Node balance

So-called plant dispersion" or extra column effects" have to be taken into account by additional mathematical models rather than including them indirectly in the model parameters of the column, e.g. by altering the dispersion coefficient. The combination of peripheral and column models is easily implemented in a modular simulation approach. In a flowsheeting approach the boundary conditions of different models are connected by streams (node balances) and all material balances are solved simultaneously. 

Concentrations c]n are the inlet boundary conditions (Eq. 6.92 or 6.93) of the columns at the beginning of each section while cout are the outlet concentrations calculated at the end of each section. Intermediate node balances consist of setting equal volume flows and assigning the outlet concentration to the inlet concentration of the subsequent column. Since SMB is a periodic process, the boundary conditions for every individual column are changed after a switching period t,hm. If SMB modifications such as VariCol, ModiCon, etc. (Chapter 5) are used the boundary conditions are modified accordingly. 

By combining the node balance and a reactive column model as described in Section 8.3.1, an SMBR model can be created. Each section consists of one or more columns. The affiliation of the single columns and the boundary conditions are changed periodically. The flowsheet of the SMBR process is similar to that for the TMBR process given in Fig. 8.7. Please note that the origin of the axis system is located in the feed port. Internal flow rates are related to the external streams by balances of the inlet-and outlet nodes. A mass balance for each node is also necessary to calculate the concentration at the inlet and outlet of each single section. 

The subtitle of the book reads From microscopic balances to large plants . The first part of the promise is fulfilled in Appendices C and D. If the reader is more deeply interested in the physics underlying the ( macroscopic ) node balances, before pemsing Chapters 4 and 5 he can begin with Appendix C,... 

Let us arrange the subgraphs G° in the manner that G , , G - K < K) are those subgraphs which are not isolated nodes observe that if G is an isolated node, the corresponding (scalar) equation (node balance) in (3.2.2) becomes automatically one of the node balances of the reduced graph G. Having selected a reference node in each G for k = I, , K , let B be the reduced incidence matrix of G, thus B is of full row rank let further A, be the corresponding... 

One of the node balances has been eliminated by graph reduction (as the merged-nodes balance) and the remaining two read... 

Let us begin with the node balance (4.2.1). On adding the accumulation term, it takes the form... 

We have chosen, in Section 5.3, a classical technology where available thermodynamic data allow us to compute the balance according to the general scheme (5.2.11). The formulation (5.2.11) of the enthalpy (approximate steady-state energy) balance is, however, not the only possible, and not always the most convenient one. In practice, one must frequently put up with a number of empirical data specific to the system and not found in thermodynamical tables. The data can even be more reliable than those found by a hypothetical thermodynamic path involving not precisely known items. For example the standard enthalpies of the components are, in fact, certain extrapolations computed backwards from reaction heats under realistic experimental conditions for instance [see (5.3.9)] sulphur is certainly not burnt at, say, 273 K. Because the theoretical items are subtracted in the input - otput node balance, a relatively small error in the data may cause a significant relative error in the result. 

Finally, Example 8 is more complex in that it comprises entropy production due to chemical reaction in combination with heat transfer, and also diffusion the role of the latter appears as marginal. The example can also be regarded as an example of complex single-node balancing, a kind of thermodynamic analysis included. Concerning the entropy production (or loss of exergy), it turns out that the chemical portion, thus the term -(Gr / Tj ) Wr in (6.2.109) can represent an enormous item in the exergy balance it can be computed, but this is usually all that we can do in practice. In other terms, what kind of work has been actually lost, is a matter of theoretical speculations only. 

In our simple examples according to Fig. 8-1, we have considered the graph of the technological system restricted to one technological unit (node), plus the environment node. The presence of other node balances can change the classification. To Fig. 8-1, let us add a splitter and another heat exchanger. 

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