On LBE natural convection and its water experimental simulation

Progress in Nuclear Energy xxx (2014) 1e8

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On LBE natural convection and its water experimental simulation Xue-Nong Chen* Karlsruhe Institute of Technology (KIT), Institute for Nuclear and Energy Technologies, P.O.B. 3640, D-76021 Karlsruhe, Germany

a r t i c l e i n f o

a b s t r a c t

Article history: Received 23 August 2013 Received in revised form 7 February 2014 Accepted 5 March 2014

This paper deals with stability and distribution of coolant flow in a pool type of lead-bismuth eutectic (LBE) cooled accelerator driven system (ADS) under natural convection conditions. Two approaches are proposed for solving the problems, namely, (i) theoretical model for stability analyses; (ii) water experimental simulation for the flow distribution in the natural convection state. The stability analysis is carried out based on a simplified integral flow model, where hot and cold free surface oscillations are taken into account. There is a transition from an absolutely stable state under the forced convection condition to a damped oscillation state under the natural convection condition. Reynolds, Richardson (Grashof) and thermal power similitude laws are revealed for the water experimental simulation. The water model scale, power and other factors can be determined by the similitude laws. By choosing the inlet and outlet temperatures even the local Reynolds number law can be approximately satisfied. By the variation of the inlet temperature in the water experiment several similitude solutions are presented. In particular, they are shown in detail for the model scale one to one. It is concluded that the water experiment can be applied for the simulation of LBE natural convection flow in a pool type of reactor. Ó 2014 Elsevier Ltd. All rights reserved.

Keywords: Leadebismuth eutectic (LBE) coolant Natural convection Stability analysis Flow similitude laws Water experimental simulation

1. Introduction Heavy metal liquids, e.g. liquids of lead or leadebismuth eutectic (LBE), are considered world-widely to be used as coolant for future fast reactors, especially for accelerator driven systems (ADS). Their natural convection ability has been recognized and applied to cool passively the reactor in operational and accident conditions (De Bruyn et al., 2011; Wallenius and Suvdantsetseq, 2011). Its flow distribution and stability are two of most crucial issues in the pool type reactor. Small-scale loop experiments cannot solve this problem. The experiments with heavy metal liquids in a large scale are very expensive and demand a high technology, mainly because of their high material costs, especially for LBE, their high melting points and corrosion problems. Unfortunately such experiments are still necessary for studying flow stability and distribution in the natural convection state. In this paper, two approaches are proposed for solving the problems, namely, (i) a theoretical model for transient and stability analyses; (ii) a water experimental simulation for the flow distribution in the natural convection state.

* Tel.: þ49 721 608 25985; fax: þ49 721 608 23824. E-mail address: [email protected].

The theoretical model is set up based on a set of integral equations (time developing ordinary differential equations). It can be applied to the loss of flow analysis and the stability analysis in the forced and natural convection states. The result of the theoretical model confirms the flow oscillation phenomenon after the pump shuts off, found by numerical simulations (Chen et al., 2010). Its stability analysis shows that the natural convection state is in the region of the damped oscillation for the MYRRHA design (De Bruyn et al., 2011). It was discovered (Chen and Zhang, 2012) that there exist the laws of similitude between water and LBE, where the Reynolds law and the Grashof law can hold at the same time. This indicates we can use water experiments to simulate LBE ones in forced and natural convection conditions. In this paper we propose such a water experiment in detail to simulate the whole reactor flow. This is of high importance for the technology development, since the water experimental facility, especially in a large scale, would be much cheaper and more feasible. This paper deals with the global flow pattern and its stability. The local flow instability and the similarity due to the heat transfer have not been discussed. In fact, as the anonymous reviewer pointed out, the major difference between water and LBE is the difference of their Prandtl numbers, where the LBE’s Prandtl number could be two-magnitude-order lower than water’s. Therefore the local heat transfer processes of water and LBE are

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X.-N. Chen / Progress in Nuclear Energy xxx (2014) 1e8

Nomenclature

Symbol A cp DH g Gr K L LW _ m P p Re Ri u t Dz

flow area [m2] specific heat capacity [J/(kg K)] hydraulic diameter acceleration due to gravity [m/s2] Grashof number total pressure drop coefficient effective kinetic length [m] wet perimeter around the flow area [m] mass flow rate [kg/s] power [W] pressure [Pa] Reynolds number Richardson number coolant velocity [m/s] time [s] height [m]

quite different. The free surface motion could be another important point in transient flow problems. Its similarity between water and LBE is associated with Froude number, which needs to be studied in more detail.

l l m n r s z

flow decay coefficient (with subscript) [1/s] model scale factor (without subscript) dynamic viscosity [kg/(m s)] kinematic viscosity [m2/s] coolant density [kg/m3] eigenvalue [1/s] free surface height [m]

Subscript C cold hot HX melt pump

core cold leg hot leg heat exchanger melting point pump

Acronyms ADS accelerator driven system LBE leadebismuth eutectic

"

#

  m _ j _ jm DzC d _ C ¼  pC;out  pC;in þ C C KC þ rC gDzC ; m AC dt 2rC A2C

(1)

"

2. Theoretical model and its stability analysis A remarkable phenomenon was observed by the numerical simulation of unprotected loss of flow (ULOF) in (Liu et al., 2010) for the design of European Facility for Industrial Transmutation (EFIT), where the flow rate decreases to lower level and increases again, then begins to oscillate and finally stabilizes at an asymptotic level. It has been shown numerically with SIMMER (Chen et al., 2010) that it occurs in the XT-ADS as well. As understood from numerical results this phenomenon is associated with the free surface motion in hot and cold coolant plena (legs). But a complete understanding of this phenomenon, e.g. the dependence on the main parameters, is difficult to be achieved by numerical calculations. To develop a simple theoretical, if possible, even analytical model for understanding this phenomenon is the purpose of this section.

  m _ j _ jm DzHX d A _ HX ¼ HX pHX;out  pHX;in  HX 2HX KHX m DzHX AHX dt 2rHX AHX #

þ rHX g DzHX þ Dppump ;

(2)

where subscripts C and HX denote coolant channels of the reactor _ stands for core including the bypass and the heat exchanger (HX), m the mass flow rate, A the coolant flux area, the average coolant R Dz density, defined as r ¼ 1=Dz 0 r dz, Dz the coolant channel

2.1. Integral equation model The geometrical model is given in Fig. 1. By modeling, the whole reactor primary coolant flow region is divided into four parts: reactor core, heat exchange and pump unit, cold and hot coolant plena. In the reactor core region, filled with subassemblies, two types of coolant channels of the hot active core channel and the cold bypass channel are merged together into one region, i.e. the reactor core one. Since we consider the problem of steady state stability of an ADS system, we can assume the power to be constant and the quasi-static energy equation to be valid, although the transient energy equation can be included in the theoretical model easily (Chen et al., 2010). If the density transient change in the channels is neglected, the mass conservation gives that the flow rate in each channel is independent of the channel height, i.e. they are equal in different height positions in each channel and only functions of time. Therefore we can have integral momentum equations as

Fig. 1. Schematic of theoretical modeling of an LBE cooled ADS.

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X.-N. Chen / Progress in Nuclear Energy xxx (2014) 1e8

3

height (length), K the total pressure drop coefficient including friction and form change effects and Dppump the pump head. The sign of the right hand side of Eq. (2) is different from that of Eq. (1), because the mass flow rate of HX is defined as downward as positive. Mass conservation in the cold and hot coolant plena holds as

d2 d A g _ C þ lC Dm _ C þ C ðDm _ C  Dm _ HX Þ ¼ 0; Dm DzC A dt dt 2

_ HX _ m d m ðrcold zcold Þ ¼  C ; dt Acold

with

 A ¼

(4)

where zcold and zhot are the cold leg and hot leg heights, Acold and Ahot the cold and hot free surface areas, rcold and rhot the cold and hot coolant densities, respectively. Since the velocities in the hot and cold plena are negligibly small, the pressure differences between inlet and outlet of core and HX can be evaluated approximately by the static pressure differences as,

pC;out  pC;in ¼ rC;out gzhot  rcold gzcold ;

(5)

pHX;out  pHX;in ¼ rcold gðzcold  DzCH Þ  rhot gðzhot þ DzC  DzCH  DzHX Þ;

(6)

To complete the equation system we still need the energy conservation equation for obtaining coolant temperatures and consequently the coolant densities. Although an integral transient energy equation can be derived and solved (Chen et al., 2010), it is sufficient to use the following quasi static energy equation for the stability analysis,

_ C DT ¼ P; cp m

(7)

where DT ¼ Thot  Tcold and P is the reactor thermal power. Since the density change is linearly proportional to the temperature change under the assumption of constant power, Eq. (7) can be written in terms of the density change as,

_ C;0 Dr0 ; _ C Dr ¼ m m

(8)

_ C;0 and Dr0 are their steady state values. where Dr ¼ rhot  rcold , m The LBE coolant density correlation of Alchagirov in OECD/NEA (2007) is taken as

h

d2 d A g _ HX þ lHX Dm _ HX  HX ðDm _ C  Dm _ HX Þ ¼ 0; Dm DzHX A dt dt 2

(3)

_ HX _ m d m ðr z Þ ¼ C ; dt hot hot Ahot

i

r ¼ rmelt 1  1:07978  104 ðT  Tmelt Þ ; where T is in K and Tmelt ¼ 398.15 K and rmelt ¼ 10,529 kg/m3 are the melting temperature and its associated density. The set of Eqs. (1)e(4) with Eqs. (5)e(7) for Dp and Dr forms a dynamical system of four first-order time derivative equations for _ C, m _ HX , zcold and zhot . The pressure loss four unknown variables m coefficient K of each channel is assumed here to be constant, which includes all pressure loss effects, e.g. the pressure loss due to the friction and the channel shape change. It can be derived from design values of the pressure drop in the steady state.

2.2. Stability analysis Based on the steady state and in terms of the mass flow rates, we _ CB obtain linearized equations for the perturbed mass flow rates Dm _ HX from Eqs. (1)e(6) as and Dm

(9)

KC ¼

1 Acold

þ

1

1 Ahot

; lC ¼

KC uC K u ;l ¼ HX HX ; DzC HX DzHX

(10)

(11a)

_C _ HX Dp Dp m m  C ; uC ¼ ;K ¼  HX 2 ; uHX ¼ ; rC AC HX rHX AHX 1 2rC u2C 1 2rHX uHX (11b)

From these linear equations, by setting the exponential ansatz of exp(s t), we obtain immediately the eigenvalue equation as



s2 þ lC s þ

g LC



s2 þ lHX s þ

g LHX

 ¼

g2 LC LHX ;

(12)

where the kinetic lengths of the core and HX are defined as

LC ¼

A A Dz ; L ¼ Dz : AC C HX AHX HX

(13)

There is a trivial eigenvalue (s ¼ 0) in Eq. (12). As it is extracted, Eq. (12) becomes finally



s3 þ ðlC þ lHX Þs2 þ lC lHX þ

   lC lHX g g sþ g ¼ 0: þ þ LC LHX LHX LC (14)

This stability analysis is devoted to the flow circulation under influences of free surface motions. It is somehow similar to a U-tube oscillation. It can be understood in such a way. The cold and hot free surfaces act as two free surfaces in the U-tube, while in the hot region the forced or natural convections (circulations) take place. It can be called as a complicated U-tube system. Its eigenvalue problem becomes also more complicated with respect to the simple U-tube one. It can be recognized from Eq. (12) that l corresponds to damping coefficient and g/L to square of circular frequency. In Eq. (14) the coefficient of s2 implies a total damping coefficient, the coefficient of s implies a total circular frequency and the last inhomogeneous term implies a decaying effect due to the combination of the damping and oscillation in this complicated U tube. The damping coefficient is related to the pressure drop. If it is small, we will have a weak damping oscillation. If it is large, we will have a strong damping system, where we do not have oscillating behavior. This is similar to a classic damped free oscillation system. In addition to the pair of complex conjugate eigenvalues we have a third eigenvalue, which is always negative and coming from the inhomogeneous decaying effect. 2.2.1. Steady state The total core thermal power of XT-ADS is 57 MW and the coolant heat capacity cp ¼ 145 J=ðkg KÞ. The coolant inlet temperature is 300  C, the active core outlet temperature 387.4  C, but the average outlet temperature is equal to the HX inlet temperature 359  C. Correspondingly we have the total mass flow rate as

_ C;0 ¼ _ HX;0 ¼ m m Then

the

P ¼ 6663 kg=s: cp DTHX average

coolant

rC ¼ rHX ¼ r0 ¼ 10298:4 kg=m3 at T ¼ 329.5  C.

density

is

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X.-N. Chen / Progress in Nuclear Energy xxx (2014) 1e8

The design values of pressure drops in the core and HX are 0.8529 and 0.3241 bar, respectively (Chen et al., 2010). The pressure loss coefficients are derived as

KC;0

¼ 57:189; KHX;0

¼ 17:484;

oscillation nature. The ULOF natural convection state is in the damped oscillation region and shown by a thick point. 3. Flow similitude laws between water and LBE

where the following mean velocities and flow areas are used

3.1. Natural convection problem and its modeling

uC ¼ 0:5382 m=s; uHX ¼ 0:60 m=s;

We consider a nuclear fission reactor that produces a certain amount of thermal power and is cooled by liquid lead or LBE (see Fig. 1). The coolant is heated up, as it goes through the reactor core, and takes the heat with it away from the core and is cooled down, as it goes through the heat exchanger and gives the heat to the socalled secondary circuit. The density difference between hot and cold sides leads to a static pressure difference, which is a driving force (a kind of buoyancy) that makes the coolant be circulated by itself as long as the reactor core is still producing heat, even if there is no pump. This is the so-called natural convection, which can be used to cool the reactor under operational or accident conditions (De Bruyn et al., 2011; Wallenius and Suvdantsetseq, 2011; Chen et al., 2010). The natural convection is actually a balance (equilibrium) between two effects, namely the pressure drop mainly due to viscous friction and the driving pressure due to the difference of density between hot and cold sides. Therefore two principal nondimensional parameters, Reynolds number and Richardson (or Grashof) number, come into play. The problem can be formulated and solved in an integral equation model, as given in the following. As already described in the last section, the transient integral model can be formulated. For the sake of simplicity we consider now only the steady state. _ is The mass conservation requires that the mass flow rate m constant in a flow channel,

AC ¼ 1:2022 m2 ; AHX ¼ 1:0785 m2 :

2.2.2. Eigenvalues of steady state The parameters in Eq. (14) are derived for the steady sate as

lC ¼ 15:3888 s1 ; lHX ¼ 5:24516 s1 ; LC ¼ 5:38915 m; LHX ¼ 6:00727 m: The three eigenvalues are obtained from Eq. (14) as

s1 ¼ 15:2709 s1 ; s2 ¼ 4:89976 s1 ; s3 ¼ 0:463323 s1 : All of them are real and negative. This means the steady state is absolutely stable. 2.2.3. Eigenvalues of natural convection state In the case of natural convection the coolant velocity is reduced to 20% of the steady state. Therefore the associated parameters in Eq. (14) changes as

lC ¼ 3:07777 s1 ; lHX ¼ 1:04903 s1 while the values of KC, KHX, LC and LHX do not change. The three eigenvalues are obtained from Eq. (14) as

s1 ¼ 2:57864 s1 ; s2 ¼ 0:774078  1:44555i s1 ; s3 ¼ 0:774078 þ 1:44555i s1 :

_ ¼ ruA ¼ const: m

(15)

where u is the mean flow velocity. Since different channels have different mass flow rates, we may consider an average channel for deriving similitude laws and extend their results approximately to other channels. The momentum conservation gives the pressure balance, i.e. the dynamic pressure drop Dpd is equal to the static buoyancy pressure, as

The first eigenvalue remains a negative real and the second and third ones are merged together as the coolant velocity decreases and then bifurcate into a pair of complex conjugates with a negative real part during the coolant velocity change from the steady state to the natural convection state. This means that the system is still stable, but possesses a damped oscillation nature. The imaginary part is actually the circular frequency u ¼ 1.44555 s1 of the oscillation, which corresponds to a period of 4.35 s. This result confirms the oscillation phenomenon found in the ULOF numerical simulation (Chen et al., 2010). 2.2.4. Eigenvalues of natural convection state The stability analysis model is rather simple, but its parameter diagram could be complicated. Here we just take two parameters as variables, the normalized coolant velocity u=u0 and the normalized free surface area A=A0 , while other parameters are fixed. The reference values of u0 and A0 are those of steady state. The bifurcation line between the stable state and the damped oscillation state is obtained from the solution of Eq. (14) by variation of A=A0 and u=u0 , as shown in Fig. 2. Above this line the flow is absolutely stable and below this line the flow possesses the damped

Fig. 2. The bifurcation between the stable state (right-upper region) and the damped oscillation state (left-lower region) in the parameter map (u/u0, A=A0 ), where the point presents the natural convection state.

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X.-N. Chen / Progress in Nuclear Energy xxx (2014) 1e8

Dpd ¼ DrgH

(16)

where H is the height difference between two vertical positions of mean temperatures in core and heat exchanger, Dr the density change between cold and hot sides and g the gravity acceleration. The energy conservation leads to an equation for obtaining the temperature as,

_ p ðTÞ mc

dT dP ¼ 0 dx dx

ZTout cp ðTÞdT ¼

P _ m

(17)

Tin

where P is the heat power produced in the channel, Tin and Tout the core inlet and outlet temperatures, respectively. In order to obtain the laws of similitude we have to non_ is dimensionalize our formulation. Because the mass flow rate m constant, it can be used as a basic quantity for the nondimensionalization. The coolant velocity is expressed as

u ¼

_ m

rA

:

(18)

As usual the pressure Eq. (16) is divided by ru2, where r and u are the average coolant density and velocity in the core, respectivley. Thus, its right-hand side becomes the so-called Richardson number, which represents the non-dimensionalized driving pressure and the importance of natural convection relative to the forced convection, as

Ri ¼

Dr gH : r u2

(19)

The left-hand side of Eq. (16) can be evaluated in principle by Reynolds number dependent correlations of pressure losses (Waltar and Reynolds, 1981) as

8

9



= X 1 L 1 2 Dpd 1
(20)

where index i indicates a certain flow section and index j a certain position where the flow area changes significantly, ReDH is the local Reynolds number defined as

ReDH ¼

uDH

n

;

(21)

and DH the hydraulic diameter defined as

DH ¼

4A ; LW

(22)

where A is the flow area and Lw the associated wet perimeter. The power in the energy equation can be normalized as, as Eq. (17) is divided by cp DT,

1 cp DT

ZTout cp ðTÞdT ¼ Tin

P ; _ p DT mc

(23)

where cp is the average coolant heat capacity, DT ¼ Tout  Tin , Tin and Tout are the core inlet and outlet temperatures. Since the average coolant heat capacity can be defined as

cp ¼

1 DT

5

ZTout cp ðTÞdT;

(24)

Tin

the dimensionless power becomes unity, i.e.,

P ¼ 1: _ p DT mc

(25)

It should be mentioned that a combination of Richardson number and Reynolds number gives the Grashof number, which indicates the ratio of buoyancy to viscous force acting on a fluid and appears more often in the literature, as

Gr ¼

Dr gH3 uH ¼ Ri  Re2H ; ReH ¼ : r n2 n

(26)

This means that the Grashof similarity is equivalent to the Richardson one under the condition that the Reynolds law holds.

3.2. Establishment of similitude laws The geometrical similarity is a basis for the physical similarity. Assume that the water experiment facility is similar to the LBE one with a model scale factor l defined as

l ¼

Lwater ; LLBE

(27)

where Lwater and LLBE are reference lengths in the water and LBE experiments, respectively. It seems at the moment that it can be chosen arbitrarily for a global parameter similitude. Later on we will see that it will be determined to be able to achieve a more accurate local similitude The physical similarity will be achieved (guaranteed) by establishing the equality between LBE and water experiments in the three non-dimensional numbers, i.e. Reynolds number, Richardson (Grashof) number and the non-dimensional power number. We call processes of equating these three numbers as Reynolds similitude law, Richardson (Grashof) similitude law and power similitude law. These three similitude laws can be satisfied by suitably choosing water experiment parameters. Since we use water to simulate an LBE experiment, we may call the LBE experiment the original (target) experiment and the water one the model experiment. Starting point is that the LBE experiment parameters are fixed according to a certain design, such as the core thermal power, the coolant inlet and outlet temperatures and the coolant mean velocity. For instance the temperature range in the LBE experiment is determined mainly due to corrosion limits of materials, e.g. from 270  C to 430  C. First of all we fix a certain model scale l. By Reynolds similitude law, where we may first consider its mean value defined as,

ReDH ¼

uDH

n

:

(28)

We can determine the coolant average velocity in the water experiment, if the average kinematic viscosity is known, which depends on the choice of the temperature range. Then, by Richardson similitude law Eq. (19), in which the part gH=u2 is associated with the coolant velocity as discussed before, we obtain Dr=r for the water experiment. Consequently we can determine the temperature range for the water experiment. Notice that we still

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X.-N. Chen / Progress in Nuclear Energy xxx (2014) 1e8

have the possibility of choosing parameter Tin, which is very important for the further discussion on the local similitude. Finally by the power similitude law Eq. (25), we determine the power for the water experiment. Up to this step we realized the global similitude between LBE and water experiments. Naturally the question is arising here, how about the local similitude. This is particularly important for the local Reynolds similitude, because the viscous friction depends on the local Reynolds number. We notice that the viscosity and density of LBE and water have similar trends with temperature variation. Therefore we might choose a suitable value of Tin in the water experiment, so that the variation of the viscosity is also in similarity, i.e. Dm=m is almost same in the both experiments. This can be realized as well and will be discussed in detail in the next subsection. Since the density change is linear with temperature in the considered similitude temperature ranges, the global Richardson similitude will cover the local Richardson similitude, meaning the local similitude concerning Dr=r has already been guaranteed by the global similitude. The local power similitude will be approximated, since cp varies very slightly with temperature, where Dcp =cp is just a few percent.

3.3. Thermo-physical properties and local similitude The density of LBE liquid depends almost solely on temperature. Its density variation with pressure can be totally neglected, not only because it is very small, but also the cover gas pressure in the LBE system is just about 1 bar or less. An empirical formula for it has been given in OECD/NEA (2007), as repeated here as,

h

i

rLBE kg=m3 ¼ 11096  1:3236 T½K:

(29)

The dynamic viscosity of LBE liquid has been given in OECD/NEA (2007) as well, as repeated here as



mLBE ½Pa s ¼ 4:94  104 exp

754:1 : T½K

(30)

The kinematic viscosity of LBE liquid can be calculated by using Eqs. (29) and (30) in the following formula

nLBE

m ¼ LBE : rLBE

(31)

The heat capacity of LBE liquid is repeated here as well (OECD/ NEA, 2007) as

cp; LBE ½J=ðkg KÞ ¼ 159:0  0:0272T þ 7:12  106 T 2

(32)

where the temperature T is in Kelvin There are a lot of open sources for obtaining water thermophysical properties. We just use the website of the U.S. National Institute of Standards and Technology (NIST, 2012) for obtaining data. Excepting the boiling point of water, the thermophysical properties change very slightly with pressure. For the sake of simplicity, we might assume they are independent of pressure in its range considered here. Since the water density change with temperature is not linear, we may need a larger temperature range for searching our similitude solution. Therefore we consider in this paper the temperature range from 0  C to 200  C at pressure of 16 bar. We use the following fitting functions

h

i

rwater kg=m3 ¼ 1001:0934  0:0796868 T  0:00375086T 2 þ 3:615219  106 T 3 ; (33) 0:001779 ; 1 þ 0:03368T þ 0:0002210T 2

mwater ½Pa s ¼

(34)

cp; water ½J=ðkg  CÞ ¼ 4204:87  0:961226T þ 7:85734  103 T 2 þ 2:0525  105 T;

(35)

where the temperature T is in  C. The local Richardson/Grashof and Reynolds similitudes demand the local Richardson/Grashof and Reynolds numbers defined in Eqs. (19), (26) and (21) to be equal in the both experiments. Since the mass flow rate is constant through the flow channel, it is more accurate to discuss two numbers in a form based on the mass flow rate as

Ri ¼

Drr _2 m

2

gHA ; ReDH ¼

_ DH m : m A

(36)

We eliminate the scale factor l by multiplying ðReDH A=DH Þ5 and 2 Ri=ðHA Þ to get a new dimensionless number, i.e.

   5 _3 m 2 ¼ 5 Drrg: HA ReDH A=DH Ri

m

(37)

By the similitude laws it implies

 3 _ m

m5



Drr

 ¼ water

_3 m

m5



Drr

:

(38)

LBE

This equation includes only the flow and thermo-physical properties without the scale factor l. Thus, we can first make the similitude of the local dynamical viscosity change before determining the scale factor. This can be approached by suitably choosing Tin and Tout in the water experiment so that

mðTin Þwater mðTout Þwater m ¼ ¼ water ; mðTin ÞLBE mðTout ÞLBE mLBE

(39)

Then we calculate

ðDrrÞwater ; ðDrrÞLBE

(40)

with

Dr ¼ rðTout Þ  rðTin Þ; r ¼ ½rðTout Þ þ rðTin Þ=2;

(41)

By Eq. (38) we obtain

_ water m ¼ _ LBE m

"

mwater mLBE

#1 5  3 ðDrrÞwater : ðDrrÞLBE

(42)

Returning to the Reynolds similitude law in the form of Eq. (36), we obtain the model scale factor as

l ¼

_ water Lwater m ¼ _ LBE LLBE m



mwater : mLBE

(43)

The power similitude gives

_ water DTwater cp; water m Pwater ¼ : _ LBE DTLBE cp; LBE m PLBE

(44)

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X.-N. Chen / Progress in Nuclear Energy xxx (2014) 1e8

7

Table 1 Results of model experiments by variation of inlet temperature, where 1:1 model scale results are highlighted by bold values. Case

1

2

3

4

5

Tinlet, C Toutlet,  C DT,  C Model scale factor Model power factor Model velocity factor

60 81.8 21.8 2.166 2.006 1.1419

100 129.0 29.0 1.229 0.888 1.206

118.45 150.7 32.28 1.0001 0.661 1.225

130 164.3 34.3 0.891 0.559 1.236

140 176.1 36.1 0.811 0.490 1.246

So now we have completed the local similitude laws, i.e. Reynolds, Richardson (Grashof) and energy similitude laws hold not only globally but also locally. It is amazing and very useful for using water to simulate LBE circulations. 3.4. Similarity examples For the sake of simplicity we fix the temperature range in the LBE experiment for the whole discussion in this section as Tin, LBE ¼ 270  C (543 K) and Tout, LBE ¼ 430  C (703 K). The power and the mass flow rate, which are associated with each other as given in Eq. (17), are actually parameters that are searched and determined in the natural convection experiments representative for a certain reactor design. Therefore they are not fixed here. 3.4.1. Inlet temperature variation in water experiments We choose several inlet temperatures as input data and go through the local similitude laws described in the last section, especially by Eqs. (36)e(44). The results are shown in Table 1, where, of course, the temperature ranges of various cases correspond to different working pressures in order to be sure no boiling takes place. With the increase of inlet temperature the model scale factor decreases, the model power factor decreases as well and the model velocity factor increases only slightly. If the temperature range is below 100  C, e.g. in Case 1, it can be carried out at atmospheric pressure, but its size scale and power are more than doubled in the water model experiment. Case 3 shows a particular situation, where the model is one-to-one to the real one and the heat power is reduced by a factor of 0.672. The model scale and the power can be even reduced by choosing a higher inlet temperature, as shown in Cases 4 and 5. If we want to even reduce the model size, we can still go on to increase the inlet temperature. This means higher pressure is needed for avoiding that the temperature reaches the boiling point. This is definitively feasible with today’s water technology.

Fig. 3. Relative temperature distributions for water and LBE experiments.

Fig. 4. Relative density change distributions for water (lower dashed one) and LBE (upper solid one) experiments.

3.4.2. Theoretical results of 1:1 scale water experiment As a particular example we choose Case 3, where the model geometric scale is identical to the original LBE one, for presenting our similitude results. For the sake of simplicity we assume that the linear power rate is constant along the core height. Therefore the temperature distribution along the core height can be solved easily by integrating (3). Since the heat capacities of water and LBE are almost constant, the temperature distributions are almost linear along the core height and they are overlapped, as shown in Fig. 3. The maximal discrepancy of DT=DTmax ¼ ðT  Tin Þ=ðTout  Tin Þ between water and LBE is about 0.005. The comparison of relative density changes is shown in Fig. 4. Their difference is zero at the two ends of the reactor and their maximal difference about 0.02 occurring at the core mid-plane. This implies that the local Richardson similitude holds as well as the global one. The local Reynolds number ratio Rewater/ReLBE is shown in Fig. 5. As we are free to choose the inlet and outlet temperatures in the water experiment, we made this ratio equal to one at the core inlet and outlet. The maximal deviation of Reynolds number is only 1.4% occurring at the core mid-plane. Therefore the Reynolds similitude law is fulfilled very well locally everywhere in the reactor. This is very important for experimental pressure drop simulations. The local density ratio is shown in Fig. 6. It has not been required that this ratio is equal at the two reactor ends. Nevertheless the deviation of this ratio from its average value is only about 0.5%. This implies that the velocity deviation from its average value has the

Fig. 5. Reynolds number ratio of water/LBE.

Please cite this article in press as: Chen, X.-N., On LBE natural convection and its water experimental simulation, Progress in Nuclear Energy (2014), http://dx.doi.org/10.1016/j.pnucene.2014.03.001

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X.-N. Chen / Progress in Nuclear Energy xxx (2014) 1e8

Acknowledgment The author appreciates the effort and support of all the scientists and institutions involved in IP EUROTRANS, as well as the financial support of the European Commission through the contract FI6W-CT-2004-516520. He thanks Dr. E. Kiefhaber, a retired scientist at KIT, for his careful reading and helpful corrections.

References

Fig. 6. Local density ratio of water/LBE.

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Please cite this article in press as: Chen, X.-N., On LBE natural convection and its water experimental simulation, Progress in Nuclear Energy (2014), http://dx.doi.org/10.1016/j.pnucene.2014.03.001