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Hyperbolicity breaking and flooding
Nuclear Z" ELSEVIER
Nuclear Engineeringand Design 146 (1994) 225-240
Dmgn
Hyperbolicity breaking and flooding J a e Y o u n g L e e a, H e e C h e o n N o b a Dept. of Nuclear and Energy Engineering, Cheju National University, Ara 1 Dong, Cheju, South Korea b Korea Advanced Institute of Science and Technology, South Korea
Abstract A new flooding correlation is developed by studying hyperbolicity breaking near a singular point in two-phase flow. Choking and instability correspond to zero and imaginary characteristics which occur when the hyperbolicityof two-phase flow is broken, respectively. Through comparison between results predicted by this model and the experimental data of tubes with and without blockage, it is shown that they are in good agreement with data. The role of the zero characteristics is investigated by comparing the presence model with the experimental data by Celata et al..
1. Introduction Two-phase flow shows many abrupt flow regime changes such as flooding. Abrupt change can be generated at the singular point of the autnomous system [1,2]. The purpose of this paper is to find the condition of the flooding from the mathematical catastrophe in the two-fluid model of two-phase flow. As noted in the review article of Bankoff and Lee [3], Flooding has a large scatter band of data, and a number of correlations and models have been developed. In this study, the two-fluid formulations of two-phase flow are selected to study flooding, and they are changed into ordinary differential equation along the wave propagation line by introducing the characteristics. The singular point of the ordinary differential equation derived contains information about characteristics. If the characteristics at the singular point are zero or imaginary, the hyperbolicity of the equation is broken to change a regular flow pattern to another flow pattern. This bifurcation provides the motivation for this study to develop a new correlation of flooding. The present study provides a general theory of the flooding limit satisfying two-phase flow transients for both phase change and adiabatic cases. But our attention is confined to the adiabatic case only.
2. Singular point and hyperbolicity breaking In order to study a relationship between the limited flow phenomena and singular points, the topological structure of two-phase flow in the phase space must be analyzed on the basis of the 0029-5493/94/$07.00 © 1994 ElsevierScienceB.V. All rights reserved SSDI 0029-5493(93)E0231-8
J.Y. Lee, H.C No/Nuclear Engineeringand Design146 (1994)225-240
226
characteristic analysis. Let us consider a system of first-order, quasi-linear, partial differential equations given by: 0o0o" A(~r)--~- +B(cr)-~z = C(~r).
(1)
Let us transform the coordinate (t, z) into the wave coordinate of ~: ~=At+z.
(2)
Then Eq. (1) is transformed into the following form: 0o(AA + B ) - ~ = C .
(3)
If we confine our attention to solutions which are everywhere continuous but for which the derivative &r/az is discontinuous across a particular line differencing equation (3) across this line gives [12]:
The condition that Eq. (4) has a nontrivial solution gives us the characteristic as follows: IA~+BI
=0.
(5)
For a hyperbolic system, the characteristic, A, must be real and nonzero. To find the condition under which Eq. (1) becomes singular, Eq. (3) is changed into the following form: 0or
N~
a---~--- (AA + B ) - I C + --f,
(6)
where a - - ] A A +BI,
(7)
and N/creates a determinant obtained from (Alia + By) by replacing the i-th column of Cj. The phase space of Eq. (6) is constructed of the n + 1 dimension where n is the number of variables and has three distinctive points of regular points, turning points, and singular points. If A ~ 0, the points in the phase space are regular points. However, this condition depends on the characteristics, A, and Eq. (7) shows that every point in the phase space seems to be irregular. That is to say, the real irregular point occurs when both conditions of A = 0 and hyperbolicity breaking are satisfied. The hyperbolicity of equations is broken when the characteristics of the equations become imaginary or zero. As an example of the above claim, many choked-flow analyses are based on A = 0, which is one of conditions of hyperbolieity breaking [1]. The critical flow occurs at a singular point (a(,~ = 0) = 0, N~ = 0). The critical flow velocity is determined from A(A = 0) = 0, and the location of choked-flow is given by N/--0.
The conditions of hyperbolieity breaking are as follows: (a) Zero characteristic: When the characteristics of equations become zero, the propagation of information is limited or choked. This condition is named as a choking condition. (b) Imaginary characteristic: Imaginary characteristics mean the oscillatory propagation of information. This condition is named as an instability condition. The singular point with hyperbolicity breaking is a bifurcation point where the neutral stability of the equation is branched to the line of choking and to the line of instability [4]. Figure 1 shows the
J.Y. Lee, H.C No/Nuclear Engineeringand Design146 (1994) 225-240
227
Zero Characteristic I
Hypcxbolicity Breaking
~ . ~ I-D Characteristics choking
•
¥
Imaginary ~ c s Hypcrbolicity Breaking
\
2-D Characteristics
\ Saddle ~.
~
2 L N -M =0 2-D Characteristics Neumtl Stability
2
L N -M >0 2-D Qharactexistics
Instability
Fig. 1. The hyperbolicitybreaking represented by the differential geometry. geometrical meaning of the hyperbolicity breaking: the zero characteristics correspond to the dimensional shrink of the solution domain from the surface to the line, and the imaginary characteristics signify the change from the hyperbolic domain to the elliptic domain.
3. Singular points of two-phase flow Since the internal energy conservation equations simply describe the transport of temperature, these equations are removed from the analysis of singular points, and only mass and momentum equations are used. The mass conservation equations of two phases are as follows: (a) Vapor mass: a a at ( a ' p g ) + ~ ( a : g ~ ) = F; (8) (b) Liquid mass: a
at( fp,) +
a =
-r.
(9)
Momentum equations of two-phase are derived on the assumptions that the vapor pressure is equal to the interracial pressure of a phase, i.e., Pk - e k i ---~ 0 and the spatial derivative of the pressure difference of two-phases, A P = P B - I f is a function of surface tension and the curvature of the surface. Since curvature of the surface is related to void fraction, spatial derivative of the pressure force, O A P / a z ) , could be expressed as (aAP/aag)Oas/az). The momentum conservation equations of two-phases are as follows: (c) Vapor momentum:
av. av. ae. 4 ,i + a'Ps'~" + a'PgVg'~'z + ag--~z "~F(V"- Vg) + D
(10)
228
ZY. Lee, H.C No/Nuclear Engineeringand Design 146 (1994) 225-240
(d) Liquidmomentum:
aVe
aVe
oAP Oa,
aPg
@fPf'-~-- "k~fPfVf-~z "{"~f O"Z-'---Otf O~g ~ZZ = - F ( V f i -
Vf) - 4
~
4
q'i + "~q-w-{'-~fpfg.
(11)
where
fi
q'i = 2"Pf IV, Ivv
(12)
fw q-w= T p f IVe IVf,
(13)
F = Qw /fg
(14)
The above four equations are linearized for the vector of cr = (P, ag, Vs, Vf) T to obtain Eq. (1) whose matrices A and B and vector C are as follows [5]:
~ g ~
%C2 2
pg
0
o~fCf 2
--Of
0
0
0 agpg
0
0
0
@fPf]
p,V,
%0,
0
-pfVf 0
0 agpgVg
afpf
~:gg
O/f
- a f ~ag
0
~fpfVf
OtfVfCf 2 B=
i/, 0
aAP
F
-F
C=
r ( K , - K) + ~-D- ¢, + ~,p,g - r ( V f i - Vf) - 4
q'i +
-~q-w--afpfg
where C~-2 Opg/OPg and c ( 2 ape~aPe After transforming the coordinate into ~ and neglecting the elements having the sonic velocity (Cg) -2 and (Cf) -2 which are smaller than the others, the gradient of the void fraction among the gradient of ois =
0ag
=
N(ag)
(15)
J.Y Lee, H.C. No ~Nuclear Engineering and Design 146 (1994) 225-240
229
where aAP A = --Offpg()[ + Vg) 2 - O ~ g p f ( ) k + Vf) 2 "at- ¢lfgO/f
(16)
~a 8 '
N(otg) _ 4 --~-'r a~g i -- --D---,rw 4ag -- a g a f A p g - F [ a f ( A + V,)
ag(A -I- Vf)]
(17)
The conditions of hyperbolicity-breaking are simply obtained from the equation that A = 0 by checking whether the characteristics from this equation become zero or imaginary. Introducing the choking condition A = 0 and A = 0 into Eq. (16) yields
OAP
(18)
aeOgVg2 + %OfVf2 = %af 0%
The solutions of the characteristic equation A(A) -- 0 are A =u + ~u2- v,
(19)
where U~ m
afp,V, + a,pfVf
(20)
~ f p g "~" ~ g p f
and v=
afpgVg 2 + ¢]tgpfVf 2 - CltgO~f~Ap//at:l~g
(21)
olfpg + agpf
Normal soluton of the characteristic equation has three types of solutions: two real solutions (hyperbolic domain), since real solution (parabolic domain), and two complex solutions (elliptic domain). Physically, the hyperbolic domain represents regular flow, the elliptic domain represents instability, and the parabolic domain represents the neutral stable state between regular flow and unstable flow. So the parabolic domain corresponding to the single real root gives us a guideline describing the bifurcation of instability. This neutral stability is obtained by setting the square root part of Eq. (19) at zero:
OAPI( O~fPg + agPf /
(22)
The above condition is a type of the onset condition of Heimholtz instability. Let us consider separately flooding due to the singularity with zero characteristics and singularity with imaginary characteristics.
3.1. Singular points with zero characteristic The compatible relations of zero characteristic are as follows: A = 0 (A = 0) ~ Eq. (18), • N ( a s) = 0 -~ Eq. (17). let us introduce the dimensionless volumetric fluxes, j* and jr*, where •
1/2
. [
pf
~1/2
(23) and new dimensionless numbers affecting flooding,
316. Lee, H.C No/Nuclear Engineeringand Design146 (1994)225-240
230
•
Flooding wave number: Nn
(OAP/Oag) 1/2. N. =
•
g D ,a p
(24)
'
k-phase's change number: Nph D ,/2
(25) Then, Eqs. (17) and (18) become, respectively,
(j;)~- (:;)~ (26)
- + - = (N.) ~ (,~.)~ (,~f)~
and
I J;
IJ~
2/i (%)5/2
2fw
I J; IJ;
(af)2
Otf 1 = af sin/3 + (ag) 2j*Nph,g -- ~ffJf"*N.ph,f,
(27)
where the angle/3 is the inclination of the tube. The ratio of the volumetric flux number and the flooding wave number would be related with the Kutatelaze number. From F_xlS. (26) and (27) we have the following solutions for the adiabatic case: ( j , ) 2 = 1 - 2 sign(j?)fw(Nn) 2 (af)2DE
(28)
and 1 ( 2fi (jf,)2= (ag) 2DE (ag) 1/2
ae ],
°ts ]
(29)
where DE-- - -
2
(
fi - -
(~f)2 (~),:
+
sign(jr*) fw )
~
(30)
By using Eqs. (28) and (29), flooding due to choking will be predicted.
3.2. Singular points with imaginary characteristics Flooding due to instability is modeled with the imaginary characteristics at the singular point. Under this condition, the critical characteristic, A*, is given by: A* --- u. (31) From A -- 0, the necessary condition of a singular point is Eq. (22), and the sufficient condition of the singular point, N[a(A* = 0)], is Eq. (17). Introducing the dimensionless volumetric fluxes into F_xlS.(22) and (17) gives
J~*
og J?
O~g
af
N. % +,~f--, pf
J.Y Lee, H.C No/Nuclear Engineeringand Design 146 (1994) 225-240
231
and
[J:lJ*
2fi (~g)5/2
lJ; IJ;
2fw (af)2 = af sin/3 + F * ,
(33)
where =
afpg + agpf
- - "*
- j f Nph,f •
(34)
Equation (34) represents the contribution of phase change to flooding. From Eqs. (32) and (33) the following two equations are given:
j* = Ej? + F,
(35)
A ( J * ) 2 + B ( j f * ) 2 + C-- 0.
(36)
and where E =
F
(37)
Pg ~ff sign(jr*), 1/~fg
%Na V pg
2E2fi A = (ag)5/---~
4EFft
~f Pf
(38)
2fw sign(jr*) (af)-----~,
Pg+Pf
[ afENph•
B -- (O~g)5/------------------~ afpg + agpf ~
2F2fi C -- (a8)5/'------'~-"
Pg+Pf
afpg+ rvgpf
(39) Nph,f)'
(40)
aS
( afF--NNPh'g/ - af sin ~ . ag ]
(41)
Because the present model has every term in the two-fluid model, we are able to to analyze flooding in both adiabatic and phase-changing cases. 4. Pressure force and singular points
Recent reaserch [6,7] has found that the surface wave of the vertically falling film has many types of waves such as the periodic, solitary, shock, and chaotic waves, and they are generated at a certain bifurcation point affected by the wave speed. Among them, the solitary wave with high amplitude will be a strong candidate to make a singular point. Also, the generation mechanism of these waves is greatly affected by the shape of the entrance. If flooding occurs by a certain magnitude of the solitary wave, then the critical length of a test tube is required for the growth of the small wave at the entrance. Since the interfaeial pressure difference force is greatly affected by the wave shape, the force is a function of flow parameters such as the physical quantities of fluid, the shape of entry and exit, and the diameter and length of the tube:
AP =f(~r, ~, ag, Vg, Vf, d, L, entrance shape).
(42)
232
J.Y. L.ee, H.C. No/Nuckar
Engineering and Design 146 (1994) 225-240
This relationship may be fully understood when nonlinear wave dynamics are understood. In this study, we reduce our attention to quantifing the interfacial pressure difference of the solitary wave and capillary wave. 4.1. The mean curvature of the solitary wave The solitary wave is a nonlinear wave whose magnitude and propagation velocity are proportional the square of the wave number, 6(x, t) =H sech2k(x-ct),
to
(43) which is a solution of the Korteweg-de Vries equation [81. The solitary wave is generated by the third order spatial derivative of the film thickness which corresponds to the pressure force of the momentum equation when the pressure difference between phases is considered. It is a ring type, 2-dimensional
Liquid Film
e
TONS
(a) Simplitication of the solitary wave to obtain its cxuvanue
(b) ‘tle mordkte
system to &ain tbc curvatwe‘of tbc torus
Fig. 2. Curvature of the solitary wave.
J. Y. La, H.C. No /Nuclear Et@ming
and LIesign 146 (1994) 225-240
wave as shown in Fig. 2. The interfacial pressure force of the solitary wave is proportional tension with the second derivative of the wave profile AP=uV’6,
ti3
to .the surface (4)
where a is the surface tension. This term can be represented follows: AP= -a(C,
as the principal curvatures C, and C, as
+ C,).
(45)
The interfacial pressure force of the solitary wave is easily determined when the principal curvatures of the inner surface of the skewed torus in Fig. 2 is known. To simplify our derivation of the principal curvature [9], the solitary wave, whose height is a& is taken for a wave represented by the hyperbolic secant function whose radius is (a - 1)6. Figure 2 shows the coordinate system of the torus of the solitary wave. As a cross-sectional area revolves about a fixed line in the plane of the circle, it generates a, torus. Suppose that the circle is initially in the (i, k) plane with the center on the j axis at a distance U (= R - S) from the origin, and the shape of the function of the solitary wave f(x) = h sech2(kx). Now consider the circle after it has been rotated through an angle 8 about k axis, as shown in Fig. 2. If u is the vector from the origin to the center of the circle and w is the radius vector of the circle, then u = U cos fli + U sin ej,
(46)
and cos @i-f(x)
w= -f(x)
sin Bj+xk,
(47)
where x is the distance having its origin at the center of the solitary wave along the R axis. It follows that y=u+v.
(48)
The unit normal vector of its surface is cos 8i + sin’ Oj +f,R
YeXYX n=
lY,XY,l
The parameters
=
Gi77
of the first fundamental
(?9) *
forms of the surface are
E=Y,*Y,=(-f(~))~,
(50)
F=y,*y,=O,
(51)
G =yx*yx = 1 + (fJ2.
(52)
and
The parameters
of second fundamental
L =yee*n =
-(U-f(x)) Jrn’
M=y,;n
= 0,
N=y,;n
=
forms of the surface are
(53) (54)
and
-fxx
\/l+o”
(55)
234
J.Y. Leg H.C. No/Nuclear Engineeringand Design146 (1994)225-240
The principal curvatures of this torus are determined try the following equation:
( EG
-F2)C
2-
( EN + GL - 2FM)C + ( L N - M 2) + O,
(56)
whose solutions are
C1 =
-1 (57)
(U-F)~l+(fx)
2
and
c2=
-Lx + (fx)2
(58)
The pressure difference between phases is obtained by inserting Eqs. (57) and (58) into Eq. (45) for fx ~: 1: aP=~,
R-f-8
Lx.
(59)
The gradient of the pressure difference to the void fraction is:
bAP
R2
( fxxx/fx R2
a% =or 2 ( R - f - 8 )
2(R-f-8)
) 3 .
(60)
At the center of the wave (x = 0), this term is maximized as:
bag /max o" ( R - a S )
2(R_aS)3
.
(61)
Although the wave number of the flooding wave has not been fully determined yet, the present model suggests a wave number to describe the flooding as: k 2= 1 / ( 8 R h ) . The present model has information on the surface tension, tube radius, film thickness, and the ratio of the height of the solitary wave to the mean film thickness, a, so that it could interlink the dimensionless numbers of volumetric flux suggested by Wallis and Kutatelaze for flooding. The most uncertain factor, a, must have information on the tube length and the shape of the inlet and exit, which can be modeled when the generation mechanism of the solitary wave is fully understood.
4.2. The interracial pressure force for the capillary wave The interracial pressure force for the capillary wave is studied to explain the flooding of the small-diameter tube where the capillary effect is dominant. We simply model it by equalizing the imaginary-characterstic condition at the singular point to the Helmholtz instability condition with the capillary wave suggested by Jefferey [10] and Zvirin et al. [11]. The imaginary-characteristic condition at the singular point is changed to obtain the interracial pressure difference as follows: aAP
PgPf
I "U .2
~Zfpg---~gpf/t cr) ,
J.Y.
Lee, H.C. No/Nuclear Engineering and Design 146 (1994)225-240
235
where U~r is a critical relative velocity between phases to generate instability. The critical relative velocity modeled by Jefferey is
(4ckl~f) Ucr = c +
1/2
flPg tanh(k6)
'
(63)
c,
where/3 is the dimensionless sheltering coefficient, 0.3. The wave celarity, is modeled by Zvirin et al. [11] from the classical formation for capillary waves in the absence of gravity and the interracial shear stress as follows:
~rk tanh(kS)) 1/2, c= ( -~
(64)
where the wave number k is determined by the following equation: ( 22#- )
= --°'82 ~tanh(kS) .
(65)
8t'fpf
Since the model is based on the linear stability theory and cocurrent flow, applicability to the countercurrent flow is highly suspected. But the reason for accepting this model is that the interfacial pressure force of the solitary wave fits flooding well by the present theory in the case of the large-diameter tube but cannot predict the experimental data by Hewitt et al. for a 1.301-cm diameter tube as shown in the results and discussion section of this paper. S. Results and discussion Validation of the present model is achieved by comparison with other analytical models and experimental data. Particular attention is given to the applicability of the present interracial pressure force, and the roles of the zero and imaginary characteristics are examined.
1.2
The present model(a=3.0) potential flow roll wove separate flow separate cylinder --f i n i t e - a m p l i t u d e flow o o o o 0 Watlis, Hewitt~ C h u n g etal.(SISO) O C O C O Chun9 et oI.(TITO,SITO)
........
..... ....
1.0-,
~x
0.8 -\
0", ~ @~
0 0 \ ",.~
D
=~ 0 . 6 - -
"~ .....'.= ~-L" .......
\a
o
..........
"--'0.4 --
0.2--
0.0 0.0
I
0.2
I
0.4. ( j.
I
0.6 )o.s
I
0.8
1
1.0
1.2
F i g . 3. Comparison of the presL=nt model with various analytical models with air-water flow data ( D = 3 . 1 7 5 cm) (interracial
pressure force: Mean curvature at a = 3, and imaginary characteristics at singular point).
236
ZY. Lee, H. C. No/Nuclear Engineering and Design 146 (1994) 225-240 1.2 - ...... .... ---
a = 2.0 mean curvature a = 3.0 mean curvature a = 4.0 mean curvature a = 5.0 mean curvature capillary wove model O00(X) Exp. data by Hewitt
1.0
0.8
0.6
,)
0.2 0.0 0.0
I
0.2
I
0.4
I
0.6
( J'~ )°.~
1
0.8
I
1.0
1.2
Fig. 4. Flooding e s t i m a t e d by the p r e s e n t m o d e l with the m e a n curvature m o d e l and the capillary wave m o d e l for 5.1 c m d i a m e t e r t u b e (a = wave h e i g h t / f i l m thickness).
5.1. Flooding in a tube without blockage Experimental data by Hewitt [12], WaUis and Makkenchery [13], and Chung et al. [14] are used to validate the present model. As shown in Fig. 3, comparison of the present model with other analytical models and experimental data is done for air-water flow in a 3.175-cm diameter tube. There is divergence of the experimental data at small jr* for smooth versus abrupt end condition. The data of Chung et al. with the tapered inlet or outlet is higher than the sharp inlet or outlet because the smooth inlet or outlet distorts the solitary wave or generates a smaller solitary wave than the sharp inlet or outlet. Since the present model still does not consider the shape of the inlet or outlet, the comparison is done by the data for the sharp entry by WaUis and Makkencherry, Hewitt, and Chung et al. with sharp inlet and sharp outlet. The present theory with the interfacial pressure difference of the solitary wave fits the experimental data when the height of the solitary wave is three times greater than the mean film thickness. To study the tube diameter effect of the present model, the flooding data in the tubes of 5.1 cm, 3.175 cm, and 1.21 cm diameter are checked. Figure 4 shows that the present model predicts flooding data by Hewitt for air-water flow (D -- 5.1 cm). The interracial pressure force of the solitary wave, whose height is about two times greater than the mean film thickness, fits the experimental data well. As the height of the solitary wave increases, flooding occurs with low air volumetric flux since the large solitary wave easily generates flooding. The present model with the interfacial pressure force of the capillary wave underpredicts the experimental data. Figure 5 shows flooding for a 3.175-cm diameter tube. The present model with the interracial pressure force of the solitary wave, whose height is about two times and about three times heigher than the mean film thickness, predicts well the data for the tappered inlet or outlet case and for the sharp inlet or outlet case, respectively. The present model with the capillary wave still underpredicts flooding as in the case of the 5.l-era diameter tube. Figure 6 shows flooding for a 1.21-cm diameter tube. Hewitt's data fit well the present model with the interracial pressure force of the capillary wave. The mean curvature of the solitary wave cannot approach the experimental data despite the fact that the large height of the solitary wave is used. Based on this fact, we recommend that the interracial pressure force be selectively used according to the size of the tube diameter. For example, the mean curvature model of the solitary wave is recommended for a tube
Z Y. Lee, H.C No ~Nuclear Engineering and Design 146 (1994) 225-240
237
1.4 -- -- o = 2.0 m e a n curvature --o = 3.0 m e a n curvature .... o = 4.0 m e a n curvature ........ o = 5.0 m e a n curvature - capillary wave m o d e l 0 0 0 0 0 Exp. data
1.2-~\ 1.0-\
0.8--
0 0
'¢,,,, x o "'0.6
--
,.....~.4 -0.2-(10
I 0.2
0.0
I 0.4
I
0.6 ( j*, ) 0 z
I 0.8
I 1.0
I
1.2
1.4
Fig. 5. Prediction of flooding by the imaginary characteristics with the interfacial pressure force of the mean curvature and the capillary wave model for 3.175 cm diameter tube (a = wave height/film thickness).
whose diameter is greater than 2.0 cm, and the capillary wave model is used for the other small diameter tube.
5.2. Flooding with blockage Celata et al. [15] did experiments on flooding by using orifices in the middle of the test tube, They reported the experimental data for the minimum orifice diameter of 0.8 cm; 1.13 cm; 1.62 cm; and 2.0 cm. This experiment is a good tool for checking the present model of zero characteristics because the sudden area change distorts or choks the wave propagation. For the small-diameter orifice, flooding due to the zero characteristics is more dominant than the imaginary characteristics. However, as the orifice diameter increases, the effect of the imaginary characteristics also increases. Since the tube diameter is less than 2.0 cm; the inteffacial pressure force of the capillary wave is chosen, and Figs. 7 through 10 represent the prediction by the zero characteristics with a solid line and that of the imaginary characteristics with the broken line.
1.4 1.2
"-
C
0.8 ,n d I'0.6
o = 2.0 m e a n curvature a = 3.0 m e a n curvature o = 4.0 m e a n curvature a = 5.0 m e a n curvature capillary wave m o d e l Exp. data by Hewitt
'":~,',',~
",, x~ ,, "''~'N"" ~
1.0
- ---.... ........ - -
"-J0.4
0.2
o
0.0 0.0
I 0.2
I 0.4
(
I 0.6 j * )o.s
I 0.8
I 1.0
I 1.2
1.4
Fig. 6. Flooding estimated by the present model with the mean curvature model and capillary wave model for 1.125 cm diameter tube ( a = wave height/film thickness).
238
].Y. Lee, H.C No/Nuclear
Engineering and Des~
1.4
146 (1994) 225-240
1.4 ......
imaginary characteristics zero characteristics • • a D o exp, dot• by Celoto et al.
1.2--
......
1.0-
1.0-
0.8-
0,8- o ~ [] []
,q
~'~.e,-
% ,...~.4-
DD
~-'o,8-
0~0
--...
%
[] []
imaginary characteristics
- zero characteristics oo•[][] exp. data by Celata et al.
1.2-
'.-..~.4,-
"'"--. "'"-....
0.2--
0.2"-..,
0.0 0.0
I
I
0.2
I
0.4
0.6
I
( j.
0.8
)o.s
I
1.0
I
I
1.2
0.0 0.0
I
1.4
1.6
I
0.2
I
0.4
I I 0.6 0.8 ( j.f )0.5
I
1.0
I
1.2
1.4
Fig. 7. Validation of the present model with experimental data by Celata et al. for 0.8 cm diameter tube. Fig. 8. Validation of the present model with experimental data by Celata et al. for 1.13 cm diameter tube. 1,4
1.4 ......
imoginacy characteristics zero characteristics 0(30o0 exp, data by Celoto et oL
1.2'--
......
1.0--0~-
OB
O@-~. o "-.
%'o6.j
[] " - .
"'002'4t
02--
OD 0.0
imaginary characteristics
zero characteristics o 0 0 0 0 exp. dot• by Celoto et ol.
I
0.2
I
0.4
I
0.6
( j. )0.5
I
0.8
I
1.0
I
1,2
on 1A
0.0
"'"'.......""-..,,, I
I
0.2
0.4
I
0.6
( O. )0.5
I
0.8
I
1.0
I
1.2
1.4
Fig. 9. Validation of the present model with experimental data by Celata et al. for 1.62 cm diameter tube. Fig. 10. Validation of the present model with experimental data by Celata et al. for 2.0 cm diameter tube.
Figure 7 shows flooding with an 0.8-cm diameter orifice. The experimental data fit the present model with zero characteristics well. There are no data corresponding to the prediction of the imaginary characteristics. This means that the severe blockage of the liquid film creates choking in subsonic flow. Figures 8 and 9 shows the flooding data with the orifices of 1.13 cm and 1.62 cm diameters. There is a small gap between the flooding of zero characteristics and that of the imaginary characteristics. Figure 10 shows that the separation of the data by choking and instability is reduced as the tube diameter is increased to 2 cm, and that they occur competitively. 6. Conclusions Validation of the present model with experimental data of air-water flow without blockage along the test tube shows that the present model fits them well. The present analysis based on the mathematical catastrophe theory could be applicable to the limit phenomena in two-phase flow.
].Y. Lee, H.C. No/Nuclear Engineering and Design 146 (1994) 225-240
239
The interracial pressure force of the mean curvature model is applicable to flooding in a large-diameter tube. But for a small-diameter tube where the capillary effect is dominant, the interracial pressure difference based on the capillary wave is used to obtain good agreement with experimental data by Hewitt. Also, the match with the Celata's experimental data shows the competitive occurence of flooding due to choking (zero characteristics) and instability (imaginary characteristics). The present model still has no information on entrance and tube length. Also, the height of the solitary wave is still arbitrary. Further research is expected to resolve these effects by studying the generation mechanisms of the nonlinear wave such as the solitary wave and the capillary wave.
7. Acknowledgement
Financial support received from the Korea Science and Engineering Foundation is greatly appreciated.
8. Nomenclature a
c D f g i j K k P Q R t Ucr V z
ratio of the solitary-wave height to the mean film thickness, wave celarity, tube diameter, drag coefficient, gravitational constant, internal energy, volumetric flux, mean curvature of the wave, wave number, pressure, heat source, tube radius, time, critical relative velocity, velocity, space.
8.1. Greek a F 8 p A /z ~r ~-
void fraction, mass exchange rate, film thickness, density, characteristics, viscocity, variables, surface tension, shear stress, wave coordinate.
ZY. Lee, I-I.C No/Nuclear Engineeringand Design 146 (1994) 225-240
240
8.2. Subscripts f fi g gi i w
liquid, interface of liquid, v a p o r o r air, i n t e r f a c e o f v a p o r o r air, interface, wall.
9. R e f e r e n c e s [1] V.I. Arnolds, Ordinary Differential Equation (The MIT press, Cambridge, 1973) 48-94. [2] Z. Blecki et al., Trajectories and singular points in steady-state models of two-phase flows, Int. J. Multiphase Flow. 13 (1987) 511-533. [3] S.G. Bankoff and S.C. Lee, A critical review of the flooding literature, Multiphase Sci. and Tech. (Hemisphere, 1986) 95-180. [4] G. Iooss and D.D. Toesph, Elementary Stability and Bifurcation Theory (Springer-Verlag, New York, 1980). [5] J.Y. Lee and H.C. No, Effect of inteffacial drag force on the numerical stability of the two-step method in the two-fluid model, Nucl. Engrg. Des. 126 (1991) 427-438. [6] P.L. Kapitza, Wave flow of thin viscous liquid films, Zh. Expel Theo. Fiz. 18 (1983). [7] H.C. Chang, Onset of nonlinear waves on falling films, Phys. Fluid A (18) (1989) 1314-1327. [8] A.C. Newell, Soliton in mathematics and physics, Society for Industrial and Applied Mathematics, Philadelphia Pennsylvania (1985) 1-57. [9] B. O'Neill, Elementary Differential Geometry (Academi, New York, 1966). [10] H. Jefferey, On the formation of water waves by wind, Prec. Roy. Soc. A., Vol 107 (1925) 189-205. [11] Y. Zvirin, R.B. Duffey, and K.H. Sun, On the derivation of countercurrent flooding theory, Syrup. Fluid Flow and Heat Transfer over Rod or Tube bundles, ASME, New York (1979) 111-119. [12] G.F. Hewitt, Influence of end conditions, tube inclination and physical properties on flooding in gas-liquid flow, Report HTFS-RS 222, UKAEA, Harwell, England (1977). [13] G.B. Wallis and S. Makkenchery, The hanging film phenomenon in vertical annular two-phase flow, J. of Fluid Engrg. 96 (1974) 179-298. [14] S.K. Chnng, L.P. Lin, and C.L Tien, Flooding in two-phase countercurrent flows, II. Experimental investigation, Physicochem. Hydrodyn. 1 (1980) 209-220. [15] G.P. Celata et al., The influence of geometry on flooding: Channel length and diameter, National Heat Transfer 1990, 18-tpf-9 (1990) 53-58.
Nuclear Engineeringand Design 146 (1994) 225-240
Dmgn
Hyperbolicity breaking and flooding J a e Y o u n g L e e a, H e e C h e o n N o b a Dept. of Nuclear and Energy Engineering, Cheju National University, Ara 1 Dong, Cheju, South Korea b Korea Advanced Institute of Science and Technology, South Korea
Abstract A new flooding correlation is developed by studying hyperbolicity breaking near a singular point in two-phase flow. Choking and instability correspond to zero and imaginary characteristics which occur when the hyperbolicityof two-phase flow is broken, respectively. Through comparison between results predicted by this model and the experimental data of tubes with and without blockage, it is shown that they are in good agreement with data. The role of the zero characteristics is investigated by comparing the presence model with the experimental data by Celata et al..
1. Introduction Two-phase flow shows many abrupt flow regime changes such as flooding. Abrupt change can be generated at the singular point of the autnomous system [1,2]. The purpose of this paper is to find the condition of the flooding from the mathematical catastrophe in the two-fluid model of two-phase flow. As noted in the review article of Bankoff and Lee [3], Flooding has a large scatter band of data, and a number of correlations and models have been developed. In this study, the two-fluid formulations of two-phase flow are selected to study flooding, and they are changed into ordinary differential equation along the wave propagation line by introducing the characteristics. The singular point of the ordinary differential equation derived contains information about characteristics. If the characteristics at the singular point are zero or imaginary, the hyperbolicity of the equation is broken to change a regular flow pattern to another flow pattern. This bifurcation provides the motivation for this study to develop a new correlation of flooding. The present study provides a general theory of the flooding limit satisfying two-phase flow transients for both phase change and adiabatic cases. But our attention is confined to the adiabatic case only.
2. Singular point and hyperbolicity breaking In order to study a relationship between the limited flow phenomena and singular points, the topological structure of two-phase flow in the phase space must be analyzed on the basis of the 0029-5493/94/$07.00 © 1994 ElsevierScienceB.V. All rights reserved SSDI 0029-5493(93)E0231-8
J.Y. Lee, H.C No/Nuclear Engineeringand Design146 (1994)225-240
226
characteristic analysis. Let us consider a system of first-order, quasi-linear, partial differential equations given by: 0o0o" A(~r)--~- +B(cr)-~z = C(~r).
(1)
Let us transform the coordinate (t, z) into the wave coordinate of ~: ~=At+z.
(2)
Then Eq. (1) is transformed into the following form: 0o(AA + B ) - ~ = C .
(3)
If we confine our attention to solutions which are everywhere continuous but for which the derivative &r/az is discontinuous across a particular line differencing equation (3) across this line gives [12]:
The condition that Eq. (4) has a nontrivial solution gives us the characteristic as follows: IA~+BI
=0.
(5)
For a hyperbolic system, the characteristic, A, must be real and nonzero. To find the condition under which Eq. (1) becomes singular, Eq. (3) is changed into the following form: 0or
N~
a---~--- (AA + B ) - I C + --f,
(6)
where a - - ] A A +BI,
(7)
and N/creates a determinant obtained from (Alia + By) by replacing the i-th column of Cj. The phase space of Eq. (6) is constructed of the n + 1 dimension where n is the number of variables and has three distinctive points of regular points, turning points, and singular points. If A ~ 0, the points in the phase space are regular points. However, this condition depends on the characteristics, A, and Eq. (7) shows that every point in the phase space seems to be irregular. That is to say, the real irregular point occurs when both conditions of A = 0 and hyperbolicity breaking are satisfied. The hyperbolicity of equations is broken when the characteristics of the equations become imaginary or zero. As an example of the above claim, many choked-flow analyses are based on A = 0, which is one of conditions of hyperbolieity breaking [1]. The critical flow occurs at a singular point (a(,~ = 0) = 0, N~ = 0). The critical flow velocity is determined from A(A = 0) = 0, and the location of choked-flow is given by N/--0.
The conditions of hyperbolieity breaking are as follows: (a) Zero characteristic: When the characteristics of equations become zero, the propagation of information is limited or choked. This condition is named as a choking condition. (b) Imaginary characteristic: Imaginary characteristics mean the oscillatory propagation of information. This condition is named as an instability condition. The singular point with hyperbolicity breaking is a bifurcation point where the neutral stability of the equation is branched to the line of choking and to the line of instability [4]. Figure 1 shows the
J.Y. Lee, H.C No/Nuclear Engineeringand Design146 (1994) 225-240
227
Zero Characteristic I
Hypcxbolicity Breaking
~ . ~ I-D Characteristics choking
•
¥
Imaginary ~ c s Hypcrbolicity Breaking
\
2-D Characteristics
\ Saddle ~.
~
2 L N -M =0 2-D Characteristics Neumtl Stability
2
L N -M >0 2-D Qharactexistics
Instability
Fig. 1. The hyperbolicitybreaking represented by the differential geometry. geometrical meaning of the hyperbolicity breaking: the zero characteristics correspond to the dimensional shrink of the solution domain from the surface to the line, and the imaginary characteristics signify the change from the hyperbolic domain to the elliptic domain.
3. Singular points of two-phase flow Since the internal energy conservation equations simply describe the transport of temperature, these equations are removed from the analysis of singular points, and only mass and momentum equations are used. The mass conservation equations of two phases are as follows: (a) Vapor mass: a a at ( a ' p g ) + ~ ( a : g ~ ) = F; (8) (b) Liquid mass: a
at( fp,) +
a =
-r.
(9)
Momentum equations of two-phase are derived on the assumptions that the vapor pressure is equal to the interracial pressure of a phase, i.e., Pk - e k i ---~ 0 and the spatial derivative of the pressure difference of two-phases, A P = P B - I f is a function of surface tension and the curvature of the surface. Since curvature of the surface is related to void fraction, spatial derivative of the pressure force, O A P / a z ) , could be expressed as (aAP/aag)Oas/az). The momentum conservation equations of two-phases are as follows: (c) Vapor momentum:
av. av. ae. 4 ,i + a'Ps'~" + a'PgVg'~'z + ag--~z "~F(V"- Vg) + D
(10)
228
ZY. Lee, H.C No/Nuclear Engineeringand Design 146 (1994) 225-240
(d) Liquidmomentum:
aVe
aVe
oAP Oa,
aPg
@fPf'-~-- "k~fPfVf-~z "{"~f O"Z-'---Otf O~g ~ZZ = - F ( V f i -
Vf) - 4
~
4
q'i + "~q-w-{'-~fpfg.
(11)
where
fi
q'i = 2"Pf IV, Ivv
(12)
fw q-w= T p f IVe IVf,
(13)
F = Qw /fg
(14)
The above four equations are linearized for the vector of cr = (P, ag, Vs, Vf) T to obtain Eq. (1) whose matrices A and B and vector C are as follows [5]:
~ g ~
%C2 2
pg
0
o~fCf 2
--Of
0
0
0 agpg
0
0
0
@fPf]
p,V,
%0,
0
-pfVf 0
0 agpgVg
afpf
~:gg
O/f
- a f ~ag
0
~fpfVf
OtfVfCf 2 B=
i/, 0
aAP
F
-F
C=
r ( K , - K) + ~-D- ¢, + ~,p,g - r ( V f i - Vf) - 4
q'i +
-~q-w--afpfg
where C~-2 Opg/OPg and c ( 2 ape~aPe After transforming the coordinate into ~ and neglecting the elements having the sonic velocity (Cg) -2 and (Cf) -2 which are smaller than the others, the gradient of the void fraction among the gradient of ois =
0ag
=
N(ag)
(15)
J.Y Lee, H.C. No ~Nuclear Engineering and Design 146 (1994) 225-240
229
where aAP A = --Offpg()[ + Vg) 2 - O ~ g p f ( ) k + Vf) 2 "at- ¢lfgO/f
(16)
~a 8 '
N(otg) _ 4 --~-'r a~g i -- --D---,rw 4ag -- a g a f A p g - F [ a f ( A + V,)
ag(A -I- Vf)]
(17)
The conditions of hyperbolicity-breaking are simply obtained from the equation that A = 0 by checking whether the characteristics from this equation become zero or imaginary. Introducing the choking condition A = 0 and A = 0 into Eq. (16) yields
OAP
(18)
aeOgVg2 + %OfVf2 = %af 0%
The solutions of the characteristic equation A(A) -- 0 are A =u + ~u2- v,
(19)
where U~ m
afp,V, + a,pfVf
(20)
~ f p g "~" ~ g p f
and v=
afpgVg 2 + ¢]tgpfVf 2 - CltgO~f~Ap//at:l~g
(21)
olfpg + agpf
Normal soluton of the characteristic equation has three types of solutions: two real solutions (hyperbolic domain), since real solution (parabolic domain), and two complex solutions (elliptic domain). Physically, the hyperbolic domain represents regular flow, the elliptic domain represents instability, and the parabolic domain represents the neutral stable state between regular flow and unstable flow. So the parabolic domain corresponding to the single real root gives us a guideline describing the bifurcation of instability. This neutral stability is obtained by setting the square root part of Eq. (19) at zero:
OAPI( O~fPg + agPf /
(22)
The above condition is a type of the onset condition of Heimholtz instability. Let us consider separately flooding due to the singularity with zero characteristics and singularity with imaginary characteristics.
3.1. Singular points with zero characteristic The compatible relations of zero characteristic are as follows: A = 0 (A = 0) ~ Eq. (18), • N ( a s) = 0 -~ Eq. (17). let us introduce the dimensionless volumetric fluxes, j* and jr*, where •
1/2
. [
pf
~1/2
(23) and new dimensionless numbers affecting flooding,
316. Lee, H.C No/Nuclear Engineeringand Design146 (1994)225-240
230
•
Flooding wave number: Nn
(OAP/Oag) 1/2. N. =
•
g D ,a p
(24)
'
k-phase's change number: Nph D ,/2
(25) Then, Eqs. (17) and (18) become, respectively,
(j;)~- (:;)~ (26)
- + - = (N.) ~ (,~.)~ (,~f)~
and
I J;
IJ~
2/i (%)5/2
2fw
I J; IJ;
(af)2
Otf 1 = af sin/3 + (ag) 2j*Nph,g -- ~ffJf"*N.ph,f,
(27)
where the angle/3 is the inclination of the tube. The ratio of the volumetric flux number and the flooding wave number would be related with the Kutatelaze number. From F_xlS. (26) and (27) we have the following solutions for the adiabatic case: ( j , ) 2 = 1 - 2 sign(j?)fw(Nn) 2 (af)2DE
(28)
and 1 ( 2fi (jf,)2= (ag) 2DE (ag) 1/2
ae ],
°ts ]
(29)
where DE-- - -
2
(
fi - -
(~f)2 (~),:
+
sign(jr*) fw )
~
(30)
By using Eqs. (28) and (29), flooding due to choking will be predicted.
3.2. Singular points with imaginary characteristics Flooding due to instability is modeled with the imaginary characteristics at the singular point. Under this condition, the critical characteristic, A*, is given by: A* --- u. (31) From A -- 0, the necessary condition of a singular point is Eq. (22), and the sufficient condition of the singular point, N[a(A* = 0)], is Eq. (17). Introducing the dimensionless volumetric fluxes into F_xlS.(22) and (17) gives
J~*
og J?
O~g
af
N. % +,~f--, pf
J.Y Lee, H.C No/Nuclear Engineeringand Design 146 (1994) 225-240
231
and
[J:lJ*
2fi (~g)5/2
lJ; IJ;
2fw (af)2 = af sin/3 + F * ,
(33)
where =
afpg + agpf
- - "*
- j f Nph,f •
(34)
Equation (34) represents the contribution of phase change to flooding. From Eqs. (32) and (33) the following two equations are given:
j* = Ej? + F,
(35)
A ( J * ) 2 + B ( j f * ) 2 + C-- 0.
(36)
and where E =
F
(37)
Pg ~ff sign(jr*), 1/~fg
%Na V pg
2E2fi A = (ag)5/---~
4EFft
~f Pf
(38)
2fw sign(jr*) (af)-----~,
Pg+Pf
[ afENph•
B -- (O~g)5/------------------~ afpg + agpf ~
2F2fi C -- (a8)5/'------'~-"
Pg+Pf
afpg+ rvgpf
(39) Nph,f)'
(40)
aS
( afF--NNPh'g/ - af sin ~ . ag ]
(41)
Because the present model has every term in the two-fluid model, we are able to to analyze flooding in both adiabatic and phase-changing cases. 4. Pressure force and singular points
Recent reaserch [6,7] has found that the surface wave of the vertically falling film has many types of waves such as the periodic, solitary, shock, and chaotic waves, and they are generated at a certain bifurcation point affected by the wave speed. Among them, the solitary wave with high amplitude will be a strong candidate to make a singular point. Also, the generation mechanism of these waves is greatly affected by the shape of the entrance. If flooding occurs by a certain magnitude of the solitary wave, then the critical length of a test tube is required for the growth of the small wave at the entrance. Since the interfaeial pressure difference force is greatly affected by the wave shape, the force is a function of flow parameters such as the physical quantities of fluid, the shape of entry and exit, and the diameter and length of the tube:
AP =f(~r, ~, ag, Vg, Vf, d, L, entrance shape).
(42)
232
J.Y. L.ee, H.C. No/Nuckar
Engineering and Design 146 (1994) 225-240
This relationship may be fully understood when nonlinear wave dynamics are understood. In this study, we reduce our attention to quantifing the interfacial pressure difference of the solitary wave and capillary wave. 4.1. The mean curvature of the solitary wave The solitary wave is a nonlinear wave whose magnitude and propagation velocity are proportional the square of the wave number, 6(x, t) =H sech2k(x-ct),
to
(43) which is a solution of the Korteweg-de Vries equation [81. The solitary wave is generated by the third order spatial derivative of the film thickness which corresponds to the pressure force of the momentum equation when the pressure difference between phases is considered. It is a ring type, 2-dimensional
Liquid Film
e
TONS
(a) Simplitication of the solitary wave to obtain its cxuvanue
(b) ‘tle mordkte
system to &ain tbc curvatwe‘of tbc torus
Fig. 2. Curvature of the solitary wave.
J. Y. La, H.C. No /Nuclear Et@ming
and LIesign 146 (1994) 225-240
wave as shown in Fig. 2. The interfacial pressure force of the solitary wave is proportional tension with the second derivative of the wave profile AP=uV’6,
ti3
to .the surface (4)
where a is the surface tension. This term can be represented follows: AP= -a(C,
as the principal curvatures C, and C, as
+ C,).
(45)
The interfacial pressure force of the solitary wave is easily determined when the principal curvatures of the inner surface of the skewed torus in Fig. 2 is known. To simplify our derivation of the principal curvature [9], the solitary wave, whose height is a& is taken for a wave represented by the hyperbolic secant function whose radius is (a - 1)6. Figure 2 shows the coordinate system of the torus of the solitary wave. As a cross-sectional area revolves about a fixed line in the plane of the circle, it generates a, torus. Suppose that the circle is initially in the (i, k) plane with the center on the j axis at a distance U (= R - S) from the origin, and the shape of the function of the solitary wave f(x) = h sech2(kx). Now consider the circle after it has been rotated through an angle 8 about k axis, as shown in Fig. 2. If u is the vector from the origin to the center of the circle and w is the radius vector of the circle, then u = U cos fli + U sin ej,
(46)
and cos @i-f(x)
w= -f(x)
sin Bj+xk,
(47)
where x is the distance having its origin at the center of the solitary wave along the R axis. It follows that y=u+v.
(48)
The unit normal vector of its surface is cos 8i + sin’ Oj +f,R
YeXYX n=
lY,XY,l
The parameters
=
Gi77
of the first fundamental
(?9) *
forms of the surface are
E=Y,*Y,=(-f(~))~,
(50)
F=y,*y,=O,
(51)
G =yx*yx = 1 + (fJ2.
(52)
and
The parameters
of second fundamental
L =yee*n =
-(U-f(x)) Jrn’
M=y,;n
= 0,
N=y,;n
=
forms of the surface are
(53) (54)
and
-fxx
\/l+o”
(55)
234
J.Y. Leg H.C. No/Nuclear Engineeringand Design146 (1994)225-240
The principal curvatures of this torus are determined try the following equation:
( EG
-F2)C
2-
( EN + GL - 2FM)C + ( L N - M 2) + O,
(56)
whose solutions are
C1 =
-1 (57)
(U-F)~l+(fx)
2
and
c2=
-Lx + (fx)2
(58)
The pressure difference between phases is obtained by inserting Eqs. (57) and (58) into Eq. (45) for fx ~: 1: aP=~,
R-f-8
Lx.
(59)
The gradient of the pressure difference to the void fraction is:
bAP
R2
( fxxx/fx R2
a% =or 2 ( R - f - 8 )
2(R-f-8)
) 3 .
(60)
At the center of the wave (x = 0), this term is maximized as:
bag /max o" ( R - a S )
2(R_aS)3
.
(61)
Although the wave number of the flooding wave has not been fully determined yet, the present model suggests a wave number to describe the flooding as: k 2= 1 / ( 8 R h ) . The present model has information on the surface tension, tube radius, film thickness, and the ratio of the height of the solitary wave to the mean film thickness, a, so that it could interlink the dimensionless numbers of volumetric flux suggested by Wallis and Kutatelaze for flooding. The most uncertain factor, a, must have information on the tube length and the shape of the inlet and exit, which can be modeled when the generation mechanism of the solitary wave is fully understood.
4.2. The interracial pressure force for the capillary wave The interracial pressure force for the capillary wave is studied to explain the flooding of the small-diameter tube where the capillary effect is dominant. We simply model it by equalizing the imaginary-characterstic condition at the singular point to the Helmholtz instability condition with the capillary wave suggested by Jefferey [10] and Zvirin et al. [11]. The imaginary-characteristic condition at the singular point is changed to obtain the interracial pressure difference as follows: aAP
PgPf
I "U .2
~Zfpg---~gpf/t cr) ,
J.Y.
Lee, H.C. No/Nuclear Engineering and Design 146 (1994)225-240
235
where U~r is a critical relative velocity between phases to generate instability. The critical relative velocity modeled by Jefferey is
(4ckl~f) Ucr = c +
1/2
flPg tanh(k6)
'
(63)
c,
where/3 is the dimensionless sheltering coefficient, 0.3. The wave celarity, is modeled by Zvirin et al. [11] from the classical formation for capillary waves in the absence of gravity and the interracial shear stress as follows:
~rk tanh(kS)) 1/2, c= ( -~
(64)
where the wave number k is determined by the following equation: ( 22#- )
= --°'82 ~tanh(kS) .
(65)
8t'fpf
Since the model is based on the linear stability theory and cocurrent flow, applicability to the countercurrent flow is highly suspected. But the reason for accepting this model is that the interfacial pressure force of the solitary wave fits flooding well by the present theory in the case of the large-diameter tube but cannot predict the experimental data by Hewitt et al. for a 1.301-cm diameter tube as shown in the results and discussion section of this paper. S. Results and discussion Validation of the present model is achieved by comparison with other analytical models and experimental data. Particular attention is given to the applicability of the present interracial pressure force, and the roles of the zero and imaginary characteristics are examined.
1.2
The present model(a=3.0) potential flow roll wove separate flow separate cylinder --f i n i t e - a m p l i t u d e flow o o o o 0 Watlis, Hewitt~ C h u n g etal.(SISO) O C O C O Chun9 et oI.(TITO,SITO)
........
..... ....
1.0-,
~x
0.8 -\
0", ~ @~
0 0 \ ",.~
D
=~ 0 . 6 - -
"~ .....'.= ~-L" .......
\a
o
..........
"--'0.4 --
0.2--
0.0 0.0
I
0.2
I
0.4. ( j.
I
0.6 )o.s
I
0.8
1
1.0
1.2
F i g . 3. Comparison of the presL=nt model with various analytical models with air-water flow data ( D = 3 . 1 7 5 cm) (interracial
pressure force: Mean curvature at a = 3, and imaginary characteristics at singular point).
236
ZY. Lee, H. C. No/Nuclear Engineering and Design 146 (1994) 225-240 1.2 - ...... .... ---
a = 2.0 mean curvature a = 3.0 mean curvature a = 4.0 mean curvature a = 5.0 mean curvature capillary wove model O00(X) Exp. data by Hewitt
1.0
0.8
0.6
,)
0.2 0.0 0.0
I
0.2
I
0.4
I
0.6
( J'~ )°.~
1
0.8
I
1.0
1.2
Fig. 4. Flooding e s t i m a t e d by the p r e s e n t m o d e l with the m e a n curvature m o d e l and the capillary wave m o d e l for 5.1 c m d i a m e t e r t u b e (a = wave h e i g h t / f i l m thickness).
5.1. Flooding in a tube without blockage Experimental data by Hewitt [12], WaUis and Makkenchery [13], and Chung et al. [14] are used to validate the present model. As shown in Fig. 3, comparison of the present model with other analytical models and experimental data is done for air-water flow in a 3.175-cm diameter tube. There is divergence of the experimental data at small jr* for smooth versus abrupt end condition. The data of Chung et al. with the tapered inlet or outlet is higher than the sharp inlet or outlet because the smooth inlet or outlet distorts the solitary wave or generates a smaller solitary wave than the sharp inlet or outlet. Since the present model still does not consider the shape of the inlet or outlet, the comparison is done by the data for the sharp entry by WaUis and Makkencherry, Hewitt, and Chung et al. with sharp inlet and sharp outlet. The present theory with the interfacial pressure difference of the solitary wave fits the experimental data when the height of the solitary wave is three times greater than the mean film thickness. To study the tube diameter effect of the present model, the flooding data in the tubes of 5.1 cm, 3.175 cm, and 1.21 cm diameter are checked. Figure 4 shows that the present model predicts flooding data by Hewitt for air-water flow (D -- 5.1 cm). The interracial pressure force of the solitary wave, whose height is about two times greater than the mean film thickness, fits the experimental data well. As the height of the solitary wave increases, flooding occurs with low air volumetric flux since the large solitary wave easily generates flooding. The present model with the interfacial pressure force of the capillary wave underpredicts the experimental data. Figure 5 shows flooding for a 3.175-cm diameter tube. The present model with the interracial pressure force of the solitary wave, whose height is about two times and about three times heigher than the mean film thickness, predicts well the data for the tappered inlet or outlet case and for the sharp inlet or outlet case, respectively. The present model with the capillary wave still underpredicts flooding as in the case of the 5.l-era diameter tube. Figure 6 shows flooding for a 1.21-cm diameter tube. Hewitt's data fit well the present model with the interracial pressure force of the capillary wave. The mean curvature of the solitary wave cannot approach the experimental data despite the fact that the large height of the solitary wave is used. Based on this fact, we recommend that the interracial pressure force be selectively used according to the size of the tube diameter. For example, the mean curvature model of the solitary wave is recommended for a tube
Z Y. Lee, H.C No ~Nuclear Engineering and Design 146 (1994) 225-240
237
1.4 -- -- o = 2.0 m e a n curvature --o = 3.0 m e a n curvature .... o = 4.0 m e a n curvature ........ o = 5.0 m e a n curvature - capillary wave m o d e l 0 0 0 0 0 Exp. data
1.2-~\ 1.0-\
0.8--
0 0
'¢,,,, x o "'0.6
--
,.....~.4 -0.2-(10
I 0.2
0.0
I 0.4
I
0.6 ( j*, ) 0 z
I 0.8
I 1.0
I
1.2
1.4
Fig. 5. Prediction of flooding by the imaginary characteristics with the interfacial pressure force of the mean curvature and the capillary wave model for 3.175 cm diameter tube (a = wave height/film thickness).
whose diameter is greater than 2.0 cm, and the capillary wave model is used for the other small diameter tube.
5.2. Flooding with blockage Celata et al. [15] did experiments on flooding by using orifices in the middle of the test tube, They reported the experimental data for the minimum orifice diameter of 0.8 cm; 1.13 cm; 1.62 cm; and 2.0 cm. This experiment is a good tool for checking the present model of zero characteristics because the sudden area change distorts or choks the wave propagation. For the small-diameter orifice, flooding due to the zero characteristics is more dominant than the imaginary characteristics. However, as the orifice diameter increases, the effect of the imaginary characteristics also increases. Since the tube diameter is less than 2.0 cm; the inteffacial pressure force of the capillary wave is chosen, and Figs. 7 through 10 represent the prediction by the zero characteristics with a solid line and that of the imaginary characteristics with the broken line.
1.4 1.2
"-
C
0.8 ,n d I'0.6
o = 2.0 m e a n curvature a = 3.0 m e a n curvature o = 4.0 m e a n curvature a = 5.0 m e a n curvature capillary wave m o d e l Exp. data by Hewitt
'":~,',',~
",, x~ ,, "''~'N"" ~
1.0
- ---.... ........ - -
"-J0.4
0.2
o
0.0 0.0
I 0.2
I 0.4
(
I 0.6 j * )o.s
I 0.8
I 1.0
I 1.2
1.4
Fig. 6. Flooding estimated by the present model with the mean curvature model and capillary wave model for 1.125 cm diameter tube ( a = wave height/film thickness).
238
].Y. Lee, H.C No/Nuclear
Engineering and Des~
1.4
146 (1994) 225-240
1.4 ......
imaginary characteristics zero characteristics • • a D o exp, dot• by Celoto et al.
1.2--
......
1.0-
1.0-
0.8-
0,8- o ~ [] []
,q
~'~.e,-
% ,...~.4-
DD
~-'o,8-
0~0
--...
%
[] []
imaginary characteristics
- zero characteristics oo•[][] exp. data by Celata et al.
1.2-
'.-..~.4,-
"'"--. "'"-....
0.2--
0.2"-..,
0.0 0.0
I
I
0.2
I
0.4
0.6
I
( j.
0.8
)o.s
I
1.0
I
I
1.2
0.0 0.0
I
1.4
1.6
I
0.2
I
0.4
I I 0.6 0.8 ( j.f )0.5
I
1.0
I
1.2
1.4
Fig. 7. Validation of the present model with experimental data by Celata et al. for 0.8 cm diameter tube. Fig. 8. Validation of the present model with experimental data by Celata et al. for 1.13 cm diameter tube. 1,4
1.4 ......
imoginacy characteristics zero characteristics 0(30o0 exp, data by Celoto et oL
1.2'--
......
1.0--0~-
OB
O@-~. o "-.
%'o6.j
[] " - .
"'002'4t
02--
OD 0.0
imaginary characteristics
zero characteristics o 0 0 0 0 exp. dot• by Celoto et ol.
I
0.2
I
0.4
I
0.6
( j. )0.5
I
0.8
I
1.0
I
1,2
on 1A
0.0
"'"'.......""-..,,, I
I
0.2
0.4
I
0.6
( O. )0.5
I
0.8
I
1.0
I
1.2
1.4
Fig. 9. Validation of the present model with experimental data by Celata et al. for 1.62 cm diameter tube. Fig. 10. Validation of the present model with experimental data by Celata et al. for 2.0 cm diameter tube.
Figure 7 shows flooding with an 0.8-cm diameter orifice. The experimental data fit the present model with zero characteristics well. There are no data corresponding to the prediction of the imaginary characteristics. This means that the severe blockage of the liquid film creates choking in subsonic flow. Figures 8 and 9 shows the flooding data with the orifices of 1.13 cm and 1.62 cm diameters. There is a small gap between the flooding of zero characteristics and that of the imaginary characteristics. Figure 10 shows that the separation of the data by choking and instability is reduced as the tube diameter is increased to 2 cm, and that they occur competitively. 6. Conclusions Validation of the present model with experimental data of air-water flow without blockage along the test tube shows that the present model fits them well. The present analysis based on the mathematical catastrophe theory could be applicable to the limit phenomena in two-phase flow.
].Y. Lee, H.C. No/Nuclear Engineering and Design 146 (1994) 225-240
239
The interracial pressure force of the mean curvature model is applicable to flooding in a large-diameter tube. But for a small-diameter tube where the capillary effect is dominant, the interracial pressure difference based on the capillary wave is used to obtain good agreement with experimental data by Hewitt. Also, the match with the Celata's experimental data shows the competitive occurence of flooding due to choking (zero characteristics) and instability (imaginary characteristics). The present model still has no information on entrance and tube length. Also, the height of the solitary wave is still arbitrary. Further research is expected to resolve these effects by studying the generation mechanisms of the nonlinear wave such as the solitary wave and the capillary wave.
7. Acknowledgement
Financial support received from the Korea Science and Engineering Foundation is greatly appreciated.
8. Nomenclature a
c D f g i j K k P Q R t Ucr V z
ratio of the solitary-wave height to the mean film thickness, wave celarity, tube diameter, drag coefficient, gravitational constant, internal energy, volumetric flux, mean curvature of the wave, wave number, pressure, heat source, tube radius, time, critical relative velocity, velocity, space.
8.1. Greek a F 8 p A /z ~r ~-
void fraction, mass exchange rate, film thickness, density, characteristics, viscocity, variables, surface tension, shear stress, wave coordinate.
ZY. Lee, I-I.C No/Nuclear Engineeringand Design 146 (1994) 225-240
240
8.2. Subscripts f fi g gi i w
liquid, interface of liquid, v a p o r o r air, i n t e r f a c e o f v a p o r o r air, interface, wall.
9. R e f e r e n c e s [1] V.I. Arnolds, Ordinary Differential Equation (The MIT press, Cambridge, 1973) 48-94. [2] Z. Blecki et al., Trajectories and singular points in steady-state models of two-phase flows, Int. J. Multiphase Flow. 13 (1987) 511-533. [3] S.G. Bankoff and S.C. Lee, A critical review of the flooding literature, Multiphase Sci. and Tech. (Hemisphere, 1986) 95-180. [4] G. Iooss and D.D. Toesph, Elementary Stability and Bifurcation Theory (Springer-Verlag, New York, 1980). [5] J.Y. Lee and H.C. No, Effect of inteffacial drag force on the numerical stability of the two-step method in the two-fluid model, Nucl. Engrg. Des. 126 (1991) 427-438. [6] P.L. Kapitza, Wave flow of thin viscous liquid films, Zh. Expel Theo. Fiz. 18 (1983). [7] H.C. Chang, Onset of nonlinear waves on falling films, Phys. Fluid A (18) (1989) 1314-1327. [8] A.C. Newell, Soliton in mathematics and physics, Society for Industrial and Applied Mathematics, Philadelphia Pennsylvania (1985) 1-57. [9] B. O'Neill, Elementary Differential Geometry (Academi, New York, 1966). [10] H. Jefferey, On the formation of water waves by wind, Prec. Roy. Soc. A., Vol 107 (1925) 189-205. [11] Y. Zvirin, R.B. Duffey, and K.H. Sun, On the derivation of countercurrent flooding theory, Syrup. Fluid Flow and Heat Transfer over Rod or Tube bundles, ASME, New York (1979) 111-119. [12] G.F. Hewitt, Influence of end conditions, tube inclination and physical properties on flooding in gas-liquid flow, Report HTFS-RS 222, UKAEA, Harwell, England (1977). [13] G.B. Wallis and S. Makkenchery, The hanging film phenomenon in vertical annular two-phase flow, J. of Fluid Engrg. 96 (1974) 179-298. [14] S.K. Chnng, L.P. Lin, and C.L Tien, Flooding in two-phase countercurrent flows, II. Experimental investigation, Physicochem. Hydrodyn. 1 (1980) 209-220. [15] G.P. Celata et al., The influence of geometry on flooding: Channel length and diameter, National Heat Transfer 1990, 18-tpf-9 (1990) 53-58.