On the basis of experimental data on the local values of the combustion rate of condensed systems along the heat-conducting filaments placed therein, regression models were constructed to relate the value of the local combustion rate with such characteristics of heat-conducting filaments as the thermal diffusivity and melting point. The obtained regression model was used to assess a possible expansion of the variation range of the local combustion rate when using various crystalline forms of CVD diamonds as heat-conducting filaments. It was shown that a local increase in the combustion rate could exceed the baseline value by 200 times. The possibility of controlling the transformation of the combustion surface by using heat-conducting filaments with variable characteristics was confirmed.

На основе известных экспериментальных данных по локальным значениям скорости горения конденсированных систем вдоль размещенных в них теплопроводных нитей построены регрессионные модели, связывающие значение локальной скорости горения с такими характеристиками теплопроводных нитей, как коэффициент температуропроводности и температура плавления. На основе полученной регрессионной модели оценено возможное расширение диапазона изменения локальной скорости горения при использовании в качестве теплопроводных нитей различных кристаллических форм CVD-алмазов. Показано, что локальное увеличение скорости горения может достигать 200-кратного превышения над базовым уровнем. Показана возможность управления трансформацией поверхности горения за счет использования теплопроводных нитей с переменными характеристиками.

As early as the 1940s, it was discovered that the local combustion rate of condensed systems (CS) can be changed within a wide range of values by introducing heat-conducting elements (HCE) into CS. Typically, wires with diameters ranging from a few tens to 300–400 micrometres are used as heat-conducting elements. Wire HCE can be made in the form of long filaments, located along the axis of the CS array, and in the form of short sections of filaments, randomly located in the CS array.

In the 1960–70s, the question of using HCE in CS was experimentally investigated in detail [

In works [

In the present study, the known experimental data concerning the dependence of Ku on the characteristics of the applied HCE are summarised and an elaborated analytical model of the HCE impact on CS combustion has been developed. Based on the results obtained, the possibility of expanding the combustion rate adjustment range for CS with HCE is considered.

1.1. Known experimental data. Uncertainty analysis

The main source of experimental data regarding the HCE impact on the combustion rate is the study [

Table

HCE materials properties

A regression model of the CRIC dependence on the factors defining their value is to be developed.

Researchers of the problem under discussion agree that thermal HCE thermal diffusivity coefficient and HCE melting point should be used as factors defining the CRIC.

Certain problems in developing the regression model are caused by the lack of information on the accuracy of determining the CRIC in the known experimental studies. In this regard, we shall proceed from general ideas regarding the accuracy of experiments execution to determine the combustion rate of the CS.

1.2. Regression models development

We considered two regression models, differing in the number of factors used:

Ku=b0+b1*α+b2α2+b3*Тmp (1)

Ku=b0+b1* α+b2*α2, (2)

where bi are coefficients of the corresponding predictor variables included in the model.

When estimating regression model coefficients, it has to be considered that the value of Ku in the point of factor space corresponding to CS without HCE is determined with considerably higher accuracy than that for combustion of CS with HCE. The absolute values of uncertainties are unknown for Ku. The analysis of the models obtained below has shown that a change in the ratio of these uncertainties within the range from 2 to 30 generates very close values of the model coefficients. Therefore, to be definite, let us assume that the root-mean-square errors of Ku determination are the same for all materials in the table (except for CS) and are three times higher than the RMS error of Ku determination for CS. These assumptions allow us to form a weighting matrix, providing for various contributions of points in the factor space to the resultant regression relationship.

The estimated vector of the regression equation (1) under the adopted assumptions has the form of bT = (0.972, –3519, 1.069e8, 7.604e–4), and for the relation (2) bT = (1.672, 7310, 4.995e7 ). The quality of the resultant dependencies can be estimated by the graph in Figure 1 where the empirical CRIC values are plotted on X axis, and the predicted values of this quantity calculated by models (1) and (2) in the same points of factor space where experiments were conducted are plotted on Y axis.

As the data in Figure 1 show, the two-factor model (1) gives a significantly better description than the one-factor model (2). Checking the significance of the predictor variables included in this model confirms this statement. The estimate of the multiple correlation coefficient R of model (1) has a value of 0.985 and it is significant (empirical F-criterion value is 32.43 for degrees of freedom 4 and 4) at both 0.95 and 0.9. On the basis of the results provided above, let us accept model (1) as an empirical model to describe the CRIC dependence on HCE material properties for further consideration.

1.3. Extrapolating into the area of high thermal conductivity values

Earlier researchers have considered the application of HCE manufactured from almost every material available at the time. However, carbon was excluded from the analysis, while most interesting materials for the development of the discussed direction have been obtained recently on the basis of carbon. It is known that natural diamond has the highest thermal conductivity among the “natural” materials [

Let us estimate the possibility of expanding the CRIC range by using a CVD diamond filament. Let us establish the thermal diffusivity coefficient of such single crystal, optical grade material to be 0.001 m2 /s [

The application of carbon HCE is also advantageous due to the fact that combustion products of such material change the resulting composition of CS combustion products insignificantly in contrast to metal HCE, i.e. they will not cause additional losses of specific momentum even when using a considerable amount of HCE.

2.1. Analysis and adjustment of the Bakhmann – Lobanov model

In the studies [

However, in the process of forming the described relationship, the authors of [

Such a substitution noticeably complicates the resolution of the derived relations, but provides a more accurate description of the observed data.

As per the study [

where Δ – length of a wire segment protruding over the CS surface, D, ρM; CM – diameter, density and thermal capacity of the wire; λср – the CS thermal conductivity coefficient; Т0, Тs and Тlq, Тf – initial temperature, CS combustion surface temperature, melting point of wire and CS combustion gases temperature, respectively; heat exchange coefficient between the combustion gases and wire.

The coefficient is determined by the criterion relationship for heat exchange along the plate [

(4)

where Nu, Re, Pr are the Nusselt, Reynolds and Prandtl numbers, respectively. The first two quantities are calculated based on the wire diameter and combustion products parameters.

Equation (3) contains two unknown values: the local value of combustion rate w and the length of the wire section protruding into the flow Δ.

To solve this problem in accordance with the considerations suggested in [

(5)

Solving a system of relations (3–5), we obtain the following expression for the local combustion rate along the HCE:

(6)

Let us compare the values of w calculated based on (6) with the experimental data available in the literature and presented in the table.

Figure 2 compares experimental data from the table with the results of calculations as per relations (3–5) at the following values of CS parameters: D = 500 µm, λср = 0.227 W/m·C, Тf = 3000 °C, Тs = 600 °C, Т0 = 25 °C.

The graph in Figure 3 clearly shows a quadratic trend. Removing it from the theoretical relationship should provide significant incremental improvement on the predictive quality of the calculated relations.

2.2. Analysis of additional possibilities for controlling changes in the combustion surface of a charge by means of heat-conducting elements

Based on the considerations presented above, a new way of controlling the change in the combustion surface of charges with HCE can be proposed, namely, the usage of HCE with length-variable characteristics impacting the value of the local combustion rate. As shown above, there are two such quantities: wire diameter and thermal conductivity.

The latter appears to be easier for implementation than the former. It can be executed by applying a certain coating to the heat-conducting element in certain areas.

As can be seen from the graph in Figure 4, performing combustion rate control by means of varying the wire diameter requires significant changes in the diameter, which appears to be quite difficult.

In the present study, we shall consider the combustion rate control via the use of heat-conducting elements with variable thermal conductivity. Let us demonstrate the application of this technique for the purposes of controlling the nature of the combustion surface change in the engine charge of 9K38 Igla man-portable surface-to-air missile complex [

The diagrams for the missile engine and its propellant are shown in Figure 4.

The engine charge is reinforced by four silver heat-conducting filaments 6, the length of the charge is 9.5 diameters. The charge burns along the rear face 5, conical surface 4 and grooves 3. The side surface 2 and the front face 1 are armoured.

In case of uniformly structured heat-conducting filaments arranged parallel to the charge axis, the change in the charge combustion area and, accordingly, the intrachamber pressure and thrust is shown in Figure 5.

The operation of such an engine results in a significant dip in the combustion (and thrust) area in the course of switching from the launch mode to the cruise mode. It is possible to eliminate the mentioned dip by using filaments with variable thermal conductivity.

To solve the problem, several versions of the charge differing both in the length of the coated filament section and the number of coated filaments have been analysed.

Let us consider two versions of the charge in which elimination of the dip in the combustion surface diagram has been achieved.

Figure 6 shows the nature of change in the combustion surface area for the case when all four filaments are diamond-coated from their end and over the length ensuring the change of the filament material as the combustion front passes through it from the side grooves. In doing so, it has been possible to completely eliminate the subsidence of the combustion surface area. Several periodical jumps in the combustion surface are observed in the launching operation mode, however, their changes are fundamentally smaller than the dip in the graph presented in Figure 5 and, if necessary, they can be compensated by an additional adjustment of the thermal conductivity of the filament.

Figure 7 shows a variant where the diamond coating is applied only to two diametrically opposed filaments and the coating only covers the area where the filaments are exposed to the combustion front from the side grooves. In this version of the charge, there is a dip in the combustion area at the transition between the launching and cruising modes, but its magnitude is considerably smaller than that in the original design.

For descriptive purposes, the difference between the areas in the critical points (transition points between modes) is shown in Figure 8.

In case 8b, the cones occupy the entire combustion surface immediately after completion of the first combustion mode, resulting in the end face combustion at increased gas flow rate. In cases 8a and 8c, the cones do not have time to expand to the full size of the combustion surface by the time the critical point is reached, therefore, the combustion surfaces in the two cases under consideration are a combination of the cone surfaces and the ignited surfaces from the side grooves.

If required, the given modes can be additionally optimised, however, no such study has been performed since the purpose of the given paper is to demonstrate the general possibilities of the proposed methods for controlling changes in the combustion surface of solid fuel charges through the use of HCE.

The empirical data on the combustion rate increase coefficient of the CS with HCE depending on the physical properties of the applied HCE material have been analysed. One-factor and two-factor regression models for the combustion rate increase coefficient depending on the thermal diffusivity and the melting point of the HCE material have been developed. The possibility has been identified to enhance the combustion rate increase coefficient by 5–20 times when using diamond for filament production.

The Bakhmann – Lobanov method has been corrected to describe the dependence between the CS combustion rate and the use of heat-conducting filaments.

New methods of controlling changes in the combustion surface of charges through the use of heat-conducting elements have been proposed, expanding the possibilities of improving such charges.

The authors declare that there are no conflicts of interest present.