毕业设计翻译.doc

中原工学院毕业设计（论文）译文
Mathematical Modeling of Chemical Conversionin Thin-Layer Exothermic Mixtures underPeriodic Electric-Spark Discharges

B. S. Seplyarskii,1 T. P. Ivleva, and E. A. Levashov2
Translated from Fizika Goreniya i Vzryva, Vol. 40, No. 3, pp. 59–68, May–June, 2004.
Original article submitted April 23, 2003.
Abstract:
The dynamics of coating production using a reaction mixture with thermoreactive electric-spark strengthening is studied numerically. It is shown that the main parameter that determines the thermal regime of coating is the initial thickness of the mixture layer. The parameter ranges for the process in a combustion regime and in a quasivolume conversion regime are determined. The effect of discharge frequency and the thermal characteristics of the reaction mixture and the substrate being strengthened on coating time is investigated. It is established that for a particular reaction mixture, the characteristic conversion temperature can be controlled by varying the electric discharge power and, hence, the heat flux at the active stage of the process,and for coating formation at this characteristic temperature, it is necessary that the thickness of the active layer be lower than a certain critical value.
Key words: mathematical modeling, chemical conversion, mixtures, action, charge.
One of the widely used methods for the surface strengthening of dies, rolls, and cutting tools is electric-spark alloying [1]. To apply functional coatings,Podlesov et al. [2, 3] used electrodes produced by self-propagating high-temperature synthesis (SHS) [4]. In this case, for each particular problem of surface strengthening, it was necessary to develop a technology to produce electrodes of the required composition.
Levashov et al. [5–7] were the first to propose to combine the processes of electric-spark alloying and SHS in the interelectrode gap. This method was called thermoreactive electric-spark strengthening (RESS). The idea of the TRESS method is that an electric discharge of a definite power not only produces transport of the alloying agent to the substrate but also initiates a chemical exothermic reaction between the components of the reactive mixture, which is placed in a tubular electrode (cathode). Unlike in the coating method proposed in [2, 3], in the TRESS method, the production of a coating of the required composition is achieved by varying the composition of the mixture placed in the tubular electrode and by varying the energy parameters of plant operation. For successful implementation of the method, it is necessary that the chemical reaction between the mixture components stop after detachment of the electrode from the surface being alloyed, which leads to current-circuit break. Experiments [5– 7] showed that the efficiency of the process increases markedly only when the heat effect of the chemical reaction is comparable to the pulsed-discharge energy.
In the present study, the thermal regimes of a new version of the TRESS process were first studied using mathematical modeling. The essence of the method is as follows: a layer of a reactive mixture of the required composition 60–300 μm thick is applied to the surface to be strengthened. The electric discharge resulting from periodic contact of the electrode with the strengthened product through the mixture layer ensures heating of the mixture in the contact area and initiates a chemical exothermic reaction between the mixture components. The chemical conversion leads to formation of a protective layer on the sample surface. In this version of the method, the electrode material is virtually not consumed and the required technical characteristics of the coating are attained by varying the composition of the mixture, the thickness of the applied layer, and the power and duration of the spark discharge. The goal of the theoretical part of this investigation was to study how the time characteristics of the chemical conversion and the thermal regime of coat application are affected by the main parameters of the process: the time th of contact of the electrode with the surface during which a current flows in the circuit and the mixture and substrate are heated; the time tad between the contacts of the electrode with the surface when there is no current flow in the circuit and adiabatic conditions are specified on the surface of the mixture; discharge power, which determines the heat-flux magnitude; the thickness of the mixture layer, which determines the thermal regimes of the reaction of the mixture and substrate preheating.

The following model of the process (Fig. 1) is considered. At the time t = 0, the electrode (cathode) is brought into contact with the mixture layer on the surface of the substrate (anode), thus completing the circuit. The electric current flowing in the circuit heats the reactive layer. The heating of the material abruptly increases the reaction rate in the surface layers of the mixture, which can lead to ignition and combustion of the mixture. The main resistance is assumed to be concentrated on the electrode–mixture contact boundary, where most of the heat is released. It is also assumed that the rate of this heat release is constant and proportional to the electric-discharge power. Therefore, for a mathematical description of the TRESS process, one can specify a constant heat flux on the electrode– mixture contact interface (second-kind boundary conditions). At the time t = th, the electrode is detached from the surface being alloyed. The electric circuit is broken, and adiabatic boundary conditions are specified on the surface of the mixture layer. In time △t= tad, the electrode is again brought into contact with the surface being alloyed, and the heating process is repeated. Thus, mathematical modeling of the application of an alloying coat using the modified TRESS method reduces to studying the thermal regime of the chemical conversion of the reactive layer with periodic switching-on of the heat source. It should be noted that the electrodes used to produce a spark discharge usually have diameters of 2–3 mm and the powder-mixture layer, as noted above, is 40–300 μm thick. If the spark diameter is assumed to be close to the electrode diameter, the dimension of the heating region far exceeds the thickness of the powder-mixture layer. Then, to study the thermal regime of the chemical conversion, we can confine ourselves to a one-dimensional formulation of the problem. We assume that the x axis is perpendicular to the surface of the mixture. The coordinate x = 0 corresponds to the outer surface of the mixture, the coordinate x = l1 to the mixture–substrate boundary, and x = l2 to the outer boundary of the substrate. In the traditional dimensionless variables of combustion theory, this process is described by the following equations:

The initial conditions are

The boundary conditions are

The following dimensionless variables and parameters
are used:

Here Ti (i = 1, 2) is the temperature (here and below the subscript 1 refers to the mixture layer, 2 to the metal substrate), Tin is the initial temperature of the substrate and reactive mixture, x is the dimensional coordinate, a and a0 are the current and initial concentrations of the rate-determining component in the starting mixture, T* is the scaling temperature, which is chosen from physical considerations, t is time, x* and t* are the coordinate and time scales, ηis the degree of conversion of the starting mixture to the reaction　product, th and tad is the heating time and the time during which there is no electric current in the circuit,respectively, λi (i = 1, 2) is the effective thermal conductivity, ci and ρi are the heat capacity and density of the mixture and substrate, E and k0 are the activation energy and preexponent of the reaction, R is the universal gas constant, Q is the heat effect of the reaction per unit mass of the condensed mixture, l1 is the thickness of the mixture layer, l2 is the total thickness of the mixture and substrate, ai (i = 1, 2) is the thermal diffusivity, q0 is the heat flux on the surface of the mixture at the discharge stage, θi (i = 1, 2), ξ and τare the dimensionless temperature, coordinate,and time, Th and Tad are the dimensionless heating time and the time during which there is no electric current in the circuit, respectively, L1 is the dimensionless thickness of the mixture layer, L2 is the total dimensionless thickness of the mixture layer and metal substrate, βandγare small parameters of combustion theory, σ0is the dimensionless energy flux at the heating stage, P is the dimensionless duration of the period, and n is the period number.
In the above equations, it is assumed that the interaction kinetics is described by a first-order reaction and that the heat loss to the ambient medium is equal to zero.
The main goal of this study is to find determiningparameter values for which the TRESS process can be performed under controllable conditions, i.e., the starting mixture can be converted to a coating without being heated far above a prescribed level. In our calculations, this level was the ignition temperature Tign of the starting mixture layer of great (infinite in the limiting case) thickness by a constant energy flux q0. From the thermal theory of ignition, it is known [8, 9] that if the temperature of the mixture surface exceeds Tign by two or three characteristic temperature intervals RT2 ign/E, the further increase in the temperature has a progressive nature (the higher the temperature, the higher the heating rate) and is well described by analytical formulas obtained in the theory of adiabatic thermal explosion [8]. It should be noted that it is unfeasible to control the temperature rise at the stage of thermal explosion, and, as a result of heat release from the chemical reaction, the surface layers are heated to temperatures far exceeding the adiabatic combustion temperature of the mixture Tad (Tad = Tin + a0Q/c1). Therefore, heating up of the reactive mixture higher than Tign is intolerable because this can lead to burnouts in the coating and nonuniformity of its physicochemical properties. The numerical value of Tign is a function of the external energy flow rate at the discharge stage, the thermal characteristics of the mixture and reaction products, and the heat effect and kinetic characteristics of the reaction of the starting components [8, 9]. Numerical calculations performed over a wide range of determining parameters showed that when the quantity Tign is chosen as the scaling temperature T*, the heating of the mixture can be divided into two stages. At T < T*, the heating is inert, and at T > T* there is conversion of most of the starting mixture.
To determine Tign, we use methods of the thermal theory of ignition [9, 10]. However, before calculating Tign, we note that for the problem in question, the other parameter that distinguishes between the different thermal regimes of coating application is the ignition delay Tign of a layer of infinite thickness. Indeed, if Th > Tign, combustion of the mixture occurs upon one contact with the electrode and the particular value of Th hardly affects the properties of the coating. For Th < Tign, the time of complete conversion of the mixture and, hence, the thermal regime of coating application is a function ofTh. Therefore, before analyzing the TRESS process, it is necessary to determine the temperature and ignition delay of a reactive material (at L1→∞) that has the same composition as the employed reactive mixture used under the second-kind boundary conditions (Th →∞).
Numerical calculations [9, 11] revealed the following main features of the ignition process, which should be taken into account by an approximate theory: up to the ignition moment, the temperature profile in the reactive mixture is close to the temperature distribution in an inert material with the same thermal characteristics; the time interval up to the moment when the integral heat release from the chemical reaction Qch becomes comparable with the heat release from the external source of heat σ0 almost coincides with the ignition time determined from the sharp rise in the temperature on the body surface. In view of the aforesaid, as Tign we use the body surface temperature that satisfies the

In calculating the integral in (6), we take into account the features of the temperature profile near the surface and ignore the product βθ(τ, ξ) compared to unity in the exponent. As can be seen from the numerical calculations in [9, 11, 12] and in our study, this profile is nearly linear with slope −σ0 up to attainment of equality (5). In this case, the temperature variation in the surface layers of the material is described with high accuracy by the equation

where θs(τ) is the temperature on the material surface ξ= 0. Since we consider highly activated reactions, for which E(Tign − Tin)/RT2 ign>>1, the major contribution to the integral (6) is from the surface layers of the material, in which the temperature change with respect to θs(τ) is about one or two characteristic temperature intervals (one or two unities in our dimensionless variables). Therefore, in the calculation of the integral (6), the obtained linear temperature profile can formally be continued to ξ→+∝ because at low temperatures (θ≤-θin), the chemical reaction rate is negligible and, hence, such variation in the lower bound does not affect the value of the integral (6). Since the ignition temperature of the materials is chosen as the scaling temperature, the expression for Qch at the ignition moment [T= Tign, θs(τ) = 0], in view of (7), becomes

Since the temperature profile is linear, expression (5) has a simple physical meaning: at the ignition moment, the heat release from the chemical reaction is equal to the heat flux to the cold material layers, which agrees with the critical condition of a hotspot thermal explosion with a linear temperature profile.
Substituting (8) into (5), we have

However, as shown in [11], it is more convenient to determine the temperature Tign from the equality

Whereσst= is the dimensionless heat flux from the reaction zone in a stationary combustion wave with a maximum reaction-zone temperature equal to the ignition temperature Tign.

14
周期性电火花放射下的薄层放热混合物化学变换的数学模拟
摘要：
人们对由电火花放热加强的感热体混合物反应形成的表层特性已经作了定量研究。研究显示决定表层材料热学性质的主要参数是混合物层的初始厚度。在燃烧过程和半体积转变过程中参数范围也被确定。我们也研究了放射频率的影响，以及反应混合物和在薄层时被加强的底层的热特性。我们也可以确定对一种特殊的反应混合物而言，它的特性转变温度可以通过改变放电电源来进行控制，由此处于这个过程活跃阶段的热流以及及此特性温度下的薄层形成也可以被控制，另外，反应层的厚度一定要比临界值要底，这非常有必要。
关键字：数学模拟，化学变化，混合物，反应，充电。
一个广泛用于加强模具，磙子和切割刀具表层的方法是电火花合金化。为了适用于功能性的表层，Podlesov et al. [2, 3]使用了由自蔓延高温（SHS）形成的电极[4]。在这种情况下，对于表层加强过程中遇到的每一个特殊问题，都需要一种新的科技，这种科技可以制造出包含所需要成分的电极。
Levashov et al. [5–7]第一次提出了把电火花合金化过程和电极间隙间的SHS结合起来的方法。这种方法叫做热反应电火花加强（TRESS）。TRESS方法的思想是一个确定电源的放电不仅产生把合金剂运至基质的输送，而且也启动了反应混合物组成成分间的热化学反应，这些混合物被放在一个管状电极中（阴极）。与[2，3]中介绍的表层