Thermal behavior of shipping casks under fire conditions

NUCLEAR ENGINEERING AND DESIGN 8 (1968) 273-288. NORTH-HOLLAND PUBLISHING COMP., AMSTERDAM

T H E R M A L B E H A V I O R OF S H I P P I N G CASKS UNDER FIRE C O N I ) I T I O N S * G. P. W A C I - I T E L L The Franklin Institute Research Laboratories, Philadelphia, Pennsylvania 19103, USA and J . W. L A N G H A A R A t o m i c Energy Division, E.I. DuPont de Nemours and Co., Wilmington, Delaware 19898, USA Received 3 April 1968

To provide basic information for m o r e reliable prediction of t e m p e r a t u r e s in a shipping cask under external fire conditions, a 15-ton lead-shielded cask was subjected to a large, o p e n - a i r petroleum fire for 1 hr. Experimental r e s u l t s were c o r r e l a t e d by e l e c t r i c a l analog computation, using various a s sumed values for important p a r a m e t e r s . The radiant absorptivity (or) of the cask was found both by analog computations and by hand calculations to be about 0.6. Although the cask was externally finned, ot was calculated from the a r e a of the outer shell excluding the fins, which contribute little to radiant heat t r a n s f e r . Convection of molten lead in the walls was found to be such that the horizontal t e m p e r a t u r e gradient in the molten lead was negligible. On the other hand, the maximum vertical t e m p e r a t u r e g r a dient was about 8°F p e r inch of height. Hand-calculation methods a r e p r e s e n t e d for predicting the extent of melting as a function of time for lead-shielded casks of r e c t a n g u l a r or cylindrical shape, for any selected fire conditions. Methods are also suggested for e s t i m a t i n g the maximum cavity t e m p e r a t u r e in casks with e i t h e r lead or non-melting shielding.

1. INTRODUCTION The fire tests were planned and the results analyzed by the Franklin Institute Research Laboratories for the Savannah River Plant. The fire t e s t s were performed at the test site of the Sandia Corporation in Albuquerque, New Mexico. The primary objective of the t e s t s was to evaluate the apparent absorptivity, the temperature gradients, the effect of convection currents in molten lead, and the effect of cask size, to develop procedures for more reliable prediction of the thermal behavior of casks when subjected to fire. A 15-ton lead-shielded cask was tested (fig. I). Also tested were a finned box, which externally was a half-scale model of the cask, to observe the effect of size on heat flux; and some smaller objects to observe the effect of sooting on surface absorptivity. * The information contained in this article was developed during the course of work under Contract AT (07-2)-I with the US Atomic Energy Commission.

A t t h e t i m e of t h e t e s t s , t h e h y p o t h e t i c a l f i r e c o n d i t i o n s to b e s p e c i f i e d in f o r t h c o m i n g s h i p ping cask regulations were under discussion. T h e t y p e of o p e n - a i r f i r e t e s t i n v e s t i g a t e d by t h e S a n d i a C o r p o r a t i o n [1] w a s c o n s i d e r e d m o r e s e vere than, and therefore an acceptable substitute for, the ASTM one-hour fire test which was s p e c i f i e d in p r o p o s e d r e g u l a t i o n s . S i n c e t h e n , a f t e r s t u d y i n g t h e f r e q u e n c y a n d s e v e r i t y of railroad and highway fires, the IAEA and the USAEC h a v e c o n c l u d e d t h a t a m o r e a p p r o p r i a t e b a s i s f o r c a s k d e s i g n i s a f i r e t e s t a t 800°C (or 1475OF in U.S. r e g u l a t i o n s ) f o r 30 m i n . 800°C i s lower than common fire temperatures because a c a s k i s u n l i k e l y to b e c o m p l e t e l y s u r r o u n d e d by thick flames. The 30-min period recognizes that a c a s k i s u n l i k e l y to b e p o s i t i o n e d o v e r a d e e p pool of f u e l . C a l c u l a t i o n a l p r o c e d u r e s d e s c r i b e d in t h i s a r t i c l e a r e a p p l i c a b l e to t h e h y p o t h e t i c a l f i r e c o n d i t i o n s , a s i l l u s t r a t e d in A p p e n d i x A f o r a cylindrical cask. Additional details are availab l e e l s e w h e r e [2].

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THERMAL BEHAVIOR OF SHIPPING CASKS 2. DESCRIPTION OF T E S T S

Thermocouples were chromel-alumel, sheathed in s t a i n l e s s s t e e l , ~ in. o.d. In a few c a s e s , e x t e r i o r unsheathed t h e r m o c o u p l e s w e r e also attached. T h e r m o c o u p l e p r o t e c t i o n tube depths w e r e s e l e c t e d so that t h e r m o c o u p l e s w e r e i n s e r t e d with f i r m contact with the b o t t o m s of the tubes. In the c a s e of sheathed t h e r m o c o u p l e s attached to i n t e r i o r or e x t e r i o r s u r f a c e s , the sheath was left intact and welded d i r e c t l y to the s u r f a c e s . The shipping cask was held by a t h r e e - l e g g e d stand whose upper s u r f a c e was ap p r o x i r n a t e l y 13 in. above the initial fuel s u r f a c e . The l e g s w e r e fitted with load c e l l s to indicate any substantial l o s s of lead. Th e t e s t was made outdoors with the cask c e n t e r e d o v e r a 30 ft by 30 ft pool of J P - 4 fuel ( s i m i l a r to k e r o s e n e ) , and l a s t e d for 1 hr. The cask was o r i e n t e d such that the bottom, lid, and 2 trunnion s i d e s faced s i d e w a r d (as d i st i n g u i sh ed f r o m up and down). A p r e l i m i n a r y t e s t had been made with a finned box and s o m e s m a l l s p h e r e s and cubes. Th e b o x w a s a h a l f - s c a l e model of t h e e x t e r i o r of the cask, except that the fin s i z e and spacing w e r e the s a m e as for the cask in o r d e r that any effect of s i z e would not be confounded with effect of fins. The w a l l s , f l o o r and lid of

The shipping c a s k had been p r e v i o u s l y d a m aged during drop t e s t s , but was r e p a i r e d b e f o r e the f i r e t es t . As p a r t of the r e p a i r , a new b o t tom plat e was i n s t a l l e d , and a 1 - i n c h - s q u a r e spa c e was left in the lead around the edge of this pla te to help p r e v e n t s e v e r e s t r e s s of the weld due to the expansion of m e l t i n g lead. A l s o , p r o tection tubes w e r e i n s t a l l e d inside the lead f o r t h e r m o c o u p l e s at v a r i o u s positions. Fig. 2 shows the location of t h e r m o c o u p l e s in the body of the shipping c a s k and in the lid. The location of the t h e r m o c o u p l e s is d e s i g n a t e d by n u m b e r when showing d i s t a n c e into lead and by l e t t e r when showing altitude as follows, r e s p e c tively: 1, welded to in n e r s u r f a c e of c a v i ty ; 3 and 6, welded to o u t e r s u r f a c e of shipping c a s k ; 2 and 5, o n e - t h i r d the way in f r o m the o u te r s u r f a c e of lead; and 4, t w o - t h i r d s the way in; A, C and G, below c e n t e r of face (lower); B, D and H, above c e n t e r of face (upper); and I, E and F, c e n t e r of face. L o c a t i o n s 2 and 3 d i f f e r e d f r o m l o c a t i o n s 5 and 6, r e s p e c t i v e l y , by t h e i r position r e l a t i v e to fins. L o c a t i o n s 2, 4 and 5 w e r e i n s t a l l e d in p r o t e c t i o n tubes.

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Fig. 2. Thermocouple locations on 15-ton shipping cask.

276

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t h e b o x w e r e of s t e e l , 1½ in. t h i c k . T h e b o x w a s f i l l e d w i t h v e r m i c u l i t e to p r e v e n t r a d i a n t h e a t e x c h a n g e b e t w e e n i t s w a l l s . T w o p a i r s of s t e e l s p h e r e s , a l l of 7½ in. d i a m e t e r , w e r e u s e d . O n e p a i r w a s a b o u t 3 ft a b o v e t h e fuel s u r f a c e , a n d t h e o t h e r p a i r a b o u t 2 ft a b o v e t h e fuel s u r f a c e . In e a c h p a i r , o n e w a s c h r o m e - p l a t e d a n d t h e o t h e r w a s in a n o r m a l s l i g h t l y o x i d i z e d c o n d i tion. Each sphere had a thermocouple at its cent e r . P a i r s of c o p p e r c u b e s , 2 in. on a s i d e , e a c h with a thermocouple, were similarly situated w i t h r e s p e c t to t h e fuel. E a c h c u b e w a s i n s u lated except for one sideward facing surface. O n e c u b e of e a c h p a i r w a s c h r o m e - p l a t e d , a n d t h e o t h e r w a s i n i t i a l l y b l a c k e n e d in a g a s f l a m e . D e t a i l s of t h e r e s u l t s f o r t h e f i n n e d b o x , s p h e r e s , a n d c u b e s a r e not i n c l u d e d . T h e a p p a r e n t v a l u e of a f o r t h e f i n n e d b o x w a s 0.56, c o m p a r e d w i t h 0.57 f o r t h e c a s k , t h u s i n d i c a t i n g t h a t s i z e h a s no a p p r e c i a b l e e f f e c t on t h e n e t h e a t flux. T h e s p h e r e s a n d c u b e s s h o w e d no a p p r e c i a b l e e f f e c t of a n i n i t i a l l y p o l i s h e d s u r f a c e c o m p a r e d w i t h a dull o r b l a c k e n e d s u r f a c e ; h o w e v e r , t h i s w a s i n c o n c l u s i v e w i t h r e g a r d to t h e e f f e c t of s o o t i n g b e c a u s e t h e f i r e d e s t r o y e d t h e chrome plating.

3. T E S T R E S U L T S Table 1 shows fire temperatures at various times during the test, as indicated by thermoc o u p l e s p l a c e d a t s e v e r a l l o c a t i o n s in t h e f l a m e s . S i n c e h e a t f l u x i s p r o p o r t i o n a l to t h e f o u r t h p o w e r of a b s o l u t e t e m p e r a t u r e , a good Table 1 F i r e t e m p e r a t u r e readings in second test. Height above initial fuel surface (in.)

0.2

Time (hr) 0.4 0.6 0.8 T e m p e r a t u r e (OF)

1.0

15

1680

17-)40 1660

1920

1910

22 30

1820 1930

1650 1790

1610 1780

1790 1940

1820 2020

40 50

1650 2050

1960 1920

1750 1870

1540 1830

1630 1760

22 (see n o t e ) 30 (see n o t e )

1650 1430

1440 1830

1990 2050

1790 1550

1610 1550

4{)

1780

1850

1500

1680

1810

Note: The lid faced these two thermoeouples. The in-

strumented sideward facing side faced all the other thcrmocouples listed in table. Other fire t h e r m o couples a r e not listed because they were short c i r cuited at the end of the test.

average value for the fire temperature was approximately 1800°F, although considerable fluct u a t i o n s did o c c u r . T h i s t e m p e r a t u r e a g r e e s w e l l w i t h p r e v i o u s o b s e r v a t i o n s [ 1 ]. Data from the load cells showed some redist r i b u t i o n of w e i g h t d u r i n g t h e f i r s t 30 m i n , but o n l y a l o s s of w e i g h t a f t e r t h a t t i m e . T h e r e a d ing of t h e t h e r m o c o u p l e at p o s i t i o n A - 4 c h a n g e d s u d d e n l y a t 33 r a i n ; a p p a r e n t l y it w a s d a m a g e d by m o l t e n l e a d w h i c h e s c a p e d f r o m t h e lid. L a t e r examination showed that a pipe installed through t h e lid f o r t h e r m o c o u p l e w i r e s h a d f a i l e d a t t h e w e l d , p r e s u m a b l y f r o m s t r e s s e s d u e to t h e e x p a n s i o n of l e a d a s it m e l t e d . T w o - t h i r d s of t h e l e a d f r o m t h e lid e s c a p e d t h r o u g h t h e r u p t u r e d w e l d . T h e b o d y of t h e c a s k a l s o l o s t l e a d , a p parently through cracks and pinholes which dev e l o p e d in t h e w e l d j o i n i n g t h e u p p e r f a c e of t h e b o d y to t h e i n n e r c a v i t y . A l t h o u g h t h i s w e l d w a s checked by the dye-penetrant method and appeared sound before the fire, examination after the fire revealed that the weld metal had been l a r g e l y r e m o v e d in s o m e r e g i o n s by m a c h i n i n g of t h e f l a n g e s u r f a c e d u r i n g o r i g i n a l f a b r i c a t i o n . T h e w e i g h t of the s h i p p i n g c a s k w a s a p p r o x i m a t e l y 28 500 l b s b e f o r e t h e t e s t a n d 23 000 l b s a f t e r t h e t e s t . Of t h e m o r e t h a n 5000 l b s of l e a d l o s t , 3000 l b s c o u l d b e a c c o u n t e d f o r f r o m t h e lid. T h e b o d y l o s t a p p r o x i m a t e l y 2000 l b s . T h i s l o s s , t o g e t h e r w i t h b u l g i n g of t h e c a s k s i d e s , l o w e r e d t h e l e v e l of t h e f r e e s u r f a c e of t h e l e a d , in f i n a l s o l i d i f i e d f o r m , by a p p r o x i m a t e l y 7 in. N e v e r t h e l e s s , t h e r m o c o u p l e r e a d i n g s at 1 h r i n dicated that the thermocouple one-third through t h e l e a d in t h e u p w a r d f a c i n g s i d e w a s s t i l l i m m e r s e d in m o l t e n l e a d . S i n c e no l a r g e o p e n i n g d e v e l o p e d , it a p p e a r s t h a t t h e l o s s w a s f a i r l y g r a d u a l . In v i e w of t h e f i n a l c o n d i t i o n of t h e c a s k a n d t h e w e i g h t l o s s , it i s c o n c l u d e d t h a t a p a r t from initial void spaces before melting, the s i d e w a r d - f a c i n g r e g i o n s (the f l o o r a n d two s i d e s ) and the downward-facing region were completely filled with lead during the entire test. Table 2 gives the temperature indicated at v a r i o u s t i m e s by t h e r m o c o u p l e s p l a c e d at v a r i o u s l o c a t i o n s in t h e s h i p p i n g c a s k . L o c a t i o n s i n d i c a t e d a s s i d e , down a n d up w e r e s i d e w a r d facing, downward-facing, and upward-facing f i n n e d s i d e s , r e s p e c t i v e l y . In t h e t e s t o r i e n t a tion, the lid and floor were sideward-facing. Temperatures for the outer surface were higher t h a n a p p e a r s r e a s o n a b l e at 0.25 h r ; it i s l i k e l y t h a t t h e t h e r m o c o u p l e s w e r e p a r t l y h e a t e d by t h e radiation from the flame. Consequently, these t e m p e r a t u r e s w e r e not u s e d in i n t e r p r e t i n g t h e e x p e r i m e n t a l r e s u l t s . T h e l e a d t e m p e r a t u r e s in

277 Table 2 T e m p e r a t u r e s indicated on shipping container (parentheses indicate l e s s reliable readings). Time (hr) Sandia TC No.

Location (see fig. 2)

0.25

0.50

0.74

1.0

Temperature (OF) 34 35

Outer surface, lid, lower upper

(1560) (i 730)

(1540) (1430)

(1500) (1240)

36 * 37* 38 *

side, lower upper upper

(900) (I 200) (940)

(900) (l100) (980)

(1250) (1350) (1200)

(1590) (1370) (I 31O)

39 41 43 42

down up floor, lower upper

(890) (940) /1440) (1480)

(860) (1540) (1750) (1730)

(1130) (1270) (1490) (1490)

(1420) (1120) (1920) {1920)

(1013)

(993)

(1367)

(1423)

501 490

(880) 060)

(1580)

(1110)

Average of * 11

through lid, lower upper

18 * 21 * 22 *

side, lower upper upper

277 325 301

544 677 625

730 870 850

1070 1150 1130

29 * 28 *

floor, lower upper

225 211

690 750

760 870

1090 1140

17 23 25

down up up

228 342 343

421 534 531

577 960 970

1020 1200 1200

268

657

816

1116

316 290

(829) (940)

(1760) (1260)

(720) (1260)

Average of * 13 14

through, lid, lower upper

19" 20 * 27 * 26 *

side, lower upper floor, lower upper

162 192 137 157

434 469 662 726

749 850 750 870

1100 1130 1080 1150

16 24

down up

158 205

404 486

530 940

1020 1180

162

573

805

1115

Average of * . . . .

(1210) (i 170)

15 32 * 31 * 30

Cavity, lid, lower side, lower floor, upper down

Average of * t h e lid r o s e a b r u p t l y s h o r t l y b e f o r e 0.50 h r , w h i c h i n d i c a t e d s u d d e n l e a k i n g . At 0.25 h r , t h e t e m p e r a t u r e a t a n y g i v e n d i s t a n c e in f r o m t h e outer s u r f a c e is s u b s t a n t i a l l y h i g h e r for the lid than for the body. T h i s is u n d e r s t a n d a b l e b e c a u s e t h e s h a p e of t h e l i d r e s u l t e d in a l a r g e r r a t i o of e x p o s e d s u r f a c e to v o l u m e of l e a d t h a n occurred for the other regions. The lid, floor and one side ( s i d e w a r d - f a c i n g regions) were instrumented internally. The floor a n d s i d e w e r e f a i r l y s i m i l a r in t h e r m a l b e h a v -

197

(691)

-

120 75 (840)

380 375 (550)

802 900 (620)

1110 1150 (1050)

(800)

98

378

851

1130

ior, although s o m e d i f f e r e n c e s may be s e e n \ i n t a b l e 2. T h e r e a d i n g s u s e d f o r c o m p u t i n g a v e r a g e t e m p e r a t u r e s a r e i n d i c a t e d in t a b l e 2 b y an a s t e r i s k . A c o m p a r i s o n of t h e a v e r a g e t e m p e r a t u r e s f o r -~ t h r o u g h , } t h r o u g h a n d in t h e c a v i t y , s h o w s (table 2) t h a t b e l o w t h e m e l t i n g p o i n t of l e a d (621OF), t h e t e m p e r a t u r e i s h i g h e r n e a r t h e o u t e r s u r f a c e and d e c r e a s e s i n w a r d , a s e x pected. When the melting point is substantially e x c e e d e d , the t e m p e r a t u r e is about the s a m e , r e g a r d l e s s of d i s t a n c e in f r o m t h e s u r f a c e . In

278

G.P. WACHTELL and J. W. LANGHAAR

fact, this was t r u e at each altitude (upper or lower) examined s e p a r a t e l y . F o r e x a m p l e , for the ' s i d e , l o w e r ' position at 0.74 hr, the t e m p e r a t u r e is 730OF for ~ through and 749OF for ] through. S i m i l a r l y , f o r the ' s i d e , u p p e r ' p o s i tion at 0.74 hr, the t e m p e r a t u r e is 870 and 850OF for two t h e r m o c o u p l e s ½ through, and 850OF for through. T h e r e f o r e , we conclude that t h e r e is v e r y little h o r i z o n t a l t e m p e r a t u r e g r a d i e n t in the molten lead. In effect, molten lead in the s i d e w a r d - f a c i n g r e g i o n s b e h a v e s as though it has an infinitely high t h e r m a l conductivity, as f a r as horizontal heat conduction is c o n c e r n e d . The v e r t i c a l s e p a r a t i o n between ' u p p e r ' and ' l o w e r ' t h e r m o c o u p l e s in the s i d e w a r d - f a c i n g r e g i o n s was 14 in. The v e r t i c a l t e m p e r a t u r e g r a d i e n t was about 8OF/in. at 0.74 hr, and about 4OF/in. at 1.0 h r; the t e m p e r a t u r e i n c r e a s e d with altitude. The s m a l l n e s s of the h o r i z o n t a l t e m p e r a t u r e gradient is e v i d e n c e of n a t u r a l c o n vection c u r r e n t s in the molten lead. The motion of molten lead may account for the o b s e r v e d v e r t i c a l t e m p e r a t u r e g r a d i e n t , which could a l s o be due to s t r a t i f i c a t i o n in an u p w a r d - f a c i n g r e gion and convection in a s i d e w a r d - f a c i n g r e g i o n ; i.e., the w a r m e r lead floats above the c o o l e r l a y e r s b e c a u s e it is l e s s dense. At the s a m e t i m e , in a s i d e w a r d - f a c i n g side with s o m e solid lead r e m a i n i n g , the g e n e r a l convection path should be upward (with i n c r e a s i n g t e m p e r a t u r e ) on the inside of the outer shell, and downward (with d e c r e a s i n g t e m p e r a t u r e ) at the i n t e r f a c e with solid lead. A v e r t i c a l g r a d i e n t in f i r e t e m p e r a t u r e (and hence heat flux) would p r o d u c e s o m e v e r t i c a l t e m p e r a t u r e gradient in the molten lead even if convection w e r e absent, but would not account for the s m a l l n e s s of the h o r i z o n t a l gradient. The v e r t i c a l s e p a r a t i o n between t h e r m o couples ~ through and ~ through in the u p w a r d facing side was 3 in. At 0.74 and 1.0 h r , the t e m p e r a t u r e d i f f e r e n c e s w e r e about 25 and 2 0 °F , r e s p e c t i v e l y . Thus, it a p p e a r s that a v e r t i c a l t e m p e r a t u r e gradient of about 8 ° F / i n . o c c u r r e d in the molten lead at 0.74 hr, and d e c r e a s e d somewhat a f t e r w a r d . Fig. 3 shows typical c u r v e s of m e a s u r e d lead t e m p e r a t u r e v e r s u s t i m e . A c u r i o u s f e a t u r e is the ra p i d change in t e m p e r a t u r e f r o m a value below 621°F to a value c o n s i d e r a b l y above. T h i s change was o b s e r v e d both for t h e r m o c o u p l e s in p r o t e c t i o n tubes and for those welded to the c a v ity wall. The s u g g es t e d explanation is that n e a r a given point on the i n t e r f a c e between solid and liquid lead, the bulk of the molten lead is at a f a i r l y u n i f o r m t e m p e r a t u r e , higher than the

3000J .

. TC

2000

.

.

.

.

.

.

.

.

.

.

.

NO. 19

~[LOW CENTRAL PLANE, ONE THIRD TI-~ WAY; THRO~IGH LEAD FROM INNER SURFACE

20 . . . . . . . .

ABOVE CENTRAL PLANE, ONE THIRD THE WAY THROUGH L E A D FROM INNER SURFACE

21

ABOVE CENTRAL PLANE, TWO THIRDS THE WIY THROUGH LEAD FROM INNER SURFACE

~l

CC

I~[ e~

. . . .

taJ

0

0.5 TIME

. . . . .

1.0

(HOURS)

Fig. 3. Internal temperature of sideward-facing side. m e l t i n g point, b e c a u s e of n at u r al convection c u r r e n t s . T h e r e is a boundary l a y e r next to the solid lead, in which convection motion is slow. Next to this boundary l a y e r , the t e m p e r a t u r e of the bulk of molten lead e x c e e d s the m e l t i n g point. -As the i n t e r f a c e m o v e s past the t h e r m o couple by a d i s t a n c e equal to the boundary l a y e r t h i c k n e s s , the t e m p e r a t u r e r i s e s to that of the bulk of molten lead.

4. ELECTRICALANALOGCOMPUTATIONS An analog m o d el was c o n s t r u c t e d which t r e a t e d the cask as a cube. The physical c o n stants a s s u m e d in the analog model c a l c u l a t i o n s a r e given in table 3. The e l e c t r i c a l analog model c o n s i s t e d of r e s i s t o r s , c a p a c i t o r s , diodes and r e l a y s connected in an ' R - C ' network so as to s i m u l a t e the t h e r m a l b e h a v i o r of the shipping cask *. It was a s s u m e d that the c a s k is a cube 44 in. on each edge, exposed to identical heat fluxes on all s u r f a c e s . The o u t er -~ in. was taken to be s t e e l , followed by 8½ in. of lead. The c a v i ty steel was not included in the model. T h e r e a r e six s y m m e t r y planes which made it p o s s i b l e to l i m i t the model to ~8 of the c o m p l e t e cube. T h i s p a r t of the cube was r e p r e s e n t e d by 149 nodes, 21 of them in the s t e e l (external nodes) and 128 of them in the lead (internal nodes). Fig. 4 shows a typical node in the lead, a r r a n g e d to s i m u l a t e the c a s e w h e r e the molten lead is m o t i o n l e s s . An e l e c t r o n i c analog c o m p u t er g e n e r a t e d the n e c e s s a r y input v o l t a g e s and o p e r a t e d the r e l a y s . * An alternative manually-operated Liebmann network for casks has previously been described in this journal [7].

THERMAL BEHAVIOR OF SHIPPING CASKS Table 3 Physical constants assumed in analog computations. Steel *

Solid lead *

Thermal conductivity k (Btu/hr ft OF)

31.8

18.6

9.3 (no convection)

Specific heat capacity c (Btu/lb OF)

0.125

0.0325

0.038

Density p (lb/ft 3)

487

687

657

Latent heat L (Btu/lb)

10.55

Melting point (OF)

621

Liquid lead **

* Values for 400°F. ** Values for 700°F. The analog m o d el o p e r a t e s as follows: c u r r e n t r e p r e s e n t i n g the net flux is fed to each e x t e r n a l node r e p r e s e n t i n g s t e e l through an input r e s i s t o r f r o m a c o m m o n v o lt a g e s o u r c e . T h i s c o m m o n v o l t a g e is d e t e r m i n e d by the analog c o m p u t e r f r o m the v o l t a g e on one p a r t i c u l a r (exte r na l ) node ( r e p r e s e n t i n g a point n e a r the c e n t e r of a face of the s t e e l o u te r shell), the a s sume d f i r e t e m p e r a t u r e , f i r e e m i s s i v i t y and

÷ 21~'

Fig. 4. Typical lead node in analog model. Initial voltage at node: 2"/,5 V ,-~ 70OF. At ~ +0.4 V diode conducts corresponding to melting point of lead ~ 621°F, At ~, *1. V signal from logic circuit operates relay which switches all contacts shown and latches itself closed. This roughly doubles the internode resistance to simulate the doubling of lead's thermal resistance.

279

s u r f a c e a b s o r p t i v i t y , so that the net flux is s i m u l a t e d in a c c o r d a n c e with the equation Q = a e a (Tf + 460) 4 - c r a

( T + 460) 4 .

(1)

The input r e s i s t o r v a l u e s a r e s e l e c t e d so that f o r e v e r y e x t e r n a l node the input c u r r e n t d e pends on node v o l t ag e in a way that a p p r o x i m a t e s the dependence of net flux on s u r f a c e t e m p e r a t u r e. In this way, the c o r r e c t input c u r r e n t is a p p r o x i m a t e d for nodes on or n e a r the edge or c o r n e r of the cube. The r e s i s t o r s within the analog model s i m u late the heat flow r e s i s t a n c e in the shipping cask. The c a p a c i t o r l a b e l l e d ' s o l i d heat c a p a c i ty' in fig. 4 s i m u l a t e s the heat c a p a c i t y of the node m a t e r i a l {lead or steel). Voltage in the model is the analog of t e m p e r a t u r e . When the v o l t a g e on the ' s o l i d heat c a p a c i t y ' in a lead node r e a c h e s the v al u e c o r r e s p o n d i n g to the m e l t i n g point of lead, a diode (not p r e s e n t in a s t e e l node} conducts. Thus, the 'l at en t heat of fusion c a p a c i t y ' is e f f e c t i v e l y connected in p a r a l l e l with the ' s o l i d heat c a p a c i t y ' . A f u r t h e r s m a l l i n c r e a s e in the lead node v o l t ag e then i n d i c a t e s that the lead r e p r e s e n t e d by that node has m e l t e d . (In p r a c t i c e , the latent heat c a p a c i t i e s of s e v e r a l n e a r b y nodes w e r e connected t o g e t h e r , so that one r e l a y with s e v e r a l c o n t a c t s could do the switching d e s c r i b e d below.) When a given node has m e l t e d c o m p l e t e l y , the analog c o m p u t e r o p e r a t e s a r e l a y that p e r f o r m s two p r i n c i p a l functions: f i r s t , it opens the c o n t a c t s that had p r e v i o u s l y s h o r t e d out r e s i s t o r s shown in fig. 4, thus s i m u l a t i n g the i n c r e a s e in t h e r m a l r e s i s t ance (reduction in t h e r m a l conductivity) of liquid lead as c o m p a r e d with solid l ead ; second, it d i s connects the 'l at en t heat of fusion c a p a c i t y ' , r e instating the ' s o l i d heat c a p a c i t y ' . (The s p e c i f i c heat of liquid lead was taken to be a p p r o x i m a t e d sufficiently well by that of solid lead.) Analog model runs w e r e made with the t h e r m a l co n d u ct i v i t y of molten lead taken to be half that of solid lead, in a c c o r d a n c e with fig. 4. In addition, r u n s w e r e made with the t h e r m a l conductivity of molten lead equal to that of solid lead, to s i m u late the effect of s o m e mixing in t r a n s f e r r i n g heat a c r o s s the molten lead. (To p e r f o r m t h e s e r u n s, the r e l a y c o n t a c t s w e r e i n su l at ed so that the r e s i s t o r s in fig. 4 w e r e not s h o r t - c i r c u i t e d . The r e l a y s t h e m s e l v e s had to be left o p e r a t i v e for t h e i r o t h er switching function.) S e v e r a l analog model runs w e r e made, to obs e r v e the effect of v a r y i n g f l a m e t e m p e r a t u r e , f l a m e e m i s s i v i t y and s u r f a c e a b s o r p t i v i t y . T a ble 4 shows the analog t e m p e r a t u r e s c o m p a r e d with o b s e r v e d t e m p e r a t u r e s . The t e m p e r a t u r e s

28o

G . P . W A C H T E I , I , and J . W. I,ANGHAAR Table 4 C o m p a r i s o n of o b s e r v e d and a n a l o g m o d e l t e m p e r a t u r e s .

Fraction Time of d i s t . into lead {hr)

Observed ternp e r a t u r e s (OF)

A n a l o g mode l t e m p e r a t u r e s

Tf (OF) No. of Average thermo- k l i q / k s o l i d E couples ~

0 (see note)

~

1013±125

3

i

993-* 71 1267± 56 ]4.1) 1423+111 I 3

~

1~0. 2

3

~ l 2

1~0. 1.0

~

1.0

(OF)

1800 1.0 0.8

0.5 0.75 0.6

1600 1.0 1.0

1.0 0.6

0.8

0.6

I.O 0.8

720

550

560

770

630

1070 1350

890 1120

900 1120

1100 1390

910 1190

0.5 0.75 0.6

1.0 1.o

1.0 0.6

0.8

0.6

510 390

390

570

,t40

800 990

6t0 830

840 1070

670 ~50

610 830

1490 1320 ]320 1560

1410 1190 990 990 1260

5

450 : 3 4 0 360 500 630 600 620 850 1040 850 840 1230 1260 1080 1090 1470

:390 3:30 640 590 1010 760 1280 1000

15 96 55 25

4

290 565 810 1075

220 505 670 905

225 350 510 730 675 1125 895 1410

98~ 22 378~: 2 871 ~ 49 1130~ 20

2

230 540 720 1000

180 480 630 ~40

180 280 480 640 630 1060 830 1360

2681 40 657+ 58 8161 57 1 1 1 6 i 29 162~ 573± 805 ~1115±

1060

250 480 630 810

240 330 470 630 630 910 800 1150

290 550 710 940

285 595 920 1210

200 155 470 375 625 565 825 680

155 260 380 570 565 830 680 1090

205 465 640 ~70

210 550 850 1160

160 120 450 350 610 550 770 640

120 200 350 520 550 770 630 1040

160 430 610 840

Note: O b s e r v e d at o u t e r s u r f a c e of s h e l l . A n a l o g is at c e n t e r of s t e e l s h e l l .

agreed best when the effective thermal conductivity of liquid lead was taken to be equal to that of solid lead, and when Tf = 1800°F, E = 1.0 and (~ = 0.6. If runs had been made with kliquid = infinity, the temperatures would have agreed even closer. Unfortunately, defects developed in some diodes, and there was not enough time to repair the trouble and make further runs. For the c a s e T f = 1800°F, ¢ = 1 . 0 a n d ~ = 0.6, the analog model showed melting to begin at approximately 0.15 hr near a corner, 0.29 hr at the center of a face, and to be completed at 0.57 hr. In some runs with kliquid = 0.5 ksolid, E and ot were varied separately. The resulting temperature r i s e in the interior did not appear to depend appreciably on E or a separately, but only on the product E~. This may be understood from eq. (i), in which c~ appears without ~ in the t e r m cr c~ (T * 460)4. This term is negligibly small until T r i s e s to an appreciable fraction of Tf, aster which the term becomes a significant correction on Q. However, variation of the correction by a modest factor has only a minor effect on the interior temperature.

5. ESTIMATION OF ot FROM THERMAL BEHAVIOR The absorptivity a is estimated by equating the calculated and observed heat inputs up to time t3, where the calculated heat input involves the unknown value of a. For this estimate, the response of the entire body is considered to be represented by the data for the two outwardfacing sides, because these data are the most reliable. A uniform heat flux around the cask is assumed even though the downward facing side had a lower flux, and heat t r a n s f e r between lid and body is neglected; these approximations are justified because they can have only a small effect on the temperature at points where thermocouples were located in the outward facing sides. The time t 3 is taken as 0.74 hr because this exceeds slightly the time for complete melting in the outward facing sides and because a printout of data was obtained at that time. The total heat absorbed is H =QI A t I + Q 2 A (t3 - tl) (2) where the flux Q2 applies to the time interval t I

THERMAL BEHAVIOR OF SHIPPING CASKS to t 3 b e c a u s e t 3 i s o n l y a l i t t l e g r e a t e r t h a n t 2. F o r t h e b o d y of t h e c a s k , A i s 61.5 ft 2. In o r d e r to u s e t h i s e q u a t i o n to c a l c u l a t e ~ , w e now e x p r e s s Q1, Q2 a n d t l in t e r m s of ~ . An i n f i n i t e l y e x t e n d e d s o l i d s l a b w i t h c o n s t a n t thermal properties, finite thickness, an initial uniform temperature, one side insulated, and a u n i f o r m f l u x on t h e o t h e r s i d e , w i l l e x p e r i e n c e a t e m p e r a t u r e r i s e on i t s h e a t e d s u r f a c e t h a t i s i n i t i a l l y p r o p o r t i o n a l to t½ [3]. S u b s e q u e n t l y , t h e r a t e of r i s e b e c o m e s i n d e p e n d e n t of t i m e a n d i s constant throughout the slab. Sufficient time has e l a p s e d in t h e p r e s e n t c a s e f o r t h i s to b e t r u e [3]. If we n e g l e c t t h e r a t e of h e a t s t o r a g e in t h e o u t e r l a y e r of s t e e l , t h e e n t i r e n e t flux, Q, e n t e r s , t h e o u t e r s u r f a c e of t h e l e a d *. T h u s , a f t e r t h e u n i f o r m r a t e of t e m p e r a t u r e r i s e h a s b e e n e s t a b l i s h e d , we h a v e t h e f o l l o w i n g c o n d i tions: 1. T e m p e r a t u r e g r a d i e n t a t o u t e r s u r f a c e i s Q / k . w h e r e k i s t h e t h e r m a l c o n d u c t i v i t y of s o l i d lead. 2. T e m p e r a t u r e g r a d i e n t a t i n n e r s u r f a c e of l e a d i s z e r o , if we n e g l e c t t h e r a t e of h e a t s t o r a g e in t h e c a v i t y s t e e l . 3. ~ 2 T / ~ x 2 i s i n d e p e n d e n t of x, s o t h a t t h e t e m p e r a t u r e g r a d i e n t i s a l i n e a r f u n c t i o n of x. The resulting equation, after appropriate integration, is HL = ¢pxo

[

T (x o) - T O -

O ol

3k J "

(3)

Eq. (3) m a y b e u s e d to f i n d t h e h e a t s t o r e d in t h e l e a d p e r u n i t a r e a n e a r t h e c e n t e r of a s i d e of t h e s h i p p i n g c a s k , if e d g e e f f e c t s a r e n e g l e c t e d a t t i m e t 1 w h e n m e l t i n g i s a b o u t to s t a r t . A t t i m e t l , t h e a d d i t i o n a l a m o u n t of h e a t s t o r e d in t h e s t e e l i s a p p r o x i m a t e l y H s = c s P s D s [ T ( x o ) - To]

(4)

w h e r e D s i s t h e t h i c k n e s s of s t e e l , c s i t s s p e c i f i c h e a t a n d Ps i t s d e n s i t y . W i t h o n l y a s m a l l e r r o r in t h e e s t i m a t e d v a l u e of Q, i t m a y b e a s s u m e d t h a t t h e s u r f a c e t e m p e r a t u r e i n c r e a s e s l i n e a r l y w i t h t i m e up to t l . T h e eq. (1) s h o w s , a f t e r a p p r o p r i a t e i n t e g r a t i o n s , t h a t f o r Tf = 1 8 0 0 ° F a n d T o = 3 2 ° F (the values found by measurement) Q1 = 45130 ~ c~ - 850 ~ B t u / f t 2 / h r .

(5)

T h e t i m e to s t a r t m e l t i n g a t t h e c e n t e r of a f a c e is then found from HL + Hs (6) tl = Q1 * In view of heat absorbed by the steel, the flux into the lead is less than Q; and HL, eq. (3), and t 1, eq. (6), are somewhat l a r g e r than calculated here.

281

On t h e b a s i s of o b s e r v e d l e a d t e m p e r a t u r e s and an estimated temperature drop through the o u t e r s h e l l , t h e o u t e r s u r f a c e t e m p e r a t u r e of t h e s t e e l a f t e r t i m e t 1 i s a p p r o x i m a t e l y 850OF. T h e n a g a i n f r o m eq. (1) we f i n d Q2 = 45130 ~ c~- 5095 c~.

(7)

U s i n g t h e p r e c e d i n g e x p r e s s i o n s f o r Q1, Q2, t l , H L , a n d H s , a n d t a k i n g T(x o) = 6 2 1 ° F a n d T o = 32OF, eq. (2) b e c o m e s H/A

= (5095 - 850) ( c P x ° + c s P s D s ) ( 6 2 1

(45130 2

- 32)

- 850~

o=,~ e p x o

-

(5095 - o~,vj ~

+ (45130 ~ - 5095) e t 3.

(8)

Eq. (8) r e p r e s e n t s the c a l c u l a t e d heat input. Now to d e t e r m i n e the o b s e r v e d heat input, the t e m p e r a t u r e T 3 at t i m e l 3 = 0.74 h r i s the a v e r age of t e m p e r a t u r e s f o r w h i c h t h e r m o c o u p l e n u m b e r s a r e m a r k e d w i t h a n a s t e r i s k in t a b l e 2, for locations ~ through, 2 through, and ~ through the lead. Thus, T 3 = 818°F. This temperature i s a s s u m e d to b e t h e s a m e t h r o u g h o u t a l l t h e lead and steel, since the small errors from neg l e c t i n g t h e s o m e w h a t h i g h e r t e m p e r a t u r e s of t h e o u t e r s h e l l a n d f r o m n e g l e c t i n g t h e e s c a p e of s o m e m o l t e n l e a d f r o m t h e b o d y n e a r t h e e n d of t h e t e s t a r e in o p p o s i t e d i r e c t i o n s . T h e b o d y c o n t a i n e d a b o u t 3150 l b s of s t e e l a n d 20 420 l b s of l e a d . U s i n g t h e v a l u e s of p h y s i c a l c o n s t a n t s in t a b l e 3 **, H = ( 0 . 1 2 5 ) ( 3 1 5 0 ) ( 8 1 8 - 32) + ( 0 . 0 3 2 5 ) ( 2 0 4 2 0 ) ( 6 2 1 - 32) + (10.55)(20420) + ( 0 . 0 3 8 ) ( 2 0 4 2 0 ) ( 8 1 8 - 621) = 1.069 × 106 B t u .

(9)

I n s e r t i n g p r o p e r v a l u e s f o r t h e c o n s t a n t s in eq. (8), a n d c o m b i n i n g w i t h eq. (9), f o r e = 1 w e f i n d (r = 0.57 w h i c h a g r e e s w e l l w i t h t h e a n a l o g model result. T h e r e a r e s e v e r a l u n c e r t a i n t i e s in t h e f o r e going analysis. First, the estimated average f i r e t e m p e r a t u r e m a y b e in e r r o r b y a s m u c h a s 5 0 ° F , w h i c h w o u l d a f f e c t t h e c a l c u l a t e d cr b y a b o u t 0.05. S e c o n d , t h e a s s u m p t i o n of a u n i f o r m s u r f a c e t e m p e r a t u r e , w h e n in f a c t t h e c o r n e r s a n d e d g e s a r e s o m e w h a t h o t t e r , r e s u l t s in a ** Actual weights of lead and steel should be used. If these were to be calculated from dimensions, r o o m - t e m p e r a t u r e values of density would be p r e f e r a b l e to those in table 3.

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s l i g h t u n d e r e s t i m a t e of a . T h i r d , t h e i n s t a l l a t i o n of a new o u t s i d e b o t t o m p l a t e on t h e c a s k a f ter the drop tests, but before the fire test, left a s m a l l a i r gap so t h a t u n t i l t h e l e a d m e l t e d , t h i s outward facing surface was hotter than average; t h i s f a c t o r a l s o r e s u l t s in a s l i g h t u n d e r e s t i m a t e of c~. F o u r t h , t h e p o s s i b l e t r a n s f e r of h e a t b e t w e e n lid a n d body w a s n e g l e c t e d b e c a u s e t h e r e w a s no s u i t a b l e b a s i s f o r e s t i m a t i n g t h e a m o u n t , which was probably small particularly after lead e s c a p e d f r o m t h e lid. F i f t h , h e a t t r a n s f e r f r o m t h e o u t w a r d f a c i n g s i d e s to t h e s o m e w h a t c o o l e r downward facing side was neglected; this probably c a u s e d a n a d d i t i o n a l s l i g h t u n d e r e s t i m a t e of a . In v i e w of t h e s e u n c e r t a i n t i e s , t h e a c t u a l v a l u e of a i s e s t i m a t e d to b e b e t w e e n 0.55 a n d 0.65. It w a s a r g u e d p r e v i o u s l y t h a t o n l y t h e p r o d u c t c~ i s of i m p o r t a n c e , r a t h e r t h a n a a l o n e , b e c a u s e the l a s t t e r m in eq. (1) is o n l y a s m a l l c o r r e c t i o n t e r m . A c c o r d i n g to t h i s v i e w . t h e v a l u e of a f o u n d f r o m t h e a n a l o g m o d e l a n d by numerical calculations should really be regarded as Ea, neither ( nor a being determined separately from the availat)le data. However, the m e t h o d of m e a s u r i n g Tf with t h e r m o c o u p l e s u n a v o i d a b l y i n c l u d e s t h e e f f e c t of f l a m e e m i s s i v i t y , s o t h a t Tf i s a n e f f e c t i v e t e m p e r a t u r e to b e u s e d with e = 1. T h u s , t h e v a l u e of a f o u n d a b o v e s h o u l d b e r e g a r d e d a s c~ a l o n e , a n d not a s e a . F o r f u r t h e r d e t a i l s , s e e r e f . 12]. T h e e f f e c t i v e a r e a of t h e c a s k i n c l u d e d t h a t p o r t i o n s h a d o w e d by t h e r a t h e r m a s s i v e t r u n n i o n s t r u c t u r e , a n d t h e c a l c u l a t e d h e a t c o n t e n t of t h e c a s k body i n c l u d e d t h e h e a t a b s o r b e d by t h e s t r u c t u r e . O t h e r c a l c u l a t i o n s not i n c l u d e d h e r e s h o w t h a t t h e c o m p u t e d v a l u e of a i s not s i g n i f i c a n t l y a f f e c t e d by e x c l u d i n g t h i s a r e a and t h e corresponding heat.

A n o m i n a l ' t i m e to c o m p l e t e m e l t i n g ' , n e g l e c t i n g t h e c o r r e c t i o n r e q u i r e d f o r s u p e r h e a t of m o l t e n l e a d , m a y b e c o m p u t e d by c a l c u l a t i n g t h e v a l u e of H c o r r e s p o n d i n g to a u n i f o r m c a s k t e m p e r a t u r e of 621OF, a n d t h e n f i n d i n g t 3 (which is r e a l l y a n o m i n a l 12) f r o m eq. (8). F o r t h e t e s t cask, the calculatedt3 is 0.58hr, which agrees closely with the analog result. Using the estim a t e d v e r t i c a l t e m p e r a t u r e g r a d i e n t of 8OF i n . for molten lead, and the reasonable assumption t h a t t h e l o w e s t e l e v a t i o n i s j u s t at t h e m e l t i n g t e m p e r a t u r e , t h e a v e r a g e t e m p e r a t u r e of l e a d a n d s t e e l at t h e t i m e to c o m p l e t e m e l t i n g i s a b o u t 795OF. U s i n g t h i s v a l u e f o r T 3 y i e l d s a m o r e realistic time for complete melting, under unif o r m flux c o n d i t i o n s , of 0.72 h r . In t h e a c t u a l t e s t , b e c a u s e of a l o w e r flux f o r t h e d o w n w a r d f a c i n g s i d e , m e l t i n g in t h a t r e g i o n w a s c o m p l e t e d at a l a t e r t i m e .

7. PROGRESS OF MELTING

F r o m e q s . (3) a n d (4), H L = 4250 B t u a n d H s = 2240 Btu. T h u s , at t i m e l l , a p p r o x i m a t e l y o n e t h i r d of t h e h e a t a b s o r b e d n e a r t h e c e n t e r of a f a c e h a s b e e n r e t a i n e d by t h e s t e e l . T h e n , f r o m eq. (6), l 1 = 0.26 h r , w h i c h a g r e e s w e l l w i t h t h e a n a l o g m o d e l d e t e r m i n a t i o n of 0.29 h r *

T h r e e i t e m s to c o n s i d e r in t h e d e s i g n a n d a n a l y s i s of l e a d - s h i e l d e d c a s k s a r e t h e t i m e at w h i c h m e l t i n g s t a r t s in c o r n e r s o r e d g e s , t h e a m o u n t of l e a d m e l t e d u n t i l t i m e t l w h e n g e n e r a l s u r f a c e m e l t i n g b e g i n s , a n d t h e t o t a l a m o u n t of m o l t e n l e a d at s o m e l a t e r t i m e b e t w e e n t 1 a n d 12 • U n t i l e d g e e f f e c t s r e a c h t h e c e n t e r of a f a c e of a h o m o g e n e o u s c u b e w i t h c o n s t a n t flux, the l i n e a r i t y of t h e h e a t c o n d u c t i o n e q u a t i o n s s h o w s t h a t t h e t e m p e r a t u r e r i s e in a g i v e n t i m e at a c o r n e r a n d an e d g e will b e , r e s p e c t i v e l y , t h r e e a n d two t i m e s t h a t at t h e c e n t e r of a f a c e . T h e s h a p e of t h e t e m p e r a t u r e - t i m e curve (convex u p w a r d ) f o r t h e c e n t e r of a f a c e s h o w s t h a t t h e t i m e r e q u i r e d to r e a c h the m e l t i n g p o i n t of l e a d w o u l d b e l e s s t h a n I l l at a c o r n e r a n d ½ l l at an e d g e . k a n d Cp a r e g r e a t e r f o r c a r b o n s t e e l t h a n f o r l e a d , so t h a t a s t e e l s h e l l i n c r e a s e s t h e t i m e , r e l a t i v e to 11, to s t a r t m e l t i n g . S i n c e t h e s e two e f f e c t s a r e in o p p o s i t e d i r e c t i o n s , a reasonable approximation is that melting begins at a c o r n e r at t i m e ½ l 1. T h e a n a l o g m o d e l i n d i c a t e d t h a t m e l t i n g b e g i n s at t i m e ½ t l , so t h a t t h e estimate ½ l 1 is probably conservative for estim a t i n g t h e a m o u n t of l e a d t h a t h a s m e l t e d up to a g i v e n t i m e . F o r a n o u t e r s h e l l of l o w e r t h e r m a l c o n d u c t i v i t y {e.g., s t a i n l e s s s t e e l ) a s h o r t e r time, perhaps ~ tl, should be assumed rather t h a n ½ t 1. F o r a c y l i n d r i c a l l e a d - l i n e d s h i p p i n g

• 1[ in eq. (3), the value of Q is multiplied b y H L / HL ' Hs to take into account the rate of heat stored in the steel shell, the resulting quadratic equation

for H L y i e l d s H L = 5680 Btu, and the corresponding t l is 0.31 hr. For c o n s e r v a t i s m , this refinement of calculation may usually be neglected in practice.

6. T I M E T O M E L T T h e v a l u e of l1 f o r t h e s t a r t of m e l t i n g at t h e c e n t e r of a f a c e m a y b e c a l c u l a t e d f r o m eq. (6). In t h e p r e s e n t c a s e , f r o m eq. (5), Q1 = (45130 - 850)(0.57) = 25,240 B t u / f t 2 h r .

THERMAL BEHAVIOR OF SHIPPING CASKS c o n t a i n e r , the situation is s i m i l a r , except that t h e r e a r e no c o r n e r s . C o n s e q u e n t ly , m e l t i n g b e gins at e d g e s , w h e r e the c y l i n d r i c a l s u r f a c e m e e t s the end s u r f a c e s . The e s t i m a t e d t i m e to s t a r t m e l t i n g is, then, ½ t 1 if the shell is of c a r bon s t e e l , o r ~ t 1 if the shell is of s t a i n l e s s steel. F o r o t h er shaped r e g i o n s , r e a s o n a b l e ju d g ment has to be used in e s t i m a t i n g the f r a c t i o n of tl that r e p r e s e n t s the t i m e for m e l t i n g to begin. T he lid of the 15-ton cask is such a region. So a l s o is the end of a c y l i n d r i c a l shipping cask with a s p h e r i c a l end. The g e n e r a l a p p r o a c h , h o w e v e r , may be p a t t e r n e d a f t e r that outlined above. The amount of molten lead in c o r n e r s and edges at t i m e t l may be e s t i m a t e d by a heat b a l ance c a l c u l a t i o n , which r e q u i r e s e s t i m a t e s of the a v e r a g e net flux f r o m t i m e z e r o to t i m e t l , the a v e r a g e t e m p e r a t u r e of the solid lead at t i m e t l , and the t e m p e r a t u r e of the inner shell at t i m e t 1. It is c o n s e r v a t i v e to a s s u m e that the a v e r a g e net flux o v e r the e n t i r e s u r f a c e is that at the c e n t e r of a side, that the a v e r a g e t e m p e r a t u r e of all solid lead is that along a line through the c e n t e r of a side, and that the t e m p e r a t u r e of the inner shell is that f o r the c e n t e r of a side. Since the amount of s u p e r h e a t of molten lead at t i m e t 1 is not known, this f a c t o r is n e g l e c t e d . C o n s i d e r a t i o n of likely e r r o r s in the e s t i m a t e d t e m p e r a t u r e s i n d i c a t e s that the above p r o c e d u r e o v e r e s t i m a t e s the amount of molten lead at t i m e t 1 by at m o s t a few % of the total amount of lead present. F r o m eq. (3), the s p a c e mean t e m p e r a t u r e of lead in a l a r g e flat slab at t i m e t 1 is a p p r o x i mately

Qxo

TL = 621°F - 3---k-

(10)

w h e r e Q is the net a b s o r b e d heat flux, x o is the slab t h i c k n e s s , and k is the t h e r m a l c o n d u c t i v i ty. The a v e r a g e value of Q may be d e t e r m i n e d f r o m the standard radiation f o r m u l a , after e s t i mating the a v e r a g e s u r f a c e t e m p e r a t u r e as d e s c r i b e d in s ect i o n 5. The a n a l y s i s leading to eq. (3) also shows that the t e m p e r a t u r e of the i n n e r shell at t i m e t l , is a p p r o x i m a t e l y

Qxo

Ti = 621°F - 2-"-k"

(11)

C o r r e s p o n d i n g t e m p e r a t u r e s in a c y l i n d r i c a l shell would be slightly h i g h e r , but for p r e s e n t p u r p o s e s and with c o n s e r v a t i v e r e s u l t s the s a m e r e l a t i o n s h i p s may be used. The r a t e of p r o g r e s s of the i n t e r f a c e between molten lead and solid lead under steady c o n d i tions has been t r e a t e d by s e v e r a l a u t h o r s [3].

283

The v e l o c i t y of the i n t e r f a c e under t r a n s i e n t conditions is a much m o r e difficult p r o b l e m . Num e r i c a l solutions for the c a s e of a finite slab, i n i t i al l y at a u n i f o r m t e m p e r a t u r e , heated on one s u r f a c e by a constant flux, with m e l t e d m a t e r i a l r e m o v e d and with the o t h er face insulated, a r e given in ref. [4]. The i n t e g r a l method has been used by Goodman to d e r i v e an analytical solution for the s a m e p r o b l e m in ref. [5], which also quotes another solution obtained by the i n t e g r a l method by Sutton. The solutions differ in the f o r m of the t e m p e r a t u r e p r o f i l e that is a s s u m e d in c a r r y i n g out the i n t e g r a l method, which is an a p p r o x i m a t e p r o c e d u r e . A study of t h ese s o l u tions shows that the d i s p l a c e m e n t of the i n t e r face i m m e d i a t e l y a f t e r the s t a r t of m e l t i n g is p r o p o r t i o n a l to (t - t l ) 2 a c c o r d i n g to G o o d m a n ' s solution and to (t - t l ) a c c o r d i n g to Sutton's s o lution. An exact d e r i v a t i o n for the initial v e l o c i ty may be developed by r e p l a c i n g the m el t i n g p r o b l e m by a m a t h e m a t i c a l l y equivalent p r o b l e m in heat conduction with no melting. One cons i d e r s an infinite solid at t i m e (t - t l ) with the t e m p e r a t u r e p r o f i l e in the region x > Q i d en t i cal with that in the s e m i - i n f i n i t e solid, with the t e m p e r a t u r e equal to the m el t i n g point at x = 0. The t e m p e r a t u r e distribution at t i m e (t - t 1) for x< 0 is e x p r e s s e d as a power s e r i e s in x, the c o e f f i c i e n t s being adjusted so that the t e m p e r a t u r e d i s t r i b u t i o n f o r (t - t l ) > 0 changes in a suitable way with t i m e. T h i s ' s u i t a b l e way' is defined by the r e q u i r e m e n t that the t e m p e r a t u r e g r a d i e n t at the point w h e r e T = m e l t i n g t e m p e r a t u r e is r e l a t e d to the v e l o c i t y of the point by the equation g o v er n i n g the i n t e r f a c e in the m el t i n g problem, namely aT Q + k-.~ = pLy (12) w h e r e L is the latent heat and v is the v el o ci t y . T h i s equation shows that the incoming flux minus the flux ( - k T / x ) conducted away f r o m the i n t e r face into the solid lead is the r a t e at which heat is supplied to the p r o c e s s of m e l t i n g lead. When this p r o c e d u r e is c a r r i e d out, it is found that for s m a l l (t - t l ) , the d i s p l a c e m e n t of the i n t e r f a c e is p r o p o r t i o n a l to (t - tl)2 , which is the f i r s t n o n - z e r o t e r m of a power s e r i e s in (t - tl)½. Succeeding t e r m s w e r e not evaluated: h o w e v e r , heat balance c a l c u l a t i o n s , f o r v a r i o u s t i m e s b e tween t 1 and t2, with the ½ power r e l a t i o n s h i p f o r d i s p l a c e m e n t of the i n t e r f a c e , yield v a l u e s for the t e m p e r a t u r e of r e m a i n i n g solid lead which appear r e a s o n a b l e . It is t h e r e f o r e s u g gested that the amount of lead m e l t e d f r o m t i m e t l to t i m e t is a p p r o x i m a t e l y p r o p o r t i o n a l to (t - tl)~ throughout the i n t e r v a l f r o m t 1 to t 2.

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G.P. WACHTELI~ and J. W. LANGItAAR

8. MAXIMUMCAVITY TEMPERATURE F o r f i r e s l o n g e r than 15 o r 20 m i n , the h e a t l o s s to the e n v i r o n m e n t f r o m the t i m e the f i r e c e a s e s until the c a s k is at a s u b s t a n t i a l l y u n i f o r m t e m p e r a t u r e is only a s m a l l f r a c t i o n of the h e a t input. T h e m a x i m u m c a v i t y t e m p e r a t u r e m a y be a p p r o x i m a t e d by d e t e r m i n i n g the u n i f o r m c a s k t e m p e r a t u r e c o r r e s p o n d i n g to the t o t a l h e a t input. F o r t < t2, the t o t a l h e a t H is c a l c u l a t e d u s i n g e q u a t i o n s s i m i l a r to eq. (5) and (7), with the c o n s t a n t s s u i t a b l y a d j u s t e d to the g i v e n f i r e t e m p e r a t u r e . If H is l e s s than the h e a t Ho r e q u i r e d to b r i n g the c a s k to the m e l t i n g t e m p e r a t u r e of l e a d (Tmp) , then the m a x i m u m t e m p e r a t u r e in the c a v i t y w i l l be T = T O + (Tmp - T o ) . H / H o . If H o + L m L > H > H o so that a f t e r the f i r e the l e a d r e m a i n s p a r t i a l l y m e l t e d , then the m a x i m u m t e m p e r a t u r e in the c a v i t y is e q u a l to T m p . If H ~ H o + L m L , the m a x i m u m t e m p e r a t u r e is T m p + (H - H o - L r n L ) / ' C . F o r t > t2, T2 m a y be c a l c u l a t e d in the p r e c e d i n g m a n n e r , and the t e m p e r a t u r e r i s e a f t e r t i m e t 2 then m a y be d e t e r m i n e d by eq. (1) c o m bined with the r e l a t i o n s h i p d T / d ! = A Q / C . T o i n t e g r a t e the r e s u l t i n g function the c a s k is a s s u m e d to be at a u n i f o r m t e m p e r a t u r e a f t e r t2; the t e m p e r a t u r e g r a d i e n t in m o l t e n l e a d can be a c c o u n t e d f o r by a final a d j u s t m e n t . If the f i r e t e m p e r a t u r e i s d e t e r m i n e d by m e a s u r e m e n t a s in the t e s t d e s c r i b e d h e r e , E should be t a k e n as unity. T h e c o n d i t i o n s g i v e n in the A E C R e g u l a t i o n s (10 C F R P a r t 71), i . e . . a t e m p e r a t u r e of 1 4 7 5 ° F , ¢ = 0.9 and a = 0.8, a r e e q u i v a l e n t to Tf = 1425°F, ¢ = 1.0 and a = 0.8. If a s i g n i f i c a n t a m o u n t of h e a t is g e n e r a t e d within the c a v i t y by the r a d i o a c t i v e c o n t e n t s , t h i s h e a t d u r i n g the f i r e and f o r a s h o r t t i m e a f t e r the f i r e (while the c a s k is c o m i n g to an a p p r o x i m a t e l y u n i f o r m t e m p e r a t u r e ) m u s t be added to the h e a t r e c e i v e d f r o m the f i r e . If t h i s a d d i tional h e a t is r e l a t i v e l y s m a l l , a c o r r e c t i o n t e r m m a y be added to T; o t h e r w i s e , the v a r i o u s e q u a t i o n s m a y h a v e to be a d j u s t e d .

9. N O N - M E L T I N G M A T E R I A L S F o r s h i p p i n g c a s k s c o m p o s e d e n t i r e l y of s t e e l o r o t h e r n o n - m e l t i n g m a t e r i a l , r e f s . [3, 6] should p r o v e helpful f o r e s t i m a t i n g the m a x i m u m t e m p e r a t u r e of the c a v i t y and a l s o the m a x i m u m t e m p e r a t u r e d i f f e r e n c e a c r o s s the c a s k wall.

An u p p e r bound f o r the m a x i m u m c a v i t y t e m p e r a t u r e m a y be found by a s s u m i n g that at any t i m e the c a s k t e m p e r a t u r e is u n i f o r m t h r o u g h out, and then a p p l y i n g eq. (1) and the r e l a t i o n ship d T / d t = A Q , / C and i n t e g r a t i n g f o r the d u r a tion of the f i r e . T h e a s s u m p t i o n u n d e r e s t i m a t e s the a m o u n t of r e r a d i a t i o n f r o m the s u r f a c e and t h e r e f o r e o v e r e s t i m a t e s Q. T h e t e m p e r a t u r e d i f f e r e n c e a c r o s s the wall will h a v e its m a x i m u m v a l u e if the f i r e d u r a t i o n is long enough f o r a l i n e a r r a t e of r i s e (due to c o n s t a n t flux) to be e s t a b l i s h e d . The c o n d i t i o n for t h i s is that the p a r a m e t e r k t / p c x 2 e x c e e d s a p p r o x i m a t e l y 0.2 o r 0.3 in the c a s e of a flat s l a b [3]. F o r t h i c k - w a l l e d c y l i n d r i c a l and s p h e r i c a l s h e l l s , e v e n s m a l l e r v a l u e s of t h i s p a r a m e t e r will s u f f i c e , so that for r u l e - o f - t h u m b p u r p o s e s , we r e q u i r e that it e x c e e d 0.2. T h i s c o n dition will u s u a l l y be m e t . It will not be m e t if the f i r e d u r a t i o n i s too s h o r t ; in t h i s c a s e the m a x i m u m t e m p e r a t u r e d i f f e r e n c e will be s m a l l e r , and m a y be e s t i m a t e d with the help of r e f s . [3, 61. F o r a flat s l a b , the p r e v i o u s l y d e s c r i b e d r e q u i r e m e n t s , which led to eq. (3), l e a d a l s o to the c o n c l u s i o n that the m a x i m u m t e m p e r a t u r e d i f f e r e n c e is Q x o / 2 k , w h e r e x o is the wall t h i c k n e s s . F o r a c y l i n d e r with i n n e r and o u t e r r a d i i r 1 and r 2, r e s p e c t i v e l y , a s i m i l a r a n a l y s i s in cylindrical coordinates yields a temperature diff e r e n c e of 2 r2r 1 k

lo )

S i m i l a r l y , f o r a s p h e r i c a l s h e l l , the m a x i m u m t e m p e r a t u r e d i f f e r e n c e is 2 3 Q/L,~. :_ rl r2 - 2 r ~ r 2 , 2k~, r ~ +

rlr2+

r 2

~"

A s with l e a d - s h i e l d e d c a s k s , a c o r r e c t i o n is r e q u i r e d if the c a s k c o n t e n t s g e n e r a t e a s i g n i f i c a n t a m o u n t of heat.

I0. SUMMARY AND CONCLUSIONS A 15-ton s h i p p i n g c a s k was s u b j e c t e d to an open a i r f i r e of l i q u i d p e t r o l e u m fuel for 1 hr. T h e r m o c o u p l e s p l a c e d in the f i r e i n d i c a t e d a f i r e t e m p e r a t u r e of a p p r o x i m a t e l y 1800°F. T h e r m o c o u p l e s p l a c e d in the body of the c a s k s h o w e d v e r y l i t t l e h o r i z o n t a l t e m p e r a t u r e g r a d i e n t in m o l t e n l e a d but a v e r t i c a l t e m p e r a t u r e g r a d i e n t of a s m u c h a s 8OF/in. Both o b s e r v a t i o n s may be a t t r i b u t e d to n a t u r a l c o n v e c t i o n of m o l t e n lead.

THERMAL BEHAVIOR OF SHIPPING CASKS A s a r e s u l t of the v e r t i c a l t e m p e r a t u r e g r a d ie n t , the mean t e m p e r a t u r e of the molten l e a d was a p p r o x i m a t e l y 175OF above the m e l t i n g point of lead, even b e f o r e all the l e a d was melted. T h e r e a r e no e x p e r i m e n t a l data on the e f f e c t s of heat flux and cask s i z e on v e r t i c a l t e m p e r a t u r e g r a dient, but a p l a u s i b l e a s s u m p t i o n is that the g r a dient is p r o p o r t i o n a l to net heat flux. An analog model, c o m p o s e d of r e s i s t o r s , c a p a c i t o r s , diodes, and r e l a y s , was u s e d to s i m u late the t h e r m a l b e h a v i o r of the body of the cask, a p p r o x i m a t e d as a cube. The f i r e t e m p e r a t u r e , f i r e e m i s s i v i t y , s u r f a c e a b s o r p t i v i t y , and a p p a r e n t t h e r m a l conductivity of molten lead w e r e v a r i e d . The analog model t e m p e r a t u r e s a g r e e d best with the m e a s u r e d t e m p e r a t u r e s when Tf = 1800OF, E = 1, a = 0.6 and kliquid = ksoli d. The t e m p e r a t u r e s would probably have a g r e e d b e t t e r if analog model c a l c u l a t i o n s had been made with infinite e f f e c t i v e k l i q u i d . A c c o r d i n g to the analog model r e s u l t s , m e l t ing b e g i n s at the c e n t e r of a f a c e at t i m e t 1 = 0.29 h r, and at a c o r n e r at t i m e ½ tl , a p p r o x i ma te ly . Melting is c o m p l e t e , a c c o r d i n g to the analog model (which did not include the effect of supe rh eat ), at t i m e t 2 = 0.57 hr. H a n d - c a l c u l a t i o n m e th o d s a r e developed to find ot f r o m the e x p e r i m e n t a l l y d e t e r m i n e d t e m p e r a t u r e s . The r e s u l t is a = 0.57. In view of v a r i o u s e r r o r s , it is concluded that a = 0.6 ~: 0.05. The hand c a l c u l a t i o n s also show t I = 0.26 hr and the nominal t i m e to c o m p l e t e m e l ti n g (assuming no s u p e r h e a t e d lead) as t = 0.58 h r , which a g r e e s c l o s e l y with the analog model r e sult. In view of the v e r t i c a l t e m p e r a t u r e g r a dient, a m o r e r e a l i s t i c t i m e f o r p r a c t i c a l l y c o m p l e t e m e l t i n g is t = 0.74 hr. The t i m e to s t a r t m e l t i n g at a c o r n e r is a p p r o x i m a t e l y ~ t 1 for a r e c t a n g u l a r shipping c o n t a i n e r with a c a r bon s t e e l shell and a p p r o x i m a t e l y A t 1 with a s t a i n l e s s s t e e l shell. At t i m e --
285

i n t e r n a l heat g e n e r a t i o n and v e r t i c a l t e m p e r a t u r e g r a d i e n t s (for which additional a l l o w a n c e s would have to be made), is found for v a r i o u s c a s e s . To c a l c u l a t e the m a x i m u m c a v i t y t e m p e r a t u r e , it is c o n s e r v a t i v e l y a s s u m e d that the cask c o m e s to a u n i f o r m t e m p e r a t u r e with no heat l o s s to the e n v i r o n m e n t a f t e r the f i r e . C o r r e c t i o n s may be r e q u i r e d for i n t e r n a l heat g e n e r a t i o n and for a v e r t i c a l t e m p e r a t u r e g r a d i e n t in the m e l t e d lead. F o r shipping c a s k s made of n o n - m e l t i n g m a t e r i a l s , such as s t e e l , f o r m u l a s for the m a x i mum t e m p e r a t u r e d i f f e r e n c e a c r o s s the wall a r e p r e s e n t e d . In using t e m p e r a t u r e s indicated by f i r e t h e r m o c o u p l e s to r e p r e s e n t f i r e t e m p e r a t u r e s , one should use E = 1 in computing the heat flux. Although the cask which was t e s t e d did not i n c o r p o r a t e any heat shield, i n t e r m e d i a t e st eel shell, or l a r g e a i r gaps, many of the r e s u l t s a r e applicable to such sp eci al c a s e s .

APPENDIX A A1. Summary An i l l u s t r a t i v e application of the i n f o r m a t i o n in the a r t i c l e is p r e s e n t e d for a hypothetical 12ton c y l i n d r i c a l cask. As in the AEC R e g u l a t i o n s , f i r e conditions a r e 1475OF f o r 30 min, with ~ = 0.9 and ot = 0.8. A s s u m e d cask c h a r a c t e r i s t i c s are: Cavity size 24 in. d i a m e t e r by 36 in. long Shell t h i c k n e s s ¼ in. o u t e r , ½ in. inner Lead shielding 8 in. Fins 44 longitudinal fins, ~ in. thick, 3 in. high Lid r e c e s s e d , ½ in. sidewall Cask o r i e n t a t i o n vertical Cask contents n e g l e c t e d for c a l c u l a t i o n s Initial t e m p e r a t u r e 150OF T h o s e r e s u l t s b e l i e v e d to be of p r i n c i p a l i n terest are:

Body Lid T i m e to s t a r t of g e n e r a l melting, t l , hr T i m e f o r c o m p l e t e melting, t2, hr % of lead m e l t e d at t i m e tl of lead m e l t e d at 30 min Max. outside temp. (top during f i r e ) , OF Max. inside temp. (after f i r e ) , OF

0.36 0.39 0.85 0.70 10 33 24 47 841 598

685 621

286

G . P . WACHTEI.I, and J. W. I,ANGHAAR

A2. Discussion

15. C a l c u l a t e m a x i m u m after fire

inner wall temperature

Hypothetical cylindrical c a s k A3. Details o f calculations t"

4 o,g

F o r t h e p u r p o s e of i l l u s t r a t i o n , d e t a i l s of t h e c a l c u l a t i o n s a r e g i v e n b e l o w f o r t h e body of the cask. Calculations for the lid are similar.

1. Weight and area of body E f f e c t i v e e x t e r n a l a r e a , ft2 Weight, fins, lbs Weight, outer shell, lbs Weight, inner shell, lbs Weight, lead, lbs

51.8 420 1590 580 1G540

2. A v e r a g e surface t e m p e r a t u r e to t 1 Dimensions are assumed at room t e m p e r a t u r e . Col'r(_,sponding densities are: Steel 4901bs/ft 3, l,ead 710 I b s / ft3. \'alues of specific heat and latent heat as given in table 3 arc used. T h e c o n f i g u r a t i o n s h o w n f o r r e c e s s i n g t h e lid d i f f e r s f r o m c u s t o m a r y d e s i g n but i s u s e f u l f o r illustration. A shell heated from one side, such a s t h a t p a r t of t h e body a r o u n d t h e e d g e of t h e lid. r a p i d l y a t t a i n s a n i n t e r m e d i a t e t e m p e r a t u r e so a s to r a d i a t e f r o m e a c h s i d e 50% of t h e a b s o r b e d flux. T h e s i t u a t i o n i s c o m p l i c a t e d by r e r a d i a t i o n f r o m t h e s i d e of t h e lid, b u t t h i s i s m i nor and a suitable approximation is that the side of t h e l i d h a s a n e t a b s o r b e d flux e q u a l to 50% of t h a t a b s o r b e d by t h e e x t e r i o r s u r f a c e . C o n d u c t i o n of h e a t to t h e body f r o m t h e p a r t of t h e s h e l l s u r r o u n d i n g t h e lid, a n d a l s o c o n d u c t i o n b e t w e e n t h e lid a n d b o d y , a r e n e g l e c t e d . T h e v a r i o u s s t e p s in t h e c o m p u t a t i o n a r e : 1. C a l c u l a t e w e i g h t a n d a r e a of c a s k p a r t s 2. E s t i m a t e a v e r a g e s u r f a c e t e m p e r a t u r e up to tl 3. C a l c u l a t e a v e r a g e n e t a b s o r b e d flux to t 1 4. C a l c u l a t e a v e r a g e t e m p e r a t u r e of s o l i d l e a d at t 1 5. C a l c u l a t e t e m p e r a t u r e of i n n e r s h e l l at / 1 6. C a l c u l a t e v a l u e of t 1 7. C a l c u l a t e t o t a l h e a t of c a s k at t 1 8. C a l c u l a t e a m o u n t of m o l t e n l e a d at l l 9. E s t i m a t e s u p e r h e a t of m o l t e n l e a d 10. E s t i m a t e s u r f a c e t e m p e r a t u r e f o r i n t e r v a l l 1 to t 2 11. C a l c u l a t e n e t a b s o r b e d flux f o r i n t e r v a l t 1 to t2 12. C a l c u l a t e t o t a l h e a t of c a s k at t 2 13. C a l c u l a t e t 2 f r o m t o t a l h e a t a n d flux 14. C a l c u l a t e a m o u n t of m o l t e n l e a d at 0.5 h r

S u p e r h e a t of m o l t e n l e a d in c o r n e r s is n e g l e c t e d f o r t h i s c a l c u l a t i o n . T o = 150OF. T h e assumption that T varies linearly with time, c o m b i n e d w i t h eq. (1), s h o w s a f t e r i n t e g r a t i o n and algebraic manipulation that T = 417°F, approximately.

3. A v e r a g e net absorbed.flux to t i m e I 1 F i r e t e m p e r a t u r e i s 1475OF = 1935OR, w i t h ( : 0.9 a n d c~ = 0.8. Q = [(19354)(0.9) - 8774] >: ( 0 . 1 7 3 ) ( 1 0 - 8 ) ( 0 . 8 ) = 16600 B t u / h r - f t 2.

4. A v e r a g e l e m p e r a t u r e o f solid lead al l i m e l 1 F r o m eq. (10), t h i s i s e s t i m a t e d to b e 423OF w h e r e x o i s l e a d t h i c k n e s s , ft, a n d k i s t h e r m a l conductivity, Btu/hr-ft2-°F ft.

5. T e m p e r a t u r e o f inner shell al l i m e tl F r o m eq. (11), t h i s i s e s t i m a t e d to b e 324OF.

6. Calculated value o.f tl T h i s i s b a s e d on a h e a t b a l a n c e f o r a s e c t i o n of t h e s i d e w h e r e m e l t i n g i s a b o u t to s t a r t . T h e r e i s a r a t h e r w i d e r a n g e of t h e r m a l c o n d u c t i v i t i e s f o r d i f f e r e n t t y p e s of s t e e l , so t h a t in a p a r t i c u l a r c a s e t h e d e s i g n e r s h o u l d u s e an a p propriate value. Since the effect is rather small f o r p r e s e n t p u r p o s e s , t h e v a l u e 31.8, a s g i v e n in t a b l e 3, i s u s e d . T h e n t h e t e m p e r a t u r e d r o p through the shell is about (16600)(0.75) = 33OF"

(31.8)(12) T h e a v e r a g e t e m p e r a t u r e of t h e s h e l l m a y b e t a k e n a s 621 + 3 3 / 2 = 637OF. T h e fin t e m p e r a t u r e m a y b e t a k e n a s 621 + 33 = 6 5 4 ° F w h i c h i s s o m e w h a t l o w e r t h a n would a c t u a l l y b e t h e c a s e a n d r e s u l t s in a s m a l l but u n i m p o r t a n t u n d e r e s t i m a t e of t 1 . F o r o n e s q u a r e foot of t h i s s i d e ,

287 the weights are: lbs Fins 10.0 Outer shell 30.6 Inner shell 12.0 Lead 368 T h e s e w e i g h t s a r e b a s e d on t h e c y l i n d r i c a l g e o m e t r y . T h e e q u a t i o n s in s e c t i o n 6 a r e b a s e d on a f l a t s l a b ; h o w e v e r , c a l c u l a t i o n s b a s e d on t h e c h a r t s of r e f . [8] w i t h c o n s t a n t e x t e r n a l c o e f f i c i e n t h. w h i c h i s c o n s i d e r e d a s u f f i c i e n t l y good approximation for this purpose, indicate that for casks with internal radius greater than the wall thickness, the space mean temperature for a cylindrical shell is only slightly higher than for a f l a t s l a b . T h e t o t a l h e a t p e r s q u a r e foot of s i d e at time t 1 is then

Btu/ft 2 Fins Outer shell Inner shell Lead

( 1 0 . 0 ) ( 6 5 4 - 150)(0.125) (30.6) (637 - 150)(0.125) ( 1 2 . 0 ) ( 3 2 4 - 150)(0.125) (368)(423 - 150)(0.0325)

= = = =

630 1860 260 3270 6020

tl

6020 = 16600 - 0.363 h r .

M e l t i n g at t h e c e n t e r of t h e b o t t o m , c a l c u l a t e d in a s i m i l a r m a n n e r b u t w i t h t h e s l a b g e o m e t r y a n d w i t h o u t f i n s . s t a r t s a t 0.392 h r . T h e d i f f e r e n c e i s u n i m p o r t a n t , a n d t l will b e taken as 0.363 hr.

(1475 + 46014 4.3OF/in" 8OF (1800 + 460)4 = In t h e 44 in. h e i g h t , t h e a v e r a g e t e m p e r a t u r e i s a b o u t 4 . 3 ° F 4 4 / 2 = 95 ° a b o v e m e l t i n g w h i l e solid lead remains. 10. Surface t e m p e r a t u r e f r o m tl to t 2 The temperature drop through the steel is slightly less than the previously calculated 33°F b e c a u s e of t h e s l i g h t l y l o w e r n e t flux, a n d w i t h s u f f i c i e n t a c c u r a c y m a y b e 30OF. T h e f a c t t h a t the bottom is a little cooler than the sides may be neglected. Then the average surface tempera t u r e i s 621 + 95 + 30 = 746OF = 1 2 0 6 ° R . 11. Net absorbed f l u x f r o m t 1 to t 2 Q : [(19354)(0.9) - ( 1 2 0 6 4 ) ] ( 0 . 1 7 3 ) ( 1 0 - 8 ) ( 0 . 8 ) : 14500 B t u / h r - f t 2 . 12. Total heat at t i m e l 2 W i t h a v e r a g e t e m p e r a t u r e s a s i n d i c a t e d in t h e following calculations, the heat is:

Blu Fins (420)(746 - 150)(0.125) Outer shell (1590)(731-150)(0.125) Inner shell (580)(716- 150)(0.125) Lead, to m e l t (16540)(621 - 150)(0.0325) ing Lead, l a t e n t heat(16540)(10.55) Lead, s u p e r h e a t (16540)(95)(0.038)

31300 = 115400 = 41000 = 253200 = 174500 = 59700

675100

7. Heat o f body at t 1 F o r t h e a r e a of 51.8 ft 2 a n d a v e r a g e n e t f l u x of 16600 B t u / h r - f t 2, t h e h e a t i s (51.8)(0.363) = 312,000 Btu.

8. A m o u n t o f molten lead al t l

Blu (420)(654 - 150)(0.125) (1590)(637-150)(0.125) (580)(324 - 150)(0.125) (16540)(423-150)(0.0325)

= = = =

26500 96800 12600 146800

282700 T h e e x c e s s h e a t of ( 3 1 2 0 0 0 - 283000) B t u i s s t o r e d in m o l t e n l e a d . T h e h e a t of m o l t e n l e a d , n e g l e c t i n g s u p e r h e a t , e x c e e d s t h a t of s o l i d l e a d on t h e a v e r a g e by (621 - 4 2 3 ) ( 0 . 0 3 2 5 ) + 10.55 = 17.0 B t u / l b . T h e a m o u n t of m o l t e n l e a d i s t h e r e f o r e 2 9 0 0 0 / 1 7 . 0 = 1700 l b s .

9. Superheat o f molten lead a f t e r l i m e t 1 The expected vertical temperature about

t 2 - tl =

6 7 5 0 0 0 - 312000 (51.8)(14500) = 0.483 hr

t 2 = 0.483 + 0 . 3 6 3 = 0 . 8 4 6 h r

Temperatures for the shells, fins, and solid l e a d a r e t a k e n to b e t h e s a m e a s in (6). If a l l lead were solid, the total heat would be: Fins Outer shell Inner shell Lead

13. Calculated value o f t 2

gradient is

14. A m o u n t o f molten lead, Wf, at 0.5 hr Using the approximation that after /1, the m e l t i n g p r o g r e s s e s a s t h e 1.5 p o w e r of t i m e , t h e n s i n c e at t l , t h e a m o u n t m o l t e n i s 1700 l b s , 16540 - 1700 = C ( 0 . 4 8 3 1 . 5 ) Wf = 1700 + C ( 0 . 5 0 0 - 0 . 3 6 3 ) 1 . 5 Wf = 1700 + 2240 = 3940 l b s . T h e c a l c u l a t i o n s do n o t i n c l u d e a n e s t i m a t e of t h e e x t e n t to w h i c h m e l t i n g w i l l p r o g r e s s i m m e d i a t e l y a f t e r t h e f i r e b e c a u s e of e x c e s s h e a t a v a i l a b l e in t h e o u t e r s h e l l a n d in t h e p r e v i o u s l y melted lead. This transient phenomenon has not been investigated; however, an upper limit for t h i s a d d i t i o n a l m e l t i n g m i g h t b e e s t a b l i s h e d by d e t e r m i n i n g t h e a m o u n t of t i m e u n d e r f i r e c o n d i t i o n s f o r n e t h e a t a b s o r p t i o n e q u a l to t h e e x c e s s heat (above the melting point) and extrapolating t h e t i m e - m e l t i n g c u r v e to t h i s e x t e n t .

288

G.P. WACHTELI, an(l J. W. I,ANGIlAAR

15. M a x i m u m

inner wall temperature

after.fire

With t h e body at a u n i f o r m t e m p e r a t u r e , t h e h e a t p e r OF up to t h e m e l t i n g p o i n t of l e a d i s : Btu/OF

Steel Lead (16540)(0.0325)=

T(x) = t e m p e r a t u r e Tf C k

t i m e f r o m s t a r t of f i r e , h r

l

324 538

ll

: t i m e at w h i c h m e l t i n g s t a r t s in c e n t e r of

862 The heat absorbed up to 0.5 hr is: At t 1 : F r o m l 1 to 0.5 h r ( 0 . 5 0 0 - 0 . 3 6 3 ) ( 1 4 5 0 0 ) ( 5 1 . 8 ) =

t2

=

t3 x

= =

X0 (Y

= =

O/

=

p

=

283000 103000 386000

Then 386000 _ 448o F t e m p e r a t u r e rise 862

With the i n i t i a l t e m p e r a t u r e of 150OF, t h e f i n a l t e m p e r a t u r e w o u l d b e 448 + 150 = 598OF. S i n c e t h i s i s b e l o w t h e m e l t i n g p o i n t of l e a d , no f u r t h e r calculation is needed.

at p o s i t i o n x , OF, at t i m e t l

= f i r e t e m p e r a t u r e , OF = s p e c i f i c h e a t , Btu lb OF = t h e r m a l c o n d u c t i v i t y , B t u / h r ft OF

f a c e ; i . e . , t i m e f o r m e l t i n g to s p r e a d o v e r e n t i r e o u t e r s u r f a c e of l e a d t i m e at w h i c h all l e a d h a s m e l t e d ; i . e . . t i m e at w h i c h m e l t i n g a l o n g l i n e t h r o u g h c e n t e r of f a c e h a s p r o g r e s s e d to i n n e r shell some time slightly greater than t 2 d i s t a n c e f r o m i n n e r s h e l l , ft (x = 0 at i n n e r s u r f a c e of l e a d ) v a l u e of x at o u t e r s u r f a c e of l e a d S t e f a n - B o l t z m a n n c o n s t a n t = 0.173 :~ 10 -8 Btu."ft2_hr_OR4 s u r f a c e a b s o r p t i v i t y , a s s u m e d e q u a l to surface emissivity flame emissivity density, Ib/ft 3

NOMENC L A T U R E A C H HL Hs L Q Q1 Q2 T To 7"

= a r e a e x p o s e d to f i r e , e x c l u d i n g f i n s , ft 2 = t o t a l h e a t c a p a c i t y of c a s k o r a p p r o p r i a t e part, Btu/OF = t o t a l h e a t a b o v e t e m p e r a t u r e T O at any t i m e , Btu = h e a t Of l e a d a b o v e To at t i m e t l , p e r ft 2 of o u t e r s u r f a c e , B t u / f t 2 :: h e a t of o u t e r s h e l l a b o v e To at t i m e t l , p e r ft2 of o u t e r s u r f a c e , Btu 'ft 2 == l a t e n t h e a t of l e a d , B t u / l b = n e t a b s o r b e d h e a t flux, B t u / f t 2 - h r = a v e r a g e v a l u e of Q f r o m t i m e z e r o to t i m e tl = a v e r a g e v a l u e of Q f r o m t 1 to l 2 = o u t e r s u r f a c e t e m p e r a t u r e at any t i m e , OF = i n i t i a l t e m p e r a t u r e of c a s k , OF = w e i g h t e d a v e r a g e v a l u e of T f r o m t i m e z e r o to t i m e t l , f o r e s t i m a t i n g a v e r a g e reradiation from surface

RefcYcn¢CS

1. B.E. Bader, Iieat t r a n s f e r in liquid hydrocarbon fires, Proceedings. International Symposium for Packaging and Transportation of Radioactive Materials. Albuquerque, N.M.. 12-15 Jan. 1965. 2. G.P. Wachtell a n d J . W . Langhaar. F i r e test and thermal behavior of 15-ton lead shielded cask, DP1070. Sawmnah River Laboratory (1966}. 3, tI.S. Carslaw a n d J . C . J a e g e r , Conduction of Heal, in Solids, 2nd Ed. {Clarendon P r e s s . Oxford, 1959~. t. H.G.l,andau. Heat conduction in a melting solid. Quart. Appl. Math. ~ (1950) ~1. 5. T.R. Goodman, Application of integral methods to transient nonlinear heat t r a n s f e r , in: Advances in ttcat T r a n s f e r , Vol. I, eds. T . F . Irvine J r . a n d J . P. Ilartnett (Academic P r e s s . New York. 1964). 6. P . J . S c h n e i d e r . T e m p e r a t u r e Response Charts ~JohnWiley, New York, 1963). 7. C . F . Bonilla and A . I , . S t r u p c z e w s k i , An e l e c t r i c analog computer for nuclear fuel shipping cask fire tests. Nucl. Struet. Eng. 2 (1965) 40.