Fig. 1.11 schematically shows a typical slug flow i.e. a train of liquid moving plugs separated by elongated gas bubbles in microchannel. In general the bubbles nose and tail have different curvatures and the difference between them depends on flow
parameters such as Reynolds number. In most slug flows gas bubbles are separated from the microchannel wall by a thin liquid layer around them. The thickness of this layer, known as δ, depends on two phase flow Capillary number, Ca, and will be discussed in section 2.1. The length of the liquid slugs and gas bubbles are normally shown by Lsand Lb, respectively. The pitch, P , is nose to nose or tail to tail distance.
Figure 1.11: Typical slug flow.
The slug flow void fraction is defined as the ratio of the gas volume to the total two phase volume in certain length of mircotube i.e. control volume. That is:
αg =
Vg
V (1.19)
and liquid fraction is:
αl =
Vl
V (1.20)
The void and liquid fractions of two phase flows are related as follows:
The mass flux, G, is the mass flow rate per unit cross sectional area which can be calculated as follows: Gg = ˙ mg A = ρgUg (1.22) and Gl = ˙ ml A = ρlUl (1.23)
where U is superficial velocity i.e. the velocity of each phase in the microtube if that phase occupied the total cross sectional area. The actual phase velocities can be defined using phase void (or liquid) fraction as follows:
ug = Ug αg = Gg ρgαg (1.24) and ul= Ul αl = Gl ρlαl (1.25) Finally, the quality of the two phase flow is:
x= Gg
Gg + Gl
(1.26) and takes a value of 0 for single phase liquid flows and 1 for single phase gas flows.
Most studies in this area are experimental. As a result of this and many other reasons, there are significant gaps in numerical simulation analysis. One of the most important reasons can be the time-expense of the numerical methods. Numerical simulation as a method for predicting the heat and mass transfer characteristics has gained popularity due in part to the developments in the computational hardware and software.
A few review articles exist in this area which mostly contain experimental works (Shao et al. [37], Kreutzer et al. [19], Angeli and Gavriilidis [17], and Gupta et al. [18]). Recently, Talimi et al. [20] reviewed numerical studies on hydrodynamics and heat transfer in non-boiling two phase flows in microscales. They gathered and categorized much of the work conducted in recent years into two major sections: gas- liquid and liquid-liquid two phase flows. These sections are categorized further into two subcategories including circular or non-circular channels. While from the numerical point of view there is no difference between the gas-liquid and liquid-liquid two-phase
flow simulations, the role of the gas phase in heat transfer is mostly ignored due to its much lower heat capacity. This further leads to another simplification i.e. using a single phase moving frame technique for simulations. This will be discussed in the later sections. In the case of liquid-liquid two phase flow for heat removal applications both of the phases are shown to play significant roles. Figure 2.1 shows the structure of the review conducted by Talimi et al. [20]. The main focus was on flow patterns, slug formation, film thickness, and bubble shape in the literature and less consideration on heat transfer and pressure drop. Numerical simulation is a very powerful tool which gives much detail on the hydrodynamics and heat transfer and hence it helps a lot in giving deeper insight in two phase slug flows in microchannels.
Figure 2.1: Review structure, (Talimi et al. [20]).
In this section a brief review of the existing literature (experimental and numer- ical) is presented. The film thickness, pressure drop, and heat transfer are the three
important aspects which are going to be considered in the present research. The im- portance of the film thickness on the internal circulations has been shown by Taylor [27]. As Fig. 2.2 illustrates the streamlines in the liquid slug depends on the Capillary number, Ca, and a dimensionless number, m, which is a relative velocity between the two phases, as follows:
m= Ub − Ul Ub
(2.1) where b and l denote the bubble and liquid phase, respectively. Figure 2.2 shows that for high Ca values a complete bypass occurs through the liquid film around the bubbles, and in the slug flows with low Ca number internal circulations exist. As presented in the following sections the film thickness is a function of Ca, so the film thickness is an important parameter which governs the flow pattern i.e. internal circulations and this way the film thickness might have strong effects on heat transfer process. In the following sections, the experimental and numerical studies on the film thickness will be reviewed along with pressure drop and heat transfer.
2.1
Film Thickness
The film thickness has been calculated experimentally using two different approaches. In the first approach, the thickness of the liquid film around the bubbles is measured directly from the images (Aussilous and Quere [38] and Han and Shikazono [52]). In the second approach, the film thickness is calculated based on the bubble velocity and
Figure 2.2: Streamlines in liquid slugs in front of a Taylor bubble, (Taylor [27]). continuity equation. Some researchers assumed a stagnant film (Warnier et al. [40]) and some assumed a fully developed annular flow velocity profile and zero shear stress at the interface in their calculations (Thulasidas et al. [41]). The following correlations have been suggested for film thickness calculation in slug flows:
Fairbrother and Stubbs [42]:
δ
R = 0.5Ca
1/2
(2.2)