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Combustion wave in a two-layer solid fuel system
Combustion wave in a two-layer solid fuel system
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Combustion wave in a two-layer solid fuel system T.P. Miroshnichenko, E.O. Yakupov, V.V. Gubernov, V.N. Kurdyumov, A.A. Polezhaev PII: DOI: Reference:
S0307-904X(19)30568-2 https://doi.org/10.1016/j.apm.2019.09.037 APM 13041
To appear in:
Applied Mathematical Modelling
Received date: Revised date: Accepted date:
7 June 2019 18 September 2019 25 September 2019
Please cite this article as: T.P. Miroshnichenko, E.O. Yakupov, V.V. Gubernov, V.N. Kurdyumov, A.A. Polezhaev, Combustion wave in a two-layer solid fuel system, Applied Mathematical Modelling (2019), doi: https://doi.org/10.1016/j.apm.2019.09.037
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Highlights • Model of combustion wave propagation in two-layered thermally coupled system is analyzed. • Parametric study to find out affect of parameters on the process is undertaken. • The existence of two regimes of combustion wave propagation is shown. • It is revealed that superadiabatic temperatures can occur in the acceptor layer. • The importance of the results for the combustion synthesis is discussed.
1
Combustion wave in a two-layer solid fuel system T.P. Miroshnichenko∗1,2 , E.O. Yakupov1 , V.V. Gubernov1 , V.N. Kurdyumov3 , and A.A. Polezhaev1,4 1
P.N. Lebedev Physical Institute of the RAS, Leninsky prosp. 53, Moscow, 119991, Russia 2 Far Eastern Federal University, Sukhanova str. 8, Vladivostok, 690090, Russia 3 Department of Energy, CIEMAT, Avda. Complutense 40, Madrid 28040, Spain 4 Peoples Friendship University of Russia (RUDN University), 6 Miklukho-Maklaya Street, Moscow, 117198, Russia September 27, 2019
Abstract We consider the model describing propagation of a combustion wave in a system of two layers of different exothermic reacting materials under conditions of thermal contact between them through a common surface. This system is directly related to synthesis of advanced materials via the Self-propagating High temperature Synthesis technology when one of the reactants serves as a heat source (donor layer) for the other reacting material (acceptor layer) and facilitates the chemical reaction in the latter. The reaction sheet approximation is used and the parametric study of the boundaries of existence and characteristics of combustion waves in the system of layers is undertaken. The parameters of the process are identified which allow to achieve significantly superadiabatic peak temperatures of combustion in the acceptor layer. Keywords: combustion wave, flame synthesis, solid fuel, superadiabatic combustion
1
Introduction
The propagation of combustion waves in layered solid fuel samples is usually considered within the applied technology of the production of advanced materials, Self-propagating High temperature Synthesis [1] and in relation with the concept of chemical furnace [2, 3]. In [4] a system of two parallel reacting layers in thermal contact is studied. The flame front propagates along the common interface of these layers. Several cases are considered: (I) one of the layers is ∗ Corresponding
author: [email protected]
2
inert, while the other is reacting, (II) the thickness of the reacting layers is decreased and the system becomes effectively spatially homogeneous, (III) both layers exhibit exothermic reactions, (IV) one of the layers reacts exothermically, while the other – endothermically. It is discussed that in all cases it is possible to find the parameter values for which there exist multiple combustion wave solutions. The co-existence of several solutions was later confirmed for the models with thermally coupled inert and reacting layers [5] and for the homogeneous system with parallel endothermic and exothermic reactions [6]. The properties of combustion waves in a model with parallel exothermic reactions are studied in [7, 8], while the case of parallel endo- and exothermic reactions is discussed in [9, 10, 6] within the context of flame inhibition via introduction of the endothermic reaction path. In [11] the combustion of complex stacks of layered reacting materials is analyzed for the cases when the layers are arranged either vertically or horizontally. It is found that high superadiabatic temperatures of combustion can be attained for the inner layers of the stack structure. It is also pointed out that in the case of horizontal arrangement of the layers the melt in heavy products can penetrate from the upper to the lower layers. The combustion of a layered stack of reacting materials which takes into account the formation of melt and its flow into the stack is studied in [12]. The existence of both stationary and non-stationary regimes of combustion wave propagation is demonstrated. In [13] the propagation of combustion waves along the surface of the thermal contact of a multilayer system is numerically investigated in two spatial dimensions. It is supposed that there are two types of layers: either reacting and inert, or both reacting, however, having different high and low exothermicity. In the first case, it is shown that there is a critical ratio of thicknesses of the inert to reacting films, above which flame propagation is quenched. There may exist both steady and unsteady regimes of combustion wave propagation depending on the number of alternating layers in the stack. If the number of layers is increased and their thicknesses are decreased the model behavior tends to the case of a homogeneous solid fuel system. When both types of materials in a stack are reacting exothermically it is also discovered that the propagation of combustion front may be either steady or unsteady, while it depends on the overall number of layers and the relative amount of low-exothermic material. It is also noted that under different conditions the reaction fronts can propagate simultaneously in all layers or the fronts in low-exothermic layers retard with respect to fronts in the high-exothermic layers. The dynamics of flame oscillations in a system of parallel identical layers aligned along the direction of combustion wave propagation is studied in [14]. The layers have exactly the same thermochemical properties and are placed in thermal contact with each other. If the Zeldovich number is increased above the critical value for the onset of pulsation, there can occur two modes of flame oscillations: either in-phase or anti-phase pulsations. These regimes can co-exist for certain values of parameters. Combustion of multilayered fuels with the thicknesses of layers of nanometer scale is also discussed [15]. In contrast to the previous studies it is shown that once the thickness of the reactants becomes small the physical mechanism of reaction wave propagation changes. The process of mixing of reactants becomes the dominant effect governing the reaction process and the velocity of front propagation may become much higher than it is usually demonstrated for 3
combustion of solid fuels. Although there have been significant amount of studies carried out previously, it is still not clear what regimes of combustion wave propagation exist in the layered solid fuel systems and how they depend on the choice of thermophysical and chemical parameters of the reacting materials composing the layered structure as well as on geometrical properties of the latter. Investigation of these issues is the main objective of the current work.
2
Model
We consider two thin solid-fuel layers, which are in thermal contact with each other along a common plane interface. The layers are made of different reacting materials which can be characterized by thermophysical parameters: ρi - the density, λi - the thermal conductivity, cpi - the specific heat capacity at constant pressure, τi - characteristic chemical reaction time for the i-th material. The index i = 1, 2 refers to either the first or the second layer of the system. The infinitely thin reaction sheet model is used [16]. The adiabatic flame temperature of the first material is assumed to be higher than the adiabatic flame temperature of the second material without loss of generality. For this reason the latter is usually called an acceptor, while the former – a donor. The layers have finite thicknesses h1,2 in the direction normal to the common surface and the thickness L of the common surface. The stack is assumed to be infinite along the coordinate of flame front propagation. The heat losses to the ambient media are supposed to be negligible. The governing equations describing the propagation of combustion wave in such system of layers can be written in the one dimensional case as ∂ 2 θ1 ∂Y1 ∂θ1 = − − ξ(θ1 − θ2 ), ∂t ∂x2 ∂t
∂Y1 = −eZ1 (θ1 −1)/2 δ(x − φ1 ), ∂t
√ ∂Y2 = − αkeZ2 (θ2 −q)/2q δ(x − φ2 ), ∂t (1) where Yi is the mass fraction of fuel in i-th layer, φi is the coordinate of the reTi − T0 action sheet in the i-th layer, the dimensionless temperatures are θi = b . T1 − T0 Here Ti is the dimensional temperature of the i-th layer and Tib is the adiabatic flame temperature for the fuel in the i-th layer, T0 is the initial temperature of the system. The dimensionless parameters in the equations (1) are T b − T0 Zi is the Zel’dovich number for reacting material in the i-th layer; q = 2b T1 − T0 is the ratio of the temperature jumps in the reacting materials; k = τ1 /τ2 is Ei /RTib the ratio of the characteristic times scales, τi = A−1 , where Ai is the i e pre-exponential factor, Ei is the activation energy, and Tib is the adiabatic temperature of the i-th layer; R is the universal gas constant; α = κ2 /κ1 is the ratio of thermal diffusivities, κi = λi /ρi cpi , of the layers; ξ is the dimensionless heat ρ1 cp1 h1 transfer coefficient between the reacting layers; s = is the coefficient of ρ2 cp2 h2 asymmetry of the layers. ∂θ2 ∂ 2 θ2 ∂Y2 =α 2 −q − sξ(θ2 − θ1 ), ∂t ∂x ∂t
4
The time and coordinate are nondimensionalyzed in equations (1) over the q −1 characteristic time, τ1 , and length, δ = κ1 τ1 , scales for the combustion wave propagation in the first reacting material. We seek the solution to the set of equations (1) in a form of a travelling wave propagating with velocity, c, in the positive x-direction. The boundary conditions corresponding to this kind of solution are ∂θi /∂x = ∂Yi /∂x = 0
for x → −∞,
θi = Yi − 1 = 0
for x → +∞.
(2)
Introducing the coordinate x0 = x − ct co-moving with the travelling wave solution we can rewrite the set of governing equations as cY1x = eZ1 (θ1 −1)/2 δ(x),
θ1xx + cθ1x + cY1x − ξ(θ1 − θ2 ) = 0,
√
αkeZ2 (θ2 −q)/2q δ(x − δφ). (3) Here primes are omitted and it is assumed that due to the translational invariance of the travelling wave solution we can fix the position of the reaction sheet in the first layer at x = φ1 = 0 and define the location of the reaction zone in the second layer as δφ = φ2 − φ1 . If ξ = 0, then the equations (3) split into two subsystems describing the propagation of the combustion waves separately in the fuel layers 1 and 2. The solutions of the first pair of equations, subsystem 1, and the last pair of equations, subsystem 2, are known in this case. In the subsystem 1 the reaction front propagates with velocity c1 = 1 and has the adiabatic √ burned temperature θ1b = 1, while for the subsystem 2 the flame speed c2 = αk and the adiabatic flame temperature θ2b = q [17]. In the general case, ξ > 0, it can be shown by means of simple algebraic manipulations with the governing equations (3) that the burned temperature behind the travelling combustion wave for x → −∞ can be written as αθ2xx + cθ2x + qcY2x − sξ(θ2 − θ1 ) = 0,
θb = θ1b = θ2b =
cY2x =
s+q . 1+s
(4)
Since it was assumed that q < 1, then θb < 1, i.e. the temperature of the products is always smaller than the adiabatic temperature of combustion of donor material. It should be noted that in the case of very large values of ξ the difference of the temperatures of the layers becomes asymptotically small and for ξ → ∞ the model reduces to the case of the combustion of the fuel with two-step parallel reactions i.e. F1 → P1 and F2 → P2 , where F1,2 are different types of fuel leading to the formation of two types of products, P1,2 . As discussed in the introduction this system was analysed previously in a number of papers [7, 8, 9, 10, 6]. The characteristic values of parameters α, Zi and q are mostly governed by phys-chemical properties of materials and can be estimated as: α ∼ O(1), Zi ∼ O(10), and q < 1. Parameters k, ξ, s include various factors such as reaction rates, heat exchange between layers and geometry of the system and can vary in orders in magnitude. It should be noted, however, that the problem 5
can be considered as one-dimensional if the parameter of the heat exchange, ξ, is small. Since in this case the heat transfer along the direction of flame propagation is greater than the heat exchange between the layers. Here we define the standard set of parameters as α = 1, Z1 = 5, Z2 = 7, q = 0.9, s = 1 and use it as a reference case in subsequent calculations.
3
Combustion wave solution
The set of equations (3) is linear everywhere but the points x = 0 and x = δφ, where the reaction occurs and appropriate discontinuity conditions need to be applied. The x-axis can be divided into three regions depending on the sign of δφ: • (I) x < 0, (II) 0 < x < δφ, (III) x > δφ for δφ > 0; • (I) x < δφ, (II) δφ < x < 0, (III) x > 0 for δφ < 0. ∗
from the equations for Yi in the system (3) that c = eZ1 (θ1 −1)/2 = √ It Zfollows ∗ (θ −q)/2q αke 2 2 , where θ1∗ = θ1 (0) and θ2∗ = θ2 (δφ) are the temperatures at the reaction front for the first and second subsystems respectively. It is also assumed that fuel is completely consumed in the course of the reaction in each of the layers so that Y1 (x) = 1 − H(x) and Y2 (x) = 1 − H(x − δφ), where H(·) is the Heaviside step function. The temperatures at the reaction sheet can be thus written as θ1∗ = Z1−1 (ln c2 + Z1 ),
θ2∗ = qZ2−1 (ln(c2 /αk) + Z2 ).
(5)
The solution of the temperature equations in (3) θ1xx + cθ1x − ξ(θ1 − θ2 ) = 0,
αθ2xx + cθ2x − sξ(θ2 − θ1 ) = 0
(6)
in the regions (I − III) can be written as (j)
θi (x) =
4 X
(j)
βl kil exp(µl x),
(7)
l=1
where index j denotes the spatial region (I − III). The coefficients µi are the solutions of the characteristic equation µ[αµ3 + (1 + α)cµ2 + (c2 − (s + α)ξ)µ − cξ(1 + s)] = 0
(8)
while ki = {k1i , k2i } are the eigenvectors corresponding to µi . The analytic solution of the equation (8) is too cumbersome to be presented here. We sort the roots of (8) as µ1 = 0, µ2 < 0, and µ3,4 > 0. Using the boundary conditions I III (2) it can be found that β3,4 = 0 and β1,2 = 0. On the boundaries between the regions (I − III) the following matching conditions can be written • in the case δφ > 0: θ2I = θ2II , θ1I = θ1II = θ1∗ , II I Z1 (θ1∗ −1)/2 I II θ1x − θ1x = −e , θ2x = θ2x , θ1II = θ1III , II III θ1x = θ1x ,
θ2II = θ2III = θ2∗ , ∗ III II θ2x − θ2x = −α−1 qkeZ2 (θ2 −q)/2q , 6
for x = 0;
for
x = δφ;
(9)
(10)
• in the case δφ < 0: θ1I = θ1II , θ2I = θ2II = θ2∗ , ∗ II I II I θ1x − θ1x , θ2x − θ2x = −α−1 qkeZ2 (θ2 −q)/2q , θ1II = θ1III = θ1∗ , ∗ III II θ1x − θ1x = −eZ1 (θ1 −1)/2 ,
θ2II = θ2III , III II θ2x = θ2x ,
for x = 0;
(11)
x = δφ.
(12)
for
The set of 12 algebraic equations (5, 9, 10) provides a closed system of equaI II III ∗ tions for unknowns β1,2 , β1−4 , β3,4 , c, δφ, θ1,2 for the case δφ > 0. Similarly, I equations (5, 11, 12) provide a closed system of equations for unknowns β1,2 , II III ∗ β1−4 , β3,4 , c, δφ, θ1,2 for the case δφ < 0. Since it is hard to use the analytic solutions of the characteristic equation (8), we solve these sets of equations numerically. In the case α = 1 the characteristic equation (8) factorizes further and the roots can be found as p (13) µ1 = 0, µ2,3 = (−c ± c2 + 4ξ(1 + s))/2, µ4 = −c,
with the corresponding eigenvectors
k1,4 = {1, 1},
k2,3 = {1, −s}.
(14)
Substituting the expansion (7) into the matching conditions (9, 10) taking into account the relations (5) leads, after some algebraic manipulations, to the system of two equations for c and δφ eµ2 δφ [Γ(1 + s)(ln c2 + Z1 ) − Γ(s + q)Z1 − cZ1 ] + Z1 cq = 0,
(15) Γq(1 + s)(ln(c2 /αk) + Z2 ) − Z2 [Γ(q + se−cδφ ) + cs(q − eµ3 δφ )] = 0, p which is valid for δφ > 0 and here we define Γ = c2 + 4ξ(1 + s). Similarly, for δφ < 0 we can find from equations (7, 11, 12) eµ2 δφ [Γq(1 + s)(ln(c2 /αk)) + Z2 ) − Z2 (Γ(s + q) + cqs)] + cZ2 s = 0,
(16)
Γ(1 + s)(ln(c2 ) + Z1 ) − Z1 (Γ(s + qe−cδφ ) + c(1 − qeµ3 δφ )) = 0. The roots of equations (15, 16) are calculated numerically.
4
Results
The analysis of the solutions of algebraic equations for the steady propagating wave which are discussed in the section above has shown that the propagation mode of the combustion wave in two-layer systems is mainly determined by the parameter k = τ1 /τ2 , which specifies the ratio of the characteristic reaction times. In the case when k 1 the reaction rate in the second (acceptor) layer exceeds the reaction rate in the first (donor) layer. As a consequence, a leading front is formed in the acceptor layer, and it determines the properties of the combustion wave in the overall system. In Fig. 1 the solid line shows the dependence of the velocity of the common combustion wave on the parameter 7
4
C
3
2
1
0 0.01
0.1
1
k
10
Figure 1: The dependence of flame speed, c, on the ratio of the reaction times, k, in logarithmic scale for α = 1, q = 0.9, ξ = 0.1 and s = 1. The solid line represents the velocity of combustion wave in the system of layers, whereas the dependence of combustion wave velocity c2 on k for the second layer is plotted with the dashed line. k for the standard set of parameters. It is seen that as k increases, c(k) grows monotonically as well. For k 1 the velocity increases approaching the dashed line, which determines the speed √ of propagation of the combustion wave in pure fuel of the second layer i.e. c2 = αk. The distance, δφ, between the reaction fronts in layers 1 and 2 is plotted versus parameter k in Fig. 2 for standard parameter values. The larger k, the farther forward the front of combustion wave in the acceptor layer is. It can be seen from Fig. 2 that the distance between two fronts can reach values greater than the characteristic thermal thickness of the combustion front in each of the + layers by orders of magnitude. As k approaches the critical value kcrit ≈ 10 the traveling wave solution ceases to exist due to the fact that δφ grows infinitely + + for k → kcrit . At the same time, as k approaches the critical value kcrit , the velocity of the combustion wave reaches its upper limiting value, c ≈ 2.9. If the parameter k is of the order of unity, the fronts propagate almost parallel to each other, without a significant distance between them and the overall velocity of propagation of the combustion wave is close to the velocity of the combustion wave in a pure donor layer, c1 = 1. When k becomes less than one, the situation is inverted. The leading front is formed in the donor layer and the combustion front in the acceptor layer − becomes lagging. As k decreases, approaching the lower critical value kcrit , the − traveling wave solution ceases to exist for k ≤ kcrit in the same way as it takes place for the large values of k i.e. δφ → −∞. The velocity of the combustion wave monotonically decreases as k is reduced (see Fig. 1) due to the fact that
8
δφ
200
100
0
− 100
0.01
0.1
1
k
10
Figure 2: The dependence of the distance between fronts, δφ, on the ratio of the reaction times, k, in logarithmic scale for α = 1, q = 0.9, ξ = 0.1 c2 is getting smaller. In other words, the acceptor layer acts as thermal ballast in this case, and restrains the propagation velocity of the overall system. As k − approaches kcrit , the velocity monotonically decreases to its lower limit. The structure of the combustion wave front in the case when the wave in the acceptor layer is leading is illustrated in Fig. 3, where the temperature profiles θ1,2 (x) in the layers 1 and 2 of the system are plotted for k = 6 and standard values of other parameters. The solid line stands for the temperature distribution θ1 in the donor layer, while the dashed line corresponds to the temperature in the acceptor layer, θ2 . It can be seen from Fig. 3 how far ahead the combustion wave in the acceptor layer can advance with respect to the wave in the donor layer. The maximum temperature in the second layer is slightly lower than the adiabatic temperature of combustion of the acceptor material, θ2b . In contrast, the maximum temperature in the donor layer is significantly higher than the adiabatic combustion temperature in the pure donor material, θ1b . This is due to the fact that the leading wave in the layer 2 preheats layer 1 behind itself. At the trailing edge, a monotonic convergence of temperatures θ1,2 (x) to the equilibrium temperature of the products occurs, which for the selected parameters is θb = 0.95. Figure 4 illustrates the opposite situation, when the leading front is in the layer 1 and creates the condition for preheating for the wave in the second layer. This case is realized for small values of k, in particular, the curves in the figure are shown for k = 0.035. The maximum temperature in the leading wave is noticeably lower than the adiabatic one, θ1b , due to the fact that the donor layer gives much of its heat to the second layer. The maximum temperature in the second layer significantly exceeds the adiabatic combustion temperature in the pure acceptor material, θ2b . In Fig. 5 the dependence of the maximum temperature in each of the layers 9
1.5 1,2
1
0.5 1 2
0 − 40
−20
0
x
20
Figure 3: The temperature profiles, θi (x) for k = 6, α = 1, q = 0.9, ξ = 0.1 and s = 1.
1.5 1,2
1
0.5 1 2
0 −20
−10
0
10
20 x 30
Figure 4: The temperature profiles, θi (x) for k = 0.035, α = 1, q = 0.9, ξ = 0.1, and s = 1.
10
Figure 5: The dependence of the maximum temperature, θi , on the ratio of the reaction times, k, for α = 1, q = 0.9, ξ = 0.1, and s = 1. on k is shown. The solid line is the temperature max θ1 in the donor layer, the dashed line is the temperature max θ2 in the acceptor layer. The horizontal dashed lines indicate the adiabatic temperatures of combustion in the pure donor (θ1b = 1) and acceptor (θ2b = 0.9) materials, when there is no thermal interaction between the layers. The presented dependencies summarize the conclusions made above that the temperature in the leading wave falls below the adiabatic temperature for the corresponding medium due to the heat loss to the wave traveling behind. The lagging wave, in turn, receives additional thermal energy due to the traveling wave ahead of it and reaches superadiabatic values. The values of k ∼ 1 correspond to the transient region, when the temperatures in both layers become comparable. It can be concluded that for the realization of the combustion synthesis in the chemical furnace mode the regime realized at small values of the parameter k, that is when the leading wave propagates in the donor layer, is the most preferable. In this case, the donor layer, which is more exothermic, gives off a lot of its heat and contributes to the heating of the second layer significantly. So that the maximum temperature in the donor layer can exceed the adiabatic combustion temperature by more than 30%. Let us study next the dependence of the characteristics of various combustion regimes in a two-layer system on the parameter α, which determines the ratio of the thermal diffusivities of the second to the first layer. Figure 6 shows the flame speed c as a function of the parameter k for two values of parameter α. The solid line corresponds to the case α = 0.2, when the donor layer conducts heat better than the acceptor layer, and the dashed line corresponds to the opposite case α = 5. In both cases the curves are discontinued at the critical conditions for the existence of travelling combustion − + waves at k = kcrit and kcrit . It can be seen that the lower value of the critical − + parameter kcrit does not strongly depend on α, while the upper limits kcrit 11
C
3
2
1
0
0.01
0.1
1
10
100
k Figure 6: The dependence of flame speed c on k for α = 0.2 and α = 5. The other parameters are standard. + significantly differ. The larger α, the faster c(k) tends to the upper limit kcrit , when a solution in the form of a traveling wave cannot be found. It is seen that the dependence c(k) is getting steeper with the increase of α and the speed of combustion wave propagation monotonically increases as well. In Fig. 7 the distance δφ between the reaction fronts in layers 1 and 2 is plotted as a function of the ratio of reaction times, k, for two values of parameter α. It can be seen that for small values of k the distance between the fronts does not change significantly with the variation of α. This is in contrast to the situation of large values of k, when δφ(k) grows much faster for α = 5 in + comparison with the case α = 0.2. Consequently, the upper critical value, kcrit for the disappearance of the travelling wave solution is achieved for smaller k at larger values of α. Qualitatively, this behaviour can be understood if we assume that c1 does not depend on α, while the combustion wave speed in pure acceptor material, √ c2 ∼ α. The increase of α results in larger reaction front velocities of the second layer. It accelerates the propagation velocity of the overall combustion front in two-layer system and c increases with the growth of α. When parameter k is large and the wave in the second layer is leading, the increase of α accelerates the propagation of combustion wave in the acceptor material and the reaction front in the layer 2 advances farther ahead with respect to the front in the layer 1. It causes the growth of c and δφ, and the reduction of the critical value + of kcrit for the existence of the traveling wave solution in the system. In the case k 1 the front in the first layer is leading and the overall properties of the combustion wave in the layered system are mostly governed by the reaction wave in donor material. The variation of α mainly changes the characteristics
12
= 0.2
=5
10
0
−10 −20 0.01
0.1
1
10
100
k Figure 7: The dependence of the distance between fronts, δφ, on the ratio of the reaction times, k, for α = 0.2 and α = 5. The other parameter are standard. of the combustion of the layer 2, which plays a secondary role in this case, and the effect of α on the properties of the combustion wave in the layered system is less pronounced. Figure 8 shows the dependence of the maximum temperature achieved in the layers, max θ1 (solid lines) and max θ2 (dashed lines), on the parameter k for different α. Thick lines correspond to α = 0.2, thin lines – to α = 5. It can be seen that in the case of α = 0.2, compared to the case α = 5, the range of k values at which the wave in the donor layer has a superadiabatic temperature, max θ2 > θ2b , is much wider. This regime can be realized only when the reaction front in the first layer is leading. The reaction in the donor material is more exothermic than in the acceptor layer, so that the leading combustion front in the donor layer can effectively preheat the fresh mixture in the second layer along which the wave in the acceptor layer propagates. It can be concluded from Fig. 8 that for the same value of the parameter k the effectiveness of the combustion synthesis in the chemical furnace mode is higher, when α < 1 i.e. the thermal diffusivity of the first layer is larger than that of the second one. The best performance in terms of the maximum value of the − peak temperature in the acceptor layer is achieved for k → kcrit in both cases α = 0.2 and α = 5. However, the structure of the combustion wave is different in these two cases. For α = 5 the reaction front in the second layer is much wider than the reaction front in the first layer. As a result, the second layer plays the role of a thermal conduit and recuperates a part of heat towards the fresh mixture in the first layer. The situation is similar to the case of the system with inert highly heat conducting element [18, 5]. In the case of α = 0.2 the reaction front in the second layer is much narrower than it is in the first layer and the acceptor material only consumes heat from the donor layer. This leads
13
1.4 i
1.2 1
= 0.2
=5
0.8 0.6 0.01
0.1
1
10
k
100
Figure 8: The dependence of the maximum temperature, θi , on the ratio of the reaction times, k, for α = 0.2 and α = 5. The other parameters are standard. The solid and dashed lines show max θ1 and max θ2 , respectively. to a larger maximum temperature max θ2 in the system with α = 0.2 than it is for α = 5. Next we consider the change in the behavior of the system as the heat transfer coefficient ξ is varied. As ξ tends to zero, the set of equations (3) decouples into two independent subsets, each describing the propagation of combustion wave in the separate layers 1 and 2.√Since the velocities of combustion waves in pure materials are c1 = 1 and c2 = αk and differ in the general case, we can expect that the distance between the fronts in a thermally coupled system of two layers diverges as ξ → 0. Assuming ξ to be an asymptotically small parameter and expanding c in Taylor series over ξ, while δφ ∼ ξ −1 (δφ0 + ξδφ1 + ...), we can expand the equations (15) in asymptotic series over ξ. Collecting the terms of the same order and solving the corresponding equations, we can find the following asymptotic relations c2 ln 1 − (1 + s)(qZ2 )−1 ln c2 2 c = c2 − Z2 sc−1 ξ + O(ξ ) δφ = , 0 2 1+s −2 θ2 = q − 2qsc2 ξ + O(ξ 2 ), θ1 = 1 + 2 ln(c2 )/Z1 − (2Z2 s/Z1 c22 )ξ + O(ξ 2 ), (17) which are valid when the reaction front in the second layer is leading. Similarly, the following relations can be found from equations (16) c = 1 − Z1 ξ + O(ξ 2 ), θ1 = 1 − 2ξ + O(ξ 2 ),
ln (1 + 2(1 + s) ln c2 ) , 1+s −1 −1 θ2 = q − 2Z2 ln c2 − 2Z1 Z2 ξ + O(ξ 2 ),
δφ0 =
(18)
which can be applied for small values of k when the front in the donor layer is leading. 14
C
3
2
k=6
1 k=0.07 k=0.05
0 0
1 *
2
3
4
Figure 9: The dependence of flame speed c on ξ for k = 6, k = 0.07 and k = 0.05 shown with the solid lines. The short dashed and the long dashed lines represent the asymptotic behavior for ξ 1 (18, 17) and ξ 1 (19, 20), respectively. The thermal interaction between the layers becomes more intense with increasing ξ. Temperature difference |θ1 − θ2 | decreases, and in the limit ξ → ∞, model (1) tends to a single-temperature model of combustion wave propagation in a homogeneous medium with two parallel independent reactions. More formally, in the limit of large values of ξ, the asymptotic estimates of the combustion wave characteristics can be obtained from equations (15) as −Z1 (1 − q) Z1 (1 − q) 1+s c = exp , δφ = q − −2 ln c2 , (19) 2(1 + s) Z2 s Z2 s −1 −1 θ1 = θb , θ2 = q − 2Z2 ln c2 − Z1 Z2 (1 − θb ). The analysis of equations (16) shows that there is a critical value of k ∗ = exp[−(Z1 + s(1 − q)Z2 )/(1 + s)] such that for k < k ∗ the travelling wave solution exists for ξ < ξ ∗ and ceases to exist as ξ → ξ ∗ . In this limit δφ → ∞ and the characteristics of combustion wave satisfy the relations Z1 (Z2 s + 2(1 + s) ln c2 ) c2 ln c(Z1 + (1 + s) ln c) , ξ∗ = c = exp , (20) 2(1 + s)(Z1 − Z2 qs) (Z1 + 2(1 + s) ln c)2 As k tends to k ∗ the value of ξ ∗ → ∞ and the travelling wave solution exists for all values of ξ if k > k ∗ . The dependence of combustion wave velocity on ξ is plotted in Fig. 9 with the solid lines for k = 6, 0.07, and 0.05. The data is obtained by solving the equations (15) and (16). The asymptotic estimates in equations (18, 17) and (19, 20) are shown with the short dashed (ξ 1) and the long dashed (ξ 1) lines, respectively. c(ξ) is a monotonically decreasing function for all values of 15
i
1
1.2
2
k=6
1
k=0.05
0.8
k=6
0.6
k=0.05
0
1 *
2
3
Figure 10: The dependence of the maximum temperatures, θ1 and θ1 , on ξ for k = 6 and k = 0.05 shown with the solid and the dashed lines, respectively. The short dashed and the long dashed lines represent the asymptotic behavior for ξ 1 (18, 17) and ξ 1 (19, 20), respectively. parameter k. For ξ → 0 the velocity tends to asymptotic value in a linear way, while the distance between the fronts diverges, |δφ| → ∞ as it is prescribed by (18, 17). As ξ is increased c approaches the limiting values for ξ → ∞ plotted with the long dashed line in Fig. 9, if k > k ∗ ≈ 0.054. The c(ξ) curve is discontinued for k = 0.05 < k ∗ at the critical value shown with a cross symbol marked as ξ ∗ . In this case δφ → ∞ as ξ → ξ ∗ and the traveling wave solution does not exist for ξ > ξ ∗ . In Figure 10 the maximum temperature, θ1 (solid line) and θ2 (dashed line), achieved in the layers is plotted versus the heat transfer coefficient, ξ, for k = 0.05 and k = 6. The functions θi (ξ) are monotonically decreasing. The curves for k = 0.05 are discontinued as the critical value of parameter ξ = ξ ∗ for the existence of the travelling wave solution is reached. It is interesting to note that the weaker the thermal interaction between the layers, the higher the maximum temperatures in them. The maximum temperature of the leading front is always less than the adiabatic combustion temperature, however, it is approaching the adiabatic combustion temperature for ξ → 0. The maximum value of the temperature in the secondary wave can reach significantly superadiabatic values, especially for small values of the parameter ξ. In this case the distance between the fronts in the layers 1 and 2 becomes large and although the heat transfer coefficient is small the secondary wave still has enough time to gather the heat from the leading front. Since the second layer is designated as donor material in the current model, it follows from equations (18) that max θ2 can be increased above θ2b = q if Z2 and k are reduced. The influence of geometrical parameters on the propagation of combustion
16
C
2
k=5
1 k=0.05 k=0.03 -
s
s+
s+
0 0
10
20
s
Figure 11: The dependence of flame speed c on s for k = 5, k = 0.05 and k = 0.03. wave in a layered system can be understood from Fig. 11 and 12, where the velocity of reaction wave and the maximum temperature in the layers are presented as functions of s, which is a coefficient of asymmetry of the layers. Let us consider first the case of k = 5, when the leading wave is a combustion wave in the acceptor layer. In the case s → 0, the leading wave is not affected by the secondary wave in layer 1, as it can be seen from equations (3). Thus, the velocity of combustion wave c → c2 and max θ2 → θ2b . In contrast to that, the secondary wave in the donor layer is strongly influenced by the leading front and can adsorb the heat generated in it effectively, what according to equation (5) leads to a superadiabatic maximum temperature, max θ1 = 1 + 2Z1−1 ln c2 . Due to the fact that s enters only into the heat balance equation for the acceptor layer, its increase has a tiny effect on the lagging wave in the donor layer, while its effect on the leading wave in the acceptor layer is significant. From the equation for θ2 of system (3) it follows that for s 1 the temperatures θ1 and θ2 , come closer to each other and the distance between fronts decreases. As a consequence the leading front becomes inhibited to large extent by the lagging front, the maximum temperature θ1 decreases at large s, and the propagation velocity of the combustion waves tends to unity, which is the propagation velocity of the combustion wave in the pure donor layer. Thus, at large s, the effect of the combustion wave in the donor layer on the global process increases, despite this wave is lagging. The maximum temperature in the donor layer obviously tends to the adiabatic combustion temperature with increasing s (see Fig. 12). In contrast to this, the maximum temperature at the front of the leading combustion wave in the acceptor layer decreases, as a significant part of the heat is given to the wave, travelling behind. The asymptotic analysis of equations (15) for s → ∞ shows that in this limit c = 1, δφ = ln(Z2 /(qZ2 − 2q ln c2 )), 17
1.5 i
k = 0.05 1
k=5
k = 0.05
k=5
0.5
s+ 1 2
0 0
10
20 s
Figure 12: The dependence of the maximum temperature, θi , on s for k = 5 and k = 0.04. max θ1 = 1, max θ2 = q(1 − Z2−1 2 ln c2 ). For k 1 the wave in the layer 1 is the leading one, while the secondary combustion front is formed in the layer 2. The analysis of the solutions of equations (16) shows that for k > k1 ≈ 0.0452 the travelling wave solution exists for s < s+ . For s 1 the expansion of equations (16) in Taylor series near s = 0 allows to obtain the following behavior of the combustion wave speed: ! Z1 c2 qc2 p c = c2 + + 1 − q s, (21) 2 c22 + 4ξ
the maximum temperatures: max θ1 = 1 + Z −1 ln c2 and max θ2 = θ2b , while the distance between the fronts remains finite. The characteristics of combustion wave in this case are illustrated in Fig. 11 and 12 for k = 0.05. The dashed line is plotted in Fig. 11 according to equation (21). The increase of s to the critical value s+ , which is indicated in Fig. 11 and 12 with the vertical dotted line, results in the infinite growth of δφ and the travelling wave solution ceases to exist for s > s+ . If k < k1 there occurs the other critical condition s = s− for the existence of the traveling combustion wave so that s− < s < s+ . This situation is illustrated in Fig. 11, where c(s) is plotted for k = 0.03 and the lower bound of the existence of the solution is shown with the dotted vertical line marked as ‘s− ’. As k is further decreased s− approaches s+ and at k2 ≈ 0.0215, the two critical values coincide and there are no travelling wave solutions for k < k2 . The dependence of the maximum temperatures in the layers on parameter s is plotted in Fig. 12 for k = 5 and k = 0.05. It is seen that the substantially superadiabatic temperatures can be reached in the acceptor layer for k 1 and 18
20
1.2
C 10
1.0 k=2
0
0.8 k=0.04
0.6 0.4
− 10 − 20
0
0.2
0.4
0.6
0.8 q
− 30
Figure 13: The dependence of flame speed c and the distance between fronts δφ on q for k = 2 and k = 0.04. The solid lines indicate the velocity of the combustion wave, and the dashed lines indicate the distances between the reaction fronts large values of s. The effect of the ratio of heats of the reactions, q, on the characteristics of the travelling combustion wave propagating in a layered system is summarized in Fig. 13 and 14. Two dependencies are combined in Fig. 13 for the sake of clarity. The solid line indicates the velocity of the combustion wave, c(q), and the dashed lines indicate the distances between the reaction fronts, δφ(q), propagating in two media. The dependencies are presented for two values of the parameter k = 0.04 and 2, reflecting the situations when the leading wave is in the donor and the acceptor layer, respectively. For large k (more than 10 for given parameters), the propagation velocity is almost independent of q, since the distance between the waves, δφ, increases significantly, exceeding the thermal thicknesses of both fronts. If k is decreased to values of the order of O(1), the distance between the fronts is reduced and the thermal interaction between them increases (see the thick dashed line in Fig. 13 shown for k = 2). When q decreases, the secondary wave in the donor layer receives less heat, the temperature in the layer 1 becomes smaller and the rate of the reaction is reduced. The secondary wave starts to lag behind the leading wave more and more, and the leading wave, in turn, receives less heat from the relatively hot donor layer. The overall velocity decreases and at q ≈ 0.15, the distance between the waves becomes so large that they stop to influence each other, δφ → ∞ and the travelling wave solution disappears. Increasing q at k = 2 results in growth of the combustion wave velocity. The leading front achieves higher burning temperatures and preheats the secondary wave to the superadiabatic temperature. This significantly accelerates the secondary wave and it approaches the leading front closely i.e. δφ is strongly reduced. The relative share of energy that the leading front gives to the wave
19
1.5 1,2
1.0
k=0.04
k=2
0.5
0.0
0
0.2
0.4
0.6
0.8
q
Figure 14: The dependence of the maximum temperature, θi , on q for k = 2 and k = 0.04.Maximum temperatures are indicated by solid lines in the donor and by dashed lines in the acceptor travelling behind is getting smaller. This leads to the acceleration of the whole system. It is also seen from Fig. 14, where the thick lines indicate maximum temperatures in the donor (solid line) and acceptor (dashed line) layers. The situation changes completely for the case of k = 0.04. The increase of q results in a decrease of the overall propagation velocity and the growth of the distance between the fronts in layers 1 and 2. At certain q ≈ 0.95, the solution disappears, since δφ → −∞. As q is decreased, the secondary wave receives a relatively larger amount of heat from the leading wave. The second layer burns at superadiabatic temperatures and the role of superadiabacity √ increases as q is reduced, since the rate of the reaction in the layer 2: w2 ∼ k exp(Z2 (max θ2 − q)/2q). Although, k 1 it becomes compensated by the exponential temperature dependence at small values of q and the secondary wave starts to approach the leading wave as q is getting smaller, so that δφ → 0 as q ≈ 0.35. In Fig. 13 and 14 the curves for k = 0.04 are discontinued at this point. The overall velocity is increased as a result of superadiabacity effect. However, as q tends to 1 the relative influence of superadiabacity is decreased, the reaction in the layer 2 slows down and the secondary wave starts to retard. The distance between the fronts grows and the velocity decrease as a result of this.
5
Conclusions
In this paper we consider a model of combustion wave propagation in a compound system consisting of two layers of reacting materials with different geometrical and thermochemical properties placed in thermal contact. This system 20
is directly relevant to the applied technology of combustion synthesis of advanced materials, when one of the reactants serves as a heat source (donor layer) for the other reacting material (acceptor layer) and facilitates the chemical reaction in the latter. Based on the reaction sheet model we undertook a parametric study to find out which parameters affect the process of reaction wave propagation in this system. In particular, it is shown, that depending on the ratio of characteristic times of the reactions there can occur two regimes of combustion wave propagation. If the rate of the reaction on the donor layer is faster, than the rate of the reaction in the acceptor layer, the leading wave is formed in the donor layer. It can advance far ahead of the reaction front in the acceptor layer and at certain critical ratio of the reaction rates in two materials the combustion wave ceases to exist. In the same way, if the reaction rate in the acceptor material is larger, the leading wave is formed in the acceptor layer and there exists critical ratio of the reaction rates at which the combustion wave cannot propagate in both layers synchronously as a single travelling wave. The distance between the reaction fronts at the critical conditions for the existence of the travelling wave tends to infinity. Propagation of the leading front creates the conditions for preheating of the secondary wave travelling behind it in the different layer and this can lead to the significantly superadiabatic temperature of combustion in the material in which the secondary front propagates. The maximum local temperature in the secondary wave is achieved at the critical conditions for the existence of the travelling wave. In particular the largest peak value of the temperature in the acceptor layer is observed when the parameter k is small and the leading front is formed in the donor material. Obviously, from the point of view of synthesis, this is the most favorable regime of process. Thus we only discuss the case, when the leading wave is formed in the donor layer and it is desirable to maximize the peak temperature attained in the acceptor layer. The ratio of the thermal diffusivity of materials constituting the layers has strong impact on the characteristics of the travelling combustion wave. The velocity of combustion wave grows with the increase of the thermal diffusivity of the acceptor layer, since the flame speed in the pure acceptor material is increased in this case. On the other hand, if the acceptor material has much larger thermal diffusivity than the donor layer, α 1, the temperature distribution in the acceptor layer widens and it recuperates part of the heat energy towards the leading front in the donor material. In contrast to that, if α 1, then the temperature distribution in the acceptor layer is more compact and it adsorbs heat generated in the leading front in the donor material more effectively. As a result, the peak local temperature in the acceptor material is the highest in the case when the thermal diffusivity in the donor material is larger than in the acceptor material. The heat exchange between the layers may be governed by both fundamental (Kapitsa thermal resistance [19]) and technological (structure of the materials and topological properties of the surfaces, preparation of the thermal contact etc.) aspects. In the regime, when the leading wave is formed in the donor layer, it is shown that the maximum peak temperature is attained in the acceptor material in the case of small coefficient of thermal contact. In this case the leading front determines the burning rate of the system and the temperature close to the adiabatic is developed in it. The distance between the leading and secondary fronts is much larger than the flame thicknesses of reaction waves in 21
both layers and the acceptor material is preheated to substantially superadiabatic temperature. From the point of view of the heat capacity of the layers it is desirable to have the system with large values of s, for example, when the thickness of the acceptor layer is smaller in comparison to the donor layer. We have also found that the maximum temperature of the acceptor layer almost linearly depends on the ratio of the heats of the reactions at least when q is not a small number. The relative increase of the local temperature above the adiabatic flame temperature of the acceptor material weakly depends on the ratio of heats of the reaction. In terms of absolute temperature increase of the acceptor layer, it is obviously desirable to have larger heats of the reaction in both materials.
6
Acknowledgments
This work was supported by RFBR [projects numbers 18-48-700037, 17-0100070 and 18-38-00523]. The publication has been prepared with the support of the ”RUDN University Program 5-100” (recipient A.A. Polezhaev, mathematical model development).
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[8] S. Margolis and B. Matkowsky, “Flame propagation with multiple fuels,” SIAM J. Appl. Math., vol. 42, no. 5, pp. 982–1003, 1982. doi:10.1137/ 0142070. [9] A. Lazarovici, S. Kalliadasis, J. Merkin, and S. Scott, “Flame quenching through endothermic reaction,” J. Eng. Math., vol. 44, no. 3, pp. 207–228, 2002. doi:10.1023/A:1020934620657. [10] P. Simon, S. Kallidasis, and S. Scott, “Inhibition of flame propagation by an endothermic reaction,” IMA J. Appl. Math., vol. 68, pp. 537–562, 2011. doi:10.1093/imamat/68.5.537. [11] G. Ksandopulo and A. Baidel’dinova, “Combustion in a system of conjugated layers and high-temperature synthesis of materials,” Russ. J. Appl. Chem+, vol. 77, no. 3, pp. 364–368, 2004. doi:10.1023/B:RJAC. 0000031272.43669.c0. [12] V. Prokofiev and V. Smolyakov, “On the theory of self-propagating high-temperature synthesis in layered systems,” Combust. Explo. Shock+, vol. 48, no. 5, pp. 636–641, 2012. doi:10.1134/S0010508212050152. [13] B. Prokofiev and V. Smolyakov, “Gasless combustion of a system of thermally coupled layers,” Combust. Explo. Shock+, vol. 52, no. 1, pp. 62–66, 2016. doi:10.1134/S0010508216010081. [14] A. Aldushin, A. Bayliss, and B. Matkowsky, “Dynamics in layer models of solid flame propagation,” Physica D, vol. 143, no. 1-4, pp. 109–137, 2000. doi:10.1016/S0167-2789(00)00098-1. [15] F. Baras, V. Turlo, O. Politano, S. Vadchenko, A. Rogachev, and A. Mukasyan, “Shs in ni/al nanofoils: a review of experiments and molecular dynamics simulations,” Adv. Eng. Mater., vol. 20, no. 8, p. 1800091, 2018. doi:10.1002/adem.201800091. [16] B. Matkowsky and G. Sivashinsky, “Propagation of a pulsating reaction front in solid fuel combustion,” SIAM J. Appl. Math., vol. 35, no. 3, pp. 465–478, 1978. doi:10.1137/0135038. [17] R. Weber, G. Mercer, H. Sidhu, and B. Gray, “Combustion waves for gases (le = 1) and solids (le → ∞),” Proc. R. Soc. Lond. A., no. 453, pp. 1105– 1118, 1997. doi:10.1098/rspa.1997.0062. [18] V. Gubernov, V. Kurdyumov, and A. Kolobov, “Flame propagation in a composite solid energetic material,” Combust. Flame, vol. 161, no. 8, pp. 2209 – 2214, 2014. doi:10.1016/j.combustflame.2014.01.023. [19] P. Kapitsa, Collected papers of P.L. Kapitza. Pergamon Press, 1964.
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Journal Pre-proof
Combustion wave in a two-layer solid fuel system T.P. Miroshnichenko, E.O. Yakupov, V.V. Gubernov, V.N. Kurdyumov, A.A. Polezhaev PII: DOI: Reference:
S0307-904X(19)30568-2 https://doi.org/10.1016/j.apm.2019.09.037 APM 13041
To appear in:
Applied Mathematical Modelling
Received date: Revised date: Accepted date:
7 June 2019 18 September 2019 25 September 2019
Please cite this article as: T.P. Miroshnichenko, E.O. Yakupov, V.V. Gubernov, V.N. Kurdyumov, A.A. Polezhaev, Combustion wave in a two-layer solid fuel system, Applied Mathematical Modelling (2019), doi: https://doi.org/10.1016/j.apm.2019.09.037
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Highlights • Model of combustion wave propagation in two-layered thermally coupled system is analyzed. • Parametric study to find out affect of parameters on the process is undertaken. • The existence of two regimes of combustion wave propagation is shown. • It is revealed that superadiabatic temperatures can occur in the acceptor layer. • The importance of the results for the combustion synthesis is discussed.
1
Combustion wave in a two-layer solid fuel system T.P. Miroshnichenko∗1,2 , E.O. Yakupov1 , V.V. Gubernov1 , V.N. Kurdyumov3 , and A.A. Polezhaev1,4 1
P.N. Lebedev Physical Institute of the RAS, Leninsky prosp. 53, Moscow, 119991, Russia 2 Far Eastern Federal University, Sukhanova str. 8, Vladivostok, 690090, Russia 3 Department of Energy, CIEMAT, Avda. Complutense 40, Madrid 28040, Spain 4 Peoples Friendship University of Russia (RUDN University), 6 Miklukho-Maklaya Street, Moscow, 117198, Russia September 27, 2019
Abstract We consider the model describing propagation of a combustion wave in a system of two layers of different exothermic reacting materials under conditions of thermal contact between them through a common surface. This system is directly related to synthesis of advanced materials via the Self-propagating High temperature Synthesis technology when one of the reactants serves as a heat source (donor layer) for the other reacting material (acceptor layer) and facilitates the chemical reaction in the latter. The reaction sheet approximation is used and the parametric study of the boundaries of existence and characteristics of combustion waves in the system of layers is undertaken. The parameters of the process are identified which allow to achieve significantly superadiabatic peak temperatures of combustion in the acceptor layer. Keywords: combustion wave, flame synthesis, solid fuel, superadiabatic combustion
1
Introduction
The propagation of combustion waves in layered solid fuel samples is usually considered within the applied technology of the production of advanced materials, Self-propagating High temperature Synthesis [1] and in relation with the concept of chemical furnace [2, 3]. In [4] a system of two parallel reacting layers in thermal contact is studied. The flame front propagates along the common interface of these layers. Several cases are considered: (I) one of the layers is ∗ Corresponding
author: [email protected]
2
inert, while the other is reacting, (II) the thickness of the reacting layers is decreased and the system becomes effectively spatially homogeneous, (III) both layers exhibit exothermic reactions, (IV) one of the layers reacts exothermically, while the other – endothermically. It is discussed that in all cases it is possible to find the parameter values for which there exist multiple combustion wave solutions. The co-existence of several solutions was later confirmed for the models with thermally coupled inert and reacting layers [5] and for the homogeneous system with parallel endothermic and exothermic reactions [6]. The properties of combustion waves in a model with parallel exothermic reactions are studied in [7, 8], while the case of parallel endo- and exothermic reactions is discussed in [9, 10, 6] within the context of flame inhibition via introduction of the endothermic reaction path. In [11] the combustion of complex stacks of layered reacting materials is analyzed for the cases when the layers are arranged either vertically or horizontally. It is found that high superadiabatic temperatures of combustion can be attained for the inner layers of the stack structure. It is also pointed out that in the case of horizontal arrangement of the layers the melt in heavy products can penetrate from the upper to the lower layers. The combustion of a layered stack of reacting materials which takes into account the formation of melt and its flow into the stack is studied in [12]. The existence of both stationary and non-stationary regimes of combustion wave propagation is demonstrated. In [13] the propagation of combustion waves along the surface of the thermal contact of a multilayer system is numerically investigated in two spatial dimensions. It is supposed that there are two types of layers: either reacting and inert, or both reacting, however, having different high and low exothermicity. In the first case, it is shown that there is a critical ratio of thicknesses of the inert to reacting films, above which flame propagation is quenched. There may exist both steady and unsteady regimes of combustion wave propagation depending on the number of alternating layers in the stack. If the number of layers is increased and their thicknesses are decreased the model behavior tends to the case of a homogeneous solid fuel system. When both types of materials in a stack are reacting exothermically it is also discovered that the propagation of combustion front may be either steady or unsteady, while it depends on the overall number of layers and the relative amount of low-exothermic material. It is also noted that under different conditions the reaction fronts can propagate simultaneously in all layers or the fronts in low-exothermic layers retard with respect to fronts in the high-exothermic layers. The dynamics of flame oscillations in a system of parallel identical layers aligned along the direction of combustion wave propagation is studied in [14]. The layers have exactly the same thermochemical properties and are placed in thermal contact with each other. If the Zeldovich number is increased above the critical value for the onset of pulsation, there can occur two modes of flame oscillations: either in-phase or anti-phase pulsations. These regimes can co-exist for certain values of parameters. Combustion of multilayered fuels with the thicknesses of layers of nanometer scale is also discussed [15]. In contrast to the previous studies it is shown that once the thickness of the reactants becomes small the physical mechanism of reaction wave propagation changes. The process of mixing of reactants becomes the dominant effect governing the reaction process and the velocity of front propagation may become much higher than it is usually demonstrated for 3
combustion of solid fuels. Although there have been significant amount of studies carried out previously, it is still not clear what regimes of combustion wave propagation exist in the layered solid fuel systems and how they depend on the choice of thermophysical and chemical parameters of the reacting materials composing the layered structure as well as on geometrical properties of the latter. Investigation of these issues is the main objective of the current work.
2
Model
We consider two thin solid-fuel layers, which are in thermal contact with each other along a common plane interface. The layers are made of different reacting materials which can be characterized by thermophysical parameters: ρi - the density, λi - the thermal conductivity, cpi - the specific heat capacity at constant pressure, τi - characteristic chemical reaction time for the i-th material. The index i = 1, 2 refers to either the first or the second layer of the system. The infinitely thin reaction sheet model is used [16]. The adiabatic flame temperature of the first material is assumed to be higher than the adiabatic flame temperature of the second material without loss of generality. For this reason the latter is usually called an acceptor, while the former – a donor. The layers have finite thicknesses h1,2 in the direction normal to the common surface and the thickness L of the common surface. The stack is assumed to be infinite along the coordinate of flame front propagation. The heat losses to the ambient media are supposed to be negligible. The governing equations describing the propagation of combustion wave in such system of layers can be written in the one dimensional case as ∂ 2 θ1 ∂Y1 ∂θ1 = − − ξ(θ1 − θ2 ), ∂t ∂x2 ∂t
∂Y1 = −eZ1 (θ1 −1)/2 δ(x − φ1 ), ∂t
√ ∂Y2 = − αkeZ2 (θ2 −q)/2q δ(x − φ2 ), ∂t (1) where Yi is the mass fraction of fuel in i-th layer, φi is the coordinate of the reTi − T0 action sheet in the i-th layer, the dimensionless temperatures are θi = b . T1 − T0 Here Ti is the dimensional temperature of the i-th layer and Tib is the adiabatic flame temperature for the fuel in the i-th layer, T0 is the initial temperature of the system. The dimensionless parameters in the equations (1) are T b − T0 Zi is the Zel’dovich number for reacting material in the i-th layer; q = 2b T1 − T0 is the ratio of the temperature jumps in the reacting materials; k = τ1 /τ2 is Ei /RTib the ratio of the characteristic times scales, τi = A−1 , where Ai is the i e pre-exponential factor, Ei is the activation energy, and Tib is the adiabatic temperature of the i-th layer; R is the universal gas constant; α = κ2 /κ1 is the ratio of thermal diffusivities, κi = λi /ρi cpi , of the layers; ξ is the dimensionless heat ρ1 cp1 h1 transfer coefficient between the reacting layers; s = is the coefficient of ρ2 cp2 h2 asymmetry of the layers. ∂θ2 ∂ 2 θ2 ∂Y2 =α 2 −q − sξ(θ2 − θ1 ), ∂t ∂x ∂t
4
The time and coordinate are nondimensionalyzed in equations (1) over the q −1 characteristic time, τ1 , and length, δ = κ1 τ1 , scales for the combustion wave propagation in the first reacting material. We seek the solution to the set of equations (1) in a form of a travelling wave propagating with velocity, c, in the positive x-direction. The boundary conditions corresponding to this kind of solution are ∂θi /∂x = ∂Yi /∂x = 0
for x → −∞,
θi = Yi − 1 = 0
for x → +∞.
(2)
Introducing the coordinate x0 = x − ct co-moving with the travelling wave solution we can rewrite the set of governing equations as cY1x = eZ1 (θ1 −1)/2 δ(x),
θ1xx + cθ1x + cY1x − ξ(θ1 − θ2 ) = 0,
√
αkeZ2 (θ2 −q)/2q δ(x − δφ). (3) Here primes are omitted and it is assumed that due to the translational invariance of the travelling wave solution we can fix the position of the reaction sheet in the first layer at x = φ1 = 0 and define the location of the reaction zone in the second layer as δφ = φ2 − φ1 . If ξ = 0, then the equations (3) split into two subsystems describing the propagation of the combustion waves separately in the fuel layers 1 and 2. The solutions of the first pair of equations, subsystem 1, and the last pair of equations, subsystem 2, are known in this case. In the subsystem 1 the reaction front propagates with velocity c1 = 1 and has the adiabatic √ burned temperature θ1b = 1, while for the subsystem 2 the flame speed c2 = αk and the adiabatic flame temperature θ2b = q [17]. In the general case, ξ > 0, it can be shown by means of simple algebraic manipulations with the governing equations (3) that the burned temperature behind the travelling combustion wave for x → −∞ can be written as αθ2xx + cθ2x + qcY2x − sξ(θ2 − θ1 ) = 0,
θb = θ1b = θ2b =
cY2x =
s+q . 1+s
(4)
Since it was assumed that q < 1, then θb < 1, i.e. the temperature of the products is always smaller than the adiabatic temperature of combustion of donor material. It should be noted that in the case of very large values of ξ the difference of the temperatures of the layers becomes asymptotically small and for ξ → ∞ the model reduces to the case of the combustion of the fuel with two-step parallel reactions i.e. F1 → P1 and F2 → P2 , where F1,2 are different types of fuel leading to the formation of two types of products, P1,2 . As discussed in the introduction this system was analysed previously in a number of papers [7, 8, 9, 10, 6]. The characteristic values of parameters α, Zi and q are mostly governed by phys-chemical properties of materials and can be estimated as: α ∼ O(1), Zi ∼ O(10), and q < 1. Parameters k, ξ, s include various factors such as reaction rates, heat exchange between layers and geometry of the system and can vary in orders in magnitude. It should be noted, however, that the problem 5
can be considered as one-dimensional if the parameter of the heat exchange, ξ, is small. Since in this case the heat transfer along the direction of flame propagation is greater than the heat exchange between the layers. Here we define the standard set of parameters as α = 1, Z1 = 5, Z2 = 7, q = 0.9, s = 1 and use it as a reference case in subsequent calculations.
3
Combustion wave solution
The set of equations (3) is linear everywhere but the points x = 0 and x = δφ, where the reaction occurs and appropriate discontinuity conditions need to be applied. The x-axis can be divided into three regions depending on the sign of δφ: • (I) x < 0, (II) 0 < x < δφ, (III) x > δφ for δφ > 0; • (I) x < δφ, (II) δφ < x < 0, (III) x > 0 for δφ < 0. ∗
from the equations for Yi in the system (3) that c = eZ1 (θ1 −1)/2 = √ It Zfollows ∗ (θ −q)/2q αke 2 2 , where θ1∗ = θ1 (0) and θ2∗ = θ2 (δφ) are the temperatures at the reaction front for the first and second subsystems respectively. It is also assumed that fuel is completely consumed in the course of the reaction in each of the layers so that Y1 (x) = 1 − H(x) and Y2 (x) = 1 − H(x − δφ), where H(·) is the Heaviside step function. The temperatures at the reaction sheet can be thus written as θ1∗ = Z1−1 (ln c2 + Z1 ),
θ2∗ = qZ2−1 (ln(c2 /αk) + Z2 ).
(5)
The solution of the temperature equations in (3) θ1xx + cθ1x − ξ(θ1 − θ2 ) = 0,
αθ2xx + cθ2x − sξ(θ2 − θ1 ) = 0
(6)
in the regions (I − III) can be written as (j)
θi (x) =
4 X
(j)
βl kil exp(µl x),
(7)
l=1
where index j denotes the spatial region (I − III). The coefficients µi are the solutions of the characteristic equation µ[αµ3 + (1 + α)cµ2 + (c2 − (s + α)ξ)µ − cξ(1 + s)] = 0
(8)
while ki = {k1i , k2i } are the eigenvectors corresponding to µi . The analytic solution of the equation (8) is too cumbersome to be presented here. We sort the roots of (8) as µ1 = 0, µ2 < 0, and µ3,4 > 0. Using the boundary conditions I III (2) it can be found that β3,4 = 0 and β1,2 = 0. On the boundaries between the regions (I − III) the following matching conditions can be written • in the case δφ > 0: θ2I = θ2II , θ1I = θ1II = θ1∗ , II I Z1 (θ1∗ −1)/2 I II θ1x − θ1x = −e , θ2x = θ2x , θ1II = θ1III , II III θ1x = θ1x ,
θ2II = θ2III = θ2∗ , ∗ III II θ2x − θ2x = −α−1 qkeZ2 (θ2 −q)/2q , 6
for x = 0;
for
x = δφ;
(9)
(10)
• in the case δφ < 0: θ1I = θ1II , θ2I = θ2II = θ2∗ , ∗ II I II I θ1x − θ1x , θ2x − θ2x = −α−1 qkeZ2 (θ2 −q)/2q , θ1II = θ1III = θ1∗ , ∗ III II θ1x − θ1x = −eZ1 (θ1 −1)/2 ,
θ2II = θ2III , III II θ2x = θ2x ,
for x = 0;
(11)
x = δφ.
(12)
for
The set of 12 algebraic equations (5, 9, 10) provides a closed system of equaI II III ∗ tions for unknowns β1,2 , β1−4 , β3,4 , c, δφ, θ1,2 for the case δφ > 0. Similarly, I equations (5, 11, 12) provide a closed system of equations for unknowns β1,2 , II III ∗ β1−4 , β3,4 , c, δφ, θ1,2 for the case δφ < 0. Since it is hard to use the analytic solutions of the characteristic equation (8), we solve these sets of equations numerically. In the case α = 1 the characteristic equation (8) factorizes further and the roots can be found as p (13) µ1 = 0, µ2,3 = (−c ± c2 + 4ξ(1 + s))/2, µ4 = −c,
with the corresponding eigenvectors
k1,4 = {1, 1},
k2,3 = {1, −s}.
(14)
Substituting the expansion (7) into the matching conditions (9, 10) taking into account the relations (5) leads, after some algebraic manipulations, to the system of two equations for c and δφ eµ2 δφ [Γ(1 + s)(ln c2 + Z1 ) − Γ(s + q)Z1 − cZ1 ] + Z1 cq = 0,
(15) Γq(1 + s)(ln(c2 /αk) + Z2 ) − Z2 [Γ(q + se−cδφ ) + cs(q − eµ3 δφ )] = 0, p which is valid for δφ > 0 and here we define Γ = c2 + 4ξ(1 + s). Similarly, for δφ < 0 we can find from equations (7, 11, 12) eµ2 δφ [Γq(1 + s)(ln(c2 /αk)) + Z2 ) − Z2 (Γ(s + q) + cqs)] + cZ2 s = 0,
(16)
Γ(1 + s)(ln(c2 ) + Z1 ) − Z1 (Γ(s + qe−cδφ ) + c(1 − qeµ3 δφ )) = 0. The roots of equations (15, 16) are calculated numerically.
4
Results
The analysis of the solutions of algebraic equations for the steady propagating wave which are discussed in the section above has shown that the propagation mode of the combustion wave in two-layer systems is mainly determined by the parameter k = τ1 /τ2 , which specifies the ratio of the characteristic reaction times. In the case when k 1 the reaction rate in the second (acceptor) layer exceeds the reaction rate in the first (donor) layer. As a consequence, a leading front is formed in the acceptor layer, and it determines the properties of the combustion wave in the overall system. In Fig. 1 the solid line shows the dependence of the velocity of the common combustion wave on the parameter 7
4
C
3
2
1
0 0.01
0.1
1
k
10
Figure 1: The dependence of flame speed, c, on the ratio of the reaction times, k, in logarithmic scale for α = 1, q = 0.9, ξ = 0.1 and s = 1. The solid line represents the velocity of combustion wave in the system of layers, whereas the dependence of combustion wave velocity c2 on k for the second layer is plotted with the dashed line. k for the standard set of parameters. It is seen that as k increases, c(k) grows monotonically as well. For k 1 the velocity increases approaching the dashed line, which determines the speed √ of propagation of the combustion wave in pure fuel of the second layer i.e. c2 = αk. The distance, δφ, between the reaction fronts in layers 1 and 2 is plotted versus parameter k in Fig. 2 for standard parameter values. The larger k, the farther forward the front of combustion wave in the acceptor layer is. It can be seen from Fig. 2 that the distance between two fronts can reach values greater than the characteristic thermal thickness of the combustion front in each of the + layers by orders of magnitude. As k approaches the critical value kcrit ≈ 10 the traveling wave solution ceases to exist due to the fact that δφ grows infinitely + + for k → kcrit . At the same time, as k approaches the critical value kcrit , the velocity of the combustion wave reaches its upper limiting value, c ≈ 2.9. If the parameter k is of the order of unity, the fronts propagate almost parallel to each other, without a significant distance between them and the overall velocity of propagation of the combustion wave is close to the velocity of the combustion wave in a pure donor layer, c1 = 1. When k becomes less than one, the situation is inverted. The leading front is formed in the donor layer and the combustion front in the acceptor layer − becomes lagging. As k decreases, approaching the lower critical value kcrit , the − traveling wave solution ceases to exist for k ≤ kcrit in the same way as it takes place for the large values of k i.e. δφ → −∞. The velocity of the combustion wave monotonically decreases as k is reduced (see Fig. 1) due to the fact that
8
δφ
200
100
0
− 100
0.01
0.1
1
k
10
Figure 2: The dependence of the distance between fronts, δφ, on the ratio of the reaction times, k, in logarithmic scale for α = 1, q = 0.9, ξ = 0.1 c2 is getting smaller. In other words, the acceptor layer acts as thermal ballast in this case, and restrains the propagation velocity of the overall system. As k − approaches kcrit , the velocity monotonically decreases to its lower limit. The structure of the combustion wave front in the case when the wave in the acceptor layer is leading is illustrated in Fig. 3, where the temperature profiles θ1,2 (x) in the layers 1 and 2 of the system are plotted for k = 6 and standard values of other parameters. The solid line stands for the temperature distribution θ1 in the donor layer, while the dashed line corresponds to the temperature in the acceptor layer, θ2 . It can be seen from Fig. 3 how far ahead the combustion wave in the acceptor layer can advance with respect to the wave in the donor layer. The maximum temperature in the second layer is slightly lower than the adiabatic temperature of combustion of the acceptor material, θ2b . In contrast, the maximum temperature in the donor layer is significantly higher than the adiabatic combustion temperature in the pure donor material, θ1b . This is due to the fact that the leading wave in the layer 2 preheats layer 1 behind itself. At the trailing edge, a monotonic convergence of temperatures θ1,2 (x) to the equilibrium temperature of the products occurs, which for the selected parameters is θb = 0.95. Figure 4 illustrates the opposite situation, when the leading front is in the layer 1 and creates the condition for preheating for the wave in the second layer. This case is realized for small values of k, in particular, the curves in the figure are shown for k = 0.035. The maximum temperature in the leading wave is noticeably lower than the adiabatic one, θ1b , due to the fact that the donor layer gives much of its heat to the second layer. The maximum temperature in the second layer significantly exceeds the adiabatic combustion temperature in the pure acceptor material, θ2b . In Fig. 5 the dependence of the maximum temperature in each of the layers 9
1.5 1,2
1
0.5 1 2
0 − 40
−20
0
x
20
Figure 3: The temperature profiles, θi (x) for k = 6, α = 1, q = 0.9, ξ = 0.1 and s = 1.
1.5 1,2
1
0.5 1 2
0 −20
−10
0
10
20 x 30
Figure 4: The temperature profiles, θi (x) for k = 0.035, α = 1, q = 0.9, ξ = 0.1, and s = 1.
10
Figure 5: The dependence of the maximum temperature, θi , on the ratio of the reaction times, k, for α = 1, q = 0.9, ξ = 0.1, and s = 1. on k is shown. The solid line is the temperature max θ1 in the donor layer, the dashed line is the temperature max θ2 in the acceptor layer. The horizontal dashed lines indicate the adiabatic temperatures of combustion in the pure donor (θ1b = 1) and acceptor (θ2b = 0.9) materials, when there is no thermal interaction between the layers. The presented dependencies summarize the conclusions made above that the temperature in the leading wave falls below the adiabatic temperature for the corresponding medium due to the heat loss to the wave traveling behind. The lagging wave, in turn, receives additional thermal energy due to the traveling wave ahead of it and reaches superadiabatic values. The values of k ∼ 1 correspond to the transient region, when the temperatures in both layers become comparable. It can be concluded that for the realization of the combustion synthesis in the chemical furnace mode the regime realized at small values of the parameter k, that is when the leading wave propagates in the donor layer, is the most preferable. In this case, the donor layer, which is more exothermic, gives off a lot of its heat and contributes to the heating of the second layer significantly. So that the maximum temperature in the donor layer can exceed the adiabatic combustion temperature by more than 30%. Let us study next the dependence of the characteristics of various combustion regimes in a two-layer system on the parameter α, which determines the ratio of the thermal diffusivities of the second to the first layer. Figure 6 shows the flame speed c as a function of the parameter k for two values of parameter α. The solid line corresponds to the case α = 0.2, when the donor layer conducts heat better than the acceptor layer, and the dashed line corresponds to the opposite case α = 5. In both cases the curves are discontinued at the critical conditions for the existence of travelling combustion − + waves at k = kcrit and kcrit . It can be seen that the lower value of the critical − + parameter kcrit does not strongly depend on α, while the upper limits kcrit 11
C
3
2
1
0
0.01
0.1
1
10
100
k Figure 6: The dependence of flame speed c on k for α = 0.2 and α = 5. The other parameters are standard. + significantly differ. The larger α, the faster c(k) tends to the upper limit kcrit , when a solution in the form of a traveling wave cannot be found. It is seen that the dependence c(k) is getting steeper with the increase of α and the speed of combustion wave propagation monotonically increases as well. In Fig. 7 the distance δφ between the reaction fronts in layers 1 and 2 is plotted as a function of the ratio of reaction times, k, for two values of parameter α. It can be seen that for small values of k the distance between the fronts does not change significantly with the variation of α. This is in contrast to the situation of large values of k, when δφ(k) grows much faster for α = 5 in + comparison with the case α = 0.2. Consequently, the upper critical value, kcrit for the disappearance of the travelling wave solution is achieved for smaller k at larger values of α. Qualitatively, this behaviour can be understood if we assume that c1 does not depend on α, while the combustion wave speed in pure acceptor material, √ c2 ∼ α. The increase of α results in larger reaction front velocities of the second layer. It accelerates the propagation velocity of the overall combustion front in two-layer system and c increases with the growth of α. When parameter k is large and the wave in the second layer is leading, the increase of α accelerates the propagation of combustion wave in the acceptor material and the reaction front in the layer 2 advances farther ahead with respect to the front in the layer 1. It causes the growth of c and δφ, and the reduction of the critical value + of kcrit for the existence of the traveling wave solution in the system. In the case k 1 the front in the first layer is leading and the overall properties of the combustion wave in the layered system are mostly governed by the reaction wave in donor material. The variation of α mainly changes the characteristics
12
= 0.2
=5
10
0
−10 −20 0.01
0.1
1
10
100
k Figure 7: The dependence of the distance between fronts, δφ, on the ratio of the reaction times, k, for α = 0.2 and α = 5. The other parameter are standard. of the combustion of the layer 2, which plays a secondary role in this case, and the effect of α on the properties of the combustion wave in the layered system is less pronounced. Figure 8 shows the dependence of the maximum temperature achieved in the layers, max θ1 (solid lines) and max θ2 (dashed lines), on the parameter k for different α. Thick lines correspond to α = 0.2, thin lines – to α = 5. It can be seen that in the case of α = 0.2, compared to the case α = 5, the range of k values at which the wave in the donor layer has a superadiabatic temperature, max θ2 > θ2b , is much wider. This regime can be realized only when the reaction front in the first layer is leading. The reaction in the donor material is more exothermic than in the acceptor layer, so that the leading combustion front in the donor layer can effectively preheat the fresh mixture in the second layer along which the wave in the acceptor layer propagates. It can be concluded from Fig. 8 that for the same value of the parameter k the effectiveness of the combustion synthesis in the chemical furnace mode is higher, when α < 1 i.e. the thermal diffusivity of the first layer is larger than that of the second one. The best performance in terms of the maximum value of the − peak temperature in the acceptor layer is achieved for k → kcrit in both cases α = 0.2 and α = 5. However, the structure of the combustion wave is different in these two cases. For α = 5 the reaction front in the second layer is much wider than the reaction front in the first layer. As a result, the second layer plays the role of a thermal conduit and recuperates a part of heat towards the fresh mixture in the first layer. The situation is similar to the case of the system with inert highly heat conducting element [18, 5]. In the case of α = 0.2 the reaction front in the second layer is much narrower than it is in the first layer and the acceptor material only consumes heat from the donor layer. This leads
13
1.4 i
1.2 1
= 0.2
=5
0.8 0.6 0.01
0.1
1
10
k
100
Figure 8: The dependence of the maximum temperature, θi , on the ratio of the reaction times, k, for α = 0.2 and α = 5. The other parameters are standard. The solid and dashed lines show max θ1 and max θ2 , respectively. to a larger maximum temperature max θ2 in the system with α = 0.2 than it is for α = 5. Next we consider the change in the behavior of the system as the heat transfer coefficient ξ is varied. As ξ tends to zero, the set of equations (3) decouples into two independent subsets, each describing the propagation of combustion wave in the separate layers 1 and 2.√Since the velocities of combustion waves in pure materials are c1 = 1 and c2 = αk and differ in the general case, we can expect that the distance between the fronts in a thermally coupled system of two layers diverges as ξ → 0. Assuming ξ to be an asymptotically small parameter and expanding c in Taylor series over ξ, while δφ ∼ ξ −1 (δφ0 + ξδφ1 + ...), we can expand the equations (15) in asymptotic series over ξ. Collecting the terms of the same order and solving the corresponding equations, we can find the following asymptotic relations c2 ln 1 − (1 + s)(qZ2 )−1 ln c2 2 c = c2 − Z2 sc−1 ξ + O(ξ ) δφ = , 0 2 1+s −2 θ2 = q − 2qsc2 ξ + O(ξ 2 ), θ1 = 1 + 2 ln(c2 )/Z1 − (2Z2 s/Z1 c22 )ξ + O(ξ 2 ), (17) which are valid when the reaction front in the second layer is leading. Similarly, the following relations can be found from equations (16) c = 1 − Z1 ξ + O(ξ 2 ), θ1 = 1 − 2ξ + O(ξ 2 ),
ln (1 + 2(1 + s) ln c2 ) , 1+s −1 −1 θ2 = q − 2Z2 ln c2 − 2Z1 Z2 ξ + O(ξ 2 ),
δφ0 =
(18)
which can be applied for small values of k when the front in the donor layer is leading. 14
C
3
2
k=6
1 k=0.07 k=0.05
0 0
1 *
2
3
4
Figure 9: The dependence of flame speed c on ξ for k = 6, k = 0.07 and k = 0.05 shown with the solid lines. The short dashed and the long dashed lines represent the asymptotic behavior for ξ 1 (18, 17) and ξ 1 (19, 20), respectively. The thermal interaction between the layers becomes more intense with increasing ξ. Temperature difference |θ1 − θ2 | decreases, and in the limit ξ → ∞, model (1) tends to a single-temperature model of combustion wave propagation in a homogeneous medium with two parallel independent reactions. More formally, in the limit of large values of ξ, the asymptotic estimates of the combustion wave characteristics can be obtained from equations (15) as −Z1 (1 − q) Z1 (1 − q) 1+s c = exp , δφ = q − −2 ln c2 , (19) 2(1 + s) Z2 s Z2 s −1 −1 θ1 = θb , θ2 = q − 2Z2 ln c2 − Z1 Z2 (1 − θb ). The analysis of equations (16) shows that there is a critical value of k ∗ = exp[−(Z1 + s(1 − q)Z2 )/(1 + s)] such that for k < k ∗ the travelling wave solution exists for ξ < ξ ∗ and ceases to exist as ξ → ξ ∗ . In this limit δφ → ∞ and the characteristics of combustion wave satisfy the relations Z1 (Z2 s + 2(1 + s) ln c2 ) c2 ln c(Z1 + (1 + s) ln c) , ξ∗ = c = exp , (20) 2(1 + s)(Z1 − Z2 qs) (Z1 + 2(1 + s) ln c)2 As k tends to k ∗ the value of ξ ∗ → ∞ and the travelling wave solution exists for all values of ξ if k > k ∗ . The dependence of combustion wave velocity on ξ is plotted in Fig. 9 with the solid lines for k = 6, 0.07, and 0.05. The data is obtained by solving the equations (15) and (16). The asymptotic estimates in equations (18, 17) and (19, 20) are shown with the short dashed (ξ 1) and the long dashed (ξ 1) lines, respectively. c(ξ) is a monotonically decreasing function for all values of 15
i
1
1.2
2
k=6
1
k=0.05
0.8
k=6
0.6
k=0.05
0
1 *
2
3
Figure 10: The dependence of the maximum temperatures, θ1 and θ1 , on ξ for k = 6 and k = 0.05 shown with the solid and the dashed lines, respectively. The short dashed and the long dashed lines represent the asymptotic behavior for ξ 1 (18, 17) and ξ 1 (19, 20), respectively. parameter k. For ξ → 0 the velocity tends to asymptotic value in a linear way, while the distance between the fronts diverges, |δφ| → ∞ as it is prescribed by (18, 17). As ξ is increased c approaches the limiting values for ξ → ∞ plotted with the long dashed line in Fig. 9, if k > k ∗ ≈ 0.054. The c(ξ) curve is discontinued for k = 0.05 < k ∗ at the critical value shown with a cross symbol marked as ξ ∗ . In this case δφ → ∞ as ξ → ξ ∗ and the traveling wave solution does not exist for ξ > ξ ∗ . In Figure 10 the maximum temperature, θ1 (solid line) and θ2 (dashed line), achieved in the layers is plotted versus the heat transfer coefficient, ξ, for k = 0.05 and k = 6. The functions θi (ξ) are monotonically decreasing. The curves for k = 0.05 are discontinued as the critical value of parameter ξ = ξ ∗ for the existence of the travelling wave solution is reached. It is interesting to note that the weaker the thermal interaction between the layers, the higher the maximum temperatures in them. The maximum temperature of the leading front is always less than the adiabatic combustion temperature, however, it is approaching the adiabatic combustion temperature for ξ → 0. The maximum value of the temperature in the secondary wave can reach significantly superadiabatic values, especially for small values of the parameter ξ. In this case the distance between the fronts in the layers 1 and 2 becomes large and although the heat transfer coefficient is small the secondary wave still has enough time to gather the heat from the leading front. Since the second layer is designated as donor material in the current model, it follows from equations (18) that max θ2 can be increased above θ2b = q if Z2 and k are reduced. The influence of geometrical parameters on the propagation of combustion
16
C
2
k=5
1 k=0.05 k=0.03 -
s
s+
s+
0 0
10
20
s
Figure 11: The dependence of flame speed c on s for k = 5, k = 0.05 and k = 0.03. wave in a layered system can be understood from Fig. 11 and 12, where the velocity of reaction wave and the maximum temperature in the layers are presented as functions of s, which is a coefficient of asymmetry of the layers. Let us consider first the case of k = 5, when the leading wave is a combustion wave in the acceptor layer. In the case s → 0, the leading wave is not affected by the secondary wave in layer 1, as it can be seen from equations (3). Thus, the velocity of combustion wave c → c2 and max θ2 → θ2b . In contrast to that, the secondary wave in the donor layer is strongly influenced by the leading front and can adsorb the heat generated in it effectively, what according to equation (5) leads to a superadiabatic maximum temperature, max θ1 = 1 + 2Z1−1 ln c2 . Due to the fact that s enters only into the heat balance equation for the acceptor layer, its increase has a tiny effect on the lagging wave in the donor layer, while its effect on the leading wave in the acceptor layer is significant. From the equation for θ2 of system (3) it follows that for s 1 the temperatures θ1 and θ2 , come closer to each other and the distance between fronts decreases. As a consequence the leading front becomes inhibited to large extent by the lagging front, the maximum temperature θ1 decreases at large s, and the propagation velocity of the combustion waves tends to unity, which is the propagation velocity of the combustion wave in the pure donor layer. Thus, at large s, the effect of the combustion wave in the donor layer on the global process increases, despite this wave is lagging. The maximum temperature in the donor layer obviously tends to the adiabatic combustion temperature with increasing s (see Fig. 12). In contrast to this, the maximum temperature at the front of the leading combustion wave in the acceptor layer decreases, as a significant part of the heat is given to the wave, travelling behind. The asymptotic analysis of equations (15) for s → ∞ shows that in this limit c = 1, δφ = ln(Z2 /(qZ2 − 2q ln c2 )), 17
1.5 i
k = 0.05 1
k=5
k = 0.05
k=5
0.5
s+ 1 2
0 0
10
20 s
Figure 12: The dependence of the maximum temperature, θi , on s for k = 5 and k = 0.04. max θ1 = 1, max θ2 = q(1 − Z2−1 2 ln c2 ). For k 1 the wave in the layer 1 is the leading one, while the secondary combustion front is formed in the layer 2. The analysis of the solutions of equations (16) shows that for k > k1 ≈ 0.0452 the travelling wave solution exists for s < s+ . For s 1 the expansion of equations (16) in Taylor series near s = 0 allows to obtain the following behavior of the combustion wave speed: ! Z1 c2 qc2 p c = c2 + + 1 − q s, (21) 2 c22 + 4ξ
the maximum temperatures: max θ1 = 1 + Z −1 ln c2 and max θ2 = θ2b , while the distance between the fronts remains finite. The characteristics of combustion wave in this case are illustrated in Fig. 11 and 12 for k = 0.05. The dashed line is plotted in Fig. 11 according to equation (21). The increase of s to the critical value s+ , which is indicated in Fig. 11 and 12 with the vertical dotted line, results in the infinite growth of δφ and the travelling wave solution ceases to exist for s > s+ . If k < k1 there occurs the other critical condition s = s− for the existence of the traveling combustion wave so that s− < s < s+ . This situation is illustrated in Fig. 11, where c(s) is plotted for k = 0.03 and the lower bound of the existence of the solution is shown with the dotted vertical line marked as ‘s− ’. As k is further decreased s− approaches s+ and at k2 ≈ 0.0215, the two critical values coincide and there are no travelling wave solutions for k < k2 . The dependence of the maximum temperatures in the layers on parameter s is plotted in Fig. 12 for k = 5 and k = 0.05. It is seen that the substantially superadiabatic temperatures can be reached in the acceptor layer for k 1 and 18
20
1.2
C 10
1.0 k=2
0
0.8 k=0.04
0.6 0.4
− 10 − 20
0
0.2
0.4
0.6
0.8 q
− 30
Figure 13: The dependence of flame speed c and the distance between fronts δφ on q for k = 2 and k = 0.04. The solid lines indicate the velocity of the combustion wave, and the dashed lines indicate the distances between the reaction fronts large values of s. The effect of the ratio of heats of the reactions, q, on the characteristics of the travelling combustion wave propagating in a layered system is summarized in Fig. 13 and 14. Two dependencies are combined in Fig. 13 for the sake of clarity. The solid line indicates the velocity of the combustion wave, c(q), and the dashed lines indicate the distances between the reaction fronts, δφ(q), propagating in two media. The dependencies are presented for two values of the parameter k = 0.04 and 2, reflecting the situations when the leading wave is in the donor and the acceptor layer, respectively. For large k (more than 10 for given parameters), the propagation velocity is almost independent of q, since the distance between the waves, δφ, increases significantly, exceeding the thermal thicknesses of both fronts. If k is decreased to values of the order of O(1), the distance between the fronts is reduced and the thermal interaction between them increases (see the thick dashed line in Fig. 13 shown for k = 2). When q decreases, the secondary wave in the donor layer receives less heat, the temperature in the layer 1 becomes smaller and the rate of the reaction is reduced. The secondary wave starts to lag behind the leading wave more and more, and the leading wave, in turn, receives less heat from the relatively hot donor layer. The overall velocity decreases and at q ≈ 0.15, the distance between the waves becomes so large that they stop to influence each other, δφ → ∞ and the travelling wave solution disappears. Increasing q at k = 2 results in growth of the combustion wave velocity. The leading front achieves higher burning temperatures and preheats the secondary wave to the superadiabatic temperature. This significantly accelerates the secondary wave and it approaches the leading front closely i.e. δφ is strongly reduced. The relative share of energy that the leading front gives to the wave
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1.5 1,2
1.0
k=0.04
k=2
0.5
0.0
0
0.2
0.4
0.6
0.8
q
Figure 14: The dependence of the maximum temperature, θi , on q for k = 2 and k = 0.04.Maximum temperatures are indicated by solid lines in the donor and by dashed lines in the acceptor travelling behind is getting smaller. This leads to the acceleration of the whole system. It is also seen from Fig. 14, where the thick lines indicate maximum temperatures in the donor (solid line) and acceptor (dashed line) layers. The situation changes completely for the case of k = 0.04. The increase of q results in a decrease of the overall propagation velocity and the growth of the distance between the fronts in layers 1 and 2. At certain q ≈ 0.95, the solution disappears, since δφ → −∞. As q is decreased, the secondary wave receives a relatively larger amount of heat from the leading wave. The second layer burns at superadiabatic temperatures and the role of superadiabacity √ increases as q is reduced, since the rate of the reaction in the layer 2: w2 ∼ k exp(Z2 (max θ2 − q)/2q). Although, k 1 it becomes compensated by the exponential temperature dependence at small values of q and the secondary wave starts to approach the leading wave as q is getting smaller, so that δφ → 0 as q ≈ 0.35. In Fig. 13 and 14 the curves for k = 0.04 are discontinued at this point. The overall velocity is increased as a result of superadiabacity effect. However, as q tends to 1 the relative influence of superadiabacity is decreased, the reaction in the layer 2 slows down and the secondary wave starts to retard. The distance between the fronts grows and the velocity decrease as a result of this.
5
Conclusions
In this paper we consider a model of combustion wave propagation in a compound system consisting of two layers of reacting materials with different geometrical and thermochemical properties placed in thermal contact. This system 20
is directly relevant to the applied technology of combustion synthesis of advanced materials, when one of the reactants serves as a heat source (donor layer) for the other reacting material (acceptor layer) and facilitates the chemical reaction in the latter. Based on the reaction sheet model we undertook a parametric study to find out which parameters affect the process of reaction wave propagation in this system. In particular, it is shown, that depending on the ratio of characteristic times of the reactions there can occur two regimes of combustion wave propagation. If the rate of the reaction on the donor layer is faster, than the rate of the reaction in the acceptor layer, the leading wave is formed in the donor layer. It can advance far ahead of the reaction front in the acceptor layer and at certain critical ratio of the reaction rates in two materials the combustion wave ceases to exist. In the same way, if the reaction rate in the acceptor material is larger, the leading wave is formed in the acceptor layer and there exists critical ratio of the reaction rates at which the combustion wave cannot propagate in both layers synchronously as a single travelling wave. The distance between the reaction fronts at the critical conditions for the existence of the travelling wave tends to infinity. Propagation of the leading front creates the conditions for preheating of the secondary wave travelling behind it in the different layer and this can lead to the significantly superadiabatic temperature of combustion in the material in which the secondary front propagates. The maximum local temperature in the secondary wave is achieved at the critical conditions for the existence of the travelling wave. In particular the largest peak value of the temperature in the acceptor layer is observed when the parameter k is small and the leading front is formed in the donor material. Obviously, from the point of view of synthesis, this is the most favorable regime of process. Thus we only discuss the case, when the leading wave is formed in the donor layer and it is desirable to maximize the peak temperature attained in the acceptor layer. The ratio of the thermal diffusivity of materials constituting the layers has strong impact on the characteristics of the travelling combustion wave. The velocity of combustion wave grows with the increase of the thermal diffusivity of the acceptor layer, since the flame speed in the pure acceptor material is increased in this case. On the other hand, if the acceptor material has much larger thermal diffusivity than the donor layer, α 1, the temperature distribution in the acceptor layer widens and it recuperates part of the heat energy towards the leading front in the donor material. In contrast to that, if α 1, then the temperature distribution in the acceptor layer is more compact and it adsorbs heat generated in the leading front in the donor material more effectively. As a result, the peak local temperature in the acceptor material is the highest in the case when the thermal diffusivity in the donor material is larger than in the acceptor material. The heat exchange between the layers may be governed by both fundamental (Kapitsa thermal resistance [19]) and technological (structure of the materials and topological properties of the surfaces, preparation of the thermal contact etc.) aspects. In the regime, when the leading wave is formed in the donor layer, it is shown that the maximum peak temperature is attained in the acceptor material in the case of small coefficient of thermal contact. In this case the leading front determines the burning rate of the system and the temperature close to the adiabatic is developed in it. The distance between the leading and secondary fronts is much larger than the flame thicknesses of reaction waves in 21
both layers and the acceptor material is preheated to substantially superadiabatic temperature. From the point of view of the heat capacity of the layers it is desirable to have the system with large values of s, for example, when the thickness of the acceptor layer is smaller in comparison to the donor layer. We have also found that the maximum temperature of the acceptor layer almost linearly depends on the ratio of the heats of the reactions at least when q is not a small number. The relative increase of the local temperature above the adiabatic flame temperature of the acceptor material weakly depends on the ratio of heats of the reaction. In terms of absolute temperature increase of the acceptor layer, it is obviously desirable to have larger heats of the reaction in both materials.
6
Acknowledgments
This work was supported by RFBR [projects numbers 18-48-700037, 17-0100070 and 18-38-00523]. The publication has been prepared with the support of the ”RUDN University Program 5-100” (recipient A.A. Polezhaev, mathematical model development).
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