Reactor Design Equations: Batch, CSTR, and PFR Kinetics Overview

This section covers the analysis of reactor networks. For PFRs in series, the total conversion can be determined by summing all reactor volumes and treating them as a single PFR with equivalent total volume. The general formula for conversion in the nth reactor of a series is Cₙ = C₀ / [1 + (kτ₁)(1 + (kτ₂) + ... + (kτₙ))] for first-order reactions. For PFRs in parallel or series-parallel combinations, the total volume equals the sum of individual reactor volumes, and a single PFR with equivalent total volume gives the same conversion. For MFRs in series, the concentration leaving the ith reactor is Cᵢ = C₀ / (1 + kτ)ⁱ for first-order reactions. As the number of equal-sized MFRs increases, the system behavior approaches that of a single PFR with equivalent total volume. For two MFRs in series, the optimal arrangement depends on reaction order: equal-sized reactors are optimal for first-order reactions, the smaller reactor should come first for orders greater than one, and the larger reactor should come first for orders less than one.

For reactor networks: (1) PFR in series - same conversion as single PFR with same total volume; (2) Equal-sized CSTRs in series: X = 1 - (1/(1 + kτ/n))^n; (3) CSTR followed by PFR - different RTD from either alone; (4) PFR followed by CSTR - different RTD from either alone. For maximum conversion with minimum volume, plot -r_A versus X and choose arrangement with minimum area under curve. PFR gives maximum conversion for given volume; CSTR gives minimum conversion.

This section addresses reactor configuration optimization. Single-reaction systems use conversion formulation; multiple reactions require molar flow rates. Levenspiel plots graphically optimize reactor arrangements by showing FA₀/(-ra) vs conversion, where area under the curve equals catalyst weight/volume. Series arrangements minimize total catalyst by placing reactors with lower slope first; parallel arrangements average individual conversions. The systematic approach involves sequential calculation: solve first reactor to get outlet conditions, then use those as inlet for subsequent reactors. This methodology applies universally across reactor types and configurations.

This segment analyzes different reactor configurations to maximize conversion. For reactors in series (CSTR followed by PFR at 300 K), the overall conversion is 86.5%. When reversed (PFR followed by CSTR), conversion increases to 87.1%, demonstrating that reactor order affects performance. For parallel operation with equal flow splitting, the overall conversion is 73.7%, lower than series arrangements. The instructor explains that series configurations are advantageous for reactions with decreasing rate as conversion increases, while parallel configurations divide the feed but maintain constant concentration in each stream.

This comprehensive section covers parallel and series reactions in chemical engineering. For parallel reactions, instantaneous selectivity (Sdu) is defined as the ratio of desired to undesired product formation rates. Reactor selection criteria include safety, selectivity, yield, heat management, and cost. Recycle reactors are commonly used in CSTRs for liquid-phase and PFRs for gas-phase reactions. Four optimization cases are presented based on reaction orders: (1) When both α1 > α2 and β1 > β2, PFR or batch reactors maximize selectivity; (2) When α1 > α2 but β1 < β2, semi-batch reactors, membrane reactors, or series CSTRs with high CA and low CB are optimal; (3) When both α1 < α2 and β1 < β2, CSTR is preferred to maintain low concentrations; (4) When α1 < α2 but β1 > β2, the reverse of Case 2 applies. For series reactions (A → B → C), time is the most important variable. For elementary liquid-phase series reactions in batch reactors: CA = CA0 × exp(-k1 × t), CB = (k1 × CA0 / (k2 - k1)) × [exp(-k1 × t) - exp(-k2 × t)], CC = CA0 - CA - CB. The concentration of B reaches a maximum when dCB/dt = 0, giving t_max = (1/(k2 - k1)) × ln(k2/k1).

This extensive section develops the mathematical framework for solving non-isothermal reactor problems. It covers: (1) Derivation of the energy balance equation for adiabatic systems with Q=0 and Ws=0, relating temperature change to conversion; (2) Simplification by combining heat capacity terms when reactant and product heat capacities are similar; (3) The simplified form expressing temperature as a function of feed temperature, heat of reaction, and conversion; (4) A systematic step-by-step procedure for reactor design calculations: using energy balance to find temperature at specified conversions, evaluating reaction rate constant k at that temperature using Arrhenius equation, finding equilibrium constant Kc at that temperature, calculating equilibrium conversion, determining actual reaction rate using rate law, and computing F_A₀/(-r_A) for integration; (5) The recognition that this iterative process must be repeated for multiple conversion points due to the coupled nature of the equations.

For non-isothermal CSTR design, the procedure involves: (1) Calculate outlet temperature T using energy balance: T = T0 + (ΔHR × XA)/Cp; (2) Calculate rate constant k using Arrhenius equation: k = A × e^(-Ea/RT); (3) Calculate volume using mass balance: V = FA0XA/(-rA). This three-step process accounts for temperature effects on reaction rate, where temperature increases with conversion for exothermic reactions.

For adiabatic reactors, the temperature and conversion are related by: T = Tref + (ΔHr(Tref) + ∫(Tref to T) ΔCp dT) / (ΣνiCp,i(products) - ΣνiCp,i(reactants)). Reactor design requires solving both mole balance and energy balance equations simultaneously. The mole balance equations determine conversion as a function of reactor volume or residence time, while the energy balance equations determine temperature as a function of conversion. These equations are coupled through the temperature dependence of reaction rates and equilibrium constants. The reaction rate constant k depends on temperature through the Arrhenius equation: k = A exp(-Ea/RT). For reversible reactions, the net rate is the difference between forward and reverse rates: r = kfCA - krCB.

In non-isothermal reactor design, the fundamental design equations, rate laws, and stoichiometric relationships remain valid, but the key difference lies in evaluating the design equation when temperature varies along the reactor. The energy balance equation, derived from the First Law of Thermodynamics, must be coupled with mole balances, rate laws, and stoichiometry to determine reactor performance. For adiabatic reactors, the energy balance simplifies to relate temperature changes to reaction heat effects, while for reactors with heat transfer, the balance includes heat exchange terms. The temperature-dependent reaction rate constant (via Arrhenius equation) requires solving coupled differential equations to obtain temperature, conversion, and concentration profiles along the reactor.

This segment covers the mathematical framework for solving reactor design problems with temperature-dependent kinetics. The Arrhenius equation (k = A × exp(-E_a/(R×T))) describes how rate constants change with temperature. By substituting concentration expressions in terms of conversion into the CSTR design equation and applying logarithms, the operating temperature can be solved. For endothermic reactions, the energy balance equation (Q = F_A0 × [Σ(θ_i × C_p,i) × (T - T_0) + ΔH_rxn × x]) calculates the heat flow required. The example demonstrates calculating Q = 18,187.2 kJ/s for a reaction requiring 70 K temperature increase with 60% conversion.

In chemical reactions, products are classified as desired (main reaction products) or undesired (byproducts). Reactions are classified as main reactions (producing desired products) or side reactions (producing undesired products). Yield is defined as the ratio of actual moles of desired product produced to theoretical moles that would be produced with 100% conversion and no side reactions. Selectivity is defined as the ratio of moles of desired product produced to moles of undesired product produced. For multiple reactions, material balances must account for all species across all reactions. The extent of reaction for each reaction (ξ1, ξ2) is calculated separately. For ethane cracking (C2H6 → C2H4 + H2 and C2H6 + H2 → C2H8): From desired product (C2H4 = 40 moles), ξ1 = 40 moles; from side product (C2H8 = 2.4 moles), ξ2 = 2.4 moles. Composition is calculated by dividing each species' moles by total moles. Selectivity = n_C2H4 / n_C2H8 = 40 / 4.8 = 8.33 mol desired product per mol undesired product.

In chemical reaction engineering, when multiple reactions occur simultaneously, yield measures the ratio of desired product formed to the maximum possible yield assuming no side reactions and complete conversion of the limiting reactant (ranging from 0 to 1), while selectivity measures the ratio of desired product to undesired product formed (any positive value, with values greater than 1 indicating better selectivity). To analyze such systems, one uses extent of reaction equations for each reaction, applies material balances (either molecular or atomic), and calculates overall and single-pass conversion based on fresh feed versus reactor exit streams.

Instantaneous selectivity is defined as the ratio of reaction rate of desired product to undesired products. For parallel reactions A → R, A → S, A → T, selectivity = RS/(RR + RT). To maximize selectivity, differentiate with respect to concentration CA and set to zero, yielding CA_optimal = √(K1/K3). When optimization yields negative concentration, it indicates the reaction completed before the given time, so CA_optimal = 0. For zero-order reactions, maximum selectivity occurs at CA = 0. Yield determines distribution of reacted reactant among reaction steps. Yield = total moles of reactant A reacted to form specific product / total moles of A reacted. Sum of yields for all products equals one. Overall yield = total moles of desired product formed / total moles of limiting reactant A. Overall yield depends on reactor type and is independent of reaction rate. For CSTR, overall yield equals fractional yield at outlet conditions. For PFR, non-uniform concentration requires integration to find average fractional yield.

In multiple reaction systems, the overall reaction rate and product distribution depend on the rate constants of individual reactions, with the rate-controlling step being the slowest reaction in the network. For series reactions (A → R → S), the intermediate concentration first increases then decreases, while the final product concentration continuously increases. For parallel reactions (A → R and A → S), the product distribution depends on the ratio of rate constants, and temperature optimization can favor desired products by exploiting different activation energies. The fractional conversion of reactants and selectivity of desired products are calculated using stoichiometric relationships and rate equations.

Selectivity in parallel chemical reactions (A → R and A → S) is defined as the ratio of desired product formation rate to undesired product formation rate, given by S = (k1/k2) × C_A^(n1-n2), where k1 and k2 are rate constants, C_A is reactant concentration, and n1 and n2 are reaction orders. To maximize selectivity: (1) maintain high concentration when n1 > n2, low concentration when n1 < n2; (2) maintain high temperature when activation energy of desired reaction (E1) > undesired reaction (E2), and low temperature when E1 < E2; (3) remove inert gases to increase concentration, add inert gases to decrease concentration; (4) maintain high pressure for reactions with positive order, low pressure for negative order.

Residence Time Distribution (RTD) describes the distribution of time that fluid elements spend inside a chemical reactor, measured experimentally using the stimulus-response technique by injecting an inert tracer and analyzing the exit concentration profile; in ideal reactors like PFRs all molecules have identical residence time, while in CSTRs there is a wide distribution from immediate exit to indefinite retention, and non-ideal behaviors such as channeling, bypassing, dead zones, and recirculation cause deviations from these ideal patterns.

Residence Time Distribution (RTD) is a fundamental concept in chemical reaction engineering that quantifies the distribution of time that fluid elements spend inside a reactor before exiting, which is crucial for understanding and designing non-ideal flow systems where ideal plug flow assumptions do not hold; RTD provides essential information about mixing characteristics, dead zones, short-circuiting, and other flow anomalies that significantly impact reactor performance and reaction outcomes.

Non-ideal reactors operate continuously under steady-state conditions but exhibit non-uniform mixing and flow patterns unlike ideal reactors. Key causes include dead zones (insufficient agitation), channeling (fluid bypassing near walls), tortuous paths (winding flow routes), and poor catalyst contact. Residence Time Distribution (RTD) characterizes how long molecules stay in a reactor. The E curve shows probability density of molecules leaving at specific times, with total area equaling 1. Multiple peaks indicate multiple flow paths. The pulse tracer experiment injects known tracer quantity at t=0 and monitors outlet concentration. For ideal plug flow, all molecules exit simultaneously; real reactors show spread-out curves. The area under the pulse curve equals m/V (tracer mass divided by volumetric flow rate). Normalization converts raw pulse data to the E curve by dividing by this area, enabling comparison across different reactor configurations.

Space time (θ) equals reactor volume divided by volumetric flow rate. For ideal reactors with constant density, space time equals mean residence time. However, for real reactors with significant density changes, these parameters differ. RTD can be determined experimentally through tracer experiments or theoretically from reactor geometry. Understanding the relationship between space time and RTD is essential for reactor design, sizing, and optimization in chemical engineering applications.

This comprehensive section establishes the theoretical foundation for analyzing non-ideal reactor behavior. RTD alone cannot predict reactor performance; three components are essential: RTD, reactor model, and kinetics. Zero-parameter models include Segregation Model (complete mixing within elements) and Maximum Mixedness Model (perfect mixing throughout). The single-parameter Tank-in-Series model uses the number of tanks (N) to approximate real reactor behavior. The Dispersion Number (D/uL) characterizes mixing—zero for PFR, infinite for CSTR. Moments analysis provides mean residence time (first moment), variance (spread), and skewness (asymmetry). Pulse and step input methods experimentally determine RTD functions. Ideal RTD functions are E(t) = δ(t-τ) for PFR and E(t) = (1/τ)e^(-t/τ) for CSTR. These concepts form the basis for identifying reactor anomalies like dead volumes or bypassing.