Reactor Volume Calculations: CSTR, PFR & Batch Reactor Example

This comprehensive section covers the fundamental principles of non-isothermal reactor design. The equilibrium constant K is defined as the ratio of product to reactant concentrations at equilibrium (K = [C][D]/([A][B]) = k1/k-1). For exothermic reactions (ΔHR < 0), increasing temperature decreases K exponentially through the relationship involving (1/T_ref - 1/T), providing inherent safety benefits by preventing reactions from running away. The fundamental trade-off in reactor design is that higher temperatures increase reaction rates but reduce equilibrium yields for exothermic reactions. In plug flow reactors, concentration of reactant A is CA = CA0(1 - X)/(1 + εX), and when temperature varies, the rate constant kA becomes temperature-dependent through the Arrhenius equation: kA = kA0 exp[-Ea/R(1/T - 1/T0)]. The energy balance for reactor systems considers accumulation, heat transfer (Q̇), work (Ẇ), and energy flow with streams: dE/dt = Q̇ - Ẇ + ΣṅiHin - ΣṅiHout. For steady-state operation, accumulation becomes zero, and enthalpy (H) is preferred over internal energy (E). Species enthalpy at temperature T is Hi(T) = Hi°(Tref) + ∫(Tref to T) Cp,i dT, where Hi°(Tref) is enthalpy of formation at reference temperature (25°C). The heat of reaction ΔHR = ΣνiHi(T) is temperature-dependent: ΔHR(T) = ΔHR(Tref) + ∫(Tref to T) ΔCp dT, where ΔCp = ΣνiCp,i. For adiabatic reactors (Q̇ = 0), the energy balance simplifies to -FA0ΔHR + Σṅi0Hi(T0) - ΣṅiHi(T) = 0, relating heat of reaction to sensible heat changes.

Runaway reactions occur when heat generation exceeds heat removal (dRT/dT < dGT/dT), causing uncontrolled temperature rise toward an upper steady state. This happens because exothermic reactions accelerate at higher temperatures, generating more heat than can be removed. At high temperatures, side reactions, product decomposition, and reactant evaporation become problematic. The system transitions from a lower to an upper steady state, which may be undesirable for the reaction process.

Non-isothermal reactor design differs from isothermal design by requiring an additional energy balance equation to account for temperature variations that affect reaction rates; unlike isothermal systems where temperature remains constant, non-isothermal reactors experience changing temperatures along their length, which in turn causes the rate constant k to vary, necessitating a six-step algorithm (mole balance, rate law, stoichiometry, energy balance, combine, evaluate) instead of the five-step approach used for isothermal systems.

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.

In non-isothermal reactor design, the energy balance equation accounts for the heat effects of chemical reactions, where the exit temperature depends on the heat of reaction (ΔHr), inlet conditions, and conversion. For adiabatic operation, the energy balance simplifies to: T - T₀ = (-ΔHr × F_A₀ × x) / (ΣF_i × Cp_i), where T is exit temperature, T₀ is inlet temperature, F_A₀ is inlet molar flow rate of reactant A, x is conversion, and Cp_i are specific heats of all species. This equation allows calculation of reactor volume by integrating the design equation with the energy balance constraint, enabling prediction of temperature profiles and conversion in reactors where heat effects significantly influence reaction rates.

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.

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 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.

This section addresses advanced material balance problems involving multiple simultaneous reactions. Selectivity measures the preference for desired product formation over undesired products: Selectivity = (moles desired product / moles undesired product). High selectivity indicates efficient production of desired products while minimizing side reactions. For complex systems with multiple reactions: (1) Write balanced equations for all reactions; (2) Assign separate extent variables for each reaction; (3) Set up extent balances incorporating all reactions; (4) Use conversion to find unreacted limiting reactant; (5) Use selectivity to relate desired and undesired products; (6) Solve the resulting system of equations. This comprehensive approach enables analysis of industrial processes where multiple reactions compete for reactants.

Selectivity and yield are critical figures of merit for evaluating multiple reaction systems. Selectivity measures product preference as moles of desired product divided by moles of undesired product. Yield has four definitions: fractional yield (actual vs theoretical maximum), less restrictive fractional yield, ratio to reactant fed, and ratio to reactant consumed (textbook definition). Instantaneous metrics use reaction rates (rD/rU for selectivity, rD/(-rA) for yield), while overall metrics use molar flow rates. A fundamental trade-off exists between conversion and selectivity: higher conversion increases both desired and undesired products, potentially reducing selectivity. Instantaneous metrics guide reactor design, while overall metrics determine economic profitability. Optimal operation requires balancing these factors to maximize profit.

The Ergun equation calculates pressure drop in a packed bed reactor using the formula -ΔP/Δz = (150(1-ε)²μu₀)/(ε³d_p²A) + (1.75(1-ε)ρu₀²)/(ε³d_pA), where pressure decreases as distance increases (negative sign), and the equation accounts for both laminar and turbulent flow contributions; the volumetric flow rate changes along the reactor due to pressure decrease, temperature increase, and mole number changes from reactions, requiring simultaneous solution with material and energy balances.

This section covers the complete numerical solution methodology and performance analysis of PBR systems. The Ergun equation calculates pressure drop: -dP/dW = (α × F₀² × y²) / (2 × ρ₀ × P₀²), where α incorporates bed properties. The PBR model results in a system of ODEs solved numerically using Runge-Kutta fourth order, with catalyst weight as the independent variable discretized over the total catalyst amount. Simulation results reveal: reactant A decreases as consumed in both reactions; reactant B decreases more rapidly (twice as fast as A in reaction 1); product C initially increases then stabilizes (produced then consumed); product D increases as undesired product; pressure fraction decreases from 100% to approximately 80%; global conversion increases then stabilizes. The analysis demonstrates how pressure drop significantly affects reaction rates by reducing concentrations, and how conversion profiles show initial rapid increase followed by stabilization as reactants are depleted.

The Ergun equation calculates pressure drop per unit length in packed beds: ΔP/L = 150μV₀(1-ε)²/(ε³DP²) + 1.75ρV₀²(1-ε)/(ε³DP). The first term represents viscous losses (proportional to velocity and viscosity), while the second term represents kinetic energy losses (proportional to velocity squared and density). This equation combines laminar and turbulent flow regimes for comprehensive pressure drop prediction.

This video demonstrates how to calculate the exit pressure from a packed bed reactor using the Ergun equation, which accounts for both laminar and turbulent flow contributions to pressure drop. The calculation involves converting mass flow rate to molar flow rate using molecular weight, calculating volumetric flow rate with the ideal gas law (V = nRT/P), and integrating the Ergun equation over the reactor length while ensuring unit consistency (kg, m, s, Pa). The example shows a pressure drop from 12 bar at the inlet to 8.8 bar at the outlet over a 30-meter reactor with specific catalyst particle and tube dimensions.

Pressure drop in packed beds is calculated using the Ergun equation: ΔP/L = 150(1-ε)²μG/(DP²) + 1.75(1-ε)ρG²/DP. Key parameters include: superficial velocity G (calculated from volumetric flow rate and cross-sectional area), particle diameter DP, bed void fraction ε, fluid density ρ, and fluid viscosity μ. Results may need unit conversion (e.g., from ATM/ft to kPa/m). The Ergun equation combines laminar and turbulent flow contributions.

RTD analysis employs statistical moments to quantify reactor behavior: mean residence time τ = ∫₀^∞ t·E(t)dt and variance σ² = ∫₀^∞ (t-τ)²·E(t)dt. Normalization via θ = t/τ transforms dimensional data for comparison. Ideal batch reactors show delta-function RTD with zero variance. Ideal PFRs also show delta-function RTD. Ideal CSTRs exhibit exponential RTD: E(t) = (1/τ)exp(-t/τ), with variance equal to τ². Non-ideal configurations include dead zones (unreactive regions reducing effective volume) and bypassing (fluid flowing without proper contact). Dead zones cause faster-than-expected outlet responses; bypassing causes slower responses. RTD analysis reveals these flow anomalies, enabling diagnosis of reactor operational problems and optimization of reactor design.

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 are chemical reactors where molecules have different residence times, unlike ideal reactors where all molecules have the same residence time. The three main reasons for non-ideal behavior are bypassing (molecules exiting without reacting), turbulence (molecules trapped in eddies spending extra time), and dead zones (molecules trapped in corners indefinitely). RTD (Residence Time Distribution) curves are used to analyze reactor behavior through tracer experiments: pulse input experiments inject a known amount of tracer all at once and measure exit concentration over time, while step input experiments maintain constant tracer concentration. For ideal CSTRs, the RTD curve follows exponential decay (C = C₀e^(-t/τ)), while ideal PFRs show a sharp spike at the space time. Series of CSTRs progressively approaches PFR behavior, and parallel configurations show combined RTD profiles.

Non-ideal reactors exhibit bypassing, where some tracer molecules take shortcuts and exit quickly, appearing first in the output curve. Turbulence causes some molecules to mix thoroughly, experiencing average residence time and creating the main peak in the residence time distribution curve. These behaviors cause the output curve to spread out over time, unlike the sharp pulse of an ideal reactor.

Non-ideal reactors are real-world reactors that do not follow ideal design equations because reactant molecules spend different amounts of time inside the reactor, unlike ideal reactors where all molecules have identical residence times; to analyze non-ideal reactors, engineers use tracer curve experiments to obtain residence time distribution (RTD) data, which reveals how different molecular fractions spend varying times in the reactor, and then apply mixing models (such as segregation, maximum mixing, tanks-in-series, or dispersion models) along with kinetic information to predict conversions for given reactor volumes or vice versa.