Formula Sheet: Reactor Design

Fundamental Reactor Definitions and Parameters

Conversion

Conversion (X): Fraction of reactant converted in a reaction \[X = \frac{N_{A0} - N_A}{N_{A0}} = \frac{F_{A0} - F_A}{F_{A0}}\]

• X = conversion (dimensionless, 0 to 1)
• N_A0 = initial moles of reactant A
• N_A = moles of reactant A at time t or position
• F_A0 = inlet molar flow rate of A (mol/time)
• F_A = outlet molar flow rate of A (mol/time)

Reactor Space Time and Space Velocity

Space Time (τ): Time necessary to process one reactor volume of feed \[\tau = \frac{V}{\nu_0}\]

• τ = space time (time)
• V = reactor volume (volume)
• ν₀ = volumetric flow rate at inlet conditions (volume/time)

Space Velocity (SV): Reciprocal of space time \[SV = \frac{1}{\tau} = \frac{\nu_0}{V}\]

• SV = space velocity (time^-1)
• Units: hr^-1 or s^-1

Residence Time

Mean Residence Time (t̄): Average time a molecule spends in the reactor \[\bar{t} = \frac{V}{\nu}\]

• t̄ = mean residence time (time)
• ν = volumetric flow rate at reactor conditions (volume/time)
• For constant density systems: t̄ = τ
• For variable density systems: t̄ ≠ τ

Rate Laws and Kinetics

General Rate Expression

Rate of Reaction: \[-r_A = kC_A^n\]

• -r_A = rate of disappearance of A (mol/volume·time)
• k = reaction rate constant (units depend on reaction order)
• C_A = concentration of A (mol/volume)
• n = reaction order (dimensionless)

Arrhenius Equation

Temperature Dependence of Rate Constant: \[k = A e^{-E_a/(RT)}\]

• A = pre-exponential or frequency factor (same units as k)
• E_a = activation energy (energy/mol)
• R = universal gas constant = 8.314 J/(mol·K) or 1.987 cal/(mol·K)
• T = absolute temperature (K or °R)

Arrhenius Equation - Two Temperature Form: \[\ln\left(\frac{k_2}{k_1}\right) = \frac{E_a}{R}\left(\frac{1}{T_1} - \frac{1}{T_2}\right)\]

Elementary Reaction Rate Laws

First Order: \[-r_A = kC_A\]

• Units of k: time^-1

Second Order: \[-r_A = kC_A^2\]

• Units of k: volume/(mol·time)

Second Order - Two Reactants: \[-r_A = kC_AC_B\] Zero Order: \[-r_A = k\]

• Units of k: mol/(volume·time)

Stoichiometry and Concentration Relationships

General Reaction Stoichiometry

General Reaction: \[aA + bB \rightarrow cC + dD\] Stoichiometric Coefficients: \[\nu_i = \begin{cases} -a & \text{for A} \\ -b & \text{for B} \\ +c & \text{for C} \\ +d & \text{for D} \end{cases}\]

• Negative for reactants, positive for products

Batch and Constant Volume Systems

Concentration as Function of Conversion: \[C_A = C_{A0}(1 - X)\]

• C_A0 = initial concentration of A (mol/volume)

For Species B (not initially limiting): \[C_B = C_{A0}\left(\Theta_B - \frac{b}{a}X\right)\]

• Θ_B = C_B0/C_A0 = initial molar ratio
• b/a = stoichiometric ratio

For Product C: \[C_C = C_{A0}\left(\Theta_C + \frac{c}{a}X\right)\]

Flow Systems with Variable Density

Change in Total Molar Flow Rate: \[F_T = F_{T0}(1 + \varepsilon_A X)\]

• F_T = total molar flow rate (mol/time)
• F_T0 = inlet total molar flow rate (mol/time)
• ε_A = fractional change in volume upon complete conversion

Fractional Volume Change (ε_A): \[\varepsilon_A = \frac{y_{A0}\delta}{1 + y_{A0}\delta}\] where for gas phase reactions: \[\delta = \frac{\sum \nu_i}{-\nu_A} = \frac{(c + d) - (a + b)}{a}\]

• y_A0 = inlet mole fraction of A
• δ = change in total moles per mole of A reacted

Concentration in Flow Systems: \[C_A = C_{A0}\frac{(1 - X)}{(1 + \varepsilon_A X)}\] Volumetric Flow Rate: \[\nu = \nu_0(1 + \varepsilon_A X)\]

Gas Phase Concentration

Ideal Gas Law for Concentration: \[C_A = \frac{P_A}{RT} = \frac{y_A P}{RT}\]

• P_A = partial pressure of A
• y_A = mole fraction of A
• P = total pressure

Batch Reactor (BR) Design

General Design Equation

Mole Balance for Batch Reactor: \[\frac{dN_A}{dt} = r_A V\] Design Equation in Terms of Conversion: \[N_{A0}\frac{dX}{dt} = -r_A V\] Integrated Form for Constant Volume: \[t = N_{A0}\int_0^X \frac{dX}{-r_A V} = C_{A0}\int_0^X \frac{dX}{-r_A}\]

Batch Reactor - Specific Reaction Orders

Zero Order (Constant Volume): \[t = \frac{C_{A0}X}{k}\] First Order (Constant Volume): \[t = \frac{1}{k}\ln\left(\frac{1}{1-X}\right) = \frac{1}{k}\ln\left(\frac{C_{A0}}{C_A}\right)\] Second Order (Constant Volume): \[t = \frac{1}{kC_{A0}}\left(\frac{X}{1-X}\right) = \frac{1}{k}\left(\frac{1}{C_A} - \frac{1}{C_{A0}}\right)\] Second Order - Two Reactants (C_A0 ≠ C_B0): \[t = \frac{1}{k(C_{B0} - C_{A0})}\ln\left(\frac{C_{B0}(1-X)}{C_{B0} - C_{A0}X}\right)\] nth Order (n ≠ 1, Constant Volume): \[t = \frac{1}{k(n-1)C_{A0}^{n-1}}\left[\frac{1}{(1-X)^{n-1}} - 1\right]\]

Continuous Stirred Tank Reactor (CSTR) Design

General Design Equation

Mole Balance for CSTR: \[F_{A0} - F_A + r_A V = 0\] Design Equation in Terms of Conversion: \[V = \frac{F_{A0}X}{-r_A}\]

• Valid for steady-state operation
• Assumes perfect mixing (uniform composition throughout)
• Reactor operates at exit conditions

Space Time for CSTR: \[\tau = \frac{V}{\nu_0} = \frac{C_{A0}X}{-r_A}\]

CSTR - Specific Reaction Orders

Zero Order: \[\tau = \frac{C_{A0}X}{k}\] First Order: \[\tau = \frac{C_{A0}X}{kC_{A0}(1-X)} = \frac{X}{k(1-X)}\] or \[V = \frac{F_{A0}X}{kC_{A0}(1-X)}\] Second Order: \[\tau = \frac{X}{kC_{A0}(1-X)^2}\] or \[V = \frac{F_{A0}X}{kC_{A0}^2(1-X)^2}\]

Multiple CSTRs in Series

For n Equal-Volume CSTRs in Series: First Order Reaction: \[(1-X_n) = \frac{1}{(1 + k\tau_i)^n}\]

• X_n = overall conversion after n reactors
• τ_i = space time of each individual reactor

Conversion in i^th Reactor: \[X_i = \frac{X_i - X_{i-1}}{1 - X_{i-1}}\]

Plug Flow Reactor (PFR) Design

General Design Equation

Differential Mole Balance: \[\frac{dF_A}{dV} = r_A\] Design Equation in Terms of Conversion: \[V = F_{A0}\int_0^X \frac{dX}{-r_A}\] For Constant Density Systems: \[V = \nu_0 C_{A0}\int_0^X \frac{dX}{-r_A}\] Space Time for PFR: \[\tau = \frac{V}{\nu_0} = C_{A0}\int_0^X \frac{dX}{-r_A}\]

PFR - Specific Reaction Orders (Constant Density)

Zero Order: \[\tau = \frac{C_{A0}X}{k}\] First Order: \[\tau = \frac{1}{k}\ln\left(\frac{1}{1-X}\right)\] or \[V = \frac{F_{A0}}{k}\ln\left(\frac{1}{1-X}\right)\] Second Order: \[\tau = \frac{1}{kC_{A0}}\left(\frac{X}{1-X}\right)\] or \[V = \frac{F_{A0}}{kC_{A0}}\left(\frac{X}{1-X}\right)\] nth Order (n ≠ 1): \[\tau = \frac{1}{k(n-1)C_{A0}^{n-1}}\left[\frac{1}{(1-X)^{n-1}} - 1\right]\]

PFR with Pressure Drop

Ergun Equation for Packed Beds: \[\frac{dP}{dz} = -\frac{G}{\rho g_c D_p}\left[\frac{150(1-\phi)\mu}{D_p \phi} + 1.75G\right]\]

• P = pressure
• z = axial distance along reactor
• G = superficial mass velocity
• ρ = fluid density
• g_c = gravitational constant
• D_p = particle diameter
• φ = bed porosity (void fraction)
• μ = fluid viscosity

Simplified Pressure Drop (Dimensionless): \[\frac{dP}{dW} = -\frac{\alpha}{2P}\left(\frac{F_T}{F_{T0}}\right)\left(\frac{T}{T_0}\right)\]

• W = catalyst weight
• α = pressure drop parameter

Dimensionless Pressure: \[y = \frac{P}{P_0}\] \[\frac{dy}{dW} = -\frac{\alpha}{2P_0}(1 + \varepsilon_A X)\left(\frac{T}{T_0}\right)\]

Reactor Comparisons

Volume Comparison for Same Conversion

For First Order Reactions: \[V_{CSTR} > V_{PFR}\] Volume Ratio (First Order): \[\frac{V_{CSTR}}{V_{PFR}} = \frac{X/(1-X)}{\ln[1/(1-X)]}\] General Rule for Positive Order Reactions:

• For the same conversion and feed rate: V_CSTR > V_PFR
• PFR is more efficient for positive order reactions
• CSTR is advantageous for highly exothermic reactions (easier temperature control)

Optimal Reactor Configuration

For Maximum Conversion (Series Operation):

• Multiple small CSTRs approach PFR performance
• As n → ∞, CSTR series approaches PFR

Recycle Reactor

Recycle Ratio

Recycle Ratio (R): \[R = \frac{\text{volumetric recycle flow rate}}{\text{volumetric feed flow rate}} = \frac{\nu_R}{\nu_0}\] Overall Conversion: \[X_{overall} = \frac{X}{1 + R(1-X)}\]

• X = conversion per pass through reactor
• X_overall = net conversion based on fresh feed

Effect of Recycle:

• R = 0: PFR with no recycle
• R → ∞: approaches CSTR behavior

Non-Ideal Reactors and RTD

Residence Time Distribution (RTD)

Exit Age Distribution Function E(t): \[E(t) = \frac{C(t)}{\int_0^\infty C(t)dt}\]

• E(t) = exit age distribution (time^-1)
• C(t) = tracer concentration at outlet as function of time
• Normalization: ∫₀^∞ E(t)dt = 1

Mean Residence Time from RTD: \[\bar{t} = \int_0^\infty t E(t)dt\] Variance: \[\sigma^2 = \int_0^\infty (t - \bar{t})^2 E(t)dt = \int_0^\infty t^2 E(t)dt - \bar{t}^2\]

RTD for Ideal Reactors

PFR (Plug Flow): \[E(t) = \delta(t - \tau)\]

• Dirac delta function at t = τ
• All molecules have identical residence time

CSTR (Perfect Mixing): \[E(t) = \frac{1}{\tau}e^{-t/\tau}\]

• Exponential distribution
• Mean residence time = τ

Segregated Flow Model

Conversion for Segregated Flow: \[X = \int_0^\infty X(t)E(t)dt\]

• X(t) = conversion achieved in a batch reactor after time t
• Applicable when there is no mixing on molecular level

Tanks-in-Series Model

Exit Age Distribution for n Tanks in Series: \[E(t) = \frac{t^{n-1}}{(n-1)!\tau_i^n}e^{-t/\tau_i}\]

• n = number of tanks
• τ_i = mean residence time in each tank

Variance: \[\sigma^2 = \frac{\bar{t}^2}{n}\] Number of Tanks from Variance: \[n = \frac{\bar{t}^2}{\sigma^2}\]

Dispersion Model

Peclet Number: \[Pe = \frac{uL}{D} = \frac{\nu_0 L}{DA}\]

• u = linear velocity
• L = reactor length
• D = axial dispersion coefficient
• A = cross-sectional area
• Pe → ∞: plug flow
• Pe → 0: perfect mixing

Dispersion Number: \[D_a = \frac{D}{uL} = \frac{1}{Pe}\]

Multiple Reactions and Selectivity

Reaction Types

Series Reactions: \[A \rightarrow B \rightarrow C\] Parallel Reactions: \[A \rightarrow B\] \[A \rightarrow C\] Complex/Network Reactions: Combination of series and parallel

Selectivity and Yield

Instantaneous Selectivity: \[S_{B/C} = \frac{r_B}{r_C}\]

• Ratio of formation rates of desired to undesired product

Overall Selectivity: \[S_{B/C} = \frac{\text{moles of B formed}}{\text{moles of C formed}}\] Yield (φ): \[\phi_B = \frac{\text{moles of B formed}}{\text{moles of A reacted}}\] Fractional Yield: \[\phi_B = \frac{F_B - F_{B0}}{F_{A0} - F_A}\]

Parallel Reactions

For Competitive-Parallel Reactions: \[A \xrightarrow{k_1} B \quad (desired)\] \[A \xrightarrow{k_2} C \quad (undesired)\] If: \[-r_{1A} = k_1C_A^{\alpha_1}\] \[-r_{2A} = k_2C_A^{\alpha_2}\] Instantaneous Selectivity: \[S_{B/C} = \frac{k_1}{k_2}C_A^{(\alpha_1 - \alpha_2)}\] To Maximize Selectivity:

• If α₁ > α₂: use high C_A (batch or PFR preferred)
• If α₁ <>₂: use low C_A (CSTR preferred)
• If α₁ = α₂: selectivity independent of C_A

Series Reactions

For Series Reactions: \[A \xrightarrow{k_1} B \xrightarrow{k_2} C\] Concentration Profiles (Batch, First Order): \[C_A = C_{A0}e^{-k_1t}\] \[C_B = \frac{k_1C_{A0}}{k_2 - k_1}(e^{-k_1t} - e^{-k_2t})\] \[C_C = C_{A0}\left[1 - \frac{k_2e^{-k_1t} - k_1e^{-k_2t}}{k_2 - k_1}\right]\] Time for Maximum B: \[t_{max} = \frac{\ln(k_2/k_1)}{k_2 - k_1}\] To Maximize C_B:

• Use batch or PFR
• Operate at optimum time or conversion
• Remove B when maximum is reached

Energy Balance and Non-Isothermal Reactors

General Energy Balance

Energy Balance for Flow Reactor: \[\sum F_{i0}H_{i0} - \sum F_i H_i + Q - W_s = 0\]

• F_i0 = molar flow rate of species i entering
• F_i = molar flow rate of species i leaving
• H_i = enthalpy of species i
• Q = heat added to system
• W_s = shaft work

Simplified Form (No Shaft Work): \[Q = \sum F_i H_i - \sum F_{i0}H_{i0}\]

Heat of Reaction

Standard Heat of Reaction: \[\Delta H_{rxn}^0(T) = \sum \nu_i \Delta H_{f,i}^0(T)\]

• ΔH_rxn⁰ = standard heat of reaction
• ν_i = stoichiometric coefficient (negative for reactants)
• ΔH_f,i⁰ = standard heat of formation of species i

Temperature Dependence: \[\Delta H_{rxn}(T) = \Delta H_{rxn}(T_{ref}) + \int_{T_{ref}}^T \Delta C_p dT\] where: \[\Delta C_p = \sum \nu_i C_{p,i}\]

Adiabatic Operation

Adiabatic Energy Balance (Q = 0): \[\sum F_i H_i = \sum F_{i0}H_{i0}\] Adiabatic Temperature Rise (Exothermic): \[\Delta T_{ad} = \frac{-\Delta H_{rxn}(T_0) X C_{A0}}{\sum F_i C_{p,i}}\]

• ΔT_ad = adiabatic temperature rise
• Valid for small temperature changes where C_p is constant

Adiabatic Temperature Profile: \[T = T_0 + \frac{-\Delta H_{rxn}(T_0)}{\sum \Theta_i C_{p,i}}X\]

• T₀ = inlet temperature
• Θ_i = F_i0/F_A0

Non-Isothermal CSTR

Design Equation: \[V = \frac{F_{A0}X}{-r_A(X,T)}\] Energy Balance (Steady State): \[Q = F_{A0}\left[\sum \Theta_i C_{p,i}(T - T_0) + X\Delta H_{rxn}(T)\right]\] For Heat Exchange: \[Q = UA(T_a - T)\]

• U = overall heat transfer coefficient
• A = heat transfer area
• T_a = heat exchange medium temperature

Non-Isothermal PFR

Design Equation: \[\frac{dX}{dV} = \frac{-r_A(X,T)}{F_{A0}}\] Energy Balance: \[\frac{dT}{dV} = \frac{Ua(T_a - T) + (-r_A)(-\Delta H_{rxn})}{F_{A0}\sum \Theta_i C_{p,i}}\]

• a = heat transfer area per unit volume
• Coupled ODEs must be solved simultaneously

For Adiabatic PFR (Q = 0): \[\frac{dT}{dX} = \frac{-\Delta H_{rxn}(T)}{\sum \Theta_i C_{p,i}}\]

Non-Isothermal Batch Reactor

Design Equation: \[\frac{dX}{dt} = \frac{-r_A(X,T)V}{N_{A0}}\] Energy Balance: \[\frac{dT}{dt} = \frac{UA(T_a - T) + (-r_A V)(-\Delta H_{rxn})}{N_{A0}\sum \Theta_i C_{p,i}}\]

Catalytic Reactions and Heterogeneous Reactors

Catalyst Weight-Based Design

PFR Design with Catalyst Weight: \[W = F_{A0}\int_0^X \frac{dX}{-r_A'}\]

• W = catalyst weight
• r_A' = rate per unit catalyst weight (mol/mass·time)

CSTR with Catalyst Weight: \[W = \frac{F_{A0}X}{-r_A'}\]

Langmuir-Hinshelwood Kinetics

General Form (Single Reactant): \[-r_A' = \frac{kK_AC_A}{1 + K_AC_A}\]

• k = reaction rate constant
• K_A = adsorption equilibrium constant

Surface Reaction Controlling (A + B → Products): \[-r_A' = \frac{kK_AK_BC_AC_B}{(1 + K_AC_A + K_BC_B)^2}\] Competitive Adsorption: \[\theta_A = \frac{K_AC_A}{1 + K_AC_A + K_BC_B + K_CC_C}\]

• θ_A = fraction of sites occupied by species A

Effectiveness Factor

Effectiveness Factor (η): \[\eta = \frac{\text{actual rate with diffusion}}{\text{rate without diffusion}}\] For First Order Reaction in Spherical Pellet: \[\eta = \frac{3}{\phi_s}\left(\frac{1}{\tanh \phi_s} - \frac{1}{\phi_s}\right)\] where Thiele Modulus (φ_s): \[\phi_s = \frac{R}{3}\sqrt{\frac{k\rho_c}{D_e}}\]

• R = pellet radius
• k = first order rate constant
• ρ_c = catalyst density
• D_e = effective diffusivity

For Large Thiele Modulus (φ_s > 3): \[\eta \approx \frac{3}{\phi_s}\] Overall Rate with Effectiveness Factor: \[-r_A' = \eta k C_{As}\]

• C_As = concentration at external surface

External Mass Transfer

Mass Transfer to Catalyst Surface: \[-r_A' = k_c a_c(C_A - C_{As})\]

• k_c = mass transfer coefficient
• a_c = external surface area per unit mass
• C_A = bulk fluid concentration
• C_As = surface concentration

Catalyst Deactivation

Activity

Catalyst Activity: \[a(t) = \frac{-r_A'(t)}{-r_A'(t=0)}\]

• a(t) = activity at time t (dimensionless, 0 to 1)
• a = 1 for fresh catalyst
• a → 0 for completely deactivated catalyst

Rate Law with Deactivation: \[-r_A' = a(t) \cdot k \cdot f(C_A)\]

Deactivation Kinetics

General Deactivation Rate: \[-\frac{da}{dt} = k_d a^d C_A^n\]

• k_d = deactivation rate constant
• d = deactivation order with respect to activity
• n = deactivation order with respect to concentration

First Order Deactivation (d = 1, n = 0): \[a = e^{-k_d t}\] Second Order Deactivation (d = 2, n = 0): \[a = \frac{1}{1 + k_d t}\]

Moving Bed Reactor

Design Equation with Deactivation: \[W = F_{A0}\int_0^X \frac{dX}{a(t) \cdot (-r_A')}\]

• Catalyst age varies with position in moving bed

Biochemical Reactors

Cell Growth Kinetics

Monod Equation (Microbial Growth): \[\mu = \mu_{max}\frac{C_s}{K_s + C_s}\]

• μ = specific growth rate (time^-1)
• μ_max = maximum specific growth rate
• C_s = substrate concentration
• K_s = Monod constant (substrate concentration at μ = μ_max/2)

Cell Growth Rate: \[r_g = \mu C_c\]

• r_g = cell growth rate
• C_c = cell concentration

Substrate Consumption Rate: \[-r_s = \frac{\mu C_c}{Y_{c/s}}\]

• Y_c/s = yield coefficient (mass cells/mass substrate)

Chemostat (Continuous Bioreactor)

Cell Balance (Steady State): \[(\mu - D)C_c = 0\]

• D = dilution rate = ν₀/V (time^-1)

At Steady State: \[\mu = D\] Substrate Balance: \[D(C_{s0} - C_s) = \frac{\mu C_c}{Y_{c/s}}\] Washout Condition: \[D > \mu_{max}\]

• Cells wash out faster than they can grow

Reactor Stability and Dynamics

Stability Criteria for CSTR

Heat Generation Function: \[G(T) = F_{A0}X(-\Delta H_{rxn})\] Heat Removal Function: \[R(T) = F_{A0}\sum \Theta_i C_{p,i}(T - T_0) + UA(T - T_a)\] Steady State Condition: \[G(T) = R(T)\] Stability Criterion:

• Stable: slope of R(T) > slope of G(T) at intersection
• Unstable: slope of R(T) < slope="" of="" g(t)="" at="">
• Multiple steady states possible for highly exothermic reactions

Runaway Criteria

Damköhler Number: \[Da = \frac{k\tau C_{A0}}{1}\]

• Ratio of reaction rate to convective transport rate
• High Da indicates potential for runaway

Heat Generation Number: \[B = \frac{-\Delta H_{rxn} E_a C_{A0}}{RT_0^2 \rho C_p}\]

• High B indicates high sensitivity to temperature changes