Adiabatic methods

Classical Adiabatic Design Method The classical adiabatic method assumes that the heat of solution serves only to heat up the liquid stream and that there is no vaporization of solvent. This assumption makes it feasible to relate increases in the hquid-phase temperature to the solute concentration x by a simple eutnalpy balance. The equihbrium curve can then be adjusted to account For the corresponding temperature rise on an xy diagram. The adjusted equilibrium curve will become more concave upward as the concentration increases, tending to decrease the driving forces near the bottom of the tower, as illustrated in Fig. 14-8 in Example 6. 

Comparisons by von Stockar and Wilke [Ind. Eng. Chem. Fundam., 16, 89 (1977)] between the rigorous and the classical adiabatic design methods for packed towers indicate that the simple adiabatic method underestimates the packing depths by as much as a factor of 1.25 to 1.5. Thus, when using the (dassical adiabatic method, one should consider the possible need to apply a design safety factor. 

Example 6 Solvent Rate for Absorption Let us consider the absorption of acetone from air at atmospheric pressure into a stream of pure water fed to the top of a packed absorber at 25 C. The inlet gas at 35 C contains 2 percent by volume of acetone and is 70 percent saturated with water vapor (4 percent H2O by volume). The mole-fraction acetone in the exit gas is to be reduced to 1/400 of the inlet value, or 50 ppmv. For 100 kmol of feed-gas mixture, how many Idlomoles of fresh water should be fed to provide a positive-driving force throughout the pacldug How many transfer units will be needed according to the classical adiabatic method What is the estimated height of pacldug required if Hqq = 0.70 m ... 

The various methods have been applied at different temperatures isothermal calorimetric method usually at 77°, adiabatic method near room temperature, and heats of combustion at 25°C. Corrections to a common temperature would in most cases be smaller than the experimental error,... 

Hartmann-Hahn cross polarization between two low-y nuclei has been successfully used to record chemical-shift correlation spectra between 13C and 15 N nuclei. Cross polarization between two low-y nuclei suffers from a high sensitivity to the exact matching condition at one of the side bands of the Hartmann-Hahn condition [101]. Adiabatic methods (APHH-CP) can eliminate most of this sensitivity and lead to high transfer efficiencies [34, 62, 90]. 

R. Verel. Adiabatic methods for Homonuclear Dipolar Recoupling in Magic Angle Spinning Solid-State NMR. PhD thesis, ETH Zurich, Diss. Nr. 14152, 2001. 

This is to be contrasted with the adiabatic method which forces a version of the FDR which is not symmetric in the two times. 

Numerical simulations of these stochastic equations under fast temperature ramping conditions indicate that the correlations in the random forces obtained by way of the adiabatic method do not satisfy the equipartition theorem whereas the proposed iGLE version does. Thus though this new version is phenomenological, it is consistent with the physical interpretation that 0(t) specifies the effective temperature of the nonstationary solvent. 

In section 6, chemical reactivity of ethylene-like species is discussed. A post-BO scheme recently proposed by us [11] is presented and used to examine the conformational change process. A check to the general properties discussed here is presented in section 7. In section 8 connections to the standard diabatic and adiabatic methods are examined. Section 9 presents a general discussion. 

Adiabatic passage schemes are particularly suited to control population transfer between states, since the adiabatic following condition assesses the stability of the dynamics to fluctuations in the pulse duration and intensity [3]. The time evolution of the wave function does not depend on the dynamical phase, and is therefore slow in comparison with the vibrational time scale. This fact guarantees that the time variation of the observables during the controlled dynamics will be slow. Adiabatic methods can therefore be of great utility to control dynamic observables that do not commute with the Hamiltonian. We are interested in the control of the bond length of a diatomic molecule [4]. 

The "crude" adiabatic method enables us to develop another approach. If the translational motion is slower than the internal motion (a << 1) then the D state is described by eqs. 7A, 76, and 81-83. The wavefunction describing the opposite case (a ... 

As noted in Section III.C.2, the adiabatic method allows one to separate "slow" rotational motion from "fast" vibrational motion. The evaluation of vibrational distributions that has been described is based on this feature of adiabatic theory. In many cases one can also similarly ignore "slow" bending motion. However, advances in experimental methods have led to measurements of rotational distributions of photofragments (see Okabe and Jackson, this volume) and thus the evaluation of these distributions has become a timely and interesting problem. 

In order to evaluate the FC factor, one requires the nuclear wavefunctions of the reactants and the products. The similarity of the state of reactants (products) to the dissociative state of photodissociation allows us to use the same adiabatic method in the framework of the present problem (see Section III). For example, the function can be written in the form... 

The previously described adiabatic method is a rigorous approach which provides a basis for the theory of molecules and solids. According to eq. A.10 the total wavefunction can be... 

The usual BO method (see eqs. A.5, A.6, A.9, A.10 as well as the "crude" approach (eqs. A.21-A.24) are characterized by a common shortcoming. Namely, it is difficult to evaluate higher nonadiabatic corrections based on these versions of adiabatic method. Neither of these approaches is adaptable to the usual perturbation theory development. In this connection, the development of a method that would enable the application of perturbation theory is of interest. Such a method has been determined by Geilikman (47) in solid state theory. An analogous method can be developed for molecules (81). 

Several approaches may be used in modeling absorption with heat effects, depending on the job at hand (1) treat the process as isothermal by assuming a particular temperature, then add a safety factor (2) employ the classical adiabatic method, which assumes that the heat of solution manifests itself only as sensible heat in the liquid phase and that the solvent vaporization is negligible (3) use semitheoretical shortcut methods derived from rigorous calculations and (4) employ rigorous methods available from a process simulator. 

The calorific value is the heat produced by the combustion of a unit quantity of coal in a bomb calorimeter with oxygen and under a specified set of conditions (ASTM D-121 ASTM D-2015 ASTM D-3286 ISO 1928). For the analysis of coal, the calorific value is determined in a bomb calorimeter either by a static (isothermal) method or by an adiabatic method, with a correction made if net calorific value is of interest. The unit is calories per gram, which may be converted to the alternate units (1.0 kcal/kg = 1.8 Btu/lb = 4.187 kJ/kg). 

Figure 4. Comparison between the adiabatic and the reference SDs. a0 = 1.00, T = 300 K, y° — 0.200, co° — 5000cin. O — 150cm-1. Matrix dimension in the adiabatic method 30.

An adiabatic method represents the most adequate technique for determining the relative tendencies of certain coals to heat spontaneously since it simulates most closely the real phenomenon. Conceivably, a field system would be similar to the adiabatic system but with appropriate modifications to hasten the oxidation process and increase the effluent gas concentrations within a reasonable test period. This could involve a more versatile system which would allow either the study of self-heating rates, similar to a method used by Guney (10) or which may be used for adiabatic calculations of a liability index through incorporation of a constant heat input. In the latter case, the heat might be supplied exclusively from the oxidizing air stream. 

Richards, Henderson, and Frevert, using the adiabatic method mentioned on a previous page, determined the heat of combustion of benzene and found it to be 2-5342 times as... 

Adiabatic calorimeter. With the adiabatic calorimeter, exchange of heat between the calorimetric vessel and the cover is suppressed. This happens so that the temperatures of the vessel and the cover are maintained at almost the same temperature. The condition (Tc-Tfi) = 0 can be attained at constant cover temperature by heating or cooling the calorimetric vessel using an internal heater or heat sink placed inside the calorimetric vessel. This compensation method is suitable for endothermic processes. For the adiabatic method, the characteristic feature is not only the equality of temperatures of the calorimetric vessel and of the cover, but also their changing value - the measurement proceeds at dynamic conditions, where the temperature of the calorimetric cover follows the temperature of the calorimetric vessel. 

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