**7. Multidimensional models**

According to the methodology used to study the fouling process, the multidimensional models can be categorized in five groups, as **Table 3** summarizes. The first category covers those numerical studies that analyze the exhaust gas flow and its effects on the deposit formation, neglecting the simulation of any fouling layer inside the heat exchanger. The second group is formed by those models that, using Eulerian–Lagrangian approach, determine the soot particle deposition on the walls of the EGR system. The third group contains those models that, using a species transport modeling approach, compute the condensation of different hydrocarbons. The fourth category includes those models that intend to reproduce the effects of the deposit, modifying the heat exchange properties of the EGR cooler surface. And the fifth group is composed of those investigations that recreate the real growth of the deposit on the walls of the EGR cooler.

The multidimensional models included in the first category are focused on the analysis of the exhaust gas flow to assess how changes in heat exchanger shape characteristics can reduce or minimize the fouling layer formation. Knowing that, in most cases, the removal process is caused by shear force, these numerical simulations intend on determining which EGR surface structures increase the shear stress and, thus, lead to an effective deposit suppression. Analyzing different parameters, such as the wall shear stress, the velocity field, or the temperature profile along the EGR cooler, these models intend to determine the fouling propensity of several heat exchanger configurations, as Lee and Min [31] and Mohammadi and Malayeri [67] shown.

Since, in the majority of cases, these models are single-phase numerical simulations, where only the gas flow is taken into consideration, the simplicity of these models allow a detailed examination of all the gas parameters involved in the fouling process. Therefore, they provide an exhaustive examination of the gas variables that can be induced or reduce the fouling layer growth. By contrast, these numerical approaches do not bring any information about the fouling mechanisms. They do not provide the estimation of deposited particulate matter, the number of removed particles, or the amount of condensate that will be generated. For this reason, although these models give an initial estimation of the fouling phenomenon, they have a limited scope of application.

The second category is formed by those models that reproduce the soot particle deposition using an Eulerian-Lagrangian approach. Employing the Lagrangian framework, these models track the trajectory of each soot particle in order to determine the regions where they can be deposited. Computing the particle transport equation, which takes into consideration the forces of the gas flow acting on a single particle, these numerical approaches determine the movement of the particulate matter inside the EGR cooler. Considering different soot particle diameters, these models offer an in-depth analysis of the particle deposition and allow the computing of the deposition efficiency inside different EGR cooler configurations.

Just like the models of the previous group, which only analyze the gas phase, the numerical approaches of this category do not provide any information about the growth and evolution of the fouling deposit, and, although they give relevant data about the regions where deposition will occur, they do not reproduce the interaction between the soot deposit and the exhaust gas flow.

**Authors**

**341**

**modeled**

Lee and Min

—

 —

[31]

Mohammadi

—

 —

> and Malayeri

[67]

Xu et al. [68] •

Thermophoresis

*dup*

*dt* ¼ *Cd Re p* 24*τp ug* � *up*

 

þ *g ρp*�*ρg* ð Þ

þ

*FB* þ

*FS* þ

*Fth*

•

Deposition

Good conformity

Simulation of soot particle deposition

inside a plate-fin heat exchanger

efficiency

with literature

> •

Deposition

results

velocity

•

Particle

deposition

distribution

• Forces acting

Good conformity

Calculation

wavy-fin EGR coolers

 of soot particle deposition on

> with literature

on particles

> •

Deposition

results

fraction

•

Exhaust gas

velocity

• Dew point

Close agreement

Prediction of

vapor, sulfuric acid, and nitric acid

formed in the exhaust gases of diesel

engines

condensation

 of water

> •

flux

•

Deposit

Differences

Evaluation of temperature inside a cooler, changing the dependence

of the deposit thermal

 evolution

conductivity

thickness

observed with

> •

Deposition

data from

efficiency

literature

Condensation

observed with literature results

*ρp*

• •

Inertial

impaction

Nagendra

•

Thermophoresis

*dup*

*dt* ¼

*Fdrag*

þ

*Fth* þ

*Fothers*

> et al. [69]

Yang et al.

• Water vapor

\_*mcondπd* ¼ *τi*

2*νfilm δ*2*film*

[70]

condensation

• Acid condensation

Gonçalves

•

Deposition

*Vth* ¼ �

*Kth*

*Tg*

*ν<sup>g</sup>*

∇

*T*

!

Guedes [71]

• •

Condensation

*Vd* ¼

0*:*057*u*∗ 3*πμg*

2*dp*

3

 �<sup>2</sup>

*ρgkbTCc* 

Removal

•

Diffusion

Diffusion

**Mechanisms**

**Main fouling equations**

**Parameters**

**Model**

**—**

**Remarks**

**experiment**

**analyzed**

•

Exhaust gas

—

Analysis of the gas phase

velocity

•

Exhaust gas temperature

•

Exhaust gas

pressure

Wall shear stress

—

Study of various tube structures that

encourage the deposit suppression

*Numerical Modelling of Fouling Process in EGR System: A Review*

*DOI: http://dx.doi.org/10.5772/intechopen.93062*


#### *Numerical Modelling of Fouling Process in EGR System: A Review DOI: http://dx.doi.org/10.5772/intechopen.93062*

The model predictions were validated, and they were in reasonably good agreement

multidimensional models can be categorized in five groups, as **Table 3** summarizes. The first category covers those numerical studies that analyze the exhaust gas flow and its effects on the deposit formation, neglecting the simulation of any fouling layer inside the heat exchanger. The second group is formed by those models that, using Eulerian–Lagrangian approach, determine the soot particle deposition on the walls of the EGR system. The third group contains those models that, using a species transport modeling approach, compute the condensation of different hydrocarbons. The fourth category includes those models that intend to reproduce the effects of the deposit, modifying the heat exchange properties of the EGR cooler surface. And the fifth group is composed of those investigations that recreate the real growth of

The multidimensional models included in the first category are focused on the analysis of the exhaust gas flow to assess how changes in heat exchanger shape characteristics can reduce or minimize the fouling layer formation. Knowing that, in most cases, the removal process is caused by shear force, these numerical simulations intend on determining which EGR surface structures increase the shear stress and, thus, lead to an effective deposit suppression. Analyzing different parameters, such as the wall shear stress, the velocity field, or the temperature profile along the EGR cooler, these models intend to determine the fouling propensity of several heat exchanger configurations, as Lee and Min [31] and Mohammadi

Since, in the majority of cases, these models are single-phase numerical simulations, where only the gas flow is taken into consideration, the simplicity of these models allow a detailed examination of all the gas parameters involved in the fouling process. Therefore, they provide an exhaustive examination of the gas variables that can be induced or reduce the fouling layer growth. By contrast, these numerical approaches do not bring any information about the fouling mechanisms. They do not provide the estimation of deposited particulate matter, the number of removed particles, or the amount of condensate that will be generated. For this reason, although these models give an initial estimation of the fouling phenomenon,

The second category is formed by those models that reproduce the soot particle deposition using an Eulerian-Lagrangian approach. Employing the Lagrangian framework, these models track the trajectory of each soot particle in order to determine the regions where they can be deposited. Computing the particle transport equation, which takes into consideration the forces of the gas flow acting on a single particle, these numerical approaches determine the movement of the particulate matter inside the EGR cooler. Considering different soot particle diameters, these models offer an in-depth analysis of the particle deposition and allow the computing of the deposition efficiency inside different EGR cooler

Just like the models of the previous group, which only analyze the gas phase, the numerical approaches of this category do not provide any information about the growth and evolution of the fouling deposit, and, although they give relevant data about the regions where deposition will occur, they do not reproduce the interaction

According to the methodology used to study the fouling process, the

with experimental data.

**7. Multidimensional models**

*Environmental Issues and Sustainable Development*

the deposit on the walls of the EGR cooler.

they have a limited scope of application.

between the soot deposit and the exhaust gas flow.

and Malayeri [67] shown.

configurations.

**340**


**Authors**

**343**

**modeled**

Paz et al.

•

Thermophoresis

*Δδd* ¼ *Sd udi*þ*uth* ð Þ*C*

*ρ f*

� *τwδd*

*Δt*

 

> *ψ*

> > [76]

• •

Inertial

*j*

*Kgρg* ln 1�*yi*

1�*yo* 

> *i* ¼

> > impaction

•

• HC condensation

**Table 3.** *Multidimensional*

 *models.*

Removal

Diffusion

**Mechanisms**

**Main fouling equations**

**Parameters**

**Model**

**—**

**Remarks**

**experiment**

**analyzed**

• flux

•

Condensed

results

mass

•

Deposit

surface

temperature

• Areas where condensation

occurs

*Numerical Modelling of Fouling Process in EGR System: A Review*

*DOI: http://dx.doi.org/10.5772/intechopen.93062*

Condensation

Good conformity

Calculation

process considering

 local scale effects

 of the HC

condensation

with literature


**Table 3.** *Multidimensional models.*

#### *Numerical Modelling of Fouling Process in EGR System: A Review DOI: http://dx.doi.org/10.5772/intechopen.93062*

**Authors**

**342**

**modeled**

**Mechanisms**

**Main fouling equations**

*Vi* ¼ 4*:*5 10�4*u*∗ *τpu*∗ 2

*log* 10*Pvap*

> Paz et al. [72] •

Thermophoresis

*u*þ*di* ¼ 0*:*057 *Sc*�2

3 þ 4*:*5 10�4*τ*þ2

*k*

*kp*þ*C Kt n*

*νCc*∇*T*

*p*

• •

Inertial

*u*þ*th* ¼ � 2*Cs*

1þ3*CmKn*

1þ2

*kp*þ2*C Kt n*

*kg*

*u*∗ *T*

impaction

*u*þ*rem*

¼ *τwδd*

*ψu*∗

•

Abarham

•

Thermophoresis

*∂ ρgY* ð Þ

*∂t* þ

∇ *ρ<sup>g</sup>* ~*v* þ

ð

*Vth*

*Y*

¼

∇ *ρ Dg B*∇

*Y*

þ

∇

�*ρg v*00*Y*00 þ

*V*00*thY*00

 

 •

Effectiveness

Good agreement

2-D the growth of the deposit using dynamic

grids

axisymmetric

 model that computes

degradation

observed

•

Deposited

mass

•

Deposit

thickness

Deposit

—

3-D model that computes the fouling

layer evolution, considering

movement of the fouling-gas

 the

 interface

thickness

 

 

> Þ

> > et al. [73]

Paz et al.

•

Thermophoresis

*Δδd* ¼ *Sd udi*þ*uth* ð Þ*C*

*ρ f*

� *τwδd*

*Δt*

 

> *ψ*

> > [74]

• •

Inertial

impaction

•

Paz et al. [75] •

Thermophoresis

\_*mdep*

\_*mrem*

¼

*K*^*τwδd*

¼ *Sd Vth* þ

ð

*Vd* þ

*Vi*

*C*

•

Thermal

Good agreement

Detailed local fouling thickness

experimental

 validation of the

observed

efficiency

degradation

•

• Outlet gas temperature

•

Deposit

thickness

Pressure drop

 Þ

> •

•

Inertial

impaction

•

Removal

Diffusion

Removal

Diffusion

•

Diffusion

Removal

Diffusion

¼ *AA*

�

*T* þ*s CC*

*BB*

*ν<sup>g</sup>* <sup>2</sup>

**Parameters**

**Model**

**—**

**Remarks**

**experiment**

**analyzed**

• Outlet gas temperature

• rate

•

Deposit

Good agreement

Simulation of the real depth of the fouling

thickness

observed

layer and its effects on the

of the flow

hydrodynamic

*Environmental Issues and Sustainable Development*

evolution

•

Deposited

mass

Condensation

The models proposed by Xu et al. [68] and Nagendra et al. [69] use this technique in order to compute the submicron particle deposition inside plate-and-fin heat exchangers. They evaluated the particle deposition under different boundary conditions and validated their results, achieving a good agreement with the experimental measurements taken from literature.

According to the methodology used, the numerical models of this fifth group can be divided into two subcategories: those that convert fluid cells into solid cells and those that use the dynamic mesh methodology to recreate the growth of the fouling

On the one hand, the first subcategory includes those models that, to simulate the growth of the deposit, transform the fluid cells of the domain into fouling cells, as **Figure 6a** illustrates. When the thickness of the fouling layer is larger than the height of the fluid cell, this is converted into a solid cell, and it becomes part of the fouling layer domain. These numerical approaches, as the proposed by Paz et al. [72], couple the gas flow solution and the fouling layer growth and provide a local

On the other hand, to recreate the fouling layer growth, the models of the second

As **Figure 7** shows, the main advantage of these numerical approaches is that they simulate the evolution of the fouling layer in a local manner. Considering the local properties of the exhaust gas flow and taking into account the mechanisms involved in the fouling process, they provide a comprehensive solution of the fouling layer and recreate its real growth inside the heat exchanger. In contrast, these kinds of models have higher computational costs than other multidimensional

*Scheme of the fouling growth: (a) converting fluid cells into solid cells and (b) using the dynamic mesh*

final thickness of the deposit considering the hydrodynamics of the flow.

*Numerical Modelling of Fouling Process in EGR System: A Review*

*DOI: http://dx.doi.org/10.5772/intechopen.93062*

subcategory employ the dynamic mesh methodology, as the 2-D axisymmetric model proposed by Abarham et al. [73] or the 3-D model proposed by Paz et al. [74– 76]. After the fouling thickness calculation, these numerical approaches adjust the thickness of the deposit moving the fouling-fluid interface, as **Figure 6b** shows. At every time-step of the simulation, they estimate the position of the nodes of the mesh and update the fouling layer domain, allowing the possibility to determine the

layer.

deposit growth evolution.

**Figure 6.**

**Figure 7.**

**345**

*Fouling thickness computed using dynamic mesh methodology.*

*methodology.*

The third category includes the multidimensional models that use the species transport modeling approach to compute the condensation of different condensable species. Considering convection, diffusion, or even the chemical reactions that take place in the exhaust gas mixture, these numerical approaches compute the partial pressure of each species to determine their dew temperature. The corresponding condensation flux is calculated in the presence of non-condensable gases, and the thin liquid film of condensate that appears on the walls of the heat exchanger is simulated.

An example that uses this modeling approach is the study proposed by Yang et al. [70]. In order to estimate the corrosion inside the EGR system of heavy-duty trucks, they developed a numerical technique that determines the condensation of nitric and sulfuric acid. The model allows the carrying out of three-dimensional simulations, computes the heat and mass transfer processes, and calculates the amount of condensate formed on the heat exchanger walls. Using the Ansys Fluent CFD code, it computes the condensation flux of water vapor, sulfuric acid, and nitric acid, providing results under different operating conditions. Results of this numerical approach were validated, and they were in close agreement with the data from literature.

These kinds of models are based solely on the study of the condensation of acid and hydrocarbon species, generating detailed reports about the condensation process and neglecting the study of the particulate matter deposition and removal processes that occur along the EGR system. For this reason, they are suitable means to find the regions where acid condensation takes place and to detect the zones of the EGR system where corrosion problems may occur.

The fourth category is formed by the numerical models that reproduce the effects of the fouling layer, modifying the heat exchange properties of the wall. Computing the fouling thermal resistance that opposes the cooling of the flow, these numerical approaches allow the simulation of the evolution of the temperature of the gas flow inside the EGR cooler. After resolving the exhaust gas flow inside the EGR cooler, the model calculates the thickness of a virtual fouling layer and adjusts the thermal resistance of the heat exchanger surface, achieving a steadystate solution of the temperature field.

Changing the properties of the virtual fouling layer according to the computed deposit thickness, these models, as the evaluated in the study of Gonçalves Guedes [66], allow the simulation of the evolution of the exhaust gas temperature. However, their main disadvantage is that they provide poor results in the calculation of different parameters, such as the pressure drop along the heat exchanger, because they avoid the simulation of the real growth of the fouling layer inside the tube. That is why employing these models, the simulation of the changes in the hydrodynamics of the exhaust gas flow caused by the fouling layer and the local parameters of the deposit cannot be estimated reliably.

Finally, the models of the fifth category intend to recreate the real growth of the deposit on the walls of the EGR cooler. To that end, they simulate the movement of the fouling-gas interface, after computing the nonuniform thickness of the deposit. Thus, taking into consideration the local-scale effects involved in the fouling phenomenon, these numerical approaches reproduce the real formation of the fouling deposit on the heat exchanger walls, causing the reduction of the cross-sectional area of the tube.

According to the methodology used, the numerical models of this fifth group can be divided into two subcategories: those that convert fluid cells into solid cells and those that use the dynamic mesh methodology to recreate the growth of the fouling layer.

On the one hand, the first subcategory includes those models that, to simulate the growth of the deposit, transform the fluid cells of the domain into fouling cells, as **Figure 6a** illustrates. When the thickness of the fouling layer is larger than the height of the fluid cell, this is converted into a solid cell, and it becomes part of the fouling layer domain. These numerical approaches, as the proposed by Paz et al. [72], couple the gas flow solution and the fouling layer growth and provide a local final thickness of the deposit considering the hydrodynamics of the flow.

On the other hand, to recreate the fouling layer growth, the models of the second subcategory employ the dynamic mesh methodology, as the 2-D axisymmetric model proposed by Abarham et al. [73] or the 3-D model proposed by Paz et al. [74– 76]. After the fouling thickness calculation, these numerical approaches adjust the thickness of the deposit moving the fouling-fluid interface, as **Figure 6b** shows. At every time-step of the simulation, they estimate the position of the nodes of the mesh and update the fouling layer domain, allowing the possibility to determine the deposit growth evolution.

As **Figure 7** shows, the main advantage of these numerical approaches is that they simulate the evolution of the fouling layer in a local manner. Considering the local properties of the exhaust gas flow and taking into account the mechanisms involved in the fouling process, they provide a comprehensive solution of the fouling layer and recreate its real growth inside the heat exchanger. In contrast, these kinds of models have higher computational costs than other multidimensional

**Figure 6.**

The models proposed by Xu et al. [68] and Nagendra et al. [69] use this technique in order to compute the submicron particle deposition inside plate-and-fin heat exchangers. They evaluated the particle deposition under different boundary conditions and validated their results, achieving a good agreement with the exper-

The third category includes the multidimensional models that use the species transport modeling approach to compute the condensation of different condensable species. Considering convection, diffusion, or even the chemical reactions that take place in the exhaust gas mixture, these numerical approaches compute the partial pressure of each species to determine their dew temperature. The corresponding condensation flux is calculated in the presence of non-condensable gases, and the thin liquid film of condensate that appears on the walls of the heat exchanger is

An example that uses this modeling approach is the study proposed by Yang et al. [70]. In order to estimate the corrosion inside the EGR system of heavy-duty trucks, they developed a numerical technique that determines the condensation of nitric and sulfuric acid. The model allows the carrying out of three-dimensional simulations, computes the heat and mass transfer processes, and calculates the amount of condensate formed on the heat exchanger walls. Using the Ansys Fluent CFD code, it computes the condensation flux of water vapor, sulfuric acid, and nitric acid, providing results under different operating conditions. Results of this numerical approach were validated, and they were in close agreement with the data

These kinds of models are based solely on the study of the condensation of acid and hydrocarbon species, generating detailed reports about the condensation process and neglecting the study of the particulate matter deposition and removal processes that occur along the EGR system. For this reason, they are suitable means to find the regions where acid condensation takes place and to detect the zones of

The fourth category is formed by the numerical models that reproduce the effects of the fouling layer, modifying the heat exchange properties of the wall. Computing the fouling thermal resistance that opposes the cooling of the flow, these numerical approaches allow the simulation of the evolution of the temperature of the gas flow inside the EGR cooler. After resolving the exhaust gas flow inside the EGR cooler, the model calculates the thickness of a virtual fouling layer and adjusts the thermal resistance of the heat exchanger surface, achieving a steady-

Changing the properties of the virtual fouling layer according to the computed deposit thickness, these models, as the evaluated in the study of Gonçalves Guedes [66], allow the simulation of the evolution of the exhaust gas temperature. However, their main disadvantage is that they provide poor results in the calculation of different parameters, such as the pressure drop along the heat exchanger, because they avoid the simulation of the real growth of the fouling layer inside the tube. That is why employing these models, the simulation of the changes in the hydrodynamics of the exhaust gas flow caused by the fouling layer and the local parameters

Finally, the models of the fifth category intend to recreate the real growth of the deposit on the walls of the EGR cooler. To that end, they simulate the movement of the fouling-gas interface, after computing the nonuniform thickness of the deposit. Thus, taking into consideration the local-scale effects involved in the fouling phenomenon, these numerical approaches reproduce the real formation of the fouling deposit on the heat exchanger walls, causing the reduction of the cross-sectional

imental measurements taken from literature.

*Environmental Issues and Sustainable Development*

the EGR system where corrosion problems may occur.

state solution of the temperature field.

of the deposit cannot be estimated reliably.

simulated.

from literature.

area of the tube.

**344**

*Scheme of the fouling growth: (a) converting fluid cells into solid cells and (b) using the dynamic mesh methodology.*

**Figure 7.** *Fouling thickness computed using dynamic mesh methodology.*

models, and, although they provide detailed information about the fouling phenomenon, they demand more computational resources.

*d* tube diameter *dp* particle diameter

*FB* Brownian force *Fdrag* drag force

*FS* Saffman lift force *Fth* thermophoretic force *g* gravitational acceleration

HC hydrocarbon *J* mass flux

*j*

DOC diesel oxidation catalyst EGR exhaust gas recirculation

*DOI: http://dx.doi.org/10.5772/intechopen.93062*

*h* mass transfer coefficient

*K* proportionality constant

from the deposit *kcond* condensation rate constant *kevap* evaporation rate constant *K <sup>f</sup>* overall pressure loss factor

*Kg* mass transfer coefficient *kg* gas thermal conductivity

*Kth* thermophoretic coefficient

*Pg* pressure of the gas flow

*Psat* saturation pressure *Pvap* vapor pressure PM particulate matter *Rcond* condensation rate *Revap* evaporation rate *Rf* fouling resistance

*Tg* gas temperature *Ts* surface temperature *U* mean velocity *UCr* critical velocity *u*<sup>∗</sup> friction velocity

*u*<sup>þ</sup>

**347**

*Pi* vapor pressure of the ith species

*Re <sup>p</sup>* particles' Reynolds number SCR selective catalytic reduction SDG sustainable development goals *Sd* particle sticking probability

*udi* isothermal deposition velocity

*di* dimensionless isothermal deposition velocity

*Kn* Knudsen number

LNT lean NOx trap

*m* mass *m*\_ mass flow *MW* molecular weight NOx nitrogen oxides

*Gth* dimensional thermophoretic parameter

*<sup>i</sup>* mass condensation flux of the ith species

*K*<sup>2</sup> parameter characterizing the dispersion of the soot particles removed

*K*<sup>1</sup> cooler structure-related parameter

*k <sup>f</sup>* fouling layer thermal conductivity

*kp* particulate matter thermal conductivity

*Dt* mean effect of velocity and concentration fluctuations

*Numerical Modelling of Fouling Process in EGR System: A Review*
