Objectives
This lab helps students to understand the drag on objects in a fluid with the same Reynolds number. We select spheres of different masses and diameters to get a wide range for Reynolds number. We also compare the values found in literature to the values calculated through experiment to determine experimental uncertainty. Keeping all in mind our main objective is to design an experiment to reduce uncertainty.

Background and Theory
When a dense particle falls through a fluid of lesser density, the gravitational force F_g causing the particle to fall is opposed by a buoyancy force F_b and a drag force F_d as shown in figure 1. If the particle falls at a constant velocity the forces can be expressed as-
F_g= F_d+F_b………(1)
We know that gravitational force F_g is given by F_g=mg= ρ_s Vg =(1/6)nD^3 ρ_p g………(2)
The buoyance force can be derived as the following with the Archimedes Principle taken onto account
F_b=(1/6)nD^3 ρ_L g………(3)
We can solve for drag force F_d by combining the values of F_(b )& F_g in equation (1), which gives us
F_d=(1/6) nD^3 g(ρ_p- ρ_L )………(4)
The drag force is a function of 4 independent variables F_d=f (D,V,ρ,μ) where V is the velocity and μ is dynamic Viscosity of the fluid. From Buckingham Pi Theorem we get the value of C_d, which is a dimensionless quantity known as drag coefficient. Drag coefficient is a function of Reynolds number and is given by
C_d= F_d/(1/2 ρ_L V^2 π D^2/4)………(5)
The Reynolds number is defined as
Re= (ρ_L VD)/μ………(6) Figure 1
These measurements will be required to construct relationship between Absolute and Relative uncertainties.

Experimental Procedures For this lab we are measuring the Coefficient of drag for three different liquids; water, glycol and glycerin. Firstly we measure the density of each liquid using the hydrometer located in the lab. To do this we put the hydrometer in the test column for glycerin and for water and glycol hydrometers are setup on the lab bench near the sink. Then we determine the viscosity of each liquid. For this first we obtain room temperature from the thermometer in the room. Then we determine the viscosity of each liquid by interpolation from the table below

Table 1

The measurements First we selected sphere of three different materials and different diameters. We don’t want to select spheres of small diameters because it can lead to small drop times, which can increase the uncertainty. Then we measure the diameters of these spheres. Using the digital mass balance we measure the mass of these spheres. Now we make the marks on the test column at a specific distance between these columns we take measurements. For water test column we make three marks and we measure time between these three marks because the viscosity of water is very less which leads tolow coefficient of drag. Now we start the most important part of experiments that is taking the reading. First we select one liquid and drop these spheres with the help of tweezers. Using the stopwatch we measure the time a sphere takes to pass through both these marks. To recover the spheres there are two valves at the bottom of the drop tubes, which allows the sphere to be recovered. They must never be opened at the same time. Close the bottom valve and open the top valve to allow sphere to drop in the space between the valves. Then close the top valve and open the bottom valve to recover the sphere. Repeat this procedure for each test column for 5 times. Now we calculate the terminal velocity. For spheres that drop quickly, the terminal velocity can be verified by comparing the velocities in several regions of tube. A third reference line is available on the water tube for that purpose. Now we rinse the spheres in the sink and dry them using paper towel.

Results The result table

Table 2
Type of Liquid Cd teflon Re teflon Cd Aluminum Re Aluminum Cd Brass Re Brass Glycerin 84.23 0.82 45.18 1.42 18.73 3.69 Glycol 0.85 248.63 0.68 345.34 0.50 644.87 Water 0.44 6061.71 0.41 7668.09 0.20 17304.11

These value found above are the average values of C_(d )& Re. The experimental C_(d ) vs.R_e are manually plotted

Discuss the claim of a universal relationship.
For each data point for C_(d )& Re we plotted, we found that the data point lies close to the trend of the sphere. With this we can say that irrespective of the shape of the object, liquid density and material type the C_(d )& Re data points will be closer to the respective trend line.

Tabulate the uncertainties.
Table 3 and table 4 give the absolute uncertainty and the relative uncertainty respectively for each material type.
Table 3 Abs
Uncertainity in Cd for Teflon Abs
Uncertainity in Re for Teflon Abs
Uncertainity in Cd for Aluminum Abs
Uncertainity in Re for Aluminum Abs
Uncertainity in Cd for Brass Abs
Uncertainity in Re for Brass

Glycerine 1.22 0.01 1.15 0.02 1.36 0.16 1.23 0.01 1.13 0.02 1.51 0.13 1.22 0.01 1.13 0.02 1.46 0.14 1.23 0.01 1.14 0.02 1.48 0.14 1.23 0.01 1.14 0.02 1.42 0.15 Glycol 0.12 20.96 0.15 42.85 0.27 140.80 0.12 22.47 0.16 36.09 0.25 172.25 0.13 19.98 0.15 40.04 0.24 177.59 0.13 19.61 0.16 38.49 0.26 157.60 0.12 20.37 0.15 39.52 0.24 177.59 Water 0.08 1048.84 0.13 1296.43 0.15 9716.52 0.09 774.39 0.13 1272.09 0.19 6148.11 0.09 763.04 0.12 1373.82 0.15 9716.52 0.09 709.89 0.12 1347.27 0.16 8242.89 0.08 848.10 0.13 1296.43 0.15 9184.21

Table 4
Sphere Relative
Uncertainity in Cd for Teflon Relative
Uncertainity in Re for Teflon Relative
Uncertainity in Cd for Aluminum Relative
Uncertainity in Re for Aluminum Relative uncertainity in Cd for Brass Relative
Uncertainity in Re for Brass Glycerine 1.46 0.73 2.46 1.23 8.21 4.11 1.45 0.72 2.50 1.25 7.42 3.71 1.46 0.73 2.49 1.24 7.65 3.82 1.45 0.73 2.48 1.24 7.56 3.78 1.45 0.73 2.47 1.23 7.86 3.93
Average
1.45 0.73 2.48 1.24 7.74 3.87 Glycol 14.75 8.37 23.79 11.89 47.25 23.62 15.27 8.67 21.83 10.92 52.26 26.13 14.40 8.17 22.99 11.50 53.06 26.53 14.27 8.10 22.54 11.27 49.99 24.99 14.54 8.25 22.84 11.42 53.06 26.53
Average
14.64 8.31 22.80 11.40 51.12 25.56 Water 18.62 16.73 30.14 17.15 99.50 49.04 20.53 12.70 29.47 17.10 68.41 41.96 20.53 12.56 33.17 17.07 70.43 58.29 19.37 12.25 31.57 17.24 86.28 46.55 19.41 13.98 29.49 17.34 81.42 51.97
Average
19.69 13.64 30.77 17.18 81.21 49.56

Figure 2 The graph plotted above shows the experimental data of C_(d ) vs.R_e, with the error bar.

We can see from the Table 4 the relative uncertainty for brass in water is very high and reaches up to 80%. This must be because of the high velocity of sphere in water for brass, which leads to high uncertainty in recording correct time. But other then that from the data points we plotted in C_(d ) vs.R_e Graph the data line lies close to the trend line of the sphere.

Achieving an approx. 2%. But for our worst case (Brass in water) we got Average relative uncertainty of 81.21%. δC_d= √(((δC_d)/δt)^2 (δt)^2 )= √(((16F_d t)/(ρ_l πL^2 D^2 ))^2 (P)^2 )
0.02= √(((16×0.019×0.376)/(1000×π×〖0.8〗^2 (7.937/1000)^2 ))^2 (P)^2 )
P=0.019
P= t_(v,A) σ/√N
0.019= 2.571 0.15/√N
N=411(Samples)

Discussion
Formulas used to calculate uncertainity.
First order time (based on velocity)
P= t_(v,A) σ/√N=2.571 0.15s/√5=0.1724

Drag force
C_d= (8F_d )/(ρ_L v^2 πD^2 )= (8F_d)/(ρ_L πD^2 )×t^2/L^2
(δC_d)/δt= (16F_d t)/(ρ_L πL^2 D^2 ) δC_d= √(((δC_d)/δt)^2 (δt)^2 )= √(((16F_d t)/(ρ_l πL^2 D^2 ))^2 (0.1724)^2 )
Reynolds Number
Re= (ρ_L VD)/μ= (ρ_L D)/μ × L/t δRe/δt= (ρ_L DL)/(μt^2 ) δRe= √((δRe/δt)^2 (δt)^2 ) δRe= √(((ρ_L DL)/(μt^2 ))^2 (0.1724)^2 )

Conclusion
In this lab experiment, our experimental data has been determined using five different spheres and three tubes, each containing a different fluid. Using these objects, and other tools such as a hydrometer and scale, we were able to discover the forces that acted on the sphere as it travelled through each tube of fluid. After performing these trials, we were able to find the coefficient of drag and Reynolds number using basic knowledge of physics, and the formulas and relationship provided in the lab.
After examining the data and the uncertainty, there was a large error in the data with respect to our stainless steel sphere. This was most likely caused to the recorded time of the sphere that had a drop time below 0.5 seconds, and the fact that our sphere would hit the tube on its way down at least once or twice. Hence, there was an obstruction, such as frictional force, that was applied to the sphere at that moment, which offset the force balance that was originally designed for this experiment.
References
MAE 338 Lab Manual.