Pollution Prevention and Controls

Technical Information Service, U.S. Department of Commerce; 1996

Kinetic Modeling for Environmental Systems

radioactive disposal facility. Chemical Engineering Journal. 2010;157:100-112

[14] Abdel Rahman RO. Preliminary assessment of continuous atmospheric discharge from the low active waste incinerator. International Journal of Environmental Sciences. 2010;1(2):

[15] EPA. Guidance for the Development of Conceptual Models for a Problem Formulation Developed for Registration Review. Available from: https://www.e pa.gov/pesticide-science-and-asse ssing-pesticide-risks/guidance-deve lopment-conceptual-models-problem.

[16] Abdel Rahman RO, Abdel Moamen OA, Hanafy M, Abdel Monem NM. Preliminary investigation of zinc transport through zeolite-X barrier: Linear isotherm assumption. Chemical Engineering Journal. 2012;185–186:

[17] Abdel Rahman RO. Performance assessment of unsaturated zone as a part of waste disposal site [PhD thesis]. Egypt: Nuclear Engineering Dep., Faculty of Engineering, Alexandria

[18] Abdel Rahman RO, El Kamash AM, Zaki AA, El Sourougy MR. Disposal: A last step towards an integrated waste management system in Egypt. In: International Conference on the Safety of Radioactive Waste Disposal; Tokyo,

Japan. IAEA-CN-135/81; 2005.

[19] Gasser MS, El Sherif E, Abdel Rahman RO. Modification of Mg-Fe hydrotalcite using Cyanex 272 for lanthanides separation. Chemical Engineering Journal. 2017;316C:758-769

[20] Abdel Rahman RO, Ibrahim HA, Hanafy M, Abdel Monem NM.

Assessment of synthetic zeolite NaA-X as sorbing barrier for strontium in a

[Accessed: 28/11/2018]

111-122

61-70

University; 2005

pp. 317-324

12

Chapter 2

Abstract

engines.

1. Introduction

pollution [1, 2].

emissions.

15

The Diesel Soot Particles Fractal

Based on the fractal growth physical model of soot particles from large diesel agriculture machinery, this chapter simulates the morphological structure of collision for the single particles and single particles, single particle and clusters, clusters and clusters, firstly. Moreover, combining with the collision frequency, the fractal growth is controlled to agglomeration using the main environmental factors interference for diesel engine soot particles, in order to make them condensed into regular geometry or larger density particles, reduce the viscous drag for capturing by the capturer or settlement and to realize the control of the pollution of the environment. The results of numerical simulation show that the proposed method is feasible and effective, which will help to understand and analyze the physical mechanism and kinetics of non-equilibrium condensation growth behavior of the actual carbon smoke particles and provide the solution to further reduce emissions of the inhalable particulate matter from diesel

Keywords: soot particles, agglomeration, fractal growth, control, diesel engine

Large diesel agriculture machinery plays an important role in economic development, but it also brings sharp problems in environmental protection. Diesel emissions are one of the most important sources of air pollution. There are many kinds of harmful substances, such as HC, CO, NOX, and soot particles, but the emission of harmful gases from diesel engines, such as HC and CO, is quite low; NOX emissions are also in the same order of magnitude as gasoline engines. The soot particles emitted are respirable particles, causing the most serious air

Particles emitted by diesel engines are usually composed of soot, organic soluble components, and sulfides [3]. The main components of particles discharged from a typical heavy-duty diesel engine under transient conditions are shown in Figure 1. Usually, soot accounts for 50–80% of total particulate matter. It is one of the most important harmful emissions [3]. Therefore, it is of great significance to control the emission of soot particles from diesel engine

Growth Model and Its

Ping Liu and Chunying Wang

Agglomeration Control

## Chapter 2

## The Diesel Soot Particles Fractal Growth Model and Its Agglomeration Control

Ping Liu and Chunying Wang

## Abstract

Based on the fractal growth physical model of soot particles from large diesel agriculture machinery, this chapter simulates the morphological structure of collision for the single particles and single particles, single particle and clusters, clusters and clusters, firstly. Moreover, combining with the collision frequency, the fractal growth is controlled to agglomeration using the main environmental factors interference for diesel engine soot particles, in order to make them condensed into regular geometry or larger density particles, reduce the viscous drag for capturing by the capturer or settlement and to realize the control of the pollution of the environment. The results of numerical simulation show that the proposed method is feasible and effective, which will help to understand and analyze the physical mechanism and kinetics of non-equilibrium condensation growth behavior of the actual carbon smoke particles and provide the solution to further reduce emissions of the inhalable particulate matter from diesel engines.

Keywords: soot particles, agglomeration, fractal growth, control, diesel engine

## 1. Introduction

Large diesel agriculture machinery plays an important role in economic development, but it also brings sharp problems in environmental protection. Diesel emissions are one of the most important sources of air pollution. There are many kinds of harmful substances, such as HC, CO, NOX, and soot particles, but the emission of harmful gases from diesel engines, such as HC and CO, is quite low; NOX emissions are also in the same order of magnitude as gasoline engines. The soot particles emitted are respirable particles, causing the most serious air pollution [1, 2].

Particles emitted by diesel engines are usually composed of soot, organic soluble components, and sulfides [3]. The main components of particles discharged from a typical heavy-duty diesel engine under transient conditions are shown in Figure 1. Usually, soot accounts for 50–80% of total particulate matter. It is one of the most important harmful emissions [3]. Therefore, it is of great significance to control the emission of soot particles from diesel engine emissions.

2.2 The simulation method of soot structure

DOI: http://dx.doi.org/10.5772/intechopen.80851

The Diesel Soot Particles Fractal Growth Model and Its Agglomeration Control

Figure 2.

The soot generation process.

(Figure 4), the collision of the cluster with cluster (Figure 5).

Figure 3.

17

with target particle; (d) form an agglomerate).

According to the characteristics of the soot growth process, the dynamic Monte Carlo method [15] is used to establish the soot fractal growth model. As shown in Figure 3, in a two-dimensional Euclidean space with many particles, one initial particle is set as the target particle, and the other particles are candidate particles. One of the candidate particles is selected to collide with the target particle according to a randomly generated locus, and adheres according to the adhesion probability. One other candidate particle repeats the above process, and the analogy eventually forms an agglomerate. If the motion reaches the boundary of the space, the particle is absorbed by the target particle and disappears. After the particles are released, they do Brownian motion, and they are required to move to the neighboring left, right, upper, and lower surrounding squares with a probability of 1/4. The process will continue until the particles leave the boundary or reach the agglomerate. There are two kinds of collision for particles: the collision of the particle with particle

Taking the collision of the single particle as an example to illustrate the collision of the particles, the trajectory vectors of two collision particles is firstly determined

The diagram of growth process. ((a) set target particle; (b) collide with target particle; (c) analogy of collide

Figure 1. The composition of particulate emissions from the heavy-duty diesel [3].

Soot is a very fine particle formed by a complex reaction mechanism in the flame of the fuel-rich region when burning hydrocarbons in the absence of air, mainly composed of a mixture of amorphous carbon and organic matter [4]. Since the concentration and particle size of soot particles emitted by the gasoline engine is lower than that of the diesel engine [5], this chapter mainly analyzes the soot particles of the diesel engine.

At present, the study of soot particles in diesel engines has focused on optical properties, chemical composition, particle size distribution, source analysis, and human health assessment [6, 7], but research on particle morphology (morphology and surface structure) is almost blank, especially the morphological structure of the particles. Most soot particles have complex fractal morphology [8, 9], affecting the nature of the particles. By studying its fractal structure, the deposition of particles, the viscous resistance of the particles and the adsorption of toxic molecules can be deduced. Therefore, it is necessary to control the fractal condensation and growth morphology of diesel soot particles.

Based on the fractal growth physical model of soot particles from large diesel agriculture machinery, this chapter simulates the morphological structure of collision for the single-single particles, single-clusters, clusters-clusters, firstly. Moreover, combining with the collision frequency, the fractal growth is controlled to agglomeration using the main environmental factors interference for diesel engine soot particles, in order to make them condensed into regular geometry or larger density particles, reduce the viscous drag for capturing by the capturer or settlement and to realize the control of the pollution of the environment.

## 2. The condensation growth process of the soot and its simulation method

#### 2.1 The generation process of soot

The soot particles generation process undergoes complex chemical reactions and physical processes. Firstly, it undergoes gas phase reactions, phase transitions from gaseous to solid state. Then the formation of soot particles in diesel cylinders undergoes the evolution of kinetic events such as nucleation, condensation, collision fragmentation, growth, and surface oxidation [10–14]. The specific formation process described by the soot particle model is shown in Figure 2.

The Diesel Soot Particles Fractal Growth Model and Its Agglomeration Control DOI: http://dx.doi.org/10.5772/intechopen.80851

The soot generation process.

Soot is a very fine particle formed by a complex reaction mechanism in the flame of the fuel-rich region when burning hydrocarbons in the absence of air, mainly composed of a mixture of amorphous carbon and organic matter [4]. Since the concentration and particle size of soot particles emitted by the gasoline engine is lower than that of the diesel engine [5], this chapter mainly analyzes the soot

At present, the study of soot particles in diesel engines has focused on optical properties, chemical composition, particle size distribution, source analysis, and human health assessment [6, 7], but research on particle morphology (morphology and surface structure) is almost blank, especially the morphological structure of the particles. Most soot particles have complex fractal morphology [8, 9], affecting the nature of the particles. By studying its fractal structure, the deposition of particles, the viscous resistance of the particles and the adsorption of toxic molecules can be deduced. Therefore, it is necessary to control the fractal condensation and growth morphology of diesel

Based on the fractal growth physical model of soot particles from large diesel agriculture machinery, this chapter simulates the morphological structure of collision for the single-single particles, single-clusters, clusters-clusters, firstly.

2. The condensation growth process of the soot and its simulation

The soot particles generation process undergoes complex chemical reactions and physical processes. Firstly, it undergoes gas phase reactions, phase transitions from gaseous to solid state. Then the formation of soot particles in diesel cylinders undergoes the evolution of kinetic events such as nucleation, condensation, collision fragmentation, growth, and surface oxidation [10–14]. The specific formation process described by the soot particle model is shown in Figure 2.

Moreover, combining with the collision frequency, the fractal growth is controlled to agglomeration using the main environmental factors interference for diesel engine soot particles, in order to make them condensed into regular geometry or larger density particles, reduce the viscous drag for capturing by the capturer or settlement and to realize the control of the pollution of the

particles of the diesel engine.

Kinetic Modeling for Environmental Systems

The composition of particulate emissions from the heavy-duty diesel [3].

soot particles.

Figure 1.

environment.

method

16

2.1 The generation process of soot

#### 2.2 The simulation method of soot structure

According to the characteristics of the soot growth process, the dynamic Monte Carlo method [15] is used to establish the soot fractal growth model. As shown in Figure 3, in a two-dimensional Euclidean space with many particles, one initial particle is set as the target particle, and the other particles are candidate particles. One of the candidate particles is selected to collide with the target particle according to a randomly generated locus, and adheres according to the adhesion probability. One other candidate particle repeats the above process, and the analogy eventually forms an agglomerate. If the motion reaches the boundary of the space, the particle is absorbed by the target particle and disappears. After the particles are released, they do Brownian motion, and they are required to move to the neighboring left, right, upper, and lower surrounding squares with a probability of 1/4. The process will continue until the particles leave the boundary or reach the agglomerate. There are two kinds of collision for particles: the collision of the particle with particle (Figure 4), the collision of the cluster with cluster (Figure 5).

Taking the collision of the single particle as an example to illustrate the collision of the particles, the trajectory vectors of two collision particles is firstly determined

#### Figure 3.

The diagram of growth process. ((a) set target particle; (b) collide with target particle; (c) analogy of collide with target particle; (d) form an agglomerate).

Figure 4. The collision of single particles and single particle.

#### Figure 5. The collision of clusters and clusters.

in Figure 6. Two small balls are defined as B<sup>1</sup> (target particles) and Bm (random particles) to represent two separate particles, with radius R<sup>1</sup> and Rm, respectively. The coordinates of B<sup>1</sup> are given, the coordinates of Bm are random, and the radius of the concentric sphere Bs of the small ball B<sup>1</sup> is defined as Rs(Rs ¼ R<sup>1</sup> þ Rm). Then let Bm move according to a random trajectory. When Bm meets the fixed ball B1, Eq.(1) is satisfied, where x<sup>0</sup> <sup>s</sup> is the center of the ball B1.

$$|\mathfrak{x}\_{\mathfrak{z}} - \mathfrak{x}\_{\mathfrak{z}}^{0}| = R\_{\mathfrak{z}}.\tag{1}$$

The collision of two small balls can be defined Eq.(3).

The Diesel Soot Particles Fractal Growth Model and Its Agglomeration Control

DOI: http://dx.doi.org/10.5772/intechopen.80851

xm <sup>¼</sup> <sup>x</sup><sup>0</sup>

(

x0

<sup>m</sup><sup>∣</sup> <sup>¼</sup> <sup>∣</sup> <sup>x</sup><sup>0</sup>

<sup>m</sup> � <sup>x</sup><sup>0</sup> s

Solve both sides of squared Eq.(3) simultaneously to obtain a quadratic equation with unknown cn. cn is the judgment factor, if cn has no solution, two balls cannot collide; if cn has a solution, the two balls collide with two cases. One is that when there is a unique solution, the two balls just collide with each other; when there are two solutions, the smallest solution is cmin according to the physical conditions. Then the coordinates of randomly moving ball Bm are also determined as Eq.(4). This process describes a simple collision process between particles and particles. Based on this, it can be used to simulate the collision and clustering process between clusters and clusters. The collision process is still established by

<sup>m</sup> þ cminv, intersect,

<sup>m</sup> þ cminv, tangent:

The shape of aggregates formed by fractal growth of soot is closely related to the radius of gyration, soot radius, and fractal dimension, which is shown in Figure 7. The Rg characterizes the compactness of soot condensation growth, Re characterizes the size of the soot agglomerates formed by fractal growth. The fractal dimension of the surface roughness of soot agglomerates is closely related to the adsorption of particles. The calculation of fractal dimension Df of soot agglomerates is based on the box calculation method [16] and shown in Eq.(5), where Nnð Þ A is the minimum number of boxes needed to contain A, 1=Tn is the boundary of the small box. When Tn is large enough, the box dimension is approximate as Eq.(6), and the fractal dimension calculation method for soot

� � <sup>þ</sup> cnv<sup>∣</sup> <sup>¼</sup> Rs: (3)

(4)

<sup>∣</sup>xm � <sup>x</sup><sup>0</sup>

using Monte Carlo method.

The diagram of single particle collisions.

Figure 6.

agglomerates is shown in Figure 7.

19

When the random particle Bm adsorbs and condenses on the target particle B1, its random motion trajectory vector v exactly intersects with or intersects the ball B1, and can be defined as Eq.(2),

$$\boldsymbol{\infty}\_{m} = \boldsymbol{\mathfrak{x}}\_{m}^{0} + \boldsymbol{c}\_{n} \boldsymbol{v}, \qquad \boldsymbol{c}\_{n} \in [0, \infty). \tag{2}$$

where xm is the sphere center coordinate after the collision of the ball Bm, and x<sup>0</sup> m is the initial sphere center coordinate of the ball Bm.

The Diesel Soot Particles Fractal Growth Model and Its Agglomeration Control DOI: http://dx.doi.org/10.5772/intechopen.80851

Figure 6. The diagram of single particle collisions.

The collision of two small balls can be defined Eq.(3).

$$|\mathfrak{x}\_m - \mathfrak{x}\_m^0| = |\left(\mathfrak{x}\_m^0 - \mathfrak{x}\_s^0\right) + \mathfrak{c}\_n \nu| = R\_\nu. \tag{3}$$

Solve both sides of squared Eq.(3) simultaneously to obtain a quadratic equation with unknown cn. cn is the judgment factor, if cn has no solution, two balls cannot collide; if cn has a solution, the two balls collide with two cases. One is that when there is a unique solution, the two balls just collide with each other; when there are two solutions, the smallest solution is cmin according to the physical conditions. Then the coordinates of randomly moving ball Bm are also determined as Eq.(4). This process describes a simple collision process between particles and particles. Based on this, it can be used to simulate the collision and clustering process between clusters and clusters. The collision process is still established by using Monte Carlo method.

$$\boldsymbol{\kappa}\_{m} = \begin{cases} \boldsymbol{\kappa}\_{m}^{0} + \boldsymbol{c}\_{\min} \boldsymbol{v}, & \text{interest}, \\ \boldsymbol{\kappa}\_{m}^{0} + \boldsymbol{c}\_{\min} \boldsymbol{v}, & \text{tangent}. \end{cases} \tag{4}$$

The shape of aggregates formed by fractal growth of soot is closely related to the radius of gyration, soot radius, and fractal dimension, which is shown in Figure 7. The Rg characterizes the compactness of soot condensation growth, Re characterizes the size of the soot agglomerates formed by fractal growth. The fractal dimension of the surface roughness of soot agglomerates is closely related to the adsorption of particles. The calculation of fractal dimension Df of soot agglomerates is based on the box calculation method [16] and shown in Eq.(5), where Nnð Þ A is the minimum number of boxes needed to contain A, 1=Tn is the boundary of the small box. When Tn is large enough, the box dimension is approximate as Eq.(6), and the fractal dimension calculation method for soot agglomerates is shown in Figure 7.

in Figure 6. Two small balls are defined as B<sup>1</sup> (target particles) and Bm (random particles) to represent two separate particles, with radius R<sup>1</sup> and Rm, respectively. The coordinates of B<sup>1</sup> are given, the coordinates of Bm are random, and the radius of the concentric sphere Bs of the small ball B<sup>1</sup> is defined as Rs(Rs ¼ R<sup>1</sup> þ Rm). Then let Bm move according to a random trajectory. When Bm meets the fixed ball B1, Eq.(1)

<sup>∣</sup>xs � <sup>x</sup><sup>0</sup>

When the random particle Bm adsorbs and condenses on the target particle B1, its random motion trajectory vector v exactly intersects with or intersects the ball

where xm is the sphere center coordinate after the collision of the ball Bm, and x<sup>0</sup>

<sup>s</sup> ∣ ¼ Rs: (1)

m

<sup>m</sup> þ cnv, cn ∈ ½ Þ 0; ∞ : (2)

<sup>s</sup> is the center of the ball B1.

xm <sup>¼</sup> <sup>x</sup><sup>0</sup>

is the initial sphere center coordinate of the ball Bm.

is satisfied, where x<sup>0</sup>

The collision of clusters and clusters.

Figure 5.

18

Figure 4.

The collision of single particles and single particle.

Kinetic Modeling for Environmental Systems

B1, and can be defined as Eq.(2),

Figure 7. The sandbox method for the fractal dimension of soot condensed matter.

$$D\_f = \lim\_{n \to \infty} \frac{N\_n(A)}{- \ln T\_n},\tag{5}$$

∇2

DOI: http://dx.doi.org/10.5772/intechopen.80851

power system of Eq.(8) is introduced as Eq.(9).

nonlinear function F is set as Eq.(10).

item and nonlinear term

(10) satisfies the relationship Eq.(14).

introduced.

where

inequality:

21

ηð Þ¼ x; y F ηð Þ x; y ;

The Diesel Soot Particles Fractal Growth Model and Its Agglomeration Control

ηmþ1,n þ ηm�1,n þ ηm,nþ<sup>1</sup> þ ηm,n�<sup>1</sup> � 4ηm,n ¼ F ηm,n; ηmþ1,n � ηm,n

F ¼ α sin ηm,n

tion of Eq.(9), a simple control system can be obtained as Eq.(13).

For more generalized processing problems, the system Eq.(11) is hereby

∂η ∂t

þ ηm,nþ<sup>1</sup> � ηm,n

Ωð Þ¼ r α sinð Þþ Ωð Þ r � 1 Ωð Þþ r � 1 um,n, (11)

Ωð Þ¼ r η<sup>m</sup>þr,n þ η<sup>m</sup>�r,n þ ηm,nþ<sup>r</sup> þ ηm,n�r, r ¼ 1, 2, ⋯: (12)

Obviously, when r ¼ 1, Eq.(12) becomes system Eq.(9). By iterating simplifica-

Ωð Þ¼ r ru þ α sinð Þþ Ωð Þ r � 1 α sinð Þþ Ωð Þ r � 2 ⋯ þ α sinð Þþ Ωð Þ 0 Ωð Þ 0 , r ¼ 1, 2, ⋯

3.2 The control of fractal growth for diesel engines' soot particles from source

According to the control method of [18], this chapter analyzes the effect of this control method on the fractal growth of soot particles. Assuming that H is a condensed region, H is a condensed boundary, and M is the scope of control of the source item u xð Þ ; y , and satisfies MathcalM ∈ H. In addition, for any ð Þ x; y ∈ H, there is 0 ≤ ηð Þ x; y ≤ 1 established. Since the analytical function u xð Þ ; y satisfies the maximum principle in H, for any ð Þ x; y ∈ H � H, condition 0 ≤ ηð Þ xy < 1 must be true, so α and u in Eq.(11) must be as small as possible, represented by an

0 ≤ η<sup>m</sup>þr,n þ η<sup>m</sup>�r,n þ ηm,nþ<sup>r</sup> þ ηm,n�<sup>r</sup> < 1,

0 ≤ Ωð Þr < 1, where r ¼ 1, 2, ⋯. Since 0 ≤ sin ð Þ Ωð Þt < Ωð Þr < 1 holds, the system

Ωð Þr < ru þ αΩð Þþ r � 1 αΩð Þþ r � 2 ⋯ þ ð Þ α þ 1 Ωð Þ 0 : (14)

ηð Þ x; y is the condensation temperature. F represents the environmental disturbance term, called the forcing term, which is a non-linear function term. u xð Þ ; y is the initial value of the digitization, called the source term. The solution to the system Eq.(8) is very tedious. To facilitate the analysis of the solution, a discrete

Considering the boundedness and variability of soot particle agglomeration, the

; u xð Þ ; y

, (8)

ð Þ mtþ<sup>1</sup> � mt

(9)

(13)

ð Þ ntþ<sup>1</sup> � nt ; um,n�,

<sup>þ</sup> um,n, (10)

$$D\_f = slope \{ \ln \left( N\_n(A) \right), \ln \left( T\_n \right) \}. \tag{6}$$

## 3. The analysis and condensation control of soot particles fractal growth

#### 3.1 The theory of control

The formation of soot particles in diesel engines is affected by factors such as temperature, pressure, soot particle concentration, and oxidation rate [17]. According to the characteristics of free particles moving in a continuous medium state, it can be considered that the soot growth of a soot particle (condensation collision) has a distribution parameter equation of motion with boundary conditions as Eq.(7).

$$\frac{\partial \eta(\mathbf{x}, \mathbf{y}, t)}{\partial t} = \delta \left( \frac{\partial \eta^2(\mathbf{x}, \mathbf{y}, t)}{\partial \mathbf{x}^2} + \frac{\partial \eta^2(\mathbf{x}, \mathbf{y}, t)}{\partial \mathbf{y}^2} \right), \tag{7}$$

The condensation growth frequency of soot particles satisfies the distribution parameter system Eq.(8).

The Diesel Soot Particles Fractal Growth Model and Its Agglomeration Control DOI: http://dx.doi.org/10.5772/intechopen.80851

$$
\nabla^2 \eta(\mathbf{x}, \boldsymbol{y}) = F\left(\eta(\mathbf{x}, \boldsymbol{y}), \frac{\partial \eta}{\partial t}, \boldsymbol{\mu}(\mathbf{x}, \boldsymbol{y})\right), \tag{8}
$$

ηð Þ x; y is the condensation temperature. F represents the environmental disturbance term, called the forcing term, which is a non-linear function term. u xð Þ ; y is the initial value of the digitization, called the source term. The solution to the system Eq.(8) is very tedious. To facilitate the analysis of the solution, a discrete power system of Eq.(8) is introduced as Eq.(9).

$$\begin{aligned} \left(\eta\_{m+1,n} + \eta\_{m-1,n} + \eta\_{m,n+1} + \eta\_{m,n-1} - 4\eta\_{m,n} = F\left[\eta\_{m,n}, \left(\eta\_{m+1,n} - \eta\_{m,n}\right)(m\_{t+1} - m\_t)\right] \right) \\ &+ \left(\eta\_{m,n+1} - \eta\_{m,n}\right)(n\_{t+1} - n\_t), \mu\_{m,n}\big), \end{aligned} \tag{9}$$

Considering the boundedness and variability of soot particle agglomeration, the nonlinear function F is set as Eq.(10).

$$F = a \sin\left(\eta\_{m,n}\right) + u\_{m,n} \tag{10}$$

For more generalized processing problems, the system Eq.(11) is hereby introduced.

$$
\Omega(r) = a \sin(\Omega(r-1)) + \Omega(r-1) + u\_{m,n} \tag{11}
$$

where

Df ¼ lim<sup>n</sup>!<sup>∞</sup>

3.1 The theory of control

Figure 7.

parameter system Eq.(8).

20

<sup>∂</sup>ηð Þ <sup>x</sup>; <sup>y</sup>; <sup>t</sup>

The sandbox method for the fractal dimension of soot condensed matter.

Kinetic Modeling for Environmental Systems

3. The analysis and condensation control of soot particles fractal growth

The formation of soot particles in diesel engines is affected by factors such as temperature, pressure, soot particle concentration, and oxidation rate [17]. According to the characteristics of free particles moving in a continuous medium state, it can be considered that the soot growth of a soot particle (condensation collision) has a distribution parameter equation of motion with boundary conditions as Eq.(7).

∂x<sup>2</sup> þ

The condensation growth frequency of soot particles satisfies the distribution

<sup>∂</sup><sup>t</sup> <sup>¼</sup> <sup>δ</sup> <sup>∂</sup>η<sup>2</sup>ð Þ <sup>x</sup>; <sup>y</sup>; <sup>t</sup>

Nnð Þ A �lnTn

Df ¼ slopef g ln ð Þ Nnð Þ A ; ln ð Þ Tn : (6)

<sup>∂</sup>η<sup>2</sup>ð Þ <sup>x</sup>; <sup>y</sup>; <sup>t</sup> ∂y<sup>2</sup>

, (5)

, (7)

$$\mathfrak{Q}(r) = \eta\_{m+r,n} + \eta\_{m-r,n} + \eta\_{m,n+r} + \eta\_{m,n-r}, r = \mathbf{1}, \mathbf{2}, \cdots \tag{12}$$

Obviously, when r ¼ 1, Eq.(12) becomes system Eq.(9). By iterating simplification of Eq.(9), a simple control system can be obtained as Eq.(13).

$$\Delta(r) = r u + a \sin(\Omega(r-1)) + a \sin(\Omega(r-2)) + \cdots + a \sin(\Omega(0)) + \Omega(0), r = 1, 2, \cdots \tag{13}$$

## 3.2 The control of fractal growth for diesel engines' soot particles from source item and nonlinear term

According to the control method of [18], this chapter analyzes the effect of this control method on the fractal growth of soot particles. Assuming that H is a condensed region, H is a condensed boundary, and M is the scope of control of the source item u xð Þ ; y , and satisfies MathcalM ∈ H. In addition, for any ð Þ x; y ∈ H, there is 0 ≤ ηð Þ x; y ≤ 1 established. Since the analytical function u xð Þ ; y satisfies the maximum principle in H, for any ð Þ x; y ∈ H � H, condition 0 ≤ ηð Þ xy < 1 must be true, so α and u in Eq.(11) must be as small as possible, represented by an inequality:

$$\mathbf{0} \le \eta\_{m+r,n} + \eta\_{m-r,n} + \eta\_{m,n+r} + \eta\_{m,n-r} < \mathbf{1},$$

0 ≤ Ωð Þr < 1, where r ¼ 1, 2, ⋯. Since 0 ≤ sin ð Þ Ωð Þt < Ωð Þr < 1 holds, the system (10) satisfies the relationship Eq.(14).

$$
\mathfrak{Q}(r) < ru + a\mathfrak{Q}(r-1) + a\mathfrak{Q}(r-2) + \dots + (a+1)\mathfrak{Q}(0). \tag{14}
$$

According to Eq.(13) and combined with mathematical induction, Eq.(15) can be get.

$$\Delta(r) \prec \left[ \left( a + \mathbf{1} \right)^{r-1} + \left( a + \mathbf{1} \right)^{r-2} + \dots + \left( a + \mathbf{1} \right) + \mathbf{1} \right] u + \left( a + \mathbf{1} \right)^r \Omega(\mathbf{0}), r = \mathbf{1}, 2, \dots, \mathbf{1} \tag{15}$$

Eq.(16) is get by changing the inequality of Ωð Þr to Ψð Þ α; u;r

$$\Psi(a,u,r) = \left[ \left( a+\mathbf{1} \right)^{r-1} + \left( a+\mathbf{1} \right)^{r-2} + \dots + \left( a+\mathbf{1} \right) + \mathbf{1} \right] u + \left( a+\mathbf{1} \right)^{r} \Omega(0) \tag{16}$$

And because 0 < α ≤ 1, 0 < u ≤ 1, it turns out to have <sup>∂</sup><sup>Ψ</sup> <sup>∂</sup><sup>α</sup> >0 and <sup>∂</sup><sup>Ψ</sup> <sup>∂</sup><sup>u</sup> >0 is true, Ωð Þr is monotonically increasing about α and u, respectively.

The particle condensation temperature η will increase with the increase of the nonlinear term α sinð Þη and the source term u. For system Eq. (8), when the action region of source item u is circular (r is a radius), the values of u are constants and random numbers(rand represents a random number in the range (0,1), respectively, and the resulting simulated pictures are shown in Figures 8–10. Comparing with Figure 3 and Figure 4 without interference and other model [19] shown in Figure 11, this chapter simulates the morphological structure of collision for the single-single particles, single-clusters, clusters-clusters, and it is obvious that the effect of the increase of the interference term and the action region on the control of the aggregation of particles is more and more condensed than in the absence of the

Figure 8. The control of single direction.

interference term. The concentration of the particles after the condensation is greater for the settlement and the aggregation. Condensation can also have a fixed direction. The control method on the fractal growth reduces the complexity of the surface area of aggregated particles and reflects the effectiveness of the control

The fractal structure of the particles is closely related to the binding resistance and the adsorption of the particles. The literature [20] investigated the relationship between the viscous resistance and the fractal structure of the particles during the descending process. Particles with a fractal structure will have a larger fractal dimension, the smaller the viscous resistance, and the faster the sedimentation rate

4. The meaning of the soot particles condensed control

Fractal diffusion of soot particles established by others [19].

method.

23

Figure 11.

Figure 10.

Particles' center point coagulation control.

DOI: http://dx.doi.org/10.5772/intechopen.80851

The Diesel Soot Particles Fractal Growth Model and Its Agglomeration Control

Figure 9. The control of multiple direction.

The Diesel Soot Particles Fractal Growth Model and Its Agglomeration Control DOI: http://dx.doi.org/10.5772/intechopen.80851

Figure 10. Particles' center point coagulation control.

According to Eq.(13) and combined with mathematical induction, Eq.(15) can

The particle condensation temperature η will increase with the increase of the nonlinear term α sinð Þη and the source term u. For system Eq. (8), when the action region of source item u is circular (r is a radius), the values of u are constants and random numbers(rand represents a random number in the range (0,1), respectively, and the resulting simulated pictures are shown in Figures 8–10. Comparing with Figure 3 and Figure 4 without interference and other model [19] shown in Figure 11, this chapter simulates the morphological structure of collision for the single-single particles, single-clusters, clusters-clusters, and it is obvious that the effect of the increase of the interference term and the action region on the control of the aggregation of particles is more and more condensed than in the absence of the

<sup>u</sup> <sup>þ</sup> ð Þ <sup>α</sup> <sup>þ</sup> <sup>1</sup> <sup>r</sup>

<sup>u</sup> <sup>þ</sup> ð Þ <sup>α</sup> <sup>þ</sup> <sup>1</sup> <sup>r</sup>

<sup>∂</sup><sup>α</sup> >0 and <sup>∂</sup><sup>Ψ</sup>

Ωð Þ 0 , r ¼ 1, 2, ⋯:

Ωð Þ 0 (16)

<sup>∂</sup><sup>u</sup> >0 is true, Ωð Þr

(15)

<sup>Ω</sup>ð Þ<sup>r</sup> <sup>&</sup>lt; ð Þ <sup>α</sup> <sup>þ</sup> <sup>1</sup> <sup>r</sup>�<sup>1</sup> <sup>þ</sup> ð Þ <sup>α</sup> <sup>þ</sup> <sup>1</sup> <sup>r</sup>�<sup>2</sup> <sup>þ</sup> <sup>⋯</sup> <sup>þ</sup> ð Þþ <sup>α</sup> <sup>þ</sup> <sup>1</sup> <sup>1</sup> h i

Kinetic Modeling for Environmental Systems

Eq.(16) is get by changing the inequality of Ωð Þr to Ψð Þ α; u;r

h i

<sup>Ψ</sup>ð Þ¼ <sup>α</sup>; <sup>u</sup>;<sup>r</sup> ð Þ <sup>α</sup> <sup>þ</sup> <sup>1</sup> <sup>r</sup>�<sup>1</sup> <sup>þ</sup> ð Þ <sup>α</sup> <sup>þ</sup> <sup>1</sup> <sup>r</sup>�<sup>2</sup> <sup>þ</sup> <sup>⋯</sup> <sup>þ</sup> ð Þþ <sup>α</sup> <sup>þ</sup> <sup>1</sup> <sup>1</sup>

And because 0 < α ≤ 1, 0 < u ≤ 1, it turns out to have <sup>∂</sup><sup>Ψ</sup>

is monotonically increasing about α and u, respectively.

be get.

Figure 8.

Figure 9.

22

The control of single direction.

The control of multiple direction.

Figure 11. Fractal diffusion of soot particles established by others [19].

interference term. The concentration of the particles after the condensation is greater for the settlement and the aggregation. Condensation can also have a fixed direction. The control method on the fractal growth reduces the complexity of the surface area of aggregated particles and reflects the effectiveness of the control method.

## 4. The meaning of the soot particles condensed control

The fractal structure of the particles is closely related to the binding resistance and the adsorption of the particles. The literature [20] investigated the relationship between the viscous resistance and the fractal structure of the particles during the descending process. Particles with a fractal structure will have a larger fractal dimension, the smaller the viscous resistance, and the faster the sedimentation rate than spherical particles of the same volume. The fractal dimension of the particle before control shown in Figures 3 and 4 is 2.029 and 2.236, respectively. The fractal dimension of the particle after control in Figure 8 is 2.3273. Obviously, the viscous resistance of the particle in Figure 8 is small, which is conducive to the settlement of particles.

The fractal dimension of particles directly affects the surface adsorption. The relationship between the number of saturated molecules adsorbed in single layer Nm and the cross-sectional area of adsorbed molecules Sm is given by Eq. (17), where ξ is the scale factor and Df is the fractal dimension of the particle.

$$N\_m = \xi(\mathbb{S}\_m)^{\left(-\frac{D\_f}{2}\right)},\tag{17}$$

This chapter simulates the control of the aggregation fractal growth trend of diesel soot particles. The results of numerical simulation show that the proposed method is feasible and effective, which will help to understand and analyze the physical mechanism and kinetics of non-equilibrium condensation growth behavior of the actual carbon smoke particles and provide the solution to further reduce emissions of the inhalable particulate matter from diesel

The Diesel Soot Particles Fractal Growth Model and Its Agglomeration Control

DOI: http://dx.doi.org/10.5772/intechopen.80851

The work was supported by the National Natural Science Foundation of China (No. 31700644), Postdoctoral Science Foundation of China (Nos. 2015 M582122 and 2016 T90644), Key research and development project of Shandong Province(Nos. 2016ZDJS02A07 and 2017GNC12105). Agricultural machinery research and development project of Shandong Province (No. 2018YF004)." The outstanding youth talent cultivation plan" project of Shandong Agriculture University (No. 564032). The authors are grateful to all study participants. The authors declared that they have no conflicts of interest to this work. We declare that we do not have any commercial or associative interest that represents a conflict of interest in connection

engines.

Acknowledgements

with the work submitted.

α the coefficient of equation. ηð Þ x; y the condensation temperature.

u xð Þ ; y the initial value of the digitization.

College of Mechanical and Electronic Engineering, Shandong Agricultural

© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

\*Address all correspondence to: liupingsdau@126.com

provided the original work is properly cited.

cn the judgment factor.

xs the center of the ball Bm.

Nomenclature

Author details

25

Ping Liu\* and Chunying Wang

University, Taian, China

If the adsorbate molecular weight is M and the density is ρ, then the adsorption amount Q<sup>0</sup> is Eq.(18).

$$\begin{aligned} Q' &= N\_m \frac{M}{\rho}, \\ Q' &= \varepsilon \frac{M}{\rho} (\mathbf{S}\_m)^{\left(-\frac{D\_f}{2}\right)}. \end{aligned} \tag{18}$$

Obviously, the adsorption of toxic particulates by atmospheric particles is not only related to the composition and chemical properties of gas molecules but also related to the fractal dimension of the particle surface. The roughness of the surface of atmospheric particles also affects the adsorption of toxic gases in the atmosphere. The bigger fractal dimension particles have, the stronger adsorption of toxic particulate matter atmospheric particles have. Then, atmospheric particles will greatly affect human health.

The controlled particulate matter (Figure 8) can adsorb more toxic particulate matter and cause it to control the settlement and reduce the environmental pollution. In addition, if the particulate matter still cannot settle after control, it will be controlled as in Figure 6 (fractal dimension is 2.029) and Figure 7 (fractal dimension 2.021, 2.031 and 2.038, respectively) shape structure, in order to reduce the adsorption of particles on toxic particles and the harm to human health.

### 5. Conclusions

The analysis and its agglomeration control of soot particles fractal growth provides a new idea for the development of particulate matter traps and also provides a new solution for reducing environmental pollution. Based on the fractal growth physical model of soot particles from large diesel agriculture machinery, this chapter simulates the morphological structure of collision for the single particles and single particles, single particle and clusters, clusters and clusters, firstly. Moreover, combining with the collision frequency, the fractal growth is controlled to agglomeration using the main environmental factors interference for diesel engine soot particles, in order to make them condensed into regular geometry or larger density particles, reduce the viscous drag for capturing by the capturer or settlement and to realize the control of the pollution of the environment.

If the particles cannot settle, they can be controlled to reduce the adsorption of inhalable particles to toxic particles and reduce the harm to human health.

The Diesel Soot Particles Fractal Growth Model and Its Agglomeration Control DOI: http://dx.doi.org/10.5772/intechopen.80851

This chapter simulates the control of the aggregation fractal growth trend of diesel soot particles. The results of numerical simulation show that the proposed method is feasible and effective, which will help to understand and analyze the physical mechanism and kinetics of non-equilibrium condensation growth behavior of the actual carbon smoke particles and provide the solution to further reduce emissions of the inhalable particulate matter from diesel engines.

## Acknowledgements

than spherical particles of the same volume. The fractal dimension of the particle before control shown in Figures 3 and 4 is 2.029 and 2.236, respectively. The fractal dimension of the particle after control in Figure 8 is 2.3273. Obviously, the viscous resistance of the particle in Figure 8 is small, which is conducive to the settlement

The fractal dimension of particles directly affects the surface adsorption. The relationship between the number of saturated molecules adsorbed in single layer Nm and the cross-sectional area of adsorbed molecules Sm is given by Eq. (17), where ξ is the scale factor and Df is the fractal dimension of the particle.

Nm ¼ ξð Þ Sm �

Q<sup>0</sup> ¼ Nm

Q<sup>0</sup> ¼ ε

If the adsorbate molecular weight is M and the density is ρ, then the adsorption

M ρ ,

Obviously, the adsorption of toxic particulates by atmospheric particles is not only related to the composition and chemical properties of gas molecules but also related to the fractal dimension of the particle surface. The roughness of the surface of atmospheric particles also affects the adsorption of toxic gases in the atmosphere. The bigger fractal dimension particles have, the stronger adsorption of toxic particulate matter atmospheric particles have. Then, atmospheric particles will greatly

The controlled particulate matter (Figure 8) can adsorb more toxic particulate matter and cause it to control the settlement and reduce the environmental pollution. In addition, if the particulate matter still cannot settle after control, it will be controlled as in Figure 6 (fractal dimension is 2.029) and Figure 7 (fractal dimension 2.021, 2.031 and 2.038, respectively) shape structure, in order to reduce the adsorption of particles on toxic particles and the harm to

The analysis and its agglomeration control of soot particles fractal growth provides a new idea for the development of particulate matter traps and also provides a new solution for reducing environmental pollution. Based on the fractal growth physical model of soot particles from large diesel agriculture machinery, this chapter simulates the morphological structure of collision for the single particles and single particles, single particle and clusters, clusters and clusters, firstly. Moreover, combining with the collision frequency, the fractal growth is controlled to agglomeration using the main environmental factors interference for diesel engine soot particles, in order to make them condensed into regular geometry or larger density particles, reduce the viscous drag for capturing by the capturer or settlement and to realize the control of the pollu-

If the particles cannot settle, they can be controlled to reduce the adsorption of inhalable particles to toxic particles and reduce the harm to human health.

M <sup>ρ</sup> ð Þ Sm �

Df 2 

> Df 2 :

, (17)

(18)

of particles.

Kinetic Modeling for Environmental Systems

amount Q<sup>0</sup> is Eq.(18).

affect human health.

human health.

5. Conclusions

tion of the environment.

24

The work was supported by the National Natural Science Foundation of China (No. 31700644), Postdoctoral Science Foundation of China (Nos. 2015 M582122 and 2016 T90644), Key research and development project of Shandong Province(Nos. 2016ZDJS02A07 and 2017GNC12105). Agricultural machinery research and development project of Shandong Province (No. 2018YF004)." The outstanding youth talent cultivation plan" project of Shandong Agriculture University (No. 564032). The authors are grateful to all study participants. The authors declared that they have no conflicts of interest to this work. We declare that we do not have any commercial or associative interest that represents a conflict of interest in connection with the work submitted.

## Nomenclature


## Author details

Ping Liu\* and Chunying Wang College of Mechanical and Electronic Engineering, Shandong Agricultural University, Taian, China

\*Address all correspondence to: liupingsdau@126.com

© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

## References

[1] Poran A, Tartakovsky L. Performance and emissions of a direct injection internal combustion engine devised for joint operation with a highpressure thermochemical recuperation system. Energy. 2017;124:214-226

[2] Robelia B, Mcneill K, Wammer K, et al. Investigating the impact of adding an environmental focus to a developmental chemistry course. Journal of Chemical Education. 2010; 87(2):216-220

[3] Kittelson DB. Engines and nanoparticles: A review. Journal of Aerosol Science. 1998;29(5–6):575-588

[4] Tian H, Liao Z. Progress on the formation mechanism of biomass soot particles. Clean Coal Technology. 2017; 23(3):7-15

[5] Shuai J et al. Review of formation mechanism and emission characteristics of particulate matter from automotive gasoline engines. Transactions of CSICE;2016(2):105-116

[6] Maozhao XIE. The Computational Combustion Theory of the Internal Combustion Engine. Dalian: Dalian Institute of Technology Press; 2005

[7] M Balthasar, M Frenklach. Monte-Carlo simulation of soot particle coagulation and aggregation: The effect of a realistic size distribution. Proceedings of the Combustion. 2005; 30(1):1467-1475

[8] Zhang L, Liu ST. Directed control for fractal growth with environmental disturbance. Control Theory & Applications. 2011;28(12):1786-1790

[9] Liu P, Liu ST. Nonlinear generalized synchronization of two different spatial Julia sets. Control Theory & Applications. 2013;30(9):1159-1164

[10] Hu E, Hu X, Liu T, et al. The role of soot particles in the tribological behavior of engine lubricating oils. Wear. 2013;304(1–2):152-161

[19] Sun J, Qiao W, Liu S. Controlling fractal diffusion of differently-sized soot particles. International Journal of

DOI: http://dx.doi.org/10.5772/intechopen.80851

The Diesel Soot Particles Fractal Growth Model and Its Agglomeration Control

Bifurcation and Chaos. 2018

[20] Chou CK, Lee CT. On the aerodynamic behavior of fractal agglomerates. Journal of Aerosol Science. 1997;28(2):620-635

27

[11] Liu Y, Tao F, Foster DE, et al. Application of a multiple-step phenomenological soot model to HSDT diesel multiple injection modeling. In: 2005 SAE World Congress. Warrendale: SAE Transactions; 2005. pp. 1141-1156

[12] Pang KM, Ng HK, Gan S. Investigation of fuel injection pattern on soot formation and oxidation processes in a light-duty diesel engine using integrated CFD-reduced chemistry. Fuel. 2012;96(7):404-418

[13] Ming-rui W, Hui-ya Z, Liang K, Wei-dong Z. Numerical simulation for the growth of diesel particulate matter and its influential factor analysis. Advances in Natural Science. 2008; 9(18):1028-1033

[14] Koto F, Yanagimoto T, Mori K, et al. The Clarification of Fuel-Vapor Concentration on the Process of Initial Combustion and Soot Formation in a Di Diesel Engine. 2017;2003.1:1-253-1-258

[15] A Witten T, Sander LM. Diffusionlimited aggregation, a kinetic critical phenomenon. Physical Review Letters. 1981;47(19):1400 (p 4)

[16] Jizhong Z. Fractal. Beijing: Tsinghua University Press; 2011

[17] Deng X, Dav RN. Breakage of fractal agglomerates. Chemical Engineering Science. 2017;161:117-126

[18] Pfeifer P, Avnir D. Chemistry in noninteger dimensions between two and three. I. Fractal theory of heterogeneous surfaces. The Journal of Chemical Physics. 1983;79(7):3558-3565 The Diesel Soot Particles Fractal Growth Model and Its Agglomeration Control DOI: http://dx.doi.org/10.5772/intechopen.80851

[19] Sun J, Qiao W, Liu S. Controlling fractal diffusion of differently-sized soot particles. International Journal of Bifurcation and Chaos. 2018

References

87(2):216-220

23(3):7-15

[1] Poran A, Tartakovsky L.

an environmental focus to a developmental chemistry course. Journal of Chemical Education. 2010;

[3] Kittelson DB. Engines and nanoparticles: A review. Journal of Aerosol Science. 1998;29(5–6):575-588

[4] Tian H, Liao Z. Progress on the formation mechanism of biomass soot particles. Clean Coal Technology. 2017;

[5] Shuai J et al. Review of formation mechanism and emission characteristics of particulate matter from automotive gasoline engines. Transactions of

[6] Maozhao XIE. The Computational Combustion Theory of the Internal Combustion Engine. Dalian: Dalian Institute of Technology Press; 2005

[7] M Balthasar, M Frenklach. Monte-Carlo simulation of soot particle coagulation and aggregation: The effect

Proceedings of the Combustion. 2005;

[8] Zhang L, Liu ST. Directed control for fractal growth with environmental disturbance. Control Theory & Applications. 2011;28(12):1786-1790

[9] Liu P, Liu ST. Nonlinear generalized synchronization of two different spatial

of a realistic size distribution.

Julia sets. Control Theory & Applications. 2013;30(9):1159-1164

30(1):1467-1475

26

CSICE;2016(2):105-116

Performance and emissions of a direct injection internal combustion engine devised for joint operation with a highpressure thermochemical recuperation system. Energy. 2017;124:214-226

Kinetic Modeling for Environmental Systems

[10] Hu E, Hu X, Liu T, et al. The role of soot particles in the tribological behavior of engine lubricating oils. Wear. 2013;304(1–2):152-161

[11] Liu Y, Tao F, Foster DE, et al. Application of a multiple-step

[12] Pang KM, Ng HK, Gan S.

Fuel. 2012;96(7):404-418

9(18):1028-1033

phenomenological soot model to HSDT diesel multiple injection modeling. In: 2005 SAE World Congress. Warrendale: SAE Transactions; 2005. pp. 1141-1156

Investigation of fuel injection pattern on soot formation and oxidation processes in a light-duty diesel engine using integrated CFD-reduced chemistry.

[13] Ming-rui W, Hui-ya Z, Liang K, Wei-dong Z. Numerical simulation for the growth of diesel particulate matter and its influential factor analysis. Advances in Natural Science. 2008;

[14] Koto F, Yanagimoto T, Mori K, et al.

[15] A Witten T, Sander LM. Diffusionlimited aggregation, a kinetic critical phenomenon. Physical Review Letters.

[16] Jizhong Z. Fractal. Beijing: Tsinghua

[17] Deng X, Dav RN. Breakage of fractal agglomerates. Chemical Engineering

[18] Pfeifer P, Avnir D. Chemistry in noninteger dimensions between two and three. I. Fractal theory of

heterogeneous surfaces. The Journal of Chemical Physics. 1983;79(7):3558-3565

The Clarification of Fuel-Vapor Concentration on the Process of Initial Combustion and Soot Formation in a Di Diesel Engine. 2017;2003.1:1-253-1-258

1981;47(19):1400 (p 4)

University Press; 2011

Science. 2017;161:117-126

[2] Robelia B, Mcneill K, Wammer K, et al. Investigating the impact of adding [20] Chou CK, Lee CT. On the aerodynamic behavior of fractal agglomerates. Journal of Aerosol Science. 1997;28(2):620-635

Chapter 3

Abstract

operational variables.

1. Introduction

29

numerous interesting applications.

Kinetic Modeling of

Photodegradation of

Photochemical Reactor

Water-Soluble Polymers in Batch

Synthetic water-soluble polymers, well-known refractory pollutants, are abundant in wastewater effluents since they are extensively used in industry in a wide range of applications. These polymers can be effectively degraded by advanced oxidation processes (AOPs). This entry thoroughly covers the development of the photochemical kinetic model of the polyvinyl alcohol (PVA) degradation in UV/ H2O2 advanced oxidation batch process that describes the disintegration of the polymer chains in which the statistical moment approach is considered. The reaction mechanism used to describe the photo-degradation of polymers comprises photolysis, polymer chain scission, and mineralization reactions. The impact of operating conditions on the process performance is evaluated. Characterization of the polymer average molecular weights, total organic carbon, and hydrogen peroxide concentrations as essential factors in developing a reliable photochemical model of the UV/H2O2 process is discussed. The statistical moment approach is applied to model the molar population balance of live and dead polymer chains taking into account the probabilistic chain scissions of the polymer. The photochemical kinetic model provides a comprehensive understanding of the impact of the design and

Keywords: kinetic modeling, population balance, free radical-induced degradation,

The growing turnover and consumption of synthetic water-soluble polymers

generate a huge amount of wastes during production, use, and disposal off. After usage, water-soluble polymers are expected to end up in rivers, lakes, oceans and even in wastewater treatment plants, thus creating a potential pollution hazard. In contrast to biopolymers, water-soluble polymers are resistant to microorganisms-mediated biodegradation [1–3]. Synthetic water-soluble polymers cover a wide range of highly varied families of products and have

One of the concerns is the accumulation of such non-biodegradable watersoluble polymers in the environment. Particularly, polyvinyl alcohol (PVA) is one

advanced oxidation process, water-soluble polymer

Dina Hamad, Mehrab Mehrvar and Ramdhane Dhib

## Chapter 3

## Kinetic Modeling of Photodegradation of Water-Soluble Polymers in Batch Photochemical Reactor

Dina Hamad, Mehrab Mehrvar and Ramdhane Dhib

## Abstract

Synthetic water-soluble polymers, well-known refractory pollutants, are abundant in wastewater effluents since they are extensively used in industry in a wide range of applications. These polymers can be effectively degraded by advanced oxidation processes (AOPs). This entry thoroughly covers the development of the photochemical kinetic model of the polyvinyl alcohol (PVA) degradation in UV/ H2O2 advanced oxidation batch process that describes the disintegration of the polymer chains in which the statistical moment approach is considered. The reaction mechanism used to describe the photo-degradation of polymers comprises photolysis, polymer chain scission, and mineralization reactions. The impact of operating conditions on the process performance is evaluated. Characterization of the polymer average molecular weights, total organic carbon, and hydrogen peroxide concentrations as essential factors in developing a reliable photochemical model of the UV/H2O2 process is discussed. The statistical moment approach is applied to model the molar population balance of live and dead polymer chains taking into account the probabilistic chain scissions of the polymer. The photochemical kinetic model provides a comprehensive understanding of the impact of the design and operational variables.

Keywords: kinetic modeling, population balance, free radical-induced degradation, advanced oxidation process, water-soluble polymer

## 1. Introduction

The growing turnover and consumption of synthetic water-soluble polymers generate a huge amount of wastes during production, use, and disposal off. After usage, water-soluble polymers are expected to end up in rivers, lakes, oceans and even in wastewater treatment plants, thus creating a potential pollution hazard. In contrast to biopolymers, water-soluble polymers are resistant to microorganisms-mediated biodegradation [1–3]. Synthetic water-soluble polymers cover a wide range of highly varied families of products and have numerous interesting applications.

One of the concerns is the accumulation of such non-biodegradable watersoluble polymers in the environment. Particularly, polyvinyl alcohol (PVA) is one of the most commercially important water-soluble synthetic polymers with an annual worldwide production of 650,000 tons. PVA polymers are abundant in wastewater effluents due to the extensive usage in paper and textile industries that accordingly generates significant amounts of PVA in wastewater streams [4, 5]. The PVA polymers are used in industry as paper and textile coatings, and also as laundry packing materials [6]. Its iodine complexes are widely used as polarization layers in liquid crystal displays (LCDs) [7]. As the production of PVA finds new markets, its consumption grows and the volume of wastewater containing PVA increases during its production and consumption. Moreover, PVA is highly soluble in water, and it leaches readily from soil into groundwater creating environmental issues. PVA polymers act as collector reagents that can be either chemisorbed or physically adsorbed since these polymer compounds have oxygen hetero-atoms capable of binding to different metal ions effectively and increase the mobilization of heavy metals from sediments of lakes and oceans which results in accumulation of hazardous materials. [6, 8–11]. Besides, The PVA solutions exhibit high surface activity supporting the formation of foams which can hinder the transport of oxygen into water streams. Therefore, the removal of PVA from wastewater systems is essential.

Conventional biological technologies are not efficient to breakdown PVA polymer chains since the degradation capacity of most microorganisms towards PVA is very limited and requires specially adapted bacteria strains [1]. In addition, wastewaters containing PVA can cause foam formation in biological equipment which inhibits the activity of aerobic microorganisms due to oxygen absence that results in unstable operation with low performance [9]. As a result, the advanced oxidation processes are utilized as alternative treatment techniques for the treatment of polymeric wastewater systems. The advanced oxidation technologies are proven to be effective in treating industrial wastewater [10, 11]. AOPs involve the formation of strong oxidants such as hydroxyl radicals and the reaction of these oxidants with pollutants in wastewater. In wastewater treatment applications, AOPs usually refer to a specific subset of processes that involve H2O2, O3, and UV light as shown in the schematic diagram in Figure 1.

The degradation of water-soluble polymers by different AOPs is studied in the open literature whether those polymers are refractory, toxic, hazardous or recalcitrant compounds. Recent studies on the removal of PVA have focused on radiationinduced oxidation process such as photo-Fenton [7], photocatalytic processes [7, 12], radiation-induced electrochemical process [13], and UV/H2O2 process [14–17]. Even though the degradation of a polymer component must be assessed by the reduction and analysis of its molecular weights, there are only a few studies in the open literature on the devolution of the molecular weight size distributions of water-soluble polymers [9, 17]. Also, the residual hydrogen peroxide is still a challenging issue in the UV/H2O2 process which has been overlooked in some studies.

concepts is often applied in fragmentation models to describe how the distributions

Kinetic Modeling of Photodegradation of Water-Soluble Polymers in Batch Photochemical Reactor

The degradation of high molecular weight polydisperse materials results in the formation of a large number of polymeric chains with different chain lengths and various chemical compositions, i.e., the number of branches. Population balance based models have been developed to study the molecular weight decrease of polymers in a fragmentation process [5, 13, 20–22]. Population balance approach is generally employed to model the size distribution of the macromolecular compound during polymerization, depolymerization, and chain breakage. The population balance model is a balance equation of species of different sizes, and it is similar to the mass, energy, and momentum balances, to track the changes in the

Few important studies have been done to understand the chemical kinetics that

dominates the degradation of water-soluble polymers with UV radiation. Even though encouraging results on the degradation of polymers were obtained, data on the molecular weight distribution of the treated polymer need to be collected. Hence, it is worthwhile to investigate further the degradation process of synthetic polymer and the devolution of their molecular weight distributions. Other studies have theoretically analyzed the thermal degradation of synthetic polymers and provided a mathematical interpretation of the polymer chain scissions. The photochemical mechanism and kinetic modeling of the photo-oxidative degradation of water-soluble polymers have been investigated in several studies. Nevertheless, the proposed mechanisms may be complex and not well-established. The majority of mathematical approaches to polymer degradation consider only the polymer

of different size entities evolve over the time of reaction.

Schematic classification of the advanced oxidation processes.

DOI: http://dx.doi.org/10.5772/intechopen.82608

size distribution.

31

Figure 1.

Furthermore, there is little information on the photochemical mechanism of the photo-oxidative degradation of PVA polymer solutions in a UV/H2O2 process. Recently, several attempts have been made to comprehend the chemical kinetics dominating thermal degradation of water-soluble polymers and assuming constant pH [18, 19]. Besides, no data is available on the distribution of the molecular weights of the polymer being degraded.

Under UV radiation, polymer chains are broken down into oligomers (shortchain polymers), dimers and monomers. Enhanced photo-degradation of polymer can lead to a broader distribution of molecular weights, indicating that the degraded polymer becomes less and less uniform. This behavior is expected for degraded polymers, as irradiation promotes an increase in the number of polymer chains, lowering the molecular weight, and consequently increasing the polydispersity. Hence, polymer degradation is a fragmentation process in which population balance Kinetic Modeling of Photodegradation of Water-Soluble Polymers in Batch Photochemical Reactor DOI: http://dx.doi.org/10.5772/intechopen.82608

Figure 1. Schematic classification of the advanced oxidation processes.

concepts is often applied in fragmentation models to describe how the distributions of different size entities evolve over the time of reaction.

The degradation of high molecular weight polydisperse materials results in the formation of a large number of polymeric chains with different chain lengths and various chemical compositions, i.e., the number of branches. Population balance based models have been developed to study the molecular weight decrease of polymers in a fragmentation process [5, 13, 20–22]. Population balance approach is generally employed to model the size distribution of the macromolecular compound during polymerization, depolymerization, and chain breakage. The population balance model is a balance equation of species of different sizes, and it is similar to the mass, energy, and momentum balances, to track the changes in the size distribution.

Few important studies have been done to understand the chemical kinetics that dominates the degradation of water-soluble polymers with UV radiation. Even though encouraging results on the degradation of polymers were obtained, data on the molecular weight distribution of the treated polymer need to be collected. Hence, it is worthwhile to investigate further the degradation process of synthetic polymer and the devolution of their molecular weight distributions. Other studies have theoretically analyzed the thermal degradation of synthetic polymers and provided a mathematical interpretation of the polymer chain scissions. The photochemical mechanism and kinetic modeling of the photo-oxidative degradation of water-soluble polymers have been investigated in several studies. Nevertheless, the proposed mechanisms may be complex and not well-established. The majority of mathematical approaches to polymer degradation consider only the polymer

of the most commercially important water-soluble synthetic polymers with an annual worldwide production of 650,000 tons. PVA polymers are abundant in wastewater effluents due to the extensive usage in paper and textile industries that accordingly generates significant amounts of PVA in wastewater streams [4, 5]. The PVA polymers are used in industry as paper and textile coatings, and also as laundry packing materials [6]. Its iodine complexes are widely used as polarization layers in liquid crystal displays (LCDs) [7]. As the production of PVA finds new markets, its consumption grows and the volume of wastewater containing PVA increases during its production and consumption. Moreover, PVA is highly soluble in water, and it leaches readily from soil into groundwater creating environmental issues. PVA polymers act as collector reagents that can be either chemisorbed or physically adsorbed since these polymer compounds have oxygen hetero-atoms capable of binding to different metal ions effectively and increase the mobilization of heavy metals from sediments of lakes and oceans which results in accumulation of hazardous materials. [6, 8–11]. Besides, The PVA solutions exhibit high surface activity supporting the formation of foams which can hinder the transport of oxygen into water streams. Therefore, the removal of PVA from wastewater systems is essential. Conventional biological technologies are not efficient to breakdown PVA polymer chains since the degradation capacity of most microorganisms towards PVA is very limited and requires specially adapted bacteria strains [1]. In addition, wastewaters containing PVA can cause foam formation in biological equipment which inhibits the activity of aerobic microorganisms due to oxygen absence that results in unstable operation with low performance [9]. As a result, the advanced oxidation processes are utilized as alternative treatment techniques for the treatment of polymeric wastewater systems. The advanced oxidation technologies are proven to be effective in treating industrial wastewater [10, 11]. AOPs involve the formation of strong oxidants such as hydroxyl radicals and the reaction of these oxidants with pollutants in wastewater. In wastewater treatment applications, AOPs usually refer to a specific subset of processes that involve H2O2, O3, and UV light as shown in the

The degradation of water-soluble polymers by different AOPs is studied in the open literature whether those polymers are refractory, toxic, hazardous or recalcitrant compounds. Recent studies on the removal of PVA have focused on radiationinduced oxidation process such as photo-Fenton [7], photocatalytic processes [7, 12], radiation-induced electrochemical process [13], and UV/H2O2 process [14–17]. Even though the degradation of a polymer component must be assessed by the reduction and analysis of its molecular weights, there are only a few studies in the open literature on the devolution of the molecular weight size distributions of water-soluble polymers [9, 17]. Also, the residual hydrogen peroxide is still a challenging issue in the UV/H2O2 process which has been overlooked in some studies. Furthermore, there is little information on the photochemical mechanism of the

photo-oxidative degradation of PVA polymer solutions in a UV/H2O2 process. Recently, several attempts have been made to comprehend the chemical kinetics dominating thermal degradation of water-soluble polymers and assuming constant pH [18, 19]. Besides, no data is available on the distribution of the molecular

Under UV radiation, polymer chains are broken down into oligomers (shortchain polymers), dimers and monomers. Enhanced photo-degradation of polymer can lead to a broader distribution of molecular weights, indicating that the degraded polymer becomes less and less uniform. This behavior is expected for degraded polymers, as irradiation promotes an increase in the number of polymer chains, lowering the molecular weight, and consequently increasing the polydispersity. Hence, polymer degradation is a fragmentation process in which population balance

schematic diagram in Figure 1.

Kinetic Modeling for Environmental Systems

weights of the polymer being degraded.

30

molecular-weight distribution (MWD) or chain-length distribution. The treatment of the wastewater streams contaminated with PVA polymers is studied using different processes [7, 12, 13, 16]. The kinetics models proposed in these studies were validated using total organic carbon (TOC) data instead of polymer concentrations or polymer molecular weights, and a constant pH was assumed.

molecules cannot enter the pores of the packing and hence, they elute first. However, smaller molecules that can penetrate or diffuse into the pores are retained for a while in the column and then elute at a later time. Thus, a sample is fractionated by molecular hydrodynamic volume, and the resulting profile describes the molecular weight distribution. A concentration detector (e.g., differential refractometer (RI) or UV detector) is placed downstream of the columns to measure the concentration of each fraction as a function of time. The actual method for determining the molecular weight averages and the MWD depends upon the attached detectors. GPC provides a convenient, quick, and relatively easy method which can be used on

Kinetic Modeling of Photodegradation of Water-Soluble Polymers in Batch Photochemical Reactor

a routine basis for determining the various moments of molecular weight.

tion is determined using spectrophotometer method using 9-dimethyl-1,

matter exposed to AOPs, do not interfere with the DMP method.

and water as final products in case of complete mineralization.

detector, which is sensitive to low levels of TOC.

DOI: http://dx.doi.org/10.5772/intechopen.82608

4. Polymer chain scission mechanism

33

The extent of degradation reactions to CO2 is monitored by measuring the total organic carbon content of the samples. TOC analyzer is based on the oxidation of organic compounds to carbon dioxide and water, with subsequent quantities of carbon dioxide. The TOC analyzer subtracts the inorganic carbon (CO and CO2) and reports the total organic carbon, which is a close approximation of organic content. The amount of carbon dioxide generated upon oxidation of the organic carbon in the sample was determined by the non-dispersive infra-red (NDIR)

The reduction of hydrogen peroxide concentration during the degradation reac-

10-phenanthroline (DMP) method. The most common way of measuring hydrogen peroxide residual in wastewater is DMP-spectrophotometer method. 9-Dimethyl-1,10-phenanthroline (DMP) method is based on the chemical reduction of copper (II) by hydrogen peroxide in the presence of DMP, thus forming a bright yellow (copper (II) – DMP) complex that is directly determined by UV–vis spectrophotometer [24]. The DMP method appears to be simple, robust, and rather insensitive to interference. Intermediate compounds such as acetic and formic acids, formaldehyde, and acetaldehyde, which are formed by the decomposition of organic

The degradation of polymers by advanced oxidation processes is mainly due to free-radical-induced chain scission that led to successive oxidation reactions which result in lower molecular weight polymer fragments. The chain scission reaction is a

In other words, the chain scission reaction can be defined as a bond scission that

The concept of polymer degradation may be explained by chain-end scission or random chain scission mechanisms where chain breaking occurs at a random location along the chain. Therefore the molecular weight decreases continuously with the extent of reaction. Chain-end scission is considered as a special case of random chain scission where the chain scission reactions are occurring most likely at the polymer chain end that results in a release of a single monomer molecule. Random chain scission is the reverse of step-growth polymerization while chain-end scission

chemical reaction between the macromolecular compounds (polymers) and end/mid-chain radicals. As the reaction progresses, the large polymer molecules eventually break down into live and dead polymer chains of lower molecular weights. A further molecular disintegration can ultimately lead to carbon dioxide

takes place in the backbone of the polymer chain. The chain scission reaction increases the number of polymer chains and reduces the polymer molecular weight [25]. Consequently, the chain scission results in an increase in the polydispersity of

the polymer sample which represents the breadth of the distribution curve.

The photo-oxidative degradation of water-soluble polymers in laboratory scale photochemical reactors is the focus of this chapter. The photochemical kinetic model of the polymer degradation in UV/H2O2 process that describes the polymer chain scission is discussed in which the statistical moment approach is presented. The development of a photochemical kinetic model incorporates the population balance of all chemical species. Considering the probabilistic nature of the polymer fragments, the statistical moment approach is applied for modeling the population balance of live and dead polymer chains, which allows estimating the polymer average molecular weights as a function of radiation time. The model also considered the effect of process parameters on the decrease of polymer molecular weight, hydrogen peroxide residual, and the acidity of the treated solution.

## 2. UV/H2O2 system description

The critical design parameters in the UV/H2O2 process include the H2O2 dose, the UV lamp type and intensity, and the reactor contact time. Basic UV reactor design configurations used for the removal of polymers from wastewater depend mainly on the flow rate. The tower design is typically utilized for large-scale applications. In the tower configuration, multiple UV lamps are arranged horizontally within a single large reactor vessel with the contaminated water flowing perpendicularly past the UV lamps [23]. For small-scale systems, a single UV lamp per reactor vessel is arranged vertically. For example, a small-scale system may consist of three individual reactor vessels in series, each containing one UV lamp in a vertical position.

A typical laboratory-scale batch recirculation UV/H2O2 system consists of an annular photoreactor, a large volume reservoir tank, centrifugal pump, and heat exchanger. The circulation tank contains the polymer solution for treatment. The hydrogen peroxide is injected into the circulation tank. A centrifugal magnet pump is placed on the circulation line to maintain a steady flow of the aqueous polymer solution between the tank and photoreactor. A flow meter is used to measure the circulation rate. The cylindrical photoreactor is made of steel vessel of annual shape and is connected to the circulation tank. The reactor is equipped with an internal quartz glass in which a low-pressure mercury UV lamp is mounted at its centerline of the cylinder with stainless steel housing. The annular photoreactor should have a very small annular space to assure a uniform light distribution in the photoreactor. Most AOPs are modular processes. Therefore, more than one reactor can be employed in series mode to obtain higher retention times or in parallel mode to process larger volumes to achieve the desired effluent for a given flow rate.

## 3. Characterization of polymeric wastewater

Determination of the polymer molecular weight, TOC content, and residual hydrogen peroxide are crucial parameters to assess the performance of the photodegradation process. The treated samples are analyzed using gel permeation chromatography (GPC) to determine the molecular weights of the degraded polymer samples. The GPC theory depends on the principle of size exclusion; therefore, when a polymer solution is passed through a column of porous particles, large

#### Kinetic Modeling of Photodegradation of Water-Soluble Polymers in Batch Photochemical Reactor DOI: http://dx.doi.org/10.5772/intechopen.82608

molecules cannot enter the pores of the packing and hence, they elute first. However, smaller molecules that can penetrate or diffuse into the pores are retained for a while in the column and then elute at a later time. Thus, a sample is fractionated by molecular hydrodynamic volume, and the resulting profile describes the molecular weight distribution. A concentration detector (e.g., differential refractometer (RI) or UV detector) is placed downstream of the columns to measure the concentration of each fraction as a function of time. The actual method for determining the molecular weight averages and the MWD depends upon the attached detectors. GPC provides a convenient, quick, and relatively easy method which can be used on a routine basis for determining the various moments of molecular weight.

The extent of degradation reactions to CO2 is monitored by measuring the total organic carbon content of the samples. TOC analyzer is based on the oxidation of organic compounds to carbon dioxide and water, with subsequent quantities of carbon dioxide. The TOC analyzer subtracts the inorganic carbon (CO and CO2) and reports the total organic carbon, which is a close approximation of organic content. The amount of carbon dioxide generated upon oxidation of the organic carbon in the sample was determined by the non-dispersive infra-red (NDIR) detector, which is sensitive to low levels of TOC.

The reduction of hydrogen peroxide concentration during the degradation reaction is determined using spectrophotometer method using 9-dimethyl-1, 10-phenanthroline (DMP) method. The most common way of measuring hydrogen peroxide residual in wastewater is DMP-spectrophotometer method. 9-Dimethyl-1,10-phenanthroline (DMP) method is based on the chemical reduction of copper (II) by hydrogen peroxide in the presence of DMP, thus forming a bright yellow (copper (II) – DMP) complex that is directly determined by UV–vis spectrophotometer [24]. The DMP method appears to be simple, robust, and rather insensitive to interference. Intermediate compounds such as acetic and formic acids, formaldehyde, and acetaldehyde, which are formed by the decomposition of organic matter exposed to AOPs, do not interfere with the DMP method.

### 4. Polymer chain scission mechanism

The degradation of polymers by advanced oxidation processes is mainly due to free-radical-induced chain scission that led to successive oxidation reactions which result in lower molecular weight polymer fragments. The chain scission reaction is a chemical reaction between the macromolecular compounds (polymers) and end/mid-chain radicals. As the reaction progresses, the large polymer molecules eventually break down into live and dead polymer chains of lower molecular weights. A further molecular disintegration can ultimately lead to carbon dioxide and water as final products in case of complete mineralization.

In other words, the chain scission reaction can be defined as a bond scission that takes place in the backbone of the polymer chain. The chain scission reaction increases the number of polymer chains and reduces the polymer molecular weight [25]. Consequently, the chain scission results in an increase in the polydispersity of the polymer sample which represents the breadth of the distribution curve.

The concept of polymer degradation may be explained by chain-end scission or random chain scission mechanisms where chain breaking occurs at a random location along the chain. Therefore the molecular weight decreases continuously with the extent of reaction. Chain-end scission is considered as a special case of random chain scission where the chain scission reactions are occurring most likely at the polymer chain end that results in a release of a single monomer molecule. Random chain scission is the reverse of step-growth polymerization while chain-end scission

molecular-weight distribution (MWD) or chain-length distribution. The treatment of the wastewater streams contaminated with PVA polymers is studied using different processes [7, 12, 13, 16]. The kinetics models proposed in these studies were validated using total organic carbon (TOC) data instead of polymer concentrations

The photo-oxidative degradation of water-soluble polymers in laboratory scale photochemical reactors is the focus of this chapter. The photochemical kinetic model of the polymer degradation in UV/H2O2 process that describes the polymer chain scission is discussed in which the statistical moment approach is presented. The development of a photochemical kinetic model incorporates the population balance of all chemical species. Considering the probabilistic nature of the polymer fragments, the statistical moment approach is applied for modeling the population balance of live and dead polymer chains, which allows estimating the polymer average molecular weights as a function of radiation time. The model also considered the effect of process parameters on the decrease of polymer molecular weight,

The critical design parameters in the UV/H2O2 process include the H2O2 dose, the UV lamp type and intensity, and the reactor contact time. Basic UV reactor design configurations used for the removal of polymers from wastewater depend mainly on the flow rate. The tower design is typically utilized for large-scale applications. In the tower configuration, multiple UV lamps are arranged horizontally within a single large reactor vessel with the contaminated water flowing perpendicularly past the UV lamps [23]. For small-scale systems, a single UV lamp per reactor vessel is arranged vertically. For example, a small-scale system may consist of three individual reactor

A typical laboratory-scale batch recirculation UV/H2O2 system consists of an annular photoreactor, a large volume reservoir tank, centrifugal pump, and heat exchanger. The circulation tank contains the polymer solution for treatment. The hydrogen peroxide is injected into the circulation tank. A centrifugal magnet pump is placed on the circulation line to maintain a steady flow of the aqueous polymer solution between the tank and photoreactor. A flow meter is used to measure the circulation rate. The cylindrical photoreactor is made of steel vessel of annual shape and is connected to the circulation tank. The reactor is equipped with an internal quartz glass in which a low-pressure mercury UV lamp is mounted at its centerline of the cylinder with stainless steel housing. The annular photoreactor should have a very small annular space to assure a uniform light distribution in the photoreactor. Most AOPs are modular processes. Therefore, more than one reactor can be employed in series mode to obtain higher retention times or in parallel mode to process larger volumes to achieve the desired effluent for a given flow rate.

Determination of the polymer molecular weight, TOC content, and residual

hydrogen peroxide are crucial parameters to assess the performance of the photodegradation process. The treated samples are analyzed using gel permeation chromatography (GPC) to determine the molecular weights of the degraded polymer samples. The GPC theory depends on the principle of size exclusion; therefore, when a polymer solution is passed through a column of porous particles, large

or polymer molecular weights, and a constant pH was assumed.

hydrogen peroxide residual, and the acidity of the treated solution.

vessels in series, each containing one UV lamp in a vertical position.

3. Characterization of polymeric wastewater

32

2. UV/H2O2 system description

Kinetic Modeling for Environmental Systems

is the reverse of chain growth polymerization [26]. Aarthi et al. [14] studied the photodegradation of water-soluble polymers by combined ultrasonic and ultraviolet radiation and found that the degradation process is controlled by random and midpoint scission. On the other hand, Konaganti and Madras [27] investigated the photocatalytic degradation of polymethyl methacrylate, polybutyl acrylate, and their copolymers by random and chain-end scissions.

In the photodegradation of PVA polymer, the random chain scission mechanism initially dominates which experimentally proved by the rapid decrease in the polymer molecular weights. In random chain scission, all bonds may have an equal probability of being cleaved along the polymer chain. Apparently, the degradation process leads to a steep decrease in molecular weights. The chain cleavage occurs and effectively shortens the polymer chains [17]. It can be concluded that PVA degradation occurs mostly by random chain scission which explains the drastic decrease in the polymer concentration.

#### 4.1 Polymer average molecular weights

Polymer molecules are made of repeat monomer units that chemically bonded into long chains. The chain length is often expressed in terms of the molecular weight of the polymer chain, related to the relative molecular weight of the monomer and the number of monomer units connected in the chain.

The molecular weight of a polymer is described by the average values of the molecular weights of the polymer chains. The molecular weight distribution (MWD) is the distribution of sizes in a polymer sample while the polydispersity index (PDI) represents the breadth of the distribution curve. Thus, the polydispersity index is used as a measure of the broadness of molecular weight distribution of a polymer sample. Most synthetic water-soluble polymers are polydisperse since they contain polymer chains of unequal lengths. The increase in the polydispersity index results in broader molecular weight distribution. The PDI is defined as the ratio of weight average molecular weight (Mw) to the number average molecular weights (Mn). The molecular weight of a polymer is not a single value since polymer molecules even those of the same type, have different sizes, so the method of averaging mainly determines the average molecular weight. The number average molecular weight is considered as the ordinary arithmetic average of the molecular weights of the polymer while the weight average molecular weight is determined by measuring the weight of each species in the sample, rather than the number of molecules of each size.

5. Photodegradation kinetic model development

tion of the pollutant.

Figure 2.

35

the polymer molecular weights.

The principle in the AOP process is the formation of hydroxyl radicals which react immediately with organic contaminants in the wastewater streams. The hydroxyl radicals are highly reactive because of the presence of unpaired electrons. Oxidation reactions that produce radicals tend to be followed by additional oxidation reactions between the radical oxidants and the intermediate products until thermodynamically stable oxidation products are formed at complete mineraliza-

Effect of radiation time on MWD of the degraded PVA polymer by UV/H2O2 process (data from [17]).

Kinetic Modeling of Photodegradation of Water-Soluble Polymers in Batch Photochemical Reactor

DOI: http://dx.doi.org/10.5772/intechopen.82608

Usually, the mineralization starts directly with pollutant degradation, however, for PVA polymers it occurs at a later stage of the reaction. In this case, it is desired to model a specific polymer degradation as the TOC is not the right parameter to choose for the development of an adequate model for polymer disintegration in a photo-oxidation process. It is plausible to develop a model that takes into account

Under the effect of UV light of a specific wavelength and using an oxidant such as hydrogen peroxide, water-soluble polymer chains can break down into smaller chains. Under the effect of radiation energy, chemical bonds of polymer chains are destabilized and weakened. The chain scission reaction is, therefore, initiated and it is defined as a bond scission that takes place in the backbone of the polymer chain.

Enhanced photodegradation of polymer by UV radiation can lead to a wider distribution of molecular weights because the polymer chains are broken down into short-chain polymers such as oligomers, dimers, and monomers [28]. The irradiation promotes the decrease in the polymer molecular weights and the increase in the polydispersity of the molecular weight distribution of the degraded polymer as shown in Figure 2.

The shape of the molecular weight distribution changes as a function of the treatment time. The untreated PVA has a uniform narrow distribution with a polydispersity index (PDI) close to unity. During the degradation process, the distribution shifts to the left as the polymer molecular weight was considerably lowered. Song and Hyuan [29] confirmed the shifting of MWD and the generation of monomer by chain-end scission at the thermal degradation of polystyrene in a batch reactor. The broadness of the molecular weight distribution which is expressed by an increase in polydispersity is due to the fragmentation and chainscission mechanism of the polymer degradation during the UV/H2O2 process.

Kinetic Modeling of Photodegradation of Water-Soluble Polymers in Batch Photochemical Reactor DOI: http://dx.doi.org/10.5772/intechopen.82608

Figure 2. Effect of radiation time on MWD of the degraded PVA polymer by UV/H2O2 process (data from [17]).

## 5. Photodegradation kinetic model development

The principle in the AOP process is the formation of hydroxyl radicals which react immediately with organic contaminants in the wastewater streams. The hydroxyl radicals are highly reactive because of the presence of unpaired electrons. Oxidation reactions that produce radicals tend to be followed by additional oxidation reactions between the radical oxidants and the intermediate products until thermodynamically stable oxidation products are formed at complete mineralization of the pollutant.

Usually, the mineralization starts directly with pollutant degradation, however, for PVA polymers it occurs at a later stage of the reaction. In this case, it is desired to model a specific polymer degradation as the TOC is not the right parameter to choose for the development of an adequate model for polymer disintegration in a photo-oxidation process. It is plausible to develop a model that takes into account the polymer molecular weights.

Under the effect of UV light of a specific wavelength and using an oxidant such as hydrogen peroxide, water-soluble polymer chains can break down into smaller chains. Under the effect of radiation energy, chemical bonds of polymer chains are destabilized and weakened. The chain scission reaction is, therefore, initiated and it is defined as a bond scission that takes place in the backbone of the polymer chain.

is the reverse of chain growth polymerization [26]. Aarthi et al. [14] studied the photodegradation of water-soluble polymers by combined ultrasonic and ultraviolet radiation and found that the degradation process is controlled by random and midpoint scission. On the other hand, Konaganti and Madras [27] investigated the photocatalytic degradation of polymethyl methacrylate, polybutyl acrylate, and

In the photodegradation of PVA polymer, the random chain scission mechanism initially dominates which experimentally proved by the rapid decrease in the polymer molecular weights. In random chain scission, all bonds may have an equal probability of being cleaved along the polymer chain. Apparently, the degradation process leads to a steep decrease in molecular weights. The chain cleavage occurs and effectively shortens the polymer chains [17]. It can be concluded that PVA degradation occurs mostly by random chain scission which explains the drastic

Polymer molecules are made of repeat monomer units that chemically bonded into long chains. The chain length is often expressed in terms of the molecular weight of the polymer chain, related to the relative molecular weight of the mono-

The molecular weight of a polymer is described by the average values of the molecular weights of the polymer chains. The molecular weight distribution (MWD) is the distribution of sizes in a polymer sample while the polydispersity index (PDI) represents the breadth of the distribution curve. Thus, the polydispersity index is used as a measure of the broadness of molecular weight distribution of a polymer sample. Most synthetic water-soluble polymers are polydisperse since they contain polymer chains of unequal lengths. The increase in the polydispersity index results in broader molecular weight distribution. The PDI is defined as the ratio of weight average molecular weight (Mw) to the number average molecular weights (Mn). The molecular weight of a polymer is not a single value since polymer molecules even those of the same type, have different sizes, so the method of averaging mainly determines the average molecular weight. The number average molecular weight is considered as the ordinary arithmetic average of the molecular weights of the polymer while the weight average molecular weight is determined by measuring the weight of each species in the sample, rather than the number of

Enhanced photodegradation of polymer by UV radiation can lead to a wider distribution of molecular weights because the polymer chains are broken down into short-chain polymers such as oligomers, dimers, and monomers [28]. The irradiation promotes the decrease in the polymer molecular weights and the increase in the polydispersity of the molecular weight distribution of the degraded polymer as

The shape of the molecular weight distribution changes as a function of the treatment time. The untreated PVA has a uniform narrow distribution with a polydispersity index (PDI) close to unity. During the degradation process, the distribution shifts to the left as the polymer molecular weight was considerably lowered. Song and Hyuan [29] confirmed the shifting of MWD and the generation of monomer by chain-end scission at the thermal degradation of polystyrene in a batch reactor. The broadness of the molecular weight distribution which is expressed by an increase in polydispersity is due to the fragmentation and chainscission mechanism of the polymer degradation during the UV/H2O2 process.

mer and the number of monomer units connected in the chain.

their copolymers by random and chain-end scissions.

decrease in the polymer concentration.

Kinetic Modeling for Environmental Systems

4.1 Polymer average molecular weights

molecules of each size.

shown in Figure 2.

34

As the reaction progresses, the large polymer molecules Pr eventually break down into live and dead polymer chains of lower molecular weights, and consequently, new intermediate polymeric components are formed. A further molecular disintegration can ultimately lead to carbon dioxide and water as final products in case of complete mineralization according to the following reaction:

$$P\_r + H\_2O\_2 \overset{hv}{\longrightarrow} intermedates \rightarrow CO\_2 + H\_2O \tag{1}$$

Under the UV irradiation, the photolysis of hydrogen peroxide generates hydroxyl radicals as follows:

$$\text{H}\_2\text{O}\_2 \xrightarrow{hv} \text{2HO}^\* \tag{2}$$

The mechanism of degradation polymer solution using UV irradiation using hydrogen peroxide as an oxidant results in the generation of polymeric hydroxyl radicals, which undergo degradation reactions. The live polymer radicals P•

No. Reaction mechanism Rate constant Reference

Kinetic Modeling of Photodegradation of Water-Soluble Polymers in Batch Photochemical Reactor

CH3COO� <sup>þ</sup> <sup>H</sup><sup>þ</sup> 1.76 � <sup>10</sup>�<sup>5</sup> [39]

<sup>k</sup><sup>14</sup> <sup>2</sup>HCOOH <sup>þ</sup> <sup>H</sup>2O<sup>2</sup> 1.60 � <sup>10</sup><sup>7</sup> L/mol s [39]

CH2COO� <sup>þ</sup> <sup>H</sup>2<sup>O</sup> 3.20 � <sup>109</sup> L/mol s [39]

<sup>k</sup><sup>17</sup>CO<sup>2</sup> <sup>þ</sup> <sup>2</sup>H2<sup>O</sup> 1.30 � <sup>10</sup><sup>8</sup> L/mol s [39]

precursor of subsequent polymer chain breakage (Reactions (16)–(19)). The polymer radical may combine with another polymer radical to terminate the reaction (Reactions (21)). The scission products are radical and non-radical fragments

Ps <sup>þ</sup> <sup>P</sup>•

Reactions (20)–(26) represent the complete mineralization of polymer compounds. It has been experimentally proven that the acidity of the treated solution varies during the degradation reaction by the UV/H2O2 process [17, 39]. The pH decreases at the beginning of the reaction, and the solution becomes more acidic due to the formation of intermediate oxidation products such as carboxylic acids [42]. A regain in the pH of the solution is expected in case of complete mineralization as a result of the degradation of acidic compounds that are oxidized to carbon dioxide that escapes the system and water at the end of the reaction. The experimental findings indicate that there is evidence of the formation of acetic and formic acids associated with the degradation of the monomer (vinyl alcohol) produced at the complete degradation of PVA polymer. Therefore, the photochemical kinetic mechanism incorporates the acidity aspect of the solution as the polymer degradation progresses. The complete mineralization of polymer compounds and the production of byproducts with no hazard to the environment (Reactions (20)–(26)) are considered as remarkable advantages of the advanced oxidation processes. A photochemical kinetic model was developed based on the mechanism presented in Reactions (1) to (27). The polymer degradation reactions are assumed to be irreversible. Binary fragmentation is also considered to explain kinetics fragmentation in which a polymer of chain length r splits into two polymer units. The reaction rate constants are assumed independent of polymer chain length. The degradation reaction is carried out at a constant temperature with a good mixing condition. For a batch reactor with recirculation, negligible degradation of the polymer per pass and good mixing condition are assumed. Direct photolysis of the polymer without the presence of

The photochemical kinetic model describing the PVA polymer degradation by photo-oxidation comprises a radiation energy balance coupled with a molar balance of the chemical species participating in the degradation reactions of the polymer. The quantum yield of PVA is usually negligible since there is no measurable change in PVA molecular weight under UV radiation alone [17]. The molar absorptivity of PVA polymer is determined using spectrophotometer by measuring the absorbance of different concentrations of PVA aqueous solutions at a wavelength of 254 nm.

(monomer or polymer with lower molecular weight) as follows:

where s ¼ 1 f0r chain-end scission and 2≤ s≤r:

(23) CH3COOH \$

(24) CH3COOH <sup>þ</sup> <sup>2</sup>HO•

(25) HO• <sup>þ</sup> CH3COO�!

(26) HCOOH <sup>þ</sup> <sup>2</sup>HO•

Table 1.

Ka<sup>2</sup>

DOI: http://dx.doi.org/10.5772/intechopen.82608

Photolysis reactions of hydrogen peroxide and the rate constants.

<sup>2</sup>!

k<sup>15</sup> •

!

hydrogen peroxide is neglected.

37

P• <sup>r</sup> ! kp

<sup>r</sup> are the

<sup>r</sup>�<sup>s</sup> (27)

The highly reactive hydroxyl radical can undergo a series of promoted dissociation reactions. Several authors [30, 33, 40, 41] have proposed a detailed chemical kinetic mechanism of hydrogen peroxide decomposition. Photolysis reactions of hydrogen peroxide (Reactions (3)–(15)) and the rate constants are provided in Table 1.


Kinetic Modeling of Photodegradation of Water-Soluble Polymers in Batch Photochemical Reactor DOI: http://dx.doi.org/10.5772/intechopen.82608


Table 1.

As the reaction progresses, the large polymer molecules Pr eventually break down into live and dead polymer chains of lower molecular weights, and consequently, new intermediate polymeric components are formed. A further molecular disintegration can ultimately lead to carbon dioxide and water as final products in case of

Under the UV irradiation, the photolysis of hydrogen peroxide generates

H2O<sup>2</sup> !

The highly reactive hydroxyl radical can undergo a series of promoted dissociation reactions. Several authors [30, 33, 40, 41] have proposed a detailed chemical kinetic mechanism of hydrogen peroxide decomposition. Photolysis reactions of hydrogen peroxide (Reactions (3)–(15)) and the rate constants are provided in Table 1.

No. Reaction mechanism Rate constant Reference

hv intermediates ! CO<sup>2</sup> <sup>þ</sup> <sup>H</sup>2<sup>O</sup> (1)

<sup>2</sup> <sup>þ</sup> <sup>H</sup>2<sup>O</sup> 2.7 � <sup>107</sup> L/mol s [30]

<sup>2</sup> <sup>þ</sup> OH� 7.5 � <sup>10</sup><sup>9</sup> L/mol s [33]

<sup>H</sup>2O<sup>2</sup> 5.5 � <sup>10</sup><sup>9</sup> L/mol s [30]

<sup>H</sup>2<sup>O</sup> <sup>þ</sup> <sup>O</sup><sup>2</sup> 6.6 � <sup>10</sup><sup>9</sup> L/mol s [35]

<sup>H</sup>2O<sup>2</sup> <sup>þ</sup> <sup>O</sup><sup>2</sup> 8.3 � <sup>105</sup> L/mol s [34]

<sup>2</sup> <sup>þ</sup> <sup>H</sup><sup>þ</sup> 1.6 � <sup>10</sup><sup>5</sup> 1/s [34]

<sup>O</sup><sup>2</sup> <sup>þ</sup> OH� 7.0 � <sup>10</sup><sup>9</sup> L/mol s [36]

<sup>2</sup> <sup>þ</sup> <sup>H</sup><sup>þ</sup> 4.5 � <sup>10</sup>�<sup>12</sup> 1/s [37]

<sup>H</sup>2O<sup>2</sup> 2.0 � <sup>10</sup><sup>10</sup> L/mol s [37]

<sup>r</sup> <sup>þ</sup> <sup>H</sup>2<sup>O</sup> 8.06 � <sup>106</sup> L/mol s [38]

<sup>2</sup>HCOOH <sup>þ</sup> HO� 1.89 � <sup>10</sup><sup>6</sup> L/mol s [38]

CH3COOH <sup>þ</sup> HO• 1.35 � <sup>102</sup> L/mol s [38]

HCOO� <sup>þ</sup> <sup>H</sup><sup>þ</sup> 1.77 � <sup>10</sup>�<sup>4</sup> [39]

1.0 � <sup>10</sup><sup>10</sup> L/mol s [34]

9.7 � <sup>10</sup><sup>7</sup> L/mol s [34]

4.69 � <sup>10</sup>�<sup>1</sup> L/mol s [38]

3.66 � <sup>10</sup><sup>2</sup> 1/s [38]

4.44 � <sup>10</sup><sup>2</sup> L/mol s [38]

HO• <sup>þ</sup> <sup>H</sup>2<sup>O</sup> <sup>þ</sup> <sup>O</sup><sup>2</sup> 3.0 L/mol s [31]

HO• <sup>þ</sup> OH� <sup>þ</sup> <sup>O</sup><sup>2</sup> <sup>13</sup> � <sup>10</sup>�<sup>2</sup> L/mol s [32]

hv 2HO• (2)

complete mineralization according to the following reaction:

Pr þ H2O<sup>2</sup> !

k1 HO•

<sup>2</sup>! k2

<sup>2</sup> ! k3

<sup>2</sup> <sup>þ</sup> HO• ! k7

<sup>2</sup> <sup>þ</sup> HO• <sup>2</sup>! k8

> <sup>2</sup>! k9 O•�

> > <sup>2</sup> ! k<sup>11</sup>

<sup>2</sup> þ Hþ! k<sup>13</sup>

> ! kp1 P•

<sup>2</sup>! kp2 P• <sup>r</sup> þ H2O<sup>2</sup>

<sup>r</sup>�<sup>s</sup> <sup>þ</sup> <sup>P</sup>• <sup>s</sup>! ktc Pr

<sup>r</sup>! kp P• <sup>r</sup>�<sup>1</sup> þ P<sup>1</sup>

<sup>2</sup> <sup>þ</sup> <sup>O</sup>•� <sup>2</sup> ! kd1

> <sup>2</sup>! kd2

> > Ka<sup>1</sup>

k<sup>12</sup> HO�

<sup>2</sup> <sup>þ</sup> <sup>O</sup>•� <sup>2</sup> ! k<sup>10</sup>

<sup>2</sup> ! k4 HO•

<sup>2</sup> þ Hþ! k5 HO• 2

> ! k6

O<sup>2</sup> þ HO� 2

hydroxyl radicals as follows:

Kinetic Modeling for Environmental Systems

(3) HO• <sup>þ</sup> <sup>H</sup>2O2!

(4) <sup>H</sup>2O<sup>2</sup> <sup>þ</sup> HO•

(5) <sup>H</sup>2O<sup>2</sup> <sup>þ</sup> <sup>O</sup>•�

(7) <sup>O</sup>•�

(9) HO•

(10) HO•

(12) HO•

(11) HO•

(13) HO• <sup>þ</sup> <sup>O</sup>•�

(14) <sup>H</sup>2O2!

(16) Pr <sup>þ</sup> HO•

(17) Pr <sup>þ</sup> HO•

(18) <sup>P</sup>•

(19) <sup>P</sup>•

(20) <sup>P</sup><sup>1</sup> <sup>þ</sup> HO•

36

(21) <sup>P</sup><sup>1</sup> <sup>þ</sup> HO•

(22) HCOOH \$

(15) HO�

(6) HO• <sup>þ</sup> HO�

(8) HO• <sup>þ</sup> HO•

Photolysis reactions of hydrogen peroxide and the rate constants.

The mechanism of degradation polymer solution using UV irradiation using hydrogen peroxide as an oxidant results in the generation of polymeric hydroxyl radicals, which undergo degradation reactions. The live polymer radicals P• <sup>r</sup> are the precursor of subsequent polymer chain breakage (Reactions (16)–(19)). The polymer radical may combine with another polymer radical to terminate the reaction (Reactions (21)). The scission products are radical and non-radical fragments (monomer or polymer with lower molecular weight) as follows:

$$P\_r^\bullet \stackrel{k\_p}{\to} P\_s + P\_{r-s}^\bullet \tag{27}$$

where s ¼ 1 f0r chain-end scission and 2≤ s≤r:

Reactions (20)–(26) represent the complete mineralization of polymer compounds. It has been experimentally proven that the acidity of the treated solution varies during the degradation reaction by the UV/H2O2 process [17, 39]. The pH decreases at the beginning of the reaction, and the solution becomes more acidic due to the formation of intermediate oxidation products such as carboxylic acids [42]. A regain in the pH of the solution is expected in case of complete mineralization as a result of the degradation of acidic compounds that are oxidized to carbon dioxide that escapes the system and water at the end of the reaction. The experimental findings indicate that there is evidence of the formation of acetic and formic acids associated with the degradation of the monomer (vinyl alcohol) produced at the complete degradation of PVA polymer. Therefore, the photochemical kinetic mechanism incorporates the acidity aspect of the solution as the polymer degradation progresses. The complete mineralization of polymer compounds and the production of byproducts with no hazard to the environment (Reactions (20)–(26)) are considered as remarkable advantages of the advanced oxidation processes. A photochemical kinetic model was developed based on the mechanism presented in Reactions (1) to (27).

The polymer degradation reactions are assumed to be irreversible. Binary fragmentation is also considered to explain kinetics fragmentation in which a polymer of chain length r splits into two polymer units. The reaction rate constants are assumed independent of polymer chain length. The degradation reaction is carried out at a constant temperature with a good mixing condition. For a batch reactor with recirculation, negligible degradation of the polymer per pass and good mixing condition are assumed. Direct photolysis of the polymer without the presence of hydrogen peroxide is neglected.

The photochemical kinetic model describing the PVA polymer degradation by photo-oxidation comprises a radiation energy balance coupled with a molar balance of the chemical species participating in the degradation reactions of the polymer. The quantum yield of PVA is usually negligible since there is no measurable change in PVA molecular weight under UV radiation alone [17]. The molar absorptivity of PVA polymer is determined using spectrophotometer by measuring the absorbance of different concentrations of PVA aqueous solutions at a wavelength of 254 nm.

For the kinetics, the general molar balance equation (Eq. 28) [43] must be applied to the recirculating batch photoreactor.

$$\frac{\partial \mathcal{E}\_i}{\partial t} + \nabla . \mathbf{N}\_i = \mathbf{R}\_i \tag{28}$$

Assuming that the system works under the well-stirred conditions (∇:Ni ¼ 0), the ratio of the photoreactor volume to the total volume ≪ 1, and high recirculating flow rate to ensure small conversion per pass, the rate of the change of the concentration in the tank could be written as follows [44]

$$\frac{\mathbf{d}\mathbf{C}\_i}{\mathbf{d}t} = \frac{V\_{ph}}{V\_T} \sum\_{j=1}^m \mathbf{R}\_{ij}, \quad \mathbf{C}\_i(\mathbf{0}) = \mathbf{C}\_{i0} \tag{29}$$

In which Ci(t) is the ith component concentration, Vph is the volume of the photoreactor, VT is the volume of the whole system, Ci(0) is the initial molar concentration of species i, and Rij is the chemical reaction rate of component i in reaction j (j = 1,2,.,m).

According to the basic photochemical mechanism given in Table 1, the mole balance of small molecule species gives the following reaction rate equations:

$$\begin{aligned} \frac{1}{\alpha} \frac{d[H\_2O\_2]}{dt} &= -R\_{UV, H\_2O\_2} - k\_1[HO^\*][H\_2O\_2] - k\_2[HO\_2^\*][H\_2O\_2] - k\_3[O\_2^{\*-}][H\_2O\_2] \\ &+ k\_6[HO^\*]^2 + k\_8[HO\_2^\*]^2 + k\_{p2}[HO\_2^\*][P\_r] - k\_{12}[H\_2O\_2] + k\_{13}[HO\_2^{\*-}][H^+] \\ &+ k\_{14}[CH\_3COOH][HO\_2^\*] - k\_{15}[HCOO^-][H\_2O\_2] \end{aligned} \tag{30}$$

$$\begin{aligned} \frac{1}{a} \frac{d[HO^{\bullet}]}{dt} &= 2k\_{UV,H\_2O\_2} - k\_1[HO^{\bullet}] \left[H\_2O\_2\right] + k\_2 \left[HO\_2^{\bullet}\right] \left[H\_2O\_2\right] + k\_3 \left[O\_2^{\bullet -}\right] \left[H\_2O\_2\right] \\ &- k\_4 \left[HO\_2^{-}\right] [HO^{\bullet}] - 2k\_6 \left[HO^{\bullet}\right]^2 - k\_7[HO^{\bullet}] \left[HO\_2^{\bullet}\right] - k\_{p1}[HO^{\bullet}][P\_r] \\ &- k\_{11}[HO^{\bullet}] \left[O\_2^{\bullet -}\right] - k\_{d1}[HO^{\bullet}][P\_1] + k\_{d2} \left[HO\_2^{\bullet}\right][P\_1] - 2k\_{16}[HO^{\bullet}]^2 \left[HCOOH\_2^{\bullet}\right] \end{aligned} \tag{31}$$

$$\frac{1}{d\alpha} \frac{d[H^+]}{dt} = -k\_5 \left[O\_2^{\bullet -}\right][H^+] + k\_9 \left[HO\_2^{\bullet}\right] + k\_{12}[H\_2O\_2] - k\_{13}[H^+] \left[HO\_2^{-}\right] + k\_{41}[HCOOH]/\gamma$$

$$[HCOO^-] + k\_{42}[CH\_3COOH]/[CH\_3COO^-]$$

$$\frac{d[HCOOH]}{dt} = k\_{d1}[HO^\*][P\_1] + k\_{d1}^{-1}[H^+][HCOO^-] + k\_{14}[HO\_2^\*]^2[CH\_3COOH]$$

$$-k\_{16}[HO^\*]^2[HCOOH]$$

$$\mathbf{(33)}$$

1 α

1 α d HO� 2 

d O•� 2 

dt ¼ �k<sup>4</sup> HO�

DOI: http://dx.doi.org/10.5772/intechopen.82608

2

d P½ � <sup>1</sup> dt <sup>¼</sup> kpp•

� <sup>k</sup><sup>11</sup> HO• ½ � <sup>O</sup>•�

RUV,i ¼ �∅ifi

The molar balance of the macromolecules Pr and P•

dt ¼ �k<sup>3</sup> <sup>O</sup>•�

1 α

irradiation absorbed by the i

balance of all polymer species.

dt ¼ �RUV,PVA � kp<sup>1</sup> HO• ½ � pr

dt <sup>¼</sup> RUV,PVA <sup>þ</sup> kp<sup>1</sup> HO• ½ � pr

� 2ktc ∑ r s¼1 p• s p• r�s

radicals P•

expressed as:

d p• r 

1 α

1 α d pr 

39

emitted at the source, and the Ci is the i

2

HO• ½ �þ <sup>k</sup><sup>10</sup> HO•

½ �� <sup>H</sup>2O<sup>2</sup> <sup>k</sup><sup>5</sup> <sup>O</sup>•�

2 

2 O•� 2

Kinetic Modeling of Photodegradation of Water-Soluble Polymers in Batch Photochemical Reactor

<sup>H</sup><sup>þ</sup> ½ �þ <sup>k</sup><sup>9</sup> HO•

Io � exp �2:303 b ∑

where α is defined as the ratio of photoreactor volume Vph to the total volume of the system VT, ∅<sup>i</sup> is the number of moles of the pollutant transformed per number of photons of wavelength λ absorbed by the pollutant, b is the path length of the ray

2

through the medium, ε is the molar absorptivity, fi is the fraction of the UV

requires special modeling approach as the PVA polymer is randomly broken down, polymer chains species of different sizes are subsequently generated, and they are expected to degrade further. The concept of the population species is considered to express the variations of the photochemical degradation of PVA. The random degradation of polymer chains of length r can be described using breakage population

Generally, the moment operation is introduced as an easier method to transform the integro-differential equations in the continuous kinetics model or the sum in the discrete model to ordinary differential equations. McCoy and Madras [45] and Stickle and Griggs [46] provided simple mathematical expressions for the discrete model. The macromolecular reactions show that the polymer consists of degrading active polymer

<sup>r</sup> and dead polymer Pr. Polymer degradation is described by a discrete approach so that a mass balance provides a difference-differential equations. The net accumulation rate of dead polymer chains of chain length r is given as follows [38]:

Similarly, the net accumulation rate of live polymer radicals of chain length r is

Using statistical mechanics, the concept of moments was applied to determine the molecular weight distribution of a polymer population. This reaction requires the production of a specified scission product from any of a range of macromolecules, so a stoichiometric kernel Ω(r,s) is employed for a polymer chain of length r to represent

<sup>þ</sup> kp<sup>2</sup> HO•

<sup>2</sup> pr

2 pr

<sup>þ</sup> kp <sup>∑</sup>

r s¼1

� kp <sup>p</sup>•

<sup>Ω</sup>ð Þ <sup>r</sup>; <sup>s</sup> <sup>p</sup>•

r <sup>þ</sup> kp <sup>∑</sup>

<sup>s</sup> þ ktc ∑ r s¼1 p• r p• r�s

> r s¼1

<sup>Ω</sup>ð Þ <sup>r</sup>; <sup>s</sup> <sup>p</sup>• s

(40)

(41)

� kp<sup>2</sup> HO•

2 O•� 2

<sup>r</sup> � kd<sup>1</sup> HO•

<sup>þ</sup> <sup>k</sup>12½ �� <sup>H</sup>2O<sup>2</sup> <sup>k</sup><sup>13</sup> HO�

2 � <sup>k</sup><sup>10</sup> HO•

½ �� <sup>P</sup><sup>1</sup> kd<sup>2</sup> HO•

N i¼1 εi:Ci (39)

th chemical species, Io is the incident light intensity

th species concentration.

2 <sup>H</sup><sup>þ</sup> ½ �

2 O•� 2 

<sup>r</sup> in Reactions (16) to (21)

½ � <sup>P</sup><sup>1</sup> (38)

2

(36)

(37)

$$\frac{1}{a} \frac{d[\text{CH}\_3\text{COOH}]}{dt} = k\_{d2} [\text{HO}\_2^\*] [\text{P}\_1] + k\_{a2}^{-1} [\text{H}^+] [\text{CH}\_3\text{COO}^-] - k\_{14} [\text{HO}^\*]^2 [\text{CH}\_3\text{COOH}] \tag{34}$$

$$\frac{1}{\alpha} \frac{d \left[ HO\_2^{\bullet} \right]}{dt} = k\_1 [HO^{\bullet}] \left[H\_2O\_2 \right] - k\_2 \left[HO\_2^{\bullet} \right] \left[H\_2O\_2 \right] + k\_4 \left[HO\_2^{\bullet -} \right] \left[HO^{\bullet} \right] + k\_5 \left[O\_2^{\bullet -} \right] \left[H^+ \right]$$

$$-k\_7 [HO^{\bullet}] \left[HO\_2^{\bullet} \right] - 2k\_8 \left[HO\_2^{\bullet} \right]^2 - k\_9 \left[HO\_2^{\bullet} \right] - k\_{p2} \left[HO\_2^{\bullet} \right] \left[P\_r \right]$$

$$-k\_{10} \left[HO\_2^{\bullet} \right] \left[O\_2^{\bullet -} \right] + k\_{d1} [HO^{\bullet}] \left[P\_1 \right] - k\_{d2} \left[HO\_2^{\bullet} \right] \left[P\_1 \right] - 2k\_{14} \left[HO\_2^{\bullet} \right]^2 \left[CH\_3COOH \right] \tag{35}$$

1 α Kinetic Modeling of Photodegradation of Water-Soluble Polymers in Batch Photochemical Reactor DOI: http://dx.doi.org/10.5772/intechopen.82608

$$\frac{1}{a}\frac{d\left[HO\_2^-\right]}{dt} = -k\_4\left[HO\_2^-\right][HO^\*] + k\_{10}\left[HO\_2^\*\right]\left[O\_2^{\bullet-}\right] + k\_{12}[H\_2O\_2] - k\_{13}\left[HO\_2^-\right][H^+] \tag{36}$$

$$\frac{1}{a}\frac{d\left[O\_{2}^{\bullet-}\right]}{dt} = -k\_{3}\left[O\_{2}^{\bullet-}\right]\left[H\_{2}O\_{2}\right] - k\_{5}\left[O\_{2}^{\bullet-}\right]\left[H^{+}\right] + k\_{9}\left[HO\_{2}^{\bullet}\right] - k\_{10}\left[HO\_{2}^{\bullet}\right]\left[O\_{2}^{\bullet-}\right] \tag{37}$$

$$-k\_{11}\quad\left[HO^{\bullet}\right]\left[O\_{2}^{\bullet-}\right]$$

$$\frac{1}{\alpha} \frac{d[P\_1]}{dt} = k\_p p\_r^\bullet - k\_{d1} \left[ H O\_2^\bullet \right] \left[ O\_2^{\bullet -} \right] [P\_1] - k\_{d2} \left[ H O\_2^\bullet \right] [P\_1] \tag{38}$$

$$R\_{UV,i} = -\mathfrak{Q}f\_i I\_o \left( -\exp\left(-2.303\,b\sum\_{i=1}^{N} \varepsilon\_i.C\_i\right) \right) \tag{39}$$

where α is defined as the ratio of photoreactor volume Vph to the total volume of the system VT, ∅<sup>i</sup> is the number of moles of the pollutant transformed per number of photons of wavelength λ absorbed by the pollutant, b is the path length of the ray through the medium, ε is the molar absorptivity, fi is the fraction of the UV irradiation absorbed by the i th chemical species, Io is the incident light intensity emitted at the source, and the Ci is the i th species concentration.

The molar balance of the macromolecules Pr and P• <sup>r</sup> in Reactions (16) to (21) requires special modeling approach as the PVA polymer is randomly broken down, polymer chains species of different sizes are subsequently generated, and they are expected to degrade further. The concept of the population species is considered to express the variations of the photochemical degradation of PVA. The random degradation of polymer chains of length r can be described using breakage population balance of all polymer species.

Generally, the moment operation is introduced as an easier method to transform the integro-differential equations in the continuous kinetics model or the sum in the discrete model to ordinary differential equations. McCoy and Madras [45] and Stickle and Griggs [46] provided simple mathematical expressions for the discrete model. The macromolecular reactions show that the polymer consists of degrading active polymer radicals P• <sup>r</sup> and dead polymer Pr. Polymer degradation is described by a discrete approach so that a mass balance provides a difference-differential equations. The net accumulation rate of dead polymer chains of chain length r is given as follows [38]:

$$\frac{1}{a}\frac{d\left[p\_r\right]}{dt} = -R\_{UV,PWM} - k\_{p\_1}[HO^\bullet]\left[p\_r\right] - k\_{p\_2}\left[HO\_2^\bullet\left[p\_r\right] + k\_p\sum\_{s=1}^r \mathcal{Q}(r,s)p\_s^\bullet + k\_{tc}\sum\_{s=1}^r p\_r^\bullet p\_{r-s}^\bullet\right] \tag{40}$$

Similarly, the net accumulation rate of live polymer radicals of chain length r is expressed as:

$$\begin{aligned} \frac{1}{\alpha} \frac{d \left[ p\_r^\bullet \right]}{dt} &= R\_{UV, PVA} + k\_{p\_1} [HO^\bullet] \left[ p\_r \right] + k\_{p\_2} \left[ HO\_2^\bullet \right] \left[ p\_r \right] - k\_p \left[ p\_r^\bullet \right] + k\_p \sum\_{s=1}^r \Omega(r, s) p\_s^\bullet \\ &- 2k\_{\rm tr} \sum\_{s=1}^r p\_r^\bullet p\_{r-s}^\bullet \end{aligned} \tag{41}$$

Using statistical mechanics, the concept of moments was applied to determine the molecular weight distribution of a polymer population. This reaction requires the production of a specified scission product from any of a range of macromolecules, so a stoichiometric kernel Ω(r,s) is employed for a polymer chain of length r to represent

For the kinetics, the general molar balance equation (Eq. 28) [43] must be

Assuming that the system works under the well-stirred conditions (∇:Ni ¼ 0), the ratio of the photoreactor volume to the total volume ≪ 1, and high recirculating flow rate to ensure small conversion per pass, the rate of the change of the concen-

In which Ci(t) is the ith component concentration, Vph is the volume of the photoreactor, VT is the volume of the whole system, Ci(0) is the initial molar concentration of species i, and Rij is the chemical reaction rate of component i in

According to the basic photochemical mechanism given in Table 1, the mole balance of small molecule species gives the following reaction rate equations:

2

2

<sup>þ</sup> <sup>k</sup>12½ �� <sup>H</sup>2O<sup>2</sup> <sup>k</sup><sup>13</sup> <sup>H</sup><sup>þ</sup> ½ � HO�

<sup>a</sup><sup>1</sup> <sup>H</sup><sup>þ</sup> ½ � HCOO� ½ �þ <sup>k</sup><sup>14</sup> HO•

<sup>a</sup><sup>2</sup> <sup>H</sup><sup>þ</sup> ½ � CH3COO� ½ �� <sup>k</sup><sup>14</sup> HO• ½ �<sup>2</sup>

2

2

2 � kp<sup>2</sup> HO•

HO• ½ �þ <sup>k</sup><sup>5</sup> <sup>O</sup>•�

½ �� <sup>P</sup><sup>1</sup> <sup>2</sup>k<sup>14</sup> HO•

2 Pr ½ �

2

� <sup>k</sup><sup>15</sup> HCOO� ½ �½ � <sup>H</sup>2O<sup>2</sup>

½ � <sup>H</sup>2O<sup>2</sup> –k<sup>3</sup> <sup>O</sup>•�

½ �þ <sup>H</sup>2O<sup>2</sup> <sup>k</sup><sup>3</sup> <sup>O</sup>•�

2

2

2 ½ � <sup>H</sup>2O<sup>2</sup>

> 2 ½ � <sup>H</sup>2O<sup>2</sup>

� kp<sup>1</sup> HO• ½ � Pr ½ �

2

2 <sup>2</sup>

<sup>þ</sup> ka1½ � HCOOH <sup>=</sup>

½ � CH3COOH

½ �� <sup>P</sup><sup>1</sup> <sup>2</sup>k<sup>16</sup> HO• ½ �<sup>2</sup>

2 <sup>H</sup><sup>þ</sup> ½ �

(30)

½ � HCOOH (31)

(32)

(33)

(34)

½ � CH3COOH (35)

½ � CH3COOH

2 <sup>H</sup><sup>þ</sup> ½ �

2 <sup>2</sup>

Pr ½ �� <sup>k</sup>12½ �þ <sup>H</sup>2O<sup>2</sup> <sup>k</sup><sup>13</sup> HO�

þ ∇:Ni ¼ Ri (28)

Rij, Cið Þ¼ 0 Ci<sup>0</sup> (29)

∂ci ∂t

applied to the recirculating batch photoreactor.

Kinetic Modeling for Environmental Systems

tration in the tank could be written as follows [44]

reaction j (j = 1,2,.,m).

d H½ � <sup>2</sup>O<sup>2</sup>

d HO• ½ �

d H<sup>þ</sup> ½ �

1 α

1 α

1 α

38

d HO• 2 

dt ¼ �k<sup>5</sup> <sup>O</sup>•�

d HCOOH ½ �

d CH ½ � <sup>3</sup>COOH

1 α

1 α

1 α dC<sup>i</sup> dt <sup>¼</sup> Vph VT ∑ m j¼1

dt ¼ �RUV,H2O<sup>2</sup> � <sup>k</sup><sup>1</sup> HO• ½ �½ � <sup>H</sup>2O<sup>2</sup> –k<sup>2</sup> HO•

2 <sup>2</sup> <sup>þ</sup> kp<sup>2</sup> HO•

2

HO• ½ �� <sup>2</sup>k<sup>6</sup> HO• ½ �<sup>2</sup> � <sup>k</sup><sup>7</sup> HO• ½ � HO•

2

½ � HCOOH

2

2 <sup>2</sup> � <sup>k</sup><sup>9</sup> HO•

<sup>þ</sup> kd<sup>1</sup> HO• ½ �½ �� <sup>P</sup><sup>1</sup> kd<sup>2</sup> HO•

½ �þ <sup>H</sup>2O<sup>2</sup> <sup>k</sup><sup>4</sup> HO•�

HCOO� ½ �þ ka2½ � CH3COOH = CH3COO� ½ �

� kd<sup>1</sup> HO• ½ �½ �þ <sup>P</sup><sup>1</sup> kd<sup>2</sup> HO•

<sup>þ</sup> <sup>k</sup><sup>6</sup> HO• ½ �<sup>2</sup> <sup>þ</sup> <sup>k</sup><sup>8</sup> HO•

� k<sup>4</sup> HO� 2

2

dt <sup>¼</sup> kd<sup>2</sup> HO•

dt <sup>¼</sup> <sup>k</sup><sup>1</sup> HO• ½ � ½ �� <sup>H</sup>2O<sup>2</sup> <sup>k</sup><sup>2</sup> HO•

2 O•� 2

� <sup>k</sup><sup>7</sup> HO• ½ � HO•

� <sup>k</sup><sup>10</sup> HO•

� <sup>k</sup><sup>11</sup> HO• ½ � <sup>O</sup>•�

<sup>þ</sup> <sup>k</sup>14½ � CH3COOH HO•

dt <sup>¼</sup> <sup>2</sup>RUV,H2O<sup>2</sup> � <sup>k</sup><sup>1</sup> HO• ½ � ½ �þ <sup>H</sup>2O<sup>2</sup> <sup>k</sup><sup>2</sup> HO•

2

<sup>H</sup><sup>þ</sup> ½ �þ <sup>k</sup><sup>9</sup> HO•

dt <sup>¼</sup> kd<sup>1</sup> HO• ½ �½ �þ <sup>P</sup><sup>1</sup> <sup>k</sup>�<sup>1</sup>

2 � <sup>2</sup>k<sup>8</sup> HO•

� <sup>k</sup><sup>16</sup> HO• ½ �<sup>2</sup>

2 ½ �þ <sup>P</sup><sup>1</sup> <sup>k</sup>�<sup>1</sup> the probability of getting shorter polymer chain lengths r-s and s [47]. In general, polymer degradation occurs most likely by random chain scission. Therefore, it is postulated that there is a low probability of the occurrence of chain-end scission reactions. For random chain scission, the distribution of shorter polymer chains is given as follows [45, 46]:

$$\mathcal{Q}(r,s) = \mathbf{1}/r \tag{42}$$

values of 0, 1, or 2 stands for zeroth, first, and second moments, respectively. The application of the moment method allows converting the discrete differential population balance equations into ordinary differential ones. The moments of dead

Kinetic Modeling of Photodegradation of Water-Soluble Polymers in Batch Photochemical Reactor

weights of the polymer. Applying the statistical moment concept to Eqs. (43, 44) gives the following model of dead and live polymer moments for n = 0, 1, and 2,

dt <sup>¼</sup> RUV,PVA <sup>þ</sup> kp<sup>1</sup> HO• ½ �μ<sup>0</sup> <sup>þ</sup> kp<sup>2</sup> HO•

molecular weight distribution of a polymer population. The number average molecular weight Mn and the weight average molecular weight Mw are calculated

dt ¼ �RUV,PVA � kp<sup>1</sup> HO• ½ �μ<sup>0</sup> � kp<sup>2</sup> HO•

dt ¼ �RUV,PVA � kp<sup>1</sup> HO• ½ �μ<sup>1</sup> � kp<sup>2</sup> HO•

dt ¼ �RUV,PVA � kp<sup>1</sup> HO• ½ �μ<sup>2</sup> � kp<sup>2</sup> HO•

dt <sup>¼</sup> RUV,PVA <sup>þ</sup> kp<sup>1</sup> HO• ½ �μ<sup>1</sup> <sup>þ</sup> kp<sup>2</sup> HO•

dt <sup>¼</sup> RUV,PVA <sup>þ</sup> kp<sup>1</sup> HO• ½ �μ<sup>2</sup> <sup>þ</sup> kp<sup>2</sup> HO•

<sup>r</sup> are used to determine the average molecular

<sup>μ</sup><sup>1</sup> <sup>þ</sup> <sup>1</sup>=2kpλ<sup>1</sup> <sup>þ</sup> ktcλ1λ<sup>1</sup> (46)

<sup>μ</sup><sup>2</sup> <sup>þ</sup> <sup>1</sup>=3kpλ<sup>2</sup> <sup>þ</sup> ktcλ2λ<sup>2</sup> (47)

<sup>μ</sup><sup>1</sup> � <sup>1</sup>=2kpλ<sup>1</sup> � <sup>2</sup>ktcλ0λ<sup>1</sup> (49)

<sup>μ</sup><sup>2</sup> � <sup>2</sup>=3kpλ<sup>2</sup> � <sup>2</sup>ktcλ0λ<sup>2</sup> (50)

2 <sup>μ</sup><sup>0</sup> � <sup>2</sup>ktcλ<sup>2</sup>

Mn ¼ NACL : Mo (51) Mw ¼ WACL : Mo (52)

<sup>0</sup> þ kpλ<sup>0</sup> (45)

<sup>0</sup> (48)

(53)

(54)

2 <sup>μ</sup><sup>0</sup> <sup>þ</sup> ktcλ<sup>2</sup>

2

2

2

2

Using statistical mechanics, the concept of moments is applied to determine the

where Mo is the molecular weight of the monomer unit. The number average chain length (NACL) and the weight average chain length (WACL) are given by:

NACL <sup>¼</sup> <sup>μ</sup><sup>1</sup> <sup>þ</sup> <sup>λ</sup><sup>1</sup>

WACL <sup>¼</sup> <sup>μ</sup><sup>2</sup> <sup>þ</sup> <sup>λ</sup><sup>2</sup>

The validity of the kinetic model is examined by direct comparison of model predictions with experimental data of the process parameters such as polymer molecular weights, polymer concentration, hydrogen peroxide residual, and pH of the solution. The goodness-of-fit between experimental yexp and predicted ym data for each variable are then determined by calculating the root mean square error (RMSE) for n<sup>0</sup> data points. The good agreement between the model predictions and the experimental results confirms the adequacy of the developed photochemical kinetic model.

A parameter estimation scheme is typically performed for the polymer photodegradation model equations to estimate the rate constants that are not available in the open literature. The objective function is the summation of squared errors between the model predictions and experimental data for selected process variables. The parameter estimation scheme is formulated to determine the estimates of the rate constants by minimizing the objective function which is subjected

μ<sup>0</sup> þ λ<sup>0</sup>

μ<sup>1</sup> þ λ<sup>1</sup>

polymer Pr and live polymer radical P•

DOI: http://dx.doi.org/10.5772/intechopen.82608

respectively:

1 α

1 α

> 1 α

1 α

according to:

1 α d μ<sup>0</sup> ½ �

d μ<sup>1</sup> ½ �

d μ<sup>2</sup> ½ �

d½ � λ<sup>1</sup>

d½ � λ<sup>2</sup>

1 α

to the kinetic model equations.

41

d½ � λ<sup>0</sup>

#### 5.1 Polymer population balance

Polymer degradation is a fragmentation process in which population balance concepts is often applied in fragmentation models to describe how the distributions of different size entities evolve over the time of reaction. The degradation of high molecular weight polydisperse materials results in the formation of a large number of polymeric chains with different chain lengths and various chemical compositions. Population balance approach is generally employed to model the size distribution of the macromolecular compound during polymerization, polymer degradation, depolymerization, and chain breakage.

In 1971, Randolph and Larson [48] proposed a solution for the population balance equation (PBE) in a well-mixed batch system. They used the concept of moment transform to convert the population balance equations into ordinary differential equations. Population balance based models have been developed to study the molecular weight decrease of polymers in a fragmentation process by advanced oxidation processes [18, 20, 22, 49]. Microwave-assisted oxidative degradation as an emerging advanced oxidation technology was used for poly(alkyl methacrylate) degradation. Random chain scission and Continuous distribution kinetics were employed to determine the degradation rate of the polymer [50]. Photocatalytic degradation of polyacrylamide co-acrylic acid by random chain scission has been investigated by Vinu and Madras [51]. The rate coefficients were determined as a linear function of the composition of co-monomer. Madras and McCoy [52] studied the kinetics of oxidative degradation of polystyrene by di-tert-butyl peroxide provided the ratio of the rate parameters for both oxidizer and polymer decomposition by moment analysis assuming random chain scission mechanism. Population balance and moment equations are solved for rate parameters [21, 53]. The model proposed by McCoy and Wang [21] is sufficiently applicable to a variety of degradation processes. Moment equations can be applied in batch and continuous stirred tank reactor (CSTR) reactors for binary or ternary fragmentation.

The population balance model is a balance equation of species of different sizes, and it is similar to the mass, energy, and momentum balances, to track the changes in the size distribution. The benefit of the population models is that they provide a straightforward technique to derive expressions for the moments of the polymer distributions during the degradation reaction. Hulburt and Katz [54] applied the concept of moments to determine the molecular weight distribution of a polymer population for a dead and live polymer moments as follow:

$$
\mu\_n = \sum\_{r=1}^{\infty} r^n p\_r \tag{43}
$$

$$
\lambda\_n = \sum\_{r=1}^{\infty} r^n \, p\_r^\* \tag{44}
$$

where pr and p• <sup>r</sup> are the polymer and the polymer radical concentrations with chain length <sup>r</sup>, <sup>μ</sup><sup>n</sup> and <sup>λ</sup><sup>n</sup> are the <sup>n</sup>th moment of the quantities pr and <sup>p</sup>• <sup>r</sup> and n having Kinetic Modeling of Photodegradation of Water-Soluble Polymers in Batch Photochemical Reactor DOI: http://dx.doi.org/10.5772/intechopen.82608

values of 0, 1, or 2 stands for zeroth, first, and second moments, respectively. The application of the moment method allows converting the discrete differential population balance equations into ordinary differential ones. The moments of dead polymer Pr and live polymer radical P• <sup>r</sup> are used to determine the average molecular weights of the polymer. Applying the statistical moment concept to Eqs. (43, 44) gives the following model of dead and live polymer moments for n = 0, 1, and 2, respectively:

$$\frac{1}{\alpha} \frac{d[\mu\_0]}{dt} = -R\_{UV, PVA} - k\_{p1} [HO^\bullet] \mu\_0 - k\_{p2} [HO\_2^\bullet] \mu\_0 + k\_{tt} \dot{\lambda}\_0^2 + k\_p \dot{\lambda}\_0 \tag{45}$$

$$\frac{1}{\alpha} \frac{d[\mu\_1]}{dt} = -R\_{UV, PVA} - k\_{p1} [HO^\bullet] \mu\_1 - k\_{p2} [HO\_2^\bullet] \mu\_1 + 1/2k\_p \lambda\_1 + k\_{lt} \lambda\_1 \lambda\_1 \tag{46}$$

$$\frac{1}{\mathbf{a}} \frac{d[\mu\_2]}{dt} = -R\_{UV, PVA} - k\_{p1} [\text{HO}^\*] \mu\_2 - k\_{p2} [\text{HO}\_2^\*] \mu\_2 + 1/3k\_p \lambda\_2 + k\_{t\ell} \lambda\_2 \lambda\_2 \tag{47}$$

$$\frac{1}{\alpha} \frac{d[\lambda\_0]}{dt} = R\_{UV, PVA} + k\_{p1} [HO^\bullet] \mu\_0 + k\_{p2} [HO\_2^\bullet] \mu\_0 - 2k\_{tc} \lambda\_0^2 \tag{48}$$

$$\frac{1}{\alpha} \frac{d[\lambda\_1]}{dt} = R\_{UV, PVA} + k\_{p1} [HO^\*] \mu\_1 + k\_{p2} [HO\_2^\*] \mu\_1 - \mathbf{1}/2k\_p \lambda\_1 - 2k\_{lt} \lambda\_0 \lambda\_1 \tag{49}$$

$$\frac{1}{\alpha} \frac{d[\lambda\_2]}{dt} = R\_{UV, PVA} + k\_{p1}[HO^\*]\mu\_2 + k\_{p2}[HO\_2^\*]\mu\_2 - 2/3k\_p\lambda\_2 - 2k\_{t2}\lambda\_0\lambda\_2 \tag{50}$$

Using statistical mechanics, the concept of moments is applied to determine the molecular weight distribution of a polymer population. The number average molecular weight Mn and the weight average molecular weight Mw are calculated according to:

$$M\_n = \text{NaCl} \, .M\_o \tag{51}$$

$$M\_w = \text{WACL} \, . M\_o \tag{52}$$

where Mo is the molecular weight of the monomer unit. The number average chain length (NACL) and the weight average chain length (WACL) are given by:

$$\text{NACL} = \frac{\mu\_1 + \lambda\_1}{\mu\_0 + \lambda\_0} \tag{53}$$

$$\text{WACC} = \frac{\mu\_2 + \lambda\_2}{\mu\_1 + \lambda\_1} \tag{54}$$

A parameter estimation scheme is typically performed for the polymer photodegradation model equations to estimate the rate constants that are not available in the open literature. The objective function is the summation of squared errors between the model predictions and experimental data for selected process variables. The parameter estimation scheme is formulated to determine the estimates of the rate constants by minimizing the objective function which is subjected to the kinetic model equations.

The validity of the kinetic model is examined by direct comparison of model predictions with experimental data of the process parameters such as polymer molecular weights, polymer concentration, hydrogen peroxide residual, and pH of the solution. The goodness-of-fit between experimental yexp and predicted ym data for each variable are then determined by calculating the root mean square error (RMSE) for n<sup>0</sup> data points. The good agreement between the model predictions and the experimental results confirms the adequacy of the developed photochemical kinetic model.

the probability of getting shorter polymer chain lengths r-s and s [47]. In general, polymer degradation occurs most likely by random chain scission. Therefore, it is postulated that there is a low probability of the occurrence of chain-end scission reactions. For random chain scission, the distribution of shorter polymer chains is

Polymer degradation is a fragmentation process in which population balance concepts is often applied in fragmentation models to describe how the distributions of different size entities evolve over the time of reaction. The degradation of high molecular weight polydisperse materials results in the formation of a large number of polymeric chains with different chain lengths and various chemical compositions. Population balance approach is generally employed to model the size distri-

In 1971, Randolph and Larson [48] proposed a solution for the population bal-

bution of the macromolecular compound during polymerization, polymer

ance equation (PBE) in a well-mixed batch system. They used the concept of moment transform to convert the population balance equations into ordinary differential equations. Population balance based models have been developed to study the molecular weight decrease of polymers in a fragmentation process by advanced oxidation processes [18, 20, 22, 49]. Microwave-assisted oxidative degradation as an emerging advanced oxidation technology was used for poly(alkyl methacrylate) degradation. Random chain scission and Continuous distribution kinetics were employed to determine the degradation rate of the polymer [50]. Photocatalytic degradation of polyacrylamide co-acrylic acid by random chain scission has been investigated by Vinu and Madras [51]. The rate coefficients were determined as a linear function of the composition of co-monomer. Madras and McCoy [52] studied the kinetics of oxidative degradation of polystyrene by di-tert-butyl peroxide provided the ratio of the rate parameters for both oxidizer and polymer decomposition by moment analysis assuming random chain scission mechanism. Population balance and moment equations are solved for rate parameters [21, 53]. The model proposed by McCoy and Wang [21] is sufficiently applicable to a variety of degradation processes. Moment equations can be applied in batch and continuous stirred

tank reactor (CSTR) reactors for binary or ternary fragmentation.

population for a dead and live polymer moments as follow:

where pr and p•

40

The population balance model is a balance equation of species of different sizes, and it is similar to the mass, energy, and momentum balances, to track the changes in the size distribution. The benefit of the population models is that they provide a straightforward technique to derive expressions for the moments of the polymer distributions during the degradation reaction. Hulburt and Katz [54] applied the concept of moments to determine the molecular weight distribution of a polymer

> μ<sup>n</sup> ¼ ∑ ∞ r¼1 r

λ<sup>n</sup> ¼ ∑ ∞ r¼1 r <sup>n</sup> p•

chain length <sup>r</sup>, <sup>μ</sup><sup>n</sup> and <sup>λ</sup><sup>n</sup> are the <sup>n</sup>th moment of the quantities pr and <sup>p</sup>•

<sup>r</sup> are the polymer and the polymer radical concentrations with

npr (43)

<sup>r</sup> (44)

<sup>r</sup> and n having

degradation, depolymerization, and chain breakage.

Ωð Þ¼ r; s 1=r (42)

given as follows [45, 46]:

5.1 Polymer population balance

Kinetic Modeling for Environmental Systems

### 5.2 Model predictions of the process variables

The polymer average molecular weights decrease with irradiation time due to the chain cleavage that effectively shortens the polymer chains which supports the success of the degradation process. The profile of the polymer molecular weight with time during the degradation process can indicate the type and mechanism of the chain scission. For instance, the steep reduction in the molecular weights of the PVA polymer at the beginning of the degradation reaction under UV irradiation is caused by the random chain scission mechanism that dominates initially in the photo-oxidative degradation of polyvinyl alcohol. At the end of the degradation reaction, the chain scission reactions occur most likely at the polymer chain end releasing a single monomer molecule when the polymer has considerably degraded. Whereas, the PVA degradation occurs mostly by random chain scission at the beginning of the reaction which explains the drastic reduction in the polymer concentration as clearly shown in Figure 3 for initial PVA concentration of 50 mg/L. For water-soluble polymers, it is common to use a different approach, based on discrete population balance equations, to model polymer degradation involving

random scission and end-chain scission in order to predict the evolution of a popu-

Kinetic Modeling of Photodegradation of Water-Soluble Polymers in Batch Photochemical Reactor

It is worth mentioning that hydrogen peroxide has a significant effect on the performance of the degradation process. The polymer molecular weight averages decrease with an increase of hydrogen peroxide concentration up to a certain limit. Therefore, a higher level of hydrogen peroxide has an adverse effect on the molecular weight reduction which can be interpreted by the scavenging effect of H2O2 over hydroxyl radicals which hinders the radical degradation since the amount of H2O2 added to the system is proportionally high [55]. The excess amount of hydrogen peroxide acts as a scavenger of hydroxyl radicals (Reaction (3) in Table 1) thus forming hydroperoxyl radicals. As shown earlier in the photochemical kinetics mechanism, the hydroperoxyl radical reacts with the PVA polymer (Reaction (17) in Table 1) [38]. Therefore, the probability of hydroxyl radicals attacking the polymer can be significantly reduced. The hydroperoxyl radicals are less reactive than hydroxyl radicals that subsequently suppress the degradation reaction. The photochemical model takes into account the scavenging effect of hydrogen peroxide by incorporating reaction rate equations of all radicals in order to enhance the

The PVA polymers are effectively degraded in a UV/H2O2 photochemical reactor. In fact, the rates of polymer degradation and TOC removal did not match with each other. In fact, the TOC accounts for the carbon content of all chemical species, including PVA polymers. The difference between TOC and PVA removal efficiencies as shown in Figure 4 is due to the presence of intermediate oxidation products and the non-degraded polymer residuals towards the end of reaction which can

Figure 4 clearly illustrates the thresholds of the mass ratio of H2O2 and the polymer at which both the TOC removal and PVA degradation efficiencies at maximum values. In the advanced oxidation process, the amount of oxidant has to be experimentally determined according to the specified operating conditions for each pollutant so that the photochemical reaction performs at its best. Using excess hydrogen peroxide in the treatment process not only impedes the removal rate of the organic pollutants but also increase the hydrogen peroxide residual in the treated solution which can negatively affect the operating cost of the photoreactor

PVA degradation and TOC removal efficiency for PVA (500 mg/L) degradation in UV/ H2O2 photoreactor

lation of molecules undergoing different scission mechanisms.

DOI: http://dx.doi.org/10.5772/intechopen.82608

slightly increase the TOC content of the treated solution.

reliability of the model.

system.

Figure 4.

43

[data from [17]].

#### Figure 3.

Variation of the weight average Mw and number average Mn molecular weights of PVA at different [H2O2]/[PVA] mass ratios in a batch UV/H2O2 photoreactor (data from [38]).

### Kinetic Modeling of Photodegradation of Water-Soluble Polymers in Batch Photochemical Reactor DOI: http://dx.doi.org/10.5772/intechopen.82608

random scission and end-chain scission in order to predict the evolution of a population of molecules undergoing different scission mechanisms.

It is worth mentioning that hydrogen peroxide has a significant effect on the performance of the degradation process. The polymer molecular weight averages decrease with an increase of hydrogen peroxide concentration up to a certain limit. Therefore, a higher level of hydrogen peroxide has an adverse effect on the molecular weight reduction which can be interpreted by the scavenging effect of H2O2 over hydroxyl radicals which hinders the radical degradation since the amount of H2O2 added to the system is proportionally high [55]. The excess amount of hydrogen peroxide acts as a scavenger of hydroxyl radicals (Reaction (3) in Table 1) thus forming hydroperoxyl radicals. As shown earlier in the photochemical kinetics mechanism, the hydroperoxyl radical reacts with the PVA polymer (Reaction (17) in Table 1) [38]. Therefore, the probability of hydroxyl radicals attacking the polymer can be significantly reduced. The hydroperoxyl radicals are less reactive than hydroxyl radicals that subsequently suppress the degradation reaction. The photochemical model takes into account the scavenging effect of hydrogen peroxide by incorporating reaction rate equations of all radicals in order to enhance the reliability of the model.

The PVA polymers are effectively degraded in a UV/H2O2 photochemical reactor. In fact, the rates of polymer degradation and TOC removal did not match with each other. In fact, the TOC accounts for the carbon content of all chemical species, including PVA polymers. The difference between TOC and PVA removal efficiencies as shown in Figure 4 is due to the presence of intermediate oxidation products and the non-degraded polymer residuals towards the end of reaction which can slightly increase the TOC content of the treated solution.

Figure 4 clearly illustrates the thresholds of the mass ratio of H2O2 and the polymer at which both the TOC removal and PVA degradation efficiencies at maximum values. In the advanced oxidation process, the amount of oxidant has to be experimentally determined according to the specified operating conditions for each pollutant so that the photochemical reaction performs at its best. Using excess hydrogen peroxide in the treatment process not only impedes the removal rate of the organic pollutants but also increase the hydrogen peroxide residual in the treated solution which can negatively affect the operating cost of the photoreactor system.

Figure 4. PVA degradation and TOC removal efficiency for PVA (500 mg/L) degradation in UV/ H2O2 photoreactor [data from [17]].

5.2 Model predictions of the process variables

Kinetic Modeling for Environmental Systems

Figure 3.

42

The polymer average molecular weights decrease with irradiation time due to the chain cleavage that effectively shortens the polymer chains which supports the success of the degradation process. The profile of the polymer molecular weight with time during the degradation process can indicate the type and mechanism of the chain scission. For instance, the steep reduction in the molecular weights of the PVA polymer at the beginning of the degradation reaction under UV irradiation is caused by the random chain scission mechanism that dominates initially in the photo-oxidative degradation of polyvinyl alcohol. At the end of the degradation reaction, the chain scission reactions occur most likely at the polymer chain end releasing a single monomer molecule when the polymer has considerably degraded. Whereas, the PVA degradation occurs mostly by random chain scission at the beginning of the reaction which explains the drastic reduction in the polymer concentration as clearly shown in Figure 3 for initial PVA concentration of 50 mg/L. For water-soluble polymers, it is common to use a different approach, based on discrete population balance equations, to model polymer degradation involving

Variation of the weight average Mw and number average Mn molecular weights of PVA at different

[H2O2]/[PVA] mass ratios in a batch UV/H2O2 photoreactor (data from [38]).

## 6. Conclusions

The performance of the UV/H2O2 advanced oxidation process was evaluated for the degradation of polymeric wastewater in the batch photoreactor. The UV/H2O2 process can significantly modify the structure of the PVA polymer and be a potential practice for the degradation of water-soluble polymers in wastewater. Under the effect of UV light, hydrogen peroxide is readily decomposed into hydroxyl radicals of high reactivity which become oxidizing agents and can immediately attack the chains resulting in polymer disintegration.

GPC gel permeation chromatography MWD molecular weight distribution NDIR non-dispersive infra-red PBE population balance equation

DOI: http://dx.doi.org/10.5772/intechopen.82608

Kinetic Modeling of Photodegradation of Water-Soluble Polymers in Batch Photochemical Reactor

PDI polydispersity index PVA polyvinyl alcohol TOC total organic carbon

US ultrasound UV ultraviolet

Author details

\*, Mehrab Mehrvar<sup>2</sup> and Ramdhane Dhib<sup>2</sup>

\*Address all correspondence to: dhamad@ryerson.ca

provided the original work is properly cited.

1 Chemical Engineering and Pilot Plant Department, National Research Center,

2 Department of Chemical Engineering, Ryerson University, Toronto, Ontario,

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

Dina Hamad<sup>1</sup>

Cairo, Egypt

Canada

45

A theoretical description of the UV/H2O2 process incorporates a population balance of polymer system and a molar balance of all chemical species to adequately represent the degradation of PVA polymer in a UV/H2O2 batch recirculating process. Modeling the photochemical degradation of the polymers represents a new approach to investigate the variations in polymer molecular weights. Considering the importance of oxidant in the advanced oxidation process performance, the dosage of hydrogen peroxide has to be experimentally determined for each polymer in order to achieve a better photochemical degradation of water-soluble polymers in wastewater. Incorporating the scavenging effect of hydrogen peroxide and the variation of the solution acidity is essential for the predictive quality and reliability of the photochemical model for degradation of polymers by UV/H2O2 process.

The photochemical mechanism and the photochemical kinetic model provide a framework for understanding the real characterization of the UV/H2O2 process and contribute to enhancing the design of industrial UV/H2O2 processes for the treatment of wastewaters contaminated with water-soluble polymers.

## Acknowledgements

The authors would like to thank the editors for their efforts in improving the quality of the manuscript. The financial support of Ryerson University and the Natural Sciences and Engineering Research Council of Canada (NSERC) is greatly appreciated.

### Nomenclature


Kinetic Modeling of Photodegradation of Water-Soluble Polymers in Batch Photochemical Reactor DOI: http://dx.doi.org/10.5772/intechopen.82608

GPC gel permeation chromatography MWD molecular weight distribution NDIR non-dispersive infra-red PBE population balance equation PDI polydispersity index PVA polyvinyl alcohol TOC total organic carbon US ultrasound UV ultraviolet

6. Conclusions

Acknowledgements

appreciated.

Nomenclature

P1 monomer

P•

P•

44

C molar concentration, mol/L

kp rate constant of propagation, 1/s kp1 rate constant of propagation, L/mol s kp2 rate constant of propagation, L/mol s

Pr dead polymer of chain length r

<sup>r</sup> live radical of chain length r

R rate of reaction, mol/L s AOP advanced oxidation process DMP 9-dimethyl-1,10-phenanthroline

ktc rate constant of termination by coupling, L/mol s Mn number average molecular weight of the polymer, g/mol Mw weight average molecular weight of the polymer, g/mol

Pr<sup>s</sup> dead polymer of chain length r-s, where 1 ≤ s < r

<sup>r</sup><sup>s</sup> live radical of chain length r-s, where 1 <sup>≤</sup> <sup>s</sup> <sup>&</sup>lt; <sup>r</sup>

i number of species

chains resulting in polymer disintegration.

Kinetic Modeling for Environmental Systems

The performance of the UV/H2O2 advanced oxidation process was evaluated for the degradation of polymeric wastewater in the batch photoreactor. The UV/H2O2 process can significantly modify the structure of the PVA polymer and be a potential practice for the degradation of water-soluble polymers in wastewater. Under the effect of UV light, hydrogen peroxide is readily decomposed into hydroxyl radicals of high reactivity which become oxidizing agents and can immediately attack the

A theoretical description of the UV/H2O2 process incorporates a population balance of polymer system and a molar balance of all chemical species to adequately represent the degradation of PVA polymer in a UV/H2O2 batch recirculating process. Modeling the photochemical degradation of the polymers represents a new approach to investigate the variations in polymer molecular weights. Considering the importance of oxidant in the advanced oxidation process performance, the dosage of hydrogen peroxide has to be experimentally determined for each polymer in order to achieve a better photochemical degradation of water-soluble polymers in wastewater. Incorporating the scavenging effect of hydrogen peroxide and the variation of the solution acidity is essential for the predictive quality and reliability of the photochemical model for degradation of polymers by UV/H2O2 process. The photochemical mechanism and the photochemical kinetic model provide a framework for understanding the real characterization of the UV/H2O2 process and contribute to enhancing the design of industrial UV/H2O2 processes for the treat-

The authors would like to thank the editors for their efforts in improving the quality of the manuscript. The financial support of Ryerson University and the Natural Sciences and Engineering Research Council of Canada (NSERC) is greatly

ment of wastewaters contaminated with water-soluble polymers.

## Author details

Dina Hamad<sup>1</sup> \*, Mehrab Mehrvar<sup>2</sup> and Ramdhane Dhib<sup>2</sup>

1 Chemical Engineering and Pilot Plant Department, National Research Center, Cairo, Egypt

2 Department of Chemical Engineering, Ryerson University, Toronto, Ontario, Canada

\*Address all correspondence to: dhamad@ryerson.ca

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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[29] Song H, Hyun J. An optimization study on the pyrolysis of polystyrene in a batch reactor. Journal of Chemical Engineering. 1999;16(3):316-324

[30] Buxton G, Greenstock C, Helman W, Ross A. Critical review of rate constants for reactions of hydrated electrons, hydrogen atoms and hydroxyl

O) in aqueous solution.

1006

Kinetic Modeling of Photodegradation of Water-Soluble Polymers in Batch Photochemical Reactor

radicals (•

28:655-658

OH/•

Journal of Physical and Chemical Reference Data. 1988;17:513-886

W. The HaberWeiss cycle.

Society. 1979;101:58-62

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[31] Koppenol W, Butler J, Van Leeuwen

Photochemistry and Photobiology. 1978;

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[18] McCoy B, Madras G. Degradation kinetics of polymers in solution: Dynamics of molecular weight distributions. AIChE Journal. 1997;43

[19] Tayal A, Khan S. Degradation of a water-soluble polymer: Molecular weight changes and chain scission characteristics. Macromolecules. 2000;

[20] Madras G, Smith S, McCoy B. Degradation of poly (methyl

methacrylate) in solution. Industrial & Engineering Chemistry Research. 1996;

[21] McCoy B, Wang M. Continuousmixture fragmentation kinetics: Particle size reduction and molecular cracking. Chemical Engineering Science. 1994;49

[22] Kodera Y, McCoy B. Distribution kinetics of radical mechanisms: Reversible polymer decomposition. AIChE Journal. 1997:3205-3214

[23] Kommineni S, Chowdhury Z, Kavanaugh M, Mishra D, Croue J. Advanced oxidation of methyl-tertiary butyl ether: Pilot study findings and full-scale implications. Journal of Water Supply: Research and Technology. 2009;

[24] Kosaka K, Yamada H, Matsui S, Echigo S, Shishida K. Comparison among the methods for hydrogen peroxide measurements to evaluate advanced oxidation processes: Application of a spectrophotometric method using copper (II) ion and 2,9 dimethyl-1,10-phenanthroline.

Environmental Science & Technology.

[25] Ghafoori S, Mehrvar M, Chan P. Kinetic study of photodegradation of

Stability. 2014;103:75-82

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33:9488-9493

35(6):1795-1800

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57(1):403-418

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47

[9] Zhang SJ, Yu HQ. Radiation-induced degradation of polyvinyl alcohol in aqueous solutions. Water Research. 2004;38:309-316

[10] Rosario-Ortiz FL, Wert EC, Snyder SA. Evaluation of UV/H2O2 treatment for the oxidation of pharmaceuticals in wastewater. Water Research. 2010;44: 1440-1448

[11] Rozas O, Vidal C, Baeza C, Jardim WF, Rossner A. Organic micropollutants (OMPs) in natural waters: Oxidation by UV/H2O2 treatment and toxicity assessment. Water Research. 2016;98(554):109-118

[12] Chen Y, Sun Z, Yang Y, Ke Q. Heterogeneous photocatalytic oxidation of polyvinyl alcohol in water. Photochemistry and Photobiology. 2011; 142(1):85-89

[13] Kaczmarek H, Kaminska A, Swiatek M, Rabek JF. Electrochemical oxidation of polyvinyl alcohol using a RuO2/Ti anode. Angewandte Makromolekulare Chemie. 1998;4622:109-121

[14] Aarthi T, Shaama M, Madras G. Degradation of water-soluble polymers under combined ultrasonic and ultraviolet radiation. Industrial and Engineering Chemistry Research. 2007; 46:6204-6210

[15] Kim S, Kim T, Park C, Shin E. Photo-oxidative degradation of some water-soluble polymers in the presence of accelerating agents. Desalination. 2003;155(1):49-57

[16] Ghafoori S, Mehrvar M, Chan P. Photoreactor scale-up for degradation of polyvinyl alcohol in aqueous solution using UV/H2O2 process. Chemical Engineering Journal. 2014;245:133-142

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[28] Shukla B, Daraboina N, Madras G. Ultrasonic degradation of poly (acrylic acid). Polymer Degradation and Stability. 2009;94(8):1238-1244

[29] Song H, Hyun J. An optimization study on the pyrolysis of polystyrene in a batch reactor. Journal of Chemical Engineering. 1999;16(3):316-324

[30] Buxton G, Greenstock C, Helman W, Ross A. Critical review of rate constants for reactions of hydrated electrons, hydrogen atoms and hydroxyl radicals (• OH/• O) in aqueous solution. Journal of Physical and Chemical Reference Data. 1988;17:513-886

[31] Koppenol W, Butler J, Van Leeuwen W. The HaberWeiss cycle. Photochemistry and Photobiology. 1978; 28:655-658

[32] Weinstein J, Bielski B. Kinetics of the interaction of HO2 and O<sup>2</sup> radicals with hydrogen peroxide: The Haber-Weiss reaction. American Chemical Society. 1979;101:58-62

[33] Christensen H, Sehested K, Corfitzen H. Reactions of hydroxyl radicals with hydrogen peroxide at ambient and elevated temperatures. Physical Chemistry. 1982;86:1588-1590

[34] Bielski B, Cabelli D. Highlights of current research involving superoxide and perhydroxyl radicals in aqueous solutions. International Journal of Radiation Biology. 1991;59:291-319

[35] Elliot A, Buxton G. Temperature dependence of the reactions OH + O2– and OH + HO2 in water up to 200°C. Chemical Society. 1992;88:2465-2470

[36] Linden K, Sharpless C, Andrews S, Atasi K, Korategere V, Stefan M, et al. Innovative UV Technologies to Oxidize Organic and Organoleptic Chemicals. London: IWA Publishing; 2005

[37] Whittmann G, Horvath I, Dombi A. UV-induced decomposition of ozone and hydrogen peroxide in the aqueous phase at pH 2-7. Ozone Science and Engineering. 2002;24:281-291

[38] Hamad D, Mehrab M, Dhib R. Photochemical kinetic modeling of degradation of aqueous polyvinyl alcohol in a UV/H2O2 photoreactor. Journal of Polymers and the Environment. 2018;26(8):3283-3293

[39] Taghizadeh M, Yeganeh N, Rezaei M. The investigation of thermal decomposition pathway and products of poly(vinyl alcohol) by TG-FTIR. Applied Polymer Science. 2015;32(25): 42117-42129

[40] Liao C, Gurol M. Chemical oxidation by photolytic decomposition of hydrogen peroxide. Environmental Science & Technology. 1995;29:3007- 3014

[41] Stefan M, Hoy A, Bolton J. Kinetics and mechanism of the degradation and mineralization of acetone in dilute aqueous solution sensitized by the UV photolysis of hydrogen peroxide. Environmental Science & Technology. 1996;30:2382-2390

[42] Peng Z, Kong L. A thermal degradation mechanism of polyvinyl alcohol/silica nanocomposites. Polymer Degradation and Stability. 2007;92: 1061-1071

[43] Bird R, Stewart W, Lightfoot E. Transport Phenomena. 2nd ed. New York: Wiley & Sons, Inc.; 1960

[44] Labas D, Zalazar S, Brandi J, Martín A, Cassano E. Scaling up of a photoreactor for formic acid degradation employing hydrogen peroxide and UV radiation. Helvetica Chimica Acta. 2002;85:82-95

reactions: Reversible initiation, chain scission, and hydrogen abstraction. Industrial and Engineering Chemistry

DOI: http://dx.doi.org/10.5772/intechopen.82608

Kinetic Modeling of Photodegradation of Water-Soluble Polymers in Batch Photochemical Reactor

[54] Hulburt H, Katz S. Some problems in particle technology: A statistical mechanical formulation. Chemical Engineering Science. 1964;19:555-574

[55] Mehrvar M, Anderson W, Moo-Young M. Photocatalytic degradation of aqueous organic solvents in the presence

of hydroxyl radical scavengers. International Journal of Photoenergy.

2001;3(4):187-191

49

Research. 2003;42:2461-2469

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[47] Sterling J, McCoy B. Distribution kinetics of thermolytic macromolecular reactions. AIChE Journal. 2001;7:2289- 2303

[48] Randolph A, Larson M. Theory of Particulate Processes: Analysis and Techniques of Continuous Crystallization. New York: Academic Press; 1971

[49] Sezgi A, Cha S, Smith J, McCoy B. Polyethylene pyrolysis: Theory and experiments for molecular weight distribution kinetics. Industrial and Engineering Chemistry Research. 1998; 37(7):2582-2591

[50] Marimuthu A, Madras G. Continuous distribution kinetics for microwave‐assisted oxidative degradation of poly(alkyl methacrylates). AIChE Journal. 2008;54 (8):2164-2173

[51] Vinu R, Madras G. Photocatalytic degradation of polyacrylamide-coacrylic acid. Polymer Degradation and Stability. 2008;93(8):1440-1449

[52] Madras J, McCoy B. Oxidative degradation kinetics of polystyrene in solution. Chemical Engineering Science. 1997;52(16):2707-2713

[53] Smagala T, McCoy B. Mechanisms and approximations in macromolecular Kinetic Modeling of Photodegradation of Water-Soluble Polymers in Batch Photochemical Reactor DOI: http://dx.doi.org/10.5772/intechopen.82608

reactions: Reversible initiation, chain scission, and hydrogen abstraction. Industrial and Engineering Chemistry Research. 2003;42:2461-2469

[35] Elliot A, Buxton G. Temperature dependence of the reactions OH + O2– and OH + HO2 in water up to 200°C. Chemical Society. 1992;88:2465-2470

Kinetic Modeling for Environmental Systems

[44] Labas D, Zalazar S, Brandi J, Martín

A, Cassano E. Scaling up of a photoreactor for formic acid degradation employing hydrogen peroxide and UV radiation. Helvetica

Chimica Acta. 2002;85:82-95

656-659

2303

Press; 1971

37(7):2582-2591

(8):2164-2173

2008;93(8):1440-1449

1997;52(16):2707-2713

[45] McCoy B, Madras G. Chemical Engineering Science. 2001;56:2831-2836

[46] Stickle J, Griggs A. Mathematical modeling of chain-end scission using continuous distribution kinetics. Chemical Engineering Science. 2012;68:

[47] Sterling J, McCoy B. Distribution kinetics of thermolytic macromolecular reactions. AIChE Journal. 2001;7:2289-

[48] Randolph A, Larson M. Theory of Particulate Processes: Analysis and

Crystallization. New York: Academic

[49] Sezgi A, Cha S, Smith J, McCoy B. Polyethylene pyrolysis: Theory and experiments for molecular weight distribution kinetics. Industrial and Engineering Chemistry Research. 1998;

Techniques of Continuous

[50] Marimuthu A, Madras G. Continuous distribution kinetics for

microwave‐assisted oxidative degradation of poly(alkyl

methacrylates). AIChE Journal. 2008;54

[51] Vinu R, Madras G. Photocatalytic degradation of polyacrylamide-coacrylic acid. Polymer Degradation and Stability.

[52] Madras J, McCoy B. Oxidative degradation kinetics of polystyrene in solution. Chemical Engineering Science.

[53] Smagala T, McCoy B. Mechanisms and approximations in macromolecular

[36] Linden K, Sharpless C, Andrews S, Atasi K, Korategere V, Stefan M, et al. Innovative UV Technologies to Oxidize Organic and Organoleptic Chemicals. London: IWA Publishing; 2005

[37] Whittmann G, Horvath I, Dombi A. UV-induced decomposition of ozone and hydrogen peroxide in the aqueous phase at pH 2-7. Ozone Science and Engineering. 2002;24:281-291

[38] Hamad D, Mehrab M, Dhib R. Photochemical kinetic modeling of degradation of aqueous polyvinyl alcohol in a UV/H2O2 photoreactor. Journal of Polymers and the

Environment. 2018;26(8):3283-3293

[39] Taghizadeh M, Yeganeh N, Rezaei M. The investigation of thermal

decomposition pathway and products of

oxidation by photolytic decomposition of hydrogen peroxide. Environmental Science & Technology. 1995;29:3007-

[41] Stefan M, Hoy A, Bolton J. Kinetics and mechanism of the degradation and mineralization of acetone in dilute aqueous solution sensitized by the UV photolysis of hydrogen peroxide. Environmental Science & Technology.

poly(vinyl alcohol) by TG-FTIR. Applied Polymer Science. 2015;32(25):

[40] Liao C, Gurol M. Chemical

42117-42129

1996;30:2382-2390

1061-1071

48

[42] Peng Z, Kong L. A thermal degradation mechanism of polyvinyl alcohol/silica nanocomposites. Polymer Degradation and Stability. 2007;92:

[43] Bird R, Stewart W, Lightfoot E. Transport Phenomena. 2nd ed. New York: Wiley & Sons, Inc.; 1960

3014

[54] Hulburt H, Katz S. Some problems in particle technology: A statistical mechanical formulation. Chemical Engineering Science. 1964;19:555-574

[55] Mehrvar M, Anderson W, Moo-Young M. Photocatalytic degradation of aqueous organic solvents in the presence of hydroxyl radical scavengers. International Journal of Photoenergy. 2001;3(4):187-191

**51**

Section 3

Environmental Assessment

Section 3
