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Fusion Engineering and Design 85 (2010) 1736–1741

Contents lists available at ScienceDirect

Fusion Engineering and Design

journa l homepage: www.e lsev ier .com/ locate / fusengdes

ode development and validation for analyzing liquid metal MHD flow inectangular ducts

ao Zhoua,c,∗, Zhiyi Yanga, Mingjiu Nic, Hongli Chena,b

Institute of Plasma Physics, Chinese Academy of Sciences, Hefei, Anhui 230031, ChinaSchool of Nuclear Science and Technology, University of Science and Technology of China, Hefei, Anhui 230027, ChinaGraduate University of Chinese Academy of Sciences, Beijing 100049, China

r t i c l e i n f o

eywords:umerical simulationsiquid metal magnetohydrodynamicsurrent density conservative scheme

a b s t r a c t

A code named MTC-H 1.0 which can simulate 3D magnetohydrodynamics (MHD) flow in rectangularducts has been developed by FDS Team. In this code, a conservative scheme of the current density wasemployed for calculation of the induced current and the Lorentz force, and the consistent projectionmethod was employed for solving the incompressible Navier–Stokes equations with the Lorentz force

included as a source term. The code was developed on a structured collocated grid, on which velocity,pressure, and electrical potential were located in the cell center, while current fluxes were located on thecell faces.

The code MTC-H 1.0 was test by simulating MHD rectangular ducts flows, and the Shercliff’s insulatedwalls and Hunt’s conductive walls were used as benchmark models. The validation cases were conductedwith Hartmann number from 102 to 104 on structured collocated grids. The results matched well withHunt’s and Shercliff’s analytical solutions and it showed good accuracy of the code.

. Introduction

Liquid metal breeder blanket concept has been a topic of greatnterest in fusion reactor blanket design [1–5] because of manydvantages. A series of fusion reactors (named FDS series) haveeen designed and assessed by the FDS Team in China [6–10].nd a series of liquid metal blanket concepts [11–13] have beeneveloped correspondingly. In these blankets, lead–lithium alloy

s considered as tritium breeder only or as breeder and coolant.ut the motion of liquid metal in fusion reactor strong magneticeld cause serious magnetohydrodynamic (MHD) effects, whichave dramatic impacts on velocity distribution, heat transfer char-cteristics, pressure drop and the required pumping power for theooling system. Therefore, the liquid metal MHD effect under fusionelevant condition is one of the key issues in making an optimalesign for LM blanket [14–18].

Up to now, liquid metal MHD duct flow has been extensively

tudied by theory [19–22] and numerical simulation [23–28]. Dueo the complex geometry of flow channel in blankets, it is diffi-ult to study the detail of MHD flow characteristics by theoreticalnalysis. As the development of computer science, the numerical

∗ Corresponding author at: Institute of Plasma Physics, Chinese Academy of Sci-nces, Hefei, Anhui 230031, China. Tel.: +86 551 5593296; fax: +86 551 5591397.

E-mail address: tzhou@ipp.ac.cn (T. Zhou).

920-3796/$ – see front matter © 2010 Elsevier B.V. All rights reserved.oi:10.1016/j.fusengdes.2010.05.034

© 2010 Elsevier B.V. All rights reserved.

simulation has become an effective tool for analyzing MHD effectsand optimizing blanket design. For MHD duct flow, the Hartmannlayer perpendicular to the magnetic field scales with Ha−1, and theside layer parallel to the magnetic field scales with Ha−1/2. In fusionblankets, Hartmann number can be as high as 104–105, whichleads to some special numerical difficulty. Therefore, simulatingand studying liquid metal MHD flow at high Hartmann number hasbecome a hot research topic for fusion blanket design.

A number of numerical codes have been developed for ana-lyzing liquid metal MHD effects [23–28]. Smolentsev developed a2D code based on structured grid and induced magnetic equation[25]. A 3D code named HIMAG [26] and another 3D code basedon the commercial package CFX 5.6 [27] were developed basedon unstructured grid and electrical potential equation. By solvinginduced magnetic equations, it is difficult to construct a properboundary condition for induced magnetic field. For fusion reactorconditions, the magnetic Reynolds numbers is small, the electricalpotential formula can be employed for MHD with good accuracy[29–31]. Ni developed a consistent and conservative scheme [32]for calculate the current density and the Lorentz force by solvingelectrical potential equations, this scheme can solve LM MHD prob-

lem at high Hartmann number. In order to research and evaluateMHD effect in liquid metal blanket as well as make an optimaldesign for FDS series blankets, FDS team has been developing theMHD simulation code named MTC (Magnetic Thermo-hydraulicsCoupling Code) for years. MTC-F 1.0 and MTC-F 2.0 are developed

ng and Design 85 (2010) 1736–1741 1737

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Table 1Computed flow rate.

Hartmann number Calculated flow rate Error

Shercliff’s case500 3.979 0.5%

Fn

T. Zhou et al. / Fusion Engineeri

ased on the commercial code FLUENT, which can solve LM MHDroblem only at low Hartman number (Ha < 500).

In this paper, a numerical code named MTC-H 1.0 for simulatingHD flow in rectangular duct is developed based on the structure

ollocated grid, in which consistent and conservative scheme aresed to calculate the current density and the Lorentz force. We vali-ated the numerical code by simulating 2D and 3D MHD flows withnalytical solutions existed. The validation cases are conductedith Hartmann number from 102 to 104.

. Governing equations and numerical methods

.1. Governing equations

LM is a kind of incompressible fluid. Under fusion blanket con-itions, the magnetic Reynolds numbers is small, this means the

nduced magnetic field can be negligible compared to the applyagnetic field, the governing equations for LM MHD flow in dimen-

ionless form can be expressed as follows:Conservation of momentum:

∂u 1 2

∂t+ u · ∇u = −∇p +

Re∇ u + NJ × B (1)

Conservation of mass (Navier–Stokes equation):

· u = 0 (2)

ig. 1. Calculating results for Shercliff’s fully developed flow at Ha = 1000: (a) convergencormal to the Hartmann walls, and (d) velocity distribution along the middle line normal

1000 3.964 0.9%

Hunt’s case500 4.008 0.2%

1000 4.016 0.4%

Ohm’s law:

J = −∇ϕ + u × B (3)

Conservation of current density:

∇ · J = 0 (4)

From Eqs. (3) and (4),we can get the Poisson’s equation for elec-trical potential:

∇ · (∇ϕ) = ∇ · (u × B) (5)

Here t, u, p, J, B, ϕ are dimensionless time, velocity, pressure, cur-rent density, applied magnetic field, and electrical potential. Theyare scaled with L/u , u , � u2, �u B , B and Lu B , respectively,

0 0 0 0 0 0 0 0 0where L is a characteristic length, u0 is characteristic velocity, � and�0 are the conductivity and density of fluid. The Reynolds numberis Re = u0L/�, the Hartmann number is Ha = B0L

√�/�0�, and the

interaction parameter is N = Ha2/Re.

e of flow rate, (b) current in the duct, (c) velocity distribution along the middle lineto the side walls.

1738 T. Zhou et al. / Fusion Engineering and Design 85 (2010) 1736–1741

F encen ormal

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ig. 2. Calculating results for Hunt’s fully developed flow at Ha = 1000: (a) convergormal to the Hartmann walls, and (d) velocity distribution along the middle line n

.2. Numerical scheme and boundary conditions

The code was developed based on the structured grid. Since theartmann layer and the side layer are very thin, and require at least–5 grid points to resolve these layers, a non-uniform grid system

s needed to improve the computational efficiency. The control vol-me method on the structured collocated grid system was used tonsure the conservation of the discrete equations. All the unknownariables were located at the cell center, and the unknown fluxesere located at cell faces. For the current and Lorentz force calcu-

ation, the consistent scheme [32] was employed to calculate theurrent flux at cell face, and the current fluxes obtained were diver-ence free, which means the obtained current fluxes on cell facesre conservative in a control volume. Then, the Lorentz force atell center can be calculated base on the divergence free currentux using a conservative interpolation of current densities to theell centers. For the incompressible Navier–Stokes equations, four-tep projection method [33] was used to calculate the velocity andressure. This kind of projection method is a general second orderrojection method, and the SIMPLE-type methods, such as the stan-ard SIMPLE method in and the SIMPLEC method in [34,35], can be

cquired from this projection method [36,37]. Both the convectionnd diffusion term of Navier–Stokes equation were discretized withhe central difference scheme in spatial discretization, and for tem-oral updating, explicit and Crank–Nicholson scheme were utilizedespectively.

of flow rate, (b) current in the duct, (c) velocity distribution along the middle lineto the side walls.

For the velocity at walls, no-slip boundary conditions were set.We have assumed a fully developed flow at outlet, and then, wecan set Dirichlet boundary condition u = c(y,z)and Neumann bound-ary condition ∂u/∂n = 0, respectively at inlet and outlet of the duct.For electrical potential, the Neumann boundary condition ∂ϕ/∂n = 0was set at all boundaries.

2.3. Code and solving procedure

The code was written in Fortran 90. An input file is provided todefine the input data, such as geometry size of calculating domain,physical properties of fluid and duct, the number of grid points andtime step size. To begin with, the code reads the input data andinitializes the field of velocity, pressure, current, Lorentz force andelectrical potential. In a time level, the four-step projection methodis conducted to calculate the flow field. The Navier–Stokes systemis solved by using Approximation factorization (AF) technique, andalternative direction implicit (ADI) method is employed to solve thepressure Poisson’s equation. Then, we solve the electrical Poisson’sequation and use the consistent and conservative scheme to calcu-

late current density and Lorentz force. The obtained Lorentz forceis considered as source term for Navier–Stokes system in next timelevel. In order to avoid the oscillations as well as to accelerate theconvergence in the solution, a relaxation technique [38] is used forsolving the pressure and electrical potential Poisson’s equations.

T. Zhou et al. / Fusion Engineering and Design 85 (2010) 1736–1741 1739

profile

3

dttowpvup

3

lw

TT

Fig. 3. Results for Hunt’s fully developed flow at Ha = 15,000: (a) velocity

. Validating cases

Shercliff’s case and Hunt’s case are the exact solutions for fullyeveloped incompressible laminar flows in rectangular ducts withransverse magnetic fields. These two cases were usually used toest numerical code for MHD duct flows. In Shercliff’s problem, allf the duct walls are insulating. In Hunt’s problem, the Hartmannalls perpendicular to the field are conducting and the side wallsarallel to the field are insulating. We also used these two cases toalidate the MTC-H 1.0 code. All the calculations were performedsing a PC (Intel core2 Quad, 2.66 GHz, 4 GB RAM) with doublerecision.

.1. 2D fully developed flow

2D fully developed flow of Shercliff’s and Hunt’s case were simu-ated to validate the numerical scheme of the code. The calculations

ere conducted on a non-uniform grid system at Ha = 500, Re = 10,

able 2he results of flow rate and pressure gradient.

Shercliff’s case

Hartmann number 1000Calculated flow rate at outlet 4.0069The given flow rate at inlet 4Error of flow rate 0.17%Calculated pressure gradient 2.0582Analytical pressure gradient 2.0578Error of pressure gradient 0.01%

, (b) velocity distribution near the side walls, and (c) current distribution.

and Ha = 1000, Re = 10 with 65 × 65 nodes for resolving fluid regionand 9 nodes for resolving duct walls. The dimension of fluid domainis 2 × 2, and the wall thickness is 0.1. The wall conductance ratiois 0.0 for Shercliff’s case and 0.05 for Hunt’s case. For Shercliff’scase, the pressure gradient −52.08 and −102.87 were given at massflow rate 4 obtained form Shercliff’s analysis results at Ha = 500and Ha = 1000, respectively, and for Hunt’s case, the pressure gra-dient were −953.48 and −3379.82, respectively. The results werecompared with the exact solutions. Table 1 shows the computedflow rate for the case mentioned above, the error of the calculatedresults are less than 1%. Figs. 1(a) and 2(a) shows the convergence ofmass flow rate. Induced current in the quarter duct at Ha = 1000 areshown in Figs. 1(b) and 2(b), The current density in the duct traces a

closed streamline, which means the current density is conservative.Figs. 1(c) and (d) and 2(c) and (d) show the computed velocity distri-bution compared with the exact solution at Ha = 1000 for Shercliff’sand Hunt’s cases, respectively, we can see the computed resultsmatch well with exact solutions.

Hunt’s case

2000 1000 20004.0135 4.0087 4.00824 4 40.3% 0.22% 0.21%4.0956 67.6525 236.3954.0800 67.5424 236.0890.38% 0.16% 0.13%

1740 T. Zhou et al. / Fusion Engineering and

Fig. 4. Results for Hunt’s 3D case at Ha = 10,000: (a) velocity profiles at x = 0.05,x = 2.0, x = 3.95, (b) velocity distribution near the side walls, and (c) current distribu-t

3

lH7iTFtic

83 (7–9) (2008) 912–919.[2] N.B. Morley, Y. Katoh, S. Malang, B.A. Pint, A.R. Raffray, S. Sharafat, et al., Recent

research and development for the dual-coolant blanket concept in the US,

ion at x = 0.05, x = 2.0, x = 3.95.

.2. A 2D case at fusion relevant Hartmann number

Since the Hartmann number at fusion reactor conditions is veryarge. We validated the code by Hunt’s case at a fusion relevantartmann number Ha = 15,000, Re = 10. A non-uniform mesh with5 × 75 nodes was used for this calculation. The dimension of fluid

s 2 × 2 and wall thickness is 0.1. The wall conductance ratio is 0.05.he pressure gradient −390809.68 were given at mass flow rate 4.ig. 3(a) shows the velocity distribution, a great jet occurs near

he side walls, and Fig. 3(b) illustrate that the computed veloc-ty matches well with the analytical solution. Fig. 3(c) shows theonservative current distribution in the duct.

Design 85 (2010) 1736–1741

3.3. 3D MHD flow

To investigate the code in 3D simulation, the calculations ofShercliff’s and Hunt’s case were conducted at Ha = 1000, Re = 500and Ha = 2000, Re = 500. We choose here the same cross section asabove and extend it in the x-direction to 50 units, and the length inx direction is 5. A non-uniform mesh containing 51 × 55 × 55 nodesand 145,800 cells was used in this simulation. The wall conductanceratio is 0.05. Table 2 illustrates the results of flow rate and pressuregradient, and the relative errors are also given in this table. Theresults match well with the analytical solution.

3.4. 3D MHD flow at high Hartmann number

A 3D MHD flow case was tested at high Hartmann numberHa = 10,000. The non-uniform mesh containing 62 × 65 × 65 nodesand 249,865 cells was used in this simulation, and wall conductanceratio is 0.05. The dimension of fluid domain is 5 × 2 × 2, and the wallthickness is 0.1. A parabolic velocity profile was given as velocityboundary condition at inlet, and the flow rate was 4. Fig. 4(a) showsthe velocity profile in the duct at x = 0.05, x = 2.0, x = 3.95. Fig. 4(b)shows the comparison of velocity distribution between computedresult and exact solution at the fully developed region near the sidewall. We can see the numerical result match well with Hunt’s ana-lytical solution. Fig. 4(c) shows the current distribution in the ductat x = 0.05, x = 2.0, x = 3.95. Since the flow is not fully developed, a3D current occurs at x = 0.1. The calculated pressure gradient at thefully developed region is 19450.62, this results is very close to theHunt’s analytical result of 19688.50. And the flow rate at outlet is4.0092.

4. Conclusions

In order to research and evaluate MHD effect in liquid metalblanket as well as make an optimal design for FDS series blan-kets, a code named MTC-H 1.0 which can simulate 3D MHD flow inrectangular ducts has been developed by FDS Team. The code wasdeveloped based on the structured grid. The current density conser-vative scheme was employed for calculation of the induced currentand the Lorentz force, and the consistent projection method wasemployed for solving the incompressible Navier–Stokes equationswith the Lorentz force included as a source term.

We validated MTC-H 1.0 code by simulating MHD rectangularducts flows, and the Shercliff’s insulated walls and Hunt’s conduc-tive walls were used as benchmark models. The results showedthat accuracy results can be obtained for duct flow at Ha = 104. Inthe next step, in order to improve the capability for dealing withthe complex geometry problem, the unstructured grid system willbe needed, and the ability of parallel computing will be developedto promote efficiency of calculation.

Acknowledgements

This work was supported partly by the National Natural Sci-ence Foundation of China with the grants No. 10675123, 10875145,50936006, 50706015, 50676108.

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