True or False: write out TRUE or FALSE (Do not use T and F, write out entire word)
1._______
𝑃𝑔𝑎𝑔𝑒= 𝑃𝑎𝑡𝑚 − 𝑃𝑎𝑏𝑠𝑜𝑙𝑢𝑡𝑒
2._______
The definition of a Newtonian fluid is 𝜏 = 𝜇 𝑑𝑦
3._______
An intensive property of a system is the property value per unit mass.
4._______
Surface tension is the energetic cost required to create more interface between 2 fluids.
5._______
If a flow is steady state, then pathlines, streaklines, and streamlines are identical.
6._______
A fluid at rest (i.e. not moving) feels a shear stress.
7._______
A net force can exist in a fluid without a pressure gradient
𝑑𝑢
Multiple Choice: Please read questions fully, and then WRITE OUT THE LETTER in the Answer
2._The force on one side of a submerged surface in a uniform fluid is ______?
A. The pressure at the center of pressure times the plate area.
B. The pressure at the centroid of the plate times the plate’s area.
C. The pressure at the centroid of the plate times the plates moment of inertia about the x axis.
D. The pressure at the center of pressure times the plates moment of inertia about the x axis
3.The center of pressure of a submerged plate is ______?
A. The point where the resultant force due to the hydrostatic pressure on the plate acts.
B. The same as the plate’s centroid.
C. The same for all shapes.
D. Always at the top of a plate.
5._The vertical component of a force on a curved surface is equal to______?
A. The weight of the fluid column above the surface.
B. The force on the plane area formed by the projection of the surface onto the vertical axis.
C. The force on the plane area formed by the projection of the surface onto the horizontal axis.
D. A force due to changes in density in a fluid column.
9._Which of the following are the three basic laws of mechanic? Provide answers for all answers that
apply.
A. Conservation of mass.
B. Boundary conditions.
C. Conservation of momentum.
D. State relations.
E. Conservation of energy
F. All of the above
2
ME 3370 Exam 2
10/3/2011
Problem 6
𝑑𝑢
𝑑𝑢
𝑑𝑢
𝑑𝑢 −1 𝑑𝑃 𝜇 𝑑 2 𝑢 𝑑 2 𝑢 𝑑 2 𝑢
+𝑢
+𝑣
+𝑤
=
+ (
+
+
)
𝑑𝑡
𝑑𝑥
𝑑𝑦
𝑑𝑧
𝜌 𝑑𝑥 𝜌 𝑑𝑥 2 𝑑𝑦 2 𝑑𝑧 2
In words, describe the meaning/origin of each term in the Navier Stokes Equations for the x
direction, shown above. Furthermore, describe the key assumptions made to get to this equation.
Answer: The left hand side is the acceleration terms, the first is local acceleration and the other
3 are convective acceleration. These terms come from liear momentum storage and flux in a
differential control volume. The pressure term is a force term for the pressure gradient that
drives flow. The final 3 terms are force terms deriving from viscous stresses for Newtonian
fluids.
Problem 7
What is the Reynolds number? What quantities does it compare? What do changes in the
Reynolds number signify? What is the significance of either Re>>1 or Re<<1 on the navier
stokes equation?
Answer: The reynolds number is a dimensionless number that describes the ratio of inertial to
viscous forces. Changes in the reynolds numbers describe the nature of the flow. At very low
reynolds number the inertial terms in the navier stokes equation can be ignored. At high
reynolds numbers the viscous terms can be ignored.
Problem 8
Looking at a particular flow problem, you assume that the flow is fully developed,
incompressible, and steady state. Please use words (and pictures if you want) to explain what
these mean
Answer: Fully developed flow means that the flow no longer changes in the primary flow
direction (or flow is only in a single direction). Incompressible flow means the density is
constant. Steady state is that the flow does not change with time.
Problem 11
On the provided chart for drag coefficient mark where the
transitions for creeping, laminar and turbulent flow are. If the
flow is turbulent what will CD be?
Answer: Creeping flow is the linear section, laminar is the
first plateau, turbulent is the second and it has a drag
coefficeient of 1.2
3
ME 3370 Exam 2
10/3/2011
Above is pictures a monometer system. All pressures are given in absolute pressure, and all the
geometries and fluid properties are known.
A) Find P1,GAGE in terms of the known geometries and fluid properties..
B) Assuming the pipes are circular, find the force due to hydrostatic pressure at the end of
the pipe in terms of the known fluid properties, geometries and pressures. NOTE: You
may use P1 as a known variable, i.e. you do not need to use results from part A.
C) Where will the force found in part B act? What is the relation of this position in
comparison to the centroid circle? Why is this the position of the force, i.e. what causes
this force to act where it does?
4
ME 3370 Exam 2
10/3/2011
Problem 5 (20 points)
Above is pictured a large tank that is ejecting fluid from a nozzle into a free jet of fluid that
strikes a vane. The vane is attached to 2 springs that resist the motion of the vane due to the
striking fluid, and the vane’s orientation remains the same (i.e. there is no angular deflection) At
point 2 the free jet has a diameter of D2 and at point 3 the free jet has a diameter of D3. The
flow is steady state and the free jet enters and exits the vane with 1D flux. The heights of the
tank and the vane are known as are density  and viscosity .
a) Find the velocity of the fluid at point 2.
b) Find the deflection of both springs due to the fluid in terms of known parameters H,h, D2,
D3, kx,, ky, g, and .
5
ME 3370 Exam 2
10/3/2011
Problem 2
The tank above is being filled with a liquid from a circular jet with a velocity
𝒓
profile 𝑽(𝒓) = 𝑽𝒊𝒏 (𝟏 − ). The outlet is at the bottom of the tank at an angle, .
𝑹𝟏
The outlet has a circular cross section with a radius of R2. The flux at the outlet is
1D with a velocity Vout. The tank has an area, AT.
A) Find the mass flow rate into the tank due to the circular jet flowing from the
top. NOTE: ∫ 𝒅𝒂 = 𝟐𝝅 ∫ 𝒓 𝒅𝒓.
B) Assuming h1 is not constant find dh1/dt. Use conservation of mass and
show your control volume.
C) Find the sum of forces in the X and Y directions. Use conservation of
momentum and show your control volume.
6
ME 3370 Exam 2
10/3/2011
Problem 3
To the right is a picture of a pipe system which is
held in place by two reaction forces, Rx and Ry. A
liquid enters at an inlet with a known flow rate, Qin.
It exits through two outlets; the conditions between
the inlet and the two outlets are symmetric (i.e. the
flow rates at the outlets are identical). The inlet and
the outlets are at atmospheric pressure, and you may
assume steady state and 1D flux at the inlets/outlets.
1) Draw and appropriate control volume for the analysis of conservation of mass and
conservation of linear momentum?
2) Find Qout in terms of Qin?
3) Find Rx.
4) Find Ry.
7
ME 3370 Exam 2
10/3/2011
Problem 5
Above is a picture of a cylinder that is suspended in an air flow. Due to the cylinder, the velocity
profile changes, which we have estimated to fit a linear profile. ASSUMING no flow leaves
from the top or the bottom of the drawn control volume and U0 is known as well as all fluid
properties, answer the following questions.
A) Find an equation for UM in terms of the incoming velocity, U0, using the conservation
of mass and the above control volume.
B) The momentum of the fluid is reduced due to the drag force on the sphere. Find the
drag force on the sphere using conservation of linear momentum and the control
volume drawn above. Please have the answer in terms of known geometry and U0.
C) Using the drag force found above, find and equation for the drag coefficient on the
cylinder. Be clear what area you are using for the area in drag coefficient.
D) Based on your answer to C, do you think the proposed control volume and linear
velocity profile accurately represent drag on a cylinder? Why?
E) If this were a real cylinder in flow, explain how the boundary layer would change when
going from creeping flow, to laminar flow, to turbulent flow. Use pictures and words.
How do these flow changes affect drag?
8
ME 3370 Exam 2
10/3/2011
Long Answer 2 (20 points)
Above is diagramed a pipe system that ends with a nozzle in which flow is being driven by a
large mass, M, which is perfectly fitted into a tank (i.e. no liquid can escape through the gap
between the mass and the tank walls). The tank has a cross sectional area At, and all pipes have
circular cross sections. Assume all fluid properties are known.
A) Assume the mass M is in force equilibrium; find an expression for the pressure at point 1
in terms of known quantities.(Hint: Perform a force balance on the mass)
B) Identify and label locations where there would be minor frictional losses in the diagram
above. Please provide a variable for each minor loss coefficient you identify.
C) Assume the loss coefficients you identify in part B are known. Using your results from
part A and B find an equation that describes the flow rate through the pipe system using
known variables. Do not neglect frictional losses.
D) Based on the equation you have solved for in part C, describe the process you would use
to solve for the flow rate.
9
ME 3370 Exam 2
10/3/2011
Above is a picture of a rounded rectangle with depth into the board b, a height h, and length 2L
that is suspended in an air flow. Due to the cylinder, the velocity profile changes in the wake,
which we have estimated to fit a linear, profile that foes from U0 to 0 at the center. ASSUMING
U0 is known as well as all fluid properties and the geometry of the rounded rectangle, answer the
following questions.
F) Find an equation for H in terms of the incoming velocity, U0, using the conservation of
mass.
G) The momentum of the fluid is reduced due to the drag force. Find the drag force on the
sphere using conservation of linear momentum and the control volume drawn above.
Please have the answer in terms of known geometry and U0.
H) Using the drag force found above, find and equation for the drag coefficient on the
cylinder based on the frontal area of the rectangle.
I) Based on your answer to C, do you think the proposed control volume and linear
velocity profile accurately represent drag? Why?
10
ME 3370 Exam 2
10/3/2011
Problem 2 (20 Points)
The picture above shows a pipe that has a depth into the page of b. The flow at the pipe inlet enters with a uniform
velocity of U0 and exits with the velocity profile shown that goes from 0 at each wall to Umax at the center of the
pipe. Note all velocities are completely in the X direction, and there is not y flow in the system. Assume the density
of the fluid, , is known and the fluid is incompressible. The flow is steady state
a)
Using the control volume shown (dotted line) find an equation for Umax in terms of U0 and known geometry
and fluid parameters.
b) Using the control volume shown, estimate the force due to shear stress acting on the fluid.
11
ME 3370 Exam 2
10/3/2011
Problem 3
A large balloon is tethered to the ground by a chain.
The balloon has a small nozzle (diameter, d) at its
bottom letting air escape. The angle of the escaping
air and the chain are both Assuming that the center
of the balloon is like a large tank with a pressure, P1,
and the air in the balloon is incompressible with a
density of , answer the following questions.
A) Assuming that the situation is quasi-steady
state (i.e. we know in reality that the balloon
pressure and volume change, but those
changes are slow so we can pretend the
situation is quasi-steady state), use the
Bernoulli equation to find the magnitude of
the outlet velocity in terms of known
pressures, fluid properties and gravity.
B) Using the balloon as your control volume, find an equation for the change of the balloons volume (i.e. use
conservation of mass). Your answer may be in terms of the known geometry, pressures, fluid properties
and Vout. No longer assume the situation is steady state, but the fluid is still incompressible.
C) Use your control volume and conservation of linear momentum to find the tension in the chain holding the
balloon in place. Hint: Even though the problem is not steady state, note that the balloon itself (i.e. your
control volume) has no velocity. Consider how that effects the first integral in the conservation of linear
momentum equation.
12
ME 3370 Exam 2
10/3/2011
Problem 1 (20 points)
A Hydraulic jump is a phenomenon when a open
surface flow suddenly and discontinuously
increases its height, shown in the Figure to the
right. Assume that the hydraulic jump has a
depth into the board, b, and all fluid properties
are known.
a) What variables do you believe are important to this phenomenon? i.e. What geometry, flow,
and fluid properties affect the hydraulic jump height?
b) From your variables above, how many non-dimensional parameters can you find? Please use
the Buckingham PI theorem to find 2 such parameters.
c) Using the control volume in the figure, use conservation of mass to determine the relationship
between the outlet velocity and the inlet velocity.
d) Using the control volume in the figure and results from c), solve for the height of the flow after
the jump. NOTE: This equation should end up as a quadratic equation (𝑎𝑥 2 + 𝑏𝑥 + 𝑐 = 0).
Assume you know h1 and Q, and solve for h2 using the quadratic formula ( 𝑥 =
find the equation for h2.
−𝑏∓√𝑏2 −4𝑎𝑐
)to
2𝑎
13
ME 3370 Exam 2
10/3/2011
Pictured above is a basic schematic of a fire house. A hose is attached to a large
pressurized tank (labeled 0) at one end and a nozzle at the other. The hose has a
known radius of R1 and the nozzle has a known radius at its end of R2. Assuming
all geometry and fluid properties are known, the fluid is incompressible,
frictional losses are negligible, the pressure (P0) in the tank is known, and the
problem is steady state, answer the following questions.
A) Find the average velocities at points 1 and 2 (V1 and V2) in terms of
known variables. NOTE: Finding V1 in terms of V2, or vice versa, is only
acceptable if one of those velocities is solved for in terms of other known
geometries and pressures.
B) Assuming V1 and V2 are now known, solve for the pressure at point 1 in
terms of known velocities, geometry and fluid properties.
C) The velocities solved for in part A are the average velocities. Assume the
actual velocities at 1 and 2 follow this equation:
𝑟
𝑉(𝑟) = 2𝑉𝑎𝑣𝑒𝑟𝑎𝑔𝑒 ∗ (1 − )
𝑅
Using the above equation, find the force needed to keep the nozzle from
moving due to the change in momentum of the fluid. Use the nozzle as a
control volume. You do not have to substitute in results for quantities
solved for in part a or b (i.e. using V1 is fine).
NOTE: ∫? 𝒅𝒂 = 𝟐𝝅 ∫? 𝒓 𝒅𝒓.
14
ME 3370 Exam 2
10/3/2011
Pictured to the left is a basic
schematic of an inflatable air
dancer advertiser. A pump with
a total energy grade line of hp
(labeled a) is attached to a
blower (labeled B) that shoot air
into the air dancer which is
ejected through a nozzle at its
top (labeled C). The blower has
a known radius of Rb and the
nozzle has a known radius at its
end of Rc. Assuming all
geometry and fluid properties
are known, the fluid is
incompressible, frictional losses are negligible, the total energy of the pump (hp)
is known, and the problem is steady state, answer the following questions.
A) Find the average velocities at points b and c (Vb and Vc) in terms of
known variables. NOTE: Finding Vb in terms of Vc, or vice versa, is only
acceptable if one of those velocities is solved for in terms of other known
geometries, pressures, or total head.
B) Assuming Vc and Vb are now known, solve for the pressure at point b in
terms of known velocities, geometry and fluid properties.
C) The velocities solved for in part A are the average velocities. Assume the
actual velocities at b and c follow this equation:
𝑟
𝑉(𝑟) = 2𝑉𝑎𝑣𝑒𝑟𝑎𝑔𝑒 ∗ (1 − )
𝑅
Using the above equation, find the net force to keep the air dancer in
equilibrium due to the change in momentum of the fluid. Use the air
dancer as a control volume. You do not have to substitute in results for
quantities solved for in part a or b (i.e. using Va is fine).
NOTE: ∫? 𝒅𝒂 = 𝟐𝝅 ∫? 𝒓 𝒅𝒓.
15
ME 3370 Exam 2
10/3/2011
Problem 3
A large tank with a nozzle at its end is shooting water onto
a vane with a right angle. The free jet (dotted lines in the
figure, hits the vain with a diameter D1 turns 90 degrees,
and exits with a diameter D2. Assume fluid is
incompressible, and all properties are known. Answer the
following questions..
A) Use the Bernoulli equation to find the magnitude
of the jet’s velocity right before it hits the vane.
B) Find an equation for the velocity of the jet as it
leaves the vane after it right turn when its diameter
has changed (use conservation of mass). Please
mark your control volume.
C) Use your control volume and conservation of
linear momentum to find the reaction forces on the
post that holds the vane in place.
16
ME 3370 Exam 2
10/3/2011
Problem
Below are several pictures showing a box with multiple inlets and outlets. Assume the box sits
on a frictionless surface. Using the pictures, your knowledge of mass and linear momentum
conservation and Reynolds Transport theorem, identify what direction and the magnitude of
forces needed to keep each box from sliding due to the flow traveling through it. Assume the
inlet velocities and areas are known. Show all your work/justify your answer..
17
ME 3370 Exam 2
10/3/2011
Problem
Below are several pictures showing a box with multiple inlets and outlets. Assume the box sits
on a frictionless surface. Using the pictures, your knowledge of mass and linear momentum
conservation and Reynolds Transport theorem, identify what direction and magnitude of forces
needed to keep each box from sliding due to the flow traveling through it.
18
ME 3370 Exam 2
10/3/2011
Problem
To the right is pictured a square pipe with a linear
velocity profile at its inlet. Find the linear momentum flux
⃗ ∙ 𝑛⃗)𝑑𝐴 ) at its inlet. Assume the velocity is
( ∫𝐶𝑆 𝜌𝑉(𝑉
perpendicular to the inlet.
19
ME 3370 Exam 2
10/3/2011
Problem
To the right is pictured a
square pipe with a linear
velocity profile at its outlet.
Find the linear momentum flux
⃗ ∙ 𝑛⃗)𝑑𝐴 ) at its inlet.
( ∫𝐶𝑆 𝜌𝑉(𝑉
Assume the velocity is
perpendicular to the inlet.
20
ME 3370 Exam 2
10/3/2011
The pipe shown to the right has a threaded 90
degree elbow and is in a plane such that the
inlet and outlet have a z=0. Assume 1D flux
with a velocity V= 1m/s, density  = 1000
kg/m3, viscosity  = 1 Pa s, diameter D = 0.1
m, total pipe length L =1m, and gravity g is 10
m/s2. Calculate the friction factor, f, and Pinlet–
Poutlet. You final answer may be in the form of a
fraction, but please simplify as much as
possible.
Inlet
90 Degree Regular
Screwed Elbow
Outlet
21
ME 3370 Exam 2
10/3/2011
Above is a picture of a laminar fountain. This fountains use laminar flow to create smooth jets
of fluid that shoot out long distances. The most famous are those in front of the Bellagio in Las
Vegas. Assuming you know fluid properties and the geometry of the system, please answer the
following questions. The fountain base is in a large tank.
A. The nozzle of a laminar fountain is key to making sure the jet coming out stays laminar.
In our diagram the nozzle is section 1 with L1 and D1. Assuming the velocity of the fluid
is known (V1), find an equation that relates D1 to V1. HINT: The key is laminar.
B. On the diagram above, label any minor losses you see in the system. Label them by
circling them, and giving them a minor loss coefficient (K1, K2,….)
𝑃
𝑉2
C. For a known flow rate, Q, find the Energy Grade Line (𝜌𝑔 + 2𝑔 + 𝑧) for the point at the
top of the nozzle/base of the jet in terms of known geometries and fluid properties.
Include frictional losses and minor losses in your answer. Assume the pump follows the
following equation, ℎ𝑝 = ℎ0 − 𝑄 2 .
D. For part C, did you need to iterate? Why or Why not. Please explain Iteration.
E. Using the energy grade line found in part C, ding the max height of the jet.
22
ME 3370 Exam 2
10/3/2011
Long Answer 2 (20 points)
Above is shown a close up of the laminar fountain base from problem 1, which is actually
floating in the large tank. Assume the tank has a height H and a cross sectional area A. Assume
outlets 2 and 3 have a radius, R2,, and that the inlet has a radius R1. Assume there is 1D flux at
𝑟
outlets 2 and 3, but at the inlet the flow has the following velocity profile: 𝑉(𝑟) = 𝑉𝑚𝑎𝑥 − 𝑅 ,
1
where Vmax is known. Hint: ∫ 𝒅𝒂 = 𝟐𝝅 ∫ 𝒓 𝒅𝒓.
Answer the following questions
A. It was quickly realized that the original design had a flaw…as the jet exited the floating
base, the base would move in the X direction. In order to alleviate that problem, a
second outlet (labeled 3) was added to put the fountain into a force equilibrium in the X
direction (∑ 𝐹𝑥 = 0). Using conservation of mass and momentum, find Q2 and Q3.
B. If the jet is off, find Hsub assuming that the base is solid with density b.
C. If the jet is on, find Hsub. Compare to your answer in B.
23
ME 3370 Exam 2
10/3/2011
Problem 2 (25 points)
Above is pictures a rain storage tanks that is used to drive water through a hose and
out a nozzle to water grass and such. The hose has a total length Lh and a diameter
Dh. Between the hoze and the nozzle there is a valve that can be closed to stop the
flow. The nozzle has a single 90 degree bend and has a total length LN and a
diameter DN. Please answer the following questions, assume all fluid properties
are known.
A. Identify any minor losses in the system and provide them a label.
B. Assuming that the geometry is known, find an equation for the velocity at
the nozzle using the energy equation. Your answer may be in terms of
known pressures, pipe geometry, fluid properties, friction coefficients, and
loss coefficients found in part A. You may use the velocity you deem
appropriate for each of the loss coefficients you found.
C. Explain the process you would use to solve the equation found in B to
supply an actual value of the velocity.
D. For a pipe with a relative roughness of 0.002 find the value of the friction
factor, f, at a Reynolds number of 106 using the Moody diagram on page 9.
24
ME 3370 Exam 2
10/3/2011
Problem 2 (16 points)
Above is picture a large reservoir which has water pumped out of it through a series of pipes to a
higher elevation. There are 3 sections of pipe each with their own diameter(d1, d2, d3) and length
(l1, l2, l3), which are known. The radio of d2/d1 is 0.5. The pump is well characterized and has
the following characteristic equation, ℎ𝑝 = ℎ0 − 𝑄 2 . Assuming the geometry is well known,
answer the following questions.
A. Identify any minor losses in the system. Using the tables provided, find the appropriate
loss coefficient if possible. (HINT: there is 1 minor loss you need to consider).
B. Assuming that the geometry is known, find an equation for the flow rate through this
system using the energy equation.
C. Explain the process you would use to solve the equation found in B to supply an actual
value for the flow rate.
D. Assuming the net positive suction head (NPSH) for this pump is known, calculate the
NPSH in terms of the frictional losses, height changes and inlet conditions at the top of
the vat. Assume the vapor pressure of the liquid is known.
25
ME 3370 Exam 2
10/3/2011
Problem 2
Shown above is a schematic for a system used to store solar energy generated by solar panels so
the energy is available when the sun is not shining. There are 2 large tanks opened to the
atmosphere and connected by a series of pipes. When excess energy is generated with a power,
Ps, it is used to run a pump that moves water from tank1 to tank2 (valve 2 is opened and valves 1
and 3 are closed). When the sun is not shining, water is moved from tank 2 to tank1 through a
turbine to generate power, PT (valve 2 is closed and valve 1 and 3 are opened). All pipes in the
system are round and have a diameter, D.
A) Assuming all the power from the solar generator makes it to the pump (i.e. Ps=bhp) and
the pump efficiency at a flow rate, QP, is p find the head added by the pump, hp, in terms
of Ps , QP, p,  and g.
B) Assuming the total length of the pipe from tank 1 to 2 is Lp, use the energy equation to
find ZR in terms of known variables for the situation when tank 2 is being filled. Use the
equation found for hp in part A. You may assume that QP, , g and D are known. Since
geometry, flow rate, and fluid properties are known, you may use fp for the friction
factor. Ignore all minor losses.
C) Assuming the total length of the pipe from tank 2 to 1 is LT, use the energy equation to
find ht in terms of known variables for the situation when tank 2 is being drained. Use
the equation found for ZR in part B. You may assume that Qt, , g and D are known.
Since geometry, flow rate, and fluid properties are known, you may use ft for the friction
factor. Ignore all minor losses.
D) Assuming the turbine is completely efficient (i.e. all the water energy of the turbine is
turned into PT) find an overall efficiency for the system, = PT/ PS.
26
ME 3370 Exam 2
10/3/2011
Problem 2 (20 Points)
A system to take fresh water from a lake to a water
tower for later usage is shown above. The pipe has
total length is L and diameter D, starts submerged in
the lake, exits into the water tower, and has 2 90°
bends.
A) Using the dimensional performance curve
(to the right) and the diagram, find an
equation for the system head and show how
it can be used to find possible H and Q.
Assume all geometry and fluid properties
are known, and provide a result for both
laminar and turbulent flows.
B) You are unhappy with the H&Q from
part A. Luckily, this family of pumps
has more sizes, and a nondimensional
performance curve has been provided
to the right. All of these pumps operate
at 20 revolutions/second, and you will
use a flow rate of 0.1 m3/s. Mark the
best efficiency point on the graph and
use it to find an efficiency, bhp,
diameter, and head that will satisfy
your needs.
C) The net positive suction head has been
provided. Find an equation to
determine whether a pump for this
system will satisfy the requirements of
the net positive suction head at a
particular Q using only known variables
27
ME 3370 Exam 2
10/3/2011
Problem 5 (20 points)
To the left is a figure with the basic layout for a Dam
that is used to generate electricity. Water enters from
point 1 flows through some pipes past a turbine and
out at point 2. The layout to the left has a pipe with
Diameter, D1, that has a total length, L1, and 2 90
Degree turns with loss coeffcient, k90. The pipe
expants at the outlet to a Diameter, D2, that has a
total length, L2. The turbine draws power from the
flow with the following characteristic equation:ℎ𝑡 =
𝑄2
,
𝐶
where c is a constant.
a) Write out the energy equation for this Pipe system and explain what terms can be ignored and
why. Note: Ignore losses due to expansions and contractions.
b) The designers would like to know how flow rate affects the performance of the design, from the
equation in part (a), find an equation for flow rate.
c) Describe how you could calculate the flow rate? i.e. what process would you use and how would
it work.
d) If V1 =10 m/s, D1 =1m,1000kg/m3, and =1 Pa s and pipe had a roughness of D1=0.01 find
the friction factor. Find the value of V1 to have laminar flow.
e) What are the differences between the Bernoulli equation and the energy equation?
28
ME 3370 Exam 2
10/3/2011
The picture above shows a 2 reservoir system. A smaller reservoir (1) can be used to add
chemicals to the larger reservoir (2)p. The chemicals can be pumped either to 1 location or to 2
locations. Answer the following questions. Assume all fluid properties are known.
For all answers bellow, you do not need to substitute everything into a large final equation. Just
solve for each individual component you need and show the final equation you would sub
everything into.
a) Identify minor losses on the diagram by circling them and label them each with a minor loss
coefficient (i.e. K1 , K2 , etc…)
b) Assuming the valve is closed the system is designed to work at a known flow rate of Q0.
Assuming the pump has an efficiency of pump , find an equation for the brake horse power (bhp)
needed to drive the pump. You may ignore minor losses. HINT: Pw = ghpQ.
c) You know the net positive suction head (NPSH) of your pump. Assume the pump is at the
height of the first reservoir, the valve is closed, and the flow rate is still Q0. Find an equation to
determine whether your pump will operate in the given setup. Include minor losses by using the
values you find in part (a).
𝑃
𝑉2
𝑃
𝑖
Hint: 𝑁𝑃𝑆𝐻 < 𝜌𝑔𝑖 + 2𝑔
− 𝜌𝑔𝑉
(d) With the valve closed the system does not mix efficiently enough. The valve will be opened,
and the total flow rate will be kept at Q0. Assume the pump follows this equation: ℎ𝑝 =
Find equations for the flow rates Q4 and Q5. You may neglect minor losses.
(e) Describe the process you would use to find the answer to part d.
Problem 3
𝑄2
𝐶
.
29
ME 3370 Exam 2
10/3/2011
Above is pictured an orifice flow meter that can be uses the pressure drop caused by flow
accelerating through the orifice to measure flow rate. All geometry parameters are known.
Answer the following questions.
A. Assuming all fluid properties are known and the system is frictionless and has no minor
losses, find an equation for flow rate through the pipe in terms of labeled geometry and
fluid properties. Do not leave any unknowns in your equation.
B. Assume now that there are both frictional losses and also a minor loss associated with the
orifice. The minor loss has a loss coefficient of K which is associated with the velocity
through the orifice. Find an equation for the flow rate through the pipe now in terms of
labeled geometry, fluid properties, and friction factors.
C. Based on your understanding of the problem, should a flow rate calculated in part A be
bigger or smaller than in part B.
30
ME 3370 Exam 2
10/3/2011
Problem 2(25 Points)
Problem 2 (20 points)
A laminar boundary layer velocity profile is
approximated by a sine wave with
𝑢
𝑈
=
𝑦
𝛿
for
y < and u=U for y > 
a) What are the assumptions that are necessary for the boundary layer theory to be applied to a
flow field?
b) What are the boundary conditions that a boundary layer velocity profile must satisfy? Does this
velocity profile satisfy those conditions?
c) Using the above velocity profile calculate determine the boundary layer thickness as a function
of x.], i.e. = (x)
d) Find an equation for the Drag coefficient of the flat plate in terms of the boundary layer
thickness as a function of x, i.e. Cd= Cd (x)
31
ME 3370 Exam 2
10/3/2011
Problem 2
Above is pictured a flat plate of thickness b into the page with a uniform flow, U0, approaching
it. A boundary layer forms with a thickness . Using a streamline to create a control volume
and the velocity profile at the exit that is provided, use the Reynolds Transport theorem to
answer the following questions. Assume all fluid properties are known.
A. Using conservation of mass and linear momentum find an equation for the drag on the
flat plate (D) in terms of the fluid properties, boundary layer thickness (and the
approaching velocity (U0).
B. Using results from part A and you knowledge of the relationship between drag and shear
stress, find an equation for (x) in terms of fluid properties, the approaching velocity
(U0), and position, x.
The following equations will be helpful:
𝑑𝐵𝑠𝑦𝑠𝑡𝑒𝑚
𝑑
⃗ ∙ 𝑛⃗)𝑑𝐴
= ∫ 𝜌𝛽𝑑∀ + ∫ 𝜌𝛽(𝑉
𝑑𝑡
𝑑𝑡 𝐶𝑉
𝐶𝑆
𝐵 = 𝑚, 𝛽 = 1,
⃗, 𝛽=𝑉
⃗,
𝐵 = 𝑚𝑉
𝜏=𝜇
𝑑𝑢
𝑑𝑦
𝑑𝑚
=0
𝑑𝑡
⃗)
𝑑(𝑚𝑉
= ∑𝐹
𝑑𝑡
𝑑𝐷
= 𝑏𝜏𝑤𝑎𝑙𝑙
𝑑𝑥
32
ME 3370 Exam 2
10/3/2011
Problem 2
The Von Karman analysis uses the following equations and a control volume around a boundary
layer to find an expression for the boundary layer thickness, , over a plate with depth b.
𝛿(𝑥)
𝐷(𝑥) = 𝜌𝑏𝑈02 ∫
𝜏𝑤𝑎𝑙𝑙
𝑑𝑢
=𝜇
𝑑𝑦
0
𝑢(𝑦)
𝑢(𝑦)
(1 −
)
𝑈0
𝑈0
Using the above equations, find an equation for the boundary layer thickness using the following
𝑢(𝑦)
𝑈0
velocity profile
𝑦
= 𝛿 . HINT: Think about how shear stress at the wall and total drag can be related
to each other.
Problem 2
The Von Karman analysis uses the following equations and a control volume around a boundary
layer to find an expression for the boundary layer thickness, , over a plate with depth b.
𝛿(𝑥)
𝐷(𝑥) = 𝜌𝑏𝑈02 ∫
𝜏𝑤𝑎𝑙𝑙
𝑑𝑢
=𝜇
𝑑𝑦
0
𝑢(𝑦)
𝑢(𝑦)
(1 −
)
𝑈0
𝑈0
Using the above equations, find an equation for the boundary layer thickness using the following
velocity profile
𝑢(𝑦)
𝑈0
related to each other.
𝑦 2
= (𝛿 ) . HINT: Think about how shear stress at the wall and total drag can be