Adiabatic equations

Some formulas based upon the adiabatic equation and useful in compressor work are as follows ... 

Adiabatic. When the net heat lost or gained by the unit to or from the surroundings is zero. Most plant installations approach this operation, and the adiabatic equations are widely used. 

This deals with purely convective heating, but defining 6 by Equation (5.12), we take h = k/r0 and r0 = V/S of the Semenov model. Equation (5.18) approximates the constant surface temperature case. Also, here the initial temperature is taken as Too- If we allow no convective cooling, the adiabatic equation becomes... 

Equation (5.19) can also be obtained from the unsteady counterpart of Equation (5.5). If internal conduction is taken as zero in Equation (5.5), this adiabatic equation arises. Equation (5.19) also becomes the basis of an alternative method for determining the bulk kinetic properties. The method is called the adiabatic furnace used by Gross and Robertson [5]. The furnace is controlled to make its temperature equal to the surface temperature of the material, thus producing an adiabatic boundary condition. This method relies on the measurement of the time for ignition by varying either ra or Too, the initial temperature. 

We can see from this, that the adiabatic equation is as follow... 

Let us begin by proving a simple identity between the ideal gas volumes Vl5..., V4. For adiabatic step II (2 —> 3), the adiabatic equation of state (3.86) requires that P2V2 = P3 Lj, while for adiabatic step IV (4 —> 1) the corresponding relationship is P Vj = P4VJ. Dividing the latter equation by the former gives the relationship... 

Media molecular dynamics is important to the formation of the energetic profile of the electron transfer. When ET occurs faster then the medium relaxation, the process is governed by the medium dynamics with the medium relaxation time xs. In such a case the pre-exponential factor in non-adiabatic equation is described by equation (Bixon, 1992)... 

Compare this to the case of the reversible, adiabatic equation of state II - 77 and to the reduced isothermal equation of state II - rj. 

In contrast to the experimentally based work discussed above, in the most recent comprehensive theoretical discussion [21d], Bixon and Jortner state that the question of whether non-adiabatic or adiabatic algorithms describe electron-transfer reactions was settled in the 1960s, and that the majority of outer-sphere electron-transfer reactions are non-adiabatic. This is certainly true for the reactions that occur in the Marcus inverted region in which these authors are interested, but we think the question of whether reactions in the normal region are best treated by adiabatic theory that includes an electronic transmission coefficient or by non-adiabatic equations remains to be established. 

If the gas in the bubble extends and contracts adiabatically, then the gas pressure in the bubble is related to the initial pressure Po by the adiabatic equation... 

We can use Equation (8-28) to relate temperature and conversion and then proceed to evaluate the algorithm described in Example 8-1. However, unless the reaction is carried out adiabatically. Equation (8-28) is still difficult to evaluate because in nonadiabatic reactors, the heat added to or removed from the system varies along the length of the reactor. This problem does not occur in adiabatic reactors, which are frequently found in industry. Therefore, the adiabatic tubular reactor will be analyzed first. 

Under the conditions of validity of the two-electronically-adiabatic-state approximation it is possible to change from the i]/al,ad(r q) (n = i, j) electronically adiabatic representation to a diabatic one 1,ad(r q) (n = i, j) for which the VR Xn(R) terms in the corresponding diabatic nuclear motion equations are significantly smaller than in the adiabatic equation or, for favorable conditions, vanish [24-26]. Such an electronically diabatic representation is usually more convenient for scattering calculations involving two electronically adiabatic PESs, but not for those involving a single adiabatic PES. This matter will be further discussed in Sec. III.B.3 for the case in which a conical intersection between the E ad(q) and Ejad(q) PESs occurs. 

Summarizing, we needed to obtain for Eq. (4.8) totally three equations, namely the thermal equation of state, the adiabatic equation, and the scaling law. The latter turned out to be equivalent with the Gibbs - Duhem equation. This means that for the three variables S, V, n we needed three equations to get the fundamental form. 

Equation (4.13) is denoted often as the adiabatic equation for an ideal gas, hereby the left-hand side is simply put as a constant ... 

In order to get the equation of the ideal gas, we must further postulate that g(T) -f(T) = R, since d(py) = (Cp - C )d7 . However, this assumption is not sufficient to obtain the adiabatic equation properly, we must more strictly assume that both Cp and Cp are constants. Further, we have to adjust the constant of integration properly. 

The adiabatic equation for the van der Waals gas can be obtained immediately by rearranging Eq. (4.37). An equation resembling closely to that of the ideal gas is obtained, i.e.. 

In the case of an isentropic process, we have to use the adiabatic equation instead of the ideal gas law. Thus to allow an isothermal process, we have to place the column into a thermostat. In this case, not the energy is equated to zero, but rather the free energy... 

In the earth atmosphere, there are usually certain fluxes of heat, therefore both approaches that treat the gas column as a closed thermodynamic system are not correct. The adiabatic case is more suitable, if the respective process of movement of a gas to some height is fast in comparison to the rate of heat conduction, thus entropy exchange of the gas. It was pointed out by Schaefer that the isothermal assumption in the barometric formula is highly unrealistic, and the adiabatic equation should be used [13]. 

It should be noted that in general the equivalent adiabatic equations (106 III) and (124,111) cannot be represented exactly in the exponential form of the accurate rate expression (67,111) which involves the activation barrier. Only the equivalent thermodinamic formulations (107.Ill) and (130,111) of the adiabatic rate theory are quite similar to the corresponding exact thermodynamic rate expression (69 111) Approximations to (106,111) of the form of equation (67,111) will be considered in the next section. 

We drop Cj from the leading order adiabatic equations in the two-phase regime. We introduee the liquid potential = 2/3(yS ), obtaining a redueed 4x4 system for the reduced wet variables P = (C, T, ... 

For example, it a gas expands under adiabatic conditions, its temperature falls (work is done against the retreating walls of the container). The adiabatic equation describes the relationship between the pressure ip) of an ideal gas and its volume (V), i.e. pV = K, where y is the ratio of the principal specific heat capacities of the gas and K is a constant. 

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