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Absorption from bubbles of dilute gas
Chemical Engineering Science, 1965, Vol. 20, pp. 785-787. Pergamon Press Ltd., Oxford. Printed in Great Britain.
Absorption from bubbles of dilute gas (Received 25 Februury 1965) or Two types of process for absorbing a gas in a liquid are considered here : dv Kt = ” (1) (a) The gas is introduced into the liquid in the form of disa 03’3 11 - (Mvo/v)] Crete bubbles-for instance through a porous plate or perforated tube. By putting M = 0, we find that for a bubble containing no insoluble gas (b) The gas is entrained into the liquid from its free surface by an agitator, all or most of the absorption taking place from the entrained bubbles. Kt = 3(vo’/3 - d/3) (2) If the soluble gas is not diluted with an insoluble aas, the mole-fraction soluble gas in a bubble does not &dr as and that the time, I, for such a bubble to disappear completely absorption proceeds. The bubble may disappear before is 3v0113/K. Equation (1) can now be re-written, with change reaching the free surface. If an insoluble gas is present, the of variable, x = (v/v3M)1~3: mole-fraction of soluble gas in the bubble is initially equal to that in the inflowing stream (in case a) or to that in the bulk of (l/&f)'/3 x3& the gas in the space above the liquid (in case b), but it falls t/r = M’f3 (3) z=(“/vM,l’S (x3 s progressively as absorption proceeds. In the limit the gas remaining in the bubble will consist only of insoluble gas The indefinite integral has the form and the vapour of the liquid. At first sight one might not, therefore, expect the rate of 1 1 (x - 1)3 absorption of a given gas in such a system to be simply pro=x+-ln portional to its mole-fraction in the bulk or inflowing stream 6 (x2 + x + 1) - z arctan (other things being equal). The driving force for absorption from a bubble of diluted soluble gas falls progressively during the sojourn of the bubble in the liquid, its average value depending on how much of the aas is absorbed before the bubble leaves the liquid. As against this, however, the I / I,llII/ 1 :/I,,, bubbles do not shrink so much as bubbles of pure gas, and at I any time after immersion there is a larger surface area from 4.0 which absorption can take place. In the following analysis certain simplifying assumptions are made. There is assumed to be no resistance to mass transfer on the gas side of the interface, and the resistance on 30 the liquid side is assumed to be independent of the size or situation of the bubble. The gas in the bubble is always saturated with solvent vapour. The pressure is assumed to be constant. The concentration of dissolved gas in the bulk of the liquid is effectively zero. The bubbles considered are all 2.0 the same shape. A bubble is formed at t = 0, when: -z Volume of bubble = DO +I Partial volume of inerts = Mvo(1 - s) Partial volume of solvent vapour = DOS -1 I-O Partial volume of soluble gas = v0(1 - M)(l - s) where M is the initial mole-fraction of insoluble gas, calculated on a solvent-free basis. After time t, the situation is: Volume of bubble = v Partial volume of inerts = Mvo(1 - s) Partial volume of solvent vapour = us Partial volume of soluble gas = (v - Mv0)(1 - s) Mole-fraction of soluble gas = (1 - Mvo/v)(l - s). Now the rate of absorption is proportional to the molefraction of the soluble gas and to the surface-area of the bubble: i.e.
of
-(l
- s)$
= Kv3/3 (1 - M;)(1
- s)
FIG. 1
785
Shorter
communications From these expressions it is calculated that y/y’ is never less than about 0.89 in very dilute bubbles. For nearly pure bubbles (M < 1 - y)
and equation (3) becomes
$ = Ml,3[1(_l)l’3 _ +!L)l’3] Values of I(x) are shown in Fig. 1, plotted against (x - 1). It is thus possible to calculate, for any given value of Ikf, the fraction y of the soluble gas absorbed in a given time, t, and to compare it with the fraction y’ = 1 - (1 - t/~)~ absorbed from a eure bubble of the same initial volume in the same time. Simple expressions can be obtained for the behaviour under limiting conditions. Thus if the bubbles are mostly immersed for-a time much less than 7, the fraction of the soluble gas absorbed is almost exactly the same for pure and dilute bubbles, and the rate of absorption is thus approximately proportional, other things being equal, to the molefraction, (1 - M), of soluble gas in the bulk or inflowing gas. If conditions are such that if the gas were undiluted most of the bubbles would disappear before reemerging, then when the gas is diluted most of the bubbles will spend a time greater than 7 in the liquid, which implies that at least 95 per cent of the soluble gas will be absorbed from them (see Fig 2); in this case, too, the rate of absorption is approximately proportional to (1 - M). For very dilute bubbles (1 - A4 Q 1) ‘~~~ln[(l--A4)/(~-it4)] 7 whence Y
Y-; = Y Y=l-exp----, Y
1 -exp(-33/r) 1 - (1 - t17)3
(
31 7)
‘>l 7
t ’ ;
<
l
whence ;wI+M-~ Y
3M
1 - (1 Y
43’3
I
(9)
Thus the effect of small traces of diluent is correspondingly small. Fig. 2 shows some values of the ratio y/y’ for different values of M and of t/T. The ratio is never less than about 0.89, whatever the degree of dilution or the fraction of gas absorbed. Thus, in dynamically comparable situations, the rate of absorption of a diluted gas from a swarm of bubbles is equal, with an error of less than 11 per cent, to the rate of absorption of the pure gas multiplied by (1 - M) , where A4 is the mole-fraction of diluent gas in the bulk of the gas or in the inflowing gas mixture (calculated on a solvent vapour free basis); in other words, the rate of absorption is proportional to the partial pressure of soluble gas initially present in the bubbles. However, this conclusion may not be valid if the dynamical behaviour of the dilute bubbles is very different from that of the pure bubbles. Consider, for instance, conditions such that when a pure gas is used most of the bubbles disappear before leaving the liquid; if a dilute gas were used, and the dilute bubbles followed the same trajectories as the pure bubbles, most of the bubbles would spend a time greater than 7 in the liquid and hence would lose almost all their soluble gas by absorption, and the use of the initial partial pressure as driving force would give the right answer. But thedilute bubble is at all times after formation larger than a pure bubble of the same initial size; this may cause the dilute bubbles to spend a time substantially less than 7 in the liquid, and thus reduce the absorption rate.
FIG. 2 786
Shorter
Communications
In the case of very small bubbles which move so slowly relative to the liquid that transfer of dissolved gas away from the bubble is governed entirely by diffusion, to the exclusion of convection, the mass-transfer coefficient varies inversely as the diameter of the bubble. The preceding analysis does not apply if a substantial part of the absorption occurs from bubbles as small as this. It will not apply, either, if there is a substantial chance that a bubble coalesces with another before leaving the liquid, or if the depth of liquid is such that there is a substantial change in the pressure on a bubble during its immersion. The conditions in which the analysis given above is exactly applicable are somewhat circumscribed, therefore; but the general conclusion may nevertheless be valid in many circumstances of practicalinterest. Department of Chemical Engineering, Pembroke Street, Cambridge.
P. V. DANCKWERTS
NOTA-I-ION
m
Integral given in equation (4) Mass-transfer factor Initial mole-fraction of inert gas, solvent-vapour free basis s Mole-fraction of solvent vapour . t Time since immersion of bubble Volume of bubble at time t V I vo Initial volume of bubble Y Fraction of soluble gas absorbed from dilute bubble in time t Y' Fraction of soluble gas absorbed from pure bubble in time t.
K M
x
(v/Mvo)~~3
7
time taken for pure bubble to disappear.
Chemical Engineering Science, 1965, Vol. 20, pp. 787-788. Pergamon Press Ltd., Oxford. Printed in Great Britain.
Gas absorption accompanied by complex chemical reaction (Received 10 February 1965) IN A recent
paper, BRIAN and BEAVERSTOCK [I] presented film and penetration theory solutions for absorption accompanied by a two-step chemical reaction in which the absorbed species reacts with the liquid phase reagent to form an intermediate species, which in turn reacts with the reagent to form the final products. The authors suggested that their theoretical model might apply to the absorption of carbon dioxide into monoethanolamine (MEA) solutions and the absorption of chlorine into ferrous chloride solutions. The authors referred to experimental results for the former system which have been presented by EMMJXRT and PIGFORD [2]. On Fig. 6 of EMMERTand P~GFORD’Spaper, two series of results for almost identical values of Cc/C* (where CB is the concentration of reagent in the bulk liquid, and C* is the physical solubility of the solute gas) are plotted in the form kr,,/ke (ratio of mass-transfer coefficient in the presence of chemical reaction to the physical mass-transfer coefficient) against exposure time t:
where R represents -CaHaOH. The carbamic acid RNHCOOH, will be almost completely dissociated at the pH values encountered in slightly carbonated solutions, as its UK is unlikely to be much greater than 60 141. The proton formed by reaction (1) ii neutralized by a second molecule of amine in a virtually instantaneous ionic reaction: RNHz + H+ + RNH: (2) Although the reactions COa + He0 +HCO,-
+ H+
COa -t OH- + HCO;
also take place, consideration of the relevant equilibrium and rate data shows that they are insignificant compared with reactions (1) and (2). The absorption of carbon dioxide into MEA solutions at low carbonation ratios (moles carbon dioxide absorbed per mole amine) is thus an example of absorption accompanied by second order reaction with a stoichiometric coefficient (moles of reagent which react with (ii) CB = 0.5, C* = 0:0068. one mole of absorbed species) of 2.0. (This behaviour was in fact demonstrated by EMM~RTand PIGFORD, who calcuIn series (i) B/k! was observed to increase with t, and in series (ii) k&/k2 decreased with t. BRIAN and BEAVERSTOCK’s lated a stoichiometric coefficient of 1.78 from their results using an estimated value 0.59 for the ratio of the diffusivity model predicts, however, that if kL/kg decreased with t of MEA to the diffusivitv of carbon dioxide. The stoichiowhen CB = 0.5, it should also have decreased with t when CE = 1.0. The model is thus incompatible with EMMERT metric coefficient calculated from EMMERTand PIGFORD’s results becomes 1.88 if the more accurate diffusivity ratio and PIGFORD’Sresults. 0.66, calculated from the measurements reported in references It is, in fact, well established that when carbon dioxide [5-71, is used. This is in reasonably close agreement with the is absorbed into MEA solutions it undergoes a second-order predicted stoichiometric coefficient of 2.0). reaction to form a carbamic acid [3] : B~UANand BEAVERSTOCK referred to the results of GILLIRNHz + COa + RHNCOO- + H+ LANDet al. [81, who measured rates of absorption of chlorine (1)
Absorption from bubbles of dilute gas (Received 25 Februury 1965) or Two types of process for absorbing a gas in a liquid are considered here : dv Kt = ” (1) (a) The gas is introduced into the liquid in the form of disa 03’3 11 - (Mvo/v)] Crete bubbles-for instance through a porous plate or perforated tube. By putting M = 0, we find that for a bubble containing no insoluble gas (b) The gas is entrained into the liquid from its free surface by an agitator, all or most of the absorption taking place from the entrained bubbles. Kt = 3(vo’/3 - d/3) (2) If the soluble gas is not diluted with an insoluble aas, the mole-fraction soluble gas in a bubble does not &dr as and that the time, I, for such a bubble to disappear completely absorption proceeds. The bubble may disappear before is 3v0113/K. Equation (1) can now be re-written, with change reaching the free surface. If an insoluble gas is present, the of variable, x = (v/v3M)1~3: mole-fraction of soluble gas in the bubble is initially equal to that in the inflowing stream (in case a) or to that in the bulk of (l/&f)'/3 x3& the gas in the space above the liquid (in case b), but it falls t/r = M’f3 (3) z=(“/vM,l’S (x3 s progressively as absorption proceeds. In the limit the gas remaining in the bubble will consist only of insoluble gas The indefinite integral has the form and the vapour of the liquid. At first sight one might not, therefore, expect the rate of 1 1 (x - 1)3 absorption of a given gas in such a system to be simply pro=x+-ln portional to its mole-fraction in the bulk or inflowing stream 6 (x2 + x + 1) - z arctan (other things being equal). The driving force for absorption from a bubble of diluted soluble gas falls progressively during the sojourn of the bubble in the liquid, its average value depending on how much of the aas is absorbed before the bubble leaves the liquid. As against this, however, the I / I,llII/ 1 :/I,,, bubbles do not shrink so much as bubbles of pure gas, and at I any time after immersion there is a larger surface area from 4.0 which absorption can take place. In the following analysis certain simplifying assumptions are made. There is assumed to be no resistance to mass transfer on the gas side of the interface, and the resistance on 30 the liquid side is assumed to be independent of the size or situation of the bubble. The gas in the bubble is always saturated with solvent vapour. The pressure is assumed to be constant. The concentration of dissolved gas in the bulk of the liquid is effectively zero. The bubbles considered are all 2.0 the same shape. A bubble is formed at t = 0, when: -z Volume of bubble = DO +I Partial volume of inerts = Mvo(1 - s) Partial volume of solvent vapour = DOS -1 I-O Partial volume of soluble gas = v0(1 - M)(l - s) where M is the initial mole-fraction of insoluble gas, calculated on a solvent-free basis. After time t, the situation is: Volume of bubble = v Partial volume of inerts = Mvo(1 - s) Partial volume of solvent vapour = us Partial volume of soluble gas = (v - Mv0)(1 - s) Mole-fraction of soluble gas = (1 - Mvo/v)(l - s). Now the rate of absorption is proportional to the molefraction of the soluble gas and to the surface-area of the bubble: i.e.
of
-(l
- s)$
= Kv3/3 (1 - M;)(1
- s)
FIG. 1
785
Shorter
communications From these expressions it is calculated that y/y’ is never less than about 0.89 in very dilute bubbles. For nearly pure bubbles (M < 1 - y)
and equation (3) becomes
$ = Ml,3[1(_l)l’3 _ +!L)l’3] Values of I(x) are shown in Fig. 1, plotted against (x - 1). It is thus possible to calculate, for any given value of Ikf, the fraction y of the soluble gas absorbed in a given time, t, and to compare it with the fraction y’ = 1 - (1 - t/~)~ absorbed from a eure bubble of the same initial volume in the same time. Simple expressions can be obtained for the behaviour under limiting conditions. Thus if the bubbles are mostly immersed for-a time much less than 7, the fraction of the soluble gas absorbed is almost exactly the same for pure and dilute bubbles, and the rate of absorption is thus approximately proportional, other things being equal, to the molefraction, (1 - M), of soluble gas in the bulk or inflowing gas. If conditions are such that if the gas were undiluted most of the bubbles would disappear before reemerging, then when the gas is diluted most of the bubbles will spend a time greater than 7 in the liquid, which implies that at least 95 per cent of the soluble gas will be absorbed from them (see Fig 2); in this case, too, the rate of absorption is approximately proportional to (1 - M). For very dilute bubbles (1 - A4 Q 1) ‘~~~ln[(l--A4)/(~-it4)] 7 whence Y
Y-; = Y Y=l-exp----, Y
1 -exp(-33/r) 1 - (1 - t17)3
(
31 7)
‘>l 7
t ’ ;
<
l
whence ;wI+M-~ Y
3M
1 - (1 Y
43’3
I
(9)
Thus the effect of small traces of diluent is correspondingly small. Fig. 2 shows some values of the ratio y/y’ for different values of M and of t/T. The ratio is never less than about 0.89, whatever the degree of dilution or the fraction of gas absorbed. Thus, in dynamically comparable situations, the rate of absorption of a diluted gas from a swarm of bubbles is equal, with an error of less than 11 per cent, to the rate of absorption of the pure gas multiplied by (1 - M) , where A4 is the mole-fraction of diluent gas in the bulk of the gas or in the inflowing gas mixture (calculated on a solvent vapour free basis); in other words, the rate of absorption is proportional to the partial pressure of soluble gas initially present in the bubbles. However, this conclusion may not be valid if the dynamical behaviour of the dilute bubbles is very different from that of the pure bubbles. Consider, for instance, conditions such that when a pure gas is used most of the bubbles disappear before leaving the liquid; if a dilute gas were used, and the dilute bubbles followed the same trajectories as the pure bubbles, most of the bubbles would spend a time greater than 7 in the liquid and hence would lose almost all their soluble gas by absorption, and the use of the initial partial pressure as driving force would give the right answer. But thedilute bubble is at all times after formation larger than a pure bubble of the same initial size; this may cause the dilute bubbles to spend a time substantially less than 7 in the liquid, and thus reduce the absorption rate.
FIG. 2 786
Shorter
Communications
In the case of very small bubbles which move so slowly relative to the liquid that transfer of dissolved gas away from the bubble is governed entirely by diffusion, to the exclusion of convection, the mass-transfer coefficient varies inversely as the diameter of the bubble. The preceding analysis does not apply if a substantial part of the absorption occurs from bubbles as small as this. It will not apply, either, if there is a substantial chance that a bubble coalesces with another before leaving the liquid, or if the depth of liquid is such that there is a substantial change in the pressure on a bubble during its immersion. The conditions in which the analysis given above is exactly applicable are somewhat circumscribed, therefore; but the general conclusion may nevertheless be valid in many circumstances of practicalinterest. Department of Chemical Engineering, Pembroke Street, Cambridge.
P. V. DANCKWERTS
NOTA-I-ION
m
Integral given in equation (4) Mass-transfer factor Initial mole-fraction of inert gas, solvent-vapour free basis s Mole-fraction of solvent vapour . t Time since immersion of bubble Volume of bubble at time t V I vo Initial volume of bubble Y Fraction of soluble gas absorbed from dilute bubble in time t Y' Fraction of soluble gas absorbed from pure bubble in time t.
K M
x
(v/Mvo)~~3
7
time taken for pure bubble to disappear.
Chemical Engineering Science, 1965, Vol. 20, pp. 787-788. Pergamon Press Ltd., Oxford. Printed in Great Britain.
Gas absorption accompanied by complex chemical reaction (Received 10 February 1965) IN A recent
paper, BRIAN and BEAVERSTOCK [I] presented film and penetration theory solutions for absorption accompanied by a two-step chemical reaction in which the absorbed species reacts with the liquid phase reagent to form an intermediate species, which in turn reacts with the reagent to form the final products. The authors suggested that their theoretical model might apply to the absorption of carbon dioxide into monoethanolamine (MEA) solutions and the absorption of chlorine into ferrous chloride solutions. The authors referred to experimental results for the former system which have been presented by EMMJXRT and PIGFORD [2]. On Fig. 6 of EMMERTand P~GFORD’Spaper, two series of results for almost identical values of Cc/C* (where CB is the concentration of reagent in the bulk liquid, and C* is the physical solubility of the solute gas) are plotted in the form kr,,/ke (ratio of mass-transfer coefficient in the presence of chemical reaction to the physical mass-transfer coefficient) against exposure time t:
where R represents -CaHaOH. The carbamic acid RNHCOOH, will be almost completely dissociated at the pH values encountered in slightly carbonated solutions, as its UK is unlikely to be much greater than 60 141. The proton formed by reaction (1) ii neutralized by a second molecule of amine in a virtually instantaneous ionic reaction: RNHz + H+ + RNH: (2) Although the reactions COa + He0 +HCO,-
+ H+
COa -t OH- + HCO;
also take place, consideration of the relevant equilibrium and rate data shows that they are insignificant compared with reactions (1) and (2). The absorption of carbon dioxide into MEA solutions at low carbonation ratios (moles carbon dioxide absorbed per mole amine) is thus an example of absorption accompanied by second order reaction with a stoichiometric coefficient (moles of reagent which react with (ii) CB = 0.5, C* = 0:0068. one mole of absorbed species) of 2.0. (This behaviour was in fact demonstrated by EMM~RTand PIGFORD, who calcuIn series (i) B/k! was observed to increase with t, and in series (ii) k&/k2 decreased with t. BRIAN and BEAVERSTOCK’s lated a stoichiometric coefficient of 1.78 from their results using an estimated value 0.59 for the ratio of the diffusivity model predicts, however, that if kL/kg decreased with t of MEA to the diffusivitv of carbon dioxide. The stoichiowhen CB = 0.5, it should also have decreased with t when CE = 1.0. The model is thus incompatible with EMMERT metric coefficient calculated from EMMERTand PIGFORD’s results becomes 1.88 if the more accurate diffusivity ratio and PIGFORD’Sresults. 0.66, calculated from the measurements reported in references It is, in fact, well established that when carbon dioxide [5-71, is used. This is in reasonably close agreement with the is absorbed into MEA solutions it undergoes a second-order predicted stoichiometric coefficient of 2.0). reaction to form a carbamic acid [3] : B~UANand BEAVERSTOCK referred to the results of GILLIRNHz + COa + RHNCOO- + H+ LANDet al. [81, who measured rates of absorption of chlorine (1)