J. Nig. Soc. Phys. Sci. 2 (2020) 61–68

Journal of the
Nigerian Society

of Physical
Sciences

Original Research

Mathematical modelling of concentration profiles for species
transport through the single and the interconnected

multiple-compartment systems

O. Adedire∗, J. N. Ndam

Department of Mathematics, University of Jos, Nigeria.

Abstract

In this research work, we investigate the concentration profiles in the single and the interconnected multiple-compartment systems with sieve
partitions for the transport of chemical species with second order chemical reaction kinetics. With assumption of unidirectional transport of
chemical species and constant physical properties with same equilibrium contant, the developed partial differential equations representing the
two systems are spatially discretised using the Method of Lines (MOL) technique and the resulting semi-discrete system of ODEs are solved
using MATLAB ode15s solver. The results show that the interconnected multiple-compartment system has lower concentration profile than the
single-compartment system for different values of diffusivities.

Keywords: Concentration profile, species, single-compartment, interconnected, multiple-compartment

Article History :
Received: 17 March 2020
Received in revised form: 02 May 2020
Accepted for publication: 04 May 2020
Published: 14 May 2020

c©2020 Journal of the Nigerian Society of Physical Sciences. All rights reserved.
Communicated by: T. Latunde

1. Introduction

Chemical reactor engineers have certain desired concentra-
tion profiles for optimum yield of products. While high con-
centration profile is preferable for some chemical species to
yield adequate product, low concentration profile is desired for
some other chemical species. The type of chemical reactor
used is a factor that may determine the concentration profiles
in such systems due to pressure, heat and possible consumption
of the chemical species as a result of the presence of other re-
actions. Different chemical species have different diffusivities
in the medium through which they are being transported. The
medium through which a chemical species is being transported
could be solid, liquid or gas medium.

∗Corresponding author tel. no:
Email address: dharenss@gmail.com (O. Adedire )

In literature, researchers have modelled high and low con-
centration profiles for different diffusion values in different me-
dia. Gaiseanu presented analysis on diffusion of boron in sil-
icon at high surface concentration beginning from some diffu-
sion constants and his conclusion is centred on the dependence
of diffusion coefficient on concentration for some given param-
eters whose details can be found in his work [1]. Chepurniy and
Savage also investigated the effect of diffusion on concentration
profile in a solar pond and concluded that constant value of dif-
fusivity existed in the medium considered [2]. While investigat-
ing probe effects on concentration profiles in the diffusion layer,
Critelli et al. [3] solved a pure diffusion problem for a one-
component two dimensional axisymmetric model. They con-
cluded that positioning of a probe like a microelectrode close to
a working electrode interferes with local potential and concen-
tration distributions.

61



O. Adedire & J. N. Ndam / J. Nig. Soc. Phys. Sci. 2 (2020) 61–68 62

Yang and Lue examined coupled concentration-dependent
diffusivities of ethanol and water mixture through a polymeric
membrane. While determining effects on pervaporative flux
and diffusivities profiles, they concluded that the concentration
profiles exhibited a linear to concave response as a function of
depth in the considered membranous medium [4]. There have
been various works on concentration profiles and diffusivities
of chemical species in different media [5-11].

Zhou et al. [12] generated complex concentration profile
by partial diffusive mixing in multi-stream laminar flow. One
of the key insights in their work is that details in the concen-
tration profile can be tuned with geometrical and operating pa-
rameters. Adedire and Ndam investigated model of chlorine
decay through water and intermediate Pseudomonas aeruginosa
in multiple-compartment isothermal system [13]; they however
did not compare their work with single compartment system.
Researchers in literature have not compared the concentration
profiles between the single and the interconnected multiple-
compartment systems for the transport of chemical species with
second order chemical reaction kinetics.

The research question is to determine whether single com-
partment system will have higher concentration profile than in-
terconnected multiple-compartment system for chemical species
with second order chemical reaction kinetics as time progres-
sively increases. Would there be higher concentration profile
for chemical species with higher diffusivities than for those with
lower diffusivities?

In this study, we intend to compare the concentration pro-
files in both the single-compartment system and the intercon-
nected multiple-compartment system for the transport of chem-
ical species with second order chemical reaction kinetics with
respect to diffusivities. The interconnected multiple-compartment
system is such that each compartment is separated by sieve
partitions allowing transport of chemical species through each
compartment of the system.

The subsequent part of this paper is organised as follows:
section 2 deals with model development. Section 3 deals with
numerical simulation of the model equations. While in section
4, results and discussion from the perspective of the considered
systems will be addressed, conclusion comes up in section 5.

2. Model Development

The development of the model in this work is centred on
treating fluid as a continuum [14]. Let any chemical species be
a substance distributed in Rn with boundary ∂Ω and let n̂ be an
outward normal to ∂Ω and define a function

Ψϕ(x, t) : Rn × R → R (1)

where Ψϕ(x, t) is the concentration of any chemical species ϕ in
some units of measurements having x ∈ Rn with x = x1, x2, ...xn
at any time t. Let q ∈ N be total number of compartments and
γ ∈ {1, 2, 3, 4, 5} be compartment 1, 2, 3, 4 and 5 respectively.
For q = 1,γ = 1, n = 1, x1 = x we set up mass balance equa-
tions assuming that convection and diffusion are taking place
along the x-axis as shown in Figure 1. For chemical species ϕ

with flux qϕ and reaction term Rϕ over infinitesimal thickness
∆x in x -direction of γth -compartment of the reactor, the mass
balance gives

(∆x) Ψϕ(x, t)
∣∣∣
t
− (∆x)Ψϕ(x, t)

∣∣∣
t+∆t

= vΨϕ(x, t)
∣∣∣
x
∆t − v Ψϕ(x, t)

∣∣∣
x+∆x

∆t

+ qh1 x
∣∣∣
x
∆t − qh1 x

∣∣∣
x+∆x

∆t + Rϕ∆x∆t

(2)

Division of (2) by (∆x)(∆t) gives

Ψϕ(x,t)|t−Ψϕ(x,t)|t+∆t
∆t = −v

Ψϕ(x,t)|x−Ψϕ(x,t)|x+∆x
∆x

−
qh1 |x−qh1 |x+∆x

∆x + Rϕ

(3)

Take the limit of (3) as ∆x → 0, ∆t → 0 to get

∂Ψϕ(x, t)
∂t

= −v1
∂Ψϕ

∂x
−
∂qϕ
∂x

+ Rϕ (4)

Assuming that the rate of consumption of the chemical species
follows second order chemical reaction kinetics and choosing
the flux qϕ to follow Fick’s law of diffusion [15], we have

∂Ψϕ(x, t)
∂t

= −v
∂Ψϕ(x, t)

∂x
+
∂

∂x

[
D
∂[Ψϕ(x, t)

∂x

]
,

−kϕ
[
Ψϕ(x, t)

]2
xL0 6 x 6 xµ, t > 0 (5)

with initial and boundary conditions given as

Ψflϕ(x, 0) = Ψflϕ0(x) (6)

Ψflϕ(x = xL0, t) = Ψflϕ(t) (7)

Ψflϕ(x = xµ, t) = ΨflϕE (x = xµ, t) (8)

Following the same procedure used to obtain equations (5), (6),
(7) and (8), we consider the following Figure 2 . For the pa-
rameters q = 5; γ = 1, 2, 3, 4, 5; n = 1, x1 = x, we obtain the
following system of partial differential equations with assump-
tion that convection and diffusion take place along x -axis as

∂Ψ1ϕ(x, t)
∂t

= −v
∂Ψ1ϕ(x, t)

∂x
+
∂

∂x

[
D1

∂Ψ1ϕ(x, t)
∂x

]
−kϕ

[
Ψ1ϕ(x, t)

]2
, xL 6 x 6 x1, t > 0 (9)

∂Ψ2ϕ(x, t)
∂t

= −v
∂Ψ2ϕ(x, t)

∂x
+
∂

∂x

[
D2

∂Ψ2ϕ(x, t)
∂x

]
−kϕ

[
Ψ2ϕ(x, t)

]2
, x1 ≤ x ≤ x2, t ≥ 0 (10)

∂Ψ3ϕ(x, t)
∂t

= −v
∂Ψ3ϕ(x, t)

∂x
+
∂

∂x

[
D3

∂Ψ3ϕ(x, t)
∂x

]
−kϕ

[
Ψ3ϕ(x, t)

]2
, x2 ≤ x ≤ x3, t ≥ 0 (11)

∂Ψ4ϕ(x, t)
∂t

= −v
∂Ψ4ϕ(x, t)

∂x
+
∂

∂x

[
D4

∂Ψ4ϕ(x, t)
∂x

]
62



O. Adedire & J. N. Ndam / J. Nig. Soc. Phys. Sci. 2 (2020) 61–68 63

Figure 1: Schematic representation of species transport through (q=1) single-compartment chemical reactor system.

Figure 2: Schematic representation of species transport through interconnected (q=5)- multiple-compartment chemical reactor system.

−kϕ
[
Ψ4ϕ(x, t)

]2
, x3 ≤ x ≤ x4, t ≥ 0 (12)

∂Ψ5ϕ(x, t)
∂t

= −v
∂Ψ5ϕ(x, t)

∂x
+
∂

∂x

[
D5

∂Ψ5ϕ(x, t)
∂x

]
−kϕ

[
Ψ5ϕ(x, t)

]2
, x4 6 x 6 xµ, t > 0 (13)

Equation (9) has initial and boundary conditions

Ψ1ϕ(x, 0) = Ψ1ϕ0(x) (14)

Ψ1ϕ(x = xL0, t) = Ψ1ϕ(t) (15)

Ψ1ϕ(x = x1, t) = k[Ψ2ϕ(x = x1, t)] (16)

Equations (10), (11) and (12) for γ = 2, 3 and 4 have the initial
and boundary conditions

Ψγϕ(x, 0) = Ψγϕ0(x) (17)

∂Ψflϕ(x = xγ, t)
∂x

=
D(γ−1)ϕ

Dγϕ

∂Ψ(γ−1)ϕ(x = x(γ−1), t)
∂x

(18)

Ψγϕ(x = xγ, t) = k
[
Ψ(γ+1)ϕ(x = xγ, t)

]
(19)

Equation (13) for γ = 5 has the initial and boundary conditions
given as

Ψ5ϕ(x, 0) = Ψ5ϕ0(x) (20)

∂Ψ5ϕ(x = x4, t)
∂x

=
D4ϕ
D5ϕ

∂Ψ4ϕ(x = x4, t)
∂x

(21)

Ψ5ϕ(x = x5, t) = Ψ5ϕ0(t) (22)

While the Neumann boundary conditions (18) and (21) show
continuity of flux of species ϕ from γth compartment to the ad-
jacent (γ + 1)th compartment, the Dirichlet boundary conditions
(16), (19) and (22) show the link between the concentrations in
one compartment with the concentration in the preceding com-
partment in a way showing relationship at the boundaries.

2.1. Existence and Uniqueness of Solution
The solutions of the governing equations (5)-(8)

and (9)-(22) which are PDEs and their auxiliary conditions ex-
ist and are unique; hence they are well–posed system of PDEs
[16]. Detailed proofs of the existence and uniqueness of the
governing model equations leading to their well-posedness will
not be considered here to avoid repetition as they have been ex-
tensively proved elsewhere. For further analysis and detailed
proofs, the reader is referred to [16] and to [17] and references
contained therein.

3. Numerical Simulation

The governing model equation (5) together with its aux-
iliary conditions (6)-(8) for (q=1) single-compartment system

63



O. Adedire & J. N. Ndam / J. Nig. Soc. Phys. Sci. 2 (2020) 61–68 64

and the system of equations (9)-(13) together with their auxil-
iary conditions (14)-(22) for interconnected multiple-compartment
system are solved using Method of Lines (MOL) technique. We
performed numerical simulation after only the spatial deriva-
tives of the governing PDEs have been discretised with finite
difference approximations. To this end, let x = xi such that
i ∈ N be an index representing positions on grids in x. We con-
sider first order approximations to ∂Ψ

∂x for the convective part of
the PDEs as

∂Ψ(x, t)
∂x

≈
Ψi − Ψi−1

∆x
+ O(∆x) (23)

Also, for the diffusive part, second order finite difference ap-
proximations to the second order derivative ∂

2 Ψ
∂x2 are considered

as

∂2Ψ(x, t)
∂x2

≈
Ψi+1 − 2Ψi + Ψi−1

∆x2
+ O(∆x2) (24)

The terms O(∆x) and O(∆x2) are the truncation error of approx-
imation of finite difference scheme obtained from Taylor series.
The number of points on the grids in x is chosen as M such that
at the first boundary (left-end) of x a value of i=1 is assigned
and at the last boundary (right-end) of x a value of i=M is cho-
sen.

For q = 1,γ = 1 , n = 1 substitution of (23) and (24) into
(5) results in

dΨ(t)(ϕ) i
dt = −v

Ψ(t)(ϕ) i−Ψ(t)(ϕ) i−1
∆x + D

[
Ψ(t)(ϕ) i+1−2Ψ(t)(ϕ) i +Ψ(t)(ϕ) i−1

∆x2

]
−kϕ

[
Ψ(t)(ϕ) i

]2
, 1 6 i 6 M

(25)

and (6), (7) and (8) are semi-discretised as

Ψ(t = 0)(ϕ) i = Ψϕ0(x(i)) (26)

Ψ(t)(ϕ) 1 = Ψϕ(t) (27)

Ψ(t)(ϕ) M = Ψ(t)(ϕ)0 M (28)

For q = 5; γ = 1, 2, 3, 4, 5 , substitution of (23) and (24) into
(9), (10), (11), (12) and (13) results in

dΨ(t)(1ϕ) i
dt

= −v1
Ψ(t)(1ϕ) i − Ψ(t)(1ϕ) i−1

∆x

+D1

[
Ψ(t)(1ϕ) i+1 − 2Ψ(t)(1ϕ) i + Ψ(t)(1ϕ) i−1

∆x2

]
−k1ϕ

[
Ψ(t)(1ϕ) i

]2
, 1 6 i 6 M(1) (29)

dΨ(t)(2ϕ) i
dt

= −v2
Ψ(t)(2ϕ) i − Ψ(t)(2ϕ) i−1

∆x

+D2

[
Ψ(t)(2ϕ) i+1 − 2Ψ(t)(2ϕ) i + Ψ(t)(2ϕ) i−1

∆x2

]
−k2ϕ

[
Ψ(t)(2ϕ) i

]2
, 1 6 i 6 M(2) (30)

dΨ(t)(3ϕ) i
dt

= −v3
Ψ(t)(3ϕ) i − Ψ(t)(3ϕ) i−1

∆x

+D3

[
Ψ(t)(3ϕ) i+1 − 2Ψ(t)(3ϕ) i + Ψ(t)(3ϕ) i−1

∆x2

]
−k3ϕ

[
Ψ(t)(3ϕ) i

]2
, 1 6 i 6 M(3) (31)

dΨ(t)(4ϕ) i
dt

= −v4
Ψ(t)(4ϕ) i − Ψ(t)(4ϕ) i−1

∆x

+D4

[
Ψ(t)(4ϕ) i+1 − 2Ψ(t)(4ϕ) i + Ψ(t)(4ϕ) i−1

∆x2

]
−k4ϕ

[
Ψ(t)(4ϕ) i

]2
, 1 6 i 6 M(4) (32)

dΨ(t)(5ϕ) i
dt

= −v5
Ψ(t)(5ϕ) i − Ψ(t)(5ϕ) i−1

∆x

+D5

[
Ψ(t)(5ϕ) i+1 − 2Ψ(t)(5ϕ) i + Ψ(t)(5ϕ) i−1

∆x2

]
−k5ϕ

[
Ψ(t)(5ϕ) i

]2
, 1 6 i 6 M(5) (33)

where M(1), M(2), M(3), M(4), M(5) are the number of points on
the grid in x in each compartment γ = 1, 2, 3, 4, 5. The initial
and boundary conditions (14),(15) and (16) are semi-discretised
as

Ψ(t = 0)(1ϕ) i = Ψ1ϕ0(x(i)) (34)

Ψ(t)(1ϕ) 1 = Ψ1ϕ(t) (35)

Ψ(t)(1ϕ) M(1) = kΨ(t)(2ϕ) M(1) (36)

and equations (17), (18) and (19) are semi-discretised for (30),
(31) and (32) as

Ψ(t = 0)(γϕ) i = Ψγϕ0(x(i)) (37)

Ψ(t)(γϕ) M(γ−1) − Ψ(t)(γϕ) M(γ−1)−1
∆x

=
D(γ−1)ϕ

Dγϕ

Ψ(t)[(γ−1)ϕ] M(γ−1) − Ψ(t)[(γ−1)ϕ] M(γ−1)−1
∆x

(38)

Ψ(t)(γϕ) M(γ) = kΨ(t)[(γ+1)ϕ] M(γ) (39)

Finally, equations (20),(21) and (22) are semi-discretised as

Ψ(t = 0)(5ϕ) i = Ψ5ϕ0(x(i)) (40)

Ψ(t)(5ϕ) M(4) − Ψ(t)(5ϕ) M(4)−1
∆x

=
D(4)ϕ
D5ϕ

Ψ(t)4ϕ M(4) − Ψ(t)4ϕ M(4)−1
∆x

(41)

Ψ(t)(5ϕ) M(5) = kΨ(t)5ϕ0 M(5) (42)

64



O. Adedire & J. N. Ndam / J. Nig. Soc. Phys. Sci. 2 (2020) 61–68 65

Figure 3: Species concentration profile for diffusivity 5.4 × 101 m2/min γ = 1 (q=1) single-compartment system at time t = 9 minutes.

Figure 4: Species concentration profile for diffusivity 5.4×101 m2/min γ = 1, 2, 3, 4, 5 of interconnected (q=5) multiple-compartment system at time t = 9 minutes.

4. Results and Discussion

For parameters q = 1,γ = 1, n = 1 , the total length of the
system is 600m with a constant boundary value ψϕ0 = 70mg/L.
The second order reaction rate constant kϕ is 1.2 × 101 L mg
min−1, the diffusivities Dϕ are taken to be 5.4×101 m2/min and
5.4 × 102 m2/min respectively with initial concentration of the

chemical species taken as ψAϕ0 = 70mg/L.
For parameters q = 5; γ = 1, 2, 3, 4, 5; n = 1 , the length of

each compartment is 120m so that the total length gives 600m.
A constant boundary value ψ1ϕ0 = 70mg/L is used, second or-
der reaction rate constants kiϕ(i = 1, 2, 3, 4, 5) are taken to be
1.2 × 101 L mg min−1 with diffusivities Diϕ(i = 1, 2, 3, 4, 5) also

65



O. Adedire & J. N. Ndam / J. Nig. Soc. Phys. Sci. 2 (2020) 61–68 66

Figure 5: Species concentration profile for diffusivity 5.4 × 102 m2/min γ = 1 of (q = 1) single-compartment system at time t = 9 minutes.

Figure 6: Species concentration profile for diffusivity 5.4 × 102 m2/min γ = 1, 2, 3, 4, 5 of interconnected (q = 5) multiple-compartment system at time t = 9
minutes.

taken to be 5.4×101 m2/min and 5.4×102 m2/min respectively
with the initial concentration of the species in the system taken
to be 70mg/L .

After spatial discretisation of the resulting PDEs and their
auxiliary conditions using MOL techniques as shown in section

3 with grid points M(1), M(2), M(3), M(4), M(5) taken to be 101
in each compartment, we obtained 505 semi-discrete systems
of ODEs in one independent variable t for (q=5)-compartment
system and 101 semi-discrete systems for (q=1)-compartment
system. We use MATLAB ode15s solver due to its efficient way

66



O. Adedire & J. N. Ndam / J. Nig. Soc. Phys. Sci. 2 (2020) 61–68 67

of handling such systems of ODEs with variable step size, the
results are displayed for t = 9 minutes as shown in Figures 3
and 4.

Analogously, the same set of parameters used to obtain Fig-
ures 3 and 4 but with different diffusivities Dϕ and Diϕ(i =
1, 2, 3, 4, 5) taken to be 5.4 × 102 m2/min are used and the re-
sults are displayed in Figures 5 and 6 at t = 9 minutes for (q=1)
single-compartment system and interconnected (q=5) multiple-
compartment system respectively.

Simulation of the governing model equations for different
values of diffusivities are used to investigate effects of diffusiv-
ity on the concentration profile of chemical species undergo-
ing second order chemical reaction kinetics in each compart-
ment of interconnected (q=5)- multiple-compartment system.
The results from Figure 4 show that for diffusivities Diϕ(i =
1, 2, 3, 4, 5) given as 5.4×101 m2/min at t = 9 minutes, the val-
ues of the concentration at the end of the boundary of each com-
partment of the system has the concentration values of 32.87
mg/L, 18.19 mg/L, 10.49 mg/L, 4.99 mg/L and 0.00 mg/L re-
spectively. On the other hand, for diffusivities Diϕ(i = 1, 2, 3, 4, 5)
given as 5.4 × 102 m2/min measured at the same time t=9 min-
utes, the values of concentration at the end of each compart-
ment are obtained as 49.32 mg/L, 34.43 mg/L, 22.23 mg/L,
11.01 mg/L and 0.00 mg/L for γ = 1, 2, 3, 4, 5 respectively as
shown in Figure 6. The mid-point of the multiple-compartment
system of length 600m is 300m, so that interconnected (q=5)-
compartment system has its mid-point in the third compartment
at the point x = 60m which is equivalent to 300m of the en-
tire system. While concentration of the species at 300m of
the system is 13.90 mg/L for diffusivity of 5.4 × 101 m2/min,
the concentration of the species is 28.13 mg/L for diffusivity of
5.4×102 m2/min at the mid-point of the system at t=9 minutes.

From the results, there is higher concentration profile when
diffusivities Diϕ(i = 1, 2, 3, 4, 5) are given as 5.4 × 102 m2/min
than when diffusivities Diϕ(i = 1, 2, 3, 4, 5) are given as 5.4×101

m2/min at t=9 minutes. This means that chemical species with
higher diffusivities in (q=5)-compartment system have higher
concentration profile than those with lower diffusivities. As
time t progressively increases, the concentration profile of chem-
ical species with higher diffusivity still shows higher values
than the one with lower diffusivity in (q=5)-compartment sys-
tem.

We also made comparison between (q=1)-compartment sys-
tems for different values of diffusivities in order to ascertain
whether the behavior exhibited by (q=5)-compartment is con-
sistent with that of (q=1)-compartment system. From the re-
sults shown in Figure 3 at t=9 minutes, the concentration at
the mid-point of the system is 34.88 mg/L when diffusivity
is taken to be 5.4 × 101 m2/min and 34.99 mg/L when diffu-
sivity is 5.4 × 102 m2/min. From the comparison made be-
tween the (q=1)-compartment system and the interconnected
(q=5)-compartment system for different values of diffusivities,
there is higher concentration profile in chemical species with
higher diffusivity than with the one with lower diffusivity in
each system. However, the concentration profile in intercon-
nected (q=5) multiple-compartment system is lower than that
of (q=1) single-compartment system.

5. Conclusion

In this paper, we compared the concentration profile be-
tween (q=1)- single–compartment and interconnected (q=5)-
multiple-compartment systems for a chemical species with sec-
ond order chemical reaction kinetics.

From the results, the (q=5)- multiple-compartment system
has lower concentration profile than the (q=1)- single-compartment
system for different values of diffusivities. We conclude that
the multiple-compartment chemical reactor system be consid-
ered whenever lower concentration profile is desired for certain
chemical reactions. However, choice of a single-compartment
chemical reactor may be made when high concentration profile
is required for certain chemical reactions.

Acknowledgments

We thank the referees for the positive enlightening com-
ments and suggestions, which have greatly helped us in making
improvements to this paper.

References

[1] F. Gaiseanu, “On the diffusion profile of boron in silicon at high
concentrations”, Journal of physical state solid 77 (1983) 51, doi:
10.1002/pssa.2210770164.

[2] N. Chepurniy & S.B. Savage, “Effect of diffusion on concentration pro-
files in a solar pond”, Journal of solar energy, 17 (1975) 203, doi:
10.1016/0038-092x(75)90061-4.

[3] R. A. J. Critelli, M. Bertotti & R.M. Torresi, “Probe effects on con-
centration profiles in the diffusion layer: Computational modeling and
near-surface pH measurements using microelectrodes”, Journal of elec-
trochemical acta 511 (2018) 203, doi: 10.1016/j.electacta.2018.09.157.

[4] T. H. Yang & S.J. Lue, “Coupled concentration-dependent diffusivities of
ethanol/water mixtures through a polymeric membrane: Effect on perva-
porative flux and diffusivity profiles”, Journal of Membrane Science 443
(2013) 1, doi: 10.1016/j.memsci.2013.05.002.

[5] B. Michl, J. Schön, W. Warta & M.C. Schubert, “The impact of different
diffusion temperature profiles on iron concentrations and carrier lifetimes
in multicrystalline silicon wafers”, IEEE Journal of photovoltaics 71(12)
(2013) 1768, doi: 10.1109/jphotov.2012.2231726.

[6] R. Schulz, K. Yamamoto, A. Klossek, F. Rancan, A. Vogt, C. Schütte, E.
Rühl & R. R. Netz, “Modeling of Drug Diffusion Based on Concentra-
tion Profiles in Healthy and Damaged Human Skin”, Biophysical Journal
117(5) (2019) 998, doi: 10.1016/j.bpj. 2019.07.027.

[7] W. Wang & Y. Lo, “Profile estimation of high concentration Arsenic or
Boron diffusion in Silicon”, Journal of IEEE Transactions on Electron
Devices, 30 (12) (2013) 1828, doi: 10.1109/t-ed.1983.21453.

[8] A. C. Lasaga, “Influence of diffusion coupling on diagenetic concentra-
tion profiles”, American Journal of Science, 281 (1981) 553.

[9] U. Rothhaar & H. Oechsner, “Determination of diffusion induced concen-
tration profiles in Cr203 films on ceramic Al2Oa by Auger sputter depth
profiling”, Fresenius Journal of Analytical Chemistry, 353 (1995) 533,
doi: 10.1007/BF00321316.

[10] A. C. Leute, “The influence of the ideal mixing entropy on concentra-
tion profiles and diffusion paths in ternary systems”, American Journal of
Science, 281 (2010) 553, doi: 10.1016/j.jpcs.2010.08.018.

[11] G. Peev & M. Rouseva, “A model for the concentration profile of PxOy
in the interwafer gas phase on phosphorus doping of silicon using a solid
planar diffusion source”, Journal of modelling simulation material Sci-
ence Engineering, 2 (1994) 1143, doi: 10.1088/0965-0393/2/6/006.

[12] Y. Zhou, Y. Wang, T. Mukherjee & Q. Lin, “Generation of complex
concentration profiles by partial diffusive mixing in multi-stream lami-
nar flow”, Lab Chip, Royal Society of Chemistry, 9 (2009) 1439, doi:
10.1039/b818485b.

67



O. Adedire & J. N. Ndam / J. Nig. Soc. Phys. Sci. 2 (2020) 61–68 68

[13] O. Adedire & J. N. Ndam, “Mathematical modeling of chlorine decay
through water and intermediate pseudomonas aeruginosa in multiple-
compartment isothermal reactor”, Journal of Advanced Mathematical
Models and Applications 4 (2019) 167.

[14] P. A. Thompson, “Compressible-fluid dynamics”, McGraw-Hill, Inc,
New York.

[15] H. J. V. Tyrrell, ”The origin and present status of Fick’s diffusion law”,

Journal of Chemical Education 41(7) (1964) 397.
[16] J. Strikwerda, “Initial boundary value problems for incomplete parabolic

systems”, Journal of Communications on Pure and Applied Mathematics
30 (1977) 797.

[17] H. Kreiss & J. Lorenz, ”Initial-boundary value problems and the Navier-
Stokes Equation”, Academic Press, Inc, San Diego.

68