Variables internal

For polymer solutions, and taking the p, v, and u as environment variables and Pv as internal variable, the dissipation becomes 

The first two terms on the right are the dissipation (entropy production times the absolute temperature), same as in the linear nonequilibrium thermodynamics. The last term is the contribution of the internal variables. The Pv acts as an internal variable and as a rate variable. The evolution equation for Pv is 

Jongschaap et al. (1994) provided detailed examples for rheological problems with the matrix method by using the configuration function as variables. 

The theories with internal variables provide detailed description of microstructure by introducing additional variables relevant to the microstructure of the system, and enlarge the domain of application of thermodynamics. The theories of internal variables are applied in rheology, dielectric, and magnetic relaxation where the structure of the macromolecules plays a relevant role. In the theories of internal variables, it is usual to propose purely relaxational equations for the internal variables and associate the additional variables with some structure of underlying molecules. 

If we assume the configuration tensor W = (RR as an independent variable, the Gibbs equation is 

The following treatment applies to homogeneous systems with constant mass. The dissipative, time-dependent effects are caused in such homogeneous systems by disturbance of the equilibrium of the internal, molecular degrees of freedom. The 

The starting point for the description of the system considered is the Gibbs fundamental equation, Eq. (1), shown in Fig. 2.91 in terms of the specific quantities  

Equation (10) of Fig. 2.92 is the Gibbs equation for the corresponding partial quantities at constant T and p. At internal equilibrium (designated by the superscript e ) a° = 0, which leads to Eq. (11) and fixes one of the independent variables. At equilibrium, one can write with help of Eq. (10) and Eq. (12) the Eqs. (13) and (14) since da = 0. Finally, one can derive the change of the internal variable with temperature at equilibrium and constant piessme, as shown in Eq. (15). 

The specific heat capacity at constant pressure can be written as shown in Fig. 2.93, based on the relationship derived in Fig. 2.19. Insertion at constant pressure into Eq. (6) for ds (from Fig. 2.91) leads to the connection to the internal degree of freedom (. The first term of the right-hand side of Eq. (16) is the heat capacity of the so-called arrested equilibrium (C = constant). This specific heat capacity is measured when the change in temperature occurs sufficiently fast that the 

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Suppose that x [Q)) is the adiabatic eigenstate of the Hamiltonian H[q]Q), dependent on internal variables q (the electronic coordinates in molecular contexts), and parameterized by external coordinates Q (the nuclear coordinates). Since x Q)) must satisfy... 

Partieulate produets, sueh as those from eomminution, erystallization, preeipi-tation ete., are distinguished by distributions of the state eharaeteristies of the system, whieh are not only funetion of time and spaee but also some properties of states themselves known as internal variables. Internal variables eould inelude size and shape if partieles are formed or diameter for liquid droplets. The mathematieal deseription eneompassing internal eo-ordinate inevitably results in an integro-partial differential equation ealled the population balanee whieh has to be solved along with mass and energy balanees to deseribe sueh proeesses. 

Another way of removing the six translational and rotational degrees of freedom is to use a set of internal coordinates. For a simple acyclic system these may be chosen as 3N — I distances, 3N — 2 angles and 3N -3 torsional angles, as illustrated in the construction of Z-matrices in Appendix E. In internal coordinates the six translational and rotational modes are automatically removed (since only 3N — 6 coordinates are defined), and the NR step can be formed straightforwardly. For cyclic systems a choice of 3A — 6 internal variables which span the whole optimization space may be somewhat more problematic, especially if symmetry is present. 

Evidence is therefore found in the data for the Haber equilibrium that the function Kp is not strictly constant with respect to any variation whatsoever of the internal variables, the number of independent variables being given by the phase rule. 

Vibrational Properties. Figures 4 and 5 show the variation of the energy Eg and the electric dipole moment p as a function of the relevant geometrical variables for H2O and NHg respectively. For the Internal variables, the curves corresponding to the Isolated molecules are also shown (dashed lines) for comparison lhe20Sclllatlon frequencies v and dipole matrix elements <1, sre also... 

Global behavior of the system is determined by a single parameter, the reduced charge Q relative distance z is the internal variable, defined by the equilibrium condition... 

A plant simulation is the set of equations necessary to approximate the response of a chemical plant to various changes. A steady- state plant simulation is one that predicts the eventual outputs when the inputs and all the internal variables are held constant. It does not say how the outputs are reached. A dynamic plant simulation is one that predicts how the outputs of a plant will change when a known change in the input occurs. It gives the path the process follows in going from one steady state to another. 

Co-payment is an instrument that should not be used on its own. Neither efficiency in drag use nor equity nor the control of pharmaceutical expenditure can rest solely on co-payment. Its effectiveness is reinforced when it is combined with other instruments and incentives. In fact, all European countries combine, in different doses and proportions, multiple instruments that influence the behaviour of the industry, prescribes and patients. It is sufficient to recall that pharmaceutical expenditure is the product of price by quantity, and to consider the enormous international variability of drag prices,35 in order to understand the limitations of co-payment regulation in comparison with other policies that influence prices. Policies aimed at price control can be as effective as co-payment - or more so - for purposes of cost containment. 

It is often convenient to treat all the internal variables as functions of the included... 

That raises the question of what a Shape is expected to do, and that takes us into the next section on types (object specifications). In programming languages, it is common for a class to represent a type. The class may perhaps define no internal variables or operations itself but instead only list the messages it expects. The rest is defined by each subclass in its own way. 

In C++, public inheritance is used to document extension and implementation, private inheritance is used for extensions that are not implementations (apart from the simple restrictions mentioned earlier) but the usual recommendation is to use instead an internal variable of the proposed base type. 

An operation can be specified in the style detailed earlier in this chapter. You can refer to the old and new values of the internal variables (and to attributes of their types, and of the attributes types, and so on). 

The diagram above shows an interactive MIMO system, where the controlled variables, outlet flow temperature and concentration, both depend on the manipulated variables. In order to design a decentralized control, a pairing of variables should be decided. A look at the state Eq.(23) suggests the assignment of the control of the temperature to the cooling flow and the concentration control to the reactor inlet flow. In this case, the internal variable Tj may be used to implement a cascade control of the reactor temperature. Nevertheless, a detailed study of the elements of the transfer matrix may recommend another option (see, for instance, [1]). 

The relationships between structures and properties can be classified as intrinsic and extrinsic owing to the molecular arrangement and morphology, respectively. The term intrinsic refers to the 3D packing of molecules, which depends on the geometry and chemical nature of the molecules, and thus on intermolecular interactions. Extrinsic structure-property relationships are related to the formation of interfaces, e.g., grain boundaries, and the presence of defects. In both cases the role of external variables such as T, P, B as well as of internal variables such as the type of guest molecules is essential. 

There are zeolite-bearing rocks in which one mineral is apparently being replaced by another mineral under constant P-T conditions. This indicates a system in which certain chemical components appear to be perfectly mobile a system in which the total number of phases that can coexist at equilibrium is reduced as a function of the number of chemical components which ar e internal variables of the system. Two examples of this type of equilibrium concerning zeolites can be cited saline lakes and analcite-bearing soil profiles (Hay, 1966 Hay and Moiola, 1963 Jones, 1965 and Frankart and Herbillon, 1970). In both cases a montmorillonite-bearing assemblage becomes analcite or zeolite-bearing at the expense of the expandable phyllosilicate. Other phases remain constantly present. 

If we assume that alkali activity varies independently of the masses in the silicate system, the activities (more precisely, the chemical potentials) become internal variables of the system where the activity is controlled by potentials or reservoirs outside of the rock observed. 

Apple cultivars have different textures due to their internal variability of structure and composition. Some apples resist boiling and do not readily sauce. Others may undergo ready cell separation. This wide range of textural behavior illustrates the complexity due to pectins and other cell-wall materials. Select apple cultivars according to the desired processing qualities. 

Unlike other authors, we choose from the beginning to characterize each element i of a system both by a logical variable ot, ( internal variable) associated with its concentration (or, more generally, its level) and by a logical function a, associated with its rate of production (or, more generally, its flux). Variable a = 1 if the concentration exceeds a functional threshold, a = 0 if not.t Function a = 1 if the rate of production... 

Our logical equations usually describe systems by relating the logical values of the functions a, (rates of production) with the values of the internal variables at (concentrations) and of input variables such as temperature T ... 

Similar timescales for comparison. Short-term simulations can provide easily different or similar particle number size distributions than measured, just from overall model internal variability. 

Santer et al. (2000) discussed the causes of differences between trends of SAT and lower tropospheric temperature. Having analyzed the SAT data for 1925 1944 and 1978-1999, Delworth and Knutson (2000) came to the conclusion that the main cause of SAT change was a combined impact of anthropogenically induced RF and unusually substantial multi-decadal internal variability of the climate system. 

This comparison showed that calculated anomalies in heat content do not differ from observational values (between 1950 and 1990) by more than 5%, with the exception (for global averaging) of data for the 1970s, when the model does not reproduce the heat content anomaly observed in this decade. On the whole, the probability that these anomalies in heat content are explained only by internal variability of the climate system does not exceed 5%, which implies the anthropogenic nature of climate change. 

Abstract A fully coupled model of hygro-thermo-chemo-mechanical phenomena in concrete is presented. A mechanistic approach has been used to obtain the governing equations, by means of the hybrid mixture theory. The final equations are written in terms of the chosen primary and internal variables. The model takes into account coupling between hygral, thermal, chemical phenomena (hydration or dehydration), and material deformations, as well as changes of concrete properties, caused by these processes, e.g. porosity, permeability, stress-strain relation, etc. 

The behavior of complex dynamical systems can be analyzed and represented in a number of ways. Figure 1 represents one such approach, a constraint-response plot. A constraint, in this case [A], is any variable which the experimenter can control directly. A response, [X]ss in this case, is a measurable property of the system which depends upon the constraint values. The constraints are the external variables, e.g., the temperature of the bath surrounding the reactor or the reservoir concentrations, while the responses are the internal variables, e.g., the temperature or concentration of species in the reactor. The phase trajectory diagram of Fig. 4 is one type of response-response plot. Obviously, in a complex system, there will be several constraints and responses subject to independent (or coupled) variation. 

Abstract The discussion of relaxation and diffusion of macromolecules in very concentrated solutions and melts of polymers showed that the basic equations of macromolecular dynamics reflect the linear behaviour of a macromolecule among the other macromolecules, so that one can proceed further. Considering the non-linear effects of viscoelasticity, one have to take into account the local anisotropy of mobility of every particle of the chains, introduced in the basic dynamic equations of a macromolecule in Chapter 3, and induced anisotropy of the surrounding, which will be introduced in this chapter. In the spirit of mesoscopic theory we assume that the anisotropy is connected with the averaged orientation of segments of macromolecules, so that the equation of dynamics of the macromolecule retains its form. Eventually, the non-linear relaxation equations for two sets of internal variables are formulated. The first set of variables describes the form of the macromolecular coil - the conformational variables, the second one describes the internal stresses connected mainly with the orientation of segments. 

In non-equilibrium situations, local states of the deformed system are described by some internal thermodynamic variables where the label a is used for the number of a variable and its tensor indices. All the equilibrium values of the internal variables are functions of two thermodynamic variables ... 

In non-equilibrium situations, the quantity Eq includes also potential of internal variables (Wood 1975, Maugin 1999, Pokrovskii 2005), so that the differential of this function has the form... 

The quantities T,w and Sa are functions of the variables s, p, At equilibrium, when there is no external fields, all the a = 0, while the quantities T and w take their equilibrium values. The external field affects the internal variables, which determine the state of the system. 

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