Rheonics Type-SR sensors are inline instruments to measure real-time viscosity and density of a fluid, as well as temperature and derivatives from these values. Rheonics offers the SRV process viscometer to measure viscosity and the inline density meter SRD for density and viscosity values of a fluid. Both sensor probes are compact, lightweight and hermetically sealed, making them suitable for all industrial processes involving liquids.

Both SRV and SRD sensors are based on balanced torsional resonator (BTR) technology. Both sensors measure and output measurements of viscosity for the fluid they are in contact. For Newtonian fluids, you get the same viscosity irrespective of the instrument used. However, for non-Newtonian fluids, that is not the case and different instruments measure different values of viscosity - this is often not due to inaccuracy of the instrument itself but due to the shear dependence of viscosity and the fact that different instruments measure different shear rates.

**Non-Newtonian fluids are fluids ****that do not exhibit a linear relationship between shear stress and shear rate**. This means that the viscosity of a non-Newtonian fluid can change depending on the shear rate it is subjected to. Most fluids we encounter in our day-to-day lives and in industrial processes are non-Newtonian in nature.

Due to this shear dependence of viscosity for non-Newtonian fluids, and to enable some comparison between different viscometers (often between process viscometers like SRV and lab instruments like rotational viscometer or rheometer), it is desired to understand the effective shear rate at which SRV or SRD is making the measurements. The analysis below mentions SRV but is equally valid for the SRD.

**Uncertainties in the estimation of effective shear rate exist for both SRV and SRD so the estimates should be used as a guideline instead of a defined value.**

**1. How does the process viscometer SRV or density meter SRD operate?**

The SRV sensor's sensing element consists of a rod and a mass attached to its end, this rod and the tip are circular and cylindrical. The other end is connected to the body that contains the transducers for excitation and sensing.

Figure 1. SRV probe mechanical structure and schematic of the fluid interaction showing velocity profile in the fluid imposed by the Type-SR resonator oscillation

The sensor vibrates in torsion, torsional resonators are more stable and better isolated from their mechanical environment. Torsional resonators that are cylindrical vibrate parallel to their own surfaces. They are influenced by shearing forces and are therefore primarily sensitive to dissipative forces (viscous damping) rather than mass loading effects (also often referred to as inertial damping).

Figure 2. Torsional resonator under fluid stress.

**2. Why is the shear rate relevant for process viscometer?**

Viscosity of a non-Newtonian fluid can change depending on the shear rate it is subjected to. This means that a single viscosity value can’t be associated with this kind of fluids under all states (eg. static, flowing at different speeds).

Laboratory viscometers often allow users to change the shear rate or rotational speed at which viscosity is measured. Rheonics SRV and SRD have a shear rate usually much higher than those for lab instruments and users cannot change it.

It is possible to have a qualitative idea about the range of shear expected for the SRV viscosity sensors and calculations are shown in this article. This helps qualify (and to some extent quantify) conditions at which viscosity is being measured and correlate readings with other instruments.

However, actual correlations between shear of Type-SR viscosity measurements and other lab instruments are mostly empirical and may not meet the qualitative estimation. The estimated shear rate may not exactly match the viscosity value from a rheometer. Consider that Rheonics sensors are process control devices more than just a viscosity sensor with emphasis on extremely high repeatability, and reproducibility of measurements with an unmatched resolution (often 10-100X higher than lab instruments).

The difference between viscosity readings is exacerbated by the fact, that the SRV and SRD measure at a constant shear rate amplitude, whereas lab-instrument a shear rate that is constant over time. This added time-dependency on shear rate may also contribute to difference in viscosity at a given shear rate for non-Newtonian fluids.

**3. Shear Rate estimation of oscillatory process viscometer**

There are two parameters that are the most important for the shear estimation which are **velocity amplitude and the boundary layer thickness**. It is necessary to calculate the following parameters.

The shear stress is given by:

Equation 1: Shear Stress.

For a Newtonian fluid,** η** is a material constant characteristic of the fluid, **∂v/∂x **is the shear rate in the fluid Applying the Navier-Stokes equations, solving under periodic, uniaxial conditions, the solution for the velocity amplitude is:

Equation 2: Velocity Amplitude

For a Newtonian fluid, η is a material constant characteristic of the fluid, ∂v/∂x is the shear rate in the fluid Applying the Navier-Stokes equations, solving under periodic, uniaxial conditions, the solution for the velocity amplitude is:

**x:**distance from the sensor wall**V**: velocity amplitude at the sensor surface, R is radius of the tip**δ:**is the boundary layer thickness**i:**is square root of -1

The **boundary layer thickness** can be found with the equation:

Equation 3: Boundary Layer Thickness

**η**: dynamic viscosity**ω**: angular frequency**ρ**: fluid density

By considering that at **x=2δ** the velocity drops to 13 % of the value at the sensor surface. The shear rate **γ=∂v(0)/∂x** at the surface of the sensor (x=0) follows:

Equation 4: Shear Rate

Where the velocity amplitude V(R) (5) is given by:

Equation 5: Velocity Amplitude

**R**: Distance from vibrational axis to sensor surface**φ**: Angular vibration amplitude.

The tip from the SRV performs a sinusoidal rotational vibration **φ** about its axis of symmetry.

Equation 6: Sinusoidal Rotational Vibration

**For the SRV, the velocity V(R) is approximately 50 mm/s and Frequency is 7500 Hz → ω=2π x 7500**

The parameter **V(R)** is independent of viscosity, but the boundary layer thickness of the fluid **δ** increases. The following graph demonstrates the shear rate vs viscosity behavior and shows shear rate variation with both viscosity and density of the fluid under investigation.

Figure 3. Shear rate vs viscosity for Type-SR sensor

**Applications of shear rate estimates of process viscometer in Industrial processes**

The effective shear rate of an oscillatory viscometer is the shear rate that the fluid experiences during measurement. It is important to estimate the effective shear rate of an oscillatory viscometer to compare the results obtained from different viscometers and to design industrial processes involving non-Newtonian fluids.

Processes likemixing, coating, sprayingimpose high shear on the fluid. To understand the behavior of fluid under these processes and control consistency of production, sensors like SRV and SRD enable process engineers to monitor viscosity of fluid at shear rates that their fluid undergoes during operation giving them direct data to optimize process parameters. This is not possible using lab viscometers which often measure at low shear that is not representative of the actual operational conditions.

Vibrational process viscometers like the SRV and SRD are widely used to measure inline real-time viscosity of non-Newtonian fluids in a variety of industries, including food processing, cosmetics, and pharmaceuticals.

Here are some examples of non-Newtonian fluids that are commonly measured using vibrational process viscometers:

Figure 4. Common Non-Newtonian Fluids measured by Rheonics sensors