Chapter 1: Fourier Equation and Thermal Conductivity

1.1 …………. Introduction of Heat Transfer

1.2 …………. Fourier’s law of heat conduction

1.3 …………. Thermal Conductivity of material

1.4 …………. General heat conduction equation

(a) Cartesian co-ordinates

(b) Cylindrical co-ordinates

(c) Spherical co-ordinates

(d) General one dimensional conduction equation

1.1… Introduction of Heat Transfer

Heat transfer is a science that studies the energy transfer between two bodies due

to temperature difference. There can be no net heat transfer between two mediums

that are at the same temperature. Basic requirement for heat transfer : presence of

temperature difference .

Note: Heat flow occurs only in the direction of decreasing temperature

The temperature difference is the driving force for heat transfer, just as the voltage

difference is the driving force for electric current flow and pressure difference is the

driving force for fluid flow.

Introduction of Heat Transfer (continue…)

Modes of Heat Transfer

Conduction Convection Radiation

Conduction: An energy

transfer across a system

boundary due to a

temperature difference by

the mechanism of

intermolecular interactions.

Conduction needs matter

and does not require any

bulk motion of matter.

Convection: An energy

transfer across a system

boundary due to a

temperature difference by the

combined mechanisms of

intermolecular interactions

and bulk transport. Convection

needs fluid matter.

Radiation: Radiation heat

transfer involves the transfer

of heat by electromagnetic

radiation that arises due to

the temperature of the

body. Radiation does not

need matter.

4

HOT(lots of vibration)

COLD(not much vibration)

Heat travels along the rod

Conduction is the transfer of heat through a solid or from one solid to another.

When you heat a metal strip at one end, the heat travels to the other end.

As you heat the metal, the particles vibrate, these vibrations make the adjacent particles vibrate, and so on and so on, the vibrations are passed along the metal and so is the heat. We call this?

http://education.jlab.org/jsat/ [Accessed 13 November 11]

Introduction of Heat Transfer (continue…)

When the handle of a spoon stirring a cup of hot

chocolate gets hot, it’s because of conduction.

How ???????

When the particles of a solid are heated they gain

energy and vibrate more quickly. They bump into

neighbor particles and transfer the energy to them.

Introduction of Heat Transfer (continue…)

1.2… Fourier’s Law of Heat Transfer• The heat flux is proportional to the

temperature gradient:

• …… (1)

Where k=thermal conductivity (W/m°C or Btu/h ft °F) -- a measure of how fast heat flows through a material-- k(T), but we usually use the value at the average temperature

q can have x, y, and z components; it’s a vector quantity

x

hot wall

cold walldx

dT

temperatureprofileQ

q k TA

dx

dTk

A

Qq

dx

dTAkQ

"

….. (2) Fourier’s Law

…… (3) Heat FluxIn most practical situations conduction, convection, and radiation appear in combination

1.3… Thermal Conductivity of Material

The heat transfer characteristics of a solid material are measured by a property called the thermal conductivity (k) measured in W/m.K. It is a measure of a substance’s ability to transfer heat through a solid by conduction. K = Q × L / (A × ΔT)

Thermal conductivity is defined as the quantity of heat (Q) transmitted through a unit thickness (L) in a direction normal to a surface of unit area (A) due to a unit temperature gradient (ΔT) under steady state conditions and when the heat transfer is dependent only on the temperature gradient.

Note: The thermal conductivity of most liquids and solids varies with temperature. For vapors, it depends upon pressure.

Thermal conductivity values for various materials at 300 K

Thermal Conductivity of Material (continue…)

8

Silver 410 W/m. °C 237 Btu/h.ft.°FCopper 385 W/m. °C 223 Btu/h.ft.°F

Window glass 0.780 W/m. °C 0.045 Btu/h.ft.°F

Brick 0.720 W/m. °C 0.0461 Btu/h.ft.°F

Glass wool 0.038 W/m. °C 0.022Btu/h.ft.°F

Ammonia 0.147 W/m. °C 0.085 Btu/h.ft.°FWater 0.556 W/m. °C 0.327 Btu/h.ft.°F

Hydrogen 0.175 W/m. °C 0.101 Btu/h.ft.°FSteam 0.0206 W/m. °C 0.0119 Btu/h.ft.°FAir 0.024 W/m. °C 0.0138 Btu/h.ft.°F

Metals

Nonmetallic solids

Liquids

Gases

Note: 1 W/(m.K) = 1W/(m.oC) = 0.85984 kcal/(hr.m.oC) = 0.5779 Btu/(ft.hr.oF)

Quantity Text Notation SI Unit English Unit heat Q Joule (J) Btu (heat transfer) heat rate q Watt (W) Btu/hr (heat transfer rate) (heat energy rate) (rate of heat flow) heat flux q” W/m2 Btu/hr-ft2 (heat rate per unit area) heat rate per unit length q’ W/m Btu/hr-ft volumetric heat

generation .q W/m3 Btu/hr-ft3

Heat Quantities

1.4… General Heat Conduction Equation(a) Cartesian (Rectangular) Coordinates:

Consider a medium within which there is no bulk motion (advection) and the temperature distribution T(x,y,z) is expressed in Cartesian coordinates.

First define an infinitesimally small (differential or elemental) control volume, dx.dy.dz, as shown in Fig.

Cartesian Coordinates system (continue…)

Conduction Heat RatesIf there are temperature gradients, conduction heat transfer will occur across each of the control surfaces and the conduction heat rates perpendicular to each of the control surfaces at the x, y, and z coordinate locations are indicated by the terms qx , qy and qz respectively.

The conduction heat rates at the opposite surfaces can then be expressed as a Taylor series expansion with neglecting higher order terms,

dxx

qqq xxdxx

dyy

qqq yydyy

dzz

qqq zzdzz

……(4)

……(5)

……(6)

Above equations simply states that the x component of the heat transfer rate at x + dx is equal to the value of this component at x plus the amount by which it changes with respect to x times dx.

Cartesian Coordinates system (continue…)

dxdydzqEg

dxdydzt

TpCstE

stoutgin EEEE

Thermal energy generation

Energy storage

Conservation of energy

……(7)

……(8)

……(9)

dxdydzt

TCqqqdxdydzqqqq pdzzdyydxxzyx

……(10)

From equation (10),

dxdydzt

TCdxdydzqdz

z

qdy

y

qdx

x

qp

zyx

……(11)

Cartesian Coordinates system (continue…)

Where,

z

Tkdxdyq

y

Tkdxdzq

x

Tkdydzq

z

y

x

……(12)

……(13)

……(14)

Net conduction heat flux into the controlled volume,

……(15)////dxxx qqdx

x

Tk

x

Cartesian Coordinates system (continue…)

Heat (Diffusion) Equation: at any point in the medium the rate of energy transfer by conduction in a unit volume plus the volumetric rate of thermal energy must equal to the rate of change of thermal energy stored within the volume.

t

TCq

z

Tk

zy

Tk

yx

Tk

x P

……(16)

Equation (16) is final form of heat conduction equation for rectangular co-ordinates system.

t

T

k

q

z

T

y

T

x

T

1

2

2

2

2

2

2

If the thermal conductivity (k) is constant.

……(17)

Where α= k/(ρCp) is the thermal diffusivity i.e. rate of heat diffuse from system

Under steady-state condition, there can be no change in the amount of energy storage.

……(18) Poisson's equation02

2

2

2

2

2

k

q

z

T

y

T

x

T

Cartesian Coordinates system (continue…)

If the heat transfer is one-dimensional, steady state and there is no energy generation, the above equation reduces to

……(22)

If the no heat generation in volume,

……(19) Fourier's equation

If steady state heat conduction with no heat generation in volume,

t

T

z

T

y

T

x

T

1

2

2

2

2

2

2

02

2

2

2

2

2

z

T

y

T

x

T……(21)

02

2

x

T

Laplace's equation

(b) Cylindrical Coordinates:

2

1 1p

T T T Tkr k k q c

r r r z z tr

•……(23)

(c) Spherical Coordinates:

22 2 2 2

1 1 1sin

sin sinp

T T T Tkr k k q c

r r tr r r

•……(24)

(c) General one dimensional conduction equation:

Coordinate system

X value n value

Cartesian X=x 0

Cylindrical X=r 1

Spherical X=r 2

……(25)

In compact form,

t

T

k

C

k

q

X

TX

XXpn

n

1