Ind. Eng. Chem. Res. 1997, 36, 2031-2040
2031
Longitudinal Mass and Heat Dispersion in Tubular Reactors Arno H. Benneker, Alexander E. Kronberg, and K. Roel Westerterp* Chemical Reaction Engineering Laboratories, Department of Chemical Engineering, Twente University of Technology, P.O. Box 217, 7500 AE Enschede, The Netherlands
The wave model for longitudinal dispersion, presented in previous papers as an alternative to the axially dispersed plug flow model, is extended to isothermal and adiabatic chemical reactors with reaction rates described by arbitrary expressions. The extended wave model is applied to the calculation of tubular reactors with three well-known types of chemical reaction: the consecutive reaction, the exothermic first-order reaction in an adiabatic reactor, and the autocatalytic reaction. Numerical solutions of the two-dimensional equations describing laminar flow tubular reactors are used for testing the validity of the wave model and the axially dispersed plug flow model. A special type of reactor is treated to stress the fundamental differences between the wave model and the dispersed plug flow model. It is confirmed that the extended wave model has a much wider region of validity than the dispersed plug flow model and is a suitable tool in reactor modeling. Introduction The exact description of most apparatus in chemical reaction engineering in terms of the basic equations of change is either impossible or leads to very complex mathematical problems. Therefore, for the description of most chemical reactors, we have to rely on simplified models which take into account the most important features of the problem at hand. From the practical point of view, one-dimensional models are preferred over multidimensional models: we do not need to know the velocity profile, which is often unknown and difficult to obtain, and they are simpler mathematically. However, it is difficult to obtain adequate cross-sectionally averaged equations for systems with significant transverse gradients in concentrations and temperature. The commonly encountered one-dimensional model for contactors and chemical reactors is the axially dispersed plug flow model, also called the standard dispersion model (SDM). For constant physical properties and a single reaction, the equations of the SDM are
∂cs ∂2cs ∂cs +u + q ) De 2 ∂t ∂x ∂x
(1)
∂Ts ∂2Ts ∂Ts Fcp + Fcpu + ∆Hq ) λe 2 ∂t ∂x ∂x
(2)
Many theoretical investigations on different types of chemical reactors were based on this model, see, e.g., Levenspiel (1972) or Westerterp et al. (1987). The parabolic mass- and heat-transport equations are sufficiently appropriate for the description of various problems. Nevertheless, the assumptions in the SDM that axial mass and heat dispersion obey Fickian- and Fourier-type equations are very questionable, and the model is not able to describe certain simple experimental results even qualitatively; see Westerterp et al. (1995b). Physical arguments require that the differential equations of nature should be hyperbolic; see Mu¨ller and Ruggeri (1993). Hyperbolic models for momentum and heat and mass transfer were proposed * Author to whom correspondence should be addressed. Fax: +31-53-4894738. E-mail:
[email protected]. S0888-5885(96)00713-0 CCC: $14.00
long ago originally by Maxwell (1867), Fock (1926), and Cattaneo (1958). But the use of hyperbolic instead of parabolic equations did not lead to an essential change of the results for those heat- and mass-transfer problems, where the dispersive transport was governed by the molecular diffusivity and heat conductivity. Only for the description of certain, rather exotic situations, when the time of interest is very short, was the necessity to use hyperbolic instead of parabolic equations substantiated; see the reviews of Joseph and Preziosi (1989, 1990) or O ¨ zis¸ ik and Tzou (1994). This situation definitely changes when hydrodynamical mixing is the governing mechanism of mass and heat dispersion as commonly encountered in chemical reactors. Here the time, distance, and velocity scales are absolutely different from those of molecular diffusion and heat conduction and the difference between hyperbolic and parabolic models may be essential. Therefore, Westerterp et al. (1995a, 1996) have developed a new hyperbolic model for longitudinal dispersion in flow-through contactors and chemical reactors as an alternative to the commonly used SDM. A qualitative analysis of the proposed wave model has been made in a subsequent paper; see Westerterp et al. (1995b). Its applicability to describe isothermal laminar flow tubular reactors with simple single-variable reaction rate expressions has been studied by Kronberg et al. (1996). The obtained wave model equations for mass dispersion are
∂cj ∂cj ∂j +u j + + qj ) 0 ∂t ∂x ∂x
(3)
(1 + τ ∂q∂c)j + τ ∂t∂j + τ(uj + u ) ∂x∂j ) -D ∂x∂cj a
e
(4)
Equation 3 is the standard conservation equation, and eq 4 is the governing equation for the dispersion flux. The dispersion flux j has been introduced as a second independent state variable in the wave model, in addition to the concentration averaged over the cross section. Its value represents the deviation from a uniform transverse concentration distribution. If the concentration becomes uniform over the cross section, the dispersion flux diminishes to zero independently of the variation of the concentration in the axial direction. In comparison to the SDM, two additional parameters appear: the relaxation time τ, which is the character© 1997 American Chemical Society
2032 Ind. Eng. Chem. Res., Vol. 36, No. 6, 1997
istic time necessary to effectuate a transverse equalization of concentration, and the parameter of velocity asymmetry ua, which characterizes the possible difference in the positive and negative fluctuations in the velocities responsible for the longitudinal dispersion. The essence of the wave model is that it embodies the ideas of both a diffusion-like processswhere frequent exchanges take place between fluid elements with different longitudinal velocitiessand a pure convective processswhere spreading is caused by the different axial velocities only. The SDM represents only a diffusion-like process. An important feature of the wave model is the presence of the consumption term q in the governing equation (4) for the dispersion flux which becomes important for large values of ∂q/∂c. The appearance of this term was physically explained by Westerterp et al. (1995a). Equation 4 has been obtained for the isothermal case where the chemical reaction rate is a function of but one variable: the concentration of the species at hand. However, in many chemical apparatus, the chemical reactions cannot be described by such simple rate expressions. Usually the rate of reaction of a species depends also on the concentration of other components and on the temperature, since most reactors operate nonisothermally. Therefore, for practical applications, a generalization of the wave model to describe both mass and heat transfer is desirable. In this article, the wave model is extended to isothermal and adiabatic chemical reactors with reaction rates described by arbitrary expressions. Three well-known examples are elaborated using the extended wave model: consecutive first-order reactions, an exothermic first-order reaction in an adiabatic reactor, and an autocatalytic reaction. The advantages of the extended wave model compared to the traditional one-dimensional models for the design and analysis of such types of reactors are demonstrated. Problems admitting a numerical solution, which can be considered to be the exact solution, are used as examples to test the validity of the different models. It is shown that the wave model is able to predict the concentration and temperature profiles adequately in a wide range of conditions, whereas the SDM fails to do so when a correct description of upstream transport of heat and mass is required or when the chemical reaction time is comparable to or shorter than the relaxation time τ. Generalization of the Wave Model for Longitudinal Dispersion To derive a one-dimensional equation, we will start from the three-dimensional equations of change for a chemical reactor as in the previous paper by Westerterp et al. (1995a). Let us assume that the concentration and temperature fields are described by the well-known equations
∂ci ∂ci + u(y,z) + qi ) ∇‚(Dt,i∇ci) ∂t ∂x ∂T Fcp
∂t
+ Fcpu(y,z)
(5)
n
∂T + ∂x
Hiqi ) ∇‚(λt∇T) ∑ i)1
(6)
species i (i ) 1, 2, ..., n), and eq 6 is the energy balance. The boundary conditions of eqs 7 and 8 imply no mass and heat fluxes to the reactor wall (adiabatic reactor). qi is the reaction rate of species i and depends on the concentration of other components and temperature. Equations 5 and 6 as parabolic equations may not be appropriate for the description of three-dimensional concentration and temperature fields when the mechanical dispersion is the predominant mechanism of transverse mixing; see Stewart (1965). However, at the moment, we have no reasonable alternatives for the description of multidimensional transport. Moreover, in our consideration, we are not interested in the detailed, three-dimensional description of mass and heat propagation. For our purpose, eqs 5 and 6 seem to be an adequate model because they contain the main features of the longitudinal mass and heat dispersion. They include a nonuniform axial velocity of the fluid described by the function u(y,z) and mass and heat exchange between the fluid elements of different velocities described by the right-hand sides of these equations. The axial mass diffusivity and axial heat conductivity are neglected on the assumption that the axial mixing of mass and heat is completely dominated by nonuniform convection, so the operator ∇ acts only in the y,z plane. Averaging eqs 5 and 6 over the cross section perpendicular to the direction of the flow, we obtain
∂ci ∂ji ∂ci +u j + + qi ) 0 ∂t ∂x ∂x ∂T h Fcp
∂t
+ Fcpu j
∂jH
∂T h + ∂x
(9)
n
+ ∂x
Hiqi ) 0 ∑ i)1
(10)
where
j )ci ji ) (u - u
(11)
j )FcpT jH ) (u - u
(12)
Here an overbar on a quantity denotes its cross-sectionally averaged value. Equations 9 and 10 contain the unknown variables ci, T h , ji, and jH. In order to have a closed set of equations, we additionally need n + 1 equations which relate the dispersion fluxes ji and jH with the average concentrations ci and temperature T h. In order to use eqs 9 and 10 and to obtain the additional equations from eqs 5-8, the consumption rate expressions must be simplified, in particular, to express the average consumption rate qi through ck, T h , jk, and jH (k ) 1, 2, ..., n). In the case when qi(c1,c2,...,cn,T) is not a homogeneous function of the first degree, the average consumption rate qi cannot be related directly to ck and T h . If the transverse variations in the concentrations and the temperature are small compared to the mean concentrations and mean temperature, we may approximate the consumption rate with a Taylor series expansion around the mean concentrations and mean temperature in ck - ck and T - T h:
with the boundary conditions
Dt,in‚∇ci ) 0 on ∂A
(7)
λtn‚∇T ) 0 on ∂A
(8)
Equation 5 represents the component mass balance for
qi(c1,...,cn,T,x,t) ) qi(c1,...,cn,T h ,x,t) + n ∂q ∂qi i (c1,...,cn,T (c1,...,cn,T h )(ck - ck) + h )(T - T h) ∂T k)1 ∂ck (13)
∑
Ind. Eng. Chem. Res., Vol. 36, No. 6, 1997 2033
in which we display the terms only through ck - ck and T-T h to the first power. In this case,
qi ) qi(c1,...,cn,T h ,x,t) and the approximation becomes exact if qi is a homogeneous function of the first degree. Applying the procedure as described in the Appendix of Westerterp et al. (1995a)swith the assumption of constant transverse mass and heat dispersion coefficients Dti and λtsresults in the following n + 1 equations for the dispersion fluxes:
ji + τi
(
∂qi
n
∑ k)1 ∂c
jH + τH
(
n
jH + τi
Fcp ∂T
k
n
)
1 ∂qi
jk +
∂qi
Hi ∑ ∑ i)1 k)1 ∂c
jk +
k
1 ∂qi Fcp ∂T
∂ji ∂t
)
+ τi(u j + ua)
)
∂x ∂ci -Dei (14) ∂x
jH + τH
τH(u j + ua)
∂ji
∂jH + ∂t
∂jH ∂x
∂T h ) -λe
∂x
(15)
This analysis provides us the values of the transport coefficients appearing in the one-dimensional equations:
j )g1i Dei ) -(u - u
g1i2 τi ) Dei
(16)
number Le ) FcpDi/λ is about one. For liquid systems, Le ≈ 0.1-0.01 and the relaxation time and dispersion coefficient for heat transfer will be approximately a factor 10-100 smaller than those for mass transfer. Above, only area-averaged concentrations and temperatures are considered. In practice, the bulk or mixed cup concentrations and temperatures may be of interest: these are related to the area-averaged concentrations and the temperatures by cbi ) cji + ji/u j and Tb ) T h + jH /u j. Examples Consecutive First-Order Reactions. Here, the first-order consecutive reactions A f P f X with the reaction rate equations
qA ) -kPcA
are treated. For the sake of simplicity, the reactions are carried out isothermally and the axial dispersion coefficients and the relaxation times for A and P are considered to be equal. The concentration distribution of A along the reactor length for the problem considered is the same as in the case of a single reaction qA ) -kPcA. This distribution predicted by the wave model compared to those by the SDM has been studied by Westerterp et al. (1995b) and by Kronberg et al. (1996). The solution of the SDM for the yield of product P at the reactor outlet in dimensionless form is (see Westerterp et al. (1987))
CPS ) j )g1H λe ) -(u - u
τH )
g1H2 λe
(17)
(u - u j )g1i2 (u - u j )g1H2 ua ) ) Deiτi λeτH
(18)
[
4 × 1-K -h
(1 + h)2e-(1-h)XL/(2RDe*) - (1 - h)2e-(1+h)XL/(2RDe*) s
(1 + s)2e-(1-s)XL/(2RDe*) - (1 - s)2e-(1+s)XL/(2RDe*)
λt∇2g1H ) u - u j
with
gli ) g1H ) 0 and n‚∇g1i ) n‚∇g1H ) 0 on ∂A The parameters of the wave model Dei and τi in eq 14 are the same for all components in the case of pure mechanical transverse mixing because the transverse dispersion coefficients are the same for all components. However, they differ for the components in multicomponent mixtures n > 2, if molecular diffusion governs the radial mixing. In particular, for laminar flow in a round tube and dilute solutions where eqs 5 and 6 hold, the parameters are
Dei )
a 2u j2 48 Di
λe )
j 2Fcp a2u 48 λ
a2Fcp τH ) 15λ
τi )
u j ua ) 4
]
+ (20)
with
where the functions g1i and g1H satisfy the equations
j Dti∇2g1i ) u - u
qP ) kPcA - kxcP
a2 15Di
h ) x1 + 4RDe*
where XL ) kPL/u, De* ) De/u2τ, R ) kPτ, and K is the ratio of the reaction rate constants kX/kP. Here, the relaxation time τ of the wave model is incorporated in the dimensionless reaction rate constant and dispersion coefficient for a convenient comparison with the wave model. The bulk concentration for the SDM is expressed through the area average concentration by the equation
CPbs ) CPs - RDe
dCPs dX
(21)
Since dCPs/dX ) 0 at the reactor outlet, the bulk concentration at this point is equal to the area-averaged concentration. The bulk concentration of component P predicted by the wave model is
Qφ2 - Pφ1 + (P - Q)λ2 λ1X e λ1 - λ2 Qφ2 - Pφ1 + (P - Q)λ1 λ2X e + Peφ1X - Qeφ2X (22) λ1 - λ2
CPbw ) (19)
In the case of a laminar gas flow, the relaxation times and dispersion coefficients for the components and heat are of the same order, since Di ∼ Dj and the Lewis
s ) x1 + 4KRDe*
with
2034 Ind. Eng. Chem. Res., Vol. 36, No. 6, 1997
P)
[
1 × φ2 - φ1
]
(K + R-1)(Λ + φ2) + K(1 - Λ) + (1 + ua* - De*)φ1(Λ + φ2) (1 + ua* - De*)φ12 + [R-1 + K(2 + ua*)]φ1 + K(K + R-1)
Q)
[
1 × φ2 - φ1
]
(K + R-1)(Λ + φ1) + K(1 - Λ) + (1 + ua* - De*)φ2(Λ + φ1) (1 + ua* - De*)φ22 + [R-1 + K(2 + ua*)]φ2 + K(K + R-1)
where λ1 and λ2 are the roots of the equation
R(1 + ua* - De*)λ2 + [1 + RK(2 + ua*)]λ + K(RK + 1) ) 0 and φ1 and φ2 the roots of the equation
R(1 + ua* - De*)φ2 + [1 + R(2 + ua*)]φ + 1 + R ) 0 The parameter Λ is defined as Λ ) (1 + ua*)/(1 + ua* De*), and the dimensionless parameter of velocity asymmetry is ua* ) ua/u and X ) kpx/u. Note that the presented solution of the SDM, eq 20, gives only the yield at the reactor outlet, whereas the solution of the wave model, eq 22, predicts the concentration profile over the reactor length. The solutions of the wave model and the SDM are considered below for limiting values of the dimensionless reaction rate constant R. For practical purposes, the values of interest for XL do not exceed 6-8, and therefore, all asymptotic solutions below are given for the case where XL is bounded. Moreover, the ratio of the reaction rate constants K is also assumed to be bounded. For low reaction rates or R f 0, both models give the same results as the plug flow model:
C ) CPs ) CPbs ) CPbw )
1 (e-X - e-KX) (23) K-1
For high reaction rates or R f ∞, the SDM gives
CPs )
XL
and the corresponding solution of the wave model is
CPbw )
[
to the laminar flow tubular reactor is used, and the dimensionless two-dimensional equations for the concentration distribution of A and P are given as
(
1 u2(u1 - 1) -X/u2 (e - e-KX/u2) + 1 - K u1 - u2 u1(1 - u2) -X/u1 (e - e-KX/u1) (25) u1 - u2
]
where u1,2 ) 1 + (ua*/2)[1 ( (1 + 4De*/(ua*)2)1/2] are the characteristic velocities of the wave model equations. Equation 25 shows that for high reaction rates, the wave model limit corresponds to a combination of two plug flow models with velocities equal to the wave velocities, as already found for a single first-order reaction in Westerterp et al. (1995b). The form of this solution differs noticeable from the one for the SDM. To check the accuracy of these one-dimensional models to predict the yield of P, we compare their predictions to the numerical solution of the multidimensional equation, eq 5, with some velocity distribution u(y,z). The parabolic velocity profile corresponding
)
2 ∂CA 1 ∂ CA 1 ∂CA - CA ) 0 (26) + - 2(1 - F2) 2 15R ∂F F ∂F ∂X
(
)
2 ∂CP 1 ∂ CP 1 ∂CP + CA - KCP ) 0 + - 2(1 - F2) 2 15R ∂F F ∂F ∂X (27)
with the boundary conditions
X ) 0: (24)
(1 + XL)(1 + KXL)
Figure 1. Bulk exit concentration of the intermediate P in firstorder consecutive reactions in a laminar flow reactor calculated by different models vs the dimensionless reactor length; kP/kX ) 25, R ) 10, De* ) 5/16, ua* ) 1/4.
F ) 0, 1:
CA ) 1
CP ) 0
∂CA ∂CP ) )0 ∂X ∂X
For simplicity, the diffusion coefficients for A and P are chosen to be equal; the influence of DA/DP was examined in the range 0.5-3 by Nigam and Vasudeva (1977), but the effect was found to be negligible. The numerical solution of the convective diffusion equations, eqs 26 and 27, for a wide range of parameter values has been discussed by Wan and Ziegler (1973) and Nigam and Vasudeva (1977). Two limiting cases can be observed. If the radial diffusion time becomes short as compared to the time required for the reactions to take place or kpa2/D f 0, the reactor performance approaches plug flow. On the other hand, if the reaction time is small compared to the diffusion time or kpa2/D f ∞, the reactants will be converted before radial transfer has taken place: this gives completely segregated laminar flow. Typical results for one set of parameters are shown in Figure 1; the bulk concentration of component P is calculated for the different one-dimensional models and compared to the numerical solution of the convective diffusion equations, eqs 26 and 27. The curves calculated for the plug flow model, the wave model, and
Ind. Eng. Chem. Res., Vol. 36, No. 6, 1997 2035 Table 1. Maximum Yield of Component P and the Corresponding Reactor Length Calculated by Different One-Dimensional Models Compared with the Numerical Calculations of the Two-Dimensional Model for First-Order Consecutive Reactions in a Laminar Flow Reactor; r ) 5, De* ) 5/16, ua* ) 1/4 kP/kX ) 2
kP/kX ) 5
CP,max
Xmax
CP,max
Xmax
CP,max
Xmax
CP,max
1.39 1.75 1.48 1.57
0.500 0.386 0.439 0.444
2.01 2.65 2.25 2.32
0.669 0.552 0.604 0.608
2.56 3.50 2.99 3.01
0.774 0.674 0.717 0.721
3.35 4.81 4.11 4.08
0.875 0.805 0.834 0.836
the two-dimensional model can be considered both as concentration profiles over the reactor length as well as the bulk exit concentration vs the reactor length. The curve for the SDM represents only the bulk exit concentration vs the reactor length. Predictions of the wave model are better than those based on the other one-dimensional models. Of practical interest for plant economics is the maximum yield and the corresponding residence time; in Table 1, numerical values calculated by different models are given for different kP/kX reaction rate ratios. The success of the wave model to predict both the maximum yield in component P and its corresponding reactor length is obvious. The SDM gives accurate results only for reaction rates where the relaxation time is lower than the reaction time. Exothermic First-Order Reaction in an Adiabatic Laminar Flow Reactor. In practice, a reaction hardly ever proceeds entirely under isothermal conditions. Here, an exothermic, irreversible, first-order reaction is considered in an adiabatic reactor under steady-state conditions. In this example as in the previous one, we consider a laminar flow tubular reactor to test the one-dimensional models. In this reactor, there is (almost) no feedback of heat for higher flow rates, and the inlet temperature must be necessarily high enough for the reaction to start. Several investigations have been carried out on the topic of nonisothermal laminar flow reactors; see, among others, the recent work of Hopkins and Golding (1993). The reaction rate is assumed to be of the Arrhenius type and first order, q ) -kc, where k ) k0 exp(-Ea/RT). The twodimensional equations of change are in dimensionless form:
)
∂C 1 ∂2C 1 ∂C - KR(θ)C ) 0 (28) + - 2(1 - F2) 15R ∂F2 F ∂F ∂X
)
1 ∂2θ 1 ∂θ ∂θ + γKR(θ)C ) 0 (29) + - 2(1 - F2) 15RH ∂F2 F ∂F ∂X with the boundary conditions
X ) 0:
C)1
θ)1
∂C ∂θ ) )0 ∂X ∂X
F ) 0, 1:
The following dimensionless terms are used:
R ) τkR0 KR )
RH ) τHkR0
kR ) exp[-β(1/θ - 1)] k R0
γ)
kP/kX ) 25
Xmax plug flow model SDM wave model exact solution
( (
kP/kX ) 10
-∆Hc0 FcpT0
β)
Ea RT0
The two-dimensional mass and heat balances are solved numerically. The concentration and temperature profiles of the plug flow and wave model are calculated by “marching” through the reactor from the inlet to the outlet. The standard dispersion model is solved numerically using the Newton-Raphson method. Figure 2 shows the results of the three one-dimensional models and the exact solution of the problem addressed for a typical set of parameters. The calculated concentration and temperature profiles for the wave model are in close agreement with the exact profiles; the SDM is even less accurate than the plug flow model. The wave model is also able to predict both the dispersion flux of the reacting component and the heat dispersion flux with high accuracy, as can be seen in Parts c and d of Figure 2. An interesting phenomenon can be observed in the profile for the heat flux: the heat flux calculated with the two-dimensional model is initially negative and becomes positive during the flow through the tube, whereas the avaraged temperature is monotonously increasing. This occurs because of the changing radial temperature profile along the reactor length for small Lewis numbers; see also Golding and Dussault (1976). The maximum temperature initially is found at a radial position near the wall: there the fluid velocities are low and the reactant is converted over a shorter distance compared to the reactant in the center line of the reactor. This results in a higher temperature near the wall. Because of the small Lewis number, the radial temperature gradient equalizes, whereas the concentration hardly does so over the same reactor distance. After that, the reaction accelerates in the center line of the reactor and the maximum temperature is established in the reactor center line. This has an essential influence on the heat flux: the flux is negative when the maximum temperature is near the wall, whereas it is positive when the maximum temperature is near the reactor center line. The wave model is capable of describing this phenomenon, whereas the SDM is not. In the case of the exothermic first order reaction, the SDM can only predict a positive heat flux, if the effective heat conductivity λe is taken to be negative. Note the strange behavior of the heat dispersion flux predicted by the SDM near the reactor outlet in Figure 2d: because of the Danckwerts boundary condition at the outlet, dθ/dx ) 0, the dispersion flux is artificially forced to zero. This behavior has absolutely nothing in common with the physics of the problem. For the problem considered, there is no feedback of mass and heat and only one steady-state exists. This corresponds to the predictions of the plug flow model and the wave model but contradicts the predictions of the SDM, which for a certain range of model parameters leads to the existence of multiple steady states; see Westerterp et al. (1987). Such erroneous predictions of the SDM occur because it does not distinguish between axial dispersion and backmixing.
2036 Ind. Eng. Chem. Res., Vol. 36, No. 6, 1997
Figure 2. Comparison of different models for the description of an exothermic first-order reaction in an adiabatic laminar flow reactor; R ) 0.5, RH ) 0.05, γ ) 0.25, β ) 20, De* ) λe* ) 5/16, ua* ) 1/4. (a, top left) concentration profile, (b, top right) temperature profile, (c, bottom left) mass dispersion flux, (d, bottom right) heat dispersion flux.
First-Order Autocatalytic Reaction. Consider the reaction system A f P which is autocatalytic according to A + P f P + P with a conversion rate qA ) - kcAcP. With respect to the autocatalytic reaction, we discuss again the interesting problem raised by Deckwer and Ma¨hlmann (1974, 1976). They investigated both experimentally as well as theoretically the SDM for the description of a liquid-phase reactor divided into three sections with different mixing properties. Their column consisted of a sequence of a bed of glass spheres, an empty tube, and a stirred vessel. The reaction studied was the formation of a thiazolium chloride from chloropropanone and thiourea in aqueous solutions. The formation of the chloride could easily be followed by conductivity measurements. Through calculations of the concentration profiles with the SDM, the authors have shown that the conversion may depend consider-
ably on the kind of boundary conditions applied at the interfaces between the sections. The differential equations used for a first-order reaction, in dimensionless form, are
z1 ∂2C ∂C Da1 C)0 Pe1 ∂z2 ∂z z1
0 e z e z1
(30)
Da2 z2 - z1 ∂2C ∂C C)0 2 Pe2 ∂z ∂z z2 - z1
z1 e z e z2 (31)
Da3 1 - z2 ∂2C ∂C C)0 2 Pe3 ∂z ∂z 1 - z1
z2 e z e 1 (32)
where the Peclet number Pei and Damko¨hler number Dai are related to section i.
Ind. Eng. Chem. Res., Vol. 36, No. 6, 1997 2037
At the interface between sections 1 and 2, the jump condition of Danckwerts (1953)
∂C(z-) )0 ∂z C(z1-) ) C(z1+) -
z2 - z1 ∂C(z1+) Pe2 ∂z
and the nonjump condition of Wehner and Wilhelm (1956)
C(z1-) -
z1 ∂C(z1-) z2 - z1 ∂C(z1+) ) C(z1+) Pe1 ∂z Pe2 ∂z C(z1-) ) C(z1+)
were used along with similar equations at the next interface at z2. To test experimentally the model predictions with the different boundary conditions, Deckwer and Ma¨hlmann increased the temperaturesthus increased the reaction ratesin the last section only. This temperature increase did not result in a change in conversion in the previous sections. So their experiments revealed that no influence of a section on a previous section can be found. The authors concluded the jump condition of Danckwerts appears to be appropriate for their reactor, as it predicts concentration profiles which agree well with their experimental results; this is in contrast to the Wehner-Wilhelm boundary condition. However, neither Deckwer and Ma¨hlmann nor other authors have ever experimentally proven Danckwerts boundary conditions to be correct. Moreover, the authors stated that the boundary conditions of Wehner and Wilhelm appear to be more logical from a physical viewpoint. Therefore, taking into account the experimental data, it is logical to suppose that the underlying model itself fails in this situation, because it does not distinguish between apparent axial mixing and real backmixing. The problem identified by Deckwer and Ma¨hlmann is avoided when the wave model is applied to describe their reactor. Both the concentration as well as the dispersion flux are continuous functions at the section interfaces. For unidirectional flow in the first and second sections, according to the wave model, there is no backmixing here, and variations in the third section do not influence the concentration profiles in the previous sections: this was experimentally shown by Deckwer and Ma¨hlmann. Calculations with the autocatalytic rate expression q ) -kcAcP were done by Deckwer and Ma¨hlmann when the reaction took place in the first and third sections of the reactor; see Figure 4 of Deckwer and Ma¨hlmann (1976). In Figure 3, we compare the wave model to the SDM with both the Danckwerts and the WehnerWilhelm boundary conditions for their imaginary reactor. The three section lengths are chosen to be equal. For clarity, it should be noted that this reactor with its mixing parameters does not correspond to the liquidphase reactor used in the experimental work of Deckwer and Ma¨hlmann. As shown in Figure 3, the variation of Da3 does not influence the concentration profiles of the wave model in the first and second sections, whereas it does so for the profiles of the SDM with the WehnerWilhelm boundary conditions. The significant differ-
Figure 3. Influence of the Damko¨hler number of the last section on the concentration profiles for an autocatalytic reaction in a reactor divided into three sections with different properties; cP0/ cA0 ) 0.01, z1 ) 0.33, z2 ) 0.67. (...) SDM with Danckwerts boundary conditions, (- -) SDM with Wehner-Wilhelm boundary conditions, (s) wave model with τiu/L ) 5li/Pei and ua,i ) 0.
Figure 4. Calculation of an autocatalytic reaction in a reactor with fore and after sections with small relaxation times; cP0/cA0 ) 0.01, z1 ) 0.05, z2 ) 0.95. (...) SDM with Danckwerts boundary conditions, (- -) SDM with Wehner-Wilhelm boundary conditions, (s) wave model with τiu/L ) 5li/Pei and ua,i ) 0.
ence between the wave model and the SDM with Danckwerts boundary conditions is also clear. An interesting feature of the wave model is the behavior of the concentration profile at the boundaries of a reactor. Let us consider a reactor with a unidirectional flow and with fore and after sections with relaxation times short compared to the averaged residence time in these sections. This corresponds to high Peclet numbers for a specified velocity distribution. In this case, the SDM with both Danckwerts as well as Wehner-Wilhelm boundary conditions predicts a concentration jump at the entry of the reactor, whereas the wave model does not; see Figure 4. This phenomenon was discussed by Westerterp et al. (1995b). In the reactor of Figure 4, we now have assumed that no reaction takes place in the first and third sections. Furthermore, the reaction section takes 90% of the total volume. At the exit of the reactor, the standard dispersion model with both kinds of boundary conditions predicts a zero concentration gradient. As shown in Figure 4, the wave model predicts a rapid increase in the area-averaged concentration at the reactor exit of the second section. Thus, the concentration in the last section is higher than at the end of the reactor! At first sight, this behavior may seem paradoxical. It contradicts the intuition which led Danckwerts (1953) to set up his zero-gradient outlet boundary condition. How-
2038 Ind. Eng. Chem. Res., Vol. 36, No. 6, 1997
ever, the apparent concentration jump at the exit predicted by the wave model is not unrealistic and can easily be understood. Because of mass conservation, the total flux uc + j passing a cross-sectional surface is continuous, also, at the interfaces z1 and z2. For steadystate operation, its value changes solely by chemical reaction. Without chemical reaction in the last section, the total flux remains constant over the section length, so
uc(z) + j(z) ) uc(z2) + j(z2) ) u j cb(z2) for z g z2
(33)
and we are reminded that cj is the concentration averaged over the cross section and cb represents the bulk or mixed cup concentration. Note that the bulk concentration is constant in the last section, too. The dispersion flux at the reactor outlet j(z2) can be negative as well as positive as we have seen in the example of the nonisothermal first-order reaction. Generally, the sign of j is determined by the transverse concentration distribution. If this distribution approaches equilibriumsi.e., a uniform concentration in the transverse directionsthe wave model approaches the SDM and the dispersion flux j is proportional to -∂cj/∂z and thus j(z2) > 0 when the species is consumed. In this case, we may write for the last section
cj(z2) < cb(z2) ) cj(z) +
j(z) u j
z g z2
(34)
Moving away from the reactor exit, the dispersion flux disappears since the concentration will tend to equalize in the transverse direction. As a result, the areaaveraged concentration will increase and approach the bulk concentration, as follows from eq 34. If the relaxation time is short enough, the dispersion flux disappears very closely to the reactor exit and we observe a rapidly increasing, area-averaged concentration profile as shown in Figure 4. Of course, the situation described in Figure 4 is rather schematic, with the apparent concentration jump at the reactor exit for the wave model. In practice, the transition from the reactor to the last section will not be so abrupt and the concentration profile will be smoother. Nevertheless, the area-averaged concentration calculated by the wave model will increase in the last section in contrast to the concentrations calculated with the SDM with Danckwerts and Wehner-Wilhelm boundary conditions. The discussed phenomenon is not typical for only autocatalytic reactions: it may be observed in the description of all kinds of chemical reactions. Let us now consider an autocatalytic reaction in a chemical reactor with unidirectional flowsi.e., there is no convective backmixingsfor situations where the concentration of P in the feed stream approaches zero. In this case, the reaction will extinguish and there will be no conversion of reactant A to P. The plug flow and wave models predict zero conversions. However, the SDM predicts two steady states, one with zero conversion and the other with a high conversion of A depending on the start-up history of the reactor. In the case of high conversion, the presence of feedbackswhich is inherent to the SDMscauses sufficient backmixing of P to keep the reaction going: the reaction is selfsustaining. Of course, there will always be some backmixing by molecular diffusion. However, the extent of this diffusion is usually much lower than the axial
mixing caused by the nonideal convective flow, and it is not sufficient to achieve a significant conversion. So the multiplicity predicted by the SDM is an artifact. Discussion In previous papers of Westerterp et al. (1995a,b) and Kronberg et al. (1996), the advantages of the wave model have been demonstrated for nonreactive conditions or for isothermal reactive systems with simple kinetics. In this article, the concept of the wave model has been extended to nonisothermal systems with arbitrary kinetics under adiabatic conditions. We have tested the accuracy of the extended wave model by comparing it to numerical solutions of the two-dimensional equations for laminar flow tubular reactors. An analytical solution of the wave model in the case of a first-order consecutive reaction system has been obtained. The results show the wave model gives a fair approximation to the exact solutions in a wide range of situations and is definitely preferable over the SDM. Also, for the description of an exothermic first-order reaction in an adiabatic laminar flow reactor, the advantages of the wave model over the conventional dispersion model were obvious. The concentration and temperature profiles along the length of the reactor obtained by solving the two-dimensional equations numerically were well approximated by the wave model, whereas the SDM failed to describe the system. Moreover, the numerical calculations with the wave model may proceed through the reactor in the so-called stepby-step marching technique, whereas the SDM needs iterative calculations from boundary to boundary. Fickian- and Fourier-type equations with positive mass and heat dispersion coefficients are unable to describe some of the experimentally observed phenomena, not even qualitatively. Components will diffuse or disperse into the direction of negative concentration gradients as described by Fick’s law. Analogously, heat will be transported into the direction of negative temperature gradients as described by Fourier’s law. However, under certain conditions in the case of an exothermic first-order reaction in an adiabatic laminar flow reactor, the heat dispersion flux may change from a negative to a positive value, whereas the temperature is monotonously increasing; see Figure 2. The same phenomenon of dispersion fluxes having the same sign as the concentration gradient was observed when transport of mass was studied; see, for example, the flow reversal experiments of Jasti and Fogler (1992). The conventional dispersion model can only describe such behavior using negative and variable dispersion coefficients as fitting parameters: these, of course, have no physical meaning. The constitutive equations for the mass and heat dispersion fluxes in the wave model contain new terms compared to the conventional Fickian- and Fourier-type gradient laws. As a result, the wave model can predict dispersion fluxes in the direction of uphill concentration and temperature profiles as shown in Figure 2 for an exothermic first-order reaction and in Westerterp et al. (1995b) for flow reversal. These mass and heat fluxes lead to unmixing phenomena for the concentration and temperature. This peculiarity of the wave model, which is not contained in the SDM, may be important for the description and design of chemical reactors; for example, the unmixing may result in local hot spots in adiabatic packed bed reactors and eventually lead to runaway and/or deactivation of the catalyst.
Ind. Eng. Chem. Res., Vol. 36, No. 6, 1997 2039
The example with an autocatalytic reaction clearly demonstrated the fundamental differences of the wave model and the SDM. The wave model distinguishes between apparent axial dispersion and backmixing and does not predict backmixing for unidirectional flow. In many real systems of practical interest, the difference between apparent mixing and backmixing is considerable. Hiby (1962) has proven backmixing to be absent in liquid flow through packed beds at moderate and high Reynolds numbers. Also, the absence of convective backmixing in laminar flow through an empty tube is obvious. For the flow in a rotating disk contactor, the difference between apparent longitudinal dispersion and backmixing was demonstrated by Westerterp and Landsman (1962) and Westerterp and Meyberg (1962). The same difference was observed in reciprocating plate columns; see Pratt and Baird (1983). Backmixing tests by Van Swaaij et al. (1969) showed, for liquid flow through a trickle bed reactor without gas flow, that backmixing was always very low. The same analogy holds for the difference between apparent axial dispersion and backmixing of heat, although the extent of both phenomena may differ from those of mass. In the conventional dispersion models, the backflow against the main stream is of equal intensity as the apparent axial dispersion. Therefore, the use of these models in the analysis of dynamic phenomena, where the course of the process depends significantly on the intensity of feedback, is very doubtful. In the case of simple reactions, it is well-known that multiplicities can only be predicted by models with sufficient backmixing of mass and/or heat, whereas in a wide class of real systems, actually no or very little backmixing occurs. Therefore, the available conclusions of theoretical work using the standard dispersion model to describe mixing in different kinds of chemical apparatus in many cases are not reliable and incorrect. In the example of the reactor with fore and after sections presented in Figure 4, the differences in the predictions of the wave model and the SDM near the reactor boundary surfaces have been clearly demonstrated. In the case of plug flow in the fore section and unidirectional flow in the reactor sectionsthus no feedback to the reactor entrancesfluid elements passing the inlet surface with different axial velocities have equal compositions. Therefore, the dispersion flux at the inlet surface, which is a measure of the deviation from a transverse uniform concentration, is zero and no concentration jump in the model predictions should be observed. Nevertheless, the SDM predicts both with Danckwerts as well as with Wehner-Wilhelm boundary conditions concentration jumps at the reactor inlet when plug flow is assumed in the fore section. As shown in Figure 4, the wave model predicts no concentration jump. The situation is absolutely different at the reactor outlet surface. When the magnitude of axial dispersion is significant in the reactor, the fluid elements passing the outlet surface do not necessarily have equal compositions. As a result, the dispersion fluxes of the reactants and products at the reactor outlet are not zero at the reactor outlet. The disappearance of these fluxes in the after section leads to a decrease or increase of the area-averaged concentrations in the wave model predictions for initially negative and positive fluxes, respectively. For rapid disappearance, apparent concentration jumps are observed as shown in Figure 4. The SDM, however, predicts dispersion fluxes equal to zero and no concentration jumps at the outlet
with both Danckwerts as well as Wehner-Wilhelm boundary conditions. To prevent erroneous predictions of the SDM at the outlet surface, the use of semiinfinite boundary conditions was often recommended; see Gunn (1969) or Wissler (1969). Nevertheless, for high reaction rates, the predictions of the SDM with the semiinfinite boundary conditions are even worse; see, e.g., Westerterp et al. (1995b). The problems of the SDM in decribing the interaction between the system and its surrounding are inherent to the model equations. The wave model is not without flaws. As already pointed out in Kronberg et al. (1996), one of the general difficulties of the description of a two- or threedimensional physicochemical system by one-dimensional equations is the averaging of the consumption term q over the cross section. The used approximation, where the consumption rate is represented through a first-order Taylor expansion around the mean concentration or temperature, may be too rough for strongly nonlinear chemical reactions with reaction times shorter than the relaxation time. For simple reaction schemes such as first- and second-order isothermal reactions, no physical contradictions were observed when the consumption rate was approximated by a first-order Taylor expansion. However, for rapid nonisothermal or autocatalytic reactions creating essentially nonequilibrium, transverse concentration distributions, problems may arise. It was shown by Kronberg et al. (1996) that the dispersion flux of the wave model is bounded because physics require the limits -u2cj < j < u1cj. The calculations using the above-mentioned approximation of the consumption rate may result in dispersion fluxes outside of the physically realistic range. Improvement can be achieved either by including additional terms in the Taylor expansion or by a different representation of the averaged consumption rate. Conclusions The wave concept introduced in previous articles of Westerterp et al. (1995a, 1996) has been extended to isothermal and adiabatic chemical reactors with reaction rates described by arbitrary expressions. The validity of the generalized wave model was tested through a comparison with numerical solutions of the two-dimensional model for the laminar flow tubular reactor. The results obtained clearly demonstrate the advantages of the wave concept over the conventionally used standard dispersion models. Acknowledgment We express appreciation to P. A. M. Winkelman for his contribution to the project. Nomenclature a ) tube radius c ) concentration cP ) heat capacity C ) dimensionless concentration, c/c0 D ) diffusion coefficient De ) dispersion coefficient De* ) dimensionless dispersion coefficient, De/u2τ Dt ) transverse dispersion coefficient Dai ) Damko¨hler number for section i, kiLi/u Ea ) activation energy Hi ) enthalpy of component i ∆H ) heat of reaction ji ) dispersion flux of component i, defined in eq 11
2040 Ind. Eng. Chem. Res., Vol. 36, No. 6, 1997 jH ) heat dispersion flux, defined in eq 12 k ) reaction rate constant K ) inverse reaction rate ratio, kX/kP li ) dimensionless length of section i, (zi - zi-1)/L L ) reactor length n ) number of components Pei ) Peclet number for section i, uLi/De qi ) consumption rate of component i per unit of reactor volume r ) radius t ) time T ) temperature x ) axial coordinate X ) dimensionless axial coordinate, kx/u j u ) velocity ua ) parameter of velocity asymmetry ua* ) dimensionless parameter of velocity asymmetry, ua/u j z ) dimensionless axial coordinate, x/L Greek Letters R ) dimensionless constant for chemical reaction, kτ β ) reaction temperature sensitivity, Ea/RT0 γ ) dimensionless adiabatic temperature rise, -∆Hc0/FCPTO λ ) thermal conductivity λe ) effective thermal conductivity λe* ) dimensionless effective thermal conductivity, λe/u2τ λt ) effective thermal conductivity transverse to the main flow F ) dimensionless radius, r/a θ ) dimensionless temperature, T/T0 τ ) relaxation time Subscripts b ) bulk H ) heat L ) reactor length s ) standard dispersion model w ) wave model 0 ) inlet, at x ) 0 in the entering stream
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Received for review November 12, 1996 Revised manuscript received February 24, 1997 Accepted March 9, 1997X IE960713J
X Abstract published in Advance ACS Abstracts, May 1, 1997.
