population balance model for c3s...

Nov. 7, 2006

Population Balance Model for C3S Hydration

Joseph J. Biernacki and Tiantian XieDepartment of Chemical EngineeringTennessee Technological University

International Summit on Cement Hydration Kinetics and Modeling

July 27, 2009

Nov. 7, 2006June 30, 2009

Outlines

Introduction – Some Historical Perspectives

Building the Population Balance Formalism

• From Single Particles to Ensembles

• Single Particle Model for C3S Hydration

• Population Balance Modeling (PBM)

Kinetic model for C3S Hydration

Results and Discussions

Inferences and Aspirations

July 27, 2009

Nov. 7, 2006

Modeling - A Historical Perspective

July 27, 2009

Date i.p. o.p.thermochemically

consistentmicrostructural kinetic transport chemistry multi-scale

size

distributionnucleation

Population

Tracking

Jander 1927 X fixed fixed

Avrami-Erofeev 1939 X X

Ginsling-Brounshtein 1950 X fixed fixed fixed

Pommerscheim 1979 X X fixed fixed fixed

Knudsen 1984 fixed

Pommerscheim 1984 X X fixed fixed fixed X

Brown 1989 X fixed fixed

Bentz 1997 X X X psydo psydo psydo X

Livingston 2000 X

Thomas 2007 X fixed fixed X

Bishnoi-Scrivener ? X X X X X X X

Bullard 2008 X X X X X X X X X X

ip inner product oriented or dominated

op outer product oriented or dominated

thermochemically consistent collapses to thermodynamic limits

microstructural predicts structure of product at microscopic length scales

kinetic has explicit and abitrarily adjustable kinetics

transport has explicit transport of at least one species

chemistry explicitly links kinetics and/or transport to the chemistry of the pore solution

multi-scale links more than one length-scale, i.e. micro to meso

size distribution explicitly incorporates initial size distribution

nucleation explicitly incorporates nucleation event

fixed developed for a fixed assumption

psydo not fundamentally based, not intrinsic

Nov. 7, 2006

Who inspired whom and a modeling genealogy…

Biernacki-Xie

Brown

Pommersheim

Ginsling-Brounshtein

Jander

Thomas Livingston

Cahn

Avrami Kolmogorov

Bishnoi-Scrivener

Pignat

Bullard

Bentz

Garboczi

Hansen

Carino

Knudsen

July 27, 2009

Microstructural

Maturity

Nucleation

Combined Mechanisms

Nov. 7, 2006

From Single Particles to Ensembles

July 27, 2009

Nov. 7, 2006

What is a “Population Balance Model?”

July 27, 2009

Nov. 7, 2006

What is a “Population Balance Model?”

Starts with an initial population

Incorporates particle model(s)

Accounts for growth initiation (nucleation), a boundary condition

Permits the particles to interact with their surroundings including other particles

Tracks the evolution of the population.

July 27, 2009

Nov. 7, 2006June 30, 2009

Single Particle Model - Basics

A semi-analytical continuum approach

(Pommersheim-like, but not Pommersheim’s model)

Based on H2SiO42- ion transport and reaction

kinetics that account for and respond to changes in

continuous phase composition (solution chemistry)

with approximate thermochemical consistency

July 27, 2009

Nov. 7, 2006June 30, 2009

Single Particle Model - Stoichiometry

The chemical reactions include dissolution of C3S and

formation of (CaO)a·SiO2·(H2O)b with fixed, but selectable,

stoichiometry and physical properties:

OHSiOHCaOHSiOCa 4332

42

2

253

• Dissolution of C3S:

• Formation of inner product:

ii baiiii OHSiOCaOOHabOHaSiOHCaa )()()()1(2 222

2

42

2

• Formation of outer product:

oo baoooo OHSiOCaOOHabOHaSiOHCaa )()()()1(2 222

2

42

2

July 27, 2009

Nov. 7, 2006

The process of C3S hydration was modeled as four parts:

• Dissolution from C3S at the boundary of the dissolving core

• Transport of water and ions (H2SiO4-2) through both inner and outer

product layers

• Precipitation of C-S-H at the boundary of the inner product unreacted

core interface

• Precipitation of C-S-H on the boundary of the outer product pore

solution interface

Single Particle Model –Reaction-Diffusion

July 27, 2009

Nov. 7, 2006

C3S

Inner product

CSH4

core

Dissolution of C3S

Precipitation of inner product

Outer layer

Outer product

C2SH5

Contin

uous phase

ro R riInner layer

Precipitation of outer product

Continuity

Water + ions transportationWater + ions transportation

Water + ions transportation

C3S

Inner product

CSH4

core

Dissolution of C3S

Precipitation of inner product

Outer layer

Outer product

C2SH5

Contin

uous phase

ro R riInner layer

Precipitation of outer product

Continuity

Water + ions transportationWater + ions transportation

Water + ions transportation

• outer product –formed outside the radius of the original C3S particle

• inner product – formed within the boundaries of the original C3S particle

• R – Initial radius of the particle

• ri – radius of the unreacted material (core)

• ro – radius of the outer product

Single Particle Model

July 27, 2009

Nov. 7, 2006June 30, 2009

Single Particle Modelreaction-transport mechanism

outer product

inner product

unreacted core

3Ca+2 + 4OH- + H2SiO4-2

(CaO)3SiO2

3H

2O

H2O

(CaO)aoSiO2(H2O)bo

p(CaO)aiSiO2(H2O)bi

H2O

(3C

a+2

+ 4

OH

-+

H2S

iO4

-2)

Ca+2 + OH-

Ca(OH)2

July 27, 2009

Nov. 7, 2006

• Inner product rate of growth was derived by applying a pseudo-

steady-state assumption and the equality of ion flows, leading to an

analytical rate expression for inner product growth (hence semi-

analytical method since integration was numerical) that is a function

reaction-dissolution rate, pore solution chemistry and transport

properties.

• Outer product rate of growth is a function of pore solution chemistry

and rate of reaction.

• The concentration in the continuous phase (the pore solution

chemistry) is controlled by dissolution, transport from the particles and

the precipitation of outer product.

• All the functions are dimensionless.


July 27, 2009

Nov. 7, 2006

Single Particle Modelreaction-transport mechanism

outer product

inner product

unreacted core

3Ca+2 + 4OH- + H2SiO4-2

(CaO)3SiO2

3H

2O

H2O

(CaO)aoSiO2(H2O)bo

p(CaO)aiSiO2(H2O)bi

H2O

(3C

a+2

+ 4

OH

-+

H2S

iO4

-2)

Ca+2 + OH-

Ca(OH)2

July 27, 2009

Nov. 7, 2006

))ˆ1(1()ˆ1()ˆ1()ˆ1(1

)ˆ1)(ˆ1(ˆ

3

1

3

1

2,3

2

1,3

1

1,

3

2

,

ooiitiitiit

iiiri

xNxNNxNNxNN

xNYN

dt

xd

YxNNdt

xdoor

o ˆ)ˆ1(ˆ

3

2

2,

YxNNN

xNxNNxNNxNN

xNYNN

dt

Ydoorcp

ooiitiitiit

iiir

cpˆ)ˆ1(

))ˆ1(1()ˆ1()ˆ1()ˆ1(1

)ˆ1)(ˆ1(ˆ3

2

1,

'

2,

3

1

3

1

2,3

2

1,3

1

1,

3

2

,'

2,

Dimensionless Parameters

Kd, KDu – dissolution rates of C3S Nr,1= 3(1-p) i iYeqKD/( cR)

Ko – reaction rate of outer product Nr,2= 3KoYeq o o/( cR)

Dip, Dop – diffusivity coefficient for inner and outer product Ni= c/ i

Yeq – equilibrium concentration of H2SiO42- in solution relative to C3S No= c/ o

c, i, o– molar density of C3S, inner and outer product Nt,1= pKDR/Dip

ρc, ρw, ρi, ρo,– weight density of C3S, water, inner and outer product Nt,2= pKDR/Dop

xi, xo – mass of inner and outer product N’cp,1= (p/(1-p))m/(VYeq i i)

wtoc– water to cement mass ratio N’cp,2= m/(VYeq o o)

m – mass of the initial unreacted particle

^ – indicates dimensionless

V – volume of pore solution

outer product

inner product

continuous phase


July 27, 2009

Nov. 7, 2006

Single Particle Performance

0 5 100

0.2

0.4

0.6

0.8

Time (hrs)

Dim

ensi

on

less

Mas

s

0.681

0

S2

S1

tf0 S0


Nr,1= 3(1-p) i iYeqKD/( cR)

Nr,2= 3KoYeq o o/( cR)

Ni= c/ i

No= c/ o

Nt,1= pKDR/Dip

Nt,2= pKDR/Dop

N’cp,1= (p/(1-p))m/(VYeq i i)

N’cp,2= m/(VYeq o o)

Reaction rates

Stoichiometry, mass and volume

Diffusion-Reaction

Pore solution

inner product

outer product

0 5 100

0.1

0.2

0.3

Time (hrs)

Dim

ensi

on

less

[H

2S

iO4

--] 0.258

0

S3

tf0 S0

Nov. 7, 2006

Population Balance Model

Describes the particle distribution using the number density and

expresses the number balance for particles of a particular state.

• Internal coordinates (particle state vector) such as particle size, mass, chemistry, etc.:

• External coordinates: position of the particle:

• Distribution function: the number of particles with special particle state space:

Domains of internal and external coordinates:

),...,,( 21 dxxxx

),...,,( 21 drrrr

i r

),,( trxf

July 27, 2009

Nov. 7, 2006

• Continuous phase (pore solution chemistry) function:

• Birth and death function: the breakage and aggregation of the particle:

• Internal and external rate of change of the particle state vector:

• Particle flux through internal space:

• Particle flux through physical space:

),,(),...,,(),,(),,(),( 321 trYtrYtrYtrYtrY c

dt

dxtYrxX ),,,(

dt

drtYrxR ),,,(

),,,(),,( tYrxXtrxf

),,,(),,( tYrxRtrxf

),,,( tYrxh


July 27, 2009

Nov. 7, 2006

The population balance is expressed as:

June 30, 2009July 27, 2009

= +

the rate of change in

number of particles

in a size range

the rate of change

of particles grow

into a size range

net birth rate

including breakage

and aggregation

= +


number of particles

in a size range


number of particles

in a size range

the rate of change

of particles grow

into a size range

the rate of change

of particles grow

into a size range

net birth rate

including breakage

and aggregation

net birth rate

including breakage

and aggregation


July 27, 2009

Nov. 7, 2006June 30, 2009

The Reynolds transport theorem is used to express the number conservation:

0)()( t

rxr

t

x

rx

hfRfXft

dVdV

The general population balance equation is obtained:

hfRfXft

rx


July 27, 2009

Nov. 7, 2006June 30, 2009

The System of Balance Equations

0),(),(),( txftxXtxft

x

0),(),(),(

0),(),(),(

txftxXtxft

txftxXtxft

ooooxoo

iiiixi i

...

0),(),(),(

0),(),(),(

...

0),(),(),(

0),(),(),(

2,2,2,2,

1,

2,2,2,2,2,

1,

2,

1,1,,1,1,

2,

1,1,,1,1,

txftxXtxft

txftxXtxft

txftxXtxft

txftxXtxft

ooooo

o

iiiii

i

xo

oobooxo

xi

iibiixi

July 27, 2009

Nov. 7, 2006

Several assumptions for the C3S model

• The particle distribution is uniform in space so f is not a function of

the particle position

• There is no fragmentation or aggregation of particles

• The particles have different initial radius with same density (any

distribution function can be used)

• Uses semi-analytical combined reaction-diffusion model for single

particles

• Includes thermodynamically driven nucleation event

• Includes continuous phase mass balance

• All the equations are dimensionless

Kinetic Model

July 27, 2009

Nov. 7, 2006

The particles that are not yet nucleated dissolve during the process.

3

2

2, )ˆ1(ˆˆ

ooroo xNNX

dt

xd

3

2

3,ˆ

ˆuru

u xNXdt

xd

Kinetic Model

))ˆ1(1()ˆ1()ˆ1()ˆ1(1

)ˆ1)(ˆ1(ˆˆ

3

1

3

1

2,3

2

1,3

1

1,

3

2

,

ooiitiitiit

iiir

ii

xNxNNxNNxNN

xNYNX

dt

xd

July 27, 2009

Inner and outer product mass is the same as shown for the

single particle only the initial particle size will depend upon the

nucleation time:

Nov. 7, 2006June 30, 2009

Concentration of silicate ions in continuous phase

The concentration is controlled by the diffusion of not yet nucleated

particles, the transportation from nucleated particles and the precipitation

of outer product. V is the volume of the fluid phase in the system

dbdxfXNdbdxfXNdbdxXfNdt

Ydooocpiiicpuuucp

ˆˆˆˆˆˆˆ

3,2,1,

July 27, 2009

Kinetic Model

Reaction rates

Stoichiometry, mass and volume

Diffusion-Reaction

Pore solution




Nr,3= 3KDuYeq c/R

Ni= c/ i

No= c/ o

Nt,1= pKDR/Dip

Nt,2= pKDR/Dop

Ncp,2= (p/(1-p))m/(VYeq i i)

Ncp,3= m/(VYeq o o)

Ncp,1= m/(VYeq c c)

Nov. 7, 2006June 30, 2009

The System of Equations

dbdxfXNdbdxfXNdbdxXfNdt

Ydooocpiiicpuuucp

ˆˆˆˆˆˆˆ

3,2,1,

...

0),(),(),(

0),(),(),(

...

0),(),(),(

0),(),(),(

2,2,2,2,

1,

2,2,2,2,2,

1,

2,

1,1,,1,1,

2,

1,1,,1,1,

txftxXtxft

txftxXtxft

txftxXtxft

txftxXtxft

ooooo

o

iiiii

i

xo

oobooxo

xi

iibiixi

3

2

2, )ˆ1(ˆˆ

ooroo xNNX

dt

xd

3

2

3,ˆ

ˆuru

u xNXdt

xd

))ˆ1(1()ˆ1()ˆ1()ˆ1(1

)ˆ1)(ˆ1(ˆˆ

3

1

3

1

2,3

2

1,3

1

1,

3

2

,

ooiitiitiit

iiir

ii

xNxNNxNNxNN

xNYNX

dt

xd

Population Balance Equations Rate Equations

Continuous Phase Balance




Nr,3= 3KDuYeq c/R

Ni= c/ i

No= c/ o

Nt,1= pKDR/Dip

Nt,2= pKDR/Dop

Ncp,2= (p/(1-p))m/(VYeq i i)

Ncp,3= m/(VYeq o o)

Ncp,1= m/(VYeq c c)

July 27, 2009

Nov. 7, 2006

Solution Methodology – Method of characteristics (MOC)

The MOC is used to solve PBM equations when the RHS is zero (no birth or

death function). Such equations are characterized as “convection equations”

and are not easily solved using ordinary finite difference or finite element

techniques.

0),(),(),( txftxXtxft

x

For the general population balance equation:

The MOC solutions is given by mapping f(x,t) into f(x(s),t(s))…

Kinetic Model

July 27, 2009

Nov. 7, 2006

Solving distribution functions fi,b(x,t), fo,b(x,t) with MOC

The population balance equations generalized for an arbitrary initial

particle distribution become, where b is the initial particle size, “i” is

for inner product and “o” is for outer product:

Kinetic Model

July 27, 2009

Nov. 7, 2006

Model Outputs

Tracking of nucleation.

0

20

40

60

80

100

120

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Nu

mb

er

of

Par

ticl

es

Time (hrs)

Active Inner Cores Total Nucleated in Size b

0

500

1000

1500

2000

2500

3000

3500

4000

0 0.2 0.4 0.6 0.8

Dis

trib

uti

on

Fu

nct

ion

(fi

)

Dimensionless Mass of inner product

Tracking the distribution of inner

product formation with time.

July 27, 2009

Nov. 7, 2006

The concentration of H2SiO4-2 in

solution - increases initially due to

dissolution and transport but

decreases when more particles

become active (are nucleated).

Model Outputs

0.965

0.97

0.975

0.98

0.985

0.99

0.995

1

1.005

0 5 10 15 20

Dim

en

sio

nle

ss R

adiu

s

Time (hrs)

Radius of Non-Nucleated Dissolving Particles

0

0.05

0.1

0.15

0.2

0.25

0 5 10 15 20

Dim

en

sio

nle

ss C

on

cen

trat

ion

Time (hrs)

H2SiO4 Ion Concentration in Pore Solution

Radius of not yet nucleated

particles that are dissolving into

pore solution.

July 27, 2009

Nov. 7, 2006

Extent of reaction and time derivative of extent.

Results and Discussions

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0

0.2

0.4

0.6

0.8

1

1.2

0 5 10 15 20

Deri

vati

ve E

xte

nt

of

Reacti

on

Exte

nt

of

Reacti

on

Time (hrs)

Extent

dExtent/dt

July 27, 2009

Nov. 7, 2006June 30, 2009

Inferences and Aspirations

•The PB formalism for C3S hydration, using semi-analytical rate

expressions that seamlessly incorporate reaction and diffusion along

with continuous phase chemistry and limited thermodynamics,

produces relevant major features of the hydration curve including

induction (due to nucleation and dissolution) when using generally

published property data, i.e. diffusion coefficients.

•One might conclude, then, that the PB formalism offers a continuum-

based mathematical framework for modeling hydration, as an

alternative or connective opportunity for automaton and other

stochastic- based approaches.

July 27, 2009

Nov. 7, 2006June 30, 2009

Acknowledgements

National Scientific Foundation (NSF)

Grant Award No. CMS-0510854

Center of Manufactory Research

Tennessee Tech University

July 27, 2009

Nov. 7, 2006

References

• Jeffery W. Bullard, 2008, “A Determination of Hydration

Mechanisms for Tricalcium Silicate Using a Kinetic Cellular

Automation Model”, J. Am. Ceram. Soc., 91[7], pp. 2088-2097.

• Ramkrishna. Doraiswami. “Population balances: theory and

applications to particulate systems in engineering.” Academic Press

2000

• Pommersheim, “Effect of Particle Size Distribution on Hydration

Kinetics,” Mat. Res. Soc. Symp. Proc., (1987) 35, 301-307.

July 27, 2009

Nov. 7, 2006June 30, 2009

Thank you

July 27, 2009

population balance model for c3s...

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