PART I: BUOYANCY
The first part of this lab refers to buoyancy.
Key Concepts. Specific Weight, Dynamic Viscosity
Introduction: Buoyancy
The transportation and accumulation of sediment in waterways and reservoirs, the movement of
dust and other pollutants in the atmosphere, and the flow of liquids through porous media are
examples of phenomenon in which specific weight and viscosity play important roles.
Consider a sphere with diameter d and specific weight γs, falling at a constant velocity V through
a liquid with viscosity μ, specific weight γl, and density ρ. The forces acting on the sphere are
shown in Figure 1.
Figure 1 Free Body Diagram of Falling Spehere (Fluid Mechanics for Civil Engineers Lab Manual, Ruff, Muccino,
Thompson)
According to Newton’s Second Law (since the sphere is not accelerating):
F = 0
FD + FB – W = 0
3???? + ??
??
3
6
− ??
??
3
6
= 0
Algebraic manipulation yields an expression for μ in terms of γs, γl, d, and V:
9
? =
?
2
(?? − ??)
18?
The above equation is valid for a sphere falling from a wall. The wall effect occurs when the
falling sphere is close to a wall. The wall effect affects the sphere when:
??ℎ??? ???????? (?)
???? ???????? (?)
>
1
3
The observed fall velocity Vo must then be corrected using:
?
??
= 1 +
9?
4?
- (
9?
4?
)
2
The drag force on a sphere may also be calculated by:
?? = ?? ?? ?
?
2
2
Where Ap is the projected area of the sphere and CD is the coefficient of drag.
Objective
Determine the specific weight and the viscosity of water at room temperature. Also determine a
relationship between the coefficient of drag and the Reynolds number.
Anticipated Results
At a minimum, the student should be able to determine
- Determine the experimental buoyancy and drag forces on the spheres
- The specific weight and viscosity of the water
- The relationship between CD and Reynolds number
Material - Water contained in tubes.
- Tweezers
- Thermometer, meter stick, stopwatch.
- At least 3 spheres of varying density and or diameter (with a specific weight greater than
water).
10
Procedure - Record the temperature of the liquids
- Weight each sphere and measure its diameter accurately. Calculate the specific weight of
each sphere. - Measure and record the inside diameter of the tubes.
- Measure and record the vertical fall distance on each tube. Use a scribed line or masking
tape to locate the distance. There should be ample liquid above and below the lines. - Drop a sphere into the liquid record time of descent through the marked distance using
the stopwatch. Record the time. The sphere must be dropped just at the fluid level. - Repeat step 5 for each sphere. Clean each sphere when retrieved.
Results - Plot at log scale the coefficient of drag, CD, of the spheres vs the Reynolds number from
the laboratory data. - Determine the equation of the plot for item 1 and compare it to the expected value.
- Compare your calculated values of viscosity and specific weight with an authoritative
source.
Volume
Total Mass
Mass of liquid
Specific Weight
Temperature
Sphere # Sphere
Diameter
Mass Volume Specific Weight
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LABORATORY #1
PART II: FLOW BETWEEN TANKS
Introduction: Flow between Tanks
The Bernoulli equation states that the sum of the kinetic, potential, and pressure energies of a
fluid particle is constant along a streamline during steady flow.
.
2
2
gz const
p V
- + =
Pressure energy + Kinetic energy + Potential energy = constant
Multiplying the Bernoulli equation by the density gives:
.
2
2
gz const
V
P + + =
Static pressure + Dynamic pressure + Hydrostatic pressure = Total pressure (The Bernoulli
equation states that the total pressure is constant along a streamline).
Dividing each term of the Bernoulli equation by g gives:
.
2
2
z H const
g
V
g
p - + = =
The pressure head + the velocity head + the elevation head = Total head (Another alternative
form of the Bernoulli equation is: the sum of the pressure, velocity, and elevation heads is
constant along a streamline).
Hydraulic Grade Line (HGL) = P/ρg + z. The line that represents the sum of the pressure and the
elevation heads.
Energy Grade Line (EGL) = P/ρg + V2
/2g + z. The line that represents the total head of the fluid.
Dynamic (velocity) head, V
2
/2g, which is the difference between the heights of EGL and HGL.
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The gage pressure of a fluid is zero at locations where the HGL intersects the fluid, and the
pressure is negative (vacuum) in a flow section that lies above the HGL.
Example 1. Water discharge from a large tank
13
Example 2. Siphoning out gasoline from a fuel tank
14
Goals
The goals of this exercise are to a) familiarize the student with application of the energy
equation, b) learn to estimate flow rates, c) compare frictionless with more realistic flow
projections, and d) compare experimental and measured flows.
Methods
The experiment consists of an upper and lower reservoir of water and a tube connecting both
reservoirs. Students should break up into 5 different groups and each group should perform the
experiments separately. If the TA can procure enough equipment two groups may work
simultaneously.
Experiment A. Run the tube from near the bottom of the upper reservoir to near the bottom of
the lower reservoir. Measure tube length, tube inside diameter, and all the elevations. Measure
the flow rate for 15 seconds (change as needed) through the tube three times and average the
result. Since the water level in the reservoirs will change during the experiment, use the average
of the before and after water levels.
Experiment B. Repeat experiment (A) modifying the height difference between containers.
Experiment C. Repeat experiment (B) after moving the tube in the upper reservoir to near the
top of the water
Experiment D. Develop and propose experiment, proceed with the TA authorization. Note that
this experiment must be described in detail in your lab report.
Analysis
Apply the energy equation to all of the experiments, show your work and list assumptions.
Estimate the flow for all of the experiments using the Moody Diagram. Use table 1 to compare
measured and calculated flow rates. From table 1, Δ Z, Volume and time are experimental data
taken the day of the laboratory.
15
Table 1. Experiment Table using experimental data, Energy and Weishbach equation and Moody Chart.
Experime
nt
Re
p
? ?
(ft, m)
Vol.
(L,
m3
)
Tim
e
(sec)
Q (L/sec,
m3
/s)
Average Q
(ft3/sec,
m3
/s)
Measured
V2 (ft/sec,
m/s)
Theoretical
V2 (ft/sec,
m/s) no losses
F (from
Moody)
Friction
losses
(ft, m)
Theoretical V
(ft/sec, m/s)
Theoretical Q
(ft3/sec, m3
/s)
A
1
2
3
4
B
1
2
3
4
C
1
2
3
4
D
1
2
3
4
Sample Solution