Subchannel analysis of nanofluids application to VVER-1000 reactor

chemical engineering research and design 9 1 ( 2 0 1 3 ) 625–632

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Chemical Engineering Research and Design journal homepage: www.elsevier.com/locate/cherd

Subchannel analysis of nanofluids application to VVER-1000 reactor Ehsan Zarifi a , Gholamreza Jahanfarnia a,∗ , Farzad Veysi b a b

Department of Nuclear Engineering, Science and Research Branch, Islamic Azad University, Tehran, Iran Mechanical Engineering School, Razi University, Kermanshah, Iran

a b s t r a c t In this study a thermal-hydraulic analysis is performed of nanofluids as a coolants in subchannels in hot fuel assembly of VVER-1000 reactor by subchannel method. Water-based nanofluids containing various volume fractions of Al2 O3 nanoparticles are analyzed. The conservation equations and conduction heat transfer equation for fuel and clad have been derived and discretized by finite volume method. The transfer of mass, momentum and energy between adjacent subchannels are split into diversion crossflow and turbulent mixing components. The governing non linear algebraic equations are solved by numerical iteration methods. Finally the nanofluids analysis results are compared with the pure water results. To validate the applied approach, the model and COBRA-EN code results are compared for the case of pure water. Results show that the heat transfer increases with nanoparticles volume concentration in the subchannel geometry. © 2013 The Institution of Chemical Engineers. Published by Elsevier B.V. All rights reserved. Keywords: Subchannel analysis; Nanofluids; Thermal-hydraulic analysis; VVER-1000 reactor

1.

Introduction

Nanofluids are engineered fluids that contain a suspension of nanoparticles in a pure substance. Nanoparticles can be any variety of metals, metal oxides, or ceramics. From previous investigations, nanofluids have been found to possess enhanced thermo-physical properties such as thermal conductivity, thermal diffusivity, viscosity and convective heat transfer coefficients compared to those of base fluids (Lee et al., 1999; Choi et al., 2001; Das et al., 2003; Xuan and Li, 2003; Bang and Chang, 2005; Kukarni et al., 2006; Buongiorno et al., 2008; Noie et al., 2009; Zeinali Heris et al., 2009; Wu et al., 2010). Generally, the thermal conductivity of the particles, metallic or nonmetallic, is typically an order of magnitude higher than that of the base fluids even at low concentrations resulting in significant increases in heat transfer. Compared with the existing techniques for enhancing heat transfer, the nanofluids show a great potential in increasing heat transfer rates in a variety of application cases, while incurring either little or no penalty in pressure drop.

∗

Relevant literature reveals that with low nanoparticles concentrations (1–5 vol %), the thermal conductivity of the suspensions can be increased by more than 20%. Such enhancement mainly depends upon factors such as the shape and the dimensions of particles, the volume fractions of particles in the suspensions, and their thermal properties (Xuan and Roetzel, 2000). Buongiorno and Truong (2005) performed a study to assess the feasibility of nanofluids in nuclear applications by checking on the performance of any water-cooled nuclear system that is heat removal limited. Possible applications include pressurized water reactor (PWR) primary coolant and safety systems (Zarifi et al., 2012, 2013; Hadad et al., 2010; Buongiorno et al., 2009). The use of nanofluids in nuclear power plants seems like a potential future application (Buongiorno et al., 2008). Several significant gaps in knowledge are evident at this time, including, demonstration of the nanofluids thermal-hydraulic performance at reactor core and the compatibility of the nanofluids chemistry with the reactor materials. In this study, the subchannel analysis of nanofluids as a coolant in hot fuel assembly of VVER-1000 reactor is investigated. The term subchannel is usually associated with the flow passages between the fuel rods. It is better to use this method for analyzing fuel assemblies, because it gives the velocity, pressure, and fuel rod temperatures for single

Corresponding author. Tel.: +98 912 6542844; fax: +98 21 44817170. E-mail address: [email protected] (G. Jahanfarnia). Received 19 July 2012; Received in revised form 31 December 2012; Accepted 7 January 2013 0263-8762/$ – see front matter © 2013 The Institution of Chemical Engineers. Published by Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.cherd.2013.01.018

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Nomenclature A C Cba d D h H H10 K kB KG l N Nu P Pr q q R Rem Re sk sl T teff T u v V w

flow area (m2 ) specific heat capacity (J/kg K) critical boron concentration in the primary coolant loop (g/kg) nanoparticle diameter (nm) open gap width (m) enthalpy (J/kg) Meyer hardness (Pa) location of control rods of type 10 (%) thermal conductivity (W/mK) Boltzman constant (1.3807 × 10−23 ) (J/K) pressure drop coefficient Center to center distance between two adjacent channels (m) reactor thermal power (MW) Nusselt number pressure (Pa) Prandtl number heat flux (W/m2 ) volumetric heat generation (W/m3 ) Radius (m) modified Reynolds number Reynolds number lateral length in “k” gap (m) lateral closed physical area (m2 ) temperature (K) effective time (day) coolant temperature (◦ C) internal energy (J/kg) velocity (m/s) volume (m3 ) mass flow rate (kg/s)

Greek symbols ˛ molecular thermal diffusivity (m2 /s) porosity   viscosity (N s m−2 ) density (kg/m3 )  volume fraction  ˚ dissipation function (W/m3 ) ı surface roughness (m)  Shear stress (Pa) Subscripts A surface b coolant bulk base fluid f gas g l,l channel number total time step n nf nanofluids nanoparticle p volume v w wall surface

and two-phase flow in reactor cores. There are lots of subchannel codes, such as COBRA-EN code which is used for thermal hydraulic analysis of nuclear reactors. This code has several minor deficiencies to model some reactors such as VVER-1000 (Basile et al., 1999) some of these deficiencies are as follows:

1. This code is not able to model the central fuel hole which results in incorrect radial temperature distribution in fuel rods. 2. The maximum number of grid spacers considered in the code is 10 while the VVER-1000 reactor has 15 grid spacers. 3. The thermal conductance of gas gap and thermal conductivity of fuel and clad in the code is not a function of temperature. 4. It has limited radial meshes in fuel rod. 5. The two-phase models have some deficiencies in the code. This study deals with the development of a computer code for steady-state and transient VVER-1000 subchannel analysis and to improve COBRA-EN adversity by considering nanofluids properties.

2.

Methods and materials

2.1.

Subchannel analysis

The subchannel analysis is one of the key thermal-hydraulic calculations in the safety analysis of the nuclear reactor core. The main purpose of this analysis is the determination of fuel and clad temperatures, coolant state during normal operation and assumed accident conditions. Subchannel analysis can be viewed as thermal-hydraulic pin-by-pin analysis of the core. Subchannel analyses can be performed for model of isolated fuel assembly, fuel assembly with several isolated adjacent fuel assemblies or for the whole core. Initial and boundary conditions are usually obtained from system codes such as RELAP and ATHLET. For the subchannel analysis, it is necessary to get the power distribution from some other (neutronic) codes. Fuel rods can be modeled as a source of heat defined by surface heat flux or conduction model of fuel rods can be considered.

2.2.

Derivation of conservation equations

2.2.1.

Mass balance

The mass conservation equation of the fluid is noted below (Todreas et al., 1982): 1 ∂nf vA 1 ∂nf + = ∂t A ∂z A



(1)

wij

In which wij is the mass transfer between adjacent channels i and j. The density of a nanofluid can be calculated by using the mass conservation as (Velagapudi et al., 2008): nf = (1 − )f + p

2.2.2.

(2)

Axial momentum balance

The dynamic equation of fluid motion with gravity as the only force body is as follows (Todreas and Kazimi, 1990):

 

∂ p ∂((nf vA)i vzi ) ∂  vAi + = −Anf gz − Afi ∂t nf ∂z ∂z −

N 

wij (vzi − vzj ) −

F  iz

z

(3)

j=1

The first term on the left-hand side denotes the momentum storage and the second term refers to the axial momentum change. The four terms on the right-hand side represent rate of changes of gravitational force, axial change

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of pressure, the lateral momentum exchange between channel i and the adjacent channels j owing to turbulent mixing of the fluid and the subchannel circumferentially averaged force per unit length of the fluid on the solid for vertical flow over the solid surfaces in the control volume respectively.

2.2.3.

Transverse momentum balance

The balance equation of transverse momentum is written as (Basile et al., 1999):

 Xk   w∗ − U wk − wnk + Ukj w∗ k k−1 k−1 t =

sk 1 gc Pkj−1 X − lk 2



KG

Xv ∗ sl



|wk | wk

(4)

where the momentum velocity for gap k is assumed to be supplied by the arithmetic mean of the momentum velocities of channels l and l connected by gap k. Uk =

 1  Ulk + Ul/ k 2

Fig. 2 – Control volume for transverse momentum balance finite-volume equation (top view).

(5) Afi

Pkj−l = Plj−l −Pq−l is the pressure difference between channels l and l and the donor cell rule is applied to the axial inflow and outflow of transverse momentum. w∗k

= wk →

Uk 0

w∗k = wk+1 → Uk 0

∂ ∂  mi hnfi = Afi nf hnf i + ∂t ∂z

Dpi Dt



+ qi  −

N 



Wij∗H hnfi − hnfj



j=1

−

N 

Wij

 ∗ hnf

(7)

j=1

(6)

The first term on the right-hand side of Eq. (4) is the lateral pressure difference on the boundaries of the momentum control volume (Figs. 1 and 2), while the last term is the pressure drop in the lateral flow through the gap. On the contrary, there is no term representing lateral flow of transverse momentum because the cross flow velocity (at least, on the average) is assumed to vanish away from the gap, i.e., on the boundary of the control volume for transverse momentum balance.

where hnf is the nanofluid enthalpy, which can be written as: hnf = Cnf T

(8)

Also, specific heat of the nanofluid can be calculated by using energy balance as (Velagapudi et al., 2008): Cnf =

(1 − )f Cf + p Cp nf

(9)

In Eq. (7), is:

2.2.4.

Energy balance

Energy conservation equation is given by (Todreas and Kazimi, 1990):

qi  =

1 VT



→ Knf − n .∇TdA

(10)

Af

where Knf is the nanofluid thermal conductivity. Jang and Choi (2004) found that the Brownian motion of nanoparticles at the molecular and nanoscale level is a key mechanism governing the thermal behavior of nanofluids. They theoretically derived a model which considers effect of the concentration, temperature, and nanoparticles size. Thus from the above analysis one can conclude that the thermal conductivity of a nanofluid, Knf , is a function of:



Kp Knf = cRem 0.175 ϕ0.05 Kf Kf

0.2324 (11)

where Rem , modified Reynolds number of nanofluids, is equal to:

Rem =

 1   18k T  12  B vf

The term Fig. 1 – Control volume for mass, energy and axial momentum balance finite-volume equation (lateral view).

DP  Dt

=

p dp

 DP  Dt

(12)

in Eq. (7) can be written as:

∂P − → → v  − P∇. v + ∇.nf − ∂t

(13)

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the non dimensional numbers. The convective heat transfer coefficient and Nusselt number are related as: hnf =

Nunf Knf D

(16)

The heat transfer coefficient of turbulent flow through circular tube can be calculated from equation in the following form: (Nunf )c.t. = a(Renf )

0.8

(Prnf )

0.4

(17)

where Renf and Prnf are defined as:

Fig. 3 – Cross section view of fuel rod. Pak and Cho (1998) correlation for Al2 O3 + water nanofluid viscosity used in conservation equation is as below: nf = f (1 + 39.11 + 533.92 )

(14)

In the present work the above equation is found to be nearer to experimental data and current conditions.

2.2.5.

Heat transfer model

The overall heat transfer model implies a fuel heating model and a heat transfer model from a heated wall to the coolant bulk. At each axial segment, the fuel heating model computes the temperature distribution in the cylindrical fuel rods by solving the heat conduction equation with heat flux boundary conditions supplied by the flow model and by the surface heat transfer model. Heat conduction in the axial direction is neglected while the heat conduction equation in the radial direction is approximated by a first-order finite-difference scheme and solved by a numerical iteration method. The heat transfer model is involved in the external iterations because it depends on the current flow conditions and, in its turn, supplies the energy equation with the heat source term. The constitutive relations for the heat transfer model essentially concern the heat transfer coefficients and the thermo physical properties (thermal conductivity and specific heat) of the fuel materials (uranium dioxide and zircaloy). The cylindrical geometry of a fuel rod is shown in Fig. 3.

Renf =

nf vD nf

(18)

Prnf =

nf Cnf knf

(19)

The Nusselt number data of the nanofluids obtained from (Xuan et al., 2003) is subjected to non-linear regression analysis and the constant “a” is obtained as 0.0256 for Al2 O3 + H2 O nanofluid. Thus the correlations for calculation of Nusselt number are developed as follows: (Nunf )c.t. = 0.0256(Renf )

(Prnf )

0.4

(20)

The above equation covers the current conditions such as nanoparticles concentration and Reynolds number and gives good agreement with the experimental results. The Reynolds and Prandtl numbers varied in the ranges 104 –105 and 6.5–12.3, respectively. For fully turbulent flow along rod bundles, Nusselt values may significantly deviate from the circular geometry because of the strong geometric nonuniformity of the subchannels. For this reason Nusselt number and convective heat transfer coefficient are expected to depend on the position of the rod within the bundle. The usual way to represent the relevant correlation is to express the Nusselt number for fully developed conditions (Nu) as a product of (Nu)c.t. for a circular tube multiplied by a correction factor: Nu =

(21)

(Nunf )c.t.

Markoczy (1972) gave a general expression for lar arrays for rod within a finite lattice:



2.2.5.1. Convective heat transfer. Concerning the thermal

= 1 + 0.9120Re−0.1 Pr0.4 1 − 2.0043e−B

resistance in the outer surface of the clad, the heat transfer equation can be written as: q˙  q˙  Tw = (Tw − Tb ) = = hnf 2 Rco hnf

0.8

B=

√   2 3 P 2 −1

D



in triangu-

(22)

(23)

(15)

where Tw is the clad surface temperature and Tb is the bulk fluid temperature. The heat transfer model supplies either the coefficient hnf or the heat flux q and it can be considered as the interface between the fuel heating model which supplies Tw and the flow model which supplies Tb . In their turn, the fuel heating and flow models depend on q and, thus, all of the three models are involved in the iterative loop of solution. In this paper, the single phase fluid is adopted to study the thermal behaviors of nanofluids. The thermo physical properties of the nanofluid itself are considered while evaluating

2.2.5.2. Conductive heat transfer. The heat conduction equations for clad and fuel in the radial direction are given by Eqs. (24) and (25) respectively: −

1 d r dr

−

1 d r dr

 Kclad r

dT dr

Kfuel r

dT dr



 

=0

(24)

= q˙ 

(25)

In these equations Kfuel and Kclad are the thermal conductivity coefficients of the fuel and clad. They are functions of

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Table 1 – Properties of fuel. T (K)

Kfuel (W/mK)

300 500 700 900 1100 1300 1500 1700 1900 2100 2300 2500 2700 2900

Cp (J/kg K)

8.15 6.7 5.4 4.4 3.75 3.25 2.8 2.5 2.4 2.42 2.44 2.5 2.65 3

270 278 302 310 314 319 320 328 340 364 390 426 470 520

Cv

(kJ/m3

K)

Table 3 – Thermo-physical properties of nanoparticles and base fluids. Property

2862 3042.2 3201.2 3286 3328.4 3381.4 3392 3476.8 3604 3858.4 4134 4515.6 4982 5512

C (J/kg K)

 (kg/m3 )

k (W/mK)

˛ (m2 /s)

4179 765

997.1 3970

0.605 40

1.47 1317

Water Al2 O3

gf + gc = 0.718

kgas



Tgas

Pgas

0.688 kgas = 3.366 × 10−3 Tgas

(31)

(32)

In addition to conduction across an open gap, heat can flow through points of fuel/clad contact. The solid conductance term is defined as follows:

temperature. Tables 1 and 2 show these values for Bushehr nuclear power plant.

hs =

 H

2Pgas 1 kfuel

+

1 kclad



(33) ıfuel 2 +ıclad 2 2

2.2.5.3. Gap conductance model. The fuel-clad gap is a major resistance to the heat transfer from the fuel pellets to the coolant. The heat transfer in gap with radius Rg is predicted from Eq. (26): Tgap =

Rg =

q˙ 

(26)

2 Rg hg

Rfo + Rci 2

2.3.

Thermal properties of nanofluids

In this study, Al2 O3 nanofluid with 0.001–10 volume percentages have been modeled in the first cycle of VVER-1000 reactor and the thermal-hydraulic parameters are calculated. The physical and thermal properties of this nanoparticle and base fluid are shown in Table 3 (Velagapudi et al., 2008).

(27)

2.4. In these equations, thermal conductance of gap hg is a strong function of hot gap size and the composition and pressure of gases in the fuel element. In this study, the modified Ross and Stout model is used for gap conductance (Ainscough, 1982). The conductance across the gap or interface between fuel and clad is considered as the sum of three terms: heat transfer across the gap by conduction through the gas, hgas ; solid conductance across points area of contact between fuel and cladding, hs ; and finally, a radiation heat transfer term, hr ; so that: hg = hgas + hs + hr

Flowchart

As seen in the programming flowchart (Fig. 4), Mass, momentum and energy equations are applied for each channel and

(28)

In this study, the term of radiation heat transfer is neglected. The thermal conductance of the gas is given by: hgas =

kgas Dg + A(ıfuel + ıclad ) + gf + gc

(29)

A = 2.75 − (2.55 × 10−8 Pgas )

(30)

Table 2 – Properties of cladding. T (K) 373.15 473.15 573.15 673.15 773.15 873.15 973.15 1273.15

Kclad (W/mK) 18 19.3 20.1 20.5 20.9 21.8 22.9 27.8

Cp (J/kg K)

Cv (kJ/m3 K)

280 301 322 343 368 398 448 290

1834 1972 2109.1 2246.7 2410.4 2606.9 2934.4 1899.5

Fig. 4 – Programming flowchart.

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the link between channels are indicated through transverse momentum equations. All aforementioned equations are in the form of coupled non-linear partial differential equations. These equations are discretized by control volume method except the transverse momentum equation which is discretized by finite difference method. Some important features of the written program can be mentioned as follows:

1. This program is written in transient state which is suitable to analyze accidents in nuclear power plants. 2. The core geometry considered as several control volumes. With capability to model an element such as pump or steam generator etc. So the whole plant can be modeled by considering this capability. 3. The point kinetic equations and the relevant feedbacks can be applied to perform transient analyses. 4. The flow area of each control volume can be changed along the axial direction which in this case there is the possibility of modeling the flow blockage accident. 5. Grid spacer local pressure drop is considered in axial momentum equations. 6. The grid convergence solution is applied to the finitevolume approach. It is investigated that the numerical solution becomes independent of the grid as the number of axial meshes exceed 60.

3.

Results and discussion

According to neutronic album of Bushehr VVER-1000 reactor (AEOI, 2010), the reactor power is continuously varying from cold zero to nominal power. In this study, power distributions in the hundredth day (nominal power) are considered for calculations. Fig. 5 shows the one-sixth power distributions in the hundredth day.

Fig. 5 – The power distributions in the hundredth day. The arrangement of hot fuel assembly is shown in Fig. 6 (FSAR of BNPP-1, 2003). The achieved results are compared with COBRA-EN code for validation. There are only thermal properties of liquid water in the library of the COBRA-EN code and there is no possibility to insert nanofluid properties, therefore the results of the modeling are compared with COBRA-EN code only for the pure water (Figs. 7 and 8). The comparisons show good agreement without significant deviations. The same boundary conditions such as heat flux, outlet pressure, inlet temperature and mass flow rate are considered both for nanofluids and pure water.

Fig. 6 – The meshing and channels arrangement of hot fuel assembly.

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Fig. 10 – Axial velocity in the hottest channel. Fig. 7 – Axial average coolant temperature.

Fig. 11 – Axial coolant temperature in the hottest channel. Fig. 8 – Radial temperature in the center of hottest rod.

In Figs. 9–11, the axial density, velocity and temperature distributions in the hottest channel for Al2 O3 nanofluid and pure water in various volume percentages are presented and compared with each other, respectively. The axial variation of above mentioned parameters is needed for coupling the written thermo hydraulic code with a neutronic code. Fig. 9 shows that as the volume percentage of nanoparticles increases, the density of the coolant grows up according to relation (2). For equal mass flow rate in various nanoparticles concentration, the coolant velocity decreases with increasing the nanofluid density in the reactor core (Fig. 10).

Fig. 9 – Axial density in the hottest channel.

As seen in Fig. 11, by considering equal coolant mass flow rate and inlet temperature, the coolant exit temperature increases with increase of nanoparticles concentration due to preserve constant heat flux value. Radial temperature in the center of hot fuel rod is presented in Fig. 12. According to Fig. 11, the fluid temperature increases with increasing concentration of nanoparticles, which will increase the fuel temperature. Due to higher heat transfer coefficient of nanofluids, increase of the fuel temperature is less than coolant.

Fig. 12 – Radial temperature in the center of hottest fuel rod.

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Conclusions

Typically, thermal-hydraulic analysis of the nuclear reactor core is done through both the porous media approach and sub-channel method. It is recommended to use the porous media approach to analyze the reactor core, because less number of the meshes and shorter program execution time are needed. Whereas it is better to use sub-channel method for analyzing fuel assemblies, since it also gives the temperature gradient in fuel rods. The results show that in low volume percentages of nanoparticles, there is no significant difference between nanofluids and pure water parameters values, but the deviations are remarkable for concentrations more than 0.1 percent. Due to increase of nanoparticles concentration, fluid temperature increases as a result of heat transfer coefficient improvement. It can be seen that for 10 volume percentage of nanoparticles the coolant temperature difference with pure water is about 15 ◦ C. Thus, to preserve equal design temperature difference, lower coolant flow rate is necessary for cooling the reactor core. Consequently, the reactor core can be more compact and the capital cost of the power plant will be reduced.

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