Simulating Ammonia Synthesis with COMSOL © COPYRIGHT 2014, COMSOL

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.

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)

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 2 3 𝑎 𝑁 𝐻 3 2 ) 𝛼 −( 𝑎 𝑁𝐻 3 2 𝑎 𝐻 2 3 ) 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∙ 10 15 exp −40.765 𝑘𝑐𝑎𝑙 𝑚𝑜𝑙 𝑅𝑇 𝑚𝑜𝑙 𝑙𝑏∙ℎ

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 =0.93431737+0.3101804∙ 10 −3 𝑇+0.295896∙ 10 −3 𝑃−0.2707279∙ 10 −6 𝑇 2 +0.4775207∙ 10 −6 𝑃 2 𝜑 𝑁𝐻 3 =0.1438996+0.2028538∙ 10 −2 𝑇+0.4487672∙ 10 −3 𝑃−0.1142945∙ 10 −5 𝑇 2 +0.2761216∙ 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 −3.8402 𝑇 0.125 +0.541 𝑃− 𝑒 −0.1263 𝑇 0.5 −15.98 𝑃 2 +300( 𝑒 −0.011901𝑇−5.941 ) 𝑒 −𝑃 300 }

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 -17.53096 0.07696849 6.900548 -1.08279·10-4 -26.42469 4.927648·10-8 38.937 225 -8.2125534 0.03774149 6.190112 -5.35457·10-5 -20.86963 2.379142·10-8 27.88 300 -4.6757259 0.02354872 4.687353 -3.46330·10-5 -11.28031 1.540881·10-8 10.46 1. Rase, H. F: Chemical Reactor Design for Process Plants, Volume 2: Case Studies and Design Data, 1977

Model Description The equilibrium constant K follows a strongly non-linear behavior in temperature: log 𝐾 𝑎 =−2.691122 log 𝑇 −5.051925∙ 10 −5 𝑇+1.848863∙ 10 −7 𝑇 2 + 2001.6 𝑇 +2.689 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.

Model Description The heat of reaction is also a non-linear function of the temperature based on experimental measurements: 𝐾=19.99563𝑇+ 49.77119 𝑇 2 2 − 15.37599 𝑇 3 3 + 1.921168 𝑇 4 4 − 0.189174 𝑇 +38.51 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 15.72 ft rand the diameter is 7.05 ft Initial concentrations and temperatures are taken from Ref. 1

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

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

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