Conduction equation

8/10/2019 Conduction equation

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Y. A. Abakr

CONDUCTION HEAT TRANSFER

Conduction Heat transfer


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Y. A. Abakr

Applications of heat transfer

Heat exchangers: boilers

condensers,

Radiators Evaporators

Heating, Cooling ofbuildings ( passive

Active)


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Y. A. Abakr

Applications of heat transfer

cooking: Cookers, ovens,

steaming etc

Cooling of Electronic

components

Cooling of turbine

plates (aircraft

propulsion)


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Y. A. Abakr

Fouriers law of heatconduction

The temperature gradient is

negative when heat isconducted in the positivex-

direction.

(W)conddT

Q kA

dx

(1)

Heat Conduction Equation


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Y. A. Abakr

nQ

xQ

yQ

(W)ndT

Q kA

dn

(2)

Heat flux vector may be resolved into orthogonal

components

Direction of heat transfer is perpendicular to lines of

constant temperature (isotherms).


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Y. A. Abakr

In rectangular coordinates, the heat

conduction vector can be expressed interms of its components as

which can be determined from Fourierslaw as

n x y z Q Q i Q j Q k (3)

x x

y y

z z

TQ kA

x

TQ kAy

TQ kA

z

(4)


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Heat Generation

(W)gen genV

E e dV (5)


8/23Y. A. Abakr

Specific heat capacity

The specific heat(capacity) c is a

measure of the

quantity of heataccumulated per

unit temperature

rise per unit mass.

It is the amount of heatneeded to increase the

temperature of 1kg mass

by one degree


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thermal diffusivity

is the thermal conductivity Kdivided by

density and specific heat capacity cat

constant pressure.

It is the measure of the ability of a material toconductthermal energy relative to its ability

to storethermal energy.

m/s


10/23Y. A. Abakr

One-Dimensional Heat Conduction

Equation - Plane Wall

xQ

Rate of heat

conductionat x

Rate of heat

conductionat x+x

Rate of heat

generation insidethe element

Rate of change of

the energy contentof the element

- + =

,gen elementEx xQ D elementE

t

D

D

(6)


11/23Y. A. Abakr

The change in the energy content and the rateof heat generation can be expressed as

,

element t t t t t t t t t

gen element gen element gen

E E E mc T T cA x T TE e V e A x

D D DD D

D

,

elementx x x gen element

EQ Q E

tD

D

D

x x xQ Q D

(7)gene A x D

t t tT T

cA x t D

D D

1gen

T TkA e cA x x t

(8)

Dividing byADx, taking the limit as Dx0 and Dt0,

and from Fouriers law:


12/23Y. A. Abakr

The areaAis constant for a plane wallthe one dimensional

transient heat conduction equation in a plane wall is

gen

T Tk e c

x x t

Variable conductivity:

Constant conductivity:2

2

1 ;

geneT T k

x k t c

1) Steady-state:

2) Transient, no heat generation:

3) Steady-state, no heat generation:

2

20

gened T

dx k

2

2

1T T

x t

2

2 0d T

dx

The one-dimensional conduction equation may be reduces

to the following forms under special conditions

(9)


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Example2-3


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16/23Y. A. Abakr

The change in the energy content and the rateof heat generation can be expressed as

Substituting into Eq. 218, we get

,

element t t t t t t t t t

gen element gen element gen

E E E mc T T cA r T T

E e V e A r

D D DD D

D

,

elementr r r gen element

EQ Q E

tD

D

D

r r rQ Q D gene A r D

t t tT T

cA r

t

D D

D

1

gen

T TkA e c

A r r t

Dividing byADr, taking the limit as Dr0 and Dt0,

and from Fouriers law:

For cylindrical system


17/23Y. A. Abakr

Noting that the area varies with the independent variable raccording toA=2prL,

1gen

T Trk e c

r r r t

10

gened dTr

r dr dr k

The one-dimensional conduction equation may be reduces

to the following forms under special conditions

1 1geneT Tr

r r r k t

1 1T Tr

r r r t

0d dT

rdr dr


Constant conductivity:

1) Steady-state:



The one dimensional transient heat conduction equation in a plane wall becomes


18/23Y. A. Abakr

One-Dimensional Heat Conduction Equation

- Sphere

2

2

1gen

T Tr k e c

r r r t

2

2

1 1geneT Tr

r r r k t




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Y. A. Abakr

General Heat Conduction Equation

x y zQ Q Q

Rate of heat

conduction

at x,y, andz

Rate of heat

conduction

at x+x, y+y,

and z+z

Rate of heat

generation

inside the

element

Rate of change

of the energy

content of the

element

- + =

x x y y z zQ Q QD D D ,gen elementE elementE

t

D

D


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Y. A. Abakr

Repeating the mathematical approach used for the one-

dimensional heat conduction the three-dimensional heat

conduction equation is determined to be

2 2 2

2 2 2

1geneT T T T

x y z k t

2 2 2

2 2 20

geneT T T

x y z k

2 2 2

2 2 2

1T T T T

x y z t

2 2 2

2 2 2 0

T T T

x y z

Two-dimensional

Three-dimensional

1) Steady-state:





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Y. A. Abakr

Cylindrical Coordinates

21 1

genT T T T T rk k k e c

r r r r z z t

(2-43)


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Y. A. Abakr

Spherical Coordinates

2

2 2 2 2

1 1 1sin

sin sin gen

T T T Tkr k k e c

r r r r r t


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Y A Ab k

Boundary and Initial Conditions

Specified Temperature Boundary

Condition

Specified Heat Flux Boundary Condition

Convection Boundary Condition

Radiation Boundary Condition

Interface Boundary Conditions Generalized Boundary Conditions

Conduction equation

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