Exponential growth/decay

These are very similar to the equations derived in chapter 3 for the decay or growth of small perturbations in well-stirred systems. Again we can expect exponential growth or decay, depending on the relative magnitudes of the four coefficients in these equations which, in turn, depend on y, fi, k, and / . 

To describe the long-term trend we use an exponential growth or decay curve and characterize it by the rate constant a [T1] ... 

From Eq. 21-17 you can conclude that in order to keep the difference between C(t) and CJt) below 10%, the absolute value of the exponential growth (or decay) rate a must obey the condition ... 

In order to prevent vaporization effects, a low fluence approach was chosen. To consider the growth of the particles, the exponential signal decay time was determined. For this, the measurements were carried out... 

These two measures of average size show an exponential growth with time that is consistent with Eqs. 6.1 and 6.54. There is a corresponding exponential decay of M0(t), the total number of floccules at time t. (Evidently kF < 2Kd in order that Eqs. 6.59 and 6.63 describe a slower flocculation process than Eqs. 6.18, 6.20b, and 6.23.)... 

Example 5.1.2 provides a clue about how to proceed. Recall that the x and y axes played a crucial geometric role. They determined the direction of the trajectories as z —> 1. They also contained special straight-line trajectories a trajectory starting on one of the coordinate axes stayed on that axis forever, and exhibited simple exponential growth or decay along it. 

In these equations, N is the predator and R is the prey species. It is assumed that the dynamics of the prey and predator populations in the absence of the other species is given by exponential growth R = aR or decay N = —bN. Predation is taken into account in the form of a mass action RN term with rates ki and k2- Model (15.1) became famous as the Lotka-Volterra model [11] (some of the work of Volterra was preceded by that of Alfred J. Lotka in a chemical context [12]). 

Heat Transfer on the Walls With Exponentially Varying Heat Flux. Exponentially varying wall heat flux is delineated by the boundary condition and represented by q" = q" exp(mx ), where the exponent m can have both positive and negative values corresponding to the exponential growth or decay of the wall heat flux. 

Exponential growth and exponential decay, sketched in Fig. 3.6, are observed in a multitude of natural processes. For example, the population of a colony of bacteria, given unlimited nutrition, will grow exponentially in time ... 

The result for radioactive decay is easily adapted to describe organic growth, the rate at which the population of a bacterial culture will increase given unlimited nutrition. We simply change - A to - - A in Eq. (8.11), which leads to exponential growth ... 

Such a form allows for very versatile behavior exponential growth (a> > 0), decaying (co < 0) or stajnng constant (o) = 0), as well as for periodic behavior (Rery = 0, Im 0, imey 7 0), or decay (Rety < 0, imey 0). 

The time constant is another characteristic time that is sometimes used to describe processes that follow exponential growth or decay as described by Eq. (3.34). The time constant is defined as = k (sec). If t = the reaction has proceeded to 63.2% of completion. 

While Figs. 16 and 20 simply describe the transformation of the system from the initial state to the final oscillating state, more details of this process have been able to be expressed in terms of the front wave and the stimulated wave, as a result of the present experimental study. In particular, it is significant to note that these waves are due to the nonlinear processes with the exponential growth and decay and that their behaviours are not necessarily same for the common parameters such as the potential between electrodes. For example, the different behaviours of the waves for the electrode potential in Figs. 21 and 22 suggest that the processes controlling these waves are not necessarily same, as discussed previously. In spite of the fact, these waves are finally combined to be a united... 

An exponential relationship takes the form y = e or y = eA It is characteristic of exponential growth or decay. The graph in Figure 11.27 shows exponential growth. 

Another model that proposes the incorporation of primary and tertiary creep strain rates is the A- 2 model. This model proposes that the primary creep strain rate be represented as an exponential decay function of increasing creep strain, and the tertiary creep strain rate be represented as an exponential growth function of increasing creep strain. However, this model is relatively new and has not been validated with independent experimental reliability data across the industry. 

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