Asst. Prof. Dr. Hayder Mohammad Jaffal Ministry of Higher Education &scientific Research Mustansiriyah University College of Engineering Mechanical Engineering Department Drift Flux Model (Section 7) Asst. Prof. Dr. Hayder Mohammad Jaffal

Introduction: The basic concept of the drift-flux model is the consideration of two separate phases as a mixture phase. Therefore the fluid properties are represented by mixture properties making the drift-flux formulation simpler than the two-fluid formulation. This requires some drastic constitutive assumptions causing some two-phase flow characteristics to be lost. It is well used in the bubbly flow and plug flow where the velocity of the gas phase is well defined and calculated to an acceptable range. Wallis was the first to assign a drift flux solution to the flooding problem by examining the relative velocity of the phases to the drift flux. This model stipulates that the shear forces at the interface are negligible when compared to the gravity forces acting on the liquid phase.

Development of the model: From the definitions of superficial velocities:       The slip velocity is defined as:  

In terms of the superficial gas and liquid velocities, this may be written as follows;   and then:   The drift flux (JGL) in (m/s) of the gas relative to the liquid is defined by:  

And the drift velocities, of the gas relative to the mean fluid (VGJ) in (m/s), and the liquid relative to the mean fluid (VLJ) in (m/s) are defined as:     It can be concluded that the drift velocities are the difference between the actual velocity and the average velocity (j). The drift flux is the volumetric of a component relative to the surface moving at the average velocity, thus the drift flux of the gas is:  

           

For homogenous flow:        

           

      Thus the two different fluxes jGL and jLG are equal and opposite; commonly only jGL is used and is usually called simply the drift flux. In upwards flow with upwards velocities being defined as positive, jGL is positive quantity. Homogenous flow which is a flow with zero slip velocity, thus corresponds to flow with jGL=0.

The physical importance of the drift flux: For steady-state one dimensional flow shown in Figure (1). The force balances for each phase. For the liquid in the absence of wall shear stress and the flow gas:    

Figure (1) control volume for force balance Hear (F) is the drag force per unit volume of mixture excreted by one phase on the other. So, eliminating (dp/dz) from Eq. and Eq. then:  

In the absence of wall shear, (F) is a function only of the void fraction, physical properties and relative motion. Using Eq.7, (F) can be written as:   Therefore both drift flux (jGL) and slip velocity (us) are function only of (α) and of the physical properties of the system.

Example Bubbly flow in vertical pipe: The slip velocity may be expressed as;   where (uB) is the rising velocity of a single bubble (m/s). Therefore, using Eq.7   This function Eq.36 has been plotted in Figure (2-a) as a function of void fraction (α), for one particular value of (uB). The graph summarizes the idea that (jGL) is a function only of (α) and of the system physical properties which determine (uB). It can be noted that since:

  And (us) is always a finite non-zero quantity, then:   We also have different types of function for the drift flux:   Which is a consequence of a volume continuity equation for the (assumed) incompressible phase. In Eq. (37) , when :    

Figure (2-a) drift flux Eq.36 as a function of void fraction And jGL is linear in (α). Hence another graphical representation of the drift flux(jGL) is shown in Figure (2-b). Figure (2-a) drift flux Eq.36 as a function of void fraction Figure (2-b) drift flux Eq. 37 as a function of void fraction

The graph summarizes the continuity equation for the system The graph summarizes the continuity equation for the system. Combining Figure (2-a) and (2-b), the two lines for (jGL) intersect; this point give the actual value of (α). The composite graph in Figure (3-a); essentially it is a graphical solution of Eq (36) and (38). In Figure (3-a) the system represented is that of co-current up flows because both superficial velocities (jG) and (jL) are positive. It is obvious from Figure (3-a) that for this case, increasing the gas velocity (jG) leads to an increase in the void fraction, and increasing the liquid velocity (jL) leads to decrease in the void fraction. In Figure (3-b) the system represented is that of co-current down flows because both superficial velocities (jG) and (jL) are negative.

Figure (3) solutions for void fraction in vertical flow, (a) vertical concurrent up flow, (b) vertical concurrent down flow, (c) vertical countercurrent flow (liquid down , gas up), (d) vertical countercurrent flow ( liquid up , gas down).

For countercurrent flow, the following cases may be considered as well: 1- The liquid flows up and the gas flows down, Figure (3-c). Here there is no solution because the situation is not physically possible. 2- The liquid flows down and gas flows up, Figure (3-d) . Here the situation is more complicated. For first line, corresponding to a large downward liquid velocity, there is no solution. For third line corresponding to a small liquid velocity, there are two solutions, normally the one at lower void fraction is found. Second line must clearly represent a limit to the counter-current flow. The limit is known as flooding. The possible areas of operation and the flooding locus are shown in Figure (4). The possible values of the superficial velocity at flooding are:

  and   Equations (39) and (40) satisfy the continuity Eq.(38) and produce a line tangential to eq.(36). Referring to figure (4):     Point (A) corresponding to a dilute suspension of bubbles being held stationary by a down flow of liquid. Point (B) would not normally be attainable because the bubbles tend to coalesce before a void fraction of (0.5) is reached.

Figure (4) Flooding envelope in a flow pattern diagram

Example (1): A 136 kg/hr of air at 21 °C and 1.4 bar flow together with 136 kg/hr of water in a 32 mm diameter pipe. 1- What is the overall volumetric flux . 2- Find (α) and the average phase velocity if the drift flux is 3m/s. ρL=997 kg/m3 and ρG=1.6 kg/m3 solution: 1-    

         

2-