0 and 1 junctions

In order to account for the abstraction of ideal, no power consuming switching in a bond graph with invariant causalities that holds for all system modes, Umarikar extended 0- and 1-junctions by allowing for more than one bond to impose an effort on a 0-junction and more than one bond imposing a flow on a 1-junction with the constraint that only one of the causality imposing bonds is active at a time instant [12, 13], These extensions are called switched power junctions and are not to be confused with controlled junctions to be referred to subsequently. Figure 2.2 illustrates the idea. 

We now have nine basic bond labels (M)C, (M)I, (M)Se, (M)Sf, (M)R(S), (M)TF, (M)GY, (X)0, and (X)l, which, for reasons of clarity can also be introduced bottom-up, in the sense that each is defined in the simplest form possible and where ports are power ports (as mentioned before, in case ports are not power ports, bond graphs are commonly addressed as pseudo-bond graphs with pseudo-bonds). This simplest form is the minimum number of ports and a minimum number of constitutive parameters, which results in the linear form. This results in one-port C, I, Se, Sf, and R, as well as the two-port TF, GY, and RS that are all characterized by one parameter and the 0- and 1-junctions, which are always linear, have no parameter at all. Junctions should have at least two ports to be power continuous, but that would limit the possible structures to chains only. This means that the simplest form of a junction required to be able to build arbitrary structures should have a minimum of three ports. In case the constitutive parameter of an element is replaced by a function of time, the modulated and switched versions of the basic elements, i.e., MC, MI, MSe, MSf, MR(S), MTF, MGY, XO, and XI, are obtained (cf. Table 1.3). 

Note that Paynter [2] originally used only one-letter labels for the node types E instead of Se, F instead of Sf, T instead of TF, and G instead of GY. His students Karnopp and Rosenberg [8] noted that interpretation became easier when in some cases two- and three-letter labels were used and they introduced the labeling used herein. In some dialects, e.g., [12], the 0- and 1-junctions are replaced by e- and /-junctions (common effort and common flow junction, respectively) or even using the domain-dependent symbols of effort and flow, like u for common voltage junction, v for common velocity junction, and T for common temperature junction. Thoma [13] uses a dialect that violates our earlier attempt to only focus at the topological structure of relations between concepts he uses the so-called i- and p-junctions, where he relates an j -junction to a series connection and a /t-junction... 

Extensions of 0- and 1-Junctions to the Multi-bond Graph Formulation... 

The continuity of flows and efforts represented by the 0- and 1-junctions can be rapidly generalized. The continuity of flows in the 1-junction implies that... 

Fig. 9.6 Definition of the 0- and 1-junctions in multi-bond graph notation...

In (5.78), and are the chemical and the electric potentials, J. and I are the material and the electric fluxes, respectively. The potentials and fluxes change their values by infinitesimal increments dp., d, dJ., dl along the compartment. The potential-flux language is chosen instead of the concentration-flux language since the Nernst-Planck equations represent a linear theory in the sense of 1-port diffusion elements, cf. Section 4.6. The values of potentials and fluxes at the bonds of (5.78) have been chosen such that KCL and KYL at the 0- and 1-junctions are automatically satisfied. Moreover, use has been made in (5.78) of what we have learned already in Section 5.6 about the coupling of material and electric fluxes, cf. (5.73). 

An example of the latter kind of condition will be presented in this section. It says that a network consisting of capacitances, 2-port elements, 0- and 1-junctions as introduced in Chapter 4 always has a unique steady state if no closed loops appear in the network. It is clear that as a closed loop we understand any series of elements connected by bonds which lead back into the initial element. A transducer, although introduced in Section 4.5 as a shortwriting for a closed loop consisting of a 0- and a 1-junction, will not interfere with the proof to be given for the above uniqueness statement. Examples for closed loops may be found in the networks for an enzyme-catalyzed reaction or pore transport (4.38) or (5.1), for carrier transport (5.41), for active transport (5.50) and for an autocatalytic reaction (6.3). Except for the latter one, all above-quoted examples showed a unique steady... 

The symbols Ko/K, Kg/K, and K /K are used to represent the relative permeabilities to oil, gas, and water, respectively. Obviously, relative permeability values range between 0 and 1. It has been found that, for a given porous medium, the relative permeability is a junction of saturation. Consider a system in which both oil and gas... 

Cardoso et al. [13] also compared the dependency of the viscoelastic properties of mature OPE/calcium gels upon the polymer and calcium concentrations to those of the LMP. They showed that, for these variables, both pectin systems exhibited a power law dependence of the G. At pH 7, for the different concentrations of non-esterified carboxyl groups available in the pectin (o-GalA ), the PPE/calcium and citrus LMP/calcium systems exhibited similar dependencies on the calcium concentration (Fig. 8a), with a power law dependence of 2.9-3.3. Still, the gelling ability of OPE/calcium systems was more dependent on the polymer concentration than the citrus pectin. For the different calcium concentrations tested, the corresponding exponents of power law dependency were approximately 3.0 and 1.9 for OPE/calcium and citrus LMP/calcium systems, respectively (Fig. 8b). These results also confirm the lower capability of the pectic olive extracts to form, under similar ionic conditions, elastically effective junctions zones. 

The application of liquid-junction technology to photovoltaic power conversion is limited by problems associated with the semiconductor-electrolyte interface. Primary among these problems is corrosion. Efficient conversion of solar energy requires a band gap between 1.0 and 1.5 eV, and most semiconductors near this band gap corrode readily under illumination. Semiconductors with large band gaps (4-5 eV) tend to be more stable but cannot convert most of the solar spectrum. 

As early as 1974, Thoma introduced the concept of time dependent junctions [24] in order to switch off and on connections between power ports. Mosterman picked up this idea and introduced controlled Junctions [21], In contrast to switched power junctions, a local control algorithm associated with a controlled junction switches off all adjacent bonds of a controlled junction when a switching device considered as an ideal switch turns off and re-activates all bonds when the switch is closed. That is, an ideal switch in ON-mode is represented by a standard 0- or 1-junction. In OFF-mode the junction is replaced by a source of value zero as shown in Fig. 2.10. 

The determination of ARRs on a bond graph model is done by elimination of unknown variables contained in the structural constraints of junctions 0 and 1. The equations of power balance on the junctions constitute the ARRs [16]. 

A structural fault noted Fs corresponds to a new effort (or flow) source that causes a change in the structure of the model. Thus, the nominal model of the system is not conserved and its dynamic is altered by the presence of the fault. This difference between the system and the model generates an unbalance in the flow, mass and energy conservation laws, calculated from junctions 0 and 1 of the bond graph model. For example, a water leakage in the tank of Fig. 3.15b is a stfuctural fault. It can be modeled by a flow source Sf Yg. The model sfructure has changed from the bond graph model of the system without fault of Fig 3.15a. 

For the sake of simplicity of writing we have omitted in (5.8) all concentration and flux variables which follow from KCL or KVL at the 0- or 1-junctions, respectively. Clearly, X, Yp Y2 are the capacitances for empty pores and for the 1- and 2-complexes, respectively, and and R2 are the corresponding formation and dissociation reactions with fluxes... 

Values of a are shghtly larger than unity because the chains are slightly strained at the common junction points between the two phases. For practical calculations, a value of 1.2 might be assumed, since a varies between 1.0 and 1.5 for most cases of interest. 

We said in Chapter 21 that all metals except gold have a layer, no matter how thin, of metal oxide on their surfaces. Experimentally, it is found that for some metals the junction between the oxide films formed at asperity tips is weaker in shear than the metal on which it grew (Fig. 25.4). In this case, sliding of the surfaces will take place in the thin oxide layer, at a stress less than in the metal itself, and lead to a corresponding reduction in x to a value between 0.5 and 1.5. 

Kawahara et al. (2002) presented void fraction data obtained in a 100 pm micro-channel connected to a reducing inlet section and T-junction section. The superficial velocities are Uqs = 0.1-60m/s for gas, and fAs = 0.02-4 m/s for liquid. The void fraction data obtained with a T-junction inlet showed a linear relationship between the void fraction and volumetric quality, in agreement with the homogeneous model predictions. On the contrary, the void fraction data from the reducing section inlet experiments showed a non-linear void fraction-to-volumetric quality relationship ... 

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