INTRODUCTION
Liquid metals and its compounds have been proposed as coolants for fusion reactors.
They have also been proposed as coolants for limiters and diverters, the highheatflux
components of a fusion reactor. Because of its reactivity with liquid metals
(lithium, sodium, potassium), water is considered a safety risk (Piet, 1986).
It has been observed that liquid metal flows in highheatflux components both
transverse to the magnetic field and also from a direction parallel to the magnetic
field to a direction perpendicular to the magnetic field. The problem of liquid
lithium flow in a straight rectangular duct with conducting walls and transverse
magnetic field has been looked at by Kim and Abdou (1989). The problem of liquidmetal
flow in a curved bend where the magnetic field acts in a single transverse direction
to the coolant flow for a straight duct and a curved bend has been looked at
by Asad (Majid, 1999). However the MHD pressure drop in the presence of two
transverse magnetic fields for different liquid metals (lithium, sodium, potassium)
has not been looked at.

Fig. 1: 
Geometric configuration of the curved bend 

Fig. 2: 
Geometric configuration of the coolant channel 
Knowledge about the MHD pressure drop in the presence of two transverse magnetic
fields, one in the x direction and varying as B_{0}(R + x)^{–1}
(Fig. 1) and the other a constant magnetic field in they direction
(Fig. 1), is necessary to accomplish a feasible fusion reactor
design. A squarecrosssection coolant channel was selected for analysis. This
channel represents the cooling channel on the face of a limiter or diverter.
The channel analyzed would closely resemble the channel on the leading edge
of the limiter or diverter (Baker, 1980). Figure 2 shows the
geometric configuration of this channel. The coordinate system can be called
a modified toroidal coordinate system (Fig. 1) because the
cross section of the torus in modified toroidal coordinate system is rectangular
while the cross section of the torus in toroidal coordinate system is circular
(Fig. 1).
THE MHD EQUATIONS
The MHD equations, assuming the conditions of steady state, incompressible flow, constant properties, negligible viscous dissipation and negligible induced magnetic field, are expressed in vector from as follows:
Continuity:
Momentum:
Ohm’s Law:
Potential equation:
Where
B 
= 
Magnetic field 
u 
= 
Vectorial velocity 
φ 
= 
Electrical potential 
Ρ 
= 
Pressure 
ρ 
= 
Fluid density 
σ 
= 
Electrical conductivity 
μ 
= 
Absolute viscosity 
J 
= 
Electrical current 
It has been observed that when liquid metal flows in a duct with conducting
walls, the free charge available makes a path through the conducting walls.
This current interacts with the magnetic field. If the orientation of the magnetic
field is transverse to the flow, then a net force (J X B) develops, which tends
to retard the flow and thus gives rise to MHD pressure drop. It can be seen
from the momentum equation that in MHD flows in the presence of magnetic field,
three kinds of fields affect the flow. They are the pressure field, the potential
field and the magnetic field.
The vectorial MHD equations were then expressed in terms of the space variables x, y, θ of the modified toroidal coordinate system, with x representing distance in the radial direction from the duct centerline, y the distance normal to the radial direction and θ the angle around the loop in the axial direction. The differential equations in terms of the space variables x, y and θ and the major radius, are obtained by the use of metric coefficients and are given next (Note that u is the velocity in the x direction, v is the velocity in the y direction and w is the velocity in the axial direction):
Continuity:
xmomentum:
ymomentum:
θmomentum:
Potential equation:
METHOD OF SOLUTION
The differential equations described above were solved using the finite difference techniques described by Patankar, 1980; 1979b; Patankar and Spalding (1970, 1972a, b; 1974a, b; 1978). The new computer code SIMPLMHD (Majid, 1990) was upgraded to incorporate the three different liquid metal properties at 230°C (Holman, 1976; Charles, 1958). This code incorporates the extended SIMPLER method (Majid, 1990) for solving the coupled set of MHD equations.
The discretization equations were developed using the control volume approach. The numerical scheme used was the power law scheme (Patankar, 1980, 1979a, b; Patankar and Spalding, 1970, 1972a, b; 1974a, b; 1978). The potential equation is solved by ordinary finite difference techniques. The final discretization equations, where the magnetic field is assumed to operate in the xθ plane as well as in the y direction are obtained as follows:
xmomentum:
where
ΔV 
= 
Volume of control volume 
P 
= 
Location of grid point of interest 
n 
= 
Northern face of control volume 
s 
= 
Southern face of control volume 
e 
= 
Eastern face of control volume 
w 
= 
Western face of control volume 
t 
= 
Top face of control volume 
b 
= 
Bottom face of control volume 
W, E 
= 
Location of the neighboring points in the x direction 
S, N 
= 
Location of the neighboring points in the y direction 
B, T 
= 
Location of the neighboring points in the θ direction 
δA (x) 
= 
Area of the control volume face perpendicular to xdirection and where 
operator [C,K] = the greater of Majid (1999) Δx,Δy,Δθ
= dimensional parameters of the control volume w_{b} = calculation
of the w velocity at the corresponding location of the bottom face of the control
volume and similarly for others with this notation.
δx, δy, δz = distance between the grid points in the x, y and z directions, respectively.
ymomentum:
where the coefficients of the discretization equation and the nomenclature are the same as the coefficients and nomenclature for the xmomentum equation except for the fact that they now pertain to the control volume around the v velocity.
θmomentum:
Where the coefficients of the discretization equation and the nomenclature are the same as the coefficients and nomenclature for the xmomentum equation except for they pertain to the control volume around the w velocity.
Potential equation:
where
The definition of various control volume parameters is the same as given for the xmomentum equation, only now they pertain to the control volume around the potential.
The boundary conditions for solving the momentum equations evolve from the no slip condition where the velocities u, v and w are zero at the wall. The boundary condition to solve the potential equation is the thin wall boundary condition where the walls are thin enough that current cannot flow in a direction in which the wall is thin but can flow only in the direction along which walls extend (Majid, 1990).
The significant feature of these analyses was the development of the pressure equation by combining continuity and discretized momentum equations. The final set of equations to be solved consisted of three discretized momentum equations for three velocities u, v and w; one pressure equation and one electric potential equation.
The first step according to the SIMPLER algorithm was to guess a velocity field. An electric potential field was also guessed. The magnetic field was given. The various steps of the SIMPLER algorithm were then followed until a converged velocity field and pressure field were obtained. With the newly achieved converged velocity field and pressure field, the potential equation was solved and a potential field was obtained. With the newly obtained velocity field, pressure field and electric potential field, the whole procedure was repeated; i.e., the various steps of the SIMPLER algorithm were followed where in place of guessed fields newly calculated fields were used. This procedure was repeated until a converged electric potential field along with a velocity field and a pressure field was obtained.
ANALYSIS OF FLOW IN A CURVED BEND
The computer code SIMPLMHD was written, as mentioned earlier in such away that
results can be obtained for two geometries, a straight duct and a curved bend,
by controlling the major radius R. The exit condition of the flow was considered
as fully developed. As the length of the bend under consideration is not sufficient
for the flow to become fully developed, the specification of the exit condition
needs justification. The justification is that the Peclet number for axial flow
is sufficiently large (Peclet number> 100) thus the flow conditions downstream
at the exit will not significantly affect the flow conditions upstream at the
exit. The assumption of fully developed conditions at the exit will thus not
cause any significant error in the analysis. It was assumed that at the inlet,
the flow enters with a three dimensional parabolic velocity profile. The assumption
in the inlet profile is not expected to drastically effect our calculations
as we start measuring the pressure from the plane next to the inlet plane. It
has to be noted that the magnetic field force affecting the flow is so strong
as compared with the ordinary hydrodynamic forces that the flow takes up the
magnetic effects within a very small distance from the inlet. This can be seen
from the ordinary hydrodynamic pressure drop, which is only about 400 Pa m^{–1}
as compared with the MHD pressure drop, whose minimum value is about 20000 Pa
m^{–1} and soars above this value as magnetic field strength is increased.
Thus any error because of an inlet profile assumption is expected to be nulled
out within the movement of the fluid from the first inlet plane to the second
plane in the direction of the flow.
The dimensions of the channel analyzed were 1x1 cm. The bend analyzed had a radius of 5 cm. The magnetic field in the x direction in the case of the curved bend was considered to vary as B_{O}(R+x)^{–1}, as a constant magnetic field in the x direction would violate the Maxwell’s equation. The magnetic field in the y direction is considered to be constant. The wall thickness was taken to be 1 mm. The flow was analyzed at a Reynolds number of 5276 for lithium, 13928 for sodium and 16411 for potassium.
The calculation of the pressure gradient (Pascal per meter) in a 5 cm radius bend when magnetic field acts in both x and y directions was performed. The pressure gradient along the curved bend for different values of center line magnetic field and different values of constant magnetic field in y direction is shown in Fig. 3 for lithium. It is observed that pressure gradient increases for an increase in the center line magnetic field and also increases for an increase in the magnetic field in y direction. The pressure gradient along the curved bend for different values of center line magnetic field and different values of constant magnetic field in y direction is shown in Fig. 4 for sodium.
It is observed that pressure gradient increases for an increase in the center
line magnetic field and also increases for an increase in the magnetic field
in y direction.

Fig. 3: 
Lithium with varying by _{0} field 

Fig. 4: 
Sodium with varying by_{0} field 

Fig. 5: 
Potassium with varying by_{0} field 
The pressure gradient along the curved bend for different values of center
line magnetic field and different values of constant magnetic field in y direction
is shown in Fig. 5 for potassium. It is observed that pressure
gradient increases for an increase in the center line magnetic field and also
increases for an increase in the magnetic field in they direction. A comparative
plot of pressure gradient for three different fluids is shown in Fig.
6. It can be seen very clearly that MHD pressure drop is maximum for sodium
and minimum for lithium.

Fig. 6: 
Comparative plot of pressure gradient for three different
fluids 
CONCLUSIONS
The purpose of these comparative analyses was to look for a fluid for the cooling of high heat flux components of a fusion reactor, which is compatible with liquid metal lithium blanket and can also remove the 5 MW/m^{–2 } heat flux falling on the limiter or diverter plate. Thus a lowpressure system which can allow enhancement of the heat transfer through greater fluid velocities is highly desirable. As higher MHD pressure drop would lead to higherpressure systems thus a liquid metal which would give rise to lower MHD pressure drop is desirable. Our results however indicate that lithium still remains the liquid metal giving rise to minimum pressure drop as compared with sodium and potassium. We can finally conclude that from MHD pressure drop point of view liquid lithium is the best choice for cooling of high heat flux components of a fusion reactor.