Initial transient period

In fact, following a small initial transient period, temporal sections of the space-time pattern are always of the form... 

Consider a highly resolved simulation of a set of microscopic TFM equations for a fluidized suspension of particles in a large periodic domain. The filtering operation does not require a periodic domain however, as each location in a periodic domain is statistically equivalent to any other location, statistical averages can be gathered much faster when simulations are done in periodic domains. After an initial transient period that depends on the initial conditions,... 

Because the coefficients are time-dependent, the stationary PDF will only approximate fe after an initial transient period. 

If the short initial transient period with complicated behavior is ignored, the solution for the concentration profile is (Equation 3-116)... 

Shendalman and Mitchell (6) have developed an equilibrium model which assumes that the adsorption isotherm is linear. The model predicts the limiting product composition for varying operating conditions as well as the product composition as a function of cycle number for the initial transient period. 

Some materials exhibit nearly steady mass loss rates when exposed to a fixed radiant heat flux. The surface temperature for these materials reaches a steady value after a short initial transient period, and all terms in Equation 14.7 are approximately constant at a specified heat flux level. L can then be obtained by measuring steady mass loss rates at different radiant heat flux levels, and... 

The principal advantage of continuous reaction vessels is that they operate (after an initial transient period) under steady-state conditions that are conducive to the formation of a highly uniform and well-regulated product. In this section, we shall confine the discussion to continuous stirred-tank reactors (CSTRs). These reactors are characterized by isothermal, spatially uniform operation. 

Analysis During initial transient periods, heat transfer through the pipe material predominantly is In the radial direction, and thus can be approximated as being one-dimensional. Then the temperature within the pipe material changes with the radial distance rand the time f. That is, T - T r, t). 

After an initial transient period characterized by activation and deactivation processes of the catalysts [4], a steady state yield of hydrocarbons of 18% was observed with the H2/CO reaction gas on the Fe-catalyst and of 70% on the Co-catalyst (Fig. 1). 

Fig. 14.11 The reactive flux correlation function, Eq. (14.99) plotted against time. After the initial transient period this function becomes essentially constant on the time scale shown.

The catalysts show an initial transient period during which either activation or deactivation can occur (Fig. 6). After about 10 h on stream, all amorphous catalysts approach the same constant activity value. 

The results show that the output flow is coupled to the input flow quite tightly after an initial transient period. We can compare these responses to the sinusoidal input with those from the constant input case by plotting qex[t] and h[t] for both cases ... 

Show that the AP signal will be proportional to the rate of rise of Pi , after an initial transient period, when Pin is a ramp function. 

The initial transient period (ITP) of the enantioselective hydrogenation of activated ketones (ethyl pyruvate, p5Tiivaldehyde dimethyl... 

Mallat, T., Bodnar, Z., Minder, B., Borszeky, K., Baiker, A. (1997) Nature of initial transient period during enantioselective hydrogenation on Pt and Pd, J Catal 168, 183-193. 

After the initial transient period, the constant drying-rate period takes place for the sapwood board. During this period, which lasts several hours, all temperatures are equal to the wet-bulb temperature and the overpressure remains very small. At the beginning of the second drying period (around 350 min), an important overpressnre develops due to the temperature increase. It disappears only when the entire board enters the hygroscopic range. At this moment, all temperatures approach the dry-bulb temperature. 

The simplest feature or pattern that a reaction can show is that after an initial transient period all the concentrations tend to a limiting value that is called the steady state concentration, stationary concentration, equilibrium concentration (this may be criticised from the thermodynamic point of view, but is often used in the theory of ordinary differential equations), or singular point. This equilibrium is independent of (or hardly dependent on) the initial vector of concentrations, i.e. it is asymptotically stable. 

The most regular behaviour of a solution of a differential equation is if it tends to a constant in the state-space. It is a bit more complicated if, after an initial transient period, the solution tends to a periodic orbit. In more than two dimensions it may also happen that the trajectories remain in a bounded set, but they neither tend to an equilibrium point, nor to an oscillatory solution. This is a loose definition of chaos. 

Figure 3 shows the measured concentrations of nitrate, nitrite, and DOC during the denitrification process subjected to different inlet C/N ratios. The system was firstly operated to steady state using an inlet C/N ratio of 1.55. During this initial transient period (approximately 30 hours) both nitrate and nitrite accumulate, after which the concentrations of both pollutants in the treated water drop to values below the maximum admissible ones. The purpose of this preliminary experiment was to simulate the startup of the water treatment plant. 

Figure 5 represents a typical textbook drying rate curve obtained from a batch of wet particles fluidized with hot air of fixed temperature, humidity, and flow rate. After an initial transient period (generally a heatup period) if the particles are very wet, i.e., covered with a liquid film, the drying rate depends entirely on the external heat and mass transfer rates, e.g., a rise in air temperature or air flow rate or a reduction in air... 

Equation 5-22 implies that after an initial transient period, the ramp input yields a ramp output with slope equal to Ka, but shifted in time by the process time constant t (see Fig. 5.5). An unbounded ramp input will ultimately cause some process component to saturate, so the duration of the ramp input ordinarily is limited. A process input frequently will be ramped from one value to another in a fixed amount of time so as to avoid the sudden change associated with a step change. Ramp inputs of this type are particularly useful during the start-up of a continuous process or in operating a batch process. 

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