Download Basic dimensionless MHD/Heat Transfer parameters in the poloidal

Comparative scaling analysis for ITER and DEMO in the Thermofluid / MHD area
(mostly for poloidal flows !)
Sergey Smolentsev
11/03/06
A. Definition of the dimensionless parameters and their role
1. Reynolds number, Re 
U ma

In purely hydrodynamic duct flows Re is responsible for the flow development length. It is also used as a
criterion of laminar to turbulent transition. In MHD flows, using only Re to estimate the flow development
length or to judge about the flow regime is not sufficient.
2. Hartmann number, Ha  B0 b


It is the most important MHD parameter. Ha 2 is a ratio of the electromagnetic force to the viscous force.
3. Grashoff number, Gr 
gTa 3
2
The most important parameter that characterizes buoyancy effects in the flow. It is strongly affected by the
dimension a since a enters the formula for Gr as a 3 . Both a and the temperature difference T that drives the
natural convection should be properly defined in accordance with the flow physics.
4. r 
Gr
a
Ha Re 
b
2
This is another dimensionless parameter that characterizes contribution of the buoyancy effects. It is constructed
in a special way to take into account the forced flow effect, the MHD effect and the influence of the duct aspect
ratio. It appears that this parameter is the best characteristics for the mixed MHD convection. When doing
scaling analysis, this parameter should be considered first. See, S. Smolentsev, N. B. Morley, M. Abdou, R.
Munipalli, R. Moreau, Current Approaches to Modeling MHD Flows in the Dual Coolant Lead Lithium
Blanket, Magnetohydrodynamics, 42, 225-236 (2006).
5. Interaction number, N 
Ha 2
Re
It is the ratio of electromagnetic to inertia forces. It is especially important when considering flow development
effects but not important for fully developed flows.
6.
Ha
Re
It is often used in Magnetohydrodynamics to judge about the flow regime (laminar or turbulent). Usually if this
parameter is higher than its critical value (about 1/200), ordinary 3-D turbulence is suppressed but 2-D MHD
may exist. When analyzing turbulence, either 2-D or 3-D, this parameter should be considered together with
another parameter that stands for the wall electrical conductivity, since in electrically conducting ducts, stronger
turbulence suppression will occur due to Joule dissipation.
7.
Ha
Gr
It is similar to
Ha
Ha
but unlike this parameter, which is used in the analysis of forced MHD flows,
applies to
Re
Gr
buoyancy driven MHD flows.
8.
Gr
Re
In ordinary flows, it is a measure of the influence of free convection in comparison with that of forced
convection on a flow. In MHD flows, parameter r seems to be more relevant.
9. Something related to the electrical conductivity
In duct flows with electrically conducting walls, a relevant parameter is the wall conductance ratio, c w 
 wt w
.
b
However, in the flows with FCI, using just c w would not be correct, since the electric circuit is closed through
both the Fe wall and the FCI. A parameter suggested by Neil Morley is c FCI 
b FCI
.
t FCI 
10. Aspect ratio, a / b
The aspect ratio does not play a significant role, providing it is close to 1.
11. Dimensionless flow length, L / a and L / b
It is an important parameter when justifying if the flow is developing or fully developed
Here,
b is half of the duct dimension parallel to the magnetic field (toroidal dimension), m
a is half of the duct dimension perpendicular to the magnetic field (radial dimension), m
L is the channel length, m
B0 is the applied magnetic field, T
U 0 is the fluid bulk velocity, m/s
T 
q* a
is the temperature difference, which drives natural convection, K
k
q* is the neutron wall loading, W/m2
k is the fluid thermal conductivity, W/m-K
g is acceleration due to gravity, m/s2
 is the fluid thermal expansion coefficient, 1/K
 is the fluid electrical conductivity, S/m
 is the fluid kinematic viscosity, m2/s
 is the fluid density, kg/m3
B. Physical properties of PbLi
k = 20 W/m-K
 = 1.8210-4 1/K
 = 0.7106 S/m
 = 1.0810-7 m2/s
 = 9300 kg/m3
C. Dimensional blanket parameters
Parameter
2a
2b
L
q*
U0
B0
ITER
0.066 m
0.12 m
1.66 m
0.78 MW/m2
0.1 m/s
4T
DEMO Outboard/Inboard
0.22 m
0.22 m
2m
3.08/1.75 MW/m2
0.06/0.04 m/s*
4/10 T
* 0.06 was taken from the old DEMO data set for the outboard blanket. The new one may be different.
Velocity in the inboard blanket can also be different (lower). We use 0.04 m/s at the inboard.
D. Dimensionless parameters
Parameter
Re
Ha
Gr
r
N
Ha/Re
Gr/Re
ITER
30500
6350
7.22109
11.1
1320
0.208
2.36105
DEMO Outboard/Inboard
61000/40600
11640/29100
3.521012/21012
70.3/41
2217/20860
0.190/0.716
5.76107/4.92107
*cFCI
Ha/Gr
8.8010-7
3.3110-9/1.4510-8
cFCI*
8.5710-5
3.1410-3
a/b
0.55
1.0
L/a
50
18
L/b
28
18
was calculated assuming tFCI=5 mm and FCI=5 S/m (ITER), FCI=100 S/m (DEMO)
E. Conclusions related to extrapolation of ITER test results to DEMO (outboard)
1. If to consider only magnetohydrodynamic parameters, such as Re , Ha , N , Ha / Re the difference between
DEMO and ITER is not so big. This is because the magnetic field in both scenarios is the same, 4 T, but the
cross-sectional dimensions do not differ much. However purely magnetohydrodynamic regimes will occur
only in ITER HH. In ITER DT and DEMO, the flow structure will be sufficiently changed by the volumetric
heating. This means that extrapolation of results of the MHD tests obtained in ITER HH to DEMO
conditions is possible but limited to the particular sub-flows, where volumetric heating does not play a major
role. Similarity between ITER HH and DEMO is expected with regard to the inlet manifold, flow
distribution in the module, electromagnetic coupling, effectiveness FCI as electric insulator, turbulence,
MHD pressure drops and the flow development length. However, flows in the long poloidal ducts in ITER
HH will be sufficiently different from those in other two scenarios (DEMO and ITER DT), where the
volumetric heating is of primary importance.
2. Heat transfer issues will be checked in ITER DT and then the ITER test results will be extrapolated to
DEMO conditions. The heat transfer tests will mostly address such an issue as effectiveness of FCI as a
thermal insulator and the temperature field in the FCI as affected by the flow conditions in the long poloidal
ducts. The most important phenomenon here is associated with natural convection and turbulence and their
effect on the formation of high-velocity near-wall jets. The relevant dimensionless parameters are Gr ,
Gr / Re , Ha / Gr and r . The electrical conductivity of the FCI will also play a major role. The key question is
to what degree heat transfer/flow conditions in ITER DT are similar to those in DEMO. Unfortunately, the
answer is more negative. If to look for example at Gr , Gr / Re and Ha / Gr , one can see a difference from 2 to
3 orders of magnitude. In these conditions, I foresee not only large quantitative but also significant
qualitative differences (like different flow regimes) between ITER DT and DEMO. This is for sure that
under DEMO conditions, the flow will be strongly agitated by the buoyancy effects, but to a smaller degree
in ITER. Sufficiently smaller Gr values in ITER DT are related to two factors. First, volumetric heating in
ITER is roughly 4 times lower than that in DEMO. Second, the radial dimension a in ITER is smaller. This
dimension enters the formula for the Grashoff number as a 4 . Thus, the most effective way to increase the
Grashoff number in ITER tests (to be closer to DEMO conditions) is to increase radial depth of the poloidal
channel. However, still the difference with DEMO will be pronounced because of lack of neutrons in ITER
DT.
F. Observations related to implementation of reduced size modules
1. It is obvious that the full scale TBM (half port size) has some advantages over the reduced size modules,
since, generally speaking, it provides better similarity with the DEMO blanket. However, using reduced
(quarter port) modules may be beneficial from the point of view of decoupling different flow phenomena,
allowing for more accurate characterization of the key processes. This would also allow studying these key
processes in a more aggressive manner.
2. It is also obvious that different flow components in the blanket have different sensitivity to changes of the
basic dimensions. For example, reduction of the module height from 1.66 m to 0.8 m may have a strong
effect on the poloidal flows, while the flow in the inlet manifold will not be affected. When designing
reduced size modules one can consider many options leading to smaller module dimensions, for example
using less than 6 channels (current ITER design), or reducing the poloidal length. In either approaches, there
is a risk that some key MHD/Heat Transfer phenomena can be seriously modified, so that the test results
may loose their value. Below, I tried to summarize what we could expect when reducing either (A) the crosssectional dimensions of the original half port module (via reduction of the number of the poloidal channels
or via reduction of the duct cross-sectional dimensions) or (B) the poloidal length while keeping the same
cross-sectional dimensions and the number of channels.
2.1.
Reduction of the poloidal length. The main risk is associated with the poloidal flows, namely one can
expect that the reduced flow length would not be enough to have a fully developed flow here. In DEMO
L / a =18, while it is 50 in ITER (current half port design). Reducing L / a from 50 to 25 will result in the
flow length even closer to the corresponding DEMO value. However, judging about the flow
development length will also require a comparison of other parameters ( Re , Ha , etc.), which are not the
same in ITER and DEMO. Unfortunately, accurate correlations for the flow development length in duct
MHD flows are not available. A generally observed tendency is that in a stronger magnetic field the flow
development length is shorter. One can expect that under a 4 T magnetic field in both ITER and DEMO
the transitional length associated with either flow inlet or outlet is no longer than 5 characteristic duct
dimensions. This means that at least over the half of the poloidal duct length of 25 duct dimensions, the
flow will demonstrate fully developed features or will be close enough to being fully developed. From
this point of view, considering a reduced length sub-module seems to be very reasonable.
2.2.
Reduction of the number of channels or/and their cross-sectional dimensions. Reduction of the crosssectional dimensions themselves does not look too bad providing the basic dimensionless parameters are
not changed much, say, the maximum change is by a factor of 2 or less. This would assure that the flow
regime is essentially the same (compared to the current half port design), although some quantitative
variations in the flow are quite possible. Reduction of the number of channels may be associated with
higher risks of loosing information. First of all it is related to the flow distribution in the whole module
and especially in the inlet manifold, flow balancing and electromagnetic flow coupling. On the other
hand, such key phenomena as natural convection and turbulence and effectiveness of the FCI as thermal
insulator can hardly be affected.