**3.2.5 Heat transfer enhancement and vortices**

In Figs. 15(a) and 15(b) the instantaneous temperature field, *<sup>W</sup>* , and the cross-sections of vortices are shown in a horizontal (*y*-*z*) plane for the bubbly flow (Case B1) and the droplet flow (Case D1). Those for the single-phase flow are also drawn in Fig. 15(c) for comparison. Notice that the temperature field is normalized by the friction temperature of each case. Red and blue represent the regions of 0 *<sup>W</sup>* and 30 *<sup>W</sup>* , respectively. The cross-sections of vortices are represented by the contour lines of the second invariant of velocity gradient tensor of 0.0125 *Q* .

Fig. 15. Temperature distribution in *y*-*z* planes for (a) the bubbly flow (Case B1), (b) the droplet flow (Case D1) and (c) single-phase flow.

In the single-phase turbulent flow, the vortices are located only in the low-temperature regions away from the walls. In the bubbly and droplet turbulent flows, on the other hand, they are also located in the near-wall regions where the temperature gradient is relatively high. It is obvious that the vortices near the walls play a major role in the heat-transfer enhancement.

It is interesting to know how the heat-transfer enhancement depends on the continuousphase Prandtl number. As is shown in Table 11, the heat-transfer enhancement is more noticeable at higher Prandtl numbers. Since the thermal boundary layer is thinner at a higher Prandtl number, the vortices near the walls, which are generated by the bubbles or droplets, more effectively enhance the heat transfer. The ratio *Nu Nu* (Pr 2) (Pr 1) is weakly dependent on the friction Reynolds number in single-phase turbulent flows. We performed a simulation for the single-phase turbulence at Re 160 and obtained the value of 1.33, which is considerably lower than 1.40 in the bubbly flow at Re 160 .


Table 11. The ratio of the Nusselt number at Pr 2 *<sup>c</sup>* to that at Pr 1 *<sup>c</sup>* . The ratio of the Nusselt number in Case B1 to that in Case B2 is shown for the bubbly flow.

### **3.3 Performance of heat transfer enhancement**

As was shown above, the Nusselt number is increased by the injection of the bubbles or droplets in the present simulations. This heat transfer enhancement is accompanied by the increase of the wall-friction, however. Reynolds analogy provides a useful concept for the evaluating the performance of heat transfer enhancement. Colburn (1933) stated that Reynolds analogy is described by the relation

$$\text{St}\,\text{Pr}\_c^{2/3} \equiv \text{j} = \text{c}\_f / \text{2} \,\text{.}\tag{26}$$

where

138 Computational Simulations and Applications

of vortices are shown in a horizontal (*y*-*z*) plane for the bubbly flow (Case B1) and the droplet flow (Case D1). Those for the single-phase flow are also drawn in Fig. 15(c) for comparison. Notice that the temperature field is normalized by the friction temperature of

respectively. The cross-sections of vortices are represented by the contour lines of the second

Fig. 15. Temperature distribution in *y*-*z* planes for (a) the bubbly flow (Case B1), (b) the

In the single-phase turbulent flow, the vortices are located only in the low-temperature regions away from the walls. In the bubbly and droplet turbulent flows, on the other hand, they are also located in the near-wall regions where the temperature gradient is relatively high. It is obvious that the vortices near the walls play a major role in the heat-transfer

It is interesting to know how the heat-transfer enhancement depends on the continuousphase Prandtl number. As is shown in Table 11, the heat-transfer enhancement is more noticeable at higher Prandtl numbers. Since the thermal boundary layer is thinner at a higher Prandtl number, the vortices near the walls, which are generated by the bubbles or droplets, more effectively enhance the heat transfer. The ratio *Nu Nu* (Pr 2) (Pr 1) is weakly dependent on the friction Reynolds number in single-phase turbulent flows. We

Case Single Phase Bubble Droplet

Table 11. The ratio of the Nusselt number at Pr 2 *<sup>c</sup>* to that at Pr 1 *<sup>c</sup>* . The ratio of the

As was shown above, the Nusselt number is increased by the injection of the bubbles or droplets in the present simulations. This heat transfer enhancement is accompanied by the

1.31 1.40 1.37

and obtained the value

.

performed a simulation for the single-phase turbulence at Re 160

(Pr 2) (Pr 1)

*Nu Nu*

**3.3 Performance of heat transfer enhancement** 

of 1.33, which is considerably lower than 1.40 in the bubbly flow at Re 160

Nusselt number in Case B1 to that in Case B2 is shown for the bubbly flow.

*<sup>W</sup>* , and the cross-sections

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

*<sup>W</sup>* ,

**3.2.5 Heat transfer enhancement and vortices** 

invariant of velocity gradient tensor of 0.0125 *Q* .

droplet flow (Case D1) and (c) single-phase flow.

enhancement.

In Figs. 15(a) and 15(b) the instantaneous temperature field,

each case. Red and blue represent the regions of 0

$$St = \frac{q\_{\rm IV}}{\rho\_c \mathbf{C}\_{\rm Pc} \, \mathbf{U}\_m (\Theta\_m - \Theta\_{\rm IV})} = \frac{\mathbf{Nu}}{\mathbf{Re}\_m \mathbf{Pr}\_c} \, \text{} \tag{27}$$

is the Stanton number, *j* denotes the j-factor, and

$$\mathcal{L}\_f = \frac{\tau\_W}{\frac{1}{2}\rho\_c \mathcal{U}\_m^2} \tag{28}$$

is the fricition coefficient. Eq.(26) holds for laminar and turbulent flow over flat plates and turbulent flow in smooth ducts. The equation 1/3 <sup>2</sup> 2 0.25 Pr Re Re 1 *j c Nu f cm* gives a relation between friction due to surface shear and heat transfer.

As is shown in Table 12, the injection of the bubbles or droplets leads to the reduction of 2 *<sup>f</sup> j c* . The forces resulting from the interfacial surface tension (and the buoyancy) significantly contribute to the increase of the wall shear stress in addition to the convection in turbulence. Heat transfer enhancement, on the other hand, is mainly caused by the increase in turbulent heat flux. Since the effects of the surface tension are more significant in the bubbly flow, the reduction is more noticeable in the bubbly flow than in the droplet flow.

The value of 2 *<sup>f</sup> j c* is larger for higher Prandtl numbers for all cases. The reduction of 2 *<sup>f</sup> j c* due to the injection of the bubbles or droplets is less significant for higher Prandtl numbers where the convection term plays more important roles.

The above results indicate that the performance of heat transfer is not so good in the bubbly and droplet turbulent flows. In the case of bubbly flows, however, the buoyancy force exerted on bubbles, *g* , may be used as a driving force for the upflow through the channel, where is the mean void fraction. When all of this buoyancy force can be used to reduce the extra driving force, the extra wall shear stress, which balances the extra driving force, is given by ' ( ) (1 / ) *W W g h <sup>W</sup> Bu* . The values in the rightmost column of Table 12 are obtained by replacing *<sup>W</sup>* in Eq.(28) by ' *<sup>W</sup>* . These values exceed 1, suggesting that the performance of heat transfer enhancement may be improved in the bubbly flow.


Table 12. The value of 2 *<sup>f</sup> j c* .The rightmost column corresponds to the case in which the buoyant effect of bubbles is considered.

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