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Dimpled and skirted liquid drops moving through viscous liquid media
.
Shorter Communications Department of Chemical Engineering University of Minnesota Minneapolis, Minn. 55435, USA.
deviation between the effectiveness factors for shapes “A” and “B” as a per cent of one of them, i.e. 100(q4 --q8)/q8. A and B denote one or the other of the three shapes P, C or S, and the maximum deviations are:
NOTATION (The figures in the brackets give the original reference for the Notation.)
A P
B
P
-
c
+11.9
as
+16.2
a
c
s
-10.5
-14.0
-
: b ;
4.0
+4.25
S. RESTER R. ARIS
9
-
i
All these occur for values of A between 1.3 and 1.7.
R QL= a/b B= Ub Y
Acknowledgment - We are indebted to D. J. Gunn for sending ‘us his computed values and N. R. Amundson and D. Luss for the use of their data.
z
A
inner radius of hollow cylinder [2] dimension of parallelpiped[3] outer radius of hollow cylinder [2] dimension of paraUelpiped[3] dimension of parallelpiped [3] effective diffisivity of catalyst particle[2,3] length or height of cylinder [3] first order reaction rate constant [2,3] length or height of cylinder[2] radius of solid cylinder [3] radius of sphere of equal volume to the particle [3] shape parameter shape parameter luss and Amundson reference, shape parameter[3] effectiveness factor thiele modulus [2,3] normalized Thiele modulus
REFERENCES [l] [21 [3] [41
ARIS R., Chem. Engng. Sci. 1957 6 262. GUNN D. J., Chem. Engng. Sci. 1967 22 1439. LUSS D. and AMUNDSON N. R.,A.I.Ch.E. .I11967 13 759. RESTER S., JOUVEN J. and ARIS R., Chem. Engng Sci. To be published.
Chemical Engineering Science, 1969, Vol. 24, pp. 795-797.
Pergamon Press.
Printed in Great Britain.
Dimpled and skirted liquid drops moving through viscous liquid media (First received 19 February 1968; in revisedform 4 November 1968) beyond the range covered in previous papers. Figures I through 4 show the stable configurations of large 2-butanone drops moving at terminal velocity through glycerol. The drop volumes, Reynolds and Weber numbers, terminal velocities, and spherical cap radii are listed in Table 1. The dimples and skirts formed in the trailing end of the
THE STUDY of drop flow in liquid-liquid systems has generally been limited to continuous phases of low or moderate viscosity [ l-41 and chambers where wall effects would influence the shape of very large drops [3-S]. The accompanying photographs show that drops take on complex shapes as Reynolds and Weber numbers increase
Table 1. Parameters of drop motion
Figure
Drop volume (cm?
V (cmlsec)
1 2 3 4
2.57 6.30 12.2 20.6
17.5 26.9 31.0 32.2
R (cm) 0.98 1.30 1.91 2.67
Re 10.2 21.1 30.3 37.6
For 2-butanone drops dispersed into glycerol, mutually saturated.
795
K =
Rb
We
Wi
1463 3030 4340 5380
25.2 80.5 133.0 171.0
17.5 55.7 91.8 118.0
O+~O698and y = O-691 when phases are
Fig 1 Small dimple at rear stagnation Width of drop about l-8 cm
pomt.
Fig 2 Promment dimple, with small drop trapped m the wake Drop width about 2-5 cm
Fig 3 Smooth, stable skirt attached meter of drop Drop width approx.
to pen3 6 cm. (Facmg page 796)
Shorter Communications drops are believed to be due to stable vortices in the wakes of the drops, and to the circulation[6,7] within the drops. This conclusion is reinforced by the two dimensional bubble wake studies of Collins[8] and of Crabtree and Bridgwater [9], and by the three dimensional spherical drop wake studies of Magarvey and MacLatchy[2]. It appears that the toroidal circulation in the wake of a large drop increases the stability of the dimple or skirt, and significantly delays the onset of vortex shedding. This phenomenon is not observed with a continuous phase of low or moderate viscosity. The interaction between dimpled drops is somewhat different from the behavior of spherical drops. If two drops are formed a fraction of a second apart with the second drop slightly larger, they will rise independently until the second drop reaches the circulating flow pattern below the first drop. The lower drop then begins to lengthen (Fig. 5) as it is accelerated into the expanding dimpleof the upper drop (Fig 6). The reader may note in Fig. 6 that the curved liquid skirt of the upper drop magnifies the leading edge of the lower drop when it is in the skirted region. Coalescence of the two dimpled drops is shown in Fig. 7. Considerable care was taken to exclude surfactants. This included hot chromic acid cleaning of the apparatus, which was constructed of stainless steel, teflon, and glass. Thus, dimpled and skirted drops cannot be attributed to the presence of surface-active contamination. The phases were mutually saturated before photographs were taken and temperature was held within tio3”C. Hence Marangoni effects were absent, no interfacial mass transfer was occurring, and inequalities of the interfacial tension over the surface of the drops can be excluded as a cause of the dimples and skirts. All drops were photographed in a chamber 71.6 cm high and 26.5 cm in dia. Drops were allowed to rise 30 cm before they were photographed. Thus, wall effects and nonequilibrium drop shape can be eliminated as sources of the dimple and skirt phenomena. The deformation phenomena illustrated by the photographs has also been observed in our laboratory with the following materials dispersed into glycerol: acetone, hexane, 2-ethyl-lhexanol, and 145 c.p. paraffin oil. Using parat% oil as the continuous phase, the dimple has been observed with two dispersed liquids: glycerol and water. The theoretical analysis of drop and bubble shape by Pan and Acrivos [ lo] predicts the formation of dimpled drops, but unfortunately that analysis cannot be applied in its present form to the large skirted drops which we have observed, since the perturbation analysis assumes small deviations from perfect spheres.
Acknowledgments-This work was suported by a Monsanto Company Fellowship and a National Aeronautics and Space
Administration Traineeship. The equipment was provided by the National Science Foundation from a previous project, P. D. SHOEMAKER L. E. MARC DE CHAZAL Department of Chemical Engineering University of Missouri Columbia, Missouri
NOTATION a radius of a sphere having a volume equivalent to the volume of the deformed drop R radius of curvature of the leading edge of a drop Re Reynolds number of the drop, baaed on continuous phase properties,
Re=Ua v
Rf Reynolds number of the drop, based on dispersed phase (drop) properties,
Weber number of the drop, based on continuous phase properties,
we Weber number of the drop, based on dispersed (drop) properties, Be^-l)a
u2 (+
U terminal velocity of a drop y ratio of dispersed phase density to continuous density,
phase
y=; K ratio of dispersed viscosity,
phase viscosity to continuous phase Kc- D v
,,
v kinematic viscosity of the continuous phase 1; kinematic viscosity of the dispersed phase p density of the continuous phase p density of the dispersed phase (drop) m interfacial tension.
REFERENCES
111 HU SHENGEN and KINTNER, R. C.,A.Z.Ch.E. Jf 1955 142. PI MAGARVEY R. H. and MACLATCHY, C. S., A.Z.Ch.E. JI 1968 14 260.
[31 THORSEN G., STORDALEN R. M. and TERJESEN S. G., Chem. Engng Sci. 1968 23 413. 141 WELLEK R. M., AGRAWAL A. K. and SKELLAND A. H. P.,A.Z.Ch.E. JI 1966 12 854. PI HAYASHI SHIGERU and MATUNOBU YASO’O, J. phys. Sot., Japan 1967 22 905. Comot. Rend. 1911 152 1735: 1912 154 109. [61 HADAMARDJ..
796
phase
(drop)
Shorter Communications [7] [S] [9] [lo]
RYBCZYNSKI W., Bull Acad. Sci de Cracovie, A 1911 p. 40. COLLINS R.,J.fluidMech. 1966 25 469. CRABTREE J. R. and BRIDGWATER J., Chem. Engng Sci. 1967 22 1517. PAN FRANK Y. and ACRIVOS ANDREAS, Znd. Engng Chem. Fundls 1968 7 227.
Chemical Engineering Science, 1969, Vol. 24, pp. 797-799.
Pergamon Press.
Printed in Great Britain.
Isothermal solid-catalyzed zeroth-order reactions (Received 10 October 1968) The effectiveness functions in isothermal solid-catalyzed zeroth-order reactions can be used to describe related problems concerning catalytic selectivity and poisoning. Let us define the geometry of the catalyst particle (flatplate with sealed ends, cylinder with sealed ends or a sphere) by the index and exponent j: volume of particle j - RjO/ external surface of particle
(1)
the concentration
where Rjo is the half-thickness or radius of particle and where j assumes the values I,2 and 3 for the three shapes respectively. The effectiveness function Qj (xj) :
of component Al (i G n - 1) drops to zero, . . S ,_$=, I,), at the extinction
as m, > d(2jS,S,.
length [2]
RiJ < R,_,J, where Ri_lJ is the extinction length of component Ai_,. The material balance equation ofAi results in:
@I(&) - (l-x,)*
(2)
@)(xy) = 1 --x*(1 -lnx,)
(3)
$(x3)
(4)
= 1+ 2x, - 3x32’3,
Consecutive reactions. A, A Az % es*“-% Ai xi, . . . !&A&6]. Let us introduce the ith-specific selectivity as ratio between the surface rate constants of two consecutive reactions, St = K,/&+1 (S, = 1). If: L-l z I* < S-I ;: I*, s=l li=1
d2[A,] j- ld[A,] uKi -=+--6s,_,) dR1 +RjdRj ,
(8)
with 6 = 1 for R,_lJ =GRj G Rjo
when x, = 1 -Q, gives the relation between catalytic effectiveness, r),, and Thiele modulus, tn, = R,OV((aKIIAJo-‘/
6=0
O,),U-31:
forR,_, s Rj c R j
and boundary conditions: @,(I -nr) = y.
CATALYTIC
(5)
RI=&.,
SELECTIVITY
Selectivity depends on the Thiele modulus, specific selectivity and ratio between the rates of mass transfer of Al and AI by diffusion at the level of fluid-catalyst interface concentrations, 1, = [A,]@,/[A,]d%. Side-by-side reactions. A, 4 Ai (desired), A, * Ai (unwanted) 141. The selectivity, defined as ratio between rates of transport of reactants AI and& results in: 4’ where S,,‘ = K,IKI. q1.i is 1 as m,/q(S,,,Z,) following equation:
[~R,=R,o= [AJo, Ml R,+:_,.,= [A&,+-,,,
s1,21)j,Il%.2 < d(2j);
it is obtained
@j (1- 7j.i) = 2jSlJi/mj
(6) by the
Rj=Ri_,,j 1 LA,1R,=R&o
(R ,,/Rjo) 1is derived by the equation: (9) 1-1 Thus n regimes follow one another as the Thiele modulus increases: lth) chemical regime with respect to AI, A*, . . . . . A 1,. . . . . , A,,-, ([A,] > 0 at R, < RIO) as m,< d(2j) The selectivities (yields of A,),
(7)
as mj/~CktZO> d(2j). 797
l,j =
Shorter Communications Department of Chemical Engineering University of Minnesota Minneapolis, Minn. 55435, USA.
deviation between the effectiveness factors for shapes “A” and “B” as a per cent of one of them, i.e. 100(q4 --q8)/q8. A and B denote one or the other of the three shapes P, C or S, and the maximum deviations are:
NOTATION (The figures in the brackets give the original reference for the Notation.)
A P
B
P
-
c
+11.9
as
+16.2
a
c
s
-10.5
-14.0
-
: b ;
4.0
+4.25
S. RESTER R. ARIS
9
-
i
All these occur for values of A between 1.3 and 1.7.
R QL= a/b B= Ub Y
Acknowledgment - We are indebted to D. J. Gunn for sending ‘us his computed values and N. R. Amundson and D. Luss for the use of their data.
z
A
inner radius of hollow cylinder [2] dimension of parallelpiped[3] outer radius of hollow cylinder [2] dimension of paraUelpiped[3] dimension of parallelpiped [3] effective diffisivity of catalyst particle[2,3] length or height of cylinder [3] first order reaction rate constant [2,3] length or height of cylinder[2] radius of solid cylinder [3] radius of sphere of equal volume to the particle [3] shape parameter shape parameter luss and Amundson reference, shape parameter[3] effectiveness factor thiele modulus [2,3] normalized Thiele modulus
REFERENCES [l] [21 [3] [41
ARIS R., Chem. Engng. Sci. 1957 6 262. GUNN D. J., Chem. Engng. Sci. 1967 22 1439. LUSS D. and AMUNDSON N. R.,A.I.Ch.E. .I11967 13 759. RESTER S., JOUVEN J. and ARIS R., Chem. Engng Sci. To be published.
Chemical Engineering Science, 1969, Vol. 24, pp. 795-797.
Pergamon Press.
Printed in Great Britain.
Dimpled and skirted liquid drops moving through viscous liquid media (First received 19 February 1968; in revisedform 4 November 1968) beyond the range covered in previous papers. Figures I through 4 show the stable configurations of large 2-butanone drops moving at terminal velocity through glycerol. The drop volumes, Reynolds and Weber numbers, terminal velocities, and spherical cap radii are listed in Table 1. The dimples and skirts formed in the trailing end of the
THE STUDY of drop flow in liquid-liquid systems has generally been limited to continuous phases of low or moderate viscosity [ l-41 and chambers where wall effects would influence the shape of very large drops [3-S]. The accompanying photographs show that drops take on complex shapes as Reynolds and Weber numbers increase
Table 1. Parameters of drop motion
Figure
Drop volume (cm?
V (cmlsec)
1 2 3 4
2.57 6.30 12.2 20.6
17.5 26.9 31.0 32.2
R (cm) 0.98 1.30 1.91 2.67
Re 10.2 21.1 30.3 37.6
For 2-butanone drops dispersed into glycerol, mutually saturated.
795
K =
Rb
We
Wi
1463 3030 4340 5380
25.2 80.5 133.0 171.0
17.5 55.7 91.8 118.0
O+~O698and y = O-691 when phases are
Fig 1 Small dimple at rear stagnation Width of drop about l-8 cm
pomt.
Fig 2 Promment dimple, with small drop trapped m the wake Drop width about 2-5 cm
Fig 3 Smooth, stable skirt attached meter of drop Drop width approx.
to pen3 6 cm. (Facmg page 796)
Shorter Communications drops are believed to be due to stable vortices in the wakes of the drops, and to the circulation[6,7] within the drops. This conclusion is reinforced by the two dimensional bubble wake studies of Collins[8] and of Crabtree and Bridgwater [9], and by the three dimensional spherical drop wake studies of Magarvey and MacLatchy[2]. It appears that the toroidal circulation in the wake of a large drop increases the stability of the dimple or skirt, and significantly delays the onset of vortex shedding. This phenomenon is not observed with a continuous phase of low or moderate viscosity. The interaction between dimpled drops is somewhat different from the behavior of spherical drops. If two drops are formed a fraction of a second apart with the second drop slightly larger, they will rise independently until the second drop reaches the circulating flow pattern below the first drop. The lower drop then begins to lengthen (Fig. 5) as it is accelerated into the expanding dimpleof the upper drop (Fig 6). The reader may note in Fig. 6 that the curved liquid skirt of the upper drop magnifies the leading edge of the lower drop when it is in the skirted region. Coalescence of the two dimpled drops is shown in Fig. 7. Considerable care was taken to exclude surfactants. This included hot chromic acid cleaning of the apparatus, which was constructed of stainless steel, teflon, and glass. Thus, dimpled and skirted drops cannot be attributed to the presence of surface-active contamination. The phases were mutually saturated before photographs were taken and temperature was held within tio3”C. Hence Marangoni effects were absent, no interfacial mass transfer was occurring, and inequalities of the interfacial tension over the surface of the drops can be excluded as a cause of the dimples and skirts. All drops were photographed in a chamber 71.6 cm high and 26.5 cm in dia. Drops were allowed to rise 30 cm before they were photographed. Thus, wall effects and nonequilibrium drop shape can be eliminated as sources of the dimple and skirt phenomena. The deformation phenomena illustrated by the photographs has also been observed in our laboratory with the following materials dispersed into glycerol: acetone, hexane, 2-ethyl-lhexanol, and 145 c.p. paraffin oil. Using parat% oil as the continuous phase, the dimple has been observed with two dispersed liquids: glycerol and water. The theoretical analysis of drop and bubble shape by Pan and Acrivos [ lo] predicts the formation of dimpled drops, but unfortunately that analysis cannot be applied in its present form to the large skirted drops which we have observed, since the perturbation analysis assumes small deviations from perfect spheres.
Acknowledgments-This work was suported by a Monsanto Company Fellowship and a National Aeronautics and Space
Administration Traineeship. The equipment was provided by the National Science Foundation from a previous project, P. D. SHOEMAKER L. E. MARC DE CHAZAL Department of Chemical Engineering University of Missouri Columbia, Missouri
NOTATION a radius of a sphere having a volume equivalent to the volume of the deformed drop R radius of curvature of the leading edge of a drop Re Reynolds number of the drop, baaed on continuous phase properties,
Re=Ua v
Rf Reynolds number of the drop, based on dispersed phase (drop) properties,
Weber number of the drop, based on continuous phase properties,
we Weber number of the drop, based on dispersed (drop) properties, Be^-l)a
u2 (+
U terminal velocity of a drop y ratio of dispersed phase density to continuous density,
phase
y=; K ratio of dispersed viscosity,
phase viscosity to continuous phase Kc- D v
,,
v kinematic viscosity of the continuous phase 1; kinematic viscosity of the dispersed phase p density of the continuous phase p density of the dispersed phase (drop) m interfacial tension.
REFERENCES
111 HU SHENGEN and KINTNER, R. C.,A.Z.Ch.E. Jf 1955 142. PI MAGARVEY R. H. and MACLATCHY, C. S., A.Z.Ch.E. JI 1968 14 260.
[31 THORSEN G., STORDALEN R. M. and TERJESEN S. G., Chem. Engng Sci. 1968 23 413. 141 WELLEK R. M., AGRAWAL A. K. and SKELLAND A. H. P.,A.Z.Ch.E. JI 1966 12 854. PI HAYASHI SHIGERU and MATUNOBU YASO’O, J. phys. Sot., Japan 1967 22 905. Comot. Rend. 1911 152 1735: 1912 154 109. [61 HADAMARDJ..
796
phase
(drop)
Shorter Communications [7] [S] [9] [lo]
RYBCZYNSKI W., Bull Acad. Sci de Cracovie, A 1911 p. 40. COLLINS R.,J.fluidMech. 1966 25 469. CRABTREE J. R. and BRIDGWATER J., Chem. Engng Sci. 1967 22 1517. PAN FRANK Y. and ACRIVOS ANDREAS, Znd. Engng Chem. Fundls 1968 7 227.
Chemical Engineering Science, 1969, Vol. 24, pp. 797-799.
Pergamon Press.
Printed in Great Britain.
Isothermal solid-catalyzed zeroth-order reactions (Received 10 October 1968) The effectiveness functions in isothermal solid-catalyzed zeroth-order reactions can be used to describe related problems concerning catalytic selectivity and poisoning. Let us define the geometry of the catalyst particle (flatplate with sealed ends, cylinder with sealed ends or a sphere) by the index and exponent j: volume of particle j - RjO/ external surface of particle
(1)
the concentration
where Rjo is the half-thickness or radius of particle and where j assumes the values I,2 and 3 for the three shapes respectively. The effectiveness function Qj (xj) :
of component Al (i G n - 1) drops to zero, . . S ,_$=, I,), at the extinction
as m, > d(2jS,S,.
length [2]
RiJ < R,_,J, where Ri_lJ is the extinction length of component Ai_,. The material balance equation ofAi results in:
@I(&) - (l-x,)*
(2)
@)(xy) = 1 --x*(1 -lnx,)
(3)
$(x3)
(4)
= 1+ 2x, - 3x32’3,
Consecutive reactions. A, A Az % es*“-% Ai xi, . . . !&A&6]. Let us introduce the ith-specific selectivity as ratio between the surface rate constants of two consecutive reactions, St = K,/&+1 (S, = 1). If: L-l z I* < S-I ;: I*, s=l li=1
d2[A,] j- ld[A,] uKi -=+--6s,_,) dR1 +RjdRj ,
(8)
with 6 = 1 for R,_lJ =GRj G Rjo
when x, = 1 -Q, gives the relation between catalytic effectiveness, r),, and Thiele modulus, tn, = R,OV((aKIIAJo-‘/
6=0
O,),U-31:
forR,_, s Rj c R j
and boundary conditions: @,(I -nr) = y.
CATALYTIC
(5)
RI=&.,
SELECTIVITY
Selectivity depends on the Thiele modulus, specific selectivity and ratio between the rates of mass transfer of Al and AI by diffusion at the level of fluid-catalyst interface concentrations, 1, = [A,]@,/[A,]d%. Side-by-side reactions. A, 4 Ai (desired), A, * Ai (unwanted) 141. The selectivity, defined as ratio between rates of transport of reactants AI and& results in: 4’ where S,,‘ = K,IKI. q1.i is 1 as m,/q(S,,,Z,) following equation:
[~R,=R,o= [AJo, Ml R,+:_,.,= [A&,+-,,,
s1,21)j,Il%.2 < d(2j);
it is obtained
@j (1- 7j.i) = 2jSlJi/mj
(6) by the
Rj=Ri_,,j 1 LA,1R,=R&o
(R ,,/Rjo) 1is derived by the equation: (9) 1-1 Thus n regimes follow one another as the Thiele modulus increases: lth) chemical regime with respect to AI, A*, . . . . . A 1,. . . . . , A,,-, ([A,] > 0 at R, < RIO) as m,< d(2j) The selectivities (yields of A,),
(7)
as mj/~CktZO> d(2j). 797
l,j =