How to Build a TF Coil Algorithm By: Z. Dragojlovic, 8/2010
Input Parameters
1. BT – Derived from the plasma physics that corresponds to the selected operating point. 2. o - Magnetic permeability. 3. NTF - Number of coils, a design based decision. ARIES-AT has 16 coils, for example. 4. R – plasma major radius. 5. a – plasma minor radius. 6. Inboard radial build from the plasma to the TF coil, including the radial thicknesses of the power core elements and the gaps in between. 7. Materials that compose the TF coil. Here we consider YBCO at 75 K, Nb3Sn at 4.2 K, NbTi at 4.2 K and NbTiTa at 2 K.
Constraints
1. As an initial constraint, reject all the physics data points that do not satisfy the following: max BT < 21T.
2. Maximum current density. J max is located at the inboard region of the TF coil and determines the cross-sectional area of the magnet. 3. Sheath provides the structural support for the winding pack and its thickness is
determined so that s Ј s sheath , where the s sheath depends on the material used. A stainless steel casing provides the structural support for the entire TF coil and is designed so that
s < s m , where m is the allowable stress in the stainless steel.
Test Point
At each step of the procedure and for a selected test point with
BT = 5 T (toroidal magnetic field) R = 5.7 m (plasma major radius) a=1.425m (plasma minor radius) n
d i =1.185 m (inboard radial build from plasma to the TF coil) еi=1 calculate all the TF coil dimensions or attributes and verify that the numbers make sense. Procedure
1. Find the maximum magnetic field, which is located on the inner surface of the TF coil and closest to the z-axis, as shown in Fig. 1.
BT 2pR Ntotal ITF = - Number of amp-turns needed in order to generate the field BT. Note mo
that Ntotal is the total number of turns which consists of the number of
turns per coil ( Ntc ) times the number of TF coils ( NTF ).
n inner RTF = R - a - d i - Inboard radius of the TF coil that corresponds to its inner surface. еi=1 The inboard thicknesses of all the power core elements enclosed
within the TF coil are denoted by i, including the gaps in between.
max mo Ntotal ITF BT = inner - Maximum magnetic field. Reject each operating point that yields 2pRTF max BT і 21T.
Test Point Check
5Ч2p Ч5.7 8 Ntotal ITF = -6 A =1.425Ч10 A - order of magnitude agrees with Leslie’s 1.2566Ч10 calculations.
inner RTF = 5.7m -1.425m -1.185m = 3.09m
-6 8 max 1.2566Ч10 Ч1.425Ч10 BT = T = 9.2236T - falls within the range of data given in 2p Ч3.09 Leslie Bromberg’s spreadsheet. For YBCO @ 75K and Nb3Sn @ 4.2 K, that range is 6-18T. For other cases, the range is 5-9T and 8-14T, depending on the temperature of the superconductor.
2. Determine the current density that corresponds to the maximum magnetic field. Leslie Bromberg provided a data sheet as a key to calculate the current density for several different superconductors. Here is the overview: Step 1. YBCO @ 75 K
J SC - Current density of the superconductor, numerically prescribed on the range max of BT from 6 T to 18 T. The curve is numerically prescribed in the data sheet, based on experimental measurements. A safety factor of 0.5 is taken into account when using these data. J J = SC Current density in copper. This insures that the thickness of copper Cu 2 will be twice as much as the thickness of the superconductor, which provides the protection from overheating the superconductor.
Material composition:
fHe = 0.15 - Helium fraction.
max 2 (BT ) d sheath = - Fractional thickness of sheath, where 2mos sheath 2 s = 1.6 109 m = 4p10 -7 T sheath Ч Pa is the stress in sheath and o Pa is the magnetic permeability of the conductor.
2 f sheath = 1- (1- d sheath ) - Sheath fraction in the conductor.
Current density of the composition:
1 J = strands 1 1 + - Current density in strands. J SC J Cu
J effective = J strands (1- f He )(1- f sheath ) - Effective current density of the composition.
Step 2. Nb3Sn @ 4.2 K max 2 max 8 JSC = [-0.0908(BT ) +1.0224BT +12.32]Ч10 - prescribed on the range of max BT from 6 T to 18 T. 2J 2t J = - Current density in copper. J2 is proportional to the heat Cu t 16 2 dissipation and is set equal to 5Ч10 As/m on the entire range of the maximum magnetic fields. The is the dissipation time and is set to 2 s. In principal, the dissipation time should be as small as possible. The value of 2s is chosen by considerations that arose from the design of the Stelarator.
Material composition:
fHe = 0.25 - Helium fraction. max 2 (BT ) d sheath = - Fractional thickness of sheath, is evaluated the same 2mos sheath 8 way as in case (a), except that s sheath= 8Ч10 Pa is the stress in sheath for this material.
2 f sheath = 1- (1- d sheath ) - Sheath fraction in the conductor.
Current density of the composition:
1 J = strands 1 1 + - Current density in strands. J SC J Cu
J effective = J strands (1- f He )(1- f sheath ) - Effective current density of the composition.
Step 3. NbTi @ 4.2 K.
The procedure is identical to the case (b), except that the current density in the superconductor JSC is prescribed by a different curve and on a different range of max BT , which is from 5 T to 9 T.
Step 4. NbTiTa @ 2 K.
max Same procedure as in (b) and (c), except that the range of BT is now 8 T to 14 T.
Test Point Check
In case of YBCO @ 75 K, 8 2 J SC = 6.75Ч10 A/ m 8 2 J Cu = 3.38Ч10 A/ m
fHe = 0.15 -2 d sheath= 2.49Ч10 -2 fsheath= 4.91Ч10 - Volume fraction of the sheath is very small, as well as the fractional thickness. In fact this material is applied in form of a tape, according to Leslie, so the numbers make sense. 8 2 J strands = 2.25Ч10 A/ m
8 2 J effective =1.6Ч10 A/ m 3. Once the effective current density of the composition is determined, it is possible to calculate the cross sectional areas and volume fractions that correspond to different materials. The cross sectional areas are given as:
ITF Aturn = total. J effective
ITF Astrands = strands. J strands
ITF Astrands+He = strands and He together. J strands (1- f He )
ITF ACu = copper. J Cu
ITF ASC = superconductor. J SC
Asheath = Aturn - Astrands+He sheath.
The value of the current that was suggested in Leslie Bromberg’s data sheet is
ITF = 40kA . This choice has to do with the ability to dissipate the energy of the magnet out at times when quench protection may be necessary. The idea for using 40 kA came from the experience with designing the Stelarator, which has 36 electrical circles and dumps 50 GJ of energy into an external resistor. The goal was to discharge the magnet as quickly as possible. Given the voltage of 20 kV, magnet energy and circuit configuration, the choice of 40 kA was necessary in order to prevent the damage to the magnet.
Ak The relevant volume fractions can be expressed as f k = , where the subscript k Aturn stands for Cu, SC or sheath (Inconel). The cross sectional area of the TF coil excluding the supporting structure is
Ntotal Aturn Aturn (Ntotal ITF ) ATF = = NTF NTF ITF
Test Point Check
-4 2 Aturn = 2.49Ч10 m -4 2 Astrands =1.78Ч10 m ; fstrands= 0.7149 -4 2 Astrands+He = 2.37 Ч10 m ; fstrands+He = 0.9518 -4 2 ACu =1.19Ч10 m ; fCu = 0.4779 -5 2 ASC = 5.93Ч10 m ; fSC = 0.2382 - area of the superconductor is roughly half of the area of the copper, as Leslie indicated. -5 2 Asheath =1.2Ч10 m ; fsheath= 0.0482 2 ATF = 0.0554m 4. Calculate the dimensions of the coil cross-section with the external structure included.
Fig 1: Horizontal cross section of the TF coil in the (r-z plane). In the vertical cross section that corresponds to the plasma major radius, t1 and t2 should be replaced by a, as given in the equations above.
Calculation of the external structure is based on the simple beam theory and consists of the following equations:
B 2 R 2 A = T 2mo
жR2 ц R2 lnз ч- R2 + R1 и R1 ш (1) S1 = A R1 (R2 - R1 )
жR2 ц R2 - R1 - R1 lnз ч и R1 ш(2) S2 = A R2 (R2 - R1 ) S1 t1 = (3) s m
S2 t2 = Ч1.5 (4) s m S R 2 + S R 2 R = 1 1 2 2 3 R (5) Alnж 2 ц з R ч и 1 ш 3 S R (R - R ) a = 2 2 2 3 (6) 8 s b R3
Here R1, R2 are inner and outer external structure radii, respectively, t1 is the thickness of the inboard side, t2 is the thickness of the outboard side, a is the thickness of the top, m is the allowable stress and the material is the stainless steel 316. The coil casing is to be assumed square in shape, as shown in Fig. 1. The side thickness of the casing is much thinner than those defined by t1, t2 and a, and can be assumed to be a couple of centimeters.
For a given plasma major radius R and a toroidal magnetic field BT, the set of equations (1)-(6) is completely determined if R1 and R2 are known. However, at this stage of the inner calculation, only the inner inboard coil radius RTF and the coil cross sectional area ATF are known, as determined in the Steps 1 and 4, respectively. Therefore, an iterative procedure can be utilized as following: Step 1. Assume a value of the outer external structure radius R2 that satisfies maintenance requirements: R = R + a + d + d 2 е outboard ma int enance Step 2. Make an initial guess for the casing inboard, outboard and top thicknesses t1, t2, a, then calculate the inner external structure radius R1 from the following equations:
ATF = d TF wTF ,
2pR1 wTF = - b NTF inner d TF = RTF - t1 - R1
where TF is the thickness of the conductive part of the TF coil in the r
direction and wTF is the thickness of the same part in the direction. ATF inner and RTF are known from previous paragraphs of this document and NTF = 16. Step 3. Iterate the equations (1)-(6) until the t1, t2 and a converge within some small prescribed error, such as 1%. Test Point Check 5. Determine the shape of the coil in the r-z plane.
Chapter 3 of the reference [1] describes several design concepts of the TF magnet for fusion. The authors indicate that a minimum stress configuration of the coil would be the one that enforces pure tension along its perimeter. A coil shape that utilizes pure tension is known as “Princeton-D” and can be described as:
3 d 2 z 1 й dz 2 щ2 r = ұ 1+ ж ц (1) dr2 k к зdrч ъ лк и шыъ where k is a dimensionless constant proportional to the tension force in the coil, T, such 4pT that k = 2 . Assuming that the sheath will carry the tension, the value of T can be mo NTF ITF
T 9 determined from the condition Ј s sheath = 1.6 Ч10 Pa . Another constraint to Asheath consider would be the design of the maintenance port since a smaller T would mean the smaller outboard radius of the TF coil.
The equation (1) was coded in and the Princeton-D shape was numerically generated within a negligible CPU time, as shown in Figure 2.a. Comparison with parts b and c of the Figure 2 shows that the Princeton-D shape matches the current ARIES-AT design much better than the new systems code does.
6. Costing.
Known parameters:
total TF coil geometry Ю total volume VTF , obtained by numerical integration.
Material composition Ю volume fractions f SC , f Cu , f sheath , f He , obtained as shown in sections 1-3 of this document.
Material densities Ю r SC , rCu , r sheath , available from the previous systems studies, may need an update for some of the superconductors considered here.
Costs per unit mass Ю CSC , CCu , Csheath , need to verify that they are up to date.
Total cost:
total total CTF = ( fSC r SCCSC + fCur CuCCu + fsheath r sheathCsheath )VTF , to be included in the account 22.1.3, reference [2]. The cost of helium, as a primary coolant should be included in the account 22.2.1.
Figure 1. A sketch of the TF coil cross section in the r-z plane, depicting the location of max inner the maximum toroidal magnetic field BT , inboard radius of the inner coil surface RTF outer and the inboard radius of the outer coil surface RTF .
Figure 2. Three different TF coil profiles. A constant tension profile “Princeton-D” is depicted in part a. This profile was generated numerically, by solving the equation (1) at 100 locations evenly distributed along the perimeter. The TF coil profile used in the ARIES-AT CAD drawings is shown in part b. An approximation of the TF coil profile used in the new systems code is shown in part c. The new and improved TF coil algorithm will use the shape depicted in a. References
[1] Richard J. Thome, John M. Tarrh, “MHD and Fusion Magnets – Field and Force Design Concepts”, John Wiley & Sons, 1982. [2] Charles G. Bathke, Robert A. Krakowski, Ronald L. Miller, Kenneth A. Werley, “ARIES II- IV Systems Studies”, unpublished.