Chemical Reaction Engineering

Residence time distribution E-curve

The residence time distribution (RTD) describes how long different fluid elements stay inside a flow reactor. The exit-age density E(t) is defined so that E(t)\,dt is the fraction of the exit stream whose age lies between t and t+dt. It must satisfy \int_0^\infty E(t)\,dt = 1. If a curve read from a figure does not have unit area, it must be normalized by dividing by its area before any moment is computed. The mean residence time is \bar t = \int_0^\infty t\,E(t)\,dt, and the variance is \sigma^2 = \int_0^\infty (t-\bar t)^2 E(t)\,dt. An ideal PFR has a single spike at \bar t, whereas an ideal CSTR has E(t) = (1/\bar t)\,e^{-t/\bar t}.

The segregated model assumes that fluid elements do not mix with one another until they leave the reactor, so each element behaves as a small batch reactor for its own age t. The exit conversion is the E-weighted average of the batch conversions, \bar X = \int_0^\infty X_{batch}(t)\,E(t)\,dt. Each element of age t contributes in proportion to its share of the exit stream. For a first-order reaction the exit concentration is \bar C_A = C_{A0}\int_0^\infty E(t)\,e^{-kt}\,dt, which is the segregated-model form used when the RTD is known only from tracer data.

When the E-curve is a set of equal-area rectangles, each rectangle carries an equal fraction of the exit stream, so the exit value is a simple average over the rectangles. A rectangle of height h spanning [t_1,t_2] must satisfy h(t_2-t_1)=1 for the curve to be normalized.

Effectiveness factor for pore-diffusion limitation

The effectiveness factor \eta is the ratio of the observed rate in a porous catalyst pellet to the rate that would occur if the entire pellet were at the external surface concentration and temperature. Intraparticle diffusion limitation is measured by the Thiele modulus. For a first-order reaction in a sphere of radius R, \phi = (R/3)\sqrt{k/D_e}, and \eta = \dfrac{3}{\phi}\left(\dfrac{1}{\tanh 3\phi} - \dfrac{1}{3\phi}\right). For large \phi (strong pore-diffusion control) this reduces to \eta \approx 3/\phi, and for small \phi the value tends to unity.

Under strong pore-diffusion control the observed order changes. For an intrinsic order n, \eta \propto 1/\phi \propto C^{-(n-1)/2}, so the observed reaction order is (n+1)/2. Because \phi \propto d\sqrt{k/D_e} for a given pellet, the observed rate per unit catalyst volume is proportional to C_A/d for a first-order reaction. Halving or doubling the particle diameter therefore changes the observed rate in direct proportion to the inverse of the diameter.

The apparent activation energy is also reduced. Since r_{obs} \propto \sqrt{k D_e}, the observed activation energy is the average of the intrinsic value E_r and the diffusion value E_d, that is E_{obs} = (E_r + E_d)/2. Thus a measured activation energy near half the intrinsic one is a diagnostic of strong internal diffusion.

Catalyst deactivation activity decay

Catalyst activity a(t) is the ratio of the instantaneous rate on an aged catalyst to the rate on fresh catalyst under the same conditions, so a(0)=1 and a(t) = (-r_A'(t))/(-r_A'(0)). If decay affects only the number of active sites, the rate law is -r_A = a(t)\,k\,C_A^n. For first-order decay with constant k_d, da/dt = -k_d\,a gives a = e^{-k_d t}.

The three common deactivation mechanisms are the following.

• Coking (fouling) — deposition of carbonaceous material on the catalyst surface, which blocks pores and sites.
• Poisoning — irreversible chemisorption of an impurity on the active sites.
• Sintering — loss of active surface area caused by prolonged exposure to high temperature.

For a constant-volume batch reactor containing W kg of catalyst, the mole balance is \dfrac{dC_A}{dt} = -\dfrac{W}{V}\,k'\,a(t)\,C_A. Integrating with a(t) inserted gives the reactant concentration at any time, and the conversion follows from X = 1 - C_A/C_{A0}.

Langmuir adsorption on catalyst active sites

In the Langmuir model adsorption occurs on a fixed number of identical sites, and the fraction of sites covered by a species is \theta = KC/(1+KC) in the liquid phase or \theta = Kp/(1+Kp) in the gas phase. The site balance states that the total number of active sites equals the vacant sites plus the sites occupied by adsorbed species. When one adsorbed molecule occupies two sites, as in a bridging adsorption on a metal surface, the balance is N_T = N_V + 2N_{ads}.

Molecular and dissociative adsorption give different coverage expressions.

• Molecular adsorption — the species adsorbs intact, \theta_A = K_A p_A\,\theta_v, where \theta_v is the fraction of vacant sites.
• Dissociative adsorption — a diatomic molecule splits into atoms, \mathrm{A_2 + 2}* \rightleftharpoons 2\mathrm{A}*, which gives \theta_A = (K_{A_2}p_{A_2})^{1/2}\,\theta_v.

Because the coverage ratio of the two species cancels the vacant-site fraction, the ratio contains a square root of the partial pressure when one species dissociates.

Reversible reaction equilibrium and rate constants

For a reversible elementary step A \rightleftharpoons B with forward constant k_f and reverse constant k_r, the equilibrium constant is K = k_f/k_r = C_B/C_A at equilibrium. For a closed cycle of reversible steps, the principle of microscopic reversibility requires that the product of forward rate constants around the cycle equals the product of the reverse rate constants. For three steps A \rightleftharpoons B \rightleftharpoons C \rightleftharpoons A this means k_1k_2k_3 = k_{-1}k_{-2}k_{-3}, so an unknown constant can be found from the other five.

For gas-phase reactions the equilibrium constant K_p depends only on temperature, but the composition relation carries a pressure term through the change in moles. For A + 2B \rightleftharpoons C, \dfrac{y_C}{y_A\,y_B^2} = K_p P^2. The mole fractions are found from a feed basis, a stoichiometric (extent) balance, and the composition constraint on one component.

Parallel reaction selectivity

For two competing reactions that form a desired product D and an undesired product U, the instantaneous selectivity is S = r_D/r_U. For example, if r_D = k_1 C_A C_B and r_U = k_2 C_A^2 C_B, then S = k_1/(k_2 C_A), so selectivity improves when the concentration of the reactant that favors U is kept low. In a PFR fed with all reactant at the inlet, C_A is highest at the entrance, whereas distributing a reactant along the reactor keeps its concentration low at every point.

For a set of parallel reactions with extents \xi_j, each outlet molar flow equals the feed flow plus \sum_j \nu_{ij}\xi_j. Consumption of a reactant equals the sum of its stoichiometric consumptions, and a product flow equals the sum of its extents. Writing these balances for every species lets the feed flow be recovered from the outlet flows.

Enzyme kinetics in batch reactor (Michaelis-Menten)

The Michaelis-Menten mechanism gives the rate r = \dfrac{V_{max}\,S}{K_M + S}, where V_{max} = k_2E_T and K_M = (k_{-1}+k_2)/k_1. The constant K_M depends only on the rate constants and not on the total enzyme concentration E_T.

Two limits are important. For S \ll K_M, the rate is (V_{max}/K_M)\,S, which is apparent first order in S. For S \gg K_M, the rate approaches V_{max} and is nearly independent of S, that is, zero order. At S = K_M the rate is half of V_{max}, not V_{max} itself.

In a well-stirred batch reactor the time for a change from S_0 to S follows from t = \int_S^{S_0} dS/(-r), which gives t = \dfrac{1}{V_{max}}\left[K_M \ln\dfrac{S_0}{S} + (S_0 - S)\right].

Gas-phase PFR with volume change

When a gas-phase reaction changes the total number of moles, the expansion factor \varepsilon is the fractional change in volume for complete conversion of the limiting reactant. For pure A feed it equals the change in total moles per mole of A converted, and diluting the feed with inert reduces it, \varepsilon = y_{A0}\,\delta. For isothermal, isobaric operation the concentration is C_A = C_{A0}\dfrac{1-X}{1+\varepsilon X}.

For a first-order reaction in a PFR the design equation is \tau = \dfrac{C_{A0}}{1}\int_0^X \dfrac{dX}{-r_A}, which becomes k\tau = -(1+\varepsilon)\ln(1-X) - \varepsilon X. The reactor length follows from the volume and the inlet volumetric flow, since the flow changes with conversion when \varepsilon \neq 0.

For a CSTR with the same first-order rate the balance gives k\tau = X(1+\varepsilon X)/(1-X).

CSTR flowsheet material balance

A flowsheet with a reactor, a separator, a purge and a recycle is solved by writing component balances around the whole process and around individual units. For an inert component that is not consumed, steady state requires that inert entering with the fresh feed equals inert leaving in the purge. The recycle stream carries the unreacted and inert material back to the reactor, so the reactor inlet is the sum of fresh feed and recycle.

If a perfect splitter sends a pure product to the bottoms, all of that product leaves the process through the bottoms and no product appears in the distillate. The reaction rate multiplied by reactor volume gives the generation of product inside the reactor, and at steady state this equals the product rate leaving the process.

When the feed and recycle are pure, and the reactor is operated at a fixed conversion, the total recycle flow can be expressed in terms of the product rate and the recycle ratio. Minimizing the sum of recycle flows then gives an optimal ratio at which an algebraic balance on an undesired species makes its net generation zero at steady state.

Arrhenius temperature dependence of rate constant

The rate constant follows the Arrhenius law k = A\exp(-E/RT), where E is the activation energy and A is the frequency factor. Between two temperatures, \ln(k_2/k_1) = (E/R)(1/T_1 - 1/T_2), which is the form used to extract E from two measured rate constants.

The temperature sensitivity of the rate is \dfrac{d\ln k}{dT} = \dfrac{E}{RT^2}. For two reactions with the same frequency factor, the one with the larger activation energy has the larger sensitivity at every temperature, and the sensitivity decreases as T rises.

Collision theory gives k = PZ\exp(-E/RT) with a collision frequency Z \propto T^{1/2}, so k \propto T^{1/2}\exp(-E/RT). The ratio of rate constants at two temperatures then includes the factor (T_2/T_1)^{1/2} in addition to the exponential term.

Tanks-in-series RTD model variance

The tanks-in-series model represents a non-ideal reactor by N equal ideal CSTRs in series. The dimensionless variance of the RTD is \sigma^2/\bar t^2 = 1/N, so N = \bar t^2/\sigma^2. Since the variance is inversely proportional to N, it decreases monotonically as the number of tanks increases. As N \to \infty the variance tends to zero, which corresponds to plug flow.

The model gives the N that matches a measured variance from a pulse tracer experiment, after which the conversion is computed from the tank equations. For a first-order reaction in N equal CSTRs with total space time \tau, X = 1 - (1 + k\tau/N)^{-N}.

Autocatalytic reaction in CSTR

In an autocatalytic reaction the product catalyzes its own formation, as in A + B \rightarrow 2B. The rate is proportional to C_A C_B, and the total concentration stays constant because one mole of A is replaced by one mole of B. Therefore C_B = C_{A0} - C_A, and the rate can be written as a function of C_A alone.

For an ideal CSTR, the design equation is \tau = \dfrac{C_{A0}X}{-r_A}, evaluated at the exit composition. Substituting the exit values of C_A and C_B into the rate expression gives the space time.

Competitive enzyme inhibition kinetics

In competitive inhibition the inhibitor I binds to the free enzyme at the same site as the substrate, so it reduces the rate by occupying enzyme molecules that would otherwise bind S. The apparent Michaelis constant becomes K_M(1 + [I]/K_I), while V_{max} is unchanged, because a high enough substrate concentration overcomes the inhibitor.