Balance constraint

If the T and P of a multiphase system are constant, then the quantities capable of change are the iadividual mole numbers of the various chemical species / ia the various phases p. In the absence of chemical reactions, which is assumed here, the may change only by iaterphase mass transfer, and not (because the system is closed) by the transfer of matter across the boundaries of the system. Hence, for phase equUibrium ia a TT-phase system, equation 212 is subject to a set of material balance constraints ... 

This expression is minimized subject to the matetial balance constraints of equation 213, which may be written as follows, where i= 1,2, ,77 ... 

The general criterion of chemical reaction equiUbria is the same as that for phase equiUbria, namely that the total Gibbs energy of a closed system be a minimum at constant, uniform T and P (eq. 212). If the T and P of a siagle-phase, chemically reactive system are constant, then the quantities capable of change are the mole numbers, n. The iadependentiy variable quantities are just the r reaction coordinates, and thus the equiUbrium state is characterized by the rnecessary derivative conditions (and subject to the material balance constraints of equation 235) where j = 1,11,.. ., r ... 

Reconciled data must satisfy all material and energy balance constraints and those of rate equations for which coefficients and parameters are fully known. The mathematical process of reconciUation can be described schematically as in Figure 7. 

Mass Balance Constraints. Erom the schematic diagram of a continuous crystallizer shown ia Eigure 11, the foUowiag mass balance on solute can be constmcted ... 

Exploitation of Boundary Curvature A second approach to boundaiy crossing exploits boundaiy curvature in order to produce compositions in different distillation regions. When distillation boundaries exhibit extreme curvature, it may be possible to design a column such that the distillate and bottoms are on the same residue curve in one distillation region, while the feed (which is not required to lie on the column-composition profile) is in another distillation region. In order for such a column to meet material-balance constraints (i.e., bottom, distillate, feed on a straight hne), the feed must be located in a region where the boundary is concave. 

Material Balance Constraint There are two types of constraints for the unit. These are the process constraints and the equipment constraints. In each of these, there are equahty constraints such as material balances and inequality constraints sucti as temperature limits. Analysts must understand the process and equipment constraints as part of the preparation for the unit analysis. 

An example adapted from Verneuil, et al. (Verneuil, V.S., P. Yan, and F. Madron, Banish Bad Plant Data, Chemical Engineeiing Progress, October 1992, 45-51) shows the impact of flow measurement error on misinterpretation of the unit operation. The success in interpreting and ultimately improving unit performance depends upon the uncertainty in the measurements. In Fig. 30-14, the materi balance constraint would indicate that S3 = —7, which is unrealistic. However, accounting for the uncertainties in both Si and S9 shows that the value for S3 is —7 28. Without considering uncertainties in the measurements, analysts might conclude that the flows or model contain bias (systematic) error. 

Spreadsheet Structure There are three principal sections to the spreadsheet. The first has tables of as-reported and normalized composition measurements. The second section has tables for overall and component flows. These are used to check the overall and component material balance constraints. The third has adjusted stream and component flows. Space is provided for recording the basis of the adjustments. The structure changes as the breadth and depth of the analysis increases. 

Global Test The measurements do not close the constraints of the process. In the linear, material balance constraint example used above,... 

The constraints considered in the mathematical formulation are divided into two modules. The first deals with the mass balance constraints and the second with the sequencing and scheduling constraints. The mass balance constraints for the case where there is no central storage are slightly different to those for the case where there is. The mass balances for each are described in the mass balance module below. The sequencing and scheduling module will be described, for both cases, in a subsequent section. The nomenclature for all the sets, variables and parameters can be found in the nomenclature list. 

The above mass balance constraints suffice for the case where there is no intermediate storage available for wastewater. The mass balance constraints necessary for the case where there is a central storage vessel available for wastewater are presented in the following section. 

Mass Balance Constraints Including Central Storage... 

Mass balance constraints (6.1), (6.3) and (6.5) need to be reformulated to account for the water from storage. The water into a unit in this case is not only comprised of freshwater and directly recycled/reused water, but also water from storage. This... 

The mass balance constraints given above would suffice if the process were continuous. However, due to the fact that the processes dealt with are batch processes, additional constraints are required to capture the discontinuous nature of the process. This implies that the time related constraints are necessary. 

Apart from the mass balances associated with water, one also has to consider a product mass balance. Constraint (7.11) states that the amount of product leaving a unit is the amount of raw material that entered the unit less the total contaminant mass load transferred to the water stream. 

One would notice that there are a number of nonlinear terms in the above constraints, specifically in the contaminant balance constraints. The linearisation technique used to remove these nonlinearities is that proposed by Quesada and Grossman (1995), the general form of this linearization technique can be found in Appendix A. During the application of the model to the illustrative examples,... 

The constraints that comprise the mathematical formulation are presented in two modules. The first deals with the mass balance constraints and the second the scheduling constraints. 

The first constraints that are dealt with in the zero effluent scheduling formulation are the unit mass balance constraints. These constraints account for the movement of mass between the various units and storage vessels. 

The synthesis model is based on the zero effluent scheduling formulation with the addition of design specific constraints. The scheduling constraints used in the synthesis formulation are identical to those used in the scheduling formulation and the mass balance constraints are virtually identical except for a few minor changes in the limiting capacities, amount of water and amount of product. These changes will be discussed below. 

The first minor change to the mass balance constraints from the scheduling formulation is found in constraint (8.2), which defines the size of a batch. In the synthesis formulation, the batch size is determined by the optimal size of the processing unit. Due to this being a variable, constraint (8.2) is reformulated to reflect this and is given in constraint (8.59). The nonlinearity present in constraint (8.59) is linearised exactly using Glover transformation (1975) as presented in Chapter 4. 

The mathematical formulation comprises of a number of mass balance constraints and scheduling constraints. The derivation of each of the constraints in the formulation is done considering a superstructure depicting the processes concerned. The sets, variables and parameters used in this chapter are all described in the nomenclature list. 

Mass balance constraints are derived around each processing unit and the central storage vessel. The mass balance constraints around a processing unit are presented... 

Mass Balance Constraints for a Processing Unit Operating in Inherent... 

The mass balance constraints around the central storage vessel are given in constraints (9.17), (9.18), (9.19), (9.20), (9.21), (9.22), (9.23) and (9.24). These constraints are very similar to those presented by Majozi (2005) and will therefore not be discussed in great detail. 

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