Temperature distribution in a circular plate heated by a moving heat source

TEMPERATURE DISTRIBUTION IN A CIRCULAR PLATE HEATED BY A MOVING HEAT SOURCE

Jadwiga Kidawa-Kukla

Institute of Mechanics and Machine Design, Częstochowa University of Technology, Poland jk@imipkm.pcz.czest.pl

Abstract. This paper concerns the heat conduction problem in a circular thin plate sub- jected to the activity of a heat source which changes its place on the plate surface with time.

The heat source moves along an circular trajectory round centre of the plate with constant angular velocity. The solution of the problem is obtained in an analytical form by using the Green’s function method.

Introduction

The solutions of heat conduction problems in circular plates with moving heat sources are presented in papers [1-3]. An analytical approach to multi-dimensional heat conduction in composite circular cylinder subjected to generally time- dependent temperature changes has been presented in paper [1]. Boundary tem- peratures were approximated as Fourier series. The Laplace transform was adopted in deducing the solution of the problem. In paper [2] authors develop an integral transform determineing temperature distribution in a thin circular plate. The plate is subjected to a partially distributed and axisymmetric heat supply. Authors find the temperature field analytically by using the finite Fourier and the finite Hankel transforms. An inverse problem of axially symmetric transient temperature and deflection of a circular plate is solved in paper [3]. A heat flux is assumed on an internal cylindrical surface of the plate. The solution of the problem was obtained by applying the Fourier cosine and the Laplace transforms. A solution to the prob- lem of heat conduction in a rectangular plate subjected to the activity of a moving heat source is presented in paper [4]. The heat source moves along an elliptical trajectory on the plate surface. An exact solution to the problem in an analytical form is obtained by applying the Green’s function method.

This paper presents an analytical solution to the heat conduction problem in

a circular thin plate which is heated by a moving heat source. The temperature of

the plate changes because the heat source moves along circular trajectory on the

plate surface. A solution of the heat conduction problem in an analytical form is

obtained by using the Green’s function method.

1. Problem formulation

Consider a circular thin plate of thickness h and radius r = b. This plate is sub- jected to the activity of a moving heat source. The heat source moves on the plate surface along a circular trajectory at radius r

0

round centre of the plate with con- stant angular velocity. The temperature distribution T ( r , φ , z , t ) of the plate is de- scribed by the differential equation of heat conduction [5]:

t T ) a t , z , , r ( k g z

T T r r

T r r

T

∂ φ ∂

∂

∂ φ

1 1

1 1

2 2 2 2 2 2

2

= +

∂ + + ∂

∂ + ∂

∂

∂ (1)

where: T ( r , φ , z , t ) - temperature, k - thermal conductivity, a - thermal diffusivity, and g ( r , φ , z , t ) denotes a volumetric energy generation.

In this study, it is assumed that the thermal energy is provided by the moving heat source (which moves along a circular trajectory on the plate surface). The function g ( r , φ , z , t ) occurring in equation (1) has the form

) ( )) ( ( ) ( ) , , ,

(r z t r r₀ t z h

g

φ

=

θ δ

−

δ φ

−

ϕ δ

−

(2)

where θ characterises the stream of the heat, δ ( ) is the Dirac delta function, r

0

is the radius of the circular trajectory along which the heat source moves, t ϕ ( ) is the function describing the movement of the heat source

t

t ω

ϕ ( ) = (3)

where ω is angular velocity of the moving heat source.

The differential equation (1) is completed by the following initial and boundary conditions:

0 ) 0 , , ,

( r z =

T φ ⁽⁴⁾

)]

, , , ( [

₀

0

T T b z t

r k T

b r

φ

α −

−

∂ =

∂

=

(5) ]

) , , , ( [ ) , , ,

( r h t

₀

T

₀

T r h t z

k T φ α φ

∂

∂ _{= −}

(6)

] ) , 0 , , ( [ ) , 0 , ,

( r t

₀

T

₀

T r t

z

k T φ α φ

∂

∂ _{= − −}

(7)

where α

₀

is the heat transfer coefficient, T is the known temperature of the sur-

₀

rounding medium.

2. Solution of the problem

The solution of the problem in an analytical form is obtained by using the prop- erties of the Green’s function (GF). The GF of the heat conduction problem de- scribes the temperature distribution induced by the temporary, local energy im- pulse. The function is a solution to the following differential equation [5]:

( ) ( ) ( ) ( )

r

t z

' r

t G z a

G G r r

G r r

G δ ρ δ φ φ δ ζ δ τ

φ

−

−

−

= −

∂

− ∂

∂ + ∂

∂ + ∂

∂ + ∂

∂

∂ 1 1 1

2 2 2 2 2 2

2

(8)

The Green’s function satisfies the initial and homogeneous boundary condi- tions analogous to the conditions (4)-(7).

The GF for the considered heat conduction problem may be written in the form of a series:

( ^, φ ^{, ,} ) = ∑

^∞

( ^{, ,} ) ^cos ( φ − φ ^' )

−∞

=

m t z r g t

z r G

m

m

(9)

Substituting (9) into equation (8) gives

( ) ( ) ( )

r t z

r t g g a

r m z g r

g r r

g

_m

m m

m m

π ζ δ τ δ

ρ δ

2 1

1

2 2 2 2 2

2

= − − −

∂

− ∂

∂ − + ∂

∂ + ∂

∂

∂ (10)

The initial and boundary conditions are in the form

0 0 ) = , z , r (

g

_m

,

₀ 

= 0









−

∂

∂

=b r m

m

g

r

g µ

0

0

0 

=









−

∂

∂

= z m

m

g

z

g µ ,

₀ 

= 0









+

∂

∂

=h z m

m

g

z

g µ ⁽¹¹⁾

where

₀ ⁰

. k µ = α

The solution of the initial-boundary problem (9)-(11) can be presented in the form of a series

( ) ^∑

^∞

( ) ( )

=

= 1 n

n mn

m

r , z , t r , t z

g Γ ψ ⁽¹²⁾

where Ψ

_n

(z ) are eigenfunctions of the following boundary problem

( ) ⁰

2 2 2

=

∂ +

∂ z

z

ⁿ

n

β ψ

ψ (13)

0

0

0 

=









−

= z n n

z d

d ψ µ ψ

,

₀ 

= 0









+

=h z n n

z d

d ψ µ ψ

(14)

The functions ψ

n

(z) are expressed as [6]

( ) z =

_n

^cos

_n

z +

0

^sin

_n

z ^, n = ^{1 , 2 , ...}

n

β β µ β

ψ ⁽¹⁵⁾

where β

n

are roots of the equation

( ) ^{sin 0}

cos

2 µ

₀

β

_n

β

_n

h − β

_n²

− µ

₀²

β

_n

h = (16) These functions are pairwise orthogonal so that the following condition is satisfied

( ) ( )





=

= ≠

∫ ^{z z dz} _Q ⁿ _n ^m _m

n h

m

n

for

for 0

0

ψ

ψ ⁽¹⁷⁾

where _∫ ( ( ) ) ( )

_^











+ +

+

=

=

h

n n

n n

n

n

h

h dz h

z Q

0

2 2 0

2 0 2 2

0 2

2

sin

1 2

2 β

β µ

µ µ β

β

ψ (18)

The Dirac function δ ^{(z –} ζ ) may be written in the form

( ) _∑

^∞

( ) ( )

=

=

−

1

n n

n n

Q

z ζ ψ z ψ ζ

δ ⁽¹⁹⁾

Substituting Eqs. (12) and (19) into Eq. (10) gives

( ) ( ) ( )

r t r

Q t

r a m r

r r

_n

n n m n

m n

n m n

m

π τ δ ρ δ ζ Γ ψ

Γ Γ β

Γ

2 1

1

2 2 2 2

2

= − −

∂

− ∂













+

∂ − + ∂

∂

∂ (20)

The initial and boundary conditions are

Γ

_mn(r,0)=0

,

₀ 

= 0



 





−

∂

∂

=b r mn mn

r µ Γ

Γ (21)

In order to solve the problem (20)-(21), the function Γ

_mn

( t r , ) is written in the form

( ) ^∑

^∞

( ) ( )

=

= 1 k

nk m mk

mn

r , t R r T t

Γ (22)

where functions R

_mk

(r ) are obtained as solutions of the Bessel’s equation

( ) ⁰

1

2 2 2 2

2



=













−

∂ + + ∂

∂

∂ R r

r m r

R r r R

mk mk

mk

mk

γ (23)

where γ

_mk

are separate constants. Moreover, the following conditions are satisfied

( ) ^{< +∞}

→

R r

lim

_mk

r 0

, R '

_mk

( ) b − µ

0

R

_mk

( ) b = ^{0 (24)} The solution of equation (23) takes the form

( ) ^{r C J} ( ^r ) ^{C Y} ( ^r )

R

_mk

=

1 _m

γ

_mk

+

2 _m

γ

_mk

(25)

where Jν(⋅) denotes the Bessel function of the first kind of order ν. Using the first condition (24), C

2

= 0 was received. Therefore the solution of Eq. (25) is

( ) ^{r C J} ( ^r )

R

_mk

=

1 _m

γ

_mk

(26)

Substituting the functions ^R

mk

( ) ^r into the second condition (24) one obtains

( ) (

0

) ( ) ⁰

1

− + =

− mk m mk

m

mk

J b m b J b

b γ γ µ γ (27)

The equation (27) is then solved numerically with respect to the unknown γ

_mk

. Note that the functions R

_mk

(r ) satisfy the orthogonality conditions:

( ) ( )

∫ =

b

' mk m mk

m

r J r dr

J r

0

γ 0

γ ^for k ' ≠ k (28)

( ^r ) ^{dr b J} ( ^b ) ^J ( ^b )

J

r

_{m mk m mk}

mk b

mk

m

γ γ

γ γ

1

0 2

2

⁺

∫ = for k = 1, 2, ... (29)

Hence, taking into account (26) in Eq. (22), the function Γ

_mn

( t r , ) is obtained in the form

( ) ^{= ∑}

^∞

( ) ( )

=1

,

k m mk mnk

mn

r t J γ r T t

Γ ⁽³⁰⁾

where ( ) ( )

( ^γ ) ^γ ( ^{ρ γ} )

^κ

⁽

^{β γ}

⁾

^{( )τ}

γ

π

₊ ^{− + −}

−

=

^t

mk m mk m

mk m mk

mnk

e

^{n mk}

b J

b J

J b

t a

T

^{2 2}

1

Finally the Green’s function G has the form

( ) ( ) ( ) ( ) ^{r , t z cos m} ( ^' )

, Q , ' , , t , z , , r G

m n

n n mn n

φ φ ζ

ψ ψ Γ

τ ς φ ρ

φ = ∑ ∑

^∞

−

−∞

=

∞

=1

1 (31)

The temperature distribution T(r, φ ,z,t) is expressed by the Green’s function G as follows

∫ ∫ ∫

= ∫

^t

d

^b

d d

^h

g ( ,

^'

, , ) G ( r , , z , t ; ,

^'

, , ) dz )

t , z , , r ( T

0 2

0 0

0

π

φ ρ φ ς τ φ ρ φ ς τ

ρ τ

φ (32)

After evaluation of the integrals in the space domain and using Eq. (2) one obtains the temperature T ( r , φ , z , t ) in the form

= ∫

^t

G ( r , , z , t ; r , ( }, h , ) d )

t , z , , r ( T

0

0

ϕ τ τ τ

φ Θ

φ ⁽³³⁾

Substituting the Green’s function (31) into Eq. (33) gives

∑ ∑ ∑

^∞

=

∞

=

∞

=

=

1 1 1 0

m n k

n mnk n

mk mk

mnk

J ( r ) J ( r ) ( z ) ( h ) ( t ) b A

) a t , z , , r (

T γ γ Ψ Ψ I

π

φ Θ ⁽³⁴⁾

where

) b ( J ) b ( J A Q

mk m mk m

mk n

mnk

γ γ

γ

1

1

+

=

τ τ γ

β κ τ

ϕ

φ ( )) Exp [ ( )( t ) d (

m cos ) t

(

_{n mn}

t

I

mnk

⁼ ∫ ^{− −}

²

⁺

²

⁻

0

and γ

mk

are roots of the equation (27). The integrals can be evaluated through a series expansion [4] or numerically.

Conclusions

In this paper, an analytical model to describe the three-dimensional temperature field for a circular plate with a heat source which moves over its surface was estab- lished. The moving heat source causes cyclic heating of various plate areas. The temperature distribution in the considered plate in an analytical form was obtained using the time-dependent Green’s function.

References

[1] Lu X., Tervola P., Viljanen M., Transient analytical solution to heat conduction in composite circular cylinder, International Journal Heat and Mass Transfer 2006, 49, 341-348.

[2] Khobragade N.L., Deshmukh K.C., Thermal deformation in a thin circular plate due to a par- tially distributed heat supply, Sadhana 2005, 30, 555-563.

[3] Khobragade N.L., Deshmukh K.C., An inverse quasi-static thermal deflection problem for a thin clamped circular plate, Journal of Thermal Stresses 2005, 28, 353-361.

[4] Kidawa-Kukla J., Temperature distribution in a rectangular plate heated by moving heat source, International Journal Heat and Mass Transfer 2008, 51, 865-872.

[5] Beck J.V., et al, Heat Conduction Using Green’s Functions, Hemisphere Publishing Corpora- tion, London 1992.

[6] Duffy D.G., Green’s Functions with Applications - Studies in Advanced Mathematics, Boca Raton, London, New York 2001.