- Email: [email protected]
Formulation of the fluid-solid interaction force for multi-dimensional, two-phase flow within tube arrays
Nuclear Engineering and Design 58 (1980) 383-391 © North-Holland Publishing Company
FORMULATION OF THE FLUID-SOLID INTERACTION FORCE FOR MULTI-DIMENSIONAL, TWO-PHASE FLOW WITHIN TUBE ARRAYS Y. COI~FFI~ and N. TODREAS * ElectricitJ de France, Direction Des Etudes et Recherches, Laboratoire National d'Hydraulique, Chatou 78400, France
Revised version received 5 December 1979
1. Introduction
need for such a tensor formulation for rod arrays, none has yet been presented. Nor has the experimental program been specified which would be necessary to fully characterize such a tensor formulation and to determine if significant differences exist between approaches for flows at arbitrary angles over rod bundles. In this paper for two-dimensional, homogeneous, two-phase flow, we present such a formulation as a flow resistance matrix and the required experimental steps to def'me all elements of this matrix. Furthermore, an interim expression for the matrix is proposed for use pending development of the complete flow resistance matrix. As an example of the use of the interim expression, the case of a PWR steam generator is considered, and a complete definition of the formulation for this case is presented.
The development of computational models for the analysis of multi-dimensional flow fields within tube bundles has accelerated in recent years. Presently, it is generally perceived that the level of sophistication in the numerical techniques utilized needs to be balanced by complementary developments of needed constitutive models. A proper formulation for the analysis of tube bundles which has been embodied in several hydraulic codes for core, heat exchanger or steam generator like COMIX-1 [1], FAITUVIT [2], BIGEVE [3], is to view the rods in the array as an anisotropic porous medium. However the characterization of required constructive laws must also be consistent with this view of a multi-dimensional flow through an anisotropic porous medium. In this paper we deal with the characterization of the momentum transfer between the array of tubes and the fluid. Most previous analysis has simply formulated this fluid-solid interaction by applying the components of the mass flux along the longitudinal and cross-flow directions in pressure drop relations for one-dimensional flow in these directions. This component approach implies a specific, unverified dependence of the interaction force with flow direction. Conversely, the general approach of formulation of a fluid-solid interaction tensor provides the framework for proper determination of this dependence as experimental data becomes available. While several authors [4,5] have recognized the
2. General relations We require expressions for the interaction forces per unit mass, F x and Fy, which can appear in conservation equations as the terms h p F x and hpFy which are forces per unit area. These forces represent the components of the total force per unit area, hpF, which is the force exerted on the fluid by the solid surfaces. The term h represents the thickness of the unit calculational element and is included explicitly to allow the possibility for variable width of this element. Table 1 summarizes the relevant force expressions and their dimensions for the general direction z. The total force is an unknown function of the superficial velocity W or mass flux G = 0 W, the rod
* Professor of Nuclear Engineering, MIT, on Sabbatical Leave at Chatou. 383
384
Y. Coeff6, N. Todreas / F o r m u l a t i o n o f the f l u i d - s o l i d i n t e r a c t i o n / b r c e
Table 1 Expression
Dimension
hpFz, pFz ,
F/L2
hap/az ap/az
tubes>
G=
F/L 3
1 ap 1Fz ,
F/M
p 3z
- - - -
G× Fig. 1. C o m p o n e n t s of the mass flux vector G.
array geometry and the state of the fluid mainly, its density p and quality X. We choose to express that function with the variables pF and G as o F = ft-(G, geometry, p, It, X)-
(1)
We have taken the flow as hnmogeneous. Otherwise, we should add slip ratio to the arguments of ~r. Since 5r is a homogeneous function of G(~'(G) = O, where G = 0) and assuming regularity of St(G) near 0, we may write eq. (1) for a given geometry as
X->-
y-axis parallel to the tubes and the x-axis transverse to the tubes), it can be seen that ~ and K are diagonal so that D is also diagonal. Therefore we can express the total force per unit volume of eq. (5) as
°
,e
D2(X, P, It, G)
LO
x
]
[-K,(G,o, It)
O
-0
K2(G, p,
] #)
(8)
a.
where K is the general matrix for flow resistance. For two-phase flow, it is common to evaluate the two-phase pressure gradient as equal to the product of the single-phase gradient assuming the total flow as a liquid and a two-phase multiplier ~[o, i.e.,
Let us define 0 as the angle between the y-axis and the flow-direction measured clockwise. Fig. 1 illustrates this angle and the components of the mass flux vector G. No we must note that the dependence of a quantity Q on G may be expressed in either of two equivalent ways
d p / d 2 = ( ¢ ~ o X d p / d Z ) L O at total f l o w ,
Q(Gx, Gy)
pF = ~(G, p, It, X) G,
(2)
(3)
since G = IGI =
where ~ , o is generally a function ot X, P, It, G. We will similarly take the two-phase effects of F into account by introducing a two-phase multiplier matrix, ~(X, P, #, G), and the total force assuming the total flow is a liquid, FLO yielding
F=~FLo.
(4)
~,(G, p, It) G,
(5)
so that eq. (4) becomes
pF = B(X, p, It, G) K(G, p, It) G,
(6)
=
(9)
rt",2 t,,]x ± r t~211/2 Lryj .
We will now presentthe determination of the flow resistance matrix K" for one-phase flow from the available literature data. In the following development the subscript LO will be omitted from the z ~ and F symbols for simplicity. For this case eq. (8) can be reduced to the form -
F K ~ ( G , p , It)
pF=K(a,p,it) a =Lo
which defines J~ in eq. (2) as •(G, p, It, X) ~(X, P, It, 17) K'(G, p, ~).
Q(O,G),
3. Theoretical determination of the flow resistance matrix ~ for single-phase flow
For a given geometry by analogy with eq. (2) FLO can be expressed as PFLo =
or
(7)
If we take now as the reference coordinate system the axes of symmetry of the bundle (assigning the
0
K2(G, p, It)
(lO) All the experimental data from the literature gives only the values of pressure loss measured along the
Y. Coeff~, N. Todreas / Formulation o f the fluid-solM interaction force
G direction, i.e., along 0 for different values of 0 and G. We will designate that pressure loss by Ape(O, G, O, la) and the corresponding component o f F by FiT. Pressure losses in the direction perpendicular to G have not been reported. We will designate them as APzD (0, G, O,/2) and the corresponding component of F by F±G. From the definition ofF, per table 1, we can relate the four previovs quantities by 1 oFG(O, 63 = -~o ApG(O' 63,
(1 1.1)
1
oF±G(O,63 - Lo+Tr/2AP±G(O'G) ,
(1 1.2)
where the dependence of forces and pressure losses on O and gt is no longer specifically indicated and FG and F±~7 may be expressed in terms of G through eq. (10) as
oFG = oF" G G = (Ka(0, G) sin20 + Ks(O, (7) cos20) G ,
= (KI(O, G) - g 2 ( o ,
G))(sin 0 cos 0) G.
(12.1)
(12.2)
385
/~-matrix has been undertaken with one limited exception which is utilized in the next section for 0 values along the principle axis, i.e., (a) KI(O, G) determined for Kl(0, 63, i.e., longitudinal flow through rod bundles. (b) K2(O, G) determined for K2(90 °, G), i.e., crossflow over rod bundles. Consequently the K'-matrix cannot presently be expressed even for the simpler case of single-phase flow. Clearly an experimental program in which Ape(0, G) and Ap±G(O, G) are simultaneously measured over the range of (0, G) and arra_y geometry of interest is needed to evaluate the K-matrix. In the absence of these measurements, it is still desirable to estimate the value of the elements of the K'-matrix to provide the needed input to the multi-dimensional analyses being undertaken. Let us start this estimation process by summarizing the useful data which is available for which at least ApG(O, G) has been determined. 4. Available literature data
The existing data are those of Kazakevich [6] interpreted in Idel'cik [7] from which we know the values of
Aoc(0, 63 ~b ApG(90 o, 63 _
Equating the above sets of equations leads to the following relations 1
- - ape(0, 63 L0 = [KI(O, G) sin20 + K2(O, 63 cos20] G , 1
Lo+~r/2
(13.1)
¢(0) _
(13.2)
These two relations show clearly that measurement of Ap6 and Ap±G for every 0 and G would allow the complete determination of the two functions KI and K2. This result is entirely consistent with the literature on hydraulic characteristics of anisotropic media. The rod bundle is a special case of anisotropic media which in the past has been analyzed considering only one-dimensional flow. Therefore experimental determination of the elements of the
(14)
An important feature of the function ~b as reported in [7], is that it may be considered as independent of G so that we may write it as ~b(0). Eq. (13.1) gives then a new expression of ~b(0):
Ap±G(O, 63
= ([KI(O, G) - K2(O, 63] sin 0 cos 0} G.
for 0 = 30 °, 45 ° and 60 ° .
=
Lo[KI(O, 63 sin20 + K2(O, 63 cos20] Lg0oKl(90 °, G)
(15)
Also recall that the literature provides available data for both purely longitudinal and transverse flows so that values of K1(90 °, G) and K2(0 °, G) are available. 5. Evaluation of K from available data
We now have to make one hypothesis to determine the two functions KI and K2 from the single eq. (15).
Y. Coeffd,N. Todreas/ Pbrmulation of the fluid-solM interactionforce
386
Since we know that K1(90 °, G) > > K 2 ( 0 °, G) for all fixed G we will assume that K1 and K2 are not strongly dependent on 0 and that
Kl(O, G) > > K 2 ( 0 , G)
for all given G and 0.
For clarity now we will determine K~ and K2 successively over the two ranges: 02 <~ 0 < 30 ° and 30 ° < 0 < 90 °. (a) 30 ° ~< 0 ~< 90 °. From the previous hypothesis we may write: K1 sin20 > > K2 cos20 ,
(ii) Continuity with the expressions of K 1 and K2, for the range of 30 ° ~< 0 ~< 90 °. We will therefore take for K: in 0 ° ~< 0 ~< 30 °:
K2(O, G) = K2(O°, G). For K1 we will estimate the function ~o(0) by adding an extrapolated value for ~o(0°). Table 2 summarizes the results for the function ~o(0). The final expressions for Kt and K2 are
K,(O, G) = ~o(0)K1(90 ° , a ) ,
(19.1)
and
30 ° ~< 0 ~< 90 ° ,
K2(O, G) = K2(O°, G)
and eq. (15) becomes:
for 0 ~< 0 ~< 90 ° .
(19.2)
Lo KI(O, G) sin20 t~(0) - L9o °
K2(90 o, G)
(16)
Since the measurements of Kazakevich were made with a single bundle, the length Lo can be expressed as
L 0 = L90o/sin 0 .
(17)
Utilizing eq. (17) in eq. (16), we obtain the desired result of am expression for KI(O, G), i.e.,
KI(O'G)=~°(O)K~(90°'G) t- 30 ° ~<0 < 9 0 ° .
LO
D2(X, ,o,/1, G)
There are two unknown functions which may also be written as
D I(X, p, ta, O, G),
Concerning K2 in this range we have assumed above that its effect is not important. To ensure continuity with the range ( 0 o - 3 0 ° ) we will take =
The matrix of two-phase multipliers is equal to:
D2(X, P,/a, 0, G ) .
(18)
where ~o(0) = tk(0)/sin 0 )
K2(O, G) K2(O°, G),
6. Determination of
30 ° ~< 0 ~< 90 ° .
(b)O°~
The literature provides only two-phase flow multipliers for purely longitudinal or transverse flow so that we know DI(X, p,/~, 90 °, G)
and
D2(X, p,/~, 0 °, G ) .
In the absence of more complete data we will assume that the two functions D1 and D~ do not vary much with 0 so that we may take
D,(X,p, IJ, O,G)=D,(x,p,u, 90°,G),
(20.1)
D2(x,p, la, O,G)=D,(x,p, la, O°,G),
(20.2)
for 0 ° ~< O ~< 90 °. Table 2
7. Summary of results for matrix
0
~b
~o(0) = qJ/sin 0
0 30 45 60 90
0.30 0.54 0.82 1.00
0.25 0.60 0.76 0.94 1.00
Eqs. (19) and (20) give the expressions chosen to determine the complete expression o f p F based on eq. (8), i.e.,
pF =
[DI(x'P'Ia'90°'G)
O
L0
D2 (X, P, P, 0°, G)
1
Y. Coeff~, N. Todreas / Formulation of the fluid-solid interaction force × [~(f) K~(90°, G)
0
t-0
Appendix
]
K2(0 °, G)
G.
(21)
We must further specify the expressions of the four functions D1, Du, Kl, and K2 which are for purely transverse or longitudinal flow. The general form of the pressure gradient commonly used is:
~P
_ f G2 LO at total flow Lc 2OL'
(22)
where the characteristic lengthLc is in common practice different for longitudinal and transverse flow. We may then define new quantities k by the relations: Kt(90 °, G) =
G (k9oo)LO
- -
(23.1)
,
2PL
G K2(O°, G) = (k0o)LO 2p L •
387
(23.2)
In this Appendix the parameters necessary to completely define eq. (25) are specified for the conditions applicable to PWR steam generators. A.I. Transverse flow across a tube array,
(k90°)LO
(z ---x in our convention) For transverse flow in an array of tubes, it is customary to express the pressure drop in terms of: (1) the pressure drop across one row of tubes based on a coefficient (Ao)Lo which comprises the terms f/Lc ofeq. (22), (2) the number of rows of tubes in the increment dx, i.e.,
dxlp, (3) the velocity head between the rows, i.e., based on the total mass flux G: G2/2OL .
We also make a change of notation for D~ and D2 and set: DI(X, 9, U, 90°, G) = ¢~o* ,
(24.1)
D2 (X, 9, ta, 0 °, G) = ¢~o .
(24.2)
Finally the general expression for oF as defined by eq. (2) becomes
Following these conventions, eq. (22) for 0 = 90 ° may be written as ~[ LO at total flow
= (A°)LO 1p G2
0
R~
G,
(25)
where R1 = t~0%0(0)(k90°)LO G 20L
G R2 = eg°(ko°)LO - 2p L '
(A.2)
It is important to note that since (kgoO)Lo is evaluated in accordance with eqs. (19) and (23.1), our evaluation of (kg0O)Lo should be done using the superficial mass flux along 0, i.e., G = pW = p(U ~ + I,/2)1/:
(A.3)
and the associated reference liquid Reynolds number, (Ref)L (Ref)L = I) WD/IAL.
p(O) has already been determined in table 2, ¢,~oo and (kgoO)Lo will be specified in the Appendix, ¢~o and (koo)Lo will be specified as well in the Appendix. The appendix specifies results for a specific case, co~aditions applicable to PWR steam generators, to complete the definition of our recommended procedure.
(A.1)
Comparing eqs. (22) and (23.1) with eq. (A.1), we determine that (k9 0*)LO = (Ao)Lo/P •
oF = J~(G, 0) G =
2pL
(A.4)
A. 1.1. Single phase transverse drag coefficient [(A0)LO] Now [(Ao)LO] is available for square arrays from the work of Boissier et al. [8]. A. 1.2. Two-phase transverse, horizon tal flow multiplier - (~o o The two-phase multiplier for horizontal cross flow
388
Y. Coolie, N. Todreas /I~brmulation o f the fluid solid interaction Jbrce
in a triangular array of transverse P/D = 1.25 (longitudinal P/D = 1.09) has been correlated by Grant and Murray [9]. It is possible that results for a square array ofP/D = 1.46 may differ because the degree of void redistribution and recovery in passing through the rod gap contraction - expansion geometry may vary significantly enough with geometry to affect the multiplier. However, in the absence of directly applicable data, the correlations of Grant and Murray are specified since they are certainly more applicable than those surveyed by Idsinga et al. [10] for longitudinal flow. Two correlations are recommended with the selection dictated by the existing flow regime. For stratified and stratified-spray, the recommended equation is ¢,3oo = 1 + ( r ~ -
1)
X (0.25X°'769(1 - -
X) 0'769
-[" X 1 " 5 3 s )
.
(A.5)
For spray and bubbly flow, the recommended equation is:
¢,)oo=l+(r 2 -
1)
X {0.75X°'769(1 - X)0"769 + X 1'538} ,
(A.6)
where
Fz - OL/og. Stratified flow regimes are generally encountered only at low mass velocities which are generally outside the normal steam generator operating range.
A.2. Longitudinal flow in a tube array For longitudinal flow in an array of tubes (z ~ y in our convention). It is customary to express the pressure gradient in terms of: (1) A friction factor and equivalent diameter which for longitudinal flow represent the terms f and Lc of eq. (22), i.e., (AL) LO/DH • p (2) A velocity head based on the effective axial velocity o = G/np where n is the porosity. Following these conventions, eq. (22) for 0 = 0 ° can be written as
d~ LO at total flow
_ (AL)rO
DH
(AL)LO
(or) 5 _
2pL
DH
G2
2,0Ln2 "
(A.7)
Comparing eq. (22) and (23.2) with eq. (A.7), we determine that (AL)LO
(A.8)
(koO)LO- DHn2 • For our square array, D H equals: DH-
4[e 7rD
D4rI P I 2
L\DI
1
-
q
~rj.
(A.9)
Evaluation of (AL)LO and ¢~o remain. In the case of the steam generator the flow will be turbulent according to the criterion: GDH/IaL > 2100. We will then propose results for the case of turbulent flow.
A.2.1. Single phase longitudinal turbulent flow (AL)LO turbulent Now (AL)Lo is the single phase friction factor for longitudinal flow in our non-circular geometry, i.e., square array. For triangular arrays in turbulent flow, Dwyer and Berry [11] have predicted (AL)LO as a function ofP/D. For P/D ratios of 1.35 or greater, slug flow analyses [12] have demonstrated that Nusselt number results for triangular and square arrays are within 4%. Since turbulent flow condition should give comparable or less difference in Nusselt number, and since friction factor behaviour is very similar to Nusselt number behaviour (p. 183 Kays, [13]), the triangular array results [1 I] can be used for our square array case ofP/D = 1.46 with only a slight overestimate in friction factor resulting. These results express the friction factor (AL)LO as a multiple of the friction factor in the circular tube, (Act)LO, when (Act)Lo is calculated from eq. (A.IO). (Moody friction factor curve for smooth pipe.) (Act)LO = 0.0058 + 0.432 ((Re)DH)~°'3,
(A.10)
where ((Re)DH)L is the liquid phase rod array Reynolds number at the total flow, which since (k0)LO is evaluated as (koO)Lo in accordance with eq. (19)
Y. Codffd,N. Todreas / Formulation of the fluid-solid interaction force Table A. 1 Friction factors in longitudinal flow in triangular arrays (from [8])
P/D
(AL)LO
Applicability
(Act)LO 1.001 1.01 1.02
0.640 0.775 0.855
1.03 1.04 1.05
0.910 0.950 0.980
1.06
1.000
1.07 1.10 1.15 1.20 1.30
1.015 1.040 1.065 1.080 1.100
1.40 1.60 1.75 1.80 2.00
1.115 1.130 1.140 1.145 1.150
Triangular arrays only
Square and triangular arrays
work. Chisholm [ 15] has generalized his previous work on two-phase pressure gradient to cover our case of smooth tubes, and also to include the Baroczy results as a basis for his required coefficient. For predictions over wide variations of both pressure and mass flux, we would recommend Chisholm's correlation with the coefficient B evaluated per his suggestions in his table 2 as a function of F and G. The values of table 2 reflect a compromise between the correlations of Baroczy, Lockhart-Martinelli and Chisholm. For the steam generator case, however, the pressure is nearly constant so that F is a constant. Also, the G value is 800 kg/m a s or less. With this restricted range or parameters, a best estimate value of the gradient can be made using the Baroczy results in this range in the form presented by Chisholm in his fig. 4. This results in an equation for B of the form B = fiG~ = 4.08
and (23.2), should be based on the equivalent longitudinal velocity G/pn so that ((Re)DH)L =.GDH _P( U'z + V2) I/2 D H
nPL
nlaL
(A.I1)
The multiplier, m, is given in table A1 and for P/D = 1.46, m = 1.12 where (AL)LO ~- m(Act)L 0 .
m(Act)LO DHn2
(pLtO'S("gtn/2
f°r r = ~Z!
\Z!
for G.V. pressure and n = 0.25,
(A.14)
where =-0.447, = 45.92, G = p ( U 2 + 1,'2) i n .
The parameter B is used in the general Chisholm equation for ~b~o,i.e., ~b~o= ATTp
APLo
(A. 12)
= 1 + (1.`2 -
Substituting eq. (A.12) into eq. (A.8) yields (ko°)LO -
389
1){B×(2-n)/2(1 - X)(2-n) + X2-n} , (A.15)
(A.13)
A.2.2. Two-phase muln'plier - ~ o Two phase multipliers are available from a variety of correlations. In our steam generator applications because of the wide range in mass velocity and the full quality range covered, we desire an approach for the two-phase multiplier which (a) includes mass velocities effects, (b) goes to the correct limits at the all liquid and all vapor conditions. Baroczy's correlation [14] satisfies these conditions but is too cumbersome to prepare for numerical
where we take n = 0.25.
Nomenclature
Dimensions B
D D 1, D2 DH
Constant defined by eq. (A.14) Two-phase multiplier Rod diameter Diagonal elements of matrix D Hydraulic diameter
L
L
Y. Coof]~, N. Todreas / bbrmulation of the fluid-solM interaction ]brce
390
f F G G
5r Gx, Gy
Fx, Fy, Fz h K K1, K2 Lc L m n
P
ap
P Q (Ref)L
U V W W X, y Z
Friction factor Force per unit mass, vector Total mass flux = IGI Mass flux vector Function defined by eq. (1) Mass flux in x, y directions Forces per unit mass in x, y, and z directions Element variable width Single phase matrix Diagonal elements of matrix Characteristic length Length across bundle Ratio of friction factors defined by eq. (A.12) (1) Volumetric averaged rod array porosity (2) Coefficient of eq. (A.14) Pressure Pressure loss Rod array pitch (distance between rod centerlines) General quantity per eq. (9) Reference Reynolds number General matrix for flow resistance Superficial velocity in x direction Superficial velocity in y direction Superficial velocity in z direction Superficial vector velocity Coordinate directions Arbitrary direction in x - y plane
-F/M M/L2T M/LT
42 X (Act)LO
M/L2T
(AL)LO
F/M L
(Ao)Lo
FT/L L L
/a
-
0
~0 F/L 2 F/L 2
#
--
L M/LT
Deg.
Subscripts L -
L G LO G ±G 0
Liquid Gas Total flow is as a liquid In G direction Normal to G In 0-direction
L/T References L/T L/T L/T L L
Greek symbols
p:
Two-phase multiplier Flowing quality Friction factor for singlephase flow in a circular tube Longitudinal friction factor for single-phase flow in a rod array Transverse drag coefficient for single-phase flow across one row of tubes - equal tof/Lc ofeq. (4.7) Mixture viscosity Ratio of pressure drops defined by eq. (14) Angle measured clockwise from y axis to mass flux vector G Function of 0 defined by eq. (18)
Constants to evaluate constant B o f eq. (A.14) Liquid to vapor density ratio appearing in eqs. (A.5), (A.6) and (A.14) Two-phase mixture density M/L 3
[ 1] W.T. Sha et al., NUREG/CR-0785, ANL-77-96 (Sept. 1978). [2] A. Verwey, Y. Caytan, O. Daubert and D. Taillifet, Numerical model of the flow in a LMFBR heat exchanger, paper presented at the 4th Seminar of the XVIII IAHR Congress (Sept. 1979). [3] Y. Co~ff6 and N. Todreas, 2-D mathematical model of the two phase flow in a steam generator, paper presented at the 4th Seminar of the XVIII IAHR Congress (Sept. 1979). [4] D. Butterworth, Internat. J. Heat Mass Trans. (1978) 256-258. [5] W.J. Wnek, J.D. Ramshaw, J.A. Trapp, E.D. Hughes and C.W. Solbdg, ANCR-1207 (Nov. 1975). [6] F.P. Kazakevich, IZW. Vses. Inst. Imoni F.E. Dzerzhimkogo 8 (1952) 7-12. [7] I.E. Idel'cik, AEC-TR-6630 (1960). [8] A. Boissier, M. Chatillon, D. Gautier, R. Perez and A.
Y. Coeffd, N. Todreas /Formulation of the fluid-solid interaction force
[9] [10] [11] [12]
Marchand, E.D.F. Bulletin de la Direction des Etudes et Recherches, S6rie A, Nucl6aire, Hydraulique, Thermique, no 2/3 (1971). I.D.R. Grant and I. Murray, NEL Report 560 (March 1974). W. Idsinga, N. Todreas and R. Bowring, Internat. J. Multiphase Flow 3 (1977) 401-413. O.E. Dwyer and H.C. Berry, Nucl. Engrg. Des. 23 (1972) 273-294. O.E. Dwyer and H.C. Berry, Nucl. Sci. Engrg. 39 (1970) 143.
391
[13] W. Kays, Convective Heat and Mass Transfer (McGrawHill, New York, 1966). [ 14 ] C. Baroczy, A systematic correlation for two-phase pressure drop, AICHE Reprint No. 37, paper presented at 8th National Heat Transfer Conf., Los Angeles, August 1965. [15] D. Chisholm, Internat. J. Heat Mass Transfer 16 (1973) 347-358.
FORMULATION OF THE FLUID-SOLID INTERACTION FORCE FOR MULTI-DIMENSIONAL, TWO-PHASE FLOW WITHIN TUBE ARRAYS Y. COI~FFI~ and N. TODREAS * ElectricitJ de France, Direction Des Etudes et Recherches, Laboratoire National d'Hydraulique, Chatou 78400, France
Revised version received 5 December 1979
1. Introduction
need for such a tensor formulation for rod arrays, none has yet been presented. Nor has the experimental program been specified which would be necessary to fully characterize such a tensor formulation and to determine if significant differences exist between approaches for flows at arbitrary angles over rod bundles. In this paper for two-dimensional, homogeneous, two-phase flow, we present such a formulation as a flow resistance matrix and the required experimental steps to def'me all elements of this matrix. Furthermore, an interim expression for the matrix is proposed for use pending development of the complete flow resistance matrix. As an example of the use of the interim expression, the case of a PWR steam generator is considered, and a complete definition of the formulation for this case is presented.
The development of computational models for the analysis of multi-dimensional flow fields within tube bundles has accelerated in recent years. Presently, it is generally perceived that the level of sophistication in the numerical techniques utilized needs to be balanced by complementary developments of needed constitutive models. A proper formulation for the analysis of tube bundles which has been embodied in several hydraulic codes for core, heat exchanger or steam generator like COMIX-1 [1], FAITUVIT [2], BIGEVE [3], is to view the rods in the array as an anisotropic porous medium. However the characterization of required constructive laws must also be consistent with this view of a multi-dimensional flow through an anisotropic porous medium. In this paper we deal with the characterization of the momentum transfer between the array of tubes and the fluid. Most previous analysis has simply formulated this fluid-solid interaction by applying the components of the mass flux along the longitudinal and cross-flow directions in pressure drop relations for one-dimensional flow in these directions. This component approach implies a specific, unverified dependence of the interaction force with flow direction. Conversely, the general approach of formulation of a fluid-solid interaction tensor provides the framework for proper determination of this dependence as experimental data becomes available. While several authors [4,5] have recognized the
2. General relations We require expressions for the interaction forces per unit mass, F x and Fy, which can appear in conservation equations as the terms h p F x and hpFy which are forces per unit area. These forces represent the components of the total force per unit area, hpF, which is the force exerted on the fluid by the solid surfaces. The term h represents the thickness of the unit calculational element and is included explicitly to allow the possibility for variable width of this element. Table 1 summarizes the relevant force expressions and their dimensions for the general direction z. The total force is an unknown function of the superficial velocity W or mass flux G = 0 W, the rod
* Professor of Nuclear Engineering, MIT, on Sabbatical Leave at Chatou. 383
384
Y. Coeff6, N. Todreas / F o r m u l a t i o n o f the f l u i d - s o l i d i n t e r a c t i o n / b r c e
Table 1 Expression
Dimension
hpFz, pFz ,
F/L2
hap/az ap/az
tubes>
G=
F/L 3
1 ap 1Fz ,
F/M
p 3z
- - - -
G× Fig. 1. C o m p o n e n t s of the mass flux vector G.
array geometry and the state of the fluid mainly, its density p and quality X. We choose to express that function with the variables pF and G as o F = ft-(G, geometry, p, It, X)-
(1)
We have taken the flow as hnmogeneous. Otherwise, we should add slip ratio to the arguments of ~r. Since 5r is a homogeneous function of G(~'(G) = O, where G = 0) and assuming regularity of St(G) near 0, we may write eq. (1) for a given geometry as
X->-
y-axis parallel to the tubes and the x-axis transverse to the tubes), it can be seen that ~ and K are diagonal so that D is also diagonal. Therefore we can express the total force per unit volume of eq. (5) as
°
,e
D2(X, P, It, G)
LO
x
]
[-K,(G,o, It)
O
-0
K2(G, p,
] #)
(8)
a.
where K is the general matrix for flow resistance. For two-phase flow, it is common to evaluate the two-phase pressure gradient as equal to the product of the single-phase gradient assuming the total flow as a liquid and a two-phase multiplier ~[o, i.e.,
Let us define 0 as the angle between the y-axis and the flow-direction measured clockwise. Fig. 1 illustrates this angle and the components of the mass flux vector G. No we must note that the dependence of a quantity Q on G may be expressed in either of two equivalent ways
d p / d 2 = ( ¢ ~ o X d p / d Z ) L O at total f l o w ,
Q(Gx, Gy)
pF = ~(G, p, It, X) G,
(2)
(3)
since G = IGI =
where ~ , o is generally a function ot X, P, It, G. We will similarly take the two-phase effects of F into account by introducing a two-phase multiplier matrix, ~(X, P, #, G), and the total force assuming the total flow is a liquid, FLO yielding
F=~FLo.
(4)
~,(G, p, It) G,
(5)
so that eq. (4) becomes
pF = B(X, p, It, G) K(G, p, It) G,
(6)
=
(9)
rt",2 t,,]x ± r t~211/2 Lryj .
We will now presentthe determination of the flow resistance matrix K" for one-phase flow from the available literature data. In the following development the subscript LO will be omitted from the z ~ and F symbols for simplicity. For this case eq. (8) can be reduced to the form -
F K ~ ( G , p , It)
pF=K(a,p,it) a =Lo
which defines J~ in eq. (2) as •(G, p, It, X) ~(X, P, It, 17) K'(G, p, ~).
Q(O,G),
3. Theoretical determination of the flow resistance matrix ~ for single-phase flow
For a given geometry by analogy with eq. (2) FLO can be expressed as PFLo =
or
(7)
If we take now as the reference coordinate system the axes of symmetry of the bundle (assigning the
0
K2(G, p, It)
(lO) All the experimental data from the literature gives only the values of pressure loss measured along the
Y. Coeff~, N. Todreas / Formulation o f the fluid-solM interaction force
G direction, i.e., along 0 for different values of 0 and G. We will designate that pressure loss by Ape(O, G, O, la) and the corresponding component o f F by FiT. Pressure losses in the direction perpendicular to G have not been reported. We will designate them as APzD (0, G, O,/2) and the corresponding component of F by F±G. From the definition ofF, per table 1, we can relate the four previovs quantities by 1 oFG(O, 63 = -~o ApG(O' 63,
(1 1.1)
1
oF±G(O,63 - Lo+Tr/2AP±G(O'G) ,
(1 1.2)
where the dependence of forces and pressure losses on O and gt is no longer specifically indicated and FG and F±~7 may be expressed in terms of G through eq. (10) as
oFG = oF" G G = (Ka(0, G) sin20 + Ks(O, (7) cos20) G ,
= (KI(O, G) - g 2 ( o ,
G))(sin 0 cos 0) G.
(12.1)
(12.2)
385
/~-matrix has been undertaken with one limited exception which is utilized in the next section for 0 values along the principle axis, i.e., (a) KI(O, G) determined for Kl(0, 63, i.e., longitudinal flow through rod bundles. (b) K2(O, G) determined for K2(90 °, G), i.e., crossflow over rod bundles. Consequently the K'-matrix cannot presently be expressed even for the simpler case of single-phase flow. Clearly an experimental program in which Ape(0, G) and Ap±G(O, G) are simultaneously measured over the range of (0, G) and arra_y geometry of interest is needed to evaluate the K-matrix. In the absence of these measurements, it is still desirable to estimate the value of the elements of the K'-matrix to provide the needed input to the multi-dimensional analyses being undertaken. Let us start this estimation process by summarizing the useful data which is available for which at least ApG(O, G) has been determined. 4. Available literature data
The existing data are those of Kazakevich [6] interpreted in Idel'cik [7] from which we know the values of
Aoc(0, 63 ~b ApG(90 o, 63 _
Equating the above sets of equations leads to the following relations 1
- - ape(0, 63 L0 = [KI(O, G) sin20 + K2(O, 63 cos20] G , 1
Lo+~r/2
(13.1)
¢(0) _
(13.2)
These two relations show clearly that measurement of Ap6 and Ap±G for every 0 and G would allow the complete determination of the two functions KI and K2. This result is entirely consistent with the literature on hydraulic characteristics of anisotropic media. The rod bundle is a special case of anisotropic media which in the past has been analyzed considering only one-dimensional flow. Therefore experimental determination of the elements of the
(14)
An important feature of the function ~b as reported in [7], is that it may be considered as independent of G so that we may write it as ~b(0). Eq. (13.1) gives then a new expression of ~b(0):
Ap±G(O, 63
= ([KI(O, G) - K2(O, 63] sin 0 cos 0} G.
for 0 = 30 °, 45 ° and 60 ° .
=
Lo[KI(O, 63 sin20 + K2(O, 63 cos20] Lg0oKl(90 °, G)
(15)
Also recall that the literature provides available data for both purely longitudinal and transverse flows so that values of K1(90 °, G) and K2(0 °, G) are available. 5. Evaluation of K from available data
We now have to make one hypothesis to determine the two functions KI and K2 from the single eq. (15).
Y. Coeffd,N. Todreas/ Pbrmulation of the fluid-solM interactionforce
386
Since we know that K1(90 °, G) > > K 2 ( 0 °, G) for all fixed G we will assume that K1 and K2 are not strongly dependent on 0 and that
Kl(O, G) > > K 2 ( 0 , G)
for all given G and 0.
For clarity now we will determine K~ and K2 successively over the two ranges: 02 <~ 0 < 30 ° and 30 ° < 0 < 90 °. (a) 30 ° ~< 0 ~< 90 °. From the previous hypothesis we may write: K1 sin20 > > K2 cos20 ,
(ii) Continuity with the expressions of K 1 and K2, for the range of 30 ° ~< 0 ~< 90 °. We will therefore take for K: in 0 ° ~< 0 ~< 30 °:
K2(O, G) = K2(O°, G). For K1 we will estimate the function ~o(0) by adding an extrapolated value for ~o(0°). Table 2 summarizes the results for the function ~o(0). The final expressions for Kt and K2 are
K,(O, G) = ~o(0)K1(90 ° , a ) ,
(19.1)
and
30 ° ~< 0 ~< 90 ° ,
K2(O, G) = K2(O°, G)
and eq. (15) becomes:
for 0 ~< 0 ~< 90 ° .
(19.2)
Lo KI(O, G) sin20 t~(0) - L9o °
K2(90 o, G)
(16)
Since the measurements of Kazakevich were made with a single bundle, the length Lo can be expressed as
L 0 = L90o/sin 0 .
(17)
Utilizing eq. (17) in eq. (16), we obtain the desired result of am expression for KI(O, G), i.e.,
KI(O'G)=~°(O)K~(90°'G) t- 30 ° ~<0 < 9 0 ° .
LO
D2(X, ,o,/1, G)
There are two unknown functions which may also be written as
D I(X, p, ta, O, G),
Concerning K2 in this range we have assumed above that its effect is not important. To ensure continuity with the range ( 0 o - 3 0 ° ) we will take =
The matrix of two-phase multipliers is equal to:
D2(X, P,/a, 0, G ) .
(18)
where ~o(0) = tk(0)/sin 0 )
K2(O, G) K2(O°, G),
6. Determination of
30 ° ~< 0 ~< 90 ° .
(b)O°~
The literature provides only two-phase flow multipliers for purely longitudinal or transverse flow so that we know DI(X, p,/~, 90 °, G)
and
D2(X, p,/~, 0 °, G ) .
In the absence of more complete data we will assume that the two functions D1 and D~ do not vary much with 0 so that we may take
D,(X,p, IJ, O,G)=D,(x,p,u, 90°,G),
(20.1)
D2(x,p, la, O,G)=D,(x,p, la, O°,G),
(20.2)
for 0 ° ~< O ~< 90 °. Table 2
7. Summary of results for matrix
0
~b
~o(0) = qJ/sin 0
0 30 45 60 90
0.30 0.54 0.82 1.00
0.25 0.60 0.76 0.94 1.00
Eqs. (19) and (20) give the expressions chosen to determine the complete expression o f p F based on eq. (8), i.e.,
pF =
[DI(x'P'Ia'90°'G)
O
L0
D2 (X, P, P, 0°, G)
1
Y. Coeff~, N. Todreas / Formulation of the fluid-solid interaction force × [~(f) K~(90°, G)
0
t-0
Appendix
]
K2(0 °, G)
G.
(21)
We must further specify the expressions of the four functions D1, Du, Kl, and K2 which are for purely transverse or longitudinal flow. The general form of the pressure gradient commonly used is:
~P
_ f G2 LO at total flow Lc 2OL'
(22)
where the characteristic lengthLc is in common practice different for longitudinal and transverse flow. We may then define new quantities k by the relations: Kt(90 °, G) =
G (k9oo)LO
- -
(23.1)
,
2PL
G K2(O°, G) = (k0o)LO 2p L •
387
(23.2)
In this Appendix the parameters necessary to completely define eq. (25) are specified for the conditions applicable to PWR steam generators. A.I. Transverse flow across a tube array,
(k90°)LO
(z ---x in our convention) For transverse flow in an array of tubes, it is customary to express the pressure drop in terms of: (1) the pressure drop across one row of tubes based on a coefficient (Ao)Lo which comprises the terms f/Lc ofeq. (22), (2) the number of rows of tubes in the increment dx, i.e.,
dxlp, (3) the velocity head between the rows, i.e., based on the total mass flux G: G2/2OL .
We also make a change of notation for D~ and D2 and set: DI(X, 9, U, 90°, G) = ¢~o* ,
(24.1)
D2 (X, 9, ta, 0 °, G) = ¢~o .
(24.2)
Finally the general expression for oF as defined by eq. (2) becomes
Following these conventions, eq. (22) for 0 = 90 ° may be written as ~[ LO at total flow
= (A°)LO 1p G2
0
R~
G,
(25)
where R1 = t~0%0(0)(k90°)LO G 20L
G R2 = eg°(ko°)LO - 2p L '
(A.2)
It is important to note that since (kgoO)Lo is evaluated in accordance with eqs. (19) and (23.1), our evaluation of (kg0O)Lo should be done using the superficial mass flux along 0, i.e., G = pW = p(U ~ + I,/2)1/:
(A.3)
and the associated reference liquid Reynolds number, (Ref)L (Ref)L = I) WD/IAL.
p(O) has already been determined in table 2, ¢,~oo and (kgoO)Lo will be specified in the Appendix, ¢~o and (koo)Lo will be specified as well in the Appendix. The appendix specifies results for a specific case, co~aditions applicable to PWR steam generators, to complete the definition of our recommended procedure.
(A.1)
Comparing eqs. (22) and (23.1) with eq. (A.1), we determine that (k9 0*)LO = (Ao)Lo/P •
oF = J~(G, 0) G =
2pL
(A.4)
A. 1.1. Single phase transverse drag coefficient [(A0)LO] Now [(Ao)LO] is available for square arrays from the work of Boissier et al. [8]. A. 1.2. Two-phase transverse, horizon tal flow multiplier - (~o o The two-phase multiplier for horizontal cross flow
388
Y. Coolie, N. Todreas /I~brmulation o f the fluid solid interaction Jbrce
in a triangular array of transverse P/D = 1.25 (longitudinal P/D = 1.09) has been correlated by Grant and Murray [9]. It is possible that results for a square array ofP/D = 1.46 may differ because the degree of void redistribution and recovery in passing through the rod gap contraction - expansion geometry may vary significantly enough with geometry to affect the multiplier. However, in the absence of directly applicable data, the correlations of Grant and Murray are specified since they are certainly more applicable than those surveyed by Idsinga et al. [10] for longitudinal flow. Two correlations are recommended with the selection dictated by the existing flow regime. For stratified and stratified-spray, the recommended equation is ¢,3oo = 1 + ( r ~ -
1)
X (0.25X°'769(1 - -
X) 0'769
-[" X 1 " 5 3 s )
.
(A.5)
For spray and bubbly flow, the recommended equation is:
¢,)oo=l+(r 2 -
1)
X {0.75X°'769(1 - X)0"769 + X 1'538} ,
(A.6)
where
Fz - OL/og. Stratified flow regimes are generally encountered only at low mass velocities which are generally outside the normal steam generator operating range.
A.2. Longitudinal flow in a tube array For longitudinal flow in an array of tubes (z ~ y in our convention). It is customary to express the pressure gradient in terms of: (1) A friction factor and equivalent diameter which for longitudinal flow represent the terms f and Lc of eq. (22), i.e., (AL) LO/DH • p (2) A velocity head based on the effective axial velocity o = G/np where n is the porosity. Following these conventions, eq. (22) for 0 = 0 ° can be written as
d~ LO at total flow
_ (AL)rO
DH
(AL)LO
(or) 5 _
2pL
DH
G2
2,0Ln2 "
(A.7)
Comparing eq. (22) and (23.2) with eq. (A.7), we determine that (AL)LO
(A.8)
(koO)LO- DHn2 • For our square array, D H equals: DH-
4[e 7rD
D4rI P I 2
L\DI
1
-
q
~rj.
(A.9)
Evaluation of (AL)LO and ¢~o remain. In the case of the steam generator the flow will be turbulent according to the criterion: GDH/IaL > 2100. We will then propose results for the case of turbulent flow.
A.2.1. Single phase longitudinal turbulent flow (AL)LO turbulent Now (AL)Lo is the single phase friction factor for longitudinal flow in our non-circular geometry, i.e., square array. For triangular arrays in turbulent flow, Dwyer and Berry [11] have predicted (AL)LO as a function ofP/D. For P/D ratios of 1.35 or greater, slug flow analyses [12] have demonstrated that Nusselt number results for triangular and square arrays are within 4%. Since turbulent flow condition should give comparable or less difference in Nusselt number, and since friction factor behaviour is very similar to Nusselt number behaviour (p. 183 Kays, [13]), the triangular array results [1 I] can be used for our square array case ofP/D = 1.46 with only a slight overestimate in friction factor resulting. These results express the friction factor (AL)LO as a multiple of the friction factor in the circular tube, (Act)LO, when (Act)Lo is calculated from eq. (A.IO). (Moody friction factor curve for smooth pipe.) (Act)LO = 0.0058 + 0.432 ((Re)DH)~°'3,
(A.10)
where ((Re)DH)L is the liquid phase rod array Reynolds number at the total flow, which since (k0)LO is evaluated as (koO)Lo in accordance with eq. (19)
Y. Codffd,N. Todreas / Formulation of the fluid-solid interaction force Table A. 1 Friction factors in longitudinal flow in triangular arrays (from [8])
P/D
(AL)LO
Applicability
(Act)LO 1.001 1.01 1.02
0.640 0.775 0.855
1.03 1.04 1.05
0.910 0.950 0.980
1.06
1.000
1.07 1.10 1.15 1.20 1.30
1.015 1.040 1.065 1.080 1.100
1.40 1.60 1.75 1.80 2.00
1.115 1.130 1.140 1.145 1.150
Triangular arrays only
Square and triangular arrays
work. Chisholm [ 15] has generalized his previous work on two-phase pressure gradient to cover our case of smooth tubes, and also to include the Baroczy results as a basis for his required coefficient. For predictions over wide variations of both pressure and mass flux, we would recommend Chisholm's correlation with the coefficient B evaluated per his suggestions in his table 2 as a function of F and G. The values of table 2 reflect a compromise between the correlations of Baroczy, Lockhart-Martinelli and Chisholm. For the steam generator case, however, the pressure is nearly constant so that F is a constant. Also, the G value is 800 kg/m a s or less. With this restricted range or parameters, a best estimate value of the gradient can be made using the Baroczy results in this range in the form presented by Chisholm in his fig. 4. This results in an equation for B of the form B = fiG~ = 4.08
and (23.2), should be based on the equivalent longitudinal velocity G/pn so that ((Re)DH)L =.GDH _P( U'z + V2) I/2 D H
nPL
nlaL
(A.I1)
The multiplier, m, is given in table A1 and for P/D = 1.46, m = 1.12 where (AL)LO ~- m(Act)L 0 .
m(Act)LO DHn2
(pLtO'S("gtn/2
f°r r = ~Z!
\Z!
for G.V. pressure and n = 0.25,
(A.14)
where =-0.447, = 45.92, G = p ( U 2 + 1,'2) i n .
The parameter B is used in the general Chisholm equation for ~b~o,i.e., ~b~o= ATTp
APLo
(A. 12)
= 1 + (1.`2 -
Substituting eq. (A.12) into eq. (A.8) yields (ko°)LO -
389
1){B×(2-n)/2(1 - X)(2-n) + X2-n} , (A.15)
(A.13)
A.2.2. Two-phase muln'plier - ~ o Two phase multipliers are available from a variety of correlations. In our steam generator applications because of the wide range in mass velocity and the full quality range covered, we desire an approach for the two-phase multiplier which (a) includes mass velocities effects, (b) goes to the correct limits at the all liquid and all vapor conditions. Baroczy's correlation [14] satisfies these conditions but is too cumbersome to prepare for numerical
where we take n = 0.25.
Nomenclature
Dimensions B
D D 1, D2 DH
Constant defined by eq. (A.14) Two-phase multiplier Rod diameter Diagonal elements of matrix D Hydraulic diameter
L
L
Y. Coof]~, N. Todreas / bbrmulation of the fluid-solM interaction ]brce
390
f F G G
5r Gx, Gy
Fx, Fy, Fz h K K1, K2 Lc L m n
P
ap
P Q (Ref)L
U V W W X, y Z
Friction factor Force per unit mass, vector Total mass flux = IGI Mass flux vector Function defined by eq. (1) Mass flux in x, y directions Forces per unit mass in x, y, and z directions Element variable width Single phase matrix Diagonal elements of matrix Characteristic length Length across bundle Ratio of friction factors defined by eq. (A.12) (1) Volumetric averaged rod array porosity (2) Coefficient of eq. (A.14) Pressure Pressure loss Rod array pitch (distance between rod centerlines) General quantity per eq. (9) Reference Reynolds number General matrix for flow resistance Superficial velocity in x direction Superficial velocity in y direction Superficial velocity in z direction Superficial vector velocity Coordinate directions Arbitrary direction in x - y plane
-F/M M/L2T M/LT
42 X (Act)LO
M/L2T
(AL)LO
F/M L
(Ao)Lo
FT/L L L
/a
-
0
~0 F/L 2 F/L 2
#
--
L M/LT
Deg.
Subscripts L -
L G LO G ±G 0
Liquid Gas Total flow is as a liquid In G direction Normal to G In 0-direction
L/T References L/T L/T L/T L L
Greek symbols
p:
Two-phase multiplier Flowing quality Friction factor for singlephase flow in a circular tube Longitudinal friction factor for single-phase flow in a rod array Transverse drag coefficient for single-phase flow across one row of tubes - equal tof/Lc ofeq. (4.7) Mixture viscosity Ratio of pressure drops defined by eq. (14) Angle measured clockwise from y axis to mass flux vector G Function of 0 defined by eq. (18)
Constants to evaluate constant B o f eq. (A.14) Liquid to vapor density ratio appearing in eqs. (A.5), (A.6) and (A.14) Two-phase mixture density M/L 3
[ 1] W.T. Sha et al., NUREG/CR-0785, ANL-77-96 (Sept. 1978). [2] A. Verwey, Y. Caytan, O. Daubert and D. Taillifet, Numerical model of the flow in a LMFBR heat exchanger, paper presented at the 4th Seminar of the XVIII IAHR Congress (Sept. 1979). [3] Y. Co~ff6 and N. Todreas, 2-D mathematical model of the two phase flow in a steam generator, paper presented at the 4th Seminar of the XVIII IAHR Congress (Sept. 1979). [4] D. Butterworth, Internat. J. Heat Mass Trans. (1978) 256-258. [5] W.J. Wnek, J.D. Ramshaw, J.A. Trapp, E.D. Hughes and C.W. Solbdg, ANCR-1207 (Nov. 1975). [6] F.P. Kazakevich, IZW. Vses. Inst. Imoni F.E. Dzerzhimkogo 8 (1952) 7-12. [7] I.E. Idel'cik, AEC-TR-6630 (1960). [8] A. Boissier, M. Chatillon, D. Gautier, R. Perez and A.
Y. Coeffd, N. Todreas /Formulation of the fluid-solid interaction force
[9] [10] [11] [12]
Marchand, E.D.F. Bulletin de la Direction des Etudes et Recherches, S6rie A, Nucl6aire, Hydraulique, Thermique, no 2/3 (1971). I.D.R. Grant and I. Murray, NEL Report 560 (March 1974). W. Idsinga, N. Todreas and R. Bowring, Internat. J. Multiphase Flow 3 (1977) 401-413. O.E. Dwyer and H.C. Berry, Nucl. Engrg. Des. 23 (1972) 273-294. O.E. Dwyer and H.C. Berry, Nucl. Sci. Engrg. 39 (1970) 143.
391
[13] W. Kays, Convective Heat and Mass Transfer (McGrawHill, New York, 1966). [ 14 ] C. Baroczy, A systematic correlation for two-phase pressure drop, AICHE Reprint No. 37, paper presented at 8th National Heat Transfer Conf., Los Angeles, August 1965. [15] D. Chisholm, Internat. J. Heat Mass Transfer 16 (1973) 347-358.