# What determines particle size in an atomizing spray nozzle?

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• Twigg
In summary: I don't think that this gives me a good estimate of the size of the droplets produced in practice. Is that right?In summary, Chestermiller explains how the shear stress caused by the no-slip boundary condition at the wall of an oil burner atomizes the oil into a mist of small droplets. The magnitude of the shear stress is what determines the final particle size of the mist.
Twigg
Gold Member
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!

Dale
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

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?

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?

## 1. What is the relationship between nozzle design and particle size?

The design of a spray nozzle plays a crucial role in determining the particle size of the spray. The size and shape of the orifice, as well as the angle and velocity of the fluid, all affect the size of the particles produced. Nozzles with smaller orifices and higher fluid velocities tend to produce smaller particles.

## 2. How does the fluid pressure affect particle size in a spray nozzle?

The pressure of the fluid passing through the nozzle also has a significant impact on the particle size. Higher fluid pressure results in smaller particles, as the increased force helps to break up the fluid into smaller droplets. However, excessively high pressures can lead to atomization and produce an undesirable mist instead of a spray.

## 3. What role does the viscosity of the fluid play in particle size?

The viscosity, or thickness, of the fluid being sprayed can affect particle size in several ways. Thicker fluids require more force to break them into smaller droplets, resulting in larger particle sizes. Additionally, the surface tension of the fluid can also play a role in determining the size of the particles produced.

## 4. How does the size of the atomizing air or gas impact particle size?

The size and pressure of the atomizing air or gas used in the spray nozzle can also affect particle size. A larger atomizing air or gas flow can help to break up the fluid into smaller droplets, resulting in smaller particle sizes. However, too much air or gas can cause overspray and result in larger particle sizes.

## 5. What other factors can influence particle size in an atomizing spray nozzle?

Other factors that can impact particle size include the distance between the nozzle and the target surface, the temperature and humidity of the surrounding environment, and the type of fluid being sprayed. Additionally, the condition and maintenance of the nozzle can also affect particle size, as worn or clogged nozzles may produce inconsistent particle sizes.

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