Chapter 1

Introduction to Heat Transfer

1.0 Introduction

• Thermodynamics – study the effects of adding or removing a quantity of heat

(or energy) to or from a system.

• Heat Transfer – study the rate at which the heat (or energy) is transferred.

• When two systems are in contact and are at different temperatures, they will exchange thermal energy – Energy travels from the system of high temperature to the

low one – The rate of exchange is proportional to the temperature

difference

• There are three modes of energy transfer – Conduction – Convection– Radiation

q’’

ConductionT1>T2

Solid or Stagnant Fluid

T1 T2

Moving fluid T∞

ConvectionTs>T∞

Ts

q1’’

q2’’

T1

T2

Radiation

q’’

1.1 Conduction

• When there exists a temperature gradient within a body, heat energy will flow from the region of high temperature to the region of low temperature

• This mode of heat transfer occurs at the molecular level via two processes:– The energy from one molecule is transferred to an

adjacent molecule or– The energy is transferred by free electrons (Mostly

encountered in pure metallic solids).– No bulk motion

• The basic equation for conductive heat transfer is defined by Fourier’s law

• Or q

Ak T [Equation 1.1]

= heat transfer rate (W or Btu/hr)qWhereA = area normal to direction of heat flux (ft2, m2)

k = thermal conductivity, property of material, (Wm-1K-1, Btu hr-1 ft-1 oF-1)

Tkq

"

= heat flux (W/m2)"q

Where

• In one dimension equation 1.1 becomes

• If the temperature distribution is linear becomes

dx

dTkqx "

dx

dT

L

TT

xx

TT 12

12

12

T1

T2

x1 x2

L

"xq

Example 1.1Calculate the rate of heat transfer through a pane of window glass (k=0.78W/m K) 1 m high, 0.5 cm thick, and 0.5 m wide, if the outer surface temperature is 24oC and the inner surface temperature is 24.5oC.

x

y

0.5 cm0.5 m

1 m24.5oC

24oC

Solution

• Assumptions:– Steady-State conditions– One-dimensional conduction through the

window– Constant thermal conductivity

Solution

dx

dTkqx" L

TTk

xx

TTk 12

12

12

cm 0.5

C24.5-C24K 0.78W/m"

oo

xq

2W/m78m 0.005

297.5K-297KK 0.78W/m" xq

W398W/m7m5.0m1" 2 xx Aqq

1.2 Convection

• Heat energy transfers between a solid and a fluid when there is a temperature difference between the fluid and the solid

• This mode of heat transfer occurs both at the molecular level and macroscopic level:– The energy from one molecule is transferred to

an adjacent molecule– The energy is transferred by the bulk or

macroscopic motion of the fluid

• Forced convection: Flow is caused by external means, fans, wind, pumps etc.

                           

                           

• Free (natural) convection: Flow is induced by buoyancy forces, which arise from density differences caused by temperature variation in the fluid.

ρT

ρT

Hot

Cold

• Convection with latent heat exchange: Associated with a phase change between liquid and vapor (boiling and condensation)

Hot plate

Water VaporBubbles

WaterDroplets

ColdWater

Moist Air

q”

q”

• The basic equation for convection heat transfer was defined by Newton and is usually referred to as the Newton rate equation:

• Or

ThA

q

[Equation 1.2]

= heat transfer rate (W or Btu/hr)qWhereA = area normal to direction of heat flux (ft2, m2)

h =convection heat transfer coefficient, (Wm-2K-1, Btu hr-1 ft-2 oF-1)

Thq "

= heat flux (W/m2)"q

Where

•  • When using equation 1.2, the key to solving

convective heat transfer problems is the determination of h. We will devote several lectures on this problem later on. Basically h depends upon the following factors:– Type of convection: Free (natural) or

Forced– Geometry– Type of flow: Laminar: heat transfer is through

conduction between streamlines.Turbulent: heat transfer due to conduction and

macroscopic movement of fluid in the direction of the heat transfer. Therefore the convective heat transfer coefficient is usually higher than that of laminar flow.

• Note1: Equation 1.2 is a definition that simplifies the problem of convective heat transfer, this is not a law.

• Note 2: We often have to distinguish between the local and average convective heat transfer coefficient(hx, h)

• Note 3: The convective heat transfer coefficient is not an inherent property of the material. But it will depend on the density, viscosity , velocity and for free convection on the thermal coefficient of expansion of the fluid

Example 1.2Calculate the rate of heat transfer by natural convection between a shed roof of area 20m x 20m and ambient air, if the roof surface temperature is 27oC the air temperature is -3oC and the average convection heat transfer coefficient is 10W/m2K.

20 m

20 mTroof = 20 oC

Solution

q

Ah T

q h A T Tair roof roof air

WCmKmWq o 120000)327(400)/(10 22

• Assumptions:– Steady-State conditions

1.3 Radiation

• Energy emitted by matter that is at a finite temperature.

• This mode of heat transfer is attributed to changes in the atom configuration.

• Does not require the presence of a medium

• Most efficiently done in a vacuum

• The basic equation for radiation heat transfer comes from Stefan-Boltzman law, which represents the upper limit to the emissive power (emissive power of a blackbody)

Where Eb = emissive power (W/m2)Ts = absolute temperature (K)σ = Stefan-Boltzman constant 5.676x10-8 W/m2K4 or0.1714x10-8 Btu/hr ft2 oR4

4sb TE

• For a real surface the emissive power is smaller and can be calculated using

• Radiation may also be incident to the surface. G the irradiation, designates the rate of all radiation incident on a unit area of surface.

• A portion or all of the incident radiation may be absorbed based on the surface radiative property termed absorptivity αWhere ε = the emissivity of the surface

4sb TE [Equation 1.3]

10

10

G

G

G

G

E

J

• Radiation can also be reflected or transmitted

Special Case

• Radiation exchange between small surface s at temperature Ts and large enclosing surface sur at temperature Tsur.

• s is a gray surface (α = ε)

Here the net rate of radiation heat transfer from the surface can be Expressed as

44" surssbrad TTGTEA

qq

• It is often convenient to linearize the radiation rate equation and express it in a manner similar to convection:

• Note: hr depends strongly on temperature, while the temperature dependence of the convection heat transfer coefficient h is generally weak.

sursrrad TTAhq

Where 22surssursr TTTTh

Example 1.3A long, cylindrical electrically heated rod, 2 cm in diameter, is installed in a vacuum furnace as shown below. The surface of the heating rod is maintained at 1000 K, while the interior walls of the furnace are black and are at 800 K. Calculate the net rate at which heat is lost from the rod per unit length and the radiation heat transfer coefficient.

Solution

• Assumptions:– Steady-State conditions– Radiation exchange between the electrically

heated rod and the furnace is between a small surface and in much larger enclosure

– The surface emissivity and absorptivity are equal

Solution

LDTTTTAq rodsurssursr 4444

mmKKqr 0.102.08001000 Km

W 5.67x109.0 4444

428-

Wqr 1893

22surssursr TTTTh

222242

8- 80010008001000 Km

W 5.67x109.0 KKKKhr

Km

W151

2rh

1.4 Conservation of Energy for a Control Volume (C.V.)

• Application of the first law of thermodynamic.– Need to define a control volume bounded by a

control surface through which energy and matter pass

– Need to define an appropriate time basis

Energy in or out due to radiation

Energy in or out due to conduction

Energy in or out due to convection

Generation of Energy

Accumulation of Energy

Accumulation of Energy = Energy In - Energy Out + Energy Generated

General Form of the Energy Conservation Equation

outginst EEE

dt

dE

Where dt

dEst

inE

gE

outE

Rate of change of the energy stored in the C.V.

Rate at which energy enters the C.V.

Rate at which energy is generated in the C.V.

Rate at which energy leaves the C.V.

Example 1.4A fluid of density ρ and specific heat Cp flows in a circular pipe. Derive an expression for the temperature of the fluid as a function of position, given that the pipe inner wall temperature Tw is constant and uniform.

Data: temperature of the fluid at the inlet T0 (T0 < Tw)velocity of fluid v0 is constantvelocity profile is flat (plug shape)convection heat transfer coefficient h is constant

T0

Tw

v0

Solution

• Assumptions:– Steady-State conditions– No radiation effect– The fluid is well-mixed (highly turbulent), so

the temperature is uniform in the radial direction

– Thermal conduction of heat along the axis is small relative to convection

Solutiondefine a control volume

Δzz z+Δz

Rate of mass in (Tz) Rate of mass out (Tz+Δz)

Convection

Solution

Accumulation of Energy = Energy In - Energy Out + Energy Generated

0 0Energy In - Energy Out = 0

Energy In =

Energy Out =

zpin TCvRE 02

wzzpout TTzhRTCvRE 202

0202

02 wzzpzp TTzhRTCvRTCvR

Divide by zR 020

w

zzzp TThz

TTCRv

0202lim 00

0

wpwzzzp

zTTh

dz

dTCRvTTh

z

TTCRv