Heat Conduction from Donuts - IJMMM · Abstract—In this paper, an exact solution of steady heat conduction from a hot donut (a torus) placed in ... Heat Conduction from Donuts .

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:

[email protected])

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


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

128


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

129


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

[1] P. Moon and D. E. Spencer, Field Theory for Engineers, Princeton, NJ:

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.

130


Heat Conduction from Donuts - IJMMM · Abstract—In this paper, an exact solution of steady heat conduction from a hot donut (a torus) placed in ... Heat Conduction from Donuts .

Documents