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A note on residence time distributions in cyclic reactors
Shorter Communications The slopes predicted by Eq. (15) are smaller than those of Eq. (1 l), but agree within three percent for CR below 10”. The predictions described above are limited to slopes of thin men&i. There is also a need to describe the slope of the thick part of the meniscus as well as the size or location of the entire protile. Such a study is in progress. In summary, the effect of speed on slope of the upper protie in withdrawal has been given in terms of Eqs. (12) and (13). A theo~tical basis for the form of the shape equation and the effect of angle is also presented herein.
Acknowledgement-This Eastman Kodak Co.
NOTATION bath location parameters (also d) Ca capillary number, up/u D film thickness, nondimensional, ho (pg/u)“‘5 thickness, meniscus region, mm 6lm thickness, constant region, mm ; k shape parameter, l/m L hlh, slope of meniscus, nondimensional, Eq. (2) z parameter, Eq. ( 1) R X/ho T film thickness, nondimensional, ho(pg/cLu)“‘” IA watingspeed,mmlsec distance below the constant region, mm X distance above liquid bath, mm Z
b,C
work was supported in part by the
JOHN
A. TALLMADGE
Greeksymbols a withdrawal angle p viscosity p density u surface tension
Department of Chemical Engineering Drexel University, Philadelphia Pennsylvania 19104, USA.
.*t
REFERENCES
.;
COXB.G.,J.FluidMech. 19621481. B. V. and LEVI S. M., Film Coating Theory, Chaps. 2 and 5. Focal Press, New York 1964. [31 GROENVELD P., Chem. Engng Sci. 1970 25 33. r41 GROENVELD P. and VAN DORTMUND R. A., Chem. Engng Sci. 1970 25 1571. PI TALLMADGE J. A. paper presented at A.1.Ch.E. National Mtg., Portland, Oregon, USA, August 1969. 197117 243. I61 TALLMADGE J. A.,A.I.Ch.E.JI 171 WHITE D. A. and TALLMADGE J. A., Chem. Engng Sci. 1965 20 33.
ii; DERYAGIN
ChemicdEngineeting
Science,
1973, VOI. 28, pp. 3 I 3-3 IS.
Pergamor! Press.
Printed
in GreatBritain
A note on residence time distributions in cyclic reactms (Received 18 April 1972) TIME
AVERAGE RESIDENCE TIME DISTRIBUTIONS PREVIOUSpapers [ 1,2] have introduced the concept of residence time distributions in unsteady systems. For a single stirred tank, the time dependent residence time decay function is
the time dependent then
concentration
I: W)
(1) where S (0, t) gives the fraction of material leaving the reactor at real time 0 which remained in the reactor for a duration greater than t. It has been shown[l,2] that S(0, t) can be used to predict conversions in exactly the same way as with ordinary, steady-state RTD’s. Thus, the conversion can be uniquely predicted for a llrst order reaction and fairly close bounds can be established for reactions of other order. In the unsteady case, the predicted conversions will be functions of real time, e. In cyclic systems, interest usually focuses on wnversions averaged over the interval of periodicity, 7. If c(e) is
of some key component,
de
is the concentration averaged over all material leaving the reactor during one period. Note that C is a time average on the discharge stream and is not necessarily the time average concentration wit&n the reactor. Analogous to C, there is a time average residence time distribution. Conceptually. all material which leaves the reactor during a single period is collected; and the time that each fluid element spent in the reactor is determined. The RTD thus defined is given by
D(e) de J-Ii Suppose
313
now that the reactor
is cycled
with respect
to
flow rates and volume but not with respect to ti_et concentration. If the reactor is completely segregated, S(r) can be used to predict yields in exactly the same nuumer as for steady-state systems: c =-
c
Gar,,(t)
6(t)
= lo C,,(t)g(r)
dt
(4)
where g(t) is the time average frpsuency function, - dS/dt. The other extreme of micromixii is maximum mixedness, and Zwietering’_s[3] results could be used to establish a second bound on C. However, maximum mixedness is a mathematical liiit on micromixing and this limit may not be physically achievable in the real system. For periodic stirred tanks, the physical limit on micromixing occurs when the stirred tank is a perfect mixer. Thus the solution of
and the dimensionless variance of the distribution is l--a 1+a
+R=_=~ l+R
MOMENTS OF S(0, t) AND S(t) The moments of the instantaneous residence time distribution are given by
P-‘S(B, t) dr.
From Eq. (l),
with c found from E.q. (2) gives the micromixing limit which is ohvsi&v achievable. This limit can be different from that of-&&m& mixedness and will then provide a closer bound on reactor performance. Thus, although it has been shown [ 11 that perfect mixing in the stirred tank does give maximum mixedness with respect to S(@, t), perfect mixing d_oes not necessarily give maximum mixedness with respect to S(t). At tirst glance there appears little merit in introducing the concept of time average RTD’s. Even if-the system is completely segregated, an alternate to using S (1) is simply to tit use_ S(0, t) and then to find C from Eq. (2). In essence, use of S (t) just reverses the order of inJegration. However, it is suggested that S(t) provides a conceptual tool for achieving a deeper understanding of periodic systems. For the case of constant feed composition, it provides a means for expressing the consequences of cyclic operation in terms more familiar to the reactor engineer: an equivalent residence time distribution function for a steady system. For example, suppose V(0) = V(T) =
F(e) = o(e)
as
--Fw -=~s+g(e,t) ae g(e, t) = -$. Then by differentiating Eq. (1 l),
with CL,,= 1. Corresponding to Eq. (1 l), the moments average RTD are given by pX = K I; t“--ls (t) dr.
pK_~whKwd~ 67 = x o(e)
de = I;: F(e) de.
K K VCLR--I de
(7)
k = ForK=
Vml” - vml.
Vnmx
(17)
(18)
1,
(8)
Comparing this result with the work of Rippin [4] shows that Eq. (7) is the residence time frequency function for a piston flow reactor in a recycle loop. The ratio of recycle flow to feed flow is a
*
l%
where
&zE=
(16)
Thus the & are weighted time averages of the instantaneous p,(e). In the Appendix we derive the interesting result:
which represents a repeated batch cycle with only partial emptying of the reactor after each batch. The time average RTD for this situation is given by
vm& - vrll,. v max .
(IS)
(6) where
= (v,,-vti)s(T-e)
m = “5,a(l-a)“--16(m)
of the time
Equation (3) can be combined with Eq. (15) and the order of integration reversed to give
&
wmaxvti.)s(e)
a=
(12)
where
v,,
v(e)=v,,,o
(10)
Perfect mixing in the st&d tank is analogous to perfect mixing at the feed point of a recycle reactor. It thus follows from Rippin’s work that this is a case where the cyclic reac@r will be a maximum mixedness reactor with respect to S(t) as well as to S (0, t) if the stirred tank is a perfect mixer.
p,&?) = K g
y-F&-DC+r(C)
v V_’
(9)
314
,_py Or
D
(1%
so that the time average or periodic mean residence time is just the time average volume divided by the time average ffowrate. It is often reasonable to compare cyclic operation @ steady-state operation at the same average flow rate, D.
Shorter Communications For the cycle delined by Eq. (6), r = V-. Thus the residence time distribution can be varied from that corresponding to perfect mixii to that for piston flow without changing the mean residence time. In real situations, however, instantaneous flow rates are constrained to tlnite values. This
[I] [2] [3] [4]
will force P to be less than V,,. Thus, cyclic operation will normally cause a decrease in mean residence time. Union Carbide Corporation Bound Brook, NJ. 08805, U.S.A.
E. B. NAUMAN
REFERENCES NAUMAN E. B., Chem. Engng Sci. 1969 24 1461. CHEN M. S. K., Chem. EngngSci. 197126 17. ZWIETERING Th. N., Chem. EngngSci. 1959 111. RIPPIN D. W. T., Znd. Engng Chem. Fund/s 1967 6 488. APPBNBIK
DERIVATION From Eq. (14) we can write ~F~d@+~
The product p,J vanishes since we are integrating over an interval of periodicity. (If V(0) = 0, ~(0) will usually be discontinuous at 0 = 0 but the product wXV still vanishes since Vvanishes.)
OF EQ. (18)
Pk=KI;:
PMz-,de.
(A-l) ~Ffi&‘=~Dkd+~
Pdl.cK,
(A-4)
Since dV/d9 = F-D, andfrom(A-l),weobtain ~F&&=~;DCLK~+~CLXdV.
(A-2)
Integmting by parts,
j,r
PK
dV = PKV 1: -
I,’ v&k.
(A-3)
Chemical Engineering Science,1973. Vol. 28. pp. 3 15-3 18. Pergamon Press.
Substitutingthis into Eq. (16) gives Eq. (18).
Printed in Great Britain
Absorption physique d’un gaz dans un jet liquide turbulent (Received 6 December 1971) POUR rendre compte du transfert de mat&e physique entre un gaz et un film liquide turbulent en mouvement vertical, Levichill a propose un modele simple, mais ce modble n’a encore fait l’objet que de r-ares confrontations avec des resultats expkimentaux. R&urn& par Davies[2], cette theorie a et6 modifiee par Atkinson[3] et utilisb par Davies et Ting[4], puis plus kemment par Davies et Hameed[S] pour interpreter les r&hats de l’absotption dun gaz dans un jet liquide turbulent. Elle a aussi 6tk test& par Coeuret et ~011. [6] sur des rksultats de l’absorption dun gaz dans un jet liquide s’&oulant verticalement autour dune tige cylindrique. En fait, la theorie de Levich a Cte mal appliquke dans les travaux[4-61; comme le soul&tent Davies et Hameed[S], elle necessite un nouvel examen. Dans le cas de l’absorption dun gaz A par un film liquide
turbulent, Levich[ 11 admet que la tension superficielle u assure la stabiliti de la surface du liquide et il est amen6 a d&nir une ‘zone de turbulence amortie’ d’epaisseur A, en dehors de laquelle les tourbillons ne sont plus affect& par la surface libre (Fig. 1). II postule que la pression capillaire u/X qui rksulte de la deformation superlicielle est opposee a la force de frottement unitaire 70= p . q*:
f=p.Lg ou v, est la vitesse de frottement et p la densit$ du liquide. Atkinson[3], qui a repris cette theorie, pose que h = (Y. R & R est le rayon de courbure de la deformation superficielle et OL un coefficient numerique.
315
Acknowledgement-This Eastman Kodak Co.
NOTATION bath location parameters (also d) Ca capillary number, up/u D film thickness, nondimensional, ho (pg/u)“‘5 thickness, meniscus region, mm 6lm thickness, constant region, mm ; k shape parameter, l/m L hlh, slope of meniscus, nondimensional, Eq. (2) z parameter, Eq. ( 1) R X/ho T film thickness, nondimensional, ho(pg/cLu)“‘” IA watingspeed,mmlsec distance below the constant region, mm X distance above liquid bath, mm Z
b,C
work was supported in part by the
JOHN
A. TALLMADGE
Greeksymbols a withdrawal angle p viscosity p density u surface tension
Department of Chemical Engineering Drexel University, Philadelphia Pennsylvania 19104, USA.
.*t
REFERENCES
.;
COXB.G.,J.FluidMech. 19621481. B. V. and LEVI S. M., Film Coating Theory, Chaps. 2 and 5. Focal Press, New York 1964. [31 GROENVELD P., Chem. Engng Sci. 1970 25 33. r41 GROENVELD P. and VAN DORTMUND R. A., Chem. Engng Sci. 1970 25 1571. PI TALLMADGE J. A. paper presented at A.1.Ch.E. National Mtg., Portland, Oregon, USA, August 1969. 197117 243. I61 TALLMADGE J. A.,A.I.Ch.E.JI 171 WHITE D. A. and TALLMADGE J. A., Chem. Engng Sci. 1965 20 33.
ii; DERYAGIN
ChemicdEngineeting
Science,
1973, VOI. 28, pp. 3 I 3-3 IS.
Pergamor! Press.
Printed
in GreatBritain
A note on residence time distributions in cyclic reactms (Received 18 April 1972) TIME
AVERAGE RESIDENCE TIME DISTRIBUTIONS PREVIOUSpapers [ 1,2] have introduced the concept of residence time distributions in unsteady systems. For a single stirred tank, the time dependent residence time decay function is
the time dependent then
concentration
I: W)
(1) where S (0, t) gives the fraction of material leaving the reactor at real time 0 which remained in the reactor for a duration greater than t. It has been shown[l,2] that S(0, t) can be used to predict conversions in exactly the same way as with ordinary, steady-state RTD’s. Thus, the conversion can be uniquely predicted for a llrst order reaction and fairly close bounds can be established for reactions of other order. In the unsteady case, the predicted conversions will be functions of real time, e. In cyclic systems, interest usually focuses on wnversions averaged over the interval of periodicity, 7. If c(e) is
of some key component,
de
is the concentration averaged over all material leaving the reactor during one period. Note that C is a time average on the discharge stream and is not necessarily the time average concentration wit&n the reactor. Analogous to C, there is a time average residence time distribution. Conceptually. all material which leaves the reactor during a single period is collected; and the time that each fluid element spent in the reactor is determined. The RTD thus defined is given by
D(e) de J-Ii Suppose
313
now that the reactor
is cycled
with respect
to
flow rates and volume but not with respect to ti_et concentration. If the reactor is completely segregated, S(r) can be used to predict yields in exactly the same nuumer as for steady-state systems: c =-
c
Gar,,(t)
6(t)
= lo C,,(t)g(r)
dt
(4)
where g(t) is the time average frpsuency function, - dS/dt. The other extreme of micromixii is maximum mixedness, and Zwietering’_s[3] results could be used to establish a second bound on C. However, maximum mixedness is a mathematical liiit on micromixing and this limit may not be physically achievable in the real system. For periodic stirred tanks, the physical limit on micromixing occurs when the stirred tank is a perfect mixer. Thus the solution of
and the dimensionless variance of the distribution is l--a 1+a
+R=_=~ l+R
MOMENTS OF S(0, t) AND S(t) The moments of the instantaneous residence time distribution are given by
P-‘S(B, t) dr.
From Eq. (l),
with c found from E.q. (2) gives the micromixing limit which is ohvsi&v achievable. This limit can be different from that of-&&m& mixedness and will then provide a closer bound on reactor performance. Thus, although it has been shown [ 11 that perfect mixing in the stirred tank does give maximum mixedness with respect to S(@, t), perfect mixing d_oes not necessarily give maximum mixedness with respect to S(t). At tirst glance there appears little merit in introducing the concept of time average RTD’s. Even if-the system is completely segregated, an alternate to using S (1) is simply to tit use_ S(0, t) and then to find C from Eq. (2). In essence, use of S (t) just reverses the order of inJegration. However, it is suggested that S(t) provides a conceptual tool for achieving a deeper understanding of periodic systems. For the case of constant feed composition, it provides a means for expressing the consequences of cyclic operation in terms more familiar to the reactor engineer: an equivalent residence time distribution function for a steady system. For example, suppose V(0) = V(T) =
F(e) = o(e)
as
--Fw -=~s+g(e,t) ae g(e, t) = -$. Then by differentiating Eq. (1 l),
with CL,,= 1. Corresponding to Eq. (1 l), the moments average RTD are given by pX = K I; t“--ls (t) dr.
pK_~whKwd~ 67 = x o(e)
de = I;: F(e) de.
K K VCLR--I de
(7)
k = ForK=
Vml” - vml.
Vnmx
(17)
(18)
1,
(8)
Comparing this result with the work of Rippin [4] shows that Eq. (7) is the residence time frequency function for a piston flow reactor in a recycle loop. The ratio of recycle flow to feed flow is a
*
l%
where
&zE=
(16)
Thus the & are weighted time averages of the instantaneous p,(e). In the Appendix we derive the interesting result:
which represents a repeated batch cycle with only partial emptying of the reactor after each batch. The time average RTD for this situation is given by
vm& - vrll,. v max .
(IS)
(6) where
= (v,,-vti)s(T-e)
m = “5,a(l-a)“--16(m)
of the time
Equation (3) can be combined with Eq. (15) and the order of integration reversed to give
&
wmaxvti.)s(e)
a=
(12)
where
v,,
v(e)=v,,,o
(10)
Perfect mixing in the st&d tank is analogous to perfect mixing at the feed point of a recycle reactor. It thus follows from Rippin’s work that this is a case where the cyclic reac@r will be a maximum mixedness reactor with respect to S(t) as well as to S (0, t) if the stirred tank is a perfect mixer.
p,&?) = K g
y-F&-DC+r(C)
v V_’
(9)
314
,_py Or
D
(1%
so that the time average or periodic mean residence time is just the time average volume divided by the time average ffowrate. It is often reasonable to compare cyclic operation @ steady-state operation at the same average flow rate, D.
Shorter Communications For the cycle delined by Eq. (6), r = V-. Thus the residence time distribution can be varied from that corresponding to perfect mixii to that for piston flow without changing the mean residence time. In real situations, however, instantaneous flow rates are constrained to tlnite values. This
[I] [2] [3] [4]
will force P to be less than V,,. Thus, cyclic operation will normally cause a decrease in mean residence time. Union Carbide Corporation Bound Brook, NJ. 08805, U.S.A.
E. B. NAUMAN
REFERENCES NAUMAN E. B., Chem. Engng Sci. 1969 24 1461. CHEN M. S. K., Chem. EngngSci. 197126 17. ZWIETERING Th. N., Chem. EngngSci. 1959 111. RIPPIN D. W. T., Znd. Engng Chem. Fund/s 1967 6 488. APPBNBIK
DERIVATION From Eq. (14) we can write ~F~d@+~
The product p,J vanishes since we are integrating over an interval of periodicity. (If V(0) = 0, ~(0) will usually be discontinuous at 0 = 0 but the product wXV still vanishes since Vvanishes.)
OF EQ. (18)
Pk=KI;:
PMz-,de.
(A-l) ~Ffi&‘=~Dkd+~
Pdl.cK,
(A-4)
Since dV/d9 = F-D, andfrom(A-l),weobtain ~F&&=~;DCLK~+~CLXdV.
(A-2)
Integmting by parts,
j,r
PK
dV = PKV 1: -
I,’ v&k.
(A-3)
Chemical Engineering Science,1973. Vol. 28. pp. 3 15-3 18. Pergamon Press.
Substitutingthis into Eq. (16) gives Eq. (18).
Printed in Great Britain
Absorption physique d’un gaz dans un jet liquide turbulent (Received 6 December 1971) POUR rendre compte du transfert de mat&e physique entre un gaz et un film liquide turbulent en mouvement vertical, Levichill a propose un modele simple, mais ce modble n’a encore fait l’objet que de r-ares confrontations avec des resultats expkimentaux. R&urn& par Davies[2], cette theorie a et6 modifiee par Atkinson[3] et utilisb par Davies et Ting[4], puis plus kemment par Davies et Hameed[S] pour interpreter les r&hats de l’absotption dun gaz dans un jet liquide turbulent. Elle a aussi 6tk test& par Coeuret et ~011. [6] sur des rksultats de l’absorption dun gaz dans un jet liquide s’&oulant verticalement autour dune tige cylindrique. En fait, la theorie de Levich a Cte mal appliquke dans les travaux[4-61; comme le soul&tent Davies et Hameed[S], elle necessite un nouvel examen. Dans le cas de l’absorption dun gaz A par un film liquide
turbulent, Levich[ 11 admet que la tension superficielle u assure la stabiliti de la surface du liquide et il est amen6 a d&nir une ‘zone de turbulence amortie’ d’epaisseur A, en dehors de laquelle les tourbillons ne sont plus affect& par la surface libre (Fig. 1). II postule que la pression capillaire u/X qui rksulte de la deformation superlicielle est opposee a la force de frottement unitaire 70= p . q*:
f=p.Lg ou v, est la vitesse de frottement et p la densit$ du liquide. Atkinson[3], qui a repris cette theorie, pose que h = (Y. R & R est le rayon de courbure de la deformation superficielle et OL un coefficient numerique.
315