**3.2.4 Effects of thermal properties of gas inside bubbles**

Figs. 14 show the distribution of the temperature variance, *<sup>W</sup>* , in the *x-y* plane for the three cases (Case B1, B3, B4) with different thermal properties for the gas inside the bubbles. Red and blue represent the regions of 0 *<sup>W</sup>* and 30 *<sup>W</sup>* , respectively. 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 convection dominates that of conduction inside the bubbles.

Fig. 14. Temperature distribution in an *x-y* plane for the bubbly flow. (a) Case B1, (b) Case B3, (c) Case B4.

Numerical Study on Flow Structures and

where

flow.

exerted on bubbles, *g*

of Table 12 are obtained by replacing

buoyant effect of bubbles is considered.

channel, where

force, is given by

bubbly flow.

Reynolds analogy is described by the relation

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

Heat Transfer Characteristics of Turbulent Bubbly Upflow in a Vertical Channel 139

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

*W*

*f*

turbulent flow in smooth ducts. The equation 1/3 <sup>2</sup> 2 0.25 Pr Re Re 1 *j c Nu f cm*

*c*

relation between friction due to surface shear and heat transfer.

numbers where the convection term plays more important roles.

' ( ) (1 / ) *W W*

 

 

*<sup>q</sup> Nu St*

, ( ) Re Pr

*c Pc m m W m c*

1 2 2

is the fricition coefficient. Eq.(26) holds for laminar and turbulent flow over flat plates and

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

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

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

reduce the extra driving force, the extra wall shear stress, which balances the extra driving

 

suggesting that the performance of heat transfer enhancement may be improved in the

 Bubble Droplet Single-phase Bubble Pr 1 *<sup>c</sup>* 0.69 0.84 0.92 1.38 Pr 2 *<sup>c</sup>* 0.77 0.92 0.96 1.54

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

*<sup>W</sup>* in Eq.(28) by

, may be used as a driving force for the upflow through the

*g h <sup>W</sup> Bu* . The values in the rightmost column

'

is the mean void fraction. When all of this buoyancy force can be used to

*W*

*c m*

*U* 

2/3 Pr 2 , *St j c c f* (26)

(28)

*<sup>W</sup>* . These values exceed 1,

gives a

*C U* (27)
