the heat conduction equation p m v subbarao associate professor mechanical engineering department...
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The Heat Conduction Equation
P M V Subbarao
Associate Professor
Mechanical Engineering Department
IIT Delhi
An Easy Solution to Industrial Heat Transfer Problems…
The Heat Equation
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
),(. 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
2
2
2
2
2
2
tzyxgz
TT
rr
Tk
t
TC p
Steady-State One-Dimensional Conduction
• Assume a homogeneous medium with invariant thermal conductivity ( k = constant) :
• For one-dimensional steady state conduction with no energy generation, 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.
Steady-State One-Dimensional Conduction
• For one-dimensional heat conduction in a variable area geometry.
• We can devise a basic description of the process.
• The first law in control volume form (steady flow energy equation) with no shaft work and no mass flow reduces to the statement that Q = 0for all surfaces.
• From Fourier law of conduction, the heat transfer rate in at the left (at x) is:
Taylor’s Theory of Continuum
• For a function converging & well behaving…
......
!3!2
3
3
32
2
2
dx
dx
xQddx
dx
xQddx
dx
xQdxQdxxQ
n
i
n
n
nn
n
dx
dx
xQdxQdxxQ
1 !1
dx
dx
xQdxQdxxQ
x
• For a pure steady state conduction:
0 xQdxxQ
0
dxdxdT
kAd
If k is constant (i.e. if the material is homogeneous and properties of themedium are independent of temperature), this reduces to
02
2
dx
dT
dx
dA
dx
TdA
Pure radial conduction throughA Sphere.
02
2
dr
dT
dr
dA
dr
TdA
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
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
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 R = R1 + R2.
• If TL is the temperature at the left, and TR is the temperature at the right, the heat transfer rate is given by
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