# Mathematical Modelling Of Time Dependent Processes Biology Essay

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The heat and mass transfer within the monolith channels, the applications and advantages of monolithic reactor/catalyst have been discussed. The mathematical model for the reaction and diffusion processes taking place in a single channel of the monolith reactor as the fluid flows through was presented. The model was used to simulate the catalytic combustion of methane in the monolith reactor; the numerical solution of the mathematical model was obtained by means of the finite element method using COMSOL Multiphysics (Liu et al., 2005; Ghadrdan and Mehdizadeh, 2008). The temperature, concentration and velocity profile within the channel for the bulk gas and porous catalyst layer phases as the combustion reaction proceed were presented. The performance of the monolith reactor was evaluated based on the Sherwood number and the effectiveness factor of the catalyst layer. The mathematical model is significant for the design and optimization of the monolith reactor structure/configuration, synthesis of control strategies, and the prediction of the monolithic reactor behaviour under different operating conditions.

A monolithic reactor consists of a large number of parallel channels often in a form of honeycomb structure or arrangement with the wall of each channel coated with a porous catalyst layer (or washcoat). The parallel channels may have the following geometries circular, square, triangular, rectangular, hexagonal, or of other shapes cross sections, a typical cross section of a monolith reactor is shown in Fig.1. The monolith substrate material is either ceramic or metallic, and can act as a structured catalyst as well as a reactor (Tomasic, 2007). Thus for application purposes the shape, channel size, catalyst layer thickness, microstructure and porosity as well as the channel wall thickness depends on the process requirements (Cybulski and Moulijn, 1998; Tomasic, 2007). Monolithic reactors are widely used for environmental pollution control such as in the cleaning of exhaust gas from power plant, gas turbine and in automobile industry for vehicle exhausts to convert NOx and CO via fast gas phase reactions (Werner, 1994; Chen et al, 2008). Other applications include catalytic combustion, hydrocarbon processing, hydrogenation or dehydrogenation, catalytic oxidation (Chen et al, 2008). Compared to conventional catalytic bed reactors such as fixed bed, slurry, and triple bed reactors monolithic reactors are advantageous due to: high surface area to volume ratio, low pressure drop, no hot spot within the channels, elimination of external mass transfer and internal diffusion limitations, low axial dispersion and back mixing, high selectivity, elimination of plugging and fouling of catalyst, extension of catalyst life span, easy to scale-up, high temperature stability, high mechanical strength, and ease of orientation in a reactor (Tomasic, 2007; Chen et al, 2008; James et al, 2003; Tomasic and Gomzi, 2004).

Therefore, mathematical modelling of the time dependent physical, simultaneous diffusion and chemical reaction occurring inside the porous washcoat as the gas flow through the channels will enhance the understanding of the process complexity and predict the performance of the monolithic reactors (Tomasic et al, 2004). Most mathematical models of monolithic reactors are on single channel with the assumption that every channel in the monolith reactor behaves the same, therefore can represent the mean behaviour of the entire channels and account for the diffusion and reactions inside the catalyst layer and in the channel (Bercic, 2001; Chen et al., 2008). However, in some circumstance a single channel model might be inadequate therefore James et al. has developed multi-channel mathematical model of the monolithic reactor to answer many questions which single channel model could not provide answers to such as the non-uniformity of flow distribution at the inlet of the monolith reactor and the effect of channel-to-channel heat conduction (James et al., 2003). In general, it is simpler and logically reasonable to develop mathematical model for a single channel to characterise the behaviour of the whole monolithic reactor since every channel within the monolith reactor structure are identical. In view of this, depending on the purpose of the model, a single channel model in 1, 2 or 3-dimensions to simplify the complexity of the heat and mass transport within the channels can be used to describe the mean behaviour of the monolithic reactor and requires much less computation effort than multi-channel (Chen et al., 2008). Schildhauer et al. developed mathematical model of film flow monolith for reactive stripping in monolithic reactor which involves both reaction and gas-liquid separation (Schildhauer et al., 2005).

The mathematical models of monolithic reactors for pollution control such as the treatment of automotive exhaust gas with considerations to the catalytic activity due to the decomposition of nitrogen monoxide (NO), oxidation of CO and HC within the channels of the monolith reactor has been reported (Tomasic and Gomzi, 2004; Jirat et al., 1999; Tronconi et al., 1992). Joshi et al. developed a one dimensional model for the washcoat (catalytic layer) of a monolithic reactor for various geometric cross section based on the internal mass transfer coefficient estimated using Sherwood number in relation to Thiele modulus to simplify the diffusion and reaction within the washcoat (Joshi et al., 2009). The mathematical modelling and simulation of the heat and mass transfer in the monolith reactor will aid the prediction of the concentration and temperature profiles in the axial fluid flow and along the channel of the reactor. The solutions of the model equations can be obtained using efficient and effective numerical software such as MATLAB, COMSOL Multiphysics, and other computational fluid dynamics (CFD) software (e.g. Fluent) (Chen et al., 2008). Hayes et al. reported the numerical investigation of the diffusion and reaction in the washcoat of the monolithic reactor, analysing the effect of washcoat geometry and thickness on mass transfer and the effect of various channel cross section shapes using finite element solver tool COMSOL (Hayes et al., 2004). Ghadrdan and Mehdizadeh reported a finite element model for the simulation of a single channel catalytic combustion of methane in monolithic reactor using COMSOL multiphysics (Ghadrdan and Mehdizadeh, 2008). The flow of fluids in the monolithic reactor channels is mostly laminar flow (Kolaczkowski, 1999).

The mathematical model of the monolithic reactor will be useful in analysing the heat and mass transfer effects, evaluation of monolith reactor performance and optimization monolith structure and configuration, and the design of the reactor (Kolaczkowski, 1999). However, the main challenges of monolithic reactor modelling are the validation of the model which is due to the difficulties in obtaining adequate experimental data for simulation and comparison and the large number of parameters to be estimated such as heat and heat transfer coefficients, kinetic expression, physical properties (Kolaczkowski, 1999; Tomasic and Gomzi, 2004; Chen et al., 2008). Pinkas et al. investigated and simulated the CO oxidation in monolithic reactors using a one dimensional two-phase model with axial dispersion to describe the heat and mass transport during the process (Pinkas et al., 1996). The study of monolithic reactors reported in literature ranges from monolith preparation and extrusion methods, its application to numerous reaction schemes, the heat and mass transfer within the reactor, the flow regime and their hydrodynamics, mathematical modelling of the diffusion and reactions within the channels, estimation of the heat and mass transfer coefficients, and the use of computational fluid dynamics (CFD) software to simulate the system and to aid optimization and design. Ghadrdan and Mehdizadeh modelled the interaction between the fluid flow in the channel and the porous catalyst layer (washcoat) by the Brinkman-Forchheimer extended Darcy flow model in a porous medium (Ghadrdan and Mehdizadeh, 2008). Liu et al. reported a novel conceptual design and CFD simulation of monolithic reactor with enhanced mass transport characteristic, which was accomplished by inserting porous substrate on the walls of the channels (Liu et al., 2005).

The objective of this report is to present a mathematical model of the monolithic reactor for catalytic combustion of methane, the solutions to the model is also sort using COMSOL Multiphysics software, the results obtained are also discussed and the significance of the model are highlighted.

Figure 1: Typical representation of monolithic reactor cross section and channel

## 1.2 Heat and Mass transfer and reaction in monolithic reactors

The gas solid reaction in the monolith channel is controlled by external mass transport at the gas-solid interface and/or internal diffusion within the catalyst layer (Liu et al., 2005). Therefore, for reaction to occur inside the monolithic reactor washcoat (catalyst layer), the reactants have to diffuse from the channels wall surface into the porous structure of the washcoat (Chen et al., 2008). This implies a simultaneous diffusion and reaction taking place within the washcoat (catalyst layer), in which reactants diffuses in while products diffuse out and are swept away by the bulk flow through the channel. External mass transfer from the fluid in the channel to the surface of the washcoat may be rate limiting or the internal diffusion inside the catalyst layer quantified by the effectiveness factor of washcoat may be the reaction rate controlling step. However, the internal diffusion inside the monolith reactor catalyst layer is dependent on the thickness of the washcoat and its porosity (Chen et al., 2008). Hayes and Kolaczkowski mathematical model and simulation investigation of a monolith reactor with laminar flow shows that the transition from kinetic to mass transfer control of the reaction within the channel of the reactor depends on the channel size, reaction kinetics, inlet conditions, diffusion coefficient and the length of the channels (Hayes and Kolaczkowski, 1994). The heat and mass transfer coefficients in monolith reactors are estimated mainly by using dimensionless quantities Sherwood number (Sh) and Nusselt (Nu) number respectively, correlated as functions of diffusion coefficient, monolith length, Graetz number, Peclet number, etc. (Chen et al., 2008). Gonzo and Gottifredi reported that the external and internal diffusion and reaction inside the catalyst layer (washcoat) are characterised by the catalyst effectiveness factor (Gonzo and Gottifredi, 2010).

For a non-adiabatic condition in monolithic reactor, the solid and fluid temperature profile radially and axially in the channels due to heat transfer is required to evaluate the reactor performance. Also if the reaction within the monolith is highly exothermic, then metallic monolith support of good heat conduction properties is a good choice to use (Chen et al., 2008). Besides, adequate reaction kinetics is required for correct estimation of the heat released and transferred by an exothermic reaction in the monolithic reactors. Chen et al. reported that the vast reactions in the washcoat and their kinetic may be complex and therefore difficult to account for all the reaction and also measuring the time dependent temperature and concentration response inside the monolithic reactor to validate the model has made mathematical modelling of the monolithic reactors highly challenging (Chen et al., 2008). Moreover, to understand the heat transfer, diffusion and chemical reactions taking place within the channels and washcoat of the monolith reactor mathematical modelling becomes an essential tool.

## 2.1 Mathematical model of a single channel of a monolithic reactor

The design and performance of a monolithic reactor is a function of the channel geometry, length and dimension of the channel, the thickness of the washcoat, operating conditions, the properties of the catalyst species, and the kinetic of the reactions (Tomasic, 2007). Therefore, a mathematical model that will account for all these parameters will be of varying complexities, but will aid optimization of the reactor. The diffusion and chemical reaction in the channel coupled with the heat and mass transfer between the fluid and solid catalyst phase is shown in Fig.2, the models were developed based fundamental laws of mass, energy and momentum conservations with the following assumptions for simplifications:

The entire monolith channel sizes and shapes are identical for single channel as well as for the whole monolithic reactor.

Uniform fluid distributions among the entire channels.

Uniform distribution of catalyst activity within each of the channels.

The monolithic reactor material is isotropic and the porous medium is homogeneous.

Pressure drop in the channels is small and therefore negligible.

No entrance effect on flow dynamics and heat transfer by radiation is negligible compared to transfer by conduction.

The fluid flow inside the channel is laminar and fully developed.

The fluid in the channel is incompressible although there will be slight changes in density due to temperature and volumetric flow changes (Chen et al., 2008).

Figure 2: Overview of the reaction and heat and mass transfer in a monolith channel (Tomasic, 2007).

## 2.2 General model equations

The generalized partial differential equations governing the conservation of mass, heat and momentum in a flowing system is given as follows:

Continuity equation:

………………………….. (1)

Momentum balance:

……….…….. (2)

Mass balance:

…………………………………………. (3)

Heat balance:

………………………………………. (4)

## 2.3 Specific model equation

Based on the generalized partial differential equations and the assumption listed above Chen et al. presented the following general mathematical model to describe the reaction-diffusion behaviour inside a single channel of a monolithic reactor (Chen et al., 2008). There is no reaction in the bulk gas phase while the reaction-diffusion takes place in the washcoat (catalyst layer).

The mass balance for the gas phase:

………………….……… (5)

Heat balance for the gas phase:

…………….…………. (6)

The mass balance for the solid phase (washcoat):

Heat balance for the solid phase:

……….………. (8)

The above equations (5), (6), (7) and (8) are general monolithic reactor models presented by Chen et al. (2008), however the models can be simplified to fit any channel cross section geometry (i.e. square, circular, etc.) and also can be expressed 1-D, 2-D and 3-D model depending the purpose.

The heat and mass transfer coefficients in the channel is calculated from the Sherwood (Sh = kmdt/D) and Nusselt (Nu = hdt/k) numbers based on the semi-empirical correlation proposed by Hawthorn (1974) for laminar flow in the monolith reactor channel.

………………………………………… (9)

………………………………………... (10)

The Graetz number for mass transfer in the channel is defined as:

……………………………….…………………….. (11)

The Graetz number for heat transfer in the channel is defined as:

……………………………………………………….. (12)

The diffusion of reactant in the pores of the catalyst layer (washcoat) is mainly by Knudsen diffusion because the mean free path (pore radius) is less 100nm (Tomasic and Gomzi, 2004). Therefore, the effective diffusion coefficient is given as:

………………………………………………... (13)

## 2.4 Initial and boundary conditions

In order to solve the above mathematical model for Equations (5), (6), (7) and (8) of the monolithic reactor the following initial and boundary conditions are applied.

Initial conditions:

At t = 0, Ci,g = Ci,s = Co and Tg = Ts = To.

Boundary conditions:

Inlet z = 0, u = Uo, Ci,g = Ci,go, Ci,s = Ci,so, Tg = Tgo, and Ts = Tso.

At the axisymmetric line of the channel: x = y = r = 0

Outlet z = Z

At the channel surface

## ,

## ,

## 3.1 Case study: Simulation of catalytic combustion of methane using COMSOL

The mathematical models presented above were used to simulated catalytic combustion of methane in a circular monolith reactor channel using COMSOL. The geometry of the channel is an axisymmetric two dimensional geometry as shown in Fig.3. The combustion reaction is given as:

Figure 3: Single channel configuration for COMSOL simulation

CH4 + 2O2 CO2 + 2H2O ………………………… (14)

(Ghadrdan and Mehdizadeh, 2008)

The simulation parameters and conditions are presented in Table 1, 2 and 3 below

Table 1: Simulation parameters for the bulk phase (Liu et al., 2005; Ghadrdan and Mehdizadeh, 2008)

Conditions Bulk phase

Reaction rate, rk (mol/m2s)

Diffusivity, Db (m2/s)

Thermal conductivity, kj (J/m s K)

Viscosity of fluid, μg

Table 2: Simulation parameter for the porous catalyst layer (Liu et al., 2005; Ghadrdan and Mehdizadeh, 2008)

Conditions Porous catalyst layer

Reaction rate 0

Effective diffusivity, Deff (m2/s)

Effective thermal conductivity

of fluid in the catalyst layer

Where ks (W/mK) is the thermal conductivity of the porous catalyst layer

Table 3: Simulation conditions (source: Liu et al., 2005; Ghadrdan and Mehdizadeh, 2008)

Geometrical Conditions

Channel length, L (m) 0.04

Channel radius, r2 (mm) 1.0

Porous catalyst layer thickness (r2 - r1) (mm) 0.30

Inlet conditions

Composition (vol. %) 1.0% CH4, 99% air

Temperature, T (K) 700

Pressure, P (atm) 1.0

Velocity, u (m/s) 3.2

Catalyst support materials

Tortuosity, τ 4.0

Porosity, ε 0.4

Permeability, K (m2) 1 x 10-8

Thermal conductivity, ks (W/mK) 25

Heat capacity, Cps (J/kgK) 900

Density, ρs (kg/m3) 7870

## 3.2 Results and Discussions

The COMSOL results for the temperature, concentration and velocity profile based on the 2D mathematical models of equations (5), (6), (7) and (8) in cylindrical coordinate is presented in Figures 4, 5 and 6. As the reactants methane (CH4) and air (O2) flow down the channel of the monolithic reactor, the methane (CH4) and air (O2) diffuse into the porous catalyst layer and react on the active sites of the catalyst layer (solid phase), while the product from the reaction diffuse back into the bulk phase. The heat release from the combustion reaction causes rise in temperature as shown in the COMSOL result in Fig.4 below. During the combustion reaction in the channel, the temperature of the gas increases in the axial and radial direction while the higher temperature of catalyst layer is due to high thermal conductivity of the support material (i.e. 25 W/mK), heat transfer by convection from bulk phase, conduction and radiation between the internal wall surfaces of the channel.

Fig.5 shows the concentration profile in the channel and the porous catalyst layer. The COMSOL simulation in Fig.4 shows that the concentration of the reactants decreases as they diffuse into the porous catalyst layer (solid phase), while the concentration gradient in the catalyst layer is higher compared to that of the bulk phase.

Figure 4: Temperature profile in the channel (Ghadrdan and Mehdizadeh, 2008)

Figure 5: Concentration profile in the monolith channel (Ghadrdan and Mehdizadeh, 2008)

The axial velocity profile in the radial direction within the channel is shown in Fig. 6. The COMSOL simulation result shows that the velocity of the bulk phase (gas) is maximum at the centre of the channel, while a retarded viscous flow exist in the porous catalyst layer (solid phase).

Figure 6: Velocity profile within the monolith channel (Ghadrdan and Mehdizadeh, 2008)

The Sherwood number of the bulk phase is used to evaluate the mass transfer between the bulk gas phase and the catalyst layer surface in order to characterise the monolith reactor performance (Liu et al., 2005). The simulation result in Fig.7 for porous and non porous catalyst layer shows that the Sherwood number increases along the length of the channel and the Sherwood number for porous catalyst layer is higher compared to the without porous layer, this is due to the decrease in mass transfer resistance between the gas phase and the porous catalyst layer.

The effectiveness factor defined the ratio of the reaction rate to the internal mass transport within the catalyst layer, which is dependent on the pore size distribution and the catalytic active site distribution. The simulation result from Liu et al. (2005) in Fig.8 show that the catalyst effectiveness factor is higher for porous catalyst layer and increases along the channel of the monolith reactor in the axial direction; this is due to enhanced mass transfer between the bulk gas phase and catalyst layer, the large surface area within the solid phase and the viscous flow effect within the porous solid phase.

Figure 7: The Sherwood numbers at the interface along channel (Liu et al., 2005)

Figure 8: Effectiveness factor of catalyst layer along the channel (Liu et al., 2005)

## 3.3 Significance of the mathematical model

Once the mathematical model of the monolithic reactor is validated with experimental data and its adequacy and accuracy is ascertained, then the mathematical model can be used for:

The evaluation of the monolith reactor performance.

The design of monolithic reactors.

Optimizing monolith structures/configuration by manufacturers (Chen et al., 2008).

Predict and understand the behaviour of the monolith reactor for different operating conditions.

The synthesis of control strategies for the monolithic reactor.

Optimization of the monolith reactor geometry and configuration to suit a given reaction kinetic and operating conditions.

Reducing the cost and time associated with pilot plant experimentation. Therefore, after model validation, the mathematical model becomes more cost efficient way to do research instead of expensive and time consuming pilot plant.

Predicting the temperature and concentration profiles within the channels of monolith reactor to ensure save operation of the reactor.

Predicting fluid distribution within the channels of the monolith reactor.

The simulation of the monolithic reactor safe operation and evaluation of the system stability.

Analysis and understanding of the complex reaction and diffusion interaction that occur within the channels and the catalyst layer of the monolithic reactor.

## 4.1 Conclusion

A lot of mathematical models of monolithic reactor both single channel and multi-channel have been reported in the literature and used by different researchers for the monolith reactor design, optimization, performance evaluation, controller synthesis, and quantitative and qualitative analysis of the monolith reactor behaviour under different operating conditions. The use of numerical and finite element method software such as COMSOL Multiphysics and MATLAB to solve the developed mathematical models of the monolithic reactor is desirable and recommended to lessen computation effort. However, there is still need to validate the models by quantitative comparison of model predictions and experimental data to assure its adequacies and accuracies before further use. The mathematical models of the monolithic reactor have contributed to the enhancement of the understanding of the complex chemical and physical processes within the channels and prove useful in the design and optimization of the process.

Nomenclature

C concentration (mol/m3)

Ci,g, Ci,s concentration of component i in bulk gas and solid phase (mol/m3)

Ci,go, Ci,so inlet concentration of component i in bulk gas and solid phase (mol/m3)

Cp heat capacity (J/kg K)

d Channel size (m)

dt hydraulic diameter of monolith channel (m)

D Diffusion coefficient (m2/s)

Di,g, Di,eff molecular and effective diffusion coefficient of component i (m2/s)

Gz Graetz number defined

h heat transfer coefficient (W/m2 K)

ΔHr,j heat of reaction for reaction j (J/mol)

k thermal conductivity (W/m K)

kg, ks thermal conductivity of gas and solid (J/m K)

km mass transfer coefficient (m/s)

M molecular weigth (kg/mol)

Nu Nusselt number defined as: (hdt/k)

P pressure (Pa)

r radial coordinate (m)

rk reaction rate (mol/m3 s)

Sh Sherwood number defined as: (kmdt/D)

T temperature (K)

Tg, Ts temperature of gas and solid phases (K)

Tgo, Tso inlet temperature (K)

t time (s)

u velocity (m/s)

x, y, z coordinates in x, y, z directions (m)

Z/L monolith reactor length (m)

Greek letters

ε porosity of the catalyst layer

λj shotomatic coefficient of reaction j

μ viscosity (Pa.s)

ρ density (kg/m3)

τ Tortuosity