Cheatsheet: Reactor Design

1. Reactor Fundamentals

1.1 Reactor Types

1.2 Key Definitions

2. Reaction Kinetics

2.1 Rate Laws

2.2 Arrhenius Equation

• k = Ae^-E_a/RT
• A = pre-exponential factor (frequency factor)
• E_a = activation energy (J/mol or cal/mol)
• R = 8.314 J/(mol·K) or 1.987 cal/(mol·K)
• T = absolute temperature (K)
• ln(k₂/k₁) = (E_a/R)(1/T₁ - 1/T₂)

2.3 Reaction Order Determination

3. Batch Reactor Design

3.1 Design Equations

3.2 Integrated Forms for Constant Volume

4. CSTR Design

4.1 Design Equations

4.2 Multiple CSTRs in Series

• For n equal-volume CSTRs: V_total = nV_i
• V_i = F_Ai-1(X_i - X_i-1)/(-r_Ai)
• Overall conversion X_n calculated stepwise through each reactor
• First order, n equal CSTRs: X_n = 1 - 1/(1 + kτ_i)ⁿ
• Smaller individual volumes more efficient than single large CSTR for same total volume

4.3 CSTR Performance

5. PFR Design

5.1 Design Equations

5.2 Integrated Forms for Constant Density

5.3 PFR vs CSTR Comparison

• For same conversion and reaction rate: V_PFR <>_CSTR
• PFR more efficient for positive order reactions
• CSTR advantageous for negative order reactions (rare)
• CSTR better for temperature control (isothermal operation easier)
• PFR: continuous composition gradient along length

6. Packed Bed Reactor

6.1 Design Equations

6.2 Key Parameters

• W = catalyst weight (kg)
• r'_A = reaction rate per unit catalyst mass (mol/kg·s)
• φ = porosity (void fraction)
• D_p = particle diameter (m)
• G = superficial mass velocity (kg/m²·s)
• ρ₀ = inlet gas density (kg/m³)
• μ = viscosity (Pa·s)
• ε = volume expansion factor = (δ)y_A0
• δ = (total moles products - total moles reactants)/moles limiting reactant
• y = P/P₀

6.3 Pressure Drop Considerations

• For gas phase reactions, pressure drop significantly affects conversion
• Lower pressure reduces concentration, decreasing reaction rate
• Larger particles reduce pressure drop but decrease surface area
• α = 2β₀/F_A0 (pressure drop parameter)

7. Non-Isothermal Reactor Design

7.1 Energy Balance

7.2 Heat of Reaction

• ΔH_rx(T) = ΔH_rx(T_ref) + ∫_Tref^T ΔC_pdT
• ΔC_p = Σ(products)C_pi - Σ(reactants)C_pi
• For constant ΔC_p: ΔH_rx(T) = ΔH_rx(T_ref) + ΔC_p(T - T_ref)
• Exothermic: ΔH_rx < 0;="" endothermic:="">_rx > 0

7.3 Adiabatic Operation

• Q = 0, no heat exchange with surroundings
• For exothermic reactions: temperature increases along reactor
• For endothermic reactions: temperature decreases along reactor
• Energy balance: X = (Σ_iF_iC_pi(T-T₀))/(F_A0(-ΔH_rx))
• T = T₀ + ((-ΔH_rx)X)/(Σθ_iC_pi) where θ_i = F_i0/F_A0

7.4 Heat Transfer Parameters

• U = overall heat transfer coefficient (W/m²·K or J/s·m²·K)
• a = heat transfer area per unit volume (m²/m³)
• T_a = ambient/coolant temperature (K)
• Ua = heat transfer term (W/m³·K or J/s·m³·K)

8. Multiple Reactions

8.1 Reaction Types

8.2 Selectivity and Yield

8.3 Maximizing Desired Product

8.3.1 Parallel Reactions: A → D (desired, rate = k₁C_A^α₁), A → U (undesired, rate = k₂C_A^α₂)

• S_D/U = (k₁/k₂)C_A^{(α₁-α₂)}
• If α₁ > α₂: high C_A favors desired product (use PFR or batch, high concentration)
• If α₁ <>₂: low C_A favors desired product (use CSTR, low concentration)
• If E₁ > E₂: high temperature favors desired product
• If E₁ <>₂: low temperature favors desired product

8.3.2 Series Reactions: A → D → U

• Intermediate D is desired product
• Optimal conversion exists; avoid complete conversion
• PFR better than CSTR for maximizing intermediate
• For first-order series: C_D,max at t = ln(k₁/k₂)/(k₁-k₂)
• High space velocity (low residence time) if k₂ >> k₁

9. Residence Time Distribution (RTD)

9.1 RTD Functions

9.2 Mean Residence Time and Variance

• Mean: t̄ = ∫₀^∞ tE(t)dt = τ
• Variance: σ² = ∫₀^∞ (t-t̄)²E(t)dt = ∫₀^∞ t²E(t)dt - t̄²
• Normalized variance: σ_θ² = σ²/t̄²

9.3 Ideal Reactor RTD

9.4 Tracer Tests

• Pulse Input: inject tracer instantaneously, measure C(t) vs t, E(t) = C(t)/∫₀^∞ C(t)dt
• Step Input: continuous tracer injection, measure F(t) = C(t)/C₀, E(t) = dF(t)/dt
• Ideal tracers: non-reactive, same density as fluid, easily detectable

10. Non-Ideal Reactors

10.1 Dispersion Model

• Accounts for axial mixing in tubular reactors
• Dispersion coefficient D characterizes mixing intensity
• Peclet number: Pe = uL/D = L/D·τ (u = velocity, L = length)
• Low Pe (high dispersion): approaches CSTR behavior
• High Pe (low dispersion): approaches PFR behavior
• Open-open vessel: σ_θ² = 2/Pe - 2/Pe²(1 - e^-Pe)
• For large Pe: σ_θ² ≈ 2/Pe

10.2 Tanks-in-Series Model

• Models real reactor as n equal-volume CSTRs in series
• E(t) = (t^n-1/(τ/n)ⁿ(n-1)!)e^-nt/τ
• σ_θ² = 1/n
• n = t̄²/σ² = 1/σ_θ²
• n = 1: CSTR; n → ∞: PFR

10.3 Segregated Flow Model

• Assumes no mixing between fluid elements (micro-mixing absent)
• Each element acts as batch reactor with residence time from RTD
• X_seg = ∫₀^∞ X_batch(t)E(t)dt
• Conservative prediction for conversion

10.4 Maximum Mixedness Model

• Assumes complete micro-mixing (opposite of segregated flow)
• Fluid elements exchange mass continuously
• Provides upper bound on conversion for given RTD
• λ = ∫_t^∞ E(t')dt' = 1 - F(t)

11. Catalytic Reactions

11.1 Adsorption Isotherms

11.2 Langmuir-Hinshelwood-Hougen-Watson (LHHW) Rate Laws

• General form: -r_A' = (kinetic term × driving force)/(adsorption term)
• Surface reaction controlling: -r_A' = kK_AP_A/(1 + K_AP_A + K_BP_B)²
• Adsorption controlling: -r_A' = k_AP_A(1 - θ_A)
• Desorption controlling: -r_A' = k_Dθ_products

11.3 Catalyst Properties

11.4 Catalyst Deactivation

• Activity: a(t) = -r_A'(t)/(-r_A'(t=0))
• Deactivation rate: r_d = -da/dt = k_da^dC_Aⁿ
• Time-on-stream: integrate deactivation equation with mole balance

12. Diffusion Effects in Catalysis

12.1 Mass Transfer Resistances

12.2 Thiele Modulus and Effectiveness Factor

• Thiele Modulus: φ_n = (V_p/S_p)[(n+1)k'ρ_cC_As^n-1/(2D_e)]^0.5
• For sphere (radius R): φ = R[(k'ρ_c/D_e)]^0.5 for first-order
• For slab (half-thickness L): φ = L[(k'ρ_c/D_e)]^0.5 for first-order
• Effectiveness Factor: η = (actual rate)/(rate if no diffusion resistance)
• First-order sphere: η = (3/φ)(coth(φ) - 1/φ) = 3(1/φ - 1/tanh(φ))/φ
• For φ < 0.4:="" η="" ≈="" 1="" (no="" diffusion="">
• For φ > 4: η ≈ 3/φ (strong diffusion limitation)
• Observable rate: -r_A' = η·k'ρ_cC_As

12.3 Effective Diffusivity

• D_e = (φ_p/τ_t)D_AB for molecular diffusion
• D_e = (φ_p/τ_t)D_K for Knudsen diffusion
• 1/D_e = (τ_t/φ_p)(1/D_AB + 1/D_K) for combined
• Knudsen diffusivity: D_K = 97r_pore(T/M)^0.5 (r_pore in m, D_K in m²/s)
• φ_p = pellet porosity; τ_t = tortuosity

12.4 External Mass Transfer

• Rate: N_A = k_ca_c(C_Ab - C_As)
• k_c = mass transfer coefficient (m/s)
• a_c = external surface area per unit volume (m²/m³)
• C_Ab = bulk concentration; C_As = surface concentration
• Sherwood number: Sh = k_cd_p/D_AB
• For packed beds: Sh = 2 + 1.1Sc^1/3Re^0.6

12.5 Weisz-Prater Criterion

• C_WP = (-r_A'_obs)ρ_cR²/(D_eC_As)
• If C_WP < 1:="" no="" internal="" diffusion="">
• If C_WP >> 1: strong internal diffusion limitation
• Requires only observable rate (no intrinsic kinetics needed)

13. Gas-Solid Reactions

13.1 Non-Catalytic Models

13.2 Shrinking Core Model Regimes

• R = initial particle radius
• ρ_B = molar density of solid B (mol/m³)
• b = stoichiometric coefficient
• C_Ag = bulk gas concentration
• k_s = surface reaction rate constant

13.3 Conversion-Time Relationships (SCM)

14. Reactor Stability and Dynamics

14.1 Multiple Steady States

• Occur in non-isothermal CSTRs with exothermic reactions
• Heat generation curve: Q_gen = (-ΔH_rx)F_A0X(T)
• Heat removal line: Q_rem = ṁC_p(T - T₀) + UA(T - T_a)
• Intersections of curves = steady states
• Three possible steady states: lower (stable), middle (unstable), upper (stable)

14.2 Stability Criteria

• Stable steady state: slope(Q_rem) > slope(Q_gen)
• Unstable steady state: slope(Q_rem) <>_gen)
• Runaway potential high when: large (-ΔH_rx), high E_a, poor heat removal

14.3 Damköhler Number

• Da = (reaction rate)/(convective transport rate) = kτ
• For first-order: Da = kV/v₀
• High Da: reaction-controlled regime
• Low Da: mass transfer-controlled regime

14.4 Runaway Criteria

• Dimensionless adiabatic temperature rise: β = (-ΔH_rx)C_A0/(ρC_pT₀)
• Dimensionless activation energy: γ = E/(RT₀)
• High β and γ increase runaway risk
• Critical for safety in industrial reactors

15. Scale-Up Considerations

15.1 Dimensionless Groups

15.2 Scale-Up Principles

• Geometric similarity: maintain length ratios
• Dynamic similarity: match dimensionless groups (Re, Da, etc.)
• Thermal similarity: maintain heat transfer characteristics
• Mixing time: t_mix ∝ V^1/3 for stirred tanks
• Surface-to-volume ratio decreases with scale (S/V ∝ 1/L)
• Heat transfer more difficult in larger reactors
• Perfect matching of all dimensionless groups often impossible

15.3 Common Scale-Up Challenges

• Heat removal capacity (exothermic reactions)
• Mass transfer limitations (gas-liquid, solid-catalyzed)
• Mixing non-idealities (bypassing, dead zones)
• Pressure drop increase (packed beds)
• Temperature gradients (hot spots)

16. Biochemical Reactors

16.1 Microbial Growth Kinetics

• μ = specific growth rate (h^-1)
• μ_max = maximum specific growth rate
• C_S = substrate concentration
• K_S = Monod constant (substrate conc. at μ = μ_max/2)
• C_X = cell concentration
• Y_X/S = cell yield coefficient (g cells/g substrate)
• m_S = maintenance coefficient
• Y_P/X = product yield coefficient
• β = non-growth associated product formation constant

16.2 Enzyme Kinetics

• Michaelis-Menten: v = v_maxC_S/(K_M + C_S)
• v_max = maximum reaction velocity
• K_M = Michaelis constant (substrate conc. at v = v_max/2)
• Lineweaver-Burk plot: 1/v = (K_M/v_max)(1/C_S) + 1/v_max
• Competitive inhibition: v = v_maxC_S/(K_M(1 + C_I/K_I) + C_S)
• Non-competitive inhibition: v = v_maxC_S/((K_M + C_S)(1 + C_I/K_I))

16.3 Continuous Culture (Chemostat)

• Dilution rate: D = F/V (h^-1)
• At steady state: μ = D
• Washout occurs when D > μ_max
• Critical dilution rate: D_crit = μ_maxC_S0/(K_S + C_S0)
• Optimal dilution rate for maximum productivity: D_opt = μ_max(1 - (K_S/(K_S + C_S0))^0.5)

17. Design Heuristics and Practical Guidelines

17.1 Reactor Selection

17.2 Operating Conditions

• Temperature: balance between rate (higher T) and selectivity/equilibrium (depends on reaction)
• Pressure: increase for gas phase reactions with volume decrease (Le Chatelier)
• Catalyst loading: typically 50-70% of reactor volume for packed beds
• Recycle ratio: 3:1 to 10:1 for autothermal operation or conversion control
• Safety factor: typically 1.1-1.5 on reactor volume for commercial design

17.3 Performance Metrics

• Conversion: X = (F_A0 - F_A)/F_A0
• Selectivity: S = F_D/(F_A0 - F_A)
• Yield: Y = F_D/F_A0 = S·X
• Productivity: moles product/(reactor volume × time)
• Space-time yield: kg product/(m³·h)

17.4 Safety Considerations

• Exothermic runaway: monitor β and γ parameters
• Emergency cooling: design for loss of coolant scenarios
• Pressure relief: size for worst-case scenario (typically 10% above MAWP)
• Temperature monitoring: multiple thermocouples/thermowells in critical zones
• Material compatibility: check for corrosion, erosion at operating conditions