# Catalytic Pellet Reactor Design under Uncertainty

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UIC

Final Report

REU 2005 Summer Program

Catalytic Pellet Reactor Design under Uncertainty

Student’s Name: Advisors:

Celia Xue Professor Andreas Linninger Dr. Libin Zhang

Laboratory for Product and Process Design University of Illinois at Chicago

LPPD-Project Final Report: Aug.5th, 2005

Abstract Design of catalytic pellet reactor is a challenging task because of the complicated mechanism and transport phenomenon inherent to this problem. Process uncertainties such as the feed flow rate, porosity of the pellet, reaction rate coefficient, etc. present additional complexity. This paper investigates the method to quantify uncertainties of design parameters in the case study of catalytic pellet reactor design. The case study starts with developing models for a single catalytic pellet. Concentration and temperature profiles are solved numerically by collocation method for isothermal and non-isothermal pellets under steady and dynamic conditions. To account for the mass diffusion effect in the rate equation, the effectiveness factor η is introduced and solved for steady state pellets. Its sensitivity with respect to pellet surface conditions and other physical parameters are extensively studied in this paper. To solve for temperature and concentration profiles inside a reactor, reactor and pellet governing equations need to be solved simultaneously. With collocation method, we are able to obtain concentration and temperature profiles for the reactor successfully. Two important packed bed catalytic reactor behaviors concern us: “hotspot” and multiplicity. The multiple steady states in pellets are observed and analyzed in this paper. The reactor multiplicity is also studied combing the observation of “hotspots”. Finally, we are able to demonstrate with a specific case that uncertainty presented in design parameters leads to “hotspots” and possible melt down of the reactor. Uncertainty tests should be extensively studied along with the design of a reactor.

2

Table of Content

1. Introduction .....................................................................................................................4 2. Development of Mathematical Model.............................................................................5

2.1 Individual Pellet at Steady State................................................................................5 Isothermal Condition ...................................................................................................5 Non-isothermal Condition ...........................................................................................6

2.2 Individual Pellet at Dynamic State ............................................................................7 2.3 Effectiveness Factor ..................................................................................................7 2.4 Reactor Design at Steady State..................................................................................8 2.5 Reactor Design at Dynamic State............................................................................10 3. Methodology..................................................................................................................10 4. Results ...........................................................................................................................12 4.1 Individual Pellet at Steady State..............................................................................12

Isothermal Condition .................................................................................................12 Non-Isothermal Condition.........................................................................................13 Sensitivity of Effectiveness Factor ............................................................................13 4.2 Individual Pellet at Dynamic State ..........................................................................15 Sensitivity of Dynamic Pellet ....................................................................................16 4.3 Reactor Design at Steady State................................................................................17 Sensitivity of Steady State Reactor ...........................................................................18 4.4 Reactor Design at Dynamic State............................................................................19 4.5 Multiplicity ..............................................................................................................20 4.6 Reactor Hotspot .......................................................................................................22 5. Conclusion .....................................................................................................................25 7. Nomenclature.................................................................................................................26 8. References .....................................................................................................................27 9. Appendix .......................................................................................................................27

3

1. Introduction

Parameters such as the properties of the chemicals, inlet flow rate, and reaction coefficient etc. present uncertainties in their values. The uncertainties in these parameters directly influence many aspects of our design such as the safety condition or the product quality. The traditional method to handle process uncertainties is to over-design the process so that a “safety factor” is incorporated. However, such traditional method often fails to predict the magnitude of safety factor and to provide insight of the processes. The lack of understanding uncertainties in design parameters often lead to loss of revenue or unsafe design. Quantifying the uncertain parameters will enable engineers to rigorously describe the safety condition of their design, guarantee product quality and other operating conditions. The importance of studying uncertainty is demonstrated by the case study of a catalytic pellet reactor design.

Catalytic pellet reactor has commercial applications such as reducing automotive emission gas, oxidation of hydrocarbons, paraffin dehydrogenation, hydrocracking, and dehydrocyclization etc. In designing a pellet reactor, packed-bed geometry with stationary spherical catalyst pellets shown in Fig.1 is usually adopted. The boundary conditions for an individual pellet are described by its surface concentration and temperature, whereas the reactor boundary conditions are described by inlet concentration ad temperature. The pellets can be treated as a porous media, where reactant diffuses axially along the reactor, and also radially into the pellets. To fully describe a reactor, several design unknowns require extensive attention: concentration profile and temperature profile inside the reactor. These two design unknowns are directly related to the conditions of individual pellets by pellet boundary conditions and effectiveness factor η. As we focus our attention on solving pellets shown Fig. 1, catalytic pellets are porous to allow diffusion of reactant, consequently creating a concentration gradient with respect to the radius. Discussed in Weisz and Hicks [4], if the chemical reaction is accompanied with a heat effect, significant temperature gradient also develops within the pellet. In such a case, concentration and temperature profiles of the pellet must be solved simultaneously.

Fig. 1 Expanded view of a fixed-bed pellet reactor

4

2. Development of Mathematical Model

In this case study, SO2 oxidation catalyst for production of sulfuric acid is chosen. This reaction as expressed in Eq. (1) is first order with respect to SO2.

SO2 (g)+1/2O2(g) ↔SO3 (g)

(1)

SO3 gas continues to react with water in later processes to form sulfuric acid. The physical

parameters of the pellets are shown in Appendix.

To account for diffusion effect inside individual pellets, a variable known as the

internal effectiveness factor η is introduced in Weisz and Hicks [4]. Its definition is

expressed below:

η=

actual reaction rate

(2)

reaction rate at pellet surface

The expression of rate of reaction incorporating Eq. (2) have been derived by Damkohler [5], Thiele [6], Zeldowitsch [7], and Wheeler [8], leading to the simplified form shown in Eq. (3) for a first order reaction.

Re action Rate = ηk(T )CA

(3)

Where k(T) is the reaction rate coefficient and CA is the concentration of reactant A. The reaction coefficient for non-isothermal pellet can be expressed with Arrhenius equation [2] shown in Eq. (4). For isothermal pellet, it is assumed to be a constant. Refer to Nomenclature section for parameter definitions.

⎡ E ⎛ Tref ⎞⎤

k(T ) = kref exp ⎢−

⎜ −1⎟⎥

(4)

⎢⎣ RTref ⎝ T ⎠⎥⎦

2.1 Individual Pellet at Steady State Isothermal Condition

An isothermal pellet only presents mass transfer within the solid. Simplify the expression of a first order reaction to Eq.(5), a mass balance on the spherical shell control volume shown by Fig. 2 can be done easily.

A ⎯c⎯ata⎯lyst→ B

(5)

5

Fig. 2 Control volume of a spherical pellet

The concentration profile with respect to the radius of the pellet is then expressed in Eq.(6).

d 2CA + 2 dCA − kCA = 0 (6) dr2 r dr D

e

De is diffusivity coefficient of the pellet. This equation is easily solved analytically.

Non-isothermal Condition

For non-isothermal condition, the temperature profile inside the pellet needs to be developed along with concentration profile. By doing an energy balance on the same control volume, the system of differential equations is shown below:

dCA2 + 2 dCA − k(T )CA = 0 (7)

d 2r r dr

D

e

dT 2 2 dT (−∆H )

d 2r + r dr + λ k(T )C A = 0

(8)

Notice that reaction coefficient k is no longer a constant. Shown in Villadsen and Michelsen [1], converting Eq.(7) and (8) into dimensionless form and inserting Arrhenius equation for k(T), we obtain:

d

2ϕ

+

2

dϕ

−

Φ2

exp

⎡ ⎢γ

⎛⎜1−

1

⎞⎟⎤⎥ ϕ

=

0

(9)

dθ 2 θ dθ

⎣ ⎝ ζ ⎠⎦

ddθ2ζ2 + θ2 ddζθ − β exp ⎡⎢⎣γ ⎛⎜⎝1− ζ1 ⎞⎟⎠⎤⎥⎦ϕ = 0 (10)

Thiele Modulus:

Φ2

=

R2

⎡ ⎢a exp(−

Ea

⎤ )⎥

(11)

De ⎣

RTs ⎦

6

ϕ = CA ,θ = r , ζ = T , γ = Ea , β = ∆Hks

(12)

C ref

R

Tref

RT

λ

a is a dimensionless variable that accounts for reference reaction rate coefficient. Shown in Eq. (12), these dimensionless variables were discussed by Pita et al. [9]. Cref and Tref are taken to be the concentration of reactant and temperature on the surface of the pellet. From here on, we will directly indicate reference conditions to be surface conditions, which are indicated by subscript s. R is the radius of the pellet. The boundary conditions for the dimensionless equations are:

[ζ ]θ =1 =

T

= 1 and

⎡dζ ⎤ ⎢⎥

=0

(13)

Ts

⎣ dθ ⎦θ =0

[ ]ϕ = CA = 1 and ⎡ dϕ ⎤ = 0

θ =1

⎢⎥

(14)

CAs

⎣ dθ ⎦θ =0

At the surface of the pellet, reactant concentration and temperature are equal to surface conditions. At the center of the pellet, we assume the concentration and temperature are constant. Eq. (9) and (10) form a system of coupled second order non-linear differential equations. Numerical method must be employed to solve for φ and ζ.

2.2 Individual Pellet at Dynamic State

To calculate the pellet concentration and temperature profile at dynamic state, replace the right side of steady state equations with time variations:

ddθ2ϕ2 + θ2 ddϕθ − Φ2 exp ⎡⎢⎣γ (1− ζ1 )⎤⎥⎦ϕ = ddϕt (15)

ddθ2ζ2 + θ2 ddζθ − β exp ⎡⎢⎣γ (1− ζ1 )⎤⎥⎦ϕ = ddζt (16)

⎡dζ ⎤ Initial Conditions: ⎢ ⎥

=

To

,

and

⎡ dϕ ⎤ ⎢⎥

= Co

(17)

⎣ dt ⎦t=0 Ts

⎣ dt ⎦t=0 CAs

To and Co are variables to represent initial conditions for dynamic state at each radial position. Boundary conditions for Eq.(15) and (16) and are similar to those of steady state pellet governing equations shown in Eq. (13) and (14). We are free to choose the values of To and Co. By choosing different initial profiles, we can observe how the pellets approach steady state on a time scale.

2.3 Effectiveness Factor

Derived by Villadsen and Michelsen [1] , η is expressed as the ratio of volume averaged reaction rate relative to the rate at surface temperature and concentration.

7

1

∫ Φ2 rate(ϕ,ζ )drs+1 1 ∫ η = 01 = rate(ϕ,ζ )dr s+1 (18)

∫ Φ2rate(1,1)drs+1 0

0

Inserting dimensionless first order rate equation into the equation above, we obtain

the expression of effectiveness factor:

∫ η

=

1

3

ϕ (θ

)

exp

⎡⎢γ

(1 −

1

)⎤⎥θ

2dθ

(19)

0

⎣ ζ⎦

If we take into account the diffusion effect from the bulk of the reactor to the surface of the pellet, we can express effectiveness factor using bulk concentrations and temperatures. Neglecting intraparticle temperature gradient, the resulting expression is shown in Eq. (20).

η = 3BiM Φ1 coth Φ1 −1

(20)

Φ12 Φ1 coth Φ1 + BiM − 1

Φ12

=

Φ2

exp⎢⎡γ (1 −

1⎤ )⎥

(21)

⎣ ζ⎦

Where BiM is the Biot number for mass transfer. Eq. (20) uses an averaged Biot number to compensate the use of an averaged dimensionless temperature.

Discussed in Pita et al. [9], for a specified set of parameters such as β, γ and Φ, there may exist more than one feasible steady state. All variables however, satisfy the governing equation (9) and (10), with boundary conditions (13) and (14). Shown in the form of effectiveness factor, one set of β, γ and Φ may correspond to multiple effectiveness factors. Also shown by Lee et al. [11], uncertainty in parameters may initiate the switching between one steady state to another.

2.4 Reactor Design at Steady State

By doing a mass balance and energy balance on a cylindrical control volume of the reactor shown in Fig. 3, the expression for concentration and temperature is derived in Fogler [2], shown in Eq. (22) and (23).

8

Fig. 3 Control Volume of a pellet reactor

D d 2CA −U dCA + r ' ρ = 0

(22)

AB dz2

dz

Ab

k d 2T − ρ C U dT + r ' ∆H = 0

(23)

e dz2

g p dz

A

Where rA ' is the overall rate of reaction within the catalyst per unit mass of catalyst. With the expression of η derived in Eq. (20), overall rate of conversion can be expressed as Eq.(24).

−rA ' = η k(T )SaCA

(24)

Sa is the internal surface area of the pellet. For the convenience of calculation, we prefer to work with scaled equations incorporating expression for k(T). Convert Eq.(22) and Eq. (23) into scaled form with respect to inlet conditions, the resulting expression are shown Eq. (25) and (26).

d 2Cr − U

dCr

− ηSa ko ρb C

exp⎢⎡γ (1 −

1⎤ )⎥ = 0

(25)

dz 2 DAB dz

DAB

r

⎣

Tr ⎦

d 2Tr

− ρ g C pU dT

− ∆HηSaCi ρb C

exp⎢⎡γ (1 −

1⎤ )⎥ = 0

(26)

dz 2 kb r

kbTi

r

⎣

Tr ⎦

T = T C = CA r

(27)

r Ti ,

Ci

The boundary conditions are shown in Eq.(28) and (29).

[Cr ]z=0 = 1 [Tr ]z=0 = 1

(28)

⎡ dCr ⎤ ⎢⎥

=0

⎡ dTr ⎤ ⎢⎥

=0

(29)

⎣ dz ⎦ z=1

⎣ dz ⎦ z=1

9

Ti and Ci represent inlet conditions and Cr, Tr represent scaled concentration and temperature profile along the axial direction of the reactor. At the inlet of the reactor, the temperature and concentration are given to be inlet conditions. And for a sufficiently long reactor, the concentration and temperature at the end of the reactor can be assumed constant. The design of the reactor is coupled with solving pellet governing equations because effectiveness factor is a function of pellet temperature profile. Further more, pellet surface conditions are replaced by the bulk conditions inside the reactor. In order to solve reactor profiles, Eq. (9), (10), (20), (25) and (26) need to be solved simultaneously. The boundary conditions are listed in Eq. (13), (14), (28) and (29) with Ts, Cs replaced by Tr, Cr.

Industrial packed-bed reactors usually adopt cooling process in addition to the reactor to remove the heat produced. This application requires us subtracting a “heat removal” term from the energy balance shown in Eq.(26). Heat removal produces the effect of “hotspot” for typical strongly exothermic reactors. Discussed in Froment et al. [3] and Syed et al. [10], the magnitude of hot spot depends on the heat of reaction, rate of reaction, inlet conditions etc.

2.5 Reactor Design at Dynamic State

With the steady state reactor governing equations defined, their dynamic forms are easily obtained by adding time variation:

d 2Cr − U

dCr

− ηSakoρb C

exp ⎡⎢γ (1−

1⎤ )⎥ =

dCr

(30)

dz2 DAB dz

DAB r ⎣

Tr ⎦ dt

d 2T r

− ρgCpU

dT

− ∆HηSaCi ρb C

⎡ exp ⎢γ (1−

1⎤ )⎥ =

dTr

(31)

dz2 kb r

kbTi

r

⎣

Tr ⎦ dt

Initial

Conditions:

⎡ dTr ⎢

⎤ ⎥

= To

and

⎡ dCr ⎢

⎤ ⎥

= Co

(32)

⎣ dt ⎦t=0 Ti

⎣ dt ⎦t=0 Ci

Boundary conditions for Eq. (30) and (31) are identical to those of steady state reactor governing equations.

3. Methodology

The numerical method chosen to solve systems of second order differential

equations is collocation method, a type of weight residual method (WRM). Weight residual

method assumes that the analytical solution can be approximated in a piecewise fashion by

a superposition of functions with unknown coefficients aj shown in Eq.(33).

∑n

Tapproximate (x, t) = a j (t)φ j (x)

j =1

(33)

Shown in Fig. 4, solving for x at each node we choose and connect all the nodes together linearly, we obtain an approximated function.

10

Final Report

REU 2005 Summer Program

Catalytic Pellet Reactor Design under Uncertainty

Student’s Name: Advisors:

Celia Xue Professor Andreas Linninger Dr. Libin Zhang

Laboratory for Product and Process Design University of Illinois at Chicago

LPPD-Project Final Report: Aug.5th, 2005

Abstract Design of catalytic pellet reactor is a challenging task because of the complicated mechanism and transport phenomenon inherent to this problem. Process uncertainties such as the feed flow rate, porosity of the pellet, reaction rate coefficient, etc. present additional complexity. This paper investigates the method to quantify uncertainties of design parameters in the case study of catalytic pellet reactor design. The case study starts with developing models for a single catalytic pellet. Concentration and temperature profiles are solved numerically by collocation method for isothermal and non-isothermal pellets under steady and dynamic conditions. To account for the mass diffusion effect in the rate equation, the effectiveness factor η is introduced and solved for steady state pellets. Its sensitivity with respect to pellet surface conditions and other physical parameters are extensively studied in this paper. To solve for temperature and concentration profiles inside a reactor, reactor and pellet governing equations need to be solved simultaneously. With collocation method, we are able to obtain concentration and temperature profiles for the reactor successfully. Two important packed bed catalytic reactor behaviors concern us: “hotspot” and multiplicity. The multiple steady states in pellets are observed and analyzed in this paper. The reactor multiplicity is also studied combing the observation of “hotspots”. Finally, we are able to demonstrate with a specific case that uncertainty presented in design parameters leads to “hotspots” and possible melt down of the reactor. Uncertainty tests should be extensively studied along with the design of a reactor.

2

Table of Content

1. Introduction .....................................................................................................................4 2. Development of Mathematical Model.............................................................................5

2.1 Individual Pellet at Steady State................................................................................5 Isothermal Condition ...................................................................................................5 Non-isothermal Condition ...........................................................................................6

2.2 Individual Pellet at Dynamic State ............................................................................7 2.3 Effectiveness Factor ..................................................................................................7 2.4 Reactor Design at Steady State..................................................................................8 2.5 Reactor Design at Dynamic State............................................................................10 3. Methodology..................................................................................................................10 4. Results ...........................................................................................................................12 4.1 Individual Pellet at Steady State..............................................................................12

Isothermal Condition .................................................................................................12 Non-Isothermal Condition.........................................................................................13 Sensitivity of Effectiveness Factor ............................................................................13 4.2 Individual Pellet at Dynamic State ..........................................................................15 Sensitivity of Dynamic Pellet ....................................................................................16 4.3 Reactor Design at Steady State................................................................................17 Sensitivity of Steady State Reactor ...........................................................................18 4.4 Reactor Design at Dynamic State............................................................................19 4.5 Multiplicity ..............................................................................................................20 4.6 Reactor Hotspot .......................................................................................................22 5. Conclusion .....................................................................................................................25 7. Nomenclature.................................................................................................................26 8. References .....................................................................................................................27 9. Appendix .......................................................................................................................27

3

1. Introduction

Parameters such as the properties of the chemicals, inlet flow rate, and reaction coefficient etc. present uncertainties in their values. The uncertainties in these parameters directly influence many aspects of our design such as the safety condition or the product quality. The traditional method to handle process uncertainties is to over-design the process so that a “safety factor” is incorporated. However, such traditional method often fails to predict the magnitude of safety factor and to provide insight of the processes. The lack of understanding uncertainties in design parameters often lead to loss of revenue or unsafe design. Quantifying the uncertain parameters will enable engineers to rigorously describe the safety condition of their design, guarantee product quality and other operating conditions. The importance of studying uncertainty is demonstrated by the case study of a catalytic pellet reactor design.

Catalytic pellet reactor has commercial applications such as reducing automotive emission gas, oxidation of hydrocarbons, paraffin dehydrogenation, hydrocracking, and dehydrocyclization etc. In designing a pellet reactor, packed-bed geometry with stationary spherical catalyst pellets shown in Fig.1 is usually adopted. The boundary conditions for an individual pellet are described by its surface concentration and temperature, whereas the reactor boundary conditions are described by inlet concentration ad temperature. The pellets can be treated as a porous media, where reactant diffuses axially along the reactor, and also radially into the pellets. To fully describe a reactor, several design unknowns require extensive attention: concentration profile and temperature profile inside the reactor. These two design unknowns are directly related to the conditions of individual pellets by pellet boundary conditions and effectiveness factor η. As we focus our attention on solving pellets shown Fig. 1, catalytic pellets are porous to allow diffusion of reactant, consequently creating a concentration gradient with respect to the radius. Discussed in Weisz and Hicks [4], if the chemical reaction is accompanied with a heat effect, significant temperature gradient also develops within the pellet. In such a case, concentration and temperature profiles of the pellet must be solved simultaneously.

Fig. 1 Expanded view of a fixed-bed pellet reactor

4

2. Development of Mathematical Model

In this case study, SO2 oxidation catalyst for production of sulfuric acid is chosen. This reaction as expressed in Eq. (1) is first order with respect to SO2.

SO2 (g)+1/2O2(g) ↔SO3 (g)

(1)

SO3 gas continues to react with water in later processes to form sulfuric acid. The physical

parameters of the pellets are shown in Appendix.

To account for diffusion effect inside individual pellets, a variable known as the

internal effectiveness factor η is introduced in Weisz and Hicks [4]. Its definition is

expressed below:

η=

actual reaction rate

(2)

reaction rate at pellet surface

The expression of rate of reaction incorporating Eq. (2) have been derived by Damkohler [5], Thiele [6], Zeldowitsch [7], and Wheeler [8], leading to the simplified form shown in Eq. (3) for a first order reaction.

Re action Rate = ηk(T )CA

(3)

Where k(T) is the reaction rate coefficient and CA is the concentration of reactant A. The reaction coefficient for non-isothermal pellet can be expressed with Arrhenius equation [2] shown in Eq. (4). For isothermal pellet, it is assumed to be a constant. Refer to Nomenclature section for parameter definitions.

⎡ E ⎛ Tref ⎞⎤

k(T ) = kref exp ⎢−

⎜ −1⎟⎥

(4)

⎢⎣ RTref ⎝ T ⎠⎥⎦

2.1 Individual Pellet at Steady State Isothermal Condition

An isothermal pellet only presents mass transfer within the solid. Simplify the expression of a first order reaction to Eq.(5), a mass balance on the spherical shell control volume shown by Fig. 2 can be done easily.

A ⎯c⎯ata⎯lyst→ B

(5)

5

Fig. 2 Control volume of a spherical pellet

The concentration profile with respect to the radius of the pellet is then expressed in Eq.(6).

d 2CA + 2 dCA − kCA = 0 (6) dr2 r dr D

e

De is diffusivity coefficient of the pellet. This equation is easily solved analytically.

Non-isothermal Condition

For non-isothermal condition, the temperature profile inside the pellet needs to be developed along with concentration profile. By doing an energy balance on the same control volume, the system of differential equations is shown below:

dCA2 + 2 dCA − k(T )CA = 0 (7)

d 2r r dr

D

e

dT 2 2 dT (−∆H )

d 2r + r dr + λ k(T )C A = 0

(8)

Notice that reaction coefficient k is no longer a constant. Shown in Villadsen and Michelsen [1], converting Eq.(7) and (8) into dimensionless form and inserting Arrhenius equation for k(T), we obtain:

d

2ϕ

+

2

dϕ

−

Φ2

exp

⎡ ⎢γ

⎛⎜1−

1

⎞⎟⎤⎥ ϕ

=

0

(9)

dθ 2 θ dθ

⎣ ⎝ ζ ⎠⎦

ddθ2ζ2 + θ2 ddζθ − β exp ⎡⎢⎣γ ⎛⎜⎝1− ζ1 ⎞⎟⎠⎤⎥⎦ϕ = 0 (10)

Thiele Modulus:

Φ2

=

R2

⎡ ⎢a exp(−

Ea

⎤ )⎥

(11)

De ⎣

RTs ⎦

6

ϕ = CA ,θ = r , ζ = T , γ = Ea , β = ∆Hks

(12)

C ref

R

Tref

RT

λ

a is a dimensionless variable that accounts for reference reaction rate coefficient. Shown in Eq. (12), these dimensionless variables were discussed by Pita et al. [9]. Cref and Tref are taken to be the concentration of reactant and temperature on the surface of the pellet. From here on, we will directly indicate reference conditions to be surface conditions, which are indicated by subscript s. R is the radius of the pellet. The boundary conditions for the dimensionless equations are:

[ζ ]θ =1 =

T

= 1 and

⎡dζ ⎤ ⎢⎥

=0

(13)

Ts

⎣ dθ ⎦θ =0

[ ]ϕ = CA = 1 and ⎡ dϕ ⎤ = 0

θ =1

⎢⎥

(14)

CAs

⎣ dθ ⎦θ =0

At the surface of the pellet, reactant concentration and temperature are equal to surface conditions. At the center of the pellet, we assume the concentration and temperature are constant. Eq. (9) and (10) form a system of coupled second order non-linear differential equations. Numerical method must be employed to solve for φ and ζ.

2.2 Individual Pellet at Dynamic State

To calculate the pellet concentration and temperature profile at dynamic state, replace the right side of steady state equations with time variations:

ddθ2ϕ2 + θ2 ddϕθ − Φ2 exp ⎡⎢⎣γ (1− ζ1 )⎤⎥⎦ϕ = ddϕt (15)

ddθ2ζ2 + θ2 ddζθ − β exp ⎡⎢⎣γ (1− ζ1 )⎤⎥⎦ϕ = ddζt (16)

⎡dζ ⎤ Initial Conditions: ⎢ ⎥

=

To

,

and

⎡ dϕ ⎤ ⎢⎥

= Co

(17)

⎣ dt ⎦t=0 Ts

⎣ dt ⎦t=0 CAs

To and Co are variables to represent initial conditions for dynamic state at each radial position. Boundary conditions for Eq.(15) and (16) and are similar to those of steady state pellet governing equations shown in Eq. (13) and (14). We are free to choose the values of To and Co. By choosing different initial profiles, we can observe how the pellets approach steady state on a time scale.

2.3 Effectiveness Factor

Derived by Villadsen and Michelsen [1] , η is expressed as the ratio of volume averaged reaction rate relative to the rate at surface temperature and concentration.

7

1

∫ Φ2 rate(ϕ,ζ )drs+1 1 ∫ η = 01 = rate(ϕ,ζ )dr s+1 (18)

∫ Φ2rate(1,1)drs+1 0

0

Inserting dimensionless first order rate equation into the equation above, we obtain

the expression of effectiveness factor:

∫ η

=

1

3

ϕ (θ

)

exp

⎡⎢γ

(1 −

1

)⎤⎥θ

2dθ

(19)

0

⎣ ζ⎦

If we take into account the diffusion effect from the bulk of the reactor to the surface of the pellet, we can express effectiveness factor using bulk concentrations and temperatures. Neglecting intraparticle temperature gradient, the resulting expression is shown in Eq. (20).

η = 3BiM Φ1 coth Φ1 −1

(20)

Φ12 Φ1 coth Φ1 + BiM − 1

Φ12

=

Φ2

exp⎢⎡γ (1 −

1⎤ )⎥

(21)

⎣ ζ⎦

Where BiM is the Biot number for mass transfer. Eq. (20) uses an averaged Biot number to compensate the use of an averaged dimensionless temperature.

Discussed in Pita et al. [9], for a specified set of parameters such as β, γ and Φ, there may exist more than one feasible steady state. All variables however, satisfy the governing equation (9) and (10), with boundary conditions (13) and (14). Shown in the form of effectiveness factor, one set of β, γ and Φ may correspond to multiple effectiveness factors. Also shown by Lee et al. [11], uncertainty in parameters may initiate the switching between one steady state to another.

2.4 Reactor Design at Steady State

By doing a mass balance and energy balance on a cylindrical control volume of the reactor shown in Fig. 3, the expression for concentration and temperature is derived in Fogler [2], shown in Eq. (22) and (23).

8

Fig. 3 Control Volume of a pellet reactor

D d 2CA −U dCA + r ' ρ = 0

(22)

AB dz2

dz

Ab

k d 2T − ρ C U dT + r ' ∆H = 0

(23)

e dz2

g p dz

A

Where rA ' is the overall rate of reaction within the catalyst per unit mass of catalyst. With the expression of η derived in Eq. (20), overall rate of conversion can be expressed as Eq.(24).

−rA ' = η k(T )SaCA

(24)

Sa is the internal surface area of the pellet. For the convenience of calculation, we prefer to work with scaled equations incorporating expression for k(T). Convert Eq.(22) and Eq. (23) into scaled form with respect to inlet conditions, the resulting expression are shown Eq. (25) and (26).

d 2Cr − U

dCr

− ηSa ko ρb C

exp⎢⎡γ (1 −

1⎤ )⎥ = 0

(25)

dz 2 DAB dz

DAB

r

⎣

Tr ⎦

d 2Tr

− ρ g C pU dT

− ∆HηSaCi ρb C

exp⎢⎡γ (1 −

1⎤ )⎥ = 0

(26)

dz 2 kb r

kbTi

r

⎣

Tr ⎦

T = T C = CA r

(27)

r Ti ,

Ci

The boundary conditions are shown in Eq.(28) and (29).

[Cr ]z=0 = 1 [Tr ]z=0 = 1

(28)

⎡ dCr ⎤ ⎢⎥

=0

⎡ dTr ⎤ ⎢⎥

=0

(29)

⎣ dz ⎦ z=1

⎣ dz ⎦ z=1

9

Ti and Ci represent inlet conditions and Cr, Tr represent scaled concentration and temperature profile along the axial direction of the reactor. At the inlet of the reactor, the temperature and concentration are given to be inlet conditions. And for a sufficiently long reactor, the concentration and temperature at the end of the reactor can be assumed constant. The design of the reactor is coupled with solving pellet governing equations because effectiveness factor is a function of pellet temperature profile. Further more, pellet surface conditions are replaced by the bulk conditions inside the reactor. In order to solve reactor profiles, Eq. (9), (10), (20), (25) and (26) need to be solved simultaneously. The boundary conditions are listed in Eq. (13), (14), (28) and (29) with Ts, Cs replaced by Tr, Cr.

Industrial packed-bed reactors usually adopt cooling process in addition to the reactor to remove the heat produced. This application requires us subtracting a “heat removal” term from the energy balance shown in Eq.(26). Heat removal produces the effect of “hotspot” for typical strongly exothermic reactors. Discussed in Froment et al. [3] and Syed et al. [10], the magnitude of hot spot depends on the heat of reaction, rate of reaction, inlet conditions etc.

2.5 Reactor Design at Dynamic State

With the steady state reactor governing equations defined, their dynamic forms are easily obtained by adding time variation:

d 2Cr − U

dCr

− ηSakoρb C

exp ⎡⎢γ (1−

1⎤ )⎥ =

dCr

(30)

dz2 DAB dz

DAB r ⎣

Tr ⎦ dt

d 2T r

− ρgCpU

dT

− ∆HηSaCi ρb C

⎡ exp ⎢γ (1−

1⎤ )⎥ =

dTr

(31)

dz2 kb r

kbTi

r

⎣

Tr ⎦ dt

Initial

Conditions:

⎡ dTr ⎢

⎤ ⎥

= To

and

⎡ dCr ⎢

⎤ ⎥

= Co

(32)

⎣ dt ⎦t=0 Ti

⎣ dt ⎦t=0 Ci

Boundary conditions for Eq. (30) and (31) are identical to those of steady state reactor governing equations.

3. Methodology

The numerical method chosen to solve systems of second order differential

equations is collocation method, a type of weight residual method (WRM). Weight residual

method assumes that the analytical solution can be approximated in a piecewise fashion by

a superposition of functions with unknown coefficients aj shown in Eq.(33).

∑n

Tapproximate (x, t) = a j (t)φ j (x)

j =1

(33)

Shown in Fig. 4, solving for x at each node we choose and connect all the nodes together linearly, we obtain an approximated function.

10

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