• Conservation of energy principle
• Total energy
• Energy transfer with heat and work
• M echanisms of heat transfer
• Specific heats, internal energy and enthalpy.
3.1.- INTRODUCTION TO THE FIRST LAW OF
TERM ODYNAM ICS:
“The
energy
can
be
neither
creat ed
nor
destroyed ; it can only change forms”. This
principle is based on experimental observations
and is known as the first law of thermodynamics
or the conservation of energy principle.
Chapter 3
1
Energy can cross the boundary of a closed
system in two distinct forms: HE AT and WORK.
System boundary
HEAT
CLOSED
SYSTEM
m= constant
WORK
3.2.- HEAT TRANSFER
Ther mal equilibr ium
Heat
Heat
BAKED
POT ATO
The direction of energy transfer is always from
th e
higher-temperature
body
to
th e
lower-
temperature one.
Chapter 3
2
HEAT: Is defined as the f orm of energy that is
transferred between two systems (or system and
its surroundings) by virtue of a temperat ure
difference.
There can not be any heat transfer between two
systems that are at the same temperature.
The transfer of heat into a system is f requently
referred to as HE AT ADDITION and the transfer of
heat out of a system as HEAT REJECTION.
Heat is energy in transition. It is recognized only
as it crosses the boundary of a system.
2kJ
T he rmal
e ne rgy
SURROUNDI NG
AIR
2kJ
HEAT
BAKED
POT ATO
HEAT
2kJ
Thermal
energy
SYSTEM
BOUNDARY
Chapter 3
3
A process during which there is no heat transfer
is called an ADIABATIC PROCESS. There are two
ways a process can be adiabatic: A system
insulated or both the system and the surrounding
are at the same temperature. An adiabatic process
should not be confused with an ISOTHERM AL
PROCESS.
INSULATION
ADIABATIC
SYSTEM
Q=0
The amount of the heat transferred during the
process between two states ( states 1 and 2) is
Q12, or just Q . Heat t ransfer per unit
mass of a system is denoted q and is determined
denoted by
from
q=
Chapter 3
Q
m
(kJ/ kg)
4
•
The heat transfer rat e is denoted Q
The heat is directional (or vector) quantity; the
universal accepted SIGN CONVECTION for heat is
as follows: “Heat transfer to a system is positive,
and heat transfer from a system is negative”.
Heat in
Q=+5k
J
SYSTEM
Heat out
Q=-5kJ
M ODES OF HEAT TRANSFER
CONDUCTION: Is the transf er of energy from the
more energetic particles of a substance to the
adjacent less energetic ones as a result of
interactions between the particles. Conduction
can take place in solids, liquids, or gases. In
gases and liquids is due to the collisions of the
molecules during their random motion. In solids,
Chapter 3
5
it is due to the combination of vibration of the
molecules in a lattice and the energy is transport
by free electrons.
COLA
T2
∆T
T1
AIR
COLA
Heat
∆x
Wall of aluminum
•
It observed that the rate of heat conduction Qcond
through a layer of constant thickness ∆x is
proportional to the temperature difference ∆T
across the layer and the area A normal to
direction of the heat transfer, and inversely
proportional to the thickness of the layer.
Therefore,
Chapter 3
6
•
Q cond = kA
∆T
∆x
(W )
Where the constant of proportionality k is the
“THERM AL CONDUCTIVITY” of the material which
is a measure of the ability of a mat erial to conduct
heat (table 3-1).
In the limiting case of ∆ x → 0 , the equation
above reduces to the differential form
•
Q cond = − kA
dT
dx
( W ) ……………………..(3-5)
k = (W /(m.K))
Which is known as “FOURIER’S LAW” of the heat
conduction. It indicates that the rate of heat
conduction in a direction is proportional to the
“TEM PERATURE GRADIENT” in that direction.
Heat is conducted in the direction of decreasing
temperature,
and
th e
temperature
gradient
becomes negative when temperature decreases
Chapter 3
7
when increasing x. Therefore, a negative sign is
added in eq. 3-5 to make heat transf er in the
positive x direction a positive quantity.
CONVE CTION: Is the mode of energy transfer
between a solid surface and the adjacent liquid or
gas which in motion and it involves the combined
effects of “conduction and fluid motion”.
The faster the fluid motion, the greater the
convection heat transfer. In the absence of any
bulk fluid motion, heat transfer between a solid
surface and the adjacent fluid in pure conduction.
Convection is called “F ORCE D CONVE CTION” if
the fluid is “FORCED” to flow in a tube or over a
surface by external means such as a fan, pump,
or the wind. In contrast, convection is called
“FREE (OR NATURAL) CONVECTION” if the fluid
motion is caused by buoyancy forces that are
induced by density differences due to the
variation of temperature in the fluid.
Chapter 3
8
•
The rate of heat transf er by convection Q conv is
determined form “NEWTON’S LAW OF COOLING”,
which is expressed as, which is expressed as
•
Qconv = hA(Ts − Tf )
h = (W /(m2K))
Where h is the “CONVECTION HEAT TRANSFER
COEFFICIENT”, A is the surf ace area through
which heat transfer takes place, T s is the surf ace
temperature, and Tf is the bulk fluid temperature
away from the surface.
RADIATION: Is the energy emitted by matt er in the
form of electromagnetic waves (or photons) as a
result
of
th e
changes
in
th e
electronic
configurations of the atoms or molecules.
Unlike conduction and convection, the transfer of
energy by radiation does not require the presence
of an intervening medium.
Chapter 3
9
In fact, energy transfer by radiation is fastest (at
the speed of light) and it suffers no attenuation in
a vacuum.
The maximum rate of radiation that can be emitted
from a surface at an ABS OLUTE TEM PERATURE
TS is given by the STEFAN-BOLTZM ANN, LAW as
•
Q emit ,max = σ AT s
4
(W )
Where: A is the surface area and σ = 5.67x10−8 W/(m2K4 )
is the Stefan-Boltzmann constant. The idealized
surface which emits radiation at this maximum
rate is called a BLACKBODY, and the radiation
emitted by a blackbody is called BLACKBODY
RADIATION.
The radiation emitted by all real surf aces is less
than the radiation emitted by a blackbody at the
same temperature and is expressed as
•
Qemit = εσATs
Chapter 3
4
( W)
10
Where
ε is the EM ISSIVITY of the surface. The
property emissivity, whose value is in the range
0 ≤ ε ≤ 1, is a measure of how closely a surface
approximates a blackbody for which ε = 1.
Another important radiation property of a surface
is its ABSORPTIVITY
α,
which is the fraction of
the radiation energy incident on a surf ace that is
absorbed by the surface.
0 ≤ α ≤1
α =1
→
Blackbody
In general, both ε and
α of a surface dep end
on the temperature and the wavelength of the
radiation. KIRCHHOFF’S LAW of the radiation
states that the emissivity and the absorptivity of a
surface are equal at the same temperature and
wavelength.
•
•
Qabs = α Qinc ( W )
Chapter 3
11
•
Q
•
inc
Q
•
ref
•
Q
= (1 − α )Q
inc
•
abs
= α Q
inc
The net rate of radiation heat transfer between
two surfaces is determined from
•
(
Qrad = εσA Ts − Tsurr
4
4
)
(W)
LARGE EN CLOSURE
A, Ts
SMALL
BODY
Tsurr
Chapter 3
12
3.3.- WORK
Work, like heat, is an energy interaction between a
system and its surrounding. As mentioned earlier,
energy can cross the boundary of a closed
system in the form of heat or work. “IF THE
ENERGY CROSSING THE BOUNDARY OF A
CLOSED S YSTEM IS NOT HEAT, IT M UST BE
WORK”.
Work is the energy transf er associat ed with a
force acting through a distance. A rising piston, a
rotating shaft, and a electric wire crossing the
system boundaries are all associated with work
interactions.
Work is also a form of energy transferred like heat
and, therefore has energy unit such as (kJ). The
work done during a process between states 1 and
2 denoted W12 or simply W . The work done per
unit mass of a system is denoted w and is defined
as
Chapter 3
13
W
w=
m
(kJ / kg)
The work done per unit time is called POWER and
•
is denoted W . The unit of power is kJ/s or kW.
The sign convection for work adapted in this text
reflects this philosophy: “Work done by a system
is positive, and Work done on a syst em is
negative”
W= 30 kJ
m= 2kg
∆t =5 s
30kJ
work
•
W =6kW
w=15 kJ/kg
Chapter 3
14
Sign convection for heat and work
(+)
Q
(-)
SYSTEM
(-)
W
(+)
PATH FUNCTIONS have “inexact differentials”
designated
by the symbol
δ . Therefore a
differential amount of heat or work is represented
by δQ or δW .
Properties, however, are P OINT FUNCTIONS (i.e.,
they depend on the stat e only, and not on how a
system reaches that st ates), and they have “exact
differentials designated by the symbol d .
Chapter 3
15
A small change in volume, for example, is
represented by dV and the total volume change
during a process 1-2, however is
2
∫ dV = V
2
− V1 = ∆V
1
That is, the volume change during process 1- 2,
regardless of the path followed.
The total work done during process 1-2, however
is
2
∫ δW = W
12
(not ∆W )
1
That is, the total work obtained by following the
process
path
and
adding
th e
differentials
amounts of work ( δW ) done along the way.
The integral of δW is not W2-W1
Chapter 3
16
P
1
Process A
Process B
2
2m3
5m3
V
∆VA = 3m3 , W A = 8 kJ
∆VB = 3m3 , W A = 12 kJ
ELECTRICAL WORK.In an electrical field, electrons in a wire move
under the effect of electromotive forces, doing
work. When N coulombs of electrons move
through a potential difference V, the electrical
work done is
We = VN (kJ )
Which can also be expressed in the rate form as
Chapter 3
17
•
W e = VI ( kW )
•
Where
W e is the electrical power and I is the
number of electrons flowing per unit time. In
general V and I vary with time, and the electrical
work done during a time interval ∆t is expressed
as
2
We = ∫ VIdt (kJ)
1
If both V and I remain constant during the time
interval ∆t, this equation will reduce to
We = VI ∆t (kJ )
I
•
We = VI
R
V
=I R
2
= V2 R
Chapter 3
18
3.4.- M ECHANICAL FORM S OF WORK.F
F
s
There are several different ways of doing work,
each in some way related to a force acting
through a distance. In elementary mechanics, the
work done by a constant force F on a body that is
displaced a distance s in the direction on the
force is given by
W = Fs (kJ)
If the force F is not constant, the work done is
obtained
by
adding
(i.e.,
integrating)
th e
differential amounts of work (forces times the
differential displacement ds)
2
W = ∫ Fds (kJ )
1
Chapter 3
19
The work done on a system by an ext ernal force
acting in the direction of motion is NEGATIVE,
and work done by a system ag ainst an external
force acting in the opposite direction to motion is
POSITIVE.
M OVING BOUNDARY WORK.One
form
of
mechanical
work
f requently
encountered in practice is associated with the
expansion or compression of a gas in a piston
cylinder device. During this process, part of the
boundary (the inner face of the piston) moves
back and forth. Therefore, the expansion and
compression
work is oft en
called
M OVING
BOUNDARY WORK or simply BOUNDARY WORK.
GAS
Chapter 3
The moving
boundary
20
F
ds
A
P
GAS
Consider the gas enclosed in the piston-cylinder
device. The initial pressure of the gas is P, the
total volume is V, and the cross-sectional area of
the piston is A. If the piston is allowed to move a
distance ds in a quasi-equilibrium manner, the
differential work done during this process is
δWb = Fds = PAds = PdV
dv is positive during an expansion process
(volume
increasing)
and
negative
during
a
compression process (volume decreasing)
The total boundary work done during the entire
process as the piston moves is obtained by
adding all the differential works from the initial
state to the final state:
Chapter 3
21
2
W b = ∫ PdV
( kJ )
1
P
1
Process path
The area under the process curve on a
P-V diagram represent the boundary
w ork
2
dA=PdV
V1
V2
dV
V
P
P
The net w ork done during a cycle is the
difference betw een the w ork done by
1
the system and the w ork done on the
A
Wnet
B
V1
Chapter 3
system
2
V2
V
22
POLYTROPIC PROCESS.During expansion and compression process of
real gases, pressure and volume are often relat ed
by
PVn = C,
where n and C are constants. A
process of this kind is called a “P OL YTROPIC
PROCESS”.
The pressure for a polytropic process can be
expressed as
P = CV − n
P1V1n=P 2V2n
P
1
P1
PVn = const.
GAS
P2
PVn=C= const.
2
V2
V1
2
Wb =
∫ PdV
1
Chapter 3
2
=
∫ CV
1
−n
dV = C
V2
−n +1
− V1
− n +1
− n +1
=
V
P 2 V 2 − P1V1
1− n
23
Since C=P1V1n=P2V 2n. For an ideal gas (PV=mRT)
Wb =
mR ( T2 − T1 )
, n ≠1
1−n
( kJ )
The special case of n=1 equivalent to the
isothermal process.
SPRING WORK.It is common knowledge that when a force is
applied on a spring changes. When the length of
the spring changes by a differential amount dx
under the influence of a force F, the work done is
δWspring = Fdx ……………………….(a)
Rest
position
dx
x
Chapter 3
F
24
To determine the total spring work, we need to
know a functional relationship between F and x.
For linear elastic springs, the displacement x is
proportional to the force applied. That is,
F = ksx
(kN) ………………..(b)
Where ks is the spring constant and has the unit
kN/m. The displacement x is measured from the
undisturbed position of the spring (that is, x=0
when F=0). Substituting Eq. (b) into Eq. (a) and
integrating yield
Wspring
(
1
2
2
= k s x 2 − x1
2
)
(kJ)
Where x1 and x2 are the initial and the final
displacements of the spring, respectively. Both x1
and x2 are measured f rom the undisturbed
position of the spring. Note that the work done on
a spring equals the energy stored in the spring.
Chapter 3
25
Rest
position x1
x2
F1
F2
3.5.- THE FIRST LAW OF THERM ODYNAM ICS.The first law of thermodynamics, also know as the
conservation of energy principle provides a
sound basis for studying the relationships among
th e
various
forms
of
energy
and
energy
interactions. Based in experimental observations,
the first law of thermodynamics stat es that
ENERGY CAN BE NEITHER CREATED NOR
DESTROYED, IT CAN ONL Y CHANGE FORM S.
First Law of thermodynamics, or the conservation
of energy principle for a CLOSED SYSTEM or a
fixed mass, may be expressed as follow:
Chapter 3
26
net energy transfer
net increase (or decrease)
to (or from) the system
in the total energy
as heat and work
of the system
Q − W = ∆E
(kJ) .......... .(I)
Where
Q = net heat transfer across system boundaries ( = ∑ Q in − ∑ Q out )
W = net work done in a ll forms
= ∑ Wout − ∑ Win
∆E = net change in total energy of system, E 2 − E 1
Examples:
Q2=-3 kJ
∆E = Qnet = 12kJ
Q1=15 kJ
Chapter 3
27
Adiabatic
We = −8 kJ
∆E = 8 kJ
V
Adiabatic
∆ E = 5 kJ
Wpw = −5 kJ
Chapter 3
28
The total energy E of a system is considered to
consist of three part s: internal energy U, kinetic
energy KE, and potential energy PE. Then change
in total energy of a syst em during a process can
be expressed as the sum of the changes in its
internal, kinetic, and potential energies:
∆E = ∆U + ∆KE + ∆PE
(kJ )
Q − W = ∆U + ∆ KE + ∆PE
(kJ )
Where:
∆U = m (u 2 − u1 ) (kJ )
∆KE =
(
1
2
2
m υ 2 − υ1
2
)
(kJ)
∆PE = mg (z 2 − z1 ) (kJ )
Thus, for STATIONARY CL OSED S YSTEM S, the
changes in kinetic and potential energies are
negligible (that is, ∆KE = ∆PE = 0 ), and the first
law relation reduces to:
Chapter 3
29
Q − W = ∆U
(kJ )
Sometimes it is convenient to consider the work
term
in
tw o
parts:
Wother
and
Wb ,where
Wother represent all forms of work except the
boundary work.
Q − W other + Wb = ∆E
(kJ )
Other forms of the First –Law relation
Dividing by the mass of system
q − w = ∆w (kJ/ kg)
the rate form of the first law is obtained by
dividing Eq. (I) by the time interval ∆t and taking
the limit as ∆t → 0 .This yields
•
•
dE
Q− W =
dt
Chapter 3
(kW )
30
•
•
Where Q is the rat e of net heat transf er,
W i s th e
dE
power, and dt is the rate of change of total
energy.
The differential form:
δ Q − δW = dE
δ q − δ w = de
(kJ )
(kJ / kg )
For a CYCLIC PROCESS, the initial and final
states are identical, and therefore ∆E = E 2 − E1 = 0 .
Then the first-law relation for a cycle simplifies to
Q−W =0
(kJ )
3.6.- SPECIFIC HEATS.The SPECIFIC HEAT is defined as “the energy
required t o raise the temperat ure of a unit mass of
a substance by one degree”. In general, this
Chapter 3
31
energy will depend on how the process is
executed. In thermodynamics, we are interested in
two kinds of specific heats: SPECIFIC HEAT AT
CONSTANT VOLUM E
C v and SPECIFIC HEAT AT
CONSTANT PRESSURE
CP .
Now we will attempt to express the specific heats
in terms of other thermodynamics properties.
First,
consider
a
stationary
closed
system
undergoing a constant-volume process ( Wb = 0 ).
The first-law relation for this process can be
expressed in the differential form as
δq − δw other = du
The left-hand side of this equation ( δ q − δ w other )
represents the amount of energy transf erred to
the system in the form of heat /or work.
Chapter 3
32
From the definition of
C v , this energy must be
C v dT , where dT is the differential
equal to
change in temperature. Thus
C v dT = du at constant volume
or
 ∂u 
Cv =  
 ∂T  v
Similarly, an expression for the specific heat at
constant
pressure
considering
( wb + ∆u
a
C p can be obtained by
constant
pressure
process
= ∆ h ).
It yields
 ∂h 
Cp =  
 ∂T  p
These equations are property relations and such
are independent of the type of process. They are
valid for any substance undergoing any process.
Chapter 3
33
V= const.
m=1 kg
∆T = 1ºC
kJ
Cv = 3 .13
kg .º C
3.13 kJ
P= const.
m=1 kg
∆T = 1ºC
kJ
C p = 5.2
kg .º C
5.2 kJ
Constant-volume and constant pressure specific
heats
C v and C p (values given are fo r helium
gas).
AIR
m=1 kg
300 → 301K
0.718 kJ
AIR
m=1 kg
1000 → 1001K
0.855 kJ
The specific heat of a substance change with
temperature
Chapter 3
34
A common unit for specific heats is kJ / (kg.º C) or
kJ / (kg.K ) . Notice that these two units are
identical since ∆T (º C) = ∆T ( K ) .
The specific heat s are sometimes given on a
_
molar basis. They are denoted by
−
C v and C p and
have the unit kJ / (kmol.º C) or kJ / (kmol.K ) .
3.7.- INTERNAL ENERGY, ENTHALP Y, AND
ESPECIFIC HEATS OF IDEALGASES.Using the definition of enthalpy and the equation
of state of an ideal gas, we have
h = u + Pv
Pv = RT
h = u + RT
Since R is constant
u = u (T ) and, it follows that
the enthalpy of an ideal gas is also a function of
temperature only:
Chapter 3
h = h (T )
35
Since u and h depend only on temperature for an
ideal gas, the specific heats
C v and C p also
depend at most, on temperature only.
Thus for ideal gases, the partial derivatives can be
replaced
by
ordinary
derivatives.
Then
th e
differential changes in the internal energy and
enthalpy of an ideal gas can be expressed as
du = C v (T )dT
dh = Cp ( T)dT
The change in internal energy or enthalpy for an
ideal gas during a process from state 1 to state 2
is determined by integrating these equations:
2
∆u = u 2 − u1 = ∫ C v (T )dT
(kJ / kg)
1
2
∆h = h 2 − h1 = ∫ Cp ( T )dT
(kJ / kg)
1
Chapter 3
36
u2 − u1 = C v,av ( T2 − T1 ) (kJ / kg)
h2 − h1 = Cp,av (T2 − T1 ) (kJ / kg)
Specific-Heat Rel ations of Ideal Gases.A special relationship between
C v and C p for
ideal gases can be obtained by differentiating the
relation
h = u + RT , which yields
dh = du + RdT
Replacing dh by CpdT and du by CvdT and
dividing the resulting expression by dT. We obtain
Cp = Cv + R
[kJ / (kg.K )]
When the specific heats are given on a molar
basis, R in the above equation should be replaced
by the universal gas constant Ru. That is
−
−
Cp = Cv + Ru
Chapter 3
[kJ / (kmol.K )]
37
At this point, we introduce another ideal gas
property called the S PECIFIC HEAT RATIO k,
defined as
k=
Cp
Cv
3.8.- INT ERNAL ENERGY, ENT HALPY, AND
ESPEC IF IC HEAT S OF SOLIDS AND LIQUIDS.-
A substance whose specific volume (or density) is
constant
is
called
an
INCOM PRESSIBLE
SUBSTANCE. The specific volume of solids and
liquids essentially remain constant during a
process. Therefore, liquids and solids can be
approximated
as
incompressible
substances
without sacrificing much in accuracy.
Cp = Cv = C
LIQUID
υ l = const .
SOLID
υ S = const .
Chapter 3
IRON
25ºc
C = C V = Cp = 0.45 kJ /( kg º C)
38
Like those of ideal gases, the specific heats of
incompressible
substances
dep end
on
temperature only.
du = C v dT = C(T )dT
The change in internal energy between stat es 1
and 2 is then obtained by integration:
2
∆u = u 2 − u1 = ∫ C(T )dT
(kJ / kg)
1
∆u = u 2 − u1 ≈ C av (T2 − T1 ) (kJ / kg)
The
enthalpy
change
of
incompressible
substances ( solids or liquids) during process 1-2
can be det ermined from the definition of enthalpy
(h=u+Pv) to be
h2 − h1 = (u2 − u1 ) + υ(P2 − P1 )
Since υ1 = υ 2 = υ . It can also be expressed in a
compact form as
∆h = ∆u + υ∆P
Chapter 3
39