CFD modelling of slug flow in vertical tubes

Abstract

In this work we present a numerical study to investigate the motion of single Taylor bubbles in vertical tubes. A complete description of the bubble propagation in both stagnant and flowing liquids was obtained. The shape and velocity of the slug, the velocity distribution and the distribution of local wall shear stress were computed and compared favourably with the published experimental findings. The volume of fluid (VOF) method implemented in the commercial CFD package, Fluent is used for this numerical study.

Introduction

When gas and liquid flow in a pipe they tend to distribute themselves in a variety of configurations. These characteristic distributions of the fluid–fluid interface are called flow patterns or flow regimes. Much time and effort has been expended in determining these regimes for various pairs of fluids, channel geometries, and inclinations (Mandhane et al., 1974, Taitel and Duckler, 1976, Taitel, 1986, Barnea, 1987). For vertical co-current flow and at low gas flow rates, the flow pattern observed is bubbly. Here, the gas phase is distributed as discrete bubbles within the liquid continuum. At higher gas flow rates, some of the bubbles have nearly the same cross-sectional area as that of the channel. These bullet-shaped bubbles—sometimes referred to as ‘Taylor bubbles’ or ‘slugs’—move along and are separated by liquid plugs that may or may not contain a dispersion of smaller gas bubbles. An increase in the gas flow rate in a two-phase mixture flowing in slug flow will eventually result in a complete destruction of slug flow integrity with consequential churning or oscillatory action. At very high gas flow rates, the flow becomes annular, in which, adjacent to the channel wall. There is a liquid continuum and the core of the channel is a gas continuum (Hewitt and Hall-Taylor, 1970).

Slug flow is the most important of the two-phase flow regimes primarily because of the numerous industrial and practical applications. Some of these include buoyancy-driven fermenters, production and transportation of hydrocarbons, boiling and condensation processes in thermal power plants, and emergency cooling of nuclear reactors. Slug flow is characterised by its random intermittence and inherent unsteadiness. A fixed observer would see a quasi-periodic occurrence of long, bullet-shaped Taylor bubbles followed by liquid plugs sometimes carrying dispersed bubbles and taking on the appearance of bubbly flow in a pipe. The bubble region may take on stratified or annular configurations depending upon the tube inclination and flow conditions. In order to understand the complex features of intermittent slug flow, mainly experimental research has been conducted to study the motion of isolated Taylor bubbles in motionless and flowing liquids for various inclination angles. Here, vertical slug flow will be emphasised.

The shape and the velocity with which a single Taylor bubble ascending through a denser stagnant liquid is influenced by the forces acting on it, namely the viscous, inertial and interfacial forces. Dimensionless analysis based on Pi-theorem leads to the following dimensionless groups:

$$\frac{U_{\mathit{TB}}^2{\rho }_L}{{\mathit{gD}}_t({\rho }_L-{\rho }_G)}\frac{g({\rho }_L-{\rho }_G)D_t^2}{\sigma }\text{,}\frac{g{\mu }_L^4({\rho }_L-{\rho }_G)}{{\rho }_L^2{\sigma }^3},\frac{{\rho }_L}{{\rho }_G},\frac{{\mu }_L}{{\mu }_G},\frac{L_{\mathit{TB}}}{D_t}\text{,}$$

where $D_t$ is the diameter of the tube, $U_{\mathit{TB}}$ is Taylor bubble velocity, ${\rho }_L$ and ${\rho }_G$ are the density of the liquid and gas, respectively, ${\mu }_L$ and ${\mu }_G$ are the viscosity of the liquid and gas respectively, $\sigma$ is the surface tension and $L_{\mathit{TB}}$ is the length of Taylor bubble. For cylindrical bubbles, the film thickness and the bubble rise velocity are independent of the bubble length (Griffith and Wallis, 1961, Nicklin et al., 1962, Mao and Duckler, 1989, Polonsky et al., 1999). Under the assumption that inertial forces in the gas are far smaller than the inertial forces in the liquid (${\rho }_L/{\rho }_G\gg 1$), ${\rho }_L/{\rho }_G$ can be eliminated. And if the viscosity of the gas in the bubble is neglected the following set of three dimensionless groups is sufficient to characterise the motion of a single bubble rising through a motionless liquid. These are the Eötvös number $\text{<mi mathvariant="italic" is="true">Eo</mi>}=g({\rho }_L-{\rho }_G)D_t^2/\sigma$, the Morton number defined as $M=g{\mu }_L^4({\rho }_L-{\rho }_G)/{\rho }_L^2{\sigma }^3$ and the Froude number defined as $\mathit{Fr}=U_{\mathit{TB}}/\sqrt{{\mathit{gD}}_t({\rho }_L-{\rho }_G)/{\rho }_L}$. The Eötvös number represents the relative significance of surface tension and buoyancy. The Morton number is sometimes referred to as the property group. The Froude number represents the ratio of inertial to gravitational forces. Other dimensionless groups could be used, e.g., Fabre and Line (1992) in their review paper on the motion of Taylor bubbles used Froude number as a unique function of Eötvös number and a dimensionless inverse viscosity number, $N_f$, given by $N_f=({\mathit{gD}}_t^3{)}^{1/2}/\nu$. Wallis (1969) used a set of Fr, $N_f$ and Archimedes number Ar ($\text{<mi mathvariant="italic" is="true">Ar</mi>}={\sigma }^{3/2}{\rho }_L/{\mu }_L^2{g}^{1/2}({\rho }_L-{\rho }_G{)}^{1/2})$. Other and these widely used dimensionless groups can all be derived by manipulating and/or combining two or more of groups adopted in this work, e.g., $N_f=(E_o^3/M{)}^{1/4},\text{<mi mathvariant="italic" is="true">Ar</mi>}=(1/M{)}^{1/2}$ and the capillary number, $\text{<mi mathvariant="italic" is="true">Ca</mi>}=\text{<mi mathvariant="italic" is="true">Fr</mi>}(M\text{<mi mathvariant="italic" is="true">Eo</mi>}{)}^{1/4}$ which is the ratio of viscous to surface tension forces. White and Beardmore (1962) described a wide spectrum of experimental results on Taylor bubbles drifting through motionless liquids in vertical tubes. The authors presented a cross plot (Fig. 6 in their paper) showing the regions in which various retarding forces may be neglected. The density of air was neglected and instead of Fr, $\sqrt{\text{<mi mathvariant="italic" is="true">Fr</mi>}}$ is plotted to have a reasonable spread of data (White and Beardmore, 1962). In the region where surface tension dominates the bubble does not move at all where the hydrostatic forces are completely balanced by surface tension forces. This occurs at $\text{<mi mathvariant="italic" is="true">Eo</mi>}<3.37$ (Hattori, 1935, Bretherton, 1961). For inertia-controlled region when viscosity and surface tension can be neglected ($\text{<mi mathvariant="italic" is="true">Eo</mi>}>100,N_f>300$, ${\rho }_L^2{\mathit{gD}}_t/{\mu }_L^2>3\times {10}^5$), the bubble rise velocity is given solely in terms of Fr (Nicklin et al., 1962, White and Beardmore, 1962, Zukoski, 1966, Mao and Duckler, 1990). In the centre of the graph, the relative magnitude of all retarding forces, namely the viscous, inertial, and interfacial forces are significant. In this region the relationship between Fr, Eo, and M for vertical tubes has been presented by White and Beardmore (1962) as a graphical map which plots lines of constant M on Fr–Eo axis. Similar maps have been produced for non-vertical tubes by Wallis (1969) and Weber et al. (1986).

The description of the motion of single Taylor bubbles in flowing liquids dates back to the pioneering work of Nicklin's et al. (1962) who placed the corner stone of slug flow modelling by recognising the fact that the bubble velocity is a superimposition of two components:

$$U_{\mathit{TB}}=C_1{U}_m+U_0\text{.}$$

The second term represents the drift due to buoyancy (the bubble velocity in a stagnant liquid) and the first term refers to the transport by the mean flow, $U_m(U_m=U_{\mathit{SL}}+U_{\mathit{SG}}$). $C_1$ is a dimensionless coefficient that depends on the velocity profile ahead of the bubble, and can be seen as the ratio of the maximum to the mean velocity in the profile. Hence for turbulent flows, $C_1\cong 1.2$ while for laminar pipe flow, $C_1\cong 2$ (Nicklin et al., 1962, Collins et al., 1978, Grace and Clift, 1979, Bendiksen, 1985, Polonsky et al., 1999). The velocity field around the bubble has been measured by many researchers primarily because of its importance regarding the interaction between two Taylor bubbles. Several techniques have been adopted, e.g.: the photochromic dye activation (PDA) method (DeJesus et al., 1995, Kawaji et al., 1997; Ahmad et al., 1998), the particle image velocimetry (PIV) technique (Polonsky et al., 1999; Nogueria et al., 2000) and the laser doppler velocimetry (LDV) technique (Kvernvold et al., 1984). Measurement of wall shear stress in stagnant (Mao and Duckler, 1989) and flowing (Nakoryakov et al., 1989) water showed clearly the reversal of the flow in the liquid film. Experimental work on the propagation of Taylor bubbles in downward flow has been limited to a few studies on ascending bubbles in vertically downward flow (Martin, 1976; Polonsky et al., 1999) and one single study (to our knowledge) on inclined downward flow by Bendiksen (1984). Theoretical treatments of the problem for predicting the rise bubble velocity have been successful for the special case of vertical tubes. This is, however, restricted to the case where both viscous and surface tension effects are negligible (Dumitrescu, 1943; Davies and Taylor, 1950), or the case where viscous effects dominate (Goldsmith and Mason, 1962). Later, Bendiksen (1985) extended the theoretical approach of Dumitrescu (1943) and Davies and Taylor (1950) to account for the surface tension. Mao and Duckler, 1990, Mao and Duckler, 1991 and Clarke and Issa (1997) performed numerical simulations in order to calculate the velocity of the bubble and the velocity field in the liquid film. Propagation of Taylor bubbles in non-vertical tubes has been less well studied experimentally, and even less well theoretically, for the obvious reason of asymmetry.

Thus, studying the motion of a single Taylor bubble in stagnant and in moving liquid is essential in order to understand the intrinsically complicated nature of slug flow. The rise of a single bubble in both stagnant and flowing liquid inside vertical tubes has been extensively studied by numerous researchers, both experimentally and theoretically. All the published numerical methods to model slug flow are restricted only to vertical tubes and they assume either the shape of the bubble or a functional form for the shape (Dumitrescu, 1943, Davies and Taylor, 1950). These assumptions constrain the nature of the solution, while the approach adopted here (the volume of fluid method, VOF) lays no such a priori foundations. The solution domain in the present model not only includes the field around the bubble, as in the study of Mao and Duckler (1990), but also extends behind the bubble, allowing field information to be obtained in the wake region. The difficulty of obtaining local data is even more complicated in horizontal and inclined slug flow by the fact that the flow is asymmetric, and consequently, very few detailed data have been published in the open literature. Previous models available in literature also fail to provide a fully satisfactory mechanistic model of slug flow in horizontal configuration (Duckler and Hubbard, 1975; Taitel and Barnea, 1990). Thus, it is evident that more insight into slug flow is needed to attain a thorough understanding of the internal structure of the slug flow pattern. In this work, an attempt is made to calculate the shape and rising velocity of a single Taylor bubble in stagnant and in moving liquid in vertical tubes. The velocity field in a slug unit (Taylor bubble + liquid plug) and wall shear stress are also calculated. The intention of future work is to characterise the hydrodynamics of slug flow when the tube is tilted away from the vertical and also to investigate the effect of the leading bubble on the shape and velocity of the trailing one.