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Turbulent flow pressure drop model for fuel rod assemblies utilizing a helical wire-wrap spacer system
NUCLEAR ENGINEERING AND DESIGN22 (1972) 19-27. NORTH-HOLLANDPUBLISHINGCOMPANY
TURBULENT FLOW PRESSURE DROP MODEL FOR FUEL ROD ASSEMBLIES UTILIZING A HELICAL WIRE-WRAP SPACER SYSTEM E.H. NOVENDSTERN Westinghouse Electric Corporation, Advanced Reactors Division, P.O. Box 158, Madison, Pa. 15663, U.S.A. Received 27 August 1971
A semi-empiricalmodel is developed that can accurately predict pressure losses in a hexagonal array of fuel pins utilizing a wire-wrap spacer system. The model is able to predict pressure drop to within +- 14 % over a wide range of geometries in the turbulent flow regime. The model compares favorably with independent pressure drop measurements, not used to obtain the correlation, made on a FFTF 217-pin fuel pin bundle with 400 to 1100°F flowing sodium. A numerical example is presented using the new model. 1. Introduction
2. Description of the model
A wire wrap spacer system utilizes small wires spiraling around each fuel pin to position the pins within their hexagonal duct. This spacing concept has been used in many nuclear reactors and is currently the reference design for the liquid-metal cooled FFTF reactor. Because of the complex geometry caused by the wire, simple equivalent diameter techniques are not sufficient to predict accurately the pressure drop in the fuel pin region of the reactor. An accurate prediction of pressure drop is needed in order that plant parameters may be optimized. Hydraulic testing is needed to establish the pressure drop versus flow characteristics of the fuel assembly. Because testing of all possible configurations is not practical, and the pressure drop in the pan bundle is a substantial portion of the total plant pressure loss, a method to predict pressure drop in fuel assemblies utilizing wire wrap spacing is required. Available techniques of Sangster [1] and Rehme [2] that predict pressure loss were compared with experimental data of Reihman [3], Rehme [2, 4], and Baumann et al. [5]. Because of inaccuracy in these methods, a new technique has been developed, using experimental data and ideas from the previous models. Results compare favorably with pressure drop data on sodium in a prototype FFTF fuel assembly [6].
2.1. Method to predict pressure loss The model developed to predict pressure losses in a wire-wrapped fuel assembly determines the flow distribution between the fuel pins theoretically, and multiplies the pressure drop for a smooth pipe, using equivalent diameter techniques, by an empirical correction factor. The empirical correction depends on fuel pin bundle dimensions and flow rate. The velocity, V1, in a central subchannel, formed by three rods in a triangular array, is determined from assembly dimensions and average fluid velocity, V T, according to: V1 =XV T .
(1)
The flow distribution factor, X, is calculated from central, side and comer subchannels, numbered 1 through 3, as shown in fig. 1. X =
AT [D e \0.714 i D e , ] 0.714 g l a 1 + g2A2~-~e~l) "+ N3A 3 \ ~ ]
(2)
Eq.2 is derived in sec.3. Flow areas, A, and equivalent diameters, De, are calculated including the wire wrap spacer geometry. The wire wrap is "smeared" uniformly in all subchannels. For example, the flow area
2()
lz:tt. Novendstern, Turbulent flow pressure drop model .[br juel rod assemblies. TYPICAL 19 ROD BUNDLE n I = 2~ n2 =
F2
n3 = 6
TYPE 2 OR
TYPE t OR BASIC UNIT CELL
EDGE UNIT CELL
TYPE 3 OR
ROD
CORNER UNIT CELL
LEAD
I
SPIRAL WIRE SPACER
I
Fig. 1. Definition of typical subchannels and of wire spacer lead. in a central subchannel is the area contained between three rods less one-half the area taken up by the wire. Only 50% of the wire area is used because on the average the wire is in that subchannel for only half the axial distance. Similarly, the wetted perimeter is the sum of the perimeters of the three rods in the subchannel plus 50% of the wire perimeter. These subchannel dimensions are based on the rods expanding freely within the duct used to enclose the rods, rather than on a tightly packed rod bundle with larger spacing between the wires on the outer rods and the duct. This expansion occurs because fuel pins are not perfectly straight and each pin has random bow, which causes a uniform distribution of pins within their duct. This expansion is due to the "springy" nature of long slender rods, and has been experimentally investigated by Myers [6] and Balent [7]. Using the flow conditions for the central subchannel, the pressure drop is determined by: L ~gV12 z2xP= Mfsmooth De , 2 (3) The multiplication factor, M, is used to increase the pressure drop above that calculated with the smooth tube friction factor. It primarily accounts for wire lead and fuel pin pitch-to-diameter ratio
29.7(P/D)6.94ReO.0860.885 M= t (P/D)0"124+ (tt/0)2.239 t 1.034
.
(4)
The friction factor for flow in a smooth pipe, fsmooth, is approximated by a modified version of the Colebrook equation [8]:
1 E
Jsmooth = 21Oglo -- ~
logl0 \ Re ]J/
(5)
This equation accurately predicts the friction factor and can easily be utilized when the calculation is done by digital computer. However, since it is tedious to evaluate, the Blasius approximation [9] can be used for hand calculation: fsmooth
=
0.316/Re 0'25 .
(6)
Using the Blasius approximation, the pressure drop can be directly calculated based on average velocity:
=
0.316M -] 0.25
~_ ~DeT]
(De pVT2X2'1 ~g
] "
(7)
E.H. Novendstern, Turbulent flow pressure drop model for fuel rod assemblies 2.2. Comparison with data used for the correlation The method recommended in eq.(1) - (7) correlates the data to -+ 14% at the 95% confidence level. The experimental data used for the correlation are presented in ref. [2 - 5]. The data include wide ranges of flow, rod and wire dimensions and number of rods. The ranges of data used to determine the multiplication factor, M, are tabulated in table 1, along with the ranges for which the correlation is recommended. All data presented by each experimenter were not used, because of the enormous amount of data. Generally, four or five data points were selected from each different geometrical configuration, with approximately 300 points being used to determine the correlation. Pin bundles containing from 19 to 217 rods enclosed in a hexagonal duct were used.Seven-pin data were not used because of the large effect of the side channels, which may behave differently than central channels and cause additional uncertainty in the value of X, as discussed later. The ratio of numbers of side to central channels in a 7-pin bundle is unity while for a 19-pin bundle it is only onehalf. The experimental data used for the correlation presented the pressure drop data in the form of a friction factor for the complete pin bundle, fT, versus bundle Reynolds number, Re T. The pressure drop for the wire wrap pins would thus be
L :VT2 Z~P=fTDe T
2g
"
(8)
Using the approach discussed in this paper, it is necessary to determine central subchannel conditions. Us-
21
ing the X-factor, the friction factor for the central subchannel is related to the measured value by combining eq.(1) and (6) fl -
ITD
e, X 2 De T
(9)
The Reynolds number must also be adjusted:
Re 1 = X(Del/DeT) Re T •
(lO)
The accuracy of the data is graphically presented in fig. 2 by dividing the experimental friction factor of eq.9 by the multiplier M..With a perfect correlation, the data would fall directly on the smooth friction factor curve. Thus, the scatter is a measure of experimental error and imperfections in the correlation. The data points fall within the ± 14% error band at the 95% confidence level. 2.3. Comparison with sodium data Pressure drop data were obtained from a FFTF 217-pin bundle used for life test simulation in flowing sodium [6]. The wire-wrapped pins were enclosed in a hexagonal duct with 4.354 in between opposite flats, and were about 92 in long. Two pressure taps were located four feet apart within the pin region, and the differentfal pressure loss was obtained by subtracting the absolute pressure readings at the two locations. Pressure drop readings were taken at sodium isothermal temperatures of 400, 800 and 1100°F with sodium flow rates varying from 72 to 533 g/min [10-12]. Fig. 3 compares the theoretical pressure loss with
Table 1 Data used in correlation. Experiment
Number of rods
Rod diameter [in]
Pitch to diameter ratio
Lead to diameter ratio
Reihman [3]
19 to 217
0.196 to 0.300
1.06 to 1.32
24 to 96
2,600 to 185,000
Rehme [2, 4]
19 to 61
0.472
1.13 to 1.42
8 to 50
6,000 to 200,000
Baumann et al. [5] 19 to 61
0.236 to 0.261
1.17 to 1.23
15 to 23
15,000 to 200,000
Applicable range for correlation
0.196 to 0.472
1.06 to 1.42
8 to 96
2,600 to 200,000
19 to 217
Reynolds number
O. 06
0.05
-
-
D-"~T~ .l~IBl'k?
,! i
0
i
~0.0~
, •
REFERENCE2 REFERENCE3
;o
REEERE"CE_
~0.03
~0.02
~{~~~
L
0.01
I
[
I I I I I
jO 3
I
I
I
I I I I
I0 ~
S M O O T H FR C IT O IN FACTORCURVE
I
L
I
I I II
I
I0 6
IO5
REYNOLDS NUMBER
Fig. 2. Comparison of ex 3erimental data with correlation I00
the experimental results. The theoretical prediction was based on hot dimensions, including thermal expansion. If cold dimensions are used, pressure drop predictions would be about 4% too large. The sodium data obtained from the experiment were not used to develop the correlation and are independent verification of the theoretical pressure drop prediction. These data are well within the + 14% range of the model.
f-
T 2
L
1
L
I
3. Derivation and mathematics
of the model
L
='
I0
3.1. Derivation o f flow distribution factor To predict pressure drop and temperatures in a heated fuel-pin bundle it is necessary to predict the flow distribution in the bundle. The method developed in this paper is similar to Sangster's model [1]. Since the pressure drop in all subchannels is the same, the velocity in each subchannel can be determined using the friction factor relationship. For the bundle shown in fig. 1:
,J f~
~,,~/e
L
J
I
f"
~
J
AP 1 = AP 2 = AP 3
THEORETICAL
i ~,/
I
I
i
J
I oo
200
300
t~oo
500
( 11 )
or
600
L pV12
GPM
Fig. 3. Comparison of proposed model with experimental sodium pressure drop data.
2g - I2
Pig22 2g - f3
L
PV32 2g
(12)
E.H. Novendstern, Turbulent flow pressure drop model for fuel rod assemblies Assuming the friction factor for each subchannel obeys the relationship: C
fi- --
Re m
C
-
3.2. Derivation o f smooth friction factor curve multi-
plier, M (13)
pV i Dei/la m
it can be substituted into eq.(12). This assumption is somewhat inaccurate in that the corner and side channels do not have exactly the same friction factor relationship as the central channel, because of the different geometries. However, this assumption is required due to lack of experimental measurements on individual subchannels. The substitution into eq.(12) yields: <2-m) /D~Il+m)
= V~2-m) "/D(le:+m)
= V~2-m)/D(ela+m)
(14)
Eq.(14) is then combined with the continuity equation:
NIV1A 1 +N2V2A 2 +N3V3A 3 = VTAT
The basic parameters that determine M for wirewrapped pin bundles are the pitch-to-diameter ratio, the wire lead-to-diameter ratio and the Reynolds number:
M = g(P/D, H/D, Re).
V'I_ ~/
k
v, (16)
AT
i= 1
The correlation proposed herein was found to fit the data still better. It is similar to eq.(21), with the addi-
(17) -f
and the flow distribution factor for the jth subchannel is k
(21)
\
VI
= XV T
i= 1 NFt i (O~ [Dei)¢~
X=
(20)
Rehme [2] found the following empirical correlation fit the data more accurately:
where a = (1 +m)/(2-m). For m = 0.25 [as in eq.(6)], a = 0.714. This relationship could also be derived for k parallel flow channels that have the same pressure drop. The velocity for the/th subchannel can be determined by
VTA T
2
1 + [ Tr(D/P)(P/H)]
(15)
V1 [ N1A 1 + N2A 2 (De I/De 2 ) a
V/=
(19)
Since theoretical relationships are difficult to derive, empirical relationships are used to correlate the data. However, some ipsight into the pressure drop correlation can be gained by examining the velocity along a wire-wrapped pin. Assuming the flow follows the wire wrap in the central subchannels, a relation between the axial velocity and the increased velocity along the wire can be derived. Fig. 4 divides the velocities into vector components. Assuming all the flow follows the wire, an increased velocity V' 1 is given by:
V1
to eliminate V2 and V3 yielding:
+N3A3(D%/D%) a] = VTAT ,
23
VI
(18)
(Del'/Det Fig.4. Effective velocity.
3t-
T
24
1"7.H. Novendstern, Turbulent flow pressure drop model.for ./uel rod assemblies
tion of a Reynolds number effect. It is assumed that
F = ~CI-(P/D~2+ C3(P/D)C4[H/D)CSRe(67 (22) where C l through C6 are independent constants. Regression analysis of the data yielded F = l / - - 1034
(p/D)O.1~2--
. ~ a ; (P;Di 6'94 Re°'°sis-
- o. ,
(-H/D)2.2
(23)
.
Pressure drop in a wire-wrap-pin bundle, as in eq. 8 or 12 but based on V'I, may be written:
~ = f ~ pv'12 De,
2g
(24)
Substituting the Blasius approximation for the friction factor [eq.(6)] yields:
({OVlDe/u)0.25~
2g ]
"
(25)
where V' 1 was replaced by F V 1. However, F 1"75 may be considered analogous to M, the smooth tube friction factor multiplier, lndeed, they are almost identical, as the empirical correlation of the data indicated that: M = F 1"77 .
(26)
This is remarkably close agreement, considering the wide range of data used. The effect of pitch and wire lead on M is shown in fig. 5. Interesting results are revealed on plotting the ratio of eq.(23) to eq.(20) in fig. 6. This may be considered as the ratio of the calculated "increased" velocity, based on the data, to that of theory alone. At high lead-to-diameter ratios this ratio approaches unity. However, at low lead-to-diameter ratios it is considerably greater than unity. This may be caused by flow jumping the wire, creating additional pressure losses due to contractions and expansions. Thus, when used in eq.24, V' 1 would have to exceed V1 to account for the increased pressure drop.
4. Experimental data and comparison with other analytical models
4.1. Experimental data employed All data analyzed were for a single wire spiraled around each rod. All rods were located on a triangular pitch and enclosed in a hexagonal duct. Other configurations, such as scalloped or circular ducts, were not used because of the difference in geometry in the outher subchannels. Only pin bundles containing 19 or more pins were used. Nuclear reactor fuel assemblies generally contain at least 19 fuel pins.
5.0
R..O REYNOLDS NO. = 50,000
P/D = i.~
3.0
2.0 --
1.3
.2
1.0 i0
20
30 LEAD TO DIAMETER RATIO (H/D)
Fig. 5. Friction factor multiplier.
u'O
50
E.H. Novendstern, Turbulent flow pressure drop model for fuel rod assemblies
25
2.2 REYNOLDS NUMBER = 50,000
~-
2.0
o
1.8
H/D =
,y
1.6
...I
~
I.u, H/D = 2
~
-''''~
H/D : 3 o ~ . . . . ~
1.0 I.I
r
T
1.2
1.3
I.~
PITCH TO DIAMETER RATIO (P/D)
Fig. 6. Velocity ratio. The accuracy of the pressure drop data was of general concern. Rehme [2, 4] did not place pressure taps within the rodded area because cross flow due to the spiral wire wrap might cause erroneous pressure readings. Instead, he placed taps above and below the ends of the pins and used two different lengths of pins for each given geometry. Thus, the difference in pressure drop for each configuration is attributable only to the length difference. Reihman [3] on the other hand, measured the pressure differential at a given axial location using six taps located 60 ° apart around the housing, finding the variations to be small. Therefore, he felt justified connecting all six to a common header, averaging them. Baumann et al. [5] did not report the method nor results of their pressure drop measurements. Instead, they reported correlating equations for the friction factor of each geometry. They presented results for many types of spacers, but only the data for one spiral wrap per rod were used here. 4.2. Comparison with other methods Table 2 compares this method for predicting the pressure drop with the other available methods. The
Table 2 Comparison of accuracy of pressure drop models.
Author
Accuracy at 95% confidence level
Average error fcalc-J'exp - / -. × 100 Jexp
This paper
14%
+1%
Sangster [1]
53%
-18%
Rehme [2]
18%
-5%
expected error at the 95% confidence level and the average error are defined. These methods shall be discussed below. Sangster's model [1] was based on predicting the flow distribution and multiplying the smooth friction factor by two constants, Y and Z. The flow distribution factor, X, is basically the same one as used in this paper. Y and Z depend upon the rod pitch to diameter ratio, P/D, and the wire lead-to-rod diameter ratio, HID. Table 2 shows that this method generally predicts a low and inaccurate value of pressure drop. One possible reason is that the Y-factor, which determines the P/D-effect, is based on bare rod data, not wire
2t>
E.tt. Aovendstern, 7)~rbuh, nl flow pressure drop model .lor ,/uel rod assembh'cs
wrap data. Also, the Z-factor, which evaluates the
H/D-effect, was based partly on pin bundles containmg seven pins in scalloped and circular ducts. In addition, sonic of the data employed exceeded the range he recommended for the correlation. Rehme's model 12] was more accurate, though it fit his data best while not fitting the other data as well. His method does not adequately handle the flow distribution effect, using a ratio of wetted perimeters to determine the flow distribution. The pressure drop model proposed by De Stordeur [131was not checked against these data. tt is approximate in its treatment of wire wrap spacers, and Sangster indicates many reasons why this model is not accurate [1].
tile wire-wrap is "smeared" uniformly in all channels. The central and side subchannels each contain onehalf a wire while each corner subchannel contains onesixth of a wire-wrap. Using the total flow area within the assembly, the average velocity is:
Vy = m/pA T = 20.3 [ft/sec] .
(27)
Using the above dimensions in eq.(2), X can easily be evaluated. From eq.(1) the central velocity is
V 1 = XVT= 0.971 X 20.3 = 19.7 [ft/sec].
(28)
For the central channel, the Reynolds number is Re 1 ,-
pVDcI_53.5 X 19.7 X 3 6 0 0 X 0.124
5. Sample calculation
/a
0.677 X 12
-----
57,900 (29)
As an example, the pressure gradient along an FFTF assembly in the reference design region will be calculated. The assembly comprises 217 pins and fills a hexagonal duct with a 4,335 in across-flats dimension. Each fuel pin has a 0.230 in OD and one 0.056 in wire-wrap with a 12 in helical pitch. Sodium flow is 183,000 lb/hr at 750°F. The liquid sodium properties are:
By eq.(6): Jsmooth = 0.316/Re 0"25 = 0.0204.
(30)
The multiplication factor is evaluated using eq.(4). The P/D-ratio for this assembly is 0.286/0.230 or 1.243. Thus, this factor is
M = ( ~ + 7)°885, U = 0.677 [lb/hr-ft],
where f l - -1.034 - -
p = 53.5[lb/ft3].
1.243 °'124" Assuming the rods are distributed uniformly within the duct, the spacing between centers of adjacent pins is 0.2879 in. The spacing between the surface of the outer rod and the duct is 0.0579 inches. Thus, the three types of subchannels in fig. 1 have the dimensions mentioned in table 3. These dimensions assume
7=
29.7 X 1.2436.94 X 579000.086
(31)
(12/0.23) 2.235 The pressure drop per foot of fuel pin length, by equation 3, is:
L pV~ Table 3 Dimensions of the three types of subchannels in fig. l.
z3aP= MJ~sm°°th De1 2g
Subchannel no.
1
2
3
Total
= 1.05 X 0.0204 X
Number of subchannels
384
48
6
438
= 4.64 ]psi/foot of length].
Flow area [in 2]
0.0139
0.0278
0.0099
6.724
Equivalent diameter [inl
0.124
0.151
0.114
0.128
53.5 X 19.72 X 2 X 32.2 X 144
(32)
E.H. Novendstern, Turbulent flow pressure drop model for fuel rod assemblies
Nomenclature A D De f F g H L m M N P Re V V' X a ~tP p
= F l o w area, = R o d diameter, = Equivalent diameter, = Friction factor, = R a t i o o f effective to axial velocity, = Gravitational constant, = Wire lead (pitch of helix), = Length, = E x p o n e n t o f fraction factor, = F r i c t i o n factor multiplier, = N u m b e r o f subchannels, = R o d pitch ( r o d center-to-center spacing), = Reynolds number, = Axial velocity, = Effective velocity, = F l o w distribution factor, =(2-m)/(l+m), = Pressure drop, = Density,
/~
= Viscosity.
References [1 ] [2]
[3]
[4]
[5]
[6]
[7]
Subscripts
[8]
1 2 3 T
[9]
= = = =
Central subchannel, Side subchannel, Corner subchannel, Total.
[10] [11]
Acknowledgements This w o r k was p e r f o r m e d in support o f the Fast F l u x Test Facility design under A t o m i c Energy Commission, Contrac t A T ( 4 5 - 1 ) - 2171, Task 1.
27
[12] [13]
W.A. Sangster, Calculation of Rod Bundle Pressure Loss, ASME Paper 68-WA/HT-35, 1968. K. Rehme, Geometry-Dependence of the Pressure Loss in Rod Bundles With Coiled Wire Spacers and Logitudinal Flow, Translation of Dissertation Submitted for a Doctoral Degree in Engineering to the Faculty of Mechanical and Process Engineering, University of Karlsruhe, Germany, December, 1967. T.C. Reihman, An Experimental Study of Pressure Drop in Wire Wrapped FFTF Fuel Assemblies, BNWL-1207, September, 1969. K. Rehme, The Measurement of Friction Factors for Axial Flow Through Rod Bundles with Different Spacers, Performed on the INR Test Rig, EURFNR-142P, November, 1965. W. Baumann, V. Casal, H. Hoffman, R. Moeller, and K. Rust, Fuel Elements with Spiral Spacers for Fast Breeder Reactors, EURFNR-571, April, 1968. R.L. Myers, Fabrication of Mark I1 Prototypic FFTF Fuel Subassembly for Sodium Flow Tests, BNWE-I 418, June, 1970. R. Balent and W.H. Hirst, Design and Development of Fast Reactor Fuel Fuel Assemblies, ANS Trans., 13 (Supplement, Conference on Power Reactor Systems and Components) (September 1-3 1970). 42. C.F. Colebrook, Turbulent Flow in Pipes, with Particular Reference to the Transition Region Between the Smooth and Rough Pipe Laws, J.Instit.Civil Engin. (London, England), II (1938-1939), 133. H. Schlichting, Boundary Layer Theory, 4 th edition, McGraw Hill, New York, (1960) 504. R.B. Duffield et al., Reactor Development Program Progress Report, ANL-7705, (June 1970). R.B. Duffield et al., Reactor Development Program Progress Report, ANL-7726, (July 1970). R.B. Duffield et al., Reactor Development Program Progress Report, ANL-7742, (September 1970). A.N. de Stordeur, Drag Coefficients for Fuel-Element Spacers, Nucleonics. (June 1961).
TURBULENT FLOW PRESSURE DROP MODEL FOR FUEL ROD ASSEMBLIES UTILIZING A HELICAL WIRE-WRAP SPACER SYSTEM E.H. NOVENDSTERN Westinghouse Electric Corporation, Advanced Reactors Division, P.O. Box 158, Madison, Pa. 15663, U.S.A. Received 27 August 1971
A semi-empiricalmodel is developed that can accurately predict pressure losses in a hexagonal array of fuel pins utilizing a wire-wrap spacer system. The model is able to predict pressure drop to within +- 14 % over a wide range of geometries in the turbulent flow regime. The model compares favorably with independent pressure drop measurements, not used to obtain the correlation, made on a FFTF 217-pin fuel pin bundle with 400 to 1100°F flowing sodium. A numerical example is presented using the new model. 1. Introduction
2. Description of the model
A wire wrap spacer system utilizes small wires spiraling around each fuel pin to position the pins within their hexagonal duct. This spacing concept has been used in many nuclear reactors and is currently the reference design for the liquid-metal cooled FFTF reactor. Because of the complex geometry caused by the wire, simple equivalent diameter techniques are not sufficient to predict accurately the pressure drop in the fuel pin region of the reactor. An accurate prediction of pressure drop is needed in order that plant parameters may be optimized. Hydraulic testing is needed to establish the pressure drop versus flow characteristics of the fuel assembly. Because testing of all possible configurations is not practical, and the pressure drop in the pan bundle is a substantial portion of the total plant pressure loss, a method to predict pressure drop in fuel assemblies utilizing wire wrap spacing is required. Available techniques of Sangster [1] and Rehme [2] that predict pressure loss were compared with experimental data of Reihman [3], Rehme [2, 4], and Baumann et al. [5]. Because of inaccuracy in these methods, a new technique has been developed, using experimental data and ideas from the previous models. Results compare favorably with pressure drop data on sodium in a prototype FFTF fuel assembly [6].
2.1. Method to predict pressure loss The model developed to predict pressure losses in a wire-wrapped fuel assembly determines the flow distribution between the fuel pins theoretically, and multiplies the pressure drop for a smooth pipe, using equivalent diameter techniques, by an empirical correction factor. The empirical correction depends on fuel pin bundle dimensions and flow rate. The velocity, V1, in a central subchannel, formed by three rods in a triangular array, is determined from assembly dimensions and average fluid velocity, V T, according to: V1 =XV T .
(1)
The flow distribution factor, X, is calculated from central, side and comer subchannels, numbered 1 through 3, as shown in fig. 1. X =
AT [D e \0.714 i D e , ] 0.714 g l a 1 + g2A2~-~e~l) "+ N3A 3 \ ~ ]
(2)
Eq.2 is derived in sec.3. Flow areas, A, and equivalent diameters, De, are calculated including the wire wrap spacer geometry. The wire wrap is "smeared" uniformly in all subchannels. For example, the flow area
2()
lz:tt. Novendstern, Turbulent flow pressure drop model .[br juel rod assemblies. TYPICAL 19 ROD BUNDLE n I = 2~ n2 =
F2
n3 = 6
TYPE 2 OR
TYPE t OR BASIC UNIT CELL
EDGE UNIT CELL
TYPE 3 OR
ROD
CORNER UNIT CELL
LEAD
I
SPIRAL WIRE SPACER
I
Fig. 1. Definition of typical subchannels and of wire spacer lead. in a central subchannel is the area contained between three rods less one-half the area taken up by the wire. Only 50% of the wire area is used because on the average the wire is in that subchannel for only half the axial distance. Similarly, the wetted perimeter is the sum of the perimeters of the three rods in the subchannel plus 50% of the wire perimeter. These subchannel dimensions are based on the rods expanding freely within the duct used to enclose the rods, rather than on a tightly packed rod bundle with larger spacing between the wires on the outer rods and the duct. This expansion occurs because fuel pins are not perfectly straight and each pin has random bow, which causes a uniform distribution of pins within their duct. This expansion is due to the "springy" nature of long slender rods, and has been experimentally investigated by Myers [6] and Balent [7]. Using the flow conditions for the central subchannel, the pressure drop is determined by: L ~gV12 z2xP= Mfsmooth De , 2 (3) The multiplication factor, M, is used to increase the pressure drop above that calculated with the smooth tube friction factor. It primarily accounts for wire lead and fuel pin pitch-to-diameter ratio
29.7(P/D)6.94ReO.0860.885 M= t (P/D)0"124+ (tt/0)2.239 t 1.034
.
(4)
The friction factor for flow in a smooth pipe, fsmooth, is approximated by a modified version of the Colebrook equation [8]:
1 E
Jsmooth = 21Oglo -- ~
logl0 \ Re ]J/
(5)
This equation accurately predicts the friction factor and can easily be utilized when the calculation is done by digital computer. However, since it is tedious to evaluate, the Blasius approximation [9] can be used for hand calculation: fsmooth
=
0.316/Re 0'25 .
(6)
Using the Blasius approximation, the pressure drop can be directly calculated based on average velocity:
=
0.316M -] 0.25
~_ ~DeT]
(De pVT2X2'1 ~g
] "
(7)
E.H. Novendstern, Turbulent flow pressure drop model for fuel rod assemblies 2.2. Comparison with data used for the correlation The method recommended in eq.(1) - (7) correlates the data to -+ 14% at the 95% confidence level. The experimental data used for the correlation are presented in ref. [2 - 5]. The data include wide ranges of flow, rod and wire dimensions and number of rods. The ranges of data used to determine the multiplication factor, M, are tabulated in table 1, along with the ranges for which the correlation is recommended. All data presented by each experimenter were not used, because of the enormous amount of data. Generally, four or five data points were selected from each different geometrical configuration, with approximately 300 points being used to determine the correlation. Pin bundles containing from 19 to 217 rods enclosed in a hexagonal duct were used.Seven-pin data were not used because of the large effect of the side channels, which may behave differently than central channels and cause additional uncertainty in the value of X, as discussed later. The ratio of numbers of side to central channels in a 7-pin bundle is unity while for a 19-pin bundle it is only onehalf. The experimental data used for the correlation presented the pressure drop data in the form of a friction factor for the complete pin bundle, fT, versus bundle Reynolds number, Re T. The pressure drop for the wire wrap pins would thus be
L :VT2 Z~P=fTDe T
2g
"
(8)
Using the approach discussed in this paper, it is necessary to determine central subchannel conditions. Us-
21
ing the X-factor, the friction factor for the central subchannel is related to the measured value by combining eq.(1) and (6) fl -
ITD
e, X 2 De T
(9)
The Reynolds number must also be adjusted:
Re 1 = X(Del/DeT) Re T •
(lO)
The accuracy of the data is graphically presented in fig. 2 by dividing the experimental friction factor of eq.9 by the multiplier M..With a perfect correlation, the data would fall directly on the smooth friction factor curve. Thus, the scatter is a measure of experimental error and imperfections in the correlation. The data points fall within the ± 14% error band at the 95% confidence level. 2.3. Comparison with sodium data Pressure drop data were obtained from a FFTF 217-pin bundle used for life test simulation in flowing sodium [6]. The wire-wrapped pins were enclosed in a hexagonal duct with 4.354 in between opposite flats, and were about 92 in long. Two pressure taps were located four feet apart within the pin region, and the differentfal pressure loss was obtained by subtracting the absolute pressure readings at the two locations. Pressure drop readings were taken at sodium isothermal temperatures of 400, 800 and 1100°F with sodium flow rates varying from 72 to 533 g/min [10-12]. Fig. 3 compares the theoretical pressure loss with
Table 1 Data used in correlation. Experiment
Number of rods
Rod diameter [in]
Pitch to diameter ratio
Lead to diameter ratio
Reihman [3]
19 to 217
0.196 to 0.300
1.06 to 1.32
24 to 96
2,600 to 185,000
Rehme [2, 4]
19 to 61
0.472
1.13 to 1.42
8 to 50
6,000 to 200,000
Baumann et al. [5] 19 to 61
0.236 to 0.261
1.17 to 1.23
15 to 23
15,000 to 200,000
Applicable range for correlation
0.196 to 0.472
1.06 to 1.42
8 to 96
2,600 to 200,000
19 to 217
Reynolds number
O. 06
0.05
-
-
D-"~T~ .l~IBl'k?
,! i
0
i
~0.0~
, •
REFERENCE2 REFERENCE3
;o
REEERE"CE_
~0.03
~0.02
~{~~~
L
0.01
I
[
I I I I I
jO 3
I
I
I
I I I I
I0 ~
S M O O T H FR C IT O IN FACTORCURVE
I
L
I
I I II
I
I0 6
IO5
REYNOLDS NUMBER
Fig. 2. Comparison of ex 3erimental data with correlation I00
the experimental results. The theoretical prediction was based on hot dimensions, including thermal expansion. If cold dimensions are used, pressure drop predictions would be about 4% too large. The sodium data obtained from the experiment were not used to develop the correlation and are independent verification of the theoretical pressure drop prediction. These data are well within the + 14% range of the model.
f-
T 2
L
1
L
I
3. Derivation and mathematics
of the model
L
='
I0
3.1. Derivation o f flow distribution factor To predict pressure drop and temperatures in a heated fuel-pin bundle it is necessary to predict the flow distribution in the bundle. The method developed in this paper is similar to Sangster's model [1]. Since the pressure drop in all subchannels is the same, the velocity in each subchannel can be determined using the friction factor relationship. For the bundle shown in fig. 1:
,J f~
~,,~/e
L
J
I
f"
~
J
AP 1 = AP 2 = AP 3
THEORETICAL
i ~,/
I
I
i
J
I oo
200
300
t~oo
500
( 11 )
or
600
L pV12
GPM
Fig. 3. Comparison of proposed model with experimental sodium pressure drop data.
2g - I2
Pig22 2g - f3
L
PV32 2g
(12)
E.H. Novendstern, Turbulent flow pressure drop model for fuel rod assemblies Assuming the friction factor for each subchannel obeys the relationship: C
fi- --
Re m
C
-
3.2. Derivation o f smooth friction factor curve multi-
plier, M (13)
pV i Dei/la m
it can be substituted into eq.(12). This assumption is somewhat inaccurate in that the corner and side channels do not have exactly the same friction factor relationship as the central channel, because of the different geometries. However, this assumption is required due to lack of experimental measurements on individual subchannels. The substitution into eq.(12) yields: <2-m) /D~Il+m)
= V~2-m) "/D(le:+m)
= V~2-m)/D(ela+m)
(14)
Eq.(14) is then combined with the continuity equation:
NIV1A 1 +N2V2A 2 +N3V3A 3 = VTAT
The basic parameters that determine M for wirewrapped pin bundles are the pitch-to-diameter ratio, the wire lead-to-diameter ratio and the Reynolds number:
M = g(P/D, H/D, Re).
V'I_ ~/
k
v, (16)
AT
i= 1
The correlation proposed herein was found to fit the data still better. It is similar to eq.(21), with the addi-
(17) -f
and the flow distribution factor for the jth subchannel is k
(21)
\
VI
= XV T
i= 1 NFt i (O~ [Dei)¢~
X=
(20)
Rehme [2] found the following empirical correlation fit the data more accurately:
where a = (1 +m)/(2-m). For m = 0.25 [as in eq.(6)], a = 0.714. This relationship could also be derived for k parallel flow channels that have the same pressure drop. The velocity for the/th subchannel can be determined by
VTA T
2
1 + [ Tr(D/P)(P/H)]
(15)
V1 [ N1A 1 + N2A 2 (De I/De 2 ) a
V/=
(19)
Since theoretical relationships are difficult to derive, empirical relationships are used to correlate the data. However, some ipsight into the pressure drop correlation can be gained by examining the velocity along a wire-wrapped pin. Assuming the flow follows the wire wrap in the central subchannels, a relation between the axial velocity and the increased velocity along the wire can be derived. Fig. 4 divides the velocities into vector components. Assuming all the flow follows the wire, an increased velocity V' 1 is given by:
V1
to eliminate V2 and V3 yielding:
+N3A3(D%/D%) a] = VTAT ,
23
VI
(18)
(Del'/Det Fig.4. Effective velocity.
3t-
T
24
1"7.H. Novendstern, Turbulent flow pressure drop model.for ./uel rod assemblies
tion of a Reynolds number effect. It is assumed that
F = ~CI-(P/D~2+ C3(P/D)C4[H/D)CSRe(67 (22) where C l through C6 are independent constants. Regression analysis of the data yielded F = l / - - 1034
(p/D)O.1~2--
. ~ a ; (P;Di 6'94 Re°'°sis-
- o. ,
(-H/D)2.2
(23)
.
Pressure drop in a wire-wrap-pin bundle, as in eq. 8 or 12 but based on V'I, may be written:
~ = f ~ pv'12 De,
2g
(24)
Substituting the Blasius approximation for the friction factor [eq.(6)] yields:
({OVlDe/u)0.25~
2g ]
"
(25)
where V' 1 was replaced by F V 1. However, F 1"75 may be considered analogous to M, the smooth tube friction factor multiplier, lndeed, they are almost identical, as the empirical correlation of the data indicated that: M = F 1"77 .
(26)
This is remarkably close agreement, considering the wide range of data used. The effect of pitch and wire lead on M is shown in fig. 5. Interesting results are revealed on plotting the ratio of eq.(23) to eq.(20) in fig. 6. This may be considered as the ratio of the calculated "increased" velocity, based on the data, to that of theory alone. At high lead-to-diameter ratios this ratio approaches unity. However, at low lead-to-diameter ratios it is considerably greater than unity. This may be caused by flow jumping the wire, creating additional pressure losses due to contractions and expansions. Thus, when used in eq.24, V' 1 would have to exceed V1 to account for the increased pressure drop.
4. Experimental data and comparison with other analytical models
4.1. Experimental data employed All data analyzed were for a single wire spiraled around each rod. All rods were located on a triangular pitch and enclosed in a hexagonal duct. Other configurations, such as scalloped or circular ducts, were not used because of the difference in geometry in the outher subchannels. Only pin bundles containing 19 or more pins were used. Nuclear reactor fuel assemblies generally contain at least 19 fuel pins.
5.0
R..O REYNOLDS NO. = 50,000
P/D = i.~
3.0
2.0 --
1.3
.2
1.0 i0
20
30 LEAD TO DIAMETER RATIO (H/D)
Fig. 5. Friction factor multiplier.
u'O
50
E.H. Novendstern, Turbulent flow pressure drop model for fuel rod assemblies
25
2.2 REYNOLDS NUMBER = 50,000
~-
2.0
o
1.8
H/D =
,y
1.6
...I
~
I.u, H/D = 2
~
-''''~
H/D : 3 o ~ . . . . ~
1.0 I.I
r
T
1.2
1.3
I.~
PITCH TO DIAMETER RATIO (P/D)
Fig. 6. Velocity ratio. The accuracy of the pressure drop data was of general concern. Rehme [2, 4] did not place pressure taps within the rodded area because cross flow due to the spiral wire wrap might cause erroneous pressure readings. Instead, he placed taps above and below the ends of the pins and used two different lengths of pins for each given geometry. Thus, the difference in pressure drop for each configuration is attributable only to the length difference. Reihman [3] on the other hand, measured the pressure differential at a given axial location using six taps located 60 ° apart around the housing, finding the variations to be small. Therefore, he felt justified connecting all six to a common header, averaging them. Baumann et al. [5] did not report the method nor results of their pressure drop measurements. Instead, they reported correlating equations for the friction factor of each geometry. They presented results for many types of spacers, but only the data for one spiral wrap per rod were used here. 4.2. Comparison with other methods Table 2 compares this method for predicting the pressure drop with the other available methods. The
Table 2 Comparison of accuracy of pressure drop models.
Author
Accuracy at 95% confidence level
Average error fcalc-J'exp - / -. × 100 Jexp
This paper
14%
+1%
Sangster [1]
53%
-18%
Rehme [2]
18%
-5%
expected error at the 95% confidence level and the average error are defined. These methods shall be discussed below. Sangster's model [1] was based on predicting the flow distribution and multiplying the smooth friction factor by two constants, Y and Z. The flow distribution factor, X, is basically the same one as used in this paper. Y and Z depend upon the rod pitch to diameter ratio, P/D, and the wire lead-to-rod diameter ratio, HID. Table 2 shows that this method generally predicts a low and inaccurate value of pressure drop. One possible reason is that the Y-factor, which determines the P/D-effect, is based on bare rod data, not wire
2t>
E.tt. Aovendstern, 7)~rbuh, nl flow pressure drop model .lor ,/uel rod assembh'cs
wrap data. Also, the Z-factor, which evaluates the
H/D-effect, was based partly on pin bundles containmg seven pins in scalloped and circular ducts. In addition, sonic of the data employed exceeded the range he recommended for the correlation. Rehme's model 12] was more accurate, though it fit his data best while not fitting the other data as well. His method does not adequately handle the flow distribution effect, using a ratio of wetted perimeters to determine the flow distribution. The pressure drop model proposed by De Stordeur [131was not checked against these data. tt is approximate in its treatment of wire wrap spacers, and Sangster indicates many reasons why this model is not accurate [1].
tile wire-wrap is "smeared" uniformly in all channels. The central and side subchannels each contain onehalf a wire while each corner subchannel contains onesixth of a wire-wrap. Using the total flow area within the assembly, the average velocity is:
Vy = m/pA T = 20.3 [ft/sec] .
(27)
Using the above dimensions in eq.(2), X can easily be evaluated. From eq.(1) the central velocity is
V 1 = XVT= 0.971 X 20.3 = 19.7 [ft/sec].
(28)
For the central channel, the Reynolds number is Re 1 ,-
pVDcI_53.5 X 19.7 X 3 6 0 0 X 0.124
5. Sample calculation
/a
0.677 X 12
-----
57,900 (29)
As an example, the pressure gradient along an FFTF assembly in the reference design region will be calculated. The assembly comprises 217 pins and fills a hexagonal duct with a 4,335 in across-flats dimension. Each fuel pin has a 0.230 in OD and one 0.056 in wire-wrap with a 12 in helical pitch. Sodium flow is 183,000 lb/hr at 750°F. The liquid sodium properties are:
By eq.(6): Jsmooth = 0.316/Re 0"25 = 0.0204.
(30)
The multiplication factor is evaluated using eq.(4). The P/D-ratio for this assembly is 0.286/0.230 or 1.243. Thus, this factor is
M = ( ~ + 7)°885, U = 0.677 [lb/hr-ft],
where f l - -1.034 - -
p = 53.5[lb/ft3].
1.243 °'124" Assuming the rods are distributed uniformly within the duct, the spacing between centers of adjacent pins is 0.2879 in. The spacing between the surface of the outer rod and the duct is 0.0579 inches. Thus, the three types of subchannels in fig. 1 have the dimensions mentioned in table 3. These dimensions assume
7=
29.7 X 1.2436.94 X 579000.086
(31)
(12/0.23) 2.235 The pressure drop per foot of fuel pin length, by equation 3, is:
L pV~ Table 3 Dimensions of the three types of subchannels in fig. l.
z3aP= MJ~sm°°th De1 2g
Subchannel no.
1
2
3
Total
= 1.05 X 0.0204 X
Number of subchannels
384
48
6
438
= 4.64 ]psi/foot of length].
Flow area [in 2]
0.0139
0.0278
0.0099
6.724
Equivalent diameter [inl
0.124
0.151
0.114
0.128
53.5 X 19.72 X 2 X 32.2 X 144
(32)
E.H. Novendstern, Turbulent flow pressure drop model for fuel rod assemblies
Nomenclature A D De f F g H L m M N P Re V V' X a ~tP p
= F l o w area, = R o d diameter, = Equivalent diameter, = Friction factor, = R a t i o o f effective to axial velocity, = Gravitational constant, = Wire lead (pitch of helix), = Length, = E x p o n e n t o f fraction factor, = F r i c t i o n factor multiplier, = N u m b e r o f subchannels, = R o d pitch ( r o d center-to-center spacing), = Reynolds number, = Axial velocity, = Effective velocity, = F l o w distribution factor, =(2-m)/(l+m), = Pressure drop, = Density,
/~
= Viscosity.
References [1 ] [2]
[3]
[4]
[5]
[6]
[7]
Subscripts
[8]
1 2 3 T
[9]
= = = =
Central subchannel, Side subchannel, Corner subchannel, Total.
[10] [11]
Acknowledgements This w o r k was p e r f o r m e d in support o f the Fast F l u x Test Facility design under A t o m i c Energy Commission, Contrac t A T ( 4 5 - 1 ) - 2171, Task 1.
27
[12] [13]
W.A. Sangster, Calculation of Rod Bundle Pressure Loss, ASME Paper 68-WA/HT-35, 1968. K. Rehme, Geometry-Dependence of the Pressure Loss in Rod Bundles With Coiled Wire Spacers and Logitudinal Flow, Translation of Dissertation Submitted for a Doctoral Degree in Engineering to the Faculty of Mechanical and Process Engineering, University of Karlsruhe, Germany, December, 1967. T.C. Reihman, An Experimental Study of Pressure Drop in Wire Wrapped FFTF Fuel Assemblies, BNWL-1207, September, 1969. K. Rehme, The Measurement of Friction Factors for Axial Flow Through Rod Bundles with Different Spacers, Performed on the INR Test Rig, EURFNR-142P, November, 1965. W. Baumann, V. Casal, H. Hoffman, R. Moeller, and K. Rust, Fuel Elements with Spiral Spacers for Fast Breeder Reactors, EURFNR-571, April, 1968. R.L. Myers, Fabrication of Mark I1 Prototypic FFTF Fuel Subassembly for Sodium Flow Tests, BNWE-I 418, June, 1970. R. Balent and W.H. Hirst, Design and Development of Fast Reactor Fuel Fuel Assemblies, ANS Trans., 13 (Supplement, Conference on Power Reactor Systems and Components) (September 1-3 1970). 42. C.F. Colebrook, Turbulent Flow in Pipes, with Particular Reference to the Transition Region Between the Smooth and Rough Pipe Laws, J.Instit.Civil Engin. (London, England), II (1938-1939), 133. H. Schlichting, Boundary Layer Theory, 4 th edition, McGraw Hill, New York, (1960) 504. R.B. Duffield et al., Reactor Development Program Progress Report, ANL-7705, (June 1970). R.B. Duffield et al., Reactor Development Program Progress Report, ANL-7726, (July 1970). R.B. Duffield et al., Reactor Development Program Progress Report, ANL-7742, (September 1970). A.N. de Stordeur, Drag Coefficients for Fuel-Element Spacers, Nucleonics. (June 1961).