Ind. Eng. Chem. Res. 2007, 46, 3863-3870
3863
Coated-Wall Reactor ModelingsCriteria for Neglecting Radial Concentration Gradients. 1. Empty Reactor Tubes Rob J. Berger* and Freek Kapteijn Catalysis Engineering, DelftChemTech, Faculty of Applied Sciences, Delft UniVersity of Technology, Julianalaan 136, 2628 BL Delft, The Netherlands
The influence of radial mass transport on the conversion in a coated-wall reactor was investigated in a modeling study. Criteria based on the observed conversion are developed to allow for neglecting radial concentration gradients and to be able to use the simple plug-flow model to describe the reactor performance and to determine reaction kinetics. The simulations were carried out using Athena Visual Studio 10.0 and Comsol Multiphysics 3.2 for several reactor geometries (round, square, annular, rectangular, and triangular channels) and reaction orders. It was verified that the entrance effects, which enhance the radial mass transfer, can be neglected in most typical situations. The criterion for the maximum allowable conversion with a less than 5% deviation in the rate constant is a function of the modified Pe´clet number (Pe′), the reaction order (n), and two constants a and b that depend on the reactor geometry: XCWR < b/(a + nPe′). The criterion is valid for both gases and liquids. Introduction Coated-wall reactors (CWRs) are devices in which the catalyst is located in a layer attached to the wall. In most applications, the fluid, gas or liquid, flows along this wall in which the required reactions are allowed to take place. In conventional applications, the most-often-used reactor shape is tubular. Additionally, annular shapes are used to achieve a higher surface-to-volume ratio and, consequently, less probability of transport limitations. More recently, however, the number of applications of coated walls is increasing rapidly in the applications of monolith reactors,1-6 in which each channel is in fact a CWR, and also of microchannel reactors.7-13 Results of experimental investigations of CWRs were reported by, e.g., Redlingsho¨fer et al.,14 who studied the catalytic partial oxidation of propene to acrolein, Karim et al.15 for steam reforming of methanol, and Srivinasa et al.16 for preferential CO oxidation. Also, for performance testing of conventional catalysts, the CWR could be interesting if the reaction to be studied is too fast and/or too exothermic for using a conventional fixed bed. The CWR allows a much faster heat exchange with the wall than the fixed-bed reactor because of the intimate contact of the catalyst with the wall. The heat transfer from the catalyst to the reactor wall increases by 2-3 orders of magnitude,17 and isothermal conditions can be achieved, which is otherwise impossible. Redlingsho¨fer et al.14 applied this concept to obtain isothermal kinetic data for modeling commercial catalysts. For studying and developing such reactors properly, it is very important to characterize the catalyst on activity and selectivity in order to optimize the reactor performance. However, in most cases, it is hardly possible to characterize a catalytical coating outside the CWR, e.g., by crushing the tube in small pieces and testing these in a conventional fixed-bed reactor. Such a treatment may cause irreversible changes to the catalyst properties or destroy the microreactor. Therefore, the characterization of the catalyst is, by preference, performed in the configuration in which it is prepared. A potential problem in CWRs is the development of significant radial concentration gradients, as illustrated in Figure * Corresponding author. Tel.: +31-15-2784316. Fax: +31-152785006. E-mail:
[email protected].
1, because of a finite radial transport (by dispersion) compared to the axial convection. This may significantly influence the conversion and the selectivity. Only if the catalytic activity is low relative to the rate of the radial mass transport toward and through the catalytic layer, the intrinsic activity and selectivity can be measured directly. The problem is, to some extent, similar to that occurring in chemical vapor deposition (CVD) reactors in which concentration gradients need to be assessed well in order to yield homogeneous layer thickness of the deposited material.18-20 A literature study showed that there are currently no handy criteria for neglecting radial concentration gradients in a CWR. Balakotaiah and Chakraborty21 presented the results of a series of simulations of CWRs; however, they do not derive easily applicable criteria. In order to derive such a criterion, several series of CWR simulations were carried out in this study in which the influence of all relevant parameters that may affect the results were investigated. On the basis of these results, a criterion is derived for the maximum allowed conversion at which one still can use the simple plug-flow reactor (PFR) model without taking radial concentrations into account to calculate the intrinsic reaction rate with acceptable accuracy. It is noted that this study focuses only on the transport limitations in the empty volume between the coated walls; any gradients inside the coated wall, as illustrated in Figure 1, are not considered, which from a modeling point of view can be translated in a coated wall with zero thickness. The effect of mass transport limitation inside the catalytically active layer can be estimated using well-known relations from catalysis engineering textbooks, see, e.g., Kapteijn and Moulijn.22 In most cases, however, the concentration gradients inside the wall are small compared to those in the empty volume. The study was performed for various CWR geometries; besides conventional cylindrical and annular shapes, square, rectangular, and triangular shapes were included. This is of particular interest for monoliths, often consisting of parallel square channels, and microchannel reactors. In these latter types of reactors, there are also many applications in which only a part of the wall, e.g., only one of the four walls in a rectangular tube, is active.
10.1021/ie0612313 CCC: $37.00 © 2007 American Chemical Society Published on Web 02/23/2007
3864
Ind. Eng. Chem. Res., Vol. 46, No. 12, 2007
Since the mass transport by axial diffusion is, in most cases, negligible compared to that due to the convection and the radial dispersion, the problem simplifies to an initial value problem in the axial direction.
0 ) -2(1 - x2)
( ) ( )
DAmL 1 ∂ ∂y ∂y + x 2 x ∂x ∂z ∂x u0R
(7)
∂y 1 1 ∂ ∂y x + ∂z Pe′ x ∂x ∂x
(8)
which can be written as
0 ) -2(1 - x2) Figure 1. Typical flow profile and radial concentration profile in an empty CWR.
Since the criteria reported in this paper can be applied very easily in experimental research because it only requires measurable quantities, the paper will be very interesting for those working with such types of reactors. Coated-Wall Reactor (CWR) Models Model for the Empty Tube. For steady-state conditions, the molar balance for a reactant A in dimensionless form is given by the following dimensionless equation using y ) CA/CA0, x ) r/R, and z ) l/L:
( ) ( )
Drad,A 1 ∂ ∂y Dax,A ∂2y 2(1 - x2)u0 ∂y dy )0) 2 + x 2 dt L ∂z L ∂z R2 x ∂x ∂x
with
Pe′ ) Pe
Axial direction (for 0 e x e 1): z ) 0: y ) 1 z ) 1:
∂y )0 ∂z
Radial direction (for 0 e z e 1): x ) 0:
(2)
[ ]
r(y) ∂y ∂y ) 0 x ) 1: ) -Darad,0 ∂x ∂x r(1)
(3)
with
Darad,0 )
r0,wallR ; r0,wall ) rwall|CA0 CA0DAm
2
x)1
|
∂y ) -Darady ∂x x)1
(5)
kwallCn-1 A0 R Darad,0 ) DAm
(6)
which results in
Note that the rate is defined per unit wall area in these expressions.
(9)
(10)
with
Darad )
kwallR DAm
(11)
Since the model does not use any assumption specific for only gases or liquids, the model and the correlations derived from the results will be valid for both gases and liquids. The PDE system was solved using the package Athena Visual Studio.25 Model for the Annular Reactor. An annular reactor consists of two concentric tubes in which the fluid flows through the space between both tubes. The reactor model for this reactor configuration is largely similar to that for the empty tube. If the distance between the walls in an annular reactor is small compared to the reactor diameter, the shape can be approached by a flat-plate reactor. If only one of the reactor walls is assumed to be catalytically active, the reactor model becomes
0 ) -6(x - x2)
rwall|CA0 ) kwallCnA0
2
For the initial value problem, the modified Pe´clet number, Pe′, and the radial Damko¨hler number, Darad,0, are the two governing parameters that determine the solution. The first parameter represents the ratio of the axial convective transport and the radial diffusive transport, or the ratio of the characteristic times in the reactor for radial diffusion and axial transport (residence time). The larger this parameter is, the poorer is the reactor performance. For a first-order reaction, the radial Damko¨hler number is constant and the boundary condition at the wall simplifies to
(4)
and, for an nth-order reaction,
uo L
(RL) ) D (RL) Am
(1)
The velocity profile has a parabolic shape23 because the flow regime can be assumed to be laminar at Re < 2000,24 since the value of Re in laboratory reactors is, at the highest, 100-500. Additionally, the flow can be assumed to be fully developed. Since the flow is laminar, the dispersion in the axial as well as in the radial direction is equal to the bulk diffusivity, DAm. The solution of this partial differential equation (PDE) is subjected to the boundary conditions,
( )
1 2 ∂ 2y ∂y + ∂z Pe′ ∂x2
(12)
Similarly as with the cylindrical tube, the flow regime can be assumed to be laminar and fully hydrodynamically developed. R is the distance between the plates and r is the local position between the plates, where x ) 0 corresponds with the wall that is catalytically inactive and x ) 1 corresponds with the catalytically active wall. Pe′ is the modified Pe´clet number. Also in this reactor, the velocity profile has a parabolic shape. The boundary conditions are the same as for the empty tube. The same reactor model is applicable for an annular reactor in which both walls are catalytically active. Only the boundary
Ind. Eng. Chem. Res., Vol. 46, No. 12, 2007 3865 Table 1. Values of Parameters a and b for the Criterion ∆k < 0.05 Given by Eq 52
a
geometry
definition R in Pe′
active walls
a
b
cylindrical square triangular rectangular 2:1 rectangular 3:1 annulara square rectangular 2:1 rectangular 3:1 annulara
circle radius half width 1/3 × height 2/3 × short wall 3/4 × short wall distance between walls half width 2/3 × short wall 3/4 × short wall distance between walls
all all all all all all one one (long side) one (long side) one
0.16 0.105 0.07 0.11 0.13 0.30 0.023 0.042 0.055 0.10
0.23 ) 1.44a 0.16 ) 1.52a 0.105 ) 1.50a 0.165 ) 1.50a 0.176 ) 1.35a 0.43 ) 1.43a 0.035 ) 1.52a 0.062 ) 1.48a 0.078 ) 1.42a 0.145 ) 1.45a
Obtained by assuming slip-symmetry at the short walls of a rectangular reactor or by assuming a flat-plate reactor.
condition in the radial direction at z ) 0 is the same as the boundary condition at z ) 1. General Model for Other Reactor Shapes. In other tube shapes of similar sizes and flow rates, the flow pattern will be laminar and fully hydrodynamically developed. For obtaining the laminar velocity profiles in these reactors, the momentum balance, for which the incompressible Navier-Stokes equation is used, consisting of the following two equations, was solved:
FF(u‚∇)u - ∇‚{µF(∇u + (∇u)T)} ) 0
(13)
∇u ) 0
(14)
The solution of this PDE is subject to the following boundary conditions,
Entrance (z ) 0): uz ) uin (and also ux ) 0 and uy ) 0) (15) At the walls: uz ) 0 (and also ux ) 0 and uy ) 0) (16) where u is the velocity vector. The assumption of incompressible Navier-Stokes is adequate since the pressure drop in an empty tube is negligible in almost all applications. In parallel, the following mass balance holds:
u‚∇CA + ∇‚(-D∇CA) ) 0
(17)
The solution of this PDE is subject to the following boundary conditions:
Reactor inlet (z ) 0): D
∂CA ) -u0(CA0 - CA) (18) ∂z ∂CA )0 ∂z
(19)
Catalytic walls: D∇CA ) -rwall
(20)
Noncatalytic walls: D∇CA ) 0
(21)
Reactor outlet (z ) 1):
Since there is laminar flow, the diffusivity D is equal to the molecular diffusivity:
D ) DAm
(22)
Unfortunately, the Danckwerts type boundary conditions at the inlet and outlet, i.e., eqs 18 and 19, are not easy to implement in CFD packages and the Comsol Multiphysics package used in this study. Therefore, instead of these boundary conditions, the following simple boundary condition was used:
Entrance (z ) 0): CA ) CA0
(23)
However, the use of this boundary condition may, particularly at very low velocities, influence the diffusion in the axial direction, which also affects the total flux at the inlet as well as at the outlet. This can cause poor convergence and sometimes also erroneous results. This problem was solved by setting the diffusion in the axial direction (the z-direction) to zero, resulting in a problem with closed-closed boundary conditions. This is allowed since, for most applications, the effect of the axial diffusion is negligibly small compared to that of the convection. In order to make the results easily comparable with those of the cylindrical tube, the same dimensionless variables are used as independent parameters in eqs 4, 5, and 9, assuming the reaction rate to be of the nth order. Now R is set equal to the hydrodynamic radius, defined as the radius R of a cylindrical reactor having the same surfaceto-volume ratio.26 The hydrodynamic radii for the geometries used are shown in Table 1. For a square tube, this is half the width; for a triangular tube, this is one-third of the height of the triangle; and for rectangular tubes, this is equal to height times width divided by their sum (height + width). For implementation of the problem in Comsol Multiphysics, absolute dimensions and absolute numbers have to be used for the physical properties required, which are the fluid density, viscosity, and diffusivity. Since the results showed very little dependency on the fluid viscosity and the density, these values were not varied but fixed at the values for air at 500 K and atmospheric pressure (FF ) 0.703 kg/m3, µF ) 2.63 × 10-5 kg/ms). Since the direct influence of the bulk diffusivity is already incorporated in the variables Darad,0 and Pe′, the value of the bulk diffusivity itself also has no appreciable influence on the results. Therefore, a typical value of the bulk diffusivity, that of oxygen in air at atmospheric pressure and 500 K, which amounts to 5.1 × 10-5 m2/s, is used in the simulations. For the same reason, the choices of the length and the width of the tubes also do not show an appreciable effect on the results. Therefore, a typical reactor length of 0.1 m and a width of ∼0.01 m were used in the simulations. Since the results are not significantly influenced by the chosen values of the density, the viscosity, and the diffusivity, the results are applicable for both gases and liquids. The inlet concentration was fixed at 1.0 mol/m3, which yields effectively the same results as when using dimensionless concentrations. In the simulations, the typical situation that the zone with the catalytically active coated wall is preceded by the noncoated wall was arranged by adding 0.011 m of inactive wall before the catalytically active zone. This resulted in all cases in a (largely) fully developed laminar flow at the entrance of the catalytically active zone. This was confirmed by the Comsol simulation results. It can also be justified using a correlation for the hydrodynamic entrance length in a cylindrical duct, given by eq 24.27 Using the highest possible typical value of Re of
3866
Ind. Eng. Chem. Res., Vol. 46, No. 12, 2007
dy3 Darad,0 n ) -2 y dz Pe′ 3
100, the ratio Lhy/dt becomes 5.7. Since the inner diameter is ∼0.005 m, this condition is fulfilled for most applications.
Lhy 0.60 ) + 0.056Re dt 0.035Re + 1
(24)
For solving the PDEs for other reactor shapes, the default meshing generated by the package was adapted by a scaling factor of 0.4 in the axial direction. For the 3D simulations, this resulted in 1000-3000 elements, corresponding to 7 00020 000 degrees of freedom. The simulations were carried out on a Pentium M 1.5 GHz computer with 1 MB cache and 1.0 GB 333 MHz SDRAM; the simulations of a square reactor with a mesh resulting in 8276 degrees of freedom took ∼20 s. Conversion Calculations. The conversion can be calculated from the total molar flows in and out:
FA0 - FA,out XA ) FA0
(26)
Making this dimensionless and assuming no volume change, using y2 ) FA/FA0 ) FA/(πR2u0CA0) and
[ ]
CA0DABDarad,0 r(y) R r(1)
(27)
this yields
[ ]
[ ]
Darad,0 r(y) dy2 r(y) L 2 DAB ) -2 ) -2 Darad,0 dz R u0L Pe′ r(1) r(1)
(28)
For an nth-order reaction, n r(y) CA ) n ≡ yn1 r(1) CA0
(29)
Darad,0 n dy2 ) -2 y dz Pe′ 1
(30)
this yields
where y2 stands for the dimensionless total molar flow rate of A through the reactor cross-sectional area. This can be solved simultaneously with the expression for the concentration y1 at the reactor wall along the length of the reactor. For the conversion in the coated-wall reactor (CWR), the following holds:
XA,CWR )
(
y3 ) exp -2
)
Darad,0 Pe′
(33)
and thus, it follows for the conversion if plug flow is assumed, that:
xPFR )
(
)
FA0 - y3FA0 Darad,0 ) 1 - y3 ) 1 - exp -2 FA0 Pe′
(34)
For an nth-order reaction (n * 1), it follows that
dFA ) 2πRrwall dl
()
For a first-order reaction, the parameter 2Darad,0/Pe′ in this ordinary differential equation (ODE) is identical with k′τ, the Damko¨hler number for a first-order reaction in a plug-flow reactor where the rate is defined per unit reactor volume. Integration of this equation results in
(25)
For the cylindrical reactor, the balance for the total molar flow of reactant A is
rwall ) -
(32)
FA0 - FA,out FA0 - y2FA0 ) ) 1 - y2 (31) FA0 FA0
To determine the deviation from the ideal plug-flow case, a third expression was solved similarly to the second, but by assuming no concentration gradients in the radial direction. Since the relative molar flow rate is similar to the relative concentration, it follows that
(
xPFR ) 1 - y3 ) 1 - 1 + 2(n - 1)
)
Darad,0 Pe′
1/(1-n)
(35)
For the annular reactor, a similar correlation for the conversion was derived. For the other reactor shapes, the conversion was determined numerically by means of integration over the reactor outlet cross section (i.e., the mixing-cup equation). Results, Discussion, and Derivation of the Criteria Cylindrical Reactor. From the solution at the reactor exit, the relative deviation of the conversion from that at plug-flow behavior ∆X can be calculated:
∆X ≡
XPFR - XCWR (1 - y3) - (1 - y2) y2 - y3 ) ) XPFR 1 - y3 1 - y3 (36)
Figure 2 shows the contour plot of the deviation of the conversion ∆X obtained from the simulations in a cylindrical reactor as a function of Darad,0/Pe′. A frequently used criterion is that the conversion may not deviate >5% from the value in the ideal plug-flow reactor. At Darad,0/Pe′ ≈ 0.1, the maximum allowed value of Darad,0 is the lowest: 0.11. A value of Darad,0/ Pe′ of 0.1 corresponds with a plug-flow conversion of 0.18 (using eq 34). At higher conversion, the criterion for Darad,0 becomes less strict because of the approach to full conversion, which is often the case for microreactor channels with high L/R values. At a value of Darad,0/Pe′ of 1.0, which corresponds with a plug-flow conversion of 0.86, the maximum allowed value of Darad,0 is 0.33. However, if the experimental results for the CWR are used to estimate the intrinsic reaction rate constants, which is typically done in a kinetic study, it is more useful to have a similar criterion for the rate constant,
∆k ≡
kw,PFR - kw,CWR kw,PFR
(37)
where kw,CWR is the rate constant from the CWR performance and kw,PFR is the rate constant calculated from the measured conversion based on the assumption of ideal plug-flow behavior,
Ind. Eng. Chem. Res., Vol. 46, No. 12, 2007 3867
Figure 2. Contours of the relative deviation of the conversion (∆X) for a first-order reaction in an empty cylindrical reactor plotted as a function of Darad,0/Pe′ and Darad,0.
Figure 3. Contours of the relative deviation of the calculated rate constant (∆k) for a first-order reaction in an empty cylindrical reactor plotted as a function of Darad,0/Pe′ and Darad,0.
thus neglecting radial concentration gradients. For a first-order reaction and using eqs 33 and 34, kw,PFR becomes
kw,PFR ) -
u0R ln(y3) 2L
(38)
For kw,CWR, it follows similarly that
kw,CWR ) -
u0R ln(y2) 2L
(39) Figure 4. Cylindrical reactor. Maximum allowed value of Pe′p ) ln{1/(1 - XCWR)} as a function of XCWR to obey the criterion of ∆k < 0.05 for a first-order reaction.
This results in
∆k ≡
kw,PFR - kw,CWR ln(y3) - ln(y2) ) kw,PFR ln(y3)
(40)
For a first-order reaction, it follows from the criterion condition that
For an nth-order reaction, using eq 35, kw,PFR becomes
kw,PFR )
u0R
{(y3) n-1
1-n
2(n - 1)LCA0
- 1}
∆k ) (41)
ln(y3) - ln(y2) ln(y3)
) 0.05
y2 ) y30.95
(43) (44)
This results in
kw,PFR - kw,CWR ∆k ≡ kw,PFR
)
{(y3)1-n - 1} - {(y2)1-n - 1}
)
(y3)1-n - (y2)1-n
(y3)1-n - 1 (y3)1-n - 1
(42)
Figure 3 shows the results in a contour plot of ∆k as a function of the two dimensionless parameters Darad,0 and Darad,0/Pe′ that govern the solution for this problem, as apparent from eqs 11, 30, and 32. The former represents the reaction rate at the wall relative to the radial diffusion rate, and the latter represents the rate in the reactor volume relative to the flowrate and is equal to the classical Damko¨hler number for a PFR. From the figure, it follows that, where ∆k < 0.05, the criterion becomes independent of the value of Darad,0/Pe′ if that value is >0.1, which corresponds to a plug-flow conversion of 0.18. For practical utility, it is preferable to have the criterion as a function of only observable parameters. Therefore, XCWR is needed instead of XPFR, and also observable parameters have to be used instead of Darad,0 since this contains the unknown kwall. If ∆k is known, XCWR can be calculated directly from XPFR.
Since XPFR ) 1 - y3 and XCWR ) 1 - y2, it follows that
XCWR ) 1 - (1 - XPFR)0.95
(45)
For an nth-order reaction (n * 1), it follows similarly that
XCWR ) 1 - ((1 - 0.05)(1 - XPFR)1-n + 0.05)1/(1-n) (46) XPFR follows directly from eqs 34 and 35 as a function of the ratio Darad,0/Pe′. Darad,0 can also be expressed as a product of Pe′, which contains only observable quantities, and the ratio Darad,0/Pe′:
Darad,0 )
( )
Darad,0 Pe′ Pe′
(47)
It appeared that the results for a first-order reaction are easy interpretable by plotting the value of Pe′ ln{1/(1 - XCWR)} instead of Pe′ as a function of XCWR. This yields Figure 4. For almost all xCWR, the following criterion holds for a deviation of
