INTRODUCTION

• What is Heat Transfer ?

• Continuum Hypothesis

• Local Thermodynamic Equilibrium

• Conduction

• Radiation

• Convection

• Energy Conservation

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

us-1 us us+1 us+2

Vibration of crystals with an atom

s s+1 s+2 s+3s-1

us-1 us us+1 us+2 us+3

Longitudinal polarization vs. Transverse

polarization

The energy of the oscillatory motions: the heat-energy of the body

More vigorous oscillations: the increase in temperature of the body

molecular translation, vibration and rotationchange in the electronic stateintermolecular bond energy

In a gasThe storage of thermal energy:

average kinetic energy

21 3

2 2u m BE mu k T

kB = 1.3807 × 10-23 J/K

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

HEAT TRANSFERHeat 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 0

limV V

m

V

local value of density

macroscopic uncertainty

microscopic uncertainty

9 30 10 mmV

(3×107 molecules at sea level, 15°C, 1atm)

m

V

V

• macroscopic uncertainty

• microscopic uncertaintydue to molecular random motion

due to the variation associated with spatial distribution of densityIn continuum, velocity and

temperature vary smoothly. → differentiableMean free path of air at STP (20°C, 1atm)m = 66 nm,

1/ 22 468.0 m/smu

bulk motion vs molecular random motion

adi b

ati c

wal

l

cold wall at Tc

L

a) m << L : normal pressure

b) m ~ L : rarefied pressure

c) m >> L

gas

adibatic wallhot wall at Th

LOCAL THERMODYNAMIC EQUILIBRIUM

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

• Due to lattice waves induced by atomic motion

• In conductors: translational motion of free electrons as well

• In non-conductors (dielectrics): exclusively by lattice waves

Fourier’s Law

hT

cTx

xQ

A

h cT T T

heat flux

T

xk

[J/(m2s) = W/m2]

k: thermal conductivity [W/m·K]

As x → 0, x

Tq k

x

xQ [J]T

t Ax

xx

Qq

A t

T

x

Notation

Q : amount of heat transfer [J]

q : heat transfer rate [W],

: heat transfer rate per unit area [W/m2]

: heat transfer rate per unit length [W/m]

Qq

A t

q

q

Qq

t

Qq

L t

q q A q L

Heat Flux

x

y

z

q

xqyq

zqvector quantityˆ

xq q i ˆ

yq j ˆzq k

ˆx

Tq q i k

x

ˆ ,y

Tq q j k

y

ˆz

Tq q k k

z

ˆTq k i

x

ˆT

jy

ˆTk

z

k T

Ex)

( , ) (0 1, 0 1)T x y x y x y

x

y

1T

1.5T

0.5T 1

1 T = constant line or surface: isothermal lines or surfaces (isotherms)

q k T ˆT

k ix

ˆTj

y

q

ˆk i j

, x yq k q k

ˆxq i ˆ

yq j ˆki ˆkj

• temperature : driving potential of heat flow• heat flux : normal to isotherms

along the surface of T(x, y, z) = constant

T(x, y, z) = constant

ˆds dxi ˆdyj ˆdzk

0T ds

ds

0q ds

0dT T T T

dx dy dzx y z

ˆ ˆ ˆT T TT i j k

x y z

q k T

Steady-State One Dimensional Conduction

xx x x

xq x xq q T = T(x) only

x

dTq k

dx

2dT d dTk k x O x

dx dx dx

steady-state

x x xq q q 0xq

x xq q 2xx

dqq x O x

dx

20x

d dTq k x O x

dx dx

0d dT

k O xdx dx

or

As x → 0, 0d dT

kdx dx

When k = const., 2

20

d T

dx

10-2 10-1 1 10 102 103

ultra violet visibl

e

0.4 0.7

thermal radiation

RADIATION

Thermal Radiation

infrared

Characteristics of Thermal Radiation

1.Independence of existence and temperature of medium

Ex) ice lens

black carbon paper

ice lens

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

diffusion

3. Spectral and Directional Dependence• quanta • history of

path

surface emission

Blackbody spectral emissive power

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

Radiating Medium

• Transparent medium ex: air

• Participating medium emitting, absorbing and

scattering

ex: CO2, H2O

• Opaque material

Stefan-Boltzmann’s law

4 2, [W/m ]b b eE q T

Stefan by experiment (1879): 4~bE T

Boltzmann by theory (1884): 4bE T

• Blackbody emissive power

• Blackbody:

8 2 45.6696 10 W/m K

a perfect absorber

Planck’s law (The Theory of Heat Radiation, Max Planck, 1901)

spectral distribution of

hemispherical emissive power of a blackbody in vacuum

2

1

/5

2

1b C T

CE

e

21 0 2 0, /C hC C hC k

h: Planck constant: 6.6260755×10-34 J•s

C0: speed of light in vacuum: 2.9979×108 m/s

k: Boltzmann constant: 1.380658×10-23 J/K

Blackbody spectral emissive power

2

1/5

0

2

1C T

Cd

e

4T

For a real surface,4E T

: emissivityWavelength, (mm)

E,

b (W

/m2.m

),

0b bE E d

q J G

G J

q

4 ,J T G

4q T G G

4 1T G

diffuse-gray surface at T

Surface Radiation

G: irradiation [W/m2]

J: radiosity [W/m2]

1,

Kirchihoff’s law 4q T G

: reflectivity

: absorptivity

Ray-tracing method vs Net-radiation method

41q J J T

41

J J T

41T J

4

1T J

4J T G

q J G

G T4

qdiffuse-gray surface at T

4 q T G 4T G

4T G

4

1T J

4J T G

41G J T

4 4q T J T

4 4

1T J T

Ex) a body in an enclosure

T1, 1, A1

T2, 2, A2

when 1

2

1,A

A 4 4

1 1 1 1 2q A T T

4 4sursq A T T

q1

Tsur

Ts, , , A

q

Surrounding can be regarded as a blackbody.

4surT

4sq AT

4surAT

4 41 1 2

1

1

1 2 2

1 11

A T Tq

AA

[W]

4surT 4

surT

A1

A2

q T A 41 sur 1 q T A 4

2 sur 2

q q1 2

Why is the irradiation on the small object the same as ?4

surT

4surT

A1

A2

q T A 41 sur 1q1

q2 q T A 42 sur 1 F12

F: view factorF F 11 12 1

F F 21 22 1

F 22 0 F 21 1

Reciprocity:

A F A F1 12 2 21 A 2

AF

A 2

121

q T A F 42 sur 1 12

AT A

A 4 2

sur 11

T A 4sur 2

CONVECTION

energy transfer due to bulk or macro- scopic 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,

s f

Tq k

n

fk

sk

ns

Tk

n

s Th T

Newton’s Law of Cooling

h : convection heat transfer coefficient [W/m2.K]

U ,T

T

n

T

n

u ,T

sT

q qcond conv

qconv

qcond

u ,T

sT

s f

Tq k

n

sh T T

T

T

sT

Convection Heat Transfer Coefficient

f s

s s

k kT Th

T T n T T n

not a property: depends on geometry and fluid dynamics

• forced convection• free (natural) convection

• external flow• Internal flow

• laminar flow• turbulent flow

ENERGY CONSERVATIONFirst law of thermodynamics

• control volume (open system)• material volume (closed system)

control volume

g ,E stEinEoutE

in g out stE E E E

steady-state:

st 0E in g out 0E E E

In a time interval t: in g out stE E E E

Surface Energy Balance

inE outE in outE E

Ex)

n

sur.

surTsT

0T

T

cond,sqradcond, fq q

c radond, ndco , fsq qq

conv radq q

radfs

Tk

nq

Tk

n

4 4surs sT Th T T

Assumptions: 1) Steady-state conditions2) Radiation exchange between the pipe and the

room is between a small surface in a much larger enclosure.

3) Surface emissivity = absorptivity

Example 1.2

70mmD

= 200 C

= 0.8sT

air

G

2

25 C

15W/m K

T

h L

sur = 25 CT

E

Find: 1) Surface emissive power E and irradiation G2) Pipe heat loss per unit length, q

q

70mmD

= 200 C

= 0.8sT

air

G

2

25 C

15W/m K

T

h L

sur = 25 CT

Eq

4sE T

1. Surface emissive power and irradiation

4surG T

8 2 4 4 20.8(5.67 10 W/m K )(473 K) 2,270 W/m

8 2 4 4 25.67 10 W/m K (298 K) 447 W/m

2. Heat loss from the pipe

A DL

4 4surs sh DL T T DL T T

lossqq

L

2

-8 2 4 4 4 4

15 W/m K 0.07 m 200 25 C

+ 0.8 0.07m 5.67 10 W/m K 473 298 K

577 W/m 421 W/m 998 W/m

shA T T 4 4sursA T T

70mmD

= 200 C

= 0.8sT

air

G

2

25 C

15W/m K

T

h L

sur = 25 CT

Eq

lossq conv radq q

70mmD

= 200 C

= 0.8sT

air

G

2

25 C

15W/m K

T

h L

sur = 25 CT

Eq

Q: Why not ?

conv rad condloss q q qq

Conduction does not take place ?

conlo v as rs dq q q

Example 1.2

= 200 C

= 0.8sT

air

2

25 C

15W/m K

T

h sur = 25 CT

T

sTT

r

cond,sq

4 4surs f s

s f

dT dTk k T T

dr dr

4 4surs sh T T T T

conv rass dlo ,qq q

cond,sq cond,fqradq

lossq cond,f radq q

cond,f convq q

Example 1.4Hydrogen-air Proton Exchange Membrane (PEM) fuel cell

Three-layer membrane electrode assembly (MEA)

Role of electrolytic membrane1. transfer hydrogen ions2. serve as a barrier to electron transfer

Membrane needs a moist stateto conduct ions. Liquid water in cathode: block oxygen from reaching cathode reaction site → need to control Tc

sat

56.4 CcT T

15 [A] 0.6 [V]=9 [W]cP I E

sur

25 C

T T

The convection heat coefficient, h0.8 2.8 0.810.9 W s / m Kh V

0.88

Anode: +22H 4H 4e

Cathode: +2 2O 4 4H 2H Oe

2 2 22H O 2H O

(exothermic)

Find: The required cooling air velocity, V, needed to maintain steady state operation at Tc = 56.4ºC.

Assumptions: 1) Steady-state conditions2) Negligible temperature

variations within the fuel cell3) Large surroundings4) Insulated edge of fuel cell5) Negligible energy flux by the

gas or liquid flows

0.8 2.8 0.810.9 W s / m K Vh

g 11.25 WE

g conv rad11.25 WE q q

Energy balance on the fuel cell

g outE E

4 4rad surcq A T T

conv chq A T T

9.4 m/sV

in g out stE E E E

conv g radq E q

4 4surc g cA T T Eh A T T

0.8 2.8 0.810.9 W s / m K Vh

4 4g surc

c

E A T T

A T T

conv radq q

Example 1.5

Find:

Expression for time needed to melt the ice, tm

ice of mass M at the fusion temperature0 CfT

AAcubical cavity

W

section A-A

k

L

1 fT T

Assumptions: 1) Inner surface of wall is at through the process.2) Constant properties3) Steady-states, 1-D conduction through each wall4) Conduction area of one wall =

fT

2 ( )W L W

1 fT T

0 CfT

126 fT Tk W

L

st sfE Mh

sfh: latent heat of fusion

216

msf

f

Mh L

kW T Tt

section A-A

k

inE

L

1 fT T

fT

stE

in cond mq tE

in g out stE E E E

in stE EM

1cond

fT Tq kA

L

126 fsfm

Tt

TkW Mh

L

Example 1.7

1.2 W /m Kk

sur 30 CT

air

220 C

2 200 W/m KT

h

2lamp 2000 W/mG

0.8, 0.5 ?T

Find: 1) Cure temperature T for 2) Effect of air flow on the cure temperature for Value of h for which the cure temperature is 50°C.

215W/m Kh 22 200 W/m Kh

Assumptions: 1) Steady-state conditions2) Negligible heat loss from back surface of plate3) Plate is very thin and a small object in large surroundings,

coating absorptivity w.r.t. irradiation from the surroundings

0.5

Coating to be cured

1.2 W /m Kk 0.8, 0.5

coating

2lamp 2000 W/mG

airsur 30 CT

220 C

2 200 W/m KT

h

?T

in g out stE E E E in outE E

condq

in lampE G

out conv rad condE q q q 4 4surh T TT T

377 K 104 CT

215 W/m Kh

convq radqlampG

4 4lamp surh TT TG T

2( 50 C) = 51.0 W/mh T

22 200 W/m Kh

1.2 W /m Kk

sur 30 CT

air

220 C

2 200 W/m KT

h

2lamp 2000 W/mG

0.8, 0.5 ?T

Coating to be cured

T

y T

in out ,E E

4 4surf

f

TdT

k Tdy

4 4surh T TT T

conv radq q

out cond,f radE q q

in lampE G

?0

cond,s s

s

dTq k

dy

Natural System

Environment

Energy Conversi

on

Electrical & Electronics

Manufac-turing

Process

Sensors & Actuators

Heat

Transfer

Bio-System

WHY HEAT TRANSFER ?

Natural System / Temperature Distribution in the Earth