**5. 0-D models**

deposit that grows on the heat exchanger walls. Although the amount of experimental studies have been larger and more frequent, the numerical models have become relevant since 2009, as **Figure 5** depicts, due to the increase in NOx emis-

The numerical approaches intend to reproduce and simulate the formation and evolution of the deposit inside the EGR system recreating the different mechanisms involved in the fouling process. Because of their significance in the prediction of the deposit, the deposition mechanisms have been implemented in 76.9% of the main numerical models that analyze the deposit formation inside the EGR system. By contrast, the numerical approaches that recreate removal mechanisms are slightly lower (50.0%), and only the 30.8% of the models are focused on the study of the condensation of volatile species. In many cases, several kinds of mechanisms are implemented and coupled in one single numerical approach, in order to achieve

According to the complexity of the formulation of the models, they can be divided into three principal categories: the zero-dimensional (0-D), the one-

The zero-dimensional models are focused on an overall heat and material balance of the system, and they do not include any analysis of the fluid dynamics. Following several assumptions and simplifications, they evaluate the overall fouling effects, and, although these numerical approaches avoid any spatial resolution of the variables involved in the process, they can give a fair indication about the

The one-dimensional approach is the next level of complexity. In these models, only one spatial dimension is considered, dividing the fluid zone in different regions and analyzing the properties of the system in each region separately. Although this approach simplifies the number of equations, it can give a detailed evolution of the

The multidimensional models require the spatial discretization of the volume of the region and can provide a thorough analysis of the variables of the process. In this field, the use of computational fluid dynamics simulations has been increasing steadily since the 1990s, due to the availability of high-performance computing hardware and the development of user-friendly interfaces. The computer-based simulations make it possible to obtain a detailed solution of the fluid flow, both in two-dimensional (2-D) and three-dimensional (3-D) domains, and they can repro-

sion regulation requirements.

*Main numerical models published from 1997 to 2020.*

*Environmental Issues and Sustainable Development*

**Figure 5.**

more complete simulation frameworks.

spatial changes of the fouling parameters.

duce the evolution and formation of fouling layers.

fouling phenomenon.

**332**

dimensional (1-D), and the multidimensional models.

**Table 1** summarizes the 0-D models that have been proposed to analyze the fouling layer effect in the EGR system.

Abarham et al. [49] proposed an analytical model for thermophoretic particle deposition that solves the mass conservation of particles and the energy equation of the gas flow for a single turbulent pipe flow. This approach considers the submicron particle deposition due to the thermophoretic effect, neglecting the diffusion and other deposition mechanisms. The model takes into consideration the pipe diameter reduction due to the growth of the fouling layer and considers different boundary conditions, such as the inlet temperature and mass flow rate of the gas, the inlet particle concentration, or the wall temperature. In this study, the properties of the soot layer, i.e., density, porosity, and thermal conductivity, have been taken from the experimental measurements of Lance et al. [18], and the soot particle diameter has been set at 57 nm, based on the study of Maricq and Harris [51]. This model computes the total mass deposited on the tube and evaluates the degradation of the heat transfer effectiveness over time. To verify the results of this numerical approach, the data were compared with the experimental measurements obtained by the Oak Ridge National Laboratory, and an acceptable agreement was achieved between both methods.

Garrido et al. [50] presented a theoretical analysis of the thermodynamics of exhaust gas condensation. They analyzed the condensation of different species that are part of the exhaust gas produced by gasoline engines, such as water vapor, ammonium, and sulfuric, nitrous, nitric, and chloric acids. The examination of the chemical reactions that takes place along the exhaust line and the analysis of the vapor-liquid equilibrium of the condensable species under different temperatures allow the study of their behavior and the calculation of their dew point. The experimental validation of the model showed that, although the collected condensate amount was slightly lower than the model predicted results, the general tendencies were verified.

Since the 0-D models do not provide any spatial resolution of the fouling parameters, their scope is deliberately more concise. Nevertheless, they can be used as essential tools in guiding the study of the fouling phenomenon.


**Table 1.**

*0-D model.*

## **6. 1-D models**

According to the mechanisms considered by the 1-D models, they can be categorized in five groups, as **Table 2** summarizes. The first group covers those models that only analyze the condensation of water or hydrocarbons. The second group is formed by those studies that investigate the fouling layer formation solely by considering the effect of particle deposition mechanisms. The third group, which combines the characteristic of the two previous groups, contains those models that take into account both the prediction of the HC condensation and the deposition of particulate matter. The fourth group includes those models that, in addition to simulating the particulate matter deposition, also discuss the removal mechanisms. And the fifth group is composed of those investigations that take into consideration all of the mechanisms mentioned above: deposition of particulate matter, removal of particles from the deposit, and condensation of hydrocarbons.

The 1-D models included in the first group are exclusively focused on the analysis of the condensation mechanisms that occur inside the exhaust system. When the temperature of the EGR line drops below the dew point of the condensable species, the condensate—made up of water, HC, and acids—appears. This condensate interacts with the soot-deposited particles, modifying the physical structure of the fouling, and it may corrode the walls of the heat exchanger when the acid amount is high enough.

On the one hand, when the fuel sulfur content is rather high, the detection of the sulfuric acid condensation becomes relevant, and, in this field, McKinley et al. [52] proposed a 1-D model that predicts the condensation of the acid. This numerical approach allows to compute the sulfuric acid dew point considering the coolant temperature, the concentration of the acid, and the engine operating point. The acid condensation rate is calculated assuming that condensate is formed due to direct condensation on the wall and due to formation in a portion of the boundary layer. In addition, the model estimates the condensate composition inside the EGR cooler, taking into account the sulfuric acid and water vapor condensation fluxes. All of these parameters allow the analysis and detection of the sulfuric acid condensation inside the EGR cooler, and, although this is an unvalidated model, it represents an essential step in understanding the effects of the acid condensation on the fouling process.

On the other hand, during the starting of a cold engine—in the first few hundred seconds—the water condensation and evaporation can interact with the existing deposit on the EGR cooler walls and can alter the normal functioning of other exhaust after-treatment devices, such as the catalyst. Although it is a process that occurs mainly during the first few seconds of an engine service, it can cause a severe effect on the deposit evolution. Within this framework, Sharma et al. [53] proposed a 1-D model that simulates the condensation and evaporation of water inside the exhaust line. This is a mathematical model that computes the condensation and evaporation rate of water and that calculates the gas flow temperature considering the heat transfer due to phase change processes. The model provides more accurate simulations of the evolution of the temperature of the gas flow than previous models that do not consider the effect of water condensation and evaporation, and it was validated with experimental results, achieving a high level of agreement.

The second group is formed by the 1-D models that investigate the fouling layer formation solely by considering the effect of soot particle deposition mechanisms. For the sake of simplicity, these numerical approaches intend to compute the fouling buildup taking into account only the effect of particulate matter deposition mechanisms, neglecting both the removal mechanisms and the presence of

**Authors**

**335**

McKinley [52]

Sharma et al. [53]

B. Ismail [54]

• •

Abarham et al. [55]

Thermophoresis

*Vth* ¼ �

*Kth*

*Tg*

*ν<sup>g</sup>*

∇

*T*

!

Diffusion

*Gth* ¼ �2*Cs*

*k*

 

*kp*þ*C Kt n* 

1þ*Kn*

½

ð

1*:*2þ0*:*41*e*�0*:*88*Kn*

 Þ

 *ν<sup>g</sup>*

*Tg*

•

Effectiveness

degradation

•

Pressure drop

evolution

• Tube diameter

Significant

Prediction of EGR cooler fouling amount and

distribution

 across a concentric tube heat exchanger with

differences

observed

a constant wall temperature

reduction

• Soot layer

thickness

•

Deposit

interface

temperature

•

Effectiveness

degradation

•

• Tube diameter

Significant

Simulation of soot and HC deposition on a concentric

tube EGR cooler with a constant wall temperature

differences

observed

reduction

•

Deposit

interface

temperature

• Mass gain

•

• HC condensed

mass

Pressure drop

Abarham et al. [56] •

Thermophoresis

*Vth* ¼ �

*Kth*

*Tg*

*ν<sup>g</sup>*

∇

*T*

!

• HC condensation

*j*

*Kgρg* ln *yinterface*

*yo* 

> *i* ¼

Pressure drop

 �

thickness

• Soot layer

—

Calculation

phases to compute the soot deposition in diesel EGR

cooling devices

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

 of the coupling between the gas and particle

> 1þ3*Cm Kn*

 Þ

1þ2 *k*

*kp*þ2*C Kt n*

ð

Thermophoresis

 Water vapor

*Rcond*

*Revap*

¼ *kevap θ*

¼ *kcond ywater* 1 � *θ* ð Þ

condensation

 Acid

condensation \_*mcond*

¼ *ρghA MWi MWg yi* � *Pi Ts* ð Þ

*Pg*

 

• Dew point

—

Prediction of

composition

 to minimize EGR cooler corrosion

condensation

 rate and condensate

> •

flux

Condensation

Close agreement

Simulation of temperature

devices considering

 water

 profiles inside

condensation

 and evaporation

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

after-treatment

observed

evaporation

fluxes

Condensation

**Mechanisms**

**Main fouling equations**

**Parameters**

**Model**

**—**

**Remarks**

**experiment**

**analyzed**

**modeled**


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

**6. 1-D models**

*Environmental Issues and Sustainable Development*

the acid amount is high enough.

process.

**334**

According to the mechanisms considered by the 1-D models, they can be categorized in five groups, as **Table 2** summarizes. The first group covers those models that only analyze the condensation of water or hydrocarbons. The second group is formed by those studies that investigate the fouling layer formation solely by considering the effect of particle deposition mechanisms. The third group, which combines the characteristic of the two previous groups, contains those models that take into account both the prediction of the HC condensation and the deposition of particulate matter. The fourth group includes those models that, in addition to simulating the particulate matter deposition, also discuss the removal mechanisms. And the fifth group is composed of those investigations that take into consideration all of the mechanisms mentioned above: deposition of particulate matter, removal

of particles from the deposit, and condensation of hydrocarbons.

The 1-D models included in the first group are exclusively focused on the analysis of the condensation mechanisms that occur inside the exhaust system. When the temperature of the EGR line drops below the dew point of the condensable species, the condensate—made up of water, HC, and acids—appears. This condensate interacts with the soot-deposited particles, modifying the physical structure of the fouling, and it may corrode the walls of the heat exchanger when

On the one hand, when the fuel sulfur content is rather high, the detection of the sulfuric acid condensation becomes relevant, and, in this field, McKinley et al. [52] proposed a 1-D model that predicts the condensation of the acid. This numerical approach allows to compute the sulfuric acid dew point considering the coolant temperature, the concentration of the acid, and the engine operating point. The acid condensation rate is calculated assuming that condensate is formed due to direct condensation on the wall and due to formation in a portion of the boundary layer. In addition, the model estimates the condensate composition inside the EGR cooler, taking into account the sulfuric acid and water vapor condensation fluxes. All of these parameters allow the analysis and detection of the sulfuric acid condensation inside the EGR cooler, and, although this is an unvalidated model, it represents an essential step in understanding the effects of the acid condensation on the fouling

On the other hand, during the starting of a cold engine—in the first few hundred seconds—the water condensation and evaporation can interact with the existing deposit on the EGR cooler walls and can alter the normal functioning of other exhaust after-treatment devices, such as the catalyst. Although it is a process that occurs mainly during the first few seconds of an engine service, it can cause a severe effect on the deposit evolution. Within this framework, Sharma et al. [53] proposed a 1-D model that simulates the condensation and evaporation of water inside the exhaust line. This is a mathematical model that computes the condensation and evaporation rate of water and that calculates the gas flow temperature considering the heat transfer due to phase change processes. The model provides more accurate simulations of the evolution of the temperature of the gas flow than previous models that do not consider the effect of water condensation and evaporation, and it was validated with experimental results, achieving a high level of agreement.

The second group is formed by the 1-D models that investigate the fouling layer formation solely by considering the effect of soot particle deposition mechanisms. For the sake of simplicity, these numerical approaches intend to compute the fouling buildup taking into account only the effect of particulate matter deposition

mechanisms, neglecting both the removal mechanisms and the presence of


**Authors**

**337**

Kuan et al. [63]

• •

Removal

Thermophoresis

*Vth* ¼ �

*If* : *ΔP*<*c*0*Ts* þ *c*1

\_*mrem*

*Else* :

\_*mrem*

Warey et al. [64]

• • •

Inertial

*Vd* ¼

0*:*057*u*∗ 3*πμg*

2*dp*

3

 �<sup>2</sup>

*ρgkbTCc* 

impaction

• drift

•

• HC

\_*mrem*

¼ *Kτwmdep*

*ψ*

condensation

**Table 2.** *1-D models.*

*j*

*Kgρg* ln *yinterface*

*yo* 

> *i* ¼

Removal

*Vg* ¼ 1 � *ρ<sup>g</sup>*

*ρp* 

*gτp*

Gravitational

*Vi* ¼ 4*:*5 10�4*u*∗ *τ<sup>p</sup> u*∗ 2

*ν<sup>g</sup>* <sup>2</sup>

Diffusion

Thermophoresis

*Vth* ¼ �

*Kth*

*Tg*

*ν<sup>g</sup>*

∇

*T*

!

•

Deposit

Reasonably

agreement

 good

Calculation

 of soot deposition,

condensation

turbulent gas flow at constant wall temperature

 of several HC species in a circular tube with

 soot removal, and

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

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

thickness

•

Condensed

mass

•

Deposit surface

temperature

• Total soot mass

•

Cooler

effectiveness

reduction

•

Fouling

thermal

resistance

 HC

¼ *e c*2þ*c*3*T* þ*s c*4*δ*þ*c*5*ΔP*

ð

 Þ

¼ 0

*Kth*

*Tg*

*ν<sup>g</sup>*

∇

*T*

!

**Mechanisms**

**Main fouling equations**

**Parameters**

**Model**

**—**

**Remarks**

**experiment**

**analyzed**

•

Exhaust outlet

Close agreement

Prediction of the long-term fouling behavior of EGR

coolers on a

conditions

medium-duty

 diesel engine for steady-state

observed

temperature

• •

Deposit

thickness

Fouling factor

**modeled**


**Table 2.** *1-D models.*

**Authors**

**336**

Teng and Regner

• •

Removal

Thermophoresis

\_*mdep*

\_*mrem*

¼

*K*2*ρ Ug* 2

*δ<sup>d</sup>*

¼

*K*1*ηdep*

*UρgC*

[57]

Teng and Regner

• •

Removal

Thermophoresis

\_*mdep*

\_*mrem*

¼

*K*2*ρ Ug* 2

*δ<sup>d</sup>*

¼

*K*1*ηdep*

*UρgC*

[58]

Teng [59]

• •

Mehravaran

Brereton [60] Reza Razmavar and

• • •

Sticking

*Jrem*

¼

*K U*

*UCr* 

*ρ f k f Rf*

probability

Sul et al. [62]

• •

Removal

Thermophoresis

*Vth* ¼ �

\_*mrem*

¼ 1

*f δ*, *T*, *ΔP*,*ΔP*2

 

> *Δt A*

*Kth*

*Tg*

*ν<sup>g</sup>*

∇

*T*

!

Removal

Thermophoresis

*Vth* ¼ �

*Kth*

*Tg*

*ν<sup>g</sup>*

∇

*T*

!

•

Fouling

Good agreement

Analysis of soot particle deposition and three potential

removal mechanisms

observed

thermal

resistance

• •

• Total mass

deposited

•

Thermal

Good

Simulation of EGR cooler fouling considering

thermophoretic

removal function

 equation and an empirically

 derived

effectiveness

correspondence

•

Trapped soot

mass

•

Deposit

thickness

•

Deposit surface

temperature

•

Pressure drop

Removal flux

Deposition flux

Reza Malayeri [61]

 and

• • •

Removal

Diffusion

Thermophoresis

*Jr* ¼ �

\_*mrem*

*b*∝*τw*

*φ*

¼ *b mdep*

*DB* þ

ð

*Dt*

*VthC*

Deposit thickness

 Good conformity

Prediction of soot layer formation based on existing

experimental

 and numerical

observations

with literature

results

 Þ *∂C*

*∂r* þ

Removal

Thermophoresis

*Vth* ¼ �

*Δp* ¼

*K f*

*ρgU* ð Þ<sup>2</sup>

2*ρ f*

*Kth*

*Tg*

*ν<sup>g</sup>*

∇

*T*

!

**Mechanisms**

**Main fouling equations**

**Parameters**

**Model**

**—**

**Remarks**

**experiment**

**analyzed**

Cooler

Good agreement

Prediction of the cooler

considering

 the

characteristics

 of the soot deposit

effectiveness

deterioration

observed

effectiveness

degradation

Cooler

Good agreement

Calculation

heat, mass, and momentum

particle-gas

Semiempirical

degradation

 and pressure drop over fouled EGR coolers

 model that predicts cooler

effectiveness

*Environmental Issues and Sustainable Development*

 system

 of soot particle

accumulation

 transfer theories for the

 employing

observed

effectiveness

degradation

•

Cooler

Good agreement

effectiveness

observed

degradation

•

Pressure drop

**modeled**

hydrocarbon and water condensates. These simplified models are based on the assumption that thermophoretic effect is three to four orders of magnitude bigger than other deposition mechanisms, and during the first stages of the deposit growth, the removal of particles does not take place [65].

deposition, also discuss the removal mechanisms. Following the assumption of Kern and Seaton [66], which determined that the net growth of the fouling layer depends on two opposing simultaneous processes of deposition and removal, the models of

The models proposed by Teng and Regner [57, 58], Teng [59], Mehravaran and Brereton [60], Reza Razmavar and Reza Malayeri [61], Sul et al. [62], and Kuan et al. [63] belong to this fourth category. These models compute the deterioration of the heat exchanger effectiveness caused by the fouling layer growth and calculate

On the one hand, with regard to the particle deposition process, thermophoresis is, in the majority of cases, the only referred deposition mechanism. Although some of these numerical approaches take into consideration the deposition of particulate matter due both to diffusion and thermophoretic effect, such as the model of Mehravaran and Brereton [60], the simulation of the deposition phenomenon of the remaining models is, on an exclusive basis, the calculation of the thermophoretic

On the other hand, the removal of soot particles from the deposit is computed

All numerical models of this fourth category were validated with experimental

Finally, the fifth category of 1-D models covers the numerical approaches that take into consideration the deposition mechanisms, the removal mechanisms, and the condensation of hydrocarbons. In addition to the features of the models of the previous group, the approaches of this category include the simulation of the hydrocarbon condensation, implementing the three phenomena in a comprehensive

It is worth stressing that, following the methodology of the previous models, the numerical implementations of this category also assume that the deposit, which is formed by soot particles and condensates, has uniform properties. Although, as has been mentioned, the presence of condensate can alter the physical structure of the deposit changing its properties, the density and thermal conductivity of the modeled fouling layer do not change over time, regardless of the amount of con-

The model proposed by Warey et al. [64] belongs to this fifth category, and it is able to compute the total mass deposited and the fouling layer resistance over time.

data. The evolution of the overall parameters of the EGR cooler undergoing a fouling process was compared with the models' results, and, in general, they were in agreement. Although the lack of detailed information prevents a full appraisal of the performance of each mechanism involved in the process, it may be concluded that the combination of deposition and removal mechanisms is expected to provide

accurate simulations of the fouling process caused by soot particles.

using different methodologies. One of these is based on the simulation of the different mechanisms that produce the erosion of the particles, i.e., calculating the physical phenomena that is potentially responsible for the removal of deposited particles. This physical approach, as used by Reza Razmavar and Reza Malayeri [61], simulates removal mechanisms such as the shear force, the effect of incident particle impact, or the particle rolling, allowing to estimate the gas maximum critical velocity to compute the particle removal flux. The other removal approach is quite different, and it is based on empirically derived removal functions that allow the estimation of the removal trend. In this removal approach, as the one proposed by Sul et al. [62], the equation that computes the removal rate is a function of different parameters, such as the deposit thickness, the temperature, or the pressure drop, and it was derived from the data of experimental tests that cover a wide range

this category recreate the effects of the fouling deposit on the EGR cooler

the increase in pressure drop along the device.

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

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

performance.

coefficient.

of fouling conditions.

model.

**339**

densate expected.

The investigations of B. Ismail [54] and Abarham et al. [55], which proposed 1-D models that investigate the soot deposit evolution considering only the effect of particulate matter deposition mechanisms, are included in this category.

Ismail [54] developed a simplified model, based on two-phase gas-particle conservation equations, which simulates the heat transfer, pressure drop, and soot deposition in EGR cooling devices. This model takes into consideration the particle transport due to the effect of diffusion and thermophoresis and employs a quasisteady-state formulation that computes the incremental deposited layer thickness along the heat exchanger. It allows the prediction of the change in soot layer thickness, the evolution of the temperature at the outlet of the heat exchanger, and the increase in pressure drop across the EGR cooling device. The weak point of this simplified model is that, although it allows the prediction of the main effects of the soot deposit on the cooler performance, its results were not validated with experimental data.

In the same way, the model presented by Abarham et al. [55] permits to simulate the cooler effectiveness degradation and pressure drop along the EGR cooler, taking into account the particulate matter deposition caused by the thermophoretic effect. This numerical approach allows the calculation of the reduction of the cross sectional area of the tube and estimates the evolution of the temperature of the soot layer interface. In this case, the results of this 1-D model were verified using the experimental measurements of a controlled EGR cooler fouling test, and, although the predicted values for the EGR cooler effectiveness were in agreement with experimental data, the values expected in pressure drop differed significantly from the experimental measurements.

The analysis of the performance of the models of the second group shows that the simulation of the fouling process solely by considering the deposition mechanisms does not bring about the expected results regarding the evolution of the pressure drop along the EGR cooler. As Abarham et al. [55] detailed, although these simplified 1-D models reproduce the fouling growth yielding positive results, it should be expected that the addition of removal mechanisms may improve the predictive capabilities of these models.

In order to complete the features of the abovementioned numerical approach, Abraham et al. [56] added to their model the simulation of the HC condensation, and this new model belongs to the third group, i.e., the category of 1-D models that take into account both the prediction of the HC condensation and the deposition of soot particles. This numerical approach incorporates, coupling with the soot particle deposition equations, the calculation of the dew point and the total mass flux of HC that condenses and becomes part of the deposit. As their other model, it allows to compute the cooler efficiency degradation and the pressure drop evolution neglecting the changes in the physical structure and the chemical reactions that occur in the fouling layer due to the presence of condensate.

Despite the fact that another mechanism was added to the model, the comparison between experimental data and the results of the new model showed a certain mismatch. Although the predicted cooler effectiveness degradation was in agreement with the experimental measurements, the calculated pressure drop continued to display certain differences with the experimental data, and no improvements were seen in this field.

The fourth category comprises the higher number of 1-D numerical approaches, and it covers those models that, in addition to simulating the particulate matter

hydrocarbon and water condensates. These simplified models are based on the assumption that thermophoretic effect is three to four orders of magnitude bigger than other deposition mechanisms, and during the first stages of the deposit

The investigations of B. Ismail [54] and Abarham et al. [55], which proposed 1-D models that investigate the soot deposit evolution considering only the effect of particulate matter deposition mechanisms, are included in this category.

Ismail [54] developed a simplified model, based on two-phase gas-particle conservation equations, which simulates the heat transfer, pressure drop, and soot deposition in EGR cooling devices. This model takes into consideration the particle transport due to the effect of diffusion and thermophoresis and employs a quasisteady-state formulation that computes the incremental deposited layer thickness along the heat exchanger. It allows the prediction of the change in soot layer thickness, the evolution of the temperature at the outlet of the heat exchanger, and the increase in pressure drop across the EGR cooling device. The weak point of this simplified model is that, although it allows the prediction of the main effects of the soot deposit on the cooler performance, its results were not validated with experi-

In the same way, the model presented by Abarham et al. [55] permits to simulate the cooler effectiveness degradation and pressure drop along the EGR cooler, taking into account the particulate matter deposition caused by the thermophoretic effect. This numerical approach allows the calculation of the reduction of the cross sectional area of the tube and estimates the evolution of the temperature of the soot layer interface. In this case, the results of this 1-D model were verified using the experimental measurements of a controlled EGR cooler fouling test, and, although the predicted values for the EGR cooler effectiveness were in agreement with experimental data, the values expected in pressure drop differed significantly from

The analysis of the performance of the models of the second group shows that the simulation of the fouling process solely by considering the deposition mechanisms does not bring about the expected results regarding the evolution of the pressure drop along the EGR cooler. As Abarham et al. [55] detailed, although these simplified 1-D models reproduce the fouling growth yielding positive results, it should be expected that the addition of removal mechanisms may improve the

In order to complete the features of the abovementioned numerical approach, Abraham et al. [56] added to their model the simulation of the HC condensation, and this new model belongs to the third group, i.e., the category of 1-D models that take into account both the prediction of the HC condensation and the deposition of soot particles. This numerical approach incorporates, coupling with the soot particle deposition equations, the calculation of the dew point and the total mass flux of HC that condenses and becomes part of the deposit. As their other model, it allows to compute the cooler efficiency degradation and the pressure drop evolution neglecting the changes in the physical structure and the chemical reactions that

Despite the fact that another mechanism was added to the model, the comparison between experimental data and the results of the new model showed a certain mismatch. Although the predicted cooler effectiveness degradation was in agreement with the experimental measurements, the calculated pressure drop continued to display certain differences with the experimental data, and no improvements

The fourth category comprises the higher number of 1-D numerical approaches, and it covers those models that, in addition to simulating the particulate matter

growth, the removal of particles does not take place [65].

*Environmental Issues and Sustainable Development*

mental data.

the experimental measurements.

predictive capabilities of these models.

were seen in this field.

**338**

occur in the fouling layer due to the presence of condensate.

deposition, also discuss the removal mechanisms. Following the assumption of Kern and Seaton [66], which determined that the net growth of the fouling layer depends on two opposing simultaneous processes of deposition and removal, the models of this category recreate the effects of the fouling deposit on the EGR cooler performance.

The models proposed by Teng and Regner [57, 58], Teng [59], Mehravaran and Brereton [60], Reza Razmavar and Reza Malayeri [61], Sul et al. [62], and Kuan et al. [63] belong to this fourth category. These models compute the deterioration of the heat exchanger effectiveness caused by the fouling layer growth and calculate the increase in pressure drop along the device.

On the one hand, with regard to the particle deposition process, thermophoresis is, in the majority of cases, the only referred deposition mechanism. Although some of these numerical approaches take into consideration the deposition of particulate matter due both to diffusion and thermophoretic effect, such as the model of Mehravaran and Brereton [60], the simulation of the deposition phenomenon of the remaining models is, on an exclusive basis, the calculation of the thermophoretic coefficient.

On the other hand, the removal of soot particles from the deposit is computed using different methodologies. One of these is based on the simulation of the different mechanisms that produce the erosion of the particles, i.e., calculating the physical phenomena that is potentially responsible for the removal of deposited particles. This physical approach, as used by Reza Razmavar and Reza Malayeri [61], simulates removal mechanisms such as the shear force, the effect of incident particle impact, or the particle rolling, allowing to estimate the gas maximum critical velocity to compute the particle removal flux. The other removal approach is quite different, and it is based on empirically derived removal functions that allow the estimation of the removal trend. In this removal approach, as the one proposed by Sul et al. [62], the equation that computes the removal rate is a function of different parameters, such as the deposit thickness, the temperature, or the pressure drop, and it was derived from the data of experimental tests that cover a wide range of fouling conditions.

All numerical models of this fourth category were validated with experimental data. The evolution of the overall parameters of the EGR cooler undergoing a fouling process was compared with the models' results, and, in general, they were in agreement. Although the lack of detailed information prevents a full appraisal of the performance of each mechanism involved in the process, it may be concluded that the combination of deposition and removal mechanisms is expected to provide accurate simulations of the fouling process caused by soot particles.

Finally, the fifth category of 1-D models covers the numerical approaches that take into consideration the deposition mechanisms, the removal mechanisms, and the condensation of hydrocarbons. In addition to the features of the models of the previous group, the approaches of this category include the simulation of the hydrocarbon condensation, implementing the three phenomena in a comprehensive model.

It is worth stressing that, following the methodology of the previous models, the numerical implementations of this category also assume that the deposit, which is formed by soot particles and condensates, has uniform properties. Although, as has been mentioned, the presence of condensate can alter the physical structure of the deposit changing its properties, the density and thermal conductivity of the modeled fouling layer do not change over time, regardless of the amount of condensate expected.

The model proposed by Warey et al. [64] belongs to this fifth category, and it is able to compute the total mass deposited and the fouling layer resistance over time. The model predictions were validated, and they were in reasonably good agreement with experimental data.
