Name ____________________________________________________
In Class Worksheet #1 ME 303 Spring 2014
1. Find the primary dimensions of
a) T, torque
b) P, power
c) V2/2, kinetic pressure;  is the density and V is the velocity of the fluid
d
 U  , where d is the derivative with
dt
dt
respect to time,  is density, and U is velocity?
2. What is the primary dimensions of
Name ____________________________________________________
3. Calculate the cost in $ to operate a pump for 1 year. The pump power is 20 hp
and it operates for 20 hrs/day. Electricity costs $0.10 per kWh.
4. The power produced by a pump is P = mgh, where m is the mass flow rate, g is
the gravitational constant, and h is the pump head. Show this equation is
dimensionally homogeneous.
5. A 4 m3 oxygen tank is at 20 °C and 700 kPa. The valve is opened, and some
oxygen is released until the pressure in the tank is 500 kPa. Assume that the
temperature in the tank does not change during this process. What mass of oxygen
was released from the tank?
ME303 Worksheet #2, January 24, 2014 Name __________________________________________
1. Two plates are separated by a gap of 2 mm filled with a fluid. The plates are 250
mm long and 100 mm wide. The bottom plate is fixed and the top plate is moving to
the right with a constant velocity of 7 m/s in response to a force of 8N. What is the
fluid viscosity.
V
250 mm
Step 1. Draw a figure that shows the dimensions, value of the forces, velocity of the
plates etc. Also draw a figure of what is happening in the fluid gap – i.e. the velocity
profile.
Step 2. Write the appropriate equation. Use algebra to get in proper form
Step 3. Identify the known quantities with units.
Step 4. Plug into equation with units.
1
ME303 Worksheet #2, January 24, 2014 Name __________________________________________
2. A water bug is suspended on the surface of a pond by surface tension. The bug
has 6 legs and each leg is in contact with the water over a length of 5 mm. What is
the maximum mass in grams of the bug if it is to avoid sinking?
3. The velocity distribution for water near a wall is u=a(y/b)1/6, where a = 10m/s,
b = 2 mm, and y is the distance from the wall in mm. Find the shear stress in the
water at y = 1mm. Assume the water is 20 degrees C.
2
ME303 Worksheet #2, January 24, 2014 Name __________________________________________
4. A cylinder of weight 15 N and 200 mm long slides downward in a lubricated pipe, as
shown in the figure. The cylinder diameter is 100 mm. The pipe diameter is 100.5 mm. The
lubricant is SAE 20W oil at 10 C. What is the rate of descent of the cylinder?
3
ME303 Worksheet #3, January 31, 2014 Name __________________________________________
1. Determine the gage pressure at the center of Pipe A in psi and in kPa.
L4
L1
L2
L3
First solve the problem in SI system (use meters, Pa etc.)
Step 1. Start at the place where you know the pressure. Here one end is open to the
atmosphere so start there (labeled L1).
Step 2. What is the pressure of the free surface of mercury (i.e. at L1) in contact with the
atmosphere? Express as a gage pressure, call it P1.
P1 =
Step 3. To find pressure P2 at location L2, use the hydrostatic equation P1-P2=-γΔz. As the fluid is
mercury use its γ value. Δz is the difference in heights.
P2 =
Step 4. Pressure at L2 and L3 are same as they lie on the same horizontal plane. So
P3=
Step 5. Now that we know pressure P3 we can find pressureP4 at location L4. Write the hydrostatic
equation P4-P3=-γΔz. This time note that the fluid is water.
P4=
Step 6. Pressure P4 is same as pressure at the center of Pipe A (as they lie on the same horizontal
plane).
1
Step 7. As pressure P1 is expressed in gage pressure, the pressure P4 will be gage and so do the
pressure at the center of Pipe A.
Step 8. Present your answer with appropriate units (kPa, psi etc.)
Step 9. Now follow steps 1-8 to solve in traditional system (psi etc.).
2
2. Determine force P necessary to just start opening the 2m wide rectangular gate.
length of gate?
Step 1. Calculate the resultant hydrostatic force using F= ̅ A. ̅ is the pressure at the depth of the
centroid given by ̅
.
= the height of fluid level measured from the centroid
A = area of the gate.
A=
=
̅
=
F= ̅ A =
Step 2. Now calculate center of pressure (ycp) using
But first we need to calculate ̅ and .̅ Use figure below to find the slant distance ̅ ̅ is measured
from the surface of the liquid to the centroid of the panel (or gate).
3
Note: The gate we have is rectangular
The moment of inertia ̅ is calculated for a rectangular gate about a horizontal axis (out of plane of
the paper) passing through the centroid.
̅
, w- width, l – length
̅
̅=
Step 3. Draw free body diagram of the gate. Show applied force P, hinge forces, resultant hydrostatic
force etc.
Step 4. Write the moment balance equation for the gate about the hinge to find force the P
∑Mhinge=0
4
3. As shown in the figure, the mouse can use the mechanical advantage provided by the hydraulic
machine to lift up the elephant. Assume the mouse has mass of 10 g and the elephant 5000 kg.
Determine the value of D2 so that the mouse can support the elephant. Given D1=20 cm.
5
ME303 Worksheet #4, February 7, 2014 Name ____________________________________________________________
1. For the cylindrical gate shown below, what will the magnitude of the reaction at A when l is 2m?
Neglect the weight of the gate. Width is 1 m.
C
Step 1. Start by applying drawing a free body diagram of the body of fluid ABC (figure above). Show
all forces, dimensions etc.
Step 2. Now find the horizontal component of force FH acting on the side AC using FH = ̅ A. ̅ is the
pressure at the depth of the centroid (of side AC) given by ̅
.
= the height of fluid level measured from the centroid of AC
A = area of the projected gate on vertical plane AC.
A=
=
̅
=
FH = ̅ A =
1
Step 3. Now find the vertical component of force FV acting on the side BC (figure above) using FV= ̅ A.
̅ is the pressure at the depth of the centroid (of side BC) given by ̅
.
= the height of fluid level measured from the centroid of BC.
A = area of the projected gate on horizontal plane BC.
A=
=
̅
=
FV = ̅ A =
Step 3. To find the weight of water in volume ABC, we first need to know the volume of water in
ABC.
Area of section ABC =
VABC = Area of section ABC * Width =
Weight W=γVABC
Step 4. Sum all the vertical forces on the body of fluid ABC
Fv, resultant =
2
Step 5. Now, find the line of action (horizontal force FH) ycp acting on the plane AC using the equation
But first we need to calculate ̅ and .̅ Use figure below to find the slant distance ̅ ̅ is measured
from the surface of the liquid to the centroid of the panel (here it is centroid of AC).
W
̅
, w- width of AC, l – length of AC
̅
̅=
ycp =
Step 6. Now, find the line of action (vertical force FV) xcp acting on the plane BC. For this sum the
moments about point C.
Moment by force FV is MV=
Moment by force FV, resultant is MV,R= FV,Resultant* xcp
3
Moment by weight W is MW= W*xw
Take xw =0.32 m
Upon summing the moments (MW, MV, MV,Resultant) we can find the value of xcp,
xcp =
Step 7. Draw a similar sketch (shown below) to the fluid body ABC.
Step 8. Write the moment balance equation for the gate about the hinge to find reaction force at A RA
Remember we have force at A, resultant force on the gate by the fluid, and hinge force.
Note: assume there is no reaction force at A along the horizontal plane.
∑Mhinge=0
4
2. A cylindrical block of wood 1 m in diameter and 1 m long has a specific weight of 5000 N/m 3.
Will it float in water with the ends horizontal? Take specific weight of water 9810 N/m3
h=?
Water
Step 1. Draw a free body diagram of the wooden block (show all forces, center of gravity, center of
buoyancy etc.)
Step 2. Calculate weight of block and buoyancy force (in terms of diameter, h, fluid properties etc.)
Calculate the weight of the block W= specific weight of body × volume
W=
Calculate the buoyancy force FB= specific weight of water × submerged volume of body
[Hint: express submerged volume in terms of unknown variable h]
FB =
Step 3. Now write the equilibrium equation for the wooden block and solve for unknown variable h
∑Fnet=0
The value of h=
5
Step 4. The geometric center of the wooden block is its
center of gravity G. Find its value?
G
C
Step 5. The center of buoyancy C is geometric center of
the submerged section of the wooden block. Find
its value?
water
Step 6. Now that you have the locations of points C and G, find the distance between them CG
CG =
Step 7. Find the metacentric height GM using the expression
where, V = submerged volume of cylindrical block, Ioo= area moment of inertia about waterline (in
the figure along axis xx)
Submerged volume V=
Area moment of inertia Ioo=
Metacentric head GM =
Step 8. Based on the sign of metacentric head what do you think? The block will
(a) float stable with its ends horizontal
(b) not float stable with its ends horizontal
6
EXTRA CREDIT PROBLEM
3. The velocity of water flow in the nozzle shown is given by the following expression:
V = 2t2/(1-0.5x/L)2
where, V=velocity in m/s, t= time in s, x=distance along the nozzle, and L= length of nozzle=4 m.
When x=0.5L and t=3 s, what is the local acceleration along the centerline? What is the convective
acceleration? Assume quasi-one-dimensional flow prevails.
Express your acceleration in m/s2
Step 1. Find local acceleration al using
substitute the values of t, x, L, and V.
. First perform partial differentiation and later
7
Step 2. Find convective acceleration ac using
substitute the values of t, x, L, and V.
. First perform partial differentiation and later
8
9
ME303 Worksheet #6, March 7, 2014 Name ________________________________________________
1. The water (density ρ=1000 kg/m3) in this jet has a speed V1=60 m/s to the right and is deflected by a cone
that is moving to the left with a speed of 5 m/s. The diameter of the jet is 10 cm. Determine the external
horizontal force F needed to move the cone. Assume negligible friction between the water and the cone (or
vane). Neglect gravity.
Step 1. Start by drawing the cone and fluid. Show the control volume surrounding the cone and fluid. In
your drawing, show all known parameters (i.e. fluid jet velocities, cone speed, flow direction,
dimensions etc).
Step 2. We now need to choose a reference frame. Let us choose a reference frame that is fixed to the
moving cone. So draw X-Y axis on the cone or vane (in step 1). This reference frame makes the
analysis simpler.
Find the inlet jet velocity Vin entering the control volume with respect to our reference frame. Remember
our reference is moving, so find the velocity of the jet with respect to the moving cone or reference frame.
Vin=
Tip 1. Assume that v1=v2=v3. This assumption can be justified with the
Bernoulli equation. In particular, assume inviscid flow and neglect
elevation changes, and the Bernoulli equation can be used to prove that
the velocity of the fluid jet is constant.
1
Step 3. Using Tip 1. Find outlet jet velocity Vout exiting the control volume with respect to our reference
frame.
Vout=
Step 4. Calculate the inlet mass flow rate ̇
̇
Step 5. In the force diagram (below figure), draw the control volume and label all the external forces
acting on the control volume. Remember, you to include the external horizontal force F in the
force diagram.
=
Force diagram
Momentum diagram
Step 6. For the momentum diagram (above figure), you need to know the magnitudes of momentum flow
in and momentum flow out of the control volume. In our problem, we just need to calculate Xdirection momentum flow in and momentum flow out.
X-direction momentum flow in = ̇
X-direction momentum flow out= ̇
=
=
2
Step 7. Now in the momentum diagram (previous page), show the momentum flow in and momentum
flow out of control volume. Remember, momentum flow is a vector --- it has directionality.
Step 8. Let’s apply linear momentum equation by using the force diagram and momentum diagram (see
figures on page 2). Usually we need to write momentum equation for X, Y, and Z directions.
However, in this problem X-direction momentum equation will suffice to find the unknown
external force F.
X-direction momentum equation,
Assume steady flow and write the x-direction momentum equation.
The external horizontal force F is
3
2. Assume that the gage pressure P is the same at sections 1 and 2 in the horizontal bend shown in the figure
(below). The fluid flowing in the bend has density ρ, discharge Q, and velocity V. The cross-sectional area
of the pipe is A. Then the magnitude of the force F (neglecting gravity) required at the flanges to hold the
bend in place will be
a) PA
b) PA + ρQV
c) 2PA + ρQV
d) 2PA + 2 ρQV
Step 1. Start by drawing the pipe bend and show the control volume surrounding the pipe bend. Show all
known parameters (i.e. fluid jet velocities, dimensions, flow direction, section #, etc.).
Step 2. We now need to choose a reference frame. Let us choose a reference frame that is fixed to the
ground. So draw X-Y axis (in step 1).
Step 3. Calculate the inlet mass flow rate ̇
.
̇
Step 4. Calculate the pressure force at sections 1 and 2, F1 or F2= Pressure*Cross-sectional area
F1=
F2=
4
Step 5. In the force diagram (next page), draw the control volume and label all the external forces acting
on the control volume. Remember, you should include the external horizontal force F, Pressure
forces F1, F2 in the force diagram.
=
Force diagram
Momentum diagram
Step 6. For the momentum diagram (above figure), you need to know the magnitudes of momentum flow
in and momentum flow out of the control volume. So, let us calculate the momentum flow in and
momentum flow out.
X-direction momentum flow in = ̇
X-direction momentum flow out= ̇
=
=
Step 7. Now in the momentum diagram (above figure), show the momentum flow in and momentum flow
out of control volume. Remember, momentum flow is a vector --- it has directionality.
Step 8. Let’s apply linear momentum equation by using the force diagram and momentum diagram (see
above figures). Usually we need to write momentum equation for X, Y, and Z directions.
However, in this problem X-direction momentum equation will suffice to find the unknown
external force F.
X-direction momentum equation,
Assume steady flow and write the x-direction momentum equation.
5
6
ME303 Worksheet #7, March 28, 2014 Name _______________________________________________
USE TABLE 8.3 (on the last page of this worksheet) to look at definitions of Re,We, M etc
1. Oil with a kinematic viscosity of 4×10-6 m2/s flows through a smooth pipe 12 cm in diameter at 2.3 m/s.
What velocity should water have at 20°C in a smooth pipe 5 cm in diameter to be dynamically similar?
Step 1. Start by looking at π-groups (Re, M, Fr, We, CF, Cf, Cp, St) and pick the one that is appropriate. Begin
by eliminating each π-group that has no relevance with the problem. For example, ask yourselves, do
we have free surface effects? If yes, then use We #; or else try other π-groups. Please note: For most
fluid problems consider using Re # first. So what other π-groups do you think will be relevant for
flow through a pipe?
HINT: We have, no compressibility effects, no free-surface effects, no gravitation effects, no
oscillation flows in the pipe.
Appropriate π-group is:
Step 2. Now equate the π-group for model and prototype that satisfies dynamic similitude. Substitute the given
values and find the model velocity.
πmodel = πprototype
Model velocity is
1
2. A spherical balloon that is to be used in air at 60°F and atmospheric pressure is tested by towing a 1/10
scale model in a lake. The model is 2 ft in diameter, and a drag of 10 lbf is measured when the model is
being towed in deep water at 5 ft/s. What drag (in pounds force) can be expected for the prototype in air
under dynamically similar conditions? Assume that the water temperature is 60°F.
Step 1. Start by looking at π-groups (Re, M, Fr, We, CF, Cf, Cp, St) and pick the one that is appropriate. Begin
by eliminating each π-group that has no relevance with the problem. For example, ask yourselves, do
we have free surface effects? If yes, then use We #; or else try other π-groups. Please note: For most
fluid problems consider using Re # first. So what other π-groups do you think will be relevant for the
spherical balloon testing?
HINT: We have, no compressibility effects, no free-surface effects, no gravitation effects, no
oscillation flows in the pipe.
Appropriate π-groups are:
Step 2. Now equate the Re# for model and prototype that satisfies dynamic similitude, from where you find
velocity ratio Vp/Vm.
Remodel = Reprototype
The velocity ratio Vp/Vm =
2
Step 3. Now to find the drag force on the prototype, look for a π-group that has force term in it.
The appropriate π-group is:
Step 4. Now equate the π-group (step 3) for model and prototype that satisfies dynamic similitude, from where
you find force on prototype Fp.
πmodel = πprototype
The value of force on prototype Fp is
3. A 60 cm valve is designed for control of flow in a petroleum pipeline. A 1/3 scale model of the full size
valve is to be tested with water in the laboratory. If the prototype flow rate is to be 0.5 m3/s, what flow rate
should be established in the laboratory test for dynamic similitude to be established? Also the pressure
coefficient Cp in the model is found to be 1.07, what will be the corresponding Cp in the full-scale valve?
The relevant fluid properties for the petroleum are S=0.82 and μ=3×10-3 N-s/m2. The viscosity of water is
1×10-3 N-s/m2.
Step 1. Start by looking at π-groups (Re, M, Fr, We, CF, Cf, Cp, St) and pick the one that is appropriate. Begin
by eliminating each π-group that has no relevance with the problem. For example, ask yourselves, do
we have free surface effects? If yes, then use We #; or else try other π-groups. Please note: For most
fluid problems consider using Re # first. So what other π-groups do you think will be relevant for the
petroleum pipe?
HINT: We have, no compressibility effects, no free-surface effects, no gravitation effects, no
oscillation flows in the petroleum pipe.
Appropriate π-group is:
3
Step 2. Now equate the π-group for model and full-scale that satisfies dynamic similitude, from where you find
velocity ratio Vm/Vp.
πmodel = πfull-scale
The velocity ratio Vp/Vm =
Step 3. Now that you know the velocity ratio Vp/Vm,, you can find the volume flow rate ratio Qp/Qm. Use
volume flow rate Q=Velocity ×Cross-sectional area
The volume flow rate ratio Qp/Qm =
Step 4.
4
Using the statement in the box above, what will be the Cp for full-scale?
Cp for full-scale is
5
ME303 Worksheet #8, April 18, 2014 Name _______________________________________________
1. As shown, a round tube (microchannel) of diameter 0.5 mm and length 750 mm is connected to plenum. A
fan produces a negative gage pressure of -1.5 inch H2O in the plenum and draws air (20°C) into the
microchannel. What is the mean velocity of air in the microchannel? Assume that the only head loss is in
the tube.
Step 1. Start by drawing a control volume (below figure) enclosing two points where you know the
information (Point 1) and where you want to find the information (Point 2).
HINT: Point 1 could be near the mouth of the tube (entrance) where the air is at atmospheric pressure with
negligible velocity. Point 2 could be at the exit of the tube where the pressure is known to be -1.5 inch
H20 while the velocity is unknown.
Step 2. Now, plan on applying the energy balance equation to the control volume. For this, choose a reference
line (or datum).
(Energy balance equation)
1
Information at Point 1
Conversion
1 inch water = 249.2 Pa
Gage Pressure, P1 =
In SI units, Gage Pressure, P2 =
Velocity, V1 =
Elevation, z1 =
Ask yourself is there any pump between points 1 and 2?
Pump head, hp =
Information at Point 2
Gage Pressure, P2 =
In SI units, Gage Pressure, P2 =
Velocity, V2 =
Elevation, z2 =
Ask yourself is there any turbine between points 1 and 2?
Turbine head, ht =
The problem states that the head loss hL is only due to the tube. So the head loss hL=hf
Step 3. Let us assume the flow in the tube is laminar, so take α = 2. Now substitute the known values and
simplify the energy equation.
NOTE: Upon simplifying the energy equation, it will end being up
a quadratic equation in V2! Solve the quadratic equation to find
the mean velocity of air in the microchannel, V2.
2
Mean velocity of air in the microchannel V2 =
Step 4. Since we have assumed the flow is laminar in step 2, this assumption has to be verified. So, calculate
the Reynolds number in the microchannel.
3
Re =
So therefore the assumption that the flow is laminar is VALID / INVALID (circle the correct option)
If Re <2000, the flow is laminar.
If Re >3000, the flow is turbulent.
Step 5. If the flow is VALID, then we are finished with the problem. If NOT, we have to go back the Step 2
and solve without making assumptions.
2. The sketch shows a test of an electrostatic air filter. The pressure drop for the filter is 3 inches of water
when the airspeed is 10 m/s. What is the minor loss coefficient (K) for the filter? Assume air properties at
20°C.
Step 1. Start by drawing a control volume (figure on next page) enclosing two points where you know the
information (Point 1) and where you want to find the information (Point 2).
HINT: Point 1 could be near the upstream of air flow where information is known while Point 2 could be
where the information is unknown. For instance, Point 1 could be at A and Point 2 could be at B.
Ensure the control volume you draw will surround the electrostatic filter.
4
Step 2. Now, plan on applying the energy balance equation to the control volume. For this, choose a reference
line (or datum).
(Energy balance equation)
Note: You may not know the absolute/ gage pressures at A and B but all we need is the difference of
pressure at these two points which is given in the problem.
Information at Point 1
Velocity, V1 =
Elevation, z1 =
Ask yourself is there any pump between points 1 and 2?
Pump head, hp =
Information at Point 2
By applying continuity equation between Points 1 and 2 determine the Velocity V2.
A1V1 =A2V2 (where, A is cross-sectional area)
Velocity, V2 =
5
Elevation, z2 =
Ask yourself is there any turbine between Points 1 and 2?
Turbine head, ht =
Now, calculate the pressure difference between Points 1 and 2, ΔP = P1 - P2 =
In SI units, Pressure, ΔP =
Step 3. Now substitute the known values and simplify the energy equation. Take α1= α2.
NOTE: Use SI units.
Upon simplifying the value of hL=
Step 4. Now to determine the value of the minor loss coefficient (K), use the definition of head loss.
We know the value of hL from Step 3. So substitute the values of V, g, and hL to obtain the value of K.
6
The minor loss coefficient (K) is
7
ME303 Worksheet #9, April 25, 2014 Name _______________________________________________
1. How much power is required to move a spherical-shaped submarine of diameter 1.5 m through seawater at
a speed of 10 knots? Assume the submarine is fully submerged. Assume all power is being used to
overcome drag.
Step 1. Start by calculating the Reynolds number.
Re=VD/ν
Unit Conversion:
1 knot= 0.514 m/s
Step 2. Now, use a correlation to estimate the drag coefficient CD of the submarine. As the submarine is
spherical-shaped, use correlation for a sphere. One relevant correlation is given by Clift and Gauvin as
shown below.
Drag coefficient, CD=
Step 3. Now, to estimate drag force we need to first find the reference area (A). In this situation, we take the
projected area as reference area (A). What is the projected area for a sphere?
Projected area, A =
1
Step 4. Using the drag force expression find the force needed to overcome the drag.
Drag force, FD =
, where Vo is the free-stream velocity.
Drag force, FD=
Step 5. Using the power equation, Power= Force × Velocity, estimate the power needed to overcome drag.
Ensure your final answer is in kW.
Power =
2. Determine the lift of a baseball when pitched at a speed of 38 m/s and with a spin rate of 35 rps. Also
determine how much the ball deflects from its original path in time t=0.5 s. Take the circumference and mass
of ball to be 0.23 m and 0.15 kg respectively. Assume the axis of rotation is vertical.
NOTE: This plot valid for
rotating sphere.
Step 1. Start by determining the rotational parameter defined by r𝜔/Vo. Ensure the values of r, 𝜔 (radian/s),
and Vo are in SI units.
Angular velocity, 𝜔 (in radian/s) =
2
Rotational parameter= r𝜔/Vo =
Step 2. For the calculated rotational parameter, determine the value of lift coefficient CL using the chart
(previous page). Studies have shown that the CL for baseball is three times of CL obtained from the
chart!
Coefficient of lift for the baseball, CL=
Step 3. Now calculate the lift force FL given by,
where Vo is the free-stream velocity, and A is the projected area of the ball.
Lift force, FL=
Step 4. Now find the acceleration of the ball due to lift force. [HINT: Use Newton’s second law of motion.]
Acceleration, a
=
Step 5. To estimate deflection, use kinematic equation for an accelerated body, X= ut +0.5at2 [NOTE: Here
we have zero initial velocity in the direction of lift, hence u=0]
Deflection, X=
3