materials engineering (med 123) - cvut.cz

MAEN 123 – p. 1

Materials Engineering (MED 123)

Igor Medved’

Department of Materials Engineering and Chemistry, Czech Technical University, Prague

Heat transfer by conduction

Heat transfer byconduction

Introduction

Introduction

Conduction in gases

Conduction in liquids

Conduction in solids

Heat flux

Fourier’s Law

Examples of gradients


Thermal conductivity λ

MAEN 123 – p. 2

Introduction


Introduction

Introduction

Conduction in gases



Heat flux

Fourier’s Law




MAEN 123 – p. 3

� In general, heat is transferred by one or more of the following

mechanisms :

� conduction� convection� radiation

� These are known from everyday experience

� For example, when we warm our homes in the winter, heat is

transferred:

� to the inside walls by convection,� through the walls by conduction,� away from the outside walls to the surroundings by convection and possibly

radiation.

� Or, when cooked food is taken out of the oven, heat is transferred:

� to the surroundings initially by radiation and convection from the surface;� the cooling of the surface causes conduction of heat from the center to the

surface;� continued removal of heat from the surface to the surroundings

Introduction


Introduction

Introduction

Conduction in gases



Heat flux

Fourier’s Law




MAEN 123 – p. 4

� In this lecture we will be concerned only with heat conduction

� A couple of books to consult:

� J. C. Jones, Principles of Thermal Sciences and their Application to Engineering.Whittles Publishing, 2000.

� M. N. Ozisik, Heat Conduction. John Wiley & Sons, 1993.

Conduction in gases


Introduction

Introduction

Conduction in gases



Heat flux

Fourier’s Law




MAEN 123 – p. 5

� In gases heat is transferred via collisions between molecules

� What is an approximate speed of a gas molecule at room temperature?

� At temperature T = 300 K an average kinetic energy of a molecule is about

ε ∼ kBT ≈ 10−23

. 300 = 3× 10−21

J,

(here kB = 1.38× 10−23 JK−1 is the Boltzmann constant)

� The kinetic energy is ε = 1

2mv2

� So, the energy ε corresponds to the molecular speed

v ∼

√

ε

m≈

√

3× 10−21 J

5× 10−26 kg≈ 245ms

−1≈ 880 kmh

−1

(we considered an oxygen molecule whose mass is m ≈ 5× 10−26 kg)



Introduction

Introduction

Conduction in gases



Heat flux

Fourier’s Law




MAEN 123 – p. 6

� Basically similar to gases, but a liquid molecule is never in isolation

from others but form small groups or clusters

� In liquids the translation of these molecular groups and the

accompanying thermal energy transfer are the origin of heat

conduction



Introduction

Introduction

Conduction in gases



Heat flux

Fourier’s Law




MAEN 123 – p. 7

� There are two basic mechanisms of heat conduction in solids:

� by lattice vibrations

� by collisions of delocalized electrons (an “electron gas ”) if present

� Why metals are good heat conductors, while non-metals (such as ceramicsor wood) are thermal insulators?

� Due to the presence of delocalized electrons in metals and their absence fromnon-metals

� The electrons are treated as a gas of molecules each having a thermalenergy

� A metal is viewed as a vibrating lattice of positive ions in a “sea” of electrons

� An electron loses most of its thermal energy by colliding not with anotherelectron but with one of the positive ions constituting the lattice



Introduction

Introduction

Conduction in gases



Heat flux

Fourier’s Law




MAEN 123 – p. 7

� Why is it that heat conduction is so much faster in metals than in

gases?

� Since the average speed is very much greater in an “electron gas”

than in a true gas (electrons are much lighter than molecules)

� At the same thermal energy ε the average speed is v ∼

√

ε/m, so

thatvelectron gas

voxygen gas

∼

√

moxygen

melectron

∼

√

55, 000 ∼ 230

� Thus, the average speed in an electron gas is more than 200 times

higher than in an oxygen gas

Heat flux


Introduction

Introduction

Conduction in gases



Heat flux

Fourier’s Law




MAEN 123 – p. 8

� Whenever there is a temperature difference ∆T between two places,

heat flows from the hotter to the cooler place

� The measure of the amount of heat transferred is the heat flux q

� It is the amount of heat transferred through unit area per unit time

� Thus, q has the unit Jm−2s−1 = Wm

−2

Heat flux


Introduction

Introduction

Conduction in gases



Heat flux

Fourier’s Law




MAEN 123 – p. 8

� Whenever there is a temperature difference ∆T between two places,

heat flows from the hotter to the cooler place

� The measure of the amount of heat transferred is the heat flux q

� It is the amount of heat transferred through unit area per unit time

� Thus, q has the unit Jm−2s−1 = Wm

−2

� The higher ∆T , the larger the heat flux q

� The larger the difference ∆x between the places, the smaller q

Fourier’s Law


Introduction

Introduction

Conduction in gases



Heat flux

Fourier’s Law




MAEN 123 – p. 9

� Fourier’s Law states that the heat flux q is

� proportional to the temperature difference ∆T

� inversely proportional to the distance ∆x

� Thus,

q = λ∆T

∆x

� The material parameter is called the thermal conductivity λ(also denoted as k)

� λ has the unit Wm−1 K−1

Fourier’s Law


Introduction

Introduction

Conduction in gases



Heat flux

Fourier’s Law




MAEN 123 – p. 9

� Precise mathematical formulation involves derivatives:

q = λ∆T

∆x−→ q = λ

dT

dx(I.1)

� In 3D the mathematical formulation of the law is

q = −λ gradT (I.2)

� Heat flux q is a vector in 3D (it has a magnitude and a direction)

� gradT is the gradient of temperature; it is a vector pointing in the direction ofthe largest rate of temperature increase

� In the rectangular coordinates x, y, z it is given as gradT = ∂T∂x

+ ∂T∂y

+ ∂T∂z

� The negative sign arises from the fact that the heat flows in the direction oppositeto the increase in T

� So, the heat flows in the direction of the largest temperature decrease



Introduction

Introduction

Conduction in gases



Heat flux

Fourier’s Law




MAEN 123 – p. 10

0 2 4 6 8 100

1

2

3

4

5

6

0 2 4 6 8 100

1

2

3

4

5

6

0 2 4 6 8 10

0

2

4

6

8

10

0 2 4 6 8 10

0

2

4

6

8

10



Introduction

Introduction

Conduction in gases



Heat flux

Fourier’s Law




MAEN 123 – p. 11

-10 -5 0 5 10

-10

-5

0

5

10

-10 -5 0 5 10

-10

-5

0

5

10

-10 -5 0 5 10

-10

-5

0

5

10

-10 -5 0 5 10

-10

-5

0

5

10



Introduction

Introduction

Conduction in gases



Heat flux

Fourier’s Law




MAEN 123 – p. 12

� It is an important thermal property of materials — it characterizes how

well a material conducts heat

� Any substance—a solid, liquid, or gas—can be assigned a numerical

value of thermal conductivity

� Its the value changes with temperature and pressure

� Of course, in heat transfer in gases and liquids the dominant mode of

heat transfer is usually convection

� It is quite acceptable to extend the concept of thermal conductivity to

powdered, fibrous, or shredded media (e.g., a bed of sawdust, or a

bale of wool)

� Each of these is actually two-component (particles + air voids/pores), but a single(average) value of the thermal conductivity can be taken to apply



Introduction

Introduction

Conduction in gases



Heat flux

Fourier’s Law




MAEN 123 – p. 12

� For a given temperature difference ∆T and thickness ∆x, the rates of heatflow will differ according to λ

� Let a material of thickness ∆x = 2 cm have a temperature difference∆T = 45

◦C across it

� If the material is copper (Cu), then λ = 386Wm−1

K−1 and the heat flow is

q = 38645

0.02= 870 kWm

−2

� For stainless steel (λ = 15Wm−1

K−1) we get

q = 34 kWm−2

� For cardboard (λ = 0.7Wm−1

K−1) we get only

q = 1.6 kWm−2



Introduction

Introduction

Conduction in gases



Heat flux

Fourier’s Law




MAEN 123 – p. 12

� Typical range of λ of various materials + the effect of the temperature

on λ

materials engineering (med 123) - cvut.cz

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