Heat Transfer:

Chap 15: Fundamentals of Heat Transfer

Chap 16: Differential Equations of Heat Transfer

Chap 17: Steady-State Conduction

Chap 18: Unsteady-State Conduction

Chap 19: Convective Heat Transfer

What is heat transfer? Heat transfer is thermal energy transport due to a spatial temperature difference.

What is thermal energy? Thermal energy is associated with the translation, rotation, vibration and electronic states of the atoms and molecules that comprise matter. It represents the cumulative effect of microscopic activities and is directly linked to the temperature of matter.

* Thermal Energy, Temperature and Heat Transfer:

Thermal energy Energy associated with microscopic behavior of matter

E or U (J, J/kg)

Quantity Meaning Symbol (units)

Temperature T (K, C) A means of indirectly assessing the amount of thermal energy stored in matter

Thermal energy transport due to temperature gradient

Heat transfer

Amount of thermal energy transferred over a time interval

Heat Q (J, Btu)

Heat transfer rate

Thermal energy transfer per unit time

q (J/s or W)

Thermal energy transfer per unit time per unit surface area

Heat flux (vector)

q (W/m2)

(1 Btu = 252 cal)

* Modes of Heat Transfer (or Heat Transfer Mechanisms):

Conduction

Conduction:

Convection Radiation

- Heat transfer in a solid or a stationary fluid (gas or liquid) due to the random motion of its constituent atoms, molecules and /or electrons.

- No bulk fluid motion

- Dominant in solids, important in fluids

* Modes of Heat Transfer (or Heat Transfer Mechanisms): Convection:

- Heat or energy transfer by fluid motion

- Forced convection: fluid flow caused by external means (pump, etc.)

- Natural convection: fluid flow caused by energy transfer itself (i.e., density gradient in space)

Radiation: - Heat or energy transfer by photons

All materials radiate thermal energy in the form of electromagnetic waves (or photons). When this radiation falls in a second body, it may be partially reflected, transmitted or absorbed. It is only the fraction that is absorbed that appears as heat in the body.

- Conduction and convection require the presence of temperature variations in a material medium.

- Although radiation originates from matter, its transport does not require a material medium and occurs most efficiently in a vacuum.

+=

++=

equation)energy convective(DtDT

)equation Stokes-Navier(Dt

uDequation) continuity(0=u

2

2

TkC

gup

p

, , Cp, k

U, T

T0

L

* Convective Heat Transfer (Newtonian fluid):

equations) 5 and unknowns (5 :Unknowns , T up,

+= TktTC p

2DD

* Convective Energy Equation:

+=

+

TkT

tTC p

2uor

convection term

conduction term

Viscous dissipation

+= qtTC p D

D +=

+

qu T

tTC por

Tkq == "q

Fouriers law of conduction

Tkq == "q

[ ]Km

Wk

=

* Fouriers law of conduction:

- Heat flux is proportional to the temperature gradient.

- The proportionality constant, k is called thermal conductivity.

Unit of k :

- Heat flux is in the direction of decreasing temperature.

* Thermophysical Properties

+=

+

TkT

tTC p

2u

- Transport properties: k (thermal conductivity), (viscosity)

Thermal diffusivity:

volumetric heat capacity

pCk

[ ]

sm2

=

- Thermodynamic properties: , Cp

[ ]

=

=

KmJ

KkgJ

mkgC p 33

Recall kinematic viscosity: [ ]

sm2

=

(ability of a material to store thermal energy per unit volume)

* Convective Energy Equation:

+=

+

TkT

tTC p

2u

Conduction:

- Heat transfer in a solid or a stationary fluid (gas or liquid) due to the random motion of its constituent atoms, molecules and /or electrons.

- No bulk fluid motion i.e., u = 0 & = 0 - Dominant in solids, important in fluids

TktTC p

2=

TC

ktT

p

2=

TtT 2=

or or

Here is the thermal diffusivity pC

k

= [ ]s

m2=

* Heat Transfer by Conduction:

TtT 2=

Tkq == "q- Fouriers law of conduction provides the heat flux.

Boundary conditions (?)

(heat equation) Temperature profile, T(x, t)

If there is an energy source within the medium,

qTktTC p +=

2

Rate of energy generation per unit volume (W/m3)

- Boundary conditions depend on the heat transfer situation.

In Cartesian coordinate system:

+

+

==

2

2

2

2

2

22

zT

yT

xTT

tT

( )

===xT

yT

xTkqqqq zyx ,,,,"

"""q

* Heat Transfer by Conduction:

TtT 2=

(heat equation)

In cylindrical coordinate system:

+

+

==

2

2

2

2

22 11

zTT

rrTr

rrT

tT

( )

===zTT

rrTkqqqq zr ,

1,,," """

q

* One-dimensional steady-state conduction in a slab (17.1)

x

T

T2

T1

L

=

2

2

xT

tT

=

=L

TTkxTkqx 21

"

022

=

xTst.-st.

==

==

2

1,At ,0At

TTLxTTx

BCs: 2),0(,0At TxTt ==

Initial condition:

xL

TTTxT 211)(

=

Steady-state solution:

"xqAq =

(W/m2)

(heat transfer rate, W)

* One-dimensional steady-state conduction in a slab (17.1)

x

T

T2

T1

L

=

=L

TTkxTkqx 21

"

xL

TTTxT 211)(

=

Steady-state solution:

"xqAq =

(W/m2)

(heat transfer rate, W)

- Temperature profile is the same regardless of the material (e.g., wood, metal block)

- Heat flux (and heat transfer rate), however, is proportional to k.

wood: k = ~0.15 W/m-K

Aluminum: k = 237 W/m-K

Thermal Conductivity of Various Materials

Materials (300K) W/m-K gas ~0.05

liquid ~1

non-metallic solid 1 ~ 10 metallic solid ~100

Materials W/m-K air 2.63 X 10-2

Steam (380K) 2.46 X 10-2

Water 0.613 Glass 1.4 Soil 0.52 Teflon 0.35 Wood ~0.15 Plaster board 0.17 Glass fiber 0.036 Aluminum 237 Copper 401

[ ]Km

WmK

mWT

qk

==

=/

/" 2

P?& T on Dependence),( PTkk =

cnk In case of gases: * Dependence of k on T & p ?

mean free path mean molecular speed number density

c- If T increases, increases. Thus, k increases. c- If molecular mass is smaller, is larger. Thus, k is large.

heliumhydrogen kk >

- If p increases, n increases. But decreases. Thus, k is a weak function of p.

k(T, p) of liquids: not well understood

Heat Transfer: Slide Number 2Slide Number 3Slide Number 4Slide Number 5Slide Number 6Slide Number 7Slide Number 8Slide Number 9Slide Number 10Slide Number 11Slide Number 12Slide Number 13Slide Number 14Slide Number 15Slide Number 16