ML20212N398

Design Basis Transient Analyses for Low Power Research Reactors

ML20212N398
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Site:	Rensselaer Polytechnic Institute
Issue date:	03/03/1987
From:	Harris D, Jones O, Wicks F RENSSELAER POLYTECHNIC INSTITUTE, TROY, NY
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/. ..

RE'ERENCE F 2.

1 DESIGN BASIS TRANSIENT ANAL.YSES FOR LOW POWER

. RESEARCH REACTORS

' D.R. Harris, 0.C. Jones, F.E. Wicks. A.B. Harris * ,

4 F. Rodriguez-Vera, and C.F. Chuangt Rensselaer Polytechnic Institute Nuclear Engineering Department Troy, New York 12181 A85 TRACT The identification and analysis of design basis acci-dents is an important step in the nuclear. licensing and relicensing process. .Large computer programs, such as RETRAN, have been developed for transient analysis of po-i wer reactors. These programs have also been applied to low power pool type reactors. The types of accidents and the response and feedback mechanisms are substantially

. different for low power pool reactors, and in addition, new transient thermal hydraulic data continue to become available. Thus it is worthwhile to develop an improved program for performing transient analysis on low power reactors. This paper describes the development and appli-cation of such a proijram to the analysis of ATWS accidents in pool reactors. "he model includes recent developments i in transient thermal hydraulics models and correlations.

, The numerical solution method employs Runge-Kutta time stepping methods. The program has been benchmarked i e against a set of Gaussian, Nordheim-Fuchs, and SPERT type

! E bursts.

8m I

888 1. INTRODUCTION 88 4

S8 Safety Analysis Reports for low power research reactors fre-4 g quently are required to deal with ATWS power bursts. These bursts j gg are assumed to develop from a variety of initial conditions aid as a l

SS

[ Q2

• Purdue University

gg, t Argonne National Laboratory L___-____

+

2

{

result of a variety of reactivity insertion modes. It is customary to reduce this variety of special cases by bounding them by design basis transients. Design basis transients for low power reactors can 2'

result in rapid power increases by many orders of magnitude, and the main burst can be followed by complex further events. Such tran-sients generally cannot be examined by deliberate experimentation on the research reactor involved. Instead, reliance must be placed on computer simulations verified against out-of-pi.le measurements and against the limited in-pile measurements such as the SPERT tests.

, Computer programs for analysis of ATWS events in low power re-search reactors should take into account relevant developments in transient thermal-hydraulics. In addition, they should be efficient

! and flexible so as to permit certain improvements in SAR analysis procedures. First, multiple cases then can be run to provide the re-sponse surface information required for probablistic analysis in view

. of uncertainties in nucleonic and thermal-hydraulics information.

Second, numerous realistic situations then can be run both to delimit .

L design basis transients and to identify relative hazards. The compu-J ter program RKGB, developed at RPI with these objectives in view, is described here nd is illustrated by results for a low power pool type facility. I

2. NUMERICAL METHOD The RKGB code [2], in FORTRAN IV, is designed to analize desigg

. basis transients in research reactors. It is based on Runge-KuttaLJi time-scepping of any number of dependent variables on a set of fixed time meshes. The functional dependences, fj (t,yl,y2,...) appearing in the general equations l

l dy3 (t) dt "I j (t, yt, (t), y2(t), ..., y (t)), j=1,2,...,j , (1) are coded by the user into subroutines. The code has been benchmark-ed against a[gt of Gaussian Fuchs bursts .

bursts, In these cases and the against analytic a set ofare solutions Nordheim-known, j and it is found that the code is remarkably accurate. A Runge-i Kutta4111L33 algorithm, also available in thp Gode, was almost as accurate. An Adams-Moulton predictor-correctorL3J option in the code i

became unstable for the burst tests. The normal Runge-Kutta of order four is used in the following, i

3. NEUTRONICS l

The dependent neutronic variables are six delayed neutron pre-

> cursor populations and the reactor power. These satisfy, in usual terminology,

j, . .

3 dC t h=6 P-AC, 33 .j = 1,2, . . . ,6 , (2) and -

h=8-8 P+ AC 33 (3)

The reactivity p is the sum of the imposed reactivity, i

9 , = p, + pit, 4

(4) representing a step (p1=0) or a ramp (p1

• 0), and the feedback I

reactivity, P fb = ag f,- T,) - f a2 Q - T,) -a 3 f -T,)

-a 4 [v g p,,-6Vfc}

• fc . .(5)

Here the first two terms are from the measured core temperature coef-ficient[g(

reactor 1J), reactivity (particular values will be given for the RPI h = ag - a 2T , (6) assuming that the bulk of the measured isothermal coefficient is from water temperature change. Th~e next term is a fuel temperature ef-

fact, where the fuel temperature coefficient of reactivity a3 is not i well-known for highly enriched fuel. A noise analysis experiment on the Kyoto University Reactor (with MTR type }lg1) yielded about -2 x 10-5/ *C for the fuel tenerature coefficient J. Computed values of i a3 are smaller by about two orders-of-magnitude. The fuel plate ex-l pansion effect contributes only about -7 x 10-7 /*C. The final part L of equation (5) involves the measurable void coefficient of reactiv-ity, a4, per cm3 of void. Here yo represents the void volume frac-i tion, V the initial water v61ume, and 6Vfc is the change in fuel /cla5ois volume from its thermal expansion.

> As the power increases in a burst, the temperature will rise first in certain parts of the core. Assuming that the temperature

rise in an assembly is proportional to the power per fuel plate in i that assembly, a' weighted average void. coefficient of reactivity can

be calculated for the core. The core weighted average void coeffi-cient coefficient is foundfor tothebeRPI only core. 67, ['LJess Neither than the core linear average void void coefficient is known

+

with this degree of accuracy. Thus, it is concluded that the core i linear average void coefficient and the isothermal temperature coef-ficient can be used with little error.

7 4

4 THERMAL-HYDRAULICS Integrating the thermal diffusion equations over the fuel meat (UO 2 -stainless steel cermets) and the clad (stainless steel) for a typical fuel plate one obtains:

1 -

d7 dt f' ~ ^p 9fc} f* I}

dT .

dt = (Fc ' ~ ^p 9fc ~^p 9 I c* (}

where Tf is the average fuel temperature, Tc is the average clad tem-

- perature, Fa is the fractional power delivered directly into the fuel i (94%), Fe qs the fractional power delivered directly to the clad 1 (3.5%), and 2 qfe is the the heat flux (w/ m2 ) from fuel to clad. In addition, 2 qcw is the the heat flux (w/cm ) from clad surface to the water Apis the heat transfer area (cm2 ), Cf is the total heat capa-

city of fuel (J/*C), and C is the total heat capacity of clad (J/*C). Equations (7)and($)canbeconsideredtobeforthewhole core when P is taken to be the core power (watts) and A the heat

! transfer area of all fuel plates (cm2 ). Thefinaldiffere$tialequa-tion solved by the Runge-Kutta is the energy balance equation for the l

water, dH

" + '

dt w ^p 9

• where F fractional power delivered directly to the water (2.5%); wHw is the is the enthalpy of the water initially in the core.

Suppose we have just stepped ahead to a new time at which we have new values for the ten (10) dependent variables1 C ,2C , ... C 6e P. Tf, T and e H .wWe first compute the water temperature Tw from the enthalpy by using a quadratic fit for the specific enthalpy Hw /M w =C1+C2 w T+C 3T2w , (10)

The of T . quadratic In eq. fit(10),M is quite accurate and of permits water inrapid the core. computationIf Tw is w

computed by eq. (10)w to isexceed the mass 100*C, it is set to 100*C and the vapor quality and void fraction vg are computed. As the burst develops, void formation and fuel expansion will expel some of the water initially present in the core. These (weak shock) displace-ments take place rapidly and are not modeled directly. The burst '

then is assumed to take place at constant (atmospheric) pressure, and the water mass being heated is the initial mass. From the bulk water temperature Tw and the void fraction vg thus determined, and from the new value of T f, the reactivity is computed from eq. (5). The

[ .f

( 5 I

computation of qfe and q" in eqs. (7), (8), and (9) requires exami-nation of the temperature profile. For a steady-state condition the temperature profile has a quadratic dependence on position in the fuel plate, and it can be shown that Tg - T, + q 'Aj/6k g q , (11) cw 1 Ac

! h,*{

e and

~

f C AC l 9fc Af Ac +9 cw E' ( )

k in terms of the current values of the lumped parameters Tf , T and c T. Here, subscripts f, e and w refer to fuel, clad, and water, w511e af and Ac refer to the fuel and clad thicknesses. The fuel l centerline and fuel-clad interface temperatures are computed at this i point as is the clad surface-to-bulk coolant temperature drop. The

! departure from steady-state conditions is considered shortly.

~

The heat tragsfer coefficient Hcw from clad to water exhibits three regimes:LOJ (a) Heat transfer into th liquid through a by natural convection;[6] e l

laminar f Subcooled pool

, bgiling;Lgog13. andset(c) upSaturated pool boiling.E73(b) As the heat flux and the temperature drop (T s in ate transi-t on or- film boiling is eventually ew - T atw) encountered qkw , [8,hr$0.1,and fuel damage may occur.

asurements of saturated pool boiling have been made[ga 12 in t

. and a successful correlation is that of Rohsenow[6,he 7,11]steady-state,

. t Pt 9 cw " "th fg[9(g A *8v]t/2 [ ,7]3 (T , - Tsat)3 . (13) sf fg r g i This result is compatible with Reference, Ell 3 where smooth and rough

! stainless steel-water surfaces are considered (as well as many l others).

Tbg p ect of subcooling on pool boiling is not fully under-

' stood,LO 1{(J and contradictory results have been obtained, usually in f the vicinity of eq. (13). We use eq. (13) in _t_he subcooled boiling l region also, replacing (Tcw - Tsat with (Tew - Tw).

l

. ,7 -

]

. l l

l 6 >

t At heat fluxes below a few w/cm2 no bubbles are observed,[6] and the l

heat' transfer depends on details of the natural convection. We use i i l q" = 0.355906

(T, - T,)e/7 wfem2, (14) h 4

in the non-boiling region, an expression which joins eq. (13) at a value of (Tew -Tw ) equal to 5'C.

i 1

) In summary, eqs. 13) and (14) are used for the three heat I

transfer regions (A), 8) and (C) identified above. Equations (13) j and eqs. (13) or (14), both relate qcw with temperature _ drops and require iteration for their solution. Thus it is convenient to i i express the heat ransfer coefficients implicit in eqs. (13) and i (14),intermsofq},. ,

ihen -

l hg , = 0.261667(q" )2/3 for q > 2.23954 w/cm2, (15) _

and' hg , = 0.404966(q" )1/s for q < 2.23954 w/cm2, (16)

Now eq. (11) and either ieq. (15) or eq. (16) are solved simultaneous-ly by iteration. For saturated pool boiling no iteration is

! required.

i

} The actually transient nature of the burst affects this analysis in two way It has i been shown I3]that First, thethe temperature asymptotic profiles profiles, are are which modified.

obtained by  ;

solving the thermal conduction equations with an exponential timg de-l pendence e w t, reasonably represent actual transient profilesi.13],

) The asymptotic temperature profile in a uniformly heated plate, for

example, is ies cosh Df x q

T=T,+c#g dw El ~

• IIII pf f cosh x f

i

! In contrast with the quadratic dependgnce in eq. (10), the asymptotic

! temperature profile is flatter, and qcw is found to be corresponding-ly less for the asymatotic case thag for the, steady-state case (for a

wall temperature). r or example -

4 to be 0.87 for a 0.074 sec period (qcw)asy/(9cwIstea and to be 0.4 p for ais0.074computed sec to eriod. For the actual fuel-clad two-region plate the ratio '

l q"w)as /(9cw)s is even less. However, the temperature varia-tion in the fuekead$ p ate is not usually severe. Moreover, a reduced i

'.-,-.~ ,, --- . - - . - . - . _ . . - - - -

T .

3 . .

7 heat transfer in the transient case will be accompanied by a higher fuel temperature, which acts to increase the heat transfer.

Second, the clad-t heat transfer correlation should be for the transient case.L e btN3 The measured results for saturated e

pool boiling are fragmentary, but are not far from eq. (13). Appor-ently the principal transient effect is to delay the onset of transi-tion boiling by aboug half-an-order of gtitude past the steady-state value for qcw of about 210 w/cm2 ,[9, 3- Finally, it is noted that the reactivity feedback expression, eq. (5), takes account of void produced by vaporization of the bulk water, but it does not include temporary voids produced in the heat transfer layer in sub-cooled pool boiling. An upper limit on the average void thus pro-vided is givon by i

a, =

< ^a dghfg

+ rg

). ns) l, Here the denominator is the ener required to form Icm3 of steam bubbles,andisapproximatelyequafto1.35J/cm 3 over a wide range-l (above about 10 cm radius). For an assumed 7

of bubble bubbig radius rfo the average void can be computed.For example, lifetime for qc, = 10 w/cm2 Ind To = 1 ms, eq. (18) yields about 100 cm3 which contributes less than 10-3 in reactivity.

j 5. RESULTS Two RKGB transients for a low power pool type facility reactor are shown in figures 1 and 2. The first case corresponds to the transient following a 0.0117 step reactivity insertion. The second a case is for a transient from a 0.001/second ramp reactivity inser-tion. Bulk coolant boiling shut the core down in this case, but earlier feedback effects were responsible for topping-out the burst.

2 REFERENCES

1. P.R. Nelson and D.R. Harris, " Reconfiguration of the RPI Criti-cal Facility to Lower Critical Mass," Nucl. Tech., 60, 320 (1983).

l 2. D.R. Harris and A.B. Harris, "RKGPC, a Runge-Kutta-Gill program

! with an Adams-Moulton Predictor-corrector Option," RPI-NES-550,

Feb (1983).

i 1

i i

t

8

3. R.L. Burden, J.D. Fai res, A.C. Reynolds , Numerical Analysis, Prindle, Weber and Schmidt, Boston (1978).

4. G.E. Hansen, " Burst characteristics associated with a slow as-sembly of fissionable materials," LA-1441 (1952).

5. M. Utsoro and T. Shibata, " Power Noise Spectra of a Water Reac-tor in Low Frequency Region," Nucl. Sci. and Tech., 4, 267 (1967).

6. W.M. Rohsenow, " Boil.ing," Chap.13 of Handbook of Heat Transfer, W.M. Rohsenow and J.P. Hartnett, Eds., McGraW-Hill, - New York, (1973).

7. W.M. Rohsenow, "A Method of Correlating Heat Transfer Data for Surface Boiling Liquids," Trans. ASME, ,74, 969 (1952),.

8. N. Zuber, "On the Stability of Boiling Heat Transfer," Trans.

ASME,80,711,(1958).

s

9. F. Tachibana, M. Akyana and H. Kauwanura, " Heat Transfer and Critical Heat Flux in Transient Boiling," Nucl. Appl., 5,3 (1968).

10. A. Sakiran and M. Shiotsu, " Transient Pool Boiling Heat Trans-fer," ASME Preprint 74-WA/ht-41 (1974).

11. R.L. Vachon, et al., " Evaluation of Constants for the Rohsenow Pool-boiling Correlation," J. Heat Transfer, 90, 239 (1968).

12. R.T. Lahey, Jr. and F.J. Moody, The Thermal-Hydraulics of a Boiling Water Reactor, Am. Nucl. Soc., LWBrange Park, Illi.,

l (1977), p. 79 et seq.

13 W.O. Doggett ,and R.H. Shultz, Jr., " Transient Heating for a Con-vection Cooled Heterogeneous Nuclear Reactor," Proceedings 1961-62, Boulder, Colo., Am. Soc. Mech. Eng. (1975).

i

'l k

4 i

- . , _ , _ . . _ _ . _ _ . . .. . _ . ~ . _ . _ _ _ _ _ . _ , , _ _ _ _ . _ _ _ _ , _ _ _ , _ . _ _ _ _ _ _ . , _ _ . . , %____ _ . _ , , , , _ . . _ _ _ _ _ _ . . , _ . , _ . . , _ . _ . _ , _

9 8

10 , .

i 7, ,

10 i

10

6. cnw ,

t

y .

I b

5- .

10

% feet meet samerature 300 200 ~*

4 . 100 10 r .

sett water Tameerature l 60

1. {

l . , q 20 0 ar clas eter win temperatur. 10 10- 8 f 1 i .- i 6 4

2 2 . . . . . ,

1 10 .

0 0.1 0.2 0.4 0.s 0.8 1.0 Tfas (ses]

Figure 1. Transient following 0.0117 step reactivity insertion. RKGB calculation.

~

~

10 n .

I "

10 0.001/SICOND EM P REACTIVITY IESIITION -

nuns cALCouTI0di QUASISTATIC MDDEL E

0 "

10 @ .

s

. I e i

E e

i q" CIAD-UAIII 5

- 400 g N - 200 'e TEMPERATURE

~

100 5 so u varam E

" 40 l -

20 l

_ _ , - ~ ~ ~ -~ .o,ot 5 IMPOSED k

i REACTIVITY g l C II 102 i '

O.0G1 0 5 to TIME (SICONDS)

Figure 2. 0.001/second ramp reactivity insertion.

RKGB calculation. Quasistatic model.

i

f REFERENCE 4.

PLATAB, A FORTRAN CODE FOR CALCULATION OF FEW-GROUP EFFECTIVE i

w DIFFUSION PARAMETERS FOR ABSORBING SLABS l

D.R. Harris Rensselaer Polytechnic Institute Troy, New York ,

ABSTRACT PLATAB is a Standard Fortran Code for calculation of few-group effective diffusion parameters for absorbing slabs. The well-known alpha and beta parameters are calculated at each energy of an integration energy mesh. Cross sections may be input on a cross section energy mesh, or a 1/v option any be selected. One or two weighting fluxes may be input on a weighting flux energy mesh, or a Maxwellian weighting option may be selected. When both cross sections and weighting flux meshes are input, the integration mesh is the union of the two. flux i spectrum is input; In theisresonance this mitiplied region, by the a slowly varying weighting factor weighting (1+aF)- in the i narrow resonance approximation. Here F is an input parameter characteristic l of the adjacent medium.  !

1. INTRODUCTION 1

Few-energy-group neutron dif fusion theory is widely used in thermal reactor  !

design and has boon shown to be reasonably accurate in this application. The required degree of accuracy is achieved only when effective diffusion parameters are used in strong absorber regions such as control rods. These effective diffusion parameters, specifically effective diffusion constants and absorption cross sections, depend on the optical thickness of the absorber and on the special mesh point distribution in and near the absorber. Because the optical thickness of the absorber region depends on neutron energy, it is l necessary to average over appropriate neutron weighting flux spectra.

I Techniques for computation of few group equivalent, diffusion parameters have been under development for a third of a century,1- and numerous detailed problems reasin to be solved. There is, however, a basic methodology which has been widely applied with good results. This methodology, which is '

I described in Section 2, is used in the PLATAB code. The code is limited to plate absorbers. Options are available for thermal and non-thermal calculations, for various input forms for weighting spectra, and for various input cross section data. Input and output are described in Section 3, and )

examples of input and output are displayed in Appendices. l

2. METHODOLOGY The PLATAB code deals specifically with mesh point centered equivalent diffusion theory. parameters for strong absorbers in place geosecry. The basic theory is described first for one energy transport, then the many energy theory is treated, and finally special techniques of practical utility are considered.

I

_ _ _ _ _J

p .

1. ~

2 ,

Consider a strong absorbec slab imbedded in other materb1's in slab geometry (Figurs.1).- Immediately outside the absoebing slab a P1 approximation is I assumed so the directed neutron flux at ~ position x and between, angles 9 and 6&d8 from the x axis is .

/ . i

$(x,u) = f 9(x) + f j(x)u.' (1) i Here, u represents cos 0, $(x) is the angle 'intearated flux, and j(x) is the not current in the x-direction at position x. For an absorhinr non-scattarine e

slab of thickness 2t (see Figure 1),

.. - i

$(c.u) = $(-c,9) e " for uc(0,1), *

(2)

$(-t, u) = _$( c , p) e " for uc(-i ,0) . (3)

Here T is the optical thickness of the slab, 2tE, in terms of tne eacroscopic cross section for absorption. The emergent flux from the slab can nardly be P1 in angular shape, but at least one can match the nat currents at the right and left faces of the slab, duu(h$(t)+fj(t)u)+ dup (f$(-c)+fj(-t)u]e

~

j(t) M , (4) o T

j(-t)=[1 du p(f- $(t) + f j(t) u] e /u +[1 du p(f $(-c) 4 f j(-c) ug'-

The angle integrals are carried out .in terms of the E-functions, l E e2(T) " f d" """ '

(6I Equations (4) and (5) constitute two equatl.)no reisting the four quantities

$+, $ , -j+ and ,

$(t),

j, $(-c)ively., j(t),Thus, re::pect and j(j +t) andwhich j areit entering is custetury currents. to relabel After some algebraic rearrangement, we obtain two customary equations for the surface

, fluxes and currents

~ ~ ~

j = h (e-8) $+ + h (e+8) $ . (7) j+=f(a&B)$*+h(a-8)$,

~

, , (8)

Here the a and 8 parameters Japend only on the optical thickness of the niab i and are defined by, 4 -

1-2E3 C )

" " 2(1+3E4 (T)]'  ; ,

r ,

._ __ .-c,e-----,e -re-,,-... -

,.---9 -r-+~ w- r ~-,

? .

4 3

(

1+2E3 (T) l G0,

" 2[1-3E4(T)J Equations (7) and (8) could suffice as boundary conditions to determine ,

analytic . solutions - to the dif fusion equations in the two regions immediately .I outside the slab. I i I We wish, however, to use a differenced solution at mesh points xn

• Kl* *1***** l in the absorbing slab; here x0 *"d *N are n the lef t and right faces of 'the i slab. For uniform mesh intervals x -x,_t = h the diffusion equation in the slab can be differenced as

$n+1~ *n++n-1

, D h' ~U n

= 0, G1) where the local source is assumed to be zero. This differenceggquation is

- easily solved by-assuming a trial solution of the form $ = ae ,

$3 " 2 sinh cN I(#0* ~#N* (*N ~ 'O ' )* *

(1 )

1 Here e is found to be Dt2 e = arcosh (1 + 2D }* (13)

Now the net current at the lef t face of the absorbing slab can be computed as

. '$ 1 ~ $_1 j = -D *

(14) 2h l

Inserting the solutions from Eq. (12) into Eq. (14) it is found that,

' ~

j =D** " (-$ Ncsch cN + $0" I* O)

Comparing Eq. (15) with Eq. (8), and noting that $+ " 4N and $~ " $0, it follows that 0

D* " coth cN = " , (16)

-D

• csch cN = * { .

(17)

Dividing these two equations it is found that c=harcosh O

g

". (18)

From Eq. (16), the eff ective diffusion coef ficient in the strong absorber is D = (8 + a)h tanh cN, (19)

s 7

~  !

and from Eq. (13), the effective absorption cross section in the strong absorber is I= (cosh c - 1) (20) g The computation.al algorithe. now is: (a)calcula$ te a and 8 from the absorber K properties by Eqs. (9) and (10), (b) calculattic from Eq. (18), and (c) calculate the desired effective D and I from Eqs. (19) and (20).

These one energy results are extended to many energies by noting that summing

, Eqs. (7) and (8) over energy gives the energy integrated currents

~

[dE j+ = f <a + $>, [dE $+ + f <a - 8>, [dE $ ,

~ ~

[dEj = f < a - S>, [dE $+ + f < a +8>, [dE $ , (21) at the right and lef t faces of the absorber in terms of the energy integrated fluxes if and only if the effective values"of a and 6 are flux weighted averagas. Specifically <a + 6>+ and <a ' 6>+ are svarages weighted by the flux $+(E) on the right of the absorber, wP:lle <a + 8)_ and <a - S>_ are weighted by the flux $~(E) on the lef t .of the absorber. If the neutron flux spectra at the right and lef t faces of the absorber are identical- then it is l reasonable to define

<a> = [dE a $,/[dE$,, ,

(22)

I and

=[dE8$/[dE(y, (23) in terms of a weighting flux $w.

The development from Eq. (11) to Eq. (20) is now regarded as applying to a l coarse energy group, e.g. , the thermal neutron group or an epithermal neutron

! group. Thus we compute e, D, and I from Eqs. (18-20), replacing a and 8 in these equations by <a) and .

If the flux spectra on the two faces of the absorber are not identia l then a difficulty arises. In the discussion from Eq. (14) et seq we ignored the right hand currgnt j+, but by the same methods as for j- it is found to be

~

(j+ = D hi+1 m-1)'

2h j+ *

=

" [- $0each cN + N c th cN]. (24)

Comparing Eqs. (8) and (15), and Eqs. (7) and (24), yields l

. 5 h sinh c coth cN = f <a + 8) _, (25) 4

- f sinh c esch cN = < a - S>, , (26) sinhceschcN=f<a-8) _, (27) h sinh c coth cN = f <a +8>,. (28)

Differ nt neutron flux spectra $+(E) and $_(E) will result in diff erent values o f < a 2 $>+ and < a i 8)_ , so these four equations are incompatible. We conclude that a single set of diffusion parameters D and I cannot be chosen in this case.

Attention now is turned to techniques of practical utility. Many calculations using these techniques have been made and compared with Monte Carlo calculations and experiments.1,2,5 All the techniques have. approximations and '

useful results of ten involve compensating errors. In the thermal neutron range many absorbers of interest have large optical thicknesses, so both a and 6 are near _0.5 and are relatively slowly varying with energy. Thus the energy averaged <a) and , Eqs. (22) and (23), are found to be insensitive to choice of weighting flux spectrum $w. It is customary to use as $w, not the flux spectrum at the absorber surface, but instead a region-average flux spectrum somewhere naar the absorber, e.g. , a simply-buckled flux spectrum for a fuel region near a control rod blade. Moreover, when the regions to lef t and right of the absorber plate are quite different, e.g. , the outside (fuel) and inside (water) regions of a box-type control rod, an average of the asymptotic spectra of these regions may be used for $w, the weighting

• spectrum 2 The PLATAB code requires, for averaging a and S in the thermal region, one or two input weighting spectra; in the case of two input weighting j spectra the two are averaged to obtain $w.

In the epithermal range, one must deal with the possiblity of strong resonances in the absorber slab such as occur in hafnium and cadmium.

Consider a symmetric uniform moderator region around the absorbing slab. Then ,

the flux per unit lethargy is, in dif fusion tneory and in the narrow resonance '

approximation, E

$(x,u) = f - A cosh gf!(H-x), (29) a m where S is the slowing-in source per unit lethargy. Em is the moderator cross section, A is a constant, Da is the moderator diffusion coefficient, and H is i

the half-thickness of the moderator region plus that (t) of the absorber.

Then at the absorber surface, x = t,

+ T 9

=h-Acos1[m(H-t),

m (30)

I l' .-- . - . - - - .- - - _ _ - - . - - - - . . --. -- -- --.

6 e

j+ = ,A sin [ a(H - t). (31)

But the ratio j+/ $+, i.e., a is determined by the absorber properties.

Combining Eqs. (30) and (31) yields 4

aF)' (32)

Im (1 where F is (DmI m)-1/2 coth /I /D If the moderator cross sections are independent of energy then thI eIer(H-t).

simply proportional to (1 + aF)-1, gy dependence where of the weighting F is a moderator dependent function constant,is

+. ~ 1 .,-

As a flux per unit lethargy, this $w is nearly constant when a is small (small optical thickness). For large optical thicknesses, e.g., resonance peaks, then a becomes approximately 0.5 so $, saturates at (1 + F/2)-1 Some authors recommend averaging 8 by (1 + SF)-1, but Eqs. (21) show clearly that 8 as well as a should be weighted by $+, e.e., (1 + aF)-1 It also has been pointed out4 that F may be approximated by /T independent of the moderator. In PLATAB, an energy dependent F function is input; then Eq. (32) is used as weighting function.

For a completely black absorber, Eqs. (9) and (10) show that a and 8 both become 1/2. However, from the known extrapolation distance (d = 0.7104/Itr)

at a vacuum boundary it is easy to show that a should be 0.4692. For strong l absorption (T+=), -

-T E (T) + *T (1 - + ...),

(33) so 1 e" 18 a = y [1 - 5 T (1 W)) , (34) 8=h(1+5* (1 - )]. (35)

Thus to a good approximation, 1

a6 = p. (36) l The limit for 8 corresponding to a = 0.4692 then is 8 = 0.5328. The " dirty i blackness" technique consists of correcting a and 8 from Eqs. (9) and (10) by multiplying both by 0.4692, and it is claimed that this leads to improved l results for strong absorption.1 PLATAB provides for optional and separate input correction factors for a and 8

7 Henry has suggested that, from Eq. (36), S be calculated approximately as 1/4a for strong absorption. This is not done by the PLATAB code which computes a and S correctly from Eqs. (9) and (10), obtaining the required E-functio,ns from algorithms 5.1.53 and 5.1.54 in Ref. 6 plus recursion. No numerical difficulties have been encountered in this computation.

Extensive 2-group,2 4-group,2 and 5 group 1 dif fusion calculations using l equivalent diffusion parameters have been made for strong absorbers with generally good results, both for slab geometries and K-Y geometries when compared against Monte Carlo calculations and against experiment. One conclusion of interest is that 2-group calculations appear to be less sensitive to explicit inclusion of heterogeneities next to the absorber than are 4-group calculations. All investigators have noted the presence of compensating errors from use of the methodology.

3. Input and Output Input and output are identified by FORTRAN variable names. All variable names are defined in the variable dictionary listed in Table 1. Three logical input units are required for input. Job input data, read from anit 5, are listed in Table 2. Weighting flux spectral data, read from unit 7, are listed in Table

3. Cross section data, read from unit 3, are listed in Table 4. Both weighting flux data and cross section data are input by increasing energy.

All input data use a single format for integer data and a single format for decimal data. All input data are echoed on the output unit 6. All output data are on unit 6. Apart from the echoed input data the normal (KP=0) output data are listed in Table 5. The additional output data for KP=1 are listed in Table 6. Twelve input cases are available on various choices of KS(1,2, or B), KC(1 or 2), and KT(1 or 2).

Listed in Tables 7 and 8 are sample input and outputs for a boron absorber control rod case. Because KT=1 this is a thermal case, and because RS=1, a single weighting flux spectrum is read in from unit 7. This weighting flux j spectrum was obtained from a LEOPARD run. Because KC=2 the cross section is l

generated from a 1/v dependence, and because KP=0, only normal output is obtained. There are two mesh intervals in the absorber (NM-2). There are NS=34 values of input weighting flux spectrum, and NK is arbitrary because it is not used. The integration energy mesh is identical to the input weighting flux energy mesh. Figure 1 shows the energy variation of a(E) and S(E) for this case, where E is the neutron energy. Also shown is the input weighting flux spectrum $(E) per unit energy. Because the abscissa is the logarithm of E, it is more graphic to consider the flux per unit lethargy 4(u) as well.

From the plot of ((u) it can be seen that the spectrum averaged values of a and 8 are near 0.49 and 0.51, respectively. The PLATAB code computed the spectrum averaged values of a and S to be 0.486681 and 0.514664, respectively (see Table 9).

l LEOPARD 7 also provides the neutron flux per unit lethargy, so using this as

! input weighting neutron flux spectrum the ef fective dif fusion constants were calculated (KT=2) for the slowing down neutron energy group from 0.625 ev to l 5530 ev. Results are shown in Table 9. Here the input values of FL and FR were taken both to be 1.732. >

l l

l

3 References

1. M.M. Bretscher, " Blackness Coef ficients, Effective Dif fusion Parameters, and Control Rod Worths for Thermal Reactors," ANL/RERTR/TM-5, 1984.

2. A.F. Henry, "A Theroretical Method for Determining the Worth of Control l Rods ," WAPD-218, 19 59.

3. C.W. Maynard, " Blackness Theory and Coef ficients for Slab Geometry," Nucl. I Sci. and Eng. 6, 174 (1959). l

4. S. Stein, " Resonance Capture in Heterogeneous Systems," WAPD-139,1955.  !

1

5. M. Goldsmith, et al., " Theoretical Analysis of Highly Enriched Light Water Moderated Critical Assemblies," Second Int. U.N. Conf. on the Peaceful Uses of Atomic Energy, Geneva, CONF 15/P/2376 (1958).

6. M. Abramowitz and I. A. Stegun, Eds. , Randbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables, National Bureau of Standards App. Math. Series #55,1965.

7. L.E. Strawbridge and R.F. Barry, " Criticality Calculations for Uniform Water-Moderated Lattices," Nucl. Sci. Eng. 23, 58 (1965).

I

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f A T

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T P - P

, E L i o L B A g A

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N

) E V D E N

( E P

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, I 0 D 1 N A

)

l M T - U I V R NM E T UO I C R - E RF C P E DE S

- PYRS

_ G A2 N X XRP- O U UEOM R L LNEC T F FEL( U E G N N I

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4

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gg $dE*

l  !,

" < i.  ! l'. ' , I

)

9 TABLE 1. VARIABLE DICTIONARY FOR PLATAB C VARIABLES C AL= ALPHA PARAMETER AT ENERGY IE C ALA= SPECTRUM AVERAGED VALUE OF ALPHA C BA= DIRTY BLACKNESS CORRECTION FACTOR FOR ALPHA C BB= DIRTY BALCKNESS CORRECTION FACTOR FOR BETA C BE= BETA PARAMETER AT ENERGY IE C BEA= SPECTRUM AVERAGED VALUE OF BETA C CA= CONCENTRATION OF ABSORBER NUCLEI C DE= EFFECTIVE DIFFUSION CONSTANT AT ENERGY IE C DEA = SPECTRUM AVERAGED VALUE OF EFF DIFF CONSTANT C DEE= EFFECTIVE DIFFUSION CONSTANT FOR ALA AND BEA

~

C El,E2,. .=E-FUNCTIONS AT ARGUMENT 22 C ED(IE)= INTEGRATION MESH ENERGY IE C EE(IE)= ENERGY (EV) FOR IE C ES(IE)= INPUT EIGHTING FLUX MESH ENERGY IE C EX(IE)= INPUT CROSS SECTION MESH ENERGY IE C FE(IE)= INTEGRATION MESH FLUX SPECTRUM VALUE AT ENERGY IE C FF= AVERAGE OF FL AND FR C FL,FR=LEFT AND RIGHT VALUES OF F IN EQ(32) FOR RESONANCE WIGHT C FS(IE)= INPUT EIGHTING FLUX VALUE AT ENERGY IE (=1,2,.. ,NS)

C FSL= INPUT WIGHTING FLUX VALUE AT LEFT OF ABSORBER C FSR= INPUT EIGHTING FLUX VALUE AT RIGHT OF ABSORBER C HT= SPATIAL MESH INTERVAL WIDTH IN ABSORBER C KC=1 FOR INPUT CROSS SECTIONS, =2 FOR 1/V CROSS SECTION

! C KP=0 FOR NORMAL PRINTS, =l FOR DETAILED PRINTS

C KS=1 FOR INPUT ONE WIGHTING FLUX SPECTRUM (WFS), =2 FOR INPUT TWO WFS, I =3 FOR GENERATE MAXELLIAN WFS C KT=1 FOR THERMAL CASE, =2 FOR EPITHERMAL CASE C NE= NUMBER OF INTEGRATION ENERGY MESH POINTS C NM= NUMBER OF SPATIAL MESH INTERVALS IN ABSORBER C NS= NUMBER OF EIGHTING FLUX SPECTRUM ENERGY MESH POINTS C NX= NUMBER OF CROSS SECTION POINTS INPUT C SA= ABSORBER MICR0 CROSS SECTION AT 0.0253 EV FOR KC=2 C SE(IE)= EFFECTIVE ABSORPTION CROSS SECTION AT IE C SEA = SPECTRUM AVERAGED VALUE OF EFF ABS CROSS SECTION C SEE= EFFECTIVE ABS CROSS SECTION FOR ALA AND BEA C TH= ABSORBER THICKNESS (CM) l C TITLE = JOB TITLE l -- C TK=C OF EQ(18)/HT, ALSO K FROM WAPD-218 P8,EQ(27)

C TS= TEMPERATURE (K) FOR MAXELLIAN WIGHTING FLUX SPECTRUM

-C T1,T2,..= TEMPORARIES C XE(IE)= INTEGRATION MESH CROSS SECTION AT ENERGY IE .

C XX(IE)= INPUT CROSS SECTION AT ENERGY POINT IE (=l,2,..,NX)

C ZZ=0PTICAL THICKNESS OF ABSORBER AT ENERGY IE l

. l o'

10

-TABLE 2. JOB INPUT FOR PLATAB FROM UNIT 5 l

I ITEM FORMAT i

Job Title 20A4 KT,KS,KC,KP,NM,NS,NX 1214 CA,TH,BA,BB 4E12.6 TS iff KS=3 4E12.6 FL,FR iff* KT=2 4E12.6 SA iff KC=2

• tff=if and only if TABLE 3. EIGHTING FLUX SPECTRUM DATA INPUT FROM UNIT 7 ITEM FORMAT Title 20A4 (ES(IE), FS(IE),IE=1,NE) iff KS=1 4E12.6

-ES(1),FSL(1),FSR(1) 4E12.6 ES(2),FSL(2),FSR(2) 4E12.6 ES(3),FSL(3),FSR(3)

ES( 4) ,FSL( 4) ,FSR( 4)

ES(NS),FSL(NS),FSR(NS) iff KS=2 4E12.6 TABLE 4. CROSS SECTION DATA INPUT FROM UNIT 8 ITEM FORMAT l Title 20A4 (EX(IT),XX(IE),IE=1,NE) iff KC=1 4E12.6 l .

TABLE 5. OUTPUT OTHER THAN ECHOED INPUT DATA ITEM FORMAT Integration EE(IE),FE(IE),XE(IE),ED(IE) 4E13.6 for each IE=1,NE EE(IE),AL,BE,DE,SE SE13.6 for each IE=1,NE ALA,BEA, DEA, SEA,FEA SE13.6 DEE ,SEE ,T4,TK 4E13.6

g, -_. _

11 r

TABLE 6. ADDITIONAL OUTPUT DATA FOR KP=1 l

l ITEM FORMAT j ZZ,El,E2,E3,E4 5E13.6 EE(IE),AL,BE,T6 4E13.6 T4,TK,ALA,BEA,FEA 5313.6 l preceeds EE(IE),AL,3E,DE,SE 5313.6 for each IE=1,NE TABLE 7. SAMPLE INPUT l

l INPUT FROM UNIT 5 RPI CONTROL RODS 1 1 2 1 2 34 34

.0108342 .2285 1. 1.

3838 .17 INPUT FROM UNIT 7 LEU CORE THERMAL SPECTRUM l

.0004 .0967953 .0009 .206480 l .0014 .308325 .0019 .407 404

.0024 .499683 .0029 .587639

.0034 .671678 .0039 .752090

.0044 .829091 .0049 .902872

.0054 .973690 .0059 1.04162

, .0l0 1.50336 .015 1.88568 l- .020 2.L1927 .025 2.24426

. 03 0 2.28893 .035 2.27481

.040 2.22323 .045 2.14132

. 05 0 2.04426 .055 1.93507

. 06 0 1.82320 .110 .912535

.160 .521307 .210 .361301

.260 .28 0157 .310 .234704 j .360 .208808 .410 .187581 i

. 46 0 .169629 .510 .154524

.560 .141713 .610 .13 08 16 l

l

12 TABLE 8. SAMPLE OUTPUT 1 LRPI CONTROL RODS

2 OKT,KS,KC,KP,NM,NS,NX

3 L L 2 0 2 34 1

- 4 OCA,TH,BA,BB,HT

5 0.108342E-01 0.228 500E+00 0.100000E+01 0.10000E+01 0.lL4250E+00 6 ILEU CORE THERMAL SPECTRUM

7 0. 400000E-03 0.967953E-OL 0.900000E-03 0.2% 48 0E+00 8 0.140000E-02 0.32325E+00 0.19 0000E-02 0. 407 404E+00

9 0.240000E-02 0. 499683E+00 0.29 0000E-02 0.587639E+00

10 0.340000E-02 0.671678 E+00 0.390000E-02 0.7 5209 0E+00

11 0. 440000E-02 ' 0.8 29 091E+00 0. 49 0000E-02 0.902872E+00

12 0.540000E-02 0.97369 0E+00 0.59 0000E-02 0.104162E+01

13 0.100000E-01 0.150336E+0L 0.150000E-01 0.188 568 E+01

14 0.200000E-01 0.211927 E+01 0.250000E-01 0.224426E+0L

15 0.300000E-01 0.228893E+01 0.350000E-01 0.227 48 IE+01

16 0.400000E-OL 0.222323E+01 0.450000E-01 0.214132E+01

17 0.500000E-01 0.204426E+01 0.550000E-01 0.193507E+01

18 0.600000E-01 0.182320E+01 0.110000E+00 0.912535E+00

19 0.160000E+00 0.521307 E+00 0.210000E+00 0.3613 01E+00

20 0.260000E+00 0.28 0157 E+00 0.310000E+00 0.23 47 04E+00

21 0.360000E+00 0.2088 0B E+00 0.410000E+00 0.187581E+00

22 0. 460000E+00 0.169629E+00 0.510000E+00 0.154524E+00

23 0.560000E+00 0.141713E+00 0.610000E+00 0.13 0816E+00

24 OSA

25 0.383817 E+04

26 1 INTEGRATION EE( IE) , FE( IE) ,XE( IE ) , ED( IE) , IE= 1,NE

27 0. 400000E-03 0.967953E-01 0.3 05'017 E+05 0. 400000E-03

28 0.9 00000E-03 0.20648 0E+00 0.203345 E+05 0.500000E-03

29 0.140000E-02 0.308 325E+00 0.163039 E+05 0.500000E-03

30 0.190000E-02 0. 407 404E+00 0.139952E+05 0.500000E-03

31 0.240000E-02 0. 499683E+00 0.12 4523E+05 0.500000E-03

32 0.29 0000E-02 0.587639E+00 0.113281E+05 0.500000E-03

33 0.340000E-02 0.671678 E+00 0.104620E+C5 0.500000E-03

34 0.39 0000E-02 0.752090E+00 0.976837 E+04 0.500000E-03

35 0. 440000E-02 0.8 29091E+00 0.919662E+04 0.500000E-03

36 0. 49 0000E-02 0.9 02872E+00 0.871478E+04 0.500001E-03

37 0.540000E-02 0.973690E+00 0.830152E+04 0. 499997E-03

38 0.590000E-02 0.104162E+01 0.79 4197 E+04 0. 500001E-03

39 0.100000E-01 0.150336E+01 0.610035E+04 0.410000E-02

40 0.150000E-01 0.188 568 E-01 0. 498 091E+04 0.500000E-02

41 0.200000E-01 0.211927 E+01 0. 431359 E+04 0.500000E-02

~42 0.250000E-01 0.22 4426E+01 0.38 5820E+04 0.500000E-02

43 0.300000E-01 0.228893E+0L 0.352204E+04 0.500000E-02

44 0.350000E-01 0.227 481E+0L 0.326077 E+04 0.500000E-02

45 0.400000E-01 0.222323E+01 0.305017 E+04 0.500000E-02

46 0. 450000E-01 0.214132E+01 0.287 573E+04 0.500000E-02

47 0.500000E-01 0.204426 E+01 0.27 2816 E+04 0.500000E-02

48 0.550000E-01 0.193507 E+01 0.260120E+04 0.500000E-02

49 0.600000E-01 0.18 2320E+01 0.2 49 046 E+04 0.500000E-02

50 0.110000E+00 0.912535E+00 0.183932E+04 0.500000E-01

51 0.160000E+00 0. 521307 E+00 0.152509 E+04 0.500000E-01

9 13

52 0.210000E+00 0.3613 01E+00 0.133121 E+04 0.500000E-01 53 0.260000E+00 0.28 0157 E+00 0.119638 E+04 0.500000E-01

54 0.310000E+00 0.23 47 04E+00 0.109565E+04 0.500000E-01

55 0.360000E+00 0.2088 08 E+00 0.101672E+04 0.500000E-01

56 0.'410000E+00 0.187581E+00 0.952714E+03 0.500000E-01

57 0. 46 0000E+00 . 0.169629E+00 0.899 447 E+03 0.500000E-01.

58 0.510000E+00 0.154524E+00 0.8 54219 E+03 0.500000E-01

59 0.560000E+00 0.141713E+00 - 0.815193E+03 0.500000E-OL

60 0.610000E+00 0.13 0816E+00 0.781069 E+03 0.500000E-01

6i OEE(IE,AL,BE,DE,SE

62 0. 400000E-03 0.500000E+00 0.500000E+00 0.8 07871E-04 0.8 7 4036 E+0 L

63 0.900000E-03 0.500000E+00 0.500000E+00 0.8 07871E-04 0.87 4036 E+0 L

64 0.140000E-02 0.500000E+00 0.500000E+00 0.8 07871E-04 0.87 4036 E+01

65 0.19 0000E-02 0.500000E+00 0.500000E+00 0.8 07871E-04 0.87 4036 E+01

66 0.240000E-02 0.500000E+00 0.500000E+00 0.8 07871E-04 0.87 4036 E+01

67 0.29 0000E-02 0.500000E+00 0.500000t+00 0.8 07871E-04 0.87 4036E+0L

68 0.340000E-02 0.500000E+00 0.500000E+00 0.8 078 71E-04 0.87 4036 E+01

69 0.39 0000E-02 0.500000E+00 0.500000E+00 0.8 07871E-04 0.87 4036 E+01

70 0.440000E-02 0.500000E+00 0.500000E+00 0.8 07871E-04 0.87 4036E+01

71 0. 49 0000E-02 0.500000E+00 0.500000E+00 0.8 07871E-04 0.87 4036E+01

• 72 0.540000E-02 0.500000E+00 0.500000E+00 0.8 07871E-04 0.87 4036E+01

73 0.59 0000E-02 0.500000E+00 0.500000E+00 0.8 07871E-04 0.87 4036 E+01

74 0.100000E-01 0.500000E+00 0.500000E+00 0.8 07871E-04 0.87 4036E+01

75 0.150000E-01 0.500000E+00 0.500000E+00 0.654151E-04 0.87 4272E+01

76 0.200000E-01 0. 499996E+00 0.500004E+00 0.227 462E-03 0.87179 4E+01

77 0.250000E-01 0. 499987 E+00 0.500014E+00 0. 42118 2E-03 0.8688 44E+01

78 0.300000E-01 0. 499966E+00 0.500034E+00 0.662739E-03 0.865177E+01

79 0.350000E-OL 0. 499932E+00 0.500068 E+00 0.9 45142E-03 0.86 0911E+01

80 0.400000E-01 0. 499879E+00 0.500121E+00 0.125689 E-02 0.856226 E+01

81 0. 450000E-01 0. 4998 05 E+00 0.50019 4E+00 0.159351E-02 0.8 5 L L973+01

82 0.500000E-01 0. 4997 09 E+00 0.500291E+00 0.19 4969E-02 0.8 459 09 E+01

83 0.550000E-01 0.49958; 8 E+00 0.500412E+00 0.232003 E-02 0.8 40446 E+01

84 0.600000E-01 0. 499 442E+00 0.500558 E+00 0.27 0051E-02 0.83 4872E+01

85 0.110000E+00 0. 496611E+00 0.503415E+00 0.668650E-02 0.778799 E+01 -

86 0.160000E+00 0. 491796E+00 0.508360E+00 0.1048 40E-01 0.72927 0E+01

87 0.210000E+00 0. 485783 E+00 0. 514689 E+00 0.139354E-01 0. 687 467 E+01

88 0.260000E+00 0. 479133 E+00 0.52189 4E+00 0.17 0676E-01 0.652089 E+01

89 0.310000E+00 0. 472197 E+00 0.529652E+00 0.199302E-01 0.6218 06E+01

90 0.360000E+00 0. 465188 E+00 0.537752E+00 0.225688 E-01 0.595558 E+0L l  : 91 0.410000E+00 0. 4SB 235 E+00 0.546058 E+00 0.250195E-01 0.5725 48 E+01

92 0. 460000E+00 0.451416E+00 0.554478 E+00 0.273116E-01 0.552169 E+01

93 0.5l0000E+00 0. 444777 E+00 0.5629 51E+00 0.29 4681E-01 0. 533959 E+01

94 0.560000E+00 0. 43834 L E+00 0.571433E+00 0.315075 E-01 0.517558 E+01

95 0.610000E+00 0. 432122 E+00 0. 57989 4E+00 0.33 4447 E-01 0.502688 E+01

96 OALA,BEA, DEA, SEA,FEA

97 0. 486651E+00 0.5147 00E+00 0.100040E-01 0.7 47723 E+01 0.2798 46 E+00

98 ODEE,SEE,TK,T4

99 0.13727 5 E-01 0.69 0672 E+01 0.357 004E+02 0.18 6788 E+02

,<-.y m.,- - - , - . - - - , - - - - - , . . < - , - , . . , , , - - - , - , - - - - - . -

~ .

14 TABLE 9. ' SPECTRUM AVERAGED a AND 8 AND CALCUALTED EFFECTIVE DIFFUSION PARAMETERS FOR BORON CONTROL ROD BLADE l SPECTRUM AVERAGES EFFECTIVE DIFFUSION PARAMETERS ENERGY GROUP a 8 De Ee THERMAL 0.486651 0.5147 00 0.0137275 6.90672

(<.625EV) l EPITHERMAL 0.101836 9.19105 1.04424 0.893706 l (.625EV TO 5530 EV) l l

l l

l I

I l

t .