**3.1 Vapour bubble growth during rising due to the natural-convective action**

In nucleate pool boiling, Vapour bubble typically attains its maximum size at the moment of departure. After lift-off, vapour bubble, however, could continue to grow during its ascent in the presence of favourable condition – a layer of

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

superheated liquid which exists close to the heated surface and is metastable in nature [7]. Although the bubble growth rate in this region is not that significant with compared to the initial pre-departure growth, the bubble will grow for the convective action (convective boiling) with the presence of superheated liquid. Vapour bubble size, shape, and rise velocity for this superheated liquid region can significantly affect the heat and mass transfer mechanism involve. It is thus critical to understand and predict the behaviour and properties of rising and simultaneously growing vapour bubbles. In this section, the growth of a rising vapour bubble is numerically investigated in quiescent superheated water under the influence of buoyancy and surface tension forces with special emphasis given to heat and mass transfers due to the convective action.

#### *3.1.1 Numerical features*

Both the water and the vapour phases can be assumed to experience the same '*mixture velocity*' at any local point within the computational domain, and the twofluid system can be approximated as a one-fluid mixture. The mixture density and viscosity of each control volume can be calculated based on the volume fraction (*α*), which has the following values:

$$a = \begin{cases} 1 & \text{liquid phase} \\ 0 < a < 1 & \text{interface} \\ 0 & \text{gas phase} \end{cases} \tag{1}$$

The variable density and viscosity are then estimated using the *α* value:

$$
\rho = (1 - a)\rho\_{\mathcal{g}} + a\rho\_l \tag{2}
$$

$$
\mu = (\mathbf{1} - a)\mu\_{\mathbf{g}} + a\mu\_{l} \tag{3}
$$

Where: Subscripts *l* and *g* indicate liquid (water) and gas (vapour) phases.

When the mass transfer is considered, a source term needs to be added to the *α*transport equation:

$$\frac{\partial a}{\partial t} + \nabla.(aV) = \left(\frac{1}{\rho}\right) \mathbb{S}\_{\text{mass}} \tag{4}$$

Where: *Smass* is the interfacial mass transfer source term. Similarly, the continuity equation becomes:

$$\nabla.V = \left(\frac{1}{\rho\_{\mathcal{g}}} - \frac{1}{\rho\_l}\right) \mathbf{S}\_{mass} \tag{5}$$

The Momentum equation is:

$$\frac{\partial \rho V}{\partial t} + \nabla(\rho V.V) = -\nabla p + \rho \mathbf{g} + \nabla .\mu \left(\nabla V + \nabla V^T\right) + F\_\sigma \tag{6}$$

Where: *p*, *g* and *F<sup>σ</sup>* are the pressure, gravity and surface tension force respectively.

The Energy equation is:

$$\frac{\partial}{\partial t}(\rho \mathbf{E}) + \nabla(V(\rho \mathbf{E} + p)) = \nabla(k \nabla T) + \mathbf{S}\_{heat} \tag{7}$$

summing the volumes of individual triangular columns – refer to **Figure 5**. For the

**3. Simulation of rising vapour bubble in convective boiling conditions**

Here, two examples of rising vapour bubble simulations due to the convective boiling conditions are illustrated. Firstly, vapour bubble growth in superheated

**3.1 Vapour bubble growth during rising due to the natural-convective action**

In nucleate pool boiling, Vapour bubble typically attains its maximum size at the moment of departure. After lift-off, vapour bubble, however, could continue to grow during its ascent in the presence of favourable condition – a layer of

water; secondly, bubble condensation in sub-cooled boiling conditions.

surface translation and the remeshing process, see Ho et al. [4].

**using the ISM method**

**Figure 4.**

**Figure 5.** *VOF calculation.*

**124**

*ISM Interface points.*

*Heat Transfer - Design, Experimentation and Applications*

Where: *Sheat*, *E*,*T* and *k* are the interfacial heat transfer source term, energy, temperature and thermal conductivity respectively.

*Sheat* can simply be obtained by multiplying *Smass* by the change of enthalpy (*hfg*) for phase-change as [8]:

$$\mathcal{S}\_{heat} = \mathcal{S}\_{mass} \times h\_{\text{f\"g}} \tag{8}$$

Vapour bubble growth was simulated using the convective heat transfer mechanism and change in enthalpy (phase-change), and the source term can be presented as [9]:

$$S\_{mass} = a\_{\circ}.\frac{h\_{\circ} \Delta T\_{super}}{h\_{\text{fg}}}\tag{9}$$

Where *Cz* is the location of bubble centre in upward, z-direction.

*Db* ¼

**Reference Correlation Proposed (***Nuevap* **=** *Nu***) Valid For** Ranz and Marshall [11] *Nu* <sup>¼</sup> <sup>2</sup> <sup>þ</sup> <sup>0</sup>*:*6*Re*<sup>1</sup>*=*<sup>2</sup>*Pr*<sup>1</sup>*=*<sup>3</sup> <sup>0</sup> <sup>≤</sup> *Re <sup>&</sup>lt;* <sup>200</sup>

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

Hughmark [13] *Nu* <sup>¼</sup> <sup>2</sup> <sup>þ</sup> <sup>0</sup>*:*6*Re*<sup>0</sup>*:*<sup>5</sup>*Pr*<sup>0</sup>*:*<sup>33</sup> <sup>0</sup> <sup>≤</sup> *Re* <sup>&</sup>lt; 776.06

Akiyama [14] *Nu* <sup>¼</sup> <sup>0</sup>*:*37*Re*<sup>0</sup>*:*<sup>6</sup>*Pr*<sup>1</sup>*=*<sup>3</sup> Laminar Flow McAdams [15] *Nu* <sup>¼</sup> <sup>0</sup>*:*37*Re*<sup>0</sup>*:*<sup>6</sup> <sup>17</sup> <sup>&</sup>lt; *Re* <sup>&</sup>lt; 70,000

First four variables are for water properties at saturation temperature and are constant (for isothermal condition). However, the last two variables are for the vapour bubble and will change continuously for added mass onto the bubble and corresponding varying rise velocity. As such *hif* needs to be calculated at each timestep for varying bubble diameter and velocity. As a result, values for the interfacial mass transfer source term (*Smass*) will also change in each time step. This demonstrates, even for the isothermal condition, the complex physics behind a growing

Coupled with an in-house variable-density and variable-viscosity single-fluid flow solver, the ISM interface tracking method was employed to simulate single vapour bubble growth (test sizes 2.5 mm, 3 mm, 4 mm) in quiescent water under the influence of gravity and surface tension forces. Detail descriptions of all the numerical features are not the scope of this chapter. **Table 2**, however, shows the salient features used during the numerical simulation. Interested readers could get

Simulations were carried out in a computational domain of 31 � 51 � 31 *Cubic Control Volumes* (CCV) with an initial spherical bubble of radius 5 *h* (where *h* is the width of the non-dimensional cubic control volume) – see **Figure 6**. Other mesh sizes, such as 21 x 31 x 21 and 41 x 61 x 41, were also investigated, but the mesh size of 31 x 51 x 31 is maintained the same as the previous successful ISM application of Ho et al. [5] to minimise numerical error and optimise computational time. See Ho et al. [4, 5] for ISM fidelity and sensitivity testing. The centre of the bubble was located in line with the centre of the cavity, at a distance of 15.5 *h* from each side wall and at a distance of 15.5 *h* from the bottom boundary. All thermos-physical properties were taken at the saturation temperature of 100 °C. To check the effect of liquid superheat on the bubble growth, a wide variety of liquid superheat temperatures, *ΔTsuper* (1 °C, 15 °C, and 35 °C) were considered during the testing. Variable time steps (in the range

ffiffiffiffiffiffiffiffi 6*Vb π* 3 r

*hif* ¼ *f ρl*, *μl*, *kl* ð Þ , *Pr*, *Ub*, *Db* (15)

*μs*

*Nu* <sup>¼</sup> <sup>2</sup> <sup>þ</sup> <sup>0</sup>*:*27*Re*<sup>0</sup>*:*<sup>62</sup>*Pr*<sup>0</sup>*:*<sup>33</sup> 776.06 <sup>≤</sup> *Re*

� �<sup>1</sup>*=*<sup>4</sup> 3.5 ≤ *Re* ≤ 7.6 x 10<sup>4</sup>

0.71 ≤ *Pr* ≤ 380 1.0 ≤ (*μ/μs*) ≤ 3.2

0 ≤ *Pr <* 250

0 ≤ *Pr <* 200

(14)

Sphere-equivalent Bubble Diameter (*Db*) is calculated as:

Whitakar [12] *Nu* <sup>¼</sup> <sup>2</sup> <sup>þ</sup> <sup>0</sup>*:*4*Re*<sup>1</sup>*=*<sup>2</sup> <sup>þ</sup> <sup>0</sup>*:*06*Re*<sup>2</sup>*=*<sup>3</sup> � �*Pr*<sup>0</sup>*:*<sup>4</sup> *<sup>μ</sup>*

In-a-nut-shell *hif* depends on the variables below:

*Evaporative Nusselt number correlations [10].*

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

further information from the relevant references.

vapour bubble.

**127**

**Table 1.**

Where: *aif* is the interfacial bubble surface area per unit volume; *hif* is the convective heat transfer coefficient; Δ*Tsuper* is the liquid superheat. Using the ISM method's capability of calculating interfacial surface area more precisely (than the conventional VOF models), local bubble interfacial surface areas (cell-wise) were used to evaluate the total mass transfer onto the growing bubble.

Thus, Eq. (9) can be written as:

$$\mathcal{S}\_{\text{mass}} = \sum\_{n} \mathcal{S}\_{\text{mass\\_cell}(n)} = \sum\_{n} \frac{h\_{\text{if}} \times a\_{b\_{\text{all}(n)}} \times \Delta T\_{\text{super}}}{h\_{\text{fg}}} \tag{10}$$

Where: *Smass*\_*cell n*ð Þ is the 3D spatial interfacial cell-by-cell mass transfer rate onto the growing bubble. *abcell n*ð Þ is the local bubble surface (interfacial) area at the interface cell and can be obtained from the ISM simulation (see **Figures 4** and **5**). Δ*Tsuper* is the liquid superheat. *hif* can be calculated by the following evaporation correlation:

$$Nu\_{evap} = \frac{h\_{if}D\_b}{k\_l} \longrightarrow h\_{if} = \frac{k\_l}{D\_b} \times Nu\_{evap} \tag{11}$$

Evaporative Nusselt number (*Nuevap*) can be calculated from the correlations. *Nuevap* depends on the mechanism of fluid flow, the properties of the fluid, and the geometry. Numerous heat transfer correlations have been proposed for convective heat and mass transfer from the sphere for specific applications and conditions. To identify an appropriate *Nuevap*, selective and widely acceptable correlations were considered (see **Table 1**), and were plotted against the Bubble Reynolds number (*Reb*) [10]. It was found, for small *Reb*, there is not much difference among the correlations. However, for higher *Reb*, a significant discrepancy exists among the correlations. Hughmark [13] is not only chosen for its broad-application and popularity, but also for providing median range values (not too high or low) for higher Reynolds number. Bubble Reynolds number (*Reb*) in the correlation can be expressed as:

$$Re\_b = \frac{\rho\_l U\_b D\_b}{\mu\_l} \tag{12}$$

Where Bubble rise velocity (*Ub*) can be calculated as:

$$U\_b = \frac{\mathbf{C}\_x^{n+1} - \mathbf{C}\_x^n}{\Delta t} \tag{13}$$

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


#### **Table 1.**

Where: *Sheat*, *E*,*T* and *k* are the interfacial heat transfer source term, energy,

Vapour bubble growth was simulated using the convective heat transfer mechanism and change in enthalpy (phase-change), and the source term can be

*Smass* ¼ *aif :*

used to evaluate the total mass transfer onto the growing bubble.

*Nuevap* <sup>¼</sup> *hifDb*

*kl*

Bubble Reynolds number (*Reb*) in the correlation can be expressed as:

Where Bubble rise velocity (*Ub*) can be calculated as:

*Smass*\_*cell n*ð Þ <sup>¼</sup> <sup>X</sup>

the growing bubble. *abcell n*ð Þ is the local bubble surface (interfacial) area at the interface cell and can be obtained from the ISM simulation (see **Figures 4** and **5**). Δ*Tsuper* is the liquid superheat. *hif* can be calculated by the following evaporation

*n*

Where: *Smass*\_*cell n*ð Þ is the 3D spatial interfacial cell-by-cell mass transfer rate onto

! *hif* <sup>¼</sup> *kl*

Evaporative Nusselt number (*Nuevap*) can be calculated from the correlations. *Nuevap* depends on the mechanism of fluid flow, the properties of the fluid, and the geometry. Numerous heat transfer correlations have been proposed for convective heat and mass transfer from the sphere for specific applications and conditions. To identify an appropriate *Nuevap*, selective and widely acceptable correlations were considered (see **Table 1**), and were plotted against the Bubble Reynolds number (*Reb*) [10]. It was found, for small *Reb*, there is not much difference among the correlations. However, for higher *Reb*, a significant discrepancy exists among the correlations. Hughmark [13] is not only chosen for its broad-application and popularity, but also for providing median range values (not too high or low) for higher Reynolds number.

> *Re <sup>b</sup>* <sup>¼</sup> *<sup>ρ</sup>lUbDb μl*

*Ub* <sup>¼</sup> *<sup>C</sup><sup>n</sup>*þ<sup>1</sup> *<sup>z</sup>* � *<sup>C</sup><sup>n</sup>*

*z*

<sup>Δ</sup>*<sup>t</sup>* (13)

*Db*

Thus, Eq. (9) can be written as:

*Smass* <sup>¼</sup> <sup>X</sup> *n*

Where: *aif* is the interfacial bubble surface area per unit volume; *hif* is the convective heat transfer coefficient; Δ*Tsuper* is the liquid superheat. Using the ISM method's capability of calculating interfacial surface area more precisely (than the conventional VOF models), local bubble interfacial surface areas (cell-wise) were

*Sheat* can simply be obtained by multiplying *Smass* by the change of enthalpy (*hfg*)

*hifΔTsuper hfg*

*Sheat* ¼ *Smass* � *hfg* (8)

*hif* � *abcell n*ð Þ � Δ*Tsuper hfg*

� *Nuevap* (11)

(9)

(10)

(12)

temperature and thermal conductivity respectively.

*Heat Transfer - Design, Experimentation and Applications*

for phase-change as [8]:

presented as [9]:

correlation:

**126**

*Evaporative Nusselt number correlations [10].*

Where *Cz* is the location of bubble centre in upward, z-direction. Sphere-equivalent Bubble Diameter (*Db*) is calculated as:

$$D\_b = \sqrt[3]{\frac{6V\_b}{\pi}}\tag{14}$$

In-a-nut-shell *hif* depends on the variables below:

$$h\_{\circ f} = f(\rho\_l, \mu\_l, k\_l, Pr, U\_b, D\_b) \tag{15}$$

First four variables are for water properties at saturation temperature and are constant (for isothermal condition). However, the last two variables are for the vapour bubble and will change continuously for added mass onto the bubble and corresponding varying rise velocity. As such *hif* needs to be calculated at each timestep for varying bubble diameter and velocity. As a result, values for the interfacial mass transfer source term (*Smass*) will also change in each time step. This demonstrates, even for the isothermal condition, the complex physics behind a growing vapour bubble.

Coupled with an in-house variable-density and variable-viscosity single-fluid flow solver, the ISM interface tracking method was employed to simulate single vapour bubble growth (test sizes 2.5 mm, 3 mm, 4 mm) in quiescent water under the influence of gravity and surface tension forces. Detail descriptions of all the numerical features are not the scope of this chapter. **Table 2**, however, shows the salient features used during the numerical simulation. Interested readers could get further information from the relevant references.

Simulations were carried out in a computational domain of 31 � 51 � 31 *Cubic Control Volumes* (CCV) with an initial spherical bubble of radius 5 *h* (where *h* is the width of the non-dimensional cubic control volume) – see **Figure 6**. Other mesh sizes, such as 21 x 31 x 21 and 41 x 61 x 41, were also investigated, but the mesh size of 31 x 51 x 31 is maintained the same as the previous successful ISM application of Ho et al. [5] to minimise numerical error and optimise computational time. See Ho et al. [4, 5] for ISM fidelity and sensitivity testing. The centre of the bubble was located in line with the centre of the cavity, at a distance of 15.5 *h* from each side wall and at a distance of 15.5 *h* from the bottom boundary. All thermos-physical properties were taken at the saturation temperature of 100 °C. To check the effect of liquid superheat on the bubble growth, a wide variety of liquid superheat temperatures, *ΔTsuper* (1 °C, 15 °C, and 35 °C) were considered during the testing. Variable time steps (in the range


**Table 2.**

*Numerical features.*

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

> *Vb Vb*<sup>0</sup>

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

¼ 1 þ

2*hif*Δ*Tsupert <sup>ρ</sup>lDb*0*hfg* !<sup>3</sup>

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

(16)

analytically as [10]:

**Figure 7.**

**Figure 8.**

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

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

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

*3.1.2.2 Shape*

**129**

#### **Figure 6.** *Schematic diagram of test setup (not-to-scale).*

of 1 <sup>10</sup><sup>4</sup> to 1 <sup>10</sup><sup>5</sup> ) 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 with 2.2 GHz quad-core processor and 16 GB RAM.
