Dependent variables Detailed balancing

The macroscopic balance ignores all the detail within a system and consequently results in a balance about the entire system. Only time remains as an independent variable in the balance. The dependent variables, such as concentration and temperature, are not functions of position but represent overall averages throughout the entire volume of the system. In effect, the system is assumed to be sufficiently well mixed so that the output concentrations and temperatures are equivalent to the concentrations and temperatures inside the system. 

The symbols are P for profit, / for equality constraints, g for inequality constraints, x for optimization variables, y for dependent variables, and / (constant) for updated parameters. The objective function is a scalar measure of plant profit it is usually the instantaneous profit ( /hr), because the optimization variables do not involve the time value of money. Typical equality constraints include material and energy balances, heat and mass transfer relationships, and thermodynamic and kinetic models, and typical inequality constraints include equipment limitations limit compressor horsepower, and distillation tray hydraulics. The optimization variables are flow rates, pressures, temperatures, and other variables that can be manipulated directly. The dependent variables involve intermediate values required for the detailed models for example, all distillation tray compositions, flow rates, and temperatures. Because of the fundamental models often used in RTO, the number of dependent variables can be quite large, on the order of hundreds of thousands. 

The time derivative is zero at steady state, but it is included so that the method of false transients can be used. The computational procedure in Section 4.3.2 applies directly when the energy balance is given by Equation 5.27. The same basic procedure can be used for Equation 5.24. The enthalpy rather than the temperature is marched ahead as the dependent variable, and then Tout is calculated from //out after each time step. The examples that follow assume constant physical properties and use Equation 5.27. Their purpose is to explore nonisothermal reaction phenomena rather than to present detailed design calculations. 

Correct modeling of variable diffiisivity, time-dependent emission sources, nonlinear chemical reactions, and removal processes necessitates numerical integrations of the species-mass-balance equations. Because of limitations of dispersion data, emission data, or chemical rate data, this approach to the modeling of air pollution may not necessarily ensure higher fidelity, but it does hold out the possibility of the incorporation of more of these details as they become known. 

The macroscopic approach, in which it is not taken into account what happens inside the cell in detail, but only an overall view of the system is described. In fact, the system is considered as a black box from the fluid dynamic point of view and then, it is assumed that the cell behaves a mixed tank reactor (the values of the variables only depend on time and not on the position since only one value of every variable describes all positions). This assumption allows simplifying directly all the set of partial differential equations to an easier set of differential equations, one for each model species. For the case of a continuous-operation electrochemical cell, the mass balances take the form shown in (4.5), where [.S, ]... 

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