Leidenfrost phenomena

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Figure 1 Schematic of a Leidenfrost drop hovering over a solid hot surface with radius R and vapor film thickness

A drop of liquid is introduced to a solid surface of constant temperature. If the temperature of the solid is around the boiling point of the liquid, the drop will rapidly boil and evaporate. However, if the solid is held at a temperature much higher than the boiling point of the liquid, a thin film of vapor forms between the solid and the liquid (see Fig. 1). A drop that floats on its own vapor in this way is called a Leidenfrost drop, named after the German physician, Johann Gottlob Leidenfrost, who first reported the phenomenon in 1756. The thin vapor film keeps the liquid and solid from coming into direct contact with each other thus inhibiting bubble nucleation. The film also acts as an insulator for the liquid above it, thereby slowing the evaporation process such that, for example, a droplet of water with radius of 1 mm can float for an entire minute over a metallic surface at 200 °C to evaporate completely.

A variety of technological applications involve imposing liquid drops on a hot surface, therefore making the Leidenfrost phenomenon an important topic of study. Examples of such applications include spray cooling of hot metal during metallurgical production, film cooling of a rocket nozzle, mist flow heat transfer in evaporators, fuel droplet vaporization in fuel-injected engines, and reflooding of a nuclear reactor core after an accidental loss of coolant.

The Leidenfrost Effect can be easily observed by placing a few drops of water on a hot frying pan (not nonstick). When the pan reaches the Leidenfrost temperature, the water droplets will bead up and “dance” across the surface due to the immediate vaporization of the bottom of the liquid drop. Physicists have hypothesized that firewalkers (people who walk barefooted across hot beds of coal) used the Leidenfrost phenomenon to their advantage to impress crowds. However, after several injurious experiments, they lost faith in their argument and decided instead that the entertainers completed this feat due to a combination of other hidden conditions (such as a layer of insulating ash).

Leidenfrost drops are considered nonwetting and exhibit levitation characteristics. The capillary length $L c$ for a liquid drop is defined as

${L_c} = \sqrt {\frac{\sigma }{{\rho g}}} \qquad \qquad(1)$

where $σ$ is the liquid surface tension and $ρ$ is the liquid density. If the radius R is smaller than the capillary length, the drop is spherical except for a flattened bottom [Fig. 1(a)]. Contact is taken as the region of the drop interface parallel to the solid surface. The relationship between the contact size $λ 1$ radius $R 1$ and capillary length $L c$ is found by combining the correlation established by Mahadevan et al. (1999) with the geometric Hertz relation

$\lambda ~ \frac{R^2}{L_c}\qquad \qquad(2)$

If the radius R is larger than the capillary length, the drop forms a puddle flattened by gravity [Fig. 1(b)]. In this case, the contact size $λ$ is related only to the radius R:

$\lambda ~ R \qquad \qquad(3)$

The puddle’s thickness h is found by balancing the surface tension and the hydrostatic force, yielding

$h = 2{L_c} \qquad \qquad (4)$

It has been observed that the radius R of a Leidenfrost drop of water must be on the order of 1 cm. In cases where the radius R exceeded 1 cm, a bubble of vapor rises at the center and bursts at the upper interface due to a Rayleigh-Taylor instability at the lower interface (Biance et al., 2003). The largest radius a droplet of water can have without bubbles forming is named the critical radius $R c$ and it is linearly related to the capillary length $L c$ and to the puddle height according to the following two equations

${R_c} = 3.84{L_c}\qquad \qquad(5)$ ${R_c} = 1.92h\qquad \qquad(6)$

Biance et al. (2003) presented a simple dimensional scaling and analysis by combining force and energy balance to show the lifetime of a droplet for $R < L c$ and $R > L c$ , which is presented below. The thickness of the vapor film $δ$ varies with time due to the evaporation process of the liquid drop on top of it. For a puddle, $R > L c$ [Fig. 1(b)], an energy balance yields the following equation for the evaporation rate:

$\frac{{dm}}{{dt}} = \frac{{{k_v}}}{{{h_{\ell v}}}}\frac{{\Delta T}}{\delta }\pi {\lambda ^2}\qquad \qquad(7)$

where $πλ 2$ is the contact zone surface area, $k v$ is the vapor thermal conductivity, $Δ T / δ$ is the temperature gradient, m is the mass of the liquid drop and $\Delta T = {T_\infty } - {T_{sat}}$ .

Assuming fully developed vapor flow between the liquid drop and the solid surface [Fig. 1(b)], a force balance yields the following solution

$\frac{{dm}}{{dt}} = {\rho _v}\frac{{2\pi {\delta ^3}}}{{3{\mu _v}}}\Delta p\qquad \qquad(8)$

where $Δ$ p is the pressure drop imposed by the liquid drop and $μ v$ is the gas viscosity. By equating eqs. (7) and (8) and using $λ$ from eq. (3) with $\Delta p = {\rm{ }}2{\rho _\ell }g{L_c}$ , one obtains the following equation for vapor film thickness for $R > L c$ :

$\delta = {\left( {\frac{{3{k_v}\Delta T{\mu _v}}}{{4{h_{\ell v}}{\rho _v}{\rho _\ell }g{L_c}}}} \right)^{1/4}}{R^{1/2}}\qquad \qquad(9)$

For small drops, $R < L c$ [Fig. 1(a)], contact size is found using eq. (2), and the pressure drop acting on the vapor film is the Laplace pressure 2 $σ$ /R. Equations (10) and (11) predict the forms of evaporation rate and vapor film thickness for small drops, respectively:

$\frac{dm}{dt} ~ \frac{k_v}{{h_{\ell v}}} \frac{\Delta T}{R} {R^2} \qquad \qquad(10)$

$\delta ~ ({\frac{{k_v} \Delta T {\mu _v} {{\rho}_{\ell}} g}{{h_{\ell v}}{\rho _v}{\sigma ^2}})^{1/3}} {R^{4/3}}\qquad \qquad(11)$

For both small drops and puddles, the film thickness $δ$ increases monotonically with the drop radius.

To determine the lifetime of a Leidenfrost drop of a particular radius, we must assume that the radius and thickness are related by the quasi-steady conditions developed above [Eqs. (7) – (11)]. For a puddle R > Lc assuming $m{\rm{ }} = {\rm{ }}\pi {R^2}h{\rho _\ell }$ ,

$R(t) = {R_o}{\left( {1 - \frac{1}{\tau }} \right)^2}\qquad \qquad(12)$

where $R 0$ is the radius at t = 0 and the lifetime $τ$ is

$\tau = 2{\left( {\frac{{4{\rho _\ell }{L_c}{h_{\ell v}}}}{{{k_v}\Delta T}}} \right)^{3/4}}{\left( {\frac{{3{\mu _v}}}{{{\rho _v}g}}} \right)^{1/4}}R_o^{1/2}\qquad \qquad(13)$

The time dependence of the film thickness is derived from the above equations and gives the following linear relationship, which decreases as the time approaches the lifetime of the drop.

$\delta (t) = {\left( {\frac{{3{k_v}\Delta T{\mu _v}R_o^2}}{{4{h_{\ell v}}{\rho _v}{\rho _\ell }g{L_c}}}} \right)^{1/4}}\left( {1 - \frac{t}{\tau }} \right)\qquad \qquad(14)$

Similarly, for smaller drops, $R < L c$ , evaporation occurs over the whole drop surface as described earlier. Therefore, assuming $m = 4\pi {R^3}{\rho _\ell }/3$ , the time dependence of the radius is

$R(t) = {R_o}{\left( {1 - \frac{t}{\tau }} \right)^{1/2}}\qquad \qquad(15)$

where

$\tau ~ \frac{{\rho _\ell }{h_{\ell v}} } {{k_v}\Delta T}{R_o^2}\qquad \qquad(16)$

The above analysis was performed using a simple, one-dimensional analysis of evaporation of a Leidenfrost droplet. A numerical solution of the two-dimensional analysis for two boundary conditions was performed by Nguyen and Avedisian (1987). The first case is a horizontal surface maintained at a constant temperature, and in the second case, the surface is insulated while the surrounding gas is heated.

In the two-dimensional analysis, vapor flows out of the droplet in the radial direction and the streamlines bend due to the presence of the wall. This results in the creation of a pressure field that lifts the droplet off the surface. The droplet continues to levitate until it has completely evaporated.

For both cases, assume the droplet shape to be spherical (which is valid for droplets in most industrial sprays and droplets with diameters of 100 $μ$ m or less). Also, take the evaporation process to be quasi-steady and properties to be constant. Radiative effects, spatial nonuniformities of temperature in the liquid, internal liquid motion, and buoyancy induced flow are all neglected. The continuity, momentum, energy, and species equations for the steady case in the vapor phase are

$\nabla \cdot {\mathbf{V}} = 0\qquad \qquad(17)$

${\mathbf{V}} \cdot \nabla {\mathbf{V}} = - \frac{1}{{{\rho _v}}}\nabla p + \frac{{{\mu _v}}}{{{\rho _v}}}{\nabla ^2}{\mathbf{V}}\qquad \qquad(18)$

${\mathbf{V}} \cdot \nabla T = {\alpha _v}{\nabla ^2}T\qquad \qquad(19)$

${\mathbf{V}} \cdot \nabla {\omega _i} = D{\nabla ^2}{\omega _i}\qquad \qquad(20)$

where i = 1 is the droplet and i = 2 is the inert ambient, such that $ω 1 + ω 2$ = 1. Performing a force balance and neglecting the effects of droplet acceleration caused by variations in levitation height, the weight of the droplet is found to be equal to the net force acting on the droplet due to viscous stress and pressure, as follows

$\int {{\mathbf{j}} \cdot {\rm{\tau }}'dA = {\mathbf{g}}({\rho _\ell } - {\rho _v}) \cdot {\mathbf{V}}} \qquad \qquad(21)$

where $τ'$ is the total stress tensor and 'j' is the unit vector in the vertical direction. The following boundary conditions are used:

Solid surface

$\left\{ \begin{array}{l} {\mathbf{V}} = 0{\rm{ }} \\ \frac{{\partial {V_n}}}{{\partial n}} = \frac{{\partial {\omega _1}}}{{\partial n}} = 0 \\ T = {T_w}{\rm{ (case 1) }} \\ \frac{{\partial T}}{{\partial n}} = 0{\rm{ (case 2)}} \\ \end{array} \right.\qquad \qquad(22)$

Droplet surface

$\left\{ \begin{array}{l} T = {T_{sat}} \\ \omega = \omega _{1s}^{} \\ \end{array} \right.\qquad \qquad(23)$

Ambient

$\left\{ \begin{array}{l} {\mathbf{V}} \to 0 \\ {\omega _1} \to 0 \\ T \to {T_\infty } \\ \end{array} \right.\qquad \qquad(24)$

Equations (22) – (24) were classified in the bispherical coordinate system, which is made up of a family of spheres ( $β$ = constant) that are each orthogonal to a family of spindle-shaped surfaces ( $α$ = constant). The droplet evaporation time is obtained by the following mass balance:

$\frac{4}{3}\pi \frac{d}{{dt}}\left( {{\rho _\ell }{R^3}} \right) = {\rho _v}\int {{u_\beta }dA} \qquad \qquad(25)$

where a droplet is the sphere $β = β 0$ and the solid surface corresponds to $β$ = 0.

References

Biance, A.L., Clanet, C., and Quere, D., 2003, “Leidenfrost Drops,” Physics of Fluids, Vol. 15, pp. 1632-1637.

Faghri, A., and Zhang, Y., 2006, Transport Phenomena in Multiphase Systems, Elsevier, Burlington, MA

Faghri, A., Zhang, Y., and Howell, J. R., 2010, Advanced Heat and Mass Transfer, Global Digital Press, Columbia, MO.

Mahadevan, L., and Pomeau, Y., 1999, “Rolling Droplets,” Physics of Fluids, Vol. 11, pp. 2449-2453.

Nguyen, T.K., and Avedisian, C.T., 1987, “Numerical Solution for Film Evaporation of a Spherical Liquid Droplet on an Isothermal and Adiabatic Surface,” International Journal of Heat and Mass Transfer, Vol. 30, pp. 1497-1509.

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Leidenfrost phenomena

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