thermal deformations of a sphere by radiation from a point source

THERMAL DEFORMATIONS OF A SPHERE BYRADIATION FROM A POINT SOURCE

ICY HARINDFR SINGH * AND A. SINGH **

Received October 8, 1971

(Communicated by Dr. B. R. Seth, F.A.SC.)

ABSTRACT

In this paper the temperature and thermal stresses set up in an elasticsphere due to a point source at a finite distance from the centre of thesphere and outside it, have been studied. The final results are obtainedin terms of series involving Legendre polynomials. The nature of the stressand temperature distributions have been studied for two different distancesof the point source from the centre of the sphere Numerical calculationsdone on an IBM 1620 Computer have been represented in graphs.

NOTATION

_( .

where x-, y-, z- form cartesian co ordiwate system withorigin at the centre of the sphere.

D — the displacement vector.

T — the excess of temperature over that of zero stress andstrain.

A, µ — Lame's constants.= A/2 (A + µ) Poisson's ratio.

a — coefficient of linear expansion.

8 =2(1 + v)a.

a — radius of the sphere.

* Lecturer, Mathematics Department, G. N. University, Amritsar (India).

** Reader, Mathematics Department, Punjabi University, Patia'ia (India).

A2 247

248 HARINDER SINGH AND A. SINGH

d — the distance of the point source from the centre of thesphere.

= a; d.

1. INTRODUCTION

THIS paper deals with the steady-state thermal stresses which arise in anelastic sphere by radiation from a point source at a distance d (a < d < co)from the centre of the sphere. It is also assumed that the sphere is losingheat to the surroundings by radiation from all over its surface. The surfaceof the sphere is taken to be stress-free.

The particular integral used for solving the thermoelastic equilibriumequations is same as used by J. T. Holden' in his paper where the sourcehas been taken at infinity. The residual surface stresses are removed byusing the method introduced by E. Sternberg et. al. 2

The numerical calculations have been done with the help of IBM 1620Computer by taking the first thirty terms. It has been found that if d islarge as compared to the radius of the sphere the results approach those ofRef. I for the temperature. However, there is no such agreement in thecase of stresses because of some mistakes or misprints in Ref. I. Thefirst mistake is that the second series on right-hand-side for aPq [eqs. (4.3)Ref. 1] should start from 2, not from 3. The second is that in the equationfor v 1 [eq. (5.1 , Ref. 1] the expression n 2 + (1 + 4v) n + 1 + v shouldbe replaced by n 2 + (1 + 2 v) n + 1 + v. After these misprints havebeen rectified there is complete agreement in our results and thoseof Ref. 1 in the limiting case d -^ oo.

2. STATEMENT OF THE PROBLEM

With the origin at the centre of the sphere we take the z-axis in the

direction of the source and k as unit vector in that direction. To ensuresymmetry with respect to z-axis, spherical polar coordinates (r, 0, ¢) havebeen used for the description of the problem.

In the steady state the heat conduction equation and the thermoelasticequilibrium equations are

V 2 T =0 (2.1)

O(O.D)+(i — 2v) V 2 D— VT=0. (2.2)

Thermal Deformations of a Sphere by Radiation 249

The temperature boundary conditions are

—K jr =— Ecos^ll+hT 0 <0 <arccosdo

= hT arc cos d° G B < 7r (2.3)

at r = a. K denotes the thermal conductivity of the sphere; h denotesthe heat transfer coefficient of the surface; E denotes the energy absorbed

per unit area per unit time by an element of surface held normal to k, andis the angle between the incident ray and the outward normal to the

surface of the sphere.

We now have

00— cosO— d° f ( s_ s+ 2

cos — l — 2d° cos 8 4 d° 2 L 2s — 1 l̂0 2s } 3 d0 Js=o

00

>: PS (cos B) _2J lsPs (cos B) (say). (2.4)3=U

Further the elastic boundary conditions for a stress -free surface are

Urr = are = 0 (2.5)

at r = a.

While solving the problem it is convenient to use the non-dimensionalquantities

KT ,p = r/a; ®_ , k = ha/K; u - -D/a and c' = z/a. (2.6)

3. TEMPERATURE DISTRIBUTION

Using the non-dimensional quantities as introduced in (2.6), theboundary conditions (2.3) take the form at p = 1

—— =—cos0+k® 0 <6 <arccosd°^p

= k® arc cos d° < B < i. (3,,1)

250 HARINDER SINGH AND A SINGH

To combine -these two conditions into one we introduce a function definedas

f (cos 0) = cosh 0 < 0 < arc cos d0

=0 arc cos d„<0<7r.

Developing the function ,f (cos 0) as a series involving. Legendre poly-.nomials, we have

f (cos 0) -_ E bnP,, (cos 0)

where

bn = I (2n + 1) f f (cos 8) P (cos B) d (cos 0)

3 (2n +1) f cos q, Pn (cos 6) d (cos 0)d^

00 1

(2n + 1) ? is f PS (cos 0) Pn (cos 0) d (cos 0).$=0 du

Making use of the results for integrals involving Legendre polynomiais 3

we get

bn = In [1 — 2d0 (P1 2 + P22 + ... + Pn__1 2) — d0Pr ' 2

+ 2 (P1P2 + P2PA + ... + Pn-iPn))12

Co

+ 2n 2 1 ^1 ^s sPs_4Pn (s

nP nP (S (s+ -

n) dOPSPnL ).—Oxn

(3.2)n = 0, 1, 2, 3, 4, 5, .. .

where in this equation P S , Pn , etc., are functions of d0 .

The temperature boundary condition (3. 1), therefore, becomes00

—_— bnPn(cos0)+kO 0 <7r (3.3)Jp.«u

at p = 1.


The heat equation satisfied by © has the solution

e = E anPn Pn (cos 0)n-o

where an 's are constants.

Substituting this in the boundary condition (3.3) we get

= bn l (n + k) n = 0, 1, 2, 3, 4, .. .

When do = 0, the as are same as those in (1). When n is finite the

sum "n-1

+ `ith term in (3.2) is finite and therefore the behaviour of seriess=o

for bn is same as the behaviour of 2 • When n is large the Factors=n}1

In =in n 1 don i — 2nn + 2 -{- 3 don+i (3.4)

is sufficiently small, as 0 < do < 1, and tends to zero as n --* oo. So inthis case we can take

001 sPs_iPn — 1PA-, — (s — n) doPSPn

bn = 2 (2n + 1) Z lg — (s — n) (s -F- n =^- 1) --8=U

Thus it is sufficient to study the series

` sPs_1Pn- -- nPsPn--t — (s — a) doPsP^, (3.5sn+ )2(21) (s-n)(s-i- n-I-1)

a=u

for the investigation of the convergence of the fog b,1 a.id ti;c^: ^, e or

a n .

Using the value of is from (2.4) the general term of (3.5) can be broken

up into six components:

s2 dos-' Ps-,Pn — s (s + 2) dos r`

(2s-1)(s—n)(s { a+1)' ( +3)(s—n)(s +n + 1)


nsd s-1 PsPn. - 1 n (s + 2) dos+1PsPn_1,— (2s— 1)(s—n)(s+n+l)' (2s+3)(s=n) (s+n+1)

sdo' PsPn (s -i- 2\,1S+2 Pspn(2s-1)(-1)' s+n+l)'

Now consider the series

1 (2n + 1) s2 dos-' Ps-iPn

2 ^j (2s — 1) (s — n) (s + n + l )a><o

00

(2n + l) !!z s2 dos 'P5- i2 (3.6)2n jea (2s — 1) ^n — 1 ) \

S n 1 + 1

Comparing (3.6) with the series with the general term PS = s-413 we getfor large s and all n

s2 do s p-1

s-1

Ids (2 — 1) (n — 1' ( s n 1 + 1

S2 s4/3 dos-1 PS-1

<1,

(2s-1)`n -1J(s n +1 )

as do < 1 and J Ps (cos 0) < '2%(s sisin B) for s large. Since the series

his converges absolutely so does the seriesa—U

00

s2 doS-1 Ps-1

a=e (2s — 1) U — O n 1 + 1 ^

Similarly we can prove that the other five component series convergeabsolutely, and their sums are proportional to either (2n + 1)/(2n 2) P. or(2n + l)/(2n) Pn- i . Both these factors are finite for finite values of n andtend to zero as n —. oo. Hence the series (3.2) for b„ is absolutelyconvergent.


Now consider the series

00©_ E anp Pn (cos B) (3.7)

n-0

with an = bnl(n + k). Since b,,, are all finite we can find a positive numberM such that I bn I < M and therefore

E anP Pn (cos 0) 1 <M E I Pn Pry (cos 8)/(n + k) 1. (3.8)s=0 n=0

Again comparing the series (3.8) with the series n w-- get for large nn=o

n413 n I n" 3 2Pn cos B)n + ( < I n { 1c Pn Vyr;r sin d l <1.

Hence the series (3.7) is absolutely convergent.

4. STRESS DISTRIBUTION

Writing (2.2) in the non-dimensional form w:. get

V(V .u)+(1 —2v) 0u— OT=O (4.1)

where

V = a p.

co

T — p ka ® _ 1nPn Pn (cos B) (4.2)w=e

^n = $Eaan /k. (4.3)

The usual procedure to find the solution of (4. 1) is to put

u = V0 4 This method leads to intricate calculatio.is. A simplermethod developed in Ref. 5 will be used here.

254 HARINDER Sm GH AND A. SINGH

We seek a solution of the form

u=ck

where V = 0, whieb gives

(4.4)

Now T is given by (4.2). Putting

00_^nPn+iPn+i (cos 0)

no

we get ti = çtk to be the solution of (4.2). The stresses corresponding tothis solution are

00

app =f2n2n 1 fn-2Pn-2 Pn (COS'I22

00

1 PnB— 2n 1 ^nP" (cos( 0)

no

100

_I^g^ Ln + 3 pnpn Pn (cos 0)

n-e

00

2n

— 2n — L In-2Pn 2 Pn (cb5'8)• n-2

00

Q^¢ _ — nPn Pn (coS 0)n=o

00

ap® —= — —12 2n_2 Pn ( ) (cos F)2n

ww=


Co

2: Bp_ P(l) (cos 6)(n+1)(2 ^ + ,^)

Q#f =&.p 0

where

P(i) (cos B) _ (sin d) ! OS 8) Pn (cos 0); app = app/1i., etc.

To make the surface, stress-free, as required by the boundary condi-tions (2.5), we make use of the Boussinesq solutions

2u=77; 17 2 4=0. (4.5)

2u = 0 (z'O) — 4 (1 — v) Ok; '71 0 = 0. (4.6)

We consider the harmonic functions

0 = pn P. (cos B) and 0 = pn Pn (Cos 0)

and denote the resulting solutions (4.5) and (4.6) by [A n] and [Ca] respec-tively. For convenience we construct the solution

[Bn] _ — (2n + 1) [Cn] + (n — 3 + 4v) [A 1 ].

itTt is found that the surface is stress-free under the solution

-^ -,U = u + [S]

where

[S] — 2 un [An] + IE vn [Bn]R-2 n=1

— $Ea (an,an_2

un— k L 2 (n 2 -̂- n+2nv+v+1) -- (2n-1)(n-1)J' n> 2 .

2)12+1k '2(n2 ±n+2n ±v+1)(2n-I-3)(n+1)'n-0.


The stresses at any point of the sphere are given by the relations

E 00 00

Qpp — 2n^n 1 Pn-2 Pn (cos 8) — 2^ + 3 Pn (cos 0)

s

n=o

00

+ E n (n — 1) unp"i-2 Pn (cos 8)n=2

00

—E n (n2 — 3n — 2v) v np'^- ' Pn_1 (cos 8) (4.8)

n00

BBB E 2f + 3 Pn (COS 8) 2n$n-1 Pn-2 Pn (Cos 8)

N0 w=s

00

+ 2 upn-2 [P', (cos 8) — n (n — 1) Pn (cos 0)]n=2

00

+ vnPn-1 [n (n2 + 3n + 3 — 2v) Pn_1(cos 0)way

— (n+4 -4v)P'n(cos 8)] (4.9)

Q^ f = — 2+' fnPn Pn (cos 0) — unPn-2 P'n-^. (cos 0)n=o a=2

00

+ 2' vnPn-1 [(n +4- 4 v) P'n(cos8)

— n (2n + 1) (1 — 2 v) P_1 (cos B)] (4.10)

00

= 24-2 n-2 P {z> (cos 8)pI — 2n-1P nR=:

00

Snpn1 )(2n+3) Pn ^l^ (cos B)(n+

n=1

00

[(n — 1) unpn-2 Pn( l ) (cos 0)n=f


2 +2v) Vpfl-1 P(i) n-1 (Cos B)] (4.11)

og^,=acbp =0

where

P' (cos B) = dd O) pn (cos B).

5. NUMERICAL EXAMPLE

The value of h lies between 1.8 x 10-4 and 3.3 x 10-4 cal. per sec.cmz C°6 and conductivity K is found to lie between 0.001 and I .00 cal.

o•

qd 9e°

go•

FiG. 1-a

per cm.sec. cm2 for metals and stone.' If a is measured in centimetersthen the range of k is found to be

1.8 x 10-4 a <k <3.3 x 10-' a.

258 HARINDER SINGH AND A SINGH

b

0

O

90

1P

i9 0 *

FIG. 1 -b

,. 1

12 /i

e- .)

for., 2•,ft r4vair-.b.

FIG. 2-a FIG. 2-b

The temperature distribution given by (3.7) and the stresses given bythe relations (4.8) to (4.11) have been computed for k = 0.1, 0.25.do = 0.01 and 0.1 on the surface of the sphere and for p = 0•5.

•014

0 12

'008

•004

0

I ux0-

-004

-•008


The results have been represented in graphs. The isotherms have alsobeen drawn in cross-section through z-axis for d„ = 001 and d, ) = 0.1.The calculations were done with the help of a hand calculator and an IBM1620 Computer at Department of Mathematics, Punjab University,Chandigarh.

FIG. 3

6. DISCUSSION OF RESULTS

As the source comes nearer the surface of the sphere the non-dimensionaltemperature O at any point of the sphere decreases as is clear fromFigs. 2 a and 2 h, which is due to the fact that the area exposed to the radia-tion decreases as the source approaches the sphere. However, the actualtemperature T may not have this behaviour because as the source is removedto a large distance from the sphere, the amount of radiation falling on anelement of area will decrease in the inverse square proportion to its distance


from the sphere. Comparing Figs. 1 a and 1 b it becomes clear that thecurvature of isotherms is greater at d =0l than at d 0 = 0.01. FromFigs. 2 a and 2 b, it is clear that the temperature is maximum for B = 0 andminimum for B = i•. The variation of temperature is more pronouncedfor 0 < 0 < ir/2 than for ir/2 <0 < ,r, this variation being similar fordifferent values of d.

--*I*

-•as

-•014

Je<b^

- ago

d=g jo

5'c at

5'

-'010

-°l^, nla lx Z73 S"/b

n

a -- b

Fiu. 4

The curves for the stress components tend to be symmetrical about0 = 7T/2 for small values of d0 . For do = 0 they are symmetrical about0 = ir/2 as in Ref. 1 (shown in figures by dotted lines). However, this sym-metry is not observed when the source is near the surface of the sphere.The asymmetry is more pronounced in case of stress component kaosj(fEa)at p = 1 (Fig. 4). The asymmetry is due to the fact that when d 0 is large,

•012

'008

. 004

o

<(^Q°

Ir 004

-'008


the terms in the series (4.8) to (4. 11) which change sign at 0 = IT/2 havecoefficients of significant magnitude. These coefficients decrease numericallyas do decreases and they vanish all together as do --0.

The stress component kaaP/(iEa), at p = 0 5 is maximum for 0 = 0and r, and is minimum for 0 = 7r/2 (Fig. 3). The stress component ka.9 1($Ea) is maximum for 0= 2Tr/3, minimum for 0 = ir/3 and vanishes at0 = ir/2 for small do (Fig. 5).

FIG. 5

The stress component ko99/(f Ea) is maximum for 0 =r and is minimumfor 0 = Tr/2 on the surface of the sphere (Fig. 4). For d„ = 0 maximumis attained both at 0 = 0 and n.

262 14ARINDER S1NGH AND A. SING I

REFERENCES

1. Holden, J. T. .. Quart. J. Mech. and App!. Math., 1962, 15.

2. Sternberg, E., Eubanks, R. A. Proc. of 1st U.S. Nat. Cong. of App!. Mech., 1951.and Sadowsky, M. A.

3. Copson, E. T. .. Theory of Functions of a Complex Variable.

4. Goodier, J. N. .. Phil. Mag., 1937, 23.

5. Fox, N. .. Proc. of Lon. Math. Soc., 1961, 11.

6. Bosworth, R. C. L. .. Heat Transfer Phenomena, Ass. Gen. Pub!. Phy. Ltd

7. McAdams, W. H. .. Heat Transmission, McGraw-Hill.

thermal deformations of a sphere by radiation from a point source

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