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Modelling of fluidized bed reactors—VI(b)
ChsmicalEt&wing Scie,,ceVol. 36, pp. t61-6771 PergamonPnor Ud., 1961. Plinted in Gnat Britain
MODELLING OF FLUIDIZED BED REACTORS-VI(b) THE NONISOTHERMAL
BED WITH
STOCHASTIC
BUBBLES
JOHN ROBERT LIGON Department of Chemical Engineering and Materials Science, University of Minnesota, 151 Amundson Hall, 421 Washington Avenue SE., Minneapolis, MN 55455, U.S.A. and
NEAL R. AMUNDSON* Departmentof ChemicalEngineering,University of Houston, Houston, TX 77004, U.S.A. (Reeefued 19 May
1980;
accepted
IO July 1980)
Abstract-Two stochastic nonisothermal fluidiied bed reactor models are developed to investigate the significance of the fluctuating nature of fluidized beds on reactor performance. Fluctuating bubble size distributions within the bed are simulated by stochastic mass and heat transfer coefficients. Results of hybrid computer simulations indicate that randomness can enhance or inhibit reactor performance depending on the operating parameters of the nonisothermal model. Bubble and dense phase concentrationstatistics are fairly similar to those of corresponding isothermal models because dense phase temperatures are relatively insensitiveto transfercoefficientfluctuations due to the high dense phase heat capacity. However, the corresponding stochastic isothermal models predict decreases in conversion with increasingvariance in the transfer coefficients for all operating conditions. Results indicate that a deterministic system with two stable steady states may have fewer stable random stationary solutions.The existence of the stationarystates is dependenton fluctuation frequency and variance of the transfer
coefficients.
order exothermic reaction occurs. The response of the stochastic models with y distributed transfer coefficients is analyzed for various operating conditions.
INTRODUCTION
In this paper two nonisothermal stochastic fluidized bed reactor models are developed to study the effect of the fluctuating nature of fluidized beds on reactor performance. They are an extension of a previously discussed isothermal modeltl] to include temperature effects. The first model, referred to as the single nonisothermal
model, consists of two well-mixed cells, one representing the bubble phase and the other the dense phase. A single first-order exothermic reaction occurs in the dense phase. Mass and heat transfer take place between cells with R and g the overall mass and heat transfer coefficients. As in the isothermal model, the transfer coefficients are related to the distribution of bubbles in the bed. Since this relationship is similar for both coefficients, H will be assumed directly proportional to k. Ahhongh more accurate representations exist [see, e.g. Kunii and LevenspielI21. this assumption was judged satisfactory for the purposes of this investigation. Since the number and size of bubbles in the bed fluctuates with time, the transfer coefficients are best modeled as stochastic processes. In the second model, referred to as the structured model, the single well-mixed bubble phase cell of the first model is replaced by a series of three smaller cells with individual mass and heat transfer coefficients selected to reflect the increasing average bubble size with height in the bed. This provides a more realistic description of bubble phase behavior. The dense phase is again represented by a single well-mixed cell in which a fist *Author to whom correspondence
NONlsoTBEpMALFLUIDBED MODEL TEIEStMPLE deterministic model: Mass and energy balances over the bubble and dense phase cells with a first-order reaction in the dense phase produce the equations The
(1)
_&&4caw.w
(4)
P.&B R(&) = k, eewnsH.
(5)
Variatiens in gas density with temperature are neglected. The following dimensionless variables are defined
should be addressed. 661
xn = CelC,
qB = F$Fr.
xg = CnlC,
qe= FI#T
662
and N. R. AMUNDSON
J. R. LGON
vB
TE=RJ(RI-InR(T,))
= ri,lA
TB=HT~+(qeIWT~o HtqalVs .
(13)
V,=VJA
R,=ln(hk,,/F~)
For the inlet temperature
E&PO R2= RC,,(AH) t=F#T
G=-
POC,, P&O
where FT = Fst FE,3~= QB+ 3, and CO= CaO= CEa. The above equations can be cast in dimensionless form dx, _ dr-K(*~-*+(l-Xs) dx,=KVBK (XB-x,)+$1 df
(6) -.%I-R(T,)xe
~=H(T,-T.)++kJ.-T.)
%=H$
(7) (8)
Steady state solutions of xE are given in Fig. 1 as the intersections of a vertical line representing fixed feed conditions and the curve for constant RI.For the parameters chosen, multiple steady states can occur for R, 221. The stochastic model: The number and size of bubbles present in a bubbling fluidized bed fluctuate with time. Since the overall heat and mass transfer coefficients are dependent on bubble size, fluctuations in bubble size cause fluctuations in these coefficients. The simple stochastic nonisothermal mbdel is formed by replacing the dhnensikless mass and heat transfer coefficients of the deterministic model by the y distributed stochastic process, which is generated by the Stratonovich stochastic differential equation
G(TB-Te)+gG(TB,-Tg) tGR(TdxE (91
a+1-;6D-/3K)df+(26K)“2dW(f). (15)
R(TB)=~w(RI-Rz/TE).
W) The same hybrid mmputer techniques used for the isothermal gamma model[l,3] were used for the nonisothermal model. The University of Houston Hybrid computing facility and a General Radio model 1382 noise generator were used. The dimensionless model equations (11) are cast in an equivalent form to improve analog scaling by defining ~~ = TB - T&, and rE = TE - T,+ Defining K as.a gamma process, assuming the proportionality of H (12) and K,and letting T,, = TE,,, the stochastic model is
For specified values of K,H,qB. G, RIand RZ the steady state values of xg, TE and Te are related to XEby xB=Kre+&5J
K+qdVB
FEED TEMPERATUi?E. Fii. I.
To
Steady state operatingcurve for the nonisothermalmodel.
Medellii of fluidized bed reactors-VI(b) programmed on the analog computer as
663
Table 1. Parametersfor the simplenonisolhennalfluidbed model t
~~=K(TE-XB)+98(1-YB)/Va
(16)
;i,=-K~(~~-x,)+~(l-x~)-x~B exp 1 $%I 1
=4
K
=8
RI = 18
R2 = 75
VB = .45
qe = .9
G
* .004
(17) is = fiK(TE - %I”-9BTB/VB
(18) transfer coefficients. The dense phase temperature exhibited only very small amplitude fluctuations because of
~~=-~GK~(~~-a)-G*=~~V=
its + xEGB exp
(1%
~=~+l-~~sO-~K+(26K)“21$
(20)
where B = exp (RI - R,/T,) and D is the variance parameter of the Wiener process associated with the electronically generated band-limited process, k. For each case studied, the results of the stochastic nonisothermal model were compared with those of an equivalent isothermal model. The isothermal model is formed by holding the temperature integrators on the analog computer at their deterministic values. The resulting reaction rate is constant. The isothermal model is formed using the exact circuitry of the nonisothermal model and is executed in the same hybrid computer run to reduce analog computer inaccuracies, thus simplifying comparison of the two models. Discussion
Response of the stochastic model is first analyzed for RI = 18 and T, = 3.75 with other parameters specified in Table 1. Mean concentrations and temperatures of each phase and their standard deviations are plotted in Fig. 2 versus standard deviation of K. The mean bubble phase concentration is seen to increase very slightly with SK while the mean dense phase concentration decreases slightly. Mean temperatures for both phases do not vary signitkantly with increasing variance in the transfer coefficients. As expected, the standard deviations of the reactor state increase with increasing variance in the
STANMRD Fii.
DEVIATION
OF
K
high
total
heat
capacity.
Performance
of
the
equivalent isothermal stochastic fluidized bed model was not significantly different from that of the nonisothermal model depicted in Fig. 2. Performance of the nonisothermal model is presented in Fig. 3 for TO= 3.8 and other parameters from Table 1. The mean bubble phase concentration is seen to increase very slightly with increasing standard deviation of the transfer coefficients. The mean dense phase concentration is more sensitive to SK and it decreases as SK increases. The result is that reactant conversion decreases with increasing input variance. The corresponding isothermal model predicts mean concentrations which are slightly higher than those of the nonisothermal model at the higher transfer coefficient variance. Although deviation of the isothermal model from the nonisothermal is not great, the isothermal model predicts a decrease in conversion with increasing variance in the transfer coefficients, which is in qualitative disagreement with the nonisothermal model. Comparison of Figs. 2 and 3, indicates that deviation of each reactor state from its deterministic value is greater for T’ = 3.8 than for TO= 3.75. The observed state standard deviations are also higher for the higher inlet temperature. The standard deviations predicted by the isothermal model do not differ significantly from those of the nonisothermal model. Performance of the nonisothermal model is described in Fig. 4 for T, = 3.9 and other parameters as in the previous cases. As before, the mean dense phase concentration increases and mean bubble phase concentration decreases with increasing variance in the transfer
STANDARD
DEVIATION
OF
K
2. Effect of the standard deviation of the transfer coefficients on the nonisothermal model.
J. R. LEON
and N. R. AMUN~XWN
To= 3.8 p
1_[
'0.0125
;;&
,725 k
.soJ 0.0
.25
STANDARD
,000 0.0 25
. DEVIATION
OF
K
STANDARD
.50 .75
I.00 1.25
DEVIATION
OF
K
Fig. 3. Effect of the standard deviation of the transfer coefficients on the nonisothermal model.
Isothermal
X9
-j 4.525
.025
STANDARD Fii.
DEVIATION
OF
K
-
STANDARD
DEVIATION
OF
K
4. Effect of the standarddeviationof the transfercoefficientson the nonisothermalmodel.
coefficients. However, the increase in bubble phase concentration relative to the decrease in dense phase concentration is sufficiently greater for this case that reactant conversion decreases with increasing SK. The COTresponding isothermal model predicts slightly higher reactant concentrations in both phases leading to further decreased conversion. The state standard deviations are considerably higher for TO= 3.9 than for the previous cases although ST= is again low. Standard deviations predicted by the isothermal model are equivalent to those of the nonisothermal model. The inlet temperature is further increased to 3.95 for the case depicted in Fig. 5. Mean bubble phase concentration again increases and mean dense phase concentration decreases with increasing variance in the transfer coefficients. Mean concentrations of the equivalent isothermal model are very close to those of the nonisothermal means at the highest input variance levels. The mean dense phase temperature was seen to decrease t'lbe90% cut-off frequency is equal to 6.3B[3]
slightly with increasing variance which could have led to the result of reactant conversion being slightly better for the isothermal model. Both models predict conversion decreasing with increasing variance in the transfer coefficients. Predicted standard deviations were again equivalent for the two models. The effect of the fluctuation frequency of the gamma distributed transfer coefficients on reactor performance was also studied. Figure 6 depicts the mean concentrations and temperatures and their standard deviations for TO= 3.8 vs the frequency factor, x, where x = p/12.5 and p is the constant in the autocorrelation function of KVI RK(7) = u eC@ + m,‘.
(211
As /3 is increased, the transfer coefficients fluctuate more rapidly.? Deviation from the deterministic state is seen to increase with decreasing fluctuation frequency. The nonisothermal and isothermal models both predict increasing bubble phase concentration and decreasing
Modellingof fluidizedbed reactors-VI(b)
-
4600
-
4.575
-
4.550
Y
225
-
4525
iii z
k! r -203
-
4500
i? au
-
4.475
p
665
5
i5
sa g
:: .I25
x--
QO STANDARD Fii.
DEVIATION
OF
K
I
I
25
.50
I
STANDARD
.75
I
I
1.0
1.25
DEVIATION
OF
K
5. Effect of the standarddeviationof the transfercoeliicientson the nonisothermalmodel.
1.
-
I)30
-
% 5 ,025
-
= 3.6
SK = 1.25 (9
= 12.5 x
3.900
-i ,725 t
’ .OOl
I
I
.Ol
0.1
FRWUENCY
FACTOR,
l/J
1.0 x
FREOUENCY
FACTOR,
x
Fig. 6. Effect of the fluctuationfrequencyof the transfercoefficientson the nonisothermalmodel.
dense phase concentration with decreasing frequency. However, as previously found for this inlet temperature, the isothermal model predicts an increase when corn-
pared with the deterministic state. The standard deviation of the dense phase concentration increases with decreasing frequency while that of the bubble phase has a maximum at the intermediate frequency x = 0.1. A similar frequency plot is given in Fig. 7 for T, = 3.95. The mean bubble phase concentrations increase and dense phase concentrations decrease with decreasing fluctuation frequency for both the nonisothermal and isothermal models. The isothermal model predicts slightly better conversion than the nonisothermal model while both models predict a decrease in conversion with decreasing frequency. The bubble phase concentration standard deviation increases with decreasing frequency while that of the dense phase is maximum at the intermediate frequency x = 0.1. Standard deviations of the isothermal model do not vary significantly from those of the nonisothermal model.
Analysis of Figs. 2-7 reveals that the dense phase concentration is more sensitive than the bubble phase concentration to transfer coefficient fluctuations for low inlet temperatures and is less sensitive at higher inlet temperatures. The concentration nearer either limit, 0.0 or 1.0, will have more damped fluctuations than the other concentration. In each case studied, the isothermal model’predicts mean values and standard deviations of the reactor states very near those of the nonisothermal model. This is because the dense phase temperature is not very sensitive to fluctuations in the heat transfer coefficient due to its high total heat capacity. Since the bubble phase is assumed solid-free, it has a much lower total heat capacity and shows significant fluctuation. However, the reaction rate of the catalytic reaction is controlled by the dense phase temperature, and, if it remains relatively constant, the isothermal model provides a good approximation of the nonisothermal model. Results of the stochastic nonisothermal and isothermal models are similar for other operating conditions depic-
I. R. LIIXN and
,/
1
1
/r
.coI
.Ol
FREQUENCY
I
1
0.1
1.0
FACTOR,
/I I
N. R.
AHUND~ON
x
I
d ,001
x
I
I
I
.Ol
0.1
I.0
FREQUENCY
FKTOR.
x
Fig. 7. Effect of the fluctuationfrequencyof the transfercoefficientson the nonisothermalmodel.
ted in Fig. 1 with one exception. Note from the steady state operating curves in Fig. 1 that three deterministic steady states co-exist for R, = 24 and T, = 2.7. The outer two steady states are asymptotically stable and the middle one is unstable. However, when this deterministic system is subjected to fluctuating transfer coefficients with p = 0.0125 and SK = 1.25, there exists only one stable stationary solution of the stochastic model and it is in the neighborhood of the less reactive deterministic steady state. Regardless of initial condition, the stationary stochastic solution will have a mean dense phase concentration near 0.99. The corresponding isothermal model cannot predict this result. This type of pathology is uncommon for the parameters used in Fig. 1 because the dense phase temperature is relatively insensitive to fluctuations in the transfer coefficients. In the following discussion, the parameter G is increased from 0.004 to 1.0 which would occur if the two phases had the same densities and heat capacities. Although this is no longer a realistic model for a gas fluidized bed, it provides some insight into the type of pathological behavior which can be observed for stochastic systems. The steady state operating curve for this case is given in Fig. 8. Multiple steady states exist for the inlet temperature T,= 2.8. Stationary solutions using the initial conditions of the two stable states of the deterministic system were studied for various variance and frequency levels of the transfer coefficients. At low input variance levels, stationary states exist in the vicinity of both deterministic steady states. As shown in Fig. 9, as the variance is increased, a point is reached where one of the deterministic steady states does not have a stable stationary state associated with it at the intermediat frequency /3 = 0.125. Hence, at this fluctuation frequency, the reaction goes essentially to extinction regardless of the initial condition. A more reactive stationary solution can occur at both higher and lower fluctuation frequencies. With even higher variance in the transfer coefficients, two stationary solutions exist at the
1.0 r
.o
,:
’ 2.6 FEED
I 2.6 TEMPERATURE,
U 3.2
3.0 To
rig. 8. Ste_adystateoperatingcurvefor the nonisothermalmodel. K = 1.0, H = 3.0, qB= 0.83, V, = 0.6, R, = 24, R2= 75, G = 1.0. high frequencies, fi = 0.125 and 12.5. At B = 0.0125, instead of again having two stationary solutions, there is now no stationary solution in the neighborhood of the less reactive steady state. The result is that a change in frequency from p = 0.125 to 0.0125 will force the mean value of the dense phase concentration from above 0.9 to less than 0.2. Some evidence for the cause of this behavior can be obtained from Figs. 10 and 11. Mean values and standard deviations of the more reactive stationary solution are plotted in Fig. 10 against p for SK =0.23. This represents the highest variance in the transport coefficients for which the more reactive stationary solutionexistsatallfrequencies tested. At/I = 12.5and 1.25,the standard deviation of xe and Te are low. At fi = 0.125, SXB and ST,aremuchhigherwhile SxEand STBarenotfarfrom these maximum values. Moving to the lower frequency, p = 0.0125, sXBand ST, are slightly higher but Sr, drops by 50% of its value at /3 = 0.0125and ST= drops sligMy..If the shape of the standard deviation curves is simikfor higher values of Sx3 it appears that all of the
667
&deUiag of fluidized bed reactors-VI(b) MODEL
FMfAHETEFW
V6 =.6
a6E.63 R, ‘24 Y
B
0
-*
= 2 D
0
l
l
DETERMINISTIC
STEADY
STATES
I
xg*.e5
X6=.16
1,=3.e
T,-3.3E
m
x6-.97
xEs.94
T,3*2.83
1~~2.64
0
0
0.2-
@ %
R=3.0 T, 12.8
B-I.0
l 0.3 -
p
if.l.0
IQ=75
iw
0.1 STATE
I
UNSTABLE
STATE
II
UNSTABLE
3
FREOUENCY FACTOR, X BOTH STATES STABLE
/3 = 12.5 x Fig.
m
9. Stochastic stability of the nonisothermal model.
I.0 -
w
3
2
O.0-D
5
I” 0.6
x E
’
& -
_TE * xg
~
a
+---r+
0.4 -
E Y
0.2 -
o_
,XE
*
5
OOb
d 001
01
0.1
2.6
.oo XKN
1.0
FREPUENCY FACTOR, X
.Ol
0.1
1.0
FACTOR, X
FREOUENCY
Fig. 10. Epect of fluctuation frequency of the mass transfer coefficient on state I. I.0
X6
r
XE 0.0
3.4
R,
-24
g
T,
= 2.8
E f
ST = 1.0 SK -0.29
0.6
_!__I
3.2
2
3.0
I j?
2.8
r y
/3
B
0.4
TL J
0.2-
0
l6
Q
,001
.Ol
FREQUENCY
0.1
I.0
= 12.5X
/
FACTOR. X
FREQUENCY
FACTOR,
X
Fig. 11. Effect of fluctuation frequency of the mass transfercoefficienton state II. state variables may have high enough variance at & = 0.125 to force this state to be unstable while the low values of Sz, and ST, maintain stability at the lower frequency /3 = 0.0125. A similar curve for the less reactive state is shown in Pig. 11. Here standard deviations of the bubble phase properties are rather insensitive to /3 while those of the dense phase properties are greatest at the lowest frequency. This indicates that the
less reactive state should tend to become unstable at the lowest frequency. B = 0.0125. The phenomenon observed here of a system with two stable steady states id the deterministic case having only one stable stationary solution when subjected to random disturbances can be defined as a loss of stochastic stability. This concept of stochastic stability will be further discussed in a later paper [4]. It should be noted that the
J. R. LIOONand N. R. AMUNMON
668
regions of stability depicted in Fig. 9 are necessarily approximate because they are based on observation of the stationary reactor state over a finite time interval.
x3= &xe f 4
q3
-4 Vs
(33)
+ 4x1 V3
RIT,)=~(,~~(x,-x=)t(4a/V~)(l-x,)) THE
sTRucrmtEn
MODRL
NO-
Deterministic model: The structured model is formed by replacing the single bubble phase cell of the simple model by three smaller cells in series. Mass and heat are exchanged with the dense phase cells as shown in the schematic of Fig. 12. The resulting model is cast in the dimensionless form dx,ldt=K,(x~-xl)+q,(l-x,)V,
(22)
dT,/dt = H,(T, - T,)+ q~(Tto- T,) VI
(23)
dxz/dr = KAxE - xz) + qz(x, - x2)/ Vz
WI
dT,/dt = HATE - Tz)+ qz(K - Tz) V,
(25)
dx,/dt = &(xE -xa) t qdxz-xd/Vs
(26)
dZ’,/dT=H,(T,-
(34) TB = &I(& -In R(T,)) for the inlet temperature TBo = 7” -
T
1=HtT,-eq, TEC& HI t q,lV,
T =H,T,tqzT,lV~ 2 I-f*t 421v2 T _H,T,tqsT,IV~ ‘H3tqJV3
dT,Jd~=$,H.G+(T,-T.)++G(TB.-T,) E P + Gx&( TB)
(36)
The bubble phase temperatures are given by
(28)
(37) (38)
*
(39)
(29) (30)
R(Te)=exp(RI-R&W
~BVF.RIT,A crltala2+alaans+4e
where
(27)
T4tqOz-TJV3
(35)
where the dimensionless variables are similar to those of the simple model. For fixed parameters Z&,Hi, Vi, G, RI and R2, the steady state values of xl, Te and R(TE) are related to xE by (31) (32)
Steady state values of xe are shown in Fig. 13 as a function of inlet temperature for the model parameters given in Table 2. Multiple steady states can exist for R, ~21. The stochastic mode! The stochastic structuredmodel is
formed by replacing the deterministic mass transfer coefficients between each bubble phase cell and the dense phase cell with statistically independent gamma distributed processes generated by the Stratonovich stochastic differential equations dK,=(crt l-f8Di-/?K,)dt+(2SK,)‘“dWi(r).
(40)
The heat transfer coefficients are assumed to be propor-
tional to the mass transfer coefficients associated with each bubble phase cell. The mass transfer coefficients K,, K, and K3 were simultaneously generated on the analog computer with independent band-limited white Table 2. Parametervalues for the structurednonisothermal fluid&d bed model K1 = 5.0 “1 - 10.0
I
qE+ql
Fig. 12. Schematicof the struchued noniaothemmlmodel.
K* = 2.5
“*
K3 = 2.0
H3 * 4.0
q1 = 0.9
VI = 0.15
92
= 0.9
V2 = 0.15
43
= 0.9
V3 = 0.15
qE = 1.0
VE = 0.55
G
R2 = 75
-
.004
= 5.0
Modelling of fluidized bed reactors-VI(b)
FEED Fig.
13.
TEMPERATURE,
To
Steady state operating curves of the structured model.
noise inputs, R, from three separate electronic noise other parameters as specified in Table 2. The statistically generators. Three equations similar to eqn (20) result. independent mass transfer coefficients simultaneously The band-limited white noise sources were a General fluctuate at identical frequencies with SK, = 0.8. As Radio model 1390-Band two General Radio Model 1382 shown in Fig. 14, the mean concentrations of each cell of noise generators. The gamma distributed mass transfer the nonisothermal model decrease slightly with decreascoefficients were recorded on a Hewlett Packard Model ing frequency leading to an increase in conversion as 3960 four channel FM tape recorder for reuse without compared to the deterministic model. The structured the need of being regenerated for each program execu- isothermal model predicts concentrations with slightly tion. The dimensionless equations of the deterministic higher mean value than those of the nonisothermal model were transformed to an equivalent form to im- model. Although the mean outlet concentration of the prove analog scaling[31 by defining TV= T, - TIO, r2 = isothermal model is only one percent above that of the Tz - T,,,, ~3= Ts - T,o, r= = TE - Tao. nonisothermal model, it predicts a very slight decrease in conversion as compared to the deterministic model. Discussion of results Hence, as was found for the simple stochastic fluidized The stochastic structured model with gamma disbed model, there is a qualitative disagreement between tributed mass and heat transfer coefficients were studied isothermal and nonisothermal models even though the for a variety of conditions. The results were compared output concentrations do not deviate by a large amount. with results of a structured isothermal model formed by For any conditions of the structured model, holding the temperature integration at their deterministic parameters for the simple model can be chosen to provalues. Results of a typical case are shown in Fig. 14 vide the same deterministic reactor output as the strucwhere concentration means and standard deviations are tured model. An equivalent single bubble cell model is plotted against the frequency factor, X, for T, = 3.78 and formed by equating the bubble phase concentration and
IO
i
c
,975
-
.%O
-
,925
-
^
.900
XI x2
0
_
0 0
‘=
-
%
-
5x
0125 .OlOC
% z
850,825
8-12
z % .ms
H 8
9
G XE
8?5
z F g
t
F
l3.7
SK, ‘0.S
c
z P t
To
.OO! .002!
,
0.C
%V FREQUENCY
FACTOR,
X
,001
.Ol
FREOUENCY
1.0
01 FACTOR,
X
Fig. 14. Effecl of fluctuation frequency on the structured fluidized bed model for the parameters of Table 2 with T, = 3.70. CE3 Vol. 16. No.&X
I. R. LIGONand N. R. AMLINEMN
670
temperature with those of the third bubble phase cell of the structured model. Dense phase concentrations and temperatures of the two models are also assumed equivalent. For any set of mass transfer coefficients K,, K2 and K3 of the structured model, it can be shown that by choosing the mass transfer coefficient K given by
and proper choice of (Ib and V,, the two models will predict the same deterministic steady state effluent concentration. Similarly, the equivalent heat transfer coefficient is given by
H’P
B
Hl+qllV~.H,tq,lVz q,lV, q21v*
H,tq3V3 .q31v3-
1
1 *
For the parameters of the structured model given in Table 2, the equivalent simple nonisothermal model has the parameters K = 4.926, qB =0.9, H = 16.2% and V, =0.45. Calculations with the two stochastic models were made with the variance of K for the simple model equal to those of the structured model. Observed reactor outlet concentration standard deviations for the simple model were always about l/2 to 213 those of the structured model. Other predictions of the equivalent simple model, such as trends in states standard deviations as a function of frequency, were qualitatively similar for the two models. It must be noted that equal transfer coefficient standard deviations for each bubble phase cell may not provide the most realistic description of a fluidized bed. However, in the absence of a more realistic relationship, these values were chosen for convenience of comparison with the single bubble cell model. CONCLUSIONS
The degree of the deviation of the stationary random solution from the deterministic solution depends on the sensitivity of the steady state. When compared to the deterministic model, the stochastic nonisothermal fluidized bed .models predict either an increase or a decrease in conversion depending on the operating conditions. In a later paper [4], a stochastic continuous stirred tank reactor model is used to show that the direction of deviatiation from the deterministic system can be predicted (I priori for random inputs of sufficiently low frequency. With the exception of stochastically unstable states, the deviation of dense phase temperature from the deterministic value is always quite small. The dense phase temperature is not very sensitive to fluctuations in the transfer coefficients because of its hi total heat capacity. Hence, the stochastic isothermal models predict mean bubble and dense phase concentrations near those of the nonisothermal models. However, the isothermal models always predicted reactant conversion decreasing with increasing variance in the transfer coefficients. This is in disagreement with the nonisothermal models which predict either increasing or
decreasing conversion depending on the operating conditions. Over the frequency range studies, both the isothermal and nonisothermal models predict increased deviation from the deterministic system with decreasing fluctuation frequency. For certain operating conditions, concentration standard deviations can have maxima at intermediate frequencies. The relationship between stochastic reactor model performance and fluctuation frequency will be further discussed in a later paper[4]. It has been shown that the nonisothermal model with multiple stable deterministic steady states may not have multiple stationary states when subjected to random perturbations. The stability of a state to random perturbations is called stochastic stability. The stability of a stochastic system has been shown to be very sensitive to the random input fluctuation frequency. The relationship between frequency and stochastic stability will be further discussed in a later paper [4]. There it is shown that the maximum input variance that a stochastically stable system can tolerate can be estimated a priori for low frequency inputs. Since all chemical processes are subjected to random fluctuations, this concept of stochastic stability should prove to be important iu chemical entineering analysis. The structured fluidized %ed model provides a more realistic description of a fluid&d bed than does the simple nonisothermal model with a single bubble phase cell. The structured model predicts increased variance in the reactor output as compared to an equivalent single bubble cell model. This suggests that the inclusion of more than three bubble cells to more closely approximate the usual bubble phase plug flow assumption may predict still higher variance in the reactor output. Performance of ihe structured and simple models was otherwise qualitatively similar. NOTATION
dimensionless pre-exponential term in scaled reaction rate expression heat capacity variance parameter of the Wiener process activation energy volumentric flow rate dimensionless relative heat capacity parameter dimensionless heat transfer coefficient heat transfer coefficient dimensionless mass transfer coefficient mass transfer coefficient pre-exponential term in reaction rate mean value of the gamma distributed process electronically generated band-limited white noise process proportionality factor between H and K dimensionless llow rate term gas constant dimensionless reaction rate dimensionless reaction rate constant dimensionless reaction rate constant stationary autocorrelation function firstarder reaction rate
Modelling of fluidized bed reactors-VI(b) standard deviation dimensionless temperature dimensionless time time dimensionless volume volume variance of the gamma distributed process Wiener process dimensionless concentration gamma distributed dimensionless mass transfer coefficient
671
7 temperature scaling factor x frequency factor, x = 8112.5 SUbSCriptS
B bubble phase E dense phase i ith bubble phase cell 0 entering component
REFERENCES U1
Ligon __ ,*_ J. R. and Amundson N. R., Own.
Engng
Sci.
1981
.W IX. Greek letters
(I
constant in gamma distributed stochastic process B time constant in gamma process AH heat of reaction 6 constant in gamma distributed stochasic process 9 temperature
f21 Kunii D. and Levenspiel 0..
Fhidizalion
Engineering.
Wiley,
New York 1969.
[31 Ligon J. R., Ph.D. Thesis, University of Minnesota 1978. [4] L$;3 J. R. and Amundson N. R., Chem. Engng Sci. 1981 [Sl Kramdeck F. J., Katz S. and Skinner R., Chem. I!%!3 24 1497.
Engng Sci.
MODELLING OF FLUIDIZED BED REACTORS-VI(b) THE NONISOTHERMAL
BED WITH
STOCHASTIC
BUBBLES
JOHN ROBERT LIGON Department of Chemical Engineering and Materials Science, University of Minnesota, 151 Amundson Hall, 421 Washington Avenue SE., Minneapolis, MN 55455, U.S.A. and
NEAL R. AMUNDSON* Departmentof ChemicalEngineering,University of Houston, Houston, TX 77004, U.S.A. (Reeefued 19 May
1980;
accepted
IO July 1980)
Abstract-Two stochastic nonisothermal fluidiied bed reactor models are developed to investigate the significance of the fluctuating nature of fluidized beds on reactor performance. Fluctuating bubble size distributions within the bed are simulated by stochastic mass and heat transfer coefficients. Results of hybrid computer simulations indicate that randomness can enhance or inhibit reactor performance depending on the operating parameters of the nonisothermal model. Bubble and dense phase concentrationstatistics are fairly similar to those of corresponding isothermal models because dense phase temperatures are relatively insensitiveto transfercoefficientfluctuations due to the high dense phase heat capacity. However, the corresponding stochastic isothermal models predict decreases in conversion with increasingvariance in the transfer coefficients for all operating conditions. Results indicate that a deterministic system with two stable steady states may have fewer stable random stationary solutions.The existence of the stationarystates is dependenton fluctuation frequency and variance of the transfer
coefficients.
order exothermic reaction occurs. The response of the stochastic models with y distributed transfer coefficients is analyzed for various operating conditions.
INTRODUCTION
In this paper two nonisothermal stochastic fluidized bed reactor models are developed to study the effect of the fluctuating nature of fluidized beds on reactor performance. They are an extension of a previously discussed isothermal modeltl] to include temperature effects. The first model, referred to as the single nonisothermal
model, consists of two well-mixed cells, one representing the bubble phase and the other the dense phase. A single first-order exothermic reaction occurs in the dense phase. Mass and heat transfer take place between cells with R and g the overall mass and heat transfer coefficients. As in the isothermal model, the transfer coefficients are related to the distribution of bubbles in the bed. Since this relationship is similar for both coefficients, H will be assumed directly proportional to k. Ahhongh more accurate representations exist [see, e.g. Kunii and LevenspielI21. this assumption was judged satisfactory for the purposes of this investigation. Since the number and size of bubbles in the bed fluctuates with time, the transfer coefficients are best modeled as stochastic processes. In the second model, referred to as the structured model, the single well-mixed bubble phase cell of the first model is replaced by a series of three smaller cells with individual mass and heat transfer coefficients selected to reflect the increasing average bubble size with height in the bed. This provides a more realistic description of bubble phase behavior. The dense phase is again represented by a single well-mixed cell in which a fist *Author to whom correspondence
NONlsoTBEpMALFLUIDBED MODEL TEIEStMPLE deterministic model: Mass and energy balances over the bubble and dense phase cells with a first-order reaction in the dense phase produce the equations The
(1)
_&&4caw.w
(4)
P.&B R(&) = k, eewnsH.
(5)
Variatiens in gas density with temperature are neglected. The following dimensionless variables are defined
should be addressed. 661
xn = CelC,
qB = F$Fr.
xg = CnlC,
qe= FI#T
662
and N. R. AMUNDSON
J. R. LGON
vB
TE=RJ(RI-InR(T,))
= ri,lA
TB=HT~+(qeIWT~o HtqalVs .
(13)
V,=VJA
R,=ln(hk,,/F~)
For the inlet temperature
E&PO R2= RC,,(AH) t=F#T
G=-
POC,, P&O
where FT = Fst FE,3~= QB+ 3, and CO= CaO= CEa. The above equations can be cast in dimensionless form dx, _ dr-K(*~-*+(l-Xs) dx,=KVBK (XB-x,)+$1 df
(6) -.%I-R(T,)xe
~=H(T,-T.)++kJ.-T.)
%=H$
(7) (8)
Steady state solutions of xE are given in Fig. 1 as the intersections of a vertical line representing fixed feed conditions and the curve for constant RI.For the parameters chosen, multiple steady states can occur for R, 221. The stochastic model: The number and size of bubbles present in a bubbling fluidized bed fluctuate with time. Since the overall heat and mass transfer coefficients are dependent on bubble size, fluctuations in bubble size cause fluctuations in these coefficients. The simple stochastic nonisothermal mbdel is formed by replacing the dhnensikless mass and heat transfer coefficients of the deterministic model by the y distributed stochastic process, which is generated by the Stratonovich stochastic differential equation
G(TB-Te)+gG(TB,-Tg) tGR(TdxE (91
a+1-;6D-/3K)df+(26K)“2dW(f). (15)
R(TB)=~w(RI-Rz/TE).
W) The same hybrid mmputer techniques used for the isothermal gamma model[l,3] were used for the nonisothermal model. The University of Houston Hybrid computing facility and a General Radio model 1382 noise generator were used. The dimensionless model equations (11) are cast in an equivalent form to improve analog scaling by defining ~~ = TB - T&, and rE = TE - T,+ Defining K as.a gamma process, assuming the proportionality of H (12) and K,and letting T,, = TE,,, the stochastic model is
For specified values of K,H,qB. G, RIand RZ the steady state values of xg, TE and Te are related to XEby xB=Kre+&5J
K+qdVB
FEED TEMPERATUi?E. Fii. I.
To
Steady state operatingcurve for the nonisothermalmodel.
Medellii of fluidized bed reactors-VI(b) programmed on the analog computer as
663
Table 1. Parametersfor the simplenonisolhennalfluidbed model t
~~=K(TE-XB)+98(1-YB)/Va
(16)
;i,=-K~(~~-x,)+~(l-x~)-x~B exp 1 $%I 1
=4
K
=8
RI = 18
R2 = 75
VB = .45
qe = .9
G
* .004
(17) is = fiK(TE - %I”-9BTB/VB
(18) transfer coefficients. The dense phase temperature exhibited only very small amplitude fluctuations because of
~~=-~GK~(~~-a)-G*=~~V=
its + xEGB exp
(1%
~=~+l-~~sO-~K+(26K)“21$
(20)
where B = exp (RI - R,/T,) and D is the variance parameter of the Wiener process associated with the electronically generated band-limited process, k. For each case studied, the results of the stochastic nonisothermal model were compared with those of an equivalent isothermal model. The isothermal model is formed by holding the temperature integrators on the analog computer at their deterministic values. The resulting reaction rate is constant. The isothermal model is formed using the exact circuitry of the nonisothermal model and is executed in the same hybrid computer run to reduce analog computer inaccuracies, thus simplifying comparison of the two models. Discussion
Response of the stochastic model is first analyzed for RI = 18 and T, = 3.75 with other parameters specified in Table 1. Mean concentrations and temperatures of each phase and their standard deviations are plotted in Fig. 2 versus standard deviation of K. The mean bubble phase concentration is seen to increase very slightly with SK while the mean dense phase concentration decreases slightly. Mean temperatures for both phases do not vary signitkantly with increasing variance in the transfer coefficients. As expected, the standard deviations of the reactor state increase with increasing variance in the
STANMRD Fii.
DEVIATION
OF
K
high
total
heat
capacity.
Performance
of
the
equivalent isothermal stochastic fluidized bed model was not significantly different from that of the nonisothermal model depicted in Fig. 2. Performance of the nonisothermal model is presented in Fig. 3 for TO= 3.8 and other parameters from Table 1. The mean bubble phase concentration is seen to increase very slightly with increasing standard deviation of the transfer coefficients. The mean dense phase concentration is more sensitive to SK and it decreases as SK increases. The result is that reactant conversion decreases with increasing input variance. The corresponding isothermal model predicts mean concentrations which are slightly higher than those of the nonisothermal model at the higher transfer coefficient variance. Although deviation of the isothermal model from the nonisothermal is not great, the isothermal model predicts a decrease in conversion with increasing variance in the transfer coefficients, which is in qualitative disagreement with the nonisothermal model. Comparison of Figs. 2 and 3, indicates that deviation of each reactor state from its deterministic value is greater for T’ = 3.8 than for TO= 3.75. The observed state standard deviations are also higher for the higher inlet temperature. The standard deviations predicted by the isothermal model do not differ significantly from those of the nonisothermal model. Performance of the nonisothermal model is described in Fig. 4 for T, = 3.9 and other parameters as in the previous cases. As before, the mean dense phase concentration increases and mean bubble phase concentration decreases with increasing variance in the transfer
STANDARD
DEVIATION
OF
K
2. Effect of the standard deviation of the transfer coefficients on the nonisothermal model.
J. R. LEON
and N. R. AMUN~XWN
To= 3.8 p
1_[
'0.0125
;;&
,725 k
.soJ 0.0
.25
STANDARD
,000 0.0 25
. DEVIATION
OF
K
STANDARD
.50 .75
I.00 1.25
DEVIATION
OF
K
Fig. 3. Effect of the standard deviation of the transfer coefficients on the nonisothermal model.
Isothermal
X9
-j 4.525
.025
STANDARD Fii.
DEVIATION
OF
K
-
STANDARD
DEVIATION
OF
K
4. Effect of the standarddeviationof the transfercoefficientson the nonisothermalmodel.
coefficients. However, the increase in bubble phase concentration relative to the decrease in dense phase concentration is sufficiently greater for this case that reactant conversion decreases with increasing SK. The COTresponding isothermal model predicts slightly higher reactant concentrations in both phases leading to further decreased conversion. The state standard deviations are considerably higher for TO= 3.9 than for the previous cases although ST= is again low. Standard deviations predicted by the isothermal model are equivalent to those of the nonisothermal model. The inlet temperature is further increased to 3.95 for the case depicted in Fig. 5. Mean bubble phase concentration again increases and mean dense phase concentration decreases with increasing variance in the transfer coefficients. Mean concentrations of the equivalent isothermal model are very close to those of the nonisothermal means at the highest input variance levels. The mean dense phase temperature was seen to decrease t'lbe90% cut-off frequency is equal to 6.3B[3]
slightly with increasing variance which could have led to the result of reactant conversion being slightly better for the isothermal model. Both models predict conversion decreasing with increasing variance in the transfer coefficients. Predicted standard deviations were again equivalent for the two models. The effect of the fluctuation frequency of the gamma distributed transfer coefficients on reactor performance was also studied. Figure 6 depicts the mean concentrations and temperatures and their standard deviations for TO= 3.8 vs the frequency factor, x, where x = p/12.5 and p is the constant in the autocorrelation function of KVI RK(7) = u eC@ + m,‘.
(211
As /3 is increased, the transfer coefficients fluctuate more rapidly.? Deviation from the deterministic state is seen to increase with decreasing fluctuation frequency. The nonisothermal and isothermal models both predict increasing bubble phase concentration and decreasing
Modellingof fluidizedbed reactors-VI(b)
-
4600
-
4.575
-
4.550
Y
225
-
4525
iii z
k! r -203
-
4500
i? au
-
4.475
p
665
5
i5
sa g
:: .I25
x--
QO STANDARD Fii.
DEVIATION
OF
K
I
I
25
.50
I
STANDARD
.75
I
I
1.0
1.25
DEVIATION
OF
K
5. Effect of the standarddeviationof the transfercoeliicientson the nonisothermalmodel.
1.
-
I)30
-
% 5 ,025
-
= 3.6
SK = 1.25 (9
= 12.5 x
3.900
-i ,725 t
’ .OOl
I
I
.Ol
0.1
FRWUENCY
FACTOR,
l/J
1.0 x
FREOUENCY
FACTOR,
x
Fig. 6. Effect of the fluctuationfrequencyof the transfercoefficientson the nonisothermalmodel.
dense phase concentration with decreasing frequency. However, as previously found for this inlet temperature, the isothermal model predicts an increase when corn-
pared with the deterministic state. The standard deviation of the dense phase concentration increases with decreasing frequency while that of the bubble phase has a maximum at the intermediate frequency x = 0.1. A similar frequency plot is given in Fig. 7 for T, = 3.95. The mean bubble phase concentrations increase and dense phase concentrations decrease with decreasing fluctuation frequency for both the nonisothermal and isothermal models. The isothermal model predicts slightly better conversion than the nonisothermal model while both models predict a decrease in conversion with decreasing frequency. The bubble phase concentration standard deviation increases with decreasing frequency while that of the dense phase is maximum at the intermediate frequency x = 0.1. Standard deviations of the isothermal model do not vary significantly from those of the nonisothermal model.
Analysis of Figs. 2-7 reveals that the dense phase concentration is more sensitive than the bubble phase concentration to transfer coefficient fluctuations for low inlet temperatures and is less sensitive at higher inlet temperatures. The concentration nearer either limit, 0.0 or 1.0, will have more damped fluctuations than the other concentration. In each case studied, the isothermal model’predicts mean values and standard deviations of the reactor states very near those of the nonisothermal model. This is because the dense phase temperature is not very sensitive to fluctuations in the heat transfer coefficient due to its high total heat capacity. Since the bubble phase is assumed solid-free, it has a much lower total heat capacity and shows significant fluctuation. However, the reaction rate of the catalytic reaction is controlled by the dense phase temperature, and, if it remains relatively constant, the isothermal model provides a good approximation of the nonisothermal model. Results of the stochastic nonisothermal and isothermal models are similar for other operating conditions depic-
I. R. LIIXN and
,/
1
1
/r
.coI
.Ol
FREQUENCY
I
1
0.1
1.0
FACTOR,
/I I
N. R.
AHUND~ON
x
I
d ,001
x
I
I
I
.Ol
0.1
I.0
FREQUENCY
FKTOR.
x
Fig. 7. Effect of the fluctuationfrequencyof the transfercoefficientson the nonisothermalmodel.
ted in Fig. 1 with one exception. Note from the steady state operating curves in Fig. 1 that three deterministic steady states co-exist for R, = 24 and T, = 2.7. The outer two steady states are asymptotically stable and the middle one is unstable. However, when this deterministic system is subjected to fluctuating transfer coefficients with p = 0.0125 and SK = 1.25, there exists only one stable stationary solution of the stochastic model and it is in the neighborhood of the less reactive deterministic steady state. Regardless of initial condition, the stationary stochastic solution will have a mean dense phase concentration near 0.99. The corresponding isothermal model cannot predict this result. This type of pathology is uncommon for the parameters used in Fig. 1 because the dense phase temperature is relatively insensitive to fluctuations in the transfer coefficients. In the following discussion, the parameter G is increased from 0.004 to 1.0 which would occur if the two phases had the same densities and heat capacities. Although this is no longer a realistic model for a gas fluidized bed, it provides some insight into the type of pathological behavior which can be observed for stochastic systems. The steady state operating curve for this case is given in Fig. 8. Multiple steady states exist for the inlet temperature T,= 2.8. Stationary solutions using the initial conditions of the two stable states of the deterministic system were studied for various variance and frequency levels of the transfer coefficients. At low input variance levels, stationary states exist in the vicinity of both deterministic steady states. As shown in Fig. 9, as the variance is increased, a point is reached where one of the deterministic steady states does not have a stable stationary state associated with it at the intermediat frequency /3 = 0.125. Hence, at this fluctuation frequency, the reaction goes essentially to extinction regardless of the initial condition. A more reactive stationary solution can occur at both higher and lower fluctuation frequencies. With even higher variance in the transfer coefficients, two stationary solutions exist at the
1.0 r
.o
,:
’ 2.6 FEED
I 2.6 TEMPERATURE,
U 3.2
3.0 To
rig. 8. Ste_adystateoperatingcurvefor the nonisothermalmodel. K = 1.0, H = 3.0, qB= 0.83, V, = 0.6, R, = 24, R2= 75, G = 1.0. high frequencies, fi = 0.125 and 12.5. At B = 0.0125, instead of again having two stationary solutions, there is now no stationary solution in the neighborhood of the less reactive steady state. The result is that a change in frequency from p = 0.125 to 0.0125 will force the mean value of the dense phase concentration from above 0.9 to less than 0.2. Some evidence for the cause of this behavior can be obtained from Figs. 10 and 11. Mean values and standard deviations of the more reactive stationary solution are plotted in Fig. 10 against p for SK =0.23. This represents the highest variance in the transport coefficients for which the more reactive stationary solutionexistsatallfrequencies tested. At/I = 12.5and 1.25,the standard deviation of xe and Te are low. At fi = 0.125, SXB and ST,aremuchhigherwhile SxEand STBarenotfarfrom these maximum values. Moving to the lower frequency, p = 0.0125, sXBand ST, are slightly higher but Sr, drops by 50% of its value at /3 = 0.0125and ST= drops sligMy..If the shape of the standard deviation curves is simikfor higher values of Sx3 it appears that all of the
667
&deUiag of fluidized bed reactors-VI(b) MODEL
FMfAHETEFW
V6 =.6
a6E.63 R, ‘24 Y
B
0
-*
= 2 D
0
l
l
DETERMINISTIC
STEADY
STATES
I
xg*.e5
X6=.16
1,=3.e
T,-3.3E
m
x6-.97
xEs.94
T,3*2.83
1~~2.64
0
0
0.2-
@ %
R=3.0 T, 12.8
B-I.0
l 0.3 -
p
if.l.0
IQ=75
iw
0.1 STATE
I
UNSTABLE
STATE
II
UNSTABLE
3
FREOUENCY FACTOR, X BOTH STATES STABLE
/3 = 12.5 x Fig.
m
9. Stochastic stability of the nonisothermal model.
I.0 -
w
3
2
O.0-D
5
I” 0.6
x E
’
& -
_TE * xg
~
a
+---r+
0.4 -
E Y
0.2 -
o_
,XE
*
5
OOb
d 001
01
0.1
2.6
.oo XKN
1.0
FREPUENCY FACTOR, X
.Ol
0.1
1.0
FACTOR, X
FREOUENCY
Fig. 10. Epect of fluctuation frequency of the mass transfer coefficient on state I. I.0
X6
r
XE 0.0
3.4
R,
-24
g
T,
= 2.8
E f
ST = 1.0 SK -0.29
0.6
_!__I
3.2
2
3.0
I j?
2.8
r y
/3
B
0.4
TL J
0.2-
0
l6
Q
,001
.Ol
FREQUENCY
0.1
I.0
= 12.5X
/
FACTOR. X
FREQUENCY
FACTOR,
X
Fig. 11. Effect of fluctuation frequency of the mass transfercoefficienton state II. state variables may have high enough variance at & = 0.125 to force this state to be unstable while the low values of Sz, and ST, maintain stability at the lower frequency /3 = 0.0125. A similar curve for the less reactive state is shown in Pig. 11. Here standard deviations of the bubble phase properties are rather insensitive to /3 while those of the dense phase properties are greatest at the lowest frequency. This indicates that the
less reactive state should tend to become unstable at the lowest frequency. B = 0.0125. The phenomenon observed here of a system with two stable steady states id the deterministic case having only one stable stationary solution when subjected to random disturbances can be defined as a loss of stochastic stability. This concept of stochastic stability will be further discussed in a later paper [4]. It should be noted that the
J. R. LIOONand N. R. AMUNMON
668
regions of stability depicted in Fig. 9 are necessarily approximate because they are based on observation of the stationary reactor state over a finite time interval.
x3= &xe f 4
q3
-4 Vs
(33)
+ 4x1 V3
RIT,)=~(,~~(x,-x=)t(4a/V~)(l-x,)) THE
sTRucrmtEn
MODRL
NO-
Deterministic model: The structured model is formed by replacing the single bubble phase cell of the simple model by three smaller cells in series. Mass and heat are exchanged with the dense phase cells as shown in the schematic of Fig. 12. The resulting model is cast in the dimensionless form dx,ldt=K,(x~-xl)+q,(l-x,)V,
(22)
dT,/dt = H,(T, - T,)+ q~(Tto- T,) VI
(23)
dxz/dr = KAxE - xz) + qz(x, - x2)/ Vz
WI
dT,/dt = HATE - Tz)+ qz(K - Tz) V,
(25)
dx,/dt = &(xE -xa) t qdxz-xd/Vs
(26)
dZ’,/dT=H,(T,-
(34) TB = &I(& -In R(T,)) for the inlet temperature TBo = 7” -
T
1=HtT,-eq, TEC& HI t q,lV,
T =H,T,tqzT,lV~ 2 I-f*t 421v2 T _H,T,tqsT,IV~ ‘H3tqJV3
dT,Jd~=$,H.G+(T,-T.)++G(TB.-T,) E P + Gx&( TB)
(36)
The bubble phase temperatures are given by
(28)
(37) (38)
*
(39)
(29) (30)
R(Te)=exp(RI-R&W
~BVF.RIT,A crltala2+alaans+4e
where
(27)
T4tqOz-TJV3
(35)
where the dimensionless variables are similar to those of the simple model. For fixed parameters Z&,Hi, Vi, G, RI and R2, the steady state values of xl, Te and R(TE) are related to xE by (31) (32)
Steady state values of xe are shown in Fig. 13 as a function of inlet temperature for the model parameters given in Table 2. Multiple steady states can exist for R, ~21. The stochastic mode! The stochastic structuredmodel is
formed by replacing the deterministic mass transfer coefficients between each bubble phase cell and the dense phase cell with statistically independent gamma distributed processes generated by the Stratonovich stochastic differential equations dK,=(crt l-f8Di-/?K,)dt+(2SK,)‘“dWi(r).
(40)
The heat transfer coefficients are assumed to be propor-
tional to the mass transfer coefficients associated with each bubble phase cell. The mass transfer coefficients K,, K, and K3 were simultaneously generated on the analog computer with independent band-limited white Table 2. Parametervalues for the structurednonisothermal fluid&d bed model K1 = 5.0 “1 - 10.0
I
qE+ql
Fig. 12. Schematicof the struchued noniaothemmlmodel.
K* = 2.5
“*
K3 = 2.0
H3 * 4.0
q1 = 0.9
VI = 0.15
92
= 0.9
V2 = 0.15
43
= 0.9
V3 = 0.15
qE = 1.0
VE = 0.55
G
R2 = 75
-
.004
= 5.0
Modelling of fluidized bed reactors-VI(b)
FEED Fig.
13.
TEMPERATURE,
To
Steady state operating curves of the structured model.
noise inputs, R, from three separate electronic noise other parameters as specified in Table 2. The statistically generators. Three equations similar to eqn (20) result. independent mass transfer coefficients simultaneously The band-limited white noise sources were a General fluctuate at identical frequencies with SK, = 0.8. As Radio model 1390-Band two General Radio Model 1382 shown in Fig. 14, the mean concentrations of each cell of noise generators. The gamma distributed mass transfer the nonisothermal model decrease slightly with decreascoefficients were recorded on a Hewlett Packard Model ing frequency leading to an increase in conversion as 3960 four channel FM tape recorder for reuse without compared to the deterministic model. The structured the need of being regenerated for each program execu- isothermal model predicts concentrations with slightly tion. The dimensionless equations of the deterministic higher mean value than those of the nonisothermal model were transformed to an equivalent form to im- model. Although the mean outlet concentration of the prove analog scaling[31 by defining TV= T, - TIO, r2 = isothermal model is only one percent above that of the Tz - T,,,, ~3= Ts - T,o, r= = TE - Tao. nonisothermal model, it predicts a very slight decrease in conversion as compared to the deterministic model. Discussion of results Hence, as was found for the simple stochastic fluidized The stochastic structured model with gamma disbed model, there is a qualitative disagreement between tributed mass and heat transfer coefficients were studied isothermal and nonisothermal models even though the for a variety of conditions. The results were compared output concentrations do not deviate by a large amount. with results of a structured isothermal model formed by For any conditions of the structured model, holding the temperature integration at their deterministic parameters for the simple model can be chosen to provalues. Results of a typical case are shown in Fig. 14 vide the same deterministic reactor output as the strucwhere concentration means and standard deviations are tured model. An equivalent single bubble cell model is plotted against the frequency factor, X, for T, = 3.78 and formed by equating the bubble phase concentration and
IO
i
c
,975
-
.%O
-
,925
-
^
.900
XI x2
0
_
0 0
‘=
-
%
-
5x
0125 .OlOC
% z
850,825
8-12
z % .ms
H 8
9
G XE
8?5
z F g
t
F
l3.7
SK, ‘0.S
c
z P t
To
.OO! .002!
,
0.C
%V FREQUENCY
FACTOR,
X
,001
.Ol
FREOUENCY
1.0
01 FACTOR,
X
Fig. 14. Effecl of fluctuation frequency on the structured fluidized bed model for the parameters of Table 2 with T, = 3.70. CE3 Vol. 16. No.&X
I. R. LIGONand N. R. AMLINEMN
670
temperature with those of the third bubble phase cell of the structured model. Dense phase concentrations and temperatures of the two models are also assumed equivalent. For any set of mass transfer coefficients K,, K2 and K3 of the structured model, it can be shown that by choosing the mass transfer coefficient K given by
and proper choice of (Ib and V,, the two models will predict the same deterministic steady state effluent concentration. Similarly, the equivalent heat transfer coefficient is given by
H’P
B
Hl+qllV~.H,tq,lVz q,lV, q21v*
H,tq3V3 .q31v3-
1
1 *
For the parameters of the structured model given in Table 2, the equivalent simple nonisothermal model has the parameters K = 4.926, qB =0.9, H = 16.2% and V, =0.45. Calculations with the two stochastic models were made with the variance of K for the simple model equal to those of the structured model. Observed reactor outlet concentration standard deviations for the simple model were always about l/2 to 213 those of the structured model. Other predictions of the equivalent simple model, such as trends in states standard deviations as a function of frequency, were qualitatively similar for the two models. It must be noted that equal transfer coefficient standard deviations for each bubble phase cell may not provide the most realistic description of a fluidized bed. However, in the absence of a more realistic relationship, these values were chosen for convenience of comparison with the single bubble cell model. CONCLUSIONS
The degree of the deviation of the stationary random solution from the deterministic solution depends on the sensitivity of the steady state. When compared to the deterministic model, the stochastic nonisothermal fluidized bed .models predict either an increase or a decrease in conversion depending on the operating conditions. In a later paper [4], a stochastic continuous stirred tank reactor model is used to show that the direction of deviatiation from the deterministic system can be predicted (I priori for random inputs of sufficiently low frequency. With the exception of stochastically unstable states, the deviation of dense phase temperature from the deterministic value is always quite small. The dense phase temperature is not very sensitive to fluctuations in the transfer coefficients because of its hi total heat capacity. Hence, the stochastic isothermal models predict mean bubble and dense phase concentrations near those of the nonisothermal models. However, the isothermal models always predicted reactant conversion decreasing with increasing variance in the transfer coefficients. This is in disagreement with the nonisothermal models which predict either increasing or
decreasing conversion depending on the operating conditions. Over the frequency range studies, both the isothermal and nonisothermal models predict increased deviation from the deterministic system with decreasing fluctuation frequency. For certain operating conditions, concentration standard deviations can have maxima at intermediate frequencies. The relationship between stochastic reactor model performance and fluctuation frequency will be further discussed in a later paper[4]. It has been shown that the nonisothermal model with multiple stable deterministic steady states may not have multiple stationary states when subjected to random perturbations. The stability of a state to random perturbations is called stochastic stability. The stability of a stochastic system has been shown to be very sensitive to the random input fluctuation frequency. The relationship between frequency and stochastic stability will be further discussed in a later paper [4]. There it is shown that the maximum input variance that a stochastically stable system can tolerate can be estimated a priori for low frequency inputs. Since all chemical processes are subjected to random fluctuations, this concept of stochastic stability should prove to be important iu chemical entineering analysis. The structured fluidized %ed model provides a more realistic description of a fluid&d bed than does the simple nonisothermal model with a single bubble phase cell. The structured model predicts increased variance in the reactor output as compared to an equivalent single bubble cell model. This suggests that the inclusion of more than three bubble cells to more closely approximate the usual bubble phase plug flow assumption may predict still higher variance in the reactor output. Performance of ihe structured and simple models was otherwise qualitatively similar. NOTATION
dimensionless pre-exponential term in scaled reaction rate expression heat capacity variance parameter of the Wiener process activation energy volumentric flow rate dimensionless relative heat capacity parameter dimensionless heat transfer coefficient heat transfer coefficient dimensionless mass transfer coefficient mass transfer coefficient pre-exponential term in reaction rate mean value of the gamma distributed process electronically generated band-limited white noise process proportionality factor between H and K dimensionless llow rate term gas constant dimensionless reaction rate dimensionless reaction rate constant dimensionless reaction rate constant stationary autocorrelation function firstarder reaction rate
Modelling of fluidized bed reactors-VI(b) standard deviation dimensionless temperature dimensionless time time dimensionless volume volume variance of the gamma distributed process Wiener process dimensionless concentration gamma distributed dimensionless mass transfer coefficient
671
7 temperature scaling factor x frequency factor, x = 8112.5 SUbSCriptS
B bubble phase E dense phase i ith bubble phase cell 0 entering component
REFERENCES U1
Ligon __ ,*_ J. R. and Amundson N. R., Own.
Engng
Sci.
1981
.W IX. Greek letters
(I
constant in gamma distributed stochastic process B time constant in gamma process AH heat of reaction 6 constant in gamma distributed stochasic process 9 temperature
f21 Kunii D. and Levenspiel 0..
Fhidizalion
Engineering.
Wiley,
New York 1969.
[31 Ligon J. R., Ph.D. Thesis, University of Minnesota 1978. [4] L$;3 J. R. and Amundson N. R., Chem. Engng Sci. 1981 [Sl Kramdeck F. J., Katz S. and Skinner R., Chem. I!%!3 24 1497.
Engng Sci.