Thermal stresses in spheres—A basis for studying the grinding of preheated rocks

Int. J. Rock 5[ech. 3tin. Sci. Vol. 9, pp. 213-240. Pergamon Press 1972. Printed in Great Britain

THERMAL STRESSES IN SPHERES--A BASIS FOR STUDYING THE GRINDING OF PREHEATED ROCKS L . B. GELLER Mining Research Centre, D e p a r t m e n t of Energy, Mines a n d Resources, Ottawa, C a n a d a ( R e c e i v e d 15 A p r i l 1971)

A b s t r a c t - - I t is a well-established fact that heating the mill feed prior to mechanical c o m m i n u tion enhances the grindability of m a n y ores. However, n o detailed calculations have been published to date by m e a n s of which the severity of the thermal shock, which will initiate fracture in the feed particles, c a n be estimated. It is the purpose of this paper to do so. Results are presented in a graphical form. They cover the thermal stresses which are induced in spheres a n d in long cylinders w h e n these are subjected to linear heat transfer at their surface. These two shapes were selected because they provide a fair representation of the multiplicity of irregular shapes f o u n d in a n actual grinding system, since the thermal stresses induced in spheres and cylinders are believed to be of the s a m e order of m a g n i t u d e as those which arise in heated mill-feed particles. Therefore the required physical parameters, for the thermal section of a t h e r m a l - m e c h a n i c a l c o m m i n u t i o n system, such as the degree of heating a n d q u e n c h severity c a n be estimated on the basis of the present results. NOTATION a

= A/cp = thermal diffusivity = Biot n u m b e r = H R / ~ = h R

B c E

= specific heat = m o d u l u s of elasticity

h

= n/a

H q r R

= = = =

X

=

T T~ T, T(1) T(0) t a

= = = ~ ~ = = ~ ~ = = ~ ~

v p ~r ,r, % ,r* ~-

=

coefficient o f surface heat transfer heat flux radius outside radius o f a disk, cylinder or sphere. Also half-thickness of a slab or rectangular bar r / R = dimensionless radius temperature initial u n i f o r m sphere temperature temperature of s u r r o u n d i n g m e d i u m surface temperature core temperature time coefficient of linear thermal expansion thermal conductivity Poisson's ratio density thermal stress at a n y point in the b o d y radial stress or (% ~ tangential stress o (1 -- v ) / ( E a A T) = dimensionless thermal stress Fourier n u m b e r = a t / R 2 INTRODUCTION

THE effect that heating, prior to mechanical particles has been investigated

comminution,

on several occasions 213

has upon

the friability of rock

during the past 50 years or so [1-9].

214

L.B. GELLER

These experiments were performed on both laboratory and production-scale equipment. They touched upon both the fundamental and the economic aspects of the process. However, they paid little attention to calculations by means of which the severity of the thermal shock required to initiate fracture in given rock particles can be determined, These types of calculations seem to have been of greater concern to scientists, connected with the ceramic industry, who were concerned with preventing damage rather than in promoting it [10-19]. This paper provides some of the missing details so that the development of thermal stresses in spherical rock particles can be followed. These are then compared with those which obtain in other basic shapes, particularly in long cylindrical ones. It is suggested that the physical parameters of a practical thermal system can be obtained from these calculations because, if the system can develop thermal fractures in spheres and cylinders, it can also weaken other shapes likely to arise in practice. A few examples have been calculated. The numerical answers appear to be in line with experimentally achieved results. THERMAL STRESS CALCULATIONS 1. General r e m a r k s

The geometric shape of the individual particles being ground in any type of comminution equipment cannot be determined exactly. Nonetheless, it can be stated that these particles must include a large variety of shapes, and that practically none of these will be of a welldeveloped simple geometric configuration. However, for practical purposes it must be assumed that particles do in fact resemble simple shapes, because moderately straightforward stress calculations can be undertaken for only simple shapes. The comminution method of current interest involves a mechanical phase besides a thermal one. Calculations of thermal stresses in various simple geometric shapes are therefore of great interest. Spherical shapes are particularly important because an evaluation of previously published data shows that, from among a range of simple shapes, they are the hardest to destroy by quenching. Therefore, the physical parameters of a thermal system must be so chosen that the stresses induced in all particles treated by the system reach failure levels, even though the particles in question are all perfect spheres. It is assumed that the material in question obeys the critical-stress theory of failure which appears to explain one particular type of thermal failure, namely spalling [20]. Detailed thermal-stress calculations for spheres have previously been published for only the technically uninteresting cases of instantaneous temperature change in the surface layer and for constant heat flux across the surface; they have now been extended in this paper to the more practical case of linear heat transfer at the surface. This is the case wherein q, the flux of heat across the surface, is proportional to the temperature difference between the surface and the surrounding medium so that q -- H ( T -

Ts) cal/(cm2sec)

where T - instantaneous surface temperature T~ -- temperature of the surrounding medium H -- coefficient of surface heat transfer Moreover this paper also deals with the question of the desirable range for the equipment used for developing the thermal shock required to nucleate cracking.

THERMAL STRESSES IN SPHERES

215

2. Previous calculations

Although the foundation for the study of temperature and stress distributions in thermally affected bodies was laid at least as long ago as 1838 by T. M. C. Duhamel in his memoir submitted to the French Academy of Sciences, the theoretical solutions were not converted into the type of detailed results which are suitable for practical reference until very much later. This happened in the 1920's when charts were first drawn up by means of which it became possible to follow the temperature and thermal stress histories within solid shapes as they were cooled or heated. These solid shapes included slabs, square bars, cubes, cylinders (both solid and hollow, short and long ones) and spheres (both solid and hollow). The temperature distributions in these shapes have by now been well documented [21-25]. One of the most interesting results of these calculations, as far as the present problem of I

Slab = Instantaneous mater ial temperature

2 - Square bar of infinite lenc:th

T

3 - Cube 4 - Cylinder of infinile length

To-Original uniform material temperalure Ts = Conslant temperature of cooling medium T Fourier number

5 - Cylinder of length = diameter 6 - Sphere 09 I

08 b 07F

J

0.4~-

2

o3~

\'

,

\ \

z

.~

c2-

" ' , ~-~ • oO

---~" 0-1

0"2

0-3

"---5"----_ 04

05

0-6

0-7

0'8

DIMENSIONLESS TIME~ 7-

Fie,. l. Rate of cooling along central axes (or at centres) of various solid shapes, after instantaneous temperature changes of To -- T~ on their surfaces [23]. studying the preheating of mill feed is concerned, is reproduced in Fig. 1. In it the instantaneous temperature T along the central axis (or at the centre) of several solid shapes is recorded, when the temperature at the surface of these bodies (which are originally at a uniform temperature To) is instantaneously altered to Ts, i.e. to that of the surrounding medium. Subsequently the surface temperature is maintained at T~. The other symbols in Fig. 1 are: T = Fourier number = at/R 2 a = thermal diffusivity t = time, elapsed since instantaneous change in surface temperature R ~ half-thickness of slab, of square bar or of cube also outside radius of cylinder or of sphere The significance of Fig. 1 is that it illustrates the fact that, if the surface temperature of variously shaped but similarly sized bodies of identical material is suddenly altered, it is in

216

L. B. GELLER

the spherically shaped bodies that uniform temperature is most quickly re-established. True, the calculations upon which Fig. 1 is based assume an instantaneous surface temperature change. However, the foregoing conclusion regarding the relative rate of temperature equalization is just as valid in case of a surface temperature change of finite duration. Consequently it may be assumed that, among various basic shapes, the spheres are most resistant to fracturing caused by thermal shocks. If the thermal shock conditions of a system are sufficiently severe to induce failure stresses in spheres, it can be assumed that within this system particles of other shapes will also fail. Unfortunately, though transient temperature histories have been well documented for many situations of technical interest, thermal stress calculations are not so complete. For example, many of them assume the technically uninteresting case of instantaneous temperature rise in the surface of the body subjected to thermal shock; others are based on linearly varying surface temperatures and on steady flow conditions. These calculations provide thermal stress histories in bodies such as spheres, cylinders and slabs [26-29[i. Certain cases have, however, been investigated under conditions which are of practical interest. In these the material is subjected to transient thermal stresses caused by transient temperature gradients, which depend upon H (the surface heat-transfer coefficient of the medium) and on a (the thermal diffusivity) of the material--and hence on A (the material's thermal conductivity) as well. Unfortunately in these cases of practical interest detailed thermal stress histories seem to have only been published for slabs and cylinders [30--32] but not for spheres. For the latter case insufficient data seem available in that this author knows only of publications of Cm4,NOALLand GIN6 [l 3] and of HASSELMANand CRANDALL[15]-which cover the case of'convective' heating, with a surface heat,transfer coefficient independent of temperature--besides HASSELMAN'S[14] work which deals with a constant flux into the sphere. However, CRANOALLand GIN6 [13] only derived a 'semi-empirical' equation for the thermal stress, while HASSELMANand CRANDALL[15] gave hardly any details concerning the results of the relevant thermal stress calculations. As for HASSELMAN[14], while the problem is correctly formulated, the deduction [from equation (10)] that the stresses at the centre of the sphere rise to a constant terminal value, seem to be in error. Actually the stresses initially increase, and then decrease to this terminal value. In any case HASSELMAN'S work [14] is based on radiation heating, while the present ~uthor s main interest is centred on linear heat transfer. It is, therefore, necessary to augment the data published to date before a meaningful practical estimate can be made of the thermal shock conditions required to initiate fracturing in mill'feed particles.

3. Calculation of thermal stresses in spheres with linear heat transfer at their surface (a) Relevant equations. The first task in calculating the thermal stress distribution within a sphere is to determine the temperature distribution in it. For constant material properties and for the case of heat flow in a spherical geometry (Le. for the case wherein the initial and surface conditions are such that the isothermal surfaces are concentric spheres, with the temperature thus depending only on the co-ordinates r and t) the equation of conduction is ([25], p. 230): 8t where0 ~< r < R.

-

a

\6°r 2

+

-

r ~r !

(11

THERMAL STRESSES IN SPHERES

217

I f the sphere radiates into a m e d i u m of constant temperature T~ and if the constant surface heat transfer coefficient is H, then the b o u n d a r y conditions are ([25], p. 237):

8T H 8-7 + - h ( T -

T~) = 0 a t r =

Rfort

> 0

(2)

and T=T~at

t ~ 0forallr

because the sphere is initially at the uniform constant temperature T~. The solution to the differential equation (1), when subjected to the given b o u n d a r y conditions (2) and when T~ = 0, is ([25], p. 238):

-- £

/3n2 q- (Rh 1) 2 sin (/3.) sin (ry.) exp ( - - a y,,z t) ,),n2[/3n 2 -~- Rh(Rh -- 1)] -

T -- 2hT~r

-

(3)

n

or in a non-dimensional f o r m : T _ 2B ~ r,

exp (-- r/3. 2)

x

/32 -k (B -- 1) 2

sin (/3.) sin (Xp.)

/3.2[/3.2 + B ( B - - 1)1

(3a) •

n

where indicates s u m m a t i o n commencing with n = 1 tl

h = H/A = relative surface heat transfer coefficient /3. = Ry. v,, = roots of the transcendental equation: Ry. cot (Ry.) -t- (Rh -- 1) = 0

X = r/R = dimensionless radius r = Fourier n u m b e r = at/R z B = Biot number = hR I f the initial uniform sphere temperature Ti is zero and the m e d i u m temperature T, is not zero then:

T = T~ -- 2hT~r

~1

exp ( - - aT,. z t)

~n2 -+(Rh --@ Rh(Rh

~/n2[/3n 2

I) z sin (/3.) sin (ry.). -- 1)]

F o r the surface (r = R) and core (r = 0) temperatures, equation (3) can be simplified. If the sphere is initially at T~ ¢ 0 and the medium is at T~ = 0 then the surface temperature is :

2hT~ T(I) = ~

exp ( - - ay. 2 t) n

3. 2 ~- (Rh -- 1) 2 y.z[/3~ 2 -k Rh(Rh -- 1)1

sin 2 ( ~ )

(4)

sin (p,).

(5)

and the core temperature is:

T(O) = 2hTi ~ " exp ( - - ay, 2 t) tl

fi 2 _}_ (Rh -- 1) 2 y.[/32 + R h ( g h -- 1)]

The surface temperature T(1) and core temperature T(0) are plotted in Figs 2 and 3

L . B . GELLER

218

respectively. They were obtained by rearranging charts published by GROBER [2l 1- The parameter r is the Fourier number (at/R 2) while B is the Biot number (hR).

Temperature 'Ts ot surrounding medium }s zero

Ti= H~IHa[ unlfoFm sphere temperoture 1.0

0

.

8

~

0.6

o

.

o •

~

U_ 0.4 0"5 06 DIMENSIONLESS TIME,

"

"-~-7

0'8

- .... 09

~0

Flo. 2. Time dependence of sphere's surface temperature T(1) [21].

TemperahJre 'Ts of surrounding medium is zero :initi'al uniform sphere temperature

08

8:02

0'6

04

I L

~

Od

OZ

03

04 3-5 0.6 DIMENSIONLESS TIME. "C

0-7

0-8

0.9

0

FIG. 3. Time dependence of sphere's core temperature T(0) [21].

With the temperature distribution thus determined, it is possible to calculate the thermal stress distribution within a sphere. The stresses developed are radial ~, and tangvntiaI %. They are expressed by the following equations ([30], p. 302) :

R

a, :

1 - ~ "v

Tr 2 0

r

dr -- ;5

Tr 2 dr 0

(6)

THERMAL STRESSES IN SPHERES

Oo

aE =

%

=

1 -

1

2

Tr 2 dr %. ~

219

(7)

Tr 2 dr - -

v 0

0

where = thermal stress in polar co-ordinates, with r being distance from origin, 0 the latitude and ~0 the azimuth v = Poisson's ratio a = coefficient o f linear thermal expansion E = modulus o f elasticity The surface stress can be written as: or(l) = 0 R

Oo(1) = o,.o(1) --

1 --

(8)

v 0

while the core stress is: R

o,(0)= %(0)=

o~o(0) - - 1 - - v - ~

(9)

Tr2 dr - - - ~ T(O) . 0

To obtain actual values for the thermal stress distribution within a sphere, it is necessary to substitute equation (3) into equations (6)-(9) and to evaluate the indicated integrations. On integrating by parts it can be shown that: r

f

dr = 2hr,

exp ( -

t)

sin fl. [sin(ry.) 1)]

y2[fl.. %.Rh(Rh_

[

r cos(ry.) ]

y2

~

3

0

(lO) with R

f

Tr2dr

0

differing from the foregoing expression only in that R replaces r in the final term in square brackets. Therefore. since Ts = 0, the dimensionless thermal stresses in a sphere can be expressed by the following equations:

~,(1

--

at* - -

Ea~

~

v)

-- 4B r~

2

fin2 %. ( B

--

1) 2

exp (-- rfl. ) f l 4 - ~ j ~2 ~[-B _ 1)1 sin ft.

n

×

B sin 3, -- ~

[sin (X3,) -- fl, X cos (Xfl,)]

(11)

and ~0" = %* -- °°(1 -- v) EcLTe

×

fi2 %. (B

%(1 -- v) --

EaT~

-- 2B

exp ( - - ~fi2)

2B sin fi, + ~ 5 [sin (X~,) -- fi, X cos (X3,)]

/3,,4[3,, 2 +

3. z s i ~ ! X 3 . ) }

--

B(B

l) 2

--

1)]

sin fi,.

(12)

220

L. B. GELLER

where sin 3, -- 3, cos 3, = B sin 3,. The surface stress can be obtained from equations (1 l) and (12) by putting X = I. It is: ~*r (1) - 0 ~o*(1)(1

o0*(1) = % * ( 1 ) =

~)

oJ(1)(l

EaT,

2B ~_~ exp (

-

~)

EaT,

r3. 2)

3"2 + (B -- l) 2 sin B.[3B sin 3. fl4[~2 _+_ B(B 1)]

/3.2 sin/3.].

(13)

tl

To calculate the core stress it is necessary to substitute equations (10) and (5) into (9). As a result the core stress can be expressed by the following equation: ~r*O) = ~0"(0) = ~ j ( 0 ) =

= 4B

exp (--

a,(0) (1 -- v)

EaTi

f12 + (B -- 1)z sin 3,[a sin 3, T3n2) fl413. 2 -~- B(B -- 1)]

(1/3) 32].

(14)

The foregoing thermal stress equations are valid for both heating and cooling. Howeyer, the two cases differ by a change in sign of the stresses. When the sphere is cooled on its periphery the radial stress is compressive throughout and vanishes at the peripheral surface, while the tangential stress changes from tension on the peripheral surface to compression at the centre. At the centre radial and tangential stresses are equal. When the sphere is heated the sign of the stress fields is reversed. It should be noted that in this paper tension is positive and compression negative. (b) Range of parameters. In order to obtain numerical results, it is necessary to calculate a certain number of the roots 3, of the transcendental equation: 3 . c o t ( 3 . ) ~- ( B - - 1) = 0.

The first six roots were published by CAmSLAWand JAEOER([25], p. 492). Six roots, however, do not provide a meaningful answer for very low values of dimensionless time r. In this case it is estimated that at least the first 35 roots need to be known. These were calculated by Mueller's iteration method (i.e. by bisecting). They are tabulated in Appendix A. Having obtained the necessary number of 3, roots, the thermal stresses in a sphere can be calculated by means of equations (11) and (12), or by means of equations (13) and (14) if only the core and surface stresses are required. The ranges of parameters ~-and B for which this needs to be done are determined by estimating the maximum and minimum values o f H, A, R, a and t which are likely to arise within the system used for preheating the rocks and ores. These estimates are as follows: (1) Heat transfer coefficient, H, is 5, 74, 1000 and 1440 Btu/(ft 2 °F hr), i.e. 0.000678, 0.01002, 0-1356 and 0" 195 cal/(cm 2 °C sec), for heating (or cooling) in still air, salt water, boiling water and in ice water respectively ([15], [20], [32]). (2) Maximum and minimum thermal conductivities, A, are 2-66 and 1.06 Btu/(ft°F hr), i.e. 0.011 and 0.0044 cal/(cm°C sec), respectively, for a particle temperature of 200°C [20].

THERMAL STRESSES IN SPHERES

221

(3) Outside radii of the maximum and minimum rock particles of interest are about 1 in. (25.4 ram) and 0.0041 in. (0. 104 mm), i.e. 65 mesh, respectively. (4) Maximum and minimum thermal diffusivities, a, are 0.0609 and 0.0262 ftZ/hr (i.e. 0"0157 and 0.00676 cmZlsec), respectively, for a particle temperature of 200°C [20]. (5) Maximum and minimum times, t, are, say, 60 sec and 0.5 sec. Consequently, the Fourier number, z, varies from a maximum of 8720 (for a slowly cooled 65-mesh quartz particle) to a minimum of 0.000524 (for a fast-cooled 2-in. diameter anorthosite sphere), while the Biot number, B, varies from a maximum of 118.5 (for a 2-in. diameter anorthosite or syenite sphere quenched in ice water) to 0.000645 (for an aircooled 65-mesh quartz particle). Actually the programmed ~- range only extended from 0.001 to 10 because even an upper limit of r = 0-3 (as used on the graphs), is quite large enough for all practical purposes. This is so because well before this dimensionless time both the radial and the tangential stresses have peaked for all B-values of interest and in all sections of the sphere. Moreover, B was not chosen less than 0.005, because only insignificant thermal stresses are generated at very low Biot numbers. The dimensionless radius varied from 0 to 1. (c) Graphical representation of results. All relationships are in a dimensionless form in the following graphs. As mentioned before, positive stresses are tensile and negative ones are compressive. Therefore the graphs illustrate quenching of a hot sphere. For heating the given sign convention is reversed. Figures 4 and 5 illustrate the distribution of the tangential stresses across the sphere at various times for two representative heat transfer rates, namely for B = 5 and for B = 50. Stresses are positive at and near the surface and are negative near the centre. At all times the maximum positive tangential stress is at the surface and the maximum negative one is at the centre.

, ,

/

~

HR m =

.

,

<:2

r

b%

'

/I

.

o

.

.

.

.

.

:=____+~ . . . . . .

.

.

~_:_<-_I A _ _

~ .

~

y

'

r:o ooz

.-" / . . / /

+ 7 ~ - - ~ ~--

cl

- o - 5

~

O0

~8~

,j/ //.~r~o,5o

|

~:~i, -

"Z

t

OI

02

0-3

05 0.6 DIMENSIONLESS RADIUS, X = ~ -

07

08

09

0

FIG. 4. Tangential stress distribution across the sphere at various times, for B = 5.

222

L . B . GELLER

+0,7 06

05 t--

~ ~

jl

0-4

f<

i~

r:O,05

@.~ o~ ,

. ~.

Ell

.,-.o.,

0~9

I.'o

0-1

/oo ;or

.-'7 , ' / 2 '

- - . - - L - - . ~ . - - ~ _ ~ - ~

.~-~-.~-/-7/-~-.-a~_ -.

v__/~ e

--0'4

r

0,~

~

oz

0.3

0'-4

i

o.s

,

o.o

017

0'.8

DIMENSIONLESSRADIUS~X: "~ FIe. 5. Tangential stress distribution across the sphere at various times, for B = 50,

Figures 6 and 7 illustrate the distribution of the radial stresses across the sphere at various times, for B = 5 and B = 50 respectively. As may be seen the radial stresses d o n o t change sign. They increase from zero at the surface to a maximum negative value at the centre, the latter value equalling the tangential stress at the centre.

-0,30 Bier numbe(, B=

- ~ - =5.0

T:o.08

~-0-25 •_.•_•i

b=w b=

- -0-20

-045

Ik

T=O.02 c~ -010 0

-0.05 Q -O.Oi

0"0

0'1

0'2

0'3

0'4 0"5 0"6 DIMENSIONLESS RADIUS~ X = ~ -

0,7

0-8

09

!' 0

FIG. 6. Radial stress distribution across the sphere at various times, for B = 5.

THERMAL -c4o

STRESSES IN SPHERES

F Biot ~umber~ 5 : r:c 05

~15

_ ~+o

5~

-325="

*--

)

a: ~d

223

.....

'. . . . . . . . . .

H~ A

- 500

~ . . . . . . . .

-0

y,

-, OC

OI

02

O~

04

05

,2+6

07

D I M E N S I O N L E S S RADIUS, X ~ r

08

09

+C

R

FIG. 7. Radial stress distribution across the sphere at various times, for B = 50.

Figures 8 and 9 illustrate the time dependence o f the tangential stresses at various distances from the centre, for B = 5 and B = 50 respectively. As may be seen all stresses rise to a peak and then die away to zero. However, while those in, and close to, the surface and the core do this without a change o f sign, some o f those at other cross sections do change their sign once before approaching zero. The maximum positive stress is always at the surface, while the maximum negative one is always at the centre. +04

i 8Jol number, B=

HR ~- : 5 0

v]~

~L

2-o~ 0

-- 0

5

0 0

. . . . . .

~

I. . . . . . . . . . .

'05

.

~

'lO

~

t

-

_ _

L

-15

D MENSPONLESS T ; M E

.

20 '

.

_ _ .

.

.

.

.

.

[

J

25

50

T: at

R2

FIG. 8. Time dependence of tangential stresses across the sphere, for B : 5.

224

L. B. GELLER L:Iiot numbet~B = ~

:

500

+ O' 7 ~,~

0.6 f\ 0"5I / / ~ 0.4 *b..O" 0"3 ~- 0.2 /_../_...0-9

-1--

~

o.o~/. . . .

.>~----"

_:.:

- - - ~ _ ~ _

0.2 o.3t-

~--~o:~-y-o.,

--0,4L O0

l

i -I0

"05

I ,15

DIMENSIONLESS TIME i

I

i

.20 T=

I .30

'25

al

FIG. 9. Time dependence of tangential stresses across the sphere, for B = 50,

Figures 10 and 11 illustrate the time dependence of the radial stresses at various distances from the centre for B -- 5 and B -----50 respectively. As may be seen all radial stresses rise to a peak and then approach zero, as do the tangential stresses. However, they never change sign, remaining negative at all times. The maximum radial stress is always at the centre.

Biot numbe G B : ~ -

: 5.0

×=~-=o

-0.3 b~

to

-O.Z

-0-1

r o

0"0

..... I

0-0

-05

----

-I0 q5 .20 DIMENSIONLESSTIME~ T=~Z

---" ,25

Fro. 10. Time dependence of radial stresses across the sphere for B = 5,

-30

THERMAL STRESSES IN SPHERES

x:~:o

7

225

Bioi number, B = - ~ : 50.O

b ~ uJ

0-3

-0-2 a~

~l -o.~ t~

-~-

o txl

0-0

- -.L._ - ----__ _ ~ _

i

i

•05

"10

_

_

_

_

_

x

_

.....

-15

=--~_

i

r

-20

"25

_

-

_.._ ._t

'30

of

DIMENSIONLESS TIME 7 T = - ~ -

FIG. 11. Time dependence of radial stresses across the sphere for B = 50. Since the m a x i m u m positive stresses always occur at the surface (tangential stresses) while the m a x i m u m negative stresses always occur at the centre (tangential stresses and radial stresses, the two being equal), the time dependence o f the surface and o f the core stresses is o f special interest. Figure 12 illustrates the former and Fig. 13 the latter for a range o f Biot numbers B. As m a y be seen both the surface and the core stresses first peak and then die away to zero. The surface stresses do so sooner than the core stresses (since

0.9 B=IO00

0"8

Z ~

- °.,IW

~ 8

~

~

40

o6 o5 k

,~

o.~V

",\

W %__ I/

o.,

F/~

o.o L - - 0'O

--

--

-~'~ .

, -05

" .

.

" .

.

.

.

.

~----~_T

.

--,- . . . . . . . . . . . . . .10 "15 '20

-_

_/_,~.~ >"4

-~- - - - - E ~ , g o o "25 "50 O'OB

DIMENSIONLESS TIME, T = ~af-

FIo. 12. Time dependence of tangential surface stresses for range of Blot numbers B. ROCK 9 [ 2 - - E

226

L. B. GELLER / 0 " m a . ( 0 ) = - ' 3 8 6 (at T = - O 5 8 Z )

/

/

-0.40

~,

*~°~'O- - 0 " 3 5

,~

_

_

_

..o=

-0.30 x

b~ -0-25 l-co w~: ©

uJ -J

-0,20

~[

-0.is --0"10

bJ -0.05 ~'< IO00

,,

0'0 0,0

"05

-I0

.I5

DIMENSIONLESS

~-~OI

20 TIME,T=

'25

"30

at Rz

FIG. 13. Time dependence of core stresses for a range of Bjot numbers B. the sphere cools inward from its surface). Moreover curves with high B values peak far sooner than do those with low B values. These facts are brought out more clearly in Fig. 14 which records the peak stress values at t h e sphere's surface and at its centre, together with the times at which these maximum stresses occur.

~ 0J.

to

b ~

*b ~

lx i-o

• o.zso

o9

O-~a

on sur face

02ZS

o~ o7

os

o,)

~ .

_ ~ -- . . . .

-- . . . . . . . . . . .

oo~roaches 0586

- o,oz~

ol

_

_

r on sur fclce

810T N U M B E R , 8 = h R

I:~G. 14. M a x i m u m su~ace and core strc~s and theirtime of occurrence.

THERMAL STRESSES IN SPHERES

227

4. Sphere with constant flux at its surface So far in this paper thermal stresses were discussed which arise in spheres with linear heat transfer at the surface. The corresponding boundary condition was defined by equation (2). It has been suggested that under certain circumstances this type o f 'convective' heat transfer may not contribute more than about 10 per cent to the total heat flux into the heated body [14]. In this case heating by radiation (wherein the flux depends on the difference of the fourth power of the absolute temperatures) would have to be considered. This case, it was suggested [14], could then be approached by utilizing the expressions obtained for a sphere with a constant flux at its surface, because the heat flux into a body is constant within about 5 per cent as long as the relative surface temperature, i.e. the ratio of the receiving body's surface temperature, T(1), to that of the enclosure, Ts, does not exceed approximately 0.475. This condition has been found to be satisfied when thermal shock is applied by inserting the body (in air) into a heated enclosure. But the enclosure temperatures required to produce fracture by thermal shock are relatively high (1000°C or more). This process is probably not o f great practical importance for the case of current interest, namely for thermal-mechanical comminution, since to achieve worthwhile results a non-conventional heating system would be required. However, for completeness, the radial and tangential stresses induced in a sphere subjected to a constant heat flux are quoted below, as they were published by HASSELMAN [14]. (a) Radial stresses are:

~ ( 1 - - v) ~* --

Ect

qoR Il _ X 2 5A

20 ~

sin (XS,) -- Xfi, cos (X/3,)

-- X-'-'-~

/34 sin/3.

exp (-- r/3,2) I.

n

(15) (b) Tangential stresses are: a0*-- %(I -- v) Ea -- ~-

10 ~ ' (1 -- X 2/3, 2) sin (X/3,) -- X/3, cos (X/3,) 1 -- 2X 2 -- - exp (-- T/3n2)J. X a ~.~ /3,* sin/3.

q°RE

(16) (c) Core stresses are:

or*(0

2qoR (l A

10

2 3

/3, sin/?,

t

!"

(17)

All symbols in the foregoing equations have been used before, except qo which is the constant flux into the sphere for the case of zero initial temperature. For example if the enclosure is at an absolute temperature of say 1500°K, the material's absorptivity is 0.8 and the Stefan-Boltzmann constant is 1.37 × 10 -x2 (cal°K -4 cm -2 sec-l), then qo

1.37 × 10 -~2 x 0.8 x 15004

/3, are the positive roots of the equation: tan/3 = /3.

5.57 cal/(cm2sec).

228

L. B. GELLER

They are the roots listed in Appendix A for the case of B = 0, with the unlisted 35th positive root being 111.51757. However, it should be noted that the strcsses cannot be calculated by putting B ---- 0 in equations (11) and (12). EVALUATION OF RESULTS OBTAINED FOR THERMAL STRESSES IN SPHERES WITH LINEAR HEAT TRANSFER AT THEIR SURFACE

1. General remarks An evaluation o f the results presented in Figs 4-14 shows that in case o f thermally stressed spheres the critical stresses arise either on the surface or at the centre, depending on whether the spheres are quenched or heated. Perusal of Fig. 14 provides a convenient method for determining those physical parameters which any system must possess within which fractures are to be initiated in spherical shapes by means of thermal shock. This author contends that additional information is necessary before the required physical parameters can be determined in practice. This is so firstly because in order to essentially 'fail' a sphere it is not enough to merely initiate cracks at either its surface or centre. These cracks must also be propagated to a certain depth, How little practical effect shallow cracks do in fact have has been demonstrated, for example, by photoelastic experiments [33]. Secondly, although the results illustrated in Fig. 1 seem to show that if thermal conditions are sufficiently severe to fracture a sphere they are also severe enough to fail any other simple geometric shape o f practical interest, this point too needs further attention.

2. Quenching Figures 4-14 illustrate the case of thermal stresses produced by quenching. Radial stresses in this case are compressive in all sections and at all times. Tangential stresses, however, are tensile on the surface and at relatively shallow depths below it. Moreover, tensile peaks are about two to three times as great as the maximum compressive stresses at the centre. Therefore tensile tangential stresses are the critical ones. Failures will be initiated in the surface. whenever a'max(l) reaches a critical level. The conditions leading to failure initiation can, therefore, be ascertained from Fig. 14. However, to obtain more than initiation of cracking (i.e. more than just skin-deep crazing at the surface) these incipient cracks must be propagated to a certain depth. Exactly to what depth depends on the practical conditions of interest within the thermal-mechanical comminution system to be built. An indication o f the required depth may be gauged from the test results published by DAVIDGE and TAPpIN [18] for ceramics and by CHEN and MAROVELU [32] for rocks. Both show how initial surface cracking expands and deepens with increased heating of the specimens. In the case o f basalt disks, for example, CnEN and MAROVELLI [32] show that surface crazing commences at a minimum temperature difference AT (between heated rock sample and quenching medium) o f about 40°C, while some 150°C is required before a number o f these cracks are propagated near to the test disk's centre. Moreover they show that to propagate cracks clean across the test disk, i.e. for complete failure, a temperature difference o f about 250°C is required. From these tests it appears that a system which is required to 'fail' a specimen thermally should be able to raise failure stresses at depths well below the surface. On the basis o f the foregoing observations it may be postulated that failure conditions in the case o f thermally induced fractures can be calculated by using an experimentally established radius X'. This X' is to be the one at which the maximum theoretical tensile stress (neglecting modifications in the stress field caused by cracks) equals the tensile strength

THERMAL STRESSES IN SPHERES

229

o f the material for the conditions that induce complete fracture. As an example, on the basis of CHEN and MAROVELLI'S tests [32] with 2-~-in. diameter basalt disks quenched in boiling water, where AT for complete failure is 250°C, Z' is about 0.65. This is so because for these conditions R = 1.0625 in., ,~ ---- 1-32 Btu/(hr °F ft), H = 1000 Btu/(hr °F ft2), E = 14.5 x 10 6 psi, v = 0.15, a ---- 5 x 10 - 6 (1/°F) and tensile strength ---- 2450 psi. Therefore B = 67 and the dimensionless tensile strength is 0.076 (plane stress conditions). Consequently o ' * t o m a x must be 0.076 at the radius 2'. Therefore X' is approximately 0.65 ([32], Fig. 7). This provides a new criterion for the fracture of disks on quenching, that has not been studied experimentally. It should be noted that the use of the actual tensile strength value is arbitrary. Different criteria are obtained if different multiples of the tensile strength are used. The best criterion of this type can only be decided by experiment.

0 i +0 2 [

o v

o9 a:

!

r Dimensionless rodius, X= ~=O.8

~:,ooo 7

z

u~ J z o

0-5

i "O5

-03

O.O

, '10

r .15

, "20

T ,25

i -30

DIMENSIONLESS TIME~ T-~ ~'2

FIG.

15.

Time dependence of tangential stresses at depth X = 0.8.

To establish X' for the case o f spheres, experiments of the kind performed by CHEN and MAROVELLI on disks [32] should be conducted on spheres also. Unfortunately this has not been done, as yet. The tangential and radial stresses in spheres, at various radii and for a range of Biot numbers, have however been calculated. Those at X = 0.8 and X = 0 . 6 are shown in Figs 15-18. These stress functions are very similar to those which, under

b~

-0.3

Dimensionless

radius)

r X = -~- = 0 . 8

u~ Fo3

-0.2

B = ]000

,7 -0-I L.J J Z 9 O3 Z W

o O'o0

-05

i

i

i_

t

i

-I0

.15

"20

"25

'50

DIMENSIONLESS T I M E ,

T=

a~

R2

Fla. 16. Time dependence of radial stresses at depth X = 0-8.

230

L . B . GELLER Dimensionless radiu% X :

~-= 0"6

+0,2 I

-f-ooL

N -0.3

-

0-0

-

'05

-/O

.t5

'20

"25

30

DIMENSIONLESS TIME, t- = ~at

FIG. 17. Time dependence of tangential stresses at depth X

-

-

0-6.

identical circumstances, arise in cylinders. This fact is brought out more clearly by Figs 19-24, which illustrate the stresses in both spheres and cylinders at various cross sections and for various Biot numbers. X' could, therefore, be estimated in spheres using Fig. 25, if an appropriate AT were known. Although consideration of the new criterion introduced above indicates that this assumption may not be strictly true, it seems likely that a reasonable estimate of conditions required for fracture may be obtained by postulating an X' for spheres in the order of 0.75. Finally, it remains to show that if the parameters of a heating system are chosen in line with the foregoing requirements for spherical shapes, the conditions are then so severe that, ipsofacto, any other shapes will also be thermally shattered. In a thermal-mechanical comminution system such additional shapes can perhaps be represented by solid, relatively long, cylinders and by small slabs. Figure 19 illustrates the maximum stresses in each of these bodies under comparable circumstances. In every case the maximum stresses occur on the surface, because quenching is being considered. The results illustrated in Fig. 19 support the assumption that, among the shapes of practical interest, the spheres are the hardest to fracture by quenching because the lowest maximum stresses are developed in them.

o

0 .~

~:I0007

-°2r// a:

A

.

.

.

.

.

.

-O'l

g 0.0

05

-I0

,t5

.20

';)5

D MENSIONLESS TIME T= o_~I R2

FIG. 18. Time dependence of radial stresses at depth X = 0-6.

-BO

THERMAL STRESSES IN SPHERES

231

zf

L i n e I ~m'~x on s u r f a c e of slab 2 crmax on s u r f a c e of long c y l i n d e r @

L_

3 cTm~x on s u r f a c e of s p h e r e

I~ x

#.~,~#t#,tl

!4,

L i n e 4 z- on s u r f a c e of slab

SLAB

offer

BOLEY

5 ~- on s u r f a c e of long cylinder @

and W E I N E R

[30]

6 z- on s u r f a c e of sphere

SpHErE

® N B : Under plane-strain conditions o 250

LONG C Y L I N D E R

after CHEN and MAROVELLI

[52]

--------

~ -

.._.=.-_-..~__

b~

.. o2z~

oe

s

~

o6

,~

*3 LU 0.5

~E

04

©

-

3~~'~

o,oo

~" *b ~ o~ _o

;-

~<

_o

w

E

'

; ; i;i;,;

2'o

;o

4'o ;o/o7o~o;o;oo

BIOT NUMBER, B = ~ -

~

3?0 4005;0 . . . .

,~o

7" C3

or B = f f ~ -

FIG. 19. Maximum surface stresses--and their time of occurrence--in spheres, long cylinders and slabs.

3. Heating If thermal fracturing is to be obtained by heating rather than by quenching, the prevailing stress conditions can again be obtained by studying Figs 4-19, remembering though to reverse all signs. The large tangential surface stresses, therefore, now become compressive while the significant core stresses become tensile. Since the latter are now the maximum

- ----

Long cylinder, under plane strain conditions

offer PARKUS [31]

Sphere

BJot number, 8= ~HR : 5 O A

z) - - O.S

"-- ,~

~J 0-2 ,7 z 3-

0"0

'01

-02

'03

"05

-06

DIMENSIONLESS

'04

TIME ,

-07

-08

"09

-I0

T=

R~

FiG. 20. Time dependence of tangential stresses in spheres and long cylinders, for B = 5.

232

L . B . GELLER --*

Long cylinder~under plane strain conditions after PARKUS [3t]

--

Sphere

---

Biot number~ B= H_R : 5'0

A

b" -o.3 r v~

~0.~

o --

0.0

1

0-0

.OI

,02

.03

.04

-05

.06

.07

I

J - _ l

-08

,

.09

,!0

DIMENSIONLESS T I M E , .t.,°- at

FiG. 21. Time dependence of radial stresses in spheres and long cylinders, for B ~ 5. Long cylinder, under plane strain conditions CHEN and MAROVELLI [32] +0'8

--

--

Sphere

•

Biot number,

~4 ~'O"

8:

HR

" - ~ - : 5"0

0"6

%-e0.4

0"2

o.o

~

0.~ -0.4 0"0

-05

-I0

d5

-20

"25

"30

DIMENSIONLESS TIME, ~ = aj_l R2

F1G. 22. Time dependence of tangential stresses in spheres and long cylinders, for B

. . . .

-0.3

//~

°

~

~

50.

Long cylinder, under plane strain conditions offer CHEN and MAROVELLI ~32] Sphere Blot

b~

=

number, B~H-~R:s 0

r

/

-O,I

o

0.0

0-0

'

-05

"lO

~

-15

"20

'

-25

--'~

"30

D I M E N S I O N L E S S T I M E , "r= ~2

FIG. 23. Time dependence of radial stresses in spheres and long cylinders, for B = 50.

THERMAL STRESSES IN SPHERES

233

2 2" ot centre of sphere

to o

uJ"

~%

~

Line3 ~m~x o1 centre of sphere

bJ

4 %~x along centre line of long cylinder ®

o z

oi~

,o

Line I 2" along centre line of long cylinder ®

u.I

® NB: Under plane strain conditions

to

after CHEN and

~

MAROVELLI

©

~-

[32] o

0.9

"%%%

o-s

z'so

oeoo "%%

o.e

' ~ L

as

0150

~

o.r o~

opprooches 0-386 asB~

. . . . . . .

~

,

-7

= ~

it:':__

.2

5

.,-- ~

4 .5 -6 78.9k0

2

.....

opprooches 0 -235 os E]"-~"oo ~

~3

0.2

o l

0~2~

,

t

.

0050

' 3

4.

5

oo~,

20

B 75910

$0

BlOT NUMBER, B =

40 50 60 70 (lOgOlO0

~00

SO0 400 500

TO00

HR A

FtG. 24. Maximum core stresses--and their time of occurrence--in spheres and long cylinders.

+1,0

0-9

I000

"b; as io0

~ 0.7

50

~ 0.6 20

~ 0.5 Jo

~

0.4 5

0.3 0.2

t +0.1

0

05

0'7

0,8 DIMENSIONLESS

0.9 _

~'0 r

RADIUS, X- .-if-

Flo. 25. Maximum positive tangential stresses, at X = 0" 7 to X = 1"0, as a function of B.

234

L.B. GELLER

tensile stresses in the sphere (amounting to about ½ to ~ of the maximum compressive surface stresses), they are the critical stresses causing failure. Visual observation of the radial fractures and rosette fragments obtained after heating experiments on 1-3-in. diameter alumina spheres are said to have confirmed that these shapes do fail due to tensile fracturing, initiated at the centre [13]. Another point which bears out the foregoing conclusions is the experimentally well-established fact that identically shaped bodies of the same material are much harder to fragment by heating than by quenching. CHEN and MAROVELH for example [32] state that for their rock disk samples about twice as much temperature difference is required to initiate cracks by heating rather than by cooling shocks. WINKELMANN and SCHOa'T'S [34] experiments with 2-cm glass cubes showed that a ATof465°C did not produce fractures on heating, while on cooling the same cubes could only withstand a temperature difference o f 52.8°C without fracturing. This relative difficulty of fracturing by heating bears out fracture initiation at the core rather than at the surface of the spheres, firstly because the maximum tensile surface stresses in cooling are about 2-3 times as great as the maximum tensile core stresses in heating, and secondly because the surface always contains many more incipient 'Griffith' cracks than does the core of a body. Both theory and experiments, therefore, indicate that in heated spheres failure initiation is due to the maximum tensile core stresses. The magnitude of these stresses is illustrated in Fig. 14, together with their time of occurrence. Figure 13 illustrates the time dependence o f all core stresses. As an example, to shatter a 2-in. diameter quartz particle by heating, the necessary ATis 485°C and t -- 82.5 sec, assuming that H == 6.78 × 10- 4 cal/(cm 2 °C sec). For quenching it was postulated that 'failure' will not occur by simply initiating shallow surface cracks, but that these cracks must also be propagated a certain distance before the sphere has truly 'failed'. The question arises whether on heating too it is necessary to produce thermal shock conditions which are so severe that they will raise maximum tensile stresses equal to the material's tensile failure strength not only at the very core. but even at points a certain distance away from it. This author does not believe that this is so. True, in cases of quenching, surface tensile cracks can travel to limited depth. But, in cases of heating, spheres have been shown to always fracture completely, whenever an internal crack is initiated in them [13]. This fact has also been observed by the author in heated glass plates. Here too failure was always complete and explosive, whenever cracking started below the heated surface. The location o f the fracture origin could be established by visual examination o f the characteristic fracture pattern. Finally, the question remains whether it is satisfactory to base a heating system's physical parameters on results obtained from Fig. 14. As mentioned, the shapes in an actual thermalmechanical comminution system can be approximated by solid spheres, cylinders and platelets. Disregarding the latter, since they probably break down under mechanical grinding action, it is necessary to compare the thermal stresses arising within spheres and long cylinders, especially at (or near to) their centres. Figures 20 and 21 illustrate the tangential and radial stresses in these two shapes, for a heat transfer rate o f B -- 5. Figures 22 and 23 illustrate the same relationship for B = 50. These graphs display the fact that for relatively short times (below approximately r -- 0.15) the stresses in a sphere, at any distance from the centre, are algebraically less than the corresponding stresses in the long cylinder, provided that the surface stresses are regarded as positive. For longer times this is not necessarily the case, for both the tangential and especially for the radial stresses. However, it is the relatively short times which are o f interest because it is during these that both the tangential and radial stresses reach their maximum. In consequence the conclusion may be reached

THERMAL STRESSES IN SPHERES

235

that during thermal fracturing by heating the physical parameters of a system should be such that core stresses induced in spheres exceed their critical levels, in order that they may attain these values along the centre line of long cylindrically shaped bodies. Figure 24 illustrates the maximum core stresses in long cylinders, together with their time of occurrence. Therefore, when calculating the parameters of a heating system, Fig. 24 should be used rather than Fig. 14. SUMMARY AND CONCLUSIONS To date, grinding experiments with preheated mill-feed have not included the examination of the severity of the thermal shock required to initiate cracking. This may have been due to the lack of detailed calculations in this field. This paper endeavours to supply the necessary details so that the physical parameters of a cooling or heating system, required to serve as the thermal section of a thermal-mechanical comminution system, may be determined. Since relatively straightforward thermal stress calculations can only be executed for certain simple geometric shapes, it is assumed that the particles in an actual grinding system can be represented by spherical and cylindrical shapes. These shapes are likely to occur in the system for practical reasons; moreover, the thermal stresses induced in them embrace a range which is not likely to be greatly exceeded by the stresses in other shapes. A comparison between the thermal stresses induced in these two bodies reveals that in cases of quenching, the thermal system needs to be able to produce an environment in which critical stresses can be raised not only at the surface of a sphere, but also below it. Experimental work is needed to define these conditions better. In cases of heating, the requirement is to establish an environment within which critical stresses can be induced on the centreline of long cylinders. Thermal stresses were calculated only on the basis of linear heat transfer at the surface. However, constant flux conditions were also considered. These would apply if thermal shattering by heating, rather than by quenching, is desired. In practice though this is likely to involve non-conventional heating systems, such as jet flames, plasma torches, pulsating combustion, etc. which are not within the scope of the thermal-mechanical systems envisaged here. Acknowledgements--It is a pleasure to express the author's indebtedness to his colleagues, Dr W. M. GRAY for many stimulating and instructive discussions and Mr N. A. ToEws for assistance with the calculations and computer programming.

REFERENCES

1. YATESA. Effect of heating, and heating and quenching Cornish tin ores before crushing. Trans lnstn Min. Metall. 28, 41-45 (1918). 2. HOLMANB. W. Heat treatment as an agent in rock-breaking. Trans. Instn Min. Metall. 36, 219-234 (1926). 3. MYERSW. M. Calcining as an aid to grinding. J. Am. Ceram. Soc. 8, 839-842 (1925). 4. MITCHELLW. JR, SOLLENBERGERC. L. and MISKELLF. F. Factors in the economics of heat-treated taconites. Trans A I M E 193 (Min. Engng 4) 962-967 (1952). 5. BROWNJ. H., GAUDINA. M. and LOEBC. M. JR Intergranular comminution by heating. Trans A I M E 199 (Min. Engng 10) 490-496 (1958). 6. DETTMERP. B. and SOBERINGA. Pilot and Commercial Dry Autogenous Grinding of Labrador Iron Ore --Carol Lake, Transactions o f the Seventh International Mineral Processing Congress (N. Arbiter, Ed.), Vol. 1, pp. 573-593, Gordon & Breach, N.Y. (1964). 7. CHAKRAVARTIA. and JOWETTA. Aspects of Comminution by Heating, Proceedings o f the Second European Symposium on Comminution, Amsterdam (H. Rumpf and W. Pietsch, Eds), pp. 583-604 (1966). 8. YAMAGUCI-IIU. and MIYAZAKIM. A study of the strength of failure of rocks heated to high temperatures. J. Min. metall. Instn Japan 86, (986) 346-351 (1970).

236

L . B . GELLER

9. DAELLENBACHC. B., VII( R. A. and MA~tANW. M. Influence of Reduction and Thermal Shock on Nonmagnetic Taconite Grindability, Paper presented at the 1970 Meeting A I M M and Petrol Engrs, S.M.E., Denver, Colorado (1970). I0. MORGANW. R. Thermal shock effect on the transverse strength of clay bodies. J. Am. Ceram. Soc. 14, 913-923 (1931). 11. KINGERYW. D. Factors affecting thermal stress resistance of ceramic materials. J. Am. Ceram. Soc. 38, (1) 3-15 (1955). 12. MANSONS. S. and SM1THR. W. Theory of thermal shock resistance of brittle materials based on Weibull's statistical theory of strength. J. Am. Ceram. Soc. 38, (I) 18-27 (1955). 13. CgANDALLW. B. and GING J. Thermal shock analysis of spherical shapes. J. Am. Ceram. Soc. 38, (1) 44-54 (1955). 14. HASSELMAND. P. H. Thermal shock by radiation heating. J. Am. Ceram. Soc. 46, (5) 229-234 (1963). 15. HASSELMAND. P. H. and CRANDALLW. B. Thermal shock analysis of spherical shapes--II. J. Am. Ceram. Soc. 46, (9) 434-437 (1963). 16. HASSELMAND. P. H. Elastic energy at fracture and surface energy as design criteria for thermal shock. J. Am. Ceram. Soc. 46, (11) 535-540 (1963). 17. DUGDALER. A. Thermal shock by gas discharge. Trans. Br. Ceram. Soc. 64, (6) 287-322 (1965). 18. DAWDGER. W. and TAm,~ G. Thermal shock and fracture in ceramics. Trans. Br. Ceram. Soc. 66, (8) 405-422 (1967). 19. DAVIDGER. W. and TAPPIN G. The Effects of Temperature and Environment on the Strength of Two Polycrystalline Aluminas, Proceedings of the British Ceramic Society No. 15, pp. 47-60 (1970). 20. GFa.L~RL. B. A new look at thermal rock fracturing. Trans. Instn Min. Metall. 79, A133-A170 (1970). 21. GR6Bm~ H. Die Erw~'mung und Abk~ihlung Einfacher Geometrischer KOrper. Z. Ver. dt. Ing. 69, (21) 705-711 (t925). 22. SCnACK A. Zur Berechnung des Zeitlichen und Ortlichen Temperaturverlaufs beim Glfihvorgang. Stahl Eisen 50, (37) 1289-1297 (1930). 23. W I L L I ~ N E. D. and ADAMS L. H. Temperature distribution in solids during heating and cooling. Phys. Rev. 14 (Ser. 2, No. 2) 99-114 (1919). 24. HF.ISLERM. P. Temperature charts for induction and constant-temperature heating. Trans. Am. Soc. mech. Engrs 69, 227-236 (1947). 25. CARSLAWH. S. and JAFX3EgJ. C. Conduction of Heat in Solids, 2rid edn, Clarendon Press, Oxford (1959). 26. K~rcr C. H. Thermal stresses in spheres and cylinders produced by temperature varying with time. Trans. Am. Soc mech. Engrs 54, 185-196 (1932). 27. KI~NTC. H. Thermal stresses in thin-walled cylinders. Trans. Am. Soc mech. Engrs 53, 167-180 (1931). 28. HEISLERM. P. Transient thermal stresses in slabs and circular pressure vessels. J appL Mech. 20, (2) 261-269 (1953). 29. MELAN E. and PARKUSH. Wi~'rmespannungen, Springer-Verlag (1953). 30. BOLEVB. A. and WEINERJ. H. Theory o f Thermal Stresses, Wiley, New York (1960). 31. P ARKUSH. lnstationiire Wiirmespannangen, Springcr-Verlag (1959). 32. CHEN T. S. and MAROVELLIR. L. Analysis of Stresses in a Rock Disk Subjected to Peripheral Thermal Shock, U.S. Bureau of Mines, RI No. 6823 (1966). See also: MAROVELLIR. L., CHEN T. S. and Vm-rHK. F. Thermal Fragmentation of Rock, Vol. 2, pp. 253-280, Transactions of the Seventh Syn~osium on Rock Mechanics, Pennsylvania State University, Penna (1965). 33. HE~RDENVAN W. L. Potential fracture zones around boreholes with fiat and spherical ends. Int. J. Rock Mech. Min. Sci. 6, (5) 453-463 (1969). 34. WI~V,,:BLMANNA. and ScrIoyr O. Ueber Thermische Widerstandscoefficienten verschiedener Gl~iser in ihrer Abh~ingigkeit v o n d e r chemischen Zusammensetzung. Annln Phys. 51, 730-746 (1894).

APPENDIX

A

The First 35 Roots of the Function: [J, cot/3, + (B -- 1) -~ 0 B=O'O00 .00000000 4-4934095 17"220755 20.371303 32- 956389 36" 100622 48" 674144 51" 816982 64-387120 67"529435 80-(198129 83. 240192 95" 808139 98" 950063

7.7252518 23,519452 39" 244432 54" 959678 70.671686 86" 382222 102. 09197

i0.904122 26"666054 42- 387914 58-102255 73"813881 89- 524221 105. 23385

14,066194 29~8115~ 45,531134 61- 244730 76-956026 92- 666192 108" 37572

T H E R M A L STRESSES IN SPHERES

237

B = 0"020 • 24445971 4"4978604 17"221917 20"372285 32.956996 36.101176 48"674555 51-817368 64.387430 67.529731 80.098378 83'240433 95"808348 98"950265

7"7278407 23.520303 39.244942 54-960042 70"671969 86.382454 102"09216

10"905956 26.666804 42"388385 58"102599 73"814152 89-524444 105"23404

14.067616 29"812270 45"531573 6•.245057 76"956286 92-666408 108-37590

B 0"050 • 38536810 4"5045364 17.223659 20.373757 32-957906 36.102007 48"675171 51"817947 64.387896 67"530175 80-098753 83"240793 95.808661 98"950568

7-7317240 23-521578 39.245706 54"960588 70.672393 86-382801 102.09246

10'908707 26-667929 42"389093 58"103115 73"814558 89.524779 105"23433

14"069749 29"813276 45"532232 61.245547 76.956676 92.666732 108"37618

B~0.200 • 75930769 4"5378886 17.232369 20"381120 32.962458 36.106162 48.678253 51"820842 64-390226 67"532396 80.100626 83"242595 95-810226 98-952084

7"7511351 23.527956 39.249529 54.963317 70.674516 86.384537 102.09393

10"922461 26"673554 42.392632 58"105697 73"816590 89"526455 105.23575

14"080411 29"818307 45"535527 61"247996 76"958625 92-668351 108"37757

B=0.500 1"1655612 4"6042168 17"249782 20"395842 32-971559 36"114472 48.684416 51"826632 64.394885 67.536839 80"104371 83"246199 95"813358 98"955116

7"7898838 23.540708 39.257172 54.968776 70.678761 86-388010 102"09686

10"949944 26"684802 42-399709 58"110860 73"820654 89"529806 105.23860

14.101725 29-828369 45-542115 61.252894 76.962523 92"671588 108"38033

B~0.800 1"4320322 4"6695848 17.267177 20"410554 32.980659 36"122779 48.690579 51"832420 64.399544 67-541281 80.108116 83"249803 95.816489 98"958148

7"8284393 23.553454 39.264815 54-974233 70"683005 86"391483 102-09980

10"977357 26"696046 42-406785 58"116023 73"824718 89"533157 105.24145

14-123007 29-838428 45"548703 61"257792 76"966421 92"674825 108"38310

B~ 1" 5707963 17. 278760 32. 986723 48- 694686 64.402649 80.110613 95"818576

7" 8539816 23- 561945 39.269908 54. 977871 70" 685835 86. 393798 102"10176

I0" 995574 26" 703538 42" 411501 58" 119464 73" 827427 89" 535391 105"24335

14" 137167 29. 845130 45. 553093 61. 261057 76. 969020 92. 676983 108"38495

7"9786657 23-604285 39.295351 54"996053 70"699978 86-405371 102"11155

11"085538 26"740916 42"435062 58-136663 73"840969 89"546558 105"25285

14"207437 29"878587 45.575032 6•'277375 76"982009 92"687772 108"39417

1.000 4" 7123890 20" 420352 36" 1 2 8 3 1 6 51- 836279 67" 544242 83" 252205 98"960169

B~2-000 2"0287578 4"9131804 17-336378 20"469167 33-017001 36"155966 48.715211 51-855561 64.418172 67"559043 80.123093 83.264215 95.829011 98"970272

238

L.B. GELLER B =

2"2889297 17" 393244 33" 047169 48" 735701 64" 433679 80"135565 95" 839441

3"000

5"0869851 20" 517523 36" 183533 51" 874814 67" 573831 83"276217 98" 980372

8"0961636 23" 646324 39" 320728 55" 014210 70" 714110 86"416937 102" 12134

11"172706 26" 778087 42" 458571 58" 153842 73" 854501 89"557719 105" 26235

t4"276353 29- 911894 45" 596928 61- 293675 76" 994990 92"698555 108" 40339

B=5"000 2"5704316 5"3540318 17"503428 20"612031 33"106961 36"238251 48"776510 51"913179 64"464619 67"603342 80"160471 83"300188 95"860279 99"000550

8"3029292 23"728935 39"371158 55"050405 70"742318 86"440040 102-14090

11"334826 26"851418 42"505330 58"188099 73"881515 89"580014 105"28133

14.407971 29"977779 45"640512 61"326189 77.020907 92"720097 108"42182

B = 7"000 2"7164597 5"5378233 17"607188 20"702444 33" 165697 36" 292158 48"816981 51"951262 64" 495412 67" 632725 80" 185300 83" 324089 95'881072 99"020688

8"4702949 23"808812 39-420952 55"086364 70" 770414 86" 463081 102"16043

1t'477273 26"922814 42" 551583 58"222155 73" 908431 89" 602253 105"30027

14-528805 30"042255 45" 683684 61"358533 77' 046738 92" 741589 108"44022

B=10"000 2"8363004 5"7172492 17"748069 20"828226 33"251057 36"370893 48"876783 52"007633 64"541202 67"676452 80"222334 83"359754 95"912138 99"050782

8"6587047 23"921790 39"493965 55-139667 70'812254 86"497474 102"18961

1t"653208 27-025010 42"619614 58,272700 73"948538 89"635462 105"32859

14-686937 30"135350 45"747345 6~-406585 77'085248 92"773691 108"46773

B = 15"000 2"9349462 5"8852392 17"941384 21-008175 33"383811 36"494620 48"973131 52"098797 64"616016 67"748022 80"283259 83"418484 95"963443 99"100511

8"8605219 24"088420 39"609652 55"226144 70"880839 86"554158 102"23785

11,863396 27"179192 42"728130 58'354925 74"014371 89"690234 105"37544

14"891711 30,278230 45-849449 61-484938 77"148535 92.826674 108"51325

B = 20"000 2" 9857240 5" 9783432 18"088723 21"152201 33"502615 36"607067 49"064151 52"185447 64'688331 67"817403 80" 342833 83" 476003 96"013940 99"149503

8" 983• 292 24"227007 39'716123 55.308767 70"947498 86" 609752 102.28542

12" 002944 27-311359 42"829040 58"433835 74"078500 89" 744023 105"42167

15- 038429 30"403675 45,945214 61.56042i 77.210307 92,878767 108"55821

B=40"000 3"0632097 6.1273477 18"408547 21"487566 33.842802 36"940823 49'363337 52"475417 64"943453 68"064561 80-561462 83-688297 96-203720 99"334291

9-1932790 24"570544 40.042123 55"589650 71,187028 86"816008 102,46545

12"261752 27"657483 43"146465 58-705847 74"310736 89"944523 105'59714

15"333365 30.748288 46.253613 61.823834 77.435579 93-073780 108"72934

T H E R M A L STRESSES IN SPHERES

239

B = 50"000 3.0788416 18.488752 33-951581 49.475258 65.048251 80-656539 96"289305

6"1581640 21"576362 37.051682 52"586388 68"167486 83"781415 99"418094

9"2384262 24-666381 40.154202 55-699370 71.288022 86"907189 102"5475l

12.320047 27.758905 43.259045 58.814085 74.409766 90"033801 105.67752

15"403389 30.853974 46.366103 61.930415 77.532631 93.161190 108-80808

9.3075021 24.828234 40"368303 55-932588 71-520880 87.130296 102'75719

12"410549 27'934455 43"479170 59-048395 74-641188 90.254393 105-88435

15.514051 31"041529 46-591034 62"165151 77.762313 93.379158 109.01205

9'3308050 24-886466 40"452794 56"033626 71-630282 87"242244 102"86800

12'441356 27"998716 43"567725 59"151687 74"751489 90'366350 105"99464

15"552144 31"111442 46'683262 62"270387 77"873300 93-490993 109"12175

9"4059306 25"082518 40"759204 56"436048 72"113109 87"790444 103"46810

12"541243 28"217845 43"894558 59"571441 75"248553 90"925948 106-60368

15"676558 31"353177 47'029919 62"706843 78"384007 94-061466 109"73927

9"4153535 25'107614 40'799886 56'492179 72"184500 87"876855 103'56925

12'553805 28"246067 43"938343 59"630641 75"322968 91"015331 106"70774

15"692257 31"384521 47"076801 62"769104 78"461437 94"153809 109"84623

B -- 80.000 3.1023427 18.618107 34.149505 49.703895 65.282834 80.884226 96.504564

6"2048044 21"722808 37"258420 52"817750 68"401420 84.006897 99'630583

B - 100"000 3"1101870 18-663225 34-224677 49-799418 65"389723 80"995705 96"616160 B 3"1353096 18'811875 34"488514 50"165287 65-842255 81"519474 97'196998

6"2204351 21"774650 37"338452 52"916204 68"509690 84"118691 99"741834 500"000 6"2706196 21"947195 37"623856 53"300664 68'977677 84"654952 100'33254

B -~ I000"000 3.1384511 18"830709 34"522975 50"215259 65'907568 81"599908 97-292289

6"2769022 21-969161 37"661431 53"353719 69"046033 84'738381 100"43077

APPENDIX B Computer Program for Calculating Thermal Stresses in Spheres [per equations (11) and (12)] C PROGRAM TO CALCULATE STRESSES IN SOLID SPHERES. COMPRESSION IS A NEGATIVE C STRESS. CASE SHOWN IS THEREFORE THAT OF Q U E N C H I N G OF A HOT SPHERE. C X = R A D I A L R A T I O , F = F O U R I E R NO.,BE=ROOTS,SI ~ R A D I A L STRESS,S2=TANGENTIAL C STRESS C B=BIOT NUMBER PARAMETER M X N ~ 3 5 DIMENSION X(11), F(22),BE(MXN),SN(MXN),SC(MXN),E(MXN),XFl(MXN,11),X 1F2(MXN, 11 ),XF3(MXN,22) READ (1,200) X,F

240

L . B . GELLER

200 F O R M A T (3(11F5.31)) DO 10 L=1,12 READ(1,100)B,BE 100 F O R M A T (4F20.8) WRITE (3,300)B 300 FORMAT(10H BIOT NO.=,FI4.3,SX,12H F O U R I E R NO. ,9X,13H R A D I A L RATI 10 ,4X,14H R A D I A L STRESS ,7X,18H T A N G E N T I A L STRESS ,/) DO 15 I = I , M X N SN(I) = SIN(BE(I)) SC(I) = SN(I)--BE(I)*COS(BE(I)) 15 E(I) = (SN(I)*(BE(I)**2 + (B-- 1)**2))/(BE(I)**4*(BE(I)**2 + B*(B-- 1))) DO 20 J~1,11 XI = 1/X(J)**3 DO 20 I = I , M X N SNN = SIN(X(J)*BE(I)) A----(SNN--X(J)*BE(t)*COS(X(J)*BE(I)))*XI XF1 (I,J) = E(I)*(SC(I)-- A) 20 XF2(I,J) = E(I)*(2*SC(I) + A - (BE(I)**2*SNN)/X(J)) DO 25 K=1,22 DO 25 I = I , M X N 25 XF3(I,K) ~ EXP(-- F(K)*BE(I)*BE(I)) DO 30 K=1,22 DO 30 J ~ l , l l S1 =0.0 $2=0.0 DO 35 I = I , M X N S 1 = S 1 + XF3(I,K)*XF1 (I,J) 35 $2 ----$2 + XF3(I,K)*XF2(I,J) S1 =4*B*SI $2 = 2 * B ' S 2 30 WRITE (3,400) F(K),X(J),S1,S2 400 F O R M A T (32X, F10.3,F20.2,E20.8,E24,8]) 10 C O N T I N U E STOP END