2nd Law of Thermodynamics - Mechanical Engineering

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27 Οκτ 2013 (πριν από 3 χρόνια και 7 μήνες)

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Entropy and

the Second Law of
Thermodynamics


MECH3400

The essence of the
2
nd

law


1
st

law:


allows for prediction of change
of state due to energy transfer


does not point the direction of
time


does not reveal the possibility or
impossibility for a process to
occur


2
nd

law helps, as entropy:


never decreases for an isolated
system


indicates the possibility for a
process to occur


is a “signpost of time”

2
nd

law:

Entropy


is a single macroscopic
property which measures:



microscopic disorder
(randomness)


uncertainty (probability) to
determine the microscopic
state


unavailability of internal
energy

Example: mixing

......

diffusion
-

“random walk”

Entropy and
temperature


experience says that
temperature is:


an indication of the direction of
energy transfer as heat


a property that two systems have
in common when in (thermal)
equilibrium


in microscopic terms
-

associated with the energy of the
molecules


now a thermodynamic
definition of temperature

Entropy and
temperature
(continued)

2nd law suggests

that at equilibrium

S=S
max
:


Q

A

B

rigid

wall

insulating

rigid wall

Entropy and
temperature
(continued)

at equilibrium:

Entropy and
temperature
(continued)

thermodynamic definition of temperature:

Entropy and
temperature
(continued)

in order to ensure
dS>0

for A and B identical

Entropy and
pressure


mechanics says that
pressure is:


force per unit area exerted by
matter on its boundaries


a property that two systems have
in common when in
(mechanical) equilibrium


in microscopic terms
-

associated with molecular
collisions with a “wall”


now a thermodynamic
definition of pressure

Entropy and
pressure
(continued)

at equilibrium:


Q

A

B

insulating

rigid wall


W

Entropy and
pressure
(continued)

gives thermal equilibrium:

gives mechanical equilibrium:

Entropy and
pressure
(continued)

thermodynamic definition of pressure:

Gibbs equation

Reversible and
irreversible processes


reversible process:



“produces” no entropy


“backward” process possible


“leaves no footprints in the sand
of time”


uncommon


irreversible process:


“produces” entropy


“backward” process impossible


increases the “disorganised”
energy on the expense of
“organised” energy


common

Some reversible
processes

pneumatic spring

frictionless motion

Some other (nearly)
reversible processes


restrained compression or
expansion


heat transfer due to a
infinitesimal temperature
difference


magnetisation, polarisation


electric current flow through
a zero resistance


restrained chemical reaction


mixing of two identical
substances at the same state

Some irreversible
processes

motion with friction

spontaneous

chemical reaction

.....
.

mixing

heat transfer

T
1

> T
2

Q

unrestrained

expansion

P
1

> P
2

Ideal reservoirs
-

TER

Thermal Energy Reservoir
(TER)



TER is a fixed mass that can
undergo only heat interactions
with its environment


heat transfer to/from TER will
alter its internal energy


TER has uniform and constant
internal temperature


TER is always in equilibrium


TER is a source or sink of
“disorganised” energy


example: large block of copper

Ideal reservoirs
-

MER

Mechanical Energy Reservoir
(MER)


MER is a system that possesses
energy only in a fully organised
mechanical form such as raising
of a weight


the only energy transfer for a
MER is reversible work


all motions within MER are
frictionless so that work input
can be completely recovered


MER is a source or sink of
“organised” energy


example: dead weight on the end
of a frictionless pulley

Entropy change for
a TER and a MER

0

Gibbs equation:

1
st

law:

for a TER:

and for a MER:

2
nd

law:

Entropy change
for a control mass

C

TER

T

MER


W


Q

CM

T


for control mass



Q

is energy input

0

2
nd

law:

Entropy change
for a control mass (cont.)


for reversible process


in a control mass



Q

is energy input


useful in measurements


for adiabatic process


which is reversible


in a control mass


only mode of energy


transfer is work


eg adiabatic


compression

Entropy flow and
production

irreversible process


reversible process


impossible process

for a

control mass:

2
nd

law:

Entropy change
for a control volume


Q
2


W
shaft

v
1

v
2

dE
CM

A

B

1

2

v
1
dt

v
2
dt


assume: fixed boundaries, 1
-
D
transient (non
-
steady state) flow

Control volume and simultaneously

control mass at time
t

Control mass boundary at time
t+dt


Q
1


Q
3

2
nd

law:

Entropy change
for a control volume

from control mass to control volume

using Reynolds transport theorem:

......

now from the 2
nd

law for the control mass:

2
nd

law:

Entropy change
for a control volume

to the 2
nd

law for the control volume:

2
nd

law:

Entropy change
for a control volume

2
nd

law in the rate form:

Application of 2
nd

law to
energy conversion systems

isothermal

compression

adiabatic

expansion

isothermal

expansion

adiabatic

compression

T
A

T
B

1
-
2

2
-
3

3
-
4

4
-
1

Q
12

Q
34

W
12

W
23

W
34

W
41

Carnot

Engine

2T engine

Application of 2
nd

law to
energy conversion systems

Carnot

Cycle

T
A

T
B

1

2

3

4

T
A

T
B

1

2

3

4

V

V

T

T

reversible

heat engine

reversible

heat pump

R2T engine

Application of 2
nd

law to
energy conversion systems

for a cycle no change in CV so:

for a reversible process:

for an irreversible process:

Efficiency of a

Carnot engine

apply 1
st

law for this cycle:

then energy conversion efficiency is:

for a reversible process:

Efficiency of an
irreversible engine

for an irreversible process:

2
nd

law
-


other formulations


Kelvin
-
Planck statement:

“continuously operating 1T
engine is impossible”




Clausius statement:

“a zero
-
work heat pump is
impossible”


www

Pressure

thermodynamic = mechanical

Gibbs:

1st law:

for a reversible process

for an equilibrium state

compression:

Entropy for ideal gasses


GENERALLY:


S = N s(T,P)


where N is the number of moles




FOR IDEAL GASSES:


-

Standard Pressure (1atm)

-

Standard Pressure entropy