© EDP Sciences, 2016

1 Introduction

Given the necessary global reduction of greenhouse gas emissions and a compulsory saving in fossil fuel energy consumption, the use of biofuels [1] is required. European legislators have taken measures in this regard [2].

The viscosity of biofuels and of their blends with fossil fuels is not well known, despite the fact that it is an essential feature, for their transport in pipelines. Like fossil fuels, these biofuels have low viscosities (some mPa s), and according to TRAPIL, a company transporting oil via pipeline, they are used in temperature conditions that can vary from −10 °C up to +50 °C, and in variable pressure conditions going from 0.1 MPa up to 10 MPa.

The viscosity that varies according to temperature and pressure is therefore an important parameter in the transport of fuels and biofuels.

The project aims to operationalize a falling ball viscometer in order to be able to measure the viscosities of fuels and biofuels blends over a temperature range from −10 °C up to +50 °C and also with a pressure up to 10 MPa. Initially, this project is limited to a temperature range from +15 °C up to +50 °C and only at atmospheric pressure.

The ultimate goal of this research is to operationalize a falling ball viscometer so that it becomes an absolute reference, by linking the dynamic viscosity of the following different quantities: mass, length, and time.

2 State of the art

Several studies on the falling ball viscometers were conducted with Newtonian fluids [3], and wealthy fluids in suspended particles [4].

In the metrology field, the work conducted at the National Metrology Institute of Japan (NMIJ) led to the construction of an absolute viscometer for measurements at high pressures (500 MPa) [5]. The NMIJ also decided in 1998 [6] to build an absolute falling ball viscometer, in order to determine a new value of water viscosity, with a new uncertainty. The progress of both the NMIJ and LCM/LNE-CNAM viscometers was presented in the CCM Working Group on Viscosity at BIPM, in May 2011 [7,8].

3 Scientific and technical aim

In 2005, LCM/LNE-CNAM had a falling ball viscometer, which was developed and studied by Brizard [9] during his thesis.

The photograph of Figure 1 shows the measurement cell, which was operating only at laboratory temperature.

The metrological study carried out during this thesis demonstrated that it is possible to obtain relative uncertainties of 10⁻³ for viscosities equal to or higher than 10 Pa s. Furthermore, the relative uncertainties of the falling ball viscometer are becoming better than those of capillary viscometers, for viscosities higher than 1 Pa s.

The development of this viscometer, which ceased in 2005 resumed in 2010 with the aim to adapt it to the low viscosities of biofuels, adjust it to temperatures between −10 °C and +50 °C, and build a small-volume cell to perform intercomparisons.

The viscometer was modified in three main stages:

• adaptation to low viscosities (of biofuels) using specific balls;

• adaptation to controlled temperatures through the development of a thermostatic bath;

• construction of a cell for intercomparisons.

Fig. 1 Measurement cell of falling ball viscometer.

3.1 Project for adapting the viscometer to low viscosities

3.1.1 Order of magnitude for biofuel viscosity

The first results achieved by LCM/LNE-CNAM regarding biofuels in terms of kinematic viscosity and density were presented during the ENG09–Biofuel (European funded programme by EURAMET) annual meeting in Helsinki [10] in January 2011.

The kinematic viscosity measurements were performed with capillary viscometers while density measurements were performed with pycnometers.

Note: The kinematic viscosity ν is defined as the ratio of the dynamic viscosity η in relation to density ρ. $$\nu =\frac{\eta }{\rho }\text{.}$$(1)

The evolutions of kinematic viscosity and density according to temperature, of two pure biofuels made from rapeseed and soybean, are shown in Figures 2 and 3.

At 20 °C, the dynamic viscosity of these biofuels is of the order of 6 mPa s, which is 6 times higher than that of water.

Fig. 2 Evolution of kinematic viscosity depending on temperature [10] (at atmospheric pressure).

Fig. 3 Evolution of density depending on temperature [10] (at atmospheric pressure).

3.1.2 Reminding the principle of the falling ball viscometer

The dynamic viscosity η of a liquid is obtained from the measurement of the fall velocity U_∞, of d diameter and ρ_b density ball into this same liquid featuring ρ₁ density.

The dynamic equilibrium equation of the ball gives the following relation: $$\eta =\frac{d^2}{18U_{\infty }}({\rho }_b-{\rho }_l)g\text{,}$$(2) g, gravity acceleration.

This relationship was obtained using the Stokes' formula (which comes from Stokes' equation) regarding the aerodynamic drag F_stokes (we explain this law in the paragraph titled “correction of the Reynolds effect”: $$F_{\text{Stokes}}=6\pi \eta U_{\infty }d,$$(3) knowing that the formula involves the following two hypotheses:

$$\frac{d}{D}\to 0\text{.}$$(4)

$$R_e\to 0,$$(5) with

$$R_e=\frac{{\rho }_l{U}_{\infty }d}{\eta }\text{.}$$(6)

$$F_x=F_{\text{Stokes}}\times f\left(\frac{d}{D}\right)\times f(R_e)\text{.}$$(7)

$$\eta =\frac{d^2}{{18U}_{\mathrm{\infty }}}({\rho }_b-{\rho }_1)g\times k_1\left(\frac{d}{D}\right)\times k_1(R_e)\text{.}$$(8)

3.1.2.1 Correction of the wall effect

Stokes' equation for a moving sphere, in cylindrical boundary conditions, was solved by Bohlin [11] in 1960. This case implies a ball that falls along the axis of symmetry of a revolution cylinder with a D diameter and a near-zero Reynolds number.

The correction function k₁(d/D) is the following: $$k_1=1/\left(1-2.10443\left(\frac{d}{D}\right)+2.08877{\left(\frac{d}{D}\right)}^3-0.94813{\left(\frac{d}{D}\right)}^5-1.372{\left(\frac{d}{D}\right)}^6+3.87{\left(\frac{d}{D}\right)}^8-4.19{\left(\frac{d}{D}\right)}^{10}+\cdots \right)\mathrm{.}$$(9)

Its validity field is limited to the following: $$\frac{d}{D}\prec 0.6\text{.}$$(10)

3.1.2.2 Correction of Reynolds' effect

Stokes' formula expresses the force applied on the moving ball in an infinite medium with R_e → 0.

In general, the aerodynamic force applied on a body by a flow is equal to the sum of the integral pressure forces and the integral frictional forces around this body. These pressure and frictional forces are known when the flow around the body, which is governed by Navier–Stokes' law is also known [12–14]: $$\begin{array}{c}\begin{array}{c}\hfill \rho (\partial {\displaystyle \overrightarrow{U}}/\partial t)\hfill \\ \hfill (\text{a})\hfill \end{array}\begin{array}{c}+\\ \end{array}\begin{array}{c}\hfill \rho ({\displaystyle \overrightarrow{U}}\cdot \mathrm{g}\mathrm{r}\mathrm{a}\overrightarrow{\mathrm{d}})\cdot {\displaystyle \overrightarrow{U}}\hfill \\ \hfill (\mathrm{b})\hfill \end{array}\begin{array}{c}=\\ \end{array}\begin{array}{c}\hfill \rho {\displaystyle \overrightarrow{g}}\hfill \\ \hfill (\text{c})\hfill \end{array}\begin{array}{c}-\\ \end{array}\begin{array}{c}\hfill \mathrm{g}\mathrm{r}\mathrm{a}{\displaystyle \overrightarrow{\mathrm{d}}}\cdot p\hfill \\ \hfill (\text{d})\hfill \end{array}\begin{array}{c}+\\ \end{array}\begin{array}{c}\hfill \eta {\nabla }^2{\displaystyle \overrightarrow{U}}\hfill \\ \hfill (\text{e})\hfill \end{array}\end{array}\mathrm{.}$$(11)

This equation has the 5 following terms:

• (a)

  represents the amount of unsteady acceleration per volume unit;

• (b)

  represents the amount of advective acceleration per volume unit (the inertial forces);

• (c)

  represents the gravity force per volume unit;

• (d)

  represents the pressure force per volume unit; and

• (e)

  represents the viscous forces per volume unit.

The Navier–Stokes' equation has an exact analytical solution only for simple cases, mostly regarding isovolume fluids and specific boundary conditions [15].

The nondimentionalization of this equation [16,17] produces a dimensionless number, Reynolds' number. This number characterizes the importance of inertial forces on the viscous forces. When R_e → 0, the advective term disappears.

This linear equation is Stokes' equation [18]. $$\rho (\partial {\displaystyle \overrightarrow{U}}/\partial t)=\rho {\displaystyle \overrightarrow{g}}-gra{\displaystyle \overrightarrow{d}}.p+\eta {\nabla }^2{\displaystyle \overrightarrow{U}}\mathrm{.}$$(12)

In the case of a stationary flow, the left member of the equation is zero.

Stokes' solution (1851) provides the velocity and pressure field around the sphere. These two speed and pressure fields allow to calculate the stress exercised on this sphere, in order to achieve the previously introduced Stokes' formula.

A great number of scientific books present this demonstration [19–21].

Different authors have calculated analytically the drag of the sphere, by approximation (e.g. linearization) of the advective term of Navier Stokes' equation.

The graphic in Figure 4 compares the experimental results of Liao [22] to different other analytical and experimental results (the drag force is here presented in its dimensionless form, the aerodynamic drag coefficient [23]).

Fig. 4 Analytical results compared with the experimental results of Liao [22].

3.1.3 Selection of the ball

A ball with small diameter helps achieve a low Reynolds number.

A silicon ball with a diameter of 0.48 mm was selected because of its relatively low density (2329 kg/m³), in order to reduce the falling speed. This ball was also selected because of its dimensional (Grade 10 ISO 3290-1998 standard), thermal, mechanical and chemical characteristics. Other balls with a larger diameter 1.73 mm and 2 mm were also selected.

Regarding biodiesel with a viscosity of 6 mPa s, the fall velocity to the dynamic equilibrium is of the order of 30.3 mm/s and the Reynolds number is about 2.1.

In this configuration, it seems difficult to use directly the falling ball viscometer to measure the biofuels' viscosity.

However, if we choose a referenced oil with a viscosity of 14.9 mPa s and a density ρ = 835 kg/m³ at 20 °C, the approximate values of the falling speed and the Reynolds number are the following: $$U_{\infty }=12.6\text{mm/s}\hspace{0.17em}{R}_e=0.34\text{.}$$

As we can see in the previous curve (Fig. 4), this Reynolds number seems acceptable.

This creates another opportunity of connection, allowing us to calibrate in absolute a referenced oil with the falling ball viscometer. This oil will then help connect viscometers that are used for biofuels.

But, the small diameter of the ball requires the construction of a gripper, suitable for releasing and recovering the ball.

3.1.4 Design and construction of a gripper

The gripping device for the ball was designed with a dual function:

• keeping the ball below the free surface of the liquid and releasing it at a given time (“ball release” phase, Fig. 5a);

• recovering the ball at the cylinder's bottom, in order to replace it under the free surface level and release it again (“ball recovery” phase, Fig. 5b).

Figure 5a  shows the gripper (1), with its actuator (2) and the reception part (3). The gripper, here in “ball release” position, is installed on an adjusting cell (4) whose cylinder has reduced dimensions compared to the one of the viscometer.

Figure 5b shows the gripper in “ball recovery” position.

Figure 5c shows the gripper and its parts together with the voltage generator feeding the actuator.

Considering the small diameter of the balls, of the order of one millimeter, we realize the gripping system by using techniques suitable for microsystems, such as the adhesion by van der Walls type forces [24].

Indeed, at this scale (of the order of 1 mm), the surface forces (adhesion force) can become preponderant, ahead of the volume forces (gravity). These types of forces are interatomic and depend on the nature of the materials and the liquid at the interface.

Explanations on the progress of the two phases, “ball release” and “ball recovery”, are presented in the following two paragraphs.

Fig. 5 Gripper mounted in the adjusting cell.

3.1.4.1 The ball release

Figure 6 shows the lower extremity of the gripper for a 2 mm ball:

• firstly, the rod (1) goes down so that the ball (2) can be grabbed through contact with the core (3) fixed on the rod;

• secondly, when to the control for lifting the rod is activated, the two surfaces in contact (ball core) are separated. In that case, the adhesion forces between the ball and the separator (4) are insufficient to keep the ball in equilibrium, and it falls.

The upward movement of the rod is caused by the contraction of the amplified piezoelectric actuator. The amplitude of the actuator's displacement is proportional to the applied voltage.

A displacement of 0.4 mm at most is produced by a voltage of +150 V. This voltage is supplied by a generator (see Fig. 5c).

Figure 7 displays the photographs of the gripper's extremity in “adherence” and “release” position respectively, for a ball with 2 mm diameter.

By changing the core and the separator, the gripper can work for balls with a diameter of 2 mm, 1.73 mm, or 0.48 mm. All the components of the gripper are made of stainless steel.

The adhesion of the balls to their core is satisfactory thanks to the polishing performed after machining. This adjusting operation on the gripper allowed to bring the contact surfaces very close to each other, decrease their roughness, and increase the intensity of van der Walls forces.

Another type of separator with notches was tested, in order to limit the ball-separator contact forces. A photograph taken with an optical microscope, shows a separator for a ball with a 2 mm diameter. This separator is presented in Figure 8.

Figure 9 shows the ball of 0.48 mm adhering to its core, which has been removed from the gripper.

Another alternative to this type of solution was studied, by attaching the ball to the “separator” part and pushing it with a modified core, as shown in Figure 10.

Fig. 6 Gripper extremity for a ball of 2 mm.

Fig. 7 “Adherence” and “release” position respectively for a ball of 2 mm.

Fig. 8 Separator with notches (optical microscope).

Fig. 9 Adhesion of the 0.48 mm ball to its core (optical microscope).

Fig. 10 Alternative solution: pushing the ball to trigger its fall.

3.1.4.2 Ball recovery

To recover the ball, the gripper must be lowered down to the recovery system presented in Figure 11.

Figure 12 shows the recuperator. A ball of 2 mm adhering to its core was placed just above, to give an indication about the scale.

The conical receiving part was polished by hand so that the ball can roll up to the receiver's symmetry axis, if accidentally it should not fall on the axis. Adjusting the recuperator became quite difficult because van der Walls forces increase the moment of resistance against rolling considerably.

In normal operation, the ball falls on the axis and is held in this position thanks to the conical part of the receptor, during the phase of adhesion to the core.

Then, when we lift the gripper, the ball is recovered and repositioned, in order to be released again.

Note: the central conical part of the recuperator has a return spring and is able to relocate. This system can avoid possible deformations of this one or the separator, by imposing a low contact force between the ball and the recovery.

A film showing the release and recovery of the ball in linseed oil, with a viscosity around 50 mPa s, was performed for 0.48 mm and 2 mm balls.

Fig. 11 Sketch of the ball recuperator.

Fig. 12 Photograph of the ball recuperator (with a ball of 2 mm diameter, to give an indication about scale).

3.2 Adaptation to controlled temperatures (−10 °C up to +50 °C)

A thermostatic bath with optical walls was created for the immersion of the viscometer. It is presented in Figure 13. This bath is made from a semi-finished product of aluminum alloy.

This thermostatic bath has two configurations. Figure 14 shows the bath windows in “high temperatures” (a single optical borosilicate glass) and “low temperature” version (two optical made of borosilicate glass).

Both configurations were tested. First, it was tested in “high temperature” configuration, see Figure 15a.

The first tests were performed in closed circuit, because the temperature control unit has two pumps with different flow rates for the fluid input and output, thus causing fluid overflows inside the enclosure.

Operation is possible without the appearance of water vapor condensation on the windows, for temperatures between +15 °C and +50 °C.

The second configuration, reserved to low temperatures, was then tested. The bath temperature lowered down to −16.4 °C, but within prohibitive time duration, and with condensation on the windows and icing on the thermostatic bath (Fig. 15b).

This step should be continued by insulating thermally the thermostatic bath to avoid freezing the water steam and to reach −10 °C in a relatively short time.

To allow the bath to operate normally in open circuit, load losses and a liquid level maintaining system were installed.

Fig. 13 Thermostatic bath.

Fig. 14 Thermostatic bath windows.

Fig. 15 Thermostatic bath.

3.3 Construction of a cell for intercomparisons and installation in the thermostatic bath

From a practical point of view, calibrations for intercomparisons are made on small volumes of liquid. Our measuring cell had a volume of 2500 ml, which was excessive.

A measuring cell of a smaller volume was thus constructed, as illustrated in Figure 16.

The cell has an inner diameter of 50 mm and a volume of 300 ml.

The new cell needs to be suspended at the center of a micrometric moving system, which is fixed on the upper part of the thermostatic bath to allow optical adjustments, especially the adjustment of the ball's image.

The drawing of the displacement system along the directions X and Y is presented in Figure 17.

Two parallel rails allow the movement along direction X. Two other are arranged perpendicular and allow the movement along Y.

The displacement system was built and the cell was implanted in the mounting as shown in Figure 18.

In this photograph, inside this thermostatic bath (1), we can see the following: the cell (2), the gripper (3), and the laser diode with its optical (4) before its alignment to the matrix camera (5).

After this alignment, the laser beam enlightens the ball during its course, and the shadow can be recorded by the camera. A processing of this image allows the ball speed to be calculated and the fluid viscosity to be deducted.

Fig. 16 Cell for intercomparison.

Fig. 17 Drawing in perspective of a cell displacement system.

Fig. 18 Implementation of the bath in the experimental setup.

4 Conclusions

This work must continue with measurements on fluids of different viscosities, especially low viscosities, at standardized temperatures, for example 15 °C or 20 °C, and also 40 °C for hydrocarbons.

The thermal insulation of the thermostatic bath must continue in order to examine the operation at low temperatures up to −10 °C.

The second step is to perform a cell pressurized up to 10 MPa in order to measure the viscosity of liquids, with regard not only to the temperature, but also to the pressure on the measuring ranges, which concern the transport of biofuels.

Another advantage of falling ball viscometers in the measurement of viscosity of non-Newtonian fluids at low shear stresses, in order to determine the yield stress, is demonstrated in various publications, such as those mentioned in [25–27].

Along with this work, a research regarding the use of this experimental device as a falling ball rheometer was launched by EURAMET, as part of a European-funded research program, which is called ENG59-NNL.

This three-year program began on 1st May 2014. It is introduced on the website www.eng59-rheology.eu, and covers the metrology of drilling fluids in oil wells.

References

All Figures

	Fig. 1 Measurement cell of falling ball viscometer.
In the text

	Fig. 2 Evolution of kinematic viscosity depending on temperature [10] (at atmospheric pressure).
In the text

	Fig. 3 Evolution of density depending on temperature [10] (at atmospheric pressure).
In the text

	Fig. 4 Analytical results compared with the experimental results of Liao [22].
In the text

	Fig. 5 Gripper mounted in the adjusting cell.
In the text

	Fig. 6 Gripper extremity for a ball of 2 mm.
In the text

	Fig. 7 “Adherence” and “release” position respectively for a ball of 2 mm.
In the text

	Fig. 8 Separator with notches (optical microscope).
In the text

	Fig. 9 Adhesion of the 0.48 mm ball to its core (optical microscope).
In the text

	Fig. 10 Alternative solution: pushing the ball to trigger its fall.
In the text

	Fig. 11 Sketch of the ball recuperator.
In the text

	Fig. 12 Photograph of the ball recuperator (with a ball of 2 mm diameter, to give an indication about scale).
In the text

	Fig. 13 Thermostatic bath.
In the text

	Fig. 14 Thermostatic bath windows.
In the text

	Fig. 15 Thermostatic bath.
In the text

	Fig. 16 Cell for intercomparison.
In the text

	Fig. 17 Drawing in perspective of a cell displacement system.
In the text

	Fig. 18 Implementation of the bath in the experimental setup.
In the text