The rate of heat and mass transfer at the surface of acoustically levitated pure liquid
droplets is predicted theoretically for the case where an acoustic boundary layer
appears near the droplet surface resulting in an acoustic streaming. The theory is based
on the computation of the acoustic field and squeezed droplet shape by means of the
boundary element method developed in Yarin, Pfaffenlehner & Tropea (1998). Given
the acoustic field around the levitated droplet, the acoustic streaming near the droplet
surface was calculated. This allowed calculation of the Sherwood and Nusselt number
distributions over the droplet surface, as well as their average values. Then, the mass
balance was used to calculate the evolution of the equivalent droplet radius in time.
The theory is applicable to droplets of arbitrary size relative to the sound wavelength
λ, including those of the order of λ, when the compressible character of the gas flow
is important. Also, the deformation of the droplets by the acoustic field is accounted
for, as well as a displacement of the droplet centre from the pressure node. The effect
of the internal circulation of liquid in the droplet sustained by the acoustic streaming
in the gas is estimated. The distribution of the time-average heat and mass transfer
rate over the droplet surface is found to have a maximum at the droplet equator and
minima at its poles. The time and surface average of the Sherwood number was shown
to be described by the expression Sh = KB/√ω[Dscr ]0,
where B = A0e/(ρ0c0)
is a scale of the velocity in the sound wave (A0e is the
amplitude of the incident sound wave, ρ0 is
the unperturbed air density, c0 is the sound velocity
in air, ω is the angular frequency in the ultrasonic range,
[Dscr ]0 is the mass diffusion coefficient of liquid vapour in air,
which should be replaced by the thermal diffusivity of air in the computation of the
Nusselt number). The coefficient K depends on the governing parameters (the acoustic
field, the liquid properties), as well as on the current equivalent droplet radius a.
For small spherical droplets with a[Lt ]λ,
K = (45/4π)1/2 = 1.89, if A0e
is found from the sound pressure level (SPL) defined using A0e.
On the other hand, if A0e is found from the same value of the SPL,
but defined using the root-mean-square pressure amplitude
= A0e/√2), then Sh
√ω[Dscr ]0, with Brms
= √2B and Krms = K/√2 = 1.336.
For large droplets squeezed significantly by the acoustic
field, K appears always to be greater than 1.89. The evolution of an evaporating
droplet in time is predicted and compared with the present experiments and existing
data from the literature. The agreement is found to be rather good.
We also study and discuss the effect of an additional blowing (a gas jet impinging
on a droplet) on the evaporation rate, as well as the enrichment of gas at the
outer boundary of the acoustic bondary layer by liquid vapour. We show that,
even at relatively high rates of blowing, the droplet evaporation is still governed by
the acoustic streaming in the relatively strong acoustic fields we use. This makes
it impossible to study forced convective heat and mass transfer under the present
conditions using droplets levitated in strong acoustic fields.