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8/10/2019 Conduction equation
1/23
Y. A. Abakr
CONDUCTION HEAT TRANSFER
Conduction Heat transfer
8/10/2019 Conduction equation
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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)
8/10/2019 Conduction equation
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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
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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)
8/10/2019 Conduction equation
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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:
8/10/2019 Conduction equation
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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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8/10/2019 Conduction equation
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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
8/10/2019 Conduction equation
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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
Variable conductivity:
Constant conductivity:
1) Steady-state:
2) Transient, no heat generation:
3) Steady-state, no heat generation:
The one dimensional transient heat conduction equation in a plane wall becomes
8/10/2019 Conduction equation
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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
Variable conductivity:
Constant conductivity:
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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:
2) Transient, no heat generation:
3) Steady-state, no heat generation:
Constant conductivity:
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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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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