The equilibrium state of a mixture of chemically reacting perfect gases is described by applying the principle of entropy maximization to the fixed case. A stability criterion based on the concavity of the entropy function in the variable space is developed as an aid to obtain fast convergence. Equilibrium State Description 3 An Iterative Solution Technique 8 Equilibrium State Stability 13 .
Application of the elemental potential method to determine the equilibrium of the combustion system of nitrogen tetroxide and hydrazine. The differential parameters dNi are a measure of mass transfer into or out of the system. The resulting postulatory form of the entropy function is, of course, identical to Gibbs' phenomenological formulation (6).
The N oC possesses the property that. where A"' is the atomic weight of atomic species < are performed by { Xj and relaxing.
It will be illustrated that initial values of the parameters are easily estimated and quickly relaxed to the equilibrium values.
The inequality of equation 35 can be expressed in terms of elemental potentials by substituting from the version of equation 18, i.e. In the linearized scheme, the slopes of the ~ functions are extended in different coordinate directions from the estimated point. Normalization allows errors to be compared to unity to determine an absolute measure of convergence.
The optimization is one-dimensional, resulting in the fraction of predicted patch size that produces the smallest error. A logical choice for the error function to be minimized is the sum of squares LP and [ ~} • Determine the error at the nth iteration. It will be shown numerically that the choice of the enthalpy basis implicitly determines such a bias.
This was chosen because it was representative of the uncertainty in temperature of an unexplored system. The parameters appropriate for the zeroth estimate, i.e. the estimate of the elementary products of Annex I, are presented to illustrate the relative accuracy of the initial estimation procedure. When defining the error function (E2), it was noted that the size of the function depends on the selected enthalpy basis.
The dependence of the size of the error in the initial estimate on the basis chosen was. The values of the enthalpy of mixing calculated from the enthalpies of the reactants described for the repeating example are, 2224 cal./gr. This change of scale does not change the normalization or dimension of the error function.
In Part IV, the criterion for the stability of the equilibrium state was developed in element potential coordinates. For the example under investigation, equation 39 was evaluated for various estimates of the solution parameters. The rank of the correction matrix is not affected by the number of molecular species considered.
This requires the introduction of new constraints that N i must be a constant equal to a certain value for the frozen components. The effect of the additional constraints is to separate the correction equations, resulting in many zeros in the correction matrix.
N G THE SO LUTION
The necessary derivatives are the same as those calculated in the iterative scheme, except that all the predefined constraints are replaced. The presence of the prepared quantities in the correction matrix would constitute a second order correction in the solution, and consequently they were replaced by the base values. Retention of the prepared quantities will also require the reevaluation of the correction matrix at each perturbation.
These were compared with the values obtained by an independent exact solution.(9) The base point was at a pressure of 150 psia. The average properties at the base point of the perturbation are all available through the iterative solution mentioned earlier. The variation of the equilibrium state with a change in the applied pressure, as discussed above.
We would expect that the most direct method for obtaining the variation of solution parameters with pressure. The disagreement appears to be due to the relative nature of enthalpy discussed earlier. For example, in the temperature derivative it is assumed that the change of absolute temperature with absolute pressure is a function of the relative property < h >.
The magnitude of the perturbed constraints, the predicted values of the parameters and the exact values are shown in Table IV. The contents of Table IV illustrate that the perturbation results compare favorably with the exact solution values. The parameters {~~}, T were determined based on the exact solution(9) of the nitrogen tetroxide-hydrazine combustion system.
It has been stated that accurate first estimates of the parameters [XX} were simple to develop. Because of the additivity of the atomic potentials to result in molecular potentials, as described in equation 11, it is not. The initial estimates of { ACI(} are obtained by solving the A equations.
For each atomic species o( there will be an order of the prevalence of the molecular species containing o(. The first example is that of the nitrogen tetroxide-hydrazine system selected as the iterative sample calculation.
