2 Simultaneous Uncorrelated Changes of Process Variables in a Fixed-Bed Reactor 1
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A. BAIKER, M . BERGOUGNAN, and W. RICHARZ Swiss Federal Institute of Technology (ΕΤΗ), Department of Industrial and Engineering Chemistry, CH-8092 Zurich, Switzerland
A dynamic experimental method f o r the i n v e s t i g a t i o n of the behaviour of a nonisothermal-nonadiabatic f i x e d bed r e a c t o r is presented. The method is based on the a n a l y s i s of the a x i a l and r a d i a l temperature and c o n c e n t r a t i o n p r o f i l e s measured under the in fluence of f o r c e d u n c o r r e l a t e d s i n u s o i d a l changes of the process v a r i a b l e s . A two-dimensional r e a c t o r model is employed f o r the d e s c r i p t i o n of the r e a c t o r behaviour. The model parameters are estimated by statistical a n a l y s i s of the measured profiles. The e f f i c i e n c y of the dynamic method is shown f o r the i n v e s t i g a t i o n of a pilot p l a n t f i x e d bed r e a c t o r using the hydrogenation of toluene with a commercial n i c k e l c a t a l y s t as a t e s t r e a c t i o n . For proper c o n t r o l of i n d u s t r i a l f i x e d bed r e a c t o r s i t i s necessary t o know t h e i r dynamic behaviour. This behaviour may be i n v e s t i g a t e d by a s e r i e s of experiments where a s i n g l e process v a r i a b l e i s changed a t a time (1-6). In general such experiments allow f o r the development of a r e a c t o r model which d e s c r i b e s the dynamic r e a c t o r behaviour. However, very o f t e n a l a r g e number of experiments i s r e q u i r e d . In the present work a method i s described to e x t r a c t the i n formation necessary f o r modelling from only a few dynamic e x p e r i mental runs. The method i s based on the measurement of the changes of the temperature and c o n c e n t r a t i o n p r o f i l e s i n the r e a c t o r under the i n f l u e n c e of forced simultaneous s i n u s o i d a l v a r i a t i o n s of the process v a r i a b l e s . The c h a r a c t e r i s t i c features of the dynamic method are demonstrated using the behaviour of a non isothermal-nonadiabatic p i l o t p l a n t f i x e d bed r e a c t o r as an example. The t e s t r e a c t i o n a p p l i e d was the hydrogénation of toluene to methylcyclohexane on a commercial n i c k e l c a t a l y s t . 1
Current address: Produits Chimiques Ugine Kuhlmann, F-69310 Pierre-Bénite, France. 0097-6156/82/0196-0015$06.00/0 © 1982 American Chemical Society
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CHEMICAL REACTION
ENGINEERING
Experimental Equipment and Procedure. The f i x e d bed r e a c t o r p i l o t p l a n t i s shown s c h e m a t i c a l l y i n F i g u r e 1. The r e a c t o r was operated as a continuous f i x e d bed r e a c t o r , w i t h r e c y c l e of the hydrogen. The j a c k e t e d r e a c t o r tube of 2 m l e n g t h and 0.05 m inner diameter was equipped f o r the measurement of a x i a l and r a d i a l temperature and c o n c e n t r a t i o n p r o f i l e s . The r e a c t o r jacked temperature was c o n t r o l l e d by a c i r c u l a t i n g p r e s s u r i z e d water system. F i g u r e 2 i n d i c a t e s s c h e m a t i c a l l y the l o c a t i o n s of the a x i a l and r a d i a l measuring devices w i t h i n the f i x e d bed. The c o n c e n t r a t i o n and temperature measuring devices c o n s i s t e d of c a p i l l a r y tubes w i t h the NiCr/Ni thermocouple j u n c t i o n i n the center of the tube entrance. The c a p i l l a r i e s were provided w i t h magnetic v a l v e s f o r gas sampling p o s i t i o n e d at each c a p i l l a r y o u t l e t . An i n f r a - r e d gas analyzer (URAS) was u t i l i z e d f o r the automatic a n a l y s i s of the toluene c o n c e n t r a t i o n at the d i f f e r e n t l o c a t i o n s i n the r e a c t o r . In a d d i t i o n , the composition of the gas mixture was measured by gas chromatography at the r e a c t o r i n l e t and o u t l e t . A process computer (PDP 11/10) was used f o r the p l a n t c o n t r o l and the data p r o c e s s i n g . The f o l l o w i n g process v a r i a b l e s were changed simultaneously: the toluene c o n c e n t r a t i o n a t the r e a c t o r i n l e t , the r e a c t o r bath temperature and the t o t a l gas flow r a t e . R e s u l t s . The r e s u l t s of a t y p i c a l experiment w i t h uncorre l a t e d changes of the process v a r i a b l e s are presented i n the F i g u r e s 3.a)-c). F i g u r e 3.a) shows the u n c o r r e l a t e d s i n u s o i d a l changes of the process v a r i a b l e s . The r e s u l t i n g temperature and concentrations measured at d i f f e r e n t a x i a l and r a d i a l p o s i t i o n s are presented i n F i g u r e 3.b) and c ) , r e s p e c t i v e l y . Mathematical E v a l u a t i o n of the Dynamic Experiments Simulation of Reactor Behaviour A dynamic pseudo-homogeneous two-dimensional model i s employed f o r the d e s c r i p t i o n of the r e a c t o r behaviour. heat
balance:
8T at
r
9r
ΔΗ -
RG PK
(C, T)
(1)
PRODUCT
Figure 1. Fixed bed reactor pilot plant. Key: 1, fixed bed reactor; 2, metering pump; 3, compressor; 4, circulating pump; 5, flow sensor; 6, evaporator; 7, level control; 8, separator; 9, buffer volumes; 10, cooler; 11,flowcontrol valve; and 12, heat exchanger.
C L : C O O L I N G LIQUID ( - 2 0 ° C )
18
CHEMICAL REACTION ENGINEERING
5.2
19-23
m '/ζλ
Figure 2. Locations of axial and radial measuring devices in reactor. Key: @, catalyst bed; ®, inert packing; 1-23, thermocouples and sampling to gas an alyzer; and 30, 31, sampling to gas chromatograph.
2.
BAIKER ET
AL.
Process Variables in
19
Fixed-Bed Reactor
% 100
TEMPERATURES
PRESSURE
UJ ω
TOLUENE CONCENTRATION AT 1,2,3 AND 4
GAS FLOW RATE REACTOR BATH TEMPERATURE 200
Ί3θ
TIME (minutes)
Figure 3a. Time profiles of the uncor related changes of process variables. Ranges of variables: temperature of re actor bath, 0-250°C; toluene concentra tion at reactor inlet, 0-5 Vol%; total gas flow rate, 0-1200 mol/h; and total pres sure 0-2.5 bar.
100 TIME ( m i n u t e s )
Figure 3b. Resulting profiles of uncor related changes of process variables for axial measuring points.
Figure 3c. Resulting profiles of uncor related changes of process variables for radial measuring points.
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CHEMICAL REACTION ENGINEERING
mass balance:
ae
ac
Λ
= - u — + D 3t 3z e
a c
ι ac
.2 3r
r 3r
(1-ε)ρ
ι;
RG (C, T)
(2)
boundary c o n d i t i o n s : Τ (ζ = ο, r , t ) = Τ
χ
C (z - o, r , t ) = C
x
(r, t)
(3)
(r, t)
(4)
3T ac 3r (z, r = o, t ) = o; — ( z , r = o, t ) = ο 3T Έ
(ζ, r = R, t ) = -
^
( z , r = R, t ) = o
f
(5)
(6)
(T - T ) w
(7)
i n i t i a l conditions: T ( z , r , t = o) = T
2
(z, r)
(8)
C ( z , r , t = o) = C
2
(z, r)
(9)
D i s c r e t i z a t i o n o f the p a r t i a l d i f f e r e n t i a l equation system i n a x i a l (z) and r a d i a l ( r ) d i r e c t i o n by means of the orthogonal c o l l o c a t i o n method (7) leads to the f o l l o w i n g system of o r d i n a r y d i f f e r e n t i a l equations. dT. . =
NZ NR Α1.^[νΐΖ. .Τ^.] Α2.^[[ ..ν2Κ. Λ
+A3-RG(C.
dC. dt
NZ BI- Σ [VIZ
+
7
> 1 ς +
νΐ . Κ
) 1 ς
].Τ. ] Λ
T. .)
C
+B3*RG(C. ., T. .)
(10)
NR ] Β2·Σ [[y.-V2R +
,+VlR
]-C
]
(11)
2.
BAIKER ET A L .
21
Process Variables in Fixed-Bed Reactor
with: ζ
,r.2
χ -τ
A
1
;
. .
' ( 1
U
-
P
-
f P
£ ) p
k A2
φ
y -
F
BI - - ?
C
k pk
L
4
D
S ( 1
-
G ) p
B2 C
k pk
R 2
R
ΔΗ A3 =
B3 =
c . pk
4
e
2
(Ι-ε)ρ. ε
_ h D l - - * -= 2 k e boundary c o n d i t i o n s : T. . ( t ) = TAj. t) (12) C. . ( t ) - C-iy.pt) »J J >1 J NR 2*i*NZ Σ V1R. , T. = D1«(T. — Τ ) rïR,k i , k i,NR w 9
A
1
Α
TD
2*i«NZ
NR Σ V k=l
i
y C
-o
(13)
1
(14)
(15)
i n i t i a l conditions: Τ
(t-0) - T
(x.,y.)
(16)
C
J caJ.c employing Eqs. 10) and 11), r e s p e c t i v e l y . A
are c a l c u l a t e d by
By i n s e r t i o n o f Eqs. 12)-15) i n t o Eqs. 10) and 11) a system of ordinary d i f f e r e n t i a l equations i s obtained i n which only T. . and C. . a t the ten c o l l o c a t i o n p o i n t s (2*i*NZ; l^j^NR-1) remain as independent v a r i a b l e s . A nonlinear multiresponse r e g r e s s i o n program (9) was used to search f o r the parameters which y i e l d s t a t i s t i c a l l y the best accordance (maximum l i k e l i h o o d (10)) between the twenty i n t e r polated and c a l c u l a t e d responses. For the s i m u l a t i o n o f the r e a c t o r behaviour the system of ordinary d i f f e r e n t i a l equations was i n t e g r a t e d by means of a Runge-Kutta-Merson method w i t h v a r i a b l e step length, whereas the nonlinear a l g e b r a i c equations were solved by a Newton-Raphson iteration. K i n e t i c Rate Equation and Heat T r a n s f e r C o e f f i c i e n t s . K i n e t i c r a t e equations of d i f f e r e n t complexity w i t h 2 to 8 parameters were t e s t e d f o r the s i m u l a t i o n o f the r e a c t o r behaviour. F i n a l l y , the semi-empirical three parameter r a t e equation 20) was chosen f o r the s i m u l a t i o n because r a t e expressions of higher complexity y i e l d e d no b e t t e r s i m u l a t i o n o f the r e a c t o r behaviour and showed l a r g e r c o r r e l a t i o n s between the estimated parameters i n the given ranges of the process v a r i a b l e s .
3
Α ·10~ (Ρ /Ρ ) RG(T,P ) = i i — i — 1 + (Ρ /Ρ ) exp [A -10 (1/T- 1/A -10^)] J
(20)
CHEMICAL REACTION ENGINEERING
24
The heat t r a n s f e r c o e f f i c i e n t h
and k used i n the twow e dimensional model were estimated simultaneously w i t h the k i n e t i c parameters and were checked by an independent e s t i m a t i o n from experiments without r e a c t i o n i n which methylcyclohexane was s u b s t i t u t e d f o r toluene. Within the confidence l i m i t s both type of experiments l e d to s i m i l a r heat t r a n s f e r c o e f f i c i e n t s . Comparison of Measured and C a l c u l a t e d P r o f i l e s . In order to compare the measured time p r o f i l e s shown i n F i g u r e 3 w i t h the c a l c u l a t e d time p r o f i l e s , the former were a x i a l l y and r a d i a l l y i n t e r p o l a t e d to o b t a i n the corresponding p r o f i l e s at the c o l l o c a t i o n p o i n t s . F i g u r e s 5.a) and b) show the measured (I) and c a l c u l a t e d (II) time p r o f i l e s f o r the a x i a l and r a d i a l c o l l o cation points, respectively. The parameters estimated from the measured p r o f i l e s and used f o r the s i m u l a t i o n were A^ = 5.0 mol/kg s (46); A^ = 6.0 Κ (68); A
2
= 4.23 Κ (426); h = 158 J/m s Κ (27); k = 0.87 J/m s Κ J w e (26). The values given i n parentheses are the t-values of the estimated parameters. The comparative r e s u l t s shown i n F i g u r e 5 i n d i c a t e that the r e a c t o r behaviour c o u l d be simulated e x c e l l e n t l y w i t h the presented model. 0
Conclusions The presented dynamic i n v e s t i g a t i o n method employing f o r c e d u n c o r r e l a t e d changes of the process v a r i a b l e s allows a more e f f i c i e n t modelling of the dynamic behaviour of a f i x e d bed r e a c t o r p i l o t p l a n t than r e s u l t s when only one process v a r i a b l e i s changed at a time. Models f o r the r e a c t o r s i m u l a t i o n can be developed w i t h data c o l l e c t e d from only one or a few experimental runs w i t h simultaneous u n c o r r e l a t e d changes of the process v a r i a b l e s . A necessary requirement f o r the a p p l i c a t i o n of the presented method i s , however, that the temperature and concen t r a t i o n p r o f i l e s can be measured i n the r e a c t o r . The method d e s c r i b e d may be p a r t i c u l a r l y u s e f u l f o r the i n v e s t i g a t i o n of i n d u s t r i a l p i l o t p l a n t r e a c t o r s s i n c e many problems l i n k e d w i t h r e a c t o r design and c o n t r o l can be s t u d i e d more e f f i c i e n t l y .
2.
BAIKER ET A L .
Ο
1
Ï5Ô
Process Variables in Fixed-Bed Reactor
2ÔO
1
2feO
Ο
1
TIME (minutes )
°
'
Ï3Ô
'
TIME (minutes)
ϊδδ
25
'
2ÔO
250
TIME (minutes)
2ÔO
2èO
Ο
' Ï5Ô ' TIME (minutes)
2ÔO
250
Figure 5. Comparison of measured profiles interpolated at the collocation points (left) and calculated profiles (right). Ranges of variables are the same as in Figure 3. Key: a, time profiles for temperature and concentration at axial collocation points; and b, time profiles for radial collocation points.
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CHEMICAL REACTION ENGINEERING
Legend of Symbols A^
k i n e t i c parameter, mol/kg s
A^
k i n e t i c parameter, Κ
A~
k i n e t i c parameter, Κ 3 toluene c o n c e n t r a t i o n , mol/m s p e c i f i c heat of c a t a l y s t , J/kg Κ
C c ρκ. c
s p e c i f i c heat of f l u i d , J/kg Κ . 2 r a d i a l e f f e c t i v e d i f f u s i v i t y , m /s
P
h
coefficient 2
W
wall,
f o r heat t r a n s f e r
through r e a c t o r
j/m s Κ
ΔΗ^
r e a c t i o n enthalpy, J/mol
k^
r a d i a l heat c o n d u c t i v i t y i n c a t a l y s t bed, J/m s Κ
L
a x i a l d i s t a n c e between the two r a d i a l p r o f i l e measuring d e v i c e s , m
P
T
P
T
toluene p a r t i a l pressure, bar mean toluene p a r t i a l pressure, bar
r
r a d i a l coordinate, m
R
radius of r e a c t o r tube, m
RG
r e a c t i o n r a t e , mol/kg s
t
time coordinate, s
Τ
temperature, Κ
u
s u p e r f i c i a l gas v e l o c i t y ,
ζ
a x i a l coordinate, m
VIZ,
V1R, V2R
m/s
d i f f e r e n t i a t i o n weighting f a c t o r s (7)
ε
bed p o r o s i t y
p„
d e n s i t y of f l u i d , kg/m
3 F
p^
3 apparent d e n s i t y of c a t a l y s t ,
kg/m
2.
BAIKER ET A L .
Literature
Process Variables in Fixed-Bed Reactor
Cited
1. Hansen, K.W.; Jörgensen, S.B. Chem. Eng. S c i . 1976, 31, 579. 2. Hoiberg, J.A.; Lyche, B . C . ; Foss, A.S. Α . I . C h . E . J . 1971, 17, 1434. 3. Lee, R . S . H . ; Agnew, J . B . Ind. Eng. Chem., Proc. Des. Dev. 1977, 16, 490. 4. Sharma, C . S . ; Hughes, R. Chem. Eng. S c i . 1979, 34, 613. 5. Sörensen, J . P . Chem. Eng. S c i . 1976, 31, 719. 6. Baiker, Α . ; Casanova, R.; Richarz, W. Germ. Chem. Eng. 1980, 3, 112. 7. Villadsen, J.V.; Michelsen, M.L. "Solution of Differential Equation Models by Polynomial Approximation", Prentice H a l l , New Jersey, 1978. 8. Van den Bosch, B . ; Hellinckx, L . A . I . C h . E . J . 1974, 20, 250. 9. Klaus, R.; Rippin, D.W.T. Proc. 12th Symp.on Comp. Appl. in Chem. Enging., Montreux 1979, p.155. 10. Bard, Y. "Nonlinear Parameter Estimation"; Academic Press, Now York, 1974; p.61. RECEIVED April 27, 1982.
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