Download Review of Engineering Thermodynamics - Part A

Principles of Energy Conversion
Jeffrey S. Allen
Part 6. Review of Engineering Thermodynamic
September 15, 2016
Copyright © 2016 by J. S. Allen
1 What is Thermodynamics?
1.1 Typical Engineering Thermodynamic Definitions . . . . . . . . . . . . . . .
1.2 Our Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
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2 Thermodynamic Energy Types & Classifications
2.1 Closed and Open Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Efficiency and the First Law of Thermodynamics . . . . . . . . . . . . . . .
2.3 Energy Flux Through A Surface . . . . . . . . . . . . . . . . . . . . . . . . .
3
3
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3 Heat Engines
3.1 Newcomen’s Engine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Watt’s Engine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3 Evans’ Engine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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4 Work from Steam
4.1 Measuring Pressure and Volume - Engine Indicators . . . . . . . . . . . . .
4.2 Boundary Motion Work & Polytropic Processes . . . . . . . . . . . . . . . .
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8
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5 Steam from Heat
5.1 Caloric Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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6 Energy to/from a Fluid
6.1 Measures of Mass . . . . . . . . . . . . . .
6.2 But what is ∆e? . . . . . . . . . . . . . .
6.3 Internal Energy (U, u) . . . . . . . . . . .
6.3.1 Joule’s Experiment . . . . . . . . .
6.3.2 Definition of Internal Energy . . .
6.4 Enthalpy . . . . . . . . . . . . . . . . . . .
6.4.1 Flow Potential vs Boundary Work
6.4.2 Definition of Enthalpy . . . . . . .
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7 Measuring Changes in Energy
7.1 Specific Heats (c, cv , cp ) . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.2 Measurable Properties and Changes in Energy . . . . . . . . . . . . . . . . .
7.2.1 State Property Relationships . . . . . . . . . . . . . . . . . . . . . .
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References
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What is Thermodynamics?
“Thermodynamics is a funny subject. The first time you go through it,
you don’t understand it at all. The second time you go through it, you
think you understand it, except for one or two points. The third time
you go through it you know you don’t understand, but by that time you
are so used to the subject, it doesn’t bother you any more.”
– Arnold Sommerfield
“In lecturing on any subject, it seems to be the natural course to
begin with a clear explanation of the nature, purpose, and scope of the
subject. But in answer to the question ‘what is thermo-dynamics?’ I
feel tempted to reply ‘It is a very difficult subject, nearly, if not quite,
unfit for a lecture.”
– Osborn Reynolds
On the General Theory of Thermo-dynamics,
November, 1883
“Thermodynamics” was first used in a publication by Lord Kelvin in 1849. The
word thermodynamics comes from the Greek language:
therme ≡ heat
} “power of heat”
dynamis ≡ power
Typical Engineering Thermodynamic Definitions
→ the science of the relationship between heat and work and the substances related
to the interaction of heat and work
→ the science of energy and entropy
→ “Thermodynamics is the science and changes in state of physical systems and the
interactions between systems which may accompany changes in state. . . . Classical
thermodynamics is concerned with changes between equilibrium states.” Kestin
Our Definition
→ conversion of transitional thermal and mechanical energy via energy
exchange with a fluid; a subset of the field of energy conversion
2
Thermodynamic Energy Types & Classifications
Energy
Classification
Energy Type
Transitional
Stored
mechanical
work
inertial
pressure
gravitational
thermal
heat
internal energy
sensible heat
latent heat
Closed and Open Systems
Closed System – no mass flow in or out of system
work
heat
energy
state 1
time = t1
energy
state 2
time = t2
t1 < t < t2
Open System – mass flow in and/or out of system
h e a t flu x , Q
m
1
e
(1 )
(2 )
1
m
2
e
2
p o w e r, W
With a Closed System, we are analyzing a fixed mass. An Open System, however, is
a volume in space with mass passing through. Conservation Laws are only strictly
applicable to a Closed System (fixed mass, inertial frame of reference).
Therefore, Conservation of Energy on an Open System has an extra energy term
associated with mass flow in and out of the volume to compensate for the non-fixed
mass. The extra energy term will be referred to as flow energy and it looks the same
as boundary work. Though not identical, the two are equivalent mathematically.
3
Steady Energy Balance
For a steady-state, open system:
∆Q − ∆W = ∆E = (Ep + Ek + Ei + Ef )∣out − (Ep + Ek + Ei + Ef )∣in
∆Q:
∆W :
∆E :
heat into system
work produced by system
change in system energy
potential energy = mzg /gc
kinetic energy = mV 2 /2gc
internal energy = mu
flow energy = P∀ = mP/ρ
Ep :
Ek :
Ei :
Ef :
This is the first law of thermodynamics, which is really:
change in transitional energy ⇐⇒ change in stored energy
Steady implies that there is no energy or mass accumulation within the system.
That is different than the exchange of energy in the mass which passes through the
system. The mass which passes through may accumulate energy.
First Law of Thermodynamics
Steady, rate form of the 1st Law:1
∑ Q̇ − ∑ Ẇ = ∑ ṁe − ∑ ṁe
out
m
2
e
2
(2 )
(3 )
m
1
e
1
(1)
in
(1 )
h e a t flu x , Q
m
3
e
3
p o w e r, W
Heat (Q̇) & Work (Ẇ )
´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¸ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¶
Not the same as useful heat and work
The 1st Law of Thermodynamics applies to the fluid!
1 The sign convention on heat (Q) and work (W ) may change. For example, some thermodynamic textbooks will write the first law of thermodynamics (equation 1) as ∆Q + ∆W , in which
case ∆W is the work into the system. The choice of sign convention will vary between engineering
disciplines as well. It is not important which sign convention you use as long as you are consistent.
The meaning of a positive or negative work can always be ascertained by examining how the stored
energy term (∆E ) changes.
4
Efficiency and the First Law of Thermodynamics
Conservation of Energy, and by corollary 1st Law of Thermodynamics, demands that
the total change in stored energy is balanced by the transitional energies to/from
the energy converter of interest. In practice, an imbalance in heat, work and energy change is nearly always observed. Some of the energy put into the system is
converted into something other than useful work. The ratio defines efficiency.
η=
energy sought actual power output Ẇuseful
=
=
energy cost
ideal power output
∆Ė
Energy Flux Through A Surface
The sign convention for ∆E is based on the definition of flux through a surface.
1 Q̇2
− 1 Ẇ2 = +Ėout − Ėin
5
Heat Engines
Newcomen’s Engine – 1707
W
o s c illa tin g b e a m
P
Work from piston:
p is to n &
c o n d e n se r
b o ile r
m∆e = Wout + {Qout + Wlost }
Q
a tm
Q
o u t
c o ld w a te r
in
d ra in
to p u m p
Watt’s Engine – 1764
W
o s c illa tin g b e a m
P
b o ile r
Work from piston:
a tm
p is to n
c o ld w a te r
m∆e = Wout + {Wlost }
Q
to p u m p
in
Q
c o n d e n se r
o u t
d ra in
Evans’ Engine – 1805
W
for the cycle:
P
∆e = 0
∑ Q = ∑ Wout + ∑ Wlost
b o ile r
p is to n
Work from piston:
m∆e = Wout + {Wlost }
Q
a tm
p u m p
p re b o ile r
in
6
to v a c u u m
p u m p
Q
o u t
c o n d e n se r
Work from Steam
Work is the expansion (or contraction) of a pressurized gas or vapor. From the
perspective of a steam cylinder, work is related to pressure P and volume ∀.
Work = F ∆x = (Pnet Apiston ) (
∆∀
) = Pnet ∆∀
Apiston
Pnet is the net pressure difference across the piston. For the atmospheric engines, the pressure difference across the slow moving cylinders was equal to
Patmospheric − Psaturation which was essentially constant.
Psaturation is the pressure at which the steam condenses
(or evaporates).
Work
(1 )
D x
(2 )
= (Patm − Psat ) ∆∀
= (Patm − Psat ) Apiston ∆x
v a p o r
The work extracted from the atmosphere could be
measured by the force F exerted from the piston and
the piston displacement ∆x during the power stroke.
After Evan’s engine development, the work was extracted from pressurized steam
instead of the atmosphere. Thus, the net pressure difference, Psteam − Patm was
not constant during the power stroke and the work during a power stroke required
integration of the pressure and volume.
W =∫
2
Pnet d∀
1
As engine technology improved, the speed of engines increased; becoming too fast
to watch a pressure gauge or manometer and record the pressures as a function of
piston position. Some sort of device was needed to measure pressure and volume
and correlate the two values.
Such a device was developed in the late 1700’s. The theory was of an indicator
diagram, P versus ∀, was proposed by Davies Gilbert in 1792. In 1796, John
Southern, who was an associate of Watt, invents the engine indicator. [? ]
7
Measuring Pressure and Volume - Engine Indicators
Steam engine indicators from Peabody [1].
Indicator Plots
The indicator plot is a pressurevolume plot with pressure on the
vertical axis and volume on the
horizontal axis.
The work is found by integrating over
the area swept by the indicator.
W = ∫ Pd∀
The procedure was to measure height,
P, of thin, d∀, vertical bands and
sum lengths. The proper scaling had
to be applied to convert the indicator
lengths to pressure and volume.
The steam pressure becomes less than back
pressure from 6 to 10. Peabody [1]
Modern Engines
Steam engine indicators are only suitable for low-rpm engines because of the inertia
of the indicator components, such as the oscillating drum. As the engine runs
faster, the indicator does not accurately reflect pressure and volume in the cylinder.
Today, we use crank angle sensors and dynamic pressure transducers to measure
pressure and volume. The sensor signals are captured via high speed data acquisition
hardware.
8
Boundary Motion Work & Polytropic Processes
Engineers found that when the pressure-volume traces were replotted on log-log
axis, that the power and compression strokes were linear. The power and compression strokes being the portion of the P-∀ trace which represented work to or from
the fluid.
log P = −a log ∀ + b ← linear relationship on log-log plot
P∀a = e b (constant)
The value of the polytropic coefficient, a, is a measure of the efficiency the process.
Often we approximate actual expansion processes with idealizations by setting the
polytropic coefficient a to specific values. For example, if the expansion process
can be assumed to occur at constant temperature, then a = 1. We will discuss the
implications of various idealized polytropic processes later.
1 W2
=∫
∀2
∀1
Pd∀
For P∀a = constant, integration results in an analytical expression for
work based on the starting and ending fluid states.
P2 ∀2 − P1 ∀1
1−a
a ≠ 1,
1 W2
=
a = 1,
1 W2
= P1 ∀1 ln (
∀2
)
∀1
Typical pressure-volume trace for an Otto cycle;
spark-ignition internal combustion engine [2].
work process
polytropic coefficent
work
constant pressure (isobaric)
0
P(∀2 − ∀1 )
constant volume (isochoric)
∞
0
constant temperature (isothermal)
1
P1 ∀1 ln(∀2 /∀1 )
constant entropy (isentropic)
k ≡ cp /cv
(P2 ∀2 − P1 ∀1 )/(k − 1)
9
Steam from Heat
{Heat} = {Change in Fluid Energy}
v a p o r
Q = ∆E
p h a se
c h a n g e
liq u id
h e a t
The change in fluid energy, ∆E , was considered different
than that which occurred during the work on the piston.
The concept of energy and the relationship between heat
and work took a long time to become established.
Caloric Theory
Up until the mid-1800’s, the relationship between heat, work, and energy exchange
with a fluid (steam or gas) was not understood. Work was known to be related to
pressure and volume, but heat added to the fluid from which work was extracted
was a mystery. Two theories were originally put forth to explain heat.
1. Live Force: Heat is a level of randomness of atoms and molecules in a
substance; heat is a form of kinetic energy of very small particles. In other
words, heat is manifestation of motion at the molecular level.
2. Caloric: Heat is a fluid-like substance that is massless, colorless, oderless,
and tasteless. Caloric was a substance that had to be conserved; as such, an
important principle of caloric theory is that work can not be converted into
heat because work is not a substance.
Caloric theory was proposed by French chemist Antoine Lavosier in 1789 and was the
prevailing theory until the mid-1800’s. Caloric is a substance and can be “poured”
from one body to another. Since it is a substance, caloric is subject to conservation
laws; i.e., caloric is conserved. When caloric is added to a body, the temperature of
that body increases. Remove caloric and the body’s temperature decreases. “When
a body could not contain any more caloric, much the same way as when a glass of
water could not dissolve any more salt or sugar, the body was said to be saturated
with caloric. The interpretation gives rise to the terms saturated liquid, saturated
vapor, and heat flow that are still in use today.” [? ]
There were many challenges to caloric theory,
• American Engineer Benjamin Thompson (Count Rumsford, 1754–1814)
– continuous heating due to friction during boring of cannons “demonstrated
categorically” that caloric theory was not valid
– science community remained unconvinced
• English Scientist James P. Joule (1818–1889)
– published experimental results in 1843 that “definitively” proved that heat
is not a substance subject to laws of conservation
10
– science community remained unconvinced
• German Physician Robert Mayer (1814–1878)
– in 1840’s, proposed heat and work were different forms of the same thing
– science community remained unconvinced
• Joule and Kelvin performed a series of experiments between 1850 and 1860
which quantified the relationship between heat and work
– took another 50 years for caloric theory to pass
11
Energy to/from a Fluid
The piston work is related to energy transfer from the steam (fluid).
Heat + Work ∼ change in fluid energy, E
1Q2
− 1W2 = E2 − E1
The amount of energy is mass dependent (extrinsic). At a given temperature, the
total energy in a glass of water depends on the mass of water present. Here, extrinsic
properties are denoted by capital letters such as energy E , internal energy U, and
volume ∀. Mass independent properties (intrinsic) are denoted by lower case letters,
e.g. specific energy e, specific internal energy u, and specific volume v .
Measures of Mass
There are several means to quantify mass during energy conversion processes. For
closed process, mass m is often used directly. The total mass in a closed system is
proportional to the volume of the system, m ∼ ∀. The constant of proportionality
is density ρ, which has units of mass per volume [kg/m3 ]. The inverse of density
is specific volume v with units of [m3 /kg]. For an open system (mass flow in and
out), density is more convenient when considering momentum exchange. Thus, in
fluid mechanics, density is a preferred measure of mass. When considering energy
exchange, specific volume is more convenient since a key factor is the pressurevolume product. Thus, in thermodynamics with open systems, specific volume is
the preferred measure of mass. Other commonly used measures of mass are specific
gravity, SG = ρ/ρH20 , which is the fluid density normalized by the density of water
at room temperature and specific weight, γ = ρg , where g is acceleration due to
gravity.
But what is ∆e?
The term energy was first used by Thomas Young in 1807 to describe capillary
phenomena. Energy, in a thermodynamic context, was explained in 1840 by William
Thomson (a.k.a. Lord Kelvin). Modern engineering thermodynamics expand ∆E to
include internal energy and a few other forms of stored energy.
• Stored Thermal Energy
– sensible heat ≡ internal energy, U,2
• Stored Mechanical Energy
– potential (gravitational), mgz,
– kinetic (inertial), mV 2 /2,
2 Internal energy is often described as sensible heat, which is a stored form of thermal energy.
Strictly speaking, however, internal energy encompasses more than just the stored thermal energy
component. We will attempt to separate the other components of internal energy, such as the
chemical and electrical, from the thermal portion. Internal energy is not exactly a stored form of
thermal energy, but it is convenient to categorize it as such for the purposes of this discussion.
12
– flow work potential –or– flow energy, P∀,
• Stored Chemical Energy (only known form)
– chemical potential
– enthalpy of formation
Mechanical energy is easy to conceptualize and measure. We require measures of
mass, velocity, elevation, pressure and volume. Thermal energy, however, is not easy
to conceptualize nor measure. Thermal energy can only be defined by a change that
occurs during a process.
13
Internal Energy (U, u)
The first law of thermodynamics is based on the principle that energy is a conservative property. As such, the first law is what defines a thermodynamic property of
energy, or more correctly, a change in a thermodynamic property.
Internal energy3 , denoted as U, is the intrinsic energy of a substance neglecting
other forms of potential energy (gravitational, inertial, flow). Thus, for a closed
system in which the mass is fixed, the energy change ∆E is ∆U.
1 Q2
− 1 W2 = U2 − U1 = m(u2 − u1 )
The concept of internal energy is somewhat intuitive. However, how do we measure
internal energy? What other measurable properties is internal energy dependent
upon?
Joule’s Experiment
Joule determined for gases that
internal energy only depended upon
temperature. His experiment consisted of two vessels in an insulated
water bath. The vessels were left in
the water bath for a long time so
that the system was isothermal. The
vessels were connected, but isolated
from one another by a stopcock (c).
One vessel (a) was filled with air at
2 atmospheres and the second vessel
(b) was under vacuum.
(c )
(a )
a ir a t
2 a tm
(b )
v a c u u m
in s u la te d w a te r b a th
Joule’s experiment
Joule opened the stopcock allowing the air to move from vessel (a) into the vessel
(b) until the pressure equilibrated at 1 atmosphere. The water bath did not change
temperature, but pressure and volume changed. There was no heat or work transfer,
so the first law for this experiment is:
1Q2
− 1W2 = 0 = U2 − U1
Since pressure and volume changed and the temperature did not, his conclusion was
that internal energy (for a gas) is only dependent upon temperature.
u = f (T ) only!
(ideal gas only)
Definition of Internal Energy
The first law applied to a closed system defines the change in internal energy, ∆U.
The value of internal energy, U or u, at a given temperature is the change relative
to some arbitrary state at which U = 0.
3 The term internal energy, coined by Clausius and Rankine, replaced terms such as inner work,
internal work, and intrinsic energy.
14
Enthalpy
Flow Potential vs Boundary Work
w o r k
c lo s e d s y s te m
The first law of thermodynamics
only applies to a fixed mass; i.e., a
closed system. If we track a fixed
mass through a turbine (modeled as an open system), there
is boundary work associated with
the compressed, high pressure gas
moving to an uncompressed, low
pressure region.
A
tu r b in e
o p e n sy ste m
A
Another simple conceptualization is to examine the total state at time 1 and then
the total state at time 2 as shown, where total state includes the gas volume.
w o r k
e n e r g y
sta te 1
tim e = t
1
e n e r g y
sta te 2
t1 < t < t
tim e = t
2
2
w o r k
A
tu r b in e
tu r b in e
A
Recall from the engine indicators, that the total work output of the fluid passing
through the turbine is related to the change in P∀.
−1 W2 = ∫ Pd∀ = E2 − E1 ,
where E2 −E1 is the change in stored energy of the fluid. For an atmospheric engine,
∫ Pd∀ = P2 ∀2 − P1 ∀1
Often, there is also an accompanying change in temperature related to a heat
transfer to or from the fluid. Therefore, the total change in energy is becomes
(from the 1st Law):
Q − W = E2 − E1 = (U2 − U1 ) + (P2 ∀2 − P1 ∀1 )
15
For continuous flow, the transitional energies become transitional power (thermal
Q̇ and mechanical Ẇ ) and with conservation of mass applied to a single inlet and
outlet the energy balance becomes
Q̇ − Ẇ = ṁ [(u2 − u1 ) + (P2 v2 − P1 v1 )]
Definition of Enthalpy
In open systems, the combination of internal energy, pressure and volume always
appear in the energy balance in the form of U + P∀. For convenience, engineers
defined a new energy, commonly called a heat, which included internal energy plus
flow potential energy.
H = U + P∀ [kJ, BTU]
and on a per mass basis
h = u + Pv
[kJ/kg, BTU/lbm]
This property simplified looking up fluid energy values by reducing the three properties to a single property. This new property was called by many names including
total heat, heat content, and heat of liquid. In the July 1936 issue of Mechanical
Engineering, published by the American Society of Mechanical Engineers (ASME),
the term enthalpy was selected to designate the thermodynamic function defined
as internal energy plus the pressure volume product of any working substance. Enthalpy is from the Greek word “enthalpo”, which means “heat within”. The original
symbol used to designate enthalpy was I , i. The use of I is still common nomenclature in the heating, ventilation and air conditioning (HVAC) industry.
16
Measuring Changes in Energy
The first law of thermodynamics relates changes in transitional energies heat and
work to stored energies of a fluid. For both open and closed systems, the measure
of stored thermal energy in a fluid is related to the internal energy, U. Unfortunately, there is no direct measure of thermal fluid energy. We must infer changes
in fluid energy from measurable parameters that include pressure P, temperature
T , volume ∀ and mass m. Somehow we must relate these four measures to other
measures of fluid energy: u internal energy, h enthalpy, and µ̂ chemical potential.
Specific Heats (c, cv , cp )
In order to relate changes in thermal energy to something measurable, we must use
Conservation of Energy and Mass. Beginning with simple compressible substances
such as air or an ideal gas such as Helium, an experiment can be designed such that
Conservation of Energy reduces to:
Q − W = ∆Ũ = m∆ũ ,
´¹¹ ¹ ¹ ¹¸ ¹ ¹ ¹ ¹¶ ´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¸¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹¶
transitional
(2)
stored in gas
where only work and heat for the transitional energies and only stored thermal energy are considered.
An example of a simple experiment is shown in Figure 1. Heat is applied to a
constant volume container holding a gas. Temperature of the gas is measured
while heat is applied. The subsequent data shows a linear relationship between the
amount of heat added and the change in temperature.
qin =
Qin
= C ∆T ,
mgas
where C is a proportionality constant, which is simply the slope of the experimental
data. For this experiment, no work is applied. Combining the experimental observation with the reduced Conservation of Energy (equation 2) results in a relationship
between internal energy and temperature.
qin = ∆ũ = C ∆T
For infinitesimal changes,
δqin = d ũ = CdT
For this scenario, the proportionality constant C is known as the specific heat at
constant volume and labeled cv .
A common misconception is that cv can only be used when considering constant
volume processes. Specific heats relate measurable property changes, in this case
temperature, to changes in stored energy. Thus, cv relates temperature changes
17
h e a t a p p lie d , q
te m p e ra tu re
g a s
T
1
h e a t
0
o
C
te m p e ra tu re
Figure 1. Conceptual experiment for determining relationship between stored thermal
energy and temperature.
(measurable) to changes in internal energy (not measurable) regardless of the process being considered.
18
19
20
Measurable Properties and Changes in Energy
As a general rule, we can only measure changes in energy. There is no method by
which we cam measure the absolute energy; we can only measure energy relative
to another state or reference, and then only through inference. An example is the
closed system heat addition to an ideal gas discussed in §7.1. The total energy
transferred to the tank containing the ideal gas can be calculated by knowing the
heat transfer rate, or power, and the duration of applying that power.
Qin = Q̇in ∆t
An example would be heat addition via an electric heater in which the electrical
power is measured. From the First Law of Thermodynamics (Conservation of Energy), the heat input is equal to the change in the energy of the ideal gas:
Qin = U2 − U1 = mgas (u2 − u1 )
As heat is added, the measured temperature of the gas increases in proportion to
the quantity of heat.
Qin ∼ T2 − T1 = c (T2 − T1 ) ,
where c is a constant of proportionality. This particular constant of proportionality
between heat addition and temperature increase is known as specific heat. Thus,
by carefully designing and controlling the process of energy change, which in this
case is a constant volume (isochoric) process, the change in energy as expressed
using the first law of thermodynamics can be related to a change in the measurable
property temperature.
State Property Relationships
The primary properties that can be measured are pressure, temperature, volume
and mass. Other useful properties can be derived from these four such as density
(ρ = mass/volume) and specific volume (v = volume/mass). When there is no
change in energy occurring, a state known as equilibrium, then the measurable
properties can be related to one another. An example is the Ideal Gas Law, where
pressure, volume, and mass are found to be proportional to temperature for simple
compressible gases.
P∀
∼ T = RT ,
m
where R is a constant of proportionality known as the gas constant. The ideal
gas law is but one example of numerous relationships between properties of a substance in an equilibrium state, which again means that there is no change in energy
occurring. Another example of such a property relationship is the Non-Ideal Gas
Law.
P∀ = Z mRT ,
where Z is an experimentally determined factor that accounts for non-ideal compression of more complex gases such as butane. Relationships between properties
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measured during a state of equilibrium are collectively known as equations of state.
A simple equation of state for an incompressible fluid (liquid) is m ∼ ∀ = ρ∀. In this
instance ρ is a constant of proportionality that does not change with temperature,
pressure, etc. In other words, this particular equation of state indicates that the
fluid density is independent of all other properties.
There are many equations of state for gases, liquids and solids. When a substance is
near the transition between vapor-liquid, liquid-solid, or solid-gas, then defining an
equation of state becomes extremely difficult. In these cases, where the equilibrium
state is near a phase transition, the relationship between properties is experimentally measured and tabulated. Examples where tabulated property relationships are
common include steam, refrigerants and gases that are near the boiling point such
as hydrogen at 21 K.
Steam tables include three sets of relationships:
(a) vapor in equilibrium, known as superheated vapor which does not behave as an
ideal gas;
(b) vapor and liquid in equilibrium with one another, known saturated liquid/vapor;
and
(c) liquid in equilibrium, known as subcooled or compressed liquid which does not
necessarily behave as an incompressible liquid.
In contrast to an ideal gas, when a saturated liquid is heated there is a change
in the internal energy, but that energy change is not accompanied by an increase
temperature. . . . least is known about subcooled liquid
finish section with process of determining energy changes
The relationship between measurable properties and energies are developed using a
combination of:
1. fluid property relationships
2. first law of thermodynamics
3. heat transfer and fluid mechanics relationships (pitot-static measurement,
hot-wire anemometry, . . . )
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References
[1] Cecil H. Peabody. Manual of the Steam-Engine Indicator. John Wiley & Sons, Inc.,
1914.
[2] Edward F. Obert. Internal Combustion Engines and Air Pollution. Harper & Row,
Publishers, 1973.
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