Calculations now suggest that by using the high plasma pressures
possible with spherical tokamaks and the high magnetic
fields possible with high-temperature superconducting coils, it
may be possible to construct a relatively compact fusion power
plant [1, 2]. A key problem for this approach has been the
heat deposition from fusion neutrons into a superconducting
core. Algebraic estimates of this heating power have been
made by Stambaugh et al [3]. For the Vulcan project Hartwig
et al [4] also used the Monte Carlo N-particle MCNP code
but calculated the power deposition into the superconducting
core as a function of shield thickness at a fixed core and
plasma radius. A detailed discussion of the inboard shield
of ARIES-ST, with a conventional, central column, is given
by El-Guebaly et al [5]. Their choice of a copper conductor
was influenced by the cooling and shielding requirements of
a conventional, low-temperature superconducting alternative.
In contrast, the cryoplant requirements for a high-temperature
superconducting central column, operating at 20K rather than
4K, are much less severe and we show that adequate shielding
can be provided without compromising the low aspect ratio of
a spherical tokamak design. The inboard shield must avoid
any radial or other gaps that would allow neutron and gamma
streaming.
We adopt a step-wise approach here, focusing on the
shielding requirements. Rather than making all computations
at the full level of detail, the complications are added in turn and
their individual effects noted. For most of the computations the
simplest practical model of a toroidal plasma at a major radius
R0 is taken with an elliptical cross-section of minor radius a
and elongation κ around a
cylindrical superconducting core of
radius rsc, surrounded by an annular shield of
thickness d. Heat
deposition into the superconducting core is presented over a
wide range of these variables, and the results are discussed in
geometric terms. The addition of an external shield at a radius
Rext and corresponding upper and lower shields at heights of
+/-Zext , is then considered and this is shown to increase the heat
deposition by a small but significant contribution of order 6%.
A further refinement is to define a radial profile of neutron
production density described with a superposition of toroidal
shells with differing elliptical cross-sections and differing
power densities. This is shown to have heat deposition changes
at a 20% level. Several potential shielding materials are
considered. Tungsten carbide with water cooling was found
to be the most promising on account of its high density and
high atomic number, with the presence of low atomic number
materials also providing neutron moderation. The results,
corrected for the effects of the external shield and plasma
profile are interpolated using least-squares fitting into a form
that can be incorporated into the plasma system codes used
for power plant design such as that by Costley et al [2].
The particular example of a pilot plant with a major radius
R0 = 1.35 m, a core radius rsc = 0.23 m and a shield
thickness d= 0.37 m, similar to that considered by Costley
et al will be considered throughout this study. It differs from
the design reported by Costley et al by neglecting engineering
features such a 0.05 m radius central tie-bar, a 0.03 m thermal
shield insulation gap and a 0.05 m plasma-wall gap. MCNP
computations show the inclusion of these particular features
increases the heat deposition by some 89% over the simplified
model. This significant increase results principally from the
reduction in the thickness of the neutron shield to make room
for the thermal break and the tie-bar. Further work aims to
study these and other engineering factors individually to reveal
their relative magnitudes.
MCNP6.1 is the latest release of a general-purpose,
continuous-energy, general-geometry Monte Carlo code for
N particles [6, 7]. It is a radiation transport code designed to
track many particle types over a broad range of energies. It is
supplied with an extensive library of nuclear cross-sections,
allowing the computation of nuclear interactions of fast
neutrons, gamma rays and other secondary particles with most
naturally occurring isotopes of common elements. Detailed
isotopic compositions can be specified for all materials used
in a model.
With a long history of use in fusion research and
incorporating substantial validation efforts, MCNP is the
standard fusion neutronics tool with the ability to calculate
a number of important nuclear quantities, including nuclear
heating, radiation dose and damage, and tritium breeding. The
detailed design for an optimized spherical tokamak power
plant has yet to be made. The present calculations were
performed with an idealized model of such a plant, while
still maintaining full three-dimensional geometry, but with the
simplifying assumption of axisymmetry. Compared with a
full model, the results are much faster to obtain, more flexible
and appropriate for a design study where parameters are being
swept. Following such an optimization, a more detailed
computation would be made.
A more detailed design study would also need to consider
several additional factors affected by the choice and thickness
of the neutron shield material. Induction losses could occur
within a thick conducting shield. Radiation damage to the
high-temperature superconductor may depend on different
factors from those for its heat deposition. Activation and
decay-heat production after operations need to be considered.
If tritium breeding is envisaged, even on the outer shield,
then the neutron reflectivity of the inner shield needs to be
considered.
A second function of the neutron shield is to reduce
the neutron damage to the high-temperature superconducting
material. This can also be studied using MCNP but
is not considered here. Experimental studies [8] show
that at moderate levels the damage to high-temperature
superconductors kept at 20K is limited at typical neutron
fluences.
2. The starting point: a simplified cylindrical model
The simplest realistic model of a tokamak power plant will
have the following three components, as shown in figure 1:
(i) A central superconducting core of radius rsc containing
the superconducting tapes, their cooling channels and
the strengthening materials needed to restrain the large
magnetic forces produced. The width of this core is
variable within the range from 0.15 to 0.25 m, depending
on the currents needed to produce the required magnetic
fields. In these first computations no detailed design
is assumed; instead a homogeneous idealization of coil
material, cooling and strengthening material is defined.
An example coil material consists of 52.5% Hastelloy
C-276, 22.5% copper and 25% neon cooling liquid by
volume. Liquid neon has a high refrigerating capacity
compared with gaseous helium on a per unit volume basis.
The average density of this homogenized material was
computed at 6.98 g cm-3.
(ii) An annular radiation shield of thickness d varying from
0.15 to 0.3 m. Options for the shielding material are
discussed in section 6 but tungsten carbide with water
cooling was the best investigated and is used in most of
the computations. This is assumed to be in the form of
concentric cylinders, separated by concentric layers of
cooling water.
(iii) A neutron-producing plasma. The MCNP code allows
toroidal sources of elliptical cross-section to be defined
and versions of such a source composed of a superposition
of several such toroidal volumes with varying radii can
describe a source profile of some complexity. This source
representation will be discussed in section 5, where we
show that a single small elliptical torus centred close to
the magnetic axis produces over half the emitted neutrons
and deposits power in the core within 1% of the weighted
superposition of eight toroids of varying sizes describing
a typical plasma profile.
Figure 1.
The simple toroidal model with three-dimensional
cylindrical geometry. The central core contains the
high-temperature superconductors. It is surrounded by a cylindrical
neutron shield. The toroidal plasma volume is assumed to produce
neutrons uniformly within the central ellipse. The outer plasma
boundary is also shown.
The calculations assume a surrounding cylindrical outer
boundary of height +/-1.5 m and radius 1.6 m but no structural
vessel or outer shielding. The purpose of this surface is to
terminate the tracking of escaping particles that can have no
further interaction with the structural components of the model.
The effects of including a realistic outer shield over a rather
larger outer boundary are considered in section 4.
The MCNP calculation proceeds by tracking the progress
of some large number of fusion neutrons, typically around
a million. These calculations take around 2 h following 1.5
million particle histories. At each collision the possible
reactions with the particular material are evaluated and
outgoing particles followed. The objective is to capture
fully the reactions that lead to nuclear heating from the
source neutrons. This results in there being more neutrons
to follow from (n,2n) reactions, and also many gamma rays
to follow from the (n,g )
reactions. Typically the heating
from gamma rays is comparable with that from neutrons as
found in the Vulcan studies [4]. As the numbers of neutrons
and gamma rays decrease due to the shielding process, it is
possible to improve the Monte Carlo statistics by increasing the
number of tracked neutrons by artificially splitting them, and
correspondingly reducing their weights, to give roughly similar
neutron populations in each part of the material. The output
from MCNP includes neutron and gamma ray mean free paths.
In the shielding layer it is approximately 20 mm for neutrons,
increasing to approximately 25 mm in the superconducting
alloy core. Gamma ray mean free paths are about 25% shorter.
The cylindrical core and annular shield have therefore been
divided up into five radial sections with thicknesses of the
order of these mean free paths. For example with a core radius
rsc = 0.15 m, the radial steps are at 0.03, 0.06, 0.09, 0.12 and
0.15 m. With a shield thickness d= 0.15 m the five radial
steps are at 0.18, 0.21, 0.24, 0.27 and 0.30 m. The program
evaluates the statistical significance of all the computed output
quantities ("tallies" in MCNP terminology), and these statistics
are satisfactory for all the cases presented. The statistical
accuracy of the MCNP computations is estimated in the output
file to be of order 1%. Aconsiderably larger error of order 10%
must be expected on the overall reliability when the assumed
simplifications of the geometry and plasma profile are taken
into account.
A further separation has been made of the central core
and shield into separate tally zones for their upper, central and
lower parts, with the central zone being defined as +/-0.2 m from
the mid-plane. This separation is to enable the heating to be
determined in the peak mid-plane region of the core where the
heating per unit length is expected to be a maximum.
The MCNP user may define energy bins to refine the
information computed about the tallies. For the present
computations twelve bins have been used, ranging from
0-1 keV up to 20-25 MeV; a few energetic particles are created
in the latter range. Figure 2 shows the relative weights of the
neutron and gamma heating contributions for each of these
energy bands. It is seen that the neutrons have most weight
in the energy range from 1 to 10 MeV. There is only a small
contribution to the heat deposition from the energy range below
10 keV. The total heat deposition from neutrons and gammas
is about the same for these parameters, however the neutron
heating is more concentrated in the central mid-plane region.
3. The calculation of heat flow into the central core
A fusing plasma of say 174 MW total power emits some
139 MW as neutrons with the remainder as alpha particles.
The alphas are slowed down within the plasma itself and their
energy is mostly transported into the divertor region and partly
transferred by relatively low energy radiation from the plasma
onto the surface of the neutron shield and the remainder of the
first wall. The alpha energy does not contribute significantly
to the heat deposition into the body of the neutron shield or
the superconducting core. We therefore consider only the heat
deposition from the neutron production.
The MCNP program produces an output file which
contains both physics results and information about the Monte
Carlo statistics. For example in the case of a core radius
rsc = 0.15 m and shield thickness d= 0.15 m the energy
deposited into the mid-plane z = +/-0.2 m zone of the core
region is Ecc = 6.40 x 10-8 MeV per gram of this region per
incident fusion neutron. Later in the output file the energy
depositions into the upper zone z > 0.2 m, Euc and into
the lower zone z < 0.2m, Elc can be found. These three
energies Ecc, Euc and Elc
and the corresponding masses Mcc,
Muc and Mlc of the zones allow the calculation of the energy
deposition into the complete superconducting core region.
Each fusion neutron carries an energy from the D-T reaction
of EDT = 14.1 MeV and if the total neutron power generated
in the plasma is given by, for example: Pneut = 139 MW then
the total number of fusion neutrons produced per second will
be Pneut/EDT. The power deposition into the central region
will therefore be Pcc = EccMccPneut/EDT. Similarly for the
complete core volume, the power deposition will be
Ptot = (EccMcc + EucMuc
+ ElcMlc)Pneut/EDT . (1)
Figure 2.
The relative weight of neutron and gamma power deposition as a function of energy on a log scale for a geometry with 0.15 m
radius core and 0.15 m thickness shield. The relative weight is higher in the mid-plane zone closer to the centre of the source in both cases.
Most of the heat arises from neutrons and gamma rays of relatively high energies above 100 keV.
Table 1. A selection of results (with external shield correction) for
a plasma major radius R0 = 1.0 m and 174 MW fusion power as a
function of superconducting core radius rsc and shield thickness d. A
small correction for the tungsten carbide external shield with water
cooling has been made as detailed in section 4. The penultimate
italic line shows the results that were used in the simplified Costley
et al study with major radius R0 = 1.35 m also including the
external shield. The bold italic line shows the results for the full
Costley study including engineering features. The cryogenic power
Pcryo will be defined in section 8 and the time tadi for adiabatic
heating constrained to a defined temperature rise in section 9.
| rsc (m) | d (m) | Pcc (kW) | Ptot (kW) |
Pcryo (MW) | tadi (s) |
| 0.15 | 0.15 | 173.72 | 520.19 | 28.30 | 1.13 |
| 0.15 | 0.20 | 96.46 | 279.09 | 16.00 | 2.10 |
| 0.15 | 0.25 | 51.22 | 142.31 | 8.66 | 4.12 |
| 0.15 | 0.30 | 26.37 | 70.34 | 4.58 | 8.33 |
| 0.15 | 0.35 | 13.91 | 35.96 | 2.50 | 16.29 |
| 0.15 | 0.40 | 7.00 | 17.26 | 1.30 | 33.95 |
| 0.20 | 0.15 | 260.84 | 763.92 | 40.30 | 1.36 |
| 0.20 | 0.20 | 142.20 | 400.73 | 22.28 | 2.60 |
| 0.20 | 0.25 | 75.57 | 204.68 | 12.06 | 5.09 |
| 0.20 | 0.30 | 38.88 | 100.83 | 6.34 | 10.33 |
| 0.20 | 0.35 | 20.20 | 50.83 | 3.42 | 20.49 |
| 0.20 | 0.40 | 10.14 | 24.35 | 1.77 | 42.78 |
| 0.25 | 0.20 | 194.51 | 532.67 | 28.93 | 3.06 |
| 0.25 | 0.25 | 102.92 | 268.65 | 15.45 | 6.06 |
| 0.25 | 0.30 | 52.44 | 131.03 | 8.04 | 12.42 |
| 0.25 | 0.35 | 26.90 | 65.24 | 4.28 | 24.95 |
| 0.25 | 0.40 | 13.54 | 31.43 | 2.22 | 51.78 |
| 0.23 | 0.37 | 7.85 | 29.80 | 1.66 | 88.00 |
| 0.23 | 0.32 | 13.72 | 56.40 | 2.98 | - |
These simple calculations are readily incorporated into a
spreadsheet or the code reading the MCNP output file. Table 1
below shows the selection of these powers for the plasma major
radius R0 = 1 m and a fusion power Ptot = 174 MW. The main
results are the total rate of energy deposition, Ptot and the rate
of energy deposition, Pcc in the mid-plane zone z = +/-0.2 m.
The time for adiabatic heating within a defined temperature
range tadi, and the cryogenic power Pcryo will
be defined later.
Figure 3 shows plots of the natural logarithm of the
heat deposition against the three variables of shield thickness,
plasma major radius and superconducting core radius. The
points fit rather well to the linear regression lines shown in
all the plots. The slopes against any given variable are quite
similar for all plots. Including fits to all the data points
gives the mean slope against major radius SRo
= -1.276 +/- 0.070 m-1, the mean slope against superconducting core radius
Srsc = 6.752 +/- 0.819 m-1 and the mean slope against shield
thickness Sd = -13.733 +/- 0.245 m-1. The good linearity
means that a fair approximation to all the computed data can
be obtained by choosing a median logarithm of heat deposition
value at for example (R0,rsc, d) = (1.00 m, 0.2 m, 0.3 m) and
performing a linear summation of products of the distances
from this point multiplied by the mean gradients:
loge P(R0,rsc, d) = loge P(1.0, 0.2, 0.3) + (R0 - 1.0)SRo
+ (rsc - 0.2)Srsc + (d - 0.3)Sd . (2)
Figure 3.
Three plots of the natural logarithm of the power
deposition plotted against (a) the shield thickness d from plasmas at
several major radii R0 for a superconducting core radius rsc = 0.2 m,
(b) the major radius R0 for several shield thicknesses and (c) the
superconducting core radius for a major radius of R0 = 1.4 m. The
lines are least-squares fits to the data. The open circles indicate the
Costley et al [2] pilot plant.
As a test of the approximation, figure 4 shows the
estimated natural logarithms of the 93 values of the heat
depositions compared with their computed values. The
agreement has a standard deviation of 0.054 in the natural
logarithm, or 5.5%.
Figure 4.
A scatter plot of the linear fit model for the logarithm of
the heat deposition to the computed values.
Figure 5 illustrates some of the geometric factors that
contribute to the results presented. Since the heat deposition
from neutrons generated at any point at a radius R0 will be the
same, it is equivalent to view all neutrons as being generated
at a single point at (X, Y ) = (R0, 0) as indicated in the figure.
All the distances involved in the calculation can be evaluated
analytically [9]. A good fit of order 10% to the heat deposition
computations at any given major radius could be made with
the simple model that the neutrons travel through the shield
and across the core along linear paths with an exponential
attenuation depending on the material. The deposited power
increases with core radius because of the increased solid angle,
and decreases with shield thickness.
Figure 5.
The geometric factors involved in
the heat deposition into a shielded central superconducting core of radius rsc
surrounded by a
cylindrical shield of thickness d from a neutron source at radius R0.
All the distances in the figure can be evaluated analytically and
integrations made in the approximation of linear paths.
4. The effect of the plasma vessel outer surfaces
The previous section described results for the simplest situation
of an elliptical cross-section plasma with shielding only of the
central core containing the superconducting windings. All
neutrons hitting the outer, upper and lower surfaces of the
vessel were assumed lost. A subsequent calculation was
performed for the situation with a 0.5 m cylindrical thick
blanket shield starting at a radius of 1.55 m extending to 2.05 m,
together with top and bottom surfaces also 0.5 m thick starting
at +/-1.05 m and ending at +/-1.55 m. Figure 6 shows the layout
of the shield including the various internal boundaries used to
optimize the statistics in the MCNP calculation.
The results are given in table 2 below for the case of core
radius rsc = 0.2 m and inner shield thickness d= 0.4 m.
The computations with the external shield took considerably
longer - around 20 h. Looking first at the total power line
of table 2, the power deposited into the central mid-plane
region of the core for major radius R0 = 1.0 m, rsc = 0.2 m
and d= 0.40 m has increased by about 6.5%. Looking at
the mid-plane power, the inclusion of the external shield has
increased the heat input by only 0.78%. It seems possible
that the tungsten carbide and water outer shield appear rather
"black" to the neutron flux and that relatively few are reflected
back. Also, the spherical tokamak geometry means that the
solid angle subtended by the central superconducting core at
the outer wall is not large.
Table 2. The power deposition in kW of typical core and shield configurations without and with the external shields (outer, upper and lower)
for a total fusion power of 174 MW and neutron power 139 MW.
| - | rsc (m) | d(m) | No shield | With shield | Increase |
| Total deposited power Ptot | 0.2 | 0.4 | 22.97 kW | 24.46 kW | 6.50% |
| Mid-plane deposited power Pcc | 0.2 | 0.4 | 9.41 kW | 9.48 kW | 0.78% |
5. The effect of the fusion neutron source profile
The first computations considered a small, uniform, elliptical
cross-section neutron source centred near the major radius.
It is known from both experiment and theory that the fusion
neutron source profile is highly peaked within the perimeter of
the plasma. This occurs because the neutron production varies
as the square of the temperature. More precise profiles have
been calculated using the SCENE[10] magneto-hydrodynamic
(MHD) equilibrium code. The parameters used here were
evaluated for a major radius R0 = 1.0 m and minor radius
a = 0.54 m, with peaked density and temperature profiles. The
number of fusion neutrons produced is given by (NDT/2)2σ
where NDT is the density of the deuterium-tritium fuel and
σ is the nuclear reactivity between D and T. Both the
density and temperature are approximately constant on a
flux surface, so the number of fusion-produced neutrons is
also approximately constant on a flux surface. A fit to the
equilibrium flux surfaces was made using the Miller [11]
description in terms of the Shafranov shift, triangularity and
elongation as a function of minor radius, and is given in
table 3.
Table 3. The major radius R0, radial minor radius ar , vertical minor
radius az and weight of the 8 radial shells.
| Shell | R0 (m) | ar (m) | az (m) | Weight |
| 1 | 0.9984 | 0.0625 | 0.1549 | 0.5321 |
| 2 | 0.9900 | 0.1250 | 0.3160 | 0.3358 |
| 3 | 0.9768 | 0.1875 | 0.4821 | 0.1080 |
| 4 | 0.9585 | 0.2500 | 0.6487 | 0.0258 |
| 5 | 0.9354 | 0.3125 | 0.8115 | 0.0057 |
| 6 | 0.9073 | 0.3750 | 0.9707 | 0.0013 |
| 7 | 0.8743 | 0.4375 | 1.1344 | 0.0003 |
| 8 | 0.8364 | 0.5000 | 1.3229 | 0.0001 |
The intensity of fusion-produced neutrons falls off
exponentially with the minor radius; so that over 99% of the
neutrons are produced within the first half of the minor radial
distance and over 50% are generated within the first one-eighth.
Figure 6.
The inclusion of an external shield. The blue areas represent the central superconducting core, which now extends into the
external shield. The orange shield surrounds the assembly at twice the thickness surrounding the core.
Figure 7(a), on the left, shows the Miller poloidal flux
surfaces at eight radial positions. However this profile is not
straightforward to use with MCNP. Elliptical cross-section
toroidal flux surfaces are standard in MCNP and the figure 7(b)
on the right shows a similar set of flux surfaces using such
elliptical toruses (which ignores the effects of triangularity).
They differ appreciably near the top and bottom of the plasma
but are very similar near the centre where most of the neutron
intensity is concentrated. A choice has been made to centre
each torus at the correct position determined by the Shafranov
shift and to match the inner radius distance. This means that
the radius at which the peak of the vertical height distribution
occurs is in error, but this is only significant at larger minor
radii where the emitted intensity is much lower. Using these
elliptical toroidal flux surfaces it is possible to evaluate the heat
deposition from each of the annular shells in turn, and to plot
the heat deposition as a function of the shell number.
Figure 8 shows the total power deposited (circles) and
the power deposited in the central 0.4m of the central
superconducting core (triangles) for each of the shells shown
in figure 7(b) assuming that the same total power is emitted
from each shell. One by one, the volumes between adjacent
elliptical toruses are assumed to generate all the neutrons, so
that it is possible to compare how the heat deposition varies.
Figure 7. (a) The Miller [11] representation of the plasma flux surfaces for a plasma of major radius 1m, superconducting core radius
rsc = 0.15 m, and shield thickness d= 0.15 m plasma. (b) By choosing elliptical cross-sections with the same minimum radius as on the
right, a very similar distribution is obtained, especially for the inner elliptical toruses which contain the bulk of the emission.
Figure 8. The heat deposition in MW into the full length of the
superconducting magnet core region (Ptot), and into the inner
mid-plane 0.4 m high section (Pcc). The full lines and points show
the eight shells of increasing radii. The crosses Wcc and Wtot show
the sums of these values weighted by their emission intensity. Note
that, for this particular case of R0 = 1.0 m, rsc = 0.15 m and
d = 0.15 m taking the central shell approximation underestimates
the weighted full-length total deposited power by 2.3%. However,
for the mid-plane region it is overestimated by 0.8%.
The point at '0' corresponds to all the emission taking place
close to a ring at the major radius. It is seen that the total
power increases as the shell number increases towards the outer
boundary, while the power into the central region of the column
decreases. However the changes become quite small when
weighted according to a typical emission profile. The cross
and plus signs indicate the heat deposition when an average
of the eight shells weighted according to their intensity profile
is taken. The weighted total heat deposition is within 1% of
the emission from the first elliptical torus used in the earlier
computations.
The set of toroidal shells may be used to evaluate the power
deposition from any radial profile to within the approximation
inherent in having only eight shells. As an example, figure 9
shows the averaged heat deposition as a function of the width
constant in the exponential function describing the radial
source profile. For low widths the averaged heat deposition
approaches the same constant value as given by the line source.
At high widths it again approaches a constant value close to
that given by the outer shells. However the variation between
these two limits is only about 15% so the effects of profile
shape are not very large.
Figure 9.
The total deposited power Ptot in MW as a function of the
width of the exponential profile function for the same conditions as
figure 8. The cross marks the width used in the other computations.
6. The choice of shield material
Several possibilities for the choice of shielding material were
considered. Table 4 below shows the summary of results for
the case of 0.15 m core radius, shielding thickness 0.15m and
major radius of 1m with a fusion power of 174 MW. The table
shows the neutron mean free path given by MCNPand the total
deposited power.
Ignoring the shield and replacing it by a void has been
suggested, reasoning that the very fast 14.1 MeV neutrons
have a fairly long mean free path (about 34 mm) in the largely
copper core region and so will deposit little power. This is not
confirmed. The copper region absorbs a considerable fraction
of the power.
Similarly it has been suggested that awater or hydrocarbon
shield (oil or polythene) will work by moderating the fast
neutrons to thermal energies where they can be easily captured.
The cross section of these materials is quite low in the slowing
downMeVrange, and thermalization hardly occurs. Water and
hydrocarbons are good moderators but are poor at fast neutron
shielding.
Table 4. The neutron mean free path and the total deposited power
in the central superconducting core for the case of core radius
rsc = 0.15 m shield thickness d= 0.15 m, major radius R0 = 1 m
and fusion power 174 MW.
| Material | Average | Mean | Deposited |
| - | density | free path | power |
| - | (g cm-3) | (mm) | (MW) |
| Void shield | 0.00 | 8 | 2.13 |
| Water-only shield | 1.00 | 31 | 1.67 |
| Hydrocarbon, CH2 | 0.80 | 23 | 1.35 |
| MnI2, EuI2 mixture | 5.26 | 64 | 1.23 |
| Zirconium borohydride, Zr(BH4)4 | 1.18 | 48 | 1.10 |
| Tungsten carbide and water | 11.83 | 19 | 0.49 |
Cross-section data suggest that elements with a large
cross section in this slowing down region include manganese,
europium and iodine. A 50% mixture by volume of europium
iodide, EuI2 and manganese iodide, MnI2 was tried but the
mean free path remains long and the power deposition in the
central column high. Better performance was obtained with
zirconium borohydride, Zr(BH4)4.
The last combination tried was a mixture of 66% tungsten
carbide, 7.2% iron and 1.8% nickel with 25% cooling water.
This gave the shortest mean free path and the best fusion power
absorption of all the materials tried. The high atomic number
of tungsten (74), its many absorption resonances in the keV
range and the high density of tungsten carbide (15.63 g cm-3)
are likely to be the key to its good performance. This material
has also been noted by Hong et al [12] as the best material for
neutron shielding from the variety of materials they considered.
7. Interpolation between the corrected results using
linear least-squares fitting
The computations presented in section 3 have been
approximately corrected for the effect of the outer shield
discussed in section 4 and for the use of one elliptical torus
rather than a weighted sum discussed in section 5 and fitted
to a more precise linear interpolation than that given in (2).
For arbitrary values of the parameters (R0,rsc, d),
first the nearest of the computed array points
(R0i ,rsci ,di ) is found. A
linear interpolation is then made, similar to that of (2) but using
local values for the gradients with respect to
R0,rsc and d and
starting from the nearest computed point. The reliability of
these fits can be found by dividing the dataset of computed
values into a 50% training and 50% testing set, and using the
training set for performing the linear interpolation constants,
and comparing the interpolated values at the test points with
their actual values. A scatter plot of the predicted logarithm of
the heat deposition against actual values is shown in figure 10.
The test points fit to the computed values to within 2.2%. This
procedure was written into a Visual Basic macro, which has
been incorporated into the systems code described by Costley
et al [2].
Figure 10.
A scatter plot of the interpolated natural logarithms
loge Pinterp of the power deposition compared with those computed
using the MCNP code loge Pmcnp. The differences are only
about 2%.
8. The cryogenic power needed to remove the
calculated heat load
The heat deposition into the superconducting central core of
any power plant will have to be removed actively by, for
example, liquid neon, as assumed in the computations, or
more conventionally by cold helium gas from a cryoplant.
Computation shows that for the 1.35m major radius pilot plant
the heat deposition increases by 12.4% when helium gas is
substituted for the neon liquid. An important factor in any
plant optimization will be the electrical power needed to run
the cryoplant. Two important parameters in this calculation
are the desired temperature of the superconducting core Tcold,
say 20 K, and the temperature of the hot cooler into which the
heat will be dumped Thot, say 300 K. The Carnot coefficient
of performance, CoP is the maximum efficiency possible in
such a process and is given by
CoP = Tcold/(Thot - Tcold).
The Carnot coefficient of performance must be multiplied
by the practical efficiency of any given cryoplant. Strobridge in
1974 collected together many of the efficiency measurements
made at that time [13]. A clear result, which has not been
altered by more recent data, is that the efficiency depends
mostly on the capacity of the cryogenic unit and not particularly
on the temperature difference. His results can be well
fitted by the line
npractical =
(6.5 log10 E + 16)% where E is the deposition power in kW.
The overall efficiency
nl =
CoP npractical
will be the product of the Carnot coefficient
of performance and the practical efficiency. This number is
also incorporated into the Costley et al system code.
As an example the pilot plant considered by Costley et al
with major radius R0 = 1.35 m,
core radius rsc = 0.23 m and
shield thickness d= 0.37 m has a heat deposition of 29 kW.
Using the results from the system code with these parameters
it is seen that the Carnot efficiency with the conductors at 20K
and the heat sink at 320K is 0.066. The Strobridge efficiency
for 29 kW is 0.265 so the overall efficiency is 1.77%, and the
cryogenic power requirement is 1.66 MW.
Figure 11.
The specific heat of 316 stainless steel and copper over
the temperature range from 10 to 100 K. On this log-log plot the
curves in the region relevant for high temperature superconductors,
say 20K to 60K are almost identical and can be adequately fitted by
the power law Cv = C0(T /Ti)p.
9. The adiabatic rate of temperature rise in a
superconducting core
In some experimental pilot power plants, it may not be
necessary to remove all the heat deposited into the central
superconducting core at the rate at which it is delivered.
Rather, for short experiments with pulse lengths of the order of
the current equilibration time, typically ten times the plasma
energy confinement time, it may be possible to use the thermal
inertia of the large mass of the core to limit the temperature
rise during the experimental period to an acceptable value.
In addition there are also the plasma build-up and ramp-down
times lasting another ten or so confinement times, during which
there will be some power deposition. For conventional lowtemperature
superconductors even a modest temperature rise of
say 10 degrees is likely to be unacceptable, but this is no longer
true for high-temperature superconductors which can be cooled
to many degrees below their transition temperature. The key
parameters are the mass of the core, the initial temperature Ti
of the superconducting core, the final temperature Tf which
the core can be allowed to rise to, and the specific heat Cv of
the core material around the temperature of interest. For this
calculation, the core material is likely to be mostly copper with
stainless steel strengthening and cladding. Figure 11 shows
the specific heats of 316 stainless steel and OFCH copper as a
function of temperature [14].
It is seen that in the important temperature range of
between 20K and 60K they are very similar and follow
roughly a power law curve
Cv = C0(T /c)p
where C0 = 10 J kg-1K-1,
Ti = 20K and p = 2.4. This line is shown by
the dotted curve in figure 11.
In a demonstration experiment, steady-state pulses at full
power are not foreseen but the pulse duration should be long
enough that a stable plasma exists for at least a few plasma
confinement times. The relevant time for establishing steady-state
plasma conditions is the current equilibration time, which
is typically 10 to 20 times the energy confinement time. In this
case the heat deposition may be much larger than the cryogenic
cooling rate so that the process is essentially adiabatic; there is
negligible time for further cooling or heat loss by conduction,
and all the deposited heat is used in raising the temperature
of the central core. Suppose also that the temperature of
the superconducting core is initially at Ti = 20K and will
be allowed to rise to say Tf = 30K during the experiment.
During a short time dt the temperature T will rise by a small
increment dT given by
dT =
Ptotdt/(CvM). (3)
where Cv is the specific heat in JK-1 kg-1,
Ptot is the power level in watts and M is the mass in kg.
The time interval tadi
during which the temperature rises from Ti to Tf
is the integral
tadi = (M/Ptot)∫
TiTf Cv(T)dT
=(M/Ptot)∫Ti
Tf C0(T/Ti)pdT
=[(MC0)/(PtotTi)p]
[Tfp+1 - Tip+1]. (4)
It is thus proportional to the heat capacity constant C0 , the mass
M of the central column and the difference of the (p + 1)th
powers of the upper and lower temperatures. It is inversely
proportional to the pth power of the starting temperature and
the deposited power.
The last column in table 1 shows this time for a variety
of conditions. This calculation has also been incorporated
into the Costley et al systems spreadsheet. For their pilot
plant conditions the time to increase the superconducting core
temperature from 20 to 30K is 88 s, which corresponds to some
282 ITER scaling (98y2) confinement times. This time should
be sufficient to enable an adiabatic experiment to be completed
without difficulty.
10. Discussion
The heat deposition into the central superconducting core of
radius rsc of a tokamak of major radius
R0, with a shield of
thickness d is presented over a wide range of these variables.
For all three variables the heat deposition varies nearly
exponentially over the ranges considered, so that interpolation
can be carried out to within a 2% accuracy. The inclusion
of an outer shield and the likely effects of plasma profile are
shown not to have a major influence. Tungsten carbide with
water cooling is shown to combine good neutron and gamma
ray shielding properties because of its high density, atomic
number and good moderating properties.
For the R0 = 1.35 m pilot plant considered by Costley
et al [2] the heat deposition is 29 kW which could be removed
by a 1.7 MW cryoplant. Alternatively, the comparatively large
mass of the central core could enable an adiabatic experiment
to be performed where a core temperature rise from 20 to
30 K could take place over a time span of 88 seconds or 282
confinement times.
Figure 12.
Examples of the use of the interpolated computations to investigate pilot fusion plants of differing major radii using the Costley
et al system code [2]. The aspect ratio is fixed at 1.8 corresponding to spherical tokamaks and the toroidal field is fixed at 4 T. The system
code is used to generate the other parameters. The curves show the fusion power Pfus, the power into the superconducting core Ptot , and the
cryogenic power Pcryo. Open symbols show the adiabatic time for a temperature rise from 20 to 30 K.
The application of our computations to a pilot plant
optimization is illustrated in figure 12. This shows a series
of output parameters from the Costley et al [2] system code as
the plasma major radius is varied. The code assumes a central
superconducting core of rsc = 0.23 m for R0 = 1.35 m, scaling
in proportion to R0. The code then evaluates the core heat
deposition Ptot by interpolation, and from this the cryogenic
power Pcryo required. It is seen that even at R0 = 0.9 m the
deposited power is less than 100 kW and the cryogenic power
less than 5 MW. At still smaller plasma major radii, adiabatic
experiments become attractive. For example at R0 = 0.6 m,
the larger superconducting core temperature rise from 20 to
40 K takes 6.4 s or 54 ITER98 confinement times.
The effects of engineering features considered in the
paper of Costley et al [2] such as a 0.05 m steel centre rod,
0.03 m thermal shield and 0.05 m plasma-wall gap have been
investigated and they result in a significant 89% increase in the
heat deposition. They continue to be studied individually to
assess their relative importance and are being included into the
system code.
Acknowledgments
The authors are grateful to many colleagues, especially Alan
Costley, Mikhail Gryaznevich, Zach Hartwig, David Kingham,
George Smith, Alan Sykes, Dennis Whyte and Martin Wilson
for many helpful discussions.
References
[1] Sykes A., Gryaznevich M.P., Kingham D., Costley A.E.,
Hugill J., Smith G., Buxton P., Ball S., Chappell S. and
Melhem Z. 2014 Recent advances on the spherical tokamak
route to fusion power IEEE Trans. Plasma Sci. 42 482-8
[2] Costley A.E., Hugill J. and Buxton P. 2014 On the power and
size of tokamak fusion pilot plants and reactors Nucl.
Fusion at press
[3] Stambaugh R.D. et al 2011 Fusion nuclear science facility
candidates Fusion Sci. Technol. 59 279-307
[4] Hartwig Z.S., Haakonsen C.B., Mumgaard R.T. and
Bromberg L. 2012 An initial study of demountable
high-temperature superconducting toroidal field magnets
for the Vulcan tokamak conceptual design Fusion Eng. Des.
87 201-14
[5] El-Guebaly L.A. and the ARIES Team 2003 ARIES-ST
nuclear analysis and shield design Fusion Eng. Des.
65 263-84
[6] Shultis J.K. and Faw R.E. 2011 An MCNP primer Technical
Report Kansas State University and Manhattan
http://mne.ksu.edu/~jks/MCNPprmr.pdf
[7] A general Monte Carlo N-particle (MCNP) transport code
https://mcnp.lanl.gov/
[8] Eisterer M. 2014 High fluence neutron irradiation of coated
conductors Keynote Lecture: Workshop on Radiation Effects
in Superconducting Magnets and Materials (Wroclaw,
Poland, 2014) https://indico.fnal.gov/getFile.py/access?,
sessionId=40&resId=0&materialId=1&confId=7702
[9] Windsor C.G. 2014 A linear path approximation for the pilot
plant neutron shield Tokamak Energy Operations Note:
Neutron Shielding (Abingdon: Tokamak Energy, Culham
Innovation Centre)
[10] Wilson H.R. 1994 SCENE-simulation of self-consistent
equilibria with neoclassical effects Technical Report
FUS-271, UK Atomic Energy Authority (Wetherby: The
British Library Document Supply Centre)
[11] Miller R.L., Chu M.S., Greene J.M., Lin-Liu Y.R. and
Waltz R.E. 1998 Noncircular, finite aspect ratio, local
equilibrium model Phys. Plasmas 5 973-8
www.frc.gatech.edu/rotation2/millerequilibriumpop.pdf
[12] Hong B.G., Hwang Y.S., Kang J.S., Lee D.W., Joo H.G. and
Ono M. 2011 Conceptual design study of a superconducting
spherical tokamak reactor with a self-consistent system
analysis code Nucl. Fusion 51 113013
[13] Strobridge T.R. 1974 Cryogenic refrigerators, an updated
survey Technical Report Tech. Note 655, National
Bureau of Standards https://archive.org/download/
cryogenicrefrige655stro/cryogenicrefrige655stro.pdf
[14] Marquardt E.D., Le J.P. and Radebaugh R. 2002 Cryogenic
material properties database 11th Int. Cryocooler Conf.
(Keystone, CO, USA, June 2002) pp 681-7 National
Institute of Standards, Technology, Springer
http://cryogenics.nist.gov/Papers/ Cryo Materials.pdf
