1 Transient Heat Transfer • Having gotten a feel for the steady state heat transfer, let us get a feel for the transient behavior of heat transfer • To begin with, let us assume that spatial variation of temperature is negligible and the temperature of the body as a whole changes with time • This implies that the thermal conductivity is very high. • We shall evolve a criterion for the validity of this assumption as we go along • Let us look at the steps in analysis • This type of problem is called lumped analysis Lumped Analysis-I • Let the body temperature be denoted by T T T ∞ , h • The body interacts with the surroundings with the temperature at T ∞ and heat transfer coefficient h • First law implies that W Q E & & & - = ( ) ) T T ( hA mcT dt d ∞ - - = ⇒ • For constant specific heat ) T T ( hA dt dT mc ∞ - - = ⇒ Lumped Analysis-II • Defining θ = T - T ∞ , we can write θ - = θ ⇒ hA dt d mc θ - = θ ⇒ mc hA dt d Let θ = θ o at t = 0 • The solution for the above equation is t mc hA 0 e - θ = θ • If the object is sphere, we have 2 3 R 4 A ; R 3 4 m π = ρ π = R 3 c h R 3 4 R 4 c h mc hA 3 2 ρ = ρ π π = ⇒ t R 3 c h 0 e ρ - θ = θ Lumped Analysis-III • In general t V A c h 0 t c V hA 0 t mc hA 0 e e e ρ - ρ - - θ = θ = θ = θ • If the object is a cylinder of Radius R and Length L, we have ( ) RL R L 2 L R R 2 RL 2 V A 2 2 + = π π + π = ⇒ t LR ) R L ( 2 c h 0 e + ρ - θ = θ • If the cylinder has L>>R , then t R 2 c h 0 e ρ - θ = θ Lumped Analysis-III • This concept can now be extended to any geometry • To generalize the result, it is customary to introduce the characteristic length L • The logic suggests that V/A is the most obvious choice 2 2 L t k hL 0 t cL k k hL 0 t cL h 0 e e e α - ρ - ρ - θ = θ = θ = θ ⇒ • In the above expression, we have introduced the property called thermal diffusivity = k/(ρc) • The non-dimensional parameter hL/k and αt/L 2 are called the Biot Number and Fourier No respectively Lumped Analysis-IV • Thus, the temperature variation is a function of two non dimensional parameters Biot number and Fourier number. • We will appreciate these parameters, as we go into more complex cases • We can give a physical interpretation for the Biot number as follows ce tan sis Re convection ce tan sis Re conduction 1 hA KA L K hL Bi = = = • When Bi is very small, it implies that conduction resistance is very small and hence lumped analysis valid • The criterion used is Bi < 0.1
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1
Transient Heat Transfer
• Having gotten a feel for the steady state heat transfer, let
us get a feel for the transient behavior of heat transfer
• To begin with, let us assume that spatial variation of
temperature is negligible and the temperature of the body
as a whole changes with time
• This implies that the thermal conductivity is very high.
• We shall evolve a criterion for the validity of this
assumption as we go along
• Let us look at the steps in analysis
• This type of problem is called lumped analysis
Lumped Analysis-I
• Let the body temperature be denoted
by T
T
T∞, h
• The body interacts with the
surroundings with the temperature at
T∞ and heat transfer coefficient h
• First law implies that WQE &&& −=
( ) )TT(hAmcTdt
d∞−−=⇒
• For constant specific heat
)TT(hAdt
dTmc ∞−−=⇒
Lumped Analysis-II
• Defining θ = T - T∞, we can write
θ−=θ
⇒ hAdt
dmc θ−=
θ⇒
mc
hA
dt
dLet θ = θo at t = 0
• The solution for the above equation is
tmc
hA
0e−
θ=θ
• If the object is sphere, we have
23 R4A;R3
4m π=ρπ=
R
3
c
h
R3
4
R4
c
h
mc
hA
3
2
ρ=
ρπ
π=⇒
tR
3
c
h
0e ρ−
θ=θ
Lumped Analysis-III
• In generalt
V
A
c
h
0
tcV
hA
0
tmc
hA
0 eee ρ−
ρ−−
θ=θ=θ=θ
• If the object is a cylinder of Radius R and Length L,
we have
( )RL
RL2
LR
R2RL2
V
A2
2 +=
π
π+π=⇒
tLR
)RL(2
c
h
0e
+
ρ−
θ=θ
• If the cylinder has L>>R , then
tR
2
c
h
0e ρ−
θ=θ
Lumped Analysis-III
• This concept can now be extended to any geometry
• To generalize the result, it is customary to introduce
the characteristic length L
• The logic suggests that V/A is the most obvious
choice
22L
t
k
hL
0
tcL
k
k
hL
0
tcL
h
0 eee
α−
ρ−
ρ−
θ=θ=θ=θ⇒
• In the above expression, we have introduced the
property called thermal diffusivity = k/(ρc)
• The non-dimensional parameter hL/k and αt/L2 are
called the Biot Number and Fourier No respectively
Lumped Analysis-IV
• Thus, the temperature variation is a function of two
non dimensional parameters Biot number and Fourier
number.
• We will appreciate these parameters, as we go into
more complex cases
• We can give a physical interpretation for the Biot
number as follows
cetansisReconvection
cetansisReconduction
1
hA
KA
L
K
hLBi ===
• When Bi is very small, it implies that conduction
resistance is very small and hence lumped analysis valid
• The criterion used is Bi < 0.1
2
Transients with spatial effects
• If Bi > 0.1 spatial effects become important and so
more complications are involved
• Exact analytical solutions can be obtained using
separation of variables similar to 2-D steady state
analysis
• Let us look at 1-D transient analysis in a slab
geometry with no heat generation
• The governing equation for this case is
qz
T
y
T
x
Tk
t
)T(c
2
2
2
2
2
2
′′′+
∂
∂+
∂
∂+
∂
∂=
∂
∂ρ
2
2
x
T
t
T1
∂
∂=
∂
∂
α⇒
1-D Transient in a Plate-I • The governing equation