Adiabatic approach condition

The validity conditions for the semiclassic adiabatic approach in the description of the systems with orbitally non-degenerate levels are elucidated in the basic works of Bom and Oppenheimer (comprehensive discussion can be found in Refs. [6,7]). In these systems, the slow nuclear motion can be separated from the fast electronic one. The situation is quite different in the JT systems where, in general, this separation is impossible due to hybridization of the electronic and vibrational states. Nevertheless, in many important cases the adiabatic approach can serve as a relatively simple and at the same time powerful tool for the theoretical study of the JT systems giving accurate quantitative results and clear insight on the physical nature of the physical phenomena. 

The physical sense of the condition of the violation of the adiabatic approach in the representation of the wave functions can be easily understood if we expand the exponent in the expression (64) on e and turn to the usual units ... 

Because the die wall material usually has a thermal conductivity much higher than polymer melts, adiabatic conditions are not likely to be achieved. On the other hand, it is also not likely that the wall temperature will remain constant. In this case, the heat flux through the wall would be such as to maintain a perfectly constant temperature along the wall. This is referred to as an isothermal wall boundary condition. Because of the high thermal conductivity of the wall, the isothermal boundary condition is more likely to occur than the adiabatic boundary condition. Adiabatic conditions can be approached if the die is very well insulated. In most actual cases, the true thermal boundary condition will be somewhere between isothermal and adiabatic, depending on the design of the die and external conditions around the die. A typical temperature profile resulting from the velocity profiles shown in Fig. 7.106 is shown in Fig. 7.108. 

The boundary conditions that may be used with the thermochemical module include specified boundary temperature, convective heat transfer or no heat transfer (adiabatic). Different conditions (e.g., different HTCs) can be applied to each element as desired. Either explicit or implicit techniques may be chosen to solve the heat transfer (Eq. [13.1]) and cure rate equations. Using either technique, these two equations are uncoupled during each solution time-step. This approach facilitates a simplified and modular solution procedure and is sufficiently accurate if small time steps are used. 

In Section IV.A, the adiabatic-to-diabatic transformation matrix as well as the diabatic potentials were derived for the relevant sub-space without running into theoretical conflicts. In other words, the conditions in Eqs. (10) led to a.finite sub-Hilbert space which, for all practical purposes, behaves like a full (infinite) Hilbert space. However it is inconceivable that such strict conditions as presented in Eq. (10) are fulfilled for real molecular systems. Thus the question is to what extent the results of the present approach, namely, the adiabatic-to-diabatic transformation matrix, the curl equation, and first and foremost, the diabatic potentials, are affected if the conditions in Eq. (10) are replaced by more realistic ones This subject will be treated next. 

Peclet number independent of Reynolds number also means that turbulent diffusion or dispersion is directly proportional to the fluid velocity. In general, reactors that are simple in construction, (tubular reactors and adiabatic reactors) approach their ideal condition much better in commercial size then on laboratory scale. On small scale and corresponding low flows, they are handicapped by significant temperature and concentration gradients that are not even well defined. In contrast, recycle reactors and CSTRs come much closer to their ideal state in laboratory sizes than in large equipment. The energy requirement for recycle reaci ors grows with the square of the volume. This limits increases in size or applicable recycle ratios. 

The process of adiabatic saturation in Section 24.4 assumed that the spray water temperature had no effect on the final air condition. If, however, a large mass of water is used in comparison with the mass of air, the final condition will approach the water temperature. If this water is chilled below the dew point of the entering air, moisture will condense out of the air, and it will leave the washer with a lower moisture content (see Figure 24.7). 

The chapter by Haynes et al. describes the pilot work using Raney nickel catalysts with gas recycle for reactor temperature control. Gas recycle provides dilution of the carbon oxides in the feed gas to the methanator, hence simulating methanation of dilute CO-containing gases which under adiabatic conditions gives a permissible temperature rise. This and the next two papers basically treat this approach, the hallmark of first-generation methanation processes. 

The conditions existing during the adiabatic flow in a pipe may be calculated using the approximate expression Pi/ = a constant to give the relation between the pressure and the specific volume of the fluid. In general, however, the value of the index k may not be known for an irreversible adiabatic process. An alternative approach to the problem is therefore desirable.(2,3)... 

If an unsaturated gas is brought into contact with a liquid which is at the adiabatic saturation temperature of the gas, a simultaneous transfer of heat and mass takes place. The temperature of the gas falls and its humidity increases (Figure 13.2). The temperature of the liquid at any instant tends to change and approach the wet-bulb temperature corresponding to the particular condition of the gas at that moment. For a liquid other than water, the adiabatic saturation temperature is less than the wet-bulb temperature and therefore in the initial stages, the temperature of the liquid rises. As the gas becomes humidified, however, its wet-bulb temperature falls and consequently the temperature to... 

For the adiabatic condition in which RHL is suppressed, the flame response exhibits the conventional upper and middle branches of the characteristic ignition-extinction curve, with the upper branch representing the physically realistic solutions. It can be noted that the effective Le of this lean methane/air mixture is sub-unity. It can be seen from Figure 6.3.1 that, with increasing stretch rate, first increases owing to the nonequidiffusion effects (S > 0), and then decreases as the extinction state is approached, owing to incomplete reaction. Furthermore, is also expected to degenerate to the adiabatic flame temperature, when v = 0. 

There are three ways of implementing the GP boundary condition. These are (1) to expand the wave function in terms of basis functions that themselves satisfy the GP boundary condition [16] (2) to use the vector-potential approach of Mead and Truhlar [6,64] and (3) to convert to an approximately diabatic representation [3, 52, 65, 66], where the effect of the GP is included exactly through the adiabatic-diabatic mixing angle. Of these, (1) is probably the most... 

The temperature rise due to this exothermic reaction then approaches the adiabatic temperature rise. The final steady state is always characterized by conditions T = T, and c = 0. A batch reactor, in which a zero order reaction is carried out, always has a unique and stable mode of operation. This is also true for any batch and semibatch reactor with any order or combination of reactions. 

In the case of adiabatic flow we use Eqs. (9-1) and (9-3) to eliminate density and temperature from Eq. (9-15). This can be called the locally isentropic approach, because the friction loss is still included in the energy balance. Actual flow conditions are often somewhere between isothermal and adiabatic, in which case the flow behavior can be described by the isentropic equations, with the isentropic constant k replaced by a polytropic constant (or isentropic exponent ) y, where 1 < y < k, as is done for compressors. (The isothermal condition corresponds to y= 1, whereas truly isentropic flow corresponds to y = k.) This same approach can be used for some non-ideal gases by using a variable isentropic exponent for k (e.g., for steam, see Fig. C-l). 

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