fouriers law and thermal conductivity

8/12/2019 Fouriers Law and Thermal Conductivity

1/20

Chapter 1: Fourier Equation and Thermal Conductivity

1.1 . Introduction of Heat Transfer

1.2 . Fouriers 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


2/20

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 twomediums 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.Transfer Driving Force

Current (I) (Ampere) Voltage differenceFluid flow (Q) (liter) Pressure difference

Heat Flow (Q) (kJ) Temperature Difference


3/20


4/20

Conductionis 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]

Conduction

Introduction of Heat Transfer (continue)


5/20

5

Conduction

HOT

(lots of vibration)

COLD

(not much vibration)

Heat travels

along the rod


6/20

When the handle of a spoon stirring a cup of hot

chocolate gets hot, its 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)


7/20

1.2 Fouriers Law of Heat Transfer

The heat flux is proportional to the

temperature gradient:

(1)

Where k=thermal conductivity (W/mC 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; its a

vector quantity

x

hot wall

cold walldx

dT

temperature

profileQ q k TA

= = -

dx

dTk

A

Qq

dx

dTAkQ

-==

-=

"

.. (2) Fouriers Law

(3) Heat FluxIn most practical situations conduction,

convection, and radiation appear in

combination


8/20

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 substances

ability to transfer heat through a solid by conduction.

K= Q L/ (A T)

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 (T)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.


9/20

Thermal conductivity values for various materials at 300 K

Thermal Conductivity of Material (continue)

9

Silver 410 W/m. C 237 Btu/h.ft.F

Copper 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.F

Water 0.556 W/m. C 0.327 Btu/h.ft.F

Hydrogen 0.175 W/m. C 0.101 Btu/h.ft.F

Steam 0.0206 W/m. C 0.0119 Btu/h.ft.F

Air 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)


10/20


11/20

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 heatgeneration q W/m

3 Btu/hr-ft

3

Heat Quantities


12/20

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.


13/20

Cartesian Coordinates system (continue)Conduction Heat Rates

If 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 termsqx, qyand 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 x

xdxx

=

dyy

qqq y

ydyy

=

dzz

qqq z

zdzz

=

(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.


14/20

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)


15/20


Where,

z

Tkdxdyq

y

Tkdxdzq

x

T

kdydzq

z

y

x

-=

-=

-=

(12)

(13)

(14)

Net conduction heat flux into the controlled volume,

(15)

////dxxx qqdx

x

Tk

x -=


16/20


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 = k/(Cp) 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 equat ion02

2

2

2

2

2

=

k

q

z

T

y

T

x

T


17/20


If the heat transfer is one-dimensional, steady state and there is no energy

generation, the above equation reduces to

(21)

If the no heat generation in volume,

(19) Fourier's equat ion

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

tT

zT

yT

xT

=

12

2

2

2

2

2

02

2

2

2

2

2

=

z

T

y

T

x

T

(20)

02

2

=xT

Laplace'sequation


18/20

(b) Cylindrical Coordinates:

2

1 1

p

T T T T kr k k q cr r r z z t r

f f

=

(22)


19/20

(c) Spherical Coordinates:

2

2 2 2 2

1 1 1

sinsin sin p

T T T T

kr k k q cr r tr r r q f f q q q q

=

(23)


20/20

(c) General one dimensional conduction equation:

1 pn

n

CT q TX

X X k k tX

=

Coordinatesystem

X value n value

Cartesian X=x 0

Cylindrical X=r 1

Spherical X=r 2

(24)

In compact form,

fouriers law and thermal conductivity

Documents