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ORNL-TM-3145

Contract No. W-7405-eng-26

Instrumentation and Controls Division

THERMAL RADIATTON TRANSFER OF AFTERBEAT IN MSBR HEAT EXCHANGERS

J. R. Tallackson

LEGAL NOTICE I

, 1

This report was prepared as an account Of work sponsored by the United States Government. Neither the United States nor the United States Atomic Energy Commission, nor any of their employees, nor any of their contractors, subcontractors, or their employees, makes any warranty, express or implied, or assume8 any legal tiabfiity or responsibility for the accaracy. completeness or usefulness of m y infarmation, apparatus, product or process disclosed, or represents that its use would not infringe privately owned rights.

MARCH 1971

OAK RIDGE NATIONAL LABORATORY Oak Ridge, Tennessee operated by UNION CARBIDE CORPORATION for the U.S. ATOMIC ENERGY COMMISSION 1

iii

i 4llaJ ! !

CONTENTS

Page

u 2

Abstract I. II. III.

IV. v. VI.

-------_--_-----------------------------------------------""""""""""""""'-""""-'--------~

Introduction

--------------------------c-

MSBR Heat Exchanger Configuration Generation

Metallurgical Results

Appendix A.

and Recommendations -----------_------------------

Conclusions

Afterheat

and Distribution

------------------------

_-------------------____c________

Considerations

----_-------------------------------------------------

3 6 11

Computational Model and Assumptions Governing the Computations ----------------------------

Appendix B.

Directional

Appendix C.

View Factors

Appendix D.

Shell

Appendix E.

The Heat Transfer

Appendix F.

Computational

Appendix G.

Thermal Radiation Characteristics of Hastelloy N ----------------------------------------

Appendix H.

Comparison - Experiments

-----------------

Appendix I.

Method Used to Estimate the Initial Peak Temperature Transient in the 563~Mw.Reference Design Heat Exchanger ----------------

* t References

Mstribution

of Radiation

-----------------

------------------------------------------

Temperatures

_----------------------------------- Temperature Equations

61 -------------

Procedure ----_--c-----------------------

and Analyses

----------------_----------------------------------------

63 69

103

r-_

1 THERMAL RADIATION TRANSFER OF AFTERHEAT

IN MSBR HEAT EXCHANGERS J. R. Tallackson

ABSTRACT A f r a c t i o n , estimated t o be bq&,of t h e heat-producing noble-metal f i s s i o n products--niobium, molybdenum, technetium, ruthenium, rhenium, and tellurium--is expected t o deposit on t h e metal surfaces within t h e primary f u e l - s a l t loop i n a molten-salt reactor. Virtually a l l of t h i s b$ w i l l be i n t h e heat exchangers. The normal means of a f t e r h e a t removal i s t o continue t o c i r c u l a t e the primary and secondary salts." The worst abnormal s i t u a t i o n arises i f t h e heat exchangers are quickly drained of both primary and secondary salts i n circumstances such t h a t a l l a f t e r h e a t removal from the heat exchangers is, of necessity, by r a d i a t i v e heat t r a n s f e r . Whereas such an event w i l l rarely, i f ever, take place, t h e primary system must accommodate the consequences, princ i p a l l y high temperatures, without compromising containment.

Steady-state temperature calculations, based on r a d i a t i v e heat t r a n s f e r i n MSBR primary heat exchangers, a r e presented. Several s i z e s with r a t i n g s from 94 Mw t o 563 Mw and a l l with t h e same general configur a t i o n were considered. Radial temperature p r o f i l e s were computed f o r a f t e r h e a t rates corresponding t o elapsed times from 100 sec t o 11 days after r e a c t o r shutdown. The e f f e c t of the emissivity of t h e i n t e r n a l r a d i a t i n g surfaces i n t h e heat exchangers was included. The calculations show t h a t t h e p r i n c i p a l single barrier t o heat removal i s t h e i n t e r mediate s h e l l surrounding t h e tube bundle. This s h e l l i s located j u s t i n s i d e t h e outer s h e l l and, unfortunately, becomes an e f f e c t i v e thermal r a d i a t i o n shield. It i s shown t h a t heat exchangers with but one s h e l l instead of two as i n t h e MSBR reference design w i l l achieve significant,reductions i n peak temperatures, p a r t i c u l a r l y i n t h e l a r g e r sizes and a t low emissivities. No t r a n s i e n t case was computed but t h e upper l i m i t of t h e i n i t i a l t r a n s i e n t was estimated; t h e heat capacity of the exchangers a f f o r d s a cushion which, with the exception of t h e 563-~wu n i t , l i m i t s m a x i m u m temperatures t o numbers t h a t are high but not disastrous. Design changes t o increase r a d i a t i n g surface areas and t o shorten t h e radial t r a n s f e r distance through t h e tube bundle should render the 563-~w exchanger acceptable.

Keywords:

thermal radiation, noble metals, heat exchangers, emergency cooling, deposition, a f t e r h e a t .

I. INTROMTCTION

The a f t e r h e a t problem i n molten-salt reactors caused by t h e noble metals which p l a t e out on metal surfaces during reactor operation has not gone away. Reliable, consistent data are scant, but, from MSRF, data, Brims' has estimated that b$ of t h e noble-metal f i s s i o n products m i g h t p l a t e out on t h e metal surfaces i n t h e primary salt c i r c u i t . These produce substantial amounts of a f t e r h e a t and v i r t u a l l y a l l of it w i l l be developed i n t h e heat exchangers.

. ~

Normally, afterheat i n an MSER i s e a s i l y removed by continuing the circulation of t h e primary and secondary salts. The situation, a l b e i t unlikely, may arise i n which both primary and secondary salts are rapidly drained immediately after reactor shutdown. In these circumstances, afterheat removal i s , of necessity, solely by radiative t r a n s f e r and t h e maximum temperatures so developed are of considerable i n t e r e s t . I n September 1967 t h e a u t h o p presented calculated estimates of temperatures produced by afterheat from noble metals plated on t h e surfaces i n an empty primary heat exchanger of a two-region MSBR. It was assumed t h a t a l l afterheat r e j e c t i o n was by radiative t r a n s f e r . These temperatures were d i s t r e s s i n g l y high, due, i n l a r g e measure, t o the overly simplified computational model employed (see Appendix E). I am pleased, without benefit of rack and thumbscrew, t o recant. More reali s t i c calculations based on the single-region "reference design'13 MSER heat exchangers indicate t h a t peak afterheat temperatures, while s t i l l uncomfortably high, w i l l be much lower than o r i g i n a l l y anticipated.

GL i

11. CONCLUSIONS AND RECOMMENDATIONS

MSBR "reference design" type heat exchangers can be designed t o withstand t h e after-shutdown temperature rise produced by noble-metal afterheat. This worst-case study indicates t h a t the two l a r g e r sizes considered, 563 Mw and 281 Mw, may experience excessively high temperatures, but t h e excess i s small. If t h e overall diameters of these exchangers are increased, t h e peak temperatures can be limited t o acceptable values without undue penalties i n cost. The improvement obtained i s twofold: (1) The reducti6n i n annulus thickness decreases t h e radia t i o n path through t h e tube annulus, and (2) t h e radiating areas of t h e outer and intermediate shells a r e increased. The additional intermediate s h e l l between t h e tubes and t h e outside shell i s an e f f e c t i v e b a r r i e r t o r a d i a t i v e transfer. This s h e l l i s required i f t h e tube bundle i s t o be replaced i n s i t u . Significant reductions i n peak temperatures w i l l be obtained i f t h i s s h e l l i s e l i m i nated. Alternatively, if t h e e f f e c t i v e emissivity of t h e surfaces of t h e intermediate shell and t h e outer s h e l l can be made very high (3 0.8) so as t o be nearly black, t h e peak temperatures w i l l be appreciably lower.

3 Calculated estimates of radiant heat transmission and r e s u l t i n g temperatures may be extremely s e n s i t i v e t o t h e assumptions, approximations, and u n c e r t a i n t i e s on which t h e calculations a r e based. For example, e a r l i e r calculations on a not t o o dissimilar heat exchanger, using an oversimplified model,* produced discouragingly high temperature forecasts; t h e temperatures estimated by these e a r l i e r computations were as much as 2500째F higher than t h e temperatures reported herein. The present comput a t i o n a l model i s widely accepted and used. It contains no gross compromises w i t h respect t o heat exchanger geometry. It does require t h a t photons be emitted, reflected, and absorbed i n a simple pattern, and t h a t t h e i r behavior be unaffected by temperature. A casual l i t e r a t u r e search i n d i c a t e s t h a t t h e e r r o r s produced by these simplifications may be r e l a t i v e l y small and probably on t h e high side, but confirmation would be highly desirable. The v a r i a t i o n of maximum temperature with emissivity i s s u b s t a n t i a l . Generally, I conclude t h a t these calculated temperatures may err on t h e high side but h e s i t a t e t o enumerate t h e amount. It w i l l not be worthwhile t o use or develop more accurate and elegant computational approaches u n t i l r e l i a b l e experimental evidence, applicable t o t h i s part i c u l a r type of problem, has been produced. Without t h e support of t h e experimental confirmation we can expect t o produce a conservative and perhaps expensive design. If radiant heat transmission remains a dominant consideration, I recommend an experimental program t o support and confirm t h e analyses. Such experiments are not uncomplicated; they must be c a r e f u l l y designed and well planned. 111. MSBR HEAT EXCHANGm CONFIGURATION

Figure 1 i s a v e r t i c a l section through t h e "reference designtt3 MSBR primary heat exchanger rated a t 563 Mw. Four of these w i l l be required f o r a 2250-Mw(th) MSBR. This exchanger type was scaled down by r a t i n g f a c t o r s of 1/2, 1/3, 1/4, and 1/6, thereby giving heat exchangers r a t e d a t 281, 188, 141, and 94 Mw. The dimensions and d e t a i l s pertinent t o these heat t r a n s f e r computations are on Fig. 2. It should be noted t h a t t h e scaled-down dimensions of t h e four smaller exchangers do not follow any precise scaling law(s) based on s t r e s s o r flow. For example, it i s more r e a l i s t i c t o choose nominal pipe s i z e s and p l a t e thicknesses f o r t h e inner and intermediate s h e l l s instead of t h e non-standard diameters and thicknesses t h a t would r e s u l t from any exact scaling down. Also, it would be unwise t o use thicknesses less than 1/2 i n . f o r t h e outer shells. Therefore, t h e outer s h e l l s a r e 1/2 i n . i n a l l t h e exchangers

A second set of calculations was made f o r 563-m exchangers having l a r g e r outside diameters, thinner annuli, and therefore fewer tube c i r c l e s than t h e "reference design." These exchangers a r e scaled-up versions of t h e 563-m u n i t which has 3 l t u b e c i r c l e s i n the tube annulus. These calculations a r e discussed i n Section VI. *F,ach tube c i r c l e was presumed t o be a continuous impenetrable s h e l l (see ref. 2).

ORNL DWG. 69-6004R

t SECONDARY

SALT

Average Tube -22 ft

/ Length:

Tube Annulus: Contains 5549 Tubes, 3/8 0. Diam, i n 31 Concentric Circles. Radial Pitch = 0.717 i n . Circumferential . Pitch = 0.750 in.

20 in. sched 40 Pipe

36 in. diam Fig. 1.

MSBR 563-Mw "Reference Design" Primary Heat Exchanger.

5 i

:hJ

O W L DWG. 71-567 Faterial; Tubes and Shells; Eastelloy A:

I Intermediate Shell

Density Melting point Them1 conductivity, a t 13009

0.320 l b / i n ?

2470

- 2755

Specific heat

0.138

Btu

lb-9

Heat Exchanger Tubes External surface area = 0.@2 rtP/it Metal area, sectional I 0.0373 in.* = 2.59 x 10-4 it* Weight = 0.143 l b / f t Effective tube length = 22 it, a l l exchangers

----

Circumferential pitch 0.750 in. Radial pitch 0.717 in. The fraction of cross sectional ores occupied by tubes is same as for tubes having

Ratilrg Inner Shell

(h)

lB6

8 in.

sched 40

A i 4.31 io. A 1 = 2.26 itp/it

io., ached bo = 5.38 In. Ai 3-91fte/it

Iotemdiate Shell t

1.00 In.

19.50 io. 1.50 in. 10.20

= 7.33

ityit

4J 7 . 6 ft /ft

Re t Ai A.

p&. 2.

R2 = 14.00 in.

ftyit

= U.00 it /it

Rs Ai

Outer Shell

MmRIsions

15.50 in. 8.10 It ft 8.36 it /ft

Ra = 21.50 in. 11.25 itg/ft A. = 11.51 -/it Ai

%be Annulus !Cube

R, = 13.14 in. % = 5.25 in. @ = 7.89 io.

% % @

18.75 io. 7.28 io. 11.47 IO.

Magrem, With Mmensiona, of "ransverse Cross Sections 'Ihrougb

Circles

Total Tubes

924

1853

Heat Exchangers

6 IV. AFTEEEIEAT GENERATION AND DISTRIBUTION

'6, i

The mount of afterheat i n t h e heat exchanger i s based on Briggs' estimate' t h a t b$ of t h e noble-metal f i s s i o n products p l a t e out on t h e metal surfaces exposed t o the primary salt. The inside surfaces of t h e heat exchanger tubes provide 39,000 f t a of surface i n a 2250-Mw(th) MSBR system. This i s a very large area compared with t h e s h e l l and pipe surface areas i n t h e primary s a l t c i r c u i t . It can be assumed, with negligible error, t h a t t h e e n t i r e b$ i s deposited on t h e inner surfaces of the exchanger tubes.

The heat-producing noble metals a r e niobium, molybdenum, technetium, ruthenium, rhodium, and tellurium. The heat produced by t h e iodine daughters of tellurium i s included. The heating by those noble metals produced i n the drain tanks by t h e decay of non-noble parent nuclides i s not included. Figure 3 shows t h e afterheat rate i n t h e 563-Mw u n i t per f o o t of length of heat exchanger, .and Fig. 4 shows the r a t e per square foot of outside tube surface* i n any MSBR exchanger of t h i s type. The accumul a t e d afterheat curve on Fig. 5 i s t h e i n t e g r a l curve of Fig. 3. Table 1 gives numerical values of these data. Heat exchanger temperatures were computed f o r two d i f f e r e n t distributions of heat generation i n t h e exchangers. The simplest case, Type 1, i s t h a t i n which a l l heat generation i s assumed t o be confined t o the tubes and uniformly distributed. This, i n e f f e c t , says t h a t gamma radia t i o n does not generate heat i n adjacent shells nor i s the t o t a l heat generation i n t h e exchanger(s) reduced by gammas escaping t o t h e outside world. The second case, Type 2 distribution, considers the e f f e c t of gamma radiation on t h e location of i n t e r n a l heat generation. Careful calcul a t i o n ~ ~ ' of " gamma heat generation i n heat exchangers of this general design are available and from these t h e t o t a l heat generation rate was subdivided i n t o four parts:-

1. t h e 2. t h e 3. t h e 4. t h e

fraction fraction fraction fraction

i n the inner s h e l l , i n t h e tube annulus (includes a l l 8- heating). i n t h e intermediate and outer s h e l l s , escaping t h e exchanger.

*Since t h e outside of the heat exchanger tubes provides 0.098 f t a per foot of length, the heat r a t e per f o o t of tube length i s obtained, very closely, by dividing t h e data on Fig. 4 by 10. %These data were developed from ref. 1, Table 5.6, p. 63, and r e f . 5, Fig. 4, p. 10.

ORNL DWG. 71-568

103

2.78hr

27.8hr

104

los

Elapsed Time Af'ter Shutdown Fig. 3.

11.613 3

108

U6a 3

- Seconds

107

Afterheat Generation Rate Per Foot of Height i n a 563-MW M m R Heat &changer. Afterheat i s that produced by &$ of the total

noble metal fission products (including iodine daughters of tellurium) which are assumed t o plate out on metal surfaces. Refer t o MSR-68-99 Rev., Fig. 9.

ORNL DWG. 71-569

2.78hr

27.8hr

U.6d

ll6d

10"

Elapsed Time m e r Shutdown Fig.

- Seconds

Afterheat Generation Rate Based on the Outside Surface Area (radiating surfaces) of MSBR Heat Exchanger Tubes. Afterheat is that produced by JK$ of a l l noble fission pr~ductsplus the iodine daughtersof tellurium (see MSR-68-99 Rev., F i g . 9). Heat exchanger configuration per Fig. 1.

9 OfwL DWG. 71-570

Elapsed W e After Shutdown Fig. 5.

Accumulated Atterheat In a Heat Exchanger per Foot of I s that pduced by k$ of ducts which are assumed to tubes. Refer to MSR-68-gg configuration per DWG

Seconds

Perfectly f i s u l a t e d 5 6 3 MSBR ~ ~ Heat Exchanger Length. Afterheat all the noble metal f1s6ion proplate out on the heat exchanger Rev., Fig. lo. Heat exchanger

69-6aa4.

Table 1. Totala Afterheat Generation by Noble Metals Plated on Tube Surfaces i n MSBR Heat Exchangers

Heat Generation Rates Elapsed Time After Reactor

Per Foot of Height i n MSBR 563-h Heat

Exchanger

loa 3x10a

10" = 16.7 m 3x10a = 0.83 h r

lo4 = 2.78 3 ~ 1 0=~ 8.33 lo0 = 27.8

3x10' = 3.47 10' = 11.6

hr hr hr d d

3x10' = 34.7 d io7 = 3.80 m~ 3x10' = 0.9 y lo8 = 3.17 y

Per Square Foot of Per Foot of Lengthb Outside Surface of of 3/8-in0-0D Heat Heat Exchanger Tubes Exchanger Tubes

Accumulated (Integrated) Heat Per Foot of Height i n MSBR 563-MW Heat

2.52 x10 7 3 9 ~ 1 0 ~4 . 6 0 ~ 1 0 ~ 1.35 x l 0 -1 4 . 5 2 ~ 1 0 ~ 1 . 3 3 ~ 1 0 ' ~ 2 32x10~ 6.82~101 4 . 2 7 ~ 1 0 ~ 1.25 xlo -1 4.20~10' 1 23 ~ 1 1 . 9 4 ~ 1 0 ~5 .6gxio1 3 55 x10a 1 . 0 4 ~ 1 0 ' ~ 3.48~10% 1 02x10 -a 1 . 7 4 ~ 1 0 ~5 1 2 ~ 1 0 ~3.20~108 9 38x10'~ 3 12x l 0 9.21~10'' 1 24x10' 3 64x10' 2 27x108 6 67x10-a 2.23 x10 6.55

. .

Exchanger

0 0 07.00~xlOa~ 2.05 1.94 x104 5.68

. .. . 5.63xlO4 1 . 6 5 a o l .. 1 . 3 6 ~ 1 0 ~3 . 9 8 ~ 1 0 ~ 3.18~10 9.32xlO' 7.36~10+ 2 .16x10~ 1 . 3 5 ~ 1 0 ~ 3.96~10" 1 . 3 3 ~ 1 0 ~3.88~10 1. 2.1oXloa 2 . 9 7 ~ 1 0 ' ~ 7 .16 5 .63x104 1.65~10, 1.03x108 3 . 2.25 7 .81x10 2.29 a0 XlO" 4.66~10~ 4 . 2 6 ~ 1 0 ~1.25 no 1.59~lO' 7.67 1.43 2.71X104 7.95 4 . 9 8 ~ 1 0 ~ 1.46 . 3.48x10e 4.90 0

x10

0 2 ~ 1 0 ' ~ 01X l O 1

1 . 2 4 ~ 1 0 ~3.64

2.27 xl0

-a xl0 -a 6.67 x10m3

6 . 5 8 ~ 1 0 ~1.93 1.21x10' 3 .54>i1Oo3 8.95 xloo4 1.67 x10' 4.89~10'' 3.05 3 30x10; 9.66~10'~ 6 . 0 5 ~ 1 0 ' ~ 1.77x10째4 6.58~10 1 . 9 3 ~ 1 0 ' ~ 1.21xlO -l 3.54 XlO -6

1 02x10

)(10째3

2.23

1.19

6 . 5 5 ~ l O ' ~ 7 37x10'

3 48x10'~ 8. 78>clooB 5 . 9 4 ~ 1 0 ' ~ 1.74~lO" 1.19 xlo -a 3.48~lO'~ 3.OOxlO"

2.16~10~

1 . 3 6 ~ 1 0 ~3.98>c103 1.94xlO' 5.68~10' 2.36~10' 6 93no: 2.64 X 1 0 7 7*73MO

These rates and accumulated heat values include a l l gamma energy and represent t h e afterheat production by k$ of t h e noble metal fission pmducts a t saturation levels. Heat generation by t h e iodine daughters born a f t e r shutdown f r o m t h e tellurium i s included.

bNominal height (length) of MSBR heat exchangers i s 22 f t .

csi i

Figure 6 i s a t y p i c a l p r o f i l e of t h e gamma heat deposition r a t e i n an empty 563-141 heat exchanger. Figure 7 shows curves of t h e fractions, (1)t o (4) above, f o r t h e exchangers i n the s i z e range considered.* These d i s t r i b u t i o n f r a c t i o n s do not show any l a r g e variations with elapsed time, p a r t i c u l a r l y a t t h e times of i n t e r e s t , from lo3 t o 10' sec (0.3 t o 30 hours) after shutdown. These curves a r e based on-averages of lo3 and le -sec data.

Figure 6 shows t h a t , of the two outermost shells, t h e thicker i n t e r mediate s h e l l i s t h e much l a r g e r heat source; also, note t h a t gamma heat generation i n these s h e l l s i s attenuated very rapidly i n t h e r a d i a l d i rection. For these reasons, with Type 2 d i s t r i b u t i o n , a l l t h e gamma heat deposition i n both outer s h e l l s was considered t o be near the i n s i d e surface of t h e intermediate shell.

V* METALLURGICAL CONSIDERATIONS The primary concern of t h i s study i s t o determine whether o r not excessive heat exchanger temperatures w i l l jeopardize t h e i n t e g r i t y of t h e Hastelloy N primary containment envelope. Because t h e event postul a t e d seldom, i f ever, would occur and because t h e r e s u l t a n t high temperatures would be of short duration, we are not concerned w i t h t h e long-term creep-rupture behavior. We a r e concerned w i t h t h e short-term physical properties of Hastelloy N a t temperatures around 2000째F (-llOO째C) and assuming t h a t these temperatures are maintained f o r no more than 20 hours

Hastelloy N pressure vessels, piping, e t c . , a r e not expected t o s u s t a i n serious damage i f held a t low s t r e s s f o r short times (< 20 h r ) a t temperatures of 2150째F (1177"C).* A vessel subject t o any s u b s t a n t i a l fluence w i l l l o s e d u c t i l i t y . Ultimate strength a t t h i s temperature w i l l be very low.' If a component i s t o survive a t t h i s temperature, we must ensure t h a t the high temperature regions be v i r t u a l l y f r e e of stressproducing imposed loads. It i s appropriate t o point out t h a t it i s rout i n e f a b r i c a t i o n p r a c t i c e t o specify a s t r e s s - r e l i e v i n g anneal a t 2150째F f o r welded Hastelloy N pressure vessels. The foregoing suggests t h a t we evaluate preliminary designs using 2100'F as an upper temperature l i m i t f o r t h e unlikely events being considered here. This assumes t h a t t h e calculated o r estimated temperatures tend t o be on t h e high side and leaves a small margin f o r thermal s t r e s s e s and other u n c e r t a i n t i e s which w i l l be evaluated with some care during gestation of a f i n a l design.

m e s e data were developed from ref. 1, Table 5.6, p. 63 and r e f . 5, Fig. 4, p. 10.

12 ORNL- DWG 69- 12604

INTERMEDIATE r 4

TUBE ANNULUS INNER SHELL TUBE DIAMETER: 0.375 in. WALL THICKNESS: 0.035 in. TOTAL NUMBER OF TUBES: 5910

10;

RADIUS ( i n . )

Fig. 6 .

Distribution of Gamma Heat Generation Produced by Tellurium* Fission Products i n a 563-MW MSPB Heat &changer. Forty percent of a l l the tellurium i s assumed t o deposit uniformly on the inside surfaces of the tubes. W e energy spectrum of tellurium gammas is considered t o be typical of the gammas produced by the other noble m e t a l fission products.

â&#x20AC;&#x2DC;O

ORNL DWG. 71-571 1.00

0.4

0.2

Fig. 7.

100

200 300 400 MSBR Heat Exchanger Rating Mw

Type 1 Distributions of Noble Metal Afterheat

i n MSBR Heat Exchangers.

VI. RESULTS Figures 8 t o 11 ( i n c l . ) a r e steady-state r a d i a l temperature prof i l e s i n four sizes, from 94 t o 281 Mw, of MSRR-type heat exchangers having t h e dimensions shown on Fig. 2. The heat generation r a t e i s t h a t predicted a t 104 sec (2.8 h r ) after shutdown and drain. These curves a r e based on t h e assumption of Type 1 d i s t r i b u t i o n (see Section IV); i.e., a l l afterheat generation i s uniformly d i s t r i b u t e d i n t h e heat exchanger tubes and no gamma energy escapes. It was a l s o assumed t h a t the emissivity of the outside surface of t h e outer s h e l l was 0.8 and t h a t t h i s surface was radiating i n t o i n f i n i t e "black" surroundings whose temperature i s 1000째F. The e m i s s i v i t p o f the i n t e r n a l radiating surfaces, tubes and shells, i s one of t h e l a r g e r uncertainties i n a calculation of t h i s type. Both tubes and shells w i l l meet stringent quality assurance standards and we should expect them t o have an excellent surface finish, perhaps appearing almost polished. Furthermore, a f t e r exposure t o molten fluoride s a l t s these i n t e r n a l r a d i a t i n g surfaces w i l l be oxide free. All t h e f a c t o r s tending t o promote bright, low emissivity surfaces i n a material tending toward low emissivity a r e present. For these reasons, _nearly a l l t h e calculations were made a t i n t e r n a l surface emissivities of 0.1, 0.2, and 0.3. The general subject of emissivity* i s discussed i n more d e t a i l i n Appendix G. The data i n Appendix G suggest t h a t w e can expect t h e i n t e r n a l Hastelloy N surfaces t o have an emissivity of 0.2 t o 0.3 and t h a t w e s e l e c t an emissivity of 0.2 i n evaluating heat exchanger performance i n t h e s i t u a t i o n considered herein. The high emissivity (0.8) of the outer surface of t h e outer s h e l l i s j u s t i f i e d by assuming t h a t it i s coated with one of the titanates' or, alternat i v e l y , deeply convoluted by f i n s o r a gridwork.

It i s emphasized t h a t these temperature curves a r e f o r steadys t a t e conditions and do not take i n t o account the rapid decrease with time of the heat generation rate nor t h e temperature reducing e f f e c t of heat capacity of the exchanger. These temperature p r o f i l e s are, perhaps, higher than would be obtained should t h e s i t u a t i o n postulated a c t u a l l y occur. The upper l i m i t s of t h e i n i t i a l temperature t r a n s i e n t i n t h e 563-b~u n i t have been estimated and are discussed i n subsequent paragraph( s). The estimate indicates t h e temperature w i l l reach i t s maximum i n about 104 sec a f t e r reactor shutdown and an immediate drain; therefore, most of t h e data herein were calculated as i f a t steady state with the afterheat rate expected a t 104 sec after shutdown. As the calculations proceeded, s t a r t i n g with t h e smallest, 94 Mw, unit, some trends became evident; (1)t h e e f f e c t of heat capacity of the exchanger cannot be ignored, (2) t h e variation of maximum i n t e r n a l temperature w i t h emissivity i s less a t higher emissivities, ( 3 ) gamma

*As t h i s report was going t o press, t h e writer's a t t e n t i o n was directed t o ref. 27, i n which emissivity measurements of INOR-8(Hsstelloy N) are reported. Bright and matte finished specimens showed an emissivity of 0.20 a t 1000째F t o 0.25 a t 1830째F. Oxidized specimens had emissivities of approximately 0.4 t o 0.6. A value of 0.2 f o r t h e emissivity of Hastelloy N i s therefore appropriate.

ORNL DWG. 70-8546

Heat Production:

i b

12,300 Btu/hr-ft height; uniformly d i s t r i buted i n tube annulus. Equivalent to afterheat rate lo* sec a f t e r shutdown produced by 4oqb o f the noble metal f i s s i o n products plated on tube surfaces.

Heat Transfer :

By radiation only.

Environment :

"Black" surroundings a t 1OOO'F.

Outer surface emissivity=0.8*

16 ORNL DWG. 70-8542

u c

5 Fig. 9.

10 Radius

- Inches15

Steady-State Temperature P r o f i l e s i n an Empty 141-Mw Heat Exchanger a t Three Values of I n t e r n a l Surface Emissivity.

Heat Production:

18,400 Btu/hr-ft height; uniformly d i s t r i b u t e d i n tube annulus. Equivalent t o a f t e r h e a t r a t e 104 sec a f t e r shutdown produced by 40$ of t h e noble metal f i s s i o n products plated on tube surf'aces.

Heat Transfer:

By radiation only.

Environment:

Outer surface emissivity = 0.8. "Black" surroundings a t 1000OF.

ORNL DWG. 70-8541

Environment :

"Black" surroundings a t 1000'F.

ORNL DWG. 71-572

Fig. ll. Steady-State Temperature P r o f i l e s i n an Empty 2 8 1 - h Heat Exchanger a t Three Values of I n t e r n a l Surface Ehissivity. Heat Production:

36,800 Btu/hr-f't height; uniformly d i s t r i b u t e d i n tube annulus. Equivalent t o a f i e r h e a t r a t e lo4 sec a f t e r shutdown produced by 4 6 of t h e noble metal f i s s i o n products plated on tube surfaces.

Heat Transfer:

By radiation only.

Environment:

"Black" surroundings

Outer surface emissivity = 0.8, 8t

1000OF.

energy l o s s e s t o t h e outside a r e i n s u f f i c i e n t t o contribute materially toward reducing peak i n t e r n a l temperatures, and (4) the maximum temperat u r e s i n t h e 563-MW "reference design" exchanger may become unacceptably high. Finally, it developed t h a t the time-sharing computational program used t o obtain temperatures i n t h e tube annulus (Appendix E) would not run i f t h e number of tube c i r c l e s i n an exchanger exceeded 22, whereas t h e 563-m exchanger contains 31. There was not s u f f i c i e n t incentive t o spend time i n rewriting t h e program f o r a l a r g e r machine. Instead, it was decided t o produce additional computations t o i n d i c a t e the changes i n t h e 563-MW"reference design" which w i l l reduce t h e maximum i n t e r n a l temperature t o acceptable values. Therefore, scaled-up versions of t h e 563-m "reference design" u n i t with l a r g e r outside diameters and a reduced number of tube c i r c l e s were'programmed and t h e peak temperatures i n t h e "reference design" were obtained by extrapolation. The e f f e c t of emissivity, number of outer s h e l l s , heat capacity, o v e r a l l s i z e and r a t i n g a r e considered i n t h e paragraphs which follow. Temperatures i n 563-~wheat exchangers having l a r g e r outside diame t e r s and thinner tube annuli than i n the "reference design" (Fig. 1) model were computed. These computations served two purposes: (1)They indicated t h e minimum outside diameter of an exchanger which w i l l l i m i t the maximum temperature t o t h e 1900째F-Z!100"F region, and ( 2 ) they produced t h e basis f o r a good estimate, by extrapolation, of t h e peak temperatures i n t h e 563-Mw "reference design" shown on Fig. 1. The dimensions of these exchangers and t h e computed temperatures t h e r e i n a r e i n Table 2 and on Figs. 12 and 13. The extrapolated temperature p r o f i l e s , Fig. 14, assigned t o t h e 5 6 3 - h "reference design" model which has 31 tube c i r c l e s , a r e presented with considerable confidence because t h e extrapolations involved only t h e temperature d i f f e r e n t i a l s i n t h e tube annulus and t h i s repres e n t s only about 25% of t h e t o t a l temperature above t h e 1000째F ambient. The remaining 75$, t h e temperatures of t h e o u t e r and intermediate shells, has been computed accurately. It can be seen t h a t i f t h e outside diameter of t h e reference design heat exchanger i s increased from 36 i n . t o approximately 50 i n . so t h a t t h e tubes a r e arrayed i n 17 t o 20 tube c i r c l e s , t h e peak steady-state i n t e r n a l temperatures w i l l be i n t h e acceptable 2 0 0 0 " F ~ 1 0 0 " Fregion at lo4 sec after shutdown when t h e i n t e r n a l surface emissivity i s about 0.2. A f u r t h e r increase i n diameter may be necessary i f : (1)t h e i n t e r n a l surface emissivity t u r n s out t o be much less than 0.2; (2) t h e "reference design" model, with two outer s h e l l s , continues t o be t h e required design; and ( 3 ) i f we use t h e steady-state temperature calculations a t lo4 sec t o guide t h e design. It w i l l be shown t h a t eliminating one of t h e shells outside t h e tube annulus e f f e c t s a very s u b s t a n t i a l reduction i n peak i n t e r n a l temperatures should t h e i n t e r n a l surface emissivity be

low (0.1).

Table 2. Temperatures Developed by Radiative Transfer of Noble Metal Afterheat i n MSBR Heat Exchangers Rated a t 563 Mu and Raving Tube Annuli of Different Thicknesses

No. Tube

Circles

Rn tin.

(Total

Tubes)

17 (5542)

50.50

(5540)

41 3.00

21.5

(5544)

45.m 43.00

3.00 24

(5544)

40.75

(5538)

39.75

2.75

31 (5549)

io 32.75 2.50

36.00

of Internal Surfaces

@shells

2019 1691 1539

1149 1149 1149

870 542

1140 1140

390

~ 4 0

2084

1648

345 337 339

1165 1165

0.1 0.2 0.3

2179 1831

362 360

u72

367

2 m 1763 1600

0.1 0.2 0.3

2208

370" 370" 370"

2142 1788 1621

ll80 ll80 ll80

441

1169

2229 1872 17&

3 9

971 614

u.72

3 9 3 9

2u5 1798 1650

1184

17.93

0.1 0.2 0.3

32.12 10.62 21.50

0.1 0.2 0.3

2280

425* 425*

2212 1843

1198

T~~

@ann

0.1 0.2

2087 1759

317 300

0.3

1607

296

39.87 26.25 13.62

0.1 0.2 0.3

2150 lso7

37.62 22.56 15.06

35.80

44.65

19.38 16.42

2.75

T*nperatures, 9

Emisaivity

in.

33.18 11.47

3-50 20

3: hm

34.40

16.47

1668

1854 1687

1911 1737

1742 1583

e* 1669

1165 1172

1172

1184 1184 1198 1198

1169

446

1172 1172

1614

1186

645 471

1186

~.~

11%

heat r a t e of 7.36 x l e Btu/hr per f t height of on t h e i s i d e of t h e heat exchanger tubes. This Btu h r and is t h e heat rate axptcted a t

ftp tube surface it length of tube io4 sec (2.8 h r ) a i t e r shutdown. (2) Heat exchangers i n %&k, i n f i n i t e " surroundings a t lOOO9. (3) Emissivity of o d e r surface of outer s h e l l 0.8.

+These temperatures obtained by extrapolation.

21 ORNL DWG. 71-573

Reference Design 25

I 1

B R

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 I

15 * 20 Number of Tube Circles

Fig. 12. Internal Steady-State Temperatures i n RSBR Heat Exchangers Rated a t 563 Mu With Tube Annuli of Different Thicknesses and With the Rnissivity, e, of'All Internal Surfaces a Parameter. Heat exchanger configurations generally similar t o Fig. 1.

Heat Generation:

7.36 x lO* Btu/hr-ft height; all uniformly distributed

(me

1) i n the tube annulus and equivalent to the afterheat rate lo4 sec a f t e r shutdown produced by 4 6 of the noble metal fission products plated on the tube surfaces. Heat Transfer:

By radiation only.

hvimnment :

"Black, infiniten surroundings a t looO째F (a) Rnissivity af outer surface of outer s h e l l = 0.8.

*For internal emissivities from 0 . 1 t o 0.3 the variation, with emissivity, of the temperature r i s e i n the tube annulus i s negligible. used.

Averaged values are

ORNL DWG. 71-574

Reference Design, Fig. 1

i i!

15 20 Huuiber of Tube Circles

Fig. 13. Peak Steady-State Temperatures Developed in MSBR Heat Exchangers Rated a t 563 Mu W i t h me Annull of Mfferent Thicknesses and W i t h anissivity, E, of All Internal Surf'aces a Parameter. Heat exchanger configurations generally similar t o Fig. 1. Heat Generation:

7.36 x lo4 Etu/hr-ft height; all uniformly distributed (!Cype 1 ) I n the tube annulus and equivalent t o the afterheat rate lo4 sec a f t e r shutdown produced by 40$ of the noble metal fission products plated on the tube surfaces.

Heat Transfer:

By radiation only.

Envltonment:

"Black, infinite" surmmdings a t 1000째F (a) Emissivity of outer surface of outer shell

0.8.

JJ , c

t .c

4) 1 ORNLDWG.71-575

3ooo

Radius

- h&es

Fig. 14. Steady-State Temperature Profiles in an Empty 563&w Heat Exchanger at Three Values of Internal Surface Emissivity. Heat Production: 73,600 Btu/hr-ft height; uniformly distributed in tube annulus. Equivalent to afterheat rate lo4 set after shutdown produced by &$ of the noble metal fission products plated on tube surfacea. Heat Transfer: By radiation only. Outer surface emissivity = 0.8, Environment : "Black" surroundings at 1000째F.

24 The estimated temperature t r a n s i e n t i n t h e 563-m exchanger i s shown on Fig. 15. The estimate shows t h a t t h e peak temperature, 2150째F, developed i n t h i s exchanger borders on a c c e p t a b i l i t y i f we can r e l y on an int e r n a l surface emissivity of 0.2 o r better.* The simplifying assumptions and approximations used i n calculating t h i s t r a n s i e n t were made so that t h e r e s u l t s would tend toward t h e high side. A b r i e f description of the method used t o develop t h i s t r a n s i e n t i s i n Appendix H.

U a-

Figure 16 shows t h e author's version of a similar t r a n s i e n t i n t h e This curve was estimated by inspection using Fig. 15 as a guide. The t r a n s i e n t peak, s l i g h t l y above 1800째F, was located s l i g h t l y below t h e i n t e r s e c t i o n of t h e adiabatic temperature growth curve of t h e annulus and t h e steady-state peak temperature curve. This smaller u n i t can be expected t o perform w e l l i n t h e stated s i t u a t i o n .

141-Mw u n i t .

Figures 17 and 18 are temperature p r o f i l e s i n t h e 94- and 281-MW u n i t s a t 104 sec after shutdown and f o r t h e nonuniform, Type 2, heat d i s t r i b u t i o n , Fig. 7. A l l other conditions are t h e same as f o r Figs. 8 and 11, with which they may be compared t o note the e f f e c t of making calculations using t h e simplifying Type l approximation. Table 3 provides a comparison of peak temperatures if calculated f o r both types of heat d i s t r i b u t i o n a t steady-state heat rates corresponding t o those expected a t 104 sec (2.8 h r ) after shutdown. Noting t h a t t h e more r e a l i s t i c assumption, Type 2, gives lower t e m peratures, it is proper t o query, *'Why not use the nonuniform case e n t i r e l y ? " The question i s p a r t i c u l a r l y appropriate because once t h e equations are programmed, t h e r e i s l i t t l e difference i n t h e ease of obtaining numbers. The more exact nonuniform case, Type 2, requires a p r i o r and not uncomplicated nor inexpensive computation of gamma heating.6 During t h e e a r l y phase of a design study it w i l l not be worthwhile t o spend much time i n t h i s e f f o r t u n t i l a d e t a i l e d design has been confidently established. The simple, uniform case, Type 1, requiring only a knowledge of heat exchanger geometry and a f t e r h e a t generation rate, i s r e l a t i v e l y easy t o calculate and, as it seems t o provide temperatures s l i g h t l y on t h e high side, w i l l tend t o produce a conservative design. Figures 8 t o 11 and Fig. 14 show inner s h e l l temperatures s l i g h t l y l e s s than t h e temperature of t h e adjacent row of tubes. A t first glance t h i s seems contrary t o a l l accepted laws governing heat t r a n s f e r . In f a c t , and as w i l l be seen, it i s not t r u e . However, i f t h e assumption of zero heat generation i n t h e inner s h e l l were a c t u a l l y t r u e w e should expect t h i s s l i g h t temperature depression a t t h e inner s h e l l . Because t h e tube matrix i s quite open, t h e inner s h e l l i s i n thermal equilibrium not only with t h e edjacent row of tubes but with t h e combination of several sets of tube c i r c l e s a t lower temperatures f a r t h e r out i n t h e tube annulus. On Figs. 17 and 18, i n which t h e inner s h e l l s are generating 9s of t h e t o t a l afterheat, t h e i r temperature i s t h e peak temperature as expected. *Refer t o footnote on page 14.

25 ORNL DWG. 71-576

27.b

11.6d

Temperature of I n f i n i t e "Blsck" surroundings, lWoF

loa

109

Elapsed Time After Shutdown

loa

- Seconds

106

Fig. 15. Estimated Initial Temperature fiansient Caused by Noble Metal Afterheat i n an &pty 563-Mw MSER Heat Exchanger. Curve A:

Curve B:

Peak steady-state temperature computed f o r Type 1 afterheat r a t e s at the indicated times (see Table 1 ) and with t h e emissivity of a l l i n t e r n a l surfaces = 0.2 and t h e emissivity of the outer surface of the outer s h e l l = 0.8. Temperature growth in the inner s h e l l and the tube annulus computed as if . (1) the annulus and s h e l l a r e perfectly insulated; ( 2 ) they have a t o t a l heat capacity of 129 Btu/'F-ft of height (based on Table 3 ) ; and ( 3 ) generate of the t o t a l afterheat (see Table 1 and Fig. 7).

Curve C:

Temperature growth in the intermediate s h e l l computed a s if: (1)the s h e l l i s perfectly insulated; (2) it has a heat capacity of 287 Btu/'F-ft of height, and ( 3 ) generates 23% of the t o t a l afterheat.

O m DWG. 71-577 2.7811

27.811

Temperature ff I n f i n i t e Black" mroundings 1000'F

lop Fig. 16.

141-Mw

Curve A:

Curve

loc

loa a a Elapsed Time Aiter Shutdown

- Seconds106

Estimated Initial !kmperature Transient Caused by Noble Metal Afterheat in an Bnpty Heat Exchanger.

Peak steady-state temperature computed for

2 afterheat r a t e s a t t h e indicated times (see Table 1and Fig. 7) and w i t h t h e emissivity of all i n t e r n a l surfaces = 0.2 and t h e emissivity of t h e outer surface of the outer s h e l l = 0.8. annulus computed as if: (1)t h e annul u s and s h e l l are perfectly insulated; (2) they have a t o t a l heat capacity of 32 B t u / O F - f t of height (based on Table 3); and (3) generate 70$ of t h e t o t a l afterheat (see Table 1and

B: Temperature growth in the inner s h e l l and the tube Fig. 7).

Curve C:

Temperature growth in the intermediate s h e l l computed as if: (1)the s h e l l is perfectly insulated; (2) it has a heat capacity of 72 Btu/'F-ft of height, and (3) generates 23% of the total afterheat.

ORNL DWG. 71-578 --

Inner Shell

5 Radius

Fig.

Intermediate Shell

10 Inches

Outer Shell

17. Steady-State Te erature Profiles in an Rnpty *=Mw MSEB Heat Exchanger for '.r;vpe2 Heat Distribution Which Takes into Account the Effects of Gamma hergy Losses and Distribution, Fig. 7. Upper (dashed) curve is profile for Type 1 distribution (see Fig. 11) and is shown for comparison,

Total Heat Generation: E?300 , E%u/hr-ft height; equivalent to afterheat rate at 104 sec after shutdown. Heat Transfer: By radiation only. Outer surface emissivity = 0.8. Environment: "Black" surroundings at 1OOO'F.

28 ORNL DWG. 71-579

Fr I

2000

1500

1000

500

Radius

w -

Inches

Fig. 18. Steady-State Temperature P r o f i l e s i n an Empty 281-MWMSBR Heat Exchanger for Type 2 Heat M s t r i b u t i o n Which Takes I n t o Account the Effects of Gama Energy Losses and Distribution, Fig. 7. Upper (dashed) curve is p r o f i l e for Type 1d i s t r i b u t i o n (see Fig. ll) and is shown for comparison.

Total Heat Generation: 36,800 Btu/hr-ft height; equivalent t o a f t e r h e a t rate a t 104 sec a f t e r shutdown. Heat Transfer: By radiation only. Outer surface emissivity = 0.8. Environment: "Black" surroundings a t 1 O O O ~ .

Table 3. The Influence of I n t e r n a l Afterheat Distribution and Heat Exchanger Size on Peak Steady-State Temperatures i n Ebpty MSBR Heat Exchangers a t t h e Heat Generation Ratea mpected 104 Sec (2.8 H r ) After Reactor Shutdownb -

Assumptions on Distribution

94-MW Heat Exchanger

141-Mw Heat Exchanger

188-MW Heat &changer

281-MW Heat &changer

I n t e r n a l Surface

I n t e r n a l Surface missivityC

I n t e r n a l Surface h i s s i v it y c

I n t e r n a l Surface mi ssivit yc

Heat Generatio 0.2

0.1

0.3

Peak Temperatures Uniform, Type 1 (100% conf i n e d t o the tube annulus ) Nonunif o m , Type 2 (see F i g * 7) Difference,

0.1

0.3

0.1

0.2

0.3

"F h)

1995

1706

1577

2111

1810

1678

2231

1913

1771

24-06

2075

1928

1908

1635

1514

2072

1776

1645

?I83

1878

1733

2380

2050

1903

a Heat generation rate = 13.3 Btu/(hr-ft length of tube), = 135 Btu/(hr-fta of outside tube surface). bHeat exchangers i n i n f i n i t e , "black" surroundings a t 1000째F. C

0.2

Emissivity of outer surface of outer s h e l l = 0.8.

All t h e preceding figures which show temperature p r o f i l e s indicate one thing i n common; namely, as i n calorimeters and similar devices, the continuous intermediate s h e l l , with low emissivity, i s an e f f i c i e n t barr i e r t o radiative heat t r a n s f e r and i s a large f a c t o r i n producing higher i n t e r n a l temperatures. The reduction i n peak temperatures effected by eliminating one of these two s h e l l s was determined by calculations made f o r t h e same group of exchangers with the outer, 1/2-in.-thick s h e l l removed. Table 4 provides a comparison of the peak steady-state temperat u r e s i n exchangers of t h i s type designed with one and two s h e l l s external t o the tube bundle and a t t h e heat rate expected a t lo4 sec after shutdown. Figure 19 shows curves of t h e data i n Table 4 which have been extrapolated t o include estimated peak steady-state temperatures i n the 563-& exchanger. I n t e r e s t was expressed i n t h e reduction of temperature attained by increasing t h e apparent emissivity of a l l surfaces of the outer and intermediate s h e l l s . A possible method would be t o add f i n s o r a gridwork so that these surfaces take the appearance of a continuous sheet of black body cavities. No e f f o r t was spent investigating the f e a s i b i l i t y of t h i s idea, but temperature p r o f i l e s i n a 281-& exchanger were calculated as i f all t h e surfaces of t h e intermediate and outer s h e l l s , i n t e r n a l and external, had an emissivity of 0.8. A value of 0.2 f o r emissivity of t h e inner s h e l l and tube surfaces was selected. No allowance was made f o r the increases i n s h e l l diameters required. The r e s u l t s , calculated by using Ty-pe 1 heat generation a t t h e rate expected a t 1041 sec a f t e r shutdown, are on Fig. 20. The temperature reductions so obtained a r e appreciable. Table 5 gives a comparison of maximum temperature i n a 281-MW exchanger f o r four d i f f e r e n t cases described i n t h e preceding paragraphs. The e f f e c t of i n t e r n a l emissivity on peak temperature i s implicit i n many of t h e preceding figures. Figure 21 shows, e x p l i c i t l y , the influence of emissivity on peak temperatures i n t h e 94-MM unit. It i s apparent t h a t w e w i l l get a worthwhile improvement i n afterheat r e j e c t i o n by t h i s heat exchanger i f t h e emissivity of the Hastelloy N surfaces, after exposure t o molten salts, i s 0.3 r a t h e r than 0.1. It i s a l s o apparent that. increasing t h e emissivity above 0.3 produces l i t t l e additional benefit. The t r a n s f e r of radiant energy from surfaces far inside the exchanger w i l l be strongly dependent on t h e combined e f f e c t s , not separable, of i n t e r n a l geometry and emissivity. A t low values of emissivity (high ref l e c t i v i t y ) a photon w i l l have a higher probability of t r a v e l i n g f a r t h e r from i t s point of o r i g i n via multiple r e f l e c t i o n s through t h e tube bundle before being absorbed. The surfaces w i l l a l s o be a t a somewhat higher temperature i f they are radiating a t a rate which maintains temperatures constant a t a constant afterheat generation r a t e . These are o f f s e t t i n g trends, but t h e fourth power e f f e c t of temperature on heat t r a n s f e r sugg e s t s t h a t higher emissivities may produce only fringe benefits i n open tube l a t t i c e s . The quantitive extension of data from Fig. 21 t o other geometries i s not recommended since t h i s figure does not provide the i n t e r r e l a t i o n between geometry and emissivity.

tr t

Table

Afterheat Temperature Reductions Attained by Removing t h e Outer S h e l l from Empty MSEIR Type Heat Exchangers

External surface emissivity = 0.8. Heat exchanger i n i n f i n i t e , "black" surroundings a t 1000oF. ~-

Maximum Steady-State Temperatures Computed e t Heat Generation Ratea Equivalent t o 4 6 of t h e Noble-Metal Afterheat l e Sec Heat Exchanger Rating

9* 141 Mw

188 Mw

No. of Tube Circles

Surfaces

0.1 0.2 0.3

563

Reference Design (see Fig. 1)

1706 1577 2 u 1810

0.1

2231 1913

1678

1771

0.1 0.2 0.3

2406 2075

0.1 0.2

0.3

27-05, 233% 2162

1928 b

563 ~w

2f5

0.1 0.2 0.3

216% 226% 2094

0.1 0.2 0.3

257%

563 ~w

0.1

2541

563

563 ~w

563 MW

Reference Design With Outer S h e l l Omitted

1995

0.1 0.2 0.3

0.2 0.3 281 Mw

After Reactor Shutdown

Emissivity

of I n t e r n a l

222kb 2057

0.2

2191

0.3

2035

0.1 0.2

2495 2144

0.3

1987

0.1 0.2

2404 2059

0.3

1903

a A t lo( s e c h e a t generation rate = 13.3

Btu/hr

= 135

ft2 tube surface and is !Qpe 1 generation. bIndicates temperatures obtained by extrapolation.

Difference

ORNL DWG. 71-580

200 300 Heat Exchanger Rating

100

- Mu400

Fig. 19. Meximum Internal Steady-State Temperatures in Single-Region MBR Heat Exchangers w i t h One and Two Shells Outside the Tube Bundle.

---

Elapsed Time After Shutdown l@ sec E 2.8 hr. 134 Btu/hr-Ft* !bbe Surface = 13.3 ftBtu/hr of tube Afterheat Rate Rnissivity of Internal Surfaces 0.1 to 0.3 as noted. Emissivity of Outer Surface 0.8. Surroundings fnfinite, "black," at l O 0 0 ~ .

---

--- ---

3000

2500

Fr O

2000

0:s

1500

lo00

500

Radius

Fig. 20.

Inches

Temperature P r o f i l e s in a 281-MUMSBR Heat Exchanger Showing t h e Effect of Increasing t h e Emissivity of t h e I n t e r n a l Surfaces of t h e Outer and Intermediate Shells,

Heat Production: Heat Transfer: Environment :

36,800 Btu/br-ft height, at

104SeC

I n f i n i t e , "black" a t 1000째F,

hissivity of Inner shell and tubes

(a)

after shutdown.

By radiation only.

Ehissivity of intermediate s h e l l surfaces Emissivity of inner surface, outer s h e l l Emissivity of o u t e r surface, o u t e r shell

-Case A

Case B

0.2 0.2 0.2 0.8

0.2 0.8

0.8 0.8

34 5 . The Effect of t h e Outer Shells on Maximum Steady-State Afterheat Temperatures i n an Einpty 281-MW MER Heat Exchanger

Table

A comparison of four cases a t

lo4

sec. a

Cases

"Reference design" per Fig. 1. All i n t e r n a l surfaces exposed t o primary and secondary salts have an emissivity of 0.2. Ty-pe 1 heat distribution.

"Reference design, 'I with Type 2 heat d i s t r i b u t i o n

Reduction i n Maximum Temperature Referred t o 11Reference Design"

Maximum I n t e r n a l SteadyState Temperature Calculated f o r Heat Rate a t 104 Sec (2.8 H r ) After Reactor Shutdown

2075째F

25'F

2050'~

which takes i n t o account gamma energy distribution. All other conditions as i n i n Case 1 (above).

"Reference design" with outer shell removed. All other conditions as i n Case 1 (above).

281'F

1794째F

"Reference design" i n which emissivity of a l l surfaces of outer and intermediate s h e l l s i s 0.8. Ehissivity of tube and inner s h e l l surfaces i s 0.2. Type 1 heat distribution.

381'F

1694'F

a Total a f t e r h e a t load i n exchanger (Type 1 d i s t r i b u t i o n ) = 8.1 x Btu/hr a t 104 sec after shutdown. Equivalent t o :

lo5

ORNL DWG. 71-582

0.1

0.2

0.3

0.4

0.5

0.6

Emissivity of Internal Surfaces

Fig. 21. Effect of I n t e r n a l Surface Emissivity on Peak Steady S t a t e T e m p e r a t u r e s i n a 94-Mw MSBR " R e f e r e n c e Design" Type H e a t Exchanger (see Figs. 1 and 2).

Type 1Heat Generation: (a)

A t lo" sec

(b)

A t lo4 sec

29,OOO Btu/hr-ft height = 31.2

-- 12,300 Btu/hr-ft

Btu/hr f t of tube

height = 13.3

Emissivity of outer surface, outer s h e l l

= 0.8

Btu/hr ft of tube

The primary i t e m of concern i s peak temperature and i t s v a r i a t i o n with heat exchanger s i z e and t h e h e a t generation r a t e . Figures 22 t o 26, incl., show peak temperatures i n the f i v e s i z e s of exchangers l i s t e d i n Fig. 2 and how these temperatures vary with t h e heat generation rate. The temperature p r o f i l e s i n s i d e t h e exchangers w i l l have the same general p a t t e r n as those shown on Figs. 8 t o 11, incl., but with d i f f e r e n t gradi e n t s . As peak temperatures r i s e and/or as emissivity decreases, t h e radial gradients through the tube annuli w i l l tend t o f l a t t e n out.

ad 8

u e

OBNLDWG. 73-583 Elapsed Time After ,

10s

3x1@

10s

Shutdown - Seconds 3xlo4

3x10s

3x10= 10' ( loa

Fig. 22.

a r a sloa Reat Generation Bate,d Btu/hr-fta

Pe!ak Steady-State Temperatures Rate and Emisalvity of Internal (a)

In a &Mw MSBRHeat Exchanger vs Heat Generation Radiating Surfaces.

Heat transmission: By thermal radiation only. &nissivity of outer surface of outer shell: 0.8. Surroundings: Infinite, "black" at 1000째F. Heat generation Is Type 1 (all afterheat is generated in the tubes). Heat exchanger design and dimensions on Figs. 1 and 2.

108

ORIVLDWG. 71-584 Elapeea Time After

shutam

- Second8 3x1@

3x1@

3 Fig. 23.

103

3xlo4

S s 7 a 3103 Heat Generation Ratep

101

3xl.O'

lost

103

Btu/hr-ft3

103

Peak Steady-State Temperature3 in a 14144~ MSBRHeat Exchanger v3 Heat Generation Rate and Rniseivity of Internal Radiating Surfacee. a) b

Heat tran3mi33lon: By thermal radiation only. Emlsaivity of outer surface of outer shell: 0.8. Surroundinga: Infinite, "black" at lGOO?T. d"I Heat generation is Type 1 (all afterheat is generated in the tubes). e), Heat exchanger design ana dimenelone on Figs. 1 and 2.

cI‘* .

‘m. c ‘1

c 'I

ORNL DWG. 71-585

Elapsed Time mer Shutdown

3x1@

10"

3xloa

Seconds

io2

3x1@

3X10a

loa ,

4000

k @

1000 Heat Generation Rate,d Btu/hr-fta Fig. 24. Peak Steady-State Temperatures i n a 188-Mw MSBR Heat Exchanger vs Heat Generation Rate and W s s i v i t y of Internal Radiating Surfaces. Heat transmission: By t h e m 1 radiation only. Emiesivity of outer surface of outer shell: 0.8. (c) Surroundings: Infinite, "black" a t 1000째F. (a) Heat generation i s !Type 1 ( a l l afterheat i s generated i n the tubes). (e) Heat exchanger design and dimenaiona on Figs. 1an? 2.

..“..” ._..“.

..--“....I.-..“. _“..“.“....

_“,” ..- “_“..., .~._.

_)._.

__i

ORNL DWO. 71-586 Elepaed Tim? Af'ter Shutdown - Seconda 3xl@

3x10'

3XloB

3xw

1ofQ

loQ

moo a

Fig.25.

(1 I a 9 lfya Heat Generation Fkte,d Btu/h~-fi~

peak Steady-State Temperaturee in a 281&i MSBRHeat Exchanger ve Heat Generation Rate and EmSaslvity of Internal Radiating Surfaces. (a)

Heat trensmiaaion: By thermal radiation only. Rnieeivity of outer aurface of outer shell: 0.8. Surroundlnge: Infinite, 'black" at 1OOODF. Heat generation Is Type 1 (all afterheat ia genereted in the tubes). Heat exchanger design and dimensions on Figa, 1 and 2.

'10'

OmL DWG. 71-58-f Elapsed Time After 3x1@

lo6

Shutdown - Seconds 3x10’ 3x10’ uJ4

4.000

Sloa

Heat Generation Fig. 26.

Peak Steady-State

Rate and EMasivity (a\ \-. (b) (c) a) t e)

Rate,'

?3tu/hr-fta

Temperatures in a 5634~ MSBRHeat Exchanger vs Heat Generation of Internal Rediating Surfaces.

By thermal radiation only. Heat trammission: Emiseivity of outer surface of outer shell: 0.8. Surroundings: Infinite, "black" at lOOO@F. Heat genemtion Is Type 1 (all afterheat is generated In the ttibee). Heat exchanger design and dim6nsiona on Mge. 1 and 2.

NCYTE: These curves obtained

by extrapolation.

10"

APPENDIX A

COMPUTATIONAL MODEL AND ASSUMPTIONS G 0 V " I N G

THE COMPUTATIONS Geometry (see Figs. 1 and 2 ) A l l equations are written f o r i n f i n i t e c y l i n d r i c a l geometry. Tube layout i s assumed t o be an annular a r r a y consisting of an i n t e g r a l number of concentric c i r c l e s of tubes. Tube spacing ( p i t c h ) is: circumfer0.750 in.; radial p i t c h 0,717 i n . I n terms of space ential pitch o r volume occupied by t h e tubing i n t h e annulus, these tube spacings a r e equivalent t o t r i a n g u l a r l y pitched tubes having P/D = 2.1.

Physical Characteristics and Considerations The computational model i s based on f i v e assumptions o r postulates1' involving t h e r a d i a t i n g surfaces and t h e energy radiated; these are:

e a r e dealing with a multi-surfaced enclosure i n which it i s (1) W possible t o construct a heat balance f o r each surface i n t h e enclosure. &ch surface i n t h e enclosure i s considered t o be isothermal. Since r a d i a l symmetry obtains, each c i r c l e of tubes was considered t o be a s i n g l e surface.

(2) The surfaces of t h e enclosure are considered t o be gray; i.e., absorptivity, CY, i s equal t o emissivity, E, f o r a l l wave lengths of radiant energy and i s uniform on any p a r t i c u l a r surface. law.

( 3 ) The d i s t r i b u t i o n of emitted r a d i a t i o n follows Lambert's cosine Lambert's law i s outlined i n t h e next appendix.

(4) The d i s t r i b u t i o n of r e f l e c t e d r a d i a t i o n a l s o follows Lambert's cosine l a w , (3) above; i.e., when a collimated beam o r pencil of rays s t r i k e s t h e surface it i s r e f l e c t e d diffusely. A consequence of (3) and ( 4 ) above i s t h a t , i n considering t h e energy leaving an element of surface, no d i s t i n c t i o n i s made between emitted and temperature equations r e f l e c t e d r a d i a t i o n . The r e s u l t i n g heat t r a n s f e r are l i n e a r and t r a c t a b l e .

( 5 ) The r a d i a t i o n incident on any p a r t i c u l a r surface i n t h e enclosure i s uniformly d i s t r i b u t e d on t h a t surface. This assumption i s required i f t h e isothermal and gray conditions p e r (1)and (2) above a r e met.

t-i

It i s generally recognized t h a t these p o s t u l a t e s are simplifying assumptions which may deviate, sometimes quite substantially, from t h e a c t u a l physical s i t u a t i o n . They receive wide use because of the t r a c t a b l e mathematical expressions r e s u l t i n g from t h e i r use. I n most cases and i n s p i t e of t h e i r deficiencies, equations derived from these general

assumptions usually produce s a t i s f a c t o r y engineering answers. Insofar as these heat exchanger calculations a r e concerned, w e cannot j u s t i f y o r favor any other s e t of assumptions unless w e have reliable experimental data which enable u s t o evaluate: (1)emissivity, i t s temperature dependence and t h e angular d i s t r i b u t i o n of emitted radiation, (2) the degree t o which r e f l e c t e d radiation i s specular instead of diffuse and how specularity i s affected by surface f i n i s h , surface composition, and immediately adjacent sub-surface structure, temperature, and exposure t o molten s a l t .

Even i f w e had these data t h e development of solvable equations would be a formidable problem and certainly, during t h e development of a design, not worthwhile from t h e standpoint of cost and time. W e would, perhaps, consider a calculation i n which tube surfaces are divided i n two p a r t s [see ( 5 ) above], the inner and outer half c i r c l e s . This would have, as i t s only e f f e c t , increasing t h e number of equations by a f a c t o r of almost 2 but would not increase the complexi t y of t h e equations. Such a s t e p would, as of now, require programming f o r one of our l o c a l computers; vur remote t i m e sharing f a c i l i t y would not have t h e necessary capacity.

APPENDIX B DIRECTIONAL DISTRIBUTION OF RADIATION

(Lambert's Law) I

Consider an elemental black surface, d%, r a d i a t i n g i n accordance with Lambert's cosine l a w . The energy, dQ, emitted from dj$ i n t o t h e elemental s o l i d angle dQ centered about t h e direction (@,e), Fig. B1, may be expressed,

I i s t h e r a t e of emission per u n i t elemental area,

q,of

emitting sur-

f%e, i n t o a u n i t elemental s o l i d angle around t h e normal, ( @= 0), t o

Normal t o

Fig. B1

al.

46 On a hemispherical surface of radius rr centered on

and i n

spherical coordinates

r s i n p de r dg dQ =

= s i n p de dg

r and

dQ(@)= Io dA, cos g s i n

de dg

The s o l i d angle, dQ, may a l s o be specified i n terms of another element of area and i t s location elsewhere i n space; e.g., on Fig. Bl dQ =

a ra

and from (B-1)

Io dQ(P) =

B dAa

COS

The t o t a l energy emitted by over the hemisphere,

The t o t a l energy emitted by

i s obtained by i n t e g r a t i n g (B-2b)

as a black body i s

Q=dAl o T 4

i n which

o = Stefan-Boltzmann constant =

1730 x

10â&#x20AC;&#x2122;la

BtU

- - RQ

h r ft a

T = temperature,

(experimental value)

(B-4)

Therefore, by equating (B-3b) and (B-4)

-- T (r T4 I

Btu/fta-hr

f o r black body emission; correspondingly, i f Lambert's cosine law i s extended t o non-black surfaces having hemispherical t o t a l emissivity, e,

The f r a c t i o n of t h e t o t a l energy emitted by dA1 which i s i n t e r cepted by d& is, from Eqs. (B-2d) and (B-3b),

Io dA1 cos B

cos

(B-6b) .9.

This equation defines t h e view f a c t o r of one d i f f e r e n t i a l element, dAl, r a d i a t i n g t o another d i f f e r e n t i a l element, d&, with t h e proviso t h a t dA1 i s r a d i a t i n g diffusely i n accordance with Lambert's cosine l a w . It is t h e basis f o r t h e view f a c t o r determination discussed i n t h e next appendix.

VIEW FACTORS*

General Considerations '

I n an enclosure made up of two o r more surfaces t h e view factor* f o r any p a r t i c u l a r surface, t h e reference surface, t o any surface i n t h e enclosure i s defined as t h e f r a c t i o n of the t o t a l radiant e n e r a leaving t h e reference surface which i s transmitted d i r e c t l y (no r e f l e c t i o n s ) t o t h e viewed surface; obviously then, t h i s f r a c t i o n i s dependent on t h e geometrical configuration of t h e surfaces i n the enclosure and on the d i r e c t i o n a l d i s t r i b u t i o n of the radiant energy leaving t h e reference surface. "View fraction" would, perhaps, be more accurate terminology. References 11 and 12 provide excellent material on t h i s subject. In t h i s i s t h e view f a c t o r of report view f a c t o r s a r e represented thus: Fm surface m looking a t surface n. ~

It i s not d i f f i c u l t t o show" t h a t f o r surfaces, a l l composed of i n f i n i t e l y long p a r a l l e l elements, t h e view f a c t o r , assuming d i f f u s e radiation, from a s t r i p of d i f f e r e n t i a l width t o a neighboring surface i s given by Fig. C1. -Surface

Surface

Elements of surfaces A and B perpendicular t o t h e plane of the paper are i n f i n i t e l y long and p a r a l l e l .

Normal t o u , a=oo

FdA -.B

0.5 ( s i n

cy1

- s i n a2)

Fig. C 1 *Also referred t o as "configuration, i n various t e x t s and references.

"angle f a c t o r , I 1

shape factor"

This is the s i t u a t i o n which obtains i n MSBR heat exchangers. Tubes and surfaces which see each other a r e separated a t most by about 6 in., and since they a r e about 250 i n . long, the i n f i n i t e length model i s appropriate. Note t h a t concave surfaces see themselves and that view f a c t o r s of surfaces t o themselves must be included. The r e l a t i o n i n Fig. C1, evaluated graphically and integrated, was used t o determine a majority of the view f a c t o r s required by t h i s analysis. The reciprocity r e l a t i o n

Am = area of mth surface An = area of nth surface

was a l s o used. View Factors, Tube t o Adjacent Tube and Adjacent Plane The view f a c t o r s f o r simple regular geometries can often be obtained analytically and several references (13 t o 20 i n c l . ) a r e good sources of view f a c t o r formulas f o r a v a r i e t y of geometrical shapes and arrangements. I n these MSEB exchangers the unobstructed view f a c t o r s f o r a tube t o an adjacent tube, Fig. C2, were determined from

I [(P - 1) ll

- x + 2 - tan-lt(2-

va)1

(c-3)

X = P/D. NOTE: In some references t h i s view f a c t o r is referenced t o one-half the perimeter of tube I since only one-half of tube I sees tube 11. The value of w i l l be twice t h a t FI + I1 obtained from the above formula.

rpl

Also, cos-â&#x20AC;&#x2122; ( 1 / X ) may be subs t i t u t e d f o r the tan-â&#x20AC;&#x2122; term i n some formulas. The tan-l term is b e t t e r adapted f o r some computers. c

Fig. C 2

Each tube sees adjacent tubes on e i t h e r side i n t h e same row; therefore, t h e view f a c t o r of a tube row t o i t s e l f where a tube row i s considered t o b e â&#x20AC;&#x2122; a single surface, n, i n an i n f i n i t e planar array i s Fn

2Ftube

0.154 f o r P/D

adjacent tube = 2.1.

A t t h e boundaries where t h e tubes see a continuous plane surface, t h e view f a c t o r of an i n f i n i t e row of tubes t o an i n f i n i t e plane i s determined thus:

Consider an i n f i n i t e row of tubes bounded on e i t h e r side by p a r a l l e l , i n f i n i t e planes, Fig. C3. ORML DWG. 71-588

l / / / / / / / / / // // / / / / / j / / / / / / / / / / // / / / / / / / > / / / / Fig. C 3

Since t h e t o t a l view factor f o r t h e tube row i s 1.00, we can write

plane = 1.00

and

plane

1.00

- Fn

2 = 0.423 f o r P/D = 2.1.

The view f a c t o r f o r a plane t o t h e tubes i s derived from t h e generally applicable r e c i p r o c i t y r e l a t i o n ,

Fplane

n = C.n

U 4

plane

An Aplane

( C-6a) a

(C-6b) =

0.630 f o r P/D = 2.1.

V i e w Factors, Tube t o Non-Adjacent Tube and Non-Adjacent Plane

V i e w f a c t o r s from a tube o r from an inner o r outer s h e l l which sees only portions of neighboring tubes through t h e gap(s) between tubes were determined graphically as i l l u s t r a t e d by the next diagram, Fig. C4, and the procedure which follows. ORNL DWG. 71-589

Fig. C4.

Graphic Integration of Tube-to-Tube View Factors

53 Procedure f o r Calculating View Factors

1. On a large-scale layout, determine graphically t h e arc, %?, t h e p a r t of t h e periphery of t h e reference tube ( 1 C ) which sees t h e viewed tube, 3R2. i s defined by MP tangent t o 2R2 and NQ tangent t o

3R2

, a, - .. - n

Subdivide %? i n t o incremental arcs, AAJ. The number of subdivisions i s a matter of judgment. Increasing the number of subdivisions increases both accuracy and labor. For these computations t h e AA were t y p i c a l l y 10" a r c s . i

Determine t h e limits of view of each incremental a r c on 1 C t o viewed tube 3R2. On t h e diagram these view l i m i t s of Bi a r e denoted by l i n e s OX and OY, making angles c y 1 and ma with t h e normal to The average view f a c t o r of t h e element of surface represented by arc seeing 3R2 i s given by

ni.

The view f a c t o r of t h e surface represented by a r c = i s t h e simple integrated average of t h e view f a c t o r s of the incremental arcs:

Fm-. 3R2 -

i=1

J i=1

and i f a l l

zi a r e equal,

t h i s i s reduced t o

The view f a c t o r of tube , r e f e r r e d t o i t s t o t a l surface, looking a t tube surface 3R2 (or 4U, kRl, o r 2R3, e t c . ) i s h

Flc

gP-

3R2

54 This procedure i s repeated t o get t h e view f a c t o r s from tube 1 C t o t h e other tubes seen d i f f e r e n t l y by 1 C u n t i l a catalog of view f a c t o r s t o a l l tubes seen by tube 1 C i s complete. Because i n f i n i t e geometry i n a l l directions has been assumed, t h e sum of the v i e w f a c t o r s of tube 1 C t o a l l t h e tubes it sees i n Row 2 i s a l s o t h e v i e w f a c t o r of the surface represented by a l l tubes i n Row 1 t o a l l tubes i n Row 2.

L) t

A s noted i n Section I11 and Fig. 2, the assumed tube spacing w i t h

0.750 i n . and 0.717 in. c i r c u l a r and radial pitch, respectively, i s equivalent, i n terms of space occupied by t h e tubes, t o t r i a n g u l a r l y pitched tubes having a pitch/diameter r a t i o of 2.1. Specifying an a c t u a l detailed tube sheet layout i s beyond t h e proper scope of t h i s investigation as it w i l l depend on accepted fabrication practices, designers' preferences, t h e vendor's machine tools, etc. A l s o , it would be f o l l y t o believe t h a t , regardless of tube sheet layout, 3/8-in.-diam tubes over 20 f t long w i l l exactly reproduce t h e tube sheet layout a t t h e midplane. W e can only depend on t h e average tube density i n the tube annulus. Since, from symmetry, each tube i n any p a r t i c u l a r c i r c l e was assumed t o be a t t h e same temperature, each c i r c l e of tubes was considered t o be a single surface ( see Appendix A). Ideally, view f a c t o r s f o r each tube c i r c l e would be determined by a careful computation, and tube c i r c l e curvature would be taken i n t o account. Nearly as good r e s u l t s would be obtained by g e t t i n g v i e w f a c t o r s a t several radii and interpolating. Either of these methods requires a tremendous expelrliture of routine labor i n graphic computations and pieceawesome t o contemplate and wholly impracticable. meal integrations Therefore, view f a c t o r s were determined as i f t h e tubes were i n i n f i n i t e l y wide and deep s l a b geometry, t r i a n g u l a r l y pitched (P/D = 2.1) and bounded on two faces by i n f i n i t e planes. This reduced t h e labor involved t o an acceptable minimum. The v i e w f a c t o r s so determined were then modified by considering t h e v i e w f a c t o r akin t o conductivity and using conduction equations as a basis f o r computing a correction coefficient which w i l l make the slab a r r a y v i e w f a c t o r s apply t o cylindrical geometry.

Consider, f o r example, t h e effect of tube c i r c l e curvature. I n a simple enclosure consisting of two i n f i n i t e l y long concentric cylinders, it i s immediately apparent t h a t t h e view f a c t o r from t h e inner cylinder t o t h e outer cylinder i s 1. However, t h e concave inner surface of t h e outer cylinder sees i t s e l f across the annulus, and t h e view f a c t o r from t h e outer t o inner annulus w i l l be less than 1. This, of course, i s a l s o evident from t h e reciprocity relation, Fn -.m An = Fm -4 A . I n these heat exchangers t h e r a t i o s of t h e r a d i i of tube c i r c l e s which

view one another are not s u f f i c i e n t l y close t o one t o j u s t i f y neglecting curvature by using a slab approximation. An i n f i n i t e , t r i a n g u l a r l y pitched array i s completely regular, and view f a c t o r s f o r a tube o r row of tubes t o other tubes o r rows may be used r e p e t i t i v e l y throughout t h e array. An i n f i n i t e array of uniformly spaced concentric c i r c l e s of tubes w i l l not be completely regular. The small l o c a l deviations fram uniform geometry a r e expected t o average out and hence were not considered i n the determination of t h e view f a c t o r s .

W '

It w i l l become apparent, by considering the equations i n subsequent Appendix E, t h a t system geometry e n t e r s t h e temperature - heat t r a n s f e r equations only v i a t h e view factors; i.e., t h e use of slab array view f a c t o r s would be equivalent t o solving t h i s heat exchanger transmission problem i n i n f i n i t e slab geometry. For t h i n annuli having ID/OD r a t i o s close t o 1, t h e substitution of slab f o r cylindrical geometry i s probably of l i t t l e consequence. The MSBR heat exchangers do not meet t h i s condition. It i s e a s i l y shown t h a t , f o r conductive t r a n s f e r with i n t e r n a l heat generation, t h e substitution of an i n f i n i t e slab having t h e same thickness as t h e tube annulus may produce a large e r r o r . For example, t h e temperature-drop equations f o r the two cases are:as

ORNL DWG. 71-590 Infinite Cylinder Infinite Slab

/ r2

H = internal heat generation rate, Btujhr

ft=

k = thermal conductivity,

Btu/hr f t2- V/ft

56 If w e use the tube annulus dimensions, Fig. 2, of t h e MSBR 563-MW exchanger i n these equations, w e get, f o r equal values of H and k: ATcyl = 1.28 H/k;

ATslab

= 1.80 H/k

The i n f i n i t e slab equation, i n t h i s case, produces answers b$ higher (referred t o t h e cylinder). Alternatively, i f t h e conductivity of the slab material i s increased by b$,t h e temperature drops w i l l be t h e same.

It was decided--for two reasons: (1) i n t u i t i o n , and ( 2 ) t h e immediate lack of a b e t t e r proven approach--to use these conduction t r a n s f e r equations t o compute t h e correction coefficient applied t o the slab array v i e w factors. If t h e conduction t r a n s f e r Equations (C-8) and ( C - 9 ) are rearranged t o give t h e r e l a t i v e values of conductivity i n a slab and a hollow cylinder which, a l l e l s e equal, provide equal temperature d i f f e r e n t i a l s , w e can write

The bracketed expression i s t h e r a t i o of conductivities t h a t must e x i s t i f an i n f i n i t e slab of thickness Ar i s t o t r a n s f e r i t s i n t e r n a l l y generated heat across the same temperature d i f f e r e n t i a l as i n an i n f i n i t e hollow cylinder of inner radius r and annular thickness Ar. Equal values of heat generation per u n i t volume are assumed. This expression i n brackets i s always positive and greater than 1

(> 1 ) i f Ar i s taken as p o s i t i v e i n t h e r a d i a l l y outward direction. It can be regarded as t h e correction factor, applied t o conductivity, required t o make a s l a b geometry computation produce r e s u l t s applicable t o a hollow cylinder.

It i s again emphasized t h a t t h e temperature radiant heat t r a n s f e r equations do not contain e x p l i c i t terms based on system geometry; i.e., i f t h e view f a c t o r s i n these equations are those of s l a b geometry, t h e r e s u l t s a r e correct f o r slab geometry. The v i e w factors, f o r reasons noted i n t h e preceding paragraphs, were obtained from a slab model and subsequent computations based on these view f a c t o r s must be corrected so t h e r e s u l t s are applicable t o hollow cylindrical geometry.

V i e w f a c t o r s were regarded as analogous t o conductivity and values obtained from t h e slab array calculations and graphics were corrected using a f a c t o r based on t h e bracketed term i n Eq. (C-10). I n d e t a i l , the corrections were made as follows:

57 slab = view f a c t o r from any row of tubes, surface n, t o Fn (n+k) another p a r a l l e l row, the (n+k)th, where k = 1, 2, 3, etc; or, from a bounding plane, n = 1, t o a row of tubes.

---

CY1

Fn r, (n+k)

= view fqctor from the nth cylindrical surface ( c i r c l e of tubes or a s h e l l ) t o t h e (n+k)th surface; n = 1

i s assigned t o t h e outer surface of t h e inner s h e l l of t h e heat exchangers.

= radius of nth surface.

Ar = difference i n t h e r a d i i of nth and (n+k)th surface. Ar i s always positive since these corrections were applied by s t a r t i n g a t the innermost radius where n = 1. = k x ( r a d i a l tube p i t c h ) = 0.717 k (see Fig. 2 )

An = area of nth surface.

An+k = area of (n+k)th surface. The view f a c t o r s i n t h e cylinder are:

(c-11)

With t h i s ex ression ( C - 1 1 t h e corrected view f a c t o r s were established s t a r t i n g w i t t h e innermos surface a t the inner s h e l l (n=l), f o r a l l surfaces t o other view surfaces located r a d i a l l y outward. Having established t h e r a d i a l l y outward view factors, the view f a c t o r s from a l l surfaces t o other surfaces located r a d i a l l y inward were then determined from the reciprocity relation, viz.;

n' CY1

because F(n-k)

CY1

(n-k)

n-k)

-n

)

(c-12)

-, n has been determined from previous calculations per

The view factor for a surface to itself was now determined from the requirement that the sum of all view factors from any surface be 1 (1.00).

M Fn - r n

(c-13 k=l

k=l

From the large-scale graphic layout of a planar or slab array of tubes, it was seen that, for this tube spacing (pitch/diam = 2.1), any tube in the array could barely see tubes beyond the 6th row distant; i.e., the view factor from a tube row to a row beyond the 6th row away was less than 0.01. This is of the same order as the accuracy of the graphics. Therefore, all radiation passing unobstructed beyond the 5th row was arbitrarily assumed to fall on the 6th row. An interesting sidelight developed out of the view factor determinations. Figure C5 is a semi-log plot of the view factor of a continuous boundary plane versus distance, in tube row spacings, into the triangularly pitched (P/D = 2.1) tube matrix. It seems that photon attenuation at least in this array, is exponential with distance. If, more generally, it turns out that photon attenuation in tube bundles is exponential, the use or development of analytical methods for transfer through continuous media might be worthwhile. There is a substantial body of analytical and experimental work on radiant transfer in absorbing and emitting gases. The relevant mathematics should apply if the descriptions of photon absorption or attenuation and emission in tube bundles and in gaseous media are similar. Neutron transport analyses may also be applicable since the photons undergo production (emission), absorption, and scattering (reflection). Monte Carlo techniques have also been used successfully.

OR?& DWG. 71-591

+ m

- m ;

3 Tube Row Number

Fig. C5.

5 1

V i e w Factors of a Diffusely Radiating, I n f i n i t e Plane t o parallel, Triangularly Pitched, I n f i n i t e l y Long Rows of Tubes.

(Tube Pitch)/(Tube Dim) = 2.1.

61 APPENDIX D SHELL TEMPERATUFES

Calculation of the steady-state temperature profiles through the outer and intermediate shells is a simple, straightforward procedure. It involves conduction transfer through the shells and radiative transfer in the simplest of geometries; between concentric, infinitely long concentric shells and from the outer shell to an infinite, "black" sink. This computation precedes the calculation of tube enclosure temperatures since it establishes the temperatures of the inner surface of the intermediate shell, the boundary value required to compute the tube enclosure temperatures. A detailed outline follows.

Intermediate Shell

l i

APPENDME

THE HEAT TRANSFER

- TEMPERATURE

EQUATIONS

Several approachesa t o calculating temperatures and radiated heat r a t e s have been developed f o r systems i n which it i s assumed t h a t t h e f i v e postulates l i s t e d i n Appendix A a r e valid, Regardless of how t h e problem i s attacked, these various methods produce, i n essence, the same f i n a l set of equations. Tne data reported herein were calculated using the "Radiosity Method"lo which, apparently, was introduced by Eckert and Drake. The development of the equations describing radiant interchange i n multi-surface enclosures follows.

(1) Consider kth surface in an enclosure: of M t o t a l surfaces "Radiosity" = t o t a l radiation fxwn a surface k - 2

B = radiosity

(la1

reflected

(radiation

%=%*;+@Is

(1b)

= %flk4

- 51%

= t o t a l incident radiation on kth surface from a l l surfaces of the cnclosun (includes radiation from kth surface to i t s e l f )

B f i n i t l o n of SyxnbolB

B = radiosity, Btu/hr-ftp H = incident flux, Btu/hr-fte

% = area

of kth surrace

Tk = temperature of kth surface, %

'i-ck

= v i e w factor, the fraction of t o t a l radiation imm the kth surface h i c h g a s directly t o

'bi

the ith &ace

&ace,

l o s t or gained, by the kth Btu/hr, heat l o s t is positive (+).

qk = h e a t enegy per unit area, l o s t or gained, by the kth surface, Btu/hr-fte Q

absorptivity1

p = reflectivity1 0

= Stefan-Baltzmnn const?

1730 x 10-la,BtdfiR hr- ( %)4

1. For grey bodies Q

= 1

-p

. I

Table IV, pa@ and 1730 x

'k-1

Substituting the reciprocal expressions (3) into (2)

E = emissivity

= heat en-,

% I

E. a, 1712 x loqa f o r the calculated and experimental values,

174 in ref. 11e v e s two values for

lWpl3ctiVely.

1, 2,

e..

câ&#x20AC;&#x2122;

â&#x20AC;&#x2122;

With thermal equilibrium established, a heat balance on the kth surface is written

p l emitted] radiant energy

= kenergy r n a developed l t o the] surface and which escapes from the surface

[

absorbed radiant energy f r a c t i odelivered n of t h e

t o the surface from other surfaces i n the enclosure

and substituting this i n (7) and rearranging we can write

/ using

Depending either

on whether

Equation

(6)

surface or (lb)

tcmperatU*

Equation

csn be written

(gc)

surface for

to eliminate

heat

transfer

each surface

dk4

from (6),

rate

is known,

Of the enCloeWe*

we haw

A t t h e r i s k of s t a t i n g the obvious we must be able t o prescribe a t l e a s t one surface temperature. When applied t o MSBR heat exchangers we use these radioslty equations

as follows: The innermost surface, (k = l), q1 = 0 i n Equation (lob) if it I s assumed t h a t there is no heat generation i n the inner s h e l l (Type 1). (b) Tube circles, k = 2 t o M 1, incl., qk, I n Equation (le), is specified from noble metal afterheat data.

(a)

(c)

i n Fquation (6)I s established from previous computations of t h e

Intermediate shell, k = M;

temperature gradients required to t r a n s f e r the t o t a l afterheat generated wlthin t h e heat exchanger from t h e inner surface of the intermediate s h e l l t o the outside vorld.

$-, is

These equations i n which radiosity,

k = 1;

- 'l-l)%

k-2;

2 ' -1

' 1 - ~ 2 ~ 2 F1+3B3

c1q2-4)B2

...

- F2+3B3 -

F2&4B4

the variable are

written thus

- "l-jB5 -

..... - Fl-Zn

= O

- F ~ + ~ B-J

....

-92

-F2--#M

See (a)

See (b) (lla)

k = j;

- -

F 5 l 1 B1-

"3292

FjL3B3

- ( 1 4 M ) ~ b 2 ~ 2- (~-E,)F,+~B~ -

....

F5,51B5

*..

- F&#M

= 95

k *

For Type 2 heat generation, i n which the g a m heat generation r a t e In the inner shell i s considered, the 1st equation I s not set equal t o zero, qk i s then the heat generation r a t e i n the inner s h e l l per unit outer surface area of the inner heat exchanger shell.

For uniform heat generation r a t e i n the tube annulus, q2 = qa =

l h t e t h a t the cylindrical geometry i s not e x p l i c i t i n these equations; it is implied by the view factors.

%)FM-lB1

.... q j ...

= qG1

(1d14H)FMM)BH

= E#!$

In matrix ionnat, suitable for machine computation, these equations (lla) are written:

Cl,1

C 192

Clr3

c2,1

c2,2

.. . . .. . .. .. . .. . .

cl,M-1

5,H

C2,h . . . . . .

c2,M,1

c2,M

%l .

. .

(lib)

Cn,2

. . . . . . . . . . . . . . . . . . . . Cn,5

...‘..‘..cn,M-l

c5,M

B W

R n,l

%;I4 For

surfaces hevlng e known or praecrlbed rate of heat trenefer: c

C R

%5 n,l

'-F

n--5 (1 -

Fnd5,

= (1 -

FJw5)

5#n

;

5=n

“Qn

For surfaces hevlng a ktmm orprescribedtamperatun: C

n,5

- - (1 - En, Fne5 ;

n,5 p

1 - (1 - En) FnT5 -

i 5#n

1 - (1 - E5) F5+5

;5=n

Rn,l =EoT4 nn The radiosltles,

‘\ E

are computed end surface tampareturas, Tk,Jobtelnaa fmm Quatlon (9d).

‘\ I; ,C1

APPENDIX F COMPUTATIONAL PROC.&DLEB

Calculations were made i n sequence as follows: 1. View f a c t o r s were determined as outlined i n Appendix C.

Temperature p r o f i l e s i n the outer and s h e l l s were calculated as outlined i n The temperatures a t t h e inner surface mediate s h e l l a r e the boundary values, Eqs. ( l l c ) i n Appendix E.

intermediate Appendix D. of t h e i n t e r TM, used i n

Having established the view factors, boundary values f o r temperatures, and heat rates, Eqs. ( l l c ) i n Appendix E were solved f o r tube c i r c l e and inner s h e l l temperatures.

The computations were made using Extended Basic (BII) programs with t h e Reactor Division's time-sharing computer f a c i l i t y . Several programs were written f o r t h e various aspects of the problem. It has been t h i s author's experience t h a t t h e inclusion of computer program l i s t s without copious explanatory notes and instructfons i s wasted e f f o r t . Should a need develop, t h e programs w i l l be reported separately.

APPEXDIX G THERMAL RADIATION CHARACTERISTICS OF HASTELLOY N

The e m i s s i v i t y of Hastelloy N (INOR 8) i s reported i n reference 27. These d a t a suggest t h a t , f o r clean, unoxidized surfaces, we use an emiss i v i t y of 0.2 and, f o r oxidized surfaces, an e m i s s i v i t y of 0.5 o r 0.6.

The author d i d not become aware of reference 27 u n t i l t h i s r e p o r t was v i r t u a l l y completed. The remainder of t h i s appendix i s t h e r e s u l t of my attempts t o i n f e r a value of e m i s s i v i t y from measurements on a l l o y s composed of similar elements. Although t h e d a t a and r e f e r e n c e s which follow are not d i r e c t l y applicable, t h e y a r e included s i n c e they may be r e l e v a n t and u s e f u l t o persons d e a l i n g with similar problems. There i s considerable data on a l l o y s containing n i c k e l , i r o n , molybdenum, and chromium. Table G1 l i s t s values of e m i s s i v i t y f o r metals of t h i s general composition. I n general, t h e numerical values of emissivity f o r smooth, clean, unoxidized surfaces a t temperatures i n t h e region 1200"F-2000"Fare from 0.1 t o 0.3. I n t h e absence of b e t t e r information, similar values were assumed f o r Hastelloy N. Therefore, t h e temperature c a l c u l a t i o n s were made f o r i n t e r n a l e m i s s i v i t i e s of 0.1, 0.2, and 0.3. I n general, t h e d a t a i n d i c a t e , and theory s u b s t a n t i a t e s , t h a t e m i s s i v i t y increased with i n c r e a s i n g temperature. This change i s not l a r g e ; within t h e temperature d i f f e r e n c e s calculated i n t h e tube annuli, it w i l l be of t h e same order as t h e uncertainty i n t h e value of t h e emissivity. No attempt was made t o include a mild temperature dependence of e m i s s i v i t y i n t h e temperature h e a t transfer equations. These equations, l i n e a r with 'I? as t h e unknown, a r e e a s i l y solved by standard r o u t i n e programs. We are dealing with conceptual designs, s u b j e c t t o change, and a p h y s i c a l system containing u n c e r t a i n t i e s o t h e r than emissivity. The incremental elegance of s o l u t i o n and t h e r e s u l t i n g improvement i n accuracy are insuff i c i e n t t o j u s t i f y t h e very appreciable i n c r e a s e i n c o s t and t i m e required t o develop o r adapt a program which allows t h e use of temperaturedependent emissivity i n t h e system of equations.

I n connection w i t h t h e data from reference 22 i n Table G1, it i s appropriate t o point out t h a t t h i s r e p o r t , NASA CR 1431, o u t l i n e s a program now apparently under way a t Purdue University t o c o l l e c t , evaluate, and present r a d i a t i v e p r o p e r t i e s data from a l l a v a i l a b l e sources. When suff i c i e n t evidence exists concerning a p a r t i c u l a r property of a p a r t i c u l a r material, such as t h e emissivity of s t a i n l e s s s t e e l , broad band curves of It recommendedn values are presented which enable t h e reader t o g e t a general f e e l i n g f o r t h e t o l e r a n c e s o r v a r i a t i o n s t o be expected and t h e circumstances i n which t h e "recommended" values are a p p l i c a b l e . These d a t a t e n d t o develop more confidence i n t h e u s e r than anything t h i s w r i t e r has seen heretofore i n t h e r e a d i l y a v a i l a b l e t e x t s and references.

It has been pointed out t h a t by assuming gray, d i f f u s e conditions, no d i s t i n c t i o n i s made between emitted and r e f l e c t e d r a d i a t i o n ; both a r e assumed t o have d i r e c t i o n a l p r o p e r t i e s i n accordance with Lambert's cosine law. There i s evidence t h a t , whereas t h e p a t t e r n of emitted r a d i a t i o n may

Table Condition and/or Watment

mteria1

Molybarnurn

Stainless steels, cleaned

Bnittance Measuremcnt

12oO.F

m0o.F

llotes

Source

ETE

Linear variation imm 0.07520.02 a t 6oo.F to 0.25oK).W a t 35007.

0.11EO.02

O.lWkO.02

Based 011 7 references. Data 18 apparently consistent with uniform scatter.

Ref. (22) HASA CR 1431, p. 26

B. G r i t b l a s t e d

ElT

0.2&0.05 a t 134O.F; 0 . 3 a t 17OO.F.

0.2eo.05

-0.p

Eased on 2 references.

Ref.

0.38

Eased on 1 reference.

Ref.

O.m*

0.275

Baaed on 1 reference.

Based on 1 reference. Curve sheet states t h a t oxide is volatile in a vacumn above 1OOo.F (8ll-K).

Shot blasted and etched

BpE

0.22

1ooo.F to 0.40 a t 1340-

TI0o.F.

Etched

and flashed

ETE

0.25 a t

Stably oxidized.

ETE

Approxlmtely 0.82 a t 7009.

Cleaned

0.3~0.09 a t 6oo.F; o.5io.og a t la0o.F; 0.4Lt0.09 a t lj4o.F;

7.3

0.25

b 0.40 a t n00.F.

7 .

0.5

Ref. (22)

MSA CR 1431, p. 26

baaed on a t least 13 references.

Ref. (22)

NASA CR 149, p. 26

Ref. (22) MSA CR 1431, p. 44 Ref. (22) NASA CR 1431, p. 44

Eased on 1 reference.

Ref. (22) NASA CR 1431, p.

o.yao.07

0.47*

Based on 6 references.

Ref. (22) MSA CR 1431, p. 45

1ooo.F;

0.46

Baaed on 1 reference.

Ref. (22) USA CR 1 4 3 , p. 45

0.64 a t 8O0'F; 0.67 a t ll63.F; 0.70 a t 13b.F.

0.67

Based on 1 reference.

Ref. (22) MSA CR 1 4 9 , p. 45

0.67i0.02 a t 98o.F; 0.7E0.03 a t 1Jlo.F; o.a*O.@ a t 1W.F; 0.82M.06 a t 2060.F.

0.6gi0.03

0.8liO.06

Based on a t least 5 refer-

Linear variation from 0.10 a t 5o.F to 0.15 a t 116o.F rising nonlinearly to 0.3 a t V8o.F (see k t e ) .

A s hlled, cleaned

RPE

0.47 a t 6007; 0.5 a t ll6O.F.

0.1520.06 a t 2609; 0.25+0.07 a t 8O0.F; 0.293l.w a t 1ooo.F;

Oxidized in a i r a t RPE red heat for 30 min.

(22)

AASA CR 1431, p. 26

Based on 4 referencea. Aotation on curve sheet a t nonlinear portion of curve reads "Oxidation Probable."

(22)

MSA CR 149, p. 26

Appears to be

-+ rpJ0-W

B. R-l55, as received or cleaned

11. Stainless steels, oxidized

mssivltiea at

M s s i v i t i e a and Corresponding Temperatures and/or wadlengths

polished

11.

Bnissivities of Holybdenum, nickel, Chromium, and Various A l l o y s containing These Elements

01.

0.16

0.35i0.07 a t 13bO.F; 0.41 a t

1700.F; essentially linear variation imm 440.F to 1700.F.

B. Buiiea, stably oxidized a t lll2.F

0.40 a t 62o.F; 0.42 a t

0.50 a t 13409.

t8'73a1.

Shotblasted, h b l y oxidized a t

Polished a d oxidized a t high temPerattuc.

lll2.F (8'73-K). mE

a t w.F; a t l34O.F; 0.8'7 a t 14509.

o.n a t 700.F; 0.8 0.6

III.

Stainless *tee-, polished

P. stably oxidized a t high temperature.

RPE

A. -8-155,

WE-

palillhed- -

(oxidation re-

tardca)

B. Vafious polished

stainless steels

0.8p0.05 a t 7oo.F; 0.8f30.05 a t gB0.F; O.gio.01r a t 1340.F.

talaed).

As received

B. As received a d oxidized a t 1200for 48 hours. C.

Pollshed

Incone1 x

Polished and oxidized a t 12aO.F for 48 hours.

received

received ~d oxidized a t

Based on 1 reference.

o.mo.05

Based on 3 references.

o.igio.04

0.28m.06

a t 8007;

poliahed

a t l52O.p;

constant a t 0.25 imm WO.F to l2oo.F.

0.25

loot

0.55 a t 400- to 0.53 a t ll0O.P. linear variation.

0.53,

0.12 a t 300- to 0.20 a t W.F. Linear variation.

0 . 9

&t stated

0.27 a t 500- to 0 . 9 a t ll00.F.

0.3*

Hot

0.m a t to 0.23 a t ~ 0 o . F . Idnear variation.

6.23,

stated

Ibt

constant a t 0.38

0 . e

stated

ll0o.F.

0.18 a t

stated

Hot

VI.

Hone1

pollshedand oxidized a t 12oo.F for 48 hours.

As received

loot

stated

B. As received and oxidized a t 1200for 48 hours C.

loot

polished and oxi-

dized a t 12009 for 48 houm. VII.

vm.

stated

M stated

Easedm-3 rrfe~nceo--mta--Bet-(2Z~ -indicates scatter above MSA CR 1431, p. 46 13407. Based on 9 references. The curve r i s e s rapidly a t high temperature and carries notation "Oxidation Probable."

Ref. (22) &AsA CR

1431, p. 46

Ref. (23)

m.F

h ~ m 300-

0.351

---

&lo a t 200- to 0 . 9 It Il0o.F. Llnear variation.

O S ~ *

0.50 a t 200- to 0.72 a t ll00'F. Linear variation.

O.n*

0.65 a t b0o.F to 0.57 a t ll007. m a r variation.

0.56*

m.F to 0.17

a t ll00.F.

0.W

0.29 a t 3009to 0.5a t Linear variation. ,

u00.F.

Ref. (23)

Re?. (23)

llickel

Polished

0.07 It 307 to 0.16 8t 2OOo.F. llearly linear mriation.

0.11

0.16

Ref. (24)

Chrmhm

Pollshed

0.05 It 300- to 0.u 8t l.&o'P.

0.26

Ref. (24)

linear variation.

Em "E

---- Hemispherical Total Pbrittance. -- llormal Total Pmittance. Tot81 &ittance.

*Extrapolated by 8uthor of t h i s rrport.

Linear variation.

stated

Ref. (22) 1431, p. 45

AASA CR

(see Me).

12oo'F

XI.~&+---

M stated

stated

O.lTiO.06 a t 260.F. a t 6 2 0 ' ~ - 0.21io.d 0.230.06 a t a t 13kO.P; O.brn.10 0.56iO.U a t 17OO.F

Ref. (22) NASA CR 1431, p. 45

AIL~ -

(22)

NASA CR 149, p, 45

0.83

w . ~0.330.07 ;

(oxidation re-

Iv. Incone1

6203; O l u ~ t g & Y ____ ~ ; 0.18 a t 1340'F; O.2m.03 a t 17009.

D.1 a t

Ref.

ences (probably 7).

Li-

be well approximqted by t h e cosine l a w , r e f l e c t e d radiation from t h e same surface can be highly specular, p a r t i c u l a r l y when very smooth or polished surfaces predominate. I n t u i t i v e l y , t h i s i s reasonable. No attempt was made t o estimate or evaluate t h e e f f e c t of a strong specular component on temperatures developed i n these heat exchangers. Any e f f e c t of specularity would be more apparent a t low e m i s s i v i t i e s when t h e energy content of t h e t o t a l r a d i a t i o n leaving a surface contains t h e l a r g e s t f r a c t i o n of r e f l e c t e d radiation. The l i t e r a t u r e contains comparisons of computat$ons based on d i f f u s e vs specular r a d i a t i o n emanated by r e l a t i v e l y simple, regular cavity configurations such as continuous rectangular and V-shaped grooves, c y l i n d r i c a l and conical cavities, and spherical cavities. The data a r e presented by p l o t t i n g curves of apparent emissivity, which i s a measure of o v e r a l l heat t r a n s f e r c a p a b i l i t y of t h e cavity, as a fâ&#x20AC;&#x2122;unction of cavity depth-to-opening r a t i o (L/R f o r a c y l i n d r i c a l cavity). Actual emissivity i s a parameter. The e f f e c t of specularity i s t o increase t h e apparent emissivity and t h e e f f e c t i s most pronounced i n deep c a v i t i e s of low emissivity. For example, reference 12, Fig. 6.2, p. 165, shows calculated curves of apparent emissivity of c y l i n d r i c a l c a v i t i e s having depth t o radius r a t i o s , L/R, from zero t o 10. The data i n Table G2 have been taken from t h i s f i g u r e . Table G2

Actual Esni s s i v i t y of Cavity Surface 0.1 0.1 0.1

Calculated Apparent Ehissivities Diffusely Reflecting

Spec u l a r l y Reflecting

0.34 0.59

0.34 0.48 0.49

0.53

Lâ&#x20AC;&#x2122;R 2

0.2 0.2 0.2

0.3

0.3 0.3

0.72

0.63 0.63

0.56 0.79 0.88

0.66 0.72 0.72

0.83

0.70 0.94

74 The degree t o which t h e i n t e r n a l surfaces, tubes mainly, r e f l e c t specularly deserves consideration. Certainly t h e spaces between t h e tubes are not unlike deep c a v i t i e s . If specular r e f l e c t i o n s w i l l take place i n the heat exchangers, it i s reasonable t o conclude t h a t t h e diffuse calculations reported here give higher temperatures and may be regarded as conservative.

75 APPENDIX H COMPARISON

EXPERIMENTS AND ANALYSES

The calculated temperatures reported herein are not presented ent i r e l y unsupported by experimental verification. In November 1965, temperatures i n a 9l-rod bundle of 0.5-in.-diameter s t r a i g h t s t a i n l e s s steel tubular e l e c t r i c a l heaters were reported." Figure H l i s a photograph of a somewhat similar heater assembly. Figure E'shows t h e e s s e n t i a l s of t h e t e s t setup. The information required t o do temperature calculations follows. A.

System Configuration 1. See Fig. II2 2. Rod pitch/diam = 1.25. 3. Rod d i m 0.5 i n . 4. Rod surface area 0.131 f t 2 / f t length. Heated length of rod 10 i n . = 0.83 f t . 5.

---

Rod array concentric, hexagonal rings formed by triangul a r l y pitched tubes. The single, c e n t r a l tube i s considered t o be r i n g No. 1.

Thermal Conditions

1. Heat input

---

= 22.4 w/rod = 91.7 Btu/hr-ft length of rod = TOO Btu/hr-fta rod surface = 6780 Btu/hr f o r 90 active rods

(1 rod not heated) 2.

Environment

2 3 microns

(a)

Tube bundle housing evacuated t o a pressure of of mercury.

(b)

Housing i n laboratory atmosphere , protected from d r a f t s .

Etnissivities No measurements were made of emissivity o f - t h e tube surfaces or t h e i n t e r n a l surface of t h e housing. Based on the several-year-old recollection of a person somewhat familiar with t h e experiment, it can only be concluded t h a t t h e stainless s t e e l surfaces were neither new, bright and polished, nor were they heavily and deeply oxidized. Between these extremes, emissivities from 0.2 t o 0.7 are possible and values from 0.3 t o 0.5 are l i k e l y . Tube surfaces displayed a dull, satiny gloss t y p i c a l of s t a i n l e s s s t e e l after long immersion i n a high-temperature l i q u i d metal (NaK, K, Na, etc. ).

Fig. Hl.

Array of 91 Heater Rods Having Similar Geometryto Rod Wrndle Shown on Fig. H 2 .

ORNLDWG.71-592

dl, ~,r&ea-40, is pip

cmva Rob.

8:: 0.80 0.40 O.bS 0.50 0.6

- inside race temp.

*This â&#x20AC;&#x153;.

maala

ls906oftot~

heat

inplt.

- Imhei

Fig. ~2.

Comparison of Experimental and Calculated Temperatures in 8 91 Rod Array.

Temperature measurements Temperatures were measured by chromel-alumel thermocouples embedded i n t h e heater rods and placed on o r i n t h e housing.

Temperature calculations were made using t h e same general method as t h a t used t o calculate heat exchanger temperatures. Because of t h e close spacing of t h e heaters (P/D = 1.25) t h e view of any rod i s limited t o adjacent rods and t o rods i n any second row d i s t a n t . The c e n t r a l rod and each of t h e f i v e hexagonal r i n g s were considered t o be an isothermal surface so t h a t t h e "enclosure" consisted of a t o t a l of seven surfaces. This assumption i s capable of producing some e r r o r since t h e corners of t h e hexagons a r e closer t o t h e outer shell. This i s p a r t i c u l a r l y apparent on examining t h e corner rods i n t h e outermost hexagonal r i n g . Each of these six rods sees only t h r e e adjacent rods instead of t h e four seen by t h e other rods i n t h i s ring. Everything else being equal, we should expect i t s temperature t o be lower than t h e other rods i n t h a t r i n g since it has more surface d i r e c t l y viewing t h e heat sink (the outer s h e l l ) . This has been borne out i n t h e second experiment discussed subsequently, but it was not taken i n t o account i n t h e temperature computations f o r t h i s t e s t setup. The length of heated section i n t h e rods was 10 in., giving t h e assembly an approximate L/D of 1.7. The one-dimensional i n f i n i t e length computational model was used; it was recognized t h a t , applied t o t h i s short assembly, accuracy would suffer. I n a n e f f o r t t o compensate p a r t i a l l y f o r a x i a l heat l o s s e s by conduction i n t h e rods and by radiation, p a r t i c u l a r l y from the outer r i n g t o t h e s h e l l , t h e r a d i a l l y radiated heat was assumed t o be 90% of t h e t o t a l input.

Since t h e a c t u a l emissivities of t h e surfaces were not known, it was decided t o assume values f o r emissivity and attempt t o match the experimental temperature data. The results of these calculations a r e on Figs. H2 t o H5, inclusive. These f i g u r e s are, f o r t h e most p a r t , self-explanatory, but i n a l l cases note t h a t t h e observed r a d i a l temperature gradient i s steeper than t h e calculated values. The temperatures developed and measured i n t h e rod a r r a y w i l l be affected by a x i a l conduction i n the heater elements and thermocouples and by specular o r other non-diffuse components i n the emitted and r e f l e c t e d radiation. Any computation designed t o obtain precise agreement with experiment must include these f a c t o r s . They were not considered i n t h e calculations reported herein. The calculations using emissivities i n t h e neighborhood of 0.5 a r e i n t h e best agreement with t h e experiment. This may be reasonable since t h e heaters had seen considerable use. Figures H6 and H7 are included t o emphasize t h a t r a d i a t i v e transfer i n a bundle of rods should not be approximated by assuming t h a t each r i n g of rods behaves as i f a continuous, impenetrable s h e l l .

79 ORNL DWG. 71-593

Radiated Heat Rate*

1400

Btu/hr ft of rod

82.5 82.5 82.5 82 05 82.5

&! 1100

B d

Radius Fig. H3.

- Inches3

A Comparison of Experimental and Calculated Temperatures Re ired to Radiate 82.5 Btu/Hr-Ft of Rod i n a gl-Rod, Hexagonal Array of $-In.-Mam Heater Rods Spaced 518 In. Apart Enclosed i n an 8-In. Sched-40 SS Pipe (refer t o CF 65-11-68). Calculations are shown with emissivity a parameter.

P Q R

0.60 0.50 0.52 0.00 0.50

0.50 0.50 0.48 0.40 0.50 0.50

0.55 0.50 0.50 0.40

0.45 0.40 045 0.30 0.50 0.40 0.40 0.40 0.46 0.u 0.42 0.50 O.& 0.40 0.40 0.60 0.50 0.40 0.40 0.80

82.5 82.5 h.5 e.5 e.5

%ubrcripts refer to muface rmhers. Surface m&er 1 Is awhr 7 Is inside of &In. sched40 pipe.

c m t d md, &ace

EXperimcnt.1 CulTc IS fmm ng. I2 in cF-65-lld8. Reiet to ng. E2 for d e t a i l s of md @xm'try and t e s t #+-UP.

0RNTo . DWG.

71-595

Continuous Shell Model

Experiment

Multi-8uriace

Enclosure Mode1

E5 1

A Canparison of Calculated Femperatures Required t o Radiate 82.5 Btu/Hr-F%

of Rod in a gl-Rod, Hexagonal Array of 1/2-In.-Mam Heater Rods Spaced 5/8 In. Apart &closed in an &In. Sched-40 SS Pipe (refer t o CF 65-11-68). Calculations are shown for emissivities (d~. surfaces) of 0.5 and 0.6.

ORNL DWG. 71-596

This cumre indicates that, in calculating the temperatures required to get radiant heat t r a n s f e r from arrays of rods o r tubes, a continuous lnrpenetrable convex s h e l l is a poor crubstitute f o r t h e

opn, conmluted geomety o r t h e tube bundle. The curve was prepared

68 f O l h W S :

1. Temperatures, Fig. E, In t h e 91-rod array were calculated by t h e method described In Appendix C, f o r several velues of rod surface and outer s h e l l emissivity and u l t h a heat input estimated

a t 82.5 -4%

1-h

of rod.

The *-rod array was then represented, Fig. 86, as a s e r i e s of contimauus, concentric, closely spaced, I n f i n i t e l y long heZegOMl 2.

FQ.6. ~ a g n ai a ~a contirmous Shell Hodel of a Hexagonal Array of 91Heater Rods.

shells mnnnmded by an outer cylindrical shell. For this configur a t i o n t h e radiative t r a n s f e r irolp s h e l l to s h e l l I s given by t h e 6-1e relatiOU,

a = stcian-W1tnnaxm constant = 1730 x ID-"

a = area of nth surface; Q

b Tn+l

(ntl)

Awl

= area of ( n + l ) t h muface

heat transferred, Btu/hr temperature of nth surface temperature of ( n + l ) t h mrface interchange f a c t o r

tmm nth to (n+l)th surface

E,, = emissivity o r nth surface emissivity of (n+l)ta surtace

Eb+l)

fornaiLa (1)and inserting therein the numbers from the -re (n+l) was calculated vhich proaucea the .ame total heat t m s f e r a t t h e s8me temperatures (see ~ l g .E)as those computd by the =re exact method per (1) above. The emissivity of the outer, (n+l)th, shell yas assumed t o be the .ame as that assmea for the =re exact computation and,rormula (2), aof En was e&abUshea. This value of En I s the emissivity which a c o n t m s h e U muat have lf It is to r a d i a t e to an adjacent s h e l l as effectively as a hexagonal tube a r m y having an emissivity of E b+l) r a a i a t i n g to a continuous outer s h e l l having t h e same emissivity,

US-

exact calculation per (1) above, a value of Fn

Actual W s s i v i t y

Fig. W. The A p p a m t bdssivltp of a Hexagoml A m y of Rods if V i m as a single, l x a g o m l Surface YS A c t u a l Rod Ehissivlty. Rod PltCh/Rod Mam = 1.25.

E(n+l)* For example, Fig. El? indicates that if the rod &ans in t h e hexagonal nnay have an a c t u a l emissivity of 0.2, they appear, to an .ajacent plane, as much blacker plane vlth

Cmisslvity of

0.7.

83 A second experiment,' not dissimilar from t h e f i r s t , involves a hexagonal a r r a y consisting of a c e n t r a l tube p l u s eight hexagonal r i n g s of heated tubes surrounded by a close f i t t i n g hexagonal shell and a f i n a l , outer s h e l l of 10-in. sched-40 pipe, Fig. H8. The datab required f o r t h e r a d i a n t t r a n s f e r calculations are: A.

System configuration 1. See Fig. H8 Pitch/diameter 1.375 3. Tube diameter 0.25 i n . 4. Tube spacing ( p i t c h ) 0.343 i n . 0.0654 f t a / f t length 5. Tube surface area 48 i n . 6. Heated length 217 7. Total number of tubes 8. Total number of heated tubes 205 (12 tubes used f o r thermocouples)

-----

---

Thermal conditions

1. Heat input

---

5.86 1.47 5.0 76.5

w/tube w / f t of tube Btu/hr-ft Btu/hr-fta tube surface = 4100 t o t a l Btu/hr, f o r 205 a c t i v e (heated) tubes. = = =

Environment The hexagonal, s t a i n l e s s sheet s t e e l tube enclosure i s protected from a i r currents and sudden t r a n s i e n t s by t h e outer housing, a 10-in. sched-40 pipe. The system was evacuated t o 100 microns of mercury o r less f o r t h e tests discussed herein.

Finissivities similar s i t u a t i o n as w i t h t h e f i r s t experiment described i n t h e preceding paragraphs obtains,except t h a t t h e 0.25-in.-diam tubes had not been immersed f high-temperature l i q u i d metal.

Temperature measurements Temperatures i n t h e bundle are measured by thermocouples i n s e r t e d i n s i d e selected tubes. These tubes do not contain h e a t e r s and are located (see Fig. H8) a t t h e corners and i n t h e centers of t h e f l a t s of t h e hexagonal. r i n g s making up t h e bundle.

'This experiment simulates a f t e r h e a t generation by W R R fuel assemblies. b ~ C. . Young, p r i v a t e communication.

ORNL DWC. 71-597

Tubing Detail Fig. H8.

Diagram of a Cross Section Through the Bexagonal Array of 217 Tubes.

h, *

Results

a The comparison of experimental and computed temperatures, Figs. H9 and HlO showed t h e same general trend as i n t h e previous experiment, namely, t h e observed temperature gradient was steeper. The use of unheated tubes f o r t h e temperature measurements gives temperature readings below t h e average temperatures of s i m i l a r l y located heated tubes. This i s borne out by t h e calculations, curves D1 and on Fig. MO. Curve calculated with zero heat input t o t h e c e n t r a l tube i s depressed 23째F below t h e temperature computed as i f t h e tube were heated. Note a l s o t h a t curves D1 and D2 show a temperature difference of 34째F a t t h e outer ring. This i s t h e computed difference i n temperature between a heated tube c e n t r a l l y located on a face of t h e outermost hexagonal r i n g and t h e unheated corner tube. This difference has two causes: (1)t h e corner tube i s not heated and ( 2 ) by v i r t u e of i t s location more of i t s surface views t h e adjacent hexagonal shield, a heat sink. It w i l l be shown t h a t t h e l o c a l temperature depression of an unheated tube t e n d s t o go inversely with t h e t h i r d power of t h e l o c a l prevailing temperature surrounding t h e unheated tube and, a t lower e m i s s i v i t i e s (7 0.3), nearly inversely w i t h emissivity. From t h i s w e should expect t h a t t h e a c t u a l r a d i a l gradient across t h e bank of heated tubes w i l l be less steep than indicated by thermocouples i n unheated tubes. During these preliminary runs t h e temperature of t h e intermediate, hexagonal s h i e l d was not measured. This i s the heat sink temperature and i t s value i s required i n t h e calculations (see Appendix E). The calculated temperature curves i n Figs. H9 and HI0 a r e based on hexagonal s h i e l d temperatures of 700째F and 6 0 0 " ~ . These temperatures were calcul a t e d by assuming an outer s h e l l temperature of 200째F and e m i s s i v i t i e s of s h e l l and s h i e l d surfaces of 0.30 and 0.42. The experiment was i n i t s i n i t i a l s t a r t u p phase when t h e data were taken, and plans were under way t o obtain these temperatures i n f u t u r e tests. The degree t o which hexagonal s h e l l temperature a f f e c t s i n t e r n a l temperatures has not y e t been determined experimentally. Two calculations, v i r t u a l l y i d e n t i c a l but f o r hexagonal shield temperatures, may be compared with the following table, H l . Reducing t h e sink temperature from Table Hl. Effect o r i n g t h e Heat Sink Temperature on Peak Temperature i n t h e Hexagonal Array of 212 Tubes

A l l Surfaces

0.4

400

cai a H. C. Young, p r i v a t e communication

859

676

AT = 78

AT = 107

ORAL DUG. 71-598

&aiai Lxntiom" of & t e r m s of Hexagonal Rings or Tubes 9

800

700

600

-5usiag, 10-In.

Stainless Steel Sched lo pip.

400

5using, Temperature -9.

2 *dillsa

- Inches

Fig. E9 Bxperbental and Calculated lbperatures in Heated, stainless Reel Tubes (see ~ i g E . ).

Hexagonal

Array of 217

----

1. p l l r i m ~ t lo-in. pipe housing evacuated to =3 acmns B g . Heat Input 1.47 rstts/ft tube = 5 Btu/hr-rt = 76.5 Btu/hr-ite.

Bnissirity, Tubes

TeIrQerature

Shield

CURIsAlandA2

0.3

7009

CURIsBlandE

0.5

7007

CURIE C l and 12

and Hexagonal

Hexagonal

%e centerlimes and shield locations shown across cornera of hexagon.

R\be temperatures f r o m central tube acmsa f l a t and to corner tubes of hex, respectively. Tube temperatures irpm central tube acmes f l a t and to corner tubes or hex, ~spectirrly. $rperimental data. as i n

an elevation (section) r i e w

O m DWG. 71-599

*dial Iocatioo' of Centerunes of Hexagonal Rings of Tubes 3 2 3 4 5 6 7 8 9

Housing, 10-In. Stainless Steel sched 40 Pipe.

-=- h5.F.

HOUsiog tW

P Radiusa

Inches

w . m0. Erperllsental and Calculated Temperatures i n a Hexagonal A r r a y of 217 Heated, Staanlese Steel Tubes (see Fi&. E). 1. B m r i r o m t 2.

Heat Input

WSSivitY

Tubes

Hexagonal Shield

Temperature Hexagonal Shield

CumsClandC2

Curve D l

0.3

Cumre

0.3

700-

CurveEl

0.10

CUrVeQ

0.4

0.5 0.5

6009 600-

Experimentally dets-ned

%be

hexagon.

centerlines and

pipe harsin(l evacuated t o -3 microns ap. ---- l0-in. 1.4? uattS/ft tube 5 Btu/br-ft 76.5 Btu/hr-ftp.

tube temperaturra from centrsl tube (unheated) across flat and to corner tubes of hex array, reapecti~ely. Temperatures calculated a s lf a l l tubes a r heated ~ and a s if outermost corner t u b a = aame view factors a s other tubes in the outer ring. Temperatures calculated as if central tube and outer corner tube8 are d e n t e d end vith corrected vim factors for the outer corner tubes. TemperetuRs calculated vlth central tube unheated and with no v i e w factor correction f o r outer corner tubes which are heated.

locations ahown as in an eievatim (section) viw scmss cornem of

aa 600'F

t o k 0 ' F effected an appreciable reduction i n t h e peak temperature and a s l i g h t increase i n t h e radial gradient. A t higher heat inputs o r i n l a r g e r assemblies with more tube rings, t h e f o u r t h power e f f e c t of temperature w i l l dominate t h e f r a c t i o n a l improvement obtained by lowering the sink temperature. It w i l l be shown, subsequently, t h a t there were a x i a l heat l o s s e s i n t h i s experiment. The differences between observed and calculated temperatures are subject t o t h e same generql considerat i o n s mentioned i n t h e previous discussion of t h e 91-rod array.

Conductive t r a n s f e r effected by t h e s p i r a l wire wrap around t h e tubes (see Fig. H8) was not considered. If any appreciable heat i s t r a n s f e r r e d by conduction, it w i l l a f f e c t t h e radial temperature gradient. Figure 811 shows t h a t t h e r e were a x i a l heat losses, amount unknown. No r e a l l y serious attempt was made t o consider these i n t h e computations. A few calculations f o r which t h e radiated heat was reduced by -5$ (from 5.0 t o 4.73 Btu/hr-ft tube) were made using t h e same i n f i n i t e cylinder model. The e f f e c t of t h i s small heat reduction, credited t o a x i a l heat flow, was negligible.

It has been noted t h a t w e should expect t h e corner tubes t o have a lower temperature. If unheated, as these are, t h e effect will be more pronounced. The unheated tubes i n t h e inner r i n g s w i l l a l s o be a t lower temperatures than t h e i r neighbors a t o r near t h e same radius. The amount by which they w i l l be lower w i l l c e r t a i n l y be dependent on emissivity. For example, t h e temperature depression can be approximated by using t h e d i f f e r e n t i a l of t h e heat t r a n s f e r expression,

i n which

F1+2 = r a d i a t i o n interchange f a c t o r from area A1 t o Az,

u = Stefan-Boltzmann constant = 1730 ,x'''ol

Tl and T, = temperatures of Al and &,

OR.

Assume t h a t t h e unheated rod of area A1 i s surrounded by other rods having a t o t a l area, &, a t t h e same o r nearly t h e same temperature so t h a t TI 2 Ta; also, assume t h a t T2 doesn't change. By d i f f e r e n t i a t i n g ,

(H-2a) f

ORNL DWG. 71-600

-3

-1 0 Distance f r o m Midplane

+1 Feet

Fig. HLl. Axial Temperatures in Hexagonal A r r a y of XnternaUy Heated Tubes (see Fig. H8).

and

L, P

(H-2b)

f o r small changes i n TI. The single tube represented by 4 and nearly completely surrounded by six tubes i s approximated by two cylindrical surfaces having an area r a t i o , +/%= 0.5. If t h e emissivity, E, of both surfaces i s t h e same, t h e interchange f a c t o r i s then given by

1 + 2 (1- E)

1.5

- 0.5

Now, i f w e disconnect t h e power from t h i s central tube, it is, i n e f f e c t , operating as a thermocouple tube and A(\/%)

- 76.5

Btu hr-ft2

The expression, Eq. (H-2b) f o r &TI, the temperature depression i n t h i s tube, becomes

&T1

= 4 F

(1730 x 10'la) +a

4 2

T31

(H-2~)

Table H2 gives values of LE f o r t h e temperatures i n t h e range of t h e experiment and f o r emissivithes considered reasonable.

It would not and w i l l not be correct t o apply these approximate tabulated values t o t h e experiment. They are l i s t e d t o indicate t h a t t h e temperatures measured i n an unheated tube w i l l be below t h a t of t h e surrounding heated tubes i n t h e bundle. Since t h i s d e f i c i t i s greater a t lower temperatures, measurements so taken w i l l indicate a gradient steeper than a c t u a l l y exists. The i n f i n i t e cylinder model was used and, taking t h e length equal t o the length of t h e heated section of t h e tubes, 4 ft, we have an LD e 8--a b e t t e r approximation than with t h e first experiment. After assessing the calculated and observed temperatures i n these tube bundles, it was concluded t h a t t h e agreement was satisfactory. Since t h e computed values tended t o be higher, it was decided t h a t t h i s calculational approach i s a s u i t a b l e method which gives safe answers

. w -

Table H2. Approximated Temperature Depression i n a Single Unheated Tube i n an Extensive Tube Array Location i n Tube Bundle

Approximate Value, TI

Center

1000째F = 146O"R

bissivity, E 0.1 0.2

AT1 " or F

-5 5 -25

0-3

-16

0.4 0.5

-11

Center

900째F = 1360"R

0.1

Edge of Wlndle

715'F

0.1 0.2 0.3 0.4 0.5

= 1175'R

O R

-9 -63a -106

-48 -31 -22

-17

"The more exact calculation, curve El on Fig. H l O shows a temperature depression o f 39" a t t h e c e n t r a l tube.

t o use i n estimating temperatures i n MSBR heat exchangers whose i n t e r n a l configuration is, e s s e n t i a l l y , a bundle of p a r a l l e l rods o r tubes.

APPENDIX I

METHOD USED TO ESTIMATE THE INITIAL PEAK TEMPERATURE TRANSIENT I N THE 563-MW REFERENCE DESIGN HEAT EXCHANGm

1. The adiabatic temperature versus t i m e curves "B" and "C" on Fig. 1 5 were plotted. The bases f o r these curves are: A.

The t o t a l accumulated afterheat, Table 1 or Fig. 5, i s deposited thus (see Fig. 7) : (1) 77% i n t h e annulus (inner s h e l l and tubes)

(2)

23% i n the intermediate shell. This includes the heat deposition i n t h e outer s h e l l and the gamma heat l o s t t o the outside.

The annulus and intermediate s h e l l are separated and both are perfectly insulated.

The heat capacities of these regions are as l i s t e d i n Table 11.

Table 11. Heat Capacity of Bnpty 563-MW MSBR Primary Heat &changer Heat Capacity Per

Mw Rated Capacity,

Ketal VolUnc;*

ft3/ft height

Region

Heat Capacity,* Btu/OF f t height

Fraction of Total

(Btu/'F f t height)

Inner s h e l l 20 i n . sched 40 pipe

0.25

0.034

0.04

Tubes

1.43

110

0.196

0.23

Intermediate s h e l l

3.71

0.510

0.60

Outer s h e l l

0.79

287 61

log

0.13

p = 0.32 lb/in."

553 l b / f t 3

Cp = 0.14 B t d 1 b - V y u m e t r l c heat

of Elaatelloy B

capacity]

pCp = 77.4 Btu/ft3-'F

94 The steady-state temperature p r o f i l e s across t h e tube annulus, Fig. 14, show l i t t l e curvature. The p r o f i l e f o r t h e case when i n t e r n a l surface e m i s s i v i t i e s a r e 0.2 was approximated, by inspection, with a s t r a i g h t l i n e . Using temperatures taken from t h i s s t r a i g h t - l i n e approximation, an average annulus temperature, was computed as outlined on 'ann Fig. 11.

eR =

-8

temperature a t R, Ri

R s Ro

= average temperature i n t h e

tube annulus

-R = radius a t which

8 = 8 R

V = volume, per u n i t height

-e,

5 =

RoQR

2 -

. RoeR RDR Ri

If t h e temperature v a r i e s l i n e a r l y w i t h radius,

Annulus

and

Fig. I1 The location of %l was taken t o be a t

fi =

25 i n .

Using t h i s value of E = 25 i n . , a simple two-shell model of t h e heat exchanger, Fig. 12, was adopted.

. & = 32.75 i n .

= 2.73 f t (see Fig. 2 )

17.13 f t a / f t

-R = 25 i n .

A1 =

= 2.08 f t (see Fig. 11)

13.1 f t " / f t

( a ) Heat generation rate

-- 77%of

t o t a l i n t h e heat exchanger

-- 129 "F-ft

( b ) Heat capacity

BtU

of height

Intermediate Shell

( a ) Heat generation rate

---

( b ) Heat capacity

23% of t o t a l i n t h e heat exchanger 287

BtU "F-ft of height (see Table 11)

Fig. 12. Radiant t r a n s f e r from t h e annulus s h e l l , surface AI, t o t h e i n t e r mediate s h e l l , surface %, i s evaluated with t h i s equation:

&+ a = Fl+a

-4

del

~ 1 ,a AI

- ea) 4

(1-1)

B U / ~

= interchange f a c t o r f o r r a d i a t i o n t r a n s f e r from

-el = average

s temperature,

O R

&d = temperature of t h e inner surface of t h e intermediate shell,

O R

o = Stefan-Boltzmann constant = 1730

-12

Btu/hr-fta

The interchange factor, F1-,2, for this model was estimated with Eq. (1-1)by using the steady-state temperatures computed with the heat rates at 104 sec after shutdown and for the case when the emissivity of all internal surfaces is 0.2. The shell temperatures, el and h, were obtained from the averaged straight-line approximation and from Fig. 14, respectively. The computed value of F1+ was 0.178 and Eq. (1-1)becomes Qil+

2 =

(0.178)(13.1)

(it - <)

(4.03 x

- $1

lO")(e;

83 #

(1-18)

The heat balance equations involved in developing the peak temperature transient are:

Total heat generated the annulus

of the annulus

Average temperature change in the annulus

Heat transferred to from the the i n t e r d ] : annulus ate -

shell

and

[

Total heat generated ij: the intermediate shell

Heat transferrez fromthe annulus to the intermediate shell

Heat capacity of the intermediate shell

- tl -

-Average temp. change in the intermediate shell c

Heat transferred from the intermediate shell to the outer shell and thence to the surroundings

(1-3)

* f

97 In t h e above, t, and t a denote any a r b i t r a r y time i n t e r v a l . These equations, involving both temperatures and heat t r a n s f e r s , were s a t i s f i e d by cut-and-try i t e r a t i o n . On Fig. 1 5 it can be seen t h a t t h e t r a n s i e n t peak occurs a t s l i g h t l y more than lo4 sec (- 3 h r ) a f t e r shutdown and drain. This time period was subdivided i n t o several increments and t h e manner i n which t h e above equations were applied i s described below. A.

From 0 t o 1000 Sec

Eq. (1-2):

jooo

U2-jooo

1000

Eq. (1-3):

u41

jooo 0

and

During t h e f i r s t 1000 sec (17 min) after shutdown and drain, the temperat u r e differences between t h e annulus and t h e intermediate s h e l l and t h e o u t e r s h e l l are small. I n these circumstances, l i t t l e heat w i l l be t r a n s f e r r e d and t h e peak temperature i n t h e annulus w i l l tend t o follow curve B i n Fig. 15. A t 1000 sec a f t e r shutdown t h e annulus is t r a n s f e r r i n g t o t h e intermediate s h e l l about 20s of t h e heat being generated within t h e annulus. Thie approximate sate was eomputqd with Eq. ( I - l a ) with tempera t u r e s taken from curves B and C on Fig. 15.

B. From io3 Sec to 5 x io3 Sec Eq. (1-2):

uz]5*â&#x20AC;&#x2122;03

u8rx103 +

103

and

u37x103 -

ueJ5*â&#x20AC;&#x2122;@ 103

During t h i s 4000-sec interval, subdivided i n t o two equal i n t e r v a l s f o r t h e computations, t h e heat t r a n s f e r from the annulus t o t h e intermediate s h e l l was taken i n t o account. It was assumed t h a t t h e heat l o s t by t h e annulus i s transferred t o t h e intermediate s h e l l . The peak temperature i n the annulus w i l l , therefore, become less than curve B, Fig. 15, by an amount proportional t o t h e heat transferred from t h e annulus and t h e average temperature of t h e intermediate s h e l l w i l l r i s e above curve C. Note t h a t , because of i t s l a r g e heat capacity and lower i n t e r n a l generation rate, t h e average temperature of t h e intermediate s h e l l r i s e s quite slowly and it i s incapable of transf e r r i n g appreciable heat t o t h e outer s h e l l . Therefore, during t h i s period, it was assumed t h a t a l l the heat transferred from the annulus t o t h e s h e l l i s retained by t h e shell. C.

After 5 x 103sec

A l l t h e components of Eq. (1-2) and 1-3) were considered. The comput a t i o n s of heat transferred from t h e intermediate s h e l l t o t h e outer s h e l l included t h e simplifying assumption t h a t t h e outer s h e l l temperature remained constant a t 1200°F, t h e i n i t i a l temperature of t h e system. The t h i c k intermediate shell i s a good heat sink and i n t e r c e p t s a l l t h e heat from t h e annulus. Moreover, it i s a poor r a d i a t o r and must develop an appreciable temperature increase over t h e outer s h e l l temperature before it can t r a n s f e r an accountable amount of heat. The outer s h e l l , on t h e other hand, i s t h i n (0.5 i n . ) and, with an outer surface emissivity of 0.8 it i s a good transmitter and r a d i a t o r of such heat t h a t it receives on i t s inner surface. I n an a c t u a l s i t u a t i o n we would expect t h e outer s h e l l temperature t o decrease, beginning immediately a t shutdown. It would not begin t o rise u n t i l t h e intermediate shell outer surface temperature had increased t o a l e v e l t h a t t r a n s f e r s s u b s t a n t i a l heat outward. Note, on Fig. 14, t h a t the steady-state temperature computation a t 104 sec after shutdown shows adjacent surface temperatures of 1940°F and 1200°F f o r t h e intermediate and outer s h e l l s , respectively. From t h e t r a n s i e n t calculation, t h e intermediate s h e l l temperature i s estimated t o be from 1750°F t o 1800°F a t t h i s time. This lower temperature w i l l reduce the heat radiated t o t h e outer s h e l l and, therfore, t h e outer s h e l l temperature w i l l be less than t h e assumed value of 1200°F.

From 5 x le t o 1.1 x la( sec t h e computations were based on time increments of 2 x lo3 sec. A t 1.1 x lo4 sec t h e rate of change of heat generation had been reduced t o a value f o r which a f i n a l t i m e increment of one hour was reasonable. The computations were terminated a t 1.46 x sec (4 h r ) a t which time it w a s evident that t h e t r a n s i e n t had turned over and system temperatures were beginning t o decrease and approach t h e steady-state values. The generalized heat balance terms, U1 t o and (I-3), w i l l now be specified i n d e t a i l .

Ub,

incl.,

lo4

i n Eqs. (1-2)

and Uz are obtained from Table 1 and/or Fig. 5 with t h e f r a c t i o n a l amounts, 77% and 23$, respectively, from Fig. 7. The remaining terms are established by i t e r a t i o n . U1

LiJ

99 The equations for Ua and Us are simple and straightforward: â&#x20AC;&#x2DC;a

heat capacity of the annulus, Fig. 12.

A&, see Fig. 11, is the change in the average annulus temperature during time interval, t,.

Note that the peak temperature in the heat exchanger will be slightly above 3 because of the gradient in the annulus. With the annulus transferring heat at the generation rate existing lo4 sec after shutdown the increase is approximately 75Â°F. = heat capacity of the intermedi-

cis

ate shell.

A& = change in the average temperature of the intermediate shell.

The average temperature of the thick (2.5 in.) intermediate shell is, during the transient, appreciably dependent on both space and time. From Fig. 6 it is apparent that most of the gamma heat deposition takes place near the inside surface. The radiant heat received from the annulus will be deposited directly on the inner surface. The net effect, of course, i s to raise the temperature of the inner surface a substantial amount above the average value, &, that must be used to evaluate u6. However, the inner surface temperature, ea, is the effective heat sink temperature used in Eq. (1-1)when radiant transfer from the annulus is being calculated. This difference between the average and inner surface temperatures was not calculated directly as a function of elapsed time; instead, the approximate transit time for heat flow through the slab was estimated. Specifically, the case considered was an infinite slab, 2.5 in, thick, of Hastelloy N, with one face jnsulated. The uninsulated surface sees a step increase in temperature. From Fig. 10-2, p. 235, and related text material in reference 28, it was estimated that the average temperature will lag the inner surface temperature by approximately 1000 sec. This l o c a l transient effect in the intermediate shell was considered during the computations, The total heat transferred fram the annulus during any interval, (1-1).

- tl , is obtained by integrating Eq.

100

As written, this expression presupposes the desired solution, 8%(t) and Therefore, it was assumed that, during each time interval, the h(t). temperature changes in both the annulus and the intermediate shell progressed linearly, i.e.:

-h(t)

= A

b(t) = M

+ Bt + Nt

(1-51 ( 1-61

Also, at the temperatures and temperature differences involved in this calculation, the fourth power temperat~edifference in Eq. (1-1)can be approximated with negligible* error, v i z .

in which

In terms of Eqs. (1-5) and 1-6)these can be written, =F+Gt 'avg A0 = J + Kt

F = 1/2 (A

+ M)

G = 1/2 ( B

J = (A

= 1/2

( I-7a (I-8a)

[Sum of the surface temperatures at t1]

N) = 1/2 Sum of the slopes of the surface temperature curves during the interval, (ts tl).

- M) =

[Temperature difference at tl]

K = (B- N) = Difference of the slopes of the temperature curves during the interval, (& - tl). and Eq.

1 d

(1-4)becomes

*For example, if & and S, are 1800'F and lbO째F, respectively, the error introduced by using this approximation is l$.

101

Bd '

After i n t e g r a t i o n and some algebra,

+,K i n which t = ( t 2

G3t6 {+ 6 FGat4 + l?Gt3 5

(I-4b)

- tl), hr.

For t h e range of temperatures and t h e time i n t e r v a l s used i n t h i s problem t h e f i r s t two terms i n each s e t of brackets i n Eq. (I-4b) are small enough t o omit and

This equation was used t o estimate the t o t a l heat transferred from t h e tl, i f t h e temperature v a r i a t i o n s annulus during t h e time i n t e r v a l , t a a r e l i n e a r during t h e i n t e r v a l as shown i n f i g . 13.

ORNL DWG. 71-601

Temperature,

e, "R

Average t z p e r a t u r e of annulus, el = A + ~ t .

Temperature of inner surface of intermediate shell, e2 = M + N t

Elapsed time.

Fig. 13.

102 The heat t r a n s f e r r e d t o t h e outer s h e l l from t h e intermediate s h e l l i s estimated by t h e familiar equation,

= AisF

Ais

Btu/hr-fta f't of height

18.45 f t a / f t of height, t h e area of t h e o u t e r surface of t h e intermediate shell.

Aos = 18.80 â&#x201A;Ź'ta/ftof height, t h e area of t h e Xnner surface of t h e outer s h e l l . = t h e temperature of t h e outer surface of t h e i n t e r mediate s h e l l , OR.

s 1660Â°R (I2OO0F), t h e temperature of t h e inner surface of t h e o u t e r s h e l l . E

F = 1 + C(1

- E J = interchange f a c t o r f o r r a d i a t i v e

t r a n s f e r from an i n f i n i t e l y long convex inner shell concentric w i t h an i n f i n i t e l y long o u t e r s h e l l , when t h e emissivities of both surfaces are equal. = AiE/Aos 6 =

emissivity

u = .the Stefan-Boltmann constant.

REFERENCES W

1. MSR Program Semiann. Progr. Rept. Feb. 28, 1969, ORNL-4396, p. 60, Sect. 5.7.3 -'Heat Removal Systems, and p. '62, Sect. 5.8 Distribution of Noble-Metal Fission Products and Their Decay Heat.

J. R. Tallackson, Estimated Temperatures Developed by Afterheat i n MSBR Primary Heat &changer, SK-4304, Rev. 3, ORNL CF 67-9-1 (Sept. 27, 1967).

E. S. Bettis and Roy C. Robertson, '!The Design and Performance of a Single-Fluid Molten-Salt Breeder Reactor," Nuclear Applications & Technology, Vol. 8, pp. 190-207, Feb. 1970.

MSR Program Semiann. Progr. Rep%. Aug. 31, 1969, ORNL-4449, p. Sect. 5.7 Gamma Heating i n MSBR Heat Exchangers.

J. R. Tallackson, Calculations of Heat Deposition i n Empty MSBR Primary Heat Exchangers by Gamma Radiation from Noble Metal Fission Products, ORNL CF 69-8-27 (AUg. 14, 1969).

56,

6. H. E. McCoy, private communication. 7. MSR Program Semiann. Progr. Rept. Aug. 31, 1969, ORNL-4449, p. 183,

Sect. 18.2 S t a t i s t i c a l Treatment of Aging Data f o r Hastelloy N, and p. 191, Fig. 18.10.

R. E. Cleary, R. Ehanuelson, W. Luoma, C. Amman, Properties of High Rnittance Materials, NASA CR-1278, United Aircraft Corporation -(Feb. 1969).

R. B. Briggs, "Heating i n Graphite of a 2000-Mw(e) Reactor by Decay of Fission Producks After Shutdown, 'I MSR-68-8 ( I n t e r n a l Correspondence)(Jan. 8, 1968).

11. Robert Siege1 and John R. Howell, "Thermal Radiation Heat Transfer," The Black Body Electromagnetic Theory and Material Properties, Vol. I; y01. I1 ,

NASA, SP-164,

12.

â&#x201A;Ź

Ir,

E, M. Sparrow and R. D. Cess, Radiation Heat Transfer, Brooks-Cole Publishing Co., 1966.

13. H. Leuenberger and R. A, Person, Radiation Shape Factors f o r Cylindrical Assemblies, ASME Paper 56-A-144, Jan. 1955.

104

14. Carol J. Sotos and Norbert

0. Stockman, "Radiant Interchange Factors and L i m i t s of V i s i b i l i t y f o r D i f f e r e n t i a l Cylindrical Surfaces w i t h P a r a l l e l Generating Lines," NASA, L e w i s Research Center, Cleveland, Ohio

a, I

0. H. Xlepper, Radiant Interchange Factors f o r Heat Transfer i n P a r a l l e l Rod Arrays, ORNL TM-583 (Dec. 26, 1963).

16. J.

S. Watson, Heat Transfer from Spent Reactor Fuels During Shipping: A f- _o- r Predictinn Distribution i n Fuel _ _ Pronosed _ - -~ _ _ _ _ Method __ __ - -v TemDerature ~

Bundles and Comparison with Ekperimehal 'Data, ORNL-3439 (May 27, 1963)

D. C. Hamilton and W. R. Morgan, Radiant Interchange Factors, NACA TN 2836.

18. Norman T. G r i e r and Ralph D. Sommers, View Factors f o r Toroids and Their Parts, NASA TN D-5006.

James P. Campbell and Dudley G. McConnell, Radiant Interchange Configuration Factors f o r Spherical and Conical Surfaces t o Spheres, NASA TN D-4457.

20.

A Feingold, Radiant Interchange Configuration Factors Between Various Selected Plane Surfaces.

21.

E. M. Sparrow, "On t h e Calculation of Radiant Interchange Between Surfaces," Modern Developments i n Heat Transfer, edited by W. Ibele, Academic Press, 1963.

22.

D. P. DeWitt, M. C. Muinzer, gram f o r t h e Compilation and Data, NASA CR-1431, prepared Center, Purdue University, W.

and R. S. Hernicz, A Comparative ProAnalyses of Thermal Radiative Properties by Thermophysical Properties Research Lafayette, Indiana. .

Joseph Poggie, "Radiation Characteristics of Materials, Engineering, p. 205 (Sept. 1953).

24.

G. G. Gubareff, J. E. Janssen, and R. H. Torberg, Thermal Radiation Properties Survey, Honeywell Research Center, Minneapolis-Honeywell Regulator Company, 1960.

B. C. Garrett, _Bench T e s t of Rod Bundle from t h e IPS System,ORNL CF 65-11-68 (Nov. 30, 1965).

26.

H. Etherington, Nuclear Ehgineering Handbook, 1 s t Edition, 1958, Sect. 1-1, 14.1 "Steady S t a t e Conduction," p. 1-53 t o 1-57 i n c l . , McGraw-Hill.

Product

4 z

27.

D. L. McElroy and T. G. Kollic, "The Total Hemispherical Einittance of Platinum, Columbiwl$ Zirconium, and Polished and Oxidized INOR-8in the Range 100" to 12OO0C," Measurement of Thermal Radiation Properties of Solids, Paper No. 38, NASA SP-31. A symposium held at Dayton, Ohio, Sept. 5-7, 1962.

28. P. J. Schneider, Conduction Heat Transfer, Addison-Wesley Publishing Co., Inc., 1955.

U t

i t

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