LIQUID MIXTURE VISCOSITIES Introduction and Theory: Liquid viscosities are needed by process engineers for quality control, while design engineers need the property for fixing the optimum conditions for the chemical processes and operations as well as for the determination of the important dimensionless groups like the Reynolds number and Prandtl number. Liquid viscosity is also important in the calculation of the power requirements for the unit operations such as mixing, pipeline design, it cetera. The flow characteristics of liquids are mainly dependent on the viscosity and are broadly divided into Newtonian, when the viscosity of a liquid remains constant and is independent of the applied shear stress. And Non-Newtonian, when liquids, viscosity depend on the applied shear force and time. Viscosity is a fundamental characteristic property of all liquids. When a liquid flows, it has an internal resistance to flow. Viscosity is a measure of this resistance to flow or shear. Viscosity can also be termed as a drag force and is a measure of the frictional properties of the fluid. Viscosity is a function of temperature and pressure. In this experiment students will deal with the viscosity of pure liquids such as glycerin and water and its change as function of temperature. As well as student will exam the viscosity of mixture solution (glycerin + water) and its change as a function of mole fraction and temperature changes. Viscosity is expressed in two distinct forms absolute or dynamic viscosity and kinematic viscosity. Dynamic viscosity is the tangential force per unit area required to slide one layer against another as when these two layers are maintained at a unit distance. In another words, the viscosity of a fluid is defined as the measure of how resistive the fluid is to flow, and it can be described as: Shear stress = µ* (Strain or shear rate) 𝝈=𝝁∗é where µ is the dynamic viscosity σ is shear stress and é is strain rate From the definition of the é is strain rate

(1)

é=

𝟏 𝒅𝒙 𝒙 𝒅𝒕

=

𝒗

(2)

𝒙

where x is the distance, t is the time, and dx/dt is the velocity v. Therefore, the dynamic viscosity can be written as 𝝁=𝝈

𝒙

(3)

𝒗

While kinematic viscosity requires knowledge of density of the liquid (ρ ) at that temperature and pressure and is defined as ! 𝜗=! (4) Consider a Newtonian fluid laminar flow through a glass cylindrical capillary tube (viscometer) of diameter r and length l, as shown in Fig. 1 with no slip at the wall of a pressure difference ΔP between the two ends and the fluid is subjected to a force F. Then the viscosity may be calculated by applying Newton’s Law of viscosity.

r F R P1 l

P2

𝜎=𝜇

!" !"

(5)

make a force balance on the cylindrical element. 𝜎2𝜋𝑟𝑙 = ∆𝑝𝜋𝑟 !

(6)

Substitute the above Eq 6 into share stress Eq 5 yields !" !"

∆!

= !!" 𝑟

(7)

Integration of Eq. (7), using the boundary condition that at the wall of the capillary v(R) = 0, gives 𝑣=

∆!(! ! !! ! ) !!"

(8)

Equation 8 shows that the velocity distribution through the capillary tube is parabolic, hence the total flow rate can be estimated by ! 2𝜋𝑣𝑟𝑑𝑟 !

𝑄=

(9)

By combining Eqs 8&9 𝑄= 𝑄=

! !!∆! 𝑟 ! !!" !∆!! ! !!"

𝑅! − 𝑟 ! 𝑑𝑟

(10) (11)

The above equation is known as Poiseuille’s equation and it can be used for calculation of viscosity when using a capillary viscometer. For vertical tube arrangement which is the case for the capillary viscometer we use in this experiment, the hydrostatic pressure, ρgh , depends on the height, h, of the liquid. Therefore, the pressure difference, Δp , in terms of hydrostatic pressure is given by ∆𝑝 = 𝜌𝑔ℎ

(12)

If we know the volume (V) of the liquid dispensed during the experiment and the time (t) required for this volume of liquid to flow between two graduation marks in a viscometer,

𝑄=

! !

(13)

by substituting eqs 12 & 13 into eq 11 𝜇=

!"!! ! !!"

𝜌𝑡

(14)

By assuming 𝐶=

!"!! !

Eq14 will become

(15)

!!"

𝜇 = 𝐶𝜌𝑡 And the Kinematic viscosity is

(16)

𝜐 = 𝐶𝑡

(17)

where C is the viscometer constant (mm2/sec2) Students may calculate the viscometer constant C at a different temperatures other than those at the viscometers were calibrated by using the following equation 𝐶 = 𝐶! (1 + 𝐵(𝑇 − 𝑇! )

(18)

where C0 the viscometer constant when filled and tested at T0 (room temperature) B the temperature depending factor C0, T0, and B can be fond in the Viscometer certification paper. There has been no comprehensive theory on the viscosity of liquids so far because of its complex nature. However there are small number of semi-empirical and empirical methods provide reasonable results such as those proposed by Kirkwood et al.44 Where at temperatures below the normal boiling point, the logarithm of liquid viscosity varies linearly with the reciprocal of the absolute temperature as described below !

ln 𝜇 = 𝐴 + ! (19) where, A & B are constants and can be determined empirically. However, when the temperatures above the normal boiling point, the ln µ versus (1/T) relationship becomes nonlinear and in this case a number of semi empirical equation are needed. While for mixture viscosity the following equation proposed by Panchenkov which is basis on the mechanism of the flow of viscous liquid. !

𝜇! =

( ) ! 𝐴! 𝜌!! 𝑇 (!)

exp

! !"

−1

𝜌! = 𝜌𝑥 + 𝜌! (1 − 𝑥) where µm is the mixture viscosity,

(20)

As ρm T εm R x ρ Mg

is an empirical coefficient, is the density of the mixture, is the absolute temperature, is the energy of bonds between the molecules in the mixture, and is the universal gas constant. is the mole fraction is the density molecular wt of glycerin = 92.09 g

Please note that The quantities As and εm of the eq (20) must be evaluated from the mixture viscosity data, and therefore at least viscosity of the mixture at two conditions must be known for the application of this method with good accuracy.

Apparatus and Method 1. Reverse Flow Viscometer (Cannon-Fenske Opaque Viscometers) 2. Water Bath controlling temperature 3. Glycerin liquid 4. Pycno meter to measure the density A schematic diagram of the Cannon-Fenske Opaque Viscometer apparatus is shown in Fig. 2. These reverse flow type viscometers wet the timing section of the viscometer capillary only during the actual measurement. The liquid sample flows into a timing bulb not previously wetted by sample, thus allowing the timing of liquids whose thin films are opaque. Reverse flow viscometers must be cleaned, dried, and refilled prior to each measurement. Clean the viscometer using suitable solvents, and dry by passing clean dry filtered air through the instrument to remove any traces of solvent. To charge the sample into the viscometer, invert the instrument and apply suction to arm L, immersing tube N in the liquid sample, and draw liquid to bulb D, up to mark G. Turn the viscometer to the normal position, wipe and clean it. When the meniscus travels through D and fills the bulb an up to its half, use a small rubber stopper to close the limb N to stop the flow of the liquid. Place the viscometer is in a thermostat by means of a proper holder and allow 10 -15 minutes to attain the required temperature. Subsequently remove the rubber stopper and record the efflux time for the liquid to pass through the bulbs C and J. By measuring the time required by the liquid to pass between the markings E, F and I, the viscosities may be calculated from the efflux times for the two bulbs. Students may compare between the values. It should be noted that any the reverse flow viscometers must be cleaned, dried, and refilled before a repeat measurement can be made. Repeat the measurement different glycerin/ water concentrations, and different temperatures like T = 10, 20, 30, 40, 50 oC and for several concentrations (in wt %)

N L

D

G I R

J F

C E

A

Figure 2. Cannon-Fenske opaque reverse flow viscometer Table 1. Various size Cannon-Fenske opaque reverse flow viscometer Size No.

Kinematic Viscosity Range, mm2/s (cSt)

Capillary Diameter, Viscometer mm ( ± 2%) Constant (C) mm2/s2

25 50 75 100 150 200 300 350 400 450 500 600

0.4 - 2 0.8 - 4 1.6 - 8 3-15 7-35 20-100 50-200 100-500 240-1,200 500-2,500 1,600-8,000 4,000-20,000

0.31 0.42 0.54 0.63 0.78 1.02 1.26 1.48 1.88 2.20 3.10 4.00

0.002 0.004 0.008 0.015 0.035 0.1 0.25 0.5 1.2 2.5 8 20

References: 1. J. Welty, C. Wicks, and R. Wilson, “Fundamentals of Momentum, Heat, and Mass Transfer” John Wiley and Sons, Inc., New York Third edition (1983). 2. A. Dinsdale and F. Moore, Viscosity and its measurement, Chapman and Hall, London (1962). 3. M. R. Cannon and M. R. Fenske, Viscosity measurement, Ind. Eng. Chem. Analytical Edition. 10(6), 297-301 (1938). 4. M. R. Cannon and M. R Fenske, Viscosity measurement, Ind. Eng. Chem. Analytical Edition. 13(5), 299-300 (1938). 5. M. R Cannon, R. E Manning, and J. D. Bell, Viscosity measurement, the kinetic-energy correction and a new viscometer, Anal. Chem. 32, 355-358 (1960). 6. Y. A. Pinkevich, New viscometer for the determination of the viscosity of petroleum products at low temperatures, Petroleum (London), 8, 214-215 (1945). 7. ASTM (American Society for Testing and Materials) D 88, Standard Test Method for Saybolt Viscosity. 8. C. R. Duhne, Viscosity-temperature correlations for liquids, Chem.Eng. 86 (15), 83 (1979). 9. G.M. Panchenkov, Calculation of absolute values of the viscosity of liquids, Zhurn. Fiz. Khim. 24, 1390-1406 (1950). 10. M. J. Lee and M. C. Wei, Corresponding-states model for viscosity of liquids and liquid mixtures, J. Chem. Eng. Japan, 26(2), 159-165 (1993).

Saddawi January 2013