Computational modeling of the effects of support fibers on evaporation of fiber-supported droplets in reduced gravity

Transcription

1 Paper # 070HE-0020 Topic: Heterogeneous combustion, sprays & droplets 8 th US National Combustion Meeting Organized by the Western States Section of the Combustion Institute and hosted by the University of Utah May 19-22, Computational modeling of the effects of support fibers on evaporation of fiber-supported droplets in reduced gravity Narugopal Ghata Benjamin D. Shaw Department of Mechanical and Aerospace Engineering, University of California, Davis, CA 95616, USA A detailed numerical investigation of the effects of support fibers on the vaporization of a fiber-supported droplet in microgravity is presented. The volume-of-fluid (VOF) method was employed to capture the liquid-gas interface while allowing for time-dependent two-phase multidimensional flows. The calculations allowed for the inclusion of thermocapillary stresses at the liquid-gas interface. Predicted evaporation rate results have been validated by comparison with a previous study. The present computational results agree well with experimental results. 1 Introduction Evaporation of droplets has been the subject of numerous investigations in the last few decades because of its importance in the liquid fuel combustion process. Almost all practical combustion processes involve non-premixed combustion of liquid fuels where liquid fuels are converted into sprays comprising a large number of droplets. The combustion of single, isolated droplet has been of great interest for fundamental understanding of the combustion process. In order for comparison of experimental data with simplified analytical and numerical results, many droplet experiments have been conducted in micro-gravity conditions. Over the last years there have been many experimental studies on liquid fuel droplet evaporation and combustions which include drop tower experiments, fiber supported droplet combustion experiments, unsupported droplet combustion experiments, and Flame Extingushment (FLEX) experiments in the International Space Station (ISS). Many droplet evaporation and combustion experiments have been conducted with droplets supported by a fiber to avoid experimental difficulties associated with free-falling, such as capturing high resolution droplet images. Despite years of experimental, analytical and numerical studies in isolated droplet evaporation and combustion, only a very few studies on the effects of fibers are available in the literature [2, 3, 16, 24]. The fiber supporting the droplets introduces additional heat conducted to the droplets that changes flow fields inside the droplets and their surroundings and the droplet evaporation rate [4, 18]. The heat conducted by the fiber introduces temperature gradients along the droplet surface causing variations in the surface tension which causes the flow field inside the droplets to change [18]. Nomura et al. [12] have done an experimental study of n-heptane droplet evaporation for a wide range of pressure (0.1 to 5 MPa) and temperature (400 to 800K) in microgravity conditions. Their 1

2 work is useful in estimating of n-heptane evaporation rates or droplet diameters as a function of time, but they haven t done any study on the effect of the supporting fiber. Chauveau et al. [2] have done some experimental investigation of effect of heat conduction through supporting fiber for n- heptane droplet both in micro-gravity and normal gravity, but they haven t done any study on the effect of surface tension on the evaporation rates. Yang et al. [24] have done some analytical and experimental study on the effect of the supporting fiber for n-heptane and n-hexadecane droplets in weakly convective flow in microgravity environment. But their modeling is based on some major simplified assumptions like the model is simple one-dimensional, all thermo-physical properties are constant and no effect of surface tension. Dwyer et al. [3] have done some study on thermal Marangoni effect on the stability of fiber-supported methanol droplets in microgravity. Shaw and Harrison [16] have also done an analytical study on the effect of supporting fiber on the shapes for heptane-hexadecane droplets in reduced gravity. But none of the above mentioned studies have done detailed investigations on the effect of fiber and thermocapillary stress on the flow and temperature fields inside and outside of the droplet, temperature variation at the droplet surface and on the droplet shape. Shringi et al. [18] have done a numerical investigation on the effect of fiber on methanol droplet in convective flow, but they haven t done any study in stagnant environment where majority of microgravity experiments have been conducted. In the present research, we investigate the effect of fiber and surface tension variation on the flow and temperature fields and the shape of the droplet by employing a full multi-dimensional computational fluid dynamics solver. The volume of Fluid (VOF) method is employed for accurate tracking of the interface between the liquid and gas phases. A commercial CFD package AN- SYS FLUENT V14.0 has been employed for the numerical analysis. Our numerical investigation consists of three different cases: (1) unsupported droplet (2) fiber-supported droplet with no variation of the surface tension (3) fiber-supported droplet with the surface tension as a function of temperature. The numerical results are compared with the experimental study by Nomura et al. [12]. 2 Mathematical formulation 2.1 Interface Tracking The volume of Fluid (VOF) method [8] has been employed to track the interface between the liquid and gas phases. In order to distinguish the two phases, an additional field variable (α) which represents the volume fraction of the liquid phase, is introduced in the VOF method: α ( ) 0 in gas phase X, t = 0 < α < 1 in the interface 1 in liquid phase where α is controlled by the following transport equation [1] (α) + v. (α) = 0 (1) t 2

3 2.2 Species conservation in gas phase The transport equation of species i is given by ρy i t +. (ρy i v) =. (ρd i Y i ) + m i where Y i is the mass fraction of species i and the diffusivity, D i, is given by [19, 21] (2) D i = 1 X i Σ j i X j D ij (3) where X i denotes the mole fraction of species i. The source term m i in Eq. 2 is caused by the volumetric evaporation at the interface. This source term is non-zero at the interface and zero anywhere else in the computational domain. X i is the mole fraction of species and D ij is the binary diffusion coefficient of species i with respect to species j. 2.3 Continuity equation (ρ) +. (ρ v) = S c (4) t The source term S c in the continuity equation arises from evaporation of liquid phase at the interface and it exists only at the interface cells and ceases to exists at other cells in the computational domain. 2.4 Momentum equation (ρ v) +. (ρ v) v = p +. [ µ ( v + v )] T + f Γ (5) t Where the momentum source f Γ at the interface is introduced because of the surface tension. This momentum source only exists in the interface cells and vanishes everywhere else in the computational domain. 2.5 Energy equation t [ρc p (T T 0 )] +. [ρc p (T T 0 ) v] =. (k T ) + S e (6) The source term S e energy equation Eq. 6 arises from the energy removal by evaporation process at the interface cells and it does not exist anywhere else in the computational domain. 3

4 2.6 Interface Conditions Species Conservation at the interface [ṁ i Y l,i + ρ l D l,i Y l,i. n] Γ [ṁ i Y g,i + ρ g D g,i Y g,i. n] Γ = 0 ṁ i = ρ ld l,i Y l,i +ρ gd g,i Y g,i. n (7) (Y l,i Y g,i) The normal vector and the curvature can be computed from the following equations: n = α (8) α κ =. n (9) Mass/Mole Fractions at the interface The vapor pressure at the interface is assumed to be at the saturation condition, which is estimated by the Wagner equation [5]: ( Aτ + Bτ Cτ Dτ 5 ) p sat = exp P c (10) 1 τ where T r = T/T c, T c is the critical temperature, τ = 1 T r, A, B, C and D are material constants. These constants for different fluids are provided by Forero et. al. [5]. The mole fraction of saturated fuel vapor is given by X sat = p sat (11) p where p is the total gas pressure. The mass fraction of saturated vapor at the interface is calculated by M v Y sat = X sat (12) X sat M v + (1 X sat )M g where M v and M g are molar mass of vapor and air respectively Mass conservation at the interface The source terms for the liquid and gas phases are computed by S c = { ṁ ρ ρ l ṁ ρ ρ g in liquid phase in gas phase Where ṁ is computed from the species balance Eq. 7. Where ρ l and ρ g are liquid and gas densities respectively. 4

5 2.6.4 Momentum conservation at the interface A source term is introduced in the momentum equation Eq. 5 to include the effect surface tension: ρκ α f Γ = σ 0.5 (ρ l + ρ g ) where ρ l and ρ g are the densities of liquid and gas phases respectively and the volume averaged density ρ is computed by ρ = αρ l + (1 α)ρ g (14) (13) Energy conservation at the interface The source term S e at the interface cells is given by S e = h fg ṁ ρ (15) ρ l where h fg is the latent heat of evaporation. 2.7 Initial and Boundary Conditions Initially, the droplet of 1 mm diameter at K is placed in hot air (21% O 2 and 79% N 2 by mole) at temperature 741 K and 1 atm pressure. 3 Solution Method A second order upwind scheme was used to discreatize all transport equations. The pressure- Implicit with Splitting Operators (PISO) scheme was used to resolve the coupling of pressure and velocity. To accelerate the convergence, the under-relaxation factors for pressure, momentum and energy were taken as 0.3, 0.7 and 0.9 respectively. The transient formulation was performed using first order implicit method with time step of 20 micro seconds. The time step is estimated by satisfying Courant-Friedrichs-Levy (CFL) stability condition [15]. The thermo-physical properties of n-heptane are given in Table 2. The gas phase is a mixture of air (21% O 2 and 79% N 2 by mole) and n-heptane vapor. For the gas phase, the ideal gas mixture theory based formulation by Wilke [22] was used for thermal conductivity and viscosity and mixing-law based formulation was employed for the specific heat. The kinetic theory based formulation was used for the mass diffusivity [21]. The calculations were performed on NASA s Pleiades Supercomputer (Intel Xeon Processor X5670) and Windows HPC (AMD Opteron 6282SE, 2.6GHz, 32 cores) at UC Davis. 4 Results and Discussion A finite volume method (FVM) based commercial code ANSYS FLUENT v14.0 was used for the numerical simulations. For unsupported droplet simulations, we took a 2D axisymmetric geometry 5

6 Table 1: Thermo-physical properties of n-heptane Property Value/ Temperature Ref Expression Range T c K - [5] P c 2740 Kpa - [5] A, B, C, D (Eq. 10) , , , [5] h fg (J/Kg) e e 5 T [20] e 3 T T e 2 T e 5 T e 8 T K [20] σ(t ) (N/m) T K [20] C p (liquid, J/Kg-K) T T T T e 7 T e 11 T K [20] C p (gas, J/Kg-K) T T e 6 T e 10 T K [20] T T e 7 T e 12 T K [20] k (liquid, W/m-K) T T e 7 T e 10 T K [20] k (gas, W/m-K) T e 6 T e 9 T e 11 T e 15 T e 18 T 6 [20] T ambient K P ambient 1 atm consisting of a droplet of diameter 1 mm and a circular outer domain of diameter 40 mm. For the fiber-supported case, we introduced a circular fiber of diameter 0.05 mm passing through the center of the droplet. The fiber material was taken as Silicon Carbide (SiC) with the density, specific heat and thermal conductivity as 2740 kg/m 3, 670 J/Kg-K and 5.2 W/m-K respectively. The computational domain was discretized into a structured quadrilateral mesh using the ANSYS ICEM CFD v14.0 mesh generation tool. Figures 1(a)-(d) show the mesh generated on the computational domain for the unsupported and fiber-supported droplets. A number of grids were generated for both unsupported and fiber-supported configurations for a grid independence study. For the unsupported droplet geometry, we started with a mesh of 2744 elements with 99 nodes along the radial and 29 nodes along the peripheral directions. It was then further refined to generate mesh of 4746 elements (115 nodes in radial and 44 nodes in peripheral directions) and 8668 elements (200 nodes in radial and 45 nodes in peripheral directions) respectively. For fiber-supported droplet, 3 mesh configurations were generated with 2058 elements (100 nodes along the radial, 20 nodes in peripheral directions and 4 nodes along fiber thickness ), 3348 elements (110 nodes in radial, 30 nodes in peripheral directions and 5 nodes along the fiber thickness) and 4838 elements (120 nodes in radial, 40 nodes in peripheral directions and 5 nodes along 6

7 Figure 1: Mesh generated for the axisymmetric geometry (a) unsupported droplet (full computational domain) (b) fiber-supported droplet (full computational domain) (c) unsupported droplet (zoon in) (d) fiber-supported droplet (zoom in) (all dimensions are in mm) the fiber thickness ). Table 2: Grid independence study for unsupported and fiber supported cases Cases Time (s) Number of Grids Droplet Diameter Unsupported Unsupported Unsupported Supported with Marangoni effects Supported with Marangoni effects Supported with Marangoni effects

8 For all unsupported simulation the mesh with 4059 elements was taken and for all fiber supported simulation the mesh with 3348 elements was taken. Figure 2: Droplet liquid phase n-heptane volume fraction for (a) unsupported, (b) fiber supported w/o Marangoni effect and (c) fiber supported with Marangoni effect Figure 3: Temperature contour for (a) unsupported, (b) fiber supported w/o Marangoni effect and (c) with Marangoni effect 4.1 Fiber Effects Figures 2(a-c) show the droplet profiles and the volume fraction of n-heptane for the unsupported droplet, fiber supported droplets with and without Marangoni effects at 0.3 seconds. The effects 8

9 Figure 4: Velocity vectors at 0.4 seconds for (a) unsupported, (b) fiber supported w/o Marangoni effect and (c) fiber supported with Marangoni effect. The solid quarter circle indicates the initial locations of the droplet-gas interface of fiber on the shape of droplets can be found by comparing Fig 2(a) and Fig 2(b). As seen in the liquid phase heptane profile in Fig. 2(a-b), the unsupported droplet profile is spherical as expected but the fiber supported droplets are not spherical because of the surface tension and fiber effects. The temperature distributions both inside and outside the droplets at 0.3 sec are shown in Figs. 3(ac) As seen in the temperature profiles, the unsupported droplet is being heated uniformly and the temperature profile is spherically symmetric but on the other hand fiber supported droplet without Marangoni effect shows the temperature contour follows the droplet shape, except at the fiber contact region. The interface surface temperature profile plotted in Fig. 5(b) also clearly shows the interface temperature is at about 321K for the unsupported droplet but the supported droplet without Marangoni effect shows the maximum interface temperature about 345K at the contact region and it sharply changes to about 325K and remains almost constant along the interface. The velocity vectors in Fig. 4(a-b) indicate that the gas phase velocity at the interface is always higher for fiber supported droplets, but the velocity vectors are always radially outward for the unsupported droplets where as those vectors are inclined towards the interface due to the effect of surface tension in the fiber supported droplets. 9

10 Figure 5: Comparison of (a) d 2 t plots (b) interface temperature variation along angular location from the axis/supporting fiber. (The diameters of the fiber supported droplets are estimated by averaging the distance of the interface from the centroid of the droplets) 4.2 Marangoni Effects The Marangoni effect on the shape of droplets can be found by comparing Fig 2(b) and Fig 2(c). Fig. 2(b-c) show differences in the droplet profiles at the fiber-gas-liquid contact region. The Maragoni effects appears to push the interface towards the fiber wall. As seen in the temperature contours in Fig. 3(b-c), the temperature contours for the case with no Maragoni effects follow the droplet interface except the contact region where as temperature variations are seen along the droplet-gas interface with Marangoni effects. The interface surface temperature profile plotted in Fig. 5(b) also clearly shows that the maximum interface temperature without Maragoni effects is much higher (about 10 deg C) at the contact region and it sharply declines and remains almost constant along the interface. But the droplets with Marangoni effects show relatively lower maximum interface temperature at the contact region and the interface temperature varies along the interface. As seen in the valocity vectors in Fig. 4(b-c), the Maragoni effects introduce vortex flows both inside and outside the droplet which do not appear in the simulations for unsupported droplets and fiber supported without Marangoni effects. The effects of these vortex flows appears to have strong effects in slowing down the evaporation rates because the circulation reduces the temperature and species gradients along the radial direction. The velocity vectors also indicate that the gas phase velocity at the interface is always higher for fiber supported droplets with Marangoni effects, but the vortex flow in the gas phase causes the air to move towards the droplets and reduces the vaporization rates. In fact, the d 2 vs time plots in Fig. 5(a) indicate that the supported droplets without Marangoni effects evaporates faster then the ones with Marangoni effects. Fig. 5(a) also compares d 2 vs time plots with the experimental results from [12]. As seen in Fig. 5(a), the numerical results with Marangoni effects are closer to the experimental results. 10

11 5 Conclusions The vaporization rates are defined by the slope of the d 2 vs. t plot and the free droplet does not show the slowest vaporization rate. The numerical study of the unsupported droplets and fiber supported droplets with and without Marangoni effects sheds light on the importance of the effects of fiber and thermocapillary stresses on droplet evaporation rates and droplet shapes. A fiber-supported droplet without Marangoni effects evaporates faster than an unsupported droplet. However, when Marangoni effects are allowed, the droplet vaporization rate slows considerably because of enhanced convection that reduces temperature radial and species gradients near the interface. The results with the Marangoni effects are reasonably close to available experimental data on vaporization of fiber-supported droplets in reduced gravity. This study can be extended to investigate the effects of wide range of fiber and droplet diameters in stagnant and convective environments. It is also interesting to study these effects in droplets in combustion environment in very high temperature. Acknowledgments The support of the National Aeronautics and Space Administration (NASA) is greatly acknowledged. The technical monitor was Dr. D.L. Dietrich. This study would not have been possible without support of Ansys, inc. References [1] ANSYS Fluent User Manuals V14.0 (2012) ANSYS, Inc., Cannosburg, PA, USA. [2] C. Chauveau, F. Halter, A. Lalonde, I. Gokalp, An experimental study on the droplet vaporization: effects of heat conduction through the support fiber, ILASS, Como Lake, Italy, 4-1 (2008). [3] H.A. Dwyer and B.D. Shaw, Marangoni and Stability Studies on Fiber-Supported Droplets Evaporating in Reduced Gravity, Combustion Science and Technology 162(1) (2001) [4] H.A. Dwyer, D. Shringi and B. D. Shaw, A Simulation of a Fiber-Supported Droplet, Computational Fluid Dynamics Journal 13(3) (2004) [5] Luis A. Forero G., Jorge A. Velsquez J., Wagner liquidvapour pressure equation constants from a simple methodology, The Journal of Chemical Thermodynamics, 43(8) (2011) [6] H. Ghassemi, S. Baek, Q. Khan, Experimental study on binary droplet evaporation at elevated pressures and temperatures, Combustion Science and Technology 178(6) (2006) [7] H. Hiroyasu, T. Senda, T. Imamoto, Evaporation of a Single Droplet at Elevated Pressures and Temperatures: Part 1, Experimental Study, Transactions of the Japan Society of Mechanical Engineers 40(339) (1974) [8] C.W Hirt, B.D Nichols, Volume of fluid (VOF) method for the dynamics of free boundaries, Journal of Computational Physics 39(1)1 (1981) [9] H. Hiroyasu, T. Senda, T. Imamoto, Evaporation of a Single Droplet at Elevated Pressures and Temperatures: Part 2, Theoretical Study, Transactions of the Japan Society of Mechanical Engineers 19(138) (1976) [10] C.K. Law, Recent advances in droplet vaporization and combustion, Progress in Energy and Combustion Science, Volume 8, Issue 3, 1982, Pages

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