Time-invariant state

In many problems, the system ultimately will reach a time-invariant state in which the total mass and/or the masses of the components are no longer changing. This condition is called the steady state. In mass transfer problems, the math-... 

Equation (5) describes the variation of a metabolite concentration over time as proportional to the rates in which a metabolite is synthesized minus the rates at which it is consumed. A stationary and time-invariant state of metabolite concentrations S° (steady state) is characterized by the steady-state condition... 

Stoichiometric analysis goes beyond topological arguments and takes the specific physicochemical properties of metabolic networks into account. As noted above, based on the analysis of the nullspace of complex reaction networks, stoichiometric analysis has a long history in the chemical and biochemical sciences [59 62]. At the core of all stoichiometric approaches is the assumption of a stationary and time-invariant state of the metabolite concentrations S°. As already specified in Eq. (6), the steady-state condition... 

In view of all of the preceding observations concerning the formal differences between closed and open systems, what general conclusions can be drawn about the applicability of equilibrium concepts in understanding and describing the chemical behavior of the elements in natural water systems Since equilibrium is the time-invariant state of a closed system, the question is under what conditions do open systems approximate closed systems. A simple example will illustrate the relationships, which are already implicit in Equation 35. If one considers the case of a simple reaction... 

If a physical system is isolated, its state changes irreversibly to a time-invariant state in which no physical or chemical change occurs, and a state of equilibrium is reached in a finite time. Some conditions of equilibrium are (i) for a system thermally insulated with an infinitesimal change at constant volume dS 0, dV = 0, dU = 0, (ii) for a system thermally insulated with an infinitesimal change at constant pressure dS = 0,dP = 0, dH = 0, (iii) for a system thermally insulated with an infinitesimal change at constant volume and temperature dA = 0, dV = 0, dT = 0, and (iv) for a system thermally insulated with an infinitesimal change at constant pressure and temperature dG = 0,dT= 0, dP = 0. 

If we consider the change of local entropy of a system at steady state ds/dt = 0, the local entropy density must remain constant because external and internal parameters do not change with time. However, the divergence of entropy flow does not vanish div J, = . Therefore, the entropy produced at any point of a system must be removed or transferred by a flow of entropy taking place at that point. A steady state cannot be maintained in an adiabatic system, since the entropy produced by irreversible processes cannot be removed because no entropy flow is exchanged with the environment. For an adiabatic system, equilibrium state is the only time-invariant state. 

Most natural water systems are continuous, open systems. Flows of matter and energy occur in the real system. The time-invariant state of a continuous system with flows at the boundaries is the steady state. This state may be poorly approximated by the equilibrium state of a closed system. In Figure 2.2 we indicated the important features of an open-system model with material fluxes and chemical reactions. The simple reversible reaction (a model reaction). 

The term steady state as used in this context means the time-invariant state of a flow system with chemical reactions. Steady state, with respect to chemical mechanisms, means that certain intermediates in a complex reaction are of low concentration, so that dC/dt = 0. It is important to keep these usages of steady state distinct. 

The dissolved species composition of most deep, confined groundwater and of some deep unconfined groundwater may remain constant for periods of years or even thousands of years. Such constancy suggests that reactions involving those species have come to thermodynamic equilibrium. When this is the case, the water/rock system involved can be considered a closed system with respect to those species and their controlling reactions in a thermodynamic sense (the system is open to the flow of energy, but closed to the flow of matter remember that equilibrium is the time-invariant state of a closed system). For these conditions equilibrium concepts can be used to explain concentrations of the aqueous species involved. 

An important concept in thermodynamics is the equilibrium state, which will be di,s-cussed in detail in the following. sections. Here we merely note that if a system is not subjected to a continual forced flow of mass, heat, or work, the system will eventually evolve to a time-invariant state in which there are no internal or external flows of heat or mass and no change in composition as a result of chemical or biochemical reactions. This state of the system is the equilibrium state. The precise nature of the equilibrium state depends on both the character of the system and the constraints imposetl on the system by its immediate sutrounding.s and its container (e.g., a constant-volume container fixes the system volume, and a thermostatic bath fixes the system temperature see Problem 1.1). 

The diversity around us, including the biological world (life), emerges from and depends on the ability to distinguish and exploit alternatives (selectivity). This discriminative process can rest on the energetic differences in thermodynamically definable and thus time-invariant states representing, however, in many cases, a... 

Tidal forces exerted by the attraction of the moon and the sun (and to a minor extent by the planets) on the earth s body produce a wide spectrum of geodynamic phenomena, from primary luni-solar attractional effects to secondary induced effects like solid earth and ocean tides, and third order ocean loading effects. Since these phenomena affect precise geodetic observations and make them time-dependent it is necessary to reduce time-variable geodetic observables and derived quantities in order to correspond to a quasinstationary, time—invariant state. 

Open system is always in non-equilibrium. A closed system can be in non-equilibrium depending on the circumstances. It may have subsystems between which exchange of matter and energy can take place or in the system itself, thermodynamic variables may not be constant in space. A typical example of the former type is thermo-osmosis, which is discussed in Chapter 3, where the two subsystems are separated by a membrane. Example of the latter type is thermal diffusion, which has been discussed in Chapter 5. When the flows and counter-flows in opposite directions are generated by corresponding gradients, steady state is obtained. Both equilibrium and non-equilibrium steady states are time-invariant states, but in the latter case both flows and gradients are present. 

Dufour effect Establishment of steady temperature gradient due to fixed concentration gradient. There can be two types of time-invariant states. One such is equilibrium state. In the equilibrium state, thermodynamic variables such as temperature T, pressure P and chemical potentials p, are adjusted in a way so that there is no (i) flow of matter, (ii) flow of energy and current and (iii) occurring in the system. Typical examples are vapour-liquid, liquid-Uquid, solid-liquid and chemical equilibria. However, time-invariant non-equilibrium steady states are also possible when opposite flows are balanced and gradients are maintained constant. 

Assume that the dynamic behaviour of a process is within a neighbourhood of an operating point and can be described sufficiently accurate by a linear time-invariant state space model. Then sensor and actuator faults, e.g. leakage from a tank, are additional external input signals to the process. They are commonly taken into account as additive terms in the state space equations and are classified as additive faults [7, 8]. 

Consideration of the equations of a faulty LTI system and of a Luenberger state variable observer reveals that any faults affecting the system have an affect on the observer output error which, after transients have settled, can be used as a fault indicator ([34], Sect. 5.2.2). Assume that thedynamics of a system may be represented by the linear time-invariant state space model... 

Steady-state models are those sets of equations which are time invariant and describe the conditions of the system at rest (i.e., when the states of the system are not changing with time). This will automatically presuppose that the system parameters are also time invariant (i.e., input variables, heat transfer coefficients, catalyst activity, and so forth are not changing with time). Of course, this is a theoretical concept, for no real system can fulfill these requirements perfectly. However, this theoretical concept represents the basis for the design and optimization of almost all chemical/ biochemical engineering equipment. The philosophy is that we assume that the system can attain such a time-invariant state and design the system... 

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