Scaling analysis of coolant spraying process in automatic depressurized system

Annals of Nuclear Energy 72 (2014) 350–357

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

Scaling analysis of coolant spraying process in automatic depressurized system Xiangbin Li a,c,⇑, Mengchao Zhang a,c, Zheng Du b, Xiaoliang Fu b, Daogang Lu a,c, Yanhua Yang b a

School of Nuclear Science and Engineering, North China Electric Power University, Beijing 102206, China National Energy Key Laboratory of Nuclear Power Software, State Nuclear Power Software Development Centre, Xicheng District, Beijing 100029, China c Beijing Key Laboratory of Passive Safety Technology for Nuclear Energy, Beijing 102206, China b

a r t i c l e

i n f o

Article history: Received 1 February 2014 Received in revised form 20 May 2014 Accepted 5 June 2014

Keywords: Scaling laws Equation analysis Steam jetting Direct contact condensation

a b s t r a c t Through the automatic depressurized system of AP1000 reactor, the steam with high temperature and high pressure from the pressurizer sprays into the water tank, and the pressure in the primary coolant system will diminish to protect the reactor. However, as the steam continues to flow into the tank, the water temperature rises rapidly until boiling occurs, which will affect the local heat transfer process. Therefore, it is necessary to understand the corresponding heat transfer mechanism by means of experiment. To ensure that the experimental results with a scale-down model reflect the actual situations of the prototype we analyzed the process of steam jetting from the pipeline into the water tank, and summarized scaling rules based on equation analysis method, including various flow stages. The results show that the phenomena-based similarity between the model and the prototype should meet: (1) geometrically similar model and prototype; (2) equal thermal parameters and identical initial conditions, which can greatly simplify other similarity parameters; (3) at the blowdown stage, keep the steam mass flux and the nozzle diameter consistent; (4) at the natural convection stage with single phase fluid, the equality in terms of Prandtl number and the Grashof number should be met first, while assessing the relative uncertainty. Ó 2014 Elsevier Ltd. All rights reserved.

1. Introduction In recent years, the passive technology becomes a very popular technology in many industrial fields, especially in the Ap1000 reactors, the third generation reactors. For example, in AP1000 PXS (Passive Core Cooling System), the steam with high temperature and high pressure sprays into the built-in refueling water storage tank (IRWST) through ADS (Automatic Depressurized System) to condensate as it discharges from the pressurizer, which can provide overpressure protection for the primary coolant system. However, under some accident conditions, the steam continues to flow into the IRWST through ADS pipeline, this causes the water temperature rising rapidly until boiling, this also affects the local heat transfer processes and IRWST draining behaviors. Therefore, it is necessary to study the pool boiling phenomenon in IRWST under the condition of steam blowdown state, especially to understand the heat transfer mechanism thoroughly, which is of great significance to the reactor safety. ⇑ Corresponding author at: School of Nuclear Science and Engineering, North China Electric Power University, Beijing 102206, China. Tel.: +86 13522735440. E-mail address: [email protected] (X. Li). http://dx.doi.org/10.1016/j.anucene.2014.06.009 0306-4549/Ó 2014 Elsevier Ltd. All rights reserved.

As the complexity of heat transfer mechanism, people have carried out many experimental investigation and theoretical simulation to study the steam-direct-jetting condensation and boiling phenomenon, which is an important step toward the ADS discharging processes. Chun et al. (1996) designed the VAPORE experimental facility to perform thermomechanical and fluid-dynamic tests on relative nuclear components and systems, including the ADS test. It has a full-scaled and full-pressurized configuration of the AP600 ADS system. Two phases were carried out in their experiments: phase A were implemented for the ADS-1/2/3 with steam flow through a sparger into IRWST to evaluate the capacity of the sparger, and phase B focuses on the thermal hydraulic behavior on ADS valves, pipes and sparger. Based on a reduced-height and reduced-pressure integral loop test facility, Song et al. (2007) investigated the direct contact condensation of steam in a water pool and developed a condensation regime map of the steam jet for simplified spargers. In addition, many researchers focused on the fundamental mechanism study of direct contact condensation in a simplified water tank. Based on visual techniques, they observed conical, ellipsoidal and divergent shapes of steam plume, although which these shapes depends on the test conditions Giovanni

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X. Li et al. / Annals of Nuclear Energy 72 (2014) 350–357

Nomenclature a p

a q g A De e Hgs Hfs Hf f t qf qg qs qv db H CD L L0 P0

C _ gf m V Vg,ex k r z h

interfacial area per unit volume, m1 pressure, N/m2 volume fraction density of fluid, kg/m3 acceleration of gravity, m/s2 area, m2 hydraulic diameter, m total specific energy, J/kg the saturation enthalpy of the gas phase, J/kg the saturation enthalpy of the liquid phase, J/kg the average specific enthalpy, J/kg friction coefficient time, s heat flux from the liquid phase into the interface, w/m2 heat flux from the gas phase into the interface, w/m2 heat flux, w/m2 internal heat source per unit volume, w/m3 bubble diameter, m the height of the water surface in the tank, m drag coefficient characteristics length, m submerged depth from sparger to water level, m pressure at the free surface of water tank, N/m2 generate rate of liquid phase, kg/(m3 s) mass flux from gas phase into liquid phase, kg/(m2 s) velocity, m/s steam velocity at the nozzle exit, m/s drag coefficient radial direction, m axial direction, m coefficient of convection heat transfer, w/(m2 k)

et al. (1984), Chun et al. (1996), Kim et al. (2001). Using different horizontal (or vertical ) nozzles under various conditions of pool water temperature and steam mass flux, Kerney et al. (1972) and Weimer et al. (1973) studied the penetration length of sonic steam jet with experimental and theoretical approaches, they obtained the classical correlations to calculate the dimensionless penetration length. Chun et al. (1996), Kim et al. (2001), Wu et al. (2009) and Tobias et al. (2011) derived and modified the similar empirical correlations for the plume length, and With (2009) introduced a new two-dimensional steam plume length diagram to predict length accurately for a wide range of conditions. Furthermore, many researchers have provided semi-empirical correlations to evaluate the average heat transfer coefficient around the steam plume interface Simpson and Chan (1982), Chun et al. (1996), Seong et al. (2000), Kim et al. (2001), Kim et al. (2004), Wu et al. (2007), Park et al. (2007). Ajmal et al. (2010) studied the phenomenon of direct-contact condensation numerically by introducing a thermal equilibrium model, and revealed the relationship between dimensionless penetration length of steam plume and the condensation heat transfer coefficient. Other related studies also depicted the structures of the corresponding flowing field, including the instantaneous velocity field, void variation and temperature distribution Van Wissen et al. (2004), Takase et al. (2002), Dahikar et al. (2010). However, all of these experimental researches had greatly simplified the relative test section. Therefore, these experimental conclusions cannot be directly used for the condensing and pool boiling induced by the steam blowdown under the reactor accident. In general, limited by the test size, the experiments can only be implemented with a scale-down model. Therefore, it is

hf

Cp hfg kgf s

heat transfer coefficient of the liquid phase across the interface, w/(m2 k) heat transfer coefficient of the gas phase across the interface, w/(m2 k) heat exchange across the interface per unit volume, w/m3 thermal conductivity of the liquid phase, w/(m k) Nusselt number of the liquid phase Reynolds number of the liquid phase Prandtl number of the liquid phase viscosity of liquid phase, Pa s diameter of the orifice, m dimensionless subcooling of the liquid mass flux of the steam, kg/(m2 s) reference mass flux of the steam, kg/(m2 s) Grashof number Prandtl number bath temperature, K temperature of the interface between the gas phase and the liquid phase specific heat of the liquid, J/(kg K) latent of heat, J/kg interface momentum force per unit volume, N/m3 velocity ratio of the gas phase and the liquid one

Subscript f g r z 0 ex

liquid gas value at the value at the initial value value at the

hg Qgf Kf Nuf Re Pr

lf D B G0 Gm Gr Pr T1 Ts

radial direction axial direction (or reference value) orifice exit

necessary to carry out corresponding scaling analysis to validate whether the results from the experimental model can reflect the real prototype phenomenon accurately. Based on the mass and energy balances and the flow-pressure drop relationship, Hsu et al. (1990) conducted a scaling study on the test equipment at the case of small break LOCA, and concluded that the coolant capacity is a more important similarity parameter. Sonin (1981) studied the scaling-down problem on the steam jetting stage in the boiling water reactor with dimensional analysis method, obtaining some general similarity criterion, and revealing that the steam mass flux and the relative thermodynamic properties are the key factors for scaling similarity. Zuber et al. (1998) developed an integrated scaling methodology: hierarchical two-tiered scaling (H2TS), in which the relative scaling analysis on a complex system is divided into four steps: system decomposition, scale identification, system scaling analysis and process scaling analysis, and a crucial problem is to obtain the corresponding similarity criteria based on the field equations. This method has been widely used for its effectiveness on larger system research. In this study, combining with the above H2TS solution, we use a scaling method based on equation analysis to further study the similarity problem at each stage in which the steam with high temperature and high pressure injected into the cold water pool, and also summarize the relative scaling ratios for design reference. 2. Objective and process description As shown in Fig. 1, the steam with high temperature and high pressure flows into the connecting pipes as the ADS is activated, and jets into the IRWST through a sparger. At the third stage, the

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X. Li et al. / Annals of Nuclear Energy 72 (2014) 350–357

Fig. 1. Sketch of steam spraying.

main processes are as follows: (1) the wet saturated steam flows in the pipe from the ADS pressure relief valve to the sparger outlet; (2) the steam nearing the sparger outlet condenses with cold water, producing intense exchange of heat and mass transfer in local area; (3) with continuous injection of the steam, the water temperature around the sparger will rise to reach a temperature gradient as compared to other regions, which will form a phenomenon of natural convection heat transfer. The purpose of this study is to obtain the similarity criteria of the above three stages. 3. Scaling analysis and discussion 3.1. Analysis of wet saturated steam flows from the ADS pressure relief valve to the sparger outlet 3.1.1. Basic equation At this stage, the spraying process can be simplified as an internal pipe flow. We use following assumptions to simplify the analysis: The injecting fluid is dry steam, the fluid only flows along the axis direction of the pipe, and the heat loss across the pipe wall is negligible. The following conservation equations can be obtained: Continuity equation:

@ qg @ qg V þ ¼0 @z @t

ð1Þ

Momentum equation: 2 @ qg V @ qg V 2 @P qg V f þ ¼   qg g @t @z @z 2 De

ð2Þ

Energy equation:

@ qg e @ðqg VeÞ þ ¼0 @t @z

@ qg

@ qg V þ ¼0 @z @t

ð4Þ 2

Y @P Y qg V f Y @ qg V @ qg V þ ¼   qg g z @ @z 2 @t D e 1 2 3

ð5Þ

@ qg e @ðqg V eÞ þ ¼0 @z @t

ð6Þ

Q

3.2. Analysis of heat and mass transfer process between the steam and water near the sparger outlet 3.2.1. Basic equation and similarity criteria At this stage, the direct contact condensation occurs as the submerged steam jets into the subcooled water. As the sparger consists of many small orifices and the steam jets from the orifices at the same time, for simplicity we assumed that each orifice behaves as similar jetting status, and do not interfere with each other. Thus, we can use similar equations to describe the jetting phenomenon of the sparger under either single nozzle condition or that with many orifices, just noting that the mass flow flux should be equivalent to the value corresponding to a single orifice. Since the most important dynamics occurs along the nozzle center line, the circumferential variation can be ignored for simplicity. Therefore, the relative conservation equations under the cylindrical coordinate can be expressed as follows:

ð3Þ

As one dimensionless form of the parameter X, X is defined as the ratio between the real value and the reference one, the corresponding dimensionless equations can be shown as follows:

2

with same structure, as it plays an important role on the export parameters. Among these above similarity numbers, the Euler Q number 1 is an indecisive criterion since the velocity is deterQ mined by the pressure difference; the friction number 2 is relative to the friction loss, which can be combined with the local Q Q loss in actual analysis. Thus, 2 can be modified as 2 ¼ fL=De þ k(k indicates the other drag coefficient in addition to the frictional one), and similar losses (For example, the heat loss across the pipe wall) between the model and the prototype can be obtained by adjusting the relative technical parameters. The Froude number Q 3 denotes the ratio between the gravity and the inertia force, which can be neglected since the gravity plays few effect on the spraying flow. In summary, at this stage, the above similarity numbers are all indecisive ones. In general, the most important similarity requirement is to ensure the steam parameters at the sparger outlet are consistent with the model and the prototype. Firstly, both of these geometric structures are similar, the sparger parameters should be reduced with same ratio since it dominates the outlet flows. Based on the same fluid medium and the equal physical conditions, we can conclude that the similarity of the outlet parameters is guaranteed with control of the pipeline loss. Noting that we hope to understand the integral heat transfer mechanism in the IRWST under the condition of steam blowdown state, the scaling ratios of the sparger parameters should be consistent with that of the IRWST. Therefore, the relative scaling ratios will be discussed in the following stage.

Q

z0 Here, 1 ¼ qPV0 2 is the Euler number; 2 ¼ fD0 e0 represents the friction Q 0 0 gz0 Q qgz0 number; 3 ¼ V 2 (or 3 ¼ qV 2 Þ is the Froude number. 0

3.1.2. Discussion for similarity criteria To ensure phenomenon similarity at this stage, the steam parameters at the model sparger outlet should be consistent with that of the prototype. First of all, the sparger will be scaled down

3.2.2. Continuity equation

@ðqf af Þ 1 @ðrqf af V f ;r Þ @ðqf af V f ;z Þ þ þ ¼C @t @r @z r @ðqg ag Þ 1 @ðr qg ag V g;r Þ @ðqg ag V g;z Þ þ þ ¼ C @t @r @z r

ð7Þ ð8Þ

Here, Z denotes the axial direction along the nozzle centerline. As the radial dynamics is neglected, the following non-dimensionalizing equations can be obtained:

@ðqf af Þ V f ;z0 t 0 @ðqf af V f ;z Þ t0  þ ¼ CC @z z0 qf 0 af 0 0 @t @ðqg ag Þ V g;z0 t 0 @ðqg ag V g;z Þ t0  þ ¼ CC @z z0 qg0 ag0 0 @t

ð9Þ ð10Þ

Here, z0 is defined as the length of the steam plume, and the corresponding characteristics time t0 = z0/Vg,z0. C represents the generate rate of liquid phase. Obviously, the speed difference between the

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steam and the water near the steam-water interface is larger. Neglecting Vf,z0, the above equations can be simplified as follows:

@ðqf af Þ 1 z0  ¼ CC @s qf 0 af 0 V g;z0 0

ð11Þ

@ðqg ag Þ @ðqg ag V g;z Þ 1 z0  þ ¼ CC @s @z qg0 ag0 V g;z0 0

ð12Þ

Defining

Q

4

1

¼q

g0 ag0

z0 V g;z0

Gm = 275 kg/(m2 s) (Kerney et al., 1972). B represents a dimensionless driving potential for the condensation process (i.e., dimensionless subcooling of the liquid), which is calculated as:

B ¼ C pf ðT s  T 1 Þ=hfg

In addition, since the velocity at the plume trailing is zero, the following correlation can be obtained:

V g;z0 ¼ 0:5V g;ex

C0 .

ð21Þ

Discussion on relative parameters:

Or

(1) C(the generating rate of liquid phase)

V g;z0 ¼ 0:5G0 =qg;ex





Here, C is defined as C ¼ mgf a. Where mgf is the interface mass transfer from the gas phase into the liquid phase and a is the interfacial area per unit volume (Tu and Yeoh, 2002), which are defined as 

mgf ¼

¼

ð15Þ

qg ¼ hg ðT s  T g Þ

ð16Þ

where Tf and Tg are the temperatures of liquid and gas phases respectively. Ts is the interfacial temperature, assuming that is equal to the saturation temperature corresponding to the local pressure. hg is the heat transfer coefficient of the gas phase across the interface, which can be assumed as hg = 104 w/(m2 k) (Brucker and Sparrow, 1977). hf is the heat transfer coefficient of the liquid phase across the interface. Considering hf = kfNuf/db, in which the Nusslet number can be calculated with Ranz–Marshall model (Ranz and Marshall, 1952): 0:3 Nu f ¼ 2:0 þ 0:6R0:6 e pr

ð17Þ

Here, the Reynolds number Re = qfjVg  Vfjdb/lf, pr is the Prandtl number based on the liquid phase. Based on the above analysis, the generate rate of the liquid phase C can be shown as: 0:3 0:6R0:6 e pr ÞðT s

 T f Þ=db þ hg ðT s  T g Þ 6ag db Hgs  Hfs

ð18Þ

qg;ex 4:2996D0 B0:8311 ðG0 =Gm Þ0:6446 qg0 db G0 

ð14Þ

qf ¼ hf ðT s  T f Þ

½kf ð2:0 þ

Y

ð13Þ

respectively. Here, qf represents the heat flux from the interface into the liquid phase, and qg represents the heat flux from the gas phase into the interface, Hgs and Hfs are the saturation enthalpies of the gas and liquid phases respectively, and db is the bubble diameter. qf and qg can be calculated as:

ð22Þ

Here, Vg,ex is the velocity of the steam at the nozzle exit, and qg,ex is the corresponding density of the steam. Based on the above correlations, the similarity criterion number Q 4 is reduced to the following form:

4

qf þ qg Hgs  Hfs

a ¼ 6ag =db

C¼

ð20Þ

0:3 ½kf ð2:0 þ 0:6R0:6 e pr ÞðT s  T f Þ=db þ hg ðT s  T g Þ Hgs  Hfs

ð23Þ

3.2.3. Momentum equations (in axial direction)

@ðqf af V f ;z Þ @ðqf af V f ;z Þ @p þ Vz ¼ af @t @z @z " # @ 2 V f ;z 1 @V f ;z @ 2 V f ;z þl þ þ r @r @r2 @z2 þ af qf g z þ kgf

ð24Þ

@ðqg ag V g;z Þ @ðqg ag V g;z Þ @p þ Vz ¼ ag @t @z @z " # @ 2 V g;z 1 @V g;z @ 2 V g;z þl þ þ r @r @r 2 @z2 þ ag qg g z þ kgf ð25Þ Here, Kgf is the interface momentum force per unit volume, which is expressed as:

K gf ¼

3 1 C D ag qf jug  uf jðug  uf Þ 4 db

ð26Þ

where, the drag coefficient CD is calculated with Schiller–Naumann model (Schiller and Naumann, 1933):

24 ð1 þ 0:15  R0:687 Þ; Re 6 1000 e Re C D ¼ 0:44; Re P 1000

CD ¼

ð27Þ ð28Þ

(2) z0 (the length of the steam plume) Many researchers have obtained the dimensionless correlations for the plume length Kerney et al. (1972), Weimer et al. (1973), Chun et al. (1996), Kim et al. (2001), Wu et al. (2009) and Tobias et al. (2011), which is related to the nozzle diameter, the steam mass flux and the subcooling of the liquid. In this paper, we adopt a classical dimensionless formula Kerney et al. (1972):

z0 =D ¼ 0:3583B

0:8311

ðG0 =Gm Þ

0:6446

It is clear that the above two equations have similar formula, just considering to non-dimensionalize the steam one, and neglecting the radial dynamics, we can obtain:

lg0 @ðqg ag V g;z Þ @ðqg ag V g;z Þ p0 @p @2V þ Vz ¼ ag  þ lg 2g;z 2 z @z q a V z @z @ @t qg0 V g;z0 g0 g0 g;z0 0 þ

ð19Þ

Here, z0 is the plume length of the steam, D is the nozzle (or orifice) diameter, G0 is the mass flux of the steam at the nozzle exit, and Gm is the reference mass flux of the steam, which is a constant:

Defining

Q

5

¼q

p0 2 g0 V g;z0

g z0 t 0 t0 ag qg g z þ kgf 0 k V g;z0 qg0 ag0 V g;z0 gf

ð29Þ

, which is the Euler number, an indecisive crite-

rion number since the velocity is determined by the pressure difference.

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X. Li et al. / Annals of Nuclear Energy 72 (2014) 350–357

Defining

Q

lg0

¼q

6

g0 ag0 V g;z0 z0

, which can be converted into the fol-

lowing form by substituting the relative correlations:

Y

¼

6

Y

2qg;ex lg0 0:8311

qg0 ag0 G0 0:3583B

0:6446

ðG0 =Gm Þ

q l ¼ 0:17915q a

Y ð30Þ

Q t0 Defining 7 ¼ gVz0g;z0 , which express the ratio between the gravitational potential energy and kinetic one, and can be converted into the following form:

Y

¼ 1:4332g z0 D0 B0:8311 G01:3554 G0:6446 q2g;ex m

ð31Þ

7

Defining

Y

¼ kgf 0

8

¼

11

2

qf 0 af 0 Hf 0

0:3 kf ð2:0 þ 0:6R0:6 e pr ÞðT s  T f Þ=db G0 =qg;ex0

z0

qf 0 af 0 Hf 0 V g;z0

ð40Þ

Q gf 0

ð41Þ

Or Y

0:6446

0:7166D0 B0:8311 ðG0 =Gm Þ qf 0 af 0 Hf 0 G0 =qg;ex0 11 Y z0 ¼ Ch qf 0 af 0 Hf 0 V g;z0 0 fg;0 12 ¼

0:3 kf ð2:0 þ 0:6R0:6 e pr ÞðT s  T f Þ 6ag0 ð42Þ db db

ð43Þ

Or

t0

ð32Þ

qg0 ag0 V g;z0

Or

Y

¼

10

D0

0:6446 g;ex g0 Gm 0:8311 D0 G1:6446 g0 g0 B 0

Or

Y

¼

0:7166D0 B0:8311 ðG0 =Gm Þ0:6446 qf 0 af 0 Hf 0 G0 =qg;ex0



0:3 ½kf ð2:0 þ 0:6R0:6 e pr ÞðT s  T f Þ=db þ hg ðT s  T g Þ 6ag0 hfg;0 ð44Þ db Hgs  Hfs

12 0:8311 1:3554 0:6446 2 1 ¼ 1:4332q1 G0 Gm qg;ex g0 ag0 kgf 0 D0 B

ð33Þ

8

3.2.4. Energy equation

@ðqf af Hf Þ @ðqf af Hf V f ;z Þ @qf þ ¼ þ Q gf þ Chfg @s @z @z

ð34Þ

Q Q Q Here, defining R ¼ m = p , which stand for the ratio of the dimensionless parameters between the model and the prototype. If Q R ¼ 1, the parameters of the model will correspond to those of the prototype one to one. That is, the similarity between the model and the prototype can be guaranteed.

where Hf is the average specific enthalpy, which is calculated as:

Hf ¼

Z

T

C p;f dT

ð35Þ

T ref

hfg is the latent heat, Qgf is the heat exchange through the interface per unit volume:

Q gf ¼ qf a ¼ hf ðT s  T f Þa

ð36Þ

Non-dimensionalizing:

qR ¼ kR ¼ prR ¼ lR ¼ hfgR ¼ ðC pf ðT s  T 1 Þ=hfg ÞR ¼ 1

@ðqf af Hf Þ V f ;z0 @ðqf af Hf V f ;z Þ qf 0 @qf 1 þ ¼ @s @z V g;z0 qf 0 af 0 Hf 0 V g;z0 @z z0 Q Q þ qf 0 af 0 Hf 0 V g;z0 gf 0 gf z0 þ C h Ch qf 0 af 0 Hf 0 V g;z0 0 fg;0 fg

ð37Þ

Defining

Y

¼

9

V f ;z0 V g;z0

Obviously,

ð38Þ Q

9

V

f ;z0 ¼ V g;z0 ¼ 1=s, which shows the velocity ratio of the

liquid phase and the gas one. Defining

Y

¼

10

1

qf 0 af 0 Hf 0

3.2.5. Scaling ratio with same fluid Q Q It is clear that the similarity criterion group, 4  12 , cannot be satisfied easily if using unequal physical conditions. However, with same physical properties and initial conditions, all the physical property ratios can be turned to unity. Therefore, for the liquid phase and gas phase:

qf 0 V g;z0

ð39Þ

ð45Þ

The ratios of the above similarity criterion numbers can be simplified greatly as in Table 1. Apparently, the two parameters, the orifice diameter D0 and the steam mass flux G0, play a crucial role in the similarity analysis. Noting that the similarity ratios CR and hfR are all related to the Reynolds number, and in fact to the steam mass flux G0. Therefore, at this stage, we firstly choose the two parameter ratios as:

D0R ¼ G0R ¼ 1

ð46Þ

Then, D0m = D0p and G0m = G0p. It’s clear that all the above similarity ratios reduced to unit. To summarize, the physical phenomenon at the stage of sparger spraying can be tested agreeable with a scale-down model as long as the same steam mass flux and the same nozzle diameter are guaranteed, and similar geometry, same physical properties, same working medium and initial conditions must meet as well.

Table 1 The similarity ratio under equal physical conditions. Similarity ratio group Q 4R

Q

5R

Q

6R

Q

7R

Q

8R

Simplified correlation D0R G0:3554 CR 0R G2 0R 1:6446 D1 0R G0R 1:3554 D0R G0R 1:3554 D0R G0R

Similarity ratio group Q Q Q Q

Simplified correlation

9R

1

10R

G1 0R hfR

11R

D0R G0:3554 hfR 0R

12R

D0R G0:3554 CR 0R

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X. Li et al. / Annals of Nuclear Energy 72 (2014) 350–357

@T @T @T Y @ 2 T 1 @T @ 2 T þ cV þ V ¼ a þ þ r z @r @z @t @r 2 r @r @z2 15

Table 2 The similarity ratio under equal physical conditions. Similarity ratio group Q

Simplified correlation

13R

p0R L1 0R

14R

L1:5 0R

15R

L1:5 0R

16R

L0:5 0R qv R

Q Q Q

¼

13

Y

3.3. Analysis of single-phase natural convection heat transfer around the sparger 3.3.1. Basic equation and similarity criteria At this stage, as the heat transfer exchanges between steam and water around the sparger, natural convection occurs in the IRWST due to the temperature gradient. For simplicity, assuming that the physical parameters at any horizontal orientation (with the same height) are coherent, the phenomenon can be expressed with N_S equations of axis symmetric flow: Conservation equation

@V z @V z @V z 1 @p @ 2 V z 1 @V z @ 2 V z þ Vr þ Vz ¼ gbðT  T 0 Þ  þm þ þ r @r @t @r @z q @z @r 2 @z2 ! 2 2 @T @T @T @ T 1 @T @ T q þ v þ Vr þ Vz ¼a þ þ @t @r @z @r 2 r @r @z2 qc p

ð47Þ ð48Þ ! ð49Þ ð50Þ

Here, z represents the vertical direction, r denotes the horizontal direction, and the heat loss near the solid p boundary ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi is neglected. Defining the characteristics velocity V 0 ¼ gbDTL0 (Henkes and Hoogendoorn, 1993), then the corresponding characteristics time pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi t 0 ¼ L0 = gbDTL0 . Where, the characteristics length L0 is defined as the submerged depth of the sparger. Thus, under the same thermophysical parameters, the ratio between the characteristics length scale and the characteristics time scale should maintain as:L0 / t2. Non-dimensionalizing the above equations:

V r @V r @V z þ þ ¼0 r @r @ z Y 1 @p @V z @V z @V z þ cV r þ Vz ¼ gbðT  T 0 Þ    @r @z q @z @t 13 ! Y @ 2 V z 1 @V z @ 2 V z þ þ þ m r @r @z2 @r 2 14

þ

Y q v qc p 16

ð53Þ

Where

Y

ð51Þ

ð52Þ

p0

ð54Þ

q0 V 2z0

¼

1 m0 V z0 L0

ð55Þ

¼

1 m0 ¼ G0:5 r V z0 L0

ð56Þ

¼

1 a0 V z0 L0

ð57Þ

14

V r @V r @V z þ þ ¼0 r @r @z ! @V r @V r @V r 1 @p @ 2 V r 1 @V r V r @ 2 V r þ Vr þ Vz ¼ þm þ  2þ 2 2 r @r @t @r @z q @r @r r @z

!

Or

Y 14

Y 15

Or

Y

m0

¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi gbDTL0 L0 15 Y t0 q v0 ¼ T q cp0 0 0 16

a0 ¼ G0:5 p1 r r m0

ð59Þ ð60Þ

where p0 is the local static pressure, the Grashof number Gr = gbDTL3/t2, the Prandtl number Pr = Cpl/k. qv0 is defined as the ratio of the ejected steam heat per unit time and the water volume in the IRWST, and can be calculated as qv0 = G0AexDh/(HA). Here, Aex is the total nozzle areas, Dh is the enthalpy difference between the steam and the pool water, H is the height of the water surface in the tank, and A is the cross-sectional area of the tank. 3.3.2. Scaling ratio with same fluid Similarly, with same physical properties and initial conditions, the following physical property ratios are turned to unity:

qR ¼ bR ¼ prR ¼ aR ¼ tR ¼ DT R ¼ cpR ¼ 1

ð61Þ

Therefore, the ratios of the above similarity criterion numbers can be simplified as in Table 2. Q Q Obviously, the criterion numbers 13R  16R are all related Q with the characteristics length L0. In addition, 16R is proportional to qvR, and qvR = G0RAexR/UR (Here, UR is the scaling ratio of the tank volume). For the natural convection flows, the Grashof number and the Prandtl number are the decisive parameters, just as the similarQ Q ity criteria 14 and 15 described. The Prandtl number reflects the influence of fluid physical properties on the heat transfer process. This similarity number can be automatically satisfied under the

Table 3 Scaling ratio under equal physical conditions. Sequence number

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Physical description

Vertical length of the tank (m) Submerged depth of the sparger (m) Horizontal length of the tank (m) Cross-sectional area of the tank (m2) Tank volume (m3) Orifice diameter (m) Orifice number Total orifice area (m2) Steam mass flux (kg/(m2 s) Power per unit volume (w/m3) Characteristics velocity Characteristics time

Prototype parameter

8.8 3.05 – 253.9 2234 0.0126 2800 0.354 1419

Model parameter

2.2 1.9 – 2.539 8.89 0.0126 9 1.122e3 1419

Scaling ratio Symbol

Value

LzR L0R LrR AR UR D0R – AexR G0R qvR V0R t0R Q Q13R Q14R Q15R

0.25 0.623 0.1 0.01 0.0025 1

16R

3.17e3 1 1.268 0.789 0.789 1.467 2.034 2.034 1

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X. Li et al. / Annals of Nuclear Energy 72 (2014) 350–357

same physical condition. The Grashof number is mainly related with the fluid physical parameters and the characteristics length scale L0R. However, the vertical characteristic length scale can only be 1:1 to meet this similarity number, which is not realistic. Thus, the reasonable model height should be chosen considering the experimental limitation, and assessing the relative uncertainty. The relative distortion factor can be evaluated as:

Q DF ¼

p

 Q

Q

m

p

¼1

Y

ð62Þ

(2) The same working medium and the equal physical properties should be chosen as far as possible, which can greatly simplify the scaling analysis process. (3) During the steam flows in the pipe, the steam parameters at the sparger outlet of the model should be consistent with that of the prototype. (4) The steam mass flux at the sparger outlet and the nozzle diameter should be uniform at the stage of steam spraying. (5) At the natural convection stage, the Prandtl number and the Grashof number should be considered specially.

R

3.4. Discussion on scaling ratios at all stages Based on same fluids with equal pressure and initial conditions, geometric similarity is adopted firstly. Then, the following scaling ratios can be obtained as shown in Table3. Limited by the experimental scale, the vertical length scaling ratio of the tank is selected as LzR = 0.25, and the cross-sectional area of the tank is scaled down as AR = 0.01. Thus, the scaling ratio for the submerged depth of the sparger cannot be satisfied. Here, L0R = 0.623, which bring relative distortion. To ensure the phenomenon similarity of the steam spraying process at the second stage, the steam mass flux G0 and the single orifice diameter D0 should keep the same value between the model and the prototype. That is, the geometric parameters about the spargers cannot be scaled down as same as other sections, and the total orifice numbers should be adjusted accordingly. As of the natural convection stage in the tank, it can be seen that the related scaling ratios cannot be satisfied completely. Although the characteristics velocity ratio reduces, the corresponding time ratio also decreases, which state that the natural convection stage in the model is accelerated. Furthermore, the similarity ratios Q Q Q 13R ; 14R and 15R are all greater than unit, which show that the inertia force reduces, while the influence of viscous dissipation and thermal diffusion strengthen. Under the initial condition with the environment pressure 1 bar and the tank temperature 49 °C, the value of the Grashof number is about 1e13. So, the relative Q Q influence can be neglected since 14 and 15 are far less than 1. Q However, the pressure effect on the similarity criterion 13 should Q be considered. In addition, the similarity ratio 16R can be met by adjusting the parameter qv. Based on the above analysis, the phenomenon between the model and the prototype can be similar as long as the following similarity criteria are consistent: (1) similar geometry between the model and the prototype, which is most important; (2) same physical properties and same initial conditions, which can greatly simplify the analysis process. Especially, several criteria number can be automatically satisfied; (3) same steam mass flux and same nozzle diameter should be guaranteed; (4) at the natural convection stage, the Prandtl number and the Grashof number should be considered specially. However, limited by the experimental scale, the natural convection process is accelerated. Therefore, the relative distortion should be assessed according to the experimental results. 4. Conclusions Based on equation analysis, we performed the scaling analysis on the steam spraying under various stages. Our results show that to ensure the phenomenon similarity between the model and the prototype, the following criteria should be met: (1) In general, geometric similarity between the model and the prototype should be adopted.

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