*4.1.1 Drag force, FD*

From Stokes law, drag force is given as:

$$F\_D = \mathbf{3}\pi\mu\_L \mathbf{V}d\tag{24}$$

based on the fact that the flow is laminar.

Drag force is the force due to the resistance provided by the fluid to the motion of a body through it.

From Eq. (24), Drag force is directly proportional to viscosity of the fluid. Drag coefficient, *CD* is given as:

$$\mathbf{C}\_{D} = \frac{\mathbf{24}}{\mathbf{Re}}\tag{25}$$

On substituting Eq. (25) into (24) given that *Re* ¼ *ρLVd=μL*,

$$F\_D = \frac{72\pi\mu\_L^2}{\mathbf{C}\_D\rho\_L} \tag{26}$$

Eq. (26) is used to compute drag force, *FD* which is plotted against superficial gas velocity for all viscosities considered as shown in **Figure 13(a)**. This can be interpreted as follows:


To further confirm the effect of drag force, drag force was plotted against void fraction, which is the volume fraction of gas in the gas–liquid mixture as shown in **Figure 13(b)**.

**Figure 13.** *Drag force versus (a) superficial gas velocity, (b) void fraction.*

The square of the inverse square dimensionless viscosity for 5, 100, 1000 and 5000 mPa s are 41,012,747, 114051.4, 1152.37 and 46.09481, respectively. Since

 8.7645 <sup>10</sup><sup>7</sup> 1139.105 6408.267 1.11353 <sup>10</sup><sup>5</sup> 1132.374 337.9223 1061.426 1122.134 33.96732 644964.7 1111.647 6.793463

viscous effect dominates. This dominating effect of viscous force over inertia force possibly causes a decrease in structure velocity with an increase in viscosity. This can be further confirmed by obtaining a relationship between drag force and vis-

**<sup>2</sup>**

*Mo Eo N <sup>f</sup>*

<sup>&</sup>lt; **<sup>3</sup> <sup>10</sup><sup>5</sup>**

,

100 mPa s, 1000 mPa s and 5000 mPa s satisfy the condition of *Nf*

*Forces acting on Taylor bubble (surface tension force helps to hold the bubbles together).*

**Viscosity, mPa s Dimensionless numbers**

cosity taking superficial gas velocity as a parameter.

*Morton, Eotvos and inverse dimensionless numbers.*

*Vortex Dynamics Theories and Applications*

**Figure 12.**

**Table 2.**

**212**

As void fraction increases, the drag force increases. This is probably due to the increase in the volume fraction of gas (void fraction); hence, an increase in the bubble size relative to the column diameter. This increase in bubble size is due to coalescence and it subsequently requires an increase in drag force to be effective enough to oppose the motion of the bubbles through the liquid. Hence, it can be said that the drag force is exponentially proportional to the void fraction. The drag force was found to be low at low viscosities (5 and 100 mPa s) and increases as viscosity increases. This is due to viscous effect as explained earlier on. As viscosity increases, void fraction increases; hence, an increase in drag force subsequently leads to hindered rise velocity.

The relationship between drag force and drag coefficient is shown in **Figure 14**. As viscosity decreases, the curves of the respective viscosities are tending towards zero. As drag coefficient increases, the drag force decreases tending towards zero. This gives an inverse relationship, which fits well into a power law expression, with the inverse proportionality constant increasing with increase in viscosity as shown in **Table 3**.

A plot of drag coefficient against Reynolds number on a log–log plot agrees with Stokes law, which gives an inverse relationship between the drag coefficient and Reynolds number as shown in **Figure 15**, with purple, green, blue and red for 5, 100, 1000 and 5000 mPa s respectively. This further confirms the fact that the drag force is directly proportional to the liquid viscosity. At low Reynolds number, drag coefficient is high, while at high Reynolds number, drag coefficient is low. Hence, the structure velocity of the latter is greater than the former.

**Figure 14.**

*Inverse relationship between drag force and drag coefficient for all the liquid viscosities considered.*


A plot of Froude number versus the superficial gas velocity is shown in

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

1.At a superficial gas velocity of 0.01–0.17 m/s for both 5 and 100 mPa s

viscosity, *Fr*< 1 which implies subcritical flow (slow/tranquil flow) due to low superficial gas velocity; while between 0.17 and 0.361 m/s, *Fr* > 1 which implies supercritical flow (fast rapid flow). At 0.2 m/s, *Fr* = 1 which implies critical flow [42]. In the case of both 1000 and 5000 mPa s, at all superficial

**Figure 16**. The following can be inferred from the plot:

**Figure 15.**

**Figure 16.**

**215**

*Effect of Froude number.*

*Drag coefficients for the rising bubbles.*

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

**Table 3.**

*Inverse proportionality constants from the power law relationship between drag force and drag coefficient.*

*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 15.**

As void fraction increases, the drag force increases. This is probably due to the increase in the volume fraction of gas (void fraction); hence, an increase in the bubble size relative to the column diameter. This increase in bubble size is due to coalescence and it subsequently requires an increase in drag force to be effective enough to oppose the motion of the bubbles through the liquid. Hence, it can be said that the drag force is exponentially proportional to the void fraction. The drag force was found to be low at low viscosities (5 and 100 mPa s) and increases as viscosity increases. This is due to viscous effect as explained earlier on. As viscosity increases, void fraction increases; hence, an increase in drag force subsequently leads to

The relationship between drag force and drag coefficient is shown in **Figure 14**. As viscosity decreases, the curves of the respective viscosities are tending towards zero. As drag coefficient increases, the drag force decreases tending towards zero. This gives an inverse relationship, which fits well into a power law expression, with the inverse proportionality constant increasing with increase in viscosity as shown

A plot of drag coefficient against Reynolds number on a log–log plot agrees with Stokes law, which gives an inverse relationship between the drag coefficient and Reynolds number as shown in **Figure 15**, with purple, green, blue and red for 5, 100, 1000 and 5000 mPa s respectively. This further confirms the fact that the drag force is directly proportional to the liquid viscosity. At low Reynolds number, drag coefficient is high, while at high Reynolds number, drag coefficient is low. Hence,

the structure velocity of the latter is greater than the former.

*Inverse relationship between drag force and drag coefficient for all the liquid viscosities considered.*

**Viscosity (mPa s) Inverse proportionality constant**

*Inverse proportionality constants from the power law relationship between drag force and drag coefficient.*

5 6 <sup>10</sup>–<sup>6</sup> 100 0.0023 1000 0.2332 5000 5.8305

hindered rise velocity.

*Vortex Dynamics Theories and Applications*

in **Table 3**.

**Figure 14.**

**Table 3.**

**214**

*Drag coefficients for the rising bubbles.*

**Figure 16.** *Effect of Froude number.*

A plot of Froude number versus the superficial gas velocity is shown in **Figure 16**. The following can be inferred from the plot:

1.At a superficial gas velocity of 0.01–0.17 m/s for both 5 and 100 mPa s viscosity, *Fr*< 1 which implies subcritical flow (slow/tranquil flow) due to low superficial gas velocity; while between 0.17 and 0.361 m/s, *Fr* > 1 which implies supercritical flow (fast rapid flow). At 0.2 m/s, *Fr* = 1 which implies critical flow [42]. In the case of both 1000 and 5000 mPa s, at all superficial

gas velocities, *Fr*< 1, this implies subcritical flow. The viscous effect makes the flow very slow, with low rise velocity of Taylor bubbles.

where *Γ* is given as:

Condition for *m*:

*Γ* ¼ **0***:***345 1** � *e*�

*m* ¼ 69 *Nf*

*UN* <sup>¼</sup> **<sup>2</sup>***:***29 1** � **<sup>20</sup>**

**w**here *Fr* is obtained from Eqs. (27) to (32).

and the parameters ð Þ *a*, *b*, ……… , *l* are:

**217**

The modified form of Viana et al. [43] model is given as:

*Eo*

*Fr* <sup>¼</sup> *L R*<sup>½</sup> ; *<sup>A</sup>*, *<sup>B</sup>*, *<sup>C</sup>*, *<sup>G</sup>*� � *<sup>A</sup>*

*<sup>A</sup>* <sup>¼</sup> *L E*½ �¼ *<sup>o</sup>*; *<sup>a</sup>*, *<sup>b</sup>*, *<sup>c</sup>*, *<sup>d</sup> <sup>a</sup>*

*<sup>B</sup>* <sup>¼</sup> *L E*½ �¼ *<sup>o</sup>*; *<sup>e</sup>*,*f*, *<sup>g</sup>*, *<sup>h</sup> <sup>e</sup>*

*<sup>C</sup>* <sup>¼</sup> *L E*½ �¼ *<sup>o</sup>*; *<sup>i</sup>*, *<sup>j</sup>*, *<sup>k</sup>*, *<sup>l</sup> <sup>i</sup>*

*a* ¼ 0*:*34; *b* ¼ 14*:*793;*c* ¼ �3*:*06; *d* ¼ 0*:*58;*e* ¼ 31*:*08; *f* ¼ 29*:*868; *g* ¼ �1*:*96;

On the average, a good agreement exists between the structure velocity from the ECT and that measured from the high-speed camera. A reasonably fair agreement exists between the former and that estimated from the time series. The modified Viana et al. [43], modified De Cachard and Delhaye [44], and Nicklin et al. [7] model gave roughly similar pattern with structure velocity from cross-correlation, manual time series analysis and high-speed camera. Modified Viana et al. [43] and modified De Cachard and Delhaye [44] models showed good agreement with the

*h* ¼ �0*:*49; *i* ¼ �1*:*45; *j* ¼ 24*:*867; *k* ¼ �9*:*93; *l* ¼ �0*:*094; *m* ¼ �1*:*0295.

velocity obtained from cross-correlation, manual time series analysis and

*Nf* is the inverse dimensionless viscosity.

Bond number, *Bo* is given as:

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

**0***:***01***Nf*

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

*Bo* <sup>¼</sup> ð Þ *<sup>ρ</sup><sup>L</sup>* � *<sup>ρ</sup><sup>G</sup> gD***<sup>2</sup>** *σ*

*m* ¼ 25 when *Nf* <18*:*

**<sup>1</sup>** � *<sup>e</sup>*�**0***:***0125***Eo*

� � � � � � *Um* <sup>þ</sup> ffiffiffiffiffiffi *Fr gD* <sup>p</sup> (32)

**<sup>1</sup>** <sup>þ</sup> *<sup>R</sup> B*

**<sup>1</sup>** <sup>þ</sup> *Eo b*

**<sup>1</sup>** <sup>þ</sup> *Eo f*

**<sup>1</sup>** <sup>þ</sup> *Eo j*

� �*<sup>C</sup>* � �*<sup>G</sup>* (33)

� �*<sup>c</sup>* h i*<sup>d</sup>* (34)

� �*<sup>g</sup>* h i*<sup>h</sup>* (35)

� �*<sup>k</sup>* � �*<sup>l</sup>* (36)

*G* ¼ *m=C* (37)

**<sup>0</sup>***:***<sup>345</sup>** � � **<sup>1</sup>** � *<sup>e</sup>*ð Þ **<sup>3</sup>***:***37**�*Bo <sup>=</sup><sup>m</sup>* h i (28)

*m* ¼ 10 when *Nf* > 250 (30)

� ��**0***:***<sup>35</sup>** when 18 <sup>&</sup>lt; *Nf* <sup>&</sup>lt;<sup>250</sup> (31)

(29)

2.As viscosity increases, Froude number decreases which indicates the dominant effect of gravitational force over inertia force. This also has a retarding effect on the rise velocity of Taylor bubbles. Hence, this shows that the combined effect of the viscous and gravitational force causes a decrease in the rise velocity of bubbles as viscosity increases. For the high viscous liquids, 1000 and 5000 mPa s, as superficial gas velocity increases, the Froude number tends to be the same. This can be seen at 0.242 and 0.361 m/s. This implies that the dominant effect of gravitational force over inertia force tends to be the same.

#### **4.2 Comparison of structure velocity computation methods**

The structure velocity obtained from ECT was compared with the manual estimate from the time series [34], high-speed camera and empirical models such as modified Viana et al. [43], modified De Cachard and Delhaye [44], model and Nicklin et al. [7] model. These are shown in **Figure 17**.

The modified form of De Cachard and Delhaye [44] model is given as:

$$\mathbf{U}\_{N} = \left\{ \mathbf{2.29} \left[ \mathbf{1} - \frac{\mathbf{20}}{E\_{\mathbf{o}}} (\mathbf{1} - e^{-0.0125 \mathbf{E}\_{\mathbf{o}}}) \right] \right\} \mathbf{U}\_{m} + \Gamma \left( \mathbf{gD} \right)^{1/2} \tag{27}$$

**Figure 17.** *Structure velocity comparison.*

*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*

where *Γ* is given as:

gas velocities, *Fr*< 1, this implies subcritical flow. The viscous effect makes the

2.As viscosity increases, Froude number decreases which indicates the dominant effect of gravitational force over inertia force. This also has a retarding effect on the rise velocity of Taylor bubbles. Hence, this shows that the combined effect of the viscous and gravitational force causes a decrease in the rise velocity of bubbles as viscosity increases. For the high viscous liquids, 1000 and 5000 mPa s, as superficial gas velocity increases, the Froude number tends to be the same. This can be seen at 0.242 and 0.361 m/s. This implies that the dominant effect of gravitational force over inertia force tends to be the same.

The structure velocity obtained from ECT was compared with the manual estimate from the time series [34], high-speed camera and empirical models such as modified Viana et al. [43], modified De Cachard and Delhaye [44], model and

**<sup>1</sup>** � *<sup>e</sup>*�**0***:***0125***Eo*

*Um* <sup>þ</sup> *<sup>Γ</sup>* ð Þ *gD* **<sup>1</sup>***=***<sup>2</sup>** (27)

The modified form of De Cachard and Delhaye [44] model is given as:

*Eo*

flow very slow, with low rise velocity of Taylor bubbles.

*Vortex Dynamics Theories and Applications*

**4.2 Comparison of structure velocity computation methods**

Nicklin et al. [7] model. These are shown in **Figure 17**.

*UN* <sup>¼</sup> **<sup>2</sup>***:***29 1** � **<sup>20</sup>**

**Figure 17.**

**216**

*Structure velocity comparison.*

$$T = \mathbf{0}.345 \left( \mathbf{1} - e^{-\frac{0.01 N\_f}{0.345}} \right) \left[ \mathbf{1} - e^{(3.37 - B\_\mathbf{e})/m} \right] \tag{28}$$

*Nf* is the inverse dimensionless viscosity. Bond number, *Bo* is given as:

$$\mathbf{B}\_{\theta} = \frac{(\rho\_L - \rho\_G)\mathbf{g}\mathbf{D}^2}{\sigma} \tag{29}$$

Condition for *m*:

$$\mathfrak{m} = 10 \text{ when } \mathbf{N}\_f > 250 \tag{30}$$

$$\mathfrak{m} = 69 \left( \mathbf{N}\_f \right)^{-0.35} \text{ when } 18 < \mathbf{N}\_f < 250 \tag{31}$$

*m* ¼ 25 when *Nf* <18*:*

The modified form of Viana et al. [43] model is given as:

$$U\_N = \left\{ \mathbf{2.29} \left[ \mathbf{1} - \frac{\mathbf{20}}{E\_o} \left( \mathbf{1} - e^{-0.0125 E\_o} \right) \right] \right\} U\_m + \, ^F\_\sqrt{g \mathbf{D}} \tag{32}$$

**w**here *Fr* is obtained from Eqs. (27) to (32).

$$Fr = L[\mathbf{R}; \mathbf{A}, \mathbf{B}, \mathbf{C}, \mathbf{G}] \equiv \frac{\mathbf{A}}{\left(\mathbf{1} + \left(\frac{\mathbf{B}}{\mathbf{B}}\right)^{\mathbf{C}}\right)^{\mathbf{G}}} \tag{33}$$

$$A = \mathcal{L}\left[E\_o; a, b, c, d\right] = \frac{a}{\left[\mathbf{1} + \left(\frac{E\_o}{b}\right)^c\right]^d} \tag{34}$$

$$B = L[\mathbf{E}\_o; \mathbf{e}, \mathbf{f}, \mathbf{g}, h] = \frac{\mathbf{e}}{\left[\mathbf{1} + \left(\frac{\mathbf{E}\_o}{\mathbf{f}}\right)^g\right]^h} \tag{35}$$

$$\mathbf{C} = \mathbf{L}[\mathbf{E}\_{\bullet}; i, j, k, l] = \frac{i}{\left[\mathbf{1} + \left(\frac{\mathbf{E}\_{\bullet}}{\bar{J}}\right)^{k}\right]^{l}} \tag{36}$$

$$\mathbf{G} = \mathfrak{m}/\mathbf{C} \tag{37}$$

and the parameters ð Þ *a*, *b*, ……… , *l* are:

*a* ¼ 0*:*34; *b* ¼ 14*:*793;*c* ¼ �3*:*06; *d* ¼ 0*:*58;*e* ¼ 31*:*08; *f* ¼ 29*:*868; *g* ¼ �1*:*96; *h* ¼ �0*:*49; *i* ¼ �1*:*45; *j* ¼ 24*:*867; *k* ¼ �9*:*93; *l* ¼ �0*:*094; *m* ¼ �1*:*0295.

On the average, a good agreement exists between the structure velocity from the ECT and that measured from the high-speed camera. A reasonably fair agreement exists between the former and that estimated from the time series. The modified Viana et al. [43], modified De Cachard and Delhaye [44], and Nicklin et al. [7] model gave roughly similar pattern with structure velocity from cross-correlation, manual time series analysis and high-speed camera. Modified Viana et al. [43] and modified De Cachard and Delhaye [44] models showed good agreement with the velocity obtained from cross-correlation, manual time series analysis and

high-speed camera at low superficial gas velocity for 1000 and 5000 mPa s, while Nicklin et al. [7] over predicts it.

**Dedication**

**Nomenclature**

*UN* rise velocity of Taylor bubble, m*=*s

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

*g* acceleration due to gravity, m*=*s2

*N <sup>f</sup>* inverse dimensionless viscosity, dimensionless

\*, Mukhtar Abdulkadir<sup>2</sup>

1 Department of Chemical and Process Engineering, University of Surrey,

3 Department of Petroleum Engineering, University of Zakho, Zakho City,

4 Department of Chemical and Environmental Engineering, University of

\*Address all correspondence to: ot.kajero@gmail.com

provided the original work is properly cited.

2 Department of Chemical Engineering, Federal University of Technology, Minna,

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

, Lokman Abdulkareem<sup>3</sup>

*μ<sup>L</sup>* liquid viscosity, kg*=*m*:*s *D* column diameter, m *σ<sup>L</sup>* surface tension Nð Þ *=*m ρ*<sup>L</sup>* liquid density, kg*=*m<sup>3</sup>

*Um* mixture velocity, m*=*s *USG* superficial gas velocity, m*=*s *USL* superficial liquid velocity, m*=*s *Re* Reynolds number, dimensionless *Eo* Eotovos number, dimensionless *Bo* Bond number, dimensionless *Fr* Froude's number, dimensionless *Mo* Morton number, dimensionless

*ρ<sup>G</sup>* gas density, kg*=*m<sup>3</sup>

*FD* drag force, N

Olumayowa T. Kajero<sup>1</sup>

and Barry James Azzopardi<sup>4</sup>

Guildford, United Kingdom

Nottingham, United Kingdom

**Author details**

Nigeria

**219**

Northern Iraq

*CD* drag coefficient, dimensionless

This publication is in loving memory of Late Prof. Barry Azzopardi.

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

The variations observed in the agreement could be due to the viscosities of the liquids used. Viana et al. [43] used silicone oil of viscosity range 1–3900 mPa s. De Cachard and Delhaye [44] and Nicklin et al. [7] used water. Also, from the structure velocity plots using cross-correlation, the distribution coefficient is in the range 1.07 to 1.6, while that of modified Viana et al. [43] and De Cachard and Delhaye [44] is approximately 2.25. A distribution coefficient of 2.0 was used for Nicklin et al. [7] as proposed for laminar flow.

The video technique of determining the rise velocity of bubbles gave errors of 4.3, 4.6, 7.3 and 11.5% for 5, 100, 1000 and 5000 mPa s, respectively when compared with cross-correlation technique.
