solutions of the conduction equation
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Solutions of the Conduction Equation. P M V Subbarao Associate Professor Mechanical Engineering Department IIT Delhi. An Idea Generates More Mathematics…. Mathematics Generate Mode Ideas…. The Conduction Equation. Incorporation of the constitutive equation into the energy - PowerPoint PPT PresentationTRANSCRIPT
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Solutions of the Conduction Equation
P M V Subbarao
Associate Professor
Mechanical Engineering Department
IIT Delhi
An Idea Generates More Mathematics….Mathematics Generate Mode Ideas…..
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The Conduction Equation
),(''. trgqt
H
),(.. trgTkt
TC p
Incorporation of the constitutive equation into the energy equation above yields:
Dividing both sides by Cp and introducing the thermal diffusivity of the material given by
s
mm
s
m
C
k
p
2
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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.
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pp C
trgT
C
k
t
T
),(
..
This is often called the heat equation.
pC
trgT
t
T
),(
..
For a homogeneous material:
pC
txgT
t
T
),(2
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This is a general form of heat conduction equation.
Valid for all geometries.
Selection of geometry depends on nature of application.
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General conduction equation based on Cartesian Coordinates
xqxxq
yyq
yqzzq
zq
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),(. 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
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General conduction equation based on Polar
Cylindrical Coordinates
):,,(1
2
2
2
2
2tzrg
z
TT
rr
Tr
rk
t
TC p
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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
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Thermal Conductivity of Brick Masonry Walls
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Thermally Heterogeneous Materials
zyxkk ,,
),(. txgTkt
TC p
),,,( tzyxgz
zT
k
y
yT
k
xxT
k
t
TC p
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),,,(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
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Satellite Imaging : Remote Sensing
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Thermal Imaging of Brain
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One Dimensional Heat Conduction problems
P M V Subbarao
Associate Professor
Mechanical Engineering Department
IIT Delhi
Simple ideas for complex Problems…
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Desert Housing & Composite Walls
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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
tzyxgx
Tk
x
T
x
k
t
TC p
),,,(2
2
tzyxgx
Tk
t
TC p
One dimensional Transient conduction with heat generation.
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Steady Heat transfer through a plane slab
02
2
dx
TdA
0),,,(2
2
tzyxgx
Tk
No heat generation
211 CxCTCdx
dT
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Isothermal Wall Surfaces
Apply boundary conditions to solve for constants: T(0)=Ts1 ; T(L)=Ts2
211 CxCTCdx
dT
The resulting temperature distribution is:
and varies linearly with x.
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Applying Fourier’s law:
heat transfer rate:
heat flux:
Therefore, both the heat transfer rate and heat flux are independent of x.
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Wall Surfaces with Convection
2112
2
0 CxCTCdx
dT
dx
TdA
Boundary conditions:
110
)0(
TThdx
dTk
x
22 )(
TLThdx
dTk
Lx
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Wall with isothermal Surface and Convection Wall
2112
2
0 CxCTCdx
dT
dx
TdA
Boundary conditions:
1)0( TxT
22 )(
TLThdx
dTk
Lx
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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
VR
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
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q
TR
R
Tq th
th
WKmW
Kmm
kA
L
L
TTkA
TT
q
TR
ss
ss
condth /
1.2
12
21
WKmW
Km
hATThA
TT
q
TR
s
s
convth /
1.12
2
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WKmW
Km
AhTTAh
TT
q
TR
rsurrsr
surrs
radth /
1.12
2
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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
Tq
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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
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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:
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Composite Walls
• The overall thermal resistance is given by
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Desert Housing & Composite Walls