1. Simulating Ammonia Synthesis with COMSOL
© COPYRIGHT 2014, COMSOL

2. Introduction
Model of the Haber-Weiss process for the synthesis of ammonia from nitrogen and Hydrogen according to:
N2 + 3H2 → 2NH3
The reaction rate is a nonlinear function of the reactant and catalyst concentrations, temperature, and pressure
In addition to the active reaction participants, argon (Ar) and methane (CH4) make up the balance of the reactant and product streams
The ability to accurately simulate large-scale chemical processes accurately is vital to achieving an environmentally friendly, energy efficient and economically viable chemical process. Arguably one of the most important of these processes, is the so-called Haber-Weiss process for the synthesis of ammonia (NH3) from nitrogen (N2) and Hydrogen (H2).
While the actual balance equation appears simple, the thermodynamic and kinetic analysis of ammonia synthesis remains one of the most challenging and best-studied examples of non-ideal reactor behavior. Not only is the reaction rate a function of the reactant and catalyst concentrations, temperature and pressure influence the time-dependent concentrations in a strongly non-linear way. In addition to the active reaction participants, argon (Ar) and methane (CH4) make up the balance of the reactant and product streams.

3. Introduction This example shows how to:
Import thermodynamic data into COMSOL,
Create complex, non-linear functions for activities, reaction rates, and the equilibrium constant
Create useful interpolations to determine the efficiency of the catalyst
Set up the simulation of a cooled plug flow reactor (PFR)

4. Model Description
The formation of ammonia from hydrogen and nitrogen may be described by:
𝑑𝑁 𝐻 3 𝑑𝑡 = 𝑟 𝑁 𝐻 3
𝑟 𝑁 𝐻 3 =2𝑘[ 𝐾 𝑎 2 𝑎 𝑁 2 ( 𝑎 𝐻 𝑎 𝑁 𝐻 ) 𝛼 −( 𝑎 𝑁𝐻 𝑎 𝐻 ) 1−𝛼 ]
𝑟 𝑁 𝐻 3 being the formation rate, k the rate constant, Ka the equilibrium constant, and ai the activity of species i
The rate constant k is calculated using an Arrhenius equation:
𝑘=1.7698∙ exp − 𝑘𝑐𝑎𝑙 𝑚𝑜𝑙 𝑅𝑇 𝑚𝑜𝑙 𝑙𝑏∙ℎ

5. Model Description
The activities for the various species can be obtained from the fugacity coefficients φ, the pressure P and the mole fraction of species i xi via: 𝑎 𝑖 = 𝑥 𝑖 𝜑 𝑖 𝑃
The fugacity coefficients are semi-empirical correlations:
𝜑 𝐻 2
𝜑 𝑁 2 = ∙ 10 −3 𝑇 ∙ 10 −3 𝑃− ∙ 10 −6 𝑇 ∙ 10 −6 𝑃 2
𝜑 𝑁𝐻 3 = ∙ 10 −2 𝑇 ∙ 10 −3 𝑃− ∙ 10 −5 𝑇 ∙ 10 −6 𝑃 2
The reaction rate has to be scaled with an effectiveness factor, η, to account for the non-uniform concentration in the porous structure:
𝑟 𝑒𝑓𝑓 = 𝑟 𝑁𝐻 3 ∙𝜂
=exp⁡{ 𝑒 − 𝑇 𝑃− 𝑒 − 𝑇 0.5 − 𝑃 ( 𝑒 − 𝑇−5.941 ) 𝑒 −𝑃 300 }

6. Model Description
η may be obtained via interpolation of fixed points from an expression involving both the temperature T and the conversion X to:
𝜂= 𝑏 0 + 𝑏 1 𝑇+ 𝑏 2 𝑋+ 𝑏 3 𝑇 2 + 𝑏 4 𝑋 2 + 𝑏 5 𝑇 3 + 𝑏 6 𝑋 3
The b parameters were measured at various pressures in Ref. 1, see below
COMSOL can interpolate between these values using a cubic spline
P/atm b0 b1 b2 b3 b4 b5 b6
150
·10-4
·10-8
38.937
225
·10-5
·10-8
27.88
300
·10-5
·10-8
10.46
1. Rase, H. F: Chemical Reactor Design for Process Plants, Volume 2: Case Studies and Design Data, 1977

7. Model Description
The equilibrium constant K follows a strongly non-linear behavior in temperature:
log 𝐾 𝑎 =− log 𝑇 − ∙ 10 −5 𝑇 ∙ 10 −7 𝑇 𝑇
The heat generated throughout this reaction is significant
A good source of data for this sort of a temperature-dependent function is the NIST database
The values can be saved to text for a wide range of temperatures, allowing the quick and easy import of this data into COMSOL.

8. Model Description
The heat of reaction is also a non-linear function of the temperature based on experimental measurements:
𝐾= 𝑇 𝑇 2 2 − 𝑇 𝑇 4 4 − 𝑇
A cooling jacket must be included in the calculations using:
𝑄=𝑈 Δ𝑇 𝐿
Where Q is the heat removed, U is the constant heat transfer coefficient, ΔT the temperature difference between the coolant stream and the reactor, and L the reactor length.
The reactor length is ft rand the diameter is 7.05 ft
Initial concentrations and temperatures are taken from Ref. 1

9. Results and Discussion
The concentration profile for hydrogen, nitrogen and ammonia are shown in figure 1
The seemingly low final concentration of ammonia in the reactor is in fact due to the exothermic nature of the reaction and the subsequent influence on the behavior of the equilibrium constant
The exothermic behavior can be seen in figure 2
The results thus obtained satisfactorily match the cited source in Ref. 1

10. Results and Discussion
Figure 1: Mole fractions for hydrogen (green), nitrogen (blue), and ammonia (red) as a function of reactor extent.

11. Results and Discussion
Figure 2: Temperature profile within the ammonia conversion reactor.