# How to determine a Reynolds number for mixture of 2 liquids

I'm experimenting with synthesis in microreactors and I'm using 2 reactants. I'd like to estimate the Reynolds number of the flow containing two miscible fluids. I have the values of their properties (density, viscosity, flowrate).

Do I just average both property values for both liquids or is there a more intricate relation?

As a very rough estimate, you could do that and just weight them by the fluids' respective concentrations.

Depending on how ideally the two fluids mix, you may want to look up or calculate the actual properties of the particular mixture you're working with. If you can't find the values in the literature but have access to physical property estimation software such as Aspen, you could calculate these values yourself.

• You are assuming that both density and viscosity are additive. Any evidence this is true? – bobthechemist Jul 20 '13 at 1:59
• No, in fact it will be entirely untrue unless the two substances mix ideally. But as mentioned, it's a good place to start as a ballpark estimate. – CTKlein Jul 21 '13 at 17:04
• Bob thank you very much for your explanation on how to tackle the problem. The thing is that the fluids are miscible, but they also react, so it would be almost imposible to conduct an experiment having an ideal mixture. For my estimation of Re the relation you suggested (Arrhenius equation) will be good enough. Thank you once again for all the help! – Ezze Jul 22 '13 at 16:57

The Reynolds Number is the ratio of inertial to viscous forces and is calculated according to this equation: $$Re = \frac{\rho v L}{\mu}$$ where $\rho$ is the density, $\mu$ is the dynamic viscosity, $v$ is the fluid velocity and $L$ is a characteristic length that depends on the flow being modeled. When estimating the Reynolds number of a mixture, the term $v L$ would not change but we need to make the assumption that the term $\frac{\rho}{\mu}$ of the mixture can be derived from the properties of the pure components. This assumption will be valid if

• The components have similar structure
• The components are non-polar
• The components are not associated
• The difference in the kinematic viscosities is small (~15 cP)

Some of these conditions could be relaxed if one species is in significant excess.

Densities are additive so long as the volume of the mixture equals the sum of the volumes of the pure solutions. (Search for volume weighted densities for more information.) Viscosities of mixtures is a much harder thing to estimate. One method is the Grunberg-Nissan equation: $$Log\eta_m=x_1Log\eta_1+x_2Log\eta_2+x_1x_2d$$ Where $\eta_i$ is the dynamic viscosity of component $i$, $x_i$ is the mole fraction of component $i$ and $d$ is an interaction coefficient. If you assume $d=0$ then you have the Arrhenius equation for the viscosity of a mixture.

So, at the end of the day, if you state you are working under the assumptions that volume weighted densities equal mass weighted densities and use the Arrhenius equation to determine the viscosity of the mixture then you can average the weighted properties of the pure components to get an estimate of the mixture.

Just a word of caution under these assumptions you should take the mass weighted average of the densities but the mole weighted average of the viscosities.

I relied on this reference for much of this answer. It will require a subscription to the service to access the PDF, but some of it can be found on google books.