c 2006 Institute of Chemistry, Slovak Academy of Sciences DOI: 10.2478/s11696-006-0079-8
Multiple Steady States in a CSTR with Total Condenser: Comparison of Equilibrium and Nonequilibrium Models* Z. ŠVANDOVÁ, J. MARKOŠ**, and Ľ. JELEMENSKÝ
Institute of Chemical and Environmental Engineering, Faculty of Chemical and Food Technology, Slovak University of Technology, Radlinského 9, 812 37 Bratislava, Slovak Republic e-mail:
[email protected] Received 3 April 2006; Revised 14 June 2006; Accepted 11 July 2006
Comparison of equilibrium and nonequilibrium models of a CSTR with total condenser focused on the multiple steady states and dynamic behaviour was carried out. The steady-state behaviour of the model system, MTBE synthesis from methanol and isobutene in a reactive distillation column, was studied in terms of the input parameters, i.e. feed flow rate of methanol or butenes, reflux ratio, and mass of catalyst. The dynamic behaviour of the system during the start-up was investigated and perturbations of manipulated variables were found to cause transitions between the parallel steady states. Keywords: CSTR, equilibrium model, nonequilibrium model, multiple steady states, reactive distillation, MTBE
INTRODUCTION Reactive distillation is a multifunctional reactor concept, which combines separation via distillation with a chemical reaction. The expected benefits from such a synergetic interaction of the two operations carried out in one unit are an enhanced conversion due to the shift of chemical equilibrium, increased selectivity, elimination of hot spots by using the reaction heat for distillation, and overcoming of azeotropic limitations [1]. Various papers can be found regarding the reactive distillation processes and their advantages over the classical technologies with stand-alone production and separation units. A comprehensive overview was given by Taylor and Krishna [2]. The authors reported that the reactive distillation led to a complex interaction combining vapour-liquid equilibrium, vapourliquid mass transfer, intra-catalyst diffusion (for heterogeneously catalyzed processes), and chemical kinetics. Such interactions condition the generation of multiple steady states, which have been verified in laboratory and pilot plant units [2, 3]. Recently, various studies were focused on the possi-
ble multiple steady states in MTBE production plant [1, 3—12]. In Jacobs—Krishna configuration of the MTBE synthesis column multiple steady states were identified by changing the feed location [13]. Their appearance might have severe safety consequences, but more frequently technological problems, e.g. unsatisfactory low conversion or product purity, were observed. Mohl et al. [3] adverted to the similarities between the multiple steady states behaviour of a reactive distillation column and a continuous stirred tank reactor. In the present study, a CSTR with total condenser (Fig. 1) used for MTBE synthesis from methanol and isobutene was assumed. Total condenser ensured that all vapour leaving the reactor was condensed and only a part of it, depending on the preset value of the reflux ratio, was returned back to the reactor in a form of condensate. The objective of this paper was to compare the equilibrium (EQ) and nonequilibrium (NEQ) models of a CSTR with total condenser, focusing on the phenomena of multiple steady states. The steady-state behaviour of the system was studied in terms of input
*Presented at the 33rd International Conference of the Slovak Society of Chemical Engineering, Tatranské Matliare, 22—26 May 2006. **The author to whom the correspondence should be addressed.
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Chem. Pap. 60 (6) 432—440 (2006)
COMPARISON OF EQUILIBRIUM AND NONEQUILIBRIUM MODELS
Fig. 2. The equilibrium model concept. Fig. 1. Schematic diagram of the CSTR with total condenser.
parameters, i.e. feed flow rate of methanol or butenes, reflux ratio, and mass of catalyst. The dynamic behaviour during start-up of the CSTR with total condenser was studied and perturbations of manipulated variables were found to cause transitions between the parallel steady states. THEORETICAL To simulate reactive distillation columns, two distinct approaches are possible: the equilibrium (EQ) and nonequilibrium (NEQ) models. The equilibrium model includes the assumption that the streams leaving the reactor are in equilib-
rium with each other [2, 14—17]. A schematic diagram of the equilibrium concept is shown in Fig. 2. The equations describing the equilibrium model are the component Material balances, equations of phase Equilibrium, Summation equations, and Heat balance of the reactor (MESH equations) [15]. Schematic diagram of the nonequilibrium concept is shown in Fig. 3. Vapour and liquid feed streams are contacted in the reactor and allowed to exchange mass and energy across their common interface represented in the diagram by a vertical wavy line. The description of the interphase mass transfer, in either fluid phase, is almost invariably based on the rigorous Maxwell— Stefan theory for calculation of the interphase heat and mass transfer rates [2, 6, 15—20]. The equations describing the nonequilibrium model
Table 1. Assumptions Used in the Model Development Equilibrium model for CSTR
Nonequilibrium model for CSTR
Phase equilibrium in the reactor (i.e. vapour— liquid equilibrium is attained). Vapour and liquid in the reactor are perfectly mixed. The vapour and liquid in the reactor have the same temperature and concentration as the vapour and liquid leaving the reactor. Reversible reaction proceeds only in the liquid phase and its course can be described by appropriate kinetic equations. The hold-up per CSTR is equal to the liquid hold-up in the CSTR (the molar vapour holdup is negligible compared to the molar liquid hold-up). Constant volumetric hold-up in the CSTR. The molar hold-up in the condenser is negligible. Constant pressure in the reactor. The condenser is assumed to be at equilibrium. No reaction proceeds in the condenser. The reflux from the condenser is returned back to the CSTR at the condensation temperature.
Vapour—liquid equilibrium is achieved only at the vapour— liquid interface. The vapour and liquid bulks on each side of the interface are perfectly mixed. The liquid and vapour in the bulk phases have the same concentrations as the corresponding streams leaving the reactor. Reversible reactions within the interface are ignored and take place in the liquid bulk and their course can be described by appropriate kinetic equations. The hold-up per CSTR consists of the liquid hold-up and vapour hold-up. Constant volumetric liquid hold-up and vapour hold-up in the CSTR. The storage capacity for mass and energy in the adjacent films is negligibly small compared to that in the bulk fluid phases (the interfacial transfer rates can be calculated from the quasistationary interfacial transfer relations). The molar hold-up in the condenser is negligible. Constant pressure in the reactor. The condenser is operated at equilibrium. No reaction proceeds in the condenser. The reflux from the condenser is returned back to the CSTR at the condensation temperature.
Chem. Pap. 60 (6) 432—440 (2006)
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Z. ŠVANDOVÁ, J. MARKOŠ, Ľ. JELEMENSKÝ
where a is the total interfacial area, xi and yi represent mole fractions of the species i in the bulk liquid and vapour, respectively, and Nt is the total mass transfer rate NI Nt = Ni (3) i=1
The diffusion fluxes in the liquid, J L , and vapour, J , phases, relative to the corresponding averaged molar velocities are given by V
Fig. 3. The nonequilibrium model concept.
are the Material balances, Energy balances, Rate equations, Summation equations, Hydraulic equation, and eQuilibrium relations of the reactor, i.e. MERSHQ equations. Mathematical equations describing both the EQ and NEQ models, developed under the assumptions summarized in Table 1, are presented in Table 2. MESH equations used to describe the CSTR were combined with the equation for molar hold-up in the liquid phase (the molar vapour hold-up was negligible compared to the molar liquid hold-up) [14], the equation for estimation of the condensate temperature, and kinetic equations. The nonequilibrium model of a distillation column without chemical reaction presented by Krishnamurthy and Taylor [19, 20] including the chemical reaction terms was adopted for the rate-based model of reactor for MTBE synthesis. The nonequilibrium model included the assumption of three different temperatures for the liquid, gas, and gas—liquid interface (T L , T G , T I ). Thus, separate material and energy balance equations were written for each phase separately and the molar hold-up in the vapour phase was also considered. These balance equations were linked by material and energy balances around the interface [15]. The equilibrium relations were used to associate compositions on each side of the phase interface. The interface composition and temperature was, therefore, determined as a part of nonequilibrium simulation of the reactor. MERSQ equations (omitting the use of Hydraulic equation due to the constant pressure assumption) were combined with equations for the molar hold-up for both phases in the CSTR, the equation for estimation of the condensate temperature, and kinetic equations like in the EQ model. The nonequilibrium model used two sets of mass transfer Rate equations (Table 2). The mass transfer rate in each phase (NiL , NiV ) was computed from diffusive and convective contributions [17]
(4)
V V a y − yI J = cV t k
(5)
cLt and cV t being the mixture molar densities for the liquid and vapour phase, respectively. The matrices of mass transfer coefficients [k] were computed by the Maxwell—Stefan approach [15]. Note that there were only (NI − 1) × (NI − 1) elements in the [k] matrices and, therefore, only (NI − 1) equations for the mass transfer rate. A necessary complementary equation was represented by the energy balance for the interface, presenting continuity of the energy fluxes across the V-L interface (Table 2). The heat transfer rates for the two phases E L , E V were defined by the equations NI Ni HiL E L = hL a T I − T L +
(6)
i=1
E V = hV a
NI V εV I − T Ni HiV + T exp εV − 1 i=1
(7)
where hL and hV are the liquid and vapour heat transfer coefficients, respectively, HiL and HiV partial molar enthalpies of the component i in both phases, and εV heat transfer rate factor [20] defined as εV =
NI
V Ni Cpi / hV a
(8)
i=1 V Cpi being the heat capacity of the species i in the bulk vapour. For calculation of the vapour phase heat transfer coefficients the Chilton—Colburn analogy between the mass and heat transfer was used [17, 20]
V 2/3 V V h V = cV t κav Cpm Le
(9)
+ xi Nt
(1)
The liquid phase heat transfer coefficients were computed using penetration model [17]
NiV = aJiV + yi Nt
(2)
L 1/2 L hL = cLt κLav Cpm Le
NiL
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L J = cLt k L a xI − x
=
aJiL
(10)
Chem. Pap. 60 (6) 432—440 (2006)
COMPARISON OF EQUILIBRIUM AND NONEQUILIBRIUM MODELS
Table 2. Equilibrium and Nonequilibrium Model Equations for the CSTR with Total Condenser Equilibrium model for CSTR
Nonequilibrium model for CSTR
The Material balance equations: the total material balance
The Material balance equations: the total material balances for each phase
dU = dt
NF
NR
Ff + mc
f =1
rj
j=1
NI
νj,i
+ LTC − V − L
i=1
L NF
dU L FfL −mc = L− dt f =1
material balance for the component i NR
f =1
j=1
d U L xi
K i x i − yi = 0
xi = 0,
rj
j=1
νj,i
−LTC −
i=1
NI
NiL
i=1
V NF
NI
f =1
i=1
dt
L
= Lxi −
NI
1−
i=1
d U V yi
yi = 0
FfL xf,i −mc
NR
(rj νj,i )−LTC xTC,i −NiL
j=1
dt
i=1
The Heat balance*
NF f =1
The Summation equations in each phase
1−
NI
material balance for the component i in both phases
The phase Equilibrium relations for each component
NI
dU V FfV + NiV =V − dt
d (U xi ) Ff xf,i +mc (rj νj,i )+LTC xTC,i −V yi −Lxi = dt NF
NR
V
= V yi −
NF
FfV yf,i + NiV
f =1
The Energy balance equations for each phase and around the interface*
d(U H L ) = Ff Hf + mc rj (−∆r Hj ) + dt NR
NF
f =1
j=1
+ LTC HTC − V H V − LH L + Q
L NF
NR
f =1
j=1
d(U L H L ) FfL HfL − mc rj (−∆r Hj ) − = LH L − dt − LTC HTC − QL − E L
The equation for the molar hold-up in the liquid phase
V NF
d(U V H V ) FfV HfV − QV + E V = V HV − dt
NI
0 = U L − VrL /
(xi /ρi )
f =1
i=1
The equations for the total condenser: reflux ratio from the total condenser to the CSTR LTC = V (Rx /(Rx + 1)) composition of the condensate xTC xTC,i = yi temperature of the condensate TTC 0=P−
NI
i=1
Pi0 (TTC ) xTC,i γi
The initial conditions for t = 0 :
EV − EL = 0 The Rate equations Ni − NiL = 0, Ni − NiV = 0 The Summation equations for each phase and the side of interface: bulk
NI
NI
xi − 1 = 0
i=1
interface film
NI
yi − 1 = 0
i=1
xIi
− 1 = 0,
i=1
xi = x0i , yi = yi0 , T = T 0 , V = V 0 , L = L0 ,
NI
yiI − 1 = 0
i=1
The phase eQuilibrium at the interface
U = U 0 , TTC = (TTC )0
Ki xIi − yiI = 0 The equations for the molar hold up in each phase
L
0=U −
VrL
NI
(xi /ρi )
0 = U V − P VrV / RT V
i=1
Chem. Pap. 60 (6) 432—440 (2006)
435
Z. ŠVANDOVÁ, J. MARKOŠ, Ľ. JELEMENSKÝ
Table 2. (continued) Equilibrium model for CSTR
Nonequilibrium model for CSTR The equations for the total condenser: reflux rate from the total condenser to the CSTR LTC = V (Rx /(Rx + 1)) composition of the condensate xTC,i = yi temperature of the condensate 0 = P −
NI
i=1
Pi0 (TTC )xTC,i γi
The initial conditions, for t = 0
xi = x0i , yi = yi0 ,
xIi
L = L0 , T L = T L
=
TV
I 0
=
xi , yiI = 0 TV , TI
I 0
,V =V
= T
UL
yi
I 0
,
0,
=
0 UL ,
UV
=
0
,
0 UV ,
Ni = Ni0 , TTC = (TTC )0 * Reference state: pure component in the liquid phase at 273.15 K. L where κV av , κav are the averaged values of the binary mass transfer coefficients for each phase, LeV , LeL V L Lewis numbers for each phase, and Cpm , Cpm heat capacities of the vapour and liquid phase mixtures. All equations presented in Table 2 could be used for simulation of either steady state or dynamic regime of the CSTR reactor equipped with total condenser. However, in this study only equations necessary for simulations (the number of equations is equal to the number of variables) were employed. The equilibrium model contained 2NI + 5 unknown variables. Therefore, the same number of independent equations was solved, i.e. Material balance equations (NI ), Equilibrium equations (NI ), Summation equations for both phases (2), the enthalpy (H) balance (1), equation for the molar hold-up in the liquid phase (1), and equation of the condensate temperature (1). For a nonequilibrium model of the CSTR with total condenser, there were altogether 2NI + 8 unknown variables. In this case, 5NI + 8 equations were solved including Material balance equations for the bulk liquid and bulk vapour phases (2NI + 2), Energy balance equations for the liquid and vapour phases (2), Energy balance around the interface (1), mass transfer Rate equations for the liquid and vapour phases (2NI − 2), Summation equations for the liquid and vapour films (2), eQuilibrium equations for the interface (NI ), equations for the molar hold up in both phases (2), and equation of the condensate temperature (1). The mathematical models presented above, consisted of a set of ordinary differential and algebraic equations, which were solved by a proprietary FORTRAN code, using a modified Powel hybrid algorithm and a finite-difference approximation to the Jacobian from the IMSL library to solve a system of nonlinear equations for the reactor steady-state simulation, Continuation and Stability Analysis Package (CONT) [21—24] to identify the multiple steady states, and subroutine DDASKR [25] to solve a combined system
436
of differential and algebraic equations derived for the dynamic simulation. Case Study: MTBE Production As a model system, the MTBE reaction system was chosen. The reaction of 2-methylpropene (IB) with methanol (MeOH) to give methyl t-butyl ether (MTBE) is catalyzed by strongly acidic ion-exchange resins (CH3 )2 C—CH2 + CH3 OH ⇔ (CH3 )3 COCH3
(A)
The reaction rate equation, the reaction rate constant, and the equilibrium constant variation with temperature were reported by Rehfinger and Hoffmann [26]. Possible competitive reactions were ignored. The reaction is usually carried out in the presence of inert components (e.g. but-1-ene) resulting from the upstream isobutene production. The vapour—liquid equilibrium of the reaction mixture was calculated using the UNIQUAC model with the binary interaction parameters given by Rehfinger and Hoffmann [26] (for binary systems comprising MeOH, IB, and MTBE) or withdrawn from the HYSYS 2.1 (all binaries containing but-1-ene). The gas phase was supposed to behave ideally. The Antoine equation was used to calculate the vapour pressure of pure components. Two feed streams: liquid methanol (320 K) and a mixture of butenes in the vapour phase (350 K) were assumed entering the CSTR. The vapour phase introduced into the reactor consisted of 35.58 % 2methylpropene and 64.42 % but-1-ene. The working pressure was fixed at 1110 kPa. At a standard operating point, the molar flow rates of methanol and the mixture of butenes were 505 kmol h−1 and 1980 kmol h−1 , respectively. The reflux ratio was set to 15. The volume of the liquid phase and the vapour phase in the reactor was 5 m3 and 3 m3 , respectively. The catalyst
Chem. Pap. 60 (6) 432—440 (2006)
COMPARISON OF EQUILIBRIUM AND NONEQUILIBRIUM MODELS
600
40 35
500
30
400
mc / kg
Rx
25 20 15 10
300 200
5 0 490
495
500
505
510
L
515
520
100 150
525
200
250
Fig. 4. Bifurcation diagram in the parametric plane MeOH feed flow rate and reflux ratio for mc = 500 kg and F V = 1980 kmol h−1 . Regions of the multiple steady states predicted by the EQ or NEQ model and the operating point ( ).
400
450
500
550
-1
Fig. 5. Bifurcation diagram in the parametric plane MeOH feed flow rate and mass of the catalyst for Rx = 15 and F V = 1980 kmol h−1 . Regions of the multiple steady or NEQ model and states predicted by the EQ the operating point ( ).
The identification of the multiple steady states locus in two-parameter planes (remaining operation parameters were kept constant) was performed by the algorithm CONT [21—24]. Two examples of the bifurcation diagram are presented in Figs. 4 and 5. The bifurcation diagrams contain the information about the limit points and the multiplicity intervals. White area in these diagrams indicates that for the combination of both investigated parameters only one steady state is possible, meanwhile, the filled area represents the combinations of investigated parameters, for which three steady states are possible. The boundary lines of the filled area present the limit points in the investigated parametric plane. If the MeOH feed flow rate and the reflux ratio were chosen as the parameters of interest, in this parametric plane important differences of the multiplicity locus were observed for the CSTR described by the EQ model and NEQ model (Fig. 4), respectively. The EQ model predicted the zone of multiple steady states noticeably larger than that calculated by the NEQ model. These zones were not overlapped in the investigated range of chosen parameters (Fig. 4). On the other hand, by choosing the MeOH feed flow rate and the mass of catalyst as the investigated parameters for the second bifurcation diagram, the EQ- and NEQ-based multiplicity regions were almost identical (Fig. 5). In fact, the multiplicity zone predicted by the EQ model was a little bit larger than that obtained by the NEQ model. According to the analysis performed using EQ
0.9 0.8
XMeOH
RESULTS AND DISCUSSION
1.0
mass was 500 kg and its ion-exchange capacity was 4.54 meq (H+ ) g−1 . During dynamic simulations, the specific heat capacity of the catalyst was also taken into account.
Chem. Pap. 60 (6) 432—440 (2006)
350
F / (kmol h )
F / (kmol h )
300 L
-1
0.7 0.6 0.5 490
500
510 L
520
530
540
-1
F / (kmol h ) Fig. 6. Solution diagram for the MeOH feed flow rate at Rx = 15, mc = 500 kg, and F V = 1980 kmol h−1 . Stable (solid lines) and unstable (dotted lines) steady states predicted by the EQ (thick lines) and NEQ (thin lines) models and the operating point ( ).
model, the CSTR with a total condenser was operated within the region of multiple steady states. On the other hand, the NEQ model prediction situated the assumed operating point within the domain of only one steady state (Fig. 6). Each of the three possible steady states predicted by the EQ model, is characterized by different methanol conversion. The higher and lower (stable) steady states predicted by the EQ model could be possible operating points of the reactor. The steady state obtained by the NEQ model is very similar to the upper steady state given by the EQ model (Fig. 6). The possibility of multiple steady states could strongly influence the CSTR behaviour if some disturbances of the input parameters occur. For this reason, the disturbances of the MeOH feed flow rate were studied (Fig. 7). At a time of 1 h, the MeOH feed flow rate was suddenly increased from the original value of 437
Z. ŠVANDOVÁ, J. MARKOŠ, Ľ. JELEMENSKÝ 540 0.9
XMeOH
510 0.7 500 0.6
490
-1
480
0.5 0.0
L
520
0.8
F / (kmol h )
530
0.5
1.0
1.5
2.0
2.5
3.0
470 3.5
t/h Fig. 7. Conversion changes predicted by the EQ (thick solid line) and NEQ (thin solid line) models caused by the step change of the MeOH feed flow rate from 505 kmol h−1 to 520 kmol h−1 and back. Dashed line represents MeOH feed flow rate.
1.0 0.9
XMeOH
0.8 0.7 0.6 0.5 1700
1800
1900 V
2000
2100
2200
-1
F / (kmol h ) Fig. 8. Solution diagram for the butenes feed flow rate at Rx = 15, mc = 500 kg, and FL = 505 kmol h−1 . Stable (solid lines) and unstable (dotted lines) steady states predicted by the EQ (thick lines) and NEQ (thin lines) models and the operating point ( ).
0.8
1750
0.7
1500
0.6
1250
0.5
1000
0.4
750
0.3
-1
2000
V
0.9
F / (kmol h )
XMeOH
1.0
500
0.2
250
0.1
0 0.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
t/h Fig. 9. The reactor start up predicted by the EQ (thick solid line) and NEQ (thin solid line) models assuming the increase of the butenes feed flow rate from 200 kmol h−1 to 1980 kmol h−1 . Dashed line represents MeOH feed flow rate.
438
505 kmol h−1 to 520 kmol h−1 . After a certain time a new steady state of the CSTR operation was established characterized by lower MeOH conversion using both EQ and NEQ models (Fig. 6). Then, the original flow rate of MeOH was re-established. At given conditions different behaviour of the CSTR was observed taking into account the equilibrium or nonequilibrium models. After returning the MeOH flow rate to the original operation value, the system was stabilized in a lower stable steady state considering the EQ model. However, if the NEQ model was used to simulate the CSTR, only one steady state was available for the reactor operation. Therefore, after perturbation the MeOH conversion returned to the original steady state (Fig. 7). Considering Rx = 15, mc = 500 kg, and F L = 505 kmol h−1 , the solution diagram, so called steady-state diagram, representing variation of the MeOH conversion with the molar flow rate of butanes, is presented in Fig. 8. Assuming the butenes feed flow rate of 1980 kmol h−1 (dashed vertical line in Fig. 8) the EQ model predicted three steady states for the CSTR operation, while forecast based on the NEQ model showed only one stable steady state. Fig. 9 represents the reactor start-up considering a gradual increase of butenes feed flow rate. Simulation of the MTBE production based on the EQ model revealed that the start-up procedure switched the reactor to the lower stable steady state characterized by lower conversion of MeOH (61.87 %, Fig. 8). The CSTR operation described by the NEQ model corresponded to the steady state, which is characterized by higher conversion of MeOH (Fig. 9) as only one steady state was predicted for given reactor operation conditions (Fig. 8). The multiple steady states behaviour was found assuming both EQ and NEQ models of the CSTR with a total condenser used for MTBE synthesis. However, localization of the zones of multiple steady states in the two-parameter plane predicted by the EQ and NEQ models could be completely different. At the same time the multiple steady states zone was often smaller if the interphase mass and heat transfer resistances were taken into account (Fig. 4). The EQ model is simpler requiring a lower number of the model parameters. On the other hand, the assumption of equilibrium between the vapour and liquid streams leaving the reactor could be difficult to fulfil, especially if some perturbations of the process parameters occur. The NEQ model takes into account the interphase mass and heat transfer resistances, thus improving the description of the modelled system. On the other hand, it is important to point out that prediction of the reactor behaviour is strongly dependent on the quality of NEQ model parameters, depending on the equipment design.
Chem. Pap. 60 (6) 432—440 (2006)
COMPARISON OF EQUILIBRIUM AND NONEQUILIBRIUM MODELS
Acknowledgements. This work was supported by the Science and Technology Assistance Agency under the contract No. APVT-20-000804.
SYMBOLS a c Cp D E F h H ∆r H J [k] Ki L Le LTC mc NF NI NR N P Pi0 Q r R Rx t T U V VrL VrV XMeOH x y
total interfacial area m2 molar concentration mol m−3 heat capacity J mol−1 K−1 diffusion coefficient m2 s−1 energy transfer rate J s−1 feed stream mol s−1 heat transfer coefficient J s−1 m−2 K−1 molar enthalpy J mol−1 heat of reaction J mol−1 molar diffusion flux relative to the molar average velocity mol m−2 s−1 matrix of the multicomponent mass transfer coefficients m s−1 vapour-liquid equilibrium constant for component i liquid flow rate mol s−1 −1 −1 −1 Lewis number (= λρ Cp D ) liquid flow rate from the total condenser to the reactor mol s−1 mass of catalyst kg number of feed streams number of components number of reactions mass transfer rate mol s−1 pressure of the system Pa vapour pressure of pure component i Pa heating rate J s−1 reaction rate mol kg−1 s−1 gas constant J mol−1 K−1 reflux ratio time s temperature K molar hold-up mol vapour flow rate mol s−1 volumetric liquid hold-up in the CSTR m3 volumetric vapour hold-up in the CSTR m3 conversion of methanol mole fraction in the liquid phase mole fraction in the vapour phase
Greek Letters ε γ κ λ ν ρ
heat transfer rate factor activity coefficient binary mass transfer coefficient m s−1 −1 thermal conductivity W m K−1 stoichiometric coefficient molar density mol m−3
Superscripts 0
initial conditions
Chem. Pap. 60 (6) 432—440 (2006)
I L V
referring to the interface referring to the liquid phase referring to the vapour phase
Subscripts av f i j m t TC
averaged value feed stream index component index reaction index mixture property referring to the total mixture referring to the total condenser REFERENCES
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