Subsystems equations

For a system composed of two subsystems a and p separated from each other by a diathemiic wall and from the surroundings by adiabatic walls, the equation corresponding to equation (A2.1.12) is... 

Consider the situation illustrated in figure A2.1.5. with the modifieation that the piston is now an adiabatie wall, so the two temperatures need not be equal. Energy is transmitted from subsystem a to subsystem (3 only in the fomi of work obviously dF = -dF so, in applying equation (A2.1.20), is dlf- P equal to dF = dF or equal todk , or is it something else entirely One ean measure the ehanges in temperature,... 

The assumption (frequently unstated) underlying equations (A2.1.19) and equation (A2.1.20) for the measurement of irreversible work and heat is this in the surroundings, which will be called subsystem p, internal equilibrium (unifomi T, p and //f diroughout the subsystem i.e. no temperature, pressure or concentration gradients) is maintained tliroughout the period of time in which the irreversible changes are... 

Example 2. Figure 8 shows a system block diagram iadicating subsystem rehabihties. Applying equation 7 to part A of Figure 8 gives... 

KDC has a cause and effect relationship between as the primary cause leading to secondary failures. Besides its drastic operational effects on redundant systems, the numerical etlects that reduce sy.stem reliability are pronounced Equation 2.4-5 shows that the probability ut failing a redundant. system composed of n components is the component probability raised to the n-th power. If a common clement couples the subsystems. Equation 2.4-5 is not correct and the failure rate is the failure rate of the common element. KDC is very serious because the time from primary failure to secondary failures may be too short to mitigate. The PSA Procedures Guide (NUREG,/CR-2.3(X)) cl.issities this type as "Type 2."... 

Generating block files (i.e, a set of Boolean equations) for subsystems... 

Conducting common cause and dependent event analysis Dropping complemented events and performing the subsequent minimization Generating block files (i.e., a set of Boolean equations) for subsystems Eliminating mincutsets with mutually exclusive events... 

Theoretical investigations of quenched-annealed systems have been initiated with success by Madden and Glandt [15,16] these authors have presented exact Mayer cluster expansions of correlation functions for the case when the matrix subsystem is generated by quenching from an equihbrium distribution, as well as for the case of arbitrary distribution of obstacles. However, their integral equations for the correlation functions... 

If equation (2.51) is the total differential for as a function of two variables, 1 and 2, we can expect that its partial derivatives (d E/d Zi) and (<9 /c> 2)5 can be expressed as functions of only those two variables. That is, — ( , 2). Thus, derivatives of (<9 /<9 ) and (d E/d Zi)- with respect to variables other than 1 and 2 should be zero. As we consider the implications of this statement, it is important to note that a change can be made independently in the r variable of one subsystem without affecting that of the other, but a change in 0 will affect both subsystems (since 0 is the same in both subsystems). Therefore, we must consider the implications for c and 0 separately in the analysis that follows. 

The first derivative on the right-hand side of equation (2.53) must be zero because subsystems 1 and 2 have been defined to be independent of each other. Therefore, (50/5 ) in the second term of equation (2.53) must also equal zero in order for this equation to be true for all conditions. In a similar manner, starting with the equivalent equations involving the derivative of 0 /0, one can show that (50/5zi) = 0. If one substitutes this last result into equation (2.52), one gets (50 /5zi) = 0, and the conclusion that 0 is independent of ri. From a similar treatment, one can also show that (5 2/5c2) = 0, so that 02 is independent of r2. Thus, since the two ratios i/0 and 2/0 are independent of r neither 0, 0i, nor 0 can depend upon the c variables. 

Equation (5.47) gives the criterion for reversibility or spontaneity within subsystem A of an isolated system. The inequality applies to the spontaneous process, while the equality holds for the reversible process. Only when equilibrium is present can a change in an isolated system be conceived to occur reversibly. Therefore, the criterion for reversibility is a criterion for equilibrium, and equation (5.47) applies to the spontaneous or the equilibrium process, depending upon whether the inequality or equality is used. 

The resulting model would therefore consist of component balance equations for the soluble component written over each of the many solid and liquid subsystems of the packed bed, combined with the component balance equation for the coffee reservoir. The magnitude of the recirculating liquid flow will depend on the relative values of the pressure driving force generated by the boiling liquid and the fluid flow characteristics of the system. 

The above rate equations confirm the suggested explanation of dynamics of silver particles on the surface of zinc oxide. They account for their relatively fast migration and recombination, as well as formation of larger particles (clusters) not interacting with electronic subsystem of the semiconductor. Note, however, that at longer time intervals, the appearance of a new phase (formation of silver crystals on the surface) results in phase interactions, which are accompanied by the appearance of potential jumps influencing the electronic subsystem of a zinc oxide film. Such an interaction also modifies the adsorption capability of the areas of zinc oxide surface in the vicinity of electrodes [43]. 

The profile of the potential energy Ep of the reacting system in dependence on the reaction coordinate x is shown in Fig. 5.6. The curve denoted as R corresponds to the initial state of the system. The coordinates of its minimum, i.e. the ground state of the system, are xo(0, 0- The curve for the final state P is shifted by a value corresponding to the difference between the final and initial energies of the electronic subsystem, AEc. The coordinate x0(f) corresponds to the minimum on the potential curve for the final state. The potential energy of the initial state of the system is given by the equation... 

Equation (9) was obtained using the assumption that the vibrational subsystem is in the state of thermal equilibrium corresponding to the initial electron state. The expression for the effective frequency a>eff has the form5... 

The calculation of the integrals in Eq. (55) in the classical limit in the improved Condon approximation (for the nuclear subsystem) using the saddle point method leads to two coupled equations for the electron wave functions of the donor and the acceptor in the transitional configuration ... 

Equation (106) shows that the interaction of the proton with the motion of the center of mass, described by the terms proportional to fx, is formally of the same form as the interaction with the medium atoms, and the first three terms in the Hamiltonian in Eq. (106) are equivalent to addition of one more degree of freedom to the vibrational subsystem. Thus, this problem does not differ from that for the process of tunnel transfer of the particles stimulated by the vibrations which were discussed in Section IV. So we may use directly the expressions obtained previously with substitution of the appropriate parameters. 

In general, the equations for the density operator should be solved to describe the kinetics of the process. However, if the nondiagonal matrix elements of the density operator (with respect to electron states) do not play an essential role (or if they may be expressed through the diagonal matrix elements), the problem is reduced to the solution of the master equations for the diagonal matrix elements. Equations of two types may be considered. One of them is the equation for the reduced density matrix which is obtained after the calculation of the trace over the states of the nuclear subsystem. We will consider the other type of equation, which describes the change with time of the densities of the probability to find the system in a given electron state as a function of the coordinates of heavy particles Pt(R, q, Q, s,...) and Pf(R, q, ( , s,... ).74,77 80... 

We may also introduce the transition probability per unit time at fixed values of the coordinates of slower subsystems, Wlf(q9 Q) and WfXq, < ), and consider the master equations for the corresponding probability densities RXq, Q) and Rf(q, Q), etc. 

Importantly, the value of the results gained in the present section is not limited to the application to actual systems. Eq. (4.2.11) for the GF in the Markov approximation and the development of the perturbation theory for the Pauli equation which describes many physical systems satisfactorily have a rather general character. An effective use of the approaches proposed could be exemplified by tackling the problem on the rates of transitions of a particle between locally bound subsystems. The description of the spectrum of the latter considered in Ref. 135 by means of quantum-mechanical GF can easily be reformulated in terms of the GF of the Pauli equation. 

The surface BCDE represents a segment of the surface defined by the fundamental equation characteristic of a composite system with coordinate axes corresponding to the extensive parameters of all the subsystems. The plane Uo is a plane of constant internal energy that intersects the fundamental surface to produce a curve with extremum at A, corresponding to maximum entropy. Likewise So is a plane of constant entropy that produces a curve with extremum A that corresponds to minimum energy at equilibrium for the system of constant entropy. This relationship between maximum entropy... 

By way of illustration consider a binary composite system characterized by extensive parameters Xk and Xf in the two subsystems and the closure condition Xk + X k — Xk. The equilibrium values of Xk and X k are determined by the vanishing of quantities defined in the sense of equation (3) as... 

In the chemical picture, the system is formed by molecules, atoms and/or ions. Each one of them has well defined properties. For such systems, a separability hypothesis is introduced in the physical picture. The different steps leading to effective equations for the subsystems have already been discussed by several authors. Here, we outline the important points for detailed discussions we refer the reader to our original papers [1-3, 6],... 

If A22 0, the system possesses unmeasured variables that cannot be determined from the available information (measurements and equations). In such cases the system is indeterminable and additional information is needed. This can be provided by additional balances that may be overlooked, or by making additional measurements (placing a measurement device to an unmeasured process variable). Also, from the classification strategy we can identify those equations that contain only measured variables, i.e., the redundant equations. Thus, we can define the reduced subsystem of equations... 

The outlined strategy has been applied to the subsystem of Example 4.4 in Chapter 4. The flow diagram, shown in Fig. 4 of Chapter 4, consists of 7 units interconnected by 15 streams. There are 8 measured flowrates and 7 unmeasured ones. The flowrate measurements with their variances are given in Table 3. In Chapter 4 we identified the subset of redundant equations. In this case it is constituted by one equation that contains the five redundant process variables. By applying the data reconciliation procedure to this reduced set of balances, we obtain the estimates of the measured variables, which are also presented in Table 3. 

As was discussed in Chapters 3 and 4, variable classification allows us to obtain a reduced subsystem of redundant equations that contain only measured and redundant variables. These are used in the reconciliation procedure. 

Now, because of the manner in which the balances arise, the total set of algebraic equations can be partitioned into two arbitrary subsystems. The first contains (m — a) equations and the second contains the remaining a equations, where a is an arbitrary number 1 < a < m. Note that the cases a = 0 or a = m correspond to the overall reconciliation problem. 

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