**3. Rise velocity of Taylor bubbles**

In this study, the rise velocity of Taylor bubbles was obtained from ECT (via cross-correlation between signals from planes 1 and 2 as shown in **Figure 3**), manual time series analysis and the high-speed camera. This rise velocity of Taylor bubbles is also referred to as structure velocity which is from the Taylor bubble periodic structures velocity at real-time measurements.

#### **3.1 Cross-correlation**

Correlation is the measure of the degree of linear relationship between two variables. Cross-correlation is a statistical method of estimating the degree to which two variables (in this case, time series data sets) are correlated.

The structure velocity was computed from the cross-sectional time averaged void fraction data measured by the ECT for both planes 1 and 2. The cross-correlation between the signals obtained from the two planes gave the structure velocity.

Given two functions *x t*ð Þ and *y t*ð Þ, the cross-correlation function, *Rxy*ð Þ*τ* between them is given as:

$$\mathcal{R}\_{\mathbf{xy}}(\mathbf{r}) = \lim\_{T \to \infty} \frac{1}{T} \Bigg[ \varkappa(\mathbf{t}) \mathbf{y}(\mathbf{t} + \mathbf{r}) d\mathbf{r} \tag{20}$$

The correlation coefficient function is expressed as:

$$\rho\_{\mathbf{xy}}(\mathbf{r}) = \frac{\mathbf{C\_{xy}}(\mathbf{r})}{\sqrt{\mathbf{C\_{xx}}}(\mathbf{0})\mathbf{C\_{yy}}(\mathbf{0})} = \frac{\mathbf{R\_{xy}}(\mathbf{r}) - \mu\_{\mathbf{x}}\mu\_{\mathbf{y}}}{\sqrt{\left(\mathbf{R\_{xx}}(\mathbf{0}) - \mu\_{\mathbf{x}}^{2}\right)\left(\mathbf{R\_{yy}}(\mathbf{0}) - \mu\_{\mathbf{y}}^{2}\right)}}\tag{21}$$

where *τ* is the time delay, *T* is the record length (period), *Cxy*ð Þ*τ* is the crosscovariance function, *Cxx*ð Þ **0** and *Cyy*ð Þ **0** are auto-covariance functions for *x* and *y*, respectively when time delay is zero, *μ<sup>x</sup>* and *μ<sup>y</sup>* are mean of the corresponding series, and *Rxx*ð Þ **0** and *Ryy*ð Þ **0** are the auto-correlation functions at a time delay of zero [34].

In this experimental work, the two functions *x t*ð Þ and *y t*ð Þ are time series data of planes 1 and 2, respectively.

the transit time (or time lag) which when used with the distance between the two

*The Effect of Liquid Viscosity on the Rise Velocity of Taylor Bubbles in Small Diameter Bubble…*

The mean velocity for the slug structure can hence be defined as distance between centres of measurement electrodes for two planes divided by the time

From **Figure 5**, the y-axis on the cross-correlation plot is the correlation coefficient. This is in the range of 0.3 to +0.9 (though generally falls between 1 and +1). There could be perfect positive correlation (correlation coefficient of +1) or perfect negative correlation (correlation coefficient of 1). A positive correlation indicates that if a signal moves either up or down, the other signal will move in the same direction, while for a negative correlation, if a signal moves either up or down, the other signal will move by an equal amount in the opposite direction. When the correlation is 0, the movement of the signals gives no correlation and is completely

planes leads to the calculation of the mean velocity for the slug structure.

*Cross-correlation results for 5 mPa s silicone oil at a superficial gas velocity of 0.02 m/s.*

*DOI: http://dx.doi.org/10.5772/intechopen.92754*

random.

**Figure 6.**

**207**

*ECT sensor signals generation for measurement of velocity in two-phase flows.*

**Figure 5.**

delay [36–38].

The important parameters required for the computation of structure velocity using cross-correlation include:


The time taken for the bubbles to travel between the two planes is calculated which then leads to the calculation of the structure velocity. This is done via an Excel Visual Basic Macro program used for the analysis of the time series data [35].

The time series, upstream and downstream with the corresponding correlation are shown in **Figure 4**. The time delay which is the time taken for the signal to travel between the two planes 1 and 2 is in the interval � 1≤ *τ* ≤1, where *τ* is the time delay.

**Figure 4** indicates periodic structures of short slugs defined as advanced form of spherical cap bubbles gradually developing into clearly distinct slugs. These periodic structures are identified to be void waves in Taylor bubbles and liquid slugs in slug flow. Cross correlation of the time series data from the two axial locations can give

**Figure 4.** *Time series upstream and downstream for 5 mPa s silicone oil at a superficial gas velocity of 0.02 m/s.*

*The Effect of Liquid Viscosity on the Rise Velocity of Taylor Bubbles in Small Diameter Bubble… DOI: http://dx.doi.org/10.5772/intechopen.92754*

**Figure 5.** *Cross-correlation results for 5 mPa s silicone oil at a superficial gas velocity of 0.02 m/s.*

the transit time (or time lag) which when used with the distance between the two planes leads to the calculation of the mean velocity for the slug structure.

From **Figure 5**, the y-axis on the cross-correlation plot is the correlation coefficient. This is in the range of 0.3 to +0.9 (though generally falls between 1 and +1). There could be perfect positive correlation (correlation coefficient of +1) or perfect negative correlation (correlation coefficient of 1). A positive correlation indicates that if a signal moves either up or down, the other signal will move in the same direction, while for a negative correlation, if a signal moves either up or down, the other signal will move by an equal amount in the opposite direction. When the correlation is 0, the movement of the signals gives no correlation and is completely random.

The mean velocity for the slug structure can hence be defined as distance between centres of measurement electrodes for two planes divided by the time delay [36–38].

**Figure 6.** *ECT sensor signals generation for measurement of velocity in two-phase flows.*

where *τ* is the time delay, *T* is the record length (period), *Cxy*ð Þ*τ* is the crosscovariance function, *Cxx*ð Þ **0** and *Cyy*ð Þ **0** are auto-covariance functions for *x* and *y*, respectively when time delay is zero, *μ<sup>x</sup>* and *μ<sup>y</sup>* are mean of the corresponding series, and *Rxx*ð Þ **0** and *Ryy*ð Þ **0** are the auto-correlation functions at a time delay of

In this experimental work, the two functions *x t*ð Þ and *y t*ð Þ are time series data of

The important parameters required for the computation of structure velocity

The time taken for the bubbles to travel between the two planes is calculated which then leads to the calculation of the structure velocity. This is done via an Excel Visual Basic Macro program used for the analysis of the time series data [35]. The time series, upstream and downstream with the corresponding correlation are shown in **Figure 4**. The time delay which is the time taken for the signal to travel between the two planes 1 and 2 is in the interval � 1≤ *τ* ≤1, where *τ* is the

**Figure 4** indicates periodic structures of short slugs defined as advanced form of spherical cap bubbles gradually developing into clearly distinct slugs. These periodic structures are identified to be void waves in Taylor bubbles and liquid slugs in slug flow. Cross correlation of the time series data from the two axial locations can give

*Time series upstream and downstream for 5 mPa s silicone oil at a superficial gas velocity of 0.02 m/s.*

zero [34].

time delay.

**Figure 4.**

**206**

planes 1 and 2, respectively.

using cross-correlation include:

*Vortex Dynamics Theories and Applications*

b. Number of data points.

c. Sampling frequency of data.

a. Void fraction data for planes 1 and 2.

d. Distance between two planes (planes 1 and 2).

$$\text{Distance between centres}$$

$$\text{of measurement}$$

$$\text{Structure velocity } (\text{m/s}) = \frac{\text{electrons for two planes } (\text{m})}{\text{Time delay } (\text{s})} \qquad (22)$$

measurement electrode is indicated in **Figure 7**. This is compared with that

*The Effect of Liquid Viscosity on the Rise Velocity of Taylor Bubbles in Small Diameter Bubble…*

distance covered by the bubble at a given time is computed which gives a corresponding value for the velocity. A typical window of the camera control

**4.1 Effect of viscosity on the rise velocity (structure velocity)**

The camera control software of the Phantom High-Speed Camera can be used to obtain an estimate of the structure velocity. This is done via a playback in which the

The effect of liquid viscosity on the rise velocity (structure velocity) has been

obtained from the ECT for the range of viscosities considered (i.e. 5, 100, 1000 and 5000 mPa s). The physical properties of the liquids used are given in **Table 1**. A plot of structure velocity versus superficial gas velocity for all the viscosities considered is given in **Figure 9** which shows that structure velocity increases with an increase in superficial gas velocity which is in agreement with the observations of

This can be explained using the slug Reynolds number, a dimensionless param-

A plot of slug Reynolds number versus superficial gas velocity is made at various

According to Bendiksen [21], Reynolds number in the range 5000–110,000 (for low viscous fluids) give turbulent flow. As the Reynolds numbers of the viscosities considered are less than 5000, laminar flow prevails. For large slug Reynolds number, viscous effect will be negligible, while for small slug Reynolds number, viscous effect will be dominant [40]. So, since

> 100 965 20.9 2.74 1000 970 21.2 2.76 5000 970 21.4 2.76

**Figure 11** reveals that as viscosity increases, slug Reynolds number decreases

i. Occurrence and prevalence of laminar flow as viscosity increases:

**Liquid Viscosity, mPa s Density, kg***=***m3 Surface tension, mN***=***m Relative permittivity**

Silicone oil 5 915 19.7 2.60

studied by making a comparison between the respective structure velocities

Abdulkaldir et al. [39] and decreases with increase in viscosity as shown in **Figure 10** (obtained from ECT Plot3d Image reconstruction software). The structure velocity of 5 and 100 mPa s is found to be approximately the same due to similar void fraction data values. The variation from small to bigger spherical cap and developing slug in 5 and 100 mPa s, and the slug flow in 1000 and 5000 mPa s

(as shown in **Figure 9**) has been discussed by Kajero et al. [33].

tending towards zero. This can be explained as follow:

viscosities as shown in **Figure 11**, with an indication of laminar flow.

obtained via cross correlation later in this paper.

*DOI: http://dx.doi.org/10.5772/intechopen.92754*

**3.3 High-speed camera estimate**

software is shown in **Figure 8**.

**4. Results and discussion**

eter (Eq. 16).

**Table 1.**

**209**

*Physical properties of silicone oil viscosities used.*

The schematics of the ECT Sensor signals generation for measurement of velocity in two-phase flows is shown in **Figure 6**.
