Defining Reaction Rate Laws in OLI
The Arrhenius Reaction Rate Equation
Using Arrhenius Equation to Determine Rate Constants
Modeling Reaction Progress: Extent of Reaction
Objective
At its core, OLI is a thermodynamics-based software in which all chemical reactions are assumed to proceed until equilibrium is reached. In scenarios where this is not practical due to extended reaction times, OLI enables users to model the speed of the chemical reaction. This guide will elucidate the theory and variables for modeling Arrhenius reaction kinetics in OLI software, applicable to both OLI Studio and OLI Flowsheet: ESP.
Defining Reaction Rate Laws in OLI
In OLI, you can select from two types of reaction rate laws:
- Standard Rate Law: This is based on the Arrhenius equation for the reaction rate constant. It is referred to as “Arrhenius” in Flowsheet: ESP and “STD” in Studio.
- Non-Standard Rate Law. This allows users to customize the rate equation and variables. It is referred to as “User Defined” in Flowsheet: ESP and “SPEC” in Studio.
For this article, we will focus on the Arrhenius-type Standard Rate Law.
The Arrhenius Reaction Rate Equation
Let us consider a generic reaction
aA + bB -> cC + dD
- A and B are reactants
- C and D are products
- The letters a, b, c, and d represent the stoichiometric coefficients.
Batch System in OLI
First, we will consider a batch system (non-flowing), which is only applicable to OLI Studio. This describes streams for which inflows are without a time basis.
A reaction rate measures a change in the amount of reactants or products over a change in time. In OLI, the unit for a rate corresponding to a batch system is mol/hr.
OLI’s generic expression for the Standard Rate Law is:
Rate = [k_{f} (a_{A}^{r1})(a_{B}^{r2}) - k_{r} (a_{C}^{p1})(a_{D}^{p2})] * Vol
where:
- kf = forward reaction rate constant
- kr = reverse reaction rate constant
- ai = activity of species i
- r1, r2 = reaction order of reactant 1 and 2, respectively
- p1, p2 = reaction order of product 1 and 2, respectively
- Vol = Liquid product volume (m3)
Variable Units
Since the final unit on Rate must be mol/hr and activities are always unitless, the units for k_{f} and k_{r} must be mol/(hr · m^{3}). This contrasts with the common formalism of rate constant units, which depend on reaction order. In OLI, however, the units on k_{f} and k_{r }are independent of reaction order, because the rate law uses activities rather than concentrations. These units are summarized in Table 1.
Standard Rate Law Variable | Description | Units |
Rate | Final reaction rate | mol/hr |
k_{f} | Forward reaction rate constant | mol/(hr · m^{3}) |
k_{r} | Reverse reaction rate constant | mol/(hr · m^{3}) |
a_{i} | Activity of species i | Unitless |
r_{1}, r_{2}, ... | Reaction order of reactant 1, 2, … respectively | Unitless |
p_{1}, p_{2}, … | Reaction order of product 1, 2, … respectively | Unitless |
Vol | Liquid product volume | m^{3} |
Table 1. Variables in the Standard Rate Law for a batch system.
Using Arrhenius Equation to Determine Rate Constants
The Arrhenius Equation, which defines a temperature dependency, is used to determine the rate constant:
k = A exp ( − E_{a} / RT )
where:
- k = reaction rate constant
- A = Arrhenius frequency factor (same units as k)
- E_{a} = activation energy (J/mol)
- R = universal gas constant (8.314 J/(mol · K))
- T = temperature (K)
The Kinetic Variables in OLI
In OLI, users can define kinetics using variables derived from the Arrhenius Equation and Standard Rate Law. These are outlined in Table 2.
Figure 1. Dialog box for entering Arrhenius kinetic variables in OLI Flowsheet: ESP.
OLI Variable | Description | Units |
KF | Forward reaction rate constant | mol/(hr · m^{3}) |
KR | Reverse reaction rate constant | mol/(hr · m^{3}) |
AF | Forward reaction Arrhenius frequency factor | mol/(hr · m^{3}) |
AR | Reverse reaction Arrhenius frequency factor | mol/(hr · m^{3}) |
BF | Forward reaction activation energy divided by R | K |
BR | Reverse reaction activation energy divided by R | K |
ER_{i} | Reaction order of reactant species i | Unitless |
EP_{i} | Reaction order of product species i | Unitless |
Table 2. Variables to enter in the OLI kinetics entry window.
Note: the first reactant listed in the reaction (in our example, A) is taken to be reactant 1, the second reactant (B) is reactant 2, and so on. Similarly, the first product (C) is product 1, the second product (D) is product 3, and so on.
A user does not need to define every variable. Instead, they must define one of the following:
- AF, AR and BF, BR -- or --
- KF, KR
The software assumes a default value for undefined variables. The reaction rate constants are assumed to be zero and the species’ reaction order coefficients are assumed to be the reaction stoichiometric values. Additionally, the reaction temperature and initial reactant molality defined in the stream are employed in the kinetics calculation.
Flowing System in OLI
In OLI Flowsheet: ESP, all streams are constructed on a flowing (per unit time) basis. Flowing streams can also be constructed in OLI Studio.
The underlying rate equation for the Standard Rate Law in OLI is the same, but volume is expressed on a time basis (m^{3}/hr).
In turn, the unit on the final rate will be (mol/hr)/hr.
Standard Rate Law Variable | Description | Units |
Rate | Final reaction rate | (mol/hr)/hr |
k_{f} | Forward reaction rate constant | mol/(hr · m^{3}) |
k_{r} | Reverse reaction rate constant | mol/(hr · m^{3}) |
a_{i} | Activity of species i | Unitless |
r_{1}, r_{2}, ... | Reaction order of reactant 1, 2, … respectively | Unitless |
p_{1}, p_{2}, … | Reaction order of product 1, 2, … respectively | Unitless |
Vol | Liquid product volume | m^{3}/hr |
Table 3. Variables in the Standard Rate Law for a flowing system.
However, this discrepancy does not affect the kinetic variables a user enters in the software; the input variables remain the same as in a batch system.
Modeling Reaction Progress: Extent of Reaction
The extent of reaction refers to the amount of input material converted over a given time:
Extent = Rate * Time
For a batch system, the extent is measured in moles. For a flowing system, the extent is measured in moles/hr. This is consistent with the inflow units provided by the user.
The extent is equivalent to the area under a Time vs Rate curve, which is an integral. In OLI, the integral is evaluated based on two calculation parameters:
- Residence time (hold-up time): the amount of time the inflows are given to react
- Number of stages (kinetic steps): the number of discrete intervals for which to calculate the area.
These parameters allow the software to approximate the integral by summing the area of all the discrete intervals.
For example, if the residence time is 1 hour and the rate is evaluated over 1 stage (Fig 2a), the approximation is less accurate than with several stages (Fig 2b). As the number of stages increases, so does computation time.
(a) (b)
Fig 2. (a) Time vs rate curve with 1 hour residence time and 1 stage; (b) Time vs rate curve with 1 hour residence time and 5 stages.
Conclusion
In this article, we reviewed the underlying equation for Arrhenius reaction rates and described how this translates to the kinetic variables in OLI. This imparts users with flexibility to model real-world processes where equilibrium may not be attainable in practical timeframes. Further articles will guide users through specific examples in OLI Studio and OLI Flowsheet: ESP.