Formula Sheet Steady State Heat Conduction - Heat Transfer - Mechanical

 Page 1


F ormula Sheet: Steady State Heat Conduction
Introduction to Steady State Heat Conduction
• Definition : Steady-state heat conduction occurs when the temper ature dis-
tribution in a system does not change with time, and heat flows at a con-
stant r ate.
• Assumptions : Constant material properties, no heat gener ation (unless
specified), one- dimensional heat flow for simplified cases.
F ourier ’ s Law of Heat Conduction
• Gener al F orm :
?
Q =-kA
dT
dx
where
?
Q = heat tr ansfer r ate (W), k = thermal conductivity (W m
-1
K), A =
cross-sectional area (m
2
),
dT
dx
= temper ature gr adient.
Conduction Through Plane W alls
• Heat Flow (Constant Area) :
?
Q =
kA(T
1
-T
2
)
L
whereT
1
,T
2
= temper atures at the two surfaces,L = wall thickness.
• Thermal Re sistance :
R
th
=
L
kA
• T emper ature Distribution :
T(x) = T
1
-
(T
1
-T
2
)x
L
wherex = distance from surface atT
1
.
Composite Plane W alls
• Heat Flow (Series) :
?
Q =
T
1
-T
n
?
R
th
whereR
th,i
=
L
i
k
i
A
for each la yeri ,T
1
,T
n
= temper atures a t outer surfaces.
1
Page 2


F ormula Sheet: Steady State Heat Conduction
Introduction to Steady State Heat Conduction
• Definition : Steady-state heat conduction occurs when the temper ature dis-
tribution in a system does not change with time, and heat flows at a con-
stant r ate.
• Assumptions : Constant material properties, no heat gener ation (unless
specified), one- dimensional heat flow for simplified cases.
F ourier ’ s Law of Heat Conduction
• Gener al F orm :
?
Q =-kA
dT
dx
where
?
Q = heat tr ansfer r ate (W), k = thermal conductivity (W m
-1
K), A =
cross-sectional area (m
2
),
dT
dx
= temper ature gr adient.
Conduction Through Plane W alls
• Heat Flow (Constant Area) :
?
Q =
kA(T
1
-T
2
)
L
whereT
1
,T
2
= temper atures at the two surfaces,L = wall thickness.
• Thermal Re sistance :
R
th
=
L
kA
• T emper ature Distribution :
T(x) = T
1
-
(T
1
-T
2
)x
L
wherex = distance from surface atT
1
.
Composite Plane W alls
• Heat Flow (Series) :
?
Q =
T
1
-T
n
?
R
th
whereR
th,i
=
L
i
k
i
A
for each la yeri ,T
1
,T
n
= temper atures a t outer surfaces.
1
• T emper ature at Interface :
T
interface
= T
1
-
?
Q
j
?
i=1
R
th,i
wherej = la yers up to the interface.
Conduction Through Cylinders
• Heat Flow (Radial, Cylindrical) :
?
Q =
2pLk(T
i
-T
o
)
ln
(
ro
r
i
)
wherer
i
,r
o
= inner and outer r adii,T
i
,T
o
= inner and outer surface temper-
atures,L = length of cylinder .
• Thermal Re sistance :
R
th
=
ln
(
ro
r
i
)
2pLk
• T emper ature Distribution :
T(r) = T
i
-
T
i
-T
o
ln
(
ro
r
i
) ln
(
r
r
i
)
Composite Cylindrical W alls
• Heat Flow (Series) :
?
Q =
2pL(T
1
-T
n
)
? ln
(
r
i+1
r
i
)
k
i
wherer
i
,r
i+1
= r adii at interfaces,k
i
= thermal conductivity of la yeri .
• Thermal Re sistance (Series) :
R
th,total
=
?
ln
(
r
i+1
r
i
)
2pLk
i
Conduction Through Spheres
• Heat Flow (Radial, Spherical) :
?
Q =
4pkr
i
r
o
(T
i
-T
o
)
r
o
-r
i
wherer
i
,r
o
= inner and outer r adii,T
i
,T
o
= inner and outer surface temper-
atures.
2
Page 3


F ormula Sheet: Steady State Heat Conduction
Introduction to Steady State Heat Conduction
• Definition : Steady-state heat conduction occurs when the temper ature dis-
tribution in a system does not change with time, and heat flows at a con-
stant r ate.
• Assumptions : Constant material properties, no heat gener ation (unless
specified), one- dimensional heat flow for simplified cases.
F ourier ’ s Law of Heat Conduction
• Gener al F orm :
?
Q =-kA
dT
dx
where
?
Q = heat tr ansfer r ate (W), k = thermal conductivity (W m
-1
K), A =
cross-sectional area (m
2
),
dT
dx
= temper ature gr adient.
Conduction Through Plane W alls
• Heat Flow (Constant Area) :
?
Q =
kA(T
1
-T
2
)
L
whereT
1
,T
2
= temper atures at the two surfaces,L = wall thickness.
• Thermal Re sistance :
R
th
=
L
kA
• T emper ature Distribution :
T(x) = T
1
-
(T
1
-T
2
)x
L
wherex = distance from surface atT
1
.
Composite Plane W alls
• Heat Flow (Series) :
?
Q =
T
1
-T
n
?
R
th
whereR
th,i
=
L
i
k
i
A
for each la yeri ,T
1
,T
n
= temper atures a t outer surfaces.
1
• T emper ature at Interface :
T
interface
= T
1
-
?
Q
j
?
i=1
R
th,i
wherej = la yers up to the interface.
Conduction Through Cylinders
• Heat Flow (Radial, Cylindrical) :
?
Q =
2pLk(T
i
-T
o
)
ln
(
ro
r
i
)
wherer
i
,r
o
= inner and outer r adii,T
i
,T
o
= inner and outer surface temper-
atures,L = length of cylinder .
• Thermal Re sistance :
R
th
=
ln
(
ro
r
i
)
2pLk
• T emper ature Distribution :
T(r) = T
i
-
T
i
-T
o
ln
(
ro
r
i
) ln
(
r
r
i
)
Composite Cylindrical W alls
• Heat Flow (Series) :
?
Q =
2pL(T
1
-T
n
)
? ln
(
r
i+1
r
i
)
k
i
wherer
i
,r
i+1
= r adii at interfaces,k
i
= thermal conductivity of la yeri .
• Thermal Re sistance (Series) :
R
th,total
=
?
ln
(
r
i+1
r
i
)
2pLk
i
Conduction Through Spheres
• Heat Flow (Radial, Spherical) :
?
Q =
4pkr
i
r
o
(T
i
-T
o
)
r
o
-r
i
wherer
i
,r
o
= inner and outer r adii,T
i
,T
o
= inner and outer surface temper-
atures.
2
• Thermal Re sistance :
R
th
=
r
o
-r
i
4pkr
i
r
o
• T emper ature Distribution :
T(r) = T
i
-
T
i
-T
o
1
r
i
-
1
ro
(
1
r
i
-
1
r
)
Critical Thickness of Insulation
• Cylindrical S y stems :
r
cr
=
k
h
where r
cr
= critical r adius, k = thermal conductivity of insulation, h = con-
vective heat tr ansfer coefficient.
• Spherical S y stems :
r
cr
=
2k
h
Heat Conduction with Heat Gener ation
• Plane W all w ith Uniform Heat Gener ation :
T(x) =
q
''
2k
(L
2
-x
2
)+T
s
whereq
''
= heat gener ation r ate per unit volume,T
s
= surface temper ature,
L = half-thickness.
• Cylindrical S y stem with Heat Gener ation :
T(r) =
q
''
4k
(r
2
o
-r
2
)+T
o
whereT
o
= outer surface temper ature,r
o
= outer r adius.
Applications
• Used in designing heat exchangers, insulation systems, and thermal man-
agement in mecha nical systems.
• Essential for GA TE problems on conduction through walls, pipes, and com-
posite structu res.
3