Stuff equations

Figure 1.6 Schematic representation of terms appearing in the stuff equation (1.4.1). The amount of stuff accumulated in a system may change because of stuff added to the system, stuff removed from the system, stuff generated in the system, or stuff consumed in the system.

For the system in Figure 1.6, the boundary allows transfer of some quantity which, for generality, we call stujf. By identifying all ways by which the amount of stuff may change, we obtain a general balance equation, which we call the stuff equation [11],... 

In general the stuff equation is a differential equation and its accumulation term can be positive, negative, or zero that is, the amount of stuff in the system may increase, decrease, or remain constant with time. In a particular situation several kinds of stuff may need to be inventoried examples include molecules, energy, and entropy. 

The stuff equation applies to both conserved and non-conserved quantities. Conserved quantities can be neither created nor destroyed so, for such quantities the stuff equation reduces to a general conservation principle... 

The name stuff equation for the general balance equation is not ours, but we are embarrassed to report we don t know who originated the idea. 

Use the stuff equation for mass to obtain equations for the following situations. 

In this chapter we develop expressions that relate heat and work to state functions those relations constitute the first and second laws of thermodynamics. We begin by reviewing basic concepts about work ( 2.1) that discussion leads us to the first law ( 2.2) for closed systems. Our development follows the ideas of Redlich [1]. Then we rationalize the second law ( 2.3) for closed systems, basing our arguments on those originally devised by Carath odory [2-4]. Finally, by straightforward applications of the stuff equations introduced in 1.4, we extend the first and second laws to open systems ( 2.4). 

In many applications the quantities we can actually measure or manipulate are the heat and work effects on the external side of the system boundary. We call these Qgj f and Wgj f) they would be measiued at a point on the boxmdary at which the surroxmd-ings have temperature and pressxue ext- These external heat and work effects would differ from the heat and work effects felt by the system whenever the system boundary possesses a finite mass that could store energy. In such cases, the second part of the first law for closed systems generalizes (via the stuff equation (1.4.1) and Figure 1.7) to... 

The traditional formulation of the second law is given by (2.3.5) and (2.3.7) however, there is an alternative that may be useful, especially for open systems. Again, the statement is in two parts a definition of entropy plus an assertion that entropy is not conserved because we now explicitly include entropy changes in the boundary. The definition takes the form of the stuff equation (1.4.1) with Figure 1.7,... 

THERMODYNAMIC STUFF EQUATIONS 55 2.4 THERMODYNAMIC STUFF EQUATIONS... 

In 2.2 and 2.3 we presented the first and second laws for closed systems. In practice these would apply to such situations as those batch processes in which the amount of material in the system is constant over the period of interest. But many production facilities are operated with material and energy entering and leaving the system. To analyze such situations, we must extend the first and second laws to open systems. The extensions are obtained by straightforward applications of the stuff equations cited in 1.4. We begin by clarifying our notation ( 2.4.1), then we write stuff equations for material ( 2.4.2), for energy ( 2.4.3), and for entropy ( 2.4.4). These three stuff equations are always true and must be satisfied by any process, and therefore they can be used to test whether a proposed process is thermodynamically feasible ( 2.4.5). 

In addition to material and energy balances, we may also perform an entropy balance on the system in Figure 2.11. But since entropy is not conserved (entropy can be generated in the system and its boundary), we must appeal to a more general form of the stuff equation, namely (1.4.1). The balance can be written as a generalization of the first part of the second law (2.3.8), in which terms are now included to account for entropy carried by the streams ... 

Although the general stuff equations (2.4.3), (2.4.10), and (2.4.21) are always true, they may not always be useful. To be useful, sufficient information must be available from calculations or measurements. Specifically, to test whether a process satisfies the first law, we must have either (a) complete specifications of the initial and final states of the system, or (b) values for both the heat and the work. To test whether the second law is satisfied, we must know the value for the heat then we would use (2.4.21) to compute dSggfi and determine whether the second part of the second law is obeyed. 

But when these criteria are met, the thermodynamic stuff equations are powerful and versatile. In particular, they can be implemented without knowing the detailed mechanisms by which a proposed process is to accomplish its task. This occurs because the first and second laws establish equivalences between process variables (Q and W) and changes in system variables (such as u, h, and s). 

How do we use the thermodynamic stuff equations to test the feasibility of a proposed process ... 

For open systems, the first and second laws are particular forms of the general stuff equation presented in 1.4. The first law represents an energy balance on a system, and it asserts that energy is a conserved quantity. Similarly, the second law represents an entropy balance, but the second law asserts that entropy is not conserved through the actions of dissipative forces, entropy is created (but never consumed) during any irreversible process. 

In 2.4 we presented differential forms of the thermodynamic stuff equations for overall mass, energy, and entropy flows through open systems. Usually, such systems, together with their inlet and outlet streams, will be mixtures of any number of components. Individual components can contribute in different ways to mass, energy, and entropy flows, so here we generalize the stuff equations to show explicitly the contributions from individual components these generalized forms contain partial molar properties introduced in 3.4. 

Thermodynamic stuff equations are internal constraints on the variables that describe open systems. Therefore, in 3.6.2 and 3.6.3 we show how those constraints enter determinations of the number of independent quantities needed to analyze open steady-flow systems. 

Many industrial processes take place in open systems in which material enters and leaves the system through process streams and in which energy can cross system boundaries as heat and work. At any instant, a complete identification of the state requires specification of values for such variables as temperatures, pressures, compositions, and flow rates. However, because of the stuff equations in 3.6.1, not all of these quantities are independent. So we have here the same kinds of questions addressed in 3.1 How many interactions are available to change the state How many independent variables must be specified to identify the state of an open steady-flow system The discussion here extends that in 3.1 from closed systems to open ones however, the discussion remains limited to systems composed of a single homogeneous phase with no chemical reactions. The extensions to multiphase systems are given in 9.1 and to those having chemical reactions in 10.3.1... 

Stuff Equations for Material Undergoing Reactions in Oosed Systems... 

We emphasize that the reactions used in the analysis do not have to be the reactions actually occurring— we only need any convenient hypothetical path that connects products to reactants. In fact, we don t even need reactions at all, so long as we can achieve a balance on every element present. Further, "elements" need not be atoms they can be groups of atoms that may or may not constitute real molecules. Our procedure for identifying and balancing reactions reduces to the stuff equation for material, reformulated to apply to elements. We consider reactions in closed systems here and reactions in open systems in 7.5. 

With the notation and stuff equations from the previous section, we can now extend the combined first and second laws from unreacting systems ( 7.1 and 7.2) to reacting systems. To facilitate the presentation, it is useful to introduce a new set of property differences that apply to reacting systems. For any extensive property F in a reacting system, we define a change in F for each reaction j by the intensive quantity... 

Stuff Equation for Material in a Single Open Phase... 

Consider an open system having any number of inlets a and any number of outlets p. For such a system, the general stuff equation (1.4.1) can be written in terms of the number of moles of species i,... 

This is the general form of the stuff equation for an open system containing a total of C species, some or all of which are engaged in chemical reactions. 

Here S counts any additional internal constraints besides the phase-equilibrium constraints in (9.1.6)-(9.1.8). Examples include constraints imposed by critical points (certain stability relations must be obeyed) and azeotropes (certain relations must exist among T, P, and the compositions of the phases). The number given by (9.1.10) can be much less than the total number of variables given by (9.1.5). For example, a four-component system in three-phase equilibrium has = 18, but only = 6 of those are needed to identify the extensive state (with S = 0). Values for the other twelve would be computed by solving stuff equations together with the phase-equilibrium equations (9.1.6)-(9.1.8) those calculations may or may not be easily performed. 

The differential forms of the stuff equations were given in 3.6. For a system of C components, the material balances take the form... 

We now turn to processing situations in which heat effects are of primary importance examples include chemical reactors and separators that exploit phase partitioning. Thermodynamic analysis of these situations invoke the stuff equations in particular, steady-state heat effects are computed from (12.3.5). To obtain the partial molar enthalpies that appear in (12.3.5), we need enthalpies as functions of composition so in 12.4.1 we show how enthalpy-concentration diagrams can be constructed from volumetric equations of state applied to binary mixtures in phase equilibrium. Then we apply the energy balance (12.3.5) to multicomponent flash separators ( 12.4.2), binary absorbers ( 12.4.3), and chemical reactors ( 12.4.4). 

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