A Model for Determining Dipole, Quadrupole, and Combined Function Magnet Costs. Page: 3 of 5
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A MODEL FOR DETERMINING DIPOLE, QUADRUPOLE, AND
COMBINED FUNCTION MAGNET COSTS*
Robert B. Palmer, J. Scott Berg, BNL, Upton, NY, USA
Abstract COST MODELOne of the most important considerations in designing
large accelerators is cost. This paper describes a model
for estimating accelerator magnet costs, including their de-
pendences on length, radius, and field. The reasoning be-
hind the cost model is explained, and the parameters of the
model are chosen so as to correctly give the costs of a few
selected magnets. A comparison is made with earlier for-
mulae. Estimates are also given for other costs linearly de-
pendent on length, and for 200 MHz superconducting RE
INTRODUCTION
While no cost model for estimating magnet costs can
be very reliable, such formulae allow the optimization of
a machine design prior to engineering and real costing.
M. Green [1] has provided two such formulae that have
proved very useful in many applications. The first formula
is only depends on the magnetic stored energy U (in MJ),
and is, after correction for inflation,
Cost = 1.34 x 0.844 UO-459 M$. (1)
The expression was obtained from a fit of a number of
dipoles, solenoids, and toroids on a cost vs. stored energy
log-log plot. It's rms deviation from the dipole magnets
used in the fit is rather poor: near a factor of 4. The second
formula depends on the product Q of field and volume in
T m3. and. after correction for inflation, isCost = 1.34 x 0.77 Qo.631 M$.
This magnet model is designed to apply to superconduct-
ing magnets with circular apertures. The magnetic field
profile in the midplane is assumed to be linear, but the mag-
net is allowed to have any combination of dipole (Bo) and
quadrupole (B1) fields. The largest field value on a circular
aperture will be in the midplane, so we will use the fields
at the aperture in the midplane. We specify that the beam
itself requires an aperture of radius R. Because the field
quality adjacent to the coils is expected to be poor, a buffer
region must be built in beyond the radius that the beam re-
quires. To allow for this, we define the radius of the inside
edge of the coils to be kRR. ThusBa = JBol IBilkBR
(3)
gives the field values at the inside edge of the magnet coils.
The t refers to the absolute maximum and minimum fields
on the two sides. If a magnet has a finite field gradient B1,
and the coils have a finite thickness, then the maximum
field in the coil will be larger than the field at the inside
edge of the coil. The distance from the inside edge of the
coil to the peak field in the coil will generally increase with
coil thickness, and thus with the magnitude of the maxi-
mum field. We approximate the relationship between B+
and the distance from the inside edge of the coil to the peak
field to be linear, and call the constant of proportionality
kc. The maximum peak field in the coil is thusg = B+ + B1 kcB+.
(2)
The fit to the dipoles is better for this case: the log rms
error of is about factor of 2. The 1.34 in both formulae is to
correct for 2.5% inflation from the paper's publication date
of 1992 to 2004.
There are, however, several known dependencies of
magnet costs that are not represented in either of these for-
mulae: a) short dipoles are more expensive per unit length
than long ones, b) costs rise faster than linear for higher
magnetic fields, c) costs do not go to zero as the aperture
becomes small compared with the coil thickness, and d)
mass produced magnets are cheaper than the cost of one or
a few. Since such dependences can play a significant role in
finding a cost minimum, it is useful to define a formula that
attempts to include them. In addition, it is useful to extend
the method to cover quadrupoles and combined function
magnets.
* Work supported by US Department of Energy contract DE-AC02-
98CH10886(4)
Also, because the coils must have a nonzero thickness
which increases (we assume linearly) with the peak field,
we can define a maximum radius:R = kR + kMB.
(5)
Our estimate of the cost of a single magnet out of run of n
magnets will be written as a product of four factors:
Cost = fB(B)fG(N,L)fs(B /B+)fN(n). (6)
Here L is the reference length of the magnet. The first fac-
tor fB (B) gives the dependence on the magnetic field. A
simple model isfB(B) = Co + CiBkB.
(7)
This allows one to have a power law behavior for high
fields, and to take into account the fact that a magnet with
zero field still has a finite construction cost.
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Palmer, R. B. & Berg, S. J. A Model for Determining Dipole, Quadrupole, and Combined Function Magnet Costs., article, September 14, 2004; United States. (https://digital.library.unt.edu/ark:/67531/metadc1411150/m1/3/: accessed April 23, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.