Modelling of a Gas Turbine with Modelica TM

Transcription

1 ISSN ISRN LUFD/FR 5668 SE Modelling of a Gas urbine with Modelica M Antonio Alejandro Gómez Pérez Deartment of Automatic Control Lund Institute of echnology May 00

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3 Deartment of Automatic Control Lund Institute of echnology Box 8 SE- 00 Lund Sweden Author(s) Antonio Alejandro Gómez Pérez Document name MASER HESIS Date of issue May 00 Document Number ISRN LUFD/FR SE Suervisors A. Rantzer H. ummescheit Sonsoring organization itle and subtitle Modeling of a Gas urbine in Modelica (Modellering av en gasturbin i Modelica) Abstract he reort describes the develoment of a global model of a simle gas turbine in order to simulate the dynamic behaviour. he Modelica language is used for the creation of the model. he model is based on the evaorative gas turbine located in the Deartment of Heat and Power Engineering in Lund. his turbine was run as a conventional turbine before including the heat exchanger and the evaorative tower. he model can be slit u in three main arts: the comressor, the combustion chamber and the exander. hese three models are based on the equations obtained from thermodynamic literature. Furthermore, information rovided by the manufacturer was used for the imlementation of the comressor and the exander models. Hence, the thesis is focused first on creating a model of a simle gas turbine by using as many comonents of the hermoflow library as ossible and second on extending the library with reusable models for turbines and comressors. Since the model involves mechanical arts, comonents of the rotational sub-library are used for the task. Keywords Object oriented modeling, Reuse, dynamic simulation, Classification system and/or index terms (if any) Sulementary bibliograhical information ISSN and key title Language English Security classification Number of ages 77 Reciient s notes ISBN he reort may be ordered from the Deartment of Automatic Control or borrowed through: University Library, Box 3, SE- 00 Lund, Sweden Fax ub@ub.se

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5 Acknowledgements I want to exress my sincere gratitude to everyone who has heled me during the develoment of the thesis. I want to give secial thanks to: My advisor Hubertus ummeschcheit who has been suorting me with infinite atience through all my work My Sanish advisor Jose Hernandez Grau, who gave me imortant information for the develoment of the thesis he Ph.D. student Jaime Arriaga, who heled me with many technical questions Jonas Eborn who heled me with technical questions about the library he rofessor Alejandrino Gallego, who made it ossible to comlete my degree abroad. he Ph.D. student Ines Romero Navarro, who encouraged me to come to Sweden. My arents Eulogio Gomez Pastor and Juana Perez Ortega, who made real the dream of finishing my degree in a foreign country o all my mates and friends in Sweden, who made my time away from home as enjoyable as could be.

6 o my girlfriend Maria, who suorted bravely my decision of coming to Sweden for all this eriod. I hoe we can enjoy for all the time that we were searated

7 3 NOMENCLAURE...5. INRODUCION...7. BACKGROUND...7. OBJECIVES WHY A DYNAMIC MODEL? P HASES OF HE PROJEC...9. GAS URBINES.... BASIC DESCRIPION.... EVAPORAIVE GAS URBINE MODELICA LANGUAGE INRODUCION CHARACERISICS OF OBJEC-ORIENED MODELING NON-CAUSAL MODELING HERMOFLOW LIBRARY INRODUCION BASIC DESIGN IDEAS...8 Control Volumes...9 Flow Models...0 Medium Models... State variable transformation SEQUENCE OF CALCULAION IN A DYNAMIC SIMULAION COMPRESSOR INRODUCION GOVERNING EQUAIONS USE OF HE COMPRESSOR MAP COMPRESSOR MODEL IN MODELICA...40 CorrectedMass...40 CorrectedMass...40 P_maxeff...40 Maxeff...40 Efficiency...40 ComressorMa...4 IsentroicVariables...4 FlowModelBaseMD...4 PolytroicEfficiency...4 Comressor...4 ComressorMec...4 ComleteComressor URBINE GOVERNING EQUAIONS URBINE MAP URBINE MODEL IN MODELICA...49 FlowModelBaseDM...49 IsentroicVariables...49 PolytroicEfficiency...49 urbine...50 urbinemec...50 Comleteurbine...5

8 7.COMBUSION CHAMBER INRODUCION...5 Mass balance...53 Energy Balance COMBUSION CHAMBER MODEL IN MODELICA...58 MassBalance...58 Valve...58 NaturalGasResS_X...58 hreeport...59 CombustionChamber ESING MODELS OF HE URBINE SYSEM MODEL WIH HYDRAULIC BRAKE SYSEM MODEL WIH SPEED CONROLLER SIMULAION RESULS SIMULAION IN HE MODEL WIH HYDRAULIC BRAKE SIMULAION OF HE MODEL WIH SPEED CONROLLER CONCLUSIONS AND FUURE WORK CONCLUSIONS FUURE WORK...75 BIBLIOGRAPHY

9 5 Nomenclature Symbol Unit Physical Meaning A m surface area c J/(kgK) secific heat caacity F N axial force h J/kg secific enthaly I kgm/s axial momentum J - Jacobean matrix m kg/s mass flow rate M kg mass n rm rotational seed Pa ressure P W ower q J/kg heat flow Q J/s heat flow R J/(molK) universal gas constant s J/(kgK) secific entroy t s time K temerature u J/kg secific internal energy U J internal energy

10 6 v m 3 /kg secific volume V m 3 volume w kg/mol molecular weight w J/kg secific work γ - ratio of secific heats η - efficiency ρ kg/m 3 density τ Nm torque ω rad/s rotational seed U J internal energy

11 7. Introduction. Background Due to the wide range of interactions between different engineering fields, systems are getting more comlex and heterogeneous. When trying to create models of these systems, roblems arise with the interaction of the different arts. Many times, it is ossible to find simulation rograms with grahical user interfaces for creating comlex model. he main roblem using these interfaces is that they are normally very secific, and are constrained to a concrete engineering disciline. Some examles of this kind of rograms are Sice and Saber for electronics simulations, or ASPEN Plus and SeedU for simulation of chemical rocesses. herefore, they are not aroriate when dealing with interoeratibily in heterogeneous roblems. Among the recent research results in modelling and simulation, two concets have strong relevance to this roblem: Object oriented modelling languages have demonstrated how object oriented concets can be successfully emloyed to suort hierarchical structuring, reuse and evolution of large and comlex models, indeendent from the alication domain and secialised grahical formalisms. By using these languages, it is ossible to suort modularity on multile levels. It means that a model can have many submodels, which have submodels themselves. Non-causal modeling. he traditional aroach for simulation based on inut and outut blocks, is relaced by another one where interaction is not defined with inuts or oututs. his generalisation rovides both simler models and more efficient simulations, while retaining the caability to include submodels with fixed inut outut relations. During the last four years the object-oriented, multi-domain language Modelica has been develoed by an international grou of engineers and researchers. he goal of the Modelica design is to become a de-facto standard for hysical modeling languages. Another activity of the grou is to also develo basic, free model libraries for several alications domains. he library for thermo-hydraulic

12 rocesses is currently being develoed at the Deartment of Automatic Control, Lund. 8 A base library like the one in rogress in Lund can only rove its usefulness in ractical alications with industrial relevance. Hence, the thesis is focused first on creating a model of a simle gas turbine by using as many comonents of the hermoflow library as ossible and second on extending the library with reusable models for turbines and comressors. Since the model involves mechanical arts, comonents of the rotational sub-library are used for the task. his roject is erformed in close cooeration with the Deartment of Heat and Power Engineering, which runs a ilot lant for an evaorative gas turbine at Lund University.. Objectives he objective of this thesis is to develo a global model of a simle gas turbine in order to simulate the dynamic behaviour. he Modelica language is used for the creation of the model. he model is based on the evaorative gas turbine located in the Deartment of Heat and Power Engineering in Lund (Lindquist, 999). his turbine was run as a conventional turbine before including the heat exchanger and the evaorative tower. Due to the large quantity of arameters involved in the model, the goal is to reroduce the general dynamic behaviour of the turbine. It should be noted that the aroximation in the result obtained the model and the results obtained with the exeriments deend on many model arameters. he tuning of the arameters of the model is beyond the scoe the roject. he model can be slit u in three main arts: the comressor, the combustion chamber and the exander. hese three models are based on the equations obtained from thermodynamic literature, mainly in (Cohen, 996), (Cengel, 994) and (Philis, 999). Furthermore, information rovided by the manufacturer was used for the imlementation of the comressor and the exander models. A model of a hydraulic brake or another kind of ower sink needs to be imlemented in order to extract ower from the turbine. he models of comressor, exander and brake are mechanically couled..3 Why a dynamic model? When designing a gas turbine for generating electrical ower, the most imortant thing is to obtain a high efficiency of the system. herefore, an oerating

13 9 oint is chosen, and all calculations are based on this design oint. Static models are normally used for the design of gas turbines. However, it is not ossible to know the resonse of the lant during transients by using static models. A concrete case of a transient can be a change in the requested ower. It is known that in a gas turbine the highest temerature in the cycle is reached at the end of the combustion chamber, before the turbine, which is aroximately the same than at the inlet of the exander. he maximum temerature that the turbine blades can withstand limits the turbine inlet temerature (I). herefore, the maximum ressure ratio that can be used in the cycle is also limited, because the I is associated to this ressure ratio. Increasing the turbine inlet temerature has been one of the main aroaches to imrove the gas turbine efficiency. he develoment of new materials and new cooling techniques has made ossible the increase of the I. his develoment has also resulted in more exensive comonents and the necessity of controlling that these comonents are working in the right oerating range. Consequently, it is necessary to evaluate the behaviour of the lant during different transients in order to kee the I in the right range. Dynamic models are used for addressing questions like this. It should be noted that these models are more comlicated than the static models, and it could yield roblems for evaluating the equations of the model. Hence, the models have to be simle and at the same time accurate enough for the simulations. Dynamic models are used mainly in order to control the lant. hese models can be used for simulating different arts of the lant and to lug them together, for analysing the behaviour of the whole lant. hen, new controllers can be tested imroving the global behaviour of the turbine. Another imortant function of the dynamic models is the training of the ersonal in ower lant. Dynamics models are created to reroduce the behaviour of the lant. In this way, the staff of the comany can be reared by using simulation based on these models..4 Phases of the roject he roject was divided in the following hases: Study of the features of the language Modelica Study of the structure and models in the hermoflow library

14 Study of the governing equations and technical data for the main arts of the lant. 0 Develoment and imlementation of the different models of the lant Simulation of the global model of the turbine Discussion of the results obtained in the simulations

15 . Gas urbines. Basic descrition It is known that nowadays, all the society in general is deendent on the electrical ower. Most of generation of heat and ower is dominated by the use of fossil fuels. urbines are the main devices for generating electrical ower by using these fuels. here are basically two kinds of turbines: steam and gas turbines. Steam turbines are able to generate larger ower than gas turbines and with an overall efficiency over 40%, but they have the drawback that rather comlicated installations for generating steam are needed. On the other hand, gas turbines are much more comact ower lants than steam turbines, since steam does not need to be generated and hot gases are used directly to run the turbine. herefore, gas turbines have very short start-u times. Figure. Simle cycle (Cohen,996) A gas turbine can be slit u in three main arts: comressor, combustion chamber and exander. In the comressor, fresh air is taken from the atmoshere and is drawn into the comressor. he comressor uses mechanical energy to raise the ressure and the temerature of the air. his air is used as oxidizer in the combustion chamber, where fuel is burnt at constant ressure. he resulting hightemerature gases enter to the exander where they exand to the atmosheric ressure, roducing ower. Mechanical shaft ower from the exander can be converted to electricity in a generator. Part of this ower is used in the comressor. he exhausted gases are released into the atmoshere. herefore, this cycle is classified as an oen cycle

16 Figure. -s diagram for the ideal Brayton cycle (Cengel,994) his oen cycle can be modelled with a closed cycle, just using air in it. his cycle is essentially the same than the one exlained before, but the combustion chamber is relaced by a heat exchanger and there is another heat exchanger which connects the outlet of the exander to the inlet of the comressor. he closed cycle is shown in figure.. he ideal cycled obtained by closing the oen cycle is the Brayton cycle. he Brayton cycle was first roosed by George Brayton in 870. his cycle is made of four internally reversible rocesses: Isentroic comression Heat addition at constant ressure Isentroic exansion Heat rejection at constant ressure here are two main factors, which affect to the erformance of the gas turbine, comonent efficiencies and turbine working temerature. he higher values they take the better is the erformance of the lant. Historically, these two factors have comlicated the develoment of the gas turbine. For examle in 904, two

17 3 French engineers, Armegaud and Lemale, made a turbine, which could hardly turn itself. It was due to the low efficiency in the comressor, around 60%, and the limitation of the gas temerature, around 740 K. Nowadays, the efficiencies of the comonents are around 85-90% and the temeratures that the turbines can withstand exceed 650 K.. Evaorative Gas urbine he evaorative gas turbine develoed at the Deartment of Heat and Power Engineering in Lund is briefly described in this section. Figure.3 Evaorative Gas urbine he lant can be slit u in the following comonents: comressor, combustion chamber, exander, recuerator, humidification tower and economiser. he first three arts where exlained in the revious section. he scheme of the lant can be seen in the figure. he air is comressed in the comressor and then flows to the humidification tower. here, the air is brought in contact with water in a counter flow, resulting in a temerature decrease and an elevated humid airflow. herefore, the exit air temerature can always be ket low by means of the humidification tower, indeendently of the exit temerature of the comressed air in the comressor. his introduces the oortunity of using a recuerator after the humidification tower. In the recuerator, the air mass flow is heated by using the remaining energy in the exhausted gas mass flow. After the recuerator, the exhausted gases ass trough the economizer, where the water is heated before it enters in the humidification tower.

18 4 ests carried out in the ilot lant showed that the efficiency increased from.7% in the simle cycle to 35% in the evaorative cycle. he NO x emissions were reduced by 90% to under 0 m, and the UHC (uncombusted hydro carbons) and CO were not measurable when running the evaorative cycle at rated ower outut. Other advantages of the lant are that it is ossible to reach full ower outut in less than five minutes and the investment costs for the evaorative cycle are much smaller than for other cycles with the same efficiency. More information about the EvG can be found in (Lindquist,999).

19 5 3. Modelica language 3. Introduction Modelica is an object-oriented language develoed for creating large, comlex and heterogeneous hysical roblems. General equations are used for modelling the hysical henomena. No articular variable needs to be solved for manually, since the Modelica tool will have enough information to do that automatically. Object-oriented and non-causal are imortant concets in Modelica. Both concets are sometimes confused. he concet of object-oriented refers to the structuring of the models whereas the concet of non-causal refers to the underlying descrition of the behaviour of the models. Nevertheless, these concets are used together in constrast to the traditional concet based in block-oriented models, which comes more from concern with comutational asects than from user concerns. he Modelica language relies on these concets of object-oriented and non-causal model. In this chater some general ideas about these two notions are given. 3. Characteristics of object-oriented modeling he rincial oint is that object-oriented modelling is able to study a system as a set of interacting objects. he total system is decomosed into simler elements, which are easier to study. Each object encasulates data, behaviour and structure. Once each object is defined, it is necessary to define the different connections for these objects and finally their behaviour. With these simle units it is ossible to build models and submodels in an easy way. Models define facts and relations, rather that being rocedures for comuting data. In object-oriented modelling the models are treated as objects. hese objects are described by a class, which can be seen as a blue-rint of the model. A model reresentation must suort modularity on multile levels, which means that one model can have many submodels which have submodels themselves. A model is also studied as something abstract, which means that it can be used without knowing all the details about its definition. In an abstract model, it is ossible to seak about an interface and internal descrition. he interface of the model describes how the variables that are internal to the model interact with the environment. he art of the model that has no interaction with the environment is the internal definition.

20 6 3.3 Non-Causal Modeling In order to allow reuse of comonents models, the equations should be stated in a neutral form without consideration of comutational order, i.e. non-causal modeling. Most of the general-urose simulation softwares on the market assume that the systems have to be slit u into block diagrams structures. herefore, these models are exressed as an interconnection of submodels on exlicit state-sace form, ODE (Ordinary Differential Equation) : dx dt = f ( x, u) ( x u) y = f, where u is inut, y is outut and x is the state. Normally, equations of the models need to be maniulated in order to get this form. A great effort has to be sent in terms of analysis and analytical transformations. It requires a lot of engineering skills and manower and it is an error-rone rocess. here is a fundamental limitation of block diagram modelling. he blocks have an unidirectional data flow from inuts to oututs. herefore, the need of manual transformations imlies that it is rather comlicated to build hysics based model libraries with a block diagram language. A general solution for this roblem requires a shift of aradigm. In Modelica it is ossible to write the equations in their natural form, i.e. as a system of differential-algebraic equations, DAE: dx f x,, y, u = 0 dt where x is the vector of unknowns that aear differentiated in the equation and y is the vector of unknowns that do not aear differentiated. Modelica has been carefully designed in such a way that comuter algebra can be used to achieve the same efficient simulation code than if the model would have been converted to ODE form manually.

21 7 4. hermoflow library For creating the comlete model of the gas turbine, the hermoflow library was used. he library is under develoment at the Deartment of Automatic Control at the Lund University. his chater is dedicated to briefly describe the library, in order to give an idea of how it is used. Further information about the library can be obtained at 4. Introduction Since the range of different thermo-hydraulic alications is very wide, it is not feasible to try to create comlete models for all these alications. When creating a library to use in these alications, emhasis has to be ut in the construction of reusable models. herefore, the hermoflow library is designed to rovide extensibility of basic building blocks rather than for creating comlete models for secific alications. In this way, the user of the library can combine several of these basic models to obtain a comlete one in a certain alication. he basic hysics of flows, fluids and heat need to be covered by the library. Comlete hysical roerties for different mediums are also needed. For this reason a great effort is ut to develo basic flow models and control volumes. In the control volume is ossible to choose the suitable roerty model for the desired alication. he models in the library are designed for system level simulation, not for detailed simulation. he models are thus discretized in one dimension or even lum arameter aroximations he hermo-flow library follows basically the following guidelines: One unified library both for lumed and distributed models Searation of the medium submodels, which can be selected through class arameters Both bi and unidirectional flows are suorted Assumtions can be selected through class arameter he main idea of the library is to enable the user to create comlex models based on the simle models rovided.

22 8 4. Basic Design Ideas A large number of engineering roblems involve mass flow in and out of the system. In many books in thermodynamics these systems are modeled as control volumes. Inside the control volumes, energy and mass flow are set. In the hermoflow library, control volumes are the basic entity. But another model is necessary for calculating the mass flow and the convective energy associated to the mass flow. Because of this, flow models are introduced. A flow model is the result of a modeling abstraction, where the volume is neglected. hese flow models contain either an algebraic equation that relates ressure dro and mass flow, or an exression for the dynamic momentum balance. he hermoflow library is based on an alternating sequence of control volumes and flow models. It can be said that storage of mass and energy are modeled in the control volume, whereas the flow of mass and energy are modeled in the flow models. Control volumes and flow models are connected through flow connectors. he flow connector for a single medium flow without dynamic momentum balance contains the following variables: {, h, m, q c, ρ,, s, k} where the quantities are ressure, secific enthaly, mass flow, convective heat flow, density, temerature, secific entroy and ratio of secific heat, resectively. All the information for the mass and energy balance is contained in the variables m and q c, which are evaluated in the flow model. he rest of variables (, h, ρ,, s, k) are evaluated in the control volume. Figure 4. Interaction between CV and FM

23 9 In order to build u new models using the library it is very imortant to understand these two basic models, the control volume and the flow model. Control Volumes Control volumes are in fact one of the most imortant illars of the hermoflow library. As it was said before, a control volume contains energy and mass balances. But it is also necessary to include a model for calculating all the thermohysical roerties, which is called medium model. he user can choose the medium model deending on the alication. Control volumes contain also connectors, which are the links between the control volume and the environment. hrough the connectors, the control volume interacts with the rest of the system. here are two different kinds of connectors: Flow connectors, which were already exlained above. Heat transfer connectors. In this connectors there is no mass flow. hese connectors are used for modeling heat transfer between fluids and solid bodies. Figure 4. Control Volume Once the connectors have been defined, mass and energy balances in the control volume can be written as: d dt n ( M ) = i m n (4.)

24 0 d dt n ( U ) =, i + i. q conv q l i. transfer, j (4.) Where M is the total mass, U the total inner energy, n is the number of flow connections (associated to mass flow m and heat flow q conv ) and l the number of heat transfer connectors (associated to heat transfer q tran ). Positive sign is associated to flows into the control volume. For simlicity, ressure volume work and dissiative work have been neglected here. Flow Models wo volumes have to be connected with a flow model, which contains mainly the momentum balance. In the hermoflow library, two tyes of flow models are defined: Stationary ressure dro models. Dynamic momentum balances for ies with constant cross-sectional area. he user can choose the tye of flow model to use deending on the model that he wants to create. It should be ket in mind that the dynamic momentum balance is only of interest when fast wave dynamics of the system are of interest. When the main interest is in the thermal behaviour, the stationary ressure dro should be used. Figure 4.3 Flow Model he momentum equation for a ie with a constant cross-section area and with volume V is:

25 I = V ρ w dv = ρ w da dz = m z (4.3) z A. Where w is the velocity in z direction, A is the normal flow area and m is the mass flow. Using Newton s law with only ressure and friction forces acting on the CV :... d m di z = = I I + dt dt ( ) A Fwall (4.4) model: he equation can be simlified in order to obtain a stationary ressure dro 0 ( ) A Fwall = (4.5) Where F wall is the friction between the fluid and the wall. It deends on the flow characteristics. Different exressions for F wall can be found in the literature. When creating models like a comressor for examle, there are many different relationshis between mass flow rate, ressure ratio, angular seed, etc. hen the flow model can contain these exressions instead of the dynamic momentum equation. Medium Models It is imortant to have accurate medium models in order to make the library really reusable. On the other hand, for the urose of dynamic simulations, it is also imortant to have fast medium models. At the moment, the following medium models are imlemented: Pure Ideal gases Mixture of ideal gases CO Water he models imlemented in the library are all very accurate and are taken from some recommended formulations or standards like IAPWS/IF97 for water.

26 Medium models are necessary in control volumes. By using this medium models it is ossible to comute all remaining variables of interest using the mass and energy balances. State variable transformation Mass and energy balances are imlemented in the control volume model. otal mass and internal energy (M and U) are the states in these equations. For using the medium models rovided in the library, these variables are not very suitable. Different variables are chosen as states deending on the choice of the medium model used. When working with ideal gases, which are used for the turbine model, and are chosen as states, mainly for efficiency reasons. In ideal gases, all medium roerties deend on. Hence, if h were chosen as a state, there would always be a non-linear system of equations for calculating. Because of this there is a secial class in the library called Stateransformation, which changes the states M and U to different states according to the desired model. A differentiation of M = ρv and U = um for a constant volume yields: dρ V = dt dm dt (4.6) M du dt du dm = u (4.7) dt dt So the energy and mass balances described above can be rewritten as: d M = V dt U u 0 d ρ ρ dt u (4.8) Where ρ is the density and u is the secific inner energy. hese rimary equations are then transformed into secondary forms to give differential equations in the states that are best suited for the medium model. For the case of erfect gases, and are chosen as states. For simlicity, the comosition of the gas is assumed constant, so it is not considered in the transformation: d dt δρ ρ = δ u δu δ δρ δ δu δ d d (4.9)

27 3 Where: = u u J δ δ δ δ δ δρ δ δρ (4.0) And for ideal gases: R = δ δρ (4.) R = δ δρ (4.) = 0 u δ δ (4.3) v C u = δ δ (4.4) o obtain differential equations for ressure and temerature, the following exression is used: = u dt d J dt d ρ (4.5) he inverse of the Jacobian is comuted as follows: = u u J J δ δρ δ δ δ δρ δ δ det (4.6) An the determinant is: R C R R C u u J v v = = = 0 det δ δ δ δρ δ δρ δ δ (4.7) So the inverse of the Jacobian can be rewritten as:

28 4 J = C v R Cv 0 R R (4.8) Similar exressions are also imlemented in the Library for other airs of state variables, for examle (,h), (ρ,), (ρ,,x),...where x refers to the comosition of a mixture of gases. 4.3 Sequence of calculation in a dynamic simulation Once control volumes have been resented, a brief exlanation of the way of oerating is given. Pressure and temerature are assumed to be the states. It should be noted that this sequence of calculation is determined automatically by the Dymola tool, without any hel from the user First of all, initial values of the states ressure and temerature ( 0 and 0 ) are needed. he user has to suly these values. Knowing the temerature, it is ossible to evaluate other termodynamical variables ( h, ρ, s, k) by using the medium model. he flow models use the value of these variables for evaluating the mass and energy flows. he flow model accesses this information through the connectors. All the mass and energy flows calculated in the flow models surrounding the control volume, are used in the mass and energy balances. ime derivatives of total mass and total inner energy are then calculated (dm, du). he class Stateransformation is used to transform the time derivatives of total mass and inner energy (dm, du) into the time derivatives of the chosen states (d, d). he time derivative of ressure and temerature are used for evaluating the new values of the states. he sequence is reeated again. he sequence can be seen in the figure.

29 Figure 4.4 Sequence of calculation 5

30 6 5.Comressor 5. Introduction he comressor for gases has the same function that the um for liquids. Basically, mechanical energy is added to the gases and this is used to raise the ressure, hence the name comressor. his adding of energy causes the gases to flow from one unit oeration to the next. A comressor is the same than a turbine in reverse, comressing rather than exanding the gases that ass through it. he model created for the comressor is based on the general equation for steady state flow. In this way, faster dynamics are neglected, and relaced by static relationshis. his aroach is used when a model has some fast and some slow dynamics. Higher frequency transients can be neglected, since the low frequency transients dominate the resonse. When this assumtion is used, it is ossible to base a model on steady state data. A comressor ma is used in the comressor model. By using this ma, the olytroic efficiency and the mass flow can be calculated as functions of the ressure ratio and the rotor seed. he comressor model includes equation for the mechanical behaviour. Inside the model, mechanical and thermodynamic owers are connected in an equation. During the first art of the chater, the governing equations are exlained. In the second art, the way of imlementing the comressor ma is shown. Finally, a short descrition about the Modelica models created for the comressor is given.

31 7 5. Governing equations First of all, a steady state energy balance is used. his equation can be found in many books of thermodynamics (Phili, 999),( Cengel, 996): dq dw dh cdc gdz = 0 (5.) where the dq is the secific external heating, dw is the secific work, dh is the enthaly, cdc is the kinetic energy term and gdz the otential energy term. Normally a comressor can be considered as adiabatic, so the term dq can be neglected. he same consideration can be taken for the height difference, so that dz=0. here is a small difference between the inlet and the outlet velocities, but it can be also neglected, i.e. cdc=0, (Phili, 996). According to this, the equation obtained is: dw = dh (5.) And integrating over the comressor section gives: w = h h (5.3) Where the subscrits and refer to the inut and the outut sections of the comressor resectively, and w is the secific work done on the gas. From the definition of constant secific heat c (), for ideal gases, equation (5.3) can be rewritten as: w = c ( )d (5.4) where c () is a function of the temerature. In the Figure (5.) is ossible to see the variation of c () with the temerature for different gases. For the case of air, the variation of secific heat with the temerature is smooth, and may be aroximated as linear over intervals of a few hundred degrees, (Cengel, 994). hen, the secific heat in equation (5.4) can be relaced by a constant average secific heat value. An aroach is to take the average for the temeratures at comressor inlet an outlet.

32 8 Figure 5. Ideal gas constant-ressure secific heats for some gases (Cohen,996) Denoting the average secific heat with c, the following equation is obtained: w = c ( ) = c (5.5) For gases, it is ossible to use the following equation for calculating the secific heat (Philli, 999): c γ = R γ (5.6) where R is the gas constant for the gas, and γ is the ratio of the secifics heats C and C v : γ C = C v (5.7)

33 9 he secific heat ratio also varies with the temerature, but this variation is very small. For the case of air, the value of the secific heat ratio is around.4 (Cengel,996). he equation (5.5) can be rewritten as: γ w = R γ (5.8) Since R and γ are constants and is the ambient temerature, the only variable in equation (5.8) is. herefore, the secific work is a minimum when the temerature is a minimum. his occurs when the comression is isentroic (adiabatic and reversible). In an isentroic comression the relation between ressures and temeratures is the following: s = ( γ ) γ (5.9) Where the subscrit s is referred to the isentroic temerature. Substituting from equation (5.9) into equation (5.8), the isentroic secific work w s is obtained: w s γ = R γ γ γ (5.0) In order to calculate the actual secific work, the concet of isentroic efficiency is used. he isentroic efficiency is defined as the ratio between the real enthaly difference and the theoretical enthaly difference, when considering the rocess as isentroic: η s = h s h h h (5.) Where h s is the secific enthaly at the outlet of the comressor, when the rocess is isentroic. Measuring the ressures and temerature at the inlet and outlet sections of the comressor, an exerimental determination of this isentroic efficiency can be obtained. s can be calculated with the equation (5.9), and assuming the gas has a constant secific heat, the isentroic efficiency can be evaluated as:

34 30 η s = c c ( s ( ) ) = s (5.) he definition of isentroic efficency, is based on a ratio of secific work to actual secific work, across the comlete section of the comressor. When the ressure ratio changes, an overall efficiency as obtained in equation (5.) does not remain constant. In fact it is found that η s tends to decrease when the ressure ratio raises, (Cohen,996). hese considerations have led to the concet of olytroic efficiency, which is defined as the isentroic efficiency of an elemental stage in the rocess, arbitrarily defined in such a way that this efficiency is constant throughout the whole rocess: η dw = s dw = constant for the whole comressor (5.3) Assuming c constant, the equation (5.3) can be transformed into: η dh = dh c = c d s s = d d d s (5.4) Using the equation for an isentroic comression rocess: γ γ = constant (5.5) Exressions (5.4) and (5.5) can be combined obtaining: d s = γ γ d (5.6) Using the definition of olytroic efficiency: d η γ = γ d (5.7) If the exression above is integrated between inlet and outlet assuming η constant by definition, the following exression is stated:

35 3 = ln ln γ γ η (5.8) his exression allows the calculation of the value for η by using the values of and at the inlet and the outlet of the comressor. his exression can be rewritten in the following form: ( ) η γ γ = (5.9) hen, it is ossible to define a new coefficient: ( ) m η γ γ η = (5.0) So that equation (5.9) can be rewriten as: ( ) m m = (5.) Equation (5.) can be combined with equation (5.) for evaluating the actual secific work: = = = ) ( m m c c c w (5.) And dividing the equation (5.0) for the isentroic secific work by the exression (5.) for the actual secific work: ( ) ( ) = m m s c c γ γ η (5.3)

36 3 By using this exression, the isentroic efficiency can be calculated as a function of the ressure ratio and the olytroic efficiency. Figure 5. shows the decrease in the isentroic efficiency when the ressure ratio raises. Figure 5. Isentroic efficiency against ressure ratio for different olytroic efficiencies (Phili,999) he actual ower needed for the comression is the roduct of the actual secific work and the mass flow rate of gas: P = m w. (5.4) his exression can be rewritten using the isentroic efficiency as: P com =. m w η s s (5.5) exressed as: he shaft connected to the turbine rovides this ower, so it can also be Pcom η = τ ω (5.6) mec com com Where τ com is the torque, ω com is the angular velocity and η mec in the comressor. In this way, the mechanical and the thermodynamic behaviours are connected.

37 Use of the comressor ma he ma of the manufacture was used for the imlementation of the comressor model. In this ma, the ressure ratio is lotted against the corrected mass flow for a range of corrected seed and olytroic efficiency curves. he corrected mass flow and the corrected comressor seed are used in the ma to comensate for different environmental conditions under which the steady state exeriments were carried out. For the comressor ma, the corrected mass flow is defined as: m cor. m = (5.6) And the corrected rotational seed: n n cor = (5.7) It is imortant to oint out that there are two critical regions that have to be taken into account, the surge and the stall lines. Along the surge line, the rotor seed contours become nearly horizontal. o the left of the surge line, the seed contours dro with resect to the ressure ratio. his may create an unstable henomenon called surging, which can destroy the comressor (Cohen, 996). At each rotor seed, a ressure for which surge occurs can be identified. Along the stall line, the mass flow becomes choked. hese regions have to be imlemented in the model. the figure. Figure 5.3 shows an examle of comressor ma. Surge line is indicated in All the information given in the ma must to be rocessed by the model. It is necessary to translate the information from the grahical form sulied by the manufacturer to the Modelica model. here are several methodologies for transforming this information.

38 34 Figure 5.3 Ma for a centrifugal comressor (Cohen,999) One of these aroaches could be to lace all the information rovided in tables by choosing several oints of the ma and then to use these tables in a looku manner (Munns, 996). But using the information, it can cause roblems in the dynamic simulations. In these tables, data oints for several ressure ratios and seed arameters are given. When a oint for a different ressure ratio or seed arameter needs to be evaluated, linear interolation has to be used. When a dynamic simulation moves over one of the given oints, there is a discontinuous derivative caused by the linear interolation, which yields in roblems for the dynamic simulation. For this reason, the aroach was rejected. Another aroach is to imlement a coule of functions where ressure and seed arameter are given as inuts and the efficiency and the mass flow arameter are obtained as outut for each one of the functions (Gustafsson, 998). In order to get a function for evaluating the corrected mass flow, an ellisoid equation is used: z x a z y + b = c (5.8)

39 35 Figure 5.4 Ellisoid curves for different values of z In figure 5.4, a set of the curves is shown. hese curves have been obtained using the ellisoid equation for different values of the arameter z. he values of a, b and c are all equal to one. he lines on the left are for lower values of z. When the value of z increases, the shae of the curves obtained is very similar to shae of the corrected seed curves in the ma. Now, it is going to be shown how to calculate the mass flow by using the ellisoid equation. First, z and n cor (corrected rotational seed) are assumed to be fixed. he variable n cor is calculated from the angular velocity, which is a dynamic state, and can thus be regarded as known. he variable z is a function of n cor. For the value of the constant a in the ellisoid equation, the value of the mass flow when the ressure ratio equals one is given. his value corresonds to the mass flow when the comressor is stalled. For calculating the value of the constant b, the ellise equation is used. herefore, once the constant a is known, x and y are given as inuts and b is obtained as an outut. he value of x corresonds to the ressure ratio at the surge line, while y corresonds to the mass flow at the surge line. In this way, the equation takes the value of the ma.

40 36 Figure 5.5 Estimation of the value of b Once values a, b and z are known and x (ressure ratio) is given as an inut, y (mass flow) can be obtained easily by using the equation (5.8). In order to be able to use this equation over the whole range of seeds, it is necessary to have a, b, and z as functions deending on the corrected seed: a = b = z = f f f ( n ) cor ( ncor ) ( n ) cor For calculating a as a function of the seed, the oints defined in the ma for the eight constant corrected seeds were used. For each one of these seeds, the values of the corrected mass flow at the ressure ratio one are read. hen a relation between the mass flow at ressure one and the corrected seed is calculated by fitting a olynomial to the data oints. Finally it is ossible to calculate a as a function of the corrected seed: a = a ncor + a3 ncor + a ncor + a ncor + a0 (5.9) where a 0, a, a, a 3 and a 4 are the coefficients of the olynomial fitting. he same aroach is adoted to obtain a function of b deending on the corrected seed:

41 37 b = b ncor + b3 ncor + b ncor + b ncor + b0 (5.30) where b 0, b, b, b 3 and b 4 are the coefficients of the olynomial fitting. he value of z is calculated as a linear function of the corrected seed. A value z for the lowest seed and another z for the highest seed are chosen, and in between a linear interolation is used: z = z + n cor ( z z) ( n n ) max min (5.3) Once a, b and z are known, it is ossible to use the ellisoid equation for the whole ma. he ressure ratio x is given as inut and the mass flow y is obtained as outut. Figure 5.6 Valid region for calculation with ellisoidal equation he surge line of the ma needs to be considered. If the comressor reaches the surge line, the corrected mass flow is not the one calculated with the ellisoid equation (Figure 5.6).

42 38 Because of this, the corrected mass flow for a given ressure ratio has to be calculated and then it is necessary to check if the surge line has been reached. A olynomial relation can be fitted between the mass flow and the ressure ratio at the surge line by using the oints obtained in the ma. he aroach is the same that was taken for calculating a and b: m surge = m + (5.3) r + m3 r + m r + m r m0 ressure ratio r. Where m surge is the mass flow at the surge line for a given value of the In order to know if the surge line has been reached, the mass flow for a given seed and ressure ratio is calculated by using the ellisoid equation. hen the mass flow at the surge line is evaluated with equation (5.3). Once both mass flows have been calculated, they are comared: m ellise m surge surge Another function for calculating the olytroic efficiency of the comressor was also needed, but getting a function for it was much more difficult, since the information rovided by the ma was not very accurate. For this reason, it was decided to create a seed-deendent function for evaluating the value of the maximum efficiency. hen a function for the degradation of this efficiency was fitted. So first of all, a olynomial relation can be fitted between the corrected seed and the value of the maximum efficiency: η max = m 4 n 4 3 cor + m3 ncor + m ncor + m ncor + m 0 (5.33) where m 0, m, m, m 3 and m 4 are the constants for the olynomial fitting. he same can be done between the seed and the ressure ratio for maximum efficiency: 4 3 max_ eff 4 ncor + 3 ncor + ncor + ncor + = (5.34) 0 where 0,,, 3 and 4 are the constants for the olynomial fitting. he maximum olytroic efficiency and the corresonding ressure ratio for the resent corrected seed are calculated with equation (5.33) and (5.34). he actual ressure

43 39 ratio sulied may not be the otimum one, and the actual efficiency can be lower. he difference in otimum ressure ratio can be exressed as a difference in otimum flow. Once this difference is known, a correction for the efficiency is made. his correction assumes a symmetrical degradation on both sides of the otimum flow. his degradation is based on a arabolic equation: max ( m m ) η = η c (5.35) max_ eff chosen. Where c is a constant. For fitting this constant several oints on the ma were

44 Comressor model in Modelica he comressor model belongs to the class of flow models. It uses the equations exlained at the beginning of this chater in order to evaluate the mass and energy flows. hese flows rovide the control volumes the information to evaluate the mass and energy balances, as it was exlained in chater [4], which was dedicated to the hermoflow library. An equation to link the thermal ower to the mechanical ower is also needed. A brief exlanation of the imlementation of the models is given here. All the models used for the creation of the comressor are in a ackage called NewComressors. CorrectedMass It is a function used to calculate the corrected mass flow by using equation (5.8). Corrected seed and ressure ratio are the inuts of this function. CorrectedMass Corrected mass flow at the surge line is calculated here by using the equation (5.3). Pressure ratio is needed as inut. P_maxeff he value of ressure at the maximum efficiency for a given seed is calculated here by using equation (5.34). Maxeff he value of maximum efficiency for a given seed is calculated here by using equation (5.33). Efficiency he function with the degradation of the efficiency is given here (equation (5.34.)). Corrected seed and ressure ratio are the inuts and the olytroic efficiency is the outut. Values obtained in P_maxeff and maxeff are used internally in this function.

45 4 ComressorMa his class uses the functions CorrectedMass and CorrectedMass for evaluating the actual mass flow in the comressor. A Boolean variable called Surge is defined here. his variable is used to check if the comressor reaches the surge line. he difference between the corrected mass flow in the comressor and the corrected mass flow for the same ressure ratio at the surge line is defined in a variable. Hence, it is ossible to know how close is the comressor to the surge line IsentroicVariables he class IsentroicVariables is a record. A record is a restricted form of class that may not have any equations. It is used for setting different variables not defined in the main model. FlowModelBaseMD he model called FlowModelBaseMD inherits the class FlowVariablesMultiStatic, which was already available, in the hermoflow library. his class is a shell model with two connectors for flow model. he connectors contains the following variables: mass fraction for each comonent of the mixture ressure enthaly mass flow for each comonent of the mixture energy flow density Ratio of secific heat caacities entroy he class connects the variables internal to the model, to the variables at the connector a and b.

46 4 PolytroicEfficiency he class PolytroicEfficiency inheritances all the variables from the class IsentroicVariables. Polytroic efficiency is calculated In this class by using the functions P_maxeff, Maxeff and Efficiency. Once this is known, the isentroic efficiency is used to evaluate the secific work in the comressor. Comressor he class Comressor is the comlete thermodynamic model. It inherits all the classes exlained above. Figure 5.7 Icon of the class Comressor ComressorMec In the model ComressorMec, equation (5.6), which links the thermodynamical and mechanical behaviour, is added. A new connector for the mechanical information is also added. his connector contains the following variables: Absolute rotation angle of flange orque in the flange

47 43 Figure 5.8 Icon of the class ComressorMec ComleteComressor he last model in the ackage NewComressor is the model ComleteComressor. his model contains a arameter called J, which is the inertia of the comressor. he units of J are Kg*m. he user can modify this arameter. Figure 5.9 Icon of the class ComleteComressor

48 44 6.urbine 6. Governing equations One form of the energy equation for general steady state flow is given by the following equation, (Cohen,996): dq dw dh cdc gdz = 0 (6.) where the dq is the secific external heating, dw is the secific work, dh is the enthaly, cdc is the kinetic energy term and gdz the otential energy term. he terms dq and dz can be neglected, because the flow is adiabatic and the height change is very small. Integrating between the (inlet section of the turbine) and (outlet section of the turbine), the following exression is obtained: h + c h + c w = 0 (6.) Where w is the secific work obtained in the turbine and c and h are seed and enthaly. he kinetic energy term at the inlet of the turbine can be neglected, because the air flow velocity at the turbine entrance is close to zero. For neglecting the value of the kinetic term at the outut of the turbine, the following consideration can be taken. A value for the velocity in the outut of 00 m/s has associated a secific kinetic energy of 0 kj/kg. he secific enthaly of the air from tables, at the temerature of 900 C is 03.5 kj/kg. It means that the kinetic term is.97 % of the enthaly of the air. It gives us an idea about the size of both terms. After this consideration, it seems logical to neglect the kinetic terms for the dynamic model (Phili,999). herefore, the next equation arises: w = h h (6.3) A constant secific heat was also assumed for the turbine. A constant secific heat c for the model is calculated as an average for the temeratures at turbine inlet and outlet. herefore, equation (6.3) can be rewritten using the temeratures: w = c ( ) c (6.4) =

49 45 as: Using equation (5.6) of the chater [5], it is ossible to rewrite equation (6.4) γ w = R γ (6.5) If the exansion rocess occurs in isentroic conditions, the work obtained is the isentroic work: w s γ = R γ s (6.6) where s is the isentroic temerature. For an isentroic exansion, the relation between ressures and temeratures is: s = ( γ ) γ (6.7) So exression (6.6) can be rewritten as: γ γ γ s w = s R (6.8) γ he isentroic efficiency for a turbine is defined as the ratio between the actual enthaly difference and the isentroic enthaly difference: h h η s = (6.9) hs h Where h s is the enthaly at the outlet when the rocess is isentroic. As it haens in the comressor, when ressure ratio changes, the isentroic efficiency does not remain constant. For the case of the turbine, η s tends to increase when the ressure ratio grows, (Cohen, 996). In order to take this into account, olytroic efficiency is introduced: dw η = = constant for the whole turbine (6.0) dw s

50 46 Following the same aroach that was followed in chater [5], it is ossible to get a function for evaluating the isentroic efficiency as a function of the olytroic efficiency and the ressure ratio: ( ) ( ) = γ γ η c c m m s (6.) Where the coefficient m is: ( ) γ γ η γ + = m (6.) Exression (6.9) and (6.) can be combined in order to evaluate the actual secific work in the turbine: = γ γ γ γ η R w s (6.3) he actual ower released in the exansion is the roduct of the actual secific work and the mass flow rate of gas: w m P =. (6.4) For taking into account the mechanical losses, the mechanical efficiency is introduced: mec P ml P P P η = = ' (6.5) where P is the mechanical shaft ower and P ml is the term referred to mechanical losses. hen, mechanical and thermal ower can be connected: mec tur tur P P η ω τ = = ' (6.7)

51 47 6. urbine ma For the case of the comressor, the manufacturer rovided a ma for evaluating the mass flow and the efficiency. For turbines there are also mas but they have secial characteristics. A tyical ma for a turbine can be seen in figure 6.. he erformance is exressed by lotting the olytroic efficiency η and the corrected mass flow against the ressure ratio for various values of the corrected seed. For the turbine ma, corrected flow mass is defined as: m cor. m = (6.6) And the corrected seed of rotation: n n cor = (6.7) Where the subscrit is referred to the inut of the turbine. he ma shows the relative seed to the design value: Figure 6. Ma of a turbine (Cohen, 996)

52 48 he efficiency lot shows that the efficiency remains constant over a wide range of corrected seeds and ressure ratios. For the case of the corrected mass flow, the maximum value of it is reached at a ressure ratio, which roduces choking conditions at some oint in the turbine. In this situation all the constant seed lines merge into a single horizontal line as indicated on the mass flow lot. aking into account that for the dynamic simulation the turbine works in choked conditions, a good aroximation can be to assume that the corrected mass flow remains constant. In this way, the equation for the design of nozzles is used for evaluating the mass flow (Cohen,996): m A thr = γ R γ + ( γ + ) ( γ ) (6.8) Where A thr is the equivalent nozzle throat area. his equation is used for dimensioning the nozzle area based on a known design oint. herefore, A thr can be evaluated knowing the rest of variables at the design oint. hen, this area is used to evaluate the mass flow. he efficiency is assumed constant for all the range of seeds. he user can adjust the value of the efficiency.

53 urbine model in Modelica he turbine model imlemented in Modelica is essentially a flow model. he equations described in this chater are included in the model. he model is very similar to the comressor model, but the turbine model does not include a class for the imlementation of the ma. In the turbine model this ma is relaced by equation (6.8), where choked conditions are assumed. he classes used for the creation of the turbine model are located in the ackage Newurbines. Some of the classes in this ackage are described in the rest of the chater. FlowModelBaseDM FlowModelBaseMD is a shell model with two connectors for flow model. he connectors contains the following variables: mass fraction for each comonent of the mixture ressure enthaly mass flow for each comonent of the mixture energy flow density Ratio of secific heat caacities entroy he class connects the convent variables internal to the model, to the variables at the connector a and b. IsentroicVariables he class IsentroicVariables is a record. In this record, different variables, which are necessary for the turbine model, are defined. PolytroicEfficiency his class includes equations for calculating the isentroic efficiency as a function of the olytroic efficiency and the ressure ratio. Isentroic efficiency then is used to evaluate the secific work obtained in the turbine.

54 50 urbine he urbine class is the comlete thermodynamic model. All the classes that were shown above are inherited from this class. herefore, mass and energy flows are evaluated in this model. urbinemec Figure 6. Icon of the class urbine his class includes equation (6.7), which links the mechanical ower to the thermal ower. A mechanical connector is included. he mechanical connector contains the following variables: Absolute rotation angle of flange orque in the flange Figure 6.3 Icon of the class urbinemec

55 5 Comleteurbine his model includes the model Inertia, which is inherited from the Rotational library of Modelica. his model is a rotational comonent with inertia and two rigidly connected flanges. he model includes the following equation: n J a = τ i (6.9) i where J is the moment of inertia, a is the angular acceleration, τ is the torque and n is the number of mechanical connectors. Figure 6.4 Interior of the class Comlete urbine he moment of inertia is a arameter that can be sulied by the user. Figure 6.5 Icon of the class Comleteurbine

56 5 7.Combustion Chamber 7. Introduction Models resented in revious chaters were limited to be non-reacting thermodynamic systems. On the other hand, for the model of the combustion chamber, chemical reactions must be taken into account. In non-reacting systems just the notions of sensible internal energy (associated with temerature and ressure changes) and latent internal energy (associated to hase changes) were used. When dealing with reacting systems, it is necessary to consider the chemical internal energy, which is associated with the destruction and the formation of chemical bonds between the atoms. In the combustion chamber, the chemical reaction involved is called combustion. In a combustion, some molecules are destroyed for the creation of new molecules with a release of a large amount of energy. Consequently, the energy and mass balances have to include the chemical equation of the combustion. Normally, energy and mass balances are treated in control volumes, but since chemical reactions are involved a new class had to be develoed. Some additional assumtions have to be made for the derivation of the model. hese assumtions are: he combustion is going to be considered as instantaneous. his is very a logical consideration since the transients involved in the combustion are much faster than all other transients in the system. Kinetic and otential energy are going to be neglected in the energy balance. he efficiency of the combustion chamber is assumed as a constant arameter, i.e. the user can select the desired value before running the simulation.

57 53 7. Governing equations Chemical equations govern the behaviour of the combustion chamber. In this section, equations for energy and mass balances are shown. he assumtion of ideal gas roerties for all comonents is used for calculating all the roerties of the gas mixtures. Mass balance he air is considered as a mixture of CO, H O, N and O. he fuel used for the turbine is natural gas. he comosition of the natural gas is variable, deending on the rovider. Natural gas is mainly comosed of a mixture of hydrocarbon fuels, but other gases can also be found in it. For the model, natural gas is considered as a mixture of the following elements: CH 4, C H 6, C 3 H 8, C 4 H 0, N and CO. From the chemical oint of view, N, CO, which can be found either in the air or in the fuel, and H O, which can be found just in the air, are assumed to be inert. It means that they are not involved in any chemical reaction. herefore, only hydrocarbon fuels (CH 4, C H 6, C 3 H 8 and C 4 H 0 ) react. he chemical equation for the combustion of a general hydrocarbon fuel assuming the stoichiometric amount of O is: C H n n + m + O mco + H O (7.) 4 m n Where m and n deend on the kind of hydrocarbon, e.g for Methane m and n are equal to and 4. Equation (7.) imlies that each kmol of C m H n reacts with (m+n/4) kmol of O, roducing m kmol of CO and (n/) kmol of HO. According to this, the following exressions arise for CH 4, C H 6, C 3 H 8 and C 4 H 0 : CH 4 + O CO + H O (7.) C H O CO + 3 H O (7.3) C3H O 3 CO + 4 H O (7.4) C 4H O 4 CO + 5 H O (7.5) Comlete combustion is assumed, which means that there is not CO among the roducts and all hydrocarbon fuel is consumed in the reaction.

58 It is ossible to transform exressions (7.), (7.3), (7.4) and (7.5) into exressions for mass flow (kg/s) by using the molecular weight of the comonents involved in the reactions: kg of comonent molecular weight w = (7.6) kmol of comonent 54 In this way: wo wco wh O CH 4 + O CO H O wch + 4 wch 4 wch 4 (7.7) wo wco wh O C H O CO 3 H O wc H + 6 wc H 6 wc H 6 (7.8) wo wco wh O C3H O 3 CO 4 H O wc 3H + 8 wc 3H 8 wc 3H 8 (7.9) wo wco wh O C 4H O 4 CO 5 H O wc 4H + 0 wc 4H 0 wc 4H 0 (7.0) Where wco, wch 4, wc H 6, wc 3 H 8, wc 4 H 0, wh O and wo are the molecular weighs of CO, CH 4, C H 6, C 3 H 8, C 4 H 0, H O and O resectively. Equations above used the soichiometric amount of O. If the amount of O is bigger than the stoichiometric, there is O in the exhausted gases. Gas turbines oerate with excess of air, i.e. O is not totally consumed in the reaction. herefore, the comosition of the exhausted gases for the gas turbine is a mixture of CO, H O, N and O. Mass flows for each one of the comonents at the outlet of the comressor can be evaluated by using the exressions (7.7), (7.8), (7.9) and (7.0). Considering that m in [CO ], m in [H O], m in [N ] and m in [O ], are the mass flows of CO, H O, N and O.at the inlet of the combustion chamber, and m fuel [CH 4 ], m fuel [C H 6 ], m fuel [C 3 H 8 ], m fuel [C 4 H 0 ], m fuel [N ], m fuel [CO ] are the mass flows of CH 4, C H 6, C 3 H 8, C 4 H 0, N and CO, the following relation can be written:

59 55 m out + m + 4 m m out + 3 m + 5 m [ CO ] = m [ CO ] + m [ CH ] fuel fuel [ C H ] + 3 m [ C H ] [ C H ] + m [ CO ] 4 in 6 0 wco wc H wco wc 4 H 6 0 fuel 4 fuel fuel wco wch [ H O] = m [ H O] + m [ CH ] fuel fuel [ C H ] + 4 m [ C H ] [ C H ] 4 in 6 0 wh O wc H 4 6 wh O wc H 0 fuel fuel wco wc H 4 3 wh O + wch 3 8 wh O wc H (7.) (7.) m [ N ] m [ N ] m [ ] = (7.3) out in + fuel N m out [ O ] = m [ O ] m [ CH ] 3.5 m 6.5 m fuel fuel in [ C H ] 5 m [ C H ] [ C H ] wo wc H 4 fuel 6 wh O wc H 0 4 wo wch fuel wo wc H 3 8 (7.4) Where m out [CO ], m out [H O], m out [N ] and m out [O ], are the mass flows of CO, H O, N and O.at the outlet of the combustion chamber. Energy Balance Once the mass balance is calculated, the next ste is to evaluate the energy balance. Now it is necessary to use the concet of enthaly of formation. It is ossible to establish a reference for enthaly in the study of reactive systems by setting the value zero to the enthaly of the stable elements in a state called standard reference state defined for ref =98.5 K and ref = atm. Stable elements are set to zero at the standard conditions. he term stable is used in the meaning chemically stable. his means that at the standard state, the stable forms of Nitrogen, Oxygen and Hydrogen are N, O and H and not N, O and H. Using this reference, it is ossible to assign values for the enthaly of formation of the comonents. he enthaly of formation of a comonent at the standard reference state, is the value of the energy that is released or absorbed for the same when it is

60 56 created from its rimary forms (O, C, H,...), at ref and ref. If the enthaly of formation is negative, energy is released during the creation of the comonent, and when it is ositive, energy is absorbed during the rocess. If the conditions differ from the standard reference, the enthaly of formation will decrease or increase. For examle, if the temerature of the comonents is higher than ref, the released energy decreases because some energy is needed to raise the temerature. Consequently, before writing the energy balance, it is necessary to exress the enthaly of a comonent in a form suitable for use in reacting systems. he enthalies of formation of each of the comonents need to be taken into account for the energy balance in the combustion chamber. A new definition of enthaly called total enthaly is introduced. his enthaly is the sum of the enthaly of formation of the comonent at 5 C and atm and the sensible enthaly of the comonent of the comonent relative to the reference temerature: 0 Enthaly h f + ( h h ) = (7.5) Ref In hermoflow, zero Kelvin is chosen as the reference state for calculating the enthaly. he exression imlemented in hemoflow for calculating the sensible enthaly is based on the NASA tables (Gordon,994). With the enthaly defined as above, including the enthaly of formation, the energy balance can be evaluated. When the changes in kinetic and otential energies are negligible, the conservation of energy relation for a chemically reacting steady-flow system can be exressed in the following form (Cengel, 994):.. Q W =. 0 0 ( h + ( h h )) m h + ( h h ) ( f ref ) r. m f ref (7.6) r Where the subscrit is taken to refer roducts, while r refers to reactants. In the combustion chamber, the work term can be neglected. hen, the chemical energy released during a combustion rocess is either lost as heat to the surroundings or it is used internally to raise the temerature of the combustion roducts. In the limit case of no heat loss to the surroundings (Q=0), the temerature of the roducts will reach a maximum, which is called the adiabatic flame or adiabatic combustion temerature. In this case, the exression (6) can be rewritten as:

61 ( h + ( h h )) = m h + ( h h ) ( f ref ) r. m f ref (7.7) r In gas turbines, the highest temerature to which the blades can be exosed is limited by metallurgical considerations. herefore, the adiabatic temerature is an imortant factor in the design of gas turbines. he maximum temeratures which occurs in gas turbines are considerably lower than the adiabatic flame temerature due to the following reasons: When a combustion chamber oerates with excess of air, which is the normal case, it serves as a coolant he combustion is usually incomlete Some heat loss takes lace Some combustion gases dissociate at high temeratures For taking into account the first three oints, the combustion efficiency (η cc ) can be introduced, so exression (7.7) is rewritten as:. m. 0 ( ( 0 h + h h ) = m h + ( h h ) f ref ( ) η cc r f ref (7.8) r

62 Combustion chamber model in Modelica In the chater [4], it was exlained that the construction of model was based in an alternative sequence of flow models and control volumes. he mass and energy balances are calculated in the control volumes. But for the model of the combustion chamber, mass and energy balances are imlemented in a flow model. For the case of the combustion chamber, equations exlained above are used to evaluate the mass and energy balances. An equation for calculating the relation between the mass flow and the variation of the ressure is also needed. he combustion chamber model evaluates the mass and energy flows, which are then used in the control volume for evaluating the mass and energy balances. emerature in the combustion chamber is obtained there. All the models used for the creation of the combustion chamber are in the ackage called CombustionChamber. Some of the classes included in this ackage are exlained briefly. MassBalance. his class is used for evaluating the comosition of each of the comonents of the gas mixture at the outut of the combustion chamber. his class contains the equations (7.), (7.), (7.3) and (7.4) Valve his model was created for controlling the mass flow of fuel into the combustion chamber. he model Valve is a simle flow model, with two flow connectors and one connector with an inut signal. his last connector is used to regulate the fuel flow. here is a arameter called mdot_max, which can be sulied by the user. his arameter is the maximum fuel flow in kg/s that the valve can suly. It is ossible to regulate this fuel flow by using an inut signal with a value between zero and one. NaturalGasResS_X he model simulates an infinitive reservoir of natural gas. It is a control volume in which ressure and temerature can be assigned. he model also includes a medium model for natural gas, which includes the comonents CH 4, C H 6, C 3 H 8, C 4 H 0, N and CO. he user can select the natural gas fuel comosition in fuel in % of mass fraction.

63 59 hreeport his class is essentially a three ort flow model. hese three connectors are used for the incoming air, fuel and exhausted gases, resectively. his class also inherits the record called FlowVariablesSingleStatic, which contains some necessary variables for a flow model. In this class, these variables are connected to the variables in the connectors. CombustionChamber his model is the real combustion model. It inherits the classes hreeport and MassBalance. he model contains an equation that relates the mass flow m to the ressure dro in the combustion chamber. he equation is the following:.. 0 m m = (7.9) d d 0 Where d is: d = (7.0) and m 0 and d 0 are the mass flow m and ressure loss d in the combustion chamber at the design oint. he energy balance of equation (7.8) is also included in this class.

64 60 8. esting models of the turbine he three main models, comressor, combustion chamber and exander, have been resented in revious chaters. However, other models are necessary for the creation of the comlete dynamic model of the turbine. In this chater, two different comlete models of the turbine are resented. he first one corresonds to the turbine connected to a hydraulic brake without any controller. he second model includes a seed controller, which comares the rotational seed to a reference and regulates the fuel mass flow in order to maintain the seed constant. A descrition of these models, including the models that have not been resented before, is given in the chater. 8. System model with hydraulic brake he model is shown in Figure. It can be seen that the inut for this model is the fuel mass flow. A descrition of the models not resented before is given in this section. Figure 8. Comlete model of the gas turbine in oen loo

65 6 In order to define environmental conditions e.g. temerature, ressure, comosition of the air, it was necessary to develo a model of the environment. In the hermoflow library, thermodynamic reservoirs are used for this urose. A reservoir is considered as a control volume of infinite size, i.e. the values of the states do not change when energy and mass are introduced or extracted. In the turbine model, these states are temerature, ressure and comosition of the mixture of the gases CO, H O, N and O. hese states are arameters, which can be modified by the user for each simulation. In this way, it is easy to choose different environmental conditions for the simulations. However, the comressor and the turbine are not connected directly to the environment. Pies and filters cause a ressure dro between the atmosheric ressure and the ressures at the inlet of the comressor and at the outlet of the combustion chamber. In order to take this into account, a flow model with a simle equation for the ressure dro is added to the reservoir model. he comlete model created by a reservoir and a ressure dro model is called source. wo air sources are used in the turbine model, one situated at the beginning of the cycle, i.e. before the comressor and another at the end of the cycle, i.e. after the exander. Another reservoir is used for simulating the fuel tank. Nevertheless, the states for this reservoir are ressure, temerature, and comosition of the mixture of the gases CH 4, C H 6, C 3 H 8, C 4 H 0, N and CO. In this way, it is ossible to set different comositions of natural gas for the exeriments. In order to extract the ower obtained in the exander without having to control the rocess, a model of a brake is needed to dissiate the energy. A hydraulic brake model was imlemented for the turbine system model. his model is based on a relation between the torque and the seed. he relation is arabolic: τ = c + c n Where τ is the torque, n is the rotational seed and c and c are arameters, which can be modified by the user. In this way, the model can be simulated for different seeds by varying the values of c and c. he model also contains four control volumes. As it was exlained in the hermoflow chater, control volumes are used for evaluating the mass and energy balances. he area and length of the control volumes are arameters and have to be set by the user.

66 6 8. System model with seed controller his second model can be used for reroducing situations which are closer to the actual use of the turbine in the ilot lant. his model is very similar to the model exlained above, but the model of the brake is relaced by roviding a torque as an inut, which could e.g. come from a generator. When the value of the torque increases, the seed decreases. he value of the seed is comared to the reference seed and the controller increases the value of the fuel mass flows, in order to increase the seed until it reaches steady state. he model of the PI controller was taken from the Block sub-library. he comlete model can be seen in the figure. Figure 8. Comlete model of the gas turbine with a seed controller