Download Examples - Rose

Systems, Accounting and Modeling
Approach
Examples showing how students
are taught to solve problems.
Examples*
Heat Engine Performance
Monkeys on a Rope
Colliding Train Cars
Grain Conveyor Belt
*Before reading the complete solution to each problem,
you are encouraged to sketch out your own solution to
the problem.
Example - Carnot Cycle Revisited
What is the maximum thermal efficiency of
a steady-state heat engine that receives
energy by heat transfer at a surface
temperature TH and rejects energy by heat
transfer at a surface temperature T ?
Q!H,in
TH
W! net,out
L
TL
Analysis
What’s the system?
What properties should we count?
What is the time interval?
What are the important interactions?
Q!L,out
dEsys 0,SS !
=
Q
net,in
dt
0 = Q!





W!
H,in
- Q!
net,out
L,out
+ W!





= Q!
H,in
- W!
net,in
Closed System
Q!
H,in
net,out
TH
- Q!
L,out
dSsys 0,SS Q! j
=
∑T
dt
j
+ S!gen
Q!
L,out
Q!
Q!
H,in - L,out
0=
T
T
H
L








net,out
TL
j









W!

















+ S!gen
T
L + T S!
Q!
= Q!
L,out
H,in T
L gen
H









T
L + T S!
W!
= Q!
- Q!
net,out
H,in
H,in T
L gen
η=
W!
net,out =
Q!
H,in









H
T
1- L
T
H









T S!gen
- L
Q!
H,in









Closed System
Q!
H,in
TH
η=
W!






























!
T
T
S
net,out = 1 - L - L gen
T
Q!
Q!
H,in














H
T
= 1- L
η
T
max
H














H,in
W!
net,out
TL
Q!
L,out
T S!gen
L
≥O
Q!
H,in
Example - Monkeys on a Rope
Three monkeys A, B, and C with masses of 10, 12,
and 8 kg, respectively, are climbing up and down the
rope suspended from point D. At the instant shown in
the figure, A is descending the rope with an
acceleration of 1.6 m/s2, and C is pulling himself up
with an acceleration of 0.9 m/s2. Monkey B is climbing
up with a constant speed of 0.6 m/s.
D
A
B
Determine the tension T in the rope at D, in newtons.
Analysis
What’s the system?
What properties should we count?
What is the time interval?
What are the important interactions?
C
D
A
B
C
Physical System
T
D
A
WA
B
WB
C
WC
Physical System
Free-body Diagram
"
dPsys
=
dt
"
∑Fext,j
Closed System
T
j
"
"
"
"
"
∑Fext,j = W A +WB +WC +T
WA
"
"
"
"
Psys = mAVA +mBVB +mCVC
WB
j
WC
"
"
"
"
"
"
"
d  m V +m V +m V  = W + W + W + T
A
B
C
dt  A A B B C C 
" 
" 
" 


"
dVA 
 dV 
"
"
"
 dV 
C
B
 = m g + m g + m g + T
 + m 
 + m 
m




B  dt 
A
B
C  dt 
C
dt 







A 







Closed System
"
"
"





"
dVA " 
"
"
 dV

 dV

C
B




T = m
-g  + mB 
-g  + mC 
-g 
dt
 dt


 dt




A 







x
WA
WB
WC







 m 


 m 
m

T = 10 kg -1.6+9.81 2  + 12 kg 0+9.81  2  + 8 kg 0.9+9.81  2 
  s 

 

 
  s 
s 




=
+
+
82.1N
117.7 N
85.6 N




 
 



 

T
T = 285.4 N
Examples - Rail Cars on the Move
A 45-Mg railroad car moving with a velocity of
3 km/h is to be coupled to a 25-Mg car which
is at rest.
Determine
(a) the final velocity of the coupled cars
(b) the average impulsive force acting on
each car if the coupling is completed in 0.3 s.
VA
A
VB = 0
B
Part (a) - Final velocity after coupling
System: Assume an open, moving system.
Initially it contains car B only and finally it
contains both cars.
Property:
x
"
"
"
"
dPsys
! - ∑m
! eVe
= ∑F
+ ∑mV
ext,j
i i
dt
j
in
out
VA
A
VB = 0
B
Conservation of
Linear Momentum
(x-direction)
dPsys,x
!
= mV
i x,i
dt
t
t

2 
dPsys,x

!
dt
=
 mV
∫ dt
∫  i x,i  dt
t
t
1
1
2
Integrate over time
interval t1 to t2.
1 - Initial state
2 - Final state
VB,1 = 0
















P
sys,x,2










m +m V
A
B
-P
sys,x,1
=mV
A A,1
0
AB,2
-mV
B B,1
mV
V
= A A,1
AB,2 m +m
A
B
=m V
A A,1
Part (b) Coupling Force
System: Assume a closed system that only
contains car B throughout the process. This
system moves with car B.
"
0
"
"
"
dPsys
! - ∑m
! eVe
= ∑F
+ ∑mV
ext,j
i i
dt
j
in
out
Property:
x
B
Fcoupling
Conservation of
Linear Momentum
(x-direction)
dPsys,x
= Fx
dt
t
VB,1 = 0
VB,2 = VAB,2
from part (a)
t








2
dPsys,x
∫ dt dt = ∫ Fx dt
t
t
1
1
2
Integrate over time
interval t1 to t2.
1 - Initial state
2 - Final state








P
sys,x,2
mV
B B,2
-P
sys,x,1
= Fx,avg #t
0
- mV
B B,1
= Fx,avg #t
mBVB,2
Fx,avg =
#t
Example - Conveyor Belt
Grain falls from a hopper onto a conveyor belt at the rate
of 200 kg/min. The conveyor belt carries the grain away
at a constant velocity of 2 m/s.
Determine the force on the belt required to keep the belt
moving at a constant speed.
Hopper
Belt
Grain
Vbelt
Hopper
Belt
Grain
Vbelt
Grain entering
1
2
Grain leaving
3
Belt entering
Fbelt
Steady-state open system
Belt leaving
x
1
3
2
Conservation of mass
for this system.
System
Fbelt
dmsys
!
!
!
!
=m
+m
-m
-m
grain,1
belt,2
belt,3
grain,3
dt
msys =m
+m
belt,sys
0, SS
dmbelt,sys
dt
!
!
=m
-m
belt,2
belt,3
!
!
m
=m
belt,2
belt,3
grain,sys
0, SS
dmgrain,sys
dt
!
!
=m
-m
grain,1
grain,3
!
!
m
=m
grain,1
grain,3
Conservation of Linear
Momentum for this
system. ( X-direction)
0, SS
1
2
x
3
System
Fbelt
dPx,sys
!
! eVx,e
= ∑Fx,ext,j + ∑ mV
m
∑
i x,i
dt
j
in
out
!
!
0 = Fbelt + m
V
+m
V
grain,1 x,grain,1
belt,2 x,belt,2
!
!
-m
V
-m
V
belt,3 x,belt,3
grain, x,grain,3








0
0



!
!
0 = Fbelt + m
V
-V
V
 + m
V
grain x,grain,1 x,grain,3 
belt  x,belt,2
x,belt,3 

!
!
Fbelt = m
V
=m
V
grain x,grain,3
grain belt
End of Examples
For additional information about the RH Sophomore
Engineering Curricula or the Systems, Accounting, and
Modeling Approach contact --Don Richards
Rose-Hulman Institute of Technology
5500 Wabash Ave. - CM 160,Terre Haute, IN 47803
Email: [email protected]
URL: http://www.rose-hulman.edu/~richards
Phone: 812-877-8477
Or check the Foundation Coalition Web Site at
http://www.foundationcoalition.org