3.1 Introduction to the First Law of Thermodynamics 3.2 Heat Transfer

3.1 Introduction to the First Law of Thermodynamics
Simply stated, the First Law of Thermodynamics is a conservation of energy principle. In a
thermodynamic sense, the energy gained by a system is exactly equal to the energy lost by the
surroundings (look at figure below) ...
system
T=400 K
HEAT
surroundings, T=290 K
Energy can cross the boundary in a closed system in the form of heat and/or work. These energy forms
are different, making it important to be able to distinguish between them.
3.2 Heat Transfer
Heat is defined as the form of energy that is transferred between two systems (or a system and
its surroundings) by virtue of a temperature difference. That is, an energy interaction is heat if and only if
it occurs because of a temperature difference. Heat is energy in transition. It is only recognized as it
crosses the boundary of a system.
A process during which there is no heat transfer is called an adiabatic process. There are two
ways a process can be adiabatic: either the system is well-insulated so that no heat flows across the
boundary, or the system and surroundings are both at the same temperature (zero temperature
difference).
Heat transfer per unit mass, q, of a system is defined as:
(3.2.1)
The total heat transferred to/from a pure substance between states 1 and 2 can be expressed in
terms of the heat transfer rate as:
(3.2.2)
where Q . is the heat transfer rate per unit time. For constant Q . , the equation above becomes ...
Q = Q . t
(3.2.3)
ENGS205--Introductory Thermodynamics page 29

Very Important Sign Convention:
Since heat is a directional quantity (it can be lost or gained), it is very important to establish a
sign convention! Heat transfer to a system is positive and heat transfer from a system is negative!
Heat In
SYSTEM
Heat out
Q=5 kJ
Q=-5 kJ
Heat can be transferred in three different ways: conduction, convection, and radiation. Below is
a brief description to familiarize you with the basic mechanisms of heat transfer.
‚ Conduction Heat Transfer: The transfer 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 takes place in solids, liquids, and gases. The rate of heat conduction, Q . cond , is given by
Fourier's Law of heat conduction:
(3.2.4)
where k is a constant of proportionality called the thermal conductivity and A is the area normal to the
direction of heat transfer. Fourier's law indicates that the rate of heat conduction in a direction is
proportional to the temperature gradient in that direction.
Note!
Have you ever wondered why sitting in a cold steel chair feels colder than sitting in a wooden
chair at the same temperature? The cold sensation you feel when sitting in the chair is due to heat being
conducted from your body to the chair. The thermal conductivities of carbon steel and wood are,
respectively, 60.5 and 0.17 W/(m K). This means you lose 350 times more heat from your body by sitting
in the steel chair as opposed to the wooden chair, initially !!
‚ Convection Heat Transfer: The mode of energy transfer between a solid surface and an
adjacent liquid or gas which is in motion, and it involves the combined effects of conduction and fluid
motion. The faster the fluid motion, the greater the convection. In the absence of bulk fluid motion, the
heat transfer between the solid and fluid occur via conduction. There are two specific kinds of
convection:
v Forced convection: occurs when the fluid is forced to flow over a surface by external
means such as a fan, blower, or the wind.
v Free (or natural) convection: occurs if fluid motion is caused by buoyancy forces
induced
by density differences due to the variation of temperature in the fluid. This is
how a
baseboard heater works.
Heat transfer processes involving change of phase of a fluid are considered convection
processes (including boiling and condensation).
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The rate of heat transfer by convection, Q . conv , is determined from Newton's Law of Cooling, which is
expressed as:
Q . conv = hA(T s − T f )
(3.2.5)
where h is the convection heat transfer coefficient, A is the surface area through which heat transfer
takes place, T s is the surface temperature, and T f is the bulk fluid temperature away from the surface.
‚ Radiation Heat Transfer: The energy emitted by matter in the form of electromagnetic
waves (or photons) as a result of the changes in the electronic configurations of the atoms or molecules.
Unlike conduction and convection, the transfer of energy by radiation requires no intervening medium. In
heat transfer, we're interested in thermal radiation, or radiation emitted by bodies because of their
temperature.
Thermal radiation is a volumetric phenomenon, and all solids, liquids, and gases, emit, absorb,
and transmit thermal radiation to varying degrees. For opaque solids, thermal radiation is a surface
phenomenon and their maximum emission of radiant energy is given by the Stefan-Boltzmann law:
Q . emit,max = AT s
4
(3.2.6)
where A is the surface area and σ = 5.67 x 10 -8 W/(m 2 K 4 ) is the Stefan-Boltzmann constant. The
idealized surface which emits radiation at this maximum rate is called a blackbody. The radiation of real
surfaces is less than radiation emitted by a blackbody at the same temperature and is expressed as:
Q . emit = AT s
4
(3.2.7)
where ε is the emissivity of the surface. The property emissivity, whose value is in the range 0 1is
a measure of how closely a surface approximates a blackbody for which ε = 1. Another important
property of a surface is called the absorptivity, α, which is the fraction of radiant energy incident on a
surface which is absorbed by the surface. A blackbody absorbs all incident radiation (α=1). In general,
both ε and α depend on temperature and wavelength of radiation. Kirchoff's law of radiation states that
the emissivity and absorptivity of a surface are equal at the same temperature and wavelength. In most
practical applications, the temperature and wavelength dependencies of surface emissivity and
absorptivity is ignored and the average emissivity is taken as the average absorptivity.
In general, the determination of the net rate of heat transfer by radiation is a very complicated
matter since it depends on the properties of the surfaces, the orientation relative to each other, and the
intervening medium. However, the following simple equation is used to get a ballpark figure as to the net
radiative transfer between a system and its surroundings:
Q . rad,net =
A⎛ ⎝
T 4 4 s − T surr
⎞ ⎠
(3.2.8)
where T s is the surface temperature and T surr is the temperature of the surroundings. All temperatures
used in radiation calculations must be absolute!
Note!
In the winter, people in a room with the curtains open may experience a faint chilly feeling. The
window glass separating the people in the room and the cold surroundings outside is semi-transparent.
That is, the windows can partially transmit thermal radiation. This causes people in the room to radiate to
the surroundings, producing that uncomfortable feeling! (Drawing the curtains should alleviate this
problem).
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3.3 Work
Work, like heat is an energy interaction between a system and its surroundings. More
specifically, work is the energy transfer associated with a force acting through a distance( e.g. rotating
shaft, rising piston, etc ...). The work done per unit mass, w, of a system is defined as:
The work done per unit time is called power and is denoted by Ẇ.
Very Important Sign Convention:
(3.3.1)
The sign convention used by your text asserts work done by a system as being positive and
work done on a system as being negative.
(+)
Work done
by system
SYSTEM
(-)
Work done
on system
Heat transfer and work are interactions between a system and its surroundings, and there are
many similarities between the two:
‚ Both are recognized at the boundaries of the system as they cross them. That is, both heat
transfer and work are boundary phenomena.
‚ Systems possess energy, but not heat transfer or work. That is, heat transfer and work are
transient phenomena.
‚ Both are associated with a process, not a state. Unlike properties, heat transfer or work has
no meaning at a state.
‚ Both are path functions (i.e. their magnitudes depend on the path followed during a process
as well as the end states).
Path functions have inexact differentials designated by the symbol δ . Therefore, a differential
amount of heat or work is represented by δQ or δW , respectively, instead of dQ or dW. Properties (like
temperature, pressure, specific volume, etc ...), however, are point functions (i.e., they depend on the
state only, and not on how a system reaches that state), and they have exact differentials designated by
the symbol d.
The work done on or by a system between two states is represented as W 12 , not ∆W!
The same goes for heat.
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Electrical Work:
The flow of electrons, usually through a wire, across a system boundary forms the basis for
electrical work. When N coulombs of electrons move through a potential difference V, the electrical work
done is:
W e = VN
(3.3.2)
which can also be expressed in the rate form as:
Ẇ e = VI
(3.3.3)
where
Ẇ e
is the electrical power and I is the electrical current (usually measured in amperes, A).
.
W e
= VI
= I 2
R
R
2
= V /R
I
V
In general, both V and I vary with time, and the electrical work done during a time interval ∆t is:
W e =
2
1
VIdt
... and for constant V and I during ∆t ... W e = VI t (3.3.4)
3.4 Mechanical Forms of Work
There are several ways of doing work, each in some way related to a force F acting through a
distance s ...
W =
2
1
Fds
... or, for a constant force F ... W = Fs (3.4.1)
There are two requirements for a work interaction between a system and its surroundings to exist:
‚ There must be a force acting on the boundary.
‚ The boundary must move.
Some important forms of mechanical work are listed below ...
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Moving Boundary Work:
One form of work frequently 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 (as in an automobile engine). Therefore, the expansion and
compression work is often called the moving boundary work, or boundary work.
F
ds
P
The boundary work between two states is the area under the P-V diagram, or ...
W b =
2
1
PdV
(3.4.2)
... or, for an isobaric process in a piston-cylinder device ...
W b = P V
(3.4.3)
Polytropic Process:
During expansion and compression processes of real gases, pressure and volume are often
related by PV n = C, where n and C are constants. A process of this kind is a polytropic process.
From Eq. (3.4.2), the boundary work is ...
... and for ideal gases (PV=mRT), this equation can also be written as:
Question to think about!
(3.4.5)
(3.4.4)
In the boundary work expressions given above (Eqs. (3.4.4) and (3.4.5)), what happens to the
boundary work when the constant n is equal to 1?
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Gravitational Work:
Gravitational work can be defined as the work done by or against a gravitational force field.
The work required to raise a body from level z 1 to level z 2 is:
(3.4.6)
This expression is the change in potential energy! Gravitational work is positive if it is done by the
system (as the system falls) and negative if done on the system (as the system is raised).
Accelerational Work:
The work associated with the change in velocity of a system is called accelerational work. By
combining Newton's second law (F=ma) and Eq. (3.4.1), the following accelerational work expression is
obtained:
(3.4.7)
This expression is recognized as being the change in kinetic energy of mass m! As always, this form of
work is positive if done by the system and negative if done on the system.
Shaft Work:
Energy transmission via a rotating shaft is known as shaft work. The shaft work generated by
applying a constant torque τ for n revolutions of the shaft is given by:
W sh = 2 n
(3.4.8)
... and the power transmitted through the shaft is:
Ẇ sh = 2 ṅ
(3.4.9)
where ṅ is the number of revolution per unit time.
Spring Work:
The work performed when displacing a spring from its equilibrium position is known as spring
work. The spring work is obtained by combining Hooke's Law (F=kx) and Eq. (3.4.1) giving the following
expression for a linear (k is constant) elastic spring ...
W spring = 1 2 k ⎛ ⎝
x 2 2 − x 1
2 ⎞ ⎠
(3.4.10)
where x 1 and x 2 are initial and final displacements of the spring measured relative to its equilibrium
position.
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3.5 The First Law of Thermodynamics
The first law of thermodynamics , a simple re-statement of the conservation of energy
principle for a closed system or control mass, may be expressed as follows:
Q − W = E
Net energy transfer
Net increase (or
to (or from) the system decrease in the total
as heat and work energy of the system
(3.5.1)
where: Q = net heat transfer across system boundaries ⎛ ⎝
Q = Q in − Q out
⎞ ⎠
W = net work done in all forms ( W = W out − W in )
∆E = net change in total energy of system ( E = E 2 − E 1 )
As discussed previously, the total energy of a system is considered to be comprised of three parts:
internal energy U, kinetic energy KE, and potential energy PE. The change in total energy ∆E is:
E = U + KE + PE
(3.5.2)
where: U = m(u 2 − u 1 )
For stationary closed systems, the changes in kinetic and potential energies are negligible, and the
first-law relation reduces to:
Other Forms of the First-Law Relation:
Q − W =
U
(3.5.3)
... on a unit-mass basis: q − w = e
(3.5.4)
... on a time rate basis: (3.5.5)
... in differential forms: Q − W = dE
(3.5.6)
q − w = de
(3.5.7)
For a cyclic process, the initial and final states are identical (∆E=0). Therefore,
Q − W = 0
Q = W
(3.5.8)
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3.7 Specific Heats
The specific heat is defined as the energy required to raise the temperature of a unit mass of a
substance by one degree. In general, this energy depends on how the process is executed. In
thermodynamics, we are interested in two kinds of specific heats:
‚ specific heat at constant volume C v : the energy required to raise the temperature of a unit
mass of a substance by one degree as the volume is maintained constant.
V = constant
3.13 kJ
m = 1 kg
∆ T = 1 K
C v
= 3.13 kJ/(kg K)
‚ specific heat at constant pressure C p : the energy required to raise the temperature of a
unit mass of a substance by one degree as the pressure is maintained constant.
P = constant
m = 1 kg
∆ T = 1 K
C p = 5.2 kJ/(kg k)
Important Point!
5.2 kJ
The value of C p will always be greater than C v for a pure substance because the constant
pressure system is allowed to expand. Extra energy must be supplied to the system to account for the
expansion (boundary) work.
Consider the differential form of the first-law (on a per-mass basis) ...
q − w other − w b = de
(3.7.1)
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For a constant volume process (δw b =0) with no work interactions, the following must be true ...
q = C v dT = du
... or ... C v = ⎛ u ⎞ ⎝ T
(3.7.2)
⎠ v
For a constant pressure process (δw b =Pdv) with no work interactions, we have ...
q = C p dT = du + Pdv = dh
... or ... C p = ⎛ h ⎞ ⎝ T
(3.7.3)
⎠ p
Note that C v is the change in specific internal energy u per unit change in temperature at
constant volume and C p is the change in specific enthalpy h per unit change in temperature at constant
pressure. Generally, specific heats vary with temperature. That is to say, the amount of energy it takes
to raise the temperature of a unit mass of a pure substance by degree (for constant volume & pressure
processes) varies at different body temperatures. However, this difference is usually not very large.
Specific heat are sometimes given on a molar basis, and are denoted by C v and C p . Specific
heats are properties of pure substances and their values can be found in engineering tables.
3.8 Internal Energy, Enthalpy, and Specific Heats of Ideal Gases
It has been demonstrated experimentally (and proven theoretically) that the internal energy and
enthalpy of an ideal gas is a function of its temperature only. Thus for ideal gases, the partial
derivatives in Eqs. (3.7.2) and (3.7.3) can be replaced with ordinary derivatives and we get:
du = C v (T)dT u = u 2 − u 1 =
dh = C p (T)dT h = h 2 − h 1 =
2
1
2
1
C v (T)dT
C p (T)dT
(3.7.4)
(3.7.5)
To carry out these integrations, we need to have relations for C v and C p as a function of
temperature. At low pressures, all real gases approach ideal-gas behavior, and therefore their specific
heats depend on temperature only. The specific heats of real gases at low pressures are called
ideal-gas specific heats, or zero-pressure specific heats , and are often denoted C v0 and C p0 .
These zero-pressure specific heat data are available in the Appendix of your text as a function of
temperature and can be used for real gases as long as they don't deviate from ideal-gas behavior
significantly (check the compressibility factor! ).
Another way of evaluating the change in specific internal energy/enthalpy is to refer to tabulated
data. Tables A17-A20 in the Appendix contain molar internal energy/enthalpy data for air, O 2 , N 2 , CO 2 ,
H 2 , and H 2O. And yet another way involves finding an average zero-pressure specific heat value (look at
Table A-2b in the Appendix) for an average temperature of (T 1 + T 2)/2. Specific internal
energies/enthalpies are then computed using the following formulas:
u 2 − u 1 = C v0,av (T 2 − T 1 )
h 2 − h 1 = C p0,av (T 2 − T 1 )
(3.7.6)
(3.7.7)
ENGS205--Introductory Thermodynamics page 29

Specific-Heat Relations of Ideal Gases:
For ideal gases, the following relationships hold:
C p = C v + R
... and ... C p = C v + R u
(3.7.8)
where R and R u are the gas constant and universal gas constant, respectively. The specific heat ratio k
is defined as ...
(3.7.9)
3.9 Internal Energy, Enthalpy, and Specific Heats of Solids
and Liquids
A substance whose specific volume (or density) is constant is called an incompressible
substance. The specific volumes of solids and liquids essentially remain constant during a process and
are usually regarded as being incompressible (e.g. hydraulics). It can be mathematically shown that the
constant-volume and constant-pressure specific heats for incompressible substances are identical.
Therefore,
C p = C v = C
(3.9.1)
and the change in specific internal energy u of an incompressible substance is:
du = C(T)dT u = u 2 − u 1 =
2
1
C(T)dT C av (T 2 − T 1 )
(3.9.2)
The enthalpy change of incompressible substances (solids or liquids) during a process can be determined
from the definition of enthalpy (h=u+Pv) to be:
h 2 − h 1 = (u 2 − u 1 ) + v(P 2 − P 1 ) u 2 − u 1
(3.9.3)
The last term of Eq. (3.9.3) is usually very small and can be neglected.
ENGS205--Introductory Thermodynamics page 29

ENGS205--Introductory Thermodynamics page 29