OVERVIEW OF MACROSCOPIC THERMAL SCIENCES FUNDAMENTALS OF THERMODYNAMICS ▪ The First Law of...

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

Qq

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