S. Yang
Department of Mechanical Engineering, Energy and Sustainability Center, and Center for Advanced Power Systems
Florida State University, Tallahassee, FL 32310 USA
Abstract
This paper elaborates a thermal analysis of a DC electromagnet cooled by high thermal conductivity discs, via which heat is conducted away from the hot spot to the heat sinks at inner and outer disc surfaces. In this work, a dimensionless mathematical expression for the hot spot temperature in the coil was derived and solved analytically. The solution was then verified numerically with a commercial FEA software. Subsequently, the model was employed to investigate the hot spot temperature variation with respect to the number of discs for a given magnetic performance. The study also examined how the change in heat flow direction, with respect to the number of discs, affected the overall thermal performance of an electromagnet. According to the results, electromagnets become thinner and longer as the number of discs increase, which in turn lowers the hot spot temperature. Furthermore, heat is observed to flow axially with smaller number of discs, whereas radial heat flow predominates as this number increases. In summary, the results infer that a thermally efficient electromagnet can be designed by minimizing the heat flow resistance in the coil, with shorter
(or highly conductive) path for the hot spot to easily access the heat sink.
Keywords: thermal management, electromagnets, multiphysics, heat transfer, electromagnetism, Joule heating
Email address: syang@caps.fsu.edu
(S. Yang)

T
0
V r, z
G
H
0
J k
L n
P q
000 r
0 r
1 r m
T
Nomenclature
B
B
0
D magnetic B-field, T magnetic flux density at the origin, cooling disc width, m normalized cooling disc width
T
Fabry factor magnetic field strength at the origin, A/m current density, A/m
2 thermal conductivity, W/m · K dimensionless thermal conductivity half-length of the solenoid, m number of cooling discs power, W volumetric heat generation rate, W/m 3 inner solenoid radius, m outer solenoid radius, m
( r
1
− r
0
) / 2, m temperature, K dimensionless temperature heat sink temperature.
K volume, m 3 cylindrical coordinates in 2D
µ
0
σ
φ
β
λ
Greek letters
α normalized solenoid radius normalized solenoid length coil space factor magnetic constant, H/m electrical conductivity, S/m
V d
/V c
2

ϕ
ρ resistivity ratio electrical resistivity, Ω · m
Subscripts c coil d disc max maximum min r z minimum radial direction axial direction
1. Introduction
5
10
15
20
Thermal management of electromagnets has become an integral task during their design stages as the demand for compact light-weight electromagnets is growing. High-density electromagnets are capable of providing high magnetic fields at smaller volumes; they can be installed in confined spaces with less structural support. Such electromagnets, however, are not efficient in dissipating the heat generated in the coil by Joule heating. Moreover, heat accumulated in the coil may rise the temperature beyond the design limit of certain components, causing their mechanical, thermal, and material properties to degrade.
Several cooling mechanisms for DC electromagnets are discussed in [1–3].
Kroon [1] presents a thermal analysis of coils cooled by high thermal conduc-
tivity inserts that transport heat from the hot spot to their surfaces at fixed temperature T
0
. In this work, the author considers two cooling configurations: discs with heat sinks at (1) outer surfaces only and (2) both inner and outer
surfaces. Subsequently, Kroon [1] derives a mathematical model for each cool-
ing configuration to solve for the hot spot temperature in the coil. The author obtains an analytical solution for each respective model based on the following assumptions: (1) uniform current density throughout the coil; (2) heat flows axially in the coil and radially in the discs; and (3) all boundaries are adiabatic
3

25
30
35 except for the exposed cooling disc surfaces at T
0
.
Gosselin and Bejan [3] adapts the mathematical model derived in [1] for
the first cooling configuration (heat sink at the outer disc surface only), and obtains a dimensionless expression for the hot spot temperature in the coil.
Furthermore, the authors verify the analytical solution by numerically solving the 2D Laplacian in cylindrical coordinates with an internal heat generation expressed as:
∂ 2
∂z
T
2
+
1 ∂ r ∂r r
∂T
∂r
+ q
000 k c
= 0 (1) where q
000 is the volumetric heat generation rate and k is the thermal conductivity equivalent to k c in the winding and k d in the discs. For a given magnetic performance and volume, Gosselin and Bejan minimizes the hot spot temperature inside an electromagnet by selecting the optimal shape of the coil (length and outer radius), the number of cooling discs, and the amount of high thermal conductivity material.
This work presents the mathematical model for the second cooling configuration (heat sinks at both inner and outer disc surfaces) and derives a dimensionless expression for the hot spot temperature in the coil ( e max
). Subsequently, the model is employed to study the variation in T max as a function of the number of cooling discs. In order to verify the model, the problem under consideration is numerically solved in a commercial FEA software without the assumption of unidirectional heat flow; therewith, the discrepancy between analytical and numerical results are examined and discussed.
2. Mathematical formulation
40
The electromagnet considered in this work is modeled as a simple solenoid
shown in Fig. 1, where
B
0 is the magnetic flux density at the solenoid center (axis of symmetry), r
0 and r
1 are inner and outer radii of the solenoid, respectively, k c and k d are thermal conductivity in the winding and cooling discs, respectively, and 2 L is the coil length. The transversal cooling discs of width 2 D separate the solenoid into several sub-coils, and their inner and outer surface temperatures
4

45 are fixed at T
0
(heat sinks). The solenoid has an air bore and the conductor is wound in multiple layers around a hollow cylinder of radius r
0
, and it comprises homogeneous sub-coils with uniform current distribution. The electric current flowing through the wire generates magnetic field according to Biot-Savart law
[4].
Figure 1: The cross-section of a simple solenoid with transversal cooling discs.
2.1. Electromagnetism
Magnetic flux density at the axis of symmetry of a solenoid B
0 is given by:
B
0
= µ
0
H
0
(2) where µ
0 is the magnetic constant and H
0 is the field strength at the solenoid center defined as follows:
H
0
= G
P λ
ρr
0
1 / 2
(3) where G is called Fabry factor which is a function of normalized solenoid geometry, α = r
1
/r
0 and β = L/r
0
, P is the power required to generate desired magnetic field, λ is the space factor defined as the ratio of active to the total coil area, and ρ is the temperature-dependent conductor resistivity. Fabry factor is
defined as[5]:
G =
1
5
2
α 2
πβ
− 1
1 / 2 ln
α + α 2 + β 2
1 / 2
1 + ( β 2 + 1)
1 / 2
(4)
5

50
Fig. 2 shows the constant-
G curves as a function of α and β , according to which a constant Fabry factor can be achieved via multiple geometric combinations except for G max
= 0 .
179, which is obtained solely when α = 3 .
13 and
β = 1 .
9.
8
7
6
5
4
0.11
0.12
0.13
3
2
1
0
1
0.15
0.16
0.14
2
G max
= 0.179
0.14
3
0.16
4
α
5 6 7
0.15
8
Figure 2: Constant Fabry curves as a function of α and β
as defined in Eq. (4).
Temperature-dependent expression for H
0 can be obtained by rearranging
and nondimensionlizing Eq. (3) as follows:
H
0
ρ
0 r
0
P λ
1 / 2
= Gϕ
− 1 / 2
(5) where ρ
0 is the conductor resistivity at fixed reference temperature T
0 and ϕ = ρ/ρ
0
. The electrical resistivity ratio ϕ for common materials can be ap-
proximated using the following linear relationship [6]:
ϕ = 1 + % ( T c
− T
0
) (6) where % is the temperature coefficient of resistivity that is constant for a particular material and T c is the average coil temperature.
Fabry factor for homogeneous coils with one cooling disc ( n = 1) is given by
[1]:
G eq,n =1
=
G
β
β 1 / 2 − G
D e
( β − e
) 1 / 2
1 / 2
(7)
6

G
β
=
1
5
2 πβ
α 2 − 1
1 / 2 ln
α + ( β
2
+ α
2
)
1 / 2
1 + ( β 2 + 1) 1 / 2
(8a)
G
D e
=
1
5
2 π e
α 2 − 1
!
1 / 2 ln
α + ( e
2 + α 2 ) 1 / 2
1 + ( e
2 + 1) 1 / 2 where e is the normalized disc width defined as:
(8b) e
=
φβ n
(9) in which φ is the fraction of the total coil volume occupied by the cooling discs and it is fixed. Similarly, Fabry factor for n >
1 is given by [3]:

G eq,n,odd
=
1
( β − n e
) 1 / 2


G
β
β
1 / 2 − G
D e
1 / 2
+
( n − 1) / 2

X


G
2 iβ/n − D e

2 iβ n i =1
 
− e 
1 / 2

− G
2 iβ/n + D e

2 iβ n
+

1 / 2  

 
 
(10a)

G eq,n,even
=
1
( β − n e
) 1 / 2


G
β
β
1 / 2 n/ 2
+
X



G
(2 i − 1) β/n − D e

(2 i − 1) β n i =1

−

1 / 2


− G
(2 i − 1) β/n + D e

(2 i − 1) β n
+

1 / 2  

 
 
(10b)
55
2.2. Heat transfer
The hot spot temperature in the coil is obtained by solving the 2D Laplacian
in Eq. (1) with the assumptions listed in Section 1. In most cases,
k d k c
7

because the conductor insulation lowers the average bulk thermal conductivity
in the winding by orders of magnitude [1]. Thereby, it is assumed that heat flows
axially in the winding and radially through the cooling disc. In the winding,
Eq. (1) reduces to:
d 2 T c dz 2
= − q
000 k c
(11) with the boundary conditions: dT c dz z = L
= 0 (12a)
T c
| z =0
= T d
( r ) (12b) where T d
( r ) is the cooling disc temperature that varies along the radius. These
boundary conditions yield the following solution to Eq. (11):
T c
( z ) − T d
( r ) = q 000
2 k c z
(2 L − z ) (13)
The radial temperature distribution is determined by the solution of the differential equation given as: d 2 T d dr 2
+
1 r dT d dr
= − q
000
L k d
D
(14) with the boundary conditions:
T d
| r = r
0
= T d
| r = r
1
= T
0
(15)
The solution to Eq. (14) with these boundary conditions is:
T d
( r ) = q 000 Lr 2
0
4 k d
D
α
2
− 1 −
α 2 ln
− 1
α ln r r
0
+ T
0
(16)
The maximum (hot spot) temperature in the coil T max is located at z = 0 and r = r m
, where r m
is defined as [1]:
r m
= r
0
α
2 − 1
2 ln α
1 / 2
(17)
8

Combining Eq. (13) and (16) for
r = r m
, the dimensionless maximum temperature difference is derived as follows:
∆ T max
=
T max
− T
0
P/ ( r
0 k c
)
=

1
2 πn ( α 2 − 1)

β + n


α
2 − 1 −
(18)
α
2 − 1 ln α ln r
α 2 −
2 ln α
1
 
  in which ˜ = k d
/k c and q
000 has been replaced by P/V , where V = πnr
3
0
β ( α
2 − 1).
Eq. (18) is useful in computing the average coil temperature
T c which can be approximated as T c
≈ ε ( T max
− T
0
) + T
0
[1], where
ε = 0 .
58. As a result, the
electrical resistivity ratio in Eq. (6) can be simplified to
ϕ = 1 + ε% ( T max
− T
0
).
60
3. Finite element analysis
65
In order to verify the analytical solution described in the previous section, the problem was solved numerically in COMSOL Multiphysics, a commercial fi-
nite element analysis (FEA) software package [7]. In COMSOL, the two physics
present in the problem (i.e., electromagnetism and heat transfer) were coupled and simultaneously solved for the temperature and magnetic field in the electro-
magnet. All assumptions listed in Section 1 were considered in the FEA except
for the unidirectional heat flow in the coil and the discs.
3.1. Physics
The volumetric Joule heating in an electromagnet is given by: q
000
=
J · J
σ
I
C
B · d l =
Z Z
S
J · d S
(19) where J is the current density and σ is the electrical conductivity of the conductor, which is temperature dependent and the inverse of electrical resistivity.
J through a surface S is linked to the magnetic field B around closed curve C
via Ampere’s Law defined as [4]:
(20)
9

or in its differential form as:
∇ × B = µ
0
J (21)
70
According to Eq. (19)-(21), temperature increase in the coil causes electrical
conductivity to drop, requiring more current to be supplied to retain the constant magnetic field. More current, however, means higher power dissipation
according to Eq. (19) and thereby, the temperature increases until steady state
is attained. In COMSOL, steady-state temperature distribution was obtained
by solving the 2D Laplacian in Eq. (1).
75
3.2. Mesh
80
Three computational domains considered in this analysis were air (magnetic field), coil, and discs. The grid in COMSOL was discretized with triangular elements (2D) and it was adaptively refined until the residual error in the solution was of order 2. The number of domain and boundary elements increased along with the number of cooling discs to account for intricate energy interactions between the coil and the cooling discs. Final meshes for n = 1 and n = 5 are
shown in Fig. 3.
Figure 3: Adaptively refined triangular meshes of the electromagnet for n = 1 and n = 5.
10

4. Results and discussion
85
Fig. 4 shows the variations in ∆
e max for cases with 1, 2, and 3 discs, along the constant G -curve ( G = 0 .
16) as a function of the geometric ratio α = r
1
/r
0
. This
analysis holds the same assumptions made in [3], where the volume occupied by
the cooling discs φ = 0 .
01 and ˜ = 100; therefore φ
˜
= 1 in Eq. (18).
0.16
0.14
n = 1 n = 2 n = 3
0.12
0.1
0.08
0.06
0.04
0.02
0
0 2 4
α
6 8
Figure 4: ∆ e max variations with respect to n ; G = 0 .
16 and φ k = 1.
90
95
Each curve in Fig. 4 exhibits a minimum ∆
e max with respect to α (not shown is the change with respect to β required to retain a constant G ). The minimum
∆ e max shifts towards smaller α as the number of cooling discs increases. These minimized hot spot temperatures are plotted with respect to n
in Fig. 5, where
∆ e max,min features an exponential decay as n increases. The qualitative trend of a solenoid design for minimum hot spot temperature and constant G can be deduced from these curves: electromagnets become thinner and longer as the hot spot temperature is minimized for a given G .
Analytical and numerical results are plotted and compared in Fig. 5, in addi-
tion to the results for the case in which only the outer disc surfaces are fixed at
T
0
[3]. Both analytical and numerical results feature similar qualitative trends,
and they are in good agreement in light of the order-of-magnitude analysis. For
11

0.05
0.045
0.04
0.035
0.03
0.025
0.02
0.015
0.01
0.005
0
Outer surface - Analytical
Outer surface - Numerical
Both surfaces - Analytical
Both surfaces - Numerical
2 4 n
6 8 10
Figure 5: Minimum ∆ e max as a function of n , where analytical and numerical results are compared in addition to the results for the case in which only the outer disc surface is at T
0
.
100
105
110
115 the range of n considered in this study, the hot spot temperature in the coil is significantly lower when both inner and outer disc surfaces are fixed at T
0
; therewith, the same ∆ e max,min can be achieved with less number of cooling discs. Nonetheless, the curves imply that further increase in n does not significantly improve the thermal performance, owing to the fact that φ is fixed and thus, the discs become thinner as n increases. Therefore, each disc does not have enough highly conductive material to transport sufficient heat to lower the hot spot temperature further.
Special attention is given to the intersections of the curves in Fig. 5, which
are attributed to the change in heat flow direction with respect to n . Such a physical observation contradicts the assumption that heat flows axially in the coil and radially in the discs. Hence a closer scrutiny is given to temperature gradients and heat flow fields obtained with COMSOL for different values of n .
Temperature distribution at the coil cross-section with two distinct cooling
configurations is depicted and compared in Fig. 6 for
n = 1, n = 5, and n =
10. The solenoid geometry changed accordingly to obtain ∆ e max,min plotted
in Fig. 5, whereas Fabry factor remained constant at
G = 0 .
16 in all cases.
12

120
Fig. 6 clearly illustrates the changes in heat flow direction as the number of
cooling discs increases. When n
= 1 in Fig. 6a, for instance, axial heat flow
predominates in the coil until it shifts towards radial direction as n increase, e.g., n = 5 and n
= 10. Fig. 6b shows a similar transition from axial to radial
heat flow, as the hot spots shift from the inside corner towards the middle.
(a) Heat sink at the outer disc surface only
(b) Heat sinks at both inner and outer disc surfaces
Figure 6: Temperature distribution at the coil cross-section for different n .
125
Isothermal contours and heat flow vectors are exhibited in Fig. 7 for the first
four vales of n
. According to Fig. 7a, the overall magnitude of the radial vector
component becomes larger starting at n = 2; the point at which the numerical
and analytical curves intersect in Fig. 5; Fig. 7b shows a similar trend. The contour lines in Fig. 7 show how heat sinks at both ends of the discs provide
shorter path for heat to travel from the hot spots to T
0
.
13

(a) Heat sink at the outer disc surface only
(b) Heat sinks at both inner and outer disc surfaces
Figure 7: Isothermal contours and heat flow vectors at the coil cross-section for different n .
5. Conclusion
130
135
This paper presented a thermal analysis of an electromagnet cooled by high thermal conductivity discs, via which heat was conducted away from the hot spot to the heat sinks at T
0
. The results from this work infer that a thermally efficient electromagnet design is the one that minimizes the heat flow resistance by providing shorter (or high thermal conductivity) path for the hot spot to
easily access the heat sink [8]. Such an objective can be accomplished through
concurrent optimization of the electromagnet for optimal thermal and electromagnetic performances.
Acknowledgment
This work was supported by the Florida State University Graduate School under Dissertation Research Grant.
14

140
References
[1] D. Kroon, Electromagnets, Boston Technical Publishing Company, 1968.
[2] D. B. Montgomery, R. J. Weggel, Solenoid magnet design, Krieger, 1969.
[3] L. Gosselin, A. Bejan, Constructal thermal optimization of an electromagnet,
International journal of thermal sciences 43 (4) (2004) 331–338.
145
[4] D. J. Griffiths, R. College, Introduction to electrodynamics, Vol. 3, prentice
Hall Upper Saddle River, NJ, 1999.
[5] W. Gauster, Some basic concepts for magnet coil design, American Institute of Electrical Engineers, Part I: Communication and Electronics, Transactions of the 79 (6) (1961) 822–828.
150
[6] W. D. Callister, D. G. Rethwisch, Fundamentals of materials science and engineering: an integrated approach, John Wiley & Sons, 2012.
[7] A. Comsol, Comsol multiphysics users guide, Version: September.
[8] A. Bejan, Constructal law: Optimization as design evolution, Journal of
Heat Transfer 137 (6) (2015) 061003.
15