Batch Reactors

This chapter presents a quantitative description of batch reactors. It also “derives” the variables which control a chemical reaction conducted in a batch reactor. This chapter includes a discussion about specifying the proportionality constant, i.e., the reaction rate constant, which quantifies the relationship between chemical reaction rate and concentration. This chapter uses the saponification of ethyl acetate to demonstrate the various procedures for specifying the reaction rate constant.

Keywords

Batch reactor; chemical reaction rate; reaction rate constant; reaction order; initial reaction rate; initial reactant concentration

Quantitative Description of Batch Reactors

Consider the typical batch reactor, which in the laboratory is a spherical glass container with ports for reagent addition and sample removal and possibly a centerline port accommodating an agitator shaft driven by an elevated, i.e., overhead, electric motor. An alternate method of mixing is to place a magnetic stirring bar in the reactor and drive it via a rotating magnet placed beneath the reactor.

We generally do not conduct gas-phase reactions in batch reactors. For a constant liquid volume, constant density reaction or process, the component balance for a laboratory batch reactor is

[A]∂t+(vr∂[A]∂r+vθr∂[A]∂θ+vφrsinθ∂[A]∂φ)=DAB(1r2∂∂r(r2∂[A]∂r)+1r2sinθ∂∂θ(∂[A]∂θ)+1r2sin2θ∂2[A]∂φ2)+RA

where [A] is the concentration of component A (mol/m3); t is the time (seconds, s); r is the radial direction, generally taken from the geometric center point of the spherical reactor (m); θ is the polar angle taken from the vertical reference axis to the point in question (dimensionless); φ is the azimuthal angle or longitude taken around the reference axis to the point in question (dimensionless); vr is the velocity in the radial direction (m/s); vθ is the polar angular velocity (m/s); vφ is the azimuthal velocity (m/s); DAB is the diffusivity (m2/s) of component A in solvent B, if a solvent B is present; and, RA is the formation of A or the consumption of A by chemical reaction (mol/m3*s).

We assume the concentration of component A in the r, θ, and φ directions is constant due to agitation and the absence of directed convection1; therefore

vr∂[A]∂r+vθr∂[A]∂θ+vφrsinθ∂[A]∂φ)=0

If the agitator maintains continuous, thorough mixing of the reactor’s contents, then we assume molecular diffusion to be negligible during the chemical reaction; thus

AB(1r2∂∂r(r2∂[A]∂r)+1r2sinθ∂∂θ(∂[A]∂θ)+1r2sin2θ∂2[A]∂φ2)=0

The component balance for A therefore reduces to

d[A]dt)Spherical=RA

At a production facility, the typical batch reactor is a circular cylinder mounted vertically in a support structure. We generally weld hemispheric caps to the top and bottom of the cylinder. A vertical agitator shaft extends through the top cap of the cylinder. A bearing seal surrounds the vertical shaft as it passes through the top cap, thereby isolating the reactor’s contents from the environment. An electric motor powers the agitator shaft via a gearbox. Various nozzles with block valves penetrate the top cap; liquid feeds enter the reactor through these nozzles. The finished product exits the reactor through a valve generally placed at the center of the bottom cap. The component balance for such a reactor is

[A]∂t+(vr∂[A]∂r+vθr∂[A]∂θ+vz∂[A]∂z)=DAB(1r∂∂r(r∂[A]∂r)+1r2∂2[A]∂θ2+∂2[A]∂z2)+RA

where [A] is the concentration of component A (mol/m3); t is the time (s); r is the radial direction, generally taken from the geometric center line of the cylindrical reactor (m); z is the vertical axis of the cylinder (m); θ is the azimuthal angle or longitude taken around the z axis (dimensionless); vr is the velocity in the radial direction (m/s); vz is the velocity in the axial direction (m); vθ is the azimuthal velocity (m/s); DAB is the diffusivity (m2/s) of component A in solvent B; and, RA is the formation of A or the consumption of A by chemical reaction (mol/m3*s).

As for the laboratory batch reactor, we assume the concentration of component A in the r, θ, and z directions is constant due to agitation and the absence of directed convection.1 Therefore

vr∂[A]∂r+vθr∂[A]∂r+vz∂[A]∂z)=0

If the agitator maintains continuous, thorough mixing of the reactor’s contents, then we assume molecular diffusion to be negligible during the chemical reaction; thus

1r∂∂r(r∂[A]∂r)+1r2∂2[A]∂θ2+∂2[A]∂z2)=0

The component balance for a commercial-sized circular, cylindrical batch reactor reduces to

d[A]dt)Cylinder=RA

We can make two statements with regard to

d[A]dt)Cylinder=RAand(d[A]dt)Spherical=RA

First, neither rate depends upon a geometric variable, so long as the concentrations and reaction temperature in the cylindrical reactor equal the concentrations in the spherical reactor, which is generally the case for liquid-phase reactions. Thus, upscaling and downscaling a chemical reaction within a batch reactor is relatively straightforward, at least for constant volume, constant density, liquid-phase reactions.2 If the reaction volume changes or the density of the contained liquid changes during reaction, then we must consider geometric similarity during upscaling and downscaling. Second, both rates depend upon RA.

What Is RA?

To upscale or downscale a batch reactor, we need to know upon what RA depends. Consider the generalized chemical reaction

*Reactants→p*Products

where r and p are the stoichiometric quantities for Reactants and Products, respectively. We know from thermodynamics that, when a chemical reaction reaches equilibrium, the rate of product formation from reactants equals the rate of reactant formation from products. Mathematically

=kForwardkReverse

where K is the equilibrium constant for the chemical reaction; kForward is a constant describing the rate of product formation from reactants; and, kReverse is a constant describing the rate of reactant formation from products. We also know from thermodynamics that

=[Products]p[Reactants]r

ForwardkReverse=[Products]p[Reactants]r

or

Forward[Reactants]r=kReverse[Products]p

Note that for Forward[Reactants]r, kForward has units of m3/mole)r−1(1/s). Therefore, if r=1, then

m3/mole)r−1(1/s)=(m3/mole)1−1(1/s)=(m3/mole)0(1/s)=(1/s)

The same is true for Reverse[Products]p. In other words, Forward[Reactants]r and Reverse[Products]p have the same units as RA, which leads to our equating them. Thus, RA in the forward reaction direction must be a function of reactant concentration while in the reverse reaction direction it must be a function of product concentration. The kinetic theory of gases also indicates that the collision of gas molecules and their resultant reaction with each other depends upon the concentration of each component gas.3

Note that r and p in the above equation are the stoichiometric quantities for the reaction after it reaches equilibrium. However, chemical reaction rate studies, i.e., kinetic studies, do not investigate the equilibrium condition of chemical reactions; thermodynamics investigates the equilibrium condition of a chemical reaction. Kinetics investigates the rate at which a chemical reaction reaches its equilibrium condition. With regard to kinetics, our strongest assertion can only be

[Reactant]dt=−kForward[Reactant]x

or

[Product]dt=−kReverse[Product]y

where x and y are unknowns not necessarily related to the stoichiometric quantities r and p of the reaction.

The above reaction rate equations indicate that we are eventually going to integrate [Reactant] from [Reactant]t=0 at t=0, when we start the chemical reaction, to [Reactant]t at time t, when we stop the chemical reaction. That integration gives us

Reactant]t−[Reactant]t=0

which will be negative since

Reactant]t<<[Reactant]t=0

But, for chemical reactions, rates of change are positive. Therefore, we insert the negative sign in the above equation to obtain a positive rate of change. We interpret the negative sign as denoting consumption of a chemical species; we interpret a positive sign as denoting formation of a chemical species.

Thermodynamics also indicates how the equilibrium constant depends on changing temperature. The Clausius–Clapeyron...