- Email: [email protected]
Diffusional mass transfer to a growing bubble
401
Shorter Communications Concentration profile for species B b
=u~~~(~,~)+$$ l_s~h(4C-A))_A
e
41
cosh($&-AN > 5
416
IASE$511)
Departmentof Chemical Engineering Universityof Salford Salford M5 4WT, England
(30)
NOTATION
dimensionless concentration of species A, a = AIAb concentration of species A inside catalyst bulk concentration of species A dimensionless concentration of species B, b = B/Ah dimensionless concentration of species B in bulk solution, bb = BtJAb concentration of species B inside catalyst concentration of species B in bulk solution dimensionless concentration of species C, c = C/Ah dimensionless concentration of species C in bulk solution, cb = C,IAb concentration of species C inside catalyst concentration of species C in bulk solution diffusion coefficient of species A, B and C in bulk solution, assumed equal for the three species effective diiusion coefficient for species A, B and C inside the catalyst particle zero order rate constant first order rate constant mass transfer coefficient of species A, B and C in the bulk solution half thickness of flat slab radial distance inside spherical pellet distance inside the spherical particle at which the concentration of A falls to zero radius of spherical particle selectivity of species B modified Sherwood number, Sh = k,,,L/D. for tlat slab and =LRID. for spherical pellet distance from centre of flat slab distance from centre of flat slab at which the concentration of A falls to zero
A”
and b =;sinh(+,e)
106[5Al
(31)
a (sinh 41 +h
Ab b bb B Bb
where a is obtained from
C
(4, cash 4, - sinh 4,))
Cb
C C, D
=b,-$[l-sti($!l-A))-Acosh($,(l-A))
I
-$(l-A)cosh(+,(l-A)) D. + +,(A - 114,‘) sinh (&(l- 01-j.
(32)
The concentration gradient of B at the surface of the pellet is given by
1
+ sinh Q,A - 4,A cash &A).
(33)
The concentration profile and concentration gradient for component C can be obtained by use of eqns (21) and (22) and the selectivity determined from eqn (23). Fiie 2 shows the &ect of the zero order Thiele modulus on the selectivity of reaction scheme (1) for a spherical pellet. There is a sharp transition in the selectivity at the critical Thiele modulus &. Below the critical value &,., the selectivity increases in favour of B. Small spherical pellets and large bulk concentrations of A therefore favour the production of B. The effect of external mass transfer resistance, characterised by the modiied Sherwood number Sh, is also obtained in Fig. 2; this figure shows that the external mass transfer resistance plays a very important role in determining the selectivity of zero order reactions, especially for values of fjOs &. Department of Chemical Engineering Unioersityof Manchester Institute of Science and Technology Manchester M60 lQD, England
&?rwing
R. KRISHNA
Science, 1976, Vol. 31, pp. 401403.
L r r. R
= b,(o,cosh9,-sinh~,)-~(9,cosh~,-sinh~,
Chemid
P. A. RAMACHANDRAN
Pergamon Press.
Ptited
2 x 5
Greek symbols a7 P arbitrary constants 5 dimensionless distance, .$= x/L for flat slab and =r/R for spherical pellet dimensionless position, A =x,/L or =rJR characteristic Thiele modulus for zero order reaction characteristic Thiele modulus for first order reaction 1 4oc critical Thiele modulus for zero order reaction
$
REFERFNCES
HI Blackmore K., Luckett P. and Thomas W. J., Gem. Bngng
sci 197530 1285. PI Ramachandran P. A. and Krishna R., Influence of external and internal diffusion on zero order reactions, J. Appi. Chem. Biotechnol. to be published.
io Great Britain
Difhsfonal mass transferto a growingbubble (Received 21 Nooember 1975) In a supersaturated gaseous solution an initiated bubble will grow as material ditises from the surroundings to the bubble. At lirst the growth of the bubble depends very strongly on the inertia, viscosity and surface tension of the surrounding liquid. However, the growth sooner or later becomes limited by the rate at which material can diuse to the bubble surface. A solution of the
diffusion equation alone will then fully describe the growth rate of the bubble. Of the several attempts (see, e.g. Refs. [l-3]) which have been made to obtain a solution, Scriven’s[4] similarity solution is the most appropriate and has been recently found[S] to be accurate and applicable. The asymptotic diffusional bubble growth theory of Striven and
402
Shorter Communications
the two-film theory of absorption of gases may be combined to yield forms useful in the study of the growth of bubbles rising through a supersaturated solution.
In most cases of gases dissolved in liquids, p8 4p,, c ;i; 1 and pl D C.,, so eqn (2) becomes:
@=h’=Ja.
S&en’s solution The final result of Scriven’s[4] solution is: R = 2fi@t)
PS
(1)
where R is the bubble radius at time t, Id is the coefficient of diffusivity of the gas in the liquid and f3 is the “dimensionless growth coefficient”. Striven then obtained an expression relating the growth coefficient ,9 to the known physical properties of the solution thus:
(11)
When the Striven asymptotic approximation for small and large values of p (eqns 4 and 5) are substituted into eqns (IO)and (ll), the result is: Forsmallf3: @=Ja+2/3=
or Sh +2.
(12)
For large p:
or
e=l-Ps.
Sh+gJa Tr
PI
Q is a driving force parameter which is a known mathematical function of /3 and c and is solvable by numerical methods. For very small and very large growth rates p, there exist asymptotic approximations to @ thus: For small 8: WI4 P]+ZP’. (4) For large fl:
(13)
.
Equations (12) and (13) define two asymptotic values of the Sherwood number. Intermediate values are calculable and are shown graphically in Fig. 1.
lDOL
(5) Thus p, for use in eqn (l), may be found from the approximations of eqns (4) and (5) or from the fulJ numerical solution, results of which are presented in tabular and graphical form by Scriven[4]. Two -film theory
According to the Whitman two-film theory of absorption of gases[6,7], a general expression for mass transfer across the phase boundary is:
Ja -
Jacob
Number
Fig. 1. Jakob number vs Sherwood number for a growingbubble. N,=t(pi-PB)=k,(C,-Ci).
(6)
Since the concentration of dissolved gas at the interface is not generally known, it is usual to write eqn (6) in the following form, for the case where mass transfer is controlled by diffusion through the liquid film: N,=K,(C-C,,)=RAC.
(7)
Combination of the two theories Several similarities between the two-tilm and the Striven theories are apparent. Primarily both eqns (1) and (7) derive from the assumption that the rate of dithtsion (hence bubble growth) is limited by resistances on the liquid film side. Suppose the instantaneous mass transfer rate can be represented by an equation of the form of eqn (7). The rate of change of the volume of the growing bubble is: dV -=dr
a,,N, pe
(8)
or dR_N&GKJa dr ps ps
(9)
DISCUSSION
This is a particularly interesting result as eqn (12)is identical to the classical Langmuir theory for diffusion from a stationary solid sphere into an inEnite stagnant medium. This would be expected for a bubble growing intinitesimally slowly. The result suggests moreover that mass transfer to a bubble rising in supersaturated solution might be able to be dealt with by existing correlations for moving solid spheres which involve the Reynolds and Schmidt numbers. Certainly analytical or numerical solutions of the rising bubble problem are extremely dilficult and to date, there do not appear to have been any successful attempts to obtain solutions. It is thus suggested that it is likely a correlation of the form: Sh =*,I Ja
+f(Re
’
SC)]
(14)
will be found to satisfy the case of a bubble rising in a supersaturated solution. Acknowledgement-One of us (J.E.B.) wishes to thank CSR Limited, Sydney, Australia, for support during the period this work was carried out at Imperial College of Science and Technology, London.
Substitution of eqn (1) into eqn (9) yields: Sh _ 2RKi_ 4@ Ja ’ D Equation (10) gives the relationship between the instantaneous Sherwood number, Sh ; the Jakob number, Ja, and the dimensionless growth coefficient @. tPresent address: CSR Limited, Box 483, G.P.O., Sydney, N.S.W. 2001, Australia.
Department of Chemical Engineering & Chemical Technology Imperial College Prince Consort Road London SW7 2BY NOTATION
ab surface area, L2 C concentration of dissolved gas, MLm3 D coefficient of dithtsivity, L*T’
J. E. BURMANt G. J. JAMESON
Shorter Communications k, gas film mass transfer coefficient, LT’ k, liquid tilm mass transfer coefficient, L-‘T
K, N, P R t V
403
SC Schmidt number Sh Sherwood number
overall mass transfer coefficient, LT’ mass transfer flux, ML-*T-’ pressure, ML-‘Tm2 radius, L time, T volume. L”
Subscripts g gas phase i interface
I liquid phase
Greek symbols p dimensionless growth coefficient c defined in eqn (3) p density, ML-’ Cp driving force parameter
[l] Bankoff S. G., Aduan. Chem. Engng 19666 l-60. [2] Epstein P. S. and Plesset M. S., J. Chem. Phys. 195018 1505. [3] Szekely J. and Martins J. P., Chem. Engng Sci. 197126147. [4] Striven L. E., Chem Engng Sci. 195910 1. [5] Burman J. E., Ph.D. Thesis, University of London, 1974. [6] Whitman W. G., Chem Met. Engng 192329 147. [7] Co&on J. M. and Richardson J. F., Chemical Engineering, Vol. II, p. 701, Pergamon Press, London 1966.
Dimensionless groups Ja Jakob number Re Reynolds number
ChemicalEnginemingScience,1976,Vol. 31, pp. 4OM04. Pergram PN.K Printed in Great kitah
Gas bubble formation from submerged ori6ces- “simultaneous bubbling” from two orifices (Received 1 October 1975;accepted 26 Nooember 1975) The formation of gas bubbles from submerged orifices has been the object of numerous investigations[l-111. However, none of such works deals with bubble growing from a multiple-orifices plate feeded from a single chamber. In such a case, for sufficiently large values of the gas flowrate, bubbles grow simultaneously from all the orifices. For smaller flowrate values the plate works discontinously and as the number of oritices increases “simultaneous bubbling” becomes more ditficult. In this work volumes of gas (nitrogen) bubbles grown in water from a two-orifices plate have been obtained from measured values of flowrate and frequency. The experimental apparatus is equal to the one described in[ll], but for the plate in which two
o’Q’
conical or&ices were drilled, with the smaller section upwards. The minimum orifice diameter was 0.15cm and the pitch was 0.5 cm in all cases but one for reasons discussed in the following. In order to verify the equivalence between the or&es they were alternatively occluded, and the volume of the outcoming bubbles resulted the same within a few percent. Data taken when both oritices were working simultaneously (“simultaneous bubbling”) are reported in Figs. 1 and 2 together with others taken when one of the orifices was occluded. In Fig. 1 the bubble volume, V, is plotted vs the volumetric flowrate from single orifice, Q, obtained as the ratio between the volumetric flowrate coming into the feeding chamber, G, and the number of orifices. The volume of bubbles outcoming from single or&es feeded by chambers of 215and 415cm3is in Fig. 1 close to that of
_._
1v, cc
A
A 0
’
0.7 -
A
‘I
,
.
0
0
’
0.6 0.6 -
00
0.4 -
4’ 2
0.3 -
f
A
a2 -
l
.
V a V
<: < 0
-
0.0
- 0.5 -
v
0.4
- 0.3
Q, ccisec 0.2L 0
5
10
- 0.2
0
, 15
Fig. 1. Bubble volume vs volumetric gas flowrate from single or&e. +, chamber volume 215cd, single orifice; A, chamber volume 415cm3,two orifices; 0, chamber volume 415cm’, single orifice; A, chamber volume 785cm>,two orifices.
s.
I 0
200
400
000
I
cc 800
Fig. 2. Bubble volume vs the ratio between the volume of the feeding chamber and the number of or&es. 7, single orilice, pitch 1 cm; V, single or&e, pitch 0.5 cm; I, two orifices, pitch 1 cm; Cl, two orifices, pitch 0.5 cm.
Shorter Communications Concentration profile for species B b
=u~~~(~,~)+$$ l_s~h(4C-A))_A
e
41
cosh($&-AN > 5
416
IASE$511)
Departmentof Chemical Engineering Universityof Salford Salford M5 4WT, England
(30)
NOTATION
dimensionless concentration of species A, a = AIAb concentration of species A inside catalyst bulk concentration of species A dimensionless concentration of species B, b = B/Ah dimensionless concentration of species B in bulk solution, bb = BtJAb concentration of species B inside catalyst concentration of species B in bulk solution dimensionless concentration of species C, c = C/Ah dimensionless concentration of species C in bulk solution, cb = C,IAb concentration of species C inside catalyst concentration of species C in bulk solution diffusion coefficient of species A, B and C in bulk solution, assumed equal for the three species effective diiusion coefficient for species A, B and C inside the catalyst particle zero order rate constant first order rate constant mass transfer coefficient of species A, B and C in the bulk solution half thickness of flat slab radial distance inside spherical pellet distance inside the spherical particle at which the concentration of A falls to zero radius of spherical particle selectivity of species B modified Sherwood number, Sh = k,,,L/D. for tlat slab and =LRID. for spherical pellet distance from centre of flat slab distance from centre of flat slab at which the concentration of A falls to zero
A”
and b =;sinh(+,e)
106[5Al
(31)
a (sinh 41 +h
Ab b bb B Bb
where a is obtained from
C
(4, cash 4, - sinh 4,))
Cb
C C, D
=b,-$[l-sti($!l-A))-Acosh($,(l-A))
I
-$(l-A)cosh(+,(l-A)) D. + +,(A - 114,‘) sinh (&(l- 01-j.
(32)
The concentration gradient of B at the surface of the pellet is given by
1
+ sinh Q,A - 4,A cash &A).
(33)
The concentration profile and concentration gradient for component C can be obtained by use of eqns (21) and (22) and the selectivity determined from eqn (23). Fiie 2 shows the &ect of the zero order Thiele modulus on the selectivity of reaction scheme (1) for a spherical pellet. There is a sharp transition in the selectivity at the critical Thiele modulus &. Below the critical value &,., the selectivity increases in favour of B. Small spherical pellets and large bulk concentrations of A therefore favour the production of B. The effect of external mass transfer resistance, characterised by the modiied Sherwood number Sh, is also obtained in Fig. 2; this figure shows that the external mass transfer resistance plays a very important role in determining the selectivity of zero order reactions, especially for values of fjOs &. Department of Chemical Engineering Unioersityof Manchester Institute of Science and Technology Manchester M60 lQD, England
&?rwing
R. KRISHNA
Science, 1976, Vol. 31, pp. 401403.
L r r. R
= b,(o,cosh9,-sinh~,)-~(9,cosh~,-sinh~,
Chemid
P. A. RAMACHANDRAN
Pergamon Press.
Ptited
2 x 5
Greek symbols a7 P arbitrary constants 5 dimensionless distance, .$= x/L for flat slab and =r/R for spherical pellet dimensionless position, A =x,/L or =rJR characteristic Thiele modulus for zero order reaction characteristic Thiele modulus for first order reaction 1 4oc critical Thiele modulus for zero order reaction
$
REFERFNCES
HI Blackmore K., Luckett P. and Thomas W. J., Gem. Bngng
sci 197530 1285. PI Ramachandran P. A. and Krishna R., Influence of external and internal diffusion on zero order reactions, J. Appi. Chem. Biotechnol. to be published.
io Great Britain
Difhsfonal mass transferto a growingbubble (Received 21 Nooember 1975) In a supersaturated gaseous solution an initiated bubble will grow as material ditises from the surroundings to the bubble. At lirst the growth of the bubble depends very strongly on the inertia, viscosity and surface tension of the surrounding liquid. However, the growth sooner or later becomes limited by the rate at which material can diuse to the bubble surface. A solution of the
diffusion equation alone will then fully describe the growth rate of the bubble. Of the several attempts (see, e.g. Refs. [l-3]) which have been made to obtain a solution, Scriven’s[4] similarity solution is the most appropriate and has been recently found[S] to be accurate and applicable. The asymptotic diffusional bubble growth theory of Striven and
402
Shorter Communications
the two-film theory of absorption of gases may be combined to yield forms useful in the study of the growth of bubbles rising through a supersaturated solution.
In most cases of gases dissolved in liquids, p8 4p,, c ;i; 1 and pl D C.,, so eqn (2) becomes:
@=h’=Ja.
S&en’s solution The final result of Scriven’s[4] solution is: R = 2fi@t)
PS
(1)
where R is the bubble radius at time t, Id is the coefficient of diffusivity of the gas in the liquid and f3 is the “dimensionless growth coefficient”. Striven then obtained an expression relating the growth coefficient ,9 to the known physical properties of the solution thus:
(11)
When the Striven asymptotic approximation for small and large values of p (eqns 4 and 5) are substituted into eqns (IO)and (ll), the result is: Forsmallf3: @=Ja+2/3=
or Sh +2.
(12)
For large p:
or
e=l-Ps.
Sh+gJa Tr
PI
Q is a driving force parameter which is a known mathematical function of /3 and c and is solvable by numerical methods. For very small and very large growth rates p, there exist asymptotic approximations to @ thus: For small 8: WI4 P]+ZP’. (4) For large fl:
(13)
.
Equations (12) and (13) define two asymptotic values of the Sherwood number. Intermediate values are calculable and are shown graphically in Fig. 1.
lDOL
(5) Thus p, for use in eqn (l), may be found from the approximations of eqns (4) and (5) or from the fulJ numerical solution, results of which are presented in tabular and graphical form by Scriven[4]. Two -film theory
According to the Whitman two-film theory of absorption of gases[6,7], a general expression for mass transfer across the phase boundary is:
Ja -
Jacob
Number
Fig. 1. Jakob number vs Sherwood number for a growingbubble. N,=t(pi-PB)=k,(C,-Ci).
(6)
Since the concentration of dissolved gas at the interface is not generally known, it is usual to write eqn (6) in the following form, for the case where mass transfer is controlled by diffusion through the liquid film: N,=K,(C-C,,)=RAC.
(7)
Combination of the two theories Several similarities between the two-tilm and the Striven theories are apparent. Primarily both eqns (1) and (7) derive from the assumption that the rate of dithtsion (hence bubble growth) is limited by resistances on the liquid film side. Suppose the instantaneous mass transfer rate can be represented by an equation of the form of eqn (7). The rate of change of the volume of the growing bubble is: dV -=dr
a,,N, pe
(8)
or dR_N&GKJa dr ps ps
(9)
DISCUSSION
This is a particularly interesting result as eqn (12)is identical to the classical Langmuir theory for diffusion from a stationary solid sphere into an inEnite stagnant medium. This would be expected for a bubble growing intinitesimally slowly. The result suggests moreover that mass transfer to a bubble rising in supersaturated solution might be able to be dealt with by existing correlations for moving solid spheres which involve the Reynolds and Schmidt numbers. Certainly analytical or numerical solutions of the rising bubble problem are extremely dilficult and to date, there do not appear to have been any successful attempts to obtain solutions. It is thus suggested that it is likely a correlation of the form: Sh =*,I Ja
+f(Re
’
SC)]
(14)
will be found to satisfy the case of a bubble rising in a supersaturated solution. Acknowledgement-One of us (J.E.B.) wishes to thank CSR Limited, Sydney, Australia, for support during the period this work was carried out at Imperial College of Science and Technology, London.
Substitution of eqn (1) into eqn (9) yields: Sh _ 2RKi_ 4@ Ja ’ D Equation (10) gives the relationship between the instantaneous Sherwood number, Sh ; the Jakob number, Ja, and the dimensionless growth coefficient @. tPresent address: CSR Limited, Box 483, G.P.O., Sydney, N.S.W. 2001, Australia.
Department of Chemical Engineering & Chemical Technology Imperial College Prince Consort Road London SW7 2BY NOTATION
ab surface area, L2 C concentration of dissolved gas, MLm3 D coefficient of dithtsivity, L*T’
J. E. BURMANt G. J. JAMESON
Shorter Communications k, gas film mass transfer coefficient, LT’ k, liquid tilm mass transfer coefficient, L-‘T
K, N, P R t V
403
SC Schmidt number Sh Sherwood number
overall mass transfer coefficient, LT’ mass transfer flux, ML-*T-’ pressure, ML-‘Tm2 radius, L time, T volume. L”
Subscripts g gas phase i interface
I liquid phase
Greek symbols p dimensionless growth coefficient c defined in eqn (3) p density, ML-’ Cp driving force parameter
[l] Bankoff S. G., Aduan. Chem. Engng 19666 l-60. [2] Epstein P. S. and Plesset M. S., J. Chem. Phys. 195018 1505. [3] Szekely J. and Martins J. P., Chem. Engng Sci. 197126147. [4] Striven L. E., Chem Engng Sci. 195910 1. [5] Burman J. E., Ph.D. Thesis, University of London, 1974. [6] Whitman W. G., Chem Met. Engng 192329 147. [7] Co&on J. M. and Richardson J. F., Chemical Engineering, Vol. II, p. 701, Pergamon Press, London 1966.
Dimensionless groups Ja Jakob number Re Reynolds number
ChemicalEnginemingScience,1976,Vol. 31, pp. 4OM04. Pergram PN.K Printed in Great kitah
Gas bubble formation from submerged ori6ces- “simultaneous bubbling” from two orifices (Received 1 October 1975;accepted 26 Nooember 1975) The formation of gas bubbles from submerged orifices has been the object of numerous investigations[l-111. However, none of such works deals with bubble growing from a multiple-orifices plate feeded from a single chamber. In such a case, for sufficiently large values of the gas flowrate, bubbles grow simultaneously from all the orifices. For smaller flowrate values the plate works discontinously and as the number of oritices increases “simultaneous bubbling” becomes more ditficult. In this work volumes of gas (nitrogen) bubbles grown in water from a two-orifices plate have been obtained from measured values of flowrate and frequency. The experimental apparatus is equal to the one described in[ll], but for the plate in which two
o’Q’
conical or&ices were drilled, with the smaller section upwards. The minimum orifice diameter was 0.15cm and the pitch was 0.5 cm in all cases but one for reasons discussed in the following. In order to verify the equivalence between the or&es they were alternatively occluded, and the volume of the outcoming bubbles resulted the same within a few percent. Data taken when both oritices were working simultaneously (“simultaneous bubbling”) are reported in Figs. 1 and 2 together with others taken when one of the orifices was occluded. In Fig. 1 the bubble volume, V, is plotted vs the volumetric flowrate from single orifice, Q, obtained as the ratio between the volumetric flowrate coming into the feeding chamber, G, and the number of orifices. The volume of bubbles outcoming from single or&es feeded by chambers of 215and 415cm3is in Fig. 1 close to that of
_._
1v, cc
A
A 0
’
0.7 -
A
‘I
,
.
0
0
’
0.6 0.6 -
00
0.4 -
4’ 2
0.3 -
f
A
a2 -
l
.
V a V
<: < 0
-
0.0
- 0.5 -
v
0.4
- 0.3
Q, ccisec 0.2L 0
5
10
- 0.2
0
, 15
Fig. 1. Bubble volume vs volumetric gas flowrate from single or&e. +, chamber volume 215cd, single orifice; A, chamber volume 415cm3,two orifices; 0, chamber volume 415cm’, single orifice; A, chamber volume 785cm>,two orifices.
s.
I 0
200
400
000
I
cc 800
Fig. 2. Bubble volume vs the ratio between the volume of the feeding chamber and the number of or&es. 7, single orilice, pitch 1 cm; V, single or&e, pitch 0.5 cm; I, two orifices, pitch 1 cm; Cl, two orifices, pitch 0.5 cm.