Steady state definition

Stage III Maximum Rate and Steady State. Definition. To express the overall rate of a sequence of reactions, a special mathematical treatment is often used, known as the steady state treatment. This is based on the assumption that the concentration of certain intermediate compounds or complexes is never large, that their concentration rises at the beginning of the reaction and soon reaches a constant (or steady) value, and that, at this point, the rate of change in the concentration, dc/dt, can be assumed to be zero. If the overall rate of reaction depends on the concentration of this intermediate, then the rate will have reached its maximum at this time. 

Time-Dependent Cascade Behavior. The period of time during which a cascade must be operated from start-up until the desired product material can be withdrawn is called the equiUbrium time of the cascade. The equiUbrium time of cascades utilizing processes having small values of a — 1 is a very important quantity. Often a cascade may prove to be quite impractical because of an excessively long equiUbrium time. An estimate of the equihbrium time of a cascade can be obtained from the ratio of the enriched inventory of desired component at steady state, JT, to the average net upward transport of desired component over the entire transient period from start-up to steady state, T . In equation form this definition can be written as... 

When a process is continuous, nucleation frequently occurs in the presence of a seeded solution by the combined effec ts of mechanical stimulus and nucleation caused by supersaturation (heterogeneous nucleation). If such a system is completely and uniformly mixed (i.e., the product stream represents the typical magma circulated within the system) and if the system is operating at steady state, the particle-size distribution has definite hmits which can be predic ted mathematically with a high degree of accuracy, as will be shown later in this section. 

In order to have effective exchange of air in important locations in a room, the age of the air in those locations should be low. The basis for comparison is the complete mixing scenario. That scenario gives the same age for any air volume selected in the room, identical to the nominal time constant for the ventilation airflow,. A steady-state scenario is assumed. See Sutcliffe for an overview of definitions related to age of air. The various air exchange efficiency indices are presented in Table 8.6. 

The respiratory quotient (RQ) is often used to estimate metabolic stoichiometry. Using quasi-steady-state and by definition of RQ, develop a system of two linear equations with two unknowns by solving a matrix under the following conditions the coefficient of the matrix with yeast growth (y = 4.14), ammonia (yN = 0) and glucose (ys = 4.0), where the evolution of C02 and biosynthesis are very small (o- = 0.095). Calculate the stoichiometric coefficient for RQ =1.0 for the above biological processes ... 

Figure 4.22 shows the steady-state effect of current, or equivalently rate, I/2F, of O2 supply to the catalyst on the rate increase Ar during C2H4 oxidation on Pt/YSZ. According to the definition of A (Eq. 4.19), straight lines passing from the (0,0) point are constant faradaic efficiency A lines. 

Previous theoretical kinetic treatments of the formation of secondary, tertiary and higher order ions in the ionization chamber of a conventional mass spectrometer operating at high pressure, have used either a steady state treatment (2, 24) or an ion-beam approach (43). These theories are essentially phenomenological, and they make no clear assumptions about the nature of the reactive collision. The model outlined below is a microscopic one, making definite assumptions about the kinematics of the reactive collision. If the rate constants of the reactions are fixed, the nature of these assumptions definitely affects the amount of reaction occurring. 

In a steady state, there can be no net isofractionation in the excreted components (by definition) the mass-weighted sum of all the 5 values for the excreted material must equal that of the diet. It so happens that what is excreted very often is dominated by one component. 

Neal and Nader [260] considered diffusion in homogeneous isotropic medium composed of randomly placed impermeable spherical particles. They solved steady-state diffusion problems in a unit cell consisting of a spherical particle placed in a concentric shell and the exterior of the unit cell modeled as a homogeneous media characterized by one parameter, the porosity. By equating the fluxes in the unit cell and at the exterior and applying the definition of porosity, they obtained... 

Note that the steady-state plasma concentration varies proportionally with the rate of infusion and inversely with the plasma clearance C/p, the definition of... 

Compare rji and r 2 after steady-state is definitely reached. 

Figure 5. A schematic representation of superposed steady-state reservoirs of constant volumes Vi (fractional crystallization is omitted in this schema). At steady-state, Vi/xi=V2/x2=..., where x is the residence time. This is analogous to the law of radioactive equilibrium between nuclides 1 and 2 Ni/Ti=N2/T2=...A further interest of this simple model is to show that residence times by definition depend on the volume of the reservoirs.

That the terminal acceleration should most likely vanish is true almost by definition of the steady state the system returns to equilibrium with a constant velocity that is proportional to the initial displacement, and hence the acceleration must be zero. It is stressed that this result only holds in the intermediate regime, for x not too large. Hence and in particular, this constant velocity (linear decrease in displacement with time) is not inconsistent with the exponential return to equilibrium that is conventionally predicted by the Langevin equation, since the present analysis cannot be extrapolated directly beyond the small time regime where the exponential can be approximated by a linear function. 

So the terminal velocity or flux is a maximum when the terminal acceleration is zero, which implies that the terminal velocity is a constant, which implies that <2(x) is a linear function of x. By definition, the steady state is the state of constant flux. This justifies the above focus on the terms that are linear in x in the study of the steady state. The preceding equations show that the steady state... 

In continuous processes the reactants are fed to the reactor and the products withdrawn continuously the reactor operates under steady-state conditions. Continuous production will normally give lower production costs than batch production, but lacks the flexibility of batch production. Continuous reactors will usually be selected for large-scale production. Processes that do not fit the definition of batch or continuous are often referred to as... 

Thus from Equation (2.13) we see that a working definition of KM is the substrate concentration that yields a velocity equal to half of the maximum velocity. Stated another way, the Ku is that concentration of substrate leading to half saturation of the enzyme active sites under steady state conditions. 

Furthermore, conjugate poles on the imaginary axis are BIBO stable—a step input leads to a sustained oscillation that is bounded in time. But we do not consider this oscillatory steady state as stable, and hence we exclude the entire imaginary axis. In an advanced class, you should find more mathematical definitions of stability. 

For a given hydrodynamic condition near the electrode in steady state, the maximum gradient is obtained when the concentration at the electrode is zero, or virtually zero. From the definition of limiting-current density, this situation corresponds to the limiting-current condition. 

Vocabulary of Terms Used in Reactor Design. There are several terms that will be used extensively throughout the remainder of this text that deserve definition or comment. The concepts involved include steady-state and transient operation, heterogeneous and homogeneous reaction systems, adiabatic and isothermal operation, mean residence time, contacting and holding time, and space time and space velocity. Each of these concepts will be discussed in turn. 

Kinetic vs. material chain. Kinetically, a chain reaction exists throughout the "life" of the radical, that is, from the initiation of a radical up to its termination by recombination or by disproportionation. The lifetime of a radical determines the so-called kinetic chain length Lp defined as the number of monomers consumed per initiating radical. Lp, by definition, can be calculated from the ratio between the propagation rate Rp to the initiation rate R, or, using steady-state hypothesis (Equation (1)), from the ratio between propagation rate to the termination rate Rt (Equation (3)). 

It should be noted that besides being widely used in the literature definition of characteristic timescale as integral relaxation time, recently intrawell relaxation time has been proposed [42] that represents some effective averaging of the MFPT over steady-state probability distribution and therefore gives the slowest timescale of a transition to a steady state, but a description of this approach is not within the scope of the present review. 

The restrictions of the definition (5.127) are the same as before It gives correct results only for monotonically evolving functions w/(f) and i)if(t) should fastly enough approach its steady-state value m/(oo) for convergence of the integral in (5.127). 

Attempts to define operationally the rate of reaction in terms of certain derivatives with respect to time (r) are generally unnecessarily restrictive, since they relate primarily to closed static systems, and some relate to reacting systems for which the stoichiometry must be explicitly known in the form of one chemical equation in each case. For example, a IUPAC Commission (Mils, 1988) recommends that a species-independent rate of reaction be defined by r = (l/v,V)(dn,/dO, where vt and nf are, respectively, the stoichiometric coefficient in the chemical equation corresponding to the reaction, and the number of moles of species i in volume V. However, for a flow system at steady-state, this definition is inappropriate, and a corresponding expression requires a particular application of the mass-balance equation (see Chapter 2). Similar points of view about rate have been expressed by Dixon (1970) and by Cassano (1980). 

From equation 2.4-2 for steady-state operation, together with the definitions provided by equations 2.3-5 to -7, the interpretations of rA in terms of FA,/A, and ca, corresponding to equations 2.3-8 to -11, are3... 

Furthermore, at steady-state, (— rA) is also the rate of mass transfer of A across the exterior film, such mass transfer being in series with the combined intraparticle processes of diffusion and reaction hence, from the definition of kAg,... 

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