:x= : A
Fig. 2.19 Examples of TF element. a Electrical transformer. b Hydraulic piston. c Mechanical lever
42
2 Bond Graph Modeling in Process Engineering
Transformers are of two different kinds: 1. Impedance transformers, where the input and output variables are of the same energy domain or class. 2. Class transformers, where the input and output variables belong to different energy domains or classes, e.g. hydraulic cylinders and pumps connecting hydraulic variables with mechanical variables. Hydraulic power is transduced into mechanical power in the piston. Gyrator element - GY De nition 2.8. A Gyrator element, denoted by GY, is a conservative two-port bond graph element which scales dissimilar power variables, generally transforming energy from one domain into another. The input effort is proportional to the output ow and the output effort is proportional to the input ow (Equation 2.47). Gyrators are also called transducers, and they allow one to transform energy from one domain into another (gyroscope, electric motor). An example is the electric motor that transforms electric power to mechanical rotary power (Figure 2.20a). The constant of proportionality r is called gyrator ratio or modulus. The gyrator may be adjustable, and its conversion ratio can be modulated by a signal applied through an activated bond. This is represented by the symbol MGY(modulated gyrator). An example is the electric motor with variable eld strength (Figure 2.20b). Unlike the TF-element, the constitutive equation for GY is “crossed”, i.e. between dissimilar power variables: e1 = r f2 ; e2 = r f1 :
(2.47)
Fig. 2.20 a Gyrator element. b Modulated gyrator element
Note that the modulated signal (activated bond) means that no power is associated with the change in m and r (in both TF and GY elements); thus the power e1 f1 is equal to e2 f2 . This is contrary to other passive elements, e.g. C-element where the stored energy changes when the information signal changes. One example of the latter case is a capacitor with movable plates.
2.5 Causality
43
2.4.5 Information Bonds In measurement and control process, the energy transferred by the signal is negligible compared to those exchanged between physical components. This signal is represented by an information bond, which corresponds to block diagrams arrows. It is shown as a full arrow on the bond and can represent the signal transmitted by a sensor, integrator, sum member, etc. Figure 2.21 shows the representations of the effort sensor De and the ow sensor Df. The causality assignment decides the type of the sensor. The concept of causality is discussed in the next section.
Fig. 2.21 Information bond. a Effort detector. b Flow detector. c Transmitted or control signal
This means that only one variable (effort or ow) is used in a signal bond. The effort sensor for example provide to the user or to operator only information about the effort measurement. There is no reaction from the sensor to the system. This is why the ow is considered equal to zero (Figure 2.21a). The bond is denoted as activated. A bond is called activated, if one of its conjugate power variable is set to zero.
2.5 Causality 2.5.1 Introduction The energetic coupling between two interconnected physical components is represented by a bond as developed before. Such a model is acausal. Each component is described by its constraint (called constitutive equation). The physical component must behave according to its constraint de ned in the terms of the conjugate power variables. But if we need to simulate the physical phenomena (the model), we have to decide in which order the variables (efforts and ows) will be computed. Thus we will introduce the block diagram simulation which is causal. Consequently we need to make a series of cause and effect decisions, alternatively called causal assignment. This concept of causality is central to any computational model development. To make the bond graph model causal, the founders of this theory introduced
44
2 Bond Graph Modeling in Process Engineering
a perpendicular stroke on each bond. This single mark on a bond, called a causal stroke,indicates how e and f are simultaneously determined on a bond. Let us consider interaction between two sub-systems A and B (hydraulic pump and a tank) as shown in Figure 2.22. The pump may deliver as output a ow (volume ow VP ) or an effort (effort PP ). The conjugate power variables for the tank are pressure (PP ) and volume rate (VR ). Computation of the models can be performed by two causal bond graphs and their corresponding block diagrams: PP is known for B (Figure 2.22a) or VR is known for B (Figure 2.22b).
Fig. 2.22 Causality assignment rule
Based on the causality concept, the imposition of causes and effects in the system is directly portrayed on the graphical representation and shows at the same time the transition to the block diagram. The simulation algorithm is then indicated by the position of the causal stroke in the bond graph which becomes causal bond graph. The rule is as follows: Remark 2.4. The causal stroke is placed at the end of the bond which receives the effort information. The other end receives the ow information. In physical systems, effort and ow information are always counter oriented. However, we will show
2.5 Causality
45
later that in non-physical systems, such as when we use available information for diagnosis, they can be co-oriented. The causality convention is easily remembered through the graphical representation given in Figure 2.22c.
2.5.2 Sequential Causality Assignment Procedure (SCAP) In order to predict how the nal equations will turn out and to avoid inconsistency, the procedure given below must be followed in strict order: 1. The sources impose always one causality: imposed effort by effort sources and imposed ow by ow sources. Then choose any Se or Sf and assign its required causality. Immediately extend the causal implications, using all 0,1,TF and GY restrictions that apply, as shown in Table 2.6. 2. Choose any C- or I-element and assign integral causality (well adapted to numerical calculations) as shown in Table 2.6. Again extend the causal implications of this action, using 0, 1, TF and GY restrictions. 3. Choose any R-element that is unassigned and give it an arbitrary causality. In linear R-elements, the causality is in principle indifferent, but indicates whether resistance or conductance need to be entered as parameter. In non-linear Relements, equations are more comfortable in one direction according to the equation form. Extend the causal implications, using 0, 1, TF and GY restrictions. Causal forms of different bond graph elements, their corresponding causal equations and block diagrams, and the causality assignment rules are given in Table 2.6. As an example consider the causal bond graphs of the thermal conduction phenomenon described in Figure 2.10 by true and pseudo bond graphs (Equations 2.18 and 2.20). Figure 2.23 represents causal pseudo bond graph models and corresponding block diagrams for thermal resistor R. The 1-junction is used because the thermal ow, as a function of the difference between two thermal efforts (temperatures), is the same. In Figure 2.23a, T3 is known for R because the causal stroke is placed near Relement. T1 and T2 are imposed by systems ∑1 and ∑2. Based on discussed causality rules (Table 2.6), constitutive causal equations for the junction and R-element can be written as follows: 1 junction, for efforts: ∆ T = T1 T2 ; for ows: Q3 = Q1 ; Q3 = Q2 : ∆T R element, Q3 = (Fourier's law in linear case). Rt
(2.48)
From the bond graph model given in Figure 2.23b, we obtain 1 junction, for efforts: T1 = T2 + T3 ; for ows : Q3 = Q1 ; Q2 = Q1 : R element, T3 = Rt Q1 :
(2.49)
46
2 Bond Graph Modeling in Process Engineering
Table 2.6 Causality assignment for bond graph elements
2.5 Causality
47
Fig. 2.23 Causal pseudo-bond graph and block diagram models of thermal conduction
Note that 1=Rt is the heat transfer coef cient.
2.5.3 Bicausal Bond Graphs As we have seen, causality imposes a certain propagation throughout the bond graph. It implies that if effort acts in one sense, ow acts in the reverse sense, which is true for all physical systems. However, in system inversion and fault diagnosis, other rules are necessary[119]. One generalization of the causality selection proposed in [80, 83, 177] is the bicausal bond graph. There one divides the causality stroke into two. One half, on the same side as the half arrow indicating power direction, indicates the direction of ow, the other indicates the direction of effort, as shown in Figure 2.24.
Fig. 2.24 Bicausal bond graph
48
2 Bond Graph Modeling in Process Engineering
In the rst case shown in Figure 2.24, the causality of R indicates that both effort and ow are available to a block representing the constitutive relation of the Relement. This allows one to calculate the resistance R, i.e. it helps in parameter estimation. More details on bicausality notation and its use are given in Chapter 7.
2.5.4 State-space Equations In control system theory, the state equation form is well suited for control analysis and synthesis. They can be systematically deduced from a bond graph in linear or non-linear form: x = Ax + Bu ; y = Cx + Du x = F(x; u) Non-linear form: ; y = G(x; u) Linear form:
(2.50)
where A,B, C and D are matrices with appropriate dimensions, u is the input vector (sources Se and Sf), y is the vector of measured variables (effort and ow detectors) and x is the state vector. The state vector x is composed of the variables p and q, i.e. the energy variables of storage elements (C- and I-elements) in integral causality: x=
pI : qC
(2.51)
2.5.4.1 Properties of State Variables The state vector does not appear on the bond graph, but only its derivatives appear: x=
eI pI = : fC qC
(2.52)
The dimension of the state vector is equal to the number of C- and I-elements in integral causality. If among the n C and I-elements, nl are in derivative causality, then the order of the model is n nl : The state-space equations can be deduced from a bond graph model in the following steps: Write structural laws associated with junction structure (0, 1, TF, GY). Write constitutive equations of sources and each passive element (R, C, I). Combine these different laws to obtain state equation through sequential ordering and substitutions.
2.5 Causality
49
2.5.4.2 Example: State-space Equation of an Electrical System. Consider a simple electrical system example given by its schema and causal bond graph model in Figure 2.25a and 2.25b, respectively.
Fig. 2.25 a An example electrical system. b Its bond graph model
The state-space equation will be derived in the form: x(t) = Ax(t) + Bu and the outputs will be written in the form y = Cx + Du: Where the state variable x, the control input u and the measurement y are respectively the energy variables in dynamic bond graph element (C and I), the source (Se) and the effort detector (De): R
e dt p2 ; = R 2 f6 dt q6 u = Se = E(t); y = De = e6 : x=
(2.53)
The structural equations are the constitutive equations of “0” and “1” junctions (energy or power balance equations): 1 junction: 0 junction:
f1 = f2 ; f3 = f2 ; f4 = f2 ; e2 = e1 e3 e4 : e4 = e6 ; e5 = e6 ; f6 = f4 f5 :
(2.54)
The following equations are the constitutive behavioral equations of bond graph I, C and R elements:
50
2 Bond Graph Modeling in Process Engineering
8 1R > > e2 dt = f2 = > > L > > R 1 < f6 dt = e6 = C > e3 = R1 f3 ; > > > > 1 > : f5 = e5 : R2
p2 ; L q6 ; C
(2.55)
Combining Equations 2.54 and 2.55, we obtain the state-space equation: " # R1 1 p2 p2 1 L C = + (2.56) E(t); 1 1 q6 q6 0 L R2C y = e6 = 0
1 C
p2 : q6
(2.57)
More details of sequential ordering to obtain state equations are given in [172]. Note that for some speci c forms, state equations cannot be obtained as simply through sequential ordering and substitutions. These situations arise due to algebraic loops (discussed in Chapter 7) and differential causality. A causal loop [172] is another situation where it becomes impossible to eliminate unknown variables.
2.5.5 Model Structure Knowledge The bond graph is an advantageous modeling tool because it exposes both the structure and the behavior of the studied system. In order to illustrate this property let us consider a simple electrical example given by its schema and causal bond graph models (Figure 2.26). The integral causality (Figure 2.26b) is recommended for engineering simulations. Here the dynamics evolves from given initial conditions. The derivative causality (Figure 2.26c) is more used in FDI because initial conditions are treated as unknown variables in process supervision. In both the models, the current and the voltage are measured respectively by the ow sensor (Df) and effort sensor (De). The sensors, assumed to be ideal, are connected to the junction structure by means of signal bonds indicating that there is no power transfer between the system and sensors. The bond graph model can be represented in a vectorial way (Figure 2.27) showing the junction structure and the different elds it is composed of. The key vectors associated with the representation are: X : the state vector (generalized momentum “p” on “I-element” and generalized displacement “q” on “Celement”) divided into xi and xd ; the subvectors respectively associated with the components in integral and derivative causality X: the time derivative of the state vector divided into xi and xd , Z: the complementary state vector (“ f ” on “I” and “e” on “C”) divided into Zi and Zd ;U: source input vector, Y : sensor output vector, Din : input vector to “R” eld, and Dout : output vector from “R” eld. This representation leads to the junction structure matrix S such that
2.5 Causality
51
Fig. 2.26 An electrical system (a) and its bond graph model in (b) integral and (c) derivative causality
Fig. 2.27 Partitioned representation of model structure
2
3 2 xi S11 6 Zd 7 6 S21 6 7 6 4 Din 5 = 4 S31 S41 Y
S12 S22 S32 S42
S13 S23 S33 S43
32 3 Zi S14 6 7 S24 7 7 6 xd 7 5 4 S34 Dout 5 S44 U
(2.58)
The matrix S has (nC + nS ) rows, where nC and nS are respectively the numbers of components (I,R or C) and sensors in the systems (De and Df). This matrix is composed of 0; 1; +1; m; r or 1=m and 1=r, where m and r are the moduli of trans-
52
2 Bond Graph Modeling in Process Engineering
former TF and gyrator GY elements. More details on the model structure and its implications can be found in [127].
2.6 Single Energy Bond Graph 2.6.1 Bond Graphs for Mechanical Systems It is easy to construct a bond graph model for electrical and hydraulic systems. Electrical and hydraulic systems can be represented as networks and then it becomes possible to apply the method of gradual uncover, method of point potential or a mix of the two methods (see [172]) to arrive at compact bond graph representations. For mechanical systems, method of ow map and method of effort map are developed in [172]. A chapter titled `art of creating system bond graphs' in [172] details various steps to arrive at beautiful and compact bond graph structures for mechanical and electrical systems. With practice, it becomes possible to produce the bond graph model directly by looking at the system morphology, skipping all intermediate steps. For complex systems, one can follow the steps shown in Figure 2.28: 1. Fix a reference axis for velocities. 2. Consider all different velocities (absolute velocities for mass and inertia) and relative velocities for others. 3. For each distinct velocity, establish a 1-junction, attach to the 1-junction corresponding bond graph elements. 4. Express the relationships between velocities. Add 0-junction (used to represent those relationships) for each relationship between 1-junctions. 5. Place sources. 6. Link all junctions taking into account the power direction. 7. Eliminate any zero velocity 1-junctions and their bonds. 8. Simplify bond graph by condensing 0- and 1-junctions having two bonds: for example: 1 + 0 + 1 is replaced by 1 + 1 and then simply by 1: The nal bond graph obtained in the last step could as well be derived in a much simpli ed manner. One may note that the velocity of the mass point and the applied load must be the same; moreover the velocity of the mass point is equal to the rate of extension of the spring and the damper. Although these rates (velocities and strain rates) are qualitatively different, they have equal magnitude and hence they should be connected to the same 1-junction.
2.6.2 Bond Graphs for Thermal Processes Bond graph modeling methodology for thermal processes is considered in this section. Let us consider as an example a thermal process given in Figure 2.29a, where
2.6 Single Energy Bond Graph
53
Fig. 2.28 Steps for construction of bond graph models of mechanical translation
only the thermal phenomenon is considered. The coupling with hydraulic energy is developed in the Section 2.8. The process is heating of a thermal bath (3) by exchanging heat through the metallic walls (2) of temperature Tp with a warm uid (1) kept at a given temperature T f : Heat is also leaving through the free surface to the environment supposed to be at a temperature Ta: So there is a thermal resistor between the bath (3) and the external environment at temperature Ta : This thermal resistor may be linear (conduction only) or non-linear (conduction and radiation). The temperature of the bath noted Tr is measured by a temperature sensor TI. Figure 2.29b gives the analogous electrical schema, based on the (T; Q) analogy, that is a pseudo bond graph. So voltage u takes the place of temperature T and current i the place of heat ow Q. A step-by-step approach to construct the bond graph model of this system is shown in Figure 2.30. 2.6.2.1 Word Bond Graph The word bond graph of the studied process is given Figure 2.31. The used conjugated variables are temperature T and thermal ow Q.
54
2 Bond Graph Modeling in Process Engineering
Fig. 2.29 Thermal system and its electrical equivalence
Fig. 2.30 Different steps in bond graph modeling and validation
2.6.2.2 Bond Graph Model This step represents a physical level of modeling. Based on an assumed modeling hypothesis, the physical phenomena can be graphically represented. At rst, suppose that the thermal capacity of the wall, Cm ; is negligible, while the thermal capacity of the bath, Cr ; is taken into account. To make the bond graph easy to read, all bonds are numbered. The bond graph model is given Figure 2.32a. The bond graph model is constructed from an energetic consideration of the process. The temperature T f is imposed as an external effort source noted Se : T f .
2.6 Single Energy Bond Graph
55
Fig. 2.31 Word bond graph of the thermal process
Fig. 2.32 Acausal (a) and causal (b) bond graph of the thermal system where the thermal capacity of the wall is neglected
There is a thermal transfer by conduction (modeled by R p ) from the hot uid T f through the wall to the bath. Part of the transferred heat is accumulated in the thermal capacity of the water Cr and the other part is dissipated through Ra to the environment at the temperature Ta (modeled as a negative source as indicated by the orientation of the bond). All these effects are coupled by the parallel and series junctions (0- and 1-junctions). The 0-junction expresses the common temperature points. The 1-junction represents the common thermal ows. The bond graph model is then constructed as given above. The bond graph can be realized using a pedagogical procedure as follows: 1. Fix reference axis for thermal ows (generally from high to low temperature). 2. Place energy sources (temperature source Se : Ta and thermal ow source S f : Q). 3. A 0-junction is associated with each different temperature (this corresponds to nodes of different point-potentials in electricity). If a energy storage occurs in this junction, attach C-element (C : Cr , here) to this junction. 4. Place a 1-junction between two zero junctions or between 0-junction and thermal effort sources, attach R-elements to model dissipation and heat transfer. 5. Place sensors (De : Tr ) and D f : Q p ). 6. Link all junctions taking into account the power directions. 2.6.2.3 Assignment of the Causalities At this stage, the computational structure of the model which is associated with block diagram structure is deduced. The assignment of causalities is often guided by the necessity to resolve numerical problems: for simulation, preferred integral causality for dynamic elements is imposed. For model based fault detection and
56
2 Bond Graph Modeling in Process Engineering
isolation purposes derivative causality is imposed. In our case, where the model is used for simulation, let us impose integral causalities for the bond graph model considered in Figure 2.32a. We start from the compulsory causalities: they are effort sources (Se : T f and Se : Ta ). Then assign integral causality to C : Cr . In the next step, causality propagation through junction rules (e.g. only one causal stroke near a 0-junction) automatically forces causalities in the remaining elements R : R p and R : Ra . The causal bond graph model in then obtained (Figure 2.32b). Sometimes, to avoid derivative causality, spurious elements are added to the model. These elements are often given some physical meaning. Such problems usually arise from our modeling assumptions, i.e. from over complexi cations or over simpli cations. As an example, if the thermal resistance R p is neglected, we would have derivative causality on a C : Cr element. One could avoid this by a ow source instead of an effort source, or by adding a thermal resistance of some value and keep the effort source as it is. These problems require ingenuity and experience. The most common structure used to break differential causalities are coupling capacitors [258] in hydraulic and thermo- uid systems, and soft and hard pad structures [172] in mechanical systems. 2.6.2.4 Writing Mathematical Equations The mathematical structure of the model is deduced systematically from a causal bond graph. According to the assigned causalities, we have: Structural equations (conservation equations from junctions) Junction 11 : e2 = T f e5 , f1 = f5 = f9 = Q p = f2 Junction 0 : Q4 = f4 = f5 f6 , e5 = e7 = e6 = Tr = e4 , Junction 12 : e3 = e6 Ta , f8 = f6 = f3 :
(2.59)
Behavioral equations (constitutive equations for Bond graph elements) T f e5 1 e2 = : Rp Rp 1 Rt Q4 C:Cr element: e4 = f4 dt + e4 (0) = : Cr 0 Cr 1 1 (e6 Ta ) ; R: Ra element: f3 = e3 = Ra Ra R:Rp element: f2 =
(2.60)
where Cr is the global thermal capacity represented as a parameter for C-element in the bond graph model. It depends on the speci c heat capacity of the water in the bath at constant volume (cv ) and the total water mass, mr : Cr = mr cv :
(2.61)
Parameters R p and Ra are thermal resistances of the hot uid-water bath interface and the reservoir-environment interface, respectively. Note that the thermal resis-
2.6 Single Energy Bond Graph
57
Fig. 2.33 Block diagram for simulation
tances R are equal to the reciprocal of the heat transfer coef cient Kc or conductance (R = 1=Kc ). 2.6.2.5 Simulation Block diagram As developed before, a block diagram representation (Figure 2.33) can be easily arrived at from Equations 2.59 and 2.60. State equations Since we have neglected hydraulic energy and assumed the mass of the uid constant, we have only one C-element in integral causality and therefore there is only one state variable. The state variable x is the accumulated thermal energy in the tank (generalized displacement), the inputs u are thermal effort sources and the measured variables y are given by the detectors: Qr , u = T f Ta
x = Q4
T
, y = Q p Tr
T
:
(2.62)
Similar to the electrical system example given in Figure 2.25, the following state equation is obtained: Qr (t) = 2
y=4
1 1 + R pCr RaCr
Q p (t) Tr (t)
3
2
6 5=6 6 4
Qr (t) +
1 1 + R p Ra
T f (t) ; Ta (t)
2 3 1 3 2 3 1 0 T f (t) R pCr 7 6 7 Rp 7 4 7 5: 7 Qr (t) + 6 4 5 5 1 Ta (t) 0 0 Cr
(2.63)
58
2 Bond Graph Modeling in Process Engineering
2.6.2.6 Re nement of the Model Suppose that the error between the model and the real process is bigger than the admissible error, which is usually about 5%. It means that some physical phenomenon has been neglected. There is no need to write the equations from the beginning, but we just have to add a bond graph element (or set of elements forming a structure) corresponding to the newly accounted physical phenomenon. As an example, we can include the thermal capacity Cm (neglected in the rst version) of the wall of the bath 3. The revised bond graph model is then given in Figure 2.34. The thermal capacity C : Cm is shown within the doted lines.
Fig. 2.34 Re ned bond graph model which includes the thermal capacity of the wall
In this case, there is conduction heat transfer, modeled by R p ; from the hot uid T f to the wall of the heated bath. One part of this energy is accumulated by the thermal capacity of the wall, modeled by Cm . The other part is transmitted by R2 into the bath. Now the system has two state variables, which come from energy storage in the wall of the heat bath (Qm ) and from the heat bath itself (Qr ). The state equations are easily deduced from the constitutive equations of two C-elements and three Relements: 2
3 1 1 1 + 6 7 Qm Qm R1Cm R2Cm R2Cr 7 x(t) = =6 4 5 Qr 1 1 1 Qr + R2Cr RaCr 2R2Cm 3 1 0 7 Tf 6 + 4 R1 1 5 Ta ; 0 Ra 2 3 3 2 1 1 T Q p (t) 0 6 7 Qm y(t) = = 4 R11Cm 5 + 4 R1 5 f : Ta Qr Tr (t) 0 0 Cr
(2.64)
2.7 Formal Generation of Dynamic Models
59
2.7 Formal Generation of Dynamic Models 2.7.1 Bond Graph Software Bond graph models are supported by many commercial software: the model can be graphically introduced in the software and the software automatically generate the dynamic models. The model can be automatically transformed into a simulation or control or supervision program. Author Samantaray presents a review of various commercially available bond graph software in [220]. Moreover, bond graph models can be described in Modelica language as connection of submodels [68] and the constitutive equations can be introduced by the user. The 20-sim software [260] developed at the University of Twente (Netherlands) is based on the well known block oriented TUTSIM simulation program. Another software developed by A. Mukherjee and A.K. Samantaray from Indian Institute of Technology in Kharagpur, allows powerful of use bond graph as integrated computer aided design tool [174] for large and complex systems. Indeed, this software, called Symbols, is an object oriented hierarchical modeling, simulation and control analysis software. It allows users to create models using bond graph, block-diagram and equation models. Recent modules [191] have been developed in order to use this software not only for modeling but for control and monitoring [96, 97, 161, 189, 190, 225]. Thanks to a developed generic plant item database[188] which consists of a set of prede ned process, controller and sensor classes, the designer can easily build the dynamic and functional models of most thermo uid processes from the Process and Instrumentation Diagram (P&I D) of the plant, and automatically generate mathematical models. The model can also be used for design of supervision systems and associated algorithm generation. Several applications in process engineering were developed using this software in the framework of European project CHEM [1] at Ecole Universitaire Polytechnique de Lille.
2.7.2 Application For illustration, let us consider brie y how the dynamic modeling of the studied thermal system (Figure 2.29a) can be automated. In the next chapters it will be shown how the obtained model can be used for control and diagnostic design. By using the few bond graph elements (R, C, I, TF, GY, Se, Sf and Junctions) displayed in a list, the model is graphically entered (Figure 2.35a). The power directions and causalities are automatically assigned and they may be altered later by the user. The dynamic model (Figure 2.35b) is then generated. In this model, the variable f4 represents the thermal ow in the bond number 4 and corresponds to the time derivative of the generalized displacement Q4. R2 and R3 are the parameter values of the resistances R p and Ra respectively. K4=1/C4 is the capacitance of Cr .
60
2 Bond Graph Modeling in Process Engineering
Fig. 2.35 Automated modeling. a Graphical User Interface in Symbols-2000. b Dynamic model in symbolic format. c Simulation result
SE1 and SE8 are efforts sources at the bonds 1 and 8 and represent, respectively, Se : Ta and Se : T f . Tr_e2 and Qp_f2 are the measurement vector. The generated model can be rewritten as follows: K4 K4 1 1 SE1 + Q4 + ; SE8 R2 R3 2 3 R2 R3 " # K4 (2.65) 1 Qp_ f 2 SE1 6 R2 7 0 y= = 4 1 5 Q4 + R2 : Tr _e2 SE8 0 0 C4 which is identical to Equation 2.63. The numerical simulation results for speci c parameter values is given in Figure 2.35c. The given simulation scenario represents a step input of T f followed by a fall in the ambient temperature. The used bond graph software has its own Q4 =
2.7 Formal Generation of Dynamic Models
61
Fig. 2.36 Simulink R S-functions to construct block diagram from a bond graph model and its block parameters
simulation interface. Moreover, the bond graph model can be converted into a MATLAB R -Simulink R S-function [238] as shown in Figure 2.36. The obtained Simulink R model can be simulated by entering numerical values of bond graph parameters (R2, R3 and K4). If the modeling hypothesis is to be changed, then the bond graph model can be modi ed graphically and the corresponding state equations, manually obtained in Equation 2.64, are automatically generated (Figure 2.37).
Fig. 2.37 Graphical modi cation of bond graph model (thermal capacity of the wall is included)
62
2 Bond Graph Modeling in Process Engineering
2.8 Coupled Energy Bond Graph 2.8.1 Representation In thermo uid process, thermal and hydraulic energies are coupled [232]. Their coupling can be represented by a small ring around the bond (Figure 2.38a) or by a multi-bond [18, 29, 43] as shown in Figure 2.38b. Other authors [127] use a different symbolism shown in Figure 2.38c, whose corresponding decoupled version is shown in Figure 2.38d.
Fig. 2.38 Representation of coupled bond graphs
To distinguish energy domains, hydraulic power is represented in full line and thermal power is given in dotted line. The coupling is modeled by some ctive element (usually R-element). This representation is well suited for analysis. A general expression for convected energy is H = m u+
P ρv2 + ρ 2
= m h+
ρv2 2
(2.66)
: 2
Taking into account that in practice the kinetic energy term ρv2 is negligible for suf ciently low ow velocities v and that the uid is under saturated, the convected energy H is calculated from the mass ow m and the speci c thermal capacity c p : H = m:h = mc p T:
(2.67)
Finally, the thermo uid power variables (efforts and ows of a pseudo-bond graph) are
2.8 Coupled Energy Bond Graph
e = eh et
T
63
= PT
T
; f = f c ft
T
= Hm
T
:
(2.68)
2.8.2 Thermo uid Sources The thermo uid source consists of hydraulic and thermal conjugate ows, transferred by the convection of the uid. A thermo uid sources is characterized by the coupling of hydraulic and thermal energies, as de ned in Equation(2.67). In bond graph form, it can be represented as given in Figure 2.39.
Fig. 2.39 Thermo uid source. a Pump. b Its bond graph model
This representation is important from the physical and system analysis points of view. In fact, the real actuator gives hydraulic ow S fh : m: The enthalpy ow H source is considered as ctive actuator since it is a consequence of hydraulic ow variation (see Equation 2.67). The temperature source Set : T can be considered as a parameter or variable (if it is provided by an external source for instance). The source can be modulated by any external control signal, which regulates the ow rate.
2.8.3 Thermo uid Multiport R In thermodynamic processes, the dissipation phenomenon is modeled by a twoport R-element. The main phenomena modeled by such multiport R-elements are evaporation, condensation, convection through a pipe, valve, nozzle, etc. Concerning the thermo uid plant items associated with dissipation processes, the driving force which leads to energy dissipation or material transportation [38, 47] is the difference between the upstream and downstream efforts (pressure drop, chemical potential difference, temperature difference). Usually the causality for this type of plant item in thermo uid processes is the following: the input is the driving force (effort) and the output is the ow. But, in practice, the causality assignment for this component will be arbitrary. It depends
64
2 Bond Graph Modeling in Process Engineering
on the inlet and outlet signals imposed by the connected processes at upstream and downstream sides; i.e. the element could be in conductance causality or in resistance causality. These cases appear when connecting several multiport R-elements (pipes) in a serial connection.
Fig. 2.40 R multiport. a Nozzle. b Bond graph of nozzle. c Pipe with ori ce. d Bond graph of pipe with ori ce
The ows leaving the R- eld, because of continuity (Fin = Fout ), are calculated as follows: F=
m ΦRH (Tin ; Tout ; Pin ; Pout ) = ; H ΦRT (Tin ; Tout ; Pin ; Pout )
(2.69)
where the constitutive equations given by functions ΦRH and ΦRT are usually nonlinear and depend mainly on the thermodynamic state of the uid and on the modeling hypothesis. According to conductance causality imposed to R-element (Figure 2.40b), the outputs of the two port restrictor are thermal ow H and hydraulic ow m: The suf xes “in” and “out” represent the physical input and output sides of the modeled system. In the following, we consider two types of technological components modeled by R-multiport: a nozzle and a pipe with a restriction.
2.8 Coupled Energy Bond Graph
65
2.8.3.1 Nozzle The general form of bond graph model (Figure 2.40a) is useful when a restrictor can choke due to supersonic ow at its throat. Then the mass ow rate depends not only on the pressure difference, but also the pressure ratio Pr : Pr =
Pout : Pin
(2.70)
It is then possible to predict, with acceptable accuracy, the mass ow rate by using the upstream and downstream pressures [127]: s γ 1 1 2γ Pin 1 (Pr ) γ , if Pr b; m = A p (Pr ) γ R (γ 1) Tin G (2.71) s Pin m = Ap Tin
γ RG
2 γ +1
γ 1 γ 1
; if Pr
b;
where b=
2 γ +1
γ 1 γ 1
;
(2.72)
RG is the gas constant, A is the exit area, γ = c p =cv , c p is the speci c heat at constant pressure and cv is the speci c heat at constant volume. 2.8.3.2 Thermo uid Transport Phenomena in a Pipe The drawback of the aforementioned modeling approach is that in the opposite causality, the effort cannot be calculated from such a non-linear equation (Equation 2.71). In industrial process engineering, we generally assume low velocity or subsonic ow. Thus it is suf cient to assume that the mass ow rate m is a function of the pressure difference. We assume rst that the ow- eld properties are constant in a long pipe and the accumulation of mass, momentum and energy in such a component are of negligible order. The ow is regular and the uid is Newtonian. The bond graph model, with these simplifying assumptions, is given in Figure 2.40b, where the I-element (developed later) is included to take uid momentum into account. Readers may look at [173] for detailed models of hydraulic transport phenomena with ow saturation characteristics, which are derived from rst principles. The energy coupling (Equation 2.67) is modeled by the ctive thermal resistor Rc modulated (using information bond) by the hydraulic ow variable m: According to the conductive causality assigned to R-element, the commonly used form of the constitutive relation is given by the Bernoulli's law, i.e.
66
2 Bond Graph Modeling in Process Engineering
2
where
1 :sign (Pin RH
6 m 6 =6 H 4 1 :sign(Pin RH
Pout )
p jPin
p Pout )c p jPin
Pout j Pout j:Tin
3
7 7 7; 5
(2.73)
D4 π (2.74) 128`ν is the coef cient of hydraulic losses, ρ is the uid density, ν is the dynamic viscosity of the uid, D and ` are respectively the diameter and the length of the pipe, and c p is the thermal capacity of the uid at constant pressure. If the uid state is saturated, the temperature Tin is calculated as a function of the pressure Pin by using thermodynamic tables. RH =
Hydraulic momentum We assume now that the accumulation of mass is neglected, i.e. incompressible uid ow or compressible uid ow under steady state, but consider the momentum of the transported uid in the model. The inertia of the uid is modeled by I-element as shown in the bond graph model in Figure 2.40d. Assuming a linear hydraulic resistance, the constitutive equations of the model are given by 8 > < junction 1 : PI = P1 P2 PR ; R element : PR = Rh m; (2.75) > : I element : m = 1 R PI dt = p : I I The pressure momentum p has been discussed before (see Equations 2.35 and 2.36). Based on Equation 2.75, the dynamics of the uid in the pipe can be expressed by a second order differential equation: τ
dm (P1 P2 ) +m = ; dt Rh
(2.76)
I ` = is a time constant. ρRh AρRh Then the mass ow response to a step input given through pressure difference ∆ Pe becomes t 1 m(0): (2.77) m(t) = ∆ Pe 1 e τ Rh where τ =
For t ! ∞ , the steady state mass ow rate is m(t) = and m(t) =
1 p ∆ Pe in the non-linear case. RH
1 ∆ Pe in the linear case Rh
2.8 Coupled Energy Bond Graph
67
2.8.4 Thermo uid Multiport C In thermo uid storage, the hydraulic and thermal energies are stored together. In integral causality, the ows are imposed for the multiport C-element or thermo uid energy storage (Figure 2.41). The two 0-junctions model the conservation of mass and energy. Fin and Fout are respectively the inlet and outlet ows; the subscripts T and H, respectively, correspond to thermal and hydraulic ports: fin = ∑ fHin ∑ fTin
T
; fout = ∑ fHout ∑ fTout
T
:
Fig. 2.41 C multiport. a Thermo uid storage tank. b Its bond graph model
The outputs of the accumulator are calculated by the constitutive equation of the multiport C. These constitutive relations link the system states (integral of ow variables, i.e. stored mass m and total enthalpy H) to the effort variables (temperature T and pressure P) as follows: e=
R
P ΦCH (R ( fin ; fout )dt) Ψch (m; H) = = ; T ΦCT ( ( fin ; fout )dt) Ψct (m; H)
(2.78)
where ΦCH and ΦCT are strongly non-linear relations and depend on the thermodynamic state of the uid (saturated, under saturated, two phase, ...). As an example, in the case of a mixture of water and steam at saturated, but not superheated, regime, the temperature and pressure are not independent. The pressure P, temperature T and the mixture ratio X are determined for the 2-port C-element by resolving the following mixture equation: 8 H > > h = = hv (P)X + hl (P)(1 X); > < M m (2.79) > Vb X 1 X > > : νM = = + ; m ρ v (P) ρ l (P)
where vM and hM represent, respectively, the speci c volume and the speci c enthalpy of the mixture; and X is the steam quality, i.e. the ratio of the mass of steam and the total mass of the mixture.
68
2 Bond Graph Modeling in Process Engineering
Typical thermo uid storage elements are accumulators of saturated steam which are used in the industry as boiler or steam generator. Taking into account the fact that this component is among the commonly used processes, many works have been devoted to steam generator modeling using non-linear ordinary differential equations (ODEs) [9, 140, 185], partial differential equations (PDEs) solved through nite element approximation [175, 200] and lumped parameter models based on bond graph representations [188].
2.8.5 Application: Bond Graph Model of a Thermo uid Process 2.8.5.1 Bond Graph Model Here we consider a heated tank as an example of a thermo uid process. The main aim of the system is to provide a continuous water ow to a consumer. The process, as given in Figure 2.42, consists of a water tank which is heated by a modulated thermal source, Qe ; to maintain a constant uid temperature. The schematic representation of the process and its bond graph model are given in Figures 2.42 and 2.43. To explain the details in the bond graph model, bonds have been numbered.
Fig. 2.42 Hot water supply unit as an example thermo uid process
The uid is assumed to be under saturated, i.e. there is no gaseous phase. The mixture is ideal and the temperature in the reservoir and the body are uniform. The system is not insulated, i.e. there are thermal losses to the environment, which is at a given temperature Ta .
2.8 Coupled Energy Bond Graph
69
We are interested in the mass ow rate of the uid being delivered to the end user as well as its temperature. Two sensors have been installed in the process: level sensor De:L and temperature sensor De:T. There are two actuators, namely a pump (or a large reservoir) generating a constant pressure Pe (t) and a heater delivering thermal power Qe (t). The ambient pressure and temperature are Ps and Ta , respectively. The feed water (of density ρ and temperature Te ) supply is made up of the pump and a pipe of length ` and is assumed to be well insulated. The inertia of the uid in the inlet pipe is taken into account and is represented by I:`=A in the bond graph model given in Figure 2.43.
Fig. 2.43 Bond graph model of the hot water supply system
The pump is represented as an effort source. It can, of course, be modeled by a more complex bond graph by taking mechanical and hydraulic energies into account. Such models can be consulted in [188]. The water leaving the tank with mass ow ms through the controlled valve Rs is owing to an external system (consumer for example) which imposes a back pressure Ps . The valve state depends on the Boolean information signal u1 (0 or 1) provided by an “On-Off” level controller, LC as follows: u1 =
1 if L < Lmin ; 0 if L Lmax :
70
2 Bond Graph Modeling in Process Engineering
In the bond graph model, the valve is represented by element R:Rs (coef cient of hydraulic losses) and it is modulated by the control signal u1 . The dynamics of the valve, i.e. its actuation dynamics, is neglected. A detailed bond graph model of such dynamics is given in [189]. MS f : Qe is the thermal power delivered by the heater, which is modeled as a ow source modulated by a control signal u2 provided by the temperature controller, TC. The temperature controller follows similar On-Off control law as the level controller. The accumulation of energy in the tank is modeled by a two port C element, C:CR . This element is associated with storage of two coupled energies, namely hydraulic and thermal. The storage of thermal energy by the body of the reservoir is modeled by simple C-element, C:Cm . Different heat transfer phenomena are represented by R-elements. The parameters Rm and Ra are thermal resistance representing, respectively, heat transfer by conduction from the uid in the reservoir to the wall and from the wall to the environment. The pair (T Q) is used as the power variables to model such thermal losses. 2.8.5.2 Dynamic Equations Junction 11 : common ow (same mass ow through the pipe) or the continuity equation and the pressure drop across the valve: 8 e3 = e1 e2 e8 > 4 ) P3 = Pe > >8 > m1 = m3 f = f > 1 3 > > < < m2 = m3 f2 = f3 , > > m4 = m3 f = f > > > 4 3 > > : : :> m25 = m3 f25 = f3
P2
P4 ; (2.80)
R:Re element: hydraulic losses according to Bernoulli's law: e2 = ΦRe ( f2 ) = Re: f32 ) ∆ P = P2 = Re m23 :
(2.81)
Looking at the causality, the mass ow rate, m3 , is imposed to R element and it is calculated from the constitutive equation of the I element (bond number 3). I : `=A element: inertial phenomena due to mass of the uid Z
1 1 e3 dt + f3 (0) ) m3 = p3 + m3 (0): (2.82) I I Recall that p3 is a generalized impulse which represents the pressure momentum per unit surface. f3 =
Junction 01 : mass conservation law: common effort (pressure in the bottom of the tank is uniform) and a conservation of mass:
2.8 Coupled Energy Bond Graph
71
8 f7 = f4 f6 )8m7 = m4 > > 7 > :: 5 : 5 e6 = e7 P6 = P7
m6 = me
ms (2.83)
Junction 02 : energy conservation law: common effort (temperature is uniform in the tank) and a conservation of energy: 8 f13 = f16 + f14 8f12 f18 ) U13 = H16 + Qe (u2 ) > > 8 > > e12 = e13 T12 = T13 > > > > > > > > > < e14 = e13 < T14 = T13 e16 = e13 ) T16 = T13 > > > > > > > > > > > >: > e17 = e13 > T17 = T13 : : e18 = e13 T18 = T13
H12
Q18 (2.84)
The information bond (number 17) represents an effort detector and receives temperature T13 information, but the reaction ow is zero. C:CR Multiport: the multiport C-element represents two coupled capacities, namely thermal CRt and hydraulic CRh : These two capacities are associated with constitutive equations relatingRefforts (P or T ) and generalized displacements (time integral of ow): mass ( mdt = m) in the hydraulic domain and internal R energy ( Udt = U) in the thermal domain. Hydraulic capacity :
R
R
C:CRh ) e7 = ΦCh (q7 ; q13 ) = ΦCh ( f7 dt; f13 dt) ) P7 =
1 R m7 f7 dt = + P7 (0): Ch Ch
(2.85)
where Ch = Ag is the hydraulic capacity parameter, and A and g are, respectively, the cross section of the cylindrical reservoir and the gravity. The level measured by the detector De:L can be expressed as a function of the state variable for mass m7 as L=
P7 m7 = : ρg Aρ
(2.86)
Thermal capacity: R
R
C:CRt ) e13 = ΦCt (q7 ; q13 ) = ΦCt ( f7 dt; f13 dt) U13 1 R ) T13 = U13 dt = + T13 (0): Ct (m7 ) cV m7
The mean temperature in the reservoir, measured by the detector De:T, is
(2.87)
72
2 Bond Graph Modeling in Process Engineering
T13 =
U13 : cV m7
(2.88)
The internal energy, U13 , is calculated by time integration of the thermal ow to C:CRt (Equation 2.84). Remark 2.5. The global thermal capacity Ct (m7 ) = cV1m7 depends on the speci c heat capacity cV and the hydraulic state variable m7 : The hydraulic capacity Ch is constant in the under saturated regime. In saturated cases, the effort variables (P and T) are dependent (as in a boiler, for example). Such models are found in [9, 169, 188]. Junction 12 e8 = e6 + e9 ) P8 = P6 Ps ; f6 = f9 = f10 = f8 ) m6 = m9 = m10 = m8 :
(2.89)
Controlled valve R:Rs p p 1 u1 :sign(e8 ): je8 j = sign(e8 ) je8 j Rs (u1 ) Rs p u1 ) m8 = ms = sign(P8 ): jP8 j: Rs
f8 =
(2.90)
R:Rm and R:Ra elements
1 1 (e18 e20 ) ) Q19 = (T18 Rm Rm 1 1 f23 = (e22 + e24 ) ) Q23 = (T22 Ra Ra
f19 =
T20 ) ; Ta ) :
(2.91)
The thermal resistances are generally non-linear functions calculated from experiments or empirical formula (see [258]). Junctions 13 and 14 : energy conservation laws for heat transfer from the uid to the metal (13 ) and from the metal to the environment (14 ) : J13 : J14 :
e19 = e18 e20 ) T19 = T18 T20 ; f18 = f19 ; f20 = f19 ) Q18 = Q19 ; Q20 = Q19 : e23 = e22 + e24 ) T23 = T22 Ta ; f22 = f23 ; f24 = f23 ) Q22 = Q23 ; Q24 = Q23 :
C : Cm element: storage of thermal energy in the metallic wall:
(2.92)
2.8 Coupled Energy Bond Graph
1 e21 = Cm
Z
73
f21 dt + e21 (0) ) T21 =
Q21 + T21 (0): Cm
(2.93)
The thermal capacity of the metal is Cm = Vm ρ m cm ; where V m is the volume of the metallic body, ρ m is the density and cm the speci c heat of the metal. Junction 03 f21 = f20 f22 ) Q21 = Q20 Q22 ; e20 = e21 ; e22 = e21 ) T20 = T21 ; T22 = T21 :
(2.94)
Coupling elements Rc1 and Rc2 RC1 : f16 = f15 = f25 c p e15 ) H16 = He = m3 c p Te : RC2 : f11 = f12 = f10 c p e12 ) H11 = Hs = m8 c p T13 :
(2.95)
2.8.5.3 Simulation of the Global Model State equations The state variables, x, of the global model are the energy variables associated with storage elements, i.e. I and C elements in integral causality. They are: The momentum of the uid in the inlet pipe, p3 ( from element I : l=A) The stored mass in the reservoir, m7 (Ch element) The stored internal energy in the reservoir, U13 ( (Ct element) The accumulated thermal energy in the metallic body of the reservoir, Q21 (C:Cm element): x = p3 m7 U13 Q21
T
:
(2.96)
The input vector u consists of sources: u = Pe Qe Ta Ps
T
:
(2.97)
The measured variables (level L and temperature T ) are expressed in terms of the state variables: 2 m 3 7
6 Aρ 7 L 6 7 y= =6 7: T 4 U13 5 cV m7
(2.98)
74
2 Bond Graph Modeling in Process Engineering
Thus, the state equation under non-linear form (because of the coupling of the two energies) x(t) = f (x; u) can be written after minor transformations as
8 p3 2 m7 > ; p3 = Pe Re > > > I Ch > s > > > m7 u m p > 1 7 3 > sign Ps Ps ; m7 = > > > I R C Ch s h > > < p3 U13 1 u1 m7 U13 = c p Te + c p sign > > I c m R R Ch V m s 7 > > > > Q 21 > > + + Qe (u2 ); > > > RmCm > > > Q21 1 1 Ta U13 > : Q21 = + + : m7 cV Rm Cm Rm Ra Ra
Ps
s
m7 Ch
Ps
!
(2.99)
In Equation 2.84, the ratio Q=C (in the thermal domain) represents temperature, which is analogous to the electricity where the ratio of charge and capacitance gives voltage. On the other hand, the temperature (voltage) of the uid in the reservoir is the ratio of internal energy (analogous to charge) to a variable capacity which depends on mass or the hydraulic charge. The model under state space format can be easily represented in block diagram form as shown in Figure 2.44.
Fig. 2.44 Block diagram representation of state equations
2.8.5.4 Block Diagram The block f (x; u) represents the behavioral equations. It may be produced by synthesizing various terms of the equation or by following a more ef cient approach of analyzing causalities in the bond graph model given in Figure 2.43. This block diagram (Figure 2.45) can then be simulated by using MATLAB R -Simulink R interface.
Problems
75
Fig. 2.45 Complete block diagram of the hot water supply system for simulation through MATLAB R -Simulink R
Problems Modeling Problems 2.1. Consider the hydraulic process depicted in Figure 2.46. The process consists of three tanks: T1 , T2 and T3 . The water ows (Q12 ) through the valve V3 from T1 to T2 : Tank T1 is lled by a pump P1 whose ow output is controlled by a controller, LC, in order to keep the water level h1 in T1 constant. Tank T3 is supplied water through valves V1 and V2 giving mass ow rates Q1 and Q2 , respectively. Tank T3 is further used as the sump (source of supply water) of the pump. The ow Q p provided by the pump depends on the control signal u and the pressure at the bottom of tank T3 . For simplicity, the constitutive equation of the pump is assumed to be linear: Q p = kuP3
(2.100)
where k is a coef cient (modulus of a modulated gyrator element, MGY, in a bond graph). R1 , R2 and R3 are hydraulic resistances of the valves. Ai (i = 1; 2; 3) are the cross sectional areas of cylindrical tanks (m2 ), g is the gravity (m/s2 ) and ρ is the density of water (kg/m3 ).
76
2 Bond Graph Modeling in Process Engineering
Fig. 2.46 A three-tank controlled hydraulic system
The objective is to study the dynamics of water levels h1 and h2 ; which are indicated by sensors LI1 and LI2 . a. b. c. d.
Draw word bond graph of the system. Construct a causal bond graph model. Derive state equations. Draw block diagram model corresponding to the bond graph model.
2.2. Consider the two mechanical systems in Figure 2.47. In the schematic representation of the mechanical system in Figure 2.47a, the lever can be considered elastic (with a bending stiffness k) or rigid. a. Derive bond graph models considering elastic and rigid rod. b. Draw block diagram representations of both bond graph models. c. Discuss the causality problem. The second system represents a vehicle suspension system by considering a socalled quarter-car model. The anti-leak solid rubber tyre has viscoelastic nature and it is usually approximated by a nite chain of mass-spring-damper combinations. Here we make further simpli cations and assume that the viscoelastic phenomena can be described by damper bg , springs kg1 and kg2 and mass mg . M, b and k are the mass of the quarter vehicle, damping coef cient and stiffness of the suspension system. By considering only vertical dynamics (heave motion of the vehicle) and that the road imposes velocity xr as a ow source. a. Construct a bond graph model of the system. b. Draw the corresponding block diagram representation from the bond graph model.
Problems
77
Fig. 2.47 Mechanical systems. a Lever. b Vehicle suspension system
2.3. The equivalent circuit of an electrical device is shown in Figure 2.48a. a. Draw its bond graph model. b. Convert the bond graph model to the corresponding block diagram form.
Fig. 2.48 Electrical system. a Isothermal network model. b Thermal effects on the electric circuit
Consider the case when the heat generated by the resistances are not immediately dissipated to the environment. Resistances R1 and R2 generate thermal power Q1 and Q2 as shown in Figure 2.48b, depending upon the current owing through the resistors (i2 R is the dissipated thermal power). Assume Ta to be the xed environment temperature and Rth to be the overall thermal resistance between the inside of the box and the external environment. Further assume that the box is thin and
78
2 Bond Graph Modeling in Process Engineering
the temperature Tc inside the global system in uences the values of resistances, such as due to thermal expansion (the reverse concept is used in Wheatstone bridge based strain gages). The mass of air inside the box is ma : Considering only conduction heat transfer and suitably assuming nomenclature for other parameters (speci c heats, densities etc.): a. Develop the bond graph model of the system. b. Draw the corresponding block diagram. c. Discuss how the bond graph model would change if the volume inside the box is vacuum (or at a very low pressure) and thus radiation (as is usual in such cases) is the dominant mode of heat transfer. In this case, the energy stored by the mass of air inside the box is neglected, whereas heat energies stored by the resistors themselves (of masses m1 and m2 , respectively) and the metallic box (of mass mb ) must be considered in the model. The usual form of Stefan-Boltzmann law with suitable view factors and gray body emissivities should be used as the constitutive relation of non-linear R-elements modeling the heat transfer phenomena. Note that the resistance offered by other parts of the circuit (e.g. the cell) are already considered in the equivalent circuit. 2.4. The schematic of a DC motor driven rotor shaft with a ywheel and bearing damping is shown in Figure 2.49. The eld of the DC motor is externally energized such that the rotor's spinning speed is proportional to the current. By assuming that the bearing is rigid and neglecting the bending modes (whirling motion of the cantilever part of the rotor):
Fig. 2.49 An electromechanical system
a. b. c. d.
Draw the word bond graph of the system. Develop the bond graph model of the complete electromechanical system. Draw the corresponding block diagram model. Derive the state equations of the system and nd the expression for the generated back emf.
Problems
79
N.B. For detailed models of different types of electric drives, rotor dynamics and related stability and control issues, see [172].
Project Type Simulation Problems 2.5. The liquid-spring given in Figure 2.50 has piston area A p (excluding holes made through it) and the piston rod area is Ar . Height of the piston is t. The total mass of the liquid inside the spring is m, which is a known value. Determine the pressure on two sides of the liquid-spring supporting the given load, W , in equilibrium. What is the equilibrium position of piston (from top or bottom)? Find the stiffness of the spring for small displacement by assuming quasi-static conditions (no resistance to ow).
Fig. 2.50 A liquid spring
When the spring has to be used as a shock isolation system [221], holes are small and the ow through them is governed by normal ow through pipe (Bernoulli) laws. Assuming laminar ow and a total discharge coef cient of nCd for n holes and isothermal operation, draw a bond graph model for the shock absorption system. Create a Simulink R block diagram model and simulate it for 4 s with given data and initial momentum given as 10000 kg.m/s. Assume the initial piston position at the middle of the spring and neglect wall friction. Plot the displacement of the load, pressure in the two chambers and the mass ow rate through the ori ces. Data: weight W = 1000g, maximum liquid column length H t = 0:5m, gravity g = 9:81 m/s2 , piston area A p = 0:005 m2 , piston rod area Ar = 0:002 m2 , total discharge coef cient Cd = 1:5 10 4 kg1=2 .m1=2 , compressibility of silicone uid β = 7:5 108 Pa and free-space (uncompressed) density of silicone uid ρ 0 = 970 kg/m3 .
80
2 Bond Graph Modeling in Process Engineering
2.6. Consider that an air- lled chamber, as shown in Figure 2.51, is loaded by a weight to setup free vibrations of the system. Derive the constitutive relations for variation of pressure, temperature and chemical potential in terms of the state variables (volume, entropy and mass) from rst principles by assuming ideal gas law. Draw the model of thermodynamic storage C- eld with pressure-volume ow rate, temperature-entropy ow rate and enthalpy-mass ow rate ports [15, 173].
Fig. 2.51 A gas spring
a. Create a block diagram model of the entire system from a true bond graph model and by using Simulink R , simulate the system for 10 s by assuming the following data. Plot the pressure-volume diagram to show thermo-dynamic damping. Data: initial pressure is 105 kg/m2 , initial temperature is 300 K, initial volume is 0.1 m3 , load mass is 1500 kg, internal radius of cylinder is 0.25 m, external radius of cylinder is 0.3 m, length of the cylinder is 0.5 m, ambient temperature is 300 K, the gas is air and the cylinder material is steel ( nding proper value of heat transfer coef cient of the material as well as that for a cylinder of variable length is part of the project). b. Change the cylinder material to silver (high thermal conductivity) and reduce wall thickness to 0.001 m to simulate approximate isothermal behavior; and increase wall thickness to 10100 m and select the wall material as lead (low thermal conductivity) to simulate approximate adiabatic behavior. Show that ideal isothermal and adiabatic behaviors (both are reversible) do not damp the free vibrations and thus prove that damping is due to irreversible thermodynamics. Plot the obtained isothermal and adiabatic pressure-volume diagrams in a single graph for comparison.
