Transcript PowerPoint 프레젠테이션

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
 Definitions of Thermodynamic Terms
system
parameters
(external forces)
constituent
constituent
constituent
constraints
environment or surroundings
 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
interaction
between the system and its environment.
 induced change of state
change of state through interaction with other
system
in the environment
 isolated system
a system which can experience only
spontaneous
change 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 time
complete description often unknown
thermodynamic 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 process
at least one way to restore both the system and its
environment to their initial states
 irreversible process
not possible to restore both the system and its
environment to their initial states
 steady state
a state that does not change as a function of time
despite interactions between the system and other
systems in the environment
 The First Law of Thermodynamics
 conservation of energy
Energy can be transferred to or from a system but
can be neither created nor destroyed.
energy balance for a system
E  E2  E1  Enet,in  Ein  Eout
for an infinitesimal change
dE   Enet,in
kinetic energy, potential energy, internal energy
 Thermodynamic Equilibrium and the Second Law
 equilibrium :
a state that cannot change spontaneously with time
unstable
metastable
stable
:
thermodynamic
equilibrium
 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
 the second law
 Q  dE   W
integrating factor
1
T
Q
dE  W
 dS


T
T
T
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.
 S  S2  S1  Snet,in  Sgen , Sgen  0
dS   Snet,in   Sgen ,  Sgen  0
 highest entropy principle
The entropy of the system is largest in the stableequilibrium state.
 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 isolated
system, 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.
 Internal energy (U): energy of a system with
volume (V) as its only parameter
r + 2 independent variables by the state principle
entropy: a property of the system
S  S (U ,V , N 1 , N 2 ,
, Nr )
U  U ( S ,V , N 1 , N 2 ,
, Nr )
 fundamental relations
definitions of temperature, pressure and chemical
potential
 U 
 U 
 U 
T 
, P  
, i  



 S V , N
 V  S , N
 N i  S ,V , N
j's
j's
j's
( j i )
 Gibbs relation
U  U ( S ,V , N 1 , N 2 ,
, Nr )
 U 
 U 
dU  
dS  
dV 


 S V , N
 V  S , N
j ,s
j ,s
 TdS  PdV 
 U 



N
i 1 
i  S ,V , N
r

dN i
j ,s
 j i 
r
  dN
i
i
i 1
1
P
dS  dU  dV 
T
T
1  S 

,

T  U V , N
j's
r
i
 T dN
i
i 1
 S 
i
P  S 

,
 


T  V U , N
T
 N i U ,V , N
j's
j's
( j i )
 The Third Law of Thermodynamics
Unique stable equilibrium state exists at zero absolute
temperature (Nernst theorem).
spontaneous
change of state
E
adiabatic
availability
A1
lowest energy
principle
Eg
A10
T
A10 '
E
S
Tg  0 K
Sg  0
Eg > 0: ground-state energy
S
stable
equilibrium state
curve
Thermodynamic Functions and Properties
 Thermodynamic Relations
 enthalpy
H  U  PV
r
dU  TdS  PdV   i dN i
dH  dU  PdV  VdP
i 1
r
 TdS  VdP   i dN i
i 1
H  H ( S , P , N1 , N 2 ,
, N r ) : characteristic function
 H 
 H 
 H 
T 
,V  
, i  




S

P

N

 P ,N

 S ,N
i  S ,P ,N

j's
j's
j's
( j i )
 Helmholtz free energy
A  U  TS
r
dA   SdT  PdV   i dN i
i 1
 A 
 A 
 A 
S  
, P  
, i  




T

V

N

V , N

T , N
i T ,V , N

j's
j's
 Gibbs free energy
j's
( ji )
G  U  PV  TS  H  TS  A  PV
r
dG   SdT  VdP   i dN i
i 1
 G 
 G 
 G 
S  
, V 
, i  




T

P

N

 P ,N

T , N
i T , P , N

j's
j's
j's
( j i )
 homogeneous state:
All subsystems are exactly identical to each other.
 simple system:
a system that experiences only homogeneous states
for k equal-volume subsystems
 intensive property: T, P, j
 U V N1 N 2
T , ,
,
,
k k k k
Nr 
,
 T (U ,V , N 1 , N 2 ,

k 
, Nr )
 extensive property: U, S, V, N
 U V N1 N 2
S , ,
,
,
k k k k
Nr 
1
,

S (U ,V , N 1 , N 2 ,

k 
k
, Nr )
 specific property
the ratio of an extensive property to the total amount
of constituents (mass, mole, or number)
dU  TdS  PdV 
r
  dN
i
i
i 1
For a simple system, U  TS  PV 
r
 N
i
Euler relation
i
i 1
dU  TdS  SdT  PdV  VdP 
SdT  VdP 
r
  dN   N d 
i
i 1
r
 N d
i
i
0
r
i
i
i
i 1
Gibbs-Duhem relation
i 1
Euler relation for r = 1, G  U  PV  TS   N
G
 (T , P )   g(T , P )
N
The chemical potential of a pure substance is specific
Gibbs free energy.
 The Gibbs Phase Rule
 phase : collection of all subsystems that have the
same values of all intensive properties
for a q-phase heterogeneous state
independent variables T, P, i (i = 1, 2, 3, …, r)
Gibbs-Duhem relation SdT  VdP 
r
 N d
i
i
0
i 1
number of independent variable reduced to
  r  2q
Gibbs phase rule
r + 2: independent variables, q: number of equations
for a pure substance,
a single-phase state, r  1, q  1    2
a two-phase state, r  1, q  2    1
a three-phase state, T, P,  are all fixed: triple point
P – T diagram for a pure substance
S-L line
P
Liquid
Critical
point
Solid
Triple point
Vapor
T
P＞Pc
S-L region
Temperature, T
T – v diagram for a material that expands upon melting
P = Pc
(Tc,Pc)
L-V
dome
P＜Pc
Saturated Saturated
Liquid
Vapor
Solid
Triple-point line
Sublimation
P＜Pt.p.
S-V region
Specific volume v
 Specific Heats
 specific heat at constant volume
 u 
 s 
cv  

T

 T 

T

V

v
 specific heat at constant pressure
 h 
 s 
cp  
T



T

T

P

P
 heat reservoir
an idealized system that experiences only reversible
heat interactions. For any finite amount of energy
transfer, its temperature remains unchanged.

ER,2  ER,1  TR SR,2  SR,1

 equation of state
For pure substance in a single phase, all properties
can be expressed as function of T and P.
f (T , P , v )  0 or
v  v(T , P )
c p (T , P )
 s 
 s 
 v 
ds  
dT

dP

dT

dP





T
 T  P
 P  T
 T  P

 v  
dh  c p ( t , P )dT  v T , P   
 dP

 T  P 

Ideal Gas and Ideal Incompressible Models
from molecular view,
 intermolecular potential energy
associated with the forces between molecules and
depends on the magnitude of the intermolecular forces
and the position at any instant of time
 molecular kinetic energy
associated with the translational velocity of individual
molecules
 intra-molecular energy (within the individual molecules)
associated with the molecular and atomic structure and
related forces
 The Ideal Gas
intermolecular potential energy
 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 approximations
At low or moderate densities: The molecules are
relatively widely spread, so that two-molecules or twoand 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.
Ideal gas
 equation of state
PV  nRT , Pv  RT
m
kg
n

M kg/kmol
kN  m
kJ
R  8.314
 8.314
kmol  K
kmol  K
real Gas
Pv
Z
RT
compressibility factor
Z depends on the temperature & pressure.
real gas: affected by intermolecular force
 Van der Waals
a 

 P  v 2   v  b   RT


 Virial
Pv
B(T ) C (T )
 1
 2  .....
RT
v
v
 Beattie-Bridgman
RT (1   )
A
P
(v  B )  2
2
v
v
a
b
C
A  A0 (1  ), B  B0 (1  ),  
v
v
vT 3
 Mayer relation
for ideal gas
u  u(T )
 u 
 u 
du  
dT    dv  cv dT

 T  v
 v  T
 h 
 h 
dh  
dT  
dP  c p dT


 T  P
 P  T
h  u(T )  Pv  u(T )  RT
dh du
cp 

 R  cv  R
dT dT
perfect gas, cv (T ) = constant
u2  u1  cv  T2  T1  , h2  h1  c p T2  T1 
entropy
Tds  du  Pdv
dT R
du P

c
 dv
ds 
 dv
v
T
v
T T
 v1 
dT
s2  s1  1 cv
 R ln  
T
 v2 
2
Tds  dh  vdP
dT R
dh v
 dP
ds 
 dP  c p
T
P
T T
 P2 
dT
s2  s1  1 c p
 R ln  
T
 P1 
2
 Incompressible Solids and Liquids
equation of state for incompressible solids and liquids
v  constant
 h 
 s 
 v 
 P   T  P   v  T  T   v

T

T

P
 h 
 P   v  constant

T
h  h0  v  P  P0   f (T )
u  h  Pv
u  u0   h  Pv    h0  P0v0    h  h0   v  P  P0   f (T )
df (T )
 u 
cv  

,

dT
 T  v
cv (T )  c p (T )
df (T )
 h 
cp  


dT
 T  P
Heat Transfer Basics
 What is heat ?
in a solid body
crystal : a three-dimensional periodic array of atoms
oscillation of atoms about their various positions of
equilibrium (lattice vibration): The body possesses heat.
conductors: free electrons ↔ dielectics
 vibration of crystals with an atom
longitudinal polarization vs. transverse polarization
us-1
us-1
s-1
us+1
us
s
s+1
s+2
us+2
us+3
s+3
the energy of the oscillatory motions:
the heat-energy of the body
more vigorous oscillations:
the increase in temperature of the body
us
us+1
us+2
in a gas
the storage of thermal energy:
• molecular translation, vibration and rotation
• change in the electronic state
• intermolecular bond energy
Energy
average kinetic energy
electronic
state 2
dissociation
energy for state 2
vibrational state
electronic
state 1
dissociation
energy for state 1
rotational state
Internuclear separation distance
(diatomic molecule)
1
3
2
Eu  mum  k BT
2
2
kB = 1.3807 × 10-23 J/K
at T = 300 K,
air M = 28.97 kg/kmol
2 1/ 2
um = 468.0 m/s
 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
resulting from a spatial variation in temperature.
energy carriers: molecule, atom, electron, ion, phonon
(lattice vibration), photon (electro-magnetic wave)
 continuum hypothesis
m
lim
Ex) density    V  V
0 V
microscopic
 m uncertainty
V
macroscopic
uncertainty
local value of
density
 V0  10 mm
9
3
V
(3×107 molecules at sea level, 15°C, 1atm)
 microscopic uncertainty
due to molecular random motion
 macroscopic uncertainty
due to the variation associated with spatial
distribution of density
In continuum, velocity and temperature vary smoothly.
→ differentiable
mean free path of air at STP (20°C, 1 atm)
lm = 66 nm,
2 1/ 2
m
u
 468.0 m/s
bulk motion vs molecular random motion
 local thermodynamic equilibrium
hot wall at Th
L
gas
cold wall at Tc
a) lm << L : normal pressure
b) lm ~ L : rarefied pressure
c) lm >> L
 governing equations
• continuity eq.: mass conservation
D
Dm

0
Dt
Dt

D
dm 
MV
Dt

MV
 dV

MV
 dV 


dV 
CV t


dV 
CV t


CS
nˆ
dS
for a fixed volume in space
D
Dt
u
m,V
ˆ
 u  ndS
 

     u   dV
    u  dV 

CV
CV
 t



Since V can be chosen arbitrary

    u  0
t
dm, dV


    u 
 u      u  0
t
t
D
   u  0
Dt
in Cartesian tensor notation
u j
D

0
Dt
x j
incompressible flow
u j
D
0
 0    u  0 or
x j
Dt
D

  u
Dt t
• momentum eq.: Newton’s 2nd law of motion
rate change of momentum = forces exerted on
the body
• forces
nˆ

   nˆ
body force  f
[N/m3]
surface force  [N/m2]
• stress tensor
f
   (nˆ , x, t )
   nˆ or  i   ij n j
u
nˆ


D
Dt
D
udm
udm
MV
Dt



  u  dV 
CV t



dm
f


  u  dV 
CV t


CV
MV
CS
 udV
 u  u  nˆ  dS
Momentum theorem
    uu  dV 



u




uu
dV






CV  t



u

  u       uu     u  u      u     u    u 
t
t
t
Du
 u

 

 
 u   u   u 
    u   
Dt
 t

 t

D
Du
 udV 

dV
MV
CV
Dt
Dt


Du

dV 
CV
Dt



CV

CV
ˆ
 fdV    ndS

CS
 fdV      dV
CV
nˆ
u

dm
 Du




f




dV  0


CV 
Dt

f

Du

  f   
Dt
or
 ij
Dui

  fi 
Dt
x j
For a Newtonian fluid
*
   pI     pI  l    u  I   u   u  


 ui u j
uk
 ij   p ij   ij   p ij  l
 ij   


xk
 x j xi



 ij


x j x j

 ui u j
uk
 ij   

  p ij  l

x k

 x j x i

p
p ij 
,
x j
x i


x j

  u u j
   i 
  x j xi

 
 
  uk    uk
 ij  
l
l
x j  x k  x i  x k

  ui
  
 
  x j  x j
Dui
p
  uk

  fi 

l
Dt
xi xi  xk
 
 
 x j
 

 x j

,

 u j 



x
i 

 ui
 
 x j
   u j 
 



x

x
j 
i 


Du
*

  f  p    l    u       u       u 
Dt

For a constant , l fluid
 uk

 xk

  ui

 
x j  x j


  u j 
  



x

x
j 
i 

p
  ui
  fi 


xi
x j  x j



l



 
xi

 u j

 x j
Dui
p


  fi 
l
Dt
xi
xi
Du

  f  p   2 u   l        u 
Dt
2
l



Stokes’ hypothesis
3
Du

2

  f  p   u      u 
Dt
3
For an incompressible flow
Du

  f  p   2 u
Dt



• energy eq.: 1st law of thermodynamics
 Q  dE   W
u
q
nˆ

1 2

2
E
  e  v  dV , u  u  v
MV
2 

dm
1 2
  e  v : total energy
2


f
e : internal energy
q
rate equation
Q
dE  W


dt
dt
dt
DE D

Dt
Dt

dE  Q  W


dt
dt
dt
or

 dV 
  dV 

MV
CV t

D
  u  nˆ  dS 

dV
CS
CV
Dt


Q
Dt


ˆ 
q  ndS
CS

qdV
q
nˆ
u

CV
W
    nˆ  udS    f  udV
CS
CV
Dt

dm
dE  Q  W


dt
dt
dt
D

dV  
CV
Dt

f
q

ˆ 
q  ndS
CS

CV
qdV 

CS
 nˆ  udS    f  udV
CV
 nˆ  u  ij   ij n j ui   ji ni u j   ji u j ni   ij u j ni   u  nˆ  ij

ˆ      u  dV
 nˆ  udS    u  ndS

ˆ 
q  ndS
CS
CS
CS

CV
  qdV
CV
D
1 2
  e  v     q     u   q   f  u
Dt 
2 
   u    : u  u      
 : u ij    p ij   ij 
 u    
u j
ui
ui
 p
  ij
   p  u   : u  ij
x j
x j
x j
 ij
ij

 ui
 ui
 p ij   ij
x j
x j


 ij
p
  ui
 ui
   u  p  u        ij
xi
x j
• total energy equation

D
1 2
e

v     q  p  u  u  p

Dt 
2 
 : u  u        q   f  u
• Mechanical energy equation
 Du

u  
  f    
 Dt

Du
D  v2 
u 

 
Dt
Dt  2 
u         u  p  u      
D  v2 
     f  u  u  p  u      
Dt  2 
D
1 2
  e  v     q  p  u  u  p
Dt 
2 
  : u  u        q   f  u
• Thermal energy equation
De

   q  p  u   : u  q
Dt
 : u : viscous dissipation
 : u  ij
 u
 ui u j
k
 l
 ij   


 xk
 x j x i
 ui u j
 uk 
 l

   
 x k 
 x j x i
2
 uk    ui u j
 l

  
2  x j x i
 x k 
2
  ui
u j 
 



2  x j x i 
  ui
 
  x j
 1  ui u j

  
 2  x j xi



2
2
   0
If l = 0,
  ui u j

  
  x j x i

 
 

De
   q  p  u  q  
Dt
equation in terms of enthalpy h
he
p

or
e  h
p

De Dh D  p  Dh  1 Dp p D  




 2


Dt Dt Dt    Dt   Dt  Dt 
De
Dh Dp p D




   q  p  u  q  
Dt
Dt Dt  Dt
 1 D

Dh
Dp

   q 
 p
   u   q  
Dt
Dt
  Dt

Dp
   q 
 q  
Dt
• entropy equation
 1  1 
1
1 
ds   de  pd      dh  dp 
T
 
   T 
1
p 1
de  ds  d  
T
T 
 De
Ds  p D  1 
Ds
p D
Ds p





 u


T Dt
Dt T Dt   
Dt T Dt
Dt T
De

   q  p  u  q  
Dt
Ds  De p
1
p
q  p



 u   q  u 
 u
Dt T Dt T
T
T
T
T
T

Ds
1
q 
    q  
Dt
T
T
T
• thermal energy equation in terms of temperature
First Tds equation
v 
Tds  c p dT  T
dp

T  p
v : specific volume
Second Tds equation
Tds  cv dT  T
p 
dv

T  v
Volume expansion coefficient
1 v 

v T  p
Isothermal compressibility
1 v 
 
v p T
from first Tds equation
v 
T
Tds  c p dT  T
dp

c
dT

dp

p
T  p

1 v 

v T  p
Ds
DT  T Dp
T
 cp

Dt
Dt
 Dt
from entropy equation T
Ds
   q    q
Dt
DT
Ds
Dp
Dp
cp
 T
 T
   q   T
   q
Dt
Dt
Dt
Dt
cp
DT
Dp
   q   T
 q  
Dt
Dt
from second Tds equation
Tds  cv dT  T
1 v 
1 v 

,
 

v T  p
v p T
p 
dv

T  v
p 
v  p 
1
1


  v
 v




T  v
T  p v T
v 
v 
p T

T
Tds  cv dT  T dv  cv dT  2 d 


from entropy equation
T
Ds
   q    q
Dt
Ds
DT  T D
T
  cv

   q    q
Dt
Dt  Dt
DT
 T D
T


 cv
   q 
   q    q 
  u    q
Dt
 Dt

DT
T

 cv
   q 
  u    q
Dt

ideal gas
pv  RT
1 v 

,

v T  p
RT
v
,
p
1 v 
 
,

v p  T
v 
R
 ,

T  p p
v 
RT
 2 ,

p T
p
1R 1


v p T
1 RT 1


2
v p
p
 summary
DT
Dp

cp
   q   T
   q
Dt
Dt
DT
T
 cv
   q 
  u    q
Dt

q  qc  qr   kT  qr,
  q     kT     qr,
with ideal gas assumption
DT
Dp

cp
   q 
   q
Dt
Dt
DT
 cv
   q  p  u    q
Dt
 Conduction
 Gases and Liquids
• Due to interactions of
atomic or molecular
activities
• Net transfer of energy by
random molecular motion
• Molecular random motion→ diffusion
• Transfer by collision of random molecular motion
 Solids
• In non-conductors (dielectrics):
exclusively by lattice waves
• In conductors:
translational motion of free electrons as well
 Fourier’s law
Th
Qx
A
 x Tc
T  Th  Tc
Q x  T  t  A [J]
x
heat flux
Qx
T
T
qx 

 k
A  t
x
x
[J/(m2s) = W/m2]
k: thermal conductivity [W/m·K]
As x → 0,
T
qx   k
x
 Heat flux
z
vector quantity
qz
q  qx iˆ  qy ˆj  qz kˆ
qx
q
qy
y
x
T
ˆ
qx  q  i   k
x
T
ˆ
qy  q  j   k
y
T
ˆ
qz  q  k   k
z
 T ˆ T ˆ T
q   k 
i
j
y
z
 x

kˆ    k T

• temperature : driving potential of heat flow
• heat flux : normal to isotherms
along the surface of T(x, y, z) = constant
ds
ds  dxiˆ  dyjˆ  dzkˆ
T(x, y, z) = constant
T ˆ T ˆ T ˆ
T 
i
j
k
x
y
z
T
T
T
dT 
dx 
dy 
dz  0
x
y
z
 T  ds  0  q  ds  0
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
U ,T
y
u
T
Ts
x
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
U ,T
u
y
T
kf
ks
Ts


solid wall
When radiation is negligible,
nˆ
kf
ks
T 

n  
T 

n  
qs   k f
T 
T 
  ks

n  
n  
 h  Ts  T 
h : convection heat transfer
coefficient [W/m2.K]
Newton’s Law of Cooling
x
u ,T
qconv
Ts
qcond
qcond  qconv
u ,T
T
T
Ts
T 
qs   k f
 h  Ts  T 

n  
Ts
 Convection Heat Transfer Coefficient
ks
T 
T 
h



T

T

n
T

T

n
 s     s   
kf
not a property: depends on geometry and fluid dynamics
• forced convection
• free (natural) convection
• external flow
• Internal flow
• laminar flow
• turbulent flow
 Thermal Radiation
 Characteristics
1. Independence of existence and temperature
of medium
Ex) ice lens
ice lens
black carbon paper
2. Acting at a distance
Ex) sky radiation
• electromagnetic wave or photon
• photon mean free path
• ballistic transport
• volume or integral phenomena
conduction
fluid: molecular random motion
solid: lattice vibration (phonon)
free electron
diffusion or differential phenomena as long as
continuum holds
3. Spectral and directional dependence
• quanta
• history of path
il
surface emission
l
 Thermal radiation spectrum
10-2 
10-1 
0.4
1
0.7
10 
infrared
ultra violet
visible
thermal radiation
102 
103 
 Two points of view
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 distinctive modes of radiation
1. Thermal radiation through transparent media:
surface radiation
Theoretical frame work
Microphysical
properties
r, g, 
q
T
Transport
theory
Geometric
integral eq.
Solid state
theory
Surface
radiative
properties
, , a
Optical
constants n, 
EM theory
2. Thermal radiation in participating media:
gas radiation
Theoretical frame work
Molecular or
particle
parameters
Quantum
theory
Mie theory
Radiation
properties
al, l
q
Transport
theory
T
Radiative Transfer Eq. (RTE)
 Physical mechanism of absorption and emission
• 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 transition
bound-bound
bound-free
free-free
free-free
free-bound
emission
transition
EI
E4
bound-free
absorption
bound-bound
free state
ionized
energy
emission
E3
bound-bound
absorption
E2
E1 = 0
Energy transition for atom or ion
bound state
 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 = hn a photon emitted with hn
or n 
E3  E2
h
fixed frequency associated with the transition of
energy level
E1  E2, E3, E4 absorption
E2  E1 E3  E1 E4  E1
n
,
,
h
h
h
in the form of spectral lines
an
n
 Broadening Effect
• natural broadening
(Heisenberg uncertainty principle)
• Doppler broadening
• collision broadening
• Stark broadening (strong electric field)
an
n
Carbon dioxide gas at 830 K, 10 atm
 Transition of energy state
1. bound-bound transition
• molecules:
rotational states
vibrational states
electronic states
• atoms: electronic state
electronic state 2
Energy
vibrational state
dissociation energy
for state 2
Transition between
rotational levels in
rotational state
different
electronic
Transition
between
Transition
between
state
rotational levels
levels in
of
rotational
dissociation energy
same vibrational
different
vibrational
for state 1
state in
states
ofsame
same
electronic
state
1
electronic
state
electronic
state
Internuclear separation distance
(diatomic molecule)
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)
 Scattering
• Redirection of photons
reflection, refraction, diffraction
• Elastic scattering (coherent)
Inelastic scattering
• Isotropic scattering
Anisotropic scattering
• Dependent scattering
Independent scattering
 Scattering Regime
size parameter: pD/l
• Rayleigh scattering:
molecular scattering pD/l << 1
• Mie scattering:
Mie theory pD/l ~ 1
• Geometric scattering:
principle of geometric optics pD/l >> 1
 Intensity (spectral)
the amount of radiation energy streaming out
through a unit area perpendicular to the
ˆ,
direction of propagation 
per unit solid angle around the direction w,
per unit wavelength around l,
and per unit time about t
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
dAn
dw  2 (steradian, sr)
R
Rsinq
dq
dAn   R sinq d   Rdq 
 R 2 sin q dq d
dw  sinq dq d
q
R
ex) hemisphere:
w   dw



2p
0

p /2
0
sin q dq d  2p (sr)
dAn
d
• spectral intensity:
il ˆ

nˆ
il
dw
q
dl
dA
d 4Q
ˆ )  i ( x , y , z ,q ,  )
il 
, il ( r , 
l
dA cosq dw d l dt

• total intensity: i  0 il d l
l
d 4Q
il 
[J/m 2  sr  m  s]
dA cosq dw d l dt
d 4Q  il dA cosq dw d l dt [J]
4
d
Q
d 3q 
 il dA cosq dw d l [W]
dt
4
d
Q
2

d q 
 il cosq dw d l [W/m 2 ]
dAdt
d 4Q
dql 
 il cosq dw [W/m2  m]
dAdtd l
• spectral radiative heat flux:

il ˆ
nˆ
ql   il cosq dw
dw

q
ˆ  nd
ˆ w
  il 
dql  il cos q d w


2p
0

p /2
0
dA
il cosq sin q dq d [W/m 2   m]
• total radiative heat flux:
il

q   qld l
0




0
0
  i cosq dw  d l
 l
2p
p /2
0
0
 
dl
il cosq sin q dq dd l [W/m 2 ]
l
 Emissive power
il, ˆ
• directional spectral emissive power
e
ˆ  nˆ
el  il ,e cosqe  il ,e 
nˆ


2p
0

p /2
0
ˆ )cos q sin q dq d
il ,e ( r , 
e
e
e
e
• hemispherical total emissive power


0
0
e  qe   el d l  


ˆ )cos q dw d l
il ,e ( r , 
e
e
dwe
qe
• hemispherical spectral emissive power
ˆ )cosq dw
el  ql,e   il ,e ( r , 
e
e
e
dA
 Blackbody radiation
a) Blackbody: a perfect absorber for all incident
radiation
black: termed based on the visible radiation, so
not a perfect description
b) Maximum emitter in each direction and at every
wavelength
T
black
T
non-black
c) Emitted intensity from a blackbody is invariant with
emission angle.
Simulated blackbody
• Blackbody hemispherical spectral emissive power
el b  qlb ,e   il b ( r )cosq dw


2p
0

p /2
0
 il b ( r ) 
il b ( r )cos q sin q dq d
2p
0
 p il b ( r )

1
0
cos q d (cos q )d
• Planck’s law
The Theory of Heat Radiation, Max Planck, 1901
spectral distribution of hemispherical
emissive power of a blackbody in vacuum
el b  p il b 
2p C1
l 5  eC
2 / lT
1

C1  hC02 , C2  hC0 / k
C0: speed of light in vacuum
h: Planck constant
k: Boltzmann constant
in a medium with a refractive index n:
el b  p il b 
2p C1

n2 l 5 e C2 / l nT  1

n = 1 in vacuum and n = 1.00029 in air at room
temperature over the visible spectrum
el b  p il b 
2p C1
l 5  eC
2 / lT
1

el b ( l , T )
2p C1

 E (lT )
5
5
C
/
l
T
2
T
1
 lT  e


elb (W/m2.m)
el b
T5
lmaxT  2898
Blackbody spectral emissive power
lT
• Wien’s displacement law (1891)
lmax : the wavelength at which elb(l,T) is maximum
 el b
d (lT )  T 5
d

0

el b
T5
C2
1
 lmaxT 
5 1  e  C2 / lmaxT
lmaxT  C 3  2897.8  m  K
lmaxT  2898
lT
• Blackbody total intensity and total emissive power

ib   il b d l  
0
2C1T 4

C 24

0


0
l e
5
2C1
C 2 / lT
1

dl
3
2C1T 4 p 4  4
d 
 T

4
e 1
C 2 15 p
2C1p 5
8
2
4


5.6696

10
W/m

K
15C 24
• Stefan-Boltzmann’s law:

eb  qb,e   el b d l  p ib   T 4 [W/m 2 ]
0
4
Stefan by experiment (1879): eb ~ T
4
Boltzmann by theory (1884): eb   T
 Radiative Transfer Equation
• Attenuation by absorption and scattering
dil ,a  s
  l il    al   l  il
ds
• Augmentation of intensity by emission
dil ,e
 al il b
ds
• Augmentation of intensity by incoming scattering
l

ds
4p
dil ,is
w
  4p
ˆ  ) P (
ˆ , 
ˆ )dw 
il ( r , 
l
Radiative Transfer Equation
ˆ)
dil ( r , 
ˆ )  a i (r )
   al   l  il ( r , 
l lb
ds
l
ˆ  )P (
ˆ , 
ˆ )dw 

i
(
r
,

l
l
4p w 4p
• Energy equation : summary
 T

 P

cp 
 u  T    T 
 u  P    qc    qr  q  
 t

 t

  qc    ( k T )
  qr 


0
 G
l
l
 4al el b   l 
w   4p
1
ˆ
where Pl ( ) 
4p
w
 4p
ˆ , 
ˆ )dw , G ( r ) 
Pl (
l

w  4p
ˆ ) 
ˆ    a    i (r , 
ˆ )  a i (r )
il (r , 
l
l
l
l lb
l

4p
el b (T )  p il b (T ) 
w
  4p
il ( r ,  )Pl (,  )dw 
2p C1
l 5  eC

ˆ ) P (
ˆ )dw  d l
il ( r , 
l
2 / lT
1

ˆ )dw
il ( r , 