• MATHEMATICAL MODELING:AN APPLICATION TO CORROSION IN A PETROLEUM INDUSTRY

© National Mathematical Centre Abuja, Nigeria
MATHEMATICAL MODELING:
AN APPLICATION TO CORROSION IN A PETROLEUM
INDUSTRY*
Oyelami, Benjamin Oyediran1
&
Asere, Abraham A2.
1. Mathematical Sciences Programme
2. Petroleum Engineering Programme
Abubakar Tafawa Balewa University
Bauchi – NIGERIA
Abstract
Mathematical modeling is richly endowed with many analytic computational techniques for
analyzing real life situations. Recent reports have confirmed that several billon dollars were lost to
corrosion, in addition to environmental pollution and economic wastage in cleaning up the
environmental mess caused by corrosion. This paper considers application of mathematical modeling
to corrosion problems. It uses the mathematical modeling techniques to forecast the life expectancy
of industrial equipment in the refinery, petroleum reservoirs and gas pipelines= distribution. The
models considered in this direction are the heat-mass transfer equation, Zhim-Hoffman=s equation,
equations arising from electrolysis and finally gas pipeline distribution.
Keywords and Phrases:
Mathematical models, corrosion, phase transition, heat equation, galvanic corrosion.
1. Introduction
Mathematical modeling is as old as mathematics and has extended its tentacles to unforeseeable
directions. Mathematical modeling can best be described as a sandwich between mathematical
theory and applied mathematics. It is an encyclopedia of theories and techniques as applicable to real
life situations.
Corrosion in the modern society is one of the outstanding challenging problems in the industry. Most
industrial design can never be made without taking into consideration the effect of corrosion on the
life span of the equipment. Recent industrial catastrophes have it that many industries have lost
several billion of dollars as a result of corrosion. Reports around the world have confirmed that some
oil companies had their pipeline ruptured due to corrosion, oil spillages are experienced which no
doubt created environmental pollution, in addition, resources are lost in cleaning up this
environmental mess and finally large scale ecological damage resulted from corrosion effects.
 NMC Proc. Workshop on Environment : Oyelami, B.O. & Asere, A.A.
The possibility of occurrence of corrosion in an industrial plant has been posing a lot of concern to
Petroleum, Chemical, Mechanical Engineers and Chemists. It is now known that corrosion can have
some effects on the chemistry of a chosen process, and product of corrosion can affect reaction and
purity of the reaction products.
Furthermore, it is also true that a lot of profit can run out of a hole in any industrial plant if care is
not taken, but early advice from corrosion experts can prevent that from happening. The study of
corrosion is multi-disciplinary in nature. Its calculation involves knowledge of viscosity, specific
heat capacity, thermal conductivity and density of the fluid concerned, in some cases, a thorough
study of the property of the material from which the plant is fabricated is highly required. The
present paper examines the effect of corrosion on equipment in a typical oil industry. The study
elucidates the application of mathematical modeling to corrosion problems in the oil industry.
The justification or the motivation for this study is conceived from the following facts:
1. In the oil refinery, fractional distillations of crude oil into fractions are performed in
perforated trays. There could be contamination of the products by corrosion or impurities
arising from the surface as the plants age with time.
2. Corrosion may attack pipeline of crude and refined oil.
3. Most petroleum products are stored in varieties of containers and oil reservoirs whose
materials may be corrosive because of variation in temperature, heat exchange and pressure
of the petroleum products as affected by environmental factors.
4. Which materials are less corrosive that could be universally adopted for storing and
transporting petroleum products?
5. Corrosive nature of some crude and fractions that have high content of sulphur compounds
called mazcaptan or thiols.
Mathematical modeling offers several powerful intuition appealing tools for studying and analyzing
the chemical kinetics and the thermo chemistry of compounds (e.g., in petroleum crude and
products).
Mathematical modeling also offers quantitative and qualitative techniques for investigating the
material science of the industrial plant for producing, storing and transporting the petroleum
products.
Before we embark on the formulation of the mathematical models, it is pertinent to make a thorough
exposition on mathematical modeling and corrosion itself.
© National Mathematical Centre Abuja, Nigeria
2. What are Mathematical Modelling and its Usefulness?
Mathematical modeling is the act of relating abstract ideas of mathematics to real life problems. The
process involves expressing a real life situation into mathematical terms, manipulating the
mathematics and translating the mathematical results back into the real life.
It is an undisputable fact that every human activity involves one mathematical problem or the other;
the need to use mathematical modelling is increasing released in modern times. It gives us insight
into many real life processes and the interplay between or among variable(s) quantifying such
models. This process saves cost and labour that would unnecessarily have been expended.
Different researchers have expressed various steps taken to model a problem. The most outstanding
one is summarized by Ale (1981; 1986), described the process involved in a modeling a process as
follows:
The identification of the real life problem, which involves modifying and simplifying the original
problem into a reasonable precise and succinct manner.
To have a full grasp of the idea of modeling, the following flow chart, and fig. 1 states the steps to
be taken when modeling a problem. In
Real life
situation
Formulation of
model
Solving the model
Empiric verification
and production
Fig.1: Idealisation of real life problem into a model
 NMC Proc. Workshop on Environment : Oyelami, B.O. & Asere, A.A.
In real life, there is the problem whose solution is sought. This problem need to be identified, in
which case, the significant features are identified and translated into mathematical entities, leading
to the mathematical model. Once a model is constructed, it needs validation, that is:
1. The mathematical structure it represents is self- consistent (i.e., it contains no contradictory
statements) and obeys all usual mathematical laws underlying it;
2. It represents the situation it is actually designed for.
Various branches of mathematics have been created in an attempt to solve some problems or
the other. One may want to predict the weather, estimate the lost caused by corrosion and so on. If
one needs to analyze a problem, it is often a good idea to start by building up a model.
A model is nothing fanciful, it is simply the "bare bones" of the problem - what it looks like after
stripping away the unimportant details. The reduced version of the original problem is what model
represents. The importance of a model is not far fetched for:
- A model is more reliable than pure intuition.
- Mathematically, a model simplifies the analysis.
- A good model is economical (Morton, 1980). That is, it can be labour - saving devices in
more than one way.
A model used for one purpose can also be used for an entirely different purpose.
3. Why Do We Need to Study Corrosion?
Many petrol chemical plants are large-scale equipment, which could be corrosive after some time.
Mathematical model to determine the amount of contamination arising from corrosion will have to
investigate the deleterious effect of the corrosion on the process of the reaction on the product
quality.
The classic example of intergranular corrosion in chemical plant is the weld delay of unstabilized
stainless steel. This is caused by the precipitation of chromium carbides at the grain boundaries in a
zone adjacent to the weld, where the temperature has been between 5000C - 8000C during welding.
Corrosion rate and the form of attack can change if the material is under stress. For some
combination of metal, corrosive media and temperature, the phenomenon called stress cracking can
occur. This leads to premature brittle failure of the metal that constitutes the petrol chemical plant.
The conditions that cause corrosion can arise in a variety of ways. For the brief discussion on the
selection of material, it is convenient to classify corrosion into the following categories:
© National Mathematical Centre Abuja, Nigeria
1. General wastage of material - uniform corrosion;
2. Galvanic corrosion - dissimilar metals in contact;
3. Pitting - localized attack;
4. Intergranular corrosion;
5. Stress corrosion;
6. Corrosion fatigue;
7. Erosion - corrosion;
8. High temperature oxidation;
9. Hydrogen embrittlement.
Metallic corrosion is essentially an electrochemical process. Fair components are necessary to set -
up an electrochemical cell:
1. Anode - the corroding electrode;
2. A cathode - the positive, non - corroding electrode;
3. The conducting medium - the electrolyte - the corroding fluid;
4. Completion of the electrical circuit - through the material.
Cathodes areas can arise in many ways:
(i) Dissimilar metals;
(ii) Corrosion products;
(iii) Inclusions in the metal, such as slag;
(iv) Less aerated areas;
(v) Areas of differential concentration;
(iv) Differential strained areas.
Consider the simplest corrosion problem in nature where iron is exposed to the atmospheric oxygen
in the presence of moisture leading to formation of rust, iron (III) oxide as well as iron (III) chloride
respectively. The chemical reaction can be summarized as follows:
A. 4Fe(s) + 30 2 (g) → 2 Fe2 03 (s) Oxidation State of Fe is 0 6 +3
B. 2 Fe0 (s) + 3 Cl 0 2 (g) → 2 Fe+3 Cl -13 (s)
Reducing Oxidizing
Agent Agent
The processes A & B illustrates the importance of redox reaction, i.e. oxidation and reduction
processes. In these processes, iron is an oxidizing agent since it gains 3 electrons and chlorine is a
 NMC Proc. Workshop on Environment : Oyelami, B.O. & Asere, A.A.
reducing agent since it loses 1 election.
From the above example and some other ones found in the literature, we note that corrosion process
involves molecular and electronic charge exchanges in such a way that:
* There is energy depletion as a result of redox reactions
* Synthesis of compounds formed as a result of oxidation and some compounds lost as result
of reduction.
* The chemistry of corrosion involves complex interactions of compounds in form of chemical
reaction activation process.
* Some basic properties of the original material before corrosion takes place such as
malleability, luster, conductivity and ductility etc is lost may be after corrosion.
* Knowledge of viscosity, specific heat capacity, thermal conductivity and density of the fluid
concerned and even material science of the plant, where the corrosive media is kept, need be
paid special attention.
* The prediction of the life expectancy of industrial plant using equation derived need be paid
attention.
4. The Mathematical Models
In corrosion testing, the corrosion rate is measured by the reduction in weight of a specimen of
known area over a fixed period of time. This is expressed by the formula
12w
ipy =
tAP
Where
w = mass loss in time t/kg
t = time, years
p = density of material, kg/m3
A = surface area, m2
In SI units, ipy = 25mm per year.
For material cost, cost-rating equation is given by
CX p
Cost rating =
σd
Where
CXp = Cost per unit mass, $/kg
P = density, kg/m3
σd = design stress, N/mm2.
British standard on corrosion (BS18$, BS1501) resistant materials made the following classification:
© National Mathematical Centre Abuja, Nigeria
Table 1: Acceptable corrosion rates
ipy mm/y
Completely satisfactory < 0.01 0.25
Use with caution < 0.03 0.75
Use only for short exposure 0.06 1.5
We start our discussion with corrosion accompanied by mass - heat transfer:
4.1 Mass - Heat Transfer Model
Heat transfer in most industrial plants often accompanies corrosion; hence, we consider the
following heat transfer flow in the following diagram:
Cold fluid
Hot fluid
t1
t2
t3
t4
Hot film Cold film
Fig.2: Temperature distribution for heat transfer across a metallic wall
The above figure shows the mechanism of heat transfer across a metallic wall. Three separate
resistances are involved. The first is through a film of fluid, liquid or gas, adjacent to the metal walls
of the vessel or tube. The other two resistances are (a) The wall of the metal whose resistance is
function of the thermal conductivity of the metal as well as its thickness, (b) A firm that forms a thin
 NMC Proc. Workshop on Environment : Oyelami, B.O. & Asere, A.A.
boundary layer just at the surface of the metal on the other side before the moisture.
The basic heat transfer model is heat lost per unit area = U x (total temperature difference (Berry
(1990))
Heat conducted through a material between temperature differences T1 - T0 across the surface is
k
Q= (T1 -T0 ) (Conductivity)
a
Q is heat lost per unit area; k is thermal conductivity of the material; a is the thickness of the
material.
The heat lost by convection using linear model
Q = h1 ( T I - T 0 )
Q = h2 ( T 2 - T 1 )
h1 and h2 are constants called convection heat transfer coefficients.
Eliminating T1 and T2 between these three equations, we have
−1
1 a 1 
Q =  + +  (T1 − T0 )
 h1 k h2 
In general, heat loss across the surface in fig 1, using a simple model is
−1
 1 2k 1 1 
Q= + + +  (T1 − T0 )
 h1 a h1 h2 
A more general expression for Q involving the viscosity of the fluid into the model is found in the
literature and is
−1
 1 2k 1 1 
Qa =  + + + + ξ ( µ ) (T1 − T0 )
 h1 a h1 h2 
ξ (µ) is a function depending on µ, the viscosity.
Of what practical importance is the above heat transfer equation is to corrosion if expressed in terms
of temperature differences? For example, the generation of heat across the surface is closely related
© National Mathematical Centre Abuja, Nigeria
to that of a laser - drilling appliances. This is found in Beddling (1994).
A laser impinges on a material to be drilled (usually a metal) causes vaporization of the material and
result in a moving boundary as a hole formed.
Beddling emphasized that various approximations and simplifications may be made, including that
regarding the hole are one - dimensional.
The governing heat equation for the laser - drilling equation is
∂ U ∂U ∂U
2
+ + =0
∂ ε 2 ∂ε ∂T
Where
U( ε ,T) is (dimensionless) temperature T and ε dimensionless is boundary position
respectively.
The boundary conditions are
u(ε,0) = 0, ε>0
u(0,T) = 1, T$ 0
u(4,T) =0, T$0
ξ is actually Z - T where Z is the momentary position of the boundary. The solution to the model is
found by Bedding as
1 z (2 - 2T)
u(Z,T) = { erfc ( ) + exp (-Z - T) erfc ( )}
2 2 T 2 T
In relation to heat dissipation, Qa
−1
 1 2k 1 1 
Qa =  + + + + ξ ( µ ) [u ( z, T1 ) − u ( z, T2 )]
 h1 a hc h2 
Where
T1 and T2 are two times for heat to pass across the metallic surfaces.
The quantity of heat generated in the plant can be monitor-using computer or a mathematical
 NMC Proc. Workshop on Environment : Oyelami, B.O. & Asere, A.A.
machines given that the temperatures u(z,T1) and u(z,T2) are known.
To calculate the life expectancy of the plant as a result of corrosion, we use the idea of perturbation
theory. Since the thickness of the surface before corrosion takes place, is Q and after corrosion, a1
we assume that the thickness is a. Then
a = a1 + h
h is the part lost into the chemical reaction as a result of the corrosion and may be inform of
impurities.
Let
1 2k 1 1
Γa = + + + +ξ( µ )
h1 a hc h 2
T a = u(t, a),T b = u(t,b)
Then
Γa Q = u(t, a)
Γb Q = T b = u(t,b)
It follows that
- 2hk
( Γa1 - Γ a )Q = 11
Q = u(t, a) - u(t, a1 )
a ( a + h)
Therefore
1 1
a ( a + h)
Q= (u(t, a1 ) - u(t, a))
2hk
This follows that:
α
h=
βQ + γ
Where
α = α 12 , β = 2k, γ = - a1 (u(t, a1 ) - u(t, a))
By dimensional analysis, we found that the life expectancy of the plant is related to h and the
12w
Corrosion rate A = by the relation
tnp
© National Mathematical Centre Abuja, Nigeria
Const
T=
hA
Const = C is a dimensionless constant that can be determine experimentally, hence
tAPC
T= ( βα + γ ) (*)
12αw
Acceptable working condition of the plant containing carbon and low alloy steel
Time (year) Life expectancy
Completely satisfactory < 0.01 ( β + γ/α )
Use with caution < 0.03 ( β + γ/a)
Use only for short expensive < 0.06 ( β + γ/α )
Completely unsatisfactory > 0.06 ( β + γ/α )
For high alloy steel, brass and aluminum
Time (year) Life expectancy time
Completely satisfactory < 0.005 ( β + γ/α )C
Use with caution < 0.03 ( β + γ/α )C
Use only for short exposure < 0.03 ( β + γ/α )C
Completely unsatisfactory > 0.03 ( β + γ/α )C
Zhimz-Hoffman (ZH) Model
In this section, we proposed an adapted Zhimz - Hoffman's model that will take care of phase
transition that arises as a result of corrosive fluid on the solid surface of the plant. This model is
particularly useful in cracking process in the refinery; it is worthy of note to mention that corrosion
of the surfaces into globules of impurities may cause catalytic poisoning of a petrol chemical plant.
 NMC Proc. Workshop on Environment : Oyelami, B.O. & Asere, A.A.
The Zhimz - Hoffman's process is a typical non-linear parabolic partial differential equation whose
solution exists in Ω X (0,T). In the absence of phase transition, that is, n = 0, the equation reduces to
the classical heat equation. The solution of Zhimz - Hoffman's model from analytic point of view is
not generally easily obtainable.
The use of computer via numerical approach is highly recommended. This can be done through
applications of the finite difference scheme, collocation method or the finite element method to the
model. The advantage of the ZH method being its accuracy is highly guaranteed to some degree of
freedom.
One striking thing about the ZH model is that it relates the relevant constants such as k, T and L that
are in fact the properties of the surfaces of the material from which the plant is fabricated. These
constants differ from material to material; the importance of the phase transition cannot be
overemphasized.
We state the Zhimz - Hoffman's equation as
∂u 1 ∂φ 2 2
∂ u ∂ u
+ - K( + )= f
∂t 2 ∂t ∂ x2 ∂ y 2
∂φ 2 2
∂ u ∂ u
τ + ε 2 ( 2 + 2 ) = g(u) + 2u
∂t ∂x ∂y
u(x,0) = u0(x), n(x,0) = n0(x) as Ω x {0}
Where
g(n) = a(x)n + b(x)n2 - c(x)n3; c0, k, l, τ, ε2
Are prescribed positive constants, u is the temperature the phase transition, K is conductivity, and l
is the latent heat released or absorbed during phase transition ε measures sharpness of the free
boundary and τ the surface tension.
The usefulness of the above model can be buttressed by the fact that, most heat generation problem
in a corrosive media, it is often accompanied by phase changes. The problem can thus be modeled
by well-known phase field model, which describes the phase transition between two different
phases, e.g. solid and liquid. The Zhimz and Hoffman's adequately take care of this.
Using experimented data, i.e., computation of the temperature from Zhimz - Hoffman's using the
computer, the time expectancy of the industrial plant can be computed, we utilize the formula in eqn.
(*) for the computation. Different materials can be subjected to corrosion testing and isolate which
material has the best expectancy life.
© National Mathematical Centre Abuja, Nigeria
In practice, it has been found that material formed from high concentration of alloyed have greater
life expectancy, although, such materials are expensive.
6. Galvanic Corrosion
Electrolysis has played a number of practical applications e.g. "isoelectric focusing" for separating
large molecules of protein (Cho et al. (1994)) and in the hydrodimetrization of acrylonitrille for the
nylon 6,6 industry (Tutty - Denualt (1994)). In this section, the idea of galvanic corrosion is
discussed, even though, the discussion may not be as cuspicious as possible. This is an area where
corrosion actually plays some useful industrial application.
In galvanic corrosion, dissimilar metals are placed in contact in an electrolyte, the corrosion rate of
the anodic metal increases while that of cathode one decreases. This practice has had useful
industrial galvanization applications (Chol et al, 1994; Babski et al, 1983; Sawille and Palusinski,
1986; Zlvenga and Egocheage, 1989).
Tutty - Denualt (1994) discussed the use of mediators in the galvanization process; we start our
discussion with work of Choi and his associates:
They considered a single - dissociation - association reaction of the form
A ↔ p B z 2 + qC
0 z3
Where
A is the neutrally charged species and B and C are the positively and negatively charged
species, respectively.
Hence, p and q are positive integers, and z1 = 0, z2 >0, z3