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Nomenclature
cp Specific heat at constant pressure [J/kg K]
cv Specific heat at constant volume [J/kg K]
d Diameter [m]
f Friction factor [-]
0
g f Free Gibbs energy molar at standard conditions [kJ/kmol]
h Specific enthalpy [kJ/kg]
h Molar enthalpy [kJ/kmol]
k Thermal conductivity [W/mK]
m Mass [kg] – Polytrophic exponent [-]
m Mass flow rate [kg/s]
q Heat transfer rate [W]
q’ Heat transfer rate per unit length [W/m]
q’’ Heat Flux [W/m2] s Specific Entropy [kJ/kg K]
t Time [s] - Thickness [m]
u Speed [m/s] –Specific internal energy [J/kg]
u Molar internal energy [kJ/kmol]
x Vapour mass fraction [-]
A Area [m2] Bi Biot number [-]
Dh Hydraulic diameter [m]
J Inertia [kg m2] Ja Jacob number [-]
L Length [m]
Nu Nusselt number [-]
P Power [kW]
Pr Prandtl number [-]
Re Reynolds number [-]
S Surface area [m2] T Temperature [K]
U Heat exchange coefficient [W/m2K]
X Molar fraction Xtt Martinell Factor [-]
Greek symbols
α Convection heat transfer coefficient [W/m2K] – Air fuel mass ratio [-]
ε Emissivity[-], Relative error [-], Turbine pressure ratio [-]
σ Stefan-Boltzmann constant [-]
η Efficiency [-]
θ Crank angle µ Viscosity [kg/s m]
τ Torque [Nm]
φ Air fuel equivalence ratio [-]
ω Angular speed [rad/s]
φ
Abbreviations and subscripts
a Air
abs Absolute avg Average b Burning cond Conduction conv Convection
cr Critical df Dumping factor exp Experimental
f Fluid, Fin, Fuel h Hydraulic i Insulation irr Irradiation in Inlet m Mechanical
bmip Brake mean indicated pressure mod Model
n Nominal l Liquid out Outlet p Pipe prod Products
rad Radiative react Reactants t Thermal tf Transfer fluid v Vapour x Axial abscissa y Longitudinal abscissa w Wind, Wall, Water C Compressor CC Combustion Chamber
F Fuel
HTF Heat transfer fluid ICE Internal Combustion Engine MGT Micro Gas Turbine
N Negative
ORC Organic Rankine Cycle P Positive, Pump R Reduced
S Sun
T Turbine
ALIBRARY OF MODELS FOR THE DYNAMIC SIMULATION OF ENERGY SYSTEMS
47 how the single blocks can be properly bounded together creating a system of interest and results of transient simulations are provided demonstrating the capabilities of the tool.
As seen in the previous Chapter, when a physical system is being modelled two main classes of objects that can be considered:
Reservoirs, characterized by one or more states that represent the "stored" amount of level variables;
Flow Control Devices which determine the amount of properties that flow through them, typically as a result of differences between reservoir levels.
These definitions, already discussed, can fall in the broader categories of state determined and not state determined models, the first including the “reservoirs” systems and the latter including “flow control devices”, depending on whether or not it is possible to define differential equations expressed in terms of the state variables of the system.
When considering energy systems and power systems, as for example a steam plant or a gas turbine plant, it is a good practice to split the elements of the plant in two categories:
Heat exchanger devices;
Fluid machinery elements.
The distinction between these two categories is crucial and allows introducing some simplifications in the analysis of the corresponding models.
Any power cycle, besides a working fluid, requires a series of heat exchangers and fluid machines to operate. In a simple steam cycles heat exchanger devices may be assumed to be the evaporator and the condenser while the fluid machines are the turbine and the pump.
In general it can be stated that heat exchangers are all those elements where the working fluid exchanges energy with the surrounding only in thermal form, i.e. any exchange of work can be neglected.
At a first sight, also the effects of friction can be neglected hence considering the process as perfectly isobaric. This assumption will often be adopted in the analyses that follow. The thermal or mass capacitance of the system, i.e. its capacity of accumulating either thermal energy or mass can be considered or not. Taking into account these phenomena will certainly lead to start from energy or mass conservation equations that will contain time derivatives of some variables that can then be considered as state variables. Heat exchangers in this case are to be considered as state determined models where the behaviour in time of the fluid temperature (or specific enthalpy) as well as the mass flows through the boundaries is not a linear function of the actual inputs, but depends also on the history of the system since capacitances are taken into account. These systems can therefore be considered as reservoirs where level variables can be fluid temperatures.
It should be noted that a volume can be associated to heat exchangers, and hence the possibility to store mass. In this case the pressure within the heat exchanger is assumed to uniformly distributed, but its evolution in time is determined by the amount of mass stored within the component hence the pressure is a state variable of the system or, equivalently, a ‘level’ variable of the reservoir. In some cases, as in the example of the organic fluid evaporator, the capability in determining the pressure dynamics through mass and energy storage has been lumped to an associated volume (the drum) while the only level
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variables associated to the actual heat exchanger are merely the fluid temperatures and flow rates (see Par.
3.2.3 and 3.2.4).
The other main category of components identified in the power systems analysis is fluid machines.
Differently from the heat exchangers, they are assumed to be ideally adiabatic, that is the heat flux exchanged between the working fluid and the surrounding is neglected. The only significant energy flux exchanged between fluid and components occur in the form of mechanical work leading to a pressure change (either positive or negative): these components therefore cannot be considered isobaric. A common assumption in this case is that the dynamic phenomena within the fluid occur much faster than the rate of change in the thermodynamic boundary conditions which take place in the reservoirs to which the fluid machineries (or fluid control devices) may be connected. With this assumption, the processes occurring in the fluid can be modeled as quasi-steady processes and static maps can be used to describe the behaviour of these devices. This approach is typical of not state determined modeling techniques and the model outputs are determined by applying algebraic correlations, often empiric in nature and usually highly nonlinear, and no accumulation phenomena are considered. Fluid machines in fact, in system dynamic analysis, are usually considered to behave exactly as flow controlling devices and the actual mass flow rate flowing through them (flow variable) will be determined by the pressure existing at inlet and outlet of the device (the pressure as seen can be considered as a level variable determined by the dynamic processes occurring within the reservoirs, i.e. heat exchangers and their associated volumes).
Within this framework, therefore, all the models presented will be divided into the two main categories of state determined and not state determined in the broad sense definition introduced here.
According to this classification, the library of component models of energy systems will be divided into these two main categories.
The sub-library ‘state determined components’ will contain the following components:
thermal solar collector;
single phase heat exchanger;
heat exchanger with phase change;
drums;
constant pressure combustion chamber;
rotating shafts dynamics;
General fluid Receiver;
ICE intercooler.
The sub-library “not state determined” will contain models of the following components:
compressor;
turbines;
pump;
valve;
heat exchanger with no thermal dynamics;
in cylinder combustion processes (in ICE).
ALIBRARY OF MODELS FOR THE DYNAMIC SIMULATION OF ENERGY SYSTEMS
49 The complete customized library of components, whose detailed description is provided in the Paragraphs that follow, is visible in Fig. 3.1 that displays the ‘Simulink® Library Browser’ interface. The
‘Energy Systems Library’ contains the two mentioned sub-libraries referring to ‘not state determined’ and
‘state determined’ components. It is possible to notice that, analogously to any other standard Simulink® block, all the created models can be dragged and dropped in a new Simulink® workspace in order to be properly linked together to simulate the desired system. Examples of this will be provided in Chapter 4 where it is also described the sub-library ‘Complete Power Systems’ that appears in the main ‘Energy Systems Library’.
As general characterization the model referring to the ‘state determined’ sub-library will be marked with black shadows in order to make them easily distinguishable from the components referring to ‘not state determined’ sub-library.
(a) (b)
Fig. 3.1. The custom ‘Energy Systems library’ accessible from the ‘Simulink® Library Browser’: detail of (a) the ‘not state determined’ sub-library and (b) of the ‘state determined’ sub-library.
3.1.1 The state equation
In the following a general insight on the way equation of state have been considered in realizing the models proposed in the present Chapter is given. As already observed, the state equation of a fluid can be described with simple correlations only when the fluid can be considered as a perfect gas or as an
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50
incompressible liquid. The state equation cannot be defined analytically when the fluid is in the liquid to vapour phase change and usually empirical correlations are provided in this case.
Since the aim of the work was not to describe the fluid behaviour but rather to describe the system behaviour, the easiest way to implement the state equation has appeared that of using already compiled databases of fluid thermodynamic properties.
Among the many available in commerce, REFPROP® database, developed by NIST (National institute of Standard and Technology, USA) has been chosen for the scope of this work. REFPROP® is an acronym for REFerence fluid PROPerties and provides tables and plots of the thermodynamic and transport properties of fluids commonly used in industrial applications and their mixtures, with an emphasis on refrigerants and hydrocarbons. REFPROP® is based on the most accurate pure fluid and mixture models currently available. It implements three models for the thermodynamic properties of pure fluids: equations of state explicit in Helmholtz energy, the modified Benedict-Webb-Rubin equation of state, and an extended corresponding states (ECS) model [1]. Mixture calculations employ a model that applies mixing rules to the Helmholtz energy of the mixture components; it uses a departure function to account for the departure from ideal mixing. Viscosity and thermal conductivity are modeled with either fluid-specific correlations, an ECS method, or in some cases the friction theory method.
These models are implemented in a suite of Fortran® subroutines. They are written in a structured format and have been tested on a variety of compilers.
Routines are provided to calculate thermodynamic and transport properties at a given (T,ρ,x) state.
Iterative routines provide saturation properties for a specified (T,x) or (P,x) state. Flash calculations describe single- or two-phase states given a wide variety of input combinations [(P,h,x), (P,T,x), etc].
Even though a separate graphical user interface, designed for the Windows operating system, can provides a convenient means of accessing the models, and it allows generating tables and plots for user-specified mixtures or a number of predefined mixtures (air and the commercially available refrigerant blends), the software has been chosen for its compatibility with the Matlab® platform.
A specific Fortran® file, called refpropm.f90, is in fact available to link Matlab® with the routines used in REFPROP®. The Fortran® routines are called externally by typing with proper arguments the Matlab® function refpropm.m, developed by Lennart Vamling at the Chalmers University of Technology in Sweden and modified by Johannes Lux of the German Aerospace Center. This function allows to use the entire REFPROP® database from Matlab®, hence allowing calculations of fluid state properties from any Matlab® based model code. The function returns the required fluid properties given a state point (defined by two specified and known state properties) and given the pure fluid considered or the fluid mixture composition (if the substance is not a pure fluid). Examples of call of the REFPROP® routines from Matlab® are provided below:
1) cp_g=refpropm('C','T',T_g,'P',p_g,'CO2','water','nitrogen','oxygen',[0.091 0.074 0.742 0.093]);
In this case the specific heat at constant pressure is provided at a given temperature and pressure for a known mixture of gases that constitute the exhaust gas composition of a natural gas fired ICE.
2) [h4,T4]=refpropm('HT','P',p_min,'S',pt3_1.s,’toluene’);
In this second case specific enthalpy and temperature are calculated at a given pressure and specific entropy for toluene.
ALIBRARY OF MODELS FOR THE DYNAMIC SIMULATION OF ENERGY SYSTEMS
51 The coder is indeed highly flexible and suitable for the application of interest. However a trial of accuracy has been conducted before using it in order to assess the errors that arise using inverse functions.
Particularly for the three fluid states of interest within fluid power systems analysis (liquid, two-phase and vapour), given two state properties (pressure p and temperature T for liquid and vapour regions, vapour fraction x and pressure p for the two-phase region) two new properties have been calculated (as for example specific enthalpy h and density ρ). Starting from these last properties the former have been calculated, according to the following procedure:
liquid and vapour:
( ) ( ) ( )
p Tp" p' T " T ' p',T ' h,ρ p",T " ; e , e
p' T '
− −
→ → = = ;
two phase region:
( ) ( ) ( )
p xp" p' x" x' p', x' h,ρ p", x" ; e , e
p' x'
− −
→ → = = .
If the functions used by the software were not approximate the same starting values should be observed, as h and ρ describe the same state point and the relative errors would appear nought. Some errors however occur in computing inverse functions and can be observed from Tab. 3.1, derived for water.
Field Property Relative error [%]
p=100 kPa 3.4420·10-8 Liquid
T=300 K 3.2211·10-13 p=100 kPa 1.8716·10-11 Vapour
T=500 K 1.8417·10-11
p=100 kPa -0.0897
Two-phase
x=0.5 0.0064
Tab. 3.1. Relative error due to inverse functions of the REFPROP® database for water.
Similar results have been gathered for other fluid families and for other state parameters and demonstrate how the error committed by recurring to inverse functions is negligible in the single phase regions and quite more significant in the two phase region. This analysis shows that a broad use of inverse functions in this region should be done carefully.
The REFPROP® database therefore has been integrated into many of the components described and in general in all the cases when specific fluid properties were required for calculation, with the double advantage of always considering precise values of the properties (rather than introducing approximating mean values also in the liquid or gas fields of the fluids considered) and relieving from the need to upload and interpolate on wide data bases.