Post on 08-Nov-2014
description
Geometrical Variations of Heat Conduction Equation
P M V Subbarao
Professor
Mechanical Engineering Department
IIT Delhi
Same Physics… But Shape Matters…..
The Heat Conduction Equation
pC
trgT
t
T
),(
..
For Isotropic Material:
For a homogeneous Isotropic material:
pC
txgT
t
T
),(2
Mass Diffusion Equation
Using the analogy
),(.. trCDt
C
For a homogeneous & isotropic material:
),(2 trCDt
C
This is a general form of heat conduction equation.
Valid for all geometries.
Selection of geometry depends on nature of application.
General conduction equation in Cartesian Coordinate System
xq xxq
yyq
yqzzq
zq
Rate of energy generation
),(. txgTkt
TC p
For an isotropic and homogeneous material:
),(2 txgTkt
TC p
):,,(2
2
2
2
2
2
tzyxgz
T
y
T
x
Tk
t
TC p
General conduction equation based on Polar
Cylindrical Coordinates
):,,(1
2tzrg
z
Tk
z
Tk
rr
Tkr
rt
TC p
),( txgTkt
TC p
zrkk ,,
),,(1
2zrg
z
Tk
z
Tk
rr
Tkr
rt
TC p
General conduction equation based on Polar Spherical Coordinates
):,,(sin
1sin
sin
112
2
2222
2trg
T
r
T
rr
Tr
rrk
t
TC p
X
Y
Thermally Heterogeneous Materials
zyxkk ,,
),(. txgTkt
TC p
),,,( tzyxgzzT
k
y
yT
k
xxT
k
t
TC p
),,,(2
2
2
2
2
2
tzyxgz
Tk
z
T
z
k
y
Tk
y
T
y
k
x
Tk
x
T
x
k
t
TC p
More service to humankind than heat transfer rate calculations
Primitive Pottery Kilns
Time
Thermal Conductivity of Brick Masonry Walls
1,thR
1,2, 2 thth RR
3,thR
3,4, 2 thth RR
1pC
3pC
12 pp CC
34 pp CC
Geographical Applications of Thermal Mapping
One Dimensional Heat Conduction problems
P M V Subbarao
Associate Professor
Mechanical Engineering Department
IIT Delhi
Simple ideas for complex Problems…
Desert Housing & Composite Walls
),(. txgTkt
TC p
General conduction equation in Cartesian Coordinate System
• Assume a homogeneous medium with invariant thermal conductivity ( k = constant) :
• For conduction through a large wall the heat equation reduces
to:),,,(
2
2
tzyxgx
Tk
x
T
x
k
t
TC p
),,,(2
2
tzyxgx
Tk
t
TC p
One dimensional Transient conduction with heat generation.
General Steady-State One-Dimensional Conduction
Rate of energy generation
Steady Heat transfer through a plane slab
02
2
dx
TdA
0),,,(2
2
tzyxgx
Tk
No heat generation
211 CxCTCdx
dT
Wall Surfaces with Convection
2112
2
0 CxCTCdx
dT
dx
TdA
Boundary conditions:
110
)0(
TThdx
dTk
x
22 )(
TLThdx
dTk
Lx
Wall with isothermal Surface and Convection Wall
2112
2
0 CxCTCdx
dT
dx
TdA
Boundary conditions:
1)0( TxT
22 )(
TLThdx
dTk
Lx
Wall Surfaces with Convection
2112
2
0 CxCTCdx
dT
dx
TdA
Boundary conditions:
110
)0(
TThdx
dTk
x
22 )(
TLThdx
dTk
Lx
Rconv,1 Rcond Rconv,2
T1 T2
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:
Composite Walls
• The overall thermal resistance is given by
Desert Housing & Composite Walls