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A unified time delay model for dispersion in flowing media
A Unified Time Delay Model for Dispersion in Flowing Media B. A. BUFFHAM
University
L. G. GIBILARO
AND
Department of Chemical Engineering, of Technology, Loughborough, Leics. (Gt. Britain)
(Received: 20 August, 1969; in final form: 16 September,
ABSTRACT A model is proposed that unifies the treatment of several important models for dispersion in jowing media, in that these models are all special cases of the new model. Transfer functions and moments are presented for the general model, together with analytical solutions for the impulse response that apply in most cases. The model is of great Jlexibility, containing as it does many established models as subcases.
of well-mixed subregions. It is shown that impulse responses having the same variance, but otherwise rather different in form, can be accommodated by the model by varying the gamma distribution parameter. Asymmetrical curves, often obtained experimentally, can be described well by the model. The treatment is designed to unify a range of previously published models, so that their inter-relations are more readily appreciated. The models covered include : tanks in series; the cell model of Deans2 and Levich et al. 3 ; the time delay models for deterministic and completely random delay processes; the classical regenerator model which has been shown’ to be equivalent to the completely random time delay model; and the Einstein statistical model advocated by Cairns and Prausnitz.4
INTRODUCTION In a forthcoming publication’ a probabilistic description of flow in packed beds is shown to lead to a simple model of such systems, and to possess certain advantages over models based on analogies with diffusion theory. The description is based on the abstraction that material flows through the bed in what would be plug-flow were it not that elements of flow material have a chance of being delayed at all points of their passage, delayed elements eventually rejoining the main flow stream at the delay point. The model parameters may be regarded as: the minimum possible transit time (i.e. the residence time of material that is not delayed at all); the probability of delay occurring; the average delay time ; and the form of the delay time distribution. Only two delay time distributions were considered specifically. The purpose of this paper is to generalise the model in this respect, and to present the transfer functions of the time delay description for both the discrete cell model and the corresponding ‘distributed-parameter’ model. Moments are presented for the general case where individual delay times are spread according to the gamma distribution and the main flow region is divided into an arbitrary number
1969)
It is found that by formulating the model carefully the transfer functions and low order moments for the general case plus the time domain solution for the case when the main flow region is in plug-flow, could be determined without the mathematics being at all arduous. Jeffreson’ recently proposed a model that accounts for dispersion in a similar way to the present model, the difference being that the delay process is represented as unsteady-state diffusion. The relationship between Jeffreson’s model and the time-delay type will be explored in a subsequent paper. None of these models takes explicit account of velocity profiles or lateral variations of flow rate: attention is focused on transients, and the concentration must be regarded as an average taken over the cross-section in flow terms. The steady-state rivulet model of Porter6 considers such effects, but does not consider transients. 31
The Chemical
Engineering
Journal
(1) (1970)-Q
Elsevier
Publishing
Company
Ltd,
England-Printed
in Great
Britain
B. A. BUFFHAM
32
TRANSFER
FUNCTION
DERIVATIONS
The cell model Figure 1 represents the discrete cell form of the time-delay model. The main flow path is represented by N well-mixed stages in series; there is lateral flow at a rate q per unit length of bed so that for a bed of
AND L. G. GIBILARO
in this way automatically generates expressions which incorporate the conservation of mass. The inverse of the combination [F(s)]’ is: (5) It is the relative simplicity of this form which enables eqn. (4) to be incorporated easily into a time delay analysis. MOMENTS
2 Fig. 1.
N
I
The cell form of the time delay model.
length x the side flow from each cell is qx/N; the distribution of delay times in the lateral zones is characterised by the transfer function F(s); the throughput flow rate is Q and the total volume of the main flow region is V. A Laplace transformed mass balance on a tracer component in the ith cell yields:
1 [ cilCs)
-
cds)
1 + $
=
I) (1) -1
t,s + CIX- axF(s)
where t, = V/Q and GI= q/Q. So that the transfer function for the whole system is given by:
(2) with
the
exponential
Cell model Equation (2) with the time delay distribution the gamma form, eqn. (4), may be expressed:
-N(2)
t,s + LXX- axF(s) Equation distribution
The cumulants form a set of parameters useful in describing distributions in much the same way as are moments; indeed moments and cumulants are closely related (see Kendall and Stuart’). Cumulants may be obtained from the transfer function by expanding lm G(s) as a power series in s: thejth cumulant, Tj, is the coefficient of (- I)’ sj/j! The low order cumulants are simply related to the moments: T, is the mean or first moment, pi’ ; T, is the variance or second central moment, p-12; and T, is the skewness or third central moment, p3.
time
+ g (m + yrn
+ 2, (t,s)3 -.
taking
. .]
delay
F(s) = (t,s + 1>- r reduces to the model discussed dispersion in porous media.
by Levich et aI.3 for
The ‘distributed-parameter’ model Allowing N in eqn. (2) to approach infinity yields the distributed version of the time delay model as a special case : G(s) = exp{ - t,s - ax + c&‘(s)}
F(s) = (
z +1>
(6)
by utilising the series expansions for (1 In (1 + a), and carrying the expansions far to generate terms up to s3. It follows, out the appropriate coefficients, that the moments and cumulants are:
(3)
The gamma distribution of delay times As mentioned earlier, a suitable choice for F(s) is the transform of the gamma distribution:
1 2
.
+ a)” and sufficiently by picking low order
pr’ = T, = t, + axt, /.L~ = T, = ax
1 + $ (
tD2 + x (t, + axtD)2 >
(7) (8)
~3=T3=~x(l+-3(1+~)to3
-m (4)
This provides a remarkable degree of descriptive flexibility with a single parameter in addition to the mean. 7 Writing the distribution in terms of the mean
ax(t, + ctxtD)tD2 + $
(t, + axt,y
(9)
A UNIFIED TIME DELAY
MODEL FOR DlSPERSION IN FLOWING
Distributed model The cell model equations, above, are formulated in such a way that when N is increased the stage volume is decreased proportionately, and the system volume remains constant. Thus the moments for the distributed model are obtained by letting N increase indefinitely, i.e. pi’ = t, + crxt, /A2
=
pa = a(1
ax
(
1+ ;
)
+ $(l
(10) t,2
(11)
+ $3
(12)
Alternatively, these results may be obtained directly by expanding the expression for F(s) in eqn. (4), and substituting for it in eqn. (3), to give: G(s) = exp
-(t,
+ ctxt,)s
+
_ (m + l)(m + 2)c~x(t,s)~ 3 !m2
(m + l)rx~(t~s)~ 2!m + ... 1
(13)
in which the coefficients give the cumulants. Equations (8) and (9) indicate that both the variance and skewness of the impulse response increase indefinitely as m decreases to zero, irrespective of the value of N, provided it is fixed. The effect of changing N with m fixed is similar: decreasing N increases the skewness and variance. Also of significance is the skewness relative to the variance on a dimensionless basis. This only takes a simple form when N + cc : in this case : -EC= P2 3’2
TIME-DOMAIN
SOLUTIONS
The cell model Inversion of transfer functions of the form of eqn. (2) is a difficult task. One case that has been studied is that in which the delay time distribution is exponential : F(s) = (t,s + 1)-l so that eqn. (2) reduces to the model discussed by Levich et a1.3 for dispersion in porous media. The inverse transformation for this case is available’ and the sum of two terms each of which is the product of a polynomial in t and a negative exponential; the polynomial coefficients are themselves constructed from summations involving the model parameters. This suggests that the inverse of eqn. (2) with F(s) the gamma distribution would be a rather intractable form. When F(s) is an integral tanks-in-series form (m = 2, 3 . . . in eqn. (5)), the model can be solved numerically using a standard routine for solving sets of differential equations. The lack of flexibility in terms of available time-domain solutions is compensated by presence of the parameter N, which does not appear in the distributed parameter time-delay model, and presence of which means less reliance need be placed on the time-delay distribution. The distributed model The time-domain solution of the distributed parameter-model is readily obtained by expanding eqn. (3) and inverting term by term:
m+2 Jaxm(m + 1)
which increases as m decreases; the skewness creases more rapidly than the variance.
33
MEDIA
bM41 2+ ... 2r I
(14) in-
(15) and O”
Additivity of variances The form of eqn. (8) is interesting because it shows that the variance is made up of two additive parts: contributions due to the delay process, and to the model being split into cells. The former is independent of the number of cells, and the latter is independent of the delay distribution parameter and the same (cell mean time squared divided by number of cells) as the variance of the tanks-in-series mixing model. So although the mixing mechanisms are not independent in the usual sense, their effects separate in the variance. This behaviour does not extend to the third central moment, which is also additive for independent series mechanisms. The two contributions to the variance may be regarded as describing transverse and axial mixing.
G(t) = epaX 2
(Lxxy
i!
F,(t - t,),
t
2
t,
i=O
= 0,
t < t,
(16)
where F,(t) is the inverse of [F(s)]‘. We have previously shown’ that when the delay process is Markovian the probability of an element being delayed i times is:
so that the ith term of the summation is the contribution to the impulse material that has been delayed i times,
of eqn. (16) response of
34
B. A. BUFFHAM
AND L. G. GIBILARO
When the gamma distribution, eqn. (5) is used for the delay times, eqn. (16) becomes:
t 2 t, =
0,
t < t,
(17)
The zeroth term in the series in eqn. (17) is an impulse of weight e Pax at t = t,, the material which traverses the bed without being delayed. This contribution to G(t) cannot be plotted as part of the impulse response, but represents an initial sharp rise in the step response. Usually e -G(Xis small, and the effective lower limit of the sum is unity. Figure 2 shows the influence of the parameters on the impulse response.
2.0
I? t
I.0
-
-+0.8 t/T Fig. 2b
\
\
2.0
I+ > z
I.0
Fig. 2a
Special cases In the analysis presented above, m may take any positive value. In particular m = 1 represents an exponential distribution of delay times (a Markovian random delay process) and m -+ co represents an impulse distribution of delay times (a deterministic delay process).l In the trivial case in which the side capacity is absent and N is finite, the model reduces to the tanks-in-series model. It has previously been
0
t/T Fig. 2c Fig. 2. Normalised residence time distributions for the time delay model: with gamma-distributed delays. (a) to/t = 0.6, m = 1, ax as shown; (b) to/t = 0.6, LXX = 10, m as shown; (c) m = 1, ax = 10, to/t as shown.
A UNIFIED
TIME DELAY
MODEL
FOR DISPERSION
pointed out that the completely random distributed version of the model (N -+ co, m = 1) is equivalent mathematically to the Anzelius regenerator model, so that mixing by random lateral bulk flow and by interaction between a plug-flow region and a static region via a transfer coefficient, are indistinguishable by tracer experiments. When the delay process is random and the number of cells finite (N finite, m = 1) we have the Deans-Levich model.2,3 Finally, if the delay process is completely random, and the transits between delays are extremely short (N -+ co, m = 1, t, + 0), the Einstein model used by Cairns and Prausnitz,4 results. It is possible to match the dispersed plug-flow model by matching moments for a variety of parameter values, or by’ any other fitting technique. However, the dispersion model is not a special case of the present model, the most important difference being that no true backflow is incorporated in the time-delay model. Hiby’s observations” indicate that true backflow is often unimportant in practice.
CONCLUSIONS The transfer functions of the time-delay model with gamma-distributed delays have been derived for both the cell and distributed versions of the model. The gamma distribution is suitable to describe the delay process, because it can represent a wide variety of behaviour with the variation of the single parameter m; because it reduces to the special cases of deterministic and completely random delay processes; and because its mathematical form lends itself to the analysis. Moments up to third order are determined from the transfer functions via the cumulants, and these show that for a fixed variance sufficient freedom is left to make wide changes in the skewness. This is important because many models lead to impulse responses that are too symmetrical. The model response is particularly sensitive to the gamma distribution parameter m, when this is small; that is when early return to the mainstream, relative to completely random return, is favoured. The treatment presented here represents a unified form of a range of models that has been discussed in the literature previously, including the tanksin-series model (N finite, c(x = 0), the classical regenerator model (N -+ co, m = l), the DeansLevich model (N finite, m = l), the authors’ previous time-delay models (N -+ 00, m = 1; N -+ cc, m -+ CD), and Cairns and Prausnitz’s model (N + a, m = 1, 1, = 0).
IN FLOWING
MEDIA
35
NOMENCLATURE
G(s)
Laplace transform of the concentration, ci(t), of material leaving the ith cell Laplace
transform
of delay time distribution of [F(s)]’
inverse transform system transfer pulse response
G(t)
inverse transform
i
cell number,
_i
order of cumulant
m
N
; transform
function
of G(s); impulse
counting
gamma
distribution
number
of cells
of imresponse
index
parameter
lateral flow rate per unit length ;
throughput s
Laplace
t
time
flow rate
transform
tD
mean delay time
t,
V/Q; minimum model
parameter
transit
time for distributed
Ti jth cumulant V volume of main flow region X
length of bed
Greek symbols a
qlQ
J?() gamma function; PI' p2#3
r(n)
mean system residence second, third central dence time
=
co e-“x”-1 s0 time moment
dx
of system resi-
REFERENCES 1. BUFFHAM, B. A., GIBILARO, L. G., AND A.I.Ch.E.JI., in press. 2. DEANS, H. A., Sot. Petrol.
M. N. RATHOR,
Engrs. Jl., 1963 3 49.
3. LEVICH, V. G., MARKIN, V. S., AND CHISMADZHEV,Yu. A., Chem. Engng. Sci., 1967 22 1357. 4. CAIRNS, E. J., AND PRAUSNITZ, J. M., Chem. Engng. Sci., 1960 12 20. 5. JEFFRESON,C. P., Chem. Engng. Sci., 1968 23 509. 6. PORTER, K. E., Trans. Znsfn. Chem. Engrs.,
1968 46 69.
1. BUFFHAM, B. A., AND GIBILARO, L. G., A.Z.Ch. E.Jl., 1968 14 805. 8. KENDALL, M. G., AND STUART, A., The Advanced of Statistics, Vol. 1, Griffin, 1958.
Theory
9. BUFFHAM, B. A., AND GIBILARO, L. G., Chem. Engng. Sci., 1968 23 1399. 10. HIBY, J. W., The Interaction Between Fluids and Particles, in Znsfn. Chem. Engrs. Symp., Series No. 9, 1963, p. 312,
36
B. A. BUFFHAM
AND L. G. GIBILARO
ZVSAMMENFASSVNG On propose un modele rassemblant le traitement de plusieurs modeles importants de la dispersion dans les milieux en e’coulement, ces modeles n’etant alors que des cas particuliers du nouveau modzle. On donne les fonctions de transfert, les moments du modele g&&al, ainsi que les solutions analytiques de reponses a une impulsion qui s’appliquent dans la plupart des cas. De nombreux modeles classiques ne sont que des cas particuliers de ce modele d’emploi trb souple.
Es wird ein Model1 vorgeschlagen, das mehrere wichtige Modelle zur Dispersion flie flfirmiger Medien vereinigt. Die genannten Modelle sind Spezialftille des neuen Modells. Vbertragungsfunktionen und Momente werden fiir das allgemeine Model1 zusammen mit in den meisten FSillen anwendbaren analytischen Liisungen fur das Impulsresuhat angegeben. Das Model1 ist auDerordentlichflexibe1 und enthalt viele der bekannten Modelle.
University
L. G. GIBILARO
AND
Department of Chemical Engineering, of Technology, Loughborough, Leics. (Gt. Britain)
(Received: 20 August, 1969; in final form: 16 September,
ABSTRACT A model is proposed that unifies the treatment of several important models for dispersion in jowing media, in that these models are all special cases of the new model. Transfer functions and moments are presented for the general model, together with analytical solutions for the impulse response that apply in most cases. The model is of great Jlexibility, containing as it does many established models as subcases.
of well-mixed subregions. It is shown that impulse responses having the same variance, but otherwise rather different in form, can be accommodated by the model by varying the gamma distribution parameter. Asymmetrical curves, often obtained experimentally, can be described well by the model. The treatment is designed to unify a range of previously published models, so that their inter-relations are more readily appreciated. The models covered include : tanks in series; the cell model of Deans2 and Levich et al. 3 ; the time delay models for deterministic and completely random delay processes; the classical regenerator model which has been shown’ to be equivalent to the completely random time delay model; and the Einstein statistical model advocated by Cairns and Prausnitz.4
INTRODUCTION In a forthcoming publication’ a probabilistic description of flow in packed beds is shown to lead to a simple model of such systems, and to possess certain advantages over models based on analogies with diffusion theory. The description is based on the abstraction that material flows through the bed in what would be plug-flow were it not that elements of flow material have a chance of being delayed at all points of their passage, delayed elements eventually rejoining the main flow stream at the delay point. The model parameters may be regarded as: the minimum possible transit time (i.e. the residence time of material that is not delayed at all); the probability of delay occurring; the average delay time ; and the form of the delay time distribution. Only two delay time distributions were considered specifically. The purpose of this paper is to generalise the model in this respect, and to present the transfer functions of the time delay description for both the discrete cell model and the corresponding ‘distributed-parameter’ model. Moments are presented for the general case where individual delay times are spread according to the gamma distribution and the main flow region is divided into an arbitrary number
1969)
It is found that by formulating the model carefully the transfer functions and low order moments for the general case plus the time domain solution for the case when the main flow region is in plug-flow, could be determined without the mathematics being at all arduous. Jeffreson’ recently proposed a model that accounts for dispersion in a similar way to the present model, the difference being that the delay process is represented as unsteady-state diffusion. The relationship between Jeffreson’s model and the time-delay type will be explored in a subsequent paper. None of these models takes explicit account of velocity profiles or lateral variations of flow rate: attention is focused on transients, and the concentration must be regarded as an average taken over the cross-section in flow terms. The steady-state rivulet model of Porter6 considers such effects, but does not consider transients. 31
The Chemical
Engineering
Journal
(1) (1970)-Q
Elsevier
Publishing
Company
Ltd,
England-Printed
in Great
Britain
B. A. BUFFHAM
32
TRANSFER
FUNCTION
DERIVATIONS
The cell model Figure 1 represents the discrete cell form of the time-delay model. The main flow path is represented by N well-mixed stages in series; there is lateral flow at a rate q per unit length of bed so that for a bed of
AND L. G. GIBILARO
in this way automatically generates expressions which incorporate the conservation of mass. The inverse of the combination [F(s)]’ is: (5) It is the relative simplicity of this form which enables eqn. (4) to be incorporated easily into a time delay analysis. MOMENTS
2 Fig. 1.
N
I
The cell form of the time delay model.
length x the side flow from each cell is qx/N; the distribution of delay times in the lateral zones is characterised by the transfer function F(s); the throughput flow rate is Q and the total volume of the main flow region is V. A Laplace transformed mass balance on a tracer component in the ith cell yields:
1 [ cilCs)
-
cds)
1 + $
=
I) (1) -1
t,s + CIX- axF(s)
where t, = V/Q and GI= q/Q. So that the transfer function for the whole system is given by:
(2) with
the
exponential
Cell model Equation (2) with the time delay distribution the gamma form, eqn. (4), may be expressed:
-N(2)
t,s + LXX- axF(s) Equation distribution
The cumulants form a set of parameters useful in describing distributions in much the same way as are moments; indeed moments and cumulants are closely related (see Kendall and Stuart’). Cumulants may be obtained from the transfer function by expanding lm G(s) as a power series in s: thejth cumulant, Tj, is the coefficient of (- I)’ sj/j! The low order cumulants are simply related to the moments: T, is the mean or first moment, pi’ ; T, is the variance or second central moment, p-12; and T, is the skewness or third central moment, p3.
time
+ g (m + yrn
+ 2, (t,s)3 -.
taking
. .]
delay
F(s) = (t,s + 1>- r reduces to the model discussed dispersion in porous media.
by Levich et aI.3 for
The ‘distributed-parameter’ model Allowing N in eqn. (2) to approach infinity yields the distributed version of the time delay model as a special case : G(s) = exp{ - t,s - ax + c&‘(s)}
F(s) = (
z +1>
(6)
by utilising the series expansions for (1 In (1 + a), and carrying the expansions far to generate terms up to s3. It follows, out the appropriate coefficients, that the moments and cumulants are:
(3)
The gamma distribution of delay times As mentioned earlier, a suitable choice for F(s) is the transform of the gamma distribution:
1 2
.
+ a)” and sufficiently by picking low order
pr’ = T, = t, + axt, /.L~ = T, = ax
1 + $ (
tD2 + x (t, + axtD)2 >
(7) (8)
~3=T3=~x(l+-3(1+~)to3
-m (4)
This provides a remarkable degree of descriptive flexibility with a single parameter in addition to the mean. 7 Writing the distribution in terms of the mean
ax(t, + ctxtD)tD2 + $
(t, + axt,y
(9)
A UNIFIED TIME DELAY
MODEL FOR DlSPERSION IN FLOWING
Distributed model The cell model equations, above, are formulated in such a way that when N is increased the stage volume is decreased proportionately, and the system volume remains constant. Thus the moments for the distributed model are obtained by letting N increase indefinitely, i.e. pi’ = t, + crxt, /A2
=
pa = a(1
ax
(
1+ ;
)
+ $(l
(10) t,2
(11)
+ $3
(12)
Alternatively, these results may be obtained directly by expanding the expression for F(s) in eqn. (4), and substituting for it in eqn. (3), to give: G(s) = exp
-(t,
+ ctxt,)s
+
_ (m + l)(m + 2)c~x(t,s)~ 3 !m2
(m + l)rx~(t~s)~ 2!m + ... 1
(13)
in which the coefficients give the cumulants. Equations (8) and (9) indicate that both the variance and skewness of the impulse response increase indefinitely as m decreases to zero, irrespective of the value of N, provided it is fixed. The effect of changing N with m fixed is similar: decreasing N increases the skewness and variance. Also of significance is the skewness relative to the variance on a dimensionless basis. This only takes a simple form when N + cc : in this case : -EC= P2 3’2
TIME-DOMAIN
SOLUTIONS
The cell model Inversion of transfer functions of the form of eqn. (2) is a difficult task. One case that has been studied is that in which the delay time distribution is exponential : F(s) = (t,s + 1)-l so that eqn. (2) reduces to the model discussed by Levich et a1.3 for dispersion in porous media. The inverse transformation for this case is available’ and the sum of two terms each of which is the product of a polynomial in t and a negative exponential; the polynomial coefficients are themselves constructed from summations involving the model parameters. This suggests that the inverse of eqn. (2) with F(s) the gamma distribution would be a rather intractable form. When F(s) is an integral tanks-in-series form (m = 2, 3 . . . in eqn. (5)), the model can be solved numerically using a standard routine for solving sets of differential equations. The lack of flexibility in terms of available time-domain solutions is compensated by presence of the parameter N, which does not appear in the distributed parameter time-delay model, and presence of which means less reliance need be placed on the time-delay distribution. The distributed model The time-domain solution of the distributed parameter-model is readily obtained by expanding eqn. (3) and inverting term by term:
m+2 Jaxm(m + 1)
which increases as m decreases; the skewness creases more rapidly than the variance.
33
MEDIA
bM41 2+ ... 2r I
(14) in-
(15) and O”
Additivity of variances The form of eqn. (8) is interesting because it shows that the variance is made up of two additive parts: contributions due to the delay process, and to the model being split into cells. The former is independent of the number of cells, and the latter is independent of the delay distribution parameter and the same (cell mean time squared divided by number of cells) as the variance of the tanks-in-series mixing model. So although the mixing mechanisms are not independent in the usual sense, their effects separate in the variance. This behaviour does not extend to the third central moment, which is also additive for independent series mechanisms. The two contributions to the variance may be regarded as describing transverse and axial mixing.
G(t) = epaX 2
(Lxxy
i!
F,(t - t,),
t
2
t,
i=O
= 0,
t < t,
(16)
where F,(t) is the inverse of [F(s)]‘. We have previously shown’ that when the delay process is Markovian the probability of an element being delayed i times is:
so that the ith term of the summation is the contribution to the impulse material that has been delayed i times,
of eqn. (16) response of
34
B. A. BUFFHAM
AND L. G. GIBILARO
When the gamma distribution, eqn. (5) is used for the delay times, eqn. (16) becomes:
t 2 t, =
0,
t < t,
(17)
The zeroth term in the series in eqn. (17) is an impulse of weight e Pax at t = t,, the material which traverses the bed without being delayed. This contribution to G(t) cannot be plotted as part of the impulse response, but represents an initial sharp rise in the step response. Usually e -G(Xis small, and the effective lower limit of the sum is unity. Figure 2 shows the influence of the parameters on the impulse response.
2.0
I? t
I.0
-
-+0.8 t/T Fig. 2b
\
\
2.0
I+ > z
I.0
Fig. 2a
Special cases In the analysis presented above, m may take any positive value. In particular m = 1 represents an exponential distribution of delay times (a Markovian random delay process) and m -+ co represents an impulse distribution of delay times (a deterministic delay process).l In the trivial case in which the side capacity is absent and N is finite, the model reduces to the tanks-in-series model. It has previously been
0
t/T Fig. 2c Fig. 2. Normalised residence time distributions for the time delay model: with gamma-distributed delays. (a) to/t = 0.6, m = 1, ax as shown; (b) to/t = 0.6, LXX = 10, m as shown; (c) m = 1, ax = 10, to/t as shown.
A UNIFIED
TIME DELAY
MODEL
FOR DISPERSION
pointed out that the completely random distributed version of the model (N -+ co, m = 1) is equivalent mathematically to the Anzelius regenerator model, so that mixing by random lateral bulk flow and by interaction between a plug-flow region and a static region via a transfer coefficient, are indistinguishable by tracer experiments. When the delay process is random and the number of cells finite (N finite, m = 1) we have the Deans-Levich model.2,3 Finally, if the delay process is completely random, and the transits between delays are extremely short (N -+ co, m = 1, t, + 0), the Einstein model used by Cairns and Prausnitz,4 results. It is possible to match the dispersed plug-flow model by matching moments for a variety of parameter values, or by’ any other fitting technique. However, the dispersion model is not a special case of the present model, the most important difference being that no true backflow is incorporated in the time-delay model. Hiby’s observations” indicate that true backflow is often unimportant in practice.
CONCLUSIONS The transfer functions of the time-delay model with gamma-distributed delays have been derived for both the cell and distributed versions of the model. The gamma distribution is suitable to describe the delay process, because it can represent a wide variety of behaviour with the variation of the single parameter m; because it reduces to the special cases of deterministic and completely random delay processes; and because its mathematical form lends itself to the analysis. Moments up to third order are determined from the transfer functions via the cumulants, and these show that for a fixed variance sufficient freedom is left to make wide changes in the skewness. This is important because many models lead to impulse responses that are too symmetrical. The model response is particularly sensitive to the gamma distribution parameter m, when this is small; that is when early return to the mainstream, relative to completely random return, is favoured. The treatment presented here represents a unified form of a range of models that has been discussed in the literature previously, including the tanksin-series model (N finite, c(x = 0), the classical regenerator model (N -+ co, m = l), the DeansLevich model (N finite, m = l), the authors’ previous time-delay models (N -+ 00, m = 1; N -+ cc, m -+ CD), and Cairns and Prausnitz’s model (N + a, m = 1, 1, = 0).
IN FLOWING
MEDIA
35
NOMENCLATURE
G(s)
Laplace transform of the concentration, ci(t), of material leaving the ith cell Laplace
transform
of delay time distribution of [F(s)]’
inverse transform system transfer pulse response
G(t)
inverse transform
i
cell number,
_i
order of cumulant
m
N
; transform
function
of G(s); impulse
counting
gamma
distribution
number
of cells
of imresponse
index
parameter
lateral flow rate per unit length ;
throughput s
Laplace
t
time
flow rate
transform
tD
mean delay time
t,
V/Q; minimum model
parameter
transit
time for distributed
Ti jth cumulant V volume of main flow region X
length of bed
Greek symbols a
qlQ
J?() gamma function; PI' p2#3
r(n)
mean system residence second, third central dence time
=
co e-“x”-1 s0 time moment
dx
of system resi-
REFERENCES 1. BUFFHAM, B. A., GIBILARO, L. G., AND A.I.Ch.E.JI., in press. 2. DEANS, H. A., Sot. Petrol.
M. N. RATHOR,
Engrs. Jl., 1963 3 49.
3. LEVICH, V. G., MARKIN, V. S., AND CHISMADZHEV,Yu. A., Chem. Engng. Sci., 1967 22 1357. 4. CAIRNS, E. J., AND PRAUSNITZ, J. M., Chem. Engng. Sci., 1960 12 20. 5. JEFFRESON,C. P., Chem. Engng. Sci., 1968 23 509. 6. PORTER, K. E., Trans. Znsfn. Chem. Engrs.,
1968 46 69.
1. BUFFHAM, B. A., AND GIBILARO, L. G., A.Z.Ch. E.Jl., 1968 14 805. 8. KENDALL, M. G., AND STUART, A., The Advanced of Statistics, Vol. 1, Griffin, 1958.
Theory
9. BUFFHAM, B. A., AND GIBILARO, L. G., Chem. Engng. Sci., 1968 23 1399. 10. HIBY, J. W., The Interaction Between Fluids and Particles, in Znsfn. Chem. Engrs. Symp., Series No. 9, 1963, p. 312,
36
B. A. BUFFHAM
AND L. G. GIBILARO
ZVSAMMENFASSVNG On propose un modele rassemblant le traitement de plusieurs modeles importants de la dispersion dans les milieux en e’coulement, ces modeles n’etant alors que des cas particuliers du nouveau modzle. On donne les fonctions de transfert, les moments du modele g&&al, ainsi que les solutions analytiques de reponses a une impulsion qui s’appliquent dans la plupart des cas. De nombreux modeles classiques ne sont que des cas particuliers de ce modele d’emploi trb souple.
Es wird ein Model1 vorgeschlagen, das mehrere wichtige Modelle zur Dispersion flie flfirmiger Medien vereinigt. Die genannten Modelle sind Spezialftille des neuen Modells. Vbertragungsfunktionen und Momente werden fiir das allgemeine Model1 zusammen mit in den meisten FSillen anwendbaren analytischen Liisungen fur das Impulsresuhat angegeben. Das Model1 ist auDerordentlichflexibe1 und enthalt viele der bekannten Modelle.