Using the progress variable

that s fine in the abstract, but how do we represent this arrow in thermodynamics And why would we want to do it How we do it is simplicity itself and illustrates once again the difference between reality and our model or simulation of reality. 

Let s consider one of the simplest kinds of reaction, a polymorphic change such as aragonite calcite. Aragonite on museum shelves actually does not change to calcite at all, but we can do it mathematically with ease. From (18.7) we have 

If n° is the number of moles of each to start with, then integrating these equations from n° to some new value of n gives 

But we are not restricted to such simple reactions, or to only one reaction. Let s next consider a case where we have two simultaneous reactions. 

The irreversible reaction aragonite (A) calcite (C) considered as a function of the progress variable 

---

Another point that is sometimes made about delta notation is that it is commonly used in two different ways (MacDonald, 1990 Spencer, 1973). One way is the scalar difference just mentioned, as in describing the difference in V, G, etc., between two equilibrium states for example, before and after the expansion of a gas. The other is in using the progress variable, as in... 

For each MEP, VTST and VTST/OMT calculations are carried out using the progress variable s, along MEP i as the optimized reaction coordinate. (Note that s, is the variable s for ensemble member i.) The improved reaction coordinate for ensemble member i yields a recrossing transmission coefficient r given by... 

The mixing curve in Figure 18.8 is useful to show that the G function actually does have a minimum, and how the progress variable can be used to simulate stages in an irreversible reaction. In fact in progressing past the equilibrium composition, we actually drove the reaction backwards all the way to pure NH3. In that sense it is no different from our previous examples, except that now we can see which direction is spontaneous and which is backwards. ... 

While integrated rate expressions can be found for second- and higher-order rate laws, in practice this is rarely useful. To see why, consider the five second-order rate laws in Table 4.1. If one monitors the concentration of A, all five rate laws lead to the same general time dependence. In terms of the progress variable (4.7), these rate laws all have the form... 

Obviously, to reason about data fi-eshness, we should model progress of time. The event Time-Progress forcefully alternates between any cyclic events of the model and non-deterministically increases the value of the variable curr.time. Event alternation is enforced by using the boolean variable time.progressed. 

FMEA can be used to provide a quantitative measure of the risk for a design. Because it can be applied hierarchically from system through subassembly and component levels down to individual dimensions and characteristics, it follows the progress of the design into detail. FMEA also lists potential failure modes and rates their Severity (S), Occurrence (O) and Detectability ( )). It therefore provides a possible means for linking potential variability risks with consequent design acceptability and associated costs. Note that the ratings of Occurrence and Detectability are equated to probability levels. 

In such matters some progress can be achieved by combinations of the decomposition method and the method of separation of variables. For example, this can be done using the method of separation of variables for the reduced system (6) upon eliminating the unknown vectors with odd subscripts j. This trick allows one to solve problem (2) here the expenditures of time are Q 2nin2 og N2 arithmetic operation, half as much than required before in the method of separation of variables. 

This series of program statements will plot CA against T on the screen as the simulation progresses. Values of T, CA, CB, CC and RATE will also appear on the screen. Subsequently, any of the variables T, CA, CB, CC and RATE can be plotted against any other variable by using the GRAPH command. 

---