On Numerical Model of Two-Dimensional Heterogeneous Combustion in Porous Media

Transcription

1 5 th ICDERS Auust 7 05 Leeds UK On Numerical Model of Two-Dimensional Heteroeneous Combustion in Porous Media Nickolay A. Lutsenko Institute of Automation and Control Processes FEB RAS Vladivostok Russia Far Eastern Federal University Vladivostok Russia Introduction Heteroeneous combustion in porous media occurs quite often in nature. From the point of view of mechanics various objects can be modeled as porous media: soil peat rock debris of destroyed buildins and so on. Smolderin is one of the types of heteroeneous combustion in porous media as heteroeneous chemical reactions take place durin smolderin []. Many materials can sustain a smolderin reaction includin coal cotton duff peat wood and most charrin polymers. The most common type of heteroeneous combustion in porous media is peat fire []; another example of such combustion is spontaneous combustion of solid waste dumps (landfills). Besides above mentioned examples the principles of heteroeneous combustion in porous media are used in a various technoloical processes: burnin the municipal and industrial solid waste or another low-calorie fuel asification of solid fuels and so on. When researchin the heteroeneous combustion of porous objects in nature we often deal with the process under free convection. Combustion of solid porous medium under free convection was analytically investiated in [] for counter-flow reime when the as and the combustion wave in porous object move in opposite directions and in [4] for co-flow reime when the as moves in the same direction as the combustion wave. In [5] the ability of enery to be concentrated in the front of a co-flow filtration combustion wave in a porous solid was analyzed. In [6] the smolderin of polyurethane foam in natural convection was experimentally studied. Very promisin tool for investiatin the heteroeneous combustion in porous media is computational modelin. In [7] a onedimensional transient model of smolderin was presented and solved numerically. A novel computational model of smolderin combustion capable of predictin both countercurrent and cocurrent wave propaation was developed in [8]. A eneralized pyrolysis model Gpyro for simulatin the asification of a variety of combustible solids encountered in fires was presented in [9]. In [0 ] Gpyro was expanded to two-dimensional case. Recently in [] a novel numerical model for investiatin the unsteady processes of heteroeneous combustion in porous objects with unknown as flow rate and as velocity at the inlet to the object was proposed. The advantae of the proposed model is that it can examine the complex as dynamics inside the porous object and allows to describe the processes for both forced filtration and free convection when the flow rate of as reulates itself. The details of the oriinal numerical method were described in []. Usin numerical experiment it Correspondence to: NickL@inbox.ru

2 On Model of D Combustion in Porous Media was studied in [4] how the ravity field inside the porous object and the pressure difference at the object boundaries which is caused by the action of ravity on the ambient air affect the appearance of self-sustainin heteroeneous combustion waves in the object. In the present work the numerical model which was previously developed for one-dimensional case in [-4] is expanded to two-dimensional case and used for solvin plane time-dependent problems of heteroeneous combustion in porous objects. In such porous objects the flow rate of oxidant which enters into the reaction zone in porous object reulates itself. Used approach enables to solve problems of filtration combustion for both forced filtration and free convection so it can be efficiently applied for modelin the combustion zones in porous media which may arise from natural or mancaused disasters. Mathematical model and numerical method Mathematical model which is discussed in this paper can describe the processes of heteroeneous combustion in motionless porous objects which have some impermeable non-heat-conductin borders and some boundaries opened to the atmosphere. The cold as can flow into the open walls of the porous object; the as can flow throuh porous medium and flow out. Suppose that a solid porous substance consist of combustible and inert components and at the same time the solid combustible material transforms into a as in the reaction with aseous oxidizer so we have the followin expression: Solid Fuel + ( μ )Oxidizer (+ μ )Gaseous Product () where μ is the mass stoichiometric coefficient for oxidizer. The model is based on the assumption of interactin interpenetratin continua [5] usin the classical approaches of the theory of filtration combustion [-5]. The model includes equations of state continuity momentum conservation and enery for each phase (solid and as); chanin the solid phase mass and porosity as well as diffusion of the oxidizer are taken into account. Combustion processes are described by one-step heteroeneous chemical reaction of first order with respect to both aruments. An essential feature of the model is usin the equation of momentum conservation for porous media for describin the dynamics of as; this equation is more correct than the classical Darcy's equation and can be used in a reater rane of Reynolds numbers [5]. Another feature of the model is that the dynamic viscosity of as is temperature dependent by Sutherland s formula because the allowance for the temperature dependence of as viscosity in its motion throuh a porous heatevolutional medium can chane the solution both quantitatively and qualitatively [6]. In the present work we investiate the two-dimensional (plane) time-dependent as flow in porous objects with zones of heteroeneous combustion; thus we can write the system of equations for this case denotin horizontal and vertical coordinates as x and x respectively and summin over repeatin indices: Tc ( ) ( ) ( ) Tc Tc ρcf ccf + ρcicci = α Tc T + Qρcf 0 W + a λc + x x T T ρ c + v = α ( T T ) ρ ρ p j x j c v v p μ ( ( ) χ a + v = a a v ρ 0Wv j cf x j x k + v v p μ ( ) + χ a + v j = a ρ a v ρ cf t x j x k ( ) 0Wv 5 th ICDERS Auust Leeds

3 On Model of D Combustion in Porous Media ρ ρ vj ρ RT + = ρcf 0W p = () x j a M C C C ρ + vi ρ D μρcf 0W ρcf 0WC t x = i x i x i ( T / ) b = W = ( ) C k exp( E/ ( )) D D0 7 η = W ρcf ( η) ρcf 0 η R T c = a = a + a η 0 cf 0 5. T μ = cs. cs + T where a is the volume fraction b is the exponent in the expression for the diffusion coefficient C is the mass concentration of oxidizer c is the specific heat c s and c s are the constants in Sutherland s formula D is the diffusion coefficient of as E is the activation enery is the ravity acceleration k is the pre-exponential factor in the expression for the rate of reaction k is the permeability coefficient M is the molar mass of as p is the as pressure Q is the heat of reaction R is the universal as constant t is the time Т is the temperature v is the as velocity W is the rate of the chemical reaction α is the constant determinin the interphase heat transfer intensity η is the deree of conversion of the combustible component of the solid medium λ is the thermal conductivity μ is the dynamic viscosity of the as ρ is the effective density χ is the coefficient takin into account the inertial interaction of the phases in their relative motion [5]; subscripts: 0 denotes the initial moment c denotes the condensed phase (solid medium) i denotes the inert component f denotes the combustible component denotes the as p denotes values at constant pressure. A distinctive feature of the considered model is that the as flow rate and as velocity at the inlet to the porous object are unknown and have to be found from the solution of the problem. So we assume that at the open boundary where as flows into the porous object the as pressure as temperature and mass concentration for the oxidizer are known. At the open boundary where as flows out the porous object the pressure is known. The conditions of heat exchane at the open and impermeable non-heat-conductin borders are also known. Thus the boundary conditions for () are as follows: p = p0 ( x) λ T c / n = βt 0 T x G c T = T 0 and C C0 if v n 0 () x G = T / n = 0 and C/ n = 0 if v 0 n > T c / n = 0 T = 0 / n v n = 0. where G is object boundary opened to the atmosphere G is impermeable boundary of object n is outward vector directed normally to G or to G β is heat removal coefficient. For the investiation of two-dimensional unsteady as flow in porous objects with zones of heteroeneous combustion an oriinal numerical method has been developed which is based on a combination of explicit and implicit finite-difference schemes. This method is the development of earlier proposed numerical alorithm for modelin the one-dimensional problems of heteroeneous combustion in porous objects [-4]; it is similar to the alorithm for computation of the as flow throuh porous objects with heat sources when as pressure at object boundaries is known [7]. Accordin to the method the enery equations momentum conservation equation and equation for oxidizer concentration are transformed into the explicit finite difference equations. The as temperature solid phase temperature as velocity and oxidizer concentration are determined from these equations. The continuity equation is transformed into the implicit finite difference equation. From this equation takin into account the perfect as equation of state the as pressure is determined 5 th ICDERS Auust Leeds

4 On Model of D Combustion in Porous Media usin Thomas alorithm [8]. The effective as density and the remainin unknown quantities are determined trivially from the perfect as equation of state and other closure equations. The method is first-order accurate in time and second-order accurate in space. Dampin fourth-order terms [8] must be added in the finite difference equations of continuity and momentum conservation. The terms smooth the dispersion errors but the formal accuracy of the method is unaltered. The proposed numerical model can be used for solvin various problems of filtration combustion for both forced filtration and free convection for both natural and technoloical processes. Numerical Results For illustratin the developed numerical model consider the followin plane time-dependent problem. The porous object with heiht H and width L is bounded of impermeable non-heat-conductin side walls and is opened at the top (outlet) and at the bottom (inlet). Before the startin moment the pressure at the object inlet and outlet corresponds to the atmospheric pressure at the assined heihts; so there is no air motion in the object. At the startin moment the pressure at the object inlet rapidly increases up to p 0 and remains constant and in the place of inition which is located at the central part of the object bottom the temperature of the solid phase reaches a value T c0 equal to or exceedin the self-inition temperature T kr and burnin is started. We use the followin parameter values: k/m H = L = 0 m ρ cf 0 =. 0 k/m ρ ci = c cf =.84 0 J/(k K) c ci =.84 0 J/(k K) α = 0 J/(m K s) c p = 0 J/(k K) λ c =. J/(m K s) 6 / c s = k/(m s K ) s = c K = 0 k m β = 0 J/(m K s) χ = = 9.8 m/s or 0 m/s R = 8.44J/(mole K) M =.99 0 k/mole Q = 8 0 J/k 7 5 k =.6 0 /s [9] E = 0 0 J/mole [9] D 0 =.8 0 m /s b =.74 μ =.667 a 0 = 0. a cf 0 = 0. T 0 = 00 K C. 0 5 p 0 = Pa. 0 = 0. In this case the combustion wave appears and moves upward and to the side walls simultaneously burnin the solid combustible substance completely. Fiure shows the example of the solid phase temperature and the field of as velocity within the porous object. Fiure. Distributions of the solid phase temperature and the field of as velocity within the porous object in 5 hours after the time of inition As can be seen from the fiure the as movin up tends to o around the heated part of the object and prefers to flow in the cold part of the object which has not been achieved by the combustion 5 th ICDERS Auust Leeds 4

5 On Model of D Combustion in Porous Media wave. The combustion wave propaation is demonstrated in Fi. which shows the area where the combustible component of the solid medium was completely burnt. Fiure. Contours boundin the completely burnt combustible component of the solid medium at different times t after the inition: t = hour (curve ) hours () 5 hours () 7 hours (4) 9 hours (5) and hours (6). As seen from the fiure the combustion wave moves up and to the side walls of the object. The lower part of the object near the side walls remains unburned as the combustion wave cannot move across the as flow or aainst the flow. The slope of the lines which separate the burnt part of the object from zones which cannot be achieved by the combustion wave depends on the as pressure at the inlet to the porous object. When solvin the similar problem but when the place of inition is located at the center of the object we can see the same situation: the combustion wave moves upward and to the side walls simultaneously burnin the solid combustible substance completely and cannot reach the part of the object located lower the certain lines. At the same time the slope of these lines depends on the as flow rate which is determined by the as pressure at the inlet to the porous object. 4 Conclusions The time-dependent problems of heteroeneous combustion in porous objects are considered when the as pressure at object boundaries is known but the flow rate and velocity of the as filtration at the inlet to the porous objects are unknown. In such porous objects the flow rate of oxidant which enters into the reaction zone in porous object reulates itself. An oriinal numerical method based on a combination of explicit and implicit finite-difference schemes has been developed for investiatin the unsteady two-dimensional as flows in such porous objects with zones of heteroeneous reactions. Some plane time-dependent problems of heteroeneous combustion in porous objects have been solved numerically usin proposed numerical method. It is shown that as movin up tends to o around the heated part of the object and prefers to flow in the cold part of the object. When the combustion wave propaates inside the porous object burnin the solid combustible substance completely it cannot reach the part of the object located lower the certain lines. The slope of these lines depends on the as flow rate which is determined by the as pressure at the inlet to the porous object. The work was supported financially by the Ministry of education and science of Russian Federation (project 4.Y6..000) the Far-Eastern Branch of the Russian Academy of Sciences (project 5-I- 4-0) the Far Eastern Federal University. The author would like to acknowlede and thank Academician Vladimir Levin for his encouraement and valuable comments. 5 th ICDERS Auust Leeds 5

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