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Abstract—In this paper, an exact solution of steady heat
conduction from a hot donut (a torus) placed in an infinite
medium of constant temperature is obtained. The governing
energy equation is recast in a naturally-fit coordinates system
and then solved using toroidal basis functions.
Index Terms—Donuts, heat conduction, toroidal coordinates.
I. INTRODUCTION
Several industries involve cooling of ring-like blanks
(donuts). In such industries, which are not limited to food
processing, it is very important to estimate, for example, the
cooling time of such products before packaging.
The problem considered here is that of a torus (donut) with
inner radius r and outer radius R , heated to a uniform
temperature sT and then placed in a medium of constant
temperature fT , Fig. 1. Our interest is to find the temperature
distribution around the donut.
II. METHOD OF SOLUTION
A torus is generated by revolving a circle in three
dimensional space about a coplanar axis which does not
necessarily touch the circle. If the axis touches the circle, the
resulting surface is a horn torus. Furthermore, if the axis is a
chord of the circle, the resulting surface is a spindle torus. A
sphere is the degenerate case when the axis is a diameter of
the circle [1]- [3].
To suit the geometry of the problem, we use the Toroidal
Coordinate System. The toroidal coordinate system ( , , ),
is related to the cartesian coordinate system by the relations
sinh cos sinh sin, ,
cosh cos cosh cos
sin
cosh cos
c cx y
cz
(1)
with the corresponding scale factors given by:
cosh cos
ch h
,
sinh
cosh cos
ch
(2)
Manuscript received October 18, 2012; revised February 27, 2013. This
work was supported by King Fahd University of Petroleum & Minerals
(KFUPM) under Grant SF121—CS-03, Saudi Arabia.
Rajai S. Alassar is with the King Fahd University, Saudi Arabia.(e-mail:
Fig. 1. Problem configuration.
Fig. 2. Toroidal coordinates.
The coordinates satisfy [0, ) , [0,2 ) , and
[0,2 ) with c ( 2 2c R r ) being the focal distance.
The toroidal coordinate system is composed of surfaces of
constant which are given by the toroids
2 2 2 2 2 22 cothx y z c c x y , surfaces of constant
given by the spherical bowls
2 2 2 2 2( cot ) /sinx y z c c , and surfaces of constant
given by the half planes tan /y x , Fig. 2.
The steady version of the differential equation of heat
conduction for a homogeneous isotropic solid with no heat
Heat Conduction from Donuts
Rajai S. Alassar and Mohammed A. Abushoshah
126DOI: 10.7763/IJMMM.2013.V1.28
International Journal of Materials, Mechanics and Manufacturing, Vol. 1, No. 2, May 2013
generation is given by
2 0T (3)
where T is the temperature, and is the thermal
diffusivity. In a general orthogonal curvilinear coordinates
system ( 1 2 3, ,u u u ), equation (3) with scale factors 1 2,h h and
3h takes the form
2 3 1 3 1 2
1 2 3 1 1 1 2 2 2 3 3 3
1( ) ( ) ( ) 0h h h h h hT T T
h h h u h u u h u u h u
(4)
Equation (4) may be specialized for toroidal coordinates as
sinh sinh0
cosh cos cosh cos
T T
n n
(5)
The boundary conditions to be satisfied are:
0( , ) , (0,0)s fT T T T (6)
where 0 ( 1
0 coshR
r
) defines the surface of the torus.
The point ( , ) (0,0) defines the far field away from the
torus whereas the first condition in (6) indicates that that the
surface of the torus is maintained at constant temperature.
We define the dimensionless temperature as
( , )f
s f
T Tu
T T
(7)
Equation (5) and the boundary conditions (6) can now be
rewritten in terms of the dimensionless temperature as
sinh sinh0
cosh cos cosh cos
u u
n n
(8)
0( , ) 1, (0,0) 0u u (9)
Laplace equation (the energy equation (8)) is R-separable
in toroidal coordinates, [4-6]. We, therefore, assume a
solution of the form
( , ) cosh cos ( ) ( )u X Y (10)
Upon substituting (10) in (8), and performing the
necessary rituals, one finds the basis functions are
1 1
2 2
(cosh ), (cosh ), cos , sinn n
P Q n n
, where 1
2n
P
and
1
2n
Q
are the toroidal functions of the first and second kinds
respectively. Due to the boundary conditions in (9) and the
boundedness requirements, only 1
2n
P
and cosn survive.
The solution can then be written as
1
0 2
( , ) cosh cos (cosh ) cos( )nn
n
u A P n
(11)
We apply the first condition in (9) to get nA . One may
make use of the integral 2
10
2
cos( )2 2 (cosh )
cosh cos n
nd Q
(12)
to show that, [7]
1 0
2
0 1 0
2
(cosh )2 2
(1 ) (cosh )
n
n
nn
Q
AP
(13)
The solution of (11), thus, is
1 0
21
0 0 1 0 2
2
( , ) cosh cos
(cosh )2 2
(cosh ) cos( )(1 ) (cosh )
n
nn n
n
u
Q
P nP
(14)
where ij is Kronecker delta.
Contour plots of the solution (14) for 1r and
3/ 2,5/ 2,and7/ 2R are shown in Figure 3.
Fig. 3. Dimensionless temperature distribution for the cases 1r and
3/2,5/2,and7/2R .
ACKNOWLEDGMENT
The authors would like to thank King Fahd University of
Petroleum & Minerals (KFUPM) for supporting this research
under grant FT2013-30.
APPENDIX ORTHOGONAL CURVILINER COORDINATES
A Cartesian coordinate system offers the unique advantage
that all three units vectors, ˆ,i ˆ,j and ˆ ,k are constant.
Unfortunately, not all physical problems are well adapted to
solution in Cartesian coordinates. In Cartesian coordinates
we deal with three mutually perpendicular families of planes:
x constant, y constant, and z constant. We
superimpose on this system three other families of surfaces.
The surfaces of any one family need not be parallel to each
other and they need not be planes. Any point is described as
the intersection of three planes in Cartesian coordinates or as
the intersection of the three surfaces which form curvilinear
coordinates. Describing curvilinear coordinates surfaces by
1 1u c , 2 2u c , 3 3u c where 1 2, ,c c and 3c are constant,
we identify a point by three numbers 1 2 3, ,u u u , called the
curvilinear coordinates of the point.
Let the functional relationship between curvilinear
127
International Journal of Materials, Mechanics and Manufacturing, Vol. 1, No. 2, May 2013
coordinates 1 2 3, ,u u u and the Cartesian coordinates
, ,x y z be given as:
1 2 3
1 2 3
1 2 3
, ,
, ,
, ,
x x u u u
y y u u u
z z u u u
( A1)
which can be inverted as
1 1
2 2
3 3
( , , )
( , , )
( , , )
u u x y z
u u x y z
u u x y z
(A2)
Any point P in the domain, having curvilinear coordinates
1 2 3( , , )c c c , there will pass three isotimic surfaces (on which
the value of a given quantity is everywhere equal)
1 1( , , )u x y z c , 2 2( , , )u x y z c , and 3 3( , , )u x y z c . As
illustrated in Figure A.1, these surfaces intersect in pairs to
give three curves passing through ,P along each of which
only one coordinate varies; these are the coordinates curves.
Fig. A.1. A curvilinear Coordinate System 1 2 3( , , )u u u
The normal to the surface i iu c is the gradient:
ˆ ˆ ˆi i i
i
u u uu
x y z
i j k
(A3)
The tangent to the coordinate curve for iu is the vector
ˆ ˆ ˆ
i i i i
r x y z
u u u u
i j k
(A4)
We say that 1 2, ,u u and 3u are orthogonal curvilinear
coordinates whenever the vectors 1 2,u u
and 3u
are
mutually orthogonal at every point.
Each gradient vector iu
is parallel to the tangent vector
i
r
u
for the corresponding coordinate curve. And any
coordinate curve for iu intersects the isotimic surface
i iu c at right angles when 1 2 3, ,u u u form orthogonal
curvilinear coordinate. To see this, consider a coordinate
curve for 1u .
1) This curve is the intersection of two surfaces 2 2u c and
3 3u c . Hence, its tangent 1
r
u
is perpendicular to both
surface normals 2u
and 3u
.
2) The vector 1u
is also perpendicular to 2u
and 3u
.
3) This implies that 1
r
u
is parallel to 1u
.
Since both point in the direction of increasing 1u , they are
parallel. It follows, that the vectors1
r
u
, 2
r
u
, and 3
r
u
also
form a right-handed system of mutually orthogonal vectors.
Define the right-handed system of mutually orthogonal unit
vectors 1 2 3( , , )e e e by
i ii
i
i
r
u u
u r
u
e
1,2,3i (A5)
We need three functions ih known as the scale factors to
express the vector operations in orthogonal curvilinear
coordinates. The scale factor ih is defined to be the rate at
which the arc length increases on the ith coordinate curve with
respect to iu . In other words, if is denotes arc length on the
ith coordinate curve measured in the direction of
increasing iu , then
ii
i
dsh
du 1,2,3i (A6)
Since the arc length can be expressed as
1 2 3
1 2 3
r r rds dr du du du
u u u
(A7)
We see that
i
i
rh
u
1,2,3i (A8)
Hence
2 2 2 2( ) ( ) ( )i
i i i
x y zh
du du du
1,2,3i (A9)
Combine equations (A7) and (A8) shows that the
displacement vector can be expressed in terms of the scale
factors by
1 1 1 2 2 2 3 3 3d r h du h du h du e e e
(A10)
A differential length ds in the rectangular coordinate
system ( , , )x y z is given by
2 2 2 2( ) ( ) ( ) ( )ds dx dy dz (A11)
The differential lengths dx , dy and dz are obtained from
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International Journal of Materials, Mechanics and Manufacturing, Vol. 1, No. 2, May 2013
equation (A1) by differentiation
1 2 3
1 2 3
x x xdx du du du
u u u
(A12a)
1 2 3
1 2 3
y y ydy du du du
u u u
(A12b)
1 2 3
1 2 3
z z zdz du du du
u u u
(A12c)
Substitute equations (A12) into equation (A11), the
differential length ds in the orthogonal curvilinear
coordinate system 1 2 3, ,u u u can be written as
2 2 2 2 2 2 2
1 1 2 2 2 2( ) ( ) ( ) ( )ds dr dr h du h du h du
(A13)
In rectangular coordinates system a differential volume
element dV is given by
dV dxdydz (A14)
and the differential areas xdA , ydA , and zdA cut from the
planes x constant, y constant, and z constant are
given, respectively, by
,xdA dydz ,ydA dxdz zdA dxdy (A15)
In orthogonal curvilinear coordinates system, the
elementary lengths from equation (A6) are given, by
i i ids h du 1,2,3i (A16)
Then, an elementary volume element dV in orthogonal
curvilinear coordinates system given, by
1 2 3 1 2 3dV h h h du du du (A17)
The differential areas 1dA , 2dA and 3dA cut from planes
1 1u c , 2 2u c and 3 3u c are given, respectively, by
1 2 3 2 3 2 3dA ds ds h h du du
2 1 3 1 3 1 3dA ds ds h h du du (A18)
3 1 2 1 2 1 2dA ds ds h h du du
The gradient is a vector having the magnitude and
direction of the maximum space rate of change. Then, for any
function 1 2 3( , , )u u u the component of 1 2 3( , , )u u u
in the
1e direction is given, by
1
1 1 1
|d
ds h u
(A19)
Since this is the rate of change of with respect to
distance in the 1e direction. Since 1e , 2e , and 3e are
mutually orthogonal unit vector, the gradient becomes
1 2 3 1 2 3
1 2 3
1 2 3
1 1 2 2 3 3
( , , )d d d
grad u u uds ds ds
h u h u h u
e e e
e e e
(A20)
Let 1 1 2 2 3 3F F F F e e e
be a vector field, given in
terms of the unit vectors 1e , 2e and 3e . Then the divergence
of F
, denoted div F
, or F
is given, by [23]
1 2 30
F( , , ) limF
dV
dAu u u
dV
(A21)
where dV is the volume of a small region of space and
dA is the vector area element of this volume.
Fig. A2. Differential Rectangular Parallelepiped
We shall compute the total flux of the field F
outsides of
the small rectangular parallelepiped Figure A2. We, then,
divide this flux by the volume of the box and take the limit as
the dimensions of the box go to zero. This limit is the F
.
Note that the positive direction has been chosen so that
1 2 3( , , )u u u or 1 2 3( , , )e e e from a right-hand system.
The area of the face abcd is 2 3 2 3h h du du and the flux
normal is 1 1F F e
. Then the flux outward from the face
abcd is 1 2 3 2 3F h h du du . The outward unit normal to the
face efgh is 1e , so its outward flux is 1 2 3 2 3F h h du du .
Since 1F , 2h , and 3h are functions of 1u as we move along
the 1u - coordinate curve, the sum of these two is
approximately
1 2 3 1 2 3
1
( )F h h du du duu
(A22)
Adding in the similar results for the other two pairs of
faces, we obtain the net flux outward from the parallelepiped
is
1 2 3 2 2 3 3 1 2 1 2 3
1 2 3
( ) ( ) ( )F h h F h h F h h du du duu u u
(A23)
Divide equation (A23) by differential volume equation
(A17). Hence the flux output per unit volume is given by
1 1h du
1e
2e
3e
a b
cd
fe
h g
3 3h du
2 2h du
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International Journal of Materials, Mechanics and Manufacturing, Vol. 1, No. 2, May 2013
1 2 3 2 1 3 3 1 2
1 2 3 1 2 3
div F F
1( ) ( ) ( )F h h F h h F h h
h h h u u u
(A24)
By using equations (A20) and (A24), the Laplacian is
given as
2 divgrad
2 3 1 3 1 2
1 2 3 1 1 1 2 2 2 3 3 3
1( ) ( ) ( )h h h h h h
h h h u h u u h u u h u
(A25)
REFERENCES
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Van Nostrand, 1961.
[2] P. M. Morse and H. Feshbach, Methods of Theoretical Physics, Part I.
New York: McGraw-Hill, 1953.
[3] G. Arfken, Mathematical Methods for Physicists, 3rd ed. Orlando, FL:
Academic Press, 1985.
[4] N. N. Lebedev, Special Functions and Their Applications, Dover
Publications, 1972.
[5] M. Abramowit and I. A. Stegun, Handbook of Mathematical Functions,
Ninth Edition. New York: Dover, 1970.
[6] M. Andrews, “Alternative Separation of Laplace’s Equation in
Toroidal Coordinates and its Application to Electrostatics,” Journal of
Electrostatics, vol. 64, pp. 664-672, 2006.
[7] I. S. Gradshteyn and I. M. Riyzhik, Tables of Integrals, Series and
Products, Fourth (English) Edition, (Prepared by Alan Jeffrey),
Academic Press, New York, 1980.
Rajai S. Alassar is a professor in the Department of
Mathematics and Statistics at King Fahd University if
Petroleum & Minerals (KFUPM), Saudi Arabia.
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International Journal of Materials, Mechanics and Manufacturing, Vol. 1, No. 2, May 2013