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OVERVIEW OF
MACROSCOPIC THERMAL SCIENCES
FUNDAMENTALS OF THERMODYNAMICS ▪ The First Law of Thermodynamics
▪ Thermodynamic Equilibrium and the Second Law
▪ The Third Law of Thermodynamics THERMODYNAMIC FUNCTIONS AND PROPERTIES ▪ Thermodynamic Relations
▪ The Gibbs Phase Rule
▪ Specific Heats
IDEAL GAS AND IDEAL INCOMPRESSIBLE MODELS ▪ The Ideal Gas
▪ Incompressible Solids and Liquids
HEAT TRANSFER BASICS
▪ Conduction
▪ Convection
▪ Radiation
Fundamentals of Thermodynamics
system
environment or surroundings
constraints
(external forces)
parameters
constituent constitue
ntconstitue
nt
Definitions of Thermodynamic Terms
property quantities that characterize the behavior of a system at any instant of time. The property must be measurable and their values are independent of measuring device spontaneous change of state
change of state that does not involve any interactionbetween the system and its environment.
induced change of state change of state through interaction with other systemin the environment
isolated system a system which can experience only spontaneouschange of state
kinematics study of the possible and allowed states of a system
dynamics study of the time evolution of the state
equation of motion relation that describes the change of state of a system as a function of timecomplete description often unknownthermodynamic description: in terms of the end states and the modes of interaction (work and heat) process specified by the end states and the modes of interaction
reversible processat least one way to restore both the system and its environment to their initial states
irreversible processnot possible to restore both the system and its environment to their initial states
steady statea state that does not change as a function of time despite interactions between the system and other systems in the environment
2 1 net,inE E E E in outE E
net,indE E
kinetic energy, potential energy, internal energy
conservation of energyEnergy can be transferred to or from a system butcan be neither created nor destroyed.
energy balance for a system
for an infinitesimal change
The First Law of Thermodynamics
Thermodynamic Equilibrium and the Second Law equilibrium :
a state that cannot change spontaneously with time
state principle (stable-equilibrium-state principle)Among all state of system with a given set of values of energy, parameters, and constituents, there exists one and only one stable-equilibrium state. All properties are uniquely determined by the amount of energy, the value of each parameter, and the amount of each type of constituents
thermodynamic equilibrium
unstablemetastablestable :
the second law
In an isolated system, entropy cannot be destroyed but can either be created (irreversible process) or remain the same (reversible process).Entropy can be transferred from one system to another.
Q dE W 1
Tintegrating factor
Q dE W
T T T
dS
2 1 net,in gen ,S S S S S gen 0S
net,in gen , dS S S gen 0S
summary of the second law of thermodynamics
There exist a unique stable-equilibrium state for any system with given values of energy, parameters, and constituents.
Entropy is an additive property, and for an isolatedsystem, the entropy change must be nonnegative.Among all states with the same values of energy,parameters, and constituents, the entropy of the stable-equilibrium state is the maximum.
highest entropy principle The entropy of the system is largest in the stable-equilibrium state.
Internal energy (U): energy of a system with volume (V) as its only parameter
definitions of temperature, pressure and chemical potential
fundamental relations
1 2( , , , , , )rS S U V N N N
1 2( , , , , , )rU U S V N N N
' ' ', , , , ( )
, ,j s j s j s
iV N S N i S V N j i
U U UT P
S V N
r + 2 independent variables by the state principleentropy: a property of the system
Gibbs relation
1
r
i ii
TdS PdV dN
1
1 ri
ii
PdS dU dV dN
T T T
' ' ', , , , ( )
1, ,
j s j s j s
i
V N U N i U V N j i
S P S S
T U T V T N
1 2( , , , , , )rU U S V N N N
, , ,1, , , ,j s j s j s
r
iiV N S N i S V N j i
U U UdU dS dV dN
S V N
E
S0gS
gEK 0gT
S
ET
1A 10A
'10A
The Third Law of Thermodynamics
spontaneous change of statestable equilibrium state curve
lowest energy principle
adiabatic availability
Eg > 0: ground-state energy
Unique stable equilibrium state exists at zero absolute temperature (Nernst theorem).
Thermodynamic Relations
H U PV
Thermodynamic Functions and Properties
dH dU PdV VdP 1
r
i ii
dU TdS PdV dN
1
r
i ii
TdS VdP dN
1 2( , , , , , )rH H S P N N N
' ' ', , , , ( )
, ,j s j s j s
iP N S N i S P N j i
H H HT V
S P N
: characteristic function
enthalpy
A U TS
1
r
i ii
dA SdT PdV dN
' ' ', , , , ( )
, ,j s j s j s
iV N T N i T V N j i
A A AS P
T V N
Gibbs free energy
G U PV TS H TS A PV
1
r
i ii
dG SdT VdP dN
' ' ', , , , ( )
, ,j s j s j s
iP N T N i T P N j i
G G GS V
T P N
Helmholtz free energy
homogeneous state:
All subsystems are exactly identical to each other. simple system:
a system that experiences only homogeneous states
for k equal-volume subsystems
1 21 2, , , , , ( , , , , , )r
r
N N NU VT T U V N N N
k k k k k
1 21 2
1, , , , , ( , , , , , )r
r
N N NU VS S U V N N N
k k k k k k
intensive property: T, P, j
extensive property: U, S, V, N
Euler relation for r = 1,
The chemical potential of a pure substance is specific Gibbs free energy.
( , ) ( , )G
T P g T PN
For a simple system,
1
r
i ii
dU TdS PdV dN
1
r
i ii
U TS PV N
1 1
r r
i i i ii i
dU TdS SdT PdV VdP dN N d
1
0r
i ii
SdT VdP N d
Gibbs-Duhem relation
G U PV TS N
Euler relation
specific propertythe ratio of an extensive property to the total amount of constituents (mass, mole, or number)
The Gibbs Phase Rule
independent variables T, P, i (i = 1, 2, 3, …, r)
for a q-phase heterogeneous state
Gibbs-Duhem relation1
0r
i ii
SdT VdP N d
number of independent variable reduced to
2r q Gibbs phase rule
for a pure substance, 1, 1r q 2 a single-phase state,
a two-phase state, 1, 2r q 1 a three-phase state, T, P, are all fixed: triple point
r + 2: independent variables, q: number of equations
phase : collection of all subsystems that have the same values of all intensive properties
P – T diagram for a pure substance
P
T
Solid
Liquid
S-V line
L-V li
ne
Vapor
Triple point
Critical point
S-L
lin
e
T – v diagram for a material that expands upon melting
Specific volume v
Tem
pera
ture
, T
P = Pc
P> Pc
P< Pc
P < Pt.
p.
Solid
Liquid
S-L
reg
ion
L-V dome
Triple-point line
Sublimation
S-V region
Saturated
Liquid
Saturated
Vapor
(Tc,Pc)
Specific Heats
vV v
u sc T
T T
pP P
h sc T
T T
heat reservoiran idealized system that experiences only reversible heat interactions. For any finite amount of energy transfer, its temperature remains unchanged. R,2 R,1 R R,2 R,1E E T S S
specific heat at constant volume
specific heat at constant pressure
( , , ) 0f T P v or ( , )v v T P
P T
s sds dT dP
T P
( , )p
P
c T P vdT dP
T T
( , ) ,pP
vdh c t P dT v T P dP
T
equation of stateFor pure substance in a single phase, all propertiescan be expressed as function of T and P.
Ideal Gas and Ideal Incompressible Modelsfrom molecular view,
intermolecular potential energyassociated with the forces between molecules and depends on the magnitude of the intermolecular forces and the position at any instant of time
molecular kinetic energyassociated with the translational velocity of individual molecules
intra-molecular energy (within the individual molecules)associated with the molecular and atomic structure and related forces
intermolecular potential energy
Ideal gas
The Ideal Gas
Impossible to determine accurately the magnitude because either the exact configuration nor orientation of the molecules is not known at any time or the exact intermolecular potential function. Two situations which lead good approximationsAt low or moderate densities: The molecules are relatively widely spread, so that two-molecules or two- and three- molecule interactions contribute to the potential energy.At very low densities (high Temperature & very low pressure): Average intermolecular distance between molecules is so large that the potential energy may be assumed to be zero. ⇒ The particles would be independent of one another.
real Gas
Z depends on the temperature & pressure.
,PV nRT Pv RT kg
kg/kmol
mn
M
kN m kJ8.314 8.314
kmol K kmol KR
PvZ
RT compressibility
factor
equation of state
real gas: affected by intermolecular force Van der Waals
2
aP v b RT
v
Virial
2
( ) ( )1 .....
Pv B T C T
RT v v
Beattie-Bridgman
2 2
(1 )( )
RT AP v B
v v
0 0 3(1 ), (1 ),a b C
A A B Bv v vT
( )u u Tfor ideal gas
v T
u udu dT dv
T v
vc dT
P T
h hdh dT dP
T P
pc dT
( )h u T Pv ( )u T RT
p v
dh duc R c R
dT dT
perfect gas,
( )vc T = constant
2 1 2 1 ,vu u c T T 2 1 2 1ph h c T T
Mayer relation
entropy
Tds du Pdv
du Pds dv
T T v
dT Rc dv
T v
2 12 1 1
2
lnv
vdTs s c R
T v
Tds dh vdP
dh vds dP
T T p
dT Rc dP
T P
2 22 1 1
1
lnp
PdTs s c R
T P
Incompressible Solids and Liquids
equation of state for incompressible solids and liquids
T T P
h s vT v T v
P P T
constantT
hv
P
v constant
0 0 ( )h h v P P f T
u h Pv
0 0 0 0u u h Pv h P v 0 0 ( )h h v P P f T
( ),v
v
u df Tc
T dT
( )
pP
h df Tc
T dT
( ) ( )v pc T c T
What is heat ?
oscillation of atoms about their various positions of equilibrium (lattice vibration): The body possesses heat.
in a solid body
conductors: free electrons ↔ dielectics
crystal : a three-dimensional periodic array of atoms
Heat Transfer Basics
us-1 us us+1 us+2
s s+1 s+2 s+3s-1
us-1 us us+1 us+2 us+3
the energy of the oscillatory motions: the heat-energy of the body
more vigorous oscillations: the increase in temperature of the body
vibration of crystals with an atom
longitudinal polarization vs. transverse polarization
• molecular translation, vibration and rotation• change in the electronic state• intermolecular bond energy
in a gas
the storage of thermal energy:
Internuclear separation distance(diatomic molecule)
En
erg
y
dissociation energy for state 1
dissociation energy for state 2
electronic state 2
vibrational state
rotational state
electronic state 1
at T = 300 K, air M = 28.97 kg/kmol
= 468.0 m/s1 22 /
mu
average kinetic energy
kB = 1.3807 × 10-23 J/K
21 3
2 2u m BE mu k T
heat transfer
Heat transfer is the study of thermal energy transport within a medium or among neighboring media by
molecular interaction: conduction
fluid motion: convection
electromagnetic wave: radiation
energy carriers: molecule, atom, electron, ion, phonon (lattice vibration), photon (electro-magnetic wave)
resulting from a spatial variation in temperature.
continuum hypothesis
Ex) density
local value of density
macroscopic uncertainty
microscopic uncertainty
9 30 10 mmV
(3×107 molecules at sea level, 15°C, 1atm)
m
V
V
0
limV V
m
V
macroscopic uncertainty
microscopic uncertainty
due to molecular random motion
due to the variation associated with spatial distribution of density
In continuum, velocity and temperature vary smoothly. → differentiable
bulk motion vs molecular random motion
mean free path of air at STP (20°C, 1 atm)
m = 66 nm, 1/ 22 468.0 m/smu
ad
i ab
ati
c
wal l
cold wall at Tc
L
a) m << L : normal pressure
b) m ~ L : rarefied pressure
c) m >> L
gas
ad
iab
atic
w
all
hot wall at Th
local thermodynamic equilibrium
0Dm
Dt
MV
DdV
Dt
for a fixed volume in space
CV
u dVt
Since V can be chosen arbitrary
0ut
• continuity eq.: mass conservation,dm dV
,m V
dSMV
Ddm
Dt MV
DdV
Dt
ˆCV CS
dV u ndSt
CV CV
dV u dVt
governing equations
u
n
0u u ut t
0D
uDt
0j
j
uD
Dt x
in Cartesian tensor notation
incompressible flow
0D
Dt
0u
or 0j
j
u
x
Du
Dt t
rate change of momentum = forces exerted on the body
n
n
f
• stress tensor
body force [N/m3]
surface force [N/m2]
f
ˆ( , , )n x t
• forces
n
• momentum eq.: Newton’s 2nd law of motion
i ij jn or
n
f
dm
u
MVud
D
Dm
t
udm
MV
DudV
Dt
ˆCV CS
u dV u u n dSt
CV CV
u dV uu dVt
CVu uu dV
t
u uut
u
u u u u ut t
uu u u u
t t
Du
Dt
Momentum theorem
MV
DudV
Dt
CV
DudV
Dt
CV
DudV
Dt
ˆ
CV CSfdV ndS
CV CVfdV dV
0CV
Duf dV
Dt
Duf
Dt
For a Newtonian fluid
*pI pI u I u u
jk iij ij ij ij ij
k j i
uu up p
x x x
ijii
j
Duf
Dt x
or
n
f
dm
u
ij jk iij ij
j j k j i
uu up
x x x x x
,ijj i
pp
x x
,k kij
j k i k
u u
x x x x
j ji i
j j i j j j i
u uu u
x x x x x x x
ji k ii
i i k j j j i
uDu u upf
Dt x x x x x x x
*Duf p u u u
Dt
For a constant , fluid
ji k ii
i i k j j j i
uDu u upf
Dt x x x x x x x
jii
i j j i j
uupf
x x x x x
2Duf p u u
Dt
For an incompressible flow
2Duf p u
Dt
Stokes’ hypothesis
2
3
2
3
Duf p u u
Dt
• energy eq.: 1st law of thermodynamics
Q dE W
rate equationQ dE W
dt dt dt
dE Q W
dt dt dt
or
21,
2MVE e v dV
2u u v
ˆMV CV CS
DE DdV dV u n dS
Dt Dt t
CV
DdV
Dt
n
f
dm
u
q
q
e : internal energy
: total energy
21
2e v
Q
Dt
n
f
dm
u
q
q
W
Dt
ˆ ˆCV CS CV CS CV
DdV q ndS qdV n udS f udV
Dt
ˆ ˆij j i ji i j ji j i ij j iij ij
n u n u n u u n u n u n
ˆ ˆCS CS CV
n udS u ndS u dV
ˆCS CV
q ndS q dV
dE Q W
dt dt dt
ˆCS CV
q ndS qdV
ˆCS CV
n udS f udV
21
2
De v q u q f u
Dt
:u u u
: :ji iij ij ijij ij
j j j
uu uu p p p u u
x x x
iji i ij ijij
j j
u u u px x
iji i ij
i j
pu u u p u
x x
21
2
De v q p u u p
Dt
: u u q f u
• total energy equation
• Mechanical energy equation
Duu f
Dt
2
2
Du D vu
Dt Dt
u u p u
2
2
D vf u u p u
Dt
21
2
De v q p u u p
Dt
: u u q f u
:De
q p u u qDt
: u
: viscous dissipation
: jk i iijij
k j i j
uu u uu
x x x x
21
2j j jk i i i
k j i j i j i
u u uu u u u
x x x x x x x
22
2jk i
k j i
uu u
x x x
If = 0,
2
2ji
j i
uu
x x
0
• Thermal energy equation
Deq p u q
Dt
equation in terms of enthalpy h
ph e
pe h
or
2
1De Dh D p Dh Dp p D
Dt Dt Dt Dt Dt Dt
De Dh Dp p Dq p u q
Dt Dt Dt Dt
1Dh Dp Dq p u q
Dt Dt Dt
Dpq q
Dt
• entropy equation
1 1 1 1ds de pd dh dp
T T
1De Ds p D Ds p D Ds pu
T Dt Dt T Dt Dt T Dt Dt T
1 1pde ds d
T T
Deq p u q
Dt
1Ds De p p q pu q u u
Dt T Dt T T T T T T
1Ds qq
Dt T T T
• thermal energy equation in terms of temperature
First Tds equation
pp
vTds c dT T dp
T
Second Tds equation
vv
pTds c dT T dv
T
v : specific volume
Volume expansion coefficient
1
p
v
v T
Isothermal compressibility1
T
v
v p
from first Tds equation
p pp
v TTds c dT T dp c dT dp
T
p
Ds DT T DpT c
Dt Dt Dt
from entropy equation
DsT q q
Dt
p
DT Ds Dp Dpc T T q T q
Dt Dt Dt Dt
p
DT Dpc q T q
Dt Dt
1
p
v
v T
from second Tds equation
1 1
v p T
T
p v pv v
T T v vvp
vv
pTds c dT T dv
T
1
,p
v
v T
1
T
v
v p
2v v
TTds c dT T dv c dT d
v
Ds DT T DT c q q
Dt Dt Dt
from entropy equation
DsT q q
Dt
v
DT T D Tc q q q u q
Dt Dt
v
DT Tc q u q
Dt
ideal gas
1,
p
v
v T
pv RT
,RT
vp
,p
v R
T p
1 1R
v p T
1,
T
v
v p
2 ,T
v RT
p p
2
1 1RT
v p p
summary
p
DT Dpc q T q
Dt Dt
v
DT Tc q u q
Dt
with ideal gas assumption
p
DT Dpc q q
Dt Dt
v
DTc q p u q
Dt
,c r rq q q k T q ,rq k T q
Conduction Gases and Liquids
• Molecular random motion→ diffusion
• Net transfer of energy by random molecular motion
• Transfer by collision of random molecular motion
• Due to interactions of atomic or molecular activities
Solids
• In conductors: translational motion of free electrons as well
• In non-conductors (dielectrics): exclusively by lattice waves
Fourier’s law
hT
cTx
xQ
A
heat flux
[J/(m2s) = W/m2]
k: thermal conductivity [W/m·K]
As x → 0,
h cT T T
xQ [J]T
t Ax
xx
A t
T
x
T
xk
x
Tq k
x
Heat flux
x
y
z
q
xqyq
zq
vector quantity
ˆ ˆ ˆx y zq q i q j q k
ˆx
Tq q i k
x
ˆy
Tq q j k
y
ˆz
Tq q k k
z
ˆ ˆ ˆT T Tq k i j k
x y z
k T
• temperature : driving potential of heat flow
• heat flux : normal to isotherms
along the surface of T(x, y, z) = constant
T(x, y, z) = constantds
ˆ ˆ ˆds dxi dyj dzk ˆ ˆ ˆT T T
T i j kx y z
dT T T T
dx dy dzx y z
0
0T ds
0q ds q k T
Convection
energy transfer due to bulk or macroscopic motion of fluid
bulk motion: large number of molecules moving collectively
• convection: random molecular motion + bulk motion
• advection: bulk motion only
xy
u
solid wall
• hydrodynamic (or velocity) boundary layer
• thermal (or temperature) boundary layer
at y = 0, velocity is zero: heat transfer only by molecular random motion
T
sT
U ,T
xy
u
T
fk
solid wall sk sT
When radiation is negligible,
fk
sk
n
Newton’s Law of Cooling
h : convection heat transfer coefficient [W/m2.K]
U ,T
T
n
T
n
s f
Tq k
n
s
Tk
n
s Th T
u ,T
sT
qconv
qcond
u ,T
sT
T
T
sT
cond convq q
s f
Tq k
n
sh T T
Convection Heat Transfer Coefficient
not a property: depends on geometry and fluid dynamics
• forced convection• free (natural) convection
• external flow• Internal flow
• laminar flow• turbulent flow
f s
s s
k kT Th
T T n T T n
Thermal Radiation
1. Independence of existence and temperature of medium
Ex) ice lens
black carbon paper
ice lens
Characteristics
2. Acting at a distance
• electromagnetic wave or photon
conduction
• photon mean free path
• volume or integral phenomena
• ballistic transport
diffusion or differential phenomena as long as continuum holds
free electron
solid: lattice vibration (phonon)
fluid: molecular random motion
Ex) sky radiation
3. Spectral and directional dependence
• quanta
• history of path
i
surface emission
Thermal radiation spectrum
10-2 10-1 1 10 102 103
visibleultra violet
0.4
infrared
0.7
thermal radiation
1. Electromagnetic wave
• Maxwell’s electromagnetic theory
• Useful for interaction between radiation and matter
2. Photons
• Planck’s quantum theory
• Useful for the prediction of spectral properties of absorbing, emitting medium
Two points of view
EM theory
1. Thermal radiation through transparent media: surface radiation
Theoretical frame work
Micro-physical properties r, ,
Optical constants n,
Solid state theory
q
T
Surface radiative properties , ,
Transport theory
Geometric integral eq.
Two distinctive modes of radiation
Transport theory
Radiative Transfer Eq. (RTE)
2. Thermal radiation in participating media: gas radiation
Theoretical frame work
Molecular or particle parameters
Radiation properties a,
q
T
Quantum theory
Mie theory
• composition of radiating gas: molecules, atoms, ions, free electrons
• photon: basic unit of radiation energy
• emission: release of photons of energy
• absorption: capture of photons of energy
• 3 types of transitionbound-boundbound-freefree-free
Physical mechanism of absorption and emission
bound state
free state
ionized energy
Energy transition for atom or ionE1 = 0
E2
E3
E4
EI
bound-bound absorption
bound-bound emission
bound-free absorption
free-bound emission
free-free transition
Bound-bound transition
• When a photon is absorbed or emitted by an atom or a molecule and there is no ionization or recombination of ions or electrons
• Magnitude of energy transition: related to frequency of emitted or absorbed radiation
E3 E2 emission, E3 - E2 = h a photon emitted with h
fixed frequency associated with the transition of energy level
or 3 2E E
h
E1 E2, E3, E4 absorption
in the form of spectral lines
3 12 1 4 1, ,E EE E E E
h h h
Broadening Effect
• natural broadening (Heisenberg uncertainty principle)
• Doppler broadening
• collision broadening
• Stark broadening (strong electric field)
Carbon dioxide gas at 830 K, 10 atm
1. bound-bound transition
• molecules: rotational states vibrational states electronic states
• atoms: electronic state
Transition of energy state
electronic state 1
Internuclear separation distance(diatomic molecule)
En
erg
y
dissociation energy for state 1
dissociation energy for state 2
electronic state 2
vibrational state
rotational state
Transition between rotational levels of same vibrational state in same electronic state
Transition between rotational levels in different electronic stateTransition between rotational levels in different vibrational states of same electronic state
1) Rotational transition within a given vibrational state: associated energies at long wavelength 8 ~ 1000 m
2) Vibration-rotation transition: at infrared 1.5 ~ 20 m
3) Electronic transition: at short wavelength in the visible region 0.4
~ 0.7 m
Engineering industrial temperature:vibration-rotation transition
2. bound-free transition
• sufficient energy of ionization or recombination• bound-free absorption (photoionization)• free-bound emission (photorecombination)• continuous absorption coefficient
3. free-free transition
• in ionized gas (bremsstrahlung)
reflection, refraction, diffraction
• Redirection of photons
Scattering
• Elastic scattering (coherent) Inelastic scattering
• Isotropic scattering Anisotropic scattering
• Dependent scattering Independent scattering
Scattering Regime
size parameter: D/
• Rayleigh scattering: molecular scattering D/
• Mie scattering: Mie theory D/
• Geometric scattering: principle of geometric optics D/
the amount of radiation energy streaming out through a unit area perpendicular to the direction of propagation , per unit solid angle around the direction , per unit wavelength around , and per unit time about t
Intensity (spectral)
solid angle: a region between the rays of a sphere and measured as the ratio of the element area dAn on the sphere to the square of the sphere’s radius
(steradian, sr)
ex) hemisphere:
d
R
ddAn
Rsin2ndA
dR
ndA sinR d Rd2 sinR d d
sind d d
d
2 / 2
0 0sin 2 (sr)d d
• spectral intensity:
• total intensity:
d
i
dA
n
d
i
4
cos
,d Q
idA d d dt
ˆ ( , ) ( , , , , )i r i x y z
0i i d
24
[J/m sr m sc s
] o
d Qi
dA d d dt
4 cos [J] d Q i dA d d dt
43 cos [W]
d Qd q i dA d d
dt
42 2cos [W/m ]
d Qd q i d d
dAdt
42cos [W/m m]
d Qdq i d
dAdtd
• spectral radiative heat flux:
• total radiative heat flux:
dA
n
d
i
d
i
cosi d
q
ˆ ˆ i nd
2 / 2 2
0 0cos sin [W/m m]i d d
cosdq i d
0q q d
0cosi d d
2 / 2 2
0 0 0cos sin [W/m ]i d d d
• directional spectral emissive power
• hemispherical spectral emissive power
• hemispherical total emissive power
e
e
dA
n
de
i,e
Emissive power
, ,ˆ ˆcose e ee i i n
, ,ˆ( , )cose e e ee q i r d
2 / 2
,0 0
ˆ( , )cos sine e e e ei r d d
0ee q e d
,0
ˆ( , )cose e ei r d d
a) Blackbody: a perfect absorber for all incident
radiation black: termed based on the visible radiation, so not a perfect description
c) Emitted intensity from a blackbody is invariant with emission angle.
b) Maximum emitter in each direction and at every wavelength
T
black T
non-black
Blackbody radiation
Simulated blackbody
• Blackbody hemispherical spectral emissive power
, ( )cosbb e bq i r de
2 / 2
0 0( )cos sinbi r d d
2 1
0 0( ) cos (cos )bi r d d
( )bi r
• Planck’s law The Theory of Heat Radiation, Max Planck, 1901
spectral distribution of hemispherical emissive power of a blackbody in vacuum
h: Planck constant
C0: speed of light in vacuum
k: Boltzmann constant
2
1/5
2
1b b C T
Ce i
e
21 0 2 0, /C hC C hC k
in a medium with a refractive index n:
n = 1 in vacuum and n = 1.00029 in air at room temperature over the visible spectrum
2
1/2 5
2
1b b C nT
Ce i
n e
2
1/5
2
1b b C T
Ce i
e
2
155 /
( , ) 2( )
1b
C T
e T CE T
T T e
Blackbody spectral emissive power
5be
T
max 2898T
e b
(W/m
2.m
)
• Wien’s displacement law (1891)
max : the wavelength at which eb(,T) is maximum
5be
T
max 2898T
5 0( )
bed
d T T
max2max2
/
1
5 1 C T
C
eT
max 3 2897.8 m KT C
• Blackbody total intensity and total emissive power
• Stefan-Boltzmann’s law:
Boltzmann by theory (1884):
Stefan by experiment (1879): 4~be T
2
1/50 0
2
1b b C T
Ci i d d
e
4 43 4
41 14 402 2
2 2
1 15
C T C Td T
C e C
58 2 41
42
25.6696 10 W/m K
15
C
C
4 2, 0
[W/m ]b b e b be q e d i T
4be T
• Attenuation by absorption and scattering
Radiative Transfer Equation
• Augmentation of intensity by incoming scattering
• Augmentation of intensity by emission
,a sdii a i
ds
,eb
dia i
ds
,
4
ˆ ˆ ˆ( , ) ( , )4
isdii r P d
ds
Radiative Transfer Equation
ˆ( , ) ˆ( , ) ( )b
di ra i r a i r
ds
4
ˆ ˆ ˆ( , ) ( , )4
i r P d
• Energy equation : summary
p
T Pc u T T u P
t t
c rq q q
( )cq k T
rq 0 4
ˆ ˆ4 ( , ) ( )bG a e i r P d d
where 4
1ˆ ˆ ˆ( ) ( , ) ,4
P P d
4
ˆ( ) ( , )G r i r d
ˆ ˆ ˆ( , ) ( , ) ( )bi r a i r a i r
4( , ) ( , )
4i r P d
2
1/5
2( ) ( )
1b b C T
Ce T i T
e