Sources and References

Primary texts — Fogler, H.S., Elements of Chemical Reaction Engineering, 6th ed., Prentice Hall, 2020; Levenspiel, O., Chemical Reaction Engineering, 3rd ed., Wiley, 1999.

Supplementary texts — Hill, C.G. and Root, T.W., Introduction to Chemical Engineering Kinetics and Reactor Design, 2nd ed., Wiley, 2014; Rawlings, J.B. and Ekerdt, J.G., Chemical Reactor Analysis and Design Fundamentals, 2nd ed., Nob Hill, 2013.

Online resources — MIT OCW 10.37 “Chemical and Biological Reaction Engineering”; NPTEL reaction engineering lectures; NIST kinetics database; AIChE open-access reactor design concept videos.

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Chapter 1: Rate Laws and Stoichiometry

1.1 Reaction Rate

For a homogeneous reaction \( \sum_i \nu_i A_i = 0 \) in a well-mixed volume, the rate of reaction is

\[ r = \frac{1}{\nu_i V}\frac{dN_i}{dt}, \]

independent of species when extensive work is normalized by stoichiometric coefficient. Rates depend on composition and temperature:

\[ r = k(T) f(c_i), \quad k(T) = A e^{-E_a/RT}. \]

The Arrhenius parameters \( A \) and \( E_a \) are empirical; \( E_a \) has the physical interpretation of an energy barrier on the reaction path.

1.2 Elementary vs. Non-Elementary Reactions

An elementary reaction occurs in a single mechanistic step and has rate law matching stoichiometry: \( A + B \to P \) with \( r = k c_A c_B \). Most observed reactions are non-elementary: their rate laws (often fractional or non-integer orders) reflect a network of elementary steps with pseudo-steady-state intermediates.

Power-law forms \( r = k c_A^\alpha c_B^\beta \) are common empirical approximations. Negative orders and concentration dependence in denominators (Langmuir-Hinshelwood forms) indicate inhibition or adsorption mechanisms.

1.3 Stoichiometric Tables

A stoichiometric table tracks moles of each species as a function of conversion \( X \) of a chosen limiting reactant. For \( A + bB \to cC + dD \) with pure A and B feed ratio \( \Theta_B = N_{B,0}/N_{A,0} \):

Species	In	Change	Out
A	\( N_{A,0} \)	\( -N_{A,0}X \)	\( N_{A,0}(1-X) \)
B	\( N_{A,0}\Theta_B \)	\( -(b)N_{A,0}X \)	\( N_{A,0}(\Theta_B - bX) \)
C	0	\( (c)N_{A,0}X \)	\( N_{A,0} c X \)
D	0	\( (d)N_{A,0}X \)	\( N_{A,0} d X \)

For gas-phase reactions with variable moles, volume (constant pressure) or pressure (constant volume) changes accompany reaction through the expansion factor \( \varepsilon = y_{A,0} \delta \), \( \delta = (d + c - b - 1)/a \).

Chapter 2: Ideal Reactors

2.1 Batch Reactor

A batch reactor is closed: no inflow or outflow during reaction. Species mole balance:

\[ \frac{dN_A}{dt} = r_A V. \]

For constant-volume liquid-phase reactions, this simplifies to \( dc_A/dt = r_A \). Integration gives the time to reach a target conversion. For first-order kinetics \( r_A = -kc_A \): \( t = (1/k)\ln(1/(1-X)) \).

2.2 Continuous Stirred-Tank Reactor (CSTR)

Perfect mixing means outlet composition equals interior composition. Steady-state mole balance:

\[ V = \frac{F_{A,0} X}{-r_A(X)}. \]

For first-order kinetics with constant flow rate, \( V = F_{A,0}X/[kc_{A,0}(1-X)] \), giving residence time \( \tau = V/Q = X/[k(1-X)] \).

2.3 Plug Flow Reactor (PFR)

Plug flow assumes radially uniform, axially non-mixed flow. Differential mole balance:

\[ \frac{dX}{dV} = \frac{-r_A}{F_{A,0}}. \]

Integrating,

\[ V = F_{A,0}\int_0^X \frac{dX'}{-r_A(X')}. \]

For first-order kinetics, the PFR volume to reach the same conversion is always less than (or equal to) the CSTR volume—a central result: CSTRs trade volume for the luxury of operating uniformly at one (low-concentration) state.

2.4 Semi-Batch

Semi-batch (one or more feeds added over time, no product withdrawal) is common for exothermic reactions where accumulation of reactant would produce a thermal runaway, and for reactions requiring concentration control of a reactive intermediate. Mass balances are time-dependent because volume grows.

Chapter 3: Reactor Networks and Multiple Reactions

3.1 CSTRs in Series

\( N \) equal-size CSTRs in series approach PFR performance as \( N \to \infty \). For first-order reaction and equal residence times:

\[ 1 - X = \frac{1}{(1 + k\tau/N)^N} \to e^{-k\tau} \text{ as } N\to\infty. \]

The series arrangement is used when heat removal or staged feed addition is required.

3.2 PFR with Recycle

Recycle ratio \( R = \dot Q_{recycle}/\dot Q_{fresh} \) blends fresh feed with a recycled portion of the effluent. At \( R \to 0 \) the unit behaves as a PFR; at \( R \to \infty \) as a CSTR. Recycle is used for adiabatic temperature control and for increasing conversion per pass.

3.3 Multiple Reactions: Series and Parallel

For parallel reactions \( A \to D \) (desired) and \( A \to U \) (undesired):

\[ S_{D/U} = \frac{r_D}{r_U} = \frac{k_D}{k_U} c_A^{\alpha_D - \alpha_U}. \]

If \( \alpha_D > \alpha_U \), high \( c_A \) (PFR, batch) favors desired product. If \( \alpha_D < \alpha_U \), low \( c_A \) (CSTR) is preferred.

For series reactions \( A \to B \to C \) (B is desired), an intermediate conversion maximizes \( B \); quenching at that point is the engineering response.

3.4 Yield and Selectivity

Instantaneous selectivity \( S_{D/U}^\text{inst} = r_D/r_U \); overall selectivity integrates over the reactor. Yield is product formed per reactant consumed. Design decisions—feed strategy (co-feed vs. sequential), reactor type, temperature profile, catalyst selection—target selectivity as much as or more than conversion.

Chapter 4: Non-Isothermal Reactors

4.1 Energy Balance Coupled to Reaction

Steady-state CSTR energy balance:

\[ \sum_i F_{i,0} C_{p,i}(T - T_0) + \xi \Delta H_{rxn}(T) = \dot Q. \]

For a PFR:

\[ \frac{dT}{dV} = \frac{Ua(T_a - T) + r_A \Delta H_{rxn}}{\sum_i F_i C_{p,i}}. \]

4.2 Adiabatic Reactors

\( \dot Q = 0 \) gives temperature as a linear function of conversion:

\[ T = T_0 + \frac{(-\Delta H_{rxn})(F_{A,0}/\sum F_i C_{p,i})X}{1 + \varepsilon X} \]

(simplified forms apply under common assumptions). Adiabatic PFRs are used widely in industrial practice (ammonia synthesis, methanol synthesis) with inter-stage cooling to manage equilibrium limits.

4.3 Multiple Steady States

Non-isothermal CSTRs can exhibit multiple steady states: the S-shaped heat generation curve \( G(T) = (-\Delta H_{rxn}) V r(c_A(T), T) \) intersects the linear heat removal curve \( R(T) = (\sum F_i C_{p,i})(T - T_0) + UA(T - T_a) \) at one, two, or three temperatures. Stability analysis: steady states with slope of \( G \) greater than slope of \( R \) are unstable. Ignition and extinction phenomena arise when operating conditions push the system past a fold bifurcation.

4.4 Optimal Temperature Profiles

For reversible reactions, equilibrium conversion decreases with temperature for exothermic reactions. The optimal PFR temperature profile starts high (fast rate at low conversion) and decreases as conversion rises (approaching equilibrium). Industrial reactors approximate this with multistage adiabatic beds and interstage cooling.

Chapter 5: Heterogeneous Catalysis

5.1 Catalytic Rate Laws

A catalyst provides an alternative pathway with lower \( E_a \), increasing rate without being consumed. Heterogeneous (solid) catalysis involves adsorption, surface reaction, and desorption:

\[ A + S \rightleftharpoons A\cdot S, \quad A\cdot S \to P\cdot S, \quad P\cdot S \rightleftharpoons P + S. \]

Langmuir-Hinshelwood kinetics for a single rate-determining step yield expressions like

\[ r = \frac{k K_A c_A}{1 + K_A c_A + K_B c_B}, \]

with adsorption constants \( K_i \) governing competition for sites.

5.2 External Mass Transfer

Reactant must diffuse from bulk fluid through a boundary layer to the catalyst surface:

\[ r_A = k_c a (c_{A,b} - c_{A,s}). \]

At steady state \( r_A \) equals the surface reaction rate, giving an observed rate that combines reaction and mass transfer resistances in series.

5.3 Internal Diffusion: The Effectiveness Factor

Porous catalyst pellets have reaction distributed throughout; reactant is consumed as it diffuses inward. For a slab of thickness \( L \) with first-order reaction,

\[ \eta = \frac{\tanh\phi}{\phi}, \quad \phi = L\sqrt{k/D_e}, \]

where \( \phi \) is the Thiele modulus and \( \eta \) is the effectiveness factor (observed rate / surface-rate ignoring diffusion). Small \( \phi \): reaction-limited, \( \eta \to 1 \). Large \( \phi \): diffusion-limited, \( \eta \to 1/\phi \). Similar results with geometric factors apply to cylinders and spheres.

5.4 Catalyst Deactivation

Mechanisms: sintering (thermal), poisoning (chemical), coking (carbonaceous deposits), attrition. Deactivation models couple activity \( a(t) \) to rate: \( -r = a(t) r_0 \). Regeneration (combustion of coke, reduction of poisoned sites) is part of reactor/catalyst design. Moving-bed and fluidized catalytic cracking (FCC) units circulate catalyst between reactor and regenerator continuously.