Dimensionless duration

The behavior predicted by this equation is illustrated in Fig. 16-33 with N = 80. Xp = (Evtp/L)/[il — )(p K -i- )] is the dimensionless duration of the feed step and is equal to the amount of solute fed to the column divided by tne sorption capacity. Thus, at Xp = 1, the column has been supplied with an amount of solute equal to the station-aiy phase capacity. The graph shows the transition from a case where complete saturation of the bed occurs before elution Xp= 1) to incomplete saturation as Xp is progressively reduced. The lower cui ves with Xp < 0.4 are seen to be neany Gaussian and centered at a dimensionless time - (1 — Xp/2). Thus, as Xp 0, the response cui ve approaches a Gaussian centered at Ti = 1. 

Characteristic time Fo defines dimensionless duration of non-stationary process and in inverse proportion to a square of characteristic thickness of bed8ch and in direct ratio an effective temperature conductivity a. For example, for increase of capacity of heat process in HHP it is necessary to reduce time of process. Active reduction of physical time of process will be promoted by reduction of thickness of bed8Ch and increase in its temperature conductivity a. 

The dimensionless duration of reactants pass through a reactor Reactive mixture density Surface tension... 

From the same relation, we define the dimensionless duration of reaction for a grain by ... 

Introducing a dimensionless time 0 and a parameter of model A which you will define, give the dimensionless duration of the reaction of a grain and estabUsh the expression between the reduced rate and time. 

Practical probability is the limit of two ratios (Section 2.2). The numerator is the number of cases of failure of the type of interest (N) the denominator, the nonnalizing term is the time duration over which the failures occurred or the total number of challenges to the system. The former has the units of per time and may be larger than 1, hence it cannot be probability which must be less than 1. The latter is a dimensionless number that must be less than 1 and can be treated as probability. 

Calculate the positive duration of the blast at the endpoint distances. Using the combustion-scaled distance, calculated in Step 4 above, Equation 7.5b and Figure 7.2b (Reference 5), the blast durations can be calculated. Entering Figure 7.2b at the combustion-scaled distance for each endpoint overpressure, the dimensionless positive phase duration (t+) may be read from Curve 10. 

The overall effect of the preceding chemical reaction on the voltammetric response of a reversible electrode reaction is determined by the thermodynamic parameter K and the dimensionless kinetic parameter . The equilibrium constant K controls mainly the amonnt of the electroactive reactant R produced prior to the voltammetric experiment. K also controls the prodnction of R during the experiment when the preceding chemical reaction is sufficiently fast to permit the chemical equilibrium to be achieved on a time scale of the potential pulses. The dimensionless kinetic parameter is a measure for the production of R in the course of the voltammetric experiment. The dimensionless chemical kinetic parameter can be also understood as a quantitative measure for the rate of reestablishing the chemical equilibrium (2.29) that is misbalanced by proceeding of the electrode reaction. From the definition of follows that the kinetic affect of the preceding chemical reaction depends on the rate of the chemical reaction and duration of the potential pulses. 

The dimensionless time, t, for Sh to come within 100x% of the steady value indicates the duration of the unsteady state for Pe = 0, Tq.i == 31.8, and — 2.35. Diffusivities in gases are of order 10" times diffusivities in liquids hence, for particles with equal size and equal exposure, transient effects in a stagnant medium are much more significant in liquids. 

Thus, we seek an asymptotic solution of the one-dimensional gasdynamic equations for a given pressure evolution curve, f(t/r), which is characterized by a sufficiently rapid pressure decrease. In a slightly different way, we may formulate the problem thus preserving the form of the dimensionless function f(t/r), we let the pressure duration go to zero and the maximum pressure to infinity, and look for the asymptotic solution—the distribution of the velocity, pressure and other quantities—after a finite time t, at a finite distance x. 

In Fig. 3.14a, the dimensionless limiting current 7j ne(t)/7j ne(tp) (where lp is the total duration of the potential step) at a planar electrode is plotted versus 1 / ft under the Butler-Volmer (solid line) and Marcus-Hush (dashed lines) treatments for a fully irreversible process with k° = 10 4 cm s 1, where the differences between both models are more apparent according to the above discussion. Regarding the BV model, a unique curve is predicted independently of the electrode kinetics with a slope unity and a null intercept. With respect to the MH model, for typical values of the reorganization energy (X = 0.5 — 1 eV, A 20 — 40 [4]), the variation of the limiting current with time compares well with that predicted by Butler-Volmer kinetics. On the other hand, for small X values (A < 20) and short times, differences between the BV and MH results are observed such that the current expected with the MH model is smaller. In addition, a nonlinear dependence of 7 1 e(fp) with 1 / /l i s predicted, and any attempt at linearization would result in poor correlation coefficient and a slope smaller than unity and non-null intercept. 

Duration and intensity of non-stationary processes heat and mass transfer in hydride beds is determined by dimensionless criteria Fourier number - Fo and Biot number - Bi. Their relationship with physical parameters is given by formulas ... 

Computer simulations of the coagulation of equal-sized particles of unit density in air at 298 K and 1 atm were performed based on the algorithm discussed in the previous section. Since the Hamaker constant for most of the aerosol systems is of the order of 10 12 erg, this value was used for the calculation of the interaction potential between the particles. Computations were performed on a CDC 815 computer. In all the computations, the duration of the time step ts for the random force was taken to be equal to one-tenth of the relaxation time for Brownian motion, i.e., ls = O.lf, so that the condition ts -4 f l is satisfactorily fulfilled. The time step f, for the motion of the fictitious particle in the region of the potential well, i.e., region II, was taken to be 0.05. In other words, the values of the dimensionless times 6 and 0 were taken as... 

If the characteristic time is defined independently of the disk radius, and diffusion (12.26) results, the Nernst diffusion layer thickness is dependent only on the number of these time units. So if the characteristic time is r and the maximum duration of the experiment is Tmax (giving Tmax = Tmax/r), then the final diffusion layer thickness is 1JDrmax. Then, in dimensionless distance units (normalisation being division by the disk radius a), this becomes, after multiplying by 6 and noting (12.27),... 

We can estimate the number of kinetic events that are effectively rate-limiting in any given process by quantif3ung the degree to which the distribution of its duration is peaked. The dimensionless ratio of the mean duration squared over the variance, nmin = (i) y/ a quantity related to... 

The formulation in Eq. (173) shows that the dimensionless parameter 8 /DO plays a crucial role in the time dependence of the S concentration profile. This parameter compares the diffusion layer thickness 8 to (D0) ". On the other hand, we have discussed the dependence of 8 on the duration of the experiment. Thus from Eq. (149), 8 % (D0), provided that 8 5conv that is, that 0 < 5conv/h>- Then the diffusion layer extends within a small fraction of the stagnant layer and Eq. (173) reformulates as in Eq. (174),... 

Dimensionless time. Duration of free motion for a random walking" particle... 

T is the dimensionless time, Tq is the dimensionless injection duration, and tR is the limit retention time under linear conditions. Nrea characterizes the contribution of the kinetics of the retention mechanism to the column efficiency. It is equal to the ratio where Hjt, is the contribution of the kinetics of adsorption-... 

The excitation profile of soft pulses is defined by the duration of the pulse, these two factors sharing an inverse proportionality. More precisely, pulse shapes have associated with them a dimensionless bandwidth factor which is the product of the pulse duration. At, and its effective excitation bandwidth, Af, for a correctly calibrated pulse. This is fixed for any given pulse envelope, and... 

The behavior of this system depends on the magnitudes of both first-order rate constants, k k (s ), and the equilibrium constant, K, It is convenient to describe the reactions in terms of dimensionless parameters related to the rate constants of the reactions (or the characteristic reaction lifetimes) and the duration of the experiment. For the Cj-Er case in the context of a potential step experiment of duration t, these are conveniently expressed by K and A = (/ f + k)t. For different methods and mechanisms, A is defined in particular ways, as given in Table 12.3.1. 

To what time should we assign Z k)l Since our current calculation really involves dividing the integral charge passed during an iteration by the duration of the iteration, it is appropriate to assign the current to the midpoint, rather than to the end, of the iteration. Thus, we say that the dimensionless current Z k) flowed at tlty = (k — 0.5)/6. 

---