AME 60634
Int. Heat Trans.
Work Examples
[1] Sliding Block
Δx
F
CM
work done to the control mass
so it is energy gained
[2] Shear Work on a Fluid
Belt
W
shear stress × speed × area
W˙ = t × v x × A
vx
t
CM
Liquid Bath
D. B. Go
é N m 2ù éJ ù
êëm2 s m úû = êë s úû = [W ]
work done to the control mass
so it is energy gained
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AME 60634
Int. Heat Trans.
Work Examples
[3] Boundary Displacement
Gas Expansion
W
p0
boundary work
Δz
W=
CM
p1
Vf
ò p dV [ J]
Vi
work done by the control
mass so it is energy lost
Strain (Compression/Expansion)
F
boundary work
Δz
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CM1
Vf
zf
Vi
zi
(constant area)
W = ò s dV = ò s A dz
work done to the control mass so
it is energy gained
[ J]
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AME 60634
Int. Heat Trans.
Work Examples
[4] Shaft/Propeller
CM
torque × angular speed
W = T ×q
[ W]
work done to the control mass
so it is energy gained
W
[5] Electrical Work (Heat Generation)
CM
Joule (or resistive or Ohmic) heating
R
W
V
W = i ×V = = i 2 R
R
2
[ W]
work done to the control mass
so it is energy gained
+D. B. Go
V
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AME 60634
Int. Heat Trans.
Work Examples
[6] Surface Tension
CM
air
Soap
bubble
surface tension × area change
straw
CM
work done to the control mass
so it is energy gained
Soap film
inside a
wire
movable
wire
ΔA
D. B. Go
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AME 60634
Int. Heat Trans.
Work Examples
[7] Spring Compression
F = kx
dW = Fdx = kxdx
F
xf
Δx
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1
W = ò kx dx = k ( x 2f - xi2 )
2
xi
[ J]
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AME 60634
Int. Heat Trans.
Enthalpy
We can literally define a new specific property enthalpy as the summation of the
internal energy and the pressure × volume (flow work)
h = u + pv ® H =U + pV
Porter, 1922
Thus for open systems, the first law is frequently written as
æ 1 2
ö
æ 1 2
ö
dECV
= Q -Wnet + å min ç h + vx + gz ÷ - å mout ç h + vx + gz ÷
è 2
øin out
è 2
øout
dt
in
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AME 60634
Int. Heat Trans.
Property, State, and Process
• Property is a macroscopic characteristic of the system
• State is the condition of the system as described by its properties.
• Process changes the state of the system by changing the values of
its properties
– if a state’s properties are not changing then it is at steady state
– a system may undergo a series of processes such that its final and
initial state are the same (identical properties) – thermodynamic cycle
• Phase refers to whether the matter in the system is vapor, liquid, or
solid
– a single type of matter can co-exist in two phases (water and steam)
– two types of matter can co-exist in a single phase (a water/solvent
mixture)
• Equilibrium state occurs when the system is in complete
mechanical, thermal, phase, and chemical equilibrium  no
changes in observable properties
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AME 60634
Int. Heat Trans.
Properties
• extensive properties (dependent on size of system)
– U internal energy
– V volume
– S entropy
[kJ]
[m3]
[kJ/K]
H enthalpy (total energy) [kJ]
m mass
[kg]
• intensive properties (independent of size of system)
–
–
–
–
 density
T temperature
p pressure
x quality
[kg/m3]
[K]
[Pa]
[-]
• specific properties: the values of extensive properties per unit of mass
of the system [kg-1] or per unit mole of the system [kmol-1]
(inherently intensive properties)
– u specific internal energy [kJ/kg]
– v specific volume [m3/kg]
– s specific entropy [kJ/(kg-K)]
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h specific enthalpy
h = u + pv
[kJ/kg]
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AME 60634
Int. Heat Trans.
Pure Substances, Compressible Systems
p-v-T Relationship
seek a relationship between pressure, specific volume, and temperature
• from experiment it is known that temperature and specific volume are
independent
• can establish pressure as a function of the others
p = f (v,T )
p-v-T surface
water
single phase: all three properties are
independent (state fixed by any two)
two-phase: properties are dependent on
each other (state fixed by specific volume
and one other)
• occurs during phase changes
saturation state: state at which phases
begins/ends
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AME 60634
Int. Heat Trans.
Pure Substances, Compressible Systems
p-v-T Surface Projections
phase diagram
• two-phase regions are lines
• triple line is a triple point
• easily visualize saturation
pressure & temperature
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p-v diagram
• constant temperature lines
(isotherms)
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AME 60634
Int. Heat Trans.
Pure Substances, Compressible Systems
p-v-T Surface Projections
T-v diagram
• constant pressure lines (isobars)
• quality x denotes the ratio of vapor to total mass in two-phase mixture
two-phase properties from saturation
properties
m
x=
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vapor
mvapor + mliquid
v = (1- x ) v f + xvg
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AME 60634
Int. Heat Trans.
Phase Changes
• vaporization/condensation – change from liquid to gas and vice versa
• only occurs below critical point
• above critical point, the distinction between the two states is not clear
• melting/freezing – change from solid to liquid and vice versa
• only occurs above triple point
• below triple point, the liquid state is not possible and solids change directly
to gas (sublimation)
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AME 60634
Int. Heat Trans.
Evaluating Liquid Properties
For liquids, specific volume and
specific internal energy are
approximately only functions of
temperature
v(T,p) ≈ vf(T)
u(T,p) ≈ uf (T)
h(T,p) ≈ uf (T)+pvf(T)
When the specific volume v varies little
with temperature, the substance can be
considered incompressible
it follows
(saturated liquid)
æ ¶u ö du
cv = ç ÷ =
è ¶T øv dT
æ ¶h ö
du
h = u (T ) + pv ® cp = ç ÷ =
è ¶T ø p dT
cp = cv
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incompressible
thus
liquids
Changes in u and h can be found by
direct integration of specific heats
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AME 60634
Int. Heat Trans.
Compressibility Factor
Compressibility Factor
universal gas
constant
R
pv
Z=
RT
8.314 kJ/kmol∙K
1.986 Btu/lbmol∙oR
1545 ft∙lbf/lbmol∙oR
R
R=
M
(molecular weight)
At states where the pressure p is
small relative to the critical
pressure pc (where pR is small),
the compressibility factor Z is
approximately 1.
Virial equations of state:
D. B. Go
Z =1+ B̂ (T ) p + Ĉ (T ) p2 + D̂ (T ) p3 +...
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AME 60634
Int. Heat Trans.
Evaluating Gas Properties
At states where the pressure p is
small relative to the critical
pressure pc (where pR is small),
the compressibility factor Z is
approximately 1.
For ideal gas, specific internal
energy and enthalpy are
approximately only functions of
temperature
Specific heat
du
cv =
dT
Z =1® pv = RT
u(T,p) ≈ u(T)
h(T,p) ≈ u(T)+pv = u(T)+RT
and
ideal gas
≈ h(T)
dh
cp =
dT
Changes in u and h can be found by
direct integration of specific heats
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AME 60634
Int. Heat Trans.
Heat Transfer
• Heat Transfer is the transport of thermal energy due to a
temperature difference across a medium(s)
– mediums: gas, liquid, solid, liquid-gas, solid-gas, solid-liquid, solid-solid,
etc.
– Thermal Energy is simply the kinetic energy (i.e. motion) of atoms and
molecules in the medium(s)
• Atoms/molecules in matter occupy different states
– translation, rotation, vibration, electronic
– the statistics of these individual molecular-level activities will give us
the thermal energy which is approximated by temperature
• Heat Transfer, Thermal Energy, and Temperature are
DIFFERENT. DO NOT confuse them.
• Heat generation (electrical, chemical, nuclear, etc.) are not forms of
heat transfer Q but forms of work W
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– Q is the transfer of heat across the boundary of the system due to a
temperature difference
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AME 60634
Int. Heat Trans.
Definitions
Quantity
Thermal Energy
Meaning
Symbol/Units
Energy associated with
molecular behavior of
matter
U [J] – extensive property
u [J/kg] – intensive property
Temperature
Means of indirectly assessing
the amount of thermal energy
stored in matter
T [K] or [°C]
Heat Transfer
Thermal energy transport due
to a temperature gradient
(difference)
various
Heat
Heat Rate/Heat Flow
Heat Flux
D. B. Go
Thermal energy
transferred over a time
interval (Δt > 0)
Q [J]
Thermal energy
transferred per unit time
˙ [W]
q, q˙, Q
Thermal energy
transferred per unit time
per unit surface area
q¢¢ [W m2 ]
Heat
Transfer
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AME 60634
Int. Heat Trans.
Modes of Heat Transfer
• Conduction & convection require a temperature difference across a
medium (the interactions of atoms/molecules)
• Radiation transport can occur across a vacuum
D. B. Go
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