Energy conservation derivation

The energy conservation equation is not normally solved as given in (9.4). Instead, an evolution equation for internal energy is used [9]. First an evolution equation for the kinetic energy is derived by taking the dot product of the momentum balance equation with the velocity and integrating the resulting differential equation. The differential equation is... 

Thermal plumes above point (Fig. 7.60) and line (Fig. 7.61) sources have been studied for many years. Among the earliest publications are those from Zeldovich and Schmidt. Analytical equations to calculate velocities, temperatures, and airflow rates in thermal plumes over point and line heat sources with given heat loads were derived based on the momentum and energy conservation equations, assuming Gaussian velocity and excessive temperature distribution in... 

In this section we show how the fundamental equations of hydrodynamics — namely, the continuity equation (equation 9.3), Euler s equation (equation 9.7) and the Navier-Stokes equation (equation 9.16) - can all be recovered from the Boltzman equation by exploiting the fact that in any microscopic collision there are dynamical quantities that are always conserved namely (for spinless particles), mass, momentum and energy. The derivations in this section follow mostly [huangk63]. 

Since plastics are generally made from hydrocarbon feedstocks they should be recycled to conserve energy. The most effectives energy conservation is to refabricate plastic items, though this is not always technically feasible. Under circumstances where recycling is not a feasible option the use of plastics in waste-derived fuels may be an acceptable conservation measure. 

Instead of using just energy conservation, Moody (1975) derived a revised model that takes into account all the conservation laws. He found that critical flow rate is given by a determinantal equation that gives G as a function of p, X, and S. 

The equation representing conservation of energy is derived upon noting that the work, pudt, done by the pressure, p, must equal the increase in the potential and kinetic energy of the medium. If the former is denoted by e -e per unit mass, the latter being u2, with the mass, pQUdt, flowing thru unit area of discontinuity, the equation will be ... 

There is obviously a close similarity between the quantum mechanically derived result for Ea in equation (54) and the thermodynamically derived result for AG in equations (57) and (58). In the classical limit of the quantum mechanical result, the characteristic quadratic dependence on AE arose from energy conservation based on the intersection between potential curves. The dependence on AG in equation (57) arose from differences in ion-solvent interactions and the multiplier m is essentially the extent of electron transfer in the activated complex for electron transfer. 

Fourier s equal ion can be used logcther with a statement on energy conservation to derive a differential equation describing the temperature held in a medium. Fourier was the first person to develop this equation and to devise means for its solution, hi vector notation, this equation is ... 

If one adopts McLennan s [78b] interpretation, then Eq. (21) is a realization of a standard theorem of Newtonian mechanics conservation of total energy = conservation of kinetic plus potential energy (see, e.g., Chap. 4 of Kleppner and Kolenkow, [80]). The reason is simple Coulomb electric force is central, then work is path independent, and total energy is function of position only. The time derivative of total energy is of course zero, as in Eq. (21). In this interpretation Qp and Qi are manifestations of kinetic energy. 

As mentioned in Section 2.1, the usual Boltzmann equation conserves the kinetic energy only. In this sense the Boltzmann equation is referred to as an equation for ideal systems. For nonideal systems we will show that the binary density operator, in the three-particle collision approximation, provides for an energy conservation up to the next-higher order in the density (second virial coefficient). For this reason we consider the time derivative of the mean value of the kinetic energy,12 16 17... 

From energy conservation I2> it follows that b12 = b2l. From Eqs. (II. 11 )>to (11.15) the following macroscopic expressions can be derived for the coefficients buv ... 

Energy conservation can be expressed in different forms. When e defines the energy per unit mass, the following equation for the conservation of energy is derived ... 

We have developed a model to explain the time dependent change in sensitivity for ions during excitation and detection. The first step is to describe the image charge displacement amplitude, S(Ap, Az), as a function of cyclotron and z-mode amplitudes. The displacement amplitude was derived using an approximate description of the antenna fields in a cubic cell. The second step in developing the model is to derive a relationship to describe the cyclotron orbit as a function of time for an rf burst. An energy conservation... 

With the help of this last relation, the time derivatives in equ. (10.32a) can be replaced by derivatives with respect to the z coordinate, thus eliminating the time dependences. Together with equ. (10.32a) (equ. (10.32b) is taken care of by implementing energy conservation which leads to equ. (10.35)) this gives... 

The equation for temperature is derived from the energy conservation law... 

A key aspect of modeling is to derive the appropriate momentum, mass, or energy conservation equations for the reactor. These balances may be used in lumped systems or derived over a differential volume within the reactor and then integrated over the reactor volume. Mass conservation equations have the following general form ... 

It follows that if dT = 0 and dP = 0, then dG = 0. This relationship replaces the law of internal energy conservation for systems under isothermic-isobaric conditions. The derivatives of Gibbs energy give - entropy... 

Internal consistency in the data for a1, I i and f2s, l is a crucial prerequisite for particle, momentum and energy conservation in such models. The consistency amongst the rates is ensured by employing the above mentioned recursive relations. This can be achieved either by fitting expressions which can be differentiated with respect to Ta and vn, or e.g., by employing data tables only for the lowest rate Iq and B-splines with sufficient smoothness to permit evaluation of the derivatives. 

As shown for the 2D case with infinite nucleus mass in Section 111, in this subsection we shall construct the TCM for the collinear eZe case with finite masses and shall elucidate the behavior near triple collisions. We use the McGehee s original transformation [22]. The derivation of the TCM is successive application of tricky transformations to the equations of motion and the energy conservation relation. We do not show all of the derivation. The readers are strongly recommended to consult with Refs. 22 and 29 for details. 

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