Condition, steady-state

The mathematical description of the echo intensity as a fiinction of T2 and for a repeated spin-echo measurement has been calculated on the basis that the signal before one measurement cycle is exactly that at the end of the previous cycle. Under steady state conditions of repeated cycles, this must therefore equal the signal at the end of the measurement cycle itself For a spin-echo pulse sequence such as that depicted in Figure B 1.14.1 the echo magnetization is given by [17]... 

The high rate of mass transfer in SECM enables the study of fast reactions under steady-state conditions and allows the mechanism and physical localization of the interfacial reaction to be probed. It combines the usefid... 

Dir, whereas for small distances d < r), /r Did. The large effective obtainable enables fast heterogeneous reaction rates to be measured under steady-state conditions. Zhou and Bard measured a rate constant of 6 x 10 Ms for the electro-hydrodimerization of acrylonitrile (AN) and observed the short-lived intennediate AN for this process [65]. 

New radicals are introduced by thermolysis of the hydroperoxide by chain-branching decomposition (eq. 4). Radicals are removed from the system by chain-termination reaction(s) (eq. 5). Under steady-state conditions, the production of new radicals is in balance with the rate of radical removal by termination reactions and equation 8 appHes for the scheme of equations 1—5 where r. = rate of new radical introduction (eq. 4). 

A second common approximation is the steady-state condition. That arises in the example if /fy is fast compared with kj in which case [i] remains very small at all times. If [i] is small then d[I] /dt is likely to be approximately zero at all times, and this condition is commonly invoked as a mnemonic in deriving the differential rate equations. The necessary condition is actually somewhat weaker (9). Eor equations 22a and b, the steady-state approximation leads, despite its different origin, to the same simplification in the differential equations as the pre-equihbrium condition, namely, equations 24a and b. 

SJng Je Rod-Fed Electron Beam Source. The disadvantages of multiple sources for alloy deposition can be avoided by using a single wire-fed or rod-fed source (Fig. 3) (3). A molten pool of limited depth is above the soHd rod. If the equiUbrium vapor pressures of the components of an alloy A B are in the ratio of 10 1 and the composition of the molten pool is A qB, under steady-state conditions, the composition of the vapor is the same as that of the soHd being fed into the molten pool. The procedure can be started with a pellet of appropriate composition A qB on top of a rod A B to form the molten pool initially, or with a rod of alloy A B to evaporate the molten pool until it reaches composition A qB. The temperature and volume of... 

Fig. 3. AHoy evaporation from a single rod-fed source under steady-state conditions p° = 10p° AB feed rod, A B molten pool, A qB and vapor and deposit, A B, where p° = the equilibrium vapor pressure of component B B, and p° = the equilibrium vapor pressure of component A A. Part (a) shows...

Rotational viscometers often were not considered for highly accurate measurements because of problems with gap and end effects. However, corrections can be made, and very accurate measurements are possible. Operating under steady-state conditions, they can closely approximate industrial process conditions such as stirring, dispersing, pumping, and metering. They are widely used for routine evaluations and quahty control measurements. The commercial instmments are effective over a wide range of viscosities and shear rates (Table 7). 

Gas Transport. Initially, ia a vessel containing air at atmospheric pressure, mass motion takes place when temperature differences exist and especially when a valve is opened to a gas pump. Initial dow ia practical systems has been discussed (29), as have Monte Cado methods to treat shockwave, turbulent, and viscous dow phenomena under transient and steady-state conditions (5). 

Thus when an electric field is appHed to a soHd material the mobile charge carriers are accelerated to an average drift velocity v, which, under steady-state conditions, is proportional to the field strength. The proportionality factor is defined as the mobility, = v/E. An absolute mobility defined as the velocity pet unit driving force acting on the particle, is given as ... 

Most theories of droplet combustion assume a spherical, symmetrical droplet surrounded by a spherical flame, for which the radii of the droplet and the flame are denoted by and respectively. The flame is supported by the fuel diffusing from the droplet surface and the oxidant from the outside. The heat produced in the combustion zone ensures evaporation of the droplet and consequently the fuel supply. Other assumptions that further restrict the model include (/) the rate of chemical reaction is much higher than the rate of diffusion and hence the reaction is completed in a flame front of infinitesimal thickness (2) the droplet is made up of pure Hquid fuel (J) the composition of the ambient atmosphere far away from the droplet is constant and does not depend on the combustion process (4) combustion occurs under steady-state conditions (5) the surface temperature of the droplet is close or equal to the boiling point of the Hquid and (6) the effects of radiation, thermodiffusion, and radial pressure changes are negligible. 

Determination of Crystallization Kinetics. Under steady-state conditions, the total number production rate of crystals in a perfectly mixed crystallizer is identical to the nucleation rate, B. Accordingly,... 

A useful simphfication of the total energy equation applies to a particular set of assumptions. These are a control volume with fixed solid boundaries, except for those producing shaft work, steady state conditions, and mass flow at a rate m through a single planar entrance and a single planar exit (Fig. 6-4), to whi(m the velocity vectors are perpendicular. As with Eq. (6-11), it is assumed that the stress vector tu is normal to the entrance and exit surfaces and may be approximated by the pressure p. The equivalent pressure, p + pgz, is assumed to be uniform across the entrance and exit. The average velocity at the entrance and exit surfaces is denoted by V. Subscripts 1 and 2 denote the entrance and exit, respectively. 

The economics of most processes are determined by the steady-state operating conditions. Excursions from these steady-state conditions generally average out and have an insignificant effec t on the economics of the process, except when the excursions lead to off-specification produc ts. In order to enhance the economic performance of a process, the steady-state operating conditions must be altered in a manner that leads to more efficient process operation. 

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 ... 

Bucket elevators, skip hoists, and cranes are used for top feeding of the furnace. Retention and downward flow are controlled by timing of the bottom discharge. Gases are propelled by a blower or by induced draft from a stack or discharge fan. In normal operation, the downward flow of sohds and upward flow of gas are constant with time, maintaining ideal steady-state conditions. 

These differential equations are readily solved, as shown by Luyben (op. cit.), by simple Euler numerical integration, starling from an initial steady state, as determined, e.g., by the McCabe-Thiele method, followed by some prescribed disturbance such as a step change in feed composition. Typical results for the initial steady-state conditions, fixed conditions, controller and hydraulic parameters, and disturbance given in Table 13-32 are listed in Table 13-33. 

Consider azeotropic distillation to dehydrate ethanol with benzene. Initial steady-state conditions are as shown in Fig. 13-108. The overhead vapor is condensed and cooled to 298 K to form two hquid phases that are separated in the decanter. The organic-rich phase is returned to the top tray as reflux together with a portion of the water-rich phase and makeup benzene. The other portion of the water-rich phase is sent to a stripper to recover organic compounds. Ordinarily, vapor from that stripper is condensed and recycled to the decanter, but that coupling is ignored here. 

As the oxygen transfer rate under steady-state conditions must equal oxygen uptake, K a may be calculated ... 

The fact that batch processes are not carried out at steady state conditions imposes broad demands on the control system. The instrumentation and control system have to be selected to provide adequate control for a wide variety of operating conditions and a wide variety of processes. In addition, basic process control and shutdown systems have to deal with sequencing issues. This chapter presents issues and concerns related to safety of instrumentation and control in batch reaction systems, and provides potential solutions. 

During a steady-.state condition, i.e. at power frequency / X i X[ and during a transient condition, i.e. during a switching operation when... 

When collecting meaningful field fractionating column data, the column must not only have constant flow rates, but the flows must give a good material balance (no accumulation). In addition, a steady state condition must exist for the given flow rates. 

To determine if steady state conditions exist, the temperatures and pressures in the column can be tabulated to assure that they are reasonably unchanging. Laboratoiy analyses are usually too slow and expensive for checking lined out conditions. Monitoring reflux accumulator boiloff is often an effective way of noting concentration changes. Simply let a sample of the accumulator liquid boil at atmospheric pressure in a bottle with a thermometer inserted. This method is limited to light hydrocarbons and is not accurate enough for precision fractionation. 

The result of the steady-state condition is that the overall rate of initiation must equal the total rate of termination. The application of the steady-state approximation and the resulting equality of the initiation and termination rates permits formulation of a rate law for the reaction mechanism above. The overall stoichiometry of a free-radical chain reaction is independent of the initiating and termination steps because the reactants are consumed and products formed almost entirely in the propagation steps. 

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