Handling of Equilibrium Reactions
Example I
The following short example illustrates how the Reaction Engineering interface and the Chemistry interface handle equilibrium reactions in the formulation of the material balance equations.
Consider the reaction:
(2-30)
According to Equation 2-2 the reaction rate (SI unit: mol/(m3·s)) is formulated as:
where cA and cB (SI unit: mol/m3) are the concentrations of A and B, and kf and kr (SI unit: 1/s) are the forward and reverse rate constants, respectively. The net rate expressions with respect to species A and B are then:
Assuming the reaction in Equation 2-30 is at equilibrium, the reaction rate r is 0:
The relationship between the forward and reverse reaction rates in Equation 2-30 is given by the following ratio:
(2-31)
The Reaction Engineering interface also sets up mass balances that are solved. The general material balances for species A and B, respectively, are:
(2-32)
(2-33)
The rate of consumption of species A equals the production rate of species B, as shown in Equation 2-32 and Equation 2-33.
With the combined information in Equation 2-32, Equation 2-33, and Equation 2-31, the Reaction Engineering interface is able to define the mass balances for the equilibrium system without the reaction rate expressions. The equation system solved for becomes:
(2-34)
(2-35)
In general, for a system of reactions contributing to k mass balances and with j reactions being at equilibrium, the reduced system of equations to be solved is composed of k – j mass balances and j equilibrium expressions. The elimination process producing the above system of equations is automated, allowing simple modeling of chemical equilibrium reactions together with irreversible and/or reversible reactions.
Example II
This example shows how equilibrium reactions are considered in the Reaction Engineering interface using the Equilibrium Species Vector section.
If two non-equilibrium reactions are taking place in a perfectly mixed isothermal reactor of constant volume:
(2-36)
the corresponding mass balances are:
(2-37)
(2-38)
(2-39)
Now compare Equation 2-37, Equation 2-38, and Equation 2-39 with the balance equations that the physics interface sets up for the related chemistry, where the second reaction is instead an equilibrium reaction:
(2-40)
(2-41)
In contrast to the reversible reaction given by Equation 2-36, to make use of the information contained in the equilibrium relation, the mass balances must be reformulated. Mass balances set up for the reactions given by Equation 2-40 and Equation 2-41 are then:
(2-42)
(2-43)
The equilibrium expression (Equation 2-43) introduces an algebraic relationship between the species’ concentrations.
Two species, B or C, can be set as Predefined Dependent Species in the Equilibrium Species Vector section. Selecting B as the dependent species solves Equation 2-42 for the concentration of C, while B is computed from Equation 2-43.