Transfer is aided by increased interfacial area. Interfacial area per unit volume aD of a single droplet or bubble is inversely proportional to the diameter of the droplet or bubble D.
To estimate the total interfacial area in a given volume, the ad value is multiplied by the fractional holdup of dispersed phase in the total volume.
where a = interfacial area/volume and Od = fraction of volume in dispersed phase = holdup. Fractional holdup in a continuous process depends on the velocities of the two phases, as if they were flowing by themselves.
Od = (dispersed phase volume)/(volume of dispersed and continuous phases)
Example 14: Interfacial Area for Droplets/Gas in Cocurrent Flow For equal mass flow of gas and liquid and with gas density 0.001 of liquid density, the gas velocity in cocurrent flow will be 1000 times the liquid velocity. This sets ®D.
®D = 1/(1 + 1000) = 0.00099 If the droplets are 500 ^m in diameter, Eqs. (14-177) and (14-178) give a = (6/0.0005)(0.00099) = 12 m2/m3 If the droplets are 100 ^m in diameter, Eqs. (14-177) and (14-178) give a = (6/0.0001)(0.00099) = 60 m2/m3
Example 15: Interfacial Area for Droplets Falling in a Vessel
Droplet systems rarely exceed a ®D value of 0.01. At this low level, ®D in a low-velocity countercurrent contactor can be approximated by Eq. (14-179).
where UL = liquid superficial velocity Ut = terminal velocity of droplet UG = gas superficial velocity
With a gas superficial velocity of 1.5 m/s, for equal mass flow of gas and liquid, with gas density 0.001 of liquid density, and with 500-^m-diameter droplets falling at a terminal settling of 2.5 m/s, Eq. (14-179) gives a fractional holdup of liquid of
®d = (0.001)1.5/(2.5 - 1.5) = 0.0015 Equations (14-177) and (14-178) then give a = (6/0.0005)(0.0015) = 18 m2/m3
Example 16: Interfacial Area for Bubbles Rising in a Vessel
For bubble systems (gases dispersed in liquids) fractional holdup can approach 0.5 as shown by Fig. 14-104. However, before reaching this holdup, the bubble systems shift to an unstable mix of bubbles and vapor jets. Hence an exact comparison to Example 14 isn't possible because at the 1.5 m/s velocity of Example 14, the system becomes a froth. But at about one-fifth the velocity of Example 14, an estimate of interfacial area is possible.
If the bubble size is 10,000 ^m and fractional holdup is 0.4, Eqs. (14-177) and (14-178) give an interfacial area of a = (6/0.01)(0.4) = 240 m2/m3
Measured interfacial area in distillation trays is consistent with this high value.
Note the much higher interfacial area than in the droplet systems of Examples 14 and 15. The higher interfacial area when the gas is dispersed explains why bubbling and froth systems often give better performance than droplet systems. The big difference in interfacial area stems from the much larger volume per unit of mass of gas, i.e., lower density of the gas than the liquid.
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