# Developing Computational Fluid Dynamics (CFD) Models to Evaluate Available Energy in Exhaust Systems of Diesel Light-Duty Vehicles

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{3}H

_{6}under oxygen-rich conditions for a better understanding of platinum oxidation catalysts in automotive emission control systems. Terms accounting for CO, C

_{3}H

_{6}and NO inhibition were included. The isothermal reactor approach in a numerical integration-optimization method was used to fit the kinetic parameters. These expressions, especially as written by Oh et al. [18], have been widely used [19,20,21,22]. This approach has also been used to build simpler kinetic models over a range of interest [23].

_{3}H

_{6}oxidation on Pt/Al

_{2}O

_{3}, a mechanism of NO reduction on Pt, and a mechanism for NO reduction and CO oxidation on Rh. This model has been used when more accuracy is required, as in Kumar et al. [25].

## 2. Materials and Methods

#### 2.1. Materials

#### 2.1.1. Engine

#### 2.1.2. Measurement Devices

_{x}analyser, based on the chemiluminescence effect from NO oxidation by ozone (O

_{3}). The GRAPHITE 52M gas analyser measures total hydrocarbons (THCs) by flame ionization detection while a MIR 2R gas analyser measures CO and CO

_{2}species, detecting the molecules absorption in the infrared spectrum. Other species compositions are estimated via chemical balances from fuel and air consumption.

#### 2.2. Methods

#### 2.2.1. Test Plan

#### 2.2.2. Experimental Characterization of Model Input Parameters

#### 2.2.3. Three-Dimensional CFD Model

^{−6}for the thermal energy and chemical residuals and 10

^{−4}for residuals from mass, momentum, turbulence kinetic energy and turbulence energy dissipation rate. Relevant physical and chemical quantities were monitored to assure convergence.

^{5}elements grid with an average mesh size of 3 × 10

^{−3}m.

## 3. Results

#### 3.1. Pressure Loss Coefficient

^{−8}m

^{2}.

#### 3.2. Convection Heat Transfer Coefficient

#### 3.3. Kinetic Parameters

#### 3.4. Chemical Results

#### 3.5. Temperature and Heat Results

#### 3.6. Flow Temperature Distribution at the DOC Outlet

#### 3.7. Temperature Loss along the Exhaust Pipe

## 4. Discussion

#### 4.1. Accuracy of the Model

_{3}H

_{6}model, respectively. Notice also that at relative low temperature modes with very low conversion rates, the model shows a 0% conversion. This is due to exclusion of near-to-zero points while developing the kinetic model, in order to better predict the whole operating range. Similarly, the CO kinetic model tends to show near total conversion for all high conversion modes, since these points were excluded when developing it for the same reason as above.

#### 4.2. Energy Recovery Considerations for Exhaust Systems

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Nomenclature

$A$ | Pre-exponential factor (m^{3}/(mol·s)) |

$B$ | Pressure drop linear fit constant (m^{2}) |

$C$ | Reaction rate linear fit constant |

$D$ | Reaction rate linear fit constant (K) |

${c}_{p}$ | Specific heat at constant pressure (J/(kgK)) |

$d$ | Diameter (m) |

$F$ | External body forces (N) |

$g$ | Gravitational acceleration (m/s^{2}) |

$G$ | Inhibition factor (K) |

$h$ | Convection heat transfer coefficient (W/m^{2}K) |

$J$ | Diffusion term (kg/m^{2}s) |

$k$ | Arrhenius kinetic constant (m^{3}/(mol·s)) |

$K$ | Thermal conductivity (W/mK) |

$L$ | Length (m) |

$\dot{m}$ | Mass flow (kg/s) |

$p$ | Pressure (Pa) |

$\dot{Q}$ | Heat loss (W) |

$R$ | Reaction rate (mol/m^{3}s) |

$S$ | Source term |

$T$ | Temperature (°C, K) |

$u$ | Vector component of velocity (m/s) |

$x$ | Cartesian coordinate (m) |

$X$ | Horizontal axis variable |

$Y$ | Vertical axis variable |

Greek | |

$\Delta $ | Variation |

$\beta $ | Inertial forces coefficient (m) |

$\epsilon $ | Surface material emissivity |

$\kappa $ | Darcy’s constant (m^{2}) |

$\mu $ | Dynamic viscosity |

$\rho $ | Density (kg/m^{3}) |

$\sigma $ | Stefan-Boltzmann constant (W/(m^{2}K^{4})) |

Subscripts | |

$\infty $ | Ambient |

$eff$ | Effective |

$g$ | Exhaust gas |

$i$ | Cartesian coordinate |

$in$ | Inlet |

$out$ | Outlet |

$m$ | Species |

$S$ | Solid |

$T$ | Thermal |

$wall$ | Pipe wall |

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**Figure 1.**View of the engine test bench including (

**a**) the exhaust system pipe and (

**b**) the diesel oxidation catalyst (DOC).

**Figure 2.**Sketch of the exhaust with measurement points involved in the development of the model. The exhaust system is 4 m long and pipe has a diameter of 5 cm (figure not to scale).

**Figure 4.**Example of infrared images of the exhaust line in (

**a**) the exhaust pipe and (

**b**) the DOC. Temperatures in the scale are in °C. Squares seen in (

**a**) are wall temperature measurement points.

**Figure 5.**Geometry of the 2D axisymmetric DOC model for the tuning process of kinetic parameters. Darker shade of grey represents the monolith.

**Figure 7.**Mesh detail. Tetrahedral (DOC inlet and outlet cones) and hexahedral zones (monolith and pipe) can be distinguished.

**Figure 8.**Results of test runs with different wall functions. No great disagreement between the three approaches was obtained.

**Figure 10.**Plot for (

**a**) CO kinetic parameters derivation. Linear fit: C

_{CO}= 21.261, D

_{CO}= 5685.10 K (

**b**) C

_{3}H

_{6}kinetic parameters derivation Linear fit: C

_{C3H6}= 14.647, D

_{C3H6}= 2388.55 K.

**Figure 11.**Streamlines and inlet velocity distribution in the monolith. An example of usual bad flow uniformity caused by inlet cones in automotive catalysts can be seen.

**Figure 12.**Temperature distribution for the 2400 rpm–110 Nm mode in (

**a**) monolith, outlet cone and exhaust pipe and (

**b**) cross-section of DOC outlet (before exhaust pipe).

**Figure 13.**Plot for coefficient of variation in temperature distribution at DOC outlet. White dots represent engine modes in which chemical processes were not active.

Parameter | Value |
---|---|

Gradient | Least-squares cell-based |

Pressure | Standard |

Momentum | Second order upwind |

Species conservation | |

Turbulent kinetic energy |

Boundary | Type |
---|---|

Inlet | Mass-flow inlet |

Outlet | Pressure-outlet |

Walls | Wall with convection heat transfer |

Monolith | Anisotropic porous media |

Porosity: 0.74 | |

Viscous pressure loss coefficient: 5.9 × 10^{7} m^{−2} |

**Table 3.**Measured temperatures, mass flow values and outlet pressure. These values have been employed as boundary conditions for the model.

Engine Speed (min^{−1}) | 1000 | 1700 | 2400 | ||||||
---|---|---|---|---|---|---|---|---|---|

Engine Torque (Nm) | 10 | 60 | 110 | 10 | 60 | 110 | 10 | 60 | 110 |

Exhaust mass flow (kg/s) | 0.017 | 0.019 | 0.020 | 0.023 | 0.027 | 0.041 | 0.031 | 0.045 | 0.066 |

T_{gas,in} (°C) | 131.0 | 244.3 | 385.9 | 164.0 | 302.0 | 363.7 | 199.1 | 305.4 | 348.9 |

Outlet relative pressure (Pa) | 70 | 220 | 370 | 330 | 560 | 1690 | 720 | 1940 | 4540 |

**Table 4.**Modeled species and volumetric fraction range in gas inlet composition within the test matrix.

Engine Speed (min^{−1}) | 1000 | 1700 | 2400 | ||||||
---|---|---|---|---|---|---|---|---|---|

Engine Torque (Nm) | 10 | 60 | 110 | 10 | 60 | 110 | 10 | 60 | 110 |

${N}_{2}$ | 7.8 × 10^{−1} | 7.6 × 10^{−1} | 7.4 × 10^{−1} | 7.8 × 10^{−1} | 7.6 × 10^{−1} | 7.6 × 10^{−1} | 7.7 × 10^{−1} | 7.7 × 10^{−1} | 7.6 × 10^{−1} |

${O}_{2}$ | 1.7× 10^{−1} | 1.3 × 10^{−1} | 7.0 × 10^{−2} | 1.6 × 10^{−1} | 1.1 × 10^{−1} | 9.6 × 10^{−2} | 1.7 × 10^{−1} | 1.2 × 10^{−1} | 1.1 × 10^{−1} |

${H}_{2}O$ | 2.2 × 10^{−2} | 5.1 × 10^{−2} | 8.5 × 10^{−2} | 3.2 × 10^{−2} | 6.4 × 10^{−2} | 7.0 × 10^{−2} | 2.4 × 10^{−2} | 5.5 × 10^{−2} | 6.0 × 10^{−2} |

$C{O}_{2}$ | 2.2 × 10^{−2} | 5.7 × 10^{−2} | 9.9 × 10^{−2} | 3.1 × 10^{−2} | 6.7 × 10^{−2} | 7.0 × 10^{−2} | 3.5 × 10^{−2} | 5.6 × 10^{−2} | 6.6 × 10^{−2} |

$NO$ | 1.6 × 10^{−4} | 7.2 × 10^{−4} | 1.2 × 10^{−3} | 1.3 × 10^{−4} | 3.0 × 10^{−4} | 7.4 × 10^{−4} | 1.3 × 10^{−4} | 2.5 × 10^{−4} | 4.3 × 10^{−4} |

$CO$ | 2.2 × 10^{−4} | 1.1 × 10^{−4} | 1.6 × 10^{−4} | 4.1 × 10^{−4} | 2.0 × 10^{−4} | 1.9 × 10^{−4} | 5.9 × 10^{−4} | 3.8 × 10^{−4} | 1.7 × 10^{−4} |

${C}_{3}{H}_{6}$ | 8.8 × 10^{−5} | 5.4 × 10^{−5} | 9.4 × 10^{−5} | 1.3 × 10^{−4} | 6.7 × 10^{−5} | 5.0 × 10^{−5} | 1.5 × 10^{−4} | 9.0 × 10^{−5} | 5.4 × 10^{−5} |

Case | Transferred Heat (W) | Number of Mesh Cells |
---|---|---|

Without boundary layer refinement | 455.7 | 4 × 10^{5} |

With boundary layer refinement | 459.5 | 2.1 × 10^{6} |

**Table 6.**Experimentally-derived effective convection heat transfer coefficients for each operating mode in test matrix.

Engine Speed (min^{−1}) | Torque (Nm) | ${\overline{\mathit{h}}}_{\mathit{e}\mathit{f}\mathit{f}}$ (W/m^{2}K) |
---|---|---|

1000 | 10 | 11.1 |

1000 | 60 | 14.6 |

1000 | 110 | 17.9 |

1700 | 10 | 13.3 |

1700 | 60 | 15.8 |

1700 | 110 | 17.9 |

2400 | 10 | 14.1 |

2400 | 60 | 16.7 |

2400 | 110 | 18.6 |

${A}_{CO}$(m^{3}/(mol·s) | 3.42 × 10^{6} |

${E}_{CO}$(m^{3}/(mol·s) | 4.73 × 10^{4} |

${A}_{{C}_{3}{H}_{6}}$(m^{3}/(mol·s) | 2.30 × 10^{3} |

${E}_{{C}_{3}{H}_{6}}$(J/mol) | 1.98 × 10^{4} |

Engine Speed (min^{−1}) | Torque (Nm) | Measured CO Conversion (%) | Modelled CO Conversion (%) | Error in CO Conversion (%) | C_{3}H_{6} Conversion (%) | Modelled C_{3}H_{6} Conversion (%) | Error in C_{3}H_{6} Conversion (%) |
---|---|---|---|---|---|---|---|

1000 | 10 | 27.3 | 25.8 | −1.4 | 1.4 | 0.0 | −1.4 |

1000 | 60 | 90.9 | 99.2 | 8.3 | 83.3 | 88.4 | 5.1 |

1000 | 110 | 93.7 | 100.0 | 6.2 | 79.3 | 85.5 | 6.2 |

1700 | 10 | 51.2 | 39.6 | −11.6 | 3.6 | 0.0 | −3.6 |

1700 | 60 | 95.0 | 99.9 | 4.9 | 69.0 | 80.8 | 11.8 |

1700 | 110 | 99.5 | 100.0 | 0.5 | 58.0 | 69.8 | 11.8 |

2400 | 10 | 66.10 | 64.7 | −1.4 | 72.0 | 76.3 | 4.3 |

2400 | 60 | 97.37 | 99.1 | 1.8 | 73.6 | 67.4 | −6.2 |

2400 | 110 | 100.0 | 98.9 | −1.1 | 58.8 | 55.4 | −3.4 |

Engine Speed (min^{−1}) | Torque (Nm) | Measured ${\mathit{T}}_{\mathit{g},\mathit{o}\mathit{u}\mathit{t}}\left(\xb0\mathbf{C}\right)$ | Modelled ${\mathit{T}}_{\mathit{g},\mathit{o}\mathit{u}\mathit{t}}\left(\xb0\mathbf{C}\right)$ | Error in ${\mathit{T}}_{\mathit{g},\mathit{o}\mathit{u}\mathit{t}}$ (%) | Experimental $\dot{\mathit{Q}}$ (W) | Modelled $\dot{\mathit{Q}}$ (W) | Error in $\dot{\mathit{Q}}$ (%) |
---|---|---|---|---|---|---|---|

1000 | 60 | 176 | 169 | 4.0 | 1384 | 1407 | 1.6 |

1700 | 10 | 125 | 123 | 1.6 | 915 | 876 | −4.3 |

1700 | 60 | 235 | 226 | 3.7 | 2016 | 2063 | −2.3 |

1700 | 110 | 299 | 288 | 3.7 | 2977 | 3107 | −4.4 |

2400 | 60 | 263 | 256 | 2.7 | 2423 | 2508 | 3.5 |

Engine Speed (min^{−1}) | Torque (Nm) | Measured ${\mathit{T}}_{\mathit{g},\mathit{o}\mathit{u}\mathit{t}}$ (°C) | Modelled ${\mathit{T}}_{\mathit{g},\mathit{o}\mathit{u}\mathit{t}}$ (°C) | Error in ${\mathit{T}}_{\mathit{g},\mathit{o}\mathit{u}\mathit{t}}$ (%) | Experimental $\dot{\mathit{Q}}$ (W) | Modelled $\dot{\mathit{Q}}$ (W) | Error in $\dot{\mathit{Q}}$ (%) |
---|---|---|---|---|---|---|---|

1000 | 10 | 94 | 96 | 1.4 | 590 | 562 | −4.9 |

1000 | 110 | 269 | 253 | 5.9 | 2587 | 2639 | 2.0 |

2400 | 10 | 167 | 164 | 1.8 | 1287 | 1357 | 5.4 |

2400 | 110 | 310 | 300 | 3.0 | 3286 | 3364 | 2.4 |

Engine Mode | 1000 rpm–10 Nm | 2400 rpm–110 Nm | ||
---|---|---|---|---|

Convection | Natural | Forced (${v}_{\infty}$ = 20 km/h) | Natural | Forced (${v}_{\infty}$ = 120 km/h) |

$\Delta T/L$ (°C/dm) | −0.9 | −1.2 | −1.2 | −2.6 |

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**MDPI and ACS Style**

Fernández-Yáñez, P.; Armas, O.; Gómez, A.; Gil, A. Developing Computational Fluid Dynamics (CFD) Models to Evaluate Available Energy in Exhaust Systems of Diesel Light-Duty Vehicles. *Appl. Sci.* **2017**, *7*, 590.
https://doi.org/10.3390/app7060590

**AMA Style**

Fernández-Yáñez P, Armas O, Gómez A, Gil A. Developing Computational Fluid Dynamics (CFD) Models to Evaluate Available Energy in Exhaust Systems of Diesel Light-Duty Vehicles. *Applied Sciences*. 2017; 7(6):590.
https://doi.org/10.3390/app7060590

**Chicago/Turabian Style**

Fernández-Yáñez, Pablo, Octavio Armas, Arántzazu Gómez, and Antonio Gil. 2017. "Developing Computational Fluid Dynamics (CFD) Models to Evaluate Available Energy in Exhaust Systems of Diesel Light-Duty Vehicles" *Applied Sciences* 7, no. 6: 590.
https://doi.org/10.3390/app7060590