Inert Tracer Model

Instead of assuming 5ct, many authors employed the inert tracer technique to measure the time-average concentration C of an inert tracer in a simulator to find 

For the inert tracer process without mass transfer  

Obviously, by comparing the foregoing two equations, A and Atracer is not equal the difference between them is depending on the value of the source term S , which represents the rate of species mass to be transferred in the process. 

In view of the drawbacks of foregoing models in applying Sct or using experimental correlation obtained by the inert tracer technique, some dependable models have been recently developed to overcome such insufficiency as shown in subsequent sections. 

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The time that a molecule spends in a reactive system will affect its probability of reacting and the measurement, interpretation, and modeling of residence time distributions are important aspects of chemical reaction engineering. Part of the inspiration for residence time theory came from the black box analysis techniques used by electrical engineers to study circuits. These are stimulus-response or input-output methods where a system is disturbed and its response to the disturbance is measured. The measured response, when properly interpreted, is used to predict the response of the system to other inputs. For residence time measurements, an inert tracer is injected at the inlet to the reactor, and the tracer concentration is measured at the outlet. The injection is carried out in a standardized way to allow easy interpretation of the results, which can then be used to make predictions. Predictions include the dynamic response of the system to arbitrary tracer inputs. More important, however, are the predictions of the steady-state yield of reactions in continuous-flow systems. All this can be done without opening the black box. 

Transient experiments with inert tracers are used to determine residence time distributions. In real systems, they will be actual experiments. In theoretical studies, the experiments are mathematical and are applied to a d5mamic model of the system. 

Solution The dynamic model governing the flow of an inert tracer through an unsteady PFR is Equation (14.13) with =0 ... 

C(t, z) capacity model Concentration of inert tracer in an unsteady tubular Exam. 15.4... 

Concentration of inert tracer in the side tank of the Exam. 15.7 capacity model... 

Mass transport in laminar flow is dominated by diffusion and by the laminar velocity profile. This combined effect is known as dispersion and the underlying model for the theoretical derivation of a kinetic study had to be derived from the dispersion model, which Taylor [91] and Aris [92] developed. Taylor concluded that in laminar flow the speed of an inert tracer impulse initially given to a channel will have the same speed as the steady laminar carrier gas flow originally prevailing in this channel. 

The state of mixing in a given reactor can be evaluated by RTD experiments by means of inert tracers, by temperature measurements, by flow visualization and, finally, by studying in the reactor under consideration the kinetics of an otherwise well-known reaction (because its mechanism has been carefully elucidated from experiments carried out in an ideal reactor, the batch reactor being generally chosen as a reference for this purpose). From these experimental results, a reactor model may be deduced. Very often, in the laboratory but also even in industrial practice, the real reactor is not far from ideal or can be modelled successfully by simple combinations of ideal reactors this last approach is of frequent use in chemical reaction engineering. But... 

These interesting devices consist of a tube or duct within which static elements are installed to promote cross-channel flow. See Figure 8.5 and Section 8.7.2. Static mixers are quite effective in promoting radial mixing in laminar flow, but their geometry is too complex to allow solution of the convective diffusion equation on a routine basis. A review article by Thakur et al. (2003) provides some empirical correlations. The lack of published data prevents a priori designs that utilize static mixers, but the axial dispersion model is a reasonable way to correlate pilot plant data. Chapter 15 shows how Pe can be measured using inert tracers. 

The atmospheric rig is also used for fuel placement studies. Initially, these will be conducted using an inert tracer — Helium — to label a surrogate fuel flow, with mass spectrometry as the measurement method. The local fuel distribution can be applied to the numerical model to assess the effects of distortion in fuel-air... 

The normalised response curves in Fig. 3B were modelled by considering a combination of CSTR pools in parallel or in series, according to the mathematical formalism developed by Soong et al. [20] or Happel et al. [21] (scheme 1). The inlet function of the system was first obtained by modelling the response of argon as an inert tracer (curve a in... 

Figure 7.9 Models of capillary mixing within a tissue region, (a) Ideal stirred-tank model and response of the stirred tank to a step change in concentration of an inert tracer at the inlet, (b) Ideal plug-flow model and its response to a step change in concentration of tracer at the inlet.

Equations (11) and (12) enable the generation of the total isotopic transient responses of a product species given (a) the transient response that characterises hypothesized catalyst-surface behaviour and (b) an inert-tracer transient response that characterises the gas-phase behaviour of the reactor system. Use of the linear-convolution relationships has been suggested as an iterative means to verify a model of the catalyst surface reaction pathway and kinetics. I This is attractive since the direct determination of the catalyst-surface transient response is especially problematic for non-ideal PFRs, since a method of complete gas-phase behaviour correction to obtain the catalyst-surface transient response is presently unavailable for such reactor systems.1 1 Unfortunately, there are also no corresponding analytical relationships to Eqs. (11) and (12) which permit explicit determination of the catalyst-surface transient response from the measured isotopic and inert-tracer transient responses, and hence, a model has to be assumed and tested. The better the model of the surface reaction pathway, the better the fit of the generated transient to the measured transient. 

The question remains, however, which flow model should be applied to the description of the gas phase. To solve this dilemma, pulse experiments with argon as an inert tracer were conducted. For the resulting response curve, see figure below. 

Inert tracers are mainly used to characterize nonideal model parameters for stirred tanks. However, attempts were made to use the fluctuation of the tracer concentration response, measured optically or by electroconductivity, to evaluate a micromixing parameter of the model such as the degree of segregation. Leitman... 

After assuming a flow model based both on the physical structure of the reactor and the characteristic features of the experimental RTD curve, mass balance equations are written for the dispersion of an inert tracer among the various zones involved in the model. These equations are linear differential equations (ODE or PDE) owing to the linear character of mixing processes. By solving the equations in the Laplace domain, the theoretical transfer function G (s,p ) is obtained, which is nothing but the Laplace transform of the - theoretical RTD E (t,pj ) where pj are the parameters of the model... 

Transient experiments with inert tracers are used to determine residence time distributions. In real systems, they will be actual experiments. In theoretical studies, the experiments are mathematical and are applied to a dynamic model of the system. Table 1-1 lists the types of RTDs that can be measured using tracer experiments. The simplest case is a negative step change. Suppose that an inert tracer has been fed to the system for an extended period, giving Ci = Cout = Q for t < 0. At time t = 0, the tracer supply is suddenly stopped so that Cm = 0 for t > 0. Then the tracer concentration at the reactor outlet will decrease with time, eventually approaching zero as the tracer is washed out of the system. This response to a negative step change defines the washout function, W(t). The responses to other standard inputs are shown in Table 1-1. Relationships between the various functions are shown in Table 1-2. 

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