Download Thermodynamics Chapter 4

The First Law of
Thermodynamics
Meeting 6
Section 4-1
So far we’ve studied two
forms of energy transfer
Work Energy (W)
*Equivalent to raising a weight
Heat (Thermal) Energy (Q)
*Caused by a temperature difference
A note about work and heat:

2
W  W12 or W, but not W2  W1
1
and

2
Q  Q12 or Q, but not Q 2  Q1
1
Both Q and W are path dependent!
• Work and Heat are forms of energy transfers
that happens at the boundary of a system.
• As Work and Heat cross the boundary, the
system Energy changes.
• Work and Heat are not stored on the system
but the Energy yes.
The Business of the First Law
• Energy is not destroyed but it is conserved.
• In fact during a thermodynamic process it is
transformed in one type to another.
• The first law expresses a energy balance of
the system.
• The energy fluxes in a system (Work and
Heat) is equal to the change in the system
Energy.
The system energy forms
• Prior to state the First Law, is necessary
to define the system energy forms.
System energy consists of
three components:
E  U  KE  PE
• U is internal energy,
• KE is kinetic energy, and
• PE is potential energy.
NOTE: All values are changes (deltas)
Internal energy…..
Internal energy is the energy a molecule
possesses, mostly as a result of:
All these are forms of kinetic energy. We will
neglect other forms of molecular energy which
exist on the atomic level.
Translation
• Kinetic energy is possessed by a molecule as it
moves through space. It transfers this energy
to other systems by means of collisions in which
its linear momentum changes. Collisions with
such things as thermometers and
thermocouples are the basis for temperature
measurement.
• It is a characteristic of both polyatomic
molecules and atoms.
Vibration
• Molecules (not atoms) also vibrate
along their intermolecular bonds.
The molecule has vibrational
(kinetic) energy in this mode.
Rotation
• Molecules (and atoms) can also
rotate and they possess kinetic
energy in this rotational mode.
They have angular momentum
which can be changed to add or
remove energy.
We will not worry about the microscopic
details of internal energy
Internal energy is a property
of the system.
Often it shows up as a change in
temperature or pressure of the system
… but it can also show up as a change
in composition if it’s a mixture.
The kinetic energy is
given by:
1
2
2
KE  m( Vf  Vi )
2
1
2
2
 m( V2  V1 )
2
The energy change in accelerating a mass of
10 kg from Vi= 0 to Vf = 10 m/s is?
1
KE  m(Vf2  Vi2 )
2



2 
1
m 
N 
 (10 kg  100 2 )  1
 500 N  m
2
 kg  m 
s


2

s 
 J   kJ 
 500 N  m  
  0.5 kJ

 N  m   1,000 J 
Gravity is another force acting
on our system. It shows up in
the potential energy change.
PE  mg(z f  z i )
 mg(z 2  z 1 )
Work can be done by a change in
elevation of the system
NOTE THE CONVERSION
TO GET FROM m2/s2 to kJ/kg
2
m
kJ
1000 2 1
s
kg
REMEMBER IT! YOU
WILL NEED IT.
TEAMPLAY
Let’s say we have a 10 kg mass that we
drop 100 m. We also have a device
that will convert all the potential energy
into kinetic energy of an object. If the
object’s mass is 1 kg and it is initially
at rest, what would be it’s final velocity
from absorbing the potential from a 100
m drop? Assume the object travels
horizontally.
Conservation of Energy
E = U + PE + KE = Q
Change in
total energy
in system
during t
Changes during t
in the amount of the
various forms that
the energy of the
system can take
 W
Net amount Net amount
of energy
of energy
transferred transferred
to system as out of system
heat.
as work.
Some comments about this
statement of the first law
Q – W = ΔE
• All terms on the left hand side are forms of
energy that cross the boundary of the system
• Q in is positive, W out is positive
• Right hand side is a change in system energy
• Algebraic form of first law
The right hand side of the energy
equation consists of three terms:
ΔE = ΔU + ΔKE + ΔPE
• ΔKE - Motion of the system as a whole with
respect to some fixed reference frame.
• ΔPE - Position change of the system as a
whole in the earth’s gravity field.
• ΔU - Internal energy of the molecule-translation, rotation, vibration, [and energy
stored in electronic orbital states, nuclear
spin, and others].
We previously had
conservation of energy
• E = U + PE + KE
• We can change the total energy E of a
system by
• Changing the internal energy, perhaps
best exemplified by heating.
• Changing the PE by raising or lowering.
• Changing the KE by accelerating or
decelerating.
Conservation of Energy
for Stationary System
• Stationary means not moving so PE and KE are zero and
the first law becomes
Q  W  U
First Law Forms
for Stationary Systems
• Differential Form:
Q  W  dU
• Rate Form:
dU


QW 
dt
• Integrated Form:
1 Q2  1 W2  U 2  U1
We can also write the first law
in differential terms
dE = Q – W, and
dU + dPE + dKE =
Change in the
amount of energy
of the system
during some time
interval.
Q
–
Differential
amount of
energy transferred in (+)
or out (-) by
heat transfer.
W
Differential
amount of energy
transferred out
(+) or in (-) by
work interaction.
If we are analyzing a transient
process, we’ll need the rate
form of the first law
dE dU dKE dPE  



 Q W
dt
dt
dt
dt
Where:
Q
Q
dt

W
W
dt

Rate form will allow us
to calculate:
•
•
•
•
Changes in temperature with time
Changes in pressure with time
Changes in speed with time
Changes in altitude with time
Hints to set up a problem
1. Define the system carefully indicating clearly
its boundaries.
2. Enroll all the simplifying hypothesis to the
case.
3. Draw the heat and work fluxes at the
boundaries including their signals.
4. Sketch a process representation on a
thermodynamic diagram Pv or Tv.
System Energy Change
ΔE = Q  W
= (Qin  Qout)
 (Wout  Win)
ΔE = (15 - 3) + 6
= 18 kJ
Example 4-1
0.01 kg of air is compressed in a piston-cylinder.
Find the rate of temperature rise at an instant of
time when T = 400K. Work is being done at a
rate of 8.165 KW and Heat is being removed at a
rate of 1.0 KW.
Solution on the black board
Example 4-2
Isothermal Process
An ideal gas is compressed reversibly and
isothermally from a volume of 0.01 m3 and a
pressure of 0.1 MPa to a pressure of 1 MPa. How
much heat is transferred during this process?
Solution on the black board
Example 4-3
Isobaric Process
The volume below a weighted piston contains
0.01 kg of water. The piston area is of 0.01 m2
and the piston mass is of 102 kg. The top face of
the piston is at atmospheric pressure, 0.1 MPa.
Initially the water is at 25 oC and the final state
is saturated vapor (x=1). How much heat and
work are done on or by the water?
Solution on the black board
Isobaric Process
For a constant-pressure process,
Wb + ΔU = PΔV + ΔU
= Δ(U+PV) = ΔH
Thus,
Q - Wother = Δ H + Δ KE + Δ PE (kJ)
Example: Boil water at constant pressure
Example
An insulated tank is divided into two parts by a
partition. One part of the tank contains 2.5 kg of
compressed liquid water at 60oC and 600 kPa
while the other part is evacuated. The partition
is now removed, and the water expands to fill the
entire tank. Determine the final temperature of
the water and the volume of the tank for a final
pressure of 10 kPa.
Example
Evacuated
Partition
H2O
m = 2.5 kg
T1 = 60oC
P1 = 600 kPa
P2 = 10 kPa
ΔE = Q - W
Solution - page 1
First Law: Q - W = ΔE
Q = W = ΔKE = ΔPE = 0
ΔE = ΔU = m(u2 - u1) = 0
u1= u2
No Work and no Heat therefore the internal
Energy is kept constant
Solution - page 2
State 1: compressed liquid
P1 = 600 kPa, T1 = 60oC
vf = vf@60oC = 0.001017 m3/kg
uf = uf@60oC = 251.11 kJ/kg
State 2: saturated liquid-vapor mixture
P2 = 10 kPa,
u2 = u1 = 251.11 kJ/kg
uf = 191.82 kJ/Kg, ufg = 2246.1 kJ/kg
Solution - page 3
u 2  u f 251.11  191.82
x2 

 0.0264
u fg
2246.1
Thus, T2 = Tsat@10 kPa = 45.81oC
v 2 = v f + x 2v g
= [0.00101+0.0264*(14.67 - 0.00101)] m3/kg
= 0.388 m3/kg
V2 = mv2 = (2.5 kg)(0.388m3/kg) = 0.97 m3
Example
One kilogram of water is contained in a pistoncylinder device at 100oC. The piston rests on lower
stops such that the volume occupied by the water is
0.835 m3. The cylinder is fitted with an upper set of
stops. The volume enclosed by the piston-cylinder
device is 0.841 m3 when the piston rests against the
upper stops. A pressure of 200 kPa is required to
support the piston. Heat is added to the water until
the water exists as a saturated vapor. How much
work does the water do on the piston?
Example: Work
Upper stops
Lower stops
Wb
Water
Q
m = 1 kg
T1 = 100oC
V1 = 0.835 m3
V2 = 0.841 m3
T-v Diagram
T
211.3 kPa
200 kPa
101.3 kPa
v1
v2
v
Solution - page 1
3
3
V1 0.835 m
m
v1 

 0.835
m
1 kg
kg
State 1: saturated liquid-vapor mixture
T1 = 100oC,
vf=0.001044 m3/kg , vg=1.6729 m3/kg
vf < v < vg ==> saturation P1=101.35 kPa
Solution - page 2
Process 1-2: The volume stay constant until the pressure
increases to 200 kPa. Then the piston will move.
Process 2-3: Piston lifts off the bottom stop while the
pressure stays constant.
State 2: saturated liquid-vapor mixture
P2= 200 kPa , v2 = v1 = 0.835 m3
Does the piston hit upper stops before or
after reaching the saturated vapor state?
Solution - page 3
3
3
V3 0.841 m
m
v3 

 0.841
m
1 kg
kg
State 3: Saturated liquid-vapor mixture
P3 = P2 = 200 kPa
vf = 0.001061 m3/kg , vg = 0.8857 m3/kg
vf < v3 < vg ==> piston hit the upper stops
before water reaches the saturated vapor state.
Solution - page 4
Process 3-4 : With the piston against the upper
stops, the volume remains constant during the
final heating to the saturated vapor state and the
pressure increases.
State 4: Saturated vapor state
v4 = v3 = 0.841 m3/kg = vg
P4 = 211.3 kPa , T4 = 122oC
Solution - page 5
Wb ,14 

4

2

3

4
PdV  PdV  PdV  PdV
1
1
2
3
 0  mP2 ( v 3  v 2 )  0
m 3 kJ
 (1 kg )( 200 kPa)(0.841 - 0.835)
3
kg m kPa
 1.2 kJ (> 0, done by the system)
Example: Heat Transfer
Upper
stops
Lower
stops
Wb
Water
Q
Find the require
heat transfer for
the water in
previous example.
Solution - page 1
First Law: Conservation of Energy
Q - W = ΔE = ΔU + ΔKE + ΔPE
Q14 = Wb,14 + ΔU14
ΔU14 = m(u4 - u1)
Solution - page 2
State 1: saturated liquid-vapor mixture
v1  v f
0.835  0.001044
x1 

 0.4988
v g  v f 1.6729  0.001044
u1  u f  x1u fg
kJ
kJ
kJ
 418.94  0.4988( 2087.6 )  1460.23
kg
kg
kg
Solution - page 3
State 4: saturated vapor state
v4 = 0.841 m3/kg = vg
u4 = 2531.48 kJ/kg
(interpolation)
Q14  Wb ,14  m(u 4  u1 )
kJ
 1.2 kJ  (1 kg)(2531.48 - 1460.23)
kg
 1072.45 kJ (> 0, added to the water)
TEAMPLAY EX. 4-6
A pressure cooker with volume of 2 liters operates
at 0.2 MPa with water at x = 0.5. After operation
the pressure cooker is left aside allowing its
contents to cool. The heat loss is 50 watts, how long
does it take for the pressure drop to 0.1 MPa?
What is the state of the water at this point?
Indicate the process on a T-v diagram.
Ex4.6)
2atm
2litros
Q=50W
Processo a v constante
T
x=0,5
0
dQ dW dU
1º Lei :


dt
dt
dt
dU 
 Q  Mdu  Q dt
dt
M (u2  u1 )  Q dt
P1=2atm
P2=1atm
v
Vol. Estado1 = Vol. Estado2
v=(1-x)vL+xvG
v=0,5*0,001+0,5*0,8919
v1=v2=0,446m3/kg
Título estado2:
P1=0,2MPa
Tsat=120ºC
v1L=0,001m3/kg
v1G=0,8919m3/kg
u1L=503,5KJ/kg
v  vL
0,446  0,001
 0,266
u1G=2025,8KJ/kg x 2  v  v 
1,672
G
L
x=0,5
Energia interna estado2:
u2=(1-x)u2L+xu2G
P2=0,1MPa
u2=0,734*418+0,266*2087,5
Tsat=100ºC
u2=862KJ/kg
v2L=0,001m3/kg
Energia interna estado1:
v2G=1,6729m3/kg
u1=(1-x)u1L+xu1G
u2L=418,9KJ/kg
u1=0,5*503+0,5*2025,8
u2G=2087,5KJ/kg
u1=1264KJ/kg
Massa de Água
v=V/MM=V/v
M=2*10-3/0,446
M=0,004kg
Aplicando na 1ºLei:
3
 M(u2  u1 ) 4  10 (1264  862)10
dt 


Q
50
dt  32s
3
TEAMPLAY EX. 4-7
A powerful 847 W blender is used to raise 1.36kg of
water from a temperature of 20oC to 70oC. If the
water loses heat to the surroundings at the rate of
0.176 W, how much time will the process take?
Ex4.7)
1HP=745W
1lbm=0,453kg
ºC=(ºF-32)/1,8
1Btu=1,055J
Wmec0
Wmec=1,2HP=894W
68ºF=20ºC (água no estado líquido)
M=1,359kg Q<0
158ºF=70ºC (água no estado líquido)
Q=10Btu/min=0,176W
3lbm=1,359kg
dU
dT


cv=cp=4,180KJ/kgºC
QW 
 MCv
dt
dt
dT
 0,176  ( 894)
ºC

 0,157
3
dt 1,359  4,180  10
s
(70  20)
Δt 
 317s  5'17"
0,157
TEAMPLAY EX. 4-10
Air, assumed to be ideal gas with constant specific
heats, is compressed in a closed piston-cylinder
device in a reversible polytropic process with n =
1.27. The air temperature before compression is
30oC and after compression is 130oC. Compute the
heat transferred on the compression process.
Ex4.10)
W   Pdv
Pv n  const.
T2
P
T1
2
T1=30ºC(303K)
P V  P1 V1 MR(T2  T1 )
W 2 2

T2=130ºC(403K)
1n
1n
n=1,27
n=1
1
v
Trabalho  específico
R(T2  T1 )  297  100
KJ
w




110
1
2
1n
0,27
kg
Calor  1º Lei
1
q 2 1 w 2  (u 2  u1 )
1
q 2  C v ΔT 1 w 2
1 q 2  0,7165  110  110  38,3
KJ
kg
Para comprimir do estado 1 ao 2 é necessário transferir
38,3 KJ por kg de ar comprimido.