Steady state replacement

The derivation of the material-balance or continuity equation for reactant A is similar to that of equations 19.4-48 and -49 for nonreacting tracer A, except that steady state replaces unsteady state (cA at a point is not a function of t), and a reaction term must be added. Thus, using the control volume in Figure 19.15, we obtain the equivalent of equation 19.4-48 as ... 

Wang et al. [140] have studied amorphous CoSn/C alloys decorated with Pt as highly efficient electrocatalysts for ethanol oxidation. The catalysts were prepared using a two-stage chemical synthesis (sol-gel preparation and Steady-state replacement method). XRD results evidenced that the CoSn-base was in the amorphous state, but the characteristic peaks of the Pt fee crystalline structure appeared after Pt deposition. The TEM images confirmed that Pt was deposited onto CoSn/C. The electrochemical measurements showed that the mass activity of the Pt-CoSn/C catalyst was 454.6mA mgp, , which was about 1.71 and 1.74 times those of Pt/C and PtSn/C. The authors attributed this behavior to the possible modification in the Pt electronic structure elicited by the amorphous CoSn alloy. Materials based on amorphous alloys can open new perspectives regarding ethanol electro-oxidation. 

The (CEF) model (see Chapter 1) provides a simple means for obtaining useful results for steady-state viscometric flow of polymeric fluids (Tanner, 1985). In this approach the extra stress in the equation of motion is replaced by explicit relationships in terms of rate of strain components. For example, assuming a zero second normal stress difference for veiy slow flow regimes such relationships arc written as (Mitsoulis et at., 1985)... 

Other Models for Mass Transfer. In contrast to the film theory, other approaches assume that transfer of material does not occur by steady-state diffusion. Rather there are large fluid motions which constantiy bring fresh masses of bulk material into direct contact with the interface. According to the penetration theory (33), diffusion proceeds from the interface into the particular element of fluid in contact with the interface. This is an unsteady state, transient process where the rate decreases with time. After a while, the element is replaced by a fresh one brought to the interface by the relative movements of gas and Uquid, and the process is repeated. In order to evaluate a constant average contact time T for the individual fluid elements is assumed (33). This leads to relations such as... 

It is important to emphasize that direet studies sueh as those earned out on the eyelopropylmethyl radieal ean be done with low steady-state eoneentrations of the radical. In the case of the study of the eyelopropylmethyl radical, removal of the source of irradiation leads to rapid disappearance of the EPR spectrum, because the radicals react rapidly and are not replaced by continuing radical formation. Under many conditions, the steady-state concentration of a radical intermediate may be too low to permit direct detection. Failure to observe an EPR signal, therefore, cannot be taken as conclusive evidence against a radical intermediate. 

When adsorbate rates are sufficiently high to make periodic adsorbent replacement uneconomical, regeneration of the adsorbent can usually be justified. In continuous, steady-state systems, regeneration is required for economical operation. 

The desirable operating characteristics of equipment include simplicity, convenience and low cost of maintenance simplicity, convenience and low cost of assembly and disassembly convenience in replacing worn or damaged components ability to control during operation and test before permanent installation continuous operation and steady-state processing of materials without excessive noise, vibration or upset conditions a minimum of personnel for its operation and, finally, safe operation. Low maintenance often... 

The relative fluctuations in Monte Carlo simulations are of the order of magnitude where N is the total number of molecules in the simulation. The observed error in kinetic simulations is about 1-2% when lO molecules are used. In the computer calculations described by Schaad, the grids of the technique shown here are replaced by computer memory, so the capacity of the memory is one limit on the maximum number of molecules. Other programs for stochastic simulation make use of different routes of calculation, and the number of molecules is not a limitation. Enzyme kinetics and very complex oscillatory reactions have been modeled. These simulations are valuable for establishing whether a postulated kinetic scheme is reasonable, for examining the appearance of extrema or induction periods, applicability of the steady-state approximation, and so on. Even the manual method is useful for such purposes. 

In the normal acquisition process, the analyzer acquires multiple blocks of data. As part of the process, the microprocessor compares each block of data as it is acquired. If a block contains a transient that is not included in subsequent blocks, the block containing the event is discarded and replaced with a transient-free block. As a result, steady-state analysis does not detect random events that may have a direct, negative effect on equipment reliability. 

In practice, thermal cycling rather than isothermal conditions more frequently occurs, leading to a deviation from steady state thermodynamic conditions and introducing kinetic modifications. Lattice expansion and contraction, the development of stresses and the production of voids at the alloy-oxide interface, as well as temperature-induced compositional changes, can all give rise to further complications. The resulting loss of scale adhesion and spalling may lead to breakaway oxidation " in which linear oxidation replaces parabolic oxidation (see Section 1.10). 

This is then replaced by a period of steady state creep which, depending on the temperature and stress level, takes up the greatest part of the creep life. 

The new pathway, too, is a chain reaction Note that the first term of Eq. (8-31) does not give a meaningful transition state composition. Since the scheme in Eqs. (8-20M8-23) seems valid for the Cu2+-free reaction, we can seek to modify it to accommodate the new result. This approach is surely more logical than inventing an entirely new sequence. To arrive at the needed modification, we simply replace Eq. (8-23) by a new termination step, Eq. (8-30). With that, and the steady-state approximation, the rate law is... 

AT is intended to include any and all of the effects of the sorption rate of monomer on the surface, steric arrangement of active species, the addition of the monomer to the live polymer chain, and any desorption needed to permit the chain to continue growing. We assume a steady state in which every mole of propylene that polymerizes is replaced by another mole entering the shell from the gas, so that all of the fluxes are equal to Ny gmol propylene reacted per second per liter of total reactor volume. The following set of equations relates the molar flux to each of the concentration driving forces. 

If the reservoir is in a steady state (dM/df — 0) then the sources (Q) and sinks (S) must balance. In this case Q can replace S in Equation (1). 

Two rather similar models have been devised to remedy the problems of simple film theory. Both the penetration theory of Higbie and the surface renewal theory of Danckwerts replace the idea of steady-state diffusion across a film with transient diffusion into a semi-inhnite medium. We give here a brief account of surface renewal theory. 

A dynamic model should be consistent with the steady-state model. Thus, Eqs (1) and (4) should be extended to dynamic form. For the better convergence and computational efficiency, some assumption can be introduced the total amounts of mass and enthalpy at each plate are maintained constant. Then, the internal flow can be determined by total mass balance and total energy balance and the number of differential equations is reduced. Therefore, the dynamic model can be established by replacing component material balance in Eq. (1) with the following equation. 

In this and subsequent equations enters as a constant of integration such that [M-] = 0 at t = to. According to Eq. (45), 2kt[M-]s on the right-hand side of Eq. (48) may be replaced by the reciprocal of the steady state lifetime r. Making this substitution and rearranging, we obtain the compact relation... 

Steady-state reactors with ideal flow pattern. In an ideal isothermal tubular pZi/g-yZovv reactor (PFR) there is no axial mixing and there are no radial concentration or velocity gradients (see also Section 5.4.3). The tubular PFR can be operated as an integral reactor or as a differential reactor. The terms integral and differential concern the observed conversions and yields. The differential mode of reactor operation can be achieved by using a shallow bed of catalyst particles. The mass-balance equation (see Table 5.4-3) can then be replaced with finite differences ... 

Most real situations are, however, such that conditions change with respect to time. Under these circumstances, a steady-state mass balance is inappropriate and must be replaced by a dynamic or unsteady-state mass balance, expressed as... 

Again one way of dealing with this is to replace those differential rate equations, having low time constants (i.e., high K values) and fast rates of response, by quasi-steady-state algebraic equations, obtained by setting... 

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