**3.2.2 Mean temperature profiles**

The profiles of the mean temperature variance, *<sup>W</sup>* , are drawn in Figs. 11. The temperature variance is decreased in the whole region of the channel for the droplet flow. In the case of the bubbly flow, the temperature difference is decreased except in the core region of the channel. This increase in the core region indicates that the enhancement of fluid mixing due to the bubbles is rather confined to the near wall regions. The difference between the mean fluid temperature and the wall temperature is smaller in the multiphase flows than in the single-phase flow, which means that the increase of the Nusselt number exceeds that of the friction Reynolds number (see Eq.(17)).

Fig. 11. The mean temperature profiles for the single-phase flow, Case B1 and Case D1.

Numerical Study on Flow Structures and

conductivity is increased due to the bubbles (or droplets).

**3.2.4 Effects of thermal properties of gas inside bubbles**  Figs. 14 show the distribution of the temperature variance,

convection dominates that of conduction inside the bubbles.

Red and blue represent the regions of 0

increase of the turbulent heat flux near the walls.

transfer is satisfied.

B3, (c) Case B4.

Heat Transfer Characteristics of Turbulent Bubbly Upflow in a Vertical Channel 137

The profiles of these three terms are drawn in Figs. 12 for Case B1 and Case D1. The sum of the molecular and turbulent fluxes is also plotted in the figures. In both cases, the sum agrees well with the left-hand side of Eq.(25), which indicates that the overall balance of heat

Fig. 13(a) shows the profiles of the left-hand side of Eq.(25) for the single-phase flow, Case B1 and Case B2. Clear differences are not seen among three cases, indicating that the change in the profile of heat-capacity flow rate due to the bubbles or droplets has an insignificant effect on the enhancement of heat transfer. The profiles of the molecular heat flux are shown in Fig.13(b). The molecular heat flux in the bubbly (or droplet) flow is reduced near the walls compared with the case of the single-phase flow. This indicates that the effective heat

Fig. 13(c) shows the profiles of turbulent heat flux. The turbulent heat flux is increased by the effects of bubbles (or droplets) in the regions near the walls. As is shown below, this is caused by the vortices whose generation is activated by the bubbles (or droplets). This increase in the turbulent heat flux in the near-wall regions is responsible for the increase in the effective heat conductivity of the fluid near the wall, which enhances the heat transfer. In summary, the enhancement of heat transfer in the bubbly or droplet flow is caused by the

three cases (Case B1, B3, B4) with different thermal properties for the gas inside the bubbles.

Contour lines represent the cross-sections of the bubbles. The temperature is high near the walls and low in the center of the channel. The flow is going upward. It is found in Figs. 14 that the temperature field is almost uniform inside the droplet. This is due to the circulating flow inside the bubble (figures not shown). The change of the temperature distribution is small if the thermal conductivity of the gas is reduced. This is because the effect of

Fig. 14. Temperature distribution in an *x-y* plane for the bubbly flow. (a) Case B1, (b) Case

*<sup>W</sup>* and 30

*<sup>W</sup>* , in the *x-y* plane for the

*<sup>W</sup>* , respectively.

#### **3.2.3 Heat flux profiles**

Now, we examine the mechanism for the enhancement of heat transfer by considering the energy balance in the channel. The averaging of the energy equation, Eq.(8), over time and the *x* and *z* directions, and the integration of the averaged equation with respect to *y* yield

$$g\_W - G \int\_0^y \overline{\rho \mathbf{C}\_P \mu} \left( y' \right) dy' = k \frac{\overline{\partial \Theta}}{\partial y} (y) - \overline{\rho \mathbf{C}\_P \Theta} \mathbf{v} \left( y \right). \tag{25}$$

The left-hand side of Eq.(25) represents the total heat flux in the wall-normal direction, and the first and the second terms on the right-hand side represent the molecular heat flux and the turbulent heat flux, respectively. In the figures below, each term is normalized by the wall heat flux of each case.

Fig. 12. Heat flux profiles for (a) Case B1 and (b) Case D1.

Fig. 13. The profiles of (a) the left-hand side of Eq.(25), (b) the molecular heat flux and (c) turbulent heat flux.

Now, we examine the mechanism for the enhancement of heat transfer by considering the energy balance in the channel. The averaging of the energy equation, Eq.(8), over time and the *x* and *z* directions, and the integration of the averaged equation with respect to *y* yield

*W P <sup>P</sup> q G C u y dy k y C y*

The left-hand side of Eq.(25) represents the total heat flux in the wall-normal direction, and the first and the second terms on the right-hand side represent the molecular heat flux and the turbulent heat flux, respectively. In the figures below, each term is normalized by the

Heat flux profiles

Molecular heat flux

*y/*

Fig. 13. The profiles of (a) the left-hand side of Eq.(25), (b) the molecular heat flux and (c)

012


0

1

 LHS of Eq.(25) Molecular Turbulent RHS of Eq.(25)

> Single-phase Bubbly flow Droplet flow

*y/*

*y/*

Turbulent heat flux

(c)


0

1

012

Fig. 12. Heat flux profiles for (a) Case B1 and (b) Case D1.

012

<sup>0</sup> ( ') ' ( ) v ( ). *<sup>y</sup>*

*y*

 

*y/*

*y/*

 Single-phase Bubbly flow Droplet flow

(b)


0

1

(b)

012

012

 LHS of Eq.(25) Molecular Turbulent RHS of Eq.(25)

 Single-phase Bubbly flow Droplet flow

(25)

**3.2.3 Heat flux profiles** 

wall heat flux of each case.

(a)

Heat flux profiles


LHS of Eq.(25)


turbulent heat flux.

0

1

(a)

0

1

The profiles of these three terms are drawn in Figs. 12 for Case B1 and Case D1. The sum of the molecular and turbulent fluxes is also plotted in the figures. In both cases, the sum agrees well with the left-hand side of Eq.(25), which indicates that the overall balance of heat transfer is satisfied.

Fig. 13(a) shows the profiles of the left-hand side of Eq.(25) for the single-phase flow, Case B1 and Case B2. Clear differences are not seen among three cases, indicating that the change in the profile of heat-capacity flow rate due to the bubbles or droplets has an insignificant effect on the enhancement of heat transfer. The profiles of the molecular heat flux are shown in Fig.13(b). The molecular heat flux in the bubbly (or droplet) flow is reduced near the walls compared with the case of the single-phase flow. This indicates that the effective heat conductivity is increased due to the bubbles (or droplets).

Fig. 13(c) shows the profiles of turbulent heat flux. The turbulent heat flux is increased by the effects of bubbles (or droplets) in the regions near the walls. As is shown below, this is caused by the vortices whose generation is activated by the bubbles (or droplets). This increase in the turbulent heat flux in the near-wall regions is responsible for the increase in the effective heat conductivity of the fluid near the wall, which enhances the heat transfer. In summary, the enhancement of heat transfer in the bubbly or droplet flow is caused by the increase of the turbulent heat flux near the walls.
