Investigating Methods to Improve Attainable Region Computations

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UNIVERISTY OF THE WITWATERSRAND
SCHOOL OF CHEMICAL AND METALLURGICAL ENGINEERING
Course Code CHMT 4019
Course Name Chemical Engineering Laboratory Project
Assignment Details
Title: Investigating the possibility of improving
attainable region computations by existing
methods which manipulate bounding hyperplanes.
Assignment Due Date 27 – 10 -2014
Student Names
Mohammed Sayanvala
Meelan Lalla
Student Numbers
435141
322211
Declaration for Individual Work
 We are aware that plagiarism (the use of someone else’s work without their
permission and/or without acknowledging the original source) is wrong.
 We confirm that the work submitted for assessment for the above course is our
own unaided work.
 We have followed the required conventions in referencing the thoughts and ideas
of others.
 We understand that the University of the Witwatersrand may take disciplinary
action against us if there is a belief that this is not our own unaided work or that
we have failed to acknowledge the source of the ideas or words in our writing.

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Abstract
The method of attainable regions is an innovative approach to reactor network design.
This method uses a purely geometric approach to map all possible operational outputs so
that the desired output can be used to identify an optimum reactor network. This method
is currently under heavy development. This particular project research is concerned with
the investigation of two current methods that use the attainable region approach. These
methods are investigated by changing some of their steps and parameters and observing
the resulting effects on accuracy and performance.
It was found that the revised version of the two methods which uses a rotation of a
bounding hyperplane to remove unobtainable points yields the best results both in terms
of performance and accuracy. Some parameter changes are suggested specific to the older
of the two methods. These suggestions are focused on steps that may be slowing down the
method. From the literature review it is shown that the older method is currently more
robust than the newer method in that it can be applied to a wider parameter set. Hence,
both methods require significant research and development.

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Table of Contents
Abstract.................................................................................................................................i
1 Motivation and Background ........................................................................................1
1.1 General overview ..................................................................................................1
The need for studying attainable regions .......................................................1
Advantage over other optimisation techniques..............................................2
2 Research Scope ............................................................................................................3
2.1 Problem Statement ................................................................................................3
2.2 Research Question.................................................................................................4
3 General and Mathematical Definitions ........................................................................5
3.1 Shapes....................................................................................................................5
Polytopes and polygons .................................................................................5
Convexity.......................................................................................................5
Extreme points ...............................................................................................7
3.2 Vectors ..................................................................................................................7
Row and column vectors................................................................................7
Basis Vector...................................................................................................8
Rate vectors....................................................................................................8
Concentration vectors ....................................................................................8
Mixing vectors ...............................................................................................9
Tangency........................................................................................................9
Unit Vectors...................................................................................................9
Normal Vectors and Null Space ..................................................................10
3.3 Stoichiometry ......................................................................................................10
Stoichiometric matrix...................................................................................10
Extent of reaction.........................................................................................10

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Mass balance equation .................................................................................11
Stoichiometric subspace...............................................................................11
3.4 Hyperplanes.........................................................................................................11
3.5 Translation...........................................................................................................12
3.6 Rotation...............................................................................................................13
3.7 Fundamental reactor types ..................................................................................13
4 Literature Review.......................................................................................................16
4.1 Traditional optimisation methods........................................................................16
The Design Objective ..................................................................................16
Linear Programming (LP)............................................................................16
Successive Linear Programming (SLP) .......................................................17
Successive Quadratic Programming (SQP) .................................................17
Reduced Gradient method (RGM)...............................................................17
Discrete and Integer variables......................................................................17
4.2 The method of Attainable Regions......................................................................18
Conditions of the attainable region..............................................................18
Constructing a candidate attainable region using the traditional method....18
An ‘outside-in’ approach .............................................................................22
Bounding Hyperplanes.................................................................................22
5 Experimental Procedure and Set-up...........................................................................28
5.1 Definitions...........................................................................................................28
Kinetics ........................................................................................................28
Grid Type.....................................................................................................28
Grid Size ......................................................................................................28
Step Size.......................................................................................................28
Constraints and current polytope .................................................................29
Vertex representation...................................................................................29

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Angle of Rotation.........................................................................................29
5.2 Tools used for the development of the code .......................................................29
Equipment....................................................................................................29
External sources of code ..............................................................................29
5.3 Reproducing the algorithms ................................................................................30
Finding the stoichiometric subspace............................................................30
Initialising an active hyperplane ..................................................................31
Translating the active hyperplane ................................................................32
Rotating the active hyperplane.....................................................................34
Evaluating points on the active hyperplane .................................................35
5.4 Testing the algorithms.........................................................................................35
Selecting a set of kinetics to work with .......................................................35
Finding the theoretical volume of the attainable region ..............................36
Establishing default parameters ...................................................................37
Testing the rotational method ......................................................................37
Testing the translational method..................................................................38
Comparing the results from the translational and rotational methods .........39
Selecting a second set of kinetics to test......................................................39
Assessing the objective function..................................................................40
Comparing the results from the second set of kinetics ................................40
6 Hazard Score Table and probability ..........................................................................41
7 Results and Discussion (Student #435141)................................................................43
7.1 Testing the translational method .........................................................................43
Effect of decreasing the translation step size...............................................43
Effect of changing the rotation of the active hyperplane on accuracy.........44
Effect of using a variable grid size ..............................................................45
7.2 Testing the rotational method..............................................................................47

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Effect of using a fixed pivot step .................................................................47
Effect of decreasing the angle of rotation....................................................49
7.3 Comparison between the two methods................................................................50
7.4 The water gas shift reaction ................................................................................52
7.5 Conclusion and Recommendations .....................................................................53
8 Results and Discussion (Student #322211)................................................................55
8.1 Translation Method .............................................................................................55
8.2 Rotation Method..................................................................................................58
8.3 The Water gas-shift reaction........................................................................................63
8.3 Conclusions and Recommendations....................................................................64
9 Appendix....................................................................................................................65
9.1 MATLAB Code...................................................................................................65
Rotational method........................................................................................65
Translational method – Van de Vusse .........................................................69
9.2 Translational Method – Water gas shift ..............................................................73
9.3 Objective function – water gas shift....................................................................79
9.4 Water gas shift kinetics .......................................................................................79
9.5 Van de Vusse kinetics .........................................................................................79
9.6 Tables of results ..................................................................................................81
Rotational Method .......................................................................................81
Translational Method...................................................................................81
Comparison between the 2 methods ............................................................84
9.7 Sample calculations and logical background for MATLAB code ......................85
Finding the stoichiometric subspace............................................................85
Populating the incidence matrix...................................................................86
Constraining the points on the hyperplane...................................................87
Mixing vectors .............................................................................................87

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9.8 RECORD OF MEETINGS PERTAINING TO 4TH
YEAR RESEARCH
PROJECTS ....................................................................................................................89
10 References..................................................................................................................93

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1 Motivation and Background
1.1 General overview
The need for studying attainable regions
Reactor design refers to the process of detailing the construction and operation (i.e. the
setup) of a process plant or factory to produce a specified set of target products. Hence,
reactor design is commonly not confined to a single reaction unit, but to a network of
reactor vessels.
The process of reactor design is primarily concerned with finding a solution that provides
an economically sound means of maximising the production of a target item or set of
items, while obeying additional constraints that may include factors such as time, cost,
environmental impact and product quality amongst others. This framework is termed the
optimum design in this report.
Current approaches to reactor design involve determining the plant output from a
simulated setup. Whilst there are volumes of information available from current plant
operations, it is a consequence of this fact that design and development of new plants may
become confined and constrained by using available data as a basis. In the past, engineers
have used various mathematical tools to estimate the setup that satisfies the optimum
design criteria. Yet, even the most rigorous arithmetic computations provide no means of
affirming that all possible combinations of operational parameters have been considered.
In addition, it is also very difficult to establish what is optimal when considering complex
systems. In 1964, Horn introduced a novel, purely geometric approach for optimal reactor
design called the attainable region (Abraham & Feinberg, 2004). A simple understanding
of the attainable region is that it is a representation of every possible output
concentration for a given set of reaction kinetics using the basic reactor types. (Abraham
& Feinberg, 2004).
Reaction kinetics are specific to a set of chemical reactions and are usually obtained
experimentally or taken from established sources. Reaction kinetics refer to the set of
equations that incorporate the rates of reaction and the presiding equilibrium conditions.
Reaction kinetics can be disturbed by temperature, the use of catalysts and other factors.
Hence, it is convenient to be able to find the attainable region for a specific set of reaction
kinetics.

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Basic reactor types are general representations of reactor vessels that differ by their
degree of mixing and hence provide different concentration profiles. The method of
attainable regions uses three reactor types, namely the continuously stirred tank reactor
(CSTR), plug flow reactor (PFR) and differential side-stream reactor (DSR). Greater
definitions of these reactor types can be found in (Fogler, 2005).
With a close approximation to the attainable region it is possible to identify the reactor
network that will satisfy the optimum design criteria. Hence, the use of attainable regions
will allow design engineers to have a holistic view of all achievable outputs that are not
confined by a particular range of process setups. The attainable region can be used in
conjunction with objective functions which incorporate relevant factors like profitability
to find an optimum point of operation that satisfies all the design criteria.
Construction of the attainable region is complex and requires an understanding of
chemical reactors, mathematical principles and computer programming. When Horn first
introduced the concept in 1964 (Abraham & Feinberg, 2004), regular computer
processors and mathematical languages were not nearly as advanced as they are today.
With the advanced technology that exists today, it may be possible to develop the method
of attainable regions into a useful tool that will actually add much value to the relevant
industries. Yet, whilst the improved technology may significantly enhance the
development of this method, it must be noted that the framework of attainable regions is
not well understood itself. Most of the current algorithms that exist are far from perfect as
they have severe limitations due to a number of challenges. Hence, there is a need to
research and understand the attainable region at a deeper level in order to fully grasp
these challenges. Apart from industrial relevance, research into this field at an
undergraduate level will provide an invaluable learning opportunity in the fields of
computing, mathematical modelling and reactor design.
Advantage over other optimisation techniques
Many of the current optimisation techniques such Levenspiel’s graphical analysis or
Mixed Integer Non-linear Programming (MINLP) can be used to provide useful insight
into possible optimum reactor design parameters, but their validity is largely limited to
simple reaction schemes and, in most cases, become computationally intensive. The
design and optimisation of complex reactor systems generally involves rigorous trial and
error procedures that rely on previous knowledge and experience to identify the design

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that satisfies the optimum criteria. Critically; these procedures do not address the process
of selecting an optimum reactor structure and provide no means of affirming that all
possible combinations of operational parameters have been considered. In rare situations,
the best design may immediately be clear. However, in most design undertakings there
are a set of problems which require the use of more robust optimisation procedures. This
provides the basic premise behind the attainable region technique. A brief explanation of
the underlying theory and application of the common optimisation techniques are
presented in further detail in the literature review section.
2 Research Scope
It is worth mentioning at this point that the boundary of the attainable region is governed
by certain conditions to ensure that there are no possible extensions to these known
conditions (Hildebrandt & Feinberg, 1997). Also, there are no sufficiency conditions
which guarantee that the nominated region does indeed contain all possible outcomes for
all possible configurations (Hilderbrandt & Glasser, 1987). For this reason, it is common
practice to refer to the region as the candidate attainable region (Abraham & Feinberg,
2004).
2.1 Problem Statement
Since the introduction of the attainable region method by Horn in 1964 (Abraham &
Feinberg, 2004), the method has undergone extensive investigation that has led to the
development of different techniques of computing the attainable region. The steps that
lead from the specification of a reaction set and its associated kinetics to the computation
of a candidate attainable region are defined as the algorithm. Research on the
development of a sound algorithm to construct attainable regions has focused on
accuracy, robustness and performance. The accuracy of an attainable region is not easily
defined, but is analysed by benchmarking it against other candidates in terms of volume,
where a smaller volume is said to represent a more accurate region (Hildebrandt &
Feinberg, 1997). Robustness refers to the successful use of an algorithm on a wide range
of schemes that may differ by temperature-dependence, the amount of independent
reactions or some other factor. Performance evaluation is a simpler means of assessing
the improvement of an algorithm as it focuses on the time that it takes to compute an
attainable region so as to ensure that the algorithm is not heavily dependent on expensive
computer hardware.

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Currently there are two key approaches that have been used for the construction of an
attainable region aptly named the inside-out and outside-in approaches. These terms have
geometric implications that are explained later in this report. A simple definition is that
the inside-out approach grows the attainable region from the feed point by identifying all
achievable outputs and the outside-in approach removes all unachievable outputs and
defines the attainable region as the remaining region of outputs. The outside-in approach
has shown the potential to aid in obtaining a closer approximation to the attainable region,
especially when used in conjunction with the inside-out approach. Efforts on developing a
combined approach are being tackled, but in an attempt to aid those efforts, this research
project will focus on testing the stages in the algorithm of the outside-in approach and
will thereby provide an opportunity to identify possible improvements to it.
2.2 Research Question
Can improvements be made to the bounding hyperplanes algorithm for
computing attainable regions using an outside-in approach?

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3 General and Mathematical Definitions
In an attempt to aid the reader in their understanding of the literature to follow, a brief
mathematical background has been provided where certain concepts and terminologies
have been introduced which are constantly referred to within the main body of the report.
3.1 Shapes
Polytopes and polygons
A polytope is a geometric object consisting of straight lines that are all connected and
form a ‘closed’ shape. A polytope may exist in any general number of dimensions where
a dimension refers to a measurement of length in one direction (Pierce, 2014)
A polygon is a polytope represented in two-dimensions and can be classified as either
regular (all the angles and sides are equal) or irregular.
Polygon
(straight sides)
Not a Polygon
(has a curve)
Not a Polygon
(open, not closed)
Figure 3-1: Representation of a polygon (Pierce, 2014)
Regular Irregular
Figure 3-2: Representation of regular and irregular polygons (Pierce, 2014)
Convexity
A polygon P is said to be convex if:
 P is non-intersecting, i.e. a straight line extending outward from any point on the
boundary of P will not intersect P at any other point.

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 For any two points p and q on the boundary of P, segment pq lies entirely inside
P.
Figure 3-3: Distinguishing between convex and non-convex objects (Glasgow, 2000)
3.1.2.1 Convex Hull
If one considers a set of points (X), intuitively the convex hull of (X) can simply be
viewed as the shape of a rubber-band stretched around these points.
Figure 3-4: Representation of the 'rubber-band' analogy (Glasgow, 2000)
A formal definition describes the convex hull of (X) as the smallest convex polygon that
contains all the points of (X). Evaluating the convex hull of a set of points is the most
elementary interesting problem in computational geometry (Skiena, 2008)
Figure 3-5: Convex Hull (Glasgow, 2000)

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Extreme points
If one considers a set of points (X), an extreme point is defined as any point which
represents a vertex of the resultant convex hull of (X). It follows that any extreme point
may not lie within the interior of the convex hull (Ming, et al., 2010).
Figure 3-6: A convex set shaded (in blue) with its corresponding extreme points (red)
3.2 Vectors
A vector is a geometric entity used to describe physical quantities that have both
magnitude and direction. Hence, a vector can be described as a set of components that
each represent a magnitude and direction within a dimension of the concerned space.
For example, a vector v in a 3 dimensional Cartesian space can be represented as:
𝒗 = [
𝑣𝑥
𝑣 𝑦
𝑣𝑧
]
𝑣𝑥, 𝑣 𝑦, 𝑣𝑧 are scalar quantities that describe the length of the vector 𝒗 in each direction. In
this report, vectors are expressed using a bold-weighted letter.
Row and column vectors
A row or column vector is a one dimensional matrix that often (but not necessarily
always) represents the solution to a system of linear equations. Although the geometric
definitions are the same, these vector types differ in their format and yield different
arithmetic results. The arithmetic implications associated with these vector types are
beyond the scope on this text. For further information, see (Stroud & Booth, 2001). For
uniformity of calculation, all vectors in this investigation were represented as column
vectors

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Basis Vector
A basis vector represents one of a single set of vectors which can be used to represent
every vector in a given vector space (a closed set formed by vector addition and scalar
multiplication), by taking a linear combination of the basis vector(s) (Burke, 2012).
Hence, if V is a vector space and S is a subset of V, it follows that S is a basis of V subject
to the following conditions:
 S spans V (must contain the necessary amount vectors to generate the respective
vector space)
 S must be linearly independent
Rate vectors
A rate vector has the rate expressions for each chemical species as components of the
vector such that for the following reaction scheme:
𝐴 → 𝐵 + 𝐶
(1)
2𝐶 → 𝐷
The rate vector can be expressed in 2𝐷 space as:
𝒓(𝑪) = [
𝑟𝐴
𝑟𝐶
] (2)
It is convenient to work in a concentration space of the same dimension as that of the
reaction scheme. The concentration of the remaining products can be found via mass
balance calculation.
Concentration vectors
Similarly, a concentration vector has the concentrations of each chemical species as its
components. For the system described in (1), in two dimensional space the concentration
vector can be defined as:
𝑪 = [
𝐶𝐴
𝐶 𝐶
] (3)

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Mixing vectors
Figure 3-7: Lever arm rule describing the mixing vector (Glasser, 2008).
For a given set of reaction kinetics and feed point(s), it is possible to solve for the
concentration vectors over a range of residence time values using the characteristic
equations for the PFR and CSTR reactors. The reaction rates are then calculated such that
each point in the concentration space has a corresponding vector, in this case the rate
vector, r(C). Considering the case of an isothermal reactor system where there are no
volume changes with respect to mixing and reaction, it is clear that the only other
operation that can be performed is mixing. When two streams with compositions C and
C0
are mixed, the resultant composition C*
(c-star) must lie on the straight line joining the
initial points. This conclusion is derived from the Lever-arm rule, proof of which can be
found in the Appendix (see section 9).
Tangency
Two vectors x and y are said to be tangent if:
𝒙 . 𝒚 = 0 (4)
Where the (.) operator is the characteristic dot product between two vectors.
Unit Vectors
Unit vectors are vectors that have a magnitude of 1 unit. A unit vector can be found for
any given vector such that it represents a length of one unit in the same direction as the
concerned vector.

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Normal Vectors and Null Space
Normal vector (n) here refers to a vector that is perpendicular in direction to a concerned
vector or space. From the figure below, the vectors Px and Py can be said to represent the
null space of the normal vector n.
Figure 3-8: Schematic representation of a normal vector to the plane P.
3.3 Stoichiometry
Stoichiometric matrix
The stoichiometric matrix is an 𝑚 × 𝑛 matrix where m is the number of species involved
and n is the number of reactions. Each element in the matrix corresponds to the
stoichiometric coefficient of species m in reaction n. Therefore, for the system in (1), the
following can be given as the stoichiometric matrix:
𝐴 = [
−1 0
1 0
1
0
−2
1
]
Extent of reaction
The extent of a reaction refers to the degree to which the reaction approaches completion.
The extent is easily evaluated and can be used to calculate to calculate the concentration
of all the species involved in the corresponding reaction. The extent vector for a set of
reactions is given by:
𝑬 = [
𝐸1
𝐸2
]
n
PPx
Py

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The extent of a reaction is conveniently calculated using the reactant which would be
consumed first because the total consumption of this species represents the completion of
the reaction. Hence, by this definition, the extent can be calculated using the following
equation:
𝐸𝑗 =
𝑚𝑜𝑙𝑒𝑠 𝑜𝑓 𝑗 𝑓𝑜𝑟𝑚𝑒𝑑
𝑠𝑡𝑜𝑖𝑐ℎ. 𝑐𝑜𝑒𝑒𝑓𝑖𝑐𝑖𝑒𝑛𝑡 𝑜𝑓 𝑗
(5)
Mass balance equation
With the above definitions, it is possible to calculate the concentration vector of a system
using the following mass balance:
𝑪 = 𝑪 𝒇𝒆𝒆𝒅
+ 𝐴 𝑬 (6)
Stoichiometric subspace
In any system of chemical reactions one can express the corresponding reaction vectors in
terms of extents. These reaction vectors span a linear subspace referred to as the
stoichiometric subspace (Cosentino & Bates, 2011) The evaluation of this subspace
enables one to characterize all the points which can be ‘achievable’ by the system in
terms of stoichiometry and allows one to establish that the candidate attainable region
will lie within this subspace.
3.4 Hyperplanes
In an algebra text, (Beardon, 2005) introduces hyperplanes by expanding on the idea that
the solutions of a single linear homogeneous equation in 3 variables can be characterized
by a plane that passes through the origin in 𝑅3
such that the plane has a dimension that is
one less than the underlying subspace. This concept is generalized to 𝑅 𝑁
so that if S
denotes a vector space wherein a line has dimension one, a hyperplane in S will have
dimension 𝑛 − 1. Prior to developments in the method of attainable regions, (Hillier,
1969)and others experimented with the manipulation of the mathematical properties of
hyperplanes to develop algorithms for integer linear programming (ILP). The limitations
of this method have been described earlier. Further to the definition presented by
(Beardon, 2005), (Burke, 2012) denotes a hyperplane in 𝑅 𝑁
as having the form:
𝐻(𝑎, 𝛽) = {𝑥 ∶ 𝑎 𝑇
𝑥 = 𝛽} (7)

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Where 𝑎 ∈ 𝑅 𝑁
and 𝛽 𝜖 𝑅. Hence, a hyperplane can also be said to be the set of vectors
that are orthogonal to a given vector (Beardon, 2005)Translating this idea to
concentration space, (Ming, et al., 2010) notes that a hyperplane may be said to obey the
following relationship:
𝐻(𝐶, 𝐶0) = {𝐶 ∈ 𝑅 𝑁
∶ (𝐶 − 𝐶0). 𝑛 = 0} (8)
Where 𝐶 𝑎𝑛𝑑 𝐶0 are points on H and n is a normal vector (at right angles to all the
vectors lying on H). Hyperplanes divide the underlying space into two half-space (Burke,
2012), where the closed half-spaces are defined as:
 A positive half-space:
𝐻≥
= {𝐶 ∈ 𝑅 𝑁
∶ (𝐶 − 𝐶0). 𝑛 ≥ 0} (9)
 A negative half-space:
𝐻≤
= {𝐶 ∈ 𝑅 𝑁
∶ (𝐶 − 𝐶0). 𝑛 ≤ 0} (10)
3.5 Translation
In Geometry, translation refers to the movement of an object, without rotation or resizing
such that every point of the initial object must move in the same direction and by an
equivalent distance (Pierce, 2014).
Figure 3-9: Illustrating the concept of translation (Wang, 2014)

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3.6 Rotation
Rotation refers to the movement around a centre or ‘pivot’ point such that the distance
from this point to any point on the object remains unchanged. The degree of rotation is
called the angle of rotation and is measured in degrees. By convention, a clockwise
rotation is regarded as a negative angle where as an anti-clockwise rotation is regarded as
a positive angleInvalid source specified..
Figure 3-10: Illustrating the concept of rotation (Wang, 2014)
3.7 Fundamental reactor types
a) PFR – Under constant density the general design equation for a PFR (written in
terms of space time) is given by:
𝑑𝐶
𝑑𝜏
= 𝑟(𝐶) (11)
The geometric interpretation of the PFR is that the rate vector r (C) is tangential to
the trajectory for all possible PFR products (figure 3-12).
b) CSTR – The general form of the CSTR design equation (again written in terms of
space time), is given by:
𝐶𝑖 − 𝐶𝑖
0
= 𝜏. 𝑟𝑖(𝐶) (12)
The geometrical interpretation of a CSTR is that the reaction vector is co-linear
(lie on a single straight line) with the line drawn from the CSTR feed to the CSTR
product (figure 3-11).

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c) Differential side-stream reactor (DSR)
A differential side-stream reactor (DSR) can be conceptualized as a PFR which is fed
along the length of the reactor. This type of configuration is prominent when reaction and
mixing take place simultaneously. The geometric interpretation of the DSR (figure 3-10)
is that the trajectory is tangential to the plane which contains both reaction and mixing
vectors respectively (Seodigeng, 2006). The general design equation (written in terms of
space time) is given below:
𝑑𝐶
𝑑𝜏
= 𝑟(𝐶) + 𝛼(𝐶)(𝐶 𝑚 − 𝐶) (13)
Where, 𝑟(𝐶) and (𝐶 𝑚 − 𝐶) are the reaction and mixing vectors respectively.
𝐶 𝑚 is commonly referred to as the mixing point.
The value of the variable (𝛼) determines the limiting behaviour of the design equation.
If 𝛼 = 0, then the DSR design equation reduces to a PFR (equation 11). Alternatively, if
𝑑𝐶
𝑑𝜏
= 0, then it reduces to a CSTR (equation 12).
Figure 3-11: Hypothetical boundary of an attainable region (Feinberg, 1999)

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Figure 3-12: Rate vectors for a CSTR Figure 3-13: Rate vectors for a PFR

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4 Literature Review
4.1 Traditional optimisation methods
This section presents a brief overview of the traditional optimisation methods that are
typically employed in, but not limited to, the search for optimum reactor networks or the
boundaries of some feasible region.
The Design Objective
This method involves the maximisation or minimisation of a quantity called the objective
function (Towler, 2008). The objective function is made up of a set of decision variables.
It is often difficult to formulate the objective function as there is an inherent uncertainty,
either in the decision variables (caused by unsteady state plant operations) or in the
economic objectives which are dependent on the prices of materials, energy and capital
costs. Sometimes the objective function may be treated as a single variable in which case
a variety of ‘search methods’ can be used to find the optimum. These include:
 The unrestricted search method which is primarily used for unconstrained
decision variables.
 The regular search (3 point interval search) method is similar in its approach but
finds the optimum value within a certain tolerance or precision range (Towler,
2008).
 The golden section search and quasi-Newton methods can also be used and are
relatively simple to implement.
The successes of these techniques are restricted to objective functions which have only
one maximum or minimum. In most cases however, processes are dependent on a number
of decision variables, thus requiring multivariable-optimisation techniques. A brief
overview of these methods is discussed below.
Linear Programming (LP)
Linear Programming optimisation is based on the premise that a set of continuous linear
constraints always defines a convex region and if the objective function is linear then it
can be expressed as a linear program and solved for a global optimum which lies on the
boundary of the feasible region (Towler, 2008). This method can be used for a large
number of variables and constraints but is not commonly used in reactor design problems

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as they are inherently non-linear. Instead, these problems are commonly solved using
non-linear programming (NLP) techniques.
Successive Linear Programming (SLP)
This approach requires the linearization of functions at a certain point to generate an
initial solution (new point), after which the functions are linearized at this new point. The
procedure is repeated until convergence (Towler, 2008). Whilst this approach is relatively
simple to implement it does not guarantee convergence and uncertainties may arise when
discontinuous functions have been used to approximate the non-linear functions.
Successive Quadratic Programming (SQP)
This approach is similar to the SLP method but approximates the non-linear function(s) as
a quadratic function and produces favourable results with problems consisting of few
variables (Towler, 2008).
Reduced Gradient method (RGM)
The RGM is quite effective when applied to problems where there are a large number of
variables, the constraints are linear and they can be written in terms of one or two
variables (Towler, 2008). This phenomenon is a common occurrence in design problems.
Although the NLP techniques appear to be more robust then the initial methods, they may
encounter convergence and local optima problems.
Discrete and Integer variables
In process design discrete variables (such as the number of trays of a distillation column)
may also be used. These discrete variables are addressed by introducing integer variables
(Towler, 2008). This results in the formation of a MILP or mixed integer linear program
or in the case of non-linear variables, a mixed integer non-linear program (MINLP).
These problems are solved using the ‘Branch and Bound’ algorithm. In the case of a
MINLP, this may become computationally intensive as it requires the solution to a large
set of non-linear equations (Towler, 2008).
The use of integer variables can be extended to various flowsheet options. This is
apparent in the method of Superstructure optimisation which could present the user with
additional configurations or networks of reactors or heat exchangers that may also result
in a global optimum. However, as with the MINLP, it may become computationally
intensive. For purposes of clarity the underlying theory and operation of these various
techniques has been omitted as it is not the main focus of this research project. However,

18
it was essential to give the reader a brief background of the available optimisation
techniques before presenting the novel concept of Attainable Regions.
4.2 The method of Attainable Regions
The traditional method of generating a candidate attainable region involves ‘growing’ the
region from a known feed point(s). This approach requires the evaluation of concentration
profiles for the fundamental reactor types. Thereafter, the region is extended by the use of
mixing vectors until a tangent point is reach along the boundary of the current region. The
region is subsequently extended by the introduction of PFR trajectories along the
boundary of the current region until no further extension to region is achieved. This
approach gives insight into points that are achievable but it does not give any information
regarding other configurations which may lead to points outside the candidate region.
Conditions of the attainable region
A candidate attainable region must satisfy the following conditions (Hilderbrandt &
Glasser, 1987):
i. The candidate attainable region should contain all defined feed point(s) or input
states.
ii. The permitted fundamental process vectors (reaction and mixing) on the boundary
of the region must not extend outside the region.
iii. None of the fundamental process vectors may intersect the boundary of the region
when negatively extrapolated.
iv. The region must be convex. It is composed of convex curved sections (reaction
vectors) and straight lines (mixing lines).
Constructing a candidate attainable region using the traditional method
The construction of a candidate attainable region using the traditional approach of
‘growing’ the region from a feed point is presented. For a given set of reaction kinetics
and feed point(s), it is then possible to solve for the concentration vector 𝐶 =
[𝐶𝐴, 𝐶 𝐵, … ., 𝐶 𝑁] 𝑇
over a range of residence times values. It is then affirmed whether the
resulting region can be extended by means of mixing or by the use of PFR trajectories
drawn from the boundary of the current region. Once the region can be extended no more,

19
it is concluded that a candidate attainable region has been obtained. An expansion on the
application of mixing vectors and their role in constructing a candidate attainable region
is presented below.
Figure 4-1: Geometric implications of mixing by varying the amount of feed bypassed.
The black arrows represent the formation of new products as a result of mixing a stream
from the CSTR and the feed stream (point O). By varying the amount of feed bypassed,
the output concentration of the reactor system is moved from the feed point (where all the
material is bypassed) to a point on the CSTR locus which corresponds to no bypass. This
process is repeated until the line drawn from the feed point is tangential to the CSTR
locus. Hence, if the feed is mixed with CSTR products beyond this point, no new
products will be produced. This result affirms the concept mentioned as a condition for a
candidate attainable region which states that the permitted process vectors (reaction or
mixing in this case) on the boundary of the attainable region must not extend outside the
region.
The concept of feed-bypass was introduced in the previous paragraph. The geometric
interpretation of feed-bypass is perhaps more apparent when one looks at the overall
construction of a candidate attainable region. As stated previously, given a set of reaction
kinetics and a feed point(s), one can generate the PFR trajectory and CSTR locus
respectively. In the figure below (Glasser, 2008), a CSTR with bypass is considered and
this results in the line drawn from the feed point to point P which is tangential to the
CSTR locus. The green shaded region represents all the possible product concentrations
of components A and B from the CSTR with bypass.

20
Figure 4-2: CSTR with feed bypass (Glasser, 2008)
Any further mixing will not yield any new products or extend this region. Hence, the
product streams (points on the boundary of the green shaded area) can now be used as
feed streams to a PFR. It is important to note that only points on the boundary of the
shaded area are considered as feed streams to a PFR. This is due to the fact that the
resultant PFR trajectory would result in an extension of the region. Furthermore, it is clear
that a PFR trajectory drawn from the tangential point P will cover all possible products of
this system. The PFR trajectories are shown in the figure below (Glasser, 2008):
Figure 4-3: Geometric implication of extending the region using PFR trajectories (Glasser, 2008)

21
The systems considered up to this point are two dimensional (2-D) however, this
technique becomes complicated when applied to higher dimensions (when the candidate
attainable region is generated with respect to more than two species). The necessary
conditions of a candidate attainable region hold for all dimensions as does the geometry
of the CSTR and PFR reactors (Hilderbrandt & Glasser, 1987). However, the problem is
that the PFR trajectory is still represented as a straight line and does not act as a
‘boundary’, separating the regions (Hilderbrandt & Glasser, 1987). The CSTR locus may
not touch the PFR trajectory, thus the 2-D method of construction can no longer be used.
However, this can be overcome by considering a PFR from the feed point and taking
linear combinations of the possible products (Hilderbrandt & Glasser, 1987). A CSTR
locus is then drawn to check if the boundary of the region can be extended. A candidate
attainable region generated using Van de Vusse kinetics is shown below (Abraham &
Feinberg, 2004):
Figure 4-4: A candidate attainable region using the Van de Vusse kinetics (Feinberg, 2004).
The boundary of the candidate attainable region is now made up of PFR’s, CSTR’s and
also the differential side-stream reactors (DSR’s). DSR’s are similar to that of PFR’s, but
are fed along the length of the reactor. It is worth mentioning at this point that there are
no conditions which state the region displayed above is the complete attainable region for
this particular system. However, (Hildebrandt & Feinberg, 1997) suggests that the
boundary of the attainable region will always consist of these three types of reactors. The
precise combination of these reactor types to produce the boundary of the attainable
region may not always be known, however, by simply identifying the location of the
boundary itself, has tremendous implication for design and optimisation (Abraham &

22
Feinberg, 2004). The ‘inside-out’ approach is hugely important in this regard as it
indicates a range of compositions that are achievable but it does not give insight into
other configurations that may lead to effluent compositions outside the current region
(Abraham & Feinberg, 2004). This provides the motivation behind the ‘outside-in’ and
hybrid approaches thereafter.
An ‘outside-in’ approach
Following the inception of this framework, traditional methods of constructing an
attainable region have focused on growing the region from a feed point for a given set of
reaction kinetics. This technique can be thought of as an inside-out approach to forming
the attainable region. However, the region obtained by this method is limited in that it is
not possible that this region encapsulates all achievable points such that any point in the
space beyond this region is not achievable. An outside-in approach presented by
(Abraham & Feinberg, 2004) aims to arrive at the region of achievable points by cutting
away at all spaces that are unachievable. This is done by manipulation of further
geometric principles (other than those employed in traditional algorithms), specifically
the use of hyperplanes and has the opposite implication to that obtained by the inside-out
approach. The region arrived at by successively removing unachievable space is said to
encapsulate the entire true attainable region, but is not limited to achievable points and
includes points which are in effect unachievable. It follows then that the smaller the
tolerance of computational error, the more tightly this region will bound the true
attainable region. Using this technique in conjunction with the inside-out technique, it is
possible to approach the true attainable region more closely by minimizing the difference
between the results obtained from both methods. This presents a fundamental opportunity
showcasing the significance of research and development of the outside-in approach.
Bounding Hyperplanes
4.2.4.1 Relevance of hyperplanes to the attainable region
(Burke, 2012) goes on to define a convex polyhedron as any subset of 𝑅 𝑁
which may be
represented by the intersections of a finite number of closed half-spaces and such a region
is said to be the constrained region for a given situation of linear programming.
Analogous to this definition, the region enclosed in 𝐻≥
of a concentration space
encapsulates all achievable concentrations for a given set of reaction kinetics and reactor
network. By definition, a subset 𝐻 ∈ 𝑅 𝑁
is said to be convex if [x; y] ⊂ H where x; y εH
i.e. every convex combination comprised of members of H is also a member of H. Simply

23
stated, it is possible to represent all achievable concentrations in concentration space by a
convex region bounded by hyperplanes such that the bounded region is the on the side of
the closed positive half-spaces generated by the hyperplanes. It is then apparent that the
normal vector n in equation (12) may be used to check for tangency with the rate vectors
as an indication for the boundary of the attainable region. (Abraham & Feinberg, 2004)
proposed the first outside-in method which uses this principle to work toward the
attainable region by removing all unachievable points that do not satisfy the criterion.
Working with this approach however involves a number of additional complexities like
the style of proceeding from the initialized hyperplane to the successive ones (by rotation,
translation or some other progression) or the extreme points from which to proceed. An
extreme point is defined as such if it is a vertex of the convex region. Two distinguishable
algorithms using the bounding hyperplanes approach have since been developed. Both
methods are described below.
4.2.4.2 Original method
The method proposed by Abraham and Feinberg (hereafter referred to as method 1)
involves the initializing of a hyperplane H in a stoichiometric subspace, say 𝑆0, such that
H divides the space into two half-spaces where one half-space is comprised entirely of
unachievable concentrations. The polytope formed from the portion of subspace which
lies in the unachievable half-space is discarded from 𝑆0 so as to reduce its size and
advance toward the true attainable region. Successive repetitions of this step lead to a
tighter bound on the attainable region. In method 1, a hyperplane is placed at a corner of
the polytope 𝑆0 and oriented intuitively by rotating the plane at an angle in a way that its
incline is said to represent an average of the inclines of the planes that meet at the
respective corner. This rotation is fixed and the hyperplane is iteratively translated toward
the region. At each iteration, the hyperplane's validity is checked by ensuring that it meets
the following two criteria (per the definition of the hyperplane employed in this method):
 The feed vector 𝐶𝑓 lies in the positive half space 𝐻≥
 For all non-equilibrium points in the other half-space or on the hyperplane, the
rate vectors point ‘’inward’’ i.e. 𝑛. 𝑟(𝐶) ≥ 0 where n is a vector normal to the
hyperplane.

24
Figure 4-5: ‘’trimming’’ the convex polyhedron to work toward the attainable region (Abraham & Feinberg,
2004).
When these criteria cease to hold, the iteration is taken one step back and the shape
formed from the section of 𝑆0 lying in the negative half-space is discarded, thereby
forming a smaller subspace 𝑆1. These steps are repeated at another corner until the region
𝑆 𝑁 has been smoothed out. The intuitive nature of this method presents room for further
development with the aim of presenting a scientifically sound and possibly more efficient
means of implementing this approach.
Figure 4-6: Method 1 using 20 hyperplanes for a 2-D Van de Vusse kinetics (Abraham & Feinberg, 2004).

25
As the paper by (Abraham & Feinberg, 2004) itself alleges, the method described above
was ‘intended only to provide a conceptual basis for bound construction’’. Up to this
point, the method had yet to be improved and extended to more complex kinetics and
other parameters.
4.2.4.3 Revised Method
In 2010, Ming et al. proposed a revised method (hereafter referred to as method 2) for
computing a candidate attainable region via bounding hyperplanes. This method aims to
reduce the computational complexities associated with method 1 by choosing to maintain
a fixed position of a hyperplane which is then rotated about an edge of the polytope
formed from the stoichiometric subspace. Method 2 begins by selecting a hyperplane that
bounds the stoichiometric subspace and passes through the feed concentration. This
hyperplane is rotated about the feed point for an angle 𝜃 where the direction of the
rotation is chosen such that it will lead to a reduction in size of the subspace 𝑆0. The
validity of the hyperplane is checked whilst ensuring that the feed and equilibrium points
remain within the region being reduced. A rotation matrix 𝑅 ∈ 𝑅 𝑁
is used to effect the
rotations and the direction of rotation is selected so as to reduce the size of convex
polyhedron with each rotation. In a different manner to method 1, Ming et al. chose to
discretize all the concentration points 𝐶∗
lying on the plane so that the rate vector r(𝐶∗
)
associated with each point may be checked for tangency with the plane by computing n.
r(𝐶∗
). When a tangent has been found, the rotation is taken one step back and the
concentration points are recorded. The hyperplane is added to the hyperplanes already
bounding the space. From the newly found extreme point, the method may be repeated
until the subspace 𝑆0 has been ‘’smoothed’’ out.
Comparison between methods - Both methods 1 and 2 use a similar stopping criterion
based on the mathematical principles outlined earlier. Hence, the advantages of method 2
over method 1 are not immediately evident, until both methods are run and their results
compared.
Figure 4-7: Progression of the revised method using rotations of the bounding hyperplanes (Ming, et al., 2010)

26
However, (Ming, et al., 2010) points out 2 problems that arise in method 1, namely the
act of hyperplane discretization and a problem of vertex enumeration. Hyperplane
discretization becomes intensive owing to the degree of the computational process which
(as mentioned earlier) is one less than the degree of the underlying subspace. The latter
problem is said to be a common issue that has long been studied in various fields. It
involves knowing the most efficient means of identifying the extreme points of the
polytope. Finally, (Ming, et al., 2010) points out the issue of redundant hyperplanes that
do not affect the feasible region and should ideally be excluded as soon as possible to
avoid ‘swamping’ the number of planes which describe the region. It then becomes clear
that the revised method offers a significant advantage over method 1 by not being
concerned by any of the issues due to the fixed position of the hyperplane being rotated
For the simple Van de Vusse kinetics, Ming et al. showed that method 1 used 32
hyperplanes in a period of 10s whilst the revised method constructed the candidate region
using 116 hyperplanes in 6s. This result showed that a much higher degree of accuracy
may be obtained in as short a time using method 2. Furthermore, it was also shown that
method 2 may be employed to construct a candidate region using temperature-dependent
kinetics, yet more significantly it was shown to construct an unbounded region in
concentration-residence-time space. This can be used to find the minimum reactor volume
which is useful to the design of batch processes.
Limitations of the revised method - One of the challenges outlined by Ming et al.
involves expanding the method to 𝑅 𝑁
for N > 2. A complication associated with this
challenge is the n number of axes of rotation that exist and the subsequent multiple
rotational pathways to select. This is a very significant limitation as many situations
demand a greater number of reactions involving independent reacting species, hence
resulting in a higher dimensional problem. Both Abraham and Feinberg and Ming et al.
have expressed that the regions obtained by the ‘outside-in’ approach are by no means the
true attainable region, but are only approximations. The inaccuracies of the candidate
region may be minimized by using smaller tolerances or reducing the size of the angle or
translational steps used in either method. These measures inevitably lead to an increased
intensity in the computations and thereby increase the time taken and the power used to
generate a candidate region. However, as computing technology and processor speeds
improve with time, it is important to simultaneously develop the said methods in order to

27
work toward a sufficiently efficient means of constructing an accurate approximate to the
attainable region

28
5 Experimental Procedure and Set-up
It is useful to review the research question presented in section 1 of the research proposal
before outlining the experimental procedure that was undertaken. In an attempt to answer
this question, it was necessary to investigate the current two algorithms that use bounding
hyperplanes in the search for a candidate attainable region. These are the algorithms
developed by (Abraham & Feinberg, 2003) and (Ming, et al., 2010). Detailed
explanations of these algorithms can be found in previous sections of this report.
5.1 Definitions
For convenience to the reader and to allow an easier flow of ideas, some of the terms used
in the following subsections are defined below.
Kinetics
Kinetics refers to a particular reaction system that is being used. Hence it includes the
stoichiometry, the rate expression and the corresponding equilibrium constants.
Grid Type
The grid referred to is the set of points at which rate vectors are evaluated to determine if
the point lies within the attainable region. There are two grid types that have been used:
 Variable Spacing – This grid is constructed using a fixed number of points and
hence, the spacing between the points changes with the length of the grid.
 Fixed Spacing – This grid is constructed using a fixed spacing between successive
points and hence, the number of points being evaluated changes with the length of
the grid.
Grid Size
Grid size simply refers to the fixed number of points specified when a variable spacing
grid is used.
Step Size
In the translational algorithm, step size refers to the distance along which the hyperplane
is translated. Hence, the step size will typically be a scalar value that is used as a factor by
which the normal vector of a hyperplane is multiplied. See (equation 20).

29
Constraints and current polytope
In this context, a set of constraints refers also to a set of hyperplanes each described in the
form shown in (equation 16). The region enclosed by a set of such constraints is referred
to as the current polytope.
Vertex representation
In a similar manner to describing a region by the bounding hyperplanes (constraints), a
polytope can be described by the set of vertices that bound the region. This is known as
the vertex representation of the region.
Angle of Rotation
In the rotational algorithm, the angle of rotation refers to the actual angle that the
hyperplane is rotated toward the region. The angle is employed in a rotation matrix shown
in (equation 21).
5.2 Tools used for the development of the code
Equipment
The code was written and run using a student version of MATLAB R2014a on a laptop
PC with the following specifications:
 Intel Core i5 CPU at 2.5GHz
 64 bit architecture
 4GB RAM memory
 Windows 8.1
External sources of code
A number of external pieces of code were used to carry out some of the steps within the
algorithm. This was done in order to save time and minimise error associated with writing
pieces of code to carry out every sub-task. However, external source codes are sometimes
generalised and do carry out wasteful calculations. Hence, it is important to list the
external source codes used so that the reader of this report can bear in mind the effects of
these on the performance results given later.
 con2vert and vert2con
Retrieved from the Mathworks repositories online, the names of these files are
indicative of their functions. Con2vert.m was used to find the constraints of a
region (the set of bounding hyperplanes) using the vertices of the region whilst

30
vert2con.m was used to find the vertices of a region (vertex representation)
described by a set of constraints.
 allcomb
The allcomb.m function was used to create a list of all possible combinations of 2
sets of points. For example, if two sets of points are defined as:
𝑎 = (1, 2)
𝑏 = (3, 4)
The list of all possible combinations will be given by:
𝑐 = (
1 3
1
2
2
4
3
4
)
5.3 Reproducing the algorithms
The investigation began by reproducing both of the algorithms. This allowed us to
develop a greater understanding of the methods so that more effective tests could be
conducted.
Finding the stoichiometric subspace
As described previously, the outside-in approach involves the removal of unachievable
concentrations from the stoichiometric subspace. Initially an algorithm was developed
with the sole aim of finding the stoichiometric subspace of any set of reactions. The
subspace was successfully plotted in 3D and 2D concentration space. This space would be
the basis from which the candidate attainable region can be found.
The stoichiometric subspace is found by the following method:
i. Define the stoichiometric matrix as the matrix of stoichiometric coefficients
representing the concerned reactions.
For example, for the following reactions:
𝐴 → 2𝐵
1.5𝐵 → 0.35𝐶
The stoichiometric matrix is:
𝑀 = [
−1 0
2 −1.5
0 0.35
]
ii. Define a feed concentration for the concerned species.

31
The feed concentration is a vector where each element represents the
concentration of a different species, such that:
𝑪 =
[
𝐶𝐴
𝐶 𝐵
𝐶 𝐶
⋮
𝐶𝑖 ]
iii. Find the exit concentrations for a range of extents of reaction.
The exit concentrations are found using a simple mass balance over the species.
𝑪 = 𝑪 𝒇 + 𝐴 ∙ 𝑬 (14)
Where: 𝑬 is the vector of extents of reaction with each element representing a
different reaction.
iv. Find the vertices of the region formed by the resulting exit concentrations. Given
the mass balance equation above (14) as well as the fact that concentrations of
species cannot be represented by negative values, it is possible to represent the
stoichiometric subspace using the following format for the constraints:
𝑪 𝒇 + 𝐴 ∙ 𝑬 ≥ 𝟎 (15)
−𝐴 ∙ 𝑬 ≤ 𝑪 𝒇 (16)
The constraints format given in (equation 16) was used as an input to the
con2vert.m function which then computed the vertices of the region. The vertices
obtained are in extent space, so that mass balance equation was used to find the
corresponding concentration space vertices.
Initialising an active hyperplane
With the stoichiometric subspace found and represented both by vertex representation and
inequalities, it is possible to begin the search for the candidate attainable region. The first
step in this search is the selection of a bounding hyperplane to translate or rotate. Before
selecting a hyperplane, it was important to map the hyperplanes to their corresponding
vertices. This allows for (a) particular hyperplane(s) to be targeted. Mapping is achieved
by the population of an incidence matrix such that each row of the incidence matrix
represents a different vertex and each column represents a different hyperplane. By
definition, at least 2 planes intersect to form a vertex. A simple “for” loop was used to run
through each of the vertices and check whether they satisfy the equation of each of the
hyperplanes. The response to the check, being either a 1 or 0, was used to populate the
incidence matrix.

32
5.3.2.1 Rotational method
For the rotational method, a bounding hyperplane is selected as the active hyperplane to
be rotated. In order for a rotation to be valid, it must not exclude the feed or equilibrium
points and it must reduce the volume of the region. Since the stoichiometric subspace is
more often than not represented by a mass balance triangle, this leaves one bounding
plane which can be rotated. As described in the literature review, the hyperplane is
described by its normal vector and a point on the plane. The initial iteration of code for
this investigation relied on choosing the hyperplane which passes through the feed point
as the active plane and the feed point, the vertex about which to rotate (called the “pivot
point”). However, caution must be exercised with this approach as the feed point is not
always a vertex of the stoichiometric subspace.
5.3.2.2 Translational method
For the translational approach, none of the bounding hyperplanes can be used to step into
the subspace as they would most likely transcend the attainable region after a few steps, if
not the first. The method by (Abraham & Feinberg, 2003) sets a hyperplane at a vertex so
that the active hyperplane represents the average of the 2 hyperplanes that make up the
particular vertex. The active hyperplane was thus found by calculating the average normal
of the hyperplanes that make up the vertex. The average normal is calculated as a vector
addition of the 2 normal vectors.
𝑵 𝒂𝒗𝒆 = 𝑵 𝟏 + 𝑵 𝟐 (17)
For ease of calculation and to minimise translational error, the average vector was
normalised so that its magnitude is one unit.
Translating the active hyperplane
The method by (Abraham & Feinberg, 2003) set the active hyperplane at the sharpest
corner of the current polytope and translated inward from thereon. This was achieved by
the following algorithm:
i. Find the angles at all of the vertices of the current polytope.
ii. Arrange the list of vertices from sharpest to least sharp.
iii. Find the incidence matrix according to the arranged list.
iv. Set a hyperplane at the sharpest corner.

33
v. Step into the region at this corner by a particular step size until possible
achievable concentrations are found at which point; translate the plane backward
by one step.
vi. If the region has been reduced in volume/the hyperplane has stepped into the
region, define the new polytope described with the addition of the translated
hyperplane and return to the first step (i.) of this algorithm.
vii. Or else, if no stepping has taken place at this corner, move on to the next vertex in
the arranged list and continue from the second previous step (v.).
The above 7 steps are repeated until the algorithm has approached every vertex of a
current polytope and is unable to step into the region at any corner. As the number of
vertices describing the region increases, so does the accuracy.
5.3.3.1 Finding the angles at the vertices
The angles between two vectors (say A and B) are found using the following principle:
𝑨 ∙ 𝑩 = |𝑨||𝑩| cos 𝜃 (18)
Hence, for unit vectors:
𝑨 ∙ 𝑩 = cos 𝜃 (19)
5.3.3.2 Defining the translated hyperplane
Translations of a hyperplane are carried out fairly easily when one considers the fact that
the normal vector to the hyperplane will remain constant so long as the hyperplane is not
rotated. Since the hyperplane is described by its normal vector and a single point, only the
single point will change with each translation. The location of the single point along the
hyperplane is also insignificant as long as it lies on the plane. For this investigation, the
single point, called the “cursor point” was identified as the vertex at which the active
hyperplane was initialised. For each translation, the cursor point is simply moved in the
opposite direction and along the normal to the hyperplane by a specified step size.
𝑛𝑒𝑤 𝑐𝑢𝑟𝑠𝑜𝑟 = 𝑐𝑢𝑟𝑠𝑜𝑟 − (𝑠𝑡𝑒𝑝 𝑠𝑖𝑧𝑒 × 𝑛𝑜𝑟𝑚𝑎𝑙 𝑣𝑒𝑐𝑡𝑜𝑟) (20)
The direction is chosen as such because the normal to the hyperplanes are defined so that
they point away from the stoichiometric subspace at first and hence they point
consistently outward from the concerned polytope.

34
Rotating the active hyperplane
Figure 5-1: Representation of the rotation algorithm
Vector rotations are achieved by multiplication with the rotation matrix:
𝑹 = [
cos 𝛼 − sin 𝛼
sin 𝛼 cos 𝛼
] (21)
The rotational method proceeded according to the following algorithm:
i. Rotate the active hyperplane by angle 𝛼 until achievable points are found at which
point; the hyperplane is rotated backward by angle 𝛼.
ii. The point at which achievable concentrations are found is also rotated backward
by angle 𝛼 and used as the new pivot point. Later on in the report, the method
used in this step is referred to as the “bounding pivot”.
iii. Return to the first step (i.)
The above 3 steps are carried out until the hyperplane passes through or may exceed the
equilibrium point. In is interesting to point out at this stage that the number of steps
involved in the rotational method is less than half of the steps involved in the translational
method.
CB
CA

35
Evaluating points on the active hyperplane
Perhaps one of the most fundamental steps of the algorithm, evaluating points along the
hyperplane leads to the determination of whether the plane has stepped into the candidate
region or not. Points along the hyperplane can be represented as vectors that are the result
of combinations of orthonormal base vectors. The orthonormal base vectors exist within
the null space of the vector normal to the active hyperplane. Hence, in two dimensional
space, the null space to a normal vector is represented by a single perpendicular vector
and points along the plane are generated as translations of a single point along the null
space.
To avoid the futility of evaluating points that lie outside the current polytope, the
hyperplane must be constrained. This is achieved by identifying the points at which the
plane intersects the current polytope and then evaluating points between these intercepts.
The points can be variable spaced or fixed spaced. A variable spacing was used as the
default approach in this investigation, although a fixed spacing was also tested.
5.4 Testing the algorithms
With the algorithms set up, it was possible to search further for an answer to the research
question. A few parameters belonging to each of the methods were selected and reviewed.
The results are presented in the section. However, the following subsections will present
the methods used for each of the tests. The tests were designed to report on the
performance and accuracy associated with changing a certain parameter or step in an
algorithm. Performance based results are concerned with simulation time which is defined
as the total time it takes to run the algorithm which computes the candidate attainable
region. Accuracy based results are obtained by evaluating the percentage by which a
candidate region is larger in volume than the theoretical region that is approximated by a
PFR trajectory for the same reaction scheme.
Selecting a set of kinetics to work with
Both (Abraham & Feinberg, 2003) and (Ming, et al., 2010) have displayed results using
the Van de Vusse kinetics. This set of kinetics was thus chosen as a basis for conducting
the tests so that the candidate region obtained may be compared to those obtained in the
reviewed pieces of literature. The reactions using the Van de Vusse kinetics are given by:
𝐴
𝑘1,𝑘2
↔ 𝐵 (22)

36
𝐵
𝑘3
→ 𝐶
2𝐴
𝑘4
→ 𝐷
The rates of reaction are given by:
𝒓(𝑪) = [
−𝑘1 𝐶𝐴 + 𝑘2 𝐶 𝐵 − 2𝑘4 𝐶𝐴
2
𝑘1 𝐶𝐴 − 𝑘2 𝐶 𝐵 − 𝑘3 𝐶 𝐵
]
For the purpose of this investigation, a two dimensional case was simulated by setting the
rate constants as:
Table 5-1: Rate constants used with the Van de Vusse kinetics
Rate constant Value
𝒌 𝟏 1
𝒌 𝟐 0
𝒌 𝟑 1
𝒌 𝟒 0
Finding the theoretical volume of the attainable region
The theoretical attainable region was found using the inside-out principle for a PFR type
reactor where:
𝑑𝑪
𝑑𝜏
= 𝒓(𝑪) (23)
The rate vectors are integrated over a range on residence time values using the feed vector
as the initial condition to find the corresponding exit concentrations which are
theoretically the achievable concentrations for a PFR reactor. The convex hull of this
theoretical space is used as the theoretical candidate attainable region. It can be shown
that a PFR from the feed is the optimal reactor structure. For each of the tests conducted,
the closer the obtained volume is to this theoretical volume, the more accurate the
candidate region is.

37
Establishing default parameters
For the purpose of conducting the tests described below, standard parameters were
selected for both methods so that a single test would only differ from another by changing
a single objective parameter/method step.
The default settings were selected arbitrarily as the set of parameters which gave the most
accurate results in a short enough time to ender the experiments feasible.
Table 5-2: Default settings used for the translations method
Kinetics 2D Van de Vusse
Step size 5e-4
Grid type Variable spacing
Grid size 20 points
Table 5-3: Default settings used for the rotations method
Kinetics 2D Van de Vusse
Rotation angle 0.09°
Grid type Variable spacing
Grid size 20 points
Testing the rotational method
Two types of tests were considered for both methods. One type of test investigates the
effect of varying a particular input parameter and the other test investigates the possibility
and feasibility of altering a particular step in the method. For the rotational method, the
following two tests were conducted:
A. Parameter: The effect of decreasing the size of the angle of rotation on performance
and accuracy.
B. Method alteration: The manner in which the pivot point is selected each time the
hyperplane is rotated backward.
For the parameter test, the angle of rotation was varied from 18° to 0.0036°. Each time,
the simulation time and the volume of the region were recorded.

38
For the method alteration test, instead of using the “bounding pivot” as described earlier,
a fixed translation size was specified such that the pivot point is translated along the
hyperplane over a distance equal to the specified step size. The pivot point then has a new
location without the need to identify the point at which the hyperplane found achievable
points and rotating that point backward.
Testing the translational method
Similarly, for the translational method, two types of tests were conducted:
A. Parameter:
a. The effect of decreasing the translational step size on performance and
accuracy.
b. The effect of reducing the grid spacing on the active hyperplane on
performance and accuracy.
B. Method alteration:
a. The possibility of using a different hyperplane orientation as opposed to
that of the average of the 2 hyperplanes at a vertex.
b. Comparing the effects of using a variable grid spacing as opposed to a
fixed grid spacing.
 Test A.a. was conducted by varying the step size between 1e-1 and 2.5e-4, each
time recording the simulation time and volume obtained.
 Test A.b. was conducted by switching from a variable grid spacing to a fixed grid
spacing and recording the results using grid values of spacing between 1e-3 and
1e-5.
 Test B.a. was conducted by using a different proportion of the hyperplanes that
make up a vertex, as opposed to using equal proportions (equation 17).
𝑵 𝒂𝒄𝒕𝒊𝒗𝒆 = (𝑟 ∗ 𝑵 𝟏) + ((1 − 𝑟) ∗ 𝑵 𝟐) (24)
Where 𝑟 is the proportion of hyperplane 1. However, the motivation behind this
test is simply to observe the magnitude of the effect this has on the accuracy.
More research should be conducted on the mathematical implications of such a
change before any concrete conclusions are drawn. In addition, the formation of
the incidence matrix should ideally be controlled so as to have a consistent
definition of hyperplane 1 and hyperplane 2. If the results of this investigation

39
prove to be significant, it may warrant further research and coding. Alternatively,
this investigation may show that such a change in the method yields insignificant
results.
 Test B.b. is simply a comparison between the results from test A.b. and tests using
the default variable grid spacing.
Comparing the results from the translational and rotational methods
Finally, the results from both methods were compared with the aim of showing that the
method later developed by (Ming, et al., 2010) provides a significant improvement from
the initial method by (Abraham & Feinberg, 2003). Since the methods use different
approaches to translation, they were compared on performance and relative accuracy by
plotting the accuracy obtained from each method over the same simulation time range.
Selecting a second set of kinetics to test
At this point, the algorithm had been tested on a theoretical reaction set (Van de Vusse
system). In order to identify a possible application of the attainable region method, it was
decided that it should be applied to a reaction system that is commonly encountered in
chemical engineering and that can have an objective function investigated along with the
candidate attainable region. The water gas shift reaction was selected to conduct this
investigation. The water gas shift reaction is often used to upgrade synthesis gas so that it
has a higher H2:CO as required by the Fischer-Tropsch process. For more information,
refer to (Dry & Steynberg, 2004).
Although the water gas shift mechanism can be represented by a single reaction, the
individual steps are governed by different rate kinetics (Naravanan, 2004). The two
dimensional water gas shift mechanism can be described as (Naravanan, 2004):
𝐻2 𝑂 + 𝐻
𝑘1
→ 𝐻2 + 𝑂𝐻 (25)
𝑂𝐻 + 𝐶𝑂
𝑘2
→ 𝐶𝑂2 + 𝐻 (26)
Although the actual kinetics of a water gas shift system are complex and can be defined in
various ways, the purpose of this investigation is simply to show how a candidate region
representing this particular reaction scheme can be used to identify a possible point of
operation that satisfies the optimum design criteria. For this reason, the kinetics were
represented by simple elementary schemes.
𝑟 𝐻2 𝑂 = −𝑘1 𝐶 𝐻2 𝑂 𝐶 𝐻 (27)

40
𝑟𝐶𝑂 = −𝑘2 𝐶 𝐶𝑂 𝐶 𝑂𝐻 (28)
The rate constants were selected as:
Table 5-4: Rate constants used
Rate constant Value
𝒌 𝟏 10
𝒌 𝟐 1
Assessing the objective function
One the candidate region was obtained, the H2:CO was evaluated at different points
within the region. Using the ratio as a third dimension, a 3D graphic was constructed so
that it can easily be identified at which points within the region the target ratio is met.
Comparing the results from the second set of kinetics
In section [5.4.5], a method test was described for the translational method where a
different proportion of each hyperplane at a vertex is used to define the active hyperplane
as opposed to using the exact average of the two hyperplanes. The results obtained from
that test would best be analysed if conducted on separate reaction schemes instead of just
one set. For this reason, the test described above was also conducted on the water gas sift
system.

41
6 Hazard Score Table and probability
Hazard
Identification
What is the
cause of the
hazard
What are
the
consequenc
es
Assessment Before
Controls
Controls envisaged
Assessment after
controls
What is the
impact of the
hazard on the
following
items
PROBABILITY(4)
RISKRANKING
(1or2,or3)x4
What is the
impact of the
hazard on the
following
items
PROBABILITY(4)
ISKRANKING(1or2,or3)x4
Monitoring
Mechanisms
A hazard is anything
that is likely to lead to
an event which has an
adverse effect on your
objective.
List all gases, chemicals,
materials, processes
Event or
situation
leading directly
to the
hazardous event
Immediate
physical or
practical as a
result of the
hazard
Safety1
Health2
Environment3
Preventative
Controls
(Likelihood)
Controls taken to
eliminate hazards or
reduce the likelihood
of the hazard
occurring (barriers)
Reactive Controls
(Impact)
Controls taken to
reduce the immediate
impact of the hazard
occurring (e.g. gas
alarms, fire
extinguishers.)
Safety1
Health2
Environment3
How we know if we
are succeeding?
Include comments
on effectiveness.
(Usually completed
after project
completion)
Exposed electrical
wires
Negligence Electrical
shock
1 1 1 3 9 Regular
maintenance of area
and equipment
Place insulation
tape around affected
area
1 1 1 1 1 No incidents to
report, all
cables/wires
were in excellent
condition.
Tangled cables or Negligence Physical 1 1 1 2 6 Careful storage of Use of cable ties 1 1 1 1 1 No incidents to

42
wires injury computer
equipment
report, use of
cable ties was
not required at
any point.
Overheating of
laptop/desktop
computer
Restricted air
flow due to
dust
accumulation
or over-usage
Physical
injury –
burns,
possibly
resulting in a
fire
3 2 1 3 18 Regular cleaning of
exhaust ports on
laptops and fans for
desktop computers
Use of cooling pads
and additional
cooling (extra fan)
in the case of a
desktop computer
1 1 1 2 6 Regular use of
cooling pads and
cleaning of fans
ensured that no
incidents
occurred.
Power or electrical
surge
Lightning
strike or
fluctuations
in power
supply
Electrical
shock
1 1 1 3 9 Ensure that all
plug points which
are not in use are
turned off and
install surge
protection on
main electrical
board.
Use adaptor plugs
that have built-in
surge/lightning
protection
1 1 1 2 6 Adaptor plugs
had built in
surge protection.
Also, no power
surges occurred.

43
7 Results and Discussion (Student #435141)
Note that some of the figures presented below contain graphs that are representative of
data points. For clarity purposes, some of the graphs use dashed lines to join the data
points. However, the dotted lines do not represent actual data points! Instead, the actual
data points are solely represented by data markers.
Figure 7-1: MATLAB plot showing a candidate attainable region for the Van de Vusse system obtained by the
translational method using the default parameters
7.1 Testing the translational method
Effect of decreasing the translation step size
Figure 7-2: Effect of decreasing the translation step size on accuracy of the candidate attainable region
0
200
400
600
800
1000
1200
0
10
20
30
40
50
60
0.00E+001.00E-03 2.00E-03 3.00E-03 4.00E-03 5.00E-03 6.00E-03 7.00E-03 8.00E-03
Time(s)
Percentagegreaterthantheoretical
volume
Translation step size (concentration units)
Accuracy and time vs translation step size
percentage error time (s)

44
Decreasing the translation step size tended to yield a more accurate (more tightly bound)
candidate attainable region. The phenomenon was tested using the 2D Van de Vusse
kinetics with a feed point of 𝑪 = [1 0 0 0] 𝑇
= [𝐶𝐴 𝐶 𝐵 𝐶 𝐶 𝐶 𝐷] 𝑇
. However, at step sizes
smaller than 0.001 there is an undesirable trade-off with simulation time which begins to
increase rapidly (performance decrease). This significant increase in simulation time
corresponds to a very small increase in accuracy (less than 2% improvement in more than
900 additional seconds). Therefore, while there is an increase in accuracy that results
from using a smaller step size, it is not feasible to reduce the step size beyond the trade-
off point where there is a staggering decrease in performance with an insignificant
increase in accuracy.
Effect of changing the rotation of the active hyperplane on accuracy
The original method by (Abraham & Feinberg, 2004) proposed the initialisation of a
hyperplane at the sharpest corner of a polytope such that the hyperplane represents the
average of the two hyperplanes that make up the respective corner (vertex). Whilst this
approach is intuitive, it lacks a scientific backing. In an attempt to investigate the validity
of this approach as opposed to a different method of orientating the hyperplane, the active
hyperplane was initialised such that it represents a mixed proportion of the two
hyperplanes that meet at the respective vertex. The resulting data shows a slight effect on
the accuracy of the candidate region. The test was conducted on both the Van de Vusse
and the water gas shift reaction systems at various feed conditions. The results have failed
to show a clear and consistent trend, suggesting that the best orientation for the active
hyperplane differs according to the shape of the attainable region. It is evident that an
orientation using the exact average (0.5 proportion) may not result in the greatest
accuracy. Hence, it may be useful to establish which is the more suitable orientation to
work with for a given reaction scheme before applying the attainable region method.

45
Figure 7-3: The effect of changing the active hyperplane orientation on accuracy using the Van de Vusse reaction
scheme.
Figure 7-4: The effect of changing the active hyperplane orientation on accuracy using the water gas shift
reaction scheme.
Effect of using a variable grid size
In this test, we investigated the effect of using a fixed spacing between the points with a
variable grid size. The results show a significant effect on performance with little or no
effect on accuracy.
0
2
4
6
8
10
12
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
Percentagegreaterthan
theoreticalvolume
proportion of hyperplane 1
Effect of changing the hyperplane rotation
feed = [1 0 0 0] feed = [1 0.7 0 0] feed = [0.7 1 0 0]
0
2
4
6
8
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
Percentagegreaterthan
theoreticalvolume
proportion of hyperplane 1
Effect of changing the hyperplane rotation
[0.2, 0.2, 0.01, 0.2, 0.1, 0.01] [0.2, 0.2, 0.3, 0.2, 0.1, 0.01]

46
Figure 7-5: Graph showing the effect of changing the spacing between successive points (on the hyperplane) with
reference to simulation time
From the figure above, it is evident that the magnitude of the spacing between successive
points has no effect on the accuracy of the candidate region. Instead, it does have an
effect on the performance of the algorithm. The performance decreases significantly as
the spacing is reduced because this implies that a greater number of points need to be
evaluated on the hyperplane. This is expected to have an even greater impact in higher
dimensional cases where the null space of the normal to a hyperplane is characterised by
more than one vector. Although, the figure above purports to show that the grid spacing
has no impact on accuracy, it must be noted that the graph only includes a grid spacing in
the range of 0 to 0.0012. Larger values for grid spacing may result in points lying
completely outside the region.
Figure 7-6: Hypothetical situation showing that points being evaluated may lie completely outside the theoretical
region when using a large grid spacing
0
20
40
60
80
100
120
140
160
180
0
1
2
3
4
5
6
0.00E+00 2.00E-04 4.00E-04 6.00E-04 8.00E-04 1.00E-03 1.20E-03
Time(s)
Percentagegreaterthantheoreticalvolume
grid spacing (concentration units)
Accuracy and time vs variable grid spacing
percentage error time
P1
P

47
When this happens, the algorithm is designed to break so as to not produce a region that is
smaller in volume than the theoretical region for a PFR. Hence, the results that have not
been included in the figure are simply results of errors for large values of grid spacing.
7.2 Testing the rotational method
Effect of using a fixed pivot step
Figure 7-7: A bar graph comparing the accuracy obtained from using the bounding and stepping pivot methods
while the remaining parameters are set to their default values.
Using a fixed stepping size for the movement of the pivot point as opposed to rotating the
point at which achievable concentrations are found, resulted in a significantly greater
accuracy. This reason behind this result is more easily explained graphically.
Figure 7-8: The mechanism of the bounding pivot method shown in an exaggerated manner to explain why the
bounding pivot method may yield a larger (less accurate) volume.
2.845691383
1.081730769
0
0.5
1
1.5
2
2.5
3
volume
Comparison of bounding pivot and stepping pivot accuracy
bounding pivot stepping pivot
Rotated plane backward
Active hyperplane
PivotBounding pivot
New pivot

48
Figure 7-9: The mechanism of the stepping pivot method shown in an exaggerated manner to explain why the
stepping pivot method may yield a more accurate region.
However, as with the other parameters, at a certain point, there is an undesirable trade-off
with simulation time.
Figure 7-10: Graph showing the effect on accuracy of changing the step size for a fixed stepping pivot. The data
is shown with reference to simulation time.
The results from both methods were then compared by displaying the accuracy obtained
over the same simulation time range. This test is inherently an investigation as to which
method provides greater accuracy at comparable levels of performance.
0
500
1000
1500
2000
2500
0
0.2
0.4
0.6
0.8
1
1.2
0.00E+00 2.00E-03 4.00E-03 6.00E-03 8.00E-03 1.00E-02 1.20E-02
Time(s)
volume
pivot step size(concentration units)
Accuracy and time vs pivot step size
Rotated plane backward
Active hyperplane
Pivot
New pivot
Fixed Stepping
Pivot

49
Figure 7-11: Comparison of the accuracies obtained from the bounding and stepping pivot methods with
reference to simulation time
Although the stepping pivot may result in a greater number of pivot points or steps in
total, the improvement in accuracy that it provides makes it a more suitable option. From
the figure above, the stepping pivot has yielded accuracy values that are not within the
reach of the bounding pivot method for the same reaction set and parameters. Both
methods reach a maximum accuracy which does not increase significantly after a certain
time. The accuracy that the stepping pivot method “levels off” at is far greater than the
accuracy at which the bounding pivot method “levels off” for the same conditions. As a
recommendation for future research, the effects of using a stepping pivot should be
investigated for different reaction kinetics to consolidate the above results.
Effect of decreasing the angle of rotation
Figure 7-12: Graph showing the effect of decreasing the angle of rotation on speed and accuracy.
0.1
1
10
100
0 50 100 150 200 250 300 350 400
percentagegreaterthanthe
theoreticalvolume
time (s)
Accuracy vs Time
bounding pivot stepping pivot
0
50
100
150
200
250
300
0
5
10
15
20
25
30
35
0 0.5 1 1.5 2 2.5 3 3.5 4
Time(s)
volume
angle of rotation (degrees)
Accuracy and time vs rotation angle

50
Although expected, it provides confirmation that the coded algorithm is working as
expected by yielding a greater accuracy with a smaller angle of rotation. As with
translational step sizes, a smaller angle of rotation allows for a tighter reach near the
actual attainable region whereas a larger angle of rotation may pass over a significant
portion of unachievable points and find achievable points, causing it to step/rotate
backward and end the search prematurely.
7.3 Comparison between the two methods
Perhaps one of the most critical pieces of analysis is the comparison between the data
obtained from using the two outside-in algorithms (translation and rotation). The results
from this analysis could either prove or disprove that the rotational method offers a
significant improvement in performance and accuracy over the translational method.
Figure 7-13: Radial chart showing how the accuracy of the two outside-in methods changes with respect to
simulation time
How to read the radial graph – The radar axis shows accuracy such that it represents
the percentage that the obtained region is greater in volume than the theoretical volume.
Hence, a point lying closer to the centre implies a more accurate region. The angular axis
shows performance such that it represents the time taken to compute the attainable region.
Hence, the simulation time increases in a clockwise direction.
From the radial graph, it is clear that the rotational method offers both performance and
accuracy improvements from the translational method.
0
10
20
30
40
50
0.51
0.53
0.82
1.08
1.51
1.62
2.61
3.46
3.68
4.28
4.86
40.70
62.21
85.39
90.69
177.26
380.72
524.92
636.60
1006.04
translations
rotations

51
For a closer look at why the rotational method offers a performance improvement over
the translational method, it is useful to analyse the results from the MATLAB profiler
which displays the time that it takes for each line of code to run.
Figure 7-14: MATLAB profiler results for rotational method using default parameters
Figure 7-15: MATLAB profiler results for translational method using default parameters
From the MATLAB profiler, we can establish that the use of the external function
“con2vert.m” has contributed to a significant reduction in performance. The external file
is used to find the set of vertices that bound the current polytope at any time during the
algorithm. In the translational method, the vertices are critical points which are found at
each translation to ensure that the equilibrium point is still a vertex and has not been
excluded from the region. More importantly, vertices are found so that the incidence
matric can be populated. The incidence matrix maps the vertices to the hyperplanes that
form them. The incidence matrix (II) is used to calculate the angles at the vertices and

52
then arrange the list of vertices from sharpest to least sharp so that translation may occur
at the sharpest possible vertex of a current polytope. These steps are all exclusive to the
translation method resulting in far fewer calls to the con2vert function by the rotational
method. A variation of the translational method may be tested such that translation occurs
at all angles of the current polytope before forming a new one instead of only at the
sharpest corner. This may remove a significant amount of dependence on the external
function.
7.4 The water gas shift reaction
The water gas shift reaction was tested with the aim of showing the potential usefulness
of the attainable region. The region was plotted in 2 dimensional concentration space
using water and carbon monoxide as the representative species.
Figure 7-16: MATLAB plot of candidate region for the water gas shift reaction scheme (bottom) and three
dimensional plot of corresponding objective function (top).
The feed concentration vector was selected as:

53
𝑪 𝒇𝒆𝒆𝒅
=
[
2
2
0.1
2
1
0.1]
=
[
𝐶 𝐻2 𝑂
𝐶 𝐻
𝐶 𝐻2
𝐶 𝑂𝐻
𝐶 𝐶𝑂
𝐶 𝐶𝑂2 ]
Hence, the initial H2:CO ratio was 0.1. The attainable region was found using the
translational method in a similar manner to that used for the Van de Vusse kinetics with
the default parameters. After finding the candidate attainable region, the corresponding
ratio (objective function) was plotted as a third dimension. This allows us to easily
identify the range of values that would yield the desired ratio. The range of concentration
values can then the plotted against other factors like reactor volume and cost so as to find
the optimum values that satisfies all the optimum criteria.
Finding the reactor setup – After the optimum operational point is selected by
evaluating other factors, the attainable region plot can be used to identify what proportion
of the feed and rate vectors would yield the desired operational point. See section 4 for a
detailed explanation on this phenomenon.
7.5 Conclusion and Recommendations
The results show that the rotational method offers a significant improvement over the
translational method in terms of performance and accuracy. However, there are still
difficulties associated with extending the rotational method to higher dimensions and
other parameter sets (see the literature review). For this reason, the translational method
was investigated in more depth to understand if adjustments to the algorithm may
improve its performance and accuracy yields. From the tests conducted, the following
recommendations are made for further investigation:
 The orientation of the active hyperplane should not be selected as the exact
average value of the two hyperplanes that make up the concerned vertex by
default, but should be varied according to the reaction system and feed point.
 The translational step size should not be so small as to hamper the performance of
the algorithm because small step sizes do not impact the accuracy as significantly
as it is does the performance. It would be beneficial to first establish an acceptable

54
accuracy range for the specific system and then to decrease the step size so as to
ensure that the accuracy obtained is within the acceptable range.
 Finally, for the translational method, a suggestion for further research would be to
compare the effects on performance and accuracy of translating at all the corners
of a current polytope as opposed to translating only at the sharpest corner. This
suggestion is based on the profiler results which shows that the steps taken to find
the angles of each vertex and rearrange the list of vertices from sharpest to least
sharp consumes a lot of simulation time, increasingly so as the list of vertices
expands.
The following recommendations are made for the rotational method:
 A stepping pivot yields greater accuracies in shorter times and is confidently
recommended as the default method for use in the rotational algorithm.
 As with the translational method, the angle of rotation should be decreased so as
to produce acceptable accuracies that are specific to the system being developed.
Very small angles so of rotation do not have as significant an impact on accuracy
as they do on performance.
As a recommendation for the algorithms in general:
 A variable grid spacing allows for a more accurate region to be described and has
a smaller impact on performance when compared to the use of a fixed grid spacing
which may need to be decreased to a very small value in order to avoid
computational errors.

APPENDIX
55
8 Results and Discussion (Student #322211)
8.1 Translation Method
8.1.1.1 Orientation of the hyperplane
It must be ensured that the orientation of the hyperplane (although fixed) in the translation
method, is such that the division of the two regions results in one of the two half spaces
containing only unachievable concentrations. Traditionally this is accomplished by
orientating the hyperplane such that the average of the normal vectors at the point of
interest is used. However, this choice is open to interpretation and was challenged by
considering varying ratios of the normal vectors at the sharpest vertex of the subspace.
Figure 8-1: Effect of varying the hyperplane ratio on accuracy
For a feed vector of C = [1 0 0 0], the results indicate that there was no appreciable
increase in the accuracy of the algorithm when varying ratios of the normal vectors were
used. However, changing the feed-point ultimately resulted in a new candidate attainable
region being formed and it was found that as the ratio of the hyperplanes changed, the
accuracy changed significantly. However, it was also apparent that consistent results were
obtained for ratios of 0.6 and 0.4 and correspondingly, 0.4 and 0.6 with respect to feed
0
2
4
6
8
10
12
0 0.2 0.4 0.6 0.8 1
%errorofcandidateregion
Ratio of hyperplane 1
Effect of altering hyperlane ratio
Feed = [1 0 0 0]
Feed = [1 0.7 0 0]
Feed = [0.7 1 0 0]

Investigating Methods to Improve Attainable Region Computations

Investigating Methods to Improve Attainable Region Computations

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