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Fusion Engineering and Design 88 (2013) 226232

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Fusion Engineering and Design

journa l homepage: www.elsevier .com/ locate / fusengdes

Numerical analysis ofliquid metal MHD flows through circular pipes based on a

fully developed modeling

Xiujie Zhang, Chuanjie Pan, Zengyu Xu

Southwestern Institute of Physics, Chengdu, Sichuan, China

h i g h l i g h t s

2D MHD code based on a fully developed modeling is developed and validated by Samad analytical results. The results ofMHD effect ofliquid metal through circular pipes at high Hartmann numbers are given. M type velocity profile is observed for MHD circular pipe flow at high wall conductance ratio condition. Non-uniform wall electrical conductivity leads to highjet velocity in Robert layers.

a r t i c l e i n f o

Article history:

Received 28 November 2011

Receivedin revised form 20 July 2012

Accepted 8 February 2013

Available online 20 March 2013

Keywords:

MHD flow

Circular pipe

Numerical simulationVelocity profile

a b s t r a c t

Magnetohydrodynamics(MHD) laminar flows through circular pipes are studied in this paper by numer-

ical simulation under the conditions of Hartmann numbers from 18 to 10000. The code is developed

based on a fully developed modeling and validated by Samads analytical solution and Changs asymp-

totic results. After the code validation, numerical simulation is extended to high Hartmann number for

MHD circular pipe flows with conducting walls, and numerical results such as velocity distribution and

MHD pressure gradient are obtained. Typical M-type velocity is observed but there is not such a big

velocityjet as that ofMHD rectangular duct flows even under the conditions ofhigh Hartmann numbers

and big wall conductance ratio. The over speed region in Robert layers becomes smaller when Hartmann

numbers increase. When Hartmann number is fixed and wall conductance ratios change, the dimension-

less velocity is through one point which is in agreement with Samads results, the locus of maximum

value ofvelocityjet is same and effects ofwall conductance ratio only on the maximum value ofvelocity

jet. In case ofRobert walls are treated as insulating and Hartmann walls as conducting for circular pipe

MHD flows, there is big velocityjet like as MHD rectangular duct flows ofHunts case 2.

2013 Elsevier B.V. All rights reserved.

1. Introduction

Liquid metal blanket have many attractive features such as

low operating pressure, design simplicity and a convenient tri-

tium breeding cycle, but MHD effect is a remaining key issue to be

resolved. The mainly purpose of MHD effect study is to reduce the

high MHD pressure drop and understand its velocity distribution.

It is necessary to model MHD flows through Maxwells equationsand the NavierStokes equation, in order to better understand the

MHD effects of liquid metal under strong magnetic field. One of

the most basic cases is the laminar MHD flow through circular

pipe under a uniform strong magnetic field. Although there are

some theoretical results available, such as analytical solutions for

MHD pipe flows with insulating or conducting walls [13], but all

those solutions are under the form of infinite series expansions

Corresponding author. Tel.: +86 28 82850419.E-mail address: [email protected] (X. Zhang).

involving modified Bessel functions, which limit those solutions

to small Hartmann number (Ha). The square of Hartmann num-

ber is the ratio of Lorentz force comparing to viscous force, while

the Hartmann number normally ranges from 103 to 105 in mag-

netic fusion reactor. There are some approximate solutions for

high Hartmann number based on the asymptotic method [46],

but this method is not full solution, there has some difference

with analytical solution at small Ha condition which can be seenfrom Fig. 1. In the analytical solution Samad [3] considered the

case of circular pipe with finite electric conductivity and finite

wall thickness, which obtained the M-type velocity profile under

the conditions of small Ha and big wall conductance ratio. How-

ever, in approximate approaches, Chang [4] did not observe the

M-type velocity profile. Since the analytical solution of Samad

is limited to small Ha which is about 30, does there exist big

velocity jet like as MHD rectangular duct flows [9] when the Hart-

mann number becomes bigger to 104? Therefore it is important

to extend study to high Hartmann numbers through numerical

simulations.

0920-3796/$ see front matter 2013 Elsevier B.V. All rights reserved.

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X. Zhang et al. / Fusion Engineering and Design 88 (2013) 226232 227

Fig. 1. The difference of velocity profiles between analytical solutions and approx-

imate results.

In this paper, high resolution numerical simulation of MHD

flows through circular pipes based on a fully developed model-

ing under the cylindrical coordinate is done to obtain the results

of velocity distribution and MHD pressure drop under the condi-

tions of Hartmann numbers from 18 to 10000. Numerical results

with small Hartmann number are validated by Samad analytical

results, and then give the results of velocity and induced electri-

cal current distribution, MHD pressure drop under high Hartmann

number conditions.

2. Mathematicalmodeling

The unidirectional incompressible laminar flows are consideredin this modeling as shown in Fig. 2a, where the magnetic field is

alongXdirection and the flow is driven by a uniform pressure gra-

dient in the Zdirection. It can be assumed that there is only one

component vzof velocity andonly one component of induced mag-

netic field Bz due to the fluid is only flow along Zdirection. The

equations [3,5,13] governing this electrical conducting flow in the

Cartesian coordinates are shown as below:

2vz

x2+

2vz

y2

p

z+ B0

BZx = 0 (1)

1

x

1

x

Bzx

+ 1

y

1

y

Bzy

+ B0

vZx = 0 (2)

where is the constant viscosity, is the magnetic permeabilityof vacuum, x and y are the electrical conductivities in x and ydirections respectively. By introducing the dimensional variables:

X= xa

, Y= ya

, Z= za

R = ra

, V= vzv0P

, B = Bzv0P

Eqs. (1) and (2) can be rewrittenin a dimensionless form as follows:

2V

X2+

2V

Y2+ Ha B

X+ 1 = 0 (3)

2B

X2+

2B

Y2+

HaV

X=0 (4)

where a is the inner radius of the circular pipe and v0 is the mean

velocity and P= (a2)/v0, Ha = aB0

/, and = (p/z),where the non-dimensional quantity Ha is the Hartmann number

and Pis the Poiseuille number. For numerical simulations, we con-

vert Eqs. (3) and (4) into cylinder coordinate wherez-axis is along

the axis of the circular cylinder:

2V

R2+ 1

R

V

R+ 1

R22V

2+Ha

B

Rcos B

sin

R + 1 = 0 (5)

R

B

R

+

R

B

R+ 1

R

R

B

+ Ha

V

Rcos V

sin

R

= 0 (6)

where is the ratio of the electrical conductivity of liquid to itslocal value and it is assumed that electrical conductivities of liquid

and solid are isotropy.

The dimensionless electrical current is calculated through the

following equations:

jx=

1

Ha

Bz

y x (7)

jy=1

Ha

Bzxy (8)

3. Numericalmethods

3.1. Numerical scheme

Similar to Reference [13], a control volume technique based on

non-uniform collocated meshes is used to get the finite differential

equations. The velocity and induced magnetic field are defined at

the center of the control volume cell, and is taken at the sidesof the cell. Eq. (5) is only solved in liquid area while Eq. (6) is

solved in all area which contains liquid and solid. The code is writ-

ten in Fortran90, uses Alternative Direction Implicit (ADI) methodto solve the finite differential equations and contains an effective

convergence acceleration technique same as Reference [13].

3.2. Mesh and treatment of cylinder coordinate singularities

For liquid metal MHD pipe flows the boundary layer is very thin

with about Ha1 non-dimensionless thickness. Under high Hart-mann conditions it cannot use the uniform meshes for very large

computation, so the non-uniformmeshes related to Ha is employed

in numerical simulations, which can insure that there are at least 7

mesh points within the boundary layer and 342342 mesh pointsin the radial and azimuthal directions. The illustration of the com-

putational mesh is shown in Fig. 2b.

Dueto thesymmetryof thecircular pipe it is reasonable to solveonly onequarterof thepipearea,i.e.theta ()from /2to . Becauseof the singularity at the origin point where need special treatment,

we choose the grids which are half-integer in radial direction and

integer in azimuthaldirection, i.e.Ri = (i3/2)R and j = (j1),where i= 2,3,. . .,N+ 1 andj = 1,2,3,. . .,M. When R1 = 0 and Riis com-puted from 2 to Nin radial direction and from References [10,11],

it can be seen that this method can avoid the singularity in radial

direction. The integered mesh in radial direction ranges only from

0 to 0.8 since the non-uniform mesh is required to solve the thin

boundary layer at high Hartmann numbers. If we compute from

/2to in azimuthaldirection, the symmetry boundary conditionsare V(/2 + )=V(/2) and V() =V(), which is not correctfrom physical consideration and cannot get reasonable numerical

results from computational test. Therefore, the computational area

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228 X.Zhang et al. / Fusion Engineering and Design88 (2013) 226232

Fig.2. Sketch of thecomputational areaand mesh: (a)the cross-sectionof thecomputational area, where thered arearepresents liquid metal, (b)non-uniformcomputational

mesh used in numerical simulations. (For interpretation of the references to color in this figurelegend, thereader is referred to theweb version of this article.)

is changed from /2 to + and thereafter the symmetryboundary conditions are:

V(/2) = V(/2+), B(/2) = B(/2+)

V( ) = V( +), B( ) = B( +)

This treatment is reasonable from physical understanding andvalidated to be correct for high Hartman numbers even Ha= 104.

There isno slip conditionfor thevelocity atthe interfaceof liquid

metal and solid. At the interface of solid wall and air, the induced

magnetic field is set to be zero.

4. Validation of the code

4.1. Samads case

To validate the code, the numerical results are compared with

Samad analytical solutions under small ha conditions. Due to the

appearance of small differences between large numbers, Samad

limit his computation to Ha= 18. With the recent development of

computing hardware, we can extend the computation to Ha= 30.

The numerical results are compared with those of Samad analyti-cal solution for Ha= 18and Ha= 30respectively, as seen in Fig. 3. As

mentioned above, a is the inner radius of the circular pipe, alpha*a

is defined as the outer radius of the pipe, and Ctakes the electrical

Fig. 3. Comparison of thevelocity profiles between numerical results and those of Samadanalytical solutionsunder theconditions of small Ha: (a) Ha= 18 and(b) Ha= 30.

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Fig. 4. Comparison of the velocityprofilesbetweennumerical results andthose of Changs asymptotic solutionat Ha= 1000with all insulating walls: (a)comparethe velocity

profiles at = /2, (b)velocity distribution in thecross section (3Ddisplay).

conductivity of the solid to liquid metal as same as that in Samad

paper. From the comparison in Fig. 3, it can be seen that numerical

results match well with those of Samad analytical solutionsand the

M-type velocity is observed from both results.

4.2. MHDpipe flowwith all insulating walls

Inthecase ofMHD flowthrough circular pipe with allinsulating

walls, there is no difference between Samad analytical and Chang

asymptotic results. Then the code is validated against Changs

results at high Ha [12]. It is seen from the comparison in Fig. 4a

that numerical results match well with those of asymptotic solu-

tion. Fig. 4b shows the non-dimensional velocity distribution in

the cross section of circular pipe. There is an interesting resultthat in the line of x= 0 (= /2) the velocity profile is paraboladistribution, which is not similar to that of MHD rectangular duct

flow with all insulating walls but that of normal fluid duct flows.

Even the Ha changes the velocity profile in this direction is always

parabola-shape.

5. Results and discussion

Because of the computational difficulty of Bessel function in

Samads analytical solution at Hartmann number greater than 30,

the numerical method is validated by Samad analytical solution at

small Ha andthen extended for simulation of MHD flows in a circu-

larduct with velocity andpressuredrop obtained at high Hartmann

number.

5.1. Electrical current and velocity distribution in the cross

section

In Figs. 5 and 6, Cw is the wall conductance ratio defined as

follows:

Fig. 5. Induced electrical current distribution in the cross-section of the MHD pipe flows.

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Fig. 6. Velocity distribution in thecrosssection of theMHD pipe flows at differentHa (3Ddisplay).

Fig. 7. Velocity profiles in theline of= /2 changing with Hartmann numbers.

Cw= (wtw)/(fa), where w, f, tware electrical conductivityof solid wall and liquid metal, solid wall thickness, respectively.

It canbe seen from electrical current distributionresults in Fig.5

that there are two circuits if we convert the current distribution to

the whole circular pipe domain by symmetry, which is reasonable

in physical understanding.Small changes of the velocity over-speed

regions in the Robert layers [7,8,12] are seen when Ha increases, as

shown in Figs. 6 and 7. There has no big velocity jet in MHD circularpipe flows even when Ha= 104 and Cw = 1.0, which is different from

that of MHD rectangular duct flows, because the difference of elec-

trical current distributions in the crosssection between rectangular

andcircularpipes, there is no obvious area where theelectrical cur-

rent is parallel to the imposed magnetic field in MHD circular pipe

flows.

The most interesting result is that the velocity distribution

changes with different Cw at Ha = 1 000, as shown in Fig. 8. All

dimensionless velocities go through one point, which is in agree-

ment with Samad results at Ha=18. Another interesting result

is that with the fixed Ha number, the locus of the maximum

non-dimensional velocity does not change with the wall conduc-

tance ratio (Cw), which only affects the value of maximum jet

velocity.

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Fig. 8. Velocity distribution in theline of= /2 vs. wall conductance ratio(Cw) at

Ha = 1000.

5.2. Compare MHDpressure drop with theory

In numerical simulation the pressure gradient can be computed

using:

dpdz= v0

4Q (9)

derived from Q=

/2

1

0 V(R, )dRd, where Q is the dimension-

less flow rate related to the dimensional pressure gradient. To

Fig. 9. Numerical results of dimensionless pressure gradient compared with theo-

retical results.

compare the pressure gradient with numerical results, the follow-

ing formula [14] is used for the theoretical results:

dpdztheory

(dimensionless)= Cw1+ Cw

(10)

Formula (9) is then transformed to the following dimensionless

format:

dpdznumerical

(dimensionless)= 4QfB

20

(11)

where f and B0 are electrical conductivity of liquid metal andimposed magnetic field, respectively. It can be seen from com-

parison shown in Fig. 9 that the dimensionless pressure gradient

Fig. 10. MHD flows through circular pipe with non-uniform wall electrical conductivity: (a) velocity distribution in the cross-section (3D display), (b) electrical current

distribution in the cross-section, Ha= 1000,Cw= 0.016.

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change smaller with Hartmann number becoming bigger, when

Hartmann number is equal to 100 the dimensionless pressure gra-

dient is much close to that of theory values, while under Ha=18

and Ha= 30 conditions there are big differences between numer-

ical and theoretical results. The reason is because the theoretical

formula (9) is suitable when Ha1. In addition, it is obvious thatthe theoretical formula only considers the effects of wall electri-

cal conductivity and the pipe shape on the pressure gradient, not

considers effect of the changing Lorentz force on it, i.e. the dimen-

sionless pressure gradient shouldalso be the function of Hartmann

number.

6. Effect of non-uniformwall electrical conductivity

In Hunts case 2 [9] of rectangular duct flows, Hartmann walls

were treated as conducting and side walls as insulating. There also

have Hartmann and Robert walls for MHD circular pipe flows, the

Robertwalls in circular pipe arelike as the side walls in rectangular

duct. When we treated Robert walls as insulating and Hartmann

walls as conducting in MHD circular pipe flows, does there have

big velocity jet like as Hunt case 2?

In numerical simulation the wall from = /2 to = 1.175/2(i.e. 1/5.7 of whole computational wall area, approximately in

Robert wall) is treated as electrical insulating and other walls as

conducting. It can be seen from numerical results in Fig. 10 that in

Robert layer there is big velocity jet like as Hunts case 2, which

is because there has obvious electric current distribution which is

parallel to the imposed magnetic field in Robert layer where the

Lorentz force is very small, so in this area form big velocity jet like

as MHD rectangular duct flows with conducting walls.

7. Conclusions

In the case of MHD flows through circular pipes at high val-

ues of Hartmann number and wall conductance ratio (Cw), there

is a typical M-type velocity profile as observed by Hunt in Refer-

ence [9]. Compared with the case of MHD rectangular duct flows,

the maximum of jet velocity in MHD circular pipe flows is smallerthan that in rectangular duct flows. Another difference lies in the

effect of wall conductance ratio on velocity profile. In MHD circu-

lar pipe flows, if the Hartmann number is fixed, for different wall

conductance ratios the dimensionless velocity profiles all through

one point and the locus of maximum velocity jet is same. Cwonly

has the effects on the maximum value of the velocity jet in Robert

layers.

From the comparison of MHD pressure gradients, it can be

seen that numerical results have some differences from theoretical

results since the theory does not consider the effect of magnetic

field on the pressure gradient, which should be the function of

wall conductance ratio and Hartmann numbers. When the elec-

trical conductivity approximately in Robert walls are treated as

insulating and other walls as conducting, there has big velocity jet

like as MHD rectangular duct flows with conducting walls.

Acknowledgments

Part of this work is supported by China National Nature Science

Fund Grant No. 11105044; the authors would like to express grati-

tude to professor Zongze Mu and Dr Jianzhou Zhu for their helpful

suggestions.

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