The adiabatic case

The argument presented above assumes that the molecules can exchange energy between themselves more rapidly than the rate at which a molecule passes across the potential energy surface. In Fig. 6, for instance, in the reverse direction, the molecules pick up the thermal energy to populate the 

In Fig. 8 we show a set of energy levels for the adiabatic case, where we assume that, in going from R to the transition state, there is no exchange between the different levels. The third level in R must go through the third level in the transition state. The overall rate is then made up of a series of parallel reactions, each with its own individual energy of activation. So we can write (12), where [RJ is given by (13). Substitution of (13) into (12) 

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Here, the Ui no)s are the electronic eigenvalues dependent on the nuclear coordinate tiQ. Note that no = n is defined as the adiabatic case and no / n is defined as the diabatic case. 

The superaiatrix notation emphasizes the structure of the problem. Each diagonal operator drives a wavepaclcet, just as in the adiabatic case of Eq. (10), but here the motion of the wavepackets in different adiabatic states is mixed by the off-diagonal non-adiabatic operators. In practice, a single matrix is built for the operator, and a single vector for the wavepacket. The operator matrix elements in the basis set <() are... 

Sometimes for compression is expressed for the isothermal case, which is always lower than that for the adiabatic case. The difference defines... 

Operating conditions The reactor is 10 cm ID, input of ethylbenzene is 0.069 kg mol/h, input of steam is 0.69 kgmol/h, total of 2,500 kg/h. Pressure is 1.2 bar, inlet temperature is 600 C. Heat is supplied at some constant temperature in a jacket. Performance is to be found with several values of heat transfer coeff cient at the wall, including the adiabatic case. 

Both lines are broadened independently and solely by adiabatic phase shift as in Lorentz and Weisskopf theories. They are Lorentzians of width (1 — cosa) and frequency shift (sin a). In general off-diagonal elements of f are not zero though they are less than diagonal elements. Consequently, the spectrum may collapse even in the adiabatic case when A 1/tc. However, adiabatic collapse is hardly ever achieved in the gas phase where l/rc > l/t0 > jS since A > 1/tc > j8 and hence only the resolved doublet limit is available. 

When the isotermic flow of the ideal gas (17) is considered, scheme (43) can be written in simplified form, since the energy equation was disappeared because T = const. A proper iteration process is governed by the same rule as in the adiabatic case, the convergence of which can be established in a similar way without difficulties. In the isotermic case with the assigned values 7 = 1, a = lOo, 6 = 0 we deduce instead of (58) that... 

Change the program to consider the adiabatic case. Calculate the temperature drop for adiabatic operation from the equation. 

In the limit of strong vibronic coupling, F4S = 0, c = 5 /3, s=, c2 - s2 =, and the dynamic Jahn-Teller effect thus renders nugatory the orbital contributions to the angular momentum, and reduces the splitting, A, by a factor of two. Note in addition that the c and s quantities used in the vibronic treatment do not correspond to those of the adiabatic case, although the expressions are formally similar, so that the static distortion, A, cannot accurately be calculated from the c and s values deduced from the and 4 data. 

Vapor flow through pipes is modeled using two special cases adiabatic and isothermal behavior. The adiabatic case corresponds to rapid vapor flow through an insulated pipe. The isothermal case corresponds to flow through an uninsulated pipe maintained at a constant temperature an underwater pipeline is an excellent example. Real vapor flows behave somewhere between the adiabatic and isothermal cases. Unfortunately, the real case is difficult to model and no generalized and useful equations are available. 

Levenspiel13 showed that the maximum velocity possible during the isothermal flow of gas in a pipe is not the sonic velocity, as in the adiabatic case. In terms of the Mach number the maximum velocity is... 

The SCRF models should be useful for any of the adiabatic cases, but a more quantitative treatment would recognize at least three time scales for frictional coupling based on the three times scales for dielectric polarization,... 

Fig. 11.5 Schematic comparison of (a) sudden and (b) adiabatic inversion of the z-component of the polarization vector. In the sudden case a n-pulse is applied while in the adiabatic case a frequency sweep is shown. The time evolution of the z-polarization as a function of the pulse duration...

In direct analogy to the adiabatic case, the classical diabatic population probability is given by... 

The adiabatic case is the most common for accident conditions. The process is treated as an isentropicfree expansion of an ideal gas using the equation of state ... 

Even though the inlet temperature in the adiabatic case just considered was 100 K greater than in the isothermal case, the cooling produced by the reaction itself has resulted in the adiabatic reactor volume being more than 50 times greater than that of the isothermal reactor needed to achieve the same conversion. The provision of some heat from an external source would reduce, or even eliminate entirely, the temperature decrease and permit a much smaller reactor to be employed. 

In the static limit (y = 0), which may usually be called the adiabatic case, the perturbation Q is constant ( 0 ) and the excited states are split into a doublet with the energies... 

Figure 3.13 Comparison of isothermal and adiabatic expansions of a monatomic ideal gas from a common starting point P0, V0, showing the steeper fall of pressure (and temperature) in the adiabatic case.

The theory of adiabatic reaction developed in the previous article is here generalized to the case when heat transfer is present. Consideration of the heat transfer leads to the appearance of new features in the consumptiontime kinetic curves, specifically, the possibility of extinction as the residence time is increased and of self-ignition when the reaction time is decreased (in the previous article, in the adiabatic case, extinction occurred only for a decrease in the reaction time, and self-ignition only for an increase). 

Figure 23. Free energy surfaces in the adiabatic case.

Chapter 3 tries to give students the essential tools to solve lumped systems that are governed by scalar equations. It starts with the simplest continuous-start reactor, a CSTR in the adiabatic case. The first section should be studied carefully since it represents the basis of what follows. Our students should write their own codes by studying and eventually rewriting the codes that are given in the book. These personal codes should be run and tested before the codes on the CD are actually used to solve the unsolved problems in the book. Section 3.2 treats the nonadiabatic case. 

Continuous Stirred Tank Reactor The Adiabatic Case... 

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