Complex systems material balance equations

Now you can reconsider the material balance equations by adding those additional factors identified in the previous step. If necessary, estimates of unaccountable losses will have to be calculated. Note that, in the case of a relatively simple manufacturing plant, preparation of a preliminary material-balance system and its refinement (Steps 14 and 15) can usefully be combined. For more-complex P2 assessments, however, two separate steps are likely to be more appropriate. An important rule to remember is that the inputs should ideally equal the outputs - but in practice this will rarely be the case. Some judgment will be required to determine what level of accuracy is acceptable, and we should have an idea as to what the unlikely sources of errors are (e.g., evaporative losses from outside holding ponds may be a materials loss we cannot accurately account for). In the case of high concentrations of hazardous wastes, accurate measurements are needed to develop cost-effective waste-reduction options. It is possible that the material balance for a number of unit operations will need to be repeated. Again, continue to review, refine, and, where necessary, expand your database. The compilation of accurate and comprehensive data is essential for a successful P2 audit and subsequent waste-reduction action plan. Remember - you can t reduce what you don t know is therel... 

Thus, in a general case to calculate the composition of the complex system, the material balance equations are necessary. If we assume that complexes of ML, type form in the monoligand system, and protonated ligand LH is unable to form coordination bonds with M" ions, it follows from Eqs. (1.4), (1.7), (1.8), and... 

Another example is processes taking place in the system Cu Cu(II), glycolic acid [8]. Procedures carried out to evaluate surface concentrations in this system are based on the material balance equations written for the surface layer (see [8]). The NTP obtained are shown in Figure 5.3. It can be stated that neither the potential sweep rate nor acidity of the solution (and the composition) has any significant influence on the nature of these dependencies. Thus, the assumption that free Cu " ions (Cu(II) aqua complexes) are electrically active in this case is quite acceptable. Kinetic parameters of the charge transfer process Cu + -F e Cu" " can be determined with reasonable accuracy (Figure 5.3). 

Note that application of a systematic approach enables us to resolve a material-balance system into a number of independent equations equal to the number of unknowns that it needs to solve for. The following steps should be followed with any material-balance system, regardless of complexity ... 

In this chapter, we first consider uses of batch reactors, and their advantages and disadvantages compared with continuous-flow reactors. After considering what the essential features of process design are, we then develop design or performance equations for both isothermal and nonisothermal operation. The latter requires the energy balance, in addition to the material balance. We continue with an example of optimal performance of a batch reactor, and conclude with a discussion of semibatch and semi-continuous operation. We restrict attention to simple systems, deferring treatment of complex systems to Chapter 18. 

The last component of the model is a method to solve this system of (simultaneous partial differential) equations, often as a function of time as the concentration distributions evolve during the experiment. The difficulty of solving these systems depends on the complexity of the material balance... 

To follow the drift in composition of both the feed and the copolymer formed one needs to integrate the copolymer equation. The process being rather complex, the numerical or graphical approach of Skeist (1946) based on Eq. (7.17) provides a simple solution to the problem. Consider a system initially containing a total of N moles of the two monomers and choose Mi as the monomer such that F > f (i.e., the polymer being formed contains more Mi than the feed). Thus at a time when dN moles of the monomer mixture have been converted into polymer, the polymer formed will contain Fi dN moles of M i, while the M i content in feed will be reduced to N - dN)(J df ) moles. Thus a material balance for monomer Ml can be written as (Odian, 1991 Billmeyer, Jr., 1994) ... 

Our discussion thus far has indicated that during copolymerization, the composition of both the feed and the polymer vary with conversion. To follow this composition drift, it is necessary to int ate the copolymer equation — a problem that is complex. Consider a system that is composed initially of M total moles of the two monomers (M = M, + Mj) and in which the resulting copolymer is richer in M, than the feed (Fi > fj). When dM moles have been polymerized, the polymer will contain F, dM moles of Ml while the feed content of M, wUl be reduced to (M - dM) (f, - dfi) moles. Writing a material balance for Mji... 

Although Eq. (1.2-14) is sometimes used directly for solution of complex equilibrium problems, it is more often employed in equivalent algebraic forms which use explicitly the chemical potential or other related quantities. Consider a closed system containing v phases and N components. Introducing the chemical potential p of each component / in each phase p and incorporating material-balance constraints, one obtains as necessary conditions to Eq. (1.2-14) a set of N(rr - 1) equations for phase equilibrium ... 

The system resembles the complex-catalyzed carbonylation of methane (cf eqs. (33-37)). However, acetic acid is not formed in equation (21). The reaction is highly catalytic for Pd, but stoichiometric for Cu. The material balance of carbon monoxide is unclear. 

Elementary devices are general reusable elements used to model complex devices by inheritance or composition. At this level, only material and energy balance equations are taken into account. In consequence, states of such devices are only dependent on input or output flows activity. As ports provide the corresponding flow variables, they are defined as token in order to build the most general element. So much a device owns ports, so much its Petri net includes port tokens. When a port token marks a hybrid place of a device, the associated flow variable is taken into account in the DAE system. The management of these tokens is done either with internal transitions (in the case of specialized device) or with command places (when the port status is dependent on the evolution of a another device). In order to preserve the model consistency, the output port activation has to activate in chain the input port of the above connected device. Figure 5 shows an example of elementary device found in PrODHyS. 

As was shown in (4.7) in solving problems to do with the determination of the total and component charges of reactors in a complex combined system, it is necessary to select, for each of the reactors with dependent feed, one equation from the equations of the component charges of all the reactors. In order to solve system (4.8) for reactors with dependent feed, it is necessary to take the free terms in the equations of the material balance of one of the components of each reactor as zero. Thus the problem is to determine the total number of variants corresponding to this condi-... 

If a more complex mathematical model is employed to represent the evaporation process, you must shift from analytic to numerical methods. The material and enthalpy balances become complicated functions of temperature (and pressure). Usually all of the system parameters are specified except for the heat transfer areas in each effect (n unknown variables) and the vapor temperatures in each effect excluding the last one (n — 1 unknown variables). The model introduces n independent equations that serve as constraints, many of which are nonlinear, plus nonlinear relations among the temperatures, concentrations, and physical properties such as the enthalpy and the heat transfer coefficient. 

Henley and Rosen s Material and Energy Balance Computations is an undergraduate text concerned with the performance of thermodynamic calculations on mathematically complex but conceptually straightforward systems which arise in the chemical industry. The book should be of interest chiefly to the chemical enginner because it describes the setting up of equations and their solution by manual or machine methods. 

In Equation (5.14), G is the Gibbs fi ee energy, is the chemical potential and Hi is the molar amount of species i. Equation (5.15) describes the side condition where bj is the quantity of chemical element /, and is the elemental matrix assigning the elements j to the species i. Hence, it represents the conservation of material. The solution of the constrained optimization problem is rather complex and has been described elsewhere [3]. Furthermore, each calculation is carried out for constant pressure and temperature and requires an iteration with the heat balance to calculate the system equilibrium temperature. The advantages include correct prediction of trace compounds and inclusion of non-ideal... 

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