A physically based, one-dimensional three-fluid model for direct contact condensation of steam jets in flowing water

International Journal of Heat and Mass Transfer xxx (2016) xxx–xxx

Contents lists available at ScienceDirect

International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt

A physically based, one-dimensional three-fluid model for direct contact condensation of steam jets in flowing water David Heinze a,b,⇑, Thomas Schulenberg b, Lars Behnke a a b

Mechanical Engineering, Kernkraftwerk Gundremmingen GmbH, Dr.-August-Weckesser-Str. 1, 89355 Gundremmingen, Germany Institute for Nuclear and Energy Technologies, Karlsruhe Institute of Technology, Hermann-von-Helmholtz-Platz 1, 76344 Eggenstein-Leopoldshafen, Germany

a r t i c l e

i n f o

Article history: Received 1 September 2015 Received in revised form 11 September 2016 Accepted 22 October 2016 Available online xxxx Keywords: Direct contact condensation Steam jet Pipe flow Surface renewal theory Interfacial transfer

a b s t r a c t A simulation model for the direct contact condensation of steam jets in flowing water is presented. In contrast to previous empirical approaches, the model takes into account the underlying physical phenomena governing the condensation process. Condensation at the interface between the steam jet and the surrounding water is calculated according to the surface renewal theory. Entrainment of water into the steam jet is modeled based on the Kelvin–Helmholtz and Rayleigh–Taylor instability theories. The resulting steam-water two-phase flow is simulated based on a one-dimensional three-fluid model. An interfacial area transport equation is used to track changes of the interfacial area density due to droplet entrainment and steam condensation on droplets. The simulation results are in good qualitative agreement with published experimental data. In particular, the dependency of the steam jet length on the flow Reynolds number is properly reproduced. This corroborates our theory that the heat transfer coefficient at the interface of a condensing steam jet can be linked to the water flow rate via the interfacial friction factor. Ó 2016 Elsevier Ltd. All rights reserved.

1. Introduction The direct contact condensation (DCC) of a high-velocity steam jet in subcooled water offers a highly efficient means of steam condensation and is therefore used in many industrial applications, such as thermal degasification, direct contact heat exchangers or the depressurization systems of current light water reactors. Furthermore, the phenomenon is of particular importance for the operation of steam injectors, where efficient steam condensation is crucial for stable operation. Direct contact condensation of a steam jet in subcooled water is based on two different mechanisms: Condensation directly at the steam-water interface on the one hand, and entrainment, atomization and subsequent droplet condensation on the other hand. Condensation due to atomization can be divided into two parts. First, the interface between the steam jet and the water is disrupted due to the high velocity difference between the two phases. Waves arise, expand into the high-speed gas phase and atomize to form small liquid droplets. The large interfacial area density obtained by this turbulent mixing process then establishes the basis for ⇑ Corresponding author. Present address: TEAM CONSULT G.P.E. GmbH, Robert-Koch-Platz 4, 10115 Berlin, Germany. E-mail addresses: [email protected] (D. Heinze), schulenberg@kit. edu (T. Schulenberg), [email protected] (L. Behnke).

rapid steam condensation. Accordingly, the initial development of the two-phase jet flow is mainly governed by the momentum transfer from the high-velocity steam to the entrained droplets, while mass and heat transfer dominate with growing interfacial area density. Direct contact condensation in pools, i.e. in free environments, has been the subject of numerous experimental and theoretical studies. Various investigators have shown that for pool DCC, the steam plume length is mainly dependent on the steam mass flux and the temperature of the water pool [1]. In our previous work [2], we have developed a physically-based simulation model that can reproduce these dependencies. In contrast, only little research has been devoted to DCC in channels, i.e. in flowing water. Recent experiments by Xu et al. [3] highlight a strong dependency of the jet shape and length on the water flow rate, more precisely the flow Reynolds number. This is attributed to the turbulent flow of the water surrounding the jet, which enhances the heat and mass transfer at the jet interface. The authors proposed an empirical correlation for the length of a steam jet in flowing water. However, there are no physically-based models that can predict the experimentally observed trends for channel DCC, most important the interrelation between water flow rate and steam condensation.

http://dx.doi.org/10.1016/j.ijheatmasstransfer.2016.10.076 0017-9310/Ó 2016 Elsevier Ltd. All rights reserved.

Please cite this article in press as: D. Heinze et al., A physically based, one-dimensional three-fluid model for direct contact condensation of steam jets in flowing water, Int. J. Heat Mass Transfer (2016), http://dx.doi.org/10.1016/j.ijheatmasstransfer.2016.10.076

2

D. Heinze et al. / International Journal of Heat and Mass Transfer xxx (2016) xxx–xxx

Nomenclature a A aif b c C cp d E0 f h k l L _ m M Ma Nu Oh p Pr q_ r R Re Rs s t ^t T w W We x z

acceleration cross-section interfacial area density wave crest thickness velocity constant, coefficient specific isobaric heat capacity diameter entrainment coefficient friction factor enthalpy wave number jet penetration length dimensionless jet penetration length mass flux volume-specific total interfacial shear force Mach number Nusselt number Ohnesorge number pressure Prandtl number heat flux radial coordinate radius Reynolds number specific gas constant specific entropy time dimensionless time temperature speed of sound channel width in stratifed flow Weber number mass fraction axial coordinate

Greek letters a heat transfer coefficient c proportionality factor for the vorticity layer C volumetric mass source term d thickness

This article presents a simulation model for DCC in flowing water which is capable of reproducing this relationship. The approach is based on a one-dimensional three-fluid model and accounts for the dominant physical processes of droplet condensation on the one hand, and condensation at the interface between the two-phase jet and the flowing water on the other hand. Modeling of droplet condensation has been subject of our previous work [2], so the present work focuses on the exchange processes at the jet interface. In the following, we develop physically-based transfer models for these processes. Afterwards, we describe our theoretical model, the underlying conservation equations and the required closure relations. Subsequently, we give details on our numerical implementation. Finally, the simulation results obtained with our model are compared to experimental data from the literature. 2. Development of exchange models for the jet interface

 ~ k ~k

l m q r s

U

x Indices 0 2ph 32 b bbl c co crit d D drp e en g H if ini KH l m RT sat t

s tot w wall

volume fraction relative volume fraction thermal conductivity wave length dynamic viscosity kinematic viscosity density surface tension shear stress interfacial area source term amplification rate

stagnation condition two-phase flow region Sauter-averaged mean value burst bubble continuous phase condensation critical condition (at sonic velocity) dispersed phase drag droplet nozzle exit entrainment gas phase hydraulic interface property initiation Kelvin–Helmholtz liquid phase mixture property Rayleigh–Taylor saturation turbulence shear total water wall

annular condensing flow seem to be a viable alternative. However, while the geometrical configuration is indeed similar – a steam core surrounded by an annular liquid layer –, the underlying physical mechanism is different: In most experiments, the liquid annulus is formed due to steam condensation at the wall and is often dominated by gravitational effects. For this reason, it seems more appropriate to refer to experiments and models for stratified condensing flow. The geometric layout is somewhat different, but in most experiments the two phases are initially separated and thus independent of each other. Accordingly, the heat transfer coefficient at the liquid side of the jet interface is modeled in analogy to stratified flow. A widely used approach for modeling interfacial transfer in stratified flow is the surface renewal theory [4,5], which considers the fluid motion near the interface. Based on this theory, various authors have proposed correlations for the liquid side heat transfer which follow the functional form

2.1. Interfacial heat transfer coefficient

Nut
Pr 0:5 ¼ C 1
ReCt 2 :

There is little experimental data for the heat transfer at the interface of condensing jets. At first glance, experiments for

Here, Pr ¼ cp lw =kw is the Prandtl number of the liquid phase based on the specific isobaric heat capacity cp [J/(kg K)], the

ð1Þ

Please cite this article in press as: D. Heinze et al., A physically based, one-dimensional three-fluid model for direct contact condensation of steam jets in flowing water, Int. J. Heat Mass Transfer (2016), http://dx.doi.org/10.1016/j.ijheatmasstransfer.2016.10.076

3

D. Heinze et al. / International Journal of Heat and Mass Transfer xxx (2016) xxx–xxx

dynamic viscosity lw [kg/(m s)] and the thermal conductivity kw [W/(m K)], and Nut and Ret are the turbulent Nusselt and Reynolds numbers. Using appropriate turbulent velocity and length scales (ct ; lt ), these can be defined as

aif ;w lt

Nut ¼

kw c t lt ; Ret ¼

;

mw

ð2Þ

In order to apply this relation to channel DCC, a length scale equivalent to the water layer thickness is required. For the stratified flow of a gas phase g and a liquid phase w (liquid layer thickness dw ) in a rectangular channel (width W), the liquid-phase hydraulic diameter based on the wetted perimeter between liquid and gas is defined as:

ð3Þ

dH;if ;w ¼

where aif ;w is the liquid side heat transfer coefficient [W/(m2 K)] and mw is the kinematic viscosity [m2/s]. In CFD approaches, ct and lt are generally assumed proportional to the scales in the turbulence model, e.g. the turbulent kinetic energy and its rate of dissipation [6]. If this information is not available, the surface renewal theory suggests that the shear velocity

cs ¼

qffiffiffiffiffiffiffiffiffiffiffi s=qw

ð4Þ

can be used as turbulent velocity scale, where s is the shear stress [kg/(m s2)] and qw the liquid density. Due to the high steam velocity, interfacial shear stress in channel DCC is much larger than wall shear stress (sif  swall ). Nevertheless, both are taken into account in the following. The exponent for the turbulent Reynolds number in Eq. (1) is in the order of unity, assuming C 2 ¼ 1 eliminates the dependency on the turbulent length scale, and thus combination of Eqs. (1)–(4) results in

aif ;w

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  kw cp;w  ¼ C1 swall þ sif :

ð5Þ

mw

According to the surface renewal theory, the factor C 1 depends on the dimensionless mean time between turbulent bursts:

C 1 ¼ ^t 0:5 b

ð6Þ

Suggested values for C 1 range between 0.079 and 0.8 (cf. Table 1), equivalent to dimensionless renewal rates ^tb between 160 and 1.6. In the present work, ^tb ¼ 10 was chosen based on a parametric study (cf. Table 5.1). 2.2. Interfacial friction factor For stratified flow, several independent experiments [12–14] have shown a linear relation between the interfacial friction factor f if and the Reynolds number of the liquid phase Rew based on the water layer thickness dw :

f if ¼ C 1
Rew þ C 2 Rew ¼ dw
qw cw =lw

4Aw 4Wdw ¼ ¼ 4dw W W

The equivalent scale for DCC of a steam jet with radius R2ph in a circular pipe with radius Rw is:

dH;if ;w ¼

  4p R2w  R22ph 2pR2ph

¼

2ðR2w  R22ph Þ R2ph

ð7Þ

Accordingly, we suggest to define the Reynolds number for interfacial shear in channel DCC as

Rew ¼

1 dH;if ;w
qw cw =lw ; 4

ð8Þ

using the water–gas hydraulic diameter dH;if ;w according to Eq. (7). The interfacial friction factor is then determined using the correlation by Paras et al. [13] for stratified-atomization flow:

1 f ¼ 3:77  106
Rew þ 0:022 4 if

ð9Þ

3. Theoretical model The transfer coefficients derived in the previous section will now be integrated into a model describing the condensation of a steam jet in flowing water. There exists little experimental data regarding the local flow structure of a turbulent condensing twophase jet. Therefore, some simplifying assumptions have been made in the model development where necessary, in particular regarding the jet profile and the changes in the flow regime. In contrast, appropriate physical model accuracy has been sought regarding the dominant processes of water entrainment and steam condensation. When steam is injected into water flowing in a channel, a conical vapor core is formed, surrounded by the channel water. The geometrical layout is thus apparently similar to annular twophase flow. However, the flow is by no means fully developed: Initially, there is a sharp radial velocity gradient at the boundary between the vapor core and the surrounding water. Waves are formed at this boundary and liquid ligaments are entrained into the gas core, rapidly breaking up into small droplets. These droplets will cause a quick deceleration of the gas phase due to their high inertia. At the same time, steam condenses upon the

Table 1 Correlations for the turbulent Nusselt number Nut ¼ aif lt =k in stratified condensing steam-water flow based on the turbulent Reynolds number Ret . FLOW REGIME

(b) (a)

Lakehal et al. [7] Jensen [8] Murata et al. [9] Hughes and Duffey [10] Kim and Bankoff [11] Kim et al. [12]

Nut Pr0:5 ¼ C 1 ReCt 2

(b) REMARKS

C1

C2

INC., CC

0.079

1

Pr  1

H, C H, C

0.10 0.14 0.11

1 1 1

low Rel high Rel sif  swall , no waves

H, C

0.80

1

sif  swall

INC., C

0.061

1.12

H, CC

0.14

0.96

V, C

0.252

0.535

SE/LE H

= small/large eddy regime = horizontal, V = vertical, INC. = inclined;

(a)

C

= cocurrent,

CC

= countercurrent

Please cite this article in press as: D. Heinze et al., A physically based, one-dimensional three-fluid model for direct contact condensation of steam jets in flowing water, Int. J. Heat Mass Transfer (2016), http://dx.doi.org/10.1016/j.ijheatmasstransfer.2016.10.076

4

D. Heinze et al. / International Journal of Heat and Mass Transfer xxx (2016) xxx–xxx

entrained droplets and at the two-phase jet boundary and the void fraction g decreases, finally resulting in a dispersed bubbly flow with negligible slip. As shown in Fig. 1, we propose a model that divides the jet region into two areas: The two-phase jet with radius R2ph , consisting of a gas phase g and a liquid phase l, and the surrounding annular water layer w (outer radius Rw ). The volume fraction of fluid i is thus defined as

i ¼

Ai Ai ¼ A pR2w

ð10Þ

and consequently w þ l þ g ¼ 1. Here, A is the total flow crosssection and Ai the cross-section occupied by fluid i. The two-phase jet flow is at first considered as a dispersed droplet flow, which turns into a dispersed bubbly flow at lower void fractions. The transition from droplet to bubbly flow is assumed g ¼ 0:5, with the relative volume fraction defined as at ~

~g ¼

g

l þ g

;

~l ¼ 1  ~g :

ð11Þ

In analogy to our previous work on pool DCC [2], entrainment is modeled based on the Kelvin–Helmholtz and Rayleigh–Taylor instabilities, and is assumed to be perpendicular to the flow axis. Accordingly, the entrained mass is added to the two-phase jet without momentum in the axial direction. In the droplet flow regime, the diameter of entrained droplets is obtained by taking into account both primary and secondary atomization and the surface renewal theory is used to model the heat and mass transfer at the two-phase jet boundary, as outlined in Section 2. 3.1. Conservation equations A one-dimensional flow field is assumed in the model development, i.e. only changes of flow properties in the axial direction (coordinate z) are considered. Additionally, the flow is taken to be stationary, since no transient phenomena are to be studied. Moreover, gravitational forces can be neglected with regard to the high momentum of the flow, and heat conduction is negligible in comparison to the occurring turbulent heat transfer processes. Traditionally, one-dimensional two-phase flow simulations are based on a two-fluid model, where each phase is represented by

a separate fluid. In our approach, the two-fluid model is augmented by introducing a third fluid group. This multi-fluid model allows for the distinction of flow properties within a single phase, in this case between the moving water layer and the liquid droplets. The conservation equations for the mass, momentum and total enthalpy of these k fluids are derived from their general form cf. [15,16] based on above assumptions. Then, the mass conservation equation for fluid i has the form k X d ði qi ci AÞ ¼ A Cj!i ; dz j¼1

ð12Þ

where Cj!i is the volumetric mass source term [kg/(m3 s)] due to exchange processes from fluid j to fluid i. Introducing the local static pressure p, the momentum equation can be written as k X   d dp ði qi c2i AÞ þ i A ¼A cif ;ij Cj!i þ Mj!i : dz dz j¼1

ð13Þ

Here, M j!i is the volume-specific total interfacial shear force [kg/(m2 s2)] and accounts for the effects of particle drag and interfacial shear. The velocity at the interface between phase i and j cif ;ij is determined by a donor formulation:

cif ;ij ¼ ci ; cif ;ij ¼ cj ;

Cj!i < 0 Cj!i > 0

ð14Þ

This treatment is common practice in the numerical scheme development of one-dimensional system codes (e.g. RELAP5) because it offers the most realistic treatment of the momentum exchange process [17]. The energy conservation equation based on the total enthalpy is

d dz



i qi ci ðhi þ



 k  X 1 2 1 ci ÞA ¼ A Cj!i hi;j þ cif ;ij ci  c2i 2 2 j¼1  þaif ½; ijq_ j!i þ M j!i cif ;ij ;

ð15Þ

where hi is the specific enthalpy [J/kg] of fluid i, aif ;ij is the interfacial area density [m2/m3] of the interface between phase i and j, and hi;j and q_ j!i represent the specific enthalpy and the sensible heat flux [W/m2] at the i-side of the phase interface between i and j, respectively. The interfacial transfer conditions are given by

Cj!i þ Ci!j ¼ 0;

ð16Þ

Mj!i þ M i!j ¼ 0; Cj!i hi;j þ aif ;ij q_ j!i þ Ci!j hj;i þ aif ;ij q_ i!j ¼ 0:

ð17Þ ð18Þ

3.2. Interfacial area transport In addition to the conservation equations, an interfacial area transport equation [18] for the dispersed phase d is used to track the change of the interfacial area density aif due to droplet entrainment, droplet growth and bubble condensation:

" !# Pk 1 d 2 aif ;d dqd j¼1 Cj!d ðaif ;d cd AÞ ¼ Ud þ  cd A dz 3 qd d dz

Fig. 1. The one-dimensional, two-phase DCC model is based on three fluids: The gas phase g and the liquid phase l in the two-phase flow region 2ph, surrounded by an annular flowing water layer w. pw ; T w ; cw : pressure, temperature and velocity of _ e ; pe : surrounding water; l: two-phase jet length; R2ph : two-phase jet radius; m ~g : relative void fraction with respect to steam nozzle exit mass flux and pressure;  the two-phase region; cen;2ph : entrainment velocity of droplets based on the Kelvin– Helmholtz and Rayleigh–Taylor instabilities; sif : shear stress at the two-phase jet interface; q_ if : interfacial heat flux according to the surface renewal theory.

ð19Þ

Here, Ud is the interfacial area source term [1/(m s)], which is equal to zero for bubbly flow. For droplet flow, the interfacial area source term due to droplet entrainment Ud ¼ Uen is calculated as

Uen ¼

12pR2ph cen;2ph : A d32;en

ð20Þ

Please cite this article in press as: D. Heinze et al., A physically based, one-dimensional three-fluid model for direct contact condensation of steam jets in flowing water, Int. J. Heat Mass Transfer (2016), http://dx.doi.org/10.1016/j.ijheatmasstransfer.2016.10.076

D. Heinze et al. / International Journal of Heat and Mass Transfer xxx (2016) xxx–xxx

Here, cen;2ph is the entrainment velocity for the two-phase jet, described in Section 3.3 below, and d32;en is the Sauter mean diameter of entrained droplets. The latter is equal to the diameter after secondary atomization d32;2 , which is calculated using correlations by Wert [19], Hsiang and Faeth [20], cf. Appendix B.1. 3.3. Turbulent entrainment In the droplet flow regime, i.e. close to the nozzle exit, it is postulated that our previously developed entrainment model for pool DCC is applicable and the entrainment velocity can thus be determined as

cen;RT

E0;RT ~ ¼
kRT xRT : 2p

ð21Þ

A brief summary of this model and the underlying equations for the wavelength ~ kRT and amplification rate xRT can be found in Appendix A. In the bubbly flow regime, entrainment is determined according to the extension of the ‘‘classical” turbulent entrainment assumption by Ricou and Spalding [21]:

cen

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ E0 qm =qw cm :

ð22Þ

The entrainment coefficient is taken as E0 ¼ 0:8, as recommended by Ricou and Spalding [21]. The mean jet density qm and velocity cm are calculated as

g qg þ l ql ; l þ g g qg c2g þ l ql c2l : c2m ¼ l ql þ g qg

qm ¼

ð23Þ ð24Þ

The rationale for this approach is as follows: The entrainment pffiffiffiffiffiffiffiffiffiffiffiffi velocity in Eq. (22) scales as qm c2m , i.e. the square root of the jet momentum flux. By defining cm according to Eq. (24), the common definition of the mean density Eq. (23) can be used while pffiffiffiffiffiffiffiffiffiffiffiffi maintaining cen / qm c2m . In their model development for submerged gas jets, Vivaldi et al. [22] assumed droplet flow inside the jet for a relative void ~g;bbl ¼ 0:5. ~g;drp ¼ 0:8 and bubbly flow below  fraction larger than  In the transition region ~ g;drp > ~g > ~g;bbl , a weighted logarithmic average was used. This approach is adopted in the present model, and the entrainment velocity for the two-phase jet is thus obtained as

"

# ~g  ~g;bbl ~g;drp  ~g cen;2ph ¼ exp lnðcen;RT Þ þ lnðcen Þ ; ~g;drp  ~g;bbl ~g;drp  ~g;bbl  ~g ¼ max ~g;bbl ; minð~g ; ~g;drp Þ :

ð25Þ ð26Þ

This allows to calculate the volumetric mass source term due to entrainment:

Cen;w!l ¼ qw cen;2ph 2pR2ph =A

ð27Þ

3.4. Interfacial shear force

M g!w ¼ sif
2pR2ph =A; 1 1 sif ¼ f if
qg ðcg  cw Þjcg  cw j: 4 2

According to Hanratty [24], accounting for the presence of droplets in the gas phase by using a mean density in Eq. (29) produces unrealistic results. He suggests that the increase in interfacial shear stress due to droplet deposition is counterbalanced by the dampening effect of droplets on turbulence. For this reason, the gas phase density and velocity are used in Eq. (29). As described in Section 2.2, the interfacial friction factor is determined as

1 f ¼ 3:77  106
Rew þ 0:022: 4 if

Mc!d ¼ 

3 CD d qc jcd  cc jðcd  cc Þ; 8 rD

ð30Þ

where the drag coefficient C D is determined according to Ishii and Mishima [23], cf. Appendix B.3. 3.4.3. Wall shear Wall shear is calculated as

Mw!wall ¼ swall
2pRw =A; 1 1 swall ¼ f wall
qw ðcw Þjcw j; 4 2

ð31Þ ð32Þ

and the wall friction factor f wall is determined using a correlation by Fang et al. [25], cf. Appendix B.4. 3.5. Interfacial heat and mass transfer Interfacial heat and mass transfer is modeled with the tworesistance model, described in detail in our previous work [2]:

q_ g!l ¼ al ðT sat  T l Þ q_ l!g ¼ ag ðT sat  T g Þ _ g!l ¼ m

ð33Þ ð34Þ

al ðT sat  T l Þ þ ag ðT satT g Þ hg;if  hl;if

_ g!l : Cl ¼ aif
m

ð35Þ ð36Þ

Here, T sat ; T l and T g are the saturation temperature and the _ g!l is temperature of the liquid and gas phase, respectively, and m the mass flux [kg/(m2 s)] from the gas to the liquid phase. The equivalent equations for the mass flux at the jet interface _ g!w are obtained by replacing the index l with w in Eqs. (33)– m (35), the volumetric mass source term is determined by

Cw ¼

2pR2ph _ g!w :
m A

ð37Þ

3.5.1. Heat transfer coefficients at the jet interface As described in Section 2.1, the heat transfer coefficient at the liquid side of the jet interface is determined as

^t0:5 b

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi kw cp;w ðswall þ sif Þ:

mw

ð5revisitedÞ

The influence of the gas side heat transfer coefficient is largely negligible, as the steam temperature is close to saturation during steam jet DCC. An upper limit is given by molecular gas dynamics [26]:

ð28Þ ð29Þ

ð9revisitedÞ

3.4.2. Interfacial drag The interfacial drag between a continuous phase c and a dispersed phase d is given as

aif ;w ¼

3.4.1. Interfacial shear In the droplet flow regime, the interfacial shear between the gas phase g and the surrounding water w is calculated based on Ishii and Mishima [23] as

5

2

aif ;g ¼

Dhlg q 2C co pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi g ð2  0:798C co Þ 2pRs T g T g

ð38Þ

Please cite this article in press as: D. Heinze et al., A physically based, one-dimensional three-fluid model for direct contact condensation of steam jets in flowing water, Int. J. Heat Mass Transfer (2016), http://dx.doi.org/10.1016/j.ijheatmasstransfer.2016.10.076

6

D. Heinze et al. / International Journal of Heat and Mass Transfer xxx (2016) xxx–xxx

In Eq. (38), Dhlg is the latent heat of condensation, Rs is the specific gas constant [J/(kg K)] of the vapor, and a condensation coefficient of C co ¼ 1 is used according to Aya and Nariai [26]. For 6

2

atmospheric steam, this gives aif ;g  13:1  10 W/(m K).

ð39Þ

which in turn is calculated according to Hughmark [27], cf. B.2. In accordance with our previous work on pool DCC [2], the dispersed-side heat transfer coefficient for both droplet and bubbly flow is taken as

W ad ¼ 10 2 : m K 4

k X

i ¼ 1

()

i¼1

3.5.2. Heat transfer coefficients at the dispersed interface The heat transfer coefficient in the continuous phase ac (ag in the droplet flow regime, al in the bubbly flow regime) is given by the Nusselt number Nuc ,

ac ¼ kc Nuc =dd ;

2. The definition of the volume fraction

ð40Þ

3.6. Qualitative implications of the chosen model approach In the proposed model approach, condensation (and thus steam jet length) is based on two mechanisms: Entrainment and subsequent droplet condensation on the one hand, and condensation at the jet interface on the other hand. These two condensation mechanisms require specification of three parameters that are unknown a priori: The entrainment coefficient for droplet flow E0;RT , the droplet heat transfer coefficient al and the dimensionless surface renewal time ^t b . In specifying these parameters, a superposition of the two condensation mechanisms is achieved. For high values of E0;RT , entrainment becomes dominant and the dependency of the jet length on the mass flux is reproduced well, while the influence of condensation at the jet interface decreases, thus diminishing the dependency of the jet length on the water Reynolds number. A high value for ^t b has the opposite effect. Consequently, a balance between both mechanism has to be found in order to account for both the influence of the steam mass flux and the water Reynolds number. The value of ^tb has little effect on the plume length in pool DCC. Therefore, the values of E0;RT ¼ 0:16 and al ¼ 104 W=ðm2 KÞ are chosen based on our previous study on pool DCC [2]. Based on these results, ^tb is varied and the results are compared to experimental data for channel DCC (section 5.1).

k X i¼1

di ¼ 0: dz

i ¼ Ai =A requires ð42Þ

3. In a confined environment, the flow cross-section AðzÞ is given by the local pipe diameter. Using above closure relations, the ODE system is solved numerically using the explicit fourth-order Runge–Kutta-Fehlberg algorithm [28], as implemented in the GNU scientific library (GSL) [29]. The routine solves the n-dimensional first-order system

dyi ðzÞ ¼ f i ðz; y1 ðzÞ; . . . yn ðzÞÞ dz marching in z-direction, where yi are the n independent variables c1...k ; h1...k ; 1...k1 and p. The functions f i for determining the derivatives are obtained from Eqs. (12), (13), (15). The algorithm uses an adaptive step size control based on a fifth-order error estimator which will keep the local error on each step within a predefined absolute and relative error with respect to the solution yi ðzÞ. The interfacial area transport equation (Eq. (19)) can be solved in the same manner to determine the interfacial area density for the dispersed phase. 4.1. Boundary conditions The water inlet conditions (pw ; hw ; cw ; w ) are set according to the values in the particular experiment to be simulated. At the steam nozzle exit, gas dynamic phenomena due to over- and underexpansion are neglected and the effective-adapted-jet approximation is applied as boundary condition. This approach is widely used in treating two-phase jets with and without condensation [30] and assumes isentropic adaptation from the nozzle exit pressure pe to the ambient pressure pw . The nozzle exit diameter is then replaced by an equivalent diameter based on the adapted flow conditions. Details on this procedure are given in Appendix C. The adapted exit velocity, density and the equivalent exit diameter are used as initial values R2ph ðz ¼ 0Þ; cg ðz ¼ 0Þ; cl ðz ¼ 0Þ; g ðz ¼ 0Þ; hg ðz ¼ 0Þ; hl ðz ¼ 0Þ. A maximum void fraction ~g ðz ¼ 0Þ ¼ 1  108 and a minimum slip of cg =cl ¼ 1:001 are of  enforced to avoid numerical errors due to division by zero. For the same reason, the initial temperatures of the liquid and gas phase are restricted to be slightly below (T l 6 T sat  108 K) or

4. Simulation model For a setup consisting of k different fluids, the conservation equations (Eqs. (12), (13), (15)) constitute a system of 3k ordinary differential equations (ODE). This system can be solved numerically, for instance using a Runge–Kutta algorithm, if sufficient closure relations are provided. The interfacial closure relations (i.e. volumetric mass source term C, volume-specific total interfacial shear force M and sensible _ have been derived in the previous section. If these are heat flux q) known, the system contains 4k þ 2 unknown variables (k ; qk ; hk ; ck ; p; A), thus k þ 2 additional relations are required: 1. The thermodynamic state of each fluid i is determined by the two independent state variables p and h. Then, the density q can be expressed as a function of p and h and its axial derivative is given as

dqi @ qi

dp @ qi

dhi þ ¼ : dz @p hi dz @hi p dz

ð41Þ

above (T g P T sat þ 108 K) the saturation temperature. Due to the effects of supersaturation and spontaneous condensation during steam expansion, the flow at the steam nozzle exit normally consists of a continuous gas phase and dispersed droplets with diameters in the order of one hundred nanometers [31]. These droplets are in equilibrium with the gas phase and therefore do not contribute to the condensation process of the steam jet. However, their mass fraction has to be properly accounted for. It can be safely assumed that the nanodroplets created by spontaneous condensation will be at least one order of magnitude smaller than the subcooled droplets which are subsequently entrained into the steam jet and have diameters in the range of few micrometers. Consequently, it is postulated that the nanodroplets will quickly coalesce with the entrained droplets and the initial value for the interfacial area density is obtained based on the initial liquid volume fraction and the initial diameter of entrained droplets according to step 3 in the following paragraph:

aif ðz ¼ 0Þ ¼

6l

d32;en z¼0

ð43Þ

Please cite this article in press as: D. Heinze et al., A physically based, one-dimensional three-fluid model for direct contact condensation of steam jets in flowing water, Int. J. Heat Mass Transfer (2016), http://dx.doi.org/10.1016/j.ijheatmasstransfer.2016.10.076

7

D. Heinze et al. / International Journal of Heat and Mass Transfer xxx (2016) xxx–xxx

4.2. Solution procedure Once the boundary conditions have been set, each solver step consists of the following major sub-steps: 1. Thermodynamic properties are determined using the IAPWS-IF97 equation of state [32] as a function of the pressure pðzÞ and the gas and liquid phase enthalpies hg ; hl and hw . 2. The entrainment velocity into the two-phase jet cen;2ph is calculated using Eq. (25) with an entrainment coefficient of E0;RT ¼ 0:16 (droplet flow) and E0 ¼ 0:08 (bubbly flow), and the volumetric mass source term Cen;w!l is calculated according to Eq. (27). 3. For droplet flow, Eqs. (B.1)–(B.3), (B.6) are solved to obtain the mean diameter of entrained droplets d32;en based on Eq. (B.4), which is then used to calculate the interfacial area source term due to droplet entrainment Uen according to Eq. (20). 4. Dispersed interfacial heat and mass transfer _ g!l ; Cl ) is solved using Eqs. (33)–(36) with interfa(q_ g!l ; q_ l!g ; m cial heat transfer coefficients al ; ag according to Eqs. (39), (40) in combination with Eq. (B.7), (B.8). 5. Interfacial droplet/bubble drag (M c!d ) is determined using Eq. (30) in combination with Eqs. (B.9)–(B.11), shear stress at the jet interface (sif ; M g!w ) with Eq. (28), (29) based on the interfacial friction factor f if according to Eq. (9). 6. The wall shear (swall ; M w!wall ) is calculated using Eqs. (31), (32) in combination with Eqs. (B.12)–(B.14). 7. Heat and mass transfer at the jet interface _ g!w ; Cw ) is given by Eqs. (33)–(35), (37) with (q_ g!w ; q_ w!g ; m interfacial heat transfer coefficients aif ;w ; aif ;g according to Eqs. (5), (38). 8. The Runge–Kutta-Fehlberg algorithm is invoked to determine the values of the independent variables yi ðz þ DzÞ for the next step. If necessary, the step size Dz is decreased until both absolute and relative error are below the specified value of 108. Initially, the solver is invoked for dispersed droplet flow (liquid ~g ¼ 0:5 is phase l = dispersed phase d). The solver proceeds until  reached, where the solver is re-initialized for dispersed bubbly flow (gas phase g = dispersed phase d) and continues until a min~g ¼ 106 is reached. The axial disimum relative void fraction of 

by Zong et al. [33], steam was injected in the lower half of a rectangular, horizontal channel. The data obtained by these authors is therefore not directly applicable to the present work. Xu et al. [3] performed experiments in a circular, vertical pipe with a diameter of 80 mm, where initially saturated steam (xg;0 ¼ 1) was injected coaxially through an 8 mm nozzle. Water inlet temperature T w , _ e were varied and Reynolds number Rew and steam mass flux m the plume length and radial temperature distribution were measured. The measurements of the plume length will be used for validation of the present model. The authors report stable steam jets with mass fluxes at the nozzle exit between 150 kg/(m2 s) and 500 kg/(m2 s) for steam stagnation pressures between 2 bar and 7 bar. According to the authors, the steam flow rate was controlled by adjusting manual control valves, but no further information is given concerning the detailed steam injection conditions. Therefore, the following assumptions are made to reproduce the experimental setup: The static water inlet pressure pw is assumed to be atmospheric (1 bar). Steam stagnation pressures are taken as 2; 3; . . . ; 7 bar. The experiments include steam mass fluxes below the critical (sonic) flux at atmospheric pressure (279 kg/(m2 s)). For stable injection, the steam exit velocity must therefore be supersonic. The critical (smallest) diameter dcrit is presumed to be the manual control valve. This value was varied in order to obtain the reported steam mass fluxes. The nozzle exit state has been determined based on above suppositions presuming isentropic equilibrium expansion (cf. Appendix D). The injection conditions calculated in this manner are listed in Table 2. 5.1. Parametric study: Determination of ^t b A qualitative comparison of the simulation results for different values of the surface renewal rate ^tb indicates the best agreement with experimental data for ^t b ¼ 10 (Fig. 2). At higher values (i.e. lower heat transfer coefficients at the jet interface), the decrease of the dimensionless jet penetration length

L ¼ l=de

ð44Þ

tance at this point corresponds to the predicted penetration ~g ¼ 106 Þ ¼ l. length: zð

with increasing Reynolds number is not observed. At lower values, the increase in plume length with increasing steam mass flux is not reproduced properly.

5. Validation

5.2. Jet penetration length

In contrast to the extensive literature on pool DCC, there is little data on channel DCC. To our knowledge, the only experimental studies are by Xu et al. [3], Zong et al. [33]. In the experiments

The simulated dimensionless jet penetration length L for different steam flow rates, water flow rates and water temperatures is compared with the experimental measurements from Xu et al.

Table 2 Simulation results for the flow conditions at the nozzle exit. The ratio between nozzle exit and critical diameter is not provided by Xu et al. [3], it has been set to obtain the _ e . Values in parentheses indicate the deviation from the literature data. experimental mass flux m STAGNATION PRESSURE

DIAMETER RATIO

NOZZLE EXIT STATE

p0 =bar

de =dcrit

Mach No. Mae

mass flux _ e =kg=ðm2 sÞ m

pressure pe =bar

quality xg;e

2 3 4 5 6 7 7

1.341 1.415 1.456 1.48 1.497 1.51 1.424

1.73 1.81 1.85 1.88 1.90 1.91 1.82

166 221 276 332 387 442 497

0.31 0.39 0.48 0.57 0.66 0.75 0.89

0.907 0.897 0.891 0.886 0.883 0.880 0.887

( ( ( ( ( ( (

0.04 %) 0.03 %) 0.02 %) 0.05 %) 0.05 %) 0.03 %) 0.03 %)

Please cite this article in press as: D. Heinze et al., A physically based, one-dimensional three-fluid model for direct contact condensation of steam jets in flowing water, Int. J. Heat Mass Transfer (2016), http://dx.doi.org/10.1016/j.ijheatmasstransfer.2016.10.076

8

D. Heinze et al. / International Journal of Heat and Mass Transfer xxx (2016) xxx–xxx

_ e. Fig. 4. Dimensionless penetration length L vs. steam mass flux m

Fig. 2. Dimensionless penetration length L vs. water Reynolds number Rew for different surface renewal rates ^tb (experimental data by Xu et al. [3], water temperature T w ¼ 40 C).

Fig. 5. Dimensionless penetration length L vs. water inlet temperature T w .

Fig. 3. Dimensionless penetration length L vs. water Reynolds number Rew .

[3] in Fig. 3–5. All trends are observed qualitatively, however, satisfactory absolute agreement is only achieved for high steam mass _ e and low Reynolds number Rew (Fig. 5). The decrease of L fluxes m for decreasing steam mass fluxes at low Reynolds numbers is underpredicted (Fig. 4), and the virtual independence of the water temperature at high Reynolds number is not observed (Fig. 5). 5.3. Interpretation of results The developed model is capable of qualitatively reproducing all major trends observed during channel DCC, where condensation is strongly dependent on the flow Reynolds number in addition to the water temperature and the steam mass flux. This suggests that the underlying physical mechanisms have been properly identified and modeled and supports the hypothesis that exchange mechanisms at a condensing jet interface can be treated with similar methods as stratified condensing flows.

The superposition of two condensation mechanisms – condensation on droplets and at the jet interface – is achieved by using appropriate values for the entrainment coefficient (E0;RT ¼ 0:16) and the surface renewal rate (^t b ¼ 10). The employed values are reasonably close to recommended values in similar fields. It should be emphasized that these parameters were kept fixed for all simulations and not adjusted to particular experiments. 6. Conclusion The present work postulates that exchange processes at the interface of a condensing steam jet can be treated with the methodology of the surface renewal theory originally developed for stratified two-phase flows. This concept correlates the interfacial heat transfer coefficient with the interfacial shear stress. In stratified flows, several independent experiments have shown the influence of the water flow Reynolds number on the interfacial friction factor. By adapting these correlations to a condensing steam jet in flowing water, the dependency of the steam jet length on the Reynolds number of the water flow could be reproduced. The developed DCC model is in good qualitative agreement with available experimental data for direct contact condensation of steam jets in flowing water. However, additional experiments are required to confirm the general validity. In order to augment the simulation accuracy, detailed local measurements are necessary

Please cite this article in press as: D. Heinze et al., A physically based, one-dimensional three-fluid model for direct contact condensation of steam jets in flowing water, Int. J. Heat Mass Transfer (2016), http://dx.doi.org/10.1016/j.ijheatmasstransfer.2016.10.076

D. Heinze et al. / International Journal of Heat and Mass Transfer xxx (2016) xxx–xxx

to identify possible differences between stratified condensation and jet condensation that should be accounted for. Moreover, a potential interdependency between condensation and entrainment at the jet interface might offer further room for improvements and should thus be investigated. Acknowledgment This work is financially supported by the

RWE

Power

r [kg/s2] and the

~kRT ¼ 2p

x2RT ¼

ql þ qg

ðA:1Þ

:

" #2 pffiffiffiffiffiffi pffiffiffiffiffi qg ql 1 qg cg ð1  pffiffiffiffiffi pffiffiffiffiffiffiÞ  cl pffiffiffiffiffi p ffiffiffiffiffiffi ; b ql ql þ qg ql þ qg

ðB:4Þ

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Oh ¼ ll = ql d32;1 r:

ðB:5Þ

ðB:6Þ

B.2. Continuous phase Nusselt number The Nusselt number for 0 6 Pr c 6 250 is calculated according to Hughmark [27]:

( Nuc ¼

0:33 2 þ 0:6Re0:5 dc Pr c

; 0 6 Redc < 776:06

0:33 2 þ 0:27Re0:62 dc Pr c

; 776:06 6 Redc

ðB:7Þ

In Eq. (B.7), the relative Reynolds number between the dispersed and the continuous phase is defined as

Redc ¼ qc jcd  cc jdd =lc :

ðB:8Þ

B.3. Interfacial drag coefficient

ðA:2Þ

The drag coefficient C D is determined according to Ishii and Mishima [23] based on the drag Reynolds number ReD using the mixture dynamic viscosity lm :

C D ¼ 24ð1 þ 0:1Re0:75 D Þ=ReD

The wave acceleration a [m/s2] is determined as

a¼

We ¼ qg ðcg  cl Þ2 d=r;

d32;1  0:2
~kRT :

In our previously developed simulation model for pool DCC [2], we have shown that liquid entrainment and atomization in a submerged steam jet can be treated in analogy to the model approach by Varga et al. [34] for the breakup of a liquid jet surrounded by a high-speed gas stream. Briefly, this implies that entrainment is governed by the secondary Rayleigh–Taylor instability between the ambient water and the continuous gas phase of the jet, which arises in consequence of the primary Kelvin–Helmholtz instability due to velocity shear at the jet interface. The Rayleigh–Taylor instability amplifies interfacial disturbances if a dense fluid l is being accelerated into a lighter fluid g. Then, the most amplified wavelength ~ k and amplification rate x

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3r ; ðql  qg Þa h i 2 k ðql  qg Þa  k r

In Eqs. (B.1)–(B.3), the Weber and Ohnesorge numbers are defined as

The initial diameter of entrained droplets, required for Wed32;1 , is obtained according to Varga et al. [34] as

AG.

Appendix A. Entrainment model

[1/s] are obtained based on the surface tension k as wave number k ¼ 2p=~

9

ðA:3Þ

ReD ¼ 2r D qc jcd  cc j=lm ( lc ð1  ~d Þ1 ðbubbly flowÞ lm ¼ lc ð1  ~d Þ2:5 ðdroplet flowÞ

ðB:9Þ ðB:10Þ ðB:11Þ

where b is the wave crest thickness, which has been experimentally determined as b  ~ kKH =10, and the primary instability wavelength ~ kKH is given as

qffiffiffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffi ~kKH ¼ c mg =cg q =q : l g

ðA:4Þ

Here, c  0:055 m1=2 is an experimentally determined proportionality factor for the vorticity layer. Our previous work shows that the entrainment velocity can then be determined as

E0;RT ~
kRT xRT : 2p

Assuming spherical particles, the drag radius can be calculated as r D ¼ d32 =2. B.4. Wall friction factor The wall friction factor f wall is determined using a correlation by Fang et al. [25]:

h i2 f wall ¼ 0:25 logð150:39Re0:98865  152:66Re1 ; 3000 6 Re 6 108 w w Þ

ð21revisitedÞ

ðB:12Þ

The value of the entrainment coefficient has been obtained based on a parametric study as E0;RT ¼ 0:16 [2].

This correlation is recommended by Xu et al. [35] as a more accurate representation of the Nikuradse/Moody diagram [36,37] than the widely used Blasius equation, particularly for high Reynolds numbers. The Reynolds number in Eq. (B.12) is obtained as

cen;RT ¼

Appendix B. Employed correlations

Rew ¼ qw jcw jdH;wall;w =lw ;

B.1. Droplet size after atomization The final fragment size distribution after atomization is determined based on the droplet Weber number We and the dimensionless initiation and total breakup time (^t ini ; ^ttot ) using correlations by Wert [19], Hsiang and Faeth [20]:

 2=3 Wed32;2 ¼ 0:32 Wed32;1 ð^ttot  ^t ini Þ

ðB:1Þ

^t ini ¼ 1:6ð1  Oh=7Þ1

ðB:2Þ

^t tot ¼ 5ð1  Oh=7Þ

ðB:3Þ

1

dH;wall;w ¼ 2Rw w :

ðB:13Þ ðB:14Þ

Appendix C. Effective-adapted-jet approximation For a simplified treatment of the adaptation of the steam nozzle exit pressure pe to the local pipe pressure pw , the effectiveadapted-jet approximation is used. Here, the steam nozzle exit diameter is replaced by an equivalent diameter which would be required for an adapted exit state, i.e. pe ¼ pw .

Please cite this article in press as: D. Heinze et al., A physically based, one-dimensional three-fluid model for direct contact condensation of steam jets in flowing water, Int. J. Heat Mass Transfer (2016), http://dx.doi.org/10.1016/j.ijheatmasstransfer.2016.10.076

10

D. Heinze et al. / International Journal of Heat and Mass Transfer xxx (2016) xxx–xxx

The numerical procedure consists of the following steps: 1. The adapted thermodynamic state is obtained from the equation of state as a function of the exit entropy se and the adapted pressure pw . 2. The adapted velocity is determined from the total enthalpy conservation:

cadapted ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ðhe  hadapted Þ þ c2e

ðC:1Þ

3. The equivalent diameter is determined from the condition of mass continuity:

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qe c e dequiv ¼ de
qadapted cadapted

ðC:2Þ

Appendix D. Isentropic expansion of steam The nozzle exit state is determined based on the homogeneous equilibrium model assuming isentropic and isenthalpic equilibrium expansion. For this purpose, the choked flow state at the nozzle throat is first determined as follows: 1. The critical pressure and enthalpy pcrit ; hcrit are estimated using the ideal gas solution. 2. The critical entropy scrit and the homogeneous equilibrium speed of sound wh;crit are calculated based on the equation of state using 1 ¼ w2h

xg

þ

xl

!2 "

qg ql

1

q

2 2 l wl

þ xg

1

q

2 w2 g g



1

q

2 2 l wl

!

@xg  

@p s

1



1

qg ql

!# :

ðD:1Þ

Here, xg and xl are the mass fractions of the gas and the liquid phase, respectively, and wg and wl are the single-phase speeds of sound for the gas and the liquid phase, respectively. 3. The critical pressure and enthalpy pcrit ; hcrit are adjusted and step 2 is iterated until the conditions for isentropic and isenthalpic flow are fulfilled:

scrit ¼ s0 hcrit ¼ h0 

ðD:2Þ 0:5w2h;crit

ðD:3Þ

Now, the nozzle exit state can be calculated: 1. The exit pressure and velocity pe ; ce are estimated using the ideal gas solution. 2. The exit enthalpy is determined from the total enthalpy conservation:

he ¼ h0  0:5c2e

ðD:4Þ

3. The exit density qe is obtained from the equation of state. 4. The exit pressure and velocity pe ; ce are adjusted and steps 2–3 are iterated until the conditions for isentropic flow and continuity are fulfilled:

se ¼ s0

ðD:5Þ

qe ce Ae ¼ qcrit ccrit Acrit

ðD:6Þ

References [1] C.-H. Song, S. Cho, H.-S. Kang, Steam jet condensation in a pool: from fundamental understanding to engineering scale analysis, J. Heat Transfer 134 (2012) 031004-1–031004-15. [2] D. Heinze, T. Schulenberg, L. Behnke, A physically based, one-dimensional twofluid model for direct contact condensation of steam jets submerged in subcooled water, J. Nucl. Eng. Radiat. Sci. 1 (2015) 021002-1–021002-8.

[3] Q. Xu, L. Guo, S. Zou, J. Chen, X. Zhang, Experimental study on direct contact condensation of stable steam jet in water flow in a vertical pipe, Int. J. Heat Mass Transfer 66 (2013) 808–817. [4] S. Banerjee, Turbulence structure and transport mechanisms at interfaces, Proc. Ninth Int. Heat Transfer Conference, vol. 1, Hemisphere Publ., Jerusalem, Israel, 1990, pp. 395–418. [5] S.S. Gulawani, S.K. Dahikar, C.S. Mathpati, J.B. Joshi, M.S. Shah, C.S. RamaPrasad, D.S. Shukla, Analysis of flow pattern and heat transfer in direct contact condensation, Chem. Eng. Sci. 64 (2009) 1719–1738. [6] D. Lakehal, M. Labois, A new modelling strategy for phase-change heat transfer in turbulent interfacial two-phase flow, Int. J. Multiphase Flow 37 (2011) 627–639. [7] D. Lakehal, M. Fulgosi, S. Banerjee, G. Yadigaroglu, Turbulence and heat exchange in condensing vapor–liquid flow, Phys. Fluids 20 (2008) 065101. [8] R.J. Jensen, Interphase transport in horizontal stratified cocurrent flow (Ph.D. thesis), Northwestern University, Evanston, 1982. [9] A. Murata, E. Hihara, T. Saito, Prediction of heat transfer by direct contact condensation at a steam-subcooled water interface, Int. J. Heat Mass Transfer 35 (1992) 101–109. [10] E. Hughes, R. Duffey, Direct contact condensation and momentum transfer in turbulent separated flows, Int. J. Multiphase Flow 17 (1991) 599–619. [11] H.J. Kim, S.G. Bankoff, Local heat transfer coefficients for condensation in stratified countercurrent steam–water flows, J. Heat Transfer 105 (1983) 706–712. [12] H.J. Kim, S.C. Lee, S.G. Bankoff, Heat transfer and interfacial drag in countercurrent steam–water stratified flow, Int. J. Multiphase Flow 11 (1985) 593–606. [13] S.V. Paras, N.A. Vlachos, A.J. Karabelas, Liquid layer characteristics in stratified – atomization flow, Int. J. Multiphase Flow 20 (1994) 939–956. [14] J.H. Linehan, M. Petrick, M.M. El-Wakil, On the interface shear stress in annular flow condensation, J. Heat Transfer 91 (1969) 450–452. [15] A. Guelfi, D. Bestion, M. Boucker, P. Boudier, P. Fillion, M. Grandotto, J.-M. Hérard, E. Hervieu, P. Péturaud, NEPTUNE: a new software platform for advanced nuclear thermal hydraulics, Nucl. Sci. Eng. 156 (2007) 281–324. [16] M. Ishii, T. Hibiki, Thermo-fluid Dynamics of Two-phase Flow, 2 ed., Springer, New York, 2011. [17] K.E. Carlson, R.A. Riemke, S.Z. Rouhani, R.W. Shumway, W.L. Weaver, RELAP5/ MOD3 Code Manual Volume I: Code Structure, System Models and Solution Methods, Technical Report NUREG/CR-5535; INEL-95/0174, U.S. Nuclear Regulatory Commission, Washington, D. C., 1995. [18] M. Ishii, S. Kim, Development of one-group and two-group interfacial area transport equation, Nucl. Sci. Eng. 146 (2004) 257–273. [19] K.L. Wert, A rationally-based correlation of mean fragment size for drop secondary breakup, Int. J. Multiphase Flow 21 (1995) 1063–1071. [20] L.P. Hsiang, G.M. Faeth, Near-limit drop deformation and secondary breakup, Int. J. Multiphase Flow 18 (1992) 635–652. [21] F.P. Ricou, D.B. Spalding, Measurements of entrainment by axisymmetrical turbulent jets, J. Fluid Mech. 11 (1961) 21–32. [22] D. Vivaldi, F. Gruy, N. Simon, C. Perrais, Modelling of a CO2 -gas jet into liquidsodium following a heat exchanger leakage scenario in sodium fast reactors, Chem. Eng. Res. Des. 91 (2013) 640–648. [23] M. Ishii, K. Mishima, Two-fluid model and hydrodynamic constitutive relations, Nucl. Eng. Des. 82 (1984) 107–126. [24] T.J. Hanratty, Separated flow modelling and interfacial transport phenomena, Appl. Sci. Res. 48 (1991) 353–390. [25] X. Fang, Y. Xu, Z. Zhou, New correlations of single-phase friction factor for turbulent pipe flow and evaluation of existing single-phase friction factor correlations, Nucl. Eng. Des. 241 (2011) 897–902. [26] I. Aya, H. Nariai, Evaluation of heat-transfer coefficient at direct-contact condensation of cold water and steam, Nucl. Eng. Des. 131 (1991) 17–24. [27] G.A. Hughmark, Mass and heat transfer from rigid spheres, AIChE J. 13 (1967) 1219–1221. [28] E. Fehlberg, Low-order Classical Runge–Kutta Formulas with Stepsize Control and their Application to Some Heat Transfer Problems. Technical Report TR R315, National Aeronautics and Space Administration, Washington, D. C., 1969. [29] B. Gough (Ed.), GNU Scientific Library Reference Manual – Third Edition, third ed., Network Theory Ltd., Bristol, 2009. [30] E. Loth, G.M. Faeth, Structure of underexpanded round air jets submerged in water, Int. J. Multiphase Flow 15 (1989) 589–603. [31] D. Heinze, T. Schulenberg, L. Behnke, Modeling of steam expansion in a steam injector by means of the classical nucleation theory, in: Proceedings of the 15th International Topical Meeting on Nuclear Reactor Thermal Hydraulics (NURETH-15), Pisa, Italy, 2013. [32] W. Wagner, J.R. Cooper, A. Dittmann, J. Kijima, H.-J. Kretzschmar, A. Kruse, R. Mareš, K. Oguchi, H. Sato, I. Stöcker, O. Šifner, Y. Takaishi, I. Tanishita, J. Trübenbach, T. Willkommen, The IAPWS industrial formulation 1997 for the thermodynamic properties of water and steam, J. Eng. Gas Turbines Power 122 (2000) 150–185. [33] X. Zong, J.-P. Liu, X.-p. Yang, J.-J. Yan, Experimental study on the direct contact condensation of steam jet in subcooled water flow in a rectangular mix chamber, Int. J. Heat Mass Transfer 80 (2015) 448–457. [34] C.M. Varga, J.C. Lasheras, E.J. Hopfinger, Initial breakup of a small-diameter liquid jet by a high-speed gas stream, J. Fluid Mech. 497 (2003) 405–434. [35] Y. Xu, X. Fang, X. Su, Z. Zhou, W. Chen, Evaluation of frictional pressure drop correlations for two-phase flow in pipes, Nucl. Eng. Des. 253 (2012) 86–97. [36] J. Nikuradse, Strömungsgesetze in rauhen Rohren, VDI-Forschungsheft Vol. 361 (1933). [37] L.F. Moody, Friction factors for pipe flow, Trans. ASME 66 (1944) 671–684.

Please cite this article in press as: D. Heinze et al., A physically based, one-dimensional three-fluid model for direct contact condensation of steam jets in flowing water, Int. J. Heat Mass Transfer (2016), http://dx.doi.org/10.1016/j.ijheatmasstransfer.2016.10.076