# Mathematical Modelling Of Time Dependent Processes Biology Essay

The heat and mass transportation within the monolith channels, the applications and advantages of massive reactor/catalyst have been discussed. The mathematical theoretical account for the reaction and diffusion procedures taking topographic point in a individual channel of the monolith reactor as the fluid flows through was presented. The theoretical account was used to imitate the catalytic burning of methane in the monolith reactor ; the numerical solution of the mathematical theoretical account was obtained by agencies of the finite component method utilizing COMSOL Multiphysics ( Liu et al. , 2005 ; Ghadrdan and Mehdizadeh, 2008 ) . The temperature, concentration and speed profile within the channel for the majority gas and porous accelerator bed phases as the burning reaction proceed were presented. The public presentation of the monolith reactor was evaluated based on the Sherwood figure and the effectiveness factor of the accelerator bed. The mathematical theoretical account is important for the design and optimisation of the monolith reactor structure/configuration, synthesis of control schemes, and the anticipation of the massive reactor behavior under different operating conditions.

A massive reactor consists of a big figure of parallel channels frequently in a signifier of honeycomb construction or agreement with the wall of each channel coated with a porous accelerator bed ( or washcoat ) . The parallel channels may hold the undermentioned geometries round, square, triangular, rectangular, hexangular, or of other forms transverse subdivisions, a typical cross subdivision of a monolith reactor is shown in Fig.1. The monolith substrate stuff is either ceramic or metallic, and can move as a structured accelerator every bit good as a reactor ( Tomasic, 2007 ) . Therefore for application purposes the form, channel size, accelerator bed thickness, microstructure and porousness every bit good as the channel wall thickness depends on the procedure demands ( Cybulski and Moulijn, 1998 ; Tomasic, 2007 ) . Massive reactors are widely used for environmental pollution control such as in the cleansing of fumes gas from power works, gas turbine and in car industry for vehicle fumess to change over NOx and CO via fast gas stage reactions ( Werner, 1994 ; Chen et Al, 2008 ) . Other applications include catalytic burning, hydrocarbon processing, hydrogenation or dehydrogenation, catalytic oxidization ( Chen et al, 2008 ) . Compared to conventional catalytic bed reactors such as fixed bed, slurry, and ternary bed reactors massive reactors are advantageous due to: high surface country to volume ratio, low force per unit area bead, no hot topographic point within the channels, riddance of external mass transportation and internal diffusion restrictions, low axial scattering and back commixture, high selectivity, riddance of plugging and fouling of accelerator, extension of accelerator life span, easy to scale-up, high temperature stableness, high mechanical strength, and easiness of orientation in a reactor ( Tomasic, 2007 ; Chen et Al, 2008 ; James et Al, 2003 ; Tomasic and Gomzi, 2004 ) .

Therefore, mathematical modeling of the clip dependent physical, coincident diffusion and chemical reaction happening inside the porous washcoat as the gas flow through the channels will heighten the apprehension of the procedure complexness and predict the public presentation of the massive reactors ( Tomasic et al, 2004 ) . Most mathematical theoretical accounts of massive reactors are on individual channel with the premise that every channel in the monolith reactor behaves the same, hence can stand for the average behavior of the full channels and history for the diffusion and reactions inside the accelerator bed and in the channel ( Bercic, 2001 ; Chen et al. , 2008 ) . However, in some circumstance a individual channel theoretical account might be unequal therefore James et Al. has developed multi-channel mathematical theoretical account of the massive reactor to reply many inquiries which individual channel theoretical account could non supply replies to such as the non-uniformity of flow distribution at the recess of the monolith reactor and the consequence of channel-to-channel heat conductivity ( James et al. , 2003 ) . In general, it is simpler and logically sensible to develop mathematical theoretical account for a individual channel to characterize the behavior of the whole massive reactor since every channel within the monolith reactor construction are indistinguishable. In position of this, depending on the intent of the theoretical account, a individual channel theoretical account in 1, 2 or 3-dimensions to simplify the complexness of the heat and mass conveyance within the channels can be used to depict the average behavior of the massive reactor and requires much less calculation attempt than multi-channel ( Chen et al. , 2008 ) . Schildhauer et Al. developed mathematical theoretical account of movie flow monolith for reactive denudation in massive reactor which involves both reaction and gas-liquid separation ( Schildhauer et al. , 2005 ) .

The mathematical theoretical accounts of massive reactors for pollution control such as the intervention of automotive fumes gas with considerations to the catalytic activity due to the decomposition of N monoxide ( NO ) , oxidization 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 theoretical account for the washcoat ( catalytic bed ) of a massive reactor for assorted geometric cross subdivision based on the internal mass transportation coefficient estimated utilizing Sherwood figure in relation to Thiele modulus to simplify the diffusion and reaction within the washcoat ( Joshi et al. , 2009 ) . The mathematical modeling and simulation of the heat and mass transportation in the monolith reactor will help the anticipation of the concentration and temperature profiles in the axial fluid flow and along the channel of the reactor. The solutions of the theoretical account equations can be obtained utilizing efficient and effectual numerical package such as MATLAB, COMSOL Multiphysics, and other computational fluid kineticss ( CFD ) package ( e.g. Fluent ) ( Chen et al. , 2008 ) . Hayes et Al. reported the numerical probe of the diffusion and reaction in the washcoat of the massive reactor, analyzing the consequence of washcoat geometry and thickness on mass transportation and the consequence of assorted channel cross subdivision forms utilizing finite component convergent thinker tool COMSOL ( Hayes et al. , 2004 ) . Ghadrdan and Mehdizadeh reported a finite component theoretical account for the simulation of a individual channel catalytic burning of methane in massive reactor utilizing COMSOL multiphysics ( Ghadrdan and Mehdizadeh, 2008 ) . The flow of fluids in the massive reactor channels is largely laminal flow ( Kolaczkowski, 1999 ) .

The mathematical theoretical account of the massive reactor will be utile in analyzing the heat and mass transportation effects, rating of monolith reactor public presentation and optimisation monolith construction and constellation, and the design of the reactor ( Kolaczkowski, 1999 ) . However, the chief challenges of massive reactor modeling are the proof of the theoretical account which is due to the troubles in obtaining equal experimental informations for simulation and comparing and the big figure of parametric quantities to be estimated such as heat and heat transportation coefficients, kinetic look, physical belongingss ( Kolaczkowski, 1999 ; Tomasic and Gomzi, 2004 ; Chen et al. , 2008 ) . Pinkas et Al. investigated and simulated the CO oxidization in massive reactors utilizing a one dimensional two-phase theoretical account with axial scattering to depict the heat and mass conveyance during the procedure ( Pinkas et al. , 1996 ) . The survey of massive reactors reported in literature ranges from monolith readying and bulge methods, its application to legion reaction strategies, the heat and mass transportation within the reactor, the flow government and their hydrokineticss, mathematical modeling of the diffusion and reactions within the channels, appraisal of the heat and mass transportation coefficients, and the usage of computational fluid kineticss ( CFD ) package to imitate the system and to help optimisation and design. Ghadrdan and Mehdizadeh modelled the interaction between the fluid flow in the channel and the porous accelerator bed ( washcoat ) by the Brinkman-Forchheimer extended Darcy flow theoretical account in a porous medium ( Ghadrdan and Mehdizadeh, 2008 ) . Liu et Al. reported a fresh conceptual design and CFD simulation of massive reactor with enhanced mass conveyance feature, which was accomplished by infixing porous substrate on the walls of the channels ( Liu et al. , 2005 ) .

The aim of this study is to show a mathematical theoretical account of the massive reactor for catalytic burning of methane, the solutions to the theoretical account is besides sort utilizing COMSOL Multiphysics package, the consequences obtained are besides discussed and the significance of the theoretical account are highlighted.

Figure 1: Typical representation of massive reactor cross subdivision and channel

## 1.2 Heat and Mass transportation and reaction in massive reactors

The gas solid reaction in the monolith channel is controlled by external mass conveyance at the gas-solid interface and/or internal diffusion within the accelerator bed ( Liu et al. , 2005 ) . Therefore, for reaction to happen inside the massive reactor washcoat ( catalyst bed ) , the reactants have to spread from the channels wall surface into the porous construction of the washcoat ( Chen et al. , 2008 ) . This implies a coincident diffusion and reaction taking topographic point within the washcoat ( catalyst bed ) , in which reactants diffuses in piece merchandises diffuse out and are swept off by the majority flow through the channel. External mass transportation from the fluid in the channel to the surface of the washcoat may be rate modification or the internal diffusion inside the accelerator bed quantified by the effectiveness factor of washcoat may be the reaction rate commanding measure. However, the internal diffusion inside the monolith reactor accelerator bed is dependent on the thickness of the washcoat and its porousness ( Chen et al. , 2008 ) . Hayes and Kolaczkowski mathematical theoretical account and simulation probe of a monolith reactor with laminar flow shows that the passage from kinetic to mass reassign control of the reaction within the channel of the reactor depends on the channel size, reaction dynamicss, recess conditions, diffusion coefficient and the length of the channels ( Hayes and Kolaczkowski, 1994 ) . The heat and mass transportation coefficients in monolith reactors are estimated chiefly by utilizing dimensionless measures Sherwood figure ( Sh ) and Nusselt ( Nu ) figure severally, correlated as maps of diffusion coefficient, monolith length, Graetz figure, Peclet figure, etc. ( Chen et al. , 2008 ) . Gonzo and Gottifredi reported that the external and internal diffusion and reaction inside the accelerator bed ( washcoat ) are characterised by the accelerator effectivity factor ( Gonzo and Gottifredi, 2010 ) .

For a non-adiabatic status in massive reactor, the solid and unstable temperature profile radially and axially in the channels due to heat transportation is required to measure the reactor public presentation. Besides if the reaction within the monolith is extremely exothermal, so metallic monolith support of good heat conductivity belongingss is a good pick to utilize ( Chen et al. , 2008 ) . Besides, equal reaction dynamicss is required for right appraisal of the heat released and transferred by an exothermal reaction in the massive reactors. Chen et Al. reported that the huge reactions in the washcoat and their kinetic may be complex and hence hard to account for all the reaction and besides mensurating the clip dependent temperature and concentration response inside the massive reactor to formalize the theoretical account has made mathematical modeling of the massive reactors extremely disputing ( Chen et al. , 2008 ) . Furthermore, to understand the heat transportation, diffusion and chemical reactions taking topographic point within the channels and washcoat of the monolith reactor mathematical modeling becomes an indispensable tool.

## 2.1 Mathematical theoretical account of a individual channel of a massive reactor

The design and public presentation of a massive reactor is a map of the channel geometry, length and dimension of the channel, the thickness of the washcoat, runing conditions, the belongingss of the accelerator species, and the kinetic of the reactions ( Tomasic, 2007 ) . Therefore, a mathematical theoretical account that will account for all these parametric quantities will be of changing complexnesss, but will help optimisation of the reactor. The diffusion and chemical reaction in the channel coupled with the heat and mass transportation between the fluid and solid accelerator stage is shown in Fig.2, the theoretical accounts were developed based cardinal Torahs of mass, energy and impulse preservations with the undermentioned premises for simplifications:

The full monolith channel sizes and forms are indistinguishable for individual channel every bit good as for the whole massive reactor.

Uniform unstable distributions among the full channels.

Uniform distribution of accelerator activity within each of the channels.

The massive reactor stuff is isotropous and the porous medium is homogenous.

Pressure bead in the channels is little and hence negligible.

No entryway consequence on flow kineticss and heat transportation by radiation is negligible compared to reassign by conductivity.

The fluid flow inside the channel is laminal and to the full developed.

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

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

## 2.2 General theoretical account equations

The generalized partial differential equations regulating the preservation of mass, heat and impulse in a flowing system is given as follows:

Continuity equation:

………………………….. ( 1 )

Momentum balance:

……….…….. ( 2 )

Mass balance:

………………………………………… . ( 3 )

Heat balance:

……………………………………… . ( 4 )

## 2.3 Specific theoretical account equation

Based on the generalized partial differential equations and the premise listed above Chen et Al. presented the following general mathematical theoretical account to depict the reaction-diffusion behavior inside a individual channel of a massive reactor ( Chen et al. , 2008 ) . There is no reaction in the majority gas stage while the reaction-diffusion takes topographic point in the washcoat ( catalyst bed ) .

The mass balance for the gas stage:

………………….……… ( 5 )

Heat balance for the gas stage:

…………….………… . ( 6 )

The mass balance for the solid stage ( washcoat ) :

Heat balance for the solid stage:

……….……… . ( 8 )

The above equations ( 5 ) , ( 6 ) , ( 7 ) and ( 8 ) are general massive reactor theoretical accounts presented by Chen et Al. ( 2008 ) , nevertheless the theoretical accounts can be simplified to suit any channel cross subdivision geometry ( i.e. square, round, etc. ) and besides can be expressed 1-D, 2-D and 3-D theoretical account depending the intent.

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

………………………………………… ( 9 )

……………………………………… … ( 10 )

The Graetz figure for mass transportation in the channel is defined as:

……………………………….…………………….. ( 11 )

The Graetz figure for heat transportation in the channel is defined as:

……………………………………………………….. ( 12 )

The diffusion of reactant in the pores of the accelerator bed ( washcoat ) is chiefly by Knudsen diffusion because the average free way ( pore radius ) is less 100nm ( Tomasic and Gomzi, 2004 ) . Therefore, the effectual diffusion coefficient is given as:

……………………………………………… … ( 13 )

## 2.4 Initial and boundary conditions

In order to work out the above mathematical theoretical account for Equations ( 5 ) , ( 6 ) , ( 7 ) and ( 8 ) of the massive reactor the undermentioned 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, travel, Ci, s = Ci, so, Tg = Tgo, and Ts = Tso.

At the axisymmetric line of the channel: ten = Y = R = 0

Outlet z = Z

At the channel surface

## 3.1 Case survey: Simulation of catalytic burning of methane utilizing COMSOL

The mathematical theoretical accounts presented above were used to fake catalytic burning of methane in a round monolith reactor channel utilizing COMSOL. The geometry of the channel is an axisymmetric two dimensional geometry as shown in Fig.3. The burning reaction is given as:

Figure 3: Single channel constellation for COMSOL simulation

CH4 + 2O2 CO2 + 2H2O ………………………… ( 14 )

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

Table 1: Simulation parametric quantities for the majority stage ( Liu et al. , 2005 ; Ghadrdan and Mehdizadeh, 2008 )

Conditionss Bulk stage

Chemical reaction rate, rk ( mol/m2s )

Diffusivity, Db ( m2/s )

Thermal conduction, kj ( J/m s K )

Viscosity of fluid, ?g

Table 2: Simulation parametric quantity for the porous accelerator bed ( Liu et al. , 2005 ; Ghadrdan and Mehdizadeh, 2008 )

Conditionss Porous accelerator bed

Chemical reaction rate 0

Effective diffusivity, Deff ( m2/s )

Effective thermic conduction

of fluid in the accelerator bed

Where Kansas ( W/mK ) is the thermic conduction of the porous accelerator bed

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

Geometric Conditionss

Channel length, L ( m ) 0.04

Channel radius, r2 ( millimeter ) 1.0

Porous accelerator bed thickness ( r2 – r1 ) ( millimeter ) 0.30

Inlet conditions

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

Temperature, T ( K ) 700

Pressure, P ( standard pressure ) 1.0

Velocity, U ( m/s ) 3.2

Catalyst support stuffs

Tortuosity, ? 4.0

Porosity, ? 0.4

Permeability, K ( M2 ) 1 ten 10-8

Thermal conduction, Kansas ( W/mK ) 25

Heat capacity, Cps ( J/kgK ) 900

Density, ?s ( kg/m3 ) 7870

## 3.2 Consequences and Discussions

The COMSOL consequences for the temperature, concentration and speed profile based on the 2D mathematical theoretical accounts of equations ( 5 ) , ( 6 ) , ( 7 ) and ( 8 ) in cylindrical co-ordinate is presented in Figures 4, 5 and 6. As the reactants methane ( CH4 ) and air ( O2 ) flow down the channel of the massive reactor, the methane ( CH4 ) and air ( O2 ) diffuse into the porous accelerator bed and react on the active sites of the accelerator bed ( solid stage ) , while the merchandise from the reaction diffuse back into the majority stage. The heat release from the burning reaction causes rise in temperature as shown in the COMSOL consequence in Fig.4 below. During the burning reaction in the channel, the temperature of the gas additions in the axial and radial way while the higher temperature of accelerator bed is due to high thermic conduction of the support stuff ( i.e. 25 W/mK ) , heat transportation by convection from bulk stage, conductivity and radiation between the internal wall surfaces of the channel.

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

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 speed profile in the radial way within the channel is shown in Fig. 6. The COMSOL simulation consequence shows that the speed of the majority stage ( gas ) is maximal at the Centre of the channel, while a retarded syrupy flow exist in the porous accelerator bed ( solid stage ) .

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

The Sherwood figure of the majority stage is used to measure the mass transportation between the majority gas stage and the accelerator bed surface in order to characterize the monolith reactor public presentation ( Liu et al. , 2005 ) . The simulation consequence in Fig.7 for porous and non porous accelerator bed shows that the Sherwood figure additions along the length of the channel and the Sherwood figure for porous accelerator bed is higher compared to the without porous bed, this is due to the lessening in mass transportation opposition between the gas stage and the porous accelerator bed.

The effectivity factor defined the ratio of the reaction rate to the internal mass conveyance within the accelerator bed, which is dependent on the pore size distribution and the catalytic active site distribution. The simulation consequence from Liu et Al. ( 2005 ) in Fig.8 show that the accelerator effectivity factor is higher for porous accelerator bed and increases along the channel of the monolith reactor in the axial way ; this is due to enhanced mass transportation between the majority gas stage and accelerator bed, the big surface country within the solid stage and the syrupy flow consequence within the porous solid stage.

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

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

## 3.3 Significance of the mathematical theoretical account

Once the mathematical theoretical account of the massive reactor is validated with experimental informations and its adequateness and truth is ascertained, so the mathematical theoretical account can be used for:

The rating of the monolith reactor public presentation.

The design of massive reactors.

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

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

The synthesis of control schemes for the massive reactor.

Optimization of the monolith reactor geometry and constellation to accommodate a given reaction kinetic and operating conditions.

Reducing the cost and clip associated with pilot works experimentation. Therefore, after theoretical account proof, the mathematical theoretical account becomes more cost efficient manner to make research alternatively of expensive and clip devouring pilot works.

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

Predicting unstable distribution within the channels of the monolith reactor.

The simulation of the massive reactor safe operation and rating of the system stableness.

Analysis and apprehension of the complex reaction and diffusion interaction that occur within the channels and the accelerator bed of the massive reactor.

## 4.1 Decision

A batch of mathematical theoretical accounts of massive reactor both individual channel and multi-channel have been reported in the literature and used by different research workers for the monolith reactor design, optimisation, public presentation rating, accountant synthesis, and quantitative and qualitative analysis of the monolith reactor behavior under different operating conditions. The usage of numerical and finite component method package such as COMSOL Multiphysics and MATLAB to work out the developed mathematical theoretical accounts of the massive reactor is desirable and recommended to decrease calculation attempt. However, there is still need to formalize the theoretical accounts by quantitative comparing of theoretical account anticipations and experimental informations to guarantee its adequatenesss and truths before farther usage. The mathematical theoretical accounts of the massive reactor have contributed to the sweetening of the apprehension of the complex chemical and physical procedures within the channels and turn out utile in the design and optimisation of the procedure.

Terminology

C concentration ( mol/m3 )

Ci, g, Ci, s concentration of constituent I in majority gas and solid stage ( mol/m3 )

Ci, travel, Ci, so inlet concentration of constituent I in majority gas and solid stage ( mol/m3 )

Cp heat capacity ( J/kg K )

500 Channel size ( m )

dt hydraulic diameter of monolith channel ( m )

D Diffusion coefficient ( m2/s )

Di, g, Di, eff molecular and effectual diffusion coefficient of constituent I ( m2/s )

Gz Graetz figure defined

H heat transportation coefficient ( W/m2 K )

?Hr, J heat of reaction for reaction J ( J/mol )

K thermic conduction ( W/m K )

kilogram, ks thermic conduction of gas and solid ( J/m K )

kilometer mass transportation coefficient ( m/s )

M molecular weigth ( kg/mol )

Nu Nusselt figure defined as: ( hdt/k )

P force per unit area ( Pa )

R radial co-ordinate ( m )

rk reaction rate ( mol/m3 s )

Sh Sherwood figure defined as: ( kmdt/D )

T temperature ( K )

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

Tgo, Tso recess temperature ( K )

t clip ( s )

u speed ( m/s )

ten, Y, omega co-ordinates in x, Y, omega waies ( m )

Z/L monolith reactor length ( m )

Grecian letters

? porousness of the accelerator bed

?j shotomatic coefficient of reaction J

? viscousness ( Pa.s )

? denseness ( kg/m3 )

? Tortuosity