conduction

1. ETHT

2. Conduction
Montu Faldu 140080125005
Haresh Gajera 140080125006
Vishal Gajjar 140080125007
Sarthak Gokani 140080125008
Guided by :
Dr. Manish Maheta

4. Many heat transfer problems require the understanding of
the complete time history of the temperature variation. For
example, in metallurgy, the heat treating process can be
controlled to directly affect the characteristics of the
processed materials. Annealing (slow cool) can soften
metals and improve ductility. On the other hand,
quenching (rapid cool) can harden the strain boundary and
increase strength. In order to characterize this transient
behavior, the full unsteady equation is needed:
2 21
, or
k
where = is the thermal diffusivity
c
T T
c k T T
t t
ρ
α
α
ρ
∂ ∂
= ∇ = ∇
∂ ∂

5. Transient heat transfer with no internal
resistance: Lumped Parameter Analysis
Solid
Valid for Bi<0.1
Total Resistance= Rexternal + Rinternal
GE:
dT
dt
= −
hA
mcp
T − T∞( ) BC: T t = 0( )= Ti
Solution: let Θ = T − T∞, therefore
dΘ
dt
= −
hA
mcp
Θ

6. Lumped Parameter Analysis
ln
hA
mc
t
i
t
mc
hA
i
pi
ii
p
p
e
TT
TT
e
t
mc
hA
TT
−
∞
∞
−
∞
=
−
−
=
Θ
Θ
−=
Θ
Θ
−=Θ
Note: Temperature function only of time
and not of space!
- To determine the temperature at a given time,
or
- To determine the time required for the
temperature to reach a specified value.

7. )exp(T
0
t
cV
hA
TT
TT
ρ
−=
−
−
=
∞
∞
t
L
Bit
LLc
k
k
hL
t
cV
hA
ccc
c
2
11 α
ρρ
=











=






≡
c
k
ρ
α
Thermal diffusivity: (m² s-1
)
Lumped Parameter Analysis

8. Lumped Parameter Analysis
t
L
Fo
c
2
α
≡
k
hL
Bi C
≡
T = exp(-Bi*Fo)
Define Fo as the Fourier number (dimensionless time)
and Biot number
The temperature variation can be expressed as
thickness2Lawithwallaplaneissolidthewhen)thickness(halfcL
sphereissolidthewhenradius)third-one(
3cL
cylinder.aissolidthewhenradius)-(half
2
or
cL,examplefor
problemtheininvlovedsolidtheofsizethetorealte:scalelengthsticcharacteriaiscLwhere
L
or
=
=
=

9. Graphical Representation of the One-Term Approximation:
The Heisler Charts
Midplane Temperature:

10. Change in Thermal Energy
Storage
Temperature
Distribution
Assumptions in using Heisler charts:
•Constant Ti and thermal properties over the body
•Constant boundary fluid T∞ by step change
•Simple geometry: slab, cylinder or sphere

11. ),(''. trgq
t
H
+−∇=
∂
∂
{ } ),(.. trgTk
t
T
Cp +∇−−∇=
∂
∂
ρ
Incorporation of the constitutive equation into the energy
equation above yields:
Dividing both sides by ρCp and introducing the thermal
diffusivity of the material given by
s
m
m
s
m
C
k
p
×⇒=
2
ρ
α

12. Thermal Diffusivity
Thermal diffusivity includes the effects of properties like
mass density, thermal conductivity and specific heat
capacity.
Thermal diffusivity, which is involved in all unsteady heat-
conduction problems, is a property of the solid object.
The time rate of change of temperature depends on its
numerical value.
The physical significance of thermal diffusivity is
associated with the diffusion of heat into the medium during
changes of temperature with time.
The higher thermal diffusivity coefficient signifies the faster
penetration of the heat into the medium and the less time
required to remove the heat from the solid.

13. pp C
trg
T
C
k
t
T
ρρ
),(
.. +








∇−−∇=
∂
∂
This is often called the heat equation.
{ }
pC
trg
T
t
T
ρ
α
),(
.. +∇∇=
∂
∂
For a homogeneous material:
pC
txg
T
t
T
ρ
α
),(2
+∇=
∂
∂

14. This is a general form of heat conduction equation.
Valid for all geometries.
Selection of geometry depends on nature of application.

15. xq xxq δ+
yyq δ+
yq
zzq δ+
zq

16. ),(. txgTk
t
T
Cp +∇∇=
∂
∂
ρ
For an isotropic and homogeneous material:
),(2
txgTk
t
T
Cp +∇=
∂
∂
ρ
):,,(2
2
2
2
2
2
tzyxg
z
T
y
T
x
T
k
t
T
Cp +





∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
ρ

17. ):,,(
1
2
2
2
2
2
tzrg
z
TT
rr
T
r
r
k
t
T
Cp θ
θ
ρ +





∂
∂
+
∂
∂
+





∂
∂
∂
∂
=
∂
∂

18. ):,,(
sin
1
sin
sin
11
2
2
222
2
2
trg
T
r
T
rr
T
r
rr
k
t
T
Cp φθ
φθθ
θ
θθ
ρ +





∂
∂
+





∂
∂
∂
∂
+





∂
∂
∂
∂
=
∂
∂
X
Y

20. ( )zyxkk ,,=
),(. txgTk
t
T
Cp +∇∇=
∂
∂
ρ
),,,( tzyxg
z
z
T
k
y
y
T
k
x
x
T
k
t
T
Cp +
∂






∂
∂
∂
+
∂






∂
∂
∂
+
∂






∂
∂
∂
=
∂
∂
ρ

21. ),,,(2
2
2
2
2
2
tzyxg
z
T
k
z
T
z
k
y
T
k
y
T
y
k
x
T
k
x
T
x
k
t
T
Cp +
∂
∂
+
∂
∂
∂
∂
+
∂
∂
+
∂
∂
∂
∂
+
∂
∂
+
∂
∂
∂
∂
=
∂
∂
ρ
More service to humankind than heat transfer rate calculations

26. Steady-State One-Dimensional Conduction
Assume a homogeneous medium with invariant thermal conductivity ( k =
constant) :
• For conduction through a large wall the
heat equation reduces to:
),,,(2
2
tzyxg
x
T
k
x
T
x
k
t
T
Cp +
∂
∂
+
∂
∂
∂
∂
=
∂
∂
ρ
),,,(2
2
tzyxg
x
T
k
t
T
Cp +
∂
∂
=
∂
∂
ρ
One dimensional Transient conduction with heat generation.

27. 02
2
=
dx
Td
A
0),,,(2
2
=+
∂
∂
tzyxg
x
T
k
No heat generation
211 CxCTC
dx
dT
+=⇒=⇒

28. Apply boundary conditions to solve for
constants: T(0)=Ts1 ; T(L)=Ts2
211 CxCTC
dx
dT
+=⇒=⇒
The resulting temperature distribution
is:
and varies linearly with x.

29. Applying Fourier’s law:
heat transfer rate:
heat flux:
Therefore, both the heat transfer rate and heat flux are
independent of x.

30. Wall Surfaces with Convection
2112
2
0 CxCTC
dx
dT
dx
Td
A +=⇒=⇒=
Boundary conditions:
( )11
0
)0( ∞
=
−=− TTh
dx
dT
k
x
( )22 )( ∞
=
−=− TLTh
dx
dT
k
Lx

31. Wall with isothermal Surface and Convection Wall
2112
2
0 CxCTC
dx
dT
dx
Td
A +=⇒=⇒=
Boundary conditions:
1)0( TxT ==
( )22 )( ∞
=
−=− TLTh
dx
dT
k
Lx

32. Electrical Circuit Theory of Heat Transfer
Thermal Resistance
A resistance can be defined as the ratio of a driving
potential to a corresponding transfer rate.
i
V
R
∆
=
Analogy:
Electrical resistance is to conduction of electricity as thermal
resistance is to conduction of heat.
The analog of Q is current, and the analog of the temperature
difference, T1 - T2, is voltage difference.
From this perspective the slab is a pure resistance to heat transfer
and we can define

33. q
T
R
R
T
q th
th
∆
=⇒
∆
=
WK
mW
Km
m
kA
L
L
TT
kA
TT
q
T
R
ss
ss
cond
th /
1.
2
12
21
⇒=





 −
−
−
=
∆
=
( )
WK
mW
Km
hATThA
TT
q
T
R
s
s
conv
th /
1.1
2
2
⇒=
−
−
=
∆
=
∞
∞

34. ( )
WK
mW
Km
AhTTAh
TT
q
T
R
rsurrsr
surrs
rad
th /
1.1
2
2
⇒=
−
−
=
∆
=

35. The composite Wall
 The concept of a thermal
resistance circuit allows ready
analysis of problems such as a
composite slab (composite planar
heat transfer surface).
 In the composite slab, the heat
flux is constant with x.
 The resistances are in series and
sum to Rth = Rth1 + Rth2.
 If TL is the temperature at the left,
and TR is the temperature at the
right, the heat transfer rate is
given by
21 thth
RL
th RR
TT
R
T
q
+
−
=
∆
=

36. Wall Surfaces with Convection
2112
2
0 CxCTC
dx
dT
dx
Td
A +=⇒=⇒=
Boundary conditions:
( )11
0
)0( ∞
=
−=− TTh
dx
dT
k
x
( )22 )( ∞
=
−=− TLTh
dx
dT
k
Lx
Rconv,1 Rcond Rconv,2
T∞1 T∞2

37. Heat transfer for a wall with dissimilar
materials
For this situation, the total heat flux Q is made up of the heat flux in the
two parallel paths:
Q = Q1+ Q2
 with the total resistance given by:

38. Composite Walls
The overall thermal resistance is given by