Liquid Viscosity Models

--CENG 402 Project--

by Eric Lee and Rebecca Sisson

Table of Contents:
    1. Introduction
    2. Theoretical Basis
    3. Empirical Calculations
    4. Summary of Results
    5. Conclusion
    6. References



Introduction

Due to the lack of a completely accurate model of liquids, the calculation of liquid viscosity is difficult. Although some models can accurately represent certain types of liquids (e.g. liquid metals, polar compounds), there is no model that encompasses all types of liquids. This project will examine one commonly used model, developed by Eyring, as well as develop a program that uses empirical data to predict liquid viscosities over a wide range of temperatures. Eyring's model is one of the earliest attempts at predicting liquid viscosities and has the least sound theoretical basis. However, later theories based on the principals of statistical mechanics have done little to improve on the accuracy of Eyring's liquid viscosity predictions. Since all of the current models are in error by as much as thirty to fifty percent, the viscosities predicted can only be used in very rough back of the envelope calculations. These calculations, however, are still useful because they require no experimentation beyond the determination of the boiling point and the liquid molar volume (both of which are very easy to measure). In industrial applications, empirical formulas are generally used for more accurate predictive results. Hence, we have developed a matlab program that uses a four parameter empirical model to predict liquid viscosity for several common liquids included in the start301 database.





Theoretical Basis

The first step in developing a theoretical model of viscosity is to define the behavior of a single molecule in a stationary fluid. The figures above depict the restricted fashion in which a molecule of liquid can move. Although each molecule vibrates within the space available to it, translational motion occurs less often. To squeeze past its neighbors, the molecule must obtain a certain activation energy, denoted by deltaGo/N. Bird, Stuart, and Lightfoot give the frequency of this event as

[Maple Math]

In a flowing liquid, shear stress aids molecular movement concurrent with the overall velocity, and hinders it in the opposite direction. A quantity of energy, +/- (a/delta)*(Tau*V/2) is imparted to the molecule in the form of work. Here 'a' is the length of the jump, delta is the spacing between fluid layers, V is the molar volume of the liquid, and Tau is shear stress. This work term adjusts the free energy of activation so that the frequency of forward jumps is now

[Maple Math]

It is evident that the magnitude of the effect of sheer stress on jump frequency is inversely proportional to the distance between the molecules and the temperature. Once the motion of a single molecule has been modeled, it is possible to describe that of the whole fluid. The velocity of a given cross section of liquid is equivalent to the net frequency of forward and reverse jumps multiplied by the distance traveled per jump. If the velocity gradient over delta is taken to be linear, we find that

[Maple Math]

When substituting the expressions for nu(+) and nu(-) into the above expression, after expanding using a Taylor series expansion is

[Maple Math]

To see further details, follow the link to the Maple worksheet or refer to Section 1.5 in BSL(2001).

Sample Calculation from Theory





Empirical Calculations

Since the simpler theoretical models for liquid viscosity do not generate reasonably accurate results, empirical formulas are often used in their place. The matlab code below calculates viscosities using four parameter model of the form:

ln(viscosity)=A+B/T+C*T+D*T^2.

Data for eighteen liquids have been entered into the start301 database, and the Matlab Program can be used to predict liquid viscosities. Refer to the sample calculation below.

Sample Calculation 

Here are your compounds' formulae and names:
No. Formula  Name
----------------------------------------
  1 C6H6     benzene
 
Here are your reactions:
----------------------------------------
No reactions given
 
Enter the number of streams: 1
 
The variables for your compounds have now been created,
>> liqmucalc(20)

mu =

    0.6516




Summary of Results

Substance

Temperature (degC)

Experimental Viscosity (cp)

Emperical Viscosity (cp)

Eyring Viscosity

Emperical % Error

Eyring % Error

Water

0

1.787

1.726

3.978

3.4

122.6

20

1.002

1.018

2.792

1.6

178.6

Diethyl Ether

0

0.283

0.275

0.276

2.8

2.5

25

0.224

0.214

0.193

4.5

14.0

Bromine

25

0.744

0.934

0.535

25.5

28.0

Benzene

20

0.649

0.652

0.436

0.5

32.8

Ethanol

0

1.786

1.720

0.908

3.7

49.2

25

1.074

1.041

0.603

3.1

43.9

50

0.694

0.670

0.426

3.5

38.6

Sulfuric Acid

25

25.5

20.1

18.5

21.2

27.4




Conclusion

As can be seen in the above table, the theoretical predictions differ from the actual viscosities by 20-50% in most cases, with the predictions for water off by over 100%. The empirical predictions are much better, with deviations from the actual values usually less than 10%. Thus, although the theory does provide a general understanding of the phenomena that are occurring in a liquid system, the inaccuracy of the predictions show that much work is still necessary to create a comprehensive theory of liquid behavior.

References

  • Bird, Stuart and Lightfoot, Transport Phenomena. Wiley and Sons, New York, 2002
  • Reid, Prausnitz and Poling, The Properties of Gases and Liquids. McGraw Hill, Boston, 1987