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A comprehensive mathematical model is developed to simulate the interactions of the complex processes that take place in typical catalytic chemical reactors. This mathematical model includes correlations representing various modes of mass transport and chemical reactions. To illustrate the application and value of this approach for reactor optimizations, the model is applied to the case of series reactions with a desirable intermediate compound and the risk of degradation of this compound if the process conditions are not optimized. The modeling results show that in such cases, which are very common in practice, replacing the conventional uniform catalyst distribution with a novel non-uniform distribution will significantly improve the performance of the reactor and the production of the desirable compound. Various catalyst distribution options are compared, and a novel non-uniform loading of catalyst is identified that gives a much better performance compared to the conventional approach. The model is versatile and useful for both the design as well as the optimization of the catalytic fixed-bed reactors in a wide variety of reactor and reaction conditions.

Many industrial catalytic processes involve multiple reactions, often in the form of a series of steps in which an intermediate compound in the series is the product of interest. In these reactor systems, the maximum conversion of the feed compound is not necessarily the best design or operation; instead, there is a certain level or range of conversion of the primary feed that gives the optimum yield of the desired product and often the best catalyst life and performance [

Among examples of these series reactions with desired intermediate products are hydrogenation [

Reaction time is the main parameter determining the extent of reaction for these cases. Consequently, the degree of conversion is primarily controlled by adjusting the residence time in the batch or mixed reactors (such as fluidized-bed). In the fixed-bed reactors, in addition to the residence time, other design and operation control options become available due to the spatial variation of concentration and reaction rates along the reactor bed. The idea behind this research is to use this spatial variation option and explore potential advantages of non-uniform catalyst distribution in the fixed-bed reactors. Therefore, the focus has been on the development and unitization of a methodology that would allow parametric analysis and evaluation of various catalyst activity distributions in the fixed-bed reactors. To achieve this, the methodology needs to be robust and flexible to allow considering and evaluating a wide range of reactor and reaction properties. Additionally, the intended analysis method should provide information about the effect of these non-conventional new catalyst distribution schemes on the deactivation and the life of the catalyst.

The method of approach adopted for this study has been to develop a comprehensive and yet versatile process model for fixed bed catalytic reactors running typical series reactions with desirable intermediate products, and undesirable and potentially deactivating end products. In the first part, the fundamental mass conservation methodology and correlations [

Specific intended results for this process modeling include: 1) determining the concentration profiles of all compounds participating in the process, the production rate of the desirable compounds and the variation of these key properties with time and location in the reactor; 2) quantifying the potential formation and deposition of the undesirable compounds on the catalyst surface and the effect on the performance of the reactor over time.

The simplified schematic of a typical fixed-bed reactor and the process steps occurring in it are shown in

A ( g ) ↔ R ( g ) (1)

R ( g ) ↔ S ( g ) (2)

In the first reaction, the transformation of A yields the compound of interest, R. However, R can degrade by further non-catalytic reaction to an undesirable compound, S. To include the potential catalytic deactivation, typically observed in these series reactions, further transformation of S is included to represent the formation and deposition of a condensed species P on the catalyst surface; this deactivation reaction is shown by Equation (3):

S ( g ) ↔ P ( s ) (3)

The model includes convective and dispersive mass transfer steps in both bulk fluid and through the catalytic packed bed in addition to these reactions.

The rates of formation and consumption of the above species are given by Equations (4) to (7). For the fluid-phase compounds A, R, and S, the concentrations (C_{A}, C_{R}, and C_{S}) are in mol/m^{3} and the rates are in mol/m^{3}×s. For the condensed compound P, the rate is in mol/m^{2}×s. All reactions are assumed to be first order with respect to the reacting species. However, when catalyst sites are involved in the reaction, the reaction rate also depends and is proportional to the number of available catalytic sites (Z-C_{P}); this is the case for the main reaction of A going to R and its reverse as well as the deactivation reaction of S going to P. In this expression, Z is the surface concentration of active catalytic sites and C_{P} is the surface concentration of deactivated sites occupied by compound P. Both Z and C_{P} vary with time and location in the reactor and are in units of mol/m^{2}. In these equations, a represents the ratio of the catalytic surface area to the volume of the reactor in m^{−1}. The intrinsic reaction rate constants are in units of m^{2}/mol×s for k_{1}, k_{1r}, and k_{3}; and s^{−1} for k_{2}, k_{2r}, and k_{3r}.

Net rate of production of A = − k 1 C A ( Z − C P ) + k 1 r C R ( Z − C P ) (4)

Net rate of production of R = k 1 C A ( Z − C P ) − k 1 r C R ( Z − C P ) − k 2 C R + k 2 r C S (5)

Net rate of production of S = k 2 C R − k 2 r C S − k 3 C S ( Z − C P ) + a k 3 r C P (6)

Net rate of production of P = 1 a k 3 C s ( Z − C P ) − k 3 r C P (7)

The transport of species in the fluid phase by both convection and dispersion were included in the model. Considering the above steps, Equations (8) to (11) are derived to represent the mass balance for participating compounds and their variation with time, t, and location along the packed bed, x. For the following expressions, υ is the velocity of the fluid in m/s, and D_{i} is the dispersion coefficient for compound i in m^{2}/s.

∂ C A ∂ t − D A ∂ 2 C A ∂ x 2 + υ ∂ C A ∂ x = − k 1 C A ( Z − C P ) + k 1 r C R ( Z − C P ) (8)

∂ C R ∂ t − D R ∂ 2 C R ∂ x 2 + υ ∂ C R ∂ x = k 1 C A ( Z − C P ) − k 1 r C R ( Z − C P ) − k 2 C R + k 2 r C S (9)

∂ C S ∂ t − D S ∂ 2 C S ∂ x 2 + υ ∂ C S ∂ x = k 2 C R − k 2 r C S − k 3 C S ( Z − C P ) + a k 3 r C P (10)

∂ C P ∂ t = 1 a k 3 C s ( Z − C P ) − k 3 r C p (11)

The boundary conditions for the inlet (x = 0) and the outlet (x = L) are described in Equation (12) and Equation (13), whereas the initial conditions used for all cases are shown in Equation (14). In these expressions, L is the length of the reactor in m; and C_{i,o}, C_{i,in}, and C_{i} are the inlet concentrations of species i in the reactor feed, the initial concentrations of species i (t = 0) and the concentrations of species i, respectively, all of them in mol/m^{3}.

υ ⋅ C i , o = υ ⋅ C i − D i ⋅ d C i d x at x = 0 (12)

d C i d x = 0 at x = L (13)

C i = C i , i n at t = 0 (14)

Among the key performance indicators for the reactor are the net consumption and net production rates of various species, and the cumulative conversion of reactant. Similarly, the total poison at any time over the length of the reactor is proposed to describe the deactivation process. The general equations for these indicators are given by Equations (15) to (18). In these expressions, C_{i,out} is the concentration of that species at the outlet in mol/m^{3}, and d is the diameter of the reactor in m.

Net rate of production of i = π ⋅ d 2 4 ⋅ υ ⋅ ( C i , o u t − C i , o ) (15)

Net rate of consumption of i = π ⋅ d 2 4 ⋅ υ ⋅ ( C i , o − C i , o u t ) (16)

Comulative conversion = ∫ 0 t π ⋅ d 2 4 ⋅ υ ⋅ ( C i , o − C i ) d t (17)

Total poison = ∫ 0 L C p ⋅ a ⋅ π ⋅ d 2 4 d x (18)

To allow a comprehensive study of the non-uniform catalyst distribution option in the model, the initial distribution of catalyst in the reactor, Z, is considered a function of location in the reactor. This parameter represents the starting property of catalyst loading in the reactor before deactivation. _{o} is the value of Z for the uniform distribution. The total amount of catalyst was the same in all the distributions tested.

A parametric case study was performed to compare potential options for catalyst distribution, using profiles shown in

Parameter | Variable | Value |
---|---|---|

Catalytic surface area divided by reactor volume | a | 1000 m^{−1} |

Concentration of compound i in reactor feed | C_{i,o} | C_{A,o} = 20 mol/m^{3} C_{R,o} = 0 mol/m^{3 } C_{S,o} = 0 mol/m^{3} C_{P,o} = 0 mol/m^{3} |

Initial concentration of compound i | C_{i,in} | C_{A,in} = 20 mol/m^{3 } C_{R,in} = 0 mol/m^{3 } C_{S,in} = 0 mol/m^{3} C_{P,in} = 0 mol/m^{2} |

Dispersion coefficient of compound i | D_{i} | D_{A} = 1 × 10^{−7} m^{2}/s D_{R} = 1 × 10^{−7} m^{2}/s D_{S} = 1 × 10^{−7} m^{2}/s |

Forward reaction rate coefficient of compound i | k_{i} | k_{1} = 4 × 10^{−1} m^{2}/mol·s k_{2} = 1 × 10^{−2} s^{−1 } k_{3} = 4 m^{2}/mol·s |

Reverse reaction rate coefficient of compound i | k_{ir} | k_{1} = 1 × 10^{−3} m^{2}/mol·s k_{2r} = 0 s^{−1 } k_{3r} = 0 s^{−1 } |

Reactor length | L | 0.5m |
---|---|---|

Fluid velocity | υ | 0.5 m/s |

Value of Z for uniform distribution | Z_{o} | 4 mol/m^{2} |

Reactor diameter | d | 0.3 m |

Equations (8) to (16) were solved numerically [

The effect of change in catalyst distribution on the performance of the reactor (given in terms of the production rate of R, which is the desired product) is given in

The comparison of production rate of R for conventional uniform distribution with the best of the non-uniform options studied here (corresponding to m = 0, referred to as preferred distribution) can be seen in

Another key consideration in improving the performance and optimizing the design of the fixed-bed catalytic reactors is prolonging the life of catalyst and inhibiting processes that result in the loss of active sites due to various deactivation mechanisms. In many series reactions of the type studied here, a key deactivation mechanism is the ultimate formation of compounds in the sequence of reactions that would finally adsorb on catalytic sites and poison or decrease the catalytic activity. For example, in case of many hydrocarbon-processing reactions, a potential degradation reaction is the formation of heavy organic molecules that are attracted and adsorbed on the same sites—a process that destroys catalysts availability for the desirable reactions [

The best way to mitigate the wasting of catalyst and to preserve its activity and lifetime is to minimize its presence in locations where the concentration of R is

high, as previously shown in

poison (deactivated sites) in the reactor is shown and compared for the conventional uniform and the proposed non-uniform catalyst distribution cases.

The mathematical process modeling is shown to be valuable for optimization of catalyst distribution in typical fixed-bed chemical reactors. The comprehensive model developed in this study includes various modes of mass transport and chemical reactions as well as their complex interactions. To illustrate the application and the value of this model, the case of series reactions is considered which consists of a desirable intermediate compound followed by the chemical degradation of this compound if the process conditions are not optimized. The modeling shows that in such cases, which are common in practice, replacing the conventional uniform catalyst distribution with a novel non-uniform distribution will significantly improve the performance of the reactor and the production of the desirable compound. Various non-uniform options are considered and compared using the developed process model. The results show that for these series reactions, the preferred option is an unconventional approach where the catalyst loading is highest close to the inlet and decreases along the reactor bed. In addition to improving the production selectivity, the proposed distribution also prolongs the catalyst life. The model is versatile and can be used for both design of new systems as well as optimization of the operating conditions in existing reactors in a wide variety of reactor and reaction conditions. For the future work, further analysis of the deactivation process for other types of multiple reactions will be conducted and an optimization subroutine will be developed and added to the model for design and process tuning applications.

The support for this study was provided by NSF/SRC Engineering Research Center and its industrial members.

The authors declare no conflicts of interest regarding the publication of this paper.

Martínez, V.A.C. and Shadman, F. (2020) Improving the Performance of Fixed-Bed Catalytic Reactors by Innovative Catalyst Distribution. Journal of Applied Mathematics and Physics, 8, 672-683. https://doi.org/10.4236/jamp.2020.84052