The Progress Variable

Up to now, we have been most interested in reactions that reach equilibrium. Now let s look at what happens before that point is reached, i.e., while the reaction is taking place. Let s say that the reaction proceeds from left to right as written. It doesn t matter for the moment whether all the reactants and products are in the same phase (a homogeneous reaction) or in different phases (a heterogeneous reaction). During the reaction, A and B disappear and C and D appear, but the proportions of A B C D that appear and disappear are fixed by the stoichiometric coefficients. If the reaction is 

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The Eulerian approach requires a measurement of the temperature or the progress variable at many sample points at a given normal distance from the ignition plane, at a given time elapsed since ignition. The progress variable introduced here can be for instance a normalized temperature or concentration that varies from... 

The mass balance aspects of distillation and percolation processes are often confused with their kinetics. Transfer of successive mass increments usually takes place over successive time increments but how the progress variable varies with time is often unknown in natural processes. Fortunately, knowledge of time-dependence is by no mean compulsory to achieve the description of the process in terms of mass balance. 

Following the description given in the previous section, the detailed chemical system is mapped on two controlling variables the mixture fraction (Z) and a reaction progress variable (c). The mixture fraction describes the mixing of the species and enthalpy, while the progress variable follows the advance of the chemical reaction. Hence, the species mass fractions, temperature, density and species chemical source terms become functions of Z and c. The mixture fraction is defined according to [19] as ... 

In turbulent reactive flows, the chemical species and temperature fluctuate in time and space. As a result, any variable can be decomposed in its mean and fluctuation. In Reynolds-averaged Navier-Stokes (RANS) simulations, only the means of the variables are computed. Therefore, a method to obtain a turbulent database (containing the means of species, temperature, etc.) from the laminar data is needed. In this work, the mean variables are calculated by PDF-averaging their laminar values with an assumed shape PDF function. For details the reader is referred to Refs. [16, 17]. In the combustion model, transport equations for the mean and variances of the mixture fraction and the progress variable and the mean mass fraction of NO are solved. More details about this turbulent implementation of the flamelet combustion model can also be found in Ref. [20],... 

Figure 9.3. A graph of the Gibbs function G as a function of the progress variable which shows equilibrium at the minimum.

But the only natural cause of such changes in these masses is a spontaneous or irreversible reaction such as we are now considering, so it appears natural to identify the increments in (14.24) with increments in the progress variable, d. Thus... 

We have tried to show that the affinity is a convenient representation of how far a system is from stable equilibrium, and increments in the progress variable allows us to consider the system at various stages as it progresses towards stable equilibrium. This progress is a part of the model we call a quasistatic reaction—a continuous succession of metastable equilibrium states in an overall irreversible reaction. 

You can get a better intuitive feeling for the physical significance of the progress variable by rearranging (19.73) to express explicitly the change in mass of any species with extent of reaction ... 

Fig. 19.6. Number of moles of species in the aqueous phase (solid lines) and minerals (only gibbsite is shown) produced and destroyed (dashed lines) per kg of water during the hydrolysis of K-feldspar (after Helgeson et al., 1969, Helgeson, 1979). The increase in kaolinite between B and G and of muscovite between D and G is omitted for clarity. Letters ABCDG correspond to Figure 19.5. is the progress variable.

The next step is to take the derivative of each of the above 5 equations with respect to the progress variable and rearrange. We will illustrate this just for the first equation (19.93) since the method is identical for the remaining four. The derivative of (19.93) with respect to i is... 

Equation (8-7) must hold for all conceivable variations of the progress variable. If SX > 0, Eq. (8-7) demands that... 

The Progress Variable. It is possible to determine the concentrations of the various components in a reaction mixture at any time during the reaction in terms of the progress variable 2. If wj,. . . , n° are the numbers of moles of the various components initially when 2 = 0, Eq. (8-2) can be integrated to yield... 

Unless the progress variable A can be controlled, the thermodynamic functions cannot be determined at all possible compositions. This is necessary in order for the general discussion of equilibrium criteria to be operationally meaningful. The rigorous thermodynamic theory of chemical equilibrium does not apply to reactions of the type N2O4 2NO2 since neither component can be isolated. 

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