CHAPTER - 4 - Information and Library Network...

CHAPTER - 4 DIFFUSION DISPERSION

MODEL

Some of the contents of this chapter are published in

Applied Mathematics & Computation, Vol. 220, pp. 560–567 (2013).

Cellulose Chemistry & Technology, (2014) In press.

ISRN Chemical Engineering, Vol. 2013, pp. 1-6 (2013).

40

4.1 Introduction of Diffusion Dispersion Model

In chemical engineering, the removal of solutes from a packed bed using a solvent is of

great importance. The solutes occupy the space between the interstices of particles (as

displaceable fluid) and the pores of the particles (as stagnant fluid). The displacement of

solute from the void spaces of the bed takes place due to diffusion-dispersion in the

direction of flow. The phenomenon of diffusion occurs due to concentration gradient,

dispersion (due to back mixing) and adsorption-desorption of solutes towards the

particle surface. It causes the stagnant fluid (containing solute) to move out of the

particle voids. The solutes are subsequently displaced by the flowing fluid. To analyze,

simulate and optimize the solute removal from packed bed, development of a

mathematical model is an imperative necessity. The general diffusion dispersion model

is given hereunder:

2

2

1 ,Lc c c nD u

x x t tε

ε∂ ∂ ∂ − ∂ = + + ∂ ∂ ∂ ∂

(4.1)

where LD is longitudinal dispersion coefficient (m2/s), c is concentration of the solute

in the liquor (kg/m3), u is velocity of liquor in the mat (m/s), x is variable cake

thickness (m), t time (s), ε is total average porosity (dimensionless) and n is

concentration of solute on the fibers (kg/ m3). First term of this partial differential

equation represents diffusion-dispersion, second is convective flow term and other terms

are concentration gradients of the fluid and of the particles respectively. Here u and

LD are functions of x while c and n are functions of both x and t.

The concentration of solute adsorbed on particle surface and concentration of bulk fluid

are linked via linear and nonlinear (Langmuir) adsorption isotherms given below:

n kc= , (4.2)

and

41

0

01A cn

B c=

+, for 0

i

F

kNAC

= and 0F

kBC

= , (4.3)

where Ni is initial concentration of solute on the fibers (kg/ m3), k is mass transfer

coefficient (dimensionless), CF is fiber consistency (kg/m3), A0 is Langmuir constant

(1/s) and B0 is Langmuir constant (m3/kg s).

The model (4.1), (4.2) and (4.3) are solved for boundary conditions:

L Scuc D u Cx∂

− =∂

at 0x = , (4.4)

0cx∂

=∂

at x L= , (4.5)

and initial condition:

( , 0) ic x C= . (4.6)

where sC is concentration of the solute in the wash liquor (kg/m3), L is cake thickness

(m) and iC is concentration of the solute in the vat (kg/m3).

A great deal of effort is underway for exact and numerical solution of the system

described by Eqs. (4.1) to (4.6). An assortment of analytic and numerical algorithms

such as Laplace transformation (Shawaqfeh, 2003; Siyakatshana, Kudrna and Machon,

2005; Trinidad, Leon and Walsh, 2006; Kukreja and Ray, 2009), Fourier’s method

(Siyakatshana, Kudrna and Machon, 2005; Cermakova et al., 2006; Tervola, 2006),

moment method (Roininen and Alopaeus, 2011), Galerkin finite element method (Moon

et al., 2010), finite difference method (Zhang, Zhao and Wang, 2006), FlexPDE

software (Rivera et al., 2010; Diaz et al., 2012), Euler Maruyama scheme (Vianna and

Nichele, 2010), orthogonal collocation method (Junco, Bildea and Floarea, 1994; Feiz,

1997; Shirashi, 2001), fitted mesh collocation method (Liu and Bhatia, 1999), Galerkin /

Petrov Galerkin method (Liu and Bhatia, 2001), orthogonal collocation on finite

elements (Liu and Jacobsen, 2004; Arora, Dhaliwal and Kukreja, 2006a, 2006b),

42

COMSOL Multiphysics package (Abeynaike et al., 2012), iterative technique (Farooq

and Karimi, 2003), collocation method (Lefevre et al., 2000), pdepe-solver (Sule,

Lakatos and Mihalyko, 2010) are employed to solve above type of models.

4.2 Applications of Diffusion Dispersion Model

Diffusion-dispersion models have been employed to describe the pulp washing

phenomenon of paper industry. Following investigations have contributed a lot in this

direction:

Brenner (1962) used Laplace transforms to solve the equation and presented the

numeric values of exit and average solute concentrations at different times for different

values of Peclet number ranging from 0 to ∞.

Pellett (1966) has developed an extended model for two zones by dividing the combined

effects of axial dispersion and intrafiber diffusion namely, zone of external fluid and the

zone of particle diffusion. For axial dispersion model the initial and boundary

conditions were same as that of Sherman (1964). However, for particle phase, different

types of boundary conditions have been taken. Pellett has also shown that at

intermediate fluid velocities both the intrafiber diffusion and liquid phase mass transfer

resistances have strong effects on the shape of breakthrough curves.

Grahs (1974) has developed a mathematical model by using Langmuir isotherm. The

non linear model was solved using double collocation method.

Kukreja (1996) has studied an axial dispersion model by taking Laplace transform to

solve the model equations. The values of exit solute concentration, average solute

concentration and mean solute concentration were presented at different values of time.

Arora, Dhaliwal and Kukreja (2006a) presented mathematical model related to

diffusion–dispersion during flow through multiparticle system. The technique of

43

orthogonal collocation on finite elements was applied on the axial and radial domain to

solve the model. The convergence and stability of the solutions were also checked.

Apart from this diffusion, dispersion equations have numerous applications in the field

of chemical and process industries. By making suitable modifications, the Eqs. (4.1) to

(4.6) are extendible to mass transfer between the solid and liquid phases (Vikhansky and

Wang, 2011), separation of glycerol from biodiesel (Abeynaike et al., 2012), tubular

flow reactor (Roininen and Alopaeus, 2011; Farooq and Karimi, 2003; Lefevre et al.,

2000), analysis of distillation column (Qin, Yang and Yang, 2011), design and

development of airlift loop reactors (Zhang, Zhao and Wang, 2006), heat transfer

processes in gas-solid turbulent, fluidized systems (Sule, Lakatos and Mihalyko, 2010),

chromatography (Saritha and Madras, 2001), measurement of neutron flux (Feiz, 1997),

sorption-desorption (Khan and Loughlin, 2003; Cocero and Garcia, 2001), enzymatic

hydrolysis of racemic ibuprofen ester by lipase (Long, Bhatia and Kamaruddin, 2003),

annular flow in a tubular reactor (Veinna and Nichele, 2010), electrochemical reactors

(Trinidad, Leon and Walsh, 2006), gas liquid bubble column (Shawaqfeh, 2003), solid

flow in liquid suspension (Cermakova et al., 2006), cake washing process (Tervola,

2006) and mixing flow pattern (Rivera et al., 2010; Diaz et al., 2012) etc.

None of the previous investigators ever compared the results of the model for different

isotherms, in order to assess their suitability to predict the actual flow pattern in the

packed bed. In this work an attempt is made, to compare the linear and nonlinear

isotherms using cubic Hermite collocation method (CHCM) for Gauss Legendre roots

as collocation points. The method has the advantage that high-order accuracy can be

achieved using small system of equations and multistep time differencing. In addition, it

requires only a small amount of arithmetic at each time step.

44

4.3 Solution of the Model

For the solution of diffusion dispersion model, Eqs. (4.1) to (4.6) are converted into

dimensionless form using Peclet number, dimensionless concentration, dimensionless

time and dimensionless thickness given below:

S

i S

c CCC C−

=−

, S

i S

n CNC C−

=−

, L

uLPeD

= , xL

ξ = , (1 )

utk L

τµ

=+

, 1

2

kkk

= , 1 εµε−

= .

The dimensionless form of model along with initial and boundary conditions is given

hereunder:

2

2

1 C C CPe ξ ξ τ

∂ ∂ ∂= +

∂ ∂ ∂, (linear case), (4.7)

( ) 2

02 2

0 0

1 1 1(1 ) (1 )1 s s

AC C C CPe k kB C C C C

µξ µ τ µ τξ

∂ ∂ ∂ ∂= + +∂ + ∂ + ∂∂ + − +

, (nonlinear), (4.8)

( , 0) 1ξ =C , at 0τ = , for all ξ , (4.9)

(0, )(0, ) 0CPeC ττξ

∂− =

∂, at 0ξ = , for all τ , (4.10)

(1, ) 0C τξ

∂=

∂, at 1ξ = , for all τ . (4.11)

The discretized form of Eq. (4.7) can be as written as follows by making use of Eq.

(2.18) defined in Chapter 2:

1

1( , ) ( ) ( , ) ( )

N

j j j jj

a P a Qξ τ ξ ξ τ ξ+

=

′ + ∑

1 1

1 1

1 ( , ) ( ) ( , ) ( ) ( , ) ( ) ( , ) ( )N N

j j j j j j j jj j

a P a Q a P a QPe

ξ τ ξ ξ τ ξ ξ τ ξ ξ τ ξ+ +

= =

′′ ′′ ′ ′′ ′= + − + ∑ ∑ ,

Here a and 'a represents the derivatives of a and 'a with respect to time, respectively.

After rearrangement, one gets:

45

1 1 1 1 1 1 1 12

( , ) ( ) ( , ) ( ) ( , ) ( ) ( , ) ( ) ( , ) ( ) ( , ) ( )N

j j j j N N N Nj

a P a Q a P a Q a P a Qξ τ ξ ξ τ ξ ξ τ ξ ξ τ ξ ξ τ ξ ξ τ ξ+ + + +=

′ ′ ′ + + + + + ∑

1

1( , ) ( ) ( , ) ( )

N

j j j jj

a aξ τ φ ξ ξ τ ψ ξ+

=

′ = + ∑ , (4.12)

where 1( ) ( ) ( )j j jP P

Peφ ξ ξ ξ′′ ′= − and

1( ) ( ) ( )j j jQ QPe

ψ ξ ξ ξ′′ ′= − .

Since 1 1 1( ) 0, ( ) 1jP Qξ ξ′≡ = and 1( ) 0jQ ξ′ ≡ for 1j > , boundary condition at 0ξ = ,

gives:

1 11( , ) ( , ) 0a a

Peξ τ ξ τ′− = . (4.13)

Differentiating this expression with time, one gets:

1 11( , ) ( , ) 0a a

Peξ τ ξ τ′− = . (4.14)

Similarly, the boundary condition at 1ξ = implies:

1( , ) 0Na ξ τ+′ = . (4.15)

Differentiating this expression with time, one gets:

1( , ) 0Na ξ τ+′ = . (4.16)

Now substituting conditions (4.13) to (4.16) in (4.12), one gets:

[ ]

[ ]

1 1 1 1 12

1 1 1 1 12

( , ) ( ) ( , ) ( ) ( ) ( ) ( , ) ( , ) ( )

( , ) ( ) ( , ) ( ) ( ) ( ) ( , ) ( , ) ( )

N

j j j j N Nj

N

j j j j N Nj

a P a Q PeQ P a a P

a a a a

ξ τ ξ ξ τ ξ ξ ξ ξ τ ξ τ ξ

ξ τ φ ξ ξ τ ψ ξ φ ξ ψ ξ ξ τ ξ τ φ ξ

+ +=

+ +=

′ + + + +

′ = + + + +

∑

∑

(4.17)

Since each ( ), ( )j jP x Q x has its support in 1 1[ , ]j jx x− + and in particular

1( ) 0P ξ = for 1 2[ , )ξ ξ ξ∉ and 1 1( ) ( ) 0N NP Qξ ξ+ += = for 1( , ]N Nξ ξ ξ +∉ , (4.18)

46

therefore, evaluating Eq. (4.17) by using Eq. (4.18) at collocation points ,η j i , one gets: For 2, 1, 2= =j i ,

1 1 2, 1 2, 2 2, 2 2 2, 2 2,( , ) ( ) ( ) ( ) ( , ) ( ) ( , )ξ τ η η η ξ τ η ξ τ′+ + + = i i i i ia P PeQ P a Q a F , with 2, 1 1 2, 1 2, 2 2, 2 2 2, 2( , ) ( ) ( ) ( ) ( , ) ( ) ( , ),ξ τ φ η ψ η φ η ξ τ ψ η ξ τ′= + + +i i i i iF a a a For 3,... , , 1, 2= =j N i ,

1 , 1 1 , 1 , , ,( ) ( , ) ( ) ( , ) ( , ) ( ) ( , ) ( )η ξ τ η ξ τ ξ τ η ξ τ η− − − −′ ′+ + + = j j i j j j i j j j j i j j j i j iP a Q a a P a Q F , with

, 1 1 , 1 1 , , ,( , ) ( ) ( , ) ( ) ( , ) ( ) ( , ) ( ) ,ξ τ φ η ξ τ ψ η ξ τ φ η ξ τ ψ η− − − −′ ′= + + +j i j j j i j j j i j j j i j j j iF a a a a

For 1, 1, 2= + =j N i ,

1, 1, 1 1, 1 1,( ) ( , ) ( ) ( , ) ( ) ( , )η ξ τ η ξ τ η ξ τ+ + + + + +′+ + = N N i N N N i N N N i N N iP a Q a P a F ,

with 1, 1, 1, 1 1 1,( , ) ( ) ( , ) ( ) ( , ) ( )ξ τ φ η ξ τ ψ η ξ τ φ η+ + + + + +′= + +N i N N N i N N N i N N N iF a a a .

Above differential algebraic equations can be put in the matrix form as follows:

AX BX= (4.19)

where

( )1 2 2 3 3 1( , ), ( , ), ( , ), ( , ), ( , ), ... , ( , ), ( , ), ( , ) TN N NX a a a a a a a aξ τ ξ τ ξ τ ξ τ ξ τ ξ τ ξ τ ξ τ+′ ′ ′=

and

( )1 2 2 3 3 1( , ), ( , ), ( , ), ( , ), ( , ), ..., ( , ), ( , ), ( , ) TN N NX a a a a a a a aξ τ ξ τ ξ τ ξ τ ξ τ ξ τ ξ τ ξ τ+′ ′ ′=

are vectors and matrices A and B are given as follows:

47

1 2, 1 1 2, 1 2 2, 1 2 2, 1

1 2, 2 1 2, 2 2 2, 2 2 2, 2

2 3, 1 2 3, 1 3 3, 1 3 3, 1

2 3, 2 2 3, 2 3 3, 2 3 3, 2

( ) ( ) ( ) ( ) 0 0 ... 0( ) ( ) ( ) ( ) 0 0 ... 0

0 ( ) ( ) ( ) ( ) ... 00 ( ) ( ) ( ) ( ) ... 0... ... ... ... ... ... ...... ... ... ... ... ... ...

P PeQ P QP PeQ P Q

P Q P QP Q P Q

A

η η η ηη η η η

η η η ηη η η η

++

=

1, 1 1, 1 1 1, 1

1, 2 1, 2 1 1, 2

... ... ... ... ... ... ...0 0 ... 0 ( ) ( ) ( )0 0 ... 0 ( ) ( ) ( )

N N N N N N

N N N N N N

P Q PP Q P

η η ηη η η

+ + + +

+ + + +

and

1 2, 1 1 2, 1 2 2, 1 2 2, 1

1 2, 2 1 2, 1 2 2, 2 2 2, 2

2 3, 1 2 3, 1 3 3, 1 3 3, 1

2 3, 2 2 3, 2 3 3, 2 3 3, 2

( ) ( ) ( ) ( ) 0 0 ... 0( ) ( ) ( ) ( ) 0 0 ... 0

0 ( ) ( ) ( ) ( ) ... 00 ( ) ( ) ( ) ( ) ... 0... ... ... ... ... ... ...... ... ... ... ... ... ...... .

B

φ η ψ η φ η ψ ηφ η ψ η φ η ψ η

φ η ψ η φ η ψ ηφ η ψ η φ η ψ η

++

=

1, 1 1, 1 1 1, 1

1, 2 1, 2 1 1, 2

.

.. ... ... ... ... ...0 0 ... 0 ( ) ( ) ( )0 0 ... 0 ( ) ( ) ( )

N N N N N N

N N N N N N

φ η ψ η φ ηφ η ψ η φ η

+ + + +

+ + + +

The 3dim( ( )) 2 2π = +H N , therefore we need 2 2+N conditions at each time to

estimate the approximate solution. Obviously, two of these conditions are obtained from

boundary conditions and rest from differential equation. The matrix system (4.19) is

solved by MATLAB ode15s solver.

4.4 Verification of Models

For the linear case, the verification is carried out by comparing the results with the

analytic solution from literature. For nonlinear case, the experimental data is used to

show the effect of different parameters.

48

4.4.1 Comparison Between Analytic and CHCM Solutions

Analytic solution for packed bed of finite length, for linear case is given by Brenner

(1962). Numerical results from Eq. (4.7) are obtained using CHCM and are compared

with the exact ones (Brenner, 1962) in Tables 4.1, 4.2 and 4.3 for Peclet number 10, 40

and 80 respectively. A close match is found between both the results.

Table 4.4 contains numerical results for nonlinear isotherm, for the data (Arora,

Dhaliwal and Kukreja, 2006a) for 40Pe = , 0 5.20 4= −A E 1/s, 0 6.25 5B E= − m3/kg-s,

2.94 3= −k E , 3.31 2µ = −E , 0 8.33C = kg/m3. Since no exact solution is available in

literature for nonlinear isotherm, therefore in Table 4.4, it is shown that the numerical

results remain consistent and stable, even by increasing the number of elements.

For different Peclet numbers, analytic (Brenner 1962) and numerical results of exit

solute concentration are compared for linear Eq. (4.7) and nonlinear Eq. (4.8) isotherms

in Figures 4.1-4.3. The analytic (Brenner 1962) and linear isotherm results are

overlapping each other. For both the cases solute removal process is starting late than

the nonlinear isotherm.

It is also observed from Tables 4.1, 4.2, 4.3 and 4.4 that the exit solute concentration for

linear case is of the order 10-16 (almost negligible) where as for nonlinear case it is of

the order 10-4. Thus indicating that very small amount of leaching continues

concomitantly even after considerable time. It is worth mentioning that sorption

isotherm for sodium was successfully described by Langmuir equation (Crotogino,

Poirier and Trinh, 1987).

49

Table 4.1: Comparison of exit solute concentration at Pe = 10 for linear case

Time Exact Solution N = 25 N = 50 N = 100

0.0 1.00000E+00 1.00000E+00 1.00000E+00 1.00000E+00

0.2 1.00000E+00 1.00000E+00 1.00000E+00 1.00000E+00

0.4 9.99987E-01 9.99987E-01 9.99987E-01 1.00000E+00

0.6 9.87554E-01 9.87557E-01 9.87600E-01 9.87600E-01

0.8 8.18514E-01 8.18519E-01 8.18520E-01 8.18500E-01

1.0 4.57337E-01 4.56991E-01 4.56525E-01 4.56500E-01

1.2 1.73343E-01 1.73340E-01 1.73303E-01 1.73300E-01

1.4 4.95553E-02 4.95585E-02 4.95587E-02 4.95600E-02

1.6 1.16114E-02 1.16230E-02 1.16240E-02 1.16200E-02

1.8 2.37146E-03 2.37198E-03 2.37201E-03 2.37200E-03

2.0 4.38032E-04 4.38123E-04 4.38128E-04 4.38200E-04

2.2 7.52085E-05 7.52166E-05 7.52171E-05 7.51900E-05

2.4 1.22255E-05 1.22244E-05 1.22243E-05 1.22200E-05

2.6 1.90602E-06 1.90529E-06 1.90524E-06 1.90500E-06

2.8 2.87684E-07 2.87460E-07 2.87446E-07 2.87400E-07

3.0 4.23275E-08 4.22736E-08 4.22706E-08 4.22600E-08


Time Exact solution N = 40 N = 80 N = 160

0.0 1.00000E+00 1.00000E+00 1.00000E+00 1.00000E+00

0.2 1.00000E+00 1.00000E+00 1.00000E+00 1.00000E+00

0.4 1.00000E+00 1.00000E+00 1.00000E+00 1.00000E+00

0.6 1.00000E+00 9.99998E-01 9.99998E-01 1.00000E+00

0.8 9.74640E-01 9.74616E-01 9.74640E-01 9.74640E-01

1.0 4.77840E-01 4.77902E-01 4.77843E-01 4.77840E-01

1.2 4.49280E-02 4.49056E-02 4.49265E-02 4.49278E-02

1.4 9.90370E-04 9.89062E-04 9.90306E-04 9.90364E-04

1.6 7.90010E-06 7.93846E-06 7.90259E-06 7.90028E-06

1.8 3.11080E-08 3.19352E-08 3.11537E-08 3.11106E-08

2.0 7.40610E-11 7.93789E-11 7.43584E-11 7.40884E-11

50


Time Exact Solution N=100 N=200 N=400

0.0 1.00000E+00 1.00000E+00 1.00000E+00 1.00000E+00

0.2 1.00000E+00 1.00000E+00 1.00000E+00 1.00000E+00

0.4 1.00000E+00 1.00000E+00 1.00000E+00 1.00000E+00

0.6 1.00000E+00 1.00000E+00 1.00000E+00 1.00000E+00

0.8 9.97410E-01 9.97410E-01 9.97411E-01 9.97411E-01

1.0 4.84280E-01 4.84289E-01 4.84280E-01 4.84280E-01

1.2 9.28220E-03 9.28058E-03 9.28206E-03 9.28215E-03

1.4 7.61530E-06 7.62324E-06 7.61583E-06 7.61535E-06

1.6 7.11770E-10 7.18406E-10 7.12188E-10 7.11804E-10

1.8 1.50320E-14 1.86514E-14 1.61947E-14 1.60089E-14

2.0 -7.71670E-20 -8.07200E-18 -1.03347E-16 -1.46734E-16

Table 4.4: Exit concentration values at 40Pe = for nonlinear case

Time N = 50 N = 100 N = 150 N = 200 N = 250 N = 300

0.0 1.0000E+00 1.0000E+00 1.0000E+00 1.0000E+00 1.0000E+00 1.0000E+00

0.2 1.0000E+00 1.0000E+00 1.0000E+00 1.0000E+00 1.0000E+00 1.0000E+00

0.4 9.99980E-01 9.99980E-01 9.99980E-01 9.99990E-01 9.99990E-01 9.99990E-01

0.6 9.85000E-01 9.86290E-01 9.86720E-01 9.86930E-01 9.87050E-01 9.87140E-01

0.8 8.03380E-01 8.10980E-01 8.13500E-01 8.14760E-01 8.15520E-01 8.16020E-01

1.0 4.38510E-01 4.47520E-01 4.50530E-01 4.52040E-01 4.52940E-01 4.53540E-01

1.2 1.63630E-01 1.68480E-01 1.70110E-01 1.70920E-01 1.71410E-01 1.71730E-01

1.4 4.62050E-02 4.78800E-02 4.84420E-02 4.87230E-02 4.88910E-02 4.90040E-02

1.6 1.07430E-02 1.11820E-02 1.13300E-02 1.14040E-02 1.14480E-02 1.14780E-02

1.8 2.17880E-03 2.27520E-03 2.30760E-03 2.32390E-03 2.33360E-03 2.34010E-03

2.0 4.00620E-04 4.19350E-04 4.25640E-04 4.28790E-04 4.30680E-04 4.31950E-04

51

Figure 4.1: Comparison of linear and nonlinear cases for Pe = 10, L = 0.05 m, DL = 5 × 10-5 m2/s, u = 0.019 m/s



0.0 0.5 1.0 1.5 2.0 2.5 3.00.0

0.2

0.4

0.6

0.8

1.0

Dimensionless Time

Dim

ensi

onle

ss C

once

ntra

tion

Linear (CHCM)Linear (Brenner)Nonlinear (CHCM)

0.0 0.5 1.0 1.5 2.00.0

0.2

0.4

0.6

0.8

1.0

Dimensionless Time

Dim

ensi

onle

ss C

once

ntra

tion

Linear (CHCM)Linear (Brenner)Nonlinear (CHCM)

0.0 0.5 1.0 1.5 2.00.0

0.2

0.4

0.6

0.8

1.0

Dimensionless Time

Dim

ensi

onle

ss C

once

ntra

tion

Linear (CHCM)LInear (Brenner)Nonlinear (CHCM)

52

4.4.2 Verification Using Experimental Data

The model for packed bed of porous particles is simulated using experimental data of a

rotary vacuum washer reported by Arora, Dhaliwal and Kukreja (2006a). The mill was

using wheat straw as the raw material. The inlet and outlet consistencies were in the

range of 1-2 % and 10-12 % respectively. Concentration of black liquor solids inside the

vat was 8-9 kg/m3 and fresh water was sprayed to wash the pulp.

Exit solute concentration profiles for linear Eq. (4.7) and nonlinear Eq. (4.8) isotherms

are plotted for axial dispersion coefficient, cake thickness and interstitial velocity in

Figures 4.4 to 4.6 respectively. During actual washing process in a plant, the

displacement of black liquor solids from the packed bed starts instantaneously as soon

as the wash liquor comes in contact with the bed. These figures indicate that, for same

set of parameters, mathematically, such type of behaviour is best described by the

nonlinear isotherm than the linear one. Similar behaviour has been observed practically

during laboratory experiments by Crotogino, Poirier and Trinh (1987) that the leaching

of solute continues even after 72 hours. The present study verifies the experimental

results of Crotogino, Poirier and Trinh (1987) for nonlinear isotherm. It can be

concluded that solute removal process can be simulated more accurately using nonlinear

isotherm.

53

Figure 4.4: Comparison of linear and nonlinear case at DL = 5 × 10-5 m2/s, L = 0.05 m, Pe = 40, u = 0.019 m/s

Figure 4.5: Comparison of linear and nonlinear case at L = 0.05 m, Pe = 40, DL = 5 × 10-5 m2/s, u = 0.019 m/s

Figure 4.6: Comparison of linear and nonlinear case at u = 0.019 m/s, Pe = 40, DL = 5 × 10-5 m2/s, L = 0.05m

0.0 0.5 1.0 1.5 2.00.0

0.2

0.4

0.6

0.8

1.0

Dimensionless Time

Dim

ensi

onle

s C

once

ntra

tion

Linear (CHCM)Nonlinear (CHCM)

0.0 0.5 1.0 1.5 2.00.0

0.2

0.4

0.6

0.8

1.0

Dimensionless Time

Dim

ensi

onle

ss C

once

ntra

tion


0.0 0.5 1.0 1.5 2.00.0

0.2

0.4

0.6

0.8

1.0

Dimensionless Time

Dim

ensi

onle

ss C

once

ntra

tion


54

4.5 Convergence Analysis

Consider Eq. (4.7) with 1 21 , 1

ePµ µ= = i.e.

2

1 22 , (0,1)C C Cµ µ ξτ ξ ξ

∂ ∂ ∂= − ∈

∂ ∂ ∂, (4.20)

for the Dirichlet, Neumann or Robin boundary conditions as:

3 4 1, 0, 0CCµ µ κ ξ τξ∂

+ = = >∂

, (4.21)

5 6 2 , 1, 0CCµ µ κ ξ τξ∂

+ = = >∂

, (4.22)

and initial condition as:

3( , 0)C ξ κ= . (4.23)

Here ' sµ are real numbers lying between 0 and 1 and ' sκ are non negative real

numbers. ξ is bounded on real axis in an open interval (0,1) .

4.5.1 Notations and Preliminary Results

Let the operator L defined by 2

2ξ∂∂

in spatial and time domains is positive definite in

2 (0,1)L space of all real valued Lebesgue measurable functions square integrable on

(0,1) , for all 0τ > . Therefore, there exist a continuous function ( , ) ( )ξ τ ∈v D L , space

of all the functions, twice continuously differentiable function, with compact support on

the interval [0,1] as reported in literature (Onah, 2002), such that for every constant λ :

, ,λ ≤v v v vL , for (0,1)ξ ∈ and 0τ > , (4.24)

and

, , , ,ξ τ= ∀v v v vL L , (4.25)

Miller, O’Riordan and Shishkin (1996) considered a family of mathematical problems

parametrized by singular perturbation parameter δ , where δ lies in the semi open

55

interval 0 1δ< ≤ . It is assumed that each problem in the family has the unique solution

denoted by vu and each vu is approximated by a sequence of numerical solutions

1( , )MMUδ∞=Ω , where Uδ is defined on the MΩ representing the set of points in R, and

M is the discretization parameter. Therefore, the numerical solutions Uδ are said to

converge to the exact solution uδ if there exist a positive integer 0M , and positive

numbers F and p (where 0M , F and p are all independent of M and δ ) such that

for all 0M M≥

0 1sup M

pU u Mδ δδ

−Ω

< ≤− ≤ Λ

where p is the rate of convergence and Λ is the error constant.

The fundamental solution of Eqs. (4.20) to (4.22) can be written in the form (Carslaw

and Jaeger, 1959):

1

0

( , ) ( )C A exp w dξξ τ ζ ζτ

= − ∫ (4.26)

where A is some constant depending on parameter ' sµ . ζ is dummy variable of

integration. Therefore, by using Cauchy–Schwarz inequality, one obtains:

( , ) exp( ) , 0C wξ τ στ τ≤ − > (4.27)

where 1w ≤ . Equation (4.27) implies that the analytic solution of parabolic boundary

value problem (4.20) to (4.23) is asymptotically convergent, i.e. ( , ) 0C ξ τ → as τ → ∞ .

56

4.5.2 Asymptotic Convergence of CHCM

By substituting Eqs. (2.18) to (2.21) in Eq. (4.20) we get:

( )1 1

21 1

1( , ) ( ) ( , ) ( ) ( , ) ( ) ( , ) ( )N N

j j j j j j j jj j

a P a Q a ah

ξ τ ξ ξ τ ξ ξ τ φ ξ ξ τ ψ ξ+ +

= =

′ ′ + = +

∑ ∑ , (4.28)

where 1 2( ) ( ) ( )j j jp h pφ ξ µ ξ µ ξ′′ ′= − , and 1 2( ) ( ) ( )j j jQ h Qψ ξ µ ξ µ ξ′′ ′= − . (4.29)

Since each ( ), ( )j jP x Q x has its support in 1 1[ , ]j jx x− + and in particular:

1( ) 0P u = for 1 2[ , )ξ ξ ξ∉ and 1 1( ) ( ) 0N NP Qξ ξ+ += = for 1( , ]N Nξ ξ ξ +∉ . (4.30)

Therefore, evaluating Eq. (4.28) at collocation points, we get:

1 , 1 1 , 1 , , ,2

1( ) ( , ) ( ) ( , ) ( , ) ( ) ( , ) ( )η ξ η ξ ξ η ξ η− − − −′ ′+ + + = j j i j j j i j j j j i j j j i j iP a t Q a t a t P a t Q Fh

(4.31)

with

, 1 1 , 1 1 , , ,( , ) ( ) ( , ) ( ) ( , ) ( ) ( , ) ( ) .ξ τ φ η ξ τ ψ η ξ τ φ η ξ τ ψ η− − − −′ ′= + + +j i j j j i j j j i j j j i j j j iF a a a a

Therefore, Eq. (4.31) can put in the matrix form by using (2.22-2.24) as:

41Aa Bah

= −

(4.32)

where

( )1 1 2 2 3 3 1 1( , ), ( , ), ( , ), ( , ), ( , ), ( , ), ... , ( , ), ( , ), ( , ), ( , ) ,TN N N Na a a a a a a a a a aξ τ ξ τ ξ τ ξ τ ξ τ ξ τ ξ τ ξ τ ξ τ ξ τ+ +′ ′ ′ ′ ′=

( )1 1 2 2 3 3 1 1( , ), ( , ), ( , ), ( , ), ( , ), ( , ), ..., ( , ), ( , ), ( , ), ( , ) TN N N Na a a a a a a a a a aξ τ ξ τ ξ τ ξ τ ξ τ ξ τ ξ τ ξ τ ξ τ ξ τ+ +′ ′ ′ ′ ′=

are dimensional vectors and matrices A, B given below, represent the coefficients of

first and second order derivatives of trial function (cubic Hermite interpolation

polynomial) at collocation points respectively. For N elements, a system of (2 2)N +

equations is obtained from Eqs. (4.20) to (4.22). Of these, two unknowns are found by

using boundary conditions and rest 2N from the differential equation.

57

and

2 3 2 31 2 1 2 1 2 2 1

2 3 2 31 2 1 2 1 2 2 1

2 3 2 31 2 1 2 1 2 2 1

1

0 0 ... 0

0 0 ... 0

0 0 ... 0

0 0

P h P hQ h Q P h P h Q hQ



PB

β β β β γ γ γ γ

γ γ γ γ β β β β


γ

µ µ µ µ µ µ µ µ



µ

′′ ′ ′′ ′ ′′ ′ ′ ′′− − + −

′′ ′ ′′ ′ ′′ ′ ′ ′′− + + −

′′ ′ ′′ ′ ′′ ′ ′ ′′− − + −

=

2 3 2 32 1 2 1 2 2 1

2 3 2 31 2 1 2 1 2 2 1

... 0... ... ... ... ... ... ... ...... ... ... ... ... ... ... ...... ... ... ... ... ... ... ...

0 0 0 ...

0 0 0 ...

h P hQ h Q P h P h Q hQ


γ γ γ β β β β


µ µ µ µ µ µ µ


′′ ′ ′′ ′ ′′ ′ ′ ′′− + + −

′′ ′ ′′ ′ ′′ ′ ′ ′′− − + −2 3 2 3

1 2 1 2 1 2 2 1 2 2NP h P hQ h Q P h P h Q hQγ γ γ γ β β β βµ µ µ µ µ µ µ µ

+

′′ ′ ′′ ′ ′′ ′ ′ ′′− + + −

0 0 ... 00 0 ... 0

0 0 ... 00 0 ... 0... ... ... ... ... ... ... ...... ... ... ... ... ... ... ...... ... ... ... ... ... ... ...0 0 0 ...0 0 0 ...

P hQ P hQP hQ P hQ

P hQ P hQP hQ P hQ

A

P hQ P hQP hQ P hQ

β β γ γ

γ γ β β

β β γ γ

γ γ β β

β β γ γ

γ γ β β

− −

− − =

− − 2 2N+

58

Therefore from Eq. (4.32) we have:

( ) ( )R a Baτ τ= −

,

Thus ( ) ( ), 0S a I aτ τ τ= − >

, (4.33)

with initial condition 3(0)a κ= , where 1S RB−= and 4R h A= are nonsingular

matrices.

For arbitrary constants 1k and 2k , we have by the inequality (4.24):

2 21 ( , )k a S a a S a≤ ≤

, (4.34)

and

2 22 ( , )≤ − ≤ − k a Ia a I a , (4.35)

where S and I are the spectral norms of S and I , respectively.

Equation (4.33) can be written as:

( , ) ( , )S a a I a a= −

.

Now 1( , ) ( , )2

dS a a S a adτ

=

,

using Eqs. (4.34) and (4.35), one obtains:

22( , ) ( , )kd S a a S a ad Sτ

−≤

.

Therefore by using Eq. (4.34), we get:

2 23

2( , ) exp kS a a SSτκ

−≤

.

(4.36)

Using the property of Euclidean norm, we have:

23

2expikaSτκ

−≤

. (4.37)

59

This implies that solution is convergent and converges to zero as τ → ∞ , now using the

definition given by Miller, O’Riordan and Shishkin (1996), the local error will be:

224 exp ka ASδ δτ

− ≤ −

. (4.38)

Using 1S RB−= 4 1h AB−= and the definitions on exponential function Eq. (4.38)

can be written as:

4a A p hδ δ∗− ≤ , where p∗ is constant. (4.39)

Hence, it is clear that in case of CHCM, the asymptotic convergence is of order 4( )O h .

Similarly, when the Legendre roots are replaced with Chebyshevs roots in the above

process the order of convergence changes and is of 2( )O h .

4.6 Verification of Convergence In order to check the theoretical analysis of the present method, two test problems are

chosen for which exact or numerical solutions are available.

Problem 4.1

Consider a convection–diffusion problem involving adsorption–desorption, diffusion–

dispersion phenomenon in porous media e.g., washing of pulp fibers, coal etc.

2

2

14

C C CPeτ ξ ξ

∂ ∂ ∂= −

∂ ∂ ∂, in (0,1)Ω∈ , (4.40)

with boundary conditions:

1 04

CCPe ξ

∂− =

∂ at 0ξ = and 0C

ξ∂

=∂

at 1ξ = , (4.41)

and initial condition: ( , 0) 1C ξ = . (4.42)

After discretization, a system of differential algebraic equations (DAEs) identical to Eq.

(4.32) is obtained, which is solved using MATLAB ODE15s system solver.

60

Exact solution (exit concentration) for small values of Pe given by Brenner (1962) is:

2

2 21

sin(2 )exp (2 ) expk k ke

k k

C PePe Pe Pe

λ λ λ ττλ

∞

=

= − − + +

∑ ,

and for large Pe, the asymptotic expression is:

2

2 2

1 (1 ) 41 (1 ) 3 2 (1 ) exp2

1 2 (3 4 ) 4 (1 ) exp(4 ) (1 ) .2

ePe Pe PeC erfc Pe

PePe Pe Pe erfc

τ ττ ττ τ π

τ τ ττ

− = − − − + + −

+ + + + + +

Results using CHCM are compared with the exact ones in Table 4.5 for Chebyshev

roots and Table 4.6 for Legendre roots at 40Pe = and 80 respectively. It is observed that

as the numbers of elements are increased the CHCM results are approaching the exact

ones. In Table 4.7 the order of convergence of CHCM for small as well as large values

of the parameter (Peclet number) is found to be of order 2 for Chebyshev roots and

order 4 for Legendre roots respectively.

Problem 4.2

A commonly used variant of axial dispersion model in chemical engineering, involving

two parameters namely Peclet number (Pe) and Biot number (Bi) is discussed below:

2

2

1C C C Bi CPe Pe

µτ ξ ξ

∂ ∂ ∂= − −

∂ ∂ ∂ in (0,1)Ω∈ with 1 εµ

ε−

= , (4.43)

with boundary conditions:

0C = at 0ξ = and 0Cξ∂

=∂

at 1ξ = , (4.44)

and initial condition ( , 0) 1C ξ = . (4.45)

After discretizing the system (4.43) to (4.45), a system of DAEs identical to Eq. (4.32)

is obtained, which is solved using MATLAB ODE15s system solver.

61

The exact solution given by Arora, Dhaliwal and Kukreja (2006b) in terms of

complimentary error function is:

1exp 1 exp2 4 4 2 4 4

3 exp( ) .2 2 2 4 4

eBi Pe Pe Pe Pe Pe PeC erfcPe

Pe Pe Pe PePe erfc

µ τ τ τ ττ π τ

τ ττ

= − − − + − − − + + +

In Table 4.8, the results of CHCM for Chebyshev roots and exact results are compared

for three parameters namely , ,Pe Biµ and in Table 4.9 the results of CHCM for

Legendre roots with exact ones are compared. From Table 4.10, the order of

convergence of CHCM is found to be 4 for different values of Peclet number.

4.7 Conclusions

A diffusion dispersion model is solved for two isotherms using cubic Hermite

collocation method. The technique is found to be simple, elegant and stable even for

large number of collocation points. The leaching of the solute from the fibres into bulk

liquor accompanies displacement and the sorption phenomenon continues for some

time. Simulations are carried out using industrial data and it is found that for more

realistic investigation one must prefer nonlinear isotherm over the linear ones. Also the

asymptotic convergence of cubic Hermite collocation method of order O(h4) was

reported theoretically. The solution profiles are converging to zero, as time is

approaching to infinity and hence, the method is asymptotically stable. The fourth order

convergence of CHCM for parabolic PDEs is established by taking Legendre roots and

second order of convergence by taking Chebshev roots.

62

Table 4.5: Comparison of numerical and exact solution of Problem 4.1 for Chebyshev roots

τ Exact CHCM for 40Pe =

Exact CHCM for 80Pe =

40N = 80N = 160N = 100N = 200N = 400N =

0.0 1.000E+00 1.000E+00 1.000E+00 1.000E+00 1.000E+00 1.000E+00 1.000E+00 1.000E+00

0.2 1.000E+00 1.000E+00 1.000E+00 1.000E+00 1.000E+00 1.000E+00 1.000E+00 1.000E+00

0.4 1.000E+00 1.000E+00 1.000E+00 1.000E+00 1.000E+00 1.000E+00 1.000E+00 1.000E+00

0.6 1.000E+00 1.000E+00 1.000E+00 1.000E+00 1.000E+00 1.000E+00 1.000E+00 1.000E+00

0.8 9.746E-01 9.757E-01 9.749E-01 9.747E-01 9.974E-01 9.975E-01 9.974E-01 9.974E-01

1.0 4.778E-01 4.852E-01 4.797E-01 4.783E-01 4.843E-01 4.876E-01 4.851E-01 4.845E-01

1.2 4.493E-02 4.701E-02 4.540E-02 4.504E-02 9.282E-03 9.490E-03 9.330E-03 9.294E-03

1.4 9.903E-04 1.040E-03 1.001E-03 9.931E-04 7.615E-06 7.645E-06 7.624E-06 7.618E-06

1.6 7.900E-06 7.868E-06 7.897E-06 7.900E-06 7.118E-10 6.547E-10 6.988E-10 7.086E-10

1.8 3.111E-08 2.730E-08 3.029E-08 3.091E-08 1.503E-14 3.293E-15 1.555E-14 1.464E-14

2.0 7.406E-11 5.318E-11 6.927E-11 7.289E-11 -7.717E-20 -6.991E-17 -1.444E-16 -1.426E-16

63

Table 4.6: Comparison of numerical and exact solution of Problem 4.1 for Legendre roots

τ Exact CHCM for 40Pe =

Exact CHCM for 80Pe =

40N = 80N = 160N = 100N = 200N = 400N =

0.0 1.00000E+00 1.00000E+00 1.00000E+00 1.00000E+00 1.00000E+00 1.00000E+00 1.00000E+00 1.00000E+00

0.2 1.00000E+00 1.00000E+00 1.00000E+00 1.00000E+00 1.00000E+00 1.00000E+00 1.00000E+00 1.00000E+00

0.4 1.00000E+00 1.00000E+00 1.00000E+00 1.00000E+00 1.00000E+00 1.00000E+00 1.00000E+00 1.00000E+00

0.6 1.00000E+00 9.99998E-01 9.99998E-01 1.00000E+00 1.00000E+00 1.00000E+00 1.00000E+00 1.00000E+00

0.8 9.74601E-01 9.74216E-01 9.74575E-01 9.74600E-01 9.97412E-01 9.97310E-01 9.97400E-01 9.97401E-01

1.0 4.77800E-01 4.77902E-01 4.77794E-01 4.77800E-01 4.84301E-01 4.84009E-01 4.84280E-01 4.84300E-01

1.2 4.49301E-02 4.49056E-02 4.49145E-02 4.49281E-02 9.28261E-03 9.28058E-03 9.28206E-03 9.28215E-03

1.4 9.90372E-04 9.89062E-04 9.90306E-04 9.90364E-04 7.61540E-06 7.62324E-06 7.61583E-06 7.61535E-06

1.6 7.90029E-06 7.93846E-06 7.90259E-06 7.90028E-06 7.11805E-10 7.18406E-10 7.12188E-10 7.11804E-10

1.8 3.11107E-08 3.19352E-08 3.11537E-08 3.11106E-08 1.50399E-14 1.86514E-14 1.61947E-14 1.60089E-14

2.0 7.40677E-11 7.93789E-11 7.43584E-11 7.40884E-11 -7.71791E-20 -8.07201E-18 -1.03350E-16 -1.46740E-16

64

Table 4.7: Rate of convergence of CHCM for Problem 4.1

Parameter

(Pe)

Number of elements

(N)

Rate of convergence (ρ)

(Chebyshev roots)


(Legendre roots)

40 40 - -

80 1.99 3.91

160 2.00 3.74

80 100 - -

200 2.00 3.89

400 2.00 4.14

65

Table 4.8: Comparison of numerical and exact solution of Problem 4.2 for Chebyshev roots

τ Exact CHCM results for 10, 0.0142, 1.5Pe Biµ= = = Exact

CHCM results for 20, 0.033, 5Pe Biµ= = =

50N = 100N = 200N = 50N = 100N = 200N = 0.0 1.000000E+00 1.000000E+00 1.000000E+00 1.00000E+00 1.000000E+00 1.000000E+00 1.000000E+00 1.000000E+00

0.2 9.994841E-01 9.994841E-01 9.994841E-01 9.994841E-01 9.983500E-01 9.983511E-01 9.983505E-01 9.983500E-01

0.4 9.620362E-01 9.620362E-01 9.620362E-01 9.620362E-01 9.938400E-01 9.938398E-01 9.938408E-01 9.938400E-01

0.6 7.768002E-01 7.768002E-01 7.768002E-01 7.768002E-01 9.083300E-01 9.083928E-01 9.083339E-01 9.083299E-01

0.8 5.272112E-01 5.272113E-01 5.272112E-01 5.272112E-01 6.533600E-01 6.533591E-01 6.533590E-01 6.533599E-01

1.0 3.217942E-01 3.217942E-01 3.217942E-01 3.217942E-01 3.709600E-01 3.709598E-01 3.709597E-01 3.709597E-01

1.2 1.849434E-01 1.849435E-01 1.849434E-01 1.849434E-01 1.784000E-01 1.783994E-01 1.783993E-01 1.783999E-01

1.4 1.026789E-01 1.026790E-01 1.026790E-01 1.026790E-01 7.695600E-02 7.695608E-02 7.695609E-02 7.695609E-02

1.6 5.585578E-02 5.585595E-02 5.585596E-02 5.585596E-02 3.090400E-02 3.090425E-02 3.090428E-02 3.090428E-02

1.8 3.001365E-02 3.001363E-02 3.001363E-02 3.001363E-02 1.182700E-02 1.182720E-02 1.182722E-02 1.182722E-02

2.0 1.600547E-02 1.600523E-02 1.600933E-02 1.600547E-02 4.378600E-03 4.378547E-03 4.378558E-03 4.378658E-03

66

Table 4.9: Comparison of numerical and exact solution of Problem 4.2 for Legendre roots

τ Exact CHCM results for 10, 0.0142, 1.5Pe Biµ= = =

Exact CHCM results for 20, 0.033, 5Pe Biµ= = =

50N = 100N = 200N = 50N = 100N = 200N =

0.0 1.00000E+00 1.00000E+00 1.00000E+00 1.00000E+00 1.00000E+00 1.00000E+00 1.00000E+00 1.000000+00

0.2 9.994843E-01 9.994842E-01 9.994842E-01 9.994842E-01 9.983500E-01 9.983511E-01 9.983505E-01 9.983500E-01

0.4 9.620363E-01 9.620363E-01 9.620363E-01 9.620363E-01 9.938400E-01 9.938399E-01 9.938409E-01 9.938400E-01

0.6 7.768003E-01 7.768002E-01 7.768003E-01 7.768003E-01 9.083300E-01 9.083929E-01 9.083339E-01 9.083300E-01

0.8 5.272113E-01 5.272113E-01 5.272113E-01 5.272113E-01 6.533601E-01 6.533591E-01 6.533591E-01 6.533600E-01

1.0 3.217942E-01 3.217942E-01 3.217942E-01 3.217942E-01 3.709600E-01 3.709599E-01 3.709597E-01 3.709597E-01

1.2 1.849435E-01 1.849435E-01 1.849435E-01 1.849435E-01 1.784000E-01 1.783994E-01 1.783994E-01 1.784000E-01

1.4 1.026790E-01 1.026790E-01 1.026790E-01 1.026790E-01 7.695600E-02 7.695608E-02 7.695610E-02 7.695610E-02

1.6 5.585578E-02 5.585596E-02 5.585596E-02 5.585596E-02 3.090400E-02 3.090426E-02 3.090428E-02 3.090429E-02

1.8 3.001365E-02 3.001364E-02 3.001364E-02 3.001364E-02 1.182700E-02 1.182720E-02 1.182722E-02 1.182724E-02

2.0 1.600547E-02 1.600524E-02 1.600933E-02 1.600547E-02 4.378600E-03 4.378547E-03 4.378558E-03 4.378658E-03

67

Table 4.10: Rate of convergence of CHCM for Problem 4.2

Parameter

(Pe)

Number of elements

(N)


(Chebyshev roots)


(Legendre roots)

10 50 - -

100 2.43 4.03

200 2.40 4.43

20 50 - -

100 2.00 4.01

200 2.43 3.77

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