Liquids with non-Newtonian properties are presented in many engineering areas, as for example in membrane bioreactors where active sludge exhibits shear thinning properties. Therefore, the ability to determine the rheology's dependence on shear is important when optimising systems with such liquids. However, rheometers capable of determining the viscosity are often expensive and so a cheaper alternative is constructed with this exact capability. Using the principle of rotating rheometers, a low-cost rheometer was built to determine the rheology of Newtonian and non-Newtonian liquids. The general principles and background assumptions and the physics are described. The rheometer was calibrated by comparison with measurements conducted on a Brookfield viscometer for Newtonian liquids. For validation measurements on non-Newtonian liquids, xanthan gum solutions were made and compared with measurements on the Brookfield viscometer and with values from other sources. Furthermore, the effect of excluding the different shear rates in the system is discussed and good practice hereto is given.

NOMENCLATURE

     
  • Dynamic viscosity (Pa s)

  •  
  • Corrected dynamic viscosity (Pa s)

  •  
  • Shear rate (s−1)

  •  
  • Shear rate periphery (s−1)

  •  
  • Area periphery (m2)

  •  
  • Shear stress (Pa)

  •  
  • Radius bob (m)

  •  
  • Radius cup (m)

  •  
  • Radius for mounting of strain gauge (m)

  •  
  • Force measured with strain gauge (N)

  •  
  • Height bob (m)

  •  
  • Gap bottom (m)

  •  
  • Torque bob (N m)

  •  
  • Reynolds number

  •  
  • Consistency factor (Pa sn)

  •  
  • Power law exponent (-)

  •  
  • Angular velocity (rad s−1)

  •  
  • TSS

    Total suspended solids (g L−1)

INTRODUCTION

Rheology of liquids is an important parameter in respect to many engineering objectives. The rheology of liquids affects for example pressure loss in pipes and, therefore, also energy consumption when pumping the liquids (Chilton & Stainsby 1998). Many liquids show non-Newtonian properties, so it is important to be able to determine the behaviour of these. An example of a non-Newtonian liquid is active sludge (AS) (Hasar et al. 2004), which is of interest when cleaning wastewater. It is especially important for membrane bioreactors (MBRs) since they operate at high sludge concentrations compared to conventional activated sludge plants. The high concentrations of total suspended solids in the sludge result in higher viscosities and more shear thinning properties (Rosenberger et al. 2002). Because of the influence of the rheology on the flow, it is an important parameter when designing and running MBR systems. Furthermore, the rheology varies not only by concentration, but also by the general composition of the sludge, resulting in different rheology depending on age and type of wastewater (Seyssiecq et al. 2003; Hasar et al. 2004; Laera 2007). Furthermore, there is no consensus about how rheological measurements should be conducted (Ratkovich et al. 2013), which causes large uncertainties when using standard values. The available rheometers which are capable of determining the rheology over a wide range of shear rates are expensive, with prices starting around 5,000 USD. Therefore, development of a cheap rheometer is sought, capable of measuring the rheology of highly viscous liquids up to 1 Pa s with shear rates in the range of 15 to 2,000 s−1. Furthermore, AS tends to flocculate, and the equipment is made in such a way that it is able to measure flocculating liquids.

METHODS

The principle for determining the dynamic viscosity (μ) is based on Newton's law, from the shear stress () and the shear rate (): 
formula
1

The rheometer is a rotational rheometer, where the spindle is the cup containing the liquid and the bob located in the middle is stationary. The cup is mounted on a lathe (599 USD) which is able to rotate, resulting in a shear rate between the cup and bob and a shear stress on the bob. When using this design, it is possible to vary the shear rate by changing rotation speed. This can be programmed so it is possible to determine not just the viscosity at a given shear rate, but to include former shear rates. Figure 1 illustrates the set-up.

Figure 1

Sketch and picture of the rheometer.

Figure 1

Sketch and picture of the rheometer.

The dimensions of the set-up are as follows: radius of bob, , radius of cup, , height of bob, and gap between bottom of bob and cup, . The data from the rheometer are compared with measurements from a Brookfield viscometer, which has the following dimensions: radius of bob , radius of cup and a height of the bob of .

The bob is located in a homemade air bearing for minimal resistance so as to avoid influences on the results, however other bearings can be used and will just result in a different resistance. For determining the shear stress at the bob a strain gauge is connected to the bob with a string whereby it measures the force applied to it, from which the shear stress can be determined. The strain gauge used for the rheometer is of the model HBM, but the exact model is unknown (10 USD). To determine the rotation speed of the cup, a frequency indicator of the type PR5725 with sensor (400 USD) is used for set-up, see Figure 1. The frequency indicator is used since it was available in the laboratory; however, a much cheaper alternative can be constructed by magnetic generated impulses (approximately 20 USD) or laser-based tachometer, which can be found for less than 100 USD. The diameter of the bob or cup can be changed depending on the demands of the liquid on which measurements are conducted. For low viscous fluids it might be necessary to have a smaller gap to avoid turbulence. If measuring on flocculating liquids, e.g. AS, a bigger gap might be needed and a smaller bob can be used.

By rotation of the cup, a velocity profile between the cup and stator is achieved. Because of the curve of the bob and the spindle, the velocity profile will not be linear, as with a Couette flow, but it will depend on the proportion between the bob and the cup. With a sufficiently small gap, , the velocity profile can be assumed linear if the flow is kept laminar (Steffe 1996). Despite this, the gap should be kept small to keep a linear velocity profile gap, but must be kept larger than the largest particles in the fluid (Slatter 1997) if, for example, measuring flocculating liquids such as AS. With the sufficiently small gap the shear rate on the periphery of the bob can be calculated as 
formula
2
where is the rotation speed in . The shear rate at the bottom is lower, due to the lower velocity, resulting in lower shear stresses. The viscosity is determined with the assumption that the shear rate in the system is the one presented on the periphery of the bob while the lower shear rates under the bob are taken into account by a calibration.

The wall shear stress is determined from the force measured with the strain gauge (), and as well as for the shear rate, the shear stress is determined as only presented on the periphery and the bottom of the bob is included by calibration. To relate the output from the strain gauge it has been calibrated by applying a known weight and make a fit from the measured voltage, which then is related to the force. The result of the calibration is illustrated in Figure 2.

Figure 2

Measured forces with calibrated strain gauge.

Figure 2

Measured forces with calibrated strain gauge.

The calibration of the strain gauge showed an almost perfectly linear fit and is therefore not inducing significant uncertainties when measuring the force.

The torque of the bob is determined from the measured force with the strain gauge and the radius in which it is mounted () 
formula
3
From the torque and the radius of the bob the shear force on the surface of the bob () is determined as 
formula
4
The shear stress at the bob is determined as 
formula
5
The viscosity is determined by Equation (6) and calibrated by Equation (7) from the measurements performed with the Brookfield viscometer to find the correct viscosity () 
formula
6
 
formula
7

The constant is taking into account that the friction of the bearing might change as a function of the force applied as well as the fact that the force applied on the bottom is not taken into account. The constant is taking into account that there is a bigger contribution to the force from the area under the bob for low viscous liquids due to radial flows.

MEASUREMENTS

The rheometer is calibrated and validated from experiments with different liquids. The calibration is made on oil and water, which are Newtonian liquids. The validation is made with xanthan gum (XG) (Gindsted Gum 80 from Danisco) solutions, which exhibit non-Newtonian shear thinning properties.

All measurements are carried out over a period of 120 seconds with a sampling frequency of 5 Hz to minimize the effect of periodic oscillations. The data from the frequency indicator and the strain gauge are logged with a data logger and processed with a programme written in MatLab R2013A. The programme determines the viscosity from the mean of the shear rate and the measured force over the entire period of measuring. The measurements are carried out at room temperature, 22 °C. This is measured in the liquids just before and right after each measurement series and varies by ± 1 °C.

The Newtonian liquids used for the calibration have viscosities in the range to Pa s, from lamp oil, sunflower oil, hydraulic oil and mixes of lamp oil and sunflower oil in ratios of 2:1 and 1:4. The measurements are made at different shear rates to verify the rheometer's capability of measuring over a range of shear rates. Measurements of viscosity of the same liquids are conducted with a Brookfield viscometer of the type LVTVD II for comparison.

It is validated that the rheometer is capable of determining the rheology of non-Newtonian liquids by measurements on four XG solutions, which are compared with data from Zhong et al. (2012), Song et al. (2006) and Tipvarakarnkoon & Senge (2008). The solutions are made with deionised water with XG concentrations of 0.5 g/L, 1.0 g/L, 2.0 g/L and 10 g/L, respectively.

RESULTS

Newtonian liquids

The measurements on oils are conducted at three different shear rates. At low viscosities, the rheometer measures too high values while it measures too low values at higher viscosities. The constants in Equation (7) are determined with the calibration and results in the following equation: 
formula
8

With this correction, the results illustrated in Figure 3 were achieved.

Figure 3

Result of measurements conducted on the Brookfield viscometer and the rheometer on, respectively, lamp oil, sunflower oil, hydraulic oil and mixtures of lamp oil and sunflower oil.

Figure 3

Result of measurements conducted on the Brookfield viscometer and the rheometer on, respectively, lamp oil, sunflower oil, hydraulic oil and mixtures of lamp oil and sunflower oil.

With the calibration of the rheometer, R2 is found at 0.99 compared to the Brookfield viscometer, for viscosities in the range of to Pa s and with shear rates in the range of 200–800 s−1.

The results of the measurements on water and lamp oil show that the flow becomes turbulent at high Reynolds numbers, resulting in incorrect measurements.

The experiments show that the flow turns turbulent at Reynolds numbers () between 550 and 580 for water and lamp oil, calculated as 
formula
9

This can be used as design criteria when constructing a rheometer so as to make sure it is able to measure in the required range of shear rates and viscosities. It is clear that the measurement of the viscosity should be consistently independent of the shear rate, as illustrated in Figure 3, while the minor variations in Figure 4 are very likely due to a not perfectly smooth bearing, which has the highest relative impact on low viscous liquids.

Figure 4

Measurements on water and lamp oil for different Reynolds numbers.

Figure 4

Measurements on water and lamp oil for different Reynolds numbers.

Non-Newtonian liquids

One of the most commonly used models to describe shear thinning liquids is the Ostwald de Waele equation, where the viscosity at a given shear rate is given as 
formula
10

The k and n parameters have been determined in various literature as describing the rheology of XG solutions (Zhong et al. 2012; Song et al. 2006; Tipvarakarnkoon & Senge 2008). The k and n values are the foundation for the validation with non-Newtonian liquids. Furthermore, the measurements are compared with measurements conducted on the Brookfield viscometer. Since this is not capable of measuring in the same area of shear rates and viscosities, only the lowest concentrations have been measured and only for low shear rates as illustrated in Figure 4.

The constructed rheometer is not capable of measuring shear rates lower than approximately 15 s−1, with the given dimensions. In the range of shear rates from 15 to 300 s−1, R2 is 0.96, 0.99 and 0.94 for the solutions 0.5 g/L, 1.0 g/L and 2.0 g/L, respectively.

Besides the R2 values, Figure 5 illustrates the rheometer's strength in determining the correct viscosity for non-Newtonian liquids as well, in the range of shear rates from 15 to 300 s−1. It also shows that the measurements from the solution with 10 g/L are between what is given in Song et al. (2006) and Tipvarakarnkoon & Senge (2008).

Figure 5

The empty markers illustrate the measured viscosities with the rheometer, while the filled markers are measured with the Brookfield viscometer and the lines illustrate the values calculated from n and k values from Zhong et al. (2012) for the three lowest concentrations. The two lines for 10 g/L is from Song et al. (2006) and Tipvarakarnkoon & Senge (2008).

Figure 5

The empty markers illustrate the measured viscosities with the rheometer, while the filled markers are measured with the Brookfield viscometer and the lines illustrate the values calculated from n and k values from Zhong et al. (2012) for the three lowest concentrations. The two lines for 10 g/L is from Song et al. (2006) and Tipvarakarnkoon & Senge (2008).

For an XG concentration of 10 g/L, the k and n values are determined and compared with Song et al. (2006) and Tipvarakarnkoon & Senge (2008). 
formula
10
 
formula
10

Both values are in the range of the determined values in the literature where k is in the range and n is in the range . Despite this, the measurement area for the Brookfield viscometer and the constructed rheometer does not correspond but it can be seen that within the area where they can both measure, the values from the two devices correspond.

CONCLUSIONS

Despite the uncertainties when determining the shear rate at the bob, the rheometer has been able to determine the viscosity of the liquids used for calibration and validation. Nonetheless, it is important to note that measurements on low viscous liquids are connected with bigger uncertainties. When measuring on low viscous liquids, a perfectly smooth bearing is very important as well as a smaller gap, which will decrease the influence of radial flow under the bob.

The rheometer is validated with shear rates starting at 15 s−1, but it might be able to measure lower than this with a system with lower velocities or a bigger gap between the periphery of bob and cup.

Despite the spatial variation of shear rates in the system, the rheometer has been able to provide good results when measuring on shear thinning liquids. Based on this, it is recommended that the diameter of the bob is smaller than the height of the bob; the smaller this relationship is, the smaller an error can be expected.

The rheometer is not able to maintain a given temperature, but it will measure at the temperature of the room. The temperature is measured before and after each test series and no difference has been found due to the friction. Therefore, all measurements on XG are carried out at the room temperature of 22 °C, but the results are compared with values for experiments carried out at a temperature of 25 °C in Zhong et al. (2012) and 20 °C in Song et al. (2006) and Tipvarakarnkoon & Senge (2008).

It has been possible to develop a rheometer which is capable of measuring the viscosity of Newtonian liquids with viscosities ranging from to Pa s with shear rates up to 800 s−1. For shear thinning liquids, it is validated with shear rates from 15 to 300 s−1 for viscosities up to 0.987 Pa s.

The rheometer can be constructed for the following price under the assumption that pressurised air, a laptop and a data logger are available.

The prices for the materials were lathe 600 USD, strain gauge 10 USD, frequency indicator 400 USD, plastic and aluminium for cup, bob and homemade air bearing as well as different wires, etc. had a total cost of approximately 150 USD. This resulted in a total cost of 1,160 USD, but could with one of the cheaper alternatives for the frequency indicator be reduced to around 800 USD. In addition to this around 20 working hours might be necessary to construct the rheometer.

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