Adiabatic gradient

Here, x is the thermometric conductivity, v is the kinematic viscosity, is the Raleigh number = g 3 LVxv is the coefficient of thermal expansion and 3 is the temperature gradient excess over the adiabatic gradient eq. (11) finally, g is the z component of gravity... 

There are several key differences between the atmospheres of Venus and Earth. The terrestrial atmosphere has a pronounced temperature inversion at the tropopause, which is the boundary between the troposphere and the stratosphere. This inversion is due to absorption of UV sunlight by ozone, but is absent on Venus, which has too little O2 (<0.3 ppmv) to form an ozone layer. The temperature gradient of —8Kkm in Venus troposphere is very close to the dry (i.e., condensation cloud free) adiabatic gradient. This is... 

R. Boehler, Adiabats of quartz, coesite, olivine, and magnesium oxide to 50 kbar and 1000 K, and the adiabatic gradient in the Earth s mantle, J. Geophys. Res. 1982, 87, 5501-5506. 

The adiabatic gradient Vad = (F — 1)/ F, F is the adiabatic index. If this criterion is satisfied, the radiative temperature-pressure profile can be obtained if not, the pressure-temperature profile follows an adiabate. 

Here, the integrand is the off-diagonal gradient mahix element between adiabatic electronic states,... 

This makes it desirable to define other representations in addition to the electronically adiabatic one [Eqs. (9)-(12)], in which the adiabatic electronic wave function basis set used in the Bom-Huang expansion (12) is replaced by another basis set of functions of the electronic coordinates. Such a different electronic basis set can be chosen so as to minimize the above mentioned gradient term. This term can initially be neglected in the solution of the / -electionic-state nuclear motion Schrodinger equation and reintroduced later using perturbative or other methods, if desired. This new basis set of electronic wave functions can also be made to depend parametrically, like their adiabatic counterparts, on the internal nuclear coordinates q that were defined after Eq. (8). This new electronic basis set is henceforth refened to as diabatic and, as is obvious, leads to an electronically diabatic representation that is not unique unlike the adiabatic one, which is unique by definition. 

It needs mentioning that the diabatic Schrodinger equation (31) also contains a gradient term W (Rx) Vr x(R ) like its adiabatic counterpart [Eq. (15)]. 

For states of different symmetry, to first order the terms AW and W[2 are independent. When they both go to zero, there is a conical intersection. To connect this to Section III.C, take Qq to be at the conical intersection. The gradient difference vector in Eq. f75) is then a linear combination of the symmetric modes, while the non-adiabatic coupling vector inEq. (76) is a linear combination of the appropriate nonsymmetric modes. States of the same symmetry may also foiiti a conical intersection. In this case it is, however, not possible to say a priori which modes are responsible for the coupling. All totally symmetric modes may couple on- or off-diagonal, and the magnitudes of the coupling determine the topology. 

Robb, Bemaidi, and Olivucci (RBO) [37] developed a method based on the idea that a conical intersection can be found if one moves in a plane defined by two vectors xi and X2, defined in the adiabatic basis of the molecular Hamiltonian H. The direction of Xi corresponds to the gradient difference... 

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 scheme of commercial methane synthesis includes a multistage reaction system and recycle of product gas. Adiabatic reactors connected with waste heat boilers are used to remove the heat in the form of high pressure steam. In designing the pilot plants, major emphasis was placed on the design of the catalytic reactor system. Thermodynamic parameters (composition of feed gas, temperature, temperature rise, pressure, etc.) as well as hydrodynamic parameters (bed depth, linear velocity, catalyst pellet size, etc.) are identical to those in a commercial methana-tion plant. This permits direct upscaling of test results to commercial size reactors because radial gradients are not present in an adiabatic shift reactor. 

In a humidifier in which the make-up liquid is only a small proportion of the total liquid circulating, its temperature approaches the adiabatic saturation temperature 0S, and remains constant, so that there is no temperature gradient in the liquid. The gas in contact with the liquid surface is approximately saturated and has a humidity Jf... 

Adiabatic Reactors. Like isothermal reactors, adiabatic reactors with a flat velocity profile will have no radial gradients in temperature or composition. There are axial gradients, and the axial dispersion model, including its extension to temperature in Section 9.4, can account for axial mixing. As a practical matter, it is difficult to build a small adiabatic reactor. Wall temperatures must be controlled to simulate the adiabatic temperature profile in the reactor, and guard heaters may be needed at the inlet and outlet to avoid losses by radiation. Even so, it is hkely that uncertainties in the temperature profile will mask the relatively small effects of axial dispersion. 

It is also well known that there exist different extinction modes in the presence of radiative heat loss (RHL) from the stretched premixed flame (e.g.. Refs. [8-13]). When RHL is included, the radiative flames can behave differently from the adiabatic ones, both qualitatively and quantitatively. Figure 6.3.1 shows the computed maximum flame temperature as a function of the stretch rate xfor lean counterflow methane/air flames of equivalence ratio (j) = 0.455, with and without RHL. The stretch rate in this case is defined as the negative maximum of the local axial-velocity gradient ahead of the thermal mixing layer. For the lean methane/air flames,... 

Substantial heat-transfer intensification was also described for a special micro heat exchanger reactor [104]. By appropriate distribution of the gas-coolant stream, the axial temperature gradient can be decreased considerably, even under conditions corresponding to very large adiabatic temperature rises, e.g. of about 1400 °C. 

Principles and Characteristics SFC-MS is a sensitive coupled technique that can be selective or universal it was first mentioned in 1978 [396]. Further developments are given in Table 7.36. It is used in an on-line mode with open cell gas-phase interfaces, where the mobile phase is decompressed to low pressures. SFC presents a number of features which allow for easier coupling with MS than other chromatographies. In practice, however, SFC-MS coupling did not turn out to be as easy as expected, a fact which can be ascribed to the problems met in the adiabatic expansion of the mobile phase and the effects of pressure gradients in the ion... 

The first role of a reservoir is to impose on the system a gradient that makes the subsystem structure nonzero. The adiabatic flux that consequently develops continually decreases this structure, but the second role of the reservoir is to cancel this decrement by exchange of variables conjugate to the gradient. This does not affect the adiabatic dynamics. Hence provided that the flux is maximal in the above sense, then this procedure ensures that both the structure and the dynamics of the subsystem are steady and unchanging in time. (See also the discussion of Fig. 9.) A corollary of this is that the first entropy of the reservoirs increases at the greatest possible rate for any unconstrained flux. 

The end effects have been neglected here, including in the expression for change in reservoir entropy, Eq. (178). This result says in essence that the probability of a positive increase in entropy is exponentially greater than the probability of a decrease in entropy during heat flow. In essence this is the thermodynamic gradient version of the fluctuation theorem that was first derived by Bochkov and Kuzovlev [60] and subsequently by Evans et al. [56, 57]. It should be stressed that these versions relied on an adiabatic trajectory, macrovariables, and mechanical work. The present derivation explicitly accounts for interactions with the reservoir during the thermodynamic (here) or mechanical (later) work,... 

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