2.2. Drift-flux model

References

• Zuber and Findlay [ZF65]: Drift-flux model

Consider a two-phase flow in a vertical pipe and the pipe axis is taken as the \(z\) coordinate. The area-averaged volume fraction for the cross-sectional area \(A\) on the \(xy\) plane is calculated as

(2.10)\[ \langle \alpha_{k} \rangle (z, t) = \frac{1}{A} \iint_{A} X_{k} (\mathbf{x}, t) dA\]

The area average of the phase-averaged local velocity is defined by

(2.11)\[ \langle\langle v_{k} \rangle\rangle = \frac{\frac{1}{A} \iint X_{k} \mathbf{v} dA}{\frac{1}{A} \iint X_{k} dA} = \frac{\langle j_{k} \rangle}{\langle \alpha_{k} \rangle}\]

For the phase \(k = 2\) (we may use this for the gas phase),

(2.12)\[ \langle\langle v_{2} \rangle\rangle = \frac{\langle j_{2} \rangle}{\langle \alpha_{2} \rangle}\]

However, from Eq. (2.7),

(2.13)\[ j_{2} = \alpha_{2} v_{2}\]

and Eq. (2.9) becomes

(2.14)\[ v_{2} = j + v_{2j}\]

so that

(2.15)\[ \langle\langle v_{2} \rangle\rangle = \frac{\langle \alpha_{2} j + \alpha_{2} v_{2j} \rangle}{\langle \alpha_{2} \rangle} = \frac{\langle \alpha_{2} j \rangle}{\langle \alpha_{2} \rangle} + \frac{\langle \alpha_{2} v_{2j} \rangle}{\langle \alpha_{2} \rangle}\]

Rewriting the last equation gives

(2.16)\[ \langle\langle v_{2} \rangle\rangle = \frac{\langle \alpha_{2} j \rangle}{\langle \alpha_{2} \rangle \langle j \rangle} \langle j \rangle + \frac{\langle \alpha_{2} v_{2j} \rangle}{\langle \alpha_{2} \rangle}\]

Using Eq. (2.12) we have

(2.17)\[ \frac{\langle j_{2} \rangle}{\langle \alpha_{2} \rangle} = \frac{\langle \alpha_{2} j \rangle}{\langle \alpha_{2} \rangle \langle j \rangle} \langle j \rangle + \frac{\langle \alpha_{2} v_{2j} \rangle}{\langle \alpha_{2} \rangle}\]

We may write

(2.18)\[ \frac{\langle j_{2} \rangle}{\langle \alpha_{2} \rangle} = C_{0} \langle j \rangle + V_{2j}\]

where

(2.19)\[ C_{0} = \frac{\langle \alpha_{2} j \rangle}{\langle \alpha_{2} \rangle \langle j \rangle}\]

and

(2.20)\[ V_{2j} = \frac{\langle \alpha_{2} v_{2j} \rangle}{\langle \alpha_{2} \rangle}\]

Since \(\langle j_{2} \rangle = Q_{2} / A\) and \(\langle j \rangle = (Q_{1} + Q_{2})/A\), where \(Q\) is the volume flow rate, are known as the inlet condition, we can calculate the volume fraction by solving the above equation for \(\alpha_{2}\):

(2.21)\[ \langle \alpha_{2} \rangle = \frac{\langle j_{2} \rangle}{C_{0} \langle j \rangle + V_{2j}}, \]

provided that \(C_{0}\) and \(V_{2j}\) are given. They are referred to as the distribution parameter and the drift velocity, respectively. Alternatively, we can write Eq. (2.18) in the following dimensionless form:

(2.22)\[ \frac{\langle \beta_{2} \rangle}{\langle \alpha_{2} \rangle} = C_{0} + \frac{V_{2j}}{ \langle j \rangle}\]

where

(2.23)\[ \langle \beta_{2} \rangle = \frac{Q_{2}}{Q_{1} + Q_{2}}\]

If the two phases flow as a complete mixture and \(v_{1} = v_{2}\), we have \(\langle \alpha_{2} \rangle = \langle \beta_{2} \rangle\). Therefore, the drift-flux model reduces to the homogeneous model when \(C_{0} = 1\) and \(V_{2j} = 0\), and the R.H.S. of Eq. (2.22) represents how the flow is far from the homogeneous state.

In the following, we choose the liquid phase \(L\) for \(1\) and the gas phase \(G\) for \(2\), so \(\alpha_{2}\) is the void fraction and will be simply written as \(\alpha\). The distribution parameter depends on the profile of \(\alpha\). When \(\alpha\) is uniform in the cross section of a pipe, \(C_{0}\) takes a value close to unity. On the other hand, \(C_{0}\) is larger than \(1\) when the void fraction in the core region is large, e.g., a parabolic profile. \(C_{0}\) may be less than \(1\) when the void fraction accumulates in the near-wall region.

The values of \(C_{0}\) and \(V_{Gj}\) depend of the flow pattern. Some examples of drift flux parameters are given in Table 2.1 [oME06]. The drift velocities of the bubbly and slug flow regimes will be found, respectively, in Wave analogy for bubble rise velocity and Order-of-magnitude analysis for bubble velocity. As can be understood from its definition, the drift velocity represents how fast the gas phase is compared to the mixture. In bubbly flows and slug flows in a vertical pipe, the drift velocity is therefore tightly related with the bubble rise velocity in still liquid. In the annular flow pattern, the cross-sectional area is almost occupaied by the gas phase, and therefore \(C_{0}\) is close to unity. The fucntional form of drift velocity will be briefly discussed in Annular flow.

Table 2.1 Drift flux parameters (quoted from Two-Phase Flow Handbook (JSME))

Flow regime	\(C_{0}\)	\(V_{Gj}\)
Bubbly flow	\(1.2 - 0.2 \sqrt{\rho_{G}/\rho_{L}}\)	\(\sqrt{2} [\sigma \Delta \rho g / \rho_{L}^{2}]^{1/4}\)
Slug flow	\(1.2\)	\(0.35 [\Delta \rho g D / \rho_{L}]^{1/2}\)
Annular flow	\(1.0\)	\(23[\mu_{L} \langle j_{L} \rangle / \rho_{G} D]^{1/2} \Delta \rho/\rho_{L}\)