Reaction-progress variables conditional

The dimensionless vector of reaction-progress variables Y is then defined to be null in the initial and inlet conditions, and obeys... 

Using the same inlet/initial conditions as were employed for the one-step reaction, this reaction system can be written in terms of two reaction-progress variables (Fi, Y2) and the mixture fraction f. A linear relationship between c and (co, Y, f) can be derived starting from (5.162) with y = Y2 = A0B0/(A0 + B0) ... 

By conditioning on the mixture fraction (i.e., on the event where f(x, t) = f), the reaction-progress-variable transport equation can be rewritten in terms of 7(f, r) 115... 

For simple chemistry, a form for Q( x, t) can sometimes be found based on linear interpolation between two limiting cases. For example, for the one-step reaction discussed in Section 5.5, we have seen that the chemical source term can be rewritten in terms of a reaction-progress variable Y and the mixture fraction f. By taking the conditional expectation of (5.176) and applying (5.287), the chemical source term for the conditional reaction-progress variable can be found to be... 

For simple chemistry, we have seen in Section 5.5 that limiting cases of general interest exist that can be described by a single reaction-progress variable, in addition to the mixture fraction.131 For these flows, the chemical source term can be closed by assuming a form for the joint PDF of the reaction-progress variable Y and the mixture fraction . In general, it is easiest to decompose the joint PDF into the product of the conditional PDF of Y and the mixture-fraction PDF 132... 

Unlike for the mixture fraction, the initial and inlet conditions for the mean and variance of Y will be zero. Thus, the chemical source term will be responsible for the generation of non-zero values of the mean and variance of the reaction-progress variable inside the reactor. Appealing again to the assumption of independence, the mean reaction-progress variable is given by... 

Thus, the turbulent-reacting-flow problem can be completely closed by assuming independence between Y and 2, and assuming simple forms for their marginal PDFs. In contrast to the conditional-moment closures discussed in Section 5.8, the presumed PDF method does account for the effect of fluctuations in the reaction-progress variable. However, the independence assumption results in conditional fluctuations that depend on f only through Tmax(f ) The conditional fluctuations thus contain no information about local events in mixture-fraction space (such as ignition or extinction) that are caused by the mixture-fraction dependence of the chemical source term. 

Despite these difficulties, the multi-environment conditional PDF model is still useful for describing simple non-isothermal reacting systems (such as the one-step reaction discussed in Section 5.5) that cannot be easily treated with the unconditional model. For the non-isothermal, one-step reaction, the reaction-progress variable Y in the (unreacted) feed stream is null, and the system is essentially non-reactive unless an ignition source is provided. Letting Foo(f) (see (5.179), p. 183) denote the fully reacted conditional progress variable, we can define a two-environment model based on the E-model 159... 

Alternatively, one can attempt to formulate an algebraic model by assuming that the spatial/temporal transport terms are null for the conditional reaction-progress variables. However, care must be taken to ensure that the correct filtered reaction-progress variables are predicted by the resulting model. 

Adesina [14] considered the four main types of reactions for variable density conditions. It was shown that if the sums of the orders of the reactants and products are the same, then the OTP path is independent of the density parameter, implying that the ideal reactor size would be the same as no change in density. The optimal rate behavior with respect to T and the optimal temperature progression (T p ) have important roles in the design and operation of reactors performing reversible, exothermic reactions. Examples include the oxidation of SO2 to SO3 and the synthesis of NH3 and methanol CH3OH. 

Note that the reaction-progress vector in the first column is non-zero. Thus, as we suspected, the mixture-fraction basis is not a linear-mixture basis. The same conclusion will be drawn for all other mixture-fraction bases found starting from (5.118). For these initial and inlet conditions, a two-component mixture-fraction vector can be found however, it is of no practical interest since the number of conserved-variable scalars is equal to Nq,m = 1 (k e 0, 1, 2). In conclusion, although the mixture fraction can be defined for the... 

Since (5.299) is solved in mixture-fraction space, the independent variables are bounded by hyperplanes defined by pairs of axes and the hyperplane defined by X = K, = 1. At the vertices (i.e., V = = (0, and e, (/el,..., AW)), where e, is the Cartesian unit vector for the /th axis), the conditional mean reaction-progress vector is null 121... 

Next we explore the question of how the apparent thermodynamic variables change as a reaction progresses from a non-equilibrium initial condition toward equilibrium in a closed system. Imagine a closed system initially contains ATP, ADP, and PI, at concentrations of 10 mM, 1 mM, and 1 mM, respectively, at neutral pH of 7. To see how pH and related thermodynamic properties change as the reaction progresses toward equilibrium, imagine that an ATP hydrolysis enzyme is present in our system, and the reference reaction of Equation (2.13) moves in... 

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