- Email: [email protected]
Mass transfer with chemical reaction for arbitrary bodies—Infinitely fast homogeneous reaction
Shorter Communications d 2G = ds2 s=0 ( z + T°)2+T2 = Acr2+AP2
Department o f Chemical Engineering University o f N e w Brunswick Frederiction, N e w Brunswick Canada
T2
Ao-*2 = _ _ ( r + To) 2" The above method has three advantages, in addition to the fact that it can be widely applied to any flow system in which a tracer response technique can be used. Firstly, the complete solution in the s-plane is not required and hence the specification for the input function is not necessary. Secondly, it is based on the use of transfer functions, and empirical transfer function through a stimulus response technique can be easily obtained even when the transfer function is of general form. Thirdly, it is radically simpler for calculation purposes than the previous methods. Using this procedure an inter-relationship among the parameters of the various models representing a flow system can be easily obtained. For example, if the three sample models given above can each adequately represent the same flow system, then
UL
[1] [2] [3] [4] [5]
C(t) C (s) Co (t) D G (s) L N s U x t tz ~r2 tr .2
L. W. SHEMILT* S. C. MEHTA*
NOTATION concentration of tracer in fluid as a function of time, at a given location concentration of tracer in fluid as a function of Laplace variable, s initial concentration of tracer the longitudinal dispersion coefficient, (LZ/T) system transfer function length of the test section number of stirred tanks in series Laplace variable average velocity of the fluid ( L / T ) distance from the entrance of the measurement point time mean of the time concentration curve variance of the time concentration curve, T 2 variance of the C-curve (dimensionless)
*Present address: Department of Chemical Engineering, McMaster University, Hamilton, Ontario, Canada.
N
REFERENCES L E V E N S P I E L O. and SMITH W. K., Chem. Engng Sci. 1957 6 227. V A N DER LAAN, E. Th., ibid. 1958 7 187. ARIS R., Chem. Engng Sci. 1959 9 266. B I S C H O F F K. B., Chem. Engng Sci. 1960 12 69. B I S C H O F F K. B., Can.J. chem. Engng 1963 41 129.
Chemical Engineering Science, 197 I, Vol. 26, pp. 1779-1781.
Pergamon Press.
Printed in Great Britain
Mass transfer with chemical reaction for arbitrary bodies--Infinitely fast homogeneous reaction (First received 16 February 1970; accepted 1 December 1970) AN IMPORTANT limiting case of simultaneous mass transfer and chemical reaction is that where the reaction may be considered to be infinitely fast. This case is amenable to analytical solution and the boundary layer model has been employed to estimate transfer rates for two dimensional surfaces[l, 2]. The purpose of this communication is to extend the analysis to arbitrary geometry, i.e. three dimensional as well as two dimensional bodies. In dealing with the three dimensional case, the method of Stewart [31 and Lightfoot[4] is closely followed and an identical curvilinear coordinate system, x, y, z, is employed. The continuity equation is
h~hz:[~~xl(hzlul)+~yl(hxlh.lv,)+ 1
O
0
~zl (hxlWl)] = 0
where xl -- x/L, vl = ReV2v/U=,
ul = u/U®, zl = z/L,
Yl = Rell2Y/L wl = w/U=
and h~ and hz, are scale factors. Definitions of variables and symbols are listed in the nomenclature. Consider the infinitely fast reaction of qA + B ~ products.
(2)
At steady state, the boundary layer equations for the conservation of components A and B in the continuous phase
are
(1) 1779
OcA_ OCA u~ Ox----~v~ ,gy~
1 02CA Sc,~ Oy~2
(3)
Shorter Communications ac.._.~n, MI OXl "1- I)1
acB Oy~
1
O2CB
with boundary conditions
(4)
S c n Oyl 2 "
C--A= 1, Ca=O, Ca=l,
The following boundary conditions may apply: CA = CA ( 0 ) ,
Yl = O,
X1 > 0
ca=O, cA = O, cn = cn(o~) ca = 0
xl =O, y~ -'---yn(x~) y~ = Yl ~- y R ( X l )
Yl > 0
aq=O n=m, ~=oo
~-'d'~-'~Jns = ~ \ d~l ]n R
(12)
where
SCA \ OYl / ItR = - S-~c~\~r~/~"
.
_ Scn OCA
a -- 7:'_-
(5)
It will be assumed that the influence of mass transfer on the velocity profile is negligible and that the reaction zone is very close to the surface, so that the x and z components
Sh(:,,)=-,3°% \ Oy /u=o
and
fl =
cB(~) qCA (o)"
Equation (!1) can easily be solved; r/n is determined from the final boundary condition and the local Sherwood number is
( m + 2)mltm+2)(O-lhza)lffm+l)Rell2ScA ll~m+2)
J:""+' z-'m+'"+"exp (--z) d z [ f [
of the velocity profile can be expressed as Ul = or1 (xx, z,)yfl ~
In the absence o f chemical reaction, */n tends to infinity and it follows that the enhancement factor is
(6)
14'1 ~ 0 .
This latter assumption is true only for Sc -> 1, but as pointed out by Lochiel and Calderbank [5], this generally applies for the dissolution of solids, liquids and gases in liquid for R e > 0.1. It should be emphasized that the postulated boundary conditions imply that the external-phase mass transfer resistance is controlling. Also, the assumption tht w~ is zero is identically true only for two dimensional or axisymmetric flows, and in the limiting case o f S c ~ ~. When the interface is moving rapidly compared to the velocity of translation, the velocity gradient is essentially zero. For this case m = 0. Alternatively, when the surface is rigid or immobile, the velocity gradient can be assumed linear and m = 1. Realistically, m may have some value between 0 and 1. The velocity component in the y direction is obtained from Eq. (1) as
(13)
(or,h,,)~'"+'h,,h~, dx,] '''+~'
i_ Sh(xl) = F[l/(m+2)] - Sho(xa) fnnm+~z-'m+lt'+2'exp (--z) dz"
Acrivos[l] treated the case when m = 1 and hx, = hz, = 1. F o r m = 0,1 = l/erflqn. To further illustrate the application o f this analysis for the prediction of transfer rates, consider mass transfer around a fully circulating spherical bubble or drop. The characteristic length, L, is taken as the diameter and h~, = 1, h,~ = (a sin O)/2a --- (sin 0)/2 and dx~ = d0/2. If the Reynolds number is less than 1, it follows from the Hadamard stream function that the tangential velocity component can be expressed as U~
= - - pl m+l. _1 0(h~m). /)1 h x l h z , ( m + 1) Oxl
u = 2[~ (P,c +tzd) sm 0.
(7)
cdc~(~) = c~(n)
ul
or,
(9)
The boundary layer thickness, 8, is expressed as
a
I
(o'~h,,) u'"+"
= sinO( /z~ 2 \~c+#a/
(16)
and
(8)
where "q = yl/8(xl, zl).
(15)
Thus
In order to obtain an analytical solution, a similarity transformation is employed. Let C~/CA(o) = cA (n)
(14)
=sin0{
Izc ~
2 \l,te+l.~a/"
(17)
Substitution into Eq. (13) gives
f(m+2)'fZ'(h~,hzO(o,hz,)aton+,,dx,]V"+2 L---~ca .Io
/3[~'~[
2+cos0
-Sho(x,) X/~,c+~d)~,(l+cos0)
~n
)
2 re.
(18)
(10) and Eqs. (3) and (4) become
Integration o v e r t h e entire surface y i e l d s t h e o v e r a l l S h e r wood number
d__~_a+ (m+2),7~,+, d ~ . = 0 drt 2
d'q
1/2
Sho = 0"65{~ ] e e l'z \~c + la,d]
(11) d2C'n + ( m + 2)otn"+' d ¢ n = 0 dn
(19)
The form of this equation coincides with that presented by 1780
Shorter Communications many authors, and the equation is identical to that derived by Lochiel and Calderbank [5]. For mass transfer to spherical drops or bubbles at high Reynolds number, the analysis of Moore [6] and Cheh and Tobias [7] can be applied to derive 2+3/~a/tLc l'45]t/~/.___/bx.__XlJ2- 1,2 - i l Sho = 1.13 1 - 1 + (Pa/xa~ v2 -Rett2J \tzc + txa]l re '.
(20)
\pcIl~c/ U,
This same problem was studied by Cheh and Tobias[7]. They obtained values of Sho numerically, apparently not realizing that an analytical solution can be obtained. The extension of the present analysis to non-isothermal systems is straightforward and is omitted here. Acknowledgement--T. Akiyama gratefully acknowledges financial support in the form of a post-doctoral fellowship from the University of Alberta. Department o f Chemical and Petroleum Engineering University o f Alberta Edmonton, Alberta, Canada
a A, B c c(o) c (~) C
T. A K I Y A M A * F. D. OTTO
NOTATION radius of a sphere two reacting components concentration of species in solvent saturation concentration of solute in solvent concentration in bulk solvent dimensionlessconcentrationCa=ca/ca(o) CB = CB/C~(Oo)
tPresent address: Department of Chemical Engineering, Shizuoka University, Hamamatsu, Shizuoka, Japan. [1] [2] [3] [4] [5] [6] [7]
diffusion coefficient scale factors enhancement factor mass transfer coefficient in continuous phase characteristic length real number m q stoichiometric coefficient U= main stream fluid velocity 19, W component of velocity for x, y, z coordinates respectively x distance along the surface following the adjacent streamlines y distance from the surface along its local normal z distance along the surface perpendicular to the adjacent streamlines Re Reynolds number U®Lpdl~c Sc Schmidt number I~dpcD Sh Sherwood number kLL/D
D hx, h~, hz 1 kL L
Greek symbols dimensionless fluid parameter ScdSca dimensionless fluid parameter c n ( OO) / qc A( o ) 6 boundary layer thickness / 0 polar angle "1 similarity variable defined by Eq. (9) cr1 defined by Eq. (6) #, viscosity O density Subscripts i A,B C d o R
dimensionless variable components A and B continuous phase dispersed phase value without chemical reaction reaction zone
REFERENCES A C R I V O S A., Chem. Engng Sci. 1960 13 57. F R I E D L A N D E R S. K. and LITT M., Chem. Engng Sci. 1958 7 229. S T E W A R T W . E.,A.I.Ch.E.J11963 9 528. L I G H T F O O T E. N., Lectures in Transport Phenomena 1964 4 44; A.I.Ch.E. Continuing Education Series. L O C H I E L A. C. and C A L D E R B A N K P. H., Chem. Engng Sci. 1964 19 471. M O O R E D. W.,J. FluidMech. 1963 16 161. C H E H H. Y. and T O B I A S C. W., Ind. Engng Chem. Fundls 1968 7 48.
Chemical Engineering Science, 1971, Vol. 26, pp. 1781-1783.
Pergamon Press.
Printed in Great Britain
Hold-up prediction in packed columns for co-current gas-liquid downflow (Received25 March 1971) INTRODUCTION SOME industrial gas-liquid contact processes require co-current flow. The advantage of this type of flow is that it carries larger volumes of each phase through the packing or catalyst, as opposed to counter-current flow which is limited by flooding. Charpentier et al. have described experimental results and formulated semi-empirical expressions for predicting the
pressure drop[2] and hold-up[l] in co-current gas-liquid downflow inside a Raschig ring packed column. References [2] and [3] contain a detailed description of the installation, the properties of the fluids, the packing characteristics and the measuring techniques.
1781
STRATIFIED-PORE MODEL An interactionless stratifed-pore model is described in
Department o f Chemical Engineering University o f N e w Brunswick Frederiction, N e w Brunswick Canada
T2
Ao-*2 = _ _ ( r + To) 2" The above method has three advantages, in addition to the fact that it can be widely applied to any flow system in which a tracer response technique can be used. Firstly, the complete solution in the s-plane is not required and hence the specification for the input function is not necessary. Secondly, it is based on the use of transfer functions, and empirical transfer function through a stimulus response technique can be easily obtained even when the transfer function is of general form. Thirdly, it is radically simpler for calculation purposes than the previous methods. Using this procedure an inter-relationship among the parameters of the various models representing a flow system can be easily obtained. For example, if the three sample models given above can each adequately represent the same flow system, then
UL
[1] [2] [3] [4] [5]
C(t) C (s) Co (t) D G (s) L N s U x t tz ~r2 tr .2
L. W. SHEMILT* S. C. MEHTA*
NOTATION concentration of tracer in fluid as a function of time, at a given location concentration of tracer in fluid as a function of Laplace variable, s initial concentration of tracer the longitudinal dispersion coefficient, (LZ/T) system transfer function length of the test section number of stirred tanks in series Laplace variable average velocity of the fluid ( L / T ) distance from the entrance of the measurement point time mean of the time concentration curve variance of the time concentration curve, T 2 variance of the C-curve (dimensionless)
*Present address: Department of Chemical Engineering, McMaster University, Hamilton, Ontario, Canada.
N
REFERENCES L E V E N S P I E L O. and SMITH W. K., Chem. Engng Sci. 1957 6 227. V A N DER LAAN, E. Th., ibid. 1958 7 187. ARIS R., Chem. Engng Sci. 1959 9 266. B I S C H O F F K. B., Chem. Engng Sci. 1960 12 69. B I S C H O F F K. B., Can.J. chem. Engng 1963 41 129.
Chemical Engineering Science, 197 I, Vol. 26, pp. 1779-1781.
Pergamon Press.
Printed in Great Britain
Mass transfer with chemical reaction for arbitrary bodies--Infinitely fast homogeneous reaction (First received 16 February 1970; accepted 1 December 1970) AN IMPORTANT limiting case of simultaneous mass transfer and chemical reaction is that where the reaction may be considered to be infinitely fast. This case is amenable to analytical solution and the boundary layer model has been employed to estimate transfer rates for two dimensional surfaces[l, 2]. The purpose of this communication is to extend the analysis to arbitrary geometry, i.e. three dimensional as well as two dimensional bodies. In dealing with the three dimensional case, the method of Stewart [31 and Lightfoot[4] is closely followed and an identical curvilinear coordinate system, x, y, z, is employed. The continuity equation is
h~hz:[~~xl(hzlul)+~yl(hxlh.lv,)+ 1
O
0
~zl (hxlWl)] = 0
where xl -- x/L, vl = ReV2v/U=,
ul = u/U®, zl = z/L,
Yl = Rell2Y/L wl = w/U=
and h~ and hz, are scale factors. Definitions of variables and symbols are listed in the nomenclature. Consider the infinitely fast reaction of qA + B ~ products.
(2)
At steady state, the boundary layer equations for the conservation of components A and B in the continuous phase
are
(1) 1779
OcA_ OCA u~ Ox----~v~ ,gy~
1 02CA Sc,~ Oy~2
(3)
Shorter Communications ac.._.~n, MI OXl "1- I)1
acB Oy~
1
O2CB
with boundary conditions
(4)
S c n Oyl 2 "
C--A= 1, Ca=O, Ca=l,
The following boundary conditions may apply: CA = CA ( 0 ) ,
Yl = O,
X1 > 0
ca=O, cA = O, cn = cn(o~) ca = 0
xl =O, y~ -'---yn(x~) y~ = Yl ~- y R ( X l )
Yl > 0
aq=O n=m, ~=oo
~-'d'~-'~Jns = ~ \ d~l ]n R
(12)
where
SCA \ OYl / ItR = - S-~c~\~r~/~"
.
_ Scn OCA
a -- 7:'_-
(5)
It will be assumed that the influence of mass transfer on the velocity profile is negligible and that the reaction zone is very close to the surface, so that the x and z components
Sh(:,,)=-,3°% \ Oy /u=o
and
fl =
cB(~) qCA (o)"
Equation (!1) can easily be solved; r/n is determined from the final boundary condition and the local Sherwood number is
( m + 2)mltm+2)(O-lhza)lffm+l)Rell2ScA ll~m+2)
J:""+' z-'m+'"+"exp (--z) d z [ f [
of the velocity profile can be expressed as Ul = or1 (xx, z,)yfl ~
In the absence o f chemical reaction, */n tends to infinity and it follows that the enhancement factor is
(6)
14'1 ~ 0 .
This latter assumption is true only for Sc -> 1, but as pointed out by Lochiel and Calderbank [5], this generally applies for the dissolution of solids, liquids and gases in liquid for R e > 0.1. It should be emphasized that the postulated boundary conditions imply that the external-phase mass transfer resistance is controlling. Also, the assumption tht w~ is zero is identically true only for two dimensional or axisymmetric flows, and in the limiting case o f S c ~ ~. When the interface is moving rapidly compared to the velocity of translation, the velocity gradient is essentially zero. For this case m = 0. Alternatively, when the surface is rigid or immobile, the velocity gradient can be assumed linear and m = 1. Realistically, m may have some value between 0 and 1. The velocity component in the y direction is obtained from Eq. (1) as
(13)
(or,h,,)~'"+'h,,h~, dx,] '''+~'
i_ Sh(xl) = F[l/(m+2)] - Sho(xa) fnnm+~z-'m+lt'+2'exp (--z) dz"
Acrivos[l] treated the case when m = 1 and hx, = hz, = 1. F o r m = 0,1 = l/erflqn. To further illustrate the application o f this analysis for the prediction of transfer rates, consider mass transfer around a fully circulating spherical bubble or drop. The characteristic length, L, is taken as the diameter and h~, = 1, h,~ = (a sin O)/2a --- (sin 0)/2 and dx~ = d0/2. If the Reynolds number is less than 1, it follows from the Hadamard stream function that the tangential velocity component can be expressed as U~
= - - pl m+l. _1 0(h~m). /)1 h x l h z , ( m + 1) Oxl
u = 2[~ (P,c +tzd) sm 0.
(7)
cdc~(~) = c~(n)
ul
or,
(9)
The boundary layer thickness, 8, is expressed as
a
I
(o'~h,,) u'"+"
= sinO( /z~ 2 \~c+#a/
(16)
and
(8)
where "q = yl/8(xl, zl).
(15)
Thus
In order to obtain an analytical solution, a similarity transformation is employed. Let C~/CA(o) = cA (n)
(14)
=sin0{
Izc ~
2 \l,te+l.~a/"
(17)
Substitution into Eq. (13) gives
f(m+2)'fZ'(h~,hzO(o,hz,)aton+,,dx,]V"+2 L---~ca .Io
/3[~'~[
2+cos0
-Sho(x,) X/~,c+~d)~,(l+cos0)
~n
)
2 re.
(18)
(10) and Eqs. (3) and (4) become
Integration o v e r t h e entire surface y i e l d s t h e o v e r a l l S h e r wood number
d__~_a+ (m+2),7~,+, d ~ . = 0 drt 2
d'q
1/2
Sho = 0"65{~ ] e e l'z \~c + la,d]
(11) d2C'n + ( m + 2)otn"+' d ¢ n = 0 dn
(19)
The form of this equation coincides with that presented by 1780
Shorter Communications many authors, and the equation is identical to that derived by Lochiel and Calderbank [5]. For mass transfer to spherical drops or bubbles at high Reynolds number, the analysis of Moore [6] and Cheh and Tobias [7] can be applied to derive 2+3/~a/tLc l'45]t/~/.___/bx.__XlJ2- 1,2 - i l Sho = 1.13 1 - 1 + (Pa/xa~ v2 -Rett2J \tzc + txa]l re '.
(20)
\pcIl~c/ U,
This same problem was studied by Cheh and Tobias[7]. They obtained values of Sho numerically, apparently not realizing that an analytical solution can be obtained. The extension of the present analysis to non-isothermal systems is straightforward and is omitted here. Acknowledgement--T. Akiyama gratefully acknowledges financial support in the form of a post-doctoral fellowship from the University of Alberta. Department o f Chemical and Petroleum Engineering University o f Alberta Edmonton, Alberta, Canada
a A, B c c(o) c (~) C
T. A K I Y A M A * F. D. OTTO
NOTATION radius of a sphere two reacting components concentration of species in solvent saturation concentration of solute in solvent concentration in bulk solvent dimensionlessconcentrationCa=ca/ca(o) CB = CB/C~(Oo)
tPresent address: Department of Chemical Engineering, Shizuoka University, Hamamatsu, Shizuoka, Japan. [1] [2] [3] [4] [5] [6] [7]
diffusion coefficient scale factors enhancement factor mass transfer coefficient in continuous phase characteristic length real number m q stoichiometric coefficient U= main stream fluid velocity 19, W component of velocity for x, y, z coordinates respectively x distance along the surface following the adjacent streamlines y distance from the surface along its local normal z distance along the surface perpendicular to the adjacent streamlines Re Reynolds number U®Lpdl~c Sc Schmidt number I~dpcD Sh Sherwood number kLL/D
D hx, h~, hz 1 kL L
Greek symbols dimensionless fluid parameter ScdSca dimensionless fluid parameter c n ( OO) / qc A( o ) 6 boundary layer thickness / 0 polar angle "1 similarity variable defined by Eq. (9) cr1 defined by Eq. (6) #, viscosity O density Subscripts i A,B C d o R
dimensionless variable components A and B continuous phase dispersed phase value without chemical reaction reaction zone
REFERENCES A C R I V O S A., Chem. Engng Sci. 1960 13 57. F R I E D L A N D E R S. K. and LITT M., Chem. Engng Sci. 1958 7 229. S T E W A R T W . E.,A.I.Ch.E.J11963 9 528. L I G H T F O O T E. N., Lectures in Transport Phenomena 1964 4 44; A.I.Ch.E. Continuing Education Series. L O C H I E L A. C. and C A L D E R B A N K P. H., Chem. Engng Sci. 1964 19 471. M O O R E D. W.,J. FluidMech. 1963 16 161. C H E H H. Y. and T O B I A S C. W., Ind. Engng Chem. Fundls 1968 7 48.
Chemical Engineering Science, 1971, Vol. 26, pp. 1781-1783.
Pergamon Press.
Printed in Great Britain
Hold-up prediction in packed columns for co-current gas-liquid downflow (Received25 March 1971) INTRODUCTION SOME industrial gas-liquid contact processes require co-current flow. The advantage of this type of flow is that it carries larger volumes of each phase through the packing or catalyst, as opposed to counter-current flow which is limited by flooding. Charpentier et al. have described experimental results and formulated semi-empirical expressions for predicting the
pressure drop[2] and hold-up[l] in co-current gas-liquid downflow inside a Raschig ring packed column. References [2] and [3] contain a detailed description of the installation, the properties of the fluids, the packing characteristics and the measuring techniques.
1781
STRATIFIED-PORE MODEL An interactionless stratifed-pore model is described in