3 September, 2002

TO: DISTRIBUTION\
FROM: C NEUMEYER\
SUBJECT: CTF DESIGN POINT STUDIES

This memo provides documentation of the results of design point studies
for a Component Test Facility (CTF) along with a description of the
methodology used.

This work is acknowledged to rely heavily on the physics algorithms[^1]
developed by C. Kessel and S. Jardin along with numerous technical
discussions with M. Peng and C. Kessel.

The 0-d physics algorithms along with various physics constraints and
engineering equations were programmed into an EXCEL spreadsheet, and the
EXCEL Solver was used to arrive at design point optimums, given
optimization criteria and constraints. The near term objective was to
characterize possible CTF design points. The long term objective was to
identify the important issues, and to develop and document a means for
continuation of the ST development path studies.

On the engineering side, the following areas deserve attention in the
future:

1)  Develop a MATLAB version of the calculations.

2)  Perform more detailed studies of the TF inner leg, including first
    wall and joints to current return path. Investigate options for
    first wall and copper alloy materials, cooling passage design,
    fabrication technique, etc.

3)  Study options for the TF power supply including homopolar generator.

4)  Determine PF requirements, investigate options for PF coils, and
    adjust power consumption accordingly.

5)  Develop concept design for first wall/blanket/shield, perform
    neutronics calculations and/or adapt results of prior calculations
    by others to improve modeling of heat loads and first
    wall/blanket/shield geometry assumptions.

6)  Refine estimates of efficiency and power consumption of the various
    elements of the power supply, heating, power conversion, and balance
    of plant systems, over the range of typical to high-efficiency
    systems.

### Summary of Results

Several optimization scans were performed using physics constraints
derived from two separate sources, namely J. Menard[^2] and C. Wong[^3].
The Menard limits are based on ideal theoretical MHD stability whereas
the Wong limits are derived from DIID experience. The scans performed
are summarized in Table 1.

Table 1: Summary of Cases

+--------+---------+--------------+-------+--------+---------+-------+
| Case   | Elec    | Optimization | R~    | A      | N~wall~ | HH    |
|        | tricity |              | 0~(m) |        |         |       |
|        | Gen     | Criteria     |       |        | (M      |       |
|        | eration |              |       |        | W/m^2^) |       |
+--------+---------+--------------+-------+--------+---------+-------+
| 1      | No      | Maximize     | 1.5   | 1      | 1.5     | 1.8   |
|        |         | neutr        |       | .4-2.0 |         |       |
|        |         | ons/sec/watt |       |        |         |       |
+--------+---------+--------------+-------+--------+---------+-------+
| 2      | No      | Maximize     | 1.5   | 1.5    | 1.5     | 1.    |
|        |         | neutr        |       |        |         | 4-1.8 |
|        |         | ons/sec/watt |       |        |         |       |
+--------+---------+--------------+-------+--------+---------+-------+
| 3      | Yes     | Maximize net | 2.5   | 1      | 2.5     | 1.8   |
|        |         | electric     |       | .4-2.0 |         |       |
|        |         | power        |       |        |         |       |
+--------+---------+--------------+-------+--------+---------+-------+
| 4      | Yes     | Maximize net | 2.5   | 1.5    | 2.5     | 1.    |
|        |         | electric     |       |        |         | 4-1.8 |
|        |         | power        |       |        |         |       |
+--------+---------+--------------+-------+--------+---------+-------+

Results are summarized in the following figures and tables.

Case 1: Optimized for n/sec/watt R=1.5m, Nwall=1.5MW/m^2^, Varying A

![](media/image1.pct)

Fig. 1: Neutron Production Efficiency vs. A

![](media/image2.pct)

Fig. 2: Electrical Demand vs. A

Table 2: Design Points based on Menard Constraints (Optimize n/sec/watt,
Varying A)

  ------------------ ---------- ---------- ---------- ---------- ---------- ---------- ----------
  R0                 1.500      1.500      1.500      1.500      1.500      1.500      1.500

  A                  1.400      1.500      1.600      1.700      1.800      1.900      2.000

  kappa              3.612      3.489      3.382      3.288      3.204      3.128      3.060

  delta              0.600      0.600      0.600      0.600      0.600      0.600      0.600

  qMHD               12.348     9.540      7.962      6.978      6.315      5.841      5.486

  qcyl               2.94       2.80       2.72       2.66       2.63       2.60       2.58

  qcyl/qcyl(A)       100.0%     100.0%     100.0%     100.0%     100.0%     100.0%     100.0%

  Bt                 1.527      1.651      1.791      1.939      2.095      2.260      2.432

  fH20               11.0%      8.7%       7.5%       6.7%       6.1%       5.7%       5.4%

  TH20max            112.5      88.8       77.5       71.1       67.1       64.3       62.3

  TCumax             143.2      106.1      91.5       83.0       77.6       73.8       71.1

  Sigmax             48.6       38.4       33.8       31.6       30.8       30.9       31.5

  Javgtf             3.38E+07   2.46E+07   2.00E+07   1.73E+07   1.56E+07   1.45E+07   1.37E+07

  W_tf               315.8      352.9      398.9      451.6      511.1      578.1      653.4

  Ip                 13.958     12.930     11.999     11.150     10.394     9.734      9.160

  Beta_N             6.84%      6.58%      6.35%      6.18%      6.03%      5.89%      5.74%

  Beta_N/Beta_N(A)   78.3%      76.9%      76.4%      76.5%      77.0%      77.4%      77.5%

  Beta_T             58.4%      51.5%      45.4%      40.3%      35.9%      32.1%      28.8%

  Beta_P             140.7%     138.3%     138.1%     140.0%     142.7%     145.5%     147.9%

  xne                2.12E+20   2.11E+20   2.17E+20   2.28E+20   2.43E+20   2.60E+20   2.75E+20

  fGW                60.6%      56.9%      55.5%      55.9%      57.1%      58.5%      59.6%

  alpha_n(A)         0.213      0.220      0.226      0.232      0.237      0.241      0.245

  alpha_T(A)         0.213      0.220      0.226      0.232      0.237      0.241      0.245

  fBS                76.5%      75.0%      74.9%      75.8%      77.4%      79.0%      80.4%

  Tempavg            8.860      9.182      9.268      9.155      8.941      8.718      8.528

  HH                 1.800      1.800      1.800      1.800      1.800      1.800      1.800

  Tau_E\*HH          1.480      1.346      1.256      1.195      1.153      1.121      1.095

  Q                  3.837      3.730      3.638      3.558      3.494      3.447      3.418

  P_Brem             15.9       13.5       12.3       11.7       11.4       11.3       11.1

  P_aux              47.1       44.6       42.3       40.3       38.4       36.5       34.7

  P_fusion           180.7      166.3      154.1      143.4      134.1      125.9      118.6

  P_alpha            36.1       33.3       30.8       28.7       26.8       25.2       23.7

  Eta_CD             0.22       0.23       0.23       0.23       0.22       0.22       0.21

  P_CD               47.1       44.6       42.3       40.3       38.4       36.5       34.7

  P_CD/P_aux         100.0%     100.0%     100.0%     100.0%     100.0%     100.0%     100.0%

  P_aux_input        104.6      99.1       94.1       89.6       85.3       81.2       77.1

  ∑P_tf              105.3      80.7       70.4       65.8       63.9       63.9       65.0

  P_bop              20.0       20.0       20.0       20.0       20.0       20.0       20.0

  ∑P_elec input      229.9      199.8      184.6      175.3      169.3      165.0      162.1

  P_elec gen         0.0        0.0        0.0        0.0        0.0        0.0        0.0

  P_net elec         -229.9     -199.8     -184.6     -175.3     -169.3     -165.0     -162.1

  Xnwall             1.5        1.5        1.5        1.5        1.5        1.5        1.5

  n/s/W              2.92E+11   3.06E+11   3.03E+11   2.93E+11   2.81E+11   2.68E+11   2.55E+11
  ------------------ ---------- ---------- ---------- ---------- ---------- ---------- ----------

Table 3: Design Points based on Wong Constraints (Optimize n/sec/watt,
Varying A)

  ------------------ ---------- ---------- ---------- ---------- ---------- ---------- ----------
  R0                 1.500      1.500      1.500      1.500      1.500      1.500      1.500

  A                  1.400      1.500      1.600      1.700      1.800      1.900      2.000

  kappa              3.044      2.913      2.799      2.698      2.608      2.528      2.456

  delta              0.600      0.600      0.600      0.600      0.600      0.600      0.600

  qMHD               10.580     8.755      7.678      6.980      6.500      6.152      5.890

  qcyl               2.54       2.60       2.65       2.70       2.74       2.78       2.81

  qcyl/qcyl(A)       100.0%     100.0%     100.0%     100.0%     100.0%     100.0%     100.0%

  Bt                 1.705      1.972      2.249      2.533      2.820      3.112      3.407

  fH20               11.5%      9.4%       8.3%       7.6%       7.1%       6.7%       6.4%

  TH20max            111.8      91.8       81.4       75.3       71.5       69.1       67.4

  TCumax             150.0      115.0      98.2       88.9       83.1       79.6       77.0

  Sigmax             60.6       54.7       53.3       53.9       55.8       58.5       61.9

  Javgtf             3.77E+07   2.94E+07   2.51E+07   2.26E+07   2.10E+07   1.99E+07   1.92E+07

  W_tf               360.8      461.9      578.5      709.3      854.6      1014.2     1189.2

  Ip                 13.182     12.004     10.984     10.086     9.295      8.600      7.989

  Beta_N             6.76%      6.23%      5.83%      5.53%      5.31%      5.14%      5.00%

  Beta_N/Beta_N(A)   84.4%      82.0%      80.5%      80.0%      80.0%      80.5%      81.1%

  Beta_T             48.8%      37.9%      30.4%      25.0%      21.0%      18.0%      15.6%

  Beta_P             120.2%     121.3%     123.6%     126.9%     131.0%     135.7%     140.6%

  xne                2.27E+20   2.29E+20   2.37E+20   2.48E+20   2.63E+20   2.80E+20   2.98E+20

  fGW                69.7%      67.5%      67.0%      67.7%      69.4%      71.6%      74.1%

  alpha_n(A)         0.250      0.250      0.250      0.250      0.250      0.250      0.250

  alpha_T(A)         0.250      0.250      0.250      0.250      0.250      0.250      0.250

  fBS                75.7%      73.6%      72.4%      71.9%      72.0%      72.5%      73.1%

  Tempavg            8.632      8.863      8.940      8.890      8.758      8.582      8.394

  HH                 1.800      1.800      1.800      1.800      1.800      1.800      1.800

  Tau_E\*HH          1.273      1.134      1.030      0.947      0.879      0.822      0.773

  Q                  3.354      3.155      2.974      2.807      2.654      2.516      2.391

  P_Brem             15.5       13.4       12.1       11.3       10.8       10.5       10.3

  P_aux              50.4       49.2       48.2       47.5       46.8       46.3       45.8

  P_fusion           169.2      155.3      143.5      133.2      124.3      116.4      109.5

  P_alpha            33.8       31.1       28.7       26.6       24.9       23.3       21.9

  Eta_CD             0.22       0.22       0.22       0.22       0.22       0.21       0.21

  P_CD               50.4       49.2       48.2       47.5       46.8       46.3       45.8

  P_CD/P_aux         100.0%     100.0%     100.0%     100.0%     100.0%     100.0%     100.0%

  P_aux_input        112.1      109.4      107.2      105.5      104.0      102.8      101.7

  ∑P_tf              114.7      100.4      96.4       96.8       99.5       103.8      109.2

  P_bop              20.0       20.0       20.0       20.0       20.0       20.0       20.0

  ∑P_elec input      246.7      229.8      223.6      222.3      223.6      226.6      230.9

  P_elec gen         0.0        0.0        0.0        0.0        0.0        0.0        0.0

  P_net elec         -246.7     -229.8     -223.6     -222.3     -223.6     -226.6     -230.9

  Xnwall             1.5        1.5        1.5        1.5        1.5        1.5        1.5

  n/s/W              2.56E+11   2.49E+11   2.33E+11   2.16E+11   1.98E+11   1.82E+11   1.66E+11
  ------------------ ---------- ---------- ---------- ---------- ---------- ---------- ----------

Case 2: Optimized for n/sec/watt R=1.5m, Nwall=1.5MW/m^2^, Varying HH

Table 4: Menard /Wong Constraints (Optimize n/sec/watt, Fixed A, Varying
HH)

  ------------------ ---------- ---------- ---------- ---------- ---------- ----------
                     Menard     Menard     Menard     Wong       Wong       Wong

  R0                 1.500      1.500      1.500      1.500      1.500      1.500

  A                  1.500      1.500      1.500      1.500      1.500      2.000

  kappa              3.489      3.489      3.489      2.913      2.913      2.456

  delta              0.600      0.600      0.600      0.600      0.600      0.600

  qMHD               9.540      9.540      9.540      8.755      8.755      5.890

  qcyl               2.80       2.80       2.80       2.60       2.60       2.81

  qcyl/qcyl(A)       100.0%     100.0%     100.0%     100.0%     100.0%     100.0%

  Bt                 1.651      1.795      1.997      1.972      2.011      4.002

  fH20               8.7%       9.2%       9.9%       9.4%       9.5%       7.2%

  TH20max            88.8       92.6       98.4       91.8       92.8       72.4

  TCumax             106.1      111.9      121.3      115.0      116.8      83.9

  Sigmax             38.4       45.4       56.1       54.7       56.9       85.4

  Javgtf             2.46E+07   2.68E+07   2.98E+07   2.94E+07   3.00E+07   2.26E+07

  W_tf               352.9      417.8      518.2      461.9      480.4      1651.2

  Ip                 12.930     14.056     15.639     12.004     12.240     9.387

  Beta_N             6.58%      5.38%      4.28%      6.23%      6.24%      3.45%

  Beta_N/Beta_N(A)   76.9%      63.0%      50.1%      82.0%      82.1%      55.9%

  Beta_T             51.5%      42.1%      33.5%      37.9%      38.0%      10.8%

  Beta_P             138.3%     113.1%     90.0%      121.3%     121.6%     96.8%

  xne                2.11E+20   1.79E+20   1.60E+20   2.29E+20   2.75E+20   2.40E+20

  fGW                56.9%      44.4%      35.7%      67.5%      79.3%      50.9%

  alpha_n(A)         0.220      0.220      0.220      0.250      0.250      0.250

  alpha_T(A)         0.220      0.220      0.220      0.250      0.250      0.250

  fBS                75.0%      61.4%      48.8%      73.6%      73.7%      50.3%

  Tempavg            9.182      10.469     11.495     8.863      7.691      9.865

  HH                 1.800      1.600      1.400      1.800      1.600      1.400

  Tau_E\*HH          1.346      1.066      0.900      1.134      0.962      0.509

  Q                  3.730      2.988      2.483      3.155      2.254      1.605

  P_Brem             13.5       10.4       8.7        13.4       17.8       7.3

  P_aux              44.6       55.7       67.0       49.2       68.9       68.2

  P_fusion           166.3      166.3      166.3      155.3      155.3      109.5

  P_alpha            33.3       33.3       33.3       31.1       31.1       21.9

  Eta_CD             0.23       0.26       0.29       0.22       0.19       0.25

  P_CD               44.6       55.7       67.0       49.2       68.9       68.2

  P_CD/P_aux         100.0%     100.0%     100.0%     100.0%     100.0%     100.0%

  P_aux_input        99.1       123.7      148.9      109.4      153.1      151.5

  ∑P_tf              80.7       95.0       117.4      100.4      104.4      148.3

  P_bop              20.0       20.0       20.0       20.0       20.0       20.0

  ∑P_elec input      199.8      238.7      286.3      229.8      277.4      319.8

  P_elec gen         0.0        0.0        0.0        0.0        0.0        0.0

  P_net elec         -199.8     -238.7     -286.3     -229.8     -277.4     -319.8

  Xnwall             1.5        1.5        1.5        1.5        1.5        1.5

  n/s/W              3.06E+11   2.56E+11   2.13E+11   2.49E+11   2.06E+11   1.20E+11
  ------------------ ---------- ---------- ---------- ---------- ---------- ----------

Fig. 1: Maximum P~net~ vs. A

*Summary* *Engineering Assumptions*

1)  10 cm shielding on inner leg

2)  Glidcop inner leg at 87% IACS conductivity

3)  15% coolant fraction in inner leg

4)  Radius of ends of inner leg flared beyond radius of central region
    by ∆R = (1-delta)\*a

5)  Flaring approximated by step change in inner leg radius beginning at
    Z=+/- 0.9\*kappa\*a

6)  Height of inner leg equal to kappa\*a + 4.0m

7)  Current return via aluminum VV shell

8)  Outer shell horizontal to R=R0+a, then angled toward midplane ending
    at R=R0+a+2.5m, Z=+/-kappa\*a, then vertical to midplane

9)  Outer shell current density equal to 1.5kA/cm\^2 in horizontal
    section, 1.25kA/cm\^2 in angled section, and 1.0kA/cm\^2 in vertical
    section.

10) TF power supply efficiency 95%

11) Aux Heating/CD efficiency 45%

12) Electric power required for balance of plant equal to 10% of
    generated electric power

13) Superconducting PF coils (P=0, refrigeration power assumed include
    in balance of plant)

14) Neutron flux absorbed by center stack based on solid angle with
    origin at R0+a/3 (see figure following)

15) After center stack shadowing, 90% of remaining neutrons incident
    with blanket Efficiency of power conversion of thermal to electric
    power equal to 35%

> ![](media/image3.pct)
>
> Fig. 3: Fraction of 14MeV Source Neutrons Intercepted by Center Stack

## *Summary Physics Constraints*

> Table 5: Summary of Physics Constraints

+------------+----------------------------+----------------------------+
| **         | **Menard**                 | **Wong**                   |
| Variable** |                            |                            |
+------------+----------------------------+----------------------------+
| Max        | (1.46155) + (4.13281 \* ε) | 1.082+2.747/A              |
| kappa(A)   | +                          |                            |
|            |                            |                            |
|            | (-2.57812 \* ε^2^) +       |                            |
|            | (1.41016 \* ε^3^)          |                            |
+------------+----------------------------+----------------------------+
| Min        | (-0.115479) + (14.5293 \*  | 1.21+1.3\*A-0.25\*A^2^     |
| qcyl(A)    | ε) +                       |                            |
|            |                            |                            |
|            | (-27.4492 \* ε^2^) +       |                            |
|            | (18.334 \* ε^3^)           |                            |
+------------+----------------------------+----------------------------+
| alpha_n(A) | (0.64-0.3\*ε)/2            | 0.25                       |
| =          |                            |                            |
| alpha_T(A) |                            |                            |
+------------+----------------------------+----------------------------+
| Peakfactor | \[∫(1-(x)^2^)^alpha_T^(1   | \[∫(1-(x)^2^)^alpha_T^(    |
|            | -(x)^2^)^alpha_n^dx\]^-1,^ | 1-(x)^2^)^alpha_n^dx\]^-1^ |
|            | (x=r/a)                    |                            |
|            |                            | (x=r/a)                    |
+------------+----------------------------+----------------------------+
| Max        | ((6.96436) + (-14.043 \*   | (3.09+3.35/A               |
| Beta_N(A)  | ε) +                       | +3.87/A^0.5^)\*(κ/3)^0.5^/ |
|            |                            |                            |
|            | (45.5 \* ε^2^) + (-31.3086 | peakfactor^0.5^            |
|            | \* ε^3^))/100              |                            |
+------------+----------------------------+----------------------------+
| kbs        | 0.344+0.195\*A             | 0.6783+0.0446/A            |
+------------+----------------------------+----------------------------+
| fbs        | Kbs\*Beta_P\*              | Kbs\*Beta_P\*              |
|            |                            |                            |
|            | peakfactor^0.25^/A^0.5^    | peakfactor^0.25^/A^0.5^    |
+------------+----------------------------+----------------------------+
| Eta_CD     | 0.025\*\<T\>               | 0.025\*\<T\>               |
+------------+----------------------------+----------------------------+

### Detailed Methodology

Zero-d physics algorithms[^4] along with various engineering equations
were programmed into an EXCEL spreadsheet, and the EXCEL Solver was used
to arrive at design point optimums, given optimization criteria and
constraints. The following is a description of the basis for this work,
with an explanation of each of the major variables, in the same order as
the calculation sequence.

*Plasma Shape Variables (R~0~, a, A,* ε*, κ, δ)*

Plasma cross sectional shape (95% flux surface, containing 95% of the
plasma poloidal magnetic flux) is described by\...

$$\begin{array}{r}
R(q) = R_{0} + a \ast \text{cos}(q + d \ast \text{sin}(q)) \\
Z(q) = k \ast a \ast \text{sin}(q)
\end{array}$$

where (see Fig. 4) \...

R~0~ = major radius (m)

a = minor radius (m)

A = aspect ratio = *R~0~* /a

ε = inverse aspect ratio = *1/A*

κ = elongation

δ = Triangularity

θ = poloidal angle

![](media/image4.jpeg){width="2.783333333333333in"
height="2.8777777777777778in"}

Fig. 4: Plasma Cross Sectional Shape

*Elongation (κ) Aspect Ratio Dependance*

According to the MHD stability studies by Menard^1^ the maximum
elongation as a function of inverse aspect ratio is...

κ~max~(ε) = 1.46155 + 4.13281 \* ε -2.57812 \* ε^2^ + 1.41016 \* ε^3^

Similarly, according to Wong^2^ the maximum elongation as a function of
aspect ratio is...

> κ~max~(Α) = 1.082+2.747/A

A comparison of these results is given in Fig. 5.

![](media/image5.pct)

Fig. 5: Max. Elongation vs. A

# MHD Safety Factor

The MHD safety factor, number of toroidal rotations per poloidal
rotation of a field line on the 95% flux surface, is expressed in two
forms, namely q~cyl~ and q~MHD~, which are defined as follows....

$$q_{\text{cyl}} = \frac{5a^{2}B_{T}}{I_{p}R_{0}}\left\lbrack \frac{1 + k^{2}}{2} \right\rbrack$$

$$q_{\text{MHD}} = \frac{5a^{2}B_{T}}{I_{p}R_{0}}\left\lbrack \frac{1 + k^{2}\left( 1 + 2d^{2} - 1\text{.}2d^{3} \right)}{2} \right\rbrack\left\lbrack \frac{1\text{.}\text{17} - 0\text{.}\text{65}e}{\left( 1 - e^{2} \right)^{2}} \right\rbrack$$

where...

B~T~ = vacuum toroidal field at R~0~ (T)

I~p~ = plasma current (MA)

The calculations described herein rely primarily on q~cyl~ and report
q~MHD~ as an output, for information only.

*MHD Safety Factor Aspect Ratio Dependance*

According to the MHD stability studies by Menard^1^ the minimum safety
factor as a function of aspect ratio is...

q~cyl~ min(A) = 12.259-13.58\*A+6.4286\* A^2^-1.0417\* A^3^

Note: Menard prescribed β~Tmax~(ε), β~Nmax~(ε), and κ~max~(ε). The
writer extracted q~cyl~ min(A) according to ...

$$\beta_{P\text{max}} = \frac{\text{25}}{\beta_{T\text{max}}}\left\lbrack \frac{1 + k^{2}}{2} \right\rbrack\beta_{{N\text{max}}^{2}}$$

and

$$q_{\text{cyl}_{\text{min}}} = \sqrt{\frac{\beta_{P\text{max}}}{\beta_{T\text{max}}}\left\lbrack \frac{1 + k^{2}}{2A^{2}} \right\rbrack}$$

According to Wong^2^ the minimum safety factor as a function of aspect
ratio is...

q~cyl~ min(A) = 1.21+1.3\*A-0.25\*A^2^

A comparison of these results is given in Fig. 6.

![](media/image6.pct)

Fig. 6: Safety Factor vs. A

*Toroidal Field Inner Leg*

The first wall is assumed 10 cm thick, so the radial build of the TF
inner leg in the midplane region is...

$$R_{\text{TFmid}} = R_{0} - a - 0\text{.}\text{10}m$$

The radius of the inner leg in the end regions was set to...

$$R_{\text{TFend}}\text{= }R_{\text{TFmid}}\text{ +}a(1 - d)$$

The height of the inner leg above the midplane is assumed equal to the
height of the plasma plus 4.0m, allowing for divertor, blanket, and
shield...

$$Z_{\text{TF}\text{max}} = k \ast a + 4\text{.}0m$$

Glidcop AL-25 material, σ=87% IACS, was assumed for the inner leg, based
upon its radiation resistance[^5]. The inner leg is modeled using 8
sections as depicted in Fig. 8, with water flowing in one end and out
the other, heating up as the sections are traversed.

A water inlet temperature of 35^o^C and flow velocity of 10m/s were
assumed. The fraction of cross section containing water is chosen by the
optimizer. The number of water passages per unit area is assumed equal
to 500/m^2^, similar to ARIES-ST^4^, but less than prior VNS[^6] studies
which were of order 2000/m^2^. Thus with the water cross sectional area
and number of passages specified the wetted perimeter of the cooling
passages is determined.

![](media/image7.pct){width="1.3069444444444445in"
height="4.379861111111111in"}

Fig. 8: TF Inner Leg

In addition to ohmic dissipation, nuclear heating is added at a rate
equal to the average neutron wall loading times the surface area. This
is conservative because 1) much of the heating would be received and
removed via the first wall armor, and 2) the Cu is partially transparent
to the neutrons. The flaring is set to 60^o^ with respect to horizontal,
with the flaring starting at 90% of the plasma height.

The thermal resistance due to the film effect and the mass flow are
computed, using water properties which are computed as a function of
temperature. Limits on copper and water temperature were typically set
at 150^o^C.

Triaxial and combined (Von Mises) stresses in the inner leg[^7] are
computed as follows...

$$s_{\text{rr}} = \left( - \frac{B_{\text{max}^{2}}}{2m_{0}} \right)\left( \frac{3 - 2\nu}{2(1 - \nu)} \right)\left( 1 - \left( \frac{r}{a} \right)^{2} \right)$$

$$s_{\text{jj}} = \left( - \frac{B_{\text{max}^{2}}}{2m_{0}} \right)\left( \frac{3 - 2\nu}{2(1 - \nu)} - \frac{1 + 2\nu}{2(1 - \nu)}\left( \frac{r}{a} \right)^{2} \right)$$

$$s_{\text{zz}} = \left( - \frac{B_{\text{max}^{2}}}{2m_{0}} \right)\left( \frac{\nu}{1 - \nu} \right)\left( 1 - 2\left( \frac{r}{a} \right)^{2} \right)$$

$$s_{\text{VonMises}} = \sqrt{\frac{\left( s_{\text{rr}} - s_{\text{jj}} \right)^{2} + \left( s_{\text{jj}} - s_{\text{zz}} \right)^{2} + \left( s_{\text{zz}} - s_{\text{rr}} \right)^{2}}{2}}$$

where:

r = radius within conductor a = outer radius of conductor

ν = Poission's ratio = 0.34 B~max~ = B at r=a, B~ma~ = µ~0~I~TF~/2πa

Vertical tension is assumed equal to zero. However the above result was
multiplied by a factor of 2.0 to account for the cooling passages in the
conductor. An appropriate limit for Glidcop at 150^o^C is 100MPa.

*Toroidal Field Outer Leg Return*

The TF current is assumed to be returned through the outer VV shell
which is constructed of aluminum, consisting of horizontal, angled, and
vertical sections operating at current densities of 150, 125, and
100A/cm^2^, respectively. This is similar in concept to the VNS and
ARIES-ST designs. The thickness of the horizontal, angled, and vertical
sections is tailored to match the specified current density. Dimensions
of the outer leg return circuit are chosen such that a region with 2.0m
thickness is reserved for the first wall, blanket, and shield
components. See Fig. 9.

![](media/image8.jpeg){width="1.5180555555555555in"
height="3.2305555555555556in"}

Fig. 9: TF Cross Section, showing Inner Leg, Outer Leg Return,

and 2.0m Space Allocation for First Wall /Blanket/Shield

(R~0~=2.5m,A=1.5,κ=3.5,δ=0.6)

The horizontal section extends out to the plasma edge at radius R~0~+a.
The height of the vertical section is equal to the plasma height
2\*κ\*a. The radius to the centerline of the vertical section is located
2.5m beyond the plasma edge at radius R~0~+a+2.5m to allow for first
wall/blanket/shield.

# Determination of Toroidal Field, Current, Input Power

The optimizer adjusts the current density in the TF inner legs along
with the fraction of cross-sectional area containing water (f~W~) to
adjust B~T~ according to...

$$I_{\text{TF}} = J_{\text{TF}}{pR}^{2_{\text{TFmid}}}\left( 1 - f_{W} \right)$$

$$B_{T} = \frac{m_{0}I_{\text{TF}}}{2{pR}_{0}}$$

Total TF power consumption is equal to the sum of the dissipation in the
inner and outer legs divided by the efficiency of the power supply
system, assumed to be 95%\...

$$P_{\text{TF}} = \frac{P_{\text{inner}} + P_{\text{outer}}}{h_{\text{TF}}}$$

# Determination of Plasma Current

With R~0~,a,κ, and q~cyl~ determined the plasma current (in MA) is
calculated according to\...

$$I_{p} = \frac{5´\text{10}^{- 6}a^{2}B_{T}}{q_{\text{cyl}}R_{0}}\left\lbrack \frac{1 + k^{2}}{2} \right\rbrack = \frac{5´\text{10}^{- 6}R_{0}e^{2}B_{T}}{q_{\text{cyl}}}\left\lbrack \frac{1 + k^{2}}{2} \right\rbrack = \frac{{pR}_{0}e^{2}B_{T}\left( 1 + k^{2} \right)}{\text{10}^{6}q_{\text{cyl}}m_{0}}$$

# Poloidal Field Requirements

PF coils are assumed superconducting, so their input power is assumed
equal to zero. Refrigeration power requirements are assumed lumped with
balance of plant power.

#### Density and Temperature Profiles

Spatial distribution of temperature and density across the plasma cross
section is assumed to be parabolic according to...

$$\begin{array}{r}
T(r) = T_{0}\left\lbrack 1 - \left( \frac{r}{a} \right)^{2} \right\rbrack^{\alpha_{T}} \\
n(r) = n_{0}\left\lbrack 1 - \left( \frac{r}{a} \right)^{2} \right\rbrack^{\alpha_{N}}
\end{array}$$

where...

r = variable in minor radius

a = minor radius of plasma = R~0~/A

T~0~,n~0~ = central peak temperature and density at r=0

α~T~, α~N~ = shape exponents on temperature and density

It is noted that for these functions the peak to average ratios are
equal to...

$$\frac{X_{0}}{X >} = \alpha_{X} + 1$$

...and the line average to volume average ratios are equal to...

$$\frac{\overline{X}}{X >} = 1 + \frac{\alpha_{X}}{2}$$

According to Menard the appropriate values for the exponents, consistent
with his analysis, are...

$$\alpha_{\text{N }}\text{= }\alpha_{T}\text{= }\frac{(0\text{.}\text{64-0}\text{.}\text{3*}e)}{2}$$

According to the Wong assumption...

$$\alpha_{\text{N }}\text{= }\alpha_{T}\text{= }0\text{.}\text{25}$$

# Beta Limits and Aspect Ratio Dependence

According to Menard an appropriate limit on β~N~ is as follows...

$$\beta_{\text{Nmax}}(e)\text{ = }\frac{6\text{.}\text{96436-14}\text{.}\text{043}e\text{+45}\text{.}5e^{2}\text{-31}\text{.}\text{3086 }e^{3}}{\text{100}}$$

According to Wong the β~N~ limit depends on A, κ, and pressure peaking
factor. The peaking factor is as follows...

$$$\text{Peakfactor}\text{ = }\left( \int_{0}^{1}{\left\lbrack \text{1-x}^{2} \right\rbrack^{\alpha_{T}}\left\lbrack \text{1-x}^{2} \right\rbrack^{\alpha_{N}}}\text{dx} \right)^{- 1}$

...where x=r/a

Then the β~N~ limit is as follows...

$$\beta_{\text{Nmax}}(A)\text{ = }\frac{\left( 3\text{.}\text{09+}\frac{3\text{.}\text{35}}{A} + \frac{3\text{.}\text{87}}{A^{0\text{.}5}} \right)\left( \frac{k}{3} \right)^{0\text{.}5}}{\text{100} \ast \text{Peakfactor}^{0\text{.}5}}$$

A comparison of these results is given in Fig. 10.

> $$ ![](media/image9.pct){width="4.417361111111111in"
> height="2.7888888888888888in"}
>
> $$Fig. 10: Max. β~N~ vs. A

The optimizer chooses an actual value for β~N~ which must be ≤ β~Nmax~.
Then, with β~N~ and other variables previously determined, β~T~ can β~P~
can be calculated based on their definitions as follows...

$$\beta_{T} = \frac{I_{P}{A\beta}_{N}}{R_{0}B_{T}} = \frac{I_{P}\beta_{N}}{R_{0}{eB}_{T}}$$

$$\beta_{P} = \frac{2\beta_{T}q_{\text{cyl}}^{2}A^{2}}{\left( 1 + k^{2} \right)}$$

Finally, the volume averaged pressure is...

$$P\text{>=}\frac{\beta_{N}I_{P}B_{T}}{2m_{0}a}$$

# Bootstrap Current Fraction

Bootstrap current fraction f~BS~ is dependant on β~P~ and aspect ratio
as follows...

$$f_{\text{BS}} = \frac{{C\beta}_{P}}{\sqrt{A}} = {C\beta}_{P}\sqrt{e}$$

...where X is a function containing terms related to shape and profile.
In the work described herein we apply a curve fit function X =
k~BS~(A)\*Peakfactor^0.25^ which results in f~BS~ = 0.9 for the Wong
case and f~BS~ = 0.99 for the Menard case since these were the
applicable bootstrap fractions for the MHD stability analysis performed.
Therefore f~BS~ is calculated as follows...

$$f_{\text{BS}} = k_{\text{BS}} \ast \text{Peakfactor}^{0\text{.}\text{25}} \ast \beta_{P}\sqrt{e}$$

where for Menard...

$$k_{\text{BS}}(A)\text{ = 0}\text{.}\text{344+0}\text{.}\text{195*A}$$

and for Wong...

$$k_{\text{BS}}(A)\text{ = 0}\text{.}\text{6783+0}\text{.}\text{0446/A}$$

A comparison of these results is given in Fig. 11.

![](media/image10.pct){width="5.299305555555556in" height="3.34375in"}

Fig. 11: Bootstrap Coefficient vs. A

*Current Drive*

The maximum current drive efficiency is assumed to be temperature
dependant^2^ as follows\....

$$h_{\text{CD}\text{max}} = 0\text{.}\text{025} < T >$$

where\...

η~CDmax~ = maximum efficiency in units Ampere/Watt-m^2^

\<T\> = volume average temperature in keV, assume T~i~=T~e~

The current to be driven is I~p~\*(1-f~BS~), and the current drive power
(MW) requirement is\...

$$P_{\text{CD}} = \frac{n_{e}R_{0}I_{p}\left( 1 - f_{\text{BS}} \right)}{\text{10}^{\text{20}}h_{\text{CD}}}$$

...where n~e~ is the electron density. Neutral beam injection (NBI) is
assumed to provide the auxiliary heating power (P~aux~) and current
drive power. The solver is constrained in such a way that
η~CD~≤ η~CDmax~ and P~CD~ ≤  P~aux~.

*Density*

Line average electron density is related to the fraction of Greenwald
Limit f~GW~ as follows...

$$\overline{n_{e}} = \frac{f_{\text{GW}}I_{p} \ast \text{10}^{\text{20}}}{{pa}^{2}} = \frac{f_{\text{GW}}I_{p} \ast \text{10}^{\text{20}}}{{pR}_{0}^{2}e^{2}}$$

Electron density (volume average) is...

$$n_{e}\text{>=}\frac{\overline{n_{e}}}{\left( 1 + \frac{\alpha_{N}}{2} \right)}$$

Helium ash density (volume average) is calculated as follows:

For power balance, total power into the plasma equals power out ...

$$P_{\text{Tot}} = P_{\alpha} + P_{\text{aux}} - P_{\text{rad}} = P_{\alpha} + \frac{P_{\text{fusion}}}{Q} - P_{\text{rad}} = P_{\alpha}\left( 1 + \frac{5}{Q} - f_{\text{rad}} \right) = \frac{W}{t_{E}}$$

where...

P~α~ = power in α particles P~aux~ = auxiliary heating power

P~rad~ = radiated power P~fusion~ = total fusion power

Q = fusion power gain f~rad~ = radiation fraction = P~rad~/P~α~

W = stored energy τ~E~ = energy confinement time

For α particle balance, the rate of α production from fusion reactions
must equal the rate of α particle loss...

$$\frac{P_{\alpha}}{W_{\alpha}} = \frac{n_{\text{He}}}{t_{p}^{}}$$

...where

W~α~ = energy per α particle, equal to
3.52MeV\*1.6e-13Joule/MeV=5.63e-13Joule

τ^\*^~p~= α particle confinement time after adjusting for recycling

The stored energy is related to the pressure as follows...

$$W = \frac{3}{2} < P\text{>=}\frac{3}{2}\frac{\beta_{N}I_{P}B_{T}}{2m_{0}a}$$

Rearranging and combining the prior three equations yields the following
expression for helium ash density...

$$n_{\text{He}} = \frac{3}{2}\left\lbrack \frac{\beta_{N}I_{P}B_{T}}{2m_{0}a} \right\rbrack\left\lbrack \frac{t_{P}^{}}{t_{E}} \right\rbrack\left\lbrack 5\text{.}\text{63}´\text{10}^{- \text{13}}\left( 1 + \frac{5}{Q} - \text{frad} \right) \right\rbrack^{- 1}$$

The ratio τ^\*^~p~/τ~E~ is an input variable assumed equal to 5.0.

For impurities, Be is assumed the dominant species (Z~imp~=4) at a
concentration f~imp~ = 6%.

For charge neutrality, with Z=2 for He...

$$n_{e} = n_{\text{DT}} + 2n_{\text{He}} + Z_{\text{imp}}n_{\text{imp}}$$

The impurity density (volume average) is...

$$n_{\text{imp}} = n_{e} \ast f_{\text{imp}}$$

The density (volume average) of H species (50:50 DT mix) is then
calculated as follows...

$$n_{\text{DT}} = n_{e}\left( 1 - f_{\text{imp}}Z_{\text{imp}} \right) - 2n_{\text{He}}$$

The total particle density (volume average) is then...

$$$n_{\text{Total}} = n_{e} + n_{\text{DT}} + n_{\text{He}} + n_{\text{imp}} = n_{\text{DT}}\left\lbrack \frac{\left( 1 + f_{\text{imp}} \right)}{\left( 1 - f_{\text{imp}}Z_{\text{imp}} \right)} + 1 \right\rbrack + n_{\text{He}}\left\lbrack \frac{2\left( 1 + f_{\text{imp}} \right)}{\left( 1 - f_{\text{imp}}Z_{\text{imp}} \right)} + 1 \right\rbrack$

*Temperature*

With the volume average density determined the peak temperature can be
backed-out from the pressure as follows...

$$P_{0} = n_{0}T_{0}\left( 1\text{.}6´\text{10}^{- \text{19}} \right)\text{=<}P > \left( 1 + \alpha_{N} + \alpha_{T} \right)\left( 1\text{.}6´\text{10}^{- \text{19}} \right)$$

$$T_{0} = \frac{P > \left( 1 + \alpha_{N} + \alpha_{T} \right)}{n_{0}\left( 1\text{.}6´\text{10}^{- \text{19}} \right)} = \frac{P > \left( 1 + \alpha_{N} + \alpha_{T} \right)}{n > (1 + \alpha_{N})\left( 1\text{.}6´\text{10}^{- \text{19}} \right)}$$

Note: 1.6 x 10^-19^ Joule/eV

*Radiation*

The effective ionic charge Z~eff~ is...

$$Z_{\text{eff}} = \frac{\sum_{\text{ions}}^{}{Z_{j}^{2}n_{j}}}{\sum_{\text{ions}}^{}{n_{j}Z_{j}}} = \frac{\sum_{\text{ions}}^{}{Z_{j}^{2}n_{j}}}{n_{e}} = \frac{n_{\text{DT}} + 4n_{\text{He}} + Z_{\text{imp}}^{2}n_{\text{imp}}}{n_{e}} = \frac{n_{\text{DT}} + 4n_{\text{He}} + Z_{\text{imp}}^{2}f_{\text{imp}}n_{e}}{n_{e}}$$

Power loss (MW), assumed to arise exclusively from Bremsstrahlung
radiation is...

$$P_{\text{rad}}»P_{\text{brem}} = 4\text{.}8´\text{10}^{- \text{43}}Z_{\text{eff}}n_{e}^{2}\sqrt{T}a^{2}R_{0}k\left\lbrack \frac{\left( 1 + \alpha_{N} \right)^{2}\left( 1 + \alpha_{T} \right)^{\frac{1}{2}}}{1 + 2\alpha_{N} + \frac{1}{2}\alpha_{T}} \right\rbrack$$

*Alpha Power*

Alpha power is calculated per the following integral...

$$P_{\alpha} = 5\text{.}6´\text{10}^{- \text{25}}\int_{0}^{V_{p}}\left\lbrack n(V)_{\text{DT}}e^{a_{0} + a_{}T(V)^{- 0\text{.}\text{275}} + a_{1}T(V) + a_{2}T(V)^{2} + a_{3}T(V)^{3}} \right\rbrack\text{dV}$$

where...

a~0~ = -23.836 a~\*~ = -22.712 a~1~ = -0.09393 a~2~ = 7.994e-4 a~3~ =
-3.144e-6

As depicted in Fig. 12, the incremental volume ∆V is...

$$DV = k\left\lbrack 2{pR}_{0} \right\rbrack\lbrack 2pR\rbrack DR$$

where R is the radius from the plasma center at R~0.~

![](media/image11.jpeg){width="1.2493055555555554in"
height="2.8041666666666667in"}

Fig. 12: P~α~ Integration

# Plasma Stored Energy

The plasma thermal energy is ...

$$W = \frac{3}{2} < P > V = \frac{3}{2}\left\lbrack \frac{\beta_{N}\text{10}^{- 6}I_{P}B_{T}}{2m_{0}a} \right\rbrack\left\lbrack 2p^{2}R_{0}a^{2}k \right\rbrack$$

Energy Confinement Time

Energy confinement time is calculated based on the ITER 98\[y,2\]
scaling...

$$\begin{array}{r}
t_{E}^{\text{98}\lbrack y,2\rbrack} = \left\lbrack 0\text{.}\text{0562}M^{0\text{.}\text{19}}Ip^{0\text{.}\text{93}}R_{0}^{1\text{.}\text{97}}BT^{0\text{.}\text{15}}e^{0\text{.}\text{58}}k\frac{a^{0\text{.}\text{78}}{\overline{ne}}_{\text{19}^{0\text{.}\text{41}}}}{P_{\text{Tot}}^{0\text{.}\text{69}}} \right\rbrack \\
\left\lbrack \frac{0\text{.}\text{0562}M^{0\text{.}\text{19}}Ip^{0\text{.}\text{93}}R_{0}^{1\text{.}\text{97}}BT^{0\text{.}\text{15}}e^{0\text{.}\text{58}}ka^{0\text{.}\text{78}}{\overline{ne}}_{\text{19}^{0\text{.}\text{41}}}}{\left( P_{\alpha}\left( 1 + \frac{5}{Q} - f_{\text{rad}} \right) \right)^{0\text{.}\text{69}}} \right\rbrack
\end{array}$$

where ${\overline{ne}}_{\text{19}}$ is the line average electron density
in units 10^19^/m^3^ and M is the average mass number, equal to 2.5 for
a 50:50 DT mix.

A confinement enhancement factor HH is introduced and used in the power
balance equation as follows...

$$\text{HH} = \frac{W}{t_{E}^{\text{98}\lbrack y,2\rbrack}P_{\text{Tot}}} = \frac{W}{P_{\alpha}\left( 1 + \frac{5}{Q} - f_{\text{rad}} \right)}$$

So, the effective confinement time is...

$$t_{E} = \text{HH} \ast t_{E}^{\text{98}\lbrack y,2\rbrack}$$

*Neutron Wall Loading*

Neutron wall loading is based on the surface area of the plasma...

$$\begin{array}{r}
N_{\text{wall}} = \frac{P_{N}}{A} = \frac{4P_{\alpha}}{4p^{2}R_{0}a\sqrt{(1 + k/2)}} \\

\end{array}$$

*Gross Thermal Power*

Electricity production is assumed derived from the power in the fusion
neutrons and the radiated power to the first wall. This includes energy
multiplication which takes place in the blanket due to reactions
involving the neutrons and the Li isotopes Energy flow through the SOL
to the divertors, and other low grade heat such as dissipation in the
magnets is not recovered. However, a fraction of the power from the
neutrons is not captured by the blanket due to absorption by the center
stack and due to the fact that the blanket does not cover 100% of the
area of the plasma.

The fraction of neutrons incident on the narrow midplane region of the
center stack is...

$$f_{\text{CS}} = \frac{f \ast \text{sin}q}{2p}$$

where φ and θ are angles in the toroidal and poloidal planes which
correspond to the intersection of a cylinder (the center stack) and a
unit sphere with center at the effective average point source of neutron
production, which is assumed located a distance a/3 beyond R~0~. See
Fig. 13. The angles φ and θ are...

$$f = \frac{2\text{sin}^{- 1}R_{\text{fw}}}{R_{0} + a3}$$

$$q = \frac{\text{tan}^{- 1}\left( {DZ}_{\text{tfmid}}2 \right)}{\left( R_{0} + a3 \right) - R_{\text{fw}}}$$

where R~fw~ is the radius of the first wall and ∆Z~tfmid~ is the height
of the narrow midplane section of the center stack.

![](media/image12.jpeg){width="2.2583333333333333in"
height="3.7534722222222223in"}

Fig. 13: Calculation of Solid Angle of Center Stack

Then the total useful thermal power is...

$$P_{\text{th}} = P_{\text{neutron}} \ast (1 - f_{\text{CS}}) \ast f_{S} \ast f_{\text{BEM}} + \left( P_{\alpha} + P_{\text{aux}} \right) \ast f_{\text{rad}}$$

where...

f~S~ = fraction of surface area covered by blanket (assumed = 0.9)

f~BEM~ = blanket energy multiplication factor (assumed =1.2)

*Electric Power Production*

Thermal to electrical energy conversion efficiency η~EC~ is assumed 35%,
so the gross electrical power production is...

$$P_{\text{grosselec}} = \frac{P_{\text{th}}}{h_{\text{EC}}}$$

Operation of the "balance of plant" is assumed to consume a fraction
(f~BOP~=10%) of the gross electric power production, so the net electric
power is...

$$P_{\text{netelec}} = P_{\text{grosselec}} \ast (1 - f_{\text{BOP}}) - P_{\text{TFinput}} - P_{\text{auxinput}}$$

*Variables Adjustable by Solver*

Typically the Solver is set up to adjust the following variables to
obtain a solution...

> f~rad~, f~GW~, β~N~, q~cyl~, P~fusion~, η~CD~, J~TF~, f~W~

In other cases more variables may be added to this list, such as R~0~ or
A.

*Solver Constraints for Valid Solutions*

In obtaining a solution the Solver iterates to satisfy the following
constraints which are required for a mathematically valid solution:

1)  Value of f~rad~ used to calculate τ~E~, n~He~, P~tot~ and subsequent
    dependent variables must be equal to ratio of calculated values
    P~brem~/P~α~.

2)  Value of P~fusion~ used to calculate Q and subsequent dependent
    variables must be equal to calculated value 5\*P~α~.

3)  Value η~CD~ must be  ≤  η~CDmax~ (T) = 0.025\<T\>

4)  Value of P~CD~ must be ≤ P~aux~

*Other Solver Constraints*

The following additional constraints are applied to ensure that physics
and engineering limits are realistic...

1)  β~N~ ≤ β~Nmax~

2)  q~cyl~ ≥ q~cylmin~

3)  0.1 ≤ f~BS~ ≤ 0.99

4)  0.1 ≤ f~GW~ ≤ 1

5)  σ~TF~ ≤ 100MPa

6)  T~TFCu~ ≤ 150C

7)  T~TFH20~ ≤ 150C

8)  0.05 ≤ f~W~ ≤ 0.3

Additional constraints are sometimes added to limit the solution to a
particular range of interest such as:

1)  HH = or ≤  an input value

2)  N~wall~ = or ≤ an input value

[^1]: S. Jardin et al, Systems Analysis of a Compact Next Step Burning
    Plasma Experiment, PPPL-3666 (2002), to appear in Fusion Engineering
    and Design

[^2]: J. Menard e-mail correspondance of 7/23/02 and 7/25/02 and draft
    manuscript "Ideal MHD Stability Limits of Advanced Tokamak Plasmas
    and Implications for the ST"

[^3]: C. Wong et al, "Toroidal Reactor Designs as a Function of Aspect
    Ratio and Elongation", Nuclear Fusion, 42 (2002) 547-556

[^4]: S. Jardin et al, Systems Analysis of a Compact Next Step Burning
    Plasma Experiment, PPPL-3666 (2002), to appear in Fusion Engineering
    and Design

[^5]: Toroidal Field Coil Design for ARIES-ST", W Reiersen et al, 17^th^
    SOFE, San Diego

[^6]: "Special Issue: ST VNS and Applications", Fusion Engineering
    Design, Vol 45 (1999) No 3

[^7]: "Spherical Torus Center Stack Design", C Neumeyer et al, 19^th^
    SOFE