Mass Balance Model with Accumulation

Recall Chapter 3, Fig. 3-1 in Section 3.1. If N is again the set of rmits (without the environment node), let us consider the subset 

The matrix C can be also row-partitioned by N = (N -NJ u Nj,. In the subset of rows n Ny-N, we have zeros in the columns j e Jj, (no change in accumulation admitted) in the subset of rows n e, in each column j J, we have just one nonnull element (-1) corresponding to the fictitious stream node n — environment. 

The mass balancing is carried out at discrete times t. (k= 1,2, ). We shall now suppose that instead of instantaneous flowrates, certain integrated mass flowrates 

Possibly unmeasured are only certain components of the subvectors 1%, say Wji for / e J = JfJf. TTien the subgraph G° of G restricted to unmeasured streams is of reduced incidence matrix (say) B. We thus partition 

Observe that Cf is reduced incidence matrix of the graph (say) Gf [N, Jf ] representing the system of given nodes and of material streams, disregarding the accumulation. It is quite natural to assume that also 

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Other Applications of the Multiple-Core Approach. The bulk of this chapter has dealt with the specific application of multiple-core methodology to questions of atmospheric Hg deposition. Whole-basin Hg accumulation rates for seven lakes, calculated from multiple sediment cores, were used in a simple mass-balance model to estimate atmospheric fluxes and Hg transport from catchment soils. This approach can be used to answer other limnological questions, and the model is not restricted to Hg or atmospheric deposition. 

This element of the mass balance model refers to the efficiency with which the generated hydrocarbons migrate to and are trapped within the reservoir. Components of this element include the trapping mechanisms identified from the geological model as well as characterization of the reservoir seals. As discussed previously, structure often plays little or no role in the accumulation of hydrocarbons in a BCGS, so evaluation of the seals is especially critical to an accurate evaluation. Key components and data requirements include ... 

The second type of models includes a representation of the seasonal cycle, and has been used to investigate the orbital theory of Milankovitch. In this case, the seasonal variations in orbital insolation are resolved. However, as for the first type of models, past changes in ice cover are assumed to follow the simulated variations in the extent of perennial snow. This approach assumes that ice cover and the powerful ice albedo feedback are governed only by temperature, as the extent of snow in these models is fixed to the latitude with a temperature of 0°C. In reality, the growth and decay of land-based ice sheets are governed by the balance of accumulation and ablation. Therefore, when investigating changes in ice cover, it is necessary to include an appropriate representation of the dynamics and mass balance of ice sheets in the model. 

When gas phase adsorption takes place in a large column, heat generated due to adsorption cannot be removed from the bed wall and accumulated in the bed because of poor beat transfer characteristics in packed beds of particles. A typical model of this situations is an adiabatic adsorption. The fundamental relations for this case are Eqs. (8-22), (8-38), (8-39) and (8-40), which are essentially similar to those employed by Pan and Basmadjian (1970). Thermal equilibrium between particle and fluid is assumed and oidy axial dispersion of heat is taken into account while mass transfer resistance between fluid phase and particle as well as axial dispersion is considered. This situation is identical with the model employed in the previous section. For further simpliHcation, axial dispersion effect may be involved in the overall mass transfer coefficient of the linear driving force model as discussed in Chapter S. In this case, after further justifiable simplifications such as negligible heat capacity and accumulation of adsorbate in void spaces, a set of basic equations to describe heat and mass balances can be ven as follows. 

The rate of change of volume in the tank can be written as a lumped parameter model, where all the resistance to flow is assumed to be associated with the valve, and all the capacitance of the process is assumed to be associated with the tank. This model is shown in Equations 3.1 and 3.2. The basis of Equation 3.1 is the principle of conservation, mass balance in this case (i.e. what goes in must come out or get accumulated in the system). 

In Chapter 1 we developed a steady-state model for a stirred-tank blending system based on mass and component balances. Now we develop an unsteady-state model that will allow us to analyze the more general situation where process variables vary with time. Dynamic models differ from steady-state models because they contain additional accumulation terms. 

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