Experimental and Numerical Studies of Water Droplet Impact on a

9174

Ind. Eng. Chem. Res. 2008, 47, 9174–9182

Experimental and Numerical Studies of Water Droplet Impact on a Porous Surface in the Film-Boiling Regime Zhao Yu, Fei Wang, and L.-S. Fan* Department of Chemical and Biomolecular Engineering, The Ohio State UniVersity, Columbus, Ohio 43210

An experimental and numerical study of the water droplet in collision with a porous surface in the filmboiling regime is reported. The porous substrate with a porosity of 34% and pore size of 76 nm is heated to 300 °C, and the motion of the droplet is recorded by a high-speed digital camera. A new three-dimensional (3-D) numerical model is developed to account for the transport phenomenon both inside and outside the porous media, by coupling the flow field with the heat and mass transfer process. The vapor layer model is used as a subgrid model to calculate the induced vapor pressure in the narrow region between the droplet and the surface. The vapor mass transfer is modeled considering the vapor generation and transport mechanisms in different domains. Direct numerical simulation is performed under the same conditions as the experiment, and the simulation results for the droplet behavior are in good agreement with the experimental results. The collision of a water droplet on the porous surface shows similar features to those on nonporous surfaces in the film-boiling regime, probably because of the small pore size of the material used in the current study. However, the droplet has a longer residence time, and it also seems to be less stable on the porous surface. Introduction The phenomenon of a liquid droplet colliding with a hot surface occurs in a wide range of industrial applications. Because of its great importance in those applications, there have been extensive studies on droplet-surface collision, using experimental and numerical techniques. However, most studies are focused on the collision between a droplet and a nonporous surface, while only a few papers considered the collision on a hot porous surface. The contact between a droplet and a heated porous surface exists in many situations. For example, it is important in the suppression of fire using sprinklers, because fire usually involves the burning of a porous material, such as wood. It is also relevant to the combustion of liquid fuel, during which the fuel droplet collides with the ceramic-lined walls of an incinerator. Particularly, the collision of a reactant droplet with a porous catalytic particle is of great relevance to many chemical engineering processes. It is recently reported that nonvolatile hydrocarbon droplet could be flash-evaporated on a high-temperature porous catalytic particle to produce hydrogen and other chemicals. This provides a promising way to directly convert these heavy fuels to hydrogen, which is an important energy source as well as an important material for the production of many synthetic fuels, chemicals, and fertilizers.1 The impact of droplets on a porous surface shares some common features with that on an impermeable surface, but it also has its unique characteristics that require separate studies. Experimental studies of a droplet impinging on a heated nonporous surface have been extensively reported in the literature. Generally, when the surface temperature is much higher than the boiling temperature of the liquid, a vapor cushion is formed between the droplet and the surface, which prevents the direct contact of the two and, therefore, hinders the heat transfer from the surface to the droplet. In this case, the droplet is said to be in the Leidenfrost regime (or the film-boiling regime).2 Wachters and Westerling3 studied a water droplet in collision with a gold surface above the Leidenfrost temperature. They found that the Weber number, which is the ratio of the * To whom correspondence should be addressed. E-mail address: [email protected].

impact inertial force to the surface tension force (We ) 2FlV2R/ σ), had an important effect on the dynamics of the droplet. Depending on the We value, the droplet would either rebound without disintegration, rebound with disintegration, or splash on the surface after the impact. Groendes and Mesler4 studied the saturated film-boiling impact of a water droplet on a quartz surface of 460 °C. They used a thermometer to measure the fluctuation of the surface temperature, and the maximum temperature drop of the solid surface during the impact was recorded to be ∼20 °C. Hatta et al.5 conducted a series of experiments to study the film-boiling impact of water droplets on different metallic surfaces. Experimental results on such properties as the deformation and rebound height of the droplet during the impact, and the residence time of the droplet on the surface, were summarized for a range of droplet sizes and impact velocities. Several studies were also reported for the impact of water or n-heptane droplets on hot surfaces under the subcooling condition, in which the initial temperature of the droplet was below the saturation temperature.6 Karl and Frohn7 investigated the oblique impact of water and ethanol droplets on a chromiumplated copper or steel wall. Different regimes based on the secondary droplet formation were found, which were similar to the regimes reported in the normal collision by Wachters and Westerling.3 The droplet collision with a porous surface has only been scarcely reported in the literature. It was found that the impact on a porous surface might be different from an impact on an impermeable surface, because of the surface roughness and/or the potential for liquid to seep into the porous surface. The Leidenfrost temperature on a porous surface is reported to be much higher than that on a nonporous surface. Under the filmboiling condition, the droplet evaporates faster above a porous surface than above a nonporous surface at the same surface temperature, and the vapor layer thickness is found to be smaller on a porous surface. There also seems to be a critical surface porosity for levitation of the droplet at a given temperature. It was observed that, for an alumina surface with micrometersized pores and a porosity of 40%, methanol droplet could not

10.1021/ie800479r CCC: $40.75  2008 American Chemical Society Published on Web 08/09/2008

Ind. Eng. Chem. Res., Vol. 47, No. 23, 2008 9175

be levitated, even at 700 K, which was more than 200 K above the Leidenfrost temperature on a stainless-steel plate.8,9 The modeling of droplet impact on a heated surface in the Leidenfrost regime has been conducted by several researchers, using computational fluid dynamics (CFD). Fujimoto and Hatta10 simulated the impingement process of 0.3-0.5 mm water droplets on a surface at 500 °C using a single-phase twodimensional (2-D) Marker-and-Cell (MAC)-type method. The no-slip and free-slip boundary conditions were adopted on the liquid/solid interface during the spreading and recoiling of the droplet, respectively. The effects of the evaporation and the vapor flow were neglected, and a simplified temperature field was preassumed, which only affects the surface tension of the droplet. Harvie and Fletcher11,12 presented a 2-D simulation of the volatile liquid droplet impacting on a hot solid surface. An axisymmetric volume of fluid (VOF) algorithm was used to model the deformation of the droplet and was coupled with a one-dimensional vapor layer model to account for the vapor flow between the droplet and the solid surface. For We ) 30-80, their model could predict the spreading process of the droplet but not the recoiling and rebound behavior. At We > 80, their model could not reproduce the rapid disintegration of the droplet that occurs during high-velocity impacts. Recently, the VOF method was used with adaptive meshing for the collision of droplets below and above the Leidenfrost temperature.13 The vapor concentration field and the temperature field were obtained from the simulation. Ge and Fan14,15 developed a three-dimensional (3-D) numerical model based on the finitevolume method to simulate the droplet impact on a flat or curved surface in the Leidenfrost regime. The level-set method was used to track the movement of the droplet surface, and the conditions on the fluid/solid interface were specified using the immersed boundary method. A two-dimensional vapor flow model was developed to account for the vapor flow. The energy equations were solved to determine the evaporation rate and temperature distribution in each phase. Their model was capable of reproducing the dynamic and transport behavior of the droplet and surface during the impact in different situations, including droplet-particle collisions and droplet-surface collisions, for both saturated and subcooled conditions. To the best of the authors’ knowledge, no previous work exists for the direct simulation of an evaporative droplet colliding on a heated porous surface. Moreover, in most previous simulation work regarding collisions on a nonporous surface, the mass transport of the vapor during the impact has not been included. Reis et al.16 used a 2-D model to study the impact of liquid droplets on cold porous surfaces. Because, in this case, there was no vapor layer to prevent the direct contact, the droplet is partly absorbed into the porous media. A single set of continuity and momentum equations were solved both in and out of the porous domain, and the matching conditions on the interface were automatically satisfied in their model. The evolution of the free surface both inside and out of the porous substrate, which was the main focus of that study, compared well with nuclear magnetic resonance (NMR) images and photos taken in the experiments. Alam et al.17 studied the impact and spreading-absorption behavior of small droplets (50 µm in diameter) on topographically irregular porous materials, using VOF simulation. Both kinetic-energy-driven impacts and wetting-driven impacts were investigated, and the surface roughness was observed to be a crucial factor during spreading. Alleborn et al.18 also modeled the spreading and sorption of droplets on a porous substrate. The model was derived using the lubrication theory, and the saturated part of the porous substrate was

Figure 1. Schematic diagram of the experimental setup.

assumed to be governed by Darcy’s law. The droplet profile above the substrate and the wetting front inside the substrate were analyzed, and the evolution of the apparent contact angle was discussed. All of the above modeling work was conducted in the context of capillary spreading of liquid droplets on cold porous substrates, which often occurs in the process of coating or ink jet printing, in which the effect of heat transfer or evaporation is negligible. In this paper, the film-boiling impact of a water droplet on a porous surface is investigated using both experimental and simulation methods. A 3-D numerical model for the flow filed coupled with heat and mass transport in both fluid and porous material is developed and described in detail. The simulation results are compared to the experiments and good agreement is found between the two. The results for droplet dynamics as well as the heat and mass transfer features, are compared with the impact on impermeable surfaces, and the similar and different characteristics of the impacts on these two types of surfaces are analyzed. Synthesis of Porous Surface A macroporous R-Al2O3 substrate, synthesized by Dr. Henk Verweij and the Inorganic Materials Science Group in Materials Science and Engineering Department at The Ohio State University, was used as the porous surface for this study.19 In the synthesis, R-Al2O3 powder (AKP30, Sumitomo Chemical Corporation, Japan) was dispersed in an aqueous HNO3 solution of pH 2.0. The stabilized and screened dispersion then went through the vacuumassisted filtration process to form a disk-shaped compact with a diameter of 43 mm and a thickness of 2 mm. After overnight drying, the compact was heated to 950 °C for 10 h at 2 °C/min to synthesize the porous surface. The surface has an average roughness of 30 ( 5 nm and adequate room-emperature gas permeability. The pore size is sharply peaked near the mean radius of 38 nm, with a standard deviation of ∼2 nm, and the porosity of the surface is 34%. More-detailed information about the surface properties can be found in ref 19. Experimental Setup The schematic diagram of the experimental setup is shown in Figure 1. It consisted of (1) a syringe, syringe pump, and a 21-gauge needle, to form and release water droplets; (2) a water heating system, to preheat droplets to 100 °C before their release; (3) a Photron FASTCAM PCI high-speed CCD camera with a Navitar ZOOM 7000 lens and a close-up lens set; (4) a Photron FASTCAM data processing system, to record the images; (5) a light unit, to provide illumination for photography; and (6) a plate heater, to heat the porous surface, and a thermocouple, to monitor the surface temperature of the porous substrate. Descriptions of the experimental setup are given below. The main experimental apparatus consists of a syringe, a syringe pump, and a stainless steel needle. Water with a flow

9176 Ind. Eng. Chem. Res., Vol. 47, No. 23, 2008

rate of 0.40 mL/min was forced by a syringe pump from the syringe to the needle. A droplet with a diameter of 2.5 mm was formed at the needle tip and released by the gravitation force at such a low water flow rate. A segment of water tube carrying water from the syringe to the needle was preheated by a water heating system with boiling water, making the droplet at its saturated temperature before releasing. The porosity and pore size of the porous substrate are 34% and 76 nm, respectively. The porous substrate was placed on, and heated by, a plate heater, and its surface temperature was monitored by a thermocouple. A Photron FASTCAM PCI high-speed CCD camera that was capable of capturing images at a speed 500 frames per second, at a resolution of 512 × 240 and a record time of 8.7 s, equipped with a Navitar ZOOM 7000 lens and a close-up lens set offering 6× magnification power, was used to record the droplet-porous surface impact process. The high-speed CCD camera was connected to a Photron FASTCAM data processing system to record the droplet-surface impact process. A speed of 500 frames per second and an exposure time of 1/1000 s were used for the record. A light source was used to illuminate the droplet and porous surface from the front. Droplet dimensions were measured directly from the pixels of the images taken in the droplet-surface impact process, and a picture of a steel ruler captured under the same magnification was used to calibrate the length scale in the images. Based on the video frames, the droplet spread radius, the droplet height, and the droplet velocities before and after the contact with the porous surface were obtained.

deformation tensor. The third term on the right-hand side of eq 2 represents the surface tension force, which is implemented using the continuum surface model (CSM).20 In the present approach, surface tension forces are replaced by surface tension volume forces, which act in a small number of computational cells surrounding the free surface. δ(φ) is the one-dimensional delta function and κ(φ) is the curvature of the free surface, which can be calculated from the level set function. Fvapor is the vapor pressure force exerting on the droplet-surface contact area due to the effect of the evaporation, as discussed later in the vapor layer model. The equations for flow inside the porous media is derived by taking averages of the Navier-Stokes equation, and considering the solid drag in the porous media:

Numerical Model

For porous media consist of randomly packed spheres, the permeability is defined by

The flow field, temperature field, and vapor concentration field in the droplet, porous substrate, and surrounding gas are solved simultaneously using direct numerical simulation. The flow is assumed to be incompressible and is modeled based on a finitevolume method for the momentum balance equations in 3-D, Cartesian coordinates. The level-set method is used to trace the gas/liquid interface. Heat transfer is solved in all three phases with proper boundary conditions on the interfaces. The concentration of the vapor in the gas phase and inside the porous substrate is solved, with consideration of the different mechanisms for mass transfer in each domain and the vapor generation during evaporation. A 2-D vapor layer model is used to account for the effect of the vapor flow. This microscopic vapor layer model is essential to the simulation, because it is responsible for the evaporation process, which has significant effects on the heat and mass transfer, as well as the dynamics of the droplet. All the submodels are coupled in each time step and solved simultaneously, as described in greater detail in the following sections. Note that the current model only applies to the film-boiling condition in which the vapor layer is always present, and the situation in which direct contact of the liquid droplet and the solid surface occurs is beyond the scope of this study. Hydrodynamics. The equation of motion for the impacting droplet and the surrounding gas follows the Navier-Stokes equation, as given by ∇·u)0

(1)

∂u + ∇ · uu ) - ∇ p + ∇ · (2µD) + σκ(φ)δ(φ) ∇ φ + F ∂t Fvapor(2)

(

)

where u is the fluid velocity, F the fluid density defined later by eq 13, µ the fluid viscosity defined by eq 14, and D the rate of

∇ · u¯ ) 0

(3)

u¯ u¯ ∂u¯ ¯ )+B¯ +∇· ) - ∇ p¯ + ∇ · (2µD F ∂t ε

(4)

(

)

where ε is the porosity. The averaged velocity, pressure, and rate of deformation tensor are defined as u¯)εu

(5a)

p¯)εp

(5b)

¯ ) 1 [∇u¯ + (∇u¯)T] (5c) D 2 The drag per unit volume inside the porous media due to the solid is calculated using Darcy’s law: µε B¯)- u¯ Kp

Kp )

(6)

ε3dp2

(7) 150(1 - ε)2 The flow on two sides of the gas-porous media interface must obey the matching conditions for velocity and stresses:16 u¯p ) uf

∇ u¯p ) ∇ uf

p¯p ) pf

(8)

It has been suggested that the two sets of equations for the fluid and porous media domains can be combined into a single set of equations for the entire computational domain, and the matching conditions on the interface can be automatically satisfied, which could greatly simplify the problem:16 ∇ · u¯ ) 0

(9)

( ∂u∂t¯ + ∇ · u¯εu¯ ) ) - ∇ p¯ + ∇ · (2µD¯) + B +

F

σκ(φ)δ(φ) ∇ φ + Fvapor(10)

Level-Set Method for Droplet Surface. The level-set method has been proven to be a versatile method for interface capturing on fixed Eulerian grids. In the level-set method, the gas/liquid interface is taken to be the zero-level set of the level-set function φ(x,t), which is the signed distance from the location x to the interface. Generally, the level set function can be written as

{

φ(x, t) > 0 φ(x, t) < 0 φ(x, t) ) 0

(for x in the gas) (for x in the droplet) (for x at the interface)

(11)

The level-set function φ(x,t) is convected passively by the flow field, and this function is responsible for capturing the motion of the interface:

Ind. Eng. Chem. Res., Vol. 47, No. 23, 2008 9177

∂φ + uΓ · ∇ φ ) 0 (12) ∂t Here, uΓ is the fluid velocity at the interface and t is the time. When the level-set function is defined, the entire fluid domain can be treated as a single domain and regions of different phases can be distinguished using the level-set function. The density and viscosity of the entire fluid field change continuously from one phase to the other. They are defined as follows: F(φ) ) Fl - (Fl - Fg)H(φ)

(13)

µ(φ) ) µl - (µl - µg)H(φ)

(14)

where the subscripts “l” and “g” denote the liquid phase and the gas phase, respectively. H(φ) is a smoothed Heaviside function, which is defined as

{

0 φ + β sin(πφ ⁄ β) + H(φ) ) 2β 2π 1

(15)

Here, β is the thickness of the interface, which is a small value, relative to the entire grid space. In our simulation, this value is taken to be 1.5∆x, where ∆x is the grid spacing. An iterative redistancing procedure is also performed to maintain a uniform thickness of the interface and to conserve the mass.21 The curvature of the interface in eq 2 can also be calculated using the level-set function: ∇φ (16) |∇φ| Vapor Layer Model. Under the film-boiling condition, the evaporation and flow in the narrow gap between the bottom of the droplet and the solid surface have significant effects on the dynamics of the droplet and evolves strong coupling between the mass, heat, and momentum transfer. Because the thickness of the gas (∼10 µm) is several orders of magnitude smaller than the scale of the entire droplet, it would be impractical to use the same computation mesh for both the macroscopic flow and the vapor layer flow.11 Thus, a 2-D subgrid model is developed to simulate the dynamics of the vapor flow between the droplet and the surface. For the film-boiling impact, the vapor layer model would allow determination of the evaporation rate and the evaporation-induced pressure inside the vapor layer. The extent of the vapor layer is determined from the location of the droplet surface, which is described by the level-set function. Whenever the minimum distance between the droplet surface and the substrate falls below ∆x, the vapor layer model is invoked. The height of the vapor layer, denoted by h, is calculated from the level-set function and is a function of r. The radial extent of the vapor layer is defined by the maximum radius where h is smaller than ∆x. Because the height of the vapor layer is very small, the lubrication theory is applied to model the flow inside the vapor layer, which assumes that the unsteady term and the convective term are small, compared to the viscous and pressure terms. Under the assumption that the flow is axisymmetric, only radial and vertical velocity components are considered, which are both functions of r and z, as depicted in Figure 2. The variation of the induced pressure is much larger in the radial direction than in the vertical direction, so pressure pV is considered to be only a function of r.11 At the top of the vapor layer, vapor comes into the vapor layer through evaporation, and the position of the droplet surface also changes with time, so the boundary condition for uz can be expressed as κ(φ) ) - ∇ ·

(17)

where uz,drop(r) is the velocity of the droplet surface, which can be obtained from the velocity field, and uevp(r) is the velocity of the vapor generated by evaporation, which is given by uevp )

m ˙ Fv

(18)

In the aforementioned equation, Fv is the vapor density. The mass flux of the vapor generated by evaporation (m ˙ ) is calculated from the heat transfer model, as described later in this paper. At the bottom of the vapor layer, the velocity of the gas that penetrates into the porous substrate can be approximated using Darcy’s law:

( )

Kp Kp pv(r) - pp(r) ∇p≈(19) µ µ ∆x in which pv(r) is the pressure inside the vapor layer and pp(r) is the pressure in the first layer of computational cells inside the porous media. The tangential velocities at the top and bottom boundaries are assumed to be no-slip, for simplicity. Morecomplicated slip conditions can also be applied, such as those used in ref 11. With the aforementioned boundary conditions, the momentum and continuity equations for the vapor layer lead to uz0(r) ) -

(if φ e -β) (if |φ| < -β) (if φ g β)

uz(r, h) ) uzd(r) ) uz,drop(r) + uevp(r)

(

)

∂pv ∂h 1 ∂ 1 ∂ (rhurd) - uzd + uzd - uz0 (20) rh3 ) 12µr ∂r ∂r 2r ∂r ∂r with the boundary conditions ∂pv )0 ∂r r)0 pv(r ) rmax) ) 0

(21a) (21b)

The second equation above is the matching condition at the extremity of the vapor layer. The absolute pressure is the sum of the vapor-induced pressure (pv) and the reference pressure measured at the radial boundary of the vapor layer. Because both pv(r) and uz0(r) are unknowns, eqs 19 and 20 must be solved iteratively. In the actual simulations, the calculation usually converges within ∼3-5 iterations. Heat Transfer. Heat transfer occurs in all solid, liquid, and vapor phases, and on the liquid/solid and solid/vapor interfaces. Therefore, the energy balance equations for all phases and interfaces are solved to determine the heat-transfer rate and evaporation rate. Inside the porous surface, the heat conduction equation in 3-D coordinates is ∂Ts ) Rs ∇ · ∇ Ts (22) ∂t where Ts(x) is the solid temperature and Rs is the thermal diffusivity of porous media. In the gas and liquid phases, thermal energy transport is described by the following equation, which

Figure 2. Coordinates of the vapor layer model.

9178 Ind. Eng. Chem. Res., Vol. 47, No. 23, 2008

considers the convective and conductive heat transfer in the fluid, but neglects the viscous dissipation: ∂Tf + u · ∇ Tf ) Rf ∇ · ∇ Tf ∂t

De ) (23)

The thermal diffusivity (Rf) is determined from the level-set function, as well as the liquid and gas properties in the same manner as eqs 13 and 14. When the vapor layer model is invoked, using the same assumptions that were made in the momentum equations, the energy conservation equation for the vapor layer can be simplified to a one-dimensional (1-D) equation:11 ∂2Tv ∂z2

)0

and the effective diffusivity in the porous media is defined as22

(

)

Tss - Tds ) -ksnp · ∇sTs δ

where Tss and Tds are the interface temperatures of the solid surface and the droplet surface, respectively; kv and ks are the heat conductivity of the vapor phase and the solid phase, respectively; δ is the thickness of the vapor layer, which is calculated from the level-set function; np is the unit normal vector of the porous surface; and ∇pT is the temperature gradient evaluated only on the solid side. On the liquid/vapor interface, the energy balance equation is kv

(

)

Tss - Tds ˙ Lc ) -kdn · ∇dTd + m δ

(26)

where kv and kd are the thermal conductivity of the vapor and liquid; Lc is the latent heat; Tss and Tds are the temperatures of the porous surface and droplet surface, respectively; and ∇dTd is the temperature gradient evaluated only in the liquid. The parameter m ˙ is the evaporation rate, which is needed in eq 18, to calculate the vapor velocity. Mass Transfer. The mass transport of the vapor in the gas phase as well as inside the porous substrate is modeled considering the different mass-transfer mechanisms. Note that, in the momentum equations, the flow field is assumed to be incompressible, both inside and outside the vapor layer. Because of the fact that the magnitude of pressure variation is small, compared to the atmospheric pressure, the density change due to pressure variation is small in the gas phase. The temperature variation also affects the density, especially near the porous surface. However, this effect is not considered in the current model. Furthermore, for simplicity, the density change due to variation of the gas composition is neglected. The concentration of the vapor in the gas is expressed using its molar fraction (yv), which is only considered in the gas inside and outside the porous substrate, but not calculated inside the liquid droplet. In the gas outside the porous substrate, the vapor is transported by both diffusion and convection. The concentration of the vapor is governed by ∂yv + u · ∇ yv ) D∇2yv ∂t

∂yv + u¯ · ∇ yv ) De∇2yv ∂t

(30)

The molecular diffusivity is calculated using the binary diffusivity equation, as a function of temperature, pressure, and the properties of the vapor and air.23

Dm )

(

)

1 1 1⁄2 + Mv Ma 1⁄3 Tac 1⁄3 3 + Pac

0.001T1.75

[( ) ( ) ]

Tvc p Pvc

(31)

Because of the small dimension of the pore size in the porous material used in this study, Knudsen diffusion must be taken into consideration. The Knudsen diffusivity is given by23,24 1 Dk ) dp 3

 8RT πM

(32)

where dp is the diameter of the pores, R the gas constant, and M the molecular weight of the vapor. Inside the vapor layer, the vapor concentration is assumed to be only a function of r, which is consistent with the previous assumption that the pressure also varies only in the r-direction. The mass balance is expressed in the integral form, considering the change of the concentration, the change of volume, and the mass fluxes. ∂ ∂t

∫∫∫y

v

dV + I∫syvu · n dA ) 0

(33)

More explicitly, the vapor concentration change at a fixed position r inside the vapor layer is given by V

∂yv ) 2π ∂t

∫ yu r

0

v zdr

dr - 2π

[2πr∫

h(r)

0

∫ yu r

0

v z0r

dr +

] - [2πr∫

yvur dz

h(r)

r-

0

]

yvur dz r+(34)

The right-hand side terms correspond to the vapor generation on the droplet surface, the vapor penetration into the porous surface, and the radial vapor fluxes coming into and out of the volume, respectively. Numerical Method. The computation code is based on CFDLIB, which is a CFD program for multiphase flows developed by the Los Alamos National Laboratory;25 however, it has been extensively modified to include the different models previously mentioned. An explicit time-marching, conservative finite-volume numerical technique is applied to solve the governing equations. The code is second-order accurate in spatial discretization. Results and Discussion

(27)

A similar equation is applied in the porous domain: ε

(29)

1 1 1 ) + D0e Dm Dk

(24)

(25)

0 e

where ε is the porosity and τ is the tortuosity factor. D0e includes the effect of both molecular diffusion and Knudsen diffusion and is given by

The boundary condition at the vapor/porous surface interface is given by kv

( τε )D

(28)

Here, the averaged velocity u j is used for the convective term,

In the experiment, a saturated water droplet with a diameter of 2.5 mm is released from a distance of 4.0 cm above the porous surface, which has been heated to 300 °C. A separate test has been performed to ensure that 300 °C is sufficient for the impact to be in the film-boiling regime. A 2.5-mm water droplet is gently placed on top of the heated surface, and the time required for the droplet to evaporate completely is recorded.

Ind. Eng. Chem. Res., Vol. 47, No. 23, 2008 9179

It is found that the droplet is able to float above the surface for ∼55 s when the surface temperature is 270 °C, and the evaporation time at the surface temperature of 250 °C is ∼50 s. When the surface temperature is reduced to ∼200 °C, the droplet cannot be levitated, but it does show violent boiling and splash behavior. Therefore, 300 °C is considered to be above the Leidenfrost temperature in the current case. It has been reported that the Leidenfrost temperature for a water droplet on a metallic surface is ∼200 °C.2 Avedisian and Koplik8 determined that the Leidenfrost temperature for methanol droplets on 10% and 25% porous surfaces with micrometersized pores are ∼100 and ∼200 °C higher than that on the stainless steel, respectively. In their experiment, film-boiling was not observed, even at 400 °C, for the surface with 40% porosity. However, in the current study, the Leidenfrost temperature is not far from that on nonporous surfaces, which might be attributed to the small pore size, which is only 76 nm. The simulation is conducted using the models previously described, under the same conditions as the experiment. The roughness of the surface is an important factor that determines the impact behavior of the droplet, but its effect is not considered in the current simulation, because of its extremely small length scale. Cartesian grids with dimensions of 80 × 80 × 100 (in the x-, y-, and z- directions) are used, with a uniform grid spacing of 0.1 mm. As the result, the 2.5-mm water droplet has ∼13 grid points per radius, which has been proven to provide sufficient resolution for the droplet shape. The average time step is ∼12 µs, and a typical run requires ∼2500 time steps to simulate 30 ms in real time. The simulation is run on the Itanium 2 cluster supercomputer at the Ohio Supercomputer Center. The computation time for a typical case is ∼8.5 h. The impact process of the droplet on the surface is shown in Figure 3, which gives a side-by-side comparison of the experimental and simulation results. The simulated droplet is created by plotting the iso-value surface of the zero level-set function in three dimensions. The time shown on the left of the figures is the time after the initial contact. In the initial stage of the contact, the droplet falls onto the solid surface and forms a flattened disk shape. The inertial force drives the liquid to continue to spread on the solid surface, while the surface tension and the viscous forces resist the spreading of the liquid film. After ∼4 ms, all the kinetic energy of the impact is either converted to the surface energy of the deformed droplet or dissipated by friction, and the droplet reaches its maximum diameter. The images at 8-16 ms show the recoiling motion of the droplet, which forms an upward flow in the center of the droplet and results in an elongated shape in the vertical direction. At 20 ms, the droplet has detached from the surface, and the droplet shape has evolved into a small upper portion and a bigger lower portion, which are connected by a thin neck region. The two regions of the droplet finally separate from each other, as shown in the image at t ) 24 ms. For the entire duration of the impact, the simulated droplet shape is very similar to that depicted in the photographs taken in the experiment. The spreading, recoiling, rebound, and disintegration of the droplet are correctly captured in each stage of the collision. The stability of the droplet during impact has been investigated previously by some researchers on nonporous surfaces. From their experiment of water droplets on a hot polished gold surface, Wachters and Westerling3 concluded that, for Wen > 80, the droplet breaks apart during the initial part of the impact; at 30 < Wen < 80, the droplet does not disintegrate until it starts to rise upward from the surface; and for Wen < 30, the droplet never disintegrates. Under the current condition, the Wen

Figure 3. Impact of a 2.5-mm water droplet on the porous surface at 300 °C. Figures on the left are photographs taken in the experiment, and images on the right are simulation results under the same conditions.

value is calculated to be 29.5, which is on the border between the disintegration/no-disintegration regimes. Compared to the photographs taken at Wen ) 70 in Wachters and Westerling’s work, which show only a very small secondary droplet breaking away from the main droplet, the secondary droplet is much larger in the current study, as shown both in experimental and numerical results. This might suggest that it is easier for the droplet to break apart on a porous surface, and therefore the droplet is less stable. However, more-comprehensive study must be performed to fully understand the stability of droplets during the impact on porous surfaces. The spread factor of the droplet is shown in Figure 4 for both simulation and experimental results. The spread factor is a dimensionless number that is defined as the ratio between the radius of the contact area between the droplet and the porous surface (shown as R in Figure 2), and the initial radius of the droplet. Generally, the simulation curve is in good agreement with the experimental curve, except that the simulation curve reaches its peak slightly earlier than the experimental curve,

9180 Ind. Eng. Chem. Res., Vol. 47, No. 23, 2008

Figure 4. Spread factor of the droplet during collision. Figure 6. Surface temperature variation at the center of the contact region.

Figure 5. Dimensionless height of the droplet during collision.

and the predicted spread factor is slightly higher than the experimental value in the spreading process. This suggests that the predicted spreading motion is faster than the actual process, which might be related to the fact that no surface roughness is considered in the simulation. The point at which the curve reaches zero corresponds to the time when the droplet has completely left the surface; therefore, it denotes the residence time of the droplet on the porous surface. From Figure 4, both the experiment and the simulation show that the residence time in the current case is ∼18 ms. For nonporous surfaces, the first-order vibration period of the oscillating drop is considered to be equal to the residence time of the droplet;3 that is,



FlR (35) 2σ This estimation works very well for droplet collision with nonporous surfaces in the film-boiling regime, as has been demonstrated in the experiments and simulations by different authors.3,14 Using the parameters in the current study, the residence time calculated by eq 35 is 13.4 ms, which is much shorter than the observed residence time. Therefore, it is possible that the porous structures on the surface will have a significant effect on the residence time of the droplet during collision. Figure 5 shows the dimensionless height of the droplet, which is the height of the droplet divided by its equivalent diameter. The simulation curve follows the same trend as the experimentally measured curve, but with lower values both in the initial and later stages. The lower height in the initial stage again shows that the droplet in the simulation spreads faster than the droplet in the actual situation. The difference in the later stage might come from overpredicted kinetic energy loss (by dissipation in the recoiling stage) or underpredicted vapor force (which levitates the droplet). τ ) πR

Figure 7. Accumulate heat transfer from the porous surface.

The temperature at the center of the contact region on the porous surface is plotted in Figure 6. The maximum temperature drop is ∼30 °C and occurs near the end of the contact. After the droplet rebounds from the surface, the surface temperature continues to increase, because of heat conduction inside the porous substrate. The location of the minimum temperature and the overall shape of the simulated temperature curve are similar to that of the curve experimentally measured by Groendes and Mesler,4 who studied the saturate impact of water droplet on a quartz surface. The amount of heat transfer from the surface to the vapor and the droplet is shown in Figure 7. Most heat loss of the surface occurs in the early stage of the contact, as shown by the larger slope of the curve. In this stage, the droplet undergoes the spreading process, and the vapor layer has not been fully developed and provides less resistance for heat transfer. The vapor concentration at different times of the contact is shown in Figure 8, together with the velocity vectors for the flow field. Because of the small pore size that causes a high flow resistance inside the porous substrate, the velocity of gas penetrating into the porous surface is extremely small (∼2-3 orders of magnitude smaller than the external flow). Generally, the gas penetration into the surface will reduce the pressure inside the vapor layer, as suggested by eq 20, and will affect the ability of the vapor layer to levitate the droplet. However, the simulation shows that gas penetration is insignificant for a porous material with a pore size on the order of tens of nanometers. This helps to explain the lower Leidenfrost temperature observed in the current study, relative to that observed by other researchers on porous surfaces. The color in Figure 8 represents the molar fraction of the water vapor in the gas. The development of the vapor layer can

Ind. Eng. Chem. Res., Vol. 47, No. 23, 2008 9181

proximately a minute over the surface, the evaporation rate can be estimated to be ∼10-7 g/ms. Therefore, the amount of evaporation predicted in the current simulation is within the reasonable range, considering the effective contact time. Concluding Remarks

Figure 8. Velocity and vapor concentration field at different times during the collision.

In this study, a new three-dimensional (3-D) numerical model for a liquid droplet in collision with a porous surface under the film-boiling condition is presented. Compared to the model originally developed for droplet collision with nonporous surfaces, the current model has the following features: (1) The momentum, as well as the heat and mass transport inside the porous substrate, is modeled. The flow outside and inside the porous media is described by a unified set of equations for the entire domain, with the flow resistance by the porous material being modeled by the solid drag term in the momentum equation. In this way, the interface condition at the surface of the porous substrate is incorporated naturally in the equations, without the need to be specified explicitly, as in the immersed boundary method that has been applied in our previous models. (2) The modified vapor layer model considers the penetration of gas into the porous substrate, which leads to the decrease of vapor pressure in the vapor layer. (3) The vapor mass transfer is modeled both inside and outside the porous substrate, with consideration of different transfer mechanisms in each domain. The porous alumina surface with pore size of 76 nm and porosity of 34% is investigated using experimental and numerical methods. Because of the small pore size, 300 °C is sufficient for the impact of the water droplet to be in the film-boiling regime. The simulated droplet shape, spreading factor, and droplet height are in good agreement with the experimental results. Although the dynamic characteristics of the droplet and the heat transfer characteristics of the surface are similar to the impact of droplets on nonporous surfaces, the impact on the porous surface shows some distinct behaviors. The droplet has a longer residence time, and it also seems to be less stable on the porous surface. More-extensive analysis of the impact under different conditions is required to fully understand the droplet behavior on porous surfaces. The results from the ongoing investigation of the impact of droplets on different porous materials with different velocities will be reported in our future publication.

Figure 9. Accumulate evaporation in the water droplet.

be clearly viewed during the contact. The initial vapor concentration is determined from the ambient conditions of air. As the droplet starts to spread on the surface, evaporation occurs and vapor begins to accumulate under the droplet. In the spreading and recoiling process, the higher vapor concentration is only limited to a small region covering the vapor layer. Because the velocity inside the porous substrate is small, the vapor mass transport inside the porous media occurs mainly via diffusion. The penetration depth of the vapor in the porous substrate is ∼0.2 mm, whereas the vapor fraction is ∼10%-15% near the end of the contact. The accumulated evaporation of water is plotted in Figure 9. The figure suggests that most of the evaporation occurs within the first 5 ms of contact, which is also consistent with the higher heat transfer from the surface during this time period. Because the measurement of vapor concentration or evaporation rate during the contact is not performed in the current study, and no experimental data are found in the literature, no direct comparison between the experimental and numerical results can be made. However, from the fact that a 2.5- mm-diameter droplet can float for ap-

Acknowledgment The authors acknowledge the Donors of the American Chemical Society Petroleum Research Fund (under Grant No. PRF 44515-AC9) for partial support of this research. The work is also supported in part by the Ohio Supercomputer Center. We are grateful to Dr. Henk Verweij, Dr. Krenar Shqau, Matthew L. Mottern, Lanlin Zhang, and the Inorganic Materials Science Group in Materials Science and Engineering Department at The Ohio State University for their preparation of the porous sample used in this study. Literature Cited (1) Salge, J. R.; Dreyer, B. J.; Dauenhauer, P. J.; Schmidt, L. D. Renewable hydrogen from nonvolatile fuels by reactive flash volatilization. Science 2006, 314, 801–804. (2) Biance, A.; Clanet, C.; Quere, D. Leidenfrost drops. Phys. Fluids 2003, 15 (6), 1632–1637. (3) Wachters, L. H. J.; Westerling, N. A. J. The heat transfer from a hot wall to impinging water drops in the spheroidal state. Chem. Eng. Sci. 1966, 21, 1047–1056.

9182 Ind. Eng. Chem. Res., Vol. 47, No. 23, 2008 (4) Groendes, V.; Mesler, R. Measurement of transient surface temperature beneath Leidenfrost water drops. In Proceedings of the 7th International Heat Transfer Conference, Munich, Federal Republic of Germany, 1982; pp 131-136. (5) Hatta, N.; Fujimoto, H.; Kinoshita, K.; Takuda, H. Experimental study of deformation mechanism of a water droplet impinging on hot metallic surfaces above the Leidenfrost temperature. J. Fluid Eng. 1997, 119, 692–699. (6) Inada, S.; Miyasaka, Y.; Sakamoto, K.; Hojo, K. Liquid-solid contact state and fluctuation of the vapor film thickness of a droplet impinging on a heated surface. J. Chem. Eng. Jpn. 1988, 21, 463–468. (7) Karl, A.; Frohn, A. Experimental investigation of interaction process between droplets and hot walls. Phys. Fluids 2000, 12, 785–796. (8) Avedisian, C. T.; Koplik, J. Leidenfrost boiling of methanol droplets on hot porous/ceramic surfaces. Int. J. Heat Mass Transfer 1987, 30, 379– 393. (9) Chandra, S.; Avedisian, C. T. Observation of droplet impingement on a ceramic porous surface. Int. J. Heat Mass Transfer 1992, 35, 2377– 2388. (10) Fujimoto, H.; Hatta, N. Deformation and rebounding processes of a water droplet impinging on a flat surface above Leidenfrost temperature. J. Fluids Eng. 1996, 118, 142–149. (11) Harvie, D.; J, E.; Fletcher, D. F. A hydrodynamic and thermodynamic simulation of droplet impacts on hot surface, Part I. Theoretical model. Int. J. Heat Mass Transfer 2001, 44, 2633–2642. (12) Harvie, D. J. E.; Fletcher, D. F. A hydrodynamic and thermodynamic simulation of droplet impacts on hot surface, Part II. Validation and applications. Int. J. Heat Mass Transfer 2001, 44, 2643–2659. (13) Nikolopoulos, N.; Theodorakakos, A.; Bergeles, G. A numerical investigation of the evaporation process of a liquid droplet impinging onto a hot substrate. Int. J. Heat Mass Transfer 2007, 50, 303–319. (14) Ge, Y.; Fan, L.-S. Three-dimensional simulation of impingement of a liquid droplet on a flat surface in the Leidenfrost regime. Phys. Fluids 2005, 17, 027104.

(15) Ge, Y.; Fan, L.-S. 3-D modeling of the dynamics and heat transfer characteristics of subcooled droplet impact on a surface with film boiling. Int. J. Heat Mass Transfer 2006, 49, 4231–4249. (16) Reis, N. C., Jr.; Griffiths, R. F.; Santos, J. M. Numerical simulation of the impact of liquid droplets on porous surfaces. J. Comput. Phys. 2004, 198, 747–770. (17) Alam, P.; Toivakka, M.; Backfolk, K.; Sirvio¨, P. Impact spreading and absorption of Newtonian droplets on topographically irregular porous materials. Chem. Eng. Sci. 2007, 62, 3142–3158. (18) Alleborn, N.; Raszillier, H. Spreading and sorption of a droplet on a porous substrate. Chem. Eng. Sci. 2004, 59, 2071–2088. (19) Mottern, M. L.; Shqau, K.; Zalar, F.; Verweij, H. Permeation porometry: Effect of probe diffusion in the condensate. J. Membr. Sci. 2008, 313, 2–8. (20) Brackbill, J. U.; Kothe, D. B.; Zemach, C. A continuum method for modeling surface tension. J. Comput. Phys. 1992, 100, 335–354. (21) Sussman, M.; Fatemi, E.; Smereka, P.; Osher, S. An improved level set method for incompressible two-phase flows. Comput. Fluids 1998, 27, 663. (22) Kolaczkowski, S. T.; Chao, R.; Awdry, S.; Smith, A. Application of a CFD code (Fluent) to formulate models of catalytic gas phase reactions in porous catalyst pellets. Chem. Eng. Res. Des. 2007, 85, 1539–1552. (23) Natarjan, S.; Zhang, C.; Briens, C. Numerical simulation and experimental verification of gas flow through packed beds. Powder Technol. 2005, 152, 31–40. (24) Valipout, M. S.; Saboohi, Y. Modeling of multiple noncatalytic gas-solid reactions in a moving bed of porous pellets based on finite volume method. Heat Mass Transfer 2007, 43, 881–894. (25) Kashiwa, B. A.; Padial, N. T.; Rauenzahn, R. M.; Vanderheyden, W. B. A cell-centered ICE method for multiphase flow simulation, Los Alamos National Laboratory Research Report, LA-UR-93-3922, 1994.

ReceiVed for reView March 26, 2008 ReVised manuscript receiVed June 6, 2008 Accepted June 12, 2008 IE800479R