Concentration evolution

With given contaminant source and sink schedules and outdoor concentrations, concentration evolutions over time can be determined for the individual zones on the basis of the calculated airflow rate values per time step. Further postprocessing allows the determination of accumulated values such as air change rate or concentration histograms (see the later example) or inhaled dose values. 

Fig. 2 Individual effect on the normalized concentration of selected parameters in the effluents for the technologies used in this study. Red line indicates the decreasing concentration evolution. Circles remark the most efficient technology in each target...

Consequently, Bakker (1996) described the concentration evolution of a single layer subjected to the vorticity field of a single CSV by means of a onedimensional differential equation where both the nondimensional time and the nondimensional spatial coordinate contain the exponential shrinking rate. In this respect, the CSV approach differs from the various Bourne models in which the successive generation of several multiple-layer stacks is required and vortex age is a crucial element. 

Figure 1-1 Comparison of (a) reactant and (b) product concentration evolution for zeroth-order, first-order, and the first type of second-order reactions. The horizontal axis is kt, and the vertical axis is normalized reactant concentration in (a) and normalized product concentration multiplied by a parameter q so that the comparison can be more clearly seen in (b).

Similarly, for Reaction 1-1, the concentration evolution with time can be written as... 

A good example of a first-order (pseudo-first-order) chemical reaction is the hydration of CO2 to form carbonic acid. Reaction l-7f, C02(aq) + H20(aq) H2C03(aq). Because this is a reversible reaction, the concentration evolution is considered in Chapter 2. 

Figure 1-1 compares the concentration evolution with time for zeroth-, first-, and the first type of second-order reactions. Table 1-2 lists the solutions for concentration evolution of most elementary reactions. 

Table 1-2 Concentration evolution and half-life for elementary reactions... 

Except for radioactive decays, other reaction rate coefficients depend on temperature. Hence, for nonisothermal reaction with temperature history of T(t), the reaction rate coefficient is a function of time k(T(t)) = k(t). The concentration evolution as a function of time would differ from that of isothermal reactions. For unidirectional elementary reactions, it is not difficult to find how the concentration would evolve with time as long as the temperature history and hence the function of k(t) is known. To illustrate the method of treatment, use Reaction 2A C as an example. The reaction rate law is (Equation 1-51)... 

In general, for unidirectional elementary reactions, it is easy to handle non-isothermal reaction kinetics. The solutions listed in Table 1-2 for the concentration evolution of elementary reactions can be readily extended to nonisothermal reactions by replacing kt with a = j k df. The concepts of half-life and mean reaction time are not useful anymore for nonisothermal reactions. 

Figure 1-5 Determination of the order of hypothetical reactions with respect to species A. (a) The initial reaction rate method is used. The initial rate versus the initial concentration of A is plotted on a log-log diagram. The slope 2 is the order of the reaction with respect to A. The intercept is related to k. (b) The concentration evolution method is used. Because the exponential function (dashed curve) does not fit the data (points) well, the order is not 1. The solution for the second-order reaction equation (solid curve) fits the data well. Hence, the order of the reaction is 2.

Some examples of concentration evolution (Crank, 1975) are shown and explained below. These are all often-encountered problems in diffusion. The purpose of the examples and the qualitative discussion is to help readers develop familiarity and gain experience in treating diffusion in a qualitative fashion. 

If a system experienced a complicated thermal history, the rate coefficient would depend on time and the solution to the rate equation would be more complicated. For the special case of reaction kinetics described by one single rate coefficient, the concentration evolution with time can be solved relatively easily. 

Concentration evolution for first-order reversible reactions... 

The concentration evolution curves of Figures 2-la and 2-lb may be used to estimate the half-life or mean reaction time. When Figures 2-la and 2-lb are compared, the mean reaction time is found to differ by four orders of magnitude Hence, for second-order reactions, the timescale to reach equilibrium in general depends on the initial conditions. This is in contrast to the case of first-order reactions, in which the timescale to reach equilibrium is independent of the initial conditions. 

The method to find the mean time to reach equilibrium without calculating the full concentration evolution curve is as follows. By comparing the definition of mean reaction time (Equation 1-60) d /dt=—( - ao)hr and Equation 2-17, the following is obtained ... 

Even though the above example is for a specific reaction, the procedures for solving the differential equation are general for all other kinds of second-order reversible reactions. Table 2-1 lists solutions for concentration evolution of all... 

Using the solution for concentration evolution for reaction A + B C in Table 2-1, we obtain the concentration evolution for reaction H+ + OH H2O to be... 

In summary, because the forward reaction rate constant is similar to the back reaction rate constant, the concentration evolution of a second-order isotopic exchange reaction often reduces to that of a first-order reaction (exponential evolution) but the rate "constants" and mean reaction time for the reduced reaction depend on total concentrations. 

The above analytic solution has two applications (i) to investigate the concentration evolution under the special conditions given above, and (ii) to check the accuracy of numerical programs. One application is given in Example 2-1. 

The above condition of equal activity of all radioactive nuclides in a decay chain (except for branch decays) is known as secular equilibrium. More detailed solutions for the concentration evolution of intermediate species can be found in Box 2-6. 

The concentration evolution with time is shown in Figure l-7a. This solution is symmetric with respect to a = 0 and approaches zero as a approaches — cxD or cxD. The concentration at a = 0 is proportional to At t=0, the... 

Figure 5-26 The concentration evolution for a "spherical diffusion couple." The radius of the initial core is a. The initial concentration is Cl = 0.2 in the core and C2 = 0.4 in the mantle. Note that the position for the midconcentration between the two halves moves toward smaller radius, which is due to the much larger volume per unit thickness in the outer shell. From Zhang and Chen (2007).

The concentration profiles stays as a symmetric cosine function, but the amplitude decreases as Ciexp(-7i Df/a ). When Dt/a = 0.4666, the amplitude decreases 1% of the initial amplitude. The concentration evolution is shown in Figure 5-28. 

Figure A3-2-4 Concentration evolution corresponding to solution 3.2.4el and 3.2.4e3. The value for each curve is Dt/L. At Dt/L = 0.4666, the amplitude is reduced hy two orders of magnitude.

Figure 1-1 Concentration evolution for some elementary reactions 19...

---