MINIMIZATION OF WASTE GAS FLOW IN
CHEMICAL INDUSTRIES
Chemical Industries and Automobiles are extensively
contributing to the pollution of environment, Carbon monoxide, nitric oxide, ozone, etc., are understood as the some of the factors of pollution from chemical industries. The maintenance of clean and healthy atmosphere makes it necessary to keep the pollution under control which is caused by combustion waste gas. The authors have suggested theory to control waste gas pollution in environment by oil refinery using fuzzy linear programming. To the best of our knowledge the authors [43]are the first one to apply fuzzy linear programming to control or minimize waste gas in oil refinery.
An oil refinery consists of several inter linked units. These units act as production units, refinery units and compressors parts. These refinery units consume high-purity gas production units. But the gas production units produce high-purity gas along with a low purity gas. This low purity gas goes as a waste gas flow and this waste gas released in the atmosphere causes pollution in the environment. But in the oil refinery the quantity of this waste gas flow is an uncertainty varying with time and quality of chemicals used in the oil refinery. Since a complete eradication of waste gas in atmosphere cannot be made; here one aims to minimize the waste gas flow so that pollution in 89
environment can be reduced to some extent. Generally waste gas flow is determined by linear programming method. In the study of minimizing the waste gas flow, some times the current state of the refinery may already be sufficiently close to the optimum. To over come this situation we adopt fuzzy linear programming method.
The fuzzy linear programming is defined by
Maximize
z
=
cx
Such
that
Ax
≤ b
x ≤ 0
where the coefficients A, b and c are fuzzy numbers, the constraints may be considered as fuzzy inequalities with variables x and z. We use fuzzy linear programming to
determine uncertainty of waste gas flow in oil refinery which pollutes the environment.
Oil that comes from the ground is called “Crude oil”.
Before one can use it, oil has to be purified at a factory called a
“refinery”, so as to convert into a fuel or a product for use. The refineries are high-tech factories, they turn crude oil into useful energy products. During the process of purification of crude oil in an oil refinery a large amount of waste gas is emitted to atmosphere which is dangerous to human life, wildlife and plant life. The pollutants can affect the health in various ways, by causing diseases such as bronchitis or asthma, contributing to cancer or birth defects or perhaps by damaging the body’s immune system which makes people more susceptible to a
variety of other health risks. Mainly, this waste gas affects Ozone Layer. Ozone (or Ozone Layer) is 10-50 km above the surface of earth. Ozone provides a critical barrier to solar ultraviolet radiation, and protection from skin cancers, cataracts, and serious ecological disruption. Further sulfur dioxide and nitrogen oxide combine with water in the air to form sulfuric acid and nitric acid respectively, causing acid rain. It has been estimated that emission of 70 percentage of sulfur dioxide and nitrogen oxide are from chemical industries.
We cannot stop this process of oil refinery, since oil and natural gas are the main sources of energy. We cannot close down all oil refineries, but we only can try to control the amount of pollution to a possible degree. In this paper, the authors use 90
fuzzy linear programming to reduce the waste gas from oil refinery. The authors describe the knowledge based system (KBS) that is designed and incorporate it in this paper to generate an on-line advice for operators regarding the proper distribution of gas resources in an oil refinery. In this system, there are many different sources of uncertainty including modeling errors, operating cost, and different opinions of experts on operating strategy. The KBS consists of sub-functions, like first sub-functions, second sub-functions, etc.
Each and every sub-functions are discussed relative to certain specific problems.
For example: The first sub-function is mainly adopted to the compressor parts in the oil refineries. Till date they were using stochastic programming, flexibility analysis and process design problems for linear or non-linear problem to compressor parts in oil refinery. Here we adopt the sub function to study the proper distribution of gas resources in an oil refinery and also use fuzzy linear programming (FLP) to minimize the waste gas flow. By the term proper distribution of gas we include the study of both the production of high-purity gas as well as the amount of waste gas flow which causes pollution in environment.
In 1965, Lofti Zadeh [115, 116] wrote his famous paper
formally defining multi-valued, or “fuzzy” set theory. He extended traditional set theory by changing the two-values indicator functions i.e., 0, 1 or the crisp function into a multi-valued membership function. The membership function assigns a “grade of membership” ranging from 0 to 1 to each object in the fuzzy set. Zadeh formally defined fuzzy sets, their
properties, and various properties of algebraic fuzzy sets. He introduced the concept of linguistic variables which have values that are linguistic in nature (i.e. pollution by waste gas = {small pollution, high pollution, very high pollution}).
Fuzzy Linear Programming (FLP): FLP problems with
fuzzy coefficients and fuzzy inequality relations as a multiple fuzzy reasoning scheme, where the past happening of the
scheme correspond to the constraints of thee FLP problem. We assign facts (real data from industries) of the scheme, as the objective of the FLP problem. Then the solution process
consists of two steps. In the fist step, for every decision 91
variable, we compute the (fuzzy) value of the objective function via constraints and facts/objectives. At the second step an optimal solution to FLP problem is obtained at any point, which produces a maximal element to the set of objective functions (in the sense of the given inequality relation).
The Fuzzy Linear Programming (FLP) problem application
is designed to offer advice to operating personnel regarding the distribution of Gas within an oil refinery (Described in Figure 5.1) in a way which would minimize the waste gas in
environment there by reduce the atmospheric pollution .
GPUI, GPU2 and GPU3 are the gas production units and
GGG consumes high purity gas and vents low purity gas. Gas from these production units are sent to some oil refinery units, like sulfur, methanol, etc. Any additional gas needs in the oil refinery must be met by the gas production unit GPU3.
The pressure swing adsorption unit (PSA) separates the
GPU2 gas into a high purity product stream and a low purity tail stream (described in the Figure 5.1). C1, C2, C3, C4, C5, are compressors. The flow lines that dead –end is an arrow
represent vent to flare or fuel gas. This is the wasted gas that is to be minimized. Also we want to minimize the letdown flow from the high purity to the low purity header
Dead end
GPU1
C1
GCG2
C2
C3
Dead end
GCG2
GPU2
PSA
ORU
C4
GPU3
ORU
C5
CGG
ORU
Letdown
Figure: 5.1
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FLP is a method of accounting for uncertainty is used by the authors for proper distribution of gas resources, so as to minimize the waste gas flow in atmosphere. FLP allows varying degrees of set membership based on a membership function defined over a range of values. The membership function
usually varies from 0 to 1. FLP allow the representation of many different sources of uncertainty in the oil refinery. These sources may (or) may not be probabilistic in nature. The uncertainty is represented by membership functions describing the parameters in the optimization model. A solution is found that either maximizes a given feasibility measure and
maximizes the wastage of gas flow. FLP is used here to
characterize the neighborhood of solutions that defines the boundaries of acceptable operating states.
Fuzzy Linear Programming (FLP) can be stated as;
max imize z = cx⎤⎥
s.t Ax ≤ b
⎥ … (*)
x ≥ 0
⎥⎦
The coefficients A, b and c are fuzzy numbers, the
constraints may be considered as fuzzy inequalities. The decision space is defined by the constraints with c, x ∈ N, b ∈
Rm and A ∈ Rm, where N, Rm, and Rmxn are reals.
The optimization model chosen by the knowledge based
system (KBS) is determined online and is dependent on the refinery units. This optimization method is to reduce the amount of waste gas in pollution.
We aim to
1. The gas (GCG2) vent should be minimized.
2. The let down flow should be minimized and
3. The make up gas produced by the as production unit
(GPU3) should be minimized.
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Generally the waste gas emitted by the above three ways
pollute the environment. The objective function can be
expressed as the sum of the individual gas waste flows. The constrains are given by some physical limitations as well as operator entries that describe minimum and maximum desired flows.
The obtained or calculated resultant values of the decision variables are interpreted as changes in the pressure swing adsorption feed, and the rate that gas is imported to CGG and gas production unit (GPU3). But in the optimization model there is uncertainty associated with amount of waste gas from oil refinery, and also some times the current state of the refinery may already be sufficiently close to the optimum.
For example to illustrate the problem, if the fuzzy
constraints x1, the objects are taken along the x-axis are shown in the figures 5.2 and 5.3, which represent the expression.
de
de
μ
μ
Membership gra
0 6
X
Membership gra
0 6
X1
10
1
10
objects
objects
Figure: 5.2
Figure: 5.3
x1 ≤ 8 (with tolerance p = 2)
(1)
The membership function µ are taken along the y-axis i.e.
µ(x1) lies in [0, 1] this can be interpreted as the confidence with which this constraint is satisfied (0 for low and 1 for high). The fuzzy inequality constraints can be redefined in terms of their α-
cuts.
{Sα / α ε [0, 1]}, where Sα = {γ / (μ (γ) ≥ α)}.
The parameter α is used to turn fuzzy inequalities into crisp inequalities. So we can rewrite equation (1)
x1 ≤ 6 + 2 (2) (1– α)
x1 ≤ 6 + 4 (1 – α)
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where α ε [0, 1] expressed in terms of α in this way the fuzzy linear programming problem can be solved parametrically. The solution is a function on α
x* = f(α) (2)
with the optimal value of the objective function determined by substitution in equation (1).
z* = cx* = g(α).
(3)
This is used to characterize the objective function. The result covers all possible solutions to the optimization problem for any point in the uncertain interval of the constraints.
The α-cuts of the fuzzy set describes the region of feasible solutions in figures 5.2 and 5.3. The extremes (α = 0 and α = 1) are associated with the minimum and maximum values of x*
respectively. The given equation (2) can also be found this, is used to characterize the objective function. The result covers all possible solutions to the optimization problem for any point in the uncertain interval of the constraints.
Fuzzy Membership Function to Describe Uncertainty: The
feasibility of any decision (µD) is given by the intersection of the fuzzy set describing the objective and the constraints.
µD (x) = µz(x) ^µN (x)
where ^ represents the minimum operator, that is the usual operation for fuzzy set intersection. The value of µN can be easily found by intersecting the membership values for each of the constraints.
µN (x) = µ1(x) ^µ2 (x)^…^ µm (x).
The membership functions for the objective (µz) however is not obvious z is defined in (2). Often, predetermined aspiration target values are used to define this function. Since reasonable values of this kind may not be available, the solution to the FLP
equation (3) is used to characterize this function.
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⎡
1
if z(x) ≥ b(0)
⎢ z(x) − b(1)
µ
⎢
≤
≤
z(x) =
if b(1)
z(x)
b(0)
⎢
(5)
b(0) − b(1)
⎢⎣ 0
if z(x) ≤ b(0).
The result is that the confidence value increases as the value of the objective value increases. This is reasonable because the goal is to maximize this function the limits on the function defined by reasonable value is obtained by extremes of the 1
de
Membership gra
0
b(1)
b(0)
object
Figure: 5.4
objective value.
These are the results generated by the fuzzy linear
programming. Since both µN and µz have been characterized, now our goal is to describe the appropriateness of any operation state. Given any operating x, the feasibility can be specified based on the objective value, the constraints and the estimated uncertainty is got using equation (4). The value of µD are shown as the intersection of the two membership functions.
Defining the decision region based on the intersection we describe the variables and constraints of our problem. The variable x1 represents the amount of gas fed to pressure swing adsorption from the gas production unit. The variable x2
represents the amount of gas production that is sent to CGG.
This problem can be represented according to equation (*). The constraints on the problem are subjected to some degree of uncertainty often some violation of the constraints within this range of uncertainty is tolerable. This problem can be
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represented according to equation (*). Using the given refinery data from the chemical plant.
μD
e 1
d
b(1)
b(0)
μN = 1
μZ(x) = 1
Membership gra
0
6
10
object
Figure: 5.5
c = [-0.544 3]
⎡ 1
0⎤
⎢
⎥
A =
0
1
⎢
⎥ ,
⎢ 0.544
−
1⎥
⎣
⎦
⎡33.652⎤
⎢
⎥
b = 23.050
⎢
⎥
⎢ 4.743 ⎥
⎣
⎦
Using equation (*) we get
Zc = – 0.544 x1 + 3x2 it represents gas waste flow. The gas waste flow is represented by the following three equations: i.
x1 + 0x2 ≤ 33.652 is the total dead – end waste flow gas.
ii.
0x1 + x2 = 23.050 is the total (GCG2) gas consuming
gas – treaters waste flow gas.
iii.
– 0544 x1 + x2 ≤ 4.743 is the total let-down waste flow
gas.
All flow rates are in million standard cubic feet per day. (i.e. 1
MMSCFD = 0.3277 m3/s at STP). The value used for may be
considered to be desired from operator experts opinion. The 97
third constraint represents the minimum let-down flow receiving to keep valve from sticking. The value to this limit cannot be given an exact value, therefore a certain degree of violation may be tolerable. The other constraints may be subject to some uncertainty as well as they represent the maximum allowable values for x1 and x2. In this problem we are going to express all constrains in terms of α, α, ε [0, 1]. We have to chose a value of tolerance on the third constraint as p3 = 0.1, then this constraint is represented parametrically as
a3 x ≤ (b3 – p3) + 2p3 (1 - α).
For example, if we use crisp optimization problem with the tolerance value p = 0.1 we obtain the following result:
X2
23
22.5
22
32
33 33.467
33.5
X1
Figure: 5.6
where x1 represents the amount of gas fed to PSA from gas production unit which is taken along the x axis, and x2 amount of gas sent to CGG which is taken along the y axis,
we get x1 = 33.469, when x2 = 23.050
⎡33.469⎤
x* = ⎢
⎥
⎣23.050⎦
z = 50.941. Finally we compare this result with our fuzzy linear programming method.
We replace two valued indicator function method by fuzzy linear programming.
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Fuzzy Linear Programming is used now to maximize the
objective function as well as minimize the uncertainty (waste flow gas). For that all of the constraints are expressed in terms of α, α, ∈ [0, 1].
a3 x ≤ (b3 – p3) + 2p3 (1 - α). α ∈ [0, 1]
where a3 is the third row in the matrix A. i.e. = 0.544x1 + x2 ≤
4.843 – 0.2 α, when the tolerance p3 = 0.3, we fix the value of α
ε [0.9,1], when the tolerance p3 = 0.1, we see α ε [0.300, 0.600].
X2
μD
23
22.5
22
31.5
33 33.469
33.5
X1
Figure: 5.7
where x1 represents the amount of gas fed to PSA from gas production unit which is taken along the x axis, and x2 amount of gas that is sent to CGG which is taken along the y axis, When
x2 = 23.050 and α = 0.0, we get x1 = 33.469.
When
x2 = 23.050 and α = 0.4, we get x1 = 33.616
The set (µz) is defined in equation 5. Fuzzy Linear
Programming solution is
⎡33.469⎤
x* = f(α) = ⎢
⎥
⎣23.050⎦
this value is recommended as there is no changes in the
operating policy.
So we have to chose the value for α as 0.6 for the tolerance p3 = 0.1, we get the following graph where x1 represents the amount of gas fed to PSA from gas production unit which is taken along the x axis, and x2 amount of gas that sent to CGG
which is taken along the y axis,
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X2
μD
23
22.5
22
31.5
33 33.469
33.5
X1
Figure: 5.8
when x2 = 23.050 and α = 0.0 we get x1 = 33.469
when x2 = 23.050 and α = 0.6 we get x1 = 33.689.
The operating region
⎡33.689⎤
x* = f (0.6) = ⎢
⎥ .
⎣23.050⎦
Now if the tolerance on the third constraint is increased to p3 =
0.2. This results is the region shown in the following graph. As expected the region has increased to allow a larger range of operating states.
when x2 = 23.050 and α = 0.0 we get x1 = 33.285
when x2 = 23.050 and α = 0.9 we get x1 = 33.947.
The operating region is
X2
μD
23
22.5
31.5
33.285
33.5
X1
Figure: 5.9
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where x1 represents the amount of gas fed to PSA from gas production unit which is taken along the x axis, and x2 amount of gas that is sent to CGG which is taken along the y axis.
⎡33.947⎤
x* = f (0.9) = ⎢
⎥ .
⎣23.050⎦
The fuzzy linear programming solution is
⎡33.285⎤
x* = f (α) = ⎢
⎥
⎣23.050⎦
z* = 51.043.
Finally we have to take α ε [0.9, 1.00].
Choose α = 0 and when the tolerance p3 = 0.3 we get the following graph when x2 = 23.050 we get x1 = 33.101.
X2
μD
23
22.5
31.5
33.101
33.5
X1
Figure: 5.10
where x1 represents the amount of gas fed to PSA from gas production unit; and x2 amount of gas that is sent to CGG.
When α = 1 and x2 = 23.050 we get x1 = 34.204. The
operating region is
⎡33.204⎤
x* = f (1.0) = ⎢
⎥ .
⎣23.050⎦
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The fuzzy linear programming solutions are
⎡33.101⎤
x* = f (α) = ⎢
⎥ .
⎣23.050⎦
The fuzzy linear programming solutions are
z* = g (α) = 51.143 .
We chose maximum value from the Fuzzy Linear Programming method i.e. z* = 51.143.
Thus when we work by giving varying membership
functions and use fuzzy linear programming we see that we get the minimized waste gas flow value as 33.101 in contrast to 33.464 measured in million standard cubic feet per day and the maximum gas waste flow of system is determined to be 51.143
in contrast to their result of 50.941 measured in million standard cubic feet per day. Since the difference we have obtained is certainly significant, this study when applied to any oil refinery will minimize the waste gas flow to atmosphere considerably and reduce the pollution.
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