gas absorption in slurries of finite-capacity microphases

GAS ABSORPTION IN SLURRIES OF FINITE-CAPACITY MICROPHASES

ANURAG MEHRA Department of Chemical Technology, University of Bombay, Matunga, Bombay 400019, India

(First receiued 19 December 1988; accepted& public&on in revisedform 31 October 1989)

AhstractPAhsorption of a gas into slurries constituted by “fine” particles microdispersed in a liquid phase, which remove the diffusing solute gas dissolved in the liquid phase, from near the gas-liquid interface by reaction, complexation, solubilization (including surface adsorption) or catalytic action, is a relatively novel means for intensifying the specific rates of gas absorption and cvcn manipulating selectivity in the case of multiple gaseous solutes. It has been shown in this paper that it is possible to develop analytical models for describing the absorption behaviour in the presence of microphase constituents which have a finite, exhaustible capacity to “store” the diffusing solute and/or the concentration of the solute within the microphase vs that in its vicinity in the original liquid phase shows a non-linear equilibrium relationship. The non-linearities may be approximated by segmental (piecewise) linearization. Mass transfer models of this type arc conceptualized in terms of different zones of “varying” capacity wthin a liquid surface element. A theory has been developed for such exhaustible microphases including the conditions of validity for the various rate expressions and structured classifying regimes. Data on the absorption of hydrogen into slurries of lanthanum-nickel alloys in oil, reported in the literature, have been analyzed. The predicted values of the specific rate are found to be in reasonable agreement with the experimentally observed values.

INTRODUCTION

Absorption of gas into slurries constituted by fine particles is fairly common as a means of intensifying gas absorption rates and even for improving selectivity in the case of multiple gaseous solutes (Alper et al., 1980; Pal et al., 1982; Sada and Kumazawa, 1982; Janakiraman and Sharma, 1982, 1985a, b; Wimmers et al., 1984; Bhaskarwar and.Kumar, 1986; Bruining et al., 1986; Holstvoogd et al., 1986, 1988; Mehra, 1988; Mehra et al., 1988; Mehra and Sharma, 1988; Tinge et al., 1987; Wimmers and Fortuin, 1988; Bhagwat and Sharma, 1988). In a large number of cases, however, the microphase (being defined as a phase whose constituents are in the form of “fine” particles or droplets having size d, < DA/kL) is either a true catalyst which provides for enhanced contact between the diffusing gaseous species A (in the liquid phase) and a reactant B dissolved in the liquid phase or a “solubilizing” agent which shows a remarkably high affinity for the diffusing solute compared to the original liquid phase, and preferentially takes up the diffusing solute, A, from near the gas-liquid interface and transfers it to the bulk liquid (“solubilization” here includes adsorption on a solid surface limited exclusively either by equilibrium or by a combination of sorption kinetics and external diffusion).

The film theory of mass transfer and very recently the unsteady-state mass transfer models have been used to construct elaborate pseudohomogen-

eous models for these situations (Pal et al., 1982; Janakiraman and Sharma, 1982; Mehra, 1988). Ana- lytical solutions have been obtained for a variety of cases, especially those requiring the use of unsteady-state mass transfer theories such as Higbie’s penetration theory and Danckwerts’ surface renewal model, to predict the specific rates of gas absorption

in the presence of a microphase as well as variations in selectivity in the case of multiple gaseous solutes. Pseudohomogeneous models are truly rigorous for low hold ups of the microphase and the diffusivity of the solutes as well as the gas-liquid interfacial area are assumed to be unaltered in the presence of the microphase (Mehra, 1988; Mehra et al., 1988; Mehra and Sharma, 1988).

A salient feature, common to situations where exclusively unsteady models have to be devised, is the significance of accumulation of solute A within the microphase. The fundamental assumption in describing the solubilizing behaviour of a microphase, such as an emulsified liquid phase or a micelle, is the use of a constant distribution coefficient mA (ratio of concentration of A in the microphase to that in the immediate vicinity of the microdroplet), which is independent of the concentrations of solute A within the mlcrophase, A,,, as well as that in the local vicinity, A,. This, however, may not be true in general.

Figure 1 shows a schematic sketch of the concentration profiles of A within a surface element, in the presence of a microphase. Inset I shows the relevant concentrations around a spherical microparticle whereas inset II depicts the usually assumed (linear) relationship between the concentrations A,, and A,.

Typical systems which cannot be characterized by a linear equilibrium relation are constituted by hydrogen (solute) dissolved in a liquid phase and lanthanum-nickel alloy hydrides in the form of “fine” particles (solid microphase). Holstvoogd et al. (1986) and Tung et al. (1986) have reported data on the absorption of hydrogen in slurries constituted by LaNi,H, particles dispersed in solvents such as silicone oil. Here the equilibrium relationship between A,, and A, is non-linear and the microphase loading

1525

1526 ANURAC MEHRA

I Iiquld film

Fig. 1. Concentration profiles for a diffusing solute “A” in a surface element in the presence of a microphase: Mu = Ah/A, = constant > 1.

has an upper, finite limit (Holstvoogd, et al., 1986; Goode11 and Rudman, 1983). A numerical solution for the analysis representing the above situation has been given by Holstvoogd et al. (1986, 1988), the latter paper including an analysis for high solid (microphase) loadings for some simple cases. These hydrogen storage systems have relevance as storage device core components for hydrogen transport (Arnold, 1987) and may find use in enhancing rates of hydrogen transport limited hydogenations.

It will be shown in this paper that it is possible to develop analytical models for describing the behaviour of the specific gas absorption rate in the presence of such finite-capacity slurries by incorporating a combination of linear approximations to the actual non-linear equilibrium curve, representing the solubilizing behaviour of the microphase, into appro- priate mass transport models. Solubilization by the microphase is represented here as a process of par- titioning equilibration, the approch to equilibrium being constrained by kinetic and/or diffusional factors. This process is viewed under two contrasting situations, namely: (i) from the point of view of equilibrium limitations (i.e. capacity) alone, the kinetics of solubilization as well as external diffusion being assumed to be relatively rapid within the contact time t,; and (ii) for the situation when equilibrium capacities are high and only kinetic and/or external diffusion are of relevance. The case of internal diffusional limitations is excluded from the analysis in this paper. These two situations correspond to the relative solubility controlled regime and the transport-controlled regime, respectively, as will be seen later. For liquid microphases such as micelles, microemulsions and macroemulsions, solubilization kinetics are indeed rapid and internal diffusion is not constraining. For solid microphases which are either impervious or macroporous, sorption kinetics may be finite but internal diffusional restrictions are irrelevant and do not

contribute to rate control. Only for the case of microporous solid microphases may internal diffusion become significant. Even though the terminology used herein is borrowed from that used in the description of physical phenomena such as adsorption or dissolu- tion, it may include, in a wider sense, the equilibrium description of reversible chemical complexation or even reaction. It is possible, for instance, that chelat- ing complexes and even fine carbon particles when acting as adsorbents under certain conditions may show finite-capacity behaviour (Holstvoogd et al., 1988).

A theory for mass transfer in the presence of a finite-capacity or, alternatively, exhaustible microphase is developed first, assuming the absence of any independent liquid phase reaction near the gas-liquid interface. A discussion on the influence of liquid phase hydrogenations in the bulk phase or slurry hydrogenations (large substrate particles) on the basic formulation is also included. The data presented by Holstvoogd et al. (1986). have been examined along with a comparative discussion on the simulated results of this paper and the model given by the above workers.

THEORY

Figure 2 shows two important equilibrium curves with the doted lines showing the linear approximations to them. In general, therefore, the distribution coefficient m,,.defined as the ratio A,/A,, may vary considerably as a function of A,, and A,. These curves represent a phase of initiation of (low) uptake capacity, followed by a phase of normal (high) uptake capacity, and terminated by an exhaustion phase of (low) uptake capacity (which will hereafter be referred to as the initiation, activated and exhaustion phases, respectively). Curve II is a special case and may be referred to as the constant external concentration curve (CEC curve). This indicates that the second

Gas absorption in slurries of finite-capacity microphases 1527

A min AL -

Fig. 2. Some non-linear equilibrium curves (I) sigmoid (finite slopes), (II) constant external concentration.

(activated) phase of high uptake is of infinite slope, as if a sink. Each of these phases may be represented by linear isotherm segments with a discontinuity in slopes. Piecewise linearization of a given (system- specific) equilibrium curve may be done by inspection and involves a measure of choice which is, to an extent, dependent upon desired levels of accuracy.

A surface element into which A is diffusing may be divided into distinct zones depending upon the local concentration, A,, surrounding the particle and the consequent equilibrium isotherm segment which represents the microphase capacity. Consider first curve 1 of Fig. 2, as represented by its linear segments. A general formulation of the species balance equations for A and A,,, assuming the microparticles to be stationary within the surface element, as indicated in an earlier work by the author (Mehra, 1988), is as follows:

D,dZA/dx2 = (1 - l,)i?A/& + I,K,(A - A&n,) (1)

a&.,/at = &(A - ALo/mA) (2)

with the conditions

IC: t = 0, A = A,, A,, = ALob, all x

BCs: t > 0 A = A* at x = 0 (3)

A = f(r) A, atx+ 03

where K, represents a transport coefficient, for the transfer of A from the continuous phase to the microdispersed phase and for the case of pure external mass transfer control, K, = 120,/d:, assuming Sherwood number = 2. Also,f(t) is a decaying function of time and may be obtained by setting the term associated with D, in eq. (I) to zero and subsequent solution of the above equations. If the slopes of linear segments 1, 2 and 3 (of curve I) are denoted by mAl, mA2 and mAJ, respectively, then we may write

A,, = mliAr + Ci (4)

for the ith segment. Furthermore, the distribution coefficient mA becomes, from eq. (4):

m _., = Ah/A, = mAi + C,/A, (5)

where i = 1,2 or 3 depending on the value that A, (or ALo) reaches, i.e.

m_4i = mAl C, = 0 for 0 < A, < Amin

mAi = mAZ C2 # 0 for Amin c A, < A,,, (6)

mAi = mA3 C3 # 0 for A,,,< A, < A*.

The specific rate of absorption is given, from Higbie’s penetration theory, as usual by

c f R, = - D, 8 A/h 1, = o dt/t, (7) Jo

where t, = 4D,/nkt.

The rigorous solution for the specific rate of absorption may be obtained by using eqs (l)(3) along with eqs (Q-o-(). This problem is fairly formidable to solve and may not admit an analytical solution.

The transport coefficient K, usually incorporates only the external-phase resistance to uptake because the microphase shows a much higher relative solubility for the solute compared to the external phase. Further, the Sherwood number = 2 for the external mass transfer coefficient is well accepted in the literature (Pal et al., 1982; Janakiraman and Sharma, 1985b; Holstvoogd et al., 1986; Mehra, 1988) especially since turbulence intensities near the (macro) interfaces are assumed to be low. A condition as to when the intraparticle diffusion does not constrain the uptake rate, in the absence of finite sorption kinetics, is given by n2m,D,, % 3D,, which is derived from the expression for an internal mass transfer coefficient for a microsphere, based on the difference between the concentration at the drop surface and the average concentration inside it (Mehra, 1988). For liquid and macroporous solid microphases D, = D,, and

1528 ANURAG MEHRA

Mu $ 3 implies exclusive uptake rate control by the external-phase film surrounding the microparticle. For the case of microporous particles when D, %- D,, the above-mentioned condition may not be satisfied.

Yet another resistance towards equilibration may be provided by finite sorption kinetics (interaction of A with an adsorption site). In the absence of internal diffusional limitations this may be accounted for by using an overall uptake coefficient defined as K D,OY = (K,’ + k, ‘)- ‘, as has been suggested by Janakiraman and Sharma (1985b) and Holstvoogd et al. (1988). For the purposes of analysis K, may be replaced by K,,,, wherever necessary.

The physical picture, represented by eqs (l)-(7), involves the uptake of A by the microphase particles near the gas-liquid interface. The microparticles can rejuvenate themselves by the subsequent circulation of these loaded microdroplets through the (relatively) A-deficient bulk where it sheds the solute when the penetration element moves into the bulk liquid.

Relative solubility controlled regime

For the condition when the microparticle is “small” (large K,), the rate of uptake of solute by the microphase is very rapid and within a time much less than t, the microparticle becomes “saturated” to its capacity (small m,), i.e. the condition

Kotc/mAi 9 1 (i = 1, 2, 3) (8)

holds and A + A, Iocally throughout the whole length of the surface element, representing a local distribution equilibrium. The species balances are somewhat simplified (Mehra, 1988) and substituting eq. (2) into eq. (1) results in

D,~~A/~x~ = (1 - r,)aA/at + i,a.4,,fat. (9) Further substitution of eq. (4) into eq. (9) gives

D,a2A/i3x2 = (p - 1,) + iom,i] aA/at

= F,aAjat (101

upon letting A + AI.. Condition (8) implies that the transport of A to the microphase is not affected by the liquid phase resistance to local uptake. The original equations may thus be written as

D,d2A/8x2 = F,aA/at (A, < A < A,.,&

DAa’Af8x2 = F,aA/& (A,,+ < A < A,,,) (11)

D,a2A/8x2 = F,dA/& (Amax< A c A*)

with the conditions

IC: t = 0, A = A,, all x

BCs: t > 0, A = A* atx=O

4 = A,,,,, at x = A(t) (12)

A = Amin at x = w(t)

A = A, at x -+ cc (A, c A,i,)

the concentrations Amin and A,,, being the zone divider concentrations (refer to Fig. 2).

Figure 3 shows a zone model schematic sketch where the different zones are clearly marked (initiation, activated and exhaustion).

The solutions to eqs (11) along with conditions (12) may be obtained by adopting the approach used in deriving the expressions for the specific absorption rate for the case of instantaneous reactions (Danckwerts, 1970; Mehra, 1988). The solutions are of the form

A/A* = ai + bi erf [x(Fi)“‘/2(DA t)l/*] (13)

which will satisfy the differential eqs (11) (Sherwood and Pigford, 1952). The constants ai and bi for each zone (i = 1,2, 3) may be found by insertion of boundary conditions (12) into eqs (13) and we may thus

Fig. 3. Schematic sketch of concentration profiles for the three-zone model: equilibrium curve I of Fig. 2.

obtain

A A*= l+

(A,dA* - 1)

erfCz,(~3/~A)1’21


where Q1 = C(-%,,- AminMA* - 4,,,,)lt~2/~~)“*, Q2 = [(Amin - A,)/(Amax- Ami, )I (‘,/‘F2)“‘, D, = (Fa - F&DA 9 and D2 = (F, - F,)/D,. Equa- tions (19) and (20) may be solved by the Newton-Raphson method for the two variables zl

(Am, < A < A*) (14) and z2 (Carnahan et al., 1969). From the definition of the specific absorption rate, namely eq. (7), we obtain

A A,,, _- A*- A*

CtAm- Amin)IA*I erfCzlt~2/~A)“21 - erf [.zI(F2/DA)“*] - erf [z2(F2/DA)“*]

tAm,x- AminYA* + erf [zi (F2/DA)“*] - erf [z2(F2/Da)‘/*]

R, = (F,)“*k,(A* - A,,,)/erf[z,(F,/~,)“‘] (21)

and an enhancement factor:

#A = R,lC(l - &)“‘k,t4* - &)I (22)

being defined as the ratio of the specific rate in the presence of active microparticles to that in the presence of inert ones.

A Amin _- A*- A*

[(A,,,,, - 4,)IA*I erfCz2(F,PA)“*1 - erf[z2(F1/DA)1/2] - 1

t4nin - Ad/A* + erf [z2(F1/D,)‘/*] - 1

x erf{ (x/2) CF, /(DA t)l I’* >

(4. ( A < Anin )

and

n(t) = 22,(@‘2 w(t) = 2.~,(r)~/*.

Figure 4 shows equilibrium curves ad belonging to

(15) a family (represented by curve I of Fig. 2) which together cover all possible shapes of equilibrium curves excluding those which display hysteresis or periodic behaviour. Class a is a sigmoid (initiation- activatedexhaustion), b represents increasing capacity (activation leads to larger capacities), c is a de- creasing capacity curve (gradual exhaustion), and d represents mixed behaviour. It is likely that curves a and c represent more realistic forms of equilibrium behaviour compared to curves b and d, since exhaustion of the finite-capacity microphase (with respect to its loading) is indeed expected.

Figure 5 shows plots of the enhancement factor &a (16) vs mA2, with mAl as the parameter when equilibrium

behaviour is assumed to be of the type given by curves a and c, respectively (of Fig. 4).

Furthermore, since the diffusing species A is the same in all the three zones, has the same solubility in the continuous phase (irrespective of zones) and is neither consumed nor generated in the continuous phase, there should be flux continuity within the continuous phase at the zone boundaries x = n(t) and x = w(t). From this we get

(17) This three-zone model reduces to a two-zone model

when A, > Amin since the concentration of A in no part of the surface element lies in the initiation zone. This model may also prevail when two segments can provide sufficient and reasonable representation of the actual non-linear curve (only one zone divider concentration is then of relevance). Only the zones defined by the plane x = L(t) remain. In such a situation, eq. (19) reduces to

- D,aA/b (zone 3) = - D,8A/ikc (zone 2) at x = L(t)

D,cYA/& (zone 2) = - D,dA/dx (Zone 1) (18)

-

at x = w(r).

erf[z,(F,/D,)‘/*] + {erf [zl(F,/D,)“*]

- l)exp( - 4d)/Q, = 0 (23)

which needs to be solved only for zr. The following approximation emerges provided l/erf [z, (F, /Da)1/2]

is large, namely:

Application of eqs(18) to the profile eqs (14H16) gives the following coupled, non-linear, algebraic balances:

erf CzlV’JDA)“*l + (erf CzI (F2/DJ1/*3

- erf Cz2(F2/D,)“*]}

x exp ( - D, zf)/Q1 = 0

l/erf [z, (F3/DA)1/2] = (F,/F,)“*

x Cl + (A,,,- A&A* - fL,,)l (24)

which upon substitution in eq. (21) gives

and

R, = (F2)"*kL(A* -A,,,) (19)

x Cl + (A,,,- Ad/(A* - &,,)I. (25)

If (A,,,- &,)/(A* - A,,,) B 1 we obtain the simple expression

x exp( - D2zf)/Q2 - erf[z,(F2/D,)‘/2] = 0 (20) R, = (F,)“*UAmax- Ai,) (26)

erf[z,(F,/D,)“‘] + (erf[z,(F,/D,)“2] - 1)

1530 ANURAG MEHRA

mAl ) mA2 - %a AL* “‘AI) m,, AL - mhaw

Fig. 4. Various families of equilibrium curves of type I of Fig. 2: possible finite-slope orders and resulting curve shapes.

15r

Equilibrium CU~VM type a ot fig. 4

I I I 1 10 100 1000 10000

52 -

Fig. 5. Enhancement factor as a function of mA2 with mA, as parameter. Values used: D, = 6 x 1O-9 m’/s, k, = 6 x 10-4mjs, 1, = 0.06, mA3 = 0, A,,,/A* = 0.4, A,,,/A* = 0.7, A, = 0.

the enhancement factor in the specific rate of absorp- A further discussion of simplifications and limiting

tion being cases would now be in order. For the two-zone model, when A,,,2 A*, no particle is ever exposed to con-

+A = [F,/(l - 1,)]“‘[(A,,,- A,)/@* - Aa)]. (27) centrations greater than A*, which would be the con-


centration in the continuous phase throughout the with t = .t,, and A, = A,,,, which in turn yields length of a surface element spending infinite time at the gas-liquid interface, and eq. (21) reduces to (by

A* - A, = exp [ - k&‘/V) (t - tm)/(F3)‘lZ] (32)

taking limits when Q1 + large values and z1 -+ small A* - A,,, values) where t,,, is determined from the previous numerical

R, = k,(F,)“‘(A* -A*) (28)

which is also what is expected from the theory presented earlier (Mehra, 1988). In effect this implies that, for any contact time, however large, the microphase is never permitted to reach the exhaustion segment 3 (of curve I of Fig. 2).

An important consideration would now be to develop a bulk phase balance in order to find A, as a function of time (batch time scales or macrotime) and also estimates of batch time for a given volume of slurry, V, having a gas-liquid interfacial area a’. This development is shown for a two-zone model only, wherein

F,dA,fdt = R,a’/V (29)

which is to be solved along with t = 0 and A, = 0,

where 0 -Z A, -z A,,, (A,i, = 0). As long as R, is given by eq. (21) this solution may be obtained numer- ically by using any simple finite-difference approximation. However, when eq; (26) holds we get

A rn%S - A,

A = exp [ - kL(a’/V)t/(F2)“‘]. (30)

msx

procedure. Note that as A, appro&hes A,,,, eq. (26) is not valid and numerical procedure has to be resorted to. An estimate of the order of magnitude of the batch time could simply be given as [(FZ)1’2 + (F,)“2]/(k,a’/V), i.e. a sum of the time scales involved during the activated and exhaustion phases.

nansport-controlled regime

The behaviour of microphases which follow curve II of Fig. 2 (i.e. the CEC curve) may now be analyzed. Let segments 1 and 3 have slopes of mAl = mAJ = 0, the slope of segment 2 being mA2 + co _ In such a situation A_,= Amin ( = A,,,) and, in the zone A > A,,,,

a reaction-like uptake prevails, the effective local volumetric rate being given by I,K,(A - A,,,), whereas in the zone defined by A < A,,, it is identical to mere diffusion. The condition Kotr/m~z + 0 holds and implies that the uptake rate to the microparticles is totally controlled by the mass transport resistance in the external film surrounding the microparticle (Mehra, 1988).

Consider first the case when segment 3 (of curve II of Fig. 2) is removed. This is equivalent to assuming

When A, > A,,, only a single zone remains and the that the capacity of the microphase is infinite (inexl

bulk balance becomes haustible) even though it requires some minimum threshold concentration to activate uptake. Figure 6A

F3dAb/dt = kL(F3)“2(A* - Ab) (31) shows a schematic sketch of the concentration profiles

A : UNSTEADY STATE ‘REPRESENTATION

B : STEADY STATE REPRESENTATION

Fig. 6. Concentration profiles for the diffusing solute “A” near the gas-liquid interface in the presence of an inexhaustible microphase for the transport-controlled regime (equilibrium behaviour of microphase given

by first two segments of the constant external concentration curve).

1532 ANURAG MEHRA

with clearly marked zones, A,,, being the zone-dividing concentration. The general eqs (l)--(3) may be re- written as

D,82A/&c2 = (1 - l,)aA/at + l,K,(A - A,,,)

(A m cA<A*) (34)

cYA,,*& = &,(A - A,) (35)

with the conditions

IC: t = 0, all x, A = A,,

A LO - -AL00 (A, ( A,)

BCs: t > 0, x = 0, A = A* (36)

x=1(t), A=A,.

In the adjacent zone we have

D,a2A/ax2 = (1 - l,,)aA/at (Aa < A <. A,,,) (37)

the equation for A,, vanishing. The required conditions are

IC: t = 0, all x, A = A, (A, < A,,,)

BCs: t>O, x=n(t), A=A, (38)

x+ co, A=A,.

Equations (34x38) are clearly formidable in so far as analytical solutions are required to be developed. However, it is obvious that the accumulation charac- teristics of the above situation are completely unim- portant, once the loading of the microphase has been accounted for by a reaction-like term, unlike the solubility-controlled models. For such a case when only reaction- and/or transport-type terms are important, eqs (34)-(38) may be simplified to their film theory (steady-state) analogs, namely

D,d2A/dx2 = l,K,(A - A,) (A, < A < A*) (39)

with the conditions

x = 0, A = A*

x = 2, A = A,,,. (40)

For the adjacent zone we have

D,d’A/dx’ = 0 (Ab < A <A,,,) (41)

along with the conditions

x = 1, A = A,,,

x = 6, A = A, (6 = D,Jk,). (42)

Figure 6B represents the film theory sketch of the concentration profiles and these remain valid as long as A, is allowed to increase without limit (inexhaustible microphase; segment 3 of curve II of Fig. 2 ex- tends till co ). From flux continuity, namely that

( - D,dA/dx),,, 1 = ( - DAdA/d&,,= 2 at x = ),

(43)

the value of il may be located. The equation that is obtained is given by

[(l,K,)/D,]“2(A* - A,)/sinh [(l,Ko/DA)1/2j1]

= (A,,, - A,)/(6 - 2) (44)

and the specific rate of absorption may be written as

R, = (A* - A,)(D,I,K,)1’2/tanh [(I,K,/D,)1/2;1].

(45)

The flux continuity condition holds (as discussed earlier) because the diffusing species A has identical solubility in each zone (continuous phase) and at the boundary plane is neither consumed nor generated. If the bulk concentration A, 2 A,,, the distinction between zones 1 and 2 disappears. Uptake of A now commences in the bulk also and R + 6. The specific rate is now given by

R, = (DAloKo)1i2 {(A* - AJtanh [ (Z,K,/D,)“2S]

- (A* - A,)/sinh [(l,K,/D,)“26]) (46)

which is obtained from the solution to eq. (39) along with the conditions

x = 0, A = A*

x = 6, A = A,. (47)

The possibility of a microphase being inexhaustible is, however, unlikely. The equilibrium behaviour is more likely to follow the termination segment 3 as given in curve II of Fig. 2. In such a situation the sudden exhaustion of the microphase may be accounted for by using the concept of a critical concentration A,, such that the microparticles contained in the zone (thus a third zone is created) between A* and A, are completely exhausted (and hence are rendered inert). Figure 7A shows the relevant concentration zones for this absorption situation where the zone nearest the gas-liquid interface consists of “exhausted” (inert) microparticles. We thus have three zones in a surface element, namely those of exhaustion, activation and initiation in order of increasing distance from the gas-liquid interface. Equation (37) holds for the exhaustion as well as the initiation zone whereas eqs (34) and (35) are valid for the intermedi- ate activation zone. The conditions for solution, however, become

IC: t = 0, all x, A = A, (A, < A,,,)

A - AL,, Lo -

BCs: t > 0, x = 0, A = A*

x = w(t), A = A,(t) (48)

X = A(t), A = A,,,

x= +co,A-+Ab

where A,(t) is the concentration of A in the continuous phase, at a location x = w(t), at which A,, = A,, ,,,_. The planes x = w(t) and x = n(t) may be located by using the conditions of flux continuity, namely

( - D, LQ/ax),,,, 3 = ( - DAaAIax),,,, 2 at x = w(t)

(49)


Fig. 7. Concentration profiles for the diffusing solute “A” near the gas-liquid interface in the presence of an exhaustible microphase for the transport-controlled regime (equilibrium behaviour of microphase given by

the constant external concentration curve).

A 8 UNSTEADY STATE REPRESENTATION

0 W X 6

B : QUASI SfEADY STATE REPRESENTATION

and

The condition for the three-zone model to prevail is

given by

lerrh < t, (51)

where telh = (ALo ,,,,_-- A,, ,)/[&(A* - A,,,)], i.e. the time required for the microphase to reach its maximum loading (become exhausted) at the gas-liquid interface.

It may not be possible to derive analytical solutions for the above formulation. A steady-state analog for the same cannot be postulated since now the microphase behaviour is not exclusively “reaction-like” and the sudden exhaustion of the microphase capacity does not allow accumulation effects to be ignored. In the zone enclosed between x = 3. and n = w, the loading of solute A within the microphase particles varies from zero (x = J_) to its maximum value (x = w). Therefore, for t eXh 2 t, the specific rate of absorption is given by eq. (45), whereas, when condition (51) holds, unsteady balances need to be resorted to, in order to describe the situation. However, for the case when texh -+ t,, the exhaustion of the microphase may be regarded as instantaneous. The activated, middle zone (2) thus disappears (w + 2) and in a surface element only two zones remain, namely that defined by A > A, containing totally exhausted particles and A < A, full of completely unconverted particles. This situation is fully analogous to that of zone melting

(Eckert and Drake, 1972) and the transient analysis developed below provides the expression for the asymptotic, maximum possible (upper bound) specific rate of absorption, which would be achieved for ultrafine particles. For this two-zone case, the differential balance (37) holds in both the zones. The initial and boundary conditions are given as

IC: t=O, A=A,, A,,=A,,, allx

BCs: t > 0, A = A*, at x -0 (52)

A = A,,,, at x = A(t)

A = A,,, at x -+ 00 (Ab -c A,)

the form and rate of motion of the dividifig plane x = A(t) being governed by the flux balance at this plane, name1 y

( - D., aAlax)sxhsusted ZOnt J

= (- D,aA/ax) unconverted zone I + &,(A,, max

- A,, ,)dl/dt. (53)

The solutions obtained are

A - A,,,

A* - A,,, _ 1 _ erfC(x/WL t)“‘l

erf[z/(DA )]I21 (A* > A > A,)

(54)

A - A,,, = 1 _ erfc CM4D_i 01’21 A, - 4, erfc [z/(D~)“~]

M,, =- A ’ 4)

1534 ANURAG MEHRA

where 0; = DA/(1 - I,) and z is given by

(DA/n)“‘exp[- z2/(Di) {(A* - A,,,)/erf[z/(D:,)“‘]

- (A,,, - A,)/erfc [z/(D;)‘/~]}

= 4dA,, max- ALob)/(l - wz (56)

which is obtained by substituting from eqs (54) and (55) into eq. (53). Furthermore

n(t) = 2z(t)“2 (57)

and the specific rate of absorption is given by

R, = (1 - I,)1’2kL(A* - A,)/erf[z/(DA)1f2] (58)

the enhancement factor being

4A = [(A* - A,MA* - --%)l/erfC~(D~)1~21 (59)

where eqs (58) and (59) are obtained from definitions given earlier in this paper.

For a large-capacity microphase (which satisfies the condition I(t,) Q 4(D,t,) ‘I2 but where tsxh -$ t, still holds) the exhaustion plane moves very slowly. A quasi-steady-state analog for the above “zone exhaustion” may thus be formulated where we may write

D,C(A* - A,)la - (A, - At,)/@ - A)]

=.&(A,, mar - A, b )dd/dt (60)

the solution to which with t = 0 and 1 = 0 leads to

D,(A* - A&ICUA, m-x- A,, *)I

= (2A, - A* - A,,) [z, - bfA* - A,,,)]

+ cz: - (A* - A,)‘&“]/2 + (A* - A,,,)

x (A, - AS” In {z,IC(A* - A,,,)~] 1 (61)

where z, = (A* - A,)6 -(A* - A,)& and the specific absorption rate may be estimated from

R, = UA,, max- A,,)U,. (62)

Figure 8 provides a plot of the enhancement factor dA vs the non-dimensional variable Ha [ = (DAI,K,)‘/2/kJ (referred to as the Hatta number), with A,/A* as parameter. The critical Hatta number (Ha,) value is shown by a broken line with dots. The asymptotic values of +,, (maximum) that can be achieved from the instantaneous exhaustion model [eq. (59)] are also indicated towards the right side of the graph. Condition (51) may also be expressed as

Ha > Ha, (63)

where Hn, = {7~i,(A~,,,- A, ,)/[4(A* - A,)]}“2. The concept of a critical Ha, has also been used by

Holstvoogd et al. (1986) wherein the authors have developed a numerical model to simulate the behaviour of hydrogen absorption in LaNiH, slurries. The results presented in Fig. 8 are very similar to those reported by the above workers. From the plot it can be observed that the enhancement factor 4” equals Ha when A,,, = 0 and Ha, > Ha > 2. The maximum value of 4, as obtained from eq. (59) is unlikely to be

-

Fig. 8. Enhancement factor C$~ as a function of Ha with activation threshold concentration A,/A* as parameter. Values used: D, = 6 x 10e9 mZ/s. k, = 6 x 10d4 m/s. (ALomnx- A,,)/A* = 2250, 1, = 0.06,

A,=O.

Gas absorption in slurries

realized in the hydrogen absorption systems referred to before, since particle sizes in the submicron range for lanthanum-nickel alloys cannot be obtained. However, these values may be easily achieved in the presence of micellar and microemulsion constituents containing exhaustible reactants, because for such systems the microdroplets are of the order of 10-100 nm (K, = 12D,/dj$) and the condition texh 4 t, may be easily satisfied.

Figure 9 provides a plot of the maximum value of the enhancement factor @A vs the capacity of the microphase (A, ,,,BX- ALo ,,)/A* with the threshold initiation concentration A,/A* as parameter. Figure a also shows the variation of &A with A,/A* at a fixed capacity (& max- A, b )/A* (see broken line). The broken line with dots gives the prediction of 4, from the quasi-steady-state model [eq. (62)]. As may be seen, for low capacities the discrepancy between the predictions from the rigorous unsteady-state model and the quasi-steady-state model is considerable since here A(Q) is comparable to the penetration depth [ = 4(OAt,)“2], while at higher capacities the agreement between the two approaches is excellent. This aIso shows that the quasi-steady-state models will be valid, under specified conditions, and may be considerably simpler to handle than the rigorous unsteady models. There is substantial similarity between

of finite-capacity microphases 1535

the quasi-steady-state analysis developed here and the approach used in the modeling of non-catalytic gas-solid reactions (Doraiswamy and Sharma, 1984). It may be possible to solve the quasi-steady-state versions of more elaborate zone models such as the one shown in Fig. 7B using the approach prevalent in the area of gas-solid reactions (Mantri et al., 1976). For the case when A, > A,,, the initiation zone vanish- es (A -+ w) and the resulting two-zone model may be solved for following the analysis procedure given by Ishida and Wen (1968).

ANALYSIS OF EXPERIMENTAL RESULTS

Holstvoogd et al. (1986) have reported data on the specific rate of absorption of hydrogen into slurries of LaNiH,. Table 1 shows the relevant details.

It can be seen that the Ha values lie below the critical value Ha, so that the microphase effectively behaves like and inexhaustible “reaction-like” sink. The value of the specific rate may be predicted using eq. (45). Columns (7) and (9) show the calculated values of the enhancement factor from the theory developed in this paper, whereas column (6) gives the calculated values from the numerical model of Holstvoogd et al. (1986). For the first three temper- atures, it can be observed that the predicted values of

21-

24-

-UNSTEADY STATE MODEL

.---QUASI STEADY STATE MODEL

I

1 CA LO mow

I -A,,t.] /A* -

I 0 O-2

, 0.4 O-6 l

A,/A+-m( REFER TO l=LOT’_“_--, 1-o

Fig. 9. Enhancement factor $1 (A, ,,,z,.-- &,,)/A*

from the instantaneous exhaustion model as a function of capacity with activation threshold concentration A,/A* as parameter. Also d1 YS A,fA* at

a fixed capacity. Values used: D, = 6 x IO-‘m’/s, k, = 6 x 10m4m/s, I, = 0.06, A, = 0.

1536 ANURAG MEHRA

Table 1. Enhancement factors in the specific rate of absorption of hydrogen into a slurry of LaNi,H, in silicone oil’

(2) k,. X 104

(m/s) (3)

(Ha)

(4) D, x lo9

(m2/s)

(6) (7) (9) (5) d* @Jf; 4: (10) (11) 4” th. th. (8) th. &IA* 6:

exp. (numerical) (calculated) Am/A* (calculated) th. (maximum)

25 3.70 0.802 2.49 1.08 1.25 1.18 0.077 1.18 0.077 1.20 40 4.24 0.967 3.44 1.19 1.36 1.22 0.179 1.23 0.138 1.29 55 5.26 1.02 4.52 1.24 1.40 1.20 0.265 1.22 0.236 1.33 70 6.18 1.10 5.70 1.89 1.45 1.17 0.384 1.17 0.384 1.37

7 D.ata used: 1, = 0.058, d, = 7.0 @m, A,, max - A, b = 3.88 x lo4 mol/m3, A, = 0, A * = 84.0-73.0 mol/m3 (at 25-70°C). Abbreviations: exp. = experlmental, th. = theoretical, * = at Am/A* = 0 (initiation threshold concentration A,,, is zero), L = A m /A* estimated from linear interpolation between first and last values, E = A,/A* estimated from exponential interpolation between first and last values, i.e. log,, (A,,JA*) = c (temperature).

the enhancement factor from the theory in this paper match better than those from the numerical model. The calculated values in column (7) or (9) tend to vary within a narrow range because the effect of increasing Ha value is counteracted by the increase in the dis- sociations plateau pressure with temperature. The last column shows the effect of increasing Ha on 4a since A, is kept at a value of zero, i.e. the effect of increasing plateau pressure with temperature is neglected.

The data of Tung et al. (1986) are not readily amenable to quantitative analysis using the theory in this paper, primarily because the reported k,a values need to be decomposed into the mass transfer coefficient and specific interfacial (gas-liquid) area components.

DISCUSSION

The foregoing theory presents a scheme for classifying the equilibrium behaviour of microphases which are present in the vicinity of diffusional gradients, with respect to the diffusing solute.

The method of segmental linearization (alternatively, piecewise linearization) on non-linear equilibrium curves provides an effective analytical tool for making quick estimates of mass transfer rates in such systems without resorting to numerical software, in some cases by mere inspection of the equilibrium curve [see eqs (25) and (26)]. The degree of accuracy of such estimates may be incremented to the required value by increasing the number of linear segments. The additional advantages of the analytical models are that stiffness and unnecessary computational ef- fort in the numerical solution procedure can be avoid- ed and the physical significance of the terms involved in the solution expression are conveniently under- stood.

The moving boundary (exhaustion plane) solution for the asymptotic value of the specific rate of absorption provides a rigorous rate expression for the behaviour of “ultrafine” microphases such as micelles and microemulsion droplets, which contain a finite, exhaustible reactant.

Such slurries can also be effective in selectively separating a mixture of gases, the point selectivities, as a first approximation, being the ratio of specific point

rates of absorption for the two different solute gases. The equations for the specific rate developed here may be used as has been indicated elsewhere (Mehra and Sharma, 1988).

Homogeneous liquid phase hydrogenations and even solid-liquid hydrogenations, whose rates are limited by interphase hydrogen transport, may benefit immensely from the presence of a loaded microphase such as particles of these hydriding/dehydriding alloys. For the more complex situation of hydrogenation of a substrate in the form of a macroslurry, the microphase will act so as to transfer dissolved hydrogen from near the gas-liquid interface to the (macro) solid-liquid interface, since the microparticle (alloy) sizes will be much smaller than the diffusional scales of A near both the (macro) interfaces. When these intrinsic reactive effects are small, i.e. a small reaction rate constant for a homogeneous hydrogenation, such that only bulk liquid phase hydrogenation is significant compared to that occurring near the gas-liquid interface, the relevant rates may be calculated by simply modifying the material balance for A on the bulk liquid phase [such as that represented by

eq. (2911.

A

A*

ALI

AC

AL

A LO

A Lob

NOTATiON

concentration of A in the continuous phase,

kmol/m3 (cant)

solubility of A in the continuous phase, kmol/m3 (cant) concentration of A in the bulk continuous phase, kmol/m3 (cant) critical concentration of A in the continuous phase demarcating the activation*xhaustion boundary for the transport-controlled regime for an exhaustible microphase, kmol/m3 (cant) concentration of A in the immediate vicinity of the microparticle, kmol/m’ (cant) concentration of A in the microphase, kmol/m 3 (m) concentration of A in the microphase located in the bulk liquid phase, kmol/m3

(m)


A LO m&x

A,

A InBX

Amin

a a’ b C

D‘4

DA D, D2 4 F

Ha

maximum absolute capacity of a microphase, kmol/m” (m) concentration of A in the continuous phase in equilibrium with the concentration A,, in the microphase for the CEC curve, kmol/m3 (cant) activation+xhaustion threshold concentration of A in the continuous phase for the relative solubility controlled regime, kmol/m3 (cant) initiation-activation threshold concentration of A in the continuous phase for the relative solubility controlled regime, kmol/m3 (cant) constant defined in eq. (13), dimensionless total gas-liquid interfacial area, m* constant defined in eq. (13), dimensionless constant defined in eq. (4), kmol/m3 diffusivity of A in the continuous phase, m2/s constant defined in eq. (55), m2/s constant defined in eq. (19), s/m2 constant defined in eq. (20), s/m2 microphase constituent size (diameter), m ratio of the capacity of the microheterogeneous media for A to that of the continuous phase (per unit volume), dimensionless Hatta number, a measure of uptake occurring near the gas-liquid interface [(D,Z, K,)“*/k,)]. dimensionless critical value for the Hatta number [eq. (63)], dimensionless transport coefficient for uptake of A based on the continuous phase, s-l overall transport coefficient for uptake of A based on the continuous phase, s-l mass transfer coefficient for gas-liquid transfer (liquid side), m/s sorption kinetic rate constant, s-l volumetric hold up of microphase as a frac- tion of the total microheterogeneous phase, dimensionless relative solubility of distribution coefficient for A (microphase/continuous phase) [kmol/m3 (m)/kmol/m3 (cant)], dimensionless constant defined in eq. (19), dimensionless constant defined in eq. (20), dimensionless specific rate of (A) gas absorption (flux), kmol/m’ s time (macro- or microscale depending on context), s time taken to exhaust microphase at the gas-liquid interface [eq. (51)], s transition time at which bulk concentration equals the zone marking concentration

Ceq. (3m s volume of slurry in batch mode, m3 distance of the plane marking the activated/initiation zones (where A = Ami,)

from the gas-liquid interface (relative solubility controlled regime), m; or distance of plane marking exhaustion/activated zones (where A = A,) from the gas-liquid interface (transport-controlled regime), m

x spatial rectangular coordinate, m Z stoichiometric factor for reaction between

AandB(A+ZB + products): default value ofZ=l proportionality constant defined in eq. (41), m/s112 constant defined in eq. (61), kmoi/m’ proportionality constant defined in eqs (19) and (20), m/s1j2 proportionality constant defined in eqs (19) and (20), rn/sl”

Greek letters

4.4 enhancement factor in the specific rate (ratio of rate in presence of a microphase to that in presence of an inert microphase), dimensionless

6 diffusion film thickness at the gas-liquid interface (liquid side: 6 = DA/k,), m

;1 distance of the plane marking the exhaustion/activated zones (where A = A,,,) from the gas-liquid interface (relative solubility controlled regime), m; or distance of plane marking activated/initiation zones (where A = A,) from the gas-liquid interface (transport-controlled regime), m

A, value of d at r, (quasi-steady-state case), m

Subscripts i pertaining to the ith zone 1, 2, 3 pertaining to the initiation, activated and

exhaustion zones, respectively

Abbreviations m pertaining to microphase cant pertaining to continuous phase CEC constant external concentration

REFERENCES

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1538 ANURAG MIS-IRA

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gas absorption in slurries of finite-capacity microphases

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