Chapter 2 Conduction Chapter 2.

Chapter 2 1

Chapter 2

Conduction

Chapter 2 2

Conduction Heat Transfer

• Conduction refers to the transport of energy in a medium (solid, liquid or gas) due to a temperature gradient.

• The physical mechanism is random atomic or molecular activity• Governed by Fourier’s law

• In this chapter we will learn The definition of important transport properties and what governs

thermal conductivity in solids, liquids and gases The general formulation of Fourier’s law, applicable to any

geometry and multiple dimensions How to obtain temperature distributions by using the heat diffusion

equation. How to apply boundary and initial conditions

Chapter 2 3

Thermal Properties of Matter

• Recall from Chapter 1, equation for heat conduction:

LTk

LTTkqx

21"

The proportionality constant is a transport property, known as thermal conductivity k (units W/m.K)

• Usually assumed to be isotropic (independent of the direction of transfer): kx=ky=kz=k

Is thermal conductivity different between gases, liquids and solids?

Chapter 2 4

Thermal Conductivity: Solids• Solid comprised of free electrons and atoms bound in lattice• Thermal energy transported through

– Migration of free electrons, ke

– Lattice vibrational waves, kl

le kkk )( y,resistivit electrical1

eekwhere

? What is the relative magnitude in pure metals, alloys and non-metallic solids?

See Figure 2.5, Appendix Tables A.1, A.2 and A.3 text

Chapter 2 5

Thermal Conductivity: Fluids

• Intermolecular spacing is much larger• Molecular motion is random• Thermal energy transport less effective than in solids; thermal

conductivity is lower Kinetic theory of gases:

cnkwhere n the number of particles per unit volume, the mean molecular speed and l the mean free path (average distance travelled before a collision)

? What are the effects of temperature, molecular weight and pressure?

c

Chapter 2 6

Thermal Conductivity: Fluids

• Physical mechanisms controlling thermal conductivity not well understood in the liquid state

• Generally k decreases with increasing temperature (exceptions glycerine and water)

• k decreases with increasing molecular weight.

• Values tabulated as function of temperature. See Tables A.5 and A.6, text.

Chapter 2 7

Thermal Conductivity: Insulators

– Can disperse solid material throughout an air space – fiber, powder and flake type insulations

– Cellular insulation – Foamed systems Several modes of heat transfer involved (conduction, convection,

radiation) Effective thermal conductivity: depends on the thermal conductivity and

radiative properties of solid material, volumetric fraction of the air space, structure/morphology (open vs. closed pores, pore volume, pore size etc.) Bulk density (solid mass/total volume) depends strongly on the manner in which the solid material is interconnected. See Table A.3.

? How can we design a solid material with low thermal conductivity?

Chapter 2 8

Thermal DiffusivityThermophysical properties of matter:• Transport properties: k (thermal conductivity/heat transfer), (kinematic

viscosity/momentum transfer), D (diffusion coefficient/mass transfer)• Thermodynamic properties, relating to equilibrium state of a system,

such as density, and specific heat cp.

– the volumetric heat capacity cp (J/m3.K) measures the ability of a material to store thermal energy.

• Thermal diffusivity is the ratio of the thermal conductivity to the heat capacity:

pck

Chapter 2 9

The Conduction Rate EquationRecall from Chapter 1:

dxdTkAqx • Heat rate in the

x-direction

• Heat flux in the x-direction dx

dTkAqqx "

x

We assumed that T varies only in the x-direction, T=T(x)

Direction of heat flux is normal to cross sectional area A, where A is isothermal surface (plane normal to x-direction)

T1(high)

T2 (low)

qx”

x1 x2

A

Chapter 2 10

The Conduction Rate EquationIn reality we must account for heat transfer in three dimensions• Temperature is a scalar field T(x,y,z)• Heat flux is a vector quantity. In Cartesian coordinates:

""""zyxx

qqq kjiq

for isotropic medium zTkq

yTkq

xTkq zyx

""" , ,

TkzT

yT

xTk

kjiq

Where three dimensional del operator in cartesian coordinates:

zyx

kji

Chapter 2 11

Summary: Fourier’s Law

• It is phenomenological, ie. based on experimental evidence

• Is a vector expression indicating that the heat flux is normal to an isotherm, in the direction of decreasing temperature

• Applies to all states of matter

• Defines the thermal conductivity, ie.

)/(

"

xTqk x

Chapter 2 12

The Heat Diffusion Equation

• Objective to determine the temperature field, ie. temperature distribution within the medium.

• Based on knowledge of temperature distribution we can compute the conduction heat flux.

Reminder from fluid mechanics: Differential control volume.

We will apply the energy conservation equation to the differential control volumeCV

Element of volume:dx dy dz

T(x,y,z)

Chapter 2 13

Heat Diffusion Equation

Energy Conservation Equation stst

outgin Edt

dEEEE

dzzdyydxxout

zyxin

qqqE

qqqE

where from Fourier’s law

zTdxdyk

zTkAq

yTdxdzk

yTkAq

xTdydzk

xTkAq

zz

yy

xx

)(

)(

)(

(2.1)

z

y

x

xy

z

Chapter 2 14

Heat Diffusion Equation• Thermal energy generation due to an energy source:

– Manifestation of energy conversion process (between thermal energy and chemical/electrical/nuclear energy)

Positive (source) if thermal energy is generated Negative (sink) if thermal energy is consumed

) (

dV

dzdydxq

qEg

• Energy storage term– Represents the rate of change of thermal energy

stored in the matter in the absence of phase change.

) ( dzdydxtTcE pst

tTcp / is the time rate of change of the sensible (thermal) energy of the medium per unit volume (W/m3)

q is the rate at which energy is generated per unit volume of the medium (W/m3)

Chapter 2 15

Heat Diffusion EquationSubstituting into Eq. (2.1):

tTcq

zTk

yyTk

yxTk

x p

Heat Equation

Net conduction of heat into the CVrate of energy generation per unit volume

time rate of change of thermal energy per unit volume

At any point in the medium the rate of energy transfer by conduction into a unit volume plus the volumetric rate of thermal energy generation must equal the rate of change of thermal energy stored within the volume

(2.2)

Chapter 2 16

Heat Diffusion Equation- Other forms• If k=constant

tT

kq

zT

yT

xT

1

2

2

2

2

2

2 is the thermal diffusivity

pck

• For steady state conditions

0

q

zTk

yyTk

yxTk

x

• For steady state conditions, one-dimensional transfer in x-direction and no energy generation

0or 0"

dxdq

dxdTk

dxd x Heat flux is constant in

the direction of transfer

(2.3)

(2.4)

Chapter 2 17

Heat Diffusion Equation

• In cylindrical coordinates:

tTcq

zTk

zTk

rrTkr

rr p

211

• In spherical coordinates:

tTcqTk

rTk

rrTkr

rr p

sin

sin1

sin11

2222

2

(2.5)

(2.6)

Chapter 2 18

Example (Problem 2.23 textbook)

The steady-state temperature distribution in a one-dimensional wall of thermal conductivity 50 W/m.K and thickness 50 mm is observed to be T(°C)=a+bx2, where a=200°C, b=-2000°C/m2, and x is in meters.

a) What is the heat generation rate in the wall?b) Determine the heat fluxes at the two wall faces. In what manner are

these heat fluxes related to the heat generation rate?

Chapter 2 19

Boundary and Initial Conditions• Heat equation is a differential equation:

– Second order in spatial coordinates: Need 2 boundary conditions– First order in time: Need 1 initial condition

Boundary ConditionsExample: a surface is in contact with a melting solid or a boiling liquid

1) B.C. of first kind (Dirichlet condition):

x

T(x,t)

Ts

Chapter 2 20

Boundary and Initial ConditionsExample: What happens when an electric heater is attached to a surface? What if the surface is perfectly insulated?

2) B.C. of second kind (Neumann condition): Constant heat flux at the surface

xT(x,t)

xT(x,t)

qx”

Chapter 2 21

Boundary and Initial Conditions

3) B.C. of third kind: When convective heat transfer occurs at the surface

T(x,t)

T(0,t)

x

hT ,

Chapter 2 22

Example 2.3, textbookA long copper bar of rectangular cross section, whose width w is much greater than its thickness L, is maintained in contact with a heat sink (for example an ice bath) at a uniform initial temperature To. Suddenly an electric current is passed through the bar, and an airstream of temperature is passed over the top surface, while the bottom surface is maintained at To. Obtain the differential equation, the boundary and initial conditions that can be used to determine the temperature as a function of position and time inside the bar.

T

Chapter 2 23

Example (Problem 2.39, textbook)Passage of an electric current through a long conducting rod of radius ri and thermal conductivity kr results in uniform volumetric heating at a rate of . The conducting rod is wrapped in an electrically nonconducting cladding material of outer radius ro and thermal conductivity kc, and convection cooling is provided by an adjoining fluid. For steady-state conditions, write appropriate forms of the heat equations for the rod and cladding. Express appropriate boundary conditions for the solution of these equations.

q