*3.1.2.1 Size*

Normalised bubble volumes of growing bubble over time are plotted in **Figure 7**. As the interfacial mass transfer source term (*Smass*) is directly proportional to the

*Numerical Investigation of Rising Vapour Bubble in Convective Boiling Using an… DOI: http://dx.doi.org/10.5772/intechopen.96303*

**Figure 7.** *Normalised bubble volume over time (*Db0 *= 3 mm) [10].*

**Figure 8.** *Comparison with theory (*Db0 *= 3 mm,* ΔTsuper *= 35 °C) (adapted from [10]).*

liquid superheat (See Eq. (9)), the bubble was growing at higher rates for larger liquid superheats. For all cases, bubble growth started to vary from 1–2 ms, as in addition to liquid superheat bubble velocity also begun to play a critical role in the bubble growth. Bubble volume growth ratios obtained during the numerical simulation are compared with the theory in **Figure 8** and found to be in good agreement. The deviation is less than 1% and is obvious, as fixed *hif* values are used in the entire theoretical calculations; on the other hand, *hif* values in the numerical simulations are always changing for varying bubble velocities. It is to be noted, normalised bubble volume of a growing bubble due to the convective action can be evaluated analytically as [10]:

$$\frac{V\_b}{V\_{b0}} = \left(\mathbf{1} + \frac{2h\_{\text{if}}\Delta T\_{super}t}{\rho\_l D\_{b0}h\_{\text{fg}}}\right)^{\beta} \tag{16}$$

#### *3.1.2.2 Shape*

Aspect Ratio (*AR*) is used to quantify the bubble shape, and is defined by the bubble height by width. *AR* value of 1.0 indicates the bubble is in a perfect spherical shape. Values less than 1.0 designate the bubble is an oblate spheroid. *AR* for

of 1 <sup>10</sup><sup>4</sup> to 1 <sup>10</sup><sup>5</sup>

*Schematic diagram of test setup (not-to-scale).*

Surface construction, Interface advection and remeshing

*Heat Transfer - Design, Experimentation and Applications*

Coupling between ISM interface tracking method and in-house variabledensity and variable-viscosity single-

Complier for ISM interface tracking algorithm and the flow solver program

fluid flow solver

**Table 2.** *Numerical features.*

*3.1.2 Numerical results*

*3.1.2.1 Size*

**128**

**Figure 6.**

with 2.2 GHz quad-core processor and 16 GB RAM.

) were also used and found to have insignificant effects on the

numerical results. In terms of the computational efforts by using the ISM method, it took 1–3 days to simulate the transient bubble growth process on a personal computer

**Description Method/Mechanism/Platform Reference**

Surface (interfacial) tension Continuum Surface Force (CSF) [17] 3D surface curvature Paraboloid Least Square fitting method —

CFD result Visualisation Techplot —

Pressure–velocity coupling Semi-Implicit Method for Pressure-Linked

Discretisation schemes Finite volume formulation - hybrid and central

ISM method (Hybrid Eulerian–Lagrangian) [4]

Equations (SIMPLE) algorithm

schemes

Intel Visual ForTran Composer XE 2011 —

Immersed Boundary Method (IBM) [16]

[18]

—

Normalised bubble volumes of growing bubble over time are plotted in **Figure 7**. As the interfacial mass transfer source term (*Smass*) is directly proportional to the

As *Eo* is fixed for the same-sized individual bubble, during its ascend the bubble

will go through different shape remiges based on its velocity or *Re* (see [19] for details). For, 2.5 mm, 3 mm and 4 mm bubbles, this transformation is from spherical-to-ellipsoidal-to-wobbling. It is to be noted that the boundary of these shape regimes is not strictly defined, and for the small bubble, such as 2.5 mm bubble could transform from spherical shape to the wobbling direct based on *Re*. At the later stage, numerical bubble shapes also reveal unstable features and disturbance on the interface. Larger bubble, 4 mm, in this case, became reverse-heart like

*Numerical Investigation of Rising Vapour Bubble in Convective Boiling Using an…*

When an adiabatic bubble is released from a stationary position, its velocity continues to rise until it reaches to its terminal velocity regime where the bubble continues to rise at a constant velocity, i.e. no acceleration [19–21]. Growing vapour bubble also showed similar velocity profile during its ascend during the numerical simulation [10]. Initially, the velocity rose exponentially after the release of the bubble from its stationary position. The bubble then entered into a relatively stable velocity regime. Generally, with the increase of the bubble size, the bubble reaches to its stable velocity regime more quickly for the larger buoyancy force resulted

The rise velocities (*Ustb*) of growing bubbles in the stable regime are

benchmarked against similar terminal velocity (*Uter*) regime of adiabatic bubbles in **Table 4**. Overall, with the increase of the bubble size, *Ustb* decreases, and the simulation results compared well with the high-low ranges reported by Clift et al. [19]. ISM's results follow the lower limit, as for the growing bubbles, *Ustb* is lower than a comparable adiabatic bubble's *Uter*. This is because the drag force increases at a larger rate due to the deformed frontal area (i.e. bubble shape) of a growing bubble for continually added mass which reduces the bubble rise velocity [22, 23]. Sideman and Taitel's [24] experiments also exhibited similar trends for evaporated

> **Numerically obtained average rise velocity of a growing bubble in stable regime,** *UStb* **[cm/s]**

2.5 1 20.28 28–17

3 1 20.98 26–17.5

4 1 17.12 25–18

15 19.98 35 19.60

15 20.12 35 19.70

15 16.98 35 16.94

**Terminal Velocity,** *Uter* **[cm/s] of Adiabatic Bubble Experimental (High - Low) [10]**

shape with disturbance on the bottom.

*DOI: http://dx.doi.org/10.5772/intechopen.96303*

*3.1.2.3 Velocity*

from larger bubble volume.

growing bubbles.

**Initial Bubble Diameter,** *D0* **[mm]**

**Table 4.**

**131**

**Liquid Superheat,** *ΔTSuper* **[°***C***]**

*Bubble velocity comparison (adapted from [10]).*

**Figure 9.**

*Comparison of aspect ratio,* AR *(*ΔTsuper *= 1 °C).*


#### **Table 3.**

*Bubble shape validation (cases with selective simulation time are chosen for demonstration). All bubbles are on the same scale for comparison.*

various bubbles are plotted in **Figure 9**. With the increase of initial bubble size, bubble deformed at a higher rate and became flattened/oblate spheroid. The liquid superheat also has direct effects on the bubble shape. For the same size bubble, with the increase of liquid superheat, the bubble deformed at a higher rate (i.e. became oblate) due to the increased rate of mass transfer.

Numerically obtained bubble shapes are also compared with the shape regimes of Clift et al. [19] and found to be in good agreement (see **Table 3**). Here, Eötvös number (*Eo*) is defined as

$$Eo = \frac{\left(\rho\_l - \rho\_\lg\right) \text{g}D\_b^2}{\sigma} \tag{17}$$

*Numerical Investigation of Rising Vapour Bubble in Convective Boiling Using an… DOI: http://dx.doi.org/10.5772/intechopen.96303*

As *Eo* is fixed for the same-sized individual bubble, during its ascend the bubble will go through different shape remiges based on its velocity or *Re* (see [19] for details). For, 2.5 mm, 3 mm and 4 mm bubbles, this transformation is from spherical-to-ellipsoidal-to-wobbling. It is to be noted that the boundary of these shape regimes is not strictly defined, and for the small bubble, such as 2.5 mm bubble could transform from spherical shape to the wobbling direct based on *Re*. At the later stage, numerical bubble shapes also reveal unstable features and disturbance on the interface. Larger bubble, 4 mm, in this case, became reverse-heart like shape with disturbance on the bottom.

#### *3.1.2.3 Velocity*

When an adiabatic bubble is released from a stationary position, its velocity continues to rise until it reaches to its terminal velocity regime where the bubble continues to rise at a constant velocity, i.e. no acceleration [19–21]. Growing vapour bubble also showed similar velocity profile during its ascend during the numerical simulation [10]. Initially, the velocity rose exponentially after the release of the bubble from its stationary position. The bubble then entered into a relatively stable velocity regime. Generally, with the increase of the bubble size, the bubble reaches to its stable velocity regime more quickly for the larger buoyancy force resulted from larger bubble volume.

The rise velocities (*Ustb*) of growing bubbles in the stable regime are benchmarked against similar terminal velocity (*Uter*) regime of adiabatic bubbles in **Table 4**. Overall, with the increase of the bubble size, *Ustb* decreases, and the simulation results compared well with the high-low ranges reported by Clift et al. [19]. ISM's results follow the lower limit, as for the growing bubbles, *Ustb* is lower than a comparable adiabatic bubble's *Uter*. This is because the drag force increases at a larger rate due to the deformed frontal area (i.e. bubble shape) of a growing bubble for continually added mass which reduces the bubble rise velocity [22, 23]. Sideman and Taitel's [24] experiments also exhibited similar trends for evaporated growing bubbles.


#### **Table 4.** *Bubble velocity comparison (adapted from [10]).*

various bubbles are plotted in **Figure 9**. With the increase of initial bubble size, bubble deformed at a higher rate and became flattened/oblate spheroid. The liquid superheat also has direct effects on the bubble shape. For the same size bubble, with the increase of liquid superheat, the bubble deformed at a higher rate (i.e. became

*Bubble shape validation (cases with selective simulation time are chosen for demonstration). All bubbles are on*

*Eo* ¼

Numerically obtained bubble shapes are also compared with the shape regimes of Clift et al. [19] and found to be in good agreement (see **Table 3**). Here, Eötvös

> *ρ<sup>l</sup>* � *ρ<sup>g</sup>*

*gD*<sup>2</sup> *b*

*<sup>σ</sup>* (17)

oblate) due to the increased rate of mass transfer.

number (*Eo*) is defined as

*the same scale for comparison.*

**Figure 9.**

**Table 3.**

**130**

*Comparison of aspect ratio,* AR *(*ΔTsuper *= 1 °C).*

A 2.5 mm bubble, *ΔTsuper* = 1 °C, Simulation time = 0.1 ms, *Re* = 10, *Eo* = 0.997

*Heat Transfer - Design, Experimentation and Applications*

B 2.5 mm bubble, *ΔTsuper* = 1 °C, Simulation time = 31.1 ms, *Re* = 1700, *Eo* = 0.997

C 3 mm bubble, *ΔTsuper* = 35 °C, Simulation time = 10 ms, *Re* = 1800, *Eo* = 1.43

D 3 mm bubble, *ΔTsuper* = 35 °C, Simulation time = 20 ms, *Re* = 2300, *Eo* = 1.43

E 4 mm bubble, *ΔTsuper* = 35 °C, Simulation time = 10 ms, *Re* = 4200, *Eo* = 2.55

**Point Bubble Characteristics Shape obtained in**

**Numerical Simulation**

**Clift et al. [19] Shape Regime**

Spherical

Ellipsoidal

Ellipsoidal

Ellipsoidal-Wobbling

Ellipsoidal-Wobbling
