Viscometer

These devices are also known as glass capillary viscometers or Ostwald viscometers, named after Wilhelm Ostwald. Another version is the Ubbelohde viscometer, which consists of a U-shaped glass tube held vertically in a controlled temperature bath. In one arm of the U is a vertical section of precise narrow bore (the capillary). Above there is a bulb, with it is another bulb lower down on the other arm. In use, liquid is drawn into the upper bulb by suction, then allowed to flow down through the capillary into the lower bulb. Two marks (one above and one below the upper bulb) indicate a known volume. The time taken for the level of the liquid to pass between these marks is proportional to the kinematic viscosity. The calibration can be done using a fluid of known properties. Most commercial units are provided with a conversion factor.

The time required for the test liquid to flow through a capillary of a known diameter of a certain factor between two marked points is measured. By multiplying the time taken by the factor of the viscometer, the kinematic viscosity is obtained.

Such viscometers can be classified as direct-flow or reverse-flow. Reverse-flow viscometers have the reservoir above the markings, and direct-flow are those with the reservoir below the markings. Such classifications exist so that the level can be determined even when opaque or staining liquids are measured, otherwise the liquid will cover the markings and make it impossible to gauge the time the level passes the mark. This also allows the viscometer to have more than 1 set of marks to allow for an immediate timing of the time it takes to reach the 3rd mark, therefore yielding 2 timings and allowing subsequent calculation of determinability to ensure accurate results. The use of two timings in one viscometer in a single run is only possible if the sample being measured has Newtonian properties. Otherwise the change in driving head, which in turn changes the shear rate, will produce a different viscosity for the two bulbs.

Creeping flow past a sphere

Stokes' law is the basis of the falling-sphere viscometer, in which the fluid is stationary in a vertical glass tube. A sphere of known size and density is allowed to descend through the liquid. If correctly selected, it reaches terminal velocity, which can be measured by the time it takes to pass two marks on the tube. Electronic sensing can be used for opaque fluids. Knowing the terminal velocity, the size and density of the sphere, and the density of the liquid, Stokes' law can be used to calculate the viscosity of the fluid. A series of steel ball bearings of different diameter are normally used in the classic experiment to improve the accuracy of the calculation. The school experiment uses glycerol as the fluid, and the technique is used industrially to check the viscosity of fluids used in processes. It includes many different oils and polymer liquids such as solutions.

In 1851, George Gabriel Stokes derived an expression for the frictional force (also called drag force) exerted on spherical objects with very small Reynolds numbers (e.g., very small particles) in a continuous viscous fluid by changing the small fluid-mass limit of the generally unsolvable Navier–Stokes equations:

${\displaystyle F=6\pi r\eta v,}$

where

${\displaystyle F}$ is the frictional force,

${\displaystyle r}$ is the radius of the spherical object,

${\displaystyle \eta }$ is the fluid viscosity,

${\displaystyle v}$ is the particle velocity.

If the particles are falling in the viscous fluid by their own weight, then a terminal velocity, also known as the settling velocity, is reached when this frictional force combined with the buoyant force exactly balance the gravitational force. The resulting settling velocity (or terminal velocity) is given by

${\displaystyle V_{\text{s}}={\frac {2}{9}}{\frac {r^{2}g(\rho _{p}-\rho _{f})}{\mu }},}$

where:

V_s is the particle settling velocity (m/s), vertically downwards if ρ_p > ρ_f, upwards if ρ_p < ρ_f,

r is the Stokes radius of the particle (m),

g is the gravitational acceleration (m/s²),

ρ_p is the density of the particles (kg/m³),

ρ_f is the density of the fluid (kg/m³),

μ is the (dynamic) fluid viscosity (Pa·s).

Note that Stokes flow is assumed, so the Reynolds number must be small.

A limiting factor on the validity of this result is the roughness of the sphere being used.

A modification of the straight falling-sphere viscometer is a rolling-ball viscometer, which times a ball rolling down a slope whilst immersed in the test fluid. This can be further improved by using a patented V plate, which increases the number of rotations to distance traveled, allowing smaller, more portable devices. The controlled rolling motion of the ball avoids turbulences in the fluid, which would otherwise occur with a falling ball.[2] This type of device is also suitable for ship board use.

The quartz viscometer is a special type of vibrational viscometer. Here, an oscillating quartz crystal is immersed into a fluid and the specific influence on the oscillating behavior defines the viscosity. The principle of quartz viscosimetry is based on the idea of W. P. Mason. The basic concept is the application of a piezoelectric crystal for the determination of viscosity. The high-frequency electric field that is applied to the oscillator causes a movement of the sensor and results in the shearing of the fluid. The movement of the sensor is then influenced by the external forces (the shear stress) of the fluid, which affects the electrical response of the sensor.[3] The calibration procedure as a pre-condition of viscosity determination by means of a quartz crystal goes back to B. Bode, who facilitated the detailed analysis of the electrical and mechanical transmission behavior of the oscillating system.[4] On the basis of this calibration, the quartz viscosimeter was developed which allows continuous viscosity determination in resting and flowing liquids.[5]

The quartz crystal microbalance functions as a vibrational viscometer by the piezoelectric properties inherent in quartz to perform measurements of conductance spectra of liquids and thin films exposed to the surface of the crystal.[6] From these spectra, frequency shifts and a broadening of the peaks for the resonant and overtone frequencies of the quartz crystal are tracked and used to determine changes in mass as well as the viscosity, shear modulus, and other viscoelastic properties of the liquid or thin film. One benefit of using the quartz crystal microbalance to measure viscosity is the small amount of sample required for obtaining an accurate measurement. However, due to the dependence viscoelastic properties on the sample preparation techniques and thickness of the film or bulk liquid, there can be errors up to 10% in measurements in viscosity between samples.[6]

An interesting technique to measure the viscosity of a liquid using a quartz crystal microbalance which improves the consistency of measurements uses a drop method.[7][8] Instead of creating a thin film or submerging the quartz crystal in a liquid, a single drop of the fluid of interest is dropped on the surface of the crystal. The viscosity is extracted from the shift in the frequency data using the following equation

${\displaystyle \Delta f=-f_{0}^{3/2}{\sqrt {\frac {\eta _{l}\rho _{l}}{\pi \mu _{Q}\rho _{Q}}}}}$

where ${\displaystyle f_{0}}$ is the resonant frequency, ${\displaystyle \rho _{l}}$ is the density of the fluid, ${\displaystyle \mu _{Q}}$ is the shear modulus of the quartz, and ${\displaystyle \rho _{Q}}$ is the density of the quartz.[8] An extension of this technique corrects the shift in the resonant frequency by the size of the drop deposited on the quartz crystal.[7]

Rheometers and viscometers work with torque and angular velocity. Since viscosity is normally considered in terms of shear stress and shear rates, a method is needed to convert from "instrument numbers" to "rheology numbers". Each measuring system used in an instrument has its associated "form factors" to convert torque to shear stress and to convert angular velocity to shear rate.

We will call the shear stress form factor C₁ and the shear rate factor C₂.

shear stress = torque ÷ C₁.

shear rate = C₂ × angular velocity.

For some measuring systems such as parallel plates, the user can set the gap between the measuring systems. In this case the equation used is

shear rate = C₂ × angular velocity / gap.

viscosity = shear stress / shear rate.

The following sections show how the form factors are calculated for each measuring system.

${\displaystyle {\begin{aligned}C_{1}&={\frac {3}{2}}r^{3},\\C_{2}&={\frac {1}{\theta }},\end{aligned}}}$

where

r is the radius of the cone,

θ is the cone angle in radians.

${\displaystyle {\begin{aligned}C_{1}&={\frac {3}{2}}r^{3},\\C_{2}&={\frac {3}{4}}r,\end{aligned}}}$

where r is the radius of the plate.

Note: The shear stress varies across the radius for a parallel plate. The above formula refers to the 3/4 radius position if the test sample is Newtonian.

${\displaystyle {\begin{aligned}C_{1}&={\frac {1}{3}}r_{\text{a}}^{2}H,\\C_{2}&={\frac {2r_{\text{i}}^{2}r_{\text{o}}^{2}}{r_{\text{a}}^{2}\left(r_{\text{o}}^{2}-r_{\text{i}}^{2}\right)}},\end{aligned}}}$

where:

r_a = (r_i + r_o)/2 is the average radius,

r_i is the inner radius,

r_o is the outer radius,

H is the height of cylinder.

Note: C₁ takes the shear stress as that occurring at an average radius r_a.

Measuring principle of the electromagnetically spinning-sphere viscometer

The EMS viscometer measures the viscosity of liquids through observation of the rotation of a sphere driven by electromagnetic interaction: Two magnets attached to a rotor create a rotating magnetic field. The sample ③ to be measured is in a small test tube ②. Inside the tube is an aluminium sphere ④. The tube is located in a temperature-controlled chamber ① and set such that the sphere is situated in the centre of the two magnets.

The rotating magnetic field induces eddy currents in the sphere. The resulting Lorentz interaction between the magnetic field and these eddy currents generate torque that rotates the sphere. The rotational speed of the sphere depends on the rotational velocity of the magnetic field, the magnitude of the magnetic field and the viscosity of the sample around the sphere. The motion of the sphere is monitored by a video camera ⑤ located below the cell. The torque applied to the sphere is proportional to the difference in the angular velocity of the magnetic field Ω_B and the one of the sphere Ω_S. There is thus a linear relationship between (Ω_B − Ω_S)/Ω_S and the viscosity of the liquid.

This new measuring principle was developed by Sakai et al. at the University of Tokyo. The EMS viscometer distinguishes itself from other rotational viscometers by three main characteristics:

• All parts of the viscometer that come in direct contact with the sample are disposable and inexpensive.
• The measurements are performed in a sealed sample vessel.
• The EMS viscometer requires only very small sample quantities (0.3 mL).

By modifying the classic Couette-type rotational viscometer, it is possible to combine the accuracy of kinematic viscosity determination with a wide measuring range.

The outer cylinder of the Stabinger viscometer is a sample-filled tube that rotates at constant speed in a temperature-controlled copper housing. The hollow internal cylinder – shaped as a conical rotor – is centered within the sample by hydrodynamic lubrication[9] effects and centrifugal forces. In this way all bearing friction, an inevitable factor in most rotational devices, is fully avoided. The rotating fluid's shear forces drive the rotor, while a magnet inside the rotor forms an eddy current brake with the surrounding copper housing. An equilibrium rotor speed is established between driving and retarding forces, which is an unambiguous measure of the dynamic viscosity. The speed and torque measurement is implemented without direct contact by a Hall-effect sensor counting the frequency of the rotating magnetic field. This allows a highly precise torque resolution of 50 pN·m and a wide measuring range from 0.2 to 30,000 mPa·s with a single measuring system. A built-in density measurement based on the oscillating U-tube principle allows the determination of kinematic viscosity from the measured dynamic viscosity employing the relation

${\displaystyle \nu ={\frac {\eta }{\rho }},}$

where:

ν is the kinematic viscosity (mm²/s),

η is the dynamic viscosity (mPa·s),

ρ is the density (g/cm³).