8/13/2019 (9 10) Transient Heat Conduction
1/28
TRANSIENT HEAT CONDUCTION
Associate Professor
IIT Delhi
E-mail: [email protected]
8/13/2019 (9 10) Transient Heat Conduction
2/28
1.Lumped System Analysis
. ,
long cylinders, and spheres with spatial effects
3. Transient heat conduction in semi-infinite solids
8/13/2019 (9 10) Transient Heat Conduction
3/28
Lumped System
Cooling of a hot metal forging
P.Talukdar/Mech-IITD
8/13/2019 (9 10) Transient Heat Conduction
4/28
Energy balanceHeat transfer into
the body during dt
The increase in
the energy of the
body during dt
=
dTmCdt)TT(A.h ps =Vm =
)TT(ddT =
ttanconsT =
dtVC
A.h
TT
)TT(d
p
s
=
tVC.
TTtln
p
s
i =
btT)t(T A.h 1
P.Talukdar/Mech-IITD
i TT s
VCp=
8/13/2019 (9 10) Transient Heat Conduction
5/28
)s(
A.hb 1s =
b has the unit of 1/s.
p
t
i
eTT
=
This equation enables us to determine thetemperature T(t) of a body at time t, or
alternatively, the time t required for the
empera ure o reac a spec e va ue .
P.Talukdar/Mech-IITD
8/13/2019 (9 10) Transient Heat Conduction
6/28
The temperature of a body approaches the )s(A.h
b 1s =ambient temperature Texponentially.
The temperature of the body changes rapidly at
the beginning, but rather slowly later on.
p
A large value of b indicates that the body will
approach the environment temperature in a short
time.e arger e va ue o e exponen , e
higher the rate of decay in temperature.
Note that b is proportional to the surface area,
specific heat of the body. This is not surprising
since it takes longer to heat or cool a larger
mass es eciall when it has a lar e s ecific
heat.
P.Talukdar/Mech-IITD
8/13/2019 (9 10) Transient Heat Conduction
7/28
Heat TransferRate of convection heat transfer between the body and its environmentat that time
The total amount of heat transfer between the body and the surrounding medium
=
= T)t(ThA)t(Q s Watt
body:
[ ]ip T)t(TmCQ = kJ
[ ]ipmax TTmCQ = kJ
P.Talukdar/Mech-IITD
8/13/2019 (9 10) Transient Heat Conduction
8/28
LAL
LLthicknesswallPlanec
==2
)2(
Criterion for lumped
s
cA
VL =
k
hL
Bic
=
When the convectioncoefficient h is high and k is
low, large temperature
P.Talukdar/Mech-IITD
inner and outer regions of a
large solid.
8/13/2019 (9 10) Transient Heat Conduction
9/28
Transienttemperature
distribution
P.Talukdar/Mech-IITD
8/13/2019 (9 10) Transient Heat Conduction
10/28
8/13/2019 (9 10) Transient Heat Conduction
11/28
Fourier Number()
P.Talukdar/Mech-IITD
8/13/2019 (9 10) Transient Heat Conduction
12/28
The specified surface temperature corresponds
o e case o convec on o an env ronmen awith a convection coefficient h that is infinite
P.Talukdar/Mech-IITD
8/13/2019 (9 10) Transient Heat Conduction
13/28
Transient heat conduction in large plane
walls, long cylinders, and spheres with
spat a e ects
P.Talukdar/Mech-IITD
8/13/2019 (9 10) Transient Heat Conduction
14/28
2
?tx2 =
trr
rr =
-
P.Talukdar/Mech-IITD
8/13/2019 (9 10) Transient Heat Conduction
15/28
Plane WallGoverning equation
T1T2
2 =
Transient temperature profiles in a plane wallexposed to convection from its surfaces for Ti > T
Governing equation along with boundary conditions
shows that T is a function of x, t, k, , h, Ti and T
P.Talukdar/Mech-IITD
8/13/2019 (9 10) Transient Heat Conduction
16/28
- .
=
TT
TT
Governing equation (Non-dimensional form)
=
2
2
X
Initial condition Boundary conditions
, =
0=
=
.Bi
The non-dimensionalization enables us to present the
temperature in terms of three parameters only:X, Bi, and P.Talukdar/Mech-IITD
8/13/2019 (9 10) Transient Heat Conduction
17/28
Solution
=
2
2
Solution takes the form of an
The solution of 1D transient heat conduction
involves infinite series, which are difficult to dealwith. However, the terms in the solutions
infinite series ,
> 0.2, keeping the first term and neglecting all
the remaining terms in the series results in an
error under 2 percent. We are usually interested)X(cos)exp(A n2
nn =
in the solution for times with > 0.2, and thus itis very convenient to express the solution using
an one term approximation.
1n=
nn
nn
2sin2
sin4A
+
=
See next slide for this expressionsBitan nn =
P.Talukdar/Mech-IITD
8/13/2019 (9 10) Transient Heat Conduction
18/28
8/13/2019 (9 10) Transient Heat Conduction
19/28
P.Talukdar/Mech-IITD
8/13/2019 (9 10) Transient Heat Conduction
20/28
8/13/2019 (9 10) Transient Heat Conduction
21/28
8/13/2019 (9 10) Transient Heat Conduction
22/28
Transient mid-plane temperature chart for a
plane wall of thickness 2L initially at a uniform
P.Talukdar/Mech-IITD
empera ure i su ec e o convec on rom o
sides to an environment at temperature T with a
convection coefficient of h.
8/13/2019 (9 10) Transient Heat Conduction
23/28
8/13/2019 (9 10) Transient Heat Conduction
24/28
Transient heat transfer
chart for a plane wall of thickness2L initially at a uniform
temperature Ti subjected toTTVCTTC.m == kJ
P.Talukdar/Mech-IITD
environment at temperature T
with a convection coefficient of h.
8/13/2019 (9 10) Transient Heat Conduction
25/28
Transient Conduction in Semi-Infinite Solid
Consider a semi-infinite solid (extends to infinity in all
but one direction with a single identifiable surface) ( ) iTxT =Case 1
T(x,0) = TiT(0,t) = Ts
Case 2
T(x,0) = Ti
Case 3
T(x,0) = Ti
T&
0x0xx =
==
( )t,0TTh
xk
0x
=
=
T Q ,
x x x
Ts x, Tt
tt
Ti
Ti
Ti
8/13/2019 (9 10) Transient Heat Conduction
26/28
( )
=
x
erf
Tt,xT s ( ) ( )TTk
tQ iss
=&
Case 1 Constant Surface Temperature: T(0,t) = Ts
and
t2TT si
xxxtq2 2o2
1
&
Case 2 Constant Surface Heat Flux: Qs = Qo
=t2
er ckt4
expk
Tt,xT i
T ,x 0x
=
=
2
Case 3 Surface Convection:
+ + = kt2erfckkexpt2erfcTT,
2i
The Gaussian error function, erf w , is a standard 2mathematical function (see Table B.2 for values)
Complementary Error Function, erfc(w) = 1 erf(w)
( )
=0
dvewerf
8/13/2019 (9 10) Transient Heat Conduction
27/28
BA
i,BBi,AAs
mm
TmTmT
++=Ts between 2 solids
P.Talukdar/Mech-IITD
8/13/2019 (9 10) Transient Heat Conduction
28/28
Top Related