Cladding circumferential hot spot factors for fuel and blanket rods

NUCLEAR ENGINEERING AND DESIGN 35 (1975) 21-28. © NORTH-HOLLAND PUBLISHING COMPANY

CLADDING CIRCUMFERENTIAL HOT SPOT FACTORS FOR FUEL AND BLANKET RODS M.C. CHUANG, R.E. KOTHMANN Westinghouse Research Laboratories, Pittsburgh, Pennsylvania 15235, USA M.J. PECHERSKY and R.A. MARKLEY Westinghouse Advanced Reactor Division, Madison, Pennsylvania, USA Received 9 June 1975

A study to determine the cladding circumferential temperature profiles in LMFBR fuel and blanket assemblies is presented. Corresponding hot spot factors were calculated as a function of pitch-diameter ratio, rod diameter and subehannel coolant velocity for 'bare' and wire-wrapped rods using the FATHOM-II computer program. A brief description of the FATHOM-II analysis and a comparison of the results with experimental and other analytical data are also presented.

1. Introduction

To evaluate the performance of a wire-wrapped LMFBR fuel or blanket assembly, the effect of the wire wrap on the local cladding temperature profile must be ascertained. While providing the proper spacing between adjacent fuel rods, there is generally contact between the wire and one or both o f the rods. This contact causes an increase in the local cladding temperature in the vicinity o f the contact point between the rod and the wire. In this paper the results o f a study to determine the cladding circumferential temperature profiles in fuel and blanket rods are presented. Calculations were also performed for 'bare' rods to determine the net effect of the wire wrap. The calculations were performed with the FATHOM-II computer program, which solves the momentum and energy equations in the coolant surrounding the fuel rod and the conduction equations in the cladding and wire wrap. This paper studies the most severe condition of the hot spot temperature caused by the wire contacting the two neighboring rods as shown in fig, l(a). Due to the symmetric geometry o f the rod assembly, the flow

Spac,er-Wire~ ~ _~ ~'FuelRod (a)

~

F ,Mi abati No c Shear Surface Wire, Tw, kw 4~

(b) Fig. 1. Typical subchannel and spacer-wire configuration.

22

M.C. Chuang et al., Cladding circumferential hot spot factors

model in fig. l(b) is used. The region is a 60 ° sector bounded by the cladding inner surface and the adiabatic no-shear surfaces as shown. A constant heat flux boundary condition can be applied at either the inner or outer cladding surface. Figure 2 is a comparison between the axial geometry of a wire-wrapped rod and the geometry assumed by the FATHOM-II code. The code does not take the curvature of the wire into account but assumes a fully developed velocity profile based on the wire running parallel to the rod. The effect of the wire wrap curvature is partially compensated for by modifying the axial pressure drop by the semi-experimental correlation developed by Novendstern [I ] relating the friction factor between bare rod bundles and wire-wrapped bundles. As the wire lead becomes large (such as in a core fuel rod) the effect of the wire curvature diminishes and the FATHOM code representatiori comes_ into better agreement with the true physical situation.

u a-;

t(k'+eM) ~

+ r'2 ~

(P÷eM)

(2)

aT] •

The eddy diffusivityo f momentum transfer eM at an arbitrary point in the turbulent flow field can be expressed by [2] (

|0,

Lu

2

if-if- ~< m o, 2\

e : _ ~ 1 [ Lu

~ -~--v- - m ° )

'

(3) -ifLu v >m~.

The scale of turbulence L for triangular pitch rod is related to position by rd) COS~b . 2(rdw -- r d cos ¢)

. 2 r d w -- ( f +

(4)

L = (r - rd) . . . . . . . .

According to the Reynolds analogy, heat and momentum are transferred by analogous' processes in turbulent flow, thus eH = OeM, where o is the ratio of the eddy diffusivities for heat and momentum transfers. In the present calculation we set o = 1. The pressure gradient due tO wire-wrap spacer effect is given by [I ]

For fully developed turbulent flow in a lattice of rods, the momentum and energy equations of an incompressible liquid in the coolant passage can be written r ar

13[

+~- ~-~ (all + ell) ~-~

2. Mathematical analysis

Bz

r

' (1)

/" Do.~

0p _ Bz

1

M f f DH

pt72 2g'

(5)

where f= 0.316/Re °as , ~ / ./

,

Re

= pffDH/la,

dN

Mf =

1.034 poe ~

29.7 p~c.94ReO.Oa6 ] o.aas, +

(/_//00)2.239

(5a)

•

oo D. - 1 + dw/Do ,~ual FATHOM - I I

Fig. 2. Comparison o f assembly and FATHOM axial geometry.

1 Wo ]

"

For a fully developed heat transfer or temperature profile, the axial temperature gradient is constant and given by OT/~z = Qt/(m t Cp),

(6)

M.C. Chuang et al., Claddingcircumferential hot spot factors

23

aT aT w k c ~ r l ( r l =rw,~P)=kw a-~-1 (rl =rw, Ot)

where

Qt -- [q fir2 -I-tlbTr(r~ -- ra2)],

th t = PAtu ,

(6a) for a > 0.

A t = 6 r~w t a n -6- = - r6~ -

4 r

"

(19)

(3) At the clad surface:

The energy equations within the clad and the wrapped wire are related by

for constant heat flux at the clad inner surface

O2Tb

aT b ( r = r a , ~ ) = _

1 0T b O2Tb £1b - -Dr - T - + -r Dr + ~ + --kb = 0,

1 (Q~f).

a-S-

(20) '

(7) for constant heat flux at the clad outer surface

02Tw + __ 1 ~0Tw -t 1 -O2Tw --0. 0r~ r I ar t r 2 00t2

(8)

The boundary conditions are: (1) At adiabatic no-shear surface aA, AD, DI, IE, EF, FG, GH

(aulaCO(r, ¢, = Ir/3) = O,

aT(

OTb(r,~b=o)=aTw, --~-a t r l , a =0) = ~-~ r, •

a~

= 0~

r,O=

(9)

=0, (11)

an IE, EF

~

cos q~

r 0¢' sin qY ~r, EF

f o r * > 0,

aT w (r I = r w , a = 0) = ~1 Q(~-c)

(21)

for • = 0.

=0,

(22)

(4) At the center point of the wire (r 1 -~ 0, a) = 0.

(10)

r g-~ sin ~ IE, EF

±(Qt

- kc\Sc]

aT.,.,,

= 3)

=0,

an ,e, EF

-,d,*)=

aT ~-(r

(23)

The above differential equations with boundary conditions are solved by finite difference with the FATHOM-If computer program. The grid in the combined regions of clad, coolant passage and wire is constructed asshown in fig. 3. The total number of mesh points is carefully evaluated to lessen the possible error for neglecting higher order terms of the Taylor series expansion. A total of more than 200 nodal points are used to solve the problem, The FATHOM-II computer

(12) (13) (OTwlOa)(r~, a = 7r/2) = 0.

(14)

(2) At interfaces of coolant and clad, coolant and wire, and clad and wire contact point

u(r = rd, ~) = 0,

(15)

u(r I =rw, a) = 0,

(16)

kb

0r

(r = rd, 4,) = k~ -~r (r = rd, 4,)

H

Lep

for I, > 0,

j'

(17)

A

k b 0Tb(r Or =r d , ~ = 0 ) = k w ~0T w (r, = rw, a = 0), (18)

Fig. 3. Subdivision of mesh grids.

D

24

M.C. Chuang et aL, Cladding circumferential hot spot factors

2.8

program solves first the momentum equations which satisfy boundary conditions to determine the coolant velocity profile. These results are then used in the energy equation for the coolant, clad and wire wrap to determine the respective temperature profile. The average subchannel coolant velocity and the subchannel mean bulk coolant temperature are used as reference quantities. By inputing the fuel, clad and coolant properties; the rod assembly geometry; and number of mesh points to be used, the FATHOM-II computer code can perform the calculations to obtain the velocity and temperature profiles according to the user's command to have constant heat flux at clad inner or outer surface, for wire wrap or bare rod conditions.

Figures 4 - 6 compare the present study with the Dwyer and Berry results [3] and Nijsing and Eifler analysis [4] for the temperature field without wire wrap in the case of constant heat flux at the clad outer surface and at the clad inner surface. The turbulent velocity distribution for subchannel without wire wrap agrees wellwith test data [5]. Figure 7 is a plot of the circumferential cladding midwall temperature for a fuel rod (large pitchdiameter ratio). The dashed line represents the bare rod case and the solid line the wire-wrapped case. In the bare rod the maximum temperatures occur in the I

-

\

....

Present Study Owyer and Berry

E 2.0 ~-

....

%~"~ ~ 1 ,^ " % "

Dwyer and Berry ResultsD-D i ~ = O. 20, Pe = 100,

o\°o

~

1.6

~

I

Present Study

-'~

--

~1k Pr=0.004834 kb

08 -

. \ \

0.4 0

I 10

20

30

Angle, ¢, degree

0.12

i~ =

0.0s

o

t ~lL

....

,

*t

O

PresentStudy Nijsing and Eifler Analysis

k

\

0 1.0

1

(R~. 4)

?= 0. 3

i

L I J L I I ~ I I I L I I 1.04 1.08 1.12 1.15 Pitch DiameterRatio,Pc

of constant heat flux at clad inner surface without wire wrap.

Results (Ref. 3, Pe= lO0, Pr = O.004834

~i-~" 1.2 _ ~--~ 0.8

L 10 20 Angle, ~, degree

~ t t l i r l t t J l l l

Fig. 6. C o m p a r i s o n with Nijsing and Eifler analysis in the case

1.6-

0

2.4

1.0

~

~: 2.4~- . . . . ~ ~~ = L lO " ~

0

~"103 -".~N~ ~

Fig. 5. Comparison with Dwyer and Berry results,in the case of constant heat flux at clad inner surface without wire wrap for large clad thickness.

3. Numerical results

2.8 ~I

_

!

,o

30

F~g. 4. Comparison with D w y e r and Berry results i n the case o f constant heat f l u x at clad outer surface w i t h o u t wire wrap.

minimum gaps, as expected. In the wire wrap the maximum temperature occurs in the gap which contains the wire (the wire comes into contact with the rod in the gap at 0°). Fig. 8 shows a similar plot for a radial blanket rod. Here, due to the much smaller pitch-diameter ratio and larger rod diameter, there is a much greater circumferential temperature variation for the bare rod case. Correspondingly, the effect of the wire wrap. is not as large as it was for the case of a fuel rod. The dimensionless temperature T* in the plot is defined as T* = (rb, ® -

-

rm=.).

(24)

M.C. 2.0[

I

I-

7~ II

R

'.°L

I

I

(T:,s)ax

I

(2) Hot spot factor due to the combined effect of the rod spacing and the wire wrap

I

. . . .

With Wire ~

....

\Tb, 4, - Tmean

wlowire

Constant Heat FIux at Cladding Inner Surface

0.6

F w - -(

0.4 0.2 I

1

t

I

I

10

20

30

40

50

60

Angular Position ~Jegrees~, ¢

Fig. 7. Cladding midwall temperature as a function of angular position in a fuel rod with and without wire wrap, where Pc = 1.257;Di = 0.2 in:;Do = 0.23 in.;H = 12 in.;d w = 0.0591 in.;ff= 20 ft/sec; and qb =0.

2.2

I

~ : 2"0 t ~e.J~.e 1.8 ~

1,o

~

0.8

~,

0.6 ~

~

0.4

c c~

0.2 0

I

i

1

I

(26)

....

~]b = 0

I 10

Figures 9 - 1 4 show typical examples of the cladding inner, midwall and outer surface hot spot factors for fuel rod and blanket rod assemblies, respectively. In general, the bare rod hot spot factor FB decreases with p i t c h - d i a m e t e r ratio, approaching a limit o f 1 as expected, The h o t spot factor, due to the wire F w increases with the p i t c h - d i a m e t e r ratio. The total h o t spot factor Fws which is the p r o d u c t of the t w o others, 1.8 1.6 1.4 1.2 ~ 1.0 ~ 0.8 " 0.6 0.4 0.2 0 1.00

//Constant Heat Flu

~

1

I

I

I

1

I

Fws --- --'---"-~F...

I

1.05

at-c,,~i.g

i

I

I

l

1.10 1.15 1.20 1.25 Pitch to Diameter Ratio, Pc

l

1.30

Inner Surface

I l I 20 30 40 Angular Position tle(jrees}, 4,

I 50

Fig. 9. Fuel rod cladding inner surface hot spot factor, where D o = 0.23 in.;b = 0.015 in4H = 12 in.; and ff = 20 ft/sec. 60 1

Fig. 8. Cladding midwall temperature as a function of angular position in a blanket rod with and without wire, where Pc = 1.077;Di = 0.49 in.;D o = 0.52 in.;H = 4 in.;dw = 0.040 in.;ff~ 12.5 fusee; andq"b = 0.

2.0 1.9 1.8

The maximum dimensionless temperature can be looked upon as a cladding circumferential hot spot factor. Three types of hot spot factors can be obtained

~- 1.6 E 1.5 t~ 1.4 1.3 1.2 I.I 1.0 1.00

from this analysis: (1) Hot spot factor due to the p i t c h - d i a m e t e r ratio effect

( Tb'° - Tmean/ FB =(rl~are)max =

(27)

- -

// / / /~/"

With Wire W/OWire

17" '*ws)max/(T* bare)max.

2.0! l

('W~max 2.0129. . . . . . (rbare) max"

'-=1""~. 1.6 " ~ , ~ \\ " 1.4 '~ ,~.

~

ws"

(3) Hot spot factor due to the presence o f the wire wrap b u t n o t including the p i t c h - d i a m e t e r ratio effect

0,8

0

25

(Pbare)m:xL04~ = ~t.o,l.J.

1.4 [ - ~ I

Chuang et aL, Cladding circumferential hot spot factors

T b , * - Tmean/bare

•

(25)

1

T

I

1

I

1.7

1.05

1.10 1.15 1.20 1.25 Pitch to Oiameter Ratio, Pc

1.30

Fig. 10. Fuel rod cladding midwall hot spot factor, where D o = 0.23 in.;b = 0.015 in.;H = 12 in.;ff = 20 ft/sec; and q' = constant at inner cladding surface.

M.C. Chuang et aL, Cladding circumferential hot spot factors

26 3.0

i

,o

2.6

2.4

I

I

/_----

2.8

~-I'~

I

~.,.

3.0 2.8

I

I

/-Fws

FB

""" ~ .-.......~......"'-...... ~ "

~. 2.2 ta.

"6 1.8 "1- 1.6

1.2, 1.0~

1.t30

"

i

2.6

~

2.4 2. 2

~

.~..,1. m ~

~ i ~ ~ "" "Fws - ~ "" " " " " "-...-..... ~

FB

1.8

.

~

.------~

Vw

1.6

14 i.//

1.2 1.0 0.8 1.00

.If.J" I

I

1.05

I

1

"

I

I

1.10 1.15 1.20 1.25 Pitch to Diameter Ratio, Pc

1.30

Fig. 1 1. Fuel rod cladding outer surface hot spot factor, where D o = 0.2 3 in.; b = 0.0 15 in.; H = 1 2 in.; and ff = 20 ft/sec.

2.0

~

1.8

N.-..

r

i

F

i

i

l

./- ws

1.4

Fw

t

I 1.05

f I 1 I 1 1.10 1.15 1.20 1,25 l,]O Pitch to Diameter Ratio, Pc Fig. 14. Blanket rod cladding outer surface hot spot factor, where D o = 0.52 in.; b = 0.015 in.; H = 4 in.; and ff = 12.5 ft/sec.

remains relatively constant down to the pitch-diameter ratio Pc "~ 1.1 for cladding inner surface and midwaU hot spot factors, while it reaches a maximum value at Pc ~ 1.1 for the outer surface hot spot factor.

1.0 0.8

=~ 0.6

example

4. Application

0.4

^

0.; 1.00

t 1.05

I t $ t 1.10 1.15 1.20 1.25 Pitch to Diameter Ratio, Pc

I 1.30

Fig. 1 2. Blanket rod cladding inner surface hot spot factor, where Do = 0.5 2 in.; b = 0.0 15 in.; H = 4 in.; and ff = 1 2.5 ft/sec.

If, for example, one wishes to compute the cladding temperatures in a fuel rod based on the previous presented hot spot factors, the process would be as follows: (1) Compute the cladding temperature based on a fuel rod in an infinite stream. This is given by r

0

q ra kb

-- q ra T b ' * -- Tmean ~ 2.2

i

f

i

1.8

~

1.6

o 1.2

~

~

F

;

(28)

(2) Multiply by the appropriate hot spot factor to obtain the maximum cladding temperature drop

..... w

(rb,*)max

-- Tmean = Tb, o -- Ymean

= (T--b,. -

FB ]

1.O

rd

i

j /

0.8

0.6 0.4 0.2 0 1.00

i

In --.r

Tmean)[

Tb'0Tmean 1 \Tb,. -- Tmean : w s

or

(Tb,*)max--Tmean= (Tb,*--Tmean)FwsI 1.05

l .I l £ 1,10 1.15 1.20 1,25 Pitch to Diameter Ratio, Pc

I 1.30

Fig. 1 3. Blanket rod cladding midwall hot spot factor, where D O -- 0.520 in.; b = 0.015 in.; H = 4 in,; ff = 12.5 ft/sec; and q' = constant at cladding inner surface.

(29)

(3) Add the c!adding temperature drop to the bulk coolant temperature in the subchannel to obtain the absolute maximum temperature (Tb,.)max. Steps 1 and 2 were carried out for the fuel rod in fig. 7 and are presented in table I. If one were inter-

M'.C. Chuang et al., Claddingcircumferential hot spot[actors

27

Table 1. Cladding temperature drops in a fuel rod. r -rd Outer surface

1

Tb, ~ - T m e a n

[Tb, o-Tmean ~

5

[ | \Tb, • - Tmean/ws

(°F) 18.39

Tb, 0 - Tmean (°F)

2.5406

46.72

1.4813

59.46

1.2785

81.13

21.75 MidwaU

0.9348

40.14

Inner surface

0.8696

63.46

6

12.74

23.32

21.67

6 = temperature difference between clad outer surface and midwall or clad midwaU and inner surface; r d = 0.115 in.; r a = 0.100 in.; k b = 12.5~Btu]hr ft OF; h = 22860.0 Btu/hr ft 2 ° F ; and q' = 4.839 x l 0 s Btu/hr ft 2.

ested in a situation with no wire present, one would apply F B. Similarly, if the temperature variation due to the Pc effect had already been computed, one would apply F w to the maximum temperature drop to obtain the maximum cladding temperature.

DO

5. Concluding remark

Fws

It should be noted that the hot spot factors presented in this paper are independent o f the heat flux. This can be shown theoretically, and has been checked out b y running two equivalent FATHOM problems with different heat fluxes. The average subchannel coolant velocity and wire wrap lead are predicted to have a very small effect on the hot spot factor. As mentioned earlier, the FATHOM representation will give more realistic results for large wire leads. However, for the low wire lead cases o f the radial blanket, the p i t c h diameter ratio is also small, so that the relative effect o f the wire is small. Therefore, any uncertainties caused b y the FATHOM representation of the wire geometry are also considered to be small.

Di

dw /78

Fw

f H hKl h kb, kc, kw

L Lf Me

Nomenclature af

At b Cp

DH

= surface area o f the fuel rod = 27rraL f = total coolant cross-sectional area = clad thickness = specific heat of coolant at constant pressure = equivalent hydraulic diameter

rh t n n o, m o Pc ar Pe p

= rod inner diameter or clad inner diameter = rod outside diameter or clad outside diameter ---wire diameter = hot spot factor due to the p i t c h diameter ratio effect = hot spot factor due to the combined effect o f the rod spacing and the wire wrap = hot spot factor due to the presence o f the wire wrap but not including the p i t c h - d i a m e t e r ratio effect = friction coefficient o f the Blasius equation = wire lead or pitch o f helical wire = mesh spacing in the radial direction within the clad = heat transfer coefficient = thermal conductivities referring to clad, coolant and wire regions, respectively = scale o f turbulance = fuel rod length = multiplication factor of friction coefficient = total average coolant flow rate = normal direction to the surface = empirical constant ~ 7.1 = p i t c h - d i a m e t e r ratio = Prandtl number = Peclet number = Rear = pressure

28 Qf

at

;7 q

t

;Tb Re ra rd rw rdw r rl Sc T

Tb rw T1, • TI, T!,o TI,30

Tm esal Tb,~

m

Tb,~

T*

(T~'are)max

M.C. Chuang et al., Cladding circumferential hot spot factors = total heat generation rate o f the fuel rod = q lrr2aL f = total net heat transfer rate per unit fuel length in coolant region = heat generation rate per unit volume of the fuel rod = heat generation rate per unit length o f the fuel rod or heat flux o f the fuel rod = heat generation rate per unit volume in the clad due to nuclear heating = average flow Reynolds number = radius o f clad inner surface = radius o f the clad outer surface = radius of the wire = half pitch or the sum o f t a and r w = radius from the fuel center point = radius from the wire center point = clad outer surface area per unit fuel length = 21rrd = coolant temperature = temperature in the clad = temperature in the wrapped wire = clad outer surface temperature at angular position, = average value o f Tt, = clad outer surface temperature at = 0° = clad outer surface temperature at = 30 ° = mean bulk coolant temperature = Clad local (inner surface, midwall or outer surface) temperature at angular position = average cladding temperature o f surface where the local temperature is taken (average o f _Tb,~) = dimensionless temperature defined in eq. (24) = dimensionless temperature o f T* without wire wrap when • = 0

(T*s)max U ff z

~H

# p P eH eM (7

= dimensionless temperature o f T* with wire wrap when (I, = 0 = coolant velocity = average bulk coolant velocity = axial coordinates = angular coordinate in the wire region = coolant thermal diffusivity = k c / p C p = thickness o f laminar sublayer = mov/U ( 8 / M f f ) u2 = dynamic viscosity o f the coolant = kinematic viscosity o f the coolant = density o f the coolant = eddy diffusivity o f heat transfer = eddy diffusivity o f m o m e n t u m transfer = ratio o f the eddy diffusivity for heat and momentum transfers = angular coordinate starting'from 0 to 7r/3: 4, = ~ for 0 < (I, ~< lr/6, q~ = rr/3 - ¢ for Ir/6 < ~ ~< 7r/3 = angular position starting from 0 to 7r/6, or from rt/3 to lr/6.

References [1 ] E.H. Novendstern, Turbulent flow pressure drop model for fuel rod assemblies utilizing a helical wire wrap spacer system, Nucl. Eng. Des. 22 0 9 7 2 ) 19. [2] N.I. Buleev, K.N. Polosukhina and V.K. Pyshin, Hydraulic resistance and heat transfer in a turbulent liquid stream in a lattice of rods, High Temperature, USSR, vol. 2, no. 5 (1964) 673. [3] O.E. Dwyer and H.C. Berry, Turbulent-flow heat transfer for in-line flow through unbaffled rod bundles: molecular conduction only, Nucl. Sci. Eng. 46 O971) 284. [4] R. Nijsing and W. Eifler, Analysis of liquid metal heat transfer in assemblies of closely spaced fuel rods, Nucl. Eng. Des. 10 0 9 6 9 ) 21. [5 ] M.J. Pechersky, R.M. Roidt, B.J. Vegter and R.A. Markley, ! 1 : 1 scale rod bundle flow tests, parts 1 through 7,

WARD-OX-3045-6, USAEC, Feb. 0974).