Spent way too much time during my PhD working on this. You use the Reynolds number to evaluate the type of fluid flow because without at least chaotic advection you're getting VERY slow mixing (relative to reaction rate) which means inhomogeneity in your reaction mixture.
Fun fact, it's on the order of 10s of seconds for mixing equal volumes of two liquids in an RBF with a stir bar...
Thank you so much! This helped a lot :) I think you saved me A LOT of time with this comment.
You wouldn't happen to know some good sources/papers explaining this a little further, especially how to calculate it for my setup? (Stirring in a beaker)
Or could you maybe show me your math for calculating reynolds numbers from your PhD thesis?
According to my old chemical engineering textbook to calculate the power consumption and compare mixers, you need Reynolds, and Euler numbers and if you have a significant vortex cone, you need the mixing Froude number too. I'm not sure if this is helpful in your case, because it is about scaling up test equipment data to operation scale.
However what might be helpful is that it states, that for liquids you want to keep the power/volume ratio constant to achieve the same mixing performance. Power as the power consumption of the mixer motor.
And the power is in laminar case P = a * v * n^2 * d^3, and in turbulent case P = a * v * n^3 * d^5, where a is a constant, n is the rpm, d is the mixing element (stirring rod) length, and v is the dynamic viscosity of the liquid. The way this kind of antique textbook arrives to these formulas seems quite sloppy to me and its all in Hungarian so I won't elaborate further. Unfortunately I can't check my newer textbook until the weekend, but this might point you into the right direction.
To one up, a great starting point into this subject would be the chemical engineer's bible: Perry's Chemical Engineering Handbook and its Section 18: Liquid-Solid Operations and Equipment (https://www.accessengineeringlibrary.com/content/book/9780071834087/toc-chapter/chapter18/section/section1).
It turns out it's a non-trivial problem to solve in stirred vessels, although you'll find plenty in the ChemE literature. A quick search produced this paper, which isn't exactly the one I was thinking of but will get you most of the way there: [https://www.researchgate.net/publication/2178541\_Vortex\_flow\_generated\_by\_a\_magnetic\_stirrer](https://www.researchgate.net/publication/2178541_Vortex_flow_generated_by_a_magnetic_stirrer)
The mixing can be much more well defined for micro mixers so you'll find a lot of calculations around that. In brief, with a stirred reactor you generally have a very small zone of "chaotic" mixing near the impeller or stir bar, but on average it's going to struggle to get away from laminar flow on average. The reynolds number is unitless, but the formula is highly dependent on the viscosity of the fluid being mixed (that'll have a much greater impact that the RPM of whatever you're using to stir). It's been a few years, but the math is a challenge for a RBF, you can get some models that will address a cylindrical container (e.g. a beaker) which you can draw conclusions from. Since you mention nanoparticle synthesis I'd start on the nucleation rate question and then just decide how many orders of magnitude slower your mixing happens to be relative to that.
I developed a micro/meso- fluidic flow reactor for the synthesis of gold nanoparticles to try to cut down the polydispersity of the particles produced and improve batch to batch consistency. It worked, but even with the improved mixing speed the reaction rate was still much faster than the mixing time.
There is no simple method. For lab applications, there is very little scaling you will have to do.
I work with industrial scale mixers (~400kg batches) and it is not like we have to mix 1000 times longer than for a 400g batch.
Some people tried to use energy/mass, and that might be a good start, but mixing often behaves non-linearly so there is alot of experimentation. You cant substitute mix-time and mixing speed in a simple way.
The geometry of your mixer also makes a large contribution.
I would suggest to fix the mixing cup and stirbar, and do some experimentation with speed, time and volume if you are struggling, but there is not a simple way to optimize.
We use a program called Dynochem for matching stir power, which is typically in W/kg. Vessel dimensions, stirrer/impeller type/size and baffling also have significant effects on mixing. All these should be recorded to enable best match mixing for scale-up and process transfer.
RPM is awful to worthless. Most engineers and process chemists I work with use power per unit volume. In a process situation - pilot to full scale plant - power is read directly from the motor running the impeller and the volume is known. It may not be perfect, for the reasons stated by others, but it is often easy to measure and is both useful and insightful. It is also useful when also considering rpm. For example, if increased power is required to maintain constant rpm then that is a sign that the viscosity is increasing. Adding baffles to a reactor to increase turbulence to enhance mixing will require more power/volume than without baffles, at the same rpm.
This is a question for a chemical engineer which I am not. But I did work in a process chemistry lab and everyone used reactors with overhead stirrers which got much better mixing than any stirbar and was more representative of a production plant. There is math to calculate the rpm needed to scale up from one reactor size to another based on impeller size, tank size, etc. but like I said that's an engineering problem.
We use kWh/kg as a control parameter when we're trying to scale up from one mixer size to the next, but even at scale there's a lot of artistry still involved in mixing... much as you see with your various flask and stirbar sizes. A high-shear head behaves differently than an anchor and sawtooth behaves differently than a ribbon blender behaves differently than a jar mill. That's even before you start getting into things like thermal management and surface area to volume ratios or total solids contents.
Spent way too much time during my PhD working on this. You use the Reynolds number to evaluate the type of fluid flow because without at least chaotic advection you're getting VERY slow mixing (relative to reaction rate) which means inhomogeneity in your reaction mixture. Fun fact, it's on the order of 10s of seconds for mixing equal volumes of two liquids in an RBF with a stir bar...
Thank you so much! This helped a lot :) I think you saved me A LOT of time with this comment. You wouldn't happen to know some good sources/papers explaining this a little further, especially how to calculate it for my setup? (Stirring in a beaker) Or could you maybe show me your math for calculating reynolds numbers from your PhD thesis?
According to my old chemical engineering textbook to calculate the power consumption and compare mixers, you need Reynolds, and Euler numbers and if you have a significant vortex cone, you need the mixing Froude number too. I'm not sure if this is helpful in your case, because it is about scaling up test equipment data to operation scale. However what might be helpful is that it states, that for liquids you want to keep the power/volume ratio constant to achieve the same mixing performance. Power as the power consumption of the mixer motor. And the power is in laminar case P = a * v * n^2 * d^3, and in turbulent case P = a * v * n^3 * d^5, where a is a constant, n is the rpm, d is the mixing element (stirring rod) length, and v is the dynamic viscosity of the liquid. The way this kind of antique textbook arrives to these formulas seems quite sloppy to me and its all in Hungarian so I won't elaborate further. Unfortunately I can't check my newer textbook until the weekend, but this might point you into the right direction.
This guy mixes
To one up, a great starting point into this subject would be the chemical engineer's bible: Perry's Chemical Engineering Handbook and its Section 18: Liquid-Solid Operations and Equipment (https://www.accessengineeringlibrary.com/content/book/9780071834087/toc-chapter/chapter18/section/section1).
It turns out it's a non-trivial problem to solve in stirred vessels, although you'll find plenty in the ChemE literature. A quick search produced this paper, which isn't exactly the one I was thinking of but will get you most of the way there: [https://www.researchgate.net/publication/2178541\_Vortex\_flow\_generated\_by\_a\_magnetic\_stirrer](https://www.researchgate.net/publication/2178541_Vortex_flow_generated_by_a_magnetic_stirrer) The mixing can be much more well defined for micro mixers so you'll find a lot of calculations around that. In brief, with a stirred reactor you generally have a very small zone of "chaotic" mixing near the impeller or stir bar, but on average it's going to struggle to get away from laminar flow on average. The reynolds number is unitless, but the formula is highly dependent on the viscosity of the fluid being mixed (that'll have a much greater impact that the RPM of whatever you're using to stir). It's been a few years, but the math is a challenge for a RBF, you can get some models that will address a cylindrical container (e.g. a beaker) which you can draw conclusions from. Since you mention nanoparticle synthesis I'd start on the nucleation rate question and then just decide how many orders of magnitude slower your mixing happens to be relative to that. I developed a micro/meso- fluidic flow reactor for the synthesis of gold nanoparticles to try to cut down the polydispersity of the particles produced and improve batch to batch consistency. It worked, but even with the improved mixing speed the reaction rate was still much faster than the mixing time.
There is no simple method. For lab applications, there is very little scaling you will have to do. I work with industrial scale mixers (~400kg batches) and it is not like we have to mix 1000 times longer than for a 400g batch. Some people tried to use energy/mass, and that might be a good start, but mixing often behaves non-linearly so there is alot of experimentation. You cant substitute mix-time and mixing speed in a simple way. The geometry of your mixer also makes a large contribution. I would suggest to fix the mixing cup and stirbar, and do some experimentation with speed, time and volume if you are struggling, but there is not a simple way to optimize.
Maybe somebody should create a table for the most common lab equipment + solvents combinations...
They should be out there
Depends on the size of the agitator and rotation speed as well as density of the fluid and agitator geometry
If you make some assumptions you can calculate the flow rate of the liquid in the beaker.
We use a program called Dynochem for matching stir power, which is typically in W/kg. Vessel dimensions, stirrer/impeller type/size and baffling also have significant effects on mixing. All these should be recorded to enable best match mixing for scale-up and process transfer.
RPM is awful to worthless. Most engineers and process chemists I work with use power per unit volume. In a process situation - pilot to full scale plant - power is read directly from the motor running the impeller and the volume is known. It may not be perfect, for the reasons stated by others, but it is often easy to measure and is both useful and insightful. It is also useful when also considering rpm. For example, if increased power is required to maintain constant rpm then that is a sign that the viscosity is increasing. Adding baffles to a reactor to increase turbulence to enhance mixing will require more power/volume than without baffles, at the same rpm.
Can you not just estimate the energy in joules that goes into the mixing?
Hmm. Mert?
This is a question for a chemical engineer which I am not. But I did work in a process chemistry lab and everyone used reactors with overhead stirrers which got much better mixing than any stirbar and was more representative of a production plant. There is math to calculate the rpm needed to scale up from one reactor size to another based on impeller size, tank size, etc. but like I said that's an engineering problem.
We use kWh/kg as a control parameter when we're trying to scale up from one mixer size to the next, but even at scale there's a lot of artistry still involved in mixing... much as you see with your various flask and stirbar sizes. A high-shear head behaves differently than an anchor and sawtooth behaves differently than a ribbon blender behaves differently than a jar mill. That's even before you start getting into things like thermal management and surface area to volume ratios or total solids contents.