Solid Propellant Burning Equilibrium Ingredients Calculation Based on Temperature Iteration

Aiming at shortcomings of depending on experiences and inferior computation accuracy brought by using linear interpolation to calculate fixed pressure burning temperature, a new method using iteration was put forward. Moreover, the optimization model to calculate equilibrium ingredients was formulated based on the new method, and sequential quadric programming method rather than Lagrange multiplier was used to compute the model. At last, a numerical example was given to validate the method in this paper. Results make show that more perfect outcomes can be achieved through the method in this paper than classical method.


Introduction
Equilibrium ingredients and burning temperature of propellant is necessary data for ignition engine design, inner ballistic computation and performance evaluation of solid rocket motor, at present there are three classical methods to calculate equilibrium ingredients and burning temperature of propellant, which are gradual approximation method, minimum G-free energy method and Brinkley method [1]. Burning temperature must be certain beforehand in all of above-mentioned method, however burning temperature is also unknown. Therefore, a simple linear interpolation method was frequently adopted to calculate burning temperature based two presupposed temperatures in all of above-mentioned method. In this way, analyst's experiences and the two preselected temperatures would have remarkable influences on computation accuracy. To solve this problem, a new method to calculate burning temperature was put forward in this paper, which linear interpolation was substituted by iteration. Furthermore, the equilibrium ingredients optimization model was set up based on minimum G-free energy and the iteration method. Finally, the method in this paper was used to calculate equilibrium ingredients and burning temperature of a given solid propellant components.

Fixed pressure burning temperature calculation
The principle of solving fixed pressure burning temperature of c T is that solid propellant's total enthalpy of p I must be equal to burning products total enthalpy of m I , namely Where T i is initial temperature of propellant, K. T s is benchmark temperature, which is commonly 298K. c i is specific heat of the i-th component of propellant, which varies hardly with temperature, J/(kg.K). pi C is fixed pressure specific heat of the i-th burning outcome, which is function of temperature and can be calculated through reference [3].

The method of classic linear interpolate
In the method of calculating burning temperature by classic linear interpolate, two temperatures were presumed firstly. Then burning ingredients and their total enthalpy could be obtained when propellant burning at the two presumed temperatures. If total enthalpy of burning ingredients was linear function of temperature between two presumed temperatures, using interpolate could easily get the fixed pressure temperature based on conversation of energy. The process can be figured by Obviously, the two presumed temperatures are vital to computation accuracy in this method. If analysts had abundant experiences, the presumed two temperatures would approximate true fixed pressure burning temperature and computation error of linear interpolate would be little. Otherwise, linear interpolate would bring non-ignorable errors. Even though for proficient analysts, it was also difficult to presume appropriate temperatures when facing a new type of propellant. Therefore, it is necessary to find new method to solve burning temperature.

Iteration method to solve burning temperature
In practice, calculating burning temperature is to solve the following equation Here, the formula (2) When absolute value of difference between p I and m I is less than 2  , where 2  denotes iteration accuracy and is a given small decimal fraction.

Equilibrium ingredients calculation based on temperature iteration
Now, there are three classical methods to calculate equilibrium ingredients and burning temperature of propellant, which are gradual approximation method, minimum G-free energy method and Brinkley method. Gradual approximation method is simplest among three methods, but this method isn't suitable for thermodynamics calculation with complex ingredients. Minimum G-free energy method has explicit physics meanings, this method considers equilibrium ingredients are ingredients which minimize G-free energy of burning products, so the problem of calculating equilibrium ingredients is transferred to minimum optimization of G-free energy. The number of equations to be solved in Brinkley method is fewest, but Brinkley method is most complicated among three methods, the method demands to input a good many coefficient matrix, which easily creates mistake, and linearization in solving equations will bring calculation errors. So, the minimum G-free energy method would be used to calculate equilibrium ingredients in this paper. In this way, we can set up following optimization model to calculate equilibrium ingredients.
Tic G is standard molar G-free energy of agglomerate, which can be calculated through reference [3].
Ti G is standard molar G-free energy of gas, which can be calculated through reference [3]. In engineering, Lagrange multiplier was usually used to solve above optimization model. In this way, i n could be less than zero in the process of optimization, which can make optimization calculation terminate. In this paper, Lagrange multiplier was substituted by SQP (Sequence Quadratic Programming) to solve the optimization model. It is noted that burning temperature must be known when we use SQP to solve the optimization model, but burning temperature is also parameter to be solved. So, we must integrate formula (4) with the new method in section 2.2 to calculate equilibrium ingredients. The step of calculation can be expressed as following.
(1) Input initial datum of propellant; (2) Calculate assumed chemical formula and total enthalpy of propellant; (3) Assume two initial burning temperatures (  From tab.2 we can see that relative error between total enthalpy of burning ingredients achieved by the method in this paper and total enthalpy of propellant is only 6.56×10 -6 . So, the method in this paper can be used to calculate equilibrium ingredients and fixed burning temperature of solid propellant.

Conclusions
We can obtain following conclusions via above analysis. (1) Compare to classic linear interpolate, iteration has no special requirements on initial assumed burning temperature, which don't rely on analyst's experiences, and can get high calculation accuracy.
(2) When solving equilibrium ingredients optimization model, SQP avoids the problem brought by Lagrange multiplier, and is more effective method.