

5000

Fig. 5. Temperature of exhaust gas (°C).

200 0.00

Exhaust energy (kW)

9.15

180

18.30 27.45

Engine torque (N.m)

160

64.05

140

82.35

36.60 45.75

73.20

27.45

54.90 64.05

120

73.20

36.60

54.90

82.35 91.50

45.75

100 80 60

9.15

18.30

40 20 2000

3000

4000

5000

6000

Engine speed (rpm) Fig. 6. Available energy of exhaust gas (kW).

200

NEDC HWY

WLTC Operating condition B

160

Engine torque (N.m)

– kW r/min Nm kg/h °C kg/h

Content Operating condition A

Operating condition B

Rated 101.3 6000 161.2 34.85 772.48 430.27

Partial load 29.5 2500 112.7 7.55 661.13 117.63

6000

Engine speed (rpm)

140 120 100 80 60 40 20 1000

Condition characteristic Engine power output Speed Torque Fuel consumption Exhaust temperature Exhaust mass flow

Unit

4.2. Off-design performance comparison

1000

180

Parameter

455

198 327

20

1000

Table 2 Main engine parameters under two operating conditions.

2000

3000

4000

5000

6000

Engine speed (rpm) Fig. 7. Operating points analysis under three typical driving cycles.

those of system A according to the total heat exchange areas. By setting the same rotational speed at design point, the pump displacement of system B is also smaller because the mass flow rate of working fluid of system B is less than that of system A.

System performance is compared between system A and system B at off-design conditions. System performance represented by the net power output is predicted and optimized in the condition that the controllable variables are the pump rotational speed and the mass flow rate of exhaust gas. Fig. 8 illustrates the procedure of optimal performance prediction. Exhaust gas parameters under mapping characteristics are set as external input parameters. The mass flow rate and the temperature of cooling water and receiver ambient temperature keep constant. The pump rotational speed is adjusted within its range to obtain optimal performance where Wnet reaches its maximum. Exhaust gas is bypassed partially if high pressure and EIT exceed the upper boundaries. If the constraints cannot be satisfied, the corresponding engine operating condition is infeasible for the CTPC system. Note that temperature is selected for steady state judgment. The criterion is for all t, EIT variation is less than 0.05 K in a time interval of 5 s which can be expressed as |Tt  Tt+5|  0.05 K. Fig. 9 shows the optimal net power output of the CTPC system designed under operating condition A. It can be seen that the optimal net power output varies greatly with the engine operating conditions due to a difference in exhaust energy. The optimal net power output reaches its maximum and the value is 7.47 kW at the rated condition, while it is less than 1 kW under many partial operating conditions. An infeasible zone obviously exists in the map which is indicated in light grey. Further analysis tells that the infeasible zone occupies 64.37% of the whole region which means the CTPC system has to be shut off under most conditions. For this design, under low load conditions, the heat exchangers would be grossly oversized. In order to build the transcritical condition, the mass flow rate of working fluid needs to increase. This will cause an excess heat exchange with the exhaust gas and bring the exhaust outlet temperature less than 120 °C, leading great harm to the heat recovery system, especially an erosion risk in exhaust gas heat exchanger. That means the exhaust energy is not enough to drive the cycle within safe limitations under these conditions. This will lead to not only zero useful power output but also additional fuel consumption if extra weight is considered. Therefore, this design would not be suitable for gasoline engines. However, it might be a good choice if engines can operate in high-speed and high-load region continuously. Fig. 10 depicts the optimal net power output of the CTPC system designed under operating condition B. It shows that the optimal net power output also differs under different engine operating conditions as system A. However, unlike system A, the infeasible zone is fairly small, which occupied only 21.72% of the whole area, and it occurs only under the conditions where engine speed is lower than 3500 rpm and engine load is small. System B shows a great potential to operate in a wide range while still meet the constraints described earlier. In addition, there is a ‘‘saturation area” on the top right where the net power output is a little greater than

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G. Shu et al. / Energy Conversion and Management 142 (2017) 188–199 Table 3 Designed system parameters. Component

Parameter

Unit

Gas heater

Pressure CO2 outlet temperature Exhaust outlet temperature Exhaust pressure Inner tube diameter Outer tube diameter Total length Heat exchange area Condensing temperature Water inlet temperature Water mass flow rate Air side HTC Inner tube diameter Outer tube diameter Total length Heat exchange area CO2 mass flow rate Rotational speed Displacement Impeller diameter Shaft speed

MPa °C °C kPa m m m m2 °C °C kg/s W/m2 K m m m m2 kg/s rpm m3 m rpm

Condenser with receiver

Pump

Expander

System A

System B

15 370 200 101 0.0313 0.1325 79.192 7.7871 25 20 2.1773 126 0.0461 0.0748 32.02 4.6374 0.1523 500 2.629e5 0.020 144,310

15 370 200 101 0.0140 0.0667 31.904 1.4032 25 20 0.4734 126 0.0215 0.0364 12.51 0.8450 0.0331 500 5.734e6 0.020 144,310

200

Start

180

Input parameters: mg, Tgin, mwater, Twaterin, Tamb

Net power output (W) Infeasible zone

Decrease mg

Obtain steady state parameters where Wnet reaches maximum

160

System A

1075

5650

160

Engine torque (N.m)

Dynamic model in Simulink

Manipulate pump speed within its range

Value

1990

6565

4735

2905 3820

140

4735

120

5650

2905 3820

1075

6565 7480

1990

100 160

80 60 40

Infeasible

No

Tgout > 120°C P > Pcrit Pc < Pcrit

20 1000

3000

4000

5000

6000

Engine speed (rpm)

Yes P < 20MPa EIT < 400°C

2000

Fig. 9. Optimal net power output of system A.

No

Yes Output all calculated parameters

200

Draw Wnet on the map based on engine speed and engine torque

180

Net power output (W) Infeasible zone

60

System B

350 600 890

Completed Fig. 8. Procedure of optimal performance prediction (net power output).

2.29 kW. It is understandable for the following reasons: (a) the system is designed under a partial load operating condition; (b) the heat exchange area is slightly undersized and the heat transfer capacity is limited; (c) with the constraints of high pressure and EIT in the CTPC, redundant exhaust gas needs to be bypassed at higher operating conditions. Overall, though the maximum recovery potential of system B is inferior to that of system A, the disadvantage of net power output reduction of system B under higher

Engine torque (N.m)

160

Circle the infeasible zone in magenta

1180 1470

140

1750 2050 2180

120

2290

100 80

1470 2050

2180

350

60

60

890 1750

40 600

20 1000

2375

2290

2000

3000

1180

4000

5000

6000

Engine speed (rpm) Fig. 10. Optimal net power output of system B.

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G. Shu et al. / Energy Conversion and Management 142 (2017) 188–199

conditions degrades in that gasoline engines generally operate at partial load conditions more often. Typical engine operating conditions are extracted from the maps in order to further investigate the gains in effective thermal efficiency of original engine. Figs. 11 and 12 give the variations of overall effective thermal efficiency of different systems with engine speed and engine torque, respectively. From the two figures, effective thermal efficiencies of original engine only, ICE coupled with system A and ICE coupled with system B change the same in trend. Fig. 11 demonstrates that at the same engine torque, the three effective thermal efficiencies rise slowly, keep almost unchanged and finally decrease with engine speed. System A brings benefits to the original engine when engine speed is over 3000 rpm while system B creates gains as long as engine speed exceeds 1200 rpm. The maximum addition to original efficiency is 1.43% at 6000 rpm for system A and 1.79% at 3500 rpm for system B. System A is superior to system B only when engine speed exceeds 5000 rpm. Fig. 12 illustrates that when engine speed is 2500 rpm, the three effective thermal efficiencies increase with engine torque and only decrease at a higher torque. System A shows no positive effect on the original engine. System B surpasses system A and has a maximum improvement of 1.84% on engine effective thermal efficiency.

Effective thermal efficiency (%)

34

ICE ICE+system A ICE+system B

+1.79% 32

30

28

26 Note: Engine torque at each speed equals that at design point B

+1.43%

24 1000

2000

3000

4000

5000

6000

Engine speed (rpm) Fig. 11. Variation of overall effective thermal efficiency with engine speed.

Effective thermal efficiency (%)

35

+1.84%

30

ICE ICE+system A ICE+system B

25

20

Note: Engine speed at each torque equals that at design point B

15

10 0

20

40

60

80

100

120

140

160

Engine torque (N.m) Fig. 12. Variation of overall effective thermal efficiency with engine torque.

For a quantitative analysis, system performance under Highway Fuel Economy Test Cycle is investigated. Fig. 13 shows the vehicle speed profiles for the Highway Fuel Economy Test Cycle (HWY), as well as the operating conditions of the engine, the exhaust conditions and the net power output produced by the CTPC. In the simulation, the optimal CTPC net power output maps (shown in Figs. 9 and 10) are integrated into a vehicle model. As it can be observed here, the engine speed ranges between the idle speed and 3000 rpm during the HWY cycle. The engine conditions are completely located within the infeasible zone of system A, thus the net power output of system A during the HWY cycle equals zero as depicted. For system B, the net power output varies with the vehicle speed and the maximum value equals 2136.38 W. The average net power output during the HWY cycle is 524.17 W, which further indicates that the average fuel consumption reduction during the HWY cycle over the original engine is 2.84%. It should be noted that exhaust temperature and CTPC system may have slow responses with a time constant and they are not considered here. Future research will involve these factors.

4.3. Operating guidance under engine mapping characteristics In order to achieve the optimal net power output described in Figs. 9 and 10, the CTPC systems need to be adjusted carefully according to the engine conditions. The pump rotational speed and the mass flow rate of exhaust gas are adjusted. Figs. 14–16 give the operating guidance for system A and system B under the whole mapping characteristics of the engine, including the mass flow rate of working fluid and the bypass ratio of exhaust gas. From Figs. 14 and 15, we can observe that the distributions of the mass flow rate of CO2 are similar to those of the net power output discussed before. The mass flow rate of CO2 ranges from 0.056 kg/s to 0.207 kg/s in system A and from 0.017 kg/s to 0.046 kg/s in system B, which will result in a much smaller receiver in system B. In the ‘‘saturation area” of system B, the mass flow rate of CO2 changes little due to the heat capacity limitation of the gas heater. From Fig. 16, it can be seen that the exhaust gas is wholly bypassed in the infeasible zone, partially bypassed in the ‘‘saturation area” while totally entered in the gas heater in other conditions. Different proportions of exhaust gas are bypassed in the ‘‘saturation area” and the bypass ratio increases with the increase of exhaust energy. Specifically, in the ‘‘saturation area”, the bypass ratio of exhaust gas ranges from 1.19% to 67.77% and the mean value is approximately 27.63%. When viewed from the perspective of energy, the heat capacity of system B is 35 kW and excess energy needs to be bypassed. At the rated condition of the engine, almost 58 kW of exhaust energy is bypassed. In system A, the map is omitted here due to the 0–1 mode: none of exhaust gas flows into the gas heater in the infeasible zone while all enters into the gas heater under other conditions. All the three maps form an operating guidance for the CTPC systems. Providing that the condition of the gasoline engine changes from 6000 rpm and 100% load to 4000 rpm and 100% load, in system A, the pump rotational speed should be lowered to make the mass flow rate of CO2 change from 0.207 kg/s to 0.120 kg/s and meanwhile the exhaust bypass valve is closed all the time. While in system B, only the exhaust bypass valve is adjusted to make the bypass ratio be the recommended value, and there is no need to adjust the pump rotational speed since the mass flow rate of CO2 is almost the same. Nonetheless, if the engine condition changes to 2000 rpm and 100% load, the CTPC system A need to be shut off due to the safe issues, while system B can operate continuously if the pump rotational speed is reduced and the bypass valve is fully closed. The three maps can be further embedded in a control model as a regulation reference.

197

Vehicle speed (km/h)

G. Shu et al. / Energy Conversion and Management 142 (2017) 188–199

120 90 60 0 3000 2000

Engine torque (N.m)

1000 160 120

Exhaust temperature (K)

80 40 0

800 600 400 200 0

0.04

Net power output (W)

0.02 2400

System A

System B

0.00

Average of system B

1800 1200

Exhaust mass flow rate (kg/s)

Engine speed (rpm)

30

600 0 0

100

200

300

400

500

600

700

Time (s) Fig. 13. Sample: vehicle speed, engine conditions, exhaust conditions and CTPC net power output during HWY driving cycle.

200

200 System A 0.170 0.189 0.152

Engine torque (N.m)

0.115 0.078

0.133

80

0.189 0.207

0.096

100

0.056

0.029 0.033 0.037 0.041

0.021

120

0.045 0.049

100 0.041

80

40

40

0.025 0.033

4000

5000

6000

Engine speed (rpm) Fig. 14. Guidance for mass flow rate of CO2 of system A.

5. Conclusions This current work is focused on the analysis of engine operating conditions where the CTPC system is designed for waste heat recovery of gasoline engines. It will provide guidance for WHR system design for gasoline engines, not only useful for the CTPC systems, but also for other WHR systems (e.g. ORCs). Under the full speed and load range of the engine, optimal net power output is predicted by employing a dynamic model. Operating guidance has been given for system operation and further controller design. Main conclusions are listed below:

0.045

0.017

20

20 3000

0.021

140

60

2000

0.017

0.025

60

1000

System B

160

0.115

0.170

0.133

Mass flow rate of working fluid (kg/s) Infeasible zone

0.096

0.152

120

180

0.078

160 140

0.056

Engine torque (N.m)

180

Mass flow rate of working fluid (kg/s) Infeasible zone

1000

2000

3000

0.037

4000

5000

6000

Engine speed (rpm) Fig. 15. Guidance for mass flow rate of CO2 of system B.

(1) Design operating condition does actually have a great influence on system performance under the whole engine conditions. A partial load condition determined by driving cycles is recommended for WHR system design for gasoline engines because the broad range of feasible region will ensure continuity of operation. (2) By adopting the mass flow rate guided operation strategy, the WHR system can operate safely and achieve optimality at the same time when engine condition varies. (3) For system designed under a partial load condition, the average fuel consumption reduction during the Highway Fuel Economy Test Cycle over the original engine is 2.84%.

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G. Shu et al. / Energy Conversion and Management 142 (2017) 188–199

200 180

Bypass ratio of exhaust mass flow rate Infeasible zone

System B

0.00% 1.50% 12.0%

Engine torque (N.m)

160

61.4%

50.9%

140

24.0% 33.9% 42.4%

42.4% 120

12.0% 0.00%

100

50.9% 61.4%

24.0%

67.8%

33.9%

80

Table A.2 Elements of fc(xc,uc) for the condenser with receiver model. Symbol

Expression

f1 f2 f3 f4 f5 f6 f7 f8

_ rec;out _ in  m m ai1 pDi L1 ðT w1  T r1 Þ þ m_ in ðhin  hg Þ ai2 pDi L2 ðT w2  T r2 Þ þ m_ rec;out ðhg  hout Þ _ rec;out ðhout  hf Þ UAðT amb  T rec Þ þ m ai1 pDi L1 ðT r1  T w1 Þ þ ao pDo L1 ðT water1  T w1 Þ ai2 pDi L2 ðT r2  T w2 Þ þ ao pDo L2 ðT water2  T w2 Þ ao1 pDo L1 ðT w1  T water1 Þ þ m_ water hwater12  m_ water hwaterout ao2 pDo L2 ðT w2  T water2 Þ þ m_ water hwaterin  m_ water hwater12

0.00% 0.00%

60 100%

40

1.50%

20 1000

2000

3000

0 00% 5000

4000

Appendix B

6000

For the condenser model, Eqs. (1)–(3) are integrated for both the working fluid and the cooling water. The limits of integration are z1 = 0 and z2 = L1 for the superheated region, and z1 = L1 and z2 = Ltotal for the two-phase region. Mass conservation of working fluid in the superheated region:

Engine speed (rpm) Fig. 16. Guidance for bypass ratio of exhaust gas of system B.

It should be noted that current investigation is more theoretical and the optimal values are more ideal than reality. Pressure drops and heat leaks will occur in an actual scenario. Model improvement and validation need to be conducted to match and reflect an actual scenario in future work.

!   1  1  dhg dpc dL1 @q 1 @q    1  qg Þ Acs ðq þ þ Acs L1  1  dpc dt dt @pc h1 2 @ h pc   1  dhin 1 @q _ in ¼ 0 _ 12  m þm þ Acs L1   2 @ h1 pc dt

ðB:1Þ

Energy conservation of working fluid in the superheated region:

Acknowledgements This work was supported by the State Key Program of National Natural Science Foundation of China (No. 51636005). The authors gratefully acknowledge them for support of this work. Appendix A

 1  q hg Þ  1h Acs ðq g

dL1 dt

! !    1  1  dhg 1 dhg  @ q 1 @q dpc    q1 þ h1 þ þ Acs L1 1  dpc  1 dt 2 dpc @pc h1 2 @ h pc  !   1 dhin 1 @ q1  _ in hin _ 12 hg  m 1 þ h þm þ Acs L1 q 1  dt 2 @h pc

¼ ai1 pDi L1 ðT w1  T r1 Þ

See Tables A.1 and A.2.

ðB:2Þ

Energy conservation of tube wall in the superheated region:

  dT w1 dL1 cpw qw Aw L1 þ ðT w1  T w12 Þ dt dt

Table A.1 Elements of Zc(xc,uc) for the condenser with receiver model. Symbol

Expression

z11

Þ   qf ð1  c  1  qg c Acs ½q       dqg dqf       Acs L2 c dp þ ð1  cÞ dp þ L1 @@pq1  þ 12 @@qh 1 

z12

c

z13

Acs L2 ðqg  qf Þ

z14 z21

1

z22

c

c

c

z32 z33

Acs L2 ðqf hg  qf hf Þ

z34 z42

hc;out  hg

z44

z77 z78 z81 z87

1

dhg pc dpc



dqg u V dpc g g qg ug qf uf qg qf

c

Þðqg  qf Þ Acs ð1  c

dq

þ dpg qg V g þ dp f uf V f þ dpf qf V f  du

c

c

du

c

 hc;out

cpw qw Aw ðT w1  T w2 Þ cpw qw Aw L1 cpw qw Aw L2   water1 h Acswater ðq water1  qwater12 hwater12 Þ    d qwater1 dh 1  water1  A L q þh cswater 1 water1 water1 2 dT dT    d qwater1 dh 1  water1  þh water1 2 Acswater L1 qwater1 dT dT   water2 h Acswater ðqwater12 hwater12  q water2 Þ    d qwater2 dh 1  water2  A L q þ h water2 water2 2 cswater 2 dT dT

q uf qg qf

g ug  f

ðB:4Þ

Energy conservation of working fluid in the two-phase region:

c

q

  dq dq dpc dL1  g þ ð1  c Þ f þ Acs L2 c dt dpc dpc dt

_ out  m _ 12 ¼ 0 þm

 1

c

ðB:3Þ

Mass conservation of working fluid in the two-phase region:

Þðqf hg  qf hf Þ Acs ð1  c i h dðq h Þ qf hf Þ Þ dðdp Þhg ddpqf  1  dpg g þ ð1  c hg ddpqg  ð1  c Acs L2 c c c

z51 z55 z66 z71

h1

dhg pc dpc

1

  q hg Þ 1h Acs ðq 1 g     g   hg Þ @ q 1  þ 1 @ q 1   1 dh Acs L1 12 q þ ðh 1  @p  2 @h dp c

z31

h1

¼ ai1 pDi ðT r1  T w1 ÞL1 þ ao pDo ðT water1  T w1 ÞL1 

dq

g

dpc

dq

V g þ dp f V f c



dL Þðqg hg  qf hf Þ 1 Acs ð1  c dt   dðqg hg Þ dðqf hf Þ dpc  Þ þ Acs L2 c þ ð1  c 1 dpc dpc dt  dc _ 12 hg _ out hout  m þ Acs L2 ðqg hg  qf hf Þ þ m dt ¼ ai2 pDi ðT w2  T r2 ÞL2

ðB:5Þ

Energy conservation of tube wall in the two-phase region:

  dT w2 dL1 cpw qw Aw L2 þ ðT w12  T w2 Þ dt dt

¼ ai2 pDi ðT r2  T w2 ÞL2 þ ao pDo ðT water2  T w2 ÞL2 Energy conservation of water in the superheated region:

ðB:6Þ

G. Shu et al. / Energy Conversion and Management 142 (2017) 188–199

dL1 water1  q  water1 h Acswater ðq water12 hwater12 Þ dt !  water1  water1 1 dh dq dT waterout dT water12   water1 þ Acswater L1 q þ hwater1 þ 2 dT dT dt dt _ water hwater12 þ m _ water hwaterout m ¼ ao1 pDo L1 ðT w1  T water1 Þ

[10]

ðB:7Þ

dL1 dt !  water2  water2 dh d dT waterin dT water12 water2 q q water2 þh þ dT dT dt dt

water2 Þ  water2 h Acswater ðqwater12 hwater12  q

_ water hwater12  m _ water hwaterin þm ¼ ao2 pDo L2 ðT w2  T water2 Þ For the receiver model, the conservations of mass and energy for the receiver are written as follows:

dðmrec urec Þ _ rec;out hout þ UAðT amb  T rec Þ _ rec;in hin  m ¼m dt

[16] [17] [18]

[19]

ðB:10Þ

dqg dq du dug ug V g þ q V g þ f uf V f þ f qf V f dpc dpc g dpc dpc ! ! # qg ug  qf uf dqg dqf qg ug  qf uf _ rec  Vg þ V f p_ c þ m qg  qf dpc dpc qg  qf _ rec;in hin  m _ rec;out hout þ UAðT amb  T rec Þ ¼m

[14]

ðB:9Þ

Assuming the receiver pressure to be equal to that of the condenser and Eq. (B.10) can be deduced as follows:



[13]

[15]

ðB:8Þ

dmrec _ rec;out _ rec;in  m ¼m dt

[11] [12]

Energy conservation of water in the two-phase region:

1 þ Acswater L2 2

[9]

[20]

[21]

[22]

[23]

ðB:11Þ

[24]

Combine Eqs. (B.1)–(B.8), (B.9) and (B.11), the final model is obtained as Eq. (9).

[25]

References

[26]

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