The models of heat and mass transfer and phase transition for “water droplet – flame” system have been developed using non-stationary nonlinear partial differential equations. The system of differential equations was solved by the finite-difference method. The locally one-dimensional method was used to solve the difference analogous of differential equations. One-dimensional differential equations were solved using an implicit four-point difference scheme. Nonlinear equations were solved by the iteration method. The evaporation rates of water droplets (with sizes from 0.05 mm to 5 mm) in the flame zone (at the temperatures from 500 K to 1200 K) were determined. Theoretical analysis established essentially nonlinear (close to exponential) form of dependence of the water droplet evaporation rate on the temperature of the external gas area and the temperature of a droplet surface. In particular, the water droplet evaporation rate varies from 0.25 to 0.29 kg/(m2s), when the temperature of external gas area is about 1100 K. On the other hand, the water droplet evaporation rate does not exceed 0.01 kg/(m2s) when the temperature of external gas area is about 350 K. Besides, it has been found out that droplets warm up at different rates depending on their initial temperature and velocity. As a result, the integral characteristics of droplet evaporation can increase substantially, when droplets move through the external gas area at the same temperature. We performed a similar investigation or droplet streams with droplet concentration 0.001–0.005 m3 in 1 m3 of gas area (typical parameters for modern spray extinguishing systems)

Strizhak, Pavel A.

Volkov, Roman S.

Zhdanova, Alena O.

English

UPCommons

Numerical simulation of water and water emulsion droplets evaporation in flames with different temperatures

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991Numerical simulation of water and water emulsion droplets evaporation in flames with different temperaturesIV International Conference on Particle-based Methods – Fundamentals and Applications PARTICLES 2015 E. Oñate, M. Bischoff, D.R.J. Owen, P. Wriggers & T. Zohdi (Eds)    NUMERICAL SIMULATION OF WATER AND WATER EMULSION DROPLETS EVAPORATION IN FLAMES WITH DIFFERENT TEMPERATURES PAVEL A. STRIZHAK, ROMAN S. VOLKOV AND ALENA O. ZHDANOVA National Research Tomsk Polytechnic University Lenina Avenue, 30, 634050, Tomsk, Russia E-mail pavelspa@tpu.ru   Key words: Water Droplet, Evaporation, High-Temperature Gas Environment. Abstract. The models of heat and mass transfer and phase transition for “water droplet – flame” system have been developed using non-stationary nonlinear partial differential equations. The system of differential equations was solved by the finite-difference method. The locally one-dimensional method was used to solve the difference analogous of differential equations. One-dimensional differential equations were solved using an implicit four-point difference scheme. Nonlinear equations were solved by the iteration method. The evaporation rates of water droplets (with sizes from 0.05 mm to 5 mm) in the flame zone (at the temperatures from 500 K to 1200 K) were determined. Theoretical analysis established essentially nonlinear (close to exponential) form of dependence of the water droplet evaporation rate on the temperature of the external gas area and the temperature of a droplet surface. In particular, the water droplet evaporation rate varies from 0.25 to 0.29 kg/(m2s), when the temperature of external gas area is about 1100 K. On the other hand, the water droplet evaporation rate does not exceed 0.01 kg/(m2s) when the temperature of external gas area is about 350 K. Besides, it has been found out that droplets warm up at different rates depending on their initial temperature and velocity. As a result, the integral characteristics of droplet evaporation can increase substantially, when droplets move through the external gas area at the same temperature. We performed a similar investigation or droplet streams with droplet concentration 0.001–0.005 m3 in 1 m3 of gas area (typical parameters for modern spray extinguishing systems).   1 INTRODUCTION The study of sprayed water phase transitions during the motion in high-temperature gas environment has been getting more and more attention in recent years [1, 2]. The results of these studies have a large number of practical applications. For example, they can be used in the technologies of polydisperse extinguishing, two-phase gas-vapor coolant formation with the specified parameters, wastewater combustion neutralization, and others [3-5]. Currently, new optical methods have emerged, that enable visualization and video recording of heat and mass transfer fast processes during the evaporation of fine liquid flow in high-temperature gases. Thus, it became possible to develop experimental methods for studying these processes. However, it is still of topical interest to develop numerical models for calculating 992Pavel A. Strizhak, Roman S. Volkov and Alena O. Zhdanova  2 the parameters of the system “sprayed liquid – high-temperature gas environment” taking into account the experimental characteristics of liquid spray motion and evaporation (the deformation and entrainment of droplets, the evaporation rate, etc.).  The aim of this work is to develop the numerical model of heat and mass transfer of water droplets during their motion in high-temperature gas environment.  1 PROBLEM STATEMENT In formulating the problem, it was considered that a water droplet moved (in free fall) in high-temperature gas environment (Figure 1). Under the conditions of phase transition, water vapor was blown into the near-wall part of the droplet and mixed with combustion products. It was believed that mass transfer was implemented by diffusion and convection mechanisms in the external environment. It was assumed that droplet size decreased under the conditions of intensive evaporation. After some time, the droplet evaporated completely. In formulating the problem of heat and mass transfer, a cylindrical droplet (elongated in the direction of its motion) was examined (Figure 1). It was considered that gas environment was binary (“combustion products – water vapor”). It was also assumed that the thermal characteristics of reacting substances do not depend on the temperature.   Figure 1: A diagram of the solution area of the problem of heat and mass transfer: 1 – high-temperature gas mixture, 2 – water droplet  1 MATHEMATICAL MODEL AND SOLUTION METHODS The system of non-stationary nonlinear partial differential equations is derived from the physical model that has been formulated taking into account convective flows in a mixture of combustion products and water vapor. The system is as follows (0<τ<τd) [6]: 0<R<RL, 0<Z<Z1, Z2<Z<ZL; R1<R<RL, Z1<Z<Z2 Poisson’s equation: 2 22 21 RR R R Z           (1) 993Pavel A. Strizhak, Roman S. Volkov and Alena O. Zhdanova  3 vorticity equation: 112 22 21 1 12Sh ReU V UR Z R RR R R RZ                          (2) energy equation for the mixture of water vapor and combustion products: 1 1 11 12 21 1 12 21 1 1Sh Re PrU VR Z R R R Z                     (3) water vapor diffusion equation: 2 2w w w w w w21 11 1Sh Re Scγ γ γ1U VR Z R R R R               (4) balance equation: w f 1     (5) 0<R<R1, Z1<Z<Z2   heat equation for the droplet: 2 22 2 2 22 221 1Fo R R R Z           (6) m mmShv tz , m m11ReV z, 1 1 111Pr =C , 111Sc =D , 2 m2 22 2 mλFoρtC z . Initial (τ=0) conditions (Figure 1): Θ=Θ0 at 0<R<R1, Z1<Z<Z2; Θ=Θf, γf=1, γw=0, Ψ=0, Ω=0 at 0<R<RL, 0<Z<Z1, Z2<Z<ZL; R1<R<RL, Z1<Z<Z2. Boundary (0<τ<τd) conditions (Figure 1): at the interface “liquid – gas” (R=R1, Z1<Z<Z2; Z=Z1, Z=Z2, 0<R<R1) – IV type of boundary conditions was assumed for energy equations taken into account vaporization, II type of boundary conditions was specified for energy equations taken into account water vapor blowing; at the external borders (R=0, R=RL, 0<Z<ZL; Z=0, Z=ZL, 0<R<RL) – the condition of zero gradients of the respective functions was assumed for all equations. Boundary conditions were specified to emphasize the effect of water vapor blowing on the conditions of heat transfer on all droplet faces for energy equations:  R=R1, Z1<Z<Z2: 2 1 e e 3 3 e 3s 2s1 m m2 2 2( )z zQ W C vTR R         , 994Pavel A. Strizhak, Roman S. Volkov and Alena O. Zhdanova  4 Z=Z1, Z=Z2, 0<R<R1, 2 1 e e 3 3 e 3s 2s1 m m2 2 2( )z zQ W C vTZ Z         . The motion equation of the droplet under the conditions of vaporization taking into account the forces of resistance and gravity:  d 3 e ed d2 d34 2dv c v v v v gdt r    , where vd(0)=v0. The following equation was used for the drag coefficient in numerical simulation taking into account the nonsphericity of the droplet, its unsteady motion [7], evaporation on the body surface [8] and movement (of convection flows) inside the droplet: 1.2 0.03 0.635g213124.3 ( 1) Re1 1c k AB   . The geometric factor kg characterizes the form deviation from spherical (for cylindrical disks kg=1.64). The ratio 1/(B+1) is a coefficient that describes the effect of droplet evaporation on resistance force (B=C2(T3s–T2s)/(Qe+qv/We)) [8]. The ratio [1+(2μ2/3μ3)]/[(1+μ2/μ3)] is the coefficient taking into account possible convective flows inside the droplet. The ratio (A+1)1.2±0.03 characterizes the accelerated body motion (A – dimensionless complex, which describes the relative acceleration ( d2d( )dvdAv dt ) [7]. The following expression was used to determine the mass evaporation rate: net e( )1 2 /P PWk R T M  , where kβ – dimensionless coefficient (kβ≈0.4). The layer thickness of the evaporating droplet was determined by the formula ee2W tl . The following scale variables were used to convert the equations into the dimensionless form: characteristic average droplet size (zm=1 mm); time scale (tm=1 s); temperature scale (Тm=1000 K); rate scale (vm=1 m/s). Where: τ – dimensionless time; τd – dimensionless time of droplet existence; Ψ – dimensionless equivalent of the current function; Ω – dimensionless equivalent of vorticity; Θ – dimensionless temperature; U, V – dimensionless velocity components of water vapor and combustion products in the direction of the axis R and Z respectively; γf – dimensionless concentration of combustion products; γw – dimensionless water vapor concentration; λ – 995Pavel A. Strizhak, Roman S. Volkov and Alena O. Zhdanova  5 thermal conductivity, W/(m·K); Qe – thermal effect of liquid evaporation, J/kg; We – evaporation rate, kg/(m2·s); ΔT – temperature difference (ΔT=Tm-T0); C – specific heat capacity, J/(kg·K); vd – droplet velocity, m/s; ve – linear velocity of vapor outflow from the droplet surface, m/s; cχ – dimensionless drag coefficient; g – acceleration of gravity, m/s2; β – dimensionless evaporation coefficient; P – water vapor pressure near the boundary of evaporation, N/m2; Pn – saturated water vapor pressure, N/m2; Rt – universal gas constant, J/(mol·K); Te – droplet surface temperature, K; M – molar mass, kg/kmol; indexes: 0 – initial state, 1 – high-temperature gases, 2 – water droplet, 3 – water vapor, d – droplet, e – evaporation, f – fuel, m – scale, s – surface, v – vapor, w – water. The system of non-stationary partial differential equations (1)–(6) was solved by the finite difference method [9]. The difference analogues of differential equations were solved by the local one-dimensional method [9]. One-dimensional differential equations were solved by the sweep method using an implicit four-point difference scheme [9]. Nonlinear equations were solved by the iteration method. Irregular dimensionless steps in time (10-8÷10-6) and in coordinate grid (10-4÷10-2) were used to increase the accuracy when solving the system of non-stationary differential equations (1)–(6). The method for assessing the reliability of the results of numerical studies was performed, based on the verification of applied difference scheme conservativeness, similar to [6]. Numerical studies were carried out at the following parameters [10, 11]: initial temperatures of liquid droplets Θ0=0.3 and gas Θf=0.3÷1.1; thermal effect of evaporation Qe=2.26 MJ/kg; droplet sizes Rd=0.25, Zd=1 and solution area RL=10, ZL=1000; initial droplet velocity V0=0.5; water molar mass M=18 kg/kmol.  3 RESULTS AND DISCUSSION Table 1 shows the dependence of the mass rate of water droplet evaporation on the temperature of gas environment. Data in the table was obtained as a result of the calculations using the developed model of heat and mass transfer. The essentially nonlinear nature of the relation We=f(Θf) was established from the results of theoretical research.  Table 1: The dependence of the mass rate of water droplet evaporation on the temperature of gas environment Θf 0.3 0.5 0.7 0.9 1.1 0.3 We, kg/(m2·s) 0.005 0.0116 0.065 0.155 0.214 0.005  The values have been obtained that correspond to the mass rate of droplet evaporation during the motion through high-temperature gases. These values correlate well with the results of previous experiments [12]: the evaporation rate was We=0.006 kg/(m2·s) at the temperature of fuel combustion products Tf=300 K, and Tf=1100 K, We=0.270 kg/(m2·s). Figure 2 shows the isotherms at τ=0.02 (correspond to the beginning of the motion) for two schemes considering the mutual disposition of droplets. In the case of two (Figure 2a), and five (Figure 2b) droplets moving consecutively, the parameter Ln varied from 0.01 to 5. 996Pavel A. Strizhak, Roman S. Volkov and Alena O. Zhdanova  6  a  b Figure 2: Isotherms (Θ) for the system with two droplets moving consecutively at τ=0.02, Rd=0.25, Zd=1, Ln=1 (a) and five droplets at τ=0.02, Rd=0.25, Zd=1, Ln=2 (b):  1 – high-temperature gas environment, 2 – water droplet  Figure 2 demonstrates a decrease of the gas temperature in the vicinity of droplets. It can be seen that the temperature in the droplet track varies considerably. This is primarily due to the evaporation of droplets. The trajectory of each droplet also plays a decisive role. The temperature in the track changes slightly (relative to the initial droplet temperature) in contrast to that in a small neighborhood of the line corresponding to the trajectory when varying Ln in a wide range. Satisfactory compliance with the results of theoretical studies and the data obtained in the experiments allow us to conclude on the adequacy of the developed model of heat and mass transfer during the motion and evaporation of droplets and droplet liquid flow in high-temperature gas environment. 997Pavel A. Strizhak, Roman S. Volkov and Alena O. Zhdanova  7 The research results have been obtained using established numerical model; they can be used to develop new and improve existing technologies, providing the implementation of phase transitions in the system “liquid spray – high-temperature gas environment”.  This work was financially supported by the Russian Science Foundation (Project No. 14-39-00003).  REFERENCES [1] Maltsev, R.V. and Rebrov, A.K. Gas-dynamic colliders: numerical simulations. J.Appl. Mech. Tech. Phys. (2007) 48:142–151. [2] Lebedev, V.P., Lemanov, V.V., Misyura, S.Ya. and Terekhov, V.I. Effect of flow turbulence on film cooling efficiency. Int. J. Heat Mass Transfer (1995) 38:2117-2125. [3] Rodriquez, B. and Young, G. Development of international space station fine water mist portable fire extinguisher. in: 43rd International Conference on Environmental Systems, Vail, CO (2013). [4] Nikitin, M.N. Using a gas–vapor mixture at the fuel combustion. Ind. Energy (2010) 12:37-42. [5] Ma, J., Liu, D., Chen, Z. and Chen, X. Agglomeration characteristics during fluidized bed combustion of salty wastewater. Powder Technol. (2014) 253:537-547. [6] Glushkov, D.O., Kuznetsov, G.V. and Strizhak, P.A. Numerical investigation of water droplets shape influence on mathematical modeling results of its evaporation in motion through a high-temperature gas. Math. Probl. in Eng. (2014) 2014:Article ID 920480. [7] Tchen, C.M. Mean value and correlation problems connected with the motion of small particles suspended in a turbulent fluid. The Hague. Martinus Nijhoff, (1947). [8] Eisenklam, P., Arunachalam, S.A. and Weston, J.A. Evaporation rates and drag resistance of burning drops. Proc. 11th Symp. On Combustion (1967) 11: 715-728. [9] Samarskii, A.A. The Theory of Difference Schemes. Marcel Dekker, USA, (2001). [10] Vargaftik, N.B., Filipov, L. P., Tarzimanov, A. A. and Totskii, E. E. Handbook of Thermal Conductivity of Liquids and Gases. CRC Press, Boca Raton, Fla, USA, (1994). [11] Vargaftik, N.B. Tables of Thermophysical Properties of Liquids and Gases, Hemisphere Publishing, New York, (1975). [12] Terekhov, V.I., Terekhov, V.V., Shishkin, N.E. and K.Ch.Bi. Heat and mass transfer in disperse and porous media experimental and numerical investigations of nonstationary evaporation of liquid droplets. Journ. Eng. Phys. Thermophys. (2010) 83:883–890. 

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Numerical simulation of water and water emulsion droplets evaporation in flames with different temperatures

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