Question
Consider again Question 2 at the end of Section 11.1.
- Can you make a rough estimate of the proportion of the liquid used in the
experiment in which the energy is dissipated at the highest power density level, i.e.,
the level that yields the small bubbles? You may assume thatg¼ 0 :05 N?m^1. - Assuming that it is possible to let the beaters turn at twice the original
revolution rate, what would then be the bubble size obtained? What conditions then
need to be fulfilled?
Answer
- The overall power density in the liquid was 50 W/200 ml, or
25? 104 W?m^3 (the turbulent energy is virtually confined to the aqueous phase,
hence 200 ml). Sincedobtained is knownð 2? 10 ^4 mÞ, we can use Eq. (11.11) to
calculate the effective power density. This gives 2? 10 ^4 ¼ 0 : 053 =^5 ?e^2 =^5? 1000 ^1 =^5 ,
which yieldse¼ 62? 104 kW?m^3. Assuming all of the energy to be dissipated at the
highest rate, this then would occur in 25/62 or 40%of the volume. It is, however,
unlikely that the power density distribution would be so uneven. Assuming that in
the rest of the liquid the average value ofeamounts to 20%of the peak value, the
maximum value ofewould hold in about 25%of the volume. - It is remarked above that for a revolving stirrer,eis proportional ton^3 ;n
being the revolution rate. According to Eq. (11.11),d!e^0 :^4 !n^1 :^2 , which then
would lead tod¼200/21.2¼ 87 mm. Conditions are that surfactant is present in
excess (since the specific surface area, and hence the amount of adsorbed surfactant,
would be larger by a factor 2.3) and that the given time of beating (2 min) would be
long enough to obtain a steady state.
11.4 ROLE OF SURFACTANT
It is often assumed that the role of a surfactant in emulsification is the
lowering ofg, thereby facilitating the breakup of drops. Although this is true
enough, it does not explain the prime role of the surfactant. Assume that we
try to make emulsions (a) of paraffin oil and water, and (b) of triglyceride oil
and water. The interfacial tensions O–W are 50 and 30 mN?m^1 ,
respectively (see Table 10.1). To equalize the conditions, we add some
surfactant to sample (a) to obtain also hereg¼30 mN?m^1. However, the
result will be that the agitation of sample (a) produces an emulsion, whereas
this fails for sample (b), despite the equality ofg. The surfactant thus plays
at least one other role. This is theformation ofg-gradients, as discussed in
Sections 11.2.2 and 3 for foam making. The same applies in emulsification.