**3.2 Vapour bubble condensation in sub-cooled boiling condition**

Vapour bubble condensation in subcooled liquid where the water temperature is below saturation is an important physical phenomenon [25]. Subcooled boiling flow is one of such examples and can be found in boilers, steam generators, nuclear reactors, and other engineering systems. To optimise the design and to make these systems safe, it is critical to understand and predict the behaviour of bubbles behaviour in subcooled boiling flows. As the presence of vapour bubbles has a significant effect on the heat transfer characteristics of a system as well as pressure drops and flow instability [8, 26]. Vapour bubble life and collapse during condensation can be either inertia (for high liquid subcooling and high Jacob number) or heat transfer controlled (for low liquid subcooling and low Jacob number). Using the ISM method, this section discusses the heat transfer controlled vapour bubble condensation in quiescent water where the bubble reduction rate is longer, and the process is controlled by the heat transfer at the interface. In order to simulate the condensing bubble, the source terms were modelled in the CFD governing equations to account for heat and mass transfers from the bubble. During the simulation, the predicted condensing vapour bubble properties such as size reduction rate,

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

shape and velocity were compared against the past works and found to be in good agreement.

## *3.2.1 Numerical features*

Likewise, bubble growth due to the convective action in the previous section, vapour bubble size reduction (i.e. condensation) was simulated using the same convective heat transfer mechanism due to the temperature gradient between vapour and water phase. Eq. (1)–(8) are also applicable here. *Smass*, however, needs to be treated in the opposite way for mass loss from the vapour bubble.

Thus,

Although small in magnitude, it is observed, with the increase of the liquid superheat, *Ustb* decreases for a same-sized bubble. This is due to the same reason: with the increase of added mass at a higher rate for higher liquid superheat (see Eq. (9)), the bubble deformed at a faster rate causing more drag force and reduced velocity. Sideman and Taitel's [24] also overserved similar phenomena of

Below summarised the major findings based on the numerical results obtained

• As the interfacial mass transfer source term (*Smass*) is directly proportional to the liquid superheat, the bubble was growing at higher rates for larger liquid

• The bubble deforms more with the increase of the size. The liquid superheat also has a direct effect on the bubble shape. For a same-sized bubble, with the increase of liquid superheat, the bubble deforms at a higher rate due to the

• 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). A growing bubble also shows similar trends. In the stable velocity regime, however, the growing bubble velocity (*Ustb*) is generally lower than the terminal velocity (*Uter*) of an adiabatic bubble. For the continuously added mass, the growing bubble shape deforms more rapidly than an adiabatic bubble causing higher

• With the increase of the liquid superheat, *Ustb* decreases for a same-sized bubble. This is due to the same reason: with the increase of added mass at a higher rate for higher liquid superheat, the bubble deformed at a faster rate

Vapour bubble condensation in subcooled liquid where the water temperature is below saturation is an important physical phenomenon [25]. Subcooled boiling flow is one of such examples and can be found in boilers, steam generators, nuclear reactors, and other engineering systems. To optimise the design and to make these systems safe, it is critical to understand and predict the behaviour of bubbles behaviour in subcooled boiling flows. As the presence of vapour bubbles has a significant effect on the heat transfer characteristics of a system as well as pressure drops and flow instability [8, 26]. Vapour bubble life and collapse during condensation can be either inertia (for high liquid subcooling and high Jacob number) or heat transfer controlled (for low liquid subcooling and low Jacob number). Using the ISM method, this section discusses the heat transfer controlled vapour bubble condensation in quiescent water where the bubble reduction rate is longer, and the process is controlled by the heat transfer at the interface. In order to simulate the condensing bubble, the source terms were modelled in the CFD governing equations to account for heat and mass transfers from the bubble. During the simulation, the predicted condensing vapour bubble properties such as size reduction rate,

reduced bubble velocity with the increase of temperature.

*Heat Transfer - Design, Experimentation and Applications*

*3.1.3 Conclusion*

using the ISM method:

superheats.

**132**

increased rate of mass transfer.

drag force and reduced velocity.

causing more drag force and reduced velocity.

**3.2 Vapour bubble condensation in sub-cooled boiling condition**

$$\mathcal{S}\_{\text{mass}} = \sum\_{n} \mathcal{S}\_{\text{mass}\_{\text{cell}(n)}} = \sum\_{n} \frac{h\_{\text{if}} \times a\_{b\_{\text{cell}(n)}} \times \Delta T\_{\text{sub}}}{h\_{\text{fg}}} \tag{18}$$

Where: *Smass*\_*cell n*ð Þ is the 3D spatial interfacial cell-by-cell mass transfer rate from the condensing bubble; *abcell n*ð Þ is the local bubble surface area at the interface cell and can be obtained from the ISM simulation (see **Figure 4**). Δ*Tsub* is the liquid subcooling. Interfacial (convective) heat transfer coefficient (*hif*) can be calculated by the following condensation correlation:

$$\text{Nu}\_{cond} = \frac{h\_{\text{if}} D\_b}{k\_l} \longrightarrow h\_{\text{if}} = \frac{k\_l}{D\_b} \times \text{Nu}\_{cond} \tag{19}$$

Eq. (12)–(14) can also be used here to determine Reynolds number, Bubble Velocity and Bubble Diameter. Next using this *Reb*, Condensate Nusselt number (*Nucond*) can be calculated from the correlations. **Table 5** shows some of the notable relations found in the literature. A preference was given to the relations having Jacob number (*Ja*) – a dimensionless number which is usually used for boiling, evaporation and condensation applications. To check the model's sensitiveness, Condensing bubble volume reduction with time for various correlations was investigated [31]. The correlations exhibit a wide discrepancy in volume reduction rates,


#### **Table 5.** *Condensate Nusselt number correlations.*

especially with the progress of simulation time. The reasons being these co-relations were developed for specific media and test setups and are valid for a range of parameters only. Warrier et al. [30], for instance, reported an uncertainty of �25% when they benchmarked their correlation with others.

Considering the scope of current work and relevance, and for validation purposes, Kim and Park [27] is applied during the numerical simulations. Jacob number (*Ja*) is defined as:

$$J\mathbf{a}\_l = \frac{\rho\_l \mathbf{C}\_{pl} \Delta T\_{sub}}{\rho\_\mathbf{g} h\_{\hat{\mathbf{g}}}} \tag{20}$$

calculate the correlated *β* values. The interfacial (convective) heat transfer coefficient (*hif*) is proportional to the bubble velocity, so higher the bubble velocity higher the bubble condensation rate. As the bubble was released from a stationary position (with zero velocity and Reynolds number), it took some time for the bubble to reach a reasonable rise velocity (similar to the terminal velocity regime of an adiabatic bubble) and corresponding Reynolds number. Numerical results show close agreement with past correlations at the later stage of condensation after the

*Bubble history comparison between correlations and ISM simulation [31]. [*Db0 *= 4 mm,* ΔTsub *= 10 °C,*

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

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

The ISM simulation results were also compared against past experimental results –

Generally, small bubble keeps its spherical shape during its rise for high surface tension forces. Bubble shape deformation accelerates with the increase of its size and becomes ellipsoidal or spherical cap or wobbling. Since 2 mm and 3 mm condensing bubbles quickly loss their mass and become small, their shape is generally limited to spherical or ellipsoidal. A larger bubble, e.g. 4 mm condensing bubble, on

see **Figure 11**. The ISM simulation results show good agreement, and the overall bubble condensation trends closely follow other benchmarked case of Kim and Park [27] (deviation in the range of 5–15%). The differences between the experimental and ISM numerical simulation are noticeable for different initial test conditions, bubble

bubble achieved relatively higher velocity values.

*Comparison of normalised bubble volume [31]. [*Db0 *= 4.9 mm*, ΔTsub *= 12 °C].*

shapes, and so on.

*3.2.2.2 Shape*

**Figure 11.**

**135**

**Figure 10.**

Re *= 500,* Ja *= 30].*

Where: *Cpl* is the liquid (water) specific heat.

For condensation, vapour bubble size and corresponding bubble rise velocity will change continuously. As such, *hif* needs to be calculated at each time-step for varying bubble size and velocity. As a result, values for the interfacial mass transfer source term (*Smass*) will also change in each time step.

Numerical test setups are the same as the previous application. For numerical test cases, 2 mm, 3 mm and 4 mm initial sized bubbles with various liquid subcooling were considered to check the effect of liquid subcooling on the bubble condensation rates. To validate ISM simulation results further, other odd bubble sizes (e.g. 1.008 mm and 4.9 mm) were also considered.

#### *3.2.2 Numerical results*

#### *3.2.2.1 Size*

The ISM simulation results were validated against past-correlations. In literature, Bubble history (*β*) has been expressed as the transient form consisting of the Fourier number (*Fo*), and is formulated as, for instance [25, 27]:

$$\beta = \left(\mathbf{1} - \mathbf{5.67} \,\mathrm{Re}\, ^{0.61}\_{b} a^{0.328} J a^{0.692}\_{l} \mathrm{Fo\_{0}}\right)^{0.72} \tag{21}$$

$$\beta = \left(1 - 0.6695 \, Re\_b^{0.7} \text{J} a\_l^{0.7957} Pr^{0.4564} Fo\_0\right)^{0.7692} \tag{22}$$

Where: Bubble history (*β*) which is defined as:

$$
\beta = \frac{D\_b}{D\_{b0}} \tag{23}
$$

Where: the subscript 0 indicates the initial state. Therefore, *Db0* is the initial bubble diameter, and *Db* is the instantaneous bubble diameter.

The Fourier number (*Fo*0) is based on the initial bubble size and is written as:

$$Fo\_0 = \frac{at}{D\_0^2} \tag{24}$$

Where: *a* is the thermal diffusivity.

Bubble history (*β*) obtained during the numerical simulation is plotted against past correlations in **Figure 10**. Depending on the bubble size, liquid subcooling and Reynolds number (*Re*), the bubble size reduced at varying rates. The bubble was condensing at a higher rate for larger liquid subcooling and higher Reynolds number. The ISM simulation, however, shows discrepancy at the beginning of the condensation stage. The reason being: fixed Reynolds numbers were used to

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

**Figure 10.**

especially with the progress of simulation time. The reasons being these co-relations were developed for specific media and test setups and are valid for a range of parameters only. Warrier et al. [30], for instance, reported an uncertainty of �25%

Considering the scope of current work and relevance, and for validation purposes, Kim and Park [27] is applied during the numerical simulations. Jacob number

> *Jal* <sup>¼</sup> *<sup>ρ</sup>lCpl*Δ*Tsub ρghfg*

For condensation, vapour bubble size and corresponding bubble rise velocity will change continuously. As such, *hif* needs to be calculated at each time-step for varying bubble size and velocity. As a result, values for the interfacial mass transfer

Numerical test setups are the same as the previous application. For numerical

subcooling were considered to check the effect of liquid subcooling on the bubble condensation rates. To validate ISM simulation results further, other odd bubble

The ISM simulation results were validated against past-correlations. In literature, Bubble history (*β*) has been expressed as the transient form consisting of the

*<sup>b</sup> α*<sup>0</sup>*:*<sup>328</sup>*Ja*<sup>0</sup>*:*<sup>692</sup>

*<sup>b</sup> Ja*<sup>0</sup>*:*<sup>7957</sup>

*<sup>β</sup>* <sup>¼</sup> *Db Db*<sup>0</sup>

Where: the subscript 0 indicates the initial state. Therefore, *Db0* is the initial

The Fourier number (*Fo*0) is based on the initial bubble size and is written as:

*Fo*<sup>0</sup> <sup>¼</sup> *at D*2 0

Bubble history (*β*) obtained during the numerical simulation is plotted against past correlations in **Figure 10**. Depending on the bubble size, liquid subcooling and Reynolds number (*Re*), the bubble size reduced at varying rates. The bubble was condensing at a higher rate for larger liquid subcooling and higher Reynolds number. The ISM simulation, however, shows discrepancy at the beginning of the condensation stage. The reason being: fixed Reynolds numbers were used to

*<sup>l</sup> Fo*<sup>0</sup> <sup>0</sup>*:*<sup>72</sup> (21)

*<sup>l</sup> Pr*<sup>0</sup>*:*<sup>4564</sup>*Fo*<sup>0</sup> <sup>0</sup>*:*<sup>7692</sup> (22)

test cases, 2 mm, 3 mm and 4 mm initial sized bubbles with various liquid

(20)

(23)

(24)

when they benchmarked their correlation with others.

*Heat Transfer - Design, Experimentation and Applications*

Where: *Cpl* is the liquid (water) specific heat.

source term (*Smass*) will also change in each time step.

sizes (e.g. 1.008 mm and 4.9 mm) were also considered.

Fourier number (*Fo*), and is formulated as, for instance [25, 27]:

*<sup>β</sup>* <sup>¼</sup> <sup>1</sup> � <sup>5</sup>*:*<sup>67</sup> *Re* <sup>0</sup>*:*<sup>61</sup>

*<sup>β</sup>* <sup>¼</sup> <sup>1</sup> � <sup>0</sup>*:*<sup>6695</sup> *Re* <sup>0</sup>*:*<sup>7</sup>

bubble diameter, and *Db* is the instantaneous bubble diameter.

Where: Bubble history (*β*) which is defined as:

Where: *a* is the thermal diffusivity.

(*Ja*) is defined as:

*3.2.2 Numerical results*

*3.2.2.1 Size*

**134**

*Bubble history comparison between correlations and ISM simulation [31]. [*Db0 *= 4 mm,* ΔTsub *= 10 °C,* Re *= 500,* Ja *= 30].*

calculate the correlated *β* values. The interfacial (convective) heat transfer coefficient (*hif*) is proportional to the bubble velocity, so higher the bubble velocity higher the bubble condensation rate. As the bubble was released from a stationary position (with zero velocity and Reynolds number), it took some time for the bubble to reach a reasonable rise velocity (similar to the terminal velocity regime of an adiabatic bubble) and corresponding Reynolds number. Numerical results show close agreement with past correlations at the later stage of condensation after the bubble achieved relatively higher velocity values.

The ISM simulation results were also compared against past experimental results – see **Figure 11**. The ISM simulation results show good agreement, and the overall bubble condensation trends closely follow other benchmarked case of Kim and Park [27] (deviation in the range of 5–15%). The differences between the experimental and ISM numerical simulation are noticeable for different initial test conditions, bubble shapes, and so on.

#### *3.2.2.2 Shape*

Generally, small bubble keeps its spherical shape during its rise for high surface tension forces. Bubble shape deformation accelerates with the increase of its size and becomes ellipsoidal or spherical cap or wobbling. Since 2 mm and 3 mm condensing bubbles quickly loss their mass and become small, their shape is generally limited to spherical or ellipsoidal. A larger bubble, e.g. 4 mm condensing bubble, on

**Figure 11.** *Comparison of normalised bubble volume [31]. [*Db0 *= 4.9 mm*, ΔTsub *= 12 °C].*

the other hand, is going through a series of interesting shape evolution, as such, it is considered to compare against Clift et al. [19] shape regimes. A series of instantaneous bubble status points were considered, and their corresponding *Re* were calculated using bubble velocity obtained from the ISM simulation for comparison. With the increase of bubble velocity and it corresponding higher *Re*, the bubble was deforming from spherical to ellipsoidal to wobbling shape regimes. **Figure 12** demonstrates the bubble shapes obtained from the ISM simulation have excellent agreement with Clift et al. [19] shape regimes.

Bubble shapes obtained in the ISM simulation were also compared with Kamei and Hirata's [32] experimentation and found to be a good agreement – see **Table 6**. The condensing bubble was keeping its spherical shape because of small size and high surface tension forces during its rise. **Table 6** also shows the comparison of shapes with past numerical works (Zeng et al. [33] and Samkhaniani and Ansari [34]).

#### *3.2.2.3 Velocity*

Likewise growing bubble, the rise velocity of condensing bubbles is different from adiabatic bubbles [8]. For continuous reduction in bubble size (i.e. volume) in subcooled boiling flow condition, bubble rise velocity and shape are also always changing. From **Figure 13**, it is evident that with the increase of liquid Subcooling bubble rise velocity continues to increase. The findings are consistent with [8] numerical results. The deviation is the result of different test setups; however, **Figure 13** overall indicates the trends of higher the liquid Subcooling higher the bubble rise velocity. Bubble buoyancy force decreases for continuous reduction of bubble volume. The drag force is also reduced for smaller bubble frontal area; however, the resulted net effect is positive buoyancy force acting upwards, and the bubble rise velocity increases continuously.

**Time** 

> **0.0**

> > **Experiment**

**137**

**(Kamei and Hirata [32])**

**Numerical**

**(Zeng et al. [33])**

**Numerical**

**(Samkhaniani**

**Numerical**

**(ISM** 

**Table 6.** *Bubble shape comparison*

 *between past* 

*experimental/numerical*

 *results and ISM simulation*

 *(*

Db0 *= 1.008 mm,*

Δ

*Tsub =* 25°C*) (adapted from [31]).*

**Simulation)**

 **and Ansari [34])**

**0.4**

**0.8**

**1.2**

**1.6**

**2.0**

**2.4**

**2.8**

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

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

**(millisecond)**

#### **Figure 12.**

*Bubble shape validation with Clift et al. [19]. Test case:* Db0 *= 4 mm,* ΔTsub *= 25 °C, Eötvös number,* Eo = *2.55 (adapted from [31]). All bubbles are on the same scale for comparison.*

**Table 6.** *Bubble shape comparison between past experimental/numerical results and ISM simulation (* Db0 *= 1.008 mm,* Δ *Tsub =* 25°C*) (adapted from [31]).*
