Download Xie, Xiaole

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
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
2.2
Validating Hartmann Flow in COMSOL ............................................... 6
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
ii
LIST OF SYMBOLS
B0
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
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
Z
Z-direction
µf
Dynamic viscosity
(kg/ms)
ρ
Density
(kg/m3)
(m)
(m)
iii
σ
Electrical conductivity
(S/m)
iv
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
v
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
vi
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.
vii
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.
viii
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:
1
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
2
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
【10 】
of sodium and potassium, (NaK)
. The following table shows the some of the
important properties of NaK.
Table 1[9]: Liquid Metal Properties
Metal
NaK(22/78 %)
Melting
Point
-12
Density
(kg/m3)
802
Spec Heat
(J/kgK)
1058
Thermal Cond
(W/mk)
35
Viscosity
(kg/ms)
9.40E-02
Electrical
Cond
(S/m)[6]
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.
3
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 B0 and EZ.
The flow of an incompressible fluid between parallel plates is governed by the
equation of continuity [3]:
∂u ∂v
+
=0
∂x ∂y
and the Navier-Stokes equations which have the following form [6]
− ∂P
∂ 2u
+µf
0=
− FX
∂x
∂y 2
4
0=
− ∂P
∂ 2v
+µf
− FY
∂y
∂y 2
− ∂P
∂2w
+µf
0=
− FZ
∂z
∂y 2
Where
− ∂P − ∂P − ∂P
,
,
are the components of the pressure gradient in the X, Y and Z
∂x
∂y
∂z
directions respectively, µ f is the dynamic viscosity of the fluid, FX , FY and FZ 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].
0=
− ∂P
∂ 2u
+ µf
− J Z B0
∂x
∂y 2
0=
− ∂P
+ J Z B X − J X BZ
∂y
0=
− ∂P
∂2w
+µf
− J X B0
∂z
∂y 2
Where J X and J Z 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]:
J X = σ ( E X − B0 w)
J Z = σ ( E Z − B0 u )
σ stands for the electrical conductivity of the liquid metal. Here E X and EZ 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]:
5
0=
− ∂P
∂ 2u
+ µf
− σ ( E Z + B0 u ) B0
∂x
∂y 2
With this equations and the assumption of no slip condition at the wall, the analytical
solution is
【6】
:
2
u=
y0
M2
 (ch( My / y 0 ) 
σ
E Z ) 
− 1

µf
chM


 1 ∂P M
(
+
 µ f ∂x y 0

Where the Hartmann number is given by M = y 0 B0 σ / µ f . 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]:
− µ f ∇ 2 u + ρ (u ⋅ ∇)u + ∇p = F
And the equation of continuity [5]:
∇ ⋅u = 0
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
6
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.
7
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, M = y 0 B0 σ / µ f , only depends on the strength of the magnetic
field given the fluid properties. Therefore, it is sufficient to apply the magnetic field in
8
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
Length of channel
Mesh Size
H
L
Number of degrees of freedom
0.2 m
2m
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.
9
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)
Analytical SOL M=5
COMSOL SOL M=5
Velocity (m/s)
0.0014
0.0012
0.001
0.0008
0.0006
0.0004
0.0002
0
-0.1
-0.05
0
0.05
0.1
Distance from Y (m)
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.
10
Hartmann Effect on Velocity
M=0
M=5
M=10
0.0012
Velocity (m/s)
0.001
0.0008
0.0006
0.0004
0.0002
0
-0.1
-0.05
0
0.05
0.1
Distance from Y (m)
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)
Normalized Velocity (U/Umax)
1
0.8
M=0 (COMSOL)
M=5 (COMSOL)
M=10 (COMSOL)
M=0 (Analytical)
M=5 (Analytical)
M=10 (Analytical)
0.6
0.4
0.2
0
-0.1
-0.05
0
0.05
0.1
-0.2
Distance from Y (m)
Figure 10: Hartmann number effect (analytical and COMSOL solutions
comparison)
11
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.
h
S
H
Xe
Xo
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
uDρ
The Reynolds number is defined as, Re =
, where u is the inlet velocity, µf is the
µf
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
[1]
data . 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
12
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.
13
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
14
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
No MHD Effect
MHD (M=100) Effect
0.6
Pressure (pa)
0.5
0.4
0.3
0.2
0.1
0
0.02
0.04
0.06
0.08
Distance from the Inlet (m)
0.1
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)
15
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
Velocity Profile
(m/s)
Inlet Velocity (Based on Re)
Outlet Velocity with no MHD effect
0.016
0.014
0.012
0.01
0.008
0.006
0.004
0.002
0
0
0.002
0.004
0.006
0.008
0.01
Distance from the Y axis (m)
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
16
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
Velocity Profile
(m/s)
Outlet Velocity with no MHD effect
Outlet Velocity with MHD (M=100)
0.008
0.006
0.004
0.002
0
0
0.005
0.01
Distance from the Y axis (m)
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
17
because the implementation of the MHD affects the pressure distribution as well.
【8】
Separation is intimately connected with the pressure distribution
. 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).
18
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.
19
Pressure Gradient Trace (along the channel) Comparison
Pressure gradient along
the top of the channel
(Pa/m)
30.00
20.00
10.00
0.00
-10.000.00
0.02
0.04
0.06
0.08
0.10
-20.00
Top M=0
-30.00
Top M=100
-40.00
-50.00
Distance from the inlet of the channel (m)
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.
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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.
21
Hartmann Effect on Heat Transfer
Temperature (K)
Hartmann Effect (M=100)
No Hartmann Effect (M=0)
315
310
305
300
0
0.02
0.04
0.06
0.08
0.1
Length (m)
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.
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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.
23
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, I︠ U︠ 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.
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Oct. 2010. <http://www.frigprim.com/articels4/LiqMetal.html>.
11. Schlichting, Hermann, J. Kestin, Hermann Schlichting, and Hermann Schlichting.
Boundary-Layer Theory. 7th ed. New York: McGraw-Hill, 1987. Print.
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