Ind. Eng. Chem. Res. 2007, 46, 3871-3876
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Coated-Wall Reactor ModelingsCriteria for Neglecting Radial Concentration Gradients. 2. Reactor Tubes Filled with Inert Particles 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 limitation on the conversion in a coated-wall reactor filled with inert particles was investigated in a modeling study. Criteria are developed to allow for neglecting the radial concentration gradients and to use the simple plug-flow model for performance testing and kinetic studies using the measured conversion. The simulations were carried out using Athena Visual Studio 10.0 for several reaction orders in a tubular reactor filled with inert particles of different sizes. It was verified that the entrance effects, which enhance the radial mass transfer, can be neglected in most typical situations. The criterion for the conversion is a function of the modified Pe´clet number (Pe′), the reaction order (n), and a constant a that depends on the reactor geometry and the internal tube radius/particle size ratio: XCWR < 1.42a/(a + nPe′). The criterion is valid for both gases and liquids as long as the viscous contribution to the pressure drop is much larger than the turbulent contribution. This applies for most gas-phase applications and some liquidphase applications. At conditions where the turbulent contribution becomes comparable or larger than the viscous contribution, the criterion for the modified Pe´clet number becomes stricter. In that case, the criterion with assumed dp ) 0 can be used as a conservative estimate. The most relaxed criterion is obtained for relatively large particles, up to dp/dt ) 0.4. Introduction This paper is a continuation of our previous manuscript (part 1) about catalytic coated-wall reactors,1 which focuses on the deviation of the conversion in empty tubes from plug-flow behavior due to finite radial mass transport. In this paper, the same approach is extended to coated-wall tubes filled with a bed of inert particles to enhance the radial transport. The results in the previous paper showed that, in coated-wall reactors (CWRs), significant radial concentration gradients may develop, which may negatively influence the conversion and may also influence the selectivity if that is an issue. A major cause of the radial concentration gradients is the laminar flow, which has a parabolic velocity profile, as illustrated in Figure 1. This causes a kind of bypass effect for the relatively fast flowing fluid in the center of the tube. This effect can be significantly suppressed by filling the empty volume with inert particles, which flattens the velocity profile and, thus, slows down the flow in the center. If relatively large inert particles are applied, the bed porosity nearby the walls will become smaller, which will result in an increase of the flow alongside the wall, as illustrated in Figure 2. On one hand, this will further decrease the radial mass transfer limitation, but on the other hand, it will cause a broader residence time distribution, which may influence the conversion negatively. This paper focuses on the radial concentration gradients in the bed of inert particles with a catalytic coating at the reactor wall. Limitations inside the wall coating are not considered, which from a modeling point of view can be translated in a coated wall with a thickness of zero. The simulations consist of solving the momentum balance to determine the radial velocity profile, followed by solving the mass balances to determine the effect on the conversion. These results are used to derive criteria for the maximum allowed conversion at which one is still allowed to use the simple plug-flow reactor model * Corresponding author. Tel.: +31-15-2784316. Fax: +31-152785006. E-mail:
[email protected].
Figure 1. Typical flow pattern and radial concentration profile in an empty CWR.
Figure 2. Typical flow pattern and radial concentration profile in the CWR filled with inert particles.
without taking radial concentrations into account to determine the intrinsic reaction kinetics with acceptable accuracy. Reactor Filled with Inert Particles 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,
10.1021/ie061232v CCC: $37.00 © 2007 American Chemical Society Published on Web 02/23/2007
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b(x)
Ind. Eng. Chem. Res., Vol. 46, No. 12, 2007
(
)
Dax,A(x) ∂2y u(x) ∂y dy )0) + dt L ∂z L2 ∂z2 2 1 1 ∂ ∂y xDrad,A(x) (1) R x ∂x ∂x
() (
)
where b(x) is the bed porosity, u(x) is the superficial axial velocity, and Drad,A(x) is the radial dispersion, all dependent on the radial position and the local bed porosity. For steady-state conditions and after neglecting the effect of axial dispersion, the molar balance becomes
(
)
Drad,A(x) ∂y u(x) ∂y 1 1 ∂ + x 0)u0 ∂z Pe′ x ∂x D0rad,A ∂x
(2)
where the standard radial dispersion D0rad,A is defined as
D0rad,A ) Drad,A(b(x) ) ∞b , u(x) ) u0)
(3)
and ∞b ) porosity of the infinitely extended bed and u0 ) superficial velocity over the whole bed. The solution of this partial differential equation (PDE) is subjected to the following boundary conditions:
Axial direction (for 0 e x e 1): ∂y z ) 1: )0 ∂z
z ) 0: y ) 1
(4)
Radial direction (for 0 e z e 1): x ) 0: x ) 1:
∂y )0 ∂x
rwallR ∂y )∂x CA0Drad,A(x ) 1) ) -Darad,0
D0rad,A
(5)
[ ]
r(y) Drad,A(x ) 1) r(1)
r0,wallR
(6)
and r0,wall ) rwall|CA0
CA0D0rad,A
(7)
rwall|CA0 ) kwallCnA0
(8)
kwallCn-1 A0 R
(9)
D0rad,A
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.
u0L
(RL) ) D (RL) 2
0 rad,A
2
and Darad,0 )
(11)
For radial dispersion, m ) 0.1, and for axial dispersion, m ) 0.5. The static term consists of the bulk diffusivity (DAm) and the tortuosity of the bed (τb). According to Wakao et al., this tortuosity typically ranges from 1.25 to 1.67. Edwards and Richardson3 reported a similar value, 1.37, while Suzuki and Smith4 reported a slightly higher value (2.3). In the simulations, however, the correlation between the bed porosity and the bed tortuosity reported by Puncochar and Drahos5 was used:
1
(12)
xb
This yields the following correlation of the radial dispersion Drad,A as a function of the local porosity b(x) and local superficial velocity u(x):
Drad,A(x) ) {b(x)}1.5DAm + mdpb(x)u(x)
{
[
b(x) ) ∞b 1 + 1.36 exp -5.0
which results in
Pe′ ) Pe
DA DAm ) + mdpu0 b τb
(13)
Additionally, the equation from Giese6,7 is used for the radial porosity as a function of the radial position and the ratio of the particle diameter and the tube internal diameter,
For an nth-order reaction, the following applies,
Darad,0 )
transport (residence time). The larger this parameter is, the poorer is the reactor performance. To solve the balance, a correlation for the radial dispersion, Drad,A(x), is required. For this, the correlation for axial and radial dispersion in fixed beds (DA) reported by Wakao et al.2 was used:
τb )
with
Darad,0 )
Figure 3. Radial porosity profile obtained with Giese’s correlation for various R/dp ratios.
rwall,0R CA0D0rad,A (10)
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
R (1 - x) dp
]}
(14)
where x ) r/R and ∞b ) porosity of the infinitely extended bed. This equation is valid for particles that are nearly (but not perfectly) monodispersed in size and nearly (but not perfectly) spherical in shape, and it describes well the experimental results by Giese for real catalyst beds. Some bed porosity profiles resulting from this correlation are shown in Figure 3. For obtaining the velocity profile, the momentum balance or the extended Brinkman equation has to be solved by forcing the pressure drop ∂P/∂z to be constant over the bed radius,
(
)
µF,eff ∂ ∂u(r) ∂P r ) -f1u(r) - f2{u(r)}2 + ∂z r ∂r ∂r
where, using the pressure drop equation from Ergun,8
(15)
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f1 ) 150
(1 - b(r))2 µF (b(r))3
dp2
f2 ) 1.75
µF,eff ) 2 exp(0.002Rep)µF
1 - b(r) FF (b(r))3 dp (16)
Rep )
FFu0dp µF
(17)
The radial integration was performed using the finite elements method with 1 000 equidistant discretization points. Some velocity profiles at different R/dp ratios at typical reaction conditions and for the same average superficial velocity u0 are shown in Figure 4. At the selected conditions shown in the caption of Figure 4, the viscous term of the pressure drop correlation, i.e., the term with f1 in eq 15, is much larger than the turbulent term, i.e., the term with f2 in eq 15. This is typical for most gas-phase applications except for cases at very high pressure or very high flow rates. For liquids, this is only true for low flow rates or liquids significantly more viscous than water. If the turbulent term becomes comparable or even larger than the viscous term, the velocity profile will become flatter than that shown in Figure 4. The PDE system consisting of the calculation of the velocity profile, eq 15, and the molar balance, eq 2, was solved using the package Athena Visual Studio9 using orthogonal collocation with 4 internal collocation points, on 25 finite elements with breakpoints set at the following positions: 0.1; 0.2; 0.3; 0.4; 0.5; 0.6; 0.7; 0.75; 0.8; 0.84; 0.87; 0.9; 0.92; 0.94; 0.96; 0.97; 0.98; 0.985; 0.99; 0.993; 0.995; 0.997; 0.998; 0.999. The increasing breakpoint density near the wall was necessary to achieve accurate simulations at reasonable CPU times.
∆x ≡
XPFR - XCWR (1 - y3) - (1 - y2) y2 - y3 ) ) XPFR 1 - y3 1 - y3
(21)
The relative deviation of the conversion from that at plug-flow behavior ∆X and of the rate constant ∆k depend on Darad,0, Pe′, and the R/dp ratio as major variables. Additionally, the values of ∞b , the fluid viscosity, the fluid density, and the bulk diffusivity will have some influence on the result. Since the value of ∞b is typically ∼0.376 and since the influence of this value on the results appeared to be small, only the simulation results for ∞b ) 0.37 are included. The results also showed very little dependency on the fluid viscosity and the density; therefore, these values were not varied either and were fixed at the values for air at 500 K and atmospheric pressure (FF ) 0.703 kg/m3, µF ) 2.63 × 10-5 kg/ms). Although the direct influence of the bulk diffusivity is already incorporated in the variables Darad,0 and Pe′, the value of the bulk diffusivity is also influencing the results via the correlation for the radial dispersion, i.e., eq 11. Also, this influence appears to be negligible, even at conditions where the second term of eq 11 becomes large compared to the first term, which often occurs with liquids. Apparently, the effect of this second term to the radial dispersion is well-distributed over the entire cross section. Apparently, the thickness of the stagnant zone at the wall is too small to cause a significant diffusion resistance. The bulk diffusivity used in the calculations is 5.1 × 10-5 m2/s, which is the diffusivity of oxygen in air at atmospheric pressure and 500 K.
Conversion Calculations The conversion of the coated-wall reactor can be calculated from the total molar flows in and out,
XCWR )
FA0 - FA,out FA0 - y2FA0 ) ) 1 - y2 FA0 FA0
(18)
where y2 is defined as the average dimensionless concentration of the reactant A. For the conversion of the reactor in which ideal plug-flow behavior is assumed (XPFR), it follows for a first-order reaction (where y3 is defined as the dimensionless concentration of the reactant A in the imaginary plug-flow reactor) that
xPFR )
(
Figure 4. Axial superficial velocity profiles obtained from solving eq 15 for various R/dp ratios (using R ) 0.01 m, u0 ) 0.0050 m/s, ∞b ) 0.37, FG ) 1.0 kg/m3, and µF ) 2 × 10-5 Pa s).
)
FA0 - y3FA0 Darad,0 ) 1 - y3 ) 1 - exp -2 FA0 Pe′
(19)
and, for an nth-order reaction (n * 1),
(
xPFR ) 1 - y3 ) 1 - 1 + 2(n - 1)
)
Darad,0 Pe′
1/(1-n)
(20)
For the derivation of these equations, see our previous paper.1 Results, Discussion, and Derivation of the Criteria Deviation as a Function of Darad,0 and Darad,0/Pe′. From the solution at the reactor exit, the relative deviation of the conversion from that at plug-flow behavior ∆X can be calculated:
Figure 5. Contours of the relative deviation of the conversion (∆X) for a first-order reaction in a circular reactor filled with inert particles plotted as a function of Darad,0/Pe′ and Darad,0, u0 ) 0.02 m/s, ∞b ) 0.37, FF ) 0.703 kg/m3, µF ) 2.63 × 10-5 kg/ms, R ) 10 mm, and dp ) 2.0 mm.
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Figure 7. Values of the numerator (b) and the parameter a (0) in the denominator of the correlation for the criterion for ∆k < 0.05 for a reactor filled with inert particles, as a function of dp. Figure 6. Contours of the relative deviation of the calculated rate constant (∆k) for a circular reactor filled with inert particles plotted as a function of Darad,0/Pe′ and Darad,0; ∞b ) 0.37, FF ) 0.703 kg/m3, µF ) 2.63 × 10-5 kg/ms, R ) 10 mm, and dp ) 2.0 mm.
Since, besides the influence of Darad,0 and Pe′, the reactor radius R hardly influences the results if the R/dp ratio is kept constant, the reactor radius was fixed at 10.0 mm and the particle size dp was varied to obtain R/dp ratios covering the range 1.2520. Figure 5 shows the contour plot of the deviation of the conversion ∆X obtained from the simulations as a function of Darad,0/Pe′ at R/dp ) 5. The usual criterion is that the conversion may not deviate >5% from the value in the ideal plug-flow reactor. At Darad,0/Pe′ ≈ 0.2, the maximum allowed value of Darad,0 is the lowest: 0.6. According to eq 19, a value of Darad,0/ Pe′ of 0.2 corresponds with a plug-flow conversion of 0.33. At higher conversion, the criterion for Darad,0 becomes less strict because of the approach to full conversion. 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 1.4. For kinetic studies, inspection of the relative deviation in the rate constant, ∆k, is more useful,
kw,PFR - kw,CWR kw,PFR
∆k ≡
(22)
where kw,CWR is the real rate constant of the coated wall and kw,PFR is the rate constant calculated from the measured conversion based on the assumption of ideal plug-flow behavior, thus neglecting radial concentration gradients. For a first-order reaction, it follows that1
∆k )
ln(y3) - ln(y2) ln(y3)
Figure 8. Maximum allowed conversion in a filled circular reactor taking into account the variation of the bed porosity as a function of the tube radius to obey the criterion for ∆k < 0.05 in eq 25 for a first-order reaction and Pe′ ) 1. The dotted line represents the maximum allowed conversion for an empty tube.
and ∆k were larger at higher R/dp ratios, i.e., small particles, and smaller at lower R/dp ratios, i.e., large particles. Expressing the Criterion as a Function of XCWR. The criterion for the 5% limit of ∆k can be expressed as a function of only observable parameters, i.e., the conversion XCWR, the modified Pe´clet number Pe′, and R/dp, of a similar shape as for an empty tube, which fitted the simulation results well at all R/dp ratios. The optimal values of the parameters in the correlation as a function of the R/dp ratio are shown in Figure 7. The dependency on the R/dp ratio can be described well with the following correlation (valid for all R/dp and XCWR):
XCWR
0.37 R for
(24)
Figure 6 shows the results of ∆k as a function of Darad,0 and Darad,0/Pe′ in a contour plot. Like for the empty reactor case, the relative deviation ∆k becomes largely independent of the value of Darad,0/Pe′ if that value is >0.1, which corresponds to a plug-flow conversion of 0.18. The contour plots obtained at other R/dp ratios are similar to the ones shown above, although the numerical values of ∆X
Figure 8 shows the results of an application of the correlation. The figure shows the maximum allowed conversion to obey the criterion for an arbitrarily chosen value of Pe′ ) 1.0 and a reaction order of 1.0. The graph also shows the criterion for the empty tube for comparison. The results clearly show the beneficial effect of filling the tube with particles, particularly for relative large particles having a size of 0.37R or more. However, applying such large particles that only three or less particles fit in the tube diameter, i.e., dp > 0.67R, is discouraged
Ind. Eng. Chem. Res., Vol. 46, No. 12, 2007 3875
since that may easily result in significant inhomogeneities in the fluid flow along the wall.10 Apparently, the disadvantageous effect of the larger particles of broadening the residence time distribution that was mentioned in the introduction is of less importance compared to the advantageous effect on the radial transport. The correlations for the criterion reported here depend on the chosen correlations used for the porosity profile, eq 14, the pressure drop, eq 15, and that for the radial dispersion, eq 11. Since the three correlations used are widely used and accepted, and since the observed trends seem realistic, the criterion can be assumed to be valid in the whole range covered (0 e n e 2 and R/dp e 0.8). The criterion is valid for both gases and liquids as long as the viscous contribution to the pressure drop is much larger than the turbulent contribution. This applies for most gas-phase applications and some liquid-phase applications. At conditions where the turbulent contribution becomes comparable or larger than the viscous contribution, the criterion for the modified Pe´clet number becomes stricter. In that case, the criterion with assumed dp ) 0 can be used as a conservative estimate. Example of Application of the Criterion. A demonstration of the applicability of the criterion is given using a case study performed by Redlingsho¨fer et al.,11 who studied the catalytic partial oxidation of propene to acrolein on a bismuth-molybdate catalyst consisting of particles between 250 and 400 µm embedded in a 300-600 µm thick montmorillonite cement layer at the wall of a tube with a length of 8 cm and an internal diameter of 10 mm. The tube was filled with inert glass spheres (0.75 < dp < 1.0 mm; the particle size used in the simulations is 0.875 mm) to improve the radial dispersion and the plugflow behavior. The value of dp/R is, thus, 0.175. For the simulation, the experiment at Wcat/Ftot ) 16.0 kg s mol-1, ypropene ) 0.068, yO2 ) 12.2, yH2O ) 0.034 (remainder N2), 703 K, and 1.013 bar is used at which the propene conversion amounts to 60% (taken from Figure 5 in the paper). For simplicity, the reaction is assumed to be first order in propene. The gas viscosity used is 3.5 × 10-5 kg/m/s, the gas density is 0.6 kg/m3, and an estimated diffusivity of propene in the gas mixture is 5.56 × 10-5 m2/s. The composition of the washcoat layer used is 50% bismuth-molybdate catalyst with a solid density of 6 070 kg/m3 and 50% montmorillonite cement with a solid density of 2 000 kg/m3. The porosity of the coating is assumed to be 0.6. For the radial dispersion, the equations from Wakao et al.,2 i.e., eqs 11 and 12, with ∞b ) 0.37, are used. For this experiment, the average superficial velocity was calculated to be 0.0289 m/s, which results in a particle Reynolds number of 0.434. From this, the value of D0rad,A was calculated to be 1.51 × 10-5 m2/s and the value of Pe′ was calculated to be 2.39. The criterion requires that
be 0.66. This corresponds with a 10% deviation in the conversion and a 17% deviation in the rate constant. The criterion also shows that a better behavior would have been obtained if larger particles of, e.g., 2 mm (dp/R ) 0.4, thus a ) 1.1) had been used. In that case, the maximum allowed conversion would be 0.47 (with Pe′ ) 2.20, obtained from D0rad,A ) 1.64 × 10-5 m2/s), which is closer to the experimental value of 0.60. Since increasing the particle size over dp/R ) 0.4 does not help any further, other measures are required to obtain the required accuracy to allow the use of the plug-flow model, e.g., decreasing the diameter or increasing the length of the reactor tube. Obviously, even much worse results would have been obtained when the tube would have been left empty. In that case, the criterion for the empty tube results in1
XCWR
