Two-sink model

In the simplest case a compound is emitted from a sample into the chamber and removed with exhaust air. This so-called dilution-model has been introduced by Dunn and Tichenor (1988) for a source with constant and exponentially decaying emission rate, respectively. A more sophisticated model requires the compartments source, chamber, exhaust and sink. The rate constants k, describe the exchange of mass between dilFerent sections. The parameter fc2 represents the air exchange rate n. Figure 5.5 shows a scheme of the two-sink model after De Bortoli et al. (1996) as an extension of the full-model ansatz after Dunn and Tichenor (1988) with fci 5 Z 0 und ks = 0. Equations 5.5 and 5.6 are solutions of the simple dilution-model with k3 s = 0 for constant emission and exponentially decaying emission, respectively. 

In our case study, the experimental observations (i.e. concentration versus time data) were used as input to the conceptualization phase of a mathematical model with two sink compartments (the so-called two-sink model). Following the above discussion, this model (schematically represented in Fig. 2.3-1) can be eonsidered as a hybrid-empirical model. Conceptually, this model describes the test chamber kinetics of a VOC for the three types of experiments which have been carried out. The adsorption-desorption kinetics is described by the rate constants k, k, k. Given that the conceptualization... 

Figure 13-1. Concentration of n-decane versus time in the test chamber and curve fitting with the one-sink and two-sink models (experiment No. 1, dynamic mode).

Dynamic adsorption experiments with -decane in the empty 0.45-m glass chamber and in the chamber with a carpet sample have been used to test the performance of the two-sink model and compare it with the one sink model. The two-sink model provided a much better regression of data points than the previous one sink model. Figures... 

In our case study, the two-sink model fitted most experimental data very efficiently. However, it is not supposed to provide a complete physical description of the real adsorption process(es). Increasing the number of regression parameters in any equation improves its adherence to the data. However, the model is supposed to be a better approximation to the more complex reality. This reality may include adsorption sites with a distribution of adsorption enthalpies instead of simply two distinct values, and may also include diffusion processes within the sorbing material. A further development of the two-sink model could take into consideration the transfer of molecules from the faster (more superficial) to the slower (deeper) sink sites. The model in its present form can be used to estimate four unknown parameters, i.e. the adsorption and desorption rate constants for each of the two sinks. 

Desorption experiments are crucial for the validation of adsorption models and in particular for confirming the estimates of the adsoihed masses. Two desorption experiments have been carried out in order to test the reliability of the two-sink model for estimating the masses in the sinks. One of the desorption tests followed dynamic adsorption of n-dodecane, the other the static adsorption of n-decane. The results showed that (1) the masses deposited into the sinks are released very slowly (2) the two-sink model estimates the masses in the sinks reasonably well therefore, this model may be considered a satisfactory tool for estimating chamber (or test material) sinks. It coherently confirmed the existence of two sinks with different saturation rates, also giving reasonable estimates of the relative parameters. 

Overall, the two-sink model provided a reasonable description of the mass balance of the adsorption-desorption experiments performed and reproducible parameter estimates. However, the reported results have given evidence that the reproducibility of parameter estimates for adsorption of a compound on chamber walls may critically depend on residues of the compound in the sinks from earlier experiments. It highlighted the very long time needed to attain equilibrium between the vapor phase and the adsorbed phase and vice versa. This effect was particularly pronounced for adsorption on chamber walls. The chamber sink, compared with the adsorption on three test materials, showed that, in general, the former will introduce only a minor or negligible bias in the measurement of the latter, provided that a sufficiently large surface of the test material is used. 

However, the two-sink model as well as other existing adsorption (sink) models do not seem to be able to describe the strong asymmetry between the adsorption/desorption of VOCs on/from indoor surface materials (the desorption process is much slower than the adsorption process). Diffusion combined with internal adsorption is assumed to be capable of explaining the observed asymmetry. Diffusion mechanisms have been considered to play a role in interactions of VOCs with indoor sinks. Dunn and Chen (1993) proposed and tested three unified, diffusion-limited mathematical models to account for such interactions. The phrase unified relates to the ability of the model to predict both the ad/absorption and desorption phases. This is a very important aspect of modeling test chamber kinetics because in actual applications of chamber studies to indoor air quality (lAQ), we will never be able to predict when we will be in an accumulation or decay phase, so that the same model must apply to both. Development of such models is underway by different research groups. An excellent reference, in which the theoretical bases of most of the recently developed sorption models are reviewed, is the paper by Axley and Lorenzetti (1993). The authors proposed four generic families of models formulated as mass transport modules that can be combined with existing lAQ models. These models include processes such as equilibrium adsorption, boundary layer diffusion, porous adsorbent diffusion transport, and conveetion-diffusion transport. In their paper, the authors present applications of these models and propose criteria for selection of models that are based on the boundary layer/conduction heat transfer problem. 

Let us define a two-box model for a steady-stafe ocean as shown in Fig. 10-22. The two well mixed reservoirs correspond to the surface ocean and deep oceans. We assume that rivers are the only source and sediments are the only sink. Elements are also removed from the surface box by biogenic particles (B). We also assume there is mixing between the two boxes that can be expressed as a velocity Vmix = 2 m/yr and that rivers input water to the surface box at a rate of Vnv = 0.1 m/yr. The resulting ratio of F mix/V riv is 20. 

Six two-component models were tested under sink conditions (models 5.1-10.1 in Table 7.3), employing three negatively charged lipids (dodecylcarboxylic acid, phosphatidic acid, and phosphatidylglycerol). These models were also tested in the absence of the sink condition (models 5.0-10.0 in Table 7.3). 

Because of the inadequacies of the aforementioned models, a number of papers in the 1950s and 1960s developed alternative mathematical descriptions of fluidized beds that explicitly divided the reactor contents into two phases, a bubble phase and an emulsion or dense phase. The bubble or lean phase is presumed to be essentially free of solids so that little, if any, reaction occurs in this portion of the bed. Reaction takes place within the dense phase, where virtually all of the solid catalyst particles are found. This phase may also be referred to as a particulate phase, an interstitial phase, or an emulsion phase by various authors. Figure 12.19 is a schematic representation of two phase models of fluidized beds. Some models also define a cloud phase as the region of space surrounding the bubble that acts as a source and a sink for gas exchange with the bubble. 

He solved this equation, using three different boundary conditions, two of which are also used in the field of particle deposition on collectors the Perfect Sink (SINK) model, the Surface Force Boundary Layer Approximation (SFBLA) and the Electrode-Ion-Particle-Electron Transfer (EIPET) model. 

Hence, the one-box and two-box models yield the same result. There is a simple reason for that. Since the only removal processes of PCE act at the lake surface, at steady-state the surface concentration in both models (C°°for the one-box model, ClE for the two-box model) must attain the same value to compensate for the input /, tot. Furthermore, since the hypolimnion has neither source nor sink, the net exchange flux across the thermocline must be zero, and this requires C(E= C,H. 

In Section 23.1, this procedure will be applied to just one completely mixed water body. This control volume may represent the lake as a whole or some part of it (e.g., the mixed surface layer). Section 23.2 deals with the dynamics of particles in lakes and their influence on the behavior of organic chemicals. Particles to which chemicals are sorbed may be suspended in the water column and eventually settle to the lake bottom. In addition, particles already lying at the sediment-water interface may act as source or sink for the dissolved chemical. In Section 23.3, two-box models of lakes are discussed, particularly a model consisting of the water body as one box and the sediment bed as the other. Finally, in Section 23.4, one-dimensional vertical models of lakes and oceans are discussed. 

Two-zone model, M = 1, N = 1 A single volume zone completely surrounded by a single sink surface zone. 

Adsorption of vapors on test chamber walls has been previously described by means of models including two or three rate constants for adsorption/desorption processes in the ease of dynamic experiments (Dunn et al., 1988 Colombo et al., 1993) and with three adsorption/desorption constants in the case of static experiments (Colombo et al., 1993). Two rate constants describe a reversible sink whereas three rate constants describe a reversible and an irreversible (i.e. leak type) sink. However, these models did not adequately describe the sorption process(es), especially in the case of long-term tests, as resulted from two observations (Colombo et al., 1993) (a) the model with three sorption rate constants (reversible + irreversible sink) provided a better description of the experimental data than the one-sink model and (b) desorption experiments following adsorption gave strong indications that the irreversible sink was in fact slowly rever-... 

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