Physico-chemical modeling of concrete carbonation · 2.3.2 Factors influencing the speed of...

People's Democratic Republic of Algeria

Ministry of Higher Education and Scientific Research

University of Tebessa

Faculty of Science and Technology

Civil Engineering Department

Master Academic study Speciality: structures

Promo: 2015/2016

President:

Rapporteur: HAMIDANE Abdelhamied

Examiner:

Physico-chemical modeling

of concrete carbonation

ABU ALI Fakher

TABEL OF CONTENTS

CHAPTER ONE: INTRODACTION

CHAPTER TWO: LITTERATURE REVIEW

2.1 Durability issues in concrete construction Page 2

2.1.1 Physical deterioration Page 3

2.1.2 Chemical deterioration Page 3

2.1.3 Reinforcement corrosion Page 4

2.2 Corrosion induced concrete deterioration Page 4

2.3 Concrete carbonation Page 7

2.3.1 Chemistry of concrete carbonation Page 7

2.3.2 Factors influencing the speed of carbonation Page 10

2.3.2.2 Factors related to environmental condition Page 11

2.3.2.3 Factors related to curing condition Page 12

2.3.3 Consequences of the carbonation on the concrete Page 12

2.4 Models for service-life prediction of concrete Page 15

CHAPETER THREE: MODEL DESCRIPTION

3.1 Governing equations Page 18

3.2 Material parameters Page 18

3.2.1 Carbon dioxide diffusion Page 19

3.2.1.1 Pore-size distribution and specific surface area of pores. Page 19

3.2.1.2 Degree of saturation of pores Page 20

3.2.1.3 Molar concentration of carbon-table constituent of Page 22

hardened cement paste

3.2.1.4 Concrete porosity decrease Page 24

3.2.2 Moisture transfer in concrete Page 27

3.2.2.1 Humidity diffusion coefficient Page 27

3.2.2.2 Moisture capacity Page 28

3.2.3 Heat transfer in concrete Page 29

3.3 Initial and boundary conditions Page 30

3.3.1 Carbon penetration Page 30

3.3.2 Moisture transfer Page 30

3.3.3 Heat transfer Page 31

CHAPTER FOUR: MODEL FORMULATION

4.1 Galerkin method Page 33

4.1.1 Carbon penetration Page 33

4.1.2 Moisture transport Page 35

4.1.3 Heat transfer Page 35

4.2 Time integration scheme Page 35

4.3 Elements description Page 36

4.3.1 Bilinear rectangular element Page 37

4.3.2 Evaluation of Element integrals Page 38

4.4 Solution procedure Page 39

4.4.1 Nonlinear solution scheme Page 39

4.4.2 Solution algorithm Page 39

4.5 Defended of matlab Page 41

4.5.1 Elementary matrix calculations Page 42

CHAPTER FIVE: NUMERICAL PERFORMANCE OF THE

MODEL

5.1 Introduction Page 44

5.2 Effect of numerical parameters Page 45

5.2.1 Effect of mesh size Page 45

5.2.2 Effect of Time increment Page 47

CHAPTER SIX: CONCLUSION

LIST OF SYMBOLS:

Porosity of concrete fresh.

( ) Change in porosity due to hydration.

( ) Change in porosity due to carbonation.

Density.

Density of cement.

Density of aggregates.

Density of water.

Carbone dioxide concentration.

Carbone dioxide diffusion coefficient.

Isothermal moisture diffusion coefficient.

Volume fraction of the aqueous film at the walls of the pores.

Relative humidity.

Time.

Water-cement-ratio.

aggregate-cement ratio

Evaporable water content.

Molecular diameter of water.

Kelvin diameter

Weight percentages of cement paste.

Weight percentages of aggregate.

Humidity diffusion coefficients of cement paste and aggregates.

Humidity diffusion coefficients of aggregates.

Rate of dissolution of ( ) in concrete pore water.

Concentration of carbon.

Relative humidity.

Concrete moisture capacity.

Effect of heat on moisture transport .

(

)

Moisture capacities of aggregate.

(

)

Moisture capacities of cement paste.

Conductivity of concrete .

Temperature [°C].

Specific heat capacity of concrete .

Surface carbon dioxide transfer coefficient

Carbon dioxide concentration at the concrete surface.

Relative humidity in the surrounding environment.

Surface moisture transfer coefficient

Temperature in the surrounding environment

Convection heat transfer coefficient

Thickness of water in pores.

Diameters of pores.

Henry's constant for the dissolution of in water.

Concentration on the surface of the concrete( ).

Coefficient of diffusion( ).

a Capacity of connection of the concrete with ( ).

effective coefficient of diffusion of the no-slump concrete carbonated

under conditions defined (cure, environment) ( ).

Fascinating coefficient of account the influence of the carbonation on

the diffusion.

Fascinating coefficient of account the influence of the cure on Rcarb.

Fascinating coefficient of account the influence of the testing method

used for determine the effective coefficient of diffusion.

Time of reference (taken equal to 1 year).

Fascinating coefficient of account the influence of the miso climate.

List of Figures

Figure 2.1 Deterioration of reinforced concrete structures. Page 2

Figure 2.2 steel passivating layer and ion transfer. Page 5

Figure 2.3 corrosion of steel reinforcement. Page 7

Figure 2.4 carbon penetrating concrete and concrete spalling. Page 9

Figure 2.5 carbocation Mechanism. Page 10

Figure 3.1.Pore-size distribution in fully hydrated hardened cement paste or

mortar, effect of (w/c) .

Page 21

Figure 3.2 Pore-size distributions resulting from curve fitting of experimental

data.

Page 21

Figure 3.3 the yearly humidity periodicity. Page 31

Figure 3.4 the yearly Heat periodicity. Page 32

Figure 4.1 Linear triangular and bilinear rectangular elements. Page 37

Figure 4.2 Solution scheme for the proposed model. Page 40

Figure 4.3 Reinforcement in concrete. Page 41

Figure 4.4 divided by the finite element of concrete surrounding the steel and

concentration carbonation in the node.

Page 42

Figure 5.1 Steel cadres. Page 44

Figure 5.2: Carbon concentration for different mesh sizes. Page 45

Figure 5.3 Carbonation depth for different mesh sizes. Page 46

Figure 5.4 Humidity values for different mesh sizes. Page 46

Figure 5.5 Carbonation depth for different Time increments. Page 47

Figure 5.6 Carbon concentration for different Time increments. Page 47

DEGRADATION OF THE CONCRETE

The expansion of the products of corrosion exerts a pressure on the concrete causing

the cracking and the bursting of coating and sometimes the delamination of the external layer

of the concrete. Some studies suggest that the products of corrosion do not exert in their

totality a pressure on the concrete. Part of these products fills the vacuums and the pores

around the bar and another part can migrate outside the steel-concrete interface via the pores.

Once the porous zone in the neighborhoods of the bar is filled, the products of corrosion start

to exert a pressure on the concrete.

LIST OF TABLE:

Table.2.1: Relation between carbonation and pore-size [Ngala, al. 1997] Page 13

Table.3.1: Parameters of major constituents of ordinary Portland cement Page 23

Table.3.2: Approximate chemical composition of the principal types of

Portland cement as weight percentages [%].

Page 24

Table.3.3: Molar volume difference Page 25

Table 3.4 Correction factors for accounting for the cement type Page 29

Table 4.1: Correspondence between the simplified and explicit

expressions giving the element integrals

Page 36

Table 5.1. Material parameters. Page 44

Acknowledgment

First and foremost praise to almighty god, this thesis was completed after four

months of hard daily continuous work, I dedicate this humble work to my family whom

I will always be indebtedness to endure the bitterness of separation, and they was

always backing me up with strength and hope, I also want to extend my appreciation to

all my brothers and friends how escort me in the long way of study life and all of my

colleagues.

l would like to express my sincere thanks to my supervisor, HAMIDANE, for

his patience, encouragement, kindness and generosity at all stages of my research

work. He always found time to share his vast and deep knowledge in a highly

intelligible manner with me. Without my supervisor, HAMIDANE’s support,

scientific, moral and monetary, it is highly doubtful that I would have finished this

thesis.

Great thank to all working staff in the department of civil engineering for the

support and help in all the course of five years of study.

CHAPTER ONE

INTRODACTION

ABU ALI FAKHER CHAPTER ONE: INTRODACTION

INTRODUCTION Page 1

INTRODUCTION

To find the CO2 uptake we must know the amount of Portland cement in concrete

structures and the amount of this cement that has been carbonated. Carbonation is an

environmental process in time. Thus to make an accurate calculation we must know the

carbonation process and put it in a temporal context.

The carbonation process in theory is a very simple process but in reality it is far more

complex. Basically calcium hydroxide in contact with carbon dioxide ( ) forms

calcium-carbonate (CaCO3). Water is not consumed but is needed in the transformation. When

the CH is consumed the pH of the cement paste/pore solution will drop and all the other hydrate

phases will successively break down including the portlandite . When the portlandite is

completely consumed, or it is no longer sufficiently accessible, the pH drops to a value less than

9, allowing the galvanic corrosion of the steel rebar. Carbonation features also a second aspect:

the microcrystals of calcium carbonate (CaCO3) which are formed from the hydrates may

partially block the pores and thus increase its resistance to the penetration of . The final

product will consist of a mixture of carbonates together with ferrite, silicate and aluminium-

hydroxide phases.

Carbon dioxide ( ) is a gas in the atmosphere and form bicarbonate and/or carbonate

ion in water together with some dissolved carbon dioxide gas. Carbon dioxide gas and carbonate

ions can be found almost every environment on the surface of earth. The problem is thus mainly

the accessibility and mode of entering the concrete. Thus, the rate of carbonation varies

considerably and thus the ( ) uptake will depend on both the type of concrete and the

environment in which the concrete is placed.

The cement components where defined in chapter three which talk about how they react

with each other too, the notice that worth mentioning is the hydration reactions are in the peak

before 28 days, due to the presence of gypsum, C3S, C2S, C4AF, and C3A in high concentrations,

after reacting these chemicals produce Ca(OH)2 which is the main reason to cause carbonation

after reacting with the ( ) that is dissolved in pore water.

The objective of this work is to develop a model of propagation of in cement

materials taking into account the evolution of porosity, the chemical rate of carbonation reactions

of and C-S-H, as well as other factors such as the temperature, the relative humidity

and the initial concentration of principal components (portlandite and C-S-H). This model was

implemented in MATLAB to solve the too complex equations.

CHAPTER TWO

LITTERETURE REVIEO

ABU ALI FAKHER CHAPTER TWO

LITTERATURE REVIEW Page 2

2.1 DURABILITY ISSUES IN CONCRETE CONSTRUCTION

Durability of concrete can simply be defined as the resistance of concrete to physical and

chemical attack resulting from either interaction with the environment (external agents) and/or

interaction among its constituents (internal agents).

Figure 2.1Deterioration of reinforced concrete structures. [7]

As indicated in Figure 2.1, the deterioration of reinforced concrete structures can be divided

into three main types:

Physical deterioration (due to frost, fire, abrasion, cracking)

Chemical deterioration (due to sulfate and acid attack, leaching, alkali aggregate

reaction)

Reinforcement corrosion (due to carbonation, chloride attack)

Concrete Deterioration

Physical Deterioration

Cracking

Frost

Fire

Chemical Deterioration

Acide

Sulphate

Alkali-Aggregate Reaction

Reinforcement Corrosion

Carbonation

Chlorides



2.1.1. PHSICAL DETRIORATION

Cracking: crackingoccurs in concrete for a variety of reasonsbut are generally

harmless. Heat of hydration and drying shrinkage during setting can result in large

cracks in massive concrete structures such as dams. Restraint of movement in concrete

structures or lack of adequate expansion joints will cause and/or intensify cracking in

concrete structures. The expansion of corroding steel can also lead to delaminating

cracks that are rather serious. [7]

Frost: Fresh concrete cannot hydrate if the mix freezes although it can re-hydrate after

subsequent warming and re-vibration, but its ultimate strength will be affected if it

experiences frost before setting and proper curing. Hardened concrete can be affected

by freeze-thaw action, leading to large cracks and spalling. This occurs due to the

expansive stresses caused by the freezing of the pore water in moist concrete. [7]

Fire: The effect of fire is another process which can be classified under physical

deterioration: Concrete's variousnature, as the various components have different

coefficients of expansion, creates serious problems at high temperatures (above

300°C). At these temperatures, cracking and explosive spalling may result, which

could lead to a significant loss of cross sectional area and carrying capacity.

Reinforcing steel may also lose its carrying capacity and cause large deflections at

high temperatures.

2.1.2. CHEMICAL DETERIORATION

Acid: The concrete presents a basic character high armature by the hydrated

compounds of the cement paste (the interstitial phase contained in the concrete

has a very high pH). It can thus have certain reactivity with respect to the acid

solutions such as the acid rains, the waste water, water of agribusiness

industries or industrial containing organic acids, water charged in mineral acids.

Sulphate:The sulfates coming from the environment (grounds, aqueous

medium) can react with the concrete to form ettringite (salt of Candlot). This

crystallization is accompanied by a very important expansion (± 300%) and

can occur as well during the plastic phase of hardening (ettringite primary) as

after hardening (ettringite secondary).



Alkali-aggregate Reaction:The reaction alkali-aggregates (RAG) results

from an interaction between alkalis of the concrete (coming from cement, the

additions, …) and of the potentially sensitive to alkalisaggregates which

contain silica reactivates (acid silicic) being presented in the form of opal, of

chalcedony, cristobalite, tridymite and quartz cryptocristallin.

2.1.3. REINFORCEMENT CORROSION

Carbonation: Environmental penetrating the concrete can cause too

much damage if dissolved in pores water and react with due to its

ability to change the concert’s pH, resulting in steel corrosion.

Chlorides: chloride ions can easily penetrate the concrete immersed in sea

water, when the chloride concentration in the concrete reaches a certain level,

the protective environment in the concrete is destroyed, and the corrosion

process starts.

2.2. Corrosion Induced Concrete Deterioration

The corrosion of steel reinforcement in concrete is a two stages process: initiation and

propagation. During the initiation stage, aggressive agents such as chlorides or carbon dioxide

penetrate into concrete cover and accumulate till reaching a critical level at reinforcement and

cause its depassivation. This marks the beginning of the propagation stage. Depassivation of

reinforcing steel can occur because of carbonation and associated loss of alkalinity in concrete

cover or localized breakdown of the passive layer caused by chlorides. During the

propagation stage, reinforcement deteriorates and rust is formed at the concrete-reinforcement

interface. Corrosion of reinforcement results in sectional area reduction and concrete cracking

and spalling due to rust formation.

The corrosion of steel reinforcement in concrete follows basic thermodynamic

principles and kinetics that depend on the microstructure and the environmental condition

around the steel. Generally, the steel in concrete has a passivating layer (around 10 nm thick

as shown in figure 2.2) which is generally in the form of and possibly other oxides. If

the steel is in its active state where the passivating layer cannot provide a protective

environment, a series of oxidation and reduction reactions take place on the steel surface with

the concrete pore solution acting as the electrolyte. [7]



Figure 2.2steel passivating layerand ion transfer. [10]

These reactions are the iron dissolution for the anode and the oxygen reduction for the

cathode and can be written as:

(Anode) (2.1)

(2.2)

Since the concrete provides an alkali environment, the hydrogen evolution for the reduction,

which generally takes place in acidic solutions, can be ignored. The iron atoms are ionized to

ferrous ions which dissolve in the pore solution around the anodic sites. The electrons

released by the anodic reaction are deposited on the steel surface and they raise its electric

potential. These electrons then flow through or along the steel to a lower potential (cathodic)

site. The iron dissolution process will proceed only if there is a cathode reaction which will

act as a sink for the electrons produced at the anodic site. Therefore, if the oxygen and water

are not available at the cathodic sites, the corrosion will stop.

Beside the internal current taking place inside the steel (electron movement); there is also ion

movement in pore solution of the concrete surrounding the steel. This external current

consists of hydroxide ions ( ) moving from the cathode to the anode and ferrous ion

( ) moving in the opposite direction, it’s important to understand that the water in the



concrete pores is actually a dilute solution of several alkali and calcium hydroxide. It is this

fluid that serves as a medium for ionic flow. If the pores are dried out or if the structure of the

concrete is dense such that the pores are not sufficiently interconnected, the flow of ions

through the pores becomes very difficult.

The entire process leads to the formation of corrosion products. The actual loss of steel

involved in the corrosion process takes place at the anodic sites, with the reaction of

(2.3)

Combining equation (2.1), (2.2) and (2.3) yields the following total electrochemical reaction:

(2.4)

Ferrous oxide can further react with the available oxygen and water at the anode

according to the following reaction:

(2.5)

, The so-called hydrated rust is the final corrosion product whose volume is

four times as large as the volume of the steel with the same mass. Therefore the formation of

red rust may cause internal stresses and consequently the cracking and spalling of the concrete

covering the reinforcementif the oxygen concentration in the pore solution is not adequate for

the formation of the red rust according to equations (2.4) and (2.5), a thickening of the oxide

film on the steel surface can be observed with the formation of black rust ( , The

formation of black rust can be explained by the following equation:

(2.6)

The volume of the black rust is only twice as large as the volume of the steel with the same

mass; therefore it is less dangerous compared to the red rust. [7]



Figure 2.3corrosion of steel reinforcement

2.3. CONCRETE CARBONATION

The carbonation is a phenomenon of natural ageing which relates to all the concretes. It

corresponds to the reaction of the hydrated products of cement with dissolved atmospheric

CO2 in the interstitial solution. These reactions are accompanied by a reduction in the pH of

the concrete from 13 to 9 approximately causing the depassivation of the protective coating of

the reinforcements and the reduction in the porosity of the concrete (the products of the

reactions of carbonation have volumes higher than those of the reactants). One of the

consequences principal of the carbonation is to support the corrosion of the reinforcements.

From a physicochemical aspect, to explain the mechanism of carbonation of the

concrete, it is necessary first of all to present the components of the hydrated cement paste

which react with CO2 at the time of carbonation. It is also necessary to present the properties

of CO2 and its reactions in the aqueous phase. In this section, we present a summary

description of cementing minerals, carbonic gas, unfolding of the carbonation and its effect on

the characteristics of the concrete and of its reinforcements.

2.3.1. CHEMISTRY OF CONCRETE CARBONATION

Carbonation happens when the carbonate ions come close to the calcium ions in the pore

solution and form calcium carbonate. Calcium carbonate has a very low solubility in concrete,

thus all calcium compounds dissolve and finally form calcium carbonate.

Carbon dioxide exists in significant amounts in the atmosphere. However, it cannot

react directly with the hydrates of the cement paste while in its gaseous form. The gas

must first dissolve in water and form carbonate ions which react with calcium ions of the pore

water. Carbon dioxide will dissolve in water. The type of carbonate ions depends on the PH.



When comes into contact with water at neutrality it forms bicarbonate. Inside concrete,

the pH is high and as a result the bicarbonate dissociates and forms carbonate ions. Thus in

the carbonated layer bicarbonate forms but closer to the un-carbonated cement paste, this

carbonate ions form and precipitate calcium carbonate crystals (CC). Calcium carbonate

exists in three crystallographic forms, aragonite, vaterite and calcite. Calcite and vaterite are

commonly found in carbonated concrete. Presumably the metastable vaterite will transform

into stable calcite over time.

The carbonation process can be described by the following chemical equations;

(2.7)

(2.8)

The carbonate ion will react with cations in the pore solution.

(2.9)

This will lead to lower concentration of which in turn will lead to dissolution of primarily

calcium hydroxide (CH). The solubility of CC is much lower than that of CH.

(2.10)

(2.11)

Thus will dissolve and will precipitate and the process

will continue until all of the CH is consumed. Apart from CH the cement paste contains

calcium silicate hydrate (C-S-H) and ettringite/monosulphate (Aft/AFm). These components

are in equilibrium with and stabilized by high pH and calcium ions in the pore solution. Thus

when the CH is consumed the pH and the calcium ions concentration drops and the C-S-H

will dissolve correspondingly. Monosulphate (AFm) will decompose at a pH of around 11.6

into ettringite and aluminate compounds and later the ettringite (Aft) will decompose at a PH

of around 10.6 resulting in sulphate ions and aluminum hydroxide compounds. At PH < 9.2

none of the original calcium containing phases remain. Most of the Ca from the C-S-H will be

bound to calcium carbonate but some Ca will always remain in silica gel.



Figure 2.4 carbon penetrating concrete and concrete spalling.

The most complicated transformation is that of the main cement paste phase C-S-H. It

is built up of short silica chains bound together by and OH-ions. When the carbonation

lowers the content of in the pore solution this will be compensated for by the release of

from C-S-H. This will successively change the composition of the C-S-H and give it a

lower Ca/Si ratio. Eventually when the Ca/Si ratio drops to less than 1 (Stronach& Glaser

1997) and the pH is around 10 it will transform into a silica gel. However, some Ca will

always remain in the silica gel. Bary and Sellier (2004) assume that the remaining C-S-H in

the fully carbonated zone has a CaO/SiO2 ratio of 0.85 compared to 1.65 in the un-carbonated

zone. This can be described by the following equation that mainly tells us that during the

chemical reaction the C-S-H releases CH which is carbonated and that this process gives a C-

S-H with lower contents of CaO.

C-S-H (1) = C-S-H (2) +CH where Ca/Si (2) <Ca/Si (1)

Both ettringite and monosulphate are stabilized by high PH and high concentrations of

. Gabrilová et al. (1991) found that in non-equilibrium conditions the disappearance

is related to PH.

Alkali ions in the pore solution will suppress the solubility of ions but still the

CH will dissolve and CC will precipitate. Bicarbonate ions stabilize when the paste has

carbonated leading to a drop in the PH. This is an acid and thus silica gel at a PH close to

neutral will be the stable compound. Therefore a front where bicarbonate converts to

carbonate ions could be found.



Figure 2.5 carbocationMechanism. [5]

2.3.2. FACTORS INFLUENCING THE SPEED OF CARBONATION

The speed of carbonation of the concrete depends on several factors relating to the

environment and the material concrete itself. These factors were studied in details in the thesis

of Thiery [2005] and in the book [Olivier and Al, 2008] which we take as a starting point to

present the most important:

2.3.2.1. FACTORS RELATED TO MATERIAL

Ratio: influences considerably the porosity of materials containing cement

[Loo and Al, 1994]. Any water excess lead to an excess of porosity supporting the gas

penetration and in particular CO2;

: The interface transition zone (ITZ) between the aggregate and the surrounding

cement paste has been considered to be very important for all kinds of properties of

concrete. The ITZ has a higher porosity resulting in higher permeability thus a higher

diffusivity than the bulk cement paste. Tests showed that the thickness of the ITZ

varies depending on the aggregate size, average spacing between aggregates, and

surface condition of the aggregate particles. Therefore, the properties obtained from

the tests of pure cement paste should not be applied directly to the cement paste matrix

in concrete. According to the foregoing analysis, when the aggregate content is low,

the matrix is composed of bulk cement paste as well as ITZs, and the properties of the



matrix should be considered as a function of the added aggregate content. When the

aggregate content is high (e.g., above 50%), most of the matrix is ITZs, and thus the

properties of cement paste may be considered as constants. [9]

Water Content of the Concrete: the water content of the concrete or the relative

humidity of the air in balance with the concrete is a factor which presents a great

influence on the kinetics of carbonation. The speed of carbonation is maximum for

relative moisture ranging between 60 and 80%. Beyond 80%, the kinetics decreases

quickly to reach extremely low values when the pores are saturated with water

(relative humidity > 90%), knowing that the diffusion of CO2 in water is ten thousand

times weaker than in the air [Houst, 1992]. On the other hand, if a concrete is placed

in a very dry environment, the quantity of water present in the pores is insufficient to

dissolve carbon dioxide. The kinetics of carbonation is thus weak or very weak when a

concrete is immersed or when it is placed in a very dry environment.

Cement content: cement content is responsible for the quantity of matter likely to be

carbonated. The more cement proportioning is raised, the more there is matter for the

carbonation. And since the carbonation causes a reduction porosity, there will be a

braking the speed of penetration of the carbonic gas [Houst, 1992], i.e. at a given

moment, the depth of carbonation decreases for an increase in cement content;

Type of Cement: cements with secondary components (dairy, fly-ashes, pozzolana)

naturally have a content of compounds likely to be carbonated lower than that of

Portland cement. Thus, the depth of carbonation of cements with additions is higher

than that of Portland cement. However, the additions make it possible to decrease

porosity and thus the negative effect can be cancelled.

2.3.2.2. FACTORS RELATED TO ENVIRONMENTAL CONDITION

Carbon dioxide concentration: the increase in the CO2 concentration induces an

increase the speed of carbonation, since one observes an increase the speed of

carbonation in places where content CO2 is high (tunnels, garages, chimneys,…).

However, the variations of the content CO2 of the air have an influence on the

concretes of moderate resistance approximately 30 MPa. Beyond, the content CO2

does not have any more an influence on carbonation.

Temperature: the effect of the temperature on the speed of carbonation is very

important. A rise in the temperature accelerates the chemical reactions (thermo-

activation according to the law of Arrhenius). In addition, an increase in the



temperature decreases the solubility of CO2 and decreases its concentration in the gas

phase.

Humidity: the relative humidity of the air surrounding the concrete is a big element

influencing carbonation. The maximum carbonation is seen in a relative humidity

ranging from 60 to 80%. Relative humidity’s higher than that carbonation decreases to

extremely low values, contrary situations that are places in very dry environment

carbonation is very weak due to lower quantity of water that is insufficient to dissolve

carbon dioxide.

2.3.2.3. FACTORS RELATED TO CURING CONDITION

Conditions of cure: the cure influences porosity notably. Indeed, an unsuited cure leads to

an insufficient hydration of the surface layer of concrete for lack of water, which increases

porosity and, consequently, the sensitivity to carbonation. The reduction of the time of

cure can thus have long-term fatal consequences whereas a prolonged wet cure limit the

depth of carbonation.

Humidity curing: it’s worth mentioning that putting the concrete in a saturated (with

water) state, gives the concrete the sufficient time (curing) to fully hydrate, the most

recommended time for this process is 28 days. This time of the saturating state decreases

the pore-size due to the resulting hydration material are more solids than the reactants,

thus less permeability is expected for the parameter involved in carbonation.

Thermal treatment: the elements made in the factory’s are fitted in the ovens to reduce the

time of reaching fully solid fresh elements, knowing that higher temperature increases the

reactions speed in the cement paste, this mechanism decrease the time for a complete

hydrated cement.

2.3.3. CONSEQUENCES OF THE CARBONATION ON THE CONCRETE

From the point of view of the concrete alone, one considers that the carbonation has a

beneficial effect: it improves the mechanical performances and limits the penetration of

aggressive agents by reducing porosity; one speaks about an effect of “filling” of the pores.

However, the reduction in the pH of the aqueous phase of the pores which the carbonation

causes supports the de-passivation of the protective coating of steels and consequently

supports the corrosion of the reinforcement’s bars. We present below the principal

consequences of the carbonation on the reinforced concrete. [5]



REDUCTION IN POROSITY

The carbonation of the Portland cement involves an increase in volume. If one takes account

of the solid products, the increase in volume compared to is, according to the

crystalline structure of formed calcium carbonate (Calcite, Aragonite and Vaterite):

Table.2.1: Relation between carbonation and pore-size [Ngala, al. 1997]. [5]

w/c Porosity

reduction due to w/c (%)

Porosity reduction

due to materials (%)

materials

0.5 14.6 12 Calcite

0.6 13.4 3 Aragonit

e

0.7 9.9 19 Vaterite

The distribution of the size of the pores of a cement paste hardened is changed due to

carbonation. It’s known that it shows a notable reduction of the volume of the pores and thus

a reduction in porosity.

INCREASE IN THE MECHANICAL RESISTANCE OF THE CONCRETE

The reduction of porosity involves an increase in the mechanical resistance of the concrete.

The calcite which was formed consolidates the structure of the cement paste hardened. It is

well-known that the calcium carbonate which is formed by carbonation of the Portland

cement is excellent binding. The compressive strength of concretes of Portland cement

preserved in an atmosphere of CO2 can increase up to 30%. The increase in resistance is more

likely to be when the ratio w/c of the concrete is lower.

INCREASE IN THE WATER CONTENT

The carbonation locally induces an accumulation of moisture in the pores: indeed, the

carbonation of the Portland cement and the CSH releases part of the water of structure of the

hydrates. This released water can take part in the composition of the interstitial solution and

contribute to the transport of aggressive agents. It is noted that the carbonation of the Portland

cement is the principal origin of this interstitial water salting out.



EVOLUTION OF THE TRANSPORT PROPERTIES

The evolution of the microstructure, which accompanies the carbonation, has an impact on the

properties of transfer of cementing materials: effective coefficients of diffusion with ions and

with gases and permeability to gases and liquid water. The effective coefficient of diffusion to

the gases (oxygen or hydrogen) is decreased after the material carbonation containing cement

Portland. However, it effective coefficient of diffusion of the ions (chlorides for example) is

increased at the conclusion of the material carbonation containing this cement.

CARBONATION SHRINKAGE

The carbonation of the Portland cement involves an increase in volume of the solids from 3%

to 19% according to the product of reaction obtained (Vaterite, Calcite or Aragonite), which

leads us to think that it is of an enlargement and not about a shrinkage. However all the

experiments highlight shrinkage. This phenomenon can be allotted to the water loss caused by

the reaction, because of observation that the quantity of water released during this reaction is

accompanied by a withdrawal which is of the same order of magnitude as a withdrawal

inflicted by the departure of the same quantity of water without carbonation.

FALL OF THE CEMENT PH AND CORROSION OF THE REINFORCEMENTS

The interstitial solution is an alkaline solution of pH close to 13. This basic medium ensures

the passivation of reinforcing steels. However, the carbonation of the concrete involves a fall

of the pH to a value from approximately 9 for which the electric potential of the

reinforcements falls towards the negative values supporting their corrosion. The pathology

which appears is a generalized corrosion resulting in a progressive reduction in the surface of

the reinforcements. Moreover, the formation of expansive rust applies a pressure on the

concrete surrounding the reinforcement. This pressure is often sufficient to cause the bursting

of the concrete of coating.

COMPRESSIVE STRENGTH

The compressive strength is in quasi linear relationship to the depth of carbonation. When the

compressive strength of the concrete increases, the carbonation decreases because of

reduction in porosity. It is observed [Tsukayama and Al, 1980] that beyond approximately 50

MPa, the carbonation becomes negligible.



2.4. MODELS FOR SERVICE-LIFE PREDICTION OF CONCRETE

Traditionally, predicting the service lives of concrete structures have been made on the basis

of the designer’s conclusion taking account of experience with similar structures in similar

environments. Over the years, this approach has enabled engineers to develop general

guidelines and, as a result, codes of practice have evolved for the design of durable structures.

Very recently, rapid advances in the computational material science of concrete and

information technology have made it possible to develop service-life prediction models that

can be used to estimate service life of concrete with a high reliability than was presented in

the past. These models can direct the manipulating of the concrete for use in any given

structure. At present, the use of such models is stalled by the lack of experienced modellers

and by the fact that there are no standard guidelines for the acceptance and use of models.

Service-life prediction models can be grouped into three main categories a) empirical,

b) mechanistic, and c) semi-empirical. Any model from any of the three categories may be

useful as long as its limitations are acknowledged. [4]

An empirical model creates its predictions based on previously observed

relationships among service life, concrete composition, and exposure conditions,

without invoking an understanding of the scientific reasons for the relationships;

this category includes neural network models, a good examples for the empirical

models are CEB, and Duracrete. [5]

a) Duracrete model [2000]:It is about a model of deterioration based on the first law

of Fick which takes into account parameters relating to the environment, material and

the execution (conditions of concreting, of cur..). The depth of carbonation according

to this model is given by:

√

√

(2.12)

(2.13)

- : effective resistance of the concrete to the carbonation (

- : capacity of connection of ; one defines this size as being mass

necessary for the complete carbonation of a volume of concrete given.



- : effective coefficient of diffusion of the no-slump concrete carbonated under

conditions defined (cure, environment) ( ).

- : fascinating coefficient of account the influence of the carbonation on the

diffusion (history of moisture on the surface of the concrete during its use).

- : fascinating coefficient of account the influence of the cure on Rcarb.

- : fascinating coefficient of account the influence of the testing method used for

determine the effective coefficient of diffusion.

- : concentration on the concrete surface ( ).

- : time (year).

- : time of reference (taken equal to 1 year).

- : fascinating coefficient of account the influence of the méso climate (orientation

and site of the structure).

The importance of this model comes owing to the fact that it makes it possible to take

into account the majority of the parameters taking part in the carbonation and, consequently,

leads to a good representation of the mechanism. However, the identification of the

parameters requires an important experimental campaign, which exceeds the framework of

this work.

b) Model of CEB [1997]:The studies undertaken by the CEB made it possible to

propose a simplified model making it possible to determine the evolution of the face

of carbonation during time. Although this model does not take into account all the

parameters influencing the carbonation (like Duracrete models it), the parameters

considered are most relevant. Thecarbonation front is defined by the relation:

√

(2.14)

Avec:



- : concentration on the surface of the concrete ;

- : coefficient of diffusion ;

- a: capacity of connection of the concrete with

- T: time (year).

This model makes it possible to take into account the most relevant factors in the

carbonation, thus ensuring a good total representativeness of this phenomenon. This model is

retained for the modeling of the phase of initiation of the carbonation.

In contrast, a mechanistic (or physic-chemical) model provides predictions for the

service life based on the mathematical descriptions of the phenomenon

contributing to concrete degradation. Most mechanistic models require an

understanding of the microstructure of the concrete before and during the

degradation process. The biggest example for this type of models is that developed

by Papadakis (1991). The mathematical model is based on chemical reaction

kinetics and mass-balance of gaseous , solid and dissolved , CSH, and

unhydrated silicates , which account for the production, diffusion, and

consumption of these substances. In our work, we will use this model.

Semi-empirical models combine features of mechanistic and empirical models and

they tend to use simpler mathematical expressions than mechanistic models. This

kind of model was adopted by Saetta et al. (1995). Predictions in this kind of

models are made using customize parameters that are calibrated on the basis of

data on the performance, in the field or in the laboratory, of concretes of known

composition.

CHAPTER THREE

MODEL DESCRIPTION

ABU ALI FAKHER CHAPETER THREE

MODEL DESCRIPTION Page 18

3.1 GOVERNING EQUATIONS

The significant phenomena (moisture, temperature and carbon transport) that contribute to the

rate and amount of diffusion in concrete will be modelled, solving the governing partial

differential equations, by a two-dimension time dependent non-linear finite element

technique. The diffusion of , moisture and heat transfer can be described by the following

coupled system of differential equations:

Mass balance of

* ( ), -+

.

, -

/ (3.1)

Mass balance of dissolved and solid ( )

, ( ) - (3.2)

Mass balance of CSH

, - (3.3)

Mass balance of unhydrated silicates

, - (3.4)

, - (3.5)

Moisture diffusion

, ( )-(3.6)

Heat transfer

, ( )-(3.7)

In the above system of equations, each process is related to a state variable. For

penetration, (Eq.3.1) throw (3 .5 ) the carbon concentration is taken as a state variable,

pore relative humidity for moisture transport (Eq.3.6), and the temperature for the heat

transfer (Eq. 3.7).

3.2 MATERIAL PARAMETERS

It is very important to use reliable material model sin order to obtain an accurate prediction of

the coupled diffusion processes.



3.2.1 CARBON DIOXIDE DIFFUSION [8]

In which x is the distance from the surface of the concrete and t denotes time. In Eq. (1) and

(2) the term is the porosity of concrete is the volume fraction of the pores partially filled

with water, is the fraction of the pore volume entirely filled with water( );is the

fraction of the pore volume filled with gases. The ; is the diffusion coefficient of carbon

dioxide ( ), and denotes the rates of production of CSH and ( )

respectively. When ( ) ( s ) reacts with dissolved this reaction occurs at a rate

in moles of ( ) per unit volume of the gaseous phase of the pores per sec.

, -,, -,, -, and , ( ) - are the concentrations of , , ( ) at

time t. and ; is the same as ( )

3.2.1.1 PORE-SIZE DISTRIBUTION AND SPECIFIC SURFACE AREA OF

PORES.

The pore-size distribution (PSD) of concrete or hardened cement past affects its effective

diffusivity and controls the degree of saturation of the pores (for given relative humidity) and

their specific surface area. There are mainly tows methods used today in research and by

industry for the determination of the pore-size distribution of porous solids are: mercury

porosimetry and nitrogen desorption.

The carbonation reaction (and other chemical reaction related to durability) take place

at the interface of the pores and the solid volume of concrete, at a rate which is proportion to

the specific surface area of the pores ( ), the total pore surface area per unit volume of

concrete. The value of ( ), can be computed from the (PSD) and the porosity of concrete, or

alternatively from the total pore surface area per unit weight of concrete S (measured by water

vapour or by the nitrogen adsorption method), the density of the solid phase and the

porosity of concrete, as follow.

∫

( )

(3.8)

( )(3.9)

3.2.1.2 DEGREE OF SATURATION OF PORES



The degree of saturation of the pores with water is the fraction of the pore volume filled with

water. Knowledge of is important for durability, because the fraction of the pore volume

available for diffusion of gases, such as ( )which leads to carbonation and ( )that excites

corrosion, is equal to( ). Pores with a diameter (d) less than the Kelvin diameter ( )are

completely filled with water, while the walls of all pores with diameters (d) exceeding

( )will be covered by a continuous film of water, of thickness ( ). The values of ( w) and

( )can be determined if the absolute temperature T in (K°) and the relative humidity H are

known.

.

( ( ) )( )/ ( )(3.11)

.

( )/ (3.12)

in which H Relative Humidity, the constant A is equal to 0.6323 for noncarbonated material

and to 0.2968 for carbonated, C is approximately equal to 100 for noncarbonated and for

carbonated cements C equals 1, is the molecular diameter of water (equal to µ.m).

The pores completely filled with water, those diameters( ), take up the following

fraction of the total pore volume

∫

( )(3.13)

Whereas pores partially filled with water, of diameter( ), occupies the following

fraction of the total volume of the pores

.

( ( ) )( )/ {∫

( )

( )} (3.14)

The degree of saturation is the sum of ( )and ( )

(3.15)



Figure 3.1.Pore-size distribution in fully hydrated hardened cement paste or mortar, effect of

(w/c) , -.

To find the value of ( )parameters, we have used AUTOCAD to retrieve

experimental data from the work done by , - . Where w/c = 0.5 and a/c=3.

Then, this experimental data is fitted using Curve Fitting Tool on MATLAB. The developed

equation gives a very good approximation for the experimental data. The results we have

obtained are shown in the figure 3.2.

Figure 3.2Pore-size distribution resulting from curve fitting of experimental data

The best fit is given by the following equation:



( .

/

) ( .

/

) ( .

/

)

( .

/

) ( .

/

) ( .

/

)

( .

/

) (3.16)

3.2.1.3 Molar Concentration of Carbonatable Constituent of Hardened Cement

Paste

The major constituents of hardened cement paste subject to carbonation in the presence of

moisture are ( ) , calcium silicate hydrate ( ), calcium silicates and prior to

their hydration. For the other hydrated or unhydrated constituents of hardened cement paste,

carbonation seems to be limited to a surface zone, without affecting the bulk of the

crystallites. Therefore, it is the molar concentrations of ( ) and , and the yet

unhydrated amounts of and that are of interest as far as carbonation is concerned.

These constituents of cement are produced or consumed by the chemical reactions of

hydration. The hydration reactions of ordinary Portland cement (OPC) in the presence of

gypsum are as follows:

→ (3.17)

→ (3.18)

→ (3.19)

→ (3.20)

When the gypsum is totally consumed, reactions (3.3) and (3.4) become as follows:

→ (3.21)

→ (3.22)

The preceding chemical reaction occur at molar (in moles of constituent i reacting, par unit

volume of concrete/sec) in which ( ), is the corresponding recting constituent

of Portland cement. , denote the reaction rate of constituent ( ) in the presence of

gypsum. Thus the rate of production of CSH and ( ) denoted by and , are

respectively

( )( ) (3.23)

( ) ( ) ) (3.24)

Hydration reactions take place at molar rate per unit concrete



volume ( ), in( ). The presence of gypsum has only a

minor effect on the rates on the rates of hydration of and , so that reaction (3.19) and

(3.20) can be taken to have the same rate as reaction (3.21) and (3.22), the rate can be

obtained from measurements of the fraction ( ) of compound i, which has been hydrated at

time t (in sec) after mixing, such measurements are presented here:

, - , - , -

(3.25)

( ) , - , - ( ( ) ( ) (3.26)

In which , - and , - are the current and initial ( ) molar concentrations of compound( ),

respectively in( ). Fitted values of the exponents and the coefficients are listed in Table

(3.1), which can be considered to apply to OPC of normal fineness, corresponding to Type (1) cement.

In the presence of gypsum, Reactions (3.19) and (3.20) dominate over Reactions (3.21) and (3.22), so

Reactions (3.21) and (3.22) take place only after all the gypsum has been consumed. This happens at a

time( ), which can be determined from the condition

, - ( ) , - (

) , - (3.27)

Table.3.1: Parameters of major constituents of ordinary Portland cement

Compound

Exponent ( ) 2.65 3.10 3.81 2.41 ــــــ

Coefficient

( ) ( ) ( )

ــــــ 2.46 1.00 0.16 1.17

Molar weight ⁄ 228.30 172.22 485.96 270.18 172.17

The value of ( )can be found numerically, from an exponential in ( )equation, which

results from substitution of ( )and (

)from Eq. (8) in terms of( ). The second

term in Eq. (9a) can be considered negligible, because ions react preferentially with

rather than with , so the value of t* can be computed in approximation

( ) [ .

, -

, - /( )

] (3.28)

The molar concentrations of the carbonatable constituents at time t after mixing are

, ( ) -

, -

, - , - (3.29)

, ( ) -

, -

, - , - , - , -



(3.30)

, -

, -

, - (3.31)

, - , - ( ) (3.32)

, - , - ( ) (3.33)

The initial molar concentrations of compound ( ) and of gypsum,

denoted by , - and , - , respectively, in OPC concrete can be computed from: a) the

weight fraction of clinker and gypsum in OPC, denoted by and ,

respectively) the weight fraction of compound in the clinker which can be calculated from

oxide analysis) the mix proportions of concrete, i.e., the water-cement ratio , the

aggregate-cement ratio , and the volume fraction of concrete in entrapped or entrained air

, - ( )

.

/ (3.34)

, - ( )

.

/ (3.35)

In which , and in ( ) are the densities of cement, water, and aggregates,

respectively, and , are the molar weights of compound and , respect, given in

Table 3.2 in .

Table.3.2:Approximate chemical composition of the principal types of Portland cement as

weight percentages [%].

Type Common Name

1 Ordinary 50 25 12 8 5

2 Modified 45 30 7 12 5

3 High early strength 60 15 10 8 5

4 Low heat 25 50 5 12 4

5 Sulfate resistant 40 40 4 10 4

3.2.1.4 CONCRETE POROSITY DECREASE

Concrete Porosity decrease due to hydration reactions

The porosity of concrete and ( ), defined as the ratio of pore volume to the total volume of



concrete, before 28 days porosity decreases with time due to the processes of cement

hydration( ( )) and decreases due to carbonation process ( ),

( ) ( ) ( ) (3.36)

In which is the porosity of fresh concrete and ( )correspond to the reduction in

porosity due to hydration. The initial valueof porosity is the sum of the volume fractions

ofmixing water and entrapped air, denoted by is the range of a few present, depending on

the maximum aggregate size, whereas the volume fraction of water in fresh concrete equals

the mass of water per unit volume of concrete divided by the mass density of water. In terms

of the composition parameters of concrete, the initial porosity equals:

( )

.

/ (3.37)

The reduction in porosity due to hydration ( ) is due to the fact that the molar volume of

the solid products of hydration, in the Eq. (3.17) through (3.22), exceeds that of the solid

reactants, in the left-hand side of these equations. , , , ,

, are the differences in molar volumes between solid reaction products and

solid reactants in the hydration reactions, Eq. (3.17) through (3.22), respectively (given in

Table 3.2, in per reacting mole of compound ( ), then

( ) , - , - , -

, -

(3.38a)

( ) , - , - , - ( )

, - . ( )/ , - (

) , - (

( )) (3.38b)

Table.3.3: Molar volume difference

Hydration reaction (3.17) (3.18) (3.19) (3.20) (3.21) (3.22)

⁄ 53.28 39.35 220 155.86 230 149.82



Concrete Porosity decrease due to carbonation

When penetrate the concrete the term ( )(the decrease in pore-size due to

carbonation) is due to the fact that the molar volumes of the solid products of the carbonation

of ( ) and exceed that of the molar volume of the materials reacting with( ).

and , respectively. Carbonation

usually proceeds in the volume of concrete in the form of a front, separating a completely

carbonated region from the yet carbonated area. In this region, the value of is zero,

whereas in the carbonated region, ( )is approximately equal to:

, ( ) -) , - (3.39)

In which the concentrations of ( ) and CSH are those at the completion of hydration.

The porosity ( )of the hardened cement paste, which is related to the total porosity ( ), is

as follows

( ) ( ) (

)(3.40)

The rate of carbonation of ( ) is given by:

, - , -(3.41)

In which [ ] is the molar concentration of in the gas phase of the pores (in moles of

( ) per unit volume of pore air); ·( ) is Henry's constant for

the dissolution of in water. is the gas constant T is

the absolute temperature in K; and (at 25 C) is the rate constant for the

reaction of and In aqueous solutions , and therefore ( ) ( ),diffuses

from regions with higher concentration of to those with lower, with a diffusivity

( ) ( ) of the order of . This diffusion takes place in the aqueous phase

of the pores.

The carbonation of CSH and of the yet unhydrated and of hardened cement paste

takes place according to the reactions at rates ( , and (in moles of

constituent j per unit volume of concrete, per sec)). These rates are taken, equal to:

, - (42)

In which , - is the molar concentration of constituent (in moles par unit volume of

concrete), its molar volume(

) in( ), and( ) the rate constant for the reaction of constituent j with the ,



in( ).

3.2.2 MOISTURE TRANSFER IN CONCRETE

The total moisture in concrete pore space is composed of liquid water and water vapor.

Samson et al. (2001) showed that if the liquid water phase in the concrete pore space is

assumed to be continuous, the gravitational effects are neglected; the movement of the total

moisture in terms of liquid water content is described in the following equation:

, ( )- (3.43)

, - Is the effect of heat on moisture transport will be incorporated empirically in the

total moisture diffusion coefficient which contains the influence of both liquid and vapor.

Is the concrete moisture capacity, is the concentration of carbon, and pore relative

humidity for moisture transport.

The self-desiccation during the hydration process and thermal gradient has no effect on

moisture transfer. In fact, in normal strength concrete, the decrease in water content due

to self-desiccation is very small and can be neglected for practical purposes. For high

strength concrete, due to low water-cement-ratio of the concrete mix, self-desiccation

becomes significantly high and has to be taken into account (Bentz et al., 1997).

The moisture transfer problem in porous concrete has been widely studied and a significant

number of computer models have been developed over the years, which can solve the

nonlinear and/or coupled moisture transport phenomenon in porous materials.

Main purpose of the current study is not to model the moisture transport in concrete, but to

model the carbonation induced reinforcement corrosion in concrete structures. In other

words, the mode-ling of moisture transport constitutes a small but important part of the

proposed model. Therefore, the transient nonlinear moister transfer model proposed in the

current study has been designed to be robust and as accurate as the existing computer

models.

3.2.2.1 Humidity diffusion coefficient

From the composite theory, that considers the concrete as a two-phase material, the humidity

diffusion coefficient of concrete can be evaluated as follows in which gi is the aggregate

volume fraction, and are the humidity diffusion coefficients of cement paste

and aggregates.



(

[ ]

[( ) ]

) , - (3.44)

The discontinuous pores in aggregates and being enveloped by cement paste caused the

humidity diffusion coefficient to be very small and even negligible. The humidity diffusion

coefficient of cement paste can be estimated as follows:

( ( )), - (3.45)

Where , , are coefficients calibrated from test data and given as:

( ) (3.46)

( ) (3.47)

( ) (3.48)

Where w/c is the water to cement ratio.

3.2.2.2 Moisture capacity

The concrete moisture capacity can be evaluated by the average of the moisture capacities

of the aggregates and cement paste (Xi, 1995a, b) as:

.

/

.

/

(3.49)

In which W is the evaporable water content, and are the weight percentages of

aggregate and cement paste and.

/

and.

/

are moisture capacities of aggregate

and cement paste.

The water content is given by the BET adsorption isotherm, in grams of water per grams of

cementitious material, as:

( )

, - , ( ) -, - (3.50)

In which H and T are the current relative humidity and temperature, C, k and Vm (the

monolayer capacity) are given by:

.

/ (3.51)

.

/

(3.52)

{

( ⁄ ) ( ) ⁄

( ⁄ )

( ⁄ ) ⁄

( ⁄ ) ⁄

(3.53)



{

( ⁄ ) ( ) ⁄

( ⁄ )

( ⁄ ) ⁄

( ⁄ ) ⁄

(3.54)

Where is the water to cement ratio and is the equivalent hydration time, given as:

, ( ) - (3.55)

Where H is the pore relative humidity and t is the real hydration time.

The values of and are multiplied by some correction factors that account for the type of

cement used. The values of these factors are given in table (3.4).

Table 3.4 Correction factors for accounting for the cement type.

For the aggregates, the values of and n could be determined using the relations:

(3.56)

(3.57)

For dense aggregates, = 0.05 and = 1.

The moisture capacity of each of the cement paste and the aggregate can be determined

using the expression:

( ( ) ) ( ) ( )

( ( ) ) ( ) (3.58)

3.2.3 HEAT TRANSFER IN CONCRETE

Thermal conduction inside concrete for a two-dimensional problem is governed by Fourier's

law, which is represented by a two-dimensional equation:

, ( )- (3.59)

Cement Type Factor correcting Vm Factor correcting n

I 0.9 1.1

II 1 1

III 0.85 1.15

IV 0.6 1.5



Where is the conductivity of concrete, - , is the temperature [°C], is the

density of concrete, -, is the specific heat capacity of concrete , - , is the

time.

The conductivity of concrete depends on several parameters 1) density, 2) moisture content

and 3) the type of aggregate used. In a normal weight concrete, density is not be a significant

factor, but because air has a low thermal conductivity, in lightweight concrete, we have a

slight change in density affecting the conductivity considerably. Since the conductivity of the

water is higher than the conductivity of air, the moisture content affects the overall

conductivity of the concrete. The mineralogical characteristics of the aggregates used in the

concrete also have an important effect on the concrete conductivity. The thermal

conductivity usually ranges between , - in ordinary Portland cement

concrete and aggregate represents the heat capacity of the material.

3.3 INITIAL AND BOUNDARY CONDITIONS

To obtain a solution for the system of partial differential equations governing the different

mechanisms contributing to the problem of carbon dioxide penetration, the initial conditions

of the concrete member as well as the conditions at its boundaries need to be defined. Here

after, the initial and boundary conditions for the heat, moisture and carbon transport will be

specified.

3.3.1 CARBON PENETRATION

Flux may be defined as : ( ) (3.60)

: Surface carbon dioxide transfer coefficient , -

: Carbon dioxide concentration at the concrete surface.

These parameters are unknown because the boundary conditions are not defined on the

concrete surface.3.3.2 MOISTURE TRANSFER

Moisture transfer from the external environment to the concrete surface (wetting) or vice

versa (drying) is evaluated from (Saetta et al., 1993):

( ) (3.61)

Where

Moisture flux normal to the concrete surface , -

pore relative humidity at the concrete surface.



Relative humidity in the surrounding environment.

Surface moisture Transfer coefficient , -

Akita et al. (in Martin-Pérez, 1999) have reported values of to range between

, - When the value of is negative, moisture is being transferred from

the environment to the concrete surface ( ) which simulates wetting conditions

of concrete. When the value of is positive, moisture is being transferred from the

concrete surface to the environment ( ) which simulates drying conditions

of concrete. A perfectly sealed concrete surface can be simulated by assuming, .

To simulate the seasonal variation of the daily average atmospheric relative humidity

a sinusoidal variability in time of this parameter has been considered (Saetta et al.,

1993), with the periodicity established over a period of one year as illustrated in figure

(3.3). The variable is thus evaluated from:

(3.62)

Where

minimum daily average atmospheric relative humidity in a year.

maximum daily average atmospheric relative humidity in a year

time in years.

Figure 3.3 The yearly humidity periodicity

3.3.3 HEAT TRANSFER

Heat transfer across the concrete surface is calculated from (Saetta et al., 1993):

( ) (3.63)

atmospheric relative humidity and (c) the daily average temperature.

Where

Convective heat flux across the concrete surface , -

Temperature at the concrete surface , -



Temperature in the surrounding environment , -

Convection heat transfer coefficient , -

To range between , -

When the value of is negative, heat is being transferred from the environment to the

concrete surface( ), which simulates heating conditions of concrete. When the

value of is positive, heat is being transferred from the concrete surface to the

environment ( ), which simulates cooling conditions of concrete. To simulate the

seasonal variation of the daily average environmental temperature, a sinusoidal variability in

time of this parameter has been considered (Saetta et al., 1993), with the periodicity

established over a period of one year as illustrated in figure (3.4). The variable is

thus evaluated from:

( ) (3.64)

Where , minimum daily average atmospheric relative humidity in a year.

Maximum daily average atmospheric relative humidity in a year. Time in years.

Figure 3.4 The yearly Heat periodicity

CHAPTER FOUR

MODEL FORMULATION

ABU ALI FAKHER CHAPTER FOUR

MODEL FORMULATION Page 33

CHAPTER FOUR: MODEL FORMULATION

4.1 GALERKIN METHOD

It is not possible to get a closed form solution of the system of equations governing the

phenomena contributing to the carbonation ingress into concrete because the various material

properties and boundary conditions depends on the physical parameters of concrete and the

time of exposure. Nevertheless, this set of equations can be solved numerically in space as a

boundary value problem and in time as an initial value problem by means of a two-

dimensional finite element formulation, in which appropriate boundary conditions are enforced

to simulate the seasonal variations of exposure conditions. A time-step integration scheme is

applied to determine the variations in time of the different variables in concrete.

4.1.1 CARBON PENETRATION

Applying Galerkin weighted residuals method to the Eq. (3.1)

* ( ), - +

.

, -

/ (4.1)

∫

* ( ), - +

4

, -

5

(4.2)

(4.3)

, - (4.3a)

, - (4.3b)

, - (4.3c)

, - (4.3d)

∫

* ( ), - +

.

, -

/ (4.4)

Where W is a weighting function and ( ) is the domain of the problem.

Using Green’s Theorem:

∫

, - ∮

∫

(4.5)

∫

2 ( )

3 ∮ (

) ∫ ( )

∫ (4.6)



The Eq. (4.2) is solved by dividing the domain into elements, in which the field variable

is expressed in terms of its nodal values according to:

, -* +

(4.7)

Where , - is a row vector containing the element shape functions corresponding to each

node, and * + is a vector containing the unknown nodal values. Superscript (e) refers

to element values. Doing the same representation with the other field variables, i.e., and

,

, -* + (4.8)

, -* + (4.9)

The Galerkin method uses a weighting function of the form

, -* + (4.10)

∫ , -* + ( ), - * +

∮ , -* +

, -* +

∫ , -* + , -* + * + ∫ , -* + , -* +

(4.11)

, -* + * + , - (4.12)

∫ * + , - ( ), -

* +

∮ , -

* +

∫ , - , - * + ∫ , -

(4.13)

While * + is arbitrary, the equation becomes

( ) (4.14)

∫ , -

( ), - 2

3 ∮ , -

(, -* + ) ∫ (, - , - )* +

∫ , -

(4.15)

∫ (, -

( ), - ) 2

3

∫ (, - , - , - , - , -

, - )* +

∮ , -

(4.16)

If we note, - , -, and knowing that the flux is given by

∫ , -

( ), - 2

3 ∮ , -

(, -* + ) ∫ (, - , - )* +

∫ , -

(4.17)

∫ (, -

( ), - ) 2

3 ∫ (, - , - , -

, - , - , - )* +

∮ , -

(4.18)

In matrix notation, the previous equation can be written as



, - 2

3 , -* + * + (4.19)

, - ∫ , - ( ), -

(4.20)

, - ∫ (, - , - , - , - )

∫ , - , -

(4.21)

, - ∫ , - (4.22)

4.1.2 MOISTURE TRANSPORT

Following the same procedure as previously, the Eq. (3.6) can be written as:

, - * +

, -* +

* + (4.23)

, - ∫ , -

, - (4.24)

, - ∫ , -

, - ∮

, - , - (4.25)

, - ∫ , -

, - ∮

, -

(4.26)

4.1.3 HEAT TRANSFER

Following the same procedure as previously, the Eq. (3.7) can be written as:

, - * +

, -* +

* + (4.27)

, - ∫ , - , -

(4.28)

, - ∫ , -

, - ∮

, - , - (4.29)

, - ∮

, -

(4.30)

4.2 TIME INTEGRATION SCHEME

After evaluating all the elemental matrices and vectors that we have previously described,

all the element contributions are combined using a direct stiffness method, and thus, Eq.

(4.19), (4.23) and (4.27) are generalized to govern the whole field. It should be noted that

the integrals along the boundary of elements should be evaluated where the flux is

specified. The resulting system is of the form:

, -{ } , -* + * + (4.31)

In which denotes the first derivative with respect to time. In order to obtain a numerical

solution, Eq. (4.31) is integrated in time by means of a finite difference approximation

(, - , -) * + (, - ( ) , -) * +



( )* + * + ) (4.32)

Where is a parameter ranging from [0 to 1] and ∆t is a time increment. The Eq. (4.32)

evaluates physical quantities at time , * + , as a function of quantities at the

previous time step, i.e., * + . Eq. (4.31) is used to solve Eq. (4.19), (4.23) and (4.27) all of

which describe time dependent problems.

4.3 Elements Description

The formula f o r a n element shape function Derivative has been based on a linear

distribution of the field variables, ( ), ( ), and ( ), with respect to ( ),and ( ), the

two elements used to discretize the region of interest are the linear triangular element and

the bilinear rectangular element, shown in figure (4.1).

To further simplify the formulation and the evaluation of all the element integrals, Eqs.

(4.19), (4.23) and (4.27) will be written in the form of Eq. (4.31) and the element integrals

as:

, - ∫ , - , -

(4.33)

, - ∫ , - , - ∮

, - , - { } ∫ , -, - , -

(4.34)

The table (4.1) shows the correspondence between Eqs. (4.31), (4.34) and (4.35) and the

equations stated before and expressing the element integrals for each process, i.e., carbon,

moisture and heat transport.

Table 4.1: Correspondence between the simplified and explicit expressions giving the

element integrals Physical problem ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

Carbon

penetration

( )

Moisture diffusion

D

0

0

C

Heat transfer

0

0

0

0



Figure 4.1: Linear triangular and bilinear rectangular elements.

4.3.1 Bilinear Rectangular Element

The bilinear rectangular element has four straight sides with a node at each corner. It is

long and wide as shown in figure (4.1). The field variable 𝞍 is approximated over

the rectangular region by:

(4.36)

Where are the shape functions corresponding to nodes i, j, k and l

respectively, and are the corresponding nodal values of ( ) the shape

functions for the rectangular element, in terms of the local coordinate system whose origin is

located at node i, are given by:

.

/ .

/ (4.37)

.

/ (4.38)

(4.39)

.

/ (4.40)



4.3.2 Evaluation of Element integrals

The integrals in Eq. (4.37), (4.38), (4.39) and (4.40) have been evaluated in closed form for

the bilinear triangular elements and are presented as follows:

Capacitance Matrix

, - ∫

(

) (4.41)

Where 2a and 2b are the height and width of the rectangle respectively.

Property Matrix , - , - , - (4.42)

, - ∫ , - , -

(

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) )

(4.43)

, - ∮ , - , -

(

)

(

)

(

)

(

)

(4.44)

Environmental Load Vector

* + * +* + * + (4.45)

* +* + Negligible than * + * +

* + ∮ , - {

}

{

}

{

}

{

}

(4.46)

Eq. (4.44) and (4.46) are evaluated only on those sides of the rectangle where fluxes are



specified.

4.4 SOLUTION PROCEDURE

The implementation of the problems described in Chapter 3 results in a system of nonlinear

equations for the cases of moisture and carbon dioxide transport in concrete. This is because

of the strong dependence of the material parameters on the field variables, i.e., relative

humidity and concentration. Since , and were assumed to be constant, the solution

of the equation governing the heat transfer yields a system of linear equations, which can be

solved using one of the known methods such as Gauss elimination.

4.4.1 Nonlinear Solution Scheme

To solve the system of nonlinear equations resulting from the time discretization of Eq.

(4.19) and (4.23), we used the Modified Newton Raphson method (MNR). For a system of

the form:

, ( )-* + * + (4.47)

In which the property matrix is dependent on the unknown field variable {Φ}, the solution

can be obtained through an iterative procedure (MNR) as follows:

{ } { } { } (4.48)

Where the superscript denotes for the number of iteration, and

{ } , - 2* + [ ( )]{ }3 (4.49)

Where , - is the Jacobian matrix of the system and is given as:

, - , ( )- 0 , ( )-

* +1 (4.50)

The matrix in the second term of the right hand side means that each coefficient ( )

is evaluated by the product of the row of , - differentiated with respect to and the

Vector * + The subscript denotes matrix index.

The difference between the regular Newton Raphson method and the modified one is

that, in the modified procedure, the jacobian matrix is evaluated only once at the beginning

of iterations, unlike the regular one, where the jacobian matrix is evaluated for each iteration.

4.4.2 Solution Algorithm

For the numerical solution for the equations system we used an iterative procedure. If, at

time step( ), the solution vectors * + * + and * + are known and we wanting to



estimate the same vectors at time step n+1, we proceed as follows (figure (4.1)):

Solve the heat transfer equation, which is in our formulation, a linear problem

independent of humidity and carbon transport.

Figure 4.2: Solution scheme for the proposed model.

Solve the nonlinear equation describing the moisture transport, which is dependent equation

is carried out. The solution of the moisture diffusion, without including the effect of

chlorides on the process, is performed using MNR method in which * + values are

recalculated until reaching the required accuracy, defined using aconvergence norm, or a ma



Solve the equation governing the carbon penetration problem. Since * + and

* + Are known now, using the same procedure as step (2), the profile of free carbon,

{ } , is determined.

Resolve the moisture diffusion problem including the effect of carbon presence, using

{ } determined in the previous step.

Resolve the penetration problem using the adjusted * + and compare the newly

determined { } to the previous one.

Repeat steps (4) and (5) until achieving a minimal change in the carbon profile.

4.5 DEFENDED OF MATLAB

The problems of carbonation, humidity transfer, and heat transfer are solved using a

mathematical equations deployed in MATLAB, we will present here the progress of the

problem solving, the figure.1 below demonstrate the way we have divided our field of

problem (A) into a plan of x (lx: the long of the study field) and y (ly: the depth of the study

field), using the finite element as shown in the figure.2 the field of the study domain is

divided into smaller rectangles (element), where each one is having the same property and

can’t be having two different values for any property in any given time, the diminutions for

each element can be given as lx/dx for the length and ly/dy for the depth of the element,

where dx and dy are the number of elements along the length and depth for the whole field

studied, respectively.

Figure 4.3: Reinforcement in concrete.



Figure 4.4 divided by the finite element of concrete surrounding the steel and concentration

carbonation in the node.

The division made by the finite element method give us the ability to study each element

alone for all the problems considered in the study then combine all the results for all problems

in all elements to have an overall look on the studied field.

4.5.1 Elementary matrix calculations

The way to determinate the elements properties (Co2, humidity, and heat concentrations) is to

measure that property in the nodes first then generate the values for the whole element.

The element is defined as a 4*4 matrix according to the nodes on its corners as shown in

figure.2

The sum of the entire elementary matrix is defined in MATLAB as (assembling of elementary

matrix).

The total element number is the number of x elements multiplies with the y elements number,

and the nodes total number is x elements plus one multiplies with the y elements number plus

one.



Heat Transfer

, ( )- (5.1)

: Conductivity of concrete known calculation

Humidity Transfer

, ( )- (5.2)

: Humidity diffusion coefficient of concrete calculation

: Concrete moisture capacity calculation

Carbon Transfer

* ( ), - +

.

, -

/ (5.3)

After 28 day when started penetration we will proceed as follows

1. Concentrations calculation ( ) 2. Porosity calculation ( )

3. Fraction calculation ( ) 4. Coefficient calculation ( ) 5. Diffusion Coefficient calculation ( ) 6. Elementary matrix calculation after assembly ( ) 7. The time domain solution ( ) 8. Gauss solution * + 9. Concentrations difference calculation caused by the carbonation and hydration

10. Actual Concentrations calculation , - , - , - , -

11. Rate of carbonation calculation , -

, -

12. Carbonation depth calculation (, - , - ) The heat transfer and carbon transfer problem is solved in MATLAB by the integration

using the difference finite method which results in a linear solution by theorem of Gauss. This

equation has no complications and can be solved by defining the surface element geometry

with a time loop.

CHAPTER FIVE

NUMERICAL

PERFORMANCE OF THE

ABU ALI FAKHER CHAPTER FIVE

NUMERICAL PERFORMANCE OF THE MODEL Page 44

Chapter 5: NUMERICAL PERFORMANCE OF THE MODEL

5.1 . Introduction

To examine the numerical performance of the developed program according to procedure

described in chapters three and four, we considered a concrete surface slice of 3cm deep and

1cm wide (figure (5.1)). This slice is exposed to environmental conditions on the upper face,

while the other three faces are sealed (no surface flux). The material parameters used in

analysis are given in table (5.1).

Table 5.1. Material parameters.

Specific heat, 1170 [J/kg.˚C]

Thermal conductivity, 3.6 [W/m.˚C]

Convection heat transfer coefficient, 0.07 [W/m2.˚C]

Surface moisture transfer coefficient, 2.43×10-7

[m/s]

Surface carbon transfer coefficient, 1 [m/s]

Cement type I

Water to cement ratio 0.5

Aggregate to cement ratio 3

Curing time 28 days

Figure.5.1 Concrete slice analyzed.



5.2 . Effect of numerical parameters

To investigate the effect of the numerical parameters, i.e., the element size and time step, on

the performance of the finite element model, the concrete slice shown in Figure (5.1) was

exposed to a 0.5% carbon dioxide atmosphere and 85% relative humidity on the upper side.

The temperature in the surrounding environment was 25˚C. The concrete slice was supposed

to be initially free of carbon, at 50% relative humidity and 15˚C temperature. The analysis

was undertaken for a period of one year.

5.2.1 Effect of mesh size

Mesh is the finite element way of dividing the volume under study into much

smaller volumes to ease the analysis. It is commonly known that as the mesh

size decreases the analysis precision increases. In order to optimize our mesh,

the concrete slide was discretized into different mesh sizes. Carbon dioxide

concentration profiles obtained for the different mesh sizes are given in figure

(5.2).

Figure.5.2Carbon concentration for differentmesh sizes.

As it can be seen, carbon concentration is high at the surface and keeps decreasing with depth

until it’s close to zero. As the mesh size decreases, the carbon dioxide concentration in

concrete increases until a value at which it becomes invariable. Mesh sizes of 0.25cm and



0.125 cm give the same results. This means that mesh size of 0.25cm is optimal for estimating

carbonation process.

Figure.5.3. Carbon dioxide profiles in concrete for different mesh sizes.

Figure.5.4. Humidity values for different mesh sizes.

-0,1

0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0 1 2 3 4

Car

bo

nat

ion

rat

e

Depth in cm

Mesh size=1cm

Mesh size=0,5cm

Mesh size=0,25cm

Mesh size=0,125cm



5.2.2 Effect of Time increment

To investigate the effect of time increment, analyses under the same conditions as previously

with different time steps were undertaken for a period of one year. The obtained results are

shown in figures (5.6) and (5.7).

At different time increments there is almost no change in Carbon concentration or

Carbonation rate. This means that the developed program is not sensitive to time step.

Figure.5.6Carbonation depthfor differentTime increments.

Figure.5.7Carbon concentrationfor differentTime increments.

-0,2

0

0,2

0,4

0,6

0,8

1

1,2

1,4

0 1 2 3 4

Car

bo

nat

ion

de

pth

Concret depth in cm

Time increment dt=2days

dt=4days

-0,00002

0

0,00002

0,00004

0,00006

0,00008

0,0001

0,00012

0,00014

0,00016

0,00018

0,0002

0 1 2 3 4

Car

bo

n c

on

cen

trat

ion

Depth in cm

Time increment

dt=2days

dt=4days

CHAPTER SIX

CONCLUSION

ABU ALI FAKHER CHAPTER SIX

CONCLUSION Page 48

CONCLUSION

One of the most important phases in designing structures is to predict the service life, to have

a precise prediction we need to consider all parameters that play role in defining the service

life, this phases importance arise when the building is in need to be reconstructed or repaired,

knowing in details the service life and how the structure is affected and changed in this period

will be very helpful to minimize the costs which is the main concern nowadays all over the

world.

The main important and most dangerous in pathology is steel degradation; steel is the element

that provides stability and flexibility to the structure, the degradation happens after carbon or

chloride penetrating concrete and react with the internal components.

This theses is a detailed study to describe and clarify the way carbon interact and impact

concrete in the simplest mathematical model to help predict service life within the structure

can afford sufficient mechanical resistant.

REFERENCES

1_

Anna V. Saettaa,*, Renato V. Vitalianib (2004), Experimental investigation and

numerical modeling of carbonation process in reinforced concrete structures

Part II. Practical applications.

2_ Anna V. Saetta (2005), Deterioration of Reinforced Concrete Structures due to

Chemical–Physical Phenomena: Model-Based Simulation.

3_

Burkan Isgor a, A. Ghani Razaqpur (2002), Finite element modeling of coupled

heat transfer, moisture transport and carbonation processes in concrete

structures.

4_ DENIS MITCHELL AND GEOFFREY FROHNSDORFF (2004), Service-Life

Modeling and Design of Concrete Structures for Durability.

5_

Jinane Kabbara (2011), Evaluation fiabiliste de l'impact des facteurs

climatiques sur la corrosion des poutres en béton-arme : application au cas

libanais.

6_ Martin-Peres (2001), Numerical solution of mass transport equation in concrete

structures.

7_ OSMAN BURKAN ISGOR, B.Sc., M.Eng (2001), A Durability Model for

Chloride rbonation lnduced Steel Corrosion in Reinforced Concrete Members.

8_ Papadakis et al (1991), Physical and Chemical Characteristics Affecting the

Durability of Concrete.

9_ Yunping Xi et al (1999), MODELING CHLORIDE PENETRATION IN

SATURATED CONCRETE, Page (3) ITZ.

10_ http://www.cement.org/for-concrete-books-learning/concrete

technology/durability/preventing-joint-deterioration.

http://www.cement.org/for-concrete-books-learning/concrete

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