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Modelling of fluidized bed reactors—VI(a)
MODELLING OF FLUIDIZED BED REACTORS-VI(a) AN ISOTHERMAL BED WITH STOCHASTIC BUBBLES JOHN ROBERT LIGON Department of Chemical Engineering and Materials Science, University of Minnesota, 151 Amundson Hall, 421Washington Avenue SE., Minneapolis, MN 55455,U.S.A.
and NEAL R. AMUNDSON* Department of Chemical Engineering, University of Houston, Houston, TX 77004,U.S.A. (Received 30 April 1980: accepted 10 July 1980)
Ah&a&-A two-phase stochastic isothermal fluidized bed reactor model with first order reaction in the dense phase is developed to investigate the signiicance of the fluctuating nature of fluidiid beds on reactor performance. Several stochastic processes are employed as the overall mass transfer coefficient between phases. Analytical moment solutions are obtained for white noise coefficients while hybrid computer simulation was used for correlated stochasticcoefficients.Resultsindicatethat a gammadistributedcoefficientis preferred over white noise and Gaussian correlated coefficients. When compared with the deterministic model, randomness in the mass transfer coe5cient is seen to lead to a decrease in reactor performance. Deviation from the deterministic model increases with increasing variance and decreasing fluctuation frequency of the correlated stochastic coefficients,
the effect of decreasing the mean conversion with deviation from the nonfluctuating case greater at lower fluctuation frequencies. Another random fluidized bed model is due to Orcutt and Carpenter[‘l]. This model consists of a computer simulation of a vertical chain of rising bubbles. Bubbles enter a bed with random size at fixed frequency and rise through a well mixed dense phase in which an isothermal first order reaction occurs. Empirical expressions for bubble velocity interactions and coalescence are employed. Although this model does allow for variation in bubble size, a realistic bubble size distribution cannot be obtained from a single vertical chain of bubbles. Since these results were not compared with a similar model without randomness, the significance of the fluctuating distribution of bubble size was not determined. It is the objective of this work to investigate further the significance of the fluctuating nature of fluidized beds on reactor performance. The models employed consist of nonlinear vector stochastic differential equations. The reader is referred to the discussions of stochastic pcocesses and stochastic differential equations in the chemical engineering literature by Seinfeld and Lapidus[ll] and KingPI.
INTRODUCTION
The features of most fluidized bed reactor models have been summarized by Grace[II, Rowe[Zl, Pyle131 and Bukur et aL[41. These models differ in many important aspects but it is generally accepted that interphase mass transfer and solids mixing are related to the size and location of bubbles in the bed. Hence, the size and spatial distribution of bubbles throughout the bed are of considerable importance in predicting reactor performance. Small bubbles form at low levels of a fluidiied bed and randomly coalesce with neighboring bubbles as they ascend resulting in randomly fluctuating bubble size distributions throughout. However, most models consider bubble size to be uniform in the bed or to increase linearly with height. It has been found that none of these common models accurately represents observed reactor behavior[5]. The chaotic nature of a bubbling fluidized bed as observed through any small-scale glass-enclosed fluidized system implies that such beds are probably better described as stochastic systems than deterministic ones. Bukur et a1.[4] have suggested that no deterministic model will ever describe bubbling fluidized bed reactors with any precision. A few simple models have been proposed which consider the random behavior of fluidized beds. The model of Krambeck et al. [6] consists of two constant volume stirred tank reactors, one for each phase with gas interchange between phases. The interchange coefficient is modelled as a random process which fluctuates between two possible values. Although this model does not represent the reactor accurately, it provides some evidence of the effect of the unsteady nature of the transport processes on reactor performance. Krambeck et al. found that the fluctuations had
ISOTHERMAL MODEL TliED-C isothermal fluidized bed model consists of two well-mixed cells, one representing the bubble phase, the other the dense phase. Mass transfer takes place between phases with g the overall mass transfer coefficient. A schematic diagram of the system is shown in Fig. 1. A mass balance over a pair of cells with first order reaction in the dense phase gives The
+=P(c,-c.)+4k(c,-c,)
*Authorto whom correspondence should he addressed. CES Vol. 36. No. LB
653
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R. LIGON and N. R. AMUN~~ON ing nature of these distributions is not yet fully understood, it can be shown that random fluctuations in the bubble size distribution cause the dimensionless mass transfer coefficient to fluctuate. Here, the behavior of the isothermal model is explored when the mass transfer coefficient is modelled by various stochastic processes.
,
--k A
A
“E
“B
t
WHFTE NOISE MODEL In this model, the dimensionless overall mass transfer coefficient is replaced by a white noise stochastic process. The resulting _stochastic mass transfer coefficient is given by K = K t k where a is a constant and k is the Gaussian white noise stochastic process. Gaussian white noise is a Gaussian stochastic process with zero expectation and stationary auto correlation function given by .
‘F
Fig. 1. Schematicof the isothermalfiuidizedbed model.
R(T) = E[k(t)k(r + T)] = L@(T)
For convenience, we define the dimensionless quantities
where FT = FB + FE, qT = tiB + pE, and C, is the inlet concentration, CBO= C&. Equations (1) and (2) now take the dimensionless form
$f=K(xe-x~)+$(x,qo-Xe) (3) B -&)+a v, (xfL?o - XE)-R&Z.
(4)
If the bubble size distribution is assumed constant, K is
a constant and the above equations have the steady state solutions
where S(T) is the Dirac delta function. This implies that the process k, and hence the mass transfer coefficient K, can change infinitely fast. Since no process in nature can change infinitely fast, white noise is not physically realizable. Nevertheless, it is often employed as a model of random physical systems. White noise is formally related to another stochastic process, the Wiener process, W(t). The Wiener process is a continuous parameter Gaussian process with zero expectation and stationary independent increments. Although the Wiener process is not differentiable in any rigorous sense, it can be shown to formally satisfy dLV/dt= /r. The constant D in eqn (7) is defined as the variance parameter of the Wiener process associated with the white noise stochastic process. Using the above relationships, the isothermal white noise fluidized bed model can now be constructed. The additive Gaussian white noise form of the overall mass transfer coefficient is substituted into the physical model, eqns (3) and (4), along with the formal relationship between white noise and the Wiener process to form
This and equations of the general form dx = f(x, t) dt + g(x. t) d W(t)
xB = (%V,K + qsqaz + qB V&)XB, + KsB VE-G~ V,K + R V, V&C + qeqE + qB VER
(5) ~BVBKXB,+(~~VBK+%%)~~, xE=
VBKtRVBV,KtqBqBtqBV_yR’
(6)
(7)
(9)
are called stochastic differential equations. In the most general form of eqn (9), x and f are n-vectors, g is an n x m matrix, and d W is an m-vector Wiener process with covariance matrix defined by E [d W(t) d W(t)‘] = Q(r)dt.
In reality, the number and size of bubbles in any section of a fluidized bed are not constant but fluctuate with time. Considerable experimental work measuring bubble distributions in fluid&d beds has been performed (e.g. see Burgess and Claderbank[lO]). Although the ffuctuat-
An important characteristic of stochastic differential equations is that they are not governed by the rules of ordinary calculus. There exist two alternative interpretations of these equations, the Ito and Stratonotich
656
J. R. LIOONand N. R. AMUNIZSON
Hence the first stationary moments of the Stratonovich interpretation are given by
k^has a Gaussian distribution with zero expectation and the stationary autocorrelation function R(T) = i uz/3De-BT.
Coupling eqn (17) and the defining expression for xa with eqns (3) and (4) of the deterministic model gives the Gaussian correlated noise model Similarly the stationary second moments of the Stratonovich interpretation are solutions of the same equation as the Ito moments, eqn (13), with the uii’s replaced by IQ’S. Results. Stationary first and second moments of the Ito and Stratonovich interpretations of the white noise model are presented in Fig. 2 as a function of the variance parameter of the Wiener process, D. The Ito first moments are independent of D and identical to the deterministic solution. The first moments of the Stratonovich interpretation deviate from the Ito moments with increasing D. For the parameter values of Fig. 2, the Stratonovich first moments become undefined as D+ 1.25 because the denominators in eqns (15) and (16) approach zero. The fact that there are two interpretations of the white noise model which yield different solutions is due to the pathalogical nature of white noise and the Wiener process. When the more realistically correlated noise processes discussed in the following section are employed, the Ito and Stratonovich interpretations become identical.
(18)
Since the coefficient of d W in this equation is independent of the state vector, the Ito and Stratonovich interpretations can be shown to be equivalent. Gamma model The second correlated noise model is formed by
replacing the overall mass transfer coefficient in the deterministic model with a gamma distributed process. The gamma process used has the probability density function
CORRELATEDNOLYE MODELS
(n+l)/sD
PM =
Gaussian model The first correlated noise model to be discussed is the
Gaussian model. The overall mass transfer coefficient in the deterministic model is replaced by x3 = a + l, where i is a correlated Gaussian noise process generated by the stochastic differential equation (17)
(0 +I--SDI,aD
@‘sD)&
(19)
and is generated by the stochastic differential equation dx3 = (a + I- I- /3x3)df +(~SX,)“~d W
zi
a 5
.6
I
0
It*
Y
Stratancvich
Solution Solution
-
*
.O &
.2 OF
THE
.4
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.6
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(201
where [ = 0 if eqn (20) is an Ito equation and 5 = $8D if it is a Stratonovich equation. The gamma process can be
E
.I
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l),sD)
.O d
.2
.4
.6
OF THE WIENER
Fig. 2. Comparisonof the Ito and Stratonovichinterpretationsof the white
PROCESS
noise model.
Modelling of fluidized bed reactors-IV(a)
shown to have expected value (a+ I)/,9 and the stationary autocorrelation function DS(a+l) i?(r) = Fe
++
657
analog computer as
g = f(x, t) + g(x,
a+l * ( P ) *
The gamma process has no nonpositive realizations. Hence it should be preferred over the Gaussian process because the Gaussian process allows negative realizations of the mass transfer coefficient. Coupling eqn (20) with the deterministic model, eqns (3) and (4), gives the gamma model
t,A
where A is an electronically generated band-limited white noise process. High speed integration takes place on the analog while samples are sent to the digital computer for computation of the solution statistics. With high speed analog to digital conversion, statistical results based on thousands of samples can be obtained in seconds. This technique was employed in the following
investigation using the hybrid system in the Engineering Systems Simulation Laboratory at the University of Houston. It consists of an IBM 360 Model 44 digital
0
1 dW
t L
(28:,)‘~z
computer and a Hybrid System Model SSlOO analog computer interfaced with a Hybrid Systems Model IO44 Hybrid Linkage Unit. An Elgenco Inc. Model 602A Gaussian Noise Generator was used as the band-limited white noise input. Details of the technique are discussed elsewhere[l3].
(21)
where [ = 0 if eqn (21) is interpreted as an Ito equation and t = $iD if it is a Stratonovich equation. Selecting the Ito value of [ and treating eqn (21) as an Ito equation gives the same Fokker-Planck equation as that obtained by using the Stratonovich value of [ and treating it as a Stratonovich equation. Hence, as with the Gaussian correlated model, the two vector solution processes have the same probability density and are therefore equivalent. Consequently, the ambiguity of two solutions of the white noise model is lost when the mass transfer coefficient is replaced by the gamma or Gaussian process. It can also be shown that this ambiguity is lost whenever the mass transfer coefficient is replaced by any stochastic process which has a continuous autocor-
Results of the gamma and Gaussian models
King[9] has stated that the gamma process is more desirable than the Gaussian process for modelling a nonnegative physical process because an additive Gaussian process can allow negative realizations. Here, negative realizations of the mass transfer coefficient did lead to undesirable results in the Gaussian model. Figures 3 and 4 show that when driven by gamma and Gaussian inputs with similar mean and autocorrelation function, the bubble and dense phase concentrations are considerably more skewed for the Gaussian model than for the gamma model. These ligures show that although the mean values of the concentrations are similar for both models, the distributions are quite different. The negative reahzations of the Gaussian input lead to negative dense 999
relation function. Solution
of the correlated
noise models
Although the use of realistice noise processes removes the pathalogy of the white noise model, the method of solution becomes much more difficult. A method of moments solution of the Fokker-Planck equations associated with the gamma and Gaussian models cannot be used because each moment equation depends on higher order moments and there is no satisfactory truncation procedure. Such an infinite hierarchy of moment equations will always be obtained when f(x, t) in eqn (9) is nonlinear in x. A satisfactory means of generating solutions of nonlinear stochastic differential equations is by hybrid computer simulation. Bullin and Dukler[lZ] have shown that a hybrid computer can be used to solve Stratonovich stochastic differential equations in the following manner. Using the formal relationship between white noise and the Wiener process, a Stratonovich stochastic difierential equation of the form of eqn (9) is programmed on the
-
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3020 =
e
IO-
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1 0.65
I
I
I
L
0.7
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0.8
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I
0.9
I
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CONCENTRATION,
I
I.0 XB
3. Non& probability plot of the bubble phase concentrations for the gamma and Gaussian models.
J. R. LICONand N. R. AMIJNLWON
658 99.9
The autocorrelation function of the noise process in both models is of the form
-
R(T) = u e+ + rut’.
DENSE Fig.
PHASE
CONCENTRATION,
The larger the value of p, the more rapidly the noise fluctuates. Concentration statistics for both models are presented in Fig. 6 for values of /3 ranging from 0.0125 to 12.5.t The concentration mean values are similar for both models. The standard deviations are similar at the higher values of g representing the higher noise frequencies with the Gaussian deviations becoming higher at lower values of p. Both models are least sensitive to the highest frequency noise input, the highest value of B. Concentration statistics for various reaction rates are given in Fig. 7 for the gamma model. Deviation from the deterministic model is approximately the same over most of the range studied. The random value of xn is consistently higher than the deterministic value and that of xE is consistently lower. Here, the added randomness reduces the effective mass transfer and reduces reactant conversion. These results are in agreement with similar runs at other parameter values.
XE
4. Normal probability plot of the dense phase concentrations for the gamma and Gaussian models.
CONCLUSIONS
phase concentration and bubble phase concentrations greater than unity. Both models indicate that the bubble phase concentration tends to be positively skewed while the dense phase concentration is negatively skewed. Figure 5 shows the effect of standard deviation of the dimensionless mass transfer coefficient on the bubble and dense phase concentrations for both models. The concentration mean values are similar for the two models and are not strong functions of the standard deviation of the noise input. The standard deviations of the concentrations are similar for both models at low mass transfer coefficient standard deviation, SX,, while the Gaussian model standard deviations increase sharply at higher values of Sx,.
use of additive white noise in the isothermal fluidized bed model leads to the pathological situation of two interpretations, one based on the Ito theory of stochastic differential equations and the other based on the Stratonovich theory. These two theories yield different results. Replacing white noise by any noise process with a continuous autocorrelation function removes this pathology and the two interpretations are equivalent. Two such processes are the gamma ane Gaussian processes. The gamma noise process is preferred over the Gaussian process for use as a mass transfer coefficient because the Gaussian process can allow negative realizations. These negative realizations of the mass transfer coefficient can lead to the pathalogies of negative dense The
tThe 90% cut-off frequency is given by 6.38.
06 5 5
05
R’I. F
q,=.a3 = x3 1.0
XE
V3’”
/
Gausrmn
5 u
I
0
.I STANDARD
1
1
I
2
.3
.4
DEVIATION
OF
‘S 5 X3
,
IF .O
’ .I STANDARD
I
.2
L
3 DEVIATION
XE Gamma
1
I
.4 OF
.5 X3
Fig. 5. Effect of the standard deviation of the mass transfer coefficient on the gamma and Gaussian models.
659
Model&g of Euidizedbed reactors-VI(a) 1.0 -
qa- 0.83 v,-0.6 R-I.0
.I20.9
-
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0.8
-
g
0.7
-
0.6
-
sxj =o.!YG0mma Medal SX3 =0.48Gaussian Modal
c’ 2 E
0.4h
8-12.5
.&v
’
0.001
X I
I
0.I
I.0
I
0.01
.oo&,
FREQUENCYFACTOR,X
’ 0.001
1
0.01
FREOUENCY
I
I
0.1
1.0
. FACTOR,
X
Fig. 6. Effect of frequencyof the mass transfercoefficienton the gammaand Gaussianmodels. 0.10
0.09
0.08
o.or
0.06
2 0 4
0.05
g ti
0.04
the mass transfer coefficient. Both models deviate from the deterministic solution with decreasing fluctuation frequency over the range tested. These results are in qualitative agreement with those of Krambeck et al. [6]. Although there is no theoretical basis for argument of whether the Ita or Stratonovich solution of the white noise model more closely resembles reality, comparison with the correlated noise models indicates that the Stratonovich model may be favored. While the Ito white noise model concentrations are independent of the variance parameter of the Wiener process, the Stratonovich concentrations deviate from the deterministic model in a similar to that of the gamma and Gaussian
s a
NOTATION
ii
0.:
0.03
0.2
002
0. I
001
I
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1.0
1
I
I
,
2.0
3.0
4.0
5.0
DtMENSlONLESS RATE
CmJ
REACTION
Fii. 7. Effect of reactionrate on concentrations.
phase concentrations and bubble phase outlet concentrations higher than the inlet. For a11 cases tested, the concentration mean values given by the two correlated noise models are essentially equivalent when the gamma and Gaussian inputs have similar variances and fluctuation frequencies. The standard deviations, however, become increasingly different as the input variance is increased and the frequency of fluctuation is decreased. In every case, the Gaussian model gives higher concentration standard deviations than the gamma model. The Gaussian and gamma model both predict a decrease in conversion with an increase in variance of
i,jth
component of the Ito moment matrix variance parameter of the scalar Wiener process expectation operator volumetric flow rate arbitrary n-vector function in general stochastic differential equation arbitrary n x m matrix function in general stochastic differential equation dimensionless mass transfer coefficient mass transfer coefficient average value of K white noise perturbation Gaussian correlated stochastic process i, jth moment function transition probability m X m covariance matrix of an m-vector Wiener process dimensionless flow rate term dimensionless reaction rate constant first-order reaction rate constant stationary autocorrelation function standard deviation dimensionless time dimensionless volume variance operator
J. R. LIOON and N. R. AMLIND~~N
660
P
ww X
x3
volume Wiener process dimensionless concentration mass transfer coefficient
Greek symbols a constant in gamma distributed process "ij i, jth component of the Stratonovich matrix B time constant in earnma and Gaussian s constant in g&ma distributed process S(T) Dirac delta function in Gaussian distributed u constant process T time X frequency factor, x = 8112.5 Subscripts B bubble phase E dense phase 0 entering component T total
stochastic moment processes stochastic
stochastic
REFERENCES 111Grace J. R., A.LCh_E. Symp. Ser. 1971116 159. I21 Rowe P. N., Proc. 5th Europ., 2nd Int. Symp. Chemical Reactor Engineering, A9-1. Amsterdam 1972. 131Pyle D. L., Aduan. Chem. Ser. 1972109 106. 141 Buker D., Caram H. S. and Amundson N. R., Same Model Studies of Fluidized Bed Reactors, In Chemical Reactor Theory, A Review (Edited by Lapidus L. and Amundson N. R.). ,Prentice-Hall, Euglewood Cliffs, New Jersey 1977. [S] Chavarie C. and Grace J. R., hd. fingng Chem. Fundls 1975 14 75. [6] fxxxxz;k$ J., Katz S. and Shinnar R., Chem. Engng SC!. [fl Orcutt J. C. and Carpenter B. H., Chem. Bngng Sci. 197126 1049.
[8] Seiafeld J. H. and Lapidus L., Ma?hematicaf Methods in Chemical Engineering, Vol. 3. Prentice-Hall, Eaglewood Cliis, New Jersey 1974. 191King R. P., Chem. .&pg Commun. 19741 221. [IO] Burgess J. M. and Calderbank P. H., Chem. Engng Sci. 1975 30 743; 197530 3107;1975Jo 1511. [ll] Jazwinski A., Stochosfic Processes and Filtering Theory. Academic Press, New York 1970. [12] Bullin J. A. and Dukler A. E.. Chem. Engng SC!. 1975Jo 631. [13] Ligon J. R., Ph.D. Thesis, University of Minnesota 1978.
and NEAL R. AMUNDSON* Department of Chemical Engineering, University of Houston, Houston, TX 77004,U.S.A. (Received 30 April 1980: accepted 10 July 1980)
Ah&a&-A two-phase stochastic isothermal fluidized bed reactor model with first order reaction in the dense phase is developed to investigate the signiicance of the fluctuating nature of fluidiid beds on reactor performance. Several stochastic processes are employed as the overall mass transfer coefficient between phases. Analytical moment solutions are obtained for white noise coefficients while hybrid computer simulation was used for correlated stochasticcoefficients.Resultsindicatethat a gammadistributedcoefficientis preferred over white noise and Gaussian correlated coefficients. When compared with the deterministic model, randomness in the mass transfer coe5cient is seen to lead to a decrease in reactor performance. Deviation from the deterministic model increases with increasing variance and decreasing fluctuation frequency of the correlated stochastic coefficients,
the effect of decreasing the mean conversion with deviation from the nonfluctuating case greater at lower fluctuation frequencies. Another random fluidized bed model is due to Orcutt and Carpenter[‘l]. This model consists of a computer simulation of a vertical chain of rising bubbles. Bubbles enter a bed with random size at fixed frequency and rise through a well mixed dense phase in which an isothermal first order reaction occurs. Empirical expressions for bubble velocity interactions and coalescence are employed. Although this model does allow for variation in bubble size, a realistic bubble size distribution cannot be obtained from a single vertical chain of bubbles. Since these results were not compared with a similar model without randomness, the significance of the fluctuating distribution of bubble size was not determined. It is the objective of this work to investigate further the significance of the fluctuating nature of fluidized beds on reactor performance. The models employed consist of nonlinear vector stochastic differential equations. The reader is referred to the discussions of stochastic pcocesses and stochastic differential equations in the chemical engineering literature by Seinfeld and Lapidus[ll] and KingPI.
INTRODUCTION
The features of most fluidized bed reactor models have been summarized by Grace[II, Rowe[Zl, Pyle131 and Bukur et aL[41. These models differ in many important aspects but it is generally accepted that interphase mass transfer and solids mixing are related to the size and location of bubbles in the bed. Hence, the size and spatial distribution of bubbles throughout the bed are of considerable importance in predicting reactor performance. Small bubbles form at low levels of a fluidiied bed and randomly coalesce with neighboring bubbles as they ascend resulting in randomly fluctuating bubble size distributions throughout. However, most models consider bubble size to be uniform in the bed or to increase linearly with height. It has been found that none of these common models accurately represents observed reactor behavior[5]. The chaotic nature of a bubbling fluidized bed as observed through any small-scale glass-enclosed fluidized system implies that such beds are probably better described as stochastic systems than deterministic ones. Bukur et a1.[4] have suggested that no deterministic model will ever describe bubbling fluidized bed reactors with any precision. A few simple models have been proposed which consider the random behavior of fluidized beds. The model of Krambeck et al. [6] consists of two constant volume stirred tank reactors, one for each phase with gas interchange between phases. The interchange coefficient is modelled as a random process which fluctuates between two possible values. Although this model does not represent the reactor accurately, it provides some evidence of the effect of the unsteady nature of the transport processes on reactor performance. Krambeck et al. found that the fluctuations had
ISOTHERMAL MODEL TliED-C isothermal fluidized bed model consists of two well-mixed cells, one representing the bubble phase, the other the dense phase. Mass transfer takes place between phases with g the overall mass transfer coefficient. A schematic diagram of the system is shown in Fig. 1. A mass balance over a pair of cells with first order reaction in the dense phase gives The
+=P(c,-c.)+4k(c,-c,)
*Authorto whom correspondence should he addressed. CES Vol. 36. No. LB
653
(1)
654
J.
I
R. LIGON and N. R. AMUN~~ON ing nature of these distributions is not yet fully understood, it can be shown that random fluctuations in the bubble size distribution cause the dimensionless mass transfer coefficient to fluctuate. Here, the behavior of the isothermal model is explored when the mass transfer coefficient is modelled by various stochastic processes.
,
--k A
A
“E
“B
t
WHFTE NOISE MODEL In this model, the dimensionless overall mass transfer coefficient is replaced by a white noise stochastic process. The resulting _stochastic mass transfer coefficient is given by K = K t k where a is a constant and k is the Gaussian white noise stochastic process. Gaussian white noise is a Gaussian stochastic process with zero expectation and stationary auto correlation function given by .
‘F
Fig. 1. Schematicof the isothermalfiuidizedbed model.
R(T) = E[k(t)k(r + T)] = L@(T)
For convenience, we define the dimensionless quantities
where FT = FB + FE, qT = tiB + pE, and C, is the inlet concentration, CBO= C&. Equations (1) and (2) now take the dimensionless form
$f=K(xe-x~)+$(x,qo-Xe) (3) B -&)+a v, (xfL?o - XE)-R&Z.
(4)
If the bubble size distribution is assumed constant, K is
a constant and the above equations have the steady state solutions
where S(T) is the Dirac delta function. This implies that the process k, and hence the mass transfer coefficient K, can change infinitely fast. Since no process in nature can change infinitely fast, white noise is not physically realizable. Nevertheless, it is often employed as a model of random physical systems. White noise is formally related to another stochastic process, the Wiener process, W(t). The Wiener process is a continuous parameter Gaussian process with zero expectation and stationary independent increments. Although the Wiener process is not differentiable in any rigorous sense, it can be shown to formally satisfy dLV/dt= /r. The constant D in eqn (7) is defined as the variance parameter of the Wiener process associated with the white noise stochastic process. Using the above relationships, the isothermal white noise fluidized bed model can now be constructed. The additive Gaussian white noise form of the overall mass transfer coefficient is substituted into the physical model, eqns (3) and (4), along with the formal relationship between white noise and the Wiener process to form
This and equations of the general form dx = f(x, t) dt + g(x. t) d W(t)
xB = (%V,K + qsqaz + qB V&)XB, + KsB VE-G~ V,K + R V, V&C + qeqE + qB VER
(5) ~BVBKXB,+(~~VBK+%%)~~, xE=
VBKtRVBV,KtqBqBtqBV_yR’
(6)
(7)
(9)
are called stochastic differential equations. In the most general form of eqn (9), x and f are n-vectors, g is an n x m matrix, and d W is an m-vector Wiener process with covariance matrix defined by E [d W(t) d W(t)‘] = Q(r)dt.
In reality, the number and size of bubbles in any section of a fluidized bed are not constant but fluctuate with time. Considerable experimental work measuring bubble distributions in fluid&d beds has been performed (e.g. see Burgess and Claderbank[lO]). Although the ffuctuat-
An important characteristic of stochastic differential equations is that they are not governed by the rules of ordinary calculus. There exist two alternative interpretations of these equations, the Ito and Stratonotich
656
J. R. LIOONand N. R. AMUNIZSON
Hence the first stationary moments of the Stratonovich interpretation are given by
k^has a Gaussian distribution with zero expectation and the stationary autocorrelation function R(T) = i uz/3De-BT.
Coupling eqn (17) and the defining expression for xa with eqns (3) and (4) of the deterministic model gives the Gaussian correlated noise model Similarly the stationary second moments of the Stratonovich interpretation are solutions of the same equation as the Ito moments, eqn (13), with the uii’s replaced by IQ’S. Results. Stationary first and second moments of the Ito and Stratonovich interpretations of the white noise model are presented in Fig. 2 as a function of the variance parameter of the Wiener process, D. The Ito first moments are independent of D and identical to the deterministic solution. The first moments of the Stratonovich interpretation deviate from the Ito moments with increasing D. For the parameter values of Fig. 2, the Stratonovich first moments become undefined as D+ 1.25 because the denominators in eqns (15) and (16) approach zero. The fact that there are two interpretations of the white noise model which yield different solutions is due to the pathalogical nature of white noise and the Wiener process. When the more realistically correlated noise processes discussed in the following section are employed, the Ito and Stratonovich interpretations become identical.
(18)
Since the coefficient of d W in this equation is independent of the state vector, the Ito and Stratonovich interpretations can be shown to be equivalent. Gamma model The second correlated noise model is formed by
replacing the overall mass transfer coefficient in the deterministic model with a gamma distributed process. The gamma process used has the probability density function
CORRELATEDNOLYE MODELS
(n+l)/sD
PM =
Gaussian model The first correlated noise model to be discussed is the
Gaussian model. The overall mass transfer coefficient in the deterministic model is replaced by x3 = a + l, where i is a correlated Gaussian noise process generated by the stochastic differential equation (17)
(0 +I--SDI,aD
@‘sD)&
(19)
and is generated by the stochastic differential equation dx3 = (a + I- I- /3x3)df +(~SX,)“~d W
zi
a 5
.6
I
0
It*
Y
Stratancvich
Solution Solution
-
*
.O &
.2 OF
THE
.4
WIENER
.6
.8
PROCESS
(201
where [ = 0 if eqn (20) is an Ito equation and 5 = $8D if it is a Stratonovich equation. The gamma process can be
E
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e-@xj,m
l),sD)
.O d
.2
.4
.6
OF THE WIENER
Fig. 2. Comparisonof the Ito and Stratonovichinterpretationsof the white
PROCESS
noise model.
Modelling of fluidized bed reactors-IV(a)
shown to have expected value (a+ I)/,9 and the stationary autocorrelation function DS(a+l) i?(r) = Fe
++
657
analog computer as
g = f(x, t) + g(x,
a+l * ( P ) *
The gamma process has no nonpositive realizations. Hence it should be preferred over the Gaussian process because the Gaussian process allows negative realizations of the mass transfer coefficient. Coupling eqn (20) with the deterministic model, eqns (3) and (4), gives the gamma model
t,A
where A is an electronically generated band-limited white noise process. High speed integration takes place on the analog while samples are sent to the digital computer for computation of the solution statistics. With high speed analog to digital conversion, statistical results based on thousands of samples can be obtained in seconds. This technique was employed in the following
investigation using the hybrid system in the Engineering Systems Simulation Laboratory at the University of Houston. It consists of an IBM 360 Model 44 digital
0
1 dW
t L
(28:,)‘~z
computer and a Hybrid System Model SSlOO analog computer interfaced with a Hybrid Systems Model IO44 Hybrid Linkage Unit. An Elgenco Inc. Model 602A Gaussian Noise Generator was used as the band-limited white noise input. Details of the technique are discussed elsewhere[l3].
(21)
where [ = 0 if eqn (21) is interpreted as an Ito equation and t = $iD if it is a Stratonovich equation. Selecting the Ito value of [ and treating eqn (21) as an Ito equation gives the same Fokker-Planck equation as that obtained by using the Stratonovich value of [ and treating it as a Stratonovich equation. Hence, as with the Gaussian correlated model, the two vector solution processes have the same probability density and are therefore equivalent. Consequently, the ambiguity of two solutions of the white noise model is lost when the mass transfer coefficient is replaced by the gamma or Gaussian process. It can also be shown that this ambiguity is lost whenever the mass transfer coefficient is replaced by any stochastic process which has a continuous autocor-
Results of the gamma and Gaussian models
King[9] has stated that the gamma process is more desirable than the Gaussian process for modelling a nonnegative physical process because an additive Gaussian process can allow negative realizations. Here, negative realizations of the mass transfer coefficient did lead to undesirable results in the Gaussian model. Figures 3 and 4 show that when driven by gamma and Gaussian inputs with similar mean and autocorrelation function, the bubble and dense phase concentrations are considerably more skewed for the Gaussian model than for the gamma model. These ligures show that although the mean values of the concentrations are similar for both models, the distributions are quite different. The negative reahzations of the Gaussian input lead to negative dense 999
relation function. Solution
of the correlated
noise models
Although the use of realistice noise processes removes the pathalogy of the white noise model, the method of solution becomes much more difficult. A method of moments solution of the Fokker-Planck equations associated with the gamma and Gaussian models cannot be used because each moment equation depends on higher order moments and there is no satisfactory truncation procedure. Such an infinite hierarchy of moment equations will always be obtained when f(x, t) in eqn (9) is nonlinear in x. A satisfactory means of generating solutions of nonlinear stochastic differential equations is by hybrid computer simulation. Bullin and Dukler[lZ] have shown that a hybrid computer can be used to solve Stratonovich stochastic differential equations in the following manner. Using the formal relationship between white noise and the Wiener process, a Stratonovich stochastic difierential equation of the form of eqn (9) is programmed on the
-
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-
99
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98
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5
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z
90 -
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g
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e
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I = 0.5 0.1
1 0.65
I
I
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L
0.7
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0.8
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ElUBi3LE Fii.
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I
0.9
I
0.95
CONCENTRATION,
I
I.0 XB
3. Non& probability plot of the bubble phase concentrations for the gamma and Gaussian models.
J. R. LICONand N. R. AMIJNLWON
658 99.9
The autocorrelation function of the noise process in both models is of the form
-
R(T) = u e+ + rut’.
DENSE Fig.
PHASE
CONCENTRATION,
The larger the value of p, the more rapidly the noise fluctuates. Concentration statistics for both models are presented in Fig. 6 for values of /3 ranging from 0.0125 to 12.5.t The concentration mean values are similar for both models. The standard deviations are similar at the higher values of g representing the higher noise frequencies with the Gaussian deviations becoming higher at lower values of p. Both models are least sensitive to the highest frequency noise input, the highest value of B. Concentration statistics for various reaction rates are given in Fig. 7 for the gamma model. Deviation from the deterministic model is approximately the same over most of the range studied. The random value of xn is consistently higher than the deterministic value and that of xE is consistently lower. Here, the added randomness reduces the effective mass transfer and reduces reactant conversion. These results are in agreement with similar runs at other parameter values.
XE
4. Normal probability plot of the dense phase concentrations for the gamma and Gaussian models.
CONCLUSIONS
phase concentration and bubble phase concentrations greater than unity. Both models indicate that the bubble phase concentration tends to be positively skewed while the dense phase concentration is negatively skewed. Figure 5 shows the effect of standard deviation of the dimensionless mass transfer coefficient on the bubble and dense phase concentrations for both models. The concentration mean values are similar for the two models and are not strong functions of the standard deviation of the noise input. The standard deviations of the concentrations are similar for both models at low mass transfer coefficient standard deviation, SX,, while the Gaussian model standard deviations increase sharply at higher values of Sx,.
use of additive white noise in the isothermal fluidized bed model leads to the pathological situation of two interpretations, one based on the Ito theory of stochastic differential equations and the other based on the Stratonovich theory. These two theories yield different results. Replacing white noise by any noise process with a continuous autocorrelation function removes this pathology and the two interpretations are equivalent. Two such processes are the gamma ane Gaussian processes. The gamma noise process is preferred over the Gaussian process for use as a mass transfer coefficient because the Gaussian process can allow negative realizations. These negative realizations of the mass transfer coefficient can lead to the pathalogies of negative dense The
tThe 90% cut-off frequency is given by 6.38.
06 5 5
05
R’I. F
q,=.a3 = x3 1.0
XE
V3’”
/
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5 u
I
0
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1
I
2
.3
.4
DEVIATION
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‘S 5 X3
,
IF .O
’ .I STANDARD
I
.2
L
3 DEVIATION
XE Gamma
1
I
.4 OF
.5 X3
Fig. 5. Effect of the standard deviation of the mass transfer coefficient on the gamma and Gaussian models.
659
Model&g of Euidizedbed reactors-VI(a) 1.0 -
qa- 0.83 v,-0.6 R-I.0
.I20.9
-
Y 9
0.8
-
g
0.7
-
0.6
-
sxj =o.!YG0mma Medal SX3 =0.48Gaussian Modal
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8-12.5
.&v
’
0.001
X I
I
0.I
I.0
I
0.01
.oo&,
FREQUENCYFACTOR,X
’ 0.001
1
0.01
FREOUENCY
I
I
0.1
1.0
. FACTOR,
X
Fig. 6. Effect of frequencyof the mass transfercoefficienton the gammaand Gaussianmodels. 0.10
0.09
0.08
o.or
0.06
2 0 4
0.05
g ti
0.04
the mass transfer coefficient. Both models deviate from the deterministic solution with decreasing fluctuation frequency over the range tested. These results are in qualitative agreement with those of Krambeck et al. [6]. Although there is no theoretical basis for argument of whether the Ita or Stratonovich solution of the white noise model more closely resembles reality, comparison with the correlated noise models indicates that the Stratonovich model may be favored. While the Ito white noise model concentrations are independent of the variance parameter of the Wiener process, the Stratonovich concentrations deviate from the deterministic model in a similar to that of the gamma and Gaussian
s a
NOTATION
ii
0.:
0.03
0.2
002
0. I
001
I
0.0 D
1.0
1
I
I
,
2.0
3.0
4.0
5.0
DtMENSlONLESS RATE
CmJ
REACTION
Fii. 7. Effect of reactionrate on concentrations.
phase concentrations and bubble phase outlet concentrations higher than the inlet. For a11 cases tested, the concentration mean values given by the two correlated noise models are essentially equivalent when the gamma and Gaussian inputs have similar variances and fluctuation frequencies. The standard deviations, however, become increasingly different as the input variance is increased and the frequency of fluctuation is decreased. In every case, the Gaussian model gives higher concentration standard deviations than the gamma model. The Gaussian and gamma model both predict a decrease in conversion with an increase in variance of
i,jth
component of the Ito moment matrix variance parameter of the scalar Wiener process expectation operator volumetric flow rate arbitrary n-vector function in general stochastic differential equation arbitrary n x m matrix function in general stochastic differential equation dimensionless mass transfer coefficient mass transfer coefficient average value of K white noise perturbation Gaussian correlated stochastic process i, jth moment function transition probability m X m covariance matrix of an m-vector Wiener process dimensionless flow rate term dimensionless reaction rate constant first-order reaction rate constant stationary autocorrelation function standard deviation dimensionless time dimensionless volume variance operator
J. R. LIOON and N. R. AMLIND~~N
660
P
ww X
x3
volume Wiener process dimensionless concentration mass transfer coefficient
Greek symbols a constant in gamma distributed process "ij i, jth component of the Stratonovich matrix B time constant in earnma and Gaussian s constant in g&ma distributed process S(T) Dirac delta function in Gaussian distributed u constant process T time X frequency factor, x = 8112.5 Subscripts B bubble phase E dense phase 0 entering component T total
stochastic moment processes stochastic
stochastic
REFERENCES 111Grace J. R., A.LCh_E. Symp. Ser. 1971116 159. I21 Rowe P. N., Proc. 5th Europ., 2nd Int. Symp. Chemical Reactor Engineering, A9-1. Amsterdam 1972. 131Pyle D. L., Aduan. Chem. Ser. 1972109 106. 141 Buker D., Caram H. S. and Amundson N. R., Same Model Studies of Fluidized Bed Reactors, In Chemical Reactor Theory, A Review (Edited by Lapidus L. and Amundson N. R.). ,Prentice-Hall, Euglewood Cliffs, New Jersey 1977. [S] Chavarie C. and Grace J. R., hd. fingng Chem. Fundls 1975 14 75. [6] fxxxxz;k$ J., Katz S. and Shinnar R., Chem. Engng SC!. [fl Orcutt J. C. and Carpenter B. H., Chem. Bngng Sci. 197126 1049.
[8] Seiafeld J. H. and Lapidus L., Ma?hematicaf Methods in Chemical Engineering, Vol. 3. Prentice-Hall, Eaglewood Cliis, New Jersey 1974. 191King R. P., Chem. .&pg Commun. 19741 221. [IO] Burgess J. M. and Calderbank P. H., Chem. Engng Sci. 1975 30 743; 197530 3107;1975Jo 1511. [ll] Jazwinski A., Stochosfic Processes and Filtering Theory. Academic Press, New York 1970. [12] Bullin J. A. and Dukler A. E.. Chem. Engng SC!. 1975Jo 631. [13] Ligon J. R., Ph.D. Thesis, University of Minnesota 1978.