Partial steady-state conditions

Under steady-state conditions the temperature of the evaporating surface increases until the rate of sensible heat transfer to the surface equals the rate of heat removed by evaporation from the surface. To calculate this temperature, it is convenient to modify Eq. (12-26) in terms of humidity rather than partial-pressure difference, as follows ... 

The following isotopic labeling experiment was performed in order to quantify the contribution of the direct and indirect reaction routes to CO formation After steady-state reaction with CH4/02/He was achieved, an abrupt switch of the feed from CH4/02/He to an isotopic mixture of CH4/1 02/ C 02/He was made, in which the partial pressures of CH4 and 62 were kept exactly the same as in the ordinary CH4/02/He mixture, so as not to disturb the steady-state condition. However, C 02 was added to the isotopic mixture in an amount corresponding to approximately 10-15% of the CO2 produced during reaction of the mixture. The purpose was to measure the production of C 0 due to reforming of CH4 with C 02 only (indirect reaction scheme) under steady-state conditions of the working catalyst surface. Figure 3 shows the transient responses of and C O... 

Partial) dialysis in flow analysis. The sample solution flows along one side of the membrane, while the analyser solution passing (often in counter-current) on the other side takes up the diffused components from the sample. A dynamic equilibrium is reached (under steady-state conditions) in the leaving analyser solution, which is then analysed and from the result of which the analyte content can be derived via calibration with standard solutions treated in exactly the same way. This is a common procedure, e.g., in Technicon AutoAnalyzers, and has also been applied in haemoanalysis by Ammann et al.154 as described above. 

A useful application of the model is to examine the S02 and 02 concentration profiles in the trickle bed. These are shown for the steady-state conditions used by Haure et al. (1989) in Fig. 25. The equilibrium S02 concentration drops through the bed, but the 02 concentration is constant. In Haure s experiments 02 partial pressure is 16 times the S02 partial pressure. At the catalyst particle surface, however, 02 concentration is much smaller and is only about one-third of the S02 concentration. This explains why 02 transport is rate limiting and why experimentally oxidation appears to be zero-order in S02. 

In this paper we present a comparative study of the catalytic properties of bare Si02, 4% Mo03/Si02 and 5% V205/Si02 catalysts in the partial oxidation of methane to formaldehyde. The role of the acidic properties and reduced sites, evaluated in steady state conditions, is addressed. A co-operative reaction path-... 

When a steady state condition has been achieved. Equation 21 implies that the relative surface concentrations are only functions of the bulk concentrations and the sputtering coefficients. This point cannot be overemphasized. Many authors have misinterpreted their data because they did not understand the consequences of this result. Once the sputtering coefficients are known, then thermodynamic properties, such as a tendency towards surface segregation, do not affect the surface concentration. However, the sputtering yields themselves are partially determined by binding energies and the type of compounds which are present in the surface region. These parameters are, of course, influenced by thermodynamic considerations. 

Propylene Partial Pressure. The polymerization rate, under steady-state conditions (Figs. 17 and 18) is proportional to the partial pressure of propylene (30, 33). 

Fig. 18. Dependency of propylene polymerization rate in steady-state conditions on the propylene partial pressure (a-TiCli sample A). 

During inhalation anesthesia, the partial pressure of the inhaled anesthetic in the brain equals that in the lung when steady-state conditions are achieved. Therefore, at a given level (depth) of anesthesia, measurements of the steady-state alveolar concentrations of different anesthetics provide a comparison of their relative potencies. The volatile anesthetic concentration is the percentage of the alveolar gas mixture, or partial pressure of the anesthetic as a percentage of 760 mm Hg (atmospheric pressure at sea level). The minimum alveolar anesthetic concentration (MAC ) is defined as the... 

For the most part, only the steady state condition will be of concern here, in which the case the partial integral of Eq. (8.1) becomes... 

The subject of kinetics is often subdivided into two parts a) transport, b) reaction. Placing transport in the first place is understandable in view of its simpler concepts. Matter is transported through space without a change in its chemical identity. The formal theory of transport is based on a simple mathematical concept and expressed in the linear flux equations. In its simplest version, a linear partial differential equation (Pick s second law) is obtained for the irreversible process, Under steady state conditions, it is identical to the Laplace equation in potential theory, which encompasses the idea of a field at a given location in space which acts upon matter only locally Le, by its immediate surroundings. This, however, does not mean that the mathematical solutions to the differential equations with any given boundary conditions are simple. On the contrary, analytical solutions are rather the, exception for real systems [J. Crank (1970)]. 

Here M and T represent methylcyclohexane and toluene in the gas phase, and Ttt represents adsorbed toluene. The first step in the above reaction sequence represents the adsorption of methylcyclohexane with subsequent reaction to form toluene, while the second step is the desorption of toluene from the surface. Very likely the first step represents a series of steps involving partially dehydrogenated hydrocarbon molecules or radicals. However, at steady-state conditions the rates of the intermediate steps would all be equal, and the kinetic analysis is, therefore, not complicated by this factor. To account for the near zero-order behavior of the reaction, it was suggested that the active catalyst sites were heavily covered with... 

Basic Protocol 2 is for time-dependent non-Newtonian fluids. This type of test is typically only compatible with rheometers that have steady-state conditions built into the control software. This test is known as an equilibrium flow test and may be performed as a function of shear rate or shear stress. If controlled shear stress is used, the zero-shear viscosity may be seen as a clear plateau in the data. If controlled shear rate is used, this zone may not be clearly delineated. Logarithmic plots of viscosity versus shear rate are typically presented, and the Cross or Carreau-Yasuda models are used to fit the data. If a partial flow curve is generated, then subset models such as the Williamson, Sisko, or Power Law models are used (unithi.i). 

At 1=0, in a steady state condition, the sensor voltage is a constant zero and is independent of the oxygen partial pressure in the exhaust gas. As shown in the calculated results (See Figure 10), both Po2(l) and P0g(o) depend on the air-fuel ratio. But Po2(l) is always equal to Po2(0) (See Equation 38) thus, E becomes zero (See Equation 39). In this case, the internal total pressure Pq1 is equal to 1 atm. Experimental data show on-off type sensor voltage characteristics in a rich atmosphere when a current is applied to the sensor. (See Figures 5 and 6)... 

Increasing recycle flow reduces the inlet, peak, and exit temperatures of the reactor. Pressure builds until the higher partial pressures of the reactants compensate for the lower specific reaction rate because of the lower temperatures. The higher velocities in the reactor tubes also increase the heat transfer coefficient, which means that the heat transfer rate does not decrease directly with the decrease in reactor temperatures. Remember, steam pressure (and temperature) is held constant in the openloop run. The net result of the various effects is that, with the fresh feed flowrates fixed, the reactor comes to a new steady-state condition, which has lower reactor temperatures but higher pressure. The net reaction rate and the heat transfer in the reactor remain the same. The... 

At partial pressures of oxygen greater than approximately 100 Torr, chain termination occurs exclusively via the mutual destruction of two alkylperoxy radicals [reaction (6)]. The cross-termination reaction (5) may be neglected. The predicted rate expression, under steady-state conditions, is then given by... 

The plug flow reactor is increasingly being used under transient conditions to obtain kinetic data by analysing the combined reactor and catalyst response upon a stimulus. Mostly used are a small reactant pulse (e.g. in temporal analysis of products (TAP) [16] and positron emission profiling (PEP) [17, 18]) or a concentration step change (in step-response measurements (SRE) [19]). Isotopically labeled compounds are used which allow operation under overall steady state conditions, but under transient conditions with respect to the labeled compound [18, 20-23]. In this type of experiments both time- and position-dependent concentration profiles will develop which are described by sets of coupled partial differential equations (PDEs). These include the concentrations of proposed intermediates at the catalyst. The mathematical treatment is more complex and more parameters are to be estimated [17]. Basically, kinetic studies consist of ... 

The general redox mechanism of metal-oxide catalyzed oxidation of hydrocarbons involves two major stages in the catalytic process, reduction of the surface layers by hydrocarbons and their reoxidation by interaction with oxygen. While these two stages occur simultaneously in a reactor with the catalyst working under steady-state conditions, they can be carried out in two separate reaction zones in a reactor with catalyst circulation [37]. A hydrocarbon is fed into the first zone where a desirable intermediate product of partial oxidation is formed after interaction with the oxidized catalyst. In the second zone, gas phase oxygen reoxidizes the catalyst. Obviously, the residence time of the catalyst in the first zone should be short enough to prevent formation of an inactive reduced state of the catalyst. If only surface layers participate in the interaction with hydrocarbons, the time of catalyst reduction is approximately several seconds. 

In the case of galvanostatic (/ = constant) oxidation the total current (I) is a sum of partial currents. Under steady-state conditions... 

Variation of the oxygen partial pressure scarcely leads to any changes because in an open system under steady state conditions any amount of oxygen can be dissolved. Hence, all species can be oxidized almost independent of the partial pressure. The concentration of Fe(III) does not change at all, while slight changes... 

This model reduces to the two-phase model given by Eq. (288) under steady-state conditions. However, for the general case of time-varying inlet conditions this model retains all the qualitative features of the full partial differential equation model and while the traditional two-phase model which does not distinguish between cm and (c) ignores the dispersion effect in the fluid phase. 

By the definition of the steady-state condition, the first terms on the right-hand side are zero and provided the first-order partial derivative terms do not all vanish, we can ignore the additional terms which are second-order in the perturbations. Thus, we obtain a pair of linear equations for the evolution of these perturbations in the vicinity of the steady-state point... 

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