Transient kinetic analysis of multipath reactions: An educational module using the IPython software package

education for chemical engineers 1 5 ( 2 0 1 6 ) 1–18

Contents lists available at ScienceDirect

Education for Chemical Engineers journal homepage: www.elsevier.com/locate/ece

Transient kinetic analysis of multipath reactions: An educational module using the IPython software package Boris Golman School of Chemical Engineering, Institute of Engineering, Suranaree University of Technology, 111 University Avenue, Muang District, Nakhon-Ratchasima 3000, Thailand

a r t i c l e

i n f o

a b s t r a c t

Article history:

A large number of industrial catalytic reactions proceeds via multiple pathways. A transient

Received 27 September 2015

response method has been successfully utilized for the analysis of kinetics and mecha-

Received in revised form 24

nism of multipath reactions. In this paper, we describe an educational module for teaching

December 2015

kinetics of complex heterogeneously-catalyzed reactions based on the software package for

Accepted 28 December 2015

simulation of transient responses in a tubular packed-bed reactor. The module includes

Available online 13 January 2016

the reactor model description, derivation and verification of kinetic equations, analysis of

Keywords:

analysis of results utilizing the developed IPython notebooks. The use of the module helps

steady-state kinetics, numerical simulation of transient responses, and visualization and Multipath catalytic reaction

students to acquire theoretical knowledge as well as the practical and analytical skills

Transient response method

related to the kinetic analysis of multipath reactions.

Educational module

© 2016 The Institution of Chemical Engineers. Published by Elsevier B.V. All rights reserved.

IPython software Simulation

1.

Introduction

The catalytic reactions lay at the core of industrially important chemical processes. Due to the existence of different types of active sites on the catalyst surface, these reactions frequently proceed through numerous paths and could include more than hundreds of intermediate reactions before producing the desired products. For example, the Fischer–Tropsch synthesis, the process vital in the preparation of chemicals and fuels from gas, coal or biomass, is a complex multipath catalytic reaction that converts the mixture of carbon monoxide and hydrogen into long-chain hydrocarbons (Maitlis and de Klerk, 2013). Azadi et al. (2015) proposed a Fischer–Tropsch reaction mechanism with 128 elementary reactions on a Co/␥-Al2 O3 catalyst. Although multi-pathway reactions are widely utilized in industry, the kinetic analysis of multi-pathway reactions remains a challenging task. The successful analysis requires one to discriminate each reaction path, identify the surface

active species contributing to the reaction path and classify the elementary steps belonging to the path (Kobayashi et al., 1995). The analysis frequently becomes complicated because contribution of the particular reaction route into overall reaction rate varies depending on the working range of reactant concentrations. Thus, commonly utilized steady-state kinetic analysis will not be able to provide enough information to elucidate the detailed mechanism of multi-pathway reactions. To further enhance understanding of the mechanism and kinetics of catalytic reactions, the transient methods have been developed. The transient method allows to discriminate reaction pathways and estimate kinetics of elementary steps by analyzing the temporal response of product concentration to the variation of inlet composition. The step response technique (Kobayashi and Kobayashi, 1974; Bennett, 1999), pulse response procedure (Redekop et al., 2014), transient analysis with a tapered element oscillating microbalance (Berger et al., 2008) etc. were successfully applied to the analysis of

E-mail address: [email protected] http://dx.doi.org/10.1016/j.ece.2015.12.002 1749-7728/© 2016 The Institution of Chemical Engineers. Published by Elsevier B.V. All rights reserved.

2

education for chemical engineers 1 5 ( 2 0 1 6 ) 1–18

complex reactions. Recently, Metkar et al. (2011) carried out the transient experiments to demonstrate the mechanism of selective catalytic reduction of NO with NH3 on commercial Fe-zeolite monolith catalyst. Pinaeva et al. (2014) studied water gas shift and partial oxidation of CH4 over nanocrystalline CeO2 –ZrO2 (–La2 O3 ) and Pt/CeO2 –ZrO2 (–La2 O3 ) catalyst using the step response method. González and Schaub (2015) performed the step-change experiments to investigate the activity of iron-based catalyst in the Fischer–Tropsch synthesis with H2 /CO2 . Chemical reaction engineering course is a fundamental subject for the advanced training of chemical engineers. The kinetic analysis of complex chemical reactions, which includes the multi-pathways catalytic reactions, is an essential section in that class. Recent utilization of computer-aided educational modules significantly impacted the chemical engineering pedagogical field (Cartaxo et al., 2014; Hernández et al., 2014). Several programs have been developed both commercially (CHEMSIMUL, Kirkegaard and Ejegbakke, 2000; Chemical-Workbench, Deminsky et al., 2003; CHEMKIN, Coltrin et al., 2001) and individually by instructors (Tenua, Wachsstock, 2015) to help students with the basic understanding of the chemical kinetics modeling. However, the skills obtained from using these modules are not easily translated to practical experience because derivations of reaction kinetic equations are programmed to be done automatically and a student has no real exposure to the actual problem solving process. Furthermore, the experimental setup for transient analysis of catalytic reaction requires fairly complicated and expensive equipment to be implemented in the student laboratory course. Therefore, there is a need to develop an alternative tool, such as an educational software to help students learn the theoretical background of transient response method while acquiring the practical skills in applying the following method for industrially important reactions. The educational software should ideally be based on the open-source resources, be easy to learn and modify, and have an extensive capability to visualize the results of simulation. The IPython system (Pérez and Granger, 2007) is well suited for the development of educational software. It uses the Python language, which is an open-source, straightforward and at the same time a powerful programming language. The IPython notebook web-based interface allows to mix the model explanation, computer code, simulation results and plots in a single file. Thus, it provides an ideal environment for students to learn the theory, construct their own kinetic model, input the model equations into the computer code by modifying the existent ones, conduct the simulation transient experiments and analyze the obtained results. Moreover, IPython notebook also offers an easy way for the instructor to monitor students’ performance as the derived kinetic model, computer code, simulation parameters and results are all present in the same place. In this paper, the educational template software was created with the aim to introduce students to the transient response method for the analysis of complex multi-pathway catalytic reactions. The notebooks contain tutorials to assist students in their learning. Using examples, a student can derive model equations himself, and by doing so broaden his knowledge on this subject. Students can use the software code as a template for his/her project. The software also includes notebooks with graphical user interface (GUI) to study the steady-state kinetics as well as to simulate the

transient responses of multipath reactions. The software is available from the author on request.

2.

Materials and methods

2.1.

Mathematical model

The stoichiometric equation for heterogeneous catalytic reaction is given as (Yablonsky et al., 1991; Murzin and Salmi, 2005)

a +

W 

int ,w

aint ,w = 0,

(1)

w=1

where  is the stoichiometric matrix for the gas-phase reacting components, int,w is the stoichiometric matrix for the surface intermediates attached to the active sites of w type, a is the vector-column of the gas-phase components, aint,w is the vector-column of intermediates and W is the number of active site types. The dimensions of the stoichiometric matrices  and int,w are S × N and S × Nint,w , respectively, where S is the number of elementary reactions, N is the number of gas-phase reacting species and Nint,w is the number of intermediates fixed on the wth type of active sites. The reaction rates of the gas-phase components, r, and surface intermediates, rint , are calculated as r = T R

(2)

and rint = Tint R,

(3)

where T and Tint are the transposed stoichiometric matrices of gas-phase reacting components and surface intermediates, respectively, and R is the vector-column of reaction rate for elementary steps. The reaction rate for elementary step s, Rs , is defined as

Rs =

k+ s

N 

si ( sign( si )−1)/2

ci

i=1

k− s

N 

j int ,ws





cint ,j



j int ,ws

sign 

 

−1 /2

−

w=1 j=1 W Nint ,w j j i i  sign  +1 /2 s ( sign( s )+1)/2 int ,ws int ,ws

ci

i=1

W N int ,w 







  ,

cint ,j

w=1 j=1

(4) where ci is the concentration of i gas-phase component and cint,j is the concentration of j surface intermediate. The overall reactions for all paths can be written as ␯T  a = 0,

(5)

where ␯ denotes the S × P matrix for stoichiometric numbers of s reaction in p path and P is the number of basic paths. The stoichiometric numbers should satisfy the equation T int = 0

(6)

This equation ensures the absence of intermediate species in the overall equation for each path. The following assumptions are made for the modeling of the laboratory scale fixed-bed reactor.

3

education for chemical engineers 1 5 ( 2 0 1 6 ) 1–18

(1) The reactor is operating under isothermal conditions since the fixed bed is composed of catalyst pellets mixed with inert particles. (2) Gas phase components are presumed to have plug flow behavior owing to sufficiently high flow velocity. (3) External and internal mass limiting transport effects are neglected.

Here the one-dimensional pseudo-homogeneous model is used for the modeling of laboratory catalytic reactor. The mass balance for i gas-phase reacting component in the plug-flow reactor is given by −

1 c Rs T



u·

∂Pi ∂u + Pi · ∂z ∂z



+ ri =

ε ∂Pi c Rs T ∂t

(7)

where Pi is the partial pressure of i component, z is the axial distance along the catalyst bed, t is the time, u is the superficial gas velocity, c is the bulk catalyst density, ε is the void fraction of catalyst bed, T is the temperature and Rs is the gas constant. The second term within the brackets describes the variation in superficial velocity through the bed. This variation can be caused by changes in the temperature, pressure and total flow rate (Fogler, 2006). In the actual experiments, the pressure drop across a short bed was kept to less than 2% of the total pressure. Considering that the conversion of reactants was less than a few percentages, the change in velocity through the reactor was small in the present study, especially in comparison with the inlet velocity. Thus, the superficial velocity is assumed to be constant during the transient period. Then, Eq. (7) becomes u ∂Pi ε ∂Pi − + ri = c Rs T ∂z c Rs T ∂t

∂t

= rint ,wj

(14)

The initial distributions of gas-phase components and surface intermediates in the axial direction of the catalytic bed are given by Pi (z, t = 0) = f1 (z)

(15)

wj (z, t = 0) = f2 (z)

(16)

where f1 (z) and f2 (z) are the arbitrary functions of axial distance z. The concentration of i gas-phase component at the reactor inlet is defined as Pi (z = 0, t) = P0i

(17)

The transient step response method refers to the analysis of temporal response of product concentration at the reactor outlet to the step change in the reactant concentration at the reactor inlet. Eq. (17) is used to specify the step change in the i reactant concentration at the reactor inlet, P0i . The product concentration at the reactor outlet, Pi (z = L, t), can be obtained by the solution of Eq. (9). The method of lines (Schiesser and Griffiths, 2009) is utilized to solve the system of N partial differential equation, Eq. (9). The uniform spatial grid is set up in the axial direction as L nz − 1

(18)

where nz is the number of spatial points, z is the mesh size and L is the length of catalytic bed. A backward difference formula is used to discretize the spatial derivative as (9)

(10)

P (z ) − Pi (zk−1 ) ∂Pi = i k ∂z z

(19)

Thus, the partial differential Eq. (9) for i gas-phase component is converted into the system of nz ordinary differential equations. The corresponding ordinary differential equation at zk , k = 1, . . ., nz − 1, spatial point is dPi (zk ) u P (z ) − Pi (zk−1 ) c Rs T =− · i k · ri (zk ) + dt ε ε z

The balance equation for each type of active sites is I cint ,w = qmw ,

wj

j=1

(8)

The mass balance for j surface intermediate on the active site of w type is ∂cint ,wj



Nint ,w ៭ −1

w = 1 −

z0 < · · · < znz−1 , z =

Eq. (8) can be rewritten as c Rs T ∂Pi u ∂P =− · i + · ri ∂t ε ∂z ε

The fraction of empty sites of w type, w , is calculated as

(11)

where  I is the unity vector with ones in all components, I = 1 · · · 1 , and qmw is the surface sorption capacity. The dimensionless fractional coverage is defined as

(20)

Eq. (9) at k = 0 is discretized as 0 dPi (z0 ) c Rs T u Pi (z0 ) − Pi =− · + · ri (z0 ) dt ε z ε

(21)

Eq. (13) becomes wj =

cint ,wj qmw

(12)

dt

The mass balance for surface intermediates in terms of fractional coverages is ∂wj ∂t

=

rint ,wj qmw

dwj (zk )

(13)

=

1 ·r (z ) qmw int ,wj k



The system of nz · N +

(22)

W w=1



(Nint ,w − 1) ordinary differ-

ential equations, Eqs. (20)–(22), is solved numerically using a method based on backward differentiation formulas (Brown et al., 1989).

4

education for chemical engineers 1 5 ( 2 0 1 6 ) 1–18

3.

Software

3.1.

Software description

The present work is an extension of our earlier efforts aimed at the development of computer software for simulation of transient responses (Golman et al., 1994). The previous version of the software was restricted to the analysis of transient responses in the differential reactor and the concentration distributions of gaseous components were taken as linearly changing with reactor length. Moreover, the software required knowledge in Fortran programming language and utilization of Fortran compiler. In the present work, all restrictions on the concentration distributions in the reactor are removed. The software was completely rewritten using an open-source Python programming language and was implemented as the modern web-based IPython notebooks. The VODE integrator from SciPy library (Jones et al., 2001) was used to solve the stiff system of ordinary differential equations. The plots were prepared with Matplotlib library (Hunter, 2007). The main features of software were designed to reflect ease of understanding and improved usability rather than efficiency in solving differential equations. The oxidation of carbon monoxide was chosen to demonstrate the application of transient response method to the analysis of multipath reaction using the developed software. We designed a notebook template to simulate the transient responses. The user input the reactor dimensions, reaction and process parameters as well as the reaction rate equations by modifying the existent code. Then, as the transient simulation is initialized, the calculated transient response curve is displayed in the real time. After finishing simulation, the user can export the calculated results in a text file for further analysis. The user can also visualize calculated results using various 1d and 3d plots. The detailed instruction on how to customize the notebook template is given in Appendix A. The source code for the notebook is shown in Appendix B. We also developed a notebook utilizing the graphical user interface to simulate the transient responses, as shown in Fig. 1(a). A user can choose the reaction temperature and the stepwise changes in the concentration of reactant at the reactor inlet to simulate the transient responses of the product at the reactor outlet. This notebook can also be used to demonstrate the different transient responses of single path and multiple paths reactions. The GUI notebook elaborated for the steady-state kinetic analysis is shown in Fig. 1(b). A user selects the reaction temperature and the value of PO2 . Then the program calculates and visualizes the variation of the steady-state rate of CO2 formation with PCO .

3.2.

Activities using the software

The educational module developed for teaching the application of transient response method for kinetic analysis of multipath reactions consists of three major parts. At first, students derive and verify the mechanistic kinetic model of multipath reaction. Then, students perform the transient response simulation using the elaborated software. They could either modify the notebook template or use the GUI notebook. Finally, students analyze the results of transient analysis by visualizing the response for various parameters and prepare the final report. The major activities are summarized below.

Fig. 1 – IPython notebooks with GUI for CO oxidation: (a) transient response analysis, (b) steady-state analysis.

(1) Derivation of the mechanistic kinetic model and its verification. 1. Define the vector-columns of the gas-phase components and surface intermediates. 2. Write the elementary reaction steps for each reaction path. 3. Specify the stoichiometric matrices for gaseous components and surface intermediates. 4. Define the matrix of stoichiometric numbers for each path. Check if the stoichiometric numbers satisfy Eq. (6). 5. Derive the overall reaction for each path. 6. Write the rate equations for elementary steps. 7. Derive the reaction rates for gaseous components and surface intermediates. (2) Steady-state analysis 8. Derive the steady-state rate equation.

5

education for chemical engineers 1 5 ( 2 0 1 6 ) 1–18

9. Plot the steady-state rate as a function of concentrations of gaseous components. 10. Visualize the contribution of each path to the steadystate rate. (3) Simulation of transient responses using the IPython notebook template or GUI notebook. 11. Simulate the transient responses separately for each path of multipath reaction and for combined multiple paths. 12. Simulate the transient responses with various magnitudes of step-change in reactant inlet concentration, temperatures, initial surface coverages and sets of kinetic parameters. (4) Analyze the results of transient response simulations by visualizing the responses for various parameters. 13. Compare the transient responses for each reaction path with one for multipath reaction. 14. Compare the multipath reaction responses for various magnitudes of step-change in the inlet reactant concentration. 15. Study the effect of temperature on the transient responses. 16. Compare transient responses for various values of initial surface coverages.

4.

Results and discussion

over copper-chromite catalyst at low temperature: rCO2 = 1 0 −1/3 , and Yu Yao (1973) reported the power k(PCO ) (PO2 ) (PCO2 ) −0.5 1.2 (PO2 ) . law equation on chromium oxide: rCO2 = k(PCO ) Our previous studies have confirmed that the carbon monoxide oxidation on zinc oxide catalyst (Kadox 25, New Jersey Zinc Co.) proceeds through two reaction paths (Kobayashi et al., 1995, 1997). Two different active sites were identified on the catalyst surface, SI and SII , and oxygen is adsorbed on the surface to form both neutral (O · SI ) and ionized (O− · SII ) species. The first path is assumed to proceed according to the Langmuir–Hinshelwood (LH) mechanism which is a reaction between adsorbed carbon monoxide and neutral oxygen species. The reaction, which belongs to the second path, between gaseous carbon monoxide and adsorbed ionized oxygen species proceeds according to the Eley–Rideal (ER) mechanism. Based on this information, students should be able to derive the following reaction mechanism of multipath reaction: Reaction path I (Langmuir–Hinshelwood route) k+ 1

CO (g) + SI CO · SI k− 1

k+ 2

O2 (g) + 2SI 2O · SI k− 2

4.1. Application of software to transient analysis of carbon monoxide oxidation reaction The oxidation of carbon monoxide is industrially important and relatively simple reaction with diverse transient behavior depending on the range of inlet concentration. The catalytic oxidation of carbon monoxide is the major method in eliminating highly toxic gas (Royer and Duprez, 2011). The development of efficient catalyst is required to deal with the large amounts of carbon monoxide emission.

k3

CO · SI + O · SI −→CO2 (g) + 2SI Reaction path II (Eley–Rideal route) k+ 4

O2 (g) + 2SII + 2e2O− · SII k− 4

4.1.1. Formulation of kinetic model for carbon monoxide oxidation The review of literature concerning the kinetics of carbon monoxide oxidation indicates the diverse kinetic behavior depending on the catalyst used and reaction parameters. Engel and Ertl (1979) reported a maximum in the rate dependence on CO pressure over a Pt (1 1 1) surface. They also showed a hysteresis in the rate of CO2 production by increasing and decreasing the CO pressure. Nibbelke et al. (1997) confirmed that the reaction rate on supported Pt/Al2 O3 catalyst is inhibited by CO with kinetics of −1 order at low concentration of CO and a first-order with respect to oxygen. However, Peterson et al. (2014) reported a positive reaction order with respect to CO (+0.35) and O2 (+0.15) on Pd/La alumina while negative order for CO (−0.2) on Pd/alumina. Recently, Tkachenko et al. (2015) studied the kinetics of lowtemperature CO oxidation by transition metal polycation exchanged low-silica faujasites (LSF). They suggested three different kinetic models such as the Eley–Rideal mechanism with zero order for CO and the first order for O2 on single Zn-, Mn- and double MnCu-polycation exchanged LSF, the Mars–van Krevelen mechanism with the reaction rate order 0.11 for CO and zero order for O2 on CuCaLSF catalyst, and the autocatalytic model with zero orders for both reactants on CuZn-polycation exchanged LSF. Hertl and Farrauto (1973) proposed the following kinetic law for CO oxidation

k5

CO(g) + O− · SII −→CO2 (g) + SII + e The overall reaction is 2 CO + O2 = 2CO2 Then, students identify three gas-phase components (CO, O2 and CO2 ), two different types of surface active sites (SI and SII ), two surface intermediates adsorbed on the active site of the first type (CO · SI and O · SI ) and an intermediate adsorbed on the active site of the second type (O− · SII ). The information is summarized in the form of the vector-columns of the gasphase components and surface intermediates as shown below.

⎡

⎤

⎡

⎤

 −  O · SII ⎢ O2 ⎥ ⎢ ⎥ a=⎣ ⎦ , aint ,I = ⎣ O · SI ⎦ , aint ,II = CO

CO2

CO · SI

SI

SII

Next, students formulate the stoichiometric matrices for gaseous components and surface intermediates. There are five elementary reactions including three reactions that enter into

6

education for chemical engineers 1 5 ( 2 0 1 6 ) 1–18

the first path and two into the second path. The stoichiometric matrix for gaseous components is

⎡ ⎢ ⎢ ⎢ ⎢ =⎢ ⎢ ⎢ ⎣

CO O2 −1

0 0

−

−

0

⎤

1

1

2

3

4

5

−1

0

0

0

−1

0

−1

0

0

+1

0

T = ⎣ 0 −1

0⎥ ⎥ 2

⎥ ⎥ −⎥ − ⎥ 0⎦ 4

+1 ⎥ 3

0 −1 −1

⎡

CO2

0 −1 0

where the transposed stoichiometric matrix T is

+1

0

⎤ CO 0 ⎦ O2

+1

CO2

Thus, the reaction rates for gaseous reactants, carbon monoxide and oxygen, and product, carbon dioxide, are defined as

5

The matrices for surface intermediates attached to the active sites of the first and second types as well as the combined matrix of surface intermediates are CO · SI O · SI SI O− · SII SII

⎡

CO · SI O · SI SI +1

0

int ,I = ⎣ 0

−1

⎤

1

2 −2 ⎦ 2

−1

−1

+2

⎡

O− · SII SII

,

 int ,II =

3

+2 −2 −1

+1



1, 2

int

+1

0

−1

0

0

0

0

0

−1

+1

⎤

1

⎢ 0 2 −2 0 0 ⎥ 2 ⎢ ⎥ ⎢ ⎥ = ⎢ −1 −1 +2 0 0⎥ 3 ⎢ ⎥ ⎣ 0 0 0 +2 −2 ⎦ 4 5

The next step includes specification of the matrix of stoichiometric numbers of elementary reactions in two paths as

⎡ ⎡

I II 2

0

⎤

⎢

2

⎡

− (R1 + R5 )

⎤

⎥ ⎢ ⎥ ⎦ = ⎣ − (R2 + R4 ) ⎦

r = ⎣ rO2

1

⎢1 0⎥ 2 ⎢ ⎥ ⎢ ⎥  = ⎢2 0⎥ 3 ⎢ ⎥ ⎣0 1⎦ 4 0

⎤

rCO

rCO2

(24)

R3 + R5

The reaction rates for surface intermediates are derived using Eq. (3) as

5

rint ,I = Tint ,I RI and rint ,II = Tint ,II RII

Students can validate the matrix of stoichiometric numbers by confirming that Eq. (6) is satisfied. Thereupon, students construct the rate expressions for elementary reaction steps Eq. (4) as

⎡

The transposed stoichiometric matrices of surface interand T , and vector-columns of reaction rate mediates, T int ,I int ,II for elementary steps, RI and RII , are given separately for each reaction path below:reaction path I

1

2

3

+1

0

−1

=⎣ 0 T int ,I

⎤ CO · SI

−1 ⎦ O · SI

2

−1 −2

+2

⎡ ⎢

R1

⎤

⎡

⎥

⎢

− k+ 1 · PCO · SI − k1 · CO·SI

− 2 2 , RI = ⎣ R2 ⎦ = ⎣ k+ 2 · PO2 · S − k2 · O·S I

⎥ ⎦

k+ 3 · CO·SI · O·SI

R3

SI

I

⎤

reaction path II

 Tint ,II =

⎡

R1

⎤

⎡

k+ 1

· PCO · SI − k− 1

· CO·SI

R5

+2

−1



 O− · SII , RII = SII

R4 R5



 =

− 2 2 k+ 4 · PO2 · S − k4 · O− ·S II



II

k+ 5 · PCO · O− ·SII

Thus, the reaction rates for surface intermediates in path I are

⎡ (23)

· PCO · O− ·SII

The reaction rates for gas-phase components are derived using Eq. (2) as r = T R,

2

−2 +1

⎤

⎢ ⎥ ⎢ + ⎥ 2 ⎢ R2 ⎥ ⎢ k2 · PO2 · S2I − k−2 · O·S ⎥ I ⎥ ⎢ ⎥ ⎢ ⎢ ⎥ ⎢ ⎥ R = ⎢ R3 ⎥ = ⎢ k+ ⎥ 3 · CO·SI · O·SI ⎢ ⎥ ⎢ ⎥ ⎢ R ⎥ ⎢ k+ · P · 2 − k− · 2 ⎥ − ⎣ 4 ⎦ ⎣ 4 O2 SII 4 O ·SII ⎦ k+ 5

1

⎢

rCO·SI

rint ,I = ⎣ rO·SI

⎤

⎡

⎥ ⎢ ⎦=⎣

R1 − R3 2R2 − R3

⎤ ⎥ ⎦

(25)

−R1 − 2R2 + 2R3

rSI

and the reaction rates for surface intermediates in path II are

 rint ,II =

rO− ·SII rSII



 =

2R4 − R5 −2R4 + R5

 (26)

7

education for chemical engineers 1 5 ( 2 0 1 6 ) 1–18

Fig. 3 – Illustration of contribution of each reaction path to the steady-state rate.

0.15 to 0.55 atm, both paths contribute nearly equally to the steady-state rate. The reaction rate passes through the minimum point at PCO = 0.34 atm, because the input of LH path to the rate decreases, but the input of ER rises with increasing PCO.

4.1.3.

Fig. 2 – Effects of (a) CO and (b) O2 concentrations on steady-state rate of CO2 formation. Subsequently, students substitute the rate expressions for elementary reactions by Eq. (23), reaction rates for gaseous components by Eq. (24) and reaction rates for surface intermediates by Eqs. (25) and (26) in Eqs. (20)–(22).

4.1.2.

The steady-state analysis

Based on the steady-state rate analysis, students can derive the following equation: 1/2

rCO2 = rE−R + rL−H =

1/2

k5 K4 PCO PO

2

1/2 1/2

1 + K4 PO

2

1/2

+



2 2

PO (zk ) − PO2 (zk−1 ) dPO2 (zk ) = −v · 2 + cv · rO2 (zk ) dt z PCO2 (zk ) − PCO2 (zk−1 ) dPCO2 (zk ) = −v · + cv · rCO2 (zk ) dt z dCO·SI (zk ) 1 = · rCO·SI (zk ) dt qmI

 , (27)

dO− ·SII (zk )

1/2 1/2 2

1 + K1 PCO + K2 PO

dPCO (zk ) PCO (zk ) − PCO (zk−1 ) = −v · + cv · rCO (zk ) dt z

dO·SI (zk ) 1 = · rO·SI (zk ) dt qmI

1/2

k3 K1 K2 PCO PO

Transient response analysis

Substituting the reaction rates for gaseous components by Eq. (24) and for intermediates by Eqs. (25) and (26) into Eqs. (20)–(22), students obtain the following system of ordinary differential equations to describe the time variations in partial pressures of gaseous components and fractional coverages of surface intermediates at the axial position zk , k = 1, . . ., nz − 1:

where K1 , K2 and K4 are the equilibrium constants for reactions (1), (2) and (4), respectively, and k3 and k5 are the forward rate constants for reaction (3) and (5). This equation was verified with experimental data (Kobayashi et al., 1995). The values of kinetic parameters are summarized in Table 1. The steady-state rate of carbon dioxide formation as a function of partial pressure of carbon monoxide has both maximum and minimum points, as illustrated in Fig. 2(a). However, the rate increases monotonically with increasing partial pressure of oxygen, as depicted in Fig. 2(b). Such complex rate dependence on reactant concentration could only be explained by the multi-pathway reaction model. The contribution of LH and ER paths to the steady-state rate of CO2 formation is illustrated in Fig. 3. At low concentration of carbon monoxide, the rate is mainly controlled by the LH path and at high PCO by the ER path. In PCO range of

dt

=

(28)

1 · r − (z ) qmII O ·SII k

where v = u/ε and cv = c · Rs · T/ε. The compositions of gas-phase components at z0 are calculated by solving the following differential equations: PCO (z0 ) − P0CO dPCO (z0 ) = −v · + cv · rCO (z0 ) dt z PO2 (z0 ) − P0O dPO2 (z0 ) 2 = −v · + cv · rO2 (z0 ) dt z

(29)

PCO2 (z0 ) − P0CO dPCO2 (z0 ) 2 = −v · + cv · rCO2 (z0 ) dt z where P0CO , P0O and P0CO are the partial pressures of carbon 2 2 monoxide, oxygen and carbon dioxide at the reactor inlet, respectively.

8

education for chemical engineers 1 5 ( 2 0 1 6 ) 1–18

Table 1 – Kinetic parameters of CO oxidation (Kobayashi et al., 1995). T [◦ C]

K1 [atm−1 ]

130 140 150 160 170

249 198 145 122 103

K2 [atm−1 ] 382 208 113 65.1 34.8

K4 [atm−1 ]

k3 [mol g−1 min−1 ] 7.76 × 10−8 1.62 × 110−7 3.46 × 110−7 8.02 × 110−7 1.50 × 110−6

105 30.4 10.1 3.1 1.14

The fractions of active sites of the first and second types are given by the following algebraic equations: SI = 1 − (CO·SI + O·SI ) SII = 1 − O− ·SII

(30)

To simulate transient responses students could modify the template notebook or use the GUI notebook. Since GUI notebook was specifically created for the example problem it does not require any modification from a student. We recommend students to use the template notebook in order to gain more understanding in model equations and numerical algorithms. By utilizing the template notebook, students can also collect the large variety of data for further analysis. The step changes in the inlet concentration of reactant are specified separately in three regions, as illustrated in Fig. 4. P0CO is increased stepwise from 0 to P0CO,A and kept constant at P0CO,A during the time interval tA (region A). Then, P0CO is again increased to P0CO,B and kept constant at P0CO,B for tB − tA minutes (region B). Finally, P0CO,B is decreased stepwise to P0CO,A (region C). The reactor parameters and kinetic constants are summarized in Tables 2 and 3. The typical example of a transient response curve of carbon dioxide concentration at the reactor outlet is illustrated in Fig. 5. The response was calculated by changing the inlet concentration of CO from 0 to 0.01 atm in A region, then to 0.71 atm in B region and finally to 0.01 atm in C region. The inlet oxygen concentration was kept constant at 0.2 atm. The temperature was set at 160 ◦ C. The mode of response depends significantly on the reaction conditions. The CO2 concentration at the reactor outlet monotonically increases at low value of PCO in the region A. However, the transient response shows a characteristic overshoot mode with a steep initial peak and quick decrease to steady value in the region B at PCO = 0.7 atm. The decrease in PCO in the region C results in a complex mode response with abrupt decrease and then sharp increase in the CO2 concentration at the beginning of this time

Fig. 4 – Illustration of stepwise change in partial pressure of carbon monoxide at reactor inlet.

k5 [mol g−1 min−1 atm−1 ] 1.85 × 110−8 5.26 × 110−8 1.37 × 110−7 3.25 × 110−7 8.50 × 110−7

interval followed by the decline to the fixed CO2 concentration. Such complex responses can be attributed to the simultaneous progress of two reaction paths. To clarify the contribution of each path into response of multipath reaction, students simulate the transient responses of each path separately, as shown in Fig. 6. The formation of carbon dioxide is controlled mainly by the LH route in the A region. In the B region, the response is determined by the progress of two routes. The increase in CO2 concentration at the initial stage results from the contribution of both ER and LH paths. Then, the concentration quickly decreases following the LH paths up to the fixed value defined by ER path. As a result, the response curve is of overshoot type. In the C region, the abrupt decrease in the CO2 concentration due to the ER path is followed by the complex response owing to the competitive progress of ER and LH paths.

Fig. 5 – Typical response curve for CO oxidation.

Fig. 6 – Transient response curves for various reaction mechanisms of CO oxidation.

9

education for chemical engineers 1 5 ( 2 0 1 6 ) 1–18

Table 2 – Reactor parameters for CO oxidation (Kobayashi et al., 1995). Reaction temperature, T [◦ C]

130–170

Reactor length, L [cm]

58.7

Superficial gas velocity, u [cm min−1 ]

204

Total amount of active sites, 106 [mol g−1 ]

qmI

qmII

1.1

1.1

It is also advisable to study the effects of the magnitude of stepwise change in the inlet CO concentration, temperature and the surface coverage of the adsorbed species on the mode of response curve. For example, Fig. 7 illustrates the transient responses calculated for P0CO equal to 0.3, 0.5 and 0.7 atm. The concentration of carbon dioxide at the reactor outlet is mainly controlled by LH path at low P0CO . However, the contribution of the ER path increases at high P0CO , as exemplified by the high fixed value of CO2 concentration in the region B and significant concentration drop in the beginning of C time interval. The transient responses at various temperatures are shown in Fig. 8. At higher temperature, the initial peak in B region is more profound and the response quickly declines following the LH path. The effect of initial coverage on transient responses is illustrated in Fig. 9(a) and (b). Both figures were calculated by

Porosity, ε [–]

0.56

Catalyst bed density, c [g cm−3 ]

1.7

imposing the same step changes PCO = 0.2 atm. However, the initial surface coverages differ according to the PCO concentration in A region, i.e. PCO = 0.51 atm (Fig. 9(a)) and PCO = 0.31 atm (Fig. 9(b)). The response curve of distinct overshoot mode in B region is obtained at low initial CO concentration (Fig. 9(b)) due to the significant contribution of LH path. Similarly, the response in region C mainly follows ER path at high initial CO concentration. The related trends are observed in the steadystate kinetic analysis, as shown in Fig. 3.

4.2.

Transient analysis to model dual-path reactions

As an exercise to learn the functionality of the developed software, two theoretical dual-path reaction models were created for students. The first model reaction (Fig. 10a) is similar to the CO oxidation. The two different active sites are assumed to participate in reaction. The reaction between adsorbed species X1 · SI and X2 · SII proceeds along path I according to the LH mechanism. The adsorbed species X2 · SII also reacts with gaseous reactant X1 following ER mechanism in path II. Furthermore, students can assume that the product Y quickly desorbs from the catalyst surface and the surface reactions

Fig. 7 – Transient responses of CO oxidation for 0 = 0.3, 0.5 and 0.7 atm. PCO

Fig. 8 – Transient responses of CO oxidation at various temperatures.

Fig. 9 – Effect of initial coverages on transient responses of CO oxidation: (a) PCO = 0.51(A)–0.71(B)–0.51(C) atm, (b) PCO = 0.31(A)–0.51(B)–0.31(C) atm.

10

education for chemical engineers 1 5 ( 2 0 1 6 ) 1–18

Table 3 – Rate constants for reaction steps 1, 2 and 4 (Kobayashi et al., 1995). T [◦ C] 130 140 150 160 170

k1 [mol g−1 min−1 atm−1 ] 3.88 × 10−5 8.10 × 110−5 1.73 × 110−4 4.01 × 110−4 7.50 × 110−4

k2 [mol g−1 min−1 atm−1 ] 3.88 × 110−5 8.10 × 110−5 1.73 × 110−4 4.01 × 110−4 7.50 × 110−4

k4 [mol g−1 min−1 atm−1 ] 9.25 × 110−6 2.63 × 110−5 6.85 × 110−5 1.63 × 110−4 4.25 × 110−4

Fig. 10 – Model multipath reactions: (a) model 1, (b) model 2. control the reaction rates in both paths. The second model reaction (Fig. 10b) also includes the dual-path reaction with the single overall reaction. However, students should consider that this reaction proceeds on the same type of active sites in both paths. In addition, they can assume that the reactant X2 is pre-adsorbed on the catalyst surface filling the active sites. The distinct modes of transient responses for model 1 and 2 are illustrated in Fig. 11(a) and (b), respectively. The transient responses were calculated using reactor parameters summarized in Table 4 and kinetic parameters shown in Tables 5 and 6.

5.

Module feedback

To evaluate the pedagogical effectiveness of educational module and usability of the software, the students were asked to fill an anonymous questionnaire as well as to give their opinions on the module. The pedagogical effectiveness is related to statements 1–7 and the module usability is evaluated in statements 8–12 (Table 7). Total seven graduate students participated in this study. They had studied Chemical Reaction Engineering I and II undergraduate courses, but those courses did not cover the analysis of multipath reactions. They had no previous experience with Python programming language. The students were asked to reply to the questionnaire using a 5-point Likert scale: 5—strongly agree (sa); 4—agree (a); 3—neither agree or disagree (nad); 2—disagree (d); 1—strongly disagree (sd). The feedback from students was very positive regarding the module for transient kinetic analysis of multipath reactions. Fig. 12 illustrates the results of survey. Here the grades were calculated by averaging the numerical values of answers on the Likert scale (Hernández et al., 2014). The survey indicated student satisfaction with the teaching methodology that combined theoretical analysis with practical experience using the developed software. The majority of students (71.4%) considered that the developed software bolstered their knowledge on kinetic analysis of multi-path reactions (Q1). All students found this module interesting and involving (Q6). However,

Fig. 11 – Model multipath reactions: (a) model 1, (b) model 2. almost 43% students disagree or neither agree or disagree that the module is well suited to study program (Q7). More than 85% of the students appreciated the GUI version of the software (Q8) by pointing out that it allows them quickly familiarize with the methodology of transient response

11

education for chemical engineers 1 5 ( 2 0 1 6 ) 1–18

Table 4 – Reactor parameters for model 1 and 2. Reaction temperature, T [◦ C]

170

Reactor length, L [cm]

50

Total amount of active sites, 106 [mol g−1 ]

Superficial gas velocity, u [cm min−1 ]

200

qmI

qmII

1.0

1.0

Porosity, ε [–]

Catalyst bed density, c [g cm−3 ]

0.50

1.5

Table 5 – Kinetic parameters for model reaction 1. k1 2.0 × 10−4

k2

k3

k4

k5

K1

K2

K4

2.0 × 10−4

4.0 × 10−7

7.5 × 10−5

1.5 × 10−7

150

110

10

Table 6 – Kinetic parameters for model reaction 2. k+ 1 1.0 × 10−4

k− 1

k+ 2

k− 2

k+ 3

k+ 4

1.0 × 10−7

1.0 × 10−4

1.0 × 10−7

1.0 × 10−7

5.0 × 10−7

Table 7 – Survey questionnaire for assessing the pedagogical effectiveness and software usability of the module. No

Statements

sa

a

nad

d

sd

1 2 3 4 5 6 7 8 9 10 11 12

The module helped me to acquire new knowledge on kinetics of multipath reactions I was able to understand the module content using my previous knowledge Problems given to me are clearly defined I was able to use notebooks by myself The module helped me to enhance my knowledge on numerical modeling The module is interesting and involving The module is well suited to study program The GUI notebook is easy to use The notebook template is easy to modify Process and simulation parameters are easy to change Simulation results are well visualized Notebooks are well documented

           

           

           

           

           

Fig. 12 – Survey results. analysis. Students also appraised the software usability especially the real time visualization of calculated transient responses (Q11). High fraction of students (86%) provided positive comments on readability in modifying the template code for specific problems (Q12). Nevertheless, 26% of the students did not agree that the notebook template is easy to modify (Q9). The possibility of the current program to collect the large number of data and presenting them in an attractive manner in the form of 3D plots was also appreciated by the students.

6.

Conclusions

In this paper, we have presented an educational module and software package for the application of transient response method to the kinetic analysis of multipath reactions. The software was developed as a set of IPython notebooks combining the model derivation, numerical simulation, and visualization and analysis of calculated results. The combination of theoretical analysis with practical applications of the software helped students to acquire knowledge and to

12

education for chemical engineers 1 5 ( 2 0 1 6 ) 1–18

develop critical thinking skills related to the kinetic analysis of complex industrially important reactions.

Appendix A. Instructions on how to customize the notebook template. To customize the IPYTHON Notebook template, students can take the following steps: (I) Modify the function that evaluates the right side of the system of differential equations: 1. Define the notation for the inlet concentrations of gaseous components in the global area. Specify the notation for the forward and reverse reaction constants in the vector params. 2. Allocate arrays of gaseous components and surface intermediates. Split array y into arrays of gaseous components and surface intermediates. 3. Revise the balance equations for vacant active sites. 4. Revise the reaction rates for elementary steps according to the reaction mechanism. 5. Revise the production rates of gas-phase components and surface intermediates.

6. Modify the corresponding material balances for gasphase components and surface intermediates. 7. Modify the vector of left-hand sides of ODEs. (II) Modify the main program: 8. Revise the global parameters as described in (1). 9. Specify the reactor parameters and kinetic constants. 10. Set the time intervals and step changes of inlet concentrations of gas components. 11. Specify the number of types of differential equations which is equal to the sum of the number of gaseous components and surface intermediates excluding the active sites. 12. Modify the specification of inlet concentrations in region A. 13. Modify the vector params as described in (1). 14. Modify the notation for the vector to store the desired product concentration at the reactor outlet at various time. 15. Modify the specification of inlet concentrations in regions B and C. 16. Modify the label titles and intervals of transient plots. 17. Modify the notation of arrays of gas-phase components to be saved in a file.

education for chemical engineers 1 5 ( 2 0 1 6 ) 1–18

Appendix B. The source code for the notebook template.

13

14

education for chemical engineers 1 5 ( 2 0 1 6 ) 1–18

education for chemical engineers 1 5 ( 2 0 1 6 ) 1–18

15

16

education for chemical engineers 1 5 ( 2 0 1 6 ) 1–18

education for chemical engineers 1 5 ( 2 0 1 6 ) 1–18

References Azadi, P., Brownbridge, G., Kemp, I., Mosbach, S., Dennis, J.S., Kraft, M., 2015. Microkinetic modeling of the Fischer–Tropsch synthesis over cobalt catalysts. ChemCatChem 7, 137–143. Bennett, C.O., 1999. Experiments and processes in the transient regime for heterogeneous catalysis. Adv. Catal. 44, 329–416. Berger, R.J., Kapteijn, F., Moulijn, J.A., Marin, G.B., De Wilde, J., Olea, M., Chen, D., Holmen, A., Lietti, L., Tronconi, E., Schuurman, Y., 2008. Dynamic methods for catalytic kinetics. Appl. Catal., A: Gen. 342, 3–28.

17

Brown, P.N., Byrne, G.D., Hindmarsh, A.C., 1989. VODE, a variable-coefficient ODE solver. SIAM J. Sci. Stat. Comput. 10, 1038–1051. Cartaxo, S.J.M., Silvino, P.F.G., Fernandes, F.A.N., 2014. Transient analysis of shell-and-tube heat exchangers using an educational software. Educ. Chem. Eng. 9, e77–e84. Coltrin, M.E., Kee, R.J., Rupley, F.M., 2001. Surface chemkin: a general formalism and software for analyzing heterogeneous chemical kinetics at a gas-surface interface. Int. J. Chem. Kinet. 23, 1097–4601.

18

education for chemical engineers 1 5 ( 2 0 1 6 ) 1–18

Deminsky, M., Chorkov, V., Belov, G., Cheshigin, I., Knizhnik, A., Shulakova, E., Shulakov, M., Iskandarova, I., Alexandrov, V., Petrusev, A., Kirillov, I., Strelkova, M., Umanski, S., Potapkin, B., 2003. Chemical Workbench––integrated environment for materials science. Comput. Mater. Sci. 28, 169–178. Engel, T., Ertl, G., 1979. Elementary steps on catalytic oxidation of carbon monoxide over platinum metals. Adv. Catal. 28, 1–78. Fogler, S.H., 2006. Elements of Chemical Reaction Engineering, fourth ed. Prentice Hall, New Jersey, pp. 111. Golman, B., Fujisaki, S., Kanno, T., Kobayashi, M., 1994. Dynamics of fixed bed reactor with a kinetically controlled complex reaction. Mem. Kitami Inst. Technol. 26, 1–7. González, M.I., Schaub, G., 2015. Fischer–Tropsch synthesis with H2 /CO2 —catalyst behavior under transient conditions. Chem. Ing. Tech. 87, 848–854. Hertl, W., Farrauto, R.J., 1973. Mechanism of carbon monoxide and hydrocarbon oxidation on copper chromite. J. Catal. 29, 352–360. Hernández, Y.G., Haza, U.J.J., Albasi, C., Alliet, M., 2014. Development of a submerged membrane bioreactor simulator: a useful tool for teaching its functioning. Educ. Chem. Eng. 9, e32–e41. Hunter, J.D., 2007. Matplotlib: a 2D graphics environment. Comput. Sci. Eng. 9, 90–95. Jones, E., Oliphant, T., Peterson, P., et al., 2001. SciPy: Open Source Scientific Tools for Python, http://www.scipy.org/ [online; accessed 2015-09-20]. Kirkegaard, P., Ejegbakke, E., January 2000. CHEMSIMUL: a simulator for chemical kinetics. In: Riso-R-1085. Riso National Laboratory, Roskilde, Denmark. Kobayashi, H., Kobayashi, M., 1974. Transient response method in heterogeneous catalysis. Catal. Rev. 10, 139–176. Kobayashi, M., Golman, B., Kanno, T., 1995. Diversity of the kinetics of carbon monoxide oxidation on differently prepared zinc oxides. J. Chem. Soc. Faraday Trans. 91, 1391–1398. Kobayashi, M., Golman, B., Kanno, T., Fujisaki, S., 1997. Reaction pathway design and optimization in heterogeneous catalysis: I. Optimal proportion of multi-pathways designed by the transient response method. Appl. Catal., A: Gen. 151, 193–205. Maitlis, P.M., de Klerk, A. (Eds.), 2013. Greener Fischer–Tropsch Processes for Fuels and Feedstocks. Wiley-VSH Verlag, Weinheim, Germany. Metkar, P.S., Salazar, N., Muncrief, R., Balakotaiah, V., Harold, M.P., 2011. Selective catalytic reduction of NO with NH3 on iron

zeolite monolithic catalysts: steady-state and transient kinetics. Appl. Catal., B: Environ. 104, 110–126. Murzin, D., Salmi, T., 2005. Catalytic Kinetics. Elsevier, Amsterdam. Nibbelke, R.H., Campman, M.A.J., Hoebink, J.H.B.J., Marin, G.B., 1997. Kinetic study of the CO oxidation over Pt/␥-Al2 O3 and Pt/Rh/CeO2 /␥-Al2 O3 in the presence of H2 O and CO2 . J. Catal. 171, 358–373. Pérez, F., Granger, B.E., 2007. IPython: a system for interactive scientific computing. Comput. Sci. Eng. 9, 21–29. Peterson, E.J., DeLaRiva, A.T., Lin, S., Johnson, R.S., Guo, H., Miller, J.T., Hun Kwak, J., Peden, Ch.H.F., Kiefer, B., Allard, L.F., Ribeiro, F.H., Datye, A.K., 2014. Low-temperature carbon monoxide oxidation catalysed by regenerable atomically dispersed palladium on alumina. Nat. Commun. 5, doi: http://dx.doi.org/10.1038/ncomms5885. Pinaeva, L.G., Sadovskaya, E.M., Ivanova, Yu.A., Kuznetsova, T.G., Prosvirin, I.P., Sadykov, V.A., Schuurman, Y., van Veen, A.C., Mirodatos, C., 2014. Water gas shift and partial oxidation of CH4 over CeO2 –ZrO2 (–La2 O3 ) and Pt/CeO2 –ZrO2 (–La2 O3 ): performance under transient conditions. Chem. Eng. J. 257, 281–291. Redekop, E.A., Yablonsky, G.S., Constales, D., Ramachandran, P.A., Gleaves, J.T., Marin, G.B., 2014. Elucidating complex catalytic mechanisms based on transient pulse-response kinetic data. Chem. Eng. Sci. 110, 20–30. Royer, S., Duprez, D., 2011. Catalytic oxidation of carbon monoxide over transition metal oxides. ChemCatChem 3, 24–65. Schiesser, W.E., Griffiths, G.W., 2009. A Compendium of Partial Differential Equations Models: Method of Lines Analysis with Matlab. Cambridge University Press, Cambridge. Tkachenko, O.P., Greish, A.A., Kucherov, A.V., Weston, K.C., Tsybulevski, A.M., Kustov, L.M., 2015. Low-temperature CO oxidation by transition metal polycation exchanged low-silica faujasites. Appl. Catal., B: Environ. 179, 521–529. Wachsstock, D., 2015. Tenua—The Kinetics Simulator for Java, http://www.bililite.com/tenua/[online; accessed 2015-09-20]. Yablonsky, G., Bykov, V., Gorban, A., Elokhin, V., 1991. Kinetic Models of Catalytic Reactions. In: Compton, R.G. (Ed.), Comprehensive Chemical Kinetics, vol.32. Elsevier, Amsterdam. Yu Yao, Y.-F., 1973. The oxidation of hydrocarbons and CO over metal oxides. J. Catal. 28, 139–149.