1 Introduction
bvp4c
employed to tackle the implicit two-point boundary value problem and describe how we tackled the coupled channel specie equation with a Euler ODE scheme. We provide the details of our approach to the cDPF model solution so that it can be easily replicated.2 Model Development
2.1 Filter Model
2.1.1 Species Continuity Balance in the Channel
2.1.2 Filter Wall Layer
2.1.3 Chemical Reactions
S/N | Chemical reaction | Reaction rate expression | Description |
---|---|---|---|
1 | S + NH3 → (NH3)S | \( {R}_1={k}_1{C}_{NH_3}\left(1-\theta \right) \) | NH3 adsorption |
2 | (NH3)S → S + NH3 | R2 = k2θ | NH3 desorption |
3 | 2(NH3)S + NO + NO2 → 2N2 + 3H2O + 2S | \( {R}_3={k}_3\theta {C}_{NO}{C}_{NO_2} \) | Fast SCR |
4 | 4(NH3)S + 4NO + O2 → 4N2 + 6H2O + 4S | R4 = k4θCNO | Standard SCR |
5 | 4(NH3)S + 3NO2 → 3.5N2 + 6H2O + 4S | \( {R}_5={k}_5\theta {C}_{NO_2} \) | Slow SCR |
6 | 2(NH3)S + 1.5O2 → N2 + 3H2O + 2S | \( {R}_6={k}_6\theta {C}_{O_2} \) | NH3 oxidation |
7 | 2(NH3)S + 2NO2 → N2 + N2O + 3H2O + 2S | \( {R}_7={k}_7\theta {C}_{NO_2} \) | Formation of N2O |
8 | NO + 0.5O2 ↔ NO2 | \( {R}_8={k}_8\left[\left({C}_{NO}\sqrt{C_{O_2}}\right)-\frac{C_{NO_2}}{K_c}\right] \) | NO-NO2 redox equilibrium |
S/N | Soot oxidation reaction | Reaction CO selectivity | Rate of reaction |
---|---|---|---|
S-1 | C + (1 − fCO/2)O2 → fCOCO + (1 − fCO)CO2 | \( {f}_{CO}=\frac{1}{1+{k}_f{y}_{O_2}^q{e}^{\raisebox{1ex}{${E}_f$}\!\left/ \!\raisebox{-1ex}{$ RT$}\right.}} \) | \( {R}_{s1}={S}_P{A}_{th}{Y}_{O_2}{T}^{x_{th}}{e}^{\left(-\raisebox{1ex}{${Ea}_{th}$}\!\left/ \!\raisebox{-1ex}{$ RT$}\right.\right)} \) |
S-2 | C + (2 − gCO)NO2 → gCOCO + (1 − gCO)CO2 + (2 − gCO)NO | \( {g}_{CO}=\frac{1}{1+{k}_g{y}_{NO_2}^{\mu }{e}^{\raisebox{1ex}{${E}_g$}\!\left/ \!\raisebox{-1ex}{$ RT$}\right.}} \) | \( {R}_{s2}={S}_P{A}_{NO_2}{Y}_{NO_2}{T}^{x_{NO_2}}{e}^{\left(-\raisebox{1ex}{${Ea}_{NO_2}$}\!\left/ \!\raisebox{-1ex}{$ RT$}\right.\right)} \) |
2.2 Model Solution
2.2.1 Overview
bvp4c
) coupled with an Euler ODE scheme. A fourth-order Runge-Kutta method is applied to solve the first-order ammonia surface coverage, soot mass balance and wall temperature equations in time. MATLAB is the solution environment.bvp4c
solver is a method of collocation utilising a continuously differentiable cubic polynomial function to approximate the problem within the solution domain. The Simpson’s method is implemented for evaluating the interpolating function at the collocation points, and the residual is used to control mesh size, implementation efficiency and solution accuracy [26]. The implementation is handled internally within the MATLAB environment, and the performance of the bvp4c
is shown to be superior to another BVP solver, MIRKDC
, which also uses residual control in its implementation [26]. We use MATLAB as our environment for bvp4c
for convenience. There are other software environments that implement the bvp4c
method, e.g. py_bvp
in Python [27].2.2.2 Coupled Channel-Wall Equations
bvp4c
MATLAB routine coupled with an Euler ODE scheme. To reduce the number of equations, the specie equations are only solved for the trace components (CO, CO2, NOx and NH3).2.2.3 Time-Explicit First-Order ODEs
ode45,
and the ammonia surface coverage is solved with ode15s
, both in the MATLAB environment. ode45
proved acceptable for the lumped parameter filter temperature dynamics; while the option of JPattern
[28] in the ode15s
routine enabled fast simulation of the surface coverage equation.2.2.4 Difference from Other Methods
3 Model Validation
3.1 Analytical Validation
3.1.1 Analytical Model and Solution
3.1.2 Results
3.2 Validation with Published Data
3.2.1 Schrade Data
Parameter | Value |
---|---|
Filter material | Silicon carbide |
Length [m] | 0.152 |
Diameter [mm] | 19.18a |
Cell density [cpsi] | 300 |
Wall thickness [mil] | 12 |
Material porosity [%] | 58 |
Mean pore size [μm] | 22 |
Catalytic coating [−] | Cu-zeolite |
Degreening | 2 hrs@600oC |
Parameter | NH3 TPD | SS NOx |
---|---|---|
Space velocity | 20 k/h | 20 k/h |
Feed gas composition | 250 ppm NH3, 5% H2O | 250 ppm NH3, 200 ppm NOx, 8% O2, 5% H2O with NO2/NOx = [0,1] |
Temperature | 150 °C | 200 °C |
Soot load | 0 g/l | 0 g/l |
3.2.2 Ammonia TPD Results
3.2.3 Steady-State NOx Results
4 Model Application
Parameter | Base case | Low case | High case | |
---|---|---|---|---|
1 | Diffusivity coefficient | 1 | 0.1 | 10 |
2 | Channel MTC | 1 | 0.1 | 10 |
3 | Temperature | 200 °C | 200 °C | 300 °C |
4 | Space velocity | 20 k/h | 20 k/h | 40 k/h |
5 | Monolith geometry | L, WT | L, WT | 0.5 L, 1.5 WT |
4.1 Internal vs. External Diffusion
Base case | Low case | High case | |
---|---|---|---|
External diffusion | 10−2 | 10−1 | 10−3 |
Internal diffusion | 10−3 | 10−2 | 10−4 |
Reaction | 10−1 | ||
System residence time | 10−1 |
NOx conversion | NH3 conversion | |
---|---|---|
Base case | 77 | 50 |
High EDC | 77 | 51 |
Low EDC | 74 | 49 |
High MTC | 79 | 52 |
Low MTC | 66 | 44 |
4.2 Compound Effects
4.2.1 Temperature
NOx conversion | NH3 conversion | |
---|---|---|
Base case T = 200 °C (“BC”) | 76.5 | 50.2 |
BC & T = 300 °C (“BCT”) | 99.6 (+30%) | 76.5 (+52%) |
BCT & 0.1Deff | 99.0 (−0.5%) | 74.8 (−2%) |
BCT & 10Deff | 99.7 (+0.1%) | 76.7 (+0.2%) |
BCT & 0.1km1,2 | 95.1 (−2%) | 74.8 (−4%) |
BCT & 10km1,2 | 99.8(+0.3%) | 76.0 (−0.6%) |
4.2.2 Space Velocity
4.2.3 Monolith Geometry
5 Conclusions
- Overall specie conversion is improved at higher temperature due to the system being reaction kinetics controlled. Temperature did not, however, significantly enhance or hinder the effect of diffusion on system deNOx performance.
- Overall specie conversion deteriorated at higher space velocity, due to the reduced residence time for reactions. The influence of external diffusion was more significant on specie conversion compared to internal diffusion when compounded with space velocity. Space velocity marginally amplified the effect of internal and external diffusion on species conversion in the high case, and marginally limited the effect of internal and external diffusion in the low case.
- A shorter and thicker-walled monolith with the same washcoat loading improved specie conversion. The change in monolith geometry limited the impact of diffusion except for the high internal diffusion case. It is believed that the increase in wall thickness supports the stronger diffusive transport in this case.