Exergy analysis of supersonic steam jet condensed into subcooled water

Annals of Nuclear Energy 138 (2020) 107224

Contents lists available at ScienceDirect

Annals of Nuclear Energy journal homepage: www.elsevier.com/locate/anucene

Exergy analysis of supersonic steam jet condensed into subcooled water Wenjun Li ⇑, Lili Miao School of Mechanical Engineering, Xi’an Shiyou University, Xi’an 710065, PR China

a r t i c l e

i n f o

Article history: Received 18 June 2019 Received in revised form 5 November 2019 Accepted 16 November 2019

Keywords: Supersonic steam jet Exergy analysis Two-phase flow Condensation

a b s t r a c t Exergy analysis of supersonic steam jet condensed into subcooled water was conducted via a numerical method. A thermal phase-change model was employed by Ansys, and considering turbulence interaction and turbulent dispersion force, three-dimensional simulations were conducted to simulate supersonic steam jet at different test conditions. Results indicated that the exergy flow of steam jet was mainly affected by water temperature and inlet steam pressure. The total physical exergy flow decayed monotonically due to the significant irreversible losses caused by heat and momentum transfer. For underexpanded supersonic steam jet, the peak distributions of steam kinetic exergy flow were obtained. The peak distributions of the water enthalpy exergy flow were also obtained at low water temperatures (293.15 K and 303.15 K). In addition, water temperature was found to be the main factor in the damping-energy properties of the supersonic steam jet. The damping-energy ratio and decay rate of the time-averaged kinetic energy decreased with the increase in water temperature, and a correlation was proposed to predict the time-averaged kinetic energy decay ratio. Ó 2019 Elsevier Ltd. All rights reserved.

1. Introduction Direct contact condensation (DCC) between high-velocity steam jet and subcooled water is a typical condensation phenomenon. Compared with film and drop condensations, DCC has excellent heat, mass, and momentum transfer capability and has been widely used in various industries, such as steam-water injector (Cai et al., 2012; Li et al., 2014; Trela et al., 2010), nuclear reactor emergency core cooling systems (Narabayashi et al., 1997), and mixing-type heat exchanger (Chen et al., 2016). Considerable research has been conducted to study the performance, condensation patterns, flow field, and flow and transfer characteristics of the steam jet condensed into subcooled water. The patterns of steam jet condensed into subcooled water can be classified into two categories according to the operating conditions of stable jet and unsteady jet (Chan and Lee, 1982). Several operating conditions (e.g., steam pressure, steam mass flux, and water temperature) and geometric parameters (dimensions of steam nozzles) affect the condensation pattern of DCC significantly. Condensation regime maps that can describe the flow pattern of DCC have been investigated by many researchers (Xu et al., 2013; Wu et al., 2009; Zong et al., 2016). For stable condensation jet, a continuous and fixed steam region called steam plume can be formed. A phasic interface can be ⇑ Corresponding author. E-mail address: [email protected] (W. Li). https://doi.org/10.1016/j.anucene.2019.107224 0306-4549/Ó 2019 Elsevier Ltd. All rights reserved.

observed around the steam plume. As one of the most common condensation patterns, the stable condensation jet has been studied experimentally and theoretically. According to the steam nozzle types, stable condensation jets can be classified into sonic and supersonic steam jets (Wu et al., 2009). The penetration characteristics of sonic steam jet were investigated by Kerney et al. (1972) and Weimer et al. (1973), and correlations were developed to predict the dimensionless penetration length of the steam plume. Three types of steam plumes (i.e., conical, ellipsoidal, and divergent shapes) for sonic steam jet were observed by Chun et al. (1996). Meanwhile, some researchers developed condensation regime maps for the sonic steam jet condensed into subcooled water (Wu et al., 2009; Petrovic et al., 2007). Considerable research on supersonic steam jet has also been conducted. Wu et al. (2007) reported six types of steam plume shapes for supersonic steam jet and developed empirical correlations to predict the dimensions of steam plumes. The dimensions of steam plume and flow field, such as the total pressure and temperature fields, were also studied by Kim et al. (2004). Furthermore, the average heat transfer coefficient for supersonic steam jet was evaluated experimentally, and some theoretical models were developed (Chun et al., 1996). However, the measuring probe may have a significant influence on the flow field due to the supersonic flow. In other words, the exact details of the steam jet flow field are difficult to obtain. Consequently, the mechanism of DCC is also difficult to obtain. Fortunately, computer science has been continuously developing, and

2

W. Li, L. Miao / Annals of Nuclear Energy 138 (2020) 107224

Nomenclature c dcr de ef eh ek Eh Ek Ephy Ej Kj f ps Qs Qw

velocity, m/s nozzle throat diameter, mm nozzle exit diameter, mm exergy flux, W/m2 specific enthalpy exergy, J/kg specific kinetic exergy, J/kg enthalpy exergy flow, W kinetic exergy flow, W physical exergy flow, W time-averaged kinetic energy decay ratio damping-energy ratio nozzle area ratio, (de/dcr)2 steam pressure, kPa heat flux from vapor phase to the interface, W/m2 heat flux from the interface to liquid phase, W/m2

with the help of computational fluid dynamics (CFD), the flow field details of steam jet can be obtained with greater precision. Using commercial software CFX, Gulawani et al. (2006) conducted a 3D simulation to investigate the steam jet condensed into subcooled water. The DCC phenomena in the steam jet and steam-water injector were simulated by Shah et al. (2010, 2011, 2013, 2014), and the simulation results were consistent with the experimental results. The steam involved in the air condensed into subcooled water was simulated by Qu et al. (2015, 2016), and the non-condensed air deteriorated the mass transfer between steam and water. Through NEPTUNE_CFD and OpenFOAM, the DCC phenomenon with a low steam mass flux was simulated by Patel et al. (2014). The chugging flow of DCC was simulated by Tanskanen et al. (2014) and Li et al. (2015a). Meanwhile, the DCC of steam and water in a tee junction was also studied by Li et al. (2015b). Both sonic and supersonic steam jets were studied by Zhou et al. (2016, 2017a–d) based on the CFD method; they analyzed the flow pattern, flow characteristics, flow field, and influence of non-condensable gas. The steam jet through a double-hole nozzle was simulated by Wang et al. (2017), and the effect of the distance between two holes on the flow characteristics was studied. A typical application of supersonic steam jet condensation into subcooled water is the steam-water injector. In recent years, exergy analysis was performed on the DCC phenomenon in a steam-water injector. Choosing the injector as a control volume, exergy analysis based on the global physical exergy balance has been used by several researchers as a tool to evaluate the efficiency

T TEk v x x0 X

temperature, K time-averaged kinetic energy, W specific volume, m3/kg axial distance from nozzle exit, mm axial distance from jet source, mm Dimensionless axial distance, x0 /de

Greek symbols a volume fraction q density, kg/m3 Subscripts 0 nozzle exit s steam w water

of and exergy losses in steam-water injectors (Cai et al., 2012; Li et al., 2014; Trela et al., 2010; Li et al., 2018). To the best of our knowledge, previous studies were mainly based on the first law of thermodynamics, and few studies on the exergy analysis of steam jet have been conducted. In the present work, the exergy transfer and transition of steam jet condensed into subcooled water were investigated with the help of a numerical method. The physical exergy flow in the flow field was presented, and the effects of physical and geometric parameters on exergy flow were investigated. Moreover, the damping-energy properties of the supersonic steam jet were investigated according to free turbulent jet theory, and a correlation was proposed to predict the time-averaged kinetic energy decay ratio. Quantitative research of exergy transfer process would be helpful in analyzing the exergy transfer characteristics of DCC between high-velocity steam jet and subcooled water. Moreover, it could provide a theoretical basis for analyzing the lifting-pressure mechanism in a steam-water two-phase injector. 2. Physical problem description Many researchers have conducted experimental and numerical studies on the DCC of sonic and supersonic steam jets. The condensation regime for supersonic steam jet in subcooled water was studied experimentally by Wu et al. (2009). The schematics of the experimental system and nozzle are shown in Figs. 1 and 2, respectively. The apparatus of the experimental system

Fig. 1. Schematic diagram of experimental system (Wu et al., 2009).

3

W. Li, L. Miao / Annals of Nuclear Energy 138 (2020) 107224

interpenetrating continua incorporates the concept of phasic volume fractions, denoted here by aq. The volume of phase q, Vq, is defined by

Z

Vq ¼ V

aq dV

ð1Þ

where n X

aq ¼ 1

ð2Þ

q¼1

The continuity equation for phase q is Fig. 2. Schematic diagram of supersonic nozzle (Wu et al., 2009).

n    X   @  aq qq þ r  aq qq ! vq ¼ mpq  mqp þ Sq @t p¼1

ð3Þ

! where v q is the velocity of phase q and mpq characterizes the mass transfer from phase p to q, and mqp characterizes the mass transfer from phase q to p, and Sq is the source term. The momentum balance for phase q is

Fig. 3. Schematic diagram of simulation fluid domain.

   @  aq qq ! v q þ r  aq qq ! v q! vq @t ! ¼ raq rp þ r  sq þ aq qq g n h   i X ! ! _ pq ! þ K pq v p  v q þ m v pq  m_ qp ! v qp p¼1

Table 1 Test conditions in CFD simulation and experiment (Wu et al., 2009). Parameter

Experiment

Nozzle throat diameter dcr, mm 8 Nozzle exit diameter de, mm 8.8, 9.6, 10.4, 11.2 Nozzle area ratio f 1.21, 1.44, 1.69, 1.96 Inlet steam pressure ps, kPa 200–600 Inlet steam temperature Ts, K Saturated Water temperature Tw, K 293.15–343.15 Ambient pressure pa, kPa 102

CFD 8 8.8, 9.6, 10.4, 11.2, 12 1.21, 1.44, 1.69, 1.96, 2.25 200–600 Saturated 293.15–333.15 102

mainly consists of a steam generator, a water tank (3000 mm  1000 mm  1200 mm), supersonic nozzles, and several valves. Saturated steam enters the subcooled water through a horizontal supersonic nozzle. In the present work, a reduced cylindrical fluid domain was adopted instead of a full cuboid water tank, with diameter of 300 mm and length of 800 mm, as shown in Fig. 3. In this study, the dimensions of the steam nozzle are the same as those in the study of Wu et al. (2009). The effects of physical (e.g., steam pressure, steam mass flux, and water temperature) and geometric (nozzle dimensions) parameters on exergy transfer, transition, and attenuation in the flow field were studied. The operating conditions and nozzle dimensions in the present work are listed in Table 1. 3. Numerical method and verification In the present work, a 3D steady simulation of the supersonic steam jet condensed into subcooled water was performed. The realizable k-e turbulent model was adopted to simulate the turbulence characteristics of the highly-turbulent jet flow. The twophase Eulerian model was adopted for simulation, in which steam and water were treated as interpenetrating continua. The governing equations of each phase were coupled with pressure and interphase exchange coefficients. The description of multiphase flow as

! ! ! ! !  þ F q þ F lift;q þ F wl;q þ F v m;q þ F td;q

ð4Þ

! ! ! where F q is an external body force, F lift;q is a lift force, F wl;q is a ! ! wall lubrication force, F v m;q is a virtual mass force, F td;q is a turbu! lent dispersion force, R pq is an interaction force between phases. ! _ pq > 0 (phase p mass is being V pq is the interphase velocity. If m ! ! _ pq < 0 (phase q mass is being transferred to phase q), V pq ¼ V p ; if m ! ! _ qp > 0 then transferred to phase p), V pq ¼ V q . Likewise, if m ! ! ! ! _ qp < 0 then V pq ¼ V p . V pq ¼ V q , if m To describe the conservation of energy in Eulerian multiphase applications, a separate enthalpy equation can be written for each phase:

   dpq @  ! ! aq qq hq þ r  aq qq ! u q hq ¼ aq þ s q : r u q  r  q q þ Sq @t dt n X   þ Q pq þ mpq hpq  mqp mqp p¼1

ð5Þ ! where hq is the specific enthalpy of phase q, q q is the heat flux, Sq is a source term that includes sources of enthalpy, Q pq is the intensity of heat exchange between phases, and hpq is the interphase enthalpy. Details of the Eulerian multiphase model are presented in manuals ANSYS Ò Help Viewer, 18.2.0 (2017). In the present work, water was treated as the primary phase, and the properties of water were taken from Fluent database. Steam (the secondary phase) was treated as a compressible ideal gas, and ideal gas law was adopted to calculate its density. The other properties of steam such as viscosity, thermal conductivity, specific heat capacity were taken from Fluent database and assumed to be constant during the simulation. In addition, Symmetric Model was adopted to calculate the interfacial area concentration, which is defined as the interfacial area between two phases per unit mixture volume, as shown in Eq. (6).

  6ap 1  ap Ai ¼ dp

ð6Þ

4

W. Li, L. Miao / Annals of Nuclear Energy 138 (2020) 107224

3.1. Condensation model For condensation models provided by Fluent, condensation rate is determined by relaxation time, void fraction, density and temperature of each phase, which can be regulated by relaxation time. However, the high-velocity flow also has a great effect on condensation rate. Therefore, a condensation model based on thermal equilibrium was developed by Shah et al. (2010) and improved by Zhou et al. (2017a), which included three key factors: interfacial area, interfacial heat transfer coefficients, and interfacial mass transfer. In this model, the mass transfer of steam to water could be divided into two processes, namely, steam to interface and interface to water, as shown in Fig. 4. With the help of a userdefined function, this condensation model has been used by numerous researchers to study the DCC of steam jet in subcooled water (Shah et al., 2010; Zhou et al., 2016, 2017a, 2017b, 2017c, 2017d; Wang et al., 2017). Therefore, this model was also used in the present work.

3.2. Phase interaction

qp f dp Ai 6s p

ð7Þ

where f in Eq. (7) is the drag function, which includes a drag coefficient (CD) based on the relative Reynolds number.

( Cd ¼

f ¼

ð10Þ

where cq is the diffusion coefficient in the qth phase, and the   termr cq raq is turbulent dispersion term that must satisfy the n X





r cq raq ¼ 0

In order to satisfy Eq. (2) the diffusion coefficients for the secondary phases are estimated from the phase turbulent viscosities, lt,q as:

cq ¼

lt;q q¼2n rq

C d Re 24

ð9Þ

In the present work, a Symmetric Model was used for drag formulation. For the Symmetric Model, the density and the viscosity were calculated from volume averaged properties. This model is recommended for flows in which the secondary phase in one region of the domain becomes the primary phase in another, which has also been recommended by previous studies (Shah et al., 2010; Zhou et al., 2017a).

Interface

n X

Dp

ð13Þ

p¼2

3.2.3. Turbulence interaction Compared with previous studies, turbulence interaction has been included in the present work. When using a turbulence model in an Eulerian multiphase simulation, Fluent can optionally include the influence of the dispersed phase on the multiphase turbulence equations. In the present work, the Sato model was adopted for turbulence interaction. The Sato model does not add explicit source terms to the turbulence equations. Instead, in an attempt to incorporate the effect of the random primary phase motion induced by the dispersed phase in bubbly flow, Sato and Sekoguchi (1975) proposed the following relation:

v q ¼ Cl

Ts

ð12Þ

where rq = 0.75 by default. For the primary phase, q = 1, the diffusion term is

ð8Þ

Re > 1000

ð11Þ

q¼1

Dq ¼ 

  24 1 þ 0:15Re0:678 =Re Re  1000 0:44

n      X   @ _ qp þ Sq _ pq  m aq qq þ r  aq qq ! v q ¼ r cq raq þ m @t p¼1

constraint:

3.2.1. Drag force Phase interaction plays a significant role in simulating the momentum transfer between steam and water. In the present work, there are unequal amounts of two fluids (water and steam), and for this type of gas–liquid mixture, the exchange coefficient in Eq. (4) can be written in the following general form:

K pq ¼

3.2.2. Turbulent dispersion force In the previous studies (Shah et al., 2010; Zhou et al., 2017a), ! turbulent dispersion force ( F td;q in Eq. (4)) was not considered. For multiphase turbulent flows using the Eulerian model, Fluent can include the effects of turbulent dispersion forces which account for the interphase turbulent momentum transfer. The turbulent dispersion force acts as a turbulent diffusion in dispersed flows. In the present work, the diffusion in VOF model was adopt for turbulent dispersion force. Instead of treating the turbulent dispersion as an interfacial momentum force in the phase momentum equations, we could model it as a turbulent diffusion term in the governing equations of phase volume fractions. With the turbulent dispersion term, the governing equation for the volume fraction in phase q could be expressed as:

k

2

e

! !   þ C l;p ap dp  U p  U q 

ð14Þ

For the dispersed turbulence models adopted by the present work, k and e are the primary phase turbulence intensity and eddy dissipation rate, respectively. Qs Steam

Qw

Ts

Fig. 4. Schematic diagram of condensation process.

3.3. Numerical method verification Water Tw

3.3.1. Grid independence Owing to the substantial heat, momentum, and mass transfer between steam and water, grid quality is crucial to the convergence of the simulation. In this work, a structure mesh was developed through the O-block method in ICEM. Given the dramatic change in physical parameters near the nozzle exit and steamwater interphase, the grid density in these regions was relatively large. The mesh solution was shown in Fig. 5.

W. Li, L. Miao / Annals of Nuclear Energy 138 (2020) 107224

5

Fig. 5. Schematic diagram of grid solution.

Grid independence was verified first before the simulation. Three grid solutions were developed, with numbers 141673, 221457, and 295681. The axial velocity distribution of steam is presented (the inlet steam pressure is 400 kPa, the water temperature is 303.15 K, and the nozzle area ratio is 1.21) in Fig. 6. These distributions were consistent with one another. Based on the computation speed and accuracy, the 221,457 grid was adopted in the following numerical simulation.

1.21, respectively. The comparison of the axial and radial temperature distributions in the simulation and experimental (Wu et al., 2010a,b) results is presented in Fig. 8. The consistency of the simulation and experimental results in terms of the steam plume and temperature distributions indicates the high reliability of the numerical method.

3.3.2. Experimental verification To verify the numerical method, the simulation results were compared with the experimental results. The comparison of the steam plumes in the simulation and experimental (Wu et al., 2009) results is presented in Fig. 7. The inlet steam pressure, water temperature, and nozzle area ratio were 400 kPa, 303.15 K, and

4.1. Exergy analysis model

Fig. 6. Grid independency verification.

4. Analysis model

Several cross-sections (diameter: 100 mm), which were vertical to the flow direction, were investigated, as shown in Fig. 9. The

Fig. 7. Comparion of steam plume between simulation and experimental (Wu et al., 2009) results.

6

W. Li, L. Miao / Annals of Nuclear Energy 138 (2020) 107224

420

340 EXP CFD

400

EXP CFD

330

380 360

tw=293.15 K

310

ps=400 kPa tw=303.15 K

340

ps=500 kPa

T/K

T/K

320

f=1.96

300

f=1.69

320

290

300 0

50

100 x/mm

150

200

280 0

5

(a) Axial temperature distributions

10

15

r/mm

20

25

30

(b) Radial temperature distributions

Fig. 8. Comparison of temperature distributions between simulation and experimental (Wu et al., 2010a) results.

where cx was axial velocity of steam/water, ex was specific enthalpy/kinetic exergy of steam/water, vx was specific volume of steam/water. Enthalpy exergy is the total useful work of a nonflowing fluid, which delivered as a closed system undergoes a reversible process from the given state to the dead state. Kinetic energy can be converted to work entirely. Therefore, exergy of the kinetic energy is equal to the kinetic energy itself. Exergy of per-unit mass is defined as the specific exergy. Therefore, the specific enthalpy and kinetic exergy could be calculated by Eqs. (16) and (17), respectively, and the dead state was set to be 100 kPa and 293.15 K for the calculation of enthalpy exergy.

eh ¼ ðh  h0 Þ  T 0 ðs  s0 Þ ek ¼

1 2 c 2

ð16Þ ð17Þ

By integrating the physical exergy flux with the flow area of each grid node, we obtained the physical exergy flow through a cross-section as follows:

Z

Eh;s ¼

Fig. 9. Schematic diagram of cross section in flow field.

A

Z physical exergy flow through these cross-sections, including steam enthalpy exergy, steam kinetic exergy, water enthalpy exergy, and water kinetic exergy, were studied. The physical exergy flux of each grid node in a cross-section could be calculated by Eq. (15)

ef ;x ¼

cx ex

vx

ð15Þ

Ek;s ¼ A

Z Eh;w ¼ A

Z Ek;w ¼ A

aef ;h;s dA

ð18Þ

aef ;k;s dA

ð19Þ

ð1  aÞef ;h;w dA

ð20Þ

ð1  aÞef ;k;w dA

ð21Þ

Fig. 10. Schematic diagram of turbulent jet.

W. Li, L. Miao / Annals of Nuclear Energy 138 (2020) 107224

Ephy ¼ Eh;s þ Ek;s þ Eh;w þ Ek;w

ð22Þ

where Eh is enthalpy exergy flow, Ek was kinetic exergy flow, Ephy is physical exergy flow, a is steam volume fraction in each grid. 4.2. Turbulent jet analysis model The velocity distributions of the turbulent flow induced by supersonic steam jet have been studied by many researchers. The

7

dimensionless velocity profiles of the steam-driven turbulent jet flow were self-similar and consistent with the Gauss distribution (Wu et al., 2010a,b), as shown in Fig. 10. In this work, the timeaveraged kinetic energy of the steam-driven turbulent jet was investigated. The total time-averaged kinetic energy at the nozzle exit can be expressed as:

TEk;0 ¼

1 2

Z A0

c3s

vs

dA

Fig. 11. Physical exergy flow in flow filed at different water temperatures.

ð23Þ

8

W. Li, L. Miao / Annals of Nuclear Energy 138 (2020) 107224

The total time-averaged kinetic energy at each cross-section in the flow field can be expressed as:

TEk ¼ TEk;s þ TEk;w ¼

1 2

Z A

ac3s

vs

dA þ

1 2

Z A

ð1  aÞc3w

vw

dA

ð24Þ

The decay ratio of the total time-averaged kinetic energy can be expressed as:

Ej ¼

TEk TEk;0

ð25Þ

The damping-energy ratio can be expressed as:

Kj ¼

TEk;0  TEk ¼ 1  Ej TEk;0

ð26Þ

where dA is area element, a is steam volume fraction. 5. Results and discussion 5.1. Physical exergy flow in flow field The physical exergy flow at different water temperatures is shown in Fig. 11. The nozzle throat diameter, area ratio, and inlet

Fig. 12. Physical exergy flow in flow filed at different inlet steam pressures.

W. Li, L. Miao / Annals of Nuclear Energy 138 (2020) 107224

steam pressure were 8 mm, 1.21, and 400 kPa, respectively. The nozzle exit was set as the zero point of the x axis. Given the minor nozzle area ratio, the steam cannot expand completely in the nozzle, whereas the steam could expand after jetting into the water, and the enthalpy exergy of steam could be transited to kinetic exergy. Along the flow direction, the steam condensed into water gradually. Therefore, the enthalpy exergy flow of steam decreased gradually, the kinetic exergy flow of steam initially increased and

9

then decreased, and with the complete condensation of steam, the physical exergy flow decayed to zero finally. With the condensation of steam, the thermal energy of steam transferred to water, thereby increasing the enthalpy exergy of water. The convection heat transfer between hot water and cool water also occurred in the flow field due to the temperature difference, thereby generating significant exergy losses. Therefore, the enthalpy exergy flow of water initially increased and then

Fig. 13. Physical exergy flow in flow filed at different nozzle area ratios.

10

W. Li, L. Miao / Annals of Nuclear Energy 138 (2020) 107224

decreased, especially at lower water temperatures (i.e., 293.15 K and 303.15 K). Given the velocity difference, the momentum transfer between steam and water through the drag force could be taken as a ‘‘collision process,” and the kinetic exergy of steam could be transferred to water. Steam and water finally merged into one. Thus, this momentum transfer process could be taken as a ‘‘completely inelastic collision”, which could generate considerable kinetic exergy losses inevitably. Therefore, the kinetic exergy flow of water was found to be one order of magnitude lower than that of steam. Given the irreversible heat and momentum transfer between steam and water, the total physical exergy flow decreased monotonically along the flow direction, and the decay rate of the total physical exergy flow also decreased with the abatement of the heat and momentum transfer. In addition, with the increase in water temperature, the irreversibility due to heat transfer decreased, thereby decreasing the decay rate of the total physical exergy. The physical exergy flow at different steam pressures is shown in Fig. 12. The nozzle throat diameter, area ratio, and water temperature were 8 mm, 1.21, and 303.15 K, respectively. The inlet steam pressures were 200, 300, 400, 500, and 600 kPa. The steam temperature and velocity at the nozzle exit increased with the inlet steam pressure, thereby enhancing the heat and momentum transfer process. Thus, the decay rate of the steam enthalpy exergy flow and the increase rate of the water enthalpy exergy flow increased, and the peak value of the water enthalpy exergy flow migrated downstream with the increase in inlet steam pressure. In addition, due to the enhanced momentum transfer with an increased inlet steam pressure, the peak value of the steam kinetic exergy flow migrated upstream, and this peak value even faded at high inlet steam pressure (600 kPa). The physical exergy flow at different nozzle area ratios is shown in Fig. 13. The nozzle throat diameter, water temperature, and inlet steam pressure were 8 mm, 303.15 K, and 400 kPa, respectively. The nozzle area ratios were 1.21, 1.44, 1.69, 1.96, and 2.25. At a small area ratio, steam cannot expand completely in the nozzle, whereas steam could expand after jetting into the water, and the kinetic exergy flow of water increased first and then decreased as shown in Fig. 13(a)–(c). The peak value of the steam kinetic exergy flow would fade at a high area ratio. The steam plume penetration length decreases with the increase in area ratio (Wu et al., 2007), which causes the peak value of the water enthalpy exergy flow migrate upstream. In addition, the nozzle area ratio has a minimal effect on the total physical flow.

5.2. Time averaged kinetic energy decay ratio and damping-energy ratio The time-averaged kinetic energy decay and damping-energy ratios at different water temperatures are shown in Fig. 14. Due to the significant irreversible losses of the momentum transfer process in the initial section, the time-averaged kinetic energy decays sharply in this section. The decay rate of the time-averaged kinetic energy decreases with the condensation of steam. In addition, water temperature was found to have a strong influence on the time-averaged kinetic energy decay and damping-energy ratios. The damping-energy ratio decreases with the increase in water temperature. At low water temperatures (293.15–313.15 K), the damping-energy ratio increases sharply when the axial dimensionless distance (X) is in the range of 0.5–2.5, and this interval would increase to 0.5–4 when the water temperature is high (323.15– 333.15 K). The inlet steam pressure and nozzle area ratio have a minimal influence on the time-averaged kinetic energy decay and damping-energy ratios, as shown in Figs. 15 and 16. Overall, the damping-energy ratio tends to decrease slightly with the increase in inlet steam pressure and nozzle area ratio. Similarity solutions for the time-averaged kinetic energy decay ratio of the free turbulent jet flow have been developed by many researchers. The commonly used similarity solution can be expressed as:

Ej ¼

TEk 4:3 ¼ TEk;0 x0 =d

ð27Þ

According to the above results, the time-averaged kinetic energy decay ratio of the steam jet condensed into subcooled water is influenced by some factors, including the inlet steam pressure, nozzle area ratio, and water temperature. Thus, a semiempirical correlation considering these factors was developed in this work and expressed as:

Ej ¼

    0:73 ps T w  273:15 þ 0:09  0:09f  0:01 x0 =d T 0  273:15 p0

ð28Þ

where p0 and T0 are environment conditions, 100 kPa and 293.15 K, respectively. The comparison between the predicted and numerical time-averaged kinetic energy decay ratios is shown in Fig. 17, and the discrepancies are within ± 15%.

Fig. 14. Time averaged kinetic energy decay ratio and damping-energy ratio at different water temperatures.

W. Li, L. Miao / Annals of Nuclear Energy 138 (2020) 107224

11

Fig. 15. Time averaged kinetic energy decay ratio and damping-energy ratio at different steam pressures.

Fig. 16. Time averaged kinetic energy decay ratio and damping-energy ratio at different nozzle area ratios.

6. Conclusions The exergy analysis of the supersonic steam jet condensed into subcooled water was conducted through a numerical method. The exergy analysis model for steam jet was developed. Furthermore, the exergy flow and time-averaged kinetic energy of steam jet were analyzed. The main results are summarized as follows:

Fig. 17. Comparation between predicted and numerical time-averaged kinetic energy decay ratio.

(1) The exergy flow in the flow field was mainly affected by the condensation heat transfer and flow characteristics of steam jet. For the under-expanded supersonic steam jet, the kinetic exergy flow of steam initially increases and then decreases. The physical exergy of steam finally decays to zero with the complete condensation of steam. At low water temperatures (293.15 K–303.15 K), the enthalpy exergy flow of water also initially increases then decreases. Owing to the significant irreversible losses caused by the heat and momentum transfer, the total physical exergy flow decays monotonically, and the decay rate increases with the decrease in water temperature. In addition, the inlet steam pressure and water temperature are the main factors of exergy flow in the flow field.

12

W. Li, L. Miao / Annals of Nuclear Energy 138 (2020) 107224

(2) Water temperature is the main factor for the dampingenergy ratio of the supersonic steam jet condensed into subcooled water. The damping-energy ratio decreases with the increase in water temperature, thereby decreasing the decay rate of the time-averaged kinetic energy. In addition, a correlation is proposed to predict the time-averaged kinetic energy decay ratio, and the discrepancies are within ± 15%.

Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. References ANSYSÒ Help Viewer, 18.2.0, Fluent Theory Guide, SAS IP, Inc., 2017. Cai, Q., Tong, M., Bai, X., 2012. Exergy analysis of two-stage steam-water jet injector. Korean J. Chem. Eng. 29 (4), 513–518. Chan, C.K., Lee, C.K.B., 1982. A regime map for direct contact condensation. Int. J. Multiph. Flow 8 (1), 11–20. Chen, W., Zhao, Q., Wang, Y., Sen, P.K., Chong, D., Yan, J., 2016. Characteristic of pressure oscillation caused by turbulent vortexes and affected region of pressure oscillation. Exp. Thermal Fluid Sci. 76, 24–33. Chun, M., Kim, Y., Park, J., 1996. An investigation of direct condensation of steam jet in subcooled water-experimental study. Int. Commun. Heat Mass Transfer 23 (7), 947–958. Gulawani, S.S., Joshi, J.B., Shah, M.S., RamaPrasad, C.S., Shukla, D.S., 2006. CFD analysis of flow pattern and heat transfer in direct contact steam condensation. Chem. Eng. Sci. 61 (16), 5204–5220. Kerney, P., Faeth, G., Olson, D., 1972. Penetration characteristics of a submerged steam jet. AIChE J. 18 (3), 548–553. Kim, Y.-S., Park, J.-W., Song, C.-H., 2004. Investigation of the stem-water direct contact condensation heat transfer coefficients using interfacial transport models. Int. Commun. Heat Mass Transf. 31 (3), 397–408. Li, S.Q., Wang, P., Lu, T., 2015a. Numerical simulation of direct contact condensation of subsonic steam injected in a water pool using VOF method and LES turbulence model. Prog. Nucl. Energy 78, 201–215. Li, S.Q., Wang, P., Lu, T., 2015b. CFD based approach for modeling steam-water direct contact condensation in subcooled water flow in a tee junction. Prog. Nucl. Energy 85, 729–746. Li, W., Chong, D., Yan, Junjie, 2014. Experimental study on performance of steamwater injector with central water nozzle arrangement. Korean J. Chem. Eng. 31 (9), 1539–1546. Li, W., Liu, M., Yan, J., Wang, J., Chong, D., 2018. Exergy analysis of centered water nozzle steam-water injector. Exp. Therm. Fluid Sci. 94, 77–88. Narabayashi, T., Mizumachi, W., Mori, M., 1997. Study on two-phase flow dynamics in steam injectors. Nucl. Eng. Des. 175 (1–2), 147–156. Patel, G., Tanskanen, V., Kyrki-Rajamäki, R., 2014. Numerical modelling of lowReynolds number direct contact condensation in a suppression pool test facility. Ann. Nucl. Energy 71, 376–387. Petrovic, A., Calay, R., de With, G., 2007. Three-dimensional condensation regime diagram for direct contact condensation of steam injected into water. Int. J. Heat Mass Transf. 50 (9–10), 1762–1770.

Qu, X.-H., Tian, M.-C., Zhang, G.-M., Leng, X.-L., 2015. Experimental and numerical investigations on the air-steam mixture bubble condensation characteristics in stagnant cool water. Nucl. Eng. Des. 285, 188–196. Qu, X.-H., Sui, H., Tian, M.-C., 2016. CFD simulation of steam-air jet condensation. Nucl. Eng. Des. 297, 44–53. Sato, Y., Sekoguchi, K., 1975. Liquid velocity distribution in two-phase bubble flow. Int. J. Multiph. Flow 2 (1), 79–95. Shah, A., Chughtai, I.R., Inayat, M.H., 2010. Numerical Simulation of direct-contact condensation from a supersonic steam jet in subcooled water. Chin. J. Chem. Eng. 18 (4), 577–587. Shah, A., Chughtai, I.R., Inayat, M.H., 2011. Experimental and numerical analysis of steam jet pump. Int. J. Multiph. Flow 37 (10), 1305–1314. Shah, A., Khan, A.H., Chughtai, I.R., Inayat, M.H., 2013. Numerical and experimental study of steam-water two-phase flow through steam jet pump. Asia-Pac. J. Chem. Eng. 8 (6), 895–905. Shah, A., Chughtai, I.R., Inayat, M.H., 2014. Experimental and numerical investigation of the effect of mixing section length on direct-contact condensation in steam jet pump. Int. J. Heat Mass Transf. 72, 430–439. Tanskanen, V., Jordan, A., Puustinen, M., Kyrki-Rajamaki, R., 2014. CFD simulation and pattern recognition analysis of the chugging condensation regime. Ann. Nucl. Energy 66, 133–143. Trela, M., Kwidzinski, R., Butrymowicz, D., Karwacki, J., 2010. Exergy analysis of two-phase steam-water injector. Appl. Therm. Eng. 30 (4), 340–346. Wang, L., Chong, D., Zhou, L., Chen, W., Yan, J., 2017. Numerical simulation on sonic steam jet condensation in subcooled water through a double-hole nozzle. Int. J. Heat Mass Transf. 115, 143–147. Weimer, J., Faeth, G., Olson, D., 1973. Penetration of vapor jets submerged in subcooled liquids. AIChE J. 19 (3), 552–558. Wu, X.-Z., Yan, J.-J., Li, W.-J., Pan, D.-D., Liu, G.-Y., 2010b. Experimental study on a steam-driven turbulent jet in subcooled water. Nucl. Eng. Des. 240 (10), 3259– 3266. Wu, X.-Z., Yan, J.-J., Li, W.-J., Pan, D.-D., Liu, G.-Y., 2010a. Experimental investigation of over-expanded supersonic steam jet submerged in quiescent water. Exp. Therm. Fluid Sci. 34 (1), 10–19. Wu, X.-Z., Yan, J.-J., Shao, S.-F., Cao, Y., Liu, J.-P., 2007. Experimental study on the condensation of supersonic steam jet submerged in quiescent subcooled water: steam plume shape and heat transfer. Int. J. Multiph. Flow 33 (12), 1296–1307. Wu, X.-Z., Yan, J.-J., Pan, D.-D., Liu, G.-Y., Li, W.-J., 2009. Condensation regime diagram for supersonic/sonic steam jet in subcooled water. Nucl. Eng. Des. 239 (12), 3142–3150. Xu, Q., Guo, L., Zou, S., Chen, J., Zhang, X., 2013. Experimental study on direct contact condensation of stable steam jet in water flow in a vertical pipe. Int. J. Heat Mass Transf. 66, 808–817. Zhou, L., Liu, J., Chong, D., Yan, J., 2016. Numerical analysis on pressure distribution for sonic steam jet condensed into subcooled water. Int. J. Heat Mass Transf. 99, 53–64. Zhou, L., Chen, W., Chong, D., Liu, J., Yan, J., 2017a. Numerical investigation on flow characteristic of supersonic steam jet condensed into a water pool. Int. J. Heat Mass Transf. 108, 351–361. Zhou, L., Chong, D., Chen, W., Liu, J., Yan, J., 2017b. Numerical investigation on steam jet submerged in subcooled water under different ambient pressures. Int. Commun. Heat Mass 83, 48–54. Zhou, L., Chong, D., Liu, J., Yan, J., 2017c. Numerical study on flow pattern of sonic steam jet condensed into subcooled water. Ann. Nucl. Energy 99, 206–215. Zhou, L., Wang, L., Chong, D., Yan, J., Liu, J., 2017d. CFD analysis to study the effect of non-condensable gas on stable condensation jet. Prog. Nucl. Energy 98, 143– 152. Zong, X., Liu, J.-P., Yang, X.-P., Chen, Y., Yan, J.-J., 2016. Experimental study on the stable steam jet in subcooled water flow in a rectangular mix chamber. Exp. Thermal Fluid Sci. 75, 249–257.