8.4 Plug flow reactors (PFR)

Plug Flow Reactor Characteristics

Reactor Configuration and Flow Pattern

A plug flow reactor is a tubular reactor where fluid moves in one direction, and each "plug" of fluid reacts independently as it travels down the length of the tube. There's no mixing between plugs in the axial (flow) direction, though the fluid within each plug is assumed to be perfectly mixed radially (across the tube cross-section).

PFRs typically have a high length-to-diameter ratio. This geometry helps ensure that radial concentration and temperature gradients stay small, so the "no radial variation" assumption holds reasonably well.

Because there's no axial mixing, every fluid element that enters the reactor at the same time exits at the same time. This means the residence time distribution is a single spike: all fluid elements have identical residence times in an ideal PFR.

Key Assumptions

The ideal PFR model rests on several assumptions:

• Composition and reaction rate vary only in the axial direction (along the tube length), not radially
• Perfect radial mixing means no concentration gradients perpendicular to flow
• Fluid properties (density, viscosity, heat capacity) are constant across the cross-section
• The reactor operates at steady state, so nothing changes with time
• The velocity profile is uniform across the cross-section (flat, not parabolic)

That last point is worth noting: real turbulent flow in tubes approximates a flat velocity profile reasonably well, which is one reason PFR assumptions work better at high flow rates. Laminar flow, by contrast, produces a parabolic velocity profile where fluid near the walls moves much slower than fluid at the center, violating the plug flow assumption.

PFR Design Equations

Isothermal PFR Design Equation

The PFR design equation comes from writing a mole balance on a differential volume element $dV$ of the reactor. For species A reacting away:

$F_{A0} \, dX_A = (-r_A) \, dV$

Rearranging and integrating gives the core design equation:

$V = F_{A0} \int_0^{X_A} \frac{dX_A}{-r_A}$

where:

• $V$ = reactor volume
• $F_{A0}$ = molar flow rate of reactant A at the inlet
• $X_A$ = conversion of reactant A
• $-r_A$ = rate of disappearance of A (a positive quantity), which depends on concentration and temperature

To evaluate this integral, you substitute the rate law and express $C_A$ in terms of $X_A$. Here's how that works for a common case:

Example: First-order, constant-density, liquid-phase reaction ($-r_A = kC_A$)

1. Write the concentration in terms of conversion: $C_A = C_{A0}(1 - X_A)$

2. Substitute into the rate law: $-r_A = kC_{A0}(1 - X_A)$

3. Plug into the design equation: $V = F_{A0} \int_0^{X_A} \frac{dX_A}{kC_{A0}(1 - X_A)}$

4. Recognize that $F_{A0} = C_{A0} \cdot v_0$ (where $v_0$ is the inlet volumetric flow rate), so the $C_{A0}$ terms cancel

5. Integrate to get: $V = \frac{v_0}{k} \ln\left(\frac{1}{1 - X_A}\right)$

For gas-phase reactions where the total number of moles changes, you'll need to account for the volume change factor $\varepsilon$. This factor adjusts the concentration expression because the volumetric flow rate is no longer constant along the reactor.

The temperature dependence of the rate constant $k$ follows the Arrhenius equation, but for an isothermal reactor, $k$ is constant throughout, which simplifies things considerably.

Non-Isothermal PFR Design Equations

When temperature changes along the reactor (due to heat of reaction or heat exchange through the walls), you can no longer treat $k$ as constant. You need to solve the material balance and energy balance simultaneously.

The material balance stays the same:

$\frac{dX_A}{dV} = \frac{-r_A}{F_{A0}}$

The energy balance for a PFR takes the form:

$\frac{dT}{dV} = \frac{\dot{Q}' + (-r_A)(-\Delta H_{rxn})}{\sum F_i \, C_{p,i}}$

where:

• $\dot{Q}'$ = rate of heat transfer per unit volume (from a jacket or coil; positive when heat is added to the reactor)
• $\Delta H_{rxn}$ = heat of reaction (negative for exothermic, positive for endothermic)
• $F_i$ = molar flow rate of species $i$
• $C_{p,i}$ = molar heat capacity of species $i$

These two coupled ODEs (one for $X_A$, one for $T$, both as functions of $V$) are typically solved numerically using tools like MATLAB or Python. The solution gives you concentration and temperature profiles along the reactor length. For an adiabatic PFR ($\dot{Q}' = 0$), the temperature rises for exothermic reactions and drops for endothermic ones, and you can sometimes solve the energy balance algebraically to get $T$ as a function of $X_A$ directly.

PFR Performance Metrics

Conversion

Conversion ($X_A$) measures the fraction of limiting reactant A that has been consumed:

$X_A = \frac{F_{A0} - F_A}{F_{A0}}$

A conversion of 0.90 means 90% of reactant A has reacted. This is the most basic measure of how far the reaction has proceeded.

Selectivity

When multiple reactions occur, you care about more than just conversion. Selectivity ($S_B$) tells you how much of the consumed reactant went toward the desired product B versus unwanted byproducts:

$S_B = \frac{\text{moles of B formed}}{\text{moles of A consumed}}$

High selectivity means most of what reacted formed the product you actually want. PFRs tend to favor selectivity for series reactions ($A \rightarrow B \rightarrow C$) because you can stop the reaction at the right point by choosing the reactor length, whereas a CSTR's backmixing exposes some of the desired product B to further reaction.

Yield

Yield ($Y_B$) combines both conversion and selectivity into one metric:

$Y_B = \frac{\text{moles of B formed}}{\text{initial moles of A}} = X_A \times S_B$

Yield captures the overall effectiveness of the reactor. You could have high conversion but low selectivity (lots of byproducts), or high selectivity but low conversion (not much reacted). Yield penalizes both problems.

PFR vs CSTR

Mixing Characteristics

The fundamental difference: a CSTR is perfectly mixed throughout, so the outlet composition equals the composition everywhere inside the reactor. A PFR has no axial mixing, so concentration changes progressively along the length.

This means a CSTR always operates at the outlet (lowest) concentration of reactant, while a PFR starts at the inlet (highest) concentration and gradually decreases.

Reactor Volume and Conversion

For most common reaction orders (positive-order kinetics), a PFR requires a smaller volume than a CSTR to achieve the same conversion. Here's why:

• In a PFR, the reaction rate is high near the inlet (where concentration is high) and decreases along the length. You get the benefit of those high rates early on.
• In a CSTR, the entire reactor operates at the low exit concentration, so the reaction rate is uniformly low throughout.

You can visualize this with a Levenspiel plot ($1/(-r_A)$ vs. $X_A$). The PFR volume corresponds to the area under the curve (the integral), while the CSTR volume is the area of a rectangle with height $1/(-r_A)_{exit}$ and width $X_A$. For positive-order kinetics, the rectangle is always larger than the area under the curve, confirming the PFR needs less volume.

Residence Time Distribution

• In an ideal PFR, all fluid elements have the same residence time. The RTD is a delta function (a single spike at $t = \tau$).
• In a CSTR, the RTD follows an exponential decay: some fluid exits almost immediately, while some stays much longer than the mean residence time.

This difference matters for reactions where timing is critical, such as series reactions with an intermediate product that degrades if it stays in the reactor too long.

Reaction Kinetics and Performance

For simple positive-order kinetics, the PFR always outperforms the CSTR in terms of volume efficiency. But there are exceptions:

• Autocatalytic reactions (where the product catalyzes its own formation) can actually favor a CSTR, because the high product concentration throughout the CSTR keeps the rate high.
• For zero-order kinetics, the rate doesn't depend on concentration at all, so PFR and CSTR require the same volume for a given conversion.
• In practice, combinations of reactor types (e.g., a CSTR followed by a PFR) are sometimes used to get the best of both worlds.

Operating Conditions Impact on PFR

Temperature Effects

Temperature controls the rate constant through the Arrhenius equation:

$k = A \exp\left(\frac{-E_a}{RT}\right)$

where $A$ is the pre-exponential factor, $E_a$ is the activation energy, $R$ is the gas constant, and $T$ is absolute temperature.

Higher temperatures increase $k$ and speed up the reaction. But there are trade-offs: high temperatures can promote unwanted side reactions (especially those with higher activation energies), deactivate catalysts, or push reversible reactions backward if the forward reaction is exothermic.

Pressure Effects

Pressure mainly matters for gas-phase reactions:

• Increasing pressure raises gas-phase concentrations (more moles per unit volume), which increases the reaction rate for positive-order kinetics
• For equilibrium-limited reactions, higher pressure favors the side with fewer moles of gas (Le Chatelier's principle). For example, in the Haber process ($N_2 + 3H_2 \rightleftharpoons 2NH_3$), high pressure favors ammonia production because 4 moles of reactant gas become 2 moles of product gas.

For liquid-phase reactions, pressure has a negligible effect on rate and equilibrium under most conditions.

Residence Time Effects

Residence time ($\tau$) is the average time fluid spends in the reactor:

$\tau = \frac{V}{v_0}$

where $V$ is reactor volume and $v_0$ is the volumetric flow rate at the inlet.

Longer residence times give higher conversions, since reactants have more time to react. But pushing for very high conversions means larger (more expensive) reactors, and it can allow unwanted secondary reactions to become significant. There's often a practical sweet spot where you get good conversion without excessive reactor size or byproduct formation.

Optimization of Operating Conditions

Reactor design is always a balancing act. You're trying to maximize conversion, selectivity, and yield simultaneously while staying within constraints like material limits, catalyst lifetime, and cost. The PFR design equations, combined with rate law data and simulation tools, let you explore how changes in temperature, pressure, and flow rate affect performance before committing to a physical design.