Ideal and Non-Ideal Reactor Types

In chemical engineering, ideal models are applied to reactors to simplify predictions of reactor performance. Three common ideal reactor types include:

batch reactor, plug flow reactor (PFR) and continuous stirred tank reactor (CSTR).

Each of these can be classified based on their reactant concentration profiles with respect to time and space (Figure 1). Batch reactors operate under transient state, where the composition of the reaction mixture changes with respect to time. In contrast, the PFR and CSTR are both continuous reactors. Hence, after an initial transient period, they operate under steady state, where the composition of the reaction mixture is constant with respect to time.¹

(a) (b) (c)

C0 Ce C0

Ce

Ct

Figure 1. Concentration profiles of three ideal reactor types: (a) batch reactor; (b) plug flow reactor (PFR); (c) continuous stirred tank reactor (CSTR). Ct = [reactant]

at time t, C0 = initial [reactant], Ce = [reactant] in the effluent.

Both the batch reactor and CSTR are assumed to have perfect mixing, which results in a uniform composition throughout the entire reactor. As such, the effluent from a CSTR has the same composition as the mixture within the reactor. For a PFR, where the reaction mixture flows through a tube, there is no mixing in the axial direction giving rise to an infinite number of discrete reaction plugs. The composition of the reaction plugs are the same at a given length along the reactor, however, their composition varies as a function of distance travelled. As the distance travelled increases, the concentration of reactants decreases, resulting in the formation of a concentration gradient along the tube.²

The residence time distribution (RTD) of a flow reactor is defined as the distribution of times which molecules spend within the reactor. Determining the RTD reveals information regarding mixing and flow patterns within a reactor.

Therefore, the RTD of real reactors can be compared to their corresponding ideal

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model to troubleshoot for potential problems, as well as determine the reactor performance for reactions with known kinetics.

The residence time distribution function, E(t), is a probability distribution function which describes the amount of time a molecule could spend within the reactor. Hence, the area under the curve for a plot of E(t) against time, t, is unity (E curve) [Eq (1)].³ The value of E(t)dt is equal to the fraction of molecules that spend time t inside the reactor. This is known as the cumulative distribution function, F(t), and is obtained by integrating the area under the E curve between the limits t and 0 [Eq (2)].⁴

As E(t) is a probability distribution function, the mean residence time, tm, is equal to the first moment of the function i.e. the total area under the curve for a plot of tE(t) against t [Eq (3)]. Similarly, the variation around the mean, σ², is given by the second moment of the function [Eq (4)].³ It is standard practice to use the values of tm and σ² when comparing reactors.

The RTD for an ideal PFR and CSTR can be derived (Figure 2). In the instance of an ideal PFR with no axial mixing, all molecules will spend an equal amount of time within the reactor (σ² = 0). The RTD is a spike of infinite height and zero width, as defined by the Dirac Delta function [Eq (5)]. For an ideal single CSTR with perfect mixing, not all molecules spend an equal amount of time within the reactor (σ² > 0).

Instead, there is an exponential decay in E(t) with time [Eq (6)].¹ τ is the space time, calculated by dividing the reactor volume, V, by the volumetric flow rate, ν (τ = V/ν).

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Figure 2. RTD profiles for ideal reactors: (a) PFR: spike of infinite height and zero width; (b) single CSTR: exponential decay of E(t) with increasing time.

In reality the RTDs of reactors will deviate from their ideal models, for example, some axial dispersion will be observed in a tubular reactor. This is caused by frictional forces between the walls of the reactor and the fluid resulting in a non-uniform velocity profile (Figure 3). This is known as Taylor dispersion, and results in a bell-shaped RTD in contrast to the spike expected from the PFR model.

The extent of dispersion is dependent on the channel dimensions, where narrower channels reduce the extent of dispersion.^{5, 6}

νmax

Figure 3. Flow through a tube: (a) ideal plug flow where there is no axial dispersion;

(b) Taylor dispersion caused by a non-uniform flow velocity, and resulting in a parabolic profile.

Other discrepancies from the ideal models can occur as a result of reactor failures, including bypassing and dead volume (Figure 4). Bypassing is where some material passes straight through the reactor, and dead volume refers to a volume in the reactor where material becomes stagnant. For a bypassing system, ν of the

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non-bypassing material is lower than that expected. Hence, the residence time observed is greater than τ. For a PFR, this would be shown by a RTD with two spikes:

one close to the origin for the bypassing material and a second at a time greater than τ. For a CSTR, this results in a slower exponential decay of E(t). For a system with a dead volume, the V of the reactor is less that that expected. Hence, the residence time observed is lower than τ. For a PFR, this would be shown by a RTD with a spike at a time less than τ. For a CSTR, this results in a faster exponential decay of E(t).³

E(t)

Figure 4. RTD profiles for reactors deviating from ideality: (a) PFR: bypassing – two spikes, one observed at t > τ; dead volume – one spike observed at t < τ; (b) CSTR:

bypassing – slower exponential decay of E(t); dead volume – faster exponential decay of E(t).

The PFR and CSTR models highlight an important difference in terms of reactor efficiency. For reactions with an order greater than 0, the rate of reaction decreases as the conversion increases, as a result of a decreasing reactant concentration. In the PFR, the reactant concentration decreases gradually along the length of the reactor; whereas in the CSTR, the reactant concentration drops immediately to a lower value as a result of perfect mixing. Thus, the average rate of reaction for a PFR is higher than a CSTR, and a higher conversion will be achieved for reactors of the same volume. However, a stepwise drop in concentration for a mixed flow reactor can be achieved by using a number, n, of equal volume CSTRs in series, where the volume of each CSTR = V/n. This stepwise drop in concentration indicates that increasing n brings the behaviour of the system closer to that of a PFR, evident in both the concentration and RTD profiles (Figure 5).⁷

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Figure 5. Convergence in behaviour of a CSTR cascade to PFR with increasing n: (a) concentration profile: stepwise decrease in reactant concentration along the length of the reactor; (b) RTD: profile becomes sharper with increasing n. E(ϴ) = normalised distribution function, ϴ = normalised residence time.