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Classification and modeling of flooding in vertical narrow rectangular and annular channels according to channel-end geometries
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Classification and modeling of flooding in vertical narrow rectangular and annular channels according to channel-end geometries ⁎
Moon Won Song, Hee Cheon No
Korea Advanced Institute of Science and Technology (KAIST), Department of Nuclear and Quantum Engineering, 291, Daehak-ro, Yuseong-gu, Daejeon 34141, Republic of Korea
A R T I C LE I N FO
A B S T R A C T
Keywords: Flooding Narrow rectangular and annular channels Data classification Hyperbolicity breaking Large diameter annuli
In this study, data classification and modeling of the flooding phenomenon were performed in a narrow channel with a gap thickness of several millimeters. To implement the conventional classification study to a narrow channel case, three key ideas for two-phase flow in a narrow channel were proposed; characteristic length, the flow path for each phase and a concept of unit-cell that can unify both narrow rectangular and annular channels including large outer diameter. Based on the ideas, 330 flooding data points for both narrow rectangular and annular channels including large outer diameter were classified into exit flooding and entrance flooding according to the geometries of the liquid inlet. In order to develop the prediction model for flooding in a narrow channel, the hyperbolicity breaking concept was introduced. The present flooding model was derived based on the hyperbolicity breaking of the two-fluid model and produced the Root Mean Square Errors (RMSE) of 30.02% in terms of the superficial gas velocity against the 330 flooding data points from narrow rectangular and annular channels with relatively small outer diameter. Moreover, the proposed model was applied to a case of narrowlarge diameter annuli with the introduction of the number of the unit-cells producing the RMSE of 26.11% against 91 flooding data points. The RMSEs of the existing models for narrow rectangular and narrow largediameter annuli were 38.25% and 31.10%, respectively. Different from the existing models, the proposed model turned out that it is applicable to various narrow channel-types; a rectangular, an annular and an annulus with a large diameter.
1. Introduction To estimate the heat removal rate of overheated systems by gravitydriven water penetration through a narrow channel, flooding in a narrow channel is one of the important issues. Typical examples are heat pipes, cooling systems of an electronic device, and nuclear systems in a severe accident. In the severe accident of a nuclear system, overheated molten-fuel in a reactor vessel is quenched by the gravity-driven water penetration, which flows through a narrow gap. The water penetration rate is restricted by the rising vapor flow; the water penetration rate governs the heat removal rate of the overheated-molten fuel in the lower plenum (Christoph, 2006). The issue of flooding in a narrow channel has recently been studied to estimate critical heat flux in a narrow channel (Juarsa, 2014; Kim et al., 2019). Hence, several experimental studies and modeling for flooding in a narrow channel were carried out. In the above experiment in narrow channels, the geometric type of a narrow channel can be divided into two categories. The first one is flooding in a rectangular narrow channel (Mishima, 1984; Sudo and
⁎
Kaminaga, 1989; Osakabe and Kawasaki, 1989; Osakabe et al., 1994; Vlachos et al., 2001; Drosos et al, 2006; Li and Sun, 2010) Their ranges of gap thickness and width are 1.5 ~ 10 mm and 40 ~ 150 mm, respectively. The second one is flooding in a narrow annular channel, which a rod was inserted into a tube (Ueda and Suzuki, 1978; Regland et al., 1989; Osakabe and Futamata, 1996). The ranges of the gap thickness and the outer diameter are 5 mm ~ 17 mm and 50 ~ 101.6 mm, respectively. Moreover, there is cases of large-diameter annuli, whose outer diameters are 440 mm (Richter et al., 1979) and 500 mm (Jeong, 2008). The hydraulic parameters of channels of the former researcher’s work are shown in Table 1. In order to predict the flooding gas velocity, following equation and non-dimensional parameters are widely used (Wallis, 1969; Tien, 1977).
(jg ∗ )1/2 + m (jf ∗ )1/2 = c ∗
jk = Ck jk k is f(liquid) or g(gas)
Corresponding author.
https://doi.org/10.1016/j.nucengdes.2020.110539 Received 30 August 2019; Received in revised form 17 January 2020; Accepted 27 January 2020 0029-5493/ © 2020 Elsevier B.V. All rights reserved.
Please cite this article as: Moon Won Song and Hee Cheon No, Nuclear Engineering and Design, https://doi.org/10.1016/j.nucengdes.2020.110539
(1) (2)
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Table 1 Hydraulic parameters of channels of former researcher’s work. Author
Gap thickness (mm)
Channel width (mm)
Inner diameter (mm)
Outer diameter (mm)
Geometry
Mishima, 1984 Sudo and Kaminaga, 1989 Osakabe and Kawasaki, 1989 Osakabe et al., 1994 Vlachos et al., 2001 Drosos et al, 2006 Li and Sun, 2010 Ueda and Suzuki, 1978 Regland et al., 1989 Osakabe and Futamata, 1996 Richter et al., 1979 Jeong, 2008
1.5, 5.3, 8.3, 12.3 2.3, 5.3, 8.3, 12.3 2, 5, 10 1, 2 5, 10 10 5 – – – – –
100 66 100 10, 15, 20, 40, 60, 80, 100 150 120 50 – – – – –
– – – – – – – 10 90.064 16, 30, 36, 40, 45 393.7 490, 494, 496
– – – – – – – 28, 35, 40 101.6 50 444.5 500
rectangular rectangular rectangular rectangular rectangular rectangular rectangular annulus annulus annulus large annulus large annulus
flooding, while the smooth geometries were classified to the exit flooding. The main characteristic of the results of the classification is that the exit flooding has higher gas flooding velocity than the entrance flooding at the same liquid superficial velocity. It means that the interaction between liquid and gas at the liquid inlet becomes maximum due to the sharp geometries. By contrast, the interaction is minimum on the condition of smooth geometry. Moreover, Jeong and NO (1996) found that the tube length effect appears on the only exit-flooding condition because a large wave that causes flooding is generated at the exit, and the wave moves along the tube length. However, in the case of the entrance flooding, since the wave causing the flooding appears at the entrance; the tube length has a weak effect. This study adopted the classification method proposed by Jeong and NO (1994) to a narrow rectangular and a narrow annular channel with key ideas for two-phase flow in a narrow channel. Based on the classification of this study, modeling was carried out using the concept of hyperbolicity-breaking derived from the two-fluid model of liquid/gas flow in a narrow channel. Prediction results were compared with 330 data for flooding in a rectangular channel and an annular channel. Finally, the classification method and the prediction model were applied to the case of an annulus with a large diameter (around 500 mm).
1/2
ρk ⎛ ⎞ : Wallis parameter CW = ⎜ gLch (ρf − ρg ) ⎟ ⎝ ⎠
(3)
1/4
ρk 2 ⎛ ⎞ : Kutateladze parameter CK = ⎜ gσ (ρf − ρg ) ⎟ ⎝ ⎠
(4)
where jk is superficial velocity of phase k. The constants m and c are empirical constants for flooding correlation. Cw and CK are the parameters that make the superficial velocity non-dimensional. ρk, g, Lch, σ are density of k phase, gravitational acceleration, characteristic length, and surface tension, respectively. The m and c values in the Eq. (1) are empirically determined because the database of flooding have large deviations from each other. Therefore, correlations for prediction of the flooding velocity in a narrow channel have been proposed based on the Eq (1). Regardless of a number of flooding correlations, there is no correlation satisfying the whole flooding experiments of flooding both in a narrow rectangular channel and an annulus. It is commonly believed that the occurrence of flooding strongly depends on tube-end conditions though experiments were carried out on similar geometric conditions (Bankoff and Lee, 1986; Jeong and NO, 1996). They classified the flooding data into two categories; entrance flooding and exit flooding (Fig. 1.). When the tube-end geometry is sharp, flooding occurs at the liquid entrance due to the hydraulic jump on the sharp region (Fig. 1-(a)). On the other hand, there is no hydraulic jump on the smooth tube-end condition. Wave at the liquid exit grows and moves up to the entrance region (Fig. 1-(b)). For this reason, Wallis proposed the c value in Eq. (1) as 0.725 for sharp-end and 1.0 for smooth-end, respectively. In order to classify mathematically reasonably the flooding data according to tube-end geometries, Jeong and NO (1994) utilized entropy minimax principle. As a result, they successfully classified 17 types of tube-end geometries into the two types of flooding; exit flooding and entrance flooding (Fig. 2). As a result, the sharp geometries were classified to the entrance
2. Classification of flooding data in a narrow channel 2.1. Key ideas for a two-phase narrow channel This study applied the classification method for flooding in a tube proposed by Jeong and NO to the cases of narrow channels. Moreover, we added three key ideas for two-phase flow in a narrow channel. The first idea refers to the characteristic length. In Sadatomi’s study (Sadatomi et al., 1982), they measured the slug velocity and the constant, Cslug, in the vertical non-circular channel (Eq. (5)). They found that the slug constant, Cslug, can be unified 0.35 for the various noncircular channels when the characteristic length, Lch, is the width dominant value instead of the conventional hydraulic equivalent
Fig. 1. Classification of flooding according to tube-end geometries (a) entrance flooding (b) exit flooding. 2
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Fig. 2. Classification of flooding in a tube according to channel-end geometries (Jeong and NO, 1994).
debris, Okano study (Okano et al., 2003) modeled that flow path of liquid/gas in a narrow channel is separated along the width direction as shown in Fig. 3-(b), unlike Fig. 3-(a). In our study, this configuration was applied to the part of the modeling of flooding in a narrow channel. The third idea is a concept of a unit-cell suggested by Osakabe study (Osakabe and Futamata, 1996). The unit-cell concept attempt to unify the narrow rectangular data and the narrow annular data (Fig. 4). Jeong (2008) observed that the liquid/gas flow path in an annular channel is separated likewise the case of a rectangular channel. Thus, dividing the annular channel along the circumferential direction, the divided one which is called unit-cell can be dealt with similar to the rectangular data. By utilizing the concept of the unit-cell, this study unified the rectangular data and the annular data. 2.2. Classification of flooding data in a narrow channel Using the concept of the unit-cell noted in Section 2.1, this study utilized the flooding data of both a vertical narrow rectangular channel (Mishima, 1984; Sudo and Kaminaga, 1989; Osakabe and Kawasaki, 1989; Vlachos et al., 2001; Drosos et al, 2006; Li and Sun, 2010) and a vertical narrow annular channel (Ueda and Suzuki, 1978; Regland et al., 1989; Osakabe and Futamata, 1996). The Osakabe data (Osakabe et al., 1994) was not included because the channel was not straight. There was area change along the tube length direction. This study classified the flooding data according to the liquid-entrance geometries likewise Jeong’s study (Jeong and NO, 1994). Based on the first key idea, this study utilized the width as the characteristic length for Wallis parameter instead of the hydraulic equivalent diameter. The classification results and the method are shown in Figs. 5 and 6, respectively. The results well agreed with the previous study; the gas flooding velocity in the case of exit flooding was higher than that of the entrance flooding. Likewise, the porous liquid inlet classified to the exit flooding because the interaction between liquid and gas becomes minimum, whereas the protruded plate inlet was classified to the entrance flooding due to maximum interaction at the liquid inlet. In the case of the squared plate geometry, some data were classified
Fig. 3. Separated flow path for liquid/gas along (a) gap direction (b) width direction.
diameter, De (Eq. (6) and Fig. 3.).
uslug = Cslug gLch 4Aflow Pw 2(w + d ) 2w = ≈ for w > > d (instead of De = π π π Pw 4wd = ≈2d ) 2(w + d )
(5)
Lch =
(6)
where uslug, Cslug, Pw, Aflow, w and d are slug velocity, slug constant, wetted perimeter, flow area, width of channel, and gap thickness of channel, respectively. The second idea is that the flow path of liquid–gas in a narrow channel is separated along the width direction instead of the gap direction (Fig. 4) based on the observations of Osakabe’s study (Osakabe et al., 1994). Moreover, one of the studies for cooling of overheated 3
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Fig. 4. Unit-cell concept.
to the exit flooding, the others were classified to the entrance flooding. This study proposed the criterion that can determine the flooding type for the case of a narrow squared plate. Using the ratio of the width (w) to the gap thickness (d), the data of the ratios is larger than 10 was classified to the exit flooding. In the case of the ratios is smaller than 10, those data were classified to the entrance flooding. It can be explained that the smaller ratio of the width to the gap thickness tends to generate the large wave that causes the flooding at the liquid inlet. Taking the example of the case of form-loss pressure drop of a single-phase flow, the case of the protruded-plate inlet has higher formloss pressure drop than the case of the squared-plate inlet. Similarly, this study showed that the protruded-plate inlet has a larger interaction between liquid and gas than the squared-plate inlet. As well as, the porous-media inlet has been utilized to make the immediate formation of a stable liquid film (Ghiaasiaan, 2008). Thus, it is reasonable that the case of porous-media inlet was classified to the exit flooding, which makes the interaction minimum. 3. Modeling and validation for flooding in a narrow channel Fig. 5. Classification of flooding data in both a rectangular and an annular channel.
3.1. Modeling based on the hyperbolicity breaking concept In order to predict the flooding velocity, the Eqs. (1)-(4) have been widely used with empirical constants. However, there is a limitation that the empirical constants are highly affected by channel geometries. On a fixed geometry, analytic models or computational-fluid-dynamics (CFD) studies for the flooding phenomenon have been proposed based on the momentum balance equations and wave motions. This study developed a prediction model for flooding in a vertical narrow channel using the concept of hyperboilicity-breaking and singular points. Lee and NO studied the hyperbolicity-breaking concept near singular points in two-phase flow (Lee and NO, 1994). NO and Jeong developed the correlation for flooding in a vertical tube based on the hyperbolicity-breaking concept (NO and Jeong, 1996). Recently through using the hyperbolicity-breaking and singular point concept, the prediction model for flooding in a nearly horizontal tube was developed (Zhou et al., 2018). This section describes the hyperboilicity-breaking near singular points derived from the two-fluid model for a narrow channel. As explained in section 2.1, the second key idea for a two-phase narrow channel was applied; each phase is separated along the width direction (Fig. 7). Though using this physical configuration, one-dimensional two-fluid models for mass and momentum can be derived as seen in Eqs. (7–11). The wall and interfacial shear terms follow Fig. 7 as described in Eqs. (9) and (10). Based on the Eqs. (7)–(11), the first-order, quasi-linear, partial differential equations system can be derived as Eqs. (12)–(16). Vapor mass:
Fig. 6. Classification of flooding data in a narrow channel according to the liquid-entrance geometries.
∂ ∂ (αg ρg ) + (αg ρg Vg ) = Γ ∂t ∂z Liquid mass: 4
(7)
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Γ ⎛ ⎞ −Γ ⎜ ⎟ 2αg 2 C = ⎜ Γ(Vgi − Vg ) − τ − w τi, g − αg ρg g ⎟ d w, g ⎜⎜ ⎟ 2αf 2 − Γ(Vfi − Vf ) + d τw, f + w τi, f − αf ρf g ⎟ ⎝ ⎠
(16)
If we introduce wave coordinate and characteristic velocity, ξand λ, Eq. (12) becomes
(Aλ + B )
∂X = C where ξ = z + λt ∂ξ
(17)
Applying Cramer’s rule, we have
∂αg ∂ξ
=
Ni Δ
(18)
where
Δ = |Aλ + B| = αf ρg (λ + Vg )2 + αg ρf (λ + Vf )2 + αg αf Fig. 7. Descriptions for counter-current flow in a narrow channel.
∂ΔP ∂αg
(19)
Nαg = Γ(αf (Vg + λ ) + αg (Vf + λ )) − Γ(αf (Vgi − Vg ) + αg (Vfi − Vf ))
∂ ∂ (αf ρf ) + (αf ρf Vf ) = −Γ ∂t ∂z
+
(8)
2αf αg d
(τw, f + τw, g ) +
2 (αf τi, g w
+ αg τi, f ) − αf αg Δρg (20)
Vapor momentum:
αg ρg
∂Vg ∂t
∂Vg
+ αg ρg Vg
∂z
+ αg
∂Pg ∂z
= Γ(Vgi − Vg ) −
2αg d
τw, g −
If Δ is zero, the points in the phase space become singular points. The solutions of the characteristic equation, Δ = 0, are
2 τi, g − αg ρg g w (9)
∂Vf ∂t
∂Vf
+ αf ρf Vf
+ αf
∂Pf
p=
∂z ∂z 2αf 2 τw, f + τi, f − αf ρf g = −Γ(Vfi − Vf ) + d w
αf ρg + αg ρf
q=
∂ΔP ∂αg Pf = Pg + ΔP; = + ∂z ∂z ∂αg ∂z
∂Pg
∂X ∂X A +B =C ∂t ∂z
(12)
X = (Pg , αg , Vg , Vf )T
(13)
∂ρ
αf
αf
∂ΔP ∂αg
0
αf ρg + αg ρf
(24)
δ (ξ ) = hsech2 (κξ ) + δ0
(25)
1 1⎞ ΔP = −σ ⎛ + ≈ σ (2hκ 2) R R 1 2⎠ ⎝
(26)
⎜
(14)
⎞ ⎟ αf ρf ⎟ ⎟ 0 ⎟ αf ρf Vf ⎟⎟ ⎠
(23)
In Eq. (24) two parameters are necessary to predict the flooding velocity; derivative of pressure difference with respect to void fraction and void fraction. The pressure difference term was obtained using the Young-Laplace equation. It was assumed that the curvature of the peak of the wave just before the occurrence of flooding is represented by the Kortewegde Vries type of solitary wave (Coulson and Jeffrey, 1977) (Fig. 8).
∂ρ
⎛ αg Vg g ρg Vg αg ρg ∂Pg ⎜ ∂ρf ⎜ αf Vf − ρf Vf 0 ∂Pg B=⎜ αg ρg Vg 0 ⎜ αg
∂ΔP ∂αg
jf ⎞2 ⎛ ∂ΔP ⎞ ⎛ αg 1 − αg ⎞ ⎛ jg + + ⎜ ⎟ = ⎜− ⎟ ⎜ 1 − αg ⎠ ρf ⎟ ⎝ αg ⎝ ∂αg ⎠ ⎝ ρg ⎠
where
⎛ αg g ρg 0 0 ⎞ ∂Pg ⎟ ⎜ ∂ρf ⎜ 0 0 ⎟ A = αf ∂Pg − ρf ⎟ ⎜ 0 αg ρg 0 ⎟ ⎜ 0 ⎜ 0 0 0 αf ρf ⎟ ⎠ ⎝
(22)
If the λ becomes imaginary, hyperbolicity-breaking takes place: the conditions for changing from the hyperbolic domain with two real solutions to the elliptic one with two complex ones represent that the twophase flow changes from stable flow to unstable flow. Thus, the occurrence of flooding can be derived from the neutral stability condition; the square root term in Eq. (21) set to zero.
(11)
and t, z, αk, Γ, Vk, Pk, Vki, τi,k, and τw,k are time, coordinate parallel to fluid motion, volume fraction for phase k, phase change rate, flowarea-averaged velocity for each phase k, pressure for phase k, velocity at the interface for phase k, interfacial drag for phase k, and wall drag for phase k, respectively. Partial differential equations system based on two-fluid model:
⎜⎜ ⎝
αf ρg Vg + αg ρf Vf
αf ρg Vg 2 + αg ρf Vf 2 + αg αf
(10)
where
∂Pf
(21)
where
Liquid momentum:
αf ρf
p2 − q
λ=p±
0
⎟
where R1 = w /2 − (h + δ0 ), R2 =
⎛ d 2δ (ξ ) ⎜ dξ 2 ⎝
⎞ 1 = ⎟ 2 hκ 2 − ξ =0 ⎠
(27)
And δ, h, κ, δ0 are film thickness, wave amplitude, wave number, and substrate thickness, respectively. Moreover, the void fraction relation was applied to calculate the term of derivative of pressure difference.
(15) 5
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jf ⎞ 1 − αg ⎞ ⎛ αg ⎛ jg + ⎜ ⎟ = (Δρgw ) ⎜ρ + α 1 α ρf ⎟ − g⎠ ⎝ g ⎠ ⎝ g
(31)
In order to get the solution using Eq. (31), we need the relation of the liquid fraction in terms of superficial velocities. The NO study (NO and Jeong, 1996), which developed the flooding correlation using the hyperbolicity breaking, also utilized the semi-empirical correlation of the liquid fraction. Theoretical basis of their correlation for the liquid fraction is the turbulent falling-film theory (Wallis, 1969) which shows relation between the liquid fraction(αf) and a non-dimensional liquid superficial velocity (jf*) as seen in Eq. (32);
αf = cm jf ∗ce
They utilized the turbulent falling-film theory for the exponent parameter, ce. Also they empirically determined multiplier parameter, cm, because they assumed that the liquid fraction at the peak of a solitary wave is proportional to the spatially averaged liquid film thickness. We adopted similar approach to the NO study to obtain the void fraction relation at the peak of a solitary wave. However, the channel geometry of interest of this study differs from the NO study. Thus, we empirically determined the exponent and the multiplier according to the classification introduced in this study as can be seen in Eq. (33) and (34);
Fig. 8. Solitary wave in a unit-narrow channel.
2h ⎞ αg = ⎛1 − w⎠ ⎝
(28)
Consequently, the derivative term becomes
∂ΔP = −σκ 2w ∂αg
(29)
Assuming that the wavelength, λ, is proportional to the marginalfastest growing wave of Helmholtz instability, we can obtain the wave number as follows:
κ=
2π = λ
σ 1/2 Δρg
( )
αf = 0.91j∗
0.124 for f
exit flooding
(33)
αf = 1.06j∗
0.137 for f
entrance flooding
(34)
where j*f is the non-dimensional liquid superficial velocity of Wallis type with the characteristic length of w. We checked that the exponent value for a narrow channel was around 0.15 by Jeong experiment (Charlgeri and Jeong, 2019) which measured the void fraction in a vertical downward rectangular narrow channel. Fig. 9 summarized the key idea, classification, and modeling
2π 2π
(32)
(30)
Therefore, the governing equation becomes
Fig. 9. Summary of the key ideas, data classification, and modeling. 6
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Table 2 Accuracies of this study and the other model. Narrow rectangular data
jg non-dimensional jg *
Osakabe (Eq. (35))
Sudo (Eq. (36))
MAE
RMSE
MAE
RMSE
MAE
RMSE
20.61% 9.73%
32.46% 14.14%
27.83% 16.39%
35.92% 23.27%
47.75% 22.09%
74.65% 32.13%
Fig. 12. Description of perimeter equivalent concept to find the width of the unit-cell.
Fig. 10. Comparison between measured and predicted jg for both rectangular and annular channels (Error band: (jg, predicted − jg, measured)/jg,measured).
channel. Although there are several studies for the narrow-channel flooding correlation (Mishima, 1984; Sudo and Kaminaga, 1989; Osakabe et al., 1994), the Osakabe correlation (Eq. (35) and Fig. 11-(a)) and the Sudo correlation (Eq. (36) and Fig. 11-(b)) were compared. The present model predicted the data with better accuracy than the other one did as shown in Table 2. Even though the other correlation can be only applied to a rectangular channel data, the present model can cover both rectangular and annular channels with better accuracy.
suggested in this study. 3.2. Validation and Comparison with other correlation The database utilized in the classification study was compared with the calculation results using the model proposed in this study. The mean-averaged error (MAE) and the root-mean-square error (RMSE) were 20.74% and 30.02%, respectively (Fig. 10.). The applicable ranges are as follows: -
Present model
Geometry: vertical rectangular, vertical annular Gap thickness: 1.5 ~ 15 mm Width: 40 ~ 300 mm Height: 470 ~ 1235 mm Superficial velocity of liquid: 0.001 ~ 0.3 m/s Superficial velocity of gas: 0.4 ~ 12 m/s Working fluid: water and air Pressure: atmospheric pressure
Kug 1/2 + 0.8Kuf 1/2 = 0.58Bo1/8
(35)
jg ∗1/2 + (0.5 + 0.001Bo∗) jf ∗1/2 = 0.66(d/w )−0.25
(36)
where Kuk, Bo, jk*, Bo*, d, and w are non-dimensional superficial velocity of the Kutateladze type and Bond number, non-dimensional superficial velocity of the Wallis type, modified Bond number, gap thickness and channel width, respectively. 3.3. Narrow annular channel with large diameter (~500 mm) As explained in Section 2.1, the characteristic length for two-phase flow in a narrow channel becomes a width of the unit-cell instead of the hydraulic equivalent diameter. However, in a case of narrow annulus
Moreover, this study compared the proposed model with other correlations for prediction of flooding velocity in a narrow rectangular
Fig. 11. Other correlations and proposed model against database for flooding in a rectangular narrow channel (Error band: (jg, predicted − jg, measured)/jg,measured) ((a): Osakabe correlation, (b): Sudo correlation). 7
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is utilizing the criterion that the flooding type changes from Wallis type to Kutateladze type. In the flooding of the Wallis type, flooding occurs on the slug flow regime in a tube with a small diameter. However, in the Kutateladze type flooding, flooding occurs on the annular flow regime in a large diameter tube. In this study, it was suggested that the total perimeter of the one unit-cell is equivalent to the total perimeter of the changing criterion for the flooding type in a conventional tube (Fig. 12, Eq. (37) and (38)).
πDWallistoKutateladze = 2(wmax + d )
(37)
πDWallistoKutateladze −d 2
(38)
→ wmax =
where DWallistoKutateladze and wmax are the diameter that changes the flooding type from Wallis to Kutateladze and maximum width of the unit-cell. The specific criterion is as follows (Ghiaasiaan, 2008);
2<
D < 20 : Wallis type flooding λL
(39)
D > 40 : Kutateladze type flooding λL Fig. 13. Comparison between measured and predicted jg for annulus channels with a large diameter (Error band: (jg, predicted − jg, measured)/jg,measured).
σ ⎞ → DWallistoKutateladze = 20λL ~40λL , where λL = ⎜⎛ ⎟ ⎝ Δρg ⎠
(40) 0.5
(41)
The second idea is that the number of unit-cells changes according to the gas velocity. Based on Osakabe’s observations (Osakabe et al., 1994), the gas plug in a narrow channel was affected by the gas velocity. In this study, the relation between the width of unit-cell and gas velocity was suggested as Eqs. (42) and (43).
⎛ jg ⎞ w = wmax ⎜ ⎟ j ⎝ g, crit ⎠
0.15
(42) 1/4
ρg 2 ⎞ where jg, crit = Ku∗ g, crit ⎛⎜ ⎟ ⎝ Δρgσ ⎠
, Ku∗ g, crit = 1. 62 (43)
The wmax can be calculated using Eqs. (38) and (41). The constant in Eq. (41) was 20. The jg, crit was obtained from critical value of Kutateladze number, 1.6 (Eq. (43)). The exponent in the Eq (42), 0.15, was determined from the relation between the bubble size and flow rate of air in a case of bubble generation by an air injection (Al Ba’ba’s et al., 2016). The flooding model proposed in this study, Eq. (31–34), and the ideas explained in this section were validated against the experimental data for a narrow annulus with a large diameter (Richter et al., 1979; Jeong, 2008). The MAE and RMSE for predicted gas superficial velocity were 22.61% and 26.11%, respectively (Fig. 13). Applicable ranges are as follows:
Fig. 14. Comparison between measured and predicted jg for annulus channels with a large diameter – This study and Jeong’s model (Jeong, 2008), (Error band: (jg, predicted − jg, measured)/jg,measured).
-
channel with a large diameter (~500 mm), the width is larger than the other narrow channel. Based on the Jeong’s experimental study (Jeong, 2008), it was investigated that there was a number of unit-cells in a narrow annulus channel with a large diameter. In order to apply the third key idea suggested in Section 2.1, it is necessary to determine the number of the unit-cells in a narrow annulus with a large diameter. In order to find the number of unit-cells in the case of a narrow annulus with a large diameter, two ideas were suggested. The first idea
Gap thickness: 2, 3, 5, 25.4 mm Outer diameter: 445, 500 mm Height: 500 mm Superficial velocity of liquid: 0.003 ~ 1.2 m/s Superficial velocity of gas: 1.5 ~ 16 m/s Working fluid: water and air Pressure: atmospheric pressure The model proposed in this study was compared with one of the
Table 3 Accuracies of this study and Jeong’s model. Annulus with large D
This study
jg
MAE: 22.61%
Jeong (Jeong, 2008) RMSE: 26.11%
8
MAE: 24.62%
RMSE: 31.10%
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existing prediction models, Jeong’s model (Jeong, 2008) (Fig. 14). Jeong’s model was developed based on Wallis type correlation, Eq. (1), with the empirical constants that were obtained by the least-square method from experimental data. Although the accuracies are similar as can be seen in Table 3, the model proposed in this study can be utilized to various narrow geometries; rectangular, annulus, and annulus with a large diameter.
Acknowledgements
4. Conclusions
Al Ba’ba’s Hasan, B., Elgammal, T., Amano, R.S., 2016. Correlations of bubble diameter and frequency for air-water system based on orifice diameter and flow rate. J. Fluids Eng. 138 (11), 114501. Bankoff, S.G., Lee, S.C., 1986. A Critical Review of the Flooding Literature, Multiphase Science and Technology. Springer, Berlin, Heidelberg, pp. 95–180. Chalgeri, V.S., Jeong, J.H., 2019. Flow regime identification and classification based on void fraction and differential pressure of vertical two-phase flow in rectangular channel. Int. J. Heat and Mass Transfer 132, 802–816. Christoph, Muller W., 2006. Review of debris bed cooling in the TMI-2 accident. Nucl. Eng. Des. 236, 1965–1975. Coulson, C.A., Jeffrey, A., 1977. Waves: A Mathematical Approach to the Common Types of Wave Motion, second ed. Longman Inc., New York, U.S.A. Drosos, E.I.P., Paras, S.V., Karabelas, A.J., 2006. Counter-current gas-liquid flow in a vertical narrow channel – Liquid Film Characteristics and Flooding Phenomena. Int. J. Multiphase Flow 32, 51–81. Ghiaasiaan, S. Mostafa, 2008. Two-Phase Flow: Boiling and Condensation in Conventional and Miniature Systems. Cambridge University Press, New York. Jeong, J.H., 2008. Counter-current flow limitation velocity measured in annular narrow gaps formed between large diameter concentric pipes. Korean J. Chem. Eng. 25 (2), 209–216. Jeong, J.H., NO, H.C., 1994. Classification of flooding data according to type of tube-end geometry. Nucl. Eng. Des 148, 109–117. Jeong, J.H., NO, H.C., 1996. Experimental study of the effect of pipe length and pipe-end geometry on flooding. Int. J. Multiphase Flow 22 (3), 499–514. Juarsa, M., 2014. Experimental study on the effect of gap size to CCFL and CHF in a vertical of narrow rectangular channle during quenching process. Ann. Nucl. Energy 72, 391–400. Kim, H., Bak, J., Jeong, J., Park, J., Yun, B., 2019. Investigation of the CHF correlation for a narrow rectangular channel under a downward flow condition. Int. J. Heat Mass Transfer 130, 60–71. Lee, J.Y., NO, H.C., 1994. Hyperboilicity breaking and flooding. Nucl. Eng. Des. 146, 225–240. Li, X.C., Sun, Z.N., 2010. Experimental study of flooding in vertical narrow rectangular channels. AIP Conf. Proc. 1207, 471–475. Mishima, K., 1984. Boiling Burnout at Low Flow Rate and Low Pressure Conditions. Ph. D. Thesis. Research Reactor Inst., Kyoto Univ., Japan. No, H.C., Jeong, J.H., 1996. Flooding correlation based on the concept of hyperbolicity breaking in a vertical annualr flow. Nucl. Eng. Des. 166, 249–258. Okano, Y., Kohiriyama, T., Yoshida, Y., Murase, M., 2003. Modeling of debris cooling with annular gap in the lower RPV and verification based on ALPHA experiments. Nucl. Eng. Des. 223, 145–158. Osakabe, M., Futamata, H., 1996. Effect of inserted rod and cross flow on top flooding of pipe. Int. J. Multiphase Flow 22 (5), 883–891. Osakabe, M., Kawasaki, Y., 1989. Top flooding in thin rectangular and annular passages. Int. J. Multiphase Flow 15 (5), 747–754. Osakabe, M., Kubo, T., Bara, H., 1994. Top flooding in thin rectangular channel. JSME International Journal Series B 37 (3), 491–496. Regland, W.A., Minkowycz, W.J., France, D.M., 1989. Single- and double-wall flooding of two-phase flow in an annulus. Int. J. Heat Fluid Flow 10 (2), 103–109. Richter, H.J., Wallis, G.B., Speers, M.S., 1979. Effect of Scale on Two-phase Countercurrent Flow Flooding. NRC NUREG/CR-0312. Sadatomi, M., Sato, Y., Saruwatari, S., 1982. Two-phase low in vertical noncircular channels. Int. J. Multiphase Flow 8 (6), 641–655. Sudo, Y., Kaminaga, M., 1989. A CHF characteristic for downward flow in a narrow vertical rectangular channel heated from both sides. Int. J. Multiphase Flow 15 (5), 755–766. Tien, C.L., 1977. A simple analytical model for countercurruent flow limiting phenomena with condensation. Lett. Heat Mass Transfer 4, 231–237. Ueda, T., Suzuki, S., 1978. Behavior of liquid films and floodings in counter-current twophase flow. Part 2. Flow in annuli and rod bunddles. Int. J. Multiphase Flow 4, 157–170. Vlachos, N.A., Paras, S.V., Mouza, A.A., Karabelas, A.J., 2001. Visual observations of flooding in narrow rectangular channels. Int. J. Multiphase Flow 27, 1415–1430. Wallis, G.B., 1969. One-dimensional Two Phase Flow. McGraw-Hill, NewYork. Zhou, R., Yu, L., Xie, H., Qiu, L., Zhi, X., Zhang, X., 2018. Evaluation of the approach based on the concept of hyperbolicity breaking for prediction of flooding velocity of both room temperature and cryogenic fluids. Cryogenics 93, 41–47.
This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korean government (Ministry of Science and ICT) (No. NRF-2017M2B2A9072062). References
In this study, data classification and model development were studied for flooding in several geometries of a narrow channel; rectangular, annulus and annulus with a large diameter. Three key ideas for two-phase flow in a narrow channel were suggested in order to adapt existing classification study for flooding in a conventional tube. The first idea is the characteristic length is better represented by the width of a channel than its equivalent hydraulic diameter or gap size. The second idea is that the flow path for each phase is separated along a width direction. The third idea is the unit-cell concept that can unify the data for a rectangular channel and an annular channel. Based on the key ideas, 330 flooding data in a narrow channel was classified according to the liquid inlet geometries. The porous inlet, which minimizes the interaction between liquid and gas at the inlet, was classified into the exit flooding whereas protruded inlet maximizing the interaction was classified to the entrance flooding. In a case of squared plate inlet, it was classified according to the ratio of the width to the gap thickness. The present model for flooding in a narrow channel was developed based on the hyperbolicity breaking concept of the system of the quasilinear two-fluid model. Through the hyperbolicity breaking concept, the governing equation was derived with the second key idea. Utilizing typical solitary wave solution, Young-Laplace equation, and empirical void fraction relation for flooding condition, the governing equation was closed. The proposed model has 32.46% of RMSE and better accuracy than the existing model. Moreover, the proposed model can be applied to both a narrow rectangular channel and an annulus narrow channel, unlike existing prediction models. In addition, the developed model was applied to a narrow annulus channel with a large diameter. In order to determine the number of the unit-cells the criterion changing flooding type from Wallis to Kutateladze was utilized. Moreover, the relation between gas velocity and the length of the unit-cell was suggested based on the observation and experimental studies. It has similar accuracy with the existing correlation, however, the proposed model can cover various geometry types of narrow channels: narrow rectangular and annular channels including large outer diameter. CRediT authorship contribution statement Moon Won Song: Conceptualization, Methodology, Software, Validation, Data curation, Writing - original draft, Visualization. Hee Cheon No: Conceptualization, Writing - review & editing, Supervision, Project administration, Funding acquisition. Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
9
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Nuclear Engineering and Design journal homepage: www.elsevier.com/locate/nucengdes
Classification and modeling of flooding in vertical narrow rectangular and annular channels according to channel-end geometries ⁎
Moon Won Song, Hee Cheon No
Korea Advanced Institute of Science and Technology (KAIST), Department of Nuclear and Quantum Engineering, 291, Daehak-ro, Yuseong-gu, Daejeon 34141, Republic of Korea
A R T I C LE I N FO
A B S T R A C T
Keywords: Flooding Narrow rectangular and annular channels Data classification Hyperbolicity breaking Large diameter annuli
In this study, data classification and modeling of the flooding phenomenon were performed in a narrow channel with a gap thickness of several millimeters. To implement the conventional classification study to a narrow channel case, three key ideas for two-phase flow in a narrow channel were proposed; characteristic length, the flow path for each phase and a concept of unit-cell that can unify both narrow rectangular and annular channels including large outer diameter. Based on the ideas, 330 flooding data points for both narrow rectangular and annular channels including large outer diameter were classified into exit flooding and entrance flooding according to the geometries of the liquid inlet. In order to develop the prediction model for flooding in a narrow channel, the hyperbolicity breaking concept was introduced. The present flooding model was derived based on the hyperbolicity breaking of the two-fluid model and produced the Root Mean Square Errors (RMSE) of 30.02% in terms of the superficial gas velocity against the 330 flooding data points from narrow rectangular and annular channels with relatively small outer diameter. Moreover, the proposed model was applied to a case of narrowlarge diameter annuli with the introduction of the number of the unit-cells producing the RMSE of 26.11% against 91 flooding data points. The RMSEs of the existing models for narrow rectangular and narrow largediameter annuli were 38.25% and 31.10%, respectively. Different from the existing models, the proposed model turned out that it is applicable to various narrow channel-types; a rectangular, an annular and an annulus with a large diameter.
1. Introduction To estimate the heat removal rate of overheated systems by gravitydriven water penetration through a narrow channel, flooding in a narrow channel is one of the important issues. Typical examples are heat pipes, cooling systems of an electronic device, and nuclear systems in a severe accident. In the severe accident of a nuclear system, overheated molten-fuel in a reactor vessel is quenched by the gravity-driven water penetration, which flows through a narrow gap. The water penetration rate is restricted by the rising vapor flow; the water penetration rate governs the heat removal rate of the overheated-molten fuel in the lower plenum (Christoph, 2006). The issue of flooding in a narrow channel has recently been studied to estimate critical heat flux in a narrow channel (Juarsa, 2014; Kim et al., 2019). Hence, several experimental studies and modeling for flooding in a narrow channel were carried out. In the above experiment in narrow channels, the geometric type of a narrow channel can be divided into two categories. The first one is flooding in a rectangular narrow channel (Mishima, 1984; Sudo and
⁎
Kaminaga, 1989; Osakabe and Kawasaki, 1989; Osakabe et al., 1994; Vlachos et al., 2001; Drosos et al, 2006; Li and Sun, 2010) Their ranges of gap thickness and width are 1.5 ~ 10 mm and 40 ~ 150 mm, respectively. The second one is flooding in a narrow annular channel, which a rod was inserted into a tube (Ueda and Suzuki, 1978; Regland et al., 1989; Osakabe and Futamata, 1996). The ranges of the gap thickness and the outer diameter are 5 mm ~ 17 mm and 50 ~ 101.6 mm, respectively. Moreover, there is cases of large-diameter annuli, whose outer diameters are 440 mm (Richter et al., 1979) and 500 mm (Jeong, 2008). The hydraulic parameters of channels of the former researcher’s work are shown in Table 1. In order to predict the flooding gas velocity, following equation and non-dimensional parameters are widely used (Wallis, 1969; Tien, 1977).
(jg ∗ )1/2 + m (jf ∗ )1/2 = c ∗
jk = Ck jk k is f(liquid) or g(gas)
Corresponding author.
https://doi.org/10.1016/j.nucengdes.2020.110539 Received 30 August 2019; Received in revised form 17 January 2020; Accepted 27 January 2020 0029-5493/ © 2020 Elsevier B.V. All rights reserved.
Please cite this article as: Moon Won Song and Hee Cheon No, Nuclear Engineering and Design, https://doi.org/10.1016/j.nucengdes.2020.110539
(1) (2)
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Table 1 Hydraulic parameters of channels of former researcher’s work. Author
Gap thickness (mm)
Channel width (mm)
Inner diameter (mm)
Outer diameter (mm)
Geometry
Mishima, 1984 Sudo and Kaminaga, 1989 Osakabe and Kawasaki, 1989 Osakabe et al., 1994 Vlachos et al., 2001 Drosos et al, 2006 Li and Sun, 2010 Ueda and Suzuki, 1978 Regland et al., 1989 Osakabe and Futamata, 1996 Richter et al., 1979 Jeong, 2008
1.5, 5.3, 8.3, 12.3 2.3, 5.3, 8.3, 12.3 2, 5, 10 1, 2 5, 10 10 5 – – – – –
100 66 100 10, 15, 20, 40, 60, 80, 100 150 120 50 – – – – –
– – – – – – – 10 90.064 16, 30, 36, 40, 45 393.7 490, 494, 496
– – – – – – – 28, 35, 40 101.6 50 444.5 500
rectangular rectangular rectangular rectangular rectangular rectangular rectangular annulus annulus annulus large annulus large annulus
flooding, while the smooth geometries were classified to the exit flooding. The main characteristic of the results of the classification is that the exit flooding has higher gas flooding velocity than the entrance flooding at the same liquid superficial velocity. It means that the interaction between liquid and gas at the liquid inlet becomes maximum due to the sharp geometries. By contrast, the interaction is minimum on the condition of smooth geometry. Moreover, Jeong and NO (1996) found that the tube length effect appears on the only exit-flooding condition because a large wave that causes flooding is generated at the exit, and the wave moves along the tube length. However, in the case of the entrance flooding, since the wave causing the flooding appears at the entrance; the tube length has a weak effect. This study adopted the classification method proposed by Jeong and NO (1994) to a narrow rectangular and a narrow annular channel with key ideas for two-phase flow in a narrow channel. Based on the classification of this study, modeling was carried out using the concept of hyperbolicity-breaking derived from the two-fluid model of liquid/gas flow in a narrow channel. Prediction results were compared with 330 data for flooding in a rectangular channel and an annular channel. Finally, the classification method and the prediction model were applied to the case of an annulus with a large diameter (around 500 mm).
1/2
ρk ⎛ ⎞ : Wallis parameter CW = ⎜ gLch (ρf − ρg ) ⎟ ⎝ ⎠
(3)
1/4
ρk 2 ⎛ ⎞ : Kutateladze parameter CK = ⎜ gσ (ρf − ρg ) ⎟ ⎝ ⎠
(4)
where jk is superficial velocity of phase k. The constants m and c are empirical constants for flooding correlation. Cw and CK are the parameters that make the superficial velocity non-dimensional. ρk, g, Lch, σ are density of k phase, gravitational acceleration, characteristic length, and surface tension, respectively. The m and c values in the Eq. (1) are empirically determined because the database of flooding have large deviations from each other. Therefore, correlations for prediction of the flooding velocity in a narrow channel have been proposed based on the Eq (1). Regardless of a number of flooding correlations, there is no correlation satisfying the whole flooding experiments of flooding both in a narrow rectangular channel and an annulus. It is commonly believed that the occurrence of flooding strongly depends on tube-end conditions though experiments were carried out on similar geometric conditions (Bankoff and Lee, 1986; Jeong and NO, 1996). They classified the flooding data into two categories; entrance flooding and exit flooding (Fig. 1.). When the tube-end geometry is sharp, flooding occurs at the liquid entrance due to the hydraulic jump on the sharp region (Fig. 1-(a)). On the other hand, there is no hydraulic jump on the smooth tube-end condition. Wave at the liquid exit grows and moves up to the entrance region (Fig. 1-(b)). For this reason, Wallis proposed the c value in Eq. (1) as 0.725 for sharp-end and 1.0 for smooth-end, respectively. In order to classify mathematically reasonably the flooding data according to tube-end geometries, Jeong and NO (1994) utilized entropy minimax principle. As a result, they successfully classified 17 types of tube-end geometries into the two types of flooding; exit flooding and entrance flooding (Fig. 2). As a result, the sharp geometries were classified to the entrance
2. Classification of flooding data in a narrow channel 2.1. Key ideas for a two-phase narrow channel This study applied the classification method for flooding in a tube proposed by Jeong and NO to the cases of narrow channels. Moreover, we added three key ideas for two-phase flow in a narrow channel. The first idea refers to the characteristic length. In Sadatomi’s study (Sadatomi et al., 1982), they measured the slug velocity and the constant, Cslug, in the vertical non-circular channel (Eq. (5)). They found that the slug constant, Cslug, can be unified 0.35 for the various noncircular channels when the characteristic length, Lch, is the width dominant value instead of the conventional hydraulic equivalent
Fig. 1. Classification of flooding according to tube-end geometries (a) entrance flooding (b) exit flooding. 2
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Fig. 2. Classification of flooding in a tube according to channel-end geometries (Jeong and NO, 1994).
debris, Okano study (Okano et al., 2003) modeled that flow path of liquid/gas in a narrow channel is separated along the width direction as shown in Fig. 3-(b), unlike Fig. 3-(a). In our study, this configuration was applied to the part of the modeling of flooding in a narrow channel. The third idea is a concept of a unit-cell suggested by Osakabe study (Osakabe and Futamata, 1996). The unit-cell concept attempt to unify the narrow rectangular data and the narrow annular data (Fig. 4). Jeong (2008) observed that the liquid/gas flow path in an annular channel is separated likewise the case of a rectangular channel. Thus, dividing the annular channel along the circumferential direction, the divided one which is called unit-cell can be dealt with similar to the rectangular data. By utilizing the concept of the unit-cell, this study unified the rectangular data and the annular data. 2.2. Classification of flooding data in a narrow channel Using the concept of the unit-cell noted in Section 2.1, this study utilized the flooding data of both a vertical narrow rectangular channel (Mishima, 1984; Sudo and Kaminaga, 1989; Osakabe and Kawasaki, 1989; Vlachos et al., 2001; Drosos et al, 2006; Li and Sun, 2010) and a vertical narrow annular channel (Ueda and Suzuki, 1978; Regland et al., 1989; Osakabe and Futamata, 1996). The Osakabe data (Osakabe et al., 1994) was not included because the channel was not straight. There was area change along the tube length direction. This study classified the flooding data according to the liquid-entrance geometries likewise Jeong’s study (Jeong and NO, 1994). Based on the first key idea, this study utilized the width as the characteristic length for Wallis parameter instead of the hydraulic equivalent diameter. The classification results and the method are shown in Figs. 5 and 6, respectively. The results well agreed with the previous study; the gas flooding velocity in the case of exit flooding was higher than that of the entrance flooding. Likewise, the porous liquid inlet classified to the exit flooding because the interaction between liquid and gas becomes minimum, whereas the protruded plate inlet was classified to the entrance flooding due to maximum interaction at the liquid inlet. In the case of the squared plate geometry, some data were classified
Fig. 3. Separated flow path for liquid/gas along (a) gap direction (b) width direction.
diameter, De (Eq. (6) and Fig. 3.).
uslug = Cslug gLch 4Aflow Pw 2(w + d ) 2w = ≈ for w > > d (instead of De = π π π Pw 4wd = ≈2d ) 2(w + d )
(5)
Lch =
(6)
where uslug, Cslug, Pw, Aflow, w and d are slug velocity, slug constant, wetted perimeter, flow area, width of channel, and gap thickness of channel, respectively. The second idea is that the flow path of liquid–gas in a narrow channel is separated along the width direction instead of the gap direction (Fig. 4) based on the observations of Osakabe’s study (Osakabe et al., 1994). Moreover, one of the studies for cooling of overheated 3
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Fig. 4. Unit-cell concept.
to the exit flooding, the others were classified to the entrance flooding. This study proposed the criterion that can determine the flooding type for the case of a narrow squared plate. Using the ratio of the width (w) to the gap thickness (d), the data of the ratios is larger than 10 was classified to the exit flooding. In the case of the ratios is smaller than 10, those data were classified to the entrance flooding. It can be explained that the smaller ratio of the width to the gap thickness tends to generate the large wave that causes the flooding at the liquid inlet. Taking the example of the case of form-loss pressure drop of a single-phase flow, the case of the protruded-plate inlet has higher formloss pressure drop than the case of the squared-plate inlet. Similarly, this study showed that the protruded-plate inlet has a larger interaction between liquid and gas than the squared-plate inlet. As well as, the porous-media inlet has been utilized to make the immediate formation of a stable liquid film (Ghiaasiaan, 2008). Thus, it is reasonable that the case of porous-media inlet was classified to the exit flooding, which makes the interaction minimum. 3. Modeling and validation for flooding in a narrow channel Fig. 5. Classification of flooding data in both a rectangular and an annular channel.
3.1. Modeling based on the hyperbolicity breaking concept In order to predict the flooding velocity, the Eqs. (1)-(4) have been widely used with empirical constants. However, there is a limitation that the empirical constants are highly affected by channel geometries. On a fixed geometry, analytic models or computational-fluid-dynamics (CFD) studies for the flooding phenomenon have been proposed based on the momentum balance equations and wave motions. This study developed a prediction model for flooding in a vertical narrow channel using the concept of hyperboilicity-breaking and singular points. Lee and NO studied the hyperbolicity-breaking concept near singular points in two-phase flow (Lee and NO, 1994). NO and Jeong developed the correlation for flooding in a vertical tube based on the hyperbolicity-breaking concept (NO and Jeong, 1996). Recently through using the hyperbolicity-breaking and singular point concept, the prediction model for flooding in a nearly horizontal tube was developed (Zhou et al., 2018). This section describes the hyperboilicity-breaking near singular points derived from the two-fluid model for a narrow channel. As explained in section 2.1, the second key idea for a two-phase narrow channel was applied; each phase is separated along the width direction (Fig. 7). Though using this physical configuration, one-dimensional two-fluid models for mass and momentum can be derived as seen in Eqs. (7–11). The wall and interfacial shear terms follow Fig. 7 as described in Eqs. (9) and (10). Based on the Eqs. (7)–(11), the first-order, quasi-linear, partial differential equations system can be derived as Eqs. (12)–(16). Vapor mass:
Fig. 6. Classification of flooding data in a narrow channel according to the liquid-entrance geometries.
∂ ∂ (αg ρg ) + (αg ρg Vg ) = Γ ∂t ∂z Liquid mass: 4
(7)
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Γ ⎛ ⎞ −Γ ⎜ ⎟ 2αg 2 C = ⎜ Γ(Vgi − Vg ) − τ − w τi, g − αg ρg g ⎟ d w, g ⎜⎜ ⎟ 2αf 2 − Γ(Vfi − Vf ) + d τw, f + w τi, f − αf ρf g ⎟ ⎝ ⎠
(16)
If we introduce wave coordinate and characteristic velocity, ξand λ, Eq. (12) becomes
(Aλ + B )
∂X = C where ξ = z + λt ∂ξ
(17)
Applying Cramer’s rule, we have
∂αg ∂ξ
=
Ni Δ
(18)
where
Δ = |Aλ + B| = αf ρg (λ + Vg )2 + αg ρf (λ + Vf )2 + αg αf Fig. 7. Descriptions for counter-current flow in a narrow channel.
∂ΔP ∂αg
(19)
Nαg = Γ(αf (Vg + λ ) + αg (Vf + λ )) − Γ(αf (Vgi − Vg ) + αg (Vfi − Vf ))
∂ ∂ (αf ρf ) + (αf ρf Vf ) = −Γ ∂t ∂z
+
(8)
2αf αg d
(τw, f + τw, g ) +
2 (αf τi, g w
+ αg τi, f ) − αf αg Δρg (20)
Vapor momentum:
αg ρg
∂Vg ∂t
∂Vg
+ αg ρg Vg
∂z
+ αg
∂Pg ∂z
= Γ(Vgi − Vg ) −
2αg d
τw, g −
If Δ is zero, the points in the phase space become singular points. The solutions of the characteristic equation, Δ = 0, are
2 τi, g − αg ρg g w (9)
∂Vf ∂t
∂Vf
+ αf ρf Vf
+ αf
∂Pf
p=
∂z ∂z 2αf 2 τw, f + τi, f − αf ρf g = −Γ(Vfi − Vf ) + d w
αf ρg + αg ρf
q=
∂ΔP ∂αg Pf = Pg + ΔP; = + ∂z ∂z ∂αg ∂z
∂Pg
∂X ∂X A +B =C ∂t ∂z
(12)
X = (Pg , αg , Vg , Vf )T
(13)
∂ρ
αf
αf
∂ΔP ∂αg
0
αf ρg + αg ρf
(24)
δ (ξ ) = hsech2 (κξ ) + δ0
(25)
1 1⎞ ΔP = −σ ⎛ + ≈ σ (2hκ 2) R R 1 2⎠ ⎝
(26)
⎜
(14)
⎞ ⎟ αf ρf ⎟ ⎟ 0 ⎟ αf ρf Vf ⎟⎟ ⎠
(23)
In Eq. (24) two parameters are necessary to predict the flooding velocity; derivative of pressure difference with respect to void fraction and void fraction. The pressure difference term was obtained using the Young-Laplace equation. It was assumed that the curvature of the peak of the wave just before the occurrence of flooding is represented by the Kortewegde Vries type of solitary wave (Coulson and Jeffrey, 1977) (Fig. 8).
∂ρ
⎛ αg Vg g ρg Vg αg ρg ∂Pg ⎜ ∂ρf ⎜ αf Vf − ρf Vf 0 ∂Pg B=⎜ αg ρg Vg 0 ⎜ αg
∂ΔP ∂αg
jf ⎞2 ⎛ ∂ΔP ⎞ ⎛ αg 1 − αg ⎞ ⎛ jg + + ⎜ ⎟ = ⎜− ⎟ ⎜ 1 − αg ⎠ ρf ⎟ ⎝ αg ⎝ ∂αg ⎠ ⎝ ρg ⎠
where
⎛ αg g ρg 0 0 ⎞ ∂Pg ⎟ ⎜ ∂ρf ⎜ 0 0 ⎟ A = αf ∂Pg − ρf ⎟ ⎜ 0 αg ρg 0 ⎟ ⎜ 0 ⎜ 0 0 0 αf ρf ⎟ ⎠ ⎝
(22)
If the λ becomes imaginary, hyperbolicity-breaking takes place: the conditions for changing from the hyperbolic domain with two real solutions to the elliptic one with two complex ones represent that the twophase flow changes from stable flow to unstable flow. Thus, the occurrence of flooding can be derived from the neutral stability condition; the square root term in Eq. (21) set to zero.
(11)
and t, z, αk, Γ, Vk, Pk, Vki, τi,k, and τw,k are time, coordinate parallel to fluid motion, volume fraction for phase k, phase change rate, flowarea-averaged velocity for each phase k, pressure for phase k, velocity at the interface for phase k, interfacial drag for phase k, and wall drag for phase k, respectively. Partial differential equations system based on two-fluid model:
⎜⎜ ⎝
αf ρg Vg + αg ρf Vf
αf ρg Vg 2 + αg ρf Vf 2 + αg αf
(10)
where
∂Pf
(21)
where
Liquid momentum:
αf ρf
p2 − q
λ=p±
0
⎟
where R1 = w /2 − (h + δ0 ), R2 =
⎛ d 2δ (ξ ) ⎜ dξ 2 ⎝
⎞ 1 = ⎟ 2 hκ 2 − ξ =0 ⎠
(27)
And δ, h, κ, δ0 are film thickness, wave amplitude, wave number, and substrate thickness, respectively. Moreover, the void fraction relation was applied to calculate the term of derivative of pressure difference.
(15) 5
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jf ⎞ 1 − αg ⎞ ⎛ αg ⎛ jg + ⎜ ⎟ = (Δρgw ) ⎜ρ + α 1 α ρf ⎟ − g⎠ ⎝ g ⎠ ⎝ g
(31)
In order to get the solution using Eq. (31), we need the relation of the liquid fraction in terms of superficial velocities. The NO study (NO and Jeong, 1996), which developed the flooding correlation using the hyperbolicity breaking, also utilized the semi-empirical correlation of the liquid fraction. Theoretical basis of their correlation for the liquid fraction is the turbulent falling-film theory (Wallis, 1969) which shows relation between the liquid fraction(αf) and a non-dimensional liquid superficial velocity (jf*) as seen in Eq. (32);
αf = cm jf ∗ce
They utilized the turbulent falling-film theory for the exponent parameter, ce. Also they empirically determined multiplier parameter, cm, because they assumed that the liquid fraction at the peak of a solitary wave is proportional to the spatially averaged liquid film thickness. We adopted similar approach to the NO study to obtain the void fraction relation at the peak of a solitary wave. However, the channel geometry of interest of this study differs from the NO study. Thus, we empirically determined the exponent and the multiplier according to the classification introduced in this study as can be seen in Eq. (33) and (34);
Fig. 8. Solitary wave in a unit-narrow channel.
2h ⎞ αg = ⎛1 − w⎠ ⎝
(28)
Consequently, the derivative term becomes
∂ΔP = −σκ 2w ∂αg
(29)
Assuming that the wavelength, λ, is proportional to the marginalfastest growing wave of Helmholtz instability, we can obtain the wave number as follows:
κ=
2π = λ
σ 1/2 Δρg
( )
αf = 0.91j∗
0.124 for f
exit flooding
(33)
αf = 1.06j∗
0.137 for f
entrance flooding
(34)
where j*f is the non-dimensional liquid superficial velocity of Wallis type with the characteristic length of w. We checked that the exponent value for a narrow channel was around 0.15 by Jeong experiment (Charlgeri and Jeong, 2019) which measured the void fraction in a vertical downward rectangular narrow channel. Fig. 9 summarized the key idea, classification, and modeling
2π 2π
(32)
(30)
Therefore, the governing equation becomes
Fig. 9. Summary of the key ideas, data classification, and modeling. 6
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Table 2 Accuracies of this study and the other model. Narrow rectangular data
jg non-dimensional jg *
Osakabe (Eq. (35))
Sudo (Eq. (36))
MAE
RMSE
MAE
RMSE
MAE
RMSE
20.61% 9.73%
32.46% 14.14%
27.83% 16.39%
35.92% 23.27%
47.75% 22.09%
74.65% 32.13%
Fig. 12. Description of perimeter equivalent concept to find the width of the unit-cell.
Fig. 10. Comparison between measured and predicted jg for both rectangular and annular channels (Error band: (jg, predicted − jg, measured)/jg,measured).
channel. Although there are several studies for the narrow-channel flooding correlation (Mishima, 1984; Sudo and Kaminaga, 1989; Osakabe et al., 1994), the Osakabe correlation (Eq. (35) and Fig. 11-(a)) and the Sudo correlation (Eq. (36) and Fig. 11-(b)) were compared. The present model predicted the data with better accuracy than the other one did as shown in Table 2. Even though the other correlation can be only applied to a rectangular channel data, the present model can cover both rectangular and annular channels with better accuracy.
suggested in this study. 3.2. Validation and Comparison with other correlation The database utilized in the classification study was compared with the calculation results using the model proposed in this study. The mean-averaged error (MAE) and the root-mean-square error (RMSE) were 20.74% and 30.02%, respectively (Fig. 10.). The applicable ranges are as follows: -
Present model
Geometry: vertical rectangular, vertical annular Gap thickness: 1.5 ~ 15 mm Width: 40 ~ 300 mm Height: 470 ~ 1235 mm Superficial velocity of liquid: 0.001 ~ 0.3 m/s Superficial velocity of gas: 0.4 ~ 12 m/s Working fluid: water and air Pressure: atmospheric pressure
Kug 1/2 + 0.8Kuf 1/2 = 0.58Bo1/8
(35)
jg ∗1/2 + (0.5 + 0.001Bo∗) jf ∗1/2 = 0.66(d/w )−0.25
(36)
where Kuk, Bo, jk*, Bo*, d, and w are non-dimensional superficial velocity of the Kutateladze type and Bond number, non-dimensional superficial velocity of the Wallis type, modified Bond number, gap thickness and channel width, respectively. 3.3. Narrow annular channel with large diameter (~500 mm) As explained in Section 2.1, the characteristic length for two-phase flow in a narrow channel becomes a width of the unit-cell instead of the hydraulic equivalent diameter. However, in a case of narrow annulus
Moreover, this study compared the proposed model with other correlations for prediction of flooding velocity in a narrow rectangular
Fig. 11. Other correlations and proposed model against database for flooding in a rectangular narrow channel (Error band: (jg, predicted − jg, measured)/jg,measured) ((a): Osakabe correlation, (b): Sudo correlation). 7
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is utilizing the criterion that the flooding type changes from Wallis type to Kutateladze type. In the flooding of the Wallis type, flooding occurs on the slug flow regime in a tube with a small diameter. However, in the Kutateladze type flooding, flooding occurs on the annular flow regime in a large diameter tube. In this study, it was suggested that the total perimeter of the one unit-cell is equivalent to the total perimeter of the changing criterion for the flooding type in a conventional tube (Fig. 12, Eq. (37) and (38)).
πDWallistoKutateladze = 2(wmax + d )
(37)
πDWallistoKutateladze −d 2
(38)
→ wmax =
where DWallistoKutateladze and wmax are the diameter that changes the flooding type from Wallis to Kutateladze and maximum width of the unit-cell. The specific criterion is as follows (Ghiaasiaan, 2008);
2<
D < 20 : Wallis type flooding λL
(39)
D > 40 : Kutateladze type flooding λL Fig. 13. Comparison between measured and predicted jg for annulus channels with a large diameter (Error band: (jg, predicted − jg, measured)/jg,measured).
σ ⎞ → DWallistoKutateladze = 20λL ~40λL , where λL = ⎜⎛ ⎟ ⎝ Δρg ⎠
(40) 0.5
(41)
The second idea is that the number of unit-cells changes according to the gas velocity. Based on Osakabe’s observations (Osakabe et al., 1994), the gas plug in a narrow channel was affected by the gas velocity. In this study, the relation between the width of unit-cell and gas velocity was suggested as Eqs. (42) and (43).
⎛ jg ⎞ w = wmax ⎜ ⎟ j ⎝ g, crit ⎠
0.15
(42) 1/4
ρg 2 ⎞ where jg, crit = Ku∗ g, crit ⎛⎜ ⎟ ⎝ Δρgσ ⎠
, Ku∗ g, crit = 1. 62 (43)
The wmax can be calculated using Eqs. (38) and (41). The constant in Eq. (41) was 20. The jg, crit was obtained from critical value of Kutateladze number, 1.6 (Eq. (43)). The exponent in the Eq (42), 0.15, was determined from the relation between the bubble size and flow rate of air in a case of bubble generation by an air injection (Al Ba’ba’s et al., 2016). The flooding model proposed in this study, Eq. (31–34), and the ideas explained in this section were validated against the experimental data for a narrow annulus with a large diameter (Richter et al., 1979; Jeong, 2008). The MAE and RMSE for predicted gas superficial velocity were 22.61% and 26.11%, respectively (Fig. 13). Applicable ranges are as follows:
Fig. 14. Comparison between measured and predicted jg for annulus channels with a large diameter – This study and Jeong’s model (Jeong, 2008), (Error band: (jg, predicted − jg, measured)/jg,measured).
-
channel with a large diameter (~500 mm), the width is larger than the other narrow channel. Based on the Jeong’s experimental study (Jeong, 2008), it was investigated that there was a number of unit-cells in a narrow annulus channel with a large diameter. In order to apply the third key idea suggested in Section 2.1, it is necessary to determine the number of the unit-cells in a narrow annulus with a large diameter. In order to find the number of unit-cells in the case of a narrow annulus with a large diameter, two ideas were suggested. The first idea
Gap thickness: 2, 3, 5, 25.4 mm Outer diameter: 445, 500 mm Height: 500 mm Superficial velocity of liquid: 0.003 ~ 1.2 m/s Superficial velocity of gas: 1.5 ~ 16 m/s Working fluid: water and air Pressure: atmospheric pressure The model proposed in this study was compared with one of the
Table 3 Accuracies of this study and Jeong’s model. Annulus with large D
This study
jg
MAE: 22.61%
Jeong (Jeong, 2008) RMSE: 26.11%
8
MAE: 24.62%
RMSE: 31.10%
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existing prediction models, Jeong’s model (Jeong, 2008) (Fig. 14). Jeong’s model was developed based on Wallis type correlation, Eq. (1), with the empirical constants that were obtained by the least-square method from experimental data. Although the accuracies are similar as can be seen in Table 3, the model proposed in this study can be utilized to various narrow geometries; rectangular, annulus, and annulus with a large diameter.
Acknowledgements
4. Conclusions
Al Ba’ba’s Hasan, B., Elgammal, T., Amano, R.S., 2016. Correlations of bubble diameter and frequency for air-water system based on orifice diameter and flow rate. J. Fluids Eng. 138 (11), 114501. Bankoff, S.G., Lee, S.C., 1986. A Critical Review of the Flooding Literature, Multiphase Science and Technology. Springer, Berlin, Heidelberg, pp. 95–180. Chalgeri, V.S., Jeong, J.H., 2019. Flow regime identification and classification based on void fraction and differential pressure of vertical two-phase flow in rectangular channel. Int. J. Heat and Mass Transfer 132, 802–816. Christoph, Muller W., 2006. Review of debris bed cooling in the TMI-2 accident. Nucl. Eng. Des. 236, 1965–1975. Coulson, C.A., Jeffrey, A., 1977. Waves: A Mathematical Approach to the Common Types of Wave Motion, second ed. Longman Inc., New York, U.S.A. Drosos, E.I.P., Paras, S.V., Karabelas, A.J., 2006. Counter-current gas-liquid flow in a vertical narrow channel – Liquid Film Characteristics and Flooding Phenomena. Int. J. Multiphase Flow 32, 51–81. Ghiaasiaan, S. Mostafa, 2008. Two-Phase Flow: Boiling and Condensation in Conventional and Miniature Systems. Cambridge University Press, New York. Jeong, J.H., 2008. Counter-current flow limitation velocity measured in annular narrow gaps formed between large diameter concentric pipes. Korean J. Chem. Eng. 25 (2), 209–216. Jeong, J.H., NO, H.C., 1994. Classification of flooding data according to type of tube-end geometry. Nucl. Eng. Des 148, 109–117. Jeong, J.H., NO, H.C., 1996. Experimental study of the effect of pipe length and pipe-end geometry on flooding. Int. J. Multiphase Flow 22 (3), 499–514. Juarsa, M., 2014. Experimental study on the effect of gap size to CCFL and CHF in a vertical of narrow rectangular channle during quenching process. Ann. Nucl. Energy 72, 391–400. Kim, H., Bak, J., Jeong, J., Park, J., Yun, B., 2019. Investigation of the CHF correlation for a narrow rectangular channel under a downward flow condition. Int. J. Heat Mass Transfer 130, 60–71. Lee, J.Y., NO, H.C., 1994. Hyperboilicity breaking and flooding. Nucl. Eng. Des. 146, 225–240. Li, X.C., Sun, Z.N., 2010. Experimental study of flooding in vertical narrow rectangular channels. AIP Conf. Proc. 1207, 471–475. Mishima, K., 1984. Boiling Burnout at Low Flow Rate and Low Pressure Conditions. Ph. D. Thesis. Research Reactor Inst., Kyoto Univ., Japan. No, H.C., Jeong, J.H., 1996. Flooding correlation based on the concept of hyperbolicity breaking in a vertical annualr flow. Nucl. Eng. Des. 166, 249–258. Okano, Y., Kohiriyama, T., Yoshida, Y., Murase, M., 2003. Modeling of debris cooling with annular gap in the lower RPV and verification based on ALPHA experiments. Nucl. Eng. Des. 223, 145–158. Osakabe, M., Futamata, H., 1996. Effect of inserted rod and cross flow on top flooding of pipe. Int. J. Multiphase Flow 22 (5), 883–891. Osakabe, M., Kawasaki, Y., 1989. Top flooding in thin rectangular and annular passages. Int. J. Multiphase Flow 15 (5), 747–754. Osakabe, M., Kubo, T., Bara, H., 1994. Top flooding in thin rectangular channel. JSME International Journal Series B 37 (3), 491–496. Regland, W.A., Minkowycz, W.J., France, D.M., 1989. Single- and double-wall flooding of two-phase flow in an annulus. Int. J. Heat Fluid Flow 10 (2), 103–109. Richter, H.J., Wallis, G.B., Speers, M.S., 1979. Effect of Scale on Two-phase Countercurrent Flow Flooding. NRC NUREG/CR-0312. Sadatomi, M., Sato, Y., Saruwatari, S., 1982. Two-phase low in vertical noncircular channels. Int. J. Multiphase Flow 8 (6), 641–655. Sudo, Y., Kaminaga, M., 1989. A CHF characteristic for downward flow in a narrow vertical rectangular channel heated from both sides. Int. J. Multiphase Flow 15 (5), 755–766. Tien, C.L., 1977. A simple analytical model for countercurruent flow limiting phenomena with condensation. Lett. Heat Mass Transfer 4, 231–237. Ueda, T., Suzuki, S., 1978. Behavior of liquid films and floodings in counter-current twophase flow. Part 2. Flow in annuli and rod bunddles. Int. J. Multiphase Flow 4, 157–170. Vlachos, N.A., Paras, S.V., Mouza, A.A., Karabelas, A.J., 2001. Visual observations of flooding in narrow rectangular channels. Int. J. Multiphase Flow 27, 1415–1430. Wallis, G.B., 1969. One-dimensional Two Phase Flow. McGraw-Hill, NewYork. Zhou, R., Yu, L., Xie, H., Qiu, L., Zhi, X., Zhang, X., 2018. Evaluation of the approach based on the concept of hyperbolicity breaking for prediction of flooding velocity of both room temperature and cryogenic fluids. Cryogenics 93, 41–47.
This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korean government (Ministry of Science and ICT) (No. NRF-2017M2B2A9072062). References
In this study, data classification and model development were studied for flooding in several geometries of a narrow channel; rectangular, annulus and annulus with a large diameter. Three key ideas for two-phase flow in a narrow channel were suggested in order to adapt existing classification study for flooding in a conventional tube. The first idea is the characteristic length is better represented by the width of a channel than its equivalent hydraulic diameter or gap size. The second idea is that the flow path for each phase is separated along a width direction. The third idea is the unit-cell concept that can unify the data for a rectangular channel and an annular channel. Based on the key ideas, 330 flooding data in a narrow channel was classified according to the liquid inlet geometries. The porous inlet, which minimizes the interaction between liquid and gas at the inlet, was classified into the exit flooding whereas protruded inlet maximizing the interaction was classified to the entrance flooding. In a case of squared plate inlet, it was classified according to the ratio of the width to the gap thickness. The present model for flooding in a narrow channel was developed based on the hyperbolicity breaking concept of the system of the quasilinear two-fluid model. Through the hyperbolicity breaking concept, the governing equation was derived with the second key idea. Utilizing typical solitary wave solution, Young-Laplace equation, and empirical void fraction relation for flooding condition, the governing equation was closed. The proposed model has 32.46% of RMSE and better accuracy than the existing model. Moreover, the proposed model can be applied to both a narrow rectangular channel and an annulus narrow channel, unlike existing prediction models. In addition, the developed model was applied to a narrow annulus channel with a large diameter. In order to determine the number of the unit-cells the criterion changing flooding type from Wallis to Kutateladze was utilized. Moreover, the relation between gas velocity and the length of the unit-cell was suggested based on the observation and experimental studies. It has similar accuracy with the existing correlation, however, the proposed model can cover various geometry types of narrow channels: narrow rectangular and annular channels including large outer diameter. CRediT authorship contribution statement Moon Won Song: Conceptualization, Methodology, Software, Validation, Data curation, Writing - original draft, Visualization. Hee Cheon No: Conceptualization, Writing - review & editing, Supervision, Project administration, Funding acquisition. Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
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