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Thermal regimes in cocurrently cooled fixed-bed reactors for parallel reactions: application to practical design problems
Pergamon
Chemical Engineerin# Science, Vol. 52, No. 10, pp. 1577 1587, 1997 PII:
S0009-2509(96)00490-3
© 1997 Elsevier Science Ltd. All rights reserved Printed in Great Britain 0009-2509/97 S17.00 + 0.00
Thermal regimes in cocurrently cooled fixed-bed reactors for parallel reactions: application to practical design problems Ver6nica Bucalfi,* Daniel O. Borio and Jos~ A. Porras Planta Piloto de Ingenieria Quimica, UNS-CONICET, 12 de Octubre 1842, 8000-Bahia Blanca, Argentina (Received 12 July 1996; accepted 6 November 1996) Abstract--The conditions of existence of the seven feasible thermal regimes for exothermic first-order parallel reactions occurring in cocurrently-cooled fixed-bed reactors are derived here. The diversity of thermal regimes offered by the cocurrent scheme was applied to solve practical optimization problems. Theoretical optimal temperature profiles inside the reactor tubes which maximize the outlet yield of the desired product were determined. These ideal thermal trajectories were very well tracked by attainable thermal profiles generated by simulation of a cocurrently-cooled fixed-bed reactor represented by a plug-flow pseudo-homogeneous model. In fact, the objective function values found at feasible operation conditions were very close to the optimal ones. These results indicate that the conventional designs such as constant coolant temperature or the countercurrent scheme (where only hot-spot or decreasing profiles can be generated) could not handle the optimization problems as efficiently as the cocurrent configuration. Moreover, the cocurrent design allows, among other advantages, to reach the specified level of conversion (or yield) with higher selectivity and lower maximum temperature values than the common constant coolant temperature configuration. These findings imply the possibility of a significant improvement in the economics of the plant. © 1997 Elsevier Science Ltd. All rights reserved Keywords: Fixed-bed reactors; cocurrent coolant flow; parallel reactions; thermal profiles;
hot-spots; reactor optimization. INTRODUCTION Several researchers have found significant improvements in the operation of fixed-bed reactors when they are cocurrently cooled. In fact, Borio et al. (1989a, b) have demonstrated, for single first-order reactions, that for operations with equal feed and coolant inlet temperatures and equivalent production rates, the cocurrent scheme is the one which leads to the lowest values for the maximum temperature and parametric sensitivity. In addition, a given yield can be obtained with the smallest reactor length, provided an adequate selection for the coolant flowrate is made. These conclusions were experimentally verified in an industrial reactor for o-xylene oxidation (Nikolov and Anastasov, 1989). Moreover, if the inlet temperature of the coolant is considered different from that of the reacting stream, Borio et al. (1995) have proved (for single irreversible first-order reactions) that seven different thermal regimes (i.e. different shapes of thermal profiles) can be achieved. The authors also found the conditions of existence of all the feasible thermal regimes.
It is well known that industrial reactions, such as partial oxidations of hydrocarbons, correspond to complex kinetic schemes. When multiple reactions are carried out in a reactor tube, the axial temperature profile determines the reactor yield (or selectivity) of the desired product. Theoretical optimal temperature profiles (e.g. those thermal trajectories which maximize the reactor yield or selectivity) can be obtained by means of suitable optimization algorithms using only the reactor mass balances. Depending on the selected kinetic scheme, a wide diversity of shapes can be found for the theoretical profiles (Rose, 1981). Generally, the shape of these optimal trajectories cannot be reproduced by using the cooling strategies commonly found in industrial practice. For this reason, the present work, focused on exothermic first-order parallel reactions, aims at three main goals:
*Corresponding author. 1577
• To analyse the operating flexibility provided by the cocurrent flow, i.e. to determine how the shape of the thermal profiles can be influenced by manipulating the inlet temperatures of the gas and coolant streams, or/and the coolant flowrate.
1578
V. Bucalfi et al.
• To apply this flexibility to solve practical optimization problems, i.e. to track the theoretical optimal thermal profiles closely, and consequently obtain values of the objective functions (e.g. reactor yield, selectivity, profit) close to the optimal ones. • To compare the performance of the cocurrent design with that of other coolant flow schemes frequently found in industry (e.g. constant coolant temperature). MATHEMATICALMODEL The following kinetic scheme consisting of two generic parallel reactions was considered: valA + VB1B kts , veP va2A + vB2B k~, vxX. In these reactions A and B are reactants, P and X are the desired and the undesired products, respectively. Both reactions are assumed to be irreversible, of pseudo-first order, and exothermic. Therefore, the rate equations can be expressed as re = ke PA
(1)
rx = kx PA.
(2)
A basic pseudo-homogeneous one-dimensional model was used to represent the reactor behavior. Assuming that the pressure drop across the reactor is negligible, the steady-state mass balances and heat balances inside and outside the catalytic bed are dxa dz
- A (va, ke + VA2 kx~ (1 -- XA) \ ve Vx / dxp = A va~ ke (1 - XA) dz ve
dT dz
B(V~ k,+V~2 Hkx)(1 \ ve Vx
x~)
rc)
(5)
dTc
-~z = D ( T -- T~).
(6)
Assuming a diluted system and A as limiting reactant: XA ~
Pao -- Pa , Pao
vax Pv -- Pro - - , ve PAO
Xp
XX ~
X A --
THERMAL REGIMES IN COCURRENTLY-COOLED FIXED-BED REACTORS
If the inlet temperatures of the gas and coolant streams, as well as the coolant flowrate are adequately selected and an exothermic first-order reaction takes place in a cocurrently-cooled fixed-bed reactor, Borio et al. (1995) found that seven different thermal regimes can be achieved. In this paper, it will be demonstrated that all these seven regimes are also feasible when parallel reactions are carried out in a cocurrent reactor. The feasible thermal regimes are hot-spot profiles, isothermal and pseudo-adiabatic operations, maxi m u m - m i n i m u m and m i n i m u m - m a x i m u m profiles, cold-spot regimes, and continuously decreasing temperature trajectories. The conditions of existence of each thermal regime are similar to those reported for single reactions (Borio et al., 1995). These authors have found that these conditions can be defined by means of four variables: • the feed temperature (To), • the slope of the axial thermal profile at z = 0 (So), for which the case of parallel reactions can be evaluated from eq. (5):
So = B [vA, ke (to) + vA2/ kx (to)] Lye
A - Pb P M~ Go
B
PbPao( - AHe) Cpg Go
H
(--AHp) C - - 4-U Cpg Go dt
Calculation of TI: characterization of temperature extremes A necessary condition for the existence of a local extreme in the temperature profile at a finite value of z = Zm is dT/dzlz, = O. Therefore, from eq. (5), B ~VA1 vP ke(Tm) +
XA=O,
Xp=O,
and
T~ = T~o
T = To (7)
H kx(Tm)] Tc,)=O.
(9)
To characterize a local temperature extreme, the sign of the second derivative has to be studied. Thus, differentiating eq. (5) and evaluating at z = Zs: dz 2
The above model must be solved using the following boundary conditions:
T~o)
For the case of parallel reactions, both the calculation of Tt and T ~ are detailed below.
x(1-XAm)-C(Tm-
(-AHx)
l
• the limiting temperature (T0 • the temperature at z ~ oo (Too).
Xp
U ~r dt tn D - - Wc Cp~
z=0:
C(To
d
(8)
kp = kooe exp( -- Ep/RT), kx = k ~ x exp( - E x / R T ) ,
vx
(3) (4)
c(r-
The reactor yield of desired product is given by rleL = XpL. The outlet selectivity based on a generic reactant j is expressed as (aeL)j = o~ XeL/Oe XjL.
= B
ke(rm) +
H kx(rs)
(1 - XAs)
(10) The sign of the second derivative is given by the sign of the factor between braces in eq. (10).
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Cocurrently cooled fixed-bed reactors Consequently, a maximum occurs when
val ke(Tm) + VA_~2kx(rm) k ve Vx
expressed as
] > --.°
(11)
A
Conversely, when the local temperature extreme corresponds to a minimum:
~0:A7[T(XA)] dXA = ~[T(XA)]Xa. Noting T* = T(XA), eq. (16) can be written as B
C
T - To = ~ y(T *)xA - ~ (To - T~o).
r
VA1 kp(r,,) + vA--2kx(r,,) LVe Vx
< --. A
I °
For z -~ oo eq. (18) becomes
Calculation of Too: T~ is defined as the value of the temperature at z ~ oo, theoretical position where the reactant is exhausted and the reactor temperature approaches that of the cooling fluid (i.e. XA = 1, T~ = Tco~). The To~ value is a function of, among other variables, the total energy released by chemical reaction. Unlike single reactions, for multiple reactions this total amount of energy depends on the temperature profiles developed inside the reactor (which affect the values of local selectivities). Since the temperature profile cannot be predicted a priori, for parallel reactions Too cannot be predicted without solving the mathematical model. However, T~ can be satisfactorily predicted using the approximation described below. Combining eqs (3), (5) and (6), the following expression can be obtained: B
C
To+~ T~ =
To+~
B
T~ -
C
?(T~)+~
T~o (20)
C I+-D
Equation (20) is exact for isothermal operations ( T * = Ti = Too), and it becomes more accurate as H --* 1 [i.e. y ( T ) ~ 1].
Conditions of existence of the thermal regimes Given the above definitions, the conditions of existence developed by Borio et al. (1995) for single firstorder reactions can be fully applied to the present case of parallel reactions. Thus, 1. Hot-spot regime: So > O
To~ > Tt.
2. Pseudo-adiabatic operation: To~ <~ Tt,
To < Tt.
3. Maximum-minimum profiles: T~ < Tt,
To > T~.
4. Cold-spot regime: (15)
So < O,
Too <-%T~.
5. Decreasing thermal profiles:
Integrating eq. 14 from the reactor inlet to any axial position z: B fxA C TO=-Ajo ~(T)dxA--~(Tc-
(19)
T * is an unknown parameter bounded by the maximum and minimum temperature values of the thermal trajectory. As a suitable approximation, T * can be assumed to be equal to To~. From eq. (19), the following implicit equation should be solved to obtain Too:
So > O,
vA~ k r ( T ) + vA2 k x ( T ) Ve Vx
T-
Too
C I+-D
So > O,
y(T) =
C
7(T*)+~
(14)
where
vA__~k e ( T ) + VA2 H kx(T) Vp Vx
B
(13)
It is clear that Tt is a lower boundary for the maxima and an upper value for the minima. Equation (13) represents the definition of the limiting temperature for first-order parallel reactions. Particularly, if kx = 0, eq. (12) reduces to the definition of the limiting temperature for single first-order reactions derived by Borio et al. (1995).
d T = -~ 7(T) dxA -- -D dT~
(18)
(12)
The left-hand side of eqs (11) and (12) increases monotonically as T is increased. Therefore, the limiting temperature Tt can be defined as follows: kx(rl) = "~. FLvA1ve ke(rl) + VA.~2 I ° Vx
(17)
Tco). (16)
The integral in eq. (16) cannot be solved without knowing the relationship between T and xA. However, since ~,(T)= ~,[T(xA)], applying the medium value theorem the integral SoA~[T(xA)] dxA can be
So < O,
To~ >~ Tl
To > T~.
6. Minimum-maximum profiles: So < O,
Too > T~,
To < T~.
7. Isothermal operation: it is relevant at this point to remark that the cocurrent cooling allows one to obtain isothermal profiles for parallel first-order reactions. The isothermal operation verifies T ( z ) = Ti, therefore eqs (3) and (6) can be integrated from the
V. Bucalfi et al.
1580
reactor inlet to any axial position z:
:ex { Ar
+
The isothermal yield can be also calculated by integrating the mass balance (4) from z = 0 to a generic axial position z and using eqs (21) and (26):
z} (21)
T c - Ti - = exp( -- Dz). T o o - Ti
(22)
Since isothermal operations must verify d T / d z = 0 for any axial position, from eq. (5) follows: Tc -- T, = -- ~
ke(Ti)
Vx
(23) B [v~,l ke(T,) + VA2 H kx(T,)]. T~ o -- Ti = - - - ~ h ve Vx
(24)
Substituting eqs (23) and (24) in eq. (22): 1 - XA = exp ( -- Dz).
(25)
The definition of the isothermal temperature is then obtained from eqs (21) and (25):
r
va, ke(Ti) + va-22kx(T,) LVp Vx
]°
= ~.
(26)
F r o m eq. (26), it can be concluded that the temperature at which the reactor operates isothermally coincides with the limiting temperature defined by eq. (13). This coincidence was also verified for single firstorder reactions (Borio et al., 1995). The initial conditions which permit us to obtain an isothermal profile must satisfy simultaneously: So = 0 [eq. (24)] To = Ti = Tl.
The Ti value can be manipulated by modifying any of the parameters involved in the A or D factors [see eq. (26)].
A xe = -- VA1 ke(Ti) [1 -- exp( -- Dz)]. Dye
It has been found that using an analogous procedure, the existence of isothermal regimes in cocurrently cooled reactors for systems of n-parallel reactions can be proved. Prediction o f thermal regimes: operating zones The above-mentioned conditions of existence can be summarized in a T c o - T o plane. The partial oxidation of ethylene to ethylene oxide (oxygen-based oxidation with a large excess of ethylene) was chosen as an example of parallel reactions (see Table 1). Although more detailed kinetic models have been published recently (Borman and Westerterp, 1992, 1995), a system of first-order parallel reactions has been proposed as a good approximation in industrial conditions of large excess of ethylene (Westerterp and Ptasinski, 1984; Miller, 1969). For operation conditions corresponding to a given set of values of Gg, We, and PAo, the different operating zones are illustrated in Fig. 1. It can be seen that only six of the seven feasible thermal regimes can be obtained for the selected set of parameters (no matter what values for the inlet temperatures Tcoand To are chosen). However, appropriate changes on Gg, We, or PAO will lead to any of the seven thermal regimes (e.g. the m i n i m u m - m a x imum zone not displayed in Fig. 1 will be achieved for lower values of the coolant flowrate) (Borio et al., 1995). The full squares in Fig. 1 indicate the Tco values obtained by simulation for which the relationship Too = Tl is satisfied. The curve labeled as T~o = Tt was obtained by using eq. (13) and the approximation given by eq. (20). F r o m this comparison it follows that the prediction given by eq. (20) is in good agreement
Table 1. Values of system and design parameters for the ethylene oxide reactor Data from Westerterp and Ptasinski (1984) 02 + 2 C 2 H 4 ~ 2C2H40 0 2 "F ½C2H4
re = 70.4 exp( - 7200/T)cA -- AHp = 210,000 kJ/k molA ko = 5.6 x 10 -5 k J / s m K Cpg = 1.16 kJ/kg K YAO = 0.06 Go = 7.88 kg/sm 2
kx, g3CO2 + ~H20 rx = 49.4 × 103 exp( - IO,800/T)CA -- AHx = 473,000 kJ/k mOIA dp= 0.004 m Pb = 850 kg/m 3 P = 1 MPa
Additional information employed Coolant: Dowtherm A Cpc = 2.72 kJ/kg K Pc = 859.5 kg/m 3 L = 12m t, = 3000
(27)
kc = 1.80x 10-4 kJ/smK /t¢ = 3.85x 10-a kg/sm #o = 1.67x 10-5 kg/sm d, = 0.0254 m Dc = 1.951 m
Cocurrently cooled fixed-bed reactors
525
1581
I [ Hot-spot [ [Isothermal I I
[De~e~mg[
500 ¸
475
45O 425
450
475
500 525 T O(K)
550
575
600
Fig. 1. Operating zones for the thermal regimes in the Too-To plane (Yao= 0.06, Wc= 125 kg/s, Gg = 7.88 kg/sm2). with the simulation results for the whole range of practical operating conditions. For the very high inlet temperatures (runaway conditions), eq. (20) starts to fail in the prediction of the limiting conditions. The phase plane showed in Fig. 1 is a powerful tool which can be used to estimate the adequate values of inlet temperatures (To and Too)which will lead to the desired temperature profile inside the reactor tubes. The cocurrent design for the case of parallel reactions can influence the shape of the reactor temperature profile strongly, this strength can be very useful to track optimum thermal profiles required to solve practical optimization problems. Due to the enormous potential of this tool, some applications will be illustrated below. OPTIMALTEMPERATUREPROFILES
Optimal theoretical temperature profiles The optimal theoretical temperature profiles denomination will be used to refer to those ideal thermal profiles (generally not feasible in practice) which allow one to achieve the optimal product distribution at the reactor outlet (leading to maximum yield or selectivity, maximum profit, etc.) without violating the minimum and maximum allowable temperatures (Tma and TMa, respectively). Depending on the relative values of the activation energies, two cases can be considered for parallel reactions: (1) Ex < Ep. In this case, as temperature increases, the main reaction will be favored with respect to the side reaction. As is well known, the temperature profile that maximizes conversion, selectivity (and
therefore the reactor yield) corresponds to an isothermal profile at TMo. (2) Ex > Ep. For this case, the side reaction will be favored with respect to the main reaction as temperature increases. Therefore, there is a trade off between maximum conversion [achieved with isothermal profiles at T ( z ) = TMa] and maximum selectivity [obtained at minimum allowable temperature, i.e. T(z) = Tma]. For objective functions such as yield or profit the optimal theoretical temperature profiles are not trivial, so they must be calculated by means of an optimization algorithm. A strict mathematical solution to this type of problem was developed by Pontryagin et al. (1962). Using just the mass balances the Pontryagin maximum principle allows one to obtain a continuous thermal profile which leads to the maximum value of the objective function and verifies: T~a <~ T(z) <~ TMa. To illustrate a practical optimization problem, the yield of the desired product (t/eL = XpL) was selected as objective function to be maximized (the selectivity is controlled by choosing the appropriate TM, value). For parallel reactions with Ex > Ep, the optimal thermal trajectories which maximize the yield of the desired product have already been established (Rose, 1983; Grzesik et al., 1983; Jackson et al., 1972). For the specific case of ethylene partial oxidation, optimal temperature profiles which verify the constraint T (z) ~< TMa are shown in Fig. 2. Profile 1 in Fig. 2 was obtained setting the maximum allowable temperature to 700 K; this TMa value surpasses the maximum allowable temperature for the oxidation process of ethylene, however, it has been considered to exemplify the typical shapes of the optimal temperature profiles. Each curve in Fig. 2 is the optimization result for
V. Bucal/l et al.
1582
given Tu~ and reactor length values. According to the selected TMa value, the theoretical optimal profiles can be either isothermal trajectories or rising curves ending by the isotherm T = Tua (although it is not shown, profile 1 in Fig. 2 ends by the isotherm T = 700 K). The axial position z = z* at which the temperature reaches the value T = Tua (see Fig. 1) is commonly called switching point. Applying Pontryagin's maximum principle, z* can be predicted analytically.
Following the derivation reported by Grzesik and Skrzypek (1983), the switching point z* for parallel reactions in optimal profiles leading to a maximum yield can be calculated as follows:
T Ma Yidd
(28)
km and kx are here evaluated at T
1 / /
z* ~< 0;
0 < z* ~< L:
570
Tort(z) =
Tua
for 0 ~< z ~
Topt(Z )
J'non-falling curve (TMa 550
530
-
...¢ z* (switching point) 510 -
3
Y 4
12
8
= TM=. The
switching point permits one, in an easy way, to qualitatively classify the optimal profiles. In other words, the shape of the optimal temperature profiles can be predicted as follows:
610
Profile
L Vx
VA1 kpEe"]-VA2 kxEx~] vp Vx /j where
590
\ lnF va2 kx(Ex--Ee)/
1
z*=L +A(VA1 k'+VA2 Vx
z (m) Fig. 2. Theoretical optimal temperature profiles for different TM=(YAo= 0.06, Gg = 7.88 kg/s m2).
for 0 ~< z < z* for z* ~< z ~< L.
Since Ex > Ee, eq. (28) indicates that z* cannot assume values higher than L. This is the reason why, independently of the value of TM,, the optimal theoretical profiles can never adopt the shape of continuously rising curves. The representation of eq. (28) for z* = 0 in a TM, vs L plane (Fig. 3) provides useful tool to predict the shape of the theoretical optimal profiles. In fact, the curve z* = 0 splits the whole plane into two operating zones, so that, when a maximum allowable temperature and a reactor length are selected, the shape of the optimal temperature profile can easily be
600
560
TIn(K)
520
480 0
8
16
24
32
40
L (m)
Fig. 3. Prediction of the shape of theoretical optimal profiles as a function of (Gg = 7.88 kg/s m2).
Tua and
reactor length
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Cocurrently cooled fixed-bed reactors determined. Therefore, the shape of the theoretical optimal profiles maximizing the outlet yield can be a priori known. For other optimization problems (e.g. to maximize the profit), the shape of the theoretical optimal thermal profiles could be predicted [-following the same procedure used to obtain eq. (28)] provided that these ideal trajectories end by an isotherm at the maximum allowable temperature value. To achieve the maximum feasible yield in an industrial unit, it is necessary to track the theoretical optimal temperature profiles closely. A design which would allow one to follow any optimal trajectory, although technically a complex solution, would be to arrange a number of different heat-transfer medium circuits so as to achieve a stepwise approximation of the optimal temperature profile (Eigenberger, 1992). However, a desirable solution should not require major changes in the reactor design. As a simpler solution, we propose to use the operational flexibility offered by the cocurrent coolant scheme to track the theoretical optimal trajectories and therefore obtain objective function values close to the optimal ones.
Attainable optimal temperature profiles The attainable temperature profile denomination will be used to refer to those thermal profiles that can be obtained by simulation of the fixed-bed reactor, represented by a plug-flow pseudo-homogeneous model, using as a coolant scheme the cocurrent configuration and different temperature values for the gas and coolant inlet temperatures. Given the selected TMa and reactor length values (for either Ex < Ee or Ex > Ep), the shape of the theoretical optimal profiles can be a priori known. Based on this information, the challenge is to define the proper operational conditions which allow to have attainable temperature profiles close (as much as possible) to the theoretical optimal ones. When the theoretical optimal temperature profile corresponds to an isothermal trajectory the cocurrent coolant scheme allows one to generate an attainable temperature profile capable of tracking the desired profile shape exactly. Therefore, the optimal objective function value could be reached. In fact, given the TM~ value, the parameters involved in either A or D (see the mathematical model section) e.g. Gg or Wc, can be set so that the eq. (26) is verified at Ti = TM,. The inlet coolant temperature for the isothermal operation is given by eq. (24). The maximum reactor yield can be calculated from eq. (27), for Ti = TMa and z = L. For the ethylene oxidation case, Fig. 4 shows the gas and coolant temperature profiles that maximizes ~/PLfor a TMa of 533 K. Figure 5 shows, for the special case of ethylene oxidation, a theoretical optimal profile (dashed line) which leads for a TMa = 541 K to the maximum allowable value of r/pL = 0.225. This profile (rising curve ending by the isotherm T = 541 K) cannot be exactly reproduced in a cocurrent cooled reactor. However, modifying operational conditions such as inlet temperatures or coolant flowrate it is possible to
54o
530
~ 520 ~.
5~0
50o
I 3
I 6
1 9
12
z(m) Fig. 4. Isothermal operation: theoretical optimal profile and simultaneously attainable optimal profile for a TMa= 533K (YAo= 0.06, Wc = 125 kg/s, Gg = 7.88 kg/sm 2, r/eL = 0.216).
550
53o
510
490
I 3
I 6
I 9
12
z(m) Fig. 5. Theoretical optimal profile (YA0= 0.06, Gg = 7.88 kg/sm 2, qeL = 0.2250) for TMa = 541 K and an attainable profile close to the optimal one (YAo= 0.06, Wc = 90 kg/s, Gg = 7.88 kg/sm 2, r/pL= 0.2245). find attainable temperature profiles very similar to the theoretical ones. The attainable optimal profiles could be obtained by using an optimization algorithm trying to track the theoretical optimal trajectories by means of changes in the operational conditions. To avoid such calculation effort, a T o o - To plane (see Fig. 1) could be very useful to establish the ranges of operational conditions that should not be explored. In fact, if the optimal profile is similar to that described in Figure 5, for example, regimes such as hot-spot profiles, maximum-minimum or decreasing profiles could not solve the optimization problem satisfactorily. Figure 5 shows the best attainable gas
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V. Bucalfi et al.
and coolant temperature profiles (solid lines) found following the procedure described above. The results indicate that the best attainable profile leads to a yield that is only 0.2% lower than the maximum theoretical value. The previous examples illustrate the capability of the cocurrent flow scheme to allow the generation of attainable temperature profiles very close (or identical) to the optimal theoretical ones. The two conventional designs, countercurrent scheme or constant coolant temperature, could not solve practical optimization problems as efficiently as the cocurrent configuration. Indeed, only hot-spot or decreasing profiles are possible when any of those configurations are used.
F~f FB,f
COMPARISON OF THE COCURRENT DESIGN WITH CONSTANT COOLANTTEMPERATURE ARRANGEMENTS In this section, the study is focused on the comparison of the cocurrent configuration with the constant coolant temperature case which is representative of the cooling arrangements commonly found in practice (boiling liquids as coolant medium, or liquids circulating at high flowrates). This comparison is performed to investigate whether the cocurrent design can improve the reactor performance, for situations of practical interest, in a novel way. This study is based on the case of ethylene oxide production. A schematic flow diagram for an ethylene oxide plant (oxygen-based oxidation) is shown in Fig. 6. As a common basis of comparison, constant
FA
FA,o FB,0
_1 Reactor Section
q
Fp
P
Product Separation
lit Fp=(Vv/Vm)ripF.,f undesired products
X~ FA FB
np =
(opLh
Fig. 6. Schematic flow diagram for an ethylene oxide plant (oxygen-based oxidation).
550 XAL=0.75
4 540
530
520 Wc, kg/s To, K
510
500 I 0
1 2 3 4 5
--~ 122.5 100.0 80.0 100.0
524.2 533.3 025.0 515.0 500.0
Tco, K
(o~)B
~PL
Tm~, K
524.2 503.2 500.4 496.7 502.6
0.703 0.709 0.712 0.714 0.713
0.212 0.216 0.210 0.221 0.219
544.5 533.3 536.5 5412 537.9
, 4
, 8
12
z (m) Fig. 7. Axial temperature profiles at conditions of constant outlet conversion (YA0= 0.06, Gg = 7.88 kg/s m2).
1585
Cocurrently cooled fixed-bed reactors molar flows of oxygen (Fa0) and ethylene (F~o) at the reactor inlet were assumed. Two different situations are considered: Constant outlet conversion
(XAL)
Figure 7 shows the typical temperature profile that a fixed-bed reactor exhibits when constant coolant temperature is assumed (curve 1), it presents a hotspot located near the reactor inlet. Curves 2-5 were obtained for different operating conditions under cocurrent coolant flow. All the curves verify the same outlet conversion xaL = 0.75. Operating at isothermal conditions (curve 2), a 75 % oxygen conversion can be reached with the minimum value of maximum temperature. Particularly, this temperature value (Tmax = Ti = 533.3 K) can be a priori known from eq. (21) setting XAL = 0.75 and z = L = 12 m. The cocurrent
coolant flow allows the same conversion to be obtained with temperature profiles showing very different shapes (e.g. curve 3 shows a minimum near the reactor entrance). The examples selected here present higher selectivities and simultaneously lower maximum temperature values than the reference case (curve 1). Bucalh (1991) and Krishna and Sic (1994) have pointed out the benefits of cocurrent cooling for the process of ethylene oxide manufacture. Particularly, Krishna and Sic (1994) have reported that improvements of selectivity by 1% in the manufacture of ethylene oxide can be extremely significant for process licensers. To analyse the influence of the selectivity on the process economy, the plant molar flows related to curves 1 and 4 of Fig. 7 are compared in Table 2. Curve 4 presents a reactor selectivity (i.e. plant yield) 1.6% higher than the reference case (curve 1). Due to
Table 2. Molar flows described in Fig. 6 for selected operations showed in Figs 7 and 8 Curve
ql,L
xaz
(treL)n
Fao
Fno
Fa,:
Fa.:
Fe
69 69
55.5 56.1
39.0 40.7
Case A: constant oxygen conversion per pass 1 of Fig. 7 4 of Fig. 7
0.212 0.221
0.75 0.75
0.703 0.714
92 92
1435 1435
Case B: constant reactor yield 1 of Fig. 8 0.212 0.75 4 of Fig. 8 0.212 0.69
0.703 0.726
92 92
1 4 3 5 6 9 . 0 5 5 . 5 39.0 1 4 3 5 6 3 . 6 5 3 . 7 39.0
550
rlPL=0.212
/ 530-/ L -~3
520
510
-~
500 0
1 2 3 4 5
WC, kg/s
To, K
TCo, K
(o~)B
XAL
T.~o K
~ 122.5 141.0 80.0 100.0
524.2 524.7 530.5 500.0 520.0
524.2 502.3 504.5 496.5 499.0
0.703 0,721 0,716 0.726 0.724
0.750 0.705 0.718 0.691 0.689
544.5 531.5 530.5 538.6 533.8
I 4
I 8
12
z (m) Fig. 8. Axial temperature profiles at conditions of constant reactor yield (Ya0 = 0.06, Go = 7.88 kg/sm2).
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V. Bucala et al.
the reaction stoichiometry (Table 1) the increase in selectivity requires a slight increase in Fn.f to maintain the reactor load constant. Therefore, the production rate (Fe) increases by 4.36%. Assuming $0.2/lb and $0.45/lb as the prices of ethylene and ethylene oxide, respectively, and considering a standard capacity plant of 100,000 t/yr (Miller, 1969), an improvement of 4.36% in the production rate would signify an increased revenue of the order of US$3 million per year.
Constant reactor yield (rleL) Figure 8 shows different temperature profiles giving the same reactor yield (r/eL = 0.212). Curve 1 is the reference case also shown in Fig. 7 (constant coolant temperature). Curves 2-5 correspond to thermal profiles obtained with cocurrent coolant flow. The specified reactor yield is achieved by operations 2-5 with higher selectivities and lower conversion than the base case. The maximum temperature values are lower than the hot-spot exhibited by curve 1. Particularly, curve 3 is an isothermal regime operating at Ti = 530.5 K, and represents the operation which leads to r/eL = 0.212 exhibiting the lowest maximum temperature value. Indeed, Fig. 3 shows that for a TM, = 530.5 K and L = 12 m, the optimal temperature profile which maximizes the reactor yield corresponds to an isothermal trajectory. For this reason, any thermal profile starting with an inlet gas temperature lower than 530.5 K has to surpass this TM, value in order to meet the specified value for the reactor yield, The plant molar flows corresponding to the curves 1 and 4 of Fig. 8 are compared in Table 2. Maintaining a constant reactor load and reactor yield implies to have the same production rate of the plant (constant Fe). Curve 4 allows one to reach the same production rate requiring lower values of the fresh feed. In fact, the same 39 k mol/h of P can be produced with a reduction of 3.2% in fresh ethylene and 7.8% in fresh oxygen with respect to the reference case. For a standard capacity plant of 100,000 t/yr of ethylene oxide, the saving only in fresh ethylene would imply a decrease in costs of the order of US$1 million per year.
trajectories, and consequently let us obtain the maximum value of reactor yield. The shape of theoretical thermal profiles that maximize the reactor yield can be established a priori. This is an important tool to define the appropriate operational conditions to obtain feasible thermal trajectories close to the ideal ones. The cocurrent flow scheme presents significant advantages when it is compared to the constant coolant configuration. The operational flexibility given by the cocurrent configuration could significantly improve the economics of the process. The present results were obtained assuming a plugflow pseudo-homogeneous model and exothermic first-order parallel reactions. As a further work, the advantages showed by the cocurrent design should be verified using more detailed reactor models. NOTATION
A
Go ] B
Aol cv,
c C
concentration (p/RT), kmol/m 3
Cp d
specific heat, kJ/kg K diameter, m
D
wc Cpc/ Dc E F G H
shell diameter, m activation energy, kJ/kmol molar flow, kmol/h specific mass flowrate, kg/m 2 s _
(-AUg)] (-AHD ]
AH k k¢
CONCLUSIONS
When exothermic first-order parallel reactions are carried out in cocurrently-cooled fixed-bed reactors seven thermal regimes can be achieved. They are: hot-spots, isothermal and pseudo-adiabatic operations, maximum-minimum and minimum-maximum profiles, cold-spot regimes, and continuously decreasing temperature trajectories. The conditions of existence of each of these theoretically feasible regimes are derived in the present work. It has been demonstrated that the operating conditions leading to different regimes can be accurately predicted. The diversity of thermal profiles offered by the cocurrent coolant scheme allows practical optimization problems to be solved very efficiently. In fact, when the reactor yield is wanted to be a maximum, this cooling arrangement can practically reproduce the ideal thermal
[=
kg k~ L p P r R S t, T Tc U W~ x y
heat of reaction, kJ/kmol reaction constant, kmol/kg kPa s thermal conductivity of cooling fluid, kJ/s m K gas thermal conductivity, kJ/s m K pre-exponential factor, kmol/kg kPa s reactor length, m partial pressure, kPa total pressure, kPa reaction rate, kmol/kg s universal gas constant, k J / k m o l K or kPa m3/kmol K (d T/dz), K/m number of reactor tubes gas temperature, K temperature of cooling fluid, K overall heat transfer coefficient, kW/m 2 K cooling mass flowrate, kg/s conversion molar fraction
Cocurrently cooled fixed-bed reactors Z 2"*
axial coordinate, m switching point, m
Greek letters reactor yield plant yield viscosity, kg/s m stoichiometric coefficient density, kg/m 3 selectivity Subscripts A limiting reactant b bed B reactant c coolant g gas i isothermal j reactant l limiting L at axial position z = L m relative extreme (maximum or minimum) ma minimum allowable max maximum temperature value of the profile T(z) Ma maximum allowable opt optimal p particle P desired product t tube X undesired product 0 at axial position z = 0 oo at axial position z ~ oo REFERENCES
Borio, D. O., Gatica, J. E. and Porras, J. A. (1989a) Wall-cooled fixed-bed reactors: parametric sensitivity as a design criterion, A.I.Ch.E.J. 35, 287-292. Borio, D. O., Bucalfi, V., Orejas, J. A. and Porras, J. A. (1989b) Cocurrently-cooled fixed-bed reactors: a simple approach to optimal cooling design. A.I.Ch.E.J. 35, 1899-1902.
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Borio, D.O., Bucalh, V. and Porras, J. A. (1995) Thermal regimes in cocurrently-cooled fixed-bed reactors. Chem. Engng Sci. 50, 3115-3123. Borman, P. C. and Westerterp, K. R. (1992) An experimental study of the selective oxidation of ethene in a wall cooled tubular packed bed reactor. Chem. Engng Sci. 47, 2541-2546. Borman, P. C. and Westerterp, K. R. (1995) An experimental study of the kinetics of the selective oxidation of ethene over a silver on ~t-alumina catalyst. Ind. Engng Chem. Res. 34, 49-58. Bucalfi, V. (1991) Operabilidad y optimizaci6n de reactores cataliticos tubulares. Ph.D. ~thesis, Department of Chemistry and Chemical Engineering, Universidad Nacional del Sur, Bahia Blanca, Argentina. Eigenberger, G. (1992) Fixed-bed reactors. In Ullmann's Encyclopedia of Industrial Chemistry, eds. B. Elvers, A. Hawkins and G. Schulz, Vol. B4, pp. 199-238. VCH Verlagsgesellschaft mbH, D-6940, Weinheim, Germany. Grzesik, M. and Skrzypek, J. (1983) Optimal temperature profiles for heterogeneous catalytic parallel reactions. 38, 1767-1773. Jackson, R., Obando, R. and Senior, M. G. (1972) The Control of Competing Chemical Reactions. Chemical Reaction Engineering, A.C.S. Krishna, R. and Sie, S. T. (1994) Strategies for multiphase reactor selection. Chem. Engng Sci. 49, 4029-4065. Miller, S. A. (1969) Ethylene and its Industrial Derivatives. Ernest Benn Limited, London, U.K. Nikolov, V. A. and Anastasov, A. I. (1989) A study of coolant temperature in an industrial reactor for o-xylene oxidation. A.I.Ch.E.J. 35, 511-513. Pontryagin, L. S., Boltyanski, V. G., Gamkrelidze, R. V. and Mischenko, E. F. (1962) The Mathematical Theory of Optimal Processes. Interscience, New York, U.S.A. Rose, L. M. (1981) Chemical Reactor Design and Practice, ed. S. W. Churchill. Elsevier, Netherlands. Westerterp, K. R. and Ptasinski, K. J. (1984) Safe design of cooled tubular reactors for exothermic, multiple reactions; parallel reactions--II. Chem. Engng Sci. 39, 245-252.
Chemical Engineerin# Science, Vol. 52, No. 10, pp. 1577 1587, 1997 PII:
S0009-2509(96)00490-3
© 1997 Elsevier Science Ltd. All rights reserved Printed in Great Britain 0009-2509/97 S17.00 + 0.00
Thermal regimes in cocurrently cooled fixed-bed reactors for parallel reactions: application to practical design problems Ver6nica Bucalfi,* Daniel O. Borio and Jos~ A. Porras Planta Piloto de Ingenieria Quimica, UNS-CONICET, 12 de Octubre 1842, 8000-Bahia Blanca, Argentina (Received 12 July 1996; accepted 6 November 1996) Abstract--The conditions of existence of the seven feasible thermal regimes for exothermic first-order parallel reactions occurring in cocurrently-cooled fixed-bed reactors are derived here. The diversity of thermal regimes offered by the cocurrent scheme was applied to solve practical optimization problems. Theoretical optimal temperature profiles inside the reactor tubes which maximize the outlet yield of the desired product were determined. These ideal thermal trajectories were very well tracked by attainable thermal profiles generated by simulation of a cocurrently-cooled fixed-bed reactor represented by a plug-flow pseudo-homogeneous model. In fact, the objective function values found at feasible operation conditions were very close to the optimal ones. These results indicate that the conventional designs such as constant coolant temperature or the countercurrent scheme (where only hot-spot or decreasing profiles can be generated) could not handle the optimization problems as efficiently as the cocurrent configuration. Moreover, the cocurrent design allows, among other advantages, to reach the specified level of conversion (or yield) with higher selectivity and lower maximum temperature values than the common constant coolant temperature configuration. These findings imply the possibility of a significant improvement in the economics of the plant. © 1997 Elsevier Science Ltd. All rights reserved Keywords: Fixed-bed reactors; cocurrent coolant flow; parallel reactions; thermal profiles;
hot-spots; reactor optimization. INTRODUCTION Several researchers have found significant improvements in the operation of fixed-bed reactors when they are cocurrently cooled. In fact, Borio et al. (1989a, b) have demonstrated, for single first-order reactions, that for operations with equal feed and coolant inlet temperatures and equivalent production rates, the cocurrent scheme is the one which leads to the lowest values for the maximum temperature and parametric sensitivity. In addition, a given yield can be obtained with the smallest reactor length, provided an adequate selection for the coolant flowrate is made. These conclusions were experimentally verified in an industrial reactor for o-xylene oxidation (Nikolov and Anastasov, 1989). Moreover, if the inlet temperature of the coolant is considered different from that of the reacting stream, Borio et al. (1995) have proved (for single irreversible first-order reactions) that seven different thermal regimes (i.e. different shapes of thermal profiles) can be achieved. The authors also found the conditions of existence of all the feasible thermal regimes.
It is well known that industrial reactions, such as partial oxidations of hydrocarbons, correspond to complex kinetic schemes. When multiple reactions are carried out in a reactor tube, the axial temperature profile determines the reactor yield (or selectivity) of the desired product. Theoretical optimal temperature profiles (e.g. those thermal trajectories which maximize the reactor yield or selectivity) can be obtained by means of suitable optimization algorithms using only the reactor mass balances. Depending on the selected kinetic scheme, a wide diversity of shapes can be found for the theoretical profiles (Rose, 1981). Generally, the shape of these optimal trajectories cannot be reproduced by using the cooling strategies commonly found in industrial practice. For this reason, the present work, focused on exothermic first-order parallel reactions, aims at three main goals:
*Corresponding author. 1577
• To analyse the operating flexibility provided by the cocurrent flow, i.e. to determine how the shape of the thermal profiles can be influenced by manipulating the inlet temperatures of the gas and coolant streams, or/and the coolant flowrate.
1578
V. Bucalfi et al.
• To apply this flexibility to solve practical optimization problems, i.e. to track the theoretical optimal thermal profiles closely, and consequently obtain values of the objective functions (e.g. reactor yield, selectivity, profit) close to the optimal ones. • To compare the performance of the cocurrent design with that of other coolant flow schemes frequently found in industry (e.g. constant coolant temperature). MATHEMATICALMODEL The following kinetic scheme consisting of two generic parallel reactions was considered: valA + VB1B kts , veP va2A + vB2B k~, vxX. In these reactions A and B are reactants, P and X are the desired and the undesired products, respectively. Both reactions are assumed to be irreversible, of pseudo-first order, and exothermic. Therefore, the rate equations can be expressed as re = ke PA
(1)
rx = kx PA.
(2)
A basic pseudo-homogeneous one-dimensional model was used to represent the reactor behavior. Assuming that the pressure drop across the reactor is negligible, the steady-state mass balances and heat balances inside and outside the catalytic bed are dxa dz
- A (va, ke + VA2 kx~ (1 -- XA) \ ve Vx / dxp = A va~ ke (1 - XA) dz ve
dT dz
B(V~ k,+V~2 Hkx)(1 \ ve Vx
x~)
rc)
(5)
dTc
-~z = D ( T -- T~).
(6)
Assuming a diluted system and A as limiting reactant: XA ~
Pao -- Pa , Pao
vax Pv -- Pro - - , ve PAO
Xp
XX ~
X A --
THERMAL REGIMES IN COCURRENTLY-COOLED FIXED-BED REACTORS
If the inlet temperatures of the gas and coolant streams, as well as the coolant flowrate are adequately selected and an exothermic first-order reaction takes place in a cocurrently-cooled fixed-bed reactor, Borio et al. (1995) found that seven different thermal regimes can be achieved. In this paper, it will be demonstrated that all these seven regimes are also feasible when parallel reactions are carried out in a cocurrent reactor. The feasible thermal regimes are hot-spot profiles, isothermal and pseudo-adiabatic operations, maxi m u m - m i n i m u m and m i n i m u m - m a x i m u m profiles, cold-spot regimes, and continuously decreasing temperature trajectories. The conditions of existence of each thermal regime are similar to those reported for single reactions (Borio et al., 1995). These authors have found that these conditions can be defined by means of four variables: • the feed temperature (To), • the slope of the axial thermal profile at z = 0 (So), for which the case of parallel reactions can be evaluated from eq. (5):
So = B [vA, ke (to) + vA2/ kx (to)] Lye
A - Pb P M~ Go
B
PbPao( - AHe) Cpg Go
H
(--AHp) C - - 4-U Cpg Go dt
Calculation of TI: characterization of temperature extremes A necessary condition for the existence of a local extreme in the temperature profile at a finite value of z = Zm is dT/dzlz, = O. Therefore, from eq. (5), B ~VA1 vP ke(Tm) +
XA=O,
Xp=O,
and
T~ = T~o
T = To (7)
H kx(Tm)] Tc,)=O.
(9)
To characterize a local temperature extreme, the sign of the second derivative has to be studied. Thus, differentiating eq. (5) and evaluating at z = Zs: dz 2
The above model must be solved using the following boundary conditions:
T~o)
For the case of parallel reactions, both the calculation of Tt and T ~ are detailed below.
x(1-XAm)-C(Tm-
(-AHx)
l
• the limiting temperature (T0 • the temperature at z ~ oo (Too).
Xp
U ~r dt tn D - - Wc Cp~
z=0:
C(To
d
(8)
kp = kooe exp( -- Ep/RT), kx = k ~ x exp( - E x / R T ) ,
vx
(3) (4)
c(r-
The reactor yield of desired product is given by rleL = XpL. The outlet selectivity based on a generic reactant j is expressed as (aeL)j = o~ XeL/Oe XjL.
= B
ke(rm) +
H kx(rs)
(1 - XAs)
(10) The sign of the second derivative is given by the sign of the factor between braces in eq. (10).
1579
Cocurrently cooled fixed-bed reactors Consequently, a maximum occurs when
val ke(Tm) + VA_~2kx(rm) k ve Vx
expressed as
] > --.°
(11)
A
Conversely, when the local temperature extreme corresponds to a minimum:
~0:A7[T(XA)] dXA = ~[T(XA)]Xa. Noting T* = T(XA), eq. (16) can be written as B
C
T - To = ~ y(T *)xA - ~ (To - T~o).
r
VA1 kp(r,,) + vA--2kx(r,,) LVe Vx
< --. A
I °
For z -~ oo eq. (18) becomes
Calculation of Too: T~ is defined as the value of the temperature at z ~ oo, theoretical position where the reactant is exhausted and the reactor temperature approaches that of the cooling fluid (i.e. XA = 1, T~ = Tco~). The To~ value is a function of, among other variables, the total energy released by chemical reaction. Unlike single reactions, for multiple reactions this total amount of energy depends on the temperature profiles developed inside the reactor (which affect the values of local selectivities). Since the temperature profile cannot be predicted a priori, for parallel reactions Too cannot be predicted without solving the mathematical model. However, T~ can be satisfactorily predicted using the approximation described below. Combining eqs (3), (5) and (6), the following expression can be obtained: B
C
To+~ T~ =
To+~
B
T~ -
C
?(T~)+~
T~o (20)
C I+-D
Equation (20) is exact for isothermal operations ( T * = Ti = Too), and it becomes more accurate as H --* 1 [i.e. y ( T ) ~ 1].
Conditions of existence of the thermal regimes Given the above definitions, the conditions of existence developed by Borio et al. (1995) for single firstorder reactions can be fully applied to the present case of parallel reactions. Thus, 1. Hot-spot regime: So > O
To~ > Tt.
2. Pseudo-adiabatic operation: To~ <~ Tt,
To < Tt.
3. Maximum-minimum profiles: T~ < Tt,
To > T~.
4. Cold-spot regime: (15)
So < O,
Too <-%T~.
5. Decreasing thermal profiles:
Integrating eq. 14 from the reactor inlet to any axial position z: B fxA C TO=-Ajo ~(T)dxA--~(Tc-
(19)
T * is an unknown parameter bounded by the maximum and minimum temperature values of the thermal trajectory. As a suitable approximation, T * can be assumed to be equal to To~. From eq. (19), the following implicit equation should be solved to obtain Too:
So > O,
vA~ k r ( T ) + vA2 k x ( T ) Ve Vx
T-
Too
C I+-D
So > O,
y(T) =
C
7(T*)+~
(14)
where
vA__~k e ( T ) + VA2 H kx(T) Vp Vx
B
(13)
It is clear that Tt is a lower boundary for the maxima and an upper value for the minima. Equation (13) represents the definition of the limiting temperature for first-order parallel reactions. Particularly, if kx = 0, eq. (12) reduces to the definition of the limiting temperature for single first-order reactions derived by Borio et al. (1995).
d T = -~ 7(T) dxA -- -D dT~
(18)
(12)
The left-hand side of eqs (11) and (12) increases monotonically as T is increased. Therefore, the limiting temperature Tt can be defined as follows: kx(rl) = "~. FLvA1ve ke(rl) + VA.~2 I ° Vx
(17)
Tco). (16)
The integral in eq. (16) cannot be solved without knowing the relationship between T and xA. However, since ~,(T)= ~,[T(xA)], applying the medium value theorem the integral SoA~[T(xA)] dxA can be
So < O,
To~ >~ Tl
To > T~.
6. Minimum-maximum profiles: So < O,
Too > T~,
To < T~.
7. Isothermal operation: it is relevant at this point to remark that the cocurrent cooling allows one to obtain isothermal profiles for parallel first-order reactions. The isothermal operation verifies T ( z ) = Ti, therefore eqs (3) and (6) can be integrated from the
V. Bucalfi et al.
1580
reactor inlet to any axial position z:
:ex { Ar
+
The isothermal yield can be also calculated by integrating the mass balance (4) from z = 0 to a generic axial position z and using eqs (21) and (26):
z} (21)
T c - Ti - = exp( -- Dz). T o o - Ti
(22)
Since isothermal operations must verify d T / d z = 0 for any axial position, from eq. (5) follows: Tc -- T, = -- ~
ke(Ti)
Vx
(23) B [v~,l ke(T,) + VA2 H kx(T,)]. T~ o -- Ti = - - - ~ h ve Vx
(24)
Substituting eqs (23) and (24) in eq. (22): 1 - XA = exp ( -- Dz).
(25)
The definition of the isothermal temperature is then obtained from eqs (21) and (25):
r
va, ke(Ti) + va-22kx(T,) LVp Vx
]°
= ~.
(26)
F r o m eq. (26), it can be concluded that the temperature at which the reactor operates isothermally coincides with the limiting temperature defined by eq. (13). This coincidence was also verified for single firstorder reactions (Borio et al., 1995). The initial conditions which permit us to obtain an isothermal profile must satisfy simultaneously: So = 0 [eq. (24)] To = Ti = Tl.
The Ti value can be manipulated by modifying any of the parameters involved in the A or D factors [see eq. (26)].
A xe = -- VA1 ke(Ti) [1 -- exp( -- Dz)]. Dye
It has been found that using an analogous procedure, the existence of isothermal regimes in cocurrently cooled reactors for systems of n-parallel reactions can be proved. Prediction o f thermal regimes: operating zones The above-mentioned conditions of existence can be summarized in a T c o - T o plane. The partial oxidation of ethylene to ethylene oxide (oxygen-based oxidation with a large excess of ethylene) was chosen as an example of parallel reactions (see Table 1). Although more detailed kinetic models have been published recently (Borman and Westerterp, 1992, 1995), a system of first-order parallel reactions has been proposed as a good approximation in industrial conditions of large excess of ethylene (Westerterp and Ptasinski, 1984; Miller, 1969). For operation conditions corresponding to a given set of values of Gg, We, and PAo, the different operating zones are illustrated in Fig. 1. It can be seen that only six of the seven feasible thermal regimes can be obtained for the selected set of parameters (no matter what values for the inlet temperatures Tcoand To are chosen). However, appropriate changes on Gg, We, or PAO will lead to any of the seven thermal regimes (e.g. the m i n i m u m - m a x imum zone not displayed in Fig. 1 will be achieved for lower values of the coolant flowrate) (Borio et al., 1995). The full squares in Fig. 1 indicate the Tco values obtained by simulation for which the relationship Too = Tl is satisfied. The curve labeled as T~o = Tt was obtained by using eq. (13) and the approximation given by eq. (20). F r o m this comparison it follows that the prediction given by eq. (20) is in good agreement
Table 1. Values of system and design parameters for the ethylene oxide reactor Data from Westerterp and Ptasinski (1984) 02 + 2 C 2 H 4 ~ 2C2H40 0 2 "F ½C2H4
re = 70.4 exp( - 7200/T)cA -- AHp = 210,000 kJ/k molA ko = 5.6 x 10 -5 k J / s m K Cpg = 1.16 kJ/kg K YAO = 0.06 Go = 7.88 kg/sm 2
kx, g3CO2 + ~H20 rx = 49.4 × 103 exp( - IO,800/T)CA -- AHx = 473,000 kJ/k mOIA dp= 0.004 m Pb = 850 kg/m 3 P = 1 MPa
Additional information employed Coolant: Dowtherm A Cpc = 2.72 kJ/kg K Pc = 859.5 kg/m 3 L = 12m t, = 3000
(27)
kc = 1.80x 10-4 kJ/smK /t¢ = 3.85x 10-a kg/sm #o = 1.67x 10-5 kg/sm d, = 0.0254 m Dc = 1.951 m
Cocurrently cooled fixed-bed reactors
525
1581
I [ Hot-spot [ [Isothermal I I
[De~e~mg[
500 ¸
475
45O 425
450
475
500 525 T O(K)
550
575
600
Fig. 1. Operating zones for the thermal regimes in the Too-To plane (Yao= 0.06, Wc= 125 kg/s, Gg = 7.88 kg/sm2). with the simulation results for the whole range of practical operating conditions. For the very high inlet temperatures (runaway conditions), eq. (20) starts to fail in the prediction of the limiting conditions. The phase plane showed in Fig. 1 is a powerful tool which can be used to estimate the adequate values of inlet temperatures (To and Too)which will lead to the desired temperature profile inside the reactor tubes. The cocurrent design for the case of parallel reactions can influence the shape of the reactor temperature profile strongly, this strength can be very useful to track optimum thermal profiles required to solve practical optimization problems. Due to the enormous potential of this tool, some applications will be illustrated below. OPTIMALTEMPERATUREPROFILES
Optimal theoretical temperature profiles The optimal theoretical temperature profiles denomination will be used to refer to those ideal thermal profiles (generally not feasible in practice) which allow one to achieve the optimal product distribution at the reactor outlet (leading to maximum yield or selectivity, maximum profit, etc.) without violating the minimum and maximum allowable temperatures (Tma and TMa, respectively). Depending on the relative values of the activation energies, two cases can be considered for parallel reactions: (1) Ex < Ep. In this case, as temperature increases, the main reaction will be favored with respect to the side reaction. As is well known, the temperature profile that maximizes conversion, selectivity (and
therefore the reactor yield) corresponds to an isothermal profile at TMo. (2) Ex > Ep. For this case, the side reaction will be favored with respect to the main reaction as temperature increases. Therefore, there is a trade off between maximum conversion [achieved with isothermal profiles at T ( z ) = TMa] and maximum selectivity [obtained at minimum allowable temperature, i.e. T(z) = Tma]. For objective functions such as yield or profit the optimal theoretical temperature profiles are not trivial, so they must be calculated by means of an optimization algorithm. A strict mathematical solution to this type of problem was developed by Pontryagin et al. (1962). Using just the mass balances the Pontryagin maximum principle allows one to obtain a continuous thermal profile which leads to the maximum value of the objective function and verifies: T~a <~ T(z) <~ TMa. To illustrate a practical optimization problem, the yield of the desired product (t/eL = XpL) was selected as objective function to be maximized (the selectivity is controlled by choosing the appropriate TM, value). For parallel reactions with Ex > Ep, the optimal thermal trajectories which maximize the yield of the desired product have already been established (Rose, 1983; Grzesik et al., 1983; Jackson et al., 1972). For the specific case of ethylene partial oxidation, optimal temperature profiles which verify the constraint T (z) ~< TMa are shown in Fig. 2. Profile 1 in Fig. 2 was obtained setting the maximum allowable temperature to 700 K; this TMa value surpasses the maximum allowable temperature for the oxidation process of ethylene, however, it has been considered to exemplify the typical shapes of the optimal temperature profiles. Each curve in Fig. 2 is the optimization result for
V. Bucal/l et al.
1582
given Tu~ and reactor length values. According to the selected TMa value, the theoretical optimal profiles can be either isothermal trajectories or rising curves ending by the isotherm T = Tua (although it is not shown, profile 1 in Fig. 2 ends by the isotherm T = 700 K). The axial position z = z* at which the temperature reaches the value T = Tua (see Fig. 1) is commonly called switching point. Applying Pontryagin's maximum principle, z* can be predicted analytically.
Following the derivation reported by Grzesik and Skrzypek (1983), the switching point z* for parallel reactions in optimal profiles leading to a maximum yield can be calculated as follows:
T Ma Yidd
(28)
km and kx are here evaluated at T
1 / /
z* ~< 0;
0 < z* ~< L:
570
Tort(z) =
Tua
for 0 ~< z ~
Topt(Z )
J'non-falling curve (TMa 550
530
-
...¢ z* (switching point) 510 -
3
Y 4
12
8
= TM=. The
switching point permits one, in an easy way, to qualitatively classify the optimal profiles. In other words, the shape of the optimal temperature profiles can be predicted as follows:
610
Profile
L Vx
VA1 kpEe"]-VA2 kxEx~] vp Vx /j where
590
\ lnF va2 kx(Ex--Ee)/
1
z*=L +A(VA1 k'+VA2 Vx
z (m) Fig. 2. Theoretical optimal temperature profiles for different TM=(YAo= 0.06, Gg = 7.88 kg/s m2).
for 0 ~< z < z* for z* ~< z ~< L.
Since Ex > Ee, eq. (28) indicates that z* cannot assume values higher than L. This is the reason why, independently of the value of TM,, the optimal theoretical profiles can never adopt the shape of continuously rising curves. The representation of eq. (28) for z* = 0 in a TM, vs L plane (Fig. 3) provides useful tool to predict the shape of the theoretical optimal profiles. In fact, the curve z* = 0 splits the whole plane into two operating zones, so that, when a maximum allowable temperature and a reactor length are selected, the shape of the optimal temperature profile can easily be
600
560
TIn(K)
520
480 0
8
16
24
32
40
L (m)
Fig. 3. Prediction of the shape of theoretical optimal profiles as a function of (Gg = 7.88 kg/s m2).
Tua and
reactor length
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Cocurrently cooled fixed-bed reactors determined. Therefore, the shape of the theoretical optimal profiles maximizing the outlet yield can be a priori known. For other optimization problems (e.g. to maximize the profit), the shape of the theoretical optimal thermal profiles could be predicted [-following the same procedure used to obtain eq. (28)] provided that these ideal trajectories end by an isotherm at the maximum allowable temperature value. To achieve the maximum feasible yield in an industrial unit, it is necessary to track the theoretical optimal temperature profiles closely. A design which would allow one to follow any optimal trajectory, although technically a complex solution, would be to arrange a number of different heat-transfer medium circuits so as to achieve a stepwise approximation of the optimal temperature profile (Eigenberger, 1992). However, a desirable solution should not require major changes in the reactor design. As a simpler solution, we propose to use the operational flexibility offered by the cocurrent coolant scheme to track the theoretical optimal trajectories and therefore obtain objective function values close to the optimal ones.
Attainable optimal temperature profiles The attainable temperature profile denomination will be used to refer to those thermal profiles that can be obtained by simulation of the fixed-bed reactor, represented by a plug-flow pseudo-homogeneous model, using as a coolant scheme the cocurrent configuration and different temperature values for the gas and coolant inlet temperatures. Given the selected TMa and reactor length values (for either Ex < Ee or Ex > Ep), the shape of the theoretical optimal profiles can be a priori known. Based on this information, the challenge is to define the proper operational conditions which allow to have attainable temperature profiles close (as much as possible) to the theoretical optimal ones. When the theoretical optimal temperature profile corresponds to an isothermal trajectory the cocurrent coolant scheme allows one to generate an attainable temperature profile capable of tracking the desired profile shape exactly. Therefore, the optimal objective function value could be reached. In fact, given the TM~ value, the parameters involved in either A or D (see the mathematical model section) e.g. Gg or Wc, can be set so that the eq. (26) is verified at Ti = TM,. The inlet coolant temperature for the isothermal operation is given by eq. (24). The maximum reactor yield can be calculated from eq. (27), for Ti = TMa and z = L. For the ethylene oxidation case, Fig. 4 shows the gas and coolant temperature profiles that maximizes ~/PLfor a TMa of 533 K. Figure 5 shows, for the special case of ethylene oxidation, a theoretical optimal profile (dashed line) which leads for a TMa = 541 K to the maximum allowable value of r/pL = 0.225. This profile (rising curve ending by the isotherm T = 541 K) cannot be exactly reproduced in a cocurrent cooled reactor. However, modifying operational conditions such as inlet temperatures or coolant flowrate it is possible to
54o
530
~ 520 ~.
5~0
50o
I 3
I 6
1 9
12
z(m) Fig. 4. Isothermal operation: theoretical optimal profile and simultaneously attainable optimal profile for a TMa= 533K (YAo= 0.06, Wc = 125 kg/s, Gg = 7.88 kg/sm 2, r/eL = 0.216).
550
53o
510
490
I 3
I 6
I 9
12
z(m) Fig. 5. Theoretical optimal profile (YA0= 0.06, Gg = 7.88 kg/sm 2, qeL = 0.2250) for TMa = 541 K and an attainable profile close to the optimal one (YAo= 0.06, Wc = 90 kg/s, Gg = 7.88 kg/sm 2, r/pL= 0.2245). find attainable temperature profiles very similar to the theoretical ones. The attainable optimal profiles could be obtained by using an optimization algorithm trying to track the theoretical optimal trajectories by means of changes in the operational conditions. To avoid such calculation effort, a T o o - To plane (see Fig. 1) could be very useful to establish the ranges of operational conditions that should not be explored. In fact, if the optimal profile is similar to that described in Figure 5, for example, regimes such as hot-spot profiles, maximum-minimum or decreasing profiles could not solve the optimization problem satisfactorily. Figure 5 shows the best attainable gas
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and coolant temperature profiles (solid lines) found following the procedure described above. The results indicate that the best attainable profile leads to a yield that is only 0.2% lower than the maximum theoretical value. The previous examples illustrate the capability of the cocurrent flow scheme to allow the generation of attainable temperature profiles very close (or identical) to the optimal theoretical ones. The two conventional designs, countercurrent scheme or constant coolant temperature, could not solve practical optimization problems as efficiently as the cocurrent configuration. Indeed, only hot-spot or decreasing profiles are possible when any of those configurations are used.
F~f FB,f
COMPARISON OF THE COCURRENT DESIGN WITH CONSTANT COOLANTTEMPERATURE ARRANGEMENTS In this section, the study is focused on the comparison of the cocurrent configuration with the constant coolant temperature case which is representative of the cooling arrangements commonly found in practice (boiling liquids as coolant medium, or liquids circulating at high flowrates). This comparison is performed to investigate whether the cocurrent design can improve the reactor performance, for situations of practical interest, in a novel way. This study is based on the case of ethylene oxide production. A schematic flow diagram for an ethylene oxide plant (oxygen-based oxidation) is shown in Fig. 6. As a common basis of comparison, constant
FA
FA,o FB,0
_1 Reactor Section
q
Fp
P
Product Separation
lit Fp=(Vv/Vm)ripF.,f undesired products
X~ FA FB
np =
(opLh
Fig. 6. Schematic flow diagram for an ethylene oxide plant (oxygen-based oxidation).
550 XAL=0.75
4 540
530
520 Wc, kg/s To, K
510
500 I 0
1 2 3 4 5
--~ 122.5 100.0 80.0 100.0
524.2 533.3 025.0 515.0 500.0
Tco, K
(o~)B
~PL
Tm~, K
524.2 503.2 500.4 496.7 502.6
0.703 0.709 0.712 0.714 0.713
0.212 0.216 0.210 0.221 0.219
544.5 533.3 536.5 5412 537.9
, 4
, 8
12
z (m) Fig. 7. Axial temperature profiles at conditions of constant outlet conversion (YA0= 0.06, Gg = 7.88 kg/s m2).
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Cocurrently cooled fixed-bed reactors molar flows of oxygen (Fa0) and ethylene (F~o) at the reactor inlet were assumed. Two different situations are considered: Constant outlet conversion
(XAL)
Figure 7 shows the typical temperature profile that a fixed-bed reactor exhibits when constant coolant temperature is assumed (curve 1), it presents a hotspot located near the reactor inlet. Curves 2-5 were obtained for different operating conditions under cocurrent coolant flow. All the curves verify the same outlet conversion xaL = 0.75. Operating at isothermal conditions (curve 2), a 75 % oxygen conversion can be reached with the minimum value of maximum temperature. Particularly, this temperature value (Tmax = Ti = 533.3 K) can be a priori known from eq. (21) setting XAL = 0.75 and z = L = 12 m. The cocurrent
coolant flow allows the same conversion to be obtained with temperature profiles showing very different shapes (e.g. curve 3 shows a minimum near the reactor entrance). The examples selected here present higher selectivities and simultaneously lower maximum temperature values than the reference case (curve 1). Bucalh (1991) and Krishna and Sic (1994) have pointed out the benefits of cocurrent cooling for the process of ethylene oxide manufacture. Particularly, Krishna and Sic (1994) have reported that improvements of selectivity by 1% in the manufacture of ethylene oxide can be extremely significant for process licensers. To analyse the influence of the selectivity on the process economy, the plant molar flows related to curves 1 and 4 of Fig. 7 are compared in Table 2. Curve 4 presents a reactor selectivity (i.e. plant yield) 1.6% higher than the reference case (curve 1). Due to
Table 2. Molar flows described in Fig. 6 for selected operations showed in Figs 7 and 8 Curve
ql,L
xaz
(treL)n
Fao
Fno
Fa,:
Fa.:
Fe
69 69
55.5 56.1
39.0 40.7
Case A: constant oxygen conversion per pass 1 of Fig. 7 4 of Fig. 7
0.212 0.221
0.75 0.75
0.703 0.714
92 92
1435 1435
Case B: constant reactor yield 1 of Fig. 8 0.212 0.75 4 of Fig. 8 0.212 0.69
0.703 0.726
92 92
1 4 3 5 6 9 . 0 5 5 . 5 39.0 1 4 3 5 6 3 . 6 5 3 . 7 39.0
550
rlPL=0.212
/ 530-/ L -~3
520
510
-~
500 0
1 2 3 4 5
WC, kg/s
To, K
TCo, K
(o~)B
XAL
T.~o K
~ 122.5 141.0 80.0 100.0
524.2 524.7 530.5 500.0 520.0
524.2 502.3 504.5 496.5 499.0
0.703 0,721 0,716 0.726 0.724
0.750 0.705 0.718 0.691 0.689
544.5 531.5 530.5 538.6 533.8
I 4
I 8
12
z (m) Fig. 8. Axial temperature profiles at conditions of constant reactor yield (Ya0 = 0.06, Go = 7.88 kg/sm2).
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the reaction stoichiometry (Table 1) the increase in selectivity requires a slight increase in Fn.f to maintain the reactor load constant. Therefore, the production rate (Fe) increases by 4.36%. Assuming $0.2/lb and $0.45/lb as the prices of ethylene and ethylene oxide, respectively, and considering a standard capacity plant of 100,000 t/yr (Miller, 1969), an improvement of 4.36% in the production rate would signify an increased revenue of the order of US$3 million per year.
Constant reactor yield (rleL) Figure 8 shows different temperature profiles giving the same reactor yield (r/eL = 0.212). Curve 1 is the reference case also shown in Fig. 7 (constant coolant temperature). Curves 2-5 correspond to thermal profiles obtained with cocurrent coolant flow. The specified reactor yield is achieved by operations 2-5 with higher selectivities and lower conversion than the base case. The maximum temperature values are lower than the hot-spot exhibited by curve 1. Particularly, curve 3 is an isothermal regime operating at Ti = 530.5 K, and represents the operation which leads to r/eL = 0.212 exhibiting the lowest maximum temperature value. Indeed, Fig. 3 shows that for a TM, = 530.5 K and L = 12 m, the optimal temperature profile which maximizes the reactor yield corresponds to an isothermal trajectory. For this reason, any thermal profile starting with an inlet gas temperature lower than 530.5 K has to surpass this TM, value in order to meet the specified value for the reactor yield, The plant molar flows corresponding to the curves 1 and 4 of Fig. 8 are compared in Table 2. Maintaining a constant reactor load and reactor yield implies to have the same production rate of the plant (constant Fe). Curve 4 allows one to reach the same production rate requiring lower values of the fresh feed. In fact, the same 39 k mol/h of P can be produced with a reduction of 3.2% in fresh ethylene and 7.8% in fresh oxygen with respect to the reference case. For a standard capacity plant of 100,000 t/yr of ethylene oxide, the saving only in fresh ethylene would imply a decrease in costs of the order of US$1 million per year.
trajectories, and consequently let us obtain the maximum value of reactor yield. The shape of theoretical thermal profiles that maximize the reactor yield can be established a priori. This is an important tool to define the appropriate operational conditions to obtain feasible thermal trajectories close to the ideal ones. The cocurrent flow scheme presents significant advantages when it is compared to the constant coolant configuration. The operational flexibility given by the cocurrent configuration could significantly improve the economics of the process. The present results were obtained assuming a plugflow pseudo-homogeneous model and exothermic first-order parallel reactions. As a further work, the advantages showed by the cocurrent design should be verified using more detailed reactor models. NOTATION
A
Go ] B
Aol cv,
c C
concentration (p/RT), kmol/m 3
Cp d
specific heat, kJ/kg K diameter, m
D
wc Cpc/ Dc E F G H
shell diameter, m activation energy, kJ/kmol molar flow, kmol/h specific mass flowrate, kg/m 2 s _
(-AUg)] (-AHD ]
AH k k¢
CONCLUSIONS
When exothermic first-order parallel reactions are carried out in cocurrently-cooled fixed-bed reactors seven thermal regimes can be achieved. They are: hot-spots, isothermal and pseudo-adiabatic operations, maximum-minimum and minimum-maximum profiles, cold-spot regimes, and continuously decreasing temperature trajectories. The conditions of existence of each of these theoretically feasible regimes are derived in the present work. It has been demonstrated that the operating conditions leading to different regimes can be accurately predicted. The diversity of thermal profiles offered by the cocurrent coolant scheme allows practical optimization problems to be solved very efficiently. In fact, when the reactor yield is wanted to be a maximum, this cooling arrangement can practically reproduce the ideal thermal
[=
kg k~ L p P r R S t, T Tc U W~ x y
heat of reaction, kJ/kmol reaction constant, kmol/kg kPa s thermal conductivity of cooling fluid, kJ/s m K gas thermal conductivity, kJ/s m K pre-exponential factor, kmol/kg kPa s reactor length, m partial pressure, kPa total pressure, kPa reaction rate, kmol/kg s universal gas constant, k J / k m o l K or kPa m3/kmol K (d T/dz), K/m number of reactor tubes gas temperature, K temperature of cooling fluid, K overall heat transfer coefficient, kW/m 2 K cooling mass flowrate, kg/s conversion molar fraction
Cocurrently cooled fixed-bed reactors Z 2"*
axial coordinate, m switching point, m
Greek letters reactor yield plant yield viscosity, kg/s m stoichiometric coefficient density, kg/m 3 selectivity Subscripts A limiting reactant b bed B reactant c coolant g gas i isothermal j reactant l limiting L at axial position z = L m relative extreme (maximum or minimum) ma minimum allowable max maximum temperature value of the profile T(z) Ma maximum allowable opt optimal p particle P desired product t tube X undesired product 0 at axial position z = 0 oo at axial position z ~ oo REFERENCES
Borio, D. O., Gatica, J. E. and Porras, J. A. (1989a) Wall-cooled fixed-bed reactors: parametric sensitivity as a design criterion, A.I.Ch.E.J. 35, 287-292. Borio, D. O., Bucalfi, V., Orejas, J. A. and Porras, J. A. (1989b) Cocurrently-cooled fixed-bed reactors: a simple approach to optimal cooling design. A.I.Ch.E.J. 35, 1899-1902.
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Borio, D.O., Bucalh, V. and Porras, J. A. (1995) Thermal regimes in cocurrently-cooled fixed-bed reactors. Chem. Engng Sci. 50, 3115-3123. Borman, P. C. and Westerterp, K. R. (1992) An experimental study of the selective oxidation of ethene in a wall cooled tubular packed bed reactor. Chem. Engng Sci. 47, 2541-2546. Borman, P. C. and Westerterp, K. R. (1995) An experimental study of the kinetics of the selective oxidation of ethene over a silver on ~t-alumina catalyst. Ind. Engng Chem. Res. 34, 49-58. Bucalfi, V. (1991) Operabilidad y optimizaci6n de reactores cataliticos tubulares. Ph.D. ~thesis, Department of Chemistry and Chemical Engineering, Universidad Nacional del Sur, Bahia Blanca, Argentina. Eigenberger, G. (1992) Fixed-bed reactors. In Ullmann's Encyclopedia of Industrial Chemistry, eds. B. Elvers, A. Hawkins and G. Schulz, Vol. B4, pp. 199-238. VCH Verlagsgesellschaft mbH, D-6940, Weinheim, Germany. Grzesik, M. and Skrzypek, J. (1983) Optimal temperature profiles for heterogeneous catalytic parallel reactions. 38, 1767-1773. Jackson, R., Obando, R. and Senior, M. G. (1972) The Control of Competing Chemical Reactions. Chemical Reaction Engineering, A.C.S. Krishna, R. and Sie, S. T. (1994) Strategies for multiphase reactor selection. Chem. Engng Sci. 49, 4029-4065. Miller, S. A. (1969) Ethylene and its Industrial Derivatives. Ernest Benn Limited, London, U.K. Nikolov, V. A. and Anastasov, A. I. (1989) A study of coolant temperature in an industrial reactor for o-xylene oxidation. A.I.Ch.E.J. 35, 511-513. Pontryagin, L. S., Boltyanski, V. G., Gamkrelidze, R. V. and Mischenko, E. F. (1962) The Mathematical Theory of Optimal Processes. Interscience, New York, U.S.A. Rose, L. M. (1981) Chemical Reactor Design and Practice, ed. S. W. Churchill. Elsevier, Netherlands. Westerterp, K. R. and Ptasinski, K. J. (1984) Safe design of cooled tubular reactors for exothermic, multiple reactions; parallel reactions--II. Chem. Engng Sci. 39, 245-252.