Materials Engineering (MED 123) -...

MAEN 123 – p. 1

Materials Engineering (MED 123)

Igor Medved’

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

Heat transfer by conduction


Introduction

Basis of heat conduction —fluids

Basis of heat conduction —solids

Fourier’s Law

Fourier’s Law

Fourier’s Law

Fourier’s Law

Thermal conductivity λ


Equation of heat conduction




Thermal diffusivity

Initial and boundaryconditions

MAEN 123 – p. 2

Introduction


Introduction



Fourier’s Law

Fourier’s Law

Fourier’s Law

Fourier’s Law







Thermal diffusivity


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

� When we warm our homes in the winter months, heat is transferred to the inside walls by convection,through the walls by conduction, and away from the outside walls to the surroundings by convection andpossibly radiation

� When cooked food is taken out of the oven, heat is transferred to the surroundings initially by radiation andconvection from the outside surface, and the cooling of the outer part so affected causes conduction ofheat from the center to the outside with continued removal from there to the surroundings

� In the following 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.

Basis of heat conduction — fluids


Introduction



Fourier’s Law

Fourier’s Law

Fourier’s Law

Fourier’s Law







Thermal diffusivity


MAEN 123 – p. 4

� In gases heat is transferred via collisions between molecules of the gas

� A molecule of the gas at the temperature T undergoes translational motion with an average kineticenergy given as

ε =3

2kBT,

where kB = 1.38 × 10−23 JK−1 is the Boltzmann constant

� At T = 300 K the energy is ε = 6.2 × 10−21 J

� Since the kinetic energy is equal to 1

2mv2 , the energy ε corresponds to the molecular speed

v ≈

√

2ε

m∼ 500ms

−1

(the mass of an oxygen molecule m ∼ 5 × 10−26 kg)

� In liquids the translation of molecular groups and the accompanying thermal energytransfer are the origin of heat conduction

� contrary to gases, a liquid molecule is never in isolation from others

Basis of heat conduction — solids


Introduction



Fourier’s Law

Fourier’s Law

Fourier’s Law

Fourier’s Law







Thermal diffusivity


MAEN 123 – p. 5

� There are two basic mechanisms of heat conduction in solids: by lattice vibrationsand by means of the thermal energy of any delocalized electrons present

� The presence of delocalized electrons in metals and their absence from non-metals (such as ceramics orwood) is the basic reason why metals are good heat conductors, while non-metals are thermal insulators

� The electrons are treated as a gas of molecules (an “electron gas”) each having a thermal energy

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

� An electron loses its thermal energy by colliding not with another electron but with one of the positiveions constituting the lattice

� 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 an actual gas

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

ε/m, so that

velectron gas

voxygen gas

∼

√

moxygen

melectron

∼

√

5 × 10−26

1 × 10−30∼ 200

� Thus, the average speed in an electron gas is about 200 times higher than in an oxygen gas

Fourier’s Law


Introduction



Fourier’s Law

Fourier’s Law

Fourier’s Law

Fourier’s Law







Thermal diffusivity


MAEN 123 – p. 6

� Fourier’s Law states that the heat flux q (=the amount of heat transferred through unitarea per unit time) is proportional to the temperature gradient, the constant ofproportionality being the thermal conductivity λ (also denoted as k)

� So, whenever there is a temperature difference ∆T between two places, heat flowsfrom the hotter to the colder place

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

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

Fourier’s Law


Introduction



Fourier’s Law

Fourier’s Law

Fourier’s Law

Fourier’s Law







Thermal diffusivity


MAEN 123 – p. 7

� In one dimension the mathematical formulation of the law is

q = −λ∆T

∆xor, more precisely, q = −λ

dT

dx(I.1)

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

� q has the unit Jm−2s−1 = Wm−2 ; λ has the unit Wm−1K−1

� In three dimensions the mathematical formulation of the law is

~q = −λ∇T (I.2)

� Heat flux is a vector (it has a magnitude and a direction)

� ∇T = grad T is the gradient of temperature; it is a vector pointing in the direction of the greatest rate oftemperature increase

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

+ ∂T∂y

+ ∂T∂z

Fourier’s Law


Introduction



Fourier’s Law

Fourier’s Law

Fourier’s Law

Fourier’s Law







Thermal diffusivity


MAEN 123 – p. 8

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

Fourier’s Law


Introduction



Fourier’s Law

Fourier’s Law

Fourier’s Law

Fourier’s Law







Thermal diffusivity


MAEN 123 – p. 9

-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



Fourier’s Law

Fourier’s Law

Fourier’s Law

Fourier’s Law







Thermal diffusivity


MAEN 123 – p. 10

� The thermal conductivity is an important thermal property of materials, it characterizesthe amount of heat transfer rate per unit area and per unit temperature gradient

� Any substance—a solid, liquid, or gas—can be assigned a numerical value of thermalconductivity; however, the value changes with temperature and pressure

� Of course, in heat transfer within fluids (= gases and liquids) the dominant mode ofheat transfer might be convection

� It is quite acceptable to extend the concept of thermal conductivity to powdered,fibrous, or shredded media, such as a bed of sawdust or a bale of wool

� Each of these is actually two-component, comprising the particles and the air voids (pores), but a singlevalue of the thermal conductivity can be taken to apply

� For a specified temperature difference ∆T and thickness ∆x, the rates of heattransfer (= heat flow) will differ according to λ of the conducting material

� Let a material of thickness ∆x = 2 cm have one side kept at 70 ◦C and the other side at 25 ◦C

� If the material is copper (Cu), then λ = 386Wm−1K−1 and the heat flow isq = 386 70−25

0.02= 870 000Wm−2

� For stainless steel (λ = 15Wm−1K−1) we get q = 34 000Wm−2

� For cardboard (λ = 0.7Wm−1K−1) we get only q = 1 600Wm−2



Introduction



Fourier’s Law

Fourier’s Law

Fourier’s Law

Fourier’s Law







Thermal diffusivity


MAEN 123 – p. 11

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



Introduction



Fourier’s Law

Fourier’s Law

Fourier’s Law

Fourier’s Law







Thermal diffusivity


MAEN 123 – p. 12

� Expresses the conservation (balance) of energy in a given material� Consider a volume V within the material, then

Time change in internal energy(energy stored) in V

=Heat flow into V

through its surface+

Rate of energygeneration in V

� If V is very small, then


=∆U

∆t=

∆m× cp ×∆T

∆t=

ρ× V × cp ×∆T

∆t

Heat flow into V

through its surface= −~q · ~nA



Introduction



Fourier’s Law

Fourier’s Law

Fourier’s Law

Fourier’s Law







Thermal diffusivity


MAEN 123 – p. 13

� Energy generation may occur due to processes that take place in the material (such asphase changes or chemical, electrical, nuclear, and other sources)

� Rate of energy generation = amount of heat produced by such processes per unitvolume and unit time = denoted as, say, g(x, t) that may depend on position and/ortime (units Js−1m−3 = Wm−3)

� Rate of energy generation in a small volume V is equal to g V



Introduction



Fourier’s Law

Fourier’s Law

Fourier’s Law

Fourier’s Law







Thermal diffusivity


MAEN 123 – p. 14

� If V is not necessarily small, then


=ρ× V × cp ×∆T

∆t−→

∫V

ρ cp∂T

∂tdV

Heat flow into V

through its surface= −~q · ~nA −→ −

∫A

~q · ~n dA

Rate of energygeneration in V

= g V −→

∫V

g dV

� The energy balance then gives∫V

ρ cp∂T

∂tdV = −

∫A

~q · ~n dA+

∫V

g dV

� To proceed further, we need to convert the surface integral (the second term) into a volume integral� This is possible thanks to J. C. F. Gauss (German mathematician, 1777 – 1855):

∫

A

~q · ~n dA =

∫

V

(∇ · ~q) dV,

� ∇ · ~q = div ~q is the divergence of the heat flux = the heat flow from an infinitely small volume around agiven point

� In the rectangular coordinates x, y, z it is given as ∇ · ~q = ∂qx∂x

+∂qy∂y

+ ∂qz∂z

if the vector

~q = (qx, qy, qz)



Introduction



Fourier’s Law

Fourier’s Law

Fourier’s Law

Fourier’s Law







Thermal diffusivity


MAEN 123 – p. 15

� The energy balance now reads∫V

ρ cp∂T

∂tdV = −

∫V

(∇ · ~q) dV +

∫V

g dV

� Further manipulations

� Using Fourier’s Law ~q = −λ∇T , we get∫

V

ρ cp∂T

∂tdV = λ

∫

V

(∇ · ∇T ) dV +

∫

V

g dV

or∫

V

{

ρ cp∂T

∂t− λ(∇ · ∇T ) − g

}

dV = 0

� The latter equality must be true for an arbitrary volume V� This is possible only if the expression in {. . . } is equal to 0

� Finally, the heat equation reads

ρ cp∂T

∂t= λ∆T − g

� ∇ · ∇T = ∆T is called the Laplacian of T

� In the rectangular coordinates x, y, z one has ∆T = ∂2T

∂x2+ ∂2T

∂y2+ ∂2T

∂z2

Thermal diffusivity


Introduction



Fourier’s Law

Fourier’s Law

Fourier’s Law

Fourier’s Law







Thermal diffusivity


MAEN 123 – p. 16

� If no generation occurs (g = 0), then the heat equation simplifies to

ρ cp∂T

∂t= λ∆T i.e.

∂T

∂t= a∆T

� a = λρcp

= thermal diffusivity (unit m2s−1); sometimes denoted as α

� Important thermal property of materials (depends on the temperature)

� It is associated with the speed of propagation of heat into the body during temperature changes (thehigher a, the faster the propagation)

� For example, consider a rod at room temperature and cool one of its ends to, say, 0 K. How fast will thetemperature of rod drop to its half at, say, 30 cm from the cooled end?

Initial and boundary conditions


Introduction



Fourier’s Law

Fourier’s Law

Fourier’s Law

Fourier’s Law







Thermal diffusivity


MAEN 123 – p. 17

� The heat equation cannot be solved without knowing how the material is heated orcooled

� Usually we know T within the body (at all positions x, y, z) at some instant t0 (mostlythe beginning of the measurement) = initial condition :

T (x, y, z, t0) = a known function of x, y, z

� We must also know what happens with T on the surface (boundary) of the material =boundary conditions

� Temperature on the boundary is prescribed at all times:

T (x, y, z, t) = a known function for any position on the boundary

(for example, T is constant on the boundary = the surface of the material is heated/cooled at a prescribedtemperature)

� Heat flow is prescribed on the boundary at all times:

∂T

∂n= a known function for any position on the boundary,

where the derivative is along the outward normal to the surface (for example, there is no heat flow into thebody, i.e., ∂T

∂n= 0)

� Other forms of boundary conditions may apply in a given situation

Materials Engineering (MED 123) -...

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