Page 318 - Fundamentals of Magnetic Thermonuclear Reactor Design

298 Fundamentals of Magnetic Thermonuclear Reactor Design

tw″ half-length in the direction of the magnetic field and in the normal direction,
β = b/a; σ and t are the electrical conductivity and thickness of the channel
w
w
σw⊥ wall; σ w ⊥ and σ are the electrical conductivity of the ‘Hartmann’ and ‘lateral’
w
(normal and parallel to the main induction component of the magnetic field, re-
tw⊥ spectively) channel walls; and t w ⊥ and t are the thicknesses of the ‘Hartmann’
′′
w
and ‘lateral’ channel walls.
As the blanket contains a large number of hydraulically parallel channels,
the relations presented are written for each channel. After the system is up,
additional requirements are imposed on it. In particular, MHD pressure losses
should be equal in all the parallel channels and differences in the liquid metal
temperature rise over the l length should be specified.
6
Estimates for a DEMO blanket with uninsulated ducts (σ = 3 × 10 1/Ω m,
6
σ = 1.25 × 10 1/Ω m, t = 5 mm, b = 50 mm, a = 5 mm, U = 2 m/s, B = 13 T)
w
w
give a pressure linear gradient of 9.66 MPa/m. For a poloidal length of 2.9 m
this is equivalent to a pressure drop of ∼28 MPa, unacceptable by the mechani-
cal strength criterion. The use of an electrical insulation barrier with the thick-
ness of a metal layer facing the liquid metal (∼0.1 mm) allows a 50× decrease
in the pressure drop, making the latter acceptable.
For round uninsulated ducts,

k = c ( + cc1 ), = σ (r 2 − r 2 σ ) (r 2 + r 2 ),
kp=c/1+c, c=σwr02−ri2/σr02+ri2, p w 0 i 0 i
where r and r are the tube’s outer and inner radius, respectively.
0
i
For electrically insulated ducts,
3 π ⋅ Ha  3 π  −1
ξ =  1 −  .
ξ=3π⋅Ha4⋅Re1−3π2⋅Ha−1. 4 ⋅ Re  2 ⋅ Ha 
If the magnetic field is non-uniform along the flow direction (at the liquid-
metal entry and exit parts), the MHD pressure losses are estimated by the fol-
lowing integral equation:

∆ p MHD = ∫ x 2 ( dpdxdx,
⋅
)
/
∆pMHD=∫x x (dp/dx)⋅dx, x 1
1 2
where x and x are the boundary coordinates of the magnetic field non-unifor-
2
1
mity region.
The following equation is convenient for determining collector losses:
ρ ⋅ U 2
∆ p MHD = k N ⋅ 0 ,
⋅
∆p3DMHD=k⋅N⋅ρ⋅U022, 3D 2
where 0.25< k <2 is the parameter accounting for the geometry complexity (the
numerical values are valid for the case where the magnetic field is uniform and
2
the flow changes its direction in a plane normal to the magnetic field), N=Ha /Re.