1 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 [...]... the data The old adage “garbage in, garbage out” (GIGO) applies: if our data are of poor quality, they will set a ceiling on the amount of information we can glean from them No amount of post-data analysis can rectify deficiencies in data quality Statistical analysis after the fact cannot improve data quality (but it can allow us to glean the maximum information from the data and it can make data deficiencies... reference state (relative zero) Possess a true (absolute) zero Ratios have meaning Best FIGURE 1.10 Data types Factors may be classified per the above categories The highest order of data is ratio data, followed by interval data, ordinal data, and nominal data Failure to recognize the factor type can lead to serious errors in analysis are data that can take on any intermediate value over an applicable range... a1 x1 + a0 − y) = 0 Letting 2 A = a2 2, B = a2 + a1 2 x1, and C = a1 1x1 + a1 x1 + a0 − y , we may write the 2 above equation as Ax2 + Bx2 + C = 0, having the general solution x2 = − B ± B2 − 4 AC = β ± β2 − γ 2A where β = −B 2 A and γ = C A , or in terms of the original factors, © 2006 by Taylor & Francis Group, LLC 26 Modeling of Combustion Systems: A Practical Approach 2 a +a x ⎞ a +a x ⎞ a x 2 + a1 x1... Design) Ambient Factors (Uncontrollable) Available air-side pressure drop Burner-to-burner spacing Burner-to-furnace wall spacing Heat release/furnace vol ratio Ambient humidity Ambient air temperature Barometric pressure Modeling of Combustion Systems: A Practical Approach Some Potential Factors Affecting NOx Response from a Burner Introduction to Modeling 1.6 35 A Linear Algebra Primer The method of least... see that the temperature ratio is meaningful and identical on an absolute scale The zero (known as absolute zero in this case) is unique In principle, the ratio scale is continuous over some range © 2006 by Taylor & Francis Group, LLC 32 1.5.2 Modeling of Combustion Systems: A Practical Approach Data Quality We should use the highest level and quality of data possible for our analysis We have already... that level of scale refers to whether the data are measured on nominal, ordinal, interval, or ratio scales And we have ranked such data from best to worst according to level: ratio > ordinal > interval > nominal (thus level of scale is itself an ordinal scale) Quality is another ordinal scale for classifying data Quality is the level of care taken in the design of the experiment and the collection of. .. Ordinal Interval Ratio Figure 1.10 illustrates this fundamental scheme 1.5.1 Level of Scale The first basic distinction is to identify discrete vs continuous data Discrete data are those data that can take on only particular and definite values For example, a pipe is available only in discrete sizes In contrast, continuous data © 2006 by Taylor & Francis Group, LLC 30 Modeling of Combustion Systems: A Practical. .. 1.5 Basic Data Classifications We may classify data in several ways: • • • • Level of scale Quality of data collection and care in the experimental design Classification by source Classification by function We discuss each in turn With respect to scale and level, data may be either discrete or continuous and characterized by four basic levels The scales are, in order of increasing level: 1 2 3 4 Nominal... group is possible, and it is PV RT The power law model would be α ⎛ PV ⎞ ⎜ RT ⎟ = k ⎝ ⎠ © 2006 by Taylor & Francis Group, LLC 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... the data quality scale, appropriately planned experiments represent the highest level of data quality * David, F.N., “No discovery of some importance would have been missed by the lack of statistical knowledge,” as quoted by Rao, C.R., in “Statistics: Reflections on the Past and Visions for the Future,” Amstat News, American Statistical Association, Alexandria, VA, September 2004, p 2 (http:www.amstat.org) . 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. 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. 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