Effort and Flow Variables

There are two basic kinds of variables that describe the action of a physical system. Effort variables are those things that cause an action to occur. Flow variables are the responses to effort variables, usually involving movement but not always (Table 2.1.1). For the simple case of a running animal, the effort variable is the force required to propel the animal the flow variable is the velocity of movement. Heat loss from that same animal, which is the flow variable, occurs in response to a 

There are several basic relationships between effort and flow variables that should be introduced. The first is resistance. 

Resistance is the ratio of effort to flow variable amounts. Mathematically, resistance is given as 

FIGURE 2.1.1 Two cases of effort and flow variables. In the top case, the differences of pressures between the two water tanks canse finid to flow between them. In the bottom case, molecular concentrations are different between the two chambers, and thns molecules move left to right through a membrane. Flow is limited by the resistances of the pipe and membrane, respectively. The amount of water stored in the tanks and the number of molecules in the chambers represents capacity. When the heights of the liquids in the two tanks are the same, and when molecular concentrations in the two chambers are the same, capacities in the two will be eqnal and there will not be any effort variable differences. Net flow will then cease, although movement from one chamber to another can still continue as long as it is equal in both directions. 

Symmetry is fascinating to the human mind, and everyone likes objects or patterns that are in some way symmetrical. Even the animal and vegetable worlds show some degree of symmetry, although the symmetry of a flower or of a bee is not as perfect or as fundamental as is that of a crystal. 

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Systems are often made of multiple elements of different kinds. They can be any combination of resistances, capacities, and inertias (Figure 2.1.2). Thus, the effort and flow variable magnitudes are dependent on the exact nature of the system of interest. 

Why is this so It was explained in a previous section on effort and flow variables (Section 2.1) that flow always occurs from points of higher effort to lower, never the other way around. Perhaps that explanation is sufficient. We usually have enough experience with real effort variables (pressure, gravity, temperature, etc.) to expect that the impossible just won t happen. And it doesn t. 

On a smaller scale, diffusion is the chief mechanism for mass movement. Diffusion causes materials to move when a concentration difference exists between any two points. In terms of effort and flow variables, concentration difference is the effort variable, and the mass rate of flow is the flow variable. 

From our effort and flow variable perspective, there are two effort variables that act in concert, either adding or subtracting. These are concentration (or osmotic) pressure and hydrostatic pressure (see Section 2.9). Either one can canse the flow of water through the semipermeable membrane (Figure 2.8.4). 

If heat and temperature difference are substituted for electric current and voltage difference, then this equation would describe the flow of heat. If mass rate of flow and concentration difference were used instead, the equation would be equivalent to Pick s law for mass movement. Thus, Ohm s law is universal in form, and demonstrates the analogies among effort and flow variables introduced in Section 2.1. 

Can you think of additional physical principles that relate to biology If so, list them. Describe the behavior of a biological organism. In this description identify the effort and flow variables. Remember that effort variables don t describe things that move that is what flow variables do. 

In terms of effort and flow variables, electron affinity represents one effort variable, electron removal energy represents a second effort variable, and combination energy represents a third effort variable. The flow variable is liberated heat. If the sum of the three effort variables is positive, heat is liberated if the sum is negative, then heat is absorbed. Resistance, or the ratio of effort to flow variables, is proportional to the spontaneity of the chanical reaction. 

The second and the third column of Table B.l list the effort and flow variables in the various energy domains. The variables in the fourth column of Table B.l are the time integral of the efforts and the variables in the fifth column are the time integral of the flows. They are called energy variables because they quantise the amount of energy in the energy storage elements of a model. 

A (multi)bond equates the efforts of the connected ports to each other. Similarly it equates the flows of the ports it connects to each other. As a result, the effort and flow variables can also be considered to live on the (multi)bond (Fig. 1.3). This means that the property of variables being conjugated also expresses that they belong to the same bond and that the dimensions of the variables that are equated by a (multi)bond should correspond to each other. 

The ECI is developed as follows. Let ei t) denote the energy in a given bond i, which is given by the time integral of the product of the generalized effort and flow variables associated with the bond, i.e.. 

Definition 6.4 (Causal path) In a causal (or bicausal) bond graph representation, a causal path is a series of effort and flow variables successively related according to the model causality assignment [34, 55],... 

The pseudo-bond graph model of the system, in the preferred derivative causality, is given in Fig. 7.17. The pressures and the mass flow rates have been considered as the generalized effort and flow variables, respectively. Performing the substitutions defined before, we obtain a model shown in Fig. 7.18, which is called a diagnostic bond graph (DBG) model [3,4],... 

From well-known thermodynamic relations, dU/dV = —p, dU/dS = T, dU/dmi = p.1 and dU/dm% = p,2, it is evident that the internal energy of the volume of the gases changes due to four distinct power exchanges which can be represented by the products of the corresponding effort and flow variables. 

Sometimes balances given for effort variables and flow variables look somewhat similar, but have an inverted appearance. For instance, using the definitions for resistance, capacity, and inertia already given, an effort balance would be... 

FIGURE 6.6 Symbolic representation of the two possible conventions for orienting dipole differences between pole variables. In the generator convention (left), dipole effort and flow have the same orientation, whereas in the receptor convention (right), they have opposite directions. ... 

Ideal case. Note that the majority of materials do not present a constant Seebeck coefficient, which can, in principle, be translated by the indication on the previous graph of a second fector, called differential factor of coupling, for connecting the energies-per-entity (efforts and flows) as appropriate for a conversion with variable coupling. The coupling fector in this case can be seen, physically speaking, as a manifestation of an additional effect, called Thomson effect, treated in case study Jll. 

State variable quantifying the energy amount in an entity. Efforts and flows are energies-per-entity. 

It can be proven by means of a linear transformation of the conjugate variables into so-called scattering variables [9, 10] that all power continuous nodes have constitutive relations with a multiplicative form. This means that the vector of dependent port variables can be written as a product of some operator on the vector of independent port variables. When this operator only relates efforts to efforts and flows to flows, a property called non-mixing [11], the multiport is called a transformer (node label TF). If the operator is a function of one or more additional node inputs, it is called a modulated transformer (node label MTF). When this operator only relates efforts to flows and flows to efforts, a property called mixing [11], the multiport is called a gyrator (node label GY). If the operator is a function of node inputs it is called a modulated gyrator (node label MGY). 

Known constitutive relation Either the effort or the flow is an independent variable, power can always be computed as the product of effort and flow... 

Unknown constitutive relation Both effort and flow are independent variables... 

Analytical redundancy relations are balance equations of effort or flow variables, in which unknown variables have been replaced by input variables and measured output variables and in which parameters are known. Evaluation of an ARR provides a residual that theoretically should be zero. In practice, however, the residual of an ARR is within certain error bounds as long as no faults occur during system operation. The value is not exactly zero over some time interval due to noise in measurement, parameter uncertainties, and numerical inaccuracies. If, however, the numerical value of a residual exceeds certain thresholds, then this is an indicator to a fault in one of the system s components. Noise in measured output variables may result in residual values indicating a fault that does not exist. Hence, measured data should pass appropriate filters before being used in ARRs. 

The computer-generated transfer function for the voltage across the capacitor crosses two different energy domains without separation since the model is all together. The transfer function is obtained in one step in symbolic form. CAMPG generated the code for the A, B, C, D matrices which are displayed in MATLAB. Any other transfer function for the efforts and flow output variables can be obtained. More details are presented in [11]. At this point, the computer-generated model becomes so versatile that all the linear control theory operations implemented in the MATLAB Control Systems Toolbox can be used on the entire mechatronics model. 

In conclusion to this section, research in the RTD area is always active and the initial concepts of Danckwerts are gradually being completed and extended. The population balance approach provides a theoretical framework for this generalization. However, in spite of the efforts of several authors, simple procedures, easy to use by practitioners, would still be welcome in the field of unsteady state systems (variable volumes and flow rates), multiple inlet/outlet reactors, variable density mixtures, systems in which the mass-flowrate is not conserved, etc... On the other hand, the promising "generalized reaction time distribution" approach could be developed if suitable experimental methods were available for its determination. 

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