Combustion model of a pyrolyzable polymer in a high-enthalpy flow with an radially applied electrostatic field

. Polymers combustion in high-enthalpy flows is realized in many power plants. This process is energetically favorable, but there are also significant problems, the main of which is the low combustion rate. An effective method to increase the condensed component regression rate is the imposition of an electrostatic field on the polymer, the polymer reaction layer, and the combustion zone. The paper proposes a physical and mathematical model of the electrostatic field influence on the PMMA combustion rate in high-enthalpy flows. The combustion rate values calculated within the described model coincide with the experimental results within the error.


Introduction
The polymers combustion in high-enthalpy flows is realized in many power plants, including in the space industry.The use of this method is caused by the cheapness, safety and environmental friendliness of the fuel combustion process [1].However, low and irregular condensed component regression rate requires effective methods for combustion control development.One of these methods is the imposition of an electrostatic field.
The combustion parameters in a high-enthalpy flow with a radial electrostatic field investigation [2,3] has shown that this method increases the polymethylmethacrylate (PMMA) combustion rate (up to 30%), as well as a number of other power plants parameters.For further development of the combustion intensification in a high-enthalpy flow by an electrostatic field method, a physical and mathematical model that takes into account the changes introduced by the field, both in the gas and condensed phases is needed.
This work purpose is to identify the electrostatic field influence on combustion in a high-enthalpy flow mechanisms and to create a model that takes into account the field influence on the solid fuel component regression rate.

Physical and mathematical model of the electrostatic field influence on PMMA combustion
The polymer combustion process can be divided into the gas phase combustion and the condensed phase pyrolysis.The the electrostatic field influence on the gas phase is caused by the ion wind appearance [4] and a change in the combustion temperature [5].Due to the fact that an excessive amount of positively charged particles is observed in the PMMA flame with an oxidizer excess [6], during PMMA combustion in a high-enthalpy flow, the ion wind should bring the flame front closer to the material surface.This increases the heat flow into the condensed phase [7] and, as a result, increases the combustion rate.The dependence absence of the combustion rate on the electrodes polarity [8], the effect assessment of electric force on the combustion rate [9], do not allow to fully describe the detected changes only by the electrostatic field influence on the gas phase.
Due to the complexity of the processes occurring in the gas phase, and since the ion wind assessment showed its relatively small role, the study of phenomena occurring at the interface and in the condensed phase is of particular interest: the substance decomposition stimulation; processes that intensify the dispersion: electrodispersion, the instability growth in the electrostatic field.The last two are especially relevant for melting polymers, so they were not considered in this paper.
For PMMA burned in a high-enthalpy flow, the structure of the burnt fuel blocks surface was studied [10].This allowed to evaluate the electrostatic field role in the substance decomposition stimulating process.
The surface in the study was a "frozen" reaction layer.Cylinders or caverns open to the outside were observed perpendicular to the combustion surface of the channel, Fig. 1.It turned out that when a radial electrostatic field is applied, the number of caverns increases by 15-30%, depending on the applied potential difference [10].
At the same field strength, a change in the oxidizer flow rate does not cause a change in the caverns concentration and their average diameter.The main feature that occurs when the field strength increases is an increase in the caverns concentration.This happens due to the chemical reactions kinetics changes in the condensed phase in the electrostatic field presence.
The dependence of the relative change in the caverns concentration on the electrostatic field strength at the phase interface is obtained: where E is the field strength at the phases interface, V/m, α is the matching coefficient, which includes information about the fuel characteristics and combustion conditions, determined experimentally, m/V.
Thus, for a pyrolyzable polymer, the physical picture of the radially applied field effect on combustion in a high-enthalpy flow is comes down to the presence of several main factors shown in Fig. 2. In the gas phase, the flame turbulization process under the mass forces action from the field [9] can have the greatest influence, which intensifies mass transfer.This process is not considered in the work.
During the PMMA combustion the volumetric pyrolysis stage occurs with the caverns formation.Moreover, in the electrostatic field, their number is growing due to a decrease in the gasification centers formation work.These centers can be dispersion sources, therefore, there is a dispersion process and the combustion rate intensification.
An increase in the cavities number leads to an increase in the pyrolysis area, which is reflected in the solid component regression rate increase.
Based on the results obtained during the the solid component surface study, it is possible to modernize the combustion model in the channel obtained by Volkov, Mazing [11].
We assume that the combustion rate is composed of 2 parts: 1. the combustion rate determined by free combustion without dispersion; where ρv is the oxidizer flow density, dk is the channel diameter, µ is the gas viscosity, B is the injection coefficient, ρT is the solid fuel component density.The constants for calculations are taken from [11,12].The calculation results are shown in Fig. 3.It can be seen that the calculation and experiment results coincide with a fairly high accuracy, however, it is also worth taking into account the effects associated with a change in the surface layer dispersion in the field.
Based on the PMMA surface layer study results, it is possible to determine the mass of the substance dispersed from the caverns.Knowing the combustion surface area and the dispersion time, the dispersion rate is determined.
The time during which the dispersion of the mass from the observed caverns occurred is assumed to be approximately equal to the cavern formation time (this is an increase from 0 to d -diameter, which averages 13.7 microns for the field absence case and 12.6 microns when a field with 266 kV/m strength is applied).Knowing the linear pyrolysis rate determined from the equation: for PMMA (K0 = 3.3*10 13 , E = 170,000 J/mol, T = 625 K, ρ = 1180 kg/m 3 [12]): vpyr =0.00017 m/s, it is possible to determine the dispersion time from the caverns.Then t = 0.08 s for the field absence case and t =0.06 s in 266 kV/m field.
Dividing the mass dispersed from the caverns from a unit area by time, we get the mass dispersion rate, which can be (divided by the fuel density) converted into a linear one.
The dispersion rate can be quantified using the following equation: mi is the mass of the substance ejected from 1 cavern (pπr 2 h/2), t is the dispersion time, N is the cavern number per 1 m 2 .Moreover, N is a function of E (experimentally determined earlier).Then: where E is the field strength, V/m; α = 1.02*10 -6 m/V.Thus, the combustion rate part, taking into account the dispersion in the field, is obtained.Then, taking into account the dispersion in the electrostatic field change (cavities size and number change), for combustion rate we get the following result: where the index E hereafter denotes the field presence, 0 -its absence.In addition, the combustion surface area S (and, consequently, the area from which pyrolysis occurs) in the field also changes due to changes in the caverns number and their sizes.As a result: The results are shown in Fig. 4.

Conclusion
Thus, a physical and mathematical model of the radial electrostatic field effect on the pyrolyzable polymer (PMMA) channel combustion in a high-enthalpy flow has been developed.Combustion rate values calculated according to the described model coincide with the experimental results within the error.

Fig. 1 .
Fig. 1.Cylindrical caverns in the waste fuel unit of the PMMA.

Fig. 2 .
Fig. 2. The effect of a radially applied field on pyrolyzing polymer combustion in a high-enthalpy flow.

Fig. 3 .
Fig. 3. Dependence of the PMMA linear combustion rate on the oxidizer flux density in an electrostatic field (experiment and calculation).

Fig. 4 .
Fig. 4. Comparison of the PMMA combustion rates in the presence and in the absence of an electrostatic field.