Numerical proof of calorimeter chemical freezing.

Catana, Razvan ; Cican, Grigore ; Predoiu, Constantin 等

1. INTRODUCTION

The process of combustion of solid propellant samples within a small constant volume calorimeter is considered as a typical slow evolution. The motion of the developed gas mixture is fairly limited to the closed vessel space and no fast moving gasses are ordinarily considered to appear. After the inner volume is fast filled with the gas mixture at high temperature and pressure a very slow cooling of the content follows, almost down to the surrounding water, preserving at the room temperature. There is no apparent reason to consider that the chemical equilibrium is not fulfilled during this process (Kuhl et al. 2006; Beckstead, 2006; Kubota, 2007).

It was observed however, during repeated calorimetric experiments, followed by frequent chemical composition analyses, that the composition of the residual gas from the calorimeter does not obey the chemical equilibrium at room temperature, but at an unexpectedly high temperature instead (Rugescu, 2005). The observation was striking and led to numerical simulations of isochoric cooling that show that the composition of the residual gases in the vessel remains frozen to a temperature as high as 1650[degrees]K after hours of standing at the final room temperature.

2. CHEMICAL FREEZING: STATE OF THE ART

Freezing in chemical composition of the air was early suspected and experimentally detected afterwards in supersonic wind tunnels and shock tubes. A similar behaviour was later attributed to the rocket nozzle gasses that flow with supersonic speed in the diverging extension of the nozzle. This phenomenon greatly complicates the computations of high-speed flow, requiring strong numerical schemes, involving differential equations of chemical kinetics. A simplifying treatment of this kind of flows was proposed during the 50's that led to approximate description of the high-speed flow with considerable computational time savings (Bray, 1959).

Many methods of simulating fast chemical interacting flows were developed (Westenberg 1962, Williams, 1965, Hill et al, 1967, Schuricht, 2001). They confirm the sudden chemical freezing in supersonic nozzles. However, no observation is reported on any non-equilibrium in calorimetric evolutions.

Chemical freezing, when identified in nozzles, produces a constant chemical composition to occur after a certain area of the fast expanding gas flow in the nozzle (Fig. 1).

[FIGURE 1 OMITTED]

Although in the diagram the temperature is a reference, the chemical freezing in nozzles is only related to high velocities of the flow, hindering the molecules from further interacting. It is important to note however, that the temperature range where the kinetical freezing was observed to occur is sensibly similar to the temperature range where a chemical departure from equilibrium within calorimeters was observed (Rugescu, 2005).

3. THERMOCHEMICAL NUMERICAL METHOD

In order to cover all possible dissociation and formation processes during the calorimeter cooling of the combustion products, a universal method is built, based on a direct linearization procedure. The equations of conservation are first written as the conservation of chemical elements,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (1)

chemical reactions with equilibrium constants [K.sub.r]:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (2)

their gas-phase equilibrium (Q--the number of condensed spec)

ln [K.sub.r] (T) = [[n.summation over (i=1)].sup.G] ([v".sub.ri]--[v'.sub.ri]) ln [p.sub.i], r [member of] {R Q}. (3)

mono-condensed species:

ln [K.sub.q] = [v".sub.qi] [[n.summation over (i=1)].sup.G] [p.sub.i], q [member of] {Q} (4)

local chemical freezing:

[p.sub.i] / [P.sub.summation] = ([p.sub.i] / [P.sub.summation]) local, [n.sub.i.sup.*] = ([n.sub.i.sup.*]) local, i [member of] {n}, (5)

equilibrium of phases:

ln [p.sup.S.sub.k] = [K.sub.k] (T) [p.sub.k] = [delta] [p.sup.S.sub.k] + (1-5 [[delta].sub.k]) [p.sub.k], [n.sup.*.sub.k] = [[delta].sub.k] [n.sup.*.sub.k], (6)

equation of state with--for the constant volume loading

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)

Energy conservation governs the adiabat-isochoric combustion

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)

while the isochoric cooling through

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)

The direct linearization with the following corrections applies,

[P.sub.j] [left arrow] [p.sup.0.sub.j] [x.sub.j], [p.sub.j] [left arrow] [p.sup.0.sub.j] [x.sub.j], [eta] [left arrow] [[eta].sup.0] [x.sub.n]. (12)

Improvements in corrections are successively extracted, at each iteration, from the linearized system of equations

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

that give the comparison of the gas at a given temperature T.

4. EXPERIMENTAL MEASUREMENTS

A standard combustor was used for the isochoric combustion and cooling of compositions A-100 and A-300:

[A.sub.o]=33.7517 [A.sub.n]=9.62048 [A.sub.c]=22.66504 [A.sub.h]=29.62565 [A.sub.pb]=0.07938 [A.sub.ca]=0.167173

[A.sub.o]=35.697 [A.sub.n]=10.24663 [A.sub.c]=19.729682 [A.sub.h]=28.14021 [A.sub.pb]=0.0815 [A.sub.ca]=0.077884

For the first propellant the chemical composition of the residual gas in Table 1 was found through gas chromatography, while for the second propellant the composition of the residual gas given in Table 2 was similarly found:

At room temperature the water is completely condensed so that only four gas concentrations were measured.

5. COMPARISON AND CONCLUSIONS

Only the four main gases are listed below, in order to compare the results. It was surprisingly found that the following high temperatures equilibrium values equal the composition, measured at room temperature:

Temperature [H.sub.2] [N.sub.2] CO C[O.sub.2]

 2000 21.816 13.679 53.445 10.963
 1711 22.816 13.504 51.481 12.103
 1704 22.847 13,499 51.420 12.138
 1608 23.309 13.418 50.513 12.665
 1403 24.571 13.197 48.035 14.103
 700 22.783 12.923 51.582 12.892

The temperatures where the computed composition is identical with the measured one are well grouped. This shows that the phenomenon is not random. The adiabatic temperature of isochoric combustion (explosion) would have been around 2775 K, far above the values of consistency from the table. It is difficult to suppose that the temperature, where the experimental composition fits the computed one, is the actual temperature of combustion, as the far more energetic formulation A-300 provided almost equal or even a little lower temperature of freezing. This completely removes the idea that the observed temperatures are the actual temperatures of isochoric combustion. The above computations follow previous ones with very different codes and all give the same conclusion: the chemical composition greatly departs from normal equilibrium. A proof of the freezing temperatures is expected from pressure measurements during isochoric combustion.

The computations and measurements clearly show that the chemical composition of the residual gas after a long and slow cooling remains blocked at the composition that the gas went through at an early stage of cooling. The reasons for this freezing are now considered for a detailed investigation through a national CNCSIS grant, currently under evaluation.

6. REFERENCES

Beckstead, M. (2006), Recent progress in modeling solid propellant combustion, Combustion, Explosion, and Shock Waves (CESW), Vol. 42, No. 6, Nov., pp. 623-641 (19).

Bray, K. N. C. (1959), Atomic recombination in a hypersonic wind-tunnel nozzle, J. of Fluid Mech., 6(1), p. 1-32.

Hill F.; Unger H. & Dickens W. (1967), AIAA Journal, 5(5) 54.

Kubota N. (2007), Propellants and Explosives-Thermochemical Aspects of Combustion, 2nd Ed, Wiley, N. York.

Kuhl, A., Neuwald, P., and Reichenbach, H. (2006), Effectiveness of combustion of shock-dispersed fuels in calorimeters of various volumes, CESW, 42(6), p. 731-734

Rugescu, R.D. (2005), Chemical Freezing, Engineering Meridian, TUM Chisinau, Moldova

Schuricht, S. R. (2001), Numerical simulation of high speed chemically reacting flows, Purdue Univ., AAT 3075724.

Williams, J. C., III (1965), Correlation of the sudden freezing point in nonequilibrium nozzle flows, AIAA J., 3(6), p. 1169

Westenberg, A. A.; Favin, S. (1962), Nozzle Flow with Complex Chemical Reaction, Johns Hopkins Univ., Appl. Phys. Lab, AD0275464.

Tab 1. Residual calorimeter gas of the A-100 propellant.

Species [H.sub.2] [N.sub.2] CO C[O.sub.2]

Sample 1 22.927 13.657 51.467 12.669
Sample 2 22.641 13,646 51.283 13.668
Sample 3 22.829 12.898 51.513 12.766
Sample 4 22.772 12.861 51.463 12.584
Sample 5 22.783 12.923 51.582 12.892
 Mean 22.847 13.197 51.481 12.665

Tab 2. Residual calorimeter gas of the A-300 propellant.

Species Sample 1 Sample 2 Mean

[H.sub.2] 16.793 16.519 16.6560
[N.sub.2] 16,484 16.643 16.5640
CO 42.255 42.579 42.4170
C[O.sub.2] 24.868 24.255 24.5615