Final Report (X.xie)

A Study of Fluid Flow and Heat Transfer in a Liquid Metal in a

Backward-Facing Step under Combined Electric and Magnetic Fields

by

Xiaole Xie

An Engineering Project Submitted to the Graduate

Faculty of Rensselaer Polytechnic Institute

in Partial Fulfillment of the

Requirements for the degree of

Master of Engineering

Major Subject: Mechanical Engineering

Approved:

_________________________________________

Ernesto Gutierrez-Miravete, Project Adviser

Rensselaer Polytechnic Institute

Hartford, Connecticut

December 2010

ii

CONTENTS

LIST OF SYMBOLS ........................................................................................................iii

LIST OF TABLES............................................................................................................. v

LIST OF FIGURES ..........................................................................................................vi

ACKNOWLEDGMENT..................................................................................................vii

ABSTRACT....................................................................................................................viii

1. Background .................................................................................................................. 1

1.1 Introduction ........................................................................................................ 1

1.2 Background on flow pattern over a backward facing step ................................. 1

1.3 Liquid metal Properties ...................................................................................... 3

2. Methodology ................................................................................................................ 4

2.1 Hartmann Problem Theory................................................................................. 4

2.1.1 Validating Hartmann Flow in COMSOL ............................................... 6

2.2 Validating flow over a backward step in the absence of external force............. 6

3. Results/Discussion ....................................................................................................... 8

3.1 Validation of Hartmann Flow in COMSOL....................................................... 8

3.2 Validation of Backward Step Flow in COMSOL using liquid metal properties

.......................................................................................................................... 12

3.3 The MHD Effect on a Back-step Flow............................................................. 14

3.4 MHD Effect on Heat Transfer in a Backward Step Flow ................................ 20

4. Conclusion ................................................................................................................. 23

5. References.................................................................................................................. 24

iii

LIST OF SYMBOLS

0B External Magnetic Field in the Y direction (Wb)

cp Specific Heat Capacity (J/kgK)

ch Hyperbolic cosine

EZ External Electric Field in the Z direction (V/m)

D Hydraulic diameter of backwards step (m)

ER Expansion ratio

F Force (N)

h Height of inlet channel (m)

H Height of outlet (m)

J Current Density (A/m2)

M Hartmann Number

Re Reynolds number

S Step height (m)

sh Hyperbolic sine

u Fluid velocity in the X direction (m/s)

v Fluid Velocity in the Y direction (m/s)

w Fluid Velocity in the Z direction (m/s)

X X-direction

x1 Reattachment point for 1st bottom recirculation zone (m)

x2 Separation point for 2nd bottom recirculation zone (m)

x3 Reattachment point for 2nd bottom recirculation zone (m)

x4 Reattachment point for 1st top recirculation zone (m)

x5 Separation point for 2st top recirculation zone (m)

Xe Inlet channel length (m)

Xo Outlet channel length (m)

Y Y-direction

yo Half Distance between the channels (m)

Z Z-direction

µf Dynamic viscosity (kg/ms)

ρ Density (kg/m3)

iv

σ Electrical conductivity (S/m)

v

LIST OF TABLES

Table 1: Liquid Metal Properties ....................................................................................... 3

Table 2: Geometry of the Channel and Mesh .................................................................... 9

Table 3: Backward facing dimension in the model (meters) .......................................... 12

vi

LIST OF FIGURES

Figure 1[7]

: Schematic of backward facing step geometry (not to scale) ........................... 2

Figure 2[8]

: Three recirculation zones for laminar flow ..................................................... 2

Figure 3: Hartmann flow in a flat channel with imposed electric and magnetic field....... 4

Figure 4: Boundary condition for the Magetostatics Module ............................................ 8

Figure 5: Magnetic field applied in the Y direction........................................................... 8

Figure 6: Pressure drop versus channel length in COMSOL............................................. 9

Figure 7: velocity as a function of channel distance........................................................ 10

Figure 8: The COMSOL solution compares well with the analytical solution................ 10

Figure 9: Effect of increasing Hartmann number on the velocity profile........................ 11

Figure 10: Hartmann number effect (analytical and COMSOL solutions comparison) .. 11

Figure 11: The back-step geometry ................................................................................. 12

Figure 12: Flow profile for liquid NaK with Re= 389..................................................... 13

Figure 13: Two recirculating regions appear @ Re= 648................................................ 13

Figure 14: Mesh grid for the back-step geometry in COMSOL...................................... 14

Figure 15: The application of magnetic field on the back-step........................................ 14

Figure 16: MHD effect on the pressure drop in the back-step Flow................................ 15

Figure 17: Velocity profile in the back-step geometry with Re=100 (No MHD) ........... 15

Figure 18: Velocity profile for step inlet and outlet......................................................... 16

Figure 19: Recirculation region gets smaller with magnetic field effect ......................... 16

Figure 20: Velocity comparison between the outlet velocities with/without MHD ........ 17

Figure 21: Second recirculation region disappear with MHD effect with Re=648 ........ 17

Figure 22: Pressure contour with M=100 and Re=648.................................................... 18

Figure 23: Pressure contour with M=0 and Re=648........................................................ 18

Figure 24: The cross-section of the pressure along the channel @ M=0......................... 19

Figure 25: The cross-section of the pressure along the channel @ M=100..................... 19

Figure 26: Pressure gradient trace (along the channel) comparison for M=0 & M=100. 20

Figure 27: Boundary condition for the conduction and convection module.................... 21

Figure 28: Heat transfer in a back-step flow (Top M=100, Bottom M=0) ...................... 21

Figure 29: Hartmann number effect on heat transfer ....................................................... 22

vii

ACKNOWLEDGMENT

First of all, I want to thank my family for their supports in my pursuit of this master

degree. It has been a long 2 to 3 years. Their ultimate support is what makes this possible.

I also want to thank Professor Ernesto for his assistance in this project as well as guiding

me through the whole master program.

viii

ABSTRACT

This study used the finite element based software, COMSOL, as the analytical tool

to investigate the effect of applied magnetic and electric fields on the flow phenomena of

an electrically conducting fluid in a backward step configuration. The

magnetohydrodynamic (MHD) approach was validated by comparison with the existing

solution for the Hartmann problem while the back step flow was validated by

comparison with previously obtained solutions. For a normal backward step flow, the

recirculation regions downstream the step only depend on the step size and the Reynolds

number. The implementation of the magnetic and electric field (MHD) has significant

impact on the effective Reynolds number and thus leads to changes in the separation and

reattachment point downstream the step. Depending on the strength of the magnetic and

electric field, the recirculation regions downstream the step could become smaller or

completely vanish due to the change in the pressure and velocity distribution. In addition,

this change in the velocity profile due to the MHD effect in a back-step flow also affects

the heat transfer mechanism in the flow since convection is very dependent on the

velocity distribution.

1

1. Background

1.1 Introduction

Liquid metal flows subjected to combine electric and magnetic fields are part of a

larger study, which is known as the Magneto-hydrodynamics (MHD). The concept of

MHD is that the magnetic fields can induce currents in a conductive fluid that creates

force, which will affect the flow and may even change the magnetic field itself. The

study of MHD has become very important because of its growing applications. For

instance, MHD pumps are utilized for different purposes, including liquid metal cooling.

One of the primary products employing this process is the liquid metal cooled nuclear

reactor, which is used in nuclear submarines as well as many power generation

applications. Some other potential liquid metal MHD applications include, but are not

limited to, energy conversion technology and metallurgy. A liquid-metal MHD power

converter has been successfully operated with the generation of AC electrical power [4]

.

Due to its wide potentials, a basic understanding of the MHD phenomenon is

essential. The so-called Hartmann flow has been studied extensively. The Hartmann

flow is the steady flow of an electrically conducting fluid between two parallel walls

under the effect of a normal magnetic and electric fields. However, not all the flows are

between parallel walls. Many engineering applications such as flow in diffusers, airfoils

with separations, combustor, turbine blades and many other relevant systems exhibit the

behavior of separated/reattached flows. Therefore, the accurate flow pattern of separated

flow is significant. Since the effect the magnetic and electric fields on flow patterns may

be quite important, it is worthwhile to study the effect of them on separated flows. The

goal of this project is to investigate the effect of applied magnetic and electric fields on

the flow over the backward facing step.

1.2 Background on flow pattern over a backward facing step

Flow over a backward facing step is an example of unilateral sudden expansion,

which results in flow separations and reattachments. Figure 1[7]

shows the schematic of a

backward facing step flow:

2

Figure 1[7]: Schematic of backward facing step geometry (not to scale)

This project will focus on the laminar regime of backward facing step flow. Without the

effects of magnetic and electric fields, the behavior of the flow over a back facing step in

laminar regime is very dependent on the Reynolds number and the ratio between the step

height (S) to the duct height (H). For laminar flow, various re-circulation zones occur

downstream from the step, as shown schematically in Figure 2[8]

. As the Reynolds

number of the flow increases, the first region of separation occurs at the step to x1 on the

bottom wall (Zone A). Next, the second region of separation occurs between x4 and x5

on the top wall (Zone B). As the Reynolds number increases into the transition zone, a

third separation region occurs in (Zone C) on the bottom wall. Theoretically,

recirculation zones will continue to develop downstream as the Reynolds number

increases and the flow remains laminar. However, this has not been observed

experimentally and the flow will eventually become turbulent.

Figure 2[8]: Three recirculation zones for laminar flow

3

1.3 Liquid metal Properties

The fluid medium plays an important role in MHD applications. Liquid metals have

properties that make them unique for micro-device applications such as cooling for high

heat flux. The heat transfer properties of liquid metals are much better than water and

other fluids. Nevertheless, many liquid metals have other concerns such as

environmental impact or will react violently with other materials.

In nuclear energy applications, one of the best candidates is an eutectic solution

of sodium and potassium, (NaK)【10】

. The following table shows the some of the

important properties of NaK.

Table 1[9]: Liquid Metal Properties

Metal Melting Point

Density (kg/m3)

Spec Heat (J/kgK)

Thermal Cond (W/mk)

Viscosity (kg/ms)

Electrical Cond (S/m)

[6]

NaK(22/78 %) -12 802 1058 35 9.40E-02 2.41E+06

NaK has density and viscosity similar to water. Although it has a lower specific heat, it

has a much higher thermal and electrical conductivity. The properties list in Table 1

would be used in this analysis.

4

2. Methodology

COMSOL Multi-physics was used in this study to investigate the effect of magnetic and

electric field on the flow pattern over a back facing step flow with liquid metal. This was

approached as a two-step process. First, the implementation of magnetic and electric

fields in COMSOL needs to be validated using the Hartmann problem. Secondly, the

validation of a typical backward-facing flow (without the effect of magnetic and electric

field) is performed in COMSOL. The successful implementation of the two models

would allow the investigation of the MHD effect on the backward step flow.

2.1 Hartmann Problem Theory

The Hartmann problem is one of the simplest problems in Magnetohydrodynamics.

However, it gives insight into MHD generators, pumps, flow meters and bearings. It

concerns the steady viscous laminar flow of an electrically conducting liquid between

two parallel plates under the effect of imposed magnetic and electric fields (Figure 3).

Figure 3: Hartmann flow in a flat channel with imposed electric and magnetic field

The constant magnetic field acting in the +Y direction and the electric field acting in the

Z direction are the external set parameters know as 0B and EZ.

The flow of an incompressible fluid between parallel plates is governed by the

equation of continuity [3]

:

0=∂∂

+∂∂

y

v

x

u

and the Navier-Stokes equations which have the following form [6]

Xf Fy

u

x

P−

∂

∂+

∂∂−

=2

2

0 µ

5

Yf Fy

v

y

P−

∂

∂+

∂∂−

=2

2

0 µ

Zf Fy

w

z

P−

∂

∂+

∂∂−

=2

2

0 µ

Wherex

P

∂∂−

, y

P

∂∂−

, z

P

∂∂−

are the components of the pressure gradient in the X, Y and Z

directions respectively, fµ is the dynamic viscosity of the fluid, XF , YF and ZF are the

force components in the X, Y and Z directions, which are zero in the simple Poiseuille

flow in the absence of gravity (which is the case here). However, in the case of applied

magnetic and electric field, the force in the X, Y, and Z direction is known as the

Lorentz force. The Lorentz force is due to induced/imposed current and the imposed

magnetic field. With the magnetic and electric field, the Navier-Stokes equations

become as the followings [6].

02

2

0 BJy

u

x

PZf −

∂

∂+

∂∂−

= µ

ZXXZ BJBJy

P−+

∂∂−

=0

02

2

0 BJy

w

z

PXf −

∂

∂+

∂∂−

= µ

Where XJ and ZJ are the current density components (these can be imposed and/or

induced by the flow under the imposed magnetic field in the X and Z direction

respectively [6]

:

)( 0wBEJ XX −= σ

)( 0uBEJ ZZ −= σ

σ stands for the electrical conductivity of the liquid metal. Here XE and ZE represent

the electric field components in the X and Z direction.

From the schematic diagram of the problem under study, the only relevant

Navier-Stokes equation that remains is the following since the applied external electric

field is only in the Z direction and the velocity in the Z direction is 0 [2,6]

:

6

002

2

)(0 BuBEy

u

x

PZf +−

∂

∂+

∂∂−

= σµ

With this equations and the assumption of no slip condition at the wall, the analytical

solution is 【6】

:

( )

−

+

∂∂

= 1/(

)1

( 0

0

2

2

0

chM

yMychE

y

M

x

P

M

yu Z

ff µσ

µ

Where the Hartmann number is given by fByM µσ /00= . Here ch and sh denote the

hyperbolic cosine and sine, respectively.

2.1.1 Validating Hartmann Flow in COMSOL

To validate the implementation of the Hartmann flow model in COMSOL, the velocity

profile from COMSOL will be compared to the analytical solution for a range of

Hartmann numbers. The magnetic field and electric field in COMSOL would be

modeled using the Magnetostatics module in COMSOL. The solution from

Magnetostatics module is then be used in the Incompressible Navier-Stokes physics to

come with the velocity profile under the influence of the magnetic fields.

2.2 Validating flow over a backward step in the absence of external

force

There is no known exact solution for a flow over a backward step. However, much

experimental data have been published [1]

. The experimental data show the separation

and the reattachment point of the sudden expansion based on the Reynolds number and

the size of the step. The basic governing equations for flow over a step are the stationary

incompressible Navier-Stokes equations [5]

:

Fpuuuf =∇+∇⋅+∇− )(2 ρµ

And the equation of continuity [5]

:

0=⋅∇ u

The first equation is the momentum balance equation from Newton’s second law. The

second equation is the equation of continuity, which implies that the fluid is

incompressible. Since flow over a backward step has been a common benchmark

problem in CFD, the COMSOL library already has a model for it. However, the

7

properties the model used are different than the liquid metal properties. The actual

properties used by the COMSOL library are air properties. The results for the COMSOL

model using air properties have been validated against the experimental data [1]

given the

same geometry and Reynolds number. The reattachment and the separation points are

consistent with these obtained from experiments. In order to validate the model for NaK,

the liquid metal properties will be used in the COMSOL model. The model can be

validated if the separation and reattachment points are the same as the ones produced

with the air properties.

8

3. Results/Discussion

3.1 Validation of Hartmann Flow in COMSOL

The implementation of Hartmann flow in COMSOL is the first step. The magnetic field

is implemented in the Magnetostatics module. To obtain the magnetic flux in the +Y

direction, constant magnetic field is applied in all 4 boundaries as shown in Figure 4 and

the electric conductivity of the liquid metal is input into the sub-domain physics.

Figure 4: Boundary condition for the Magetostatics Module

The solution obtained by using the Magnetostatic module in COMSOL is shown in Fig 5.

Figure 5: Magnetic field applied in the Y direction

The resultant magnetic flux density is then used as input into the Incompressible

Navier-stokes flow and it results in the Lorentz force acting against the flow. The

Lorentz force depends on the velocity, external electric and magnetic field. The

Hartmann number, fByM µσ /00= , only depends on the strength of the magnetic

field given the fluid properties. Therefore, it is sufficient to apply the magnetic field in

9

order to validate the implementation of the Hartmann problem. In addition, the velocity

profile at the inlet was modeled as a parabolic shape laminar flow with no slip condition

applied at the wall. The pressure gradient is the main driver of the flow. Table 2 contains

the geometry dimension for the model. The values for the height and the length of the

channel are chosen to be the same as the ones input into the analytical calculations. The

mesh size is chosen based on its accuracy and computing time. A coarse mesh would not

be able to generate good results while a finer mesh would increase the computing time of

the model.

Table 2: Geometry of the Channel and Mesh

Height of channel H 0.2 m

Length of channel L 2 m

Mesh Size Number of degrees of freedom 5207

In order to compare the COMSOL solution to the analytical solution, the pressure

gradient would be a constant. Thus, the pressure drop is linear versus the channel length

(Fig 6).

Figure 6: Pressure drop versus channel length in COMSOL

In addition, an appropriate comparison between the analytical and COMSOL solution

can be made only if the velocity at the outlet has reached steady state in COMSOL. Thus,

it is important to ensure that the velocity has reached steady state at the outlet for every

Hartmann number before making any comparison. Figure 7 shows the velocity as a

function of channel distance when Hartmann number is 5. In this case, the outlet velocity

has stabilized.

10

Figure 7: velocity as a function of channel distance

With these settings, COMSOL is able to reproduce the analytical solutions given

a constant pressure gradient, fluid properties and Hartmann number. Figure 8 shows the

overlay between the analytical and COMSOL solution. The difference between the

absolute values is within 1%.

Comparison Between COMSOL & Analytical

Solution

(Hartmann = 5)

0

0.0002

0.0004

0.0006

0.0008

0.001

0.0012

0.0014

-0.1 -0.05 0 0.05 0.1

Distance from Y (m)

Velocity (m/s)

Analytical SOL M=5 COMSOL SOL M=5

Figure 8: The COMSOL solution compares well with the analytical solution

With the increase of the Hartmann number, the absolute maximum velocity at the center

of the channel gets smaller because the Lorentz force acting in the negative X-direction

would slow down the overall velocity as seen in Figure 9.

11

Hartmann Effect on Velocity

0

0.0002

0.0004

0.0006

0.0008

0.001

0.0012

-0.1 -0.05 0 0.05 0.1

Distance from Y (m)

Velocity (m/s)

M=0 M=5 M=10

Figure 9: Effect of increasing Hartmann number on the velocity profile

The analytical solution also suggests that as Hartmann number increases, the normalized

velocity would flatten out. The velocity gradient near the wall is much bigger due to the

combined effect of the Lorentz force and the no slip condition. The magnitude of the

Lorentz force is a function of the incoming velocity. Greater velocity will result in a

greater impeding Lorentz force. Figure 10 shows the effect of Hartmann number (M=0,

M=5 and M=10) on the velocity profile of the flow for both the analytical solution and

the solution from COMSOL.

Hartmann Number Effect

(Analytical and COMSOL Solution Comparison)

-0.2

0

0.2

0.4

0.6

0.8

1

-0.1 -0.05 0 0.05 0.1

Distance from Y (m)

Normalized Velocity (U/Umax)

M=0 (COMSOL)

M=5 (COMSOL)

M=10 (COMSOL)

M=0 (Analytical)

M=5 (Analytical)

M=10 (Analytical)

Figure 10: Hartmann number effect (analytical and COMSOL solutions

comparison)

12

3.2 Validation of Backward Step Flow in COMSOL using liquid

metal properties

The backward step flow problem has already been modeled in COMSOL. Fluid

enters from the left side of the channel with a parabolic velocity profile, passes over a

step and then leaves through the right side of the channel as shown in Figure 11. No slip

conditions are assumed at the upper and bottom of the channel.

Figure 11: The back-step geometry

This geometry has an expansion ration, ER, of 1.942, which is consistent with the

literatures[9]

. In the model, the following geometry dimensions are assumed.

Table 3 [5]: Backward facing dimension in the model (meters)

Height of inlet channel h 0.0052

Height of outlet H 0.0101

Step height S 0.0049

Inlet channel length Xe 0.02

Outlet channel length Xo 0.08

Number of degrees of freedom (Fig 14) 19722

The Reynolds number is defined as, f

uD

µρ

=Re , where u is the inlet velocity, µf is the

dynamic viscosity, ρ is the density, and D is the hydraulic diameter. The Reynolds

number has been expressed differently throughout the literature. To ensure agreement

with the experimental data [1]

, this study used D=2h.

The model in the COMSOL library has been validated against the experimental

data[1]

. However, the model is validated using the properties of air. Since liquid metal

would be the fluid medium in our study, it is essential to ensure that the model still

applies with the properties of this liquid metal, NaK.

With the same step size and the Reynolds number, the liquid metal flow is able

to regenerate the same separation and reattachment point as the ones generated by using

the properties of air. This also means that the separation and reattachment points in a

h

SH

Xe

Xo

13

back-step depend only on the Reynolds number and the step size. To generate the same

Reynolds number, a greater velocity is needed since viscosity and density of the liquid

metal is different than the air. Figure 12 shows the flow profile over a step for NaK with

a Reynolds number of 389.

Figure 12: Flow profile for liquid NaK with Re= 389

In Figure 12, there is only one recirculating region downstream of the step, one at

the lower wall (Zone A referring to the introduction). However, as mentioned in the

introduction, as the Reynolds number increases, a different recirculating region would

appear downstream of the step. This time the additional reciculation would be at the

upper wall (Zone B referring to the introduction). At Reynolds number of 648, the

second recirculating region occurs and this can be seen in Figure 13.

Figure 13: Two recirculating regions appear @ Re= 648

This is consistent with the result obtained using the air properties as well as the

experimental results. Thus, the back-step flow without MHD effect is validated.

14

3.3 The MHD Effect on a Back-step Flow

To investigate the MHD effect on a back-step flow, the magnetic field that is validated

previously in the Hartmann problem would be required to be applied to the step flow.

The back-step geometry is kept the same as the one shown in Figure 11. The mesh for

the model is shown in figure 14. The number of degrees of freedom is 19722, which is

chosen to produce accurate results in a relatively short time

Figure 14: Mesh grid for the back-step geometry in COMSOL

To investigate the MHD effect, inlet velocity and all the fluid properties remain the same.

The model would be first run without the magnetic effect. The same model is rerun with

the magnetic field applied as shown in Figure 15. The magnetic flux on region 2 is

generated the same way as the one shown in Figure 4.

Figure 15: The application of magnetic field on the back-step

The magnetic field is applied in the Y-direction on only the 2nd

region where the velocity

profile is of the interest. No magnetic field is applied to the 1st region because the inlet

velocity profile needs to be consistent to ensure an accurate comparison. A larger

pressure drop is required to maintained a consistent inlet velocity under the magnetic and

15

electric field. Figure 16 shows the pressure drop in the 2nd

region for the model with and

without the MHD effect.

Pressure Comparison @ Y=0.005 m

0

0.1

0.2

0.3

0.4

0.5

0.6

0.02 0.04 0.06 0.08 0.1

Distance from the Inlet (m)

Pressure (pa)

No MHD Effect MHD (M=100) Effect

Figure 16: MHD effect on the pressure drop in the back-step Flow

Obviously, a much larger pressure drop is required to maintain a constant inlet velocity

under the MHD effect. It is important to note that the magnitude of the Hartmann

number depends only on the strength of the magnetic field given the liquid metal

properties.

For a Reynolds number of 100 without MHD effect, the velocity profile in the

back-step geometry has one recirculation region downstream of the step as shown in

Figure 17.

Figure 17: Velocity profile in the back-step geometry with Re=100 (No MHD)

16

At the inlet, the velocity profile is parabolic. The recirculation occurs downstream of the

step. At the exit, the velocity profile becomes parabolic again. However, this time the

mean velocity and the maximum velocity is less than the inlet velocity (Figure 18) due to

the expansion in area.

Velocity Profile for Step Inlet and Outlet

00.0020.0040.0060.0080.01

0.0120.0140.016

0 0.002 0.004 0.006 0.008 0.01

Distance from the Y axis (m)

Velocity Profile

(m/s)

Inlet Velocity (Based on Re)

Outlet Velocity with no MHD effect

Figure 18: Velocity profile for step inlet and outlet

With an applied magnetic field in such that M=100, the recirculation region downstream

the step becomes much smaller as shown in Figure 19.

Figure 19: Recirculation region gets smaller with magnetic field effect

17

The velocity profile at the exit exhibits the behavior of the normal flow through the

channel under the effect of MHD. The velocity at the exit becomes flatter (Figure 20)

compares with the case for M=0. The velocity gradient becomes greater near the wall.

The effect of the flow pattern in the step flow varies depending on the strength of the

electric and magnetic fields.

Velocity Profile for Step Inlet

and Outlet

0

0.002

0.004

0.006

0.008

0 0.005 0.01

Distance from the Y axis (m)

Velocity Profile

(m/s)

Outlet Velocity with no MHD effect

Outlet Velocity with MHD (M=100)

Figure 20: Velocity comparison between the outlet velocities with/without MHD

The same effect is observed in the step flow with a higher Reynolds number. At

Reynolds number of 648, the second recirculation downstream the step vanishes with a

Hartmann number of a 100. With the disappearance of the second region, the profile

looks like the one with a lower Reynolds number (Figure 21 & 17).

Figure 21: Second recirculation region disappear with MHD effect with Re=648

After the disappearance of the second recirculation region, the velocity profile looks

similar to the velocity profile with the smaller Reynolds number. This is anticipated

18

because the implementation of the MHD affects the pressure distribution as well.

Separation is intimately connected with the pressure distribution【8】

. With M=100 and

Re=648, the pressure distribution is much more uniform vertically (channel height) at

various channel length (Figure 22) compared to (Figure 23) when M=0 and Re=648.

Figure 22: Pressure contour with M=100 and Re=648

Figure 23: Pressure contour with M=0 and Re=648

At the recirculation region, the pressure distribution along the y axis is not uniform as

shown in Figure 24 and Figure 25. In the region where recirculation exists for both M=0,

and M=100 at X=0.02 through X=0.04, the pressure changes with the channel height.

However, with the disappearance of the second recirculation region @ M=100 (Figure

25), the pressure at X=0.06 to X=0.09 along the y axis is much more uniform compared

to the case when M=0 (Figure 24).

19

Figure 24: The cross-section of the pressure along the channel @ M=0

Figure 25: The cross-section of the pressure along the channel @ M=100

The disappearance of the second recirculation region in the model is correlated with the

change in the pressure. Figure 26 shows that the pressure gradient along the top of the

channel for M=100 and M=0. For the regions where recirculation exist, the absolute

pressure gradient (change in pressure) is much smaller relatively to the region of no

recirculation. In addition, the pressure gradient for M=0 exhibits more points of

inflexion, which could be the cause of separation.

20

Pressure Gradient Trace (along the channel) Comparison

-50.00

-40.00

-30.00

-20.00

-10.00

0.00

10.00

20.00

30.00

0.00 0.02 0.04 0.06 0.08 0.10

Distance from the inlet of the channel (m)

Pressure gradient along

the top of the channel

(Pa/m)

Top M=0

Top M=100

Figure 26: Pressure gradient trace (along the channel) comparison for M=0 &

M=100

It is expected that all the recirculation regions would disappear given a large enough

magnetic and electric fields because the Lorentz force would hinder the flow and change

the pressure distribution.

3.4 MHD Effect on Heat Transfer in a Backward Step Flow

Since the magnetic and electric fields have significant effect on the velocity profile on

the step flow, the heat transfer mechanism in a backward step flow is likely to be

affected as well. The heat transfer is modeled in the conduction and convection module

with these two mechanisms as the main sources of heat transport. In our study, the inlet

is modeled to have a constant temperature at 1000 deg Kelvin. The bottom side of the

channel has a constant temperature of 400 deg Kelvin. All the other sides are assumed to

be thermal insulated. The velocity obtained using the Magnetostatic and Incompressible

Navier-Stokes fluid is input into the heat transfer module. In addition, the liquid metal

fluid, NaK, properties such as the thermal conductivity and specific heat are inputted

into the sub-domain. Figure 27 shows the schematic setup of the boundary condition for

the conduction and convection module.

21

Figure 27: Boundary condition for the conduction and convection module

In this case, the effect of convection is small and conduction is the dominant source of

heat transfer since the thermal conductivity is large for the liquid metal. Although the

computed temperature fields for the models with and without MHD effect are similar

(Figure 28), there are differences (Figure 29).

Figure 28: Heat transfer in a back-step flow (Top M=100, Bottom M=0)

As discussed in Section 3.3, the velocity profile in a back-step configuration changes

with the effect of MHD. Since convection is very dependent on the velocity of the flow,

the heat transfer is slightly different between the two cases. Figure 29 shows the

temperature trace along the topside of the back-step geometry with/without the

implementation of the magnetic and electric fields.

22

Hartmann Effect on Heat Transfer

300

305

310

315

0 0.02 0.04 0.06 0.08 0.1

Length (m)

Temperature (K)

Hartmann Effect (M=100) No Hartmann Effect (M=0)

Figure 29: Hartmann number effect on heat transfer

The slight difference in the temperature is due to the change in the velocity profile. As

presented in Section 3.1, the Hartmann effect generated a greater velocity gradient along

the walls and this led to the temperature difference seen along the topside of the

geometry.

23

4. Conclusion

The successful validation of the Hartmann problem and the simple backward step flow

in COMSOL allows the evaluation of MHD effect in the backward step flow. The result

and the analysis indicate that the Lorentz force generated by the magnetic and electric

field has a significant effect on the flow pattern in a backward step flow. Just like the

simple Hartmann problem between two parallel channels, the Lorentz force generated

under MHD in the step flow also flattens the velocity profile and increases the velocity

gradients near the wall. Depending on the Hartmann number, the overall velocity profile

becomes flatter and smaller in magnitude compared to a parabolic inlet velocity profile

shape. This effect on the velocity profile in a backward step flow leads to the change in

the separation and reattachment point. Depending on the strength of the fields, the

recirculation regions that were once there could become smaller or vanish altogether.

This is due to the change in the velocity profile and the pressure distribution in the

channel. This change in the velocity profile also alters the heat transfer mechanism in the

back-step flow since convection is affected.

24

5. References

1. Armaly, B.F., Durst, F., Pereira, J.C.F., and Schonung, B., Experimental and

theoretical investigation of backward-facing step flow, J. Fluid. Mech. 127

(1983), pp. 473–496.

2. Blums, Elmars, I︠U︠ A. Mikhailov, and R. Ozols. Heat and Mass Transfer in

MHD Flows. Ed. R.K. T. Hsieh. Vol. 3. Singapore: World Scientific, 1987. Print.

3. Cengel, Yunus A., and John M. Cimbala. Fluid Mechanics Fundamentals and

Applications. New York: McGraw-Hill, 2006. Print.

4. Cerini, DJ. Liquid- metal MHD power conversion Source: JPL Quart Tech Rev, v 1,

n 1, p 64-67, Apr 1971 Database: Compendex

5. COMSOL. "Stationary Incompressible Flow over a Back-step - Documentation -

Model Gallery - COMSOL." Multiphysics Modeling and Simulation Software -

COMSOL. 2008. Web. 12 Oct. 2010.

<http://www.comsol.com/showroom/documentation/model/94/>.

6. Hughes, William F., and F. J. Young. The Electromagnetodynamics of Fluids. New

York: Wiley, 1966. Print.

7. Jongeboled, Luke. "Numerical Study Using FLUENT of the Seperation and

Reattachment Points for Backwards-Facing Step Flow." Thesis. Hartford, CT,

Rensselaer Polytechnic Institute, 2008. Print.

8. Kliman, Gerald B. Axisymmetric Modes in Hydromagnetic Waveguide. Tech. no. 449.

Cambridge, MA: Massachusetts Institute of Technology, 1967.

9. Lima, R.C., Andrade, C.R., and Zaparoli, E.L., Numerical study of three recirculation

zones in the unilateral sudden expansion flow, International Communications in

Heat and Mass Transfer, Volume 35, Issue 9, November 2008, Pages 1053-1060.

10. "Liquid Metal Cooling." Frigus Primore - Thermal Design for Electronics. Web. 13

Oct. 2010. <http://www.frigprim.com/articels4/LiqMetal.html>.

11. Schlichting, Hermann, J. Kestin, Hermann Schlichting, and Hermann Schlichting.

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