Transient terms

However, we have only considered a steady state solid model in Equation (8.4). Consider the effect if we add the transient term (Equation (3.40)), the rate of enthalpy per unit area in the CV,... 

Since the mass of the fireball is small and changes in temperature are likely to be slow, we will ignore the transient term in Equation (10.69). These lead to a constant fireball temperature identical to the fire temperature of Equation (10.35) ... 

Under steady conditions, or where the transient term is negligible, this is simply... 

The transient term can be neglected in most cases, but any sudden change in the energy will result in a pressure change. This can be responsible for puffing effects seen in compartment fires, especially when the burning is oscillatory. 

It should be noted that, in media of low viscosity, the transient term is significant only at short times (< 100 ps at viscosities similar to that of water) and can be neglected, whereas in viscous media, this term cannot be ignored (the fluorescence decay is then no longer a single exponential). 

Equation (4.18) can be compared to the Stern-Volmer relation (4.10). The multiplying factor Y-1 accounts for the transient term and leads to a slight upward curvature of the Stern-Volmer plot. 

The method using intermolecular excimer formation is based on the same principle because this process is also diffusion-controlled. Excimers should, of course, be formed during the monomer excited-state lifetime. In Section 4.4.1, it was shown that the ratio Ie/Im of the intensities of the excimer and monomer bands is proportional to ki provided that the transient term can be neglected. When the dissociation rate of the excimer is slow with respect to de-excitation, the relationship is... 

The quencher molecules that are near the excited molecule in the moment of excitation have a much higher probability for depleting the excited molecules than the other quencher molecules. For that reason the rate coefficients used to describe the decay of excited molecules often have a so-called transient term, l a nDt) the transient term... 

Pinczewski and Sideman [63] have introduced additional transient terms in the steady-state model of Ruckenstein. Eqs. (209) to (211) were therefore replaced by the unsteady equations... 

Schmidt number, ShocSc1/3, as the experiment does. The transient term leads to a too strong dependence, ShocSc1/2. In addition, one can easily verify that, for sufficiently large values of the Schmidt number, the term in Eq. (257) due to the transient process has the dominant contribution to the value of the mass transfer coefficient and that the values predicted are much too large. 

Thus the partially reflecting boundary condition reduces the effective encounter distance by a factor of fcact (4nRD + fcact) 1 for both the steady-state and transient terms in the rate coefficient. 

Another consequence of the solvent s presence on the rate of reactant diffusion towards (and away from) each other is that solvent has to be squeezed out of ( sucked into ) the intervening space between the reactants. Because this takes time, the approach (or separation) of reactants is slowed. Effectively, the solvent diffusion coefficient is reduced at distances of separation between reactants from one to several solvent diameters. Figure 38 (p. 216) shows the diffusion coefficient as a function of reactant separation distances. This effect is known as hydrodynamic repulsion and it more than cancels the net increase of reaction rate due to the potential of mean force. It is discussed further in Chap. 8 Sect. 2.5 and Chap. 9 Sect. 3. Both the steady-state and transient terms in the rate coefficient depend on these effects. 

Despite the fact that steady-state solutions are the principal concern here, the transient terms are retained to facilitate the hybrid solution algorithm as discussed in Chapter 15 [159]. Alternative formulations for the diffusive mass flux, jkiZ, were introduced briefly in Section 3.5.2, and are discussed in more depth later in this chapter. 

In this equation a negative transition term is omitted, which vanishes at large times. The method of Hopkins and Hamming is justified by the fact that, for a description of steady flow, the upper integration limit t in eq. (2.3) can always be chosen so large that the intervals of r, where G (t) differs from zero, and where the transient term of / (t— x) differs from zero, can be separated. This is the reason, why influence of the transition term of J (t— x) disappears in the final result. 

An example of a pseudo 3-D model following the model of Celik et al. (2003) and Pakalapati (2003, 2006) and Pakalapati et al. (2006) will be presented here. This model is applicable to an SOFC. However, those for other types of cells, such as the PEMFC, are very similar except that a careful account of the liquid must be included either by solving extra transport equations for the liquid phase or by some empirical/algebraic relations that essentially keeps track of the liquid content in the cell. The details of the full 3-D transient governing equations for fuel cells can be found in Chapter 3. These equations also include the transient terms. Most models... 

Transient problems begin with an initial condition and march forward in time in discrete time steps. We have discussed space derivatives, and now we will introduce the time derivative, or transient, term of the differential equation. Although the Taylor-series can also be used, it is more helpful to develop the ED with the integral method. The starting point is to take the general expression... 

The previous section used the constant strain three-noded element to solve Poisson s equation with steady-state as well as transient terms. The same problems, as well as any field problems such as stress-strain and the flow momentum balance, can be formulated using isoparametric elements. With this type of element, the same (as the name suggests) shape functions used to represent the field variables are used to interpolate between the nodal coordinates and to transform from the xy coordinate system to a local element coordinate system. The first step is to discretize the domain presented in Fig. 9.12 using the isoparametric quadrilateral elements as shown in Fig. 9.15. 

When the catalyst maintains a constant activity level the transient term dNA/dt becomes negligible after 3 to 5 times the mean residence time r. In the case of a deactivating catalyst, however, the reactor effluent composition continues to change and steady state operation is never achieved. Since the gas phase composition is measured as a function of time, the transient term in Equation 3 may be calculated for each component. 

There is a transient term at the beginning of the scan, with E denoting the initial potential applied prior to the potential sweep, and a nontransient or constant term, vCd). When the direction of the scan is reversed in CV, another transient contribution appears and the nontransient component of changes sign. At high sweep rates it is difficult to separate the if component from ic by use of subtraction procedures. Usually the contribution from the transient part is only acceptable within the first 100 mV of the potential scan, which leads to the expression shown in Eq. 84 for the maximum applicable sweep rate, Vmax-... 

The problem (91) is splitted into a steady state term s [R] and transient term 5 [R, t]. Since both boundary conditions (92 b, 92 c) are of the second kind, the average temperature has to be included into the splitting formula [20] ... 

For tiie present problem, the quasi-steady solution of problem (11), essentially removing the transient term in eq.(ll.a) is considered... 

The origin of the nonlinear density dependences in the diffusion limit arises from the fact that for a single reactive site, the transient terms in diffusion theory become infinitely long lived for d < 2, and a steady-state rate cannot be defined for a single site. This arises from depletion of the equilibrium density for R> Rj, which becomes more severe as the dimensionality is reduced. However, for a finite reactive site density a steady state is eventually produced... 

Transient terms are approximated Transient terms are neglected... 

This problem exhibits multiple steady states. Obtain all the steady states by equating the transient term to zero in all the equations. For mathematical convenience, express steady state P, T, and Pp in terms of steady state Tp using the first three equations. Use the steady state equation for Tp (after eliminating all other dependent variables) to obtain the multiple steady states. Solve the dynamic problem using the initial conditions P(0) = 0.1, T(0) = 600, Pp(0) = 0 and Tp(0) = 761 and plot the dynamic profiles for t = 0..15. Can you change the initial conditions to obtain a different steady state (see examples 2.2.6 and 2.2.7)... 

The transient term in (1.1) can be manipulated as the order in which the partial time derivation and the integration is performed can be reversed. The convective and diffusive transport terms in (1.1) can be rewritten as a volume integral using Gauss theorem (App. A). 

The transient term on the RHS might be manipulated as the order in which the partial time derivation and the integration is performed can be reversed. 

Overall, the net inflow of molecules into the phase space drdc is balanced by the transient term, hence ... 

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