AE3450/Seitzman
Fall 2013
Solution for Problem Set #2: Work, Paths and Gas State Equations
Problem 1. Work and Paths
Given: Fixed mass system going from state 1
to 2, with p2>p1, v2>v1 in 2 paths
A: isobaric then isochoric
B: pv1/2
Find: (a) WA in or out
Due Tue., Sept. 2
Problem Set #1: Units, Significant Figures, Energy and Systems
Always indicate any assumptions you make. If you use any results or equations
from the class notes or text in you solutions, please note and reference them (but you
better b
Due Sep. 11
Problem Set #2: Work, Paths and Gas State Equations
Always indicate any assumptions you make. If you use any results or equations
from the class notes or text in you solutions, please note and reference them (but you
better be sure they are a
AE 3125
Test No.2; Spring 2012
1. For a linear elastic rod (of length L and Uniform cross sectional area) that is
subjected to uniaxial forces, obtain an expression for the strain energy, and strain
energy density.
For an Euler-Bernoulli beam of length L
AE3450/Seitzman
Fall 2013
Solution for Problem Set #5: 1st Law for Control Masses
W12
Problem 1. Adiabatic Compression
Xe
Given: Closed and insulated container with Xe being
compressed with initial and final conditions
shown in figure, (cp/R)Xe = 2.5
Find
Mass Flow Rate
m =rAV =
m=
T0
p
g
T
M gRT A =
RT
R p0
p
(
)
g
M
p0
A
g+
1
R
g - 1 2 2(g- 1) T0
M
1 +
g
p0
MA
T0
(17)
m T0
Ap0
Flow rate out of a tank depends on
tank temperature, pressure and
size of throat.
1
M
Mass Flow Rate
For a Diatomic Gas (e.g.,A
AE3450/Seitzman
Spring 2013
Due Mon., Jan. 14
Problem Set #1: Units, Energy and Systems
Homework solutions should be neat and logically presented, see format requirements
at (www.ae.gatech.edu/people/jseitzma/classes/ae3450/homeworkformat.html).
Always
AE3450/Seitzman
Fall 2013
Solution for Problem Set #4: Mass Conserv.: Control Volume Analysis
Problem 1. Low Speed Wind Tunnel
Given: Wind tunnel with test section
conditions shown in figure.
Test Section
m fan
mtest
Find: a) mass flow rate supplied by fa
AE3450/Seitzman
Fall 2013
Solution for Problem Set #8: Stagnation Properties and Isentropic Flow
Problem 1. Turbine Stagnation Properties
Q0
Given: Adiabatic turbine with conditions shown
Find: a) To1
Reversible?
b)
T1 628K
po1 14atm
Assume: 1) flow is st
AE3450/Seitzman
Spring 2013
Solution for Problem Set #8: Isentropic Flow and Nozzles
Problem 1. Supersonic Nozzle Graphs
Given: Diverging section of axisymmetric (cylindrical) supersonic nozzle flowing gas either
with =1.33 or 1.20
Plot: M, T/To , a/ao ,
AE3450/Seitzman
Fall 2013
Solution for Problem Set #9: Nozzle Flow
Problem 1. Supersonic Exhaust System
Given: Air flow facility shown in picture.
Find: a) To1 with respect to facility
b) A1/A2 required to choke facility if
isentropic
Assume: air is tpg a
AE3450/Seitzman
Spring 2013
Due Fri., Feb. 1
Problem Set #4: State Equations Liquid and Vapor (Water)
Homework solutions should be neat and logically presented, see format requirements
at (www.ae.gatech.edu/people/jseitzma/classes/ae3450/homeworkformat.h
AE3450/Seitzman
Spring 2013
Due Fri., Jan. 25
Problem Set #3: State Equations Ideal Gases
Homework solutions should be neat and logically presented, see format requirements
at (www.ae.gatech.edu/people/jseitzma/classes/ae3450/homeworkformat.html).
Alway
AE3450/Seitzman
Spring 2013
Due Fri., Jan. 18
Problem Set #2: Work, Paths and Enthalpy
Homework solutions should be neat and logically presented, see format requirements
at (www.ae.gatech.edu/people/jseitzma/classes/ae3450/homeworkformat.html).
Always i
AE3450/Seitzman
Spring 2013
Solution for Problem Set #7: Stagnation Properties, Mach Number,
Isentropic Flow
Problem 1. Nozzle Efficiency
Q0
Given: Adiabatic nozzle flowing air as shown
To1 500 K
po1 3bar
Find: Adiabatic nozzle efficiency nozzle
M 2 0.9
p
School of Aerospace Engineering
Energy Conservation for Control Volumes:
Integral Form
Start with Reynolds Transport Theorem (RTT):
r
dB
d
=
dV + CS (v rel n )dA
dt CM dt CV
Energy
1
B = Etot = U + Ekinetic = U + mv 2
2
v2
= etot = u +
uo
2
will put
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Already derived differential
forms of steady conservation eqs.
no body forces
neglect viscous work
Mass
d 1 dv 2 dA
+
+
=0
2
2 v
A
p
x
T
v
A
dx
h q
(VI.9)
p+dp
T+dT
+d
v+dv
A+dA
h+dh
Recall, 1-D valid
only for dA/dx sma
School of Aerospace Engineering
Mach Angle and Mach Number
Looking for relationship between speed of sound and flow
speed (or speed of body moving through fluid)
Consider small body (point) moving in stagnant fluid
continuously launches weak pressure d
AE3450/Seitzman
Fall 2013
(Brief) Solution for Supersonic Inlet Computer Problem
CD Diffuser
M
Design
The design areas for the M=1.75 inlet can be
found from the A/A* value required for isentropic
flow at M and for the required mass flow rate
required at
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Speed of Sound
Consider adiabatic, 1-D propagation of weak
(infinitessimal) pressure wave traveling through
initially stationary (nonmoving), simple
compressible substance
Can think of piston given
small push
p
time t1
F
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Entropy Approach to 2nd Law for CM
Start with existence of entropy as a thermodynamic
property (measure of microscopic disorder of system,
or better, energy states of system)
S=S(U,V) function of two indep. variables
Ima
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Classical Approach to 2nd Law for CM
Start with observations about the ability to build devices
(thermodynamic cycles)
Clausius Statement of 2nd Law
concerns cycles that cause heat transfer from low
temperature body to h
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Isentropic Processes
Isentropic (process or system)
no change in entropy
dS=0 or S=dS=0
2nd Law for control mass
dS= Q/T+ 3s
So two ways to get no entropy change
production (irreversibilities) balanced by cooling
(hea
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Differential Form of Energy Conservation
For Quasi-1D, Steady Flow
Also assume no work but flow work (e.g., no
viscosity), thermally/calorically perfect gas
v2
1
= c p dT + dv 2
q = dh o = dh + d
2
2
q dT
1
=
+
dv 2
School of Aerospace Engineering
Energy CVAnalysis: Example 1
Given: Steady flow in wind tunnel with air intake at
30 m/s, 300 K and exit at 300 m/s
m1
Find: T2
Assume: Adiabatic (insulated)
v1=30 m/s
No work but PV work
T1=300 K
Air is ideal gas
Unifor
School of Aerospace Engineering
2nd Law Development for Closed Systems
At least two approaches for developing
mathematical form of 2nd Law for Closed Systems
(i.e. Control Masses)
1. Entropy Approach: start with existence of
entropy as TD property, find i
School of Aerospace Engineering
Entropic State Eqns. Ideal Gases
du
p
dv
T
T
cv dT R
for ideal gas
dv
T
v
s2
T2 c dT
v2 R
v
integrate
ds
dv
s1
T1
v1 v
T
T2 c dT
v2 dv
s2 s1 v R
T1
v1 v
T
Recall Gibbs eqn.
ds
cv dT
v
R ln 2
T
v1
T2 c dT
s12 s2 s1
School of Aerospace Engineering
Equilibrium Diagrams and
Saturated Liquid/Vapor Systems
In equilibrium, different phases of matter
gas, liquid, solid (also multiple solid phases,
e.g., different crystalline structures of steel)
So far looked at individ
Equilibrium Diagrams
More than one phase (solid, liquid, gas)
may be present in equilibrium.
Consider what happens to a single
substance when it is heated at constant
pressure.
Assume solid contracts upon freezing
(not water).
Drawings by J. Seitzman.