Bubble Flow

Figure 2 shows gas bubbles and liquid in upwards co-current flow. Consider the plane A-A, drawn so that it lies entirely in the liquid. If U_l is the mean upwards liquid velocity across AA, continuity requires that:

(1)

(Over a relatively shot vertical span, the pressure varies little and the gas bubbles have an essentially constant volume.) Thus, the gas bubbles just below A-A are rising relative to a liquid that is already moving at a velocity U_l, so that the velocity of the gas bubbles is:

(2)

in which U_b is the bubble velocity rising into a stagnant liquid. But the total volumetric flow rate of gas is:

(3)

So that the void fraction is given by:

or

(4)

This relation for the void fraction holds within certain limits for G and/or L negative - that is, for downflow of one or both phases.

Concerning the pressure gradient in the upwards vertical direction, note that the density of the liquid, which occupies a fraction (1-e) of the total volume, is much greater than that of the gas. Also, for the relatively low liquid velocities likely to be encountered in the bubble-flow regime, friction is negligible. Therefore, the pressure gradient is given fairy accurately by considering only the hydrostatic effect:

(5)

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