Concentrations versus time

At higher flow rates the dispersion of the tracer and the lower limit for time resolution of the concentration versus time distribution limits the accuracy to better than dt2%. 

Figure 3. Tracers concentration versus time at input and output of a heat exchanger.

Fig. 12-3. Concentration versus time profiles of propene, NO, NO, -NO, and Oj from smog chamber irradiation /Cj = 0.16 min. Source Akimoto, H., Sakamaki, F., Hoshino, M, Inoue, G., and Oduda, M., Emiiron. Set. Technol. 13, 53-58 (1979).

Figure 10. Plot of interface height and solids concentration versus time.

For an autocatalytic reaction some product B must be present if the reaction is to proceed. Starting with a small concentration of B, the rate rises as B is formed, and when A is used up, the rate must drop to zero. A plot of concentration versus time gives a straight line through the origin as shown in Figure 3-12. 

This method is based on differentiating the concentration versus time data in order to obtain the actual rate of reaction to be tested. All the terms in the equation including the derivative (dCj/dt) are determined, and the goodness of fit are tested with the experimental... 

From the experimental data of concentration versus time, determine the reaction rate at various times. 

Figure 5-7. Concentrations versus time of A, B, and C in a series reaction...

Simulation of a semibatach reactor concentrations versus time of the type A + B —kl—> C... 

A CONTINUOUS FLOW STIRRED TANK REACTOR OUTLET CONCENTRATION VERSUS TIME... 

Simulritiun of a nonisothermal batch reactor Concentration versus time... 

Figure 6-5a Concentration versus time in an adiabatic batch reactor.

A recommended method of characterizing the RTD in flow systems is by using their moments. These are known as the mean, variance, and skewness. The mean value or the centroid of distribution for a concentration versus time curve is... 

Fig. 9-3. Concentration versus time in the extract for SMBS at cyclic steady state. — more retained component — less retained component.

Determination of the instantaneous rate at a particular concentration. To determine the rate of reaction, plot concentration versus time and take the tangent to the curve at the desired point. For the reaction N20s(g) — 2N02(g) + 02(g), it appears that the reaction rate at [N205] = 0.080 M is 0.028 mol/L - min. 

To find the rate of decomposition of N2Os, it is convenient to use Figure 11.2, which is a magnified version of a portion of Figure 11.1. If a tangent is drawn to the curve of concentration versus time, its slope at that point must equal A[N205]/Af. But because the reaction rate is —A[N205]/At, it follows that... 

Fig. E.10.1. Plot of natural logarithm of the cell dry weight concentration versus time in the batch bioreactor...

They varied only the values of the adsorption and desorption rate constants of the reaction intermediate B, and by using the simplest Langmuir kinetics, they calculated time-concentration curves of compounds A, B, and C shown in Fig. 5. Also from this example, which does not consider any step as clearly rate determining, it is evident how very different concentration versus time plots can be obtained for the same sequence of surface reactions if adsorption and desorption of the intermediate B proceed by different rates, which are, however, comparable with the rate of surface reactions. In particular, the curves in the first and second columns of Fig. 5 simulate the parallel formation of substances B and C, at least... 

Many real reaction systems are not amenable to normal mathematical treatments that give algebraic expressions for concentration versus time, but by no means is the situation hopeless. Such systems need not be avoided. The numerical methods presented... 

Figure 7.2. A logarithmic plot of the exponential decay of Figure 7.1 showing the linear dependence of In(concentration) versus time...

Analysis of most (perhaps 65%) pharmacokinetic data from clinical trials starts and stops with noncompartmental analysis (NCA). NCA usually includes calculating the area under the curve (AUC) of concentration versus time, or under the first-moment curve (AUMC, from a graph of concentration multiplied by time versus time). Calculation of AUC and AUMC facilitates simple calculations for some standard pharmacokinetic parameters and collapses measurements made at several sampling times into a single number representing exposure. The approach makes few assumptions, has few parameters, and allows fairly rigorous statistical description of exposure and how it is affected by dose. An exposure response model may be created. With respect to descriptive dimensions these dose-exposure and exposure-response models... 

In the total plasma response approach, the bioavailability of a compound is determined by measuring its plasma concentration at different times (up to weeks) after single or long-term ingestion of the compound from supplements or food sources. Generally, a plasma concentration-versus-time plot is generated, from which is determined the area-under-curve (AUC) value used as an indicator of the absorption of the componnd. Here, the term relative bioavailability is more appropriate since AUC valnes of two or more treatments are usually compared. This is in contrast to absolnte bioavailability for which the AUC value of the orally administered componnd is compared to that obtained with intravenous administration taken as a reference (100% absorption). 

Figure 5.3-14. Concentration versus time in a batch reactor.

In integral analysis concentration-versus-time (or equivalently concentration-versus-distance from the inlet of the integral flow reactor) data are known. Kinetic expressions to be determined are incorporated into the differential material balance equations ... 

Referring to Fig. 1.4, the solution begins with the initial concentration conditions Aq, Bq, Cq and Dq, defined at time t = 0. Knowing the magnitudes of the kinetic rate constants k], k2, k3 and k4, thus enables the initial rates of change dCA/dt, dCfi/dt, dCc/dt and dCo/dt, to be determined. Extrapolating these rates over a short period of time At, from the initial conditions, Aq, Bq, Cq and Do, enables new values for A, B, C and D to be estimated at the new time, t = t -I- At. If the incremental time step At is sufficiently small, it is assumed that the error in the new estimated values of the concentration. A, B, C and D, will also be small. This procedure is then repeated for further small increments of time until the entire concentration versus time curves have been determined. 

Figure 5.2. Profiles of concentrations versus time as obtained from the rate constants in the program.

Figure 5.32 A change in program TANKD was made at TIME = 5, such that the dimensionless parameter PAR varied from 9 to 5, and the results were plotted as concentration versus time. The dimensionless plot looks quite different.

Graphing Data On a sheet of graph paper, make a graph of temperature versus time, using the data from Part A. Then make a graph of concentration versus time, using the data from Part B. Were your hypotheses supported Explain. 

The values of the rate constants are estimated by fitting equations 1.4a and 1.4b to the concentration versus time data. It should be noted that there are kinetic models that are more complex and integration of the rate equations can only be done numerically. We shall see such models in Chapter 6. An example is given next. Consider the gas phase reaction of NO with 02 (Bellman et al. 1967) ... 

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