UNIVERSIT `
A DI PISA
Dipartimento di Ingegneria Civile e Industriale
Corso di Laurea Magistrale in Ingegneria Chimica
Tesi di Laurea Magistrale
SYSTEM IDENTIFICATION OF A MILD COMBUSTION FURNACE ORIENTED TO CONTROL SYSTEM DESIGN
RELATORE: CANDIDATO:
Prof. Gabriele Pannocchia Matteo Savarese
SUPERVISORE:
Prof. Alessandro Parente
CONTRORELATORE: Prof. Chiara Galletti
Acknowledgements
I would like to express my special thanks to professor Pannocchia and pro-fessor Galletti, for their guide and useful advice about this work.
I would also like to thank professor Parente, that with great availability gave me the possibility of doing this research project at ULB.
I would like to extend my gratitude to all the ATM group, for their sup-port and help with experimental activities and their useful advice.
Finally, I wish to thank my parents that made this experience possible and to give me all the support I need.
ed agli amici e colleghi che mi hanno sempre sostenuto
Abstract
The main objective of this thesis is to identify a simplified dynamic model of a semi-industrial furnace operating in MILD combustion regime with a nominal maximal power of 20 kW for control system design application by the use of System Identification techniques.
The identification of the model has been carried out performing tests on an experimental facility located at the laboratory of the Aero-Thermo-M´ecanique Department, Universit´e Libre de Bruxelles.
Along with the experimental campaign, transient CFD simulations has been conducted on a model of that furnace and the obtained results has been validated on experimental data-sets, in order to assess the feasibility of a CFD-assisted System Identification procedure.
Introduction 8
1 State of the art 10
1.1 MILD combustion . . . 10
1.2 MILD combustion modeling . . . 13
1.3 System Identification . . . 17
1.3.1 Linear System Identification . . . 20
1.4 Challenges . . . 22
1.5 Objective and Methodology . . . 23
2 Case Study 26 2.1 Description of the experimental facility . . . 26
2.1.1 Geometry and configuration . . . 26
2.1.2 Instrumentation and measurements . . . 29
2.2 Description of experimental test . . . 31
2.2.1 Selection of MVs and CVs . . . 31
2.2.2 Test Design . . . 32
2.3 CFD Model . . . 33
2.4 Initial Operating Point . . . 35
3 Experimental Campaign 37 3.1 Step tests . . . 37
CONTENTS
3.2.1 Fuel response . . . 38
3.2.2 Air response . . . 41
3.2.3 Cooling response . . . 43
3.3 Final test design . . . 44
3.4 Final test results . . . 46
4 Numerical Campaign 49 4.1 CFD simulation of initial point . . . 49
4.2 Transient CFD model . . . 53
4.3 Time step sensitivity analysis . . . 57
5 Model Identification 62 5.1 Model estimation with experimental data . . . 63
5.2 Model estimation with CFD data . . . 70
5.3 Basic application: PI controller . . . 76
6 Conclusions 83 Appendix A CFD Mathematical Model 86 A.1 Reactive Turbulent Flows . . . 86
A.2 Turbulence Model . . . 88
A.3 Turbulence-chemistry interactions . . . 89
A.4 Radiation Model . . . 90
Appendix B Identification Algorithms 91 B.1 Prediction Error Methods . . . 91
Appendix C Identified Models 93 C.1 Furnace Temperature Models . . . 93 C.2 Chemical Species Model . . . 94
List of Figures
1.1 Flameless VS Flame mode . . . 11
1.2 Conceptual visualization of PaSR model . . . 15
1.3 SI algorithm . . . 18
1.4 PRBS Signal shape . . . 19
1.5 CFD-based system identification procedure . . . 25
1.6 Work Methodology . . . 25
2.1 Furnace configuration . . . 27
2.2 Flow arrangement . . . 28
2.3 3D model of the furnace . . . 30
2.4 Conceptual black box scheme of the furnace . . . 32
2.5 2D Mesh of the furnace symmetry plane . . . 34
3.1 Qualitative representation of the step test . . . 38
3.2 Tf g and Tf urn response to fuel step . . . 39
3.3 Wall temperature response to fuel step . . . 40
3.4 CO2 response to fuel step . . . 40
3.5 Flue gases and furnace temperature response to air step . . . . 42
3.6 Temperature of flue gases at the exit of the heat exchanger . . 42
3.7 Tf urn and Tf g response to cooling step . . . 43
3.8 Final Identification Test Profile . . . 45
3.9 Final test result for Tf urn . . . 46
4.1 Temperature field in the furnace . . . 52
4.2 Velocity field in the furnace . . . 52
4.3 2D Mesh . . . 53
4.4 Time step sensitivity analysis, CO2 response . . . 57
4.5 Comparison between experimental data and CFD estimated model . . . 58
4.6 Distribution of Cu number, 0.01s time step simulation . . . . 60
4.7 Distribution of Cu number, 0.02s time step simulation . . . . 60
4.8 Distribution of Cu number, 0.05s time step simulation . . . . 61
5.1 Simulated response with first and second order transfer func-tion models . . . 63
5.2 Simulated response of a 1st order transfer function model com-pared to experimental multi-step data-set . . . 64
5.3 Simulated response of a 2nd order transfer function model compared to experimental multi-step data-set . . . 65
5.4 PEM estimated model comparison with experimental data . . 66
5.5 Simulated response of a 2nd order discrete time state space model compared to experimental multi-step data-set . . . 67
5.6 Simulated response of a 3rd order transfer function model on Tout data . . . 68
5.7 Simulated response of a 3rd order transfer function model on Tout multi-step data . . . 69
5.8 Unsteady simulation of a PRBS input profile of fuel flowrate, CO2 response . . . 70
LIST OF FIGURES
5.9 Performance of estimated transfer function model from un-steady simulation data . . . 71 5.10 Simulated step response of estimated model and comparison
with experimental data, CO2 . . . 72
5.11 Performance of a 3rd order transfer function model on CO 2
response data . . . 73 5.12 Step response of the 3rd order transfer function model . . . . . 73
5.13 Unsteady simulation of a PRBS input profile of air flowrate . . 74 5.14 Performance of a 1st and a 2nd order transfer function models
on CO2 to air response . . . 75
5.15 Simulation of a 1st and a 2nd order transfer function model on
an experimental step test data-set, CO2 response . . . 75
5.16 Block scheme of the SISO feedback control system . . . 76 5.17 20 ◦C set point step response with control action, open-loop
vs close-loop . . . 78 5.18 Temperature and control action response to 20 ◦C set point
step variation . . . 79 5.19 Temperature and control action response to 2.5 Nm3/h cooling
flowrate step disturbance . . . 80 5.20 Temperature and control action response to a 20◦C set point
step variation plus white noise disturbance . . . 80 5.21 CL simulation of the control algorithm, comparison between
2.1 Starting initial conditions . . . 36
2.2 Starting operating point . . . 36
3.1 Step test values in net variation . . . 38
3.2 Relative input amplitude . . . 47
4.1 CFD modelling approach . . . 50
4.2 Boundary conditions . . . 50
4.3 Comparison between experimental data and CFD results, tem-peratures . . . 51
4.4 Comparison between experimental data and CFD, chemical species . . . 51
4.5 Transient simulation general set-up . . . 56
4.6 Cu average values . . . 59
5.1 AIC value for different estimated models . . . 68
5.2 Skogestad tuning rule . . . 77
6.1 Pros and issues of integrated CFD system identification pro-cedure . . . 85
Introduction
MILD combustion, in the last few years, has become very attractive due to its numerous benefits compared to traditional combustion regimes. In fact, MILD or ”flameless” combustion can combine high thermal efficiency, very low pollutants emission and high fuel flexibility [5, 27]. However, this thesis will not focus on a detailed physico-chemical characterization of such regime, that is still an open research challenge, but on the development of a simpli-fied dynamic model that can be suitable for control system design purposes. All the procedures, techniques and numerical methods that are used to built such dynamic model from raw data go under the name of System Identifica-tion.
In order to design an automatic control system, either simple or advance in structure and algorithms, system identification is of crucial importance, because such control systems are often based on a process model, that must represent correctly the dynamic of the process but have to be also simple in structure, allowing to reduce computational cost. For example, a full-scale transient CFD model can represent very well the dynamic of a process, but it’s not suitable for control purposes due to its really high computational cost. So, by the use of System Identification techniques, it is possible to es-timate a simpler structure model (i.e. transfer functions, state space, ecc...) from raw input-output dataset, that can be obtained through specific exper-imental campaigns or rigorous simulations (such as CFD).
In the following chapters, a more detailed description of some System Iden-tification techniques and an overview of MILD combustion features is given.
After this initial review of literature, the experimental campaign on the furnace under study has been described, together with data treatment, model estimation and validation. In the last part the results of transient two-dimensional CFD simulations of the experimental system are reported, in order to check the feasibility of a CFD assisted control system design proce-dure [41], that might have some industrial interest.
Chapter 1
State of the art
1.1
MILD combustion
Combustion has always had a very important role in the energy and indus-trial sector, accomplishing energetic needs mainly by the use of non-renewable and fossil fuels. Traditional combustion systems are therefor characterized by consistent emissions of greenhouse gases, mainly CO2, and other
pollu-tants, such as NOx and SOx. This led to very serious climate issues, so
nowadays any efforts in combustion research is mainly oriented to reducing environmental impact achieving at the same time high thermal efficiency. MILD combustion seems to be a very attractive solution, due to its very low amount of pollutants produced and its large fuel flexibility, combined with high thermal efficiency.
The acronym MILD states for Medium or Intense Low Oxygen Dilution, re-ferring to the low Oxygen concentration at which combustion reactions occur in this regime. An extensive description of physical and chemical phenomena behind MILD combustion is provided by Cavaliere and De Joannon [5], but a complete understanding of this regime is still an open research challenge.
This combustion regime is characterized by an uniform temperature field in the combustion chamber, with the suppression of high temperature peaks. A
Figure 1.1: Flameless VS Flame mode
stable flame front is then no longer visible and MILD is sometimes referred as colorless or flameless combustion, as shown in Figure 1.1. This is possible by means of two key factors:
• Pre-heating reactants above their self-ignition temperature (Tsi)
• Recirculating a large amount of flue gases into the reaction zone Pre-heating reactants above Tsi is necessary to create a process that is
ther-mally self-sustained, while the high degree of recirculation limits the temper-ature increase due to combustion reactions. According to Cavaliere and De Joannon [5], a more accurate definition is: “A combustion process is named Mild when the inlet temperature of the reactant mixture is higher than mix-ture self-ignition temperamix-ture whereas the maximum allowable temperamix-ture increase with respect to inlet temperature during combustion is lower than mixture self-ignition temperature (in Kelvin)”. The formation on NOx and
soot, that occurs at very high temperature, is then suppressed to very low amount.
metal-CHAPTER 1. STATE OF THE ART
in terms of geometrical configuration. A high flexibility is also obtained in terms of fluid-dynamic configurations as there is no need to stabilize a flame front because ignition and extinction phenomena do not occur.
The high thermal efficiency is then achieved combining the combustion pro-cess with heat-recovery, such as recuperative or regenerative heat-exchangers, that can be used to pre-heat the reactants.
Summarizing, the main advantage of MILD combustion are: - Low level of pollutants emission
- Large fuel flexibility
- More uniform temperature field
- More choice in fluid-dynamic configuration, shape and materials - No need of flame stabilization
To achieve MILD combustion is required: - Pre-heat reactants above Tsi
- Large degree of inert flue gas recirculation
Those factors make flameless combustion suitable for industries requiring uniform heating and controlled thermal treatment, i.e. for glass and iron manufacturing.
1.2
MILD combustion modeling
Along with practical development of MILD combustion technology and appli-cations, a particular attention has been paid also on its mathematical mod-eling. Computational Fluid Dynamic (CFD) techniques have been widely used for this purpose, because they can give an accurate representation of physical and chemical phenomena occurring in real combustion systems, ei-ther with complicated geometry. A physical domain is discretized into small regions and a set of non-linear partial differential equations for momentum, heat, mass transfer and transport/reaction of chemical species is numerically solved in every small spaces (computational cells). This allows to calculate the velocity and other scalar fields in the discretized geometry.
More details about modeling of reactive turbulent flows are given in ap-pendix A, but it is here important to focus on what makes MILD combus-tion more difficult to model with respect to other combuscombus-tion regimes, i.e. the turbulence-chemistry interaction. In particular, a combustion model is needed to determine the average source or sink of a chemical species due to chemical reactions for the solution of Favre-averaged Navier-Stokes equa-tions, which are still preferred in MILD combustion over Large Eddy Simula-tion or Direct Numerical SimulaSimula-tions because of the low computaSimula-tional cost. In traditional combustion, chemical reactions usually occur with a time-scale which is much lower than the one of the turbulent mixing, hence it is possible to consider infinitely fast kinetic and hence the reaction rates are determined by the turbulent mixing rate. In MILD regime, due to lower temperatures and to the high degree of dilution, the reaction rates are slow, with a chemical
CHAPTER 1. STATE OF THE ART
time-scale which becomes comparable with the turbulent mixing scale. The competition between chemistry and turbulence is measured by the Damk¨oler number, that is defined as: Da = τt/τcwhere τtand τc are the turbulent and
the chemical time-scale, respectively. In traditional systems Da >> 1 while in MILD regime it has been observed that Da ∼ 1 [10].
Among the combustion models, the Eddy Dissipation Concept (EDC) was observed to be very promising and able to deal with such low Da [22], also because of its ability to handle detailed reaction kinetics. However, some-times EDC fails to reproduce with accuracy the shape and position of the reactive zone, and hence modifications of EDC constants has been suggested. For instance Aminian et al. [1] and De et al. [7] both suggested to increase the constant used to determine the residence time in the EDC fine structures to 1.5 and 3 respectively, in order to account the increase in fine structure volume and residence time, due to the more homogeneous distribution of the reaction zone in a larger volume typical of MILD combustion. Later Parente et al. [28, 29] proposed a model to relate the EDC constant to the local Reynolds turbulent and Damk¨ohler numbers.
Also the Partially Stirred Reactor (PaSR) model [18] seems to be a valid alternative. It assumes that the computational cell is spitted in a reactive zone and in a mixing zone. The final species concentration is given by the mass exchange between these two regions, driven by turbulence. Referring to Figure (1.2), Y0
i is the initial mass fraction of the i species, ˜Yi is the final
average mass fraction and Yi∗ is the mass fraction in the reaction zone. In the computational cell, the mean reaction rate for the species i is calculated
Figure 1.2: Conceptual visualization of PaSR model as: ¯˙ ωi = κ∗ ¯ ρ(Yi∗− ˜Yi) τ∗(1 − κ∗) (1.1)
Where κ∗ is the mass fraction of the reactive zone in the cell, τ∗ is the residence time in the reaction zone and ¯ρ the mean density. The mass fraction of reactive zone is computed as:
κ∗ = τc τmix+ τ c
(1.2)
Where τc and τmix are the chemical and mixing time-scale. τc is estimated
in each cell evaluating the rate of change of chemical species, focusing on species that are controlling the main oxidation process. For the estimation of τmix several approaches has been suggested [9, 28]: it can be based on
the Kolmogorov scale, or by means of a constant τmix = Cmixk or by more
sophisticated approaches based on the concept of fractal structure in turbu-lence or by dynamic computation using a flow-field approach [9].
Regarding chemical kinetics, for an accurate modeling of MILD combustion detailed kinetic schemes are required, but the aim of this thesis is to find
CHAPTER 1. STATE OF THE ART
a suitable compromise between accuracy and computational cost (transient simulation are very time consuming), so detailed kinetic mechanism will not be used in this work. More details about the modeling approach in this thesis are given in the following chapters.
1.3
System Identification
System Identification, according to Zhu [43], is “the field in mathematical modeling of systems (processes) from test or experimental data”. A dynamic model, instead, is a mathematical relation between input and output vari-ables in form of difference or differential equation that describes the behaviour of a system during time. So System Identification represent the way to ob-tain a model from data.
System Identification (SI) essentially requires three different “ingredients”: 1. Data-Set: a sequence of measurement of some selected manipulated
variables (MVs), disturbing variables (DVs) and controlled variables (CVs).
2. Model Set: a set of candidate models in which the best model will be selected.
3. Identification Algorithm: a numerical method to calculate the model parameters.
MVs are the input variables that can be measured and manipulated through actuators. They are selected according to their effect on the CVs, that are the variables that we want to control in the process. DVs, instead, are disturb-ing variables that can be measured but not manipulated (like environment temperature) and they can have some effect on the CVs.
The typical System Identification procedural algorithm is shown in Figure 1.3. Once the list of variables is completed, it is possible to carry out specific
CHAPTER 1. STATE OF THE ART
Figure 1.3: SI algorithm
input signal is superimposed to the MVs, the output response is measured and data are collected. A typical test is to superimpose various step of dif-ferent amplitude and duration to an input variable per time with the system operating in open-loop (OL), that is called step test. A step test allows to know some important information about the process, such as main time con-stant and static gain, while through a qualitative analysis of the response we can understand the nature of system dynamic. Amplitude and duration of the steps are decided on the basis of prior knowledge of the process or through preliminary tuning tests.
However, a step test sometimes could be not so informative about the system dynamic, because it does not excite the process at various frequency, leading to models that are not able to represent correctly the dynamic behaviour.
So, more often input signals such as GBN, PRBS (Figure 1.4) or multiple sinusoid are preferred, because they have also a frequency power spectrum able to excite the system in a desired frequency range.
Figure 1.4: PRBS Signal shape
Once the data has been collected, they have to be pre-treated, removing anomalous peaks, outliers and checking if some portion of data is corrupted. After the pre-treatment, it is possible to start estimating a suitable model from a family of possible candidates. Models can be differs a lot in terms of structure: a first classification is between linear models and non-linear mod-els. Here, in this thesis, only linear time invariant models have been used to describe the system dynamics, because of the complexity of the latter. Linear models can be also classified in SISO or MIMO, continuous or discrete time, and they can have a lot of possible forms and structures.
Once a set of candidates is selected, through a numerical algorithm is possible to estimate the parameters of a model, and such model can be tested sim-ulating it on another indipendent data-set and measuring its performance. We can repeat this procedure for all the candidates and at the end we can select the “optimal” one, in terms of fitting, explaination of phenomena and
CHAPTER 1. STATE OF THE ART
simplicity. Some examples of models and parameters estimation algorithms are given below, to better understand the main idea of the numerical process.
1.3.1
Linear System Identification
Most of industrial processes are time-continuous and they can be modelled through differential equation. For example, we can see a SISO model in the form:
y(n)(t) + a1y(n−1)(t) + ... + any(t) = b0u(m)(t) + b1u(m−1)(t) + ... + bmu(t) (1.3)
Where y(t) represent the output variable at time t, u(t) represent the input variable while the exponent n represent the nth order of derivative. This is
an example of linear time-invariant system, because the model parameters are all constant, nor time or other variables dependent. The (1.3) can be expressed in other forms, like transfer functions if we perform the Laplace transform, or in State Space form if we transform the equation in a system of first order differential equations. But, basically, (1.3) can be seen as:
y(t) = x1(t)θ1+ x1(t)θ2+ ... + xn(t)θn (1.4)
Here y(t) is the observed variable, x1(t), ..., xn(t) are known functions and
[θ1, ..., θn] are a set of constant parameters we want to estimate. Let us
suppose that we have a sequence of measure of y and x1, ..., xnat time 1, ..., N
so that:
So we can arrange the (1.5) in a matrix form: y = Φθ (1.6) Where: y = y(1) y(2) ... y(N ) , Φ = x1(1) x2(1) ... xn(1) x1(2) x2(2) ... xn(2) ... ... ... ... x1(N ) x2(N ) ... xn(N ) , θ = θ1 θ2 ... θn (1.7)
The system in (1.7) has a unique solution for θ if n = N , but usually, when we deal with experimental measurements, N >> n, and it is known from linear algebra that the system:
ˆ
θ = Φ−1y (1.8)
Has not a unique solution. Here ˆθ represent an estimation of the parameters vector, so that:
ˆ
y = Φˆθ (1.9)
Where ˆy is the estimated value of y at each sample time. Now we can define the error between the measured and the estimated value of y(t) as:
(t) = |y(t) − ˆy(t)| = |y(t) − ϕ(t)θ| (1.10)
CHAPTER 1. STATE OF THE ART errors: VLS = N X k=1 2k= N X k=1 |y(k) − ϕ(k)θ|2 (1.11)
The θ vector obtained through the minimization of the function VLS is called
the least square estimation of the parameter set. The least square estima-tor is the most widely used in linear system identification because of its low computational cost, simplicity and applicability to various different model structures.
Also different algorithms are available and some of them have been tested on the case study of this work, but the concept behind those mathematical procedure is slightly the same: select a model family, define an error and a loss function that should be minimized and solve the optimization problem in order to find the parameters. Then, algorithms can differs by applicabil-ity to model families, definition of the loss function and by the optimization algorithm used.
1.4
Challenges
As mentioned before, MILD combustion is still an open research challenge, both in experimental and numerical characterization, but its potentials and features has been extensively studied. An aspect that is not well documented in literature is MILD combustion dynamic oriented to control system design. Control systems are of crucial importance in the industrialization of such combustion devices, because they allow accuracy, reduce fuel and energy usage and advanced algorithms can potentially minimize the production of
pollutants during operational changes.
Dynamic characterization, however, is strongly related to the specific exper-imental or industrial system, it means that different systems have different dynamic response. So, every system needs specific experiments.
The experimental procedure to obtain a valid data-set for model identifica-tion has been extensively studied and is described in the previous chapter. However, on those complex systems such as furnaces, especially when they reach huge dimension, experiments become long and expensive, they could be difficult to design and to apply without exceeding safety conditions. In this way, a CFD-assysted system identification technique could help to overcome some of those difficulties. Very few examples of this procedure has been reported in literature and no experimental evidences of the valid-ity of this technique are reported. An objective of this thesis is to apply this procedure on a furnace case study and at the end validating the results with experimental data. The importance of doing so is to give experimental evidence that this procedure is accurate, giving the basis for further develop-ment. A description of this methodology is reported in the following section.
1.5
Objective and Methodology
The aim of this thesis is to apply system identification techniques on a semi-industrial furnace operating in MILD combustion regime. Raw data for the identification have been obtained through experimental campaigns, with test designed specifically for this purpose. Along with experiments, another
ob-CHAPTER 1. STATE OF THE ART
jective of this work is to test the feasibility of a CFD-based system identifi-cation procedure. A CFD model of the furnace is available in steady-state conditions, however system identification requires a time-dependent data-set with input variable variation. For this reason, CFD transient simula-tions with time varying boundary condisimula-tions are required, thus leasing to very time-consuming and computationally expensive calculations if complex modeling approaches and huge kinetic mechanisms are employed. So, it is important to assess how much a simulation can be “simplified” in order to achieve results useful for model identification with good accuracy and in rea-sonable time. This will be discussed along with the case study CFD model description.
Few example of this relatively new approach are available in literature [13, 40], where the procedure to follow is the following:
- Find a starting operating point of your system through a steady-state simulation
- Switch to transient mode, apply a time-varying boundary condition and run the transient simulation
- Apply linear System Identification to data-set obtained
Other works [41, 24] highlighted also the possibility, once a control system algorithm is constructed, of implementing the control system into the CFD environment by the use of a User Defined Function (UDF). In this way, it is possible to test the control system performance in an environment that is capable to capture all system non-linearities.
new control system design approach. Here, in this thesis, we have the oppor-tunity to compare CFD data with real experiments. The CFD-based system identification procedure and the workflow of this thesis are schematically shown in Figure 1.5 and 1.6, respectively.
Figure 1.5: CFD-based system identification procedure
Chapter 2
Case Study
In this chapter an extensive description of the experimental facility is given, together with details about experimental set-up and the available CFD model. Pure methane has been used as fuel and atmospheric air as oxidant.
2.1
Description of the experimental facility
2.1.1
Geometry and configuration
The system under study, shown in Figure 2.1, is a semi-industrial furnace with a maximal nominal power of 20 kW that can operate either in tradi-tional (flame) or in MILD (flameless) conditions [27].
The combustion chamber has a cubic section of 700mm, it is made of stain-less steel with an internal ceramic fiber insulation layer that helps to reduce heat losses to the environment. Fuel and air are fed co-axially through two injectors of 8.2 and 36 mm of OD. The chamber is also equipped with four cooling finned tubes with an OD of 80 mm and a length of 630mm entering the chamber and simulating an external load, by extracting power from the system. Atmospheric air is used as cooling medium.
In order to increase the thermal efficiency, the furnace is equipped with a recuperative heat exchanger which allows to pre-heat air with hot flue gases,
thus rising the reactive mixture temperature above Tsi. This is the necessary
condition to achieve MILD regime. The chamber is also provided with a glass window for optical access and measurements. A schematic view of the furnace is reported in Figure 2.2.
The furnace is fed with gaseous fuel that can be a mixture of CH4, H2, CO2,
CO and N2, while atmospheric air is used as oxidant. It is possible to select
the fuel composition because the pure gases, which are stored individually, can be mixed before entering the combustion chamber. The system is also provided with a natural gas feeding line, that is taken directly from the gas network. Regarding the flow dynamic configuration, the chamber is
charac-Figure 2.1: Furnace configuration
terized by an internal degree of flue gases recirculation, that are entrained into the reaction zone by backflow that is produced by impinging gases into
CHAPTER 2. CASE STUDY
Figure 2.2: Flow arrangement
the walls. We can thus define a recirculation ratio:
kR = ˙ mf g ˙ mf + ˙ma (2.1)
where ˙mf g, ˙mf and ˙maare flue gases, fuel and air mass flowrate, respectively.
kR can be estimated by a geometrical correlation [27]:
kR= 0.19α 1 1.36 where α = AC AN − 1 2 (2.2)
Here AC stands for the cross-sectional area of the chamber while AN for the
combined area of the fuel and air nozzles. With the actual configuration, the internal recirculation ratio is found to be kR ≈ 15, that is sufficient to
achieve MILD combustion.
Exhaust gases exit co-axially with respect to air and fuel nozzles and enter the finned heat exchanger, pre-heating air while fuel is not pre-heated.
2.1.2
Instrumentation and measurements
For temperature measurement, six thermocouples are positioned along the height of the chamber with a spacing distance of 100 mm, as we can see from Figure 2.3, allowing wall temperature profile measurements. Another thermocouple is positioned near the outlet, and it is assumed to be the tem-perature of the flue gases before entering the heat exchanger. The temper-ature of the chamber, instead, is assumed to be the measured near the top of the furnace. It is also possible to measure the temperature of the exhaust gases exiting the heat-exchanger and the temperature of the cooling air at the outlet.
Regarding chemical species, a FTIR (Fourier Transform Infrared Spectroscopy) gas analyzer is available and allows continuous measurements of flue gases composition. However, sometimes, the amount of pollutants produced (such as NOx) is so low that is not detectable by the analyzer, being lower than
the measure uncertainty of the instrument itself.
The amount of fuel injected is controlled via LabView interface and can be adjusted through a PC and the data are stored, while combustion air and cooling air are manipulated through manual actuator and data cannot be recorded.
So, summarising, the list of available measurement is: – Wall temperature profiles
– Chamber temperature – Flue gas temperature
CHAPTER 2. CASE STUDY
Figure 2.3: 3D model of the furnace
– Outlet flue gas temperature (after the heat exchanger) – Cooling temperature
– Inlet temperature of air and fuel, but not the temperature of the pre-heated air (but can be estimated)
– Concentration measurement of major species, NOx, CO, unless they
2.2
Description of experimental test
2.2.1
Selection of MVs and CVs
The input variables that we can manipulate are essentially: • Fuel flowrate
• Combustion air flowrate • Cooling air flowrate
And they will be the selected MVs. The cooling air flowrate can be seen in two ways, depending on the application of the control system we want to build: we can see the cooling as a way to control the chamber temperature improving precision for laboratory application or we can see that as a heat load, like in an industrial heating application, and so we can be more in-terested in controlling the cooling temperature, either the chamber one. So cooling flowrate can be seen either as a MV or a DV.
As a first CVs, the temperature is of course the most important. Regarding which temperature precisely, it is possible to keep all the temperature mea-surements initially, and in a second time decide which one is convenient to keep, referring to possible control configuration. The last CVs are the major chemical species concentrations: i.e., in an industrial application, we can be interested in controlling the O2 concentration, that is an important value
for pollutants emission and furnace performance control. So we can see the furnace as a “black box” with the simplified scheme shown in Figure 2.4:
CHAPTER 2. CASE STUDY
Figure 2.4: Conceptual black box scheme of the furnace
2.2.2
Test Design
The main objective of the test is to investigate the dynamic behaviour of the furnace responding to a variation of the input variables (MVs). The system is operating in open-loop without any PID controller, allowing to identify the free dynamic of the system. A first important test to carry on is a multi step test, that helps to tune amplitude and duration of the steps for a future and final identification test. To carry on this type of test the following procedure has been followed:
1. Select a starting, steady-state, operating point.
2. Impose a first step to a selected MV, followed by a negative one and by a last one that bring the system on the initial operating point, keeping the other constant.
3. Repeat the test with another MV.
The starting point is selected in a way that allows flexibility, especially in terms of temperature variations without exceeding safety conditions: for example, if a positive step to fuel is imposed, the furnace power will increase
with its temperature, so we have to be careful with temperature rising, but we need also to have sensible variation of that variable, with high signal-to-noise ratio.
Once a first complete step test will be done, we will be able to decide a proper shape, amplitude and duration of the identification signal for the final test. In fact, from step tests the main time constants such settling time (tS) will
be available or at least estimated and an empirical rule, according to Zhu [43] is that the average switching time of a GBN signal, in order to “correctly” excite the system is:
TSW ≈
98%tS
3 (2.3)
A proper duration of an identification test should be between 6 and 18 times tS, but can be reduced to 5 times tS if the signal-to-noise ratio is high and
the number of input is low.
In the final identification test simultaneous perturbing signals have been imposed to all the MVs, in order to obtain a data-set for multivariable system identification.
2.3
CFD Model
In order to limit the computational cost, a 2D computational domain was adopted, representing representing half of the symmetry plane of the furnace. The computational grid is fully structured with 31553 cells and is shown in Figure 2.5.
Transient Favre-averaged Navier Stokes equations for continuity, momentum, energy and transport/reaction of chemical species were solved with the finite
CHAPTER 2. CASE STUDY
Figure 2.5: 2D Mesh of the furnace symmetry plane
volume method of FLUENT 19.2 by ANSYS Inc by using a second order dis-cretization method and a SIMPLE algorithm for pressure-velocity coupling. The standard k − ε model has been used to determine Reynolds stresses for turbulence, while chemistry has been modeled with a global reaction for methane oxidation available in FLUENT library, taken from [36]:
CH4+ 2O2 → CO2+ 2H2O (2.4)
The reason of this choice is because with experimental measurements only major species can be monitored and the accuracy obtained with this single-step mechanism is reasonable. Turbulence-chemistry interactions has been modeled using PaSR approach by a UDF implemented by the ULB group [9]. For radiation the discrete ordinate model has been used with the Weighted Sum of Gray Gases model to estimate spectral properties of the gaseous mix-ture.
more detailed description is provided in section 4.2. As boundary condi-tions, mass-flow inlet was set for the fuel and the air inlet, while pressure outlet was set at the gas outlet. On the lateral walls a constant negative heat flux was imposed, taking into account heat losses through the environment and heat transferred to the cooling medium. The heat losses are estimated by an heat balance on the real system, as the starting operating conditions are known after the first experimental test. Further description and expla-nation are given in chapter 4.
2.4
Initial Operating Point
The variables we want to manipulate in experimental tests are: fuel flowrate ( ˙Vf), air flowrate ( ˙Va) and cooling flowrate ( ˙Vc). It is known that if we give a
positive step to ˙Vf the furnace temperature will raise, same behaviour when
reducing ˙Vaand ˙Vc. On the other way, we need a sufficiently high value of ˙Vf
to sustain the MILD combustion process. The initial operating point must be chosen in order to not exceed safety conditions, i.e. too high temperature or rich conditions, but allowing a certain degree of flexibility in terms of in-put variation.
CHAPTER 2. CASE STUDY
Nominal Power 17 kW
Air excess 20 %
Cooling rate 25 Nm3/h
Table 2.1: Starting initial conditions
The nominal power is the value obtained multiplying the Lower Heating Value (LHV) of the fuel by its mass-flowrate ( ˙mf). Selecting the power, we
are able to calculate ˙mf. The air excess is defined as:
a=
( ˙ma/ ˙mf)ef f
( ˙ma/ ˙mf)stoich
− 1 (2.5)
Going in rich conditions means burning with an a < 1, we need to stay in
lean condition, a > 1, while in stoichiometric condition we have a = 1. At
the end, cooling rate has been chosen based on previous experience.
The resulting operating condition at the initial point are shown in Table 2.2:
Starting Operating Condition ˙
Vf V˙a V˙c Tf urn Tf g N Ox
[N m3/h] [N m3/h] [N m3/h] ◦C ◦C ppm
1.709 19.53 25.00 1010 970 4
Experimental Campaign
In this chapter the description of the experimental procedure that has been followed is described. First of all, preliminary step tests have been performed, in order to get an estimation of the main time constants of the system, necessary to proper design the final test. At the end, the final identification test is reported, which is an experiment with multiple input signals with a PRBS shape and profile specifically designed.
3.1
Step tests
In this type of test, starting from the initial operating point described in the previous chapter, every MVs is perturbed, once at time, keeping the other MVs constant, with steps, as shown in Figure 3.1.
Steps amplitude, for each MVs, has been kept in a small percentage of the initial value of the variable itself (in range ±10%): this is because we want to identify a linear model of the process, that means that the process must be kept in its linear range around the operating point. Despite the small steps amplitude, the output must vary with sufficient signal-to-noise ratio. Step duration, instead, has been chosen directly during test, in order to have an idea of the settling time of the main outputs.
CHAPTER 3. EXPERIMENTAL CAMPAIGN
Figure 3.1: Qualitative representation of the step test
Summarizing, the input profile imposed are shown in Table 3.1:
Var Initial value Step I Step II Step III Unit
˙ Vf 1.709 +0.085 -0.171 +0.085 Nm3/h ˙ Va 19.53 -1.63 +3.26 -1.63 Nm3/h ˙ Vc 25.00 -2.50 +5.00 -2.50 Nm3/h
Table 3.1: Step test values in net variation
3.2
Step tests results
3.2.1
Fuel response
Various temperatures and concentrations of major species have been mea-sured in response to a step of the fuel flowrate. The experiment highlighted a huge difference in terms of time constant between temperatures and chemical species. Chemical species reach a new steady-state value in a considerably
less time interval (minutes), while temperatures need more than one hour. This big difference in settling time made not possible to conduct the entire multi-step test with all the variables manipulated in the same day, because the furnace must be shutted down during the night and it takes an entire morning to reheat.
As we can see from Figure 3.2, a positive step to fuel flowrate, which means
Figure 3.2: Tf g and Tf urnresponse to fuel step
in this case raising the nominal power by +5% of the original value, leads to higher temperatures, as expected. The dynamic seems to be linear and quite simple, like a first order system. The wall temperature seems to have a very similar behaviour, as shown in Figure 3.3.
Regarding chemical species, as mentioned above, we observed a more rapid change, referring to Figure 3.4. As we can see the new steady-state value for CO2 is reached in few seconds, about two minutes. Why this huge difference
CHAPTER 3. EXPERIMENTAL CAMPAIGN
Figure 3.3: Wall temperature response to fuel step
Figure 3.4: CO2 response to fuel step
material has an high thermal inertia which takes lot of time to heat. But re-garding flue gas and furnace temperature the reason is not clear, considering the fluid dynamic configuration of the chamber with only an internal recircu-lation. An explanation could be that thermocouples are positioned too close to the wall or in stagnation point, where evolution of variables happens in
longer time.
This difference in time constants brings some difficulty in designing a correct final identification test, because to properly excite both species and temper-ature very different frequency spectra of the excitation signal are needed: an high frequency signal for species and lower frequency for temperature. Because the species tracked in the experiment are all major species, they can be predicted with stoichiometric calculation, while temperature is more difficult to predict and, for industrial application, it may be more important to control. For this reason, final test has been focused on the latter.
3.2.2
Air response
Combustion air has, from a quantitative point of view, the smallest effect on temperature, if we consider as a term of comparison with other variables the modified value in percentage of the initial one. Air has, instead, an effect on the diluition of the major species and, in optical of an industrial con-trol system, may act as a manipulated variable to adjust their value without modifying so much the temperature.
We can see from Figure 3.5 that the behaviour of the temperature is slightly the same compared with fuel. Also in this case the response seems to be lin-ear, similar to a first order system, with a settling time longer than one hour. For time issues, it had not been possible to allow the system settle completely on its new steady-state value.
This experiment differs from the others for one variable that exhibited a slightly different transient behaviour, closer to a second order response. As
CHAPTER 3. EXPERIMENTAL CAMPAIGN
Figure 3.5: Flue gases and furnace temperature response to air step
Figure 3.6: Temperature of flue gases at the exit of the heat exchanger
we can see from Figure 3.6, the response has a comparable settling time with the others, but here there is the presence of a “zero” that give a different and faster transient behaviour. So, in the final identification test, we expect not to have higher order in the model of this input-output pair.
3.2.3
Cooling response
Cooling air has only an effect on temperature, because is not mixed with gases in the chamber and the major species concentration depends only on a stoichiometric balance. As we can see in Figure 3.7 the experiment started before the furnace had reached the steady-state value. In other words, when
Figure 3.7: Tf urn and Tf g response to cooling step
the step was imposed, the chamber was not heated up completely. This conditions brought non-linearities in the dynamic and it seems that also hystheresis is present. Non-linearities could arise if we think to the heat transfer coefficient of the finned tube, but hystheresis is no longer possible in a heat transfer system. In fact, we can see from Figure 3.7 that at the end of the test temperatures are going on the steady-state value measured in previous tests. This highlighted the importance of a stable steady-state initial point, otherwise “apparent” non-linearities could arise, corrupting the
CHAPTER 3. EXPERIMENTAL CAMPAIGN
identification.
It is interesting to study the behaviour of the system responding to this variable, because it can be seen as an external load that we need to heat for a possible industrial application, or it can become the most important manipulated variable to increase the accuracy of the chamber for laboratory application and reducing considerably the time required to heat and cool to reach a new desired temperature.
3.3
Final test design
We observed during preliminary test that the tsettling for almost all the
tem-peratures is close to 1.5h. For chemical species, instead, we observed a set-tling time of about one minute. If we chose a tclock comparable to the time
constant of species, that would be:
tswitch '
ts,species
3 ' 40s (3.1)
With this tswitch it would have been very difficult to manage the experiment,
because we have to impose random step every 40 s and it is very easy to make mistake with the instrumentation provided in the facility, and also with this tswitch it is impossible to observe a satisfactory response for the identification
of temperature, that in case of methane combustion and for industrial inter-est might be more important, so we will design the experiment in order to identify the temperature response. It means that a correct mean switching time to choose is about 30min and a proper duration of the test about 10h, according to the discussion in previous chapters, in order to have also the
possibility to divide the final data-set into an estimation section and an in-dependent validation section. Due to time constraints mentioned before, the mean switching time of the signal has been reduced to about 20min with a tclockof 10min and the duration of the test has been around 5h. The data-set
obtained will be used for the estimation and the validation of the model. The difference between tclock and tswitch is that the latter can be defined as the
number of steps imposed divided the total time of the test, the first is the minimum time between one step and another.
Such signals can be generated in MATLAB with the System Identification Toolbox, they will have zero mean, the tclock selected and an oscillation value
between ±1, so it is necessary to multiply the signal by the desired step amplitude and add the starting value of the selected MVs. The signals that have been imposed are shown in Figure 3.8:
CHAPTER 3. EXPERIMENTAL CAMPAIGN
3.4
Final test results
The first thing to do is to give a visual inspection to data obtained, to iden-tify if the test has been carried out correctly and if there is some outlier or missing data. It is a common practice in data analysis and also for the following identification procedure to “detrend” the data, removing the mean value in every variable response profile. In this way is also possible to better compare all the variables measured from a dynamical point of view. How a final data-set looks like is shown in Figure 3.9:
Figure 3.9: Final test result for Tf urn
strong effect on output temperatures, while excess air has a lower effect and the cooling an intermediate one. It is not possible to say that only looking to the absolute variation of every input, but we can see it from the relative amplitude that has been imposed, shown in Table 3.2:
Fuel ± 3%
Air ± 5%
Cooling ± 5%
Table 3.2: Relative input amplitude
Now we can give a look to differences in terms of dynamic response for vari-ous temperatures examined:
Figure 3.10: Comparison between various temperature response
We can immediately notice that all the temperatures measured, except one, have a very similar output profile. This is an important feature, because
CHAPTER 3. EXPERIMENTAL CAMPAIGN
perature in the entire chamber, mantaining its value on the initial (and maybe desired) one. The only difference in dynamic response is highlighted with the temperature at the outlet of the recuperative heat exchanger, as reported in the previous chapter. Probably, this is due to the strong interaction be-tween the air flow and the heat transfer, that increase simultaneously when a positive step is imposed. In this way, air has a strong effect on this output variable that have also a “faster” dynamic than the others. Now it is possible to estimate linear models as discussed in chapter 5.
Numerical Campaign
In the previous chapter, an experimental way to obtain a dynamic identi-fication of the system is described. Of course, experiments are always the most accurate choice in modelling real systems, but they can be very time consuming, energetically expensive and they can also represent a consistent economic loss. So, it is interesting to see if other routes are possible to avoid a comprehensive experimental campaign. As mentioned before, CFD rep-resents a rigorous and “white box” modelling approach that is capable to capture most of the system non-linearities, but it is too complex to be used in a control system algorithm. So, the main idea is to use complex transient CFD simulations to obtain data-set on which the estimation of a linearized model can be done. In this chapter, this different approach has been ex-plained and analyzed and at the end a linear model has been estimated on a CFD data-set and validated on an experimental one.
4.1
CFD simulation of initial point
As a first thing, the starting operating point must be simulated in order to have a stable CFD solution which is the initial condition of the transient simulations. RANS equations are used.
CHAPTER 4. NUMERICAL CAMPAIGN
Turbulence Standard k − ε
Chemistry Single-step global mechanism
Turbulence-chemistry interactions PaSR
Radiation Discrete Ordinate
Table 4.1: CFD modelling approach
As boundary conditions we have (Table 4.2):
Name Type Value Unit
Fuel Inlet Mass-flow inlet 3.3983e-4 kg/s
Fuel Temperature 310 K
Air Inlet Mass-flow inlet 6.9842e-3 kg/s
Air Temperature 876 K
Outlet Pressure Outlet n.a. n.a.
Lateral Walls Constant Heat Flux -5600 W/m2
Table 4.2: Boundary conditions
Fuel inlet temperature has been measured, while preheated inlet air tem-perature should be estimated by a thermal balance on the recuperative heat exchanger. We know the temperature difference of the flue gases from inlet to outlet of the heat exchanger and the fresh air temperature. Supposing an efficiency of the heat exchanger, it is possible to estimate the pre-heating temperature.
Heat flux, instead, is more difficult to predict and it has to be tuned as a pa-rameter in CFD simulations. The value of heat flux takes into account heat losses of the chamber, that can be estimated with a thermal balance once a
first experiment is done. The main results of the simulation are reported in Table 4.3 and compared with experimental data:
Wall temperature profile [◦C]
Tw1 Tw2 Tw3 Tw4 Tw5
Experimental data 967 978 989 992 1003
CFD simulation results 900 905 968 985 999
Relative error 6.88 % 7.47 % 2.08 % 0.73 % 0.36 %
Table 4.3: Comparison between experimental data and CFD results, temperatures
As we can see from the table above, the steady-state simulation gave good results, as the relative error is below 10% for every parameter. Regarding chemical species, instead, we have a bigger relative error, but is inside the uncertainty of measurements of the instrumentation. Results are reported in Table 4.4.
Chemical species vol %
CO2 H2O O2 N2
Experimental data 8.78% 18.58% 3.35% 69.29 %
CFD simulation 8.05% 16.09% 3.00% 72.70 %
Relative error 8.37% 13.40% 5.39% 4.91%
Table 4.4: Comparison between experimental data and CFD, chemical species
Unless a higher degree of accuracy can be obtained, the results of CFD simu-lation seem to be in agreement with experimental data, so this has been
con-CHAPTER 4. NUMERICAL CAMPAIGN
of temperature and velocity are reported in Figures 4.1 and 4.2, respectively.
Figure 4.1: Temperature field in the furnace
Figure 4.2: Velocity field in the furnace
As we can notice, the temperature field is very uniform and, more than a flame front, a confined reaction zone is present. The amount of N Ox is so
difficult to predict in CFD, because at this temperature thermal N Ox are
negligible (under 4-5 ppm).
4.2
Transient CFD model
Figure 4.3: 2D Mesh
The main idea of those simulations is to describe the dynamic behaviour of the furnace under the perturbation of some selected input, around a se-lected initial operating point. The steady-state solution found in the previous section will be the starting point of all the transient simulations.
In practice, when the steady-state solution of the starting point is converged, it is possible to switch the FLUENT solver from a steady to transient mode. At this point, a discussion about transient modelling and solving options is needed.
CHAPTER 4. NUMERICAL CAMPAIGN
by the geometrical mesh, while the discretization in time is decoupled from space and it is represented by a time step that could be fixed or adapting, depending by the solver method. The choice of a correct time step or time stepping is of crucial importance for multiple reasons.
The transient formulation of the mathematical problem can be expressed by the following generic form:
∂φ
∂t = F (φ) (4.1)
where φ is a generic scalar quantity and F is a generic function that in-corporates the spatial discretization. F in CFD problems is very complex and a direct integration of the previous equation is not possible, a numer-ical method is needed. Explicit methods, for example, have the following formulation:
φn+1− φn
∆t = F (φ
n
) (4.2)
where φnand φn+1are the value of φ evaluated at time tnand tn+ ∆t = tn+1.
The word explicit means that the unknown value of φn+1, that is what we
want to calculate, can be expressed as a function of known values at time tn
and directly evaluated as:
φn+1 = φn+ ∆tF (φn) (4.3)
It is important to mention that φ represent the value of the variable calculated in every computational node of the space discretization, so more precisely we have:
where i denotes the generic spatial node, and this transient calculation must be made for every node, so the entire field is updated at every time step. The explicit formulation is very simple and is the less computationally expensive, but its use is limited by numerical stability constraint. The accuracy of a transient method is increased as the time step size is decreased, but at the same time the computational cost becomes higher. But if we increase the time step in an explicit method above a certain threshold, the method become unstable leading to an unbounded solution. Sometimes the time step size required are really small and at the end those method are impracticable. Method that are unconditionally stable are the implicit methods, that have the following formulation:
φn+1− φn
∆t = F (φ
n+1) (4.5)
In this case the solution from one time step to the following is iterative, because the function F depends by the future value of φ, that is unknown. It means that to integrate numerically from one step to the next a certain number of iterations of the solver is needed, and the number of iterations per time step is also a slightly important parameter of the simulation setup. So, this method, respect to the explicit formulation, has the advantage to be unconditionally stable, but its more computationally expensive.
For the purpose of this work, an implicit method must be chosen, in order to have more flexibility with the time step selection. A general criteria about the time step selection is according to the flow Courant number, defined as:
CHAPTER 4. NUMERICAL CAMPAIGN
that represents the ratio between the time step and the time required by the convective flow to cross a computational cell. Even though an implicit method does not suffer of stability problem, as a general rule of thumb, Cu should not exceed the value of 20. Above that value a consistent loss in accuracy can occur.
In FLUENT also adaptive methods are available: those methods have not a fixed time step but they adapt its size step after step following the evolution of Cu. However, for System Identification, its better to work with a fixed time step, because its easier to handle the data-set obtained.
Once the transient method has been chosen, a time-varying boundary condi-tions can be loaded into FLUENT by the use of a transient table. A transient table is .csv file that contains two column vectors: time and the profile of a selected variable over time.
Resuming, the transient set-up used for simulations is shown in Table 4.5: Transient set-up
Transient Formulation First Order Implicit
Time step Fixed
Iterations per time step 50
Table 4.5: Transient simulation general set-up
At this point, a sensitivity analysis for the correct choice of the time step is needed and described in the following chapter.
4.3
Time step sensitivity analysis
This analysis is aimed at answering the following questions: how much the dynamic response is influenced by the time step size? How much the time step can be increased without consistent loss of accuracy? Accordingly, a sensitivity analysis with time step size is mandatory. The same simulation has been carried out with three different time steps and responses have been compared with experimental data. Here, as an example, a simulation with a multi-step profile superimposed to the air flow is reported:
Figure 4.4: Time step sensitivity analysis, CO2 response
The tracked variable is the CO2 concentration and the manipulated variable
is the air flowrate. The time step used are 0.01, 0.02 and 0.05 seconds. Dou-bling the time step means halving the time required for the simulation; this is a key issue, as, despite the 2D feature, the simulation took more than a
CHAPTER 4. NUMERICAL CAMPAIGN
week to simulate 350 s of real time with a 0.01s time step. It can be observed that the 0.01s simulation presented some numerical outliers, whose reason is not fully clarified, but probably due to numerical issue in case of reduced time step.
To test the quality of those simulations, it is not possible to compare di-rectly numerical with experimental data, because the profile imposed to air is different either in size and duration with respect to the experimental test. So, a transfer function model is estimated on FLUENT data-set and then the estimated model is simulated on an experimental data-set, in order to assess the accuracy of the simulations. In this way, we can not only test the quality of simulations, but we can have also an idea of the generality of the procedure. This is what we obtain if a first order transfer function model is estimated on the data-set (Figure 4.5):
We can see from Figure 4.5 that the simulation with the smallest time step is the one that better capture the system dynamic. There is also an offset between the experimental and the estimated new steady-state value, but a difference of 0.1 % is acceptable.
The average flow Courant number obtained with those time steps are re-ported in Table 4.6:
Time Step Cu
0.01s 0.86
0.02s 1.87
0.05s 4.71
Table 4.6: Cu average values
It seems that Cu is low and the time step can be further increased, but those reported are average value over the entire 2D surface, while inside the fur-nace we can have some critical situations with higher Cu related to specific areas of high velocity, like injection or turbulence zones, so it is interesting to report also the Cu contour inside the furnace. Looking at Figures 4.6, 4.7 and 4.8 it can be seen that the areas with the critical value of Cu are the end of the injector and the upper angle of the wall, where there is turbulence due to the impingement of the flow. In the last simulation, a local value of Cu of about 50 in that region is reported, so could be not safe for accuracy to increase the time step above 0.05s.
CHAPTER 4. NUMERICAL CAMPAIGN
Figure 4.6: Distribution of Cu number, 0.01s time step simulation
Figure 4.8: Distribution of Cu number, 0.05s time step simulation
The results in Figure 4.5 showed that the time step of 0.02s is the best trade-off between computational cost and accuracy, reaching the Cu value of 20 only in the high velocity regions, so this time step size has been used for other simulations that are shown in the following chapter.
Chapter 5
Model Identification
From previous chapters, various data-sets have been obtained in two different ways: experiments and transient CFD. In this chapter, those data-sets have been treated and a model estimation has been performed. Various model structures, orders and estimation algorithms have been tested and compared and at the end the estimated models have been simulated on step tests data-set, in order to validate them. At the end, the models with the best trade-off between representation of phenomena and complexity have been chosen and the dynamic behaviour of the furnace has been finally described by merging temperature and species models into a multi-input multi-output final model.
5.1
Model estimation with experimental data
The model estimation is carried out in MATLAB by the use of the System Identification ToolboxTM[20]. It is known that a good model must be simple
and representative at the same time, so we can start estimating simple and low order transfer function models, for example a first and a second order thinking at the results obtained in the step test. Let us select for example the temperature of the furnace and estimate a first order and a second order transfer function model (Figure 5.1).
Figure 5.1: Simulated response with first and second order transfer function models
If we focus on fitting, as shown in Figure 5.1, it seems that the second order transfer function performs better than the first order. We can also be inter-ested in seeing how those models performs on other data-sets, for example we can test those models on step-tests experimental data-sets and see how they behaves with separated single step input, as reported in Figure 5.2.
CHAPTER 5. MODEL IDENTIFICATION
Figure 5.2: Simulated response of a 1st order transfer function model compared to experimental multi-step data-set
of every single input-output pair. This is due to the fact that probably the system under investigation is really a first order system, and not only well approximated by this model. In Figure 5.3 we can see how a second order model perform on other data-set. We can notice that the second order model has some numerical instabilities and does not represent well the dynamic be-haviour of the system on this multi-step data-set. This strange fact could have various reasons: first, when doing an identification test with multiple signals with a particular frequency content, the order of the system may become higher, due to multiple interactions and other more complex phe-nomena occurring; another reason could be that the system response is slow, so the transfer function has poles really close to zero (the higher the time constant, the smaller the pole value) and when adding a zero to the transfer function the system become numerically sensitive and this may generate
in-Figure 5.3: Simulated response of a 2nd order transfer function model compared to experimental multi-step data-set
stabilities.
Now we are interested in testing other identification algorithms and different models structures. The Prediction Error Minimization (PEM) method, for example, performs quite well, as we can see in Figure 5.4.
The PEM method estimates a discrete time model in a state-space form, that is different from the transfer function. However, the model can also be converted in continuous time or in a trasnfer function model too by the use of specific Matlab commands. The PEM estimated method performs also quite well on multi-step data-set, as we can see from Figure 5.5. When various order and structures models have been estimated and tested, an occurring question is: which is the best model between the candidates? As mentioned before, a model should be either representative and simple, so the best model
CHAPTER 5. MODEL IDENTIFICATION
Figure 5.4: PEM estimated model comparison with experimental data
will be a trade-off between complexity and performance. A more objective comparison can be made by the use of the Akaike Information Criterion (AIC). The AIC is defined as:
AIC = N log det 1 N N X 1 ε(t, θN)ε(t, θN)T + 2nP + N (nylog(2π) + 1) (5.1)
Figure 5.5: Simulated response of a 2nd order discrete time state space model compared to experimental multi-step data-set
Where:
− N is the number of sample in the data-set − θN is the vector of estimated parameters
− ε(t) is the prediction error at time t − nP is the number of parameters
− ny is the number of output
According to Akaike’s theory, the most accurate model has the smallest value of AIC. The value itself does not mean anything about the model quality, but it is useful if compared with others AIC value. A comparison of the raw
CHAPTER 5. MODEL IDENTIFICATION
Model raw AIC value
1st order transfer function model 0.9675
2nd order transfer function model -0.7629 2nd order state space PEM estimated model -7.0432
Table 5.1: AIC value for different estimated models
In this case, the smallest value of AIC is achieved by the last model, it means that it is the best trade off between complexity and fitting of data.
Regarding the temperature with a different dynamic behaviour, the temper-ature at the outlet of the heat exchanger, it can be modelled with a 3rdorder
transfer function model (Figure 5.6):
Figure 5.6: Simulated response of a 3rd order transfer function model on Toutdata
We can also compare the simulated response on multi-step data-set (Figure 5.7).
Figure 5.7: Simulated response of a 3rdorder transfer function model on Tout
multi-step data
of this data-set.
Summarizing, different models have been estimated on the final identification data-set and they have been also simulated on different data-sets, to test the transient behaviour in response to isolated steps for every input-output pair. At this point it will be the end use of these identified model that will bring the definitive choice: if a more sophisticated control system algorithm has to be built, maybe a state-space form is more comfortable, while for simple SISO feedback controllers a transfer function model is more suitable.
