chapter 2 literature review 2.1 introduction to porous media 2.1.1

Chapter 2: LITERATURE REVIEW 4

CHAPTER 2

LITERATURE REVIEW

2.1 Introduction to Porous Media

2.1.1 Description of Porous Media

A porous medium is a solid which contains an interconnected void space which is filled with one

or more fluids (Bear, 1990; Nield and Bejan, 1999). Typically, naturally occurring porous media

exhibits the irregular geometry, such as soil, depicted in Figure 2.1.

Figure 2.1: A naturally occurring porous medium.

Single phase transport phenomena in porous medium found in engineering applications

typically consist of particles such as spheres (Ergun, 1952), parallelepipedal particles (Comiti et

al., 1989), ceramic foams (Richardson et al., 2000), circular cylinders (Lee and Yang, 1997), and

ring and saddle packing (Perry et al., 1997). A porous medium composed of spheres is illustrated

in Figure 2.2. Porous medium composed of large solid particles find numerous applications in

Solid PhaseVoid Space

Fluid Phase


cannot contribute to transport of matter across the porous medium, only the interconnected or

effective pore space can (Dullien, 1992).

chemical engineering, the most important of which are packed bed reactors. In addition, porous

media composed of gravel has energy storage applications. More detail on porous media

applications can be found in Section 2.2. Alternatively, porous media can also consist of fibrous

materials. Many examples of these types of media have both natural and man-made applications.

Fluid Phase

Spherical Particles

Void Space

Figure 2.2: A porous medium encountered in engineering applications.

2.1.2 Porosity or Voidage

Porosity (also called voidage or void fraction) is the fraction of the bulk volume of the porous

sample that is occupied by pore or void space.

There are two types of void space. One which forms a continuous phase within the

porous medium, called “interconnected” or “effective” pore space, and the other which consists

of “isolated” or “non-interconnected” pores or voids. Non-interconnected void or pore space


sed to determine the porosity or voidage between the

1992). The most common methods

e the bulk volume of a piece of porous material and then to compact the body so

(b)

icroscope. The disadvantage is that it is not

(c)

2.1.3 Measurement of Porosity (Void Space)

They are various experimental methods u

particles (Collins, 1961; Scheidegger, 1974; Dullien,

available are:

(a) Direct method: The most direct method of determining the porosity or voidage is to

measur

as to destroy all its voids, and to measure the difference in volumes. Unfortunately, this

can be done only if the body is very soft.

Optical method: Another direct way to determine the porosity is simply to look at a

section of the porous medium under a m

always possible to make sections of a porous medium conveniently. Difficulties will be

encountered especially if the porous medium is dispersed.

Density method: If the density of the solid material, sρ , making up the porous medium

is known, then the bulk density ρ of the medium is relateb lε

by

d to the fractional porosity

s

bl ρ

ρε −=1 (2.1)

The bulk density can be obtained in several ways. If measuring the outside dimensions

and weighing the piece of material do not give accurate results, a volumetric

(d)

Then, solid particles are added to create

displacement method can be applied. Determining their volume by a displacement

method thereupon yields the effective porosity.

Water displacement methods: In the first method, porosity is measured by first filling

the test chamber with a known volume of water.


the porous medium. When the solid particles are added, the water is displaced. Hence the

porosity can be computed from

V i

dl V

−=1ε (2.2)

where Vi and Vd are the initial and displaced volumes, respectively.

In the second method, one can fill the test chamber with the solid particles and then add

ained and the drained

the water to make up the total volume, Vt. Afterward, the water is dr

water volume, Vf, is measured. Then the porosity is computed from

tl V

V f=ε (2.3)

The limitation of the second method is that some water may be trapped between the solid

particles after the water is drained. In this case the computed porosity will be less than

2.2

Phenom ort in porous media are encountered in many engineering

pli Dullien, 1992). Understanding the nature of flow in

the actual porosity. Another concern is the formation of gas bubbles in the measurement

system because bubbles will occupy the void space. Therefore one should be very careful

to make sure that no bubbles form during the measurement.

Application of Porous Media

ena of single phase transp

disci nes (Scheidegger, 1979; Bear, 1990;

porous media is essential for many science and engineering applications including chemical

engineering, petroleum engineering (e.g. oil and gas flow), biotechnology and biomechanics,

material science, soil science, mechanical engineering, groundwater hydrology, powder

metallurgy, and environmental engineering.


s solids or bed of particles. Some typical examples

include

ples are the flow of gas/liquid through a tubular reactor containing catalyst particles,

•

d the products flow. In some cases the bed of

•

ydrosulfide

•

ctrolytic cells play important technological roles in improving current

•

s, and dolomites. To be recovered, the oil or gas must flow

There are numerous physical processes of interest to chemical and biological engineers

that involves the flow of fluids through porou

:

• Packed columns – flow through packed beds occurs in several areas of chemical engineering.

Exam

and the flow of water through cylinders packed with ion-exchange resin in order to produce

deionized water. In all cases, it is necessary to be able to predict the corresponding pressure

drop as a function of flow rate (Wikles, 1999).

Reactor engineering – a common type of chemical reactor consists of a fixed bed of solid

catalyst particles through which the reactants an

particles moves slowly countercurrent to the reacting stream (Levenspiel, 1999).

Pulp digester – a complex vertical cylindrical reactor, used in the pulp and paper industry to

remove lignin from wood chips. Aqueous solutions of sodium hydroxide and h

ion are used to react with porous and wet wood chips. (Walkush and Gustafson, 2002;

Harkonen, 1987).

Electrochemical engineering – porous electrodes and permeable and semipermeable

diaphragms for ele

efficiencies (Mantell, 1960).

Oil and gas production – oil and gas deposits usually occur in porous structures such as

sands, sandstones, limestone

through these porous structures to a well hole. In the later stages of reservoir production, the

oil or gas must often be displaced from the porous solid by water or miscible solvents.

Chapter 2: LITERATURE REVIEW 9 • Filtration – solids can be separated from a suspension by a porous medium that retains the

solids but allows the liquid to pass. Examples include water treatment by sand beds and the

recovery of solids products by various types of filters (Han and Ingmanson, 1967).

• Biophysics and biomechanics – the flow of blood and gases in the lung, and the flow of

blood in the kidney are examples of life processes involving flow in porous media (Glaser,

2001). In biomechanics, the bone can be represented as a porous medium infiltrated by blood

(Ambrosi and Preziosi, 2000).

• Hydrology – underground water resources usually occur in stratified layers of sand called

acquifers. The recovery of water for drinking or irrigation, the movement of trace pollutants

into acquifers, salt water encroachment into the fresh water acquifers, and the dissolution of

underground structures such as salt domes are some of the problems in hydrology dealing

with flow through porous materials (Adler, 1992).

2.3 Flow Resistance in Porous Media

Blake (1922) was the first researcher who successfully treated the flow in porous media problem

by an approach analogous to pressure drop in cylindrical pipes. Blake employed dimensional

analysis to correlate pressure drop data through packed beds of crushed particles and obtained

the following equations,

2

3

1 U

dLPf p

l

l

ρεε

∆∆

−= (2.4)

and

)(Ud

Rel

pPM εµ

ρ−

=1

(2.5)

Chapter 2: LITERATURE REVIEW 10 where U is the superficial velocity (m/s), lε is the liquid void fraction, dp is equivalent diameter

of particle (m), ρ is fluid density (kg/m3), and µ is fluid viscosity (Pa·s).

Equation (2.4) is recognized as the modified friction factor and Equation (2.5) is the

modified Reynolds number. Blake (1922) suggested that modified friction factor should be

plotted against modified Reynolds number.

It is known that pressure drop is caused by simultaneous viscous and inertia (kinetic

energy) losses. Theoretical considerations (Burke and Plummer, 1928; Ergun and Orning, 1949;

Ergun, 1952) indicate that dependency of each energy loss upon fractional void volume is

different. Burke and Plummer (1928) proposed that the total resistance of the packed bed can be

treated as the sum of the separate resistance of the individual particles in it. They studied gas

flow through packed beds of spheres and attempted to give a theoretical basis to the equation of

Blake (1922) with essentially a “drag model”. Burke and Plummer found that viscous energy

loss was to be proportional to( and kinetic loss to( . For viscous flow,

Kozeny (1927) derived an equation that depended on the void fraction into( . This

factor is different by a fraction of

21 ll /) εε−

l )(

31 ll /) εε−

321 ll /) εε−

l/ εε−1

21 l /)ε−

from the one derived by Burke and Plummer

(1928) for viscous flow. Carman (1937) recommended the plot of Blake’s dimensional groups as

a general correlation for all flow rates. However, Leva and Grummer (1947) found that the

pressure drop was proportional to( at lower flow rates and to ( at higher

flow rates. In addition, Leva (1951) treated the effect of roughness of packing as of secondary

importance.

3lε

3ll /) εε−1

At low fluid flow rates, the method of Blake leads to the Kozeny equation (Carman,

1937), hence to the factor of ( and the pressure drop equation becomes 321 ll /) εε−

( )23

21

pl

l

dUA

LP µ

εε

∆∆ −

= (2.6)


On the other hand, at high flow rates Blake’s method (1922) gives rise to the equation of

Burke and Plummer (1928) for turbulent flow:

pl

l

dUB

LP 2

3

1 ρεε

∆∆ −

= (2.7)

In contrast to the drag model (Burke and Plummer, 1928), Ergun and Orning (1949) used

a model of equal size and parallel channels and found that the pressure drop was a sum of

viscous and kinetic losses, given by

( ) ( ) 23

23

2 18

12 USUS

LP

vl

lv

l

l ρεεβµ

εε

α∆∆ −

+−

= (2.8)

where α and β are constants and Sv is specific surface (surface of the solids per unit volume of

the solids).

In Equation (2.8), Sv is a characteristic dimension, the diameter of a sphere having the

same specific surface area per unit volume of the particle (Ergun, 1952), expressed as

vp S

d 6= (2.9)

Substituting Equation (2.9) into Equation (2.8) gives

( ) ( ) 2323

2 11U

dBU

dA

LP

pl

l

pl

l ρεεµ

εε

∆∆ −

+−

= (2.10)

which is the standard form of the Ergun equation, with A and B constants.

A linear form of Equation (2.10) is:

B)(Ud

AU

d

)(LP

lp

p

l

l +−=−

ερ

µρε

ε∆∆ 1

1 2

3

(2.11)

Equation (2.11) can be written as


BRe

Af l +−

=ε1 (2.12)

where

2

3

1 U

dLPf p

l

l

ρεε

∆∆

−= (2.13)

and

µρUd

Re p= (2.14)

Equation (2.12) gives a good correlation for numerous experimental data from the

literature. The value of the constants A and B have been evaluated for spheres by curve-fitting of

Equation (2.12) and found to be 150 and 1.75, respectively (Ergun, 1952). However,

MacDonald et al. (1979) stated that the constants A and B from the Ergun equation must be a

function of the particle geometries rather than universal constants. Their results showed that for

the viscous flow region, a single-valued constant with a value of 180 was obtained for A. For the

inertia flow region, the parameter B is not a single-value; it depends on particle internal surface

roughness and for the media investigated (MacDonald et al. 1979) appears to lie in the range of

1.8-4.0.

Ahmed and Sunada (1969) considered the pressure drop equation as both a

phenomenological and a Navier-Stokes type model. They proposed an equation as

"ReH 11+= (2.15)

where

2ULPH

βρ∆∆

= (2.16)

Chapter 2: LITERATURE REVIEW 13 and

αµβρURe" = (2.17)

where α and β are model parameters to be established empirically. In practice, the values of α

and β are determined by rearranging Equation (2.15) with (2.16) and (2.17) to the form

µρβα

µ∆∆ U

ULP

+= (2.18)

The equation is simply a rearrangement of the Forchheimer equation (1901) and

equivalent expressions have been studied by Ergun (1952). There are a number of drawbacks to

the Ahmed and Sunada (1969) equation. The most serious shortcoming of the equation is the

lack of parameters that characterize the porous medium. Therefore, the parameters of α and β

must be functions of the particle geometries rather than universal constants, and accordingly,

must be empirically established for each separate medium. A second drawback is that the

dimensionless variables, H and , contain the to-be-determined parameters "Re α and β as well

as experimental measurable quantities (Macdonald et al., 1979).

For packed beds of fibrous material, the friction factor versus Reynolds number

correlation is found to be different from those mentioned above (Kyan et al., 1970). The most

peculiar finding for fluid flow through a fibrous bed is the unexpectedly high pressure drop

measured in spite of the high void fraction of the bed. The causes of this high pressure drop are

postulated as follows: first, only a fraction of the free space as calculated from the bulk density

of the bed is available for fluid flow, the rest being occupied by stagnant fluid. Second, some

energy is absorbed by the deflection of individual fibers, causing an additional pressure drop

other than those of a fluid dynamics nature. In contrast to the Ergun (1952) equation, the

correlated pressure drop for fibrous bed consists of three energy losses: pressure drop due to

Chapter 2: LITERATURE REVIEW 14 viscous flow losses, pressure drop caused by form drag, and pressure drop due to deflection of

fibers. More detailed derivation on this pressure drop can be found in Kyan (1969) and Kyan et

al. (1970).

Experimental studies show that fluid flow through packed beds of parallelepipedal

particles of low thickness-to-length ratio, such as wood chips used in the pulp and paper

industry, is very different from that of the flow through beds of spherical particles (Comiti and

Renaud, 1989). Comiti and Renaud (1989) argued that using the equation that Ergun (1952) and

MacDonald et al. (1979) proposed is not sufficient to characterize the pressure drop on

parallelepipedal particles. Two reasons are given (Comiti and Renaud, 1989) for this:

1. A significant part of the surface area of the particles may not be reached by the flow

because they mutually overlap.

2. For tightly packed beds in a vertical cylindrical column, the main orientation of the plates

is nearly horizontal and a layered structure appears. In this case, the fluid path for a given

thickness is longer through this type of bed than through beds of isotropic particles.

Comiti and Renaud (1989) proposed a model that takes into account wall effects, bed

tortuosity and the overlap between particles:

2UMUNHP ** +=

∆ (2.19)

where

( )( )

3

2222 1

1412

l

l

lvdvd

*

DAAN

εε

εµτ

−

−

+= (2.20)

and


( )3

322

11096801041301

l

lvd

pp* ADd

.Dd

.Mεε

ρτ−

−+

−−= (2.21)

where D is the diameter of column, dp is the diameter of particle, τ is the tortuosity and Avd is

the dynamic surface area of particles and can be different from geometrical surface area if

particles overlap mutually.

N* term takes into account the surface area contributing to viscous friction which is the

bracketed term in Equation 2.20. M* term takes into account the mean friction factor for the

whole bed, weighted by the ratio of the corresponding particle diameter to the total column

cross-section area as shown in the curly bracket in Equation 2.21. More detail on the Equations

2.19, 2.20 and 2.21 can be found in Comiti and Renaud (1989). The tortuosity τ is defined as

Ly

=τ (2.22)

where y is the length of the mean fluid path and L is the bed height.

Experimental studies showed that this model provides a description of pressure drop in

non-consolidated beds of spheres, parallelepipedal particles (Comiti and Renaud, 1989) and

short cylinders (Brunjail and Comiti, 1990). A single equation is used to represent the

dependence of tortuosity on voidage in various types of packed beds:

+=

llnP

ετ 11 (2.23)

This relationship satisfies the requirement that 1=τ for .l 1=ε P is a parameter depending on

the shape of the particles and their mean orientation in the bed. P values for different particle

shapes are shown in Table 2.1.

The values of P obtained for fixed beds of plates, wood chips and cubes are correlated

with the following equation (Comiti and Renaud, 1989),


+−=

ltP 18.055.0ln (2.24)

Where, t is the thickness of the plate and l is the length of the plates.

Table 2.1: P value for different particle shapes.

Author(s) Particles P value Wyllie and Gregory (1955) Spheres and a mixture of spheres 0.41 Wyllie and Gregory (1955) Cubes 0.63 Peck (1985) Wood chips 1.60

Plates: t/l = 0.102 3.20 Plates: t/l = 0.209 1.66

Comiti and Renaud (1989)

Plates: t/l = 0.440 0.86

Equation (2.23) with Equation (2.24) allows an estimation of the tortuosity, particularly

for fixed beds of parallelepipedal particles and the mean velocity, Um, in the pore

l

mUετ

=U (2.25)

where U is the superficial velocity, τ is the tortuosity and lε is the liquid void fraction.

Table 2.2 summarizes published coefficients for the Ergun equation. From Table 2.2, it is

shown that even for rigid particles the resistance coefficients (A and B) are not constant but

depend on the structure of the bed. The values of B for plates are greater than spheres. This may

be explained by the flow pattern through fixed beds of flat plates (Comiti and Renaud, 1989).

Their explanation is as follows: “For tightly packed beds of flat plates in a vertical cylindrical

column, the mean orientation of the parallelepidedal particles is nearly perpendicular to the flow

direction. One can therefore assume that a jet-type flow occurs when the fluid meets the main

face of the plates – the thinner the flat plates, the greater the number of particle layers and,

consequently, the greater the jet frequency – which causes large important kinetic energy

losses.”


Niven (2002) argued that the non-linearity in the Ergun equation arises from “local’

losses (sometimes called “shock” or “minor” losses) produced during laminar flow through the

expansions, contractions and changes in flow direction within the packed beds. Equivalently,

these losses can be attributed to the gradual onset of pressure drag, due to flow separation behind

each solid particle. This non-linearity is referred to as pressure drag or inertial loss rather than

turbulence loss. Niven commented that great caution must be exercised in the interpretation of

dimensional analysis. He stated that one cannot (without direct evidence) attribute a Reynolds

number transition to the onset of turbulence, only to an increase in inertial forces relative to

viscous forces.

Table 2.2: Coefficients of Ergun equation for different types of particles studied.

Authors Type of particles lε A B Ergun (1952) Spheres -- 150 1.75 Leva (1959) --- -- 200 1.75 MacDonald et al. (1979) Various shapes -- 180 1.8 - 4.0

Plates t/l = 0.102 0.46 216 12.2 Plates t/l = 0.209 0.35 161 6.69

Cylinders h/dc = 5.49 0.39 166 3.20 Spheres dp = 1.12 mm 0.36 140 1.68

Comiti and Renaud (1989)

Spheres dp = 4.99 mm 0.36 142 1.59

2.4 Compressibility of Porous Media

The volume reduction of a porous medium subjected to a compacting pressure is due to

deformation of the solid material, bending and slipping of individual particles and/or

disintegration of solid particles (Jonsson and Jonsson, 1992).

Many studies have shown that the bulk density of a porous medium compressed under a

static pressure obeys the compressibility equation (Wilder, 1960; Jones, 1963; Han, 1969; Miles

and May, 1990; Senger, 1998):

(2.26) NMσρ =

Chapter 2: LITERATURE REVIEW 18 where M and N are empirical constants for a given material and ρ is the bulk packing density

(kg/m3) and σ is the stress or compacting pressure (Pa).

2.5 Review of Pulp Digester s

In Canada, there are about 44 kraft mills in operation. The distribution of these 44 Kraft mills is

shown in Table 2.3 (MacLeod, 2001). About one third of the kraft mills are located in British

Columbia. Worldwide, pulp production is around 52 million tons per year with approximately

400 Kamyr-type digesters in operation.

Table 2.3: Locations of 44 kraft mills distribute across Canada (MacLeod, 2001).

Province Number of Mills British Columbia 16 Alberta 4 Saskatchewan 1 Manitoba 1 Ontario 9 Quebec 9 New Brunswick 3 Nova Scotia 1

Kraft pulping, which utilizes a digester, is the most common modern chemical pulping

process. In digesters, wood chips are cooked with white liquor (sodium hydroxide (NaOH) and

sodium sulfide (Na2S)) to remove lignin from the wood and release individual pulp fibers for

further processing. There are two basic digesters: batch digesters and continuous digesters.

Batch digesters range from 70 m3 to 340 m3 capacity, with a standard capacity of 170 m3

to 230 m3 for most modern mills. Batch digesters fall into two categories: directly heated and

indirectly heated. The principal operations in batch digesting include chip packing and steaming,

liquor filling, slow temperature rise to assure complete liquor penetration of the chips, relief of

gases, cooking at maximum temperature, relief of pressure, and blowing the digester. Each of

these operations affects pulp properties and qualities (Smook, 1992).

Chapter 2: LITERATURE REVIEW 19 In direct heating, steam is injected through a valve in the bottom of the digester. The

difference in temperature between the top and bottom makes the liquor circulate by convection,

and hot liquor rises through the middle of the digester, while colder liquor at the top flows down

the walls to the bottom where it meets hot stream and is heated. However, there are some

disadvantages in direct heating. The cooking liquor becomes diluted with steam condensate,

putting an additional load on the evaporators. Also, the heating is non-uniform, resulting in

temperature differences in large digesters. Non-uniform heating results in uneven cooking that

lowers the quality of the pulp.

Indirectly heated digesters require more equipment, including a circulation system with a

pump, an external heat exchanger, and a strainer section in the digester walls, as shown in Figure

2.3. Indirect heating with forced liquor circulation avoids liquor dilution and a more uniform

temperature profile throughout the digester is achieved.

Continuous digesters, as represented by the Kamyr system, separate the principal

operations between different vessels, and between different sections within the digester. A

typical continuous digester has the bottom section wider than its top section (the widest has a

diameter of 9.15 m). A typical height is between 60 and 70 m (Gullichsen, 1999). Chip charging

and liquor circulation patterns are different. In digesters, washing is also standard in these

continuous units. Liquor is removed earlier from the digester, while the chips are removed at the

base along with wash liquor.

Today over 65% of the world’s chemical pulp is produced using continuous systems,

with single digester capacities in excess of 2000 ADT/day (Gullichsen, 1999). The original

digester systems have been continuously modified over the years with major improvements, such

as 1) HI-HEAT™ washing, 2) single vessel and two vessel systems, 3) Modified Continuous

Chapter 2: LITERATURE REVIEW 20 Cooking (MCC®), and Extended Modified Continuous Cooking (EMCC®), 4) Isothermal®

Cooking, 5) atmospheric presteaming and 6) Lo-Solids® cooking.

Most continuous digesters consist of three basic zones: an impregnation zone, one or

more cooking zones, and a wash zone. A typical one vessel Kamyr digester is shown in Figure

2.4. Wood chips and white liquor are fed to the top of the digester. After entering the digester,

the wood chips form a chip column that continuously moves vertically downwards. The driving

force for downward movement of the chip column is the difference in density between the chip

column and the free liquor (external liquor phase).

In the impregnation zone, white liquor penetrates into the wood chips. The white liquor

and wood chips are then heated in the cooking zone to reaction temperatures at about 170 0C.

After the chips reach 170 0C, the temperature is maintained for 1.5 to 2.5 hours depending on

many factors including the wood furnish, chemical charge, cooking temperature and desired

degree of delignification. The free liquor is in either (1) concurrent or (2) countercurrent flow

with respect to the wood chips in the cooking zones. This is the region where the majority of the

delignification reactions occur. The wash zone is located at the bottom of the digester where the

countercurrent flow of free liquor washes the dissolved lignin and other compounds from the

pulp. The rate of reaction depends primarily on the cooking temperature, the alkali

concentration, and the lignin content. Some factors affecting the overall process are the

composition and moisture content of the wood, the sulfidity of the cooking liquor, the liquor-to-

wood ratio, the chip size and size distribution, the dynamic movement of the chip column and

the liquor, and the temperature distribution. This wash flow also cools the pulp to quench the

reaction and reduce damage to the cellulose fibers from continued reaction.

Digesters are very capital intensive. In 2000, the cost of a typical continuous digester is

about Cdn$75-$150 million (Luis et al., 2000). Because of this, much research has been

Chapter 2: LITERATURE REVIEW 21 conducted on digesters, as shown in Table 2.4, including pulping kinetics, digester control and

chip/liquor flow dynamics. It is important to optimize digester performance to maximize the

produced pulp quality and yield, reduce the overall operating costs and minimize the

environmental impact of pulp mills. With more pulp and paper mills having modern fiberlines

using continuous digesters to meet increasing competitiveness in the global market and tighter

environmental regulations, there is an increasing need for improved control of continuous

digesters (Luis et al., 2000). Understanding and predicting the flow distribution, flow

channeling, stagnation regions, temperature profile, and chemical species within the digester is

essential for several reasons. First, fiber properties of the pulp are dependent on the fluid

dynamics occurring inside the digester. In the cooking zone, the flow and temperature

distribution have an effect on the residence time of chips. Secondly, high corrosion rates in

digesters have been reported (He et al., 1999) which may be caused or exacerbated by liquor

flow.


Figure 2.3: Schematic of indirect heating system in a batch digester (Smook, 1992).


Figure 2.4: Schematic of a typical one vessel Kamyr digester. (Luis et al., 2000)


Table 2.4: Literature studies on aspects of kraft digester performance.

Author(s) Description and Comments Vroom (1957) Vroom developed the H-factor concept that describes the

reaction rate of a kraft cook in which the parameters of time and temperature are varied.

Hatton (July 1973; August 1973; 1976)

Equations were developed relating total pulp yield and kappa number with H-factor and applied effective alkali (EA) for three softwood and one hardwood species (July 1973). These equations were applied to kraft cooking control.

Akhtaruzzaman et al. (1979) A series of works on the influence of chip dimensions in kraft pulping:

I. Mechanism of movement of chemicals into chips.

II. Effect on delignification and a mathematical model for predicting the pulping parameters.

III. Effect on screened pulp yield and effective alkali consumption; predictive mathematical models.

Gustafson et al. (1983) Jimenez et al. (1989) Agarwal and Gustafson (1997)

Studies on the effect of liquor penetration, diffusion, and chip size on the reaction kinetics in kraft pulping.

Smith et al. (1974) The digester was approximated by a series of continuous stirred tank reactors (CSTRs) with external flows entering and exiting those CSTRs where the heaters and extraction screens were located. Their digester model is known as the Purdue model.

Christensen et al. (1982) Christensen et al. used the Purdue model to predict the reactant concentrations in the free liquor at various locations along the digester and the blow-line Kappa number for an industrial digester, even during hardwood/softwood swings.

Harkonen (1987) Harkonen studied and developed a flow model (2-D) in the digester. This model calculates the flow as being 2-D and irrotational. Mass, momentum and heat conservation equations are written separately for the solid phase and liquid phase and inertial terms are ignored. Interaction between phases is calculated using Ergun’s equation.

Michelsen (1995) Michelsen combined and extended the work of Harkonen and Christensen to develop a detailed digester model from mass, momentum, and energy balances. His model assumes radial uniformity and rotational symmetry.

Wisnewski and Francis (1997) The digester model presented in their paper is an extension of the well-known Purdue digester model. This extended model calculates chip porosity and liquor densities as a function of the extent of delignification.

He et al. (1999) Three-dimensional (3-D) coupled two-phase computer model of a continuous digester simulating the flow distribution and delignification process was developed.


The Kraft pulping process is modeled by using a wood chip degradation, alkali consumption, and diffusion model through the chip. The kinetics of Kraft pulping is divided into the initial, bulk, and residual stages. The chemical reactions in the digester are mainly controlled by the flow conditions of the cooking circulation and the wash circulation.

Pageau and Marcoccia (2001) Introduced an improved filtrate addition method to address circumferential temperature gradients. The results show a corresponding decrease in reject content and significant improve in cooking uniformity and screened yield.

Walkush and Gustafson (2002) They developed a pulping model by using WinGEMS™

program to look at the operation of commercial continuous digesters operate particularly in EMCC® and LoSolids® cook modes.

2.6 Non-Uniform Flow in Digesters

Current operating strategies rely on contacting chips with liquors of different temperatures and

compositions throughout a cook to optimize pulp quality and strength. This requires creation of

uniform liquor flow throughout a chip column. The typical superficial velocity in a commercial

digester reaches approximately 7.5 mm/s (equivalent to 11 gal/min·ft2) of the digester’s cross-

sectional area (Horng et al., 1987).

In batch digesters, liquor must reach all chips in a timely manner as cooking conditions

are changed. In continuous digesters, creation of distinct reaction zones (liquor flow zones)

through which the descending chip mass column moves permits effective chip-liquor contacting.

However, recent findings have shown that liquor flow is likely non-uniform, particularly in

larger digesters. Indications of this in continuous digesters include corrosion in the lower

digester (Kiessling, 1995; Wensley, 1996 and 2002), variability in the exit kappa number (He et

al., 1999), and circumferential temperature gradients in the digester wash and cook zones

(Pageau and Marcoccia, 2001). A review of Canadian continuous digester installations (Crellin,

1982) shows that larger digesters usually have low recirculation flows in relation to their cross-

Chapter 2: LITERATURE REVIEW 26 sectional area. Thus these digesters may be limited in their ability to produce pulps at full

potential strength (Horng et al., 1987). However, Tikka (1992) found that the cooking

uniformity is related to the way the chemicals and heat are introduced. With a well controlled

system, the distribution of chemicals and heat is very uniform. Therefore, digester circulation is

not a factor that affected the pulp strength.

Limited work in the open literature has been focused on flow in digesters. In early work,

radioactively labeled “chips” were followed as they traveled through an operating digester

(Hamilton, 1961). Later work avoided experimental complexities and focused on the

development of computational models of increasing sophistication (Harkonen, 1987; He et al.,

1999). The most recent of these models includes hydrodynamic interactions between the fluid

and chip phases, as well as chemical reactions. The flow solutions have been obtained for

steady-state operation and show radial uniformity of liquor flow. However, both upset operating

conditions and flow non-uniformity are possible and of concern in industrial digesters. Recently,

it has been shown that the temperature non-uniformity at the periphery (outer wall) of a digester

(indicative of liquor flow near the vessel walls) can be markedly improved by adjusting the way

liquor is added to the digester (Pageau and Marcoccia, 2001).

Horng et al. (1987) studied the performance efficiency of continuous digesters and found

that the rate of liquor circulation is one of the most important factors affecting pulp quality. They

used mill data to support their findings. They used a correlation between digester performance

and the recirculating liqour’s flow density to quantify how liquor flow affects pulp quality.

The uniformity of filtrate distribution influence chip column and liquor flow patterns

(plug vs. channel flow) in the cooking zones above the blow dilution zone (Pageau and

Marcoccia, 2001). Channel flow of chips and/or liquor causes variations in the residence time,

cooking chemical profile, and temperature profile experienced by chips passing through different

Chapter 2: LITERATURE REVIEW 27 zones of the digester. This in turn causes variability in extent of reaction. In the extreme case,

this specific type of non-uniformly cooking results in higher reject contents and lower screened

yield. Gross non-uniformity of filtrate distribution has been found to be directly related to

circumferential temperature gradients in the digester wash and cook zones, and also in the blow

line.

Many continuous digesters operate well above their original design production capacities

without experiencing severe operating problems. However, some mills have reported operating

problems when operating above design capacity (increasing from a designed 1250 ADT/day to

1680 ADT/day) (Lorinez and Marcoccia, 2001). These increased production rates have resulted

in decreased overall brownstock washing performance. In particular, as rates increased, the

upflow in the digester’s counter-current wash zone has decreased. This has resulted in less

digester washing capacity and lower washing efficiency (Lorinez and Marcoccia, 2001).

Moreover, this increased production requires increasing the volumetric output of pulp (which

decreases their retention time in the digester) as well as increasing chemical application. It may

also be necessary to increase the cooking temperature. Liqour flow must also increase, although

several factors can limit the extraction capacity of an overloaded digester (Gullichsen, 1999),

including:

1. Extraction screen fouling by fibrous material and/or scale deposits. Fouling rates

increases with increased liquor flow, which can further limit extraction capacity.

2. Excessive chip column compaction at the extraction screens decreases column

permeability and increases the radial pressure drop for the free liquor. The compaction

forces peak at this point (due to convergence of the downward and upward flowing

liquors from the concurrent and countercurrent liquor zones). This can cause problems

with chip column movement and can cause the column to “hang up”. “Hang up” or


column floating could occur if the drag created by the counter-flowing liquor (upflow)

exceeds the net gravitational driving force of the column. This phenomenon represents a

limitation in upward flow capacity rather than a limitation in extraction capacity.

3. The increased drag caused by the increased superficial velocity of the liquor in the

countercurrent flow zone can also cause the chip column to “float” or “hang”.

2.7 Pressure Drop (Flow Resistance) Studies in Digesters

A number of researchers have measured the compressibility and flow resistance of chip columns.

Harkonen (1987) developed a 2-D flow model of the digester using the Ergun equation (1952) to

account for interactions between the liquor and chip phases and measured pressure drop in

support of his modeling. Other researchers, including Lindqvist (1994), Lammi (1996), and

Wang and Gullichsen (1998), have experimentally measured pressure drop as a function of flow

velocity and void fraction for different chip furnishes and size distributions.

Linqvist (1994) examined the flow conditions (e.g. flow resistance, void fraction, and

flowrate) and chip pressure on delignification and pulp characteristics. Different flow conditions

were created by varying the chip size distribution and the compacting pressure exerted on the

chips with kappa numbers in between 40 to 15.

Lammi (1996) studied the flow resistance of chip column by using hardwoods, namely

Scandinavian Birch and Eucalyptus Camaldulensis, with the Superbatch process.

Wang and Gullichsen (1998) studied the effect of the chipping technique on the

compressibility and flow resistance. They produced chips which had the uniform length of 40

mm and thickness of 4 mm and were called “new” chips. They also made “reference” chips

which were defined as chips passing through a 13 mm round hole and retained on a 7 mm round

Chapter 2: LITERATURE REVIEW 29 hole. Their results showed that “new” chips and “reference” chips have the same pressure drop

under similar conditions.

For wood chips, the particle geometry is non-uniform and covers a wide range of sizes.

Therefore it is difficult to obtain a good equivalent particle diameter to use in the Ergun equation

(Equation 2.10). Because of this, Harkonen (1987) rewrote the Ergun equation, grouping the

equivalent particle diameter and the fluid properties into coefficients of R1 and R2. Thus

( ) ( ) 2323

2

111

URURdLdP

l

l

l

l

εε

εε −

+−

= (2.27)

where dP/dL is pressure drop in Pa/m, lε is the void fraction, U is the superficial velocity in m/s

and R1 and R2 empirical constants. Equation 2.27 can be linearized to:

( )( )

212

3 111

RU

RUdL

l

l+

−=

−ε

εdP lε (2.28)

where

21pd

AR µ= (2.29)

and

pd

BR ρ=2 (2.30)

Values of R1 and R2 have been measured for several wood species (Harkonen, 1987; Linqvist,

1994; Lammi, 1996; Wang and Gullichen, 1999) as summarized in Table 2.5. A wide range of

values are given, and we note that some of the R2 values from literature are negative. However,

there is no physical reason why R2 would be negative. Theoretically, R2 should be positive

according to Equation 2.30 from curve fitting experimental data to Equation 2.27.

Chapter 2: LITERATURE REVIEW 30 Harkonen (1987) also correlated the void fraction of the chip column to the compacting

pressure applied Pc, and the kappa number, κ , using:

( )ln(kkP

kk

cl κε 320

1

10+−

+= ) (2.31)

Here k0 is equivalent to the initial void fraction (when no compacting pressure is applied) while

the other constants (k1, k2, and k3) are fitted parameters. Where the compacting pressure, Pc (in

kPa), is the contact force acting on the chip column divided by the cross-sectional area of the

column and κ is the kappa number. k0, k1, k2, and k3 have been measured for several wood

species (Harkonen, 1987; Linqvist, 1994; Lammi, 1996; Wang and Gullichen, 1999) as

summarized in Table 2.6.

This empirical correlation shows that the void fraction depends on both the compacting

pressure and kappa number. The second term of Equation (2.31) should be negative as the void

fraction decreases with increased compacting pressure and decreased kappa number.


Table 2.5: Summary of the available literature on flow resistance of chip columns by using Equation 2.27 with dP/dL in Pa/m and U in m/s.

Wood species and chip size distribution Reference R1 R2 Scandinavian Pine (reference chips) Defined as chips passed through a 13 mm hole but retained on a 7 mm hole.

Wang and Gullichen (1999) 0.52×105 1.5×106

Scandinavian Pine (“new” chips) “New” chips: 4 mm thick, 40 mm length

Wang and Gullichen (1999) 0.82×105 -0.11×106

Scandinavian Birch + 45 mm hole 1.1% + 8 mm bar 4.8% + 13 mm hole 78.8% + 7 mm hole 12.6% + 3 mm hole 2.2% + fines 0.5%

Lammi (1996) 2.8×105 -1.2×106

Eucalyptus Camaldulensis + 45 mm hole 0.6% + 8 mm bar 9.2% + 13 mm hole 51.4% + 7 mm hole 31.2% + 3 mm hole 6.4% + fines 1.1%

Lammi (1996) 5.5×105 0.75×106

Scandinavian Pine (mix 1) + 6 mm bar 22.1% + 4 mm bar 44.2% + 2 mm bar 29.1% + 3 mm hole 4.6%

Linqvist (1994) 0.28×105 -0.13×106

Scandinavian Pine (mix 2) + 6 mm bar 23.2% + 4 mm bar 46.3% + 2 mm bar 30.5%

Linqvist (1994) 0.51×105 -0.35×106

Scandinavian Pine (mix 3) + 4 mm bar 60.3% + 2 mm bar 39.7%

Linqvist (1994) 0.055×105 -0.006×106

Scandinavian Pine (Distribution not specified.) Harkonen (1987) 0.046×105 3.9×106

Note: “+” means retained on the plate.

Chapter 2: LITERATURE REVIEW 32 Table 2.6: Summary of the available literature on void fraction equation constants by using Equation 2.31. More detail on chip size distribution can be found in Table 2.5.

Wood species and chip size distribution

Reference k0 k1 k2 k3

Scandinavian Pine (reference chips)

Wang and Gullichen

(1999) 0.663 0.56 0.788 0.133

Scandinavian Pine (“new” chips) Wang and Gullichen

(1999) 0.840 0.39 1.092 0.161

Scandinavian Birch Lammi (1996) 0.630 0.64 0.697 0.151

Eucalyptus Camaldulensis Lammi (1996) 0.591 0.56 0.645 0.148

Scandinavian Pine (mix 1) Linqvist (1994) 0.604 0.63 0.956 0.191 Scandinavian Pine (mix 2) Linqvist (1994) 0.615 0.79 0.910 0.181 Scandinavian Pine (mix 3) Linqvist (1994) 0.647 0.74 1.021 0.205 Scandinavian Pine (Distribution not specified.)

Harkonen (1987) 0.644 0.59 0.831 0.139

2.8 Wood Chips

Presently, the major goal in the pulp and paper industry is to develop environmentally friendly

processes as well as to improve end product quality. Wood handling, at the beginning of the

process, is essential to achieving these goals. Wood chips are very important in pulp mills

because wood constitutes more than 50% of the pulp manufacturing costs and basically whole

quality depends on the quality of raw material, i.e. the chips (Tahkanen, 1994).

2.8.1 Chip Quality

Chip quality, particularly the chip thickness, has attracted growing attention for many years.

Many modifications have been made in the wood handling process from chipping (Tahkanen,

1994) to chip screening (Luxardo and Javid, 1992) to achieve the common slogan in the pulp and

paper industry: “High quality pulp starts with high quality chips” (Wang and Gullichsen, 1998).

Chapter 2: LITERATURE REVIEW 33 The results of poor chip quality will be: poor usability of the fiber source, non-uniform pulping,

off-grade pulp and yield losses, and high production costs.

2.8.2 Chip Thickness

Chip thickness has been recognized as one of the key factors for efficient pulping and good pulp

quality (Tikka and Tahkanen, 1992). Depending on the penetration and diffusion mechanism,

chip thickness has the greatest effect on liquid transfer (Gustafson et al., 1988). Due to the fact

that kraft delignification is effective only to certain depth in wood chips, cores of thick chips do

not delignify properly. However, chips that are sufficiently long and thin will make the kraft

cook more uniform (Gullichsen, 1992 and 1995).

2.8.3 Oversize and Undersize Fractions

Oversize and overthick chips do not cook completely and uniformly. These chips leave

uncooked residues, which appear in the pulp as shives, knots and knotter rejects (Christie, 1987).

Screening systems have been developed for efficient isolation of each critical fraction

from the unscreened whole to allow separate chip processing prior to pulp production. The

separation of the overthick fraction, which is reduced in thickness by slicing or destructing, and

typically returned to accepts stream, eliminates the undesirable effect of overthick chips in

pulping (Luxardo and Javid, 1992).

Fines are usually eliminated from the pulp mill chips and diverged to a power boiler for

combustion. This elimination of fines has a number of advantages, such as improvement in pulp

yield and quality, and reduced chemical usage (Kreft and Javid, 1990). However, if it is not

economical to reject the fines, they can be cooked in a separate digester.


In some applications, the separation of small pin chips for alternative processing, or

metered feeding prior to a continuous digester, can improve liquor circulation in the digester. In

other cases, retention of pin chips is important due to fibre value. Tikka and Tahkanen (1992)

reported that pin chips give a reasonable yield, for example yield at kappa number 25 is about

47%.

2.8.4 Evaluation of Chip Size Distribution

A narrow chip size distribution is thought to be important to ensure uniform liquor penetration

and even cooking conditions. The chip size distribution is evaluated with screening systems

called classifier. The laboratory or mill chip screening system classifies the incoming chips as

follows:

• The top screen has 45 mm wide round holes. The chips retained on this screen are called

overlarge.

• The next screen has 10 mm wide slots for softwoods (8 mm for hardwoods). The chips

retained on this screen are overthick.

• The third screen has 7 mm round holes. On this screen the accept fraction is retained.

• The fourth screen with 3 mm round holes will retain the pin chips.

• What passes all four screens and collects on the bottom is the fines fraction, or pan fines.

Table 2.7 shows a typical chip size distribution for the mill input to the digester (Walkush and

Gustafson, 2002).

Table 2.7: Typical mill produces chip size distribution.

Size (mm) Weight % Over fraction > 8 5.8 Accept chips 7 79.2 Pin chips 3 - 7 14.1 Pan fines < 3 0.9


All currently available devices classify chips based on just one or a combination of two

criteria – chip thickness, defined by passage through a slot or a hole on planar surface. By using

different criteria, fundamentally different attributes of the chips are measured. This means data

from one device may not be directly comparable to data from another device (Sacia and Marrs,

1994). Table 2.8 lists many of the commonly used classifiers and the primary and secondary

criteria they use as a basis for separation.

Table 2.8: Commonly used chip size classification devices in mills (Sacia and Marrs, 1994).

Company Name

Machine Name Slot Opening Criteria

(Thickness)

Round Hole Criteria

Combination Thickness & Round Hole

BM&M Chip & Sawdust Primary Consilium Automatic Chip Classifier Primary Lorentzen & Wettre

STFI Chip Classifier Primary

Price Rotary Chip Classifier Primary Secondary Rader CC2000 Primary Secondary Rotex GradexTM Primary SWECO Vibro-Energy Separator® Square Hole TMI Auto Domtar Chip

ClassifierTM Primary Secondary

TMI The Williams Chip Classification

Primary

Tompkins-Beckwith

EMP Chip Classifier Primary

Weyerhaeuser ChipClassTM Primary


2.9 Research Objectives

The literature shows that pressure drop has been widely studied for a wide range of chemical

engineering system and is of great importance for flow through chip beds and consequently for

digester design. However, very limited work has been published on pressure drop through chip

columns (Harkonen, 1987; Lindqvist, 1994; Lammi, 1996; Wang and Gullichsen, 1998). In order

to better understand the liquid flow in pulp digesters, we developed an experimental program to

measure the pressure drop and liquor flow through packed beds of cooked wood chips. The

objectives of this thesis are:

• To improve our understanding of pressure drop and liquor flow through packed beds of

cooked wood chips and to develop pressure drop models for cooked wood chips, particularly

as a function of chip size and chip size distribution. White spruce chips, produced by a Chip-

N-Saw chipper, are used in this study.

• To identify whether previous correlations can be used to predict pressure drop for our data

and to investigate the applicability of the Ergun equation to describe the flow resistance in

packed bed of wood chips.

• To develop a model that correlates the void faction of a chip column as a function of

compacting pressure and kappa number for different chip size distributions.

chapter 2 literature review 2.1 introduction to porous media 2.1.1

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