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Ind. Eng. Chem. Res. 2003, 42, 5312-5317
GENERAL RESEARCH Theoretical and Experimental Investigation of the Flooding Velocity of an Immiscible Vapor Mixture Condensing in Vertical Tubes Jian-Guo Wang,†,‡ Xue-Hu Ma,*,† Jia-Bin Chen,† Song-Ping Li,† and Ji-Fang Lin† Institute of Chemical Engineering, Dalian University of Technology, Dalian 116012, China, and China Huanqiu Chemical Engineering Corporation, Beijing 10029, China
A mathematical model accounting for the pressure oscillation when flooding occurs is presented to predict the velocity limit of an immiscible vapor mixture condensing in a vertical tube. This model incorporated the parameters of vapor inlet velocity, pressure drop, interfacial friction stress, tube inside diameter, and the fluid physical properties. The in-tube condensation experiments of an oil-steam vapor mixture with various mass ratios have been conducted using an 18 mm i.d. brass tube and a 20 mm i.d. glass tube, respectively. The model predicted that data agreed very well with the experimental data, indicating that this model is valid to predict the velocity limit of the immiscible vapor mixture condensing in a vertical tube. Introduction
Experiments
The countercurrent flow phenomenon of vapor and liquid films exists in a vertical tube condenser, a cooling tower, a falling-film chemical reactor, the urgent accident treatment of a nuclear reactor, and so on. Flooding will take place when the vapor velocity is so high that the condensate hardly flows downward from the tube wall. At this occasion, the well-balanced operation will be broken down, giving rise to the unstable two-phase flow in the tube and hence deteriorating the heat and mass transfer as well as the normal operation of a specific system. The previous investigations generally emphasize the empirical correlations1,2 or semiempirical correlations3-5 for the single-component vapor system. Palen and Yang6 had reviewed the prediction models for the reflux condensation flooding. The mechanisms of flooding phenomena had been thoroughly reviewed by Hewitt7 and Whally.8 The immiscible vapor mixture is widely used in petrochemical processes and waste heat recovery systems. For example, a steam-stripping vapor mixture, such as the waste heat carrier, condensed in the tube side of the evaporator and generator to drive the absorption heat transformer.9 Consequently, it is of significance to investigate the flooding velocity of the immiscible vapor mixture in a vertical tube, to specify the system design and operation. The objective of the present paper is to investigate theoretically the factors dominating the flooding velocity of an immiscible vapor mixture condensing in a vertical tube and confirm its validation by comparing the model results with experimental data.
The experimental setup is schematically shown in Figure 1. The oil, a mixture of pentane, hexane, and heptane with a mass ratio of 20.3:79:0.7, is pumped via a proportional pump into a mixing vessel and heated by steam from an electrically heated boiler. The vapor mixture flowing upward condenses in a vertical tube with the countercurrent flow of cooling water in the annulus. A brass tube and a glass tube were used in this experiment. The condensate is collected and measured in a measuring pipe. The excessive vapor is condensed in a secondary condenser. Copper-constantan thermocouples are used to measure the temperatures of the vapor, condensate, and cooling water inlet and exit. All thermocouples were calibrated in a constanttemperature bath with an accuracy of (0.1 °C. The temperature signals were collected by a HP3852A data acquisition/control unit and then transmitted to a personal computer for data reduction. The pressure drop between the inlet and exit of the vertical tube was measured using a U-type manometer of mercury. The condensation states at the exit of the test tube can be observed through a section of the glass tube. When obvious flooding phenomena occurred for a specific ratio of oil and water in the vapor mixture, the temperatures, the pressure drop, and the condensate mass flux are recorded, and thus the flooding vapor velocity is obtained.
* To whom correspondence should be addressed. Tel.: +86 411 3676164. Fax: +86 411 3633080. E-mail: xuehuma@ dlut.edu.cn. † Dalian University of Technology. ‡ China Huanqiu Chemical Engineering Corp.
Mathematical Model For the case of annular flow of liquid along a vertical tube, the friction stress at the interface of the liquid film and vapor phase is of little significance when the liquidvapor interface is smooth. However, it will, in fact, increase dramatically when the interface becomes wavy. On the other hand, a larger interfacial friction stress can promote the liquid film to be much more wavy than ever. When the crest of the wavy liquid film increases
10.1021/ie0208016 CCC: $25.00 © 2003 American Chemical Society Published on Web 09/09/2003
Ind. Eng. Chem. Res., Vol. 42, No. 21, 2003 5313
Figure 1. Flow diagram of the flooding experiment: 1, oil tank; 2, measuring pump; 3, steam generator; 4, mixer; 5, flowmeter; 6, test section; 7, measuring pipe; 8, secondary condenser; 9, collecting tank; 10, measuring pipe.
fi dp 4 - Fgg ) Fgug2 dz 2 DxR
(3)
where ug refers to the actual vapor velocity. Because it is difficult to measure ug experimentally, the apparent vapor velocity jg is used to replace the actual vapor velocity.
jg ) ujR
(4)
As a result, eq 3 can be rearranged as
1 4 dp - Fgg ) fiFg jg2 5/2 dz 2 DR
For simplification of the calculation, the following dimensionless parameters are introduced:
Figure 2. Schematic diagram of vapor flow.
high enough to produce a liquid bridge within the tube, the vapor channel may be blocked and, consequently, the flooding phenomenon occurs. On the Vapor Phase. Considering an increment dz near the tube entrance, as shown in Figure 2, the force balance over dz can be expressed as eq 1.10
dp π(D - 2δ)2/4 - Fggπ(D - 2δ)2/4 dz τiπ(D - 2δ) dz ) 0 (1) Because of the wavy liquid film, we can regard the vapor-liquid film interface as a rough wall. The interfacial friction stress can be obtained:
1 τi ) fiFgug2 2
(2)
Substituting eq 2 into eq 1 and introducing a void fraction:
R)
(D -D 2δ)
2
then the following equation can be obtained:
(5)
∆p* )
jg* )
dp/dz - Fgg g(Ff - Fg) jgFg1/2
[gD(Ff - Fg)]1/2
(6)
(7)
When eqs 5-7 are combined, the following can be obtained:
jg*2 ∆P* ) 2fi 5/2 R
(8)
On the Liquid Film. The physical and mathematical model of the liquid film is presented based on the following assumptions: (1) Compared to the tube radius, the film thickness is negligible, so it is assumed that the liquid film flows downward one-dimensionally along the vertical plane without considering the influence of tube curvature. (2) Because the liquid film thickness and the vapor velocity at the tube entrance are so significant, flooding takes place first at the entrance of tube. Therefore, the mathematical model here is mainly focused on the film thickness at the tube bottom section. At the same time, it is appropriate not to consider the vapor condensation
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Flooding Criterion of Vapor Condensing in a Vertical Tube Wall. By introduction of dimensionless film thickness ∆* ) δ/D, eq 8 is altered as
∆P* ) 2jg*2fi(1 - 2∆*)-5
(14)
When just the right flooding occurs, the flow pattern of the liquid film will be in the unstable state; as a result, the assumption of the ideal smooth film will not be validated, and hence the actual liquid film thickness cannot be identified any more. According to the assumption (3), the imaginary film thickness at which just the right flooding takes place can be withdrawn from eq 13: Figure 3. Schematic diagram of the liquid film.
and the momentum transfer between the vapor and liquid films because heat transfer is considerably deteriorated when the flooding phenomenon occurs. (3) The effect of the liquid film wave is combined with the vapor-liquid interfacial friction factor, and hence an imaginary film thickness is used to replace the actual film thickness. So, a smooth liquid film assumption is used in the mathematical analysis. On the basis of the above assumptions, the N-S equation along coordinate z, Figure 3, can be written:
Ff
(
)
∂uz ∂uz ∂uz ∂uz + ux + uy + uz ) ∂t ∂x ∂y ∂z -
(
∂2uz
∂2uz
∂x
2
+ Ffg )
)
∂p ∂z
(10)
The boundary conditions are
x ) 0, uz ) 0 x ) δ, -µ
)
∂p ∂p Ffg Fgug2 Ffgδ - δ ∂z ∂z 2 uz ) -fi x+ x (11) 2µ µ 2µ Consequently, the average velocity within the liquid film is
∂p 2 Ffg δ Fgug2δ ∂z u dx ) f + (12) i 0 z 4µ 3µ
∫
(
δ
)
When U E 0, it is identified that a flooding phenomenon definitely takes place, i.e.
∂p 2 Ffg δ Fgug2δ ∂z g fi 4µ 3µ
(
(15)
3 ∆* ) fi jg*2(1 - 2∆*)-4(1 - ∆P*)-1 4
(16)
)
The dimensionless film thickness becomes
When eq 14 is combined with eq 16, the dimensionless pressure drop is obtained:
)
8∆* 3 + 2∆*
(13)
(17)
From the experimental observations, it can be found that the pressure drop between the inlet and exit of the tested tube will increase to a certain value when the flooding phenomenon occurs. So, the dimensionless pressure drop can be defined as an indicator to identify when the flooding takes place. The dimensionless pressure drop varies with the experimental conditions. Accordingly, the following equation can be used to forecast the vapor flooding velocity:
8∆* 3 + 2∆*
∆P* ) 2jg*2fi(1 - 2∆*)-5
When eq 10 is integrated with the boundary conditions mentioned above, the following can be obtained:
1 U) δ
)
∆P* )
duz Fgug2 ) fi dx 2
(
(
(
∂2uz
From the assumption (1) and regarding the stable state of the system, eq 9 can be simplified as
∂2uz
3Dfi 3fiFg jg2 D - 2δ -4 ) jg*2 ∂p 2 4(1 - ∆P*) D 4 Ff g R ∂z
∆P* )
∂p + 2 + 2 + Ffg (9) +µ ∂z ∂x2 ∂y ∂z
µ
δ)
(18) (19)
It also can be found that ∆P* increases sharply with an increase in the dimensionless film thickness ∆*.7,8 The increase of the wave amplitude resulting from the increase of the liquid film thickness gives rise to an increase of the interfacial friction stress, i.e., an increase of the pressure drop. When ∆* approaches to a certain point, i.e., the step change point on the curve of ∆P*-jg, the flooding phenomenon will occur. Consequently, the two-phase flow pattern may be altered, and hence the resultant ∆P* goes up abruptly. On the other hand, because ∆* is associated with jg and increases with an increase in jg when vapor is completely condensed, the increase of jg can lead to flooding at the step change point on the curve of ∆P*-jg, and the vapor flooding velocity can be obtained accordingly. Friction Factor. The previous investigations have paid much more attention to the friction factor of fluids flowing in channels with different geometries, and some empirical or semiempirical correlations were presented. For simplifications, the following correlation was used
Ind. Eng. Chem. Res., Vol. 42, No. 21, 2003 5315
to express the friction factor for the annular flow in a vertical tube:3
fi ) 0.005[1 + 300δ/D]
(20)
where δ is the liquid film thickness. The friction factor for a rough pipe is written as
fi ) 0.005[1 + 75Ks/D]
(21)
where Ks is the absolute roughness of the pipe wall. It is noticed that eq 19 is very similar to eq 20, substituting 4δ with Ks. During condensation in a vertical tube with the vapor flowing upward, especially at the occurrence of flooding, the friction stress at the vapor-liquid interface is dominated by the effect of an interfacial wave, treated as a rough surface. (a) Friction Factor for a Single-Component or Miscible Mixture System. The viscosity and surface tension of miscible mixtures vary with the mixture components and play an importance part in the flow pattern at the liquid-vapor interface. The larger the viscosity and surface tension, the smaller the wave amplitude of the liquid film, and hence the lower the friction factor. Obviously, it is not appropriate to define the interfacial friction factor directly using eq 21. So, the interfacial friction factor is corrected by taking the viscosity and surface tension into account, as follows:
fi ) 0.005[1 + 300kδ/D]
(22)
k ) 6.1e - 7/µσ4/3
(23)
Figure 4. Steam velocity dependence of dimensionless pressure.
where
(b) Friction Factor for an Immiscible Mixture System. For an immiscible mixture system such as a water and oil system, the oil and water phases coexist in the liquid film at a certain proportion. According to a shared-surface model,11 the rectifying coefficient of the interfacial friction factor, k, is associated with the area ratio of the oil and water phases. The molecular affinity between water and oil is less than that between the same molecules, resulting in liquid film instability. So, the effect of the interaction of oil and water phases on the interfacial friction factor is taken into account to modify the rectifying coefficient k:
k ) 6.1 × 10-7
(
xs1 4/3
+
µ1 σ1
xs2 µ2 σ2
4/3
+
xs1xs2
Figure 5. Steam velocity dependence of dimensionless pressure before flooding.
)
xµ1σ14/3µ2σ24/3
(24) Results and Discussion There are three variables, ∆*, ∆P*, and jg, in eqs 18 and 19 of the present model, where ∆P* changes with jg. However, an implicit condition reveals that ∆P* changes abruptly when jg approaches the flooding point, as shown in Figure 4. Therefore, ∆P* can be calculated with different steam velocities, working out to a ∆P*jg curve. It can be found, from the curve of ∆P*-jg in Figure 6, that there is step change point near jg ) 5.5 m/s, indicating an abrupt change of the pressure drop. So, it is reasonable to deduce that flooding begins at this “just right” point. This also agrees with Imora and Kusuda’s idea12 that flooding is attributed to the liquid bridge phenomenon. It should be pointed out here that
Figure 6. Vapor velocity dependence of dimensionless pressure for different x values in a brass tube.
the equations mentioned above are derived at the condition prior to the flooding point. Therefore, it is not valid after the occurrence of flooding. Clearly, the flow pattern has been altered from annular flow to plug or slug flow. From Figure 4, ∆P* prior to the step change point seems to be zero. However, it is an actual illusion resulting from large values at the flooding point. Zooming up the curve before the flooding point, it will be found that ∆P* increases rapidly with an increase in jg, as shown in Figure 5. As a result, we can get the
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Figure 7. Comparison of flooding model data with experimental and Wallis’ results for condensation in a brass tube.
Figure 9. Comparison of flooding model data with experimental and Wallis’ results for condensation in a glass tube. Table 1. Flooding Vapor Velocities in a Brass Tube x
present model, m/s
Wallis’ model, m/s
0.0 0.2 0.4 0.6 0.8 1.0
5.55 4.15 3.40 2.85 2.40 1.90
4.27 3.68 3.09 2.49 1.87 1.20
Table 2. Flooding Vapor Velocities in a Glass Tube
Figure 8. Vapor velocity dependence of dimensionless pressure for different x values in a glass tube.
flooding vapor velocity, i.e., the value of jg at the step change point, from the ∆P*-jg curve. From eq 18 or eq 19, the ∆P* of an oil-water vapor mixture condensing in a brass tube having 18 mm o.d. diameter and 1 mm wall thickness is obtained. The vapor density is calculated with ASPEN software, the surface tension of the oil phase with the MaceodSugden equation, and the density of oil with the TongJingshan method.13 Therefore, the ∆P*-jg curve, as shown in Figure 6, can be drawn by calculating data at a variety of oil-water mass ratios, x. Whally8 recommended that the Wallis correlation is suitable for the cases if the tube diameter is less than 50 mm. For comparison, the resulting flooding vapor velocities from the present model and Wallis’ model are shown in Table 1. The general form of the Wallis correlation is as follows:3
jg*1/2 + mjf*1/2 ) c
(25)
where the constants m and c are 1.0 and 0.6, respectively. The calculated vapor flooding velocities based on the present model and Wallis’ model are demonstrated in Figure 7 in terms of the oil-water ratio, x, to compare with the experimental data. Similarly, the results for oil-water vapor condensation in a glass tube having 20 mm o.d. and 1 mm wall thickness are obtained, as demonstrated in Figures 8 and 9 and Table 2. From Figures 7 and 9, it can be found that the experimental data and the results predicted by the present model agreed well and the value from Wallis’ model is smaller than the experimental data. The main
x
present model, m/s
Wallis’ model, m/s
0.0 0.2 0.4 0.6 0.8 1.0
5.70 4.25 3.50 2.95 2.50 1.95
4.40 3.79 3.18 2.57 1.93 1.24
deviation of the data predicted by the present model from the experimental results attributes to the fact that the mathematical model cannot precisely elucidate the complex flow patterns of the flooding phenomenon. For example, it is not always true that the area ratio of oil and water is equal to their volume ratio. During the experiment, it was observed that water droplets forming on the oil film often slide downstream when the proportion of water in the condensate is small and water droplets are enchased deeply or shallowly in the liquid film when the proportion is large. Only at certain oilwater proportion ranges do the oil and water films exist together as described by the shared-surface model.14 The present mathematical model agrees well with the shared-surface model. Conclusions The flooding velocity of an immiscible vapor mixture condensing in vertical tubes was investigated theoretically and experimentally. The following conclusions can be withdrawn from the results: (1) A mathematical model based on the virtue of the oscillatory pressure at the flooding point is developed incorporating the vapor inlet velocity, the pressure drop, the interfacial friction stress, the tube inside diameter, and the fluid physical properties. (2) The vapor flooding velocities predicted by eq 18 agreed well with the experimental data of an immiscible vapor mixture in a vertical tube.
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(3) Compared with the experimental data, the Wallis correlation underestimated the flooding velocity of an immiscible vapor mixture to certain content. Acknowledgment This work was partly supported by the National Natural Science Foundation of China through Contract 59906002 and the Scientific Research Foundation for the Returned Oversea Chinese Scholars, State Education Ministry of China, through Contract 1999-747. Nomenclature D ) internal diameter, m fi ) friction factor j ) superficial velocity, m/s P ) pressure, Pa U ) actual velocity, m/s x ) oil-water mass ratio xs ) area percentage Greek Letters R ) sectional vapor area percentage δ ) liquid film thickness, m F ) density, kg/m3 σ ) surface tension, N/s τi ) surface friction stress, N µ ) viscosity, Pa‚s Superscript * ) dimensionless Subscripts 1 ) oil phase 2 ) water phase f ) liquid phase g ) vapor phase
Literature Cited (1) English, K. G.; Jones, W. T.; Spillers, R. C.; Orr, V. Flooding in a vertical updraft partial condenser. Chem. Eng. Prog. 1963, 59 (7), 51-53. (2) Diehl, J. E.; Koppany, C. R. Flooding velocity correlation for gas-liquid counter flow in vertical tubes. Chem. Eng. Symp. Ser. 1965, 92/95, 77-83. (3) Wallis, G. B. One-Dimensional Two-Phase Flow; McGrawHill: New York, 1969. (4) Kutatcladze, S. S. Elements of the hydrodynamics of gasliquid systems. Fluid Mech.-Sov. Res. 1972, 4, 29-103. (5) Bankoff, S. G.; Lee, S. G. A comparison of flooding models for air-water and steam-water flow. Adv. Two-Phase Flow Heat Transfer 1983, 2, 745-780. (6) Palen, J.; Yang, Z. H. Reflux condensation flooding prediction: review of current status. Trans. Inst. Chem. Eng. 2001, 79, 463-469. (7) Hewitt, G. F. In search of two-phase flow. Max Jakob Award Lecture. 30th National Heat Transfer Conference, Portland, OR, 1995. (8) Whally, P. B. Two-Phase Flow and Heat Transfer; Oxford Science Publications: Oxford, U.K., 1996. (9) Ma, X. H.; Chen, J. B.; Li, S. P.; Sha, Q. Y.; Liang, A. M.; Li, W.; Zhang, J. Y.; Zheng, G. J.; Feng, Z. H. Application of absorption heat transformer to recover waste heat from a synthetic rubber plant. Appl. Therm. Eng. 2003, 23 (7), 797-806. (10) Collier, J. G.; Thome, J. R. Convective Boiling and Condensation, 3rd ed.; Oxford University Press: Oxford, U.K., 1996. (11) Bernhardt, S. H.; Sheridan, J. J.; Westwater, J. W. Condensation of immiscible mixtures. AIChE Symp. Ser. 1972, 68, 21-37. (12) Imora, H.; Kusuda, H. Flooding velocity in a countercurrent annular two-phase flow. Chem. Eng. Sci. 1977, 32, 7987. (13) Tong, J. S. Thermophysical Property of Fluids; China Petrochemical Engineering Press: Beijing, China, 1996. (14) Huang, S. Y.; Wei, B. T. Two-Phase Flow and Heat Transfer of Gas and Liquid; Press of East China University of Technology: Shanghai, China, 1988.
Received for review October 9, 2002 Revised manuscript received July 22, 2003 Accepted August 4, 2003 IE0208016
