Estrategias de enseñanza-aprendizaje que implican el

3.2. A m odel sy stem for foam fra c tio n a tio n 43

p ro v em en ts to o u tp u t an d m ay reduce w aste of m a te ria ls and energy (a pressing concern in m o d e rn tim es).

3.2

A m o d e l s y s te m for foam fr a c tio n a tio n

T h e m odel we chose for stu d y is an in v erted U -tu b e system , show n in schem atic form in F igu re 3.1. A n in verted U -sh ap ed tu b e con nects two liquid reservoirs. Foam is g en era ted by a gas sp arg er in th e left-h an d reservoir, rises th ro u g h th e le ft-h an d (inflow) leg an d th ro u g h th e U -bend, an d flows dow nw ards in th e rig ht- h a n d leg in to th e rig h t-h a n d collection reserv o ir^ As th e foam is in co n tact w ith a liquid surface at b o th ends of th e tu b e , we have w ell-defined b o u n d a ry co nditions. T h is allows us to sidestep th e challenge of m a th e m a tic a lly describing an overflowing foam , a n d op ens th e jjroblem of co ntin uou s foam fractio n atio n to m a th e m a tic a l ex am in atio n . U -tui)e system s are also am en ab le for ex p erim en tal work (used, for exam ple, by M a rtin et al. [45]). T h e U -tu b e m akes it easier to collect outflow ing foam a n d m ake a c c u ra te m easu rem en ts of th e foam . T h is will be fu rth e r d etaile d in S ection 3.2.5, w hich describes ou r ex p erim en ts on a real U -tube.

We assum e th a t th e re is a stead y flow of b o th gas a n d li(iuid from left to rig ht, an d th a t th e gas flow ra te is c o n stan t. T h e key (juestion, therefore, is: what is the

liquid flo w rate that is delivered? T h is m ust vary w ith th e gas flow ra te a n d o th e r

physical a n d chem ical p a ra m e te rs of th e foam , such as b u b b le size, surface ten sio n an d viscosity.

'I n a real fra c tio n atio n colum n of th is type, th e left reservoir would hold th e so lution co n tain ­ ing surface-active m olecules an d th e right reservoir would co n tain a m ore c o n cen trated solution (th e c o n c en tratio n difference would d epend on th e colum n set-u p ). As will be explained, our m odel does not consider th is co n c e n tra tio n directly.

44 C h a p te r 3. F o a m F r a c tio n a tio n

X (position)

B = Ji R

Foam flow

Gas (sparging)

--- ►

Foam

Solution

L

_Concentrated

solution

F i g u r e 3.1: Schem atic illu stratio n of th e inverted U -tube set-up for th e stu d y of foam fractionation. G as is sparged into a surfactant solution reservoir a t a con­ s ta n t rate, g en erating foam which flows through th e tu b e. T h e foam prefer­ entially carries th e surface-active com ponents of th e solution, leading to an increase in co n centration in the outflow reservoir. T his m ode of o p eration is term ed “sim ple m ode” [46] as there is no independent liquid feed.

3.2. A m odel system for foam fractio n atio n 45

A p ro p o rtio n of th e surface-active m olecules in th e inflow reservoir becom e tra p p e d a t th e film surfaces as th e foam is g en erated. T hese m olecules are carried th ro u g h th e tu b e w ith th e m oving foam and, hence, are delivered a t a r a te d e te r­ m ined by th e gas flow. However, th e in te rs titia l d ilu te liquid does not flow th ro u g h th e tu b e a t th e sam e ra te (due to d rain ag e th ro u g h th e foam ). W hen considering th e efficiency of a colum n, we m ay seek to ensure as m any of th e surface-active co m p o n en ts are carried w ith th e foam as possible, w hile m inim ising th e delivery of th e d ilu te solution. T h is will be discussed in g reater d e ta il in Section 3.3

We shall exam ine th e b eh av io u r of th e U -tu b e using th e s te a d y -s ta te version of th e F D E in tro d u ced in C h a p te r 2.2. (A d eriv atio n of th e full tim e-d ep en d en t F D E can be foim d in A pj)endix A .) A g reat ad v an tag e of th e elem entary F D E is th a t, w ith it, m an y problem s m ay be tre a te d analytically in a relatively straig h t-fo rw ard fashion. W ith a full an aly tic al th e o ry in place, we can begin to u n d e rs ta n d th e process of fractio n atio n in g reater detail. M uch of th e analysis can also be ex ten d ed to th e m ore general case and, eventually, to o th e r fractio n atio n colum ns.

We will p resent im m erical solutions w hich co rro b o ra te th e findings of th e an ­ alytic theory, an d p relim in ary ex p erim en ts to te st it. We will also analyse th e depen dence of th e resu lts on th e len g th of th e tw o legs. W'e v/ill derive a m etric of p erform an ce for fractio n atio n cohnnns, an d show how it m ay be used to m axim ise th e efficiency of U -tu b e set-up s. Finally, we will o u tlin e how to ex ten d th ese resu lts to o th e r ty p e s of fra c tio n a tio n colunni, w ith exam ples.