Modeling of Combustion Systems A Practical Approach 9 pdf

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1
Introduction to Modeling
Chapter Overview
We begin this chapter by looking at various model categories and
associated kinds of experiments. We then survey various analyt-
ical methods such as qualitative analysis and dimensional anal-
ysis and introduce function shape analysis. Because many times
we consider multidimensional data, we devote a section to per-
ceiving greater than three dimensions. We then define and discuss
basic data classifications, i.e., nominal, ordinal, interval, and ratio
data. Distinguishing among these data types is important — some
mathematical operations have no meaning for certain data types.
In other cases, we must change our analytical methodology. In
closing, we provide a primer on linear algebra and least squares,
and some proofs regarding a generalized mean.
1.1 Model Categories
We consider three possible kinds of mathematical models, each having two
subdivisions:
• Theoretical models
– Fundamental
– Simulations
• Semiempirical models
– General models with adjustable parameters
– Dimensionless models with adjustable parameters
• Empirical models
– Quantitatively empirical
– Qualitatively empirical
© 2006 by Taylor & Francis Group, LLC
2 Modeling of Combustion Systems: A Practical Approach
1.1.1 Model Validation

Validation is the testing of the model with data from the situation of interest.
All models must be validated, but for different reasons. Theoretical models
require validation to define their applicable range. For example, the ideal
gas law is wrong at high pressures and low temperatures. Newton’s second
law of motion is wrong at high speeds approaching the velocity of light. But
within their spheres of applicability they are highly accurate. Simulations
require validation because there may be errors in the computer code, ill-
conditioned problems, improper convergence, etc. Semiempirical models
require some data to determine values of the adjustable parameters. They
cannot even begin without some valid data. But a well-formulated semiem-
pirical model may or may not be valid for new situations, so one must
validate these. Empirical models build from particular data. As a rule, extrap-
olation to new conditions will yield erroneous results. However, if one is
within the bounds of the original data set, empirical models are excellent
interpolators and give valid estimations. Extrapolation is not always obvious
Another way of looking at things is to consider that extrapolations of the
various kinds of models reference different domains. For fundamental phys-
ical models, anything within the system of physics is an interpolation. For
semiempirical models, data representing similar systems are an interpola-
tion. For empirical models, interpolation is constrained to values bounded
by the original data set. Therefore, regardless of the model type, validation
is always a good practice and one should carefully assess whether or not
the results represent extrapolation.
1.1.2 Fundamental Theoretical Models
If we have a very good understanding of the system, it may be possible to
formulate a theoretical model. By theoretical model, we mean a model with
no adjustable parameters, i.e., an a priori model with known form, factors,
and coefficients. A valid theoretical model represents the highest level of
understanding. A theoretical model, insofar as it represents the actual phys-
ics involved, can make accurate generalizations about new situations.

1.1.3 Simulations
A simulation is a computer-generated result based on the solutions of fun-
damental physical equations. An example is computational fluid dynamics
(CFDs), which solves simultaneous equations for mass, momentum, and heat
transfer. It is not possible to do this without some simplifications, so one
must validate the results. However, once validated, the simulations are quite
accurate and good extrapolators for new problems of the same kind. For
example, CFD has shown itself to be valid for fluid flow and heat transfer
© 2006 by Taylor & Francis Group, LLC
for a complex multidimensional data set (termed hidden extrapolation; Chap-
ter 3 gives some tools for finding it).
Introduction to Modeling 3
in combustion systems. However, it is not generally valid for quantitative
combustion kinetics (e.g., as NOx and CO formation), though it may indicate
trends. At present, simulation science is a dedicated profession requiring
detailed domain knowledge.
1.1.4 Semiempirical Models
The next level of modeling is semiempirical modeling — the main subject
of this text. A semiempirical model is a model whose factors and form are
known a priori, but where some or all of the coefficients are determined from
the data. The form of the model may arise from theoretical considerations
or dimensional considerations, or both. Semiempirical models can also cor-
relate simulation data in order to make a smaller, faster, and more portable
model. This kind of empirical fitting requires some special considerations
because simulation data have no random error, only bias error.
1.1.5 Dimensionless Models
Dimensionless models are semiempirical models that arise from a consider-
ation of the units involved in the system physics. Dimensions in this sense
are synonymous with fundamental units, e.g., mass [M], moles [N], length
[L], temperature [T], and time [θ]. (This text uses Arial or Greek typeface

enclosed in square brackets to denote a fundamental unit dimension. If we
wish to specifically note a dimensionless quantity, we will use empty square
brackets [ ].) Dimensionless models assess the system physics only in the
sense of understanding the units associated with the phenomena. Moreover,
we must presume the model form — generally it is a power law relation.
That is, the model has the form
(1.1)
where Y is the dependent variable, a dimensionless group, whose behavior
we wish to correlate; C is a constant; k indexes the n dimensionless coefficients;
X is a dimensionless independent factor or factor group; and a
1
to a
n
are the
exponents for the dimensionless groups.
If the dimensionless model is valid, it provides a scaling law for testing,
enabling a study of system behavior at other than full scale.
1.1.6 Empirical Models
Finally, we may consider empirical models. We distinguish two types. In a
quantitatively empirical model, we know most or all of the factors a priori, but
we do not know the form of the model or the coefficient values. However,
YC X CXXX X
k
a
k
n
aaa
n
a
kn

==
=
∏
1
12
3
123

© 2006 by Taylor & Francis Group, LLC
4 Modeling of Combustion Systems: A Practical Approach
we do know qualitatively which factors belong to the model. One could dare
to define an even lesser type of an empirical model: in a qualitatively empirical
model, we know nothing but the response with certainty; we do not even
know qualitatively most of the factors that are important, though we may
have a menu of possibilities (candidate factors). The data themselves deter-
mine the model form a posteriori in some post hoc procedure.
1.1.7 Problems with Post Hoc Models
All post hoc procedures have pitfalls; there is a good chance that one may
develop a senseless model. Consider the analogy of the drunken shooter
who takes the bet that even in his inebriated state, he can hit a target with
good accuracy during the pitch-black evening. Amid the company’s sporadic
laughter, all agree that the drunken shooter will fire five rounds into his
vacant barn and all will retire for the evening to examine the results at
daylight. About 10:00
A
.
M
. they awaken. Upon inspection, and to all but the
shooter’s amazement, three of five rounds are in the center of the target, one
is not too far off the mark, and only one has strayed a considerable distance.

He wins the bet handily. Now only the shooter is laughing. What happened?
At first light the shooter walked outside and drew a target around his best
three-shot group before stumbling back to bed to reawaken with the others.
This is the pitfall of any post hoc procedure. If we decide what the model is
after the fact, we are prone to commit this kind of error. The problem grows
worse with smaller data sets and larger numbers of candidate factors, as
these elevate the probability of finding a senseless model that fits the data.
Senseless factors or coefficients with the wrong signs often betray this fallacy.
1.2 Kinds of Testing
There are three kinds of testing: no testing, scale testing, and full-scale testing.
1.2.1 No Physical Testing
A thorough and complete theoretical knowledge requires no additional test-
ing. This is the case for certain fluid flow problems, freefall of bodies, etc.
These mathematical models signify the highest level of modeling and rep-
resent great cost, design, and time advantages. Simulations based on rigor-
ous solution of fundamental physics also fall into this category in the sense
that they model behavior without physical testing. However, no simulation
perfectly captures all the physics, and therefore, we must validate such
models with some confirmatory data. Depending on the speed of coding the
problem into the computer, the time for convergence of the simulation, and
© 2006 by Taylor & Francis Group, LLC
Introduction to Modeling 5
the cost of computer time, it may actually be less expensive to perform
physical experiments.
1.2.2 Scale Testing
Scale models are physical models at other than actual size. If the investigator
is successful in specifying the necessary and sufficient factors that determine
the system response, then one may determine appropriate dimensionless
groups or other similarity parameters. If one understands which similarity
parameters are actually necessary and sufficient to characterize the system,

then one may construct a physically similar system at other than actual size
(scale model). Scale testing often inures significant cost and time advantages.
While an accurate theoretical model represents the ultimate in terms of
reduced cost and time saving, scale testing often represents a significant cost
and time advantage over full-scale testing. Moreover, a scale unit has greater
flexibility than the actual unit.
1.2.3 Full-Scale Testing
Full-scale testing represents the lowest level of understanding in the sense
that we are so unsure of the results that we will only believe a full-scale test.
However, full-scale testing does not always represent the greatest test bur-
den. For example, many full-scale units are sufficiently instrumented and
historical data so well preserved that one requires only limited additional
data. This may (or may not) be the case for so-called plant data. By plant data,
we mean the historical data records of an operating process unit. Plant data
often suffer from a number of statistical maladies that we will address later.
Notwithstanding, very often plant data play a very useful role in model
development, and the actual process unit is the ultimate target for the model
development in the first place.
Full-scale testing is not always the most expensive alterative. For many of
the processes we consider, full-scale testing is required because a theoretical
model is simply intractable and scale testing is not credible. Although process
units are dumb and mute, they can physically solve what we can barely
formulate: highly coupled nonlinear systems of differential equations com-
prising simultaneous chemistry and heat, mass, and momentum transfer.
1.3 Analytical Methods
Associated with and derivative of our degree of knowledge are several
analytical methods. A theoretical analysis is an a priori analysis from first
principles. It will consider first principles, physics, and domain knowledge
© 2006 by Taylor & Francis Group, LLC
6 Modeling of Combustion Systems: A Practical Approach

(detailed knowledge of a particular physical system) in order to come up
with a theoretical model, or at least a model form. We make a determined
effort to provide such models for most features of combustion, including
fuel flow and airflow, emissions such as NOx and CO, flame length, and
A dimensional analysis is an assessment of a plausible system model based
on dimensional consistency. Usually we will need some understanding of
the system in order to arrive at appropriate candidate factors. A first step
for dimensional analysis is often a qualitative analysis. A qualitative analysis
is a derivation of the model factors based on domain knowledge. If we know
at least the general shape of the factor response relation, we may perform a
function shape analysis. A function shape analysis is a derivation of a plausible
model form based on response behavior. In this section we first treat quali-
tative analysis, then dimensional analysis, and finally introduce the reader
to function shape analysis.
1.3.1 Qualitative Analysis
A qualitative analysis will seek to identify the important factors and assign some
sign to the candidates (+, 0, –) based on what we know about the system; that
is, will an increase in the factor increase the value of the response (+), leave it
unchanged (0), or decrease it (–)? If the candidate does not change the response
(0), then we remove the factor from the model. This is a valuable analysis to
perform prior to any modeling effort because it forces one to think about the
system and advance a hypothesis. Now one of several things will happen:
1. The data support the model and the investigator has confirmed his
intuition and understanding of the system.
2. The model coefficients are not what the investigator expected (the
wrong sign or unexpected magnitudes), in which case:
a. The model is wrong and the investigator must revise it; thereby
learning occurs
b. The model is right and the design of the experiment or the col-
lection of the data has errors, which the investigator must find.

All of these outcomes are beneficial.
Example 1.1 Qualitative Analysis
Problem statement: Consider NOx as a function of the following
factors:
• x
1
, the furnace temperature
• x
2
, the furnace excess oxygen concentration
© 2006 by Taylor & Francis Group, LLC
heat flux. We introduce these as the need arises beginning in Chapter 2.
Introduction to Modeling 7
• x
3
, the hydrogen content in the fuel
• x
4
, the fuel pressure
• x
5
, the burner spacing, centerline to centerline
• x
6
, the air preheat temperature
• x
7
, the absolute humidity
Qualitatively, what will be the effect of these factors?
Solution: Solving such a problem requires expert knowledge. For

the practitioner that is new to the field, this will require brief
interviews with experts. This is usually an extremely valuable
exercise. In the case where interviewees agree, the interviewer
can say that he has established consensus. Although consensus is
not always right, it is usually right, so we start here.
Mixed opinions force a more critical evaluation of the system. In
such cases, open disagreement should not only be tolerated but
embraced, because only what we do not know constitutes new
learning and knowledge. Even the seasoned engineer should con-
sider outside opinions. The opinions of juniors or new employees
can be a good source of new and creative thinking. Such employ-
ees have had less exposure to the status quo. In such encounters,
at least one party will learn something. Table 1.1 comprises the
author’s opinion.
Consider another example of qualitative analysis regarding a common
plant operation: fluid flow in a pipe.
Example 1.2 Qualitative Analysis for Fluid Flow in a Pipe
Problem statement: Postulate candidate factors that might be
important to correlate pressure drop associated with friction of a
flowing fluid.
TABLE 1.1
Qualitative Analysis of NOx Formation
Factor Factor Description Sign Strength
1 Furnace temperature + Strong
2 Excess oxygen concentration + Moderate
3 Hydrogen content in the fuel + Moderate
4 Fuel pressure – Weak
5 Burner spacing – Weak
6 Air preheat temperature + Strong
7 Absolute humidity – Weak

© 2006 by Taylor & Francis Group, LLC
8 Modeling of Combustion Systems: A Practical Approach
Solution: Most engineers have adequate exposure to fluid flow
and conservation of mechanical energy. We delay these topics
the following factors are important and that they affect the value
of the response (pressure drop):
• v, the velocity of the fluid [L/θ]
• ρ, the density of the fluid [M/L
3
]
• D, the diameter of the pipe [L]
• μ, the viscosity of the fluid [M/Lθ]
• δ, the roughness of the pipe [L]
• L, the length of the pipe [L]
One might imagine other fluid factors to be important, such as
surface tension, diffusion coefficient, etc. However, it turns out
that these are not important considerations for macroscopic pipe
flow.
1.3.2 Dimensional Analysis
Dimensional analysis takes an appropriate qualitative analysis a step farther
by refining the equation form. A dimensional analysis is a method of estab-
lishing a model from consideration of the dimensions (units) of the important
combustion-related factors. Dimensional analysis finds dimensionless factor
ratios. The method can drastically reduce the number of factors required for
fitting and correlation. Buckingham Pi theory gives the degrees of freedom
of the system as a function of the number of factors (f ) and dimensions (d)
according to
(1.2)
If the dimensionless parameters are the proper ones, then we have discov-
ered similarity and we gain an ability to perform scale testing. This combi-

nation of reducing the scale of our testing and the scope (by reducing the
required number of factors needing investigation) has great economic ben-
efits. Generally, our experimentation (cost and time, C) will be proportional
to some base (b) to the exponent of the number of factors (n
f
),
(1.3)
with b = 1.4 as a typical value. To see the kind of reductions that are possible,
consider the following example.
Ffd=−
Cb
n
f
∝
© 2006 by Taylor & Francis Group, LLC
until Chapter 2. However, intuitively, we may understand that
factors. See Appendix C, especially Table C.2, for dimensions of common
Introduction to Modeling 9
Example 1.3 Reduction in the Degrees of Freedom
from a Dimensional Analysis
Problem statement: Use a Buckingham Pi theory to decide if
dimensional analysis can reduce the system of Example 1.2 to
fewer factors. If so, how many dimensionless factors will there
be? Estimate the possible savings in cost and time.
Solution: From Equation 1.2 we need to find f and d. We have six
factors (ΔP is not a factor but a response). We find d by listing the
unit dimensions for all factors and the response. A matrix of
exponents organizes these most conveniently:
For example, the first row of the above matrix specifies .
Collectively, the above factors comprise three dimensions (d = 3:

M, L, and θ). Therefore, we can combine these six factors into F =
f – d = 6 – 3 = 3 groups. So yes, dimensional analysis can dramatically
reduce the system from six factors to three. The approximate reduc-
tion in cost and time is . In other words, if
the foregoing is true, we can cut our experiments by nearly two
thirds.
1.3.3 Raleigh’s Method
The Raleigh method is a technique to determine dimensionless groups. The
general idea is as follows:
1. Use a qualitative analysis to develop a system of candidate factors.
2. Write a power law model according to Equation 1.1.
MLθ
μ
δ
ρ
111
010
011
010
130
010
112
−−
−
−
−−
v
D
L
PΔ

μθ= ML
(. . ) . %14 14 14 64
636
−=
© 2006 by Taylor & Francis Group, LLC
10 Modeling of Combustion Systems: A Practical Approach
3. Construct a dimensional matrix with undetermined exponents for
all n factors and the response.
4. For each dimension, write an equation in the exponents.
5. Choose f – d independent exponents to comprise as many dimen-
sionless groups. Select exponents corresponding to factors you
believe will constitute separate dimensionless groups.
6. Solve for the remaining exponents in terms of the f – d exponents.
7. Group terms under each exponent.
8. Determine the independent exponent values from experiment.
We illustrate with an example.
Example 1.4 The Raleigh Method for Dimensional Analysis
Problem statement: For the pipe flow problem of Example 1.2, use
the Raleigh method to determine the proper form for the exponents.
Solution:
Step 1: From Example 1.2 we obtain the candidate factors.
Step 2: We construct the following power law model:
Step 3: We construct the following matrix, augmented with coef-
ficients:
Step 4: For each dimension, we obtain the following equations:
For M: 1 = a
1
+ a
5
For L: –1 = – a

1
+ a
2
+ a
3
+ a
4
– 3a
5
+ a
6
For θ: –2 = – a
1
– a
3
or 2 = a
1
+ a
3
ΔPC vD L
aaa aaa
=μδ ρ
123 456
MLθ
μ
δ
ρ
111
010
011

010
130
010
1
2
3
4
5
6
−−
−
−
a
a
va
Da
a
La
ΔΔP 112−−
© 2006 by Taylor & Francis Group, LLC
Introduction to Modeling 11
Step 5: There are six factors and three dimensions, leaving 6 – 3 =
3 degrees of freedom. We select μ, δ, and L as factors that we
wish to have in separate dimensionless groups. These corre-
spond to the exponents a, a
2
, and a
6
, respectively. The selection
is arbitrary. We could have selected any three factors. In fact,

we can multiply any two dimensionless groups we choose to
create a new dimensionless group after the fact if we desire.
Step 6: From the first equation, we obtain a
5
= 1 – a
1
, and a
3
= 2 – a
1
from the third. Solving the second equation for a
4
gives a
4
= –1
+ a
1
– a
2
– a
3
+ 3a
5
– a
6
. Substituting the above equations into
this gives a
4
= –1 + a
1

– a
2
– (2 – a
1
) + 3(1 – a
1
) – a
6
or a
4
= –(a + a
2
+ a
6
).
Step 7: Collecting factors under common exponents we obtain
The response is the Euler number N
Eu
, the second dimensionless
group is the reciprocal of the Reynolds number, 1/N
Re
, the third
quantity is the relative pipe roughness, and the fourth is the
length-to-diameter ratio of the pipe. Since the exponents are unde-
termined anyway, for convenience we will substitute a
1
for –a
1
and write the following:
(1.4)

If a power law is applicable and we have selected the appropriate
factors, we should be able to correlate pressure drop due to fric-
tion with only three factors: N
Re
, δ/D, and L/D. Moreover, sys-
tems having identical values of these three factors should behave
identically. That is, N
Re
, δ/D, and L/D are the similarity param-
eters and define similarity for any system scale.
1.3.3.1 Cautions Regarding Dimensional Analysis
This is quite a substantial reduction in experimental effort. It is all the more
amazing considering we have derived a model knowing nothing but the
fundamental units of the important parameters. However, let us list our
heretofore tacit modeling assumptions:
• We have specified all of the important factors.
• A power law model is appropriate.
• The model has physical significance.
Δp
v
C
Dv D
L
D
a
a
ρ
μ
ρ
δ

2
1
2
⎛
⎝
⎜
⎞
⎠
⎟
=
⎛
⎝
⎜
⎞
⎠
⎟
⎛
⎝
⎜
⎞
⎠
⎟
⎛
⎝
⎜
⎞
⎠
⎟
aa
6

NCN
D
L
D
a
aa
Eu
=
⎛
⎝
⎜
⎞
⎠
⎟
⎛
⎝
⎜
⎞
⎠
⎟
Re
1
26
δ
© 2006 by Taylor & Francis Group, LLC
12 Modeling of Combustion Systems: A Practical Approach
By no means should we take these assumptions for granted. It turns out
from experiment and theoretical considerations that
where
Here, N

F
is the Darcy–Weisbach form of the friction factor — a function of
the Reynolds number and the relative roughness of the pipe — and φ()
denotes an implicit function of the enclosed factors with respect to the
response, N
F
. That is, Equation 1.4 conforms to the present case when C =
1/2 and h = 1 if
(1.5)
We can correlate data with the above equation, so in the present case the
method will work. However, the equation
is the most widely used equation to correlate the friction factor data.* Invert-
ing gives
(1.6)
Clearly, Equation 1.6 requires an iterative solution for N
F
; it is not conform-
able to Equation 1.5.
In the present case, we have only correlated the pressure drop due to
friction. It turns out that Equation 1.7 correlates the total pressure drop,
friction + fluid motion:
(1.7)
* This is known as the Colebrook equation.
NN
L
D
FEu
=
⎛
⎝

⎜
⎞
⎠
⎟
1
2
NN
D
F
=
⎛
⎝
⎜
⎞
⎠
⎟
φ
δ
Re
,
NN
D
F
a
b
=
⎛
⎝
⎜
⎞

⎠
⎟
Re
δ
1
2
1
37
251
ND
NN
F
F
=− +
⎛
⎝
⎜
⎞
⎠
⎟
log
.
.
Re
δ
N
D
NN
F
F

=
−
+
⎛
⎝
⎜
⎞
⎠
⎟
1
2
1
37
251
log
.
.
Re
δ
NKNN
D
FEu
=+
()
+
⎛
⎝
⎜
⎞
⎠

⎟
∑
1
2
1
1
2
Re
,
δ
© 2006 by Taylor & Francis Group, LLC
Introduction to Modeling 13
where ΣK is the sum of K-factors (friction factors) for all flow disturbances,
terms, it is not strictly a power law model and a dimensional analysis may
not arrive at this form.
Furthermore, unless we know a lot about the system we are studying,
there is a very real possibility that we will miss some important factors. For
example, we know that the molar volume of a gas is a function of the
pressure, and temperature only over quite a wide range. Can we perform a
dimensionless analysis and reduce the system?
Example 1.5 The Raleigh Method Applied to an Ideal Gas
Problem statement: For ideal gases, the molar volume of a gas is
a function of the pressure and temperature only. Use a dimen-
sional analysis to propose useful dimensionless groups for corre-
lating the data.
Solution: We organize our data in a matrix:
However, the matrix has more dimensions than factors. This
means we have F = 3 – 4 = –1 degrees of freedom. But unless F ≥ 0,
the matrix is overspecified and insoluble. How can this be? It can
be because we have neglected a very important parameter — the

universal gas constant, R. Without it, we cannot develop a dimen-
sionally consistent equation. Augmenting the matrix with R, we
obtain a soluble matrix, F = 4 – 4 = 0:
We have no degrees of freedom; therefore, only one dimensionless
group is possible, and it is . The power law model would
be
MLT θ
P
V
T
1102
1300
0010
−−
−
MLT θ
P
V
T
R
1102
1300
0010
0212
−−
−
−−

PV RT


PV
RT
k

⎛
⎝
⎜
⎞
⎠
⎟
=
α
© 2006 by Taylor & Francis Group, LLC
such as elbows, tees, sudden contractions and enlargements, etc. (See Table
B.3 in the Appendices for such K-factors.) Since Equation 1.7 has additive
14 Modeling of Combustion Systems: A Practical Approach
and we would evaluate the constants, k and α, from the data. If
the data were from a combustion system, the ideal gas law would
be adequate and our data would generate α = 1 and k = 1.
In the above example, we could not form the proper dimensionless rela-
tionship because we did not have all of the starting parameters. However,
even if the starting factors can form a consistent dimensional equation, that
does not guarantee that the dimensionless equation will be appropriate: for
real gases at low temperatures or high pressures, the dimensionless formu-
lation in the above example would not be adequate. For example, if we define
we can rewrite the ideal gas law as From statistical
mechanics, one may write a virial equation:
where a
0
– a

k
are constant coefficients. If the virial equation is the governing
equation, then we have improperly specified our slate of starting factors
because we have excluded terms having powers of reciprocal volumes.
Below is another model. The principle of corresponding states says that real
gases behave very much like one another if they comprise the same ratio
with their respective critical pressures (P
c
) and temperatures (T
c
). Such critical
properties are unique for each pure gas, and Z = φ(T
r
, P
r
), where
and φ( ) designates an implicit (unspecified) functionality. We refer to T
r
and
P
r
as the reduced temperature and pressure, respectively. Since it did not
occur to us to list the critical pressure and temperature in our factor list for
the dimensional analysis, we did not derive these dimensionless parameters.
We conclude our treatise on dimensional analysis with a final caution:
cramming four factors together in a dimensionless group does not hold the
universe hostage. Unless one has done experiments validating all the factors
over several levels, one has not established general similarity. While it is true
that a valid equation must be dimensionally consistent, the converse is not
true; dimensionally consistent equations may be nonsense. For example,

is dimensionally consistent, but does not have physical meaning
for macroscopic flow in a pipe.
The author recommends dimensional analysis — it can be a big time and
cost saver. Use the technique, but be aware of its pitfalls and let the data
validate your conclusions. Dimensional analysis is a very powerful tool for
establishing similar systems and correlating data. Many engineers use it in
ZPVRT=

, PV ZRT

= .
Za
a
V
a
V
a
a
V
k
k
k
=++ +=+
=
∞
∑
0
12
2
0

1
ˆˆ

T
T
T
P
P
P
r
c
r
c
==,
PL=μ 2
© 2006 by Taylor & Francis Group, LLC
Introduction to Modeling 15
a wide variety of disciplines. But it is only accurate if our starting model is
appropriate (e.g., a power law model) and provided that we have specified
and varied all the physically significant factors. Table 1.2 summarizes a few
dimensionless groups that are often important in combustion.
1.3.4 Function Shape Analysis
Here we introduce the reader to a new kind of analysis — function shape
analysis. By function shape analysis, we mean a technique for developing
equations by direct inspection of the curvilinear form. This also is a weak
form of analysis because we do not consider any underlying physics, except
as expressed by the shape of the function. Notwithstanding, function shape
analysis can provide some insight or at least obviate an incorrect model form.
dependent variable (y) on the vertical axis (positive direction up) and the
independent variable (x) on the horizontal axis (positive direction right). It

has two asymptotes: a vertical one at x = 0 and a horizontal one at y = 0
given by the dashed lines. Obviously, the data are not linear or even para-
bolic; e.g., models of the form or are not appro-
priate. Traditionally, practitioners characterize functions by the monikers
linear, second order, third order, etc., referring to the highest power of the
independent variables. These are quite familiar shapes (lines, parabolas,
cubics, etc.), and they are used in all technical disciplines. However, this
TABLE 1.2
Some Dimensionless Groups
Dimensionless Number Ratio Symbol
Euler Pressure/inertial N
Eu
= ΔP/ρv
2

Reynolds Inertial/viscous N
Re
= Lvρ/μ
Froude Inertial/gravitational N
Fr
= v
2
/gL
Prandlt
=
N
Pr
= C
p
μ/k

Force Group
Buoyant ΔρgL
Gravitational ρgL
Inertial ρv
2
Pressure ΔP
Thermal
Viscous
Note: P is pressure [M/Lθ
2
]; ρ is density [M/L
3
]; v is velocity [L/θ]; L is characteristic length,
height, or diameter [L]; μ is viscosity [ML/θ]; g is gravitational acceleration [L/θ
2
]; C
p
is
isobaric heat capacity [L
2
/θ
2
T]; and k is thermal conductivity [ML/θ
3
T].
Viscous
Thermal
Momentum diffusivity
Thermal diffusivit
y

k
C
v
L
p
ρ
ρ
⋅
μ
ρ
ρ
⋅
v
L
ya ax=+
01
ya axax=+ +
01 2
2
© 2006 by Taylor & Francis Group, LLC
Consider the graph of Figure 1.1. Per the usual convention, we graph the
16 Modeling of Combustion Systems: A Practical Approach
kind of characterization is not what we are after. Function shape analysis
uses a different nomenclature based only on the shape of the function, not
the underlying equation. The presumption is that we do not know the
underlying equation; we know the function shape. We shall consider 12 basic
FIGURE 1.1
A graph with asymptotes. The graph has asymptotes indicated by the dashed lines — the curve
approaches but never reaches these lines for any value of x or y. In the present case, the vertical
line corresponds to x = 0 and the horizontal line corresponds to y = 0.

FIGURE 1.2
Basic function shapes. The chart shows shapes for functions that do not exceed second order
and have no more than one relative extremum. In some cases the graphs have been scaled for
convenience of viewing. For graphs with poles, the pertinent part is shown.
y
x
x
xf
xfy
x
)(
)(
0
∞<<
=
=
1
vh(a)
vmh(e) vl(f) (g) vu
(i) hmh
(j) vml
(k) vmu
(l) vmv
v
2
(h)
l(b) umu(c)
m
2
(d)

(x)
=
(x)
=
(x)
=
(x)
=
(x)
= –x
(x)
(x)
= x
2
=±
√
1–x
2
1
x
x
1
1
x
(x)
=
1
x
(x)
=

(x)
=
1
x
x
1
x
2
0
<< ∞
x
0
<< ∞
x
0
<<
∞ –
∞
< x < 0
x
0
<<
∞
x
<<
∞–∞
x
<<
∞–∞
–1

≤
x
≤
1
–1
≤
x
≤
1
–1
≤
x
≤
1
– x
+ x
2
1
x
+ x
2
–1+x
2
1+x
2
(x)
=
–∞ < x < ∞
–1 < x < ∞
x

2
1+x
(x)
=
x
2
1–x
2
–
© 2006 by Taylor & Francis Group, LLC
shapes per Figure 1.2.
Introduction to Modeling 17
We use the following five descriptors:
• v indicates a vertical asymptote.
• u indicates an unbounded curve.
• m indicates a maximum or minimum point in the curve (i.e., maxima),
either local or absolute.
• l indicates a linear asymptote or straight line.
• h indicates a horizontal asymptote.
vertical asymptote followed by a horizontal asymptote. The 12 figures shown
simple, we mean continuous curves with no more than one descriptor each for
near-, middle-, and far-field behavior. The above nomenclature is sufficient to
characterize more complicated forms, but the simple shapes are more than
enough for our purposes. In addition, the function may undergo var-
ious transformations. If a function shape retains its nomenclature after a trans-
formation, we refer to the function as being invariant under that transformation.
The above function shapes are invariant under the following transformations:
• Translation:
– Vertical translation by +a units:
– Horizontal translation by +a units:

• Reflection:
– Across the y axis:
– Across the x axis:
– Across the line y = x:
• Rotation:
– By any counterclockwise angle θ :
– where φ and ξ are the new vertical and horizontal axes, respec-
tively. One may substitute –θ for clockwise angles.
– For the special case of θ = 90° rotation (π/2 radians): .
– For the special case of a 45° (π/4 radians) rotation:
yx=φ()
yxa=+φ()
yxa=−φ()
yx=−φ()
yx=−φ()
xy=φ()
ξθφθ
θφ
() cos() ()sin()
() sin() ()c
xx x
xx x
=− +
=− −Φ oos( )θ
xy=−φ()
ξφ
φ
() ()
() ()
xxx

xxx
=−
⎡
⎣
⎤
⎦
=− +
⎡
⎣
⎤
⎦
2
2Φ
© 2006 by Taylor & Francis Group, LLC
Therefore, we characterize the curve of Figure 1.1 as vh because it has a
in Figure 1.2 represent all the simple two-dimensional function shapes. By
18 Modeling of Combustion Systems: A Practical Approach
• Scaling:
– Vertical scaling [stretch (a > 0) or shrink (a < 0) vertically by a
times]:
– Horizontal scaling [stretch (a > 0) or shrink (a < 0) vertically by
a times]:
However, functions with arguments greater than zero remain invariant
under positive power transformations.
The function shapes are not necessarily invariant under the following
transformations:
• Power scaling by a:
• Exponential scaling:
In general, a modified Laurent series having terms in order ranging from
Laurent series, we mean specifically Equation 1.8:

(1.8)
We can fit the constants b
2
, b
1
, a
0
, a
1
, and a
2
with least squares if we know
c
1
and c
2
. If either b
1
or b
2
is nonzero, then y will have a vertical asymptote
at x = c
1
or x = c
2
, respectively; thus, we can estimate c
1
and c
2
by inspection

of the curve. That is not to say that other functional forms cannot give the
same shapes — they can. However, the Laurent series provides the most
convenient method for generating them.
1.3.5 The Method of Partial Fractions
In order to arrive at the modified Laurent series, it may be necessary to use
the method of partial fractions. For those readers unfamiliar with the
method, we show it here. It may come about that we wish to transform an
equation having a product in the denominator to one having separate
denominators, e.g., transform
to
The reader may verify that these two equations are identical for all x. We
may find the latter expression via the method of undetermined coefficients.
That is, we presume the identity
ya x=⋅φ()
yxa=
()
φ
yx
a
=φ()
yex=
φ
()
y
b
xc
b
xc
aaxax=
−

()
+
−
()
++ +
2
2
2
1
1
01 2
2
y
xx
=
+
()
−
()
1
12
y
xx
=
−
()
−
+
()
1

32
1
31
© 2006 by Taylor & Francis Group, LLC
–2 to 2 can produce all the basic shapes given by Figure 1.2. By modified
Introduction to Modeling 19
and then solve for A and B. According to a theorem in linear algebra, this
identity always exists. Starting with
we multiply by the denominator at left, . Now if the
foregoing relationship is true for all x, then we may select convenient values
at our choosing. Thus, if x is 2, then the first term vanishes, and when x is
–1, the second term vanishes. For x = 2, we obtain , or ,
and when x = –1, we obtain , or . In general, we can
write
(1.9)
or
(1.10)
where
and
In the above equations φ
1
(x), φ
2
(x), and φ
3
(x) are polynomial expressions
having no repeated roots, and , , and are the inverse func-
tions. In all cases, the degree of the denominator must exceed that of the
numerator. If it does not, we simply divide by the denominator to obtain a
number plus a remainder. Then we expand the remainder as a partial frac-

tion. For repeated roots we may write
(1.11)
and use the method of undetermined coefficients to solve for A
1
, and B
1
to B
n
.
1
12
12
xx
A
x
B
x
+
()
−
()
≡
+
+
−
1
12
12
xx
A

x
B
x
+
()
−
()
≡
+
+
−
121=−
()
++
()
Ax Bx
121=+
()
B B = 13
112=−−
()
A A =−13
1
12 1 2
φφ φ φ() () () ()xx
A
x
B
x
≡+

φ
φ
φ
φ
φ
φ
3
1
3
1
3
2
()
()()
()
()
()
()
x
xgx
A
x
x
B
x
x
≡+
A =
()
⎡

⎣
⎤
⎦
−
1
0
21
1
φφ
B =
()
⎡
⎣
⎤
⎦
−
1
0
12
1
φφ
φ
1
1−
()x φ
2
1−
()x
φ
3

1−
()x
φ
φφ
φ
φ
φ
φ
3
12
1
3
1
1
3
2
()
() ()
()
()
()
()
x
xx
A
x
x
B
x
x

n
≡++BB
x
x
B
x
x
n
n
2
3
2
2
3
2
φ
φ
φ
φ
()
()
()
()
++
© 2006 by Taylor & Francis Group, LLC
20 Modeling of Combustion Systems: A Practical Approach
Example 1.6 Partial Fraction Expansion
Problem statement: Use Equation 1.9 to determine the partial
fraction expansion of
Solution: We rewrite the above equation as

to conform to Equation 1.9. Therefore, and .
We find the inverse functions by solving φ
1
(x) and φ
2
(x) for x:
and . Then and
. Then according to Equation 1.9,
and
leading to
or more succinctly,
Now, it is obvious that vmh is equivalent to hmv, being only a reflection
across the y axis. So vmh and hmv are synonymous, i.e., two different terms
with the same meaning. This differs from equivocal — calling two different
things by the same name. Synonyms present fewer problems than equivo-
cations, but we can eliminate even these by adopting a reverse alphabetic
order. Notwithstanding, m will always need to come between two letters
when it occurs.
y
xx
=
−
()
3
21
y
xx
=
⎡
⎣

⎢
⎤
⎦
⎥
3
1
12
φφ() ()
φ
1
()xx= φ
2
21()xx=−
φφ
1
1
1
−
==xx() φφ
2
1
2
12
−
== +xx[() ] φ
1
1
00
−
=()

φ
2
1
012
−
=()
A =
()
⎡
⎣
⎤
⎦
=
()
=
−
=−
−
1
0
1
0
1
1
1
21
1
2
φφ
φ

B =
()
⎡
⎣
⎤
⎦
=
⎛
⎝
⎜
⎞
⎠
⎟
=
−
1
0
1
1
2
2
12
1
1
φφ
φ
3
21
3
12

21
6
21
3
xx
x
x
xx
−
()
=−+
−
()
⎡
⎣
⎢
⎢
⎤
⎦
⎥
⎥
=
−
−
y
xx
xx
=
−
()

=
−
−
3
21
6
21
3
© 2006 by Taylor & Francis Group, LLC
Introduction to Modeling 21
Example 1.7 Function Shape Analysis
Problem statement: For a process heater combusting fuel in air,
the excess air (ε) and oxygen (y
O2
) have the following general
relations: ε = 0 when y
O2
→ 0 and ε → ∞ as y
O2
→ 21%. So the
function shape is vh. Figure 1.3 gives an example for CH
4
com-
bustion in air. Use function shape analysis to select an appropriate
explicit function when .
h
shape is . We will use the generic function shape
variables, letting and . However, we must transform
it to the desired form. First, we reflect it across the y axis:
. Then we translate it so that it has a vertical

asymptote at 21%: . As x →
– ∞, y → 0; this is a horizontal asymptote. However, at x = 0,
, which is not the proper value. To obtain the proper
value we can adjust it with a vertical translation. For the vertical
translation we have . This has the
limiting value of as x → ∞. However, this is not a
significant limitation, as we have already decided that .
FIGURE 1.3
Excess air vs. oxygen for CH
4
.
Excess Air ε
20%
18%
16%
14%
12%
10%
8%
6%
4%
2%
0%
90%
85%
80%
75%
70%
65%
60%

0% 20% 30% 40% 50% 60% 70% 80% 90% 100%10%
0021
2
<<y
O
.
yx x==φ() 1
y =ε
xy=
O2
yx xx=−=−=−φ()1 1
yxa x x=−+=−+ = −φφ()(.)(.)021 1 021
y = 100 21
yxa x=−= −−φ() (. ) .1 0 21 1 0 21
y =−1021.
0021<<x .
© 2006 by Taylor & Francis Group, LLC
Solution: From Figure 1.2, one possible solution for a v function
22 Modeling of Combustion Systems: A Practical Approach
Another way to force the function to zero at x = 0 is to multiply
by x. That is, . From either of these forms, we may
add adjustable parameters to force the function to intermediate
behaviors. For example,
or
of fuel in air resembles the first of these equation forms. Specifically,
where K is a function of the fuel composition.
1.3.5.1 Limitations of Function Shape Analysis
One should always view the concept of something for nothing with great suspi-
cion. Earlier we found that dimensional analysis was not a something-for-
nothing technique. Both the power and poverty of dimensional analysis lie in

its presumptions; likewise, for function shape analysis. We know nothing about
the system that the shape of the curve does not reveal, and we are examining
it only qualitatively. Consider the normal probability curve given by .
In our terminology, this would be an hmh curve. But going backwards, we
may not arrive at this equation. For example, also gives the
same qualitative curve (hmh). Other equations of this type are
We can fit a
1
to the data using the method of least squares described later in
this chapter, if the log transform is appropriate, i.e.,
where a
′
0
= ln a
0
.
.
The agreement is not spectacular, but the curves are qualitatively the same
hmh form. The point is that qualitative analysis will arrive at a qualitatively
yx x=−(. )021
ya
x
x
=
−
⎛
⎝
⎜
⎞
⎠

⎟
1
021.
ya
x
=
−
−
⎛
⎝
⎜
⎞
⎠
⎟
1
1
021
1
021
ε=
−
⎛
⎝
⎜
⎞
⎠
⎟
K
y
y

O
O
2
2
021.
ye
x
=
−(/)
2
2
yx=+11
2
()
ya
x
a
=
+
⎛
⎝
⎜
⎞
⎠
⎟
0
2
1
1
1

ln lnya a x=
′
−+
()
01
2
1
2 x−(/)
2
2
© 2006 by Taylor & Francis Group, LLC
Figure 1.4 compares and contrasts the equations yx=+11() and ye=
In fact, from theory (Chapter 2) the actual relation for combustion
Introduction to Modeling 23
correct form, but there may be important quantitative differences between
the source and target curves. This is an important limitation of the method.
Therefore, it is always preferable to develop a theoretical model or to use
function shape analysis in concert with theoretical modeling. Notwithstand-
ing, a straight-line fit for this curve would have been futile. A polynomial
of second or higher order would have approximated the near-field behavior
well, but not the far-field behavior. In contrast, the proper qualitative curve
is likely to have better results over all.
1.4 Perceiving Higher Dimensionality
One may extend the method of function shape analysis to higher dimensions.
In three dimensions, the curves become surfaces. Greater than three dimensions
generates hypersurfaces. The number and variety of possible surfaces grow
exponentially as the dimensionality increases. The method loses convenience
in greater than two dimensions. Notwithstanding, it is a general benefit to
have some perception of higher dimensions, so we collect some methods
here. Dedicated texts have addressed this problem in a variety of ways.

1–4
With some standard perceptual tricks one may envision data in greater than
three dimensions.
1.4.1 A View from Flatland
Probably the most famous treatise about multiple dimensions is Edwin A.
Abbott’s Flatland: A Romance of Many Dimensions.
5
We will follow Abbott’s
FIGURE 1.4
Two hmh curves. The solid line is the normal ordinate curve y = exp(–x
2
/2), while the dashed
line is an imposter, y = a
0
+ a
1
/(1 + x
2
). The agreement is not spectacular, but both curves share
the same qualitative features.
1.2
1.0
0.8
0.6
0.4
0.2
0.0
-2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5
© 2006 by Taylor & Francis Group, LLC
24 Modeling of Combustion Systems: A Practical Approach

initial approach in expanding our dimensional thinking by first imagining
a two-dimensional creature who has no notion of a three-dimensional world
until a sphere enlightens him. Figure 1.5 shows such a creature as a sphere
penetrates his world.
The sphere enters the plane at a point, expanding as a circle. Once the
sphere’s equator passes through the plane, the circle appears to shrink to a
point and then vanishes. Thus, the two-dimensional creature does not appre-
hend the sphere as a whole, but as a process in time; time functions as a
third dimension. In the same way, a four-dimensional hypersphere entering
our three-dimensional world would appear as a point growing as a sphere,
then shrinking to a point and vanishing. In our mind’s eye, we can replay
this “movie” at will using time as a fourth dimension.
1.4.2 Contour Surfaces
A contour map has one less dimension than the object it represents. Thus,
two-dimensional contour lines represent a surface having extension in three
dimensions. The most familiar is the typical hiking map with contour lines
for elevation. The contour lines represent vertical slices of terrain at equal
intervals of elevation. Close lines mean steep hills; far apart lines mean
shallow ones. So, the three-dimensional contour surfaces for a hypersphere
the contour surfaces for a four-dimensional hyperellipsoid.
FIGURE 1.5
An observer in Flatland. (1) The sphere is traveling at a slow but steady speed on its way
toward a two-dimensional world. (2) The sphere first contacts the plane at t
0
. The observer
(happy face) will see a point in his plane. (3) The point will grow into a circle as the sphere
continues its journey. At t
1
the observer will see a growing circle. (4) At t
2

the circle will reach
its maximum diameter and then begin to shrink. At t
3
the observer will see the diameter shown
on the upper circle. (5) Finally, the circle will shrink back to a point at t
4
just before disappearing.
The sphere will have now passed completely through the two-dimensional world.
t
4
t
3
t
2
t
1
t
0
X
© 2006 by Taylor & Francis Group, LLC
would resemble a series of spheres of various dimensions. Figure 1.6 shows
Introduction to Modeling 25
We may write analytical expressions up to second-order in multiple dimen-
sions without too much trouble. For example, for y = φ(x
1
, x
2
, x
3
, x

4
), one
solves for the variable of interest, say x
1
, as x
1
= φ(x
2
, x
3
, x
4
, y). Each constant
contour surface is found by setting y to a desired value (constant) and then
tracing x
1
= φ(x
2
, x
3
, x
4
) to produce a three-dimensional contour of the hyper-
surface; x
1
= φ(x
2
, x
3
) creates contour lines for a surface extending in three

dimensions.
Example 1.8 Contour Lines for y = f(x
1
, x
2
)
Problem statement: Find the general expression to generate con-
tour lines for
(1.12)
Solution: We may choose either or as our
contour equation. If we choose the latter, the expression becomes
. We may rearrange
this to . Letting
, , and , we may write the
above equation as , having the general solution
where and , or in terms of the original factors,
FIGURE 1.6
Contour surfaces for a hyperellipsoid in four dimensions. A hyperellipsoid is represented as a
series of three-dimensional contour surfaces. The surface is spherical in three dimensions and
elliptical in a fourth dimension.
fourth
dimension
ya ax ax ax axx ax=+ + + + +
01122111
2
12 1 2 22 2
2
xxy
12
=φ(,)

xxy
221
=φ (,)
aaxaxaxaxxaxy
01122111
2
12 1 2 22 2
2
0+++ + + −=
() ( ) ( )ax a axx ax ax a y
22 2
2
21212 111
2
11 0
0++ + + +−=
Aa=
2
2
Ba ax=+
2121
Cax ax a y=++−
11 1
2
11 0
A
xBxC
2
2
2

0++=
x
BB AC
A
2
2
2
4
2
=
−± −
=± −ββγ
β=−BA2 γ=CA
© 2006 by Taylor & Francis Group, LLC