Experimental Investigation of Fluid Viscosity and Its Dependencies Using Falling Ball Viscometer

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Introduction

Viscosity of a fluid is the measure of the fluid’s resistance to gradual deformation by shear and tensile stress. In informal terms, it is a measurable quantity denoting the ’thickness’ of a fluid. This was discovered in 1938 by French physicist Jean LÃľonard Marie Poiseuille. He discovered this while studying blood flow through tubes.

This led him to formulate, the now famous, Poiseuille’s Law, which could be applied to fluids of any viscosity. Aim of this experiment is to learn how to use a falling ball viscometer along with understanding how the fluid mechanics worked in case of Newtonian and Non-Newtonian fluids.

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The relationship between the viscosity of the fluids and their concentration was to be determined as well, with the fluids being sugar solutions of differ- ent concentration. The effect of temperature on the viscosity of a fluid of constant concentration was studied.

All these effects were studied by measuring the time taken for a ball to fall through the fluid and using fluid mechanics equations to determine the viscosity of the fluid used.

A solution of unknown concentration was also 1 provided, where the mass and volume of the solution was measured and used to derive its density. After the density calculation, the viscometer was used to find the viscosity of the fluid, which was then used to calculate the concentration of the fluid.

Theory

This experiment involves the use of a ball of radius r, falling through a variety of viscous fluids at a constant terminal velocity.

For a body to move with a constant velocity, the forces experienced by the body must be balanced. There are three forces acting on the ball: Weight of the ball, bouyant force and drag or fluid resistance. When the ball is falling downwards, the weight of ball is the only force acting downwards, while the other two forces are acting upwards.

A the forces are to be balanced, the magnitude of forces in the upward and downward direction must be equal, giving rise to the following equation: mballg = 6πηrv +mfluidg (1) [1][2] where mball is the mass of the ball, g is the acceleration due to gravity, η is the viscosity of the fluid, v is the velocity of the ball and mfluid is the mass of the fluid displaced by the ball. The mass of the ball and air can be written in terms of density and radius as follows: m = 43πr 3ρ (2) [2][1] where, ρ is the density of the material of the body or fluid. Rewriting Equation (1) using Equation (2), the following equation was deduced: 4 3πr 3ρballg = 6πηrv + 4 3πr 3ρfluidg (3) [2][1] Equation (3) can be rearranged to give the following equation: η = 2r 2g(ρball − ρfluid) 9v (4) [2] The density and radius of the balls used along with the density of the fluid were provided in the data sheet.

The only unknown quantity, terminal velocity v, was measured through experimental methods. Equation (4) has to be corrected as it assumes the conditions for measurement of the velocity is perfect. It assumes that the fluid extends infinitely in all directions, which is unrealistic and does not take into account that the velocity distribution of the fluid particles relative to the surface of the sphere is affected by the finite dimensions of the fluid. Also, if the fluid was infinite, the flow would not be laminar as the falling velocity would be too large. The use of the cylinders, with almost the same radius as the balls used, was so that the falling velocities were reasonable and not too large.

Hence, the experimental velocities are going to be lesser than the theoretical values. The ratio of the radius of both the ball and the cylinder of fluid can help in determining the terminal velocity of the ball. Closer the ratio of rR is to 1, smaller is the velocity of the ball. This is in agreement with the perfect condition of the ball falling in infinite fluid, with r«R being a condition, where, r is the radius of the ball and R is the radius of the cylinder of fluid. This means that Equation (4) has to be corrected, using a correction factor Kp, where Kp.v = ve.Kp as follows: Kp = ( 1− 0.475 dD 1− dD )4 (5) [2]

The corrected version of Equation (4) is as follows: η = 2r 2g(ρball − ρfluid)t 9Kps (6) 2 [2] In this experiment, a ball bearing is to fall through a chosen fluid, covering a distance of 100 mm in time t. Sugar solutions used had varying concentrations from 10% to 60%. This, in turn implies that, density of the solutions increase with increase in sugar concentration in the solutions, resulting in increase in viscosity as per the following equation: η = K(ρball − ρfluid)t (7) [2] where, η is the dynamic viscosity of the solution used and K is the adjustment factor that depends on the ball used for the experiment. This equation takes into account all of the dependencies stated in Equation (4).

In the other parts of the experiment, the fluid was kept constant with the temperature being changed. Vis- cosity of a fluid is dependent on temperature and decreases with increase in temperature. The sugar solutions used were treated as Newtonian solutions. Newtonian fluids are isotropic, i.e, their density and viscosity stay constant throughout their volume and the shear stress applied on the fluid is directly proportional to the shear rate, as can be seen from the following two equations: τ ∝ du dx (8) [3] τ = η du dx (9) [3] where η is the viscosity of the fluid and acts as the proportionality constant between shear stress and shear rate. Viscosity (η) is related to concentration and temperature through the ’Arrhenius Equation’, which is stated below: η = K0e( Ea RT +BC) (10) [3] where, K0 and B are constants, C is concentration of the fluid used, Ea is the activation energy, R is the ideal gas constant and T is the temperature of the solution.

Method

Experimental Setup

The experimental setup of the viscometer was tilted at an angle of 10◦ to the vertical. The tube had three white line markers on it, denoting a starting and finishing point for the ball along with a point in the middle. The starting and finishing markers were separated by a distance of 100 mm. The tube was surrounded by a water jacket which was connected to a reservoir. When the reservoir was turned on, it heated and pumped the water into the water jacket at a constant rate.

This in turn heats up the solution in the viscometer. The temperature of the water could be controlled by a dial attached to the reservoir. The viscometer is tilted at an angle of 10◦ to the vertical to allow the ball to roll down one side of the tube, resulting a more consistent and uniform velocity for the ball to travel, without slowing down significantly. There are 6 balls available, with different radii, density and materials. The balls were supposed to be used depending on the solution used and whether it provided a reasonable time measurement.

For instance, a light and large ball will take a very long time to fall through a highly viscous fluid, compared to a small and heavy ball. When the ball was chosen, the ball is to be loaded into the tube. At first the tube and ball are to be kept clean. Then the fluid is to be poured slowly and uniformly such that there are no air bubbles. The ball is then introduced into the tube. This is followed by the hopper being put on the top of the tube and the cap being screwed on. The ball was allowed to travel twice through the fluid to ensure the consistency of the density of the fluid.

Varying Concentrations

In this part of the experiment, sugar solutions with concentration varying from 10% to 60% along with a sugar solution of unknown concentration were used. At first, the experiments were carried out using a boron silica glass ball of radius 15.81 ± 0.01 mm. However, for the solutions of higher concentrations the boron silica ball was not falling down in time. Hence, the measurements for all the solutions were taken by using a iron nickel alloy ball of radius 15.60 ± 0.01 mm.

Three time measurements were taken, by rotating the viscometer, so that the ball starts again at the top of the tube. Time taken by the ball to cover a distance of 100 mm was noted down and the corresponding viscosity was calculated. This was repeated for all the solutions. 3.3 Varying Temperature In this part of the experiment, a sugar solution of constant concentration (40 % wt. sugar solution) was used. The ball used was the boron silica glass ball of radius 15.81 ± 0.01 mm. At first, the measurement was taken at room temperature (20◦ C), with the temperature being increased by 10◦C from 20◦C to 80◦C. Three time measurements were taken for each temperature, by rotating the viscometer such that the ball starts falling from the top of the tube everytime. The time measurements were noted and the corresponding viscosity was calculated.

Non-Newtonian Solution

In this part of the experiment, shampoo, which is a non-Newtonian solution, was used. The ball used at room temperature was an iron alloy ball of radius 11 ± 1 mm. However, for the rest of the temperatures, a nickel iron alloy ball with radius 15.2 ± 0.1 mm was used. At first, the measurement was taken at room temperature (20◦ C), with the temperature being increased by 10◦C from 20◦C to 70◦C. Three time measurements were taken for each temperature, by rotating the viscometer such that the ball starts falling from the top of the tube everytime. The time measurements were noted and the corresponding viscosity was calculated.

Results and Analysis

Error Calculation

Time

There are 2 errors associated with the time measurement: Human error in pressing the stopwatch button, which is ± 0.28 s and rounding error, which is ± 0.005 s. Hence, σtot = √ 0.0052 + 0.282 + σ2n (11) [3] where σn is the standard error involved in measurement of time, due to the process of measurement involving multiple measurements of time.

Viscosity

When the viscosity is calculated, it is dependent on two factors, density and time, as can be seen in Equation (7). Both these factors have errors associated with them. The error in each measurement changes due to these dependencies. Hence: ση = √( δη δt δt )2 + ( δη δρ δρ )2 = √ (K(∆ρ)δt)2 + (Ktδρ)2 (12)

Sugar Solutions of known concentration

Varying Concentrations

In this experiment, the aim was to find the viscosity of different solutions with varying concentrations. The viscosities were found using Equation (7). It can be deduced that the viscosity of the solutions increases exponentially with increase in concentration/density of the solution. The equation of the fitting curve was y = 0.0059 e0.1447x where y is the viscosity and x is the concentration. Error bars present on the graph are around 1.7% of the recorded values. Hence, they are really small and not very visible.

Most of the data points fall on the fitted curve, with the exception of few outliers. The pres- ence of these anomalous points can be attributed to errors involved in the experimental process, such as, delay in response time when dealing with time measurement, presence of air bubbles which hinder the movement of the ball in the fluid and parallax error when determining the position of the ball.

Varying Temperatures

In this part of the experiment, the relationship between temperature and viscosity for 40 % wt. sugar solution was determined. A boron silica glass ball (diameter = 15.81 ± 0.01 mm) was used for this part of the experiment, whilst varying the temperature from 20◦C to 70◦C. Figure 2: Ln of Viscosity vs Temperature for 40 % wt. sugar solution From Figure 2, it can be noticed that the relationship between natural log of the viscosity of a fluid and 1RT is a linear relationship as shown below: 5 η = K0e( Ea RT +BC) (13) ln(η) = ln(K0) + Ea RT +BC (14) ln(η) = Ea.( 1 RT ) + [ln(K0) +BC] (15) This equation represents an equation of a straight line where Ea is the gradient of the line and ln(K0) + BC is the y-intercept as ln(K0), B, C are all constants.

Hence, the gradient of this line gives us the activation energy. From Figure 2, the gradient of the line is equal to 18644 J or 18.644 KJ, which is the activation energy for the viscous flow of the 40% wt. sugar solution. Error bars present on the graph are around 0.73% of the recorded values. Hence, they are really small and not very visible. The inverse relationship between viscosity and temperature can be understood by delving into the molecular interactions involved in viscous flow. When temperature in increased, the random thermal motion increases. This reduces trans-laminar interaction, al- lowing from smooth motion of one lamina over the other. This reduces the dynamic resistance between two laminae, hence reducing viscosity.[1]

Unknown Solution

In this part of the experiment, the viscosity and concentration of a solution of unknown concentration was determined. At first, the density was calculated by measuring the mass of 10 cm3 equivalent of the solution, which was 10.59 g. Hence, the density of the solution was 1.059 g/cm3. The viscometer was used to get the time taken by a ball to fall 0.1m in the unknown solution. The ball used was a nickel iron alloy ball of diameter 15.60 ± 0.05 mm. The average time measured was 3.75 ± 0.114 s. Using this, the viscosity of the solution was found to be 2.789 ± 0.0855 mPas. Using the fitting curve equation from Figure 1, the concentration was calculated: y = 0.0059e0.1447x =⇒ 2.789 = 0.0059e0.1447x =⇒ x = concentration = ln( 2.7890.0059 ) 0.1447 = 42.56% (16)

Non-Newtonian Solution

It can be determined that for both Newtonian and non-Newtonian fluids, the relationship with temperature is an inverse relationship. However, for the non-Newtonian fluid, a small plateauing was noticed for lower temperatures, whilst for Newtonian fluids there was no plateauing. Error bars present on the graph are around 1.2% of the recorded values. Hence, they are really small and not very visible.

Labscript Questions

As Equation (6) and Equation (7) are both formulas to determine the viscosity (η), they can be equated. Once equated, we get the following equation: K = 2r 2g 9Kps (17) [2] where, r is the radius of the ball used, whose diameter (d) = 11 ± 1 mm , the diameter of the cylinder (D) is roughly 15.83 mm, Kp is the correction factor dependent on d and D, s is the distance covered by the ball in the liquid. Using Equation (5), Kp was determined to be 23.392. All of the values when put into Equation (17), gives us the following calculation: K = 2× (5.5× 10 −3)2 × 9.81 9× 23.392× 0.1 = 28.27mPascm 3g−1s−1 (18) However, error calculations for K were as follows: σK = √( δK δr δr )2 + ( δK δKp δKp )2 + ( δK δs δs ) = √ 2.822 + 0.812 + 0.232 ≈ 2.94 (19) [3] Including the errors, the theoretically calculated value of K for the smallest ball is 28.27± 2.94mPascm3g−1s−1, which is close to the practical value of 33.88 mPascm3g−1s−1 with an accuracy of 92.11%. The inaccuracy can be attributed to incorrect measurement of the physical parameters such as the diameters of the ball and the cylinder.

Conclusion

The purpose of this experiment was to understand the working of a falling ball viscometer and also to study the effect of different parameters like concentration of a fluid and temperature on the viscosity of both Newto- nian and non-Newtonian fluids. It was deduced that the relationship between viscosity and concentration for a Newtonian fluid is a near exponential relationship. The relationship between viscosity and temperature for a Newtonian fluid was found to be exponential as well, while the relationship between the same two parameters were found to be an inverse relationship for a non-Newtonian fluid.

However, for both Newtonian and non- Newtonian fluids, with increasing temperature, viscosity was found to decrease. The concentration of an unknown solution was also found by calculating the density and viscosity of the fluid and utilising the equation for the fitting curve for our data in Figure 1. The concentration was found to be 42.56% The activation energy for the viscous flow of the 40% wt. sugar solution was determined graphically from Figure 2 by finding the slope of the straight line plot of ln(η) vs 1RT .

The value of the activation energy was found to be 18.644 KJ. This experiment had few errors involved with it. The main sources of error were the use of a stopwatch to determine the time and the use of a thermometer to determine the temperature, both of which include human errors. It was also assumed that the densities of the balls used were accurate as the values were provided by the manufacturers directly. The use of digital apparatus to eliminate human error would greatly improve the accuracy of this experiment.