one dimensional non-homogeneous conduction equation
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One Dimensional Non-Homogeneous Conduction Equation
P M V Subbarao
Associate Professor
Mechanical Engineering Department
IIT Delhi
A truly non-homogeneous ODE….…..A Basis for Generation of Tremendous Power….
Homogeneous ODE
• How to obtain a non-homogeneous ODE for one dimensional Steady State Heat Conduction problems?
• Blending of Convection or radiation effects into Conduction model.
0
drdrdT
Ad
0)(22
2
TTmdx
Td
Define: TT
022
2
m
dx
d
How to get strictly non-homogeneous Equation?
Conduction with Thermal Energy Generation
• A truly non-homogenous ODE.• Consider the effect of a process occurring within a
medium such as thermal energy generation, qg , e.g.,
• Conversion of electrical to thermal energy in an electric rod.
• Curing of concrete brides and dams.• Nuclear fuel rod.• Solid Propellant Rockets.
Heat Transfer in Rocket Solid Propellent
Fast Construction of Bridges : RCC Technology
'''Iq
Convection & Radiation
Convection & Radiation
The term ‘curing’ is used to include maintenance of a favorable environment for the continuation of chemical reactions, i.e. retention of moisture within, or supplying moisture to the concrete from an external source and protection against extremes of temperature.
ReleaseHeat Hydration of Rate:'''Iq
Plane Wall with Thermal Energy Generation
For one-dimensional, steady-state conduction in an isotropic medium properties:
homogeneous medium with constant properties:
0'''2
2
qdx
Tdk
0'''
qdx
dxdT
kd
q’’’
212
'''
2)( CxCx
k
qxT
The temperature distribution is parabolic in x.
Applying Fourier’s law:
heat transfer rate:
1
'''
)( Cxk
qkA
dx
dTkAxq
Thus, dT/dx is a function of x, and therefore both the heat transfer rate and heat flux are dependent on x for a medium with energy generation.
Boundary Conditions
Case 1: Simple Dirichlet Boundary Conditions:
1)( sTLT 2)( sTLT
21 ss TT q’’’
02
)( 212
'''
CxCxk
qxTSolution :
&
022
12
)( 1212
22'''
ssss TT
L
xTT
L
x
k
LqxT
Maximum temperature occurs inside the slab close the higher temperature surface.
Boundary conditions
Case 1: Convection Boundary Conditions:
111)( sTTAhLq
At x = -L :
222)( sTTAhLq
At x = L :
2121 & hhTT
02
)( 212
'''
CxCxk
qxTSolution :
q’’’
heat transfer rate:
1
'''
)( Cxk
qkA
dx
dTkAxq
)()( 111
'''
LTTAhCLk
qkALq
)()( 221
'''
LTTAhCLk
qkALq
2
)()( 22111
LTThLTThAC
212
'''
2)( CLCL
k
qLT
212
'''
2)( CLCL
k
qLT
2
)()( 22111
LTThLTThAC
with
Boundary Conditions
Case 3: Symmetric Dirichlet Boundary Conditions:
sTLTLT )(
q’’’0
2)( 21
2'''
CxCxk
qxTSolution :
012
)(22'''
sT
L
x
k
LqxT
Maximum temperature occurs at the center of the slab.The left and right parts of the slab are isolated at the axis.
Modified Boundary Conditions
Case 3: Symmetric Dirichlet Boundary Conditions:
axisAdiabaticdx
xdT
x
0)(
0
02
)( 212
'''
CxCxk
qxTSolution :
012
)(22'''
sT
L
x
k
LqxT
sTLT ```
Boundary Conditions
Case 3: Symmetric Convection Boundary Conditions:
axisAdiabaticdx
xdT
x
0)(
0
02
)( 212
'''
CxCxk
qxTSolution :
sTThALq )(At x = L :
Radial Systems
For one-dimensional, steady-state conduction in an isotropic homogeneous medium with constantproperties:
q’’’
0'''
qdr
dTr
dr
d
r
k
21
2'''
ln4
)( CrCk
rqrT
Cartridge Heaters
A Reliable heater should
continue to provide superior heat transfer,
uniform temperatures and
resistance to oxidation and corrosion even at high temperatures.
Central Condition for a solid cylinder:
00
rdr
dT
One dimensional conduction is possible only if there is axial Symmetry of temperature profile.
21
2'''
ln4
)( CrCk
rqrT
002
)(1
0
1'''
0
Cr
C
k
rq
dr
rdT
rr
q’’’
2
2'''
4)( C
k
rqrT
Surface boundary condition:
Dirichlet Boundary Condition: At r = rO T(rO) = TO
k
rqTCC
k
rqT
OO
OO 44
2'''
22
2'''
22'''
4)( rr
k
qTrT OO
Maximum temperature occurs at center.
2'''
max 4 OO rk
qTT
Surface Convection Boundary Condition:
q’’’
All the heat generated in the cylinderof length L, is transferred to ambient fluidby Convection heat transfer.
TTLrhLqr OOO '''2
h
qrTT O
O 2
'''
2''''''
2'''
max 424 OO
OO rk
q
h
qrTr
k
qTT
Current Carrying Conductor
• An important practical application.
• Cooling of current carrying conductors enhances their current carrying capacity.
• Knowledge of temperature distribution is required to make sure that the conductor is not reaching its burn out condition.
• Uniform internal heat generation occurs due to Joule heating.
• Rate of heat generation per unit volume:
2
2
2'''
CC
C
C A
i
LA
Al
i
LA
Riq
Current Density : i/AC
222
4)( rr
kA
iTrT O
CO
Hollow Cylinder with Heat Generation
• Hollow cylinder geometry has significant applications as a nuclear fuel rods.
• Nuclear fuel rods are made of hollow cylinder where the heat generated is carried away by a liquid metal coolant flowing either on the inside or outside the tubes.
• Hollow electrical conductors of cylindrical shape are used for high current carrying applications, where the cooling is done by a fluid (Hydrogen) flowing on the inside.
• Annual reactors insulated from inside or outside are used in Chemical Processes.
21
2'''
ln4
)( CrCk
rqrT
Solid Propellant Rocket
Solid propellant rockets burn a solid block made of fuel, oxidizer, and binder (plastic or rubber). The block is called grain.
Hybrid RocketSection A-A
Section A-A
Sphere with Uniform Heat Generation
BombsGF- 3 US Baseball Grenade
0'''22
qdr
dTr
dr
d
r
k
0'''22
qdr
dTr
dr
d
r
k
k
rq
dr
dTr
dr
d 2'''2
21
'''
3 r
C
k
rq
dr
dT
21
2'''
6)( C
r
C
k
rqrT
Solid Sphere with heat generation:
22'''
6)( rr
k
qTrT Ow
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