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Solving for your own Sutherland Coefficients using Python

Sutherland’s equation is a useful model for the temperature dependence of the viscosity of gases. I give a few details about it in this post: https://curiosityfluids.com/2019/02/15/sutherlands-law/

The law given by:

\mu=\mu_o\frac{T_o + C}{T+C}\left(\frac{T}{T_o}\right)^{3/2}

It is also often simplified (as it is in OpenFOAM) to:

\mu=\frac{C_1 T^{3/2}}{T+C}=\frac{A_s T^{3/2}}{T+T_s}

In order to use these equations, obviously, you need to know the coefficients. Here, I’m going to show you how you can simply create your own Sutherland coefficients using least-squares fitting Python 3.

So why would you do this? Basically, there are two main reasons for this. First, if you are not using air, the Sutherland coefficients can be hard to find. If you happen to find them, they can be hard to reference, and you may not know how accurate they are. So creating your own Sutherland coefficients makes a ton of sense from an academic point of view. In your thesis or paper, you can say that you created them yourself, and not only that you can give an exact number for the error in the temperature range you are investigating.

So let’s say we are looking for a viscosity model of Nitrogen N2 – and we can’t find the coefficients anywhere – or for the second reason above, you’ve decided its best to create your own.

By far the simplest way to achieve this is using Python and the Scipy.optimize package.

Step 1: Get Data

The first step is to find some well known, and easily cited, source for viscosity data. I usually use the NIST webbook (
https://webbook.nist.gov/), but occasionally the temperatures there aren’t high enough. So you could also pull the data out of a publication somewhere. Here I’ll use the following data from NIST:

Temparature (K)Viscosity (Pa.s)
200
0.000012924
4000.000022217
6000.000029602
8000.000035932
10000.000041597
12000.000046812
14000.000051704
16000.000056357
18000.000060829
20000.000065162

This data is the dynamics viscosity of nitrogen N2 pulled from the NIST database for 0.101 MPa. (Note that in these ranges viscosity should be only temperature dependent).

Step 2: Use python to fit the data

If you are unfamiliar with Python, this may seem a little foreign to you, but python is extremely simple.

First, we need to load the necessary packages (here, we’ll load numpy, scipy.optimize, and matplotlib):

import numpy as np
import matplotlib.pyplot as plt
from scipy.optimize import curve_fit

Now we define the sutherland function:

def sutherland(T, As, Ts):
    return As*T**(3/2)/(Ts+T)

Next we input the data:

T=[200,
400,
600,
800,
1000,
1200,
1400,
1600,
1800,
2000]

mu=[0.000012924,
0.000022217,
0.000029602,
0.000035932,
0.000041597,
0.000046812,
0.000051704,
0.000056357,
0.000060829,
0.000065162]

Then we fit the data using the curve_fit function from scipy.optimize. This function uses a least squares minimization to solve for the unknown coefficients. The output variable popt is an array that contains our desired variables As and Ts.

popt = curve_fit(sutherland, T, mu)
As=popt[0]
Ts=popt[1]

Now we can just output our data to the screen and plot the results if we so wish:

print('As = '+str(popt[0])+'\n')
print('Ts = '+str(popt[1])+'\n')

xplot=np.linspace(200,2000,100)
yplot=sutherland(xplot,As,Ts)

plt.plot(T,mu,'ok',xplot,yplot,'-r')
plt.xlabel('Temperature (K)')
plt.ylabel('Dynamic Viscosity (Pa.s)')
plt.legend(['NIST Data', 'Sutherland'])
plt.show()

Overall the entire code looks like this:

import numpy as np
import matplotlib.pyplot as plt
from scipy.optimize import curve_fit

def sutherland(T, As, Ts):
    return As*T**(3/2)/(Ts+T)

T=[200, 400, 600,
800,
1000,
1200,
1400,
1600,
1800,
2000]

mu=[0.000012924,
0.000022217,
0.000029602,
0.000035932,
0.000041597,
0.000046812,
0.000051704,
0.000056357,
0.000060829,
0.000065162]

popt, pcov = curve_fit(sutherland, T, mu)
As=popt[0]
Ts=popt[1]
print('As = '+str(popt[0])+'\n')
print('Ts = '+str(popt[1])+'\n')

xplot=np.linspace(200,2000,100)
yplot=sutherland(xplot,As,Ts)

plt.plot(T,mu,'ok',xplot,yplot,'-r')
plt.xlabel('Temperature (K)')
plt.ylabel('Dynamic Viscosity (Pa.s)')
plt.legend(['NIST Data', 'Sutherland'])
plt.show()

And the results for nitrogen gas in this range are As=1.55902E-6, and Ts=168.766 K. Now we have our own coefficients that we can quantify the error on and use in our academic research! Wahoo!

Summary

In this post, we looked at how we can simply use a database of viscosity-temperature data and use the python package scipy to solve for our unknown Sutherland viscosity coefficients. This NIST database was used to grab some data, and the data was then loaded into Python and curve-fit using scipy.optimize curve_fit function.

This task could also easily be accomplished using the Matlab curve-fitting toolbox, or perhaps in excel. However, I have not had good success using the excel solver to solve for unknown coefficients.

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