Advanced Heat Transfer (MEC508)
Dr. Pawan K. Singh
Asst. Professor, Mech. Engg.,
IIT (ISM) Dhanbad
Department of Mechanical Engineering, IIT (ISM) Dhanbad
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Module 1
Steady State Heat Conduction
References:
1. F. Incropera, and D. J. Dewitt, Fundamentals of Heat and Mass Transfer –Wiley &
Sons Inc., 7th Edition, 2011.
2. Latif M. Jiji., “Heat Conduction”, Springer, 3rd Edition, 2009.
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Review of basic modes of Heat Transfer
Thermodynamics: talks about the end state of process but it does not provide
any information about the nature of interaction.
Heat Transfer: is thermal energy in transit due to a spatial temperature
difference.
Conduction: Heat Transfer that will occur across a stationary medium.
Convection: Heat Transfer that will occur between surface and a moving fluid
with different temperature.
Thermal radiation: In absence of an intervening medium, there is a net heat
transfer by radiation between surface at different temperatures by
electromagnetic waves.
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Conduction: Physical origin and rate equation
Conduction may be viewed
as the transfer of energy
from the more energetic to
the less energetic particles
of a substance due to
interactions between the
particles.
• Consider a gas with temperature gradient and no bulk macroscopic motion.
• Temperature at any point can be associated with the energy of gas molecules
(due to random translational motion, as well as to the internal rotational and
vibrational motions, of the molecules) in proximity to the point.
• In case of collision, the energy will be transferred form higher energy
molecules to lower energy molecules, essentially transferring energy in the
direction of deceasing temperature.
• This net transfer of energy by random molecular motion can be called as a
diffusion of energy.
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Conduction: Physical origin and rate equation__cont.
• The situation is much the same in liquids, although the molecules are more
closely spaced and the molecular interactions are stronger and more frequent.
• In a solid, conduction may be attributed to atomic activity in the form of lattice
vibrations. The modern view is to ascribe the energy transfer to lattice waves
induced by atomic motion.
• Non Conductor: exclusively due to Lattice waves only
• Conductor: Also due to the translational motion of the free electrons
• For heat conduction, the rate equation is known as Fourier’s law. For the one
dimensional plane wall shown in Figure, having a temperature distribution
T(x), the rate equation is expressed as
𝒒𝒙"
𝒅𝑻
=−𝒌
𝒅𝒙
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Convection: Physical Origin and Rate Equation
• In addition to energy transfer due to random molecular motion (diffusion),
energy is also transferred by the bulk, or macroscopic, motion of the fluid.
• Convection = Diffusion + Advection
• Convection heat transfer is related to
• (i) Velocity Boundary layer
• (ii) Thermal Boundary layer
• The contribution due to random molecular
motion (diffusion) dominates near the surface where the fluid velocity is low.
• For convection heat transfer process, the appropriate rate equation is of the
𝒒" = 𝒉 (𝑻𝒔 − 𝑻∞) and is known as Newton’s law of cooling
𝒒" the convective heat flux (W/m2), is proportional to the
where,
difference between the surface and fluid temperatures, Ts and 𝑻∞ respectively. The
form:
parameter h (W/m2K) is termed the convection heat transfer coefficient.
It depends on conditions in the boundary layer, which are influenced by surface
geometry, the nature of the fluid motion, and an assortment of fluid
thermodynamic and transport properties.
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Convection: Physical origin and rate equation__cont
•
•
•
•
•
Type of Convection:
(a) Forced Convection
(b) Free/natural Convection
(c) Boiling
(d) Condensation
Typical values of the convection heat transfer coefficient
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Thermal Radiation: Physical Origin and Rate Equation
• Thermal radiation is energy emitted by matter that is at a nonzero temperature. It
may occur from solid surfaces, liquids and gases.
• The energy of the radiation field is transported by electromagnetic waves (or
alternatively, photons).
• Radiation that is emitted by the surface originates from
the thermal energy of matter bounded by the surface,
and the rate at which energy is released per unit area
(W/m2) is termed the surface emissive power, E. There is
an upper limit to the emissive power, which is prescribed
by the Stefan–Boltzmann law
𝐸𝑏 = 𝜎𝑇𝑆 4, where TS is surface temperature and
σ is Stefan-Boltzmann constant - 5.67×10-8 W/m2K.
For real surface ,
𝐸 = 𝜀𝜎𝑇𝑆 4, where ε is a radiative
property of the surface termed the emissivity. 0 ≤ ε ≤ 1
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Thermal Radiation: Physical origin and rate equation___cont.
• Irradiation (G): All incident radiation on unit surface
area
• Absorptivity (α): A portion, or all, of the irradiation
may be absorbed. 0 ≤ α ≤ 1
• Reflectivity: for opaque surface
• Transmissivity: for semi-transparent surface
Radiation exchange between a small surface at Ts and a much larger,
isothermal surface that completely surrounds the smaller one:
For a grey surface (α=ε),the net rate of
radiation heat transfer from the surface,
expressed per unit area of the surface, is:
𝑞
=
= 𝜀𝐸𝑏(𝑇𝑆) − 𝛼𝐺
𝐴
= 𝜀𝜎(𝑇𝑆 4 − 𝑇∞4)
𝑞𝑟𝑎𝑑"
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Heat Diffusion equation
What determines temperature distribution in a region?
The temperature is adjusted in such a way that
conservation of energy is applied at each point; so
differential formulation may be applied.
• Figure shows a region described by rectangular
coordinates in which heat conduction is
three-dimensional.
• The material is assumed to be moving.
• Energy is generated throughout the material at a
rate per unit volume.
We select an infinitesimal element dxdydz and apply the principle of conservation of
energy (first law of thermodynamics).
Rate of energy added + Rate of energy generated - Rate of energy removed
= Rate of energy change within element
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Heat Diffusion equation____cont..
˙ 𝑜𝑢𝑡 𝑎𝑛𝑑 𝐸
˙ respectively,
Denoting these terms by the symbols 𝐸˙𝑖𝑛, 𝐸˙𝑔, 𝐸
˙𝒈−𝑬
˙ 𝒐𝒖𝒕 = 𝑬
˙ . − − − − − − − − − − − − − − − − − − − (𝟏)
we obtain, 𝑬˙𝒊𝒏+ 𝑬
Due to
conduction
Due to mass
motion
Assumptions:
1. uniform velocity,
2. constant pressure,
3. constant density
4. negligible changes in potential energy.
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Heat Diffusion equation____cont.
^ 𝒅𝒙 𝒅𝒚 − (𝒂)
˙ 𝒊𝒏 = 𝒒𝒙"𝒅𝒚𝒅𝒛 + 𝒒𝒚"𝒅𝒙𝒅𝒛 + 𝒒𝒛"𝒅𝒙 𝒅𝒚 + 𝝆𝑼 𝒉^ 𝒅𝒚 𝒅𝒛 + 𝝆𝑽 𝒉^ 𝒅𝒙 𝒅𝒛 + 𝝆𝑾 𝒉
𝑬
˙ 𝑔 is
Energy generation 𝐸
˙ 𝒈 = 𝒒′′′𝒅𝒙 𝒅𝒚 𝒅𝒛 − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − (𝒃)
𝑬
For the energy leaving using Taylor series expansion
˙ 𝒐𝒖𝒕 = (𝒒 " +
𝑬
𝒙
^
^
^
𝝏𝒒𝒚"
𝝏𝒒𝒛"
𝝏𝒒𝒙"
𝝏𝒉
𝝏𝒉
𝝏𝒉
𝒅𝒙)𝒅𝒚𝒅𝒛 + (𝒒𝒚" +
𝒅𝒚)𝒅𝒙𝒅𝒛 + 𝒒𝒛" +
𝒅𝒙 𝒅𝒙 𝒅𝒚 + 𝝆𝑼 𝒉^ +
𝒅𝒙 𝒅𝒚 𝒅𝒛 + 𝝆𝑽 𝒉^ +
𝒅𝒚 𝒅𝒙 𝒅𝒛 + 𝝆𝑾 𝒉^ +
𝒅𝒛 𝒅𝒙 𝒅𝒚 − − (𝒄)
(
)
𝝏𝒙
𝝏𝒚
𝝏𝒛
𝝏𝒙
𝝏𝒚
𝝏𝒛
where U, V, W are constant since motion is uniform.
˙ 𝐢𝐬 𝐞𝐱𝐩𝐫𝐞𝐬𝐬𝐞𝐝 𝐚𝐬
Energy change within the element 𝑬
^
𝝏𝒖
˙
𝑬=𝛒
𝒅𝒙 𝒅𝒚 𝒅𝒛 − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − (𝒅)
𝝏𝒕
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Heat Diffusion equation____cont.
Substituting a-d in equation 1
^
^
^
𝝏𝒒𝒙" 𝝏𝒒𝒚" 𝝏𝒒𝒛"
𝝏𝒉
𝝏𝒉
𝝏𝒉
−
−
−
− 𝝆𝑼
− 𝝆𝑽
− 𝝆𝑾
+ 𝒒′′′ =
𝝏𝒙
𝝏𝒚
𝝏𝒛
𝝏𝒙
𝝏𝒚
𝝏𝒛
^
𝝏𝒖
𝛒
− − − − − − − − − (𝐞)
𝝏𝒕
𝑷
^
^
^
Enthalpy 𝒉 is defined as , 𝒉 = 𝒖 + , here P is pressure assumed constant. Putting this h in
𝝆
eqn e,
^
^
^
^
𝝏𝒒𝒙" 𝝏𝒒𝒚" 𝝏𝒒𝒛"
𝝏𝒉
𝝏𝒉
𝝏𝒉
𝝏𝒉
−
−
−
+ 𝒒′′′ = 𝝆
+𝑼
+𝑽
+𝑾
− − − − − − − − − − − − (𝒇 )
𝝏𝒙
𝝏𝒚
𝝏𝒛
𝝏𝒕
𝝏𝒙
𝝏𝒚
𝝏𝒛
^
Now writing enthalpy in term of temperature, 𝐝 𝒉 = 𝒄𝒑𝒅𝑻 𝒂𝒏𝒅 𝒔𝒖𝒃𝒔𝒕𝒖𝒕𝒊𝒏𝒈 𝒊𝒏 𝒆𝒒𝒏 𝒇
𝝏
𝝏𝑻
𝝏
𝝏𝑻
𝝏
𝝏𝑻
𝝏𝑻
𝝏𝑻
𝝏𝑻
𝝏𝑻
𝒌
+
𝒌
+
𝒌
+ 𝒒′′′ = 𝝆𝒄𝒑(
+𝑼
+𝑽
+𝑾
)
(
)
(
)
(
)
𝝏𝒙
𝝏𝒙
𝝏𝒚
𝝏𝒚
𝝏𝒛
𝝏𝒛
𝝏𝒕
𝝏𝒙
𝝏𝒚
𝝏𝒛
For a constant conductivity and stationary material, this equation simplifies to
𝝏 𝟐𝑻 𝝏 𝟐𝑻 𝝏 𝟐𝑻
𝒒′′′
𝝏𝑻
𝜶( 𝟐 + 𝟐 + 𝟐 ) +
=
𝝏𝒙
𝝏𝒚
𝝏𝒛
𝝆𝒄𝒑
𝝏𝒕
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The Heat Conduction Equation in Cylindrical
Coordinates
𝐶𝑦𝑙𝑖𝑛𝑑𝑟𝑖𝑐𝑎𝑙 𝐶𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒𝑠 (𝑟, 𝜃, 𝑧):
1 𝜕
𝜕𝑇
1 𝜕 2𝑇 𝜕 2𝑇
𝑞′′′
𝜕𝑇
𝜕𝑇 𝑉𝜃 𝜕𝑇
𝜕𝑇
𝛼
𝑟
+ 2 2 + 2 +
=
+ 𝑉𝑟
+
+ 𝑉𝑧
𝜕𝑧 ] 𝜌𝑐𝑝 [ 𝜕𝑡
𝜕𝑟
𝑟 𝜕𝜃
𝜕𝑧 ]
[ 𝑟 𝜕𝑟 ( 𝜕𝑟 ) 𝑟 𝜕𝜃
Where Vr, Vθ and Vz are velocity component in r, θ and z
direction respectively.
For a stationary medium the equation simplifies to
1 𝜕
𝜕𝑇
1 𝜕 2𝑇 𝜕 2𝑇
𝑞′′′
𝜕𝑇
𝛼
𝑟
+ 2 2 + 2 +
=
𝜕𝑧 ] 𝜌𝑐𝑝
𝜕𝑡
[ 𝑟 𝜕𝑟 ( 𝜕𝑟 ) 𝑟 𝜕𝜃
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The Heat Conduction Equation in Polar
Coordinates
𝐹𝑜𝑟 𝑃𝑜𝑙𝑎𝑟 𝐶𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒𝑠 (𝑟, 𝜃, 𝜑) :
𝛼
𝑉𝜃 𝜕𝑇
1 𝜕
1
𝜕 2𝑇
1
𝜕
𝜕𝑇
𝑞′′′
𝜕𝑇
𝜕𝑇 𝑉𝜑 𝜕𝑇
2 𝜕𝑇
𝑟
+
+
(𝑠𝑖𝑛𝜑
)
+
=
+
𝑉
+
+
𝑟
2
2
2
2
𝑟 𝑠𝑖𝑛𝜑 𝜕𝜑
𝜕𝜑 ] 𝜌𝑐𝑝
𝜕𝑡
𝜕𝑟
𝑟 𝜕𝜑 𝑟𝑠𝑖𝑛𝜑 𝜕𝜃
[ 𝑟 𝜕𝑟 ( 𝜕𝑟 ) 𝑟 2𝑠𝑖𝑛 𝜑 𝜕𝜃
Where Vφ are velocity component in φ direction.
For a stationary medium the equation simplifies to
2
1 𝜕
𝜕𝑇
1
𝜕
𝑇
1
𝜕
𝜕𝑇
𝑞′′′
𝜕𝑇
2
𝛼 2
𝑟
+
+
(𝑠𝑖𝑛𝜑
) +
=
𝜕𝜑 ] 𝜌𝑐𝑝
𝜕𝑡
[ 𝑟 𝜕𝑟 ( 𝜕𝑟 ) 𝑟 2𝑠𝑖𝑛2𝜑 𝜕𝜃 2 𝑟 2𝑠𝑖𝑛𝜑 𝜕𝜑
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One dimensional Steady state Conduction
Governing equation:
𝒅
𝒅𝑻
𝒌
+ 𝒒′′′ = 𝟎
(
)
𝒅𝒙
𝒅𝒙
Where k=Thermal conductivity, 𝒒′′′=rate of energy generation per unit
volume, T is temperature and x is independent variable
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Example: Plate with Energy generation and variable thermal
conductivity
𝒊𝒇 𝒌 = 𝒌𝟎(𝟏 − γ𝑻 )
𝒘𝒉𝒆𝒓𝒆 γ 𝒊𝒔 𝒄𝒐𝒏𝒔𝒕𝒂𝒏𝒕.
For one dimensional, steady state, stationary and uniform energy generation,
the governing equation can be written as
𝒅
𝒅𝑻
𝒌
+ 𝒒′′′ = 𝟎
(
)
𝒅𝒙
𝒅𝒙
𝒅
𝒅𝑻
𝒌𝟎(𝟏 − γ𝑻 ))
+ 𝒒′′′ = 𝟎 − − − − − (𝟏)
(
[
]
𝒅𝒙
𝒅𝒙
𝒅
𝒅𝑻
𝒒′′′
𝟏
−
γ𝑻
+
=𝟎
(
)
𝒅𝒙 [
𝒅𝒙 ] 𝒌𝟎
Boundary condition: T(0)=0 and T (L) =0
Integrating equation 1
𝒅𝑻 𝒒′′′
+
. 𝒙 = 𝑪𝟏
(𝟏 − γ𝑻 )
𝒅𝒙 𝒌𝟎
𝑻𝟐
𝒒′′′ 𝒙 𝟐
=−
+ 𝑪𝟏.x+𝑪𝟐 − − − − − − − (𝟐)
Again, 𝑻 − 𝜸
𝟐
𝒌𝟎 𝟐
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Example: Plate with Energy generation and variable thermal
conductivity___cont.
𝑻𝟐
𝒒′′′ 𝒙 𝟐
𝑻−𝜸
=−
+ 𝑪𝟏.x+𝑪𝟐 − − − − − − − (𝟐)
𝟐
𝒌𝟎 𝟐
Applying Boundary conditions,
At x=0, T=0
𝑞′′′
0 − 𝛾0 = − 0 + 𝐶1 . 0 + 𝐶2 ⇒ 𝐶2 = 0
𝑘0
At x=L, T=0
𝑞′′′ 𝐿 2
𝑞′′′𝐿
0 − 𝛾0 = −
+ 𝐶1 . 𝐿 + 0 ⇒ 𝐶1 =
𝑘0 2
2𝑘0
Putting in eqn 2
𝑇2
𝑞′′′ 2 𝑞′′′
𝑇−𝛾
+
𝑥 −
𝐿𝑥 = 0
2
2𝑘0
2𝑘0
2
1
𝑥
⇒𝑇 − 𝑇+
𝑞′′′𝐿𝑥 1 −
=0
(
)
𝛾
𝛾𝑘0
𝐿
2
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Example: Plate with Energy generation and variable thermal
conductivity___cont.
2
1
𝑥
𝑇 − 𝑇+
𝑞′′′𝐿𝑥 1 −
=0
(
)
𝛾
𝛾𝑘0
𝐿
2
⇒𝑇 =
2
+γ ±
1
⇒𝑇 = ±
γ
4
γ2
−
4𝑞′′′𝐿𝑥
𝛾𝑘0
𝑥
(1 − 𝐿 )
2
1
𝑞′′′𝐿𝑥
𝑥
−
(1 − )
2
γ
𝛾𝑘0
𝐿
For + sign, T at x=0 is not satisfied,
1
So 𝑇 = −
γ
1
𝑞′′′𝐿𝑥
𝑥
−
(1 − )
2
γ
𝛾𝑘0
𝐿
Heat Transfer rate,
𝑑𝑇(0)
𝑞(0) = − 𝐴𝑘0[1 − γ𝑇 (0)]
𝑑𝑥
𝑑𝑇(𝐿)
𝑞(𝐿) = − 𝐴𝑘0[1 − γ𝑇 (𝐿)]
𝑑𝑥
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Example: Radial Conduction in a Composite Cylinder with
interface friction
Description: A shaft of radius Rs rotates inside a sleeve of inner radius Rs and outer
radius R0. Frictional heat is generated at the interface and is represented by heat flux
qi”. The outside surface of the sleeve is cooled by convection with an ambient fluid at
T∞. The heat transfer coefficient is h. Consider one-D steady state conduction in
radial direction and determine the temperature distribution in shaft and sleeve.
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Example: Radial Conduction in a Composite Cylinder with
interface friction___cont.
Solution: Since the sleeve is free from any external effect and all effect are being
applied at boundaries only,
𝒅
𝒅𝑻
𝒓
= 𝟎 − − − − (𝟏)
(
)
𝒅𝒓
𝒅𝒓
Boundary conditions:
1. At the sleeve interface, r= Rs
𝒅𝑻 (𝑹𝑺 )
and 𝒒𝒊 = − 𝒌𝟏
𝒅𝒓
"
Where k1 is thermal conductivity of sleeve and 𝒒𝒊" is interface frictional heat flux
2. At the outer surface at r=Ro
−𝒌𝟏
𝒅𝑻 (𝑹𝒐)
𝒅𝒓
= 𝒉[𝑻 (𝑹𝒐) − 𝑻∞]
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Example: Radial Conduction in a Composite Cylinder with
interface friction___cont.
𝑰𝒏𝒕𝒆𝒈𝒓𝒂𝒕𝒊𝒏𝒈 𝒆𝒒𝒖𝒂𝒕𝒊𝒐𝒏 𝟏
𝑻 = 𝑪𝟏ln𝒓 + 𝑪𝟐 − − − − − (𝟐)
𝒒𝒊"𝑹𝑺
𝑪𝟏
⇒ 𝐶1 = −
At r=Rs, 𝒒𝒊" = − 𝒌𝟏
𝒌𝟏
𝒓
𝑹𝑺
Putting the value in equation and applying the 2nd boundary condition
𝑪𝟏
−𝒌𝟏
= 𝒉[𝑪𝟏𝒍𝒏 𝑹𝒐 + 𝑪𝟐 − 𝑻∞]
𝑹𝟎
𝒒𝒊"𝑹𝑺
𝒒𝒊"𝑹𝑺
𝒒𝒊"𝑹𝑺 𝒒𝒊"𝑹𝑺
+
𝒍𝒏 𝑹𝒐 + 𝑻∞
+𝒌𝟏
=𝒉 −
𝒍𝒏 𝑹𝒐 + 𝑪𝟐 − 𝑻∞ ⇒ 𝑪𝟐 =
[
]
𝒉𝑹𝟎
𝒌𝟏
𝒌𝟏𝑹𝟎
𝒌𝟏
𝒒𝒊"𝑹𝑺 𝒌𝟏
⇒ 𝑪𝟐 =
(
+ 𝒍𝒏 𝑹𝒐) + 𝑻∞
𝒌𝟏 𝒉𝑹𝟎
𝒒𝒊"𝑹𝑺
𝒒𝒊"𝑹𝑺 𝒌𝟏
Finally ⇒ 𝑇 = −
𝒍𝒏 𝒓 +
(
+ 𝒍𝒏 𝑹𝒐) + 𝑻∞
𝒌𝟏
𝒌𝟏 𝒉𝑹𝟎
𝑅𝑜
𝑇 − 𝑻∞
𝒌𝟏
⇒ 𝒒 "𝑹 = ln
+
𝒊 𝑺
𝑟
𝒉𝑹𝟎
𝒌𝟏
Department of Mechanical Engineering, IIT (ISM) Dhanbad
22
22
Example: Composite wall with energy generation (Home
Assignment)
A plate of thickness L1 and conductivity k1 generates heat at a volumetric rate of q’’’.
The plate is sandwiched between two plates of conductivity k2 and thickness L2 each.
The exposed surfaces of the two plates are maintained at constant temperature T0.
Determine the temperature distribution in the three plates.
Department of Mechanical Engineering, IIT (ISM) Dhanbad
23
23