Chapter 10: Catalysis and Catalytic Reactors


Topics

  1. Octane Rating
  2. Steps in Catalytical Reaction
  3. Rate Limiting Step
  4. Regulation for Automotive Exhaust Emissions
  5. Chemical Vapor Deposition
  6. Types of Catalyst Deactivation
  7. Temperature-Time Trajectories
  8. Moving Bed Reactors & Straight Through Transport Reactors

Octane Rating

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A Typical Engine Piston

Diagram of an internal combustion engine cylinder with a spark plug at the top. The image includes labeled regions: '1' showing red curved lines near the spark plug and '2' with red circular waves near the piston, indicating flame propagation.
  1. a uniform burning front

  2. spontaneous combustion producing
    detonation waves and knock
          

Determine the compression ratio, CR,
to achieve the standard knock intensity.

The more compact molecules are (for a given
number of carbon atoms), the greater the
octane number they will have.

Graph showing the relationship between compression ratio (CR) and octane number. The x-axis represents the octane number, ranging from 0% n-heptane to 100% iso-octane, while the y-axis represents the compression ratio, increasing non-linearly.
 
100% isooctane = 100 octane number
100% heptane = 0 octane number

Link Reaction Video: Some multiple reaction systems oscillate, recreating the reactants when certain conditions are met.

Diagram illustrating the steps in a heterogeneous catalytic reaction. The process involves external diffusion, internal diffusion, surface reaction, and product diffusion. Reactant A diffuses to the catalytic surface, undergoes adsorption and reaction to form product B, which then desorbs and diffuses away. The figure is labeled with steps 1 through 7, representing each stage of the reaction process.

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Molecular Adsorption

\(\text{H}_2 + S \rightleftharpoons \text{H}_2 \cdot S\)

At equilibrium:

\(r_{AD} = 0 = k \left[ P_{\text{H}_2} C_V - \frac{C_{\text{H}_2 \cdot S}}{K_{\text{H}_2}} \right]\)

\(C_{\text{H}_2 \cdot S} = K_{\text{H}_2} P_{\text{H}_2} C_V\)

\(C_t = C_V + C_{\text{H}_2 \cdot S}\)

\(C_{\text{H}_2 \cdot S} = \frac{K_{\text{H}_2} P_{\text{H}_2}}{1 + K_{\text{H}_2} P_{\text{H}_2}} C_t\)


Langmuir Isotherms

Graph showing the relationship between the volume of gas adsorbed and pressure (P_A). Two curves, labeled T1 and T2, represent different temperatures, with T2 being greater than T1. The red arrow indicates that an increase in temperature decreases gas adsorption. Equation notation shows the volume of gas adsorbed is proportional to C_A.s.

Dissociative Adsorption

\(\text{H}_2 + 2S \rightleftharpoons 2\text{H} \cdot S\)

At equilibrium:

\(r_{AD} = 0 = k \left[ P_{\text{H}_2} C_V^2 - \frac{C_{\text{H}_2 \cdot S}^2}{K_H} \right]\)

\(C_{\text{H} \cdot S} = C_T \frac{\sqrt{P_{\text{H}_2} K_H}}{1 + \sqrt{P_{\text{H}_2} K_H}}\)


Steps in a Catalytic Reaction

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\( A \leftrightarrow B \)

Adsorption on Surface

\( A + S \leftrightarrow A \bullet S \)

Surface Reaction

Single Site

\( A \bullet S \leftrightarrow B \bullet S \)

Dual Site

\( A \bullet S + S \leftrightarrow B \bullet S + S \)

Desorption from Surface

\( B \bullet S \leftrightarrow B + S \)

  


\( A + B \leftrightarrow C + D \)

Adsorption on Surface

\( A + S \leftrightarrow A \bullet S \)

\( B + S \leftrightarrow B \bullet S \)

Surface Reaction

Dual Site

\( A \bullet S + B \bullet S \leftrightarrow C \bullet S + D \bullet S \)

\( A \bullet S + B \bullet S' \leftrightarrow C \bullet S + D \bullet S' \)

Eley-Rideal

\( A \bullet S + B (g) \leftrightarrow C \bullet S + D (g) \)

Desorption from Surface

\( C \bullet S \leftrightarrow C + S \)

\( D \bullet S \leftrightarrow D + S \)


Example: Catalytic Reaction to Improve the Octane Number of Gasoline

\( \text{n-pentane} \leftrightarrow \text{i-pentane} \)

\( \text{75% Pt} \)

\( \text{on } \text{Al}_2\text{O}_3 \)

Rationale:

n-pentane: Octane No. = 62
i-pentane: Octane No. = 90

The difference in octane ratings provides
an economic incentive for carrying out this
reaction.

Steps in this reaction:

\( \text{n-pentane} \overset{\text{Pt}, -H_2}{\leftrightarrow} \text{n-pentene} \overset{\text{Al}_2O_3}{\leftrightarrow} \text{i-pentene} \overset{\text{Pt}, +H_2}{\leftrightarrow} \text{i-pentane} \)

Focusing on the second reaction:

\( \text{n-pentene} \leftrightarrow \text{i-pentene} \)

\( \text{Al}_2O_3 \)

\( N \leftrightarrow I \)


Rate Limiting Steps

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Adsorption

\( N + S \leftrightarrow N \bullet S \)

\( r_{AD} = k_A \left[ P_N C_{\vee} - \frac{C_{N \bullet S}}{K_N} \right] \)

Surface Reaction

\( N \bullet S \leftrightarrow I \bullet S \)

\( r_S = k_S \left[ C_{N \bullet S} - \frac{C_{I \bullet S}}{K_S} \right] \)

Desorption

\( I \bullet S \leftrightarrow I + S \)

\( r_D = k_D \left[ C_{I \bullet S} - \frac{P_I C_{\vee}}{K_{DI}} \right] = k_D \left[ C_{I \bullet S} - K_R P_I C_{\vee} \right] \)

Assume surface reaction is rate limiting

\( -r_S = k_S \left[ C_{N S} - \frac{C_{I \bullet S}}{K_S} \right] \)

If the surface reaction is limiting then:

\( \frac{r_{AD}}{k_A} \approx 0 \quad \therefore \quad C_{N \bullet S} = K_N P_N C_{\vee} \)

see also stirctly speaking link

\( \frac{r_D}{k_D} \approx 0 \quad \therefore \quad C_{I \bullet S} = K_I P_I C_{\vee} \)

  see also strictly speaking link

\( -r_N = k_S \left[ K_N P_N - K_I P_I \right] C_{\vee} = k_S K_N \left[ P_N - \frac{P_I}{K_P} \right] C_{\vee} \)

\( \text{where } K_P = \frac{K_S K_N}{K_I} \)

Site balance:

\( C_T = C_{\vee} + C_{N \bullet S} + C_{I \bullet S} \)

Substituting for CN-S, CI-S, and CV into CT = CV (1 + KN PN + KI PI) :

\( -r_N' = \frac{k_S K_N C_T \left[ P_N - \frac{P_I}{K_P} \right]}{1 + K_N P_N + K_I P_I} \)

where KP is the thermodynamic equilibrium constant for the reactor.

Linearizing the Initial Rate:

\( -r_N' = \frac{k P_{N0}}{1 + K_N P_{N0}} \quad \Rightarrow \quad \frac{P_{N0}}{-r_N'} = \frac{1}{k} + \frac{K_N}{k} P_{N0} \)

A mathematical plot with a y-axis labeled as P_N0 / -r'_N and an x-axis labeled as P_N0. The plot shows a straight red line with an upward slope. An arrow points to the line, and a red annotation states 'Slope = K_N / k'. Another annotation in blue states 'Intercept = 1 / k'.


Strictly speaking on setting rad/ka=0


Single Site
A)

\( A \cdot S \rightarrow B \cdot S \)

\(-r'_A = \frac{k P_A}{1 + K_A P_A + K_B P_B} \)

 Plot
Dual Site
B)

\( A \cdot S + S \rightarrow B \cdot S + S \)

\(-r'_A = \frac{k P_A}{(1 + K_A P_A + K_B P_B)^2} \)

 Plot
C)

\( A \cdot S + B \cdot S \rightarrow C \cdot S + S \)

\(-r'_A = \frac{k P_A P_B}{(1 + K_A P_A + K_B P_B + K_C P_C)^2} \)

 Plot
Eley-Rideal
D)

\( A \cdot S + B(g) \rightarrow C \cdot S \)

\(-r'_A = \frac{k P_A P_B}{1 + K_A P_A + K_C P_C} \)

 Plot


\( 2CO + O_2 \mathrel{\substack{\xrightarrow{} \\ \xleftarrow{\text{cat}}}} 2CO_2 \)

Regulations for Automotive Exhaust Emissions

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\( CO + NO \xrightarrow{\text{cat}} CO_2 + \frac{1}{2} N_2 \)
A table displaying emission levels in grams per mile for three pollutants (HC, CO, and NO) across three years (1981, 1993, and 2004). HC values are 0.41 in 1981, 0.25 in 1993, and 0.125 in 2004. CO values are 3.4 in 1981, 3.4 in 1993, and 1.7 in 2004. NO values are 1.0 in 1981, 0.4 in 1993, and 0.2 in 2004. The labels HC, CO, and NO are in red, blue, and green respectively.
Principle Reactions:

\( CO + S \leftrightarrow CO \bullet S \)

\( NO + S \leftrightarrow NO \bullet S \)

\( CO \bullet S + NO \bullet S \rightarrow CO_2 + S + N \bullet S \)

\( N \bullet S \leftrightarrow \frac{1}{2} N_2 + S \)
Surface reaction limiting:

\( -r_{CO} = r_s = \frac{k_s C_T^2 K_{NO} K_{CO} P_{NO} P_{CO}}{(1 + K_{NO} P_{NO} + K_{CO} P_{CO})^2} \)

A mathematical equation showing the rate of reaction for CO. The equation is given as -r_CO = r_s = (k_s * C_T^2 * K_NO * K_CO * P_NO * P_CO) / ( (1 + K_NO * P_NO + K_CO * P_CO)^2 ). The numerator includes reaction constants and partial pressures, while the denominator represents the squared sum of adsorption terms. The equation includes a bracketed notation for k, indicating a grouping of terms.

Derive Derive this equation
A plot with K_CO2 on the y-axis and P_CO on the x-axis. The curve peaks at an optimal P_CO, labeled as 'P_CO OPT' in green. To the right of the plot, there are two mathematical statements: In red, 'Low P_CO Then r_CO ≈ P_CO'. In blue, 'High P_CO Then r_CO ≈ P_CO / P_CO^2 ≈ 1 / P_CO'.

\( \frac{d \, r_{CO_2}}{d P_{CO}} = 0 = \frac{k P_{NO} (1 + K_{CO} P_{CO} + K_{NO} P_{NO})^2 - k P_{NO} P_{CO} \cdot 2(1 + K_{CO} P_{CO} + K_{NO} P_{NO}) K_{CO}}{(1 + K_{CO} P_{CO} + K_{NO} P_{NO})^4} \)

\( P_{NO} (1 + K_{NO} P_{NO} + K_{CO} P_{CO}) = 2 P_{NO} P_{CO} K_{CO} \)

\[ \left. P_{CO} \right|_{\text{opt}} = \frac{1 + K_{NO} P_{NO}}{K_{CO}} \]

  

Example

Let's see what fraction of sites are covered by CO at the optimum:

\( K_{CO} P_{CO} = 1 + K_{NO} F_{NO} \)
Multiplying by CV:

\( C_{\vee} K_{CO} P_{CO} = C_{\vee} + C_{\vee} K_{NO} P_{NO} \)
(A)

\( C_{CO \bullet S} = C_{\vee} + C_{NO \bullet S} \)
(B)

\( C_t = C_{\vee} + C_{NO \bullet S} + C_{CO \bullet S} \)

\( \frac{C_{CO \bullet S}}{C_t} = \frac{1}{2} \)
Example Dimethyl Ether Examples
Example Example Exam Questions

Chemical Vapor Deposition

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Manufacturing of a Silicon Layer

A detailed diagram illustrating the microelectronic fabrication steps. It begins with the Czochralski crystal growth process, where a silicon ingot is pulled from molten silicon in a quartz crucible. The ingot is sliced into wafers, which are then cleaned and polished using a polishing slurry. Next, a silicon dioxide layer is deposited via Chemical Vapor Deposition (CVD), followed by photoresist application. UV irradiation through a mask exposes the photoresist, which is then developed, followed by etching and photoresist removal. Additional steps include silicon dioxide etching, phosphorus diffusion doping, and multiple cycles of CVD, masking, etching, and stripping to form the final microelectronic layers.



We see that a number of the key steps in the microelectronic fabrication involve CVD, we shall consider the CVD of silicon.


I Mechanism
(1) \( \text{SiH}_4 (g) \leftrightarrow \text{SiH}_2 (g) + \text{H}_2 (g) \)

\( -\tilde{r}_{\text{SiH}_4} = k_g \left[ P_{\text{SiH}_4} - P_{\text{SiH}_2} P_{\text{H}_2} / K_P \right] \)
(2) \( \text{SiH}_2 (g) + S \leftrightarrow \text{SiH}_2 \bullet S \)

\( r_{AD} = k_A \left[ P_{\text{SiH}_2} f_v - \frac{f_{\text{SiH}_2}}{K_{\text{SiH}_2}} \right] \)
(3) \( \text{SiH}_2 \bullet S \rightarrow \text{Si} + \text{H}_2 (g) \)

\( r_S = k_S f_{\text{SiH}_2} \)
Diagram showing the chemical vapor deposition (CVD) of silicon from silane (SiH2). The top and side views illustrate silicon atoms (Si) on a substrate surface. Silane gas (SiH2) adsorbs onto the surface, with hydrogen atoms attached. In the side view, a silicon atom from SiH2 bonds to the surface, and H2 gas is released. This results in the formation of a new silicon layer. The final image shows a step-like structure where a new silicon atom has been incorporated into the surface lattice.
II Rate Limiting Step (Reaction 3)
\( r_{\text{dep}} = r_S = k_S f_{\text{SiH}_2} \)
III Expressing fSiH2 in Terms of Partial Pressures
\( \frac{r_{AD}}{k_A} \approx 0 \quad \Rightarrow \quad f_{\text{SiH}_2} = K_{\text{SiH}_2} P_{\text{SiH}_2} f_v \)
\( -\tilde{r}_{\text{SiH}_2} = r_S = k_S f_{\text{SiH}_2} = k_S K_{\text{SiH}_2} P_{\text{SiH}_2} f_v \)
IV Site / Surface Area Balance:

\( f_{\text{SiH}_2} + f_v = 1 \quad \Rightarrow \quad K_{\text{SiH}_2} P_{\text{SiH}_2} f_v + f_v = 1 \)

\( f_v = \frac{1}{1 + K_{\text{SiH}_2} P_{\text{SiH}_2}} \)

\[ -\tilde{r}_{\text{SiH}_2} = k_S K_{\text{SiH}_2} \frac{P_{\text{SiH}_2}}{1 + K_{\text{SiH}_2} P_{\text{SiH}_2}} \]

A graph showing the relationship between the rate of silane decomposition, -r''_SiH2, on the y-axis and the partial pressure of silane, P_SiH2, on the x-axis. The red curve rises steeply at first and then levels off, indicating a saturation behavior where the rate increases with pressure but eventually reaches a maximum limit.

 
 
For the homogeneous reaction:  
then

\[ -\tilde{r}_{\text{SiH}_4} = \frac{k_1 K_{\text{SiH}_2} \left( P_{\text{SiH}_4} \right)^{1/2}}{1 + K_1 \left( P_{\text{SiH}_4} \right)^{1/2}} \]

Derive Derive this equation
 
where:

\[ k_1 = k_S k_{\text{SiH}_4} \sqrt{K_P} \]

\[ K_1 = \sqrt{K_P K_{\text{SiH}_4}} \]

Types of Catalyst Deactivation

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Separable Kinetics: \( -\dot{r}_A(t) = a(t) \left[ -\dot{r}_A(\text{fresh catalyst}) \right] \)

Types of Decay

1.) Sintering \( -\frac{da}{dt} = k_D a^2 \)
2.) Coking \( a = \frac{1}{1 + B \cdot t^n} \)
3.) Poisoning
  		\( -\frac{da}{dt} = k_D a^n f\left[ C_i \right] \)
4.) Slow Decay Temperature-Time Trajectories
5.) Moderate Decay Moving Bed
6.) Rapid Decay STTR

Temperature-Time Trajectories

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The catalyst decay rate is a function of temperature, so you can vary the temperature with time to keep the rate of decay as constant as possible.

\( -r_A(T_0,\, t = 0) = r_A(T, t) \)
Then:

\( k_o = a(t) \, k(T) \)

\( t = \frac{1 - \exp \left[ \frac{E_A - n E_A + E_d}{R} \left( \frac{1}{T} - \frac{1}{T_0} \right) \right]} {k_{40} \left( 1 - n + \frac{E_d}{E_A} \right)} \)

or solving for \( T = f(t) \)
\( T = \frac{(E_A - n E_A + E_d) T_0} {(E_A - n E_A + E_d) - R T_0 \ln \left[ \frac{1}{1 - k_D t \left( 1 - n + \frac{E_d}{E_A} \right)} \right]} \)


Moving Bed Reactors & Straight Through Transport Reactors

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Catalyst Decay Example

The gas-phase, irreversible reaction \( \text{A} \xrightarrow{\text{cat}} \text{B} \) is elementary with first order decay. The reaction is carried out at constant temperature and pressure.

Parameters Batch Reactor Moving Bed Reactor Straight Through Transport Reactor
Mole Balance

\(\frac{dX}{dt} = \frac{-r_A W}{N_{A0}}\)

\(\frac{dX}{dW} = \frac{-r_A}{F_{A0}}\)

\(\frac{dX}{dz} = \frac{-r_A}{F_{A0}}\)
Rate Law

\(-r_A = a(t)kC_A\)

\(-r_A = a(t)kC_A\)

\(-r_A = \rho_B a(t)kC_A\)
Decay Law

\(-\frac{da}{dt} = k_D a\)

\(a = e^{-k_D t}\)

\(W = U_s t\)

\(-\frac{da}{dW} = \frac{k_D}{U_s} a\)

\(a = e^{-\frac{k_D W}{U_s}}\)

\(z = U_p t\)

\(-\frac{da}{dz} = \frac{k_D}{U_p} a\)

\(a = e^{-\frac{k_D z}{U_p}}\)
Stoichiometry

\(C_A = C_{A0}(1 - X)\)

\(C_A = C_{A0}(1 - X)\)

\(C_A = C_{A0}(1 - X)\)
Combine

\(\frac{dX}{dt} = \frac{e^{-k_D t} k W C_{A0}(1 - X)}{N_{A0}}\)

\(N_{A0} = C_{A0} v = C_{A0} \frac{W}{\rho_B}\)

\(\frac{dX}{dt} = k \rho_B e^{-k_D t}(1 - X)\)

\(\frac{dX}{1 - X} = k \rho_B e^{-k_D t} dt\)

\(\ln{\frac{1}{1 - X}} = \frac{k \rho_B}{k_D} \left[ 1 - e^{-k_D t} \right]\)

\(\frac{dX}{dW} = \frac{e^{-\frac{k_D W}{U_s}} k C_{A0}(1 - X)}{F_{A0}}\)

\(F_{A0} = C_{A0} v_0\)

\(\frac{dX}{1 - X} = \frac{k}{v_0} e^{-\frac{k_D W}{U_s}} dW\)

\(\ln{\frac{1}{1 - X}} = \frac{k U_s}{v_0 k_D} \left[ 1 - e^{-\frac{k_D W}{U_s}} \right]\)

\(\frac{dX}{dz} = \frac{\rho_B a k C_{A0}(1 - X)}{F_{A0}}\)

\(F_{A0} = C_{A0} v_0 A_C\)

\(\frac{dX}{1 - X} = \frac{\rho_B k}{v_0} e^{-\frac{k_D z}{U_p}} dz\)

\(\ln{\frac{1}{1 - X}} = \frac{\rho_B U_p k}{v_0 k_D} \left[ 1 - e^{-\frac{k_D z}{U_p}} \right]\)

Gas phase, but \(\delta = 0\), \(T = T_0\), and \(P = P_0\)


Another Catalyst Decay Example

The second-order, irreversible reaction \( \text{A} \xrightarrow{\text{cat}} \text{B} \) is carried out in a moving bed reactor. The catalyst loading rate is 1 kg/s to a reactor containg 10 kg of catalyst. The rate of decay is second order in activity and first order in concentration for the product, B, which poisons the catalyst. Plot the conversion and activity as a function of catalyst weight down the reactor.

 
Additional information:

\( C_{A0} = 0.1 \; \text{mol/dm}^3 \)   \( \nu_0 = 10 \; \text{dm}^3/\text{mol/s} \)

\( k_D = 50 \; \text{dm}^3/\text{mol/s} \)   \( k_R = 50 \; \text{dm}^6/(\text{mol} \cdot \text{kg} \cdot \text{s}) \)


 

Solution:

Polymath
Mole Balance: \( \frac{dX}{dW} = \frac{-r_A}{F_{A0}} \)
Rate Law: \( -r_A = a(t) \, k_R \, C_A^2 \)
Decay Law: \( -\frac{da}{dt} = k_D \, a^2 \, C_B \)
\( W = U_S \, t \)
\( -\frac{da}{dW} = \frac{k_D}{U_S} \, a^2 \, C_B \)
Stoichiometry: \( C_A = C_{A0} (1 - X) \)
\( C_B = C_{A0} \, X \)
Combine: \( \frac{dX}{dW} = a \left( \frac{k_R C_{A0}^2}{F_{A0}} \right) (1 - X)^2 = a \, k_{1R} (1 - X)^2 \)
\( \frac{da}{dW} = -\left( \frac{k_D C_{A0}}{U_S} \right) a^2 X = -k_{2D} a^2 X \)
\( k_{1R} = \frac{k_R C_{A0}^2}{F_{A0}} = \frac{(50)(0.1)^2}{10} = 0.05 \; \text{kg}^{-1} \)
\( k_{2D} = \frac{k_D C_{A0}}{U_S} = \frac{(50)(0.1)}{1} = 5.0 \; \text{kg}^{-1} \)



Conversion vs. Catalyst Weight
A line graph showing the variable X on the y-axis and W on the x-axis. The curve starts at the origin and increases with a decreasing slope, indicating a saturation trend. X increases rapidly at first and then more slowly as W approaches 10. The left side of the graph includes a label 'KEY:' but no legend is provided.



Catalyst Activity vs. Catalyst Weight
A line graph with variable 'a' on the y-axis and 'W' on the x-axis. The curve starts at a high value of approximately 1.0 and decreases steadily as W increases from 0 to 10, showing a downward trend. The graph suggests an inverse relationship between 'a' and 'W'. A label 'KEY:' appears on the left side but no legend is provided.


 

Example 10-7: Strictly Speaking


When there is a change in the velocity due to a change in the number of moles up through the STTR, one cannot directly substitute t = z/U in the coking activity equation:

\( a = \frac{1}{1 + A t^{1/2}} \)

 (1)
Instead, one must add another equation to the Polymath program. We know that at any location, the gas velocity up the column is:

\( \frac{dz}{dt} = U \)

 (2)
Then:

\( \frac{dt}{dz} = \frac{1}{U} \)

 (3)
where t = 0 at z = 0.

You can use either Polymath or MatLab to solve this equation and substitute it for t in the activity equation:

\( a = \frac{1}{1 + A t^{1/2}} \)

  
Along with:

\( \frac{dX}{dz} = \frac{-r_A}{U \, C_{A0}} \)

  

\( \frac{dt}{dz} = \frac{1}{U} \)

  

\( U = U_0 (1 + \varepsilon X) \)

  
etc.

(same as the program in Table E10-7.1)

 
Self Test Finding the rate law and mechanism for
Self Test Finding the rate law and mechanism for A+B<=>C+D
Self Test CO + NO; -r'CO = f(PCO , PNO , PN2 , PCO2)
Self Test Calculating Fractional Coverage
Self Test Decaying Catalyst in a Batch Reactor
Self Test Moving Bed Reactor

Example Catalyst Decay in a Packed Bed
Self Assess Object Assessment of Chapter 10

  * All chapter references are for the 4th Edition of the text Elements of Chemical Reaction Engineering .

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