What determines particle size in an atomizing spray nozzle?

[Twigg]

TL;DR Summary

Consider a simple atomizing spray nozzle: just a circular hole with liquid being atomized by a choked air flow. Can you use first principles to make a rough estimate of the particle size of liquid swept up in the gas jet? Where does the shear force that atomizes the drop come from?

Hi all

Another random, kinda open-ended question here. Sorry for that. I found myself reading about atomizing nozzles in oil burners, and got curious about the physics of atomized sprays. I didn't have much luck researching this on my own, so I'm turning to you all. It was the kind of situation where I either found resources that were super hand-wavy or super specialized.

Specifically, what forces are involved in breaking a macroscopic body of liquid into a mist of fine droplets? How does the gas flow generate these forces? What are the parameters of the gas flow that influence the particle size of the mist? Can you estimate the particle size, even crudely?

Just to show that I've made some effort, here's what I think I understand so far. I think that the force of surface tension resists atomization, since the spray of tiny particles has more surface area than a bulk liquid. My gut feeling says that it takes a shear force to break apart big drops into little drops, as opposed to a uniform force (like drag on the droplet due to the gas flow) which would only accelerate the droplet's center of mass. So I figure that the flow of air (I figure the air flow would be choked at the orifice) somehow causes a shear force on the liquid that splits it up into a mist of tiny drops. I feel like the magnitude of that shear force is what determines the final particle size, since the force of surface tension depends on the diameter of a drop in the mist, and since those forces should balance for the final particle size. But I have no idea what this shear force would be. How does a one-dimensional flow of gas induce a large shear force? It can't be the pressure of the gas on the drop, because that would be uniform over the cross-sectional area of a droplet, so it wouldn't shear the droplet. Are there just really high pressure gradients over small length scales? I'm pretty confused on this point.

Any kind of help is appreciated here, be it direct explanation or just pointing me to resources. Thanks all!

 

[Chestermiller]

The viscous shear stress in the fluid resulting from the no-slip boundary condition at the wall causes the drop to deform against the restoring force of surface tension. When it extends and distorts beyond a certain point, an instability develops which allows the surface tension to pinch off the drop into smaller droplets.

https://pubs.acs.org/doi/abs/10.1021/i160028a009

 

[Twigg]

That makes a lot of sense Chestermiller, much appreciated!

So, I tried to take this new information and see if I could estimate the particle size of a soybean oil spray from a choked air flow. I'm approximating 300m/s as the flow speed, so the effect of the oil on the momentum balance (acting as a negative thrust) is negligible. My answer for the droplet size is waaay small. Any thoughts on where I goofed?

So I start with an estimate for the boundary layer thickness based on dimensional analysis, taken from an argument on the wikipedia page: $$\rho u^2 /L≈\mu u/\delta^2$$ where $\delta$ is the boundary layer thickness, $L$ is the characteristic length (here, I interpret this as the diameter of a droplet, or at least something of that order), $\rho$ is the oil's density, and $\mu$ is the oil's viscosity. This yields $$\delta \approx \sqrt{\frac{\mu L}{\rho u}}$$ Then I approximate the shear stress as $$\sigma \approx \mu \frac{u}{\delta} = \frac{\mu u}{L} \sqrt{Re}$$

So then I get that the shear force is the cross sectional area times the shear stress, which gives $$F = A\sigma = \pi \left( \frac{L}{2} \right)^2 \sigma$$ Then I use the standard result for the force of surface tension on a spherical surface $$F = \gamma \times 2\pi \left(\frac{L}{2}\right)$$ and the force balance leaves $$\sqrt{Re} = \frac{2\gamma}{\mu u}$$ and finally $$L = \frac{16\gamma^2}{\mu \rho u^3}$$

Dimensionally, this does give me a length scale, but it's way too small. For $u=300m/s$, $\rho = 900kg/m^3$, $\mu = 60mPa*s$, $\gamma = 30mN/m$ (values for soybean oil taken from here), I get $L \approx 10^{-11}m$, and there's no way the droplets are smaller than a hydrogen atom.

I feel like my mistake was that I misinterpreted the characteristic length in the scale argument for the boundary layer thickness?? Is that plausible? Can anyone point out the correct interpretation if so?

 

[Twigg]

I'm a dummy, and totally did not see the link in Chestermiller's reply. Sorry! It camouflaged on me! Based on equation (3) of that paper, I see that the minimum droplet size you'd expect for soybean oil in air ($\frac{\mu'}{\mu}\rightarrow \infty$) is $\frac{16\gamma}{19G\mu}$ where $\gamma$ is the surface tension of the oil, $G$ is the shear rate of the surrounding air flow, and $\mu$ is the viscosity of the air.

I'm very pleased with this understanding, but (correct me if I'm wrong) estimating the shear rate $G$ in a choked orifice flow sounds like a headache on top of a root canal. Is there a way to even get in the ballpark estimating $G$? Something at the level of depth of the scale argument in my previous post?