Microscopic mass balance models

There are two general types of aerosol source apportionment methods dispersion models and receptor models. Receptor models are divided into microscopic methods and chemical methods. Chemical mass balance, principal component factor analysis, target transformation factor analysis, etc. are all based on the same mathematical model and simply represent different approaches to solution of the fundamental receptor model equation. All require conservation of mass, as well as source composition information for qualitative analysis and a mass balance for a quantitative analysis. Each interpretive approach to the receptor model yields unique information useful in establishing the credibility of a study s final results. Source apportionment sutdies using the receptor model should include interpretation of the chemical data set by both multivariate methods. 

Three generic types of receptor model have been identified, chemical mass balance, multivariate, and microscopical identification. Each one has certain requirements for input data to provide a specified output. An approach which combines receptor and source models, source/ receptor model hybridization, has also been proposed, but it needs further study. 

Receptor models presently in use can be classified into one of four categories chemical mass balance, multivariate, microscopic, and source/receptor hybrids. Each classification will be treated individually, though it will become apparent that they are closely related. 

Microscopic Identification Models. Many different optical and chemical properties of single aerosol particles can be measured by microscopic identification and classification in order to distinguish particles originating in one source type from those originating in another. The microscopic analysis receptor model takes the form of the chemical mass balance equations presented in Equation 1. 

The microscopic receptor model can include many more aerosol properties than have been used in the chemical mass balance and multivariate models. The data inputs required for this model are the ambient properties measurements and the source properties measurements. To estimate the confidence Interval of the calculated source contributions the uncertainties of the source and receptor measurements are also required. Microscopists generally agree that a list of likely source contributors, their location with respect to the receptor, and windflow during sampling are helpful in confirming their source assignments. 

The micro-inventory should be a prerequisite for aerosol mass balance or microscopic model applications to identify the most likely sources for sampling, analysis and inclusion in the balance. 

The solute is disseminated in a solid matrix in the most of the supercritical extractions of natural products. If the interactions between solute and solid matrix are not important, the mass transfer models can be developed from the equations of microscope balances to a volume element of the extractor. If the mass transfer resistance is in its solid phases, the mathematical models must consider the solute transport within the solid particles or the surface phenomena. 

We shall be interested in a broad, macroscopic, view of reactors—one where elementary mass balances are applied over entire process units. That is, we are not interested in modeling the system in detail, and attainable region (AR) theory does not demand the use of microscopic transport equations for instance. 

Microscopic modeling considers a small but statistically representative volume element of the absorber (or reactor), that is, a "point" in the equipment. Recently, Thoenes (8) grouped such considerations as "volume element modeling". It is necessary to make energy and component mass balances in the reactive liquid-phase (it is assumed that no reaction takes place in the gas phase). Fortunately for most systems, the isothermal approximation is often justified. Thus, the components mass balance yields ... 

A single cylindrical pore of length L and radius of r (=d) located in a microscopic section of the catalyst particle is generally used for modeling the diffusion-reaction process (Figure 2.3). The steady-state component mass balance for a control volume extending over the cross section of the pore includes diffusion of reactant into and out of the control volume as well as reaction on the inner wall surface. The simple case taken as an example is that of an isothermal, irreversihle first-order reaction ... 

A recent model (1988) was published by J. Guttierrez Gonzalez et al. According to the authors, although the liquid fiow is laminar, due to the high Schmidt number in the liquid phase, eddy mass transfer can be significant and eddy diffusion cannot be disregarded with respect to molecular diffusion. Eddy thermal diffusion in the liquid phase is much smaller than thermal diffusion, so that it is not introduced in the microscopic heat balance. The Spanish authors needed to validate their results on practical data. The pressure drop over the reactor, heat- and mass transfer were fitted. 

Second, the balance is taken over an incremental space element, Ac, Ar, or AV. The mass balance equation is then divided by these quantities and the increments allowed to go to zero. This reduces e difference quotients to derivatives and the mass balance now applies to an infinitesimal point in space. We speak in this case of a "difference" or "differential" balance, or alternatively of a "microscopic" or "shell" balance. Such balances arise whenever a variable such as concentration undergoes changes in space. They occur in all systems that fall in the category of the device we termed a 1-D pipe (Figure 2.1b). When the system does not vary with time, i.e., is at steady state, we obtain an ODE. When variations with time do occur, the result is a partial differential equation (PDE) because we are now dealing with two independent variables. Finally, if we discard the simple 1-D pipe for a multidimensional model, the result is again a PDE. 

In particular cases simplified reactor models can be obtained neglecting the insignificant terms in the governing microscopic equations (without averaging in space) [9]. For axisymmetrical tubular reactors, the species mass and heat balances are written in cylindrical coordinates. Himelblau and Bischoff [9] give a list of simplified models that might be used to describe tubular reactors with steady-state turbulent flow. A representative model, with radially variable velocity profile, and axial- and radial dispersion coefficients, is given below ... 

---